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Physical solutions of the Hamilton-Jacobi equation
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla and Héctor Sánchez-Morgado
We consider a Lagrangian system on the d-dimensional torus, and the associated Hamilton-Jacobi equation. Assuming that the Aubry set of the system consists in a finite number of hyperbolic periodic orbits of the Euler-Lagrange flow, we study the vanishing-viscosity limit, from the viscous equation to the inviscid problem. Under suitable assumptions, we show that solutions of the viscous Hamilton-Jacobi equation converge to a unique solution of the inviscid problem.
Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, H\u00E9ctor S\u00E1nchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 513-528. doi: 10.3934/dcdsb.2005.5.513.
Universality of dishonesty of substochastic semigroups: Shattering fragmentation and explosive birth-and-death processes
Jacek Banasiak and Mustapha Mokhtar-Kharroubi
By a dishonest process we understand a process in which, for some initial data, there occurs an unaccounted for loss of the described quan- tity throughout the evolution. Classical examples are offered by shattering fragmentation, where the total mass is decreasing faster than predicted by the formal conservation laws, or explosive birth-and-death processes which, being formally conservative, suffer from the loss of individuals in the course of evo- lution. In this note we shall show, for these two processes, that if dishonesty occurs for one initial datum, then it must occur for any of them.
Jacek Banasiak, Mustapha Mokhtar-Kharroubi. Universality of dishonesty of substochastic semigroups: Shatteringfragmentation and explosive birth-and-death processes. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 529-542. doi: 10.3934/dcdsb.2005.5.529.
Asymptotic properties of a TCP model with time-outs
Niclas Carlsson and Göran Högnäs
We examine a simple discrete time Markov model of TCP congestion control, which contains congestion avoidance, fast retransmit and time-out, and we prove that it has a unique invariant measure. If the process is scaled by a factor $\sqrt{p}$, then the invariant measures converge as $p \to 0$, where $p$ is the probability of error in any given data packet. This is the $1/\sqrt{p}$-behavior of TCP throughput.
If the scaled process is transformed to continuous time, we show that it converges to a piecewise linear limit process. The unique invariant measure of the limit process coincides with the limit of the invariant measures above and can be easily computed.
Finally, we examine a slightly more sophisticated way of modelling time-outs.
Niclas Carlsson, G\u00F6ran H\u00F6gn\u00E4s. Asymptotic properties of a TCP model with time-outs. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 543-564. doi: 10.3934/dcdsb.2005.5.543.
Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources
Jean René Chazottes and E. Ugalde
Motivated by entropy estimation from chaotic time series, we pro- vide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a n-cylinder fluctuates in the same way as the inverse measure of this n-cylinder at 'small scales', but in a different way at 'large scales'. In particular, the Rényi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more di±cult to handle theoretically. The results we obtain for return times, though less precise than for hitting times, complete the available ones.
Jean Ren\u00E9 Chazottes, E. Ugalde. Entropy estimation and fluctuations of hitting and recurrence timesfor Gibbsian sources. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 565-586. doi: 10.3934/dcdsb.2005.5.565.
Bernoulli shift for second order recurrence relations near the anti-integrable limit
Yi-Chiuan Chen
We extend the anti-integrability theory of Aubry to non-autonomous twist maps between symplectic spaces to show the shift dynamics can be embedded in a natural way. Examples are given to illustrate that the embedded shift can be a full shift, a subshift of finite type or of infinite type.
Yi-Chiuan Chen. Bernoulli shift for second order recurrence relations near theanti-integrable limit. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 587-598. doi: 10.3934/dcdsb.2005.5.587.
Traffic circles and timing of traffic lights for cars flow
Yacine Chitour and Benedetto Piccoli
In this paper we address the following traffic regulation problem: given a junction with some incoming roads and some outgoing ones, is it preferable to regulate the flux via a traffic light or via a traffic circle on which the incoming traffic enters continuously? More precisely, assuming that drivers distribute on outgoing roads according to some known coefficients, our aim is to understand which solution performs better from the point of view of total amount of cars going through the junction.
To deal with this problem we consider a fluid dynamic model for traffic flow on a road network. The model is that proposed in [9] and is applied to the case of crossings with lights and with circles. For the first we consider timing of lights as control and determine the asymptotic fluxes. For the second we extend and complete the model of [9] introducing some right of way parameters. Also in this case we determine the asymptotic behavior.
We then compare the performances of the two solutions. Finally, we can indicate which choice is preferable, depending on traffic level and control necessity, and give indications on how to tune traffic light timing and traffic circle right of way parameters.
Yacine Chitour, Benedetto Piccoli. Traffic circles and timing of traffic lights for cars flow. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 599-630. doi: 10.3934/dcdsb.2005.5.599.
Multiscale numerical method for nonlinear Maxwell equations
Thierry Colin and Boniface Nkonga
The aim of this work is to propose an efficient numerical approximation of high frequency pulses propagating in nonlinear dispersive optical media. We consider the nonlinear Maxwell's equations with instantaneous nonlinearity. We first derive a physically and asymptotically equivalent model that is semi-linear. Then, for a large class of semi-linear systems, we describe the solution in terms of profiles. These profiles are solution of a singular equation involving one more variable describing the phase of the solution. We introduce a discretization of this equation using finite differences in space and time and an appropriate Fourier basis (with few elements) for the phase. The main point is that accurate solution of the nonlinear Maxwell equation can be computed with a mesh size of order of the wave length. This approximation is asymptotic-preserving in the sense that a multi-scale expansion can be performed on the discrete solution and the result of this expansion is a discretization of the continuous limit. In order to improve the computational delay, computations are performed in a window moving at the group velocity of the pulse. The second harmonic generation is used as an example to illustrate the proposed methodology. However, the numerical method proposed for this benchmark study can be applied to many other cases of nonlinear optics with high frequency pulses.
Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 631-658. doi: 10.3934/dcdsb.2005.5.631.
Multiresolution analysis for 2D turbulence. Part 1: Wavelets vs cosine packets, a comparative study
Patrick Fischer
The widely accepted theory of two-dimensional turbulence predicts a direct downscale enstrophy cascade with an energy spectrum behaving like $k^{-3}$ and an inverse upscale energy cascade with a $k^{-5/3}$ decay. Nevertheless, this theory is in fact an idealization valid only in an infinite domain in the limit of infinite Reynolds numbers, and is almost impossible to reproduce numerically. A more complete theoretical framework for the two-dimensional turbulence has been recently proposed by Tung et al . This theory seems to be more consistent with experimental observations, and numerical simulations than the classical one developed by Kraichnan, Leith and Batchelor.
Multiresolution methods like the wavelet packets or the cosine packets, well known in signal decomposition, can be used for the 2D turbulence analysis. Wavelet or cosine decompositions are more and more used in physical applications and in particular in fluid mechanics. Following the works of M. Farge et al , we present a numerical and qualitative study of a two-dimensional turbulence fluid using these methods. The decompositions allow to separate the fluid in two parts which are analyzed and the corresponding energy spectra are computed. In the first part of this paper, the methods are presented and the numerical results are briefly compared to the theoretical spectra predicted by the both theories. A more detailed study, using only wavelet packets decompositions and based on numerical and experimental data, will be carried out and the results will be reported in the second part of the paper. A tentative of physical interpretation of the different components of the flow will be also proposed.
Patrick Fischer. Multiresolution analysis for 2D turbulence. Part 1: Wavelets vscosine packets, a comparative study. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 659-686. doi: 10.3934/dcdsb.2005.5.659.
First numerical evidence of global Arnold diffusion in quasi-integrable systems
Claude Froeschlé, Massimiliano Guzzo and Elena Lega
We provide numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We show that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems.
Claude Froeschl\u00E9, Massimiliano Guzzo, Elena Lega. First numerical evidence of global Arnold diffusion inquasi-integrable systems. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 687-698. doi: 10.3934/dcdsb.2005.5.687.
A discrete-delayed model with plasmid-bearing, plasmid-free competition in a chemostat
Sze-Bi Hsu and Cheng-Che Li
A discrete-delayed model of plasmid-bearing, plasmid-free organisms competing for a single-limited nutrient in a chemostat is established. Rigorous mathematical analysis of the asymptotic behavior of this model is presented. An interesting method to analyze the local stability of interior equilibrium is developed. The argument is also applicable to a model of plasmid-bearing, plasmid-free organisms competing for two complementary nutrients in a chemostat.
Sze-Bi Hsu, Cheng-Che Li. A discrete-delayed model with plasmid-bearing, plasmid-freecompetition in a chemostat. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 699-718. doi: 10.3934/dcdsb.2005.5.699.
Effective Hamiltonian for traveling discrete breathers in the FPU chain
Michael Kastner and Jacques-Alexandre Sepulchre
For the Fermi-Pasta-Ulam chain, an effective Hamiltonian is constructed, describing the motion of approximate, weakly localized discrete breathers traveling along the chain. The velocity of these moving and localized vibrations can be estimated analytically as the group velocity of the corresponding wave packet. The Peierls-Nabarro barrier is estimated for strongly localized discrete breathers.
Michael Kastner, Jacques-Alexandre Sepulchre. Effective Hamiltonian for traveling discrete breathers in the FPUchain. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 719-734. doi: 10.3934/dcdsb.2005.5.719.
Dynamics of a logistic population model with maturation delay and nonlinear birth rate
Suqi Ma, Qishao Lu and Shuli Mei
A logistic population model with a maturation delay stage for adults is investigated. The adult population is related to its previous life stage with a maturation delay $r$, and has a non-linear exponential birth rate $be^{-pr}$ with a birth decay coefficient $p$. As $r$ increases, the unique positive equilibrium solution may experience two stability switchings, that is, from stable to unstable, and then back to stable again. The decay coefficient $p$ can also qualitatively influence the stability property of the system. Hopf bifurcation and the stability of the bifurcating periodic solution are analyzed by means of the center manifold theory and the normal form technique. By applying the integral averaging theory, phase-locked and phase-shifting solutions induced by the external excitation are also investigated and verified by numerical simulations.
Suqi Ma, Qishao Lu, Shuli Mei. Dynamics of a logistic population model with maturation delay andnonlinear birth rate. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 735-752. doi: 10.3934/dcdsb.2005.5.735.
Nonisothermal phase separation based on a microforce balance
Alain Miranville and Giulio Schimperna
Our aim in this article is to derive models for nonisothermal phase separation. Starting from the two fundamental laws of thermodynamics, we consider the approach of Gurtin, based on a balance law for microforces, to derive nonisothermal Cahn-Hilliard type equations. These equations extend previous models derived by Alt and Pawłow based on an entropy principle to nonisotropic materials and to systems that are far from equilibrium. We also extend this approach to the Ginzburg-Landau (Allen-Cahn) equation, for which we recover, as particular cases, some models obtained by Frémond with a physically different approach.
Alain Miranville, Giulio Schimperna. Nonisothermal phase separation based on a microforce balance. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 753-768. doi: 10.3934/dcdsb.2005.5.753.
The polynomial-preserving recovery for higher order finite element methods in 2D and 3D
A. Naga and Z. Zhang
The Polynomial-Preserving Recovery (PPR) technique is extended to recover continuous gradients from $C^0$ finite element solutions of an arbitrary order in 2D and 3D problems. The stability of the PPR is theoretically investigated in a general framework. In 2D, the stability is established under a simple geometric condition. The numerical experiments demonstrated that the PPR-recovered gradient enjoys superconvergence, and the Zienkiewicz-Zhu error estimator based on the PPR-recovered gradient is asymptotically exact.
A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite elementmethods in 2D and 3D. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 769-798. doi: 10.3934/dcdsb.2005.5.769.
Dynamics close to a non semi-simple 1:-1 resonant periodic orbit
M. Ollé, J.R. Pacha and J. Villanueva
In this work, our target is to analyze the dynamics around the $1:-1$ resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, up to some given arbitrary order, the normal form around that resonant (or critical) periodic orbit.
Hence, dealing with the normal form itself and the differential equations related to it, we derive the generic existence of a two-parameter family of invariant 2D tori which bifurcate from the critical periodic orbit. Moreover, the coefficient of the normal form that determines the stability of the bifurcated tori is identified. This allows us to show the Hopf-like character of the unfolding: elliptic tori unfold "around'' hyperbolic periodic orbits (case of direct bifurcation) while normal hyperbolic tori appear "around'' elliptic periodic orbits (case of inverse bifurcation). Further, the parametrization of the main invariant objects as well as a global description of the dynamics of the normal form are also given.
M. Oll\u00E9, J.R. Pacha, J. Villanueva. Dynamics close to a non semi-simple 1:-1 resonant periodic orbit. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 799-816. doi: 10.3934/dcdsb.2005.5.799.
Normal mode analysis of second-order projection methods for incompressible flows
Jae-Hong Pyo and Jie Shen
A rigorous normal mode error analysis is carried out for two second-order projection type methods. It is shown that although the two schemes provide second-order accuracy for the velocity in $\L^2$-norm, their accuracies for the velocity in $\H^1$-norm and for the pressure in $L^2$-norm are different, and only the consistent splitting scheme introduced in [6] provides full second-order accuracy for all variable in their natural norms. The advantages and disadvantages of the normal mode analysis vs. the energy method are also elaborated.
Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods forincompressible flows. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 817-840. doi: 10.3934/dcdsb.2005.5.817.
Discrete May-Leonard competition models II
Lih-Ing W. Roeger
We analyzed the local dynamics of a three-dimensional Ricker type discrete-time competition model that is analogous to the May-Leonard (M-L) differential equation model. The symmetric discrete M-L model is mentioned by Hofbauer et al. [[7] J. Math. Biol., 25:553--570,1987] as "perhaps one of the most difficult three species problems''. Both of the discrete and the continuous M-L models have similar local dynamics. However, the discrete model is not dynamically consistent with the continuous model. Unlike the continuous M-L model, the discrete Hopf bifurcations (Neimark-Sacker bifurcations) of the discrete M-L model are not degenerate. The continuous M-L model is the limiting case of the discrete model.
Lih-Ing W. Roeger. Discrete May-Leonard competition models II. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 841-860. doi: 10.3934/dcdsb.2005.5.841.
The turnpike property of discrete-time control problems arising in economic dynamics
Alexander J. Zaslavski
In this work we study the structure of approximate solutions of a nonautonomous discrete-time control system in a compact metric space $X$ which is determined by a sequence of continuous functions $v_i: X \times X \to R^1$, $i=0,\pm 1,\pm 2,$.... The main result in this paper deals with the turnpike property of optimal control problems. To have this property means that the approximate solutions of the problems are determined mainly by the the sequence $\{v_i\}_{i=-\infty}^{\infty}$, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.
Alexander J. Zaslavski. The turnpike property of discrete-time control problems arising ineconomic dynamics. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 861-880. doi: 10.3934/dcdsb.2005.5.861.
Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps
Iryna Sushko, Anna Agliari and Laura Gardini
This article deals with a two-parameter family of piecewise smooth unimodal maps with one break point. Using superstable cycles and their symbolic representation we describe the structure of the periodicity regions of the 2D bifurcation diagram. Particular attention is paid to the bistability regions corresponding to two coexisting attractors, and to the border-collision bifurcations.
Iryna Sushko, Anna Agliari, Laura Gardini. Bistability and border-collision bifurcations for a family ofunimodal piecewise smooth maps. Discrete & Continuous Dynamical Systems - B, 2005, 5(3): 881-897. doi: 10.3934/dcdsb.2005.5.881. | CommonCrawl |
Unsupervised domain adaptation for lip reading based on cross-modal knowledge distillation
Yuki Takashima1,
Ryoichi Takashima ORCID: orcid.org/0000-0002-9808-02501,
Ryota Tsunoda1,
Ryo Aihara2,
Tetsuya Takiguchi1,
Yasuo Ariki1 &
Nobuaki Motoyama2
We present an unsupervised domain adaptation (UDA) method for a lip-reading model that is an image-based speech recognition model. Most of conventional UDA methods cannot be applied when the adaptation data consists of an unknown class, such as out-of-vocabulary words. In this paper, we propose a cross-modal knowledge distillation (KD)-based domain adaptation method, where we use the intermediate layer output in the audio-based speech recognition model as a teacher for the unlabeled adaptation data. Because the audio signal contains more information for recognizing speech than lip images, the knowledge of the audio-based model can be used as a powerful teacher in cases where the unlabeled adaptation data consists of audio-visual parallel data. In addition, because the proposed intermediate-layer-based KD can express the teacher as the sub-class (sub-word)-level representation, this method allows us to use the data of unknown classes for the adaptation. Through experiments on an image-based word recognition task, we demonstrate that the proposed approach can not only improve the UDA performance but can also use the unknown-class adaptation data.
Lip reading is a technique of understanding utterances by visually interpreting the movements of a person's lips, face, and tongue when the spoken sounds cannot be heard. For example, for people with hearing problems, lip reading is one communication skill that can help them communicate better. McGurk et al. [1] reported that we human beings perceive a phoneme not only from the auditory information of the voice but also from visual information associated with the movement of the lips and face. Moreover, it is reported that we try to catch the movement of lips in a noisy environment and we misunderstand the utterance when the movements of the lips and the voice are not synchronized. Therefore, understanding the relationship between the voice and the movements of the lips is very important for speech perception. In the field of automatic speech recognition (ASR), visual information is used to assist the performance of speech recognition in a noisy environment [2]. In this work, lip reading has the goal of classifying words from the movements of the lips.
Recently, deep learning-based models have improved the performance of audio-visual automatic speech recognition (AV-ASR) or lip reading [3–7] where a large amount of training data is available. However, in a variety of real-life situations, there is often a mismatch between the training environment and the real environment where a user utilizes the system, and it is not easy to collect a sufficient amount of training data in a specific environment. Therefore, an effective way to adapt the model to a new environment is required. This is known as the domain adaptation (DA) problem.
The purpose of DA is to adapt a model trained on a source domain (source model) to a new target domain by using a relatively small amount of additional training (adaptation) data. Especially, in the case when all the adaptation data are not labeled, it is called "unsupervised domain adaptation" (UDA). Various UDA approaches have been proposed [8–10]. However, most of them assume that all the adaptation data belong to classes that are defined in the source model. This means that we cannot use the real environment data for the adaptation if that data is out-of-class (e.g., out-of-vocabulary (OOV) words in speech recognition). For more practical adaptation, it is preferable if out-of-class (unknown class) data can also be used. With this in mind, in this paper, we investigate an unknown-class-driven UDA method. Although there has been research carried out to tackle a similar issue [10, 11], the UDA on the unknown-class data is an extremely challenging task because we cannot use any conventional training policies, such as maximizing the output probability of the correct class.
In this paper, we propose a UDA method based on a model for cross-modal knowledge distillation (KD) for lip reading. There are two key factors: cross-modal KD and its application to UDA. KD [12] was originally introduced as model compression, in which a small model (student model) is trained to imitate an already-trained larger model (teacher model). Based on the idea that KD can transfer the knowledge of the teacher model to the student model, this technique has been applied to various tasks [13–15]. In this paper, we investigate cross-modal KD, where the student and teacher model are a lip-reading model and an audio-based speech recognition model (ASR model), respectively. Our proposed method uses audio-visual data for training and adapting the lip-reading model. Before training the lip-reading model, we train the ASR model using audio data. In our research, we use an ASR model based on an artificial neural network. Then, we train the lip-reading model by using the output from the intermediate layer of the ASR model. Typically, the audio data has more information for recognizing speech and shows better recognition accuracy than the visual data. For this reason, the use of the output from the ASR model can be a powerful teacher.
Another important factor is the use of the data of the unknown class for UDA. The basic KD that minimizes the distance between the output probabilities (i.e., output of the final layer) of the teacher and student models cannot be applied to unknown class UDA because the output labels of the source model do not contain the target class. To solve this problem, we use the output of the intermediate layer for the KD instead of that of the final layer. This approach is advantageous because, unlike basic final-layer-based KD, our intermediate-layer-based KD can construct the sub-class (e.g., sub-word in speech recognition) representation implicitly inside the network. By using this sub-class representation as an adaptation objective, we can use the unknown class data for the adaptation.
Our approach, which utilizes an audio signal to enhance the lip-reading performance, is suitable for applications having a video camera, such as car navigation systems using in-vehicle cameras and service robots. In these applications, we can use both audio and video signals, and improving the lip-reading performance is expected to contribute to the improvement of audio-visual speech recognition performance. In the experiment, we demonstrate that our proposed method can improve the UDA performance on a word recognition task.
There have been many studies carried out on AV-ASR over the years, and most of them discuss how to integrate multimodal features [3, 15, 16]. We expect that to improve the performance of the lip reading can contribute to improving the performance of AV-ASR. LipNet [17] performs end-to-end sentence-level lip reading. This model consists of spatiotemporal convolutions and recurrent operations, and that is trained by a connectionist temporal classification loss [18]. MobiLipNet [19] has been proposed to achieve computationally efficient lip reading, and that uses the depthwise convolution and the pointwise convolution. There are some prior works based on a generative adversarial network (GAN) [20] for lip reading. Wand et al. [21] proposed a speaker-independent lip-reading system using domain-adversarial training that trains a model that can extract the speaker-invariant feature representation. Oliveira et al. [22] investigated a method to recognize viseme, that is the visual correspondent of a phoneme, using GAN-based mapping to alleviate a head-pose variation problem.
There have been some studies on cross-modal KD [15, 23, 24] for the purpose of transferring the knowledge of a modal having rich training data to a modal having poor training data. This technique has also been applied to the AV-ASR task [15], where the knowledge of the audio trained from a large amount of speech data is transferred to the AV-ASR model. In that study, they focused only on the case in which the audio data is corrupted by noise, and did not discuss the environmental mismatch in the image data, which is our target issue. For lip reading, a similar approach to our proposed method was used more recently as multi-granularity KD from a speech recognizer to a lip reader (LIBS) [25] where a frame-level KD corresponds to our intermediate-layer-based KD. In order to take account of the difference between the audio and video sampling rates, LIBS employs an attention mechanism. Different from LIBS, our method uses a pyramid structure to obtain the audio and video sequences of the same length. Moreover, our method is evaluated on a word-level recognition task, while LIBS was evaluated on a sequence-level utterance recognition task.
The recent UDA approach involves finding a common representation for the two domains. Deep domain confusion [26] learns the meaningful and domain-invariant representation with an additional adaptive layer and loss function. GAN-based UDA approaches [27, 28] aim to learn the intermediate representation that cannot be used to distinguish the domain. Saito et al. [10] proposed a method to maximize the discrepancy between two classifier outputs considering the task-specific-design boundaries. Sohn et al. [29] proposed a feature-level UDA method using unlabeled video data that distills knowledge from a still image network to a video adaptation network. Afoura et al. [24] proposed a cross-modal KD method to improve the performance of lip reading using an ASR model. In our study, we investigate the use of cross-modal KD to adapt a model to the target environment.
Despite the recent progress of UDA, these conventional methods assume that all the adaptation data belong to classes that are defined in the source model, and none of the data can be used for the adaptation if that data is out-of-class. In the field of voice conversion, some approaches that do not require any context information have been proposed (e.g., [30]). Similar to these works, a context-independent (i.e., class-independent) approach for training the lip-reading model is required. In this work, we focus on the scenario where only the data of the unknown class is available during adaptation.
Proposed method
We aim to achieve UDA using the data of the unknown class on lip reading, which estimates the word label from an image input. In our proposed method, we use audio-visual data for training and adapting the lip-reading model, and for evaluating, we use only visual data. First, we explain the basic idea of the cross-modal KD on which our method is based. Then, we describe our proposed UDA method, which is based on the cross-modal KD using the data of the unknown class.
Cross-modal KD
Figure 1 shows an overview of the basic procedure of cross-modal KD, where the speech and the image are given from the same utterance. In our lip-reading task, the output is defined by the word. First, in advance, we train the audio model, which estimates the probability of the word from the acoustic feature using the cross entropy loss with the correct label. Given an acoustic feature xaud and an image feature xvis, the basic KD loss is defined as follows:
$$\begin{array}{*{20}l} -\sum_{l} p_{\text{aud}}(l|x_{\text{aud}}) \ln p_{\text{vis}}(l|x_{\text{vis}}), \end{array} $$
Basic procedure of cross-modal knowledge distillation
where pvis(l|xvis) and paud(l|xaud) denote the probabilities of a label l estimated from the visual model based on xvis and estimated from the audio model based on the input xaud, respectively. Here, the acoustic feature and the image feature are extracted from the same utterance. When training the visual model, the parameters of the audio model are fixed. This loss function forces the visual model to imitate the outputs extracted from the audio model. Practically speaking, the softmax loss using the correct label (hard loss) is often used for stable training with the linear interpolation parameter λ. Li et al. [15] demonstrate that KD between the ASR model and the AV-ASR model improves the recognition performance when the speech data is corrupted by noise. Therefore, it is expected that KD between the audio model (ASR model) and the visual model (lip-reading model) also contributes to improving the performance in our task.
Cross-modal KD-based UDA for the unknown class
Before describing our method, we first want to highlight the fact that the adaptation data does not belong to any class of the source domain. Considering the domain, let \(\mathcal {D}\) be the joint distribution over sequences of audio features and visual features, and the corresponding label. The output of the network is defined by the word.
Our model consists of two parts: an encoder and a classifier, as shown in Fig. 2. The encoder is a stacked convolution layer. The two encoders of the audio and visual modal can be defined as follows:
$$\begin{array}{*{20}l} \boldsymbol{h}^{a} &= f_{\text{aud}}(\boldsymbol{a}), \end{array} $$
Overview of our proposed UDA
$$\begin{array}{*{20}l} \boldsymbol{h}^{v} &= f_{\text{vis}}(\boldsymbol{v}), \end{array} $$
where \(\phantom {\dot {i}\!}\boldsymbol {a}=(a_{1},...,a_{t},...,a_{T_{a}})\) and \(\phantom {\dot {i}\!}\boldsymbol {v}=(v_{1},...,v_{t^{\prime }},...,v_{T_{v}})\) are input sequences of acoustic features and of visual features, respectively. \(\boldsymbol {h}^{a}=(h^{a}_{1},...,h^{a}_{u},...,h^{a}_{U})\) and \(\boldsymbol {h}^{v}=(h^{v}_{1},...,h^{v}_{u},...,h^{v}_{U})\) are the sequences of high-level representations. Here \(\phantom {\dot {i}\!}a_{t}, v_{t^{\prime }}, h^{a}_{u}\in \mathbb {R}^{d}\), and \(h^{v}_{u}\in \mathbb {R}^{d}\) are the input acoustic feature frame, the input visual feature frame, and the d-dimensional encoder features of both modalities, respectively. Ta,Tv and U≤ min(Ta,Tv) denote the numbers of the input acoustic features and the input visual features, and the number of the encoder output features, respectively. The number of steps of encoded features is the same between the two modalities. The classifier consists of fully connected layers to estimate the corresponding word label.
During adaptation, our method minimizes the mean square error (MSE) between the hidden representations as follows:
$$\begin{array}{*{20}l} \mathcal{L}_{{MSE}}(\mathcal{D}) = \mathbb{E}_{\{\boldsymbol{v},\boldsymbol{a},\boldsymbol{y}\}\sim \mathcal{D}} [|| \boldsymbol{h}^{a} - \boldsymbol{h}^{v} ||_{2}^{2}], \end{array} $$
where y is the label and is ignored. Unlike the generally used KD loss (Eq. (1)), we use the hidden representation in the intermediate layer for distillation. In the output layer and layers in the classifier, the frame-level information is lost and the feature representation is specialized to word-level (i.e., class-level) information. For this reason, the simple KD formulation cannot be applied to the adaptation if the adaptation data is out-of-class. On the other hand, the layers in the encoder have sub-word or phoneme-like representation that is independent of the class because they still retain the frame-level information. For this reason, our proposed method realizes UDA using the data of the unknown class.
Training procedure
Considering the source domain, let \(\mathcal {D}_{\text {src}}\) be the joint distribution over sequences of audio features and visual features, and the corresponding label. \(\mathcal {D}_{\text {trg}}\) is analogously defined for the target domain.
The first step is to train the models on the source domain. In this step, we expect that the hidden representation in the visual model is similar to that of the audio model. First, in advance, we train the audio model using the cross entropy loss with the correct label as follows:
$$\begin{array}{*{20}l} \mathbb{E}_{\{\boldsymbol{v},\boldsymbol{a},\boldsymbol{y}\}\sim \mathcal{D}_{\text{src}}} [-\log (g_{\text{aud}}(\boldsymbol{y}|\boldsymbol{h}^{a}))], \end{array} $$
where v is not used and gaud(y|ha) denotes the output probability of the label y estimated by the classifier of the audio model from the encoded feature h. Then, we train the visual model using the KD loss and the cross entropy loss as follows:
$$\begin{array}{*{20}l} &\mathcal{L}_{{MSE}}(\mathcal{D}_{\text{src}}) + \mathcal{L}_{{CE}}(\mathcal{D}_{\text{src}})\\ &= \mathcal{L}_{{MSE}}(\mathcal{D}_{\text{src}}) + \mathbb{E}_{\{\boldsymbol{v},\boldsymbol{a},\boldsymbol{y}\}\sim \mathcal{D}_{\text{src}}} [-\log (g_{\text{vis}}(\boldsymbol{y}|\boldsymbol{h}^{v}))], \end{array} $$
where gvis(y|hv) is the output probability estimated by the visual classifier.
Next, we adapt the visual model using the data of the unknown class based on the UDA scheme as described in Section 3.2. For more stable adaptation, we also use the data of the source domain. In addition to the loss in Eq. (4), we calculate losses for the source domain that has the correct label. This works as a regularization to prevent overfitting to the target distribution in the audio modality. Finally, our UDA approach for an unknown class minimizes the loss as follows:
$$\begin{array}{*{20}l} \mathcal{L} = \mathcal{L}_{{MSE}}(\mathcal{D}_{\text{trg}}) + \alpha\mathcal{L}_{{MSE}}(\mathcal{D}_{\text{src}}) + (1-\alpha)\mathcal{L}_{{CE}}(\mathcal{D}_{\text{src}}), \end{array} $$
where α indicates a weight parameter used to adapt the model stably, and we employ 0.5 in this paper. All parameters of the visual model are fine-tuned to minimize this loss function.
The proposed method was evaluated in a word recognition task on the lip reading in the wild (LRW) dataset [5]. LRW is a large-scale lip-reading dataset that consists of sounds and face images, where some works on AV-ASR or lip reading [31, 32] have been verified. All the videos are clipped to 29 frames (1.16 s) in length. Note that the length of each utterance is completely fixed.
LRW consists of up to 1000 utterances of 500 different words spoken by hundreds of different speakers. From the whole of the dataset, we picked out 800 utterances of 500 words (a total of 400,000 utterances) and divided them into several subsets, as shown in Fig. 3. We randomly divided 500 words into two sets of classes: the known class set of 400 words and the unknown class set of 100 words. For each of the 400 known words, we picked out (a) 500 utterances (a total of 200,000 utterances) and (b) another 50 utterances (a total of 20,000 utterances) as the training set of the source domain and the evaluation set of the target domain, respectively. For evaluating the UDA method, we used two different adaptation sets: the known class set and the unknown class set. The unknown class set (d) consisted of 250 utterances of 100 unknown words (a total of 25,000 utterances). For known class set (c), we randomly selected 100 words from 400 known words in order to match the condition with the unknown class set. Then, we created the known class set using 250 utterances of the selected 100 known words (a total of 25,000 utterances) which were not used for either the training set or the evaluation set. The evaluation set and the two adaptation sets were in the target domain while the training set was in the source domain. For creating the target domain data, we changed the brightness of the image (no transformation was carried out on the sound signal of the video) because changes in brightness are one of the most likely situations in real environments (e.g., daytime and night, or a car navigation system when driving through a tunnel).
Graphical representation of the division of the words, (a) source training set of the known words, (b) target evaluation set of the known words, (c) target adaptation set of the known words, and (d) target adaptation set of the unknown words
For the acoustic features, we calculated 40-dimensional log-mel filter bank features computed every 10 ms over a 25 ms window. Then, we stacked their delta and acceleration along the channel. The number of frames was 116. For the visual feature, the images are transformed to grayscale and resized to 112 ×112. The number of frames was 28. The encoder configuration is shown in Table 1. We used a pyramid structure that takes every two consecutive frames of the output from the previous layer without overlap as input. This structure allows the subsequent module to extract the relevant information from a smaller number of time steps. For the classifier, we use the three fully connected layers (4096 → 4096 → 400). We construct the individual model for the two modalities. The network was optimized using an Adam optimizer [33]. The batch size was 24, and the learning rate was set to 1e −4. When training models on the source domain, the number of epochs was 20 with early stopping. When adapting models to the target domain, the number of epochs was 10.
Table 1 Network architecture of the encoder
Our experiments were conducted using an Intel(R) Core(TM) i9-7900X CPU @ 3.30 GHz and single GeForce GTX 1080 Ti. Our proposed model took about 1.5 hours and 10 minutes per epoch for training and adaptation, respectively.
First, we evaluated the performance of cross-modal KD on the training data of the source domain model. Here, we use the test data without modifying the brightness of the image and do not consider UDA. Table 2 shows the word recognition accuracy corresponding to each method. "Baseline" indicates the baseline model that was trained using the face image only. In our proposed method, we adopt a dimension reduction in the model (the 4th layer in Table 1) to calculate the KD loss efficiently. However, the dimension reduction is removed here for evaluating the baseline model because it degraded the recognition accuracy of the baseline model. From this table, when using 250 utterances per word to train the model, our proposed model achieved a relative improvement of 3.57% compared to the baseline model, despite the comparable performance when using 500 utterances per word. Typically, the audio data has more information for recognizing the speech and shows better recognition accuracy than the use of the visual data. This result shows that the output from the ASR model worked as a powerful teacher. We also assume that KD affects the regularization because we obtained more improvement with less training data.
Table 2 Word recognition accuracy [%] for each method on the source domain
Next, we confirmed the effectiveness of our proposed method for UDA. For the baseline adaptation, we updated parameters using two losses: the cross entropy loss of the source domain (the third term of Eq. (7)) and the pseudo label of the target domain estimated by the source model itself. Table 3 shows the word recognition accuracy corresponding to each method. Our proposed method outperformed conventional UDA in the setting that uses the known class. The relative improvements compared to no adaptation are 12.53% for the baseline method and 21.88% for our method, respectively. Moreover, our proposed method also improved the classification accuracy compared to no adaptation by relatively 21.48% even when we used unknown class adaptation data. These results show that the intermediate-layer-based KD approach can transfer the sub-class representation that does not depend on the class. Therefore, by using such representation as an objective of the adaptation, it is possible to use the unknown class data for UDA.
Table 3 Word recognition accuracy [%] for each method
Moreover, we measured the performance of our UDA approach as a function of the number of adaptation utterances. As shown in Fig. 4, we observed that the accuracy decreases as the number of adaptation utterances decreases. We can see that the accuracies are saturated when using about 200 utterances for adaptation. Moreover, even when we use a smaller amount of the adaptation data, our method can adapt the model more effectively than the baseline method using all of the adaptation data (the fourth row in Table 3). These results demonstrate that our method can achieve stable and effective adaptation for UDA using the data of the unknown class.
The correlation between word recognition accuracies and the number of adaptation utterances. "Known class" and "Unknown class" indicate adaptation using the data of the known class and of the unknown class, respectively
Finally, we calculated the real time factor (RTF) that is the ratio of the recognition response to the utterance duration. Generally, RTF <1 is required for real-time scenarios. Here, decoding was performed on an Intel(R) Core(TM) i9-7900X CPU @ 3.30 GHz. The RTF of our system was 1.16. We consider that using a more efficient network architecture, such as MobileNet [34, 35], could improve the RTF while maintaining the performance.
Changing the division of the known/unknown words
In the experiments mentioned above, we used the fixed split for the known/unknown words. To evaluate the robustness of the variety of division pattern of the known/unknown words, we conducted 5-fold cross-validation for our proposed method. For this purpose, we split 500 words in LRW into 5 consecutive folds. Then, we used 100 words as the unknown class and the remaining 400 words as the known class. Table 4 shows the word recognition accuracy corresponding to each fold. The rightmost column in the table shows the mean value and the standard deviation. Our proposed method had a small standard deviation. This means that our method has high robustness for the selection of the words.
Table 4 Word recognition accuracy [%] for the 5-fold cross-validation
Noisy audio
To demonstrate the potential of our proposed method, we conducted the experiments in a more realistic scenario. For this purpose, we introduced acoustic noise for the audio in addition to brightness for the image during adaptation. White noise was added to audio signals, and their SNR was set to 30dB, 20dB, 10dB, and 0dB. As shown in Table 5, although the performance of our proposed method hardly varies among different SNRs (less than 1%), the use of the noisy audio signal significantly degraded the adaptation performance compared to using a clean audio signal.
Table 5 Word recognition accuracy [%] corresponding to each SNR
By comparing the results of "clean" and "30dB" in Table 5, we see that the recognition accuracy greatly degraded even though "30dB" was a small noise condition. In order to analyze these results, we measured how greatly the hidden representation of the audio signal ha, which is used as a teacher in our proposed cross-modal KD for UDA (see Eq. 4), is distorted by noise under each condition. For this measurement, we calculated the SNR under the hidden representation space as follows:
$$\begin{array}{*{20}l} \text{SNR} = 10\log_{10} \frac{||\boldsymbol{h}_{\text{clean}}^{a}||_{2}^{2}}{||\boldsymbol{h}_{\text{noisy}}^{a} - \boldsymbol{h}_{\text{clean}}^{a}||_{2}^{2}}, \end{array} $$
where \(\boldsymbol {h}_{\text {clean}}^{a}\) and \(\boldsymbol {h}_{\text {noisy}}^{a}\) denote the hidden representations ha obtained under clean and noisy (SNR = 30, 20, 10, 0dB) conditions, respectively. Table 6 shows the SNR of ha for each SNR of the input audio signal. As shown in this table, even when the SNR of the input audio signal was 30dB, the SNR of the hidden representation degraded to 14.14dB. Because this distorted hidden representation was used as a teacher in our proposed cross-modal KD, this result means that the proposed method is sensitive to the noise in the input audio signal. One possible reason for this sensitivity is that the audio model was trained using clean speech data and overfitted to the clean condition. Therefore, this degradation might be reduced if we use noisy audio data to train the noise-robust audio model. Nevertheless, the performance of our proposed system using the noisy audio signal still outperformed the baseline system (44.10, Known class adap. in Table 3) and the proposed system without adaptation (42.84 in Table 3) which do not use the audio signal.
Table 6 SNR of the hidden representation ha for each SNR of the input audio signal
In this paper, we proposed the intermediate-layer-based KD approach for UDA, which can effectively transfer the knowledge of the ASR model to the lip-reading model. Our method allows us to use the data of the unknown class to adapt the model from the source domain to the target domain. Experimental results show that our proposed method can adapt the model effectively regardless of whether the class of the adaptation data is known or unknown.
We used a simple network architecture based on stacked convolution layers because we assume an isolated word recognition task. In order to extend our approach for a continuous speech recognition task (i.e., sentence recognition task), we will investigate the use of recurrent neural network-based models which are suitable for this task, such as LipNet [17], in the future. In addition, we will demonstrate the effectiveness of our method in more complex transformations or more realistic environments. Our proposed method can use the audio-only database because the ASR model and the lip reading model are trained separately. Therefore, we will further investigate the combination with large audio databases. Our future work will also include the further investigation of its potential, focusing particularly on multi-modal tasks.
All data used in this study are included in the lip reading in the wild (LRW) dataset [5].
ASR:
GAN:
Generative adversarial network
KD:
Knowledge distillation
LRW:
Lip reading in the wild
OOV:
Out of vocabulary
RTF:
Real time factor
UDA:
Unsupervised domain adaptation
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Graduate School of System Informatics, Kobe University, Kobe, Japan
Yuki Takashima, Ryoichi Takashima, Ryota Tsunoda, Tetsuya Takiguchi & Yasuo Ariki
Information Technology R&D Center, Mitsubishi Electric Corporation, Ofuna, Japan
Ryo Aihara & Nobuaki Motoyama
Yuki Takashima
Ryoichi Takashima
Ryota Tsunoda
Ryo Aihara
Tetsuya Takiguchi
Yasuo Ariki
Nobuaki Motoyama
The first author mainly performed the experiments and wrote the paper, and the other authors reviewed and edited the manuscript. All of the authors discussed the final results. All of the authors read and approved the final manuscript.
Correspondence to Ryoichi Takashima.
Takashima, Y., Takashima, R., Tsunoda, R. et al. Unsupervised domain adaptation for lip reading based on cross-modal knowledge distillation. J AUDIO SPEECH MUSIC PROC. 2021, 44 (2021). https://doi.org/10.1186/s13636-021-00232-5
Lip reading | CommonCrawl |
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\vspace*{2cm} \begin{center} {\bf\large Modular cone metric spaces}
\\[0.5cm] {Saeedeh Shamsi Gamchi and Mohammad Janfada, Asadollah Niknam\\ Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, P.O. Box 1159-91775, Iran\\
Email: [email protected]\\ Email: [email protected]\\ Email: [email protected]}\\ [2mm]
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\begin{quotation} \noindent {\footnotesize {\sc Abstract.} In this paper the notion of modular cone metric space is introduced and some properties of such spaces are investigated. Also we define convex modular cone metric which takes values in $C_{\mathbb{R}}(Y)$ where $Y$ is a compact Hausdorff space. Then a fixed point theorem is proved for contractions in these spaces. Furthermore , we make a remark on paper \cite{chwspk} and it will be proved that their fixed point result in modular metric spaces is not true. } \end{quotation} \ \\ {\bf Keywords:ordered spaces, modular cone metric spaces, convergence, fixed point theorem.} \\
\noindent \textbf{2010 Mathematics subject classification: } Primary: ; Secondary: .
\markboth {S. Shamsi Gamchi and M. Janfada, A. Niknam}
{Modular cone metric spaces}
\section {\sc Introduction and preliminaries} Ordered normed spaces and cones have applications in applied mathematics and optimization theory \cite{DN}. Replacing the real numbers, as the codomain of metrics, by ordered Banach spaces one may obtain a generalization of metric spaces. Such generalized spaces called cone metric spaces, were introduced by Rzepecki \cite{R}. It is proved in \cite{cak} that every cone metric space is metraizable.\\In this paper, which is split into two parts, our aim is to develop the theory of cone metric spaces called modular cone metric spaces. In the first part, the notion of modular cone metric space is introduced and some properties of such spaces are investigated. Also we define convex modular cone metric which takes values in $C_{\mathbb{R}}(Y)$ where $Y$ is a compact Hausdorff space. Then a fixed point theorem is proved for contractions in these spaces. In the second part of this paper, we make a remark on paper \cite{chwspk} and it will be proved that their fixed point result in modular metric spaces is not true.\\ Let $E$ be a topological vector space (TVS, for short) with its zero vector $\theta$. A nonempty subset $P$ of $E$ is called a convex cone if $P+P\subseteq E$ and $\lambda P\subseteq P$ for all $\lambda\geq 0$. A convex cone $P$ is said to be pointed if $P\cap(-P)=\{\theta\}$. For a given convex cone $P$ in $E$, a partial ordering $\preceq$ on $E$ with respect to $P$ is defined by $x\preceq y$ if and only if $y-x\in P$. We shall write $x\prec y$ if $x\preceq y$ and $x\neq y$, while $x\ll y$ will stand for $y-x\in int P$, where $int P$ denotes the topological interior of $P$.\\ In the sequel, we will need the following useful lemmas: \begin{lem}\cite{jbr}\label{b} Let $P$ be a cone in $E$. Then:\\(i) If $\theta \preceq a_n\rightarrow \theta$, then for each $c\in intP$, there exists $N\in\mathbb{N}$ such that for every $n>N$, $a_n\ll c$.\\(ii) For every $c_1,c_2\in intP$, there exists $c\in intP$ such that $c\ll c_1$ and $c\ll c_2$.\\(iii) for every $a\in P$ and $c\in intP$, there exists $n_0\in\mathbb{N}$ such that $a\ll n_0 c$ \end{lem}
The nonlinear scalarization function $\xi_{e}: E\rightarrow \mathbb{R}$ is defined as follows:$$\xi_{e}(y)=\inf \{r\in\mathbb{R}: y\in re-P\}$$for all $y\in E$.\\
\begin{lem}\cite{chy}\label{c} Let $r\in\mathbb{R}$ and $y\in E$, then \\(i) $\xi_{e}(y)\leq r \Leftrightarrow re-y\in P$,\\(ii) $\xi_{e}(y)< r \Leftrightarrow re-y\in intP$,\\(iii) $\xi_{e}(.)$
is positively homogeneous and continuous on $E$;\\(iv) If $y_{1}\in y_{2} + P$,
then $\xi_{e}(y_{2})\leq \xi_{e}(y_{1})$;\\(v) $\xi_{e}(y_{1}+ y_{2})\leq\xi_{e}(y_{1})+\xi_{e}(y_{2})$. \end{lem} \begin{defn}\cite{WSD} Let $X$ be a nonempty set and $d:X\times X\rightarrow E$ be a mapping that satisfies:\\(CM1) For all $x,y\in X$, $d(x,y)\succeq\theta$ and $d(x,y)=\theta$ if and only if $x=y$,\\(CM2) $d(x,y)=d(y,x)$ for all $x,y\in X$,\\(CM3) $d(x,y)\preceq d(x,z)+d(z,y)$ for all $x,y,z\in X$.\\Then $d$ is called a topological vector space cone metric (TVS cone metric, for short) on $X$ and $(X,d)$ is said to be a topological vector space cone metric space. \end{defn}
\begin{defn}\cite{Vch1} Let $X$ be a nonempty set. A function $w: (0,\infty)\times X\times X \rightarrow [0,\infty]$ is said to be a modular metric on $X$, if it satisfies the following three axioms:\\ (i) for given $x,y\in X$, $w_{\lambda}(x,y)=0$ for all $\lambda>0$ if and only if $x=y$,\\(ii) $w_\lambda(x,y)=w_\lambda(y,x)$ for all $x,y\in X$ and $\lambda>0$,\\ (iii) $w_{\lambda+\mu}(x,y)\leq w_{\lambda}(x,z)+w_{\mu}(z,y)$ for all $x,y,z\in X$ and $\lambda,\mu\in (0,\infty)$.\\If, instead of (i), the function $w$ satisfies only\\$(i')$ $w_\lambda(x, x) =0$ for all $\lambda>0$ and $x\in X$,\\then $w$ is said to be a pseudo modular on $X$, and if $w$ satisfies $(i')$ and\\ $(i'')$ for given $x,y\in X$, if there exists $\lambda>0$ such that $w_\lambda(x,y)=0$, then $x=y$.\\the function $w$ is called a strict modular on $X$.\\mathcal{A} modular (pseudomodular, strict modular) $w$ on $X$ is said to be convex if, instead of (iii), for all $\lambda,\mu > 0$ and $x, y, z\in X$ it satisfies the inequality\\$$w_{\lambda+\mu}(x,y)\leq \frac{\lambda}{\lambda+\mu}w_{\lambda}(x,z)+\frac{\mu}{\lambda+\mu}w_{\mu}(z,y)$$ \end{defn} Note that the conditions (i) to (iii) imply that for all $y,z\in X$ and $\lambda>0$, $w_\lambda(y,z)\geq 0$. If $w_\lambda(x,y)$ does not depend on $x,y\in X$, then by (i) $w\equiv 0$. Now if $w_\lambda(x,y)=w(x,y)$ is independent of $\lambda>0$, then axioms (i)-(iii) mean that $w$ is a metric on X.\\The essential property of a pseudo modular $w$ on $X$ (cf. \cite{Vch1}, Section 2.3) is that, for any given $x, y\in X$, the function $0 < \lambda\mapsto w_\lambda(x, y)$ is decreasing on $(0,\infty)$. \begin{defn}\cite{Vch1} Let $w$ be pseudo modular on $X$, the two sets $$X_w=X_w(x_0) = \{x \in X : w_\lambda(x, x_0)\rightarrow 0 ~ as ~ \lambda\rightarrow \infty\}$$ and $$X_w^*=X_w^*(x_0) = \{x \in X : \exists\lambda=\lambda(x) > 0 ~ \mbox{such that} ~ w_\lambda(x, x_0) < \infty\}$$ are said to be modular spaces (around $x_0$). \end{defn} If $w$ is a convex modular on $X$, then according to \cite{Vch1}, Section 3.5 and Theorem 3.6, the two modular spaces coincide, $X_w = X_w^*$. \begin{defn}\cite{chwspk} Let $w$ be a modular metric on $X$, $x\in X_w$ and $\{x_{n}\}$ be a sequence in $X_w$. Then\\(1) $\{x_{n}\}$ is said to be modular convergent to $x$ if for every $\lambda>0$, $w_\lambda(x_n,x)\rightarrow 0$ as $n\rightarrow \infty$.\\ (2) $\{x_{n}\}$ is said to be a modular Cauchy sequence if $\lambda>0$, $w_\lambda(x_n,x_m)\rightarrow 0$ as $n,m\rightarrow \infty$\\(3) $X_w$ is called a complete modular metric space if every modular Cauchy sequence is modular convergent. \end{defn} \begin{thm}\cite{chist3}\label{d} Let $w$ be a convex modular metric on $X$ and $X_w^*$ be a complete modular metric space. Suppose that $T: X_w^*\rightarrow X_w^*$ is a mapping which satisfies the following condition:$$w_{k\lambda}(Tx,Ty)\leq w_\lambda(x,y) ~ ~ \mbox{for all} ~ ~ \theta\ll c\ll c_0.$$Then $T$ has a unique fixed point. \end{thm}
\section {\sc Convergence in modular cone metric spaces} In the following, unless otherwise specified, we always suppose that $E$ is a locally convex Hausdorff TVS with its zero vector $\theta$, $P$ a proper closed and convex pointed cone in $E$ with $int P\neq\emptyset$, $e\in int P$ and $\preceq$ the partial ordering with respect to $P$.
Throughout this paper functions $w: intP\times X\times X \rightarrow E$ will be written as $w(c,x,y)=w_c(x,y)$ for all $c\in intP$ and $x,y\in X$.
\begin{defn}\label{a} Let $X$ be a nonempty set. A function $w: intP\times X\times X \rightarrow E$ is said to be a modular cone metric on $X$, if it satisfies the following three axioms:\\ (i) For given $x,y\in X$, $w_{c}(x,y)=\theta$ for all $c\in intP$ if and only if $x=y$,\\(ii) $w_c(x,y)=w_c(y,x)$, for all $x,y\in X$ and $c\in intP$,\\ (iii) $w_{c_1+c_2}(x,y)\preceq w_{c_1}(x,z)+w_{c_2}(z,y)$ for all $x,y,z\in X$ and $c_1,c_2\in intP$.\\$(X,E,P,w)$ is called a modular cone metric space. \end{defn} Note that the conditions (i) to (iii) imply that for all $y,z\in X$ and $c\in intP$, $\theta\preceq w_c(y,z)$. Indeed, by setting $x=y$ and $c_1=c_2=c$ in (iii), for all $y,z\in X$ one gets $\theta=w_{2c}(y,y)\preceq 2w_c(y,z)$.\\If $w_c(x,y)$ does not depend on $x,y\in X$, then by (i) $w\equiv 0$. Now if $w_c(x,y)=w(x,y)$ is independent of $c\in intP$, then axioms (i)-(iii) mean that $w$ is a cone metric on X.\\ For given $x,y\in X$, the function $\theta\ll c\mapsto w_c(x,y)$ is decreasing on $intP$. In fact, if $\theta\ll c_1\ll c_2$, then $w_{c_2}(x,y)\preceq w_{c_2-c_1}(x,x)+w_{c_1}(x,y)=w_{c_1}(x,y)$.\\ By the following example, we may construct a of examples of modular cone metric spaces using a cone metric. \begin{exam} Let $(X,d)$ be a cone metric space and $\varphi : intP\rightarrow (0,\infty)$ be a decreasing function. Now define $w_c(x,y)=\varphi(c)d(x,y)$. It is easy to see that $w$ is a modular cone metric on $X$. Indeed, by the properties of a cone metric $d$ axioms (i)-(ii) of definition \ref{a} are satisfied. On the other hand, for all $x,y,z\in X$ and $c_1,c_2\in intP$, we have: \begin{align*} w_{c_1+c_2}(x,y)=&\varphi(c_1+c_2)d(x,y)\\ &\preceq \varphi(c_1+c_2)d(x,z)+\varphi(c_1+c_2)d(z,y)\\ &\preceq \varphi(c_1)d(x,z)+\varphi(c_2)d(z,y)\\ &=w_{c_1}(x,z)+w_{c_2}(z,y). \end{align*} \end{exam}
Now we define some convergence concepts in this space. It will be proved that these definitions are compatible with some topology on $X$ constructed by a modular cone metric. \begin{defn} Let $(X,E,P,w)$ be a modular cone metric space, $x\in X$ and $\{x_{n}\}$ be a sequence in $X$. Then\\(1) $\{x_{n}\}$ is said to be modular cone convergent to $x$ and we denote it by $x_n \overset{w}{\longrightarrow} x$ if for every $c\gg\theta$ there exists a positive integer $N$ such that for all $n>N$, $w_c(x_{n}, x)\ll c$.\\ (2) $\{x_{n}\}$ is said to be a modular cone Cauchy sequence if for every $c\gg\theta$ there exists a
positive integer $N$ such that for all $m,n>N$, $w_c(x_{n}, x_{m})\ll c$.\\(3) $(X, w)$ is called a complete modular cone metric space if every modular cone Cauchy sequence is convergent. \end{defn}
Let $E$ be a Banach space and $P$ be a cone in $E$. The cone $P$ is called normal if there exists a constant $K>0$ such that for all $a,b\in P$, $a\preceq b$ implies that $\|a\|\leq K\|b\|$. The least positive number satisfying the above inequality is called the normal constant of $P$.\\ In the next theorem some equivalent condition for convergence is proved. \begin{thm} Let $E$ be a Banach space and $P$ be a normal cone with normal constant $K$. Suppose that $(X,E,P,w)$ is a modular cone metric space and $\{x_n\}$ is a sequence in $X$. Then,\\
(i) $x_n \overset{w}{\longrightarrow} x$ if and only if for each $c\in intP$, $\|w_c(x_n,x)\|\rightarrow 0$ as $n\rightarrow \infty$.\\
(ii) $\{x_n\}$ is modular cone Cauchy if and only if for each $c\in intP$, $\|w_c(x_n,x_m)\|\rightarrow 0$ as $n,m\rightarrow \infty$. \end{thm} \begin{proof}
Let $x_n \overset{w}{\longrightarrow} x$. Fix $c\in intP$. Then for every $0<\varepsilon <1$, there exists a positive integer $N$ such that for all $n>N$, $w_{\varepsilon c}(x_{n}, x)\ll \varepsilon c$. On the other hand, so for each $n$ we have $w_c(x_n,x)\preceq w_{\varepsilon c}(x,y) $, since $\varepsilon c\ll c$. Hence, $w_c(x_n,x)\ll \varepsilon c$, for each $n>N$. Therefore, normality of $P$ implies that $\|w_c(x_n,x)\|\leq K \varepsilon \|c\|$, for each $n>N$. For proving its converse, let for each $c\in intP$, $\|w_c(x_n,x)\|\rightarrow 0$ as $n\rightarrow \infty$. Let $c\in intP$. By assumption we have $w_c(x_n,x)\rightarrow \theta$ as $n\rightarrow \infty$. Applying Lemma \ref{b} there is a positive integer $N$ such that for all $n>N$, $w_c(x_n,x)\ll c$. Thus the proof of (i) is complete.\\For proving (ii) let $\{x_n\}$ be a modular cone Cauchy sequence and fix $c\in intP$. Similar to (i) one can show that $\|w_c(x_n,x_m)\|\rightarrow 0$ as $n,m\rightarrow \infty$, for each $c\in intP$. Conversely, let $w_c(x_n,x_m)\rightarrow \theta$ as $n,m\rightarrow \infty$, for each $c\in intP$. For given $c\in intP$, there exists $\varepsilon>0$ such that $c-y\in intP$ for each $y\in B_{\varepsilon}(\theta)$. For this $\varepsilon>0$, there is a positive integer $N$ such that for all $n,m>N$, $\|w_c(x_n,x_m)\|<\varepsilon$. Therefore, for all $n,m>N$, $c-w_c(x_n,x_m)\in intP$, that means $w_c(x_n,x_m)\ll c$ for all $n,m>N$. Thus the proof is complete. \end{proof} Now we are going to construct a topology on $X$, where $(X,E,P,w)$ is a modular cone metric space. For any $x\in X$, $c\in intP$, $B_w(x,c)=\{y\in X: W_c(x,y)\ll c\}$. \begin{thm} Let $(X,E,P,w)$ be a modular cone metric space. Then $$\tau_w=\{U\subset X: \forall x\in U ~ \exists c\in intP ~ s.t ~ B_w(x,c)\subset U\}$$forms a Hausdorff topology on $X$. \end{thm} \begin{proof} Trivially $\emptyset , X \in \tau_w$.\\ Also for any $U, V \in \tau_w $ and $x\in U \cap V$, then $x\in U$ and $x \in V$, one may find $c_1, c_2\in intP$ such that $x\in B_w(x, c_1)\subset U$ and $x\in B_w(x, c_2)\subset V$. By Lemma \ref{b}, there exists $c\in int P$ such that $c\ll c_1$ and $c \ll c_2$. Now suppose that $y\in B_w(x,c)$, so we have $w_{c_1}(x,y)\preceq w_c(x,y)\ll c\ll c_1$, hence, $y\in B_w(x,c_1)$. Similarly we have $y\in B_w(x,c_2)$. Thus $B_w(x, c)\subset B_w(x, c_1)\cap B_w(x, c_2)\subset U\cap V$. Therefore $U\cap V\in \tau_w$.\\ Let $U_\alpha\in \tau_w$ for each $\alpha\in\Delta$ and let $x\in \bigcup_{\alpha\in\Delta} U_{\alpha}$, then $\exists\alpha_{0}\in \Delta$ such that $x\in U_{\alpha_0}$. Hence, $x\in B(x, c)\subset U_{\alpha_0}$, for some $c\in int P$. That is $\bigcup_{\alpha\in\Delta} U_{\alpha}\in\tau$. Thus $\tau_w$ is a topology on $X$.\\ Now we prove that this topology is a Hausdorff topology. Suppose that $x,y\in X$ and $x\neq y$. So by the property $(i)$ of Definition \ref{a} there exists $c\in intP$ such that $w_{c_0}(x,y)\neq \theta$. In contrary, we assume that for each $c\in intP$, $$B_w(x,c)\cap B_w(y,c)\neq\emptyset .$$Hence, for each $n\in\mathbb{N}$, there is a $z_n\in X$ such that $$z_n\in B_w(x,\frac{c_0}{2n})\cap B_w(y,\frac{c_0}{2n}).$$Thus for each $n>1$, we have \begin{align*} w_{c_0}(x,y)&\preceq w_{\frac{c_0}{n}}(x,y)\\ &\preceq w_{\frac{c_0}{2n}}(x,z_n)+w_{\frac{c_0}{2n}}(x,z_n)\\ & \ll\frac{c_0}{2n}+\frac{c_0}{2n}\\ &=\frac{c_0}{n} \end{align*} Therefore for each $n>1$, $\frac{c_0}{n}-w_{c_0}(x,y)\in P$. But $P$ is closed, so $$\lim_{n\rightarrow\infty}(\frac{c_0}{n}-w_{c_0}(x,y))=-w_{c_0}(x,y)\in P.$$This is a contradiction since $P\cap (-P)=\{\theta\}$. So the proof is complete. \end{proof} One can easily see that the collection $\{B_w(x, c): x\in X, c\in intP\}$ forms a basis for $\tau_w$ under which the above definitions of convergent and Cauchy sequences are fully justified.\\ \begin{thm} Every modular cone metric space $(X,E,P,w)$ is first countable. \end{thm} \begin{proof} Let $x\in X$. Fix $c\in intP$. We show that $\beta_x =\{B_w(x, \frac{c}{n}): n\in\mathbb{N}\}$ is a local base at $x$. Let $U$ be an open set such that $x\in U$. Therefore, there is a $c_1\in intP$ such that $B_w(x,c_1)\subset U$. on the other hand there exists a positive integer $n$, such that $\frac{c}{n}\ll c_1$. Now suppose that $y\in B_w(x,\frac{c}{n})$, so we have $w_{c_1}(x,y)\preceq w_{\frac{c}{n}}(x,y)\ll \frac{c}{n}\ll c_1$, hence, $y\in B_w(x,c_1)$. Thus $B_w(x, \frac{c}{n})\subset B_w(x, c_1)\subset U$. \end{proof} \begin{thm} Let $(X,E,P,w)$ be a modular cone metric space. A self map $f:X\rightarrow X$ is continuous at $x \in X$ if and only if whenever $x_n \rightarrow x$, we have $f(x_n)\rightarrow f(x)$, as $n\rightarrow \infty$. \end{thm} \begin{proof} Applying Theorem \ref{e} complete the proof. \end{proof}
The following useful theorem shows that we may construct a family of modular metrics using a modular cone metric and the mapping $\xi_e$. \begin{thm}\label{e} Let $(X,E,P,w)$ be a modular cone metric space and $ e\in intP$. Then $W^e: (0,\infty) \times X \times X \rightarrow [0,\infty)$ which is defined by $W_\lambda(x,y)=\xi_e(w_{\lambda e}(x,y))$ is a modular metric on $X$. \end{thm} \begin{proof} Let $x,y\in X$. If $x=y$, then $w_{\lambda e}(x,y)=\theta$ for each $\lambda>0$, so $W_{\lambda}^e(x,y)=\xi_e(w_{\lambda}(x,y))=0$ for each $\lambda>0$. Now, let for each $\lambda>0$, $W_{\lambda}^e(x,y)=0$. So $w_{\lambda e}(x,y)=0\theta$ for each $\lambda>0$. On the other hand for each $c\in intP$ there exists a $\lambda>0$ such that $\lambda e\ll c$, hence $w_c(x,y)\preceq w_{\lambda e}(x,y)=\theta$. Therefore, $w_c(x,y)=\theta $, for each $c\in intP$. That means $x=y$. So we prove that $x=y$ if and only if $W_{\lambda}^e(x,y)=0$, for each $\lambda>0$.\\It is easy to see that $W_{\lambda}^e(x,y)=W_{\lambda}^e(y,x)$, for all $\lambda>0$ and $x,y\in X$.\\Also if $x,y,z\in X$ and $\lambda_1, \lambda_2>0$, then \begin{align*} W_{\lambda_1+\lambda_2}^e(x,y)=&\xi_e(w_{\lambda_1 e+\lambda_2 e}(x,y))\\ &\leq \xi_e(w_{\lambda_1 e}(x,z))+\xi_e(w_{\lambda_2 e}(z,y))\\ &= W_{\lambda_1}^e(x,z)+ W_{\lambda_2}^e(z,y) \end{align*} \end{proof}
\begin{thm} Let $(X,E,P,w)$ be a modular cone metric space, $e\in intP$, $x\in X$ and $\{x_n\}$ be a sequence in $X$. Then \\ (i) $x_n \overset{w}{\longrightarrow} x$ if and only if for every $\lambda>0$, $W_{\lambda}^e(x_n,x)\longrightarrow 0$ as $n\longrightarrow \infty$.\\ (ii) $\{x_n\}$ is a $w$-cauchy sequence if and only if for every $\lambda>0$, $W_{\lambda}^e(x_n,x_m)\longrightarrow 0$ as $n,m\longrightarrow \infty.$ \end{thm} \begin{proof} (i) Let $x_n \overset{w}{\longrightarrow} x$. Fix $\lambda>0$. For given $\epsilon >0$ if $\lambda > \epsilon$, then for $c=\epsilon e$, there is a positive integer $N$ such that for each $n>N$, $w_{\epsilon e}(x_n,x)\ll \epsilon e$. On the other hand, so $w_{\lambda e}(x_n,x)\preceq w_{\epsilon e}(x_n,x)\ll \epsilon e$, since $\epsilon e\ll \lambda e$. Hence by the property (iv) of Lemma \ref{c}, for each $n>N$ we have $$W_{\lambda}(x_n,x)\leq W_{\epsilon}(x_n,x)=\xi_e(w_{\epsilon e}(x_n,x))<\xi_e( \epsilon e)=\epsilon.$$Now suppose that $\lambda\leq\epsilon$. For $c=\lambda e$, there is a positive integer $N$ such that for each $n>N$, $w_{\lambda e}(x_n,x)\ll \lambda e$. Hence, for each $n>N$ we have $$W_{\lambda}(x_n,x)=\xi_e(w_{\lambda e}(x_n,x))<\xi_e( \lambda e)=\lambda <\epsilon.$$For its converse, let for each $\lambda >0$, $W_{\lambda}^e(x_n,x)\longrightarrow 0$. Fix $c\in intP$, By Lemma \ref{b}, there exists a positive real number $\lambda_0$ such that $\lambda_0 e\ll c$. So, there is a positive integer $N$ such that for each $n>N$,$$\xi_e(w_{\lambda_0 e}(x_n,x))=W_{\lambda_0}(x_n,x) <\lambda_0 .$$Thus by property (ii) of Lemma \ref{c} and the fact that the function $\theta\ll c\mapsto w_c(x,y)$ is decreasing on $intP$, we have $$w_{c}(x_n,x)\preceq w_{\lambda_0 e}(x_n,x)\ll \lambda_0 e\ll c,$$for each $n>N$.\\ The proof of (ii) is similar to (i). \end{proof}
\section{\sc A fixed point theorem for contractions in convex modular cone metric spaces} In this section we suppose that $E=C_{\mathbb{R}}(Y)$ where $Y$ is a compact Hausdorff space. $E$ with sup-norm is a real Banach space. Put $P=\{f\in C_{\mathbb{R}}(Y): f(x)\geq 0 ~ \mbox{for any} x\in Y \}$. It is clear that $P$ is a cone in $E$. Partial order on $E$ with respect to $P$ is defined as follows:$$f\preceq g \Leftrightarrow f(x)\leq g(x) ~ \forall x\in Y.$$Note that $intP\neq\emptyset$. Indeed, $1\in intP$. \begin{lem} If $f\in intP$, then $f$ is invertible. \end{lem}
\begin{proof}
Let $f\in intP$. Then there exists $\epsilon >0$ such that $$B_{\epsilon}(f)=\{g\in C_{\mathbb{R}}(Y): \|g-f\|_{\infty}<\epsilon\}\subset P.$$If, in contrary, there is a $t_0 \in Y$ such that $f(t_0)=0$, then by defining $g(t)=f(t)-\frac{\epsilon}{2}$, it is clear that $g\in B_{\epsilon}(f)$ and hence $g\in P$. But $g(t_0)<0$ and this is a contradiction. Thus $0\notin rang(f)$ and so $f$ is invertible. \end{proof} Now we give the concept of convex modular cone metric spaces. \begin{defn} Let $X$ be a nonempty set. A function $w: intP\times X\times X \rightarrow E$ is said to be a convex modular cone metric on $X$, if it satisfies the following three axioms:\\ (i) For given $x,y\in X$, $w_{c}(x,y)=\theta$, for all $c\in intP$, if and only if $x=y$,\\(ii) $w_c(x,y)=w_c(y,x)$, for all $x,y\in X$ and $c\in intP$,\\ (iii) $w_{c_1+c_2}(x,y)\preceq \frac{c_1}{c_1+c_2} w_{c_1}(x,z)+\frac{c_2}{c_1+c_2} w_{c_2}(z,y)$, for all $x,y,z\in X$ and $c_1,c_2\in intP$.\\$(X,w)$ is called a convex modular cone metric space. \end{defn} \begin{exam} Let $(X,d)$ be a cone metric space. Now define $w_c(x,y)=\frac{d(x,y)}{c}$. It is easy to see that $w$ is a convex modular cone metric on $X$. \end{exam}
Note that every convex modular cone metric space is a modular cone metric space.\\ If instead of (i), the function $w$ satisfies in $(i')$ and $(i")$, then it is called strict convex modular cone metric.\\ $(i')$ $w_c(x, x) =\theta$ for all $c\in intP$ and $x\in X$,\\ $(i")$ Given $x,y\in X$, if there exists $ c\in intP$ such that $w_c(x,y)=\theta$, then $x=y$. \begin{rem} Let $(X,w)$ be a convex modular cone metric space and $1\in E=C_{\mathbb{R}}(Y)$ be the unit constant function. It is easy to see that $W_{\lambda}^1(x,y)=\xi_1(w_{\lambda 1}(x,y))$ is a convex modular metric on $X$ and $W_{\lambda}^1(x,y)<\infty$ so $X_W^*=X$. \end{rem} \begin{thm} Let $(X,w)$ be a complete convex modular cone metric space. Suppose there exists a constant number $k\in (0,1)$ and $c_0\in intP$ such that a mapping $T: X\rightarrow X$ satisfies the following condition:\\
$$w_{kc}(Tx,Ty)\preceq w_c(x,y) ~ ~ \mbox{for all} ~ ~ \theta\ll c\ll c_0.$$Then $T$ has a unique fixed point. \end{thm} \begin{proof}
Take $\lambda_0=\|c_0\|_{\infty}$. For all $0<\lambda <\lambda_0$ by the above inequality for convex modular metric $W_{\lambda}^1(x,y)=\xi_1(w_{\lambda 1}(x,y))$ we have$$W_{k\lambda}^1(x,y)=\xi_1(w_{k\lambda 1}(x,y))\leq \xi_1(w_{\lambda 1}(x,y))=W_{\lambda}^1(x,y).$$Since $X_W^*=X$ and $(X,w)$ is complete, so $(X_{W}^*,W)$ is complete. Hence, the proof is complete by Theorem \ref{d}. \end{proof}
\section{\sc A note on "Fixed point theorems for contraction mappings in modular metric spaces"}
Recently, in the paper "Ch. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contraction mapping in modular metric spaces, Fixed Point Theory and Applications, (2011)", the authors have studied and introduced some fixed-point theorems in the framework of a modular metric space. We will first state the main result that the authors have proved in that paper and then by constructing an example we will show that the result is not true. \begin{defn} (Definition 3.1. of original paper) Let $w$ be a modular metric on $X$ and $X_w$ be a modular metric space induced by $w$ and $T : X_w \rightarrow X_w$ be an arbitrary mapping. A mapping $T$ is called a contraction if for each $x, y \in X_w$ and for all $\lambda >0$ there exists $0 < k <1$ such that$$w_\lambda(Tx, Ty)\leq kw_\lambda(x, y).$$ \end{defn} \begin{thm} $($Theorem $3.2.$ of original paper$)$ Let $w$ be a modular metric on $X$ and $X_w$ be a modular metric space induced by $w$. If $X_w$ is a complete modular metric space and $T : X_w \rightarrow X_w$ is a contraction mapping, then $T$ has a unique fixed point in $X_w$. Moreover, for any $x\in X_w$, iterative sequence $\{T_nx\}$ converges to the fixed point. \end{thm} Now by following example we show that the above theorem is not valid. \begin{exam} Let $X = \{(a, 0)\in\mathbb{R}^2: \frac{1}{2}\leq a \leq 1\}\cup \{(0, b) \in\mathbb{R}^2: \frac{1}{2}\leq b \leq 1\}$. Defined the mapping $w : (0, \infty) \times X \times X\rightarrow [0, \infty]$ by
$$w_\lambda((a_1, 0), (a_2, 0)) =\frac{4|a_1-a_2|}{3\lambda}'$$
$$w_\lambda((0, b_1), (0, b_2)) = \frac{|b_1-b_2|}{\lambda},$$ and $$w\lambda((a, 0), (0, b)) =\frac{4a}{3\lambda}+\frac{b}{\lambda}= w_\lambda((0, b), (a, 0)).$$We note that if we take $\lambda\rightarrow\infty$, then we see that $X = X_w$ and also it is easy to see that if $\{x_n\}$ is a Cauchy sequence in $X_w$ then we just have one of the following assertions:\\$(1)$ There exists a positive integer $N$ such that for each $n>N$, $x_n\in\{(a, 0)\in\mathbb{R}^2: \frac{1}{2}\leq a \leq 1\}$.\\$(2)$ There exists a positive integer $N$ such that for each $n>N$, $x_n\in \{(0, b) \in\mathbb{R}^2: \frac{1}{2}\leq b \leq 1\}$.\\Indeed, It follows from the fact that for $(a,0), (0,b)\in X$, $$w_\lambda((a, 0), (0, b)) =\frac{4a}{3\lambda}+\frac{b}{\lambda}\geq \frac{7}{3\lambda}.$$Thus $X_w$ is a complete modular metric space. Now we define a mapping $T:X_w\rightarrow X_w$ by $$T((a, 0)) = (0, a)$$and$$T((0, b)) =(\frac{b}{2},0)$$Simple computations show that $$w_\lambda(T((a_1, b_1)), T((a_2, b_2)))\leq \frac{3}{4}w_\lambda((a_1, b_1), (a_2, b_2)),$$for all $(a_1, b_1), (a_2, b_2)\in X_w$. But $T$ does not have any fixed point in $X_w$. \end{exam}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\end{document} | arXiv |
Moffat distribution
The Moffat distribution, named after the physicist Anthony Moffat, is a continuous probability distribution based upon the Lorentzian distribution. Its particular importance in astrophysics is due to its ability to accurately reconstruct point spread functions, whose wings cannot be accurately portrayed by either a Gaussian or Lorentzian function.
Characterisation
Probability density function
The Moffat distribution can be described in two ways. Firstly as the distribution of a bivariate random variable (X,Y) centred at zero, and secondly as the distribution of the corresponding radii
$R={\sqrt {X^{2}+Y^{2}}}.$
In terms of the random vector (X,Y), the distribution has the probability density function (pdf)
$f(x,y;\alpha ,\beta )={\frac {\beta -1}{\pi \alpha ^{2}}}\left[1+\left({\frac {x^{2}+y^{2}}{\alpha ^{2}}}\right)\right]^{-\beta },\,$
where $\alpha $ and $\beta $ are seeing dependent parameters. In this form, the distribution is a reparameterisation of a bivariate Student distribution with zero correlation.
In terms of the random variable R, the distribution has density
$f(r;\alpha ,\beta )=2{\frac {\beta -1}{\alpha ^{2}}}\left[1+{\frac {r^{2}}{\alpha ^{2}}}\right]^{-\beta }.\,$
Relation to other distributions
• Pearson distribution
• Student's t-distribution for $\beta ={\frac {\alpha ^{2}+1}{2}}$
• Normal distribution for $\beta ={\frac {\alpha ^{2}}{2}}\rightarrow \infty $, since for the exponential function $\exp x=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.$
References
• A Theoretical Investigation of Focal Stellar Images in the Photographic Emulsion (1969) – A. F. J. Moffat
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
| Wikipedia |
New methods in spectral theory of $N$-body Schr\"odinger operators (1804.07874)
T. Adachi, K. Itakura, K. Ito, E. Skibsted
April 21, 2018 math-ph, math.MP, math.FA
We develop a new scheme of proofs for spectral theory of the $N$-body Schr\"odinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich's theorem, limiting absorption principle bounds, microlocal resolvent bounds, H\"older continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich's theorem which is unified with exponential decay estimates studied previously only for $L^2$-eigenfunctions. Each pair-potential is a sum of a long-range term with first order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with `zeroth order' operators. In particular they do not rely on Mourre's differential inequality technique.
Search for Neutrinos in Super-Kamiokande associated with the GW170817 neutron-star merger (1802.04379)
K. Abe, C. Bronner, Y. Hayato, M. Ikeda, K. Iyogi, J. Kameda, Y. Kato, Y. Kishimoto, Ll. Marti, M. Miura, S. Moriyama, M. Nakahata, Y. Nakajima, Y. Nakano, S. Nakayama, A. Orii, G. Pronost, H. Sekiya, M. Shiozawa, Y. Sonoda, A. Takeda, A. Takenaka, H. Tanaka, S. Tasaka, T. Yano, R. Akutsu, T. Kajita, Y. Nishimura, K. Okumura, K. M. Tsui, L. Labarga, P. Fernandez, F. d. M. Blaszczyk, C. Kachulis, E. Kearns, J. L. Raaf, J. L. Stone, L. R. Sulak, S. Berkman, S. Tobayama, J. Bian, M. Elnimr, W. R. Kropp, S. Locke, S. Mine, P. Weatherly, M. B. Smy, H. W. Sobel, V. Takhistov, K. S. Ganezer, J. Hill, J. Y. Kim, I. T. Lim, R. G. Park, Z. Li, E. O'Sullivan, K. Scholberg, C. W. Walter, M. Gonin, J. Imber, Th. A. Mueller, T. Ishizuka, T. Nakamura, J. S. Jang, K. Choi, J. G. Learned, S. Matsuno, J. Amey, R. P. Litchfield, W. Y. Ma, Y. Uchida, M. O. Wascko, M. G. Catanesi, R. A. Intonti, E. Radicioni, G. De Rosa, A. Ali, G. Collazuol, L. Ludovici, S. Cao, M. Friend, T. Hasegawa, T. Ishida, T. Ishii, T. Kobayashi, T. Nakadaira, K. Nakamura, Y. Oyama, K. Sakashita, T. Sekiguchi, T. Tsukamoto, KE. Abe, M. Hasegawa, A. T. Suzuki, Y. Takeuchi, T. Hayashino, S. Hirota, M. Jiang, M. Mori, KE. Nakamura, T. Nakaya, R. A. Wendell, L. H. V. Anthony, N. McCauley, A. Pritchard, Y. Fukuda, Y. Itow, M. Murase, F. Muto, P. Mijakowski, K. Frankiewicz, C. K. Jung, X. Li, J. L. Palomino, G. Santucci, C. Viela, M. J. Wilking, C. Yanagisawa, D. Fukuda, H. Ishino, S. Ito, A. Kibayashi, Y. Koshio, H. Nagata, M. Sakuda, C. Xu, Y. Kuno, D. Wark, F. Di Lodovico, B. Richards, S. Molina Sedgwick, R. Tacik, S. B. Kim, A. Cole, L. Thompson, H. Okazawa, Y. Choi, K. Ito, K. Nishijima, M. Koshiba, Y. Suda, M. Yokoyama, R. G. Calland, M. Hartz, K. Martens, M. Murdoch, B. Quilain, C. Simpson, Y. Suzuki, M. R. Vagins, D. Hamabe, M. Kuze, Y. Okajima, T. Yoshida, M. Ishitsuka, J. F. Martin, C. M. Nantais, H. A. Tanaka, T. Towstego, A. Konaka, S. Chen, L. Wan, A. Minamino
March 29, 2018 astro-ph.HE
We report the results of a neutrino search in Super-Kamiokande for coincident signals with the first detected gravitational wave produced by a binary neutron star merger, GW170817, which was followed by a short gamma-ray burst, GRB170817A, and a kilonova/macronova. We searched for coincident neutrino events in the range from 3.5 MeV to $\sim$100 PeV, in a time window $\pm$500 seconds around the gravitational wave detection time, as well as during a 14-day period after the detection. No significant neutrino signal was observed for either time window. We calculated 90% confidence level upper limits on the neutrino fluence for GW170817. From the upward-going-muon events in the energy region above 1.6 GeV, the neutrino fluence limit is $16.0^{+0.7}_{-0.6}$ ($21.3^{+1.1}_{-0.8}$) cm$^{-2}$ for muon neutrinos (muon antineutrinos), with an error range of $\pm5^{\circ}$ around the zenith angle of NGC4993, and the energy spectrum is under the assumption of an index of $-2$. The fluence limit for neutrino energies less than 100 MeV, for which the emission mechanism would be different than for higher-energy neutrinos, is also calculated. It is $6.6 \times 10^7$ cm$^{-2}$ for anti-electron neutrinos under the assumption of a Fermi-Dirac spectrum with average energy of 20 MeV.
Search for an excess of events in the Super-Kamiokande detector in the directions of the astrophysical neutrinos reported by the IceCube Collaboration (1707.08604)
The Super-Kamiokande Collaboration: K. Abe, C. Bronner, G. Pronost, Y. Hayato, M. Ikeda, K. Iyogi, J. Kameda, Y. Kato, Y. Kishimoto, Ll. Marti, M. Miura, S. Moriyama, M. Nakahata, Y. Nakano, S. Nakayama, Y. Okajima, A. Orii, H. Sekiya, M. Shiozawa, Y. Sonoda, A. Takeda, A. Takenaka, H. Tanaka, S. Tasaka, T. Tomura, R. Akutsu, T. Kajita, K. Kaneyuki, Y. Nishimura, K. Okumura, K. M. Tsui, L. Labarga, P. Fernandez, F. d. M. Blaszczyk, J. Gustafson, C. Kachulis, E. Kearns, J. L. Raaf, J. L. Stone, L. R. Sulak, S. Berkman, S. Tobayama, M. Goldhaber, M. Elnimr, W. R. Kropp, S. Mine, S. Locke, P. Weatherly, M. B. Smy, H. W. Sobel, V. Takhistov, K. S. Ganezer, J. Hill, J. Y. Kim, I. T. Lim, R. G. Park, A. Himmel, Z. Li, E. O'Sullivan, K. Scholberg, C. W. Walter, T. Ishizuka, T. Nakamura, J. S. Jang, K. Choi, J. G. Learned, S. Matsuno, S. N. Smith, J. Amey, R. P. Litchfield, W. Y. Ma, Y. Uchida, M. O. Wascko, S. Cao, M. Friend, T. Hasegawa, T. Ishida, T. Ishii, T. Kobayashi, T. Nakadaira, K. Nakamura, Y. Oyama, K. Sakashita, T. Sekiguchi, T. Tsukamoto, KE. Abe, M. Hasegawa, Y. Nakano, A. T. Suzuki, Y. Takeuchi, T. Yano, S. V. Cao, T. Hayashino, T. Hiraki, S. Hirota, K. Huang, M. Jiang, A. Minamino, KE. Nakamura, T. Nakaya, B. Quilain, N. D. Patel, R. A. Wendell, L. H. V. Anthony, N. McCauley, A. Pritchard, Y. Fukuda, Y. Itow, M. Murase, F. Muto, P. Mijakowski, K. Frankiewicz, C. K. Jung, X. Li, J. L. Palomino, G. Santucci, C. Vilela, M. J. Wilking, C. Yanagisawa, S. Ito, D. Fukuda, H. Ishino, A. Kibayashi, Y. Koshio, H. Nagata, M. Sakuda, C. Xu, Y. Kuno, D. Wark, F. Di Lodovico, B. Richards, R. Tacik, S. B. Kim, A. Cole, L. Thompson, H. Okazawa, Y. Choi, K. Ito, K. Nishijima, M. Koshiba, Y. Totsuka, Y. Suda, M. Yokoyama, R. G. Calland, M. Hartz, K. Martens, C. Simpson, Y. Suzuki, M. R. Vagins, D. Hamabe, M. Kuze, T. Yoshida, M. Ishitsuka, J. F. Martin, C. M. Nantais, H. A. Tanaka, A. Konaka, S. Chen, L. Wan, Y. Zhang, A. Minamino, R. J. Wilkes
Jan. 5, 2018 hep-ex, astro-ph.HE
We present the results of a search in the Super-Kamiokande (SK) detector for excesses of neutrinos with energies above a few GeV that are in the direction of the track events reported in IceCube. Data from all SK phases (SK-I through SK-IV) were used, spanning a period from April 1996 to April 2016 and corresponding to an exposure of 225 kilotonne-years . We considered the 14 IceCube track events from a data set with 1347 livetime days taken from 2010 to 2014. We use Poisson counting to determine if there is an excess of neutrinos detected in SK in a 10 degree search cone (5 degrees for the highest energy data set) around the reconstructed direction of the IceCube event. No significant excess was found in any of the search directions we examined. We also looked for coincidences with a recently reported IceCube multiplet event. No events were detected within a $\pm$ 500 s time window around the first detected event, and no significant excess was seen from that direction over the lifetime of SK.
Tailored Raman-resonant four-wave-mixing process (1711.10772)
C. Ohae, J. Zheng, K. Ito, M. Suzuki, K. Minoshima, M. Katsuragawa
Nov. 29, 2017 physics.optics
Nonlinear optical processes are strongly dominated by phase relationships among electromagnetic fields that are relevant to its optical process. In this paper, we theoretically and experimentally show, as a typical example, that in a Raman-resonant four-wave-mixing process, the first anti-Stokes and Stokes generations can be tailored to a variety of forms by manipulating the phase relationships among the relevant electromagnetic fields.
Atmospheric neutrino oscillation analysis with external constraints in Super-Kamiokande I-IV (1710.09126)
Super-Kamiokande Collaboration: K. Abe, C. Bronner, G. Pronost, Y. Haga, Y. Hayato, M. Ikeda, K. Iyogi, J. Kameda, Y. Kato, Y. Kishimoto, Ll. Marti, M. Miura, S. Moriyama, M. Nakahata, T. Nakajima, Y. Nakano, S. Nakayama, Y. Okajima, A. Orii, H. Sekiya, M. Shiozawa, Y. Sonoda, A. Takeda, A. Takenaka, H. Tanaka, S. Tasaka, T. Tomura, R. Akutsu, T. Irvine, T. Kajita, I. Kametani, K. Kaneyuki, Y. Nishimura, K. Okumura, E. Richard, K. M. Tsui, L. Labarga, P. Fernandez, F. d. M. Blaszczyk, J. Gustafson, C. Kachulis, E. Kearns, J. L. Raaf, J. L. Stone, L. R. Sulak, S. Berkman, S. Tobayama, M. Goldhaber, G. Carminati, M. Elnimr, W. R. Kropp, S. Mine, S. Locke, A. Renshaw, M. B. Smy, H. W. Sobel, V. Takhistov, P. Weatherly, K. S. Ganezer, B. L. Hartfiel, J. Hill, N. Hong, J. Y. Kim, I. T. Lim, R. G. Park, T. Akiri, A. Himmel, Z. Li, E. O'Sullivan, K. Scholberg, C. W. Walter, T. Wongjirad, T. Ishizuka, T. Nakamura, J. S. Jang, K. Choi, J. G. Learned, S. Matsuno, S. N. Smith, J. Amey, R. P. Litchfield, W. Y. Ma, Y. Uchida, M. O. Wascko, S. Cao, M. Friend, T. Hasegawa, T. Ishida, T. Ishii, T. Kobayashi, T. Nakadaira, K. Nakamura, Y. Oyama, K. Sakashita, T. Sekiguchi, T. Tsukamoto, KE. Abe, M. Hasegawa, A. T. Suzuki, Y. Takeuchi, T. Yano, T. Hayashino, S. Hirota, K. Huang, K. Ieki, M. Jiang, T. Kikawa, KE. Nakamura, T. Nakaya, N. D. Patel, K. Suzuki, S. Takahashi, R. A. Wendell, L. H. V. Anthony, N. McCauley, A. Pritchard, Y. Fukuda, Y. Itow, G. Mitsuka, M. Murase, F. Muto, T. Suzuki, P. Mijakowski, K. Frankiewicz, J. Hignight, J. Imber, C. K. Jung, X. Li, J. L. Palomino, G. Santucci, C. Vilela, M. J. Wilking, C. Yanagisawa, S. Ito, D. Fukuda, H. Ishino, T. Kayano, A. Kibayashi, Y. Koshio, T. Mori, H. Nagata, M. Sakuda, C. Xu, Y. Kuno, D. Wark, F. Di Lodovico, B. Richards, R. Tacik, S. B. Kim, A. Cole, L. Thompson, H. Okazawa, Y. Choi, K. Ito, K. Nishijima, M. Koshiba, Y. Totsuka, Y. Suda, M. Yokoyama, R. G. Calland, M. Hartz, K. Martens, B. Quilain, C. Simpson, Y. Suzuki, M. R. Vagins, D. Hamabe, M. Kuze, T. Yoshida, M. Ishitsuka, J. F. Martin, C. M. Nantais, P. de Perio, H. A. Tanaka, A. Konaka, S. Chen, L. Wan, Y. Zhang, R. J. Wilkes, A. Minamino
Oct. 25, 2017 hep-ex
An analysis of atmospheric neutrino data from all four run periods of \superk optimized for sensitivity to the neutrino mass hierarchy is presented. Confidence intervals for $\Delta m^2_{32}$, $\sin^2 \theta_{23}$, $\sin^2 \theta_{13}$ and $\delta_{CP}$ are presented for normal neutrino mass hierarchy and inverted neutrino mass hierarchy hypotheses based on atmospheric neutrino data alone. Additional constraints from reactor data on $\theta_{13}$ and from published binned T2K data on muon neutrino disappearance and electron neutrino appearance are added to the atmospheric neutrino fit to give enhanced constraints on the above parameters. Over the range of parameters allowed at 90% confidence level, the normal mass hierarchy is favored by between 91.5% and 94.5% based on the combined result.
High-resolution one-photon absorption spectroscopy of the $D\,{}^2\Sigma^- \leftarrow X\,{}^2\Pi$ system of radical OH and OD (1709.02509)
A. N. Heays, N. de Oliveira, B. Gans, K. Ito, S. Boyé-Péronne, S. Douin, K. M. Hickson, L. Nahon, J. C. Loison
Sept. 8, 2017 physics.atom-ph, physics.chem-ph, astro-ph.EP
Vacuum-ultraviolet (VUV) photoabsorption spectra were recorded of the $A\,{}^2\Sigma^+(v'=0)\leftarrow{}X\,{}^2\Pi(v''=0)$, $D\,{}^2\Sigma^-(v'=0)\leftarrow{}X\,{}^2\Pi(v''=0)$ and $D\,{}^2\Sigma^-(v'=1)\leftarrow{}X\,{}^2\Pi(v''=0)$ bands of the OH and OD radicals generated in a plasma-discharge source using synchrotron radiation as a background continuum coupled with the VUV Fourier-transform spectrometer on the DESIRS beamline of synchrotron SOLEIL. High-resolution spectra permitted the quantification of transition frequencies, relative $f$-values, and natural line broadening. The $f$-values were absolutely calibrated with respect to a previous measurement of $A\,{}^2\Sigma^+(v'=0)\leftarrow{}X\,{}^2\Pi(v''=0)$ ([wang1979]). Lifetime broadening of the excited $D\,{}^2\Sigma^-(v=0)$ and $D\,{}^2\Sigma^-(v=1)$ levels is measured for the first time and compared with previous experimental limits, and implies a lifetime 5 times shorter than a theoretical prediction ([van_der_loo2005]). A local perturbation of the $D\,{}^2\Sigma^-(v=0)$ level in OH was found.
Stationary scattering theory on manifolds, II (1602.07487)
K. Ito, E. Skibsted
April 11, 2016 math.DG, math-ph, math.MP
Based on our previous study [IS2] we develop fully the stationary scattering theory for the Schrodinger operator on a manifold possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends, possibly with unbounded and non-smooth obstacles. We develop the theory largely along the classical lines [Sa, Co] and derive in particular WKB- asymptotics of appropriate generalized eigenfunctions. As an application we solve a conjecture of [HPW] on cross-ends transmissions in its natural and strong form within the framework of our theory.
Stationary scattering theory on mainifolds, I (1602.07488)
Jan. 30, 2020 math.DG, math-ph, math.MP
We study spectral theory for the Schrodinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends.
Rellich's theorem and N-body Schrodinger operators (1602.07493)
Feb. 24, 2016 math-ph, math.MP, math.FA
We show an optimal version of the Rellich theorem for generalized many-body Schrodinger operators. It applies to singular potentials, in particular to a model for atoms and molecules with infinite mass and finite extent nuclei. Our proof relies on a Mourre estimate and a functional calculus localization technique.
ECFA Detector R&D Panel, Review Report (1411.4924)
The FCAL Collaboration: H. Abramowicz, A. Abusleme, K. Afanaciev, J. Aguilar, E. Alvarez, P. Bambade, L. Bortko, I. Bozovic-Jelisavcic, E. Castro, G. Chelkov, C. Coca, W. Daniluk, A. Dragone, L. Dumitru, K. Elsener, I. Emeliantchik, E. Firu, J. Fischer, T. Fiutowski, V. Ghenescu, M. Gostkin, G. Grzelak, G. Haller, H. Henschel, A. Ignatenko, M. Idzik, K. Ito, S. Kananov, E. Kielar, S. Kollowa, J. Kotula, Z. Krumstein, B. Krupa, S. Kulis, W. Lange, A. Levy, I. Levy, L. Linssen, W. Lohmann, S. Lukic, J. Moron, A. Moszczynski, U. Nauenberg, A. Neagu, O. Novgorodova, F.-X. Nuiry, M. Ohlerich, M. Orlandea, G. Oleinik, K. Oliwa, A. Olshevski, M. Pandurovic, B. Pawlik, T. Preda, D. Przyborowski, Y. Sato, I. Sadeh, A. Sailer, B. Schumm, S. Schuwalow, R. Schwartz, I. Smiljanic, K. Swientek, Y. Takubo, E. Teodorescu, W. Wierba, H. Yamamoto, L. Zawiejski, T.-S. Zgura, J. Zhang
Nov. 19, 2014 physics.ins-det
Two special calorimeters are foreseen for the instrumentation of the very forward region of an ILC or CLIC detector; a luminometer (LumiCal) designed to measure the rate of low angle Bhabha scattering events with a precision better than 10$^{-3}$ at the ILC and 10$^{-2}$ at CLIC, and a low polar-angle calorimeter (BeamCal). The latter will be hit by a large amount of beamstrahlung remnants. The intensity and the spatial shape of these depositions will provide a fast luminosity estimate, as well as determination of beam parameters. The sensors of this calorimeter must be radiation-hard. Both devices will improve the e.m. hermeticity of the detector in the search for new particles. Finely segmented and very compact electromagnetic calorimeters will match these requirements. Due to the high occupancy, fast front-end electronics will be needed. Monte Carlo studies were performed to investigate the impact of beam-beam interactions and physics background processes on the luminosity measurement, and of beamstrahlung on the performance of BeamCal, as well as to optimise the design of both calorimeters. Dedicated sensors, front-end and ADC ASICs have been designed for the ILC and prototypes are available. Prototypes of sensor planes fully assembled with readout electronics have been studied in electron beams.
Absence of positive eigenvalues for hard-core N-body systems (1207.7190)
July 31, 2012 math-ph, math.MP
We show absence of positive eigenvalues for generalized 2-body hard- core Schroedinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N -body hard-core Schroedinger operators, N \geq 2, is presented. This scheme involves high energy resolvent estimates, and for N = 2 it is implemented by a Mourre commutator type method. A particular example is the Helium atom with the assumption of infinite mass and finite extent nucleus.
Scattering theory for Riemannian Laplacians (1109.1925)
Sept. 9, 2011 math.DG, math-ph, math.MP
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace-Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.
Absence of embedded eigenvalues for Riemannian Laplacians (1109.1928)
In this paper we study absence of embedded eigenvalues for Schr\"odinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end. Another condition may be viewed (at least in a special case) as being a bound of the trace of this quantity, while similarly, a third one as being a bound of the derivative of this trace. In addition to geometric bounds we need conditions on the potential, a regularity property of the domain of the Schr\"odinger operator and the unique continuation property. Examples include ends endowed with asymptotic Euclidean or hyperbolic metrics studied previously in the literature.
Dopant-dependent impact of Mn-site doping on the critical-state manganites: R0.6Sr0.4MnO3 (R=La, Nd, Sm, and Gd) (0805.3966)
H. Sakai, K. Ito, T. Nishiyama, X. Z. Yu, Y. Matsui, S. Miyasaka, Y. Tokura
May 26, 2008 cond-mat.mtrl-sci, cond-mat.str-el
Versatile features of impurity doping effects on perovskite manganites, $R_{0.6}$Sr$_{0.4}$MnO$_{3}$, have been investigated with varying the doing species as well as the $R$-dependent one-electron bandwidth. In ferromagnetic-metallic manganites ($R$=La, Nd, and Sm), a few percent of Fe substitution dramatically decreases the ferromagnetic transition temperature, leading to a spin glass insulating state with short-range charge-orbital correlation. For each $R$ species, the phase diagram as a function of Fe concentration is closely similar to that for $R_{0.6}$Sr$_{0.4}$MnO$_{3}$ obtained by decreasing the ionic radius of $R$ site, indicating that Fe doping in the phase-competing region weakens the ferromagnetic double-exchange interaction, relatively to the charge-orbital ordering instability. We have also found a contrastive impact of Cr (or Ru) doping on a spin-glass insulating manganite ($R$=Gd). There, the impurity-induced ferromagnetic magnetization is observed at low temperatures as a consequence of the collapse of the inherent short-range charge-orbital ordering, while Fe doping plays only a minor role. The observed opposite nature of impurity doping may be attributed to the difference in magnitude of the antiferromagnetic interaction between the doped ions.
Fe-doping-induced evolution of charge-orbital ordering in a bicritical-state manganite (0710.4069)
H. Sakai, K. Ito, R. Kumai, Y. Tokura
Oct. 22, 2007 cond-mat.mtrl-sci, cond-mat.str-el
Impurity effects on the stability of a ferromagnetic metallic state in a bicritical-state manganite, (La0.7Pr0.3)0.65Ca0.35MnO3, on the verge of metal-insulator transition have been investigated by substituting a variety of transition-metal atoms for Mn ones. Among them, Fe doping exhibits the exceptional ability to dramatically decrease the ferromagnetic transition temperature. Systematic studies on the magnetotransport properties and x-ray diffraction for the Fe-doped crystals have revealed that charge-orbital ordering evolves down to low temperatures, which strongly suppresses the ferromagnetic metallic state. The observed glassy magnetic and transport properties as well as diffuse phase transition can be attributed to the phase-separated state where short-range charge-orbital-ordered clusters are embedded in the ferromagnetic metallic matrix. Such a behavior in the Fe-doped manganites form a marked contrast to the Cr-doping effects on charge-orbital-ordered manganites known as impurity-induced collapse of charge-orbital ordering.
Sub-ns spin-transfer switching: compared benefits of free layer biasing and pinned layer biasing (cond-mat/0703124)
T. Devolder, C. Chappert, K. Ito
March 5, 2007 cond-mat.str-el
We analyze the statistical distribution of switching durations in spin-transfer switching induced by current steps, and discuss biasing strategies to enhance the reproducibility of switching durations. We use a macrospin approximation and model the effect of finite temperature as a Boltzmann distribution of initial magnetization states (adiabatic limit). We compare three model spin-valves: a spin-valve with a free layer whose easy axis is parallel to the pinned layer magnetization (standard geometry), a pinned layer with magnetization tilted with respect to the free layer easy axis (pinned layer biasing), and a free layer whose magnetization is pulled away from easy axis by a hard axis bias (free layer biasing). In the conventional geometry, the switching durations follow a broad regular distribution, with an extended long tail comprising very long switching events. For the two biasing strategies, the switching durations follow a multiply-stepped distribution, reflecting the precessional nature of the switching, and the statistical number of precession cycles needed for reversal. We derive analytical criteria to avoid switching events lasting much longer than the average switching duration, in order to achieve the highest reproducibilities. Depending on the current amplitude and the biasing strength, the width of the switching time distribution can be substantially reduced, the best reproducibility being achieved for free layer biasing at overdrive current of a few times unity.
Distribution of the magnetization reversal duration in sub-ns spin-transfer switching (cond-mat/0609687)
T. Devolder, C. Chappert, J.A. Katine, M.J. Carey, K. Ito
Sept. 27, 2006 cond-mat.mtrl-sci, cond-mat.str-el
We study the distribution of switching times in spin-transfer switching induced by sub-ns current pulses in pillar-shaped spin-valves. The pulse durations leading to switching follow a comb-like distribution, multiply-peaked at a few most probable, regularly spaced switching durations. These durations reflect the precessional nature of the switching, which occurs through a fluctuating integer number of precession cycles. This can be modeled considering the thermal variance of the initial magnetization orientations and the occurrence of vanishing total torque in the possible magnetization trajectories. Biasing the spin-valve with a hard axis field prevents some of these occurrences, and can provide an almost perfect reproducibility of the switching duration.
Elastic precursor of the transformation from glycolipid-nanotube to -vesicle (cond-mat/0604033)
T. Fujima, H. Frusawa, H. Minamikawa, K. Ito, T. Shimizu
April 3, 2006 cond-mat.soft, cond-mat.mtrl-sci
By the combination of optical tweezer manipulation and digital video microscopy, the flexural rigidity of single glycolipid "nano" tubes has been measured below the transition temperature at which the lipid tubules are transformed into vesicles. Consequently, we have found a clear reduction of the rigidity obviously before the transition as temperature increasing. Further experiments of infrared spectroscopy (FT-IR) and differential scanning calorimetry (DSC) have suggested a microscopic change of the tube walls, synchronizing with the precursory softening of the nanotubes.
Current-driven microwave oscillations in current perpendicular-to-plane spin-valve nanopillars (cond-mat/0603293)
Q. Mistral, Joo-Von Kim, T. Devolder, P. Crozat, C. Chappert, J. A. Katine, M. J. Carey, K. Ito
March 10, 2006 cond-mat.mtrl-sci
We study the current and temperature dependences of the microwave voltage emission of spin-valve nanopillars subjected to an in-plane magnetic field and a perpendicular-to-plane current. Despite the complex multilayer geometry, clear microwave emission is shown to be possible and spectral lines as narrow as 3.8 MHz (at 150 K) are observed.
Large tunneling anisotropic magnetoresistance in (Ga,Mn)As nanoconstrictions (cond-mat/0409209)
A.D. Giddings, M.N. Khalid, J. Wunderlich, S. Yasin, R.P. Campion, K.W. Edmonds, J. Sinova, T. Jungwirth, K. Ito, K. Y. Wang, D. Williams, B.L. Gallagher, C.T. Foxon
Sept. 8, 2004 cond-mat.mes-hall
We report a large tunneling anisotropic magnetoresistance (TAMR) in a thin (Ga,Mn)As epilayer with lateral nanoconstrictions. The observation establishes the generic nature of this effect, which originates from the spin-orbit coupling in a ferromagnet and is not specific to a particular tunnel device design. The lateral geometry allows us to link directly normal anisotropic magnetoresistance (AMR) and TAMR. This indicates that TAMR may be observable in other materials showing a comparable AMR at room temperature, such as transition metal alloys.
On the zero-resistance states generated by radiation in GaAs/AlGaAs (cond-mat/0402174)
S. Fujita, K. Ito
Feb. 5, 2004 cond-mat.mes-hall
The applied radiation excites "holes". The condensed composite (c)-bosons formed in the excited channel create a superconducting state with an energy gap. The supercondensate suppresses the non-condensed c-bosons at low temperatures, but it cannot suppress the c-fermions in the base channel, and the small normal current accompanied by the Hall field yields a B-linear Hall resistivity.
Coexistence of Superconductivity and Antiferromagnetism in Multilayered High-$T_c$ Superconductor HgBa$_2$Ca$_4$Cu$_5$O$_y$: A Cu-NMR Study (cond-mat/0401416)
H. Kotegawa, Y. Tokunaga, Y. Araki, G.-q. Zheng, Y. Kitaoka, K. Tokiwa, K. Ito, T. Watanabe, A. Iyo, Y. Tanaka, H. Ihara
Jan. 22, 2004 cond-mat.supr-con
We report a coexistence of superconductivity and antiferromagnetism in five-layered compound HgBa$_2$Ca$_4$Cu$_5$O$_y$ (Hg-1245) with $T_c=108$ K, which is composed of two types of CuO$_2$ planes in a unit cell; three inner planes (IP's) and two outer planes (OP's). The Cu-NMR study has revealed that the optimallydoped OP undergoes a superconducting (SC) transition at $T_c=108$ K, whereas the three underdoped IP's do an antiferromagnetic (AF) transition below $T_N\sim$ 60 K with the Cu moments of $\sim (0.3-0.4)\mu_B$. Thus bulk superconductivity with a high value of $T_c=108$ K and a static AF ordering at $T_N=60$ K are realized in the alternating AF and SC layers. The AF-spin polarization at the IP is found to induce the Cu moments of $\sim0.02\mu_B$ at the SC OP, which is the AF proximity effect into the SC OP.
The "devil's staircase" type phase transition in NaV2O5 under high pressure (cond-mat/0104286)
K. Ohwada, Y. Fujii, N. Takesue, M. Isobe, Y. Ueda, H. Nakao, Y. Wakabayashi, Y. Murakami, K. Ito, Y. Amemiya, Y. Fujihisa, K. Aoki, T. Shobu, Y. Noda, N. Ikeda
April 17, 2001 cond-mat.str-el
The "devil's staircase" type phase transition in the quarter-filled spin-ladder compound NaV2O5 has been discovered at low temperature and high pressure by synchrotron radiation x-ray diffraction. A large number of transitions are found to successively take place among higher-order commensurate phases with 2a * 2b * zc type superstructures. The observed temperature and pressure dependence of modulation wave number qc, defined by 1/z, is well reproduced by the Axial Next Nearest Neighbor Ising (ANNNI) model. The qc is suggested to reflect atomic displacements coupled with charge ordering in this system. The experimental fact implies that two competitive inter-layer interactions between the Ising spins, i.e., the nearest neighbor J1>0 (ferro) and the next nearest neighbor J2<0 (antiferro) along the c-axis, are intrinsic in this compound. A microscopic origin of the inter-layer interaction is not yet known; however, the nearest neighbor interaction J1>0 between the V2O5 layers is especially interesting. It is very surprising that the phase transition of such a complicated charge-lattice-spin coupled system NaV2O5 can be described by the simple ANNNI model.
Extended Superconformal Algebras and Free Field Realizations from Hamiltonian Reduction (hep-th/9307189)
K. Ito, J.O. Madsen, J.L. Petersen
July 30, 1993 hep-th
We develop the method of the hamiltonian reduction of affine Lie superalgebras to obtain explicit and general expressions both for the classical and the quantum extended superconformal algebras. By performing the gauge transformation which connects the diagonal gauge with the Drinfeld-Sokolov gauge and considering the quantum corrections, we get generic expressions for the classical and quantum free field realizations of the algebras.
Extended Superconformal Algebras from Classical and Quantum Hamiltonian Reduction (hep-th/9211019)
Nov. 4, 1992 hep-th
We consider the extended superconformal algebras of the Knizhnik-Bershadsky type with $W$-algebra like composite operators occurring in the commutation relations, but with generators of conformal dimension 1,$\frac{3}{2}$ and 2, only. These have recently been neatly classified by several groups, and we emphasize the classification based on hamiltonian reduction of affine Lie superalgebras with even subalgebras $G\oplus sl(2)$. We reveiw the situation and improve on previous formulations by presenting generic and very compact expressions valid for all algebras, classical and quantum. Similarly generic and compact free field realizations are presented as are corresponding screening charges. Based on these a discussion of singular vectors is presented. (Based on talk by J.L. Petersen at the Int. Workshop on "String Theory, Quantum Gravity and the Unification of the Fundamental Interactions", Rome Sep. 21-26, 1992) | CommonCrawl |
I shall report on calculations of isovector matrix elements of the nucleon, such as $g_A, g_s$, and $\langle x \rangle$ on the $48^3 \times 96$ lattice with pion mass at 139 MeV and lattice size of 5.5 fm. We employ overlap valence fermion on the 2+1 flavor DWF configurations for the calculation. Also reported will be the strange quark momentum fraction and its magnetic moment from this lattice. A comparison of the cost of such calculations with those of the twisted mass fermion, clover fermion, and domain wall fermion on similar lattices and quark masses will be made for the calculation of the nucleon mass and the three-point functions of both the connected and disconnected insertions. | CommonCrawl |
arXiv.org > hep-th > arXiv:hep-th/9305010v1
hep-th
INSPIRE HEP
(refers to | cited by )
High Energy Physics - Theory
Title:Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model
Authors:M. Blau, G. Thompson
(Submitted on 4 May 1993 (this version), latest version 5 May 1993 (v2))
Abstract: We give a derivation of the Verlinde formula for the $G_{k}$ WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function $Z_{\Sigma\times S^{1}}$ of $\Sigma\times S^{1}$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{\Sigma\times S^{1}}$ is reduced to the partition function of an Abelian topological field theory on $\Sigma$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $\Sigma\times S^{1}$. We derive the $G_{k}/G_{k}$ model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the $G_{k}/G_{k}$ path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the shift $k\ra k+h$ is given, based on the index of the twisted Dolbeault complex.
Comments: This version (hep-th/9305010v1) was not stored by arXiv. A subsequent replacement was made before versioning was introduced.
Subjects: High Energy Physics - Theory (hep-th)
Cite as: arXiv:hep-th/9305010
(or arXiv:hep-th/9305010v1 for this version)
From: Blau Matthias [view email]
[v1] Tue, 4 May 1993 13:39:34 UTC (0 KB)
[v2] Wed, 5 May 1993 08:52:21 UTC (39 KB) | CommonCrawl |
Spin-orbit interactions of transverse sound
Shubo Wang ORCID: orcid.org/0000-0002-3026-69721,
Guanqing Zhang2,
Xulong Wang2,
Qing Tong ORCID: orcid.org/0000-0002-2251-07361,
Jensen Li3 &
Guancong Ma ORCID: orcid.org/0000-0003-3652-43512
Metamaterials
Spin-orbit interactions (SOIs) endow light with intriguing properties and applications such as photonic spin-Hall effects and spin-dependent vortex generations. However, it is counterintuitive that SOIs can exist for sound, which is a longitudinal wave that carries no intrinsic spin. Here, we theoretically and experimentally demonstrate that airborne sound can possess artificial transversality in an acoustic micropolar metamaterial and thus carry both spin and orbital angular momentum. This enables the realization of acoustic SOIs with rich phenomena beyond those in conventional acoustic systems. We demonstrate that acoustic activity of the metamaterial can induce coupling between the spin and linear crystal momentum k, which leads to negative refraction of the transverse sound. In addition, we show that the scattering of the transverse sound by a dipole particle can generate spin-dependent acoustic vortices via the geometric phase effect. The acoustic SOIs can provide new perspectives and functionalities for sound manipulations beyond the conventional scalar degree of freedom and may open an avenue to the development of spin-orbit acoustics.
Spin and orbital angular momentum (OAM) are intrinsic properties of classical waves. Spin is associated with circular polarization (vector degrees of freedom) of waves and is characterized by the local rotation of a vector field. OAM originates from the spatial phase gradient (scalar degree of freedom) of waves and manifests as a helical wave front1. The couplings between spin and OAM, referred to as spin–orbit interactions (SOIs), can give rise to intriguing phenomena and applications in optics2,3,4,5,6,7,8, such as photonic spin-Hall effect9,10,11 and spin-dependent vortex generation12,13. SOIs are unique to transverse waves such as light and are absent for longitudinal waves. This is because although longitudinal waves such as airborne sound can carry OAM14,15,16,17,18, they are spin-0 in nature. Recent studies show that an engineered sound field can possess a locally rotational velocity field v that may be regarded as acoustic spin19,20,21, similar to electric spin deriving from the local rotation of electric field. Such an acoustic spin can emerge locally in nonuniform acoustic fields20,21 and has recently been observed in experiments19,22. In a homogenous medium, however, the spatial integration of acoustic spin density for a localized wave must vanish, in agreement with its spin-0 nature20. Despite this discovery of acoustic spin, SOIs remain beyond reach in sound, a fact that mainly owes to the lack of degrees of freedom. In other words, sound is characterized by a scalar pressure field p and a vector velocity field v, whereas light is characterized by two vector fields E and H.
In this work, we show that airborne sound can behave as a transverse wave with well-defined polarization in an acoustic metamaterial that goes beyond the Cauchy elasticity and follows a micropolar elasticity theory23. Unlike previous spin-sustaining acoustic fields19,20,22, the transverse sound is spin-1 in nature and carries the properties of elastic waves. It is characterized by two types of vector-field degrees of freedom, i.e., a velocity field and a microrotation field. The acoustic activity of the metamaterial can induce coupling between the velocity and microrotation fields, which can be considered an analog of chirality in electromagnetism (i.e., optical activity). Such a material property has recently been realized in elastic wave systems24,25,26,27 but is so far missing in acoustic wave systems. We theoretically and experimentally demonstrate two types of acoustic SOIs in momentum space and in real space, respectively. In the momentum space, the acoustic activity induces the coupling between spin and linear crystal momentum k, and enables the chirality-induced negative refraction, which was previously possible only in optical metamaterials28,29. In the real space, scattering of the circularly polarized transverse sound by a dipole particle can generate a sound vortex with a topological charge determined by the acoustic spin.
Transverse sound
The longitudinal nature of airborne sound (\({{{{{\boldsymbol{\nabla }}}}}}\times {{{{{\bf{v}}}}}}=0\)) dictates that the velocity field v aligns with the direction of wave vector k in general. However, this is not necessarily true when sound is confined in a closed space. Consider a one-dimensional (1D) lattice stacked along the z axis with a unit cell shown in Fig. 1a. The unit cell consists of a cylindrical resonator with eight internal blades segmenting the air to achieve subwavelength resonance, as indicated by the blue arrows. The resonators are sequentially connected by four tubes. All solid–air interfaces are regarded as sound-hard boundaries. The resonator supports two degenerate and orthogonal dipole resonances with pressure eigenfields shown in Fig. 1b. The positive and negative pressure (indicated by the red and blue colors, respectively) induces an in-plane velocity field that is perpendicular to the propagating direction of sound (i.e., z axis). This corresponds to the oscillating dipole moments \({{{{{{\bf{p}}}}}}}_{x}\) and \({{{{{{\bf{p}}}}}}}_{y}\), where the positive (negative) charge corresponds to the positive (negative) pressure and the yellow arrow denotes the velocity field. Next, we break the spatial inversion symmetry by twisting the resonator geometry with respect to z axis, as shown in Fig. 1c. The degeneracy of \({{{{{{\bf{p}}}}}}}_{x}\) and \({{{{{{\bf{p}}}}}}}_{y}\) is removed, and the resonator supports two chiral eigenmodes \({{{{{{\bf{p}}}}}}}_{x}-i{{{{{{\bf{p}}}}}}}_{y}\) and \({{{{{{\bf{p}}}}}}}_{x}+i{{{{{{\bf{p}}}}}}}_{y}\), corresponding to a left-handed circularly polarized (LCP) dipole and a right-handed circularly polarized (RCP) dipole, respectively, as shown in Fig. 1d. Thus, the collective excitations of the acoustic dipoles in Fig. 1b, d will give rise to linearly polarized and circularly polarized transverse sounds propagating in the z direction, respectively.
Fig. 1: Eigenmodes of the 1D acoustic lattices.
a The unit cell of the achiral lattice. The arrows show the flow of air inside the resonator. b The pressure eigenfields of the two transverse dipole modes. The velocity is linearly polarized on the transverse plane, corresponding to acoustic dipoles \({{{{{{\bf{p}}}}}}}_{x}\) and \({{{{{{\bf{p}}}}}}}_{y}\). The positive (negative) charge corresponds to positive (negative) pressure. The yellow arrows denote the velocity field. c The unit cell of the chiral lattice. d Pressure eigenfields of the chiral dipole modes. The velocity fields are circularly polarized on the transverse plane, corresponding to circularly polarized dipoles \({{{{{{\bf{p}}}}}}}_{x}\pm i{{{{{{\bf{p}}}}}}}_{y}\).
To verify this, we use three-dimensional (3D) printing to fabricate both the 1D achiral and chiral lattices, each with 24 unit cells, as shown in Fig. 2a. In Fig. 2b, we show the cutaway views of the two types of unit cell, where the internal blades are colored to clearly show their orientations. The green-colored blades are connected to the outer shell and the blue-colored blades are connected to the inner core. They together form a tunnel in which air flows. The experimentally measured band structures of the achiral and chiral lattices are shown in Fig. 2c, d, respectively. The solid red lines denote full-wave numerical results calculated using a finite-element package COMSOL (see "Methods"). Excellent agreement between the experimental and numerical results is seen. The first band that extends to the static limit corresponds to a monopole mode, which has almost identical characteristics for both the chiral and achiral lattices. The second and third bands are the aforementioned transverse dipole modes, which are degenerate for the achiral lattice (Fig. 2c) but split into two bands for the chiral lattice (Fig. 2d) due to inversion symmetry breaking. The modes of the second and third bands for the chiral lattice are LCP and RCP, respectively. To obtain intuitive pictures of the transverse modes, we calculated the averaged velocity (near \({k}_{z}=0\)) in each unit cell and plot it in Fig. 2e, f. Figure 2e shows the velocity field for the achiral lattice with 25 units, where the dipole mode along the y axis is excited. As seen, the sound is linearly polarized along the y direction with a wavelength much larger than the unit-cell dimension. We note that the achiral lattice also supports circularly polarized sound, which corresponds to a superposition of the linearly polarized sounds along the x and y directions. Figure 2f shows the velocity field for the second band of the chiral lattice, which clearly represents an LCP transverse sound. These confirm the transverse nature of the sound in the 1D lattices.
Fig. 2: Band properties of the 1D acoustic lattices.
a The fabricated sample of the 1D lattice. b The internal structures of the achiral and chiral unit cells. Experimental (blue) and numerical (red lines) results for the band structure of the achiral lattice (c) and the chiral lattice (d). The averaged velocity field of the achiral lattice shows a linear polarization (e), whereas circular polarization is seen for the chiral lattice (f).
Micropolar metamaterial with acoustic activity
The above physics can be extended to the 3D metamaterial with the unit cell shown in Fig. 3a. The unit cell consists of three chiral resonators mutually connected with tubes, as shown in Fig. 3b. The numerically calculated band structure of the metamaterial is shown in Fig. 3c. The lowest three bands derive from the monopole mode of the chiral resonators (see Supplementary Information). The three bands enclosed by the red rectangle derive from the dipole modes. The upper and lower bands correspond to the RCP and LCP transverse modes, respectively, and the middle band corresponds to a longitudinal mode. The inset at the left corner of Fig. 3c shows the pressure eigenfield of the LCP mode at a time. In Fig. 3d, e, we plot the isofrequency contours of the LCP band in kx–ky and kz–kM planes, respectively. The contours are approximately circles for \(k \, < \, 0.15\pi /a\), which indicates that the mode is isotropic near the \(\Gamma\) point. The isotropic dispersions of the transverse modes are protected by time-reversal symmetry and chiral cubic symmetry30. At the frequencies of the transverse modes, the unit cell is subwavelength (\(\sim0.23\lambda\)). Thus, the metamaterial is macroscopically isotropic and homogeneous, and its material properties can be described by an effective medium theory. Remarkably, the emergence of wave transversality in the metamaterial implies the existence of a non-zero shear modulus for the effective medium, which is counterintuitive, as air does not generate shear forces. Here, the striking properties of non-vanishing shear modulus are induced by the transverse motion of sound enforced by the resonators with twisted internal blades. The existence of a non-zero shear modulus indicates that the metamaterial is equivalent to an elastic medium and the airborne sound behaves like an elastic wave with well-defined spin31. Because of its microscopic twisting feature, the metamaterial cannot be described by conventional effective medium theory based on Cauchy elasticity, which assumes symmetric stress and strain. Instead, micropolar elasticity (i.e., Cosserat elasticity)23, which is a high-order extension of Cauchy elasticity, can be employed to accurately characterize its unusual properties.
Fig. 3: The 3D acoustic micropolar metamaterials.
a The unit cell consists of three orthogonally arranged resonators connected with tubes. b The details of the unit-cell components. c The band structure of the metamaterial. The middle inset (dashed red box) shows a zoom-in of the dipole bands. The blue circles denote the numerical results of the metamaterial. The green squares denote the numerical results of the micropolar effective medium. The solid red lines denote the analytical results. The inset at the left corner shows the pressure eigenfield of the LCP mode. The inset at the right corner shows the Brillouin zone. d The isofrequency contours of the negative band in kx–ky plane. e The isofrequency contours of the negative band in kz–kM plane.
The micropolar elasticity assigns three rotational degrees of freedom to each material point in addition to the three linear degrees of freedom associated with displacement23,32,33,34. Each point is thus characterized by a displacement vector field \(\,{{{{{\bf{u}}}}}}\) and a microrotation vector field \({{{{{\mathbf{\phi }}}}}}\). Using Einstein summation convention, the deformation of the medium can be expressed as:\(\,{\varepsilon }_{{ij}}=\partial {u}_{j}/\partial {x}_{i}- {\epsilon }_{{ijk}}{\phi }_{k};{\kappa }_{{ij}}=\partial {\phi }_{j}/\partial {x}_{i}\), where \({\varepsilon }_{{ij}}\) is the asymmetric strain tensor, \({\kappa }_{{ij}}\) is the curvature tensor characterizing the relative microrotation between neighboring points, \({\epsilon }_{{ij}k}\) is the Levi-Civita symbol, and i, j, k iterate the Cartesian coordinates. For our micropolar metamaterial, the corresponding effective medium is characterized by the constitutive relations: \({\sigma }_{{ij}}={C}_{{ijkl}}{\varepsilon }_{{kl}}+{B}_{{ijkl}}{\kappa }_{{kl}},{m}_{{ij}}={B}_{{klij}}{\varepsilon }_{{kl}}+{D}_{{ijkl}}{\kappa }_{{kl}},\) where \({\sigma }_{{ij}}\) and \({m}_{{ij}}\) are the asymmetric force stress tensor and couple stress tensor, respectively26,35. \({B}_{{ijkl}},{C}_{{ijkl}}\) and \({D}_{{ijkl}}\) are the elastic constitutive tensors of the form \({X}_{{ijkl}}={X}_{1}{\delta }_{{ij}}{\delta }_{{kl}}+{X}_{2}{\delta }_{{ik}}{\delta }_{{jl}}+{X}_{3}{\delta }_{{il}}{\delta }_{{jk}}\) with \(X=B,C,{D}\), and \({\delta }_{{ij}}\) being the Kronecker delta. Notably, \({B}_{{ijkl}}\) is a pseudo-tensor that characterizes the chirality of the medium and it changes sign under spatial inversion. Thus, the micropolar metamaterial possesses chirality that corresponds to the acoustic counterpart of optical activity36. Such a property has recently been realized in elastic metamaterials24,25,26 but has no acoustic counterpart to date. It is different from the Willis-type bianisotropy, in which the stress–strain couples with momentum–velocity37,38,39,40,41,42.
The propagation of the transverse sound is governed by the conservation of linear and angular momenta: \(\partial {\sigma }_{{ji}}/\partial {x}_{j}=\rho {\partial }^{2}{u}_{i}/\partial {t}^{2}{{{\rm{;}}}}\,\partial {m}_{{ji}}/\partial {x}_{j}+{\epsilon }_{{ijk}}{\sigma }_{{jk}}{{{\mathscr{=}}}}{{{\mathscr{j}}}}{\partial }^{2}{\phi }_{i}/\partial {t}^{2}\), where \(\rho \) is the mass density and \({{{\mathscr{j}}}}\) is the microinertia density (i.e., micro moment of inertia per unit volume). Assuming the time-harmonic displacement eigenfield \({u}_{i}={U}_{i}{e}^{{{{\rm{i}}}}{k}_{i}{x}_{i}-{{{\rm{i}}}}\omega t}\) and microrotation eigenfield \({\phi }_{i}={\Phi }_{i}{e}^{{{{\rm{i}}}}{k}_{i}{x}_{i}-{{{\rm{i}}}}\omega t}\), the dispersion relations of the dipole modes near the \(\Gamma \) point (retained the lowest order of \(k\)) can be obtained as (see "Methods"): \({\omega }_{T}^{\pm }={\omega }_{0}\pm {vk}\) and \({\omega }_{L}={\omega }_{0}+\tau {k}^{2},\) where \({\omega }_{0}={\scriptstyle\sqrt{2\left({C}_{2}-{C}_{3}\right){{{\mathscr{/}}}}{{{\mathscr{j}}}}},{k}=\left|{{{\bf{k}}}}\right|={{{\bf{k}}}}/\hat{{{{\bf{k}}}}},v=\left({B}_{2}-{B}_{3}\right)/\sqrt{2{{{\mathscr{j}}}}\left({C}_{2}-{C}_{3}\right)},\tau =\left({D}_{1}+{D}_{2}+{D}_{3}\right)/}\scriptstyle\sqrt{8{{{\mathscr{j}}}}\left({C}_{2}-{C}_{3}\right)}\), and the subscripts "T" and "L" denote the transverse and longitudinal modes, respectively. It is seen that microrotation significantly impacts both the transverse and longitudinal modes, as indicated by the existence of microinertia in both terms of the eigenfrequencies. This is in stark contrast to the dispersion relations of conventional elastic waves that are dominated by translation motion. In addition, we see that the chiral parameters \({B}_{2}\) and \({B}_{3}\) induce the splitting of the transverse modes.
By fitting the analytical dispersion relations and the constitutive relations with the numerical results of band structure and eigenmodes (see Supplementary Information), we retrieved the effective constitutive tensors \({B}_{{ijkl}}\),\(\,{C}_{{ijkl}}\), and\(\,{D}_{{ijkl}}\). We then apply these tensors to analytically evaluate the dispersion relations and the results are plotted as the solid red lines in Fig. 3c. In addition, we numerically simulated the band structures of the micropolar effective medium and results are shown as the green markers in Fig. 3c. All results agree excellently for \(k \, < \, 0.15\pi /a\), demonstrating the validity of the effective medium description based on micropolar elasticity.
Under the effective medium description, the transverse modes are circularly polarized plane waves propagating in a homogeneous micropolar medium with acoustic activity. They carry well-defined spin and allows the possibility of achieving SOIs. In what follows, we demonstrate two SOI phenomena via numerical simulations and experiments. Effective medium theory based on micropolar elasticity is also applied to understand the results.
SOI in momentum space
The transverse sound near the \(\Gamma\) point in Fig. 3c can be described by an effective Hamiltonian \(H=-v{{{{{\bf{S}}}}}}\cdot {{{{{\bf{k}}}}}}\) with S being the spin-1 operator defined as \({({S}_{i})}_{{jk}}=-{{{{{\rm{i}}}}}}{\epsilon }_{{jki}}\). The Hamiltonian indicates a coupling between the spin and linear crystal momentum k, which induces splitting of the eigenfrequencies \(\triangle \omega \propto k\) and leads to a "negative band" for the LCP sound with spin \(s=\langle {{{{{\rm{LCP}}}}}}|{{{{{\bf{S}}}}}}\cdot \hat{{{{{{\bf{k}}}}}}}|{{{{{\rm{LCP}}}}}}\rangle =+1\), as shown in Fig. 3c. Near the \(\Gamma\) point, the group velocity and phase velocity take opposite signs, indicating negative refraction for a sound wave passing the metamaterial-air interface. We can define an effective refractive index \(n={-v}_{0}k/{\omega }_{T}^{-}\) with \({v}_{0}\) being the speed of sound in air28. This acoustic activity-induced negative index is different from those derived from overlapped monopolar and dipolar resonances43,44 or from multipole scattering45. It was proposed and verified in optics28,29, but has been long considered impossible for sound, as longitudinal waves cannot distinguish material chirality. Next, we numerically and experimentally demonstrate negative refraction in the 3D micropolar metamaterial.
In the numerical simulation, we consider the metamaterial consisting of 5 unit cells along z direction and 30 unit cells along x direction, as shown in Fig. 4a. A periodic boundary condition is applied in the y direction. A Gaussian beam obliquely incidents on the metamaterial at 70°. As the sound beam is longitudinal in the air but transverse in the metamaterial, impedance mismatch happens at the interfaces. For the efficient excitation of transverse sound, we engineered the surface impedance by adding acoustic tubes (see Supplementary Information). This also guarantees that only the \(s=+1\) sound is excited in the metamaterial. Figure 4a, b show the real part and the amplitude of the pressure field, respectively. The negative refraction is clearly observed. To verify the effective medium description of this phenomenon, we apply the effective parameters (same as those in Fig. 3c) to simulate the propagation of the same Gaussian beam in the micropolar effective medium. Negative refraction is seen again, as shown in Fig. 4c.
Fig. 4: Negative refraction induced by SOI in momentum space.
The real part (a) and the amplitude (b) of the pressure field for a Gaussian beam that propagates through the acoustic metamaterial with an incident angle of 70°. c The amplitude of pressure field in the corresponding micropolar effective medium system.
Experimentally, we fabricated a smaller sample consisting of \(11\times 4\) unit cells, as shown in Fig. 5a. This one-layer metamaterial can also induce negative refraction, as expected from the band structure of the 1D lattice system in Fig. 2d. We indeed observed the phenomenon by measuring the transmitted pressure field in the yellow zone of Fig. 5a. Figure 5b, c, respectively, show the amplitude and the real part of the pressure field. The beam with an incident angle of 40° is generated by an array of speakers. The simulation results are shown in Fig. 5d, e, where the region of experimental measurement is marked by the rectangle. Good agreement between the simulation and experimental results is seen, which confirms the negative refraction phenomenon induced by SOI.
Fig. 5: Experimental demonstration of the negative refraction.
a A photograph of the metamaterial lattice and the measurement area (yellow colored). b The amplitude and (c) the real part of the measured pressure field. d The amplitude and (e) the real part of the simulated pressure field. The dashed boxes indicate the corresponding measurement area in the experiment.
SOI in real space
The SOIs of transverse waves can also happen in real space. One intriguing phenomenon induced by such SOIs is the spin-dependent vortex generation in the scattering of subwavelength particles, which leads to the conversion of spin to OAM with important applications in optics such as optical manipulations and imaging2,46,47,48. It is commonly believed that airborne sound does not have this remarkable property. Here we demonstrate the real-space SOI for the transverse sound in the micropolar metamaterial.
We consider the micropolar metamaterial consisting of \(19\times 19\times 4\) unit cells under the normal incidence of a Gaussian beam at f = 655 Hz (corresponding to the frequency of the "negative band"), as shown in Fig. 6a. We remove one unit cell from the center of the metamaterial to create a subwavelength defect, as shown by the blue cube in Fig. 6b. This defect then serves as an acoustic dipole particle. Figure 6c shows the amplitude of the transmitted pressure field obtained by simulations. We notice a spiral pattern with two arms, which is a signature of an optical vortex with topological charge \(q=+2.\) This phenomenon can be understood as a result of SOI mediated by the dipole particle. The longitudinal sound in air excites the transverse sound in the metamaterial that carries spin \(s=+1\). The transverse sound has a velocity field \({{{{{{\bf{v}}}}}}}_{0}\,\) and a negative wave vector \({{{{{{\bf{k}}}}}}}_{0}\). It is scattered by the dipole particle, which generates scattered fields \({{{{{{\bf{v}}}}}}}_{{{{{{\rm{s}}}}}}}\) with a negative wave vector \({{{{{\bf{k}}}}}}\), as shown in Fig. 6b. The scattered field can be considered a spherical projection of the incident field: \({{{{{{\bf{v}}}}}}}_{{{{{{\rm{s}}}}}}}{{{{\propto }}}}-\hat{{{{{{\bf{r}}}}}}}\times (\hat{{{{{{\bf{r}}}}}}}\times {{{{{{\bf{v}}}}}}}_{0})\), where \(\hat{{{{{{\bf{r}}}}}}}\) is the unit radial vector. The projection induces noncommutative SO(3) rotations of the incident field and leads to geometric phases that account for the spin-to-OAM conversion5. This process can be expressed as \(|s\rangle \to {c}_{1}|s\rangle +{c}_{2}{e}^{2{{{{{\rm{i}}}}}}s\varphi }|-\!s\rangle\), where \(\varphi\) is the azimuthal angle, \({c}_{1}\) and \({c}_{2}\) are the coefficients characterizing the efficiency of the SOI49. The second term indicates the flip of spin and the emergence of an optical vortex with topological charge \(q=2s\). At the output interface, the background Gaussian beam and the scattered field are both converted to longitudinal sound, and their interference gives rise to the spiral pattern of pressure amplitude shown in Fig. 6c. To verify the results, we simulate the phenomenon in the micropolar effective medium using the same effective parameters as in Fig. 4. Similar interference pattern of the velocity field is obtained inside the micropolar effective medium, as shown in Fig. 6d. Figure 6e shows the real part of the scattered velocity field with \(s=-1\), which clearly shows a \(4\pi\) phase variation in the azimuthal direction and confirms the optical vortex with charge \(q=+2\).
Fig. 6: Spin-dependent vortex generation enabled by SOI in real space.
a The schematic of the scattering system. One unit cell is removed from the center of the metamaterial to create a dipole scatterer. A Gaussian beam is normally incident on the metamaterial. b The schematic of the scattering of transverse sound inside the metamaterial. The blue cube denotes the scatterer. c The amplitude of the transmitted pressure field. d The velocity amplitude in the micropolar effective medium due to the interference of \(s=-1\) scattered field with the background field. e The real part of the \(s=-1\) scattered velocity field in the micropolar effective medium.
We have demonstrated a mechanism that transforms airborne sound into a transverse wave with rich phenomena of SOIs. The SOIs are in contrast to the pseudo-SOIs in acoustic topological insulators where hybridization of modes are employed to construct "pseudo-spins"50. Our idea relies on engineering acoustic resonances at the subwavelength level to emulate shear responses, thereby giving rise to a fully vectorial transverse sound that carries a spin. From a microscopic perspective, this mechanism is similar to the emergence of induced dipole moments in a dielectric medium. Notably, dipole responses have been widely leveraged for anomalous effective mass density51. However, those dipole moments are parallel to the propagation direction, whereas in our micropolar metamaterial, the dipoles undergo microrotation in the plane orthogonal to the propagation direction. Consequently, a total of six degrees of freedom are needed to fully characterize the transverse sound in 3D, thereby bringing richer functionalities for sound manipulations. We note that the acoustic resonators in the metamaterial unit cell also support higher-order modes (e.g., quadrupole), which can endow sound with similar transverse properties. However, these modes exist at higher frequencies where an effective medium description may encounter difficulties and diffraction effects at the interface can affect the SOI phenomena.
We anticipate more explorations of the intriguing properties of the spin-1 transverse sound. For example, an interface formed by two micropolar metamaterials can support surface acoustic waves, which may have a topological origin and interesting non-Hermitian properties under a Weyl-type representation similar to electromagnetic surface waves52,53. In particular, the presence of micropolar material parameters can significantly enrich the properties of the surface "acoustic plasmons"54. In addition, the reflection/refraction of the transverse sound at an interface can give rise to an acoustic spin-Hall effect. The canonical momentum and spin densities of the transverse sound can induce radiation forces and torques on small particles in configurations similar to the one shown in Fig. 6a55,56, which could give rise to counterintuitive mechanical effects that can be experimentally probed using interference methods57,58. The acoustic activity can enable chiral sound–matter interactions with many applications, such as chiral discrimination and sensing, acoustic manipulations of chiral particles, and acoustic circular dichroism, etc. The spin-1 sound demonstrated here can also realize the bosonic analog of Kramers doublet. We thus expect a variety of applications and extensions of the results in spin–orbit acoustics, topological acoustics, and acoustic metamaterials.
Micropolar effective medium theory
Near the \(\Gamma (k=0)\) point, the acoustic metamaterial is approximately equivalent to a homogeneous and isotropic micropolar medium. Each point of the medium is characterized by a displacement field u and a microrotation field \({{{{{\mathbf{\phi }}}}}}\). Using Einstein summation convention, the strain tensor and curvature tensor can be expressed as23
$${\varepsilon }_{{ij}}=\frac{\partial {u}_{j}}{\partial {x}_{i}}-{\epsilon }_{{ijk}}{\phi }_{k},$$
$${\kappa }_{{ij}}=\frac{\partial {\phi }_{j}}{\partial {x}_{i}}.$$
The constitutive relations are26,35
$${\sigma }_{{ij}}={C}_{{ijkl}}{\varepsilon }_{{kl}}+{B}_{{ijkl}}{\kappa }_{{kl}},$$
$${m}_{{ij}}={B}_{{klij}}{\varepsilon }_{{kl}}+{D}_{{ijkl}}{\kappa }_{{kl}},$$
where the elastic constitutive tensors can be expressed as
$${C}_{{ijkl}}={C}_{1}{\delta }_{{ij}}{\delta }_{{kl}}+{C}_{2}{\delta }_{{ik}}{\delta }_{{jl}}+{C}_{3}{\delta }_{{il}}{\delta }_{{jk}},$$
$${B}_{{ijkl}}={B}_{1}{\delta }_{{ij}}{\delta }_{{kl}}+{B}_{2}{\delta }_{{ik}}{\delta }_{{jl}}+{B}_{3}{\delta }_{{il}}{\delta }_{{jk}},$$
$${D}_{{ijkl}}={D}_{1}{\delta }_{{ij}}{\delta }_{kl}+{D}_{2}{\delta }_{{ik}}{\delta }_{{jl}}+{D}_{3}{\delta }_{{il}}{\delta }_{{jk}}.$$
In terms of conventional notation, we have
$${C}_{1}=\lambda ,{C}_{2}=\mu +\kappa ,{C}_{3}=\mu -\kappa ,$$
$${B}_{1}=\eta ,{B}_{2}=\zeta +\xi ,{B}_{3}=\zeta -\xi ,$$
$${D}_{1}=\alpha ,{D}_{2}=\beta +\gamma ,{D}_{3}=\beta -\gamma .$$
Here, \(\lambda\) and \(\mu\) are the Lame constants; \(\kappa ,{{{{{\rm{\alpha }}}}}},{{{{{\rm{\beta }}}}}}\), and \(\gamma\) are the micropolar elastic constants; and \(\eta ,\zeta\), and \(\xi\) are the elastic constants due to material chirality. The equations governing the propagation of the sound wave in the chiral micropolar medium are given by the conservation of linear momentum and angular momentum:
$$\frac{\partial {\sigma }_{{ji}}}{\partial {x}_{j}}=\rho \frac{{\partial }^{2}{u}_{i}}{\partial {t}^{2}},$$
$$\frac{\partial {m}_{{ji}}}{\partial {x}_{j}}+{\epsilon }_{{ijk}}{\sigma }_{{jk}}={{{{{\mathscr{j}}}}}}\frac{{\partial }^{2}{\phi }_{i}}{\partial {t}^{2}},$$
where \(\rho\) is the mass density and \({{{{{\mathscr{j}}}}}}\) is the microinertia density. Assuming time-harmonic forms of the displacement field \({u}_{i}={U}_{i}{e}^{{{{{{\rm{i}}}}}}{k}_{i}{x}_{i}-{{{{{\rm{i}}}}}}\omega t}\) and microrotation field \({\phi }_{i}={\Phi }_{i}{e}^{{{{{{\rm{i}}}}}}{k}_{i}{x}_{i}-{{{{{\rm{i}}}}}}\omega t}\), and using the constitutive relations, the above governing equations can be reduced to
$$-{k}_{j}{k}_{k}{C}_{{jikl}}{U}_{l}+({{{{{\rm{i}}}}}}{k}_{j}{\epsilon }_{{nkl}}{C}_{{jikn}}-{k}_{j}{k}_{k}{B}_{{jikl}}){\Phi }_{l}=-\rho {\omega }^{2}{U}_{i},$$
$$(i{k}_{n}{\epsilon }_{{ijk}}{C}_{{jknl}}-{k}_{j}{k}_{k}{B}_{{klji}}){U}_{l}+{{{{{\rm{i}}}}}}{k}_{j}({\epsilon }_{{nkl}}{B}_{{knji}}+{\epsilon }_{{ink}}{B}_{{nkjl}}){\Phi }_{l}$$
$$-({k}_{j}{k}_{k}{D}_{{jikl}}-{\epsilon }_{{ijk}}{\epsilon }_{{nml}}{C}_{{jkmn}}){\Phi }_{l}={{{{{\mathscr{-}}}}}}{{{{{\mathscr{j}}}}}}{\omega }^{2}{\Phi }_{i}.$$
Expressing \({U}_{i}\) in terms of \({\Phi }_{i}\) by using Eq. (9) and substituting it into Eq. (10), we obtain
$$H{{{{{\boldsymbol{\Phi }}}}}}=\left[-v{{{{{\bf{S}}}}}}\cdot {{{{{\bf{k}}}}}}+{a}_{1}{{{{{\bf{kk}}}}}}{{{{{\boldsymbol{+}}}}}}{a}_{2}{k}^{2}+O\left({k}^{3}\right)\right]{{{{{\boldsymbol{\Phi }}}}}}=\delta \omega {{{{{\boldsymbol{\Phi }}}}}},$$
where we have expanded the equation at \(k\to 0\) and \(\omega -{\omega }_{0}=\delta \omega \to 0\) with \({\omega }_{0}=\sqrt{2\left({C}_{2}-{C}_{3}\right){{{\mathscr{/}}}}{{{\mathscr{j}}}}}\). Here, H is the effective Hamiltonian, S is the spin-1 matrix operator defined as \({\left({S}_{i}\right)}_{{jk}}=-{{{\rm{i}}}}{\epsilon }_{{jki}},{v}=\left({B}_{2}-{B}_{3}\right)/\sqrt{2{{{\mathscr{j}}}}\left({C}_{2}-{C}_{3}\right)},{a}_{1}=\left({D}_{1}+{D}_{3}\right)/\sqrt{8{{{\mathscr{j}}}}\left({C}_{2}-{C}_{3}\right)}{{{\boldsymbol{-}}}}\sqrt{2{{{\mathscr{j}}}}\left({C}_{2}-{C}_{3}\right)}/8\rho \), and \({a}_{2}={D}_{2}/\sqrt{8{{{\mathscr{j}}}}\left({C}_{2}-{C}_{3}\right)}+\sqrt{2{{{\mathscr{j}}}}\left({C}_{2}-{C}_{3}\right)}/8\rho \). It is noted that the leading order of the effective Hamiltonian describes the SOI. The above equation gives three eigenmodes that are dominated by the microrotation of mass points, among which two are transverse waves and one is a longitudinal wave. Their dispersion relations (retained the lowest order of \(k\)) are
$${\omega }_{T}^{\pm }={\omega }_{0}\pm {vk},\,\,{\omega }_{L}={\omega }_{0}+\tau {k}^{2},$$
where \(\tau =\left({D}_{1}+{D}_{2}+{D}_{3}\right)/\sqrt{8{{{{{\mathscr{j}}}}}}\left({C}_{2}-{C}_{3}\right)}\). In the low-frequency limit, microrotation vanishes in the metamaterial due to the cut-off frequencies of the resonators. Thus, \(\mu ,\kappa ,\alpha ,\beta ,\gamma ,\eta ,\zeta\), and \(\xi\) all vanish, only \({C}_{1}=\lambda\) (i.e., bulk modulus) remains. In this case, the metamaterial reduces to conventional acoustic metamaterial without bianisotropy.
Effective parameters retrieval
We retrieved the effective parameters based on the numerically computed band structures and the eigenmodes. Among the total 11 material parameters, only 9 parameters (i.e., \({B}_{2},{B}_{3},{C}_{2},{C}_{3},{D}_{1},{D}_{2},{D}_{3},\rho ,{{j}}\)) contribute to the microrotation-dominated waves that are responsible for the SOI phenomena. \({B}_{1}\) and \({C}_{1}\) do not play a role in the effective properties of the metamaterial for these waves. A three-step approach is applied to retrieve the effective parameters. We first evaluated the total force and torque acting on the unit cell, and applied Newton's second law to calculate the effective mass density \(\rho\) and microinertia density \({{{{{\mathscr{j}}}}}}\). Then, we fit the analytical dispersion relations with high-order corrections to the numerically computed band structures, from which the values of \(\,{C}_{2}-{C}_{3},{B}_{2},{B}_{3},{D}_{2},{D}_{1}+{D}_{3}\) can be determined. To further determine the values of \({C}_{2},{C}_{3},{D}_{1}\), and \({D}_{3}\), we employ the constitutive relations of Eqs. (3) and (4), where the strain and coupling stress can be obtained via boundary averaging of the eigenmode fields. The details about the parameter retrieval can be found in Supplementary Information. For a narrow frequency region near the \(\Gamma\) point, the retrieved effective parameters are approximately constants: \(\rho =0.637{{{{{\rm{kg}}}}}}/{{{{{{\rm{m}}}}}}}^{3}\), \({{j}}{{{{{\mathscr{=}}}}}}5.64\times {10}^{-4}{{{{{\rm{kg}}}}}}/{{{{{\rm{m}}}}}}\), \({B}_{2}=5.91{{{{{\rm{N}}}}}}/{{{{{\rm{m}}}}}}\), \({B}_{3}=55.0{{{{{\rm{N}}}}}}/{{{{{\rm{m}}}}}}\), \({C}_{2}=-1.68\times {10}^{4}{{{{{\rm{Pa}}}}}}\), \({C}_{3}=-2.16\times {10}^{4}{{{{{\rm{Pa}}}}}}\), \({D}_{1}=20.3{{{{{\rm{N}}}}}}\), \({D}_{2}=2.69{{{{{\rm{N}}}}}}\), and \({D}_{3}=-16.2{{{{{\rm{N}}}}}}\). These material parameters are then used in full-wave numerical simulations of the micropolar effective medium to verify the band structures and SOI phenomena of the metamaterial systems.
Full-wave numerical simulations are performed by using the finite-element package COMSOL Multiphysics (www.comsol.com) for both the metamaterial systems and the micropolar effective medium. For the resonators in Figs. 1, 3, 4, and 6, we set the radius R = 5 cm and height h = 2 cm. The period in both 1D and 3D metamaterials is \(a=12.1\,{{{{{\rm{cm}}}}}}\). The tubes have radii r = 0.2 cm. For the chiral resonator, the upper surface is twisted \(\pi /2\) with respect to the bottom surface. The Gaussian beam in Figs. 4 and 6 has a beam width \(w=1.2\lambda\). A sound-hard boundary condition is applied on all the boundaries of the resonators and tubes. Floquet periodic boundary conditions are applied to the 1D lattice and 3D metamaterials to compute the band structures. To compute the band structures of the micropolar effective medium and to simulate the associated SOIs phenomena, we developed weak-form formulations for the micropolar constitutive relations and momentum conservation equations, which are then implemented using COMSOL. The band structures of the micropolar medium are calculated by considering a unit cell made of homogenous and isotropic micropolar medium with the retrieved effective parameters.
The 1D lattice and 3D metamaterial were fabricated by using 3D printing. The resonators and connecting tubes are made of acrylonitrile butadiene styrene plastics, which were then assembled to form the structures in Figs. 2a and 5a. The fabricated units correspond to a scaled version of the units in Figs. 1a, c and 3a with \(R=3.5\,{{{{{\rm{cm}}}}}},{h}=1.75\,{{{{{\rm{cm}}}}}},{r}=0.4\,{{{{{\rm{cm}}}}}}\), and \(a=10.7\,{{{{{\rm{cm}}}}}}\). For the band structures of the 1D lattices, we excite the lattice using a loudspeaker at one end. The signal is generated by a waveform generator (Keysight 33500B) as a short pulse covering the frequency range of interest. We then measure the pressure responses with a microphone and a digital oscilloscope (Keysight DSO2024A) at all 24 unit cells with one measurement point per cell. Then, we perform a 2D Fourier transform to obtain the dispersion curves, which show the band structures (Fig. 2c, d). For the negative refraction experiment, we used an array of 11 loudspeakers to generate an obliquely incident Gaussian beam. Each speaker was driven by an independent channel of a computer sound interface (MOTU 16A). Both the amplitudes and phases of the output signal from each channel were precisely controlled by a PC (via a MATLAB program) to generate the targeted Gaussian beam with a tilted phase profile, in order to emulate oblique incidence at the chosen angle. On the far side of the metamaterial, small horns are connected to each unit cell, to improve impedance matching between the metamaterial and the air. The tabletop and a top plate (removed in Fig. 5a to show the metamaterials) form a two-dimensional waveguide in the output region for the better observation of the negative refracted field profile. A microphone is carried by a translational stage to raster-map the output beam profiles.
The authors declare that all data supporting the findings of this study are available within the paper and its Supplementary Information files. Additional data related to this paper are available from the corresponding authors upon reasonable request.
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The work described in this study was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Projects number CityU 21302018 and number C6013-18G). G.M. is supported by National Natural Science Foundation of China Excellent Young Scientist Scheme (Hong Kong and Macau) (number 11922416) and Youth Program (number 11802256). We thank Professors C. T. Chan, Z. Q. Zhang, Y. Wu, and Dr. R. Y. Zhang for helpful discussions, and Dr. Y. Chen for assistance in the implementation of weak form.
Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China
Shubo Wang & Qing Tong
Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
Guanqing Zhang, Xulong Wang & Guancong Ma
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
Jensen Li
Shubo Wang
Guanqing Zhang
Xulong Wang
Qing Tong
Guancong Ma
S.W. conceived the idea and designed the metamaterial, and conducted the numerical simulations. G.Z. and Q.T. assisted in numerical simulations. S.W. and Q.T. developed the effective medium theory. G.Z. and X.W. performed the experiments. J.L. assisted in theoretical interpretations. S.W. wrote the manuscript with input from all authors. S.W. and G.M. supervised the project. All authors contributed to the data analysis and the polishing of the manuscript.
Correspondence to Shubo Wang or Guancong Ma.
Peer review information Nature Communications thanks the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Wang, S., Zhang, G., Wang, X. et al. Spin-orbit interactions of transverse sound. Nat Commun 12, 6125 (2021). https://doi.org/10.1038/s41467-021-26375-9 | CommonCrawl |
# Understanding dimensionality and its impact on machine learning
Dimensionality refers to the number of features or variables in a dataset. In machine learning, high-dimensional datasets can pose challenges. As the number of features increases, the complexity of the data also increases, making it harder to analyze and interpret. This is known as the "curse of dimensionality."
The curse of dimensionality can lead to several problems in machine learning. One problem is increased computational complexity. As the number of features grows, the time and resources required to process and analyze the data also increase. This can make training and testing models slower and less efficient.
Another problem is overfitting. When the number of features is high compared to the number of samples, models can become overly complex and fit the training data too closely. This can lead to poor generalization and performance on new, unseen data.
Dimensionality reduction techniques aim to address these challenges by reducing the number of features while preserving important information. These techniques can help simplify the data, improve computational efficiency, and prevent overfitting.
For example, let's say we have a dataset with 100 features and 1000 samples. Each feature represents a different aspect of a product, such as its price, size, color, and so on. Analyzing and modeling this dataset directly would be challenging due to its high dimensionality.
## Exercise
Think about a real-world scenario where high dimensionality could pose challenges in machine learning. What are some potential problems that could arise?
### Solution
One potential problem could be the increased computational complexity and resource requirements. With a high-dimensional dataset, training and testing models could take a long time and require significant computational resources. Additionally, overfitting could be a problem, as models may become overly complex and fit the training data too closely, leading to poor generalization on new data.
# Feature selection techniques for dimensionality reduction
Feature selection is a technique used to select a subset of relevant features from a larger set of features. It aims to reduce the dimensionality of the data by eliminating irrelevant or redundant features. By selecting only the most informative features, we can simplify the data and improve the performance of machine learning models.
There are several feature selection techniques that can be used for dimensionality reduction. Some common techniques include:
1. Filter methods: These methods rank features based on statistical measures such as correlation or mutual information. Features with high rankings are considered more informative and are selected for further analysis.
2. Wrapper methods: These methods evaluate the performance of a machine learning model using different subsets of features. Features are selected based on their impact on the model's performance.
3. Embedded methods: These methods incorporate feature selection into the model training process. They select features based on their importance or contribution to the model's performance.
For example, let's say we have a dataset with 100 features and we want to select the top 10 most relevant features. We can use a filter method such as correlation coefficient to rank the features based on their correlation with the target variable. The top 10 features with the highest correlation coefficients can then be selected for further analysis.
## Exercise
Think about a real-world scenario where feature selection could be useful for dimensionality reduction. What are some potential benefits of feature selection in that scenario?
### Solution
One potential scenario could be in medical diagnosis. In a dataset with hundreds of medical features, feature selection could help identify the most relevant features for predicting a certain disease. By selecting only the most informative features, we can simplify the diagnosis process and improve the accuracy of the prediction model. Additionally, feature selection can help identify biomarkers or risk factors that are most strongly associated with the disease, providing valuable insights for further research and treatment development.
# Principal Component Analysis (PCA) for dimensionality reduction
Principal Component Analysis (PCA) is a widely used technique for dimensionality reduction. It is a statistical method that transforms a dataset into a new set of variables called principal components. These principal components are linear combinations of the original features and are chosen in such a way that they capture the maximum amount of variance in the data.
PCA works by finding the directions in the data that have the maximum variance. These directions, called principal axes or eigenvectors, represent the new coordinate system in which the data is transformed. The corresponding eigenvalues represent the amount of variance explained by each principal component.
To perform PCA, we follow these steps:
1. Standardize the data: PCA is sensitive to the scale of the features, so it is important to standardize the data by subtracting the mean and dividing by the standard deviation.
2. Compute the covariance matrix: The covariance matrix measures the relationship between each pair of features. It is computed by taking the dot product of the standardized data matrix.
3. Compute the eigenvectors and eigenvalues: The eigenvectors and eigenvalues of the covariance matrix represent the principal components and the amount of variance explained by each component, respectively. These can be computed using matrix decomposition techniques such as singular value decomposition (SVD) or eigenvalue decomposition.
4. Select the top k eigenvectors: The top k eigenvectors with the largest eigenvalues represent the most important principal components. These can be selected to reduce the dimensionality of the data.
5. Transform the data: The original data can be transformed into the new coordinate system defined by the selected principal components. This can be done by taking the dot product of the standardized data matrix with the selected eigenvectors.
For example, let's say we have a dataset with two features, height and weight, and we want to reduce the dimensionality to one dimension using PCA. We first standardize the data by subtracting the mean and dividing by the standard deviation. Then, we compute the covariance matrix and find its eigenvectors and eigenvalues. Let's say the first eigenvector has an eigenvalue of 2 and the second eigenvector has an eigenvalue of 1. We select the first eigenvector as the principal component, as it explains more variance in the data. Finally, we transform the data by taking the dot product of the standardized data matrix with the selected eigenvector.
## Exercise
What are the advantages of using PCA for dimensionality reduction? What are some potential use cases for PCA?
### Solution
Some advantages of using PCA for dimensionality reduction are:
- It reduces the dimensionality of the data while retaining most of the information.
- It can help visualize high-dimensional data by projecting it onto a lower-dimensional space.
- It can improve the performance of machine learning models by reducing the noise and redundancy in the data.
Some potential use cases for PCA include:
- Image compression: PCA can be used to reduce the dimensionality of image data, making it easier to store and transmit.
- Genetics: PCA can be used to analyze genetic data and identify patterns or clusters.
- Finance: PCA can be used to analyze financial data and identify key factors that drive stock prices or other financial variables.
# Linear Discriminant Analysis (LDA) for dimensionality reduction
Linear Discriminant Analysis (LDA) is another popular technique for dimensionality reduction. Unlike PCA, which is an unsupervised method, LDA is a supervised method that takes into account class labels in the data.
LDA works by finding a linear combination of features that maximizes the separation between different classes while minimizing the variance within each class. It does this by projecting the data onto a lower-dimensional space that maximizes the between-class scatter and minimizes the within-class scatter.
To perform LDA, we follow these steps:
1. Standardize the data: As with PCA, it is important to standardize the data by subtracting the mean and dividing by the standard deviation.
2. Compute the class means: Calculate the mean vector for each class in the data.
3. Compute the scatter matrices: Compute the within-class scatter matrix and the between-class scatter matrix. The within-class scatter matrix measures the variance within each class, while the between-class scatter matrix measures the variance between classes.
4. Compute the eigenvectors and eigenvalues: Compute the eigenvectors and eigenvalues of the matrix inverse of the within-class scatter matrix multiplied by the between-class scatter matrix. These represent the linear discriminants and the amount of variance explained by each discriminant, respectively.
5. Select the top k eigenvectors: Select the top k eigenvectors with the largest eigenvalues as the linear discriminants.
6. Transform the data: Transform the original data into the new coordinate system defined by the selected linear discriminants.
For example, let's say we have a dataset with two features, height and weight, and two classes, male and female. We want to reduce the dimensionality to one dimension using LDA. We first standardize the data, compute the class means, and then compute the within-class scatter matrix and the between-class scatter matrix. We then find the eigenvectors and eigenvalues of the matrix inverse of the within-class scatter matrix multiplied by the between-class scatter matrix. Let's say the first eigenvector has an eigenvalue of 2 and the second eigenvector has an eigenvalue of 1. We select the first eigenvector as the linear discriminant, as it explains more variance between classes. Finally, we transform the data by taking the dot product of the standardized data matrix with the selected eigenvector.
## Exercise
What is the main difference between PCA and LDA for dimensionality reduction? When would you choose to use PCA over LDA, and vice versa?
### Solution
The main difference between PCA and LDA for dimensionality reduction is that PCA is an unsupervised method that only considers the variance in the data, while LDA is a supervised method that takes into account the class labels.
You would choose to use PCA over LDA when you want to reduce the dimensionality of the data without considering the class labels. PCA is useful for tasks such as data visualization, noise reduction, and feature extraction.
On the other hand, you would choose to use LDA over PCA when you want to reduce the dimensionality of the data while considering the class labels. LDA is useful for tasks such as classification, where the goal is to maximize the separation between different classes.
In summary, PCA is a general-purpose dimensionality reduction technique, while LDA is specifically designed for supervised classification tasks.
# Non-linear dimensionality reduction techniques
While linear dimensionality reduction techniques like PCA and LDA are effective in many cases, they are limited in their ability to capture non-linear relationships in the data. In situations where the underlying structure of the data is non-linear, non-linear dimensionality reduction techniques can be more appropriate.
There are several non-linear dimensionality reduction techniques that have been developed to address this limitation. Some of the popular ones include kernel PCA, locally linear embedding (LLE), Isomap, and t-distributed stochastic neighbor embedding (t-SNE).
Kernel PCA is an extension of PCA that uses kernel functions to map the data into a higher-dimensional space where it can be linearly separated. This allows for non-linear relationships to be captured. LLE, on the other hand, seeks to preserve the local relationships between neighboring data points in the lower-dimensional space. It does this by finding a linear representation of each data point based on its neighbors.
Isomap is a technique that uses the geodesic distances between data points to construct a low-dimensional embedding. It is based on the idea that the distance between two points on a manifold is not the same as the Euclidean distance between them in the high-dimensional space. Finally, t-SNE is a technique that aims to preserve the pairwise similarities between data points in the lower-dimensional space. It is particularly effective at visualizing high-dimensional data.
Let's say we have a dataset of handwritten digits, where each digit is represented by a high-dimensional vector. We want to visualize the data in a lower-dimensional space while preserving the similarities between the digits. We can use t-SNE to achieve this. By applying t-SNE to the dataset, we can obtain a two-dimensional representation of the data where similar digits are grouped together.
## Exercise
Which non-linear dimensionality reduction technique would you choose to use if you wanted to preserve the local relationships between neighboring data points in the lower-dimensional space?
### Solution
I would choose to use locally linear embedding (LLE) if I wanted to preserve the local relationships between neighboring data points in the lower-dimensional space. LLE seeks to find a linear representation of each data point based on its neighbors, which allows for the preservation of local relationships.
# K-means clustering for unsupervised learning
K-means clustering is a popular unsupervised learning algorithm used to group data points into clusters based on their similarity. The goal of K-means clustering is to partition the data into K clusters, where each data point belongs to the cluster with the nearest mean.
The algorithm works by iteratively assigning data points to the cluster with the nearest mean and updating the cluster means based on the newly assigned data points. The process continues until the cluster assignments no longer change significantly.
To illustrate how K-means clustering works, let's consider a simple example. Suppose we have a dataset of customer transactions, where each transaction is represented by two features: the amount spent and the number of items purchased. We want to group the transactions into three clusters based on their spending and purchasing patterns.
We start by randomly initializing three cluster centroids. Then, for each data point, we calculate the distance to each centroid and assign the data point to the cluster with the nearest centroid. After all data points have been assigned to clusters, we update the cluster centroids by calculating the mean of the data points in each cluster. We repeat this process until the cluster assignments no longer change significantly.
Let's say we have the following dataset:
| Transaction | Amount Spent | Number of Items |
|-------------|--------------|-----------------|
| 1 | $50 | 5 |
| 2 | $30 | 2 |
| 3 | $70 | 7 |
| 4 | $20 | 1 |
| 5 | $60 | 6 |
| 6 | $40 | 3 |
We randomly initialize three cluster centroids:
| Cluster | Centroid (Amount Spent) | Centroid (Number of Items) |
|---------|------------------------|----------------------------|
| 1 | $50 | 5 |
| 2 | $30 | 2 |
| 3 | $70 | 7 |
We then calculate the distance between each data point and each centroid and assign the data points to the nearest cluster:
| Transaction | Amount Spent | Number of Items | Nearest Cluster |
|-------------|--------------|-----------------|-----------------|
| 1 | $50 | 5 | 1 |
| 2 | $30 | 2 | 2 |
| 3 | $70 | 7 | 3 |
| 4 | $20 | 1 | 2 |
| 5 | $60 | 6 | 3 |
| 6 | $40 | 3 | 2 |
We then update the cluster centroids by calculating the mean of the data points in each cluster:
| Cluster | Centroid (Amount Spent) | Centroid (Number of Items) |
|---------|------------------------|----------------------------|
| 1 | $50 | 5 |
| 2 | $30 | 2 |
| 3 | $65 | 6.5 |
We repeat the process of assigning data points to clusters and updating the centroids until the cluster assignments no longer change significantly.
## Exercise
Consider the following dataset:
| Data Point | Feature 1 | Feature 2 |
|------------|-----------|-----------|
| 1 | 2 | 3 |
| 2 | 4 | 5 |
| 3 | 6 | 7 |
| 4 | 8 | 9 |
| 5 | 10 | 11 |
| 6 | 12 | 13 |
Perform K-means clustering with K=2 on this dataset. Initialize the cluster centroids randomly.
### Solution
Here are the steps to perform K-means clustering with K=2 on the given dataset:
1. Randomly initialize two cluster centroids.
2. Calculate the distance between each data point and each centroid.
3. Assign each data point to the cluster with the nearest centroid.
4. Update the cluster centroids by calculating the mean of the data points in each cluster.
5. Repeat steps 2-4 until the cluster assignments no longer change significantly.
For example, the initial cluster centroids could be:
| Cluster | Centroid (Feature 1) | Centroid (Feature 2) |
|---------|---------------------|---------------------|
| 1 | 3 | 4 |
| 2 | 8 | 9 |
After one iteration, the cluster assignments could be:
| Data Point | Feature 1 | Feature 2 | Nearest Cluster |
|------------|-----------|-----------|-----------------|
| 1 | 2 | 3 | 1 |
| 2 | 4 | 5 | 1 |
| 3 | 6 | 7 | 1 |
| 4 | 8 | 9 | 2 |
| 5 | 10 | 11 | 2 |
| 6 | 12 | 13 | 2 |
The updated cluster centroids would be:
| Cluster | Centroid (Feature 1) | Centroid (Feature 2) |
|---------|---------------------|---------------------|
| 1 | 4 | 5 |
| 2 | 10 | 11 |
After another iteration, the cluster assignments and centroids might not change, indicating convergence of the algorithm.
# Hierarchical clustering for unsupervised learning
Hierarchical clustering is another popular unsupervised learning algorithm used to group data points into clusters. Unlike K-means clustering, which requires the number of clusters to be specified in advance, hierarchical clustering builds a hierarchy of clusters.
The algorithm starts by treating each data point as a separate cluster. Then, it iteratively merges the two closest clusters based on a distance metric, until all data points belong to a single cluster.
To illustrate how hierarchical clustering works, let's consider a simple example. Suppose we have a dataset of customer transactions, where each transaction is represented by two features: the amount spent and the number of items purchased. We want to group the transactions into clusters based on their similarity in spending and purchasing patterns.
We start by treating each transaction as a separate cluster. Then, we calculate the distance between each pair of clusters and merge the two closest clusters. We repeat this process until all transactions belong to a single cluster.
Let's say we have the following dataset:
| Transaction | Amount Spent | Number of Items |
|-------------|--------------|-----------------|
| 1 | $50 | 5 |
| 2 | $30 | 2 |
| 3 | $70 | 7 |
| 4 | $20 | 1 |
| 5 | $60 | 6 |
| 6 | $40 | 3 |
We can calculate the distance between each pair of clusters using a distance metric such as Euclidean distance. For example, the distance between clusters 1 and 2 is calculated as the Euclidean distance between the centroids of the two clusters:
$$\sqrt{(50-30)^2 + (5-2)^2} = \sqrt{400 + 9} = \sqrt{409}$$
We then merge the two closest clusters and update the centroid of the merged cluster. We repeat this process until all transactions belong to a single cluster.
## Exercise
Consider the following dataset:
| Data Point | Feature 1 | Feature 2 |
|------------|-----------|-----------|
| 1 | 2 | 3 |
| 2 | 4 | 5 |
| 3 | 6 | 7 |
| 4 | 8 | 9 |
| 5 | 10 | 11 |
| 6 | 12 | 13 |
Perform hierarchical clustering on this dataset.
### Solution
Here are the steps to perform hierarchical clustering on the given dataset:
1. Treat each data point as a separate cluster.
2. Calculate the distance between each pair of clusters using a distance metric such as Euclidean distance.
3. Merge the two closest clusters.
4. Update the centroid of the merged cluster.
5. Repeat steps 2-4 until all data points belong to a single cluster.
For example, the initial clusters could be:
| Cluster | Data Points |
|---------|-------------|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
After one iteration, the two closest clusters could be merged:
| Cluster | Data Points |
|---------|-------------|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 1, 2 |
The updated clusters would be:
| Cluster | Data Points |
|---------|-------------|
| 1 | 3 |
| 2 | 4 |
| 3 | 5 |
| 4 | 6 |
| 5 | 1, 2 |
After another iteration, the two closest clusters could be merged:
| Cluster | Data Points |
|---------|-------------|
| 1 | 3 |
| 2 | 4 |
| 3 | 5 |
| 4 | 6 |
| 5 | 1, 2 |
| 6 | 3, 4 |
The updated clusters would be:
| Cluster | Data Points |
|---------|-------------|
| 1 | 5 |
| 2 | 6 |
| 3 | 1, 2 |
| 4 | 3, 4 |
We repeat this process until all data points belong to a single cluster, resulting in a hierarchical clustering of the dataset.
# Density-based clustering for unsupervised learning
Density-based clustering is a type of unsupervised learning algorithm that groups data points based on their density. It is particularly useful for discovering clusters of arbitrary shape and handling noise in the data.
The main idea behind density-based clustering is that a cluster is a dense region of data points, separated by regions of lower density. The algorithm starts by selecting a core point, which is a data point with a sufficient number of neighboring points within a specified distance.
To illustrate how density-based clustering works, let's consider a simple example. Suppose we have a dataset of customer transactions, where each transaction is represented by two features: the amount spent and the number of items purchased. We want to group the transactions into clusters based on their similarity in spending and purchasing patterns.
We start by selecting a core point and finding all its neighboring points within a specified distance. We repeat this process for each core point until we have identified all the dense regions in the dataset. Points that do not belong to any dense region are considered noise.
Let's say we have the following dataset:
| Transaction | Amount Spent | Number of Items |
|-------------|--------------|-----------------|
| 1 | $50 | 5 |
| 2 | $30 | 2 |
| 3 | $70 | 7 |
| 4 | $20 | 1 |
| 5 | $60 | 6 |
| 6 | $40 | 3 |
We can start by selecting a core point, for example, transaction 1. We then find all the neighboring points within a specified distance, let's say 10. In this case, transactions 2, 3, and 5 are within a distance of 10 from transaction 1.
Next, we select another core point, for example, transaction 4. We find all the neighboring points within a distance of 10 from transaction 4, which in this case is transaction 6.
After repeating this process for each core point, we have identified two dense regions: {1, 2, 3, 5} and {4, 6}. Transaction 4 and transaction 6 are considered noise, as they do not belong to any dense region.
## Exercise
Consider the following dataset:
| Data Point | Feature 1 | Feature 2 |
|------------|-----------|-----------|
| 1 | 2 | 3 |
| 2 | 4 | 5 |
| 3 | 6 | 7 |
| 4 | 8 | 9 |
| 5 | 10 | 11 |
| 6 | 12 | 13 |
Perform density-based clustering on this dataset with a specified distance of 5.
### Solution
Here are the steps to perform density-based clustering on the given dataset:
1. Select a core point, for example, data point 1.
2. Find all the neighboring points within a specified distance, for example, 5. In this case, data points 2 and 3 are within a distance of 5 from data point 1.
3. Repeat steps 1 and 2 for each core point until all dense regions are identified. In this case, there are no other core points, so we have identified one dense region: {1, 2, 3}.
4. Points that do not belong to any dense region are considered noise. In this case, data points 4, 5, and 6 are considered noise.
Thus, the density-based clustering of the dataset with a specified distance of 5 results in one dense region: {1, 2, 3}, and three noise points: {4, 5, 6}.
# Evaluating clustering performance
Evaluating the performance of clustering algorithms is essential to determine the quality of the clusters generated and to compare different algorithms. There are several metrics commonly used to evaluate clustering performance, including:
1. **Silhouette coefficient**: The silhouette coefficient measures how well each data point fits into its assigned cluster. It takes into account both the distance between a data point and other data points in its cluster (cohesion) and the distance between a data point and data points in other clusters (separation). The silhouette coefficient ranges from -1 to 1, with values closer to 1 indicating better clustering.
2. **Calinski-Harabasz index**: The Calinski-Harabasz index measures the ratio of between-cluster dispersion to within-cluster dispersion. It considers both the separation between clusters and the compactness of each cluster. Higher values of the Calinski-Harabasz index indicate better clustering.
3. **Davies-Bouldin index**: The Davies-Bouldin index measures the average similarity between each cluster and its most similar cluster, taking into account both the separation and compactness of clusters. Lower values of the Davies-Bouldin index indicate better clustering.
4. **Rand index**: The Rand index measures the similarity between two data partitions, such as the ground truth partition and the clustering result. It compares the pairs of data points and counts the number of pairs that are either in the same cluster or in different clusters in both partitions. The Rand index ranges from 0 to 1, with values closer to 1 indicating better clustering.
It is important to note that no single metric can capture all aspects of clustering performance, and different metrics may be more appropriate for different datasets and clustering objectives. It is recommended to use multiple metrics and consider the overall performance of the clustering algorithm.
In addition to these metrics, visual inspection of the clustering result can also provide valuable insights. Plotting the data points and their assigned clusters can help identify any patterns or anomalies.
Let's consider an example to illustrate the evaluation of clustering performance. Suppose we have a dataset of customer transactions and we want to cluster the transactions based on their spending patterns. We apply a clustering algorithm and obtain the following clusters:
Cluster 1: {transaction 1, transaction 2, transaction 3}
Cluster 2: {transaction 4, transaction 5}
Cluster 3: {transaction 6, transaction 7, transaction 8}
To evaluate the clustering performance, we can calculate the silhouette coefficient, Calinski-Harabasz index, Davies-Bouldin index, and Rand index. Let's assume that the silhouette coefficient is 0.8, the Calinski-Harabasz index is 150, the Davies-Bouldin index is 0.5, and the Rand index is 0.9.
Based on these metrics, we can conclude that the clustering algorithm has performed well, with high values of the silhouette coefficient, Calinski-Harabasz index, and Rand index, and a low value of the Davies-Bouldin index.
## Exercise
Consider the following clustering result:
Cluster 1: {data point 1, data point 2, data point 3}
Cluster 2: {data point 4, data point 5}
Cluster 3: {data point 6, data point 7}
Calculate the silhouette coefficient, Calinski-Harabasz index, Davies-Bouldin index, and Rand index for this clustering result.
### Solution
To calculate the silhouette coefficient, Calinski-Harabasz index, Davies-Bouldin index, and Rand index, we need additional information, such as the pairwise distances between data points and the ground truth partition (if available). Without this information, we cannot calculate these metrics accurately.
However, based on the given clustering result, we can visually inspect the clusters and make qualitative assessments. For example, if the data points within each cluster are close to each other and well-separated from data points in other clusters, we can infer that the clustering result is good. Similarly, if the clusters align with our prior knowledge or expectations, we can consider the clustering result to be reliable.
Remember that these metrics are just tools to assist in evaluating clustering performance, and they should be used in conjunction with other techniques, such as visual inspection and domain knowledge.
# Implementing dimensionality reduction and clustering algorithms in Python
To begin, we need to install scikit-learn. Open your terminal or command prompt and run the following command:
```
pip install scikit-learn
```
Once scikit-learn is installed, we can start implementing dimensionality reduction and clustering algorithms.
**Implementing Dimensionality Reduction**
Let's start by implementing Principal Component Analysis (PCA) for dimensionality reduction. PCA is a widely used technique for reducing the dimensionality of high-dimensional data while preserving the most important information.
First, we need to import the necessary libraries:
```python
from sklearn.decomposition import PCA
```
Next, we need to create an instance of the PCA class and specify the number of components we want to keep:
```python
pca = PCA(n_components=2)
```
Then, we can fit the PCA model to our data and transform the data to its principal components:
```python
principal_components = pca.fit_transform(data)
```
Here, `data` is the input data matrix. The `fit_transform` method performs both the fitting and transformation steps.
Finally, we can access the explained variance ratio, which tells us the proportion of the dataset's variance that is explained by each principal component:
```python
explained_variance_ratio = pca.explained_variance_ratio_
```
This information can help us determine how many principal components to keep.
Let's consider an example to illustrate the implementation of PCA for dimensionality reduction. Suppose we have a dataset of images, where each image is represented as a high-dimensional vector. We want to reduce the dimensionality of the images while preserving the most important features.
```python
from sklearn.decomposition import PCA
# Create an instance of the PCA class
pca = PCA(n_components=2)
# Fit the PCA model to the data and transform the data to its principal components
principal_components = pca.fit_transform(images)
# Access the explained variance ratio
explained_variance_ratio = pca.explained_variance_ratio_
```
In this example, `images` is the input data matrix containing the images. After applying PCA, `principal_components` will contain the reduced-dimensional representation of the images, and `explained_variance_ratio` will provide information about the importance of each principal component.
## Exercise
Implement Principal Component Analysis (PCA) for dimensionality reduction on the given dataset. Set the number of components to 3.
```python
from sklearn.decomposition import PCA
# Create an instance of the PCA class with 3 components
# Fit the PCA model to the data and transform the data to its principal components
# Access the explained variance ratio
```
### Solution
```python
from sklearn.decomposition import PCA
# Create an instance of the PCA class with 3 components
pca = PCA(n_components=3)
# Fit the PCA model to the data and transform the data to its principal components
principal_components = pca.fit_transform(data)
# Access the explained variance ratio
explained_variance_ratio = pca.explained_variance_ratio_
```
# Case studies and real-world examples
**Case Study 1: Image Compression**
One common application of dimensionality reduction is image compression. Images often have a high dimensionality due to the large number of pixels. By applying dimensionality reduction techniques, we can reduce the dimensionality of the image while preserving the important features.
For example, let's consider a dataset of grayscale images. Each image is represented as a matrix of pixel values. We can use techniques like PCA to reduce the dimensionality of the images. This can be useful for tasks like image storage, transmission, and processing.
To illustrate this case study, let's consider a dataset of grayscale images. Each image is a 2D matrix of pixel values. We can apply PCA to reduce the dimensionality of the images.
```python
from sklearn.decomposition import PCA
# Load the dataset of grayscale images
images = load_images()
# Flatten each image into a 1D array
flattened_images = flatten_images(images)
# Create an instance of the PCA class
pca = PCA(n_components=100)
# Fit the PCA model to the flattened images and transform the images to their principal components
principal_components = pca.fit_transform(flattened_images)
# Reconstruct the images from the principal components
reconstructed_images = pca.inverse_transform(principal_components)
```
In this example, `load_images()` is a function that loads the dataset of grayscale images. `flatten_images()` is a function that flattens each image into a 1D array. After applying PCA, `principal_components` will contain the reduced-dimensional representation of the images. We can then reconstruct the images using `inverse_transform()`.
## Exercise
Consider a dataset of color images. Each image is represented as a 3D matrix of RGB values. Apply PCA for dimensionality reduction to the dataset. Set the number of components to 50.
```python
from sklearn.decomposition import PCA
# Load the dataset of color images
# Flatten each image into a 1D array
# Create an instance of the PCA class with 50 components
# Fit the PCA model to the flattened images and transform the images to their principal components
# Reconstruct the images from the principal components
```
### Solution
```python
from sklearn.decomposition import PCA
# Load the dataset of color images
images = load_color_images()
# Flatten each image into a 1D array
flattened_images = flatten_images(images)
# Create an instance of the PCA class with 50 components
pca = PCA(n_components=50)
# Fit the PCA model to the flattened images and transform the images to their principal components
principal_components = pca.fit_transform(flattened_images)
# Reconstruct the images from the principal components
reconstructed_images = pca.inverse_transform(principal_components)
``` | Textbooks |
Bienz, Stefan; Hesse, Mauhad (1987). Synthese makrocyclischer, $\alpha, \beta$-ungesättigter y-Oxolactone durch Ringerweiterungsreaktionen; ein neuer Weg zum makrocyclischen Lacton-Antibiotikum A 26771 B. Helvetica Chimica Acta, 70(5):1333-1340.
A new synthetic route to the a$- unsaturated y-oxolactones 2a and 2b, involving two ring-enlargement reactions, is described. Ring opening of bicyclic a- nitroketones of the type 3 gave ring-enlarged compounds of the type 4 which were converted to monoprotected diketones of the type 10 by using a variation of the Nefreaction as a key step. Macrocyclic lactones of the type I1 were obtained by Baeyer-Viffiger oxidation and converted into compounds of the type 2. The conversion of 2b to the macrocyclic lactone antibiotic A 26 771 B (1) is already described in the literature.
Download PDF 'Synthese makrocyclischer, $\alpha, \beta$-ungesättigter y-Oxolactone durch Ringerweiterungsreaktionen; ein neuer Weg zum makrocyclischen Lacton-Antibiotikum A 26771 B'. Item availability may be restricted. | CommonCrawl |
proof $e^{x}<1+x+x^{2}$ (mean value theorem preferred)
How it can be shown that:
$$e^{x}<1+x+x^{2}$$ For all $x<0.5$ I tried to use mean value theorem, but I have some problem, any idea or hint if highly appreciated. Clearly $e^x$ is continuous over $\left[x,x+1\right]$ and differentiable over $\left(x,x+1\right)$, hence there exist $c∈\left(x,x+1\right)$, such that: $$\frac{f\left(x+1\right)-f\left(x\right)}{x+1-x}=f^{'}\left(c\right)$$ Hence; $$e^{\left(x+1\right)}-e^{\left(x\right)}=e^c$$ or $$\ln\left(e^{\left(x+1\right)}-e^{\left(x\right)}\right)=c$$ but this is not helpful.
AbsurdAbsurd
$\begingroup$ What are your assumptions about $x$? The strong inequality is wrong when $x=0$. $\endgroup$ – Christian Blatter Dec 5 '19 at 14:43
$\begingroup$ $e^x = \sum_{k=2}^{\infty}\frac{x^k}{k!}+x+1< \sum_{k=2}^{\infty}\frac{x^2}{k!}+x+1 = x^2(e^1 - 2)+x+1)<x^2+x+1$ $\endgroup$ – fGDu94 Dec 5 '19 at 14:43
This is not a complete answer, but I thought it provides a simple method to find some values of x.
That expression is not true for all $x<0.5$ . Try $0$, and you obtain $1>1$ . For $x<-0.5$ the inequality holds, because we can write $e^x -1 < x(x+1)$ and if we differentiate we obtain $e^x$ and $2x+1$. Then, for all $x<-1/2$, the derivative of $e^x -1 $ is positive while the one for $x(x+1)$ is negative. Since $e^{(-1/2)} - 1 < 1+1/2+(1/2)^2$ then the inequality holds for $x< - 0.5$.
Actually, it is true for $ x \in ]-∞;0[ \; \cup \; ]0;1.79328[$ , but I'm not sure how to prove it analytically, neither of where the $1.79328$ comes from (aproximated value). I hope my method is close enough to what you wanted.
RicardoMMRicardoMM
$\begingroup$ the inequality is equivalent to $e^x-x^2-x-1<0$. So the $1.79328$ is one of the roots of $e^x-x^2-x-1$ $\endgroup$ – Zacharias Zarowski Dec 5 '19 at 16:42
$\begingroup$ Yeah, I knew, I just don't know a algebraic expression for it. But one could express it that way, yes. $\endgroup$ – RicardoMM Dec 5 '19 at 17:55
$\begingroup$ Ok I'm also not sure how to find a closed form :) but in this case its enough since its greater than $0.5$ $\endgroup$ – Zacharias Zarowski Dec 5 '19 at 18:31
Let us consider the function $$ g(x) = e^x - 1 - 2x $$ We have that $g(0)=0$. Moreover, since $g(1)=e-3<0$ while $g(2)=e^2-5>0$, by the theorem of zeros, there is a value $1<a<2$ such that $g(a)=0$. Since the function $f(x)=e^x$ is strictly convex, it must be $e^x<1+2x$ or each $0<x<a$. Therefore $$ \int\limits_0^x {e^t dt} < \int\limits_0^x {\left( {1 + 2t} \right)dt} $$ for each $0<x<a$. It means that $$ e^x - 1 < x + x^2 $$ for such values of $x$. This prove the inequality for $0<x<0.5$. Now, let be $x<0$. Then, again by convexity, it is $e^t>1+t$ for each $t\in \mathbb R$, $t\neq 0$. Therefore $$ \int\limits_x^0 {e^t dt} > \int\limits_x^0 {\left( {1 + t} \right)dt} $$ so that $$ 1 - e^x > - x - \frac{{x^2 }} {2} $$ thus $$ 1 + x + \frac{{x^2 }} {2} > e^x $$ and, a fortiori, the inequality follows.
You can also prove that $ e^t <1+2t $ for each $ 0< t< 1/2$ also by means of mean value theorem. Whith $0 < t <1/2$ you have that $$ \frac{{e^t - e^0 }} {{t - 0}} = e^c < e^{\frac{1} {2}} < 2 $$ thus $e^t< 1+2t$ if $0 < t < 1/2$.
Luca Goldoni Ph.D.Luca Goldoni Ph.D.
Two partial integrations give the following version of Taylor's theorem: $$\eqalign{e^x&=1+\int_0^x 1\cdot e^t\>dt=1+(t-x)e^t\biggr|_{t=0}^{t=x}-\int_0^x(t-x)e^t\>dt\cr &= 1+x-{(t-x)^2\over2}e^t\biggr|_{t=0}^{t=x}+\int_0^x{(t-x)^2\over2}e^t\>dt\cr &=1+x+{x^2\over2}+\int_0^x{(t-x)^2\over2}e^t\>dt\ .\cr}$$ We therefore have to show that $$\int_0^x{(t-x)^2\over2}e^t\>dt\leq{x^2\over2}\qquad(-\infty<x\leq1)\ .$$ When $x\leq0$ this is obviously true. When $0<x\leq1$ we have $$\int_0^x{(t-x)^2\over2}e^t\>dt\leq{e\over2}\int_0^x\tau^2\>d\tau={e\over2}{x^2\over3}<{x^2\over2}\ .$$
Christian BlatterChristian Blatter
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Mean value theorem inequality problem. | CommonCrawl |
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PROBT - Bayesian Programming -
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WebCHURCH, WebPPL, and OpenBUGS
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References and Further Reading
Pi by Monte-Carlo-Simulation
Hoeffding Bound
Example 1a: Two Fair Coins
Example 1b: Two fair coins with an observational constraint
Example 2: Two fair coins and one counter
Example 3: Loopy Probabilistic Program
Naive Bayesian Classifier
Example 5: Bayesian Network 'Student Model'
Example 6a: Bayesian Network 'Student Model' with Evidence
Example 6b: Bayesian Network 'Student Model' with more evidence
Example 6c: Bayesian Network 'Student Model' with further evidence
Example 6d: Bayesian Network 'Student Model' : P( I | D=0, L=1, S=0)
Example 7: The Fair Die (Discrete Time Markov Chain)
Example 8: Bayesian Skill Rating
Example 9: Lotka-Volterra Dynamic Population Model
BCM-Ch03.1: Inferring a Rate
BCM-Ch03.2: Difference between Two Rates
BCM-Ch03.3: Inferring a Common Rate
BCM-Ch03.4: Prior and Posterior Prediction
BCM-Ch03.6: Joint Distributions - The Survey Model -
BCM-Ch04.1: Inferring a mean and a Standard Deviation
BCM-Ch04.2: The Seven Scientists
Mixture Models of pairs of Binomial Models
Mixture Model for the 'Seven Scientists' Problem
BCM-Ch06.1: Latent Class Model "Exam Scores"
Latent Class Mixture Model with Dirichlet Allocation
Appendix 1: Gamma-Distribution
Appendix 2: Factor
Example 9: Prob-C Code for the Lotka-Volterra Dynamic Population Model
In their overview article about probabilistic programming the authors present a series of paradigmatical examples (Gordon, Henzinger, Nori & Rajamani, Probabilistic Programming, 2014). Example 9 deals with the classical predator-prey population model known as Lotka-Volterra model. The convential model is formulated as system of deterministic differential equations. Here the authors offer a stochastic solution in form of a probabilistic C-like computer code (Prob-C). The underlying mathematical model is a Continous-Time Markov Chain (CTMC).
"The Lotka-Volterra predator-prey model (Lotka, 1925; Volterra, 1926) is a population model which describes how the population of a predator and prey species evolves over time. It is specified using a system of so called 'stoichiometric' reactions as follows:
$$G \longrightarrow 2G$$
$$G+T \longrightarrow 2T$$
$$T \longrightarrow 0$$
We consider an ecosystem with goats (denoted by G) and tigers (denoted by T). The first reaction models goat reproduction. The second reaction models consumption of goats by tigers and consequent increase in tigers. The third reaction models death of tigers due to natural causes.
It turns out that this system can equivalently modelled as a Continuous Markov Chain (CTMC) whose state is an ordered pair (G, T) consisting of the number of goats G and the number of tigers T. The first reaction can be thought of as a transition in the CTMC from state (G, T) to (G + 1, T) and this happens with a rate equal to $$c_1 \cdot G$$ where c1 is some rate constant, and is enabled only when G > 0. Next, the second reaction can be thought of as a transition in the CTMC from state (G, T) to (G-1, T+1) and this happens with a rate equal to $$c_2 \cdot G \cdot T $$, where c2 is some rate constant, and is enabled only when G > 0 and T > 0. Finally , the last reaction can be thought of as a transition in the CTMC from state (G, T) to (G, T-1) and this happens with a rate equal to $$c_3 \cdot T $$ where c3 is some rate constant, and is enabled only when T > 0.
Using a process called uniformization, such a CTMC can be modeled using an embedded Discrete-Time Markov Chain (DTMC), and encoded as a probabilistic program, shown in Figure 1. The program starts with an initial population of goats and tigers and executes the transitions of the Lotka-Volterra model until a prescribed time limit is reached, and returns the resulting population of goats and tigers.
Since the executions are probabilistic, the program models the output distribution of the population of goats and tigers. The body of the while loop is divided into 3 conditions:
The first condition models the situation when both goats and tigers exist, and models the situation when all 3 reactions are possible.
The second condition models the situation when only goats exist, which is an extreme case, where only reproduction of goats is possible.
The third condition models the situation when only tigers exist, which is another extreme case, where only death of tigers is possible." (Gordon et al., 2014, p.6)
Fig 1: Lotka-Volterra Prob-C code of Gordon et al., (Fig.10, 2014), modelling a Continous-Time Markov Chain (CTMC)
Example 9: Formal Reconstruction of Gordon's et al. Prob-C Code
We think that the so called 'stoichiometric' reactions are misleading because they are rather different from chemical stoichiometric 'equations' or 'patterns'. According to Gordon et al. agent 'molecules' can appear out of nothing or disappear into nothing. This is not allowed in classical stoichiometry (Erban & Jonathan, 2020).
So we prefer to model the Lotka-Volterra-Scenario directly as a competitive Continuous Time Markov Chain (CTMC) with countable state space along its definition (e.g. Pishro-Nik, 2014, p.666; Ross, 2010, 10/e, p.373) with two subprocesses:
a holding-time process, when in state i
a jump process from state i to a different state j
Besides their standard use in natural (Erban & Chapman, 2020) and computational (Gordon, 2014; Mitzenmacher & Upfahl, 2018; Cassandras & Lafortune, 2008) sciences CTMCs are a non-standard modelling tool in psychology (Wickens, 1982, McGill, 1967; Diederich & Mallahi-Karai, 2020) and sociology (Coleman, 1964; Bartholomew, 1967; Tuma & Hannan, 1984).
The holding-time Poisson counting process is waiting for an event of type birth (of prey) or fatal attack (of predator), or death (of predator). In this kind of process it is assumed that holding or waiting times are exponential distributed with rates:
$$\lambda(G,T) = \lambda_1(G) +\lambda_2(G,T) + \lambda_3(T).$$
$$\lambda_1(G) = c_1 \cdot G \text{ ; birth process with birth events}$$
$$\lambda_2(G, T) = c_2 \cdot G \cdot T \text{ ; birth-death process with attack events}$$
$$\lambda_3(T) = c_3 \cdot T \text{ ; death process with death events}$$
Because rates are state-dependent the Poisson counting processes are inhomogeneous. The total rate is a sum of the rates of the three individual and independent Poisson counting subprocesses (birth, birth-death, and death) running parallel and in competion to each other. Poisson race processes are widely used in such diverse fields as cognitive reaction time (RT), choice processes or reliability studies (Ibe, 2013, ch. 2.7.5; 2014, ch. 12.5.9). In independent Poisson race processes the total rate is the sum of the rates of the subprocesses and the probability of winning this race can be determined by the ratio:
$$ p_{j} = \frac{\lambda_{j}}{\lambda} \text{ ; j = 1,...,3} \text{ ; probability of j is winning the process race}.$$
As the interarrival times of a Poisson counting process have an exponential distribution we can sample holding times ('dwellTime') of the merged Poisson process
$$ \Delta t(G, T) \sim DExp(\lambda(G,T)) = DExp('rate')$$
add them up to the current time ('curTime'), and compare the current time with our TIMELIMIT.
At the end of the holding-time interval the CTMC has to determine which of the three subprocesses generated the event. The probability of winning the race between the three Poisson counter processes and the jump-to probability is the simple standardized rate parameter (Ibe, 2013, ch. 2.7.5; 2014, ch. 12.5.9):
$$ p_{ij}(G,T) = \frac{\lambda_{ij}(G,T)}{\lambda_i(G,T)} \text{ ; i = 1, 2, ..., ; j = 1,...,3}.$$
Since the rates and probabilities for all i are constant ("uniform"), we suppress the state index i from now on.
$$ p_{j}(G,T) = \frac{\lambda_{j}(G,T)}{\lambda(G,T)} \text{ ; j = 1,...,3} \text{ ; probability that j is winning the process race}.$$
The jump-to index j is sampled from discrete distribution with the jump probabilities as parameter vector
$$j(G,T) \sim Discrete(p_1(G,T), p_2(G,T), p_3(G,T)) \text{ ; j = 1, ..., 3}$$
According to the selected jump-to index j the three processes generate at most two simultaneous actions. According to Gordon et al. these actions are:
$$(G, T) \stackrel{c_1 \cdot G}{\longrightarrow} (G+1, T) \text{ ; if j = 1, then birth process}$$
$$(G, T) \stackrel{c_2 \cdot G \cdot T}{\longrightarrow} (G-1, T+1) \text{ ; if j = 2, then birth-death process}$$
$$(G, T) \stackrel{c_3 \cdot T}{\longrightarrow} (G, T-1) \text{ ; if j = 3, then death process}$$
The simultaneous actions of the birth-death process (j = 2) are definitional for this kind of process, though they seem to be a bit unnatural in the Lotka-Volterra context. It is a bit unrealistic that during an exponential distributed attack episode a goat's disappearance is accompanied in the same episode by a birth of a tiger.
These transitions reach every state i in two-dimensional state-space S.
$$ S \subseteq G \times T $$ $$ G, T \subset \mathbb{N} $$
Transitions of the Lotka-Volterra CTMC are visualized in a transition-rate diagram (Fig. 2). This 2-dimensional transition space is typical for a competition process (Reuter, 1961; Iglehart, 1964) . It deviates from the conventional one-dimensional transition space typically used in CTMC definitions.
Empirical research has shown that the maximum of the predator frequency distribution appears most times later than the maximum of the prey distribution. This is not true for Gordon et al's model, when we use their rate-constants c as we demonstrate with our WebPPL-coded reconstruction (see below).
The classical Lotka-Volterra model is valuable for educational purposes but too too simple for empirical predictions and as an agent-based model it is not embedded in an ecological context (Charles J. Krebs, Rudy Boonstra, Stan Boutin, A.R.E. Sinclair, 2001). This is the fundamental deficit.
Fig. 2: transition-rate diagram of Gordon et al.'s Lotka-Volterra competitive CTMC model
WebPPL-Implementation of the Lotka-Volterra Continous-Time Markov Chain (CTMC) Model
Lotka-Volterra Simulation Run with WebPPL Script and Mateos' Parameters
Mateos' parameters were obtained from his 2018 lecture slides (visited 2020/07/27)
$$c_1 = 1.0$$
$$c_2 = 0.005 $$
$$c_3 = 0.6 $$
$$Goats(0) = 50$$
$$Tigers(0) = 100 .$$
The WebPPL-Code was developed independently. Results are similar though some of our simulations runs end in an ex- or implosion of the state space.
Lotka-Volterra Simulation Run with WebPPL Script and Gordon et al.'s Parameters and Rates
Gordon et al.'s parameters and rates were obtained from the authors 2014 publication (visited 2020/07/28).
$$Goats(0) = 100$$
$$Tigers(0) = 4 .$$
The pure functional WebPPL-Code is an independently developed reimplementation of the imperative Prob-C code . Results demonstrate that the model does not show the typical Lotka-Volterra behavior with its cyclic trajectories. A model run with Mateos' start parameters and rate are much more typical for the Lotka-Volterra model trajectories (see above).
Bartholomew, D.J., Stochastic Models for Social Processes, John Wiley, 1/e 1967, 3/e 1982
Cassandras Ch.G. & Lafortune, St., Introduction to Discrete Event Systems, 2/e, Springer, 2008
Diederich, A. & Mallahi-Karai, K., Stochastic Methods for Modelling Decision-Making, p.1-70, in: Batchelder, W.H., Colonius, H. & Dzhafarov, E.N., New Handbook of Mathematical Psychology, Vol.2: Modeling and Measurement, Cambrige University Press, 2018
Erban, R. & Chapman, J., Stochastic Modelling of Reaction-Diffusion Processes, Cambrige University Press, 2020
Gordon, A. D., et al. Probabilistic programming. In: Proceedings of the on Future of Software Engineering. 2014. S. 167-181.
Ibe, O., Markov Processes for Stochastic Modeling, 2/e, Elsevier, 2013
Ibe, O., Fundamentals of Applied Probability and Random Processes, Academic Press, 2014
Iglehart, D. L., et al. Multivariate competition processes. The Annals of Mathematical Statistics, 1964, 35. Jg., Nr. 1, S. 350-361.
Krebs, Ch. J., et al. What Drives the 10-year Cycle of Snowshoe Hares? The ten-year cycle of snowshoe hares—one of the most striking features of the boreal forest—is a product of the interaction between predation and food supplies, as large-scale experiments in the Yukon have demonstrated. BioScience, 2001, 51. Jg., Nr. 1, S. 25-35.
Lotka, A., Elements of physical biology, Williams & Wilkins company, Baltimore, 1925.
Mateos, G., Predator-Prey Population Dynamics, Lecture Slides, 2018/11/13, Dept. of ECE and Goergen Institute for Data Science, University of Rochester (visited 2020/07/28)
McGill, W.J., Stochastic Latency Mechanisms, p.309-360, in: Luce, R.D., Bush, R.R. & Galanter, E. (eds), Handbook of Mathematical Psychology, Vol. I, John Wiley & Sons, 2/e, 1967
Mitzenmacher, M. & Upfahl, E., Probability and Computing; Randomization and Probabilistic Techniques in Algorithms and Data Analysis, 2/e, 2017
Pishro-Nik, H., Probability, Statistics, and Random Processes, Kappa-Research LLC, 2014
Reuter, G. E. H. Competition processes. In: Proc. 4th Berkeley Symp. Math. Statist. Prob. 1961. S. 421-430.
Ross, S.M, Introduction To Probability Models, Elsevier, 2010, 10/e
Ruan, Sh., Poisson Race Models: Theory and Application in Conjoint Choice Analysis. 2007. PhD-Thesis. The Ohio State University.
Tuma, N.B. & Hannan, M.T., Social Dynamics: Models and Methods, Academic Press, 1984
Volterra, V., Fluctuations in the abundance of a species considered mathematically, Nature, 118: 558–560, 1926.
Wickens, Th.D., Models For Behavior: Stochastic Processes in Psychology, Freeman & Co., 1982 | CommonCrawl |
\begin{document}
\title{On Pebbling Graphs by their Blocks}
\author{ Dawn Curtis\thanks{[email protected]}, Taylor Hines\thanks{[email protected]}, Glenn Hurlbert\thanks{[email protected]}, Tatiana Moyer\thanks{[email protected]}\\
Department of Mathematics and Statistics\\
Arizona State University,
Tempe, AZ 85287-1804 }
\maketitle
\begin{abstract}
Graph pebbling is a game played on a connected graph $G$. A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of $G$ (called a configuration) and chooses a target vertex $r$. The player may make a pebbling move by taking two pebbles off of one vertex and moving one pebble to a neighboring vertex. The player wins the game if he can move $k$ pebbles to $r$. The value of the game $(G,k)$, called the $k$-pebbling number of $G$ and denoted ${\pi}_k(G)$, is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer $m$ of pebbles so that, from every configuration of size $m$, one can move $k$ pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the $k$-pebbling number of diameter-two graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two.
\end{abstract}
\section{Introduction}\label{Intro}
Graph pebbling is a game played on a connected graph $G=(V,E)$. \footnote{We assume the notation and terminology of \cite{W} throughout.} A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of $G$ (called a {\it configuration}) and chooses a target, or {\it root} vertex $r$. The player may make a {\it pebbling move} by taking two pebbles off of one vertex and moving one pebble to a neighboring vertex. The player wins the game if he can move $k$ pebbles to $r$, in which case we say that $r$ is $k$-{\it pebbled}. Another common terminology calls the configuration $k$-{\it fold} $r$-{\it solvable}. The {\it value} of the game $(G,k)$, called the $k$-{\it pebbling number} of $G$ and denoted ${\pi}_k(G)$, is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer $m$ of pebbles so that, from every configuration of size $m$, one can move $k$ pebbles to any root. If $k$ is not specified, it is assumed to be one.
For example, by the pigeonhole principle we have ${\pi}(K_n)=n$, where $K_n$ is the complete graph on $n$ vertices. From there, induction shows that ${\pi}_k(K_n)=n+2(k-1)$. Induction also proves that ${\pi}_k(P_n)=k2^{n-1}$, where $P_n$ is the path on $n$ vertices. These two graphs illustrate the tightness of the two main lower bounds ${\pi}(G)\ge\max\{n(G),2^{{\rm diam}(G)}\}$, where ${\rm diam}(G)$ is the {\it diameter} of $G$, the number of edges in a maximum induced path. Another fundamental result uses the path fact and induction to calculate the $k$-pebbling number of trees (see \cite{Chung}). The survey \cite{H} contains a wealth of information regarding pebbling results and variations.
Complete graphs and paths are examples of {\it greedy} graphs. That is, the most efficient pebbling moves are directed towards the root. More formally, a pebbling move from $u$ to $v$ is greedy if $dist(v,r) < dist(u,r)$, where $dist(x,y)$ denotes the distance between $x$ and $y$. A greedy solution uses only greedy moves. A graph $G$ is greedy if every configuration of size ${\pi}(G)$ can be greedily solved. If a graph is greedy, then we can assume every pebbling move is directed towards the root. The greedy property of trees follows from the No-Cycle Lemma of \cite{Moe} (see also \cite{CCF,MC}), which states that the digraph whose arcs represent the pebbling moves of a minimal solution contains no directed cycles. A {\it cut vertex} of a graph is a vertex that, if removed, disconnects the graph. The {\it connectivity} ${\kappa}$ of a graph is the minimum number of vertices whose deletion disconnects the graph or reduces it to only one vertex. Two important results relate diameter and connectivity to pebbling numbers. Pachter, Snevily, and Voxman proved the first.
\begin{res}\label{psv}\cite{PSV}. If $G$ is a connected graph on $n$ vertices with ${\rm diam}(G)\le 2$ then ${\pi}(G)\le n+1$. \end{res}
Clarke, Hochberg, and Hurlbert \cite{CHH} characterized which diameter two graphs have pebbling number $n$ and which have pebbling number $n+1$. We will use the graphs that describe that characterization in Section \ref{Cliques}. Motivated by the characterization, Czygrinow, Hurlbert, Kierstead, and Trotter proved the second.
\begin{res}\label{chkt}\cite{CHKT}. If $G$ is a connected graph on $n$ vertices with ${\rm diam}(G)\le d$ and ${\kappa}(G)\ge 2^{2d+3}$ then ${\pi}(G)=n$. \end{res}
This result states that high connectivity compensates for large diameter in keeping the pebbling number to a minimum. In this paper we exploit graph structures further to investigate pebbling numbers. A {\it block} of a graph $G$ is a maximal subgraph of $G$ with no cut vertex. Let ${\cal B}$ be the set of all blocks of $G$ and ${\cal C}$ be the set of all cut vertices of $G$. Then the {\it block-cutpoint} graph of $G$, denoted $B(G)$, has vertices ${\cal B}\cup{\cal C}$, with edges $(B,C)$ whenever $C\in V(B)$. Note that $B(G)$ is always a tree (see \cite{W}). Figure \ref{BofG} shows an example.
\begin{figure}
\caption{A graph and its block-cutpoint graph}
\label{BofG}
\end{figure}
Here we instigate a line of research into using the $k$-pebbling numbers of $B(G)$ and of the blocks of $G$ to give upper bounds on ${\pi}_k(G)$. To begin, we generalize Chung's tree result to weighted trees in Section \ref{Trees}. We then present the exact $k$-pebbling number of $G$ when every block of $G$ is complete in Section \ref{Cliques}. Also in Section \ref{Cliques}, we prove the following theorem, and show that there is a diameter-2 graph $G$ on $n\ge 6$ vertices with ${\pi}_k(G)=n+4k-3$ for all $k$ (Theorem \ref{bigconfig}). Thus Theorem \ref{kpd2} is not known to be tight.
\begin{thm}\label{kpd2} If $G$ is a graph on $n$ vertices with ${\rm diam}(G)\le 2$ then ${\pi}_k(G)\le n+7k-6$. \end{thm}
Section \ref{Remarks} provides some further conjectures, questions, and possibilities for future research.
\section{Trees and General Pebbling}\label{Trees}
A {\it tree} is a connected, acyclic graph, and a {\it forest} is a union of pairwise vertex-disjoint trees. A {\it leaf} of a tree is a vertex of degree one. An $r$-{\it path partition} of a particular tree $T$ is a partition of the edges of $T$ into paths, constructed by carrying out the following algorithm. Construct the sequence of pairs ${(T_i, F_i)}$, where each $T_i$ is a tree and each $F_i$ is a forest, with $E(T_i) \cup E(F_i) = E(T)$, and $E(T_i) \cap E(F_i) = \emptyset$. Begin with $T_0 = r$, $F_0 = T$ and end with $T_t = T$, and $F_t = \emptyset$. At each stage, for some path $P_i$ we have $P_i = T_i - T_{i-1} = F_{i-1} - F_i$, with the property that for each i, the intersection $V(P_i)\cap V(T_{i-1})$ is a leaf of $P_i$. The path partition is {\it r-maximal} if each $P_i$ is the longest such path in $F_{i-1}$. An $r$-maximal path partition is {\it maximal} if $r$ is one of the leaves of the longest path in $T$. An r-path partition of a tree is depicted in Figure \ref{NotMaxTree}, and a maximal path partition of a tree is depicted in Figure \ref{Max}.
\begin{figure}
\caption{A non-maximal $r$-path partition of a tree, with its corresponding unsolvable configuration}
\label{NotMaxTree}
\end{figure}
\begin{figure}
\caption{An $r$-maximal path partition of a tree, with its corresponding unsolvable configuration}
\label{Max}
\end{figure}
Define $x_i$ to be the leaf of $P_i$ in $T_{i-1}$ and $y_i$ to be the leaf of $P_i$ not in $T_{i-1}$, and let $a_i = |E(P_i)|$.
\begin{lem}\label{Unsolvable}. The configuration $C$ on $T$ defined by each $C(y_i)=2^{a_i}-1$ and $C(v)=0$ for all other $v$ is $r$-unsolvable. \end{lem}
{\noindent {\it Proof}.\ \ } We use induction. Let $C_i$ be the restriction of $C$ to $T_i$. The case in which $i=0$ is trivial since the root has no pebbles. Now, assume that $C_k$ is $r$-unsolvable on $T_k$. We know that the configuration on $P_{k+1}$ is $x_{k+1}$-unsolvable because the pebbling number of a path of length $l$ is $2^l$. Thus, no pebbles can be moved to from $P_{k+1}$ to $T_k$ since $V(T_k) \cap V(T_{k+1}) = x_{k+1}$. Since we already know $T_k$ is unsolvable, $T_{k+1}$ is unsolvable also. Thus, by induction, the configuration $C$ on $T$ is $r$-unsolvable. {
$\Box$}
Chung's result generalizes this idea for k-pebbling.
\begin{res}\label{chung}\cite{Chung}. If $T$ is a tree and $a_1, a_2,\ldots,a_t$ is the sequence of the path size (i.e. the number of vertices in the path) in a maximum path partition of $T$, then ${\pi}_k(T)=k2^{a_1}+ \sum_{i=2}^{t} 2^{a_i} - t + 1$. \end{res}
Chung's proof of this result uses induction performed on the vertices of $T$ by fixing and then removing the root, thus dividing $T$ into subtrees in order to use induction. We give a different proof of the more general Theorem \ref{ourtree}, relying on the fact that trees are greedy.
First we consider a more general form of pebbling. For each edge $e$ of a graph $G$ we can assign a weight $w_e$. The weight is intended to signify that it takes $w_e$ pebbles at one end of $e$ to place 1 pebble at its other end. Hence the pebbling considered to this point has $w_e=2$ for all $e$. We define the weighted pebbling number ${\pi}_k^w(G,r)$ to be the minimum number $m$ so that every configuration of size $m$ can $k$-pebble $r$ by using $w$-weighted pebbling moves on $G$.
\begin{figure}
\caption{An edge-weighted tree}
\label{Weighted}
\end{figure}
Given a weight function $w:E(G){\rightarrow}{\mathbb N}$, we extrapolate to a weight function on the set of all paths of $G$, where $w(P)$ is the product of edge weights over all edges of the path $P$. Now when constructing maximal path partitions, we replace the condition ``longest path" by ``heaviest path" (greatest weight). This is equivalent for constant weight 2 pebbling. Nothing in the proof of Chung's theorem changes for weighted trees, but we introduce a new proof of the pebbling number of a weighted tree.
Let $P_1,\ldots,P_t$ be an $r$-maximal path partition of $T$, with $w(P_1) \ge \ldots \ge w(P_t)$. Let $f_k^w(T,r)=kw(P_1)+ \sum_{i=2}^t w(P_i) - t + 1$. For vertices $x$ and $y$ on a path $P$, denote by $P[x,y]$ the subpath of $P$ from $x$ to $y$.
\begin{thm}\label{ourtree} Every weighted tree T satisfies ${\pi}_k^w(T,r) \le f_k^w(T,r)$. \end{thm}
{\noindent {\it Proof}.\ \ } The theorem is trivially true when $t=1$ since $T$ is a path.
For $t \ge 1$, define $T'=T-P_t$. Then $f_k(T,r)=f_k(T',r)+w(P_t)-1.$ Let $P_j$ be a path containing the non-leaf endpoint $x_t$ of $P_t$, and let vertex $y_j$ be the leaf of $T$ on $P_j$. Define $W = w(P_j[x_t,y_j])$. Thus we know from the maximal $r$-path construction that $W \ge w(P_t)$.
Let $C$ be an unsolvable configuration on $T$ with $|C|=f_k(T,r)$. Without loss of generality, we can assume that all the pebbles are on the leaves of a tree because the maximum sized unsolvable configuration sits on the leaves only. Let $s \ge 0$ be the number of pebbles $P_t$ contributes to the vertex $x_t$, so we have $sw(P_t) \le |C(P_t)| < (s+1)w(P_t)$.
Now define the configuration $C'$ on $T'$ by $C'(y_j)=C(y_j)+sW$ and $C'(v)=C(v)$ otherwise. Then, \begin{eqnarray*}
|C'|& = &|C|-[(s+1)w(P_t)-1] + sW\\ &\ge& f_k(T,r)-w(P_t)+1\\ &=& f_k(T',r)\ . \end{eqnarray*} Hence $C'$ is $k$-fold solvable on $T'$. Now define $C^*$ on $T'$ by $C^*(x_t)=C(x_t)+s$ and $C^*(v)=C(v)$ otherwise. In particular, because of greediness, $C^*$ is $k$-fold $r$-solvable on $T'$ because moving at most $sw(P_t)$ pebbles from $y_j$ to $x_t$ converts $C'$ to a solvable subconfiguration of $C^*$. Now, since $C(P_t) \ge sw(P_t)$, the base case says we can move $s$ pebbles from $P_t$ to $x_t$, and in doing so we arrive again at $C^*$ on $T'$. Hence $C$ is $k$-fold $r$-solvable. {
$\Box$}
We will use Theorem \ref{ourtree} to upper bound the pebbling number of graphs composed of blocks. The technique utilizes the block-cutpoint graph.
For a graph $G$ and its block-cutpoint graph $B(G)$, let $b_i$ denote the vertex of $B(G)$ that corresponds to the block $B_i$ in $G$. For each block $B_i$, let $x_i$ denote the cut vertex of $G$ in $B_i$ that is closest to the root (it is possible that some $x_i=x_j$). Let $e_i$ denote the edge of $B(G)$ between $b_i$ and $x_i$, and define its weight by $w(e_i)={\pi}(B_i, x_i)$. Let all other edges have weight 1. For a root $r$ of $G$, let $B$ denote the block containing it, represented by the vertex $b$ in $B(G)$. Let $B'(G)$ be the graph obtained from $B(G)$ by adjoining to $b$ by an edge of weight 1 a new vertex $r'$. Then we arrive at the following theorem.
\begin{thm}\label{BlockCut} Every graph $G$ satisfies ${\pi}_k(G,r) \le {\pi}_k^w(B'(G),r')$ \end{thm}
{\noindent {\it Proof}.\ \ } For a set $U$ of vertices, denote by $C(U)$ the sum $\sum_{v \in U} C(v)$. Let $x(B_i)$ denote all the cut vertices of $G$ in the block $B_i$. Given a configuration $C$ on $G$, define $C'$ on $B'(G)$ by \begin{itemize} \item $C'(x_i) = C(x_i)$ for all cut vertices $x_i$, and \item $C'(b_i) = C(B_i) - C(x(B_i))$ for all blocks $B_i$. \end{itemize} Given an $r'$-solution $S'$ of $C'$ on $B'(G)$, which exists because
$|C'|=|C|={\pi}_k^w(B'(G),r')$, define the $r$-solution $S$ of $C$ on $G$ by the following: replace every pebbling step along $e_i$ in $S'$ by some $x_i$-solution of some ${\pi}(B_i)$ of the pebbles in $B_i$. Then $S$ is an $r$-solution. {
$\Box$}
\section{Larger Blocks}\label{Cliques}
In this section we consider the cases in which all blocks are cliques or all have bounded diameters. The following proposition is well known.
\begin{prp}\label{super} If $H$ is a connected spanning subgraph of $G$ then ${\pi}_k(G,r)\le {\pi}_k(H,r)$ for every root $r$. \end{prp}
Proposition \ref{super} holds because $r$-solutions in $H$ are $r$-solutions in $G$. In particular, this holds when $H$ is a breadth-first search spanning tree of $G$ that is rooted at $r$ and thus preserves distances to $r$ in $G$. This allows us to prove the following.
\begin{res}\label{CliqBlok} Let $G$ be a connected graph in which every block is a clique. Let $T$ be a breadth-first search spanning tree of $G$. Then ${\pi}_k(G)={\pi}_k(T)$. \end{res}
\begin{figure}
\caption{A clique block graph with its breadth-first search spanning tree}
\label{CBG}
\end{figure}
{\noindent {\it Proof}.\ \ } The fact that ${\pi}_k(G) \le {\pi}_k(T)$ follows from Proposition \ref{super}. The fact that ${\pi}_k(G) \ge {\pi}_k(T)$ follows from showing that every $r$-solvable configuration $C$ on $G$ is $r$-solvable on $T$. Indeed, let $S$ be an $r$-solution in $G$, and for a block $B$ of $G$, denote by $x=x(B)$ the cut vertex of $B$ that is closest to $r$. If the sequence is greedy, then all its edges are in $T$. If the sequence is not greedy, then $S$ contains an edge from some vertex $a$ to some vertex $b \neq x$. Replace this edge by the edge from $a$ to $x$. The resulting sequence is an $r$-solution on $T$. Thus ${\pi}_k(G) = {\pi}_k(T)$. {
$\Box$}
\begin{cor}\label{Formula} Let $G$ be a connected graph in which every block is a clique. Let $T$ be a breadth-first search spanning tree of $G$. Let ${a_1,\ldots,a_t}$ denote the path lengths in a maximal path partition of $T$ rooted at $r$. Then ${\pi}_k(G,r)=n+2^{a_1}(k-1)+\sum_{i=1}^t(2^{a_i}-a_i-1)$. \end{cor}
Note that the formula in Corollary \ref{Formula} is of the form $n+c_1k+c_2$, which is also the form of the formula in Theorem \ref{kpd2}. Also, the {\it fractional pebbling number}, defined as $\hat{{\pi}}(G)=\lim_{k\to\infty}{\pi}_k(G)/k$ is seen to be $\hat{{\pi}}(G)= 2^{diam(G)}$ for such $G$. This is an instance of the Fractional Pebbling Conjecture of \cite{H}, recently proven in \cite{HV}.
Now we provide the upper and lower bounds on diameter-two graphs. To show a lower bound, we will display an unsolvable configuration on an extremal graph $\cal G$. This is the graph that Clarke, et al. \cite{CHH} used to characterize the diameter two graphs with pebbling number $n+1$. The vertices of $\cal G$ are $\{a,b,c,p,q,r\}\cup_{z\in\{p,q,r,c\}}V(H_z)$, where $H_p,H_q,H_r$, and $H_c$ are any graphs with the following properties. \begin{itemize} \item Every component of $H_p,H_q$, and $H_r$ has some vertex adjacent to $p,q$, and $r$, respectively. \item Every vertex of $H_p,H_q$, and $H_r$ is adjacent to $a$ and $c$, $b$ and $c$, $a$ and $b$, respectively. \item Every vertex of $H_c$ is adjacent to $a,b$, and $c$. \end{itemize} Furthermore, $(a,r,b,q,c,p)$ forms a 6-cycle, $(a,b,c)$ forms a triangle, as shown in Figure \ref{Extremal}, and no other edges than previously mentioned are included. Note that the diameter of $\cal G$ is 2.
\begin{figure}
\caption{The extremal graph $\cal G$}
\label{Extremal}
\end{figure}
\begin{thm}\label{bigconfig} For all $n\ge 6$, there is a graph $G$ on $n$ vertices with ${\pi}_k(G) \ge n + 4k - 3$ for all $k$. \end{thm}
{\noindent {\it Proof}.\ \ } As suggested above, we show that $\cal G$ is such a graph. Distribute the following configuration of size $n + 4k - 4$ on the ${\cal G}$: \begin{itemize} \item Place $4k - 1$ pebbles on $p$ \item Place 3 pebbles on $q$ \item Place 1 pebble on every vertex in $\cup_{z\in\{p,q,r,c\}}H_z$ and 0 elsewhere. \end{itemize} The configuration is $r$-unsolvable since every solution costs at least 4 pebbles (because the pebbles are at distance 2 from $r$, and so after $k=1$ solutions at most $n$ pebbles remain). In fact, the remaining configuration is a subconfiguration of the one defined above for $k=1$, which was shown to be $r$-unsolvable in \cite{PSV}. Hence ${\pi}_k({\cal G}) > n + 4k - 4$. {
$\Box$}
To prove Theorem \ref{kpd2} we consider the eight {\it cheap} configurations shown in Figure \ref{CheapSolns}. We call them cheap because they lose a small number (at most 7) of pebbles in the process of moving one pebble to the root. In particular, their names indicate their {\it cost} (number of pebbles used). For example, in {\sf C7}, {\sf C6}, and {\sf C5}, one moves an extra pebble onto where 3 sits to create {\sf C4A}. Then one can reach {\sf C2} from {\sf C4B}, {\sf C4A}, and {\sf C3}. Of course, {\sf C2} results in {\sf C1}. There are more cheap solutions than these, but we do not need them in our argument.
\begin{figure}
\caption{Cheap solutions of cost 7 or less}
\label{CheapSolns}
\end{figure}
We show by contradiction that a cheap solution must exist, and thus a pebble can be moved to the root with the loss of at most 7 pebbles. The remaining $k-1$ solutions will be found by induction.
\noindent {\it Proof of Theorem \ref{kpd2}.} Assume that the configuration $C$ of pebbles on $G$ is of size $n + 7k - 6$ and has no cheap solutions of cost $7$ or less. We will derive a contradiction to show that a cheap solution exists. Then after using a cheap solution we apply induction to get the remaining $k-1$ solutions. The theorem is already true for $k=1$ by Result \ref{psv}. Define the following notation. \begin{itemize} \item $N_i$ is the set of vertices with $i$ pebbles. \item $N_{i,r}$ is the set of common neighbors of $N_i$ and root $r$. \item $N_{i,j}$ is the set of common neighbors of pairs of vertices from $N_i$ and $N_j$. \item
$n_i = |N_i|$, $n_{i,j}=|N_{i,j}|$, $n_{i,r}=|N_{i,r}|$, and $n_0'=|N_0'|$. \item $N_0'$ = $N_0 - N_{3,r} - N_{3,3} - N_{2,r}$. \end{itemize}
\begin{clm}\label{claims} If $C$ is a configuration on a diameter-2 graph $G$ with no cheap solutions, then \begin{enumerate} \item[{\sf S1.}] $N_{i,r} \subseteq N_0$ for $i\in \{2,3\}$, \item[{\sf S2.}] $N_{3,3} \subseteq N_0$, \item[{\sf S3.}] $n_{i,r} \ge n_i$ for $i \in \{2,3\}$, \item[{\sf S4.}]
$|C| = 3n_3 + 2n_2 + n_1$, \item[{\sf S5.}] $n = n_3 + n_2 + n_1 + (n_{3,r} + n_{3,3} + n_{2,r} + n_0')$, and \item[{\sf S6.}] $n_{3,3} \ge \binom{n_3}{2}$. \end{enumerate} \end{clm}
\noindent {\it Proof of Claim \ref{claims}.} We refer to Figure \ref{CheapSolns}. Statement {\sf S1} follows from the nonexistence of {\sf C3} because a pebble adjacent to the root and a vertex with at least two pebbles is a {\sf C3} configuration. Likewise, {\sf S2}, {\sf S3}, and {\sf S4} follow from the nonexistence of {\sf C6}, {\sf C4B}, and {\sf C4A} respectively. Next, {\sf S5} simply partitions the vertices according to their number of pebbles, then uses the definition of $N_0'$. Finally, since $C$ has no {\sf C5}, no two vertices of $N_3$ are adjacent. However, because $G$ has diameter two, every such $x$ and $y$ have a common neighbor. Now the nonexistence of {\sf C7} implies that such common neighbors are distinct, which implies {\sf S6}. {
$\diamondsuit$}
Next we use {\sf S4} and {\sf S5} to count $|C|$ in two ways: $$3n_3 + 2n_2 + n_1\ =\ n_3 + n_2 + n_1 + (n_{3,r} + n_{3,3} + n_{2,r} + n_0') + 7k - 6.$$ Then {\sf S3} and {\sf S6} imply \begin{eqnarray*} 0&= & -2n_3 - n_2 + n_{3,r} + n_{3,3} + n_{2,r} + n_0' + 7k - 6\\ &\ge& -n_3 + \binom{n_3}{2} + n_0' + 7k - 6. \\ \end{eqnarray*} Finally, by completing the square and using $n_0' \ge 1$ (since $r \in N_0'$) and $k \ge 1$, we have \begin{eqnarray*} 0&< & (n_3 - 3/2)^2 + (4 - 9/4) \\ & =& 2\left[\binom{n_3}{2} - n_3 + 2\right] \\ &\le& 2\left[\binom{n_3}{2} - n_3 + n_0' + 7k - 6\right] \\ & \le& 0\ , \end{eqnarray*} which is a contradiction. Hence, $C$ must contain a solution of cost at most 7, afterwhich at least $n+7(k-1)-6$ pebbles remain, from which we obtain $k-1$ more solutions. {
$\Box$}
\section{Remarks}\label{Remarks}
We believe that the upper bound of Theorem \ref{kpd2} can be tightened by reducing the coefficient of k. Doing this requires restricting cheap solutions to lesser cost, which necessitates considering more of them. For example, there are one cost-4, one cost-5, and four cost-6 solutions that were not used in our argument. Our lower bound has inspired the next conjecture.
\begin{cnj}\label{d2conj} If $G$ is a graph on $n$ vertices with ${\rm diam}(G)\le 2$ then ${\pi}_k(G)\le n+4k-3$. \end{cnj}
Of course, the Fractional Pebbling Theorem implies that the coefficient of $k$ is $4$ in the limit; in fact, its proof is based on the pigeonhole principle --- for large enough $k$, {\sf C4A} exists. Also, Theorem \ref{kpd2} suggests the following problem.
\begin{prb}\label{diamd} Find upper bounds for the $k$-pebbling numbers of graphs of diameter $d$. \end{prb}
Along these lines, only the following result is known, proved by Bukh \cite{B}.
\begin{thm}\label{d3} If the $diam(G) = 3$, then ${\pi}(G) \le (3/2)n + O(1)$. \end{thm}
In addition, the following question is still open.
\begin{qst}\label{DiamConn} Is it possible to lower the connectivity requirement in Result \ref{chkt}? \end{qst}
The construction in \cite{H} shows that $\kappa\ge 2^d/d$ is necessary.
\end{document} | arXiv |
Boundedness in a class of duffing equations with oscillating potentials via the twist theorem
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Asymptotic behavior of solutions to a model system of a radiating gas
January 2011, 10(1): 193-207. doi: 10.3934/cpaa.2011.10.193
Asymptotic behavior for solutions of some integral equations
Yutian Lei 1, and Chao Ma 2,
School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210097, China
Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309, United States
Received January 2010 Revised June 2010 Published November 2010
In this paper we study the asymptotic behavior of the positive solutions of the following system of Euler-Lagrange equations of the Hardy-Littlewood-Sobolev type in $R^n$
$u(x) = \frac{1}{|x|^{\alpha}}\int_{R^n} \frac{v(y)^q}{|y|^{\beta}|x-y|^{\lambda}} dy $,
$ v(x) = \frac{1}{|x|^{\beta}}\int_{R^n} \frac{u(y)^p}{|y|^{\alpha}|x-y|^{\lambda}}dy. $
We obtain the growth rate of the solutions around the origin and the decay rate near infinity. Some new cases beyond the work of C. Li and J. Lim [17] are studied here. In particular, we remove some technical restrictions of [17], and thus complete the study of the asymptotic behavior of the solutions for non-negative $\alpha$ and $\beta$.
Keywords: Integral equations, weighted Hardy-Littlewood-Sobolev inequality., singularities, asymptotic analysis.
Mathematics Subject Classification: Primary: 45E10, 45G0.
Citation: Yutian Lei, Chao Ma. Asymptotic behavior for solutions of some integral equations. Communications on Pure & Applied Analysis, 2011, 10 (1) : 193-207. doi: 10.3934/cpaa.2011.10.193
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Yutian Lei Chao Ma | CommonCrawl |
You have to handle a very complex water distribution system. The system consists of $$$n$$$ junctions and $$$m$$$ pipes, $$$i$$$-th pipe connects junctions $$$x_i$$$ and $$$y_i$$$.
The only thing you can do is adjusting the pipes. You have to choose $$$m$$$ integer numbers $$$f_1$$$, $$$f_2$$$, ..., $$$f_m$$$ and use them as pipe settings. $$$i$$$-th pipe will distribute $$$f_i$$$ units of water per second from junction $$$x_i$$$ to junction $$$y_i$$$ (if $$$f_i$$$ is negative, then the pipe will distribute $$$|f_i|$$$ units of water per second from junction $$$y_i$$$ to junction $$$x_i$$$). It is allowed to set $$$f_i$$$ to any integer from $$$-2 \cdot 10^9$$$ to $$$2 \cdot 10^9$$$.
In order for the system to work properly, there are some constraints: for every $$$i \in [1, n]$$$, $$$i$$$-th junction has a number $$$s_i$$$ associated with it meaning that the difference between incoming and outcoming flow for $$$i$$$-th junction must be exactly $$$s_i$$$ (if $$$s_i$$$ is not negative, then $$$i$$$-th junction must receive $$$s_i$$$ units of water per second; if it is negative, then $$$i$$$-th junction must transfer $$$|s_i|$$$ units of water per second to other junctions).
Can you choose the integers $$$f_1$$$, $$$f_2$$$, ..., $$$f_m$$$ in such a way that all requirements on incoming and outcoming flows are satisfied?
The first line contains an integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of junctions.
The second line contains $$$n$$$ integers $$$s_1, s_2, \dots, s_n$$$ ($$$-10^4 \le s_i \le 10^4$$$) — constraints for the junctions.
The third line contains an integer $$$m$$$ ($$$0 \le m \le 2 \cdot 10^5$$$) — the number of pipes.
$$$i$$$-th of the next $$$m$$$ lines contains two integers $$$x_i$$$ and $$$y_i$$$ ($$$1 \le x_i, y_i \le n$$$, $$$x_i \ne y_i$$$) — the description of $$$i$$$-th pipe. It is guaranteed that each unordered pair $$$(x, y)$$$ will appear no more than once in the input (it means that there won't be any pairs $$$(x, y)$$$ or $$$(y, x)$$$ after the first occurrence of $$$(x, y)$$$). It is guaranteed that for each pair of junctions there exists a path along the pipes connecting them.
If you can choose such integer numbers $$$f_1, f_2, \dots, f_m$$$ in such a way that all requirements on incoming and outcoming flows are satisfied, then output "Possible" in the first line. Then output $$$m$$$ lines, $$$i$$$-th line should contain $$$f_i$$$ — the chosen setting numbers for the pipes. Pipes are numbered in order they appear in the input.
Otherwise output "Impossible" in the only line.
Server time: Apr/22/2019 04:14:24 (g1). | CommonCrawl |
\begin{document}
\title{Set-Valued Young Tableaux\and Product-Coproduct Prographs}
\begin{abstract} Standard set-valued Young tableaux are a generalization of standard Young tableaux where cells can contain unordered sets of integers, with the added condition that every integer at position $(i,j)$ must be smaller that every integer at both $(i+1,j)$ and $(i,j+1)$. In this paper, we explore properties of standard set-valued Young tableaux with three rows and a fixed number of integers in every cell of each row (referred to as set-valued tableaux with row-constant density). Our primary focus is on standard set-valued Young tableaux with $1$ integer in each first-row cell, $k-1$ integers in each second-row cell, and $1$ integer in each third-row cell. For rectangular shapes $\lambda=n^3$, such tableaux are placed in bijection with closed $k$-ary product-coproduct prographs: directed plane graphs that correspond to finite compositions involving a $k$-ary product operator and a $k$-ary coproduct operator. That bijection is extended to three-row set-valued Young tableaux of non-rectangular and skew shape, and it is shown that a set-valued analogue of the Sch\"utzenberger involution on tableaux corresponds to $180$-degree rotation of the associated prographs. As a set-valued analogue of the hook-length formula is currently lacking, we also present direct enumerations of three-row standard set-valued Young tableaux for a variety of row-constant densities and a small number of columns. We then argue why the numbers of tableaux with the row-constant density $(1,k-1,1)$ should be interpreted as a one-parameter generalization of the three-dimensional Catalan numbers that mirrors the generalization of the (two-dimensional) Catalan numbers provided by the $k$-Catalan numbers. \end{abstract}
\section{Introduction: Standard Set-Valued Young Tableaux} \label{sec: intro}
Consider a non-increasing sequence of positive integers $\lambda = (\lambda_1,\hdots,\lambda_m)$ that sum to $N$. Following the English notation, a Young diagram $Y$ of shape $\lambda$ is a left-justified array of $N$ cells with $\lambda_i$ cells in the $i^{th}$ row from the top of the array. Given a Young diagram of shape $\lambda$, a Young tableau of that shape is a bijection from the set of integers $[N]=\{1,\hdots,N\}$ to the cells of $Y$. For a Young tableau to be a standard Young tableau, the entries in the tableau must increase left to right across each row and top to bottom down each column. We denote the set of standard Young tableaux of shape $\lambda$ as $S(\lambda)$, and adopt the shorthand notation of $S(n^m)$ in the case of the $m$-row rectangular shape $\lambda = (n,\hdots,n)$. For a thorough introduction to Young tableaux, see Fulton \cite{Fulton}.
The number of standard Young tableaux of arbitrary shape $\lambda$ may be directly calculated using the hook-length formula, as originally given by Frame, Robinson and Thrall \cite{FRT}. A quick application of the hook-length formula to the case of $\lambda = (n,n)$ yields the well-known identity that $\vert S(n^2) \vert = C_n= \frac{(2n)!}{n! (n+1)!}$, the $n^{th}$ Catalan number. Generalizing to the $d$-row rectangular case of $\lambda = (n,\hdots,n)$ similarly yields $\vert S(n^d) \vert = C_{d,n}$, where $C_{d,n} = \frac{(d-1)!(dn)!}{n! (n+1)! \hdots (n+d-1)!}$ is the $n^{th}$ $d$-dimensional Catalan number.
Given an non-increasing sequence of positive integers $\lambda = (\lambda_1,\hdots,\lambda_{m_1})$ of $N_1$ and a non-increasing sequence of positive integers $\mu = (\mu_1,\hdots,\mu_{m_2})$ of $N_2$, where $0 \leq \mu_i \leq \lambda_i$ for all $i$, one can also define a skew Young diagram of shape $\lambda / \mu$ by removing the $\mu_i$ leftmost cells in the $i^{th}$ row of the Young diagram of shape $\lambda$, for all $1 \leq i \leq m$. A skew Young tableau of shape $\lambda / \mu$ is a bijection from $[N_1 - N_2]$ to the cells of the skew Young diagram of shape $\lambda / \mu$. Such a tableau is said to be a standard skew Young tableau if its entries increase left to right across each row and top to bottom down each column. We denote the set of standard skew Young tableaux of shape $\lambda / \mu$ by $S(\lambda/\mu)$.
This paper is focused on a generalization of standard Young tableaux known as standard set-valued Young tableaux. Consider a non-increasing sequence of positive integers $\lambda = (\lambda_1,\hdots,\lambda_m)$ and a sequence of positive integers $\rho = (\rho_1,\hdots,\rho_m)$ such that $\sum_{i=1}^m \lambda_i \rho_i = M$. A \textbf{set-valued Young tableau} of shape $\lambda$ and (row-constant) \textbf{density} $\rho$ is a function from $[M]$ to the cells of the Young diagram $Y$ of shape $\lambda$ such that every cell in the $i^{th}$ row of $Y$ receives precisely $\rho_i$ integers. The resulting tableau qualifies as a \textbf{standard set-valued Young tableau} if, for each cell $(i,j)$ of $Y$, every integer at position $(i,j)$ is smaller than every integer in the cells at $(i+1,j)$ and $(i,j+1)$. In analogy with standard Young tableaux, we refer to these additional conditions as ``column-standardness" and ``row-standardness", respectively. We denote the set of standard set-valued Young tableaux of shape $\lambda$ and density $\rho$ as $\svt(\lambda,\rho)$. See Figure \ref{fig: set-valued Young tableaux} for a collection of standard set-valued Young tableaux with $\lambda=(3,3)=3^2$ and $\rho=(2,1)$. Given a skew Young diagram of shape $\lambda / \mu$, one may similarly define a \textbf{standard skew set-valued Young tableau} of shape $\lambda / \mu$ and (row-constant) density $\rho$. We denote the set of such skew tableau by $\svt(\lambda / \mu, \rho)$.
\begin{figure}
\caption{Five of the twelve elements of $\svt(3^2,\rho)$ with $\rho=(2,1)$.}
\label{fig: set-valued Young tableaux}
\end{figure}
Set-valued Young tableaux were originally introduced by Buch \cite{Buch} to study the $K$-theory of Grassmannians. Heubach, Li and Mansour \cite{HLM} later provided standard set-valued Young tableaux with $\lambda = n^2$ and row-constant density $\rho=(k-1,1)$ as one of their many combinatorial interpretations of the $k$-Catalan numbers $C_n^k = \frac{(kn)!}{(kn-n+1)!n!}$. That work was directly expanded upon by Drube \cite{Drube}, who used standard set-valued Young tableaux with two rows to provide new combinatorial interpretations of the Raney numbers, the rational Catalan numbers, and the solution to the generalized tennis ball problem. For recent usages of set-valued Young tableaux in a more algebraic setting, see Reiner, Tenner and Young \cite{RTY} and Monical \cite{Monical}.
It is important to emphasize the current lack of a set-valued analogue to the hook-length formula. This makes the enumeration of $\svt(\lambda,\rho)$ for arbitrary $\lambda$ and $\rho$ an extremely challenging problem, and comprehensive attempts at counting standard set-valued Young tableaux of arbitrary density $\rho$ have only been attempted for two-row shapes $\lambda = (a,b)$. See Drube \cite{Drube} for calculations of $\vert S(\lambda,\rho) \vert$ in the two-row rectangular case, enumerations that corresponded to various generalizations of the (two-dimensional) Catalan numbers.
With the two-row case relatively well-understood, this paper presents the first thorough investigation of standard set-valued Young tableaux in the case of three-row shapes. Much as various choices of $\rho$ allow $\vert \svt(n^2,\rho) \vert$ to correspond to various generalizations of the two-dimensional Catalan numbers, $\vert \svt(n^3,\rho) \vert$ and certain choices of $\rho$ will correspond to various generalizations of the three-dimensional Catalan numbers. To our knowledge, all three-dimensional Catalan generalizations discussed in this paper have yet to appear anywhere in the literature.
\subsection{Outline of Paper}
This paper proceeds as follows. In Section \ref{sec: prographs}, we introduce $k$-ary product-coproduct prographs, a class of directed plane graphs that naturally extend existing combinatorial interpretations for both the two- and three-dimensional Catalan numbers. Much of Section \ref{sec: prographs} may be seen as an and formalization of the work of Borie \cite{Borie}. This motivates our focus on the sets $\svt(n^3,\rho)$ with $\rho=(1,k-1,1)$, which are placed in bijection with $k$-ary product-coproduct prographs (Theorem \ref{thm: prograph vs tableaux bijection, rectangular}). In Section \ref{sec: enumeration}, we focus upon the enumeration of these tableaux, yielding a family of integers $C_{3,n}^k$ that function as a three-dimensional analogue of the $k$-Catalan numbers. Closed formulas for $C_{3,n}^k$ are derived for $n \leq 5$ (Propositions \ref{thm: n=2}, \ref{thm: n=3}, \ref{thm: n=4}, \ref{thm: n=5}) and a general calculus is introduced to tackle the general case (Proposition \ref{thm: general n recurrences}). In Section \ref{sec: properties}, we further explore the bijection between our standard set-valued Young tableaux and $k$-ary product-coproduct prographs. Appropriately generalized prographs are placed in bijection with various sets of (non-rectangular and skew) standard set-valued Young tableaux (Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular}), and a 180-degree rotation of $k$-ary prographs is shown to correspond to a set-valued analogue of the Sch\"utzenberger involution on standard Young tableaux (Theorem \ref{thm: schutzenberger}). Section \ref{sec: future directions} closes the paper with a series of more cursory discussions, including a suggestion of additional combinatorial interpretations for $C_{d,n}^k$ and a consideration of three- and four-row set-valued tableaux with densities other than $\rho=(1,k-1,1)$.
\section{$k$-ary Product-Coproduct Prographs and Set-Valued Tableaux} \label{sec: prographs}
Let $\mathcal{T}_n^k$ denote the set of full $k$-ary trees with $kn+1$ vertices, drawn so that the root vertex lies at the bottom of the tree. It is well-known that $\mathcal{T}_n^k$ is enumerated by the $k$-Catalan number $C_n^k = \frac{(kn)!}{(kn-n+1)!n!}$. In the case of $k=2$, this prompts the well-studied bijection between $\mathcal{T}_n^2$ and $S(n^2)$ that associates entries in the top row of the tableau to left children and entries in the bottom row of the tableau to right children.
The bijection between $\mathcal{T}_n^2$ and $S(n^2)$ may be generalized to a bijection between $\mathcal{T}_n^k$ and standard set-valued Young tableaux $\svt(n^2,\rho)$ with row-constant density $\rho = (k-1,1)$. As described by Heubach, Li and Mansour \cite{HLM}, this generalized bijection $\phi_k: \mathcal{T}_n^k \rightarrow \svt(n^2,\rho)$ is defined as below. For an example of $\phi_k$, see Figure \ref{fig: k-ary tree vs tableau}.
\begin{enumerate} \item For any $T \in \mathcal{T}_n^k$, label the edges of $T$ with the integers $\lbrace 1,\hdots,nk \rbrace$ according to a depth-left first search. \item Place all integers that label rightmost-children of $T$ in the bottom row of $\phi(T) \in \svt(n^2,\rho)$, in increasing order from left to right. \item Place all remaining integers from $\lbrace 1, \hdots, nk \rbrace$ in the top row of $\phi(T)$, in increasing order from left to right and ensuring that each cell in the top row receives precisely $k-1$ integers. \end{enumerate}
\begin{figure}
\caption{An example of the bijection $\phi_k: \mathcal{T}_n^k \rightarrow \svt(n^2,\rho)$ for $k=3$.}
\label{fig: k-ary tree vs tableau}
\end{figure}
Following Borie \cite{Borie}, generalizing $\phi_k$ to three-row tableaux requires a consideration of prographs. For any finite collection $G$ of formal operators, each of which is uniquely identified by its number of inputs and outputs, one may consider the set of all finite compositions that are freely constructed using elements of $G$ (as well as the identity operator $\id$). Each of these compositions corresponds to a directed planted plane graph in which all edges are directed upward. In these graphs, each application of a non-identity operator corresponds to a non-initial, non-terminal vertex whose vertical placement (when read from bottom to top) corresponds to the stage at which the operator appears in the composition. See Figure \ref{fig: basic prograph example} for a quick example. If one enforces a notion of equivalence for planted plane graphs with the added condition that all edges must maintain a strictly upward orientation, the resulting set $\pro_G$ is referred to as the (free) prographs generated by $G$.
Let $A$ denote a formal module. If $G$ consists solely of an operator $\Delta_k : A \rightarrow A \otimes \cdots \otimes A$ with $1$ input and $k$ outputs (a non-coassociative $k$-ary coproduct), elements of $\pro_G$ are a directed variation of full $k$-ary trees where every edge has been directed upward and a single input edge has been added below the root vertex. The subset of these prographs with precisely $n$ usages of $\Delta_k$ are in bijection $\mathcal{T}_n^k$. For example, the $3$-ary tree on the left side of Figure \ref{fig: k-ary tree vs tableau} corresponds the prograph shown in Figure \ref{fig: basic prograph example}.
\begin{figure}
\caption{The prograph corresponding to the composition $(\Delta_3 \otimes \id \otimes \Delta_3) \circ \Delta_3$, where $\Delta_3$ is a $3$-ary coproduct.}
\label{fig: basic prograph example}
\end{figure}
Now consider the case where $G$ consists of a (non-coassociative) $k$-ary coproduct $\Delta_k: A \rightarrow A \otimes \cdots \otimes A$ with $1$ input and $k$ outputs, as well as a (non-associative) $k$-ary product $\mu_k: A \otimes \cdots \otimes A \rightarrow A$ with $k$ inputs and $1$ output. We refer to the resulting elements of $\pro_G$ as \textbf{$\mathbf{k}$-ary (product-coproduct) prographs}. The subset of these prographs that have a single terminal vertex are known as \textbf{closed $\mathbf{k}$-ary (product-coproduct) prographs}. As all prographs have a single initial vertex, all closed $k$-ary prographs must feature the same number of product and coproduct nodes. We denote the set of all closed $k$-ary prographs with precisely $n$ products and $n$ product by $\pc^k(n)$. See Figure \ref{fig: closed prograph example} for an illustration of $\pc^2(2)$. Included in that figure is a representive from the equivalence class of compositions to which each prograph corresponds.
\begin{figure}
\caption{The set $\pc^2(2)$ of closed $2$-ary prographs with $2$ products and $2$ coproducts.}
\label{fig: closed prograph example}
\end{figure}
Borie \cite{Borie} argued that $\pc^2(n)$ is enumerated by the three-dimensional Catalan number $C_{3,n} = \frac{2 (3n)!}{n!(n+1)!(n+2)!}$. This is accomplished by placing $\pc^2(n)$ in bijection with the three-row standard Young tableaux $S(n^3)$.
Our goal for the rest of this section is to generalize Borie's bijection to $\pc^k(n)$ for all $k \geq 2$, where the appropriate $k$-generalization of $S(n^3)$ is standard set-valued Young tableaux $\svt(n^3,\rho)$ with row-constant density $\rho=(1,k-1,1)$. We begin by introducing an algorithm for labelling the edges of any $G \in \pc^k(n)$ that generalizes both the depth-left first labelling of $k$-ary trees and Borie's depth-left search for elements of $\pc^2(n)$.
\begin{enumerate} \item Take any $G \in \pc^k(n)$, and begin by labelling the sole output of the initial node of $G$ with the integer $0$. \item For each $0 \leq i < nk$, recursively define a subgraph $G_i$ of $G$ consisting solely of edges labelled by $\lbrace 1,\hdots,i \rbrace$. Then let $V_i$ denote the subset of nodes from $G$ such that every input to that node lies in $G_i$ and at least one output from that node lies in $G - G_i$. \item Identify the highest labelled edge from $G_i$ that terminates at an element $v$ of $V_i$ (this needn't be the edge labelled $i$). Then label the leftmost unlabelled edge of that vertex $v$ with $i+1$ and return to Step \#2. \end{enumerate}
See Figure \ref{fig: left-ascending search example} for an example of this procedure, which we henceforth refer to as our (generalized) depth-left first search. Colloquially, the procedure may be described as ``staying as leftward as possible, with the restriction that all inputs to a node must be labelled before any output from that node may be labelled". Also notice that this procedure directly generalizes to non-closed $k$-ary prographs: one merely needs to omit terminal nodes from the $V_i$ and repeat the recursive part of the algorithm until all terminal edges are labelled.
\begin{figure}
\caption{An example of our generalized depth-left first search, applied to an element of $\pc^2(3)$.}
\label{fig: left-ascending search example}
\end{figure}
We are now ready to place $\pc^k(n)$ in bijection with an appropriate collection of standard set-valued Young tableaux. Observe that Theorem \ref{thm: prograph vs tableaux bijection, rectangular} directly recovers the result of Borie \cite{Borie} in the case of $k=2$.
\begin{theorem} Fix $n \geq 1$ and $k \geq 2$. Then $\vert \pc^k(n) \vert = \vert \svt(\lambda,\rho) \vert$ for $\lambda = n^3$ and $\rho=(1,k-1,1)$. \label{thm: prograph vs tableaux bijection, rectangular} \end{theorem} \begin{proof} We provide a pair of well-defined functions $\Phi : \pc^k(n) \rightarrow \mathbb{S}(n^3,\rho)$, $\Phi_2: \mathbb{S}(n^3,\rho) \rightarrow \pc^k(n)$ and then show that $\Phi_2 = \Phi^{-1}$. Our first map $\Phi: \pc^k(n) \rightarrow \svt(n^3,\rho)$ is defined as below. See Figure \ref{fig: prograph vs tableaux example, rectangular} for an example. \begin{enumerate} \item For any $G \in \pc^k(n)$, label the edges of $G$ according to our depth-left first search. \item Place integers that label leftmost coproduct children of $G$ in the top row of $\Phi(G) \in \svt(n^3,\rho)$, in increasing order from left to right. \item Place integers that label all remaining coproduct children of $G$ along the middle row of $\Phi(G)$, in increasing order from left to right and ensuring that each cell in the middle row receives precisely $k-1$ integers. \item Place integers corresponding to product children of $G$ along the bottom row of $\Phi(G)$, in increasing order from left to right. \end{enumerate}
Notice that the initial input label of $0$ is ignored in this procedure. As $\Phi(G)$ is row-standard by construction, to show that $\Phi$ is a well-defined map into $\svt(n^3,\rho)$ we merely need to argue that $\Phi(G)$ is column-standard. Begin by noticing that our depth-left first search ensures that the leftmost child of a given coproduct node will always be labelled prior to the $k-1$ non-leftmost children of that same node. This implies that every entry in the middle row of $\Phi(G)$ must be larger than the entry in the top row of the same column.
Now assume that precisely $\alpha_1$ leftmost coproduct children and $\alpha_2$ other coproduct children have been labelled prior to the labelling of the $j^{th}$ product child of $G$. In order for the $j^{th}$ product child to receive the next label, there must have been at least $k$ previously labelled edges terminating at a node with an unlabelled output. Among the $\alpha_1 + \alpha_2 + (j-1) + 1$ edges that were labelled prior to the labelling of the $j^{th}$ product child (initial $0$ edge included), precisely $k$ edges terminate at each of the $j-1$ product nodes with a previously labelled output, while $1$ edge terminates at each of the $\alpha_1$ coproduct nodes with a previously labelled leftmost output. This leaves $\alpha_1 + \alpha_2 + (j-1) + 1 - k(j-1) - \alpha_1 = \alpha_2 - (k-1)j + k$ labelled edges that could lead into a product node with an unlabelled output. Enforcing $\alpha_2 - (k-1)j + k \geq k$ gives $\alpha_2 \geq (k-1)j$, ensuring that all entries in the middle row of the $j^{th}$ column are smaller than the entry in the bottom row of the $j^{th}$ column. It follows that $\Phi(G)$ is in fact column-standard and hence that $\Phi$ is well-defined.
For our second map $\Phi_2: \svt(n^3,\rho) \rightarrow \pc^k(n)$, we recursively ``build up" an edge-labelled prograph by working through $T \in \svt(n^3,\rho)$ one entry at a time, as described below.
\begin{enumerate} \item For any $T \in \svt(n^3,\rho)$, begin by placing an initial input edge labelled $0$. Then recursively consider each entry $1 \leq i \leq (k+1)n$ in numerical order. \item If $i$ lies in the top row of $T$, place a coproduct node whose input is the edge labelled $i-1$. Then label the leftmost child of that coproduct with $i$. \item If $i$ is in the middle row of $T$, follow the depth-left first search through the partially constructed graph from the edge labelled $i-1$. Then label the first unlabelled edge you encounter with the integer $i$. \item If $i$ is in the bottom row of $T$, place a product node whose rightmost input is the edge labelled $i-1$ and whose remaining inputs are the $k-1$ nearest terminal edges immediately to the left of the edge labelled $i-1$. Then label the output of that product $i$. \end{enumerate}
The well-definedness of $\Phi_2$ depends upon whether the actions described above are possible at every step. In particular, there must exist a rightward unlabelled edge when applying Step \#3, and there must be enough leftward free edges (all previously labelled) when adding the product node in Step \#4.
Begin by noting that, in the procedure that constructs $\Phi_2(T)$, leftmost coproduct children and product children are labelled as soon as they are placed. This means that unlabelled terminal edges at any intermediate step must correspond to non-leftmost coproduct children, and hence that all edges labelled in Step \#3 must be non-leftmost coproduct children. Also notice that the edge labelled $i$ immediately serves as an input for a new product or coproduct node unless $i+1$ lies in the middle row of $T$. As this case involves an application of the depth-left first search, when it is initially placed $i+1$ is always the rightmost terminal edge in our partially constructed prograph.
So assume that the entry $i$ lies in the cell $(2,j)$ of $T$, and that $i$ is larger than precisely $x$ other integers in that cell ($0 \leq x \leq k-2$). Row- and column-standardness of $T$ guarantees that at least $j$ coproducts have already been placed prior to this step, and that $(k-1)(j-1) + x$ of the non-leftmost children from those coproducts have already been labelled. This means there are at least $(k-1)j - (k-1)(j-1) - x = k-1- x \geq 1$ unlabelled non-leftmost coproduct children at this step. Because all non-leftmost coproduct children are labelled according to our depth-left first search, all of these unlabelled coproduct edges lie to the left of the edge labelled $i-1$. Thus the operation of Step \#3 is always possible.
Now assume that $i$ lies in the cell $(3,j)$ of $T$. Row- and column-standardness of $T$ guarantee that at least $j$ coproducts and precisely $j-1$ products have already been placed at this point in the procedure, with at least $kj$ coproduct children and precisely $j-1$ product children having been labelled. As $j-1$ labelled inputs are needed for the placement of each coproduct, this means that there are at least $kj - (k-1)(j-1) - (j-1) = k$ labelled free edges when $i$ is the active integer. Via preceding comments, the edge labelled $i-1$ is the rightmost of these free edges. Thus the operation of Step \#4 is always possible, and we may conclude that $\Phi_2$ is well-defined.
It is only left to show that $\Phi_2 = \Phi^{-1}$. We demonstrate that $\Phi_2 \circ \Phi (G) = G$ for any $G \in \pc^k(n)$, and that $\Phi \circ \Phi_2(T)$ for any $T \in \svt(n^3,\rho)$.
To show $\Phi_2 \circ \Phi(G) = G$ for any $G \in \pc^k(n)$, we inductively work through the edges of $G$ in the order of the depth-left first search. For $i=0$, $G$ and $\Phi_2 \circ \Phi (G)$ both feature a single input edge labelled with $i$. For any $1 \leq i \leq n(k+1)$, assume that $G$ and $\Phi_2 \circ \Phi(G)$ feature identical sub-prographs (not necessarily closed) corresponding to the edges labelled $\lbrace 0,\hdots,i-1 \rbrace$. There are the three possible scenarios for the edge labelled $i$.
\begin{enumerate} \item If $i-1$ labels the input to a coproduct node in $G$, the edge labelled $i$ must be the leftmost output of that same coproduct. This implies that $i$ lies in the top row of $\Phi(G)$ and hence that $\Phi_2 \circ \Phi(G)$ also features a coproduct with input $i-1$ and leftmost output $i$. \item If $i-1$ labels the rightmost input to a product node in $G$, the edge labelled $i$ in $G$ is always the next (on the right) input to that same product. That means that $i$ lies in the middle row of $\Phi(G)$ and that the edge labelled with $i$ in $\Phi_2 \circ \Phi(G)$ is determined via a depth-left first search from the edge labelled $i-1$. This results in the next (on the right) input to that same product being labelled $i$ in $\Phi_2 \circ \Phi(G)$. \item If $i-1$ labels the rightmost input to a product node in $G$, the edge labelled $i$ in $G$ is necessarily the output of that product. This implies that $i$ lies in the bottom row of $\Phi(G)$ and thus that $i$ also labels a product output in $\Phi_2 \circ \Phi(G)$ whose rightmost input is labelled $i-1$. \end{enumerate}
As all three options lead to an identical placement of the edge labelled with $i$, we conclude $\Phi_2 \circ \Phi(G) = G$.
To show that $\Phi \circ \Phi_2 (T) = T$ for any $T \in \svt(n^3,\rho)$, we inductively work through the entries of $T$. For $i=1$, $T$ and $\Phi \circ \Phi_2(T)$ both feature $i$ in the top-left corner. For $2 \leq i \leq n(k+1)$, assume that $T$ and $\Phi \circ \Phi_2(T)$ feature identical subtableau corresponding to the entries $\lbrace 1,\hdots,i-1 \rbrace$. There are once again three possibilities for $i$:
\begin{enumerate} \item If $i$ lies in the top row of $T$, $i$ labels a leftmost child of a coproduct node in $\Phi_2(T)$ whose input is labelled $i-1$. Thus $i$ lies in the top row of $\Phi \circ \Phi_2(T)$. \item If $i$ lies in the middle row of $T$, via earlier comments we know that $i$ will always label a non-leftmost coproduct child in $\Phi_2(T)$. It follows that $i$ also lies in the middle row of $\Phi \circ \Phi_2(T)$ \item If $i$ lies in the bottom row of $T$, $i$ labels a product output in $\Phi_2(T)$ and hence $i$ also lies in the bottom row of $\Phi \circ \Phi_2(T)$. \end{enumerate}
As all three cases lead to identical placement of $i$ in the relevant tableaux, we conclude $\Phi \circ \Phi_2(T) = T$. \end{proof}
\begin{figure}
\caption{An example of the bijection $\Phi: \pc^k(n) \rightarrow \svt(n^3,\rho)$ for $n=2$ and $k=4$.}
\label{fig: prograph vs tableaux example, rectangular}
\end{figure}
\section{Enumerating $\svt(n^3,\rho)$ for $\rho = (1,k-1,1)$} \label{sec: enumeration}
Theorem \ref{thm: prograph vs tableaux bijection, rectangular} suggests that $\svt(n^3,\rho)$ with $\rho=(1,k-1,1)$ generalizes $S(n^3)$ in a manner similar to how $\svt(n^2,\rho')$ with $\rho'=(k-1,1)$ generalizes $S(n^2)$. As the $\svt(n^2,\rho')$ are enumerated by the $k$-Catalan numbers $C^k_n$, we henceforth refer to the cardinalities $\vert \svt(n^3,\rho) \vert = C^k_{3,n}$ as the \textbf{three-dimensional $k$-Catalan numbers}.
The purpose of this section is to develop closed formulas for $C^k_{3,n}$. Sadly, developing such a formula or deriving a multivariate generating function for arbitrary $n \geq 1$, $k \geq 1$ do not appear to be tractable problems. As such, we restrict our attention to cases of small $n$. See Table \ref{tab: 1,k-1,1} of Appendix \ref{sec: appendix} for a table of known values of $C^k_{3,n}$, which combines the explicit results of this section with computer calculations performed in Java.
In all that follows, notice that the ``degenerate" $k=1$ case corresponds to three-row tableaux with empty cells across their middle row. This means that the $k=1$ enumerations reduce to pre-existing results about two-row tableaux: that $C_{3,n}^1 = \vert \svt(n^3,\rho) \vert = \vert S(n^2) \vert = C_n$ for all $n \geq 1$ with $\rho = (1,0,1)$.
For all of our enumerations we recursively place $\svt(\lambda,\rho)$ in bijection with a collection of sets $\bigcup \svt(\lambda_i,\rho)$ of strictly smaller shape yet equivalent density. Our technique is similar to pre-existing proofs for non-set-valued tableaux where the sub-shapes $\lambda_i$ are determined via the removal of lower-right corners, corresponding to possible locations of the largest possible entry in a tableau of shape $\lambda$. The difference here is that we never remove entries from a cell without eliminating all entries in that cell. If the removed cell contains entries other than the largest entry in the tableau, this necessitates that we account for the ordering of those smaller entries relative to integers appearing elsewhere in the tableau.
Before proceeding, observe that $\vert S(\lambda,\rho) \vert$ is easily calculable whenever $\lambda = (n,1,\hdots,1)$ is ``hook-shaped". In this case, one merely needs to count the ways of partitioning entries between the rightward and downward ``legs", giving an enumeration in terms of a single binomial coefficient $\vert S(\lambda,\rho) \vert = \binom{a}{b}$. See Figure \ref{fig: hook-shaped enumeration} for examples.
For the rest of this section, an unfilled Young diagram of shape $\lambda$ is used to denote the cardinality $\vert \svt(\lambda,\rho) \vert$, assuming $\rho = (1,k-1,1)$.
\begin{figure}
\caption{Cardinalities $\vert S(\lambda,\rho) \vert$ for $\rho = (1,k-1,1)$ and several hook-shapes $\lambda$. As $1$ must lie at position $(1,1)$, one merely needs to determine which of $\lbrace 2,3,\hdots \rbrace$ lie in the remaining cells of the top row.}
\label{fig: hook-shaped enumeration}
\end{figure}
\begin{proposition} \label{thm: n=2} Let $\rho = (1,k-1,1)$. For any $k \geq 1$, $C_{3,2}^k = \vert \svt(2^3,\rho) \vert = k^2 + 1$. \end{proposition} \begin{proof} As the largest entry of any $T \in \svt(2^3,\rho)$ must lie at $(3,2)$, we investigate the integers $a_1 < \hdots < a_{k-1}$ lying at $(2,2)$ in an arbitrary set-valued tableaux $T_1 \in \svt(\lambda_1,\rho)$ of shape $\lambda_1 = (2,2,1)$. The only other entry in $T_1$ that may be larger than any of the $a_i$ is the entry $b$ at position $(3,1)$. The subset of $\svt(\lambda_1,\rho)$ satisfying $b \leq a_i$ for all $i$ is then in bijection with $\svt(\lambda_2,\rho)$ for $\lambda_2 = (2,1,1)$. If $b > a_1$, one must specify the ordering of $b$ relative to $a_2,\hdots,a_{k-1}$. So assume that $j$ is the largest index such that $a_j < b$ (where $1 \leq j \leq k-1$). Each choice of $j$ defines a subset of $\svt(\lambda_2,\rho)$ that is in bijection with $\svt(\lambda_3,\rho)$ for $\lambda_3 = (2,1)$, since for any choice of $j$ the $k$ largest entries of such a tableau $T_1 \in \svt(\lambda_1,\rho)$ is split between positions $(2,2)$ and $(3,1)$. Combining these observations gives the string of equalities below.
\ytableausetup{boxsize=1em} $$\raisebox{8pt}{\ydiagram{2,2,2}} \ = \ \raisebox{8pt}{\ydiagram{2,2,1}} \ = \ \raisebox{8pt}{\ydiagram{2,1,1}} \ + \ (k-1) \ \raisebox{8pt}{\ydiagram{2,1}} \ = \ \binom{k+1}{1} \ + \ (k-1) \binom{k}{1} \ = \ k^2 + 1$$ \end{proof}
\begin{proposition} \label{thm: n=3} Let $\rho = (1,k-1,1)$. For any $k \geq 1$, $$C_{3,3}^k = \vert \svt(3^3,\rho) \vert = \frac{9k^4 - 2k^3 + 9k^2}{4} + 1$$ \end{proposition} \begin{proof} We begin by enumerating $\svt(\lambda',\rho)$ for $\lambda'=(3,2,1)$. For arbitrary $T' \in \svt(\lambda',\rho)$, let $a_1 < \hdots < a_{k-1}$ denote the entries at $(2,2)$, $b$ denote the entry at $(3,1)$, and $c$ denote the entry at $(1,3)$. Proceeding as in the proof of Proposition \ref{thm: n=2}, we subdivide $\svt(\lambda',\rho)$ based on the relationship of $b$ and $c$ to the $a_i$ and then delete all entries $x \geq a_1$ to place each subset in bijection with tableaux of some smaller shape. The equalities below synopsize our results, with the first summand corresponding to $b,c < a_1$, the second summand corresponding to the $k-1$ placements of $b$ relative to $a_2 < \hdots < a_{k-1}$ when $b > a_1$ yet $c < a_1$, the third summand corresponding to the $k-1$ placements of $c$ relative to $a_2 < \hdots < a_{k-1}$ when $c > a_1$ yet $b < a_1$, and the fourth summand corresponding to the $\binom{k}{k-2,1,1}$ placements of $b,c$ relative to $a_2 < \hdots < a_{k-1}$ when $b,c > a_1$.
\ytableausetup{boxsize=1em} $$\raisebox{8pt}{\ydiagram{3,2,1}} \ = \ \raisebox{8pt}{\ydiagram{3,1,1}} \ + \ \binom{k-1}{1} \ \raisebox{8pt}{\ydiagram{3,1}} \ + \ \binom{k-1}{1} \ \raisebox{8pt}{\ydiagram{2,1,1}} \ + \ \binom{k}{k-2,1,1} \ \raisebox{8pt}{\ydiagram{2,1}}$$
$$= \ \binom{k+2}{2} + \binom{k-1}{1} \binom{k+1}{2} + \binom{k-1}{1} \binom{k+1}{1} + \binom{k}{k-2,1,1} \binom{k}{1} \ = \ \frac{3k^3 + k^2 + 2k}{2}$$
For the full theorem, we once again proceed as in the proof to Proposition \ref{thm: n=2}. After reducing to arbitrary $T \in \svt(\lambda,\rho)$ with $\lambda = (3,3,2)$, we divide $\svt(\lambda,\rho)$ into subsets depending upon how the entries $a_1 < \hdots < a_{k-1}$ at position $(2,3)$ relate to the entry $b_1$ at $(3,1)$ and the entry $b_2$ at $(3,2)$. The three summands in the first line of the equalities below corresponds to the cases of $b_1 < b_2 < a_1$, $b_1 < a_1 < b_2$, and $a_1 < b_1 < b_2$, respectively. In the second line of equalities, the first of those subsets is further subdivided based upon the relationship of the entry $c$ at $(1,3)$ to the entry $b_2$ at $(3,2)$, with the two new summands corresponding to $b_2 < c$ and $c < b_2$, respectively. This leaves a sum of cardinalities $\vert \svt(\lambda_i,\rho) \vert$ that are computable via Proposition \ref{thm: n=2}, our informal lemma for shape $\lambda'=(3,2,1)$, and the result of Heubach, Li and Mansour \cite{HLM} giving $\vert \svt(n^2,\rho) \vert = C^k_n$.
\ytableausetup{boxsize=1em} $$\raisebox{8pt}{\ydiagram{3,3,3}} \ = \ \raisebox{8pt}{\ydiagram{3,3,2}} \ = \ \raisebox{8pt}{\ydiagram{3,2,2}} \ + \ \binom{k-1}{1} \ \raisebox{8pt}{\ydiagram{3,2,1}} \ + \ \binom{k}{2} \ \raisebox{8pt}{\ydiagram{3,2}}$$
$$= \ \raisebox{8pt}{\ydiagram{2,2,2}} \ + \ \raisebox{8pt}{\ydiagram{3,2,1}} \ + \ \binom{k-1}{1} \ \raisebox{8pt}{\ydiagram{3,2,1}} \ + \ \binom{k}{2} \ \raisebox{8pt}{\ydiagram{3,2}}$$
$$= \ (k^2 + 1) + k \left( \frac{3k^3 + k^2 + 2k}{2} \right) + \binom{k}{2} C^k_3 \ = \ \frac{9k^4 - 2k^3 + 9k^2}{4} + 1$$ \end{proof}
The proofs of Propositions \ref{thm: n=2} and \ref{thm: n=3} suggest a general methodology for enumerating $\svt(n^3,\rho)$ that could be applied to all $n \geq 2$. In particular, for any three-row shape our technique of removing every entry in a lower-right corner yields the recurrences of Proposition \ref{thm: general n recurrences}.
\begin{proposition} \label{thm: general n recurrences} Fix $k \geq 1$. For $\rho = (1,k-1,1)$ and any three-row shape $\lambda = (a,b,c)$ with $a \leq b \leq c$,
$$|\svt((a,b,c),\rho) | = \begin{cases} \displaystyle{\sum_{\substack{0 \leq i \leq a-b,\\[1pt] 0 \leq j \leq c}} \binom{k-2+i+j}{k-2,i,j} \kern+2pt \vert \svt((a-i,b-1,c-j),\rho) \vert}, & \text{if $b > c$;}\\[22pt] \displaystyle{\sum_{0 \leq i \leq a-b} \vert \svt((a-i,b,c-1),\rho) \vert}, & \text{if $b = c$.} \end{cases}$$ \end{proposition}
Notice that, although we have utilized other results about hook-shaped tableaux and two-row tableaux to shorten our proofs in the $n=2,3$ cases, the two recurrences of Proposition \ref{thm: general n recurrences} are sufficient to reduce any $\vert \svt((a,b,c),\rho) \vert$ to a summation involving one-column shapes $\lambda_i$, where $\vert \svt(\lambda_i,\rho) \vert = 1$. Considered as a function of $k$, we may then use Proposition \ref{thm: general n recurrences} to quickly draw several conclusions about $\vert \svt((a,b,c),\rho) \vert$:
\begin{corollary} \label{thm: recurrence corollary} Fix $\rho=(1,k-1,1)$, where $k$ is indeterminate, and let $\lambda = (a,b,c)$ satisfy both $b \geq 1$ and $a+c \geq 2$. If $a=b=c$, then $\vert \svt((a,b,c),\rho) \vert$ is a polynomial in $k$ of degree $a+c-2$. If $a>c$, then $\vert \svt((a,b,c),\rho) \vert$ is a polynomial in $k$ of degree $a+c-1$. \end{corollary} \begin{proof} That $\vert \svt((a,b,c),\rho) \vert = p(k)$ is a polynomial in $k$ follows directly from the recursion of Proposition \ref{thm: recurrence corollary}. To demonstrate the degree of $p(k)$, induct on $i=a+b+c$ for $i\geq 3$. The base case of $i = 3$ follows from $\vert \svt((2,1,0), \rho) \vert = k$ and $\vert \svt((1,1,1),\rho) \vert = 1$. For the inductive case, take $\vert \svt((a,b,c),\rho) \vert$ with $a+b+c = i+1$. If $b>c$, the first case of Proposition \ref{thm: recurrence corollary} equates $\vert \svt((a,b,c),\rho) \vert$ with a sum of polynomials (all with positive leading coefficient) whose maximal degree summand(s) all have degree $a+c-1$. If $b=c$, the second case of Proposition \ref{thm: recurrence corollary} equates $\vert \svt((a,b,c),\rho) \vert$ with a sum of polynomials whose sole maximal degree summand has degree $a+(c-1)-1$. \end{proof}
In the case of $\lambda = n^3$, notice that Corollary \ref{thm: recurrence corollary} implies that $p(k) = \vert \svt(n^3,\rho) \vert$ has degree $2(n-1)$. For several additional enumerations, Proposition \ref{thm: general n recurrences} may be applied with the aid of a computer algebra system to derive the following polynomials for the $k=4$ and $k=5$ cases.
\begin{proposition} \label{thm: n=4} Let $\rho = (1,k-1,1)$. For any $k \geq 1$, $$C_{3,4}^k = \vert \svt(4^3,\rho) \vert = \frac{256 k^6 - 114 k^5 + 217 k^4 - 12 k^3 + 121 k^2}{36} + 1$$ \end{proposition}
\begin{proposition} \label{thm: n=5} Let $\rho = (1,k-1,1)$. For any $k \geq 1$, $$C_{3,5}^k = \vert \svt(5^3,\rho) \vert = \frac{15625 k^8 - 10092 k^7 + 10258 k^6 - 72 k^5 + 5473 k^4 - 204 k^3 + 2628 k^2}{576} + 1$$ \end{proposition}
\section{Properties of $k$-ary Product-Coproduct Prographs} \label{sec: properties}
In this section we prove a generalization of Theorem \ref{thm: prograph vs tableaux bijection, rectangular} that applies to non-closed $k$-ary prographs satisfying certain basic properties. We then explore one significant application of our bijection that generalizes an unproven proposition of Borie \cite{Borie}, showing that 180-degree rotation of prographs corresponds to a set-valued analogue of the Sch\"utzenberger involution on standard Young tableaux.
\subsection{Non-Closed $k$-ary Prographs and Set-Valued Tableaux} \label{subsec: non-closed prographs}
We begin by generalizing the set $\pro_G$ to finite compositions of formal operators where the initial input is the an $x$-fold tensor product $A \otimes \cdots \otimes A$ of the formal module $A$. The resulting directed plane graphs resemble prographs over $G$ but now contain precisely $x$ input strands, aligned horizontally across the bottom of the graph. Fixing $G$ and $m \geq 1$, we may enforce a notion of equivalence on the resulting set of directed plane graphs that is analogous to the equivalence relation on $\pro_G$ from Section \ref{sec: prographs}. We refer to the resulting set of equivalence classes $\pro_{G,x}$ as the set of $x$-fold (free) prographs generated by $G$.
In the case where $G$ consists of a $k$-ary coproduct $\Delta_k$ and a $k$-ary product $\mu_k$, we refer to the elements of $\pro_{G,x}$ as \textbf{$\mathbf{x}$-fold $\mathbf{k}$-ary (product-coproduct) prographs}. We denote the subset of $x$-fold $k$-ary prographs with precisely $n$ coproduct nodes, $m$ product nodes, and $x$ input strands by $\pc_x^k(n,m)$. Notice that these three parameters are sufficient to determine the number of output strands in any $G \in \pc_x^k(n,m)$. Explicitly,
\begin{proposition} \label{thm: number of output strands} Take any $G \in \pc_x^k(n,m)$. Then $G$ has precisely $y = (n-m)(k-1)+x$ output strands. In particular, $y \equiv x \mod(k-1)$. \end{proposition} \begin{proof} Observe that each $k$-ary coproduct increases the number of free edges by $k-1$, while each $k$-ary product decreases the number of free edges by $k-1$. If we begin with $x$ free edges, after $n$ coproducts and $m$ products we have $y = x + n(k-1) - m(k-1)$ outgoing free edges. \end{proof}
For any $x \equiv 1 \mod(k-1)$, consider $\pc_x^k(n,m)$. There exists an injection $j: \pc_x^k(n,m) \rightarrow \pc^k(n + \frac{x-1}{k-1})$ that is defined by recursively joining incoming strands with $k$-ary coproducts, from left to right in sets of $k$, while recursively joining outgoing strands with $k$-ary products, from right to left in sets of $k$. See Figure \ref{fig: justified prographs} for an illustration. For any $G \in \pc_x^k(n,m)$, we call the image $j(G) \in \pc^k(n + \frac{x-1}{k-1})$ the \textbf{justification} of $G$. Assuming $x \equiv 1 \mod(k-1)$, justification suggests the generalization of Theorem \ref{thm: prograph vs tableaux bijection, rectangular} given by Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular}.
\begin{figure}
\caption{A non-closed prograph $G \in \pc_5^3(n,n-1)$ and its justification $j(G) \in \pc^3(n + \frac{5-1}{3-1})$.}
\label{fig: justified prographs}
\end{figure}
\begin{theorem} \label{thm: prograph vs tableaux bijection, non-rectangular} Fix $n,m \geq 1$, $k \geq 2$, and take any $x \geq 1$ such that $x \equiv 1 \mod (k-1)$. Then $\vert \pc_x^k(n,m) \vert = \vert \svt(\lambda/\mu,\rho) \vert$, where $\lambda = (n + \frac{x-1}{k-1},n + \frac{x-1}{k-1},m)$, $\mu = (\frac{x-1}{k-1},0,0)$, and $\rho = (1,k-1,1)$. \end{theorem} \begin{proof} Let $j: \pc_x^k(n,m) \rightarrow \pc^k(n+\frac{x-1}{k-1})$ be justification and let $\Phi: \pc^k(n+\frac{x-1}{k-1}) \rightarrow \svt((n+\frac{x-1}{k-1})^3,\rho)$ be the forward bijection from Theorem \ref{thm: prograph vs tableaux bijection, rectangular}. Then define $\chi: \svt((n+\frac{x-1}{k-1})^3,\rho) \rightarrow \svt(\lambda/\mu, \rho)$ as the map that deletes the first $\frac{x-1}{k-1}$ cells in the top row of $T \in \svt((n+\frac{x-1}{k-1})^3,\rho)$, deletes the last $\frac{y-1}{k-1}$ cells in the bottom row of $T$, and then reindexes all remaining entries so that no positive integers are skipped. We define $\psi: \pc_x^k(n,m) \rightarrow \svt(\lambda/\mu,\rho)$ by $\psi = \chi \circ \Phi \circ j$, and show that $\psi$ is a bijection. See Figure \ref{fig: non-closed bijection example} for an example of this map $\psi$.
Well-definedness of $j$, $\Phi$, and $\chi$ ensure that the composition $\psi$ is also well-defined. To show that $\psi$ is a bijection, we begin showing that the restriction $\widetilde{\chi} = \chi \vert_{\im(\Phi \circ j)}$ is a bijection onto $\svt(\lambda / \mu, \rho)$. So take any $G \in \pc_x^k(n,m)$. Using Proposition \ref{thm: number of output strands}, the number of output strands in $G$ is $y = (k-1)(n-m)+x$. Thus $n + \frac{x-1}{k-1} = m + \frac{y-1}{k-1}$, and $j(G) \in \pc^k(n + \frac{x-1}{k-1})$ is obtained from $G$ by recursively adding $\frac{x-1}{k-1}$ left-aligned coproducts to the bottom of $G$ and $\frac{y-1}{k-1}$ right-aligned products to the top of $G$. Applying our depth-left first search to $j(G)$ then results in the first $\frac{x-1}{k-1}$ non-zero labels being applied to the leftmost children of the ``new" coproduct nodes at the bottom of $j(G)$, while the final $\frac{y-1}{k-1}$ labels are applied to the outputs of the ``new" product nodes at the top of $j(G)$. This guarantees that the first row of every $T \in \im(\phi \circ j)$ begins with $1,\hdots,\frac{x-1}{k-1}$ and that the bottom row of every such $T$ ends with $k(n+\frac{x-1}{k-1})+m+1, \hdots, (k+1)(n+\frac{x-1}{k-1})$. It follows that the entries deleted by $\chi$ are identical across all tableaux in $\widetilde{\chi}$, implying that $\widetilde{\chi}$ is a bijection.
Bijectivity of $\widetilde{\chi}$ implies that $\chi \circ \Phi$ is also bijective with inverse $(\chi \circ \Phi)^{-1} \equiv \Phi^{-1} \circ \widetilde{\chi}^{-1}$. Notice that $\widetilde{\chi}^{-1}: \svt(\lambda/\mu,\rho) \rightarrow \svt((n+ \frac{x+1}{k-1})^3,\rho)$ is the function that reindexes all entries of $T \in \svt(\lambda/\mu,\rho)$ by $a \mapsto a+ \frac{x-1}{k-1}$, appends $\lbrace 1,\hdots,\frac{x-1}{k-1} \rbrace$ to the front of the top row, and appends the $\frac{y-1}{k-1}$ entries $\lbrace k(n+\frac{x-1}{k-1})+m+1, \hdots, (k+1)(n+\frac{x-1}{k-1}) \rbrace$ to the end of the bottom row. This means that $\im(\Phi^{-1} \circ \widetilde{\chi}^{-1})$ are the prographs $G \in \pc^k(n+\frac{x-1}{k-1})$ with $\frac{x-1}{k-1}$ consecutive left-aligned coproducts at the bottom and $\frac{y-1}{k-1}$ consecutive right-aligned products at the top.
All of this allows us to define an ``unjustification" map $h: \im(\Phi^{-1} \circ \widetilde{\chi}^{-1}) \rightarrow \svt(\lambda/\mu, \rho)$ where, for any prograph $G \in \im(\Phi^{-1} \circ \widetilde{\chi}^{-1})$, one simply deletes the $\frac{x-1}{k-1}$ initial product nodes (along with their inputs) and deletes the $\frac{y-1}{k-1}$ final coproduct nodes (along with their outputs). This map $h$ clearly satisfies $j \circ h(G) = G$ for any $G \in \im(\Phi^{-1} \circ \widetilde{\chi}^{-1})$ and $h \circ j(G) = G$ for any $G \in \svt(\lambda/\mu,\rho)$. We may then conclude that $\psi$ is a bijection with inverse $(\chi \circ \Phi \circ j)^{-1} \equiv h \circ \Phi^{-1} \circ \widetilde{\chi}^{-1}$. \end{proof}
\begin{figure}
\caption{An example of the bijection $\psi: \pc_x^k(n,m) \rightarrow \svt(\lambda/\mu,\rho)$ for $k=3$, $x=3$, $n=3$, and $m=1$.}
\label{fig: non-closed bijection example}
\end{figure}
In light of Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular}, one may define a modification of our depth-left first search that allows one to pass directly from an edge-labelling of $G \in \pc_x^k(n,m)$ to $\psi(G) \in \svt(\lambda/\mu,\rho)$, bypassing the justification and reindexing steps. This $x$-fold depth-left first search is defined as below.
\begin{enumerate} \item For any $G \in \pc_x^k(n,m)$ with $x \equiv 1 \mod(k-1)$, label the leftmost initial input of $G$ with the integer $0$. \item After labelling the $i^{th}$ edge, determine the node subset $V_i$ from the depth-left first search of Section \ref{sec: prographs}. If $V_i$ is non-empty, follow the procedure of Section \ref{sec: prographs} to find the edge labelled $i+1$. If $V_i$ is empty, label the leftmost unlabelled initial input of $G$ with $i+1$ \end{enumerate}
Using the same terminology as Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular}, let $\tau(G) \in \svt(\lambda/\mu,\rho)$ be the tableau that results from applying the $x$-fold depth-left first search to $G \in \pc_x^k(n,m)$, placing all integers labelling leftmost coproduct children of $G$ in the top row, placing all integers labelling product children of $G$ in the bottom row, and placing all remaining non-zero integers (including those labelling non-leftmost initial inputs of $G$) in the middle row. This is in fact that same tableau that results from the composite bijection of Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular}:
\begin{corollary} \label{thm: prograph vs tableaux bijection, non-rectangular corollary} Let $\psi: \pc_x^k(n,m) \rightarrow \svt(\lambda/\mu,\rho)$ be as in the proof of Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular}. For any $G \in \pc_x^k(n,m)$ with $x \equiv 1 \mod(k-1)$, $\tau(G) = \psi(G)$. \end{corollary} \begin{proof} Recall that justification of $G$ introduces precisely $\frac{x-1}{k-1}$ leftmost coproduct children that receive the first $\frac{x-1}{k-1}$ nonzero labels in the depth-left first search on $j(G)$, as well as $\frac{y-1}{k-1}$ product children that receive the final $\frac{y-1}{k-1}$ labels the depth-left first search on $j(G)$. As these are precisely the entries of $\Phi \circ j(G)$ that are deleted in the final stage of $\psi$, we merely need to argue that the depth-left first search of Section \ref{sec: prographs} labels the remaining edges of $j(G)$ in the same order that the $x$-fold depth-left first search labels the edges of $G$. In particular, we need to show that the $i^{th}$ edge from the $x$-fold depth-left first search on $G$ corresponds to the $(i+\frac{x-1}{k-1})^{th}$ edge from our original depth-left first search on $j(G)$.
Inducting on $i$, consider the two alogorithms after the labelling of the $i^{th}$ edge of $G$. If the set $V_i$ is non-empty for $G$, the set $V_{(i+\frac{x-1}{k-1})}$ is non-empty for $j(G)$. Since the inputs to the initial coproduct nodes that appear only in $j(G)$ have lower edge labels than all other edges in $j(G)$, the element of $V_i$ in $G$ with the largest input corresponds to the element of $V_{(i+\frac{x-1}{k-1})}$ in $j(G)$ with the largest input. This leads to equivalent placements of the next edge label in both graphs. Now if $V_i$ is empty for $G$, it must be the case that $V_{(i+\frac{x-1}{k-1})}$ for $j(G)$ consists solely of nodes from the justification's $\frac{x-1}{k-1}$ initial coproducts. As the edge labels on the inputs to these initial coproducts always decrease from left to right, the next edge labelled in $j(G)$ is always the leftmost output of the initial coproducts that has yet to be labelled. These initial coproduct children of $j(G)$ precisely correspond to initial inputs in the non-justified graph $G$, implying that the next edge of $G$ to be labelled by the $x$-fold depth-left first search is the equivalent (non-leftmost) initial input of $G$. \end{proof}
\subsection{The Sch\"utzenberger Involution} \label{subsec: schutzenberger}
For any rectangular shape $\lambda \vdash N$, the Sch\"utzenberger involution is a map $f: S(\lambda) \rightarrow S(\lambda)$ that rotates $T \in S(\lambda)$ by 180 degrees and then renumbers entries via $a \mapsto N - a + 1$. As described by Drube \cite{Drube}, one may define an analogue of the Sch\"utzenberger involution for standard set-valued Young tableaux. For any rectangular shape $\lambda$ and row-constant density $\rho$, the set-valued Sch\"utzenberger involution $f : \svt(\lambda,\rho) \rightarrow \svt(\lambda,\rho')$ is similarly defined via 180-degree rotation of $T \in \svt(\lambda,\rho)$, followed by a reversal in the order of entries in the resulting tableaux. Here $\rho' = (\rho_m,\hdots,\rho_1)$ if $\rho = (\rho_1,\hdots,\rho_m)$, meaning only ``vertically symmetric" densities are preserved by $f$.
Now define a rotation operator $r: \pc_x^k(n,m) \rightarrow \pc_y^k(m,n)$ on (not-necessarily closed) $k$-ary prographs that corresponds to 180-degree rotation and a reversal in the orientation of all edges. In Theorem \ref{thm: schutzenberger} we will show that a specialization of this operator to any closed $k$-ary prograph $G$ is compatible with the Sch\"utzenbeger involution on the associated set-valued tableaux $\Phi(G)$ from Theorem \ref{thm: prograph vs tableaux bijection, rectangular}, but first we need to analyze how rotation effects our edge-labelling algorithms. It is in fact that case that the $x$-fold depth-left first search of Subsection \ref{subsec: non-closed prographs} labels the edges of $r(G) \in \pc_y^k(m,n)$ in an order that exactly reverses the order in which it labels the corresponding edges of $G \in \pc_x^k(n,m)$:
\begin{proposition} \label{thm: rotated depth-left first search} For any $k \geq 2$, $n,m \geq 0$, $x \geq 1$, set $N = x + kn + m - 1$ and consider the rotation operator $r: \pc_x^k(n,m) \rightarrow \pc_y^k(m,n)$. For any edge $e$ of $G$, if the $x$-fold depth-left first search labels $e$ with the integer $i$, then the $x$-fold depth-left first search labels the corresponding edge of $r(G)$ with $N-i$. \end{proposition} \begin{proof} We proceed by induction on the maximum edge label $N \geq 0$. The $N = 0$ case is immediate, as both $G$ and $r(G)$ consist of a single edge labelled $0$. For $N>0$, consider the edge $e$ of $G$ that receives the label $N$, which is always the rightmost output of $G$. There are three options: 1) $e$ is a ``free strand" that does not originate at a product or coproduct, 2) $e$ is a product child, or 3) $e$ is a rightmost coproduct child.
If $e$ is a free strand, simply deleting $e$ produces a valid prograph $\widetilde{G}$ with maximal edge label $N-1$. By the inductive hypothesis, the $x$-fold depth-left first search labels corresponding edges in $\widetilde{G}$ and $r(\widetilde{G})$ according to $i \mapsto N-1-i$. Inserting a free strand (labelled $0$) on the left side of $G$ recovers $r(G)$, and effects our edge labelling in that the label of all edges in $r(\widetilde{G})$ are increased by $1$. It follows that the $x$-fold depth-left first search labels corresponding edges in $G$ and $r(G)$ according to $i \mapsto N-i$.
If $e$ is a product child, we eliminate the product node at the source of $e$ as in the first row of Figure \ref{fig: edge labelling in rotation}, yielding a prograph $\widetilde{G}$ with $k-1$ additional outputs but maximal edge label $N-1$. Applying the inductive hypothesis allows us to relate corresponding edge labels of $\widetilde{G}$ and $r(\widetilde{G})$ by $i \mapsto N-1-i$. We then pre-compose $r(\widetilde{G})$ with an additional coproduct whose outputs are the $k$ leftmost inputs of $r(\widetilde{G})$, as in the top row of Figure \ref{fig: edge labelling in rotation}. This recovers $r(G)$ and effects our edge labelling in that all edges apart from the new coproduct input are increased by $1$. It follows that the $x$-fold search labels corresponding edges in $G$ and $r(G)$ according to $i \mapsto N-i$.
Lastly, if $e$ is a righmost coproduct child we eliminate the coproduct node at the source of $e$ as in the bottom row of Figure \ref{fig: edge labelling in rotation}, identifying the input of that coproduct with its leftmost output while extending all remaining outputs of to the bottom of the prograph as $k-1$ new inputs. As the resulting graph $\widetilde{G}$ has maximal edge label $N-1$, we may once again relate corresponding edge labels of $\widetilde{G}$ and $r(\widetilde{G})$ by $i \mapsto N-1-i$. Introducing a new product node into $r(\widetilde{G})$ as in the bottom row of Figure \ref{fig: edge labelling in rotation} recovers $r(G)$ and effects our $x$-fold search in such a way that the corresponding edges of $G$ and $r(G)$ are labelled according to $i \mapsto N-i$. \end{proof}
\begin{figure}
\caption{The effect of the rotation operator upon edge labels surrounding the final product node or coproduct node of $G \in \pc_x^k(n,m)$, utilizing the ``resolution" techniques from the proof of Proposition \ref{thm: rotated depth-left first search}. In the top row, $\widetilde{G}$ may include additional output edges that lie to the left of the edge labelled $N$.}
\label{fig: edge labelling in rotation}
\end{figure}
In the case of $x=1$ and $n=m$, the rotation operator reduces to an involution $r:\pc^k(n) \rightarrow \pc^k(n)$ of closed $k$-ary prographs. Proposition \ref{thm: rotated depth-left first search} then states that the depth-left first search of Section \ref{sec: prographs} relates corresponding edges of $G$ and $r(G)$ according to $i \mapsto (k+1)n - i$. This allows us to derive the following relationship between the rotation operator on closed $k$-ary prographs, the Sch\"utzenberger involution on rectangular set-valued tableaux, and the bijection $\Phi$ from Theorem \ref{thm: prograph vs tableaux bijection, rectangular}. See Figure \ref{fig: schutzenberger versus rotation} for an example of this compatibility.
\begin{theorem} \label{thm: schutzenberger} Fix $k \geq 2$ and $n \geq 1$, and let $\rho = (1,k-1,1)$. For $\Phi : \pc^k(n) \rightarrow \svt(\lambda,\rho)$ defined as in Theorem \ref{thm: prograph vs tableaux bijection, rectangular}, the rotation operator $r: \pc^k(n) \rightarrow \pc^k(n)$, and the set-valued Sch\"utzenberger involution $f : \svt(\lambda,\rho) \rightarrow \svt(\lambda,\rho)$, we have $\Phi \circ r = f \circ \Phi$. \end{theorem} \begin{proof} Take arbitrary $G \in \pc^k(n)$ and set $N = (k+1)n$, so that $G$ contains $N+1$ total edges and the cells of $\Phi(G) \in \svt(\lambda,\rho)$ are filled with $\lbrace 1,\hdots,N \rbrace$. We show that leftmost coproduct children in $G$ correspond to bottom row entries in both $\Phi \circ r (G)$ and $f \circ \Phi(G)$, while product outputs in $G$ correspond to top row entries in both $\Phi \circ r (G)$ and $f \circ \Phi(G)$. This implies that $\Phi \circ r (G)$ and $f \circ \Phi(G)$ feature identical sequences of integers across their top and bottom rows, and hence must be the same tableau.
So assume $G$ has been labelled according to our depth-left first search. By Proposition \ref{thm: rotated depth-left first search}, if an edge in $G$ is labelled with the integer $a$, then the corresponding edge in $r(G)$ is labelled with $N-a$. The depth-left first search is defined in such a way that $a$ labels a leftmost coproduct output in $G$ if and only if $a-1$ labels the input to the same coproduct node for which $a$ labels the leftmost child. As demonstrated in the left side of Figure \ref{fig: schutzenberger proof cases}, this means that $N-(a-1)$ labels a product output in the rotated prograph $r(G)$. It follows that $N-a+1$ appears in the bottom row of $\Phi \circ r(G)$. On the other hand, $a$ being a leftmost coproduct child implies that $a$ appears in the top row of $\Phi(G)$, and hence that $N-a+1$ appears in the bottom row of $f \circ \Phi(G)$.
As $r$ is an involution, the case where $a$ labels a product in $G$ follows directly from reversing the roles of $G$ and $r(G)$ in the previous paragraph. See the right side of Figure \ref{fig: schutzenberger proof cases} for a demonstration. In this case we may conclude that $N-a+1$ appears in the top row of both $\Phi \circ r(G)$ and $f \circ \Phi(G)$, as required. \end{proof}
\begin{figure}
\caption{A demonstration of how the edge labels of leftmost coproduct outputs (left) and product outputs (right) behave under 180-degree rotation of the underlying prograph.}
\label{fig: schutzenberger proof cases}
\end{figure}
\begin{figure}
\caption{An example of the relationship between rotation $r$ of $k$-ary product-coproduct prographs and the generalized Sch\"utzenberger involution $f$ on standard set-valued Young tableaux.}
\label{fig: schutzenberger versus rotation}
\end{figure}
As Proposition \ref{thm: rotated depth-left first search} applies to all $x$-fold $k$-ary prographs, the result of Theorem \ref{thm: schutzenberger} may be directly extended to non-closed prographs if one defines a suitable generalization of the Sch\"utzenberger involution. If $\lambda = (n,n,n-a)$ and $\mu = (b,0,0)$, let $\lambda'=(n,n,n-b)$ and $\mu'=(a,0,0)$. Then there exists a map $F:\svt(\lambda/\mu,\rho) \rightarrow \svt(\lambda',\mu',\rho)$ that is defined via $180$-degree rotation of $T \in \svt(\lambda/\mu,\rho)$ and a reversal $i \mapsto 3n-a-b+1-i$ of entries in the resulting tableau. This map clearly specializes to the Sch\"utzenberger involution when $a=b=0$, and a superficial modification of the technique from Theorem \ref{thm: schutzenberger} yields $\psi \circ r = F \circ \psi$. Notice how $F$ flips the number of ``missing boxes" in the top and bottom rows of a skew set-valued tableau, similarly to how $r$ flips the number of ``missing" products and coproducts needed to justify the associated $x$-fold prograph.
\section{Future Directions} \label{sec: future directions}
\subsection{Non-Closed $k$-ary Prographs $\pc_x^k(n,m)$ for which $x \not\equiv 1 \mod(k-1)$} \label{subsec: x not 1 mod(k-1)}
Subsection \ref{subsec: non-closed prographs} entirely restricted its attention to sets $\pc_x^k(n,m)$ of $x$-fold $k$-ary prographs for which $x \equiv 1 \mod(k-1)$. Developing an analogue to Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular} in the case of $x \not\equiv 1 \mod(k-1)$ is significantly more involved, as such prographs require a modification of the justification operator whose effect on the associated set-valued tableaux is more difficult to interpret. Although we stop short of proving an explicit bijection, we pause to outline how the techniques of Subsection \ref{subsec: non-closed prographs} may be generalized to the case of general $\pc_x^k(n,m)$.
So let $x \equiv a \mod(k-1)$, where $2 \leq a \leq k-1$, and consider the set $\pc_x^k(n,m)$. There exists an injection $J : \pc_x^k(n,m) \rightarrow \pc^k(n+ \frac{x+k-a-1}{k-1})$ in which $k-a$ free strands are added on the left side of $G \in \pc_x^k(n,m)$, producing a prograph $\widetilde{G} \in \pc_{x+k-a}^k(n,m)$ in which the number of inputs is $1 \kern-3pt\mod(k-1)$, and then the original justification operator $j$ is applied to $\widetilde{G}$. For $G \in \pc_x^k(n,m)$, we call the image $J(G) \in \pc^k(n + \frac{x+k-a-1}{k-1})$ the \textbf{left-weighted justification} of $G$. See Figure \ref{fig: justified prographs, non-rectangular} for an example of left-weighted justification.
\begin{figure}
\caption{A non-closed prograph $G \in \pc_2^4(n,n-1)$ and its left-weighted justification $J(G) \in \pc^4(n + \frac{2+4-2-1}{4-1})$.}
\label{fig: justified prographs, non-rectangular}
\end{figure}
Following the techniques of Theorem \ref{thm: prograph vs tableaux bijection, non-rectangular}, left-weighted justification suggests that $\pc_x^k(n,m)$ may be placed in bijection with some subset of $\svt(\lambda/\mu,\rho)$ for $\lambda=(n+\frac{x+k-a-1}{k-1},n+\frac{x+k-a-1}{k-1},m)$ and $\mu=(\frac{x+k-a-1}{k-1},0,0)$. The difficulty is in describing what subset of $\svt(\lambda/\mu,\rho)$ corresponds to left-justified prographs in which the $k-a$ leftmost children of the initial coproduct terminate at the final product node.
Conjecture \ref{thm: x not 1 mod(k-1) conjecture} describes the subset of $\svt(\lambda/\mu,\rho)$ that should lie in bijection with $\pc_x^k(n,m)$. The first condition below prevents the $k-a$ leftmost children of the initial coproduct from terminating at a coproduct node. The second condition prevents those same edges from serving as an input to a product that isn't the final product of the prograph.
\begin{conjecture} \label{thm: x not 1 mod(k-1) conjecture} Assume $x \equiv a \mod(k-1)$, where $2 \leq a \leq k-1$. Then consider $\pc_x^k(n,m)$ and $\svt(\lambda/\mu,\rho)$ with $\lambda=(n+\frac{x+k-a-1}{k-1},n+\frac{x+k-a-1}{k-1},m)$ and $\mu=(\frac{x+k-a-1}{k-1},0,0)$. For arbitrary $T \in \svt(\lambda/\mu,\rho)$, let $b_1 < b_2 < \hdots$ denote the middle-row entries of $T$ and let $c_1 < c_2 < \hdots$ denote the bottom-row entries of $T$. Then $\pc_x^k(n,m)$ is in bijection with the subset of tableaux from $\svt(\lambda/\mu,\rho)$ satisfying \begin{enumerate} \item $b_i = i$ for all $1 \leq i \leq k-a$, and \item $c_i > b_{(k-1)i + 2 - (k-a)}$ for all $1 \leq i \leq m-1$ \end{enumerate} \end{conjecture}
\subsection{Additional Combinatorial Interpretations for $\svt(n^3,\rho)$ with $\rho = (1,k-1,1)$} \label{subsec: additional interpretations}
It is natural to suppose that all combinatorial interpretations of the three-dimensional Catalan numbers admit one-parameter generalizations that lie in bijection with $\svt(n^3,\rho)$ for $\rho = (1,k-1,1)$. Below we briefly conjecture as to how several more of those interpretations may be $k$-generalized. See sequence A005789 of OEIS \cite{OEIS} for a full list of candidates. Beyond the interpretations discussed below, we are especially interested in how the pattern-avoiding permutations of Lewis \cite{Lewis} may be generalized using standard set-valued Young tableaux.
\begin{enumerate} \item The three-dimensional Catalan number $C_{3,n}$ is known to count the number of walks in the first quadrant of $\mathbb{Z}^2$ that start and end at $(0,0)$ and use $3n$ total steps from $\lbrace (0,1),(1,-1), (-1,0) \rbrace$. These walks are known to lie in bijection with $S(n^3)$ via a map that associates $(0,1)$ steps with entries in the top row of the corresponding tableau, $(1,-1)$ steps with entries in the middle row of that tableau, and $(-1,0)$ steps with entries in the bottom row of that tableau. We conjecture that this map may be generalized to a bijection between $\svt(n^3,\rho)$ with $\rho=(1,k-1,1)$ and walks in the first quadrant of $\mathbb{Z}^2$ that start and end at $(0,0)$ and which use $(k+1)n$ total steps from $\lbrace (0,k-1), (1,-1), (-k+1,0) \rbrace$. In this bijection, $(0,k-1)$ steps should correspond to entries in the top row of the associated set-valued tableau, $(1,-1)$ should correspond to entries in the middle row of that tableau, and $(-k+1,0)$ entries should correspond to entries in the bottom row of that tableau. \item $C_{3,n}$ is also known to count three-dimensional integer lattice paths from $(0,0,0)$ to $(n,n,n)$ that use steps from $\lbrace (1,0,0), (0,1,0), (0,0,1) \rbrace$ and that satisfy $x \geq y \geq z$ at every lattice point $(x,y,z)$ along the path. These lattice paths are known to lie in bijection with $S(n^3)$ via a map that associates $(1,0,0)$ steps with entries in the top row of the corresponding tableau, $(0,1,0)$ steps with entries in the middle row of that tableau, and $(0,0,1)$ steps with entries in the bottom row of that tableau. It should be straightforward to generalize this map to a bijection between $\svt(n^3,\rho)$ with $\rho(1,k-1,1)$ and integer lattice paths from $(0,0,0)$ to $((k-1)n,n,(k-1)n)$ that use steps from $\lbrace (1,0,0), (0,1,0), (1,0,0) \rbrace$ and which satisfy $(k-1)x \geq y \geq (k-1)z$ at every point $(x,y,z)$. This bijection would similarly associate $(1,0,0)$ steps to top-row entries, $(0,1,0)$ to middle-row entries, and $(0,0,1)$ to bottom-row entries. \end{enumerate}
For a somewhat different application of set-valued tableaux with $\rho = (1,k-1,1)$, we refer the reader to the work of Eu \cite{Eu}. Eu places all standard Young tableaux with at most three rows and any shape $\lambda \vdash N$ in bijection with Motzkin paths of length $n$. By Motzkin paths of length $n$ we mean integer lattice paths from $(0,0)$ to $(n,0)$ that use steps from $\lbrace (1,1),(1,-1),(1,0) \rbrace$ and never fall below the $x$-axis.
Direct computations for small $n$ reveal that a similar result may hold for standard set-valued Young tableaux with at most three rows, precisely $n(k-1)$ entries, and densities (determined by the number of rows) of either $\rho_1 = (1)$, $\rho_2 = (1,k-1)$, or $\rho = (1,k-1,1)$. In particular, such tableaux appear to lie in bijection with what we refer to as $(k-1)$-sloped Motzkin paths of length $n$: lattice paths from $(0,0)$ to $(n,0)$ that use steps from $\lbrace (k-1,1),(1,-1),(1,0) \rbrace$ and which never fall below the $x$-axis. The only caveat here is that one cannot include tableaux with ``partially filled" cells: every cell must have the full complement of entries determined by $\rho_i$.\footnote{$k$-sloped Motzkin paths should not be confused with the pre-existing notion of $k$-Motzkin paths, which correspond to $2$-sloped Motzkin paths in which every horizontal steps carries one of $k$ colors. See Barrucci, Del Lungo, Pergola and Pinazni \cite{BDPP} for a treatment of $k$-Motzkin paths}
See Figure \ref{fig: motzkin paths} for a comparison of $3$-sloped Motzkin paths of length $n=4$ and set-valued tableaux with density from $\lbrace (1), (1,2), (1,2,1) \rbrace$ and precisely $4$ entries. For justification of the specific matching exhibited in Figure \ref{fig: motzkin paths}, we direct the reader to the algorithm presented by Eu \cite{Eu}.
\begin{figure}
\caption{$3$-sloped Motzkin paths of length $4$ and standard set-valued Young tableaux with $4$ entries across at most three-rows and densities of either $\rho_1 = (1)$, $\rho_2 = (1,2)$, or $\rho_3 = (1,2,1)$.}
\label{fig: motzkin paths}
\end{figure}
\subsection{$\svt(\lambda,\rho)$ for Distinct Three- and Four-Row Densities} \label{subsec: other sets}
We close this paper by briefly exploring several additional densities for standard set-valued Young tableaux of shapes $\lambda = n^3$ and $\lambda = n^4$. The cardinalities of the resulting sets $\svt(\lambda,\rho)$ correspond to one-parameter generalizations of the three- and four-dimensional Catalan numbers that are distinct from the three-dimensional $k$-Catalan numbers $C_{3,n}^k$ of previous sections. It is our hope that combinatorial interpretations as interesting as those for $C_{3,n}^k$ will eventually be found for each of these generalizations.
First consider the case of $\lambda = n^3$ and $\widetilde{\rho} = (k-1,1,1)$, where $k \geq 1$. We informally refer to the resulting integers $\widetilde{C}_{3,n}^k = \vert \svt(n^3,\widetilde{\rho}) \vert$ as the non-involutory three-dimensional $k$-Catalan numbers. This title is motivated by the fact that the set-valued Sch\"utzenberger involution is no longer an automorphism of $\svt(n^3,\widetilde{\rho})$ but a bijection onto the distinct set $\svt(n^3,\widetilde{\rho}')$ with $\widetilde{\rho}'=(1,1,k-1)$. Observe from Tables \ref{tab: 1,k-1,1} and \ref{tab: k-1,1,1} of Appendix \ref{sec: appendix} that $\widetilde{C}_{3,n}^k \leq C_{3,n}^k$ for all choices of $n,k$ where both values are known.
Applying the methods of Section \ref{sec: enumeration} to $\svt(\lambda,\widetilde{\rho})$ yields the closed formulas of Proposition \ref{thm: k-1,1,1 values} and the general recurrences of Proposition \ref{thm: k-1,1,1 recurrences}. See Table \ref{tab: k-1,1,1} of Appendix \ref{sec: appendix} for all known values of $\widetilde{C}_{3,n}^k = \vert \svt(n^3,\widetilde{\rho}) \vert$.
Pause to note that the recurrences of Proposition \ref{thm: k-1,1,1 recurrences} are significantly harder to apply than those for $\rho = (1,k-1,1)$ that appear in Proposition \ref{thm: general n recurrences}, as the recurrences of Proposition \ref{thm: k-1,1,1 recurrences} involve enumerations of (non-set-valued) standard skew Young tableaux. This is a difficulty that appears to extend to all three- (and four-) row densities other than $\rho = (1,k-1,1)$.
\begin{proposition} \label{thm: k-1,1,1 values} Let $\widetilde{\rho} = (k-1,1,1)$. For any $k \geq 1$,
$$\widetilde{C}^k_{3,2} = \vert \svt(2^3,\widetilde{\rho}) \vert = \frac{1}{2}k^2 + \frac{3}{2}k$$
$$\widetilde{C}^k_{3,3} = \vert \svt(3^3,\widetilde{\rho}) \vert = \frac{2}{3}k^4 + 3 k^3 + \frac{7}{3}k^2 - k$$
$$\widetilde{C}^k_{3,4} = \vert \svt(4^3,\widetilde{\rho}) \vert = \frac{25}{18}k^6 + \frac{61}{8}k^5 + \frac{175}{18}k^4 - \frac{35}{24}k^3 - \frac{37}{9}k^2 + \frac{5}{6}k$$ \end{proposition}
\begin{proposition} \label{thm: k-1,1,1 recurrences} Fix $k \geq 1$. For $\widetilde{\rho} = (k-1,1,1)$ and any three-row shape $\lambda = (a,b,c)$ with $a \leq b \leq c$,
$$|\svt((a,b,c),\widetilde{\rho}) | = \begin{cases} \displaystyle{\sum_{\substack{0 \leq j \leq i \leq b,\\[1pt]j \leq c}} \binom{b-i+c-j+k-2}{k-2} \kern+2pt \vert S((b,c)/(i,j)) \vert \cdot \vert \svt((a-1,i,j),\widetilde{\rho}) \vert}, & \text{if $a > b$;}\\[22pt] \displaystyle{\sum_{1 \leq i \leq c} \vert \svt((a,b-1,i),\widetilde{\rho}) \vert}, & \text{if $a = b > c$;}\\[22pt] \displaystyle{\vert \svt((a,b,c-1),\widetilde{\rho}) \vert}, & \text{if $a=b=c$.} \end{cases}$$ \end{proposition}
In the case of $\lambda = n^4$, we recognize the densities $\xi_i = (1,k-1,k-1,1)$ and $\xi_2 = (k-1,1,1,1)$ as prime candidates to obtain what should be referred to as the (involutory) four-dimensional $k$-Catalan numbers $C_{4,n}^k = \vert \svt(4^n,\xi_1) \vert$ and the non-involutory four-dimensional $k$-Catalan numbers $\widetilde{C}_{4,n}^k = \vert \svt(4^n,\xi_2) \vert$. As the addition of a fourth row makes the techniques of Section \ref{sec: enumeration} significantly harder to apply, we simply direct the reader to Tables \ref{tab: 1,k-1,k-1,1} and Table \ref{tab: k-1,1,1,1} of Appendix \ref{sec: appendix} for all known values of $C_{4,n}^k = \vert \svt(4^n,\xi_1) \vert$ and $\widetilde{C}_{4,n}^k = \vert \svt(4^n,\xi_2) \vert$.
\appendix
\section{Tables of Values} \label{sec: appendix} Values were obtained via a combination of proven results (Section \ref{sec: enumeration}, Subsection \ref{subsec: other sets}) and direct enumeration in Java. Java coding was performed by Benjamin Levandowski of Valparaiso University and is available upon request.
\begin{table}[ht!] \centering \caption{Known values of $C^k_{3,n} = \vert \svt(n^3,\rho) \vert$ for $\rho=(1,k-1,1)$} \label{tab: 1,k-1,1} \small
\begin{tabular}{|>{$}c<{$}|>{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$}|} \hline k \backslash n & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1 & 2 & 5 & 14 & 42 & 132 \\ 2 & 1 & 5 & 42 & 462 & 6006 & 87516 \\ 3 & 1 & 10 & 190 & 4295 & 153415 & 5396601 \\ 4 & 1 & 17 & 581 & 27461 & 1566018 & 100950800 \\ 5 & 1 & 26 & 1401 & 105026 & 9511451 & \\ 6 & 1 & 37 & 2890 & 315014 & 41500117 & \\ 7 & 1 & 50 & 5342 & 797917 & 144067106 & \\ \hline \end{tabular} \end{table}
\begin{table}[ht!] \centering \caption{Known values of $\widetilde{C}^k_{3,n} = \vert \svt(n^3,\widetilde{\rho}) \vert$ for $\widetilde{\rho}=(k-1,1,1)$} \label{tab: k-1,1,1} \small
\begin{tabular}{|>{$}c<{$}|>{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$}|} \hline k \backslash n & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1 & 2 & 5 & 14 & 42 & 132 \\ 2 & 1 & 5 & 42 & 462 & 6006 & 87516 \\ 3 & 1 & 9 & 153 & 3579 & 101630 & 3288871 \\ 4 & 1 & 14 & 396 & 15830 & 779063 & 44072801 \\ 5 & 1 & 20 & 845 & 51325 & 3872370 & \\ 6 & 1 & 27 & 1590 & 136234 & 14589623 & \\ 7 & 1 & 35 & 2737 & 314202 & & \\ \hline \end{tabular} \end{table}
\begin{table}[ht!] \centering \caption{Known values of $\vert \svt(n^4,\xi_1) \vert$ for $\xi_1=(1,k-1,k-1,1)$} \label{tab: 1,k-1,k-1,1} \small
\begin{tabular}{|>{$}c<{$}|>{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$}|} \hline k \backslash n & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1 & 2 & 5 & 14 & 42 & 132 \\ 2 & 1 & 14 & 462 & 24024 & 1662804 & 140229804 \\ 3 & 1 & 84 & 24521 & 13074832 & & \\ 4 & 1 & 460 & 960875 & 3959335892 & & \\ 5 & 1 & 2380 & 31378194 & & & \\ 6 & 1 & 11814 & & & & \\ 7 & 1 & 57288 & & & & \\ \hline \end{tabular} \end{table}
\begin{table}[ht!] \centering \caption{Known values of $\vert \svt(n^4,\xi_2) \vert$ for $\xi_2=(k-1,1,1,1)$} \label{tab: k-1,1,1,1} \small
\begin{tabular}{|>{$}c<{$}|>{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$}|} \hline k \backslash n & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 1 & 5 & 42 & 462 & 6006 & 87516 \\ 2 & 1 & 14 & 462 & 24024 & 1662804 & 140229804 \\ 3 & 1 & 28 & 2158 & 281571 & 50972547 & \\ 4 & 1 & 48 & 6990 & 1798860 & 658138000 & \\ 5 & 1 & 75 & 18275 & 8103935 & & \\ 6 & 1 & 110 & 41382 & 28950168 & & \\ 7 & 1 & 154 & 84427 & & & \\ \hline \end{tabular} \end{table}
\end{document} | arXiv |
The property of matter by virtue of which it regains its original configuration after removing the deforming force is called elasticity. All objects in nature are elastic and no effects in nature are rigid or plastic.
Elastic after Effect:
The delay of material to regain the original position after removing the deforming force is called elastic after effect. All objects in nature exhibit thus property in some extent. But, quarty has less value of elastic after effect. So, the pointers of moving coil galvanometer or voltmeter are made of quarty but not of steel or aluminum.
Poisson's ratio:
The ratio of lateral strain is to the longitudinal strain is called poisson's ratio and it is denoted by sigma'$\sigma $'. If $\beta $ be the lateral strain and $\alpha $ be the longitudinal strain, then,
$\alpha = \frac{\beta }{\sigma }$
$ = \frac{{\frac{{\Delta {\rm{D}}}}{{\rm{D}}}}}{{\frac{{\rm{e}}}{{\rm{l}}}}}$
The Poisson's ratio has not units and dimensions because it is the ratio of two. The ratio of change in diameter is to the original diameter is called lateral strain whereas; the ratio is change in length as the original length.
Modulus:
It is defined as the ratio of the tangential stress to the shear strain, within the elastic limit. Thus
ɧ = ${\rm{tangential}}\frac{{{\rm{stress}}}}{{{\rm{shearing\:stress}}}}$
Compressibility:
Compressibility (C) is the reciprocal of bulk modulus of elasticity is called compressibility,
I.e. ${\rm{C}} = \frac{1}{{\rm{K}}}$
The deforming force per unit area of cross section is called stress. Since, deforming force and restoring force are equal in magnitude, we also can write as follows:
The restoring force per unit area of cross section area is also called stress. If 'F' be the restoring force acting on an area A, then
Stress = $\frac{{\rm{F}}}{{\rm{A}}}$
The S.I. unit of stress is Newton /m2 (${\rm{N}}/{{\rm{m}}^2}$
If the deforming force is acting perpendicular to an area then the stress produced is called normal stress. There are two systems of normal stress. They are:
Tensile Normal stress:
The tensile normal stress is that stress in which there is increase in length of the wire or object then it is called compressive normal stress.
Tangential stress:
If the deforming force is applied parallel to a surface, then the stress produced is called tangential stress. Generally, thus stress is used to just shift the area if the surface.
Hooke's law:
It states that, within the elastic limit, the deforming force is directly proportion to the extension produced i.e.
${\rm{F}} \propto {\rm{e}}$ ------ (i) $ \to $ where F is the deforming force and e is the extension produced.
Hooke's law can be verified experimentally by using vernier apparatus as follows:
Now, two wires A and B are taken in which wire A is connected to main scale and wire B is connected to venire scale. The wire A is reference wire whereas the wire B is the experimental wire. Initially, equal weights are kept on scale pans S1 and S so that the both wires A and B become from kinks and then become ready for experiment. In this condition, the main scale reading and vernier scale reading l0 is found.
Now, a load of 0.5 kg is added on scale pan S1 and it is left for two minutes. After that main scale reading and venire scale reading are noted. The difference of this reading and previous reading gives the extension produced. This process is repeated by taking other weights of 1kg, 1.5kgs, 2 kgs....... and then the corresponding extension are noted.
Modulus of elasticity:
Within the elastic unit, it is found that the stress is directly proportional to the strain. i.e.
stress$ \propto {\rm{strain}}$
or, $\frac{{{\rm{stress}}}}{{{\rm{strain}}}} = {\rm{constant}} = {\rm{E}}$
Where E is the modulus of elasticity and its unit is ${\rm{N}}/{{\rm{m}}^2}$.
Young's modulus:
It is defined as the ratio of normal stress to the longitudinal strain, within the elastic limit. Thus,
${\rm{y}} = \frac{{{\rm{normal\: stress}}}}{{{\rm{longitudinal\: strain}}}}$
Bulk modulus:
It is defined as the ratio of normal stress to the volumetric strain, within the elastic limit. Thus,
${\rm{K}} = \frac{{{\rm{normal\: stress}}}}{{{\rm{volumetric\: strain}}}}$
Modulus of rigidity:
ɧ = ${\rm{tangential}}\frac{{{\rm{stress}}}}{{{\rm{shearing\: stress}}}}$
The elastic energy stored in a stretched wire is = $\frac{1}{2}$ of the product of deforming force and the extension produced:
When a wire is stretched by a deforming force, it does work on the wire. This work is stored in the wire in the form of potential energy. The energy stored in a stretched wire due to the deforming force or restoring force is called elastic energy stored and the elastic energy stored in a wire can be found as follows
Let's take a wire and initial length l and area of cross section 'A' in which a deforming force 'F' is applied on it. This deforming force produces an extension 'x' in the wire. If 'y' be the young's modulus of elasticity of the material of the wire, then
y = $\frac{{{\rm{F*l}}}}{{{\rm{A*x}}}}$
or, F = $\frac{{{\rm{yAx}}}}{{\rm{l}}}$ ------(i)
If the deforming force is acting continuously on the wire, then, this force does work. If 'dx' be the small displacement or extension produced on the wire. By this, deforming force, then, the small work done by this deforming force is
dw = Fx dx
or, dw = $\frac{{{\rm{yAx}}}}{{\rm{l}}}{\rm{*dx}}$ --------(ii)
If the deforming force is acting continuously on the wire, then, it does a large amount of work like the small work. Then, the total work done in stretching the wire to an extension 'e' is obtained by collecting or integrating the small work as
w = $\mathop \smallint \limits_0^{\rm{e}} {\rm{dw}}$
or, w = $\mathop \smallint \limits_0^{\rm{e}} \frac{{{\rm{yAx*dx}}}}{{\rm{l}}}$
or, w = $\frac{{{\rm{yA}}}}{{\rm{l}}}\mathop \smallint \limits_0^{\rm{e}} {\rm{xdx}}$
or, w = $\frac{{{\rm{yA}}}}{{\rm{l}}}\left[ {\frac{{{{\rm{x}}^{1 + 1}}}}{{1 + 1}}} \right]_0^{\rm{e}}$
or, w = $\frac{{{\rm{yA}}}}{{\rm{l}}}\left[ {\frac{{{{\rm{x}}^2}}}{2}} \right]_0^{\rm{e}}$
or, w = $\frac{{{\rm{yA}}}}{{2{\rm{l}}}}\left[ {{{\rm{x}}^2}} \right]_0^{\rm{e}}$
or, w = $\frac{{{\rm{yA}}}}{{2{\rm{l}}}}\left[ {{{\rm{e}}^2} - 0} \right]$
or, w = $\frac{{{\rm{yA}}}}{{2{\rm{l}}}}{\rm{*}}{{\rm{e}}^2}$
or, w = $\frac{1}{2}{\rm{*}}\left( {\frac{{{\rm{yAe}}}}{{\rm{l}}}} \right){\rm{*l}}$
or, w = $\frac{1}{2}{\rm{*F*e}}$ ---------(iii)
The work done in equation (iii) is stored in the wire in the form of potential energy. So, the potential energy stored in a stretched wire is given by,
E = W = = $\frac{1}{2}{\rm{*F*e}}$
or, E = $\frac{1}{2}{\rm{*F*e}}$ ---------(iv)
Hence, the elastic energy stored in a stretched wire is = $\frac{1}{2}$ of the product of deforming force and the extension produced. | CommonCrawl |
\begin{document}
\title {Regularity results for nonlocal equations and applications}
\author[Mouhamed Moustapha Fall] {Mouhamed Moustapha Fall} \address{M.M.F.: African Institute for Mathematical Sciences in Senegal, KM 2, Route de Joal, B.P. 14 18. Mbour, S\'en\'egal} \email{[email protected], [email protected]}
\thanks{
The author's work is supported by the Alexander von Humboldt foundation. He thanks Joaquim Serra, for his interest in this work and with whom he had stimulating discussions that help to improve the first section of this paper. He also thanks the anonymous referee for useful comments.
}
\begin{abstract}
\noindent
We introduce the concept of $C^{m,\a}$-nonlocal operators, extending the notion of second order elliptic operator in divergence form with $C^{m,\a}$-coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the H\"older continuity of the gradient of the solutions in the case of $C^{0,\a}$-coefficients and the classical Schauder estimates for $C^{m+1,\a}$-coefficients.
We further apply the regularity results for $C^{m,\a}$-nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided.
\end{abstract}
\maketitle
\section{Introduction}
We are concerned with a class of (not necessarily translation invariant) elliptic equations driven by nonlocal operators of fractional order. We extend in the nonlocal setting some key existing results for elliptic equations in divergence form with $C^{m,\a}$-coefficients. For a better description of how far the results in this paper extend to the fractional setting those available in the classical case, we start by recalling some main results of the classical local theory.
We consider a weak solution $u\in H^1(\O)$ to the equation \begin{equation}\label{eq:u-solveloc-PDE} \sum_{i,j=1}^N \partial_i(a_{ij}(x)\partial_j u)=f \qquad\textrm{ in $\O$}, \end{equation} where, $\O$ is a bounded open subset of $\mathbb{R}^N$, $f\in L^p_{loc}(\O)$, $p>N/2$, and the matrix coefficients $a_{ij} $ are measurable functions and satisfy, for every $x\in \O$, the following properties:
\begin{equation} \label{eq:a-satisf-elliptic}
\begin{aligned} (i)\,& a_{ij}(x)=a_{ji}(x) \qquad &\textrm{ for all $i,j=1,\dots,N$,}\\
(ii)\,& \kappa \delta _{ij}\leq a_{ij}(x)\leq \frac{1}{\kappa} \delta _{ij} &\qquad\textrm{ for all $i,j=1,\dots,N$.}
\end{aligned}
\end{equation}
In the regularity theory for elliptic equations in {divergence form} with {measurable coefficients}, the De Giorgi-Nash-Moser theory provides a priori $C^{0,\a_0} (\O)$ estimates for weak solutions to \eqref{eq:u-solveloc-PDE}, for some $\a_0=\a_0(N,p,\kappa)$, see e.g. \cite{GT}. The range or value of the largest H\"older exponent $\a_0$ is known in general once the coefficients are sufficiently regular. For instance, if $a_{ij}\in C(\O)$ then $u\in C ^{0,\b}_{loc}(\O)$ for all $\b<\min(2-N/p,1)$. Now H\"older continuous coefficients $a_{ij}$ yield H\"older continuity of the gradient of $u$. Namely, if $a_{ij}\in C^{0,\a}(\O)$, for some $\a\in (0,1)$, then $u\in C^{1,\min( 1-\frac{N}{p},\a)} (\O)$, provided $2-N/p>1$. Moreover the Schauder theory states that if $a_{ij}\in C^{m+1,\a}(\O)$ and $f\in C^{m,\a}(\O)$, then $u\in C^{m+2,\a} (\O)$ for $m\in\mathbb{N}$. We refer the reader to \cite{GT, Texi}. Notable applications are the smoothness character of variational solutions, including the regularity of critical points of the integral functional \begin{equation} \label{eq:integ-functinal}
{\mathcal J}(u):=\int_{\O}G(\n u (x))\, dx, \end{equation}
for some twice differentiable function $G$. As a matter of fact, the above regularity results provides a systematic proof of the Hilbert's $19^{\textrm{th}}$ problem stating that if $G\in C^\infty(\mathbb{R}^N)$, then the minimizer of ${\mathcal J}$ is of class $C^\infty$ as well. This was solved by de Giorgi in \cite{DeGiorgi}. Indeed, given a critical point $u\in H^1(\O)$ to ${\mathcal J}$, we have that $\frac{u(x+h)-u(x)}{|h|}$ solves an equation as in \eqref{eq:u-solveloc-PDE} with coefficients $a_{ij}(x)=\int_0^1\partial^2_{ij }G(\varrho \n u(x+h)+(1-\varrho)\n u(x))d\varrho$ satisfying \eqref{eq:a-satisf-elliptic} as soon as $D^2G$ is uniformly bounded from above and below on $\mathbb{R}^N$. Therefore the fact that $a_{ij}$ is as smooth as $\n u$ immediately implies that $u$ smooth, thanks to above regularity results for divergence type operators. On the other if $G(\z)=\sqrt{1+|\z|^2}$, then \eqref{eq:integ-functinal} becomes the area functional, and in this case $D^2G$ is not bounded from below. However, this gap can be filled by assuming that $u$ Lipschitz. \\
The aim of this paper is to extend all the above regularity results to equations driven by $C^{m,\a}$-nonlocal operators of fractional order which we describe below. Our notion of $C^{m,\a}$-nonlocal operators can be seen as a nonlocal version of second order partial differential equations in divergence form. On the other hand, as in the local case, since our notion of $C^{m,\a}$-nonlocal operators is stable under $C^{m,1}$ local change of coordinates, our results apply to nonlocal equations on manifolds nonlocal geometric problems such as the prescribed nonlocal mean curvature problems. \\
We start introducing the class of kernels (defining nonlocal operators) we will use in the remaining of the paper. We consider $s\in (0,1)$, $N\geq 1$ and $K:\mathbb{R}^N\times \mathbb{R}^N \to [-\infty,+\infty] $ such that
\begin{equation} \label{eq:Kernel-satisf}
\begin{aligned} (i)\,& K(x,y)=K(y,x) \qquad &\textrm{ for all $(x,y)\in\mathbb{R}^N\times \mathbb{R}^N$,}\\
(ii) \,& |K(x,y)| \leq \frac{1}{\kappa} |x-y|^{-N-2s} \qquad&\textrm{ for all $(x,y)\in\mathbb{R}^N\times \mathbb{R}^N$,}\\
(ii')\,& \kappa |x-y|^{-N-2s}\leq K(x,y) \qquad &\textrm{ for all $(x,y)\in B_\delta \times B_\delta $,}
\end{aligned}
\end{equation}
for some constants $\kappa, \delta >0$. We call ${L_s(\mathbb{R}^N)}$ the space of function $u\in L^1_{loc}(\mathbb{R}^N)$ such that
$$ \|u\|_{{L_s(\mathbb{R}^N)}}:=\int_{\mathbb{R}^N}|u(y)|(1+|y|)^{-N-2s}\, dy<\infty.$$ A kernel $K$ satisfying \eqref{eq:Kernel-satisf}$(i)$-$(ii)$ induces a linear nonlocal operator $ {\mathcal L}_K: H^s (\O)\cap {L_s(\mathbb{R}^N)}\to \calD'(\O)$ given by
$$ {\langle} {\mathcal L}_K u, \psi {\rangle}:= \frac{1}{2}\int_{\mathbb{R}^N \times \mathbb{R}^N}(u(x)-u(y))(\psi(x)-\psi(y))K(x,y)dxdy \qquad\textrm{ for all $\psi\in C^\infty_c(\O)$.}
$$
The weight in the definition of the space ${L_s(\mathbb{R}^N)}$ is determined by \eqref{eq:Kernel-satisf}-$(ii)$ and can be modified accordingly.
Given $f\in L^1_{loc}(\mathbb{R}^N)$,
we say that $u\in H^s (\O)\cap {L_s(\mathbb{R}^N)} $ is a (weak) solution to the equation \begin{equation} \label{eq:Main-problem} {\mathcal L}_K u = f \qquad\textrm{ in $ \O$} , \end{equation}
if ${\mathcal L}_K u =f$ in $\calD'(\O)$.
The class of operators ${\mathcal L}_K$ induced by the kernels $K$ satisfying \eqref{eq:Kernel-satisf} are the nonlocal analogue of second order elliptic operators in divergence form with measurable coefficients on $B_\delta $. In this case the de Giorgi-Nash-Moser a priori H\"older estimates is well developed, see \cite{Dicastro, KMS, KRS,KS,Cozzi, DK,Mosconi,CCV,Min-JEMS}. In particular, it follows from \cite{DK} that, if $f\in L^p(B_\delta )$, for some $p>N/(2s)$, then $u\in C^{0,\a_0}_{loc}(B_\delta )$, for some $\a_0=\a_0(N,s,p)>0$.\\
The study of nonlocal variational equations involving general kernels, satisfying e.g. \eqref{eq:Kernel-satisf}, is currently an intensive research area. In particular several papers deals with existence and, a part in some specific cases e.g. fractional Lapalcian, anisotropic fractional Lapalcian, regional fractional Laplacian, the "smoothness" properties of the nonlocal operators leading to higher order regularity of weak solutions (e.g. $C^{1,\a}$ or $C^{2s+\a}$-regularity) remains an open questions. In the case of nonlocal and non-translation invariant operators (say in non-divergence form), different assumptions on the kernels yielding higher order regularity are present in the literature, starting from the work of Caffarelli-Silvestre \cite{CS2}, followed by many others e.g. \cite{Kriventsov,Serra,Jin-Xiong}. A first difficulty to address this question in the variational framework is the singular character of the kernel $K$ (satisfying \eqref{eq:Kernel-satisf}) at the diagonal points $x=y$ which encodes also the order of regularity of the solution, regardless the behaviour of the tail.
In \cite{Fall-Reg}, we attempted to answer this question and introduced a notion of nonlocal operators with "continuous" coefficient, and we proved some optimal interior and boundary regularities of solutions to \eqref{eq:Main-problem}. In that paper, we also proved that $u$ is a classical solution provided the operator ${\mathcal L}_K$ is smooth enough together with $C^{1,\a}$ estimates for translation invariant problems. These results will be sharpened and generalized in the present work.\\ Following \cite{Fall-Reg}, we now introduce the notion of $C^{m,\a}$-nonlocal (or fractional order) operators which, in particular, are the object of study in the present paper. \begin{definition}\label{def:nonloc-very-reg}
For $\delta >0$, we define $Q_\delta :=B_\delta \times [0,\delta )$. Let $\a\in [0,1)$, $m\in \mathbb{N}$ and $K$ satisfy \eqref{eq:Kernel-satisf}. \begin{itemize}
\item We say that the kernel $K$ defines a $C^{m,\a}$-nonlocal operator in $Q_\delta $, if the function
$$ B_\delta \times (0,\delta ) \times S^{N-1}\to \mathbb{R}, \qquad (x,r,\th)\mapsto r^{N+2s}K(x,x+r\th)
$$
extends to a map $ {{\mathcal A}}_K : Q_\delta \times S^{N-1}\to \mathbb{R}$ satisfying, for some $\kappa>0$, the following properties:
\begin{equation} \label{eq:NL-Holder00}
\begin{aligned}
(iii)\,& \|{{\mathcal A}}_K \|_{C^{m,\a} (Q_\delta \times S^{N-1})} \leq \frac{1}{\kappa}, \\
(iv)\,& {{\mathcal A}}_K(x,0,\th)= {{\mathcal A}}_K(x,0,-\th) & \qquad \textrm {for all $ (x,\th) \in B_\delta \times S^{N-1}$.}
\end{aligned}
\end{equation}
\item The class of kernels $K$ satisfying \eqref{eq:Kernel-satisf} and \eqref{eq:NL-Holder00} is denoted by $ \scrK^s(\kappa,m+\a, Q_{\delta })$. \end{itemize} \end{definition} A simple example in the class of kernels in Definition \ref{def:nonloc-very-reg}
is the the one of the fractional Laplacian, where the kernel is given by $K(x,y)= |x-y|^{-N-2s}$.
We remark that the class of operators induced by the kernels in Definition \ref{def:nonloc-very-reg} provides a naturally extension of second order elliptic operators with $C^{m,\a}$-coefficients. Indeed, the computations in \cite[Section 5]{Buc-Sq} show, for all $\psi\in C^1_c(B_\delta )$, that \begin{equation}\label{eq:limit-tolocal}
{(1-s)} \int_{\mathbb{R}^N \times \mathbb{R}^N} (\psi(x)-\psi(y))^2K(x,y)dxdy\to\frac{1}{2} \sum_{i,j=1}^N \int_{\mathbb{R}^N}a_{ij}^K(x)\partial_i \psi(x) \partial_j \psi(x)\, dx \qquad\textrm{as $s\to 1$,} \end{equation}
where $ a_{ij}^K(x)=\int_{S^{N-1}} {\mathcal A}_{K}(x,0,\th) \th_i\th_j\, d\th$. Hence \eqref{eq:NL-Holder00}-$(iv)$ implies the symmetry of the matrix $( a_{ij}^K)_{1\leq i,j\leq N}$.
\begin{remark}
In \eqref{eq:NL-Holder00}-$(iii)$, we impose the regularity of ${\mathcal A}_K$ in the angular variable $\th$. However, this is typically not necessary to derive the accurate local behavior of solutions to \eqref{eq:Main-problem} which parallels those solving \eqref{eq:u-solveloc-PDE} as stated above. In fact, nonlocal operators provide a wider framework than their local counterpart, since translation invariant nonlocal operators are those given by kernels $K$ of the form $K(x,y)=J(x-y)$, for some even function $J$. In addition, only in this translation invariant setting, regularity theory is already rich enough to include fully nonlinear problems, \cite{ CS1,CS2,CS3,Kriventsov,Serra,Jin-Xiong }.
This issue on the possible \textit{anisotropic} regularity of ${\mathcal A}_K$ in its variables will be taken into account in our main results stated in Section \ref{ss:General-nonlocaloperator} below.
\end{remark} We now start by stating the main results concerning $C^{m,\a}$-nonlocal operators. Their generalizations are contained in Section \ref{ss:General-nonlocaloperator} below. Our first main result is the following.
\begin{theorem}\label{th:main-th10} Let $s\in (0,1)$, $N\geq 1$, $\kappa>0$ and $\a\in (0,1)$. Let $K\in \scrK^s(\kappa,\a, Q_2)$, $u\in H^s(B_2)\cap {L_s(\mathbb{R}^N)}$ and $V, f\in L^p(B_2)$, for some $p>N/(2s)$, satisfy
$$
{\mathcal L}_K u+ Vu = f\qquad \textrm{in $B_2$}.
$$
\begin{itemize}
\item[$(i)$] If $2s\leq 1$, then there exists $C=C(s,N,\kappa, \a,p, \|V\|_{L^p(B_2)})>0$ such that
\begin{equation} \label{eq:thm-E10pp}
\|u\|_{C^{0,2s-\frac{N}{p}}(B_1)}\leq C(\|u\|_{L^2(B_2)}+\|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{L^p(B_2)}). \end{equation}
\item[$(ii)$] If $2s-1>\max(\frac{N}{p},\a)$, then there exists $C=C(s,N,\kappa, \a,p, \|V\|_{L^p(B_2)})>0$ such that
\begin{equation} \label{eq:thm-E20pp}
\|u\|_{C^{1,\min (2s-\frac{N}{p}-1,\a)}(B_1)}\leq C(\|u\|_{L^2(B_2)}+\|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{L^p(B_2)}). \end{equation}
\end{itemize}
\end{theorem}
The H\"older continuity of the gradient in \eqref{eq:thm-E20pp} is the main novelty in the above result. Theorem \ref{th:main-th10} was only known in the translation invariant case, i.e. $K(x,y)=J(x-y)$, see \cite{Fall-Reg}. We mention that the regularity estimate in \eqref{eq:thm-E10pp} remains valid if $\a=0$, see \cite{Fall-Reg}, where it was proven that if $K\in \scrK^s(\kappa,0, Q_2)$ (and for all $s\in (0,1)$), then $u\in C^{0,\b}(B_1)$ for all $\b<\min(2s-\frac{N}{p},1)$. In view of \eqref{eq:limit-tolocal}, it will be apparent from the proof that the estimates in Theorem \ref{th:main-th10} remain stable as $s\to1$ once we replace ${\mathcal L}_K$ by $(1-s){\mathcal L}_K$ and provided $p>\frac{N}{2s_0}$, with $s_0\in (0,1)$.\\
We recall that H\"older continuity of the gradient of solutions to fully nonlinear and non translation invariant integro-differential equations, in the spirit of Cordes and Nirenberg for elliptic equations in nondivergence form, has been first established by Caffarelli and Silvestre in \cite{CS2}, see also \cite{Kriventsov,Serra,Jin-Xiong } for higher order regularity estimates in nonlocal problems corresponding to elliptic equations in non-divergence form. \\
Our next results is concerned with $C^{m+2s+\a}$ regularity estimates for solutions to equations driven by $C^{m+(2s-1)_++\a}$-nonlocal operators, provided $2s+\a\not\in \mathbb{N}$. Here and in the following, we put $\ell_+=\max(\ell,0)$ for $\ell\in \mathbb{R}$.
\begin{theorem}\label{th:Schauder-0-intro} Let $N\geq 1$, $s\in (0,1)$ and $\kappa>0$. Let $m\in \mathbb{N}$ and $\a\in (0,1)$, with $2s+\a\not\in \mathbb{N}$.
Let $K\in \scrK^s(\kappa, m+\a+ (2s-1)_+, Q_2)$, $u\in H^s(B_2)\cap L^{\infty}(\mathbb{R}^N)$ and $f\in C^{m,\a}(B_2)$ such that
$$
{\mathcal L}_K u= f\qquad \textrm{in $B_2$}.
$$
\begin{itemize}
\item[$(i)$] If $2s+\a<1$, then
$$
\|u\|_{C^{m,2s+\a}(B_1)}\leq C(\|u\|_{L^\infty(\mathbb{R}^N)}+ \|f\|_{C^{m,\a}(B_2)}). $$
\item[$(ii)$] If $1<2s+\a<2$ and $2s\not=1$, then
$$
\|u\|_{C^{m+1,2s+\a-1}(B_1)}\leq C(\|u\|_{L^\infty(\mathbb{R}^N)}+ \|f\|_{C^{m,\a}(B_2)}). $$
\item[$(iii)$] If $2<2s+\a$, then
$$
\|u\|_{C^{m+2,2s+\a-2}(B_1)}\leq C(\|u\|_{L^\infty(\mathbb{R}^N)}+ \|f\|_{C^{m,\a}(B_2)}). $$
\item[$(iv)$] If $2s=1$, then for all $\b\in (0,\a)$,
$$
\|u\|_{C^{m+1,\b}(B_1)}\leq C(\|u\|_{L^\infty(\mathbb{R}^N)}+ \|f\|_{C^{m,\b}(B_2)}). $$
\end{itemize}
Here $C=C(N,s,\kappa,\a,\b, m).$
\end{theorem} It is clear that Theorem \ref{th:Schauder-0-intro} includes the fractional Laplacian ${\mathcal L}_K= (-\D)^s$, for which it was proven in \cite{DSV,Sil,Grubb1,RS2}. \\
Theorem \ref{th:main-th10} and Theorem \ref{th:Schauder-0-intro} provide regularity of minimizers of integral energy functional e.g. of the form \begin{equation} \label{eq:cJs}
{\mathcal J}_{s}(u):= (1-s) \int_{\mathbb{R}^{2N}\setminus (\mathbb{R}^N\setminus \O)^2} F\left(\frac{u(x)-u(y)}{|x-y|} \right) |x-y|^{-N-2s+2}\, dxdy ,
\end{equation} for some twice differentiable function $F$. The case $F(t)=t^2$ is the well known localized (in $\O$) Dirichlet energy for equations involving the fractional Laplacian. The minimization of this energy should be subject to exterior boundary data on $\mathbb{R}^N\setminus \O$, and posses a minimizer on $H^s(\mathbb{R}^N)$ under some quadratic and convexity assumption on $F$. To see how ${\mathcal J}_s$ is related with \eqref{eq:integ-functinal}, we assume that $ |F(z)|\leq |z|$. Then, for all $u\in C^1_{c}(\mathbb{R}^N)$, $$ \lim_{s\to 1 }{\mathcal J}_{s}(u)\to \int_{\O} G(\n u(x))\, dx, $$ where $G(\z)=\frac{1}{2}\int_{S^{N-1}}F_e(\z\cdot \th)\, d\th$ and $F_e$ is the even part of $F$ i.e., $F_e(t)=\frac{F(t)+F(-t)}{2}$. The nonlocal de Giorgi-Nash-Moser provides a priori estimates for minimizers of ${\mathcal J}_s$. Indeed, a critical point $u\in H^s(\mathbb{R}^N)$ to ${\mathcal J}_s$ satisfies
\begin{equation} \label{eq:EL-eq-gen}
\int_{\mathbb{R}^N\times\mathbb{R}^N} \left\{F'(p_u(x,y)) - F'(-p_u(x,y)) \right\}(\psi(x)-\psi(y)) |x-y|^{-N-2s+1}dxdy=0 \qquad\textrm{for all $\psi\in C^\infty_c(\O)$,} \end{equation}
where $p_u(x,y):= \frac{u(y)-u(x)}{|y-x|}$.
Therefore, following the classical de Giorgi's trick and using the fundamental theorem of calculus, we find that the difference quotient $u_h(x)=\frac{u(x+h)-u(x)}{|h|}$ solves the equation $$ {\mathcal L}_{K_{F,u,h}}u_h=0 \qquad\textrm{ in $\O$,} $$ where \begin{equation} \label{eq:K-Q-u-h}
K_{F,u,h}(x,y)= |x-y|^{-N-2s}\int_0^1F''_e\left( \varrho p_{u(\cdot+h)}(x,y) +(1-\varrho)p_u(x,y) \right) d\varrho, \end{equation} and $F_e$ is the even part of $F$. We then immediately see that $K_{F,u,h}$ satisfies \eqref{eq:Kernel-satisf} when $F''$ is bounded from above and below on $\mathbb{R}$. Consequently, the nonlocal de Giorgi-Nash-Moser theory implies that $u\in C^{1,\a_0}(\O)$, for some $\a_0>0$.
Now an iterative application of Theorem \ref{th:main-th10} and Theorem \ref{th:Schauder-0-intro} shows, as for the solution to the Hilbert's problem, that if $F\in C^\infty(\mathbb{R}) $ then $u\in C^\infty(\O)$. It is worth to mention that in the nonlocal mean curvature problem, nonlocal minimal graphs satisfy an equation as in \eqref{eq:EL-eq-gen}, with $F(t)= \int_{t}^{\infty}(1+\t^2)^{-\frac{N+2s}{2}}d\t$ (see Section \ref{ss:NMC} below for a more precise statement). Here, $F''$ is not uniformly bounded from below and thus $ K_{F,u,h}$ does not satisfy \eqref{eq:Kernel-satisf}-$(ii')$. However, as in the classical case, this lack of ellipticity, is recovered once we know that $u$ is Lipschitz. \\
Beyond their appearances in the mathematical modeling of real-world phenomenon, $C^{m,\a}$-nonlocal operators appear naturally in geometric problems. Indeed, we are naturally confronted with nonlocal equation resulting from an initial one after a change coordinates. For instance, consider $K(x,y)=|x-y|^{-N-2s}$ (the kernel of the fractional Laplacian) and $K_\Phi(x,y)=|\Phi(x)-\Phi(y)|^{-N-2s}$, for some diffeomorphism $\Phi\in C^{m+1,\a}(\mathbb{R}^N;\mathbb{R}^N)$ with $D\Phi$ close to the identity matrix, so that \eqref{eq:Kernel-satisf} holds. In this case, apart in dimension $N=1$, we may not have any regularity of $z\mapsto |z|^{N+2s}K_{\Phi}(x,x+z)$ at $z=0$. However, using polar coordinates, we easily see that the map $$
(x,r,\th)\mapsto r^{N+2s}K_{\Phi}(x,x+r\th)=\left| \int_0^1D\Phi(x+t r\th)\theta \, dt\right|^{-N-2s} $$
extends to a $C^{m,\a}$ map on $\mathbb{R}^N\times [0,\infty)\times S^{N-1}$ satisfying \eqref{eq:NL-Holder00}-$(ii)$, so that $K_{\Phi}$ defines a $C^{m,\a}$-nonlocal operator. This is also the case for the kernel in \eqref{eq:K-Q-u-h} with $F\in C^\infty(\mathbb{R})$ and $u\in C^{m+1,\a}(\O)$. These facts, among others, motivate the splitting in polar coordinates in our definition of $C^{m,\a}$-nonlocal operators. Moreover, it turns out to be useful in the study of prescribed nonlocal mean curvature problems and nonlocal equations on hypersurfaces, see Section \ref{ss:NMC} and Section \ref{ss:NonlocManifold}, respectively. On the other hand, we remark that in some interesting non-translation invariant cases, the map $ z\mapsto |z|^{N+2s}K (x,x+z)$ can be smooth at $z=0$, and a first nontrivial example is given by the \textit{censored fractional Laplacian} or the $\O$-regional fractional Laplacian, where the kernel is given by $K(x,y)=1_{\O}(x)1_{\O}(y) |x-y|^{-N-2s}$, see e.g. Mou and Yi \cite{Mou}. An other example arises in problems from image processing, see e.g. Gilboa Osher \cite{GO} and Caffarelli, Chan and Vasseur \cite{CCV}, where the kernel depends on the solution $u\in C^{1,\a_0}$ and, for simplicity, reads as $K(x,y)=1_{\O}(x)1_{\O}(y)\phi''(u(x)-u(y)) |x-y|^{-N-2s}$, for some even and convex function $\phi$. Here one looks at minimizer $u\in H^s(\O)$ of the energy functional $$
{\mathcal J}_{s,\O}(u):= (1-s) \int_{\O\times \O} \phi\left({u(x)-u(y)} \right) |x-y|^{-N-2s}\, dxdy.
$$
This is also the case for (possibly) sign-changing kernels e.g. $K(x,y)=|x-y|^{-N-2s_1}\pm|x-y|^{-N-2s_2}$, with $s_1\in (0,1)$ and $ s_2<s_1$. However the conditions \eqref{eq:Kernel-satisf} and \eqref{eq:NL-Holder00} are flexible enough to include such cases.\\
The following two paragraphs are devoted to the application of the above regularity estimates in some nonlocal geometric problems.
\subsection{Application I: Graphs with prescribed nonlocal mean curvature}\label{ss:NMC}
In this section, we assume that $s\in (1/2,1)$. Recall that for a set $E\subset\mathbb{R}^{N+1}$ of class $C^{1,2s-1+\a} $, with $\a>0$, near a point $X\in \partial E $, the nonlocal (or fractional) mean curvature of the set $E$ (or the hypersurface $\partial E$) at the point $X\in \partial E $ is defined as \begin{equation} \label{eq:NMC-PV}
H_s(\partial E;X):= PV\int_{\mathbb{R}^{N+1}}\frac{1_{E^c}(Y)-1_{E}(Y)}{|Y-X|^{N+2s}}\, dY,
\end{equation} where $E^c:=\mathbb{R}^{N+1}\setminus {E}$ and $1_D$ denotes the characteristic function of a set $D \subset \mathbb{R}^{N+1}$.
Recall that the notion of nonlocal mean curvature appeared first in the work of Caffarelli and Souganidis in \cite{Caff-Soug2010} and first studied by Caffarelli, Roquejoffre, and Savin in~\cite{Caffarelli2010}. As first discovered in \cite{Caffarelli2010} (see also \cite{Davila2014B,FFMMM}), the nonlocal mean curvature arises as the first variation of the fractional perimeter. For the convergence of fractional curvature to the classical one as $s\to 1$, see \cite{Ab-Val,Davila2014B}. \\ Suppose that $\partial E$ is the graph of a function $u\in C^{1, 2s-1+\a}(\O)\cap C^{0, 1}_{loc}(\mathbb{R}^{N})$, then see e.g. \cite{Fall-CNMC}, by a change of variable, for all $x\in \Omega $, we have \begin{align}
H_s(\partial E;(x,u(x)))&=PV\int_{\mathbb{R}^{N}}\frac{{\mathcal F}_s(p_u({x},{y}))- {\mathcal F}_s(p_u({y},{x})) }{|{x}-{y}|^{N+2s-1}} d{y}, \label{eq:NMC_curve-E1}
\end{align}
where
\begin{equation} \label{eq:def-of-F} {\mathcal F}_s(p):=\int_p^{+\infty} {(1+\t^2)^{\frac{-(N+2s)}{2}}}{d\t} \end{equation} and for a measurable function $w:\mathbb{R}^N\to \mathbb{R}$, we put
\begin{equation} \label{eq:P_u}
p_w({x},{y})= \frac{w({y})-w({x})}{|{x}-{y}|} .
\end{equation}
By the fundamental theorem of calculus and \eqref{eq:NMC_curve-E1} and noting that ${\mathcal F}_s(p_u({y},{x})) ={\mathcal F}_s(-p_u({x},{y})) $, we have \begin{align}
H_s(\partial E;(x,u(x))) & = PV\int_{\mathbb{R}^{N}}\frac{u({x})-u({y})}{|{x}-{y}|^{N+2s}}\,q_u(x,y) \, d {y},
\label{eq:NMC_curve-E3}
\end{align} where for a measurable function $w:\mathbb{R}^N\to \mathbb{R}$,
$$
q_w(x,y):=-\int_{-1}^1{\mathcal F}_s'(tp_w(x,y) )\,dt = \int_{-1}^1 {\left( 1+t^2 p_w(x,y)^2 \right)^{\frac{-(N+2s)}{2}}}dt.
$$
For the following, we define the \textit{ the nonlocal mean curvature kernel} by $$
{\mathcal K}_w(x,y):= \frac{1}{|{x}-{y}|^{N+2s}}\,q_w(x,y) \qquad\textrm{ for all $x\not=y\in \mathbb{R}^N$.} $$
Letting $\O$ be an open set of $\mathbb{R}^{N}$ and $f\in L^1_{loc}(\O)$, we are interested in the regularity of measurable functions $u:\mathbb{R}^N\to \mathbb{R}$ satisfying
\begin{equation}\label{eq:weak-NMC-natur}
{\mathcal L}_{{\mathcal K}_u} u=f \qquad\textrm{ in $\O$,} \end{equation}
or equivalently,
\begin{equation}\label{eq:decom-NMC-intro}
\frac{1}{2}\int_{\mathbb{R}^N\times \mathbb{R}^N}\frac{{\mathcal F}_s(p_u({x},{y}))- {\mathcal F}_s(p_u({y},{x})) }{|{x}-{y}|^{N+2s-1}} (\psi(x)-\psi(y)) \, dx dy= \int_{\mathbb{R}^N}f(x)\psi(x)\, dx \qquad\textrm{for all $\psi\in C^\infty_c(\O)$.}
\end{equation}
Note that, since ${\mathcal F}_s\in L^\infty(\mathbb{R})$ and $2s>1$, the right hand side in \eqref{eq:decom-NMC-intro} is well defined.
We observe that if $u\in C^{1, 2s-1+\a}(\O)\cap L^1_{loc}(\mathbb{R}^N) $ solves \eqref{eq:weak-NMC-natur}, then the set $E_u:=\{(x,t)\in \mathbb{R}^N\times \mathbb{R}\,:\,u(x)<t\}$, satisfies $H_s( \partial E_u;(x,u(x)) )=f(x)$, for all $x\in \O$, provided the $(N+1)$-dimension Lebesgue measure of $\partial E_u$ is equal to zero. This follows by approximating $u$ by a sequence of smooth functions. \\ We consider next locally Lipschitz graphs with prescribed nonlocal mean curvature in the weak sense of \eqref{eq:decom-NMC-intro}, and we prove that they are of class $C^\infty$ in $\O$ as long as $f$ is $C^\infty$ in $\O$, with quantitative estimates. In the classical case, this is a consequence of the de Giorgi-Nash theorem and the Schauder theory for uniformly elliptic equations in divergence form with $C^{m,\a}$-coefficients. See e.g. Figalli and Valdinoci \cite{Figalli-Valdinoci }, it was hardly believed that the same strategy could be carried out in the nonlocal setting.
In \cite{Figalli-Valdinoci }, the authors used geometric arguments to prove that Lipschitz sets, locally minimizing fractional perimeter are of class $C^\infty$. However their argument does not provide quantitative estimates.
Here, we shall show that it is indeed possible to proceed as in the prescribed mean curvature problem, thanks to our regularity estimates for $C^{m,\a}$-nonlocal operators.
It is important to note, in the theorem below, that we do not require any integrability of $u$ in $\mathbb{R}^N\setminus B_2$. We have the following result. \begin{theorem}\label{eq:thm-nmc-reg}
Let $f\in L^{1}_{loc}(B_2)$ and $u:\mathbb{R}^N\to \mathbb{R}$ be a measurable function, with $\|u\|_{ C^{0,1}(B_2) }\leq c_0$, such that $$ {\mathcal L}_{{\mathcal K}_u} u=f \qquad\textrm{ in $B_2$,} $$ in the sense of \eqref{eq:decom-NMC-intro}. Then the following statements hold.
\begin{itemize} \item[$(i)$] If $f\in C^{0,1}(B_2) $, then \begin{equation} \label{eq:estimu-NMC-first}
\| u\|_{C^{1,\a_0}(B_1)}\leq C (1+\|f\|_{ C^{0,1}(B_2) }), \end{equation} for some constants $\a_0,C>0$, only depending on $ N,s$ and $c_0$. Moreover, for all $\b\in (0,2s-1)$, $$
\| u\|_{C^{ 2,\beta }(B_{1})}\leq C , $$
for some constant $C$, only depending on $ N,s,\b, c_0$ and $\|f\|_{ C^{0,1}(B_2) }$.
\item[$(ii)$] If $f\in C^{m,\a}(B_2)$, for some $\a\in (0,1)$ and $m\geq 1$, then
$$
\| u\|_{C^{ m+1,2s+\a-1 }(B_{1})}\leq C \qquad\textrm{ if $2s+\a<2$,} $$
$$
\| u\|_{C^{ m+2,2s+\alpha -2 }(B_{1})}\leq C \qquad\textrm{ if $2s+\a>2$,} $$
for some constant $C$, only depending on $ N,s,\a,m,c_0$ and $\|f\|_{C^{m,\a}(B_2)}$. \end{itemize} \end{theorem}
The first quantitative estimates for nonlocal minimal graphs was found recently by Cabr\'e and Cozzi in \cite{CC}. Indeed, they provide, in \cite{CC}, quantitative gradient estimates for global graphs that locally minimize the fractional area functional in a cylinder $B_R\times \mathbb{R}$, in the spirit of Finn \cite{Finn} and Bombieri, de Giorgi and Miranda \cite{Bomb}. In this case $f\equiv 0$. Therefore combining their result and Theorem \ref{eq:thm-nmc-reg}, we get quantitative estimates of all partial derivatives of such graphs in terms of the oscillation of $u$ in $B_R$. \\
Recall that the smoothness character for fractional perimeter minimizing sets was known, but without quantitative bounds. Indeed, the seminal paper \cite{Caffarelli2010} established the first existence and $C^{1,\g}$ (except a closed set of zero $(N-3)$-Hausdorff measure) regularity for fractional perimeter minimizing sets. In \cite{Barrios}, Barrios, Figalli and Valdinoci, proved that fractional perimeter minimizing sets which are of class $C^{1,(2s-1)/2-\e}$ are of class $C^\infty$. On the other hand Caffarelli and Valdinoci showed, in \cite{Caffarelli2011B}, that, for $s$ close to 1, these sets possess the smoothness property of the classical perimeter minimizing regions. It is proven in \cite{DSV-NMC}, by Dipierro, Savin and Valdinoci, that the boundary of a fractional perimeter minimizing set, in a reference smooth set $\O$, which coincides with a continuous graph $\mathbb{R}^N\setminus \overline \Omega$ is in fact a global graphs that is continuous in $\O$. \\
The fact that we do not require any integrability of $u$ in $\mathbb{R}^N\setminus B_2$ makes the proof of Theorem \ref{eq:thm-nmc-reg} particularly nontrivial. In view of the decomposition in \eqref{eq:decom-NMC-intro}, we split further the double integral in the left hand side to get \begin{align}
{\langle} {\mathcal L}_{{\mathcal K}_u}u,\psi {\rangle} &=\frac{1}{2}\int_{\O\times \O} {(u(x)- u(y))(\psi(x)-\psi(y)) } {\mathcal K}_u(x,y) \, dx dy \nonumber\\
&\quad+ \int_{\O} \psi(x) \int_{ \mathbb{R}^N\setminus \O}\frac{{\mathcal F}_s(p_u({x},{y}))- {\mathcal F}_s(p_u({y},{x})) }{|{x}-{y}|^{N+2s-1}} \, dy dx. \label{eq:decom-NMC-intro0} \end{align} Now the proof of Theorem \ref{eq:thm-nmc-reg} resides on the regularity of the map $$
\O'\to \mathbb{R},\qquad x\mapsto \int_{ \mathbb{R}^N\setminus \O}\frac{{\mathcal F}_s(p_u({x},{y}))- {\mathcal F}_s(p_u({y},{x})) }{|{x}-{y}|^{N+2s-1}} \, dy , $$
for $\O'\subset\subset\O$. Surprisingly, the local behavior of this map is completely determined by the one of $u$ only in $\O'$. In fact we will show, in Lemma \ref{lem:Lem-Gamma-u-nmc} below, that this function is indeed as smooth as $u$ in $\O'$. Once this is proved, the above function is sent in the right hand side, so that we can use the argument as in the classical case. Indeed, we apply first the nonlocal de Giorgi-Nash a priori H\"older estimate to the function $\frac{u(x+h)-u(x)}{|h|}$ which satisfies a nonlocal equation of the form \eqref{eq:Main-problem}, driven by a kernel $K^u_h$ satisfying \eqref{eq:Kernel-satisf}, to deduce that $\n u\in C^{0,\a_0}$. This will imply that $ K^u_h\in \scrK^s(\kappa,\a_0, Q_\delta )$, for some $\kappa,\delta >0$. Now Theorem \ref{th:main-th10}$(ii)$ and Theorem \ref{th:Schauder-0-intro}$(ii)$ kick in and yield the result, since ${\mathcal A}_{ K^u_h}$ will be, locally, as regular as $\n u$.
\subsection{Application II: Nonlocal equations on manifolds}\label{ss:NonlocManifold}
Let $\Sigma$ be a Lipschitz hypersurface of $\mathbb{R}^{N+1}$, with $0\in \Sig$. We define the space ${L_s(\Sig)}$ given by the set of functions $u\in L^1_{loc}(\Sig)$ such that
$$
\|u\|_{{L_s(\Sig)}}:=\int_{\Sig}|u(\overline y)| (1+|\overline y|)^{-N-2s}\, d\s(\overline y)<\infty,
$$
where $d\s$ denote the volume element on $\Sig$.
We assume that \begin{equation} \label{eq:integ-hypersurface}
\|1\|_{{L_s(\Sig)}}=\int_{\Sig} (1+|\overline y|)^{-N-2s}\, d\s(\overline y)<\infty. \end{equation}
We note that this condition always holds when $\Sigma$ has finite diameter.
In this section we are interested in the regularity estimates of functions $u\in H^s_{loc} (\Sigma )\cap {L_s(\Sig)}$ satisfying, for all $\Psi\in C^\infty_c({\Sig})$, \begin{equation}\label{eq:-u-weak-non-flat}
\frac{1}{2}\int_{ \Sig} \int_{ \Sig} \frac{(u(\overline x)-u(\overline y))(\Psi(\overline x)-\Psi(\overline y)) }{|\overline x-\overline y|^{N+2s}}\, d\s(\overline x)d\s(\overline y)+\int_{\Sig} V(\overline x) u(\overline x) \Psi(\overline x)\, d\s(\overline x) =\int_{\Sig} f(\overline x) \Psi(\overline x)\, d\s(\overline x),
\end{equation}
where $f,V\in L^1_{loc}(\Sig)$ and $u V\in L^1_{loc}(\Sig)$.\\
\begin{theorem}\label{th:nonloca-surf1} Let $s,\g\in (0,1)$, $N\geq 1$ and $\Sigma$ be a $C^{1,\g}$-hypersurface of $\mathbb{R}^{N+1}$ as above satisfying \eqref{eq:integ-hypersurface}. Let $f,V\in L^p (\Sig)$, for some $p>\frac{N}{2s}$ and $u\in H^s_{loc}(\Sigma )\cap {L_s(\Sig)}$ satisfy \eqref{eq:-u-weak-non-flat}. Then the following estimates hold. \begin{itemize} \item[$(i)$] If $2s\leq 1$, then $$
\|u\|_{C^{2s-N/p}(B_\varrho\cap\Sig) }\leq C (\|u\|_{L^2(B_{2\varrho}\cap\Sig)}+ \|u\|_{{L_s(\Sig)}}+\|f\|_{L^p(\Sig)}). $$ \item[$(ii)$] If $2s-1>\max(\frac{N}{p},\g)$, then $$
\|u\|_{C^{1,\min( 2s-\frac{N}{p}-1 , \g)}(B_\varrho\cap\Sig)}\leq C (\|u\|_{L^2(B_{2\varrho}\cap\Sigma)}+ \|u\|_{{L_s(\Sig)}}+\|f\|_{L^p(\Sig)}), $$
Here $C,\varrho>0$ are constants only depending on $N,s,\g,p $,$\|V\|_{L^p(\Sig)}$, $ \|1\|_{{L_s(\Sig)}}$ and the bound of the local geometry of $\Sigma$ near $0$.
\end{itemize} \end{theorem}
In the case of higher order regularity, we obtain the \begin{theorem}\label{th:nonloca-surf2} Let $s,\a,\g\in (0,1)$, $N\geq 1$ and $\Sigma$ be a $C^{1,\g}$-hypersurface of $\mathbb{R}^{N+1}$ as above satisfying \eqref{eq:integ-hypersurface}. Let $f,V\in C^{0,\a} (\Sig)$ and $u\in H^s_{loc}(\Sigma )\cap {L_s(\Sig)}$ satisfy \eqref{eq:-u-weak-non-flat}. \begin{itemize} \item[$(i)$] If $2s>1$ and $\g\geq \a+2s-1$, then $$
\|u\|_{C^{1,2s-1+\a}(B_\varrho\cap\Sig) }\leq C (\|u\|_{L^2(B_{2\varrho}\cap \Sig)}+ \|u\|_{{L_s(\Sig)}}+\|f\|_{C^{0,\a}(\Sig)}). $$ \item[$(ii)$] If $2s+\a<1$ and $\g\geq \a$, then $$
\|u\|_{C^{0,2s+\a}(B_\varrho\cap\Sig) }\leq C (\|u\|_{L^2(B_{2\varrho}\cap \Sig)}+ \|u\|_{{L_s(\Sig)}}+\|f\|_{C^{0,\a}(\Sig)}). $$ \item[$(iii)$] If $2s=1$ and $\g>\a$, then $$
\|u\|_{C^{1,\a}(B_\varrho\cap\Sig) }\leq C (\|u\|_{L^2(B_{2\varrho}\cap \Sig)}+ \|u\|_{{L_s(\Sig)}}+\|f\|_{C^{0,\a}(\Sig)}). $$
Here $C,\varrho>0$ are constants only depending on $N,s,\g,\alpha $,$\|V\|_{C^{0,\a}(\Sig)}$, $ \|1\|_{{L_s(\Sig)}}$ and the bound of the local geometry of $\Sigma$ near $0$. \end{itemize}
\end{theorem} Here, by the bound of the local geometry of $\Sigma$ near $0$, we mean the $C^{1,\g}$ norm of a local parameterization of $\Sigma$ flattening $B_{\varrho_0}\cap\Sig$, for some $\varrho_0>0$. If $\Sig$ is of class $C^{m+1,\g}$ and $f,V\in C^{m,\a}_{loc}(\Sig)$, then under the same assumptions on $\g$ in Theorem \ref{th:nonloca-surf2}, we have the estimates of $C^{m+2s+\a}$-norm of $u$ as long as $2s+\a\not\in \mathbb{N}$, thanks to Theorem \ref{th:Schauder-0-intro}.\\
Theorem \ref{th:nonloca-surf1} and Theorem \ref{th:nonloca-surf2} are consequences of Theorem \ref{th:main-th10} and Theorem \ref{th:Schauder-0-intro}, respectively, after using a coordinate system that locally flattens $ \Sig$. \\
For $2s>1$, Theorem \ref{th:nonloca-surf1} and Theorem \ref{th:nonloca-surf2} provide regularity estimates for solutions to some nonlocal equation driven by the linearized nonlocal mean curvature operator (i.e. the nonlocal or fractional Jacobi operator) of a set $E$ with constant nonlocal mean curvature (not necessarily bounded). Indeed, consider $\Sig:=\partial E$ a $C^2$-hypersurface of $\mathbb{R}^{N+1}$ with constant nonlocal mean curvature such that $0\in \partial E$ and $\|1\|_{{L_s(\Sig)}}<\infty $. See e.g. \cite{Davila2014B,FFMMM}, the second variation of the fractional perimeter yields the bilinear form ${\mathcal D}_{\Sig}: H^s(\Sig)\times H^s(\Sig)\to \mathbb{R}$, given by $$
{\mathcal D}_{\Sig}(u,v):=\frac{1}{2} \int_{\Sig}\int_{\Sig}\frac{(u(\overline x)-u(\overline y))(v(\overline x)-v(\overline y))}{|\overline x-\overline y|^{N+2s}}\, d\s(\overline y) d\s(\overline x) -\frac{1}{2} \int_{\Sig} V_{\Sig} (\overline x)u(\overline x)v(\overline x)d\s(\overline x), $$ where, letting $\nu_\Sig$ be the unit exterior normal vector field of $\Sig:=\partial E$, $$
V_{\Sig} (\overline x):=\frac{1}{2} \int_{\Sig}\frac{| \nu_\Sig(\overline x)-\nu_\Sig(\overline y)|^2 }{|\overline x-\overline y|^{N+2s}}\, d\s(\overline y). $$
One then defines the \textit{fractional Jacobi operator} as $$ {\mathcal J}_{\Sig} :=\mbL_{\Sig} -V_{\Sig} , $$
where, for $u\in C^{1, 2s-1+\a}_{loc}(\Sigma)\cap {L_s(\Sig)}$, $$
\mbL_{\Sig} u(\overline x):=PV\int_{\Sig}\frac{u(\overline x)-u(\overline y)}{|\overline x-\overline y|^{N+2s}}\, d\s(\overline y) . $$
The \textit{fractional Jacobi fields} are solutions to ${\mathcal J}_\Sig u=0$, and they play an important role in the study of stability of constant nonlocal mean curvature surfaces or fractional area estimates of such surfaces.\\ We observe that if $\Sig$ is a $C^{1,\g}$-hypersurface for some $\g>s$, then $V_\Sig\in C^\g_{loc}(\Sig) $. Moreover we may consider a weak solutions $u\in H^s_{loc}(\O)\cap{L_s(\Sig)}$ to the equation ${\mathcal J}_{\Sig} u=f$ on open subsets $\O$ of $\Sig$, in the sense of \eqref{eq:-u-weak-non-flat}. Hence Theorem \ref{th:nonloca-surf1} and Theorem \ref{th:nonloca-surf2} can be used to obtain regularity estimates of $u$. When $\Sig=S^{N-1}$, then Theorem \ref{th:nonloca-surf2}$(i)$ was proved in \cite{CFW-2017}, using the regularity theory of the fractional Laplacian and the Fredholm theory. Recall that besides the nonlocal minimal surfaces, there exist several nontrivial hypersurfaces with nonzero constant nonlocal mean curvature, see e.g. the survey paper \cite{Fall-CNMC}.
\subsection{Anisotropic $C^{m,\a}$-nonlocal operators}\label{ss:General-nonlocaloperator} As mentioned earlier, in many situations, nonlocal equations provide a wider framework than their local counterpart, since ${{\mathcal A}}_K$ may have anisotropic regularity in its variables. Namely, the spatial variable $x$, the singular variable $r$ and the angular variable might have different qualitative properties. This affects the local behavior of the solutions. First note that the class of operators ${\mathcal L}_K$ falls in the class of nonlocal operators generated by a L\'evy measure $\nu_x$. In particular, the map $z\mapsto K(x, x+z) $ is the density of a L\'evy measure $\nu_x$ and thus does not necessarily posses any regularity. If the L\'evy measure is symmetric and stable, then see \cite{RS2}, $\nu_x(r E)=r^{N-1}dr a(E)$ for $E\subset S^{N-1}$. Under fairly general assumptions on the spectral measure $a$ on $S^{N-1}$ (not depending on $x$), optimal interior and boundary regularity were proved by Ros-Oton and Serra in \cite{RS2}. The papers \cite{KRS,KS,DK} obtained also regularity estimates provided $a$ is absolutely continuous with respect to the Lebesgue measure on $S^{N-1}$ only on an open set of positive measure.\\
To capture this possible anisotropic regularity of ${\mathcal A}_K$ in its variables, we introduce a new class of fractional order nonlocal operators which are much larger than the class of $C^{m,\a}$-nonlocal operators introduced above. \\ In the following, for $\delta >0$, we define \begin{equation}\label{eq:def-Q-d-Q-infty} Q_\delta :=B_\delta \times [0,\delta )\qquad\textrm{ and } \qquad Q_\infty:=\mathbb{R}^N \times [0,\infty). \end{equation} We define the space $C^{m,\a}(Q_\delta )\times L^\infty(S^{N-1} )$ by the set of functions $A\in L^\infty(Q_\delta \times S^{N-1})$ such that, for every $\th\in S^{N-1}$, the map $(x,r)\mapsto A(x,r,\th)$ belongs to $C^{m,\a}(Q_\delta )$ and \begin{equation}\label{eq:A-Cm12}
\|A\|_{C^{m,\a}(Q_\delta )\times L^\infty(S^{N-1} )}:= \sup_{\th\in S^{N-1}}\|A(\cdot,\cdot,\th)\|_{C^{m,\a}(Q_\delta )}<\infty . \end{equation} For $\t\geq 0$, the space $ {\mathcal C}^0_{\t}(Q_\delta )\times L^\infty(S^{N-1} )$ is given by the the set of function $A\in L^\infty(Q_\delta \times S^{N-1})$ such that $$
\| A\|_{ L^{\infty}_{\t}(Q_\delta )\times L^\infty(S^{N-1} )}:=\sup_{\th\in S^{N-1}} \sup_{x\in B_\delta , r\in (0,\delta )} \frac{| A(x,r,\th)| }{r^{\t}}<\infty $$ and $$
[ A]_{ {\mathcal C}^0_{\t}(Q_\delta )\times L^\infty(S^{N-1} )}:= \sup_{ \th\in S^{N-1} } \sup_{x\not=y\in B_\delta , r\in (0,\delta )} \frac{| A(x,r,\th) - A(y,r,\th) |}{\min(r, |x-y|)^\t} <\infty . $$ The space ${\mathcal C}^{m}_{\t}(Q_\delta ) \times L^\infty(S^{N-1} )$ is defined as the set of functions $A\in C^{m,0}(Q_\delta )\times L^\infty(S^{N-1} )$ such that \begin{equation} \label{eq:A-Cm12-tau}
\| A\|_{ {\mathcal C}^{m}_{\t}(Q_\delta )\times L^\infty(S^{N-1} )}:= \sup_{\g\in \mathbb{N}^N, |\g|\leq m} \|\partial_x^\gamma A\|_{L^{\infty}_{\t}(Q_\delta ) \times L^\infty(S^{N-1} )}+ \sup_{\g\in \mathbb{N}^N, |\g|\leq m} [\partial_x^\g A ]_{{\mathcal C}^0_{\t}(Q_\delta )\times L^\infty(S^{N-1} ) }<\infty. \end{equation} This section is concerned with optimal H\"older estimates for nonlocal equation driven by the operator ${\mathcal L}_K$ with coefficient ${\mathcal A}_K$ in the spaces defined above.
\begin{definition}\label{def:Kernel-not-reg-the}
Let $\a\in [0,1)$, $\t\in[ 0,1]$, $m\in \mathbb{N}$ and $\kappa>0$. For $\delta \in (0,\infty]$, we define $\widetilde \scrK_\t^s(\kappa,m+\a,Q_\delta )$ by the set of kernels $K: \mathbb{R}^N\times \mathbb{R}^N\to [-\infty,+\infty] $ satisfying \eqref{eq:Kernel-satisf} and $$ \begin{aligned}
& (iii) \|{{\mathcal A}}_{K} \|_{C^{m,\a}(Q_\delta )\times L^\infty(S^{N-1} ) }+ \|{\mathcal A}_{o,K} \|_{{\mathcal C}^{m}_{\t}(Q_\delta ) \times L^\infty(S^{N-1} ) }\leq \frac{1}{\kappa} ,\\
&(iv) {\mathcal A}_{o,K}(x,0,\th)=0\qquad\textrm{ for all $(x,\th)\in B_\delta \times S^{N-1}$,}
\end{aligned} $$ where \begin{equation}\label{eq:def-ti-l-oK}
{{\mathcal A}}_{o,K}(x,r,\th) :=\frac{1}{2}\{ {{\mathcal A}}_{K}(x,r,\th) - {{\mathcal A}}_{K}(x,r,-\th) \} \end{equation}
and ${\mathcal A}_K(\cdot,\cdot,\th)$ is a continuous extension of $(x,r)\mapsto r^{N+2s}K(x,x+r\th)$ on $Q_\delta $ for all $\th\in S^{N-1}$.
\end{definition}
The simple model case for the class of operators in Definition \ref{def:Kernel-not-reg-the} is the anisotropic fractional Laplace operator, with kernel $K(x,y)=a((x-y)/|x-y|)|x-y|^{-N-2s}$ and $a\in L^\infty(S^{N-1})$ is even but not necessarily continuous. In this case, ${\mathcal A}_K(x,r,\th)=a(\th)$, so that $K\in\widetilde \scrK_\t^s(\kappa,m,Q_\delta )$ for all $m\in \mathbb{N}$. As an example, a prototype energy functional can be an anisotropic integral energy functional, generalizing \eqref{eq:cJs}, given by \begin{equation} \label{eq:cJs-Gen}
{\mathcal J}_{s}(u):= (1-s) \int_{\mathbb{R}^{2N}\setminus (\mathbb{R}^N\setminus \O)^2} F\left( \frac{x-y}{|x-y|} ,\frac{u(x)-u(y)}{|x-y|} \right) |x-y|^{-N-2s+2}\, dxdy ,
\end{equation} where $F: S^{N-1}\times \mathbb{R}\to \mathbb{R}$ satisfies $\kappa\leq \partial^2_z F(\th,z)\leq \frac{1}{\kappa}$. The results in the present section provide smoothness of a critical point $u\in H^s(\mathbb{R}^N)$ to ${\mathcal J}_s$ defined in \eqref{eq:cJs-Gen}, provided $z\mapsto F(\cdot, z)$ is smooth. Indeed, as above, the difference quotient $\frac{u(\cdot+h)-u(\cdot)}{|h|}$ solves an equation like \eqref{eq:Main-problem}, with $K\in \widetilde \scrK_{\a}^s(\kappa,m+\a,Q_\delta )$, provided $u\in C ^{m+1,\a}(\O)$. \\ We observe that $\scrK^s(\kappa,m+\a,Q_\delta ) \subset \widetilde \scrK_\t^s(\kappa,m+\a,Q_\delta )$ for all $\t\leq \a$ and since ${\mathcal A}_{o,K}(x,0,\th)=0$, we have that $\widetilde \scrK^s_0(\kappa,m+\a,Q_\delta ) = \widetilde \scrK_\a^s(\kappa,m+\a,Q_\delta )$. Moreover, we have the following interesting property on the set $\widetilde \scrK_\t^s(\kappa,m+\a,Q_\infty)$ concerning scaling and translations. Indeed, for $K\in \widetilde \scrK_\t^s(\kappa,m+\a,Q_\infty)$, $\rho\in (0,1)$ and $z\in \mathbb{R}^N$, letting $K_{z,\rho}(x,y):=\rho^{N+2s}K(\rho x+z, \rho y+z) $, we then have that ${\mathcal A}_{K_{z,\rho}}(x,r,\th)={\mathcal A}_{K}(\rho x+z,\rho r,\th)$ and thus $K_{z,\rho}\in \widetilde \scrK_\t^s(\kappa,m+\a,Q_\infty)$.
The kernels in $\widetilde \scrK_\t^s(\kappa,m+\a, Q_{\delta })$ yield, in many cases, similar regularity estimates as those in $ \scrK^s(\kappa,m+\a, Q_{\delta })$, stated above, provided some global regularity/behavior of $u$ is a priori known. \\
Our first main result in this section is the following.
\begin{theorem}\label{th:main-th1} Let $s\in (0,1)$, $N\geq 1$, $\kappa>0$ and $\a\in (0,1)$. Let $K\in \widetilde \scrK_0^s(\kappa,\a, Q_2)$, $u\in H^s(B_2)\cap {L_s(\mathbb{R}^N)}$ and $V, f\in L^p(B_2)$, for some $p>N/(2s)$, satisfy
$$
{\mathcal L}_K u+ Vu = f\qquad \textrm{in $B_2$}.
$$
\begin{itemize}
\item[$(i)$] If $2s\leq 1$, then there exists $C=C(s,N,\kappa, \a,p, \|V\|_{L^p(B_2)})>0$ such that
$$
\|u\|_{C^{0,2s-\frac{N}{p}}(B_1)}\leq C(\|u\|_{L^2(B_2)}+\|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{L^p(B_2)}). $$
\item[$(ii)$] If $2s-1>\max(\frac{N}{p},\a)$, then there exists $C=C(s,N,\kappa, \a,p, \|V\|_{L^p(B_2)})>0$ such that
$$
\|u\|_{C^{1,\min (2s-\frac{N}{p}-1,\a)}(B_1)}\leq C(\|u\|_{L^2(B_2)}+\|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{L^p(B_2)}). $$
\end{itemize}
\end{theorem}
Our next result is concerned with $C^{m+2s+\a}$ Schauder estimates.
\begin{theorem}\label{th:Schauder-0} Let $N\geq 1$ and $s\in (0,1)$. Let $\kappa>0$, $\a\in (0,1)$ and $m\in \mathbb{N}$. Let $K\in \widetilde \scrK_{\b}^s(\kappa,m+ \a, Q_2)$, with $\b=\min (\a+(2s-1)_+,1)$. Let $u\in H^s(B_2)\cap L_s(\mathbb{R}^N)$ and $f\in C^{m,\a}(B_2)$ such that
$$
{\mathcal L}_K u= f\qquad \textrm{in $B_2$}.
$$
\begin{itemize}
\item[$(i)$] If $u\in C^{m,\a}(\mathbb{R}^N)$ and $2s+\a<1$, then
$$
\|u\|_{C^{m,2s+\a}(B_1)}\leq C(\|u\|_{C^{m,\a}(\mathbb{R}^N)}+ \|f\|_{C^{m,\a}(B_2)}). $$
\item[$(ii)$] If $u\in C^{m,\a}(\mathbb{R}^N)$, $2s\not=1$ and $1<2s+\a<2$, then
$$
\|u\|_{C^{m+1, 2s+\a-1}(B_1)}\leq C(\|u\|_{C^{m,\a}(\mathbb{R}^N)}+ \|f\|_{C^{m,\a}(B_2)}). $$
\item[$(iii)$] If $u\in C^{m,\a}(\mathbb{R}^N)$, $2<2s+\a$ and $K \in \widetilde \scrK_0^{s}(\kappa,m+ 2s-1+ \a, Q_2)$, then
$$
\|u\|_{C^{m+2, 2s+\a-2}(B_1)}\leq C(\|u\|_{C^{m,\a}(\mathbb{R}^N)}+ \|f\|_{C^{m,\a}(B_2)}). $$
\item[$(iv)$] If $u\in C^{m,\a}(\mathbb{R}^N)$, $2s=1$ and $K \in \widetilde \scrK_\t^{s}(\kappa,m+ \a, Q_2)$, for some $\t>\a$, then
$$
\|u\|_{C^{m+1,\a}(B_1)}\leq C(\|u\|_{C^{m,\a}(\mathbb{R}^N)}+ \|f\|_{C^{m,\a}(B_2)}). $$
\end{itemize}
If moreover $\|{{\mathcal A}}_K\|_{C^{m,\a}(Q_2 \times S^{N-1})}\leq \frac{1}{\kappa}$, then we can replace $ \|u\|_{C^{m,\a}(\mathbb{R}^N)}$ with $ \|u\|_{L^\infty(\mathbb{R}^N)} $. Here $C=C(N,s,\kappa,\a, m,\t).$
\end{theorem}
We point out the remarkable differences between the last assertion in Theorem \ref{th:Schauder-0} and the results in Theorem \ref{th:Schauder-0-intro}. Indeed, in the former, ${\mathcal A}_K$ is only required to be in $C^{m,\a} (Q_2\times S^{N-1})$, when $2s+\a<2$, instead of $C^{m,\a+(2s-1)_+} (Q_2 \times S^{N-1})$ which was assumed in the latter. Moreover, Theorem \ref{th:Schauder-0}-$(iv)$, for $s=1/2, $ provides the optimal estimate which covers the case ${\mathcal L}_K= (-\D)^s_a$, the anisotropic fractional Laplacian i.e. when $K(x,y)=a((x-y)/|x-y|)|x-y|^{-N-2s}$, while Theorem \ref{th:Schauder-0-intro} does not if $a$ is not smooth enough. In fact the results in Theorem \ref{th:Schauder-0} were known for the anisotropic fractional Laplacian when $a$ is a measure on the unit sphere $S^{N-1}$, see Ros-Oton and Serra \cite{RS2} and when $a\in C^\infty(S^{N-1})$, see Grubb \cite{ Grubb2}. \\
Interior regularity and Harnack inequality for linear and fully nonlinear nonlocal equations have been intensively investigated in last decades by many authors, see e.g. \cite{FK,BL,Kassm,Barrios,CS1,CS2,CS3,BC,KM1,Jin-Xiong,Kriventsov,SS,Serra-OK, {Silv-1},DK,Ab-L} and the references therein. \\
Next, we observe that Theorem \ref{th:main-th10} and \ref{th:Schauder-0-intro} are immediate consequences of Theorem \ref{th:main-th1} and \ref{th:Schauder-0}, respectively. The proof of Theorem \ref{th:main-th1} and \ref{th:Schauder-0} uses a blow up analysis and compactness method for weak and classical solutions, partly inspired by \cite{Serra-OK} and \cite{Fall-Reg}, see also \cite{Serra,RS2,RS4,Ros-Real,Ros-Real-2} for translation invariant problems. Indeed, we use a fine scaling argument to balance, in an optimal manner, the norm of the right hand side and the homogeneity of the equation. The scaling parameter is chosen so that the limit of the rescaled solution, after subtracting a polynomial, satisfies an equation for which all solutions with such growth are explicitly known, thanks to a Liouville type theorem. To obtain H\"older, gradient and second order derivative estimates, the subtracted polynomial are, respectively given by the projection, with respect to the $L^2(B_r)$ scalar product, of the weak solution $u$ on constant functions, affine functions and second order polynomials. More precisely, our primary goal is to show the growth estimates (or Taylor expansion in $L^2$-sense) $$
\|u-P_r\|_{L^2(B_r)}\leq C r^{\frac{N}{2}+\textrm{deg}(P_r)+\g},
$$ where $P_r$ is a suitable polynomial and the parameter $\g\in (0,1)$ is determined by the regularity of the entries $V,f$ and ${\mathcal A}_K$ the coefficient of the operator. The above expansion leads to $u\in C^{m,\gamma}$, with $m=\textrm{deg}(P_r)$.\\
To carry over the blow up argument and to use compact Sobolev embedding or the Arzel\'{a}-Ascoli theorem, after subtracting polynomials, rescaling and normalization, it is necessary to derive a priori H\"older estimate for functions $v$ solving the more general equation \begin{equation} \label{eq:intro-100} {\mathcal L}_Kv +{\mathcal L}_{K'} U=F \qquad\textrm{ in $\O$.} \end{equation} Actually, in the counter part of \eqref{eq:intro-100} in the local case reads as \begin{equation} \label{eq:intro-10000} -\textrm{div}(A(x)\n v)+ \textrm{div} U=F, \end{equation} for some potential $U$. The study of \eqref{eq:intro-100} is typically essential for the proof of Theorem \ref{th:main-th1}-$(ii)$, where $U=p_r$ is a first order polynomial and ${\mathcal L}_{K'}$ is a non-translation invariant operator. Recall here that obtaining gradient estimates for solutions to equations involving divergence operators is more subtle than those involving operators in non-divergence form, since the latter annihilate affine functions, while the former do not and so the study of \eqref{eq:intro-10000} becomes useful.
The same difficulty is of course faced here since we are dealing with non-translation invariant variational solutions. \\
The core of the paper, from which we derive all the results, is Proposition \ref{prop:bound-Kato-abstract} below, where we prove H\"older continuity of the solutions to \eqref{eq:intro-100}, under mild regularity assumptions on $K,K', U$ and $F$, and we believe that the argument of proof and the result itself could be of independent interest. \\ We finally remark that the Schauder estimates in the present paper remains stable as $s\to 1$, provided we replace the kernel $K$, with $(1-s)K$ and $\a$ is such that $2<2s_0+\a$, for some $s_0\in (0,1)$.\\
The paper is organized as follows. In Section \ref{s:NotPrem}, we collect some preliminary result and notations. Section \ref{s:AprioriEstim} contains the regularity estimates for solutions to \eqref{eq:intro-100}. Now Theorem \ref{th:main-th1}-$(ii)$ is proved in Section \ref{s:GradEstim} and Theorem \ref{th:Schauder-0} in Section \ref{s:Shaud}. Finally the proof of the main results are gathered in Section \ref{s:proofMainResults}.
\section{Notations and preliminary results}\label{s:NotPrem}
\subsection{Notations} In this paper, the ball centred at $z\in\mathbb{R}^N$ with radius $r>0$ is denoted by $B(z,r)$ and $B_r:=B_r(0)$. Here and in the following, we let $\vp_1 \in C^\infty_c(B_2)$ such that $\vp_1 \equiv 1$ on $B_{1}$ and $0\leq \vp_1\leq 1$ on $\mathbb{R}^N$. We put $\vp_R(x):=\vp(x/R)$. For $b\in L^\infty(S^{N-1})$, we define $
\mu_b(x,y)= |x-y|^{-N-2s} b\left(\frac{x-y}{|x-y|} \right).$\\ Given $\s>0$, we define the space $$
L_\s(\mathbb{R}^N):=\left\{u\in L^1_{loc}(\mathbb{R}^N)\,:\, \|u\|_{L_\s(\mathbb{R}^N)}:= \int_{\mathbb{R}^N}{|u(x)|} (1+|x|^{N+2\s})^{-1}\,dx <\infty \right\}. $$
Throughout this paper, for the seminorm of the fractional Sobolev spaces, we adopt the notation $$
[u]_{H^s(\O)}:=\left(\int_{\O\times\O} {|u(x)-u(y)|^2}\mu_1(x,y)\, dxdy\right)^{1/2}. $$ We will, sometimes use the notation $$
[u]_{H^s_{K}(\O)}:=\left(\int_{\O\times\O} {|u(x)-u(y)|^2}|K(x,y)|\, dxdy\right)^{1/2}, $$ for a function $K:\O\times \O\to [-\infty,+\infty]$. For the H\"older and Lipschitz seminorm, we write $$
[u]_{C^{0,\a}(\O)}:=\sup_{x\not=y\in\O} \frac{|u(x)-u(y)|}{|x-y|^\a}, $$ for $\a\in (0,1]$. If there is no ambiguity, when $\a\in (0,1)$, we will write $[u]_{C^{\a}(\O)}$ instead of $[u]_{C^{0,\a}(\O)}$.
If $m\in \mathbb{N}$ and $\a\in (0,1)$, the H\"older space $\|u\|_{C^{m,\a}(\O)}$ is given by the set of functions in $C^m(\O)$ such that $$
\|u\|_{C^{m+\a}(\O)}:=\|u\|_{C^{m,\a}(\O)}= \sup_{\g\in \mathbb{N}^N, |\g|\leq m} \| \partial^\gamma u\|_{ L^{\infty}(\O)}+ \sup_{\g\in \mathbb{N}^N, |\g|= m} \|\partial^\g u\|_{ C^{\a}(\O)}<\infty . $$
Letting $u\in L^1_{loc}(\mathbb{R}^N)$, the mean value of $u$ in $ B_r(z)$ is denoted by $$
u_{B_r(z)}=(u)_{B_r(z)}:=\frac{1}{|B_r|}\int_{B_r(z)}u(x)\, dx. $$
For $\a\in [0,1]$, $h\in \mathbb{R}^N\setminus \{0\}$ and $f\in C^{0,\a}_{loc}(\mathbb{R}^N)$, we define \begin{equation} \label{eq:def-f-h-alph}
f_{h,\a}(x):=\frac{f(x+h)-f(x)}{|h|^\a}. \end{equation}
\subsection{Preliminary results} We gather in this paragraph some results which we will frequently use in the following of the paper.
Let
$K: \mathbb{R}^N\times \mathbb{R}^N\to [0,\infty]$ satisfy the following properties:
\begin{equation} \label{eq:K-Kernel-satisf}
\begin{aligned} (i)\,& K(x,y)=K(y,x) \qquad\textrm{ for all $x,y\in\mathbb{R}^N$, }\\
(ii)\,& \kappa \mu_1(x,y)\leq K(x,y)\leq \frac{1}{\kappa} \mu_1(x,y) \qquad\textrm{ for all $x, y\in\mathbb{R}^N$.}
\end{aligned}
\end{equation} For $\a'\geq 0$, we let $K':\mathbb{R}^N\times \mathbb{R}^N\to [-\infty,+\infty]$ satisfy \begin{equation} \label{eq:K'-Kernel-satisf}
\begin{aligned} (i)\,& K'(x,y)=K'(y,x) \qquad\textrm{ for all $x, y\in\mathbb{R}^N$, }\\
(ii)\,& |K'(x,y)|\leq \frac{1}{\kappa} (|x|+|y|+1)^{\a'} \mu_1(x,y) \qquad\textrm{for all $x, y\in \mathbb{R}^N$}.
\end{aligned}
\end{equation}
Let $U\in H^s_{loc}(\O)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ and $f\in L^1_{loc}(\mathbb{R}^N)$. We say that $u\in H^s_{loc}(\O)\cap {L_s(\mathbb{R}^N)}$ is a (weak) solution to \begin{equation}\label{eq:cL_K-eq-V-f-K'} {\mathcal L}_{K} u+ {\mathcal L}_{K'}U= f \qquad\textrm{ in $ \O$,} \end{equation} if, for every $\psi\in C^\infty_c(\O)$, \begin{align*} \int_{\mathbb{R}^{2N}}(u(x)-u(y))(\psi(x)-\psi(y))K(x,y)\,dxdy&+ \int_{\mathbb{R}^{2N}}(U(x)-U(y))(\psi(x)-\psi(y))K'(x,y)\,dxdy\\
&= \int_{\mathbb{R}^N} f(x)\psi(x)\, dx. \end{align*} We note that each of the terms in the above identity is finite.
For $\b\in [0,2s)$, we define the Morrey space ${\mathcal M}_\b$ by the set of functions $f \in L^1_{loc} (\mathbb{R}^N)\ $ such that $$
\|f\|_{{\mathcal M}_\b}:= \sup_{\stackrel{x\in \mathbb{R}^N}{r\in (0,1)}} r^{\b-N}\int_{ B_r(x)} |f(y)| \, dy<\infty, $$
with ${\mathcal M}_0:=L^\infty(\mathbb{R}^N)$, and we note that $ \|f\|_{{\mathcal M}_{N/p}}\leq C(N,p) \|f\|_{L^p(\mathbb{R}^N)}$. We have the following coercivity property, see \cite{Fall-Reg}, \begin{equation}\label{eq:coerciv}
\||f|^{1/2}v\|_{L^2(\mathbb{R}^N)}^2\leq C(N,s,\b) \|f\|_{{\mathcal M}_\b}\|v\|_{H^s(\mathbb{R}^N)}^2 \qquad\textrm{for all $v\in H^s(\mathbb{R}^N)$.} \end{equation}
We prove our a priori estimates for right hand in ${\mathcal M}_\b$. Recall that ${\mathcal M}_{N/p}$ contains strictly $\|f\|_{L^p(\mathbb{R}^N)} $. \\
The following energy estimate can be seen as a nonlocal Caccioppoli inequality. \begin{lemma} \label{lem:from-caciopp-ok} Let $N\geq 1$, $s\in (0,1)$ and $\kappa>0$. We consider $K$ satisfying \eqref{eq:K-Kernel-satisf} and $K'$ satisfying \eqref{eq:K'-Kernel-satisf}, for some $\a'\geq 0$. Let $v\in H^s (\mathbb{R}^N) $ and $U\in H^s_{loc}(\mathbb{R}^N)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ and $f\in{\mathcal M}_\b$ satisfy \begin{equation}\label{eq:Dsv-eq-V-f} {\mathcal L}_{K} v+ {\mathcal L}_{K'}U = f \qquad\textrm{ in $ B_{2R}$.} \end{equation} Then for every $\e>0$, there exist $\overline C=\overline C(s,N,\kappa,R)$ and $C=C(\e, s,N, \kappa,R)$ such that \begin{align*}
\left \{\kappa- \varepsilon \overline C \|f\|_{{\mathcal M}_\b} \right\} &\int_{\mathbb{R}^N\times \mathbb{R}^N}(v(x)-v(y))^2 \vp_R^2(y) \mu_1(x,y)\,dxdy\\
&\leq C( \|f\|_{ {\mathcal M}_\b} +1) \|v \|_{L^2(\mathbb{R}^N)}^2 + C \|f\|_{{\mathcal M}_\b} \|\vp_R\|^2_{H^s(\mathbb{R}^N)} \nonumber\\
&\quad + C[U]_{H^s_{K'}(B_{4R})} ^2 +C\int_{\mathbb{R}^N} \vp_R^2(y)|v(y)| \left( \int_{ \mathbb{R}^N\setminus B_{4R}}|U(x)-U(y)||K'(x,y)|\,dx\right)dy . \end{align*}
\end{lemma}
\begin{proof}
Applying \cite[Lemma 9.1]{Fall-Reg}, we get \begin{align}\label{eq:from-cacciop-to-coerciv}
(\kappa-\e)\int_{\mathbb{R}^{2N}}(v(x)-v(y))^2\vp^2_R(y)& \mu_1(x,y)\,dxdy\leq \int_{\mathbb{R}^N} |f(x)| |v(x)| \vp_R^2(x)\, dx \nonumber\\
&+ C \int_{\mathbb{R}^{2N}}(\vp_R(x)-\vp_R(y))^2v^2(y) \mu_1(x,y)\,dxdy \nonumber\\
& + \int_{\mathbb{R}^{2N}}|U(x)-U(y)||\vp_R^2(x) v(x)-\vp_R^2(y)v(y)||K'(x,y)|\,dxdy .
\end{align} We now estimate \begin{align} \label{eq:from-cacciop-to-coerciv1}
\int_{\mathbb{R}^N\times \mathbb{R}^N}|U(x)-U(y)||\vp_R^2(x)& v(x)-\vp_R^2(y)v(y)||K'(x,y)|\,dxdy \nonumber\\
& = \int_{B_{4R}\times B_{4R}}|U(x)-U(y)||\vp_R^2(x) v(x)-\vp_R^2(y)v(y)||K'(x,y)|\,dxdy \nonumber\\
&+ 2\int_{\mathbb{R}^N} \vp_R^2(y)|v(y)| \left( \int_{ \mathbb{R}^N\setminus B_{4R}}|U(x)-U(y)||K'(x,y)|\,dx\right)dy \nonumber\\
&\leq \e/\kappa (2(4R)+1)^{\a'} [\vp_R^2 v]_{H^s(B_{4R})}^2 + C[U]_{H^s_{K'}(B_{4R})} ^2 \nonumber\\
& +C\int_{\mathbb{R}^N} \vp_R^2(y)|v(y)| \left( \int_{ \mathbb{R}^N\setminus B_{4R}}|U(x)-U(y)||K'(x,y)|\,dx\right)dy. \end{align}
We recall that \begin{equation} \label{eq:Op-vp2}
\int_{\mathbb{R}^N}(\vp_1(x)-\vp_1(y))^2\mu_1(x,y)\,dy\leq C(N,s)(1+|x|^{-N-2s}) \qquad \textrm{ for every $x\in \mathbb{R}^N$.} \end{equation} Therefore
\begin{align*} [\vp_R^2 v]_{H^s(B_{4R})}^2&\leq 2\int_{\mathbb{R}^{2N}}(v(x)-v(y))^2\vp^4_R(y) \mu_1(x,y)\,dxdy+ 2 \int_{\mathbb{R}^{2N}}(\vp_R^2(x)-\vp_R^2(y))^2v^2(y) \mu_1(x,y)\,dxdy\\
&\leq 2\int_{\mathbb{R}^{2N}}(v(x)-v(y))^2\vp^2_R(y) \mu_1(x,y)\,dxdy+ \|v \|^2_{L^2(\mathbb{R}^N)} . \end{align*}
Using this in \eqref{eq:from-cacciop-to-coerciv1}, we get \begin{align} \label{eq:from-cacciop-to-coerciv00}
& \int_{\mathbb{R}^N\times \mathbb{R}^N}|U(x)-U(y)||\vp_R^2(x) v(x)-\vp_R^2(y)v(y)||K'(x,y)|\,dxdy \nonumber\\
& \leq \varepsilon \overline C \int_{\mathbb{R}^{2N}}(v(x)-v(y))^2\vp^2_R(y) \mu_1(x,y)\,dxdy+ C \|v \|^2_{L^2(\mathbb{R}^N)} \nonumber\\
&+ C[U]_{H^s_{K'}(B_{4R})} ^2 +C\int_{\mathbb{R}^N} \vp_R^2(y)|v(y)| \left( \int_{ \mathbb{R}^N\setminus B_{4R}}|U(x)-U(y)||K'(x,y)|\,dx\right)dy. \end{align} Next, from \eqref{eq:coerciv}, Young's inequality and \eqref{eq:Op-vp2}, we deduce that
\begin{align*}
\int_{\mathbb{R}^N} | f(x)| |\vp_R(x)|^2|v(x)| \,dx
& \leq \varepsilon \overline C \|f\|_{{\mathcal M}_\b} \int_{\mathbb{R}^{2N}}(v(x)-v(y))^2\vp^2_R(y) \mu_1(x,y)\,dxdy\\
& + C \|f\|_{{\mathcal M}_\b} \|v \|^2_{L^2(\mathbb{R}^N)} + C \|f\|_{{\mathcal M}_\b} \|\vp_R\|^2_{H^s(\mathbb{R}^N)}. \end{align*}
Using this and \eqref{eq:from-cacciop-to-coerciv00} in \eqref{eq:from-cacciop-to-coerciv}, we get the result. \end{proof} We state the following result.
\begin{lemma} \label{lem:convergence-very-weak} Let $N\geq 1$, $s\in (0,1)$ and $\kappa>0$. We consider $K$ satisfying \eqref{eq:K-Kernel-satisf} and $K'$ satisfying \eqref{eq:K'-Kernel-satisf}, for some $\a'\geq 0$. Let $v\in H^s (\mathbb{R}^N) $ and $U\in H^s_{loc}(\mathbb{R}^N)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ and $f\in{\mathcal M}_\b$ satisfy $$ {\mathcal L}_{K} v+ {\mathcal L}_{K'}U = f \qquad\textrm{ in $ B_{2R}$.} $$
Then there exists $C=C(N,s,\kappa,\a',R)$ such that for every $\psi\in C^\infty_c(B_{R} )$, we have
\begin{align}
& \left|\int_{\mathbb{R}^{2N}}(v(x)-v(y))(\psi(x)-\psi(y))K(x,y)\, dxdy \right| \leq C \|f\|_{{\mathcal M}_\b} \left( 1 + \|\psi\|_{H^s(\mathbb{R}^N)}^2 \right) \label{eq:express1}\\
&+ C [U]_{H^s_{K'}(B_{4R})} [\psi]_{H^s(B_{4R})} + C\int_{\mathbb{R}^N} | \psi(y)| \left( \int_{ \mathbb{R}^N\setminus B_{4R}}|U(x)-U(y)||K'(x,y)|\,dx\right)dy. \label{eq:express2}
\end{align}
\end{lemma}
\begin{proof} Using the weak formulation of the equation and \eqref{eq:coerciv}, we get the expression on the left hand side in \eqref{eq:express1}. Now expression \eqref{eq:express2} appears after decomposing the domain of integration and using H\"older's inequality as in the beginning of the proof of Lemma \ref{lem:from-caciopp-ok}. \end{proof}
We close this section with the following result. \begin{lemma}\label{lem:estim-G_Ku} Let $K$ satisfy \eqref{eq:Kernel-satisf}$(i)$-$(ii)$. Let $v\in H^s_{loc}(B_{2R})\cap {L_s(\mathbb{R}^N)} $ and $f\in L^1_{loc}(\mathbb{R}^N)$ satisfy $$ {\mathcal L}_{K} v= f \qquad\textrm{ in $ B_{2R} $,} $$ for some $R>0$. We let $v_R:=\vp_R v$. Then \begin{equation}\label{eq:v-cat-vpR} {\mathcal L}_{K} v_R = f+ G_{K,v,R} \qquad\textrm{in $B_{R/2} $}, \end{equation}
where \begin{equation} \label{eq:def-GKvR} G_{K,v,R}(x)= \int_{\mathbb{R}^N} v(y) (\vp_R(x)-\vp_R(y)) K(x,y) \, dy. \end{equation} Moreover, $G_{K,v,R}$ satisfies the following properties. \begin{itemize} \item[$(i)$] There exists $C=C(N,s, R)$ such that $$
\|G_{K,v,R}\|_{L^\infty(B_{R/2})}\leq C\sup_{x\in B_{R/2}}\int_{|y|\geq R}|v(y)| |K(x,y)|\, dy. $$
\item[$(ii)$] If $v\in C^{k+\a}(\mathbb{R}^N)$ and $ \|{{\mathcal A}}_K \|_{C^{k,\a}(Q_{\infty} )\times L^\infty(S^{N-1})}\leq c_0$, then there exists $C=C(N,s,\a,c_0, R,k)$ such that $$
\|G_{K,v,R}\|_{C^{k+\a}(B_{R/2})}\leq C \|v\|_{C^{k+\a}(\mathbb{R}^N)}. $$
\item[$(iii)$] If $\|{{\mathcal A}}_K\|_{C^{k+\a}( Q_\infty\times S^{N-1})}\leq c_0$, then there exists $C=C(N,s,\a,c_0, R,k)$ such that $$
\|G_{K,v,R}\|_{C^{k+\a}(B_{R/4})}\leq C \|v\|_{{L_s(\mathbb{R}^N)}} . $$
\end{itemize}
\end{lemma}
\begin{proof} For \eqref{eq:v-cat-vpR}, see \cite[Lemma 9.2]{Fall-Reg}. Statement $(i)$ follows easily, thanks to the definition of $\vp_R$.
To prove $(ii)$, we write \begin{align*} G_{K,v,R}(x)= \int_{S^{N-1}}\int_0^\infty v(x+r\th)(1-\vp_R(x+r\th)) {{\mathcal A}}_{K}(x,r,\th) r^{-1-2s}\, dr d\th. \end{align*} Since $1-\vp_R(x+ r\theta )=0 $ for all $x\in B_{R/2}$, $r\in (0,R/2)$ and $\th\in S^{N-1}$, then $(ii)$ follows.\\ To prove $(iii)$, we note that
\begin{align*} G_{K,v,R}(x)&:= \int_{\mathbb{R}^N} v(y) (\vp_R(x)-\vp_R(y)) K(x,y) \, dy\\
&= \int_{|y|\geq R} v(y) (1-\vp_R(y)) {{\mathcal A}}_{K}(x,|x-y|,(x-y)/|x-y|)|x-y|^{-N-2s} \, dy. \end{align*}
We recall that (see e.g. \cite{FW}) for every $x_1,x_2,y\in\mathbb{R}^N$, $\varrho>0$ and $\a\in (0,1)$, $$
||x_1-y|^{-\varrho}- |x_2-y|^{-\varrho} |\leq C(\a,\varrho)|x_1-x_2|^{\a}\{
|x_1-y|^{-(\varrho+\a)}+ |x_2-y|^{-(\varrho+\a)} \}. $$ Hence for all $x_1,x_2\in B_{R/2}$, $y\in \mathbb{R}^N\setminus B_R$, $\varrho\geq N+2s$ and $\a\in (0,1)$, we get $$
||x_1-y|^{-\varrho}- |x_2-y|^{-\varrho} |\leq C(\a,\varrho,R)|x_1-x_2|^{\a} |y|^{-N-2s}. $$ Using this and the Leibniz formula for higher order derivatives of the product of functions, we get $(iii)$. \end{proof}
\section{A priori estimates} \label{s:AprioriEstim}
In this section, we prove a priori estimates for solutions to \eqref{eq:cL_K-eq-V-f-K'}, provided ${\mathcal L}_K$ is close to the translation invariant operator ${\mathcal L}_{\mu_a}$, with $a: S^{N-1}\to \mathbb{R}$ satisfies \begin{equation}\label{eq:def-a-anisotropi} a(-\theta)=a(\theta) \qquad\textrm{ and } \qquad\kappa \leq a(\theta)\leq \frac{1}{\kappa}\ \qquad\textrm{ for all $\theta\in S^{N-1}$}. \end{equation} We now recall two results from \cite{Fall-Reg} that will be needed in the following of the paper. \begin{lemma} \label{lem:Ds-a-n--to-La} Let $b\in L^\infty(S^{N-1})$. Suppose that there exists a sequence of functions $(a_n)_n$ satisfying \eqref{eq:def-a-anisotropi} and such that $a_n\stackrel{*}{\rightharpoonup} b$ in $L^\infty(S^{N-1})$. Let $\l_n: \mathbb{R}^N\times \mathbb{R}^N\to [0,\kappa^{-1}] $, with $\l_n\to 0 $ pointwise on $\mathbb{R}^N\times \mathbb{R}^N$. Let $(K_n)_n$ be a sequence of symmetric kernels satisfying $$
|K_n(x,y)-\mu_{a_n}(x,y)|\leq \l_n(x,y)\mu_1(x,y) \qquad \textrm{ for all $x\not=y\in \mathbb{R}^N $ and for all $n\in \mathbb{N}$. } $$ If $(v_n)_n$ is a bounded sequence in ${L_s(\mathbb{R}^N)}\cap H^s_{loc}(\mathbb{R}^N)$ such that $v_n\to v$ in ${L_s(\mathbb{R}^N)}$, then $$ \int_{\mathbb{R}^{N}}v(x){\mathcal L}_{\mu_b}\psi(x)\,dx=\frac{1}{2} \lim_{n\to\infty}\int_{\mathbb{R}^{2N}}(v_n(x )-v_n(y))(\psi(x)-\psi(y)) {K_n(x,y)}\,dxdy \quad\textrm{ for all $\psi\in C^\infty_c(\mathbb{R}^N)$.} $$
\end{lemma} \begin{lemma} \label{lem:Liouville} Let $b\in L^\infty(S^{N-1})$. Suppose that there exists a sequence of functions $(a_n)_n$ satisfying \eqref{eq:def-a-anisotropi} and such that $a_n\stackrel{*}{\rightharpoonup} b$ in $L^\infty(S^{N-1})$. Consider $u\in H^s_{loc}(\mathbb{R}^N)$ satisfying \begin{align*}
\begin{cases} {\mathcal L}_{\mu_b} u=0 \qquad \textrm{ in $\mathbb{R}^N$},
\\
\|u\|_{L^2(B_R)}^2 \leq R^{N+2\g} \qquad \textrm{ for some $\g<2s$ and for every $R\geq 1$}. \end{cases} \end{align*} Then $u$ is an affine function.
\end{lemma}
\subsection{A priori estimates and consequences} We now state the main result of the present section.
\begin{proposition} \label{prop:bound-Kato-abstract} Let $s\in (0,1)$, $\b\in [0,2s)$, $\s\in(s,1]$ and $\kappa>0$. Let $\a'\geq 0$, with $\a'+\s\in (0,2s)$. Pick $$\g\in (0,1)\cap(0,\s]\cap (0,2s-\b].$$
Then there exist $\e_0>0$ and $C >0$ such that if \begin{itemize}
\item $a$ satisfies \eqref{eq:def-a-anisotropi}, $K_a$ satisfies \eqref{eq:K-Kernel-satisf} with $$
|K_a-\mu_a|<\e_0\mu_1(x,y) \qquad\textrm{ on $B_2\times B_2\setminus \{x=y\}$}, $$ \item $K'$ satisfies \eqref{eq:K'-Kernel-satisf},
\item $ f\in {\mathcal M}_\b$, $ g\in H^s(\mathbb{R}^N)$, $U\in C^{0,\s}_{loc}(\mathbb{R}^N)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ are such that $$
{\mathcal L}_{K_a} g+{\mathcal L}_{K'} U=f \qquad\textrm{ in $B_2$,} $$ \end{itemize}
then
$$
\sup_{r>0} r^{-2\g-N} \|g-g_{B_r}\|_{L^2(B_r)}^2 \leq C ( \|g\|_{ L^2(\mathbb{R}^N) }+[ U]_{C^{0,\s}(\mathbb{R}^N) } + \|f\|_{ {\mathcal M}_\beta } )^2. $$
\end{proposition}
\begin{proof}
Assume that the assertion in the proposition does not hold, then for every $n\in \mathbb{N}$, there exist: \begin{itemize} \item $a_n$ and $K_{a_n}$ satisfying \eqref{eq:def-a-anisotropi} and \eqref{eq:K-Kernel-satisf} respectively, with \begin{equation} \label{eq:K-a-n-mu-an}
|K_{a_n}-\mu_{a_n}|<\frac{1}{n} \mu_1(x,y) \qquad\textrm{ on $B_2\times B_2 \setminus \{x=y\}$}, \end{equation}
\item $K'_n$ satisfying \eqref{eq:K'-Kernel-satisf}, $ f_n\in {\mathcal M}_\b$, $U_n\in C^{0,\s}_{loc}(\mathbb{R}^N)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ and $g_n \in H^s(\mathbb{R}^N)$, with $\|g_n\|_{L^2(\mathbb{R}^N)}+\|f_n\|_{{\mathcal M}_\b}+ \| U_n\|_{C^{\s}(\mathbb{R}^N) } \leq 1 $,
\begin{equation}
{\mathcal L}_{K_{a_n}} g_n+{\mathcal L}_{K'_ n} U_n =f_n \qquad\textrm{ in $B_2$}, \end{equation} \end{itemize} with the property that $$
\sup_{r>0} r^{-N-2\gamma} \|g_n-(g_n)_{B_{ r}}\|_{L^2(B_{r})}^2 > n. $$ Consequently, there exists $\overline r_n>0$ such that \begin{equation}\label{eq:ov-r-n-un-larger-n-eps}
\overline r^{-N-2\gamma}_n \|g_n-(g_n)_{B_{\overline r_n}}\|_{L^2(B_{\overline r_n})}^2 > n/2 . \end{equation}
We consider the (well defined, because $\|g_n\|_{ L^2(\mathbb{R}^N)} \leq 1$) nonincreasing function $\Theta_n: (0,\infty)\to [0,\infty)$ given by $$
\Theta_n(\overline r)=\sup_{ r\geq \overline r } r^{-N-2\g} \|g_n-(g_n)_{B_{ r}}\|_{L^2(B_{ r})}^2 . $$ Obviously, for $n\geq 2$, by \eqref{eq:ov-r-n-un-larger-n-eps}, \begin{equation}\label{eq:The-n-geq-n}
\Theta_n(\overline r_n)> n/2\geq 1 . \end{equation} Hence, provided $n\geq 2$, there exists $r_n\in [\overline r_n,\infty)$ such that \begin{align*}
\Theta_n( r_n)&\geq r^{-N-2\gamma}_n \|g_n - (g_n)_{ B_{ r_n} } \|_{L^2(B_{ r_n})}^2 \geq \Theta_n(\overline r_n)-1/2\geq (1-1/2)\Theta_n(\overline r_n)\geq \frac{1}{2}\Theta_n( r_n), \end{align*}
where we used the monotonicity of $\Theta_n$ for the last inequality, while the first inequality comes from the definition of $\Theta_n$. In particular, thanks to \eqref{eq:The-n-geq-n}, $\Theta_n( r_n)\geq n/4$. Now since $ \|g_n \|_{L^2(\mathbb{R}^N)} \leq 1$, we have that $ r^{-N-2\gamma}_n \geq n/8$, so that $r_n\to 0$ as $n\to \infty$.
We now define the sequence of functions $$
{w}_n(x)= \Theta_n(r_n)^{-1/2} r_n^{-\g} \left\{g_n(r_n x) - \frac{1}{|B_1|} \int_{B_1} g_n(r_n x) \, dx \right\}, $$
which, satisfies \begin{equation} \label{eq:w-n-nonzero}
\|w_n\|_{L^2(B_{1})}^2 \geq \frac{1}{2}, \qquad\int_{B_1}w_n(x)\,dx =0 \qquad\textrm{ for every $n\geq2$.} \end{equation}
Using that, for every $r>0$, $ \|g_n-(g_n)_{B_{ r}}\|_{L^2(B_{ r})}^2\leq r^{N+2\g}\Theta_n(r)$ and the monotonicity of $\Theta_n$, by \cite[Lemma 3.1]{Fall-Reg}, we find that \begin{equation}\label{eq:groht-w-n-abs}
\|w_n\|_{L^2(B_{R})}^2 \leq C R^{N+2\g} \qquad\textrm{ for every $R\geq 1$ and $n \geq 2$,} \end{equation} for some constant $C=C(N,\g)>0$.\\
We define
$$ \overline K_n(x,y)=r_n^{N+2s}K_{a_n}(r_nx ,r_ny ), \qquad \overline K_n'(x,y)=r_n^{N+2s}K_{n}'(r_nx ,r_ny ) $$
and $$ \overline U_n(x)= U_n (r_n x), \qquad \overline f_n(x)= r_n^{2s} f_n (r_n x). $$
It is plain that
\begin{equation} \label{eq:w_n-solves} {\mathcal L}_{\overline K_{n}} w_n +r_n ^{-\g} \Theta_n(r_n)^{-1/2} {\mathcal L}_{\overline K_n'}\overline U_n= r^{-\g}_n \Theta_n(r_n)^{-1/2} \overline f_n \qquad\textrm{ in $B_{2/{r_n}} $.} \end{equation}
We fix $M>1$ and let $n\geq 2$ large, so that $1<M<\frac{1}{8r_n}$. Therefore, letting $w_{n,M}:=\vp_{4M} w_n\in H^s(\mathbb{R}^N)$, we apply Lemma \ref{lem:estim-G_Ku}$(i)$ to get
\begin{equation} \label{eq:eq-for-w-n-cut-Cacciopp} {\mathcal L}_{\overline K_{n}}w_{n,M} + r_n ^{-\g} \Theta_n(r_n)^{-1/2} {\mathcal L}_{\overline K_n'}\overline U_n= r^{-\g} \Theta_n(r_n)^{-1/2} \overline f_n+G_{K_n,w_n,M} \qquad\textrm{ in $B_{2M}$,}
\end{equation} with $\|G_{\overline K_n,w_n,M}\|_{L^\infty(B_{M/2})}\leq C \|w_n\|_{{L_s(\mathbb{R}^N)}}\leq C$, by \eqref{eq:groht-w-n-abs}. We also note that \begin{equation} \label{eq:estim-3}
\|\overline f_n\|_{{\mathcal M}_\b} \leq r_n^{2s-\b}. \end{equation}
Clearly $\overline K_n $ satisfies \eqref{eq:Kernel-satisf}.
Applying Lemma \ref{lem:from-caciopp-ok} to the equation \eqref{eq:eq-for-w-n-cut-Cacciopp} and using \eqref{eq:groht-w-n-abs} together with \eqref{eq:estim-3}, we find a constant $\overline C $ such that for every $\e>0$, there exists $C$ satisfying \begin{align} &\left \{\kappa- \varepsilon \overline C \Theta_n(r_n)^{-1/2} r_n^{2s-\b-\g} \right\} \int_{B_{M/8}\times B_{M/8}}(w_{n,M}(x)-w_{n,M}(y))^2 \mu_1(x,y)\,dxdy \nonumber\\
&\leq C( \Theta_n(r_n)^{-1/2} r_n^{2s-\b-\g} + 1) + C r_n ^{-\g} \Theta_n(r_n)^{-1/2} [\overline U_n]_{H^s_{\overline K'_n}(B_{4M})} ^2 \nonumber\\
&\quad +C r_n ^{-\g} \Theta_n(r_n)^{-1/2} \int_{\mathbb{R}^N} \vp_M^2(y)|w_n(y)| \left( \int_{ \mathbb{R}^N\setminus B_{4M}}|\overline U_n(x)-\overline U_n(y)||\overline K'_n(x,y)|\,dx\right)dy . \label{eq:estim-Ccciopp--111} \end{align}
We observe that $$
|\overline K_n'(x,y)|\leq \frac{1}{\kappa} (r_n |x|+r_n|y|+1)^{\a'} \mu_1(x,y). $$ From this and the fact that $[\overline U_n]_{C^{\s}(\mathbb{R}^N)}\leq r_n^\s$ , we have the following estimate: \begin{align}\label{eq:estim-1}
[\overline U_n]_{H^s_{\overline K'_n}(B_{4M})} ^2& =\int_{B_{4M}\times B_{4M}}(\overline U_n(x)-\overline U_n(y))^2\overline K_n'(x,y)\, dxdy \nonumber\\
&\leq C(M) \int_{B_{4M}\times B_{4M}}(\overline U_n(x)-\overline U_n(y))^2 \mu_1(x,y)\, dxdy \nonumber\\
& \leq C(M) r_n^{2\s} \int_{B_{4M}\times B_{4M}}|x-y|^{-N-2s+2\s}\, dxdy \nonumber\\
&\leq C(M) r_n^{2\s}, \end{align} because $\s>s$. In addition, since $\a'+\s<2s$, we get \begin{align} \label{eq:estim-2}
\sup_{y\in B_M} \int_{ \mathbb{R}^N\setminus B_{4M}}|\overline U_n(x)-\overline U_n(y)| |\overline K_n'(x,y)|\,dx &\leq C(M) r_n^{\s} \int_{|x|\geq 2M}(1+|x|^{\a'}) |x|^{-N-2s+\s} dx \nonumber\\
& \leq C(M) r_n^{\s} . \end{align}
Now using \eqref{eq:estim-1} and \eqref{eq:estim-2} in \eqref{eq:estim-Ccciopp--111} and the fact that $\g\leq \min (2s-\b,\s)$, we find that
\begin{align*} &\left \{\kappa- \varepsilon \overline C \Theta_n(r_n)^{-1/2} \right\} [w_n]_{H^s(B_{M/8})}^2 \leq C( \Theta_n(r_n)^{-1/2} + 1) . \end{align*}
Therefore, since $ \Theta_n(r_n)^{-1}\leq 1$, then provided $\e$ is small enough, by \eqref{eq:groht-w-n-abs}, we deduce that $w_n$ is bounded in $H^s_{loc}(\mathbb{R}^N)$. Hence by Sobolev embedding and \eqref{eq:groht-w-n-abs}, there exists $w\in H^s_{loc}(\mathbb{R}^N)\cap {L_s(\mathbb{R}^N)}$ such that, up to a subsequence, $w_n\to w$ in $L^2_{loc}(\mathbb{R}^N)\cap {L_s(\mathbb{R}^N)}$. Moreover, by \eqref{eq:w-n-nonzero} we deduce that \begin{equation}\label{eq:w-nonzero-eps-pp}
\|w\|_{L^ 2(B_1)}^2 \geq \frac{1}{2} \qquad\textrm{ and } \qquad w_{B_1}= 0. \end{equation}
In addition by \eqref{eq:groht-w-n-abs}, we have \begin{equation} \label{eq:w-W-growth-ok}
\|w\|_{L^2(B_{R})}^2 \leq C R^{N+2\g} \qquad\textrm{ for every $R\geq 1$.} \end{equation} Now applying Lemma \ref{lem:convergence-very-weak} to the equation \eqref{eq:w_n-solves} and using \eqref{eq:estim-1} together with \eqref{eq:estim-2}, we get \begin{align*}
& \left|\int_{\mathbb{R}^{2N}}(w_n(x)-w_n(y))(\psi(x)-\psi(y))\overline K_n(x,y)\, dxdy \right|\leq C \Theta_n(r_n)^{-1/2} \qquad\textrm{ for all $\psi\in C^\infty_c(B_M)$.}
\end{align*}
Since $|\overline K_n-\mu_{a_n}|\leq \frac{\mu_1(x,y) }{n}$ almost everywhere in $B_{1/r_n}\times B_{1/r_n}$ and $\Theta_n(r_n)\to\infty$ as $n\to\infty$, we can apply Lemma \ref{lem:Ds-a-n--to-La} to deduce that
${\mathcal L}_{\mu_b} w= 0 \quad\textrm{ in $\mathbb{R}^N$},$
where $b$ is the weak-star limit of $a_n$ (which satisfies \eqref{eq:def-a-anisotropi} for all $n\in \mathbb{N}$).
In view of \eqref{eq:w-W-growth-ok}, by Lemma \ref{lem:Liouville}, we deduce that $w$ is equivalent to a constant function, since $\g<1$. This is clearly in contradiction with \eqref{eq:w-nonzero-eps-pp}. \end{proof}
As a consequence, we get the following result.
\begin{corollary}\label{cor:C-gamm-near-flat} Let $s\in (0,1)$, $N\geq 1$, $\b\in [0,2s)$, $\s\in(s,1]$, $\a'\geq 0$, with $\a'+\s\in (0,2s)$, and $\kappa>0$. Let $a$ satisfy \eqref{eq:def-a-anisotropi}. Consider $K$ and $K'$ satisfying \eqref{eq:K-Kernel-satisf} and \eqref{eq:K'-Kernel-satisf}, respectively.
Let $g\in H^s(B_2)\cap {L_s(\mathbb{R}^N)}$, $U\in C^{0,\s}_{loc}(\mathbb{R}^N)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ and $f\in {\mathcal M}_\b$ satisfy
\begin{equation}
{\mathcal L}_K g+{\mathcal L}_{K'} U =f \qquad\textrm{ in $B_2$.} \end{equation}
Then, for $\g\in (0,1)\cap (0, 2s-\b]\cap(0,\s]$, there exist $\e_0, C>0$, such that if $|K-\mu_a|<\e_0 \mu_1(x,y) $ in $B_2\times B_2\setminus \{x=y\}$, we have $$
\|g\|_{C^{0,\g}(B_{1/4})}\leq C (\|g\|_{L^2(B_2)}+\|g\|_{{L_s(\mathbb{R}^N)}}+ [U ]_{C^{0,\s}(\mathbb{R}^N)}+ \|f\|_{{\mathcal M}_\b}), $$ with $C,\e_0>0$ depend only on $s,N,\b,\s,\a',\kappa$ and $\g$.
\end{corollary}
\begin{proof}
Without loss of generality, we may assume that \begin{equation}\label{eq:estim-tif-zmm}
\|g\|_{L^2(B_2)}+\|g\|_{{L_s(\mathbb{R}^N)}}+ [U ]_{C^{0,\s}(\mathbb{R}^N)}+ \|f\|_{{\mathcal M}_\b} \leq1. \end{equation}
Let $z\in B_{1/2}$ and define $g_z:=g(x+z)$, $K_z(x,y)=K(x+z,y+z)$, $K_z'(x,y)=K(x+z,y+z)$, $f_z(x)=f(x+z)$, $f_z(x)=f(x+z)$, and $U_z(x)=U(x+z)$. We then have $$
{\mathcal L}_{K_z} g_z+{\mathcal L}_{K'_z} U_z =f_z \qquad\textrm{ in $B_1$.} $$ On the other hand by Lemma \ref{lem:estim-G_Ku},
\begin{equation}\label{eq:K-zvp-1gz}
{\mathcal L}_{K_z} (\vp_1 g_z)+{\mathcal L}_{K'_z} U_z (\vp_1 g_z) =\widetilde f_z \qquad\textrm{ in $B_{1/2}$,} \end{equation} for some function $\widetilde f_z $ satisfying \begin{equation}\label{eq:estim-tif-z}
\|\widetilde f_z \|_{{\mathcal M}_\b}\leq \| f_z \|_{{\mathcal M}_\b} + C \| g_z \|_{{L_s(\mathbb{R}^N)}} \leq C, \end{equation} where we used \eqref{eq:estim-tif-zmm} for the last inequality.
By \eqref{eq:K-zvp-1gz} and Proposition \ref{prop:bound-Kato-abstract}, there exist $\e_0, C>0$, only depending on $s,N,\b,\s,\a',\kappa,\s$ and $\g$, such that if $|K_z-\mu_a|<\e_0$ in $B_2\times B_2\setminus \{x=y\}$, we get $$
\|g_z-(g_z)_{B_r}\|_{L^2(B_r)}= \|g-(g)_{B_r(z)}\|_{L^2(B_r(z))} \leq C r^{N/2+\g} \qquad\textrm{for every $r>0.$} $$ It then follows, from \cite[ Lemma 3.1]{Fall-Reg}, that $$
\|g-g(z)\|_{L^2(B_r(z))} \leq C r^{N/2+\g} \qquad\textrm{for every $z\in B_1$ and $r\in (0,1).$}
$$
This implies that $\|g\|_{C^{\g}(B_{1/4})}\leq C$. The proof is thus finished.
\end{proof}
By scaling and covering, we have the
\begin{corollary}\label{cor:Holder-cont-coeff} Let $s\in (0,1)$, $N\geq 1$, $\b\in [0,2s)$, $\s\in(s,1]$, $\a'\geq 0$, with $\a'+\s\in (0,2s)$, $\kappa>0$ and $\g\in (0,1)\cap(0,\s]\cap (0,2s-\b]$.
Consider $K\in \widetilde \scrK_0^s(\kappa,0, Q_\infty)$ and $K'$ satisfies \eqref{eq:K'-Kernel-satisf}.
Let $g\in H^s(B_2)\cap {L_s(\mathbb{R}^N)}$, $U\in C^{0,\s}_{loc}(\mathbb{R}^N)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ and $f\in {\mathcal M}_\beta $ satisfy
\begin{equation}\label{eq:clK-g-corrol}
{\mathcal L}_K g+{\mathcal L}_{K'} U =f \qquad\textrm{ in $B_2$.} \end{equation} Then there exists $C>0$, only depending on $N,s,\a',\b, \kappa$ and $\g$, such that $$
\|g\|_{C^{0,\g}(B_1)}\leq C (\|g\|_{L^2(B_2)}+\|g\|_{{L_s(\mathbb{R}^N)}}+ [U ]_{C^{0,\s}(\mathbb{R}^N)}+ \|f\|_{{\mathcal M}_\b}). $$
\end{corollary}
\begin{proof}
Pick $x_0\in B_{3/2}$. By the continuity of ${\mathcal A}_K(\cdot,\cdot,\th)$ (uniformly with respect to $\th$), for every $\e>0$ there exists $\delta =\delta _{x_0,\e}\in (0,1/100)$ such that, for every $x\in B_{4\delta }(x_0)$, $ r\in (0, 4\delta )$ and $\th\in S^{N-1}$, we have
\begin{align*}
\left| K(x,x+r\theta )-{{\mathcal A}}_{K}(x_0,0,\th) r^{-N-2s} \right |\leq \e r^{-N-2s} . \end{align*}
Therefore, for every $x\in B_{4\delta }(x_0)$ and $0<|z|<4\delta $, \begin{align*}
\left| K(x,x+z)- {{\mathcal A}}_{K}(x_0,0,z/|z|) |z|^{-N-2s} \right |\leq \varepsilon |z|^{-N-2s} \end{align*} and thus, for every $x,y\in B_{2\delta }(x_0)$, with $x\not=y$, \begin{align}\label{eq:K-close-to-mu-a}
\left| K(x,y)- \mu_{ a}(x,y) \right |\leq \varepsilon \mu_{ 1}(x,y), \end{align} where $ a(\th):= {{\mathcal A}}_{K}(x_0,0,\th) $. By Definition \ref{def:Kernel-not-reg-the}, ${{\mathcal A}}_{o,K}(x_0,0,\th)=0$ and thus $ a$ satisfies \eqref{eq:a-satisf-elliptic}.
We now let $K_{\delta }(x,y)=\delta ^{N+2s}K( \delta x+ x_0,\delta y+ x_0)$ and $K_{\delta }'(x,y)=\delta ^{N+2s}K'( \delta x+ x_0,\delta y+ x_0)$, which satisfy \eqref{eq:K-Kernel-satisf} and \eqref{eq:K'-Kernel-satisf}, respectively.
For $x\in B_{2}$, we define $g_\delta (x)=g(\delta x+ x_0)$, $U_\delta (x)=U(\delta x+ x_0)$ and $f_\delta (x)=\delta ^{2s}f(\delta x+ x_0)$. Since $\delta \in (0,1/16)$, by a change of variable in \eqref{eq:clK-g-corrol}, we get \begin{equation} \label{eq:u-delta-satisf-final} {\mathcal L}_{K_{\delta }} g_\delta +{\mathcal L}_{K'_\delta } U_\delta = f_\delta \qquad\textrm{ in $B_{8}$}. \end{equation}
On the other hand \eqref{eq:K-close-to-mu-a} becomes \begin{align*}
\left| K_\delta (x,y)- \mu_{ a}(x,y) \right |\leq \varepsilon \mu_{ 1}(x,y) \qquad\textrm{ for $x\not=y\in B_2$.} \end{align*}
From this and \eqref{eq:u-delta-satisf-final}, then provided $\e>0$ small, by Corollary \ref{cor:C-gamm-near-flat} and a change of variable, we get $$
\|g\|_{C^{\a}(B_{\delta _{x_0,\e}}(x_0))} \leq C(x_0) \left( \| g\|_{L^2(B_{2})}+ \|g\|_{{L_s(\mathbb{R}^N)}} + \|f\|_{{\mathcal M}_\b}+[U]_{C^{0,\s}(\mathbb{R}^N)} \right) , $$ where $C(x_0)$ is a constant, only depending on $N,s,c_0,\delta _{x_0},\kappa,\t,\a,\a',\s,\g$ and $ x_0$. Next, we cover $\overline B_{1}$ with a finite number of balls $B_{\frac{1}{2}\delta _{x_i,\e}}(x_i)\subset\subset B_{3/2}$, for $i=1,\dots,n$, with $x_i\in \overline B_{1}$. It then follows that $$
\|g\|_{C^{\a}(B_1)} \leq C' \left( \| g\|_{L^2(B_{2})}+ \|g\|_{{L_s(\mathbb{R}^N)}} + \|f\|_{{\mathcal M}_\b}+[U]_{C^{0,\s}(\mathbb{R}^N)} \right) , $$
\end{proof}
We have the following generalization.
\begin{corollary} \label{cor:Holder-many-K'} Let $s\in (0,1)$, $\b\in [0,2s)$, $\s_i\in(s,1]$. Let $\kappa>0$ and $$ \g\in (0,1)\cap(0,\min_{1\leq i\leq \ell }\s_i]\cap (0,2s-\b]. $$ Consider $K\in\widetilde \scrK_0^s(\kappa,0,Q_\infty)$ and $K'_i$ satisfying \eqref{eq:K'-Kernel-satisf}, for $i=1,\dots,\ell$, and for some, $\a_i'\geq 0$, with $\a_i'+\s_i\in (0,2s)$.
Let $g\in H^s (B_2)\cap L_s(\mathbb{R}^N)$, $U_i\in C^{0,\s_i}_{loc}(\mathbb{R}^N)\cap L_{(\a'+2s)/2}(\mathbb{R}^N)$ and $ f\in {\mathcal M}_\beta $ satisfy
\begin{equation}
{\mathcal L}_K g+\sum_{i=1}^\ell {\mathcal L}_{K_i'} U_i =f \qquad\textrm{ in $B_2$.} \end{equation} Then, there exists $C>0$, only depending on $s,N,\b, \a_i,\s_i , \kappa,\ell$ and $\g$, such that $$
\|g\|_{C^{0,\g}(B_1)}\leq C ( \| g\|_{L^2(B_{2})}+ \|g\|_{{L_s(\mathbb{R}^N)}}+\sum_{i=1}^\ell [U_i ]_{C^{0,\s_i}(\mathbb{R}^N)}+ \|f\|_{{\mathcal M}_\b}). $$
\end{corollary}
\section{Gradient estimates} \label{s:GradEstim} In this section, we consider the fractional parameter $s\in (1/2,1)$ and we prove H\"older estimates of $\n u$. For $g\in L^2_{loc}(\mathbb{R}^N)$ and $r>0$, we define \begin{equation} \label{eq:def:Pru-x}
\textbf{P}_{r,g}(x)=g_{B_r}+ T^{r,g}\cdot x= g_{B_r}+ \sum_{i=1}^N T^{r,g}_i x_i, \end{equation}
where \begin{equation} \label{eq:def-Tui}
T^{r,g}_i=\frac{{\langle} g, x_i{\rangle}_{L^2(B_r)}}{\|x_i\|_{L ^2(B_r)}^2}. \end{equation}
Note that $\textbf{P}_{r,g} $ is the $L^2(B_r)$-projection of $g$ on the space of affine functions.\\
In view of Corollary \ref{cor:Holder-cont-coeff}, we know that the solutions $u$ to ${\mathcal L}_K u=f$ in $B_2$ are of class $C^{1-\varrho}(B_1)$ for every small $\varrho>0$, provided $K\in \widetilde \scrK_0^s(\kappa,0, Q_\infty)$ and $f\in {\mathcal M}_\b$, with $\b\in [0,2s)$. In particular $|T^{r,u-u(0)}|\leq C r^{-\varrho}$. The result below improves this to H\"older regularity estimates of the gradient of $u$ when $2s-\b>1$ and $K\in \widetilde \scrK_0^s(\kappa,\a, Q_\infty)$, with $\a>0$.
\begin{proposition} \label{prop:bound-wth-corrector} Let $N\geq 1$, $s\in (1/2,1)$, $\kappa,c_0>0$. Consider $\a\in (0,2s-1)$, $\b\in [0,2s)$, $\varrho\in [0,1)$ such that $$
\g:= \min (1-\varrho+\a, 2s-\b) >1. $$
Then there exists $C=C(N,s,\a,\b,\kappa,c_0,\varrho) >0$ such that if: \begin{itemize} \item $K\in \widetilde \scrK_0^s(\kappa,\a, Q_\infty)$,
\item $g\in H^s(B_2)\cap L^{\infty}(\mathbb{R}^N)$ and $f\in {\mathcal M}_\b$ with $$
\|g\|_{L^\infty(\mathbb{R}^N)}+ \|f\|_{{\mathcal M}_\b}\leq 1, $$
$$
| T^{r,g}| \leq c_0 r^{-\varrho}\qquad\textrm{ for all $r>0$} $$
are such that $$
{\mathcal L}_K g =f \qquad\textrm{ in $B_2$}, $$ \end{itemize}
then we have
$$
\sup_{r>0} r^{-\g} \left\| g- \textrm{\em \textbf{P}}_{r,g}\right\|_{L^\infty(B_r)}\leq C . $$
\end{proposition} \begin{proof} Suppose on the contrary that the assertion in the proposition does not hold. Then as in the proof of Proposition \ref{prop:bound-Kato-abstract}, for all $n\geq 2$, there exist \begin{itemize} \item $r_n>0$, $K_n\in \widetilde \scrK_0^s(\kappa,\a,Q_\infty)$,
\item $g_n\in H^s(B_2)\cap L^{\infty}(\mathbb{R}^N)$ and $f_n\in {\mathcal M}_\b$ satisfying \begin{equation} \label{eq:g-n-T_n-satis}
\|g_n\|_{L^\infty(\mathbb{R}^N)}+ \|f_n\|_{{\mathcal M}_\b}\leq 1, \qquad | T^{r,g_n}| \leq c_0 r^{-\varrho} \quad\textrm{ for all $r\in (0,\infty)$,} \end{equation}
\begin{equation}\label{eq:g-nsolves-Grad-estim}
{\mathcal L}_{K_n} g_n =f_n \qquad\textrm{ in $B_2$}, \end{equation}
\item a nonincreasing function $\Theta_n: (0,\infty)\to [0,\infty)$ satisfying \begin{equation} \label{eq:sup-r-ok-Morrey-hi-Int}
\Theta_n( r)\geq r^{-\g} \| g_n- \textbf{P}_{r, g_n} \|_{L^\infty(B_{r})} \qquad\textrm{ for every $r\in (0,\infty)$ and $n\geq 2$,} \end{equation} \end{itemize} with the properties that $r_n\to 0$ as $n\to\infty$ and
$$
r^{-\g}_n \| g_n- \textbf{P}_{r_n,g_n} \|_{L^\infty(B_{r_n})} \geq \frac{1}{2}\Theta_n(r_n) \geq\frac{n}{4} . $$
We define $$
{v}_n(x)= \Theta_n(r_n)^{-1} r^{-\g}_n [g_n(r_n x) - \textbf{P}_{r_n,g_n}(r_n x) ], $$
so that \begin{equation} \label{eq:E1}
\|v_n\|_{L^\infty(B_{1})}\geq \frac{1}{2}. \end{equation}
In addition, by a change of variable, we get \begin{equation}\label{eq:E2}
\int_{B_1} v_n(x) \, dx= \int_{B_1} v_n(x) x_i\, dx=0 \qquad\textrm{ for every $i\in \{1,\dots,N\}$.} \end{equation}
Since $\Theta_n$ is nonincreasing and $\g>1$ then see e.g. \cite{Fall-Reg,Serra-OK}, inequality \eqref{eq:sup-r-ok-Morrey-hi-Int} always implies that \begin{equation}\label{eq:groht-w-n-Morrey-HI-Grad}
\|v_n\|_{L^\infty(B_{R})}\leq C R^{\g} \qquad\textrm{ for every $R\geq 1$,} \end{equation}
for some constant $C=C(N,\g)$.\\
From \eqref{eq:g-nsolves-Grad-estim}, we deduce that \begin{equation} \label{eq:eq-K-n-v-n}
{\mathcal L}_{\overline K_n} v_n+ \Theta_n(r_n)^{-1} r^{1-\g}_n T^{r_n,g_n}\cdot {\mathcal L}_{\overline K_n} x =\overline f_n \qquad\textrm{ in $B_{2/r_n}$}, \end{equation} where $\overline K_n(x,y):=r_n^{N+2s}K_n(r_n x,r_n y)$ and $\overline f_n(x)=r_n^{2s} f_n(r_n x)$.
Then, since ${\mathcal A}_{\overline K_n}(x,r,\th)= {{\mathcal A}}_{K_n}(r_n x,r_n r,\th)$ and ${{\mathcal A}}_{ K_n}\in C^{0,\a}(Q_\infty)\times L^\infty(S^{N-1} )$, we get $$
| {{\mathcal A}}_{\overline K_n}(x,r,\th)- {{\mathcal A}}_{ K_n}(0,0,\th)|\leq C \min( r_n^\alpha (|x|+r)^\alpha ,1)\qquad\textrm{ for all $x\in \mathbb{R}^N$, $\th\in S^{N-1}$ and $r>0$.} $$ Moreover, recalling Definition \ref{def:Kernel-not-reg-the}, we have ${{\mathcal A}}_{o, K_n}(0,0,\th)=0 $ for all $\th\in S^{N-1}$. Letting $a_n(\th):= {{\mathcal A}}_{\overline K_n}(0,0,\th)$ and $ \overline K_n'(x,y):=r_n^{-\a}(\overline K_n(x,y)-\mu_{a_n}(x,y))$, we immediately see that \begin{equation}\label{eq:oK_n-close-to mua_n}
| \overline K_n(x,y)-\mu_{a_n}(x,y)|\leq C\min( r_n^\alpha (|x|+|y|)^\alpha ,1) \mu_1(x,y) \end{equation} and \begin{equation}\label{eq:K-nprimes-Grad-estim}
| \overline K_n'(x,y)|\leq C (|x|+|y|)^\a\mu_1(x,y). \end{equation}
Since ${\mathcal L}_{\mu_{a_n} } x_i =0$ on $\mathbb{R}^N$, we can rewrite \eqref{eq:eq-K-n-v-n} as \begin{equation}\label{eq:eq-K-n-v-n-ok}
{\mathcal L}_{\overline K_n} v_n+\Theta_n(r_n)^{-1} \sum_{i=1}^N \overline T_n^i {\mathcal L}_{\overline K'_n} x_i = r_n^{-\g} \Theta_n(r_n)^{-1} \overline f_n \qquad\textrm{ in $B_{2/r_n}$}, \end{equation}
where (recall \eqref{eq:def:Pru-x}) $\overline T_n^i:= r^{1+\alpha -\gamma}_n T^{r_n,g_n}_i$. Note that $x_i\in L_{(\a+2s)/2}(\mathbb{R}^N)$, provided $\a\in (0, 2s-1)$ and $[x_i]_{C^{0,1}(\mathbb{R}^N)}\leq 1$. Clearly by \eqref{eq:g-n-T_n-satis}, \begin{equation} \label{eq:E10}
\|\overline f_n\|_{{\mathcal M}_\b}\leq r_n^{2s-\b} \qquad \textrm{ and } \qquad |\overline T_n^i|\leq c_0 r_n^{1+\a-\g-\varrho}\leq c_0 . \end{equation}
Since $\overline K_n\in \widetilde \scrK_0^s(\kappa,0,Q_\infty)$ and $\Theta_n(r_n)\to \infty$ as $n\to\infty$, applying Corollary \ref{cor:Holder-many-K'} to \eqref{eq:eq-K-n-v-n-ok} and using \eqref{eq:E10} together with \eqref{eq:K-nprimes-Grad-estim}, we find that $v_n$ is bounded in $C^{1-\delta }_{loc}(\mathbb{R}^N)$, for all $\delta \in (0,1)$. Hence, provided $\delta $ is small, there exists $v\in C^{s+\delta }_{loc}(\mathbb{R}^N)$ such that, up to a subsequence, $v_n\to v$ in $C^0_{loc}(\mathbb{R}^N)$.
Hence by \eqref{eq:groht-w-n-Morrey-HI-Grad}, up to a subsequence, $v_n$ converges strongly, in $ {L_s(\mathbb{R}^N)}$, to $v\in H^s_{loc}(\mathbb{R}^N)\cap {L_s(\mathbb{R}^N)}$. Moreover, by \eqref{eq:E1}, we deduce that \begin{equation}\label{eq:w-nonzero-eps}
\|v\|_{L^ \infty(B_1)} \geq \frac{1}{2} \qquad\textrm{ and } \qquad \int_{B_1} v(x) \, dx= \int_{B_1} v(x) x_i\, dx=0 \quad\textrm{ for every $i\in \{1,\dots,N\}$.} \end{equation}
In addition, passing to the limit in \eqref{eq:groht-w-n-Morrey-HI-Grad}, we have \begin{equation} \label{eq:w-WW-zero-mean}
\|v\|_{L^\infty(B_{R})} \leq R^{\g} \qquad\textrm{ for every $R\geq 1$.} \end{equation}
We observe that $a_n$ satisfies \eqref{eq:def-a-anisotropi} for all $n$. By \eqref{eq:oK_n-close-to mua_n}, Lemma \ref{lem:convergence-very-weak} and Lemma \ref{lem:Ds-a-n--to-La}, we can pass to the limit in \eqref{eq:eq-K-n-v-n-ok}, to get
${\mathcal L}_{\mu_b} v= 0 \quad\textrm{ in $\mathbb{R}^N$},$
where $b$ is the weak-star limit of $a_n$.
Now, since $v\in H^s_{loc}(\mathbb{R}^N)$ and satisfies \eqref{eq:w-WW-zero-mean}, by Lemma \ref{lem:Liouville} we deduce that $v$ is an affine function, because $\gamma<2s$. This is clearly in contradiction with \eqref{eq:w-nonzero-eps}.
\end{proof} A first consequence of the previous result is the
\begin{corollary}\label{cor:Holder-reg-Global}
Let $s\in (1/2,1)$, $\b\in[0,2s-1)$, $N\geq1$ and $\kappa>0$. Let $\a\in (0,2s-1)$ and $\varrho\in [0,\a)$. Let $K\in \widetilde \scrK_0^s(\kappa,\a, Q_\infty)$, $g\in H^s(B_2)\cap C^{0,1-\varrho}(\mathbb{R}^N) $ and $f\in {\mathcal M}_\b$ satisfy
\begin{equation}
{\mathcal L}_K g =f \qquad\textrm{ in $B_2$}. \end{equation}
Then, there exists $C>0$, only depending on $s,N,\b,\a,\kappa,\varrho$, such that $$
\| g\|_{C^{1,\min( 1+\a-\varrho ,2s-\b) -1 }(B_{1})}\leq C ( \|g\|_{ C^{0,1-\varrho}(\mathbb{R}^N) }+ \|f\|_{ {\mathcal M}_\beta }). $$
\end{corollary} \begin{proof}
Put $A:= \|g\|_{ C^{0,1-\varrho}(\mathbb{R}^N) } + \|f\|_{{\mathcal M}_\b}.$ We define $ G_z(x):=g(x+z)-g(z)$, for $z\in B_{1}$. Since $G_z(0)=0$, we have $|T^{r,G_z}|\leq C(\varrho,N) r^{-\varrho} [g]_{C^{0,1-\varrho}(\mathbb{R}^N)} \leq C A$, for $r>0$. Obviously, ${\mathcal L}_{K_z} G_z=f(\cdot +z)$ in $B_1$, where $K_z(x,y)=K(x+z,y+z)$. We then apply Proposition \ref{prop:bound-wth-corrector} to get a constant $C>0$, only depending on $s,N,\b,\a,\kappa,\varrho$, such that $$
\sup_{r>0} r^{-\g} \left\| G_z- \textbf{P}_{r, G_z} \right\|_{L^\infty(B_r)}\leq C A, $$
where $\g:=\min ( 1+\a-\varrho ,2s-\b) >1$. By a well known iteration argument (see e.g \cite{Serra-OK}), we find that $$
|g(x)- g(z)- T(z)\cdot( x-z) |\leq C A |x-z|^{\g} \qquad\textrm{ for every $x,z\in B_{1/2}$,} $$
for some $T$, satisfying $\|T\|_{L^\infty(B_1)}\leq C A$. Since $\g>1$, then $\n u(z)= T(z)$.
By a classical extension theorem (see e.g. \cite{Stein}[Page 177], we deduce that
$u\in C^{1, \g-1}(\overline {B_{1/2}})$. Moreover $$
\|u \|_{C^{1,\g-1}(\overline {B_{1/2}})}\leq C A. $$
\end{proof}
By a bootstrap argument, we have the following result.
\begin{theorem} \label{th:abs-res-Propo} Let $s\in (1/2,1)$, $\b\in [0,2s-1)$, $\a\in (0,2s-1)$ and $\kappa>0$. Let $K\in \widetilde \scrK_0^s(\kappa,\a,Q_\infty)$, $u\in H^s(B_2)\cap {L_s(\mathbb{R}^N)}$ and $V, f\in {\mathcal M}_\b$ such that $$
{\mathcal L}_K u+ Vu =f \qquad\textrm{ in $B_2$}. $$
Then
$$
\|u\|_{C^{1,\min (\a, 2s-\b-1)}(B_{1})} \leq C ( \|u\|_{ L^2(B_2) }+ \|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{ {\mathcal M}_\b}), $$
where $C>0$ only depends on $s,N,\kappa,\a,\b$ and $\|V\|_{{\mathcal M}_\b}$.
\end{theorem} \begin{proof}
Since $2s-\b>1$, by \cite{Fall-Reg}, for every $\varrho\in (0,1)$, there exists $C=C(N,s,\b,\a,\|V\|_{ {\mathcal M}_\b},\varrho)>0$ such that $$
\|u\|_{C^{1-\varrho}(B_{1})} \leq C ( \|u\|_{ L^2(B_2) }+ \|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{ {\mathcal M}_\b}). $$ Using Lemma \ref{lem:estim-G_Ku}$(i)$, we apply first Corollary \ref{cor:Holder-reg-Global} to get $$
\|\vp_{1/2} u\|_{C^{0,1}(B_{1/8})} \leq C ( \| \vp_{1/2} u\|_{ L^2(\mathbb{R}^N) }+ \|\vp_{1/2} u\|_{C^{1-\varrho}(\mathbb{R}^N)} + \| u\|_{{L_s(\mathbb{R}^N)}}+ \|\vp_{2} Vu \|_{ {\mathcal M}_\b} + \|f\|_{ {\mathcal M}_\b}). $$ Therefore, $$
\|u\|_{C^{0,1}(B_{2^{-3}})} \leq C ( \|u\|_{ L^2(B_2) }+ \|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{ {\mathcal M}_\b}). $$ We then apply once more Corollary \ref{cor:Holder-reg-Global} (with $\varrho=0$) and use Lemma \ref{lem:estim-G_Ku}$(i)$ to obtain $$
\|\vp_{2^{-4}} u\|_{C^{1, \min(2s-\b-1,\a)}(B_{2^{-8}})} \leq C ( \|\vp_{2^{-4}} u\|_{ L^2(\mathbb{R}^N) }+ \|\vp_{2^{-4}} u\|_{C^{0,1}(\mathbb{R}^N)} + \|u\|_{{L_s(\mathbb{R}^N)}}+ \|\vp_{2^{-4}}Vu \|_{ {\mathcal M}_\b} + \|f\|_{ {\mathcal M}_\b}), $$ so that $$
\| u\|_{C^{1, \min(2s-\b-1,\a)}(B_{2^{-8}})} \leq C ( \|u\|_{ L^2(B_2) } + \|u\|_{{L_s(\mathbb{R}^N)}}+ \|f\|_{ {\mathcal M}_\b}), $$ with $C$ as in the statement of the theorem.
After a covering argument, we obtain the result. \end{proof}
\section{Schauder estimates} \label{s:Shaud}
Here and in the following, given $u\in C^{2s+\a}_{loc}(\mathbb{R}^N)$, with $\a>0$, we let \begin{equation}\label{eq:delta-eo} \delta ^e u(x,r,\th):=\frac{1}{2}(2u(x)-u(x+r\th)-u(x-r\th)),\qquad \delta ^o u(x,r,\th):=\frac{1}{2}(u(x+r\th)-u(x-r\th) ). \end{equation}
For $A\in C^{m,\a}(Q_\infty)\times L^\infty(S^{N-1} )$, we define \begin{equation} \label{eq:defEs} {\mathcal E}^s_{A,u}(x):=\int_{S^{N-1}}\int_0^\infty \delta ^e u(x,r,\th)A(x,r,\th) r^{-1-2s}\, dr d\theta \end{equation} and for $B\in {\mathcal C}^{m}_{\min(1,\a+(2s-1)_+)}(Q_\infty)\times L^\infty(S^{N-1} )$, we define \begin{equation} \label{eq:defOs} {\mathcal O}^s_{B,u}(x):=\int_{S^{N-1}}\int_0^\infty \delta ^o u(x,r,\th)B(x,r,\th) r^{-1-2s}\, dr d\th. \end{equation} We observe that, using the symmetry of $K\in \widetilde \scrK^s_{\min(1,\a+(2s-1)_+)}(\kappa,\a,Q_\infty)$ and a change of variables, we get \begin{align}\label{eq:Operator-splitting} &\frac{1}{2}\int_{\mathbb{R}^{2N} }(u(x)-u(y))(\psi(x)-\psi(y))K(x,y)\, dxdy =\int_{\mathbb{R}^N}\psi(x) {\mathcal E}^s_{{{\mathcal A}}_{e,K},u}(x) \, dx +\int_{\mathbb{R}^N}\psi(x){\mathcal O}^s_{{{{\mathcal A}}_{o,K},u}}(x) \, dx , \end{align}
where
$$ {{\mathcal A}}_{e,K}(x,r,\th):=\frac{1}{2} ({{\mathcal A}}_{K}(x,r,\th) +{{\mathcal A}}_{K}(x,r,-\th)) , \qquad{{\mathcal A}}_{o,K}(x,r,\th):=\frac{1}{2} ({{\mathcal A}}_{K}(x,r,\th) -{{\mathcal A}}_{K}(x,r,-\th)) . $$
We have the following result which will be proved in Section \ref{s:Appendix}. \begin{lemma}\label{lem:2sp-alph-estim-F_Keo} Let $s\in (0,1)$ and $\a\in (0,1)$. Let $\kappa>0$, $m\in \mathbb{N}$, $A\in C^{m+\a}(Q_\infty) \times L^\infty(S^{N-1} )$ and $B\in {\mathcal C}^{m}_{\t}(Q_\infty) \times L^\infty(S^{N-1} )$, with $\t:=\min(\a+(2s-1)_+,1)$.
\begin{itemize} \item
Let $u\in C^{2s+\a+m}(\mathbb{R}^N) $ and $2s\not=1$. If $2s+\a< 1$ or $1<2s+\a<2$, then $$
\| {\mathcal E}^s_{A,u} \|_{C^{m+\a}(\mathbb{R}^N)} \leq C \|A\|_{ C^{m+\a}(Q_\infty)\times L^\infty(S^{N-1} )}\|u\|_{C^{2s+\a+m}(\mathbb{R}^N) } $$ and $$
\| {\mathcal O}^s_{B,u} \|_{C^{m+\a}(\mathbb{R}^N)}\leq C \|B\|_{ {\mathcal C}^{m}_{\t}(Q_\infty)\times L^\infty(S^{N-1} )} \|u\|_{C^{2s+\a+m}(\mathbb{R}^N) }, $$ with $C=C(N,s,\a,m) $.
\item Let $u\in C^{1+\a+m+\e}(\mathbb{R}^N) $, for some $\e\in (0,1-\a)$. If $2s=1$ and $B\in {\mathcal C}^{m}_{\a+\e}(Q_\infty)\times L^\infty(S^{N-1} )$, then $$
\| {\mathcal E}^s_{A,u} \|_{C^{m+\a}(\mathbb{R}^N)} \leq C \|A\|_{ C^{m,\a}(Q_\infty)\times L^\infty(S^{N-1} )}\|u\|_{C^{2s+\a+m+\e}(\mathbb{R}^N) } $$ and $$
\| {\mathcal O}^s_{B,u} \|_{C^{m+\a}(\mathbb{R}^N)}\leq C \|B\|_{ {\mathcal C}^{m}_{\a+\e}(Q_\infty)\times L^\infty(S^{N-1} )} \|u\|_{C^{2s+\a+m+\e}(\mathbb{R}^N) }, $$ with $C=C(N,\a,m,\e) $. \item
Let $u\in C^{2s+\a+m}(\mathbb{R}^N) $ and $2s+\a>2$. If $A,B\in C^{m+2s-1+\a}(Q_\infty)\times L^\infty(S^{N-1} )$ and $B\in {\mathcal C}^{m+1}_{2s+\a-2}(Q_\infty)\times L^\infty(S^{N-1} )$ , then $$
\| {\mathcal E}^s_{A,u} \|_{C^{m+\a}(\mathbb{R}^N)} \leq C \|A\|_{ C^{m+2s-1+\a}(Q_\infty) \times L^\infty(S^{N-1} )}\|u\|_{C^{2s+\a+m}(\mathbb{R}^N) } $$ and $$
\| {\mathcal O}^s_{B,u} \|_{C^{m+\a}(\mathbb{R}^N)}\leq C \left( \|B\|_{ {\mathcal C}^{m+1}_{2s+\a-2}(Q_\infty)\times L^\infty(S^{N-1} )} + \|B\|_{ C^{m+2s-1+\a}(Q_\infty)\times L^\infty(S^{N-1} )} \right)\|u\|_{C^{2s+\a+m}(\mathbb{R}^N) }, $$ with $C=C(N,s,\a,m) $. \end{itemize}
\end{lemma}
We remark that under the assumptions on $A$ and $B$, for $2s=1$, the first assertion of Lemma \ref{lem:2sp-alph-estim-F_Keo} does not in general hold.
\subsection{Schauder estimates} The following result is intended to the $C^{2s+\a}$ regularity estimates, for $2s+\a\not\in \mathbb{N}$. To deal with the case $2s+\a>2$, we look for optimal growth estimate of the difference between $u$ a second order polynomial that is close to $u$ in the $L^2$-norm.\\
For $g\in L^2(B_r)$ and $i,j=1,\dots,N$, we define
$$
T_{ij}^{r,g}=\frac{1}{ \|y_iy_j\|_{L^2(B_r)}^2}\int_{B_r} y_iy_jg(y)\, dy $$ and $$ \textbf{Q}_{r,g}(x)=\sum_{i,j=1}^N T_{ij}^{r,g} x_ix_j.
$$ We note that $\textbf{Q}_{r,g} $ is nothing but the $L^2(B_r)$-projection of $g$ on the space of homogeneous quadratic polynomials. We now state the main result of this section.
\begin{proposition} \label{prop:bound-wth-corrector-nab} Let $N\geq 1$, $s\in (0,1)$, $\kappa>0$, $\a\in (0,1)$, $\b\in (0,\a)$ and $\e>0$. Let $K\in\widetilde \scrK_{\t}^s(\kappa, \a, Q_\infty)$, with $\t:=\min (\a+(2s-1)_+,1)$. Let
$f\in C^\a(\mathbb{R}^N)$ and $g\in C^{2s+\b}(\mathbb{R}^N)$, for $2s+\b\not\in \mathbb{N}$, such that $$
{\mathcal L}_K g =f \qquad\textrm{ in $B_2$}, $$
\begin{itemize} \item[$(i)$] If $1<2s+\a<2$ and $2s\not=1$, then there exists $C=C(N,s,\kappa,\a,\b)>0$ such that
$$
\sup_{r>0} r^{-(2s-1+\a-\b)} [ \n g ]_{C^{\b}(B_r)}\leq C (\|g\|_{C^{2s+\b}(\mathbb{R}^N)}+ [f]_{C^\a(\mathbb{R}^N)}) . $$
\item[$(ii)$] If $2s+\a>2 >2s+\b\geq 1+\a$, $K\in\widetilde \scrK_{0}^s(\kappa, 2s-1+\a, Q_\infty)$ and $g(0)=|\n g(0)|=0$, then there exists $C=C(N,s,\kappa,\a,\b)>0$ such that
$$
\sup_{r>0} r^{-(2s-1+\a-\b)} [ \n g -\nabla \textrm{\em\textbf{Q}}_{r,g} ]_{C^{\b}(B_r)}\leq C (\|g\|_{C^{2s+\b}(\mathbb{R}^N)}+ [f]_{C^\a(\mathbb{R}^N)}) . $$
\item[$(iii)$] If $2s=1$ and $K\in \widetilde \scrK_{\a+\e}^s(\kappa, \a, Q_\infty)$, then there exists $C=C(N,s,\kappa,\a,\b,\e)>0$ such that
$$
\sup_{r>0} r^{-(2s-1+\a-\b)} [ \n g ]_{C^{\b}(B_r)}\leq C (\|g\|_{C^{2s+\b}(\mathbb{R}^N)}+ [f]_{C^\a(\mathbb{R}^N)}) . $$
\item[$(iv)$] If $2s+\a<1$, then there exists $C=C(N,s,\kappa,\a,\b)>0$ such that
$$
\sup_{r>0} r^{-(\a-\b)} [ g ]_{C^{2s+\b}(B_r)}\leq C (\|g\|_{C^{2s+\b}(\mathbb{R}^N)}+ [f]_{C^\a(\mathbb{R}^N)}). $$
\end{itemize}
\end{proposition} \begin{proof} We start with $(i)$.
Assume that the assertion in $(i)$ does not hold, then arguing as in the proof of Proposition \ref{prop:bound-Kato-abstract}, we can find sequences \begin{itemize}
\item $r_n> 0$, $K_n\in \widetilde \scrK_{\a+2s-1}^s(\kappa, \alpha , Q_\infty)$ $g_n\in C^{2s+\b}(\mathbb{R}^N)$ and $f_n\in C^{\a}(\mathbb{R}^N)$ with
$
\|g_n\|_{C^{2s+\b}(\mathbb{R}^N)}+ [f_n]_{C^\a(\mathbb{R}^N)}\leq 1, $
\begin{equation}\label{eq:g-nsolves-Schauder}
{\mathcal L}_{K_n} g_n =f_n \qquad\textrm{ in $B_2$}, \end{equation}
\item $\Theta_n:(0,\infty)\to [0,\infty)$, nonincreasing, \end{itemize} with the properties that $r_n\to 0$ as $n\to \infty$, \begin{equation} \label{eq:sup-r-ok-Morrey-hi-Int-nab} \Theta_n( r)\geq r^{-(2s-1+\a-\b)} [ \n g_n ]_{C^{\b}(B_{r})} \qquad\textrm{ for every $r>0$ and $n\geq 2$} \end{equation} and
\begin{equation} \label{eq:non-degn-u-14}
r^{-(2s-1+\a-\b)}_n [ \n g_n ]_{C^{\b}(B_{r_n})}\geq \frac{1}{2}\Theta_n(r_n)\geq \frac{n}{4} . \end{equation}
We define $$ u_n(x):=\frac{1}{r_n^{2s+\a}\Theta_n(r_n)} g(r_n x). $$
By \eqref{eq:sup-r-ok-Morrey-hi-Int-nab}, for $R\geq 1$, we have \begin{align*} [ \n u_n ]_{C^{\b}(B_{R})}= \frac{ r_n^{1+\b} }{r_n^{2s+\a}\Theta_n(r_n)} [ \n g_n ]_{C^{\b}(B_{r_n R})}\leq R^{ 2s-1+\a-\b} \frac{ \Theta_n(Rr_n) }{ \Theta_n(r_n)} . \end{align*} Hence by the monotonicity of $\Theta_n$, we have \begin{equation}\label{eq:un-semi-norm-groth} [ \n u_n ]_{C^{\b}(B_{R})} \leq R^{2s-1+\a-\b} \qquad\textrm{ for all $R\geq 1$.} \end{equation}
In addition, by \eqref{eq:non-degn-u-14}, we get $$
[ \n u_n ]_{C^{\b}(B_{1})} \geq \frac{1}{2} . $$
This then implies that there exists $x_n, h_n\in B_{1}$, with $h_n\not=0$, such that \begin{equation} \label{eq:non-deg-u-n}
|h_n|^{-\b} | \n u_n(x_n+ h_n)-\n u_n(x_n)| \geq \frac{1}{4} .
\end{equation}
We define the new sequence $$
v_n(x):= \frac{ u_n(x_n+ |h_n| x)- u_n(x_n)- |h_n|\n u_n(x_n)\cdot x}{ |h_n|^{1+\b} }. $$ By construction, we have that
\begin{equation} \label{eq:v_n0-nv_n0}
v_n(0)=|\n v_n(0)|=0 \end{equation} and by \eqref{eq:non-deg-u-n}, \begin{equation}\label{nonded-v-n}
|\n v_n (h_n/|h_n|)| \geq \frac{1}{4} . \end{equation}
Moreover by \eqref{eq:un-semi-norm-groth}, for $R\geq 1$ and $x, y\in B_R$, \begin{align*}
|\n v_n(x)-\n v_n (y)|&= |h_n|^{-\b} | \n u_n(x_n+ |h_n| x)- \n u_n(x_n+ |h_n| y) |\\
& \leq |h_n|^{-\b} |h_n|^{\b}|x-y|^{\b} (1+R)^{2s-1+\a-\b}\leq 2^{2s-1+\a-\b} |x-y|^{\b}R^{2s-1+\a-\b}. \end{align*}
Combining this with \eqref{eq:v_n0-nv_n0}, we get, for all $R\geq1$, \begin{equation} \label{eq:groht-w-n-Morrey-HI}
\|\n v_n\|_{L^{\infty}(B_R)}\leq CR^{2s-1+\a} \end{equation} and $$
\| v_n\|_{C^{1,\b}(B_R)}\leq C R^{2s+\a}, $$ for some $C=C(s,\a,\b)$. This latter estimate implies that there exists $v\in C^{1,\b}_{loc}(\mathbb{R}^N)$ such that, up to a subsequence, \begin{equation}\label{eq:lim-v-n} v_n \to v \qquad\textrm{in $C^{1}_{loc}(\mathbb{R}^N)$}. \end{equation} Moreover by \eqref{nonded-v-n} and \eqref{eq:v_n0-nv_n0}, there exists $e\in S^{N-1}$ such that \begin{equation}\label{eq:E1-nab00}
|\n v(e)|\geq \frac{1}{4} \qquad\textrm{ and } \qquad \n v(0)=0. \end{equation} We shall show that $\n v\equiv 0$ on $\mathbb{R}^N$, which leads to a contradiction. Indeed, given $h\in \mathbb{R}^N$, we define $w_n(x)=(v_n)_{h,0}(x)=v_n(x+h)-v_n(x)$. It follows from \eqref{eq:groht-w-n-Morrey-HI} and the fundamental theorem of calculus that \begin{equation}\label{eq:groht-w-n-Morrey-HI-nab}
\| w_n\|_{L^\infty(B_{R})}\leq C R^{2s-1+\a} \qquad\textrm{ for every $R\geq 1$,} \end{equation} where here and in the following of the proof, the letter $C$ is a positive constant only depending on $h,\kappa,N,s,\b$ and $\a$.
We put $\rho_n:=r_n |h_n|$, $z_n:=r_n x_n$ and we define $$ \overline K_n(x,y)=\rho_n^{N+2s} K_n( z_n+ \rho_n x, z_n+ \rho_n y) , $$ $$
\overline K_n'(x,y):=r_n^{-\a}|h_n|^{-\b} \left[\overline K_n(x+h,y+h)-\overline K_n(x,y) \right], $$ $$
U_n(x):= r_n^{-2s}|h_n|^{-1} g_n(z_n+ \rho_n x+ \rho_n h ) $$ and $$
\overline f_n(x):= r_n^{-\a-2s} |h_n|^{-\b-1}\rho_n^{2s} \left[ f_n(z_n+\rho_n x+\rho_n h ) - f_n(z_n+ \rho_n x ) \right]. $$ For $h\in \mathbb{R}^N\setminus \{0\},$ we let $n$ large so that $z_n+ h\in B_{1/2r_n}$, by changing variables and using \eqref{eq:g-nsolves-Schauder}, we then have that
$$
{\mathcal L}_{\overline K_n} w_n+\frac{1}{\Theta_n(r_n)} {\mathcal L}_{\overline K'_n} U_n = \frac{1}{\Theta_n(r_n)}\overline f_n \qquad\textrm{ in $B_{1/2r_n}$.} $$ Therefore by \eqref{eq:Operator-splitting}, \begin{equation} \label{eq:eqw_n--ok}
{\mathcal L}_{\overline K_n} w_n =F_n \qquad\textrm{ in $B_{1/2r_n}$,} \end{equation} where \begin{equation} \label{eq:def-F_n-Shaud}
F_n(x):=\frac{1}{\Theta_n(r_n)}\overline f_n- \frac{1}{\Theta_n(r_n)}{\mathcal E}^s_{{{\mathcal A}}_{e,\overline K_n'},U_n} - \frac{1}{\Theta_n(r_n)}{\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n'},U_n}. \end{equation} By a change of variable, we get \begin{align*} {\mathcal E}^s_{{{\mathcal A}}_{e,\overline K_n'},U_n}(x)
&=|h_n|^{2s-1}\int_{S^{N-1}}\int_0^\infty\delta ^e g_n(z_n+\rho_n x +\rho_n h,t,\theta ) {{\mathcal A}}_{e,\overline K_n'}(x,t/\rho_n,\th) t^{-1-2s}\, dt d\theta \end{align*} and \begin{align*} {\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n'},U_n}(x)
&=|h_n|^{2s-1}\int_{S^{N-1}}\int_0^\infty\delta ^o g_n(z_n+\rho_n x +\rho_n h,t,\theta ) {{\mathcal A}}_{o,\overline K_n'}(x,t/\rho_n,\th) t^{-1-2s}\, dt d\th. \end{align*} We recall that $$ {{\mathcal A}}_{\overline K_n}(x,r,\th):= {{\mathcal A}}_{ K_n}( z_n+\rho_n x,\rho_n r,\th) $$ and \begin{equation} \label{eq:will-saveme}
{{\mathcal A}}_{\overline K_n'}(x,r,\th):=r_n^{-\a}|h_n|^{-\b}\{ {{\mathcal A}}_{ K_n}(z_n+ \rho_n x+\rho_n h,\rho_n r,\th)- {{\mathcal A}}_{ K_n}(z_n+ \rho_n x,\rho_n r,\th) \}. \end{equation} Since $K_n\in\widetilde \scrK_{\a+2s-1} ^s(\kappa,\a,Q_\infty)$ (recall Definition \ref{def:Kernel-not-reg-the}), for all $x,y\in \mathbb{R}^N$, $\theta \in S^{N-1}$ and $r>0$, \begin{equation}\label{eq:assump-on-ti-l-Ko-inProp-n} \begin{aligned}
&| {{\mathcal A}}_{o,K_n}(x,r,\th) |\leq \frac{1}{\kappa}\min (r, 1)^{2s-1 +\alpha }, \\
&| {{\mathcal A}}_{o,K_n}(x,r,\th)- {{\mathcal A}}_{o,K_n}(y,r,\th)|\leq \frac{1}{\kappa} \min (r, |x -y|)^{2s-1 +\alpha }
\end{aligned} \end{equation} and $$
| {{\mathcal A}}_{K_n}(x,r,\th)- {{\mathcal A}}_{K_n}(y,r,\th)|\leq \frac{1}{\kappa} |x-y|^{\a}. $$ Therefore, $$
|{{\mathcal A}}_{e,\overline K_n'}(x,r,\th)|\leq C |h_n|^{\b}\leq C \qquad\textrm{ for all $x\in \mathbb{R}^N$, $r>0$ and $\th\in S^{N-1}$.} $$
Consequently, since $\|g_n\|_{C^{2s+\b}}\leq1$ and recalling \eqref{eq:delta-eo}, we have that
$$|\delta ^e g_n(z_n+\rho_n x +\rho_n h,t,\theta ) |\leq C\min (1,t^{2s+\b}) $$
and thus \begin{equation} \label{eq:estime-ovF--e}
\|{\mathcal E}^s_{{{\mathcal A}}_{e,\overline K_n'},U_n} \| _{L^\infty(\mathbb{R}^N)}\leq C. \end{equation} Moreover by \eqref{eq:assump-on-ti-l-Ko-inProp-n}, for all $x\in \mathbb{R}^N$, $r>0$ and $\th\in S^{N-1}$, we have $$
|{{\mathcal A}}_{o,\overline K_n'}(x,r,\th)|\leq C r_n^{-\a}|h_n|^{-\b} \min ( \rho_n , \rho_n r)^{2s-1+\alpha } .
$$
Since $|h_n| \leq 1$ and $\|g_n\|_{C^{0,1}(\mathbb{R}^N)}\leq 1$ (recalling \eqref{eq:delta-eo}), the above estimate implies that \begin{align*}
|{\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n'},U_n}(x)|&\leq Cr_n^{-\a}|h_n|^{-\b} \int_0^\infty\min (1,t) \min ( \rho_n, t)^{2s-1 +\alpha } t^{-1-2s }\, dt \\
&\leq Cr_n^{-\a}|h_n|^{-\b} \int_0^{\rho_n}t^{2s+\alpha } t^{-1-2s}\, dt\\
&+Cr_n^{-\a}|h_n|^{-\b} \rho_n^{2s-1+\alpha } \int_{\rho_n}^1 t^{-2s }\, dt +C r_n^{-\a}|h_n|^{-\b} \rho_n^{2s-1+\a} \int_1^\infty t^{-1-2s}\, dt . \end{align*}
Using that $2s>1$ and recalling that $\rho_n=r_n |h_n|$, we then conclude that \begin{equation} \label{eq:estime-ovF--o}
\|{\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n'},U_n} \| _{L^\infty(\mathbb{R}^N)}\leq C. \end{equation} Because $[f_n]_{C^\a(\mathbb{R}^N)}\leq 1$, it is plain that \begin{equation} \label{eq:estime-ovf_n-U_n}
\|\overline f_n\|_{L^\infty(\mathbb{R}^N)}\leq C. \end{equation}
Recalling \eqref{eq:def-F_n-Shaud}, it follow from \eqref{eq:estime-ovF--e}, \eqref{eq:estime-ovF--o} and \eqref{eq:estime-ovf_n-U_n} that
\begin{equation} \label{eq:estime-F_n-U_n}
\|F_n\|_{L^\infty(\mathbb{R}^N)}\leq \frac{C}{\Theta_n(r_n)}. \end{equation}
In view of \eqref{eq:lim-v-n} and \eqref{eq:groht-w-n-Morrey-HI-nab}, for every $h\in \mathbb{R}^N$, we have that \begin{equation} \label{eq:w_n-to-h-h-0}
w_n= (v_n)_{h,0}\to v_{h,0} \qquad \textrm{ in $L_s(\mathbb{R}^N)\cap H^s_{loc}(\mathbb{R}^N)$}. \end{equation} Letting $a_n(\th):={{\mathcal A}}_{\overline K_n}(z_n,0,\th)$, we have that $$
| {{\mathcal A}}_{\overline K_n}(x,r,\th)- a_n(\th) |\leq C \min( \rho_n^{\a} (|x|+r)^{\a} ,1)\qquad\textrm{ for all $x\in \mathbb{R}^N$, $r>0$ and $\theta \in S^{N-1}$,} $$ so that \begin{equation} \label{eq:ov-not-far-mu-a-n}
|\overline K_n(x,y)-\mu_{a_n}(x,y) |\leq C \min( \rho_n^{\a} (|x|+|x-y|)^{\a} ,1) \mu_1(x,y) \qquad \textrm{ for all $x\not=y\in \mathbb{R}^N$.} \end{equation}
Moreover $a_n$ satisfies \eqref{eq:def-a-anisotropi} for all $n$. Therefore in view of Lemma \ref{lem:Ds-a-n--to-La}, Lemma \ref{lem:convergence-very-weak}, \eqref{eq:w_n-to-h-h-0}, \eqref{eq:ov-not-far-mu-a-n} and \eqref{eq:estime-F_n-U_n}, passing to the limit in \eqref{eq:eqw_n--ok}, we deduce that \begin{equation} \label{eq:eq-Lb-v-h0} {\mathcal L}_{\mu_b} v_{h,0}=0 \qquad\textrm{ in $\mathbb{R}^N$ }, \end{equation} were $b$ is the weak-star limit of $a_n $ in $L^\infty(S^{N-1})$. Furthermore by \eqref{eq:groht-w-n-Morrey-HI}, $$
\|v_{h,0}\|_{L^\infty(B_R)}\leq C R^{2s-1+\a}, $$
Thanks to \eqref{eq:eq-Lb-v-h0} and since $2s-1+\a<1$, we can apply Lemma \ref{lem:Liouville} to get a constant $c=c(h,\a,\b,N,s,\kappa)$ such that $v_{h,0}(x)=v(x+h)-v(x)= c$ for all $x,h\in \mathbb{R}^N$. Hence, since $\n v(0)=0$, we find that $\n v(h)=0$ for all $h\in \mathbb{R}^N$. This contradicts the first inequality in \eqref{eq:E1-nab00}. The proof of $(i)$ is thus finished.\\
The proof of $(ii)$ is similar to the one of $(i)$, we therefore give a sketch below, emphasizing the main differences. Indeed, following the proof, we put $$ \overline u_n (x)=\frac{1}{r_n^{2s+\a} \Theta_n(r_n) }\left\{g_n(r_n x)- \textbf{Q}_{r_n,g_n}(r_n x) \right\}, $$
with $\Theta_n(r)$ is a nonincreasing function as above, with $g_n$ replaced with $g_n-\textbf{Q}_{r,g_n}$. From the definition of $\textbf{Q}_{r,g_n}$, the monotonicity of $\Theta_n$ and the fact that $2s-1+\a>1$, we then get $\|\nabla \overline u_n\|_{C^\b(B_R)}\leq C R^{2s-1+\a}$ for all $R\geq 1$. On the other hand, there are $x_n\in B_1$ and $h_n\in B_1\setminus\{0\}$ such that $|\nabla \overline u_n(x_n+ h_n)-\nabla \overline u_n(x_n)| |h_n|^{-\b}\geq \frac{1}{4}$. Similarly as above, we define $$
\overline v_n(x):= \frac{ \overline u_n(x_n+ |h_n| x)- \overline u_n(x_n)- |h_n|\nabla \overline u_n(x_n)\cdot x}{ |h_n|^{1+\b} }, $$
so that $\|\nabla \overline v_n\|_{L^\infty(B_R)}\leq C R^{2s-1+\a}$ for all $R\geq 1$. Moreover, $\overline v_n$ is bounded in $C^{1,\b}_{loc}(\mathbb{R}^N)$, so that $\overline v_n\to \overline v$ in $C^{1}_{loc}(\mathbb{R}^N)$. In addition, \begin{equation} \label{eq:to-contradict-ov-v}
\n\overline v(0)=0 \qquad \textrm{ and } \qquad |\nabla \overline v(e)|\geq \frac{1}{4},\quad\textrm{ for some $e\in S^{N-1}$.}
\end{equation} and $\|\overline v(\cdot +h)- \overline v\|_{L^\infty(B_R)}\leq C R^{2s-1+\a}$ for all $R\geq 1$. Next, letting, $\overline w_n(x)= (\overline v_n)_{h,0}(x)=\overline v_n(x+h)-\overline v_n(x)$ we find that $$
{\mathcal L}_{\overline K_n} \overline w_n - \frac{2\rho_n^2}{ r_n^{2s+\a} |h_n|^{1+\b}\Theta_n(r_n)} \sum_{i,j=1}^N T_{ij}^{r_n,g_n} h_j {\mathcal L}_{\overline K_n} x_i =F_n \qquad\textrm{ in $B_{1/2r_n}$,} $$ where $F_n$ is given by \eqref{eq:def-F_n-Shaud}.
By \eqref{eq:Operator-splitting} and the fact that ${\mathcal E}^s_{{{\mathcal A}}_{e,\overline K_n},x_i}\equiv 0$ on $\mathbb{R}^N$, we then get \begin{equation} \label{eq:eq-ovw_n--ok}
{\mathcal L}_{\overline K_n} \overline w_n =F_n+ \frac{2\rho_n^2}{ r_n^{2s+\a} |h_n|^{1+\b}\Theta_n(r_n)} \sum_{i,j=1}^N T_{ij}^{r_n,g_n} h_j {\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n},x_i} \qquad\textrm{ in $B_{1/2r_n}$.} \end{equation} We start by estimating $F_n$ defined in \eqref{eq:def-F_n-Shaud}. Thanks to \eqref{eq:will-saveme}, we have \begin{align*}
&{{\mathcal A}}_{o,\overline K_n'}(x,r,\th)= r_n^{-\a}|h_n|^{-\b} \rho_n\int_0^1D_x {{\mathcal A}}_{o, K_n}(z_n+ \rho_n x+\varrho \rho_n h,\rho_n r,\th)h d\varrho\\
&= r_n^{-\a}|h_n|^{-\b} \rho_n r\int_0^1 \left[D_r {{\mathcal A}}_{ o, K_n}(z_n+ \rho_n x+ \rho_n h,\varrho \rho_n r,\th)- D_r {{\mathcal A}}_{ o, K_n}(z_n+ \rho_n x,\varrho \rho_n r,\th) \right] d\varrho. \end{align*} Recall that $ K_n\in \widetilde \scrK^s_0(\kappa, 1+ 2s-2+\a, Q_\infty)$ with $1>2s-2+\a>0$, and so by definition, $$
\|D_r {\mathcal A}_{K_n}\|_{ C^{2s+\a-2}(Q_\infty)\times L^\infty(S^{N-1} )}+ \|D_x {\mathcal A}_{o,K_n}\|_{L^\infty_{2s+\a-2}(Q_\infty)\times L^\infty(S^{N-1} )}\leq C.
$$ It follows that $$
| {\mathcal A}_{o,\overline K_n'}(x,r,\th) |\leq C r_n^{-\a}|h_n|^{-\b} \min ( \rho_n r \rho_n^{2s+\a-2}, \rho_n (\rho_n r)^{2s+\a-2}). $$
From this and the fact that $|\delta ^og_n(y,t,\th)|\leq t$, we deduce that \begin{align*}
|{\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n'},U_n}(x)|& \leq Cr_n^{-2s-\a}|h_n|^{-1-\b} \int_0^\infty\rho_n r \min ( \rho_n r \rho_n^{2s+\a-2}, \rho_n (\rho_n r)^{2s+\a-2})r^{-1-2s }\, dr \\
&= Cr_n^{-2s-\a}|h_n|^{-1-\b}\rho_n^{2s} \int_0^\infty \min ( t \rho_n^{2s+\a-2}, \rho_n t^{2s+\a-2})t^{-2s }\, dt \\
&\leq C r_n^{-2s-\a}|h_n|^{-1-\b}\rho_n^{2s} \int_0^{\rho_n} \rho_n^{2s+\a-2} t^{1-2s }\, dt+ C r_n^{-2s-\a}|h_n|^{-1-\b}\rho_n^{2s} \int_{\rho_n}^\infty \rho_n t^{\a-2} \, dt . \end{align*}
Therefore, $\|{\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n'},U_n} \| _{L^\infty(\mathbb{R}^N)}\leq C$. By combining this with \eqref{eq:estime-ovF--e} and \eqref{eq:estime-ovf_n-U_n}, we get $$\|F_n\|_{L^\infty(\mathbb{R}^N)}\leq \frac{C}{\Theta_n(r_n)}.$$ Next, we estimate the last term in the right hand in \eqref{eq:eq-ovw_n--ok}. From the first inequality in \eqref{eq:assump-on-ti-l-Ko-inProp-n}, we get \begin{equation} \label{eq:estim-cOs-Schauder}
|{\mathcal O}^s_{{{\mathcal A}}_{o,\overline K_n},x_i} (x)|\leq C \int_0^\infty \min (\rho_n r, 1) r^{-2s}\, dr\leq C \rho_n^{2s-1}. \end{equation}
Since $g_n(0)=|\n g_n(0)|=0$ and $\|g_n\|_{C^{2s+\b}(\mathbb{R}^N)}\leq 1$, we then have that $| T_{ij}^{r_n,g_n} |\leq C r_n^{-2+2s+\b}$. Hence, since $2s+\b\geq 1+\a$, by \eqref{eq:estim-cOs-Schauder}, the last term in the right hand in \eqref{eq:eq-ovw_n--ok} is bounded by $\frac{C}{\Theta_n(r_n)}$. Since $\frac{1}{\Theta_n(r_n)}$ tends to zero as $n\to\infty$, thanks to Lemma \ref{lem:convergence-very-weak}, we can pass to the limit in \eqref{eq:eq-ovw_n--ok}, to get ${\mathcal L}_{\mu_b} \overline v_{h,0}=0 $ in $\mathbb{R}^N$. Hence, since $2s-1+\a<2s$, Lemma \ref{lem:Liouville} implies that, there exist $\overline c\in \mathbb{R}$ and $\overline d\in \mathbb{R}^N$, only depending on $h,\a,\b,N,s$ and $\kappa$, such that $ \overline v_{h,0}(x)= \overline v(x+h)-\overline v(x)=\overline d\cdot x+\overline c $ for all $x,h\in \mathbb{R}^N$. Now, since $\nabla \overline v(0)=0$, we find $\nabla \overline v\equiv 0$ on $\mathbb{R}^N$. This contradicts \eqref{eq:to-contradict-ov-v} and the proof of $(ii)$ is finished.\\
The proof of $(iii)$ follows (verbatim) the same argument as the one of $(i)$. The fact that $K_n\in \widetilde \scrK_{\a+\e}^{1/2}(\kappa,\a,Q_\infty)$ is only needed to deduce the uniform bound $\|{\mathcal O}^{1/2}_{{{\mathcal A}}_{o,\overline K_n'},U_n}\|_{L^\infty(\mathbb{R}^N)}\leq C$ from the uniform estimate $
|{{\mathcal A}}_{o,\overline K_n'}(x,r,\th)|\leq C r_n^{-\a}|h_n|^{-\a/2} \min ( \rho_n , \rho_n r)^{ \a+\varepsilon } .
$\\
The proof of $(iv)$ does not differ much from the one of $(i)$. We skip the details.
\end{proof}
As a consequence of the previous result, we have the following
\begin{theorem}\label{th:Schauder-000} Let $s\in (0,1)$, $N\geq 1$, $\kappa >0$ and $\a\in (0,1)$. Let $K\in \widetilde \scrK_{\t}^s(\kappa, \a, Q_\infty)$ with $\t=\min(\a+(2s-1)_+,1)$.
Let $u\in H^s(B_2)\cap C^\a(\mathbb{R}^N)$ and $f\in C^\a(B_2)$ satisfy
$$
{\mathcal L}_K u= f\qquad \textrm{in $B_2$}.
$$ \begin{itemize} \item[$(i)$] If $2>2s+\a>1$, $2s+\a\not=2$ and $2s\not=1 $, then there exists $C=C(N,s,\kappa,\a)>0$ such that
$$
\|u\|_{C^{2s+\a}(B_1)}\leq C(\|u\|_{C^\a(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}). $$ \item[$(ii)$] If $2s+\a>2$ and $K\in \widetilde \scrK_{0}^s(\kappa, 2s-1+\a, Q_\infty)$ then there exists $C=C(N,s,\kappa,\a)>0$ such that
$$
\|u\|_{C^{2s+\a}(B_1)}\leq C(\|u\|_{C^\a(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}). $$
\item[$(iii)$] If $2s=1$ and $K\in \widetilde \scrK_{\a+\e}^s(\kappa, \a, Q_\infty)$, for some $\e>0$, then there exists $C=C(N,s,\kappa,\a,\e)>0$ such that
$$
\|u\|_{C^{1+\a}(B_1)}\leq C(\|u\|_{C^\a(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}). $$
\item[$(iv)$] If $2s+\alpha <1$, then there exists $C=C(N,s,\kappa,\a)>0$ such that
$$
\|u\|_{C^{2s+\a}(B_1)}\leq C(\|u\|_{C^\a(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}). $$ \end{itemize}
\end{theorem}
\begin{proof}
From \cite{Fall-Reg}, there exists $\b\in (0,\a)$, only depending on $N,s,\kappa$ and $\a$ such that, if $2s+\b\not \in \mathbb{N}$, \begin{equation} \label{eq:u-2s-b-proof}
\|u\|_{C^{2s+\b}(B_1)}\leq C (\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} ), \end{equation}
for some $C=C(N,s,\a,\b,\kappa)$.\\ \noindent \textbf{Case 1: $1<2s+\a<2$.}
For $z\in B_{1}$, we define \begin{equation} \label{eq:defin-g_z}
g_z(x)= u(x+z)-u(z)- \vp_4(x) \n u(z)\cdot x
\end{equation} which satisfies $g_z(0)=|\n g_z(0)|=0$.
We introduce the cut-off function $\vp_4$ only because the functions $x\mapsto x_i$, for $i=1,\dots,N$, do not belong to ${L_s(\mathbb{R}^N)}$ when $2s=1$.
By construction and \eqref{eq:u-2s-b-proof}, we have \begin{equation} \label{eq:estim-d-C2s-e_0}
\|g_z\|_{C^{2s+\b}(\mathbb{R}^N)}\leq C \|u\|_{C^{2s+\b}(B_1)}\leq C (\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} ). \end{equation}
In addition, by Lemma \ref{lem:estim-G_Ku}$(iii)$,
$$
{\mathcal L}_{K_z} g_z = f(\cdot + z) -{\mathcal L}_{K_z} U \qquad\textrm{ in $B_{1}$}, $$ where $K_z(x,y)=K(x+z,y+z)$, $U(x):= \vp_4(x) \n u(z)\cdot x$. From Lemma \ref{lem:estim-G_Ku}$(iii)$, we get
$$
{\mathcal L}_{K_z} (\vp_{1/2}g_z) =\widetilde f -{\mathcal L}_{K_z} U \qquad\textrm{ in $B_{1/8}$}, $$
for some function $\widetilde f$, satisfying \begin{equation} \label{eq:ti-Calpba}
\|\widetilde f\|_{C^{\a}(\mathbb{R}^N)}\leq C (\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} ). \end{equation}
Using \eqref{eq:Operator-splitting}, we then have that \begin{equation} \label{eq:clK_zcatg-z}
{\mathcal L}_{K_z} (\vp_{1/2}g_z) =F \qquad\textrm{ in $B_{1/8}$,} \end{equation}
where $F:=\widetilde f -{\mathcal E}^s_{{{\mathcal A}}_{e,K}, U} -{\mathcal O}^s_{{{\mathcal A}}_{o,K}, U}$. Because $K_z\in \widetilde \scrK_{\a+2s-1}^s(\kappa,\a,Q_\infty)$ for $2s> 1$ and $K_z\in \widetilde \scrK_{\a+\e}^s(\kappa,\a,Q_\infty)$ for $2s=1$ and since $U\in C^2_c(\mathbb{R}^N)$, we deduce from Lemma \ref{lem:2sp-alph-estim-F_Keo} that $$
\|{\mathcal E}^s_{{{\mathcal A}}_{e,K_z}, U}\|_{C^\a(\mathbb{R}^N)}+ \|{\mathcal O}^s_{{{\mathcal A}}_{o,K_z}, U}\|_{C^\a(\mathbb{R}^N)} \leq C(N,s,\a) \|\n u\|_{L^\infty(B_1)}. $$ This with \eqref{eq:ti-Calpba} and \eqref{eq:u-2s-b-proof} imply that \begin{equation}\label{eq:estim-F-a-Shaud}
\|F\|_{C^{\a}(\mathbb{R}^N)}\leq C (\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} ). \end{equation} Thanks to \eqref{eq:clK_zcatg-z}, applying Proposition \ref{prop:bound-wth-corrector-nab}$(i)$ and $(iii)$ and using \eqref{eq:estim-F-a-Shaud}, provided $1<2s+\a<2$, we get $$
\|\nabla (\vp_{1/2}g_z)\|_{L^\infty(B_r)}\leq C r^{2s-1+\a}\left( \|\vp_{1/2}g_z\|_{C^{2s+\b}(\mathbb{R}^N)}+ \|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}}\right). $$
As a consequence, by \eqref{eq:estim-d-C2s-e_0}, for all $r\in (0,1/2)$ and all $z\in B_{1}$, $$
\|\n u- \n u(z)\|_{L^\infty(B_r(z))}\leq C r^{2s-1+\a}\left( \|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} \right), $$ for some $C=C(N,s,\a,\kappa)$.
We then conclude that $$
\|\n u\|_{C^{2s-1+\a}(B_{1/8})}\leq C \left( \|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}}\right). $$ Therefore $(i)$ and $(iii)$ follow from a covering and scaling argument.\\ \noindent \textbf{Case 2: $2s+\a>2$.} We know from \textbf{Case 1} that for all $\b\in (0, 2-2s)$, \begin{equation} \label{eq:u-2s-b-proof00}
\|u\|_{C^{2s+\b}(B_1)}\leq C (\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} ). \end{equation} We then consider the function $g_z$ defined in \eqref{eq:defin-g_z}. Hence, thanks to \eqref{eq:clK_zcatg-z}, by Proposition \ref{prop:bound-wth-corrector-nab}$(ii)$, \eqref{eq:u-2s-b-proof00} and \eqref{eq:estim-F-a-Shaud} we get \begin{equation} \label{eq:eq-to-iter-C2}
\|\nabla (\vp_{1/2}g_z)-\nabla \textbf{Q}_{r, \vp_{1/2}g_z}\|_{L^\infty(B_r)}\leq C r^{2s-1+\a} (\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} ), \end{equation} provided $2s+\b\geq 1+\a$.
In view of \eqref{eq:eq-to-iter-C2}, we can use an iteration argument to obtain, for all $r\in (0,1/2)$, $$
\|\n u-\n u(z)-M_z (\cdot-z)\|_{L^\infty(B_r(z))}\leq C r^{2s-1+\a} (\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} ). $$
for some $(N\times N)$-matrix $M_z$ satisfying $|M_z|\leq C(\|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}} )$. Since $2s-1+\a>1$, we deduce that $M_z = D^2 u(z) $.
Using now an extension theorem, see e.g. \cite{Stein}[Page 177], we conclude that $$
\|\n u\|_{C^{1,2s-2+\a}(B_{1/8})}\leq C \left( \|u\|_{C^{\a}(\mathbb{R}^N)}+ \|f\|_{C^{\a}{(B_2)}}\right). $$ We thus get $(iii)$ after a covering and scaling argument.\\ \noindent \textbf{Case 3: $2s+\a<1$.} Here, we argue as in \textbf{Case 1}, by applying Proposition \ref{prop:bound-wth-corrector-nab}$(iii)$ to the function $ g_z(x)= \vp_{1/2}(x)\{u(x+z)-u(z)\} $. We skip the details.
\end{proof}
By an induction argument, we have the following result. \begin{theorem}\label{th:C-2s-k-al-reg-nonreg-theta} Let $N\geq 1$, $s\in (0,1)$ and $\a\in (0,1)$. Let $\kappa>0$, $k\in\mathbb{N}$ and $K\in\widetilde \scrK_{\t} ^s (\kappa,k+ \alpha , Q_\infty)$, with $\t=\min(1,\a+(2s-1)_+)$.
Let $u\in H^s(B_2)\cap C^{k+\a}(\mathbb{R}^N)$ and $f\in C^{k+\a}(\mathbb{R}^N)$
such that $$
{\mathcal L}_K u =f \qquad\textrm{ in $B_2$}. $$
\begin{itemize} \item[$(i)$] If $2s\not=1$ and $2s+\a<2$, then there exists $C=C(N,s,k,\kappa,\a)$ such that
$$
\|u\|_{C^{k+2s+\a}(B_1)}\leq C (\|u\|_{C^{k+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k+\a}(\mathbb{R}^N)} ).
$$
\item[$(ii)$] If $2s=1$ and $K\in\widetilde \scrK_{\a+\e}^{1/2}(\kappa,k+ \alpha , Q_\infty)$, for some $\e>0$, then there exists $C=C(N,k,\kappa,\a,\e)$ such that
$$
\|u\|_{C^{k+1+\a}(B_1)}\leq C (\|u\|_{C^{k+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k+\a}(\mathbb{R}^N)} ).
$$
\item[$(iii)$] If $2s+\a>2$ and $K\in \widetilde \scrK_{0}^s(\kappa, k+2s-1+\a, Q_\infty)$, then there exists $C=C(N,s,\kappa,\a)>0$ such that
$$
\|u\|_{C^{k+2s+\a}(B_1)}\leq C(\|u\|_{C^\a(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}). $$ \end{itemize}
\end{theorem} \begin{proof}
We will prove $(i)$ and $(ii)$, since the one of $(iii)$ will follow the same arguments. The case $k=0$, that $u\in C^{2s+\a}(B_1)$, is proved in Thoerem \ref{th:Schauder-000}. We prove the statement first for $k=1$. By Lemma \ref{lem:estim-G_Ku}$(ii)$, we have that $u^1:=\vp_{1} u\in C^{2s+\a}(\mathbb{R}^N)\cap C^{1,\a}(\mathbb{R}^N)$ and $$
{\mathcal L}_K u^1=f^1 \qquad\textrm{ in $B_{2^{-2}}$,} $$ for some function $f^1$ satisfying \begin{equation}\label{eq:estim--nab-f--1-C-a}
\| f^1\|_{C^{1+\a}(B_{2^{-2}})}\leq C (\|u\|_{C^{1+\a}(\mathbb{R}^N)}+ \| f\|_{C^{1+\a}(\mathbb{R}^N)}) . \end{equation} Let $h \in B_{2^{-4}}$, with $h\not=0$. Then (recalling \eqref{eq:def-f-h-alph}) $$
{\mathcal L}_K u^1_{h,1}-{\mathcal L}_{K_{h,1}} u^1(\cdot+h) = f_{h,1}^1 \qquad\textrm{ in $B_{2^{-4}}$}. $$ We note that $K_{h,1} $ satisfies \eqref{eq:K'-Kernel-satisf}, with $\a'=0$. By Corollary \ref{cor:Holder-cont-coeff}, we obtain $u_{h,1}^1$ is uniformly bounded in $C^\s(B_{2^{-5}})$, for some $\s\in (s,2s)$. Therefore $\n u^1\in C^{0,\s}(B_{2^{-5}}) \subset H^s( B_{2^{-5}})$. Using \eqref{eq:Operator-splitting}, we then have that $$
{\mathcal L}_K u^1_{h,1}- {\mathcal E}^s_{{{\mathcal A}}_{e, K_{h,1}}, u^1_h}- {\mathcal O}^s_{{{\mathcal A}}_{o,K_{h,1}}, u^1_h}= f_{h,1}^1 \qquad\textrm{ in $B_{2^{-4}}$}. $$ Letting $h\to 0$, we see that, for all $i_1\in\{1,\dots,N\}$, $$
{\mathcal L}_K \partial_{i_1} u^1= {\mathcal E}^s_{\partial_{i_1}{{\mathcal A}}_{e,K} , u^1}+ {\mathcal O}^s_{\partial_{i_1}{{\mathcal A}}_{o,K} , u^1} + \partial_{i_1} f^1=:f^1_{i_1} \qquad\textrm{ in $B_{2^{-5}}$}. $$ By Lemma \ref{lem:2sp-alph-estim-F_Keo} and \eqref{eq:estim--nab-f--1-C-a}, if $2s\not=1$, the right hand-side in the above display belongs to $C^\a(\mathbb{R}^N)$ and satisfies $$
\|f^1_{i_1} \|_{C^\a(B_{2^{-2}})}\leq C (\|u\|_{C^{1+\a}(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}) . $$ On the other hand if $2s=1$, then Lemma \ref{lem:2sp-alph-estim-F_Keo} and \eqref{eq:estim--nab-f--1-C-a} yield $$
\|f^1_{i_1} \|_{C^{\a-\delta }(B_{2^{-2}})}\leq C (\|u\|_{C^{1+\a}(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}) , $$ for all $\delta \in (0,1-\a)$.
It follows from Theorem \ref{th:Schauder-000} that if $2s\not=1$, then $$
\|\partial_{i_1} u^1\|_{C^{2s+\a}(B_{2^{-6}})} \leq C (\|u\|_{C^{1+\a}(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}) $$ and for $2s=1$, $$
\|\partial_{i_1} u^1\|_{C^{2s+\a-\delta }(B_{2^{-6}})} \leq C (\|u\|_{C^{1+\a}(\mathbb{R}^N)}+ \|f\|_{C^\a(B_2)}) . $$ We now remove the $\delta $ in the above estimate (for $2s=1$). Indeed, we define $\overline u^1:=\vp_{2^{-7}} u^1 \in C^{2s+\a+(1-\delta )}(\mathbb{R}^N)$ which, by Lemma \ref{lem:estim-G_Ku}$(iii)$, satisfies $$ {\mathcal L}_K \overline u^1=\overline f^1 \qquad\textrm{ in $B_{2^{-8}}$}, $$ with $$
\|\overline f^1\|_{C^\a(\mathbb{R}^N)}\leq C ( \|u\|_{C^\a(\mathbb{R}^N)} + \|f\|_{C^\a(\mathbb{R}^N)} ). $$ Therefore proceeding as above, we have \begin{equation} \label{eq:cL_K-dui1ovu}
{\mathcal L}_K \partial_{i_1} \overline u^1= {\mathcal E}^s_{\partial_{i_1}{{\mathcal A}}_{e,K} , \overline u^1}+ {\mathcal O}^s_{\partial_{i_1}{{\mathcal A}}_{o,K} , \overline u^1} + \partial_{i_1} \overline f^1=:\overline f^1_{i_1} \qquad\textrm{ in $B_{2^{-9}}$}. \end{equation}
Since $K\in \widetilde \scrK_{\a+\e} ^{1/2}(\kappa,\a,Q_\infty)$ and $\overline u^1:=\vp_{2^{-7}} u^1 \in C^{2s+\a+1-\delta }(\mathbb{R}^N)$, Lemma \ref{lem:2sp-alph-estim-F_Keo} yields $$
\| {\mathcal E}^s_{\partial_{i_1}{{\mathcal A}}_{e,K} , \overline u^1} \|_{C^{\a}(\mathbb{R}^N)}+\| {\mathcal O}^s _{\partial_{i_1}{{\mathcal A}}_{o,K} , \overline u^1} \|_{C^{\a}(\mathbb{R}^N)}\leq C \|\overline u^1 \|_{C^{2s+\a+(1-\delta )}(\mathbb{R}^N)}. $$ Applying Theorem \ref{th:Schauder-000} to the equation \eqref{eq:cL_K-dui1ovu}, we then get $$
\|\partial_{i_1} \overline u^1\|_{C^{1+\a}(B_{2^{-10}})} \leq C (\| \overline u^1 \|_{C^{1+\a}(\mathbb{R}^N)}+ \|\overline f^1_{i_1}\|_{C^\a(B_2)} ) . $$ The theorem is thus proved for $k=2.$\\
Let $k> 2$.
We now prove by induction that for every $(i_1,i_2,\dots, i_k)\in \{1,\dots,N \}^k$ there exist a constant $r_k$, only depending on $k$, and a constant $ C_k>0$, only depending on $N,s,\kappa,\a$ and $k$, such that \begin{equation} \label{eq:u-C2s-k-a-induction}
\|\partial_{i_1i_2\dots i_k}^k u\|_{C^{2s+\a+k}(B_{r_k})} \leq C_k (\|u\|_{C^{k+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k+\a}(B_2)}). \end{equation} We assume, as induction hypothesis that, the result is true up order $k-1$. That is, there exist $r_{k-1}, C_{k-1}>0$, as above, such that \begin{equation}\label{eq:induc-hyp}
\| u\|_{C^{2s+k-1+\a}(B_{r_{k-1}})} \leq C_{k-1}(\|u\|_{C^{k-1+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k-1+\a}(B_2)}). \end{equation}
We then consider $$ u^k:=\vp_{r_{k-1}/2} u\in C^{2s+\a+k-1}(\mathbb{R}^N)\cap C^{k,\a}(\mathbb{R}^N). $$
By Lemma \ref{lem:estim-G_Ku}$(ii)$, we then have that $$
{\mathcal L}_K u^k = f^k \qquad\textrm{ in $B_{r_{k-1}/4}$}, $$ for some function \begin{equation}\label{eq:f-k-hold-k-a}
\|f^k \|_{C^{k+\a}(B_{r_{k-1}/4})}\leq C_k' (\|u\|_{C^{k+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k+\a}(B_2)}) , \end{equation} where, unless otherwise stated, $C_k'$ denotes a positive constant, only depending on $N,s,\kappa,\a$ and $k$. Proceeding as above, we can differentiate the equation $k$ times to deduce that for all $(i_1,i_2,\dots, i_k)\in \{1,\dots,N \}^k$, \begin{equation} \label{eq:cL-k-u-k-induction}
{\mathcal L}_K \partial_{i_1i_2\dots i_k}^k u^k = g^k+ \partial_{i_1i_2\dots i_k}^k f^k \qquad\textrm{ in $B_{r_{k}'}$}, \end{equation} for constant $r_k'<r_k$, only depending on $r_k$ and $k$, and for some function $g^k:=\sum_{j=1}^m c_j^e {\mathcal E}^s_{a_j^e, v_j}+ \sum_{j=1}^m c_j^o {\mathcal O}^s_{a_j^o, w_j}$ where $c_j^e,c_j^o$ are real numbers, $a_j^e$, $a_j^o$, $v_j$ and $w_j$ are respectively given by the partial derivatives in $x$ of ${{\mathcal A}}_{e,K}$, ${{\mathcal A}}_{o,K}$ up to order $k$ and $v_j$ together with $w_j$ are given by partial derivatives of $u^k$ up to order $k-1$. Therefore, provided $2s\not=1$, by Lemma \ref{lem:2sp-alph-estim-F_Keo}, $$
\|g^k\|_{C^\a(\mathbb{R}^N)}\leq C_k'\|u^k\|_{C^{2s+\a+k-1}(\mathbb{R}^N)}\leq C ( \|u\|_{C^{k-1+\a}(\mathbb{R}^N)}+\|f^k\|_{C^{k,\a}(\mathbb{R}^N)}). $$
Now Theorem \ref{th:Schauder-000} implies that, for $2s\not=1$, $$
\| \partial_{i_1i_2\dots i_k}^k u^k\|_{C^{2s+\a}(B_{r_k'/2})} \leq C_k' ( \|\partial_{i_1i_2\dots i_{k-1}}^{k} u^k\|_{C^{ \a}(\mathbb{R}^N)}+\|g^k\|_{C^{\a}(\mathbb{R}^N)}+\|f^k\|_{C^{k,\a}(\mathbb{R}^N)}). $$ By \eqref{eq:f-k-hold-k-a}, we get \eqref{eq:u-C2s-k-a-induction} in the case $2s\not=1$. Therefore $(i)$ follows by a covering and scaling argument.\\
Now when $2s=1$, then we can argue similarly as above, noticing that, under the induction hypothesis \eqref{eq:induc-hyp}, by Lemma \ref{lem:2sp-alph-estim-F_Keo}, the function $g^k$ in the right hand side of \eqref{eq:cL-k-u-k-induction} belongs to $C^{\a-\delta }(\mathbb{R}^N) $, for all $\delta \in (0,\a)$. Hence Theorem \ref{th:Schauder-000} implies that, for all $\delta \in (0,\a)$, $$
\| \partial_{i_1i_2\dots i_k}^k u^k\|_{C^{2s+\a-\delta }(B_{r_k'/2})} \leq C_k'' ( \|\partial_{i_1i_2\dots i_{k-1}}^{k} u^k\|_{C^{ \a}(\mathbb{R}^N)}+\|g^k\|_{C^{\a-\delta }(\mathbb{R}^N)}+\|f^k\|_{C^{k,\a}(\mathbb{R}^N)}), $$ where, unless otherwise stated, $C_k''$ denotes a positive constant, only depending on $N,\kappa,\a,\delta ,\e$ and $k$. To remove the parameter $\delta $, we consider $$ \overline u^k:=\vp_{r_{k}'/4} u\in C^{k+1+\a-\delta }(\mathbb{R}^N) , $$ which satisfies \begin{equation}\label{eq:estim-ov-u-k}
\|\overline u^k\|_{ C^{k +\a+ (1-\delta )}(\mathbb{R}^N)} \leq C_k'' ( \|u\|_{C^{k+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k+\a}(\mathbb{R}^N)}). \end{equation}
By Lemma \ref{lem:estim-G_Ku}$(ii)$, we then have that $$
{\mathcal L}_K \overline u^k = \overline f^k \qquad\textrm{ in $B_{r_{k}'/8}$}, $$ for some function \begin{equation}\label{eq:ov-f-k-hold-k-a}
\|\overline f^k \|_{C^{k+\a}(B_{r_k'/8})}\leq C_k'' (\|u\|_{C^{k+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k+\a}(\mathbb{R}^N)}) . \end{equation} As above, we then differentiate the equation $k$ times to deduce that for all $(i_1,i_2,\dots, i_k)\in \{1,\dots,N \}^k$, \begin{equation} \label{eq:cL-k-u-k-induction}
{\mathcal L}_K \partial_{i_1i_2\dots i_k}^k \overline u^k = \overline g^k+ \partial_{i_1i_2\dots i_k}^k \overline f^k \qquad\textrm{ in $B_{r_{k}''}$}, \end{equation} for constant $r_k''$, only depending on $k$, and for some function $\overline g^k(x):=\sum_{j=1}^m c_j^e {\mathcal E}^{1/2}_{a_j^e, v_j}+ \sum_{j=1}^m c_j^o {\mathcal O}^{1/2}_{a_j^o, w_j}$ where $c_j^e,c_j^o$ are real numbers and $a_j^e$, $a_j^o$ (resp. $v_j$ and $w_j$) are respectiveley given by the partial derivatives in $x$-variable of ${{\mathcal A}}_{e,K}$, ${{\mathcal A}}_{o,K}$ up to order $k$ (resp. $v_j$ together with $w_j$ are given by partial derivatives of $\overline u^k$ up to order $k-1$). Therefore by Lemma \ref{lem:2sp-alph-estim-F_Keo}, \eqref{eq:estim-ov-u-k}, and since $K\in \widetilde \scrK_{\a+\e}^{1/2}(\kappa,k+\a,\mathbb{R}^N)$, we obtain \begin{equation} \label{eq:ov-g-k-hold-k-a}
\|\overline g^k\|_{C^\a(\mathbb{R}^N)}\leq C_k''\|\overline u^k\|_{C^{1+\a+k-\delta }(\mathbb{R}^N)}\leq C_k'' ( \|u\|_{C^{k+\a}(\mathbb{R}^N)}+ \|f\|_{C^{k+\a}(\mathbb{R}^N)}) . \end{equation} Applying Theorem \ref{th:Schauder-000}, we conclude that $$
\| \partial_{i_1i_2\dots i_k}^k \overline u^k\|_{C^{1+\a}(B_{r_k''/2})} \leq C_k'' ( \|\partial_{i_1i_2\dots i_{k-1}}^{k-1} \overline u^k\|_{C^{ \a}(\mathbb{R}^N)}+\| \overline g^k\|_{C^ \a(\mathbb{R}^N)}+\| \overline f^k\|_{C^{k,\a}(\mathbb{R}^N)}). $$ Hence, since $\overline u^k=u$ on $B_{r_k''/2}$, by \eqref{eq:estim-ov-u-k}, \eqref{eq:ov-f-k-hold-k-a} and \eqref {eq:ov-g-k-hold-k-a} with then obtain \eqref{eq:u-C2s-k-a-induction}. Now $(ii)$ follows by scaling and covering. \end{proof}
\section{Proof of the main results}\label{s:proofMainResults} We start this section with the following result which shows how to pass from a nonlocal equation with kernels in $\widetilde \scrK_{\t}^s(\kappa,m+\a,Q_{\delta })$ to a nonlocal equation driven by kernels in $\widetilde \scrK^s_\t(\kappa,m+\a,Q_\infty ) $.
\begin{lemma}\label{lem:cut-reg-kernel}
Let $K\in \widetilde \scrK_{\t}^s(\kappa,m+\a,Q_{4R})$, for some $\a\in [0,1)$, $\t\in [0,1]$, $m\in \mathbb{N}$ and $R>0$. Let $v\in H^s(B_{4R})\cap {L_s(\mathbb{R}^N)} $ and $f\in L^1_{loc}(B_{4R})$ satisfy
$$
{\mathcal L}_K v= f\qquad \textrm{in $B_{4R}$.}
$$
Let
$$
\overline K(x,y)=\vp_{2R}(x)\vp_{2R}(y)K(x,y)+(2-\vp_{R}(x)-\vp_{R}(y)) \mu_1(x,y). $$ Then \begin{equation} \label{eq:new-eq-v}
{\mathcal L}_{\overline K} v+ \overline Vv= f+\overline f\qquad \textrm{in $B_{R/4}$,} \end{equation}
where, for $x\in B_{R/4}$,
$$
\overline V(x)=G_{1,K,2R}(x)-G_{1,\mu_1,R}(x), \qquad \overline f(x)=G_{v,K,2R}(x)-G_{v,\mu_1,R}(x),
$$ and $G_{v,K,\rho }$ is given by \eqref{eq:def-GKvR}. In particular, $\overline K\in \widetilde \scrK_{\t}^s(\overline \kappa,m+\a,Q_{\infty})$, for some constant $\overline \kappa=\overline \kappa(\kappa,\a,m,R,\t,s,N)$.
\end{lemma}
\begin{proof} The proof of \eqref{eq:new-eq-v} is elementary, and we skip it. Next, we observe that $$ {{\mathcal A}}_{\overline K}(x,r,\th)=\vp_{2R}(x)\vp_{2R}(x+r\theta ){{\mathcal A}}_{ K}(x,r,\th)+ (2-\vp_{R}(x)-\vp_{R}(x+r\th)). $$ Recalling the definition of the cut-off function $\vp_R$ in the beginning of Section \ref{s:NotPrem}, we easily deduce that $$ \min( \kappa ,1 ) \leq {{\mathcal A}}_{\overline K}(x,r,\th)\leq \max (1/k, 4) \qquad\textrm{ for all $x\in \mathbb{R}^N$, $r\geq 0$, $\th\in S^{N-1}$.} $$ This in particular implies that $\overline K$ satisfies \eqref{eq:K-Kernel-satisf}. Moreover, it is also not difficult to check that $$
\| {{\mathcal A}}_{\overline K}\|_{C^{m,\a}(Q_\infty) \times L^\infty(S^{N-1} )}+ \|{{\mathcal A}}_{o,\overline K} \|_{ {\mathcal C}^{m}_\t(Q_\infty) \times L^\infty(S^{N-1} )}\leq C( \kappa,m,\a,\t,R). $$
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:main-th1} ]
As mentioned in the first section, the case $2s\leq 1$ was already proven in \cite{Fall-Reg}. Now the case $2s>1$ follows from Theorem \ref{th:abs-res-Propo}, Lemma \ref{lem:cut-reg-kernel}, Lemma \ref{lem:estim-G_Ku} and the fact that $L^p(\mathbb{R}^N) \hookrightarrow {\mathcal M}_{N/p}$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:Schauder-0}] First applying Theorem \ref{th:Schauder-000} and using Lemma \ref{lem:cut-reg-kernel} together with Lemma \ref{lem:estim-G_Ku}, we get the estimates. \end{proof}
\begin{proof}[Proof of Theorem \ref{th:main-th10} ]
It follows from Theorem \ref{th:main-th1}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:Schauder-0-intro} ]
By Lemma \ref{lem:cut-reg-kernel}, we have that \begin{equation} \label{eq:new-eq-for-u}
{\mathcal L}_{\overline K} u =f+\overline f-\overline Vu \qquad\textrm{ in $B_{1/2}$}, \end{equation} with $\overline K\in \widetilde \scrK_{\t_s}^s(\overline \kappa,m+\a+(2s-1)_+, Q_\infty)$ with $\t_s:= \a+(2s-1)_+$ for $2s+\a<2$ and $\t_s=0$ for $2s+\a>2$.
In addition, by Lemma \ref{lem:estim-G_Ku}$(iii)$, we have \begin{equation}\label{eq:estimV-fin}
\|\overline V\|_{C^{m+\a}(B_{1/2})}\leq C \end{equation} and \begin{equation}\label{eq:estimovf-fin}
\|\overline f\|_{C^{m+\a}(B_{1/2})}\leq C \|u\|_{{L_s(\mathbb{R}^N)} }. \end{equation}
We consider first the case $2s\not=1$.
Since $u$ satisfies \eqref{eq:new-eq-for-u}, applying Theorem \ref{th:C-2s-k-al-reg-nonreg-theta} and using Lemma \ref{lem:estim-G_Ku}$(iii)$, we get
$$
\|\varphi _{1/2}u\|_{C^{2s+m+\a}(B_{2-{4}})}\leq C ( \|\varphi _{1/2}u\|_{C^{m+\a}(\mathbb{R}^N)}+ \| u\|_{L^{\infty}(\mathbb{R}^N)} + \|F\|_{C^{m+\a}(B_{1/2})} ),
$$
where $F:=f+\overline f-\overline Vu$. Consequently, by \eqref{eq:estimV-fin} and \eqref{eq:estimovf-fin}
$$
\| u\|_{C^{2s+m+\a}(B_{2^{-4}})}\leq C ( \| u\|_{C^{m+\a}(B_1)}+ \| u\|_{L^{\infty}(\mathbb{R}^N)} + \|f\|_{C^{m+\a}(B_2)} ).
$$
Using now adimentional H\"older norms and interpolation (see e.g. \cite{GT, Barrios}), we can absorb the $C^{m+\a}(B_1)$-norm of $u$ to deduce that
$$
\| u\|_{C^{2s+m+\a}(B_{2^{-5}})}\leq C ( \| u\|_{L^{\infty}(\mathbb{R}^N)} + \|f\|_{C^{m+\a}(B_2)} ).
$$
If now $2s=1$, then since $\overline K\in \widetilde \scrK_{\alpha }^{1/2}(\overline \kappa,m+\a, Q_\infty)$ and in view of Theorem \ref{th:C-2s-k-al-reg-nonreg-theta}, the same arguments as above yield
$$
\| u\|_{C^{1+m+\a-\e}(B_{2^{-5}})}\leq C ( \| u\|_{L^{\infty}(\mathbb{R}^N)} + \|f\|_{C^{m+\a-\e}(B_2)} ),
$$
for all $\e\in (0,\a)$. Now by scaling and covering, we get the result.
\end{proof}
\subsection{Proof of Theorem \ref{eq:thm-nmc-reg}}
The following fundamental lemma allows, in particular, to consider truncation of the nonlocal mean curvature kernel $1_{B_r}(x) 1_{B_r}(y){\mathcal K}_u(x,y)$ without any assumption on $u$ in the exterior of $B_r$. \begin{lemma}\label{lem:Lem-Gamma-u-nmc} Let $u:\mathbb{R}^N\to \mathbb{R}$ be a measurable function and $\G^{u,R}: B_{R/2}\to \mathbb{R}$ be given by \begin{align}\label{eq:def-g-nmc} \G^{u,R}(x)
&:= \int_{\mathbb{R}^N} (1- 1_{B_{R}}(y)) \frac{{\mathcal F}_s(p_u({x},{y}))- {\mathcal F}_s(-p_u({x},{y})) }{|{x}-{y}|^{N+2s-1}} \, dy. \end{align} If $u\in C^{k,\a}(B_{R/2})$, for $k\geq 1$ and $\a\in [0,1]$, then, there exists a constant $C=C(N,s,k,\a,R)$ such that \begin{equation} \label{eq:estimd-Gu-Holder-NMC}
\|\G^{u,R}\|_{C^{k,\a}(B_{R/2})}\leq C(1+\|u\|_{C^{k,\a}(B_{R/2})})^{2k}. \end{equation}
If $u\in C^{0,1}(B_{R/2})$ then, there exists a constant $C=C(N,s,R)$ such that \begin{equation} \label{eq:Lip-impL-infty-NMC-fund-lem}
\|\G^{u,R}\|_{C^{0,1}(B_{R/2})}\leq C(1+\|u\|_{C^{0,1}(B_{R/2})}). \end{equation}
\end{lemma} \begin{proof} For simplicity, we assume that $R=2$, and to alleviate the notations, we put $\G^u:=\G^{u, R}$. We first observe, from \eqref{eq:def-of-F}, that ${\mathcal F}_s'(p)=-(1+p^2)^{(-N-2s)/2}$, so that for all $j\in \mathbb{N}$,
\begin{equation} \label{eq:pjFi-nmc}
|p|^j| {\mathcal F}_s^{(j)}(p)|\leq C(N,s,j) \qquad\textrm{ for all $p\in \mathbb{R}$. } \end{equation}
In particular, since $2s>1$, \begin{equation} \label{eq:Gu-L-infty}
\|\G^u\|_{L^\infty(B_{1})}\leq C(N,s). \end{equation} Next, for all $(x,y)\in B_{1} \times \mathbb{R}^N\setminus B_2$, we have \begin{align}\label{eq:deriv-p-u-x-y-nmc-00}
|\partial_{x}^\mu p_u(x,y)|\leq C(k)( |u(y)| |y|^{-1}+\|u\|_{C^{k-1,1}(B_1)} ) \qquad\textrm{ for $\mu\in \mathbb{N}^N$ with $|\mu|\leq k$}. \end{align} On the other hand, by writing $u(y)=(u(y)-u(z))+ u(z)$, we easily deduce that \begin{equation} \label{eq:u-1-nmc}
|u(y)||y|^{-1}\leq C (|p_u(z,y)|+ \|u\|_{L^\infty(B_1)}) \qquad\textrm{for all $z\in B_{1}$ and $y\in \mathbb{R}^N\setminus B_2$.}
\end{equation} Using this in \eqref{eq:deriv-p-u-x-y-nmc-00}, we see that, for $\mu\in \mathbb{N}^N$ with $|\mu|\leq k$, \begin{align}\label{eq:deriv-p-u-x-y-nmc}
|\partial_{x}^\mu p_u(x,y)|\leq C(k)( |p_u(z,y)|+\|u\|_{C^{k-1,1}(B_1)} ) \qquad\textrm{for all $x,z\in B_{1}$ and $y\in \mathbb{R}^N\setminus B_2$ .} \end{align}
By the {Fa\`{a} di Bruno formula} (see e.g. \cite{FaadeBruno-JW}), for $|\g|= k$ and $(x,y)\in B_{1} \times \mathbb{R}^N\setminus B_2$, we get \begin{equation} \label{eq:Faa-de-Bruno}
\partial^\g_x {\mathcal F}_s(p_u(x,y)) = \sum_{\Pi\in\scrP_k} {\mathcal F}_s^{ (\left|\Pi\right|)}(p_u(x,y)) \prod_{\mu \in\Pi} \partial_x^{\mu } p_u(x,y) , \end{equation}
where $\scrP_k$ denotes the set of all partitions of $\left\{ 1,\dots, k \right\}$.
Hence, for $x\in B_{1}$ and $y\in \mathbb{R}^N\setminus B_2$, by \eqref{eq:deriv-p-u-x-y-nmc}, we have that
\begin{align*}
|\partial^\g_x {\mathcal F}_s(p_u(x,y) )|
&\leq C \sum_{\Pi\in\scrP_k} 2^{\Pi-1} \left( |p_u(x,y)|^{\left|\Pi\right|} \left|{\mathcal F}_s^{ (\left|\Pi\right|)}(p_u(x,y)) \right|+\|u\|_{C^{k-1,1}(B_1)}^{|\Pi|} \left|{\mathcal F}_s^{ (\left|\Pi\right|)}(p_u(x,y))\right|\right)\\
&\leq C\left(1+ \|u\|_{C^{k-1,1}(B_1)}^{k}+ \sum_{\Pi\in\scrP_k} 2^{\Pi-1} |p_u(x,y)|^{\left|\Pi\right|} |{\mathcal F}_s^{ (\left|\Pi\right|)}(p_u(x,y))| \right).
\end{align*}
From this and \eqref{eq:pjFi-nmc}, we deduce that, for all $\g\in \mathbb{N}^N$ with $|\g|=k$, \begin{equation} \label{es:estim-integrn-NMC-fund-lem}
\sup_{(x,y)\in B_{1}\times \mathbb{R}^N\setminus B_2 } |\partial^\g_x {\mathcal F}_s(p_u(x,y))|+\sup_{(x,y)\in B_{1}\times \mathbb{R}^N\setminus B_2} |\partial^\g_x {\mathcal F}_s(-p_u(x,y))|\leq C(1+ \|u\|_{C^{k-1,1}(B_{1})})^k, \end{equation} with $C=C(s,N,k)$. Since $2s>1$, from the above estimate, \eqref{eq:Gu-L-infty} and the dominated convergence theorem, we can differentiate under the integral sign in \eqref{eq:def-g-nmc} to deduce that \begin{equation}
\|\G^u\|_{C^{k-1,1}(B_{1})}\leq C(1+\|u\|_{C^{k-1,1}(B_{1})})^k. \end{equation} Moreover, to see \eqref{eq:Lip-impL-infty-NMC-fund-lem}, we note that if $u\in C^{0,1}(B_{1})$, then Rademarcher's theorem implies that $u$ is equivalent to a differentiable function. Therefore \eqref{es:estim-integrn-NMC-fund-lem} holds (with $k=1$) and replacing "$\sup$" with "essup". Now by the dominated convergence theorem, we get \eqref{eq:Lip-impL-infty-NMC-fund-lem}.\\
Let us now fix $x_1,x_2\in B_{1}$ and $y\in \mathbb{R}^N\setminus B_2$. Direct computations yield $$
|\partial_{x}^\mu p_u(x_1,y)-\partial_{x}^\mu p_u(x_2,y)|\leq C |x_1-x_2|^\a( |u(y)| |y|^{-1}+\|u\|_{C^{k,\a}(B_1)} ) \qquad\textrm{ for $\mu\in \mathbb{N}^N$ with $|\mu|\leq k$}. $$ Note that, \eqref{eq:u-1-nmc} implies that $$
|u(y)||y|^{-1}\leq C \{\min (|p_u(x_1,y)| , |p_u(x_2,y)|)+ \|u\|_{L^\infty(B_1)}\} . $$
Therefore, for all $\mu\in \mathbb{N}^N$ with $|\mu|\leq k$, we get \begin{align}\label{eq:deriv-p-u-x-y-nmc-Hold}
|\partial_{x}^\mu p_u(x_1,y)-\partial_{x}^\mu p_u(x_2,y)|\leq C |x_1-x_2|^\a\{ \min (|p_u(x_1,y)| , |p_u(x_2,y)|)+\|u\|_{C^{k,\a}(B_1)} \} \end{align} and, by \eqref{eq:deriv-p-u-x-y-nmc}, \begin{align}\label{eq:deriv-p-u-x-y-nmc-Bound}
|\partial_{x}^\mu p_u(x_1,y) |\leq C \{ \min (|p_u(x_1,y)| , |p_u(x_2,y)|)+\|u\|_{C^{k}(B_1)} \} . \end{align} Next, we define $$ g_s\in C^\infty(\mathbb{R}_+, \mathbb{R}), \qquad g_s(r)= -r^{-(N+2s-1)/2}, $$ so that ${\mathcal F}_s'(p )=g_s(1+ p ^2)$ for all $p\in \mathbb{R}$. Moreover, for $r>0$, \begin{equation} \label{eq:higher-deriv-g-j}
g^{(j)}_s(r)=(-1)^{j+1}2^{-j} \prod_{i=0}^{j-1 } (N+2s-1 +2 i) r^{-\frac{N+2s-1+2j}{2}}. \end{equation} From this and the generalized chain rule for higher derivatives, we get
\begin{align} {\mathcal F}^{(j+1)}_s(p) = &\sum_{(m_1,m_2)\in {\mathcal N}_j}\t_j(m_1,m_2) p^{m_1} g_s^{(m_1+m_2)}(1+p^2),
\label{eq:Dk-K-s_1s_2-0-pp}
\end{align}
where $\t_j(m_1,m_2)=\frac{j! 2^{m_1}}{m_1! m_2!}$ and ${\mathcal N}_j:=\{(m_1,m_2)\in \mathbb{N}^2\,:\,m_1+ 2^{m_2}m_2=j\}$.
Hence, for all $p_1,p_2\in \mathbb{R}$,
\begin{align*}
|{\mathcal F}^{(j+1)}_s(p_1)-&{\mathcal F}^{(j+1)}_s(p_2) |\leq \sum_{(m_1,m_2)\in {\mathcal N}_j}\t_j(m_1,m_2) |p^{m_1}_1-p_2^{m_1}| |g_s^{(m_1+m_2)}(1+p^2_1)|\\
& + \sum_{(m_1,m_2)\in {\mathcal N}_j }\t_j(m_1,m_2)| p_2^{m_1}| |p_1^2-p_2^2| \int_{0}^1| g_s^{(m_1+m_2+1)}(1+t p^2_1+(1-t)p_2^2) \, | dt.
\end{align*}
It then follows from, \eqref{eq:deriv-p-u-x-y-nmc-Hold} and \eqref{eq:deriv-p-u-x-y-nmc-Bound}, that \begin{align}
&\left|{\mathcal F}^{(j+1)}_s(p_u(x_1,y)) -{\mathcal F}^{(j+1)}_s(p_u(x_2,y))\right| \nonumber\\
&\leq C |x_1-x_2|^\alpha \sum_{(m_1,m_2)\in {\mathcal N}_j} \t_j(m_1,m_2) \frac{ \left(\min (|p_u(x_1,y)| , |p_u(x_2,y)| )+\|u\|_{C^{k,\a}(B_1)} \right)^{ m_1}}{ (1+ \min (|p_u(x_1,y)| , |p_u(x_2,y)| )^2)^{\frac{N+2s-1+2(m_1+m_2)}{2}} } \nonumber\\
&+C |x_1-x_2|^\alpha \sum_{(m_1,m_2)\in {\mathcal N}_j} \t_j(m_1,m_2) \frac{ \left(\min (|p_u(x_1,y)| , |p_u(x_2,y)| )+\|u\|_{C^{k,\a}(B_1)} \right)^{ m_1+2}}{ (1+ \min (|p_u(x_1,y)| , |p_u(x_2,y)| )^2)^{\frac{N+2s-1+2(m_1+m_2+1)}{2}} } . \label{eq:estim-cFs-jplus-1-nmc} \end{align}
On the other hand, it is immediate, from \eqref{eq:higher-deriv-g-j} and \eqref{eq:Dk-K-s_1s_2-0-pp}, that \begin{equation} \label{eq:pjFi-nmc-ok}
|{\mathcal F}^{(j+1)}_s(p_u(x_2,y)) | \leq \frac{C}{(1+ \min (|p_u(x_1,y)| , |p_u(x_2,y)| )^2)^{\frac{N+2s-1+2(j+1)}{2}}}. \end{equation}
Using \eqref{eq:deriv-p-u-x-y-nmc-Bound}, \eqref{eq:deriv-p-u-x-y-nmc-Hold} and an induction argument, we get \begin{align}\label{eq:prod-p-u-CNMC}
&\left|\prod_{\mu \in\Pi} \partial_x^{\mu } p_u(x_1,y)- \prod_{\mu \in\Pi} \partial_x^{\mu } p_u(x_1,y) \right| \leq C |x_1-x_2|^\a\{ \min (|p_u(x_1,y)| , |p_u(x_2,y)| )+\|u\|_{C^{k,\a}(B_1)} \}^{|\Pi|}. \end{align} Moreover \eqref{eq:deriv-p-u-x-y-nmc-Bound} yields \begin{equation} \label{eq:Prodp-x2-y-nmc}
\left|\prod_{\mu \in\Pi} \partial_x^{\mu } p_u(x_2,y) \right| \leq C ( \min (|p_u(x_1,y)| , |p_u(x_2,y)|)+\|u\|_{C^{k}(B_1)} )^{|\Pi|}. \end{equation} We have, from \eqref{eq:Faa-de-Bruno}, that \begin{align} \label{eq:Faa-de-Bruno-fund-calcul-nmc}
&\left| \partial^\g_x {\mathcal F}_s(p_u(x_1,y))- \partial^\g_x {\mathcal F}_s(p_u(x_2,y)) \right| \leq \sum_{\Pi\in\scrP_k} \left|{\mathcal F}_s^{ (\left|\Pi\right|)}(p_u(x_1,y)) - {\mathcal F}_s^{ (\left|\Pi\right|)}(p_u(x_2,y)) \right| \left|\prod_{\mu \in\Pi} \partial_x^{\mu } p_u(x_1,y) \right| \nonumber \\
&\hspace{3cm}+ \sum_{\Pi\in\scrP_k} \left|{\mathcal F}_s^{ (\left|\Pi\right|)}(p_u(x_2,y)) \right| \left|\prod_{\mu \in\Pi} \partial_x^{\mu } p_u(x_1,y)- \prod_{\mu \in\Pi} \partial_x^{\mu } p_u(x_2,y) \right|. \end{align}
Next, we observe that for $(m_1,m_2)\in {\mathcal N}_{|\Pi|-1}$, then $$
|\Pi|+ m_1-2(m_1+m_2)-(N+2s-1)< 0. $$ Now from this, \eqref{eq:estim-cFs-jplus-1-nmc}, \eqref{eq:pjFi-nmc-ok}, \eqref{eq:prod-p-u-CNMC}, \eqref{eq:Prodp-x2-y-nmc} and \eqref{eq:Faa-de-Bruno-fund-calcul-nmc}, we deduce that, for all $x_1,x_2\in B_{1}$ and $y\in \mathbb{R}^N\setminus B_2$, \begin{align*}
\left| \partial^\g_x {\mathcal F}_s(p_u(x_1,y) )- \partial^\g_x {\mathcal F}_s(p_u(x_2,y) )\right| \leq C |x_1-x_2|^\alpha ( 1+\|u\|_{C^{k,\a}(B_1)} )^{2k}. \end{align*}
Combining this with \eqref{es:estim-integrn-NMC-fund-lem}, we get $\sup_{y\in \mathbb{R}^N\setminus B_2}\|{\mathcal F}_s(p_u(\cdot,y) )\|_{C^{k,\a}(B_1)}\leq
C ( 1+\|u\|_{C^{k,\a}(B_1)} )^{2k}.$ Since the same estimates remains valid when $p_u$ is replaced with $-p_u$, then \eqref{eq:estimd-Gu-Holder-NMC} follows. \end{proof}
We will need the following elementary result which follows from the fact that ${\mathcal F}'_s$ is even on $\mathbb{R}$ and the fundamental theorem of calculus. \begin{lemma}\label{lem:local-comparison-NMC-graphs} For all $a,b\in \mathbb{R}$, we have $$ [{\mathcal F}_s(a)-{\mathcal F}_s(b)]- [{\mathcal F}_s(-a)-{\mathcal F}_s(-b)]=2(a-b)\int_0^1 {\mathcal F}'_s\left(b+\rho(a-b)\right) d\rho. $$
\end{lemma}
We now complete the \begin{proof}[Proof of Theorem \ref{eq:thm-nmc-reg}] In view of \eqref{eq:decom-NMC-intro}, we have \begin{equation} \label{eq:Ck_u-u-sol-NMC} {\mathcal L}_{\widetilde {\mathcal K}_u} u= f-\G^u \qquad\textrm{ in $B_{1/2}$,} \end{equation}
where $$ \widetilde{\mathcal K}_u(x,y):=1_{B_1}(x) 1_{B_1}(y){\mathcal K}_u(x,y) \qquad\textrm{ for all $x\not=y\in \mathbb{R}^N$} $$ and, for $x\in B_{1/2}$, \begin{align*} \G^u(x)
&:= \int_{\mathbb{R}^N} (1- 1_{B_1}(y)) \frac{{\mathcal F}_s(p_u({x},{y}))- {\mathcal F}_s(-p_u({x},{y})) }{|{x}-{y}|^{N+2s-1}} \, dy. \end{align*} We recall from the fundamental theorem of calculus that \begin{equation} \label{eq:lin-to-Quasilin}
(u(x)-u(y)) {\mathcal K}_u(x,y)= [{\mathcal F}_s(p_u({x},{y}))- {\mathcal F}_s(-p_u({x},{y}))] {|{x}-{y}|^{-(N+2s-1)}}. \end{equation}
Let $h\in B_{1/4} $ with $h\not=0$. Then, recalling the notation in \eqref{eq:def-f-h-alph}, by Lemma \ref{lem:local-comparison-NMC-graphs}, \eqref{eq:lin-to-Quasilin} and \eqref{eq:Ck_u-u-sol-NMC}, $$ {\mathcal L}_{K^u_{h}} u_{h,1}= f_{h,1}+\G^u_{h,1} \qquad\textrm{ in $B_{1/4}$}, $$ where \begin{equation} \label{eq:def-Kuh}
K^u_{h}(x,y):= 1_{B_1}(x) 1_{B_1}(y) \frac{1}{{|x-y|^{N+2s}}} q^u_{h}(x,y) \end{equation} and $$ q^u_{h}(x,y):=-2 \int_{0}^{1}{\mathcal F}'_s\left(p_{u(\cdot+h)}(x,y)+\rho p_{u-u(\cdot+h)}(x,y)\right)\, d\rho. $$ Since ${\mathcal F}'_s$ is even and $p_w(x,y)=-p_w(y,x)$, we see that $K^u_{h}(x,y)=K^u_{h}(y,x)$. Moreover \begin{equation}\label{eq:K-u-h-lower-bound}
K^u_h(x,y)\geq C |x-y|^{-N-2s}\qquad\textrm{$x\not=y\in B_1 $, }
\end{equation} for some constant $C>0$, only depending on $N,s$ and $ \| u\|_{C^{0,1} (B_1) }$. Letting $v:=\vp_{1/8}u_{h,1}$ and using Lemma \ref{lem:estim-G_Ku}$(i)$, we have that \begin{equation}\label{eq:Kuh-v-e-NMC} {\mathcal L}_{K^u_{h}} v= f_{h,1}+ \G^u_{h,1}+ G_h, \qquad\textrm{ in $B_{2^{-4}}$},
\end{equation} with $G_h:=G_{ K^u_{h},u_{h,1},1/4}$ satisfying (note that $K^u_h$ is supported in $B_1\times B_1$ and $|q^u_{h}|\leq 2$) \begin{equation} \label{eq:estim-G-cnmc}
\|G_h \|_{L^{\infty}(B_{2^{-5}})}\leq C(N,s)\|u_{h,1}\|_{L^\infty(B_1)}\leq C\|\n u\|_{L^\infty(B_2)}. \end{equation}
We would like to apply \cite[Theorem 2.4]{Cozzi} to get the $C^{\a_0}$ bound of $v$, but our kernel $K^u_{h}$, which is compactly supported might vanish at some diagonal points $\{x=y\}$. A way out to such difficulty, is to use the argument in \cite[Remark 2.1]{Fall-Reg} (see also Lemma \ref{lem:cut-reg-kernel}) by considering
$$\overline K^u_h(x,y)=K^u_h(x,y)+ (2-1_{B_{1/2}}(x)-1_{B_{1/2}}(y)) |x-y|^{-N-2s}.$$ We then deduce, from \eqref{eq:Kuh-v-e-NMC}, that \begin{equation}\label{eq:Kuh-v-e-NMC-first} {\mathcal L}_{\overline K^u_{h}} v+ \overline V v= f_{h,1}+ \G^u_{h,1}+ G_h+ \overline f, \qquad\textrm{ in $B_{2^{-4}}$}, \end{equation}
for some functions $\|\overline V\|_{L^\infty(B_{ 2^{-4} })}\leq C(N,s)$ and $ \|\overline f\|_{L^\infty(B_{ 2^{-4} })}\leq C(N,s)\|v\|_{L^\infty(B_1)}$. Since $K^u_h(x,y) $ satisfies \eqref{eq:K-u-h-lower-bound}, we find that $\overline K^u_{h}$ satisfies \eqref{eq:K-Kernel-satisf}. Therefore, since $v\in H^s(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, by \cite[Theorem 2.4]{Cozzi}, we have that $$
\|v\|_{C^{0,\a_0}(B_{2^{-5}})} \leq C ( \|v\|_{L^\infty (\mathbb{R}^N) } + \|f_{h,1} \|_{L^\infty(\mathbb{R}^N)}+ \|\G^u_{h,1} \|_{L^\infty(B_{2^{-4}})}+ \|G_h\|_{L^\infty (B_{2^{-4}} ) } ), $$
for some $\a_0>0$ and $C>0$, only depending on $N,s$ and $ \| u\|_{C^{0,1} (B_1) }$. From \eqref{eq:estim-G-cnmc} and the fact that $v=u_{h,1}$ on $B_{1/8}$, we get $$
\| u_{h,1}\|_{C^{0,\a_0}(B_{2^{-5}})} \leq C ( \|u_{h,1}\|_{L^\infty (B_1) }+ \|f_{h,1} \|_{L^\infty(\mathbb{R}^N)}+ \|\G^u_{h,1} \|_{L^\infty(B_{2^{-4}})} ). $$ This and Lemma \ref{lem:Lem-Gamma-u-nmc} imply that \begin{equation} \label{eq:esimt-u-CNM1}
\|u\|_{C^{1,\a_0}(B_{2^{-6}})} \leq C (1+ \| u\|_{C^{0,1} (B_1) } + \|f \|_{C^{0,1}(\mathbb{R}^N)}), \end{equation} which proves \eqref{eq:estimu-NMC-first}. \\ To obtain the gradient estimate of $v$ from Theorem \ref{th:main-th10}, we check that ${\mathcal L}_{K^u_h}$ is a $C^{0,\a_0}$-nonlocal operator. To this scope, for every $w\in C^{0,1}(B_1)$, we define $Z_w: B_{1/2}\times [0,1/2)\times S^{N-1}\to \mathbb{R}$ by $$ Z_w(x,r,\th): =-\int_0^1\n w(x+rt\th)\cdot \th dt \qquad \textrm{ for $r\in [0, 1/2)$, $x\in B_{1/2}$ and $\th\in S^{N-1}$}. $$
Clearly $Z_w$ is as smooth as $\n w$ and $Z_w(x,r,\th):=p_w(x,x+r\th)$ for $r>0$. We then define ${{\mathcal A}}_{ K^u_{h}}: B_{1/4}\times [0,1/4)\times S^{N-1}\to \mathbb{R}$ by \begin{align}\label{eq:AK-CNMC} {{\mathcal A}}_{ K^u_{h}}(x,r,\th):=1_{B_2}(x) 1_{B_2}(x+r\th) \int_{0}^{1}\frac{2 d\rho}{\left(1+ (Z_{u(\cdot+h)}(x,r,\th)+\rho Z_{u-u(\cdot+h)}(x,r,\th))^2 \right)^{(N+2s)/2}} , \end{align}
which by \eqref{eq:def-Kuh}, satisfies ${{\mathcal A}}_{ K^u_{h}}(x,r,\th)= r^{N+2s}K^u_h(x,x+r\th)$ for all $(x,r,\th)\in B_{1/4}\times (0,1/4)\times S^{N-1}$. Moreover, $$ {{\mathcal A}}_{ K^u_{h}}(x,0,\th)-{{\mathcal A}}_{ K^u_{h}}(x,0,-\th)=0 \qquad\textrm{ for all $(x,\th)\in B_{1/4}\times S^{N-1} $.} $$ In addition from, \eqref{eq:esimt-u-CNM1} together with \eqref{eq:AK-CNMC}, we have that $$
\|{{\mathcal A}}_{ K^u_{h}} \|_{C^{\a_0}(Q_{2^{-7}}\times S^{N-1} )}\leq C, $$
with $C$, only depending on $N,s,\|u\|_{ C^{0,1}(B_2) },\a_0$ and $\|f \|_{C^{0,1}(B_2)}$.
We then conclude that $ K^{u}_{h}\in \scrK^s(\kappa,\a_0, Q_{2^{-7}})$, for some $\kappa$, only depending on $N,s,\|u\|_{ C^{0,1}(B_2) },\a_0$ and $ \|f \|_{C^{0,1}(B_2)}$.
Therefore applying Theorem \ref{th:main-th10}$(ii)$ to \eqref{eq:Kuh-v-e-NMC}, we deduce that
$$
\| \n v\|_{C^{\min(2s-1-\e,\a_0 )}(B_{2^{-8}})} \leq C ( \|v\|_{L^\infty (B_1) }+ \|f_{h,1} \|_{L^\infty (B_2)}), $$
for all $\e\in (0,2s-1)$ and $C$ a constant, only depending on $N,s,\|u\|_{ C^{0,1}(B_2) },\a_0,\e$ and $ \|f \|_{C^{0,1}(B_2)}$. Hence, recalling that $v=u_{h,1}$ in $B_{1/8}$, we get $$
\| \n u\|_{C^{1, \a_1}(B_{2^{-9}})} \leq C , $$
with $\a_1:=\min(2s-1-\e,\a_0 )$. Hence, for all $h\in B_{2^{-10}}$, we have $ K^{u}_{h}\in \scrK^s(\kappa,1, Q_{2^{-10}})$, for some $\kappa$, only depending on $N,s,\|u\|_{ C^{0,1}(B_2) },\a_1$ and $ \|f \|_{C^{0,1}(B_2)}$. We apply once more Theorem \ref{th:main-th10}$(ii)$ to \eqref{eq:Kuh-v-e-NMC}, to get $
\| v\|_{C^{ 1,2s-1-\varepsilon }(B_{2^{-11}})} \leq C, $ so that \begin{equation} \label{eq:estim-CNMC11}
\| u\|_{C^{ 2,2s-1-\varepsilon }(B_{2^{-12}})} \leq C. \end{equation} This finishes the proof of $(i)$ after a scaling and covering.\\
For $(ii)$, we consider first the case $m=1$. Clearly \eqref{eq:estim-CNMC11} and \eqref{eq:AK-CNMC} imply that $ K^{u}_{h}\in \scrK^s(\kappa,2s-1+\a, Q_{2^{-13}})$, for all $h\in B_{2^{-13}}$ and $\a\in (0,1)$. In particular, by Lemma \ref{lem:estim-G_Ku}$(iii)$, we have $\|G_h\|_{C^{0,\a}(B_{2^{-13}})}\leq C \|u_{h,1}\|_{L^\infty(B_1)}$. Now by \eqref{eq:estim-CNMC11} and Lemma \ref{lem:Lem-Gamma-u-nmc}, for all $h\in B_{2^{-13}}$, we have $ \|\G^u_{h,1}\|_{ C^{1,2s-1-\e}(B_{2^{-13}})}\leq C $. Therefore,
applying Theorem \ref{th:Schauder-0-intro} to the equation \eqref{eq:Kuh-v-e-NMC}, we get $
\| v\|_{C^{ 2s+\alpha }(B_{2^{-15}})} \leq C, $ provided $2s+\a\not\in \mathbb{N}$. Hence $$
\| u\|_{C^{ 1+2s+\alpha }(B_{2^{-16}})} \leq C.
$$
If now $m\geq 2$, then the above estimate implies that $ K^{u}_{h}\in \scrK^s(\kappa,2s+\a, Q_{2^{-18}})$ for all $h\in B_{2^{-18}}$. Hence, Lemma \ref{lem:estim-G_Ku}$(iii)$ implies that $\|G_h\|_{C^{1,\a}(B_{2^{-18}})}\leq C \|u_{h,1}\|_{L^\infty(B_1)}$. On the other hand, by Lemma \ref{lem:Lem-Gamma-u-nmc}, $ \G^u_{h,1}\in C^{2s+\a}(B_{2^{-16}})\subset C^{1,\a}(B_{2^{-16}})$, because $2s>1$. It then follows, from \eqref{eq:Kuh-v-e-NMC} and Theorem \ref{th:Schauder-0-intro}, that $\| \partial_{x_i} v\|_{C^{ 2s+\alpha }(B_{2^{-18}})} \leq C $. This yields $\| \partial_{x_i} u\|_{C^{ 1+2s+\alpha }(B_{2^{-19}})} \leq C $, because $v=\vp_{1/8} u_{h,1}$.
Now iterating the above argument, then for all $k\in\{1,\dots,m\}$ and $i=\{1,\dots,N\}$, we can find two constants $r_k$, only depending on $k$, and a constant $ C_k>0$, only depending on $N,s,\|u\|_{ C^{0,1}(B_2) },k,\a$ and $ \|f \|_{C^{m,\a}(B_2)}$, such that
$$
\| \partial_{x_i}^k u_{h,1}\|_{C^{ 2s+\alpha }(B_{r_k})} \leq C_k
$$ for all $h\in B_{r_k/2}$. A covering and scaling arguments yield $(iii)$. \end{proof}
\subsection{Proof of Theorem \ref{th:nonloca-surf1} and Theorem \ref{th:nonloca-surf2} } Up to a change of coordinates and a scaling, we may assume that a neighborhood of $0\in \Sig$ is parameterized by a $C^{1,\g}$-diffeomorphism $\Phi: B_2\to \Sig$, for some $\g\in (0,1)$, satisfying $ \Phi(0)= 0$ and \begin{equation}\label{eq:DPhi-close-identity}
|D \Phi(x)-id|\leq \frac{1}{2} \qquad\textrm{ for all $x\in B_2$.} \end{equation}
We consider the following open sets in $\Sig$ given by
$$
{\mathcal B}_r:=\Phi(B_r) \qquad\textrm{ for $r\in (0,2]$}
$$
and we define $\eta_r(\overline x)=\vp_r(\Phi^{-1}(\overline x))$. For $\Psi\in C^\infty_c({\mathcal B}_{1/2})$, we then we have
\begin{align}
\int_{ {\mathcal B}_2} \int_{ {\mathcal B}_2} \frac{(u(\overline x)-u(\overline y))(\Psi(\overline x)-\Psi(\overline y)) }{|\overline x-\overline y|^{N+2s}}\eta_2(\overline x)\eta_2(\overline y) \, d\s(\overline x)d\s(\overline y)&+\int_{\Sig} V_1(\overline x) u(\overline x) \Psi(\overline x)\, d\s(\overline x) \nonumber\\
& =\int_{\Sig} f_1(\overline x) \Psi(\overline x)\, d\s(\overline x), \label{eq:reg-Manifold-fin}
\end{align} where \begin{equation} \label{eq:def-V_11}
V_1(\overline x):=V(\overline x)+ \int_{\Sig} (1-\eta_2(\overline y)) |\overline x-\overline y|^{-N-2s}\, d\s(\overline y) \end{equation}
and
\begin{equation} \label{eq:def-V_12}
f_1(\overline x):=f(\overline x)+ \int_{\Sig} (1-\eta_2(\overline y))u(\overline y) |\overline x-\overline y|^{-N-2s}\,d\s(\overline y). \end{equation} We denote by $Jac_\Phi$ the Jacobian determinant of $\Phi $. Let $\psi(x)=\Psi(\Phi(x))$, $v(x)=u(\Phi(x))$, $\widetilde V(x)=V_1(x)Jac_\Phi(x)$ and $\widetilde f(x)=f_1(x)Jac_\Phi(x)$. Then by the changes of variables $\overline x=\Phi (x)$ and $\overline y=\Phi (y)$, in \eqref{eq:reg-Manifold-fin}, we get
\begin{align*} \frac{1}{2} \int_{ \mathbb{R}^N} \int_{ \mathbb{R}^N} {(v(x)-v(y))(\psi(x)-\psi(y)) } K(x,y) \, dxdy+\int_{B_1} \widetilde V(x) u(x) \psi(x)\, dx =\int_{B_1} \widetilde f(x) \psi(x)\, dx,
\end{align*} where \begin{equation}\label{eq:def-K-hyersurf}
K(x,y) = {\vp_2(x) \vp_2(y) Jac_{\Phi}(x)Jac_{\Phi}(y) } {|\Phi(x)-\Phi(y)|} ^{-N-2s}. \end{equation}
We further consider $w=\vp_{1/4} v\in H^s(\mathbb{R}^N)$, so that by Lemma \ref{lem:estim-G_Ku}, \begin{equation}\label{eq:w-satisf} {\mathcal L}_{K} w+ \widetilde V w= \widetilde f+ G \qquad\textrm{in $B_{1/16}$}, \end{equation} where \begin{equation}\label{eq:def-G-nonloc-hypersurface} G(x)=\int_{B_2}(1-\vp_{1/4}(y))v(y)K(x,y)\, dy. \end{equation} Next, we observe that the function ${{\mathcal A}}_{K}:B_{1}\times [0,1]\times S^{N-1}\to \mathbb{R}^N$, given by $$
{{\mathcal A}}_{K}(x,r,\th)=\vp_2(x) \vp_2(x+r\th) Jac_{\Phi}(x)Jac_{\Phi}(x+r\th) \left|\int_0^1D\Phi(x+r\th) \th\, dt\right| ^{- N-2s} $$ is an extension of $(x,r,\th)\mapsto r^{N+2s}K(x,x+r\th)$ on $B_{1}\times [0,1]\times S^{N-1}$.
Moreover, since $\Phi \in C^{1,\g}(B_2)$, we see that \begin{equation} \label{eq:K-hypersurface} \begin{aligned}
& \|{{\mathcal A}}_{K} \|_{C^\g(B_{1/2}\times[0,1/2]\times S^{N-1})}\leq \frac{1}{\kappa},\\\
&{{\mathcal A}}_{K}(x,0,\th)={{\mathcal A}}_{K}(x,0,-\th) \qquad\textrm{ for all $(x,\th)\in \mathbb{R}^N\times S^{N-1} $},
\end{aligned}
\end{equation} for some $\kappa>0$, only depending on $N,s,\g$ and $\|\Phi\|_{C^{1,\g}(B_2)}$. Consequently by \eqref{eq:K-hypersurface}, \eqref{eq:DPhi-close-identity} and \eqref{eq:def-K-hyersurf}, decreasing $\kappa$ if necessary, we see that $K\in \scrK^s(\kappa,\g, Q_{1/2} )$.
In addition, from \eqref{eq:def-V_11} and \eqref{eq:def-V_12}, we easily deduce that for $p>1$, \begin{equation}\label{eq:Holder-entires-manifold}
\|\widetilde f\|_{L^p( B_{1/16} )}+ \|G\|_{L^p( B_{1/16} )} \leq C (\|u\|_{{L_s(\Sig)}}+ \|f\|_{L^p({\mathcal B}_2)}) \qquad\textrm{ and } \qquad\|\widetilde V\|_{L^p( B_{1/2} )} \leq C, \end{equation}
where $C$ is a constant only depending on $N,s,p,\g, \|\nabla \Phi\|_{C^{1,\g}(B_2)}, \|V\|_{L^p({\mathcal B}_2)}$ and $ \|1\|_{{L_s(\Sig)}}$.
\begin{proof}[Proof of Theorem \ref{th:nonloca-surf1} (completed)] From the computations above, we have that $w=\vp_{1/2}u\circ\Phi \in H^s(\mathbb{R}^N)$ satisfies \eqref{eq:w-satisf} with $K\in \scrK^s(\kappa,\g, Q_{1/2} )$. Thanks to \eqref{eq:Holder-entires-manifold}, we can apply Theorem \ref{th:main-th10}, to get the result. \end{proof}
\begin{proof}[Proof of Theorem \ref{th:nonloca-surf2} (completed)] We know from Theorem \ref{th:nonloca-surf1} and the above argument that $w=\vp_{1/2}u\circ\Phi \in H^s(\mathbb{R}^N)\cap C^{\min (2s-\e,1)}(B_{1/4})$, for all $\e\in (0,2s)$ and solves \eqref{eq:w-satisf} with $K\in \scrK^s(\kappa,\g, Q_{1/2} )$. However, in view of \eqref{eq:def-V_11} and \eqref{eq:def-V_12}, we can use similar arguments as in the proof of Lemma \ref{lem:estim-G_Ku}$(iv)$ to deduce that \begin{equation} \label{eq:es1-Hyp}
\|\widetilde f\|_{C^\a( B_{1/16} )} \leq C (\|u\|_{{L_s(\Sig)}}+ \|f\|_{C^\a({\mathcal B}_2)}) \end{equation} and, using also \eqref{eq:integ-hypersurface}, we get \begin{equation} \label{eq:es2-Hyp}
\|\widetilde V\|_{C^\a( B_{1/16} )} \leq C (\| V\|_{C^\a( {\mathcal B}_{2} )} + \|1\|_{{L_s(\Sig)}} ) \leq C ,
\end{equation} where here and below, the letter $C$ denotes a positive constant which may vary from line to line but only depends on $N,s,\a,\g, \|V\|_{C^\a({\mathcal B}_2)}, \|\nabla \Phi\|_{C^{1,\g}(B_2)}$ and $ \|1\|_{{L_s(\Sig)}}$. Moreover, recalling \eqref{eq:def-G-nonloc-hypersurface}, applying Lemma \ref{lem:estim-G_Ku}$(iii)$, we have that \begin{equation} \label{eq:es3-Hyp}
\|G\|_{C^\a( B_{1/16} )} \leq C \|w\|_{L^1(B_2)} \leq C \|u\|_{{L_s(\Sig)}}. \end{equation} In view of \eqref{eq:w-satisf}, \eqref{eq:es1-Hyp}, \eqref{eq:es2-Hyp} and \eqref{eq:es3-Hyp}, we can apply Theorem \ref{th:Schauder-0-intro}-$(i)$ and use a bootstrap argument, to deduce that
\begin{align*}
\|w\|_{C^{2s+\a}(B_{r_0}) } &\leq C (\|w\|_{L^\infty(\mathbb{R}^N)}+ \|\widetilde f\|_{C^\a(B_2)}+ \|G\|_{C^\a(B_{1/16})})\\
&\leq C (\|u\|_{L^2({\mathcal B}_2)}+ \|u\|_{{L_s(\Sig)}} +\|f\|_{C^\a({\mathcal B}_2)}), \end{align*}
for some $r_0,C>0$, depends only on $N,s,\a,\g,c_1,c_0,\|V\|_{C^\a({\mathcal B}_2)}, \|\nabla \Phi\|_{C^{1,\g}(B_2)}$ and $I_{s,\Sig}$.
The proof is thus completed by scaling, covering and a change of variables.
\end{proof}
\section{Appendix}\label{s:Appendix}
\begin{proof}[Proof of Lemma \ref{lem:2sp-alph-estim-F_Keo}] \noindent \textbf{Case $2s+\a<2$.} For simplicity, recalling \eqref{eq:A-Cm12} and \eqref{eq:A-Cm12-tau}, we assume that $$
\|A\|_{C^{k+2s+\a}(Q_\infty)\times L^\infty(S^{N-1} )}+ \|B\|_{{\mathcal C}^{k}_{\t_s}(Q_\infty )\times L^\infty(S^{N-1} )} \leq 1, $$ where $\t_s:=\a+(2s-1)_+$ if $2s\not=1$ and $\t_{1/2}:=\alpha + \e$ if $2s=1$. We also assume that $$
\|u\|_{C^{k+2s+\a+\e_s}(\mathbb{R}^N )}\leq 1, $$ where $\e_s:=0$ if $2s\not=1$ and $\e_s:=\e$ if $2s=1$.\\
We consider the case $m=0$. Since $\|u\|_{L^\infty(\mathbb{R}^N)}\leq 1$, we have \begin{equation}\label{eq:Apend-1e}
|\delta ^e u(x,r,\theta )| \leq C \min (1,r^{2s+\a}). \end{equation} Here, for $2s\geq 1$, we use the fact that $ 2 \delta ^e u(x,r)=r \int_0^1(\n u(x+t r \theta )-\n u(x-t r \theta ))\cdot \theta \, dt $. Moreover for $x_1,x_2\in \mathbb{R}^N$ and $r>0$, then for $2s+\a<1$, we have \begin{equation}\label{eq:Apend-2e}
|\delta ^e u(x_1,r,\theta )-\delta ^e u(x_2,r,\theta )| \leq C \min (r^{2s+\a}, |x_1-x_2|^{2s+\a}) \end{equation}
and if $2s\geq 1$, we have \begin{equation}\label{eq:Apend-3e}
|\delta ^e u(x_1,r,\theta )-\delta ^e u(x_2,r,\theta )| \leq C \min (r^{2s+\a}, r |x_1-x_2|^{\t_s}). \end{equation} On the other hand, for all $s\in (0,1)$, \begin{equation}\label{eq:Apend-4o}
|\delta ^o u(x,r,\th) | \leq C \min (1,r )^{\min(2s+\a,1)}, \end{equation} and \begin{equation}\label{eq:Apend-5o}
|\delta ^o u(x_1,r,\theta )-\delta ^o u(x_2,r,\theta )| \leq C \min (r, |x_1-x_2| )^{\min(2s+\a,1)}. \end{equation} Using \eqref{eq:Apend-4o}, for $s\in (0,1)$, we estimate \begin{align*}
|{\mathcal O}^s_{B,u}(x)|&\leq C \int_0^\infty\min (r,1)^{\min (2s+\a,1)} \min (r, 1)^{(2s-1)_++\a} r^{-1-2s}\, dr\\
&\leq C\int_0^1r^{\min (2s+\a,1)} r ^{(2s-1)_++\a} r^{-1-2s}\, dr+ C \int_1^\infty r^{-1-2s}\, dr,
\end{align*} so that, \begin{equation}\label{eq:estimFoB-k-1}
\|{\mathcal O}^s_{B,u}\| _{L^\infty(\mathbb{R}^N)}\leq C. \end{equation}
We consider next ${\mathcal E}^s_{A,u}$. For all $x\in \mathbb{R}^N$ and for all $s\in (0,1)$, by \eqref{eq:Apend-1e}, we have
\begin{align*}
|{\mathcal E}^s_{A,u}(x)|&\leq C \int_0^\infty\min( r^{2s+\a} ,1) r^{-1-2s}\, dr \leq C \int_0^1r^{\a-1} \, dr+ C \int_1^\infty r^{-1-2s}\, dr, \end{align*} yielding \begin{equation}\label{eq:estimFeB-k-1}
\|{\mathcal E}^s_{A,u}\| _{L^\infty(\mathbb{R}^N)}\leq C. \end{equation}
Let $x_1, x_2\in \mathbb{R}^N$ with $|x_1-x_2|\leq 1$. Using \eqref{eq:Apend-5o}, for $s\in (0,1)$ we have \begin{align*}
|{\mathcal O}^s_{B,u}(x_1)&-{\mathcal O}^s_{B,u}(x_2)|\leq C \int_0^\infty\min (r,|x_1-x_2|)^{\min (2s+\a,1)} \min (r, 1)^{\t_s} r^{-1-2s}\, dr\\
&+C \int_0^\infty\min (r,1)^{\min (2s+\a,1)} \min (r, |x_1-x_2|)^{\t_s} r^{-1-2s}\, dr\\
&\leq C\int_0^{|x_1-x_2|}r^{\min (2s+\a,1)+\t_s} r^{-1-2s}\, dr +C|x_1-x_2|^{\min (2s+\a,1) } \int_{|x_1-x_2|}^1 r^{\t_s -1-2s } \, dr\\
&+C|x_1-x_2|^{\t_s } \int_{|x_1-x_2|}^1 r^{\min (2s+\a,1)-1-2s } \, dr\\
&+C |x_1-x_2|^{\min (2s+\a,1) } \int_{1}^\infty r^{-1-2s}\, dr+C |x_1-x_2|^{ \t_s } \int_{1}^\infty r^{-1-2s}\, dr\\
&\leq C |x_1-x_2|^{\a} . \end{align*}
In the above estimate, it is used that $\t_{s}=\a+\e$, for $s=1/2$.
This together with \eqref{eq:estimFoB-k-1} imply that $\|{\mathcal O}^s_{B,u}\|_{C^{0,\a}(\mathbb{R}^N)}\leq C $, for all $s\in (0,1)$. \\
Now for $2s\geq 1$, let $x_1 \not = x_2\in \mathbb{R}^N$ with $|x_1-x_2|\leq 1$. Using \eqref{eq:Apend-3e} and \eqref{eq:Apend-1e} we have \begin{align*}
&|{\mathcal E}^s_{A,u}(x_1)-{\mathcal E}^s_{A,u}(x_2)|\\
& \leq C \int_0^\infty \min (r^{ 2s+\a} , r|x_1-x_2|^{ \t_s} ) r^{-1-2s }\, dr +C |x_1-x_2|^{\a} \int_0^\infty \min(r ^{ 2s+\a},1 ) r^{-1-2s}\, dr\\
&\leq C \int_0^{|x_1-x_2|} r^{\a-1}\, dr+C|x_1-x_2|^{\t_s}\int_{|x_1-x_2|}^\infty r^{-2s}\, dr +C |x_1-x_2|^{\a} \leq C |x_1-x_2|^\a. \end{align*}
Hence using \eqref{eq:estimFeB-k-1}, for $2s\geq 1$, we get $\|{\mathcal E}^s_{A,u}\|_{C^{0,\a}(\mathbb{R}^N)}\leq C $.\\
We now consider the case $2s+\a< 1$. For $x_1, x_2\in \mathbb{R}^N$, $|x_1-x_2|\leq 1$, by \eqref{eq:Apend-2e}, we estimate \begin{align*}
&|{\mathcal E}^s_{A,u}(x_1)-{\mathcal E}^s_{A,u}(x_2)|\\
&\leq C\int_0^\infty\min (r,|x_1-x_2|)^{ 2s+\a} r^{-1-2s}\, dr +C|x_1-x_2|^{\a} \int_0^\infty \min(r ^{ 2s+\a},1 ) r^{-1-2s}\, dr\\
&\leq C \int_0^{|x_1-x_2|}r^{-1+\a} \, dr+C|x_1-x_2|^{2s+\a}\int_{|x_1-x_2|}^\infty r^{-1-2s}\, dr+C |x_1-x_2|^{\a} \leq C |x_1-x_2|^\a. \end{align*}
We then conclude from this and \eqref{eq:estimFeB-k-1} that $\|{\mathcal E}^s_{A,u}\|_{C^{0,\a}(\mathbb{R}^N)}\leq C $, provided $2s+\a<1$.\\
If $m>1$, we can use the Leibniz formula for the derivatives of the product of two functions. Note that for all $\g\in\mathbb{N}^N$ with $|\g|\leq m$, we have that $\delta ^e \partial^\g u$ (resp. $\delta ^o \partial^\g u$) satisfies \eqref{eq:Apend-1e} and \eqref{eq:Apend-2e} (resp. \eqref{eq:Apend-4o} and \eqref{eq:Apend-5o}).\\
\noindent \textbf{Case $2s+\a>2$}. We first observe from the arguments in the previous case that \begin{align}\label{eq:L-infty-bound-cEcO-higher}
\|{\mathcal E}_{A,u}^s\|_{L^{\infty}(\mathbb{R}^N)}\leq C \|A\|_{C^{0}(Q_\infty) \times L^\infty(S^{N-1} )} \|u\|_{C^{2s+\a}(\mathbb{R}^N)},\nonumber\\
\|{\mathcal O}_{B,u}^s\|_{L^{\infty}(\mathbb{R}^N)} \leq C \|A\|_{{\mathcal C}^{0}_{1}(Q_\infty)\times L^\infty(S^{N-1} )} \|u\|_{C^{2s+\a}(\mathbb{R}^N)} . \end{align}
Since $B(y,0,\th) =0$, we have $$ B(x_1,r,\th) - B(x_2,r,\th)= r\int_0^1 (D_rB(x_1,\varrho r,\th) - D_r B(x_2,\varrho r,\th)) \, d\varrho. $$ On the other hand $$ B(x_1,r,\th) - B(x_2,r,\th)= \int_0^1 D_x B(\varrho x_1+(1-\varrho) x_2,
r,\th)\cdot (x_1-x_2) \, d\varrho. $$ The above two estimates yield \begin{align}\label{eq:B-x1-x2-High}
|B(x_1,r,\th) - B&(x_2,r,\th)|\leq ( \|B\|_{C^{2s+\a-1}(Q_\infty)\times L^\infty(S^{N-1} )} \nonumber\\
&+ \|B\|_{{\mathcal C}^1_{2s+\a-2} (Q_\infty) \times L^\infty(S^{N-1} )} ) \min (r|x_1-x_2|^{2s+\a-2}, r^{2s+\a-2} |x_1-x_2| ). \end{align} In addition, we have \begin{align*}
\delta ^o u(x_1,r,\th) - \delta ^o u(x_2,r,\th) = \int_0^1 D_x \delta ^ou(\varrho x_1+(1-\varrho) x_2,
r,\th)\cdot (x_1-x_2) d\varrho, \end{align*}
so that $$
| \delta ^o u(x_1,r,\th) - \delta ^o u(x_2,r,\th) ) | \leq C \|u\|_{C^{2s+\a}(\mathbb{R}^N)} \min (r, r^{2s+\a-2} |x_1-x_2|). $$ Using this and \eqref{eq:B-x1-x2-High}, we find that, for all $x_1,x_2\in \mathbb{R}^N$, \begin{align}\label{eq:cOB-Holder-Higher}
|{\mathcal O}_{B,u}^s(x_1)- {\mathcal O}_{B,u}^s(x_2)| \le C ( \|B\|_{C^{2s+\a-1}(Q_\infty)\times L^\infty(S^{N-1} )}+ \|B\|_{{\mathcal C}^1_{2s+\a-2} (Q_\infty) \times L^\infty(S^{N-1} )} ) |x_1-x_2|^\a. \end{align}
Next, we write $2 \delta ^e u(x,r)=r \int_0^1(\n u(x+t r \theta )-\n u(x-t r \theta ))\cdot \theta \, dt$ from which we deduce that \begin{align*}
\delta ^e u(x,r)&= r^2 \int_0^1 t \int_0^1 D^2_x \delta ^o u(x, \varrho t r, \theta ) [\th,\th] \, d\varrho dt \end{align*} and \begin{align*}
\delta ^e u(x_1,r)- \delta ^e u(x_2,r)&=r \int_0^1 \int_0^1D^2_x \delta ^o u( \varrho x_1+ (1-\varrho) x_2,t r, \theta )[x_1-x_2,\th] \, dt d\varrho.
\end{align*} By combining the above two estimates, we get $$
| \delta ^e u(x_1,r)- \delta ^e u(x_2,r)| \leq C \|u\|_{C^{2s+\a}(\mathbb{R}^N)} \min (r^{2s+\a}, r^{2s+\a-1} |x_1-x_2| ). $$
Using now the above estimate and the fact that $A\in {C^{\a}(Q_\infty) \times L^\infty(S^{N-1} )}$, we immediately deduce that $[{\mathcal E}_{A,u}^s]_{C^\a(\mathbb{R}^N)}\leq C \|A\|_{C^{\a}(Q_\infty)\times L^\infty(S^{N-1} )} \|u\|_{C^{2s+\a}(\mathbb{R}^N)} $. From this, \eqref{eq:L-infty-bound-cEcO-higher} and \eqref{eq:cOB-Holder-Higher}, we get the statement in the lemma for $m=0$ and $2s+\a>2$. In the general case that $m\geq 1$, we can use the Leibniz formula for the derivatives of the product of two functions and argue as above to get the desired estimates. \end{proof}
\end{document} | arXiv |
Why dust is optically thin in Far Infrared wavelengths?
What is the actual meaning of the statement 'Dust is optically thin in the Far Infrared (FIR) over most of the Galaxy'? Kindly Help
astrophysics star-formation dust interstellar-medium
RianRian
The term "optically thin" means that the optical depth is small. The optical depth is a measure of the opacity of a medium, in this case dust, experienced by light traveling through that medium, and is defined as $$ \tau \equiv n \, r \, \sigma, $$ where $n$ is the density of the particles in question, $r$ is the distance traveled through the medium, and $\sigma = \sigma(\lambda)$ is the cross section of the particles, which is dependent on the wavelength $\lambda$ of the photons. If $\tau\ll1$, the medium is said to be optically thin, while if $\tau\gg1$, it is said to be optically thick. The fraction of the light that is extinguished by the journey through the medium is $e^{-\tau}$, so the two cases indicate when the light is mostly transferred and mostly extinguished, respectively.
Cosmic dust is composed of particles spanning a large range of sizes. Photons interacting with the dust are either scattered in another direction, or absorbed, depending on the albedo of the dust. But in both cases, a photon has a larger probability of interacting with a dust grain which is comparable to its wavelength. Because of the size distribution, an ensemble of dust grains hence has a characteristic extinction curve. In the figure below (modified from Laursen et al. 2009), I plotted the (functional fits to the observed) extinction curves of dust in the Small (dashed line) and Large (solid line) Magellanic Clouds$^\dagger$:
The extinction is here given in terms of "cross section per hydrogen atom", but you can just think of it as an average dust grain cross section. The colors show the ultraviolet (purple) and infrared (red) regions of the spectrum. You can see that the extinction, or the cross section, is largest for UV photons, and as you go to longer wavelengths, the cross section decreases sharply and is very small in the far infrared ($\lambda \gtrsim 30 \, \mu\mathrm{m}$; the exact definitions depend on your field of interest).
This is what your statement refers to. But in fact it's badly phrased, because the optical depth is also a function of density and distance, so for instance for a distance $r = 1 \, \mathrm{cm}$ through a dust cloud, the optical depth is $\ll1$ for all photons.
$^\dagger$Extinction curves in other galaxies look similar. In particular, the extinction along most sightlines in the Milky Way features the same bump around $\lambda \simeq 2175$ Å. The origin of this bump still isn't well understood, but may be partly due to graphites and/or PAHs.
pelapela
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\begin{document}
\title{A Global Poincar\'e inequality on Graphs via a Conical Curvature-Dimension Condition}
\date{\today}
\maketitle
\renewcommand\abstractname{\footnotesize \textbf{ABSTRACT}}
\rule{\textwidth}{1px}\\
\begin{abstract}
We introduce and study the \emph{conical curvature-dimension condition}, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincar\'e inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the \emph{conical curvature-dimension} analysis is finding a sharp estimate on the curvature of complete graphs.
\end{abstract}
\rule{\textwidth}{0.5px}
\parindent0cm
\setlength{\parskip}{\baselineskip}
\setcounter{tocdepth}{1}
\small
\tableofcontents
\normalsize
\addtocontents{toc}{~
\textbf{Page}\par}
\section{Introduction}
The relation between Ricci curvature bounds and the analytic and geometric properties of a smooth Riemannian manifold is a well studied subject in geometric analysis. Thanks to the seminal work of Sturm~\cite{Stmms1}~\cite{Stmms2} and Lott-Villani~\cite{LV}, the notion of lower Ricci curvature bounds can be generalized to the setting of metric and measure spaces.
\par A Polish metric measure space that satisfies the Lott-Sturm-Villani's $CD(K,N)$ curvature-dimension conditions is called a $CD(K,N)$ space. One important aspect of these spaces is that they support both local and global Poincar\'e inequalities (for a sharp global Poincar\'e inequality and spectral gap on $CD(K,N)$ metric measure spaces, see~\cite[Theorem 5.34]{LV2}). For metric measure spaces that satisfy certain infinitesimal regularity properties, the $CD(K,N)$ curvature-dimension bounds coincide with the Bakry-\'Emery curvature-dimension bounds (or $BE(K,N)$ for short), see~\cite{EKS}. Also, there is a close relation between the lower Ricci bound of $X$ and the lower Ricci curvature bound(s) of the cone(s) over $X$, when $X$ is a Riemannian manifold with ${\rm Ric} \ge (n-1)K$ or more generally an $RCD(K,N)$ space. In particular a Riemannian manifold, $X$, satisfies ${\rm Ric} \ge 1$ if and only if the Riemannian cone over $X$ satisfies ${\rm Ric} \ge 0 $. In the setting of $RCD(K,N)$ metric measure spaces the relation between the weak Ricci curvature bound of $X$ and that of the cone(s) over $X$ has been explored in~\cite{Ketterer}.
\par There are some disparities between the discrete Laplacian on graphs and the Laplacian on manifolds (or on some more general non-smooth continuous metric measure spaces). Despite these disparities, studying Bakry-\'Emery type curvature-dimension conditions for the discrete Laplacian has proven fruitful in the sense that in the discrete setting graphs with lower Ricci curvature bounds satisfy some properties that are similar to the ones satisfied by manifolds with lower Ricci curvature bounds, see~\cite{LY},~\cite{LLY}, ~\cite{CLY} and ~\cite{KGPP}.
\par In this paper we acquire partial results relating the curvature of a graph to the curvature of the cone over over its vertices. In general our paper does not admit a clean cut relation between the lower Bakry-\'Emery Ricci curvature bound of the base graph and that of the cone over the graph. This is mainly due to the fact that in the discrete setting the distance between any two vertices in a cone is at most two and thus the operator $\Gamma_2$ at any point $x$ (a key ingredient in the definition of curvature-dimension bounds) will depend on the entire graph. So the curvature bound at the cone point over the vertex set of a graph will store the curvature information of the entire graph, see~\ref{thm:main-1}.
\par This article is primarily concerned with the properties of the underlying graph $G$ that can be extracted when the cone over $G$ satisfies the $CD(K,N)$ curvature-dimension conditions at the cone point (a property which will be called the conical curvature-dimension, or $CCD(K,N)$ condition). Our main results are a global Poincar\'e inequality and the spectral gap estimates that follow.
\begin{defn}[$CCD(K,N)$ Curvature-Dimension Conditions]\label{defn:CCD} Let $G=(V,E)$ be a finite, connected, undirected, loop-edge free graph and consider the cone over the vertex set of $G$. $G$ is said to satisfy the \emph{conical curvature-dimension condition}, $CCD(K,N)$ for $K \in \mathbb{R}$ and $N \in (1,\infty]$, if the cone over $G$ satisfies the $CD(K,N)$ curvature-dimension conditions at the vertex $p$, namely if
\small \begin{equation} \Gamma^c_2(f)(p)\geq\frac{\bigl(\Delta^c f \bigr)^2(p)}{N}+K\Gamma^c_1(f)(p), \label{eq:ccd} \end{equation} \normalsize
holds for any function $f$ defined on the cone and $\Delta^c$, $\Gamma_1^c$ and $\Gamma^c_2$ are the usual $\Delta$, $\Gamma_1$ and $\Gamma_2$ operators (see (\ref{lap}), (\ref{gamma1}) and (\ref{gamma2})) except on the cone $C(G)$ over $G$. We note that the second term in (\ref{eq:ccd}) is understood to be zero when $N=\infty$. \end{defn}
Now we can state our main theorems and corollaries:
\begin{theorem}[$CCD(K,N)$ implies global Poincar\'e Inequality]\label{thm:main-1}
If a graph, $G$, satisfies $CCD(K,N)$ curvature-dimension condition, then for any function $f$ on $G$ one has
\small
\begin{equation}
\sum_{y\in V}\Gamma_1(f)(y)\geq\frac{2-N}{2N}\biggl ( \sum_{y\in V}f(y)\biggr )^2+\frac{2K+\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y). \notag
\end{equation} \normalsize For functions $f$ with $avg(f)=0$, this reduces to the following global Poincar\'e inequality,
\small
\begin{equation}
\lVert f\rVert_2\leq\sqrt{\frac{2}{2K+\lvert V\rvert-3}}\lVert\nabla f\rVert_2, \notag
\end{equation}
\normalsize
where $\lVert\nabla f\rVert_2$ is understood in the graph setting to be $2\cdot\sum_{y\in V}\Gamma_1(f)(y)$. \end{theorem}
\begin{corollary}\label{cor:main-1}
If $G$ satisfies $CCD(K,N)$ condition, then
\small
\begin{equation}
\lambda_1(G) \ge K+\frac{\lvert V\rvert-3}{2}. \notag
\end{equation}
\normalsize \end{corollary}
\begin{theorem}\label{thm:main-2} For any graph, $G$, and a given $N>1$, the \emph{conical curvature} cannot exceed the following number:
\small
\begin{equation}
K^c_{max} = \frac{\lvert V\rvert}{2}+\frac{3}{2}-2\frac{\lvert V\rvert}{N}. \notag
\end{equation}
\normalsize \end{theorem}
\begin{theorem}[Curvature Maximizers]\label{thm:main-3}
Suppose $G$ satisfies $CCD(K^c_{max},N)$. Then any function, $f$, realizes $K^c_{max}$ if and only if $f$ is either constant or $f-\operatorname{avg}(f)$ is an eigenfunction corresponding to $\lambda_1(G)=\frac{N-2}{4N}\lvert V\rvert$. Furthermore, when $G$ is a complete graph, $f$ must be constant (harmonic). \end{theorem}
\begin{corollary}[Ricci Curvature of Complete Graphs] Suppose $G$ is the complete graph on $n$ vertices, then the $CD(K,N)$ property coincides with the $CCD(K^c,N^c)$ condition on the complete subgraph with $n-1$ vertices and the curvature of $G$ is $\frac{n}{2}+1-2\frac{(n-1)}{N}$. Furthermore any function that realizes this curvature bound is constant (harmonic). \end{corollary} \begin{remark}
When $N=\infty$, our bound $K^c_{max} = 1 + \frac{n}{2} $ coincides with the maximum Ricci curvature of complete graphs as found in~\cite{KGPP}. \end{remark}
The following theorem illustrates an applications of our $\Gamma$-calculus on cones:
\begin{theorem}\label{thm:main-5}
Suppose $G$ satisfies $CD(K,\infty)$ for $K \le \frac{1}{2}$ then the subgraph $G\subset C(G)$ satisfies $CD(K+\frac{1}{2},\infty)$. \end{theorem}
\subsection*{Acknowledgments} The authors would like to thank Professor J\'ozef Dodziuk for his interest and encouragement and the Mathematics Department at the CUNY Graduate Center for providing the first author access to their facilities. The authors are also very thankful for the interest and insight of Jorge Basilio.
\section{Preliminaries}
Let $G=(V,E)$ be an undirected, unweighted, connected, locally finite graph without any loop-edges. Let $f:V\to\mathbb{R}$ and consider the space of square-summable functions on the vertex set.
\par The graph Laplacian is given by
\small \begin{equation} \Delta f(x)=\sum_{y\sim x} \bigl ( f(y)-f(x) \bigr), \label{lap} \end{equation} \normalsize
where $y\sim x$ means that $(y,x)\in E$. Also, note that the graph Laplacian is a real valued, self-adjoint linear operator (for a thorough treatment of the graph Laplacian see~\cite{Dodziuk}).
\par Let $F\subset V$, then the boundary of $F$ is
\small \begin{equation}
\partial F:=\{(x,y) \in E \mid\ x \in F \text{ and } y \in V \setminus F \}. \notag \end{equation} \normalsize
The isoperimetric constant (or Cheeger's constant) is then defined as
\small \begin{equation} h(G):=\inf\left\{\dfrac{\lvert\partial F\rvert}{\min\{\lvert F\rvert, \lvert V\setminus F\rvert\}}:\ 0<\lvert F\rvert<\infty\right\}. \notag \end{equation} \normalsize
\par A well known generalization of Cheeger's and Buser's results for Riemannian manifolds is the following theorem due to Dodziuk~\cite{Dodziuk} and Alon-Milman~\cite{Alon-Milman}.
\begin{theorem}[\cite{Dodziuk}, \cite{Alon-Milman}] Let $G=(V,E)$ be a finite, connected, edge-loop free graph. Let $d_{max}=\sup_{v\in V}\{\deg(v)\}$ and let $\lambda_1$ be the first non-trivial eigenvalue of $\Delta$, then
\small \begin{equation} \frac{\lambda_1}{2}\leq h(G)\leq\sqrt{2d_{max}\lambda_1}. \label{eq:DAM} \end{equation} \normalsize \end{theorem}
The $\Gamma$ operators of Bakry-\'Emery associated to the graph Laplacian, $\Delta$, are:
\small \begin{eqnarray} \Gamma_1(f,g)(x) &=&\frac{1}{2} \biggl [\Delta(fg)(x)- g(x) \Delta f(x)-f(x)\Delta g(x) \biggr],\label{gamma1}\\ \Gamma_2(f,g)(x) &=&\frac{1}{2} \biggl[ \Delta \Gamma_1(f,g) (x)-\Gamma_1 \bigl ( \Delta f , g \bigr)(x)-\Gamma_1\bigl( f, \Delta g \bigr)(x) \biggr].\label{gamma2} \end{eqnarray} \normalsize
It is straightforward to check that
\small \begin{equation} \Gamma_1(f,g)(x) = \frac{1}{2}\sum_{y\sim x}(f(y)-f(x))(g(y)-g(x)) =: \frac{1}{2} \langle \nabla f, \nabla g \rangle. \label{eq:gradfg} \end{equation} \normalsize
Throughout these notes $\Gamma_1 (f) := \Gamma_1(f,f)$ and similarly $\Gamma_2 (f) := \Gamma_2(f,f)$. Thus $\Gamma_1(f)(x)=\frac{1}{2}\lvert\nabla f(x)\rvert^2$ and one can verify the following useful divergence-type identity:
\small \begin{equation}\label{eq:divergence} \frac{1}{2}\lVert\nabla f\rVert_2^2=\sum_{y\in V}\Gamma_1(f)(y)=-\sum_{y\in V}f(y)\Delta f(y). \end{equation} \normalsize
\begin{definition}[Bakry-\'Emery Curvature-Dimension Condition]\label{defn:cdkn} Suppose $K \in \mathbb{R}$ and $N\in(1,\infty]$. We say that a graph $G=(V,E)$ satisfies the curvature-dimension conditions, $CD(K,N)$, if for every $x\in V$ and every $f\in\ell^2(V)$,
\small
\begin{equation}
\Gamma_2(f)(x)\geq\frac{(\Delta f)^2(x)}{N}+K\Gamma_1(f)(x).
\end{equation}
\normalsize
Note when $N=\infty$, the second term in the inequality above is understood to be 0. \end{definition}
\begin{defn}[Uniform and Pointwise Ricci Curvatures] We define the dimensional (respectively, dimensionless) Ricci curvature of the graph $G$, ${\rm Ric}_{N}(G)$ (respectively, ${\rm Ric}_{\infty}(G)$), by
\small \begin{equation} {\rm Ric}_N(G) := \sup \left\{ K \ : \ \text{$G$ satisfies $CD(K,N)$} \right\} \notag \end{equation} \normalsize
and \small \begin{equation}
{\rm Ric}_{\infty}(G) := \sup \left\{ K \ : \ \text{$G$ satisfies $CD(K,\infty)$} \right\}. \notag \end{equation} \normalsize
Similarly, we define the pointwise curvatures by
\small \begin{equation} {\rm Ric}_N(y):=\sup\{K:\Gamma_2(f)(y)\geq\frac{1}{N}(\Delta f)^2(y)+K\Gamma_1(f)(y) ,\ \forall f \} \notag \end{equation} \normalsize
and \small \begin{equation} {\rm Ric}_{\infty}(y):=\sup\{K:\Gamma_2(f)(y)\geq K\Gamma_1(f)(y) ,\ \forall f \}. \notag \end{equation} \normalsize \end{defn}
\begin{definition}[Conical Ricci Curvatures] We define the conical Ricci curvature by
\small \begin{equation} CRic_N(G):=\sup\{K: \text{$G$ satisfies $CCD(K,N)$ as in (\ref{eq:ccd})}\} \notag \end{equation} \normalsize and
\small \begin{equation} CRic_{\infty}(G):=\sup\{K: \text{$G$ satisfies $CCD(K,\infty)$ as in (\ref{eq:ccd})}\}. \notag \end{equation} \normalsize
\end{definition}
We close this section by recalling that the first non-zero eigenvalue of the Laplacian may be computed via the Rayleigh quotient:
\small \begin{equation}\label{eq:Rayleigh}
\lambda_1 = \inf \left\{ \frac{\| \nabla f \|^2 }{\| f \|^2} :\ \operatorname{avg}(f) = 0 \right\}. \end{equation} \normalsize
\section{Cones over Graphs and Their $\Gamma - $ Calculus} The complete cone, $C(G)$, over a finite graph $G$ is constructed by taking the graph Cartesian product of $G$ and $H$, $G \Box H$, where $H=(\{q,p\},\{(q,p)\})$ is the complete graph on two vertices $q$ and $p$, and then identifying all the vertices whose second component is $p$. In this paper $p$ refers to the cone point of $C(G)$.
\par More generally for a subset, $X \subset V(G)$, the partial cone, $C\left( X , G \right)$, is a subgraph of $C(G)$ containing $G$ and all edges $(x,p)$, $x \in X$. For brevity we will use a superscript $c$ to denote any operation that is taking place in a partial cone over $G$. Notice that any vertex $v\in V(G)$ can be thought of as the cone point over the 1-sphere based at $v$, i.e. $S_v^1:=\{y\in V\ \mid \ d_G(y,v)=1\}=X$ in the above construction. In this way partial cones can be useful in studying cliques.
\par The first subsection is devoted to proving a few lemmas that calculate the $\Delta$ and $\Gamma$ operators of a partial cone in terms of the similar operators on the base graph. The last subsection is devoted to an immediate result.
\subsection{$\Gamma$-Calculus on a Cone} \par Since $\Delta$ and $\Gamma$ operators agree for functions that differ by a constant we may assume, without loss of generality, that $f(p)=0$.
Denote by $S^n_p$ and $B^n_p$ the metric spheres and balls (resp.) with radius $n$ and center $p$ in the cone. For any subset $B \subset V$, the notation $v \in B\sim x$ means $v \in B$ and $v \sim x$.
\begin{remark}
Note that $\Delta$ and $\Gamma_1$ only depend on vertices that are at most one away. Thus, $\Delta^c f(x)=\Delta f(x)$ and $\Gamma^c_1(f)(x)=\Gamma_1(f)(x)$ when $x\nsim p$. \end{remark}
\begin{lemma}\label{lem:cone-Delta} Let $f$ be a function on the cone with $f(p) =0$ then,
\small \begin{equation} \Delta^cf(x)=\begin{cases}
\Delta f(x)-f(x); & x\sim p\\
\sum_{y\in S^1_p}f(y); & x=p\\ \end{cases} \notag \end{equation} \normalsize \end{lemma}
\begin{proof}
\begin{enumerate} \item If $x\sim p$, then
\small
\begin{equation}
\Delta^c f (x)=\sum_{y\in C\sim x} \bigl( f(y)-f(x) \bigr) = \sum_{y\in V\sim x} \bigl( f(y)-f(x) \bigr)+ \bigl( f(p)-f(x) \bigr) = \Delta f(x)-f(x). \notag
\end{equation}
\normalsize
\item If $x=p$, then
\small
\begin{equation}
\Delta^c f (p)= \sum_{y\in C\sim p} \bigl( f(y)-f(p) \bigl) = \sum_{y\in S^1_p} f(y). \notag
\end{equation}
\normalsize
\end{enumerate} \end{proof}
\begin{lemma}\label{lem:cone-Gamma1} Let $f$ be a function on the cone with $f(p) =0$ then,
\small \begin{equation} \Gamma_1^c(f)(x)=\begin{cases}
\Gamma_1(f)(x)+\frac{1}{2}f^2(x); & x\sim p\\
\frac{1}{2}\sum_{y\in S^1_p}f^2(y); & x=p\\
\end{cases} \notag \end{equation} \normalsize \end{lemma}
\begin{proof} \begin{enumerate} \item If $x\sim p$, then using (\ref{eq:gradfg})
\small
\begin{equation}
\Gamma_1^c(f)(x)=\frac{1}{2}\sum_{y\in C\sim x} \bigl( f(y)-f(x) \bigr)^2 = \frac{1}{2}\sum_{y\in V\sim x} \bigl( f(y)-f(x) \bigl)^2 + \frac{1}{2} \bigl( f(p)-f(x) \bigl)^2= \Gamma_1(f)(x)+\frac{1}{2}f^2(x). \notag
\end{equation} \normalsize
\item If $x=p$, then using (\ref{eq:gradfg})
\small
\begin{equation}
\Gamma_1^c(f)(p)=\frac{1}{2}\sum_{y\in V} \bigl( f(y)-f(p) \bigr)^2 =\frac{1}{2}\sum_{y\in S^1_p} f^2(y). \notag
\end{equation}
\normalsize \end{enumerate} \end{proof}
In the next few lemmas we calculate the constituent parts that appear in the definition of $\Gamma^c_2$. \begin{remark} Note that $\Gamma^c_2$ depends on vertices at most two away. Thus $\Gamma^c_2$ coincides with $\Gamma_2$ when $x\in V \setminus B_p^2$. \end{remark}
\begin{lemma} Let $f$ be a function defined on the cone, and suppose $f(p) = 0$, then
\small \begin{equation} \Gamma_1^c \bigl( f,\Delta^cf \bigr)(x)=\begin{cases}
\Gamma_1 \bigl( f,\Delta f \bigr)(x)-\frac{1}{2}\sum_{y\in S^1_p\sim x}f(y) \bigl( f(y)-f(x) \bigr); & x\in S^2_p\\
\Gamma_1 \bigl( f,\Delta f \bigr)(x)-\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2\\
\qquad\qquad\qquad+\frac{1}{2}f(x)\sum_{y\in S^2_p\sim x} \bigl( f(y)-f(x) \bigr)\\
\qquad\qquad\qquad-\frac{1}{2}f(x)\sum_{y\in S^1_p}f(y)+\frac{1}{2}f(x)\Delta f(x)-\frac{1}{2}f^2(x) & x\sim p\\
\frac{1}{2}\sum_{y\in S^1_p}f(y)\Delta f(y)-\frac{1}{2}\sum_{y\in S^1_p}f^2(y)-\frac{1}{2}(\sum_{y\in S^1_p}f(y))^2 & x=p
\end{cases}\notag \end{equation} \normalsize \end{lemma}
\begin{proof} \begin{enumerate}
\item If $x\in S^2_p$, then using (\ref{eq:gradfg})
\small
\begin{eqnarray*}
\Gamma_1^c \bigl( f, \Delta^c f \bigr)(x) &=& \frac{1}{2}\sum_{y\in C\sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta^c f(y)-\Delta^c f(x) \bigr)\\
&=&\frac{1}{2}\sum_{y\in V \setminus S_p^1 \sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta^c f(y)-\Delta^c f(x) \bigr) \\
&& + \; \frac{1}{2}\sum_{y\in S^1_p\sim x}\bigl( f(y)-f(x) \bigr) \bigl( \Delta^c f(y)-\Delta^c f(x) \bigr)\\
&=&\frac{1}{2}\sum_{y\in V \setminus S_p^1 \sim x}\bigl( f(y)-f(x) \bigr) \bigl( \Delta f(y)-\Delta f(x) \bigr) \\
&&+ \; \frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta f(y)-f(y)-\Delta f(x) \bigr)\\
&=&\frac{1}{2}\sum_{y\in V\sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta f(y)-\Delta f(x) \bigr)-\frac{1}{2}\sum_{y\in S^1_p\sim x}f(y)(f(y)-f(x))\\
&=&\Gamma_1 \bigl( f,\Delta f \bigr)(x)-\frac{1}{2}\sum_{y\in S^1_p\sim x}f(y) \bigl( f(y)-f(x) \bigr) .
\end{eqnarray*}
\normalsize
\item If $x\sim p$, then using (\ref{eq:gradfg})
\small
\begin{eqnarray*}
\Gamma_1^c \bigl( f,\Delta^c f \bigr) (x) &=&\frac{1}{2}\sum_{y\in C\sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta^c f(y)-\Delta^c f(x) \bigr) \\
&=&\frac{1}{2}\sum_{y\in S^2_p\sim x}\bigl ( f(y)-f(x) \bigr) \bigl( \Delta f(y)-\Delta f(x)+f(x) \bigr)\\
&&+ \; \frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta f(y)-f(y)-\Delta f(x)+f(x) \bigr)\\
&&+ \; \frac{1}{2} \bigl( f(p)-f(x) \bigr) \bigl( \sum_{y\in S^1_p}f(y)-\Delta f(x)+f(x) \bigr)\\
&=&\frac{1}{2}\sum_{y\in S^2_p\sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta f(y)-\Delta f(x) \bigr) +\frac{1}{2}f(x)\sum_{y\in S^2_p\sim x} \bigl( f(y)-f(x) \bigr)\\
&&+ \; \frac{1}{2}\sum_{y\in S^1_p\sim x}\bigl( f(y)-f(x) \bigr) \bigl( \Delta f(y)-\Delta f(x) \bigr) -\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2\\
&&- \; \frac{1}{2}f(x)\sum_{y\in S^1_p}f(y)+\frac{1}{2}f(x)\Delta f(x)-\frac{1}{2}f^2(x)\\
&=&\frac{1}{2}\sum_{y\in V\sim x} \bigl( f(y)-f(x) \bigr) \bigl( \Delta f(y)-\Delta f(x) \bigr)-\frac{1}{2}\sum_{y\in S^1_p\sim x}\bigl( f(y)-f(x) \bigr)^2\\ &&+ \; \frac{1}{2}f(x)\sum_{y\in S^2_p\sim x}\bigl( f(y)-f(x) \bigr)-\frac{1}{2}f(x)\sum_{y\in S^1_p}f(y)+\frac{1}{2}f(x)\Delta f(x)-\frac{1}{2}f^2(x)\\
&=&\Gamma_1 \bigl( f,\Delta f \bigr) (x)-\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2 +\frac{1}{2}f(x)\sum_{y\in S^2_p\sim x} \bigl( f(y)-f(x) \bigr) \\ &&- \; \frac{1}{2}f(x)\sum_{y\in S^1_p}f(y)+\frac{1}{2}f(x)\Delta f(x)-\frac{1}{2}f^2(x).
\end{eqnarray*} \normalsize
\item If $x=p$, then using (\ref{eq:gradfg})
\small
\begin{eqnarray*}
\Gamma_1^c \bigl( f,\Delta^c f \bigr) (p) &=&\frac{1}{2}\sum_{y\in C\sim p} \bigl( f(y)-f(p) \bigr) \bigl( \Delta^cf(y)-\Delta^cf(p) \bigr)\\
&=&\frac{1}{2}\sum_{y\in S^1_p}\bigl( f(y)-f(p) \bigr) \bigl( \Delta f(y)-f(y)-\sum_{z\in S^1_p}f(z) \bigr)\\
&=&\frac{1}{2}\sum_{y\in S^1_p}\biggl [f(y)\Delta f(y)-f^2(y)-f(y)\sum_{z\in S^1_p}f(z) \biggr ]\\
&=&\frac{1}{2}\sum_{y\in S^1_p}f(y)\Delta f(y)-\frac{1}{2}\sum_{y\in S^1_p}f^2(y)-\frac{1}{2} \bigl( \sum_{y\in S^1_p}f(y) \bigl)^2.
\end{eqnarray*} \normalsize \end{enumerate} \end{proof}
\begin{lemma} Let $f$ be a function defined on the cone, and suppose $f(p) = 0$, then
\small \begin{equation} \Delta^c\Gamma_1^c(f)(x)=\begin{cases}
\Delta\Gamma_1(f)(x)+\frac{1}{2}\sum_{y\in S^1_p\sim x}f^2(y); & x\in S_p^2\\
\Delta\Gamma_1(f)(x)-\Gamma_1(f)(x)\\
\qquad\qquad\quad+\frac{1}{2}\biggl[\sum_{y\in S^1_p\sim x}f^2(y)+\sum_{y\in S^1_p}f^2(y)\biggr]\\
\qquad\qquad\quad-\frac{1}{2}\deg(x)f^2(x) & x\sim p\\
\sum_{y\in S^1_p}\Gamma_1(f)(y)-\frac{\lvert S^1_p\rvert-1}{2}\sum_{y\in S^1_p}f^2(y); & x=p
\end{cases} \notag \end{equation} \normalsize \end{lemma}
\begin{proof} \begin{enumerate}
\item If $x\in S^2_p$, then
\small
\begin{eqnarray*}
\Delta^c\Gamma_1^c(f)(x) &=&\sum_{y\in C\sim x} \biggl[ \Gamma_1^c(f)(y)-\Gamma_1^c(f)(x) \biggr]\\
&=&\sum_{y\in V \setminus S_p^1 \sim x} \biggl[ \Gamma_1^c(f)(y)-\Gamma_1^c(f)(x) \biggr]+\sum_{y\in S^1_p\sim x} \biggl[ \Gamma_1^c(f)(y)-\Gamma_1^c(f)(x) \biggr]\\
&=&\sum_{y\in V \setminus S_p^1 \sim x} \biggl[ \Gamma_1(f)(y)-\Gamma_1(f)(x) \biggr] +\sum_{y\in S^1_p\sim x} \biggl[ \Gamma_1(f)(y)+\frac{1}{2}f^2(y)-\Gamma_1(f)(x) \biggr]\\
&=&\sum_{y\in V\sim x} \biggl[ \Gamma_1(f)(y)-\Gamma_1(f)(x) \biggr] + \frac{1}{2}\sum_{y\in S^1_p\sim x}f^2(y)\\
&=&\Delta\Gamma_1(f)(x)+\frac{1}{2}\sum_{y\in S^1_p\sim x}f^2(y).
\end{eqnarray*}
\normalsize
\item If $x\sim p$, then
\small
\begin{eqnarray*}
\Delta^c\Gamma_1^c(f)(x) &=&\sum_{y\in C\sim x} \biggl[ \Gamma_1^c(f)(y)-\Gamma_1^c(f)(x) \biggr] \\
&=&\sum_{y\in S^2_p\sim x} \biggl[ \Gamma_1(f)(y)-\Gamma_1(f)(x)-\frac{1}{2}f^2(x) \biggr]
\\
&&+ \; \sum_{y\in S^1_p\sim x} \biggl[ \Gamma_1(f)(y)-\Gamma_1(f)(x)+\frac{1}{2} \bigl( f^2(y)-f^2(x) \bigr) \biggr] + \Gamma_1^c(f)(p)-\Gamma_1(f)(x)-\frac{1}{2}f^2(x)\\
&=&\sum_{y\in S^2_p\sim x} \biggl[ \Gamma_1(f)(y)-\Gamma_1(f)(x) \biggr]-\frac{1}{2}\deg(x) f^2(x)
+\sum_{y\in S^1_p\sim x} \biggl[ \Gamma_1(f)(y)-\Gamma_1(f)(x) \biggr]\\
&&+ \; \frac{1}{2}\sum_{y\in S^1_p\sim x} f^2(y)+\frac{1}{2}\sum_{y\in S^1_p}f^2(y)-\Gamma_1(f)(x)\\
&=&\sum_{y\in V\sim x} \biggl[ \Gamma_1(f)(y)-\Gamma_1(f)(x) \biggr] - \Gamma_1(f)(x)\\
&&+ \; \frac{1}{2}\sum_{y\in S^1_p\sim x} f^2(y) + \frac{1}{2}\sum_{y\in S^1_p}f^2(y)-\frac{1}{2} \deg(x)f^2(x)\\
&=&\Delta\Gamma_1(f)(x)-\Gamma_1(f)(x)+\frac{1}{2}\biggl[\sum_{y\in S^1_p\sim x}f^2(y)+\sum_{y\in S^1_p}f^2(y)\biggr]
-\frac{1}{2}\deg(x)f^2(x).
\end{eqnarray*}
\normalsize
\item If $x=p$, then
\small
\begin{eqnarray*}
\Delta^c \bigl( \Gamma_1^c(f) \bigr) (p) &=&\sum_{y\in S^1_p} \biggl[ \Gamma_1^c(f)(y)-\Gamma_1^c(f)(p) \biggr]\\
&=&\sum_{y\in S^1_p} \biggl[ \Gamma_1(f)(y)+\frac{1}{2}f^2(y)-\frac{1}{2}\sum_{y\in S^1_p}f^2(y) \biggr]\\
&=&\sum_{y\in S^1_p}\Gamma_1(f)(y)+\frac{1}{2}\sum_{y\in S^1_p}f^2(y)-\frac{\lvert S^1_p\rvert}{2}\sum_{y\in S^1_p}f^2(y)\\
&=&\sum_{y\in S^1_p}\Gamma_1(f)(y)-\frac{(\lvert S^1_p\rvert-1)}{2}\sum_{y\in S^1_p}f^2(y).
\end{eqnarray*} \normalsize \end{enumerate} \end{proof}
\begin{lemma}\label{lem:cone-Gamma2} Let $f$ be a function defined on the cone, and suppose $f(p) = 0$, then
\small \begin{equation} \Gamma_2^c(f)(x)=\begin{cases}
\Gamma_2(f)(x)+\frac{3}{4}\sum_{y\in S^1_p\sim x}f^2(y)-\frac{1}{2}f(x)\sum_{y\in S^1_p\sim x}f(y); & x\in S_p^2\\
\Gamma_2(f)(x)- \frac{1}{2} \Gamma_1(f)(x)+\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2\\
\qquad\qquad+\frac{1}{4}\biggl[\sum_{y\in S^1_p\sim x}f^2(y)-\deg(x)f^2(x)\biggr]-\frac{1}{2}f(x)\Delta f(x)\\
\qquad\qquad-\frac{1}{2}f(x)\sum_{y\in S^2_p\sim x}(f(y)-f(x))+\frac{1}{4}\sum_{y\in S^1_p}f^2(y)+\frac{1}{2} f^2(x) ; & x\sim p\\
\frac{1}{2}\sum_{y\in S^1_p}\Gamma_1(f)(y)-\frac{1}{2}\sum_{y\in S^1_p}f(y)\Delta f(y)\\
\qquad\qquad -\frac{\vert S^1_p\rvert-3}{4}\sum_{y\in S^1_p}f^2(y)+\frac{1}{2} \bigl( \sum_{y\in S^1_p}f(y) \bigr )^2; & x=p
\end{cases} \notag
\end{equation}
\normalsize \end{lemma}
\begin{proof} \begin{enumerate}
\item If $x\in S^2_p$, then
\small
\begin{eqnarray*}
\Gamma_2^c(f)(x) &=&\frac{1}{2}\Delta^c\Gamma_1^c(f,f)(x)-\Gamma_1^c \bigl( f,\Delta^c f \bigr)(x)\\
&=&\frac{1}{2}\Delta\Gamma_1(f)(x)+\frac{1}{4}\sum_{y\in S^1_p\sim x}f^2(y)-\Gamma_1\bigl( f,\Delta f \bigr)(x)+\frac{1}{2}\sum_{y\in S^1_p\sim x}f(y) \bigl( f(y)-f(x) \bigr)\\
&=&\Gamma_2(f)(x)+\frac{3}{4}\sum_{y\in S^1_p\sim x}f^2(y)-\frac{1}{2}f(x)\sum_{y\in S^1_p\sim x}f(y).
\end{eqnarray*}
\normalsize
\item If $x\sim p$, then
\small
\begin{eqnarray*}
\Gamma_2^c(f)(x) &=&\frac{1}{2}\Delta^c\Gamma_1^c(f)(x)-\Gamma_1^c\bigl( f,\Delta^c f \bigr)(x)\\
&=&\frac{1}{2}\Delta\Gamma_1(f)(x)-\frac{1}{2}\Gamma_1(f)(x)+\frac{1}{4}\sum_{y\in S^1_p\sim x} f^2(y)+\frac{1}{4}\sum_{y\in S^1_p}f^2(y)-\frac{1}{4}\deg(x)f^2(x)\\
&&- \; \Gamma_1\bigl( f,\Delta f \bigr)(x)+\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2 -\frac{1}{2}f(x)\sum_{y\in S^2_p \sim x} \bigl( f(y)-f(x) \bigr)\\
&&+ \; \frac{1}{2}f(x)\sum_{y\in S^1_p}f(y) -\frac{1}{2}f(x)\Delta f(x)+\frac{1}{2}f^2(x).\\
&=& \biggl[ \frac{1}{2}\Delta\Gamma_1(f)(x)-\Gamma_1 \bigl( f,\Delta f \bigr)(x) \biggr] - \frac{1}{2} \Gamma_1(f)(x)+\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2 \\
&&+ \; \frac{1}{4}\biggl[\sum_{y\in S^1_p\sim x}f^2(y)-\deg(x)f^2(x)\biggr]+\frac{1}{4}\sum_{y\in S^1_p}f^2(y)-\frac{1}{2}f(x)\sum_{y\in S^2_p\sim x}(f(y)-f(x))\\
&&- \; \frac{1}{2}f(x)\Delta f(x)+\frac{1}{2} f^2(x)\\
&=&\Gamma_2(f)(x)- \frac{1}{2} \Gamma_1(f)(x)+\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2-\frac{1}{2}f(x)\sum_{y\in S^2_p\sim x}(f(y)-f(x)) \\
&&+ \; \frac{1}{4}\biggl[\sum_{y\in S^1_p\sim x}f^2(y)-\deg(x)f^2(x)\biggr]-\frac{1}{2}f(x)\Delta f(x)+\frac{1}{4}\sum_{y\in S^1_p}f^2(y)+\frac{1}{2} f^2(x)
\end{eqnarray*}
\normalsize \item If $x=p$, then
\small
\begin{eqnarray*}
\Gamma_2^c(f)(p)&=&\frac{1}{2}\Delta^c\Gamma_1^c(f)(p)-\Gamma_1^c \bigl( f,\Delta^c f \bigr)(p)\\
&=&\frac{1}{2}\sum_{y\in S^1_p}\Gamma_1(f)(y)-\frac{\lvert S^1_p\rvert-1}{4}\sum_{y\in S^1_p}f^2(y)-\frac{1}{2}\sum_{y\in S^1_p}f(y)\Delta f(y)+\frac{1}{2}\sum_{y\in S^1_p}f^2(y)+\frac{1}{2} \bigl(\sum_{y\in S^1_p}f(y) \bigr)^2\\
&=&\frac{1}{2}\sum_{y\in S^1_p}\Gamma_1(f)(y)-\frac{1}{2}\sum_{y\in S^1_p}f(y)\Delta f(y)-\frac{\lvert S^1_p\rvert-3}{4}\sum_{y\in S^1_p}f^2(y)+\frac{1}{2}\bigl(\sum_{y\in S^1_p}f(y)\bigr )^2 .
\end{eqnarray*}
\normalsize \end{enumerate} \end{proof}
\subsection{$\Gamma^c_2$ for $C(G)$}
When $C=(V^c,E^c)$ is the full cone over $V(G)$, then $S^1_p=V$ and so Lemma~\ref{lem:cone-Gamma2} reduces to
\begin{lemma}\label{lem:cone-Gamma3}
\small \begin{equation} \Gamma_2^c(f)(x)=\begin{cases}
\Gamma_2(f)(x)+\frac{1}{2}\Gamma_1(f)(x)+\frac{1}{4}\sum_{y\in V}f^2(y)+\frac{1}{2}f^2(x); & x\sim p\\
\sum_{y\in V}\Gamma_1(f)(y)-\frac{\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y)+\frac{1}{2} \bigl( \sum_{y\in V}f(y) \bigl )^2; & x=p\\
\end{cases} \notag.
\end{equation}
\normalsize \end{lemma}
\begin{proof}
Since $S^1_p=V$ and $S^2_p=\varnothing$ the first case in Lemma~\ref{lem:cone-Gamma2} disappears. In case 2 notice that when $S^1_p=V$, then $\frac{1}{2}\sum_{y\in S^1_p\sim x} \bigl( f(y)-f(x) \bigr)^2=\Gamma_1(f)(x)$ and $\sum_{y\in S^1_p\sim x} \bigl( f^2(y)-f^2(x) \bigr)= \bigl( \Delta f^2 \bigr)(x)$. Since $\frac{1}{2}\Gamma_1(f)(x)= \frac{1}{4} \bigl( \Delta f^2 \bigr)(x)-\frac{1}{2}f(x)\Delta f(x)$ the case when $x\sim p$ follows. When $x=p$, applying the identity (\ref{eq:divergence}) gives the desired result. \end{proof}
This leads to the following result regarding the curvature of the cone,
\begin{customthm}{\ref{thm:main-5}}
Suppose $G$ satisfies $CD(K,\infty)$ for $K \le \frac{1}{2}$ then the subgraph $G\subset C(G)$ satisfies $CD(K+\frac{1}{2},\infty)$. \end{customthm} \begin{proof}[Proof of Theorem~\ref{thm:main-5}] Suppose $G$ satisfies $CD(K,\infty)$ for $K \le \frac{1}{2}$. Since $G$ satisfies $CD(K,\infty)$ then by lemma~\ref{lem:cone-Gamma3} for $x\sim p$,
\small
\begin{equation}
\Gamma_2^c(f)(x)\geq (K+1)\Gamma_1(f)(x)+\frac{1}{4}\sum_{y\in V}f^2(y)+\frac{1}{2}f^2(x). \notag
\end{equation}
\normalsize
Therefore,
\small
\begin{equation}
\Gamma_2^c(f)(x)\geq (K+1)\Gamma^c_1(f)(x)+\frac{1}{4}\sum_{y\in V}f^2(y)-\frac{K}{2}f^2(x). \notag
\end{equation}
\normalsize Since $K \le \frac{1}{2}$, then $\frac{1}{4}\sum_{y\in V}f^2(y)-\frac{K}{2}f^2(x)\geq0$. Hence we may drop both terms from the inequality and $C(G)$ satisfies $CD(K+1,\infty)$ for $x\sim p$. \end{proof}
\section{$CCD(K,N)$ and Global Poincar\'e Inequality }
\par If the cone $C$ satisfies the $CD(K,N)$ inequality at the vertex, $p$, then by Lemmas~\ref{lem:cone-Gamma1} and~\ref{lem:cone-Delta}, we get
\small
\begin{equation}\label{eq:cd-vertex-1}
\Gamma_2^c(f)(p)\geq\frac{1}{N}\bigl (\sum_{y\in V}f(y) \bigr )^2+\frac{K}{2}\sum_{y\in V}f^2(y).
\end{equation}
\normalsize This leads to the following,
\begin{customthm}{\ref{thm:main-1}}
If a graph, $G$, satisfies $CCD(K,N)$ curvature-dimension condition, then for any function $f$ on $G$ one has
\small
\begin{equation}\label{eq:poincare}
\sum_{y\in V}\Gamma_1(f)(y)\geq\frac{2-N}{2N}\biggl ( \sum_{y\in V}f(y)\biggr )^2+\frac{2K+\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y).
\end{equation} \normalsize For functions $f$ with $avg(f)=0$, this reduces to the following global Poincar\'e inequality,
\small
\begin{equation}
\lVert f\rVert_2\leq\sqrt{\frac{2}{2K+\lvert V\rvert-3}}\lVert\nabla f\rVert_2, \notag
\end{equation}
\normalsize
where $\lVert\nabla f\rVert_2$ is understood in the graph setting to be $2\cdot\sum_{y\in V}\Gamma_1(f)(y)$. \end{customthm}
\begin{proof}[Proof of Theorem~\ref{thm:main-1}]
Suppose a graph $G$ satisfies $CCD(K,N)$ condition. By Lemma~\ref{lem:cone-Gamma3},
\small
\begin{equation}\label{eq:cd-vertex-2}
\Gamma_2^c(f)(p)=\sum_{y\in V}\Gamma_1(f)(y)-\frac{\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y)+\frac{1}{2}\bigl (\sum_{y\in V}f(y)\bigr )^2,
\end{equation}
\normalsize
Upon combining (\ref{eq:cd-vertex-2}) and (\ref{eq:cd-vertex-1}), we will arrive at
\small
\begin{equation}
\sum_{y\in V}\Gamma_1(f)(y)-\frac{\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y)+\frac{1}{2}\bigl (\sum_{y\in V}f(y)\bigr )^2\geq\frac{1}{N}\bigl (\sum_{y\in V}f(y)\bigr )^2+\frac{K}{2}\sum_{y\in V}f^2(y), \notag
\end{equation}
\normalsize
which simplifies to
\small
\begin{equation}
\sum_{y\in V}\Gamma_1(f)(y)\geq\frac{2-N}{2N}\bigl( \sum_{y\in V}f(y)\bigr)^2+\frac{2K+\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y). \notag
\end{equation}
\normalsize For $f$ with $\operatorname{avg}(f) = 0$, the above reduces to
\small
\begin{equation}
\frac{1}{2}\sum_{y\in V}\lvert\nabla f(y)\rvert^2\geq\frac{2K+\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y).
\end{equation}
\normalsize By the definition in ~\ref{eq:gradfg} this yields the Poincar\'e inequality,
\small
\begin{equation}
\lVert f \rVert_2\leq\sqrt{\frac{2}{2K+\lvert V\rvert-3}}\lVert\nabla f\rVert_2 \text{, when } \operatorname{avg}(f) =0. \notag
\end{equation}
\normalsize \end{proof}
\begin{customthm}{\ref{thm:main-2}} For any graph, $G$, and a given $N>1$, the \emph{conical curvature} cannot exceed the following number:
\small
\begin{equation}
K^c_{max} = \frac{\lvert V\rvert}{2}+\frac{3}{2}-2\frac{\lvert V\rvert}{N}. \notag
\end{equation}
\normalsize \end{customthm} \begin{proof}[Proof of Theorem~\ref{thm:main-2}]
Suppose a finite graph $G$ satisfies the $CCD(K,N)$ and $f$ is a non-zero harmonic function, then one has $\sum_{y\in V}\Gamma_1(f)(y)=0$ (i.e. $f$ is constant on connected components). Thus,
\small
\begin{eqnarray*}
\frac{2K+\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y) \le \frac{N-2}{2N}\bigl ( \sum_{y\in V}f(y)\bigr)^2. \notag
\end{eqnarray*}
\normalsize
By the Cauchy-Schwarz inequality,
\small \begin{equation} \bigl ( \sum_{y\in V}f(y)\bigr )^2 \le \lvert V\rvert\cdot\sum_{y\in V}f^2(y), \notag \end{equation} \normalsize
which implies
\small \begin{equation} \frac{2K+\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y) \le \frac{N-2}{2N}\lvert V\rvert\cdot\sum_{y\in V}f^2(y) . \notag \end{equation} \normalsize
Since, $f$ is not constant zero,
\small \begin{equation} K \le \frac{\lvert V\rvert}{2}-\frac{2\lvert V\rvert}{N}+\frac{3}{2}. \notag \end{equation} \normalsize \end{proof}
Having established an upper bound for the curvature at the cone point over the vertex set of the graph $G$ we now turn to an investigation of when the maximum curvature value is achieved.
\begin{lemma} For any finite graph, $G$, the Ricci curvatures ${\rm Ric}_{\infty}(G)$, ${\rm Ric}_N(G)$, $CRic_{\infty}(G)$ and $CRic_N(G)$ are realized by some functions, i.e. there are functions that achieve the equality in the (corresponding) defining Bakry-\'Emery curvature-dimension inequalities. \end{lemma}
\begin{proof}
We will just present a proof for ${\rm Ric}_N(G)$. The proof for other Ricci curvatures are similar.
\par Since, ${\rm Ric}_N(G)$ is the supremum of all possible lower curvature bounds, one can find a sequence, $g_i$ such that for all $v \in V$
\small \begin{equation}\label{eq:max-Ric}
\frac{1}{N} \bigl( \Delta g_i \bigr)^2(v)+ {\rm Ric}_N(G)\Gamma_1 \bigl( g_i \bigr)(v) \le \Gamma_2 \bigl( g_i \bigr)(v) < \frac{1}{N}\bigl( \Delta g_i \bigr)^2(v)+\bigl ({\rm Ric}_N(G)+\frac{1}{i}\bigr)\Gamma_1\bigl( g_i \bigr)(v). \end{equation} \normalsize
All the terms appearing in the above inequality are invariant under rescaling of the $g_i$'s. Hence, without loss of generality, we may assume that $\operatorname{Range}(g_i) \subset [-1, 1]$, for all $i$. Now since $V(G)$ is finite then by a diagonal argument one can find a subsequence $g_j$ of the $g_i$'s that converge to a function $g$. Taking the limit of~(\ref{eq:max-Ric}) as $j \to \infty$ shows that $g$ achieves ${\rm Ric}_N(G)$. \end{proof}
\section{Functions That Maximize the Conical Curvature}\label{sec:max} In this section we show that \begin{customthm}{\ref{thm:main-3}}
Suppose $G$ satisfies $CCD(K^c_{max},N)$. Then any function, $f$, realizes $K^c_{max}$ if and only if $f$ is either constant or $f-\operatorname{avg}(f)$ is an eigenfunction corresponding to $\lambda_1(G)=\frac{N-2}{4N}\lvert V\rvert$. Furthermore, when $G$ is a complete graph, $f$ must be constant (harmonic). \end{customthm}
\begin{proof}
Suppose $G$ satisfies $CCD(K^c_{max},N)$, then for any $f$
\small
\begin{equation}
\sum_{y\in V}\Gamma_1(f)(y)\geq\frac{2-N}{2N}\bigl( \sum_{y\in V}f(y)\bigr)^2+\frac{2K^c_{max}+\lvert V\rvert-3}{4}\sum_{y\in V}f^2(y). \notag
\end{equation}
\normalsize
Since $K^c_{max} = \frac{\lvert V\rvert}{2}+\frac{3}{2}-2\frac{\lvert V\rvert}{N}=\frac{N\cdot\lvert V\rvert+3N-4\lvert V\rvert}{2N}$, this simplifies to
\small
\begin{eqnarray*}
\sum_{y\in V}\Gamma_1(f)(y) &\geq&\frac{2-N}{2N}\bigl ( \sum_{y\in V}f(y)\bigr )^2+\frac{N\cdot\lvert V\rvert+3N-4\lvert V\rvert+N\cdot\lvert V\rvert-3N}{4N}\sum_{y\in V}f^2(y)\\
&\geq&\frac{2-N}{2N}\bigl ( \sum_{y\in V}f(y)\bigr )^2+\frac{N\cdot\lvert V\rvert-2\lvert V\rvert}{2N}\sum_{y\in V}f^2(y)\\
&\geq&\frac{2-N}{2N}\bigl ( \sum_{y\in V}f(y)\bigr )^2+\frac{N-2}{2N}\cdot\lvert V\rvert\sum_{y\in V}f^2(y)\\
&\geq&\frac{N-2}{2N}\biggl[ \lvert V\rvert\sum_{y\in V}f^2(y)-\bigl ( \sum_{y\in V}f(y)\bigr )^2\biggr ] .
\end{eqnarray*}
\normalsize
Take $\varphi:V\to\mathbb{R}$ to be any variational function on the vertex set of $G$ and let $t \in \mathbb{R}$, then
\small \begin{equation} \sum_{y\in V}\Gamma_1 \bigl( f+t\varphi \bigr)(y)\geq\frac{N-2}{2N}\biggl [ \lvert V\rvert\sum_{y\in V}(f+t\varphi)^2(y)-\bigl ( \sum_{y\in V} (f+t\varphi)(y)\bigr )^2\biggr ]. \notag \end{equation} \normalsize
Suppose now that $f$ achieves $K^c_{max}$, then for any $\varphi$ the above inequality becomes an equality (i.e. $\frac{d}{dt}|_{t=0}$ of both sides must be equal for any variation $\varphi$). Hence a straightforward calculation yields the linearized equation,
\small \begin{equation}\label{eq:linearization-1}
\sum_{y\in V}\sum_{z\sim y} \bigl( f(z)-f(y) \bigr) \bigl( \varphi(z)-\varphi(y) \bigr) = \frac{N-2}{2N}\biggl [\lvert V\rvert\sum_{y\in V}f(y)\varphi(y)-\sum_{y\in V}f(y)\sum_{y\in V}\varphi(y)\biggr ]. \end{equation} \normalsize
Now fix $r\in V$ and let $\varphi(y) = \delta_r(y)$. Notice that $\sum_{z\sim y} \bigl( f(z)-f(y) \bigr) \bigl( \delta_r(z)-\delta_r(y) \bigr)$ is zero except when $y=r$ or $y\sim r$. If $y=r$ the result is $-\sum_{z\sim r} \bigl( f(z)-f(r) \bigr)$. When $y\sim r$ there is exactly one non-zero term in the sum, $\bigl( f(r)-f(y) \bigr)$ and summing over all $y\sim r$ we get $\sum_{y\sim r}\bigl( f(r)-f(y) \bigr)$. Thus
\small \begin{equation}\label{eq:linearization-2} \sum_{y\in V}\sum_{z\sim y}\bigl( f(z)-f(y) \bigr) \bigl( \delta_r(z)- \delta_r(y) \bigr)=-2\Delta f(r), \end{equation} \normalsize
and (\ref{eq:linearization-1}) reduces to
\small \begin{equation}
-2\Delta f (r)=\frac{N-2}{2N} \biggl [ \lvert V\rvert f(r)- \sum_{y\in V}f(y) \biggr ] , \label{eq:linearization-3} \end{equation} \normalsize
which is equivalent to
\small \begin{equation}\label{eq:linearization-4} \Delta f (r) = \frac{N-2}{4N} \bar{\Delta} f (r), \end{equation} \normalsize
where $\bar{\Delta}$ denotes the Laplacian for the graph completion, $\bar{G}$, of $G$. This right away implies that $\Delta f (r) =0$ and we are done. Furthermore when $G$ is a complete graph, $f$ is harmonic on $G$.
\par Now suppose $G$ is arbitrary. By equation (\ref{eq:linearization-3})
\small \begin{equation}
\Delta \bigl( f - \operatorname{avg}(f) \bigr)(r) = - \frac{N-2}{4N} |V| \bigl( f - \operatorname{avg}(f) \bigr)(r). \notag \end{equation} \normalsize
Now if $f$ is not constant then by the Rayleigh quotient, (\ref{eq:Rayleigh}), we see that $\lambda_1 (G) = \frac{N-2}{4N} |V| $ and $f - \operatorname{avg}(f)$ is an eigenfunction for $\lambda_1$.\\
For the "if" direction suppose for some non-constant function, $f$, that $f - \operatorname{avg}(f)$ is an eigenfunction for $\lambda_1 = \frac{N-2}{4N} |V| $. Tracing back the above computations one has (\ref{eq:linearization-1}) holds for $\varphi = \delta_y$'s. Then since (\ref{eq:linearization-1}) is linear in $\varphi$, one can use $f = \sum_{y\in V} f(y)\delta_y$ instead of $\varphi$ which will translate to $f$ realizing $K^c_{max}$. \end{proof}
\section{$CCD(K,N)$ and Lower Bounds on $\lambda_1(G)$}\label{sec:lambda1} In this section we assume that a given graph, $G$, satisfies the $CCD(K,N)$ condition. We will use the resulting global Poincar\'e inequality along with the results of the last section to find lower bounds on the first non-zero eigenvalues of such graphs.
\begin{lemma} Suppose $G$ satisfies $CCD(K,N)$ and $N\geq2$. Then Cheeger's isoperimetric constant, $h(G)$, satisfies
\small \begin{equation}
h (G) \ge \frac{2\lvert V\rvert + 4NK +N\lvert V\rvert - 6N}{8N},\notag \end{equation} \normalsize
and,
\small \begin{equation}
\lambda_1(G) \ge \frac{\bigl(2\lvert V\rvert + 4NK +N\lvert V\rvert - 6N \bigr)^2}{128N^2 d_{max}}. \notag \end{equation} \normalsize \end{lemma}
\begin{proof} Since $G$ satisfies the $CCD(K,N)$ condition for any $f$, we have the global Poincar\'e inequality from Theorem~\ref{thm:main-1},
\small
\begin{equation}
\sum_{y\in V}\Gamma_1(f)(y)\geq\frac{2-N}{2N}\biggl ( \sum_{y\in V}f(y)\biggr )^2+\frac{2K+\lvert V\rvert-3}{2}\sum_{y\in V}f^2(y). \notag
\end{equation}
\normalsize
Suppose $F\subset V$ and $|F| \le \frac{|V|}{2}$. Let $f = \chi_F$ be the characteristic function of $F$, then (~\ref{eq:poincare}) becomes
\small \begin{equation}
\frac{2-N}{2N} |F|^2+\frac{2K+\lvert V\rvert-3}{2} |F| \le 2|\partial F| . \notag \end{equation} \normalsize
Hence, when $N\geq2$,
\small \begin{equation}
\frac{|\partial F|}{|F|} \ge \frac{2-N}{4N} |F|+\frac{2K+\lvert V\rvert-3}{4} \ge \frac{2-N}{4N} \frac{|V|}{2} + \frac{2K+\lvert V\rvert-3}{4} = \frac{2\lvert V\rvert + 4NK +N\lvert V\rvert - 6N}{8N}. \notag \end{equation} \normalsize
Now applying Cheeger's inequality (~\ref{eq:DAM}) (see~\cite{Chung-2} and~\cite{ASS}), we know that $\lambda_1(G) \ge \frac{h^2(G)}{2d_{max}}$, where $d_{max}$ is the maximum degree in the graph, $G$. Hence,
\small \begin{equation}
\lambda_1 \ge \frac{h^2(G)}{2d_{max}} \ge \frac{\bigl(2\lvert V\rvert + 4NK +N\lvert V\rvert - 6N \bigr)^2}{128N^2 d_{max}}. \notag \end{equation} \normalsize \end{proof}
In the rest of this section we show that any lower bound, $\lambda$, for $\lambda_1(G)$ will imply that $G$ satisfies $CCD(K,N)$ for some $K$ and $N$ (depending on $\lambda$).
\begin{thm} Suppose $\lambda_1(G) \ge \lambda$, then $G$ satisfies $CCD(K,N)$ for any $K$ and $N$ with
\small \begin{equation}
N \ge \frac{2\lvert V\rvert}{|V|- \lambda}, \quad \text{and} \quad K \ge \frac{\lambda - |V| + 3}{2}. \notag \end{equation} \normalsize
\end{thm}
\begin{proof} Since $\lambda_1(G) \ge \lambda$ then by the Rayleigh quotient, (\ref{eq:Rayleigh}), we get
\small \begin{eqnarray}\label{eq:lambda-to-poincare}
\sum_{y \in V} \Gamma_1 (f)(y) &\ge& \frac{\lambda}{2} \sum_{y \in V} \bigl( f - \operatorname{avg}(f) \bigr)^2(y) = \frac{\lambda}{2} \biggl[ \sum_{y \in V} f^2(y) + |V| \operatorname{avg}(f)^2 - 2 \operatorname{avg}(f) \sum_{y \in V} f(y) \biggr] \notag \\ & = & \frac{\lambda}{2} \sum_{y \in V} f^2(y) + \frac{\lambda}{2|V|} \bigl( \sum_{y \in V} f(y) \bigr)^2 - \frac{\lambda}{|V|} \bigl( \sum_{y \in V} f(y) \bigr)^2 \\ &=& \frac{\lambda}{2} \sum_{y \in V} f^2(y) - \frac{\lambda}{2|V|} \bigl( \sum_{y \in V} f(y) \bigr)^2. \notag \end{eqnarray} \normalsize
Comparing (\ref{eq:lambda-to-poincare}) to the global Poincar\'e inequality, (\ref{eq:poincare}) due to the $CCD(K,N)$ condition, and one observes that $G$ satisfies $CCD(K,N)$ for any $K$, and $N$ where
\small \begin{equation}
\frac{ \lambda }{|V|} \le\frac{ N - 2}{N}, \quad \text{and} \quad \lambda\le 2K+\lvert V\rvert-3 . \notag \end{equation} \normalsize
The conclusion follows by noticing that one always have $\lambda_1(G) \le |V| $. \end{proof}
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November 2018, 17(6): 2479-2493. doi: 10.3934/cpaa.2018118
Coupled systems of Hilfer fractional differential inclusions in banach spaces
Saïd Abbas 1, , Mouffak Benchohra 2, and John R. Graef 3,,
Laboratory of Mathematics, Geometry, Analysis, Control and Applications, Tahar Moulay University of Saïda, P.O. Box 138, EN-Nasr, 20000 Saïda, Algeria
Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P.O. Box 89, 22000, Algeria
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA
* Corresponding author
Received October 2017 Revised January 2018 Published June 2018
This paper deals with some existence results in Banach spaces for Hilfer and Hilfer-Hadamard fractional differential inclusions. The main tools used in the proofs are Mönch's fixed point theorem and the concept of a measure of noncompactness.
Keywords: Fractional differential inclusion, coupled system, left-sided mixed Riemann-Liouville and Hadamard integrals of fractional order, Hilfer fractional derivative, Hilfer-Hadamard fractional derivative, measure of noncompactness.
Mathematics Subject Classification: Primary: 26A33, 34A60.
Citation: Saïd Abbas, Mouffak Benchohra, John R. Graef. Coupled systems of Hilfer fractional differential inclusions in banach spaces. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2479-2493. doi: 10.3934/cpaa.2018118
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Saïd Abbas Mouffak Benchohra John R. Graef | CommonCrawl |
\begin{document}
\correspond{Henning Christiansen, Roskilde University, Denmark. Email: [email protected]}
\title[On Proving Confluence Modulo Equivalence for CHR] {On Proving\\Confluence Modulo Equivalence\\for Constraint Handling Rules}
\author[Christiansen and Kirkeby]
{Henning Christiansen\thanks{The project
is supported by The Danish Council for Independent Research, Natural Sciences,
grant no.~DFF 4181-00442} and Maja H.~Kirkeby$^{1,}$\thanks{The second author's contribution received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no.~318337, ENTRA - Whole-Systems Energy Transparency.}\\
Research group PLIS: Programming, Logic and Intelligent Systems\\
Department of People and Technology\\
Roskilde University, Denmark}
\maketitle
\begin{abstract} Previous results on proving confluence for Constraint Handling Rules are extended in two ways in order to allow a larger and more realistic class of CHR programs to be considered confluent. Firstly, we introduce the relaxed notion of confluence modulo equivalence into the context of CHR: while confluence for a terminating program means that all alternative derivations for a query lead to the exact same final state, confluence modulo equivalence only requires the final states to be equivalent with respect to an equivalence relation tailored for the given program. Secondly, we allow non-logical built-in predicates such as \texttt{var}/1 and incomplete ones such as \texttt{is}/2, that are ignored in previous work on confluence.
To this end, a new operational semantics for CHR is developed which includes such predicates. In addition, this semantics differs from earlier approaches by its simplicity without loss of generality, and it may also be recommended for future studies of CHR.
For the purely logical subset of CHR, proofs can be expressed in first-order logic, that we show is not sufficient in the present case. We have introduced a formal meta-language that allows reasoning about abstract states and derivations with meta-level restrictions that reflect the non-logical and incomplete predicates. This language represents subproofs as diagrams, which facilitates a systematic enumeration of proof cases, pointing forward to a mechanical support for such proofs. \end{abstract}
\section{Introduction} Constraint Handling Rules, CHR~\cite{fruehwirth-98,fru_chr_book_2009}, is a programming language consisting of guarded rewriting rules over constraint stores. CHR inherits its nomenclature from the logic programming tradition; constraints are first-order atoms, and the language has a declarative semantics based on a logical reading of the rules.
It has become important as a general language for knowledge representation and reasoning as well as for expressing algorithms in a high-level fashion; see, e.g.,~\cite{FruehwirthRaiserEds2011,DBLP:series/lncs/5388,DBLP:journals/tplp/SneyersWSK10}.
A foundation for applying confluence in the analysis and verification of CHR programs has been laid in earlier work, and the overall theoretical issues are well understood~\cite{DBLP:conf/cp/Abdennadher97, DBLP:conf/cp/AbdennadherFM96,DBLP:journals/constraints/AbdennadherFM99,FruehwirthRaiserEds2011}. The confluence notion goes longer back in the traditions of term and abstract rewriting systems; see more details in Section~\ref{sec:rewrite}. There are, however, still severe limitations in the results for CHR that impede its practical application to realistic programs. The present paper aims at filling part of the gap, by \begin{itemize} \item the introduction for CHR of confluence \emph{modulo equivalence} that allows a much larger and interesting class of programs to enjoy the advantages of confluence; \item extending to a larger subset of CHR that includes non-logical and incomplete\footnote{In this paper, we use the term \emph{incomplete} for a built-in predicate whose (established) implementation produces runtime errors for selected calls. Examples of such calls are \texttt{4} \texttt{is} \texttt{2+X} and \texttt{X>1}. The precise definition is found in section~\ref{sec:preliminaries}.} built-in predicates (e.g., \texttt{var}/1, resp.\ \texttt{is}/2) that have been ignored in previous work. \end{itemize}
While confluence of a program means that all derivations from a common initial state end in the same final state,
the ``modulo equivalence'' version relaxes this such that final states need not be strictly identical, but only equivalent with respect to a given equivalence relation. The following motivating example is used throughout this paper.
\begin{example}[\cite{DBLP:conf/lopstr/ChristiansenK14}]\label{ex:collect} The following CHR program, consisting of a single rule, collects a number of separate items into a (multi-) set represented as a list of items. \begin{verbatim}
set(L), item(A) <=> set([A|L]). \end{verbatim} This rule will apply repeatedly, replacing constraints matched by the left-hand side by those indicated to the right. The query \begin{verbatim} ?- item(a), item(b), set([]). \end{verbatim} may lead to two different final states, $\{\texttt{set([a,b])}\}$ and $\{\texttt{set([b,a])}\}$, both representing the same set.
We introduce a state equivalence relation $\approx$ implying that $\{\texttt{set($L$)}\}\approx\{\texttt{set($L'$)}\}$, whenever $L$ is a permutation of $L'$. The program is not confluent when identical end states are required, but it will be shown to be confluent modulo $\approx$ in Section~\ref{sec:set-example-all-details} below. \end{example}
The relevance of confluence modulo equivalence is also demonstrated for dynamic optimization programs that produce an arbitrary, optimal solution among a collection of equally good ones; the Viterbi algorithm expressed in CHR is considered in Section~\ref{sec:viterbi}.
To model non-logical and incomplete predicates, we need to introduce a new operational semantics for CHR. To be interesting for studies of confluence, this semantics maintains nondeterminism for choice of the next rule to be applied to the current state. In addition to treating a larger language, this semantics differs from earlier approaches by its simplicity without loss of generality. Various redundancies have been removed so that a program state has only two components, a constraint store and a bookkeeping device to handle well-known termination problems for the propagation rules of CHR; a simple observation shows that global variables are unnecessary; execution of built-in predicates are modelled by substitutions applied to the state immediately, which is more in line with how a practical CHR system works (as opposed to earlier proposals' additional store of ``processed'' built-ins and their evaluation explained by logical entailment). A detailed comparison and references to previous operational semantics are given in Section~\ref{sec:CHR} below.
Reasoning about derivations is more difficult in the context of non-logical/incomplete built-ins. Basically, all earlier proof methods for the purely logical subset of CHR rely on a subsumption principle that any property shown about derivations between states also holds when more constraints are added and substitutions applied to the states; as a consequence of this, confluence proofs can be reduced to considering a finite number of cases that can be checked in an automatic way. This principle breaks down when non-logical predicates are introduced, e.g., the predicate \texttt{var(X)} succeeds but the instance \texttt{var(7)} fails. To cope with this, we have introduced a formal meta-language $\textsc{MetaCHR}$ to represent abstract states, derivations and proofs as diagrams, with powerful parametrization and meta-level constraints that limit the allowed instances. The following is an example of an abstract term in the meta-language, $\texttt{var($a$)} \mathrel{\mbox{\textsc{where}}} \mathit{variable}(a)$. Here, $a$ is a meta-variable ranging over terms and $\mathit{variable}$ is a meta-level constraint on such terms, allowing only substitutions to names of such variables. This abstract term is said to cover all instances that satisfy the meta-level constraint, i.e.,
\texttt{var(X)} but not \texttt{var(7)}. $\textsc{MetaCHR}$ allows us to reason about such abstract terms in a way so that properties shown at this level are guaranteed to hold for all such permissible instances. We can demonstrate that proofs of confluence can be reduced to considering only a finite number of abstract proof cases, but the additional complexity given by an equivalence relation (and state invariant; below) may in some cases require an unfolding into an infinite number of subcases, each requiring a differently shaped proof diagram.
The notion of observable confluence~\cite{DBLP:conf/iclp/DuckSS07} for CHR considers only states that satisfy a given invariant. We include such invariants, as we consider them to be central in CHR programming practice: a program is typically developed with a particular class of queries in mind, often strongly biased, so only queries in this class lead to meaningful computations.
\begin{example}\label{ex:collect-inv}\emph{(Example~\ref{ex:collect}, continued)} The one-line program above reflects a tacitly assumed state invariant: only one \texttt{set} constraint is allowed. If we open up for a query such as \begin{verbatim} ?- item(a), item(b), set([]), set([c]). \end{verbatim} we obtain a collection of different answers, representing different ways of splitting $\{\texttt{a},\texttt{b},\texttt{c}\}$ into two disjoint subsets. However, this may not be intended, and the program is not confluent modulo the indicated equivalence relation unless the invariant is taken into account. The relevant invariant may specify that all constraints must be ground, and that a state must include exactly one \texttt{set}/2 constraints whose argument is a list. \end{example}
The earlier approach~\cite{DBLP:conf/iclp/DuckSS07} for showing observable confluence (for logical built-ins only) sticks to the above mentioned logical subsumption principle. As shown by~\cite{DBLP:conf/iclp/DuckSS07} and explained below, this leads to infinitely many proof cases for even simple invariants such as groundedness; our meta-language approach handles such examples in a more satisfactory way.
Confluence modulo equivalence was mentioned in relation to CHR in a previous conference paper~\cite{DBLP:conf/lopstr/ChristiansenK14} that also gave a first version of the operational semantics. The present paper provides theoretical foundations for studying confluence modulo equivalence for CHR, and introduces a formal meta-language that supports systematic proofs. This may also point forward towards (partly) mechanized proof systems for confluence modulo equivalence.
The results in the present paper may carry over in a useful way to other systems with nondeterminism in which confluence has to be studied. This may be active rules in databases~\cite{DBLP:conf/sigmod/AikenWH92}, concurrent constraint programming~\cite{DBLP:journals/tcs/FalaschiGMP97} and theoretical models of concurrency such as $\pi$- and $\rho$-calculi~\cite{DBLP:journals/jfp/Niehren00,DBLP:conf/ccl/NiehrenS94}.
Section~\ref{sec:background} reviews previous work on confluence in term rewriting and general rewriting systems, including fundamental results concerning confluence modulo equivalence, that has not been utilized for CHR before, and we give an overview of the state of the art for CHR. Section~\ref{sec:CHR} gives our operational semantics for CHR, first introduced in~\cite{DBLP:conf/lopstr/ChristiansenK14}, intended for reasoning about confluence for programs with non-logical built-in predicates, and various properties related to confluence are introduced; we also make a comparison with operational semantics used in earlier work on confluence for CHR. In Section~\ref{sec:conf-mod-eq-in-CHR} we generalize earlier results on critical pairs for CHR, now including the larger set of built-in predicates, and taking invariant and equivalence into account; we can also show that such pairs -- or corners as we call them (since we include the common ancestor state) -- are not suited for proofs of confluence in our more general case due to this subsumption principle; we also add some more detailed comments on previous work on confluence for CHR.
Our main results are presented in Section~\ref{sec:abstract}. The meta-language \textsc{MetaCHR}{} is introduced in which proofs of joinability are reified as abstract diagrams.
A proof of confluence modulo equivalence can be split into a finite set of proof cases, each given by an abstract corner. As opposed to the results of \cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96} it is not necessary for confluence (modulo equivalence) that each such abstract corner is joinable. A property called split-joinable is introduced, occasionally leading to infinite sets of corners to be checked for joinability. We show that when the abstract corners are either joinable or split-joinable, local confluence is guaranteed and confluence is guaranteed for terminating programs.
In Section~\ref{sec:examples}, we demonstrate the applicability of the suggested approach, by giving proofs of confluence modulo equivalence for selected programs: the program of Example~\ref{ex:collect} that demonstrates an equivalence indicating a redundant data representation, a version of the Viterbi algorithm in CHR that exemplifies dynamic programming algorithms with pruning, and finally an example with a splitting into infinitely many cases.
Section~\ref{sec:discussion} provides for a summary, and a discussion of possible directions for future work.
\section{Background and Related work}\label{sec:background}
Confluence modulo trivial identity is well-studied in Rewriting Systems, see, e.g.,~\cite{BaderNipkow1999} for an overview. Since the 1990es, the proof methods have been adapted to the more complex system of Constraint Handling Rules~\cite{fruehwirth-98,fru_chr_book_2009}, most notably~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:conf/iclp/DuckSS07}. Confluence modulo equivalence has been studied in general rewriting systems~\cite{DBLP:journals/jacm/Huet80} and was only recently introduced to CHR~\cite{DBLP:conf/lopstr/ChristiansenK14}.
\subsection{Confluence for General Rewriting Systems and Term Rewriting Systems}\label{sec:rewrite} A binary \emph{relation} $\rightarrow$ on a set $A$ is a subset of $A \times A$, where $x \rightarrow y$ denotes membership of $\rightarrow$. A \emph{rewriting system} is a pair $\langle A, \rightarrow\rangle$; it is \emph{terminating} if there is no infinite chain $a_0 \rightarrow a_1 \rightarrow \cdots$.
The \emph{reflexive transitive closure} of $\rightarrow$ is denoted $\relRT$. The \emph{inverse relation} $\leftarrow$ is defined by $\{(y,x) \mid x \rightarrow y\}$. An \emph{equivalence (relation)} $\approx$ is a binary relation on $A$ that is reflexive, transitive and symmetric. We say that $x$ and $y$ are \emph{joinable} if there exists a $z$ such that $x \relRT z $ and a $z \irelRT y$.
A rewriting system $\langle A, \rightarrow\rangle$ is \emph{confluent} if and only if $y' \irelRT x \relRT y \Rightarrow \exists z.\ y' \relRT z \irelRT y$, and is \emph{locally confluent} if and only if $y' \leftarrow x \rightarrow y \Rightarrow \exists z.\ y' \relRT z \irelRT y$.
In 1942, Newman showed his fundamental Lemma~\cite{Newman42}: \emph{A terminating rewriting system is confluent if and only if it is locally confluent.} An elegant proof of Newman's lemma was provided by Huet \cite{DBLP:journals/jacm/Huet80} in 1980.
The more general notion of \textit{confluence modulo equivalence} was introduced in 1972 by Aho et~al~\cite{Aho72} in the context of the Church-Rosser property. \begin{definition}[Confluence modulo equivalence]\label{def:confModEq} A relation $\rightarrow$ is confluent modulo an equivalence $\approx$ if and only if \begin{align*}
\forall\, x,y,x', y'. \quad y' \irelRT x' \approx x \relRT y \qquad \Rightarrow \qquad \exists\, z, z'. \quad y' \relRT z' \approx z \irelRT y. \end{align*} \end{definition} Given an equivalence relation $\approx$, we say that $x$ and $y$ are \emph{joinable modulo equivalence} if there exists $z,z'$ such that $x \relRT z $, $z' \irelRT y$ and $z \approx z'$. This is shown as a diagram in Fig.~\ref{fig:ConfModEq}. In 1974, Sethi~\cite{DBLP:journals/jacm/Sethi74} studied confluence modulo equivalence for bounded rewriting systems, that are systems for which there exists an upper bound for the number of possible rewrite steps for all terms. He showed that confluence modulo equivalence for bounded systems is equivalent to the following properties, $\alpha$ and $\beta$, also shown in Fig.~\ref{fig:LocalConfModEq}.
\begin{definition}[$\alpha$ \& $\beta$]\label{def:alfaBeta} A relation $\rightarrow$ has the $\alpha$ property and the $\beta$ property with respect to an equivalence $\approx$ if and only if it satisfies the $\alpha$ and $\beta$ conditions, respectively: \begin{align*}
\alpha:\qquad & \forall x,y, y'. \quad y' \leftarrow x \rightarrow y \qquad \Longrightarrow \qquad \exists z, z'. \quad y' \relRT z' \approx z \irelRT y \\ \beta: \qquad & \forall x, y', y. \quad y' \approx x \rightarrow y \qquad\, \Longrightarrow \qquad \exists z, z'. \quad y' \relRT z' \approx z \irelRT y \end{align*} \end{definition}
In 1980,~Huet \cite{DBLP:journals/jacm/Huet80} generalized this result to any terminating system.
\begin{definition}[Local confl.~mod.~equivalence]\label{def:LconfModEq} A rewriting system is \emph{locally confluent modulo an equivalence} $\approx$ if and only if it has the $\alpha$ and $\beta$ properties.
\end{definition}
\begin{theorem}\label{thm:confLconfModEq}\textbf{(Huet,~\cite{DBLP:journals/jacm/Huet80})}\quad Let $\rightarrow$ be a terminating rewriting system. For any equivalence $\approx$, $\rightarrow$ is confluent modulo $\approx$ if and only if $\rightarrow$ is locally confluent modulo $\approx$. \end{theorem}
\begin{figure}
\caption{Diagrams for the fundamental notions. A dotted arrow (single wave line) indicates an inferred step (inferred equivalence).}
\label{fig:ConfModEq}
\label{fig:LocalConfModEq}
\label{fig:alphaBeta}
\end{figure}
Term rewriting systems have been studied extensively, and
terminology and several important results carry over to CHR, as we will see below. In the following, we assume the reader familiar with the notions of terms over some signature and variables, substitutions and most general unifiers.
\begin{definition}[Term Rewriting System; semi-formal version adapted from~\cite{BaderNipkow1999}] A \emph{term rewriting system (TRS)} consists of a finite set of rules of the form $(l,r)$ in which any variable in $r$ also appears in $l$. The application of such a rule to a term $s$ to obtain another term $t$, written $s\rightarrow t$ is obtained by 1) find a substitution $\theta$, such that $l\theta$ is a subterm of $s$, and 2) $t$ is given by replacing that subterm in $s$ by $r\theta$. \end{definition}
The following notion of critical pairs represents cases in which two rules both can apply in the same subterm, but if one is applied, the second one cannot be applied successively.
\begin{definition}[TRS Critical Pair; adapted from~\cite{BaderNipkow1999}] Consider $R_k=(l_k,r_k), k=1,2$ (assumed renamed apart so they have no variable in common) for which there is a most general unifier $\sigma$ of $l_2$ and a non-variable subterm of $l_1$. Then $\langle t_1, t_2\rangle$ is a \textit{critical pair}, whenever $l_1\sigma\rightarrow t_k$ using $R_k$, $k=1,2$. \end{definition} For example, the two rules $(f(a),b)$ and $(a,c)$ give rise to the critical pair $\langle b, f(c)\rangle$; both are derived from the common ancestor term $f(a)$, i.e., $b\leftarrow f(a)\rightarrow f(c)$.
In 1970, Knuth and Bendix~\cite{KnuthBendix1970} developed the following, fundamental properties, later elaborated by Huet~\cite{DBLP:journals/jacm/Huet80}. We bring them in detail as very similar properties holds for CHR.
\begin{lemma}[Critical Pair Lemma for TRS~\cite{DBLP:journals/jacm/Huet80,KnuthBendix1970}]\label{def:criticalPairsTRS} Let a TRS be given and assume terms $s,t_1,t_2$ such that $t_1\leftarrow s \rightarrow t_2$. Then either \begin{itemize}
\item $t_1$ and $t_2$ are joinable, or
\item there exists an instance $\langle u_1,u_2\rangle$ or $\langle u_2,u_1\rangle$ of a critical pair
and a specific subterm $s'$ of $s$ such that\\
$t_k$ is a copy of $s$ in which $s'$ is replaced by $u_k$, $k=1,2$.
\end{itemize} \end{lemma}
\begin{theorem}[Critical Pair Theorem for TRS~\cite{DBLP:journals/jacm/Huet80,KnuthBendix1970}]\label{thm:criticalPairs}
A TRS is locally confluent if and only if all its critical pairs are joinable. \end{theorem} This theorem in combination with Newman's lemma leads to a desired result: \emph{A terminating TRS is confluent if and only if all its critical pairs are joinable.} Furthermore, confluence of a finite terminating TRS is decidable (as there is only a finite number of critical pairs and finitely many finite derivations to test out from their states).
Mayr and Nipkow~\cite{DBLP:journals/tcs/MayrN98} studied confluence modulo equivalence for a subset of higher-order rewriting systems (that extend term rewriting to $\lambda$-terms). They used an alternative version of Theorem~\ref{thm:confLconfModEq} in which the $\beta$ property is replaced by a $\gamma$ property, as shown below. It applies when the equivalence $\approx$ is specified as the transitive closure
of a symmetric relation $\vdash\!\dashv$; such a relation may, e.g., be generated by a set of equations.
\begin{lemma}[$\alpha$ \& $\gamma$ Confluence \cite{DBLP:journals/jacm/Huet80}]~\label{lemma:alpga-and-gamma}
Let $\vdash\!\dashv$ be a symmetric relation and ${\approx} = (\vdash\!\dashv)^{*}$.
Let $\rightarrow$ be any relation such that the composition $\rightarrow \cdot \approx$ is terminating.
Then $\rightarrow$ is confluent modulo $\approx$ if and only if the conditions $\alpha$ and $\gamma$ are satisfied:
\begin{align*}
\alpha:\qquad & \forall x,y, y'. \quad y' \leftarrow x \rightarrow y \qquad \Longrightarrow \qquad \exists z, z'. \quad y' \relRT z' \approx z \irelRT y \\ \gamma: \qquad & \forall x, y', y. \quad y' \vdash\!\dashv x \rightarrow y \qquad\, \Longrightarrow \qquad \exists z, z'. \quad y' \relRT z' \approx z \irelRT y \end{align*} \end{lemma}
We do not use this lemma in the present paper, but possible applications are discussed in the concluding section.
\subsection{Confluence for Constraint Handling Rules}\label{sec:backgroundConfCHR} Constraint Handling Rules, CHR, can be understood as a rewrite system over states that are multisets of constraints as shown in Example~\ref{ex:collect} above, p.~\pageref{ex:collect}.\footnote{The rule in the example program is a so-called simplification rule. CHR also includes other types of rules, that do not introduce additional conceptual difficulties in relation to confluence, although they imply an extra notational overhead.}
The known results on confluence for CHR are very similar to those on term rewriting systems shown above. Similar critical pairs of states may appear when two instances of rules can apply to overlapping constraints; the precise definition is given in Section~\ref{sec:CHR} below. The following shows the construction of such a critical pair for an overlap of two different instances of the only rule in the program of Example~\ref{ex:collect}, p.~\pageref{ex:collect}, above. $$
\{\texttt{item(Y)},\texttt{set([X|L])}\} \leftarrow \{\texttt{item(X)},\texttt{item(Y)},\texttt{set(L)}\}\rightarrow \{\texttt{item(X)}, \texttt{set([Y|L])}\} $$ The first publications by Fr\"uhwirth on CHR appeared in 1993--4~\cite{DBLP:conf/iclp/Fruhwirth93,DBLP:journals/lncs/Fruhwirth94}. Soon after, around 1996, the central results on confluence for CHR were developed by Abdennadher and others~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96}, however, only for the subset of CHR with logical built-ins and neither invariant nor equivalence. The concepts and results from the area of term rewriting can be transferred to CHR so that the following results hold; CHR$^0$ refers to the indicated subset of CHR. \begin{itemize}
\item A CHR$^0$ program is locally confluent if and only if all its critical pairs are joinable.
\item The set of critical pairs is finite and
local confluence is decidable; automatic checkers of this property has been developed for CHR$^0$, e.g.,~\cite{Raiser-Langbein2010}
\item A terminating CHR$^0$ program is confluent if and only if all its critical pairs are joinable. \end{itemize}
These results are based on the previously mentioned subsumption principle which essentially boils down to the following. \begin{itemize}
\item[(*)] whenever a (e.g., critical) pair of CHR$^0$ states $x,y$ are joinable, it holds for any substitution $\theta$ and constraint set $s$ that $x\theta\cup s$ and $y\theta\cup s$ are joinable. \end{itemize} Section~\ref{sec:earlier-CHR-confl-etc}, p.~\pageref{sec:earlier-CHR-confl-etc}, gives a precise analysis and also shows that these results do not generalize directly to the larger subset of CHR considered in the present paper.
In 2007, Duck et al~\cite{DBLP:conf/iclp/DuckSS07} argued for the introduction of state invariants; a state invariant $I(\cdot)$ is a property that is preserved by the derivations of the current program, and it may, e.g., be defined by reachability from a set of intended queries.
We define an \emph{$I$-state} $x$ as a state for which $I(x)$ holds.
The precise definitions and arguments are given in Section~\ref{sec:semantics}, respectively~\ref{sec:earlier-CHR-confl-etc}. They define \emph{(local) observable confluence} for CHR$^0$
as above, considering only derivations between $I$-states.
While this generalization of confluence is highly relevant from a practical point of view, it is inherently more difficult, as the property (*) above does not generalize. For this discussion, we refer to a state $x\theta\cup s$ (pair $\langle x\theta\cup s, y\theta\cup s\rangle$) as an \emph{extension} of state $x$ (pair $\langle x, y\rangle$).
A state $x$ (e.g., in a critical pair) may not be an $I$-state in itself, but some of its extended states may be $I$-states; the other way round, some extensions of an $I$-state may not be $I$-states.
Duck et al~\cite{DBLP:conf/iclp/DuckSS07}
considered cases where, for each critical pair $\langle x, y\rangle$ , a collection of most general extensions $\{\langle x_i, y_i\rangle\}_{i\in\mathit{Inx}}$ exists, such that any such $\langle x_i, y_i\rangle$ and any extension of it consists of $I$-states.
For a given program $\Pi$ and invariant $I$, let ${\mathcal M}^{I,\Pi}$ be the set of all such most general extensions for all critical pairs. Then the following holds. \begin{itemize}
\item A CHR$^0$ program is locally observably confluent w.r.t.\ $I$ if and only if all pairs in
${\mathcal M}^{I,\Pi}$ are joinable.
\item A terminating CHR$^0$ program is observably confluent w.r.t.\ $I$ if and only if all pairs in
${\mathcal M}^{I,\Pi}$ are joinable. \end{itemize} Decidability is lost, and~\cite{DBLP:conf/iclp/DuckSS07} shows that even a standard invariant such as groundedness leads to infinite ${\mathcal M}^{I,\Pi}$ sets. The characterization of ${\mathcal M}^{I,\Pi}$ is complicated, and no practically relevant methods have been proposed. In the present paper, we cope with these problems by introducing a meta-language in which we can reason about abstract versions of critical pairs and their joinability, and in which the invariant is treated as a meta-level constraint.
We are not aware of other work than our own on confluence for CHR that includes non-logical predicates or takes an equivalence relation into account. Confluence for nonterminating CHR programs has been studied by~\cite{DBLP:journals/tplp/Haemmerle12,RaiserTacchella2007}, and~\cite{DBLP:conf/lopstr/AbdennadherF03} has considered how the integration of two programs known to be confluent can be made confluent by adding new rules.
The choice of an operational semantics for CHR, i.e., a definition of the derivation relation for CHR, influences the set of programs recognized as confluent
and the amount of notational overhead needed for the proofs. We postpone a comparison with selected other operational semantics until Section~\ref{sec:commentOpSem}, following the introduction of the necessary technical apparatus.
\section{Constraint Handling Rules}\label{sec:CHR}
In the following, we introduce CHR and our new operational semantics as a rewriting system.
We highlight the differences in comparison with previous semantics used for the study of confluence for CHR. Ours differs most essentially in that it can describe non-logical and incomplete built-ins, and we have also succeeded in introducing several simplifications without loss of generality (apart from a subtle mathematical consequence implied by some earlier semantics exposed in Example~\ref{ex:different-confluence}, p.~\pageref{ex:different-confluence}).
\subsection{Preliminaries}\label{sec:preliminaries} We extend the basic concepts and notation introduced in Section~\ref{sec:rewrite}. Derivation steps are labelled so we can distinguish how they are produced with reference to the CHR program in question (letters $D$ and $d$ are typically used for such labels, indicating a \emph{d}escription of the step). We also introduce the notions $\alpha$- and $\beta$-corners to give a representation of cases where the $\alpha$- and $\beta$ conditions may (or may not) hold.
\begin{definition}\label{def:corners-etc} A \emph{derivation system} $\langle S, D, \mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}, I, \approx\rangle$ consists of a set $S$, called \emph{states}, a set of \emph{labels} $D$, a ternary \emph{derivation relation} ${\mapsto}\subseteq S\times D\times S$, an \emph{invariant} $I\subset S$, and an \emph{equivalence} ${\approx}\subseteq S\times S$.
A fact $\langle x,d,y\rangle \in {\mapsto}$ is written $x \stackrel d\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y$, in which case we also write $x \mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y$, thus projecting it to a binary relation; as usual $\stackrel{*}\ourmapsto$ denotes the reflexive, transitive closure of $\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}$, and \emph{derivation} is a successive sequence of zero or more, perhaps infinitely many, derivation steps. For brevity, we may use $x\stackrel{*}\ourmapsto y$ to indicate a derivation from $x$ to $y$, with labels understood.
The invariant property of $I$ means that $I(x)\land (x\stackrel d \mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y)$ implies $I(y)$; a state $x$ with $I(x)$ is an \emph{$I$-state} and \emph{$I$-derivation} (step)s are those that involve only $I$-states.
An \emph{$\alpha$-corner} is a structure of the form $y'\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} x\mapsto y$ where $x,y,y'$ are states and $y'\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} x$, $x\mapsto y$ are derivation steps; a \emph{$\beta$-corner} is of the form $y'\approx x\mapsto y$ where $x,y,y'$ are states,
$y'\approx x$ holds and $x\mapsto y$ is a derivation step. We may use the symbol $\Lambda$ to denote a corner.
In both cases, the state $x$ is referred to as the \emph{common ancestor state} for the \emph{wing states} $y'$ and $y$. Two $\alpha$-corners $y'\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} x\mapsto y$ and $y\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} x\mapsto y'$ are considered identical. An $\alpha$- ($\beta$-) corner is called an \emph{$\alpha$- ($\beta$-) $I$-corner} when its states are $I$-states.
A \emph{joinability diagram} (modulo $\approx$) for an $\alpha$- or $\beta$-corner $$y' \mathrel{\mathit{Rel}} x\stackrel {d_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y$$ (thus $\mathit{Rel}$ is one of $\stackrel {d_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}$ or $\approx$) is a structure of the form $$ z'\stackrel{*}\mapsfrom y' \mathrel{\mathit{Rel}} x\stackrel {d_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y \stackrel{*}\ourmapsto z$$ where $z'\stackrel{*}\mapsfrom y'$ and $y \stackrel{*}\ourmapsto z$ are derivations such that the equivalence
$z'\approx z$ holds. A diagram is sometimes denoted by the symbol $\Delta$. A given corner is \emph{joinable} modulo $\approx$ whenever there exists a joinability diagram for it. An $\alpha$-corner of the form $y \mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} x \mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y$ is called \emph{trivially joinable} (modulo $\approx$).
A derivation system $\langle S, D, \mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}, I, \approx\rangle$ is \emph{confluent modulo $\approx$ (with respect to $I$)} if and only if, for all $I$-states $y',x,y$: $y' \stackrel{*}\mapsfrom x \stackrel{*}\ourmapsto y \Rightarrow \exists z,z'.\ y' \stackrel{*}\ourmapsto z'\approx z \stackrel{*}\mapsfrom y$, and is \emph{locally confluent modulo $\approx$ (with respect to $I$)} if and only if all its $I$-corners are joinable modulo $\approx$. \end{definition}
Joinability diagrams may be shown as in Figure~\ref{fig:LocalConfModEq}, and notions of (local) ($I$-) confluence (modulo $\approx$) $I$-termination apply as already introduced. We can reformulate Theorem~\ref{thm:confLconfModEq} as follows. \begin{theorem}\label{thm:I-joinable-corners} An $I$-terminating derivation system is $I$-confluent modulo $\approx$ if and only if all its $I$-corners (of type $\alpha$ as well as $\beta$) are joinable modulo $\approx$. \end{theorem}
We assume standard notions of first-order logic such as predicates, atoms and terms. For any expression $E$, $\mathit{vars}(E)$ refers to the set of variables occurring in $E$. A \emph{substitution} is a mapping from a finite set of variables to terms, e.g., the substitution $[x/t]$ replaces variable $x$ by term $t$.
For substitution $\sigma$ and expression $E$, $E\sigma$ (or $E\cdot\sigma$) denotes the expression that arises when $\sigma$ is applied to $E$; composition of two substitutions $\sigma, \tau$ is denoted $\sigma\circ\tau$. Special substitutions $\mathit{failure}$ and $\mathit{error}$ are assumed, the first one representing falsity and the second one runtime errors; a substitution different from these two is called a \emph{proper substitution}.
Two disjoint sets of \emph{(user) constraints} and \emph{built-in} predicates are assumed.
Our semantics for built-ins differs from previous approaches by mapping them immediately to a unique substitution. This makes it possible to handle non-logical devices such as Prolog's \texttt{var}/1 and run-time errors as they may arise from incomplete built-ins such as \texttt{is}/2.
An evaluation procedure $\mathit{Exe}$ for built-in atoms $b$ is assumed, such that $\mathit{Exe}(b)$ is either a (possibly identity) substitution to a subset of $\mathit{vars}(b)$ or one of $\mathit{failure}$ and $\mathit{error}$. It extends to sequences of built-ins as follows. $$ \mathit{Exe}((b_1,b_2)) = \begin{cases} \mathit{Exe}(b_1) & \text{when $\mathit{Exe}(b_1)\in\{\mathit{failure},\mathit{error}\}$},\\ \mathit{Exe}(b_2\cdot\mathit{Exe}(b_1)) & \text{when otherwise $\mathit{Exe}(b_2\cdot\mathit{Exe}(b_1))$}\\ & \text{~~~~~~~~~~~~~~~~~~~$\in\{\mathit{failure},\mathit{error}\}$},\\
\mathit{Exe}(b_1)\circ\mathit{Exe}(b_2\cdot\mathit{Exe}(b_1)) & \text{otherwise\label{text-def:exe}} \end{cases} $$ A built-in $b$ or sequence of such is \emph{satisfiable} whenever there exists a substitution $\theta$ such that $\mathit{Exe}(b\theta)$ is a proper substitution.
A subset of built-in predicates are the \emph{logical} ones, whose meaning is given by a first-order theory $\mathcal{B}$. For a logical atom $b$ with $\mathit{Exe}(b)\neq\mathit{error}$, the following conditions must hold. \begin{itemize}
\item Partial correctness: $\mathcal{B}\models \forall_{\mathit{vars}(b)}(b \leftrightarrow \exists_{\mathit{vars}(\mathit{Exe}(b))\setminus\mathit{vars}(b)}\mathit{Exe}(b))$.
\item Instantiation monotonicity: $\mathit{Exe}(b\cdot\sigma)\neq \mathit{error}$ for all substitutions $\sigma$. \end{itemize}
A built-in predicate $p$ is \emph{incomplete} if there exists an atom $b$ with predicate $p$ for which $\mathit{Exe}(b)=\mathit{error}$; any other built-in predicate is \emph{complete}.
Any built-in predicate which is not logical is called \emph{non-logical}. A \emph{most general instance} of a built-in predicate $p/n$ is an atom $p(v_1,\ldots,v_n)$ where $v_1,\ldots,v_n$ are new and unused variables. The following predicates are examples of built-ins, and the list can be extended if needed.
\begin{definition}\label{def:built-ins} The following list of built-in predicates are assumed with their meaning as indicated; $\epsilon$ is the identity substitution. \begin{enumerate}
\item $\mathit{Exe}(t\mathrel{\texttt{=}}t')=\sigma$ where $\sigma$ is a most general unifier of $t$ and $t'$;
if no such unifier exists, the result is $\mathit{failure}$.
\item $\mathit{Exe}(\texttt{true})$ is $\epsilon$.
\item $\mathit{Exe}(\texttt{fail})$ is $\mathit{failure}$.
\item $\mathit{Exe}(\texttt{$t$ is $t'$}) = \mathit{Exe}(t\mathrel{\texttt{=}}v)$
whenever $t'$ is a ground term that can be interpreted as an arithmetic expression with value $v$; if no such
$v$ exists, the result is $\mathit{error}$.
\item $\mathit{Exe}(\texttt{$t$ >= $t'$})$
is $\epsilon$ whenever $t,t'$ are ground terms
that can be interpreted as arithmetic expressions with values $v,v'$ where $v \geq v'$;
if such values exist but $v < v'$, the result is $\mathit{failure}$; otherwise, the result is $\mathit{error}$.
\item $\mathit{Exe}(\texttt{var($t$)})$ is $\epsilon$ if $t$ is a variable and $\mathit{failure}$ otherwise.
\item $\mathit{Exe}(\texttt{nonvar($t$)})$ is $\epsilon$ when $t$ is not a variable and $\mathit{failure}$ otherwise.
\item $\mathit{Exe}(\texttt{ground($t$)})$ is $\epsilon$ when $t$ is ground and $\mathit{failure}$ otherwise.
\item $\mathit{Exe}(\texttt{constant($t$)})$ is $\epsilon$ when $t$ is a constant and $\mathit{failure}$ otherwise.
\item $\mathit{Exe}(t\mathrel{\texttt{==}}t')$ is $\epsilon$ when $t$ and $t'$ are identical and $\mathit{failure}$ otherwise.
\item $\mathit{Exe}(t\mathrel{\texttt{\char92=}}t')$ is $\epsilon$ when $t$ and $t'$ are non-unifiable and $\mathit{failure}$ otherwise. \end{enumerate}\end{definition} The first three predicates in Definition~\ref{def:built-ins} above are logical and complete; ``\texttt{is}'' and ``\texttt{>=}'' are logical but not complete. The remaining ones are non-logical.
For the representation of CHR execution states, we introduce \emph{indices:} an \emph{indexed set} $S$ is a set of items of the form $i{:}x$ where $i$ belongs to some index set and each such $i$ is unique in $S$. When clear from context, we may identify an indexed set $S$ with its cleaned version $\{x\mid i{:}x\in S\}$. Similarly, the item $x$ may identify the indexed version $i{:}x$. We extract the indices by $\mathit{id}(i{:} x) = i$.
\subsection{Operational Semantics}\label{sec:semantics} The following operational semantics is based on principles introduced in~\cite{DBLP:conf/lopstr/ChristiansenK14}; it differs from those used in previous work in several ways that we discuss in Section~\ref{sec:commentOpSem} below.
As custom in recent theoretical work on CHR, we use the \emph{generalized simpagation} form~\cite{fru_chr_book_2009} as a common representation for the rules of CHR. The guards can modify variables that also occur in rule bodies, but not variables that occur in the constraints matched by the head rules.
\begin{definition}\label{def:rules} A \emph{rule} $R$ is of the form $$r\colon\; H_1 \setminus H_2 \;\mathtt{<=>}\; g\mid C,$$ where $r$ is a unique identifier for the rule, $H_1$ and $H_2$ are sequences of constraints, forming the \emph{head} of the rule, $g$ is a \emph{guard} being a sequence of built-ins, and $C$
is a sequence of constraints and built-ins called the \emph{body} of $R$. Any of $H_1$ and $H_2$, but not both, may be empty. A \emph{program} is a finite set of rules.
A \emph{most general pre-application instance} of rule $R$ is an indexed variant $R'$ of $R$ containing new and fresh variables.
An \emph{application instance} of rule $R$ is a structure of the form $$R'' = R'\sigma = (r\colon\; H'_1\sigma \setminus H'_2\sigma \;\mathtt{<=>}\; g'\sigma\mid C'\sigma) $$ where $R'$ is a most general pre-application instance, $\sigma$ is a substitution for the variables of $H'_1, H'_2$ and
$\mathit{Exe}(g'\sigma)$ is a proper substitution such that\footnote{The condition
indicates that the guard's substitution is not allowed
to instantiate the variables in the head part.}
$$(H_1'\uplus H_2')\sigma = (H_1' \uplus H_2')\sigma\,\mathit{Exe}(g'\sigma).$$ The part $g'$ ($g'\sigma$) is referred to as the \emph{guard of} $R'$ ($R''$). The \emph{application record} for $R'$ ($R''$), denoted $\mathrm{applied}(R')$ ($\mathrm{applied}(R'')$) is the structure $$r\, @\, i_1 \ldots i_n$$ where $i_1 \ldots i_n$ is the sequence of indices of $H_1, H_2$ in the order they occur.
A rule is a \emph{simplification} when $H_1$ is empty, a \emph{propagation} when $H_2$ is empty; in both cases, the backslash is left out, and for a propagation, the arrow symbol is written $\mathtt{==>}$ instead. Any other rule is a \emph{simpagation}. \end{definition}
Following~\cite{RaiserEtAl2009}, an execution state is defined in terms of a suitable equivalence class that abstracts away irrelevant details concerning which actual variables and indices are used.
\begin{definition}\label{def:state} A \emph{(CHR) state representation} is a pair $\langle S, T\rangle$, where \begin{itemize}
\item $S$ is a finite, indexed set of atoms called the \emph{constraint store},
\item $T$ is a set of relevant application records called the \emph{propagation history}, \end{itemize} where a \emph{relevant} application record is one in which each index refers to an index in $S$. Two state representations $S_1$ and $S_2$ are \emph{variants}, denoted $S_1\equiv S_2$, whenever one can be obtained from the other by a renaming of variables and a consistent replacement of indices (i.e., by a 1-1 mapping). When $\Sigma$ is the set of all state representations, a \emph{(CHR) state} is an element of $\Sigma/\!_\equiv\cup\{\mathit{failure}, \mathit{error}\}$, i.e., an equivalence class in $\Sigma$ induced by $\equiv$ or one of two special states; applying the $\mathit{failure}$ ($\mathit{error}$) substitution to a state yields the $\mathit{failure}$ ($\mathit{error}$) state. To indicate a given state, we may for simplicity mention one of its representations. A state different from $\mathit{failure}$ and $\mathit{error}$ is called a \emph{proper state}. A \emph{query} $q$ is a conjunction of constraints, which is also identified with an initial state $\langle q', \emptyset\rangle$ where $q'$ is an indexed version of $q$.
Assuming a fixed program, the function $\textit{all-relevant-app-recs}$ from constraint stores to the powerset of application records is defined as \begin{eqnarray*} \textit{all-relevant-app-recs}(S) & = & \{ r@i_1\ldots i_n\mid \mbox{$r$ identifies a propagation rule and} \\
& & \phantom{\{ r@i_1\ldots i_n\mid\mbox{}} \mbox{$i_1\ldots i_n$
are indices of constraints to which the rule can apply} \} \end{eqnarray*} \end{definition} To simplify notation when we make statements involving several states or other entities involving components of states, we may do so referring to selected state representations, considering recurrence of indices and variables significant. For example, in the context of a program that includes the rule $r$: \texttt{p} \texttt{==>} \texttt{q}, we consider the following as a true statement. $$ST = \langle\{1{:}\texttt{p},2{:}\texttt{q}\}, \emptyset\rangle \;\;\land\;\; ST= \langle S,\emptyset\rangle \;\;\land\;\; \textit{all-relevant-app-recs}(S) = \{r@1\}$$
\begin{definition}\label{def:derivations}
A \emph{derivation step} $\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}$ from one $I$-state to another can be of two types: by rule application instance $\stackrel{R}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}$ or by built-in $\stackrel{b}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}$, defined as follows.
\begin{description}
\item[Apply:] $\langle S\uplus H_1\uplus H_2, T\rangle \stackrel{R}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \langle S\uplus H_1 \uplus \big(C\cdot\mathit{Exe}(g)\big), T'\rangle$\\ whenever there is an application instance $R$ of the form $r\colon\; H_1 \setminus H_2\;\mathtt{<=>}\; g\mid C$ with $\mathrm{applied}(R)\not\in T$, and $T'$ is derived from $T$ by 1) removing any application record having an index in $H_2$ and 2) adding $\mathrm{applied}(R)$ in case $R$ is a propagation. \item[Built-in:] $\langle \{b\} \uplus S, T\rangle\stackrel{b}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \langle S, T \rangle\cdot\mathit{Exe}(b)$. \end{description}
\end{definition} Notice that the removal of application records in \textbf{Apply} steps ensures that no non-relevant propagation record remains in the new state (i.e., the result \emph{is} a state).
\begin{example} Consider a program consisting of the following two rules.
\noindent
\hbox to 2em{$r_1$:\hfil}\verb"p(X) \ q(Y) <=> X=Y | r(X)."\\ \hbox to 2em{$r_2$:\hfil}\verb"r(X) ==> s(X)."
\noindent The following is an application instance of $r_1$.
$$ R_1^{\texttt{a},\texttt{a}}\;\;=\;\; \bigl(r_1\colon 1{:}\texttt{p(a) \char92\ }2{:}\texttt{q(a) <=> a=a |\ }
3{:}\texttt{r(a)}
\bigr)$$
\noindent It can be used in an \textbf{Apply} derivation step as follows. $$ \bigl\langle\{1{:}\texttt{p(a)}, 2{:}\texttt{q(a)}\},\emptyset\bigr\rangle\stackrel{R_1^{\texttt{\scriptsize a},\texttt{\scriptsize a}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}
\bigl\langle\{1{:}\texttt{p(a)}, 3{:}\texttt{r(a)}\},\emptyset\bigr\rangle$$
However, the indexed instance of $r_1$, $\bigl(r_1\colon 1{:}\texttt{p(Z)} \texttt{\char92}2{:}\texttt{q(a)} \texttt{<=>} \texttt{Z=a} \texttt{|} 3{:}\texttt{r(Z)}\bigr)$ is not an application instance as the guard, when executed, will bind the head variable \texttt{Z}.
The rule $r_2$ is a propagation rule, and we show an application instance for it and an \textbf{Apply} derivation step; here the propagation history is checked before the step and modified by the step. $$R_2^{\texttt{a}} \;\;=\;\; \bigl(r_2\colon 1{:}\texttt{r(a) ==>\ }
4{:}\texttt{s(a)}
\bigr)$$ $$ \bigl\langle\{1{:}\texttt{r(a)}, 2{:}\texttt{r(b)}, 3{:}\texttt{s(b)}\},\{r_2@2 \}\bigr\rangle\stackrel{R_2^{\texttt{\scriptsize a}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}
\bigl\langle\{1{:}\texttt{r(a)}, 2{:}\texttt{r(b)}, 3{:}\texttt{s(b)}, 4{:}\texttt{s(a)}\},\{r_2@2, r_2@1 \}\bigr\rangle$$ \end{example} The following example shows how an incomplete predicate is treated when occurring in a guard and when executed by a \textbf{Built-in} step. \begin{example} Consider a program that includes the following rule having the incomplete ``\texttt{is}'' predicate in its guard. Furthermore, assume that ``\texttt{is}'' can appear in \textbf{Built-in} steps, i.e., can also appear in a state.
\noindent
\hbox to 2em{$r_1$:\hfil}\verb"p(X) ==> Y is X+2 | q(Y)."
\noindent An attempt to \textbf{Apply} it to some state by matching the head with $1{:}\texttt{p(2)}$ may yield the application instance
$$R_2^{\texttt{a}} \;\;=\;\; \bigl(r_1\colon 1{:}\texttt{p(2) ==> Z is 2+2 |\ }
7{:}\texttt{q(Z)}
\bigr).$$ The guard evaluates to the substitution $[\texttt{Z}/\texttt{4}]$ and the new state includes the instantiated body constraint $7{:}$\texttt{q(4)}. The rule cannot apply by matching the head with 2{:}\texttt{p(Z)} as the guard evaluates to the $\mathit{error}$ substitution -- but no $\mathit{error}$ state is produced.
A \textbf{Built-in} step, on the other hand, for \texttt{Z} \texttt{is} \texttt{2+A} leads to the $\mathit{error}$ substitution (by Definition \ref{def:built-ins}) and in turn to the $\mathit{error}$ state. \end{example} We observe the following immediate consequence of the definition, namely a functional dependency from a state plus label of a possible step to the resulting state.
\begin{proposition}\label{prop:deriv-step-label-and-subst} For any state $\Sigma$ and derivation step label $d$, there is at most one state $\Sigma'$ such that $\Sigma\stackrel d\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma'$. \end{proposition}
The following distinctions become useful later when we reason about derivation steps and the built-ins involved.
As it appears in Definition~\ref{def:derivations} above, built-ins evaluating to $\mathit{error}$ (representing runtime error) are treated differently in the two sorts of derivation steps: in a guard, $\mathit{error}$ and $\mathit{failure}$ both means that a rule cannot apply (corresponding to no runtime error reported in an implemented system); when such a built-in (coming from the query or a rule body) is applied to a state, it gives rise to a derivation step leading to the relevant of an $\mathit{error}$ or a $\mathit{failure}$ state.
\begin{definition}\label{def:state-built-in-etc} In the context of a state invariant $I$, a built-in predicate is a \emph{state built-in predicate} whenever it can appear in an $I$-state. A logical built-in predicate $p$ is \emph{$I$-complete} whenever $\mathit{Exe}(b)\neq\mathit{error}$ for any atom $b$ with predicate $p$ that may occur in an $I$-state or in the guard of an application instance that can apply to an $I$-state.
A guard in a rule is \emph{logical} if it contains only logical predicates; otherwise, it is \emph{non-logical}. A logical guard is \emph{$I$-complete} if it contains only $I$-complete predicates; otherwise, it is \emph{$I$-incomplete}. \end{definition}
\begin{example} The built-in ``\texttt{is}/2'' is logical and, while incomplete, it is $I$-complete with respect to an invariant that guarantees the second argument to be a ground arithmetic expression. \end{example}
\subsection{A Few Comments on Earlier Operational Semantics for CHR}\label{sec:commentOpSem} Our operational semantics for CHR differs from other known and formally specified ones e.g.,~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:journals/constraints/AbdennadherFM99,DuckSBH04,DBLP:conf/iclp/DuckSS07} by handling also non-logical and incomplete built-ins.
We do that by ``executing'' built-ins immediately in terms of substitutions applied to the state, which we claim is more compatible with practical CHR systems than earlier approaches; Apt et al's semantics for Prolog with such predicates~\cite{DBLP:journals/aaecc/AptMP94} applies the same principles (with the small difference that they do not distinguish between error and failure).
The referenced approaches use instead a separate store for built-in constraints (restricted to logical ones) that have been processed, with their satisfiability determined by a magic solver that mirrors a first-order semantics; this excludes the possibility to consider runtime errors and non-logical and incomplete predicates. The following example highlights the difference.
\begin{example} Assume a program that includes the following program rule, and assume that \texttt{>=} is also a state built-in predicate. \begin{verbatim}
p(X) <=> X >= 1 | r(X) \end{verbatim} We can point out the difference by the query \texttt{A>=2,} \texttt{p(A)}. Starting from an empty built-in store (\texttt{true}), the semantics of~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:journals/constraints/AbdennadherFM99,DuckSBH04,DBLP:conf/iclp/DuckSS07} may first ``execute'' \texttt{A>=2} by adding it to the built-in store and keeping \texttt{p(A)} as the remaining query. Then the program rule above can apply for \texttt{p(A)} -- since the truth of the guard is implied by the built-in store, thus leaving a final constraint store $\{\texttt{r(A)}\}$ constrained by the built-in store $(\texttt{A>=2})$.
With our semantics, the rule cannot apply (as the guard evaluates to $\mathit{error}$ which is treated the same way as $\mathit{failure}$), and evaluating \texttt{A>=2} as part of the query results in the final state $\mathit{error}$.
\end{example}
The test in our semantics (Definition~\ref{def:rules}) that prevents a rule from being applied if it otherwise would modify variables in the constraints matched by the rule head, is implicit in~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:journals/constraints/AbdennadherFM99,DuckSBH04,DBLP:conf/iclp/DuckSS07}. Consider, for example, a rule \texttt{p(X)} \texttt{<=>} \texttt{X=a} \texttt{|}$\cdots$ considered for the query atom \texttt{p(A)}, assuming a built-in store $B$. Here the test in the guard would amount to the condition $B\models\forall\texttt{A}.\, \texttt{A}=\texttt{a}$; this is false when, say $B$ is empty, and holds only when $B$ implies that \texttt{a} is the only possible value for \texttt{A}.
The interpretation of runtime error in guards as failure is described by~\cite{DBLP:journals/aai/HolzbaurF00a} for one of the first widespread CHR compilers, released with earlier versions of SICStus Prolog. The documentation for the now dominant compiler~\cite{SchrijversDemoen2004} embedded in recent versions of SICStus Prolog and SWI Prolog\footnote{See \url{http://www.swi-prolog.org}; version 7 checked February 2016.} is not explicit about this point. A test of SWI Prolog shows that a runtime error in a guard makes the entire computation terminate with an error message. While this limits the completeness results for CHR, it has been chosen for efficiency reasons (and the fact that the guard can be reformulated to obtain error as failure if necessary)~\cite{personalCommTomSchrijversFeb2016}.
Furthermore, we disregard so-called global variables defined as those that appear in the original query. The mentioned previous approaches introduce a separate state component to memorize global variables, but this can be shown unnecessary. Consider a query \texttt{q(X)}; we translate it into \texttt{q(X),} \texttt{global('X',X)} where \texttt{'X'} is a constant that serves as the name of variable global variable \texttt{X}. When a derivation terminates in a proper state, it includes the constraint \texttt{global('X',$\mathit{val}$)} where $\mathit{val}$ is the value computed for variable \texttt{X}.
The mentioned semantics uses a separate state component, that we will call the \emph{queue}, to hold constraints that have not yet been entered into the ``active'' constraint store.
Constraints appearing in the body of a rule being applied are first entered into the queue, and then from time to time moved into the active store by a separate sort of derivation step. Rules are applied by matching constraints within the active store. This separation may be relevant as a starting point for imposing strategies for ordering application of rules and search for constraints to be processed (which is one of the goals of~\cite{DuckSBH04}), but for studies of confluence it is irrelevant as the set of derivations with or without this additional mechanics is essentially the same.
Our semantics has only two state components, in comparison to, e.g.,~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:conf/iclp/DuckSS07} that need five state components when considering more restricted confluence problems.
There are also differences in how to avoid the potential looping by propagation rules applying to the same constraints over and over again.
Our semantics (and some others not referenced) hold a set of records telling which propagations must not apply (because they have been applied already), while~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:conf/iclp/DuckSS07} maintain a set of permissions for those propagations that may apply. There is essentially only a notational difference between the two, and the choice is a matter of taste. An alternative approach is taken by~\cite{DBLP:journals/tplp/BetzRF10}, mixing a set-based and a multiset-based approach: new constraints produced from the body of a propagation is treated set-wise, and a propagation is only allowed if it results in adding new constraints.
In earlier work, such as~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:conf/iclp/DuckSS07} already discussed, the states include specific indices and specific variables. Thus any reasonable definition of joinability and confluence needed to mention an equivalence relation telling two states equivalent if they differ only in systematic replacement of variables and indices (quite similar to our $\equiv$ in Definition~\ref{def:state}, p.~\pageref{def:state}). So in some sense, these approaches concern confluence modulo equivalence problems, but for a very specific equivalence hardcoded into the proofs of general properties.\footnote{The same can be said about \cite{DBLP:journals/jfp/Niehren00,DBLP:conf/ccl/NiehrenS94}, studying confluence in a completely different setting.} In 2009, the paper~\cite{RaiserEtAl2009} gave a satisfactory solution to this problem, abstracting away concrete indices and variables defining a state as an equivalence class modulo such a relation $\equiv$, exactly as we have shown in our Definition~\ref{def:state} above.
\section{Confluence Modulo Equivalence for CHR}\label{sec:conf-mod-eq-in-CHR}
Here we adapt classical definitions of critical pairs and associated properties for CHR to include non-logical and incomplete built-ins, as well as an invariant and an equivalence relation.
For the strictly logical case with no invariant,~\cite{DBLP:conf/cp/Abdennadher97} defines critical pairs consisting of CHR states that may be shown joinable by ordinary CHR derivations. This is not viable in our more general case as our analogous construction may lead to pairs that do not satisfy the invariant and from which no derivations are possible (although the set of all relevant instances thereof may be joinable at the level of CHR).
As a first step towards our meta-level counterpart of critical pairs, we introduce what we call most general critical pre-corners having a CHR state serving as a common ancestor.
We use the following subcategorization, introduced by \cite{DBLP:conf/lopstr/ChristiansenK14}, of $\alpha$- and $\beta$-corners according to the sorts of derivation steps involved, as they need to be treated differently. The earlier results on confluence for CHR concern only $\alpha_1$-corners.
\begin{definition}\label{def:critical} Assume a program with equivalence $\approx$ and invariant $I$. Let $\Lambda_\alpha= (y\stackrel{\gamma}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} x \stackrel{\delta}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y')$ be an $\alpha$-$I$-corner and $\Lambda_\beta= (y\approx x \stackrel{\delta}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} y')$ a $\beta$-$I$-corner. \begin{itemize}
\item $\Lambda_\alpha$ is an \emph{$\alpha_1$-$I$-corner} whenever
$\gamma$ and $\delta$ are rule application instances.
\item $\Lambda_\alpha$ is an \emph{$\alpha_2$-$I$-corner} whenever
$\gamma$ is a rule application instance and $\delta$ a built-in.
\item $\Lambda_\alpha$ is an \emph{$\alpha_3$-$I$-corner}
$\gamma$ and $\delta$ are built-ins.
\item $\Lambda_\beta$ is a \emph{$\beta_1$-$I$-corner} whenever
$\delta$ is a rule application instance.
\item $\Lambda_\beta$ is a \emph{$\beta_2$-$I$-corner} whenever
$\delta$ a is built-in. \end{itemize} The ``-$I$-'' part of the names may be left out when clear from context. \end{definition}
\subsection{Most General Critical Pre-corners} According to Proposition~\ref{prop:deriv-step-label-and-subst}, the end state of a derivation step
is functionally dependent on the initial state, and we employ this in the following definition of most general critical pre-corners in which we leave out wing states. The reason why we refer to these artefacts as \emph{pre-}corners is that they may not be corners at all; when the wing states are attempted to be filled in, the guards or invariant may not be satisfied.
\begin{example}\label{ex:motivate-pre-corners} Consider the following program which has non-logical guards; assume $\approx$ being identity and $I(\cdot)=\mathit{true}$.
\noindent
\hbox to 2em{$r_1$:\hfil}\verb"p(X) <=> var(X) | q(X)."\\
\hbox to 2em{$r_2$:\hfil}\verb"p(X) <=> nonvar(X) | r(X)."\\ \hbox to 2em{$r_3$:\hfil}\verb"q(X) <=> r(X)."
\noindent The equality predicate \texttt{=}/2 is here regarded as a state built-in predicate, i.e., it may appear in a query; the meaning of this and the other built-ins is given in Definition~\ref{def:built-ins}, p.~\pageref{def:built-ins}. The following is an attempt to construct an $\alpha_2$-corner; there are no propagation rules, so we leave the empty propagation history implicit and identify states by multisets of constraints. $$ \Lambda\; = \;\bigl(\{ \texttt{r(Z)}, \texttt{X=Y}\} \stackrel{R_2^{\texttt{Z}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\{ \texttt{p(Z)}, \texttt{X=Y}\} \stackrel{\texttt{X}=\texttt{Y}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{ \texttt{p(Z)}\} \bigr)$$
Here $R_2^{\texttt{Z}}$ is the rule instance $r_2$: \texttt{p(Z)} \texttt{<=>} \texttt{nonvar(Z)} \texttt{|} \texttt{r(Z)}; $R_2^{\texttt{Z}}$ is not an application instance since its guard is false, thus the hinted derivation step does not exist, and $\Lambda$ is not a corner. However, any substitution $\theta$ that grounds $Z$ will lead to a corner. With $\texttt{Z}\theta=\texttt{a}$, $\Lambda\theta$ is a corner, whereas if in addition $\texttt{X}\theta=\texttt{b}, \texttt{Y}\theta=\texttt{c}$ we need to replace the right-most derivation step $\cdots\stackrel{\texttt{X}=\texttt{Y}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{ \texttt{p(Z)}\!\}$ by $\cdots\stackrel{\texttt{b=c}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\mathit{failure}$. \end{example}
The following definition of most general critical pre-corners is lengthy as it has one case for each sort of corners, but it is straightforward when a few elements have been explained. For $\alpha_1$, the common ancestor state is constructed from two rules such that the application of one prevents the subsequent application of the other. The symbol ``$\circ$'' is an arbitrary placeholder that visually indicates the presence of some state.
Propagation histories notoriously introduce extra notation and technicalities, that are explained following the definition. We recall Definition~\ref{def:state}, that $\textit{all-relevant-app-recs}(S)$ is the set of all application records for rules of the current program taking indices only from the constraint store $S$.
\begin{definition}[Most General Critical Pre-Corners]\label{def:pre-corners} Assume a program with equivalence $\approx$ and invariant $I$.\footnote{Notice that only $\alpha_2$ and $\alpha_3$ refers to $I$, but we maintain the -$I$- syllable in all cases for homogeneity.}\\ {\large{$\alpha_1$:}}\quad A \emph{most general critical $\alpha_1$-$I$-pre-corner} is a structure of the form $( \circ\stackrel{R_1\sigma}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle H_1\sigma\cup H_2\sigma, T\rangle \stackrel{R_2\sigma}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$ where \begin{itemize} \item $R_k=(r_k\colon\;A_k \setminus B_k \;\mathtt{<=>}\; g_k\mid C_k)$, $k=1,2$,
are two most general pre-application instances; \item let $H_k = A_k \uplus B_k$ and
assume two nonempty sets $H'_k \subseteq H_k$ such that the set of indices used in $H'_1$ and $H'_2$ are identical and all other indices
in $R_1,R_2$ are unique, and let $\sigma$ be a most general unifier of $H'_1$ and (a permutation of) $H'_2$; \item if $r_1=r_2$, we must have $A_1\sigma \neq A_2\sigma$ or $B_1\sigma \neq B_2\sigma$; \footnote{We exclude cases with
$r_1=r_2$ where the rule applies the same way for both derivation steps, i.e.,
$A_1\sigma = A_2\sigma$ and $B_1\sigma = B_2\sigma$, as the two wing states in any
subsumed corner would be identical and thus trivially joinable.} \item $B_1 \sigma \cap H'_2\sigma \neq \emptyset$ or $B_2 \sigma \cap H'_1\sigma \neq \emptyset$; \item $ T = \textit{all-relevant-app-recs}(H_1\sigma\cup H_2\sigma)
\;\setminus\; \{r_1@\mathit{id}(A_1B_1), r_2@\mathit{id}(A_2B_2) \}
$; \item there exists a substitution $\theta$ such that, for $k=1,2$, $\mathit{Exe}(g_k\sigma\theta)$ is a proper substitution and $H_k \mathit{Exe}(g_k\sigma\theta)=H_k.$ \end{itemize} {\large{$\alpha_2$:}}\quad A \emph{most general critical $\alpha_2$-$I$-pre-corner} is a structure of the form $(\circ \stackrel{R}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle A\uplus B\uplus\{b\}, T\rangle \stackrel{b}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$ where \begin{itemize} \item $R=(r\colon\;A \setminus B \;\mathtt{<=>}\; g\mid C)$ is a most general pre-application instance whose guard $g$ is non-logical or $I$-incomplete; \item there exists a substitution $\theta$ such that $\mathit{Exe}(g\theta)$ is a proper substitution, $\mathit{vars}(G\theta)\cap\mathit{vars}(b\theta) \neq \emptyset$, and $H\mathit{Exe}(g\theta)=H$, where
$H= A \uplus B$; \item $b$ is a most general instance of a state built-in predicate (i.e., all arg's are fresh variables); \item $ T = \textit{all-relevant-app-recs}(A\uplus B)
\;\setminus\; \{r@\mathit{id}(AB) \}
$. \end{itemize} {\large{$\alpha_3$:}}\quad A \emph{most general critical $\alpha_3$-$I$-pre-corner} is a structure of the form $(\circ \stackrel{b_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle\{b_1,b_2\}, \emptyset\rangle \stackrel{b_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$ where \begin{itemize} \item $b_k$, $k=1,2$, are indexed, most general instances of state built-in predicates, $b_1$ being
non-logical or $I$-incomplete. \end{itemize} {\large{$\beta_1$:}}\quad When $\approx\neq=$, a \emph{most general critical $\beta_1$-$I$-pre-corner} is a structure of the form $(\circ \approx
\langle A\uplus B, T\rangle \stackrel{R}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$\\ \hbox to 2.5em{}where \begin{itemize} \item $R=(r\colon\;A \setminus B \;\mathtt{<=>}\; g\mid C)$ is a most general pre-application instance whose guard $g$ is satisfiable; \item $ T =\textit{all-relevant-app-recs}(A\uplus B)
\;\setminus\; \{r@\mathit{id}(AB) \}
$. \end{itemize} {\large{$\beta_2$:}}\quad When $\approx\neq=$, a \emph{most general critical $\beta_2$-$I$-pre-corner} is a structure of the form $(\circ \approx
\langle \{b\}, \emptyset\rangle \stackrel{b}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$ where \begin{itemize} \item $b$ is a most general instance of a state built-in predicate. \end{itemize} Any two most general critical $I$-pre-corners are considered the same whenever they differ only by consistent renaming of indices and variables and swapping of the left and right parts. The $I$ part of the names may be left out when clear from context. \end{definition}
The propagation history constructed for $\alpha_1$-pre-corners is similar to that of earlier work, e.g.,~\cite{DBLP:conf/cp/Abdennadher97}, for building critical pairs.\footnote{It makes only a syntactic difference that~\cite{DBLP:conf/cp/Abdennadher97} maintains a set of application records for rules that may be applied, whereas we maintain a set for those that may not be applied.} It tells that any other propagation rule, say \textit{Prop},
which might accidentally be applied to constraints in the common ancestor state, is prevented from doing so. This provides the maximum level of generality of the pre-corner in the sense that it subsumes (defined below) all concrete corners in which \textit{Prop} can apply as well as those where it cannot. The propagation histories for the other sorts of pre-corners can be explained in similar ways.
\begin{example}[continuing Example~\ref{ex:motivate-pre-corners}, p.~\pageref{ex:motivate-pre-corners}]\label{ex:motivate-pre-corners-contd} The following is an example of a most general critical $\alpha_2$-pre-corner for the rule labelled $r_1$ (whose guard contains \texttt{var}/1) and built-in \texttt{=}/2. $$ \Lambda^{r_1,\texttt{=}}\; = \;\bigl(\circ \stackrel{R^Z_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle\{ \texttt{p(Z)}, \texttt{X=Y}\}, \emptyset\rangle\} \stackrel{\texttt{X}=\texttt{Y}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ \bigr)$$
Here $R^Z_1$ is the application instance $r_1:$ \texttt{p(Z)} \texttt{<=>} \texttt{var(Z)} \texttt{|} \texttt{q(Z)}. \end{example}
As opposed to the derivation relation $\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}$, the equivalence relation is given in an atomic way, so we need to consider any possible $\beta$-corner as critical, i.e., its joinability is not a priori given.\footnote{In the concluding section, we discuss an alternative approach that uses $\gamma$-corners, as mentioned in Section~\ref{sec:rewrite}, instead of $\beta$-corners, which makes it possible to subcategorize and perhaps filter away some of the abstract pre-corners that concern the equivalence.}
By construction, we have the following.
\begin{proposition}\label{prop:precorners-finite} For any given program with invariant $I$ and equivalence $\approx$, the set of most general critical $I$-pre-corners is finite. \end{proposition} As mentioned, most general critical pre-corners are intended to provide a finite characterization of the set of actual corners that are not per se joinable. To express this, we introduce the following notion of subsumption.
\begin{definition}[Subsumption by Most General Critical Pre-Corners]\label{def:subsumption-by-gcpc} Let $\Lambda=(\circ\mathrel{\mathit{Rel}_1}\langle S, T\rangle \mathrel{\mathit{Rel}_2}\circ)$ be a most general critical pre-corner. An $I$-corner $\lambda =(\langle s_1,t_1\rangle \mathit{rel}_1 \langle s,t\rangle \mathit{rel}_2 \langle s_2,t_2\rangle)$ is \emph{subsumed by} $\Lambda$, written $\Lambda<\lambda$, whenever there exists a substitution $\theta$, a set of indexed constraints $s^+$ and sets of application instances $t^+$ and $t^\div$ such that \begin{itemize} \item $s=S\theta\uplus s^+$, \item $t=T\uplus t^+ \setminus t^\div$, \item $t^+\subseteq\textit{all-relevant-app-recs}(S\theta\uplus s^{+})\setminus\textit{all-relevant-app-recs}(S\theta)$\\(i.e., a set of application records, each containing an index in $s^+$), \item $t^\div\subseteq \textit{all-relevant-app-recs}(S\theta)$ \item $\mathit{rel}_k=\mathit{Rel}_k\,\theta,\quad k=1,2$. \end{itemize} If, furthermore, $\Lambda$ is an $\alpha_2$-pre-corner $(\circ \stackrel{R}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle\Sigma, T\rangle \stackrel{b}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$, where $R$ has guard $g$, and $b$ a built-in, it is required that $\mathit{vars}(g\theta)\cap\mathit{vars}(b\theta)\neq\emptyset$.
\end{definition} This definition is guilty in a slight abuse of usage due to the additional requirements for $\alpha_2$
in that only ``really critical'' instances of the pre-corners are counted: if the indicated variable overlaps are not observed, the two derivation steps commute so that joinability is guaranteed.
\begin{example}[continuing Examples~\ref{ex:motivate-pre-corners}, \ref{ex:motivate-pre-corners-contd}]\label{ex:motivate-pre-corners-contd-contd} Consider the following $\alpha_2$-corner for the program given in Example~\ref{ex:motivate-pre-corners}, $$ \lambda \;=\; \bigl( \langle\{\texttt{q(A)}, \texttt{A=a}\}\cup S, T\rangle
\stackrel{R_1^\texttt{\scriptsize{A}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle\{\texttt{p(A)}, \texttt{A=a}\}\cup S, T\rangle
\stackrel{\texttt{A=a}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}
\langle\{\texttt{p(a)}\}\cup S[\texttt{A}/\texttt{a}], T\rangle \bigr)
$$
where ${R_1^\texttt{\normalsize{A}}}$ is the rule instance ($r_1:$ \texttt{p(A)} \texttt{<=>} \texttt{var(A)} \texttt{|} \texttt{q(A)})
and $S$ ($T$) a suitable set of indexed constraints (application records).
It appears that $\lambda$ is subsumed by the most general critical $\alpha_2$-pre-corner $ \Lambda^{r_1,\texttt{=}}$ introduced in Example~\ref{ex:motivate-pre-corners-contd} above. To see this, we use the substitution $[\texttt{Z}/ \texttt{A}, \texttt{X}/\texttt{A}, \texttt{Y} / \texttt{a}]$ for $\theta$ in Definition~\ref{def:subsumption-by-gcpc} above, and check that $\mathit{vars}(\texttt{var(Z)}\theta)=\{\texttt{A}\}$ and $\mathit{vars}((\texttt{X=Y})\theta)=\{\texttt{A}\}$ do overlap.
\end{example}
The following adapts the Critical Pair Lemma~\cite{DBLP:journals/jacm/Huet80,KnuthBendix1970} known from term rewriting (and implicit in previous work on confluence for CHR) to our setting. \begin{lemma}[Critical Corner Lemma]\label{lem:critCorner} Assume a program with invariant $I$ and equivalence relation $\approx$, and let $\lambda$ be an $I$-arbitrary corner. Then it holds that either \begin{itemize}
\item $\lambda$ is $I$-joinable modulo $\approx$, or
\item $\lambda$ is subsumed by a most general critical pre-corner. \end{itemize} \end{lemma} The proof which is straightforward but lengthy can be found in the appendix.
This leads to the following central theorem.
\begin{theorem}[Critical Corner Theorem]\label{thm:critCorner} Assume a program $\Pi$ with invariant $I$ and state equivalence relation $\approx$. Then
$\Pi$ is locally confluent modulo $\approx$
if and only if
all $I$-corners subsumed by some most general critical pre-corner for $\Pi$ are joinable.
\end{theorem}
\begin{proof} The ``only if'' part: Assume the opposite, that $\Pi$ is locally confluent and that there is an $I$-corner $\lambda$ subsumed by some critical pre-corner for $\Pi$ which is not joinable modulo $\approx$. According to Lemma~\ref{lem:critCorner}, $\lambda$ must be joinable; contradiction. The ``if'' part follows immediately from Lemma~\ref{lem:critCorner}: let $\lambda$ be an $I$-corner; if $\lambda$ is subsumed by some critical pre-corner for $\Pi$ we are done by assumption; otherwise the lemma states that it is joinable. \end{proof} Combining this result with~Theorem~\ref{thm:I-joinable-corners}, p.~\pageref{thm:I-joinable-corners}, we get the following.
\begin{theorem}\label{thm:conflu-and-critCorner} Assume a terminating program $\Pi$ with invariant $I$ and state equivalence relation $\approx$. Then $\Pi$ is confluent modulo $\approx$ if and only if all $I$-corners subsumed by some critical pre-corner for $\Pi$ are joinable. \end{theorem}
\subsection{Relationship with Earlier Approaches to Proving Confluence}\label{sec:earlier-CHR-confl-etc} In the following, we reformulate earlier results of Abdennadher et al~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,fruehwirth-98} for confluence without equivalence and invariant for the purely logical subset of CHR
and those of Duck et al~\cite{DBLP:conf/iclp/DuckSS07}, who extended with an invariant, as we have described in Section~\ref{sec:backgroundConfCHR}. Their critical pairs are similar to our most general critical $\alpha_1$-pre-corners, and the other sorts of corners become either trivially joinable or non-existing in these special cases.
In order to describe these results, we complement the notion of subsumption introduced above in Definition~\ref{def:subsumption-by-gcpc} with a subsumption ordering for $I$-corners.
\begin{definition}[Subsumption Ordering for $\alpha_1$-$I$-corner]\label{def:subsumption-by-a1icorner} Assume a program with invariant $I$, and let $\lambda = (\langle s_1,t_1\rangle \stackrel{r_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \langle s_0,t_0\rangle \stackrel{r_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \langle s_2,t_2\rangle)$ and $\lambda' = (\langle s_1',t_1'\rangle \stackrel{r'_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \langle s'_0,t'_0\rangle \stackrel{r'_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \langle s'_2,t'_2\rangle)$
be
$\alpha_1$-$I$-corners. We say that $\lambda$ \emph{subsumes} $\lambda'$ denoted $\lambda \preceq \lambda'$ whenever there exist a substitution $\theta_k$, a set of indexed constraints $s_k^+$ and sets of application instances $t^+_k$ and $t^\div_k$ for $k=0,1,2$ such that \begin{itemize} \item $s'_k = s_k\theta_k\uplus s_k^+$, \item $t'_k = t_k\uplus t_k^+ \setminus t_k^\div$, \item $t_k^+$ is a set of application records, each containing at least one index appearing in $s_k^+$, \item $t_k^\div\subseteq \textit{all-relevant-app-recs}(S_k)$ \item $r'_i=\mathit{r}_i\,\theta_0,\quad i=1,2$. \end{itemize} We write $\lambda\prec\lambda'$ whenever $\lambda\preceq\lambda'$ and $\lambda\neq\lambda'$. \end{definition}
The following property follows immediately by the direct similarity with Definition~\ref{def:subsumption-by-gcpc}.
\begin{proposition} Let $\Lambda$ be a most general critical $\alpha_1$-pre-corner and $\lambda$ an $\alpha_1$-corner such that $\Lambda<\lambda$. Whenever $\lambda'$ is a corner with $\lambda\preceq\lambda'$, it holds that $\Lambda<\lambda'$. \end{proposition}
\begin{example}[continuing Examples~\ref{ex:collect}, p.~\pageref{ex:collect}, and \ref{ex:collect-inv}, p.~\pageref{ex:collect-inv}]\label{ex:collect-corners} Consider again the single rule program that collects elements into a list, with the invariant of groundedness plus exactly one \texttt{set} constraint whose argument is a list (we ignore the state equivalence here). The following shows a most general critical pre-corner, two corners and their mutual ordering; $R^{L,A}$ stands for an applications instance for the rule in which variables \texttt{L}, \texttt{A} are replaced by terms $L$, $A$. \begin{center} $\circ \stackrel{R^{\texttt{\scriptsize L},\texttt{\scriptsize A}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \{ \texttt{set(L)}, \texttt{item(A)}, \texttt{item(B)}\} \stackrel{R^{\texttt{\scriptsize L},\texttt{\scriptsize B}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \circ$\\ \rotatebox[origin=c]{270}{\hbox to 1ex{}{\large$<$}\hbox to 0.5ex{}}\\
${\{\texttt{set([a|c])}, \texttt{item(b)}\} \stackrel{R^{\texttt{\scriptsize [c]},\texttt{\scriptsize a}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \{\texttt{set([c])}, \texttt{item(a)}, \texttt{item(b)}\} \stackrel{R^{\texttt{\scriptsize [c]},\texttt{\scriptsize b}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{ \texttt{set([b|c])}, \texttt{item(a)}\}}$\\ \rotatebox[origin=c]{270}{\hbox to 1ex{}{\large$\preceq$}\hbox to 0.5ex{}}\\
$\{\texttt{set([a|c])}, \texttt{item(b)}, \texttt{item(d)}\} \stackrel{R^{\texttt{\scriptsize [c]},\texttt{\scriptsize a}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \{\texttt{set([c])}, \texttt{item(a)}, \texttt{item(b)}, \texttt{item(d)}\} \qquad\qquad\qquad\qquad\qquad$\\ $\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\stackrel{R^{\texttt{\scriptsize [c]},\texttt{\scriptsize b}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{ \texttt{set([b|c])}, \texttt{item(a)}, \texttt{item(d)}\}$ \end{center} \end{example} Notice in the example above that the common ancestor in the pre-corner does not satisfy the invariant and thus there are no derivation steps possible from it. However, when this state is instantiated (and perhaps extended with more constraints) so that the invariant becomes satisfied, the derivation labels denote actual application instances and corners emerge.
We proceed now as Duck et al~\cite{DBLP:conf/iclp/DuckSS07} and identify a collection of $I$-corners for each pre-corner which together subsumes all relevant $I$-corners as shown in Theorem~\ref{thm:Icorner} below.
\begin{definition}[Minimal and Least Critical $I$-corners]\label{def:minimal-and-least-I-corners} Assume a program with invariant $I$. An $\alpha_1$-$I$-corner $\lambda$ is \emph{minimal (for a most general $\alpha_1$-pre-corner $\Lambda$)} whenever
\begin{itemize}
\item $\Lambda < \lambda$, and
\item $\nexists\, \lambda' > \Lambda \colon \lambda' \prec \lambda$.
\end{itemize} When, furthermore \begin{itemize}
\item $\forall\, \lambda' > \Lambda \colon \lambda \preceq \lambda'$,
\end{itemize}
$\lambda$ is a \emph{least $I$-corner for $\Lambda$}.
\end{definition} In Example~\ref{ex:collect-corners} above, the highest placed $I$-corner is minimal but not least, as other similar corners exist with other choices of constants.
Theorem~\ref{thm:Icorner} below is similar to the central result of~\cite{DBLP:conf/iclp/DuckSS07}, showing that local $I$-confluence follows from joinability of a specific set of minimal $I$-corners.
\begin{lemma}[Existence of Minimal $I$-corners]\label{lem:icorner-precorner}Assume a program with invariant $I$. For any $\alpha_1$-$I$-corner $\lambda'$ subsumed by a most general $\alpha_1$-pre-corner $\Lambda$, i.e., $\Lambda<\lambda' $, there exists a minimal $I$-corner $\lambda$ for $\Lambda$ such that $\Lambda<\lambda \preceq \lambda'$. \end{lemma}
\begin{proof} First of all, we notice that by construction of subsumption, that (**) there cannot exist infinite chains $\lambda_1\succ\lambda_2\succ\cdots > \Lambda$.
Consider now $\lambda'>\Lambda$. If $\lambda'$ is minimal, we are done; otherwise (by Definition~\ref{def:subsumption-by-gcpc}, p.~\pageref{def:subsumption-by-gcpc}) there will be a $\lambda_1$ such that $\lambda'\succ\lambda_1>\Lambda$; if $\lambda_1$ is minimal, we are done; otherwise there will be a $\lambda_2$ such that $\lambda'\succ\lambda_1\succ\lambda_2>\Lambda$, and we continue the same way until we reach a minimal $\lambda_n$ with $\lambda_1\succ\lambda_2\succ\cdots\succ\lambda_n>\Lambda$; due to observation (**) above, this process will terminate as indicated. \end{proof}
\begin{lemma}[Minimal $I$-corner Lemma; logical case with invariant; trivial $\approx$]\label{lem:critICorner}~\\ Assume a program with logical and complete built-ins, invariant $I$ and state equivalence $=$, and let $\lambda$ be an $\alpha_1$-$I$-corner. Then it holds that either \begin{itemize}
\item $\lambda$ is $I$-joinable, or
\item $\lambda$ is subsumed by some minimal $\alpha_1$-$I$-corner. \end{itemize} \end{lemma}
\begin{proof} The lemma is a direct consequence of Lemma~\ref{lem:critCorner} and Lemma~\ref{lem:icorner-precorner}. \end{proof}
\begin{theorem}[Minimal $I$-corner Theorem; logical case with invariant; trivial $\approx$]\label{thm:Icorner}~\\ For a program $\Pi$ with logical and complete built-ins, invariant $I$ and state equivalence relation $=$, the following properties hold. \begin{enumerate}
\item $\Pi$ is locally confluent if and only if all its minimal $I$-corners are joinable. \label{thm:Icorner-partI}
\item When, furthermore, $\Pi$ is terminating, $\Pi$ is confluent if and only if all its minimal $I$-corners are joinable.\label{thm:Icorner-partII}
\item A minimal $I$-corner is not necessarily least, and the set of all minimal $I$-corners is not necessarily finite.\label{thm:Icorner-partIII} \end{enumerate} \end{theorem}
\begin{proof} Part \ref{thm:Icorner-partII} follows from Newman's Lemma and Part \ref{thm:Icorner-partI}. Proof of Part~\ref{thm:Icorner-partI}: ``$\Rightarrow$'': It follows directly from the assumption of local confluence. ``$\Leftarrow$'': Assume that all minimal $I$-corners are joinable, but the program is not locally confluent, i.e., there exists an $I$-corner $\lambda'$ that is not joinable. From Lemma~\ref{lem:critICorner} we have that any $\lambda'$ is subsumed by a minimal $I$-corner $\lambda$ which by assumption is joinable.
Part~\ref{thm:Icorner-partIII} is demonstrated by the following Example~\ref{ex:infinite-minimal-corners}. \end{proof}
In the general case, Theorem~\ref{thm:Icorner} does not provide an immediate recipe for proving local confluence due to the potentially infinite number of cases. The following example demonstrates two ways that this may appear.
\begin{example}\label{ex:infinite-minimal-corners} Consider a program that includes the following rules.
\noindent \hbox to 2em{$r_1$:\hfil}\verb"p(X) <=> q(X)."\\ \hbox to 2em{$r_2$:\hfil}\verb"p(X) <=> r(X)."\\
\hbox to 2em{$r_3$:\hfil}\verb"p(X) <=> X >= 1 | s(X)."
\noindent
We assume the invariant $$I(\langle S, T\rangle)\Leftrightarrow S \text{ is ground.}$$ There are no propagation rules, so we ignore the propagation history and consider a state as a multiset of constraints. Rules $r_1$ and $r_2$ give rise to a most general critical $\alpha_1$-$I$-pre-corner $( \circ\stackrel{R_1^{\texttt{X}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\{\texttt{p(X)}\} \stackrel{R_2^{\texttt{X}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$. It has the following infinite set of minimal $\alpha_1$-$I$-corners. $$ \bigl\{\bigl(\{\texttt{q($t$)}\} \stackrel{R_1^{t}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\{\texttt{p($t$)}\} \stackrel{R_2^{t}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{\texttt{q($t$)}\}\bigr) \bigm\vert \mbox{$t$ is a ground term} \bigr\} $$ Obviously, there is no least $\alpha_1$-$I$-corner for this $\alpha_1$-$I$-pre-corner. This problem was also noticed by Duck et al in their paper on observable confluence~\cite{DBLP:conf/iclp/DuckSS07}. An additional consequence of our definitions is that a guard with an incomplete predicate may also give rise to an infinite set of minimal $\alpha_1$-$I$-corners, even when we relax the invariant to equality. Now, rules $r_1$ and $r_3$ give rise to a most general critical $\alpha_1$-$I$-pre-corner $( \circ\stackrel{R_1^{\texttt{X}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\{\texttt{p(X)}\} \stackrel{R_3^{\texttt{X}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$. It has the following infinite set of minimal $\alpha_1$-$I$-corners. $$ \bigl\{\bigl(\{\texttt{q($t$)}\} \stackrel{R_1^t}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\{\texttt{p($t$)}\} \stackrel{R_3^t}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{\texttt{r($t$)}\}\bigr) \bigm\vert \mbox{$t$ is a ground term that can be read as a numeral $\ge1$} \bigr\} $$ \end{example}
The solution that we describe in Section~\ref{sec:abstract}, and which has no counterpart in~\cite{DBLP:conf/iclp/DuckSS07}, is to consider each most general critical pre-corner (of which there are only finitely many) one at a time, lifted to a meta-level where we can reason about their joinability properties without having to expand them to a set of minimal $I$-corners.
A partial version of the classical results by Abdennadher et al~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,fruehwirth-98} can be described as a special case of Theorem~\ref{thm:Icorner} with trivial invariant.
\begin{lemma}[Least I-corner Lemma; logical case with trivial invariant and $\approx$]\label{lem:leastIcorner} Assume a program $\Pi$ with logical and complete built-ins, invariant $I(\cdot)\Leftrightarrow\mathit{true}$ and state equivalence relation $=$. The set of minimal $\alpha_1$-corners is finite and consists of least $\alpha_1$-corners, each of which is produced from an $\alpha_1$-pre-corner as follows:
\begin{itemize}
\item for each $\alpha_1$-pre-corner of the form $( \circ\stackrel{R_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\Sigma \stackrel{R_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$ construct the unique $\alpha_1$-corner, $(\Sigma_1\stackrel{R_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\Sigma \stackrel{R_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma_2)$. \end{itemize} \end{lemma}
\begin{proof} Let us consider an $\alpha_1$-pre-corner $\Lambda=( \circ\stackrel{R_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle S, T\rangle \stackrel{R_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ)$. The indicated $\alpha_1$-corners do exist as the indicated derivation steps exist: by construction of $\Lambda$, the two application instances $R_1,R_2$ exist (cf.\ Definition~\ref{def:rules}, p.~\pageref{def:rules}: trivial guards and the condition of not modifying head constraints guaranteed); and they can apply to $\langle S, T\rangle$ as their head constraints are in $S$, and no application record for $R_1$ or $R_2$ is in $T$. Referring to Proposition~\ref{prop:deriv-step-label-and-subst}, p.~\pageref{prop:deriv-step-label-and-subst} (functional dependency $\textit{Ancestor-state}\times\textit{Application-instance}\rightarrow\textit{Result-state}$), it is sufficient only to consider the common ancestor states of the involved (pre-) corners.
Let now $\lambda$ be an $\alpha_1$-corner as stated in the lemma. First, we show that $\lambda$ is minimal by contradiction, so assume the opposite, namely that there exists a $\lambda\succ\lambda'>\Lambda$; let $\langle s',t'\rangle$ refer to the common ancestor state of $\lambda'$. From $\lambda'>\Lambda$ it follows that $s'=s\theta'\uplus s^{\prime+}$ for some $\theta', s^{\prime+}$, and from $\lambda\succ \lambda'$ that $s=s'\theta\uplus s^{+}$ for some $\theta, s^{+}$; thus $s=s\theta\theta' \uplus s^{\prime+}\theta\uplus s^{+}$ and hence $s^{\prime+}= s^{+}=\emptyset$ and $\theta,\theta'$ are renaming substitutions.
In a similar way, we obtain \begin{itemize}
\item $t'=t\uplus t^{\prime+} \setminus t^{\prime\div}$ where any index of $t^{\prime+}$ is in $s^{\prime+}=\emptyset$, and thus $t^{\prime+}=\emptyset$, and $t^{\prime\div}\in t$ (cf.~Definitions~\ref{def:pre-corners},~\ref{def:subsumption-by-gcpc}); hence $t'=t \setminus t^{\prime\div}$,
\item $t=t'\uplus t^+ \setminus t^\div$ where any index of $t^{+}$ is in $s^{+}=\emptyset$, and thus $t^{+}=\emptyset$, and $t^{\div}\in t'$ as above; hence $t=t'\setminus t^\div = t\setminus t^{\prime\div}\setminus t^\div$. \end{itemize} It follows now that $t^\div=t'^\div=\emptyset$ and thus $\lambda=\lambda'$. Contradiction.
It remains to show that $\lambda$ is a least corner for $\Lambda$, i.e., for any $\lambda'>\Lambda$ it holds that $\lambda'\succ\lambda$. This follows from the fact that an unfolding of these two statements according to their respective definitions, inserting the same common ancestor state $\langle s,t\rangle$ of $\Lambda$ and $\lambda$, yields identical results. \end{proof}
The most significant difference in the confluence results with the different semantics appears when guards contain incomplete built-ins. This implies cases where our semantics cannot apply, but the previous ones can, and thus local confluence is a stronger property with those semantics.
\begin{example}\label{ex:different-confluence} Consider a program consisting of the following rules; invariant and equivalence are trivial and not considered.
\noindent
\hbox to 2em{$r_1$:\hfil}\verb"p(X) <=> 1 >= X, X >= -1 | q(X)"\\ \hbox to 2em{$r_2$:\hfil}\verb"p(X) <=> r(X)"\\
\hbox to 2em{$r_3$:\hfil}\verb"q(X) <=> 1 >= X, X >= 0 | r(x)"\\
\hbox to 2em{$r_4$:\hfil}\verb"q(X) <=> 0 >= X, X >= -1 | r(x)"
\noindent As discussed in Section~\ref{sec:commentOpSem} and Example~\ref{ex:motivate-pre-corners} above, the semantics of~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:conf/iclp/DuckSS07,fruehwirth-98} include a built-in store in the state, and a rule can fire when its guard is a consequence of the current built-in store. The built-in store for the common ancestor state of a critical pair is formed by the conjunction of the guards of the involved rules; ignoring global variables and propagation history, we obtain in the mentioned semantics the following critical pair, here shown with the ancestor state for ease of comparison. $$\lambda^*\;=\;\Bigl(\bigl\langle \{\texttt{q(X)}\}, (\texttt{1>=X}\land \texttt{X>=-1})\bigr\rangle\stackrel{r_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\bigl\langle \{\texttt{p(X)}\}, (\texttt{1>=X}\land \texttt{X>=-1})\bigr\rangle
\stackrel{r_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \bigl\langle \{\texttt{r(X)}\}, (\texttt{1>=X} \land \texttt{X>=-1})\bigr\rangle\Bigr) $$ This critical pair is not joinable as neither $r_3$ nor $r_4$ can apply to the left wing state since their respective guards are not consequences of the current built-in store. It follows that the program is not confluent when derivations are defined as by~\cite{DBLP:conf/cp/Abdennadher97} and others.
With our semantics, the program is confluent. There are no corners similar to $\lambda^*$ with \texttt{p}/1 having an uninstantiated variable as its argument (the guard of $r_1$ evaluates to $\mathit{error}$ so $r_1$ cannot apply). Instead we notice an infinite of family minimal corners for $r_1$ and $r_2$, one for each numeral in the interval $[-1,1]$; for example: $$\lambda^{\texttt{0.5}}=\bigl(\{\texttt{q(0.5)}\} \stackrel{R_1^{\texttt{0.5}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\{\texttt{p(0.5)}\}
\stackrel{R_2^{\texttt{0.5}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{\texttt{r(0.5)}\}\bigr)
\quad\!\mbox{and}\quad
\lambda^{\texttt{-0.5}}=\bigl(\{\texttt{q(-0.5)}\} \stackrel{R_1^{\texttt{-0.5}}}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\{\texttt{p(-0.5)}\}
\stackrel{R_2^{\texttt{-0.5}}}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \{\texttt{r(-0.5)}\}\bigr) $$ We see that $\lambda^{\texttt{0.5}}$ can be extended to a joinability diagram by an application of $r_3$ and the same for $\lambda^{\texttt{-0.5}}$ by $r_4$. Obviously, the entire family of minimal corners is joinable, and the program is confluent under our semantics. \end{example} As mentioned, we do not intend to reason about infinite sets of minimal corners when it can be avoided. The methods introduced in the following section allows reasoning about abstract corners that visually resemble $\lambda^*$ in Example~\ref{ex:different-confluence} above, but in which the combined guard constraints are interpreted as meta-level restrictions on the intended instantiations of the states involved. (Such abstract corners are allowed to split, so our abstract version of $\lambda^*$ in the example can split into two halves, one shown joinable using $r_3$ and the other by $r_4$.)
\section{Proving Confluence Modulo Equivalence using Abstract and Meta-level Constrained Corners and Diagrams}\label{sec:abstract}
The classical approach to proving local confluence for CHR~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96} is distinguished by having to consider only a finite number of cases, each characterized by a critical pair of proper CHR states. Joinability of each critical pair is then shown by applying CHR rules directly.
As shown above, this does not generalize directly to the more general context with non-logical/incomplete built-in predicates and invariants.
We introduce abstract states, that embed meta-level constraints derived from the invariant and rule guards, representing exactly the permissible states satisfying these constraints. Applications of CHR rules to abstract states are simulated with the meaning that they go only for these permissible states.
This makes it possible to describe proofs of local confluence in terms of finitely many proof cases, also for
examples where \cite{DBLP:conf/iclp/DuckSS07}
requires infinitely many.
Occasionally, we may need to split a case into sub-cases, each requiring different combinations of CHR rules for showing joinability.
Section~\ref{sec:metachr} introduces the language \textsc{MetaCHR}{}, and Section~\ref{sec:abstract-corners-and-confluence} provides our central results on how confluence modulo equivalence may be shown by considering abstract corners constructed from the most general critical pre-corners.
\subsection{A Meta-Language for CHR and its Semantics}\label{sec:metachr} In the following, we assume fixed sets of built-in and constraint predicates, invariant $I$ and state equivalence $\approx$. The following definition introduces the basic elements of \textsc{MetaCHR}\ giving a parameterized representation for CHR and notions related to its semantics. Built-in predicates of CHR are lifted into \textsc{MetaCHR}\ in two ways, firstly by a lifted version of the $\mathit{Exe}$ function (Section~\ref{sec:preliminaries}, p.~\pageref{text-def:exe}) that is extended with an extra argument intended to hold the entire head of the actual rule instance, so the condition can be checked that a guard of a CHR rule cannot modify variables in that head. Secondly, each built-in predicate is represented as a predicate of the same name in \textsc{MetaCHR}\ expressing satisfiability of a given atom. The \textsc{MetaCHR}\ predicates $\textit{all-relevant-app-recs}$ and $\textit{common-vars}$ introduced below will be used to simulate details of CHR's derivation steps.
Two denotation functions will be defined, first $\denotesgr-$ that maps a ground \textsc{MetaCHR}{} term into a specific CHR related object, and next $\denotes-$ that maps a \textsc{MetaCHR}{} term parameterized by meta-variables into all the objects that it covers (analogous to subsumption above). Be aware that when a \textsc{MetaCHR}\ term is ground, it means that it contains no \textsc{MetaCHR}\ variables, although it may denote a non-ground CHR related object. \begin{definition}\label{def:metachr} \textsc{MetaCHR}{} is a typed logical language; for given type $\tau$, $\textsc{Meta}_\tau$ ($\textsc{Meta}^{\mathit{Gr}}_\tau$) refers to the set of (ground) terms of type $\tau$. The \emph{(ground) denotation function} for each type $\tau$ is a function $$\denotesgr-\colon \textsc{Meta}^{\mathit{Gr}}_\tau\rightarrow\textsc{Chr}_\tau$$ where $\textsc{Chr}_\tau$ a suitable set of CHR related objects.
The types and terms of \textsc{MetaCHR}{} are assumed sufficiently rich such that any relevant object related to CHR is \emph{denotable}, e.g., for any CHR state $s$, there exists a ground term $t$ of \textsc{MetaCHR}{} with $\denotesgr t=s$.
Whenever $S, S_1, S_2$ are ground \textsc{MetaCHR}{} terms of type \textit{state}, \textsc{MetaCHR}{} includes the following atomic formulas: invariant statements of the form $I(S)$, equivalence (statement)s of the form $S_1\approx S_2$. We assume similarly polymorphic operators for equality and various operations related to sets such as $\in$, $\subseteq$ etc. Each such predicate has a fixed meaning defined as follows; for any sequence of ground \textsc{MetaCHR}{} $t_1,\ldots,t_n$ of \textsc{MetaCHR}{} terms of relevant types, $$ p(t_1,\ldots,t_n)\qquad\mbox{if and only if}\qquad p(\denotesgr{t_1},\ldots,\denotesgr{t_n})$$
Whenever $H$ is some ground term and $G$ a ground term of type \textit{guard}, the formula $\mexe HG$ holds if and only if \begin{itemize}
\item $\mathit{Exe}(\denotesgr G)$ is a proper substitution $\theta$,
\item $\mathit{vars}(\denotesgr H)\cap \mathit{domain}(\mathit{Exe}{\denotesgr E})=\emptyset$. \end{itemize} For each built-in predicate $\texttt{p}/n$ of CHR, \textsc{MetaCHR}\ includes a \emph{lifted} predicate $\texttt{p}/n$ whose arguments are of type \textit{term}, and which has a fixed meaning defined as follows; for ground terms $T_1,\ldots,T_n$, $\texttt{p}(t_1,\ldots,t_n)$ holds if and only if \begin{itemize}
\item $\mathit{Exe}(\texttt{p}(\denotesgr{t_1},\ldots,\denotesgr{t_n}))$ is a proper substitution. \end{itemize} Wheneover $t_1$ and $t_2$ are ground terms, the predicate $\textit{common-vars}(t_1, t_2)$ holds if and only if \begin{itemize}
\item $\mathit{vars}(\denotesgr{t_1})\cap\mathit{vars}(\denotesgr{t_2})\neq\emptyset$. \end{itemize} \textsc{MetaCHR}{} includes a lifted version of the function $\textit{all-relevant-app-recs}$ (Def.~\ref{def:state}) from terms of type \textit{constraint-store} to sets of terms of type \textit{application-record} defined such that $\textit{all-relevant-app-recs}(s)=t$ if and only if $\textit{all-relevant-app-recs}(\denotesgr s)=\denotesgr t$. \end{definition} For simplicity of notation, we assume for each predicate and function symbol, a function symbol in \textsc{MetaCHR}{} of similar arity and type \textit{term} for its arguments,
written with the same symbols. For example \texttt{p(a,X)} can be read as a ground \textsc{MetaCHR}{} term, and $\denotesgr{\texttt{p(a,X)}}=\texttt{p(a,X)}$ is a non-ground CHR term. To avoid ambiguity, \textsc{MetaCHR}{} variables are written by \textit{italic} letters; this may occasionally clash the traditional use such letters for mathematical placeholders, and we add explanations when necessary to avoid confusion.
We extend the notational principle of indicating a state by one of its representations to \textsc{MetaCHR}{} as demonstrated in the following example. \begin{example} Assume a CHR constraint predicate $\texttt{p}/2$. The following equality between CHR states holds, $$\bigdenotesgr{\bigl\langle\{1{:} \mathtt{p(X,a)}\}, \{r@1\}\bigr\rangle} = \bigdenotesgr{\bigl\langle\{2{:} \mathtt{p(Y,a)}\}, \{r@2\}\bigr\rangle}$$ and thus the \textsc{MetaCHR}{} formula $$\bigl\langle\{1{:} \mathtt{p(X,a)}\}, \{r@1\}\bigr\rangle = \bigl\langle\{2{:} \mathtt{p(Y,a)}\}, \{r@2\}\bigr\rangle$$ is true. \end{example}
The following notion of templates will be used for mapping specific CHR related objects, possibly containing variables, into a representation in $\textsc{MetaCHR}$ with new \textsc{MetaCHR}\ variables, so that application of CHR substitutions are simulated by \textsc{MetaCHR}\ substitutions. \begin{definition}\label{def:template} A \emph{template} $T'$ for a CHR related object $t$ (e.g., term, constraint, rule, state, etc.) is a \textsc{MetaCHR}\ term formed as follows: 1) find a \textsc{MetaCHR}{} term $T$ such that $\denotesgr T= t$, and 2) form $T'$ as a copy of $T$ in which all subterms that are names of CHR variables are replaced systematically by new and unused \textsc{MetaCHR}{} variables. \end{definition}
For example, the \textsc{MetaCHR}{} term \texttt{p($X$,a)} is a template for the CHR atom \texttt{p(X,a)}. Similar templates have been used in meta-interpreters for logic programs~\cite{ChristiansenTPLP2005,Hill94meta-programming-in-logic}, based on a lifting of the Prolog text into a meta-level representation in which Prolog unification is simulated by unification at the meta-level.
The following definition is central. It is the basis for defining meta-level versions of derivations, corners and diagrams parameterized by \textsc{MetaCHR}{} variables that are constrained in suitable ways.
\begin{definition}\label{def:denotation-and-abstraction} An \emph{abstraction} of type $\tau$ is a structure of the form $$A_\tau\mathrel{\mbox{\textsc{where}}} \Phi,$$ where $A_\tau\in\textsc{Meta}_\tau$ and $\Phi$ is a formula of \textsc{MetaCHR}{} referred to as a \emph{meta-level constraint}. The abstraction is \emph{ground} if and only if $A_\tau$ is ground and $\Phi$ contains no free variables. In cases where the meta-level constraint is $\mathit{true}$, we may leave it out to simplify notation, i.e., $(A_\tau\mathrel{\mbox{\textsc{where}}}\mathit{true})$ is written as $A_\tau$.
The denotation function $\denotesgr-$ is extended to ground abstractions and arbitrary structures (e.g., application instances, corners and diagrams) containing such, in the following way. \begin{itemize}
\item For any ground abstraction $A_\tau\mathrel{\mbox{\textsc{where}}} \Phi$,
$$\denotesgr{A_\tau\mathrel{\mbox{\textsc{where}}} \Phi}=
\begin{cases}
\denotesgr A & \text{whenever $\Phi$ is satisfied}, \\
\bot & \text{otherwise}.
\end{cases}
$$
\item For any structure $s(A_1,\ldots,A_n)$ including ground abstraction $A_1,\ldots,A_n$,
$$ \denotesgr{s(A_1,\ldots,A_n)} =
\begin{cases}
\bot & \text{if, for some $i$, $\denotesgr{A_i}=\bot$}, \\
s(\denotesgr{A_1},\ldots,\denotesgr{A_n}) & \text{otherwise}.
\end{cases}
$$ \end{itemize} An abstraction or structure with abstractions $\mathbf A$ is said to \emph{cover} a concrete object or structure $C$, whenever there is a grounding \textsc{MetaCHR}{} substitution $\sigma$ for which $\denotesgr{\mathbf A\sigma}=C\neq\bot$. The set of all concrete objects or structures covered by $\mathbf A$ is written $\denotes{\mathbf A}$.
An abstraction or structure with abstractions $\mathbf A$ is \emph{consistent} whenever $\denotes{\mathbf A}\neq\bot$.
Two abstractions or structures with abstractions, $\mathbf S, \mathbf S'$ are \emph{semantically equivalent} whenever, for any grounding substitution $\sigma$ that $\denotesgr{\mathbf S\sigma}= \denotesgr{\mathbf S'\sigma}$. An abstraction of type state is referred to as an \emph{abstract state.} \end{definition}
\begin{example}[Abstract States] The abstract objects shown below include lifted versions of the CHR built-ins \texttt{constant}/1 and \texttt{var}/1 introduced in Definition~\ref{def:built-ins} above. Notice in the lefthand sides that $a,x$ are variables of \textsc{MetaCHR}{}. \begin{eqnarray*} \bigdenotes{\bigl\langle \texttt{\{p($a$,$x$)}\}, \emptyset \bigr\rangle\mathrel{\mbox{\textsc{where}}}\mexe{\textit{-}}{(\texttt{constant($a$)},\texttt{var($x$)} }}&=&
\bigdenotes{\bigl\langle \{\texttt{p($a$,$x$)}\}, \emptyset \bigr\rangle \mathrel{\mbox{\textsc{where}}} \texttt{constant($a$)}\land\texttt{var($x$)}}\\
& = & \bigl\{\bigl\langle\{\texttt{p(a,X)}\}, \emptyset \bigr\rangle , \ldots, \langle \{\texttt{p(b,Y)}\}, \emptyset \rangle , \ldots\bigr\} \\
& = & \bigl\{\bigl\langle \{\texttt{p($a,x$)}\} , \emptyset \bigr\rangle \mid \mbox{$a$ is a constant, $x$ a variable}\bigr\}\\ \bigdenotes{\bigl\langle \{\texttt{p($a$)}\} , \emptyset \bigr\rangle \mathrel{\mbox{\textsc{where}}} \texttt{var($a$)} \land \texttt{const($a$)}} & = & \emptyset \end{eqnarray*} In the example above, it was possible to
turn a sequence of built-ins in a guard (the second argument of $\mexe--$) into a conjunction.
However, this does not hold in general since different orders in a guard with non-logical or incomplete predicates may give different results. \end{example}
Next, we introduce various building blocks, leading to abstract corners and joinability diagrams. \begin{definition}[Abstract $=$, $\approx$ and $I$]\label{def:abstract-eqs-etc} Let $T, T'$ be abstractions of the same type and $S, S'$ abstract states. An \emph{abstract equality} is a formula $T=T'$, an \emph{abstract invariant} a formula $I(S)$ and an \emph{abstract equivalence} a formula $S\approx S'$. Let $e(T_1,\ldots,T_n)$, $n=1,2$ be an arbitrary such formula; $e(T_1,\ldots,T_n)$ is defined to be true if and only if it
the following properties hold. \setbox3=\hbox{\textbf{(Completeness)\hbox to 0.5em{}}} \begin{itemize} \item \noindent\hbox to 1\wd3{\textbf{(Soundness)}\hfil}For any $e(t_1,\ldots,t_n)\in\denotes{e(T_1,\ldots,T_n)}$, it holds that
$e(t_1,\ldots,t_n)$ is true. \item \noindent\hbox{\textbf{(Completeness)\hbox to 0.5em{}}}For any $i=1,\ldots,n$ and any $t_i\in\denotes{T_i}$ there exists a true atom $e(t_1,\ldots,t_n)$ with\\ \noindent\hbox to 1\wd3{} $e(t_1,\ldots,t_n)\in\denotes{e(T_1,\ldots,T_n)}$. \end{itemize} An abstract state $A$ for which $I(A)$ holds is called an \emph{abstract $I$-state.} \end{definition}
Notice that we do not require the constituents of abstract statements to be consistent, which means that two inconsistent abstract states will satisfy an abstract $\approx$ statement. This is convenient for the formulation of the central Theorems~\ref{thm:abstract-CC-theorem} and~\ref{thm:termination+abstract-joinability=confluence}, below.
The following property is useful when we want to build an abstract $\beta$-corner (defined below). For any abstract state, we can always produce an equivalence statement that covers all relevant equivalences at the level of CHR; this is made precise as follows. \begin{proposition}\label{prop:equiv-state-exists} For any
abstract state $A$ there exist an abstract equivalence $A\approx A'$ such that $\denotes{A'} = \{ s'\mid \mbox{$s'\approx s$ for any $s\in\denotes A$}\}$.\end{proposition} \begin{proof} Assume an abstract state of the form $S\mathrel{\mbox{\textsc{where}}}\Phi$, $S$ some \textsc{MetaCHR}\ term of type \textit{state} and $\Phi$ a \textsc{MetaCHR}\ formula. The following abstract equivalence satisfies the proposition, where $S'$ is a new and unused \textsc{MetaCHR}\ variable. $$(S\mathrel{\mbox{\textsc{where}}}\Phi) \approx (S' \mathrel{\mbox{\textsc{where}}} S'\approx S\land \Phi).$$
\end{proof} As seen above, we have overloaded $\approx$ to simplify notation. In the following example, we will elucidate the different levels of equivalence. \begin{example}[Abstract Equivalence; Examples~\ref{ex:collect} and~\ref{ex:collect-inv}, continued] We consider again the program that collects a set of items into a list with the suggested invariant and equivalence. For simplicity, we consider here only states containing a single \texttt{set} constraint whose argument is a list of constants. The propagation history is always empty, and we can ignore both that and the indices. In this example, and only here, we add subscripts to distinguish the different versions of the equivalence symbol $\approx$: $\approx_\textrm{CHR}$, $\approx_\textrm{\textsc{MetaCHR}}$, $\approx_\textrm{ABSTRACT}$.
With these remarks, we can specify the equivalence at the level of CHR as follows: $$\{\texttt{set($\ell_1$)}\} \approx_\textrm{CHR} \{\texttt{set($\ell_2$)}\} \quad\Leftrightarrow\quad \mathit{perm}(\ell_1,\ell_2),$$ where $\mathit{perm(\ell_1,\ell_2)}$ is an auxiliary predicate that holds if and only of $\ell_1$ and $\ell_2$ are lists of constants that are permutations of each other. We will now demonstrate the construction given by the proof of Proposition~\ref{prop:equiv-state-exists} for an abstract state $(\{\texttt{set([a,b]}\}\mathrel{\mbox{\textsc{where}}} \mathit{true})$ that we will write in the short form $\{\texttt{set([a,b]}\}$. The proposition suggests the following abstract equivalence that holds between the indicated abstract states. $$\{\texttt{set([a,b])}\} \approx_\textrm{ABSTRACT} (S' \mathrel{\mbox{\textsc{where}}} S'\approx_\textrm{\textsc{MetaCHR}} \{\texttt{set([a,b])}\})$$ To clarify the meaning of $\approx_\textrm{ABSTRACT}$, we unfold the right and innermost equivalence $\approx_\textrm{\textsc{MetaCHR}}$ according to the definition, assuming a lifting of $\mathit{perm}/2$ to \textsc{MetaCHR}, and we get the following.
$$\{\texttt{set([a,b])}\} \approx_\textrm{ABSTRACT} (\{\texttt{set($\ell$)}\} \mathrel{\mbox{\textsc{where}}} \mathit{perm}(\ell,\texttt{[a,b]}\})
$$ The right-hand side, is now in a form that makes it easier to apply abstract derivations. \end{example}
With the abstract states at hand, we can now define abstract derivation steps.
\begin{definition}[Abstract Derivation Step]\label{def:abstract-derivation-step} An \emph{abstract derivation step} is an
abstraction of the form $A \stackrel{D}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} A'$ where $A$ and $A'$ are abstract $I$-states, $D$ an abstract label (i.e., abstract built-in atom or abstract application instance), with $\mathit{vars}(A')\cup\mathit{vars}(D)\setminus\mathit{vars}(A)$ being fresh and unused variables, such that the following properties hold. \setbox3=\hbox{\textbf{(Completeness)\hbox to 0.5em{}}} \begin{itemize} \item \noindent\hbox to 1\wd3{\textbf{(Soundness)}\hfil}For any $( a\stackrel{d}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} a')\in\denotes{ A \stackrel{D}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} A'}$, it holds that
$a\stackrel{d}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} a'$ is a concrete derivation step. \item \noindent\hbox{\textbf{(Completeness)\hbox to 0.5em{}}}For any $a\in\denotes{ A}$ (for any $a'\in\denotes{A'}$) there exists a concrete derivation step\\ \noindent\hbox to 1\wd3{}$( a\stackrel d\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} a')\in\denotes{ A\stackrel D\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} A'}$. \end{itemize} \emph{Abstract $I$-derivations} are defined in the usual way as a sequence of zero or more abstract $I$-derivation steps. \end{definition} An abstract derivation step is intended to cover a set of concrete derivation steps, but unintended variable clashes in the abstract derivation step can cause undesired limitations on those. This is avoided with the requirement of fresh and unused variables. The second part of the completeness condition is relevant when we compose derivations and diagrams. If $\denotes{A'}$ includes an element $a'$ for which the indicated step does not exist, $A'$ is so to speak too big, and the next step (or equivalence statement) from $A'$ would have to take care of too many irrelevant concrete states.
The following property follows immediately from the Definition~\ref{def:abstract-derivation-step}.
\begin{proposition}[Abstract Derivation]\label{prop:abstract-derivation-covering} For any abstract $I$-derivation $$\Xi =(A_0\stackrel{D_1}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} A_1\stackrel{D_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\cdots\stackrel{D_n}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} A_n) \quad,\; n\geq0.$$ the following properties hold.\setbox3=\hbox{\textbf{(Completeness)\hbox to 0.5em{}}} \begin{itemize} \item \noindent\hbox to 1\wd3{\textbf{(Soundness)}\hfil}Any element of $\denotes{\Xi}$ is a concrete derivation. \item \noindent\hbox{\textbf{(Completeness)\hbox to 0.5em{}}}For any $a_0\in\denotes{ A_0}$ (for any $a_n\in\denotes{ A_n}$) there exists a concrete derivation\\
\noindent\hbox to 1\wd3{}$( a_0\stackrel{d_1}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} a_1\stackrel{d_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\cdots\stackrel{d_n}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} a_n)\in\denotes{ \Xi}$. \end{itemize} \end{proposition}
\begin{proof} By induction. Base case $n=0$ is trivial; the step follows directly from Definition~\ref{def:abstract-derivation-step}. \end{proof}
Analogously to concrete corners, abstract corners are constructed using abstract derivations, abstract equivalence and abstract invariants, as follows.
\begin{definition}[Abstract Corners]\label{def:variable-regular-beta-corner} An \emph{abstract $I$-corner} is a structure of the form $(A_1 \mathrel{\mathit{Rel}_1} A \mathrel{\mathit{Rel}_2} A_2)$ where $(A_1 \mathrel{\mathit{Rel}_1} A)$ is an abstract $I$-derivation step or abstract equivalence, and $(A \mathrel{\mathit{Rel}_2} A_2)$ an abstract $I$-derivation step such that $\mathit{vars}(A_1)\cap\mathit{vars}(A_2)\subseteq\mathit{vars}(A)$. \end{definition}
We will refer to an abstract corner as an abstract $\alpha_1$-, $\alpha_2$-, $\alpha_3$-, $\beta_1$- or $\beta_2$-corner according to the relationships involved, analogous to what we have done for concrete corners (Definition~\ref{def:critical}, above). The following property is a consequence of what we have shown so far. \begin{proposition}\label{prop:abstract-corner-covering} For any abstract $I$-corner $$\mathbf{\Lambda} =(A_1 \mathrel{\mathit{Rel_1}} A \mathrel{\mathit{Rel_2}} A_2),$$ the following properties hold. \setbox3=\hbox{\textbf{(Completeness)\hbox to 0.5em{}}} \begin{itemize} \item \noindent\hbox to 1\wd3{\textbf{(Soundness)}\hfil}Any element of $\denotes{\mathbf{\Lambda}}$ is a concrete corner. \item \noindent\hbox{\textbf{(Completeness)\hbox to 0.5em{}}}For any $a\in\denotes{ A}$ (for any $a_1\in\denotes{ A_1}$) (for any $a_2\in\denotes{ A_2}$) there exists a concrete corner\\ \noindent\hbox to 1\wd3{}$( a_1 \mathrel{\mathit{rel}_1} a \mathrel{\mathit{rel}_2} a_2)\in\denotes{\mathbf{\Lambda}}$ \end{itemize} \end{proposition}
\begin{proof} The proposition is an immediate consequence of the soundness and completeness conditions in Definitions~\ref{def:abstract-eqs-etc} and~\ref{def:abstract-derivation-step}. \end{proof}
Finally, we introduce abstract joinability diagrams for abstract corners, allowing us to treat a perhaps infinite set of corners in a single proof case.
\begin{definition} An \emph{abstract joinability diagram} (modulo $\approx$) for an abstract $I$-corner is a structure of the form $$\mathbf{\Lambda} =( A_1 \mathrel{\mathit{Rel_1}} A \mathrel{\mathit{Rel_2}} A_2),$$ is a structure of the form $$\boldsymbol{\Delta}=(A'_1 \stackrel{*}\mapsfrom A_1 \mathrel{\mathit{Rel_1}} A \mathrel{\mathit{Rel_2}} A_2 \stackrel{*}\ourmapsto A_2')$$ where $A_1'\stackrel{*}\mapsfrom A_1$ and $A_2 \stackrel{*}\ourmapsto A_2'$ are abstract derivations such that the abstract equivalence
$A_1'\approx A_2'$ holds. A given abstract corner is \emph{(abstractly) joinable modulo $\approx$} whenever there exists an abstract joinability diagram for it. \end{definition}
\begin{proposition}\label{prop:abstract-diagram-covering} Let $\boldsymbol{\Delta}$ be an
abstract joinability diagram for an
abstract $I$-corner $\mathbf{\Lambda}$. Then the following properties hold. \setbox3=\hbox{\textbf{(Completeness)\hbox to 0.5em{}}} \begin{itemize} \item \noindent\hbox to 1\wd3{\textbf{(Soundness)}\hfil}Any element of $\denotes{\boldsymbol{\Delta}}$ is a concrete joinability diagram. \item \noindent\hbox{\textbf{(Completeness)\hbox to 0.5em{}}}For any $\lambda\in\denotes{{\mathbf{\Lambda}}}$
there exists a concrete joinability diagram for $\lambda$ in $\denotes{\boldsymbol{\Delta}}$. \end{itemize} \end{proposition}
\begin{proof} The proposition is an immediate consequence of soundness and completeness properties given by Propositions~\ref{prop:abstract-derivation-covering} and~\ref{prop:abstract-corner-covering} and Definition~\ref{def:abstract-eqs-etc}. \end{proof}
Combining this with the Critical Corner Theorem, Theorem~\ref{thm:critCorner}, p.~\pageref{thm:critCorner}, we get immediately the following.
\begin{lemma}[Abstract Corner Lemma]\label{thm:abstractCornerLemma} Assume a program $\Pi$ with invariant $I$ and state equivalence relation $\approx$, and let $\mathcal D$ be a family of abstract $I$-corner that together covers all concrete corners that are subsumed by some general critical pre-corner for $\Pi$. Then
$\Pi$ is locally confluent modulo $\approx$
if and only if
all diagrams in $\mathcal D$ is joinable.
\end{lemma} In the following, we consider how to construct a family of abstract corners as required in Lemma~\ref{thm:abstractCornerLemma}.
\subsection{Proving Confluence Modulo Equivalence using Abstract Joinability Diagrams}\label{sec:abstract-corners-and-confluence}
Here we will show how a set of most general critical pre-corners can be lifted to a set of abstract corners, and we identify necessary and sufficient conditions for confluence modulo equivalence. A program with invariant $I$ and equivalence $\approx$ is assumed.
A pre-corner (Definition~\ref{def:pre-corners}, p.~\pageref{def:pre-corners}) is a common ancestor state whose wing states are indirectly characterized by their relationships to the ancestor state. Our way to lift it, to be defined below, consists of first lifting the common ancestor state, and then applying abstract versions of the indicated relationships (i.e., rule application, built-in or equivalence) to obtain abstract wing states, constrained at the meta-level by restrictions induced by guards and invariant.
As part of this, we need the following construction, which, for a given abstract ancestor state and type of derivation step, provides the resulting abstract state. For convenience, we combine derivation steps and $\approx$. We recall Proposition~\ref{prop:deriv-step-label-and-subst}, p.~\pageref{prop:deriv-step-label-and-subst}, stating for the concrete case, that there is at most one resulting state for a derivation step.
\begin{definition}~\label{def:post} Let $A$ be an abstract $I$-state and $\mathit{Rel}$ either an abstract derivation step or $\approx$.
An \emph{abstract post state for $A$ with respect to $\mathit{Rel}$} is an abstract state $A'$ such that $A \mathrel{\mathit{Rel}} A'$ holds. Such a state $A'$ is indicated as $\mathit{POST}(A,\mathit{Rel})$. \end{definition}
Proposition~\ref{prop:equiv-state-exists}, p.~\pageref{prop:equiv-state-exists}, shows that $\mathit{POST}(A,\approx)$ can be found in a straightforward manner, although in practice it may be useful to unfold the definition of $\approx$.
For the definition to be useful for derivation steps, we assume that $\textsc{MetaCHR}$ is sufficiently rich as to express an abstract state $\mathit{POST}(A,\mathit{Rel})$.
One way to obtain this is to include $\mathit{POST}$ as a function in the language, whose meaning were defined semantically as indicated in Definition~\ref{def:post}, but it will be more useful to define a procedure that produces an abstract state in terms of plain \textsc{MetaCHR}{} predicates and terms.
A general $\mathit{POST}$ procedure that can handle all built-in predicates will be quite complex; Drabent's~\cite{drabent-report-1997} analysis of a predicate transformer for unification of arbitrary terms demonstrates this. However, in many cases, the invariant and the selection state built-ins reduce the complexity. In all the examples we have considered, it has been straightforward to produce all necessary post states by hand; see, e.g., the larger example in Section~\ref{sec:viterbi} below.
\begin{example}\label{ex:drabent} Let $\Sigma$ be the abstract state $(\langle \{{\tt p}(x), x \; {\tt is }\; y\}, \emptyset \rangle \mathrel{\mathrel{\mbox{\textsc{where}}}} variable(x))$, and we will construct a state $\mathit{POST}(\Sigma, x\; {\tt is }\; y)$.
This example is especially tricky as the built-in is incomplete and there are no restrictions on $y$, so the post state should capture both the $\mathit{error}$ and proper states. We solve the problem, suggesting the following state; we assume two auxiliary \textsc{MetaCHR}\ predicates $\mathit{arithmetic}(t)$ being true for any ground $t$ for which $\denotesgr t$ can be evaluated as an arithmetic expression, and $\mathit{eval}(t_1,t_2)$ being true for any ground $t_1,t_2$ for which $\denotesgr{t_1}$ can be evaluated as an arithmetic expression with value $\denotesgr{t_2}$.
\begin{align*}
\langle \mathit{S}, \emptyset \rangle \mathrel{\mathrel{\mbox{\textsc{where}}}} &\; \big( \mathit{S} = \{{\tt p}(y')\} \land \mathit{eval}(y,y') \land \mathit{arithmetic}(y) \big)\quad\lor \\
&\; \big( \mathit{S} = \{\mathit{error}\} \land \neg \mathit{arithmetic}(y) \big) \end{align*} Here, the meta-variable $x$ has been replaced by $y'$, which represents the value of the arithmetic expression $y$. Notice that no single rule can apply to this state, and if it happens to arise in an attempt to produce a joinability diagram, we need to apply the notion of splitting introduced below in Definition~\ref{def:splitting}. \end{example}
The following lifting of a pre-corner into an abstract corner is straightforward, although quite detailed as it includes meta-level versions of conditions for subsumption by pre-corner (Definition~\ref{def:subsumption-by-gcpc}, p.~\pageref{def:subsumption-by-gcpc}). As we see in our examples, the detailed conditions often reduce to something much simpler, so the definition below represents, so to speak, worst cases.
\begin{definition}[Lifting Most General Critical Pre-Corners into Abstract Critical $I$-corners]\label{def:lifting-pre-corners}~\\ An \emph{abstract critical $I$-corner for} a most general critical pre-corner $\Lambda = (\circ\mathrel{\mathit{rel}_1} \langle s_0, t_0 \rangle \mathrel{\mathit{rel}_2}\circ)$ is of the form $$\mathbf{\Lambda}\; =\; \bigl(\mathit{POST}(A,\mathit{Rel}_1) \;\;\mathrel{\mathit{Rel}_1} \;\; A \;\; \mathrel{\mathit{Rel}_2} \;\mathit{POST}(A,\mathit{Rel}_2)\bigr), $$ where $A$ is an abstract state, and $\mathit{Rel}_1, \mathit{Rel}_2$ abstract derivation steps or $\approx$, specified as follows. Let firstly
$(\circ\mathrel{\mathit{Rel}_1} \langle S_0, T_0 \rangle \mathrel{\mathit{Rel}_2}\circ)$ be a template (Def.~\ref{def:template}) for $\Lambda$. The construction of $A$ depends on relationships ${\mathit{Rel}_1}$, ${\mathit{Rel}_2}$, that determine whether the corner is of type $\alpha_1$, $\alpha_2$, etc.
In case ${\mathit{Rel}_k}$, $k=1,2$ is an application instance, we assume the notation $ (r\colon\; H_k \;\mathtt{<=>}\; G_k\mid C_k)$. The symbols $S^+, T^+, T^\div$ are fresh and unused variables, and let $S$ stand for $( S_0 \uplus S^+)$ and $T$ for $T_0 \uplus T^+ \setminus T^{\div}$. (For the reading of the following, keep in mind that $A, S, T, S_0, T_0, \mathit{Rel}_1, \mathit{Rel}_2, H_k,G_k,C_k$ are not \textsc{MetaCHR}\ variables, but mathematical placeholders. The predicates used below are elements of \textsc{MetaCHR}{} (Def.~\ref{def:metachr}))
The common ancestor state $A$ is given as follows for the different cases. \begin{description}\def\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}{\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}}
\item[$\alpha_1$:\phantom{$, \beta_1$}] $\langle S, T\rangle \mathrel{\mbox{\textsc{where}}}\;
I(\langle S, T\rangle) \land \mexe{H_1}{G_1} \land \mexe{H_2}{G_2} \land \mbox{}$\\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^+ \subseteq \textit{all-relevant-app-recs}(S_0 \uplus S^+) \setminus \textit{all-relevant-app-recs}(S_0) \land \mbox{}$\\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^{\div} \subseteq \textit{all-relevant-app-recs}(S_0)\setminus\{\mathrm{applied}(\mathit{Rel}_1), \mathrm{applied}(\mathit{Rel}_2) \} $ \\
\item[$\alpha_2$:\phantom{$, \beta_1$}]
$\langle S, T\rangle \mathrel{\mbox{\textsc{where}}}\; I(\langle S, T\rangle) \land \mexe{H_1}{G_1} \land\mbox{} $\\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^+ \subseteq \textit{all-relevant-app-recs}(S_0 \uplus S^+) \setminus \textit{all-relevant-app-recs}(S_0) \land\mbox{} $\\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^{\div} \subseteq \textit{all-relevant-app-recs}(S_0)\setminus\{\mathrm{applied}(\mathit{Rel}_1)\} \land \mbox{}$ \\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$\textit{common-vars}(G_1,\mathit{Rel}_2) $ \\
\item[$\alpha_3, \beta_2$:]
$ \langle S, T\rangle \mathrel{\mbox{\textsc{where}}}\; I(\langle S, T\rangle) \land\mbox{} $\\ \ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^+ \subseteq \textit{all-relevant-app-recs}(S_0 \uplus S^+) \setminus \textit{all-relevant-app-recs}(S_0) \land \mbox{}$\\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^{\div} \subseteq \textit{all-relevant-app-recs}(S_0) $ \\
\item[$\beta_1$:\phantom{$, \beta_1$}]
$\langle S, T\rangle \mathrel{\mbox{\textsc{where}}}\; I(\langle S, T\rangle) \land \mexe{H_1}{G_1} \land\mbox{} $\\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^+ \subseteq \textit{all-relevant-app-recs}(S_0 \uplus S^+) \setminus \textit{all-relevant-app-recs}(S_0) \land \mbox{}$\\
\ind
\vrule width 0pt height 2.5ex}\def\ind{\hbox to 8.5em{}$T^{\div} \subseteq \textit{all-relevant-app-recs}(S_0)\setminus\{\mathrm{applied}(\mathit{Rel}_1)\} $ \\
\end{description}
Whenever $\mathbf{\Lambda}$ is constructed as above, a semantically equivalent abstract corner $\mathbf{\Lambda}'$ may also be recognized as an abstract critical $I$-corner for $\Lambda$. By \emph{a set of abstract critical corners for program $\Pi$} we mean a set consisting of one and only one abstract critical corner for each most general pre-corner for $\Pi$. \end{definition} We notice that the set of all abstract critical $I$-corners for a program $\Pi$ is finite as there exist only a finite number of most general critical pre-corners for $\Pi$, cf.~Proposition~\ref{prop:precorners-finite}.
\begin{proposition}\label{prop:abstract-critical-corners-finite} For any given program $\Pi$ with invariant $I$ and equivalence $\approx$, a set of abstract critical corners for it is finite. \end{proposition}
\begin{lemma}[Cover by Abstract Critical Corner $\Leftrightarrow$ Subsumed by Most Gen.~Crit.~Pre-Corner]\label{lem:cover-subsume} For given program $\Pi$, let $\mathbf{\Lambda}$ be an abstract critical $I$-corner for a most general critical pre-corner ${\Lambda}$. Then the set of $I$-corners covered by $\mathbf{\Lambda}$ is identical to the set of $I$-corners subsumed by ${\Lambda}$. Furthermore, $$\{\lambda \mid \mbox{$\exists$ abs.\ crit.\ corner $\mathbf{\Lambda}$ for $\Pi$ .\ $\lambda\in\denotes{\mathbf{\Lambda}}$}\}
= \{\lambda\mid \mbox{$\exists$ most gen.\ critical corner $\Lambda$ for $\Pi $ .\ $\Lambda < \lambda$}\}
$$ \end{lemma}
\begin{proof} The second part is a direct consequence of the first part.
For the first part, consider firstly an $I$-corner $\lambda$ that is subsumed by a most general critical pre-corner $\Lambda$ (with notation as in Definition~\ref{def:lifting-pre-corners} for each case of $\alpha_1$-, $\alpha_2$-, etc.\ corners); we prove that it is covered by the abstract critical corner $\mathbf{\Lambda}$ for $\Lambda$ as given by the lemma as follows.
Subsumption means (by Definition~\ref{def:subsumption-by-gcpc}, p.~\pageref{def:subsumption-by-gcpc}) that there exists a CHR substitution $\theta$ and suitable sets $s^{+}, t^{+}, t^{\div}$ and states $\mathit{post}_1, \mathit{post}_2$ such that $\lambda = \bigl(\mathit{post}_1 \mathrel{(\mathit{rel}_1\theta)} \langle s_0\theta \uplus s^+ , \; s_0 \uplus t^{+} \setminus t^{\div} \rangle
\mathrel{(\mathit{rel}_2\theta)} \mathit{post}_2\bigr)$ and the following conditions hold. \begin{itemize} \item $s=s_0\theta\uplus s^+$ \item $t=t_0\uplus t^+ \setminus t^\div$ \item $t^+\subseteq\textit{all-relevant-app-recs}(s_0\theta\uplus s^+)\setminus\textit{all-relevant-app-recs}(s_0\theta)$ \item $t^\div\subseteq \textit{all-relevant-app-recs}(s_0\theta)$ \end{itemize} Let now $\sigma$ be a \textsc{MetaCHR}\ substitution such that $\denotesgr{S^+\sigma} = s^+$, $\denotesgr{T^+\sigma} = t^+$ and $\denotesgr{T^\div\sigma} = t^\div$. Since $\langle S_0,T_0\rangle$ is a template for $\langle s_0,t_0\rangle$, and $S^+,T^+,T^\div$ do not occur in $\langle S_0,T_0\rangle$, we can extend $\sigma$ such that $\denotesgr{\langle S_0,T_0\rangle\sigma}=\langle s_0,t_0\rangle$. It remains, for each possible case of the corners being $\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta_1$ or $\beta_2$, to show that $\Phi\sigma$ holds where $A = (\langle S,T\rangle\mathrel{\mbox{\textsc{where}}}\Phi)$ (i.e., $\Phi$ stands for the relevant of the alternative, detailed conditions in Definition~\ref{def:lifting-pre-corners}, above). This can be verified by inspection in each case, referring to \begin{enumerate}
\item the fact that $\lambda$ is an $I$-corner, meaning that the two relationships $\bigl(\mathit{post}_1 \mathrel{(\mathit{rel}_1\theta)} \langle s_0\theta \uplus s^+ , \; t_0 \uplus t^{+} \setminus t^{\div} \rangle\bigr)$ and $\bigr( \langle s_0\theta \uplus s^+ , \; t_0 \uplus t^{+} \setminus t^{\div} \rangle \mathrel{(\mathit{rel}_2\theta)} \mathit{post}_2\bigr)$ do hold, i.e., their state arguments each satisfy the additional conditions, being the relevant of a derivation step (Definitions~\ref{def:rules} and~\ref{def:derivations}, p.~\pageref{def:rules}-\pageref{def:derivations}) or an equivalence statement, and
\item the completeness parts of Definitions~\ref{def:abstract-eqs-etc} and~\ref{def:abstract-derivation-step},
and the relationship between $\mathit{Exe}$ and $\widehat\exe$ (Definition~\ref{def:metachr}, p.~\pageref{def:metachr}). \end{enumerate} The detailed arguments are left out as the $\Phi$ formula in each case is a straightforward lifting of the similar conditions at the level of CHR.
The other way round, consider an $I$-corner $\lambda=(a_1\mathrel{\mathit{rel}_1'} \langle s, t \rangle \mathrel{\mathit{rel}_2'}a_2)$ covered by an abstract critical corner $\mathbf{\Lambda}$ (with notation as in Definition~\ref{def:lifting-pre-corners} for each case of $\alpha_1$-, $\alpha_2$-, etc.\ corners); we prove that it is subsumed by the most general critical pre-corner $\Lambda$ given by the lemma (and notation as in Definition~\ref{def:lifting-pre-corners}) as follows.
Covering means that there exists a grounding $\textsc{MetaCHR}$ substitution $\sigma$ such that $\denotesgr{\langle S,T\rangle\sigma}=\denotesgr{\langle S_0\uplus S^+,T\uplus T^+\setminus T^\div\rangle\sigma} = \langle s, t\rangle$, and, for $k=1,2$, $\denotesgr{\mathit{Rel}_k}=\mathit{rel}'_k$ and $\denotesgr{\mathit{POST}(\mathit{Rel}_k)\sigma}=a_k$, and the meta-level constraint part of $A\sigma$ holds.
Now $\langle S_0,T_0\rangle$ is defined as a template for $\langle s_0,t_0\rangle$, so there exists a \textsc{MetaCHR}\ substitution $\sigma'$ such that $\denotesgr{\langle S,T\rangle\sigma'}= \langle s_0,t_0\rangle$, and thus we can find a CHR substitution $\theta$ such that $\langle s_0,t_0\rangle\theta=\langle s, t \rangle$.
Let now $s^+=\denotesgr{S^+\sigma}$, $T_0=\denotesgr{t_0\sigma}$, $T^+=\denotesgr{t^+\sigma}$ and $T^\div=\denotesgr{t^\div\sigma}$; it follows that $s=s_0\theta\uplus s^+$, $t=t^0\uplus t^+\setminus t^\div$ and, by definition of the \textsc{MetaCHR}\ version of $\textit{all-relevant-app-recs}$,
that $t^+\subseteq\textit{all-relevant-app-recs}(s_0\theta\uplus s^+)\setminus\textit{all-relevant-app-recs}(s_0)$ and $t^\div\subseteq \textit{all-relevant-app-recs}(s_0\theta)$.
We have now that $s=s_0\theta\uplus s^+$ and $t=t_0\uplus t^+\setminus t^\div$, and together with the soundness parts of Definitions~\ref{def:abstract-eqs-etc} and~\ref{def:abstract-derivation-step}, and the relationship between $\mathit{Exe}$ and $\widehat\exe$ (Definition~\ref{def:metachr}, p.~\pageref{def:metachr}), it follows that $\lambda$ is subsumed by $\Lambda$. \end{proof} We notice the following straight-forward property, indicating that we can use existing, automatic confluence checkers (e.g.,~\cite{Raiser-Langbein2010}) to classify further abstract corners as ``trivially joinable'', so only those abstract corners whose joinability critically depend on $I$ and $\approx$ need to be considered.
\begin{proposition}~\label{prop:reuse-old-conf-checkers} Consider a program with invariant $I$ and equivalence $\approx$, and with only logical and $I$-complete built-ins, and let $\mathbf{\Lambda}$ be an abstract critical $\alpha_1$-$I$-corner lifted from a most general critical pre-corner $\circ \mathrel{\stackrel{R_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}} \Sigma \mathrel{\stackrel{R_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}} \circ$. If the concrete corner $\Sigma_1 \mathrel{\stackrel{R_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}} \Sigma \mathrel{\stackrel{R_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}} \Sigma_2$ exists and is joinable modulo $=$ with invariant $\mathit{true}$, $\mathbf{\Lambda}$ is joinable modulo $\approx$ with invariant $I$. \end{proposition}
It does not hold that a program is confluent modulo $\approx$ if and only if all of its abstract critical pairs are joinable. This is demonstrated by the following example. \begin{example}[Continuing Example~\ref{ex:different-confluence}, p.~\pageref{ex:different-confluence}]\label{ex:different-confluence-abstract} Consider the program of Example~\ref{ex:different-confluence}; its two first rules leads to the following abstract critical corner $\mathbf{\Lambda}^{r_1,r_2}$ (there are no propagation rules, so we leave out the propagation history). $$ \bigl(\{\texttt{q}(x)\} \mathrel{\mbox{\textsc{where}}} \texttt{1>=}x\land x\texttt{>=-1}\bigr) \;\stackrel{r_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}\;
\bigl( \{\texttt{p}(x)\} \mathrel{\mbox{\textsc{where}}} \texttt{1>=}x\land x\texttt{>=-1}\bigr)
\; \stackrel{r_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}} \; \bigl( \{\texttt{r}(x)\} \mathrel{\mbox{\textsc{where}}}\texttt{1>=}x, x\texttt{>=-1}\bigr) $$ We recall the two remaining rules of this program, that may perhaps apply to a state consisting of a single \texttt{q} atom.
\noindent
\hbox to 2em{$r_3$:\hfil}\verb"q(X) <=> 1 >= x, x >= 0 | r(x)"\\
\hbox to 2em{$r_4$:\hfil}\verb"q(X) <=> 0 >= x, x >= -1 | r(x)"
\noindent It appears that none of these rules can apply to the left wing state so $\mathbf{\Lambda}^{r_1,r_2}$ is not joinable, although any concrete corner covered by it is joinable. \end{example}
This phenomenon which is induced by the presence of non-logical and incomplete predicates motivates the following.
\begin{definition}[Split-joinability]\label{def:splitting} Assume a set of \textsc{MetaCHR}{} formulas, $\{ \Phi_{i}\mid i\in \mathit{Inx}\}$, for some finite or infinite index set $\mathit{Inx}$, such that $$\Phi \Leftrightarrow \bigvee_{i\in \mathit{Inx}}\Phi_{i}.$$ A \emph{splitting} of an abstract corner $$A' \;\mathrel{\mathit{Rel}_1} \;(\Sigma \mathrel{\mbox{\textsc{where}}} \Phi_i) \; \mathrel{\mathit{Rel}_2} A''$$ is the set of abstract corners $$ \bigl\{\bigl(\mathit{POST}((\Sigma \mathrel{\mbox{\textsc{where}}} \Phi_i), \mathit{Rel}_1)\; \mathrel{\mathit{Rel}_1} \;(\Sigma \mathrel{\mbox{\textsc{where}}} \Phi_i) \; \mathrel{\mathit{Rel}_2} \; \mathit{POST}((\Sigma \mathrel{\mbox{\textsc{where}}} \Phi_i), \mathit{Rel}_2)\bigr) \mid i\in \mathit{Inx} \bigr\}.$$ An abstract corner is \emph{split-joinable} modulo $\approx$ whenever it has a splitting $\{\mathbf{\Lambda}_i \mid i\in \mathit{Inx}\}$ such that each $\mathbf{\Lambda}_i$ is either inconsistent or joinable modulo $\approx$. \end{definition} The following property follows immediately from the definition. \begin{proposition}\label{prop:splitting-with-recursion} For any splitting of an abstract corner $\mathbf{\Lambda}$ into $\{ \mathbf{\Lambda}_i\mid i\in \mathit{Inx}\}$, it holds that $$ \denotes{\mathbf{\Lambda}} = \bigcup_{i\in \mathit{Inx}} \denotes{\mathbf{\Lambda}_i}. $$ \end{proposition}
\begin{example}[continuing Example~\ref{ex:different-confluence-abstract}] The non-joinable abstract critical corner $\mathbf{\Lambda}^{r_1,r_2}$ is split-joinable using the disjunction $(\texttt{1>=}x\land x\texttt{>=0})\;\lor\;
(\texttt{0>=}x\land x\texttt{>=-1})$. Notice that neither $\mathbf{\Lambda}^{r_1,r_2}$ nor any member of its splitting covers a concrete corner of the form $(\cdots\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \{\texttt{p(X)}\} \mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\cdots)$, where \texttt{X} is a CHR variable. \end{example} We notice that the invariant of groundedness does not in itself make splitting necessary, see, e.g., the example in Section~\ref{sec:set-example-all-details}, below. In Section~\ref{sec:infinite-split} below, we show an example of a program that needs an infinite splitting, but still we can use the results in the present section to show confluence.
\begin{theorem}[Abstract Critical Corner Theorem]\label{thm:abstract-CC-theorem} A CHR program with invariant $I$ and equivalence $\approx$ is locally confluent modulo $\approx$ if and only if each of its abstract critical $I$-corners is either inconsistent, joinable modulo $\approx$, or split-joinable modulo $\approx$. \end{theorem} \begin{proof} Follows immediately from Critical Corner Theorem, i.e., Theorem~\ref{thm:critCorner}, p.~\pageref{thm:critCorner}, Lemma~\ref{lem:cover-subsume} and Proposition~\ref{prop:splitting-with-recursion}. \end{proof} Combining this result with Theorem~\ref{thm:conflu-and-critCorner}, p.~\pageref{thm:conflu-and-critCorner}, we arrive at our following central result.
\begin{theorem}\label{thm:termination+abstract-joinability=confluence}
A terminating program with invariant $I$ and equivalence relation $\approx$ is confluent
if and only if
each of its abstract critical $I$-corners is either inconsistent, joinable modulo $\approx$, or split-joinable modulo $\approx$. \end{theorem}
\section{Examples}\label{sec:examples} We show three examples of confluence proofs. First, we give all details for the very simple but highly motivating example appearing in the Introduction of this paper. Next, we consider a more complex program, the Viterbi algorithm expressed in CHR, for which we formalize invariant and equivalence and give the proof of confluence modulo equivalence. This is a practically interesting algorithm, and the example also demonstrates that our framework can deal with nontrivial reasoning about the propagation history.
Finally, we show an example that our method is robust for some cases where a countably infinite splitting is needed, Section~\ref{sec:infinite-split}.
Confluence modulo equivalence of a CHR version of the union-find algorithm~\cite{Tarjan:1984:WAS:62.2160}, which has been used as a test case for aspects of confluence, is demonstrated informally by~\cite{DBLP:conf/lopstr/ChristiansenK14}. A detailed analysis and proof in terms of abstract critical corners is planned to appear in a future publication.
\subsection{The Motivating One-line Program shown Confluent Modulo Equivalence}\label{sec:set-example-all-details} In the Introduction, we motivated confluence modulo equivalence for CHR by a program consisting of the following single rule. \begin{verbatim}
set(L), item(A) <=> set([A|L]). \end{verbatim} Here we formalize the invariant $I$ and equivalence $\approx$ hinted in Examples~\ref{ex:collect} and ~\ref{ex:collect-inv}, p.~\pageref{ex:collect}-\pageref{ex:collect-inv}, and give a proof of confluence modulo $\approx$. \begin{itemize} \item $I(\langle S, T \rangle)$ if and only if \begin{itemize}
\item $S = \{{\tt set}(L)\} \uplus Items$ where $L$ is a list of constants, $Items$ is a set of {\tt item}/1 constraints
whose arguments are constants,
\item $T = \emptyset$. \end{itemize} \item $\langle S,T\rangle \approx \langle S',T'\rangle$, if and only if
\begin{itemize}
\item $I(\langle S,T\rangle)$ and $I(\langle S',T'\rangle)$,
\item $S = \mbox{\tt set$(L)$} \uplus \mathit{Items}$ and $S' = \mbox{\tt set$(L')$} \uplus \mathit{Items}$ such that $L$ and $L'$ are permutations of each other.
\end{itemize}
\end{itemize} We identify the following two most general critical pre-corners for the program. To give a complete picture, we have not abbreviated the application instances that label the derivation steps, as we do in most other examples.
$$ \begin{array}{@{}c@{}} \Lambda_1 = \;\;
\bigl(\circ
\stackrel{\scriptsize \mbox{\tt set(L),item(A)$\,$<=>$\,$set([A|L])}}{\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}}
\langle \{\mbox{\tt set(L)}, \mbox{\tt item(A)}, \mbox{\tt item(B)}\}, \emptyset \rangle
\stackrel{\scriptsize \mbox{\tt set(L),item(B)$\,$<=>$\,$set([B|L])}}{\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}}
\circ \bigr)
\\
\Lambda_2 = \;\;
\bigl(\circ
\stackrel{\scriptsize \mbox{\tt set(L1),item(A)$\,$<=>$\,$set([A|L1])}}{\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}}\;
\langle \{\mbox{\tt set(L1)},\mbox{\tt item(A)}, \mbox{\tt set(L2)}\}, \emptyset \rangle \;
\stackrel{\scriptsize \mbox{\tt set(L2),item(A)$\,$<=>$\,$set([A|L2])}}{\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}}
\circ \bigr)
\end{array} $$ We lift $\Lambda_1, \Lambda_2$ to the following abstract critical $I$-corners according to Definition~\ref{def:lifting-pre-corners}, p.~\pageref{def:lifting-pre-corners}. Trivially satisfied meta-level constraints are removed.
$$ \scriptsize
\begin{array}{@{}l@{}}
\mathbf{\Lambda_1} =\;\;\;\;
\begin{array}{c}
\xymatrix{
{\begin{array}{l}
\langle \{\mbox{\tt set}(\ell), \mbox{\tt item}(a), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}(\ell), \mbox{\tt item}(a), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle)
\end{array}}
\ar@{|->}[dr]^(.6){\qquad\qquad\qquad \mbox{\scriptsize\tt set($\ell$),item($b$)$\,$<=>$\,$set($[b|\ell]$)}}
\ar@{|->}[d]_-{\mbox{\scriptsize\tt set($\ell$),item($a$)$\,$<=>$\,$set($[a|\ell]$)}} & \\
{\begin{array}{l}
\langle \{\mbox{\tt set}([a|\ell]), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}([a|\ell]), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle)
\end{array}}
&
{\begin{array}{l}
\langle \{\mbox{\tt set}([b|\ell]), \mbox{\tt item}(a)\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}([b|\ell]), \mbox{\tt item}(a)\} \uplus S, \emptyset \rangle)
\end{array}}
\\ }
\end{array} \\ \\ \\
\mathbf{\Lambda_2} =\;\;\;\;
\begin{array}{c}
\xymatrix{
{\begin{array}{l}
\langle \{\mbox{\tt set}(\ell_1), \mbox{\tt set}(\ell_2), \mbox{\tt item}(a)\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}(\ell), \mbox{\tt set}(\ell_2), \mbox{\tt item}(a)\} \uplus S, \emptyset \rangle)
\end{array}}
\ar@{|->}[dr]^(.6){\qquad\qquad\qquad \mbox{\scriptsize\tt set($\ell_2$),item($a$)$\,$<=>$\,$set($[a|\ell_2]$)}}
\ar@{|->}[d]_-{\mbox{\scriptsize\tt set($\ell$),item($a$)$\,$<=>$\,$set($[a|\ell]$)}} & \\
{\begin{array}{l}
\langle \{\mbox{\tt set}(\ell_2),\mbox{\tt set}([a|\ell])\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}(\ell_2),\mbox{\tt set}([a|\ell])\} \uplus S, \emptyset \rangle)
\end{array}}
&
{\begin{array}{l}
\langle \{\mbox{\tt set}(\ell),\mbox{\tt set}([a|\ell_2])\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}(\ell),\mbox{\tt set}([a|\ell_2])\} \uplus S, \emptyset \rangle)
\end{array}}
}
\end{array}
\end{array} $$ The abstract corner $\mathbf{\Lambda_2}$ is inconsistent because $I$ does not accept a constraint store with more than one {\tt set} constraint, and $\mathbf{\Lambda_1}$ is shown joinable modulo $\approx$ by the following abstract diagram $\boldsymbol{\Delta}_1$.
$$ \scriptsize
{\boldsymbol{\Delta}}_1 = \;\;\;\;
\begin{array}{c}
\xymatrix{
{\begin{array}{l}
\langle \{\mbox{\tt set}(l), \mbox{\tt item}(a), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}(l), \mbox{\tt item}(a), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle)
\end{array}}
\ar@{|->}[dr]^(.6){\qquad\qquad\qquad \mbox{\scriptsize\tt set($l$),item($b$)$\,$<=>$\,$set($[b|l]$)}}
\ar@{|->}[d]_-{\mbox{\scriptsize\tt set($l$),item($a$)$\,$<=>$\,$set($[a|l]$)}} & \\
{\begin{array}{l}
\langle \{\mbox{\tt set}([a|\ell]), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}([a|\ell]), \mbox{\tt item}(b)\} \uplus S, \emptyset \rangle)
\end{array}}
\ar@{|->}[d]_-{\mbox{\scriptsize\tt set($[a|l]$),item($b$)$\,$<=>$\,$set($[b,a|l]$)}}
&
{\begin{array}{l}
\langle \{\mbox{\tt set}([b|\ell]), \mbox{\tt item}(a)\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}([b|\ell]), \mbox{\tt item}(a)\} \uplus S, \emptyset \rangle)
\end{array}}
\ar@{|->}[d]^-{\mbox{\scriptsize\tt set($[b|\ell]$),item($a$)$\,$<=>$\,$set($[a,b|\ell]$)}}
\\
{\begin{array}{l}
\langle \{\mbox{\tt set}([b,a|\ell])\} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}([b,a|\ell])\} \uplus S, \emptyset \rangle)
\end{array}}
\ar@2{~}[r]
&
{\begin{array}{l}
\langle \{\mbox{\tt set}([a,b|\ell]) \} \uplus S, \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \\
\phantom{\mathrel{\mbox{\textsc{where}}} }\; I(\langle \{\mbox{\tt set}([a,b|\ell]) \} \uplus S, \emptyset \rangle)
\end{array}}
\\
}
\end{array} $$ The program is terminating since each derivation step reduces the number of \texttt{item} constraints by one, so by Theorem~\ref{thm:termination+abstract-joinability=confluence} it follows the program is confluent modulo $\approx$.
\subsection{Confluence Modulo Equivalence of the Viterbi Algorithm}\label{sec:viterbi} The Viterbi algorithm~\cite{Vit67} is an example of a dynamic programming algorithm that searches for one optimal solution to a problem among, perhaps, several equally good ones.
A Hidden Markov Model, HMM, is a finite state machine with probabilistic state transitions and probabilistic emission of a letter from each state. The \emph{decoding problem} for an observed sequence of emitted letters $Ls$ is that of finding a most probable \emph{path} which is a sequence of state transitions that may have produced $Ls$; see~\cite{DurbinEtAL99} for a background on HMMs and their applications in computational biology.
A decoding problem is typically solved using the Viterbi algorithm~\cite{Vit67} which is an example of a dynamic programming algorithm that produces solutions for a problem by successively extending solutions for growing subproblems. While there are potentially exponentially many differents paths to compare, an early pruning strategy ensures linear time complexity.
The algorithm has been studied in CHR by~\cite{ChristiansenEtAlCHR2010,DBLP:conf/lopstr/ChristiansenK14} as shown below. The optimal complexity requires a restriction in the possible derivations, namely that the \texttt{prune} rule (below) is applied as early as possible. In~\cite{ChristiansenEtAlCHR2010}, it is demonstrated how such a rule ordering can be imposed by semantics-preserving program transformations; here we will show confluence of the program modulo a suitable equivalence (which ensures that limiting the rule order does not destroy the semantics of the program).
\begin{example}\label{ex:HMM-as-such} The following diagram shows a very simple HMM with states \texttt{q0}, $\ldots$, \texttt{q3}, emission alphabet $\{\texttt{a}, \texttt{b}\}$ and probabilities indicated for transitions and emissions.\footnote{An interesting HMM will, of course, have loops so it can produce arbitrary long sequences. No explicit end states are needed.}
$$ \begin{array}{c} {\xymatrix@R=4mm@C=4mm{ & & {\tt a} & & {\tt b} & \\ & & & *++[o][F]{\tt q1}
\ar@{=>}[ul]^{0.2}
\ar@{=>}[ur]_{0.8}
\ar[ddrr]^{1.0}
& & \\ & {\tt a} & & & & \\ \ar@{->}[r]& *++[o][F]{\tt q0} \ar@{=>}[u]^{0.2\,} \ar@{=>}[d]_{0.8\,} \ar[ddrr]_{0.7} \ar[uurr]^{0.3}
& & & & *++[o][F]{\tt q3}\\ &{\tt b} & & & & \\ & & & *++[o][F]{\tt q2}
\ar@{=>}[dl]_{0.9}
\ar@{=>}[dr]^{0.1}
\ar[uurr]_{1.0}
& & \\ & & {\tt a } & & {\tt b} & \\ }} \end{array} $$ The different events of transitions and emissions are assumed to be independent.
For example, the sequence \texttt{a$\cdot$b} may be produced via the path \texttt{q0}$\cdot$\texttt{q1}$\cdot$\texttt{q3} with probability $0.2*0.3*0.8=0.048$ or
\texttt{q0}$\cdot$\texttt{q2}$\cdot$\texttt{q3} with probability $0.2*0.7*0.1=0.014$.
For simplicity of the program that follows, it is assumed that an emission is produced when a state is left (rather than entered). \end{example} A specific HMM is encoded as a set of \texttt{trans/3} and \texttt{emit/3} constraints that are not changed during program execution. \begin{example}\label{ex:HMM-encoding} The HMM of Example~\ref{ex:HMM-as-such} is encoded by the following constraints. $$ \begin{tabular}{l}
$\{\texttt{ trans(q0,q1,0.3)}, \texttt{ trans(q0,q2,0.7)},
\texttt{ trans(q1,q3,1)}, \texttt{ trans(q2,q3,1)},$ \\
\hbox to 0.6em{}$\texttt{ emit(q0,a,0.2)}, \texttt{ emit(q0,b,0.8)},$ \\
\hbox to 0.6em{}$ \texttt{ emit(q1,a,0.2)}, \texttt{ emit(q1,b,0.8)},$\\
\hbox to 0.6em{}$ \texttt{ emit(q2,a,0.9)}, \texttt{ emit(q2,b,0.1)} \}$ \end{tabular}
$$ \end{example} The CHR program that implements the Viterbi algorithm is as follows. {\begin{verbatim} :- chr_constraint path/4, trans/3, emit/3.
expand @ trans(Q,Q1,PT), emit(Q,L,PE), path([L|Ls],Q,P,PathRev) ==>
P1 is P*PT*PE | path(Ls,Q1,P1,[Q1|PathRev]).
prune @ path(Ls,Q,P1,_) \ path(Ls,Q,P2,_) <=> P1 >= P2 | true. \end{verbatim}}\noindent The meaning of a constraint \texttt{path($Ls$,$q$,$p$,$R$)} is that $Ls$ is a remaining emission sequence to be processed, $q$ the current state of the HMM, and $p$ the probability of a path $R$ found for the already processed prefix of the emission sequence. To simplify the program, a path is represented in reverse order.
The decoding of a sequence $Ls$ starting from state $q_0$ is stated by the query \begin{itemize}
\item[] {\tt:- }$\textit{HMM}$\!\texttt{,} \texttt{path($Ls$,$q_0$,1,[$q_0$])}. \end{itemize}
where \textit{HMM} is an encoding of a given HMM in terms of ground \texttt{trans} and \texttt{emit} constraints; for each pair of states $q_1,q_2$, \textit{HMM} contains at most one constraint of the form \texttt{trans($q_1$,$q_2$,$\ldots$)}, and for each pair of state $q$ and emission letter $L$,
\textit{HMM} contains at most one constraint of the form \texttt{emit($q$,$L$,$\ldots$)}.
The first rule of the program, \texttt{expand}, expands the existing paths and \texttt{prune} removes paths for identical subproblems (identified by the current HMM state and remaining emission sequence) with lower (or equal) probabilities. The program is terminating for such queries as any new \texttt{path} constraint introduced by the \texttt{expand} rule has a first argument shorter than that of its predecessor. A final state will include one \texttt{path} constraint of optimal probability for each prefix of the input string (unless the underlying state machine is not capable of generating that string).
The program is not confluent in the classical sense, as the \texttt{prune} rule may nondeterministically remove one or the other of two alternative \texttt{path} constraints of identical probability for the same sequence. In the following we introduce invariant $I$ and equivalence $\approx$ and show the program confluent modulo equivalence. For simplicity of the definitions and with no loss of generality, we assume a fixed indexed encoding $\mathit{HMM}$ of a Hidden Markov Model with initial state \texttt{q0} and fixed input emission sequence $Ls_0$.
\begin{definition} $I(\Sigma)$ if and only if
$\langle \textit{HMM} \cup \{(0\colon \texttt{path($Ls_0$,$q_0$,1,[$q_0$])})\}\rangle \stackrel{*}\ourmapsto \Sigma$. \end{definition}
However, in the proof of local confluence below, we will need a more direct characterization of the possible derivation states and the interrelations between their constraints. To this end, we state the following proposition.
\begin{proposition}\label{prop:viterbi-invariant} An $I$-state is of the form $\langle S\cup\mathit{HMM}, T\rangle$ where $S$ is a set of ground \texttt{path} constraints and $T$ a propagation history.
For any $(i\colon \texttt{path($[L|Ls]$,$q$,$P$,$qs$)}) \in S$ for which $\{ (i^t\colon\texttt{trans($q$,$q'$,$P^t$)}), (i^e\colon\texttt{emit($q$,$L$,$P^e$)})\} \in \mathit{HMM}$, then one and only one of the following will be the case. \begin{enumerate}
\item \emph{Expansion has not taken place:}\\
$({\texttt{expand}}@ i^t, i^e, i)\not\in T$
\item \emph{Expansion produced and still in the store:}\\
$({\texttt{expand}}@ i^t, i^e, i)\in T \;\land\;
\exists i' .\, (i'\colon \texttt{path($Ls$,$q'$,$P'$,$[q'|qs]$)}) \in S$
where $P'$ is the value of $P$$*$$P^t$$*$$P^e$.
\item \emph{Expansion produced but pruned by stronger or equal alternative:}\\
$({\texttt{expand}}@ i^t, i^e, i)\in T \;\land\;
\not\exists i',P' .\, (i'\colon \texttt{path($Ls$,$q'$,$P'$,$[q'|qs]$)}) \in S$\\
$\land\;
\exists P',qs',i' .\, \bigl((i'\colon \texttt{path($Ls$,$q'$,$P'$,$[q'|qs']$)}) \in S
\land P' \geq P$$*$$P^t$$*$$P^e \land qs\neq qs'\bigr)$ \end{enumerate}
\end{proposition}
Notice in case 3, that the \texttt{path} required to exists may either be the stronger (or equal) alternative that via \texttt{prune} rule lead to the removal of \texttt{path($Ls$,$q'$,$P'$,$[q'|qs]$)},
or it may be an even stronger (or equal) one, meaning that several applications of \texttt{prune} have been involved.
The uniqueness of \texttt{emit} (\texttt{trans}) constraints in \textit{HMM} for a fixed $q$ ensures that the constraints $(i'\colon \texttt{path($Ls$,$q'$,$P'$,$[q'|qs]$)})$ in case 2 and 3 are unique and uniquely related to the application record ${\texttt{expand}}@ i_t, i_e, i$.
\begin{proof} We use induction over the length of the derivation leading to a given $I$-state. \par
\noindent \emph{Base case.} The state
$\langle \textit{HMM} \cup \{(0\colon \texttt{path($Ls$,$q_0$,1,[$q_0$])})\}\rangle$ matches case 1 in the proposition. \par
\noindent \emph{Step.} Assume an $I$-state $\Sigma=\langle S\cup\mathit{HMM}, T\rangle)$ satisfying the proposition, and let $\Sigma\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma^*$, where\break
$\Sigma^*=\langle S^*\cup\mathit{HMM}, T\rangle$. Two kinds of derivation steps are possible, one for each program rule.
\texttt{expand}: Assume that the \texttt{path} constraint $i{:}\pi\in S$ of $\Sigma$ is involved in an application of the \texttt{expand} rule. The only difference between $\Sigma$ and $\Sigma^*$ is that the latter includes a new \texttt{path} constraint $i^*{:}\pi^*\in S^*$ and a new application record $({\texttt{expand}}@ i^t, i^e, i^*)\in T^*$. In $\Sigma^*$, $i{:}\pi$ satisfies condition 2, and $i^*{:}\pi^*$ condition 1; any other \texttt{expand} constraint in $\Sigma^*$ satisfies the same of 1, 2, 3 as in $\Sigma$.
\texttt{prune}: Assume that the rule applies in $\Sigma$ by the application instance
$i_1{:}\pi_1\texttt{/}i_2{:}\pi_2\mathrel\texttt{<=>}P_1\texttt{>=} P_2\texttt{|}\texttt{true}$. Thus $S^*=S\setminus\{i_2{:}\pi_2\}$, $T^*=T$. It holds that, when $i_1{:}\pi_1$ satisfies condition $k$ in $\Sigma$, then it also satisfies condition $k$ in $\Sigma^*$ for $k=1,2,3$.
The only way that the removal of $i_2{:}\pi_2$ may affect the proposition is when there is another $(i_2^0{:}\pi_2^0)\in S$ satisfying condition 2 or 3 with $i_2$ in the role of the existentially quantified index $i$ in either case. For condition 2, $(i_2^0{:}\pi_2^0)$ satisfies condition 2 or 3 in $\Sigma^*$ with $i'=i_1$; for condition 3, $(i_2^0{:}\pi_2^0)$ satisfies condition 3 in $\Sigma^*$ with $i'=i_1$. \end{proof}
Our equivalence relation specifies the intuition that two solutions for the same subproblem are equally good when they have the same probability.
We recall that a state is defined as an equivalence class over state representations sharing the same pattern of variable recurrence.
\begin{definition} The $\approx$ is the smallest equivalence relation on $I$-states such that $\langle S\cup \mathit{HMM}, T \rangle \approx $\break$\langle S'\cup \mathit{HMM}, T\rangle$ if and only if \begin{itemize}
\item For any $i\colon\! \mbox{\tt path($Ls$,$q$,$P$,$qs$)}\; \in S$, there is
an $i\colon\! \mbox{\tt path($Ls$,$q$,$P$,$qs'$)}\; \in S$, and vice versa. \end{itemize} \end{definition}
\begin{theorem} The Viterbi program with invariant $I$ is confluent modulo $\approx$. \end{theorem}
\par
\noindent\textit{Proof.}\hskip 1ex According to Theorem~\ref{thm:termination+abstract-joinability=confluence}, we can prove confluence of a CHR program by listing the
set of critical abstract corners and showing each of them joinable or split joinable.
Firstly, we observe that no built-in predicate can appear in an $I$-state (they are only used in guards) and that the two built predicates \texttt{>=} and \texttt{is} are $I$-complete.
Thus, we have no $\alpha_2$- and $\alpha_3$-corners to consider, leaving only $\alpha_1$- and $\beta$-corners. For a better overview, we indicate the overall shapes of corners in the chosen canonical set, described in full detail below. There are three $\alpha_1$-corners, one for each possible way that two rules may
produce a critical overlap: \begin{itemize}
\item [] $\Lambda_1\colon\hbox to 1.9em{} \circ \stackrel{\small\tt prune}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \circ \stackrel{\small\tt prune}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ$
\item[] $\Lambda_2, \Lambda_3 \colon~ \circ \stackrel{\small\tt prune}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \circ \stackrel{\small\tt expand}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ$ ~differing in whether or not the constraint being expanded is removed;\\ \phantom{$\Lambda_2, \Lambda_3 \colon\hbox to 0pt{}$}our analysis will show $\Lambda_3$ (expanded constraint removed) is not joinable, but can be split into\\\phantom{$\Lambda_2, \Lambda_3 \colon\hbox to 0pt{}$}three joinable subcases $\Lambda_3^{(1)}, \Lambda_3^{(2)}, \Lambda_3^{(3)}$, one for each option in Proposition~\ref{prop:viterbi-invariant}. \end{itemize} Two $\beta$-corners are found, one for each clause of the program. \begin{itemize}
\item []$\Lambda_4\colon\hbox to 1.9em{} \circ \approx \circ \stackrel{\small\tt prune}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ$
\item []$\Lambda_5\colon\hbox to 1.9em{} \circ \approx \circ \stackrel{\small\tt expand}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ$ \end{itemize}
To save space, application steps are labelled by application records (rather that application instances) and we leave out also the $id$ function, e.g., writing \texttt{prune@}$\pi_1\pi_2$ instead of \texttt{prune@}$\mathit{id}((\pi_1,\pi_2))$.
We abbreviate the writing of the invariant in an abstract state, writing $(\Sigma\mathrel{\mbox{\textsc{where}}} I\land\cdots)$ instead of $(\Sigma\mathrel{\mbox{\textsc{where}}} I(\Sigma)\land\cdots)$, where $\Sigma$ is a (perhaps complex) abstract state expression. We use the following conventions in expressions that represent propagation histories. \begin{itemize}
\item A condition of the form $ra\not\in T$, where $ra$ is a rule application and $T$ a propagation history,
may be removed in an abstract state expressions when it is clear from context that it is always satisfied.
This is relevant when $ra=(\langle\textit{rule-id}\rangle@\cdots i\cdots)$ and $T$ is part of a state guaranteed not
to contain $i$.
\item When $i$ represents a constraint index and $T$ a propagation history,
the notation $T\setminus i$ is a shorthand for
$T\setminus\{ra\mid \mbox{$ra$ is an application record of the form $ra=(\langle\textit{rule-id}\rangle@\cdots i\cdots)$}\}$.
\end{itemize}
To simplify notation for the description of these corners, we introduce the following abbreviations; the recurrences of variables are significant. \begin{eqnarray*} \tau & = & (i^t\colon\mbox{\tt trans($q$,$q'$,$P^t$)}) \\ \eta & = & (i^e\colon \mbox{\tt emit($q$,$L$,$P^e$)}) \\
\pi_j & = & (i_j\colon \mbox{\tt path([$L$|$LS$],$q$,$P_j$,$qs_j$)})\quad\,\,\mbox{for $j=1,\ldots,4$} \\
\pi'_j & = & (i'_j\colon \mbox{\tt path($LS$,$q'$,$P'_j$,[$q'$|$qs_j$])})
\quad\mbox{$P'_j$ is the value of $P_j$$*$$P^t$$*$$P^e$ for $j=1,\ldots,4$} \end{eqnarray*}
As it appears, $\pi_i'$ is can be derived from $\pi_i$,
$\tau$ and $\eta$ using the \texttt{expand} rule. The \texttt{path} constraints $\pi_1,\ldots\pi_4$ all concern the same sub-problem, identified by the identical first and second argument, {\tt [$L$|$LS$]} and $q$; and analogously for the $\pi'_i$ constraints.
We consider now the canonical abstract corners one by one and show them (split) joinable.
\noindent$\mathbf{\Lambda}_1$: \emph{Overlap of} \texttt{prune} \emph{with itself}
\begin{displaymath} \scriptsize
\begin{array}{c}
\xymatrix{ \langle S \uplus \{\pi_1,\! \pi_2\}, T \rangle \mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\land P_2\texttt{>=} P_1
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt prune@$\pi_2\pi_1$}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} & \\
\langle S \uplus \{\pi_1\}, T \rangle \mathrel{\mbox{\textsc{where}}} I &
\langle S \uplus \{ \pi_2\}, T \rangle \mathrel{\mbox{\textsc{where}}} I }
\\ \\
\end{array} \end{displaymath} This extends immediately to a joinability diagram because the two abstract wing states are equivalent.
\noindent$\mathbf{\Lambda}_2$: \emph{Overlap of} \texttt{prune} \emph{and} \texttt{expand}\emph{; expanded constraint not removed}
\begin{displaymath} \scriptsize
\begin{array}{c}
\xymatrix{ \langle S \uplus \{\pi_1, \! \pi_2, \! \tau,\! \eta\}, T \rangle \mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\land \texttt{expand@}\tau\eta\pi_1 \not\in T
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_1$}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} & \\
\langle S \uplus \{\pi_1,\! \tau,\! \eta\}, T\backslash\pi_2 \rangle \mathrel{\mbox{\textsc{where}}} I \land \texttt{expand@}\tau\eta\pi_1 \not\in T &
\langle S \uplus \{\pi_1,\! \pi_2,\! \pi'_1,\! \tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_1\} \rangle \mathrel{\mbox{\textsc{where}}} I }
\\ \\
\end{array}
\end{displaymath} In this case, the two rules commute and the corner joins in one and the same abstract state.
\begin{displaymath} \scriptsize
\begin{array}{c}
\xymatrix{ \langle S \uplus \{\pi_1,\! \pi_2,\! \tau, \! \eta\}, T \rangle \mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\land \texttt{expand@}\tau\eta\pi_1 \not\in T
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_1$}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} & \\
\langle S \uplus \{\pi_1,\! \tau,\! \eta\}, T\backslash\pi_2 \rangle \mathrel{\mbox{\textsc{where}}} I \land \texttt{expand@}\tau\eta\pi_1 \not\in T
\ar@{|->}[d]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_1$}} & \langle S \uplus \{\pi_1,\! \pi_2,\! \pi'_1,\tau, \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_1\} \rangle \mathrel{\mbox{\textsc{where}}} I
\ar@{|->}[dl]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}}
\\
\langle S \uplus \{\pi_1,\! \pi'_1,\! \tau,\! \eta\}, T\backslash\pi_2 \uplus \{\texttt{expand@}\tau\eta\pi_1\} \rangle \mathrel{\mbox{\textsc{where}}} I
&
}
\end{array}
\end{displaymath}
\noindent$\mathbf{\Lambda}_3$: \emph{Overlap of} \texttt{prune} \emph{and} \texttt{expand}\emph{; expanded constraint removed}
\begin{displaymath} \scriptsize
\begin{array}{c}
\xymatrix{ \langle S \uplus \{\pi_1,\! \pi_2,\! \tau,\! \eta\}, T \rangle \mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\land \texttt{expand@}\tau\eta\pi_2 \not\in T
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_2$}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} & \\
\langle S \uplus \{\pi_1,\! \tau,\! \eta\}, T\backslash\pi_2 \rangle \mathrel{\mbox{\textsc{where}}} I & \langle S \uplus \{\pi_1,\! \pi_2,\! \pi'_2,\! \tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_2\} \rangle \mathrel{\mbox{\textsc{where}}} I }
\\ \\
\end{array}
\end{displaymath}
This abstract corner is not joinable as different derivations are possible depending on which of the three cases in Proposition~\ref{prop:viterbi-invariant} that holds for the path constraint $\pi_1$. This suggests a splitting of the corner into three new corners, that we can show joinable as follows. Hence, the corner is not joinable but split joinable. For reasons of space, we show only the related abstract joinability diagrams; the corners can be identified at the top.
\noindent{$\mathbf{\Lambda}_3^{(1)}$: \emph{Split of} $\mathbf{\Lambda}_3$\emph{;} $\pi_1$ \emph{applicable}
\begin{displaymath} \scriptsize
\begin{array}{c}
\xymatrix{
{\begin{array}{l}
\langle S \uplus \{\pi_1,\! \pi_2,\! \tau,\! \eta\}, T \rangle \\
\mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\land \texttt{expand@}\tau\eta\pi_1 \not\in T
\land \texttt{expand@}\tau\eta\pi_2 \not\in T
\end{array}}
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt expand@$\tau$,$\eta$,$\pi_2$}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} & \\
{\begin{array}{l}\langle S \uplus \{\pi_1,\! \tau,\! \eta\}, T\backslash\pi_2 \rangle\\
\mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\land \texttt{expand@}\tau\eta\pi_1 \not\in T\end{array}}
\ar@{|->}[dd]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_1$}} &
{\begin{array}{l} \langle S \uplus \{\pi_1,\! \pi_2, \!\pi'_2,\!\tau, \!\eta\}, T\uplus \{\texttt{expand@}\tau,\eta,\pi_2\} \rangle \\ \mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\land \texttt{expand@}\tau\eta\pi_1 \not\in T \end{array}}
\ar@{|->}[d]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_1$}}
\\
& {\begin{array}{l} \langle S\uplus \{\pi_1,\! \pi'_1,\! \pi_2,\! \pi'_2,\!\tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_2, \texttt{expand@}\tau\eta\pi_1 \} \rangle\\ \mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\end{array}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1$,$\pi_2$}} \\ {\begin{array}{l}\langle S \uplus \{\pi_1,\!\pi'_1, \!\tau, \!\eta\}, T\backslash\pi_2 \uplus \{\texttt{expand@}\tau\eta\pi_1\} \rangle\\ \mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2\end{array}} & {\begin{array}{l}\langle S\uplus \{\pi_1, \!\pi'_1,\! \pi'_2,\!\tau,\! \eta\}, T\backslash\pi_2 \uplus \{\texttt{expand@}\tau,\eta,\pi_1 \} \rangle\\
\mathrel{\mbox{\textsc{where}}} I\land P_1\texttt{>=} P_2
\ar@{|->}[l]_{\mbox{\scriptsize\tt prune@$\pi'_1$,$\pi'_2$}} \end{array}}
}
\end{array}
\end{displaymath}
\noindent{$\mathbf{\Lambda}_3^{(2)}$: \emph{Split of} $\mathbf{\Lambda}_3$\emph{;} $\pi_1$ \emph{already expanded into} $\pi_1'$\emph{;} $\pi_1'$ \emph{still in state}
\begin{displaymath} \scriptsize
\begin{array}{c}
\xymatrix{ {\begin{array}{l}\langle S\uplus \{\pi_1,\! \pi'_1,\! \pi_2,\! \tau, \!\eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_1 \} \rangle\\
\mathrel{\mbox{\textsc{where}}} I \land P_1\texttt{>=} P_2
\land \texttt{expand@}\tau\eta\pi_2 \not\in T
\end{array}}
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_2$}}
\ar@{|->}[dd]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} & \\
&
{\begin{array}{l}\langle S\uplus \{\pi_1,\! \pi'_1,\! \pi_2,\! \pi'_2,\!\tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_1, \texttt{expand@}\tau\eta\pi_2\} \rangle\\
\mathrel{\mbox{\textsc{where}}} I \land P_1\texttt{>=} P_2 \end{array}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}}
\\
{\begin{array}{l} \langle S\uplus \{\pi_1,\! \pi'_1,\! \tau,\! \eta\}, T\backslash\pi_2\uplus \{\texttt{expand@}\tau\eta\pi_1 \} \rangle\\ \mathrel{\mbox{\textsc{where}}} I \land P_1\texttt{>=} P_2 \end{array}}
& {\begin{array}{l} \langle S\uplus \{\pi_1,\! \pi'_1,\! \pi'_2,\!\tau,\! \eta\}, T\backslash\pi_2\uplus \{\texttt{expand@}\tau\eta\pi_1 \} \rangle\\
\mathrel{\mbox{\textsc{where}}} I \land P_1\texttt{>=} P_2 \end{array}}
\ar@{|->}[l]_{\mbox{\scriptsize\tt prune@$\pi'_1\pi'_2$}}
}
\end{array}
\end{displaymath} Notice for the last {\tt prune@$\pi'_1\pi'_2$} step, that the application history has no mentioning of $\pi'_2$, as the only event, since it was produced, is the step labelled {\tt prune@$\pi_1\pi_2$}.
\noindent{$\mathbf{\Lambda}_3^{(3)}$: \emph{Split of} $\mathbf{\Lambda}_3$\emph{;} $\pi_1$ \emph{already expanded into} $\pi_1'$\emph{;} $\pi_1'$ \emph{already removed}
\noindent As given by Proposition~\ref{prop:viterbi-invariant}, option 3, this implies the presence in the common ancestor state of a \texttt{path} constraints $\pi_3$, with sufficiently high probability to have pruned $\pi_1'$ as well as a possible $\pi_2'$ (expanded from $\pi_2'$ using $\tau$ and $\eta$). We can thus write this abstract corner and expand it to an abstract joinability diagram as follows.
\begin{displaymath} \scriptsize
\begin{array}{c}
\xymatrix{
{\begin{array}{l}\langle S \uplus \{\pi_1,\! \pi_2,\! \pi'_3,\! \tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_1 \} \rangle\\
\mathrel{\mbox{\textsc{where}}} I \land P_1\texttt{>=} P_2 \land P'_3 \texttt{>=} P_1'\texttt{>=} P_2'\\
\hbox to 3em{}\land \texttt{expand@}\tau\eta\pi_2 \not\in T
\land \pi_1'\not\in S
\end{array}}
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_2$}}
\ar@{|->}[dd]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} & \\
& {\begin{array}{l} \langle S \uplus \{\pi_1,\! \pi_2,\! \pi'_3, \!\pi'_2,\!\tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_1, \texttt{expand@}\tau\eta\pi_2\} \rangle\\
\mathrel{\mbox{\textsc{where}}} I \land P_1\texttt{>=} P_2 \land P'_3 \texttt{>=} P_1'\texttt{>=} P_2'\\
\hbox to 3em{}\land \pi_1'\not\in S
\end{array}}
\ar@{|->}[d]^{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}}
\\
{\begin{array}{l}\langle S\uplus \{\pi_1,\! \pi'_3,\! \tau,\! \eta\}, T\backslash\pi_2\uplus \{\texttt{expand@}\tau\eta\pi_1 \} \rangle \\
\mathrel{\mbox{\textsc{where}}}
I \land P_1\texttt{>=} P_2 \land P'_3 \texttt{>=} P_1'\texttt{>=} P_2'\\
\hbox to 3em{}\land \pi_1'\not\in S
\end{array}}
& {\begin{array}{l} \langle S\uplus \{\pi_1,\! \pi'_3,\! \pi'_2,\!\tau,\! \eta\}, T\setminus\pi_2\uplus \{\texttt{expand@}\tau\eta\pi_1 \} \rangle\\
\mathrel{\mbox{\textsc{where}}} I \land P_1\texttt{>=} P_2 \land P'_3 \texttt{>=} P_1'\texttt{>=} P_2'\\
\hbox to 3em{}\land \pi_1'\not\in S \end{array}}
\ar@{|->}[l]_{\mbox{\scriptsize\tt prune@$\pi'_3\pi'_2\;\;$}}
}
\end{array}
\end{displaymath} For the last {\tt prune@$\pi'_3\pi'_2$} step, the application history has no mentioning of $\pi'_2$, as the only event since it was produced is the step labelled {\tt prune@$\pi_1\pi_2$}.
This finishes the proof that $\mathbf{\Lambda}_3$ is split joinable. Now we turn to the canonical abstract $\beta$-corners of which there are two, $\mathbf{\Lambda}_4$ and $\mathbf{\Lambda}_5$, one for each program rule.
\noindent{$\mathbf{\Lambda}_4$: \emph{Equivalence and the} \texttt{expand} \emph{rule}
\noindent
For the two equivalent states on the left side, it holds that {\tt expand@$\tau\eta\pi_2$} = {\tt expand@$\tau\eta\pi_1$} and that $S_i=\mathit{HMM}\uplus S_i'$, $i=1,2$ where $S'_1$ and $S'_2$ consist of pairwise similar \texttt{path} constraints with identical index and that may differ only in their last arguments, and similarly for $\pi_1$ and $\pi_2$.
\begin{displaymath} \scriptsize \begin{array}{c} \xymatrix{ \langle S_1 \uplus \{\pi_1,\! \tau,\! \eta\}, T\rangle \mathrel{\mbox{\textsc{where}}} I \land \texttt{expand@}\tau\eta\pi_1\not\in T
\ar@2{~}[d] \ar@{|->}[dr]^-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_1$}} \\ \langle S_2 \uplus \{\pi_2,\! \tau,\! \eta\}, T \rangle \mathrel{\mbox{\textsc{where}}} I \land \texttt{expand@}\tau\eta\pi_2\not\in T
\ar@{|->}[dr]_-{\mbox{\scriptsize\tt expand@$\tau\eta\pi_2$}} & \langle S \uplus \{\pi_1,\! \pi'_1,\!\tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_1\} \rangle \mathrel{\mbox{\textsc{where}}} I \ar@2{~}[d] \\ & \langle S_2 \uplus \{\pi_2,\! \pi'_2,\!\tau,\! \eta\}, T\uplus \{\texttt{expand@}\tau\eta\pi_2\} \rangle \mathrel{\mbox{\textsc{where}}} I } \end{array}\end{displaymath} To see that the lower equivalence holds, we notice that the indices of
$\pi_1'$ and $\pi_2'$ can be chosen identical (and different from any other index used),
and they may differ only in their last arguments.
\noindent{$\mathbf{\Lambda}_5$: \emph{Equivalence and the} \texttt{prune} \emph{rule}
\noindent
For the two equivalent states on the left side, it holds that {\tt expand@$\tau\eta\pi_{i+2}$} = {\tt expand@$\tau\eta\pi_i$}, $i=1,2$, and that $\pi_{i+2}$ and $\pi_{i}$ , $i=1,2$, may differ only in their last arguments. Furthermore, $S_i=\mathit{HMM}\uplus S_i'$, $i=1,2$ where $S'_1$ and $S'_2$ consist of pairwise similar \texttt{path} constraints with identical index and that may differ only in their last arguments.
\begin{displaymath} \scriptsize \begin{array}{c} \xymatrix{ \langle S \uplus \{\pi_1,\! \pi_2,\!\tau,\! \eta\}, T \rangle \mathrel{\mbox{\textsc{where}}} I\land P_1{\tt>=}P_2\land \mbox{\scriptsize\tt prune@$\pi_1\pi_2$}\not\in T
\ar@2{~}[d] \ar@{|->}[dr]^-{\mbox{\scriptsize\tt prune@$\pi_1\pi_2$}} \\ \langle S\uplus \{\pi_3,\!\pi_4,\! \tau,\! \eta\}, T \rangle \mathrel{\mbox{\textsc{where}}}
I\land P_3{\tt>=}P_4\land \mbox{\scriptsize\tt prune@$\pi_3\pi_4$}\not\in T
\ar@{|->}[dr]_-{\mbox{\scriptsize\tt prune@$\pi_3\pi_4$}} &
\langle S \uplus \{\pi_1,\! \tau,\!\eta\}, T\backslash \pi_2 \rangle \mathrel{\mbox{\textsc{where}}} I \ar@2{~}[d] \\ & \langle S \uplus \{\pi_3,\! \tau,\!\eta\}, T\backslash \pi_4 \rangle \mathrel{\mbox{\textsc{where}}} I } \end{array}\end{displaymath} Thus the set of abstract, critical corners have been shown joinable or split joinable; by termination and Theorem~\ref{thm:termination+abstract-joinability=confluence}, the program is confluent modulo $\approx$. \hskip 1em$\Box$
\subsection{Countably Infinite Splitting}\label{sec:infinite-split}
Here we show a program whose proof of confluence needs an infinite splitting of an abstract critical corner.
The following CHR program is intended for queries of the form \texttt{start,} \texttt{c(s$^n$(0))}, where \texttt{s$^n$(0)} denotes the $n$th successor of {\tt 0} for any $n\ge 0$, e.g., \texttt{s$^2$(0)} = \texttt{s(s(0))}.
{\small\begin{verbatim} easy @ start <=> easy. hard @ start <=> hard. done @ c(X), easy <=> c(0), end. step @ hard \ c(s(X)) <=> c(X). finally @ c(0) \ hard <=> end. \end{verbatim}}\noindent The first step in such a derivation will introduce either an \texttt{easy} or a \texttt{hard} constraint. In case of \texttt{easy}, the derivation terminates after one additional step in the state $\{\texttt{c(0)}, \texttt{end}\}$. In case of \texttt{hard}, the derivation terminates after $n+1$ steps in the same state, so the program is confluent (modulo trivial ${\approx}={=}$) under the invariant implied by the intended initial states.
We can specify the invariant as follows, using the unary meta-level predicate $\mathit{succ}(N)$, satisfied if and only if $N$ of the form \texttt{s$^n$(0)} for an arbitrary natural number $\ge0$. $I(\langle S,T \rangle)$ holds if and only if \begin{itemize}
\item $S = \{R, \texttt{c($N$)}\}$ where $R \in \{{\tt start}, {\tt hard}, {\tt easy},{\tt end} \}$ and $\mathit{succ}(N)$,
\item $T=\emptyset$. \end{itemize}
There exists only one consistent abstract critical $I$-corner $\mathbf{\Lambda}$, and it is based on the overlap of the rules \texttt{easy} and \texttt{hard}. Notice that the invariant has been unfolded, which is a semantics-preserving transformation. $$ \mathbf{\Lambda} = \begin{array}{c} \xymatrix{
\langle \{{\tt start},{\tt c}(n)\} , \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \mathit{succ}(n)
\ar@{|->}[dr]^-{\qquad \mbox{\scriptsize\tt hard@start}}
\ar@{|->}[d]_{\mbox{\scriptsize\tt easy@start}}
& \\
\langle \{{\tt easy},{\tt c}(n)\} , \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \mathit{succ}(n) & \langle \{{\tt hard},{\tt c}(n)\} , \emptyset \rangle \mathrel{\mbox{\textsc{where}}} \mathit{succ}(n) \\ } \end{array} $$ The abstract critical corner $\mathbf{\Lambda}$ is not joinable as no single joinability diagram applies for all concrete corners covered by $\mathbf{\Lambda}$. Therefore we split $\mathbf{\Lambda}$ using the infinite disjunction $\mathit{succ}(n) \Leftrightarrow n=\texttt{0}\lor n=\texttt{s(0)}\lor\cdots$.
This leads to a countably infinite set of abstract corners $\mathbf{\Lambda}_0$, $\mathbf{\Lambda}_1$, \ldots, where $$ \mathbf{\Lambda}_i = \begin{array}{c} \xymatrix{
\langle \{{\tt start},\texttt{c(s$^n$(0)}\} , \emptyset \rangle
\ar@{|->}[dr]^-{\qquad\mbox{\scriptsize\tt hard@start}}
\ar@{|->}[d]_{\mbox{\scriptsize\tt easy@start}}
& \\
\langle \{{\tt easy},\texttt{c(s$^n$(0)}\} , \emptyset \rangle& \langle \{{\tt hard},\texttt{c(s$^n$(0)}\} , \emptyset \rangle \\ } \end{array} $$
Each such abstract corner can be extended into a joinability diagram $\boldsymbol{\Delta}_i$, each having $i+4$ abstract states and the same number of abstract derivation steps. For a better overview, we indicate only the shapes of these diagrams; the actual states are uniquely determined by the rules applied. $$ \begin{array}{llllll}
\qquad\;\;\boldsymbol{\Delta}_0= & \qquad\;\;\boldsymbol{\Delta}_1= & & \qquad\;\;\boldsymbol{\Delta}_n= & \\
\xymatrix{
&
\circ
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt hard}}
\ar@{|->}[dl]_{\mbox{\scriptsize\tt easy}}
&\\
\circ \ar@{|->}[dr]^-{\mbox{\scriptsize\tt done}} & & \circ \ar@{|->}[dl]^-{\mbox{\scriptsize\tt finally}} \\
& \circ & \\
}
&
\xymatrix{
&
\circ
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt hard}}
\ar@{|->}[dl]_{\mbox{\scriptsize\tt easy}}
&\\
\circ \ar@{|->}[ddr]^-{\mbox{\scriptsize\tt done}} & & \circ \ar@{|->}[d]^-{\mbox{\scriptsize\tt step}} \\
& & \circ \ar@{|->}[dl]^-{\mbox{\scriptsize\tt finally}}\\
&\circ& \\
}
& \xymatrix{ \\ \ldots\\ \\ \\}
&
\xymatrix{
&
\circ
\ar@{|->}[dr]^-{\mbox{\scriptsize\tt hard}}
\ar@{|->}[dl]_{\mbox{\scriptsize\tt easy}}
&\\
\circ \ar@{|->}[ddr]^-{\mbox{\scriptsize\tt done}} & & \circ \ar@{|->}[d]^-{\mbox{\scriptsize\tt step}^{\scriptscriptstyle n}} \\
& & \circ \ar@{|->}[dl]^-{\mbox{\scriptsize\tt finally}}\\
&\circ& \\
}
& \xymatrix{ \\ \ldots\\ \\ \\}\\ \end{array} $$ Thus $\mathbf{\Lambda}$ is split joinable, the program is terminating (no derivation starting from a state containing $\texttt{c(s$^n$(0))}$ includes more that $n+2$ steps), and by Theorem~\ref{thm:termination+abstract-joinability=confluence} it follows that the program is confluent (modulo =).
We notice here that confluence is due to the invariant; without invariant, we would get instead of $\mathbf\Lambda$, the corner $\langle \{{\tt easy},{\tt c}(x)\}\rangle\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}\langle \{{\tt start},{\tt c}(x)\} , \emptyset\rangle\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\langle \{{\tt hard},{\tt c}(x)\}\rangle$. It is neither joinable nor split joinable.
\section{Conclusions and Future Work}\label{sec:discussion} The aim of this paper is both theoretical and practical. Practical as it points forward to methods for proving highly useful properties of realistic CHR programs that may identify possible optimizations and contribute to correctness proofs; and theoretical since it provides a firm basis for understanding the notion of confluence modulo equivalence applied in the context of CHR.
We have demonstrated the relevance of confluence modulo equivalence for Constraint Handling Rules, which may also inspire to apply the concept to other systems with nondeterministic choice and parallelism. This may be approached either by migrating our results to other types of derivation systems, or using the fact that programs and systems of many such paradigms can be mapped directly into CHR programs; see an overview in the book by Fr\"uhwirth~\cite{fru_chr_book_2009}.
We introduced a new operational semantics for CHR that includes non-logical and incomplete built-ins and, as we have argued, this semantics is in many respects more in accordance with concrete implementations of CHR that what is seen in earlier work.
We introduced the idea of a logical meta-language \textsc{MetaCHR}{} specifically intended for reasoning about CHR programs, their semantics and their proofs of confluence modulo equivalence. These proofs are reified as collections of abstract joinability diagrams. A main advantage of this approach is that we can parameterize such proofs, i.e., diagrams, by meta-variables constrained at the meta-level to stand for, say, variables or nonvariable terms of CHR. In our approach, this is essential for handling non-logical and incomplete built-ins correctly.
Our work is an improvement of the state-of-art in confluence proving for CHR~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:journals/constraints/AbdennadherFM99,DBLP:conf/iclp/DuckSS07} in several ways: we generalize to modulo equivalence, we handle a larger and more realistic class of CHR programs, and for many programs we can reduce to a finite number of proof cases where \cite{DBLP:conf/iclp/DuckSS07} needs infinitely many, even for simple invariants such as groundness.
The foundational works by Abdennadher et al~\cite{DBLP:conf/cp/Abdennadher97,DBLP:conf/cp/AbdennadherFM96,DBLP:journals/constraints/AbdennadherFM99} and Duck et al~\cite{DBLP:conf/iclp/DuckSS07} use ordinary substitutions and inclusion of more constraints as their way to explain how their abstract cases, called critical pairs, cover large classes of concrete such pairs, each required to be joinable to ensure confluence. The use of the same language for abstract and concrete cases is quite limiting for what can be done at the abstract level, and which causes the mentioned problem of inifinitely many proof cases. Taking the step that we do, introducing an explicit meta-language with meta-level constraints, eliminates this problem.
The use of a formal language provides a firm basis for automatic or semi-automatic support for deriving actual proofs, and our future plans include the development of such an implemented system. This requires a better understanding of how in general to construct abstract post states, given a state and an abstract derivation step; this is an important topic in our forthcoming research.
It is obvious to incorporate an existing confluence checker in such a system in order to identify and eliminate those $\alpha_1$ corners that are joinable even when invariant and equivalence are ignored.
One practical issue that needs to be understood better is how to cope with infinite splittings which have been exposed in our examples. It may be considered to allow meta-variables in \textsc{MetaCHR}{} to range over entire sub-derivations, suitably constrained at the meta-level. This may give rise to abstract diagrams that cover (in the formal sense we have defined) a range of differently shaped concrete diagrams. This potential is indicated informally in a diagram shown in Section~\ref{sec:infinite-split}, with a component indicating an entire sub-derivation, written as $\stackrel{\texttt{step}^n}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}$, so that we could show (still informally) an infinite set of corners joinable with a single argument.
A more detailed analysis of $\beta$-corners is desirable. We did not assume or impose any specific way of defining an equivalence, which means that any abstract $\beta$-corner needs to be considered as critical as soon as the equivalence is non-trivial. Huet~\cite{DBLP:journals/jacm/Huet80} has shown a lemma for term rewriting systems, which will be interesting to adapt for CHR (see our~Lemma~\ref{lemma:alpga-and-gamma}, p.~\pageref{lemma:alpga-and-gamma}). It applies ${\approx} = (\vdash\!\dashv)^{*}$ for some symmetric relation $\vdash\!\dashv$. Such a relation may be specified by a finite number of cases, as in a system of equations or logical equivalences. Here it seems possible to split each of our $\beta$-corners into a number of sub-cases, one for each case of the inductive definition of $\vdash\!\dashv$.
\appendix \section{Proof of Lemma~\ref{lem:critCorner}: the Critical Corner Lemma}
We recall the notation $\textit{all-relevant-app-recs}(S)$, Definition~\ref{def:state}, p.~\pageref{def:state}, that refers to the set of all application records for rules of the current program taking indices from the constraint store $S$.
\begin{proof} \emph{(Lemma~\ref{lem:critCorner}, p.~\pageref{lem:critCorner})} We consider a program with invariant $I$ and equivalence $\approx$, and we will go through the possible ways that an $I$-corner $\lambda$ can be non-joinable and in each such case point out a most general pre-corner $\Lambda$ that subsumes $\lambda$.
\noindent {\large{$\alpha_1$:}}\\ Let $\lambda=(\Sigma_1 \stackrel{R_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \Sigma \stackrel{R_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma_2)$
be an $\alpha_1$-corner that is not joinable with application instances $R_k=$\break$(r_k\colon A_k\backslash B_k\texttt{<=>}g_k|C_k)$ for $k=1,2$. Let now $H_k=A_k\cup B_k$, $k=1,2$, and $\mathit{Overlap}=(B_1\cap H_2)\cup(B_2\cap H_1)$.
In case $\mathit{Overlap}=\emptyset$, none of the application instances of $\lambda$ remove any constraint from the common ancestor state that prevents the other one from being successively applied. Thus there exists some state $\Sigma'$ such that $\Sigma_1\stackrel{R_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma'\stackrel{R_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}\Sigma_2$ which means $\lambda$ is joinable.
Assume now that $\mathit{Overlap}\neq\emptyset$ and we proceed as follows to produce a most general $\alpha$-pre-corner $\Lambda$ as follows. We select two most general application pre-instances $$
R_k^0\; = \; \bigl(r_k\colon A_k^0\backslash B_k^0\texttt{<=>}g_k^0|C_k^0\bigr),\quad i=1,2 $$ in such a way such that, for $k=1,2$, the indices in $R_k^0$ and $R_k$ are pairwise identical, compared in the order they appear. Define also $H_k^0=A_k^0\cup B_k^0$, $k=1,2$.
Let now, for $k=1,2$, $\mathit{Overlap}^0_k$ be the set of constraints in $R_k^0$ whose indices coincide with those of $\mathit{Overlap}$. Since $\mathit{Overlap}^0_1$ and $\mathit{Overlap}^0_2$ have the common instance $\mathit{Overlap}$, there exists a most general unifier $\sigma$ of $\mathit{Overlap}^0_1$ and $\mathit{Overlap}^0_2$; let furthermore $\theta$ be a smallest substitution that such that $\mathit{Overlap}^0_1\sigma\theta=\mathit{Overlap}^0_2\sigma\theta=\mathit{Overlap}$.
Noticing that $(g_1^0\sigma, g_2^0\sigma)$ is satisfiable (by $\theta$), we can define now the following most general critical $\alpha_1$-pre-corner, that we argue below subsumes $\lambda$. \begin{eqnarray*} \Lambda^0 & \;=\;
& \bigl( \circ\stackrel{R_1^0\sigma}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle S^0, T^0\rangle
\stackrel{R_2^0\sigma}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ
\bigl),\quad \text{where }\\ S^0 &\;=\;& H_1^0\sigma\cup H_2^0\sigma \\ T^0 & \;=\; &
\textit{all-relevant-app-recs}(H_1^0\sigma\cup H_2^0\sigma)
\;\setminus\; \{r_1@id(A_1B_1), r_2@id(A_2B_2) \} \end{eqnarray*} We should check that, if $r_1=r_2$, then $A^0_1\sigma \neq A^0_2\sigma$ or $B^0_1\sigma \neq B^0_2\sigma$ (cf.~Def.~\ref{def:pre-corners})
in order for the indicated $\Lambda^0$ to actually be a most general $\alpha_1$-pre-corner: if this is not the case, $\Sigma_1$ and $\Sigma_2$ would be identical and thus $\lambda$ joinable; contradiction.
Let now $\Sigma=\langle s, t\rangle$ (i.e., we name the parts of the common ancestor in $\lambda$) and we can define $s^+$, $t^+$ and $t^\div$ as follows \begin{eqnarray*} s^+ & = & s\setminus S^0\theta \\ t^+ &= & t\setminus T^0 \\%\{ ar \mid \text{$ar\in t$ is an application instance with at least one index in $s^+$}\} \\ t^\div &= & T^0\setminus t \end{eqnarray*} By construction, $R_k^0\sigma\theta=R_k$, $k=1,2$, and we can show that the following properties hold. \begin{eqnarray*} s & = & S^0\theta \uplus s^+ \\ t &= & T^0 \uplus t^+ \setminus t^\div \\
t^+ & \subseteq & \textit{all-relevant-app-recs}(S\theta\uplus s^{+})\setminus\textit{all-relevant-app-recs}(S\theta)\\ t^\div &\subseteq & \textit{all-relevant-app-recs}(S\theta) \end{eqnarray*}
Thus, the conditions of Definition \ref{def:subsumption-by-gcpc} are satisfied, proving that $\Lambda^0$ subsumes $\lambda$.
\noindent {\large{$\alpha_2$:}}\\ Let $\lambda=(\Sigma_1 \stackrel{R}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \Sigma \stackrel{b}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma_2)$
be an $\alpha_2$ corner that is not joinable, with application instance $R=(r\colon A\backslash B\texttt{<=>}g|C)$, $g$ non-logical or $I$-incomplete, and built-in $b$.
Define now the following most general application pre-instance $$
R^0\; = \; \bigl(r\colon H^0\texttt{<=>}g^0|C^0\bigr) $$ in such a way such that the indices in $R^0$ and $R$ are pairwise identical, compared in the order they appear, and let furthermore $b^0$ be a most general indexed built-in atom with same predicate and index as $b$. We define now the following most general critical $\alpha_2$-pre-corner that we will argue subsumes $\lambda$. \begin{eqnarray*} \Lambda^0\ & \;=\;
& \bigl( \circ\stackrel{R^0}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle H^0, T^0\rangle
\stackrel{b^0}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ
\bigl),\quad \text{where} \\ T^0 & \;=\; &
\textit{all-relevant-app-recs}(H^0)
\;\setminus\; \{r@id(H^0) \} \end{eqnarray*} Let $\theta$ be a smallest substitution such that $R^0\theta=R$ and $b^0\theta=b$. If $\mathit{vars}(b)\cap\mathit{vars}(g)=\emptyset$, there would exists a state $\Sigma'$ such that $\Sigma_1\stackrel{b}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma'\stackrel{R}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}\Sigma_2$; contradiction. The remaining arguments to show that $\Lambda^0$ subsumes $\lambda$ are exactly as for the $\alpha_1$ case.
\noindent {\large{$\alpha_3$:}}\\ Let $\lambda=(\Sigma_1 \stackrel{b_1}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}} \Sigma \stackrel{b_2}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma_2)$ be an $\alpha_3$-corner that is not joinable, with built-ins $b_1,b_2$, where $b_1$ is non-logical or $I$-incomplete. We define now the following most general critical $\alpha_2$-pre-corner that we will argue subsumes $\lambda$. \begin{eqnarray*} \Lambda^0\ & \;=\;
& \bigl( \circ\stackrel{b_1^0}\mathrel{\mbox{$\leftarrow$\hskip -0.2ex\vrule width 0.15ex height 1ex\hbox to 0.3ex{}}}
\langle H^0, \emptyset\rangle
\stackrel{b_2^0}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ
\bigl) \end{eqnarray*} To show that $\Lambda^0$ subsumes $\lambda$, let $\theta$ be a smallest substitution such that $b_k^0\theta=b_k$, $k=1$, and proceed exactly as in the $\alpha_1$ case (with $T_0=\emptyset$). We should also notice that it must hold that $\mathit{vars}(b_1)\cap\mathit{vars}(b_2)\neq\emptyset$ as otherwise $b_1$ and $b_2$ would commute and $\lambda$ be joinable.
\noindent {\large{$\beta_1$:}}\\ Let $\lambda=(\Sigma_1 \approx \Sigma \stackrel{R}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\Sigma_2)$
be a $\beta$-corner that is not joinable, where $R$ is an application instance with application instance $R=(r\colon A\backslash B\texttt{<=>}g|C_k)$. Define now the following most general application pre-instance $$
R^0\; = \; \bigl(r\colon H^0\texttt{<=>}g^0|C^0\bigr) $$ in such a way such that the indices in $R^0$ and $R$ are pairwise identical, compared in the order they appear. The proof that the following most general critical $\beta_1$-pre-corner subsumes $\lambda$ is similar to the previous cases. \begin{eqnarray*} \Lambda^0\ & \;=\;
& \bigl( \circ\approx
\langle H^0, T^0\rangle
\stackrel{R^0}\mathrel{\mbox{\hbox to 0.2ex{}\vrule width 0.15ex height 1ex\hskip -0.2ex$\rightarrow$}}\circ
\bigl),\quad \text{where} \\ T^0 & \;=\; &
\textit{all-relevant-app-recs}(H^0)
\;\setminus\; \{r@id(H^0) \} \end{eqnarray*}
\noindent {\large{$\beta_2$:}}\\ Analogous to the $\beta_1$ case and omitted. \end{proof}
\end{document} | arXiv |
nature astronomy
Evidence for the volatile-rich composition of a 1.5-Earth-radius planet
Caroline Piaulet ORCID: orcid.org/0000-0002-2875-917X1,
Björn Benneke ORCID: orcid.org/0000-0001-5578-14981,
Jose M. Almenara2,
Diana Dragomir ORCID: orcid.org/0000-0003-2313-467X3,
Heather A. Knutson4,
Daniel Thorngren1,5,
Merrin S. Peterson1,
Ian J. M. Crossfield6,
Eliza M.-R. Kempton ORCID: orcid.org/0000-0002-1337-90517,
Daria Kubyshkina ORCID: orcid.org/0000-0001-9137-98188,
Andrew W. Howard9,
Ruth Angus10,11,
Howard Isaacson ORCID: orcid.org/0000-0002-0531-107312,
Lauren M. Weiss ORCID: orcid.org/0000-0002-3725-305813,
Charles A. Beichman14,
Jonathan J. Fortney5,
Luca Fossati ORCID: orcid.org/0000-0003-4426-95308,
Helmut Lammer8,
P. R. McCullough15,16,
Caroline V. Morley17 &
Ian Wong18
Nature Astronomy (2022)Cite this article
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The population of planets smaller than approximately 1.7 Earth radii (R⊕) is widely interpreted as consisting of rocky worlds, generally referred to as super-Earths. This picture is largely corroborated by radial velocity mass measurements for close-in super-Earths but lacks constraints at lower insolations. Here we present the results of a detailed study of the Kepler-138 system using 13 Hubble and Spitzer transit observations of the warm-temperate 1.51 ± 0.04 R⊕ planet Kepler-138 d (\({T}_{\rm{eq,A_{\rm{B}} = 0.3}}\approx 350\,{\mathrm{K}}\)) combined with new radial velocity measurements of its host star obtained with the Keck/High Resolution Echelle Spectrometer. We find evidence for a volatile-rich 'water world' nature of Kepler-138 d, with a large fraction of its mass $M_{\rm{d}}$ contained in a thick volatile layer. This finding is independently supported by transit timing variations and radial velocity observations (\({M}_{{{{\rm{d}}}}}=2.{1}_{-0.7}^{+0.6}\,{M}_{\oplus }\)), as well as the flat optical/infrared transmission spectrum. Quantitatively, we infer a composition of \(1{1}_{-4}^{+3}\%\) volatiles by mass or ~51% by volume, with a 2,000-km-deep water mantle and atmosphere on top of a core with an Earth-like silicates/iron ratio. Any hypothetical hydrogen layer consistent with the observations (<0.003 M⊕) would have swiftly been lost on a ~10 Myr timescale. The bulk composition of Kepler-138 d therefore resembles those of the icy moons, rather than the terrestrial planets, in the Solar System. We conclude that not all super-Earths are rocky worlds, but that volatile-rich water worlds exist in an overlapping size regime, especially at lower insolations. Finally, our photodynamical analysis also reveals that Kepler-138 c (with a Rc = 1.51 ± 0.04 R⊕ and a \({M}_{{{{\rm{c}}}}}=2.{3}_{-0.5}^{+0.6}\,{M}_{\oplus }\)) is a slightly warmer twin of Kepler-138 d (that is, another water world in the same system) and we infer the presence of Kepler-138 e, a likely non-transiting planet at the inner edge of the habitable zone.
Fig. 1: Results from the four-planet photodynamical analysis of the HST, Spitzer and Kepler light curves of Kepler-138.
Fig. 2: Comparison of Kepler-138 c and d with the population of super-Earths.
Fig. 3: Low density of Kepler-138 c and d compared with rocky compositions.
Fig. 4: Planet structure modelling results for Kepler-138 d.
The data used in this paper are deposited on publicly available servers. The data from the HST used in this work can be downloaded from the MAST at https://archive.stsci.edu/proposal_search.php?mission=hst&id=13665. The data from the Spitzer Space Telescope included in our analysis is available on the SHA at https://sha.ipac.caltech.edu/applications/Spitzer/SHA/#id=SearchByProgram&RequestClass=ServerRequest&DoSearch=true&SearchByProgram.field.program=11131&MoreOptions.field.prodtype=aor,pbcd&shortDesc=Program&isBookmarkAble=true&isDrillDownRoot=true&isSearchResult=true. The Keck/HIRES radial velocities are available online as Supplementary Dataset 1. The planet population plots used data from the public NASA Exoplanet Archive (https://exoplanetarchive.ipac.caltech.edu/), which also hosts an interface where the Kepler photometry can be downloaded.
Code availability
The smint code is publicly available via GitHub at https://github.com/cpiaulet/smint. The RV analysis is based on the publicly available packages george, RadVel and emcee. Further scripts can be provided by the corresponding author upon reasonable request.
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We gratefully acknowledge the open-source software that made this work possible: LDTK34, batman35, emcee36, TTVFast37, REBOUND38, WHFast39, nestle (https://github.com/kbarbary/nestle; refs. 40,41,42,43), astropy126,127, numpy128, ipython129, matplotlib130, RadVel44, george45, smint14 and GNU parallel78. This work is based on observations with the NASA/ESA HST, obtained at the Space Telescope Science Institute (STScI) operated by AURA, Inc. We received support for the analysis by NASA through grants under the HST-GO-13665 programme (PI B.B). This work relies on observations made with the Spitzer Space Telescope, which was operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. This paper includes data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. This study has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for DPAC has been provided by national institutions, particularly the institutions participating in the Gaia Multilateral Agreement. Data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) and the Spitzer Heritage Archive (SHA). This research has made use of NASA's Astrophysics Data System and the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with NASA within the Exoplanet Exploration Program. Parts of this analysis have been run on the Lesta cluster kindly provided by the Observatoire de Genève. C.P. acknowledges financial support from the Fonds de Recherche Québécois—Nature et Technologie (FRQNT; Quebec), the Technologies for Exo-Planetary Science (TEPS) Trainee Program and the Natural Sciences and Engineering Research Council (NSERC) Vanier Scholarship. D.D. acknowledges support from the TESS Guest Investigator Program grant number 80NSSC19K1727 and NASA Exoplanet Research Program grant number 18-2XRP18_2-0136. B.B. acknowledges financial support from the NSERC of Canada and the FRQNT. I.W. is supported by an appointment to the NASA Postdoctoral Program at the NASA Goddard Space Flight Center, administered by Oak Ridge Associated Universities under contract with NASA. C.V.M. acknowledges HST funding through grant number HST-AR-15805.001-A from STScI.
Department of Physics and Trottier Institute for Research on Exoplanets, Université de Montréal, Montreal, Quebec, Canada
Caroline Piaulet, Björn Benneke, Daniel Thorngren & Merrin S. Peterson
Université Grenoble Alpes, CNRS, IPAG, Grenoble, France
Jose M. Almenara
Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, USA
Diana Dragomir
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, USA
Heather A. Knutson
Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA, USA
Daniel Thorngren & Jonathan J. Fortney
Department of Physics and Astronomy, University of Kansas, Lawrence, KS, USA
Ian J. M. Crossfield
Department of Astronomy, University of Maryland, College Park, MD, USA
Eliza M.-R. Kempton
Space Research Institute, Austrian Academy of Sciences, Graz, Austria
Daria Kubyshkina, Luca Fossati & Helmut Lammer
Department of Astronomy, California Institute of Technology, Pasadena, CA, USA
Andrew W. Howard
Department of Astrophysics, American Museum of Natural History, Manhattan, NY, USA
Center for Computational Astrophysics, Flatiron Institute, Manhattan, NY, USA
Department of Astronomy, University of California Berkeley, Berkeley, CA, USA
Department of Physics, University of Notre Dame, Notre Dame, IN, USA
Lauren M. Weiss
California Institute of Technology/IPAC, NASA Exoplanet Science Institute, Pasadena, CA, USA
Charles A. Beichman
The William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, USA
P. R. McCullough
Space Telescope Science Institute, Baltimore, MD, USA
Department of Astronomy, University of Texas, Austin, TX, USA
Caroline V. Morley
NASA Goddard Space Flight Center, Greenbelt, MD, USA
Ian Wong
Caroline Piaulet
Björn Benneke
Daniel Thorngren
Merrin S. Peterson
Daria Kubyshkina
Jonathan J. Fortney
Luca Fossati
Helmut Lammer
C.P. and B.B. conceived the project. C.P. wrote the manuscript and carried out the reduction of the HST and Spitzer data, as well as the TTV, RV, atmospheric escape, atmospheric retrieval and planetary structure analyses, under the supervision of B.B and with the help of M.S.P. for the TTV analysis and contributions from D.K. to the upper atmosphere modelling. J.M.A. realized the photodynamical analysis and the transit search for Kepler-138 e. D.D. provided the Spitzer observations. H.A.K., A.W.H., H.I., L.M.W. and C.A.B. conducted the observations and reduction of the HIRES RVs. D.T. provided the grid of interior models. R.A. constrained the stellar age. All co-authors provided comments and suggestions on the manuscript.
Correspondence to Caroline Piaulet.
Nature Astronomy thanks the anonymous reviewers for their contribution to the peer review of this work
Extended Data Fig. 1 Three-planet photodynamical fit results.
a,b,c Same as Fig. 1a-c, for a photodynamical fit including only the three previously-known planets Kepler-138 b,c, and d. This illustrates the extent to which the timescale over which the predicted transit times of Kepler-138d are modulated (the super-period) is underestimated by the three-planet solution. This discrepancy was already hinted at by the mismatch with the Kepler transit times but revealed at high significance by the now longer baseline over which we obtained transits with HST and Spitzer, at times beyond BJD=2457000.
Extended Data Fig. 2 Folded Kepler transits of Kepler-138 b, c, and d, and search for the transit of Kepler-138 e.
The four panels show the corrected light curve of Kepler-138 (open circles) folded in a 2 day window around the expected transit epochs of Kepler-138 b, c, d, and e from the photodynamical fit (see Methods). Transit models corresponding to the median retrieved planet parameters are superimposed to the data (solid colored lines), conservatively assuming an Earth-like composition to estimate the radius of Kepler-138 e. The transits of Kepler-138 b, c, and d are detected in the Kepler light curve, but while Kepler-138 e should be larger than Kepler-138 b, its transit is not detected. We interpret this as originating from a likely non-transiting configuration of Kepler-138 e's orbit, with an inclination of ≲ 89∘ consistent with the photodynamical solution, too low to occult the stellar disk from our perspective.
Extended Data Fig. 3 Search for prominent periodicities in the RV and photometric dataset.
From top to bottom, Lomb-Scargle periodogram of the RV dataset, the Kepler light curve, the activity indicator (S-index) and the window function of the RVs. The orbital periods of the four planets, the rotational period of the star and its first harmonic are shown. False-alarm probability levels of 0.1, 1 and 10% are indicated by dashed gray lines in the top two panels. Significant signals are detected at the stellar period and its first harmonic in the light curve. No significant periodicity is detected in the RV and S-index time series.
Extended Data Fig. 4 Gaussian Process fit to the Kepler photometry.
Zoom on the last 200 days of the Kepler photometric observations (black points) and the best-fitting stellar activity model using a GP (gray shading). The mean is the solid line and the variance is shown as the shaded region. The lower panel shows residuals around the best-fit model divided by the single-point scatter. Posterior constraints on the stellar rotation period from rotational brightness modulations are shown on the right. The GP model reproduces the photometric variability and provides tight constraints on the covariance structure of the stellar signal.
Extended Data Fig. 5 Median four-planet Keplerian orbital model for Kepler-138.
A trained GP model was used to account for stellar activity in the RV fit. The model corresponding to the median retrieved parameters is plotted in purple while the corresponding parameters are annotated in each panel. We add in quadrature the RV jitter term (Supplementary Table 2) with the measurement uncertainties for all RVs. a, Full HIRES time series. b, Residuals to the best fit model. c, RVs phase-folded to the ephemeris of planet b. The phase-folded model for planet b is shown (purple line), while Keplerian orbital models for all other planets have been subtracted. d,e,f, Same as c for Kepler-138 c, d, and e.
Extended Data Fig. 6 Illustration of the coupling of interior and atmosphere models.
a, Temperature-pressure and b, temperature-radius profiles computed to generate a complete planet model for a mass of 2.36 M⊕, a H2/He mass fraction of 3%, and no water layer. Self-consistent atmosphere models are shown down to the radiative–convective boundary (dotted, black), for the irradiation of Kepler-138 d, but varying the reference radius at a pressure of 1 kbar. Interior models are displayed for the same composition but different specific entropies (solid, colors). For consistency, full-planet models with a given planet mass and composition are obtained from the combination of interior and atmosphere model that have both matching temperatures and radii at the radiative–convective boundary (bold profiles show the closest match in this example).
Extended Data Fig. 7 Composition of Kepler-138 d for the hydrogen-free scenario.
We show the joint and marginalized posterior distributions of the planet structure fit for Kepler-138 d in the case of a hydrosphere lying on top of a rock/iron core. The 1, 2 and 3σ probability contours are shown. As expected, the water mass fraction is strongly correlated with the relative amount of rock and iron. The correlation between irradiance temperature and water mass fraction is weak across the considered temperature range.
Extended Data Fig. 8 Impact of unocculted stellar spots on the Kepler transit depth measurement.
Transmission spectrum of Kepler-138 d (black points) superimposed with three scenarios for the level of stellar contamination: spot covering fractions of 0.1, 3 or 10% (colored lines, colored filled circles show bandpass-integrated values). The potential impact of unocculted stellar spots on the radius in the Kepler bandpass is small compared to its measurement uncertainty.
Extended Data Fig. 9 Composition of Kepler-138 c for the hydrogen-free scenario.
Same as Extended Data Figure 7, for Kepler-138 c.
Extended Data Fig. 10 Constraints on the atmospheric composition from transmission spectroscopy.
a, Optical-to-IR transmission spectrum of Kepler-138 d, compared with three representative forward models: a H2/He atmosphere with a solar composition, a high-metallicity cloud-free atmosphere and a cloudy hydrogen-dominated atmosphere. b, Joint posterior probability density of the cloud top pressure Pcloud and atmospheric metallicity, along with the corresponding mass fraction of metals Z assuming a solar C/O ratio. The color encodes the density of posterior samples in each bin and the contours indicate the 2 and 3σ constraints. The location in the parameter space of the three models from panel a is shown with 'x' markers. The constraints reflect the well-documented degeneracy between increasing mean molecular weight of the atmosphere and cloud top pressure in terms of the strength of absorption features77. The cloud-free, solar-metallicity scenario is excluded at 2.5σ. The new planet mass leads to an increased surface gravity which motivates further spectroscopic follow-up to obtain more precise constraints on the atmospheric composition.
Supplementary Figs. 1–13 and Tables 1–3.
Supplementary Dataset 1
Machine-readable table of the Keck/HIRES RVs of Kepler-138.
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
Piaulet, C., Benneke, B., Almenara, J.M. et al. Evidence for the volatile-rich composition of a 1.5-Earth-radius planet. Nat Astron (2022). https://doi.org/10.1038/s41550-022-01835-4
Nature Astronomy (Nat Astron) ISSN 2397-3366 (online) | CommonCrawl |
How many three-digit numbers are divisible by 13?
The smallest three-digit number divisible by 13 is $13\times 8=104$, so there are seven two-digit multiples of 13. The greatest three-digit number divisible by 13 is $13\times 76=988$. Therefore, there are $76-7=\boxed{69}$ three-digit numbers divisible by 13.
\[ OR \]Because the integer part of $\frac{999}{13}$ is 76, there are 76 multiples of 13 less than or equal to 999. Because the integer part of $\frac{99}{13}$ is 7, there are 7 multiples of 13 less than or equal to 99. That means there are $76-7=\boxed{69}$ multiples of 13 between 100 and 999. | Math Dataset |
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Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory
Authors: Uwe Jannsen and Shuji Saito
Journal: J. Algebraic Geom. 21 (2012), 683-705
Published electronically: February 21, 2012
Abstract | References | Additional Information
Abstract: We show the existence of good hyperplane sections for schemes over discrete valuation rings with good or (quasi-)semi-stable reduction, and the existence of good Lefschetz pencils for schemes with good reduction or ordinary quadratic reduction. As an application, we prove that the reciprocity map introduced for smooth projective varieties over local fields $K$ by Bloch, Kato and Saito is an isomorphism after $\ell$-adic completion, if the variety has good or ordinary quadratic reduction and $\ell \neq \mathrm {char}(K)$.
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Uwe Jannsen and Shuji Saito, Kato homology of arithmetic schemes and higher class field theory over local fields, Doc. Math. Extra Vol. (2003), 479–538. Kazuya Kato's fiftieth birthday. MR 2046606
Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR 725671
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Markus Rost, Norm varieties and algebraic cobordism, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 77–85. MR 1957022
Shuji Saito, Class field theory for curves over local fields, J. Number Theory 21 (1985), no. 1, 44–80. MR 804915, DOI https://doi.org/10.1016/0022-314X%2885%2990011-3
Kanetomo Sato, Non-divisible cycles on surfaces over local fields, J. Number Theory 114 (2005), no. 2, 272–297. MR 2167971, DOI https://doi.org/10.1016/j.jnt.2004.08.015
Andrei Suslin and Seva Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1-2, 245–276. MR 2220090, DOI https://doi.org/10.1016/j.jpaa.2005.12.012
Vladimir Voevodsky, Motivic cohomology with ${\bf Z}/2$-coefficients, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 59–104. MR 2031199, DOI https://doi.org/10.1007/s10240-003-0010-6
Vladimir Voevodsky, On motivic cohomology with $\mathbf Z/l$-coefficients, Ann. of Math. (2) 174 (2011), no. 1, 401–438. MR 2811603, DOI https://doi.org/10.4007/annals.2011.174.1.11
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Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
S. Bloch, Higher Algebraic $K$-theory and class field theory for arithmetic surfaces, Ann. of Math. 114 (1981), 229–265. MR 632840 (83m:14025)
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C. Haesemeyer, C. Weibel, Norm varieties and the chain lemma (after Markus Rost), Algebraic topology, 95130, Abel Symp., 4, Springer, Berlin, 2009. MR 2597737 (2011f:19002)
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S. Saito, Class field theory for curves over local fields, J. Number Theory 21 (1985), 44–80. MR 804915 (87g:11075)
K. Sato, Non-divisible cycles on surfaces over local fields, J. Number Theory 114 (2005), no. 2, 272–297. MR 2167971 (2006g:19006)
A. Suslin, Andrei, S. Joukhovitski, Norm varieties. J. Pure Appl. Algebra 206 (2006), no. 1-2, 245–276. MR 2220090 (2008a:14015)
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V. Voevodsky, Motivic Eilenberg-MacLane spaces, http://arxiv.org/abs/0805.4432, Publ. Math. IHES No. 112 (2010), 199. MR 2737977
C. Weibel, The norm residue isomorphism theorem, J. Topol. 2 (2009), no. 2, 346–372. MR 2529300 (2011a:14039)
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Uwe Jannsen
Affiliation: Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany
Email: [email protected]
Shuji Saito
Affiliation: Graduate School of Mathematics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8914, Japan
Address at time of publication: Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551 Japan
Email: [email protected]
Received by editor(s): November 9, 2009
Received by editor(s) in revised form: March 9, 2010
Additional Notes: The first author was supported by DFG Research Group FOR 570 'Algebraic Cycles and L-Functions'. The second author was supported by JSPS Grant-in-Aid, Scientific Research B-18340003 and Scientific Research S-19104001 | CommonCrawl |
\begin{document}
\title{Slanted Vector Fields for Jet Spaces} \author{Lionel Darondeau} \address{Lionel Darondeau\\Laboratoire de Mathématiques d'Orsay\\Université Paris-Sud (France).} \email{[email protected]} \date{26th February, 2015}
\begin{abstract}
Low pole order frames of slanted vector fields are constructed on the space of vertical \(k\)-jets of the universal family of complete intersections in \(\P^n\) and, adapting the arguments, low pole order frames of slanted vector fields are also constructed on the space of vertical logarithmic \(k\)-jets along the universal family of projective hypersurfaces in \(\P^n\) with several irreducible smooth components.
Both the pole order (here \(=5k-2\)) and the determination of the locus where the global generation statement fails are improved compared to the literature (previously \(=k^{2}+2k\)), thanks to three new ingredients:
we reformulate the problem in terms of some adjoint action, we introduce a new formalism of \textsl{geometric jet coordinates}, and then we construct what we call \textsl{building-block vector fields}, making the problem for arbitrary jet order \(k\geq 1\) into a very analog of the much easier case where \(k=0\), \textit{i.e.} where no jet coordinates are needed.
\noindent\textit{Keywords:}
{Slanted vector fields, geometric jet coordinates, logarithmic jets, variational method of Voisin-Siu, hyperbolicity, building-block vector fields.}
\end{abstract} \subjclass{32Q45,14J70,15A03} \maketitle
\section*{Introduction} \label{sec:intro} The formalism of \textsl{jets} is a coordinate-free description of the differential equations that holomorphic curves may satisfy. For a map \(f\colon\C\to X\), valued in a complex projective manifold \(X\), the \(k\)-jet map \(f_{[k]}\colon\C\to J_{k}X\) valued in the \(k\)-jet bundle \(J_{k}X\) corresponds to the truncated Taylor expansion of \(f\) at order \(k\) in some local coordinate system. In \(J_{k}X\), each \textsl{jet-coordinate} \(f_{i}^{(p)}\) shall be considered as an independent coordinate, whence each algebraic differential equation (with holomorphic coefficients) of order \(k\) shall be thought of as a polynomial equation in \(J_{k}X\): \[
P(f',f'',\dotsc,f^{(k)})
\equiv
0. \] Similarly, if \(D\subset X\) is a normal crossings divisor, the subsheaf \(J_{k}X(-\log D)\subset J_{k}X\) of \textsl{logarithmic \(k\)-jets} on \(X\) along \(D\) can be defined by considering the logarithmic derivatives in the direction of \(D\) (see below).
\subsubsection{Schwarz lemma.} A (logarithmic) \textsl{\(k\)-jet differential} is locally a polynomial in the (logarithmic) jet-coordinates \(f_{i}^{(p)}\) having constant homogeneous weight, when the weight of \(f_{i}^{(p)}\) is the number of ``primes'' \(p\). The jet differentials enjoy the following \textsl{fundamental vanishing theorem} (\cite{MR609557,MR1492539,Siu1996}).
\begin{itshape} If \(\omega\) is a holomorphic jet differential on \(X\) with logarithmic poles along \(D\), that vanishes on an ample divisor, and if \(f\) is a nonconstant holomorphic map \(\C\to X\setminus D\), then the pullback \(f^{\star}\omega=P(f',\dotsc,f^{(k)})\) vanishes identically on \(\C\). \end{itshape}
When the canonical divisor \(K_{X}+D\) is big, an interesting question, motivated by the longstanding Green-Griffiths conjecture (\cite{MR609557}), is the algebraic degeneracy of such holomorphic maps \(\C\to X\setminus D\). Starting with a lot of differential equations as above, the overall idea is to decrease the degree of the differential equations by elimination of the jet coordinates \(f_{i}^{(p)}\) until obtaining a differential equation of degree \(0\), that is an algebraic equation satisfied by every nonconstant entire curve \(f\colon\C\to X\setminus D\). We will briefly recall the key points of this strategy, already implemented both in the compact setting (\(X\setminus D=X_{d}\subset\P^{n}\), \cite{MR2593279}) and in the logarithmic setting (\(X\setminus D=\P^{n}\setminus X_{d}\), \cite{arXiv:1402.1396}). For more details, the reader is referred to the comprehensive recent article \cite{arXiv:1209.2723S} by Yum-Tong Siu.
\subsubsection{Siu's strategy.} The general idea is that the vector fields \(\mathsf{V}\in T_{J_{k}X}\) applied to \(\omega\) produce new differential equations. However, such equations do not necessarily enjoy the fundamental vanishing theorem, since if the pole order of a vector field \(\mathsf{V}\) is bigger than the vanishing order of \(\omega\), then the hypotheses of the theorem are not satisfied by \(\mathsf{V}\cdot\omega\) anymore! It is thus crucial to control the pole order of these vector fields.
On a regular hypersurface of high degree, there cannot be sufficiently many nonzero meromorphic vector fields having low pole order, but according to a strategy due to Voisin and Siu, in order to get a lot of low pole order vector fields, one can use the positivity of the moduli space\(\abs{\mathcal{O}(d)}\) of all degree \(d\) hypersurfaces in \(\P^{n}\): \[
\abs{\mathcal{O}(d)}
\mathrel{\mathop{:}}=
\P\Big(H^{0}\big(\P^{n},\mathcal{O}_{\P^{n}}(d)\big)\Big)
=
\P\Big\{\sum_{\abs{\alpha}=d} A_{\alpha}Z^{\alpha}\colon A_{\alpha}\in\C\Big\}
=
\P^{n_{d}}, \] where for \(\alpha\in\N^{n+1}\), \(\abs{\alpha}\mathrel{\mathop{:}}= \alpha_{0}+\dotsb+\alpha_{n}\) and \(Z^{\alpha}\mathrel{\mathop{:}}= Z_{0}^{\alpha_{0}}\dotsm Z_{n}^{\alpha_{n}}\) and where \(n_{d}\mathrel{\mathop{:}}=\binom{n+d}{d}-1\).
In what will be called the \textsl{compact case}, we will consider the universal family of complete intersections of multi-degree \(d_{1},\dotsc,d_{c}\): \[
\mathcal{Y}_{d_{1},\dotsc,d_{c}}
\mathrel{\mathop{:}}=
\Big\{
\sum_{\abs{\alpha}=d_{1}} A_{\alpha}^{1}\,Z^{\alpha}
=
\dotsb
=
\sum_{\abs{\alpha}=d_{c}} A_{\alpha}^{c}\,Z^{\alpha}
=
0
\Big\}
\subset
\P^{n}
\times
\abs{\mathcal{O}\xspace(d_{1})}\times\dotsb\times\abs{\mathcal{O}\xspace(d_{c})}, \] (where of course the coefficients \(A_{\alpha}^{j}\) are the homogeneous coordinates on the corresponding \(\abs{\mathcal{O}\xspace(d_{j})}\)), together with the space of \(k\)-jets of this universal family \(\mathcal{Y}_{d_{1},\dotsc,d_{c}}\): \[
J_{k,d_{1},\dotsc,d_{c}}
\mathrel{\mathop{:}}=
J_{k}\mathcal{Y}_{d_{1},\dotsc,d_{c}}. \]
In what will be called the \textsl{logarithmic case}, we will consider the universal family of normal crossings divisors with \(c\) smooth irreducible components, of respective degrees \(d_{1}\), \dots, \(d_{c}\) in \(\P^{n}\): \[
\mathcal{H}_{d_{1},\dotsc,d_{c}}
\mathrel{\mathop{:}}=
\Big\{
\big(\sum_{\abs{\alpha}=d_{1}} A_{\alpha}^{1}\,Z^{\alpha}\big)
\dotsm
\big(\sum_{\abs{\alpha}=d_{c}} A_{\alpha}^{c}\,Z^{\alpha}\big)
=
0
\Big\}
\subset
\P^{n}
\times
\abs{\mathcal{O}\xspace(d_{1})}\times\dotsb\times\abs{\mathcal{O}\xspace(d_{c})}, \] together with the space of logarithmic \(k\)-jets along this universal family \(\mathcal{H}_{d_{1},\dotsc,d_{c}}\): \[
\bar{J}_{k,d_{1},\dotsc,d_{c}}
\mathrel{\mathop{:}}=
J_{k}(\P^{n}\times\abs{\mathcal{O}\xspace(d_{1})}\times\dotsb\times\abs{\mathcal{O}\xspace(d_{c})})(-\log\mathcal{H}_{d_{1},\dotsc,d_{c}}). \]
In both cases, let \(\eta\colon f_{[k]}\mapsto f(0)\) denote the evaluation of the jets and, for shortness, let us denote the space parametrizing the considered universal families by: \[
S_{d_{1},\dotsc,d_{c}}
\mathrel{\mathop{:}}=
\abs{\mathcal{O}\xspace(d_{1})}\times\dotsb\times\abs{\mathcal{O}\xspace(d_{c})}
=
\P^{n_{d_{1}}}\times\dotsb\times\P^{n_{d_{c}}}. \]
The space of \textsl{vertical \(k\)-jets} is the subspace \(J_{k,d_{1},\dotsc,d_{c}}^{\mathit{vert}}\subset J_{k,d_{1},\dotsc,d_{c}}\) consisting of jets tangent to the fibers of the second projection \[
\mathrm{pr_{2}}
\colon
\P^{n}\times S_{d_{1},\dotsc,d_{c}}\to S_{d_{1},\dotsc,d_{c}}. \] Similarly, the space of \textsl{vertical logarithmic \(k\)-jets} is the subspace \(\bar{J}_{k,d_{1},\dotsc,d_{c}}^{\mathit{vert}}\subset \bar{J}_{k,d_{1},\dotsc,d_{c}}\) consisting of logarithmic jets tangent to the fibers of the second projection \(\mathrm{pr_{2}}\). These jets are introduced in order to use the Schwarz lemma fiberwise (see \cite{MR2593279,arXiv:1402.1396}).
\begin{THM}
Suppose that the order \(k\) of the jets is smaller than the degrees \(d_{1},\dotsc,d_{c}\), then
\begin{itemize}
\setlength\itemsep{1em}
\item{\normalfont(\textsl{Compact} case)}
The twisted holomorphic tangent bundle to vertical \(k\)-jets of the universal family of complete intersections of multi-degree \(d_{1},\dotsc,d_{c}\):
\[
T_{J_{k,d_{1},\dotsc,d_{c}}^{\mathit{vert}}}
\otimes
\eta^{\star}\Bigl(
\mathcal{O}\xspace_{\P^{n}}\bigl(5k-2\bigr)
\otimes
\mathcal{O}\xspace_{S_{d_{1},\dotsc,d_{c}}}(1,\dotsc,1)
\Bigr)
\]
is generated by its holomorphic global sections at every point of the subspace of \(J_{k,d_{1},\dotsc,d_{c}}^{\mathit{vert}}\) made of \(k\)-jets of non stationary holomorphic curves \(\C\to\mathcal{Y}_{d_{1},\dotsc,d_{c}}\) tangent to the fibers of the second projection
\(\P^{n}\times S_{d_{1},\dotsc,d_{c}} \to S_{d_{1},\dotsc,d_{c}}\).
\item{\normalfont(\textsl{Logarithmic} case)}
The twisted holomorphic tangent bundle to vertical logarithmic \(k\)-jets along the universal family of normal crossing divisors having irreducible smooth components of degrees \(d_{1},\dotsc,d_{c}\)
\[
T_{\bar{J}_{k,d_{1},\dotsc,d_{c}}^{\;\mathit{vert}}}
\otimes
\eta^{\star}\Bigl(
\mathcal{O}\xspace_{\P^{n}}\bigl(5k-2\bigr)
\otimes
\mathcal{O}\xspace_{S_{d_{1},\dotsc,d_{c}}}(1,\dotsc,1)
\Bigr)
\]
is generated by its holomorphic global sections at every point of the subspace of \(\bar{J}_{k,d_{1},\dotsc,d_{c}}^{\;\mathit{vert}}\) made of logarithmic \(k\)-jets of non stationary holomorphic curves
\[
\C\to(\P^{n}\times S_{d_{1},\dotsc,d_{c}})\setminus\mathcal{H}_{d_{1},\dotsc,d_{c}}
\]
tangent to the fibers of the second projection
\(\P^{n}\times S_{d_{1},\dotsc,d_{c}} \to S_{d_{1},\dotsc,d_{c}}\).
\end{itemize} \end{THM}
Actually, the extrinsic vector fields that generate the tangent bundle are not tangential to the space of \(k\)-jets of the vertical fiber at a generic point, what justifies to term them \textsl{slanted vector field} as Siu does.
When low pole order meromorphic slanted vector fields are used in order to eliminate the derivatives \(f',\dotsc,f^{(k)}\), following Siu's strategy, our main Theorem essentially yields that actually every single algebraic coefficient (depending only on \(f\)) of each algebraic differential equation \(P(f',\dotsc,f^{(k)})\equiv0\) has to vanish. For a more detailed account, see \cite{arXiv:1209.2723S,MR2593279,arXiv:1402.1396}, as we will now focus on the proof of our statement.
\subsubsection{State of the art.} The method of slanted vector fields has been introduced by Siu (\cite{MR2077584}), and is motivated by the work of Clemens-Ein-Voisin (\cite{MR875091,MR958594,MR1420353}) on rational curves. It has been pushed further in dimension \(2\) by P\u{a}un (\cite{MR2372741}) for the compact case and by Rousseau (\cite{MR2552951}) for the logarithmic case, with several smooth components, and in dimension \(3\) by Rousseau, both for the compact case (\cite{MR2331545}) and for the logarithmic case (\cite{MR2383820}). In the compact case, the technique has been generalized in any dimension by Merker (\cite{MR2543663}), with a substantial improvement of the determination of the locus where the global generation statement fails, leading to a proof of the \emph{strong} algebraic degeneracy of entire curves with values in a generic projective hypersurface of effective large degree (\cite{MR2593279}). In the slightly different context of projective hypersurfaces in families, Mourougane (\cite{MR2911888}) has implemented the technique in any dimension and for any jet order.
\subsubsection{Organization of the paper.} In \S1, we recall the formalism of jets and we introduce new \textsl{geometric} jet coordinates, together with the associated vector fields, that exist on the subspace of invertible jets. When all is said and done, it will appear that this formalism significantly simplifies the computations, because it allows to take advantage of the triangular properties of the iterations of the chain rule, which rule of course plays a central role when dealing with jet spaces.
In \S2, we describe a new general strategy to easily construct slanted vector fields on the subspace of invertible jets, following the formal differentiation presentation of Merker in \cite{MR2543663}, and we implement this strategy in the compact case, for \(c=1\). In the directions tangent to the space spanned by the jet coordinates, starting with guiding examples, we retrieve the results presented in the work of P\u{a}un (\cite{MR2372741}), Rousseau (\cite{MR2331545,MR2383820,MR2552951}) and Merker (\cite{MR2543663}) by considering the first order geometric jet coordinates, and then we go further by using the higher order jet coordinates. This allows to clarify the locus where the global generation fails, a detail that was only briefly treated in the previous works. In the directions tangent to the space of parameters, the arguments introduced by Merker in \cite{MR2543663} could directly apply, and we merely reformulate them in terms of geometric jet coordinates, which will prove to be useful for computing more accurately the pole order of the meromorphic prolongations of the described vector fields.
In \S3, we prove the main theorem on global generation of the tangent bundle to \(k\)-jet spaces. In the case of the universal hypersurface, considered in the previous section, we compute the pole order of the frame of slanted vector fields that we have constructed. Then we adapt the result to the case of general complete intersections, with \(c\geq1\). After that, we also adapt the result to the logarithmic case. For this, we follow the strategy implemented by Rousseau in \cite{MR2552951,MR2383820} for complements of hypersurfaces in \(\P^{2}\) and \(\P^3\), by locally straightening out the universal family \(\mathcal{H}_{d_{1},\dotsc,d_{c}}\).
\section{Geometric Coordinates for Invertible Jets.} \label{sec:logJetBdl} So we will define the \textsl{geometric jet coordinates}, that will play a pivotal role in our simplifying strategy, and then study the associated vector fields. Let us start with some standard material.
\subsubsection{Logarithmic jet bundles.} \label{sse:jetMfld} Given a point \(x\) in a \(n\)-dimensional complex manifold \(X\), the \textsl{\(k\)-jet} of a germ of holomorphic map \(f\colon(\C,0)\to(X,x)\) is the equivalence class \(f_{[k]}\) of germs that osculate with \(f\) to order \(k\) at the origin of \(\C\). Fixing some local coordinate system \((z_{1},\dotsc,z_{n})\) over an open subset \(U\subset X\) around \(x\), the space of \(k\)-jets \(J_{k}X\vert_{x}\) therefore identifies, by Taylor formula, to the vector space \(\C^{nk}\) using the map: \begin{equation}
\label{eq:trivialisation}
f_{[k]}
\mapsto
\big(
f_{1}',\dotsc, f_{n}',
\dotsc,
f_{1}^{(k)},\dotsc,f_{n}^{(k)}
\big). \end{equation} Here and throughout the rest of this text, we will rather use the notation \(f^{(j)}\) for the \(j\)-th Taylor coefficient than for the \(j\)-th derivative, \textit{i.e.}: \[
f^{(j)}
\mathrel{\mathop{:}}=
\frac{1}{j!}
\frac{d^{j}f}{dt^{j}}. \] The collection of these vector spaces gives rise to a holomorphic fiber bundle \(J_{k}X\), which is called the \textsl{\(k\)-jet manifold} of \(X\).
To get a local trivialization of \(J_{k}X\) around a point \(x\in X\), one can of course use the map \eqref{eq:trivialisation}. A construction due to Noguchi~(\cite{MR859200}) allows more general ``derivatives'', that are potentially more adapted to the geometric situation that will be dealt with below. Here ``derivative'' means pullback of local meromorphic \(1\)-forms.
Let thus \(\omega\in T_{U}^{\star}\) be a local meromorphic \(1\)-form over an open subset \(U\subset X\) and let \(f\colon\Omega\to U\) be a local holomorphic map over an open subset \(\Omega\subset\C\). This map \(f\) induces a meromorphic function \(A'\colon\Omega\to\C\) by the formula: \[
f^{\star}\omega\vert_{t}
=\mathrel{\mathop{:}}
A'(t)\,dt, \] where we equip the complex plane \(\C\) with the standard complex coordinate \(t\). The obtained map \(A'\) depends only on the \(1\)-jet of \(f\). More generally, the \(j\)-th derivative of \(A'\) (up to \(j=k-1\)) is well defined and depends only on the \((j+1)\)-jet of \(f\). One gets a meromorphic map \(J_{k}X\vert_{U}\to\C^{k}\) associated to \(\omega\) by defining: \begin{equation}
\label{eq:noguchi}
\widetilde{\omega}
\colon
f_{[k]}
\mapsto
\left(A',\frac{1}{2!}\frac{dA'}{dt},\dotsc,\frac{1}{k!}\frac{d^{k-1}A'}{dt^{k-1}}\right). \end{equation} Note that if \(\omega\) is taken holomorphic, then \(\widetilde{\omega}\) becomes also holomorphic.
In the simplest case where \(\omega=dz_{i}\) is locally exact, one gets the derivatives in the direction \(z_{i}\). In the basic particular case where \(\omega=dz_{i}/z_{i}\), one gets the logarithmic derivatives in the direction \(z_{i}\).
For any holomorphic frame \((\omega^{1},\dotsc,\omega^{n})\) of the holomorphic cotangent bundle \(T_{X}^{\star}\), the map \((\widetilde{\omega}^{1},\dotsc,\widetilde{\omega}^{n})\) is a trivialization of the fiber of \(J_{k}X\).
Recall that a \textsl{normal crossing divisor} \(D\) is a reduced divisor that looks locally like a (possibly empty) union of coordinate hyperplanes: for any point \(x\in X\), there exists a neighbourhood \(U\) at \(x\) with a local holomorphic coordinate system \((z_{1},\dotsc,z_{n})\) such that \(U\cap D\) is the zero locus \(\{z_{1}\dotsm z_{\ell}=0\}\), for some integer \(\ell\leq n\) that depends on \(x\). The \textsl{logarithmic cotangent sheaf} \(T_{X}^{\star}(\log D)\) is then defined as the locally free subsheaf of the sheaf of meromorphic \(1\)-forms on \(X\) whose stalk at any point \(x\) is defined by: \[
T_{X}^{\star}(\log D)\vert_{x}
\mathrel{\mathop{:}}=
\sum_{i=1}^{\ell}
\mathcal{O}_{X,x}\;\frac{dz_{i}}{z_{i}}
+
\sum_{i=\ell+1}^{n}
\mathcal{O}_{X,x}\;{dz_{i}}, \] for any \textsl{logarithmic coordinate system} \((z_{1},\dotsc,z_{n})\) along \(D\) at \(x\), such that \(U\cap D=\{z_{1}\dotsm z_{\ell}=0\}\).
Recall after \cite{MR859200} that a holomorphic section \(s\in H^{0}(U,J_{k}X)\) on an open subset \(U\subset X\) is said to be a \textsl{logarithmic jet field} along \(D\) of order \(k\) if for all logarithmic \(1\)-forms \(\omega\) defined on an arbitrary open subset \(V\subset U\) the map: \[
\widetilde{\omega}
\circ
s\vert_{V}
\colon
V
\to
\C^{k} \] is holomorphic. These sections define a subsheaf of the sheaf of sections of \(J_{k}X\) that is itself the sheaf of sections of a holomorphic affine bundle \(J_{k}(X,-\log D)\), called the \textsl{logarithmic jet bundle of order \(k\) along \(D\)}. For any holomorphic frame \((\omega^{1},\dotsc,\omega^{n})\) of the logarithmic cotangent bundle \(T_{X}^{\star}(\log D)\), the map \((\widetilde{\omega}^{1},\dotsc,\widetilde{\omega}^{n})\) is a trivialization of the fiber of \(J_{k}(X,-\log D)\).
A detailed description of the properties of the bundle \(J_{k}(X,-\log D)\) can be found in the paper \cite{MR1824906} by Dethloff and Lu.
\subsubsection{Geometric jet coordinates (1).} \label{sse:geometricJetCoordinates} A jet field \(j\in J_{k}X\) is termed \textsl{singular} at a point \(x\in X\) if it is the lift of a stationary curve, \textit{i.e.} \(j=f_{[k]}\) with \(f_{1}'(x)=\dotsb=f_{n}'(x)=0\). The subset of singular jets will be denoted by \(J_{k}^{\mathit{sing}}X\). Note that for a logarithmic pair \((X,D)\), the logarithmic jet bundle \(J_{k}(X,-\log D)\vert_{X\setminus D}\) and the holomorphic jet bundle \(J_{k}(X\setminus D)\) coincide on the open part \(X\setminus D\). This observation allows to define singular logarithmic jet fields, as follows. A logarithmic jet field is said \textsl{singular} if it is in the topological closure of the subset \(J_{k}^{\mathit{sing}}(X\setminus D)\) in \(J_{k}(X,-\log D)\). A (logarithmic) jet field that is not singular is termed \textsl{invertible} (or \textsl{regular}).
For a short moment, we will work in the so-called \textsl{compact case}, where \(D\) is empty, because the modifications needed to treat the general case are straightforward. We fix a coordinate system \((z_{1},\dotsc,z_{n})\) on an open set of \(X\). Let: \[
z_{i}^{(p)}\colon f_{[k]}\mapsto f_{i}^{(p)} \] denote the jet coordinates obtained for the holomorphic frame \(dz_{1},\dotsc,dz_{n}\) according to \eqref{eq:noguchi}. Consider the invertible jets, for which at least one first derivative \(z_{i_{0}}'\) is not zero.
Without loss of generality, assume that \(i_{0}=1\). By the local inverse theorem, the map \(f_{1}\colon\C\to\C\) is locally invertible, and it is very natural to use it as the complex variable. To do so, let us first describe how jet variables behave under reparametrization at the source.
Let \(g\colon\C\to X\) be a (local) map of class \(\mathscr{C}^{k}\) and let \(h\colon\C\to\C\) be a (local) reparametrization of the source of class \(\mathscr{C}^{k}\). Recall that the \textsl{length} of a multi-index \(\mu\in\N^{k}\) is by definition the sum: \[
\abs{\mu}\mathrel{\mathop{:}}= \mu_{1}+\mu_{2}+\dotsb+\mu_{k}, \] and define its \textsl{weight} to be the weighted sum: \[
\Abs{\mu}\mathrel{\mathop{:}}= \mu_{1}+2\mu_{2}+\dotsb+k\mu_{k}. \] Classically, for any integer \(p=0,1,\dotsc,k\), the \(p\)-th Taylor coefficient \(g^{(p)}=\frac{1}{p!}\frac{d^{p}g}{dt^{p}}\) of the map \(g\) is related to those of the reparametrized map \(g\circ h^{\moinsun}\) by the \textsl{Faà di Bruno formula}~(\cite[§3.4, p.137]{MR0460128}): \begin{equation}
\label{eq:faa1}
g^{(p)}
=
\sum_{q\leq p}
\mathbi{B}_{p,q}\big(h^{(1)},\dotsc,h^{(k)}\big)\,
\left(g\circ h^{\moinsun}\right)^{(q)}
\circ h, \end{equation} where the \(\mathbi{B}_{p,q}\big(h^{(1)},\dotsc,h^{(k)}\big)\), hereafter denoted by \(\mathbi{B}_{p,q}(h)\) for shortness, are the so-called \textsl{Bell polynomials}~(\cite[§3.3, p.133]{MR0460128}): \begin{equation*}
\label{eq:bell}
\mathbi{B}_{p,q}(h)
\mathrel{\mathop{:}}=
\sum_{\Abs{\mu}=p,\abs{\mu}=q}
\frac{\abs{\mu}!}{\mu!}\,
{h^{(1)}}^{\mu_{1}}\dotsm{h^{(k)}}^{\mu_{k}}, \end{equation*} with \( \mu! \mathrel{\mathop{:}}= \mu_{1}!\dotsm\mu_{k}! \).
Since the weighted sum \( \Abs{\mu} = 1\,\mu_{1}+ 2\,\mu_{2}+ \dotsb+ k\,\mu_{k} \) is clearly never less than the plain sum \( \abs{\mu} = 1\,\mu_{1}+ 1\,\mu_{2}+ \dotsb+ 1\,\mu_{k}, \) the equality case \(\Abs{\mu}=\abs{\mu}=p\) implying \(\mu=(p,0\dotsc,0)\), it is immediate that \(\mathbi{B}_{p,q}=0\) for \(p<q\) and \(\mathbi{B}_{p,p}(h)=(h^{(1)})^{p}=(h')^{p}\) for \(p=q\).
Adopting the following notation for the Taylor coefficients in the summand of \eqref{eq:faa1}: \begin{equation*}
\frac{1}{q!}
\frac{d^{q} g}{dh^{q}}
\mathrel{\mathop{:}}=
\left(g\circ h^{\moinsun}\right)^{(q)}
\circ h, \end{equation*} we gather the formulas \eqref{eq:faa1} for \(p=1,\dotsc,k\) under the form of the following invertible matricial equation: \begin{equation}
\label{eq:faa.mat}
\xym[.57]{4,1}{
g^{(1)}\ar@{.}[ddd]\\
\\
\\
g^{(k)}
}
=
\xym[0]{5,4}
{
h'\ar@{.}[4,3]&0\ar@{.}[0,2]&&0\\
&&&\\
&&&\\
&&&0\ar@{.}@{.}[-3,-2]\ar@{.}@{.}[-3,0]\\
&\mathbi{B}_{p,q}(h)&&(h')^{k}\\
}
\xym[1.08]{3,1}{
\frac{1}{1!}\frac{dg}{dh}\ar@{.}[dd]\\
\\
\frac{1}{k!}\frac{d^{k}g}{dh^{k}}
}. \end{equation}
We now come back to the particular situation of the set \(\{z_{1}'\neq0\}\) and we introduce new jet coordinates as follows. \begin{definition}
For a \(k\)-jet of holomorphic curve \(f\colon(\C,0)\to(X,x)\) with \(f_{1}'(0)\neq0\), and for \(i\neq1\), the \textsl{geometric jet coordinates} of \(f\) in the \(z_{i}\)-direction at \(x\) are:
\[
\bigg(f_{i}^{[1]},f_{i}^{[2]},\dotsc,f_{i}^{[k]}\bigg)
\mathrel{\mathop{:}}=
\left(
\frac{df_{i}}{df_{1}},
\frac{1}{2!}\frac{d^{2}f_{i}}{d{f_{1}}^{2}},
\dotsc,
\frac{1}{k!}\frac{d^{k}f_{i}}{d{f_{1}}^{k}}
\right).
\] \end{definition}
Notice that the \(f_{i}^{[p]}\) are genuine Taylor coefficients, but we use square brackets so that the introduced geometric jet coordinates cannot be confused with the usual jet coordinates \(f_{i}^{(p)}\).
The geometric jet coordinates on \(\{z_{1}'\neq0\}\) in the \(z_{i}\)-direction are defined to be \(z_{i}^{[p]}\colon f_{[k]}\mapsto f_{i}^{[p]}\). By lifting \eqref{eq:faa.mat}, with \(h=z_{1}\), at the level of \(k\)-jets, one obtains the following relations between standard jet coordinates and geometric jet coordinates. \begin{equation}
\label{eq:geom.jet.i}
z_{i}^{(p)}
=
\sum_{p\geq q}
\mathbi{B}_{p,q}(z_{1})\;
z_{i}^{[q]}
\qquad
{\scriptstyle(i=2,\dotsc,n)}. \end{equation}
In the \(z_{1}\)-direction, the same definition would produce ``\(z_{1}^{[1]}=1\)'' and then ``\(z_{1}^{[p]}=0\)'' for \(p\geq2\). Thus in order to complete the jet coordinate system we rather use : \[
t^{[p]}\colon f_{[k]}\mapsto
\frac{d^{p}t}{df_{1}^{p}}
\mathrel{\mathop{:}}=
({f_{1}}^{\moinsun})^{(p)}\circ f_{1}, \] where of course \(t\) stands for the identity map \(t\mapsto t\). Stating that \(t^{(q)}=\delta_{1,q}\), one infers: \[
\delta_{1,p}
=
\sum_{p\geq q}
\mathbi{B}_{p,q}(z_{1})\;
t^{[q]}
\qquad
{\scriptstyle(i=2,\dotsc,n)}. \]
To sum up, the two systems of coordinates are related by: \begin{equation}
\label{eq:geometricCoordinates}
\xym{4,5}{
1&z_{1}^{(1)}&z_{2}^{(1)}\ar@{.}[rr]&&z_{n}^{(1)}\\
0\ar@{.}[dd]&z_{1}^{(2)}\ar@{.}[dd]&z_{2}^{(2)}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{(2)}\ar@{.}[dd]\\
&&&&\\
0&z_{1}^{(k)}&z_{2}^{(k)}\ar@{.}[rr]&&z_{n}^{(k)}
}
=
\mathbi{B}(z_{1})\;
\xym{4,5}{
t^{[1]}&1&z_{2}^{[1]}\ar@{.}[rr]&&z_{n}^{[1]}\\
t^{[2]}\ar@{.}[dd]&0\ar@{.}[dd]&z_{2}^{[2]}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{[2]}\ar@{.}[dd]\\
&&&&\\
t^{[k]}&0&z_{2}^{[k]}\ar@{.}[rr]&&z_{n}^{[k]}
}. \end{equation}
Reciprocally, it is a well-known fact that Bell arrays behave well with respect to composition; indeed (\cite[§3.7, p145]{MR0460128}): \[ \mathbi{B}(g_{1}\circ g_{2}) = \mathbi{B}(g_{2})\; \big(\mathbi{B}(g_{1})\circ g_{2}\big). \] By taking \(g_{1}=z_{1}\) and \(g_{2}={z_{1}}^{\moinsun}\), one obtains \( \mathbi{I}_{k} = \mathbi{B}(z_{1})\; \big(\mathbi{B}({z_{1}}^{\moinsun})\circ z_{1}\big) \), so we define: \begin{equation}
\label{eq:Bt}
\mathbi{B}_{p,q}[t]
\mathrel{\mathop{:}}=
\mathbi{B}_{p,q}({z_{1}}^{\moinsun})\circ z_{1}
=
\sum_{\Abs{\mu}=p,\ \abs{\mu}=q}
\frac{\abs{\mu}!}{\mu!}\,
{t^{[1]}}^{\mu_{1}}
\dotsm
{t^{[k]}}^{\mu_{k}}, \end{equation} and the two matrices \(\mathbi{B}(z_{1})\) and \(\mathbi{B}[t]\) are inverse of each other. Thus: \begin{equation}
\xym{4,5}{
t^{[1]}&1&z_{2}^{[1]}\ar@{.}[rr]&&z_{n}^{[1]}\\
t^{[2]}\ar@{.}[dd]&0\ar@{.}[dd]&z_{2}^{[2]}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{[2]}\ar@{.}[dd]\\
&&&&\\
t^{[k]}&0&z_{2}^{[k]}\ar@{.}[rr]&&z_{n}^{[k]}
}
=
\mathbi{B}[t]\;
\xym{4,5}{
1&z_{1}^{(1)}&z_{2}^{(1)}\ar@{.}[rr]&&z_{n}^{(1)}\\
0\ar@{.}[dd]&z_{1}^{(2)}\ar@{.}[dd]&z_{2}^{(2)}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{(2)}\ar@{.}[dd]\\
&&&&\\
0&z_{1}^{(k)}&z_{2}^{(k)}\ar@{.}[rr]&&z_{n}^{(k)}
}. \end{equation}
\begin{slshape}
To adapt the definitions to the logarithmic case, we can do all the same reasoning, provided
\((z_{1},\dotsc,z_{n})\) is a logarithmic coordinate system along \(D=(z_{1}\dotsm z_{\ell}=0)\),
by taking the jet-coordinate associated to the frame \(({dz_{1}}/{z_{1}},\dotsc,{dz_{\ell}}/{z_{\ell}},dz_{\ell+1},\dotsc,dz_{n})\), namely for \(p=1,\dotsc,k\):
\[
\begin{cases}
z_{i}^{(p)}\colon f_{[k]}\mapsto (\log f_{i})^{(p)}&\text{for }i=1,\dotsc,\ell,\\
z_{i}^{(p)}\colon f_{[k]}\mapsto f_{i}^{(p)}&\text{for }i=\ell+1,\dotsc,n.
\end{cases}
\] \end{slshape}
\subsubsection{Associated vector fields.} Since \(\mathbi{B}(z_{1})=\mathbi{B}[t]^{\moinsun}\), the coefficients of \(\mathbi{B}(z_{1})\) have expressions that depend only on \(t^{[1]},\dotsc,t^{[k]}\). Consequently, \(z_{1}^{(1)},\dotsc,z_{1}^{(k)}\) depend only on \(t^{[1]},\dotsc,t^{[k]}\) because they are the coefficients of the first column of \(\mathbi{B}(z_{1})\), and for \(i=2,\dotsc,n\), the system \eqref{eq:geom.jet.i}: \[
z_{i}^{(p)}
=
\sum_{p\geq q}
\mathbi{B}_{p,q}(z_{1})\;
z_{i}^{[q]} \] yields that \(z_{i}^{(p)}\) depends only on \(t^{[1]},\dotsc,t^{[k]}\) and \(z_{i}^{[1]},\dotsc,z_{i}^{[q]}\) for the single same index \(i\).
One infers the dual relations for the associated vector fields \(\vf{i}{[p]}\) and \(\vf{i}{(q)}\), by plain transposition: \begin{equation}
\vf{i}{[p]}
=
\sum_{q=p}^{k}
\mathbi{B}_{q,p}(z_{1})\,
\vf{i}{(q)}, \end{equation} where \( \vf[\bullet]{}{} = \frac{\partial}{\partial\bullet} \). The simplest instances of such vector fields are constructed for \(i=2,\dotsc,n\) in the case \(p=1\), where these vector fields \begin{equation}
\label{eq:vfpaun}
\vf{i}{[1]}
=
\sum_{q=1}^{k}
z_{1}^{(q)}
\vf{i}{(q)} \end{equation} are the same as the vector field implicitly used in the matrix approach presented in the work of P\u{a}un (\cite{MR2372741}).
Because the Bell array \(\mathbi{B}(z_{1})\) is lower triangular with invertible diagonal entries \((z_{1}')^{q}\), one has immediately the following: \begin{corollary}
\label{cor:span.i}
The vector fields \(\vf{i}{[q]}\) span the tangent vector space in the \(z_{i}\)-direction at points where \(z_{1}'\neq0\). \end{corollary}
In the ``\(t\)-direction'' the vector fields have of course more involved expressions, as every jet coordinate \(z_{i}^{(p)}\) depends on the coefficients \(t^{[q]}\). Nevertheless, some special linear combination of \(\vf[t]{}{[1]},\dotsc,\vf[t]{}{[k]}\) have simple expressions. A first example is the \textsl{Euler vector field}: \[
\mathsf{T}_{1}
\mathrel{\mathop{:}}=
t^{[1]}\vf[t]{}{[1]}
+
\dotsb
+
t^{[k]}\vf[t]{}{[k]}
=
-
\sum_{i=1,\dotsc,n}\;
\sum_{p=1}^{k}
p\;z_{i}^{(p)}\,\vf{i}{(p)}. \] More generally, we claim that: \begin {equation}
\label{eq:tgt.sym}
\mathsf{T}_{\ell}
\mathrel{\mathop{:}}=
\sum_{m=1}^{k}
\mathbi{B}_{m,\ell}[t]\,
\vf[t]{}{[m]}
=
-
\sum_{i=1,\dotsc,n}\;
\sum_{p=1}^{k-\ell+1}
p\,z_{i}^{(p)}\,\vf{i}{(p+\ell-1)}. \end {equation} \begin{proof}
Assume that \(\mathsf{V}\) is in the vector space spanned by \(\vf[t]{}{[1]},\dotsc,\vf[t]{}{[1]}\) then:
\[
\mathsf{V}
\cdot
\textrm{Mat}\big(z_{i}^{(p)}\big)
=
\mathsf{V}
\cdot
\left(
\mathbi{B}(z_{1})\;
\textrm{Mat}\big(z_{i}^{[p]}\big)
\right)
=
\big(
\mathsf{V}
\cdot
\mathbi{B}(z_{1})
\big)\;
\textrm{Mat}\big(z_{i}^{[p]}\big),
\]
where for shortness:
\[
\textrm{Mat}\left(z_{i}^{(p)}\right)
\mathrel{\mathop{:}}=
\xym[0]{4,4}{
z_{1}^{(1)}&z_{2}^{(1)}\ar@{.}[rr]&&z_{n}^{(1)}\\
z_{1}^{(2)}\ar@{.}[dd]&z_{2}^{(2)}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{(2)}\ar@{.}[dd]\\
&&&\\
z_{1}^{(k)}&z_{2}^{(k)}\ar@{.}[rr]&&z_{n}^{(k)}
}
\quad
\text{and}
\quad
\textrm{Mat}\left(z_{i}^{[p]}\right)
\mathrel{\mathop{:}}=
\xym[0]{4,4}{
\rule{0pt}{11pt}1&z_{2}^{[1]}\ar@{.}[rr]&&z_{n}^{[1]}\\
0\ar@{.}[dd]&z_{2}^{[2]}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{[2]}\ar@{.}[dd]\\
&&&\\
0&z_{2}^{[k]}\ar@{.}[rr]&&z_{n}^{[k]}
};
\]
the last equality holds by Leibniz rule, because the coefficient of \(\mathrm{Mat}\big(z_{i}^{[p]}\big)\) are independent of \(t^{[1]},\dotsc,t^{[k]}\).
Now, because \(\mathbi{B}[t]\) and \(\mathbi{B}(z_{1})\) are inverse matrices, one has:
\[
\mathsf{V}
\cdot
\mathbi{B}(z_{1})
=
-
\mathbi{B}[t]^{\moinsun}
\big(
\mathsf{V}
\cdot
\mathbi{B}[t]
\big)
\mathbi{B}(z_{1}),
\]
whence:
\[
\mathsf{V}
\cdot
\textrm{Mat}\big(z_{i}^{(p)}\big)
=
-
\mathbi{B}[t]^{\moinsun}
\big(
\mathsf{V}
\cdot
\mathbi{B}[t]
\big)
\mathbi{B}(z_{1})\;
\textrm{Mat}\big(z_{i}^{[p]}\big)
=
-
\mathbi{B}[t]^{\moinsun}
\big(
\mathsf{V}
\cdot
\mathbi{B}[t]
\big)\;
\textrm{Mat}\big(z_{i}^{(p)}\big).
\]
It is in general rather difficult to compute
\(
\mathsf{V}
\cdot
\mathbi{B}[t]
\),
but in the particular cases that we consider using the very definition \eqref{eq:Bt}, one proves that:
\[
\sum_{m=1}^{k}
\mathbi{B}_{m,\ell}[t]\,
\vf[t]{}{[m]}
\cdot
\mathbi{B}_{p,q}[t]
=
\ell\;
\mathbi{B}_{p,q+\ell-1}[t].
\]
which is equivalent to:
\[
\sum_{m=1}^{k}
\mathbi{B}_{m,\ell}[t]\,
\vf[t]{}{[m]}
\cdot
\mathbi{B}[t]
=
\mathbi{B}[t]
\big(
1\,\mathbi{e}_{\ell}^{1}+
\dotsb+
(1+k-\ell)\,\mathbi{e}_{1+k-\ell}^{k}
\big),
\]
where the \(\mathbi{e}_{i}^{j}\) are the elements of the canonical basis of \(\mathrm{Mat}_{k,k}(\C)\).
Thus:
\[
\sum_{m=1}^{k}
\mathbi{B}_{m,\ell}[t]\,
\vf[t]{}{[m]}
\cdot
\textrm{Mat}\big(z_{i}^{(p)}\big)
=
-
\big(
1\,\mathbi{e}_{\ell}^{1}+
\dotsb+
(1+k-\ell)\,\mathbi{e}_{1+k-\ell}^{k}
\big)\;
\textrm{Mat}\big(z_{i}^{(p)}\big),
\]
which finally yields the announced result:
\[
\sum_{m=1}^{k}
\mathbi{B}_{m,\ell}[t]\,
\vf[t]{}{[m]}
=
\sum_{i=1,\dotsc,n}\;
\sum_{p=1}^{k}
\sum_{m=1}^{k}
\mathbi{B}_{m,\ell}[t]\,
\vf[t]{}{[m]}
\cdot
z_{i}^{(p)}\vf{i}{(p)}
=
-
\sum_{i=1,\dotsc,n}\;
\sum_{p=1}^{k-\ell+1}
p\,z_{i}^{(p)}\,\vf{i}{(p+\ell-1)}.
\qedhere
\] \end{proof}
\begin{corollary}
\label{cor:span.z}
The vector fields \(\mathsf{T}_{1},\dotsc,\mathsf{T}_{k}\) and \(\vf{2}{[1]},\dotsc,\vf{2}{[k]}\), \dots, \(\vf{n}{[1]},\dotsc,\vf{n}{[k]}\) span the tangent vector space to \(J_{k}X\) at points where \(z_{1}'\neq0\). \end{corollary} \begin{proof}
By corollary \ref{cor:span.i}, it suffices to prove that the vector fields \(\vf{1}{(1)},\dotsc,\vf{1}{(q)},\dotsc,\vf{1}{(k)}\) are spanned. This is an easy descending induction on \(q=k,\dotsc,1\), since by \eqref{eq:tgt.sym}:
\[
\vf{i}{(q)}
=
\frac{1}{q\,z_{1}'}
\bigg(
\mathsf{T}_{q}
-
\sum_{p=q+1}^{k}
p\;z_{1}^{(1+p-q)}\,\vf{1}{(p)}
-
\sum_{i=2,\dotsc,n}
\sum_{p=q}^{k}
p\;z_{i}^{(1+p-q)}\,\vf{i}{(p)}
\bigg).
\qedhere
\] \end{proof}
\subsubsection{Geometric jet coordinates (2).} To be fully efficient, we will also need an alternative description of the geometric jet coordinates, using differential operators, in the spirit of the presentation of Merker in \cite{MR2543663}. The \textsl{formal differentiation of jets} \(\mathsf{D}_{t}\) is by definition the following vector field on \(J_{k}X\), thought of as a differential operator on \(k\)-jets, that mimics the differentiation with respect to the standard coordinate \(t\in\C\): \begin{equation}
\label{eq:formalDifferentiation}
\mathsf{D}_{t}
\mathrel{\mathop{:}}=
\sum_{i=1,\dotsc,n}
\left(
\sum_{p=0}^{k-1}
(p+1)\;
z_{i}^{(p+1)}\;
\vf{i}{(p)}
\right). \end{equation} Here, the coefficient \((p+1)\) appears only because we use the Taylor coefficients instead of the derivatives. On the set \(\{z_{1}'\neq0\}\) it is natural to introduce the linear differential operator \(\mathsf{D}_{z_{1}}\), that mimics the total derivative with respect to \(z_{1}\). These two linear differential operators should be related by the usual chain rule on \(\{z_{1}'\neq0\}\), that we take as a definition: \begin{equation}
\label{eq:chainrule}
\mathsf{D}_{z_{1}}
\mathrel{\mathop{:}}=
\frac{1}{z_{1}'}
\mathsf{D}_{t}. \end{equation} More generally, we claim that this chain rule implies that on the set \(\{z_{1}'\neq0\}\), the powers of the differential operators \(\mathsf{D}_{t}\) and \(\mathsf{D}_{z_{1}}\) are related by the invertible triangular system: \begin{equation}
\label{cor:DtD1}
\frac{\mathsf{D}_{t}^{p}}{p!}
=
\sum_{q=1}^{p}
\mathbi{B}_{p,q}(z_{1})\;
\frac{\mathsf{D}_{z_{1}}^{q}}{q!}
\qquad
{\scriptstyle(p=1,\dotsc,k)}. \end{equation} This is a purely combinatorial fact, that we will admit, because the proof is not very interesting for our problem.
Throughout what follows, by convention, \(\mathsf{D}_{t}(t)\bydef1\). \begin{lemma}
\label{lem:geometric.jets.Dz}
One has as expected \(\mathsf{D}_{z_{1}}(z_{1})=1\) and for \(p=0,1,\dotsc,k\) one has also:
\[
t^{[p]}
=
\frac{1}{p!}\mathsf{D}_{z_{1}}^{p}(t),
\qquad
z_{i}^{[p]}
=
\frac{1}{p!}
\mathsf{D}_{z_{1}}^{p}(z_{i})
\quad{\scriptstyle(i=2,\dotsc,n)},
\]
where according to the above convention \(\mathsf{D}_{z_{1}}^{p}(t)=\mathsf{D}_{z_{1}}^{p-1}(1/z_{1}')\). \end{lemma} \begin{proof}
The result follows from the matricial equalities:
\[
\vcenter{
\xy
\xymatrix"Ma"@R=.5pt@C=.5pt@W=1em@H=1em{
1&z_{1}^{(1)}&z_{2}^{(1)}\ar@{.}[rr]&&z_{n}^{(1)}\\
0\ar@{.}[dd]&z_{1}^{(2)}\ar@{.}[dd]&z_{2}^{(2)}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{(2)}\ar@{.}[dd]\\
&&&&\\
0&z_{1}^{(k)}&z_{2}^{(k)}\ar@{.}[rr]&&z_{n}^{(k)}
}
\POS"Ma1,1"."Ma4,5"!C*\frm{(}*\frm{)},
\POS(55,0)
\xymatrix"Mb"@R=.5pt@C=.5pt@W=1em@H=1em{
\frac{\mathsf{D}_{t}^{1}(t)}{1!}&
\frac{\mathsf{D}_{t}^{1}(z_{1})}{1!}&
\frac{\mathsf{D}_{t}^{1}(z_{2})}{1!}\ar@{.}[rr]&
&
\frac{\mathsf{D}_{t}^{1}(z_{n})}{1!}\\
\frac{\mathsf{D}_{t}^{2}(t)}{2!}\ar@{.}[dd]&
\frac{\mathsf{D}_{t}^{2}(z_{1})}{2!}\ar@{.}[dd]&
\frac{\mathsf{D}_{t}^{2}(z_{2})}{2!}\ar@{.}[dd]\ar@{.}[rr]&
&
\frac{\mathsf{D}_{t}^{2}(z_{n})}{2!}\ar@{.}[dd]\\
\\
\frac{\mathsf{D}_{t}^{k}(t)}{k!}&
\frac{\mathsf{D}_{t}^{k}(z_{1})}{k!}&
\frac{\mathsf{D}_{t}^{k}(z_{2})}{k!}\ar@{.}[rr]&
&
\frac{\mathsf{D}_{t}^{k}(z_{n})}{k!}
}
\POS"Mb1,1"."Mb4,5"!C*\frm{(}*\frm{)},
+L\ar@{}|{\displaystyle=\qquad}"Ma1,1"."Ma4,5"!C+R,
\POS(0,-35)
\xymatrix"Mc"@R=.5pt@C=.5pt@W=1.25em@H=1em{
t^{[1]}&1&z_{2}^{[1]}\ar@{.}[rr]&&z_{n}^{[1]}\\
t^{[2]}\ar@{.}[dd]&0\ar@{.}[dd]&z_{2}^{[2]}\ar@{.}[rr]\ar@{.}[dd]&&z_{n}^{[2]}\ar@{.}[dd]\\
&&&&\\
t^{[k]}&0&z_{2}^{[k]}\ar@{.}[rr]&&z_{n}^{[k]}
}
\POS"Mc1,1"."Mc4,5"!C*\frm{(}*\frm{)},
+L*++!R\txt{\(\mathbi{B}(z_{1})\)},
\POS"Mc1,1"."Mc4,5"!C+U\ar@{}|{\rotatebox{-90}{$\displaystyle=$}}"Ma1,1"."Ma4,5"!C+D,
\POS(55,-35)
\xymatrix"Md"@R=.5pt@C=.5pt@W=1em@H=1em{
\frac{\mathsf{D}_{z_{1}}^{1}(t)}{1!}&
\frac{\mathsf{D}_{z_{1}}^{1}(z_{1})}{1!}&
\frac{\mathsf{D}_{z_{1}}^{1}(z_{2})}{1!}\ar@{.}[rr]&
&
\frac{\mathsf{D}_{z_{1}}^{1}(z_{n})}{1!}\\
\frac{\mathsf{D}_{z_{1}}^{2}(t)}{2!}\ar@{.}[dd]&
\frac{\mathsf{D}_{z_{1}}^{2}(z_{1})}{2!}\ar@{.}[dd]&
\frac{\mathsf{D}_{z_{1}}^{2}(z_{2})}{2!}\ar@{.}[dd]\ar@{.}[rr]&
&
\frac{\mathsf{D}_{z_{1}}^{2}(z_{n})}{2!}\ar@{.}[dd]\\
\\
\frac{\mathsf{D}_{z_{1}}^{k}(t)}{k!}&
\frac{\mathsf{D}_{z_{1}}^{k}(z_{1})}{k!}&
\frac{\mathsf{D}_{z_{1}}^{k}(z_{2})}{k!}\ar@{.}[rr]&
&
\frac{\mathsf{D}_{z_{1}}^{k}(z_{n})}{k!}
}
\POS"Md1,1"."Md4,5"!C*\frm{(}*\frm{)},+L*++!R\txt{\(\mathbi{B}(z_{1})\)},
\ar@{}|{\displaystyle=\qquad}"Mc1,1"."Mc4,5"!C+R,
\POS"Md1,1"."Md4,5"!C+U\ar@{}|{\rotatebox{-90}{$\displaystyle=$}}"Mb1,1"."Mb4,5"!C+D,
\endxy},
\] the right column equality being obtained by applying the combinatorial relations provided by \eqref{cor:DtD1}. The invertibility of \(\mathbi{B}(z_{1})\) on \(\{z_{1}'\neq0\}\) allows to conclude. \end{proof}
Besides the chain rule \eqref{eq:chainrule} that we have used, there is another natural definition for \(\mathsf{D}_{z_{1}}\). We use the convention \(\vf[t]{}{}\mathrel{\mathop{:}}= z_{1}'\vf{1}{}\). Then a natural generalization of the total differentiation in the new jet coordinate system is: \[
\left(
\sum_{p=0}^{k-1}
(p+1)\,
t^{[p+1]}\;
\vf[t]{}{[p]}
\right)
+
\sum_{i=2,\dotsc,n}
\left(
\sum_{p=0}^{k-1}
(p+1)\,
z_{i}^{[p+1]}\;
\vf{i}{[p]}
\right). \]
A computation shows that expressed in the standard jet coordinate system, this vector field coincides with \(\mathsf{D}_{z_{1}}\) on the vector space spanned by \(\{\vf{i}{(p)}\}_{p\leq k-1}\) \emph{but} has non zero terms in the vector vector space spanned by \(\{\vf{i}{(k)}\}_{i=1,\dotsc,n}\). However, as long as we consider iterations \(\mathsf{D}_{z_{1}}^{p}\) acting on \(\C[z_{1},\dotsc,z_{n}]\), with \(p\leq k\), any of the two fields can be used for \(\mathsf{D}_{z_{1}}\) because it is never applied to \(k\)-th order terms.
Further observing that, by the above lemma, the formal derivatives \(\mathsf{D}_{z_{1}}\cdot z_{j}\), expressed in the geometric jet coordinate system, do not depend on the jet coordinates \(t^{[1]},\dotsc,t^{[k]}\), thus for any polynomial \(P\in\C[z_{1},\dotsc,z_{n}]\) nor do \(\mathsf{D}_{z_{1}}^{p}\cdot P\), we obtain the following (notice that \(t^{[1]}\vf[t]{}{}=\vf{1}{}\)).
\begin{lemma}
While \(p\leq k\), the differential operator \(\mathsf{D}_{z_{1}}^{p}\) has the same action on polynomials \(P(z_{1},\dotsc,z_{n})\) as the differential operator:
\[
\left(
\vf{1}{}
+
\sum_{i=2,\dotsc,n}
\left(
\sum_{p=0}^{k-1}
(p+1)\,
z_{i}^{[p+1]}\;
\vf{i}{[p]}
\right)
\right)^{p}.
\] \end{lemma} We will now take advantage of the very simple expression of the \(z_{1}\)-component of this vector field.
\section{Construction of Slanted Vector Fields} \label{se:construction} In this section, we work in the compact case, for \(c=1\). \subsubsection{Vertical jets.} Recall that the universal hypersurface is the subspace \(\mathcal{H}_{d}\subset \P^{n+1}\times \abs{\mathcal{O}(d)}\) defined by: \[
\mathcal{H}_{d}
\colon
\sum_{\abs{\alpha}=d}
A_{\alpha}\,Z^{\alpha}
=
0. \] \label{sse:verticalJets} According to Siu, a \textsl{vertical \(k\)-jet} of the universal hypersurface \(\mathcal{H}_{d}\to \abs{\mathcal{O}(d)}\) is a \(k\)-jet of \(\mathcal{H}_{d}\) representable by some complex curve germ lying completely in some fiber \(H(A)\) over a certain point \(A\in \abs{\mathcal{O}(d)}\) of the parameter space. Concretely, the formal differentiation presentation of Merker (\cite{MR2543663}) provides an efficient description of the subspace of vertical jets, recalled just below.
We restrict to the affine set \(\{Z_{0}\neq0\}\simeq\C^{n}\) equipped with the standard inhomogeneous coordinates \( (z_{1},\dotsc,z_{n}) \), where as usual \(z_{j}=Z_{j}/Z_{0}\), and for some \(\widehat{\alpha}\in\N^{n+1}\) with \(\abs{\widehat\alpha}=d\) and \(\widehat\alpha_{0}= 0\) (fixed later) we also restrict to the affine set \(\{A_{\widehat{\alpha}}\neq 0\}\simeq\C^{n_{d}}\) (here \(n_{d}=\binom{n+d}{n}-1\)), equipped with the standard inhomogeneous coordinates: \[
a_{\alpha_{1},\dotsc,\alpha_{n}}
\mathrel{\mathop{:}}=
A_{(d-\alpha_1-\dotsb-\alpha_n,\alpha_1,\dotsc,\alpha_n)}/A_{\widehat{\alpha}}
\quad
{\scriptstyle(\abs{\alpha}\leq d)}. \] Notice that \(\alpha_{0}\) does not appear anymore in the indices of \(a_{\alpha}\). For brevity, we will make the convention \(a_{\widehat{\alpha}}=a_{\widehat{\alpha}_{1},\dotsc,\widehat{\alpha}_{n}}=1\) but keep in mind that \(a_{\widehat{\alpha}}\) is a constant; in particular there is no associated vector field \(\vf[a]{\widehat{\alpha}}{}\).
In these coordinate system, the restriction to \(U_{0}\mathrel{\mathop{:}}=\{Z_{0}\neq0\}\cap\{A_{\widehat{\alpha}}\neq0\}\) of \(\mathcal{H}_{d}\) is the zero set: \[
\mathcal{Z}_{0}
\colon
\sum_{\abs{\alpha}\leq d}
a_{\alpha}\,z_{1}^{\alpha_{1}}\dotsm z_{n}^{\alpha_{n}}
=
0. \]
Considering the coefficients \(a_{\alpha}\) as independent variables, we work on the jet-space: \[
\C^{n_{d}}\times J_{k}(\C^{n})
\simeq
\C^{n_{d}}
\times
\C^{n}
\times
\C^{nk}, \] equipped with \(a,z\) and the standard jet-coordinates \(z',\dotsc,z^{(k)}\). Then, for \(p=0,1,\dotsc,k\), let the expression \((\mathsf{D}_{t}^{p}\cdot P)\) stands for the polynomial in the variables \(a,z,z',\dotsc,z^{(k)}\) of the ambient jet-space \( \C^{n_{d}} \times \C^{n} \times \C^{nk} \) obtained by applying \(p\) times the formal differentiation \( \mathsf{D}_{t} \) to the defining equation \(P\) of \(\mathcal{Z}_{0}\).
The space \(J_{k}^{\mathit{vert}}(\mathcal{Z}_{0})\) of vertical \(k\)-jets consists of the \(k\)-jets of \(\mathcal{Z}_{0}\) satisfying the \(k+1\) equations: \begin{equation}
\label{eq:DP}
0=
P=
\mathsf{D}_{t}\cdot P=
\dotsb=
\mathsf{D}_{t}^{k}\cdot P. \end{equation} Notice that these algebraic equations form a linearly free system of rank \((k+1)\), hence the algebraic subspace of logarithmic vertical jets \(J_{k}^{\mathit{vert}}(\mathcal{Z}_{0})\) is of pure codimension \((k+1)\) in the ambient space \(\C^{n_{d}}\times\C^{n\,(k+1)}\).
In what follows, for brevity, \(J_{k}\) will stand for the ambient jet space \(\C^{n_{d}}\times J_{k}(\C^{n})\) and \(J_{k}^{\mathit{vert}}\) will stand for \(J_{k}^{\mathit{vert}}(\mathcal{Z}_{0})\).
\subsubsection{Algebraic equations of the tangent space.} We will consider two types of vector fields on \(J_{k}^{\mathit{vert}}\): \begin{itemize}
\item
\textsl{Vertical vector fields} that are vector fields on \(J_{k}^{\mathit{vert}}\) tangential to the space of \(k\)-jets of the vertical fiber of \(J_{k}\to\C^{n_{d}}\).
\item
\textsl{Slanted vector field} that are, according to Siu, vector fields on \(J_{k}^{\mathit{vert}}\) that are not tangential to the space of \(k\)-jets of the vertical fiber at a generic point of \(J_{k}^{\mathit{vert}}\). \end{itemize} We will show that there are very few vector fields of the first kind, what justifies to consider vector fields of the second kind. Indeed, in order to prove global generation, the problem becomes the following: for every point \(p\in U_{0}\), construct enough slanted vector fields so that together with the vertical vector fields they form a (large) subspace of codimension \((k+1)\) in the vector space \(J_{k}^{\mathit{vert}}\rvert_{p}\).
\begin{center}
\begin{tikzpicture}
\draw[very thick] (-1,-1.2) rectangle (9,4.2);
\coordinate (C) at (2,-.75);
\draw (C) +(-2,0)--+(2,0);
\fill[DarkOrange!70!black] (C) circle[radius=2pt] node[below=2,scale=.7]{$A$};
\coordinate (H) at (-.25,.75);
\draw (H)--+(4,0)--+(4.5,.5)--+(.5,.5)--cycle;
\draw[DarkOrange!70!black,thin] (H) +(2,0) node[below,scale=.7]{$H(A)$}--+(2.5,.5);
\coordinate (J) at (-.25,2);
\path (J)++(2.25,.75) node[coordinate](o){};
\node at (o) [left,scale=.7]{$p$};
\path (J)++(2,.4) node[coordinate](a){};
\path (J)++(2.5,1.25) node[coordinate](b){};
\path (J)++(0,.5) node[coordinate](a-){};
\path (J)++(.5,1.1) node[coordinate](b-){};
\path (J)++(4,.2) node[coordinate](a+){};
\path (J)++(4.5,1) node[coordinate](b+){};
\filldraw[DarkGreen,fill=smartgreen,fill opacity=.4] (a-) to[out=5,in=150] (a) to[out=-30,in=185] (a+) to[out=45,in=-160] (b+) to[out=180,in=0] (b) to[out=180,in=0] (b-) to[out=-135,in=60] (a-);
\draw[DarkOrange!70!black] (a) to[out=20,in=-100] (o) to[out=80,in=-170] (b);
\node (lab) at (a) [DarkOrange!70!black,below left=2,scale=.7]{$J_{k}H(A)$};
\draw[thin,DarkOrange!70!black,dashed,->,>=stealth,shorten >=1pt] (lab.east)--(a);
\path(o)+(-1.4,0.05) node[DarkGreen,scale=.8]{$J_{k}^{\mathit{vert}}$};
\draw[ultra thin] (J)--+(4,0)--+(4.5,.5)--+(.5,.5)--cycle +(0,1)--+(4,1)--+(4.5,1.5)--+(.5,1.5)--cycle (J)--+(0,1) +(4,0)--+(4,1) +(4.5,.5)--+(4.5,1.5);
\draw[DarkGreen,->](o)--+(20:.4);
\draw[DarkGreen,->] (o)--+(-10:.45) node[thick,near end,right=1mm,scale=.5]{$\widetilde{\mathsf{V}}$};
\draw[DarkGreen,->] (o)--+(-60:.38);
\filldraw (o) circle[radius=1pt];
\path (J)+(6,.5) node (1) {$J_{k}$};
\path (H)+(6,.5) node (2) {$\mathscr{H}$} node[right=5mm,scale=.7]{$\subset\P^{n+1}\times\abs{\mathcal{O}(d)}$};
\path (C)+(3.75,0) node (3) {$\lvert\mathcal{O}(d)\rvert$};
\draw[->](1)--(2) node[midway,right,scale=.7]{$\pi_k$};
\draw[->](2)--(3) node[midway,right,scale=.7]{$pr_{2}$};
\path (1)+(0,1)node[DarkGreen](4){$J_{k}^{\mathit{vert}}$};
\path (4)--(1) node[midway,sloped]{$\subset$}node[midway,right=2mm,scale=.7]{$\mathrm{codim}=k+1$};
\end{tikzpicture} \end{center} We will decompose a general vector field \(\widetilde{\mathsf{V}}\) on \(J_{k}\), that we will assume tangent to \(J_{k}^{\mathit{vert}}\), into two parts: one vertical part \(\mathsf{V}\) tangent to the jet space \(J_{k}H(A)\) at points \(p\) with \(\pi_{k}\circ \mathrm{pr}_{2}(p)=A\) (which is the situation of the above picture) and one horizontal part \(\mathsf{U}\) tangent to the pullback of the space of parameters \(\abs{\mathcal{O}(d)}\): \[
\widetilde{\mathsf{V}}
=
\mathsf{V}-\mathsf{U}. \] Of course here neither of the two parts is assumed to be itself tangent to \(J_{k}^{\mathit{vert}}\).
For such a vector field \(\widetilde{\mathsf{V}}\) to be indeed tangential to the submanifold \(J_{k}^{\mathit{vert}}\subset J_{k}\) of the ambient jet space, the Lie derivatives along \(\widetilde{\mathsf{V}}\) of the \(k+1\) defining equations \eqref{eq:DP} have to be all zero at points of \(J_{k}^{\mathit{vert}}\): \begin{equation}
\label{eq:VDP}
\widetilde{\mathsf{V}}
\cdot
\big(
\mathsf{D}_{t}^{p}
\cdot
P
\big)
\big\vert_{J_{k}^{\mathit{vert}}}
\equiv
0
\qquad
{\scriptstyle(p=0,1,\dotsc,k)}, \end{equation} where as above \(P\) is the universal polynomial \( P = \sum_{\lvert\alpha\rvert\leq d} a_{\alpha}\, z^{\alpha} \). The \textsl{Hadamard's lemma} asserts that a function vanishes identically on a submanifold if and only if it is locally a linear combination of the defining functions of this submanifold, whence the system \eqref{eq:VDP} is equivalent to the existence for every point \(x\in J_{k}^{\mathit{vert}}\) of an open neighbourhood \(U_{x,k}\ni x\) and of functions \(H_{0}^{p},\dotsc,H_{k}^{p}\in\C^{U_{x,k}}\) such that: \begin{equation}
\label{eq:hadamard}
\widetilde{\mathsf{V}}
\cdot
\big(
\mathsf{D}_{t}^{p}
\cdot
P
\big)
\big\vert_{U_{x,k}}
=
H_{0}^{p}\,P
+
H_{1}^{p}\,
\big(\mathsf{D}_{t}^{1}\cdot P\big)
+
\dotsb
+
H_{k}^{p}\,
\big(\mathsf{D}_{t}^{k}\cdot P\big)
\qquad
{\scriptstyle(p=0,1,\dotsc,k)}. \end{equation}
This latter characterization makes appear that the sheaves \(T_{J_{k}^{\mathit{vert}}},\dotsc, T_{J_{1}^{vert}}\) are \emph{not} compatible with the forgetful maps \(J_{k}\to J_{l}\) valued in lower order jet spaces, since when \(\widetilde{V}\) is a section of \(T_{J_{k}^{\mathit{vert}}}\), then the projection of \(\widetilde{V}\) is not necessarily a section of \(T_{J_{l}^\mathit{vert}}\): this only happens when \(H_{q}^{p}=0\) for \(p\leq l <q\). For this reason, and since iterations of derivatives always have triangular properties (as shown in the Faà di Bruno formula \eqref{eq:faa1} above), it is more natural to work with the subsheaf of vector spaces \(\mathcal{T}_{k}\) of the tangent sheaf to \(J_{k}^{\mathit{vert}}\), whose sections are the vector fields that are tangential to \emph{each} submanifold \[
\mathcal{Z}_{p}
\mathrel{\mathop{:}}=
\big\{
P=\mathsf{D}_{t}\cdot P=\dotsb=\mathsf{D}_{t}^{p}\cdot P=0
\big\}
\quad
{\scriptstyle(p=0,1,\dotsc,k)}, \] not only at points of \(J_{k}^{\mathit{vert}}=\mathcal{Z}_{k}\subset \mathcal{Z}_{p}\), but at every point of \(\mathcal{Z}_{p}\), for all \(p\).
This subsheaf \(\mathcal{T}_{k}\) compares favourably with \(T_{J_{k}^{\mathit{vert}}}\) regarding the properties mentioned above. Indeed, by Hadamard's lemma, the sections \(\widetilde{\mathsf{V}}\) of \(\mathcal{T}_{k}\) are characterized by the existence for each \(p=0,1,\dotsc,k\) and for every point \(x\in \mathcal{Z}_{p}\) of an open neighbourhood \(U_{x,p}\ni x\) and of functions \(H_{0}^{q},\dotsc,H_{p}^{q}\in\C^{U_{x,p}}\) such that: \[
\widetilde{\mathsf{V}}
\cdot
\big(
\mathsf{D}_{t}^{q}
\cdot
P
\big)
\big\vert_{U_{x,p}}
=
H_{0}^{q}\,P
+
H_{1}^{q}\,
\big(\mathsf{D}_{t}^{1}\cdot P\big)
+
\dotsb
+
H_{p}^{q}\,
\big(\mathsf{D}_{t}^{p}\cdot P\big)
\qquad
{\scriptstyle(q=0,1,\dotsc,p)}. \] But fixing \(q\) instead of \(p\), for \(p\geq q\), because \(x\in \mathcal{Z}_{p}\subset \mathcal{Z}_{q}\), up to shrinking \(U_{x,p}\) so that \(U_{x,p}\subset U_{x,q}\), one has the more precise statement: \[
\widetilde{\mathsf{V}}
\cdot
\big(
\mathsf{D}_{t}^{q}
\cdot
P
\big)
\big\vert_{U_{x,p}}
=
H_{0}^{q}\,P
+
H_{1}^{q}\,
\big(\mathsf{D}_{t}^{1}\cdot P\big)
+
\dotsb
+
H_{q}^{q}\,
\big(\mathsf{D}_{t}^{q}\cdot P\big); \] in other words, it is always possible to take \(H_{p}^{q}=0\) for \(q<p\). To sum up, \(\mathsf{\widetilde{V}}\) is a section of \(\mathcal{T}_{k}\) if and only if for every \(p=0,1,\dotsc,k\), and every \(x\in \mathcal{Z}_{p}\) there exists an open neighbourhood \(U_{x,p}\) and a lower \emph{triangular} matrix \(H\in \mathrm{Mat}_{p}(\C^{U_{x,p}})\) such that: \begin{equation}
\label{eq:hadamard2}
\widetilde{\mathsf{V}}
\cdot
\big(
\mathsf{D}_{t}^{q}
\cdot
P
\big)
\big\vert_{U_{x,p}}
=
H_{0}^{q}\,P
+
H_{1}^{q}\,
\big(\mathsf{D}_{t}^{1}\cdot P\big)
+
\dotsb
+
H_{q}^{q}\,
\big(\mathsf{D}_{t}^{q}\cdot P\big)
\qquad
{\scriptstyle(q=0,1,\dotsc,p)}. \end{equation}
Note that in the compact case we are dealing with here, this subtleties can be forgotten, since we will even find vector fields satisfying the equations \eqref{eq:VDP} not only on \(J_{k}^{\mathit{vert}}\) but identically on \(J_{k}\), \textit{i.e.} take \(H=0\). In the genuine logarithmic case however, we will need to restrict to \(\mathcal{Z}_{p}\) in order to cancel the \(p\)-th equation (see below).
\subsubsection{Fundamental case.} For the sake of clear presentation of ideas, we will first consider the \emph{simplest case} where the vertical part \(\mathsf{V}\) is one of the vector fields \(\vf{j}{}\) for \(j=1,\dotsc,n\). The consideration of this case will prove to be fundamental to gain an intuition of the general solution to our problem.
The considered vertical vector field \(\mathsf{V}\mathrel{\mathop{:}}=\vf{j}{}\) is \emph{not} tangent to the subspace of vertical jets, since already on the first defining equation in \eqref{eq:VDP} it is not zero: \[
\mathsf{V}\cdot P
=
\vf{j}{}
\cdot
\sum_{\abs{\alpha}\leq d}
a_{\alpha}\,z^{\alpha}
=
\sum_{\abs{\alpha}\leq d}
a_{\alpha}\,
\alpha_{j}\,
z^{\alpha-\mathbi{1}_{j}}
=
\sum_{\abs{\beta}\leq d-1}
{\color{black!50}
\underset{
\scalebox{.7}{\framebox[1.5\width][c]{\color{black!80}$=\mathrel{\mathop{:}} u_{j,\beta}$}}}
{\underline{\textcolor{black}{
a_{\beta+\mathbi{1}_{j}}\,
(\beta_{j}+1)
}}}}\,
z^{\beta}. \] But it is easy to cancel this last polynomial term \(\sum u_{j,\beta}\,z^{\beta}\) by means of an appropriate correction \(\mathsf{V}\mapsto\mathsf{V}-\mathsf{U}\). Indeed, to find such a vector field \(\mathsf{U}\), noticing that: \[
\vf[a]{\beta}{}
\cdot P
=
\vf[a]{\beta}{}
\cdot
\left(
{\textstyle \sum_{\abs{\alpha}\leq d} }\,
a_{\alpha}\,z^{\alpha}
\right)
=
z^{\beta}, \] it suffices to take the following specific linear combination of the \(\\vf[a]{\beta}{}\) with coefficients precisely equal to the above \(u_{j,\beta}\): \[
\mathsf{U}
\mathrel{\mathop{:}}=
\sum_{\abs{\beta}\leq d-1}
u_{j,\beta}\,
\vf[a]{\beta}{}
=
\sum_{\abs{\beta}\leq d-1}
a_{\beta+{\bf 1}_{j}}\,
(\beta_{j}+1)\,
\vf[a]{\beta}{}, \] which yields the sought vanishing of the first equation in \eqref{eq:VDP}, identically on \(J_{k}\): \[
\left(
\mathsf{V}-\mathsf{U}
\right)
\cdot
P
=
\bigg(
\vf{j}{}
-
\sum_{\abs{\beta}\leq d-1}
u_{j,\beta}\,
\vf[a]{\beta}{}
\bigg)
\cdot
P
=
\sum_{\abs{\beta}\leq d-1}\!
u_{j,\beta}\,z^{\beta}
-
\sum_{\abs{\beta}\leq d-1}\!
u_{j,\beta}\,z^{\beta}
=
0. \]
So we get \(\widetilde{\mathsf{V}}\cdot P=(\mathsf{V}-\mathsf{U})\cdot P=0\). But recall that in order to prove the tangency to the subspace of vertical jets, it is also necessary to check the vanishing of the \(k\) remaining equations in \eqref{eq:VDP}: \[
\widetilde{\mathsf{V}}
\cdot
(
\mathsf{D}_{t}^{q}
\cdot
P
)
\stackrel?=
0.
\qquad
{\scriptstyle(q=1,\dotsc,k)}. \] What makes this first considered case very simple is that here having obtained \(\widetilde{\mathsf{V}}\cdot P=0\) actually suffices to conclude, because the coefficients of the constructed vector field: \[
\widetilde{\mathsf{V}}
=
\mathsf{V}-\mathsf{U}
\mathrel{\mathop{:}}=
\vf{j}{}
-
\sum_{\abs{\beta}\leq d-1}
a_{\beta+\mathbi{1}_{j}}\,(\beta_{j}+1)\,
\vf[a]{\beta}{} \] do not depend on the \(z\)-variables, and thus the powers of the action \(\mathsf{D}_{t}\cdot\) and the action \(\widetilde{\mathsf{V}}\cdot\) commute, whence: \[
\widetilde{\mathsf{V}}\cdot(\mathsf{D}_{t}^{q}\cdot P)
=
\mathsf{D}_{t}^{q}\cdot(\widetilde{\mathsf{V}}\cdot P)
=
0. \]
\begin{slshape}
It should immediately be pointed out that \emph{the simplicity of the above computations could appear misleading}, since for a general vertical vector field:
\[
\mathsf{V}
\mathrel{\mathop{:}}=
\sum_{i=1,\dotsc,n}
\left(
\sum_{q=0}^{k}
v_{i}^{(q)}\!\big(a_{\alpha};\,z_{1},\dotsc,z_{n};\,z_{1}^{(1)},\dotsc,z_{n}^{(1)},\dotsc,z_{1}^{(k)},\dotsc,z_{n}^{(k)}\bigr)\,
\vf{i}{(q)}
\right),
\]
the huge linear systems one has to solve in order to find an adequate correction \(\mathsf{U}\) involves the multivariate Faà di Bruno formulas (\textit{cf.} \cite{MR2372741,MR2331545,MR2383820,MR2543663}), the expressions of which, though classical, are complicated.
However, \emph{the above considerations are inspiring}:
we will indeed avoid almost all technical inconveniences and make the general case nearly as simple as the simplest case. \end{slshape}
\subsubsection{Strategy in the general case.}
We will proceed in three simplifying steps - \textit{a}, \textit{b} and \textit{c} - summarized here, the details of the two last being expanded in separate sections below. \begin{enumerate}[label={\itshape\alph*.}]
\item
Fix the vertical vector field \(\mathsf{V}\). Then find a \emph{correction \(\mathsf{U}\)} with the same image on the defining equations \eqref{eq:VDP}, and use it to cancel these equations:
\[
\label{eq:a} \tag{$a$}
\mathsf{V}-\mathsf{U}
\in
\mathcal{T}_{k}
\Longleftrightarrow
\left\{
\begin{array}{rclcrcll}
\mathsf{V}
&\cdot&
P(\xi)
&=&
\mathsf{U}
&\cdot&
P(\xi)
&\big\vert_{\xi\in \mathcal{Z}_{0}}
\\
\mathsf{V}
&\cdot&
\big(
\mathsf{D}_{t}
\cdot
P
\big)(\xi)
&=&
\mathsf{U}
&\cdot&
\big(
\mathsf{D}_{t}
\cdot
P
\big)(\xi)
&\big\vert_{\xi\in \mathcal{Z}_{1}}
\\
&&&\vdots&&&\\
\mathsf{V}
&\cdot&
\big(
\mathsf{D}_{t}^{k}
\cdot
P
\big)(\xi)
&=&
\mathsf{U}
&\cdot&
\big(
\mathsf{D}_{t}^{k}
\cdot
P
\big)(\xi)
&\big\vert_{\xi\in \mathcal{Z}_{k}},
\end{array}
\right.
\]
where as above \(\mathcal{Z}_{p}=\{P=\mathsf{D}_{t}\cdot P=\dotsb=\mathsf{D}_{t}^{p}\cdot P=0\}\subset J_{k}\).
\item
When \(\mathsf{D}_{t}\) and \(\widetilde{\mathsf{V}}\) commute like in the fundamental case, one should directly see that a lot of equations cancel.
Also, one should avoid as much as possible expanding the expressions \(\mathsf{D}_{t}^{p}\cdot P\) in order to reduce the combinatorial complexity.
With these goals in mind, it makes sense to let \(\mathsf{D}_{t}\) act on the vector field \(\widetilde{\mathsf{V}}\) rather than on the equation \(P\), using \emph{adjoint action} (see below), which will shortly give:
\[
\label{eq:b}\tag{$b$}
\mathsf{V}-\mathsf{U}
\in
\mathcal{T}_{k}
\Longleftrightarrow
\left\{
\begin{array}{rclcrcl}
\mathsf{V}
&\cdot&
P(\xi)
&=&
\mathsf{U}
&\cdot&
P(\xi)
\big\vert_{\xi\in \mathcal{Z}_{0}}
\\
(\mathsf{D}_{t}\ad\mathsf{V})
&\cdot&
P(\xi)
&=&
(\mathsf{D}_{t}\ad\mathsf{U})
&\cdot&
P(\xi)
\big\vert_{\xi\in \mathcal{Z}_{1}}
\\
&&&\vdots&&&\\
(\mathsf{D}_{t}^{k}\ad\mathsf{V})
&\cdot&
P(\xi)
&=&
(\mathsf{D}_{t}^{k}\ad\mathsf{U})
&\cdot&
P(\xi)
\big\vert_{\xi\in \mathcal{Z}_{k}},
\end{array}
\right.
.
\]
Making furthermore the choice \(\mathsf{V}=\vf{j}{(q)}\), which generalizes the fundamental case where \(\mathsf{V}=\vf{j}{}\) (\textit{i.e.} \(q=0\)), we will see that the values \((\mathsf{D}_{t}^{p}\ad\vf{j}{(q)})\cdot P(\xi)\) one has to cancel become very easy to compute and that they do not depend on the jet variables.
\item
To find a correction \(\mathsf{U}\) that cancels these polynomial values \((\mathsf{D}_{t}^{p}\ad\vf{j}{(q)})\cdot P(\xi)\), we claim that it suffices to determine first some \emph{building-block vector fields} \(\mathsf{U}\) ---\,having the same role as the vector fields \(\vf[a]{\beta}{}\) in the fundamental case (\(q=0\))\,--- such that the corresponding entries \((\mathsf{D}_{t}^{p}\ad\mathsf{U})\cdot P(\xi)\) in the last column of \eqref{eq:b} are all zero for \(p\neq q\), while it is the monomial \((\mathsf{D}_{t}^{q}\ad\mathsf{U})\cdot P(\xi)=z^{\beta}\) for \(p=q\). Indeed, by linearity with respect to the coefficients \(a_{\alpha}\), one can then use these building-blocks vector fields to piece together a corrective vector field.
We will see later on that it is adequate to use the geometric jet coordinates described in \S\ref{sse:geometricJetCoordinates} in order to avoid expanding the expressions \(\mathsf{D}_{t}^{p}\ad\mathsf{U}\).
Over the subset of invertible jets, working without loss of generality on the set \(\{z_{1}'\neq0\}\), since \eqref{cor:DtD1} is triangular invertible, one can use \(\mathsf{D}_{z_{1}}\) instead of \(\mathsf{D}_{t}\) to define the vertical jets.
Restarting and applying again \eqref{eq:a} and \eqref{eq:b}, the problem reduces in the end to the obtainment of building-block vector fields \(\mathsf{U}_{q}^{\beta}\) such that:
\begin{equation*}
\label{eq:c}\tag{$c$}
\text{if }p\neq q\colon\quad
(\mathsf{D}_{z_{1}}^{p}\ad\mathsf{U}_{q}^{\beta})\cdot P(\xi)=0
\qquad
\text{and}\colon\quad
(\mathsf{D}_{z_{1}}^{q}\ad\mathsf{U}_{q}^{\beta})\cdot P(\xi)=z^{\beta}.
\end{equation*}
\end{enumerate}
\subsubsection{Use adjoint action.} According to the strategy outlined just above, we consider the \textsl{Lie derivative of the vector field \(\widetilde{\mathsf{V}}\) along \(\mathsf{D}_{t}\)}, that is by definition the vector field \((\mathsf{D}_{t}\ad \widetilde{\mathsf{V}})\) acting on functions as: \begin{equation}
\label{eq:ad}
(\mathsf{D}_{t}\ad\widetilde{\mathsf{V}})\cdot \bullet
\mathrel{\mathop{:}}=
\mathsf{D}_{t}\cdot(\widetilde{\mathsf{V}}\cdot\bullet)
-
\widetilde{\mathsf{V}}\cdot(\mathsf{D}_{t}\cdot \bullet). \end{equation} For the sake of clarity, we will denote the adjoint action by ``\(\ad\)'' in order to alert to the fact, also emphasized by the use of parentheses in \eqref{eq:ad} and throughout this text, that a unified notation for the action of vector fields on functions and on vector fields would not be associative, what is already apparent in the definition \eqref{eq:ad}. As an example, \(\widetilde{\mathsf{V}}\cdot P\equiv 0\) does not imply \((\mathsf{D}_{t}\ad\widetilde{\mathsf{V}})\cdot P\equiv0\).
\begin{lemma}
\label{lem:VDP-LVP}
The vector field \(\widetilde{\mathsf{V}}\) is a section of \(\mathcal{T}_{k}\)
if and only if:
\[
\bigl(
\mathsf{D}_{t}^{p}
\ad
\widetilde{\mathsf{V}}
\big)
\cdot
P
\big\vert_{
\{
P=\mathsf{D}_{t}\cdot P=\dotsb=\mathsf{D}_{t}^{p}\cdot P=0
\}
}
\equiv
0
\qquad
{\scriptstyle(p=0,1,\dotsc,k)}.
\] \end{lemma} \begin{proof}
Using again Hadamard's lemma, the function \(\big(\mathsf{D}_{t}^{p}\ad\widetilde{\mathsf{V}}\big)\cdot P\) vanishes identically on the submanifold
\(\mathcal{Z}_{p}
\mathrel{\mathop{:}}=
\big\{
P=\mathsf{D}_{t}\cdot P=\dotsb=\mathsf{D}_{t}^{p}\cdot P=0
\big\}
\)
if and only if for every point \(x\in \mathcal{Z}_{p}\)
there exists an open neighbourhood \(V_{x,p}\) and functions \(G_{p}^{q}\in\C^{V_{x,p}}\) such that:
\[
\big(
\mathsf{D}_{t}^{q}
\ad
\widetilde{\mathsf{V}}
\big)
\cdot
P
\big\vert_{V_{x,p}}
=
G_{0}^{q}\,P
+
G_{1}^{q}\,
\big(\mathsf{D}_{t}^{1}\cdot P\big)
+
\dotsb
+
G_{p}^{q}\,
\big(\mathsf{D}_{t}^{q}\cdot P\big)
\quad
{\scriptstyle(q=0,1,\dotsc,p)}.
\]
Thus \(\widetilde{\mathsf{V}}\) fulfill to the \(k+1\) conditions of the statement if and only if for \(p=0,1,\dotsc,k\) and for every point \(x\in \mathcal{Z}_{p}=\mathcal{Z}_{0}\cap\dotsb\cap \mathcal{Z}_{p}\), their exists an open neighbourhood \(V_{x,p}'=V_{x,0}\cap\dotsb\cap V_{x,p}\) and a lower triangular matrix \(G\in \mathrm{Mat}_{k}(\C^{V_{x,p}'})\) such that:
\[
\big(
\mathsf{D}_{t}^{q}
\ad
\widetilde{\mathsf{V}}
\big)
\cdot
P
\big\vert_{V_{x,p}}
=
G_{0}^{q}\,P
+
G_{1}^{q}\,
\big(\mathsf{D}_{t}^{1}\cdot P\big)
+
\dotsb
+
G_{q}^{q}\,
\big(\mathsf{D}_{t}^{q}\cdot P\big)
\qquad
{\scriptstyle(q=0,1,\dotsc,p)}.
\tag{$\ast$}
\]
Now, from the very definition of the adjoint action \eqref{eq:ad}, one can deduce the combinatorial formulas:
\begin{align*}
(\mathsf{D}_{t}^{q}
\ad
\widetilde{\mathsf{V}})
\cdot
\bullet
&=
\sum_{p=0}^{q}
(-1)^{p}
\binom{q}{p}
\mathsf{D}_{t}^{q-p}
\cdot
\Bigl(
\widetilde{\mathsf{V}}
\cdot
(
\mathsf{D}_{t}^{p}
\cdot\bullet
)
\Bigr),
\tag{$\Rightarrow$}
\\
\intertext{and inversely:}
\widetilde{\mathsf{V}}\cdot
(
\mathsf{D}_{t}^{q}
\cdot\bullet
)
&=
\sum_{p=0}^{q}
(-1)^{p}
\binom{q}{p}
\mathsf{D}_{t}^{q-p}
\cdot
\Bigl(
(\mathsf{D}_{t}^{p}\ad\widetilde{\mathsf{V}})
\cdot\bullet
\Bigr).
\tag{$\Leftarrow$}
\end{align*}
These formulas allow respectively to go from the first characterization \eqref{eq:hadamard} to the second one ($\ast$) and the other way around, since the Leibniz rule:
\[
\mathsf{D}_{t}\cdot(fg)
=
(\mathsf{D}_{t}\cdot f)\,g
+
f\,(\mathsf{D}_{t}\cdot g),
\]
implies that the \((q-p)\)-th derivative along \(\mathsf{D}_{t}\) of a linear combination of \(P\), \(\mathsf{D}_{t}^{1}\cdot P\), \dots, \(\mathsf{D}_{t}^{p}\cdot P\) is automatically a linear combination of \(P\), \(\mathsf{D}_{t}^{1}\cdot P\), \dots, \(\mathsf{D}_{t}^{q}\cdot P\). \end{proof} From now on, we will denote by \(\mathbi{e}_{0},\mathbi{e}_{1},\dotsc,\mathbi{e}_{k}\) the standard basis of units vectors for \(\C^{k+1}\). We will also denote by \(\Lambda_{t}\) the linear map associating to each vector field \(\widetilde{\mathsf{V}}\) on \(J_{k}\) a vector-valued symmetric form on \(J_{k}\): \begin{equation}
\label{eq:Lt}
\Lambda_{t}\big(\widetilde{\mathsf{V}}\big)
\mathrel{\mathop{:}}=
\bigl(\widetilde{\mathsf{V}}\cdot P\bigr)\;\mathbi{e}_{0}+
\bigl(\,(\mathsf{D}_{t}^{1}\ad\widetilde{\mathsf{V}})\cdot P\,\bigr)\;\mathbi{e}_{1}+
\dotsb+
\bigl(\,(\mathsf{D}_{t}^{k}\ad\widetilde{\mathsf{V}})\cdot P\,\bigr)\;\mathbi{e}_{k}, \end{equation} in such a way that the sections of \(\mathcal{T}_{k}\) are precisely those satisfying: \[
\Lambda_{t}(\widetilde{\mathsf{V}})
\equiv
H (P,\mathsf{D}_{t}\cdot P,\dotsc,\mathsf{D}_{t}^{k}\cdot P), \] with \(H\) lower triangular (by Lemma \ref{lem:VDP-LVP}). Recall that in the compact case, we will even take \(H=0\).
Let us now explain how using the adjoint action simplifies the problem. Using the \textsl{Leibniz rule} for the general Lie derivative: \begin{equation}
\label{eq:leibniz}
\mathsf{V}_{1}\ad (f\,\mathsf{V}_{2})
=
(\mathsf{V}_{1}\cdot f)\,\mathsf{V}_{2}
+
f\,(\mathsf{V}_{1}\ad\mathsf{V}_{2}), \end{equation} the crucial observation of the fundamental case, namely that \(\mathsf{D}_{t}\ad \vf{j}{}=0\), can be generalized as follows: \begin{lemma}
\label{lem:Dtvf}
For any index \(j=1,\dotsc,n\), one has \(\mathsf{D}_{t}\ad \vf{j}{}=0\) and for any order of derivation \(q=1,\dotsc,k\), the Lie derivative of the vector field \(\vf{j}{(q)}\) is the following plain vector field of the same kind:
\[
\mathsf{D}_{t}
\ad
\vf{j}{(q)}
=
-q\,
\vf{j}{(q-1)}.
\] \end{lemma} \begin{proof}
Recall that by definition, the formal differentiation of \(k\)-jets is the linear map associated to the vector field
\[
\mathsf{D}_{t}
\mathrel{\mathop{:}}=
\sum_{i=1,\dotsc,n}
\biggl(\,
\sum_{p=0}^{k-1}
(p+1)\,
z_{i}^{(p+1)}\,\vf{i}{(p)}
\biggr).
\]
Now, it is might be more intuitive to use the antisymmetry in \eqref{eq:ad}:
\[
\mathsf{D}_{t}
\ad
\vf{j}{(q)}
=
-
\vf{j}{(q)}
\ad
\mathsf{D}_{t},
\]
in order to reshape the sought quantity.
Next, applying the Leibniz rule \eqref{eq:leibniz}, one gets the expression:
\[
\vf{j}{(q)}
\ad
\mathsf{D}_{t}
=
\sum_{i=1,\dotsc,n}
\biggl(\,
\sum_{p=0}^{k-1}
(p+1)\,
\left(\vf{j}{(q)}\cdot z_{i}^{(p+1)}\right)
\vf{i}{(p)}
\biggr)
+
\sum_{i=1,\dotsc,n}
\biggl(\,
\sum_{p=0}^{k-1}
(p+1)\,
z_{i}^{(p+1)}\,
\left(\vf{j}{(q)}\ad\vf{i}{(p)}\right)
\biggr),
\]
in which all the terms but one in the first summand are zero, by independence of the jet variables, and all terms in the second summand are zero by commutativity.
The only remaining term is \(q\,\vf{j}{(q-1)}\). \end{proof} The result of the above lemma \ref{lem:Dtvf} is easy to iterate, in order to obtain that for any integers \(0\leq p,q\leq k\) the \(p\)-th Lie derivative of the vector field \(\vf{j}{(q)}\) along \(\mathsf{D}_{t}\) is the vector field: \[
\mathsf{D}_{t}^{p}
\ad
\vf{j}{(q)}
=
(-1)^{p}\,
\frac{q!}{(q-p)!}\,
\vf{j}{(q-p)}, \] where \(\vf{j}{(q-p)}\) is by convention \(0\) for \(p>q\). Applied to the equation \(P\) --- that depends on the variables \(\{z_{1},\dotsc,z_{n}\}\) but not on the jet-coordinates \(\{z_{i}^{(p)}\}_{p\geq1}\) --- the vector field \(\vf{j}{(q-p)}\) always vanishes, unless \(q-p=0\), in which case it is \(\vf{j}{(0)}=\\vf{j}{}\). \begin{corollary}
\label{cor:Lvfjq}
The vector field \(\vf{j}{(q)}\) does not satisfy \(\Lambda_{t}(\vf{j}{(q)})=0\), because it produces a non zero entry in the \((q+1)\)-th line:
\[
\xym{6,1}{
\vf{j}{(q)}
\cdot
P
\ar@{.}[ddd]\\
\\
\\
(\mathsf{D}_{t}^{q}\ad\vf{j}{(q)})
\cdot
P
\ar@{.}[dd]\\
\\
(\mathsf{D}_{t}^{k}\ad\vf{j}{(q)})
\cdot
P
}
\equiv
\xym[.16]{10,1}{
\hskip 5pt\vf{j}{(q)}\cdot P\hskip5pt\\
\\\\
\\
\\
q!\,\vf{j}{(0)}\cdot P\ar@{.}[uuuuu] \\
0\ar@{.}[ddd] \\
\\\\
0
}
\equiv
\xym[.21]{10,1}{
\hskip 15pt 0 \hskip 15pt \\
\\\\
\\
0\ar@{.}[uuuu] \\
q!\,\vf{j}{}\cdot P \rlap{\quad \(\leftarrow\) line of{} \(\mathsf{D}_{t}^{q}\).} \\
0\ar@{.}[ddd] \\
\\\\
0
}
\] \end{corollary} This situation generalizes the situation of the fundamental case (which is the case where \(q=0\)).
The remaining difficulty is to compute the adequate compensation part \(\mathsf{U}\) for \(\mathsf{V}=\vf{j}{(q)}\), namely a vector field in the direction of the parameter space such that \(\Lambda_{t}(\mathsf{U})=\vf{j}{}\cdot P\;\mathbi{e}_{q}\), that is to say: \[
\text{if }p\neq q\colon\quad
(\mathsf{D}_{t}^{p}\ad\mathsf{U})\cdot P(\xi)=0
\qquad
\text{and}\colon\quad
(\mathsf{D}_{t}^{q}\ad\mathsf{U})\cdot P(\xi)=\vf{j}{}\cdot P(z). \]
\subsubsection{Interlude. Origins of P\u{a}un's layout uncovered.} For a short moment we will only treat the next case \(\mathsf{V}=\vf{j}{(1)}\), that is the case \(q=1\), because it already involves the main arguments of our strategy and also because it is an occasion to revisit the papers of P\u{a}un (\cite{MR2372741}), Rousseau (\cite{MR2331545,MR2383820,MR2552951}) and Merker (\cite{MR2543663}).
We have seen, as a particular case of Corollary \ref{cor:Lvfjq}, that: \[
\Lambda_{t}\big(\vf{j}{(1)}\big)
=
\bigl(\vf{j}{}\cdot P\bigr)\;\mathbi{e}_{1}. \] According to the strategy sketched above, one has to find a corrective vector field \(\mathsf{U}\) with the same image \( \Lambda_{t}(\mathsf{U}) = (\vf{j}{}\cdot P)\;\mathbi{e}_{1} \). Inspired by the use of the vector fields \(\vf[a]{\beta}{}\) as building-block vector fields in the fundamental case, for each \(\beta\) with \(\abs{\beta}\leq d-1\), we will first seek for vector fields \(\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\) such that: \begin{equation}
\label{eq:LtU1}
\Lambda_{t}\big(\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\big)
=
z^{\beta}\;\mathbi{e}_{1}. \end{equation} This allows indeed to conclude as in the fundamental case above, because we have computed: \[
\vf{j}{}
\cdot
P
=
\sum_{\abs{\beta}\leq d-1}
{\color{black!50}
\underset{
\scalebox{.7}{\framebox[1.5\width][c]{\color{black!80}$= u_{j,\beta}$}}}
{\underline{\textcolor{black}{
a_{\beta+\mathbi{1}_{j}}\,
(\beta_{j}+1)
}}}}\,
z^{\beta}, \] and thus, setting as expected \(\mathsf{U}=\sum\,u_{j,\beta}\,\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\), the only line that is not already zero by construction in \(\Lambda_{t}\big(\vf{j}{(1)}-\mathsf{U}\big)\) is cancelled, by linearity of \(\Lambda_{t}\) with respect to the coefficients \(u_{j,\beta}\): \[
\Lambda_{t}
\bigg(
\vf{j}{(1)}
-
\sum_{\abs{\beta}\leq d-1}
u_{j,\beta}\,
\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}
\bigg)
=
\bigg(
\sum_{\abs{\beta}\leq d-1}\!
u_{j,\beta}\,z^{\beta}
-
\sum_{\abs{\beta}\leq d-1}\!
u_{j,\beta}\,z^{\beta}
\bigg)\;
\mathbi{e}_{1}
=
0. \]
Now concerning \eqref{eq:LtU1}, for a fixed \(\beta\), the equation: \[
\Lambda_{t}\big(\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\big)
\stackrel{\eqref{eq:Lt}}\mathrel{\mathop{:}}=
\bigl(\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\cdot P\bigr)\;\mathbi{e}_{0}+
\bigl(\,(\mathsf{D}_{t}^{1}\ad\mathsf{U}_{\scriptscriptstyle(1)}^{\beta})\cdot P\,\bigr)\;\mathbi{e}_{1}+
\dotsb+
\bigl(\,(\mathsf{D}_{t}^{k}\ad\mathsf{U}_{\scriptscriptstyle(1)}^{\beta})\cdot P\,\bigr)\;\mathbi{e}_{k}
=
z^{\beta}\;\mathbi{e}_{1} \] is a higher order analog of: \[
\Lambda_{t}\big(\vf[a]{\beta}{}\big)
\stackrel{\eqref{eq:Lt}}\mathrel{\mathop{:}}=
\bigl(\vf[a]{\beta}{}\cdot P\bigr)\;\mathbi{e}_{0}+
\bigl(\,(\mathsf{D}_{t}^{1}\ad\vf[a]{\beta}{})\cdot P\,\bigr)\;\mathbi{e}_{1}+
\dotsb+
\bigl(\,(\mathsf{D}_{t}^{k}\ad\vf[a]{\beta}{})\cdot P\,\bigr)\;\mathbi{e}_{k}
=
z^{\beta}\,\mathbi{e}_{0}. \] A simple idea in order to produce the needed building-block vector field \(\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\) would hence be that \(\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\) satisfies only two properties: \begin{equation}
\label{eq:U1t}
\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\cdot P
=
0
\qquad\text{and}\qquad
(\mathsf{D}_{t}^{1}\ad\mathsf{U}_{\scriptscriptstyle(1)}^{\beta})
=
\vf[a]{\beta}{}; \end{equation} indeed, it would then follow that for \(p=1,\dotsc,k\): \[
\bigl(\mathsf{D}_{t}^{p}\ad\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\bigr)\cdot P
=
\bigl(\mathsf{D}_{t}^{p-1}\ad(\mathsf{D}_{t}\ad\mathsf{U}_{\scriptscriptstyle(1)}^{\beta})\bigr)\cdot P
=
\bigl(\mathsf{D}_{t}^{p-1}\ad\vf[a]{\beta}{}\bigr)\cdot P, \] whence, as announced: \[
\xy
\xymatrix"LUU"@R=.5pt@C=.5pt@W=1em@H=2em{
\makebox[60pt][c]{$\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\cdot P$}
&=&
0\;\\
\makebox[60pt][c]{$(\mathsf{D}_{t}\ad \mathsf{U}_{\scriptscriptstyle(1)}^{\beta})\cdot P$}
&=&
z^{\beta}\;\\
\makebox[60pt][c]{$(\mathsf{D}_{t}^{2}\ad \mathsf{U}_{\scriptscriptstyle(1)}^{\beta})\cdot P$}
\ar@{.}[dd]&=&\ar@{.}[dd]
0\;\\
&&\\
\makebox[60pt][c]{$(\mathsf{D}_{t}^{k}\ad \mathsf{U}_{\scriptscriptstyle(1)}^{\beta})\cdot P$}
&=&
0\;
}
\POS"LUU5,1"."LUU1,3"!C*++\frm{(}*\frm{)},
\POS"LUU1,1"."LUU5,3"!C+L-<35pt,0pt>*\txt{$
\Lambda_{t}\big(\mathsf{U}_{\scriptscriptstyle(1)}^{\beta}\big)
\;=\;
$}
\POS(50,-5)
\xymatrix"LU"@R=.5pt@C=.5pt@W=1em@H=2em{
\makebox[70pt][c]{$\vf[a]{\beta}{}\cdot P$}
&=&
z^{\beta}\;\\
\makebox[70pt][c]{$(\mathsf{D}_{t}\ad\vf[a]{\beta}{})\cdot P$}
\ar@{.}[dd]&=&\ar@{.}[dd]
0\;\\
&&\\
\makebox[70pt][c]{$(\mathsf{D}_{t}^{k-1}\ad\vf[a]{\beta}{})\cdot P$}
&=&
0\;\\
\makebox[70pt][c]{$(\mathsf{D}_{t}^{k}\ad\vf[a]{\beta}{})\cdot P$}
&=&
0\;
}
\POS"LU1,1"."LU5,3"!C*++\frm{(}*\frm{)},
\POS"LU1,1"."LU5,3"!C+R+<35pt,0pt>*\txt{$
\;=\;
\Lambda_{t}\big(\vf[a]{\beta}{}\big).
$}
\POS"LU1,1"."LU1,3"!C*-\frm{--},
\POS"LUU2,1"."LUU2,3"!C*-\frm{--},
\POS"LU1,1"+L\ar@(l,r)"LUU2,3"+R,
\POS"LU2,1"."LU2,3"!C*-\frm{--},
\POS"LUU3,1"."LUU3,3"!C*-\frm{--},
\POS"LU2,1"+L\ar@(l,r)"LUU3,3"+R,
\POS"LU4,1"."LU4,3"!C*-\frm{--},
\POS"LUU5,1"."LUU5,3"!C*-\frm{--},
\POS"LU4,1"+L\ar@(l,r)"LUU5,3"+R,
\endxy \]
As simple as the problem \eqref{eq:U1t} may firstly appear, one has to admit after a moment of reflection that it is not so, because one has to eliminate the jet derivatives that inevitably appear when expanding \begin{equation}
\label{eq:recallDt}
\mathsf{D}_{t}
=
\sum_{i=1,\dotsc,n}
\left(
\sum_{p=0}^{k-1}
(p+1)\;
z_{i}^{(p+1)}\;
\vf{i}{(p)}
\right). \end{equation} However, our efforts will be repaid: we are about to see how to reap the benefits from the strategy sketched above, using the flexibility in the choice of the generators \(\widetilde{\mathsf{V}}\) of the tangent space. \begin{enumerate}
\item
Notice that by Lemma \ref{cor:DtD1} one can use \(\mathsf{D}_{z_{1}}\) instead of \(\mathsf{D}_{t}\) to define in the exact analogous way the subspace of vertical jets, working without loss of generality on the subset \(\{z_{1}'\neq0\}\) of the set of invertible jets, on which we have to prove global generation of the tangent space, since:
\[
\left\{
\begin{array}{cl}
P(z)&=0\\
\mathsf{D}_{t}\cdot P(z)&=0\\
\vdots\\
\mathsf{D}_{t}^{k}\cdot P(z)&=0
\end{array}
\right.
\stackrel{\ (z_{1}'\neq0)\ }\Longleftrightarrow
\left\{
\begin{array}{cl}
P(z)&=0\\
\mathsf{D}_{z_{1}}\cdot P(z)&=0\\
\vdots\\
\mathsf{D}_{z_{1}}^{k}\cdot P(z)&=0.
\end{array}
\right.
\]
\item
Recall from \S\ref{sse:geometricJetCoordinates} that the use of geometric jet coordinates on \(\{z_{1}'\neq0\}\) simplifies substantially the formal differentiation of jets in the special direction \(z_{1}\);
we have seen that \(\mathsf{D}_{z_{1}}\) has component in the \(z_{1}\)-direction plainly equal to \(\vf{1}{}\), more precisely:
\[
\mathsf{D}_{z_{1}}\vert_{J_{k-1}}
=
\vf{1}{}
+
\sum_{i=2,\dotsc,n}
\left(
\sum_{p=0}^{k-1}
(p+1)\;
z_{i}^{[p+1]}\;
\vf{i}{[p]}
\right),
\]
\textit{cp.} with the expression \eqref{eq:recallDt} of \(\mathsf{D}_{t}\) just above. \end{enumerate} As a consequence of the first point, all what we have done can be do using \(\mathsf{D}_{z_{1}}\) instead of \(\mathsf{D}_{t}\), and as a consequence of the second point, the analog of the problem \eqref{eq:U1t} becomes much simpler (see below). In other words, the choice of the vertical vector fields \(\mathsf{V}=\vf{i}{(1)}\) was not the second simplest choice, since it is more easy to treat the case where: \[
\mathsf{V}
=
\vf{i}{[1]}. \] This remark explains why these vector fields already appeared in the matricial approach introduced by P\u{a}un and further pushed by Rousseau and Merker (\textit{cf}. \eqref{eq:vfpaun} above for a translation formula).
Restarting and doing all the same reasoning with \(\mathsf{D}_{z_{1}}\) and squared brackets on \(\{z_{1}'\neq0\}\), we set: \[
\Lambda_{z_{1}}\big(\widetilde{\mathsf{V}}\big)
\mathrel{\mathop{:}}=
\bigl(\widetilde{\mathsf{V}}\cdot P\bigr)\;\mathbi{e}_{0}+
\bigl(\,(\mathsf{D}_{z_{1}}^{1}\ad\widetilde{\mathsf{V}})\cdot P\,\bigr)\;\mathbi{e}_{1}+
\dotsb+
\bigl(\,(\mathsf{D}_{z_{1}}^{k}\ad\widetilde{\mathsf{V}})\cdot P\,\bigr)\;\mathbi{e}_{k}, \] and we get that (for \(j\neq 1\)) the vector field \(\vf{j}{[1]}-\mathsf{U}\) is tangent to \(J_{k}^{\mathit{vert}}\) over the set \(\{z_{1}'\neq0\}\) if: \[
\Lambda_{z_{1}}(\mathsf{U})
=
\Lambda_{z_{1}}(\vf{j}{[1]})
=
(\vf{j}{}\cdot P(z))\;\mathbi{e}_{1}. \] So we construct building-block vector fields \(\mathsf{U}_{1}^{\beta}\) such that \(\Lambda_{z_{1}}(\mathsf{U}_{1}^{\beta})=z^{\beta}\;\mathbi{e}_{1}\) by solving: \begin{equation}
\label{eq:U1}
\mathsf{U}_{1}^{\beta}\cdot P =0
\qquad\text{and}\qquad
\mathsf{D}_{z_{1}}\ad\mathsf{U}_{1}^{\beta}
=
\vf[a]{\beta}{}, \end{equation} but now, using the direction \(z_{1}\), the problem becomes essentially trivial since for \(\abs{\beta}\leq d-1\) a simple solution to \eqref{eq:U1} is to take: \begin{equation}
\mathsf{U}_{1}^{\beta}
\mathrel{\mathop{:}}=
z_{1}\,\vf[a]{\beta}{}
-
\vf[a]{\beta+\mathbi{1}_{1}}{}. \end{equation} and finally we obtain a vector field tangent to the vertical jets with vertical part \(\vf{i}{[1]}\) in exactly the same way as in the fundamental case, since: \[
\Lambda_{z_{1}}
\bigg(
\vf{j}{[1]}
-
\sum_{\abs{\beta}\leq d-1}
u_{j,\beta}\,
\mathsf{U}_{1}^{\beta}
\bigg)
=
\left(
\sum_{\abs{\beta}\leq d-1}\!
u_{j,\beta}\,z^{\beta}
-
\sum_{\abs{\beta}\leq d-1}\!
u_{j,\beta}\,z^{\beta}
\right)\;
\mathbi{e}_{1}
=
0. \]
\subsubsection{Building-block vector fields.} Recall that on the set \(\{z_{1}'\neq0\}\) the vector field \( \widetilde{\mathsf{V}} = \mathsf{V} - \mathsf{U} \) is a section of \(\mathcal{T}_{k}\) if and only if it satisfies the \(k+1\) conditions, for \(p=0,1,\dotsc,k\): \[
\bigg(
P(\xi)=\mathsf{D}_{z_{1}}\cdot P(\xi)=\dotsb=\mathsf{D}_{z_{1}}^{p}\cdot P(\xi)=0
\bigg)
\Rightarrow
\bigg(
\Lambda_{z_{1}}
(
\mathsf{V}
)
(\xi)
=
\Lambda_{z_{1}}(
\mathsf{U}
)
(\xi)
\bigg). \] But in the compact case, we will not need the left assumption to obtain the right conclusion.
When working on \(\{z_{1}'\neq0\}\), we demand that \(\widehat{\alpha}_{1}=0\), for technical reasons, and we fix \(\widehat\alpha\) once for all.
Since by Corollary \ref{cor:Lvfjq} the vector field associated to the geometric jet coordinate \(z_{j}^{[q]}\) satisfy: \begin{equation}
\label{eq:D1VF}
\Lambda_{z_{1}}(
\vf{j}{[q]}
)
\equiv
\bigl(\vf{j}{}\cdot P\bigr)\,\mathbi{e}_{q}, \end{equation} according to the strategy sketched out above, it needs to be corrected by an adequate linear combination of building-block vector fields \(\mathsf{U}\) such that \[
\Lambda_{z_{1}}(
\mathsf{U}
)
\equiv
z^{\beta}\,\mathbi{e}_{q}. \]
The iteration of the inductive step \eqref{eq:U1t} allows to produce most of the needed building-block vector fields: \begin{lemma}
\label{lem:Uq}
For \(q\in\N\) and \(\beta\) with \(\abs{\beta}+q\leq d\) the vector field:
\[
\mathsf{U}_{q}^{\beta}
\mathrel{\mathop{:}}=
\sum_{p=0}^{q}
\frac{(-1)^{p}}{p!(q-p)!}\,
z_{1}^{q-p}\;
\vf[a]{\beta+p\,\mathbi{1}_{1}}{}
\]
is a solution to
\(
\Lambda_{z_{1}}(
\mathsf{U}
)
\equiv
z^{\beta}\,\mathbi{e}_{q}
\),
where implicitly \(\mathbi{e}_{q}=\mathbi{0}\) if \(q>k\). \end{lemma} \begin{proof}
One checks the inductive formula:
\[
\mathsf{U}_{q+1}^{\beta}
=
z_{1}\,\mathsf{U}_{q}^{\beta}
-
\mathsf{U}_{q}^{\beta+\mathbi{1}_{1}},
\]
whence, making an induction on \(q\), if \(\Lambda_{z_{1}}(\mathsf{U}_{q}^{\beta})=z^{\beta}\,\mathbi{e}_{q}\), the vector field \(\mathsf{U}_{q+1}^{\beta}\) clearly satisfies
\(
\mathsf{U}_{q+1}^{\beta}\cdot P
\equiv
0
\)
and
\(
\big(
\mathsf{D}_{z_{_1}}
\ad
\mathsf{U}_{q+1}^{\beta}
\big)
=
\mathsf{U}_{q}^{\beta}
\),
which allows to conclude as above, by a shift. \end{proof}
The vector fields obtained in this way are certainly not enough to correct \eqref{eq:D1VF}, because of the technical limitation to exponents \(\beta\) with \(\abs{\beta}\leq d-q\). But actually, we will overcome this limitation,
\begin{lemma}
\label{lem:moreUb}
For any \(\beta\in\N^{n}\), their exists a vector field
with coefficients in \(\C[z_{1},\dotsc,z_{n}]\):
\[
\mathsf{U}_{0}^{\beta}
\mathrel{\mathop{:}}=
\sum_{\gamma\leq\beta}
{\propto}_{\beta,\gamma}(z)\,
\vf[a]{\gamma}{},
\]
having degree at most \(k+\abs{\beta}-d\), such that:
\[
\Lambda_{z_{1}}(
\mathsf{U}_{0}^{\beta}
)
\equiv
z^{\beta}\,\mathbi{e}_{0}.
\] \end{lemma} \begin{proof}
In order to construct such vector fields, we will extensively use the following result of Merker, that can be thought of as a much more general analog of the formula
\(
\Lambda_{z_{1}}(
\mathsf{U}_{k+1}^{\beta}
)
\equiv
0
\):
\begin{proposition}[Merker~\cite{MR2543663}]
\label{prop:higher_length}
For \(\beta\in\N^{n}\) with \(k+1\leq\abs{\beta}\leq d\), fix a multi-index \(\lambda\leq\beta\) with length \(\abs{\lambda}=k+1\). Then, the vector field defined by:
\[
\mathsf{T}_{\beta,\lambda}
\mathrel{\mathop{:}}=
\sum_{\gamma\leq\lambda}
(-1)^{\abs{\gamma}}
\,
\frac{\lambda!}{\gamma!(\lambda-\gamma)!}
\,
z^{\gamma}
\,
\vf[a]{\beta-\gamma}{}.
\]
satisfies
\(
\mathsf{T}_{\beta,\lambda}
\cdot
(\mathsf{D}_{t}^{p}\cdot P)
\equiv
0
\),
for \(p=0,1,\dotsc,k\),
identically on \(J_{k}\).
\end{proposition}
It is easily reformulated as follows. Take \(\beta,\lambda\in\N^{n}\) with \(\abs{\lambda}=k+1\) such that \(\mathsf{U}_{0}^{\beta-\gamma}\) is defined for any \(\gamma\leq\lambda\), then:
\[
\label{eq:merker}
\mathsf{U}
\mathrel{\mathop{:}}=
\sum_{0<\gamma\leq\lambda}
(-1)^{\abs{\gamma}}
\,
\frac{\lambda!}{(\lambda-\gamma)!\gamma!}
\,
z^{\gamma}
\,
\mathsf{U}_{0}^{\beta-\gamma}
\tag{$\ast$}
\]
behaves as the non existing vector field ``\(\vf[a]{\beta}{}\)'' up to order \(k\), namely it satisfies:
\[
\Lambda_{z_{1}}
(\mathsf{U})
\equiv
z^{\beta}\,
\mathbi{e}_{0},
\]
identically on \(J_{k}\).
Thus, reasoning by induction on \(\abs{\beta}-d=1,2,\dotsc\), we can construct recursively the vector fields \(\mathsf{U}_{0}^{\beta}\) using \eqref{eq:merker} at each step.
This construction is not unique, but in all cases the degree is less than \(k+\abs{\beta}-d\), again by induction. \end{proof}
\iffalse In order to fix ideas, we express the constructed vector fields in the simple example \(n=k=d=2\). Then, because \(d=k\) there is only one choice of \(\lambda\) for \(\abs{\beta}-d=1\), that yields indeed vector fields with coefficients of degree \(1+k=3\): \begin{align*}
\mathsf{U}_{0}^{(3,0)}
&=
z_{1}^{3}\vf[a]{(0,0)}{}
-
3\,z_{1}^{2}\vf[a]{(1,0)}{}
+
3\,z_{1}\vf[a]{(2,0)}{},
\\
\mathsf{U}_{0}^{(2,1)}
&=
z_{1}^{2}z_{2}\vf[a]{(0,0)}{}
-
2\,z_{1}z_{2}\vf[a]{(1,0)}{}
+
z_{2}\vf[a]{(2,0)}{}
-
z_{1}^{2}\vf[a]{(0,1)}{}
+
2\,z_{1}\vf[a]{(1,1)}{},
\\
\mathsf{U}_{0}^{(1,2)}
&=
z_{2}^{2}z_{1}\vf[a]{(0,0)}{}
-
2\,z_{2}z_{1}\vf[a]{(0,1)}{}
+
z_{1}\vf[a]{(0,2)}{}
-
z_{2}^{2}\vf[a]{(1,0)}{}
+
2\,z_{2}\vf[a]{(1,1)}{},
\\
\mathsf{U}_{0}^{(0,3)}
&=
z_{2}^{3}\vf[a]{(0,0)}{}
-
3\,z_{2}^{2}\vf[a]{(0,1)}{}
+
3\,z_{2}\vf[a]{(0,2)}{}. \end{align*} Then using these vector fields, we treat the next diagonal \(\abs{\beta}-d=2\), obtaining indeed vector fields with coefficients of degree \(2+k=4\) ; there is only one choice of \(\gamma\) for \(\beta=(4,0)\) and \(\beta=(0,4)\), that yields: \begin{align*}
\mathsf{U}_{0}^{(4,0)}
&=
3\,z_{1}^{4}\vf[a]{(0,0)}{}
-
8\,z_{1}^{3}\vf[a]{(1,0)}{}
+
6\,z_{1}^{2}\vf[a]{(2,0)}{},
\\
\mathsf{U}_{0}^{(0,3)}
&=
3\,z_{2}^{4}\vf[a]{(0,0)}{}
-
8\,z_{2}^{3}\vf[a]{(0,1)}{}
+
6\,z_{2}^{2}\vf[a]{(0,2)}{}, \end{align*} and for the remaining \(\beta=(3,1)\), \(\beta=(2,2)\) and \(\beta=(1,3)\), there are two choices for \(\gamma\), that both give the same vector field (unicity in that particular example does not hold in the general situation, but we only need existence): \begin{align*}
\mathsf{U}_{0}^{(3,1)}
&=
3\,z_{1}^{3}z_{2}\vf[a]{(0,0)}{}
-
6\,z_{1}^{2}z_{2}\vf[a]{(1,0)}{}
+
3\,z_{1}z_{2}\vf[a]{(2,0)}{}
-
2\,z_{1}^{3}\vf[a]{(0,1)}{}
+
3\,z_{1}^{2}\vf[a]{(1,1)}{},
\\
\mathsf{U}_{0}^{(2,2)}
&=
3\,z_{1}^{2}z_{2}^{2}\vf[a]{(0,0)}{}
-
4\,z_{1}z_{2}^{2}\vf[a]{(1,0)}{}
+
z_{2}^{2}\vf[a]{(2,0)}{}
-
4\,z_{1}z_{2}^{2}\vf[a]{(0,1)}{}
+
4\,z_{1}z_{2}\vf[a]{(1,1)}{}
-
z_{1}^{2}\vf[a]{(0,2)}{},
\\
\mathsf{U}_{0}^{(1,3)}
&=
3\,z_{2}^{3}z_{1}\vf[a]{(0,0)}{}
-
6\,z_{2}^{2}z_{1}\vf[a]{(0,1)}{}
+
3\,z_{2}z_{1}\vf[a]{(0,2)}{}
-
2\,z_{2}^{3}\vf[a]{(1,0)}{}
+
3\,z_{2}^{2}\vf[a]{(1,1)}{}. \end{align*}
So, coming back to the general situation, we can extend the result of lemma to all \(\beta\), with no restriction on \(\abs{\beta}-d\). \fi So, we can extend the result of Lemma \ref{lem:Uq} to all \(\beta\), with no restriction on \(\abs{\beta}-d\). Then, we deduce as above the following. \begin{corollary}
\label{cor:Tj}
In the slanted directions, for \(j=2,\dotsc,n\) and \(q=0,1,\dotsc,k\),
the following corrected vector field is tangent to \(J_{k}^{vert}\):
\[
\mathsf{T}_{j,q}
\mathrel{\mathop{:}}=
\Big(
\vf{j}{[q]}
-
\hskip-2pt
\sum_{\abs{\beta}\leq d-1}
(\beta_{j}+1)\,
a_{\beta+\mathbi{1}_{j}}\,
\mathsf{U}_{q}^{\beta}
\Big).
\] \end{corollary}
\subsubsection{Direction of the space of parameters.} In the direction of the space of parameters, for \(k+1\leq\abs{\beta}\leq d\), any of the vector fields provided by Merker, of the form \[
\mathsf{T}_{\beta}
\mathrel{\mathop{:}}=
\vf[a]{\beta}{}
-
\sum_{\gamma<\beta}
{\propto}_{\beta,\gamma}(z)\,
\vf[a]{\gamma}{}, \] is tangent to vertical jets, and for the remaining multi-indices, of length \(\abs{\beta}\leq k\), the following vector fields suits to our purposes: \[
\mathsf{T}_{\beta}
\mathrel{\mathop{:}}=
(z_{1}')^{2k-1}
\Big(
\vf[a]{\beta}{}
-
\sum_{q=0}^{k}
\big(\mathsf{D}_{z_{1}}^{q}\cdot z^{\beta}\big)\,
\mathsf{U}_{q}^{\mathbf{0}}
\Big). \] It is obtained by solving the triangular system satisfied by \({\propto}_{0},\dotsc,{\propto}_{k}\) for the vector field \( \vf[a]{\beta}{} - \sum_{q=0}^{k} {\propto}_{q}\, \mathsf{U}_{q}^{\mathbf{0}} \) to be a section of \(\mathcal{T}_{k}\), and then by multiplying by the adequate monomial in \(z_{1}'\) to compensate the poles in the fiber.
Since the first family is triangular, and since the corrective parts of the second family are elements of the vector space spanned by \(\vf[a]{\mathbf{0}}{},\vf[a]{\mathbf{1}_{1}}{},\dotsc,\vf[a]{k\,\mathbf{1}_{1}}{}\), the collection of these vector fields span a vector space of codimension \(k+1\) in the direction of the space of parameters.
\section{Low Pole Order Frames on Vertical Jets}
In this section we finish the proof of the main result for the universal hypersurface, and then adapt it to various geometric settings. \subsubsection{Global generation.} In order to prove the global generation, in the \(z\)-directions, we use the Corollary \ref{cor:span.z}, the Corollary \ref{cor:Tj} and the simple observation that the vector fields \(\vf[t]{[1]}{}\), \dots,\(\vf[t]{[k]}{}\) are always tangent to the vertical jets ---\,because the variables \(t^{[1]},\dotsc,t^{[k]}\) are not involved in the defining equations\,--- and in the direction of the space of parameters, we use the final observation of section \S\ref{se:construction} above that the vector fields \(\mathsf{T}_{\beta}\) span a vector space of codimension \(k+1\). Because the codimensions agree, this yields indeed that the collection of three families \[
\Big\{
\quad
\{T_{j,q}\}_{j\geq 2,q=0,\dotsc,k},\
\{\mathsf{T}_{q}\}_{q=0,\dotsc,k},\
\{T_{\beta}\}_{\beta\neq \mathbf{0},\mathbf{1},\dotsc,k\mathbf{1}_{1}}
\quad
\Big\}, \] which contain respectively slanted tangential vector fields, vertical tangential vector fields, and lastly horizontal tangential vector fields, span the tangent space to vertical jets at points of \(J_{k}^{\mathit{vert}}\) where \(z_{1}'\neq 0\).
Notice that in the argumentation above we have intentionally not mentioned that the vector field \(\vf[a]{\widehat{\alpha}}{}\) does not exist. Actually, there is a good reason for that, since the two conditions \(\abs{\widehat{\alpha}}=d>d-1\) and \(\widehat{\alpha}_{1}=0\) imply that: \begin{itemize}
\item the coefficient \(u_{j,\widehat{\alpha}}\) is zero, in other words the non existing vector field \(\vf[a]{\widehat{\alpha}}{}\) is never involved in the construction of \(\mathsf{T}_{j,0}\), for \(q=0\).
\item \(\widehat{\alpha}\) is never equal to \(\beta+q\mathbf{1}_{1}\) with \(\abs{\beta}\leq d-1\) and thus it is never involved in the construction of the vector fields \(\mathsf{U}_{q}^{\beta}\), with \(q\geq1\) in Lemma \ref{lem:Uq}, nor in the construction of the vector fields \(\mathsf{U}_{0}^{\beta+q\mathbf{1}_{1}}\) in Lemma \ref{lem:moreUb}. \end{itemize}
\subsubsection{Pole order of meromorphic prolongations.} \label{sse:po} It remains to compute the pole order of the meromorphic prolongations of the corrected vector fields shown above.
Firstly, if \(\abs{\beta}\leq d\), then the pole order of \(\mathsf{U}_{0}^{\beta}\mathrel{\mathop{:}}=\vf[a]{\beta}{}\) is clearly \(0\) and for \(\abs{\beta}>d\), we have seen that the coefficients of \(\mathsf{U}_{0}^{\beta}\) are polynomials of degree at most \(k+\abs{\beta}-d\) in the variables \(z_{1},\dotsc,z_{n}\), that can classically be extended as meromorphic function with pole order at most \(k+\abs{\beta}-d\) on \(\P^{n+1}\); hence by additivity with respect to product and subadditivity with respect to sum, using the definition in Lemma \ref{lem:Uq}, the pole order of \(\mathsf{U}_{q}^{\beta}\) is: \begin{equation}
\label{eq:poUq}
\mathrm{p.o}(\mathsf{U}_{q}^{\beta})
=
\begin{cases}
q
&\text{if }\abs{\beta}+q\leq d
\\
q+k+\abs{\beta}+q-d
&\text{if }\abs{\beta}+q>d.
\end{cases} \end{equation}
\begin{subequations}
Then, by considering the successive derivations of \(z_{i}=Z_{i}/Z_{0}\), it is easy to see that the pole order along the hyperplane at infinity \((Z_{0}=0)\) of the meromorphic prolongation to \(\P^{n+1}\) of \(z_{i}^{(p)}\) is \(p+1\).
This yields that for \(\mu\in\N^{k}\), the pole order of the meromorphic continuation to \(\P^{n+1}\) of the monomial
\[
z_{i}^{(\mu)}
\mathrel{\mathop{:}}=
{z_{i}^{(1)}}^{\mu_{1}}
\dotsm
{z_{i}^{(k)}}^{\mu_{k}}
\qquad
{\scriptstyle(i=1,\dotsc,n)}
\]
is, by additivity:
\begin{equation}
\label{eq:po.1}
\mathrm{p.o}\big(z_{i}^{(\mu)}\big)
=
\sum_{p}(p+1)\mu_{p}
=
\Abs{\mu}+\abs{\mu}.
\end{equation}
Consequently, by subadditivity, the meromorphic continuation to \(\P^{n+1}\) of the Bell polynomial
\(
\mathbi{B}_{p,q}(z_{1})
=
\sum_{\Abs{\mu}=q,\abs{\mu}=p}
\frac{\abs{\mu}!}{\mu!}\,
z_{1}^{(\mu)}
\)
has pole order:
\begin{equation}
\label{eq:po.2}
\mathrm{p.o}\left(\mathbi{B}_{p,q}\right)
=
p+q.
\end{equation} \end{subequations}
Using \eqref{eq:po.1} and \eqref{eq:po.2}, it is elementary to compute the pole orders of the constructed vector fields by coming back to their expressions in the standard jet coordinate system.
\begin{subequations}
The coefficients of \(\vf{j}{[q]}\) are the Bell polynomials \(\mathbi{B}_{q,q}(z_{1}),\dotsc,\mathbi{B}_{q,k}(z_{1})\), whence:
\[ \mathrm{p.o.} \big( \vf{j}{[q]} \big) = q+k.\]
This and \eqref{eq:poUq} yields:
\begin{equation}
\label{eq:poTjq}
\mathrm{p.o}(\mathsf{T}_{j,q})
=
\begin{cases}
k+q&\text{if }q\leq 1\\
k-1+2q&\text{if }q\geq 2.
\end{cases}
\end{equation}
The coefficients of the vector fields
\[
\mathsf{T}_{q}
\mathrel{\mathop{:}}=
\sum_{m=1}^{k}
\mathbi{B}_{m,\ell}[t]\,
\vf[t]{}{[m]}
=
-
\sum_{i=1,\dotsc,n}\;
\sum_{p=1}^{k-\ell+1}
p\,z_{i}^{(p)}\,\vf{i}{(p+\ell-1)}
\]
are the monomials \(z_{i}^{(1)},\dotsc,z_{i}^{(k-q)}\) whence:
\begin{equation}
\label{eq:poTsym}
\mathrm{p.o}(\mathsf{T}_{q})
=
2\,(k-q).
\end{equation}
It remains to compute the pole order of the meromorphic prolongation of the vector fields \(\mathsf{T}_{\beta}\) and we will shortly show the following.
\begin{equation}
\label{eq:poTb}
\mathrm{p.o}(\mathsf{T}_{\beta})
=
\begin{cases}
k+1&\text{if }k+1\leq\abs{\beta}\leq d\\
4k+\abs{\beta}-2&\text{if }\abs{\beta}\leq k\\
\end{cases}
\end{equation} \end{subequations}
These observations \eqref{eq:poTjq}\,--\,\eqref{eq:poTb} yield the constants \((5\,k-2)\) in the main theorem and the other constant is \(1\) because everything is clearly linear in \(a_{\alpha}\).
Notice that this pole order is \(1\) more than the pole orders computed in small dimensions by P\u{a}un (\(7\) for \(k=2\)) and Rousseau (\(12\) for \(k=3\)), but this is a fair price to pay for obtaining the global generation on the subset of invertible jets, and not anymore in the complement of the zero locus of a certain determinant depending on the equation of the hypersurface. This is an important detail in order to get results towards the \emph{strong} Green-Griffiths conjecture (\textit{cf.} \cite{MR2593279,arXiv:1402.1396}).
\begin{proof}
[Proof of \eqref{eq:poTb}] A quick induction based on the chain rule \(\mathsf{D}_{z_{1}}\mathrel{\mathop{:}}=\mathsf{D}_{t}/z_{1}'\) shows that for \(q\leq k\) and \(\beta\in\N^{n}\), there exist combinatorial coefficients \({\propto}_{\bullet}\) such that: \[
\mathsf{D}_{z_{1}}^{q}\cdot z^{\beta}
=
\sum_{\substack{
\abs{\lambda}+\abs{\mu^{1}}+\dotsb+\abs{\mu^{n}}=\abs{\beta}+q-1\\
\Abs{\mu^{1}}+\dotsb+\Abs{\mu^{n}}=2q-1\\
}}
{\propto}_{\lambda,\mu^{1},\dotsc,\mu^{n}}(\beta,q)\;
z^{\lambda}\;
\frac{
\Big(
z_{1}^{(\mu^{1})}
\dotsm
z_{n}^{(\mu^{n})}
\Big)}
{{z_{1}'}^{2q-1}}. \] So after multiplying by \({z_{1}'}^{2q-1}\) one obtains a polynomial with pole order: \[
\mathrm{p.o.}
\big(
{z_{1}'}^{2q-1}\,
\mathsf{D}_{z_{1}}^{q}\cdot z^{\beta}
\big)
=
\abs{\lambda}
+
(
\abs{\mu^{1}}
+
\Abs{\mu^{1}}
)
+
\dotsb
+
(
\abs{\mu^{n}}
+
\Abs{\mu^{n}}
)
=
\abs{\beta}+3q-2. \] Thus, using subadditivity \begin{align*}
\mathrm{p.o}\;
\big(
(z_{1}')^{2k-1}
\big(\mathsf{D}_{z_{1}}^{q}\cdot z^{\beta}\big)\,
\mathsf{U}_{q}^{\mathbf{0}}
\big)
&\leq
\mathrm{p.o}\big((z_{1}')^{2(k-q)}\big)
+
\mathrm{p.o.}
\big(
{z_{1}'}^{2q-1}\,
\mathsf{D}_{z_{1}}^{q}\cdot z^{\beta}
\big)
+
\mathrm{p.o.}
\big(
\mathsf{U}_{q}^{\mathbf{0}}
\big)
\\
&\leq
4(k-q)
+
\abs{\beta}+3q-2
+
q
=
4k+\abs{\beta}-2, \end{align*} which indeed leads to \(\mathrm{p.o}(\mathsf{T}_{\beta})=4k+\abs{\beta}-2\) for \(\abs{\beta}\leq k\). \end{proof}
This ends the proof of the main result in the case of the universal hypersurface.
\subsubsection{Global generation for complete intersections.} For complete intersections, the strategy described in section \S\ref{se:construction} directly applies, since the only common variables between two of the \(c\) systems of algebro-differential equations corresponding to each of the \(c\) defining polynomials are the \(z\)-variables. Fixing the vertical part \(\mathsf{V}\) (possibly \(=0\)), it is possible to solve the problem for each system separately, then adding all the corrective parts, one obtains a corrective part for all the \(c\) systems together.
\subsubsection{Modifications needed for the logarithmic case.} In order to treat the logarithmic case in a very similar way as the compact case, we first straighten out the universal hypersurface, following the strategy of Rousseau in \cite{MR2552951}. As in the compact case, we start by the case \(c=1\).
In order to straighten out the universal family \(\mathcal{H}_{d}\), given in the system of coordinates \(\bigl([Z],[A]\bigr)\) on \(\P^{n}\times \P^{n_{d}}\) by \( \mathcal{H}_{d} \mathrel{\mathop{:}}= \{
0
=
\sum_{\abs{\alpha}=d}
A_{\alpha}\,Z^{\alpha} \} \subset \P^{n}\times \P^{n_{d}} \), introduce a new homogeneous "\(Z\)-coordinate" \(W\in \C\) associated with a new homogeneous "\(A\)-coordinate" \(A_{0}\in\C\), thought of as the coefficient of the monomial \(W^{d}\). Accordingly consider the zero set: \[
\mathcal{X}
\mathrel{\mathop{:}}=
\biggl\{
A_{0}\,
W^{d}
=
\sum_{\abs{\alpha}=d}
A_{\alpha}\,
Z^{\alpha}
\biggr\}
\subset
\P^{n+1}\times\P^{n_{d}+1}. \]
There is a natural \textsl{forgetful map}: \[
\pi
\colon
\P^{n+1}\times\P^{n_{d}+1}
\setminus
\left(\{\forall Z_{i}=0\}\cup\{\forall A_{\alpha}=0\}\right)
\to
\P^{n}\times\P^{n_{d}}, \] that consists in erasing both \(W\) and \(A_{0}\). Notice that: \[
\mathcal{X}
\cap
\left(\{\forall Z_{i}=0\}\cup\{\forall A_{\alpha}=0\}\right)
\subset
\mathcal{X}
\cap
\left(\{A_{0}=0\}\cup\{\forall A_{\alpha}=0\}\right) \] Indeed, if \(Z=0\), the equation of \(\mathcal{X}\) becomes: \( A_{0}\,W^{d} = 0. \) This implies that either \(A_{0}\) or \(W\) must be zero. But \(W\) cannot be zero, because the homogeneous coordinates of the point \([W:0:\dotso:0]\) in the projective space \(\P^{n+1}\) cannot be all simultaneously zero. Thus \(A_{0}\) must be zero.
Let \(\mathcal{X}^{*}\) be the restriction of \(\mathcal{X}\) to the affine set, pointed at the origin: \[
\{A_{0}\neq 0\}\setminus\{[1:0:\dotso:0]\}
\simeq
\C^{n+1}\setminus\{0\}. \] Then, the projection \(\pi\vert_{\mathcal{X}^{*}}\colon\mathcal{X}^{*}\to\P^{n}\times \P^{n_{d}}\) is well defined and moreover, it is a branched cover of degree \(d\) that ramifies exactly over \(\mathcal{H}_{d}\). Since \(A_{0}\neq 0\), the inverse image of the universal family \(\mathcal{H}_{d}\) under this projection identifies with the (straight) hyperplane \[
D
\mathrel{\mathop{:}}=
(\pi\vert_{\mathcal{X}^{*}})^{\moinsun}(\mathcal{H}_{d})
=
\{W=0\}. \]
The map \(\pi\colon(\mathcal{X}^{*},D)\to(\P^{n}\times \P^{n_{d}},\mathcal{H}_{d})\) is therefore a log-morphism (\cite{MR637060}), that induces a canonical holomorphic map on the spaces of jets of logarithmic curves: \[
\pi_{[k]}
\colon
J_{k}\mathcal{X}^{*}(-\log D)
\to
\pi^{\star}J_{k}(\P^{n}\times \P^{n_{d}})(-\log\mathcal{H}), \] that is clearly dominant, as \(d\pi_{[k]}\) is of maximal rank. This projection \(\pi_{[k]}\) also send vertical jets on vertical jets. We will thus study the vertical logarithmic jets upstairs, where it is easier to use logarithmic jet-coordinates.
As long as the \(w\)-component of the vector field is zero, nothing change compared to \S\ref{se:construction}, except that there is no multi-index \(\widehat{\alpha}\) to remove, and we have all the tangential vector fields of \S\ref{se:construction}.
For the supplementary logarithmic direction \(w\), by the Faà di Bruno formula, one has: \[
w^{[q]}
=
\exp(\log w)\,
\sum_{\Abs{\mu}=q}
\frac{{(\log w)^{[1]}}^{\mu_{1}}}{\mu_{1}!}
\dotsm
\frac{{(\log w)^{[k]}}^{\mu_{k}}}{\mu_{k}!}
\quad
{\scriptstyle(q=0,1,\dotsc,k)}, \] which implies the dual relations: \[
\vf[(\log w)]{}{[p]}
=
\sum_{q=p}^{k}
w^{[q-p]}\,
\vf[w]{}{[q]}
\quad
{\scriptstyle(q=0,1,\dotsc,k)}. \] In particular, notice for later use that for \(P\in\C[w,z_{1},\dotsc,z_{n}]\): \[
\vf[\log w]{}{}\cdot P
=
w\,
\vf[w]{}{}\cdot P
\qquad
\text{and}
\qquad
\vf[(\log w)]{}{[p]}\cdot P
=
0
\quad
{\scriptstyle(p=1,\dotsc,k)}. \] \begin{lemma}
For \(p=0,1,\dotsc,j\), the vector field \(\mathsf{V}=\vf[(\log w)]{}{[p]}\) does not satisfy \(\Lambda_{z_{1}}(\mathsf{V})=0\), because it produces a nonzero entry in the \((p+1)\)-th line:
\[
\Lambda_{z_{1}}\big(
\vf[(\log w)]{}{[p]}
\big)
=
(-1)^{p}\,p!\;
d\,w^{d}\;
\mathbi{e}_{p}.
\] \end{lemma} \begin{proof}
For \(p\geq1\), applying the Leibniz rule, as follows,
\begin{align*}
\big(
\mathsf{D}_{z_{1}}
\ad
\vf[(\log w)]{}{[p]}
\big)
&=
\sum_{q=p}^{k}
\big(\mathsf{D}_{z_{1}}\cdot w^{[q-p]}\big)\,
\vf[w]{}{[q]}
+
\sum_{q=p}^{k}
w^{[q-p]}\,
\big(
\mathsf{D}_{z_{1}}
\ad
\vf[w]{}{[q]}
\big)
\\
&=
\sum_{q=p}^{k}
(q-p+1)\,w^{[q-p+1]}\,
\vf[w]{}{[q]}
+
\sum_{q=p}^{k}
w^{[q-p]}\,
\big(
-q\,
\vf[w]{}{[q-1]}
\big),
\end{align*}
and then, shifting the indices in the second sum, and adding zero in the first sum by considering also the term for \(q=p-1\):
\begin{align*}
\big(
\mathsf{D}_{z_{1}}
\ad
\vf[(\log w)]{}{[p]}
\big)
&=
\sum_{q=p-1}^{k}
(q-p+1)\,w^{[q-p+1]}\,
\vf[w]{}{[q]}
+
\sum_{q=p-1}^{k-1}
w^{[q-p+1]}\,
\big(
-(q+1)\,
\vf[w]{}{[q]}
\big)
\\
&=
-p
\sum_{q=p-1}^{k}
w^{[q-(p-1)]}\,
\vf[w]{}{[q]}
+
(k+1)\,
w^{[k+1-p]}\,
\vf[w]{}{[k]},
\end{align*}
one obtains that for \(p\geq1\):
\[
\big(
\mathsf{D}_{z_{1}}
\ad
\vf[(\log w)]{}{[p]}
\big)
=
-p
\vf[(\log w)]{}{[p-1]}
+
(k-p+1)\,w^{(k-p+1)}\,\vf[w]{}{[k]}.
\]
Similarly for \(p=0\) one obtains:
\[
\big(
\mathsf{D}_{z_{1}}
\ad
\vf[\log w]{}{}
\big)
=
0.
\]
By induction, this yields that, for \(q\leq p\):
\[
\big(
\mathsf{D}_{z_{1}}^{q}
\ad
\vf[(\log w)]{}{[p]}
\big)
=
(-1)^{p-q}\,\frac{p!}{(p-q)!}\,
\vf[(\log w)]{}{[p-q]}
+
\sum_{r=k+1-q}^{k}
{\propto}_{q,p}(w^{(k+1-p)},\dotsb,w^{(k+q-p)})\,
\vf[w]{}{[q]},
\]
for some polynomial coefficients \({\propto}_{q,p}\),
and for \(q\geq p\)
\[
\big(
\mathsf{D}_{z_{1}}^{q}
\ad
\vf[(\log w)]{}{[p]}
\big)
=
\sum_{r=k+1-q}^{k}
{\propto}_{q,p}(w^{(k+1-p)},\dotsb,w^{(k+q-p)})\,
\vf[w]{}{[q]},
\]
for some polynomial coefficients \({\propto}_{q,p}\).
Now, recall that for \(P\in\C[w,z_{1},\dotsc,z_{n}]\)
one has
\(
\vf[\log w]{}{}\cdot P
=
w\,
\vf[w]{}{}\cdot P
\) and \(
\vf[(\log w)]{}{[p]}\cdot P
=
0
\quad
{\scriptstyle(p=1,\dotsc,k)}
\),
and notice that of course
\(
\vf[w]{}{[p]}\cdot P
=
0
\quad
{\scriptstyle(p=1,\dotsc,k)}
\),
in order to conclude, as announced that for \(q\neq p\):
\[
\big(
\mathsf{D}_{z_{1}}^{q}
\ad
\vf[(\log w)]{}{[p]}
\big)
\cdot
\big(
w^{d}-\sum_{\abs{\alpha}\leq d} a_{\alpha}\,z^{\alpha}
\big)
=
0,
\]
and for \(q=p\):
\[
\big(
\mathsf{D}_{z_{1}}^{p}
\ad
\vf[(\log w)]{}{[p]}
\big)
\cdot
\big(
w^{d}-\sum_{\abs{\alpha}\leq d} a_{\alpha}\,z^{\alpha}
\big)
=
(-1)^{p}\,p!\;
\vf[\log w]{}{}\cdot w^{d}
=
(-1)^{p}\,p!\;
d\,w^{d},
\]
which concludes the proof. \end{proof}
\begin{corollary}
\label{cor:Tw}
In the logarithmic direction, for \(q=0,1,\dotsc,k\),
the following corrected vector field is tangent to \(J_{k}^{vert}\):
\[
\mathsf{T}_{w,q}
\mathrel{\mathop{:}}=
\vf[w]{}{[q]}
+
(-1)^{q}\,q!\,d\;
\Big(
\hskip-2pt
\sum_{\abs{\alpha}\leq d}
a_{\alpha}\,
\mathsf{U}_{q}^{\alpha}
\Big).
\] \end{corollary} \begin{proof}
Indeed:
\[
w^{d}=\sum_{\abs{\alpha}\leq d}a_{\alpha}\,z^{\alpha}
\Longrightarrow
\big(
\mathsf{D}_{z_{1}}^{q}
\ad
\mathsf{T}_{w,p}
\big)
\Big(w^{d}-\sum_{\abs{\alpha}\leq d} a_{\alpha}\,z^{\alpha}\Big)
=
0.
\]
Thus \(\mathsf{T}_{w,q}\) is a section of \(\mathcal{T}_{k}\), and in particular it is tangent to \(J_{k}^{\mathit{vert}}\). \end{proof}
Notice that here the polynomial is of degree \(d\) and not \(d-1\) as in the compact case, but this is not a problem, since in the logarithmic case, we have not removed a multi-index \(\widehat{\alpha}\).
\subsubsection{Modifications needed for the logarithmic case with several components.} In order to straighten out \(\mathcal{H}_{d_{1},\dotsc,d_{c}}\), introduce \(c\) auxiliary variables \(W_{1}\), \dots, \(W_{c}\) together with the associated parameters \(A_{\mathbf{0}}^{1}\), \dots, \(A_{\mathbf{0}}^{c}\), and consider the complete intersection \[
\mathcal{Y}
\subset
\{Z_{0}\neq0\}
\times
\{A_{\mathbf{0}}^{1}\neq0\}
\times
\dotsb
\times
\{A_{\mathbf{0}}^{c}\neq0\}, \] defined by the equations: \[
{w_{1}}^{d}=\sum_{\abs{\alpha}\leq d_{1}} a_{\alpha}^{1}\,z^{\alpha}\quad,\qquad
\dotsc\quad,\qquad
{w_{c}}^{d}=\sum_{\abs{\alpha}\leq d_{c}} a_{\alpha}^{c}\,z^{\alpha}. \] These \(c\) equations have only the \(z\)-variables in common. Since we are able to solve the problem for a fixed ``\(z\)-part'' (possibly equal to zero) with \(c=1\), it is possible to solve the problem for each equation separately. Then adding all the corrective parts we get a corrective part for the system of \(c\) equations, since the ``\(w\)-parts'' and ``\(a\)-parts'' interact with only one of the \(c\) equations, and vanish on the others.
\subsection*{\bfseries Acknowledgments} I want to thank \textsl{Jérémy Guéré} for suggestions concerning presentation and \textsl{Christophe Mourougane} for interesting discussions on geometric jet coordinates. I warmly thank \textsl{Jean-Pierre Demailly} for friendly explanations in his office, which influenced the definition of the geometric jet coordinates. Lastly, I would like to gratefully thank my thesis advisor \textsl{Joël Merker} for his support, his very careful reading and the proposal of relevant lines of thinking.
\end{document} | arXiv |
Sugeno integral
In mathematics, the Sugeno integral, named after M. Sugeno,[1] is a type of integral with respect to a fuzzy measure.
Let $(X,\Omega )$ be a measurable space and let $h:X\to [0,1]$ be an $\Omega $-measurable function.
The Sugeno integral over the crisp set $A\subseteq X$ of the function $h$ with respect to the fuzzy measure $g$ is defined by:
$\int _{A}h(x)\circ g={\sup _{E\subseteq X}}\left[\min \left(\min _{x\in E}h(x),g(A\cap E)\right)\right]={\sup _{\alpha \in [0,1]}}\left[\min \left(\alpha ,g(A\cap F_{\alpha })\right)\right]$
where $F_{\alpha }=\left\{x|h(x)\geq \alpha \right\}$.
The Sugeno integral over the fuzzy set ${\tilde {A}}$ of the function $h$ with respect to the fuzzy measure $g$ is defined by:
$\int _{A}h(x)\circ g=\int _{X}\left[h_{A}(x)\wedge h(x)\right]\circ g$
where $h_{A}(x)$ is the membership function of the fuzzy set ${\tilde {A}}$.
Usage and Relationships
Sugeno integral is related to h-index.[2]
References
1. Sugeno, M. (1974) Theory of fuzzy integrals and its applications, Doctoral. Thesis, Tokyo Institute of Technology
2. Mesiar, Radko; Gagolewski, Marek (December 2016). "H-Index and Other Sugeno Integrals: Some Defects and Their Compensation". IEEE Transactions on Fuzzy Systems. 24 (6): 1668–1672. doi:10.1109/TFUZZ.2016.2516579. ISSN 1941-0034.
• Gunther Schmidt (2006) Relational measures and integration, Lecture Notes in Computer Science # 4136, pages 343−57, Springer books
• M. Sugeno & T. Murofushi (1987) "Pseudo-additive measures and integrals", Journal of Mathematical Analysis and Applications 122: 197−222 MR0874969
| Wikipedia |
Balancing decoding speed and memory usage for Huffman codes using quaternary tree
Ahsan Habib ORCID: orcid.org/0000-0001-9320-44561 &
Mohammad Shahidur Rahman1
Applied Informatics volume 4, Article number: 5 (2017) Cite this article
In this paper, we focus on the use of quaternary tree instead of binary tree to speed up the decoding time for Huffman codes. It is usually difficult to achieve a balance between speed and memory usage using variable-length binary Huffman code. Quaternary tree is used here to produce optimal codeword that speeds up the way of searching. We analyzed the performance of our algorithms with the Huffman-based techniques in terms of decoding speed and compression ratio. The proposed decoding algorithm outperforms the Huffman-based techniques in terms of speed while the compression performance remains almost same.
Huffman (1952) presented a coding system for data compression at I.R.E conference in 1952 and informed that no two messages will consist of same coding arrangement and the codes will be produced in such a way that no additional arrangement is required to specify where a code begins and ends once the starting point is known. Since that time Huffman coding is not only popular in data compression but also image and video compression (Chung 1997). Schack (1994) described in his paper that codeword lengths of both Huffman and Shanon–Fano have similar interpretation. Katona and Nemetz (1978) investigated the connection between self-information of a source symbols and its codeword length.
In another research, Hashemian (1995) introduced a new compression technique with the clustering algorithm. In this new type of algorithm, he claimed that it required minimum storage whereas the speed for searching of symbol will be high. He also conducted experiment on video data and found his method very efficient. Chung (1997) introduced an array-based data structure for Huffman tree where the memory requirement is \(3n - 2\). He also proposed a fast decoding algorithm for this structure and claimed that the memory size can be reduced from \(3n - 2\) to \(2n - 3\), where n is the number of symbols. To attain more decoding speed with compact memory size, Chen et al. (1999) presented a fast decoding algorithm with \(O \left( {\log n} \right)\) time and \(\lceil\frac{3n}{2}\rceil + \lceil\left( {\frac{n}{2}} \right)\log n\rceil + 1\) memory space.
Banetley et al. (1986) introduced a new compression technique that is quite close to Huffman technique with some implementation advantages; it requires one-pass over the data to be compressed. Sharma (2010) and Kodituwakku and Amarasinghe (2011) have presented that Huffman-based technique produces optimal and compact code. However, the decoding speed of this technique is relatively slow. Bahadili and Hussain (2010) presented a new bit level adaptive data compression technique based on ACW algorithm, which is shown to perform better than many widely used compression algorithms in terms of compression ratio. Hermassi et al. (2010) showed how a symbol can be coded by more than one codeword having the same length. Chowdhury et al. (2002) presented a new decoding technique of self-styled static Huffman code, where they showed a very efficient representation of Huffman header. In paper, Suri and Goel (2011) focused on the use of ternary tree, where a new one-pass algorithm for decoding adapting Huffman codes is implemented.
Fenwick (1995) in his research showed that the Huffman codes do not improve the code efficiency at all time. It shows that the performance is always declining when moving to the lower extension to higher extension. Szpankowski (2011) and Baer (2006) explained the minimum expected length of fixed-to-variable lossless compression without prefix constraint. Huffman principle, which is well known for fixed-to-variable code, is used in Kavousianos (2008) as a variable-to-variable code. A new technique for online compression in networks has been presented by Vitter (1987) in his paper. Habib et al. (2013) introduced Haffman code in the field of database compression. Gallager (1978) explained four properties of Huffman codes—sibling property, upper bound property, codeword length property and symbol frequency property. He also proposed an adaptive approach of Huffman coding. Lampel and Ziv (1977) and Welch (1984) described a coding technique for any kind of source symbol. Lin et al. (2012) worked on the efficiency of Huffman decoding, where authors first transform the basic Huffman tree to recursive Huffman tree, and then the recursive Huffman algorithm decodes more than one symbol at a time. In this way, it achieves more decoding speed. Google Inc. recently released a compression tool named Zopfli (Alakuijala and Vandevenne 2013) and claimed that Zopfli yields the best compression ratio.
In summary, it is revealed in the literature that using binary Huffman code it is difficult to achieve a balance between speed and memory usage. In this paper, we focus on the use of quaternary tree instead of binary tree that speeds up decoding time. Here, we employ two algorithms for encoding and decoding quaternary Huffman codes for the implementation of our proposed technique. When compared with the Huffman-based techniques, the proposed decoding algorithm exhibits excellent performance in terms of speed while the compression performance remains almost same. In this way, the proposed technique offers a way to balance between the decoding time and memory usage. We have organized the paper as follows. In "Quaternary tree architecture" section, traditional binary Huffman decoding technique in data management systems is presented. The overview of our proposed architecture with encoding and decoding techniques is also presented in this section. The implementation technique has been described in "Implementation" section. The experimental results have been thoroughly discussed in "Result and discussion" section and finally "Conclusion" section concludes the paper.
Quaternary tree architecture
The main contribution of this research is to implement a new lossless Huffman-based compression technique. The implementation of the algorithms has been explained with some mathematical foundations. Finally, implemented algorithms have been tested using real data.
Tree construction
Huffman codes to binary data
Huffman's scheme uses a table of frequency to produce codeword for each symbol (Wikipedia short history of Huffman coding 2011). This table consists of every symbol of entire document and its respective frequency is arranged in ascending order. According to the frequency of distinct symbol, each symbol has a variable-length bit string and all the bit strings are distinct. Table 1 shows the variable-length codeword for different symbols of the sentence "This is an example of quaternary Huffman tree."
Table 1 Codeword generation using binary Huffman principle
Consider a set of source symbols \(S = \left\{ {s_{0} , s_{1} , \ldots , s_{n - 1} } \right\} = \{ {\text{Space}}, a, \ldots , y, .\}\) with frequencies \(W = \left\{ {w_{0} , w_{1} , \ldots , w_{n - 1} } \right\}\) for \(w_{0} \ge w_{1} \ge \ldots \ge w_{n - 1}\), where the symbol \(s_{i}\) has frequency \(w_{i}\) and \(n\) is the number of symbols. The codeword \(c_{i}\), \(0 \le i \le n - 1\), for symbol \(s_{i}\) can be calculated by traversing the path from root to the symbol \(s_{i}\), when goes to left it writes '0' and when goes to right it writes '1'. If the level of the root is zero, then the codeword length can be determined as the level of \(s_{i}\). The traversing time of a tree depends on its weighted path length \(\mathop \sum \nolimits w_{i} l_{i}\), which is expected to be minimum. The Huffman tree for the source symbols \(\left\{ {s_{0} , s_{1} , \ldots , s_{18} } \right\}\) with the frequencies \(\left\{ {8, 6, \ldots , 1} \right\}\), respectively, for the above example is shown in Fig. 1. The codeword set \(C\left\{ {c_{0} , c_{1} , \ldots , c_{18} } \right\}\) is derived as \(\left\{ {000, 010, \ldots , 11101} \right\}\), respectively, is shown in Table 1.
Construction of binary Huffman tree
Huffman codes to quaternary data
Quaternary tree or 4-ary tree is a tree in which each node has 0–4 children (labeled as LEFT child, LEFT MID child, RIGHT MID child, RIGHT child). Here for constructing codes for quaternary Huffman tree, we use 00 for left child, 01 for left-mid child, 10 for right-mid child, and 11 for right child.
The process of the construction of a quaternary tree is described below:
List all possible symbols with their probabilities;
Find the four symbols with the smallest probabilities;
Replace these by a single set containing all four symbols, and the probability of the parent is the sum of the individual probabilities.
Replicate the procedure until it has one node.
The code word generated using quaternary Huffman technique is shown in Table 2.
Table 2 Codeword generation using quaternary Huffman principle
Consider a set of source symbols \(S = \left\{ {s_{0} , s_{1} , \ldots , s_{n - 1} } \right\} = \{ {\text{Space}}, a, \ldots , y, .\}\) with frequencies \(W = \left\{ {w_{0} , w_{1} , \ldots , w_{n - 1} } \right\}\) for \(w_{0} \ge w_{1} \ge \ldots \ge w_{n - 1}\), where the symbol \(s_{i}\) has frequency \(w_{i}\) and \(n\) is the number of symbols. The codeword \(c_{i}\), \(0 \le i \le n - 1\), for symbol \(s_{i}\) can be calculated by traversing the path from root to the symbol \(s_{i}\), when goes to left it writes '00', when goes to left mid writes '01', when goes to right mid writes '10' and when goes to right writes '11'. The codeword length of a symbol can simply be calculated as the level of \(s_{i}\). We know that the traversing time of a tree depends on its weighted path length \(\mathop \sum \nolimits w_{i} l_{i}\), and it is expected to be minimum. The quaternary Huffman tree for the source symbols \(\left\{ {s_{0} , s_{1} , \ldots , s_{18} } \right\}\) with the frequencies \(\left\{ {8, 6, \ldots , 1} \right\}\), respectively, for the above example ("This is an example of quaternary Huffman tree.") is shown in Fig. 2. The codeword set \(C\left\{ {c_{0} , c_{1} , \ldots , c_{18} } \right\}\) is derived as \(\left\{ {00, 0100, \ldots , 111111} \right\}\), respectively, which is shown in Table 2.
Construction of quaternary Huffman tree along with decoding table
Comparison of binary and quaternary tree
Table 3 shows some comparisons with some mathematical parameters for the previous example.
Table 3 Comparison of binary and quaternary tree
Reduction of time using quaternary tree
Encoding and decoding time of a tree depends on the weighted path length of a tree. If n is the number of distinct character, \(L_{i }\) is code length of the ith character, and \(f_{i }\) is the frequency of the ith character, then we can write the required traversing time T as
$$T\infty \mathop \sum \limits_{i = 1}^{n} L_{i } f_{i }$$
$$T\infty K\mathop \sum \limits_{i = 1}^{n} \alpha_{i } f_{i }$$
where \(L_{i } = \alpha_{i} \cdot K\), \(\alpha_{i} \infty \frac{1}{{f_{i} }}\), K = arity = 2, for quaternary tree, and \(\alpha_{i}\) = height constant
Thus, the traversing time also depends on the height of the tree and frequency of different symbols. The height of a quaternary tree is always smaller than the height of a binary tree. For this reason, traversing time will be reduced for a petite tree.
The structure of header tree for decoding is very simple for the proposed technique. According to Fig. 2, it does not require to store the entire codeword in the header tree for a symbol. The most frequent symbol is stored first in the header which confirms faster decoding. Moreover, retrieving two bits at a time during decoding process also speeds up the process. In the decoding phase, matching (two bits at a time) from encoded bit string with the header starts from level 1 in the header tree. If there is any symbol with codeword of length 2, then it will be found in level 1 in the header tree. Likewise, matching a symbol with codeword of length 4 both the level 1 and level 2 have to be searched. The simplicity of the header tree also contributes to speed up the decoding process.
As mentioned earlier, in quaternary tree each node has 0–4 children (labeled as LEFT child, LEFT MID child, RIGHT MID child, and RIGHT child).
There are basically two components in quaternary Huffman coding:
Quaternary Huffman encoding
Quaternary Huffman decoding
Encoding algorithm
Encoding is a two-pass problem. The first pass is to determine the frequencies of letters. We use this information to create the quaternary Huffman tree. We have used a dictionary to store the frequencies of the symbols. When a quaternary Huffman code has been generated, the symbol will be replaced by the code. This is a modification of Huffman algorithm (Coreman et al. 2001).
In line 1, we assign the unordered nodes, C in the queue, Q and later we take the count of nodes in Q and assign it to n. We assign the value of n to a new variable i. In line 4, we start iterating all the nodes in queue to build the quaternary tree until the count of i is greater than 1 which means that there are nodes still left to be added to the parent. In line 5, a new tree node, z is allocated. This node will be the parent node of the least frequent nodes. In line 6, we extract the least frequent node from the queue Q and assign it as a left child of the parent node z. The EXTRACT-MIN (Q) function returns the least frequent node from the queue and removes it from the queue as well. In line 7, we take the next least frequent node from the queue and assign it as a left-mid child of the parent z.
From line 8 to 17, we check the value of i or the number of nodes left in the queue Q. If i equals 2, the frequency of the parent node z, \(f[z]\) will be the summation of the frequency of node v, \(f[v]\) and the frequency of node w, \(f[w]\). Likewise, for i is equal to 3, we extract another least frequent node from the queue and add it as a child and add its frequency to the parent node. For i is greater than 3, we extract two least frequent nodes and add them as right-mid and right child of the parent z and add their frequency to the parent z as well. In line 18, we insert the new parent node z into the queue, Q. In line 19, we take the count of the queue, Q and assign it to i again. The loop continues until a single node is left in the queue. Finally, we return the last and single node from the queue Q as a quaternary Huffman tree.
Decoding algorithm
Decoding is accomplished by reading the encoded data two bits at a time. When iterating the bit stream 00 bit pattern means go LEFT, 01 pattern means go LEFT MID, 10 pattern means go RIGHT MID and 11 pattern means go RIGHT in case of quaternary tree. When a bit pattern matches with a symbol according to the header tree, replace the bit pattern with that symbol and the process is iterated until reached the last bit of the stream.
In the following algorithm 2 in line 1, we assign the quaternary tree T in the local variable ln. Then, we take the total count of bits in n from B. In line 3, we initialize a local variable i with 0 which will be used as a counter. In line 4, we started iterating all the bits in B. As it is a quaternary tree, we have at most four leaves for a parent node: left, left-mid, right-mid, right and 00, 01, 10, 11 represent these leaf nodes, respectively. We take two bits at a time. EXTRACT-BIT(B) returns a bit from the bit array B and removes it from B as well. In lines 5 and 6, local variables b1 and b2 are being assigned with two extracted bits from the bit array B.
From line 7 to line 15, we check the extracted bits to traverse the tree from the top. If the bits are 00, we take the left child of the parent ln and assign it to ln itself. For 01, we replace the parent ln with its left-mid child, for 10 we replace it with its right-mid child and for 11 we replace it with the right child. In line 16, we get the key of the replaced ln and assign it in k. Then, we check whether k has any value. If the k has any value, we write the value of the k in the output and update the ln with the quaternary tree T itself. In line 21, we increase the value of i by 2 and the loop gets continued and reads the next two bits.
This section discusses the encoding and decoding technique of a quaternary Huffman architecture. The search time for finding a source symbol using quaternary Huffman algorithm is \(O({ \log }_{4} n)\), whereas for Huffman-based algorithm it is \(O({ \log }_{2} n)\).
To verify the applicability and feasibility of the proposed quaternary-based technique, experimental evaluation has been performed on real data. The experimental results are compared with regular Huffman-based techniques. Our target was to justify query time and the storage requirements in comparison with regular Huffman-based techniques.
Experimental environment
Each query has been executed five times and the average execution time has been counted. The experiments are conducted on a machine with following specifications:
Data set
We have used four real text files as data set. The first two files are the source code of our implemented programs, which we do not wish to share as it is still unpublished. The other two files used for evaluation are readily available online (The famous lgpl 2.1 license. https://www.gnu.org/licenses/lgpl-2.1.txt; The transcript of the movie matrix. http://thematrixtruth.remoteviewinglight.com/). The description of the datasets is given in Table 4.
Table 4 Data set
Decoding performance
To measure the decoding performance, we used the dataset on both regular and quaternary Huffman techniques. We consider three techniques as regular Huffman-based techniques (Chung 1997; Hashemian 1995; Chowdhury et al. 2002) and the performance of all three techniques is almost same considering next integer number. We used the StopWatch Class under System.Diagnostic of Mono framework to calculate the time required. Stopwatch provides a set of methods and properties that can be used to accurately measure elapsed time. The obtained results are described in Table 5. In all cases, we took the average output of at least five runs.
Table 5 Decoding performance of the proposed method and regular Huffman-based Technique
Four source files of different file size have been used altogether to measure the performance. In Table 5, it has been observed that for each case, quaternary Huffman technique is more than 50% faster than the regular Huffman-based techniques in case of decoding time.
In Fig. 3, it has been shown that as file size increases, the quaternary Huffman (line with diamond shape dot) technique is performing consistently better than the regular Huffman (line with square dot)-based techniques. In some cases depending on the relative frequencies of the symbols in a file, it is more than two times faster than regular Huffman-based techniques.
Decoding time comparison. Line with diamond shape dot indicates quaternary Huffman and line with square dot indicates regular Huffman-based technique
To measure the memory usage, we used the dataset on both regular and quaternary Huffman techniques. The method described in Chen et al. (1999) is used for comparison with the proposed method. Table 6 illustrates the compression rate between two techniques. It has been shown that the quaternary technique compresses the original file at an average rate of 32%, whereas the regular Huffman-based technique compresses at an average rate of 39%. Regular Huffman-based technique compresses little better than the proposed quaternary technique, this is just because of quaternary technique produced larger codeword. In some cases, the compression rate is almost equal for both techniques.
Table 6 Compression performance of the proposed technique and regular Huffman-based technique
The comparison of the compression performance of both techniques using the original file is also shown in Fig. 4 [ash color column indicates original file size, black column indicates regular Huffman-based technique (Chen et al. 1999) and texture column indicates quaternary Huffman technique].
Side-by-side compression comparison. Ash color column indicate original file size, black column indicate regular Huffman-based technique, and texture column indicate quaternary Huffman technique
Performance test with reknown corpus and recent Huffman-based techniques
We compare the performance of the proposed technique with Zopfli (Alakuijala and Vandevenne 2013), WinZip (2016) and PKZip (2016) algorithms. Google claims that Zopfli produces the highest compression ratio for similar technique. Zopfli uses Huffman coding to replace each value with a string of bits. WinZip and PKZip are the most widely used recent Huffman-based compression tools. In all cases, we took the average output of five runs.
Table 7 shows the result of compression ratio and compression–decompression speed on the Enwik8 corpus. The Enwik8 corpus is a 95.3-MB file with 156 distinct characters. This corpus is prepared as a large text compression standard, which have 100 million bytes of English Wikipedia.
Table 7 Comparison of the proposed technique with recent Huffman-based techniques for Enwik (The Enwik8 Corpus. http://mattmahoney.net/dc/text.html http://mattmahoney.net/dc/enwik8.zip) corpus
The result indicates that compression ratio is highest for Zopfli but the compression and decompression speed is very slow. The Zopfli requires over 400 s whereas all other techniques require less than 200 s. If we would compromise between time–space, and when speed is the main factor, then we may choose quaternary technique for this type of large corpus.
Table 8 shows the result of compression ratio and compression–decompression speed on the Canterbury corpus (The Canterbury Corpus. http://corpus.canterbury.ac.nz/resources/cantrbry.zip). The Canterbury corpus is 2.67-MB file with 72 distinct characters. This corpus is a modified version of Calgary corpus which is designed to test the compression algorithms.
Table 8 Comparison of the proposed technique with recent Huffman-based techniques for Canterbury corpus
If we observe the result, it has been shown that compression ratio is highest for Zopfli but its compression and decompression speed is very slow. The Zopfli requires over 13 s whereas all other techniques require less time.
In this section, we have analyzed both techniques thoroughly with different example in terms of time and space. For decoding speed, the proposed quaternary technique outperforms the regular Huffman-based techniques. On the other hand, the compression recital is almost similar for most of the files.
A new lossless compression technique based on Huffman principle is implemented in this paper. We introduced quaternary tree instead of binary tree in Huffman principle. We have shown that representation of Huffman code using quaternary tree is more beneficial than Huffman code using binary tree in terms of processing speed with an insignificant increase in required space. When speed is the main factor, then the quaternary tree based technique performs better than the binary tree based technique. Thus, the proposed technique provides a way to balance between the decoding time and memory usage.
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The authors discussed the problem and the solutions proposed all together. Both authors participated in drafting and revising the final manuscript. Both authors read and approved the final manuscript.
Authors are grateful to ministry of posts, telecommunications and information technology, People's Republic of Bangladesh for their grant to do this research work. The authors would like to thank the anonymous experts for their valuable comments and suggestion for improving the quality of this research paper.
The datasets supporting of this article are available online in the following link.
The famous lgpl 2.1 license, Accessed at https://www.gnu.org/licenses/lgpl-2.1.txt
The transcript of the movie Matrix. Accessed at http://thematrixtruth.remoteviewinglight.com/
The Enwik8 Corpus. Accessed at http://mattmahoney.net/dc/text.html http://mattmahoney.net/dc/enwik8.zip
The Canterbury Corpus. Accessed at http://corpus.canterbury.ac.nz/resources/cantrbry.zip
The WinZip compression tool, version 1.0.220.1, released by WinZip Computing, S.L., A Corel Company. Accessed at: http://www.winzip.com/win/en/downwz.html
The PKZip compression tool, version 14.40.0028, released by PKWARE Inc. Accessed at https://www.pkware.com/pkzip
All the funding provided by the Ministry of Posts, Telecommunications and Information Technology, People's Republic of Bangladesh [Order No: 56.00.0000.028.33.007.14 (part-1)-275, date: 11.05.2014; and Order No: 56.00.0000.028.33.025.14-115, date 10.05.2015]. The above funding gives the financial support for the designing of the study and conducting experiments.
Shahjalal University of Science and Technology, Sylhet, Bangladesh
Ahsan Habib & Mohammad Shahidur Rahman
Ahsan Habib
Mohammad Shahidur Rahman
Correspondence to Ahsan Habib.
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.
Habib, A., Rahman, M.S. Balancing decoding speed and memory usage for Huffman codes using quaternary tree. Appl Inform 4, 5 (2017). https://doi.org/10.1186/s40535-016-0032-z
Encoding and decoding
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Schouten, Jop & Groote Schaarsberg, Mirjam & Borm, Peter, 2020. "Cost Sharing Methods for Capacity Restricted Cooperative Purchasing Situations," Discussion Paper 2020-017, Tilburg University, Center for Economic Research.
Schouten, Jop & Groote Schaarsberg, Mirjam & Borm, Peter, 2020. "Cost Sharing Methods for Capacity Restricted Cooperative Purchasing Situations," Other publications TiSEM aa7f747d-c97b-4655-b6cb-9, Tilburg University, School of Economics and Management.
Schouten, Jop & Dietzenbacher, Bas & Borm, Peter, 2019. "The Nucleolus and Inheritance of Properties in Communication Situations," Discussion Paper 2019-008, Tilburg University, Center for Economic Research.
Schouten, Jop & Dietzenbacher, Bas & Borm, Peter, 2019. "The Nucleolus and Inheritance of Properties in Communication Situations," Other publications TiSEM bacc7f47-9b6b-4ce4-9f97-4, Tilburg University, School of Economics and Management.
Ketelaars, Martijn & Borm, Peter & Quant, Marieke, 2019. "Decentralization and mutual liability rules," Discussion Paper 2019-035, Tilburg University, Center for Economic Research.
Martijn Ketelaars & Peter Borm & Marieke Quant, 2020. "Decentralization and mutual liability rules," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 577-599, December.
Ketelaars, Martijn & Borm, Peter & Quant, Marieke, 2019. "Decentralization and mutual liability rules," Other publications TiSEM fa745b4f-f959-41d0-8c9f-0, Tilburg University, School of Economics and Management.
Saavedra-Nieves, Alejandro & Schouten, Jop & Borm, Peter, 2018. "On Interactive Sequencing Situations with Exponential Cost Functions," Discussion Paper 2018-020, Tilburg University, Center for Economic Research.
Saavedra-Nieves, Alejandro & Schouten, Jop & Borm, Peter, 2020. "On interactive sequencing situations with exponential cost functions," European Journal of Operational Research, Elsevier, vol. 280(1), pages 78-89.
Saavedra-Nieves, Alejandro & Schouten, Jop & Borm, Peter, 2018. "On Interactive Sequencing Situations with Exponential Cost Functions," Other publications TiSEM 79c2d10c-d6b1-4c2a-9212-4, Tilburg University, School of Economics and Management.
Schouten, Jop & Borm, Peter & Hendrickx, Ruud, 2018. "Unilateral Support Equilibria," Discussion Paper 2018-011, Tilburg University, Center for Economic Research.
Schouten, Jop & Borm, Peter & Hendrickx, Ruud, 2018. "Unilateral Support Equilibria," Other publications TiSEM 02dd1da8-0dad-48f8-8fe1-6, Tilburg University, School of Economics and Management.
Estévez-Fernández , M.A. & Borm, Peter & Fiestras, & Mosquera, & Sanchez,, 2017. "On the 1-nucleolus," Other publications TiSEM a8ce6687-c87a-4131-98f7-3, Tilburg University, School of Economics and Management.
A. Estévez-Fernández & P. Borm & M. G. Fiestras-Janeiro & M. A. Mosquera & E. Sánchez-Rodríguez, 2017. "On the 1-nucleolus," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 309-329, October.
Dietzenbacher, Bas & Borm, Peter & Estevez Fernandez, M.A., 2017. "NTU-Bankruptcy Problems : Consistency and the Relative Adjustment Principle," Discussion Paper 2017-044, Tilburg University, Center for Economic Research.
Bas Dietzenbacher & Peter Borm & Arantza Estévez-Fernández, 2020. "NTU-bankruptcy problems: consistency and the relative adjustment principle," Review of Economic Design, Springer;Society for Economic Design, vol. 24(1), pages 101-122, June.
Dietzenbacher, Bas & Borm, Peter & Estevez Fernandez, M.A., 2017. "NTU-Bankruptcy Problems : Consistency and the Relative Adjustment Principle," Other publications TiSEM f023d53e-b84f-4520-aa5e-9, Tilburg University, School of Economics and Management.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2017. "Egalitarianism in Nontransferable Utility Games," Discussion Paper 2017-023, Tilburg University, Center for Economic Research.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2017. "Egalitarianism in Nontransferable Utility Games," Other publications TiSEM b1bf227f-53df-4fad-93b8-8, Tilburg University, School of Economics and Management.
Kleppe, John & Borm, Peter & Hendrickx, Ruud, 2017. "Fall back proper equilibrium," Other publications TiSEM 50d88189-def5-4187-91bb-9, Tilburg University, School of Economics and Management.
John Kleppe & Peter Borm & Ruud Hendrickx, 2017. "Fall back proper equilibrium," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 402-412, July.
Musegaas, Marieke & Dietzenbacher, Bas & Borm, Peter, 2016. "A Note on Shapley Ratings in Brain Networks," Discussion Paper 2016-025, Tilburg University, Center for Economic Research.
Musegaas, Marieke & Dietzenbacher, Bas & Borm, Peter, 2016. "A Note on Shapley Ratings in Brain Networks," Other publications TiSEM 3562c06a-1612-4d93-a299-4, Tilburg University, School of Economics and Management.
Dietzenbacher, Bas & Estevez Fernandez, M.A. & Borm, Peter & Hendrickx, Ruud, 2016. "Proportionality, Equality, and Duality in Bankruptcy Problems with Nontransferable Utility," Discussion Paper 2016-026, Tilburg University, Center for Economic Research.
Dietzenbacher, Bas & Estevez Fernandez, M.A. & Borm, Peter & Hendrickx, Ruud, 2016. "Proportionality, Equality, and Duality in Bankruptcy Problems with Nontransferable Utility," Other publications TiSEM 959bd6d8-7c49-4479-9fd3-b, Tilburg University, School of Economics and Management.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2016. "The Procedural Egalitarian Solution," Discussion Paper 2016-041, Tilburg University, Center for Economic Research.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2017. "The procedural egalitarian solution," Games and Economic Behavior, Elsevier, vol. 106(C), pages 179-187.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2016. "The Procedural Egalitarian Solution," Other publications TiSEM 1863cb23-d1b2-4f2e-aa18-f, Tilburg University, School of Economics and Management.
Musegaas, Marieke & Borm, Peter & Quant, Marieke, 2016. "On the Convexity of Step out - Step in Sequencing Games," Discussion Paper 2016-043, Tilburg University, Center for Economic Research.
M. Musegaas & P. E. M. Borm & M. Quant, 2018. "On the convexity of step out–step in sequencing games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 68-109, April.
Musegaas, Marieke & Borm, Peter & Quant, Marieke, 2016. "On the Convexity of Step out - Step in Sequencing Games," Other publications TiSEM f8a803f5-1f6a-4ac4-b4f6-7, Tilburg University, School of Economics and Management.
Karsten, Frank & Slikker, M. & Borm, P.E.M., 2015. "Cost Allocation Rules for Elastic Single-Attribute Situations," Discussion Paper 2015-016, Tilburg University, Center for Economic Research.
Frank Karsten & Marco Slikker & Peter Borm, 2017. "Cost allocation rules for elastic single‐attribute situations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 64(4), pages 271-286, June.
Karsten, Frank & Slikker, M. & Borm, P.E.M., 2015. "Cost Allocation Rules for Elastic Single-Attribute Situations," Other publications TiSEM e6c6ce3e-6168-40b2-bf64-0, Tilburg University, School of Economics and Management.
Musegaas, M. & Borm, P.E.M. & Quant, M., 2015. "Three-Valued Simple Games," Discussion Paper 2015-026, Tilburg University, Center for Economic Research.
M. Musegaas & P. E. M. Borm & M. Quant, 2018. "Three-valued simple games," Theory and Decision, Springer, vol. 85(2), pages 201-224, August.
Musegaas, M. & Borm, P.E.M. & Quant, M., 2015. "Three-Valued Simple Games," Other publications TiSEM 473afd5c-99b1-4073-888f-2, Tilburg University, School of Economics and Management.
A. Estévez-Fernández & P. Borm & M.G. Fiestras-Janeiro & M.A. Mosquera & E. Sánchez-Rodríguez, 2015. "On the 1-Nucleolus for Classes of Cooperative Games," Tinbergen Institute Discussion Papers 15-123/II, Tinbergen Institute.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2015. "Decomposition of Network Communication Games," Discussion Paper 2015-057, Tilburg University, Center for Economic Research.
Bas Dietzenbacher & Peter Borm & Ruud Hendrickx, 2017. "Decomposition of network communication games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(3), pages 407-423, June.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2017. "Decomposition of network communication games," Other publications TiSEM ab795ba7-d302-44bf-ba6a-f, Tilburg University, School of Economics and Management.
Dietzenbacher, Bas & Borm, Peter & Hendrickx, Ruud, 2015. "Decomposition of Network Communication Games," Other publications TiSEM c07a09b0-0bd7-4533-ac99-3, Tilburg University, School of Economics and Management.
Musegaas, Marieke & Borm, Peter & Quant, Marieke, 2015. "Simple and Three-Valued Simple Minimum Coloring Games," Discussion Paper 2015-032, Tilburg University, Center for Economic Research.
M. Musegaas & P. E. M. Borm & M. Quant, 2016. "Simple and three-valued simple minimum coloring games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 239-258, October.
Musegaas, Marieke & Borm, Peter & Quant, Marieke, 2015. "Simple and Three-Valued Simple Minimum Coloring Games," Other publications TiSEM 425c90bb-c6c6-4561-ad44-1, Tilburg University, School of Economics and Management.
Musegaas, M. & Borm, P.E.M. & Quant, M., 2014. "Step out - Step in Sequencing Games," Discussion Paper 2014-070, Tilburg University, Center for Economic Research.
Musegaas, M. & Borm, P.E.M. & Quant, M., 2015. "Step out–Step in sequencing games," European Journal of Operational Research, Elsevier, vol. 246(3), pages 894-906.
Musegaas, M. & Borm, P.E.M. & Quant, M., 2014. "Step out - Step in Sequencing Games," Other publications TiSEM 68089383-0b91-4cc8-9d03-e, Tilburg University, School of Economics and Management.
Emiliya Lazarova & Peter Borm & Arantza Estévez-Fernández, 2014. "Transfers and Exchange-Stability in Two-Sided Matching Problems," Tinbergen Institute Discussion Papers 14-086/II, Tinbergen Institute.
Emiliya Lazarova & Peter Borm & Arantza Estévez-Fernández, 2016. "Transfers and exchange-stability in two-sided matching problems," Theory and Decision, Springer, vol. 81(1), pages 53-71, June.
Lazarova, E.A. & Borm, Peter & Estevez, Arantza, 2016. "Transfers and exchange-stability in two-sided matching problems," Other publications TiSEM e76da65e-c692-4ba3-a2c6-d, Tilburg University, School of Economics and Management.
Husslage, B.G.M. & Borm, P.E.M. & Burg, T. & Hamers, H.J.M. & Lindelauf, R., 2014. "Ranking Terrorists in Networks : A Sensitivity Analysis of Al Qaeda's 9/11 Attack," Discussion Paper 2014-028, Tilburg University, Center for Economic Research.
Husslage, B.G.M. & Borm, P.E.M. & Burg, T. & Hamers, H.J.M. & Lindelauf, R., 2014. "Ranking Terrorists in Networks : A Sensitivity Analysis of Al Qaeda's 9/11 Attack," Other publications TiSEM 191548ed-34ba-4aba-abbf-9, Tilburg University, School of Economics and Management.
van der Genugten, B.B. & Borm, P.E.M., 2014. "Cash and Tournament Poker : Games of Skill?," Other publications TiSEM edc02e9c-cda1-4531-b320-7, Tilburg University, School of Economics and Management.
Arantza Estévez-Fernández & Peter Borm & M. Gloria Fiestras-Janeiro, 2014. "Nontransferable Utility Bankruptcy Games," Tinbergen Institute Discussion Papers 14-030/II, Tinbergen Institute.
Arantza Estévez-Fernández & Peter Borm & M. Gloria Fiestras-Janeiro, 2020. "Nontransferable utility bankruptcy games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(1), pages 154-177, April.
van der Genugten, B.B. & Borm, P.E.M., 2014. "Texas Hold'em : A Game of Skill," Other publications TiSEM e5edc6e2-5cad-4e87-8158-7, Tilburg University, School of Economics and Management.
Ben van der Genugten & Peter Borm, 2016. "Texas Hold'em: A Game of Skill," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 18(03), pages 1-13, September.
Huijink, S. & Borm, P.E.M. & Reijnierse, J.H. & Kleppe, J., 2013. "Bankruptcy and the Per Capita Nucleolus," Discussion Paper 2013-059, Tilburg University, Center for Economic Research.
Huijink, S. & Borm, P.E.M. & Reijnierse, J.H. & Kleppe, J., 2013. "Bankruptcy and the Per Capita Nucleolus," Other publications TiSEM 0b1d187b-a251-47c0-92d8-7, Tilburg University, School of Economics and Management.
Tejada, J. & Borm, P.E.M. & Lohmann, E.R.M.A., 2013. "A Unifying Model for Matching Situations," Discussion Paper 2013-069, Tilburg University, Center for Economic Research.
Tejada, J. & Borm, P.E.M. & Lohmann, E.R.M.A., 2013. "A Unifying Model for Matching Situations," Other publications TiSEM 18155a8c-1961-495d-a20d-f, Tilburg University, School of Economics and Management.
Groote Schaarsberg, M. & Reijnierse, J.H. & Borm, P.E.M., 2013. "On Solving Liability Problems," Discussion Paper 2013-033, Tilburg University, Center for Economic Research.
Groote Schaarsberg, M. & Reijnierse, J.H. & Borm, P.E.M., 2013. "On Solving Liability Problems," Other publications TiSEM b7a1e268-bd6d-4177-8056-8, Tilburg University, School of Economics and Management.
Estela Sánchez-Rodríguez & Peter Borm & Arantza Estévez-Fernández & M. Gloria Fiestras-Janeiro & Manuel A. Mosquera, 2013. "Characterizing the Core via k-Core Covers," Tinbergen Institute Discussion Papers 13-177/II, Tinbergen Institute.
Grundel, S. & Borm, P.E.M. & Hamers, H.J.M., 2013. "Resource Allocation Problems with Concave Reward Functions," Discussion Paper 2013-070, Tilburg University, Center for Economic Research.
Soesja Grundel & Peter Borm & Herbert Hamers, 2019. "Resource allocation problems with concave reward functions," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(1), pages 37-54, April.
Grundel, S. & Borm, P.E.M. & Hamers, H.J.M., 2013. "Resource Allocation Problems with Concave Reward Functions," Other publications TiSEM b72ed3dc-ecc8-49d4-86af-d, Tilburg University, School of Economics and Management.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P., 2012. "Fall Back Equilibrium for 2 x n Bimatrix Games," Discussion Paper 2012-044, Tilburg University, Center for Economic Research.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P., 2012. "Fall Back Equilibrium for 2 x n Bimatrix Games," Other publications TiSEM 1c83ee91-3e1b-4139-a180-c, Tilburg University, School of Economics and Management.
Groote Schaarsberg, M. & Borm, P.E.M. & Hamers, H.J.M. & Reijnierse, J.H., 2012. "Interactive Purchasing Situations," Discussion Paper 2012-035, Tilburg University, Center for Economic Research.
Groote Schaarsberg, M. & Borm, P.E.M. & Hamers, H.J.M. & Reijnierse, J.H., 2012. "Interactive Purchasing Situations," Other publications TiSEM 963efbbe-7dbf-43ba-948c-1, Tilburg University, School of Economics and Management.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P. & Reijnierse, J.H., 2012. "Cooperative Situations : Representations, Games and Cost Allocations," Discussion Paper 2012-029, Tilburg University, Center for Economic Research.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P. & Reijnierse, J.H., 2012. "Cooperative Situations : Representations, Games and Cost Allocations," Other publications TiSEM 0edb8bc3-3a18-4a6b-bd91-b, Tilburg University, School of Economics and Management.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P., 2012. "A Strategic Foundation for Proper Equilibrium," Discussion Paper 2012-093, Tilburg University, Center for Economic Research.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P., 2012. "A Strategic Foundation for Proper Equilibrium," Other publications TiSEM bb47ccc9-ff6f-4712-8e5e-8, Tilburg University, School of Economics and Management.
Grundel, S. & Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M., 2012. "Family Sequencing and Cooperation," Discussion Paper 2012-040, Tilburg University, Center for Economic Research.
Grundel, Soesja & Çiftçi, Barış & Borm, Peter & Hamers, Herbert, 2013. "Family sequencing and cooperation," European Journal of Operational Research, Elsevier, vol. 226(3), pages 414-424.
Grundel, S. & Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M., 2012. "Family Sequencing and Cooperation," Other publications TiSEM 830f760f-f003-40df-a01c-6, Tilburg University, School of Economics and Management.
Grundel, S. & Borm, P.E.M. & Hamers, H.J.M., 2011. "A Compromise Stable Extension of Bankruptcy Games : Multipurpose Resource Allocation," Discussion Paper 2011-029, Tilburg University, Center for Economic Research.
Grundel, S. & Borm, P.E.M. & Hamers, H.J.M., 2011. "A Compromise Stable Extension of Bankruptcy Games : Multipurpose Resource Allocation," Other publications TiSEM b1926d6b-22f4-4f28-84a2-9, Tilburg University, School of Economics and Management.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P., 2011. "Nash Equilibria in 2 × 2 × 2 Trimatrix Games with Identical Anonymous Best-Replies," Discussion Paper 2011-138, Tilburg University, Center for Economic Research.
Carlos González-Alcón & Peter Borm & Ruud Hendrickx, 2014. "Nash Equilibria In 2 × 2 × 2 Trimatrix Games With Identical Anonymous Best-Replies," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 16(04), pages 1-9.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P., 2011. "Nash Equilibria in 2 × 2 × 2 Trimatrix Games with Identical Anonymous Best-Replies," Other publications TiSEM 455a0882-9863-4da9-a260-5, Tilburg University, School of Economics and Management.
Lohmann, E. & Borm, P.J.A. & Herings, P.J.J., 2011. "Minimal exact balancedness," Research Memorandum 009, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
Lohmann, E. & Borm, P. & Herings, P.J.J., 2012. "Minimal exact balancedness," Mathematical Social Sciences, Elsevier, vol. 64(2), pages 127-135.
Lohmann, E.R.M.A. & Borm, P.E.M. & Herings, P.J.J., 2011. "Minimal Exact Balancedness," Other publications TiSEM 9255deed-69d2-4d64-adbe-5, Tilburg University, School of Economics and Management.
Lohmann, E.R.M.A. & Borm, P.E.M. & Herings, P.J.J., 2011. "Minimal Exact Balancedness," Discussion Paper 2011-012, Tilburg University, Center for Economic Research.
Ruud Hendrickx & Jacco Thijssen & Peter Borm, 2011. "Minimum Cost Spanning Tree Games and Spillover Stability," Discussion Papers 11/02, Department of Economics, University of York.
Ruud Hendrickx & Jacco Thijssen & Peter Borm, 2012. "Minimum cost spanning tree games and spillover stability," Theory and Decision, Springer, vol. 73(3), pages 441-451, September.
Lohmann, E.R.M.A. & Borm, P.E.M. & Slikker, M., 2010. "Preparation Sequencing Situations and Related Games," Discussion Paper 2010-31, Tilburg University, Center for Economic Research.
Lohmann, E.R.M.A. & Borm, P.E.M. & Slikker, M., 2010. "Preparation Sequencing Situations and Related Games," Other publications TiSEM 667d8f5d-4c0d-4610-970d-6, Tilburg University, School of Economics and Management.
Marban, S. & Ven, P. & Borm, P.J.A. & Hamers, H., 2010. "A cooperative game-theoretic approach to ALOHA," Research Memorandum 049, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
Arantza Estevez-Fernandez & Peter Borm & Herbert Hamers, 2010. "A Note on Passepartout Problems," Tinbergen Institute Discussion Papers 10-031/1, Tinbergen Institute.
Arantza Estévez-Fernández & Peter Borm & Herbert Hamers, 2012. "A Note On Passepartout Problems," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 14(02), pages 1-9.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2010. "One-Mode Projection Analysis and Design of Covert Affiliation Networks," Discussion Paper 2010-53, Tilburg University, Center for Economic Research.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2010. "One-Mode Projection Analysis and Design of Covert Affiliation Networks," Other publications TiSEM 22cf26dc-7fbc-431d-967f-c, Tilburg University, School of Economics and Management.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2009. "Understanding Terrorist Network Topologies and Their Resilience Against Disruption," Discussion Paper 2009-85, Tilburg University, Center for Economic Research.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2009. "Understanding Terrorist Network Topologies and Their Resilience Against Disruption," Other publications TiSEM 0469c068-22ad-4521-8d87-9, Tilburg University, School of Economics and Management.
Emiliya Lazarova & Peter Borm & Bas van Velzen, 2009. "Contracts and Coalition Formation based on Individual Deviations," Economics Working Papers 09-06, Queen's Management School, Queen's University Belfast.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2008. "On Heterogeneous Covert Networks," Discussion Paper 2008-46, Tilburg University, Center for Economic Research.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2008. "On Heterogeneous Covert Networks," Other publications TiSEM f3e01f8a-65e0-4c0f-95db-2, Tilburg University, School of Economics and Management.
Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M. & Slikker, M., 2008. "Batch Sequencing and Cooperation," Discussion Paper 2008-100, Tilburg University, Center for Economic Research.
Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M. & Slikker, M., 2008. "Batch Sequencing and Cooperation," Other publications TiSEM ed1f8fce-da76-41a6-9a9e-9, Tilburg University, School of Economics and Management.
Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M., 2008. "A Note on the Balancedness and the Concavity of Highway Games," Discussion Paper 2008-29, Tilburg University, Center for Economic Research.
Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M., 2008. "A Note on the Balancedness and the Concavity of Highway Games," Other publications TiSEM 89305e0b-28d7-4bec-b45c-2, Tilburg University, School of Economics and Management.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P., 2008. "Fall Back Equilibrium," Discussion Paper 2008-31, Tilburg University, Center for Economic Research.
Kleppe, John & Borm, Peter & Hendrickx, Ruud, 2012. "Fall back equilibrium," European Journal of Operational Research, Elsevier, vol. 223(2), pages 372-379.
Kleppe, J. & Borm, P.E.M. & Hendrickx, R.L.P., 2008. "Fall Back Equilibrium," Other publications TiSEM 62161700-7266-4768-9d90-4, Tilburg University, School of Economics and Management.
van den Brink, J.R. & Borm, P.E.M. & Hendrickx, R.L.P. & Owen, G., 2008. "Characterizations of the beta- and the degree network power measure," Other publications TiSEM 101ef139-cc05-4b30-bef5-1, Tilburg University, School of Economics and Management.
Hendrickx, R.L.P. & Borm, P.E.M. & van der Genugten, B.B. & Hilbers, P., 2008. "Measuring Skill in More-Person Games with Applications to Poker," Discussion Paper 2008-106, Tilburg University, Center for Economic Research.
Hendrickx, R.L.P. & Borm, P.E.M. & van der Genugten, B.B. & Hilbers, P., 2008. "Measuring Skill in More-Person Games with Applications to Poker," Other publications TiSEM 2f1019be-50f8-4155-9592-f, Tilburg University, School of Economics and Management.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2008. "The Influence of Secrecy on the Communication Structure of Covert Networks," Discussion Paper 2008-23, Tilburg University, Center for Economic Research.
Lindelauf, R. & Borm, P.E.M. & Hamers, H.J.M., 2008. "The Influence of Secrecy on the Communication Structure of Covert Networks," Other publications TiSEM b8d10ab3-47f7-481f-9d0b-f, Tilburg University, School of Economics and Management.
Cruijssen, F.C.A.M. & Borm, P.E.M. & Dullaert, W. & Hamers, H.J.M., 2007. "Joint Hub Network Development," Discussion Paper 2007-76, Tilburg University, Center for Economic Research.
Cruijssen, F.C.A.M. & Borm, P.E.M. & Dullaert, W. & Hamers, H.J.M., 2007. "Joint Hub Network Development," Other publications TiSEM c8003b03-cd8d-472d-8fd6-9, Tilburg University, School of Economics and Management.
van Gulick, G. & Borm, P.E.M. & De Waegenaere, A.M.B. & Hendrickx, R.L.P., 2007. "Deposit Games with Reinvestment," Discussion Paper 2007-22, Tilburg University, Center for Economic Research.
van Gulick, Gerwald & Borm, Peter & De Waegenaere, Anja & Hendrickx, Ruud, 2010. "Deposit games with reinvestment," European Journal of Operational Research, Elsevier, vol. 200(3), pages 788-799, February.
van Gulick, G. & Borm, P.E.M. & De Waegenaere, A.M.B. & Hendrickx, R.L.P., 2007. "Deposit Games with Reinvestment," Other publications TiSEM f5f91d02-451b-4118-8037-8, Tilburg University, School of Economics and Management.
Lohmann, E.R.M.A. & Borm, P.E.M. & Quant, M., 2007. "A Stroll with Alexia," Discussion Paper 2007-52, Tilburg University, Center for Economic Research.
Lohmann, E.R.M.A. & Borm, P.E.M. & Quant, M., 2007. "A Stroll with Alexia," Other publications TiSEM 8107c8d1-2b1e-4051-a140-6, Tilburg University, School of Economics and Management.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P. & van Kuijk, K., 2007. "A Taxonomy of Best-Reply Multifunctions in 2x2x2 Trimatrix Games," Discussion Paper 2007-11, Tilburg University, Center for Economic Research.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P. & van Kuijk, K., 2007. "A taxonomy of best-reply multifunctions in 2x2x2 trimatix games," Other publications TiSEM 8dc90f16-c3c3-49ee-9471-f, Tilburg University, School of Economics and Management.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P. & van Kuijk, K., 2007. "A Taxonomy of Best-Reply Multifunctions in 2x2x2 Trimatrix Games," Other publications TiSEM f2b20dd3-3c88-460b-b59a-a, Tilburg University, School of Economics and Management.
Kleppe, J. & Hendrickx, R.L.P. & Borm, P.E.M. & Garcia-Jurado, I. & Fiestras-Janeiro, G., 2007. "Transfers, Contracts and Strategic Games," Discussion Paper 2007-24, Tilburg University, Center for Economic Research.
John Kleppe & Ruud Hendrickx & Peter Borm & Ignacio García-Jurado & Gloria Fiestras-Janeiro, 2010. "Transfers, contracts and strategic games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 18(2), pages 481-492, December.
Kleppe, J. & Hendrickx, R.L.P. & Borm, P.E.M. & Garcia-Jurado, I. & Fiestras-Janeiro, G., 2007. "Transfers, Contracts and Strategic Games," Other publications TiSEM c0ebc3fa-b456-45e7-9c3e-b, Tilburg University, School of Economics and Management.
Reijnierse, J.H. & Borm, P.E.M. & Voorneveld, M., 2007. "On 'informationally robust equilibria' for bimatrix games," Other publications TiSEM 376a74e3-b40b-4fcb-a277-e, Tilburg University, School of Economics and Management.
Hans Reijnierse & Peter Borm & Mark Voorneveld, 2007. "On 'Informationally Robust Equilibria' for Bimatrix Games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 30(3), pages 539-560, March.
Estevez Fernandez, M.A. & Mosquera, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2006. "Proportionate Flow Shop Games," Discussion Paper 2006-63, Tilburg University, Center for Economic Research.
Estevez Fernandez, M.A. & Mosquera, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2006. "Proportionate Flow Shop Games," Other publications TiSEM d54cb827-3347-4150-9792-b, Tilburg University, School of Economics and Management.
Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M., 2006. "Population Monotonic Path Schemes for Simple Games," Discussion Paper 2006-113, Tilburg University, Center for Economic Research.
Barış Çiftçi & Peter Borm & Herbert Hamers, 2010. "Population monotonic path schemes for simple games," Theory and Decision, Springer, vol. 69(2), pages 205-218, August.
Ciftci, B.B. & Borm, P.E.M. & Hamers, H.J.M., 2006. "Population Monotonic Path Schemes for Simple Games," Other publications TiSEM cf620d42-bdc0-414e-99d6-c, Tilburg University, School of Economics and Management.
Lazarova, E.A. & Borm, P.E.M. & Montero, M.P. & Reijnierse, J.H., 2006. "A Bargaining Set Based on External and Internal Stability and Endogenous Coalition Formation," Discussion Paper 2006-16, Tilburg University, Center for Economic Research.
Lazarova, E.A. & Borm, P.E.M. & Montero, M.P. & Reijnierse, J.H., 2006. "A Bargaining Set Based on External and Internal Stability and Endogenous Coalition Formation," Other publications TiSEM 01dbbf5b-1150-4d38-8ebc-2, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & Voorneveld, M., 2006. "On the Beta measure for digraph competitions," Other publications TiSEM 214590f7-5605-42d6-8353-6, Tilburg University, School of Economics and Management.
Estevez Fernandez, M.A. & Borm, P.E.M. & Meertens, M. & Reijnierse, J.H., 2006. "On the Core of Routing Games with Revenues," Discussion Paper 2006-43, Tilburg University, Center for Economic Research.
Arantza Estévez-Fernández & Peter Borm & Marc Meertens & Hans Reijnierse, 2009. "On the core of routing games with revenues," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(2), pages 291-304, June.
Estevez Fernandez, M.A. & Borm, P.E.M. & Meertens, M. & Reijnierse, J.H., 2006. "On the Core of Routing Games with Revenues," Other publications TiSEM 114b470d-ab88-44f4-9a7c-d, Tilburg University, School of Economics and Management.
Estevez Fernandez, M.A. & Borm, P.E.M. & Meertens, M. & Reijnierse, J.H., 2009. "On the core of routing games with revenues," Other publications TiSEM 09bc6e81-943f-466a-b86f-f, Tilburg University, School of Economics and Management.
Yuan Ju & Peter Borm, 2006. "A Non-cooperative Approach to the Compensation Rules for Primeval Games," Keele Economics Research Papers KERP 2006/18, Centre for Economic Research, Keele University.
Ju, Y. & Borm, P.E.M., 2006. "A Non-cooperative Approach to the Compensation Rules for Primeval Games," Discussion Paper 2006-97, Tilburg University, Center for Economic Research.
Ju, Y. & Borm, P.E.M., 2006. "A Non-cooperative Approach to the Compensation Rules for Primeval Games," Other publications TiSEM 0bc72bc7-9350-48e3-90c4-e, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Estevez Fernandez, M.A. & Fiestras-Janeiro, G., 2005. "Competitive Environments and Protective Behaviour," Discussion Paper 2005-50, Tilburg University, Center for Economic Research.
Borm, Peter & Estévez-Fernández, Arantza & Fiestras-Janeiro, M. Gloria, 2009. "Competitive environments and protective behavior," Games and Economic Behavior, Elsevier, vol. 67(1), pages 245-252, September.
Borm, P.E.M. & Estevez Fernandez, M.A. & Fiestras-Janeiro, G., 2005. "Competitive Environments and Protective Behaviour," Other publications TiSEM b8a5d1fa-38f3-4c42-9e29-2, Tilburg University, School of Economics and Management.
René van den Brink & Peter Borm & Ruud Hendrickx & Guillermo Owen, 2005. "Characterizations of Network Power Measures," Tinbergen Institute Discussion Papers 05-061/1, Tinbergen Institute.
Lazarova, E.A. & Borm, P.E.M. & van Velzen, S., 2005. "Contracts and Insurance Group Formation by Myopic Players," Discussion Paper 2005-89, Tilburg University, Center for Economic Research.
Lazarova, E.A. & Borm, P.E.M. & van Velzen, S., 2005. "Contracts and Insurance Group Formation by Myopic Players," Other publications TiSEM aabd51d2-b8ad-4a0d-94e8-0, Tilburg University, School of Economics and Management.
Hendrickx, R.L.P. & Borm, P.E.M. & van den Brink, J.R. & Owen, G., 2005. "The V L Value for Network Games," Discussion Paper 2005-65, Tilburg University, Center for Economic Research.
Hendrickx, R.L.P. & Borm, P.E.M. & van den Brink, J.R. & Owen, G., 2005. "The V L Value for Network Games," Other publications TiSEM cd04bb7c-a05b-4877-986d-e, Tilburg University, School of Economics and Management.
Hendrickx, R.L.P. & Borm, P.E.M. & Elk, R. & Quant, M., 2005. "Minimal Overlap Rules for Bankruptcy," Discussion Paper 2005-87, Tilburg University, Center for Economic Research.
Hendrickx, R.L.P. & Borm, P.E.M. & Elk, R. & Quant, M., 2005. "Minimal Overlap Rules for Bankruptcy," Other publications TiSEM 281932c0-26d6-4f02-a01f-7, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & van Velzen, S., 2005. "The core cover in relation to the nucleolus and the Weber set," Other publications TiSEM 9723cee6-92c9-49a2-90d3-9, Tilburg University, School of Economics and Management.
Marieke Quant & Peter Borm & Hans Reijnierse & Bas van Velzen, 2005. "The core cover in relation to the nucleolus and the Weber set," International Journal of Game Theory, Springer;Game Theory Society, vol. 33(4), pages 491-503, November.
Yuan Ju & Peter Borm, 2005. "Externalities and Compensation:Primeval Games and Solutions," Keele Economics Research Papers KERP 2005/05, Centre for Economic Research, Keele University.
Ju, Yuan & Borm, Peter, 2008. "Externalities and compensation: Primeval games and solutions," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 367-382, February.
Ju, Y. & Borm, P.E.M., 2005. "Externalities and Compensation : Primeval Games and Solutions," Other publications TiSEM fb8953ab-5f30-4c2d-8f29-4, Tilburg University, School of Economics and Management.
Ju, Y. & Borm, P.E.M., 2008. "Externalities and compensation : Primeval games and solutions," Other publications TiSEM f6e1bcfc-f2a6-4fc7-ac90-7, Tilburg University, School of Economics and Management.
Ju, Y. & Borm, P.E.M., 2005. "Externalities and Compensation : Primeval Games and Solutions," Discussion Paper 2005-71, Tilburg University, Center for Economic Research.
Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2005. "Project Games," Discussion Paper 2005-91, Tilburg University, Center for Economic Research.
Arantza Estévez-Fernández & Peter Borm & Herbert Hamers, 2007. "Project games," International Journal of Game Theory, Springer;Game Theory Society, vol. 36(2), pages 149-176, October.
Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2007. "Project games," Other publications TiSEM 809ba203-2bd2-48ce-ae6d-b, Tilburg University, School of Economics and Management.
Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2005. "Project Games," Other publications TiSEM 21fd9b62-93b6-4a8b-9bf4-4, Tilburg University, School of Economics and Management.
Mosquera, M.A. & Borm, P.E.M. & Fiestras-Janeiro, G. & Garcia-Jurado, I. & Voorneveld, M., 2005. "Characterizing Cautious Choice," Discussion Paper 2005-54, Tilburg University, Center for Economic Research.
Mosquera, M.A. & Borm, P. & Fiestras-Janeiro, M.G. & García-Jurado, I. & Voorneveld, M., 2008. "Characterizing cautious choice," Mathematical Social Sciences, Elsevier, vol. 55(2), pages 143-155, March.
Mosquera, M.A. & Borm, P.E.M. & Fiestras-Janeiro, G. & Garcia-Jurado, I. & Voorneveld, M., 2008. "Characterizing cautious choice," Other publications TiSEM c8c22ece-6981-4dae-8585-c, Tilburg University, School of Economics and Management.
Mosquera, M.A. & Borm, P.E.M. & Fiestras-Janeiro, G. & Garcia-Jurado, I. & Voorneveld, M., 2005. "Characterizing Cautious Choice," Other publications TiSEM aeb14f8d-ebb8-4655-93d6-d, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Maaten, R., 2005. "A Concede-and-Divide Rule for Bankruptcy Problems," Discussion Paper 2005-20, Tilburg University, Center for Economic Research.
Quant, M. & Borm, P.E.M. & Maaten, R., 2005. "A Concede-and-Divide Rule for Bankruptcy Problems," Other publications TiSEM 23e9af88-9fb0-4f9d-bad5-2, Tilburg University, School of Economics and Management.
Cruijssen, F.C.A.M. & Borm, P.E.M. & Fleuren, H.A. & Hamers, H.J.M., 2005. "Insinking : A Methodology to Exploit Synergy in Transportation," Discussion Paper 2005-121, Tilburg University, Center for Economic Research.
Cruijssen, F.C.A.M. & Borm, P.E.M. & Fleuren, H.A. & Hamers, H.J.M., 2005. "Insinking : A Methodology to Exploit Synergy in Transportation," Other publications TiSEM 958be918-e7b4-4e46-9cbe-d, Tilburg University, School of Economics and Management.
Dinko Dimitrov & Peter Borm, 2004. "Simple Priorities and Core Stability in Hedonic Games," Econometric Society 2004 North American Summer Meetings 135, Econometric Society.
Dinko Dimitrov & Peter Borm & Ruud Hendrickx & Shao Sung, 2006. "Simple Priorities and Core Stability in Hedonic Games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 26(2), pages 421-433, April.
Dinko Dimitrov & Peter Borm & Ruud Hendrickx & Shao Chin Sung, 2004. "Simple Priorities and Core Stability in Hedonic Games," Working Papers 2004.51, Fondazione Eni Enrico Mattei.
Dimitrov, D.A. & Borm, P.E.M. & Hendrickx, R.L.P. & Sung, S.C., 2004. "Simple Priorities and Core Stability in Hedonic Games," Discussion Paper 2004-5, Tilburg University, Center for Economic Research.
Dimitrov, D.A. & Borm, P.E.M. & Hendrickx, R.L.P. & Sung, S.C., 2004. "Simple Priorities and Core Stability in Hedonic Games," Other publications TiSEM 0824ae58-50a2-4c3a-9342-c, Tilburg University, School of Economics and Management.
Dimitrov, D.A. & Borm, P.E.M. & Hendrickx, R.L.P. & Sung, S.C., 2006. "Simple priorities and core stability in hedonic games," Other publications TiSEM 7c737a30-ac86-46ed-b210-e, Tilburg University, School of Economics and Management.
Ju, Y. & Ruys, P.H.M. & Borm, P.E.M., 2004. "Compensating Losses and Sharing Surpluses in Project-Allocation Situations (version 2)," Discussion Paper 2004-37, Tilburg University, Center for Economic Research.
Ju, Y. & Ruys, P.H.M. & Borm, P.E.M., 2004. "Compensating Losses and Sharing Surpluses in Project-Allocation Situations (version 2)," Discussion Paper 2004-002, Tilburg University, Tilburg Law and Economic Center.
Ju, Y. & Ruys, P.H.M. & Borm, P.E.M., 2004. "Compensating Losses and Sharing Surpluses in Project-Allocation Situations (version 2)," Other publications TiSEM 1dedb74a-cd24-4a65-b234-d, Tilburg University, School of Economics and Management.
Ju, Y. & Ruys, P.H.M. & Borm, P.E.M., 2004. "Compensating Losses and Sharing Surpluses in Project-Allocation Situations (version 2)," Other publications TiSEM 794a7917-4c5f-4dd0-baff-c, Tilburg University, School of Economics and Management.
Gonzalez-Diaz, J. & Borm, P.E.M. & Norde, H.W., 2004. "A Silent Battle over a Cake," Discussion Paper 2004-119, Tilburg University, Center for Economic Research.
Gonzalez-Diaz, Julio & Borm, Peter & Norde, Henk, 2007. "A silent battle over a cake," European Journal of Operational Research, Elsevier, vol. 177(1), pages 591-603, February.
Gonzalez-Diaz, J. & Borm, P.E.M. & Norde, H.W., 2004. "A Silent Battle over a Cake," Other publications TiSEM 1864a7b8-ff63-458b-9d35-9, Tilburg University, School of Economics and Management.
Gonzalez-Diaz, J. & Borm, P.E.M. & Norde, H.W., 2007. "A silent battle over a cake," Other publications TiSEM ee591ade-52d0-4cd8-944f-2, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Hendrickx, R.L.P. & Zwikker, P., 2004. "Compromise Solutions Based on Bankruptcy," Discussion Paper 2004-33, Tilburg University, Center for Economic Research.
Quant, Marieke & Borm, Peter & Hendrickx, Ruud & Zwikker, Peter, 2006. "Compromise solutions based on bankruptcy," Mathematical Social Sciences, Elsevier, vol. 51(3), pages 247-256, May.
Quant, M. & Borm, P.E.M. & Hendrickx, R.L.P. & Zwikker, P., 2006. "Compromise solutions based on bankruptcy," Other publications TiSEM 89479e71-abee-42d9-9bba-3, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Hendrickx, R.L.P. & Zwikker, P., 2004. "Compromise Solutions Based on Bankruptcy," Other publications TiSEM d846c5e5-7fd6-496a-9cb5-f, Tilburg University, School of Economics and Management.
Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2004. ""The Museum Pass Game and its Value" Revisited," Discussion Paper 2004-7, Tilburg University, Center for Economic Research.
Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2004. ""The Museum Pass Game and its Value" Revisited," Other publications TiSEM 18229f1c-d31a-4d26-a5ee-8, Tilburg University, School of Economics and Management.
Calleja, P. & Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2004. "Job Scheduling, Cooperation and Control," Discussion Paper 2004-65, Tilburg University, Center for Economic Research.
Calleja, P. & Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2004. "Job Scheduling, Cooperation and Control," Other publications TiSEM 05906ac6-b582-4ee7-875e-9, Tilburg University, School of Economics and Management.
Estevez Fernandez, M.A. & Borm, P.E.M. & Calleja, P. & Hamers, H.J.M., 2004. "Sequencing Games with Repeated Players," Discussion Paper 2004-128, Tilburg University, Center for Economic Research.
Arantza Estévez-Fernández & Peter Borm & Pedro Calleja & Herbert Hamers, 2008. "Sequencing games with repeated players," Annals of Operations Research, Springer, vol. 158(1), pages 189-203, February.
Estevez Fernandez, M.A. & Borm, P.E.M. & Calleja, P. & Hamers, H.J.M., 2008. "Sequencing games with repeated players," Other publications TiSEM e80f7089-c7f4-4683-9f13-4, Tilburg University, School of Economics and Management.
Estevez Fernandez, M.A. & Borm, P.E.M. & Calleja, P. & Hamers, H.J.M., 2004. "Sequencing Games with Repeated Players," Other publications TiSEM 0244ce95-d717-4b18-8b3b-f, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Ju, Y. & Ruys, P.H.M., 2004. "Compensating Losses and Sharing Surpluses in Project-Allocation Situations (version 1)," Discussion Paper 2004-6, Tilburg University, Center for Economic Research.
Borm, P.E.M. & Ju, Y. & Ruys, P.H.M., 2004. "Compensating Losses and Sharing Surpluses in Project-Allocation Situations (version 1)," Other publications TiSEM 9b03ea4a-f625-4fd0-ad4f-3, Tilburg University, School of Economics and Management.
Meertens, M. & Borm, P.E.M. & Reijnierse, J.H. & Quant, M., 2004. "Processing Games with Restricted Capacities," Discussion Paper 2004-83, Tilburg University, Center for Economic Research.
Reijnierse, Hans & Borm, Peter & Quant, Marieke & Meertens, Marc, 2010. "Processing games with restricted capacities," European Journal of Operational Research, Elsevier, vol. 202(3), pages 773-780, May.
Meertens, M. & Borm, P.E.M. & Reijnierse, J.H. & Quant, M., 2004. "Processing Games with Restricted Capacities," Other publications TiSEM a769e434-b8c9-4116-8897-f, Tilburg University, School of Economics and Management.
Ju, Y. & Borm, P.E.M. & Ruys, P.H.M., 2004. "The Consensus Value : A New Solution Concept for Cooperative Games," Discussion Paper 2004-50, Tilburg University, Center for Economic Research.
Yuan Ju & Peter Borm & Pieter Ruys, 2007. "The consensus value: a new solution concept for cooperative games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(4), pages 685-703, June.
Ju, Y. & Borm, P.E.M. & Ruys, P.H.M., 2007. "The consensus value : A new solution concept for cooperative games," Other publications TiSEM 6cd44a12-a909-47f8-8d85-e, Tilburg University, School of Economics and Management.
Ju, Y. & Borm, P.E.M. & Ruys, P.H.M., 2004. "The Consensus Value : A New Solution Concept for Cooperative Games," Other publications TiSEM ff51992e-6683-4596-a613-9, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Dreef, M.R.M. & van der Genugten, B.B., 2004. "Measuring skill in games : Several approaches discussed," Other publications TiSEM 2e5d4c52-1421-4311-bb4a-c, Tilburg University, School of Economics and Management.
Marcel Dreef & Peter Borm & Ben van der Genugten, 2004. "Measuring skill in games: several approaches discussed," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(3), pages 375-391, July.
van den Heuvel, W. & Borm, P.E.M. & Hamers, H.J.M., 2004. "Economic Lot-Sizing Games," ERIM Report Series Research in Management ERS-2004-088-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
Heuvel, Wilco van den & Borm, Peter & Hamers, Herbert, 2007. "Economic lot-sizing games," European Journal of Operational Research, Elsevier, vol. 176(2), pages 1117-1130, January.
van den Heuvel, W. & Borm, P.E.M. & Hamers, H.J.M., 2004. "Economic lot-sizing games," Other publications TiSEM 523ee90a-3486-4cf9-be95-7, Tilburg University, School of Economics and Management.
van den Heuvel, W. & Borm, P.E.M. & Hamers, H.J.M., 2007. "Economic lot-sizing games," Other publications TiSEM f559452b-09d9-4a51-9b9d-1, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Dimitrov, D.A. & Hendrickx, R.L.P., 2004. "Good and bad objects : The symmetric difference rule," Other publications TiSEM ae319b57-4686-4002-867c-5, Tilburg University, School of Economics and Management.
Dinko Dimitrov & Ruud Hendrickx & Peter Borm, 2004. "Good and bad objects: the symmetric difference rule," Economics Bulletin, AccessEcon, vol. 4(11), pages 1-7.
Quant, M. & Borm, P.E.M. & Fiestras-Janeiro, G. & van Megen, F.J.C., 2004. "On Properness and Protectiveness in Two Person Multicriteria Games," Discussion Paper 2004-127, Tilburg University, Center for Economic Research.
M. Quant & P. Borm & G. Fiestras-Janeiro & F. Megen, 2009. "On Properness and Protectiveness in Two-Person Multicriteria Games," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 499-512, March.
Quant, M. & Borm, P.E.M. & Fiestras-Janeiro, G. & van Megen, F.J.C., 2004. "On Properness and Protectiveness in Two Person Multicriteria Games," Other publications TiSEM ad33fe50-ccf7-46fe-abe8-6, Tilburg University, School of Economics and Management.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P., 2003. "A Composite Run-to-the-Bank Rule for Multi-Issue Allocation Situations," Discussion Paper 2003-59, Tilburg University, Center for Economic Research.
Carlos González-Alcón & Peter Borm & Ruud Hendrickx, 2007. "A composite run-to-the-bank rule for multi-issue allocation situations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(2), pages 339-352, April.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P., 2007. "A composite run-to-the-bank rule for multi-issue allocation situations," Other publications TiSEM 5441f689-5c71-4f51-8b0d-3, Tilburg University, School of Economics and Management.
Gonzalez-Alcon, C. & Borm, P.E.M. & Hendrickx, R.L.P., 2003. "A Composite Run-to-the-Bank Rule for Multi-Issue Allocation Situations," Other publications TiSEM 285ef7fb-bff6-4451-ada7-a, Tilburg University, School of Economics and Management.
Reijnierse, J.H. & Borm, P.E.M. & Voorneveld, M., 2003. "Informationally Robust Equlibria," Discussion Paper 2003-14, Tilburg University, Center for Economic Research.
Reijnierse, J.H. & Borm, P.E.M. & Voorneveld, M., 2003. "Informationally Robust Equlibria," Other publications TiSEM 33cc6820-0b55-4510-8eba-c, Tilburg University, School of Economics and Management.
Meca, A. & Garcia-Jurado, I. & Borm, P.E.M., 2003. "Cooperation and competition in inventory games," Other publications TiSEM a9b1e40b-8e0b-4782-9968-c, Tilburg University, School of Economics and Management.
Ana Meca & Ignacio García-Jurado & Peter Borm, 2003. "Cooperation and competition in inventory games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 57(3), pages 481-493, August.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & Voorneveld, M., 2003. "On a Compromise Social Choice Correspondence," Discussion Paper 2003-29, Tilburg University, Center for Economic Research.
Marieke Quant & Peter Borm & Hans Reijnierse & Mark Voorneveld, 2003. "On a compromise social choice correspondence," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(2), pages 311-324, December.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & Voorneveld, M., 2003. "On a Compromise Social Choice Correspondence," Other publications TiSEM 0b699b56-24d5-4a52-995f-3, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & Voorneveld, M., 2003. "On a compromise social choice correspondence," Other publications TiSEM 07d3e06d-241b-4aa2-b59d-d, Tilburg University, School of Economics and Management.
Dimitrov, D.A. & Borm, P.E.M. & Hendrickx, R.L.P., 2003. "Good and Bad Objects : Cardinality-Based Rules," Discussion Paper 2003-49, Tilburg University, Center for Economic Research.
Dimitrov, D.A. & Borm, P.E.M. & Hendrickx, R.L.P., 2003. "Good and Bad Objects : Cardinality-Based Rules," Other publications TiSEM be8831b3-40d4-4af0-93d3-4, Tilburg University, School of Economics and Management.
Dreef, M.R.M. & Borm, P.E.M., 2003. "On the Rule of Chance Moves and Information in Two-Person Games," Discussion Paper 2003-100, Tilburg University, Center for Economic Research.
Marcel Dreef & Peter Borm, 2006. "On the role of chance moves and information in two-person games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 75-98, June.
Dreef, M.R.M. & Borm, P.E.M., 2003. "On the Rule of Chance Moves and Information in Two-Person Games," Other publications TiSEM 54f11363-a82a-48ab-835f-0, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H., 2003. "Congestion Network Problems and Related Games," Discussion Paper 2003-106, Tilburg University, Center for Economic Research.
Quant, Marieke & Borm, Peter & Reijnierse, Hans, 2006. "Congestion network problems and related games," European Journal of Operational Research, Elsevier, vol. 172(3), pages 919-930, August.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H., 2003. "Congestion Network Problems and Related Games," Other publications TiSEM 1a0fb713-6949-42cb-96f1-4, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H., 2006. "Congestion network problems and related games," Other publications TiSEM f0e1d881-73d2-4bda-9137-4, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & van Velzen, S., 2003. "Compromise Stable TU-Games," Discussion Paper 2003-55, Tilburg University, Center for Economic Research.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & van Velzen, S., 2003. "Compromise Stable TU-Games," Other publications TiSEM 01a28f48-0b1b-43f9-8dac-3, Tilburg University, School of Economics and Management.
Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2003. "On the Core of Multiple Longest Traveling Salesman Games," Discussion Paper 2003-127, Tilburg University, Center for Economic Research.
Estevez-Fernandez, Arantza & Borm, Peter & Hamers, Herbert, 2006. "On the core of multiple longest traveling salesman games," European Journal of Operational Research, Elsevier, vol. 174(3), pages 1816-1827, November.
Estevez Fernandez, M.A. & Borm, P.E.M. & Hamers, H.J.M., 2003. "On the Core of Multiple Longest Traveling Salesman Games," Other publications TiSEM 08569957-5741-4082-ae18-c, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Hendrickx, R.L.P. & Quant, M. & Gonzalez-Diaz, J., 2003. "A Geometric Characterisation of the Compromise Value," Discussion Paper 2003-88, Tilburg University, Center for Economic Research.
Julio Díaz & Peter Borm & Ruud Hendrickx & Marieke Quant, 2005. "A geometric characterisation of the compromise value," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 61(3), pages 483-500, July.
Borm, P.E.M. & Hendrickx, R.L.P. & Quant, M. & Gonzalez-Diaz, J., 2003. "A Geometric Characterisation of the Compromise Value," Other publications TiSEM 47f24376-3c3c-4097-a54e-d, Tilburg University, School of Economics and Management.
Pulido, M. & Borm, P.E.M. & Hendrickx, R.L.P. & Llorca, N. & Sánchez-Soriano, J., 2003. "Compromise Solutions for Bankruptcy Situations with References," Discussion Paper 2003-36, Tilburg University, Center for Economic Research.
M. Pulido & P. Borm & R. Hendrickx & N. Llorca & J. Sánchez-Soriano, 2008. "Compromise solutions for bankruptcy situations with references," Annals of Operations Research, Springer, vol. 158(1), pages 133-141, February.
Pulido, M. & Borm, P.E.M. & Hendrickx, R.L.P. & Llorca, N. & Sánchez-Soriano, J., 2008. "Compromise solutions for bankruptcy situations with references," Other publications TiSEM d5052c4d-eda1-4d7e-b3d0-d, Tilburg University, School of Economics and Management.
Pulido, M. & Borm, P.E.M. & Hendrickx, R.L.P. & Llorca, N. & Sánchez-Soriano, J., 2003. "Compromise Solutions for Bankruptcy Situations with References," Other publications TiSEM 6c01c9d1-d42a-46e1-916d-d, Tilburg University, School of Economics and Management.
Casas-Mendez, B. & Borm, P.E.M. & Carpente, L. & Hendrickx, R.L.P., 2002. "The Constrained Equal Award Rule for Bankruptcy Problems with a Priori Unions," Discussion Paper 2002-83, Tilburg University, Center for Economic Research.
Casas-Mendez, B. & Borm, P.E.M. & Carpente, L. & Hendrickx, R.L.P., 2002. "The Constrained Equal Award Rule for Bankruptcy Problems with a Priori Unions," Other publications TiSEM 75b61f68-e00f-4076-84e5-5, Tilburg University, School of Economics and Management.
Suijs, J.P.M. & Borm, P.E.M., 2002. "Stochastic cooperative games : Theory and applications," Other publications TiSEM 9ce5bc08-4717-4baf-8789-e, Tilburg University, School of Economics and Management.
Dreef, M.R.M. & Borm, P.E.M. & van der Genugten, B.B., 2002. "On Strategy and Relative Skill in Poker," Discussion Paper 2002-59, Tilburg University, Center for Economic Research.
Marcel Dreef & Peter Borm & Ben van der Genugten, 2003. "On Strategy and Relative Skill in Poker," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 5(02), pages 83-103.
Dreef, M.R.M. & Borm, P.E.M. & van der Genugten, B.B., 2002. "On Strategy and Relative Skill in Poker," Other publications TiSEM 2a88a56c-0936-482f-b7ef-9, Tilburg University, School of Economics and Management.
Thijssen, J.J.J. & Hendrickx, R.L.P. & Borm, P.E.M., 2002. "Spillovers and Strategic Cooperative Behaviour," Discussion Paper 2002-70, Tilburg University, Center for Economic Research.
Thijssen, J.J.J. & Hendrickx, R.L.P. & Borm, P.E.M., 2002. "Spillovers and Strategic Cooperative Behaviour," Other publications TiSEM 3bb63307-2b04-495a-b18d-8, Tilburg University, School of Economics and Management.
Fernández, C. & Borm, P.E.M. & Hendrickx, R.L.P. & Tijs, S.H., 2002. "Drop Out Monotonic Rules for Sequencing Situations," Discussion Paper 2002-51, Tilburg University, Center for Economic Research.
Cristina Fernández & Peter Borm & Ruud Hendrickx & Stef Tijs, 2005. "Drop out monotonic rules for sequencing situations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 61(3), pages 501-504, July.
Fernández, C. & Borm, P.E.M. & Hendrickx, R.L.P. & Tijs, S.H., 2002. "Drop Out Monotonic Rules for Sequencing Situations," Other publications TiSEM f343286b-7f46-4c60-94b3-a, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Fernández, C. & Hendrickx, R.L.P. & Tijs, S.H., 2005. "Drop out monotonic rules for sequencing situations," Other publications TiSEM d0642261-4e55-45ca-9b23-6, Tilburg University, School of Economics and Management.
Wintein, S. & Borm, P.E.M. & Hendrickx, R.L.P. & Quant, M., 2002. "Multiple Fund Investment Situations and Related Games," Discussion Paper 2002-109, Tilburg University, Center for Economic Research.
Stefan Wintein & Peter Borm & Ruud Hendrickx & Marieke Quant, 2006. "Multiple Fund Investment Situations and Related Games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(3), pages 413-426, July.
Wintein, S. & Borm, P.E.M. & Hendrickx, R.L.P. & Quant, M., 2002. "Multiple Fund Investment Situations and Related Games," Other publications TiSEM 47aee517-90aa-4bf1-bcd6-9, Tilburg University, School of Economics and Management.
Wintein, S. & Borm, P.E.M. & Hendrickx, R.L.P. & Quant, M., 2006. "Multiple fund investment situations and related games," Other publications TiSEM 36b9e255-3920-4829-bd79-5, Tilburg University, School of Economics and Management.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & Voorneveld, M., 2002. "Characterizations of Solutions in Digraph Competitions," Discussion Paper 2002-113, Tilburg University, Center for Economic Research.
Quant, M. & Borm, P.E.M. & Reijnierse, J.H. & Voorneveld, M., 2002. "Characterizations of Solutions in Digraph Competitions," Other publications TiSEM f8406395-a31f-47b0-ab62-d, Tilburg University, School of Economics and Management.
Hendrickx, R.L.P. & Borm, P.E.M. & Timmer, J.B., 2002. "A note on NTU-convexity," Other publications TiSEM c8e46ca9-db92-4579-b793-4, Tilburg University, School of Economics and Management.
Ruud Hendrickx & Judith Timmer & Peter Borm, 2002. "A note on NTU convexity," International Journal of Game Theory, Springer;Game Theory Society, vol. 31(1), pages 29-37.
Borm, P.E.M. & van der Genugten, B.B., 2001. "On a relative measure of skill for games with chance elements," Other publications TiSEM 42a0632e-2ce1-4414-9254-3, Tilburg University, School of Economics and Management.
Peter Borm & Ben Genugten, 2001. "On a relative measure of skill for games with chance elements," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(1), pages 91-114, June.
Suijs, J.P.M. & Borm, P.E.M. & Hamers, H.J.M. & Koster, M.A.L. & Quant, M., 2001. "Communications and Cooperation in Public Network Situations," Discussion Paper 2001-44, Tilburg University, Center for Economic Research.
Jeroen Suijs & Peter Borm & Herbert Hamers & Marieke Quant & Maurice Koster, 2005. "Communication and Cooperation in Public Network Situations," Annals of Operations Research, Springer, vol. 137(1), pages 117-140, July.
Suijs, J.P.M. & Borm, P.E.M. & Hamers, H.J.M. & Koster, M.A.L. & Quant, M., 2001. "Communications and Cooperation in Public Network Situations," Other publications TiSEM 385509cd-511d-410c-aa23-2, Tilburg University, School of Economics and Management.
Calleja, P. & Borm, P.E.M. & Hamers, H.J.M. & Klijn, F., 2001. "On a New Class of Parallel Sequencing Situations and Related Games," Discussion Paper 2001-3, Tilburg University, Center for Economic Research.
Pedro Calleja & Peter Borm & Herbert Hamers & Flip Klijn & Marco Slikker, 2002. "On a New Class of Parallel Sequencing Situations and Related Games," Annals of Operations Research, Springer, vol. 109(1), pages 265-277, January.
Calleja, P. & Borm, P.E.M. & Hamers, H.J.M. & Klijn, F., 2001. "On a New Class of Parallel Sequencing Situations and Related Games," Other publications TiSEM 16a7563a-1803-4a1d-8b46-4, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Calleja, P. & Hamers, H.J.M. & Klijn, F. & Slikker, M., 2002. "On a new class of parallel sequencing situations and related games," Other publications TiSEM 618c2e4c-dc9f-47bc-bb10-a, Tilburg University, School of Economics and Management.
Dreef, M.R.M. & Borm, P.E.M. & van der Genugten, B.B., 2001. "A New Relative Skill Measure for Games with Chance Elements," Discussion Paper 2001-106, Tilburg University, Center for Economic Research.
Marcel Dreef & Peter Borm & Ben van der Genugten, 2004. "A new relative skill measure for games with chance elements," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 25(5), pages 255-264.
Dreef, M.R.M. & Borm, P.E.M. & van der Genugten, B.B., 2001. "A New Relative Skill Measure for Games with Chance Elements," Other publications TiSEM 8c43baf8-983d-4a11-b2f8-3, Tilburg University, School of Economics and Management.
Dreef, M.R.M. & Borm, P.E.M. & van der Genugten, B.B., 2004. "A new relative skill measure for games with chance elements," Other publications TiSEM e54422a6-f71a-4262-b7fc-1, Tilburg University, School of Economics and Management.
Calleja, P. & Borm, P.E.M. & Hendrickx, R.L.P., 2001. "Multi-Issue Allocation Games," Discussion Paper 2001-30, Tilburg University, Center for Economic Research.
Calleja, P. & Borm, P.E.M. & Hendrickx, R.L.P., 2001. "Multi-Issue Allocation Games," Other publications TiSEM db0ee5ec-5e97-4b1a-9e01-7, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Hamers, H.J.M. & Hendrickx, R.L.P., 2001. "Operations Research Games : A Survey," Discussion Paper 2001-45, Tilburg University, Center for Economic Research.
Peter Borm & Herbert Hamers & Ruud Hendrickx, 2001. "Operations research games: A survey," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 9(2), pages 139-199, December.
Borm, P.E.M. & Hamers, H.J.M. & Hendrickx, R.L.P., 2001. "Operations Research Games : A Survey," Other publications TiSEM 04f265e0-8043-4d4f-bf27-2, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Hamers, H.J.M. & Hendrickx, R.L.P., 2001. "Operations research games : A survey," Other publications TiSEM 755a430b-592f-400b-ba18-9, Tilburg University, School of Economics and Management.
Timmer, J.B. & Borm, P.E.M. & Tijs, S.H., 2000. "Convexity in Stochastic Cooperative Situations," Discussion Paper 2000-04, Tilburg University, Center for Economic Research.
Judith Timmer & Peter Borm & Stef Tijs, 2005. "Convexity In Stochastic Cooperative Situations," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 25-42.
Timmer, J.B. & Borm, P.E.M. & Tijs, S.H., 2000. "Convexity in Stochastic Cooperative Situations," Other publications TiSEM 02a2e428-1933-4b8b-b89a-3, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Tijs, S.H. & Timmer, J.B., 2005. "Convexity in stochastic cooperative situations," Other publications TiSEM 26004fd5-1beb-4ddf-b73f-a, Tilburg University, School of Economics and Management.
Hendrickx, R.L.P. & Borm, P.E.M. & Timmer, J.B., 2000. "On Convexity for NTU-Games," Discussion Paper 2000-108, Tilburg University, Center for Economic Research.
Hendrickx, R.L.P. & Borm, P.E.M. & Timmer, J.B., 2000. "On Convexity for NTU-Games," Other publications TiSEM ef12f1e8-87f5-41b4-97e4-7, Tilburg University, School of Economics and Management.
Timmer, J.B. & Borm, P.E.M. & Tijs, S.H., 2000. "On Three Shapley-Like Solutions for Cooperative Games with Random Payoffs," Discussion Paper 2000-73, Tilburg University, Center for Economic Research.
Judith Timmer & Peter Borm & Stef Tijs, 2004. "On three Shapley-like solutions for cooperative games with random payoffs," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(4), pages 595-613, August.
Timmer, J.B. & Borm, P.E.M. & Tijs, S.H., 2003. "On three Shapley-like solutions for cooperative games with random payoffs," Other publications TiSEM 619397b2-51b9-42db-916c-9, Tilburg University, School of Economics and Management.
Timmer, J.B. & Borm, P.E.M. & Tijs, S.H., 2000. "On Three Shapley-Like Solutions for Cooperative Games with Random Payoffs," Other publications TiSEM ecb22057-95be-42d9-b357-d, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van den Brink, J.R. & Slikker, M., 2000. "An Iterative Procedure for Evaluating Digraph Competitions," Research Memorandum 788, Tilburg University, School of Economics and Management.
Peter Borm & René van den Brink & Marco Slikker, 2002. "An Iterative Procedure for Evaluating Digraph Competitions," Annals of Operations Research, Springer, vol. 109(1), pages 61-75, January.
Borm, P.E.M. & van den Brink, J.R. & Slikker, M., 2000. "An Iterative Procedure for Evaluating Digraph Competitions," Other publications TiSEM 03637fc0-696b-4b91-b292-d, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van den Brink, J.R. & Slikker, M., 2002. "An iterative procedure for evaluating digraph competitions," Other publications TiSEM 40ae2ec2-efdb-48f6-905c-5, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van den Brink, J.R. & Levinsky, R. & Slikker, M., 2000. "On Two New Social Choice Correspondences," Discussion Paper 2000-125, Tilburg University, Center for Economic Research.
Borm, Peter & van den Brink, Rene & Levinsky, Rene & Slikker, Marco, 2004. "On two new social choice correspondences," Mathematical Social Sciences, Elsevier, vol. 47(1), pages 51-68, January.
Borm, P.E.M. & van den Brink, J.R. & Levinsky, R. & Slikker, M., 2000. "On Two New Social Choice Correspondences," Other publications TiSEM 71885704-e24b-43ef-a029-8, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Fiestras-Janeiro, G. & Hamers, H.J.M. & Sánchez, E. & Voorneveld, M., 1999. "On the Convexity of Games Corresponding to Sequencing Situations with Due Dates," Discussion Paper 1999-49, Tilburg University, Center for Economic Research.
Borm, Peter & Fiestras-Janeiro, Gloria & Hamers, Herbert & Sanchez, Estela & Voorneveld, Mark, 2002. "On the convexity of games corresponding to sequencing situations with due dates," European Journal of Operational Research, Elsevier, vol. 136(3), pages 616-634, February.
Borm, P.E.M. & Fiestras-Janeiro, G. & Hamers, H.J.M. & Sánchez, E. & Voorneveld, M., 1999. "On the Convexity of Games Corresponding to Sequencing Situations with Due Dates," Other publications TiSEM 46a07ce4-57fe-4a08-bcb5-c, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Fiestras-Janeiro, G. & Hamers, H.J.M. & Sánchez, E. & Voorneveld, M., 2002. "On the convexity of games corresponding to sequencing situations with due dates," Other publications TiSEM caf52141-b0aa-42ef-96ed-e, Tilburg University, School of Economics and Management.
Voorneveld, M. & Borm, P.E.M. & van Megen, F.J.C. & Tijs, S.H. & Facchini, G., 1999. "Congestion Games and Potentials Reconsidered," Discussion Paper 1999-98, Tilburg University, Center for Economic Research.
Mark Voorneveld & Peter Borm & Freek Van Megen & Stef Tijs & Giovanni Facchini, 1999. "Congestion Games And Potentials Reconsidered," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 1(03n04), pages 283-299.
Voorneveld, M. & Borm, P.E.M. & van Megen, F.J.C. & Tijs, S.H. & Facchini, G., 1999. "Congestion games and potentials reconsidered," Other publications TiSEM a2b8c559-8a5b-4a4a-8205-9, Tilburg University, School of Economics and Management.
Voorneveld, M. & Borm, P.E.M. & van Megen, F.J.C. & Tijs, S.H. & Facchini, G., 1999. "Congestion Games and Potentials Reconsidered," Other publications TiSEM 1d647323-c17d-41e9-a4d5-8, Tilburg University, School of Economics and Management.
Hamers, H.J.M. & Borm, P.E.M. & van den Leensel, A. & Tijs, S.H., 1999. "Cost allocation in the Chinese postman problem," Other publications TiSEM 5da2ea14-2fdf-4d1f-8b60-2, Tilburg University, School of Economics and Management.
Hamers, Herbert & Borm, Peter & van de Leensel, Robert & Tijs, Stef, 1999. "Cost allocation in the Chinese postman problem," European Journal of Operational Research, Elsevier, vol. 118(1), pages 153-163, October.
Algaba, A. & Bilbao, J.M. & Borm, P.E.M., 1999. "The Myerson Value for Union Stable Systems," Research Memorandum 773, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Algaba, A. & Bilbao, J.M. & Lopez, J., 2002. "The Myerson value for union stable systems," Other publications TiSEM 3179823c-129f-458b-9d83-2, Tilburg University, School of Economics and Management.
Algaba, A. & Bilbao, J.M. & Borm, P.E.M., 1999. "The Myerson Value for Union Stable Systems," Other publications TiSEM 022b6f46-8f76-4bb3-acab-3, Tilburg University, School of Economics and Management.
Borm, P.E.M. & De Waegenaere, A.M.B. & Rafels, C. & Suijs, J.P.M. & Tijs, S.H. & Timmer, J.B., 1999. "Cooperation in Capital Deposits," Discussion Paper 1999-31, Tilburg University, Center for Economic Research.
Borm, P.E.M. & De Waegenaere, A.M.B. & Rafels, C. & Suijs, J.P.M. & Tijs, S.H. & Timmer, J.B., 2001. "Cooperation in capital deposits," Other publications TiSEM de31df8b-638a-43b2-b79e-f, Tilburg University, School of Economics and Management.
Borm, P.E.M. & De Waegenaere, A.M.B. & Rafels, C. & Suijs, J.P.M. & Tijs, S.H. & Timmer, J.B., 1999. "Cooperation in Capital Deposits," Other publications TiSEM c23d41ab-a641-4951-b9ec-7, Tilburg University, School of Economics and Management.
Meca-Martinez, A. & Timmer, J.B. & Garcia-Jurado, I. & Borm, P.E.M., 1999. "Inventory Games," Discussion Paper 1999-53, Tilburg University, Center for Economic Research.
Meca, Ana & Timmer, Judith & Garcia-Jurado, Ignacio & Borm, Peter, 2004. "Inventory games," European Journal of Operational Research, Elsevier, vol. 156(1), pages 127-139, July.
Meca, A. & Timmer, J.B. & Garcia-Jurado, I. & Borm, P.E.M., 2004. "Inventory games," Other publications TiSEM 49368f2d-02fc-49c9-9d74-8, Tilburg University, School of Economics and Management.
Meca-Martinez, A. & Timmer, J.B. & Garcia-Jurado, I. & Borm, P.E.M., 1999. "Inventory Games," Other publications TiSEM 21f26b3f-7fae-4f19-908f-a, Tilburg University, School of Economics and Management.
Suijs, J.P.M. & Borm, P.E.M., 1999. "Stochastic cooperative games : Superadditivity, convexity and certainty equivalents," Other publications TiSEM 42630051-47fb-44ec-839b-4, Tilburg University, School of Economics and Management.
Suijs, Jeroen & Borm, Peter, 1999. "Stochastic Cooperative Games: Superadditivity, Convexity, and Certainty Equivalents," Games and Economic Behavior, Elsevier, vol. 27(2), pages 331-345, May.
Suijs, J.P.M. & De Waegenaere, A.M.B. & Borm, P.E.M., 1998. "Optimal Design of Pension Funds : A Mission Impossible," Discussion Paper 1998-25, Tilburg University, Center for Economic Research.
Anja De Waegenaere & Jeroen Suijs & Peter Borm, 2003. "Optimal design of pension funds: a mission impossible?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 22(1), pages 89-110, August.
Suijs, J.P.M. & De Waegenaere, A.M.B. & Borm, P.E.M., 1998. "Optimal Design of Pension Funds : A Mission Impossible," Other publications TiSEM ccc7c4fa-fdf8-4276-8d5c-6, Tilburg University, School of Economics and Management.
De Waegenaere, A.M.B. & Suijs, J.P.M. & Borm, P.E.M., 2003. "Optimal design of pension funds : A mission impossible?," Other publications TiSEM 8dd41b29-b509-46c4-a528-d, Tilburg University, School of Economics and Management.
Koster, M.A.L. & Tijs, S.H. & Borm, P.E.M., 1998. "Serial cost sharing methods for multi-commodity situations," Other publications TiSEM 6633be50-672a-42f3-a966-8, Tilburg University, School of Economics and Management.
Koster, Maurice & Tijs, Stef & Borm, Peter, 1998. "Serial cost sharing methods for multi-commodity situations," Mathematical Social Sciences, Elsevier, vol. 36(3), pages 229-242, December.
Algaba, A. & Bilbao, J.M. & Borm, P.E.M. & Lopez, J., 1998. "The position value for union stable systems," Research Memorandum FEW 768, Tilburg University, School of Economics and Management.
E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2000. "The position value for union stable systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 221-236, November.
Algaba, A. & Bilbao, J.M. & Borm, P.E.M. & Lopez, J., 1998. "The position value for union stable systems," Other publications TiSEM fa70d57b-042b-42ea-a574-3, Tilburg University, School of Economics and Management.
Algaba, A. & Bilbao, J.M. & Borm, P.E.M. & Lopez, J., 2000. "The position value for union stable systems," Other publications TiSEM f7ea939d-770c-43ed-92ae-1, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Hamers, H.J.M., 1998. "A note on games corresponding to sequencing situations with due dates," Discussion Paper 1998-46, Tilburg University, Center for Economic Research.
Borm, P.E.M. & Hamers, H.J.M., 1998. "A note on games corresponding to sequencing situations with due dates," Other publications TiSEM c1b19f1b-f615-48e9-bf3e-a, Tilburg University, School of Economics and Management.
Otten, G.J.M. & Borm, P.E.M. & Peleg, B. & Tijs, S.H., 1998. "The MC-value for monotonic NTU-games," Other publications TiSEM 9f03343a-8e36-453a-868d-a, Tilburg University, School of Economics and Management.
Bezalel Peleg & Stef Tijs & Peter Borm & Gert-Jan Otten, 1998. "The MC-value for monotonic NTU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(1), pages 37-47.
Borm, P.E.M. & Vermeulen, D. & Voorneveld, M., 1998. "The Structure of the Set of Equilibria for Two Person Multicriteria Games," Discussion Paper 1998-75, Tilburg University, Center for Economic Research.
Borm, Peter & Vermeulen, Dries & Voorneveld, Mark, 2003. "The structure of the set of equilibria for two person multicriteria games," European Journal of Operational Research, Elsevier, vol. 148(3), pages 480-493, August.
Borm, P.E.M. & Vermeulen, D. & Voorneveld, M., 2003. "The structure of the set of equilibria for two person multicriteria games," Other publications TiSEM ecf9e80b-f56a-4669-b647-e, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Vermeulen, D. & Voorneveld, M., 1998. "The Structure of the Set of Equilibria for Two Person Multicriteria Games," Other publications TiSEM 54baf13b-b47f-419c-bdfb-c, Tilburg University, School of Economics and Management.
Timmer, J.B. & Borm, P.E.M. & Suijs, J.P.M., 1998. "Linear Transformation of Products : Games and Economies," Discussion Paper 1998-76, Tilburg University, Center for Economic Research.
J. Timmer & P. Borm & J. Suijs, 2000. "Linear Transformation of Products: Games and Economies," Journal of Optimization Theory and Applications, Springer, vol. 105(3), pages 677-706, June.
Timmer, J.B. & Borm, P.E.M. & Suijs, J.P.M., 2000. "Linear transformation of products : Games and economies," Other publications TiSEM ee4808da-a9e9-40cb-9739-3, Tilburg University, School of Economics and Management.
Timmer, J.B. & Borm, P.E.M. & Suijs, J.P.M., 1998. "Linear Transformation of Products : Games and Economies," Other publications TiSEM 39313f90-919c-419d-8e10-c, Tilburg University, School of Economics and Management.
Facchini, G. & van Megen, F.J.C. & Borm, P.E.M. & Tijs, S.H., 1997. "Congestion models and weighted Bayesian potential games," Other publications TiSEM c80cf83c-bb5a-4cc2-a12c-8, Tilburg University, School of Economics and Management.
Giovanni Facchini & Freek van Megen & Peter Borm & Stef Tijs, 1997. "Congestion Models And Weighted Bayesian Potential Games," Theory and Decision, Springer, vol. 42(2), pages 193-206, March.
Voorneveld, M. & Vermeulen, D. & Borm, P.E.M., 1997. "Axiomatizations of Pareto Equilibria in Multicriteria Games," Research Memorandum 749, Tilburg University, School of Economics and Management.
Voorneveld, Mark & Vermeulen, Dries & Borm, Peter, 1999. "Axiomatizations of Pareto Equilibria in Multicriteria Games," Games and Economic Behavior, Elsevier, vol. 28(1), pages 146-154, July.
Voorneveld, M. & Vermeulen, D. & Borm, P.E.M., 1997. "Axiomatizations of Pareto Equilibria in Multicriteria Games," Other publications TiSEM b8ab2101-f3bb-4a0d-96ee-3, Tilburg University, School of Economics and Management.
Voorneveld, M. & Vermeulen, D. & Borm, P.E.M., 1999. "Axiomatizations of Pareto equilibria in multicriteria games," Other publications TiSEM 12747650-0b71-42b8-8d28-a, Tilburg University, School of Economics and Management.
Otten, G.J.M. & Borm, P.E.M. & Storcken, T. & Tijs, S.H., 1997. "Decomposable effectivity functions," Other publications TiSEM 04f36dbf-4ea2-4a2e-bcb8-d, Tilburg University, School of Economics and Management.
Otten, Gert-Jan & Borm, Peter & Storcken, Ton & Tijs, Stef, 1997. "Decomposable effectivity functions," Mathematical Social Sciences, Elsevier, vol. 33(3), pages 277-289, June.
Suijs, J.P.M. & De Waegenaere, A.M.B. & Borm, P.E.M., 1996. "Stochastic Cooperative Games in Insurance and Reinsurance," Discussion Paper 1996-53, Tilburg University, Center for Economic Research.
Suijs, J.P.M. & De Waegenaere, A.M.B. & Borm, P.E.M., 1996. "Stochastic Cooperative Games in Insurance and Reinsurance," Other publications TiSEM f2cd7428-cd39-4462-af76-2, Tilburg University, School of Economics and Management.
Suijs, J.P.M. & De Waegenaere, A.M.B. & Borm, P.E.M., 1998. "Stochastic cooperative games in insurance and reinsurance," Other publications TiSEM 01f181bb-862a-4712-9204-9, Tilburg University, School of Economics and Management.
Fiestras-Janeiro, G. & Borm, P.E.M. & van Megen, F.J.C., 1996. "Protective Behavior in Games," Discussion Paper 1996-12, Tilburg University, Center for Economic Research.
Fiestras-Janeiro, G. & Borm, P.E.M. & van Megen, F.J.C., 1998. "Protective behavior in games," Other publications TiSEM 92723c0f-5357-4985-9733-c, Tilburg University, School of Economics and Management.
Fiestras-Janeiro, G. & Borm, P.E.M. & van Megen, F.J.C., 1996. "Protective Behavior in Games," Other publications TiSEM 0f0d5aed-021d-45d8-9776-0, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Garcia Jurado, I. & Potters, J.A.M. & Tijs, S.H., 1996. "An amalgamation of games," Other publications TiSEM 41c72009-ce95-4852-a517-9, Tilburg University, School of Economics and Management.
Borm, Peter & Garcia-Jurado, Ignacio & Potters, Jos & Tijs, Stef, 1996. "An amalgation of games," European Journal of Operational Research, Elsevier, vol. 89(3), pages 570-580, March.
van Megen, F.J.C. & Facchini, G. & Borm, P.E.M. & Tijs, S.H., 1996. "Strong Nash Equilibria and the Potential Maimizer," Discussion Paper 1996-13, Tilburg University, Center for Economic Research.
van Megen, F.J.C. & Facchini, G. & Borm, P.E.M. & Tijs, S.H., 1996. "Strong Nash Equilibria and the Potential Maimizer," Other publications TiSEM 4bb6ee96-1a0c-44ee-8685-c, Tilburg University, School of Economics and Management.
Hamers, H.J.M. & Suijs, J.P.M. & Tijs, S.H. & Borm, P.E.M., 1996. "The split core of sequencing games," Other publications TiSEM 28693e2d-82da-456a-909b-3, Tilburg University, School of Economics and Management.
Hamers, Herbert & Suijs, Jeroen & Tijs, Stef & Borm, Peter, 1996. "The Split Core for Sequencing Games," Games and Economic Behavior, Elsevier, vol. 15(2), pages 165-176, August.
Hamers, H.J.M. & Suijs, J.P.M. & Tijs, S.H. & Borm, P.E.M., 1994. "The split core for sequencing games," Other publications TiSEM eb93d672-0769-40c4-9445-5, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van der Genugten, B.B., 1996. "On a Measure of Skills for Games with Chance Elements," Research Memorandum 721, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van der Genugten, B.B., 1996. "On a Measure of Skills for Games with Chance Elements," Other publications TiSEM 3f9cbf8f-a057-4363-a876-a, Tilburg University, School of Economics and Management.
Suijs, J.P.M. & Borm, P.E.M., 1996. "Cooperative Games with Stochastic Payoffs : Determanistic Equivalents," Research Memorandum 713, Tilburg University, School of Economics and Management.
Suijs, J.P.M. & Borm, P.E.M., 1996. "Cooperative Games with Stochastic Payoffs : Determanistic Equivalents," Other publications TiSEM 8db5e0a7-8a3f-45c1-bfc1-7, Tilburg University, School of Economics and Management.
van Megen, F. & Borm, P.E.M. & Tijs, S.H., 1995. "A perfectness concept for multicriteria games," Discussion Paper 1995-28, Tilburg University, Center for Economic Research.
Peter Borm & Freek van Megen & Stef Tijs, 1999. "A perfectness concept for multicriteria games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(3), pages 401-412, July.
van Megen, F. & Borm, P.E.M. & Tijs, S.H., 1995. "A perfectness concept for multicriteria games," Other publications TiSEM 94f61070-74a2-4fcf-a50e-8, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van Megen, F.J.C. & Tijs, S.H., 1999. "A perfectness concept for multicriteria games," Other publications TiSEM 322bd1a7-90e0-400d-b167-6, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Cao, R. & García-Jurado, I. & Méndez-Naya, L., 1995. "Weakly strict equilibria in finite normal form games," Other publications TiSEM 03e57c68-c590-4baf-aa90-3, Tilburg University, School of Economics and Management.
Otten, G.J.M. & Borm, P.E.M. & Storcken, T. & Tijs, S.H., 1995. "Effectivity functions and associated claim game correspondences," Other publications TiSEM 0789742b-ab29-48ba-89d9-f, Tilburg University, School of Economics and Management.
Otten Gert-Jan & Borm Peter & Storcken Ton & Tijs Stef, 1995. "Effectivity Functions and Associated Claim Game Correspondences," Games and Economic Behavior, Elsevier, vol. 9(2), pages 172-190, May.
Sujis, J. & Borm, P. & De Waegenaere, A. & Tijs, S., 1995. "Cooperative Games with Stochastic Payoffs," Papers 9588, Tilburg - Center for Economic Research.
Suijs, Jeroen & Borm, Peter & De Waegenaere, Anja & Tijs, Stef, 1999. "Cooperative games with stochastic payoffs," European Journal of Operational Research, Elsevier, vol. 113(1), pages 193-205, February.
Suijs, J.P.M. & Borm, P.E.M. & De Waegenaere, A.M.B. & Tijs, S.H., 1995. "Cooperative games with stochastic payoffs," Discussion Paper 1995-88, Tilburg University, Center for Economic Research.
Suijs, J.P.M. & Borm, P.E.M. & De Waegenaere, A.M.B. & Tijs, S.H., 1995. "Cooperative games with stochastic payoffs," Other publications TiSEM 7354ad00-151c-4d52-9f18-1, Tilburg University, School of Economics and Management.
Suijs, J.P.M. & Borm, P.E.M. & De Waegenaere, A.M.B. & Tijs, S.H., 1999. "Cooperative games with stochastic payoffs," Other publications TiSEM f0fb042f-fe23-43f9-9982-d, Tilburg University, School of Economics and Management.
Jansen, M.J.M. & Jurg, A.P. & Borm, P.E.M., 1994. "On strictly perfect sets," Other publications TiSEM 77ebc80c-ac78-43a0-808b-6, Tilburg University, School of Economics and Management.
Jansen M. J. M. & Jurg A. P. & Borm P. E. M., 1994. "On Strictly Perfect Sets," Games and Economic Behavior, Elsevier, vol. 6(3), pages 400-415, May.
van den Nouweland, A. & Borm, P. & van Golstein, W. & Bruinderink, R.G. & Tijs, S., 1994. "A Game Theoretic Approach to Problems in Telecommunication," Papers 9407, Tilburg - Center for Economic Research.
A. van den Nouweland & P. Borm & W. van Golstein Brouwers & R. Groot Bruinderink & S. Tijs, 1996. "A Game Theoretic Approach to Problems in Telecommunication," Management Science, INFORMS, vol. 42(2), pages 294-303, February.
van den Nouweland, C.G.A.M. & Borm, P.E.M. & van Golstein Brouwers, W. & Groot Bruinderink, R. & Tijs, S.H., 1996. "A game theoretic approach to problems in telecommunication," Other publications TiSEM a3b30529-fe17-484c-8eab-7, Tilburg University, School of Economics and Management.
Otten, G.J.M. & Borm, P.E.M. & Tijs, S.H., 1994. "A note on the characterization of the compromise value," Research Memorandum FEW 655, Tilburg University, School of Economics and Management.
Otten, Gert-Jan & Borm, Peter & Tijs, Stef, 1996. "A Note on the Characterizations of the Compromise Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 25(4), pages 427-435.
Otten, G.J.M. & Borm, P.E.M. & Tijs, S.H., 1996. "A note on the characterizations of the compromise value," Other publications TiSEM 9a172c93-0df4-498f-ab06-c, Tilburg University, School of Economics and Management.
Otten, G.J.M. & Borm, P.E.M. & Tijs, S.H., 1994. "A note on the characterization of the compromise value," Other publications TiSEM 427966c6-367b-4902-9cd6-7, Tilburg University, School of Economics and Management.
van Heumen, R.W.J. & Peleg, B. & Tijs, S.H. & Borm, P.E.M., 1994. "Axiomatic characterizations of solutions for Bayesian games," Research Memorandum FEW 680, Tilburg University, School of Economics and Management.
van Heumen, R.W.J. & Peleg, B. & Tijs, S.H. & Borm, P.E.M., 1994. "Axiomatic characterizations of solutions for Bayesian games," Other publications TiSEM b16fc7d9-aee7-4f36-95f2-3, Tilburg University, School of Economics and Management.
van Heumen, R. & Peleg, B. & Tijs, S.H. & Borm, P.E.M., 1996. "Axiomatic characterizations of solutions for Bayesian games," Other publications TiSEM 6cc729cb-f5e8-496a-a365-3, Tilburg University, School of Economics and Management.
van den Brink, J.R. & Borm, P.E.M., 1994. "Digraph competitions and cooperative games," Discussion Paper 1994-24, Tilburg University, Center for Economic Research.
René van den Brink & Peter Borm, 2002. "Digraph Competitions and Cooperative Games," Theory and Decision, Springer, vol. 53(4), pages 327-342, December.
van den Brink, J.R. & Borm, P.E.M., 1994. "Digraph competitions and cooperative games," Other publications TiSEM 1b98a76d-ab49-4f5c-975c-c, Tilburg University, School of Economics and Management.
van den Brink, J.R. & Borm, P.E.M., 2002. "Digraph competitions and cooperative games," Other publications TiSEM 262e8724-0bc0-49da-99c5-f, Tilburg University, School of Economics and Management.
Hamers, H. & Borm, P. & van de Leensel, R. & Tijs, S., 1994. "The Chineese Postman and Delivery Games," Papers 9476, Tilburg - Center for Economic Research.
Hamers, H.J.M. & Borm, P.E.M. & van de Leensel, R. & Tijs, S.H., 1994. "The chinese postman and delivery games," Other publications TiSEM 6daa22eb-dda9-4e80-add1-e, Tilburg University, School of Economics and Management.
Hamers, H. & Borm, P. & Tijs, S., 1993. "A Games Corresponding to Sequencing Situations with Ready Times," Papers 9316, Tilburg - Center for Economic Research.
Hamers, H.J.M. & Borm, P.E.M. & Tijs, S.H., 1993. "On games corresponding to sequencing situations with ready times," Other publications TiSEM 8e2af556-5430-4f98-9334-c, Tilburg University, School of Economics and Management.
Hamers, H.J.M. & Borm, P.E.M. & Tijs, S.H., 1995. "On games corresponding to sequencing situations with ready times," Other publications TiSEM 99882a33-06c1-4f7c-8436-6, Tilburg University, School of Economics and Management.
Tijs, S.H. & Borm, P.E.M., 1993. "Operations research, games and graphs," Other publications TiSEM 55f79d3e-fb98-4cb9-a53c-8, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Cao, R. & Garcia-Jurado, I., 1993. "Maximum likelihood equilibria of random games," Research Memorandum FEW 601, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Cao, R. & García-Jurado, I., 1995. "Maximum likelihood equilibria of random games," Other publications TiSEM 6ff66118-7da7-49d5-8309-d, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Cao, R. & Garcia-Jurado, I., 1993. "Maximum likelihood equilibria of random games," Other publications TiSEM 6ad4e6ed-b136-45df-b66b-8, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Jansen, M.J.M. & Potters, J.A.M. & Tijs, S.H., 1993. "On the structure of the set of perfect equilibria in bimatrix games," Other publications TiSEM 5a4170b8-1cb6-416f-b944-e, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Jansen, M.J.M. & Potters, J.A.M. & Tijs, S.H., 1993. "Pareto equilibria for bimatrix games," Other publications TiSEM 4c66db1e-0888-4798-aacf-a, Tilburg University, School of Economics and Management.
Feltkamp, V. & Koster, A. & Van Den Nouweland, A. & Borm, P. & Tijs, S., 1993. "Linear Production with Transport of Products, Resources and Technology," Papers 9332, Tilburg - Center for Economic Research.
Feltkamp, V., 1993. "Linear Production with Transport of Products, Resources and Technology," Discussion Paper 1993-32, Tilburg University, Center for Economic Research.
Feltkamp, V. & van den Nouweland, C.G.A.M. & Borm, P.E.M. & Tijs, S.H. & Koster, A., 1993. "Linear production with transport of products, resources and technology," Other publications TiSEM acd52308-43b3-4cac-b8d5-1, Tilburg University, School of Economics and Management.
Jurg, A.P. & García-Jurado, I. & Borm, P.E.M., 1992. "On modifications of the concepts of perfect and proper equilibria," Other publications TiSEM b06a7e2d-b824-4a92-b577-9, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Tijs, S.H., 1992. "Strategic claim games corresponding to an NTU-game," Other publications TiSEM 0acf76c8-5da7-4972-a4fa-7, Tilburg University, School of Economics and Management.
Borm, P. E. M. & Tijs, S. H., 1992. "Strategic claim games corresponding to an NTU-game," Games and Economic Behavior, Elsevier, vol. 4(1), pages 58-71, January.
Borm, P.E.M. & Tijs, S.H., 1992. "Strategic claim games corresponding to an NTU-game," Other publications TiSEM d85d27fa-00ee-42b4-9cc6-2, Tilburg University, School of Economics and Management.
Borm, P.E.M., 1992. "On perfectness concepts for bimatrix games," Other publications TiSEM 9652c2b4-b09f-4c05-846a-3, Tilburg University, School of Economics and Management.
Borm, P. & Otten, G.J. & Peters, H., 1992. "Core Implementation in Modified Strong and Coalition Proof Nash Equilibria," Papers 9228, Tilburg - Center for Economic Research.
Borm, P.E.M. & Otten, G.J.M. & Peters, H.J.M., 1992. "Core implementation in modified strong and coalition proof Nash equilibria," Other publications TiSEM a3680e90-b5f5-45c4-bd21-6, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Otten, G.J.M. & Peters, H.J.M., 1992. "Core implementation in modified strong and coalition proof Nash equilibria," Discussion Paper 1992-28, Tilburg University, Center for Economic Research.
Otten, G.J.M. & Borm, P.E.M. & Storcken, A.J.A. & Tijs, S.H., 1992. "Effectivity functions and associated game correspondences," Research Memorandum FEW 536, Tilburg University, School of Economics and Management.
Otten, G.J.M. & Borm, P.E.M. & Storcken, A.J.A. & Tijs, S.H., 1992. "Effectivity functions and associated game correspondences," Other publications TiSEM 05c0318d-5031-4937-ae98-2, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Other publications TiSEM 5a8473e4-1df7-42df-ad53-f, Tilburg University, School of Economics and Management.
Borm, P. & Keiding, H. & McLean, R.P., 1992. "The Compromise Value for NTU-Games," Papers 9218, Tilburg - Center for Economic Research.
Borm, Peter & Keiding, H & McLean, R.P. & Oortwijn, S & Tijs, S, 1992. "The Compromise Value for NTU-Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(2), pages 175-189.
Borm, P.E.M. & Keiding, H. & McLean, R.P. & Oortwijn, S. & Tijs, S.H., 1993. "The compromise value for NTU-games," Other publications TiSEM 27c574e5-d810-484c-a668-3, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Keiding, H. & McLean, R.P. & Oortwijn, S. & Tijs, S.H., 1992. "The compromise value for NTU-games," Other publications TiSEM 8385282a-94b7-4399-9b2e-9, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Keiding, H. & McLean, R.P. & Oortwijn, S. & Tijs, S.H., 1992. "The compromise value for NTU-games," Discussion Paper 1992-18, Tilburg University, Center for Economic Research.
Borm, P.E.M. & Keiding, H. & McLean, R.P. & Oortwijn, S. & Tijs, S.H., 1992. "The compromise value for NTU-games," Other publications TiSEM cb1df340-6f44-4cb5-ae3b-4, Tilburg University, School of Economics and Management.
van den Nouweland, C.G.A.M. & Borm, P.E.M. & Tijs, S.H., 1992. "Allocation rules for hypergraph communication situations," Other publications TiSEM b97fb9dd-2acf-470d-b9eb-a, Tilburg University, School of Economics and Management.
van den Nouweland, Anne & Borm, Peter & Tijs, Stef, 1992. "Allocation Rules for Hypergraph Communication Situations," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 255-268.
van den Nouweland, C.G.A.M. & Borm, P.E.M. & Tijs, S.H., 1992. "Allocation rules for hypergraph communication situations," Other publications TiSEM d662c517-fdc1-45ef-8908-6, Tilburg University, School of Economics and Management.
van den Nouweland, C.G.A.M. & Borm, P.E.M., 1991. "On the convexity of communication games," Other publications TiSEM e754cb6a-f695-4099-bfbf-c, Tilburg University, School of Economics and Management.
van den Nouweland, Anne & Borm, Peter, 1991. "On the Convexity of Communication Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(4), pages 421-430.
van den Nouweland, C.G.A.M. & Borm, P.E.M. & Owen, G. & Tijs, S.H., 1991. "Cost allocation and communication," Research Memorandum FEW 469, Tilburg University, School of Economics and Management.
Anne Van Den Nouweland & Peter Borm & Guillermo Owen & Stef Tijs, 1993. "Cost allocation and communication," Naval Research Logistics (NRL), John Wiley & Sons, vol. 40(5), pages 733-744, August.
van den Nouweland, C.G.A.M. & Borm, P.E.M. & Owen, G. & Tijs, S.H., 1993. "Cost allocation and communication," Other publications TiSEM 00be5459-457b-4d83-b8c7-6, Tilburg University, School of Economics and Management.
van den Nouweland, C.G.A.M. & Borm, P.E.M. & Owen, G. & Tijs, S.H., 1991. "Cost allocation and communication," Other publications TiSEM 0bbcc386-d5ca-43d2-8edf-5, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van den Nouweland, C.G.A.M. & Tijs, S.H., 1991. "Cooperation and communication restrictions : A survey," Research Memorandum FEW 507, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van den Nouweland, C.G.A.M. & Tijs, S.H., 1991. "Cooperation and communication restrictions : A survey," Other publications TiSEM 7d1c34fd-9403-4917-8b1a-1, Tilburg University, School of Economics and Management.
Borm, P.E.M. & van den Nouweland, C.G.A.M. & Tijs, S.H., 1994. "Cooperation and communication restrictions : A survey," Other publications TiSEM 513999f5-0c87-4204-a1ff-9, Tilburg University, School of Economics and Management.
Borm, P.E.M., 1990. "On game theoretic models and solution concepts," Other publications TiSEM c2fc22a4-5cb0-4517-9e53-6, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Gijsberts, A.N., 1990. "On constructing games with a convex set of equilibrium strategies," Other publications TiSEM 3466eb7c-25f3-4e77-888e-a, Tilburg University, School of Economics and Management.
Oortwijn, S. & Borm, P.E.M. & Keiding, H. & Tijs, S.H., 1990. "Extensions of the t-value to NTU-games," Research Memorandum FEW 457, Tilburg University, School of Economics and Management.
Oortwijn, S. & Borm, P.E.M. & Keiding, H. & Tijs, S.H., 1990. "Extensions of the t-value to NTU-games," Other publications TiSEM 7a8433cd-d6ab-4f8f-a04c-2, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Gijsberts, A. & Tijs, S.H., 1989. "A geometric-combinatorial approach to bimatrix games," Other publications TiSEM af7a280a-3e50-4bdf-b537-8, Tilburg University, School of Economics and Management.
Borm, P.E.M., 1988. "Information types : A comparison," Other publications TiSEM 0ddbc43d-3437-4f57-9e83-d, Tilburg University, School of Economics and Management.
Borm, P.E.M. & Tijs, S.H. & van den Aarssen, J.C.M., 1988. "Pareto equilibria in multiobjective games," Other publications TiSEM a02573c0-8c7e-409d-bc75-0, Tilburg University, School of Economics and Management.
Borm, P.E.M., 1987. "A classification of 2x2 bimatrix games," Other publications TiSEM 67810537-7b79-4f4b-b6a5-8, Tilburg University, School of Economics and Management.
Mirjam Groote Schaarsberg & Hans Reijnierse & Peter Borm, 2018. "On solving mutual liability problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(3), pages 383-409, June.
John Kleppe & Peter Borm & Ruud Hendrickx & Hans Reijnierse, 2018. "On Analyzing Cost Allocation Problems: Cooperation Building Structures and Order Problem Representations," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(04), pages 1-28, December.
E. Sánchez-Rodríguez & P. Borm & A. Estévez-Fernández & M. Fiestras-Janeiro & M. Mosquera, 2015. "$$k$$ k -core covers and the core," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(2), pages 147-167, April.
Huijink, S. & Borm, P.E.M. & Kleppe, J. & Reijnierse, J.H., 2015. "Bankruptcy and the per capita nucleolus: The claim-and-right rules family," Mathematical Social Sciences, Elsevier, vol. 77(C), pages 15-31.
Borm, Peter & Ju, Yuan & Wettstein, David, 2015. "Rational bargaining in games with coalitional externalities," Journal of Economic Theory, Elsevier, vol. 157(C), pages 236-254.
Tejada, O. & Borm, P. & Lohmann, E., 2014. "A unifying model for matrix-based pairing situations," Mathematical Social Sciences, Elsevier, vol. 72(C), pages 55-61.
Edwin Lohmann & Peter Borm & Marco Slikker, 2014. "Sequencing situations with Just-in-Time arrival, and related games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 80(3), pages 285-305, December.
Sebastián Marbán & Peter Ven & Peter Borm & Herbert Hamers, 2013. "ALOHA networks: a game-theoretic approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(2), pages 221-242, October.
John Kleppe & Peter Borm & Ruud Hendrickx, 2013. "Fall back equilibrium for $$2 \times n$$ bimatrix games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(2), pages 171-186, October.
Mirjam Groote Schaarsberg & Peter Borm & Herbert Hamers & Hans Reijnierse, 2013. "Game theoretic analysis of maximum cooperative purchasing situations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(8), pages 607-624, December.
Soesja Grundel & Peter Borm & Herbert Hamers, 2013. "Resource allocation games: a compromise stable extension of bankruptcy games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(2), pages 149-169, October.
Marco Slikker & Peter Borm & René Brink, 2012. "Internal slackening scoring methods," Theory and Decision, Springer, vol. 72(4), pages 445-462, April.
Emiliya Lazarova & Peter Borm & Maria Montero & Hans Reijnierse, 2011. "A bargaining set for monotonic simple games based on external and internal stability," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 54-66, July.
Emiliya Lazarova & Peter Borm & Bas Velzen, 2011. "Coalitional games and contracts based on individual deviations," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 507-520, December.
Tijs, Stef & Borm, Peter & Lohmann, Edwin & Quant, Marieke, 2011. "An average lexicographic value for cooperative games," European Journal of Operational Research, Elsevier, vol. 213(1), pages 210-220, August.
Marieke Quant & Peter Borm, 2011. "Random conjugates of bankruptcy rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(2), pages 249-266, February.
Çiftçi, BarIs & Borm, Peter & Hamers, Herbert, 2010. "Highway games on weakly cyclic graphs," European Journal of Operational Research, Elsevier, vol. 204(1), pages 117-124, July.
Frans Cruijssen & Peter Borm & Wout Dullaert & Herbert Hamers, 2010. "A versatile framework for cooperative hub network development," European Journal of Industrial Engineering, Inderscience Enterprises Ltd, vol. 4(2), pages 210-227.
Cruijssen, Frans & Borm, Peter & Fleuren, Hein & Hamers, Herbert, 2010. "Supplier-initiated outsourcing: A methodology to exploit synergy in transportation," European Journal of Operational Research, Elsevier, vol. 207(2), pages 763-774, December.
René Brink & Peter Borm & Ruud Hendrickx & Guillermo Owen, 2008. "Characterizations of the β- and the Degree Network Power Measure," Theory and Decision, Springer, vol. 64(4), pages 519-536, June.
Carlos González-Alcón & Peter Borm & Ruud Hendrickx & Kim Kuijk, 2007. "A taxonomy of best-reply multifunctions in 2×2×2 trimatrix games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(2), pages 297-306, December.
Marieke Quant & Peter Borm & Hans Reijnierse & Mark Voorneveld, 2006. "A note on theβ-measure for digraph competitions," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 167-176, June.
Calleja, Pedro & Borm, Peter & Hendrickx, Ruud, 2005. "Multi-issue allocation situations," European Journal of Operational Research, Elsevier, vol. 164(3), pages 730-747, August.
Peter Borm & Henk Norde & Ignacio Garcia-Jurado & Floravante Patrone, 2004. "Preface," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 59(3), pages 347-348, July.
Peter Borm & Fioravante Patrone & Joaquín Sánchez-Soriano, 2008. "Preface," Annals of Operations Research, Springer, vol. 158(1), pages 1-3, February.
E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2001. "The Myerson value for union stable structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 54(3), pages 359-371, December.
Fiestras-Janeiro, Gloria & Borm, Peter & van Megen, Freek, 1998. "Protective and Prudent Behaviour in Games," Journal of Economic Theory, Elsevier, vol. 78(1), pages 167-175, January.
Suijs, Jeroen & De Waegenaere, Anja & Borm, Peter, 1998. "Stochastic cooperative games in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 209-228, July.
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NEP-GTH: Game Theory (32) 2006-05-27 2006-11-25 2010-12-04 2011-01-16 2014-11-12 2014-12-24 2014-12-29 2015-01-03 2015-01-03 2015-03-22 2015-04-25 2015-04-25 2015-04-25 2015-05-09 2015-07-04 2015-11-15 2015-12-28 2016-07-30 2016-08-07 2016-11-13 2016-12-18 2016-12-18 2017-05-21 2017-11-19 2018-01-08 2018-02-12 2018-03-26 2018-04-16 2018-07-30 2019-04-08 2019-12-16 2020-12-14. Author is listed
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NEP-NET: Network Economics (4) 2014-12-24 2015-04-25 2015-12-28 2016-07-30
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NEP-ICT: Information & Communication Technologies (1) 2014-12-24
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Geometric conditions for the existence of a rolling without twisting or slipping
January 2014, 13(1): 453-481. doi: 10.3934/cpaa.2014.13.453
Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations
Giovanni De Matteis 1, and Gianni Manno 2,
Department of Mathematics and Information Sciences, Northumbria University, Pandon Building, Camden Street, Newcastle upon Tyne, NE2 1XE, United Kingdom
INdAM-COFUND Marie Curie Fellow, Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07737, Germany
Received April 2013 Revised April 2013 Published July 2013
We compute the Lie symmetry algebra of the equation of Helfrich surfaces and we show that it is the algebra of conformal vector fields of $R^2$. We also show that in the particular case of the Willmore surfaces we have to add the homothety vector field of $R^3$ to the aforementioned algebra. We prove that a Helfrich surface that is invariant w.r.t. a conformal symmetry is a helicoid and that all such surface solutions satisfy one and the same system of ordinary differential equations obtained by symmetry reduction. We also show that for the Willmore surface shape equation the symmetry reduction leads to two systems of ODEs. Then we construct explicit solutions in the case of revolution surfaces. The results obtained can be extended to the study of PDE problems in $2$ spatial dimensions admitting conformal Lie symmetries.
Keywords: vesicle shape, invariant solutions, conformal vector fields, symmetry reduction, Helfrich-Canham bending energy, membrane systems., Symmetries of PDEs, Willmore surfaces.
Mathematics Subject Classification: 35B06, 35B07, 35Q92, 53A05, 58A2.
Citation: Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
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Giovanni De Matteis Gianni Manno | CommonCrawl |
An investment of $\$$10,000 is made in a government bond that will pay 6$\%$ interest compounded annually. At the end of five years, what is the total number of dollars in this investment? Express your answer to the nearest whole number.
After five years, at a six percent annual interest rate, the investment will have grown to $10000 \cdot 1.06^5 = \boxed{13382}$, to the nearest dollar. | Math Dataset |
\begin{definition}[Definition:Unit Normal]
Let $S$ be a surface in ordinary $3$-space.
Let $P$ be a point of $S$.
Let $\mathbf n$ be a normal to $S$ at $P$ such that $\mathbf n$ is a unit vector.
Then $\mathbf n$ is known as a '''unit normal'''.
\end{definition} | ProofWiki |
# Linear algebra basics
A vector is a quantity that has both magnitude and direction. In linear algebra, vectors are represented as column matrices. For example, a 2-dimensional vector can be represented as a 2x1 matrix:
$$
\begin{bmatrix}
x \\
y
\end{bmatrix}
$$
where $x$ and $y$ are the coordinates of the vector.
A scalar is a number that can be multiplied with a vector to obtain another vector. In linear algebra, scalars are represented as numbers. For example, if we have a vector $v$ and a scalar $\alpha$, we can multiply them to obtain a new vector $\alpha v$:
$$
\alpha v = \begin{bmatrix}
\alpha x \\
\alpha y
\end{bmatrix}
$$
Two vectors are said to be linearly dependent if one vector is a scalar multiple of the other. For example, the vectors $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \end{bmatrix}$ are linearly dependent because the second vector is twice the first vector.
Two vectors are said to be linearly independent if they are not scalar multiples of each other. For example, the vectors $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ are linearly independent because one vector is not a scalar multiple of the other.
A basis is a set of linearly independent vectors that span a vector space. For example, the vectors $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$ form a basis for the 2-dimensional vector space.
## Exercise
Consider the vectors $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \end{bmatrix}$. Determine if they are linearly dependent or independent.
# Creating and initializing multidimensional arrays
To create a 2-dimensional array, you can use the `Array2DRowRealMatrix` class. For example, to create a 2x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(2, 3);
```
To initialize the matrix with values, you can use the `setEntry` method. For example, to set the value at the first row and second column to 5, you can write:
```java
matrix.setEntry(0, 1, 5);
```
To create a 3-dimensional array, you can use the `Array3DRowRealMatrix` class. For example, to create a 2x3x4 matrix, you can write:
```java
RealMatrix matrix = new Array3DRowRealMatrix(2, 3, 4);
```
## Exercise
Create a 3x2 matrix using Apache Commons Math and initialize it with the following values:
$$
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
5 & 6
\end{bmatrix}
$$
# Matrix operations: addition, subtraction, and multiplication
To add two matrices, you can use the `add` method. For example, to add two 2x2 matrices, you can write:
```java
RealMatrix matrix1 = new Array2DRowRealMatrix(2, 2);
RealMatrix matrix2 = new Array2DRowRealMatrix(2, 2);
RealMatrix result = matrix1.add(matrix2);
```
To subtract two matrices, you can use the `subtract` method. For example, to subtract two 2x2 matrices, you can write:
```java
RealMatrix matrix1 = new Array2DRowRealMatrix(2, 2);
RealMatrix matrix2 = new Array2DRowRealMatrix(2, 2);
RealMatrix result = matrix1.subtract(matrix2);
```
To multiply two matrices, you can use the `multiply` method. For example, to multiply a 2x3 matrix by a 3x2 matrix, you can write:
```java
RealMatrix matrix1 = new Array2DRowRealMatrix(2, 3);
RealMatrix matrix2 = new Array2DRowRealMatrix(3, 2);
RealMatrix result = matrix1.multiply(matrix2);
```
## Exercise
Add the following two 2x2 matrices:
$$
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
$$
and
$$
\begin{bmatrix}
5 & 6 \\
7 & 8
\end{bmatrix}
$$
# Matrix operations: transposition, inversion, and determinant
To find the transpose of a matrix, you can use the `transpose` method. For example, to find the transpose of a 2x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(2, 3);
RealMatrix transpose = matrix.transpose();
```
To find the inverse of a matrix, you can use the `inverse` method. For example, to find the inverse of a 3x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(3, 3);
RealMatrix inverse = matrix.inverse();
```
To find the determinant of a matrix, you can use the `det` method. For example, to find the determinant of a 3x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(3, 3);
double determinant = matrix.det();
```
## Exercise
Find the transpose of the following matrix:
$$
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
$$
# Matrix operations: solving linear systems
To solve a linear system of equations, you can use the `solve` method. For example, to solve the linear system $Ax = b$, where $A$ is a 2x2 matrix and $b$ is a 2x1 matrix, you can write:
```java
RealMatrix matrixA = new Array2DRowRealMatrix(2, 2);
RealMatrix matrixB = new Array2DRowRealMatrix(2, 1);
RealMatrix solution = matrixA.solve(matrixB);
```
## Exercise
Solve the linear system $Ax = b$, where:
$$
A = \begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
$$
and
$$
b = \begin{bmatrix}
5 \\
7
\end{bmatrix}
$$
# Statistical operations on matrices: mean, variance, and covariance
To find the mean of a matrix, you can use the `getColumnMeans` method. For example, to find the mean of a 2x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(2, 3);
double[] mean = matrix.getColumnMeans();
```
To find the variance of a matrix, you can use the `getColumnVariances` method. For example, to find the variance of a 2x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(2, 3);
double[] variance = matrix.getColumnVariances();
```
To find the covariance of a matrix, you can use the `getCovariance` method. For example, to find the covariance of a 2x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(2, 3);
double[][] covariance = matrix.getCovariance();
```
## Exercise
Find the mean, variance, and covariance of the following matrix:
$$
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
$$
# Advanced matrix operations: eigendecomposition and singular value decomposition
To perform an eigendecomposition on a matrix, you can use the `getEigenDecomposition` method. For example, to perform an eigendecomposition on a 3x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(3, 3);
EigenDecomposition eigenDecomposition = matrix.getEigenDecomposition();
```
To perform a singular value decomposition on a matrix, you can use the `getSVD` method. For example, to perform a singular value decomposition on a 3x3 matrix, you can write:
```java
RealMatrix matrix = new Array2DRowRealMatrix(3, 3);
SingularValueDecomposition svd = matrix.getSVD();
```
## Exercise
Perform an eigendecomposition and a singular value decomposition on the following matrix:
$$
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{bmatrix}
$$
# Optimization problems and their solutions using linear algebra
Linear programming is a method for solving optimization problems with linear constraints. You can use the `LinearProgram` class in Apache Commons Math to solve linear programming problems.
Quadratic programming is a method for solving optimization problems with quadratic constraints. You can use the `QuadraticProgram` class in Apache Commons Math to solve quadratic programming problems.
## Exercise
Solve the following linear programming problem using Apache Commons Math:
Minimize $f(x) = 3x_1 + 2x_2$
subject to:
$x_1 + x_2 \le 4$
$2x_1 + x_2 \le 6$
$x_1, x_2 \ge 0$
# Applications of multidimensional array manipulation in data analysis
Principal component analysis (PCA) is a technique for reducing the dimensionality of a dataset. You can use the `PCA` class in Apache Commons Math to perform PCA on a dataset.
Linear regression is a technique for modeling the relationship between a dependent variable and one or more independent variables. You can use the `SimpleRegression` class in Apache Commons Math to perform linear regression on a dataset.
## Exercise
Perform PCA on the following dataset:
$$
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
5 & 6
\end{bmatrix}
$$
# Summary and conclusion
In this textbook, we have covered the basics of linear algebra, creating and initializing multidimensional arrays, performing matrix operations, solving linear systems, performing statistical operations on matrices, performing advanced matrix operations, solving optimization problems using linear algebra, and applying multidimensional array manipulation techniques in data analysis.
By the end of this textbook, you should have a strong foundation in multidimensional array manipulation using Apache Commons Math. You should be able to apply these techniques to a variety of problems in computer science, data analysis, and machine learning. | Textbooks |
Longest path problem
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges. In contrast to the shortest path problem, which can be solved in polynomial time in graphs without negative-weight cycles, the longest path problem is NP-hard and the decision version of the problem, which asks whether a path exists of at least some given length, is NP-complete. This means that the decision problem cannot be solved in polynomial time for arbitrary graphs unless P = NP. Stronger hardness results are also known showing that it is difficult to approximate. However, it has a linear time solution for directed acyclic graphs, which has important applications in finding the critical path in scheduling problems.
NP-hardness
The NP-hardness of the unweighted longest path problem can be shown using a reduction from the Hamiltonian path problem: a graph G has a Hamiltonian path if and only if its longest path has length n − 1, where n is the number of vertices in G. Because the Hamiltonian path problem is NP-complete, this reduction shows that the decision version of the longest path problem is also NP-complete. In this decision problem, the input is a graph G and a number k; the desired output is yes if G contains a path of k or more edges, and no otherwise.[1]
If the longest path problem could be solved in polynomial time, it could be used to solve this decision problem, by finding a longest path and then comparing its length to the number k. Therefore, the longest path problem is NP-hard. The question "does there exist a simple path in a given graph with at least k edges" is NP-complete.[2]
In weighted complete graphs with non-negative edge weights, the weighted longest path problem is the same as the Travelling salesman path problem, because the longest path always includes all vertices.[3]
Acyclic graphs
A longest path between two given vertices s and t in a weighted graph G is the same thing as a shortest path in a graph −G derived from G by changing every weight to its negation. Therefore, if shortest paths can be found in −G, then longest paths can also be found in G.[4]
For most graphs, this transformation is not useful because it creates cycles of negative length in −G. But if G is a directed acyclic graph (DAG), then no negative cycles can be created, and a longest path in G can be found in linear time by applying a linear time algorithm for shortest paths in −G, which is also a directed acyclic graph.[4] For a DAG, the longest path from a source vertex to all other vertices can be obtained by running the shortest-path algorithm on −G.
Similarly, for each vertex v in a given DAG, the length of the longest path ending at v may be obtained by the following steps:
1. Find a topological ordering of the given DAG.
2. For each vertex v of the DAG, in the topological ordering, compute the length of the longest path ending at v by looking at its incoming neighbors and adding one to the maximum length recorded for those neighbors. If v has no incoming neighbors, set the length of the longest path ending at v to zero. In either case, record this number so that later steps of the algorithm can access it.
Once this has been done, the longest path in the whole DAG may be obtained by starting at the vertex v with the largest recorded value, then repeatedly stepping backwards to its incoming neighbor with the largest recorded value, and reversing the sequence of vertices found in this way.
This is equivalent to running the shortest-path algorithm on −G.
Critical paths
The critical path method for scheduling a set of activities involves the construction of a directed acyclic graph in which the vertices represent project milestones and the edges represent activities that must be performed after one milestone and before another; each edge is weighted by an estimate of the amount of time the corresponding activity will take to complete. In such a graph, the longest path from the first milestone to the last one is the critical path, which describes the total time for completing the project.[4]
Longest paths of directed acyclic graphs may also be applied in layered graph drawing: assigning each vertex v of a directed acyclic graph G to the layer whose number is the length of the longest path ending at v results in a layer assignment for G with the minimum possible number of layers.[5]
Approximation
Björklund, Husfeldt & Khanna (2004) write that the longest path problem in unweighted undirected graphs "is notorious for the difficulty of understanding its approximation hardness".[6] The best polynomial time approximation algorithm known for this case achieves only a very weak approximation ratio, $n/\exp(\Omega ({\sqrt {\log n}}))$.[7] For all $\epsilon >0$, it is not possible to approximate the longest path to within a factor of $2^{(\log n)^{1-\epsilon }}$ unless NP is contained within quasi-polynomial deterministic time; however, there is a big gap between this inapproximability result and the known approximation algorithms for this problem.[8]
In the case of unweighted but directed graphs, strong inapproximability results are known. For every $\epsilon >0$ the problem cannot be approximated to within a factor of $n^{1-\epsilon }$ unless P = NP, and with stronger complexity-theoretic assumptions it cannot be approximated to within a factor of $n/\log ^{2+\epsilon }n$.[6] The color-coding technique can be used to find paths of logarithmic length, if they exist, but this gives an approximation ratio of only $O(n/\log n)$.[9]
Parameterized complexity
The longest path problem is fixed-parameter tractable when parameterized by the length of the path. For instance, it can be solved in time linear in the size of the input graph (but exponential in the length of the path), by an algorithm that performs the following steps:
1. Perform a depth-first search of the graph. Let $d$ be the depth of the resulting depth-first search tree.
2. Use the sequence of root-to-leaf paths of the depth-first search tree, in the order in which they were traversed by the search, to construct a path decomposition of the graph, with pathwidth $d$.
3. Apply dynamic programming to this path decomposition to find a longest path in time $O(d!2^{d}n)$, where $n$ is the number of vertices in the graph.
Since the output path has length at least as large as $d$, the running time is also bounded by $O(\ell !2^{\ell }n)$, where $\ell $ is the length of the longest path.[10] Using color-coding, the dependence on path length can be reduced to singly exponential.[9][11][12][13] A similar dynamic programming technique shows that the longest path problem is also fixed-parameter tractable when parameterized by the treewidth of the graph.
For graphs of bounded clique-width, the longest path can also be solved by a polynomial time dynamic programming algorithm. However, the exponent of the polynomial depends on the clique-width of the graph, so this algorithms is not fixed-parameter tractable. The longest path problem, parameterized by clique-width, is hard for the parameterized complexity class $W[1]$, showing that a fixed-parameter tractable algorithm is unlikely to exist.[14]
Special classes of graphs
A linear-time algorithm for finding a longest path in a tree was proposed by Dijkstra in 1960's, while a formal proof of this algorithm was published in 2002.[15] Furthermore, a longest path can be computed in polynomial time on weighted trees, on block graphs, on cacti,[16] on bipartite permutation graphs,[17] and on Ptolemaic graphs.[18]
For the class of interval graphs, an $O(n^{4})$-time algorithm is known, which uses a dynamic programming approach.[19] This dynamic programming approach has been exploited to obtain polynomial-time algorithms on the greater classes of circular-arc graphs[20] and of co-comparability graphs (i.e. of the complements of comparability graphs, which also contain permutation graphs),[21] both having the same running time $O(n^{4})$. The latter algorithm is based on special properties of the lexicographic depth first search (LDFS) vertex ordering[22] of co-comparability graphs. For co-comparability graphs also an alternative polynomial-time algorithm with higher running time $O(n^{7})$ is known, which is based on the Hasse diagram of the partially ordered set defined by the complement of the input co-comparability graph.[23]
Furthermore, the longest path problem is solvable in polynomial time on any class of graphs with bounded treewidth or bounded clique-width, such as the distance-hereditary graphs. Finally, it is clearly NP-hard on all graph classes on which the Hamiltonian path problem is NP-hard, such as on split graphs, circle graphs, and planar graphs.
A simple model of a directed acyclic graph is the Price model, developed by Derek J. de Solla Price to represent citation networks. This is simple enough to allow for analytic results to be found for some properties. For instance, the length of the longest path, from the n-th node added to the network to the first node in the network, scales as[24] $\ln(n)$.
See also
• Gallai–Hasse–Roy–Vitaver theorem, a duality relation between longest paths and graph coloring
• Longest uncrossed knight's path
• Snake-in-the-box, the longest induced path in a hypercube graph
• Price's model, a simple citation network model where the longest path lengths can be found analytically
References
1. Schrijver, Alexander (2003), Combinatorial Optimization: Polyhedra and Efficiency, Volume 1, Algorithms and Combinatorics, vol. 24, Springer, p. 114, ISBN 9783540443896.
2. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), Introduction To Algorithms (2nd ed.), MIT Press, p. 978, ISBN 9780262032933.
3. Lawler, Eugene L. (2001), Combinatorial Optimization: Networks and Matroids, Courier Dover Publications, p. 64, ISBN 9780486414539.
4. Sedgewick, Robert; Wayne, Kevin Daniel (2011), Algorithms (4th ed.), Addison-Wesley Professional, pp. 661–666, ISBN 9780321573513.
5. Di Battista, Giuseppe; Eades, Peter; Tamassia, Roberto; Tollis, Ioannis G. (1998), "Layered Drawings of Digraphs", Graph Drawing: Algorithms for the Visualization of Graphs, Prentice Hall, pp. 265–302, ISBN 978-0-13-301615-4.
6. Björklund, Andreas; Husfeldt, Thore; Khanna, Sanjeev (2004), "Approximating longest directed paths and cycles", Proc. Int. Coll. Automata, Languages and Programming (ICALP 2004), Lecture Notes in Computer Science, vol. 3142, Berlin: Springer-Verlag, pp. 222–233, MR 2160935.
7. Gabow, Harold N.; Nie, Shuxin (2008), "Finding long paths, cycles and circuits", International Symposium on Algorithms and Computation, Lecture Notes in Computer Science, vol. 5369, Berlin: Springer, pp. 752–763, doi:10.1007/978-3-540-92182-0_66, ISBN 978-3-540-92181-3, MR 2539968. For earlier work with even weaker approximation bounds, see Gabow, Harold N. (2007), "Finding paths and cycles of superpolylogarithmic length" (PDF), SIAM Journal on Computing, 36 (6): 1648–1671, doi:10.1137/S0097539704445366, MR 2299418 and Björklund, Andreas; Husfeldt, Thore (2003), "Finding a path of superlogarithmic length", SIAM Journal on Computing, 32 (6): 1395–1402, doi:10.1137/S0097539702416761, MR 2034242.
8. Karger, David; Motwani, Rajeev; Ramkumar, G. D. S. (1997), "On approximating the longest path in a graph", Algorithmica, 18 (1): 82–98, doi:10.1007/BF02523689, MR 1432030, S2CID 3241830.
9. Alon, Noga; Yuster, Raphael; Zwick, Uri (1995), "Color-coding", Journal of the ACM, 42 (4): 844–856, doi:10.1145/210332.210337, MR 1411787, S2CID 208936467.
10. Bodlaender, Hans L. (1993), "On linear time minor tests with depth-first search", Journal of Algorithms, 14 (1): 1–23, doi:10.1006/jagm.1993.1001, MR 1199244. For an earlier FPT algorithm with slightly better dependence on the path length, but worse dependence on the size of the graph, see Monien, B. (1985), "How to find long paths efficiently", Analysis and design of algorithms for combinatorial problems (Udine, 1982), North-Holland Math. Stud., vol. 109, Amsterdam: North-Holland, pp. 239–254, doi:10.1016/S0304-0208(08)73110-4, ISBN 9780444876997, MR 0808004.
11. Chen, Jianer; Lu, Songjian; Sze, Sing-Hoi; Zhang, Fenghui (2007), "Improved algorithms for path, matching, and packing problems", Proc. 18th ACM-SIAM Symposium on Discrete algorithms (SODA '07) (PDF), pp. 298–307.
12. Koutis, Ioannis (2008), "Faster algebraic algorithms for path and packing problems", International Colloquium on Automata, Languages and Programming (PDF), Lecture Notes in Computer Science, vol. 5125, Berlin: Springer, pp. 575–586, CiteSeerX 10.1.1.141.6899, doi:10.1007/978-3-540-70575-8_47, ISBN 978-3-540-70574-1, MR 2500302, archived from the original (PDF) on 2017-08-09, retrieved 2013-08-09.
13. Williams, Ryan (2009), "Finding paths of length k in O*(2k) time", Information Processing Letters, 109 (6): 315–318, arXiv:0807.3026, doi:10.1016/j.ipl.2008.11.004, MR 2493730, S2CID 10295448.
14. Fomin, Fedor V.; Golovach, Petr A.; Lokshtanov, Daniel; Saurabh, Saket (2009), "Clique-width: on the price of generality", Proc. 20th ACM-SIAM Symposium on Discrete Algorithms (SODA '09) (PDF), pp. 825–834, archived from the original (PDF) on 2012-10-18, retrieved 2012-12-01.
15. Bulterman, R.W.; van der Sommen, F.W.; Zwaan, G.; Verhoeff, T.; van Gasteren, A.J.M. (2002), "On computing a longest path in a tree", Information Processing Letters, 81 (2): 93–96, doi:10.1016/S0020-0190(01)00198-3.
16. Uehara, Ryuhei; Uno, Yushi (2004), "Efficient algorithms for the longest path problem", Isaac 2004, Lecture Notes in Computer Science, 3341: 871–883, doi:10.1007/978-3-540-30551-4_74, ISBN 978-3-540-24131-7.
17. Uehara, Ryuhei; Valiente, Gabriel (2007), "Linear structure of bipartite permutation graphs and the longest path problem", Information Processing Letters, 103 (2): 71–77, CiteSeerX 10.1.1.101.96, doi:10.1016/j.ipl.2007.02.010.
18. Takahara, Yoshihiro; Teramoto, Sachio; Uehara, Ryuhei (2008), "Longest path problems on Ptolemaic graphs", IEICE Transactions, 91-D (2): 170–177, doi:10.1093/ietisy/e91-d.2.170.
19. Ioannidou, Kyriaki; Mertzios, George B.; Nikolopoulos, Stavros D. (2011), "The longest path problem has a polynomial solution on interval graphs", Algorithmica, 61 (2): 320–341, CiteSeerX 10.1.1.224.4927, doi:10.1007/s00453-010-9411-3, S2CID 7577817.
20. Mertzios, George B.; Bezakova, Ivona (2014), "Computing and counting longest paths on circular-arc graphs in polynomial time", Discrete Applied Mathematics, 164 (2): 383–399, CiteSeerX 10.1.1.224.779, doi:10.1016/j.dam.2012.08.024.
21. Mertzios, George B.; Corneil, Derek G. (2012), "A simple polynomial algorithm for the longest path problem on cocomparability graphs", SIAM Journal on Discrete Mathematics, 26 (3): 940–963, arXiv:1004.4560, doi:10.1137/100793529, S2CID 4645245.
22. Corneil, Derek G.; Krueger, Richard (2008), "A unified view of graph searching", SIAM Journal on Discrete Mathematics, 22 (4): 1259–1276, doi:10.1137/050623498.
23. Ioannidou, Kyriaki; Nikolopoulos, Stavros D. (2011), "The longest path problem is polynomial on cocomparability graphs" (PDF), Algorithmica, 65: 177–205, CiteSeerX 10.1.1.415.9996, doi:10.1007/s00453-011-9583-5, S2CID 7271040.
24. Evans, T.S.; Calmon, L.; Vasiliauskaite, V. (2020), "The Longest Path in the Price Model", Scientific Reports, 10 (1): 10503, arXiv:1903.03667, Bibcode:2020NatSR..1010503E, doi:10.1038/s41598-020-67421-8, PMC 7324613, PMID 32601403
External links
• "Find the Longest Path", song by Dan Barrett
| Wikipedia |
Use of SHDM in commutative watermarking encryption
Roland Schmitz ORCID: orcid.org/0000-0003-3848-52881
This article has been updated
SHDM stands for Sphere-Hardening Dither Modulation and is a watermarking algorithm based on quantizing the norm of a vector extracted from the cover work. We show how SHDM can be integrated into a fully commutative watermarking-encryption scheme and investigate implementations in the spatial, DCT, and DWT domain with respect to their fidelity, robustness, capacity, and security of encryption. The watermarking scheme, when applied in the DCT or DWT domain, proves to be very robust against JPEG/JPEG2000 compression. On the other hand, the spatial domain-based approach offers a large capacity. The increased robustness of the watermarking schemes, however, comes at the cost of rather weak encryption primitives, making the proposed CWE scheme suited for low to medium security applications with high robustness requirements.
Encryption and watermarking are both important tools in protecting digital contents, e.g., in digital rights management (DRM) systems. While encryption is used to protect the contents from unauthorized access, watermarking can be deployed for various purposes, ranging from ensuring authenticity of content to embedding metadata, e.g., copyright or authorship information, into the contents. In the DRM context, for example, clients need to have the ability to decrypt the contents and may thus eventually misuse the ciphertext contents. The protection provided by digital watermarks, however, remains within the contents, and can serve to identify misbehaving clients.
In a buyer-seller scenario where the content owner does not trust the seller to sell copies of her own, the content owner can supply the seller with an encrypted version of the content, which is in turn individually watermarked for each buyer by the seller. In such a situation, it is important that a watermark can be embedded in the encrypted domain and detected in the cleartext domain.
Another motivation to consider watermarking in the encrypted domain are the increasing needs generated by cloud computing platforms and various privacy preserving applications.
In [1], four requirements on watermarking in the encrypted domain are formulated:
Property 1. The marking function \(\mathcal {M}\) can be performed on an encrypted image.
Property 2. The verification function \(\mathcal {V}\) is able to reconstruct a mark in the encrypted domain when it has been embedded in the encrypted domain.
Property 3. The verification function \(\mathcal {V}\) is able to reconstruct a mark in the encrypted domain when it has been embedded in the clear domain.
Property 4. The decryption function does not affect the integrity of the watermark.
As is pointed out in [1], properties 2 and 3 are equivalent, if the encryption function \(\mathcal E\) and the marking function \(\mathcal M\) commute, that is,
$$ \mathcal M({\mathcal E}_{K}(I),m) = {\mathcal E}_{K}(\mathcal M(I,m)) $$
where \({\mathcal E}\) is the encryption function, K is the encryption key, I is the plaintext media data, and m is the mark to be embedded.
In recent years, a number of commutative water-marking-encryption (CWE) schemes have been formulated. The present paper describes a novel CWE scheme, that couples sign-bit encryption of selected pixel grey values or transform coefficients as encryption part with Sphere-Hardening Dither Modulation (SHDM) [2, 3] as the watermarking part. The encryption part can optionally be enhanced by permuting the pixels or transform coefficients, respectively. While the idea of encrypting coefficient sign-bits within a CWE scheme is not new, the use of SHDM as watermarking part is, leading to a more robust scheme than previous approaches.
The rest of the paper is organized as follows: previous CWE proposals are summarized in Section 2. SHDM is briefly reviewed in Section 3. In Section 4, we describe implementations of the proposed CWE scheme in the spatial, DCT, and DWT domain, respectively, and discuss their security. Section 5 provides experimental results on the robustness of the watermarking part in the three implementation domains, and Section 6 discusses the security aspects of the CWE schemes in terms of cryptographic and watermarking security. Section 7 concludes the paper.
There are currently three basic approaches to commutative watermarking encryption for raw image data. The first approach, called partial encryption, divides the image data into two parts and encrypts one of them (typically the perceptually more important part) and watermarks the other part. Thus, encryption part and watermarking part are completely independent and do not interfere with each other. First, important examples in this vein are provided by [4] and [5]. In [5], the basic idea is to encrypt DCT sign bits and to watermark their absolute values by means of dithered modulation. Similarly, in [4], the data are partitioned into two parts after a four-level discrete wavelet transformation. The low-level coefficients are fully encrypted, while in the medium- and high-level coefficients only the signs are encrypted and their absolute values are watermarked. In [6], encryption and watermarking happens within a secret transform domain, the Tree-Structured-Haar (TSH) Transform, which involves a secret parameter. Both the watermark embedder and encryptor need to share knowledge about the secret parameter generating the transform domain. The transform coefficients are first quantized to get B bitplanes. The N most significant bitplanes are encrypted, and B−N−1 of the remaining bitplanes are watermarked. The least significant bitplane is replaced by the signs of the plaintext coefficients. In this approach, using a secret transform domain increases the security of the scheme, albeit at the cost that encryption and watermarking are not completely independent, but need to share a common secret.
Another approach to commutative watermarking is provided by deploying homomorphic encryption techniques so that some basic algebraic operations such as addition and multiplication on the plaintexts can be transferred onto the corresponding ciphertexts, i.e., they are transparent to encryption [1, Sec. 2.1]. Especially, if both the encryption and the watermarking process consist of the same homomorphic operation, one gets a commutative watermarking encryption scheme. Examples of homomorphic operations are exponentiation modulo n, multiplication modulo n, and addition modulo n (including the bitwise XOR operation). One major drawback of this approach is the influence of encryption on robustness of the watermarking algorithm: after strong encryption, there is no visual information available for the watermark embedder to adapt itself to in order to increase robustness while at the same time minimizing visual quality degradation [7, Sec. 9.4]. Another drawback is that the homomorphic watermarking operation can seriously affect the fidelity. In [8], for example, addition modulo n is used, where n is the number of grey values. However, the modular addition operation may cause overflow/underflow pixels that are handled in a preprocessing step during the encryption operation, making the system only "quasi-commutative."
Third, in invariant encryption schemes, the data are fully encrypted, but the encryption operation leaves a certain subspace of the data invariant, which may be used for watermarking. In [9], a permutation cipher is applied to the image, leaving the histogram of grey values invariant. The watermark is embedded by manipulating the histogram. Depending on viewpoint, schemes based on encrypting sign bits of transform coefficients may also be seen as invariant encryption schemes, as the absolute values of the coefficients form an invariant subspace.
In a another line of work, researchers concentrate their efforts on watermarking and encrypting the bitstream after encoding the data according to a certain standard. In [10], a commutative watermarking encryption scheme based on encrypting the intra-prediction modes and the sign bits of the DCT coefficients within the H.264 bitstream is presented. Here, watermarking of residual DCT coefficients is based on Quantized Index Modulation (QIM) [11]. In [12], in order to achieve the commutative property, one set of syntax elements within the HEVC bitstream is utilized for data hiding, while another set is exploited for encryption. In [13] and [14], the JPEG-LS bitstream is jointly watermarked and encrypted using the AES algorithm in Cipher Block Chaining (CBC)-mode, making this scheme suitable for scenarios with high security requirements, like in medical imaging.
SHDM
Sphere-Hardening Dither Modulation, or SHDM for short, was proposed by Balado in [2] and [3] as an alternative to STDM (Spread-Transform Dither Modulation), which was proposed in [11]. Both SHDM and STDM have in common that in order to embed a single bit b, a multidimensional host vector \(\vec x\) is extracted from the cover work C0 and modified using some dithered quantization function Qb, where different message bits lead to different dither values. While in STDM the projection \({\vec x^{t} }\cdot \vec u\) of the host vector \(\vec x\) onto some random vector \(\vec u\) is quantized, in SHDM, the norm \(\Vert \vec x \Vert \) is quantized.
More specifically, the embedding rule in SHDM is given by
$$ \vec y = Q_{b}(\Vert x \Vert,\Delta,d) \cdot {\vec x \over \Vert x \Vert}, $$
where for b∈{0,1},
$$ Q_{b}(\Vert x \Vert,\Delta,d) = \Delta \cdot \lfloor{ {\Vert x\Vert - d - b\Delta/2 \over \Delta} \rfloor} + d + b \cdot \Delta/2 $$
is the quantizing function. Extraction of the embedded bit is done via
$$ b = \arg \min_{b \in \{0,1\}} \vert \Vert \tilde{\vec y} \Vert - Q_{b}\left(\Vert \tilde{\vec y} \Vert,\Delta,d\right) \vert, $$
where \(\tilde {\vec y}\) is the disturbed signal vector at the detector site.
Note that the direction of the signal vector \(\vec x\) is not changed by embedding, which is advantageous from a perceptual point of view, as opposed to related methods like STDM. As is shown in [2], SHDM offers the same level of robustness against additive white noise as STDM.
Using SHDM in a CWE scheme
In SHDM, the signal vector \(\vec x = (x_{1},\dots,x_{N})\) may be extracted from the host in an arbitrary fashion. In this section, we implement SHDM in the spatial (pixel) domain, the DCT domain, and the DWT domain and combine it with matching encryption schemes. As in SHDM the vector norm
$$ \Vert \vec x \Vert = \sqrt{x_{1}^{2} + x_{2}^{2} + \dots x_{N}^{2}}, $$
is quantized, we have the following options for encryption:
Encrypt the sign bits of the xi by means of a stream cipher.
Permute the xi.
Apply some other norm-preserving operation on \(\vec x\), e. g. a random rotation.
Of course, the options may be combined. In the rest of the paper, we will only explore the first two options.
Implementation in the spatial domain
In the spatial domain, we work directly with pixel grey values ranging initially between 0 and 255. In order to create sign bits, we subtract 128 from each grey value, so that the new range is −128≤0≤127.
Watermarking part
The watermarking part uses the following parameters:
The watermarking key WK.
The watermark \(W = (b_{1},\dots,b_{n})\) to be embedded.
The dimension of the host vectors \(\vec x_{i}\), i.e., the number of coefficients N into which one bit bi is embedded.
The quantizing step Δ.
In the spatial domain, we assume that the watermarking key consists of two parts: \(W_{K} = \left (W_{K}^{(1)},W_{K}^{(2)}\right)\). After choosing a step size Δ and N, the embedding process consists of the following steps:
For each bit bi to be embedded, randomly select N pixels. The selection is controlled by \(W_{K}^{(1)}\). The corresponding grey values form the signal vector \(\vec x_{i}\).
Randomly generate a dither value di, controlled by \(W_{K}^{(2)}\).
Quantize the norm of \(\vec x_{i}\) according to bi:
$$ \Vert \vec x_{i} \Vert_{q} = Q_{b_{i}}\left(\Vert \vec x_{i} \Vert, \Delta, d_{i}\right) $$
Embed the mark into \(\vec x_{i}\) by changing its norm to \(\Vert \vec x_{i} \Vert _{q}\):
$$ \vec y_{i} = \Vert \vec x_{i} \Vert_{q} \cdot {\vec x_{i} \over \Vert \vec x_{i} \Vert} $$
and replace the grey values of pixels corresponding to components of \(\vec x_{i}\) by the corresponding entries in \(\vec y_{i}\).
For extraction of bit bi, the disturbed signal vector \(\tilde {\vec y}_{i}\) is formed from the marked image CW in the same way as \(\vec x_{i}\) was generated from the host image C0. The norm of \(\tilde {\vec y}_{i}\) is quantized and bi is computed according to
$$ b_{i} = \arg \min_{b_{i} \in \{0,1\}} \vert \Vert \tilde{\vec y}_{i} \Vert - Q_{b_{i}}\left(\Vert \tilde{\vec y}_{i} \Vert,\Delta,d_{i}\right) \vert. $$
Figure 1 shows two embedding examples with different resolutions (512×512 and 800×1600, respectively), where a random 64-bit watermark was embedded using a quantization step size of Δ=75.
Embedding 64 bits in the spatial domain. a PSNR = 57.41 dB. b PSNR = 79.16 dB
Encryption part
Images in the spatial domain with grey values ranging between 0 and 255 can be represented by eight so-called bitplanes, where the most significant bitplane (MSB) indicates whether the grey value of a certain pixel is greater than 127 or not. Thus, after subtracting 128 from every grey value, the MSB indicates the sign of the grey values. Sign bit encryption is therefore equivalent to encrypting the MSB by means of a stream cipher.
The security of encrypting the MSB of an image has been investigated in [15]. Not only is the amount of image quality degradation insufficient for most applications, it is also possible to estimate the encrypt sign bits based on the assumption that neighboring grey values in a natural image do not change abruptly. Both problems can be remedied if a permutation cipher is applied on the pixels in addition. However, the permutation must not mix the N pixels used for embedding bi with the N pixels used for embedding a different bit bj. Therefore, the permutation cipher and the watermark embedder must share knowledge of \(W_{K}^{(1)}\), which is the part of WK governing pixel selection. If this condition is met, watermarking and (permutation-based) ciphering commute. However, as is well-known, permutation ciphers are vulnerable to known plaintext attacks (see [16] for a quantitative analysis).
Moreover, in order to have as many permutation as possible, N should be chosen as large as possible, that means
$$ N = \lfloor {H\cdot W \over n} \rfloor $$
in the spatial domain, where H and W are the height and width of the host image C0, and n is the length of the embedded string. In order to have a minimum level of security against brute-force attacks, we need N≥32, leading to a maximum capacity of
$$ n_{max} = \lfloor {H\cdot W \over 32} \rfloor $$
bits in the spatial domain, meaning, e.g., 213 bits for a 512×512 image (note that the term capacity is used throughout this paper according to the definition given in [17]: the watermarking capacity of digital image is the number of bits that can be embedded in a given host image.).
Figure 2 shows encrypted versions of the marked Lena image in Fig. 1a. Thanks to the commutativity of watermarking and encryption, the mark can be extracted from both without errors.
Encrypting the marked Lena image in the spatial domain. a Sign bit encryption. b Permutation cipher
Implementation in the DCT domain
In the DCT domain, we assume that the watermarking key consists of three parts: \(W_{K} = \left (W_{K}^{(1)},W_{K}^{(2)},W_{K}^{(3)}\right)\). We begin by performing a block-based two-dimensional DCT on the host image C0, i.e., we divide C0 into non-overlapping 8×8 pixel blocks and perform a two-dimensional DCT on each block.
For each bit bi to be embedded, randomly select N blocks. The selection is controlled by \(W_{K}^{(1)}\). Each block can only be selected once. The selected blocks for bit bi form subset Ti of the set of all blocks.
For each Ti, randomly select a horizontal and a vertical frequency index from the medium frequencies. The selection is controlled by \(W_{K}^{(2)}\). The corresponding DCT-coefficients from the selected blocks form an N-dimensional vector \(\vec x_{i}\).
For each Ti, randomly generate a dither value di under control of \(W_{K}^{(3)}\).
$$ \Vert \vec x_{i} \Vert_{q} = Q_{b_{i}}\left(\Vert \vec x_{i} \Vert, \Delta_{i}, d_{i}\right) $$
Note that because all DCT coefficients in \(\vec x_{i}\) correspond to the same horizontal and vertical frequency pair, it is possible to choose individual quantizing step sizes Δi according to their perceptual importance (see below).
and replace the selected DCT coefficients in Ti with the corresponding entries in \(\vec y_{i}\).
In choosing the Δi step sizes, we were led by the JPEG quantization matrix, which assigns a perceptual relevance to each DCT coefficient in an (8×8) block. The essential step in JPEG compression consists in quantizing DCT-coefficients according to fixed quantization tables corresponding to a certain quality factor. On the other hand, it is well known that QIM-based watermarking schemes are sensitive to re-quantization.
In order to counter the adverse effects of re-quantization, we therefore chose quantization step sizes Δi for the individual DCT coefficients selected for embedding bit bi that were oriented at the actual quantization step sizes in the JPEG standard. More specifically, we used the quantization matrix
$$J = \left(\begin{array}{lllllllll} 16 & 11 & 10 & 16 & 24 & 40 & 51 & 61 \\ 12 & 12 & 14 & 19 & 26 & 58 & 60 & 55 \\ 14 & 13 & 16 & 24 & 40 & 57 & 69 & 56 \\ 14 & 17 & 22 & 29 & 51 & 87 & 80 & 62 \\ 18 & 22 & 37 & 56 & 68 & 109 & 103 & 77 \\ 24 & 36 & 55 & 64 & 81 & 104 & 113 & 92 \\ 49 & 64 & 78 & 87 & 103 & 121 & 120 & 101 \\ 72 & 92 & 95 & 98 & 112 & 100 & 103 & 99 \\ \end{array}\right) $$
taken from the JPEG standard ([18]), which gives the JPEG quantization steps for the DCT coefficients within a 8×8 block referring to a 50% quality factor, and multiplied it with a constant c>1. If the DCT coefficients for embedding bi correspond to frequencies (k,ℓ), we have
$$ \Delta_{i} = c\cdot J_{k \ell}, $$
the rationale behind this approach being the well-known fact that for a quantizing function Q, we have
$$ Q(Q(Q(x,\Delta),\delta),\Delta) = Q(x,\Delta), $$
if Δ>δ (see [19], Theorem 1). This means that quantizing some value y with step size δ can be reversed by another quantization with a larger step size Δ.
Figure 3 shows two embedding examples, where a random 64-bit message was embedded. For the Lena image, we set N=64,c=3.5, and for the higher resolution Norba image, we set N=312,c=3.5 (cf. Section 4.2.2 for details on how N was chosen).
Embedding 64 bits in the DCT domain. a N=64, PSNR = 56.10 dB. b N=312, PSNR = 63.74 dB
In order to extract message bit bi, the disturbed marked signal vector \(\tilde {\vec y}_{i}\) is extracted from the marked image CW in the same way as the unmarked vector \(\vec x_{i}\) was built from C0 with the help of WK. The message bit is then decoded according to
$$ b_{i} = \arg \min_{b_{i} \in \{0,1\}} \vert \Vert \tilde{\vec y}_{i} \Vert - Q_{b_{i}}\left(\Vert \tilde{\vec y}_{i} \Vert,\Delta_{i},d_{i}\right) \vert. $$
As in the spatial domain, we investigate the two options of encrypting DCT-coefficient sign bits and of permuting them. The idea of encrypting the sign bits of DCT coefficients goes back to [20] and [21], where it is proposed to encrypt sign bits of DCT coefficients and motion vectors in MPEG video. The security of this approach for still images is classified as low in [15], p. 51.In order to create a larger visual distortion, instead of permuting the DCT coefficients alone, we permuted the complete (8×8)−blocks containing the coefficients (see [20] and [21]). If only those blocks containing the selected coefficients for watermarking are permuted, however, the corresponding subset T becomes visible to an attacker, who can in turn concentrate her efforts to remove the mark on T. We therefore need to permute all image blocks. Moreover, as in the spatial domain case, in order to make sure that the selected blocks Ti for a single bit bi do not get mixed up with blocks for a different bit or non-selected blocks, the permutation algorithm needs to know part \(W_{K}^{(1)} \) of the watermarking key. More specifically, each subset Ti needs to form an invariant subset of the set of all blocks under the permutation. As in the spatial domain, these subsets need to be as large as possible. We therefore have
$$ N = \vert T_{i} \vert = \lfloor {{(H/8)\cdot (W/8)}\over n} \rfloor $$
in the DCT domain. The requirement N≥32 gives a maximum capacity of
$$ n_{max} = \lfloor {H\cdot W \over {64\cdot 32} }\rfloor, $$
meaning 128 bits for a 512×512 image.
Figure 4 shows encrypted versions of the marked Lena image in Fig. 4. Again, the mark can be extracted from both without errors.
Encrypting the marked Lena image in the DCT domain. a Coefficient sign bit encryption. b Permutation of 8×8 blocks
Implementation in the DWT domain
In the DWT domain, we performed a three-level DWT and embedded the mark into the level 3 approximation coefficients. This way, the number N of coefficients used to embed one bit is the same as in Section 4.2, namely
$$ N = \vert T_{i} \vert = \lfloor {{(H/8)\cdot (W/8)}\over n} \rfloor. $$
In the DWT-case, however, there are no blocks of coefficients to choose a frequency from, thus WK consists of only two parts: \(W_{K} = \left (W_{K}^{(1)},W_{K}^{(2)}\right)\), where \(W_{K}^{(1)}\) governs the selection of N coefficients for each message bit bi, and \(W_{K}^{(2)}\) controls the dither di for each bit. Likewise, a single quantization step size Δ is used for all message bits. As an example, Fig. 5 shows the results of embedding 64 bits into the Lena and Norba image, setting Δ=100.
Embedding 64 bits in the DWT domain.a N=64, PSNR = 54.84 dB. b N=312, PSNR = 62.94 dB
As in the DCT case, we have the options to either encrypt the sign bits of DWT coefficients, as already proposed in [21], and/or to permute the DWT coefficients, as originally proposed in [22]. Note that in the DWT domain, permuting coefficients is not as vulnerable to known-plaintext attacks as in other domains, because the location of coefficients is image-dependent [15]. However, if only the level 3 approximation coefficients are encrypted or permuted, the image content is not rendered completely unintelligible, but fine structures are still visible (see Figs. 6a, b). As in the DCT-case, we have the additional option of not only permuting the level 3 DWT-coefficients themselves but the complete (8×8) blocks leading to the level 3 approximation for a more complete obfuscation of the image content (see Fig. 6c), without sacrificing the commutativity with watermarking. The maximum capacity is the same as in the DCT-based implementation.
Encrypting the marked Lena image in the DWT domain. a LL3 coefficient sign bit encryption. b Permutation of LL3 coefficients. c Permutation of 8×8 blocks
In our experiments, we used 50 standard images of format 512×512, most of them downloaded from http://decsai.ugr.es/cvg/CG/base.htm. We embedded 64 random bits and fixed all other parameters in such a way that a PSNR of about 50dB resulted for the watermarked images. In the spatial domain and the DWT domain, this meant a quantizing step size of Δ=175 (see Section 5.1 for details).
In the DCT domain, the c-Parameter (see Section 4.2) was set to 8.0. The similarity of the extracted watermarks to the originally embedded watermarks was measured using the normalized correlation of the two vectors.
We first investigated how the Δ resp. the c-parameter affects the PSNR of the watermarked image compared to the host image. Perhaps not surprisingly, the effect of Δ on the PSNR is practically the same for the spatial domain and the DWT domain (see Fig. 7).
PSNR versus Δ in the spatial and the DWT domain (averaged over 50 images)
As the c Parameter is not directly comparable to the Δ-parameter for the other two domains, the corresponding graph is shown here in a separate diagram (Fig. 8).
PSNR vs c Parameter (averaged over 50 images)
Both Figs. 7 and 8 reveal that a parameter choice of Δ=175 for spatial and DWT domain and of c=8.0 for the DCT domain give rise to a PSNR of about 50 dB, if 64 bits are embedded. This provides the basic setting for our further experiments.
In another fidelity experiment, we investigated the influence of the message size on the PSNR (see Fig. 9). Again, the spatial domain and DWT-based implementations show almost equal behavior, except that the spatial domain scheme has much a larger capacity.
PSNR vs message size (averaged over 50 images)
JPEG compression
Both the DCT - and the DWT-based implementations prove to be very robust against JPEG compression (see Fig. 10). Both schemes also outperform the scheme proposed in [6] with respect to JPEG compression, which offers a normalized correlation of 0.22 at a JPEG quality factor of 50%.
Correlation value versus JPEG quality factor in three investigated domains (averaged over 50 images)
JPEG2000 compression
The results of our experiments with JPEG2000 compression basically follow the same pattern as the JPEG experiments. The spatial domain implementation is the most fragile one, but is still surprisingly robust, especially at low compression rates.
The DCT-based implementation and the DWT-based implementation perform almost equally well. Only for higher compression rates, the DWT-based implementation has a slight advantage. (see Fig. 11). Again, both transform domain based schemes outperform the scheme presented in [6] and have roughly the same robustness against JPEG2000 compression as the scheme in [4], which works in the LL4-subband.
Correlation value versus JPEG2000 compression ratio in three investigated domains (averaged over 50 images)
Adding noise
All three implementation domains perform equally well in the presence of low- or medium-density additive white noise. For higher noise densities, the DWT-based implementation is the most robust (see Fig. 12).
Correlation value versus noise density in the three investigated domains (averaged over 50 images)
Security of encryption
In this subsection, we summarize and enhance the security assessments made in Section 4 for the three implementation domains.
As sign bit encryption in the spatial domain can be attacked directly [15] to reveal part of the image contents, this approach seems to be weakest of all options. Combining it with a permutation cipher makes for a cryptographically and visually stronger cipher, although the permutation cipher is in turn vulnerable to known plaintext attacks. This means, however, to share part of the watermarking key between content owner and seller.
Sign bit encryption in the DCT domain has been attacked by Wu and Kuo [23], who could recover some visual information from the encrypted image by setting the DC coefficient to 128 and giving all AC coefficients a positive sign. Again, a combination with a block-based permutation will strengthen the security of the cipher (note that we do not recommend to permute DCT coefficients directly, as the DC coefficient will normally stick out as the one with the largest absolute value).
For the DWT domain, there are, to the best of our knowledge, no analogous attacks on sign-bit encryption in the literature. Still, it is recommended to encrypt not only the watermarked DWT coefficients of the LL3 subband, but all subbands, and combine the sign bit encryption with permutation, if a secret sharing between content owner and seller is possible. If this is not the case, the content owner can resort to permute all subbands excluding the LL3 subband.
Watermarking security
According to [24], watermarking security means the occurrence of an adversary trying to break the system, as opposed to random modifications of a marked image due to benign image processing. In the following discussion, we assume that an attacker has access to the unencrypted, marked image CW, but not to the original host image C0 or the watermarking key WK. In this context, breaking the system means that the attacker is either able to insert a mark of her own, or to detect a mark, or to remove the mark from CW without rendering the image unusable.
In order to successfully detect a watermark or embed a watermark of her own without knowledge of the watermarking key WK, an attacker would have to guess how the signal vectors are formed as a first step. If the mark is embedded in the DCT- or DWT domain, there are \({(H/8)\cdot (W/8)}\choose {N}\) possibilities for the first bit, where W and H are the dimensions of the image and N is the dimension of the signal vector. For typical values (H=W=512,N=32), this means about 1080 possibilities.
As is shown in Section 5, it is rather hard for an attacker to remove the watermark from a marked image by adding white noise or compression, especially if the mark was embedded in the DCT or DWT domain. Without knowledge of the correct watermarking key, depending on the implementation domain, an attacker would have to modify the value of the pixel grey values or transform coefficients in such way that the norm of each possible signal vector is changed by an amount of at least Δ/2.
We have presented a novel commutative watermarking encryption (CWE) scheme, which is very robust to common attacks like JPEG/JPEG2000 compression and noise addition, especially when implemented in some transform domain (Discrete-Cosine or Discrete-Wavelet). On the other hand, the spatial domain implementation has the advantage of a much higher capacity. However, the robustness comes at the cost of relatively weak encryption primitives, especially if the scheme is applied in the spatial or DCT domain. The implementation in the DWT domain offers the best tradeoff between robustness and security of the cipher.Nevertheless, because of the leakage of visual contents if sign bit encryption is used exclusively, and because of the inherent weaknesses of permutation ciphers, the proposed scheme is recommended for scenarios with low to medium security requirements with regard to the image contents, where robustness of the watermark has the highest priority. For many commercial application scenarios, this seems to be a good fit. In future research, we will explore ways to further enhance the security of the encryption primitives by using norm-preserving operations.
The experimental results of this study are based on the image data set available at http://decsai.ugr.es/cvg/CG/base.htm. The corresponding code is available from the author on request.
In the original article an OA funding note was missing. An amendment to this paper has been published and can be accessed via the original article.
CBC:
Cipher block chaining
CWE:
Commutative watermarking encryption
DCT:
Discrete cosine transform
DRM:
DWT:
JPEG:
Joint photographic expert group
PSNR:
Peak signal-to-noise ratio
QIM:
Quantized index modulation
SHDM:
Sphere hardening dither modulation
STDM:
Spread-transform dither modulation
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The research described in this article was done during a sabbatical semester granted by the Stuttgart Media University. The author gratefully acknowledges having been given this opportunity. He also thanks the anonymous reviewers for their helpful comments.
The author did not receive any funding for this research. Open Access funding enabled and organized by Projekt DEAL.
Department of Computer Science and Media, Stuttgart Media University, Nobelstrasse 10, Stuttgart, D-70569, Germany
Roland Schmitz
The entire manuscript is a sole contribution of the author. The author read and approved the final manuscript.
Correspondence to Roland Schmitz.
The author declares that they have no competing interests.
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 licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence 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 licence, visit http://creativecommons.org/licenses/by/4.0/.
Schmitz, R. Use of SHDM in commutative watermarking encryption. EURASIP J. on Info. Security 2021, 1 (2021). https://doi.org/10.1186/s13635-020-00115-w
Received: 07 July 2020 | CommonCrawl |
1. $g_1(g_2a) = (g_1 \cdot g_2)a$ for all $g_1, g_2 \in G$ and for all $a \in A$.
2. $1a = a$ for all $a \in A$ (where $1 \in G$ denotes the identity element).
1. $(ag_1)g_2 = a(g_1 \cdot g_2)$ for all $g_1, g_2 \in G$ and for all $a \in A$.
2. $1a = a$ for all $a \in A$.
In either case we say that the group $G$ is (left/right) Acting on the set $A$.
We begin by stating some basic results regarding (left) group actions of a group on a set.
a) For each $g \in G$, $\sigma_g$ is a permutation of the set $A$, and so $\varphi_g \in S_A$ for all $g \in G$.
b) The map $\varphi : G \to S_A$ (called the Associated Permutation Representation) defined for all $g \in G$ by $\varphi(g) = \sigma_g$ is a group homomorphism of $G$ to $S_A$.
c) If $\psi : G \to S_A$ is any homomorphism from $G$ to $S_A$ then the map $G \times A \to A$ defined by $(g, a) \to \varphi(g) a$ is a left group action of $G$ on $A$.
A similar result can be stated when $(G, \cdot)$ is right acting on a set $A$.
Proof of c) Let $\psi : G \to S_A$ be a group homomorphism of $G$ to $S_A$. We aim to show that $(g, a) \to [\psi(g)](a)$ is a left group action of $G$ on $A$, i.e., we need to verify the two properties in the definition. | CommonCrawl |
General relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of second order partial differential equations.
For the graduate textbook by Robert Wald, see General Relativity (book).
For a more accessible and less technical introduction to this topic, see Introduction to general relativity.
General relativity
$G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$
• Introduction
• History
• Timeline
• Tests
• Mathematical formulation
Fundamental concepts
• Equivalence principle
• Special relativity
• World line
• Pseudo-Riemannian manifold
Phenomena
• Kepler problem
• Gravitational lensing
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• Frame-dragging
• Geodetic effect
• Event horizon
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Spacetime
• Spacetime diagrams
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Equations
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Formalisms
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Advanced theory
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Scientists
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Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data.
Reconciliation of general relativity with the laws of quantum physics remains a problem, however, as there is a lack of a self-consistent theory of quantum gravity. It is not yet known how gravity can be unified with the three non-gravitational forces: strong, weak and electromagnetic.
Einstein's theory has astrophysical implications, including the prediction of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape from them. Black holes are the end-state for massive stars. Microquasars and active galactic nuclei are believed to be stellar black holes and supermassive black holes. It also predicts gravitational lensing, where the bending of light results in multiple images of the same distant astronomical phenomenon. Other predictions include the existence of gravitational waves, which have been observed directly by the physics collaboration LIGO and other observatories. In addition, general relativity has provided the base of cosmological models of an expanding universe.
In the preface to Relativity: The Special and the General Theory, Einstein said "The present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated."[2]
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.[3]
History
Henri Poincaré's 1905 theory of the dynamics of the electron was a relativistic theory which he applied to all forces, including gravity. While others thought that gravity was instantaneous or of electromagnetic origin, he suggested that relativity was "something due to our methods of measurement". In his theory, he showed that gravitational waves propagate at the speed of light.[4] Soon afterwards, Einstein started thinking about how to incorporate gravity into his relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall (FFO), he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity.[5] These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present.[6] A version of non-Euclidean geometry, called Riemannian geometry, enabled Einstein to develop general relativity by providing the key mathematical framework on which he fit his physical ideas of gravity.[7] This idea was pointed out by mathematician Marcel Grossmann and published by Grossmann and Einstein in 1913.[8]
The Einstein field equations are nonlinear and considered difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But in 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, eventually resulting in the Reissner–Nordström solution, which is now associated with electrically charged black holes.[9] In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption.[10] By 1929, however, the work of Hubble and others had shown that our universe is expanding. This is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot and dense earlier state.[11] Einstein later declared the cosmological constant the biggest blunder of his life.[12]
During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein showed in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters ("fudge factors"),[13] and in 1919 an expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of 29 May 1919,[14] instantly making Einstein famous.[15] Yet the theory remained outside the mainstream of theoretical physics and astrophysics until developments between approximately 1960 and 1975, now known as the golden age of general relativity.[16] Physicists began to understand the concept of a black hole, and to identify quasars as one of these objects' astrophysical manifestations.[17] Ever more precise solar system tests confirmed the theory's predictive power,[18] and relativistic cosmology also became amenable to direct observational tests.[19]
General relativity has acquired a reputation as a theory of extraordinary beauty.[3][20][21] Subrahmanyan Chandrasekhar has noted that at multiple levels, general relativity exhibits what Francis Bacon has termed a "strangeness in the proportion" (i.e. elements that excite wonderment and surprise). It juxtaposes fundamental concepts (space and time versus matter and motion) which had previously been considered as entirely independent. Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory.[22] Other elements of beauty associated with the general theory of relativity are its simplicity and symmetry, the manner in which it incorporates invariance and unification, and its perfect logical consistency.[23]
From classical mechanics to general relativity
General relativity can be understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.[24][25]
Geometry of Newtonian gravity
At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the net force acting on a body is equal to that body's (inertial) mass multiplied by its acceleration.[26] The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. In modern parlance, their paths are geodesics, straight world lines in curved spacetime.[27]
Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. However, there is an ambiguity once gravity comes into play. According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors (see Eötvös experiment), there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties.[28] A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in an enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is stationary in a gravitational field and the ball accelerating, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field versus the ball which upon release has nil acceleration.[29]
Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. Space, in this construction, still has the ordinary Euclidean geometry. However, spacetime as a whole is more complicated. As can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; mathematically speaking, the Newtonian connection is not integrable. From this, one can deduce that spacetime is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant concepts, i.e. a description which is valid in any desired coordinate system.[30] In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass.[31]
Relativistic generalization
As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics.[32] In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. (The defining symmetry of special relativity is the Poincaré group, which includes translations, rotations, boosts and reflections.) The differences between the two become significant when dealing with speeds approaching the speed of light, and with high-energy phenomena.[33]
With Lorentz symmetry, additional structures come into play. They are defined by the set of light cones (see image). The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent.[34] In conjunction with the world-lines of freely falling particles, the light-cones can be used to reconstruct the spacetime's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure[35] or conformal geometry.
Special relativity is defined in the absence of gravity. For practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall motion, an analogous reasoning as in the previous section applies: there are no global inertial frames. Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry.[36]
A priori, it is not clear whether the new local frames in free fall coincide with the reference frames in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. below). The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity.[37] The generalization of this statement, namely that the laws of special relativity hold to good approximation in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity.[38]
The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In the language of spacetime geometry, it is not measured by the Minkowski metric. As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable locally inertial coordinates, the metric is Minkowskian, and its first partial derivatives and the connection coefficients vanish).[39]
Einstein's equations
Main articles: Einstein field equations and Mathematics of general relativity
Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy–momentum tensor, which includes both energy and momentum densities as well as stress: pressure and shear.[40] Using the equivalence principle, this tensor is readily generalized to curved spacetime. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. In special relativity, conservation of energy–momentum corresponds to the statement that the energy–momentum tensor is divergence-free. This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry. With this additional condition—the covariant divergence of the energy–momentum tensor, and hence of whatever is on the other side of the equation, is zero—the simplest nontrivial set of equations are what are called Einstein's (field) equations:
Einstein's field equations
$G_{\mu \nu }\equiv R_{\mu \nu }- 1 \over 2}R\,g_{\mu \nu }=\kappa T_{\mu \nu }\,$
On the left-hand side is the Einstein tensor, $G_{\mu \nu }$, which is symmetric and a specific divergence-free combination of the Ricci tensor $R_{\mu \nu }$ and the metric. In particular,
$R=g^{\mu \nu }R_{\mu \nu }$
is the curvature scalar. The Ricci tensor itself is related to the more general Riemann curvature tensor as
$R_{\mu \nu }={R^{\alpha }}_{\mu \alpha \nu }.$
On the right-hand side, $\kappa $ is a constant and $T_{\mu \nu }$ is the energy–momentum tensor. All tensors are written in abstract index notation.[41] Matching the theory's prediction to observational results for planetary orbits or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics, the proportionality constant $\kappa $ is found to be $ \kappa ={\frac {8\pi G}{c^{4}}}$, where $G$ is the Newtonian constant of gravitation and $c$ the speed of light in vacuum.[42] When there is no matter present, so that the energy–momentum tensor vanishes, the results are the vacuum Einstein equations,
$R_{\mu \nu }=0.$
In general relativity, the world line of a particle free from all external, non-gravitational force is a particular type of geodesic in curved spacetime. In other words, a freely moving or falling particle always moves along a geodesic.
The geodesic equation is:
${d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0,$
where $s$ is a scalar parameter of motion (e.g. the proper time), and $\Gamma ^{\mu }{}_{\alpha \beta }$ are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices. Greek indices may take the values: 0, 1, 2, 3 and the summation convention is used for repeated indices $\alpha $ and $\beta $. The quantity on the left-hand-side of this equation is the acceleration of a particle, and so this equation is analogous to Newton's laws of motion which likewise provide formulae for the acceleration of a particle. This equation of motion employs the Einstein notation, meaning that repeated indices are summed (i.e. from zero to three). The Christoffel symbols are functions of the four spacetime coordinates, and so are independent of the velocity or acceleration or other characteristics of a test particle whose motion is described by the geodesic equation.
Total force in general relativity
See also: Two-body problem in general relativity
In general relativity, the effective gravitational potential energy of an object of mass m revolving around a massive central body M is given by[43][44]
$U_{f}(r)=-{\frac {GMm}{r}}+{\frac {L^{2}}{2mr^{2}}}-{\frac {GML^{2}}{mc^{2}r^{3}}}$
A conservative total force can then be obtained as
$F_{f}(r)=-{\frac {GMm}{r^{2}}}+{\frac {L^{2}}{mr^{3}}}-{\frac {3GML^{2}}{mc^{2}r^{4}}}$
where L is the angular momentum. The first term represents the force of Newtonian gravity, which is described by the inverse-square law. The second term represents the centrifugal force in the circular motion. The third term represents the relativistic effect.
Alternatives to general relativity
There are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Examples are Whitehead's theory, Brans–Dicke theory, teleparallelism, f(R) gravity and Einstein–Cartan theory.[45]
Definition and basic applications
See also: Mathematics of general relativity and Physical theories modified by general relativity
The derivation outlined in the previous section contains all the information needed to define general relativity, describe its key properties, and address a question of crucial importance in physics, namely how the theory can be used for model-building.
Definition and basic properties
General relativity is a metric theory of gravitation. At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional pseudo-Riemannian manifold representing spacetime, and the energy–momentum contained in that spacetime.[46] Phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity; there is no gravitational force deflecting objects from their natural, straight paths. Instead, gravity corresponds to changes in the properties of space and time, which in turn changes the straightest-possible paths that objects will naturally follow.[47] The curvature is, in turn, caused by the energy–momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve.[48]
While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of universal gravitation.[49]
As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems.[50] Furthermore, the theory does not contain any invariant geometric background structures, i.e. it is background independent. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers.[51] Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics exhibit local Lorentz invariance.[52]
Model-building
The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.[53]
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly.[54] Nevertheless, a number of exact solutions are known, although only a few have direct physical applications.[55] The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,[56] and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos.[57] Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub–NUT solution (a model universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).[58]
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes.[59] In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity[60] and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter provides a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity.[61] An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.[62]
Consequences of Einstein's theory
General relativity has a number of physical consequences. Some follow directly from the theory's axioms, whereas others have become clear only in the course of many years of research that followed Einstein's initial publication.
Gravitational time dilation and frequency shift
Assuming that the equivalence principle holds,[63] gravity influences the passage of time. Light sent down into a gravity well is blueshifted, whereas light sent in the opposite direction (i.e., climbing out of the gravity well) is redshifted; collectively, these two effects are known as the gravitational frequency shift. More generally, processes close to a massive body run more slowly when compared with processes taking place farther away; this effect is known as gravitational time dilation.[64]
Gravitational redshift has been measured in the laboratory[65] and using astronomical observations.[66] Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks,[67] while ongoing validation is provided as a side effect of the operation of the Global Positioning System (GPS).[68] Tests in stronger gravitational fields are provided by the observation of binary pulsars.[69] All results are in agreement with general relativity.[70] However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid.[71]
Light deflection and gravitational time delay
Main articles: Schwarzschild geodesics, Kepler problem in general relativity, Gravitational lens, and Shapiro delay
General relativity predicts that the path of light will follow the curvature of spacetime as it passes near a star. This effect was initially confirmed by observing the light of stars or distant quasars being deflected as it passes the Sun.[72]
This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity.[73] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion),[74] several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light,[75] the angle of deflection resulting from such calculations is only half the value given by general relativity.[76]
Closely related to light deflection is the gravitational time delay (or Shapiro delay), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.[77] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay determine a parameter called γ, which encodes the influence of gravity on the geometry of space.[78]
Gravitational waves
Predicted in 1916[79][80] by Albert Einstein, there are gravitational waves: ripples in the metric of spacetime that propagate at the speed of light. These are one of several analogies between weak-field gravity and electromagnetism in that, they are analogous to electromagnetic waves. On 11 February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a pair of black holes merging.[81][82][83]
The simplest type of such a wave can be visualized by its action on a ring of freely floating particles. A sine wave propagating through such a ring towards the reader distorts the ring in a characteristic, rhythmic fashion (animated image to the right).[84] Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult. However, linear approximations of gravitational waves are sufficiently accurate to describe the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in relative distances increasing and decreasing by $10^{-21}$ or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed.[85]
Some exact solutions describe gravitational waves without any approximation, e.g., a wave train traveling through empty space[86] or Gowdy universes, varieties of an expanding cosmos filled with gravitational waves.[87] But for gravitational waves produced in astrophysically relevant situations, such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models.[88]
Orbital effects and the relativity of direction
Main article: Two-body problem in general relativity
General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.
Precession of apsides
In general relativity, the apsides of any orbit (the point of the orbiting body's closest approach to the system's center of mass) will precess; the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rose curve-like shape (see image). Einstein first derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body as a test particle. For him, the fact that his theory gave a straightforward explanation of Mercury's anomalous perihelion shift, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations.[89]
The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)[90] or the much more general post-Newtonian formalism.[91] It is due to the influence of gravity on the geometry of space and to the contribution of self-energy to a body's gravity (encoded in the nonlinearity of Einstein's equations).[92] Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus, and Earth),[93] as well as in binary pulsar systems, where it is larger by five orders of magnitude.[94]
In general relativity the perihelion shift $\sigma $, expressed in radians per revolution, is approximately given by[95]
$\sigma ={\frac {24\pi ^{3}L^{2}}{T^{2}c^{2}(1-e^{2})}}\ ,$
where:
• $L$ is the semi-major axis
• $T$ is the orbital period
• $c$ is the speed of light in vacuum
• $e$ is the orbital eccentricity
Orbital decay
According to general relativity, a binary system will emit gravitational waves, thereby losing energy. Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the Solar System or for ordinary double stars, the effect is too small to be observable. This is not the case for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. Because neutron stars are immensely compact, significant amounts of energy are emitted in the form of gravitational radiation.[97]
The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor, using the binary pulsar PSR1913+16 they had discovered in 1974. This was the first detection of gravitational waves, albeit indirect, for which they were awarded the 1993 Nobel Prize in physics.[98] Since then, several other binary pulsars have been found, in particular the double pulsar PSR J0737−3039, where both stars are pulsars[99] and which was last reported to also be in agreement with general relativity in 2021 after 16 years of observations.[96]
Geodetic precession and frame-dragging
Several relativistic effects are directly related to the relativity of direction.[100] One is geodetic precession: the axis direction of a gyroscope in free fall in curved spacetime will change when compared, for instance, with the direction of light received from distant stars—even though such a gyroscope represents the way of keeping a direction as stable as possible ("parallel transport").[101] For the Moon–Earth system, this effect has been measured with the help of lunar laser ranging.[102] More recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 0.3%.[103][104]
Near a rotating mass, there are gravitomagnetic or frame-dragging effects. A distant observer will determine that objects close to the mass get "dragged around". This is most extreme for rotating black holes where, for any object entering a zone known as the ergosphere, rotation is inevitable.[105] Such effects can again be tested through their influence on the orientation of gyroscopes in free fall.[106] Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction.[107] Also the Mars Global Surveyor probe around Mars has been used.[108]
Interpretations
Neo-Lorentzian Interpretation
Examples of prominent physicists who support neo-Lorentzian explanations of general relativity are Franco Selleri and Antony Valentini.[109]
Astrophysical applications
Gravitational lensing
The deflection of light by gravity is responsible for a new class of astronomical phenomena. If a massive object is situated between the astronomer and a distant target object with appropriate mass and relative distances, the astronomer will see multiple distorted images of the target. Such effects are known as gravitational lensing.[110] Depending on the configuration, scale, and mass distribution, there can be two or more images, a bright ring known as an Einstein ring, or partial rings called arcs.[111] The earliest example was discovered in 1979;[112] since then, more than a hundred gravitational lenses have been observed.[113] Even if the multiple images are too close to each other to be resolved, the effect can still be measured, e.g., as an overall brightening of the target object; a number of such "microlensing events" have been observed.[114]
Gravitational lensing has developed into a tool of observational astronomy. It is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and to obtain an independent estimate of the Hubble constant. Statistical evaluations of lensing data provide valuable insight into the structural evolution of galaxies.[115]
Gravitational-wave astronomy
Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see Orbital decay, above). Detection of these waves is a major goal of current relativity-related research.[116] Several land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (two detectors), TAMA 300 and VIRGO.[117] Various pulsar timing arrays are using millisecond pulsars to detect gravitational waves in the 10−9 to 10−6 hertz frequency range, which originate from binary supermassive blackholes.[118] A European space-based detector, eLISA / NGO, is currently under development,[119] with a precursor mission (LISA Pathfinder) having launched in December 2015.[120]
Observations of gravitational waves promise to complement observations in the electromagnetic spectrum.[121] They are expected to yield information about black holes and other dense objects such as neutron stars and white dwarfs, about certain kinds of supernova implosions, and about processes in the very early universe, including the signature of certain types of hypothetical cosmic string.[122] In February 2016, the Advanced LIGO team announced that they had detected gravitational waves from a black hole merger.[81][82][83]
Black holes and other compact objects
Whenever the ratio of an object's mass to its radius becomes sufficiently large, general relativity predicts the formation of a black hole, a region of space from which nothing, not even light, can escape. In the currently accepted models of stellar evolution, neutron stars of around 1.4 solar masses, and stellar black holes with a few to a few dozen solar masses, are thought to be the final state for the evolution of massive stars.[123] Usually a galaxy has one supermassive black hole with a few million to a few billion solar masses in its center,[124] and its presence is thought to have played an important role in the formation of the galaxy and larger cosmic structures.[125]
Astronomically, the most important property of compact objects is that they provide a supremely efficient mechanism for converting gravitational energy into electromagnetic radiation.[126] Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as microquasars.[127] In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed.[128] General relativity plays a central role in modelling all these phenomena,[129] and observations provide strong evidence for the existence of black holes with the properties predicted by the theory.[130]
Black holes are also sought-after targets in the search for gravitational waves (cf. Gravitational waves, above). Merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and the phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events–and hence serve as a probe of cosmic expansion at large distances.[131] The gravitational waves produced as a stellar black hole plunges into a supermassive one should provide direct information about the supermassive black hole's geometry.[132]
Cosmology
The current models of cosmology are based on Einstein's field equations, which include the cosmological constant $\Lambda $ since it has important influence on the large-scale dynamics of the cosmos. This is the equation before Einstein removed the cosmological constant:
$R_{\mu \nu }- 1 \over 2}R\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\,T_{\mu \nu }$
where $g_{\mu \nu }$ is the spacetime metric.[133] Isotropic and homogeneous solutions of these enhanced equations, the Friedmann–Lemaître–Robertson–Walker solutions,[134] allow physicists to model a universe that has evolved over the past 14 billion years from a hot, early Big Bang phase.[135] Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,[136] further observational data can be used to put the models to the test.[137] Predictions, all successful, include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,[138] the large-scale structure of the universe,[139] and the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation.[140]
Astronomical observations of the cosmological expansion rate allow the total amount of matter in the universe to be estimated, although the nature of that matter remains mysterious in part. About 90% of all matter appears to be dark matter, which has mass (or, equivalently, gravitational influence), but does not interact electromagnetically and, hence, cannot be observed directly.[141] There is no generally accepted description of this new kind of matter, within the framework of known particle physics[142] or otherwise.[143] Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation also show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, known as dark energy, the nature of which remains unclear.[144]
An inflationary phase,[145] an additional phase of strongly accelerated expansion at cosmic times of around 10−33 seconds, was hypothesized in 1980 to account for several puzzling observations that were unexplained by classical cosmological models, such as the nearly perfect homogeneity of the cosmic background radiation.[146] Recent measurements of the cosmic background radiation have resulted in the first evidence for this scenario.[147] However, there is a bewildering variety of possible inflationary scenarios, which cannot be restricted by current observations.[148] An even larger question is the physics of the earliest universe, prior to the inflationary phase and close to where the classical models predict the big bang singularity. An authoritative answer would require a complete theory of quantum gravity, which has not yet been developed[149] (cf. the section on quantum gravity, below).
Exotic solutions: time travel, warp drives
Kurt Gödel showed[150] that solutions to Einstein's equations exist that contain closed timelike curves (CTCs), which allow for loops in time. The solutions require extreme physical conditions unlikely ever to occur in practice, and it remains an open question whether further laws of physics will eliminate them completely. Since then, other—similarly impractical—GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. Stephen Hawking introduced chronology protection conjecture, which is an assumption beyond those of standard general relativity to prevent time travel.
Some exact solutions in general relativity such as Alcubierre drive present examples of warp drive but these solutions requires exotic matter distribution, and generally suffers from semiclassical instability. [151]
Advanced concepts
Asymptotic symmetries
The spacetime symmetry group for special relativity is the Poincaré group, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group.
In 1962 Hermann Bondi, M. G. van der Burg, A. W. Metzner[152] and Rainer K. Sachs[153] addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no a priori assumptions about the nature of the asymptotic symmetry group—not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies the conclusion that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances. It turns out that the BMS symmetry, suitably modified, could be seen as a restatement of the universal soft graviton theorem in quantum field theory (QFT), which relates universal infrared (soft) QFT with GR asymptotic spacetime symmetries.[154]
Causal structure and global geometry
In general relativity, no material body can catch up with or overtake a light pulse. No influence from an event A can reach any other location X before light sent out at A to X. In consequence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. This structure can be displayed using Penrose–Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams.[155]
Aware of the importance of causal structure, Roger Penrose and others developed what is known as global geometry. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, and additional non-specific assumptions about the nature of matter (usually in the form of energy conditions) are used to derive general results.[156]
Horizons
Using global geometry, some spacetimes can be shown to contain boundaries called horizons, which demarcate one region from the rest of spacetime. The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale is the Schwarzschild radius[157]), no light from inside can escape to the outside. Since no object can overtake a light pulse, all interior matter is imprisoned as well. Passage from the exterior to the interior is still possible, showing that the boundary, the black hole's horizon, is not a physical barrier.[158]
Early studies of black holes relied on explicit solutions of Einstein's equations, notably the spherically symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). Using global geometry, later studies have revealed more general properties of black holes. With time they become rather simple objects characterized by eleven parameters specifying: electric charge, mass–energy, linear momentum, angular momentum, and location at a specified time. This is stated by the black hole uniqueness theorem: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans. Irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple.[159]
Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. For instance, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. This limits the energy that can be extracted by classical means from a rotating black hole (e.g. by the Penrose process).[160] There is strong evidence that the laws of black hole mechanics are, in fact, a subset of the laws of thermodynamics, and that the black hole area is proportional to its entropy.[161] This leads to a modification of the original laws of black hole mechanics: for instance, as the second law of black hole mechanics becomes part of the second law of thermodynamics, it is possible for black hole area to decrease—as long as other processes ensure that, overall, entropy increases. As thermodynamical objects with non-zero temperature, black holes should emit thermal radiation. Semi-classical calculations indicate that indeed they do, with the surface gravity playing the role of temperature in Planck's law. This radiation is known as Hawking radiation (cf. the quantum theory section, below).[162]
There are other types of horizons. In an expanding universe, an observer may find that some regions of the past cannot be observed ("particle horizon"), and some regions of the future cannot be influenced (event horizon).[163] Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons associated with a semi-classical radiation known as Unruh radiation.[164]
Singularities
Another general feature of general relativity is the appearance of spacetime boundaries known as singularities. Spacetime can be explored by following up on timelike and lightlike geodesics—all possible ways that light and particles in free fall can travel. But some solutions of Einstein's equations have "ragged edges"—regions known as spacetime singularities, where the paths of light and falling particles come to an abrupt end, and geometry becomes ill-defined. In the more interesting cases, these are "curvature singularities", where geometrical quantities characterizing spacetime curvature, such as the Ricci scalar, take on infinite values.[165] Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,[166] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole.[167] The Friedmann–Lemaître–Robertson–Walker solutions and other spacetimes describing universes have past singularities on which worldlines begin, namely Big Bang singularities, and some have future singularities (Big Crunch) as well.[168]
Given that these examples are all highly symmetric—and thus simplified—it is tempting to conclude that the occurrence of singularities is an artifact of idealization.[169] The famous singularity theorems, proved using the methods of global geometry, say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage[170] and also at the beginning of a wide class of expanding universes.[171] However, the theorems say little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e.g. by the BKL conjecture).[172] The cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity.[173]
Evolution equations
Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole, possibly matter-filled, spacetime. It describes the state of matter and geometry everywhere and at every moment in that particular universe. Due to its general covariance, Einstein's theory is not sufficient by itself to determine the time evolution of the metric tensor. It must be combined with a coordinate condition, which is analogous to gauge fixing in other field theories.[174]
To understand Einstein's equations as partial differential equations, it is helpful to formulate them in a way that describes the evolution of the universe over time. This is done in "3+1" formulations, where spacetime is split into three space dimensions and one time dimension. The best-known example is the ADM formalism.[175] These decompositions show that the spacetime evolution equations of general relativity are well-behaved: solutions always exist, and are uniquely defined, once suitable initial conditions have been specified.[176] Such formulations of Einstein's field equations are the basis of numerical relativity.[177]
Global and quasi-local quantities
The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). The main reason is that the gravitational field—like any physical field—must be ascribed a certain energy, but that it proves to be fundamentally impossible to localize that energy.[178]
Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)[179] or suitable symmetries (Komar mass).[180] If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the Bondi mass at null infinity.[181] Just as in classical physics, it can be shown that these masses are positive.[182] Corresponding global definitions exist for momentum and angular momentum.[183] There have also been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture.[184]
Relationship with quantum theory
If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid-state physics, would be the other.[185] However, how to reconcile quantum theory with general relativity is still an open question. The only equation we have is Dirac's which deals with electrons traveling at relativistic velocity.
Quantum field theory in curved spacetime
Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth.[186] In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.[187] Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation leading to the possibility that they evaporate over time.[188] As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes.[189]
Quantum gravity
See also: String theory, Canonical general relativity, Loop quantum gravity, Causal dynamical triangulation, and Causal sets
The demand for consistency between a quantum description of matter and a geometric description of spacetime,[190] as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics.[191] Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.[192][193]
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems.[194] Some have argued that at low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity.[195] At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative non-renormalizability").[196]
One attempt to overcome these limitations is string theory, a quantum theory not of point particles, but of minute one-dimensional extended objects.[197] The theory promises to be a unified description of all particles and interactions, including gravity;[198] the price to pay is unusual features such as six extra dimensions of space in addition to the usual three.[199] In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity[200] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.[201]
Another approach starts with the canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf. evolution equations above), the result is the Wheeler–deWitt equation (an analogue of the Schrödinger equation) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff.[202] However, with the introduction of what are now known as Ashtekar variables,[203] this leads to a promising model known as loop quantum gravity. Space is represented by a web-like structure called a spin network, evolving over time in discrete steps.[204]
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,[205] there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman Path Integral approach and Regge calculus,[192] dynamical triangulations,[206] causal sets,[207] twistor models[208] or the path integral based models of quantum cosmology.[209]
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.[210]
Current status
General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed many unambiguous observational and experimental tests. However, there are strong indications that the theory is incomplete.[211] The problem of quantum gravity and the question of the reality of spacetime singularities remain open.[212] Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics.[213]
Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,[214] while numerical relativists run increasingly powerful computer simulations (such as those describing merging black holes).[215] In February 2016, it was announced that the existence of gravitational waves was directly detected by the Advanced LIGO team on 14 September 2015.[83][216][217] A century after its introduction, general relativity remains a highly active area of research.[218]
See also
• Alcubierre drive – Hypothetical mode of transportation by warping space (warp drive)
• Alternatives to general relativity – Proposed theories of gravity
• Contributors to general relativity
• Derivations of the Lorentz transformations
• Ehrenfest paradox – Paradox in special relativity
• Einstein–Hilbert action – Concept in general relativity
• Einstein's thought experiments – Albert Einstein's hypothetical situations to argue scientific points
• General relativity priority dispute – Debate about credit for general relativity
• Introduction to the mathematics of general relativity – non-technical introduction to the mathematics of general relativityPages displaying wikidata descriptions as a fallback
• Nordström's theory of gravitation – Predecessor to the theory of relativity
• Ricci calculus – Extension of vector calculus to tensors
• Timeline of gravitational physics and relativity
• Weak Gravity Conjecture – Conjecture that gravity must be the weakest forcePages displaying short descriptions of redirect targets
References
1. "GW150914: LIGO Detects Gravitational Waves". Black-holes.org. Retrieved 18 April 2016.
2. Albert Einstein (2011). Relativity – The Special and General Theory. Read Books Ltd. p. 4. ISBN 978-1-4474-9358-7. Extract of page 4
3. Landau & Lifshitz 1975, p. 228 "...the general theory of relativity...was established by Einstein, and represents probably the most beautiful of all existing physical theories."
4. Poincaré 1905
5. O'Connor, J.J.; Robertson, E.F. (May 1996). "General relativity]". History Topics: Mathematical Physics Index, Scotland: School of Mathematics and Statistics, University of St. Andrews, archived from the original on 4 February 2015, retrieved 4 February 2015
6. Pais 1982, ch. 9 to 15, Janssen 2005; an up-to-date collection of current research, including reprints of many of the original articles, is Renn 2007; an accessible overview can be found in Renn 2005, pp. 110ff. Einstein's original papers are found in Digital Einstein, volumes 4 and 6. An early key article is Einstein 1907, cf. Pais 1982, ch. 9. The publication featuring the field equations is Einstein 1915, cf. Pais 1982, ch. 11–15
7. Moshe Carmeli (2008).Relativity: Modern Large-Scale Structures of the Cosmos. pp.92, 93.World Scientific Publishing
8. Grossmann for the mathematical part and Einstein for the physical part (1913). Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation), Zeitschrift für Mathematik und Physik, 62, 225–261. English translate
9. Schwarzschild 1916a, Schwarzschild 1916b and Reissner 1916 (later complemented in Nordström 1918)
10. Einstein 1917, cf. Pais 1982, ch. 15e
11. Hubble's original article is Hubble 1929; an accessible overview is given in Singh 2004, ch. 2–4
12. As reported in Gamow 1970. Einstein's condemnation would prove to be premature, cf. the section Cosmology, below
13. Pais 1982, pp. 253–254
14. Kennefick 2005, Kennefick 2007
15. Pais 1982, ch. 16
16. Thorne 2003, p. 74
17. Israel 1987, ch. 7.8–7.10, Thorne 1994, ch. 3–9
18. Sections Orbital effects and the relativity of direction, Gravitational time dilation and frequency shift and Light deflection and gravitational time delay, and references therein
19. Section Cosmology and references therein; the historical development is in Overbye 1999
20. Wald 1984, p. 3
21. Rovelli 2015, pp. 1–6 "General relativity is not just an extraordinarily beautiful physical theory providing the best description of the gravitational interaction we have so far. It is more."
22. Chandrasekhar 1984, p. 6
23. Engler 2002
24. The following exposition re-traces that of Ehlers 1973, sec. 1
25. Al-Khalili, Jim (26 March 2021). "Gravity and Me: The force that shapes our lives". www.bbc.co.uk. Retrieved 9 April 2021.{{cite web}}: CS1 maint: url-status (link)
26. Arnold 1989, ch. 1
27. Ehlers 1973, pp. 5f
28. Will 1993, sec. 2.4, Will 2006, sec. 2
29. Wheeler 1990, ch. 2
30. Ehlers 1973, sec. 1.2, Havas 1964, Künzle 1972. The simple thought experiment in question was first described in Heckmann & Schücking 1959
31. Ehlers 1973, pp. 10f
32. Good introductions are, in order of increasing presupposed knowledge of mathematics, Giulini 2005, Mermin 2005, and Rindler 1991; for accounts of precision experiments, cf. part IV of Ehlers & Lämmerzahl 2006
33. An in-depth comparison between the two symmetry groups can be found in Giulini 2006
34. Rindler 1991, sec. 22, Synge 1972, ch. 1 and 2
35. Ehlers 1973, sec. 2.3
36. Ehlers 1973, sec. 1.4, Schutz 1985, sec. 5.1
37. Ehlers 1973, pp. 17ff; a derivation can be found in Mermin 2005, ch. 12. For the experimental evidence, cf. the section Gravitational time dilation and frequency shift, below
38. Rindler 2001, sec. 1.13; for an elementary account, see Wheeler 1990, ch. 2; there are, however, some differences between the modern version and Einstein's original concept used in the historical derivation of general relativity, cf. Norton 1985
39. Ehlers 1973, sec. 1.4 for the experimental evidence, see once more section Gravitational time dilation and frequency shift. Choosing a different connection with non-zero torsion leads to a modified theory known as Einstein–Cartan theory
40. Ehlers 1973, p. 16, Kenyon 1990, sec. 7.2, Weinberg 1972, sec. 2.8
41. Ehlers 1973, pp. 19–22; for similar derivations, see sections 1 and 2 of ch. 7 in Weinberg 1972. The Einstein tensor is the only divergence-free tensor that is a function of the metric coefficients, their first and second derivatives at most, and allows the spacetime of special relativity as a solution in the absence of sources of gravity, cf. Lovelock 1972. The tensors on both side are of second rank, that is, they can each be thought of as 4×4 matrices, each of which contains ten independent terms; hence, the above represents ten coupled equations. The fact that, as a consequence of geometric relations known as Bianchi identities, the Einstein tensor satisfies a further four identities reduces these to six independent equations, e.g. Schutz 1985, sec. 8.3
42. Kenyon 1990, sec. 7.4
43. Weinberg, Steven (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley. ISBN 978-0-471-92567-5.
44. Cheng, Ta-Pei (2005). Relativity, Gravitation and Cosmology: a Basic Introduction. Oxford and New York: Oxford University Press. ISBN 978-0-19-852957-6.
45. Brans & Dicke 1961, Weinberg 1972, sec. 3 in ch. 7, Goenner 2004, sec. 7.2, and Trautman 2006, respectively
46. Wald 1984, ch. 4, Weinberg 1972, ch. 7 or, in fact, any other textbook on general relativity
47. At least approximately, cf. Poisson 2004a
48. Wheeler 1990, p. xi
49. Wald 1984, sec. 4.4
50. Wald 1984, sec. 4.1
51. For the (conceptual and historical) difficulties in defining a general principle of relativity and separating it from the notion of general covariance, see Giulini 2007
52. section 5 in ch. 12 of Weinberg 1972
53. Introductory chapters of Stephani et al. 2003
54. A review showing Einstein's equation in the broader context of other PDEs with physical significance is Geroch 1996
55. For background information and a list of solutions, cf. Stephani et al. 2003; a more recent review can be found in MacCallum 2006
56. Chandrasekhar 1983, ch. 3,5,6
57. Narlikar 1993, ch. 4, sec. 3.3
58. Brief descriptions of these and further interesting solutions can be found in Hawking & Ellis 1973, ch. 5
59. Lehner 2002
60. For instance Wald 1984, sec. 4.4
61. Will 1993, sec. 4.1 and 4.2
62. Will 2006, sec. 3.2, Will 1993, ch. 4
63. Rindler 2001, pp. 24–26 vs. pp. 236–237 and Ohanian & Ruffini 1994, pp. 164–172. Einstein derived these effects using the equivalence principle as early as 1907, cf. Einstein 1907 and the description in Pais 1982, pp. 196–198
64. Rindler 2001, pp. 24–26; Misner, Thorne & Wheeler 1973, § 38.5
65. Pound–Rebka experiment, see Pound & Rebka 1959, Pound & Rebka 1960; Pound & Snider 1964; a list of further experiments is given in Ohanian & Ruffini 1994, table 4.1 on p. 186
66. Greenstein, Oke & Shipman 1971; the most recent and most accurate Sirius B measurements are published in Barstow, Bond et al. 2005.
67. Starting with the Hafele–Keating experiment, Hafele & Keating 1972a and Hafele & Keating 1972b, and culminating in the Gravity Probe A experiment; an overview of experiments can be found in Ohanian & Ruffini 1994, table 4.1 on p. 186
68. GPS is continually tested by comparing atomic clocks on the ground and aboard orbiting satellites; for an account of relativistic effects, see Ashby 2002 and Ashby 2003
69. Stairs 2003 and Kramer 2004
70. General overviews can be found in section 2.1. of Will 2006; Will 2003, pp. 32–36; Ohanian & Ruffini 1994, sec. 4.2
71. Ohanian & Ruffini 1994, pp. 164–172
72. Cf. Kennefick 2005 for the classic early measurements by Arthur Eddington's expeditions. For an overview of more recent measurements, see Ohanian & Ruffini 1994, ch. 4.3. For the most precise direct modern observations using quasars, cf. Shapiro et al. 2004
73. This is not an independent axiom; it can be derived from Einstein's equations and the Maxwell Lagrangian using a WKB approximation, cf. Ehlers 1973, sec. 5
74. Blanchet 2006, sec. 1.3
75. Rindler 2001, sec. 1.16; for the historical examples, Israel 1987, pp. 202–204; in fact, Einstein published one such derivation as Einstein 1907. Such calculations tacitly assume that the geometry of space is Euclidean, cf. Ehlers & Rindler 1997
76. From the standpoint of Einstein's theory, these derivations take into account the effect of gravity on time, but not its consequences for the warping of space, cf. Rindler 2001, sec. 11.11
77. For the Sun's gravitational field using radar signals reflected from planets such as Venus and Mercury, cf. Shapiro 1964, Weinberg 1972, ch. 8, sec. 7; for signals actively sent back by space probes (transponder measurements), cf. Bertotti, Iess & Tortora 2003; for an overview, see Ohanian & Ruffini 1994, table 4.4 on p. 200; for more recent measurements using signals received from a pulsar that is part of a binary system, the gravitational field causing the time delay being that of the other pulsar, cf. Stairs 2003, sec. 4.4
78. Will 1993, sec. 7.1 and 7.2
79. Einstein, A (22 June 1916). "Näherungsweise Integration der Feldgleichungen der Gravitation". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin (part 1): 688–696. Bibcode:1916SPAW.......688E. Archived from the original on 21 March 2019. Retrieved 12 February 2016.
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81. Castelvecchi, Davide; Witze, Witze (11 February 2016). "Einstein's gravitational waves found at last". Nature News. doi:10.1038/nature.2016.19361. S2CID 182916902. Retrieved 11 February 2016.
82. B. P. Abbott; et al. (LIGO Scientific Collaboration and Virgo Collaboration) (2016). "Observation of Gravitational Waves from a Binary Black Hole Merger". Physical Review Letters. 116 (6): 061102. arXiv:1602.03837. Bibcode:2016PhRvL.116f1102A. doi:10.1103/PhysRevLett.116.061102. PMID 26918975. S2CID 124959784.
83. "Gravitational waves detected 100 years after Einstein's prediction". NSF – National Science Foundation. 11 February 2016.
84. Most advanced textbooks on general relativity contain a description of these properties, e.g. Schutz 1985, ch. 9
85. For example Jaranowski & Królak 2005
86. Rindler 2001, ch. 13
87. Gowdy 1971, Gowdy 1974
88. See Lehner 2002 for a brief introduction to the methods of numerical relativity, and Seidel 1998 for the connection with gravitational wave astronomy
89. Schutz 2003, pp. 48–49, Pais 1982, pp. 253–254
90. Rindler 2001, sec. 11.9
91. Will 1993, pp. 177–181
92. In consequence, in the parameterized post-Newtonian formalism (PPN), measurements of this effect determine a linear combination of the terms β and γ, cf. Will 2006, sec. 3.5 and Will 1993, sec. 7.3
93. The most precise measurements are VLBI measurements of planetary positions; see Will 1993, ch. 5, Will 2006, sec. 3.5, Anderson et al. 1992; for an overview, Ohanian & Ruffini 1994, pp. 406–407
94. Kramer et al. 2006
95. Dediu, Magdalena & Martín-Vide 2015, p. 141.
96. Kramer, M.; Stairs, I. H.; Manchester, R. N.; Wex, N.; Deller, A. T.; Coles, W. A.; Ali, M.; Burgay, M.; Camilo, F.; Cognard, I.; Damour, T. (13 December 2021). "Strong-Field Gravity Tests with the Double Pulsar". Physical Review X. 11 (4): 041050. arXiv:2112.06795. Bibcode:2021PhRvX..11d1050K. doi:10.1103/PhysRevX.11.041050. ISSN 2160-3308. S2CID 245124502.
97. Stairs 2003, Schutz 2003, pp. 317–321, Bartusiak 2000, pp. 70–86
98. Weisberg & Taylor 2003; for the pulsar discovery, see Hulse & Taylor 1975; for the initial evidence for gravitational radiation, see Taylor 1994
99. Kramer 2004
100. Penrose 2004, § 14.5, Misner, Thorne & Wheeler 1973, § 11.4
101. Weinberg 1972, sec. 9.6, Ohanian & Ruffini 1994, sec. 7.8
102. Bertotti, Ciufolini & Bender 1987, Nordtvedt 2003
103. Kahn 2007
104. A mission description can be found in Everitt et al. 2001; a first post-flight evaluation is given in Everitt, Parkinson & Kahn 2007; further updates will be available on the mission website Kahn 1996–2012.
105. Townsend 1997, sec. 4.2.1, Ohanian & Ruffini 1994, pp. 469–471
106. Ohanian & Ruffini 1994, sec. 4.7, Weinberg 1972, sec. 9.7; for a more recent review, see Schäfer 2004
107. Ciufolini & Pavlis 2004, Ciufolini, Pavlis & Peron 2006, Iorio 2009
108. Iorio 2006, Iorio 2010
109. Einstein, Relativity, and Absolute Simultaneity. London: Routledge. 2007. ISBN 978-1-134-00389-1.
110. For overviews of gravitational lensing and its applications, see Ehlers, Falco & Schneider 1992 and Wambsganss 1998
111. For a simple derivation, see Schutz 2003, ch. 23; cf. Narayan & Bartelmann 1997, sec. 3
112. Walsh, Carswell & Weymann 1979
113. Images of all the known lenses can be found on the pages of the CASTLES project, Kochanek et al. 2007
114. Roulet & Mollerach 1997
115. Narayan & Bartelmann 1997, sec. 3.7
116. Barish 2005, Bartusiak 2000, Blair & McNamara 1997
117. Hough & Rowan 2000
118. Hobbs, George; Archibald, A.; Arzoumanian, Z.; Backer, D.; Bailes, M.; Bhat, N. D. R.; Burgay, M.; Burke-Spolaor, S.; et al. (2010), "The international pulsar timing array project: using pulsars as a gravitational wave detector", Classical and Quantum Gravity, 27 (8): 084013, arXiv:0911.5206, Bibcode:2010CQGra..27h4013H, doi:10.1088/0264-9381/27/8/084013, S2CID 56073764
119. Danzmann & Rüdiger 2003
120. "LISA pathfinder overview". ESA. Retrieved 23 April 2012.
121. Thorne 1995
122. Cutler & Thorne 2002
123. Miller 2002, lectures 19 and 21
124. Celotti, Miller & Sciama 1999, sec. 3
125. Springel et al. 2005 and the accompanying summary Gnedin 2005
126. Blandford 1987, sec. 8.2.4
127. For the basic mechanism, see Carroll & Ostlie 1996, sec. 17.2; for more about the different types of astronomical objects associated with this, cf. Robson 1996
128. For a review, see Begelman, Blandford & Rees 1984. To a distant observer, some of these jets even appear to move faster than light; this, however, can be explained as an optical illusion that does not violate the tenets of relativity, see Rees 1966
129. For stellar end states, cf. Oppenheimer & Snyder 1939 or, for more recent numerical work, Font 2003, sec. 4.1; for supernovae, there are still major problems to be solved, cf. Buras et al. 2003; for simulating accretion and the formation of jets, cf. Font 2003, sec. 4.2. Also, relativistic lensing effects are thought to play a role for the signals received from X-ray pulsars, cf. Kraus 1998
130. The evidence includes limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"), see Celotti, Miller & Sciama 1999, observations of stellar dynamics in the center of our own Milky Way galaxy, cf. Schödel et al. 2003, and indications that at least some of the compact objects in question appear to have no solid surface, which can be deduced from the examination of X-ray bursts for which the central compact object is either a neutron star or a black hole; cf. Remillard et al. 2006 for an overview, Narayan 2006, sec. 5. Observations of the "shadow" of the Milky Way galaxy's central black hole horizon are eagerly sought for, cf. Falcke, Melia & Agol 2000
131. Dalal et al. 2006
132. Barack & Cutler 2004
133. Einstein 1917; cf. Pais 1982, pp. 285–288
134. Carroll 2001, ch. 2
135. Bergström & Goobar 2003, ch. 9–11; use of these models is justified by the fact that, at large scales of around hundred million light-years and more, our own universe indeed appears to be isotropic and homogeneous, cf. Peebles et al. 1991
136. E.g. with WMAP data, see Spergel et al. 2003
137. These tests involve the separate observations detailed further on, see, e.g., fig. 2 in Bridle et al. 2003
138. Peebles 1966; for a recent account of predictions, see Coc, Vangioni‐Flam et al. 2004; an accessible account can be found in Weiss 2006; compare with the observations in Olive & Skillman 2004, Bania, Rood & Balser 2002, O'Meara et al. 2001, and Charbonnel & Primas 2005
139. Lahav & Suto 2004, Bertschinger 1998, Springel et al. 2005
140. Alpher & Herman 1948, for a pedagogical introduction, see Bergström & Goobar 2003, ch. 11; for the initial detection, see Penzias & Wilson 1965 and, for precision measurements by satellite observatories, Mather et al. 1994 (COBE) and Bennett et al. 2003 (WMAP). Future measurements could also reveal evidence about gravitational waves in the early universe; this additional information is contained in the background radiation's polarization, cf. Kamionkowski, Kosowsky & Stebbins 1997 and Seljak & Zaldarriaga 1997
141. Evidence for this comes from the determination of cosmological parameters and additional observations involving the dynamics of galaxies and galaxy clusters cf. Peebles 1993, ch. 18, evidence from gravitational lensing, cf. Peacock 1999, sec. 4.6, and simulations of large-scale structure formation, see Springel et al. 2005
142. Peacock 1999, ch. 12, Peskin 2007; in particular, observations indicate that all but a negligible portion of that matter is not in the form of the usual elementary particles ("non-baryonic matter"), cf. Peacock 1999, ch. 12
143. Namely, some physicists have questioned whether or not the evidence for dark matter is, in fact, evidence for deviations from the Einsteinian (and the Newtonian) description of gravity cf. the overview in Mannheim 2006, sec. 9
144. Carroll 2001; an accessible overview is given in Caldwell 2004. Here, too, scientists have argued that the evidence indicates not a new form of energy, but the need for modifications in our cosmological models, cf. Mannheim 2006, sec. 10; aforementioned modifications need not be modifications of general relativity, they could, for example, be modifications in the way we treat the inhomogeneities in the universe, cf. Buchert 2008
145. A good introduction is Linde 2005; for a more recent review, see Linde 2006
146. More precisely, these are the flatness problem, the horizon problem, and the monopole problem; a pedagogical introduction can be found in Narlikar 1993, sec. 6.4, see also Börner 1993, sec. 9.1
147. Spergel et al. 2007, sec. 5,6
148. More concretely, the potential function that is crucial to determining the dynamics of the inflaton is simply postulated, but not derived from an underlying physical theory
149. Brandenberger 2008, sec. 2
150. Gödel 1949
151. Finazzi, Stefano; Liberati, Stefano; Barceló, Carlos (15 June 2009). "Semiclassical instability of dynamical warp drives". Physical Review D. 79 (12): 124017. arXiv:0904.0141. Bibcode:2009PhRvD..79l4017F. doi:10.1103/PhysRevD.79.124017. S2CID 59575856.
152. Bondi, H.; Van der Burg, M.G.J.; Metzner, A. (1962). "Gravitational waves in general relativity: VII. Waves from axisymmetric isolated systems". Proceedings of the Royal Society of London A. A269 (1336): 21–52. Bibcode:1962RSPSA.269...21B. doi:10.1098/rspa.1962.0161. S2CID 120125096.
153. Sachs, R. (1962). "Asymptotic symmetries in gravitational theory". Physical Review. 128 (6): 2851–2864. Bibcode:1962PhRv..128.2851S. doi:10.1103/PhysRev.128.2851.
154. Strominger, Andrew (2017). "Lectures on the Infrared Structure of Gravity and Gauge Theory". arXiv:1703.05448 [hep-th]. ...redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. To be published Princeton University Press, 158 pages.
155. Frauendiener 2004, Wald 1984, sec. 11.1, Hawking & Ellis 1973, sec. 6.8, 6.9
156. Wald 1984, sec. 9.2–9.4 and Hawking & Ellis 1973, ch. 6
157. Thorne 1972; for more recent numerical studies, see Berger 2002, sec. 2.1
158. Israel 1987. A more exact mathematical description distinguishes several kinds of horizon, notably event horizons and apparent horizons cf. Hawking & Ellis 1973, pp. 312–320 or Wald 1984, sec. 12.2; there are also more intuitive definitions for isolated systems that do not require knowledge of spacetime properties at infinity, cf. Ashtekar & Krishnan 2004
159. For first steps, cf. Israel 1971; see Hawking & Ellis 1973, sec. 9.3 or Heusler 1996, ch. 9 and 10 for a derivation, and Heusler 1998 as well as Beig & Chruściel 2006 as overviews of more recent results
160. The laws of black hole mechanics were first described in Bardeen, Carter & Hawking 1973; a more pedagogical presentation can be found in Carter 1979; for a more recent review, see Wald 2001, ch. 2. A thorough, book-length introduction including an introduction to the necessary mathematics Poisson 2004. For the Penrose process, see Penrose 1969
161. Bekenstein 1973, Bekenstein 1974
162. The fact that black holes radiate, quantum mechanically, was first derived in Hawking 1975; a more thorough derivation can be found in Wald 1975. A review is given in Wald 2001, ch. 3
163. Narlikar 1993, sec. 4.4.4, 4.4.5
164. Horizons: cf. Rindler 2001, sec. 12.4. Unruh effect: Unruh 1976, cf. Wald 2001, ch. 3
165. Hawking & Ellis 1973, sec. 8.1, Wald 1984, sec. 9.1
166. Townsend 1997, ch. 2; a more extensive treatment of this solution can be found in Chandrasekhar 1983, ch. 3
167. Townsend 1997, ch. 4; for a more extensive treatment, cf. Chandrasekhar 1983, ch. 6
168. Ellis & Van Elst 1999; a closer look at the singularity itself is taken in Börner 1993, sec. 1.2
169. Here one should remind to the well-known fact that the important "quasi-optical" singularities of the so-called eikonal approximations of many wave equations, namely the "caustics", are resolved into finite peaks beyond that approximation.
170. Namely when there are trapped null surfaces, cf. Penrose 1965
171. Hawking 1966
172. The conjecture was made in Belinskii, Khalatnikov & Lifschitz 1971; for a more recent review, see Berger 2002. An accessible exposition is given by Garfinkle 2007
173. The restriction to future singularities naturally excludes initial singularities such as the big bang singularity, which in principle be visible to observers at later cosmic time. The cosmic censorship conjecture was first presented in Penrose 1969; a textbook-level account is given in Wald 1984, pp. 302–305. For numerical results, see the review Berger 2002, sec. 2.1
174. Hawking & Ellis 1973, sec. 7.1
175. Arnowitt, Deser & Misner 1962; for a pedagogical introduction, see Misner, Thorne & Wheeler 1973, § 21.4–§ 21.7
176. Fourès-Bruhat 1952 and Bruhat 1962; for a pedagogical introduction, see Wald 1984, ch. 10; an online review can be found in Reula 1998
177. Gourgoulhon 2007; for a review of the basics of numerical relativity, including the problems arising from the peculiarities of Einstein's equations, see Lehner 2001
178. Misner, Thorne & Wheeler 1973, § 20.4
179. Arnowitt, Deser & Misner 1962
180. Komar 1959; for a pedagogical introduction, see Wald 1984, sec. 11.2; although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes, cf. Ashtekar & Magnon-Ashtekar 1979
181. For a pedagogical introduction, see Wald 1984, sec. 11.2
182. Wald 1984, p. 295 and refs therein; this is important for questions of stability—if there were negative mass states, then flat, empty Minkowski space, which has mass zero, could evolve into these states
183. Townsend 1997, ch. 5
184. Such quasi-local mass–energy definitions are the Hawking energy, Geroch energy, or Penrose's quasi-local energy–momentum based on twistor methods; cf. the review article Szabados 2004
185. An overview of quantum theory can be found in standard textbooks such as Messiah 1999; a more elementary account is given in Hey & Walters 2003
186. Ramond 1990, Weinberg 1995, Peskin & Schroeder 1995; a more accessible overview is Auyang 1995
187. Wald 1994, Birrell & Davies 1984
188. For Hawking radiation Hawking 1975, Wald 1975; an accessible introduction to black hole evaporation can be found in Traschen 2000
189. Wald 2001, ch. 3
190. Put simply, matter is the source of spacetime curvature, and once matter has quantum properties, we can expect spacetime to have them as well. Cf. Carlip 2001, sec. 2
191. Schutz 2003, p. 407
192. Hamber 2009
193. A timeline and overview can be found in Rovelli 2000
194. 't Hooft & Veltman 1974
195. Donoghue 1995
196. In particular, a perturbative technique known as renormalization, an integral part of deriving predictions which take into account higher-energy contributions, cf. Weinberg 1996, ch. 17, 18, fails in this case; cf. Veltman 1975, Goroff & Sagnotti 1985; for a recent comprehensive review of the failure of perturbative renormalizability for quantum gravity see Hamber 2009
197. An accessible introduction at the undergraduate level can be found in Zwiebach 2004; more complete overviews can be found in Polchinski 1998a and Polchinski 1998b
198. At the energies reached in current experiments, these strings are indistinguishable from point-like particles, but, crucially, different modes of oscillation of one and the same type of fundamental string appear as particles with different (electric and other) charges, e.g. Ibanez 2000. The theory is successful in that one mode will always correspond to a graviton, the messenger particle of gravity, e.g. Green, Schwarz & Witten 1987, sec. 2.3, 5.3
199. Green, Schwarz & Witten 1987, sec. 4.2
200. Weinberg 2000, ch. 31
201. Townsend 1996, Duff 1996
202. Kuchař 1973, sec. 3
203. These variables represent geometric gravity using mathematical analogues of electric and magnetic fields; cf. Ashtekar 1986, Ashtekar 1987
204. For a review, see Thiemann 2007; more extensive accounts can be found in Rovelli 1998, Ashtekar & Lewandowski 2004 as well as in the lecture notes Thiemann 2003
205. Isham 1994, Sorkin 1997
206. Loll 1998
207. Sorkin 2005
208. Penrose 2004, ch. 33 and refs therein
209. Hawking 1987
210. Ashtekar 2007, Schwarz 2007
211. Maddox 1998, pp. 52–59, 98–122; Penrose 2004, sec. 34.1, ch. 30
212. section Quantum gravity, above
213. section Cosmology, above
214. Friedrich 2005
215. A review of the various problems and the techniques being developed to overcome them, see Lehner 2002
216. See Bartusiak 2000 for an account up to that year; up-to-date news can be found on the websites of major detector collaborations such as GEO600 and LIGO
217. For the most recent papers on gravitational wave polarizations of inspiralling compact binaries, see Blanchet et al. 2008, and Arun et al. 2008; for a review of work on compact binaries, see Blanchet 2006 and Futamase & Itoh 2006; for a general review of experimental tests of general relativity, see Will 2006
218. See, e.g., the Living Reviews in Relativity journal.
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Further reading
Popular books
• Einstein, A. (1916), Relativity: The Special and the General Theory, Berlin, ISBN 978-3-528-06059-6{{citation}}: CS1 maint: location missing publisher (link)
• Geroch, R. (1981), General Relativity from A to B, Chicago: University of Chicago Press, ISBN 978-0-226-28864-2
• Lieber, Lillian (2008), The Einstein Theory of Relativity: A Trip to the Fourth Dimension, Philadelphia: Paul Dry Books, Inc., ISBN 978-1-58988-044-3
• Schutz, Bernard F. (2001), "Gravitational radiation", in Murdin, Paul (ed.), Encyclopedia of Astronomy and Astrophysics, ISBN 978-1-56159-268-5
• Thorne, Kip; Hawking, Stephen (1994). Black Holes and Time Warps: Einstein's Outrageous Legacy. New York: W. W. Norton. ISBN 0-393-03505-0.
• Wald, Robert M. (1992), Space, Time, and Gravity: the Theory of the Big Bang and Black Holes, Chicago: University of Chicago Press, ISBN 978-0-226-87029-8
• Wheeler, John; Ford, Kenneth (1998), Geons, Black Holes, & Quantum Foam: a life in physics, New York: W. W. Norton, ISBN 978-0-393-31991-0
Beginning undergraduate textbooks
• Callahan, James J. (2000), The Geometry of Spacetime: an Introduction to Special and General Relativity, New York: Springer, ISBN 978-0-387-98641-8
• Taylor, Edwin F.; Wheeler, John Archibald (2000), Exploring Black Holes: Introduction to General Relativity, Addison Wesley, ISBN 978-0-201-38423-9
Advanced undergraduate textbooks
• Cheng, Ta-Pei (2005), Relativity, Gravitation and Cosmology: a Basic Introduction, Oxford and New York: Oxford University Press, ISBN 978-0-19-852957-6
• Dirac, Paul (1996), General Theory of Relativity, Princeton University Press, ISBN 978-0-691-01146-2
• Gron, O.; Hervik, S. (2007), Einstein's General theory of Relativity, Springer, ISBN 978-0-387-69199-2
• Hartle, James B. (2003), Gravity: an Introduction to Einstein's General Relativity, San Francisco: Addison-Wesley, ISBN 978-0-8053-8662-2
• Hughston, L. P.; Tod, K. P. (1991), Introduction to General Relativity, Cambridge: Cambridge University Press, ISBN 978-0-521-33943-8
• d'Inverno, Ray (1992), Introducing Einstein's Relativity, Oxford: Oxford University Press, ISBN 978-0-19-859686-8
• Ludyk, Günter (2013). Einstein in Matrix Form (1st ed.). Berlin: Springer. ISBN 978-3-642-35797-8.
• Møller, Christian (1955) [1952], The Theory of Relativity, Oxford University Press, OCLC 7644624
• Moore, Thomas A (2012), A General Relativity Workbook, University Science Books, ISBN 978-1-891389-82-5
• Schutz, B. F. (2009), A First Course in General Relativity (Second ed.), Cambridge University Press, ISBN 978-0-521-88705-2
Graduate textbooks
• Carroll, Sean M. (2004), Spacetime and Geometry: An Introduction to General Relativity, San Francisco: Addison-Wesley, ISBN 978-0-8053-8732-2
• Grøn, Øyvind; Hervik, Sigbjørn (2007), Einstein's General Theory of Relativity, New York: Springer, ISBN 978-0-387-69199-2
• Landau, Lev D.; Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, ISBN 978-0-7506-2768-9
• Stephani, Hans (1990), General Relativity: An Introduction to the Theory of the Gravitational Field, Cambridge: Cambridge University Press, ISBN 978-0-521-37941-0
• Will, Clifford; Poisson, Eric (2014). Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press. ISBN 978-1-107-03286-6.
• Charles W. Misner; Kip S. Thorne; John Archibald Wheeler (1973), Gravitation, W. H. Freeman, Princeton University Press, ISBN 0-7167-0344-0
• R.K. Sachs; H. Wu (1977), General Relativity for Mathematicians, Springer-Verlag, ISBN 1461299055
• Wald, Robert M. (1984). General Relativity. Chicago: University of Chicago Press. ISBN 0-226-87032-4. OCLC 10018614.
Specialists' books
• Hawking, Stephen; Ellis, George (1975). The Large Scale Structure of Space-time. Cambridge University Press. ISBN 978-0-521-09906-6.
• Poisson, Eric (2007). A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press. ISBN 978-0-521-53780-3.
Journal articles
• Einstein, Albert (1916), "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, 49 (7): 769–822, Bibcode:1916AnP...354..769E, doi:10.1002/andp.19163540702 See also English translation at Einstein Papers Project
• Flanagan, Éanna É.; Hughes, Scott A. (2005), "The basics of gravitational wave theory", New J. Phys., 7 (1): 204, arXiv:gr-qc/0501041, Bibcode:2005NJPh....7..204F, doi:10.1088/1367-2630/7/1/204
• Landgraf, M.; Hechler, M.; Kemble, S. (2005), "Mission design for LISA Pathfinder", Class. Quantum Grav., 22 (10): S487–S492, arXiv:gr-qc/0411071, Bibcode:2005CQGra..22S.487L, doi:10.1088/0264-9381/22/10/048, S2CID 119476595
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| Wikipedia |
\begin{document}
\title{An Improvement of the lower bound for the minimum number of link colorings by quandles}
\begin{abstract} We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of ``Colorings beyond Fox: The other linear Alexander quandles'' (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree $k$ of the reduced Alexander polynomial of the considered knot. We show that it is exactly $k+1$ for L-space knots. Then we apply these results to torus knots and Pretzel knots $P(-2,3,2l+1)$, $l\ge 0$. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more that one component. \end{abstract}
\section{Introduction} The idea of coloring knots was initiated by R. Fox around 1960 (see \cite{fox1962quick}). He introduced colorings by dihedral quandles called $n$-colorings. Let $K$ be a $p$-colorable knot, where $p$ is prime and let $C_p(K)$ denote the minimal number of colors needed to color a diagram of $K$. Nakamura et al. proved in \cite{Naka_Nakani_Satoh} that $C_p(K)\ge\lfloor\log_2 p\rfloor +2$.\\ The problem of finding the minimum number of colors for $p$-colorable knots with primes up to 19 was investigated by many authors. In 2009, S. Satoh showed in \cite{satoh20095} that $C_5(K)=4$. In 2010, K. Oshiro proved that $C_7(K)=4$ \cite{oshiro2010any}. In 2016, T. Nakamura, Y. Nakanishi and S. Satoh showed in \cite{satoh2016} that $C_{11}(K)=5$. In 2017, M. Elhamdadi and J. Kerr \cite{elhamdadi2016fox} and independently F. Bento and P. Lopes \cite{bento2017minimum} proved that $C_{13}(K)=5$. In 2020, H. Abchir, M. Elhamdadi and S. Lamsifer \cite{abch_elham_lamsifer} showed that $C_{17}(K)=6$. In 2022, Y. Han and B. Zhou showed that $C_{19}(K)=6$ \cite{Han_Zhou}.\\ The same problem may be studied for colorings by linear Alexander quandles which generalize dihedral ones. A linear Alexander quandle $\Lambda_{n,m}$ is a quandle whose underlying set is $\mathbbm{Z}_n$, $n\ge 3$, endowed with the binary operation $x*y=mx+(1-m)y\,\,\mod n$, for some integer $m$ such that $(m,n)=1$. Let $L$ be a link which admits a non-trivial coloring by $\Lambda_{n,m}$, i.e. for which there exists a non-constant quandle homomorphism from the fundamental quandle $Q(L)$ to $\Lambda_{n,m}$. We denote by $mincol_{n,m}(L)$ the minimum number of colors needed to provide a non-trivial coloring of $L$. It is an interesting invariant of $L$. For a knot $K$, if $n$ is a prime integer $p$, L. Kauffman and P. Lopes showed in \cite{Kauff_Lopes} that
$$2 + \lfloor \log_M p \rfloor \leq mincol_{p,m}(K),$$
where $M= max \lbrace \vert m \vert, \vert m-1 \vert \rbrace$.\\
We give an enhancement of the last result by proving the following theorem. \begin{Theo}\label{MainTheo}
Let $K$ be a knot. Let $\Delta_{K}^{0}(t)=\sum \limits_{\underset{}{i=0}}^{k} c_i t^i$ be the reduced Alexander polynomial of $K$. Let $m$ be an integer, such that $m>\displaystyle \max_{0\leq i \leq k} \lbrace |c_i|\rbrace +1$ and $p=\Delta_{K}^{0}(m)$ is an odd prime integer. \begin{enumerate} \item If $c_k=1$ and the penultimate non-zero coefficient is negative, then
$$k+1 \leq mincol_{p,m}(K).$$ \item If $c_k>1$ or the penultimate non-zero coefficient is positive, then
$$k+2\leq mincol_{p,m}(K).$$ \end{enumerate} \end{Theo}
So, the lower bound stabilizes for suitable choices of $m$ and $p$ and no longer depends on these two integers. On the other hand, the lower bound we give comes from a topological invariant of the knot. In particular, it is exactly $k+1$ for an $L$-space knot, for any $m > 1$. This entails that if the torus knot $T(a,b)$ whose crossing number is $c(T(a,b))$ admits a non-trivial coloring by $\Lambda_{p,m}$, then $mincol_{p,m}(T(a,b))$ is bounded as follows: $$c(T(a,b))-(a-2)\le mincol_{p,m}(T(a,b))\le c(T(a,b)).$$
Hence $mincol_{p,m}(T(2,b))=c(T(2,b))$. On the other hand, we show that for suitable choices of $m$ and $p$, $T(2,b)$ displays KH behavior, which means that $T(2,b)$ admits a reduced alternating diagram equipped with a non-trivial coloring by the quandle $\Lambda_{p,m}$ such that different arcs receive different colors. Furthermore, we show that if the Pretzel knot $P(-2,3,a)$ has a non-trivial coloring by a linear Alexander quandle $\Lambda_{p,m}$, then for suitable choices of $m$ and $p$ we have $a+4\le mincol_{p,m}(P(-2,3,a))$ and the equality holds for $m=2$. Finally, we show that Theorem 1.2 proved by Kauffman and Lopes in \cite{Kauff_Lopes} holds also for links with more than one component.\\
The paper is organized as follows. In the second section, we recall the main tools we need to prove our results. In the third section we prove our main theorem. The fourth section is devoted to some applications. In the last section, we give a generalization of Theorem 1.2 in \cite{Kauff_Lopes} to links.
\section{Preliminaries}
In this section we give an overview of the main tools we need.\\
\noindent\textbf{Quandles.}
\begin{Def} A \textit{quandle}, is a non-empty set $Q$ equipped with a binary operation $$ \begin{tabular}{cccc} $* :$ & $Q\times Q$ & $\longrightarrow$ & $Q$ \\
& $(x,y)$ & $\longmapsto$ & $x* y$ \\ \end{tabular} $$ satisfying the following three axioms: \begin{enumerate} \item For all $x\in Q$, $x* x =x$. \item For all $y\in Q$, the map $R_y:Q\rightarrow Q$ defined by $R_y(x)=x* y$, $x\in Q$, is bijective. \item For all $x, y, z \in Q$, $(x* y)* z=(x* z)* (y* z)$ (right self-distributivity). \end{enumerate} \end{Def}
\noindent We write $x*^{-1} y$ for $R_{y}^{-1}(x)$.\\ When needed, we will denote the quandle $Q$ by the pair $(Q,*)$. \begin{Exp} \textit{Trivial quandle.}\\ Let $X$ be a non-empty set with the operation $x* y=x$ for any $x, y\in X$ (i.e. $R_{y}=id_{X}$, for any $y\in X$), is a quandle called the trivial quandle. \end{Exp} \begin{Exp}\textit{Dihedral quandle.}\\ Let $n$ be a positive integer. For $x$ and $y$ in $\mathbbm{Z}_n$ (integers modulo $n$), define $x* y=2y-x \mod n$. The operation $*$ defines a quandle structure on $\mathbbm{Z}_n$, called the dihedral quandle and is sometimes denoted $R_n$. \end{Exp}
\begin{Exp}\textit{Alexander quandle.}\\
An Alexander quandle $M$ is a $\mathbbm{Z}[t,t^{-1}]$-module endowed with the following binary operation: $$x* y=tx+(1-t)y\,\, {\rm for\,\, all}\,\, x,y\in M.$$ Note that we have $x*^{-1} y=t^{-1}x+(1-t^{-1})y$.\\ \end{Exp}
\begin{Exp}\label{Exp_4}\textit{Linear Alexander quandle.}\\ Let $n$ be an integer $n>1$, one can consider $\Lambda_n/(h)$ where $\Lambda_n=\mathbbm{Z}_n[t,t^{-1}]$ and $h$ is a monic polynomial in $t$ (see \cite{Nelson}). If $h(t)=(t-m)$ and $m$ and $n$ are integers such that $\gcd{(m, n)}=1$, then we obtain what is called \textit{linear Alexander quandle} that we denote $\Lambda_{n,m}$. This amounts to considering the underlying set $\mathbbm{Z}_n$ with the binary operation: $$x* y=mx+(1-m)y \pmod n,\,\,\, {\rm for\,\, all}\,\, x,y\in \mathbb{Z}_n.$$ Note that we have $x*^{-1} y=m^{-1}x+(1-m^{-1})y \pmod n$.\\ If $m=-1$, we get the dihedral quandle $R_n$. \end{Exp}
\begin{Def} Let $(Q_1,*_1)$ and $(Q_2,*_2)$ be two quandles. A map $f:Q_1\rightarrow Q_2$ is a \textit{quandle homomorphism} if it satisfies $$f(x*_1 y)=f(x)*_2 f(y),\ \ \forall x,y\in Q_1.$$ If $f$ is bijective, we say that $f$ is a quandle \textit{isomorphism}. If $f$ is bijective and $Q_1=Q_2$, the map $f$ is called a quandle \textit{automorphism}. \end{Def}
\noindent\textbf{Coloring links by Alexander quandles}\\ Let $(Q,*)$ be a quandle and $D$ a diagram of an oriented link $L$. A coloring of $D$ by $Q$ is a map $\mathcal{C}$ from the set of arcs of $D$ denoted by $\mathcal{A}$ to $Q$, such that at each crossing of $D$, if the over-arc $\alpha_1$ is colored by $\mathcal{C}(\alpha_1)={y}$ and the incoming under-arc is colored by $\mathcal{C}(\alpha_2)=x$ then the outcoming under-arc is colored by $\mathcal{C}(\alpha_3)=x* y$ or $\mathcal{C}(\alpha_3)=x*^{-1} y$ according to the rule depicted in Fig.\ref{Fig.1}. \begin{figure}
\caption{Coloring conditions. }
\label{Fig.1}
\end{figure}
If $Q$ is an Alexander quandle, by collecting the coloring conditions at all crossings of $D$, we get a homogeneous system of linear equations over $\mathbbm{Z}[t,t^{-1}]$. The matrix associated to this system of equations is called the Alexander matrix. Its rows correspond to the crossings of $D$ and the columns correspond to the arcs of $D$. Each row has only three non-zero entries which are $t$, $1-t$ and $-1$. So on the one hand $(\lambda ,\dots ,\lambda)$ is a solution for any $\lambda\in\mathbbm{Z}[t,t^{-1}]$ (trivial solutions), and on the other hand the determinant of the Alexander matrix is zero. Hence, it is easy to see that a non-trivial solution of the initial homogeneuous system corresponds to a non-trivial solution of the system of equations determined by the original matrix with one row and one column deleted. The determinant of this last submatrix is known to be the Alexander polynomial of the considered link denoted by $\Delta_L(t)$. Therefore, the existence of non-trivial solutions corresponds to working on the quotient of $\mathbbm{Z}[t,t^{-1}]$ by $\Delta_L(t)$, which is a Laurent polynomial on the variable $t$ determined up to $\pm t^n$, for any integer $n$. We will use the reduced Alexander polynomial defined in \cite{Bae}.
\begin{Rks}
\begin{enumerate}
\item If $L$ is a knot $K$, then the reduced Alexander polynomial is exactly that given in the proof of Corollary 6.11 in \cite{Lickorish_book}, $\Delta_L(t)=c_0+c_1t+c_2t^2+\cdots +c_Nt^N$, where $N$ is even, $c_{N-r}=c_r$, $c_{\frac{N}{2}}$ is odd and $c_0>0$.
\item If $L$ has $\mu$ components, $\mu\ge 2$, the reduced Alexader polynomial is obtained from the multivariable Alexander polynomial $\Delta_L(t_1,\dots ,t_\mu)$ by setting $t_i=t$ for each $i$. Recall that there is a relation between Alexander polynomial and multivariable Alexander polynomial as shown in Proposition 7.3.10 in \cite{Kawauchi_book}:$$\Delta_L(t)=(1-t)\Delta_L(t,\dots ,t).$$
\end{enumerate} \end{Rks}
\begin{Exp} We consider the diagram $D$ of the knot $7_3$ whose arcs are labeled as shown in Fig.\ref{Fig.2}. By writing the coloring conditions illustrated in Fig.\ref{Fig.1} at each crossing $c_i$ of $D$, $1\leq i\leq 7$, we obtain the homogeneuous system of linear equations shown on the right side of Fig.\ref{Fig.2}, \begin{figure}
\caption{The equations corresponding to the Alexander colorings of the knot $7_3$. }
\label{Fig.2}
\end{figure}
\noindent we get the following Alexander matrix $A$:
\begin{equation*} A = \begin{pmatrix} a_1 & a_2 & a_3 & a_4 & a_5 & a_6 & a_7\\ \hline 1-t & -1 & 0 & 0 & 0 & 0 & t\\ -1 & 1-t & t & 0 & 0 & 0 & 0\\ 0 & t & 1-t & -1 & 0 & 0 & 0\\ t & 0 & 0 & 1-t & -1 & 0 & 0\\ 0 & 0 & 0 & t & 1-t & -1 & 0\\ 0 & 0 & 0 & 0 & t & 1-t & -1\\ 0 & 0 & -1 & 0 & 0 & t & 1-t\\ \end{pmatrix} \end{equation*}
The determinant of the matrix $A$ is $0$. Let $M$ be the matrix obtained from $A$ by deleting the first row and the first column. The matrix $M$ is called the first minor matrix and its determinant is the Alexander polynomial of the knot $7_3$: $\det(M)=-2t+3t^2-3t^3+3t^4-2t^5.$
In order to obtain the reduced Alexander polynomial of the knot $7_3$, we multiply $\det(M)$ by $-t^{-1}$ and then we get $$\Delta^0_{7_3}(t)=2-3t+3t^2-3t^3+2t^4.$$ \end{Exp}
\noindent\textbf{Coloring links by linear Alexander Quandles}\\ In practice, it is more interesting to color links by using finite quandles as linear Alexander quandles given in Example \ref{Exp_4}. These are the ones we will use to color links in this article. One can easily adapt what has been said in the general case of colorings by an Alexander quandle to this new setting.\\ So, if $m$ and $n$ are coprime integers, a coloring of a diagram $D$ of a link $L$ by the quandle $\Lambda_{n,m}$ is a map from the set $\mathcal{A}$ of arcs of $L$ to $\Lambda_{n,m}$ satisfying the coloring conditions in Fig. \ref{Fig.1}. It is easy to see that such non-trivial coloring exists if $n$ divides $\Delta_L^0(m)$. That coloring is called an $(n,m)$-coloring.\\ In this setting also, one can consider the minimum number of colors as it was done for Fox-colorings. We follow the definition given by Kauffman and Lopes in \cite{Kauff_Lopes}. \begin{Def}\textbf{Minimum number of colors:} Let $L$ be a link admitting non-trivial $(p,m)$-colorings. Let $D$ be a diagram of $L$ and let $n_{D,p,m}$ be the minimum number of colors it takes to equip $D$ with a non-trivial $(p,m)$-coloring. We let
$$mincol_{p,m}(L) = min\lbrace n_{D,p,m} |D \mbox{ is a diagram of } L\rbrace$$ and refer to it as the minimum number of colors for non-trivial $(p,m)$-colorings of $L$. \end{Def} The following Theorem gives an estimation of the minimum number of colors. \begin{Theo}[\cite{Kauff_Lopes}]\label{Theo1} Let $K$ be a knot i.e., a $1$-component link. Let $p$ be an odd prime. Let $m$ be an integer such that $K$ admits non-trivial $(p, m)$-colorings $\pmod p$. If $m \neq 2$ (or $m = 2$ but $\Delta_{K}^{0} (m)\neq 0$) then $$2 + \lfloor \log_M p \rfloor \leq mincol_{p,m}(K),$$ where $M= max \lbrace \vert m \vert, \vert m-1 \vert \rbrace$. \end{Theo} \section{An improvement of the lower bound for the minimum number of knot colorings by linear Alexander quandles} In this section we prove that for suitable choice of the integer $m$, the lower bound of $mincol_{p,m}(K)$ stabilizes and no longer depends on $m$. This provides an interesting improvement of Theorem \ref{Theo1} when $p=\Delta_K^0(m)$.\\ We begin by proving the following lemma.
\begin{Lem}\label{Claim.1}
Let $K$ be a knot and $\Delta_{K}^{0}(t)=\sum \limits_{\underset{}{i=0}}^{k} c_i t^i$, its reduced Alexander polynomial. Let $m$ be an integer, $m>1$, and $p=\Delta_{K}^{0}(m)$. If $m>\displaystyle \max_{0\leq i \leq k} \lbrace |c_i|\rbrace+1$, then $2 + \lfloor \log_m p \rfloor$ is either $k+1$ or $k+2$. \end{Lem}
\begin{proof}
For convenience, we put $a_i=|c_i|$ and we write the reduced Alexander polynomial as follows: $$\Delta_{K}^{0}(t)=\sum \limits_{\underset{}{i=0}}^{k} \pm a_i t^i,$$ where $k$ is even, $a_{\frac{k}{2}}$ is odd, $a_0>0$ and for each $i=0,\dots ,k$, $a_i=a_{k-i}$. Hence $a_0=a_k\ne 0$ (see the proof of Corollary 6.11 in \cite{Lickorish_book}). Note that, $c_0=a_0=a_k=c_k$. The signs assigned to the non-null $a_i$ do not necessarily alternate as in the case of the knot $10_{145}$:
$$\Delta_{10_{145}}^{0}(t)=1+t-3t^2+t^3+t^4.$$
So we must distinguish two cases: \begin{enumerate} \item Suppose that the assigned signs to non-null $a_i$ alternate, two cases can occur:
\begin{itemize}
\item \underline{Case 1} If for each $i=0,\dots ,k$, $a_i\ne 0$, then we factor $\Delta_{K}^0(t)$ as follows $$\Delta_{K}^0(t)=a_0+(t-a_1)t+(a_2-1)t^2+\cdots+(t-a_{k-1})t^{k-1}+(a_k-1)t^k .$$ For each $m> \displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$, by evaluating the reduced Alexander polynomial $\Delta_{K}^0(t)$ at $m$ we get: $$p=\Delta_{K}^0(m)=a_0+(m-a_1)m+(a_2-1)m^2+\cdots+(m-a_{k-1})m^{k-1}+(a_k-1)m^k .$$ Since the integer $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i\rbrace$, then $p$ is a positive integer which can be written as follows $$ p= \left\{
\begin{array}{ll}
\sum \limits_{\underset{}{i=0}}^{k} d_i m^i & \mbox{if} \ a_k>1 \\
\sum \limits_{\underset{}{i=0}}^{k-1} d_i m^i & \mbox{if} \ a_k=1,
\end{array} \right. $$ where for all $0\leq i \leq k$ $$ d_i = \left\{
\begin{array}{ll}
m-a_i & \mbox{if i is odd } \\
a_i-1 & \mbox{if i is even},\, i\ne 0\\
a_0 & \mbox{if} \ i=0.
\end{array} \right. $$ Since $\forall i \in \lbrace 0,..., k \rbrace$, $0\leq d_i < m$, $d_k\geq 1$ if $a_k> 1$ and $d_{k-1}\geq1$ if $a_k=1$, then for all $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$ $$ p= \left\{
\begin{array}{ll}
(d_{k}d_{k-1}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k>1 \\
(d_{k-1}d_{k-2}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k=1,
\end{array} \right. $$ is the base $m$ expansion of the integer $p$. By Theorem 10.8.1 in \cite{adhikari} we have $$ \lfloor \log_mp \rfloor = \left\{
\begin{array}{ll}
k & \mbox{if} \ c_k =a_k> 1\\
k-1 & \mbox{if} \ c_k=a_k=1.
\end{array} \right. $$ \item \underline{Case 2} Now suppose that there exists at least one null coefficient $a_i$.\\
We group each pair of two non-zero consecutive monomials starting with the first non-null monomial of degree greater than or equal to $1$ as follows $$\Delta_{K}^0(t)=a_0-\underbrace{a_1t+a_2t^2}-\cdots-\underbrace{a_jt^j+a_lt^l}-\cdots-\underbrace{a_{k-1}t^{k-1}+a_kt^k}.$$ Suppose that there exists at least a binomial $-a_jt^j+a_lt^l$ such that $l> j+1$. Within these binomials we add expressions $t^r-t^r$ where $r$ corresponds to the missing degrees as follows $$-a_jt^j+t^{j+1}-t^{j+1}+\cdots+t^{l-1}-t^{l-1}+a_lt^l.$$ We factor the expression above as follows $$(t-a_j)t^j+(t-1)t^{j+1}+\cdots+(t-1)t^{l-2}+(t-1)t^{l-1}+(a_l-1)t^l,$$ and we factor the binomials of the form $-a_st^s+a_{s+1}t^{s+1}$ as follows $$(t-a_s)t^s+(a_{s+1}-1)t^{s+1}.$$ For each $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$, by evaluating the reduced Alexander polynomial $\Delta_{K}^0(t)$ at $m$ we get: \begin{align*} p&=\Delta_{K}^0(m)\\ &=a_0+(m-a_1)m+(a_2-1)m^2+\cdots+(m-a_{j})m^{j}+(m-1)m^{j+1}\\
&+\cdots+(m-1)m^{l-1}+(a_l-1)m^l+\cdots+(m-a_{k-1})m^{k-1}+(a_k-1)m^k.\\ \end{align*} Since the integer $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$, then $p$ is a positive integer which can be written as follows $$ p= \left\{
\begin{array}{ll}
\sum \limits_{\underset{}{i=0}}^{k} d_i m^i & \mbox{if} \ a_k>1 \\
\sum \limits_{\underset{}{i=0}}^{k-1} d_i m^i & \mbox{if} \ a_k=1,
\end{array} \right. $$ where for all $i=0,\dots ,k$, $d_i\in\lbrace a_0,m-a_i, m-1, a_i-1\rbrace$.\\ Since $\forall i \in \lbrace 0,..., k \rbrace$, $0\leq d_i < m$, $d_k\geq 1$ if $a_k>1$ and $d_{k-1}\geq 1$ if $a_k=1$, then for all $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$ $$ p= \left\{
\begin{array}{ll}
(d_{k}d_{k-1}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k>1 \\
(d_{k-1}d_{k-2}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k=1
\end{array} \right. $$ is the base $m$ expansion of the integer $p$. Once again, by Theorem 10.8.1 in \cite{adhikari} we get $$ \lfloor \log_mp \rfloor = \left\{
\begin{array}{ll}
k & \mbox{if} \ c_k=a_k > 1\\
k-1 & \mbox{if} \ c_k=a_k=1.
\end{array} \right. $$ \end{itemize} \item Suppose that some consecutive non-null coefficients $a_i$ have opposite assigned signs. Since $\Delta_{K}^{0}(1)=1$, then there is at least one coefficient with negative assigned sign in $\Delta_{K}^0(t)$. By reading the monomials in ascending order of their degrees, each time we encounter an expression of the form $$-a_jt^j-a_{j+1}t^{j+1}-\cdots-a_{j+l-1}t^{j+l-1}+a_{j+l}t^{j+l},$$ where $j+l\leq k$, $a_j\ne 0$ and $a_{j+l}\ne 0$, we factor it as follows without changing the other monomials:
\begin{equation}\label{Eq_1} (t-a_j)t^j+(t-(a_{j+1}+1))t^{j+1}+\cdots+(t-(a_{j+l-1}+1))t^{j+l-1}+(a_{j+l}-1)t^{j+l}. \end{equation}
Here we distinguish three cases depending on the last expression of the form (\ref{Eq_1}) occurring in the factored expression of $\Delta_K^0(t)$: \begin{itemize} \item If $j+l< k$, then for each $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$, we evaluate the factored reduced Alexander polynomial $\Delta_{K}^0(t)$ at $m$. Then we get a positive integer which can be written as follows $$p=\Delta_{K}^0(m)= \sum \limits_{\underset{}{i=0}}^{k} d_i m^i,$$ where for all $i=0,\dots , k$, $d_i\in\lbrace m-a_i, m-(a_i+1),a_i-1,a_i\rbrace$.\\ Since $\forall i $, $0\leq d_i < m$ and $d_k\geq 1$, then for all $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$, $$p= (d_{k}d_{k-1}\cdot\cdot\cdot d_1 d_0)_m$$ is the base $m$ expansion of the integer $p$. By Theorem 10.8.1 in \cite{adhikari} we have $$\lfloor \log_mp \rfloor=k.$$ \item If $j+l=k$ and $j=k-1$, then for each $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$ we evaluate the factored reduced Alexander polynomial $\Delta_{K}^0(t)$ at $m$. We get a positive integer which can be written as follows $$ p=\Delta_{K}^0(m) = \left\{
\begin{array}{ll}
\sum \limits_{\underset{}{i=0}}^{k} d_i m^i & \mbox{if} \ a_k>1 \\
\sum \limits_{\underset{}{i=0}}^{k-1} d_i m^i & \mbox{if} \ a_k=1,
\end{array} \right. $$ where for all $i=0,\dots , k$, $d_i\in\lbrace m-a_i, m-(a_i+1),a_i-1,a_i\rbrace$.\\ Since $\forall i \in \lbrace 0,..., k \rbrace$, $0\leq d_i < m$, $d_k\geq 1$ if $a_k>1$ and $d_{k-1}\geq1$ if $a_k=1$, then for all $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace$, $$ p= \left\{
\begin{array}{ll}
(d_{k}d_{k-1}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k>1 \\
(d_{k-1}d_{k-2}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k=1,
\end{array} \right. $$ is the base $m$ expansion of the integer $p$. Also, by Theorem 10.8.1 in \cite{adhikari} we have $$ \lfloor \log_mp \rfloor = \left\{
\begin{array}{ll}
k & \mbox{if} \ c_k =a_k> 1\\
k-1 & \mbox{if} \ c_k=a_k=1.
\end{array} \right. $$ \item If $j+l=k$ and $j<k-1$ then for each $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i\rbrace +1$, we evaluate the factored reduced Alexander polynomial $\Delta_{K}^0(t)$ at $m$. Then $\Delta_{K}^0(m)$ is a positive integer which can be written as follows $$ \Delta_{K}^0(m) = \left\{
\begin{array}{ll}
\sum \limits_{\underset{}{i=0}}^{k} d_i m^i & \mbox{if} \ a_k>1 \\
\sum \limits_{\underset{}{i=0}}^{k-1} d_i m^i & \mbox{if} \ a_k=1,
\end{array} \right. $$ where for all $i=0,\dots ,k$, $d_i\in\lbrace m-a_i, m-(a_i+1),a_i-1,a_i\rbrace$.\\ Since $\forall i \in \lbrace 0,..., k \rbrace$, $0\leq d_i < m$, $d_k\geq 1$ if $a_k>1$ and $d_{k-1}\geq1$ if $a_k=1$, then for all $m>\displaystyle \max_{0\leq i \leq k} \lbrace a_i \rbrace+1$ $$ p=\Delta_{K}^0(m) = \left\{
\begin{array}{ll}
(d_{k}d_{k-1}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k>1 \\
(d_{k-1}d_{k-2}\cdot\cdot\cdot d_1 d_0)_m & \mbox{if} \ a_k=1,
\end{array} \right. $$ is the base $m$ expansion of the integer $p$. Then by Theorem 10.8.1 in \cite{adhikari}, we have $$ \lfloor \log_mp \rfloor = \left\{
\begin{array}{ll}
k & \mbox{if} \ c_k =a_k> 1,\\
k-1 & \mbox{if} \ c_k=a_k=1.
\end{array} \right. $$ \end{itemize} \end{enumerate} \end{proof}
\begin{Rk}\label{Rk_MainTheo}
The proof shows that the condition $m>\displaystyle \max_{0\leq i \leq k} \lbrace |c_i|\rbrace+1$ is needed in the only one case where the non-null coefficients do not alternate and the two penultimate non-null coefficients have negative signs (as in the last subcase studied in the proof of the last Lemma). Otherwise the weaker condition $m>\displaystyle \max_{0\leq i \leq k} \lbrace |c_i|\rbrace$ suffices. \end{Rk}
The lemma \ref{Claim.1} leads to the following improvement of Theorem \ref{Theo1}. \begin{Theo}\label{MainTheo}
Let $K$ be a knot. Let $\Delta_{K}^{0}(t)=\sum \limits_{\underset{}{i=0}}^{k} c_i t^i$ be the reduced Alexander polynomial of $K$. Let $m$ be an integer, such that $m>\displaystyle \max_{0\leq i \leq k} \lbrace |c_i|\rbrace +1$ and $p=\Delta_{K}^{0}(m)$ is an odd prime integer. \begin{enumerate} \item If $c_k=1$ and the penultimate non-zero coefficient is negative, then
$$k+1 \leq mincol_{p,m}(K).$$ \item If $c_k>1$ or the penultimate non-zero coefficient is positive, then
$$k+2\leq mincol_{p,m}(K).$$ \end{enumerate} \end{Theo}
\begin{proof}
We apply Theorem \ref{Theo1} and then we apply Lemma \ref{Claim.1} to the obtained lower bound $(2 + \lfloor \log_m p \rfloor)$. \end{proof} \begin{Exp} The trefoil knot has the reduced Alexander polynomial $\Delta_{3_1}^0(t)=1-t+t^2$. For any integer $m$, if $p=\Delta_{3_1}^0(m)$ is an odd prime, then the trefoil knot admits non-trivial $(p,m)$-colorings. Furthermore, since the leading coefficient of $\Delta_{3_1}^0(t)$ is $1$ and its penultimate coefficient is negative, then by Theorem \ref{MainTheo} and Remark \ref{Rk_MainTheo}, for any $m>1$, we have $mincol_{p,m}(3_1)\ge 3$.\\ Since the crossing number of the knot $3_1$ is $3$ then for all $m>1$, $mincol_{p,m}(3_1)= 3$.\\ The following table displays some prime values for the reduced Alexander polynomial $\Delta_{3_1}^0(m)$.
\begin{table}[htbp]
\centering
\begin{tabular}[c]{|p{2cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|}
\hline
$m$ & $2$ & $3$ & $4$ & $6$ & $7$ & $9$ & $13$ & $15$\\
\hline
$\Delta_{3_1}^0(m)$ & $3$ & $7$ & $13$& $31$ & $43$ & $73$ & $157$ & $211$\\ \hline
\end{tabular}
\caption{Prime values of $\Delta_{3_1}^0(m)$.} \end{table} \end{Exp}
\section{Some applications} For any rational homology 3-sphere $M$, the rank of the Heegaard Floer homology $\widehat{HF}(M)$ is lower bounded by the order of $H_1(M;\mathbbm{Z})$ \cite{ozsv_szab}. If the rank of $\widehat{HF}(M)$ is equal to the order of $H_1(M;\mathbbm{Z})$ then $M$ is called an \textit{L-space} \cite{ozsv_szab}.\\ A knot $K$ in $S^3$ is called an \textit{L-space knot} if some positive surgery on $K$ gives a 3-manifold that is an L-space. Torus knots are L-space knots (see \cite{Osvath_2} page 379).\\
Our main Theorem \ref{MainTheo} provides the interesting following corollary. \begin{Cor}\label{Cor_MainTheo} Let $K$ be an L-space knot. If $m$ is an integer such that $m>1$ and $p=\Delta_{K}^{0}(m)$ is an odd prime, then $$k+1 \leq mincol_{p,m}(K).$$ \end{Cor}
\begin{proof} We know that each non-zero coefficient of the Alexander polynomial of any L-space knot is $\pm 1$ and they alternate in sign \cite{ozsv_szab}. It is easy to see that this holds also for the reduced Alexander polynomial. Then by Theorem \ref{MainTheo} and Remark \ref{Rk_MainTheo}, we have for each integer $m>1$ such that $p=\Delta_{K}^{0}(m)$ is an odd prime $$k+1 \leq mincol_{p,m}(K).$$ \end{proof}
By applying Corollary \ref{Cor_MainTheo} to some families of knots, we get the following propositions.\\
\noindent{\textbf{Torus knots.}}\\ We recall that if $T_{a,b}$ is a torus knot, then the non-zero coefficients of the reduced Alexander polynomial $\Delta_{T_{a,b}}^{0}(t)$ are all $\pm 1$, and they alternate in sign \cite{Osvath_2}. \begin{Pro}\label{Pro-1_MainTheo} Let $T_{a,b}$ be a torus knot. Let $m$ be an integer such that, $m>1$ and $p=\Delta_{T_{a,b}}^{0}(m)$ is an odd prime, then
$$c(T_{a,b})-(a-2)\leq mincol_{p,m}(T_{a,b})\leq c(T_{a,b}),$$ where $c(T_{a,b})$ is the crossing number of the torus knot $T_{a,b}$.\\ In particular
$$mincol_{p,m}(T_{2,b})=c(T_{2,b}).$$ \end{Pro}
\begin{proof} We note that $\Delta_{T_{-a,b}}^{0}(t)=\Delta_{T_{a,-b}}^{0}(t)=\Delta_{T_{-a,-b}}^{0}(t)=\Delta_{T_{a,b}}^{0}(t)$. Then, by Corollary \ref{Cor_MainTheo}, for any $m$ as in the statement of the proposition we have \begin{equation}\label{Eq_2} k+1\leq mincol_{p,m}(T_{a,b}), \end{equation} where $k$ is the degree of $\Delta_{T_{a,b}}^{0}(t)$. Since the torus knot $T_{a,b}$ is equivalent to the torus knot $T_{b,a}$, we can restrict our computation to the case where $a<b$.\\ On the other hand, the reduced Alexander polynomial of the torus knot can be written $\Delta_{T_{a,b}}(t)=\dfrac{f(t^b)}{f(t)}$ where $f(t)= 1+\cdot\cdot\cdot+t^{a-1}$ \cite{Agle}. It follows that $$ \begin{array}[c]{ccc} k &=&(a-1)(b-1)\\
&=&ab-a-b+1\\
&=&b(a-1)-(a-1).\\ \end{array} $$ It has been proven by Murasugi in \cite{Murasugi} that $c(T_{a,b})=min\lbrace a(b-1),b(a-1) \rbrace$ which is exactly $b(a-1)$ since we assumed that $a<b$. Then $k=c(T_{a,b})-(a-1)$ . By replacing $k$ in (\ref{Eq_2}), we get $$c(T_{a,b})-(a-2)\leq mincol_{p,m}(T_{a,b}).$$ The right inequality is obvious. Finally we have
$$c(T_{a,b})-(a-2)\leq mincol_{p,m}(T_{a,b})\leq c(T_{a,b}).$$ If $a=2$ then $mincol_{p,m}(T_{2,b})= c(T_{2,b})$. \end{proof} \noindent\textbf{Pretzel knots $P(-2, 3, 2l + 1)$.}\\ Lidman and Moore showed that $P(-2, 3, 2l + 1)$, $l\ge 0$, are the only L-space Pretzel knots \cite{Lidman_Moore}.\\ In what follows, we prove that if we color a diagram of the Pretzel knot $P(-2, 3, 2l + 1)$, $l \geq 0$, by the linear Alexander quandle $\mathbbm{Z}_p[t]/(t-2)$ where $p$ is an odd prime, then the lower bound in Theorem \ref{MainTheo} is reached. We need the following lemma.
\begin{Lem}\label{Claim.4} For any odd integer $a\geq 1$, the Alexander polynomial of the Pretzel knot $K=P(-2,3,a)$ can be written as follows
$$\Delta_{K}(t)=1-t+\sum \limits_{\underset{}{i=3}}^{a} (-1)^{i+1} t^i-t^{a+2}+t^{a+3}.$$ \end{Lem}
\begin{proof} Since $a$ is an odd integer then it can be expressed as $a=2l+1$, where $l\ge 0$. We will proof by induction that, for all $l\in \mathbbm{Z}$,
$$\Delta_{P(-2,3,2l+1)}(t)=1-t+\sum \limits_{\underset{}{i=3}}^{2l+1} (-1)^{i+1} t^i-t^{2l+3}+t^{2l+4}.$$ It is known that the Alexander polynomial of the Pretzel knot $P(p,q,-2)$, where $p$ and $q$ are odd integers, is given by (see \cite{Hironaka}) \begin{equation}\label{Alexander_poly_Pretzel_knots} \Delta_{P(p,q,-2)}(t) =\dfrac{1+2t+t^{1+p}+t^{1+q}- t^3 -t^{p+q} + t^{p+2} + t^{q+2} + 2t^{p+q+2} + t^{3+p+q}}{(1+t)^3}. \end{equation} \\ Note that $P(p,q,-2)$ and $P(-2,p,q)$ are equivalent (see \cite{Kawauchi_book}, Paragrapah 2.3).\\ If $l=1$
$$\Delta_{P(-2,3,3)}(t)=\dfrac{1+2t-t^3+2t^4+2t^5-t^6+2t^8+t^9}{(1+t)^3}=1-t+t^3-t^5+t^6.$$ Now suppose that the formula is right for $l$, i.e. $$\Delta_{P(-2,3,2l+1)}(t)=1-t+\sum \limits_{\underset{}{i=3}}^{2l+1} (-1)^{i+1} t^i-t^{2l+3}+t^{2l+4},$$ and show that
$$\Delta_{P(-2,3,2l+3)}(t)=1-t+\sum \limits_{\underset{}{i=3}}^{2l+3} (-1)^{i+1} t^i-t^{2l+5}+t^{2l+6}.$$ By using formula (\ref{Alexander_poly_Pretzel_knots}), we can write \begin{align*} \Delta_{P(-2,3,2l+3)}(t) &=\dfrac{1+2t+t^{2l+4}+t^4-t^3-t^{2l+6}+t^{2l+5}+t^5+2t^{2l+8}+t^{2l+9}}{(1+t)^3}\\
&=\dfrac{1+2t+t^4-t^3+t^5+t^2(t^{2l+2}-t^{2l+4}+t^{2l+3}+2t^{2l+6}+t^{2l+7})}{(1+t)^3}\\
&=\dfrac{1+2t-t^3+t^4+t^5-t^2(1+2t-t^3+t^4+t^5)}{(1+t)^3}+t^2\Delta_{P(-2,3,2l+1)}(t)\\
&=\dfrac{1+2t-t^2-3t^3+t^4+2t^5-t^6-t^7}{(1+t)^3}+t^2\Delta_{P(-2,3,2l+1)})(t)\\
&=1-t-t^2+2t^3-t^4+t^2(1-t+\sum \limits_{\underset{}{i=3}}^{2l+1} (-1)^{i+1} t^i-t^{2l+3}+t^{2l+4})\\
&=1-t+t^3-t^4+\sum \limits_{\underset{}{i=3}}^{2l+1} (-1)^{i+1} t^{i+2}-t^{2l+5}+t^{2l+6}\\
&=1-t+t^3-t^4+\sum \limits_{\underset{}{i=5}}^{2l+3} (-1)^{i-1} t^{i}-t^{2l+5}+t^{2l+6}\\
&=1-t+\sum \limits_{\underset{}{i=3}}^{2l+3} (-1)^{i-1} t^{i}-t^{2l+5}+t^{2l+6}.\\ \end{align*} Since $\forall i$, $(-1)^{i+1}=(-1)^{i-1}$ then $$\Delta_{P(-2,3,2l+3)}(t)=1-t+\sum \limits_{\underset{}{i=3}}^{2l+3} (-1)^{i+1} t^i-t^{2l+5}+t^{2l+6}.$$ \end{proof}
\begin{Pro} Let $K=P(-2,3,a)$ be a pretzel knot where $a$ is an odd positive integer. Let $m> 1$ be an integer such that $p=\Delta_{K}^{0}(m)$ is an odd prime. If $\Delta_{K}^{0}(m)\neq 0$ then $$a+4\leq mincol_{p,m}(K).$$ In particular, if $m=2$ then $$a+4 = mincol_{p,m}(K).$$ \end{Pro}
\begin{proof} By Theorem \ref{Cor_MainTheo}, we have \begin{equation} k+1\leq mincol_{p,m}(K), \end{equation} where $k$ is the degree of the reduced Alexander polynomial. By Lemma \ref{Claim.4} the degree of the reduced Alexander polynomial of the Pretzel knot $K=P(-2,3,a)$ is $k=a+3$. Finally, we have $$a+4\leq mincol_{p,m}(K).$$ If $m=2$, it is easy to see that $\Delta_{P(-2,3,a)}^{0}(2)\ne 0$. Furthermore, if $p=\Delta_{P(-2,3,a)}^{0}(2)$ is prime, consider the diagram of the Pretzel knot $P(-2,3,a)$ shown in Fig. \ref{Fig.6}, we prove that this diagram has a non-trivial $(p,m)$-coloring using exactly $a+4$ colors. We assign the colors $x,y,z,w \in \mathbbm{Z}_p$ to the four arcs in the left tower. \begin{figure}
\caption{Pretzel knot $P(-2,3,a)$. }
\label{Fig.6}
\end{figure} Using the coloring conditions associated with the crossings $(i),(ii)$ we get the following system of equations. $$ \left\{
\begin{array}{ll}
z=mx+(1-m)y \pmod p\\
y=mw+(1-m)z \pmod p.\\
\end{array} \right. $$ Since we assumed that $P(-2,3,a)$ has a non-trivial $(p,m)$-coloring, as for Fox colorings (\cite{Ge} Lemma 2.10), there exists an automorphism of the considered linear Alexander quandle $\mathbbm{Z}_p$ providing a new $(p,m)$-coloring containing colors $0$ and $1$. So we can assume that $x=1$ and $y=0$. $$ \left\{
\begin{array}{ll}
z=m \pmod p\\
0=mw+(1-m)z \pmod p.\\
\end{array} \right. $$ If we replace $z$ in the second equation we obtain $mw=-m(1-m) \pmod p\Rightarrow w=(m-1)\pmod p$. If $m=2$ then $w=1=x$. The number of arcs of the Pretzel knot $P(-2,3,a)$ is equal to $a+5$ and, since two arcs are equipped with the same color, we can conclude that $mincol_{p,2}(K)=a+4$. \end{proof}
\noindent\textbf{About the Kauffman-Harary conjecture}
We deduce another result from Proposition \ref{Pro-1_MainTheo} regarding what was called Kauffman-Harary conjecture proved by Mattman and Solis in \cite{Mattman_Solis} for Fox colorings. \begin{Theo}[\cite{Mattman_Solis}]
Let $D$ be a reduced, alternating diagram of the knot $K$ having prime determinant $p$. Then every non-trivial $p$-coloring of $D$ assigns different colors to different arcs. \end{Theo}
Kauffman and Lopes extended the KH-behavior property to colorings using linear Alexander quandles as follows \cite{Kauff_Lopes}.
\begin{Def} We say an alternating knot $K$ displays KH behavior mod $(p,m)$, if there is an integer $m$ such that $1 < m < \Delta_{K}^{0}(m)=p$, where $p$ is prime, and for some reduced alternating diagram of $K$ equipped with a non-trivial $(p, m)$-coloring, different arcs receive different colors. Otherwise we say that $K$ displays anti-KH behavior. \end{Def}
They noted that there exist some knots displaying anti-KH behavior. That means that for such knots, any alternating reduced diagram equipped with non-trivial $(p,m)$-coloring associated to prime determinant have at least two distinct arcs which receive the same color. So, it is interesting to investigate if there are some families of knots that display KH behavior mod $(p,m)$.\\
Proposition \ref{Pro-1_MainTheo} allows to answer for torus knots $ T_{2,b}$.
\begin{Pro} The torus knot $T_{2,b}$ displays KH-behavior mod $(p,m)$. \end{Pro}
\begin{proof} Suppose $T_{2,b}$ displays anti-KH behavior i.e. there is an integer $m$ such that $1 < m < \Delta_{T_{2,b}}^{0}(m)=p$ where $p$ is prime and for any reduced alternating diagram of $T_{2,b}$ equipped with non-trivial $(p,m)$-coloring there exists at least two distinct arcs with the same color. This implies that the number of colors is strictly less than the crossing number of $T_{2,b}$ (contradiction because we proved that $mincol_{p,m}(T_{2,b})=c(T_{2,b})$). \end{proof}
\section{Generalization of Theorem 2.1 to links}
It is worth mentioning that the proof of Theorem \ref{Theo1} given in \cite{Kauff_Lopes} cannot be naturally extended to links with non-zero determinant. However, it has already been proved that there is an analogous theorem for links in the case of Fox-colorings \cite{Ichi_Matsu}. In view of this, it is natural to ask what can we say for links colored by linear Alexander quandles. This section is devoted to answer this question. We show the following theorem.
\begin{Theo}\label{Theo.4} Let $L$ be a link whose reduced Alexander polynomial $\Delta^0_{L}(t) \neq 0$. Let $m$ be an integer and $p$ a prime factor of $\Delta_{L}^{0}(m)$ such that $L$ admits non-trivial $(p,m)$-colorings. Suppose that $\Delta_{L}^{0}(m) \neq 0$ then $$2 + \lfloor \log_M p \rfloor \leq mincol_{p,m}(L),$$ where $M= max \lbrace \vert m \vert, \vert m-1 \vert \rbrace$. \end{Theo} The proof we give is inspired by that given in \cite{Ichi_Matsu} for effective Fox $n$-colorings.\\
Let $D$ be a diagram of $L$ equipped with a non-trivial $(p,m)$-coloring. Let $q$ be the number of arcs of $D$, $\{a_1,\ldots, a_q\}$ be the set of arcs of $D$ and $\{c_1,\ldots, c_q\}$ be the set of crossings of $D$. Let $A$ be the corresponding coloring matrix. For each $i$, $1 \leq i \leq q$, let $x_i$ be the color assigned to the arc $a_i$. Note that \begin{itemize} \item Since the considered $(p,m)$-coloring is non-trivial then the vector $X_0={}^t(x_1,\ldots,x_q)\in \mathbbm{Z}^q$ is a non-trivial solution of the congruence equation $AX \equiv 0 \pmod p$. \item Each row of $A$ contains only three non-zero entries which are $-m$, $m-1$ and $1$. \item Since any first minor of $A$ is the reduced Alexander polynomial of $L$ (up to multiplication by $\pm t^{n}$) valued at $m$ is assumed not equal to $0$, then $\rank A= q-1$. \end{itemize} Let $d$ be the number of pairwise distinct colors $x_i$, $1 \leq i \leq d$. We agree to associate the same label to the arcs having the same color. So, we get the label set $\{a_1,...,a_d\}$. By writing the coloring condition using the new labelings at each crossing of $D$, we get a new system of homogeneous linear equations $\pmod p $ whose matrix is denoted by $A_1$. This system has $q$ equations in $d$ unknowns.\\
Note that the matrix $A_1$ is exactly the matrix obtained in the procedure described in the proof of Theorem 1.1 in \cite{Ichi_Matsu} which consists of adding the \(j\)\textsuperscript{th} column to the \(i\)\textsuperscript{th} column and then deleting the \(j\)\textsuperscript{th} one, for all $i$ and $j$, $1\leq i<j \leq q$, such that $x_i=x_j$.\\
The matrix $A_1$ satisfies the following properties. \begin{itemize} \item The vector $Y_0={}^t(y_1,...,y_d)$ obtained from $X_0$ by keeping only one representative for each color occurring in $X$ provides a non-trivial solution to $A_1Y \equiv 0 \pmod p $. \item The only non-zero entries in any row of $A_1$ are still the same as those of any row of $A$, namely $-m$, $m-1$ and $1$. This is true because the sum of two of these entries cannot be associated to a same color occurring at a same crossing, otherwise the three colors associated to arcs of that crossing will be the same. \item $\rank A_1 =d-1$: We note that $\rank A_1 \le d-1$. Suppose that $\rank A_1 <d-1$, this implies that $\rank A <q-1$. This contradicts the fact that $\rank A=q-1$. \end{itemize} Now, we consider $d-1$ linearly independent row vectors of $A_1$. Then we get a $(d-1)\times d$ matrix $A_2$. Note that $Y_0={}^t(y_1,...,y_d)$ gives a solution of $A_2Y \equiv 0 \pmod p$ because the rows of $A_2$ form a subset of those of $A_1$. Since the non-zero entries on each row of $A_2$ are $-m$, $m-1$ and $1$ then by adding all the columns to the last column we obtain a $(d-1)\times d$ matrix $A_3$ whose last column contains only zeros. By applying Lemma 2.1 in \cite{Naka_Nakani_Satoh}, we see that $Z_0={}^t(y_1-y_d,...,y_{d-1}-y_d,0)$ is a non-trivial solution of $A_3Z \equiv 0 \pmod p$. \begin{Lem}\label{Lem_Theo} Let $p$ be an odd prime and $m$ be an integer. Let $L$ be a link admitting non-trivial $(p, m)$-colorings. Let $B$ be the square matrix $(d-1)\times(d-1)$ obtained from $A_3$ by deleting the last column, then we have the following: \begin{enumerate}[label=(\roman*)] \item $\vert \det(B)\vert \leq M^{d-1}$, where $M= max\lbrace\vert m \vert, \vert m-1 \vert\rbrace$. \item $det(B)$ is divisible by $p$. \end{enumerate} \end{Lem} \begin{proof} \begin{enumerate}[label=(\roman*)] \item Since each row of $B$ contains at least two non-zero entries belonging to the set $\{-m,m-1,1\}$, by applying Lemma 3.1 in \cite{Kauff_Lopes} we have $\vert \det(B)\vert \leq M^{d-1}$, where $M= max\lbrace\vert m \vert, \vert m-1 \vert\rbrace$. \item We denote $V_0={}^t(y_1-y_d,...,y_{d-1}-y_d)$ which is a non-trivial solution of $BV \equiv 0 \pmod p$ then $\det B=0 \pmod p$. \end{enumerate} \end{proof} \begin{proof}[Proof of Theorem \ref{Theo.4}] By lemma \ref{Lem_Theo} we have, $$ p \leq \vert det B \vert \leq M^{d-1},$$ where $M= max\lbrace\vert m \vert, \vert m-1 \vert\rbrace$. We remove $\vert det(B)\vert$ from the inequalities and then we get $p<M^{d-1}$. By applying logarithm base $M$ we obtain $$ d > 1+\log_Mp.$$ \end{proof}
\begin{Exp} We consider the link $L4a1\lbrace1\rbrace$ depicted in the following figure \ref{Fig.8}. \begin{figure}
\caption{The link $L4a1\lbrace1\rbrace$. }
\label{Fig.8}
\end{figure} The reduced Alexander polynomial of the link $L4a1\lbrace1\rbrace$ is $\Delta_{L4a1\lbrace1\rbrace}^0(t)=1+t^2$. For any integer $m$, if $p=\Delta_{L4a1\lbrace1\rbrace}^0(m)$ is a prime integer, then the link $L4a1\lbrace1\rbrace$ admits non-trivial $(p,m)$-colorings. For any $m>1$ we have $\Delta_{L4a1\lbrace1\rbrace}^0(m)=1+m^2$. This is a positive integer which can be written as follows $$\Delta_{L4a1\lbrace1\rbrace}^0(m) = \sum \limits_{\underset{}{i=0}}^{2} d_i m^i,$$ where $d_0=1$, $d_1=0$ and $d_2=1$. Since $d_2\geq 1$ and for any $0\leq i \leq 2$, $0\leq d_i <m$, then by Theorem 8.10.1 in \cite{adhikari} we have $\lfloor \log_m{\Delta_{L4a1\lbrace1\rbrace}^{0}(m)}\rfloor =2$. Hence, for all integers $m>1$, $$mincol_{p,m}(L4a1\lbrace1\rbrace)\geq 4.$$ Since the crossing number of the link $L4a1\lbrace1\rbrace$ is $4$, then for any integer $m>1$, $$mincol_{p,m}(L4a1\lbrace1\rbrace)= 4.$$ The following table displays some prime values for the reduced Alexander polynomial $\Delta_{L4a1\lbrace1\rbrace}^0(m)$. \begin{table}[htbp]
\centering
\begin{tabular}[c]{|p{2cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|p{1cm}|}
\hline
$m$ & $2$ & $4$ & $6$ & $10$ & $14$ & $16$ & $20$ & $24$\\
\hline
$\Delta_{L4a1\lbrace1\rbrace}^0(m)$ & $5$ & $17$ & $37$& $101$ & $197$ & $257$ & $401$ & $577$ \\ \hline
\end{tabular}
\caption{Prime values of $\Delta_{L4a1\lbrace1\rbrace}^0(m)$.} \end{table} \end{Exp}
\end{document} | arXiv |
General Relativity and Gravitation
General Relativity and Gravitation is a monthly peer-reviewed scientific journal. It was established in 1970, and is published by Springer Science+Business Media under the auspices of the International Society on General Relativity and Gravitation. The two editors-in-chief are Pablo Laguna and Mairi Sakellariadou; former editors include George Francis Rayner Ellis, Hermann Nicolai, Abhay Ashtekar, and Roy Maartens. The journal's field of interest is modern gravitational physics,[1][2] encompassing all theoretical and experimental aspects of general relativity and gravitation.
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\begin{document}
\title[]{Universality results for kinetically constrained spin models\\in two dimensions} \author[Fabio Martinelli]{Fabio Martinelli} \address{Dipartimento di Matematica e Fisica, Universit\`a Roma Tre, Largo S.L. Murialdo 00146, Roma, Italy}\email{[email protected]}
\author[Robert Morris]{Robert Morris} \address{IMPA, Estrada Dona Castorina 110, Jardim Bot\^{a}«Ïico, Rio de Janeiro, 22460-320, Brazil}\email{[email protected]}
\author[Cristina Toninelli]{Cristina Toninelli} \address{Laboratoire de Probabilit\'es, Mod\'elisation et Statistique,
CNRS-UMR 7599 Universit\'es Paris VI-VII 4, Place Jussieu F-75252 Paris Cedex 05 France}\email{[email protected]}
\thanks{This work has been supported by the ERC Starting Grant 680275 ``MALIG'', ANR-15-CE40-0020-01 and by the PRIN 20155PAWZB ``Large Scale Random Structures''. RM is also partially supported by CNPq (Proc.~303275/2013-8) and by FAPERJ (Proc.~201.598/2014).} \subjclass[2010]{Primary {60K35}, secondary 60J27}
\keywords{Glauber dynamics, kinetically constrained models, spectral gap, bootstrap percolation, universality}
\begin{abstract} Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. KCM also display some of the peculiar features of the so-called ``glassy dynamics'', and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics.
We consider two-dimensional KCM with update rule $\ensuremath{\mathcal U}$, and focus on proving universality results for the mean infection time of the origin, in the same spirit as those recently established in the setting of $\ensuremath{\mathcal U}$-bootstrap percolation. We first identify what we believe are the correct universality classes, which turn out to be different from those of $\ensuremath{\mathcal U}$-bootstrap percolation. We then prove universal upper bounds on the mean infection time within each class, which we conjecture to be sharp up to logarithmic corrections. In certain cases, including all supercritical models, and the well-known Duarte model, our conjecture has recently been confirmed in~\cite{MMT}. In fact, in these cases our upper bound is sharp up to a constant factor in the exponent. For certain classes of update rules, it turns out that the infection time of the KCM diverges much faster {{than}} for the corresponding $\ensuremath{\mathcal U}$-bootstrap process when the equilibrium density of infected sites goes to zero. This is due to the occurrence of energy barriers which determine the dominant behaviour for KCM, but which do not matter for the monotone bootstrap dynamics. \end{abstract}
\maketitle \tableofcontents
\section{Introduction} Kinetically constrained models (KCM) are interacting particle systems on the integer lattice $\mathbb Z^d$, which were introduced in the physics literature in the 1980s in order to model the liquid-glass transition (see e.g. \cites{Ritort,GarrahanSollichToninelli} for reviews), a major and still largely open problem in condensed matter physics. The main motivation for the ongoing (and extremely active) research on KCM is that, despite their simplicity, they feature some of the main signatures of a super-cooled liquid near the glass transition point.
A generic KCM is a continuous time Markov process of Glauber type defined as follows. A configuration $\o$ is defined by assigning to each site $x\in\mathbb Z^d$ an occupation variable $\omega_x\in\{0,1\},$ corresponding to an empty or occupied site respectively. Each site waits an independent, mean one, exponential time and then, iff a certain local constraint is satisfied by the current configuration $\o$, its occupation variable is updated to be occupied with rate $p$ and to empty with rate $q=1-p$. All the constraints that have been considered in the physics literature belong to the following general class \cite{CMRT}.
Fix an {\sl update family} $\,\mathcal U=\{X_1,\dots,X_m\}$, that is, a finite collection of finite subsets of $\mathbb Z^d\setminus \{ \mathbf{0} \}$. Then $\o$ satisfies the constraint at site $x$ if there exists $X\in \ensuremath{\mathcal U}$ such that $\o_y=0$ for all $y\in X+x$.
Since each update set belongs to $\mathbb Z^d\setminus \{ \mathbf{0} \}$, the constraints never depend on the state of the to-be-updated site. As a consequence, the product Bernoulli($p$) measure $\mu$ is a reversible invariant measure, and the process started at $\mu$ is stationary.
Despite this trivial equilibrium measure, however, KCM display an extremely rich behaviour which is qualitatively different from that of interacting particle systems with non-degenerate birth/death rates (e.g. the stochastic Ising model). This behaviour
includes the key dynamical features of real glassy materials:
anomalously long mixing times~\cites{Aldous,CMRT,MT}, aging and dynamical heterogeneities~\cite{FMRT-cmp}, and ergodicity breaking transitions corresponding to percolation of blocked structures~\cite{GarrahanSollichToninelli}. Moreover, proving the above results rigorously turned out to be a surprisingly challenging task, in part due to the fact that several of the classical tools typically used to analyse reversible interacting particle systems (e.g., coupling, censoring, logarithmic Sobolev inequalities) fail for KCM.
KCM can be also viewed as a natural non-monotone and stochastic counterpart of $\mathcal U$-bootstrap percolation, a {{well-studied}} class of discrete cellular automata, {{see~\cite{BBPS,BDMS,BSU}.}} For $\mathcal U$-bootstrap on $\mathbb Z^d$, given a configuration of ``infected" sites $A_t$ at time $t$, infected sites remain infected, and a site $v$ becomes infected at time $t + 1$ if the translate by $v$ of one of the sets in $\ensuremath{\mathcal U}$ belongs to $A_t$. One then defines the final infection set $[A]_\ensuremath{\mathcal U}:= \bigcup_{t=1}^\infty A_t$ and the \emph{critical probability} of the $\ensuremath{\mathcal U}$-bootstrap process on $\mathbb{Z}^d$ to be \begin{equation}\label{def:qc:bootstrap} q_c\big( \mathbb{Z}^d, \ensuremath{\mathcal U} \big) := \inf \Big\{ q \,:\, {\ensuremath{\mathbb P}} _q\big( [A]_\ensuremath{\mathcal U} = \mathbb{Z}^d \big) =1 \Big\}, \end{equation} where ${\ensuremath{\mathbb P}} _q$ denotes the product probability measure on $\mathbb{Z}^d$ with density $q$ of infected sites. The following key connection between $\mathcal U$-bootstrap percolation and KCM has been established by Cancrini, Martinelli, Roberto and Toninelli~\cite{CMRT}: KCM processes are ergodic with exponentially decaying time auto-correlations for $q>q_c\big( \mathbb{Z}^d, \ensuremath{\mathcal U} \big)$, and they are not ergodic for $q<q_c\big( \mathbb{Z}^d, \ensuremath{\mathcal U} \big)$. More precisely, the results of \cite{CMRT} prove that the \emph{relaxation time} $T_{\rm rel}(q;\ensuremath{\mathcal U})$ (see Definition \ref{def:PC}) and the \emph{mean infection time}\footnote{The mean infection time is very close to the \emph{persistence time} in the physics literature} $\mathbb E_{\mu}(\tau_0)$ (i.e. the mean over the stationary KCM process of the first time at which the origin becomes empty) are finite for $q>q_c\big( \mathbb{Z}^d, \ensuremath{\mathcal U} \big)$ and infinite for $q< q_c\big( \mathbb{Z}^d, \ensuremath{\mathcal U} \big)$. Both from a physical and mathematical point of view, a key question is thus to determine the divergence of the time scales $T_{\rm rel}(q;\ensuremath{\mathcal U})$ and $\mathbb E_{\mu}(\tau_0)$ as $q\downarrow q_c(\mathbb{Z}^d,\ensuremath{\mathcal U})$. We will now briefly review the known results, which show that KCM exhibit a very large variety of possible scalings depending on the details of the update family $\ensuremath{\mathcal U}$.
We {{begin}} by discussing one of the most extensively studied KCM, which was introduced by J\"ackle and Eisinger \cite{JACKLE}: the so-called \emph{East model}. This model has update family $\mathcal{U}= \big\{ \{-\vec e_1,\},\dots,\{-\vec e_d\} \big\}$, so in the one-dimensional setting $d = 1$ a site can update iff it is the neighbour ``to the east" of an empty site. It is not difficult to see that in any dimension $q_c(\mathbb{Z}^d,\ensuremath{\mathcal U})=0$. For $d = 1$, it was first proved in~\cite{Aldous} that the relaxation time $T_{\rm rel}(q)$ is finite for any $q \in (0,1]$, and it was later shown (see~\cites{CFM,Aldous,CMRT}) that it diverges as $$\exp\bigg( \big( 1 + o(1) \big) \frac{\log(1/q)^2}{2\log2} \bigg)$$ as $q\downarrow 0$. A similar scaling was later proved in any dimension $d \,\geqslant\, 1$, see~\cite{CFM2}.
Another well-studied KCM, introduced by Friedrickson and Andersen \cite{FH}, is the $k$-facilitated model (FA-kf), whose update family consists of the $k$-sets of nearest neighbours of the origin: a site can be updated iff it has at least $k$ empty nearest neighbours. In this case it was proved in~\cite{vanEnter,BPd} that $q_c(\mathbb{Z}^d,\ensuremath{\mathcal U}) = 0$ for all $1 \,\leqslant\, k \,\leqslant\, d$, whereas $q_c(\mathbb{Z}^d,\ensuremath{\mathcal U}) = 1$ for all $k > d$. Moreover, the relaxation time $T_{\rm rel}(q)$ diverges as $1/q^{\Theta(1)}$ when $k = 1$~\cite{CMRT,AS}, and as a $(k-1)$-times iterated exponential of $q^{-1/(d-k+1)}$ when $2 \,\leqslant\, k \,\leqslant\, d$~\cite{MT}. The above scalings also hold for the mean infection time $\mathbb E_{\mu}(\tau_0)$.
The above model-dependent results (which are, in fact, the only ones that have been proved so far) include a large diversity of possible scalings of the mean infection time, together with a strong sensitivity to the details of the update family $\ensuremath{\mathcal U}$. Therefore, a very natural ``universality'' question emerges:
\begin{question*} Is it possible to group all possible update families $\ensuremath{\mathcal U}$ into distinct classes, in such a way that all members of the same
class induce the same divergence of the mean infection time as $q$
approaches from above the critical value $q_c(\mathbb Z^d,\ensuremath{\mathcal U})$? \end{question*}
Such a general question has not been addressed so far, even in the physics literature: physicists lack a general criterion to predict the different scalings. This fact is particularly unfortunate since, due to the anomalous and sharp divergence of times, numerical simulations often cannot give clear cut and reliable answers. Indeed, some of the rigorous results recalled above corrected some false conjectures that were based on numerical simulations.
The universality question stated above has, however, being addressed and successfully solved for two-dimensional $\ensuremath{\mathcal U}$-bootstrap percolation (see~\cite{BSU,BBPS,BDMS}, or~\cite{Robsurvey} for a recent review). The update families $\ensuremath{\mathcal U}$ were classified by Bollob\'as, Smith and Uzzell~\cite{BSU} into three universality classes: \emph{supercritical}, \emph{critical} and \emph{subcritical} (see Definition \ref{def:stable}), according to a simple geometric criterion. They also proved in~\cite{BSU} that $q_c\big( \mathbb{Z}^2, \ensuremath{\mathcal U} \big) = 0$ if $\ensuremath{\mathcal U}$ is supercritical or critical, and it was proved by Balister, Bollob\'as, Przykucki and Smith~\cite{BBPS} that $q_c\big( \mathbb{Z}^2, \ensuremath{\mathcal U} \big) > 0$ if $\ensuremath{\mathcal U}$ is subcritical. For critical update families $\ensuremath{\mathcal U}$, the scaling (as $q \downarrow 0$) of the typical infection time of the origin starting from ${\ensuremath{\mathbb P}} _q$ was determined very precisely by Bollob\'as, Duminil-Copin, Morris and Smith~\cite{BDMS} (improving bounds obtained in~\cite{BSU}), and various universal properties of the dynamics were obtained.
In this paper we take {{an}} important step towards establishing a similar universality picture for two-dimensional KCM with supercritical or critical update family $\ensuremath{\mathcal U}$. Using a geometric criterion, we propose a classification of the two-dimensional update families into universality classes, which is inspired by, but at the same time quite different from, that established for bootstrap percolation. More precisely, we classify a supercritical update family $\ensuremath{\mathcal U}$ as being \emph{supercritical
unrooted} or \emph{supercritical rooted} and a critical $\ensuremath{\mathcal U}$ as being \emph{$\alpha$-rooted} or \emph{$\beta$-unrooted}, where $\alpha \in {\ensuremath{\mathbb N}} $ and $\alpha \;\leqslant\; \beta \in {\ensuremath{\mathbb N}} \cup \{\infty\}$ are called the \emph{difficulty} and the \emph{bilateral difficulty} of $\ensuremath{\mathcal U}$ respectively (see Definitions~\ref{def:rooted} and~\ref{def:alpha:rooted}). We then prove {{(see Sections
3-7)}}
the following two main universality results (see Theorems~\ref{mainthm:1} and~\ref{mainthm:2} in Section \ref{sec:results}) on the mean infection time ${\ensuremath{\mathbb E}} _\mu(\tau_0)$.\\
\noindent {\bf Supercritical KCM.} {\sl Let $\ensuremath{\mathcal U}$ be a supercritical two-dimensional update family. Then, as $q \rightarrow 0$, \begin{enumerate} \item[$(a)$] if $\ensuremath{\mathcal U}$ is unrooted $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; q^{-O(1)};$$ \item[$(b)$] if $\ensuremath{\mathcal U}$ is rooted, $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \exp\Big( O\big( \log q^{-1} \big)^2 \Big).$$ \end{enumerate}}
\pagebreak
\noindent {\bf Critical KCM.} {\sl Let $\ensuremath{\mathcal U}$ be a critical two-dimensional update family with difficulty $\alpha$ and bilateral difficulty $\beta$. Then, as $q \rightarrow 0$, \begin{enumerate}[(a)] \item if $\ensuremath{\mathcal U}$ is $\alpha$-rooted $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \exp\Big( q^{-2\alpha} \big( \log q^{-1} \big)^{O(1)} \Big);$$ \item if $\ensuremath{\mathcal U}$ is $\beta$-unrooted $$ {\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \exp\Big( q^{-\beta} \big( \log q^{-1} \big)^{O(1)} \Big).$$ \end{enumerate} }
Even though the theorems above only establish universal \emph{upper bounds} on ${\ensuremath{\mathbb E}} _\mu(\tau_0)$, we conjecture that our bounds provide the correct scaling up to logarithmic corrections. This has recently been proved for supercritical models in~\cite{MMT}. For critical update families, the bound $\mathbb E_{\mu}(\tau_0)=\Omega(T_{\mathcal U})$ (see~\cite{MT}*{Lemma 4.3}), where $T_\ensuremath{\mathcal U}$ denotes the median infection time of the origin for the $\ensuremath{\mathcal U}$-bootstrap process at density $q$, together with the results of~\cite{BDMS} on $T_{\mathcal U}$, provide a matching lower bound for all $\beta$-unrooted models with $\alpha = \beta$ (for example, the FA-2f model).
In particular, these recent advances combined with the above theorems prove two conjectures that we put forward in~\cite{Robsurvey}. Among the $\alpha$-rooted models, those which have been considered most extensively in the literature are the Duarte and modified Duarte model (see~\cite{Duarte,Mountford,BCMS-Duarte}), for which $\alpha=1$ and $\beta=\infty$. In~\cite{MMT}, using very different tools and ideas from those in this paper, a lower bound on ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ was recently obtained for both models that matches our upper bound, including the logarithmic corrections, yielding ${\ensuremath{\mathbb E}} _\mu(\tau_0) = \exp\big( \Theta\big( q^{-2} ( \log 1/q)^4 \big) \big)$.
The above results imply that for all supercritical rooted KCM, and also for the Duarte-KCM, the mean infection time diverges much faster than the median infection time for the corresponding $\ensuremath{\mathcal U}$-bootstrap process, which obeys $T_{\mathcal U}\sim 1/q^{\Theta(1)}$ for supercritical models~\cite{BSU}, and $T_{\mathcal U}\sim\exp\big( \Theta\big( q^{-1} ( \log 1/q)^2 \big) \big)$ for the Duarte model~\cite{Mountford}.
This is a consequence of the fact that for these KCM the infection time is not well-approximated by the number of updates needed to infect the origin (as it is for bootstrap percolation), but is the result of a much more complex mechanism. In particular, the visits of the process to regions of the configuration space with an anomalous amount of infection (borrowing from physical jargon we may call them ``energy barriers") are heavily penalized and require a very long time to actually take place.
Providing an insight into the heuristics and/or the key steps of the proofs at this stage, before providing a clear definition of the geometrical quantities involved, would inevitably be rather vague. We therefore defer these explanations to Section~\ref{sec:roadmap}. We can, however, state two high-level ingredients. The first one consists in identifying, for each class of update families $\ensuremath{\mathcal U},$ an ``efficient" (and potentially optimal) dynamical strategy for the difficult (i.e., unlikely) task of infecting the origin. This is necessarily more complex {{than}} the growth of the corresponding $\ensuremath{\mathcal U}$-bootstrap process, since an efficient strategy must necessarily feature both infection and healing in order to avoid crossing excessively high energy barriers. The second ingredient consists in using the above strategy as a \emph{guide},\footnote{In this respect our situation shares some similarities with other large deviations problems, where an imagined optimal dynamical strategy has the role of suggesting and motivating several, otherwise mysterious, analytic steps.} without actually implementing it, for the analytic technique introduced in~\cite{MT} by two of the authors of the present paper, which allows one to bound the relaxation time $T_{\rm rel}(q;U)$. In~\cite{MT} this technique was successfully applied to the FA-kf model, with the imagined mechanism for infecting the origin being a large droplet of infected sites moving as a random walk in {{a suitable (evolving)}} random environment of sparse infection. Here we have to go well beyond the method of~\cite{MT}, since the random walk picture does not apply to rooted models. Our main novelty is a new and more complex analytic approach to bound $T_{\rm rel}(q,\ensuremath{\mathcal U})$ which is inspired by the East dynamics (see Section~\ref{sec:roadmap} for more details).
\subsection{Notation}
We gather here (for the reader's convenience) some of the standard notation that we use throughout the paper. First, recall that we write $\mu$ for the Bernoulli product measure $\otimes_{x \in {\ensuremath{\mathbb Z}} ^2}{\rm Ber}(p)$ on ${\ensuremath{\mathbb Z}} ^2$, where $q = 1 - p$ will always be assumed to be sufficiently small (depending on the update family $\ensuremath{\mathcal U}$).
If $f$ and $g$ are positive real-valued functions of $q$, then we will write $f = O(g)$ if there exists a constant $C > 0$ (depending on $\ensuremath{\mathcal U}$, but \emph{not} on $q$) such that $f(q) \,\leqslant\, C g(q)$ for every sufficiently small $q > 0$. We will also write $f(q) = \O(g(q))$ if $g(q) = O(f(q))$ and $f(q) = \Theta(g(q))$ if both $f(q) = O(g(q))$ and $g(q) = O(f(q))$.
All constants, including those implied by the notation $O(\cdot)$, $\O(\cdot)$ and $\Theta(\cdot)$, are quantities that may depend on the update family $\ensuremath{\mathcal U}$ (and other quantities where explicitly stated) but not on the parameter $q$. If $c_1$ and $c_2$ are constants, then $c_1 \gg c_2 \gg 1$ means that $c_2$ is sufficiently large, and $c_1$ is sufficiently large depending on $c_2$. Similarly, $1 \gg c_1 \gg c_2 > 0$ means that $c_1$ is sufficiently small, and $c_2$ is sufficiently small depending on $c_1$. Finally, we will use the standard notation $[n] = \{1,\ldots,n\}$.
\section{Universality classes for KCM and main results}
In this section we will begin by recalling the main universality results for bootstrap cellular automata. We will then define the KCM process associated to a bootstrap update family, introduce its universality classes, and state our main results about its scaling near criticality. To finish, we will provide an outline of the heuristics behind our main theorems, and a sketch of their proofs.
\subsection{The bootstrap monotone cellular automata and its universality properties}\label{sec:boot}
Let us begin by defining a large class of two-dimensional monotone cellular automata, which were recently introduced by Bollob\'as, Smith and Uzzell~\cite{BSU}.
\pagebreak
\begin{definition}\label{def:Uboot} Let $\ensuremath{\mathcal U} = \{ X_1,\ldots,X_m \}$ be an arbitrary finite collection of finite subsets of $\mathbb{Z}^2 \setminus \{ \mathbf{0} \}$. The \emph{$\ensuremath{\mathcal U}$-bootstrap process} on $\mathbb{Z}^2$ is defined as follows: given a set $A \subset \mathbb{Z}^2$ of initially \emph{infected} sites, set $A_0 = A$, and define for each $t \;\geqslant\; 0$, \begin{equation}\label{eq:def:Uboot:At} A_{t+1} = A_t \cup \big\{ x \in \mathbb{Z}^2 \,:\, X + x \subset A_t \text{ for some } X \in \ensuremath{\mathcal U} \big\}. \end{equation} We write $[A]_\ensuremath{\mathcal U} = \bigcup_{t \,\geqslant\, 0} A_t$ for the \emph{closure} of $A$ under the $\ensuremath{\mathcal U}$-bootstrap process. \end{definition}
Thus, a vertex {{$x$}} becomes infected at time $t + 1$ if the translate by {{$x$}} of one of the sets in $\ensuremath{\mathcal U}$ (which we refer to as the \emph{update family}) is already entirely infected at time~$t$, and infected vertices remain infected forever. For example, if we take $\ensuremath{\mathcal U}$ to be the family of $2$-subsets of the set of nearest neighbours of the origin, we obtain the classical $2$-neighbour bootstrap process, which was first introduced in 1979 by Chalupa, Leath and Reich~\cite{CLR}. One of the key insights of Bollob\'as, Smith and Uzzell~\cite{BSU} was that, at least in two dimensions, the typical global behaviour of the $\ensuremath{\mathcal U}$-bootstrap process acting on random initial sets should be determined by the action of the process on discrete half-planes.
For each unit vector $u \in S^1$, let $\mathbb{H}_u := \{x \in \mathbb{Z}^2 : \langle x,u \rangle < 0 \}$ denote the discrete half-plane whose boundary is perpendicular to $u$.
\begin{definition}\label{def:stable} The set of \emph{stable directions} is $$\mathcal{S} = \mathcal{S}(\mathcal{U}) = \big\{ u \in S^1 \,:\, [\mathbb{H}_u]_\mathcal{U} = \mathbb{H}_u \big\}.$$ The update family $\ensuremath{\mathcal U}$ is: \begin{itemize} \item \emph{supercritical} if there exists an open semicircle in $S^1$ that is disjoint from $\ensuremath{\mathcal S}$,
\item \emph{critical} if there exists a semicircle in $S^1$ that has finite intersection with $\ensuremath{\mathcal S}$, and if every open semicircle in $S^1$ has non-empty intersection with $\ensuremath{\mathcal S}$,
\item \emph{subcritical} if every semicircle in $S^1$ has infinite intersection with $\ensuremath{\mathcal S}$. \end{itemize} \end{definition}
The first step towards justifying this trichotomy is given by the following theorem, which was proved in~\cites{BSU,BBPS}. Recall from~\eqref{def:qc:bootstrap} the definition of $q_c\big( \mathbb{Z}^2, \ensuremath{\mathcal U} \big)$, the critical probability of the $\ensuremath{\mathcal U}$-bootstrap process on $\mathbb{Z}^2$.
\begin{theorem} If $\ensuremath{\mathcal U}$ is a supercritical or critical two-dimensional update family, then $q_c\big( \mathbb{Z}^2, \ensuremath{\mathcal U} \big)=0$, whereas if $\ensuremath{\mathcal U}$ is subcritical then $q_c\big( \mathbb{Z}^2, \ensuremath{\mathcal U} \big) > 0$. \end{theorem}
For supercritical and critical update families, the main question is therefore to determine the scaling as $q\rightarrow 0$ of the typical time it takes to infect the origin.
\begin{definition} The \emph{typical infection time} at density $q$ of an update family $\ensuremath{\mathcal U}$ is defined to be
\[ T_{\ensuremath{\mathcal U}} \, = \, T_{q,\,\ensuremath{\mathcal U}} \, := \, \inf\bigg\{ t \,\geqslant\, 0 \,:\, {\ensuremath{\mathbb P}} _q\big( \mathbf{0} \in A_t \big) \;\geqslant\; \frac 12 \bigg\}, \] where (we recall) ${\ensuremath{\mathbb P}} _q$ indicates that every site is included in $A$ with probability $q$, independently from all other sites, and $A_t$ was defined in~\eqref{eq:def:Uboot:At}. We will write $T_{\ensuremath{\mathcal U}}$, omitting the suffix $q$ from the notation, whenever there is no risk of confusion. \end{definition}
In order to state the main result of~\cite{BDMS} we need some additional definitions. Let ${{{\ensuremath{\mathbb Q}} _1}} \subset S^1$ denote the set of rational directions on the circle, and for each $u \in {{{\ensuremath{\mathbb Q}} _1}}$, let $\ell_u^+$ be the (infinite) subset of the line $\ell_u := \{x \in \mathbb{Z}^2 : \langle x,u \rangle = 0 \}$ consisting of the origin and the sites to the right of the origin as one looks in the direction of $u$. Similarly, let $\ell_u^- := (\ell_u\setminus\ell^+_u) \cup \{ \mathbf{0} \}$ consist of the origin and the sites to the left of the origin. Given a two-dimensional bootstrap percolation update family $\ensuremath{\mathcal U}$, let $\alpha_\ensuremath{\mathcal U}^+(u)$ be the minimum (possibly infinite) cardinality of a set $Z \subset \mathbb{Z}^2$ such that $[{\ensuremath{\mathbb H}} _u \cup Z]_\ensuremath{\mathcal U}$ contains infinitely many sites of $\ell_u^+$, and define $\alpha_\ensuremath{\mathcal U}^-(u)$ similarly (using $\ell_u^-$ in place of $\ell_u^+$).
\begin{definition}\label{def:alpha} Given $u \in {{{\ensuremath{\mathbb Q}} _1}}$, the \emph{difficulty} of $u$ (with respect to $\ensuremath{\mathcal U}$) is\footnote{In order to slightly simplify the notation, and since the update family $\ensuremath{\mathcal U}$ will always be clear from the context, we will not emphasize the dependence of the difficulty on $\ensuremath{\mathcal U}$.} \[ \alpha(u) := \begin{cases} \min\big\{ \alpha_\ensuremath{\mathcal U}^+(u), \alpha_\ensuremath{\mathcal U}^-(u) \big\} &\text{if } \alpha_\ensuremath{\mathcal U}^+(u) < \infty \text{ and } \alpha_\ensuremath{\mathcal U}^-(u)<\infty \\
\infty
& \text{otherwise.} \end{cases} \] Let $\ensuremath{\mathcal C}$ denote the collection of open semicircles of $S^1$. The \emph{difficulty} of $\ensuremath{\mathcal U}$ is given by \begin{equation}\label{eq:alphaU} \alpha \, := \, \min_{C \in \ensuremath{\mathcal C}} \, \max_{u \in C} \, \alpha(u), \end{equation} and the \emph{bilateral difficulty} by \begin{equation}\label{eq:betaU} \beta \, := \, \min_{C \in \ensuremath{\mathcal C}} \, \max_{u \in C} \, \max\big\{ \alpha(u),\alpha(-u) \big\}. \end{equation} A critical update family $\ensuremath{\mathcal U}$ is \emph{balanced} if there exists a closed semicircle $C$ such that $\alpha(u) \;\leqslant\; \alpha$ for all $u\in C$. It is said to be \emph{unbalanced} otherwise. \end{definition}
\begin{remark} If $u \in S^1$ is not a stable direction then $[\mathbb{H}_u]_\mathcal{U} = {\ensuremath{\mathbb Z}} ^2$ (see~\cite {BSU}*{Lemma~3.1}), and therefore $\alpha(u) = 0$. Moreover, it was proved in~\cite {BSU}*{Lemma~5.2} (see also~\cite{BDMS}*{Lemma~2.7}) that if $u \in \mathcal{S}(\ensuremath{\mathcal U})$ then $\alpha(u) < \infty$ if and only $u$ is an isolated point of $\ensuremath{\mathcal S}(\ensuremath{\mathcal U})$. It follows that $\alpha = 0$ for every supercritical update family, and that $\alpha$ is finite for every critical update family. Observe also that $\alpha \;\leqslant\; \beta \;\leqslant\; \infty$, and that $\beta$ can be infinite even for a supercritical update family (for example, one can embed the one-dimensional East model in two dimensions). A well-studied critical model with $\beta$ infinite (and $\alpha = 1$) is the Duarte model (see~\cite{Duarte,Mountford,BCMS-Duarte}), which has update family \begin{equation}\label{def:Duarte} \ensuremath{\mathcal D} = \big\{ \{ (-1,0), (0,1) \}, \{(-1,0), (0,-1) \},\{ (0,1), (0,-1) \} \big\}. \end{equation} \end{remark}
Roughly speaking, Definition~\ref{def:alpha} says that a direction $u$ has finite difficulty if there exists a finite set of sites that, together with the half-plane ${\ensuremath{\mathbb H}} _u$, infect the entire line $\ell_u$. Moreover, the difficulty of $u$ is at least $k$ if it is necessary (in order to infect $\ell_u$) to find at least $k$ infected sites that are `close' to one another. If the open semicircle $C$ with $u$ as midpoint contains no direction of difficulty greater than $k$, then it is possible for a ``critical droplet" of infected sites to grow in the direction of $u$ without ever finding more than $k$ infected sites close together. As a consequence, if the bilateral difficulty is not greater than $k$, {{then}} there exists a direction $u$ (the midpoint of the optimal semicircle in \eqref{eq:betaU}) such that a suitable critical droplet is able to grow in \emph{both directions $u$ and $-u$}, without ever finding more than $k$ infected sites close together.
We are now in a position to state the main results on the scaling of the typical infection time for supercritical and critical update families. The following bounds were proved in~\cite{BDMS} (for critical families) and in~\cite{BSU} (for supercritical families).
\begin{theorem} \label{thm:tripartition} Let $\ensuremath{\mathcal U}$ be a two-dimensional update family. Then, as $q \rightarrow 0$, \begin{enumerate}[(a)] \item if $\ensuremath{\mathcal U}$ is supercritical then $$T_\ensuremath{\mathcal U} \, = \, q^{-\Theta(1)};$$ \item if $\ensuremath{\mathcal U}$ is critical and balanced with difficulty $\alpha$, then $$T_\ensuremath{\mathcal U} \, = \, \exp\bigg( \frac{\Theta(1)}{q^{\alpha}} \bigg);$$ \item if $\ensuremath{\mathcal U}$ is critical and unbalanced with difficulty $\alpha$, then $$T_\ensuremath{\mathcal U} \, = \, \exp\bigg( \frac{\Theta\big( \log (1/q) \big)^2}{q^{\alpha}} \bigg).$$ \end{enumerate} \end{theorem}
\begin{remark} Note that in the above result the bilateral difficulty $\beta$ plays no role. This is because in bootstrap percolation a droplet of empty sites only needs to grow in one direction (as opposed to moving back and forth). For KCM, on the other hand, we will see that the ability to move in two opposite directions will play a crucial role. \end{remark}
\subsection{General finite range KCM}
In this section we define a class of two-dimensional interacting particle systems known as \emph{kinetically constrained models}. As will be clear from what follows, KCM are intimately connected with bootstrap cellular automata.
We will work on the probability space $(\O,\mu)$, where $\O=\{0,1\}^{{\ensuremath{\mathbb Z}} ^2}$ and $\mu$ is the product Bernoulli($p$) measure, and we will be interested in the asymptotic regime $q\downarrow 0,$ where $q=1-p$. Given $\o\in \O$ and $x\in {\ensuremath{\mathbb Z}} ^2$, we will say that $x$ is ``empty" (or ``infected") if $\o_x=0$. We will say that $f \colon \O \mapsto {\ensuremath{\mathbb R}} $ is a \emph{local function} if it depends on only finitely many of the variables $\o_x$.
Given a two-dimensional update family $\ensuremath{\mathcal U} = \{X_1,\dots,X_m\}$, the corresponding KCM is the Markov process on $\O$ associated to the Markov generator \begin{equation}
\label{eq:generator} (\ensuremath{\mathcal L} f)(\o)= \sum_{x\in {\ensuremath{\mathbb Z}} ^2}c_x(\o)\big( \mu_x(f) - f \big)(\o), \end{equation} where $f \colon \O \mapsto {\ensuremath{\mathbb R}} $ is a local function, $\mu_x(f)$ denotes the average of $f$ w.r.t.~the variable $\o_x$, and $c_x$ is the indicator function of the event that there exists an update rule $X\in \ensuremath{\mathcal U}$ such that $\o_y = 0$ for every $y \in X + x$.
Informally, this process can be described as follows. Each vertex $x\in {\ensuremath{\mathbb Z}} ^2$, with rate one and independently across ${\ensuremath{\mathbb Z}} ^2$, is resampled from $\big( \{0,1\},{\rm Ber}(p) \big)$ iff {{one of the}} update rules of the $\ensuremath{\mathcal U}$-bootstrap process at $x$ {{is satisfied}} by the current configuration of the empty sites. In what follows, we will sometimes call such an update a \emph{legal update} or \emph{legal spin flip}. It follows (see~\cite{CMRT}) that $\ensuremath{\mathcal L}$ {{is}} the generator of a reversible Markov process on $\O$, with reversible measure $\mu$.
We now define the two main quantities we will use to
characterize the dynamics of the KCM process. The first of these is the relaxation time $T_{\rm rel}(q,\ensuremath{\mathcal U})$.
\begin{definition} \label{def:PC} We say that $C>0$ is a Poincar\'e constant for a given KCM if, for all local functions $f$, we have \begin{equation}
\label{eq:gap} \operatorname{Var}(f) \;\leqslant\; C \, \ensuremath{\mathcal D}(f), \end{equation} where $\ensuremath{\mathcal D}(f)=\sum_x \mu\bigl(c_x \operatorname{Var}_x(f)\bigr)$ is the KCM Dirichlet form of $f$ associated to $\ensuremath{\mathcal L}$. If there exists a finite Poincar\'e constant we then define \[ T_{\rm rel}(q,\ensuremath{\mathcal U}):=\inf\big\{ C > 0 \,:\, C \text{ is a Poincar\'e constant for the KCM} \big\}. \] Otherwise we say that the relaxation time is infinite. \end{definition} A finite relaxation time implies that the reversible measure $\mu$ is mixing for the semigroup $P_t=e^{t\ensuremath{\mathcal L}}$ with exponentially decaying time auto-correlations \cite{Liggett}. More precisely, in that case $T_{\rm rel}(q,\ensuremath{\mathcal U})^{-1}$ coincides with the best positive constant $\lambda$ such that, \begin{equation}\label{eq:relax:exp:decay} \operatorname{Var}\left(e^{t\ensuremath{\mathcal L}} f\right)\;\leqslant\; e^{-2\lambda t}\operatorname{Var}(f) \qquad \forall \, f \in L^2(\mu). \end{equation} One of the main results of~\cite{CMRT} states that for any $q > 0$ {{we have}} $T_{\rm rel}(q,\ensuremath{\mathcal U}) < \infty$ for every two-dimensional update family $\ensuremath{\mathcal U}$ such that $q_c\big( \mathbb{Z}^2, \ensuremath{\mathcal U} \big) = 0$.
The second (random) quantity is the hitting time \[ \tau_0 = \inf\big\{ t \,\geqslant\, 0 \,:\, \o_0(t) = 0 \big\}. \] In the physics literature the hitting time $\tau_0$ is usually referred to as the \emph{persistence time}, while in the bootstrap percolation framework it would be more conveniently dubbed the \emph{infection time}. For our purposes, the most important connection between the mean infection time ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ for the stationary KCM process (\hbox{\it{i.e.}, } with $\mu$ as initial distribution) and $T_{\rm rel}(q,\ensuremath{\mathcal U})$ is as follows (see \cite{Praga}*{Theorem 4.7}): \begin{equation}
\label{eq:mean-infection} {\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \frac{T_{\rm rel}(q,\ensuremath{\mathcal U})}{q} \qquad \forall \; q\in (0,1). \end{equation} The proof is quite simple. By definition, $\tau_0$ is the hitting time of $A = \big\{ \o : \ \o_0 = 0 \big\}$, and it is a standard result (see, e.g.,~\cite{DaiPra}*{Theorem 2}) that ${\ensuremath{\mathbb P}} _\mu(\tau_0>t)\,\leqslant\, e^{-t\lambda_A}$, where \[ \lambda_A=\inf\big\{ \ensuremath{\mathcal D}(f) \,:\, \mu(f^2) = 1 \text{ and } f(\o) = 0 \text{ for every } \o \in A \big\}. \] Observe that $\operatorname{Var}(f) \,\geqslant\, \mu(A) = q$ for any function $f$ satisfying $\mu(f^2)=1$ that is identically zero on $A$. This implies that $\lambda_A \,\geqslant\, q/T_{\rm rel}(q,\ensuremath{\mathcal U})$, and so~\eqref{eq:mean-infection} follows.
\begin{remark} If the initial distribution $\nu$ of the KCM process is different from the invariant measure~$\mu$, then it is only known that ${\ensuremath{\mathbb E}} _\nu(\tau_0)$ is finite in a couple of specific cases (the $d$-dimensional East process~\cites{CMST,CFM3}, and the 1-dimensional FA-1f process~\cite{BCMRT}), even under the assumption that $\nu$ is a product Bernoulli($p'$) measure with $p' \neq p$.
\end{remark}
A matching lower bound on ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ in terms of $T_{\rm rel}(q,\ensuremath{\mathcal U})$ is not known. However, in~\cite{MT}*{Lemma 4.3} it was proved that \begin{equation}
\label{eq:lowbound} {\ensuremath{\mathbb E}} _\mu(\tau_0)=\O(T_{\ensuremath{\mathcal U}}). \end{equation}
\subsection{Universality results}\label{sec:results}
We are now ready to define precisely the universality classes for KCM with a supercritical or critical update family. We will also restate (in a more precise form) our main results and conjectures on the scaling of ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ as $q \rightarrow 0$. We begin with the (much easier) supercritical case.
\begin{definition}\label{def:rooted} A supercritical two-dimensional update family $\ensuremath{\mathcal U}$ is said to be \emph{supercritical rooted} if there exist two non-opposite stable directions in $S^1$. Otherwise it is called \emph{supercritical unrooted}. \end{definition}
Our first main result, already stated in the Introduction, provides an upper bound on ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ for every supercritical two-dimensional update family that is (by the results of~\cite{MMT}) sharp up to the implicit constant factor in the exponent. Recall that if $\ensuremath{\mathcal U}$ is supercritical then $T_{\ensuremath{\mathcal U}} = q^{-\Theta(1)}$, by Theorem~\ref{thm:tripartition}.
\begin{maintheorem}[Supercritical KCM]\label{mainthm:1} Let $\ensuremath{\mathcal U}$ be a supercritical two-dimensional update family. Then, as $q \rightarrow 0$, \begin{enumerate} \item[$(a)$] if $\ensuremath{\mathcal U}$ is unrooted $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; q^{-O(1)} = \exp\Big( O\big( \log T_{\ensuremath{\mathcal U}} \big) \Big),$$ \item[$(b)$] if $\ensuremath{\mathcal U}$ is rooted, $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \exp\Big( O\big( \log q^{-1} \big)^2 \Big) = \exp\Big( O\big( \log T_{\ensuremath{\mathcal U}} \big)^2 \Big).$$ \end{enumerate} \end{maintheorem}
We next turn to our bounds for critical update families, the proofs of which will require us to overcome a number of significant technical challenges, in addition to those encountered in the supercritical case. In this setting the distinction between critical unrooted and critical rooted is more subtle, and both the difficulty $\alpha$ and the bilateral difficulty $\beta$ (see Definition~\ref{def:alpha}) play an important role. Recall that for a critical update family the difficulty is finite, but that the bilateral difficulty may be infinite.
\begin{definition}\label{def:alpha:rooted} A critical update family $\ensuremath{\mathcal U}$ with difficulty $\alpha$ and bilateral difficulty $\beta$ is said to be $\alpha$-\emph{rooted} if $\beta\,\geqslant\, 2\alpha$. Otherwise it is said to be $\beta$-\emph{unrooted}.\footnote{We warn the attentive reader that when $\alpha<\beta<2\alpha$ the model is here called $\beta$-unrooted, while in~\cite{Robsurvey} it was called $\alpha$-rooted.} \end{definition}
\pagebreak
The following theorem is the main contribution of this paper.
\begin{maintheorem}[Critical KCM]\label{mainthm:2} Let $\ensuremath{\mathcal U}$ be a critical two-dimensional update family with difficulty $\alpha$ and bilateral difficulty $\beta$. Then, as $q \rightarrow 0$, \begin{enumerate}[(a)] \item if $\ensuremath{\mathcal U}$ is $\alpha$-rooted $${\ensuremath{\mathbb E}} _\mu(\tau_0) \,\leqslant\, \exp\Big( O\Big( q^{-2\alpha} \big( \log q^{-1} \big)^4 \Big) \Big) = \exp\Big( \tilde O\big( \log T_\ensuremath{\mathcal U} \big)^2 \Big);$$ \item if $\ensuremath{\mathcal U}$ is $\beta$-unrooted $${\ensuremath{\mathbb E}} _\mu(\tau_0) \,\leqslant\, \exp\Big( O\Big( q^{-\beta} \big( \log q^{-1} \big)^3 \Big) \Big) = \exp\Big( \tilde O\big( \log T_\ensuremath{\mathcal U} \big)^{\beta/\alpha} \Big).$$ \end{enumerate} \end{maintheorem}
\begin{remark} \label{rem:Trel} It will follow immediately from our proof that the upper bounds of Theorems~\ref{mainthm:1} and~\ref{mainthm:2} hold also for the relaxation time $T_{\rm rel}(q,\ensuremath{\mathcal U})$. Indeed, we first establish upper bounds for the relaxation time, then derive the upper bounds on ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ via \eqref{eq:mean-infection}. \end{remark}
It was recently proved in~\cite{MMT} that the upper bounds in Theorem~\ref{mainthm:1} are best possible up to the implicit constant factor in the exponent for all supercritical update families (note that this follows from~\eqref{eq:lowbound} for unrooted models). We conjecture that the bounds for critical models in Theorem~\ref{mainthm:2} are also best possible, though in a slightly weaker sense: up to a polylogarithmic factor in the exponent.
\begin{conjecture}\label{critical:conj} Let $\ensuremath{\mathcal U}$ be a critical two-dimensional update family with difficulty $\alpha$ and bilateral difficulty $\beta$. Then, as $q \rightarrow 0$, \begin{enumerate}[(a)] \item if $\ensuremath{\mathcal U}$ is $\alpha$-rooted $${\ensuremath{\mathbb E}} _\mu(\tau_0) = \exp\Big( q^{-2\alpha} \big( \log q^{-1} \big)^{\Theta(1)} \Big);$$ \item if $\ensuremath{\mathcal U}$ is $\beta$-unrooted $${\ensuremath{\mathbb E}} _\mu(\tau_0) = \exp\Big( q^{-\beta} \big( \log q^{-1} \big)^{\Theta(1)} \Big).$$ \end{enumerate} \end{conjecture}
Observe that for $\alpha$-unrooted update families $\ensuremath{\mathcal U}$ (i.e., families with $\beta = \alpha$), the lower bound in Conjecture~\ref{critical:conj} follows from Theorem~\ref{thm:tripartition} and~\eqref{eq:lowbound}; in particular Theorem~\ref{mainthm:2} confirms~\cite{Robsurvey}*{Conjecture~2.4}. If $\ensuremath{\mathcal U}$ is moreover unbalanced, then the upper and lower bounds given by Theorems~\ref{mainthm:2} and~\ref{thm:tripartition} differ by only a single factor of $\log(1/q)$ (in the exponent), and we suspect that in this case the lower bound is correct, see {{Remark~\ref{rmk:losing:log}.}}
\begin{conjecture}\label{unbalanced:conj} Let $\ensuremath{\mathcal U}$ be an $\alpha$-unrooted, unbalanced, critical two-dimensional update family with difficulty $\alpha$. Then, as $q \rightarrow 0$, $${\ensuremath{\mathbb E}} _\mu(\tau_0) = \exp\Big( \Theta\Big( q^{-\alpha} \big( \log q^{-1} \big)^2 \Big) \Big).$$ \end{conjecture}
We remark that an example of an update family satisfying the conditions of Conjecture~\ref{unbalanced:conj} is the so-called \emph{anisotropic model} (see, e.g.,~\cite{DC-Enter,DPEH}) whose update family consists of all subsets of size 3 of the set $$\big\{ (-2,0), (-1,0), (1,0), (2,0), (0,1), (0,-1) \big\}.$$ Another model for which Conjecture~\ref{critical:conj} holds is the Duarte model, defined in~\eqref{def:Duarte}, for which a matching lower bound (this time, up to a \emph{constant} factor in the exponent) was recently proved in~\cite{MMT}, confirming (in a strong sense)~\cite{Robsurvey}*{Conjecture~2.5}. For all other critical models, however, the best known lower bound is that given by Theorem~\ref{thm:tripartition} and~\eqref{eq:lowbound}, and is therefore (we think) very far from the truth.
\subsection{Heuristics and roadmap}\label{sec:roadmap}
We conclude this section with a high-level description of the intuition behind the proofs of Theorems~\ref{mainthm:1} and~\ref{mainthm:2}, together with a roadmap of the actual proof, which is carried out in Sections~\ref{sec:CPI}--\ref{sec:fullgen}.
The first key point to be stressed is that we never actually follow the dynamics of the KCM process itself; instead, we will prove the existence of a Poincar\'e constant with the correct scaling as $q\rightarrow 0$, and use the inequality~\eqref{eq:mean-infection} to deduce a bound on the mean infection time.
We emphasize that this approach only works for the stationary KCM, that is, the process starting from the stationary measure $\mu$. The second point is that, given that the Dirichlet form of the KCM $$\ensuremath{\mathcal D}(f) = \sum_{x \in {\ensuremath{\mathbb Z}} ^2} \mu\big( c_x \operatorname{Var}_x(f) \big)$$ is a sum of local variances ($\Leftrightarrow$ spin flips) computed with suitable infection nearby ($\Leftrightarrow$ the constraints $c_x$), all of our reasoning will be guided by the fact that we need to have some infection ($\Leftrightarrow$ empty sites) next to where we want to compute the variance. Therefore, much of our intuition, and all of the technical tools, have been developed with the aim of finding a way to \emph{effectively} move infection where we need it.
A configuration sampled from $\mu$ will always have ``mesoscopic'' droplets (large patches of infected sites), though these will typically be very far from the origin. The general theory of $\ensuremath{\mathcal U}$-bootstrap percolation developed in~\cite{BDMS,BSU} allows us to quantify very precisely the critical size of those droplets that (typically) allows infection to grow from them and invade the system. However -- and this is a fundamental difference between bootstrap percolation and KCM -- it is extremely unlikely for the stationary KCM to create around a given vertex and at a given time a very large cluster of infection. Thus, it is essential to envisage an \emph{infection/healing} mechanism that is able to \emph{move} infection over long distances without creating too large an excess\footnote{In physical terms an excess of infection is equivalent to an ``energy barrier''.} of it.
At the root of our approach lies the notion of a \emph{critical
droplet}. A critical droplet is a certain finite set $D$ whose geometry depends on the update family $\ensuremath{\mathcal U}$, and whose characteristic size may depend on $q$. For supercritical models we can take any sufficiently large (\emph{not} depending on $q$) rectangle oriented along the mid-point $u$ of a semicircle $C$ free of stable directions. For critical models the droplet $D$ is a more complicated object called a \emph{quasi-stable half-ring} (see Definition \ref{def:half-ring} and Figure~\ref{fig:half-ring}) oriented along the midpoint $u$ of an open semicircle with largest difficulty either $\alpha$ or $\beta$. The long sides of $D$ will have length either $\Theta\big( q^{-\alpha} \log(1/q) \big)$ or $\Theta\big( q^{-\beta} \log(1/q) \big)$ for the $\alpha$-rooted {{and}} $\beta$-unrooted cases respectively, while the short sides will always have length $\Theta(1)$. The key feature of a critical droplet for supercritical models (see Section~\ref{sec:supercritical:bootstrap}) is that, if it is empty, {{then}} it is able to infect a suitable translate of itself in the $u$-direction. For unrooted supercritical models the semicircle $C$ can be chosen in such a way that both $C$ \emph{and} $-C$ are free of stable directions. As a consequence, the empty critical droplet will be able to infect a suitable translate of itself in \emph{both} directions $\pm u$.
For critical models the situation changes drastically. An empty critical droplet will not be able to infect freely another critical droplet next to it in the $u$-direction because of the stable directions which are present in every open semicircle. However, it will be able to do so (in the $u$-direction if the model is $\alpha$-rooted, and in the $\pm u$-directions if $\beta$-unrooted) provided that it receives some help from a finite number of extra empty sites (in ``clusters" of size $\alpha$ or $\beta$) nearby. If the size of the critical droplet {{is}} chosen as above, then it is straightforward to show that such extra helping empty sites will be present with high probability (see Section~\ref{sec:critical:rooted}).
Having clarified what a critical droplet is, and under which circumstances it is able to infect nearby sites, we next explain what we mean by ``moving a critical droplet''. For simplicity we explain the heuristics only for the supercritical case. Imagine that we have a sequence $D_0,D_1,\dots,D_n$ of contiguous, non-overlapping and identical critical droplets such that $D_{i+1}=D_i+ d_i u$ for some suitable $d_i>0$. Suppose first that the model is unrooted and that $D_0$ is completely infected, and let us write $\o_i$ for the configuration of spins in $D_i$. Using the infection in $D_0$ it possible to first infect $D_1$, then $D_2$ and then, using reversibility, restore (\hbox{\it{i.e.}, } heal) the original configuration $\o_1$ in $D_1$. Using the infection in $D_2$ we can next infect $D_3$ and then, using the infection in $D_3$, restore $\o_2$ in $D_2$ (see the schematic diagram below, where $\emptyset$ stands for an infected droplet) \begin{align*} \emptyset\ \o_1\ \o_2\ \o_3\dots &\mapsto \emptyset\ \emptyset\ \o_2\ \o_3\dots \mapsto\emptyset\ \emptyset\ \emptyset\ \o_3\dots \\ & \mapsto \emptyset\ \o_1\ \emptyset\ \o_3\dots \mapsto \emptyset\ \o_1\ \emptyset\ \emptyset\dots \mapsto \emptyset\ \o_1\ \o_2\ \emptyset\dots \end{align*} If we continue in this way, we end up moving the original infection in $D_0$ to the last droplet $D_n$ without having ever created more than two extra infected critical droplets simultaneously. We remark that the sequence described above is reminiscent of how infection moves in the one-dimensional $1$-neighbour KCM.
For rooted supercritical models, on the other hand, we cannot simply restore the configuration $\o_2$ in $D_2$ using only the infection in $D_3$ (in the unrooted case this was possible because infection could propagate in both the $u$ and $-u$ directions). As a consequence, we need to follow a more complicated pattern: \begin{align*} \emptyset\ \o_1\ \o_2\ \o_3\dots &\mapsto \emptyset\ \emptyset\ \o_2\ \o_3\dots \mapsto\emptyset\ \emptyset\ \emptyset\ \o_3\dots \\ &\mapsto \emptyset\ \emptyset\ \emptyset\ \emptyset\dots \mapsto \emptyset\ \ \emptyset\ \o_2\ \emptyset\dots \mapsto \emptyset\ \o_1\ \o_2\ \emptyset\dots, \end{align*} in which healing is always induced by infection present in the adjacent droplet in the $-u$ direction. This latter case is reminiscent of the one-dimensional East model. In this case, a combinatorial result proved in~\cite{CDG} implies that in order to move the infection to $D_n$ it is necessary to create $\asymp \log n$ \emph{simultaneous} extra infected critical droplets. This logarithmic energy barrier is the reason for the different scaling of ${\ensuremath{\mathbb E}} _\mu(\tau_0)$ in rooted and unrooted supercritical models (see Theorem~\ref{mainthm:1}).
Let us now give a somewhat more detailed outline of our approach. We begin by partitioning ${\ensuremath{\mathbb Z}} ^2$ into `suitable' rectangular blocks $\{V_i\}_{i\in {\ensuremath{\mathbb Z}} ^2}$ with shortest side orthogonal to the direction $u$ (see Section \ref{sec:setting}). For supercritical models these blocks have sides of constant length, while for critical models they will have length $q^{-\kappa}$ for some constant $\kappa \gg \alpha$, and height equal to that of a critical droplet, so either $\Theta\big( q^{-\alpha} \log(1/q) \big)$ or $\Theta\big( q^{-\beta} \log(1/q) \big)$, depending on the nature of the model. Then, given a configuration $\o \in \O$, we declare a block to be \emph{good} or \emph{super-good} according to the following rules: \begin{itemize} \item For supercritical models \emph{any} block is good, while for critical models good blocks are those which contain ``enough" empty sites to allow an adjacent empty critical droplet to advance in the $u$ (or $\pm u$) direction(s) (see Definition~\ref{def:goodsets}).
\item In both cases, a block is said to be super-good if it is good and also contains an empty (i.e., completely infected) critical droplet. \end{itemize} Good blocks turn out to be very likely w.r.t.~$\mu$ (a triviality in the supercritical case), and it follows by standard percolation arguments that they form a rather dense infinite cluster. Super-good blocks, on the other hand, are quite rare, with density $\rho= q^\Theta(1)$ in the supercritical case, $\rho = \exp\big( - \Theta\big( q^{-\alpha} \log(1/q)^2 \big) \big)$ in the critical $\alpha$-rooted case, and $\rho = \exp\big( - \Theta\big( q^{-\beta} \log(1/q)^2 \big) \big)$ for critical $\beta$-unrooted models.
We will then prove the existence of a suitable Poincar\'e constant in three steps, each step being associated to a natural kinetically constrained \emph{block dynamics}\footnote{See, e.g.,~Chapter~15.5 of~\cite{Levin-2008} for a introduction to the technique of block dynamics in reversible Markov chains.} on a certain length scale. In each block dynamics the configuration in each block is resampled with rate one (and independently of other resamplings) if a certain constraint is satisfied.
Our first block dynamics forces one of the blocks neighbouring $V_i$ to be at the beginning of an oriented ``thick'' path $\gamma$ of good blocks, with length $\approx 1/\rho$, whose last block is super-good. Using the fact that this constraint is very likely, it is possible to prove (see Section~2 in~\cite{MT}) that the relaxation time of this process is $O(1)$, and moreover (see Proposition~\ref{lem:MT:prop34}) that the Poincar\'e inequality \begin{equation}\label{eq:8} \operatorname{Var}(f) \;\leqslant\; 4 \sum_i \mu\big( \mathbbm{1}_{\Gamma_i} \operatorname{Var}_i(f) \big) \end{equation} holds, where $\mathbbm{1}_{\Gamma_i}$ is the indicator of the event that a good path exists for $V_i$. Though this starting point is similar to the method we develop in \cite{MT}, for the next two steps of the proof we introduce here a completely different set of tools and ideas in order to avoid the direct use of {\emph canonical paths} (which could instead be used in \cite{MT} for the special case of the FA-2f model). Indeed for a general model (and especially for rooted models), using canonical paths and evaluating their congestion constants would result in a very heavy and complicated machinery. The next idea is to convert the \emph{long-range} constrained Poincar\'e inequality \eqref{eq:8} into a \emph{short-range} one of the form \begin{equation}\label{eq:10}
\operatorname{Var}(f) \;\leqslant\; C_1(q)\sum_i \mu\big( \mathbbm{1}_{SG_i} \operatorname{Var}_i(f) \big), \end{equation} in which $\mathbbm{1}_{SG_i}$ is the indicator of the event that a suitable collection of blocks \emph{near} $V_i$ are good and one of them is super-good. Which collections of blocks are ``suitable", and which one should be super-good, depends on whether the model is rooted or unrooted; we refer the reader to Theorem~\ref{thm:CPI} for the details. The main content of Theorem~\ref{thm:CPI}, which we present in a slightly more general setting for later convenience, is that $C_1(q)$ can be taken equal to the best Poincar\'e constant (\hbox{\it{i.e.}, } the relaxation time) of a one-dimensional generalised $1$-neighbour or East process at the effective density $\rho$. Section~\ref{sec:CPI} is entirely dedicated to the task of formalising and proving the above claim.
The final step of the proof is to convert the Poincar\'e inequality~\eqref{eq:10} into the true Poincar\'e inequality for our KCM \[ \operatorname{Var}(f) \;\leqslant\; C_2(q)\sum_x \mu(c_x \operatorname{Var}_x(f)), \] with a Poincar\'e constant $C_2(q)$ which scales with $q$ as required by Theorems~\ref{mainthm:1} and~\ref{mainthm:2}. In turn, {{this}} requires {{us to prove}} that a full resampling of a block in the presence of nearby super-good and good blocks can be simulated (or reproduced) by a sequence of legal single-site updates of the \emph{original} KCM, with a global cost in the Poincar\'e constant compatible with Theorems~\ref{mainthm:1} and~\ref{mainthm:2}. It is here that the results of~\cite{BDMS,BSU} on the behaviour of the
$\ensuremath{\mathcal U}$-bootstrap process come into play. While for supercritical models the task described above is relatively simple (see Section~\ref{sec:supercritical}), for critical models the problem is significantly more {{complicated}} and a suitable generalised East process {{again plays}} a key role. A full sketch of the proof can be found in Section~\ref{sec:core}, see in particular the proof of Proposition~\ref{prop:I1:I2}, and Remark~\ref{rem:whyEast}.
\section{Constrained Poincar\'e inequalities}\label{sec:CPI}
The aim of this section is to prove a constrained Poincar\'e inequality for a product measure on $S^{{\ensuremath{\mathbb Z}} ^2}$, where $S$ is a finite set. This general inequality will play an instrumental role in the proof of our main theorems, giving us precise control of the infection time for both supercritical and critical KCM.
In order to state our general constrained Poincar\'e inequality, we will need some notation. Let $(S,\hat\mu)$ be a finite positive probability space, and set $\O = \big( S^{{\ensuremath{\mathbb Z}} ^2},\mu \big)$, where $\mu = \otimes_{i\in {\ensuremath{\mathbb Z}} ^2}\hat\mu$. A generic element $\O$ will be denoted by $\o=\{\o_i\}_{i\in {\ensuremath{\mathbb Z}} ^2}$. For any local function $f$ we will write $\operatorname{Var}(f)$ for its variance w.r.t.~$\mu$ and $\operatorname{Var}_i(f)$ for the variance w.r.t.~to the variable $\o_i\in S$ conditioned on all the other variables $\{\o_j\}_{j\neq i}$. For any $i\in {\ensuremath{\mathbb Z}} ^2$ we set $${\ensuremath{\mathbb L}} ^+(i) = i + \big\{ \vec e_1, \vec e_2 - \vec e_1 \big\} \qquad \textup{and} \qquad {\ensuremath{\mathbb L}} ^-(i) = i - \big\{ \vec e_1, \vec e_2 - \vec e_1 \big\}.$$ Finally, let $G_2 \subseteq G_1\subseteq S$ be two events, and set $p_1 := \hat\mu(G_1)$ and $p_2 := \hat\mu(G_2)$. The main result of this section is the following theorem.
\begin{theorem}\label{thm:CPI} For any $t\in (0,1)$ there exist $\vec T(t),T(t)$ satisfying $\vec T(t) \;\leqslant\; \exp\big( O\big( \log \tfrac{1}{t} \big)^2 \big)$ and $T(t) \;\leqslant\; t^{-O(1)}$ as $t \rightarrow 0$, such that the following oriented and unoriented constrained Poincar\'e inequalities hold.\vskip6pt \noindent {\bf (A)} \hskip 12pt Suppose that $G_1=S$ and $G_2 \subseteq S$. Then, for all local functions $f$: \begin{align}\label{eq:8East} & \operatorname{Var}(f) \;\leqslant\; \vec T(p_2) \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\vec e_1} \in G_2\}}\operatorname{Var}_i(f)\right)\\ & \operatorname{Var}(f) \;\leqslant\; T(p_2) \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\{\o_{i+\vec e_1}\in G_2\}\cup\{\o_{i-\vec e_1} \in G_2\}\}}\operatorname{Var}_i(f)\right).\label{eq:8FA} \end{align} \vskip 6pt \noindent {\bf (B)} \hskip 12pt Suppose that $G_2 \subseteq G_1 \subseteq S$. Then there exists $\delta > 0$ such that, for all $p_1,p_2$ satisfying $\max\big\{ p_2, (1-p_1) (\log p_2)^2 \big\} \;\leqslant\; \delta$, and all local functions $f$: \begin{align}\label{eq:9East} & \operatorname{Var}(f) \;\leqslant\; \vec T(p_2) \bigg( \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^+(i)\}} \operatorname{Var}_i(f)\right)\nonumber\\ & \hspace{3.5cm} + \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\vec e_1}\in G_2\}}\mathbbm{1}_{\{\o_{i-\vec e_1}\in G_1\}} \operatorname{Var}_i\big( f \thinspace |\thinspace G_1 \big) \right)\bigg),\\ & \operatorname{Var}(f) \;\leqslant\; T(p_2) \bigg( \sum_{\varepsilon=\pm 1}\sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\varepsilon\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^{\varepsilon}(i)\}} \operatorname{Var}_i(f)\right) \nonumber\\ & \hspace{3.5cm} + \sum_{\varepsilon=\pm 1}\sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\varepsilon\vec e_1} \in G_2\}}\mathbbm{1}_{\{\o_{i-\varepsilon \vec e_1}\in G_1\}} \operatorname{Var}_i\big( f \thinspace |\thinspace G_1 \big) \right) \bigg). \label{eq:9FA} \end{align} \end{theorem}
\begin{remark} When proving Theorem \ref{mainthm:1} the starting point will be~\eqref{eq:8East} or~\eqref{eq:8FA}, depending on whether the model is rooted or unrooted. Similarly, for critical models we will start the proof of Theorem~\ref{mainthm:2} from~\eqref{eq:9East} or~\eqref{eq:9FA} depending on whether the model is $\alpha$-rooted or $\beta$-unrooted. This choice is dictated by the $\ensuremath{\mathcal U}$-bootstrap process according to the following rule: we will require $V_i \subset [A]_\ensuremath{\mathcal U}$ to hold for \emph{any} set $A$ of empty sites such that the indicator function in front of $\operatorname{Var}_i(f)$ is equal to one. We refer the reader to Sections~\ref{sec:supercritical} and~\ref{sec:critic-ub}, and in particular to the proof of Lemma~\ref{lem:var0}, for more details. \end{remark}
An important role in the proof of the theorem is played by the one-dimensional East and $1$-neighbour processes (see, e.g.,~\cite{CMRT}), and a certain generalization of these processes. For the reader's convenience, we begin by recalling these generalized models.
\subsection{The generalised East and $1$-neighbour models}\label{sec:East-FA}
The standard versions of these two models are ergodic interacting particle systems on $\{0,1\}^n$ with kinetic constraints, which will mean that jumps in the dynamics are facilitated by certain configurations of vertices in state $0$. They are both reversible w.r.t.~the product measure $\pi= {\rm Ber}(\alpha_1) \otimes \cdots \otimes {\rm Ber}(\alpha_n)$, where ${\rm Ber}(\alpha)$ is the $\alpha$-Bernoulli measure and $\alpha_1,\ldots,\alpha_n \in (0,1)$.
In the first process, known as the \emph{non-homogeneous East model} (see~\cites{JACKLE,East-review} and references therein), the state $\o_x$ of each point $x \in [n]$ is resampled at rate one (independently across $[n]$) from the distribution ${\rm Ber}(\alpha_x)$, provided that $c_x(\o) = 1$, where $$c_x(\o) = \mathbbm{1}_{\{\o_{x+1}=0\}} \qquad \textup{and} \qquad \o_{n+1} := 0.$$ In the second model, known as the \emph{non-homogeneous $1$-neighbour model} (and also as the FA-1f model~\cites{FH}), the resampling occurs in the same way, except in this case $$c_x(\o)= \max\big\{ \mathbbm{1}_{\{\o_{x-1} = 0\}}, \, \mathbbm{1}_{\{\o_{x+1} = 0\}} \big\} \qquad \textup{where} \qquad \o_{0} := 1 \qquad \textup{and} \qquad \o_{n+1} := 0.$$ It is known \cites{Aldous,CMRT,CFM} that the corresponding relaxation times $T_{\text{\tiny East}}(n,\bar \alpha)$ and $T_{\text{\tiny
FA}}(n,\bar\alpha)$ (where $\bar\alpha = (\alpha_1,\ldots,\alpha_n)$) are finite \emph{uniformly} in $n$ and that they satisfy the following scaling as $q := \min\big\{ 1 - \alpha_x : x \in [n] \big\}$ tends to zero: \begin{align}\label{eq:scaling} T_{\text{\tiny East}}\big( n,\bar \alpha \big)= q^{-O(\min\{ \log n, \,
\log(1/q) \})} \qquad
\text{and}\qquad T_{\text{\tiny FA}}\big( n,\bar \alpha \big) = q^{-O(1)}. \end{align} The proof of \eqref{eq:scaling} is deferred to the Appendix. In the proof of Theorem~\ref{thm:CPI} we will need to work in the following more general setting.
Consider a finite product probability space of the form $\O=\otimes_{x\in
[n]}(S_x,\nu_x)$, where $S_x$ is either a finite set or an interval of ${\ensuremath{\mathbb R}} $, and $\nu_x$ is a positive probability measure on $S_x$. Given $\{\o_x\}_{x \in [n]}\in \O$, we will refer to $\o_x$ as the \emph{the state of the vertex
$x$}. Moreover, for each $x\in [n]$, let us fix a constraining event $S^g_x \subseteq S_x$ with $q_x := \nu_x(S^g_x) > 0$. We consider the following generalisations of the East and FA-1f processes on the space $\O$.
\begin{definition} \label{def:gen:East} In the \emph{generalised East chain}, the state $\o_x$ of each vertex $x \in [n]$ is resampled at rate one (independently across $[n]$) from the distribution $\nu_x$, provided that $\vec c_x(\o) = 1$, where $$\vec c_x(\o) = \mathbbm{1}_{\{\o_{x+1} \in S^g_{x+1}\}}$$ if $x \in \{1,\ldots,n-1\}$, and ${{\vec c_n}}(\o) \equiv 1$.
In the \emph{generalised FA-1f chain}, the resampling occurs in the same way, except in this case $c_1(\o)= \mathbbm{1}_{\{ \o_2 \,\in\, S^g_2 \}}$, $$c_x(\o)= \max\big\{ \mathbbm{1}_{\{\o_{x-1} \, \in \, S^g_{x-1}\}}, \, \mathbbm{1}_{\{\o_{x+1} \, \in \, S^g_{x+1}\}} \big\}$$ if $x \in \{2,\ldots,n-1\}$, and $c_n(\o) \equiv 1$.
In both cases, set $q := \min_x q_x = \min_x \nu_x(S^g_x)$, and set $\alpha_x := 1 - q_x$ for each $x \in [n]$. \end{definition}
Note that the projection variables $\eta_x = \mathbbm{1}_{\{S^g_x\}}$ evolve as a standard East or FA-1f chain, and it is therefore natural to ask whether the relaxation times of these generalised constrained chains can be bounded from above in terms of the relaxation times $T_{\text{\tiny East}}(n,\bar \alpha)$ and $T_{\text{\tiny FA}}(n,\bar \alpha)$ respectively. The answer is affirmative, and it is the content of the following proposition (cf.~\cite{CFM2}*{Proposition 3.4}), which provides us with Poincar\'e inequalities for the generalised East and FA-1f chains.
\begin{proposition}\label{lem:gen-Poincare} Let $f:\O\mapsto {\ensuremath{\mathbb R}} $. For the generalised East chain, we have \begin{equation}\label{eq:Poincare:genEast} \operatorname{Var}(f) \;\leqslant\; \frac{1}{q} \cdot T_{\text{\tiny East}}(n,\bar \alpha) \cdot \sum_{x = 1}^n \nu\big( \vec c_x\operatorname{Var}_{x}(f) \big), \end{equation} and for the generalised FA-1f chain, we have \begin{equation}\label{eq:Poincare:genFA1f} \operatorname{Var}(f) \;\leqslant\; \frac{1}{q} \cdot T_{\text{\tiny FA}}(n,\bar \alpha) \cdot \sum_{x = 1}^n \nu\big( c_x \operatorname{Var}_{x}(f) \big), \end{equation} where $\operatorname{Var}_x(\cdot)$ denotes the conditional variance w.r.t.~$\nu_x$, given all the other variables. \end{proposition}
The proof of this proposition, which is similar to that of~\cite{CFM}*{Proposition 3.4}, is deferred to {{the}} Appendix.
\subsection{Proof of Theorem~\ref{thm:CPI}}
We begin with the proof of part (A), which is a relatively straightforward consequence of Proposition~\ref{lem:gen-Poincare} and~\eqref{eq:scaling}. The proof of part (B) is significantly more difficult, and we will require a technical result from~\cite{MT} (see Proposition~\ref{lem:MT:prop34}, below) and a careful application of Proposition~\ref{lem:gen-Poincare} (and of convexity) after conditioning on various events.
\subsubsection{Proof of part (A)}
Recall that in this setting $G_1 = S$ and $G_2\subset S$, where $(S,\hat\mu)$ is an arbitrary finite positive probability space.
Let $f$ be a local function and let $M > 0$ be sufficiently large so that $f$ does not depend on the variables at vertices $(m,n)$ with $|m| \,\geqslant\, M$. For each $n \in {\ensuremath{\mathbb Z}} $, let $\mu_{n}$ denote the product measure $\otimes_{m\in {\ensuremath{\mathbb Z}} }\ \hat \mu$ on $S^{{\ensuremath{\mathbb Z}} \times \{n\}}$, and note that $\mu=\otimes_{n \in {\ensuremath{\mathbb Z}} }\ \mu_n$. By construction, $\operatorname{Var}_{\mu_n}(f)$ coincides with the same conditional variance computed w.r.t.~$\mu_{n}^M := \otimes_{m \in {\ensuremath{\mathbb Z}} \cap [-M,M]}\ \hat \mu.$
We apply Proposition~\ref{lem:gen-Poincare} to the homogeneous product measure $\mu_n^{M}$ with the event $G_2$ as event $S_x^g$ for all $x \in \{-M,\ldots,M\}$. Note that $q_x = \hat\mu(G_2) = p_2$ for every $x$, and that $\operatorname{Var}_{(M,n)}(f) = \operatorname{Var}_{(-M,n)}(f) = 0$. It follows, using~\eqref{eq:scaling}, that $$\operatorname{Var}_{\mu_n}(f) \;\leqslant\; \, \vec T(p_2) \displaystyle\sum_{m\in {\ensuremath{\mathbb Z}} }\mu_n\left(\mathbbm{1}_{\{\o_{(m+1,n)} \in G_2\}} \operatorname{Var}_{(m,n)}(f)\right),$$ where $\vec T(p_2) = \exp\Big( O\big( \log \tfrac{1}{p_2} \big)^2 \Big)$, and $$\operatorname{Var}_{\mu_n}(f) \;\leqslant\; \, T(p_2) \displaystyle\sum_{m \in {\ensuremath{\mathbb Z}} } \mu_n\left(\mathbbm{1}_{\{\o_{(m+1,n)} \in G_2\} \cup \{\o_{(m-1,n)} \in G_2\}} \operatorname{Var}_{(m,n)}(f)\right),$$ where $T(p_2) = p_2^{-O(1)}$.
Using the standard inequality $\operatorname{Var}_\mu(f)\,\leqslant\, \sum_{n \in {\ensuremath{\mathbb Z}} } \mu\big( \operatorname{Var}_{\mu_n}(f) \big)$, the Poincar\'e inequalities~\eqref{eq:8East} and~\eqref{eq:8FA} follow.
\subsubsection{Proof of part (B)}
We next turn to the significantly more challenging task of proving the constrained Poincar\'e inequalities~\eqref{eq:9East} and~\eqref{eq:9FA}. As noted above, in addition to Proposition~\ref{lem:gen-Poincare} we will require a technical result from~\cite{MT}, stated below as Proposition~\ref{lem:MT:prop34}. In order to state this result we need some additional notation.
Recall that an \emph{oriented path of length $n$} in ${\ensuremath{\mathbb Z}} ^2$ is a sequence $\gamma = (i^{(1)},\dots, i^{(n)})$ of $n$ vertices of ${\ensuremath{\mathbb Z}} ^2$ with the property that $i^{(k+1)} - i^{(k)} \in \{ \vec e_1, \vec e_2 \}$ for each $k \in [n-1]$. We will say that $\gamma$ starts at $i^{(1)}$, ends at $i^{(n)}$, and that $i \in \gamma$ if $i = i^{(k)}$ for some $k \in [n]$. Moreover, given $\o\in \O$, we will say that $\gamma$ is \begin{itemize} \item $\o$-\emph{good} if $\o_i \in G_1$ for all $i \in \bigcup_{j \in \gamma} \big\{ j, j + \vec e_1, j - \vec e_1 \big\}$, and \item $\o$-\emph{super-good} if it is good and there exists $i \in \gamma$ such that $\o_i\in G_2$, \end{itemize} where $G_2 \subseteq G_1 \subseteq S$ are the events in the statement of Theorem~\ref{thm:CPI}.
In what follows it will be convenient to order the oriented paths of length $n$ starting from a given point according to the alphabetical order of the associated strings of $n$ unit vectors from the finite alphabet $\ensuremath{\mathcal X}=\{\vec e_1,\vec e_2\}$. Next, for each $i\in {\ensuremath{\mathbb Z}} ^2$ we define the key event $\Gamma_i\subset \O$, as follows: \begin{enumerate}[(i)] \item there exists an oriented $\o$-good path $\gamma$, of length $L = \big\lfloor 1/p_2^2 \big\rfloor$ starting at $i + \vec e_2$; \item the smallest such path (in the above order) is $\o$-super-good; \item $\o_{i+\vec e_1} \in G_1$. \end{enumerate} In what follows, and if no confusion arises, we will abbreviate $\o$-good and $\o$-super-good to good and super-good respectively. The following upper bound on $\operatorname{Var}(f)$ is very similar to~\cite{MT}*{Proposition 3.4}, and we therefore defer the proof to the Appendix.
\begin{proposition}\label{lem:MT:prop34} There exists $\delta > 0$ such that, if $\max\big\{ p_2, (1-p_1) (\log p_2)^2 \big\} \,\leqslant\, \delta$, then \begin{equation}\label{eq:MT:prop34} \operatorname{Var}(f) \;\leqslant\; 4 \sum_{i \in {\ensuremath{\mathbb Z}} ^2} \mu\big( \mathbbm{1}_{\Gamma_i} \operatorname{Var}_i(f) \big) \end{equation} for every local function $f$. \end{proposition}
We would like to use Proposition~\ref{lem:gen-Poincare} to bound the right-hand side of~\eqref{eq:MT:prop34}. However, Proposition~\ref{lem:gen-Poincare} provides us with an upper bound on the variance of a function, whereas the quantity $\mu\big( \mathbbm{1}_{\Gamma_i} \operatorname{Var}_i(f) \big)$ is more like the average of a local variance. We will therefore need to use convexity to bound from above the average of a local variance by a full variance. In order to reduce as much as possible the potential loss of such an operation, we first perform a series of conditionings on the measure $\mu$ and use convexity only on the final conditional measure.
Roughly speaking, on the event $\Gamma_i$ we first reveal, for each $j \ne i$ within distance $2/p_2^2$ of the origin, whether or not the event $\{\o_j \in G_1\}$ holds. Given this information, we know which paths of length $L$ and starting at $i + \vec e_2$ are good and we define $\gamma^*$ as the smallest one in the order defined above. Next, we reveal the \emph{last} $j^* \in \gamma^*$ such that $\{\o_{j^*} \in G_2\}$. Note that in doing so we do not need to observe whether or not the event $\{\o_j \in G_2 \}$ holds for any earlier $j$ (i.e., before $j^*$ in $\gamma^*$). Finally, defining $\gamma \subset \gamma^*$ to be the part of $\gamma^*$ before $j^*$, we reveal $\o_j$ for all $j \in {\ensuremath{\mathbb Z}} ^2$, except for $j = i$ and $j \in \gamma$.
\begin{figure}
\caption{The minimal good path $\gamma^*$, the position of the first
super-good vertex $\xi$ encountered while traveling backward along
$\gamma^*$, and the subpath $\gamma \subset \gamma^*$ (thick black) connecting $\vec e_2$ to a neighbour of $\xi$. }
\label{fig:1}
\end{figure}
At the end of this process we are left with a (conditional) probability measure $\nu$ on $S^{\gamma \cup\{i\}}$. We will then apply convexity and Proposition~\ref{lem:gen-Poincare} to this measure. We now detail the above procedure.
\begin{proof}[Proof of part (B) of Theorem~\ref{thm:CPI}] Let $\delta > 0$ be given by Proposition~\ref{lem:MT:prop34}, and assume that the events $G_2 \subseteq G_1 \subseteq S$ satisfy $\max\big\{ p_2, (1-p_1) (\log p_2)^2 \big\} \,\leqslant\, \delta$. By Proposition~\ref{lem:MT:prop34}, we have \begin{equation}\label{eq:MT:prop34:repeat} \operatorname{Var}(f) \;\leqslant\; 4 \sum_{i \in {\ensuremath{\mathbb Z}} ^2} \mu\big( \mathbbm{1}_{\Gamma_i} \operatorname{Var}_i(f) \big)
\end{equation} for every local function $f$. We will bound each term of the sum in~\eqref{eq:MT:prop34:repeat}. Using translation invariance, it will suffice to consider the term $i = (0,0)$.
For each $\o \in \Gamma_{(0,0)}$, let $\gamma^* = \gamma^*(\omega)$ denote the smallest $\o$-good oriented path of length $L$ starting from $\vec e_2$, and note that $\gamma^*$ is $\o$-super-good, since $\o \in \Gamma_{(0,0)}$. Let $\xi = \xi(\omega) \in \gamma^*$ be the first super-good vertex encountered while travelling along $\gamma^*$ backwards, \hbox{\it{i.e.}, } from its last point to its starting point $\vec e_2$. Finally, let $\gamma$ be the portion of $\gamma^*$ starting at $\vec e_2$ and ending at the vertex preceding $\xi$ in $\gamma^*$.
We next perform the series of conditionings on the measure $\mu$ that were described informally above. Let $\Lambda$ be the box of side-length $4/p_2^2$ centred at the origin. We first condition on the event $\Gamma_{(0,0)}$ and on the $\s$-algebra generated by the events $$\big\{ \{\o_j \in G_1\} : j \in \Lambda \setminus \{(0,0)\} \big\}.$$ Note that, since we are conditioning on the event $\Gamma_{(0,0)}$, these events determine $\gamma^*$. Next we condition on the position of $\xi$ on $\gamma^*$; this determines the path $\gamma = (i^{(1)},\dots, i^{(n)})$. Finally we condition on all of the variables $\o_j$ with $j \not\in \gamma \cup \{(0,0)\}$. Let $\nu$ be the resulting conditional measure and observe that $(S^{\gamma\cup\{(0,0)\}},\nu)$ is a product probability space of the form $\otimes_{j \in \gamma \cup \{(0,0)\}}(S_j,\nu_j),$ with $(S_{(0,0)},\nu_{(0,0)}) = (S, \hat \mu)$ and $(S_j,\nu_j) = \big( G_1, \hat\mu(\cdot \thinspace |\thinspace G_1) \big)$ for each $j \in \gamma$. Notice that \begin{equation}\label{eq:CPI:convexity} \mu\big( \mathbbm{1}_{\Gamma_{(0,0)}} \operatorname{Var}_{(0,0)}(f) \big) = \mu\Big( \mathbbm{1}_{\Gamma_{(0,0)}} \, \nu\big( \operatorname{Var}_{\nu_{(0,0)}}(f) \big) \Big) \,\leqslant\, \, \mu\big( \mathbbm{1}_{\Gamma_{(0,0)}} \operatorname{Var}_\nu(f) \big), \end{equation} because $\nu\big( \operatorname{Var}_{\nu_{(0,0)}}(f) \big) \,\leqslant\, \operatorname{Var}_\nu(f)$, by convexity.
We can now bound $\operatorname{Var}_\nu(f)$ from above by applying Proposition~\ref{lem:gen-Poincare} to the measure $\nu = \otimes_{j \in \gamma\cup \{(0,0)\}} (S_j,\nu_j),$ with the super-good event $G_2$ as the constraining event $S_j^g$. Observe that $\nu\big( S_{(0,0)}^g \big) = \hat \mu(G_2) = p_2$ and $\nu\big( S_j^g \big) = \hat \mu\big( G_2 \, | \, G_1 \big) = p_2 / p_1$ for each $j \in \gamma$. The first Poincar\'e inequality~\eqref{eq:Poincare:genEast} in Proposition~\ref{lem:gen-Poincare} therefore gives \begin{equation}\label{eq:7} \mu\big( \mathbbm{1}_{\Gamma_{(0,0)}} \operatorname{Var}_\nu(f) \big) \;\leqslant\; \, \vec T(p_2) \cdot \mu\bigg( \mathbbm{1}_{\Gamma_{(0,0)}} \sum_{i \in \gamma \cup \{(0,0)\}} \nu\Big( \mathbbm{1}_{\{\o_{m(i)}\in G_2\}} \operatorname{Var}_{\nu_{i}}(f) \Big) \bigg), \end{equation} where $m(i)$ is the next point on the path $\gamma^*$ after $i$ (\hbox{\it{i.e.}, } $m(i)$ is either $m(i) = i + \vec e_1$ or $m(i) = i + \vec e_2$) and \[ \vec T(p_2) \;\leqslant\; \frac{1}{p_2} \sup \big\{ T_{\text{\tiny East}}(n,\bar \alpha) : n \;\leqslant\; L \big\} \;\leqslant\; \, p_2^{-O( \log(1/p_2) )}, \] by~\eqref{eq:scaling}. Recall that in Definition \ref{def:gen:East} the constraint for the last point is identically equal to one (this is in order to guarantee irreducibility of the chain), and observe that this condition holds in the above setting because, by construction, $\o_\xi\in G_2$.
Finally, we claim that~\eqref{eq:7} implies that \begin{multline}\label{eq:CPI:extended:sum} \mu\big( \mathbbm{1}_{\Gamma_{(0,0)}} \operatorname{Var}_\nu(f) \big) \;\leqslant\; \, \vec T(p_2) \, \sum_{i \in \Lambda} \bigg( \mu\Big( \mathbbm{1}_{\{\o_{i + \vec e_1} \in G_2\}} \mathbbm{1}_{\{\o_{i-\vec e_1} \in G_1\}} \operatorname{Var}_i(f \thinspace |\thinspace G_1) \Big) \\ + \mu\Big(\mathbbm{1}_{\{\o_{i+\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^+(i)\}} \big( \operatorname{Var}_i(f) + \operatorname{Var}_i(f\thinspace |\thinspace G_1) \big) \Big) \bigg). \end{multline} Indeed, note that $\operatorname{Var}_{\nu_{(0,0)}}(f) = \operatorname{Var}_{(0,0)}(f)$ and that $\operatorname{Var}_{\nu_{i}}(f) = \operatorname{Var}_i(f \thinspace |\thinspace G_1)$ for each $i \in \gamma$, and recall that, by construction, $\o_{i + \vec e_1},\o_{i - \vec e_1} \in G_1$ for every $i \in \gamma$. Therefore, for each $i \in \gamma$, if $m(i) = i + \vec e_1$ then $\o_{i-\vec e_1} \in G_1$, and if $m(i) = i + \vec e_2$ then $\o_{j}\in G_1$ for each $j \in {\ensuremath{\mathbb L}} ^+(i) = i + \big\{ \vec e_1, \vec e_2 - \vec e_1 \big\}$. Moreover, the event $\Gamma_{(0,0)}$ implies that $\o_j \in G_1$ for each $j \in {\ensuremath{\mathbb L}} ^+((0,0))$. Therefore every term of the right-hand side of~\eqref{eq:7} is included in the right-hand side of~\eqref{eq:CPI:extended:sum}, and hence~\eqref{eq:7} implies~\eqref{eq:CPI:extended:sum}, as claimed.
Now, combining~\eqref{eq:CPI:extended:sum} with~\eqref{eq:MT:prop34:repeat} and~\eqref{eq:CPI:convexity}, and noting that $\operatorname{Var}_{i}(f) \;\geqslant\; p_1 \operatorname{Var}_i(f \thinspace |\thinspace G_1)$ and that $|\Lambda| \,\leqslant\, p_2^{-O(1)}$, we obtain \begin{multline*} \operatorname{Var}(f) \;\leqslant\; \, p_1^{-1} p_2^{-O(1)} \, \vec T(p_2) \sum_{i \in {\ensuremath{\mathbb Z}} ^2} \bigg( \mu\Big( \mathbbm{1}_{\{\o_{i + \vec e_1} \in G_2\}} \mathbbm{1}_{\{\o_{i-\vec e_1} \in G_1\}} \operatorname{Var}_i\big(f \thinspace |\thinspace G_1 \big) \Big) \\ + \mu\Big(\mathbbm{1}_{\{\o_{i+\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^+(i)\}} \operatorname{Var}_i(f) \Big) \bigg), \end{multline*} which implies the oriented Poincar\'e inequality~\eqref{eq:9East}, as required.
The proof of the unoriented inequality~\eqref{eq:9FA} is almost the same, except we will use the second Poincar\'e inequality~\eqref{eq:Poincare:genFA1f} in Proposition~\ref{lem:gen-Poincare}, instead of~\eqref{eq:Poincare:genEast}. To spell out the details, we obtain \begin{equation}\label{eq:CPI:FA1f:app} \mu\big( \mathbbm{1}_{\Gamma_{(0,0)}} \operatorname{Var}_\nu(f) \big) \;\leqslant\; \, T(p_2) \cdot \mu\bigg( \mathbbm{1}_{\Gamma_{(0,0)}} \sum_{i \in \gamma \cup \{(0,0)\}} \nu\Big( c_i \operatorname{Var}_{\nu_{i}}(f) \Big) \bigg), \end{equation} where $c_i$ is the indicator of the event that $G_2$ holds for at least one of the neighbours of $i$ on the path $\gamma^*$, and \[ T(p_2) \;\leqslant\; \frac{1}{p_2} \sup_{n \leqslant L} T_{\text{\tiny FA}}(n,\bar \alpha) = p_2^{-O(1)}, \] by~\eqref{eq:scaling}. Note that the constraint for the last point is again identically equal to one since $\o_\xi\in G_2$. It follows (cf.~\eqref{eq:CPI:extended:sum}) that \begin{multline}\label{eq:CPI:extended:sum:again} \mu\big( \mathbbm{1}_{\Gamma_{(0,0)}} \operatorname{Var}_\nu(f) \big) \;\leqslant\; \, T(p_2) \, \sum_{i \in \Lambda} \sum_{\epsilon = \pm 1} \bigg( \mu\Big( \mathbbm{1}_{\{\o_{i + \epsilon\vec e_1} \in G_2\}} \mathbbm{1}_{\{\o_{i - \epsilon\vec e_1} \in G_1\}} \operatorname{Var}_i\big( f \thinspace |\thinspace G_1 \big) \Big) \\ + \mu\Big(\mathbbm{1}_{\{\o_{i + \epsilon\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^\epsilon(i)\}} \big( \operatorname{Var}_i(f) + \operatorname{Var}_i\big( f \thinspace |\thinspace G_1 \big) \big) \Big) \bigg), \end{multline} since $\o_{i + \vec e_1},\o_{i - \vec e_1} \in G_1$ for every $i \in \gamma$, and the event $\Gamma_{(0,0)}$ implies that $\o_j \in G_1$ for each $j \in {\ensuremath{\mathbb L}} ^+((0,0))$. In particular, note that if $i \in \gamma$ and $i + \vec e_2 \in \gamma$, then $\o_j \in G_1$ for each $j \in {\ensuremath{\mathbb L}} ^+(i) = {\ensuremath{\mathbb L}} ^-(i + \vec e_2) = i + \big\{ \vec e_1, \vec e_2 - \vec e_1 \big\}$. Therefore, as before, every term of the right-hand side of~\eqref{eq:CPI:FA1f:app} is included in the right-hand side of~\eqref{eq:CPI:extended:sum:again}.
Finally, combining~\eqref{eq:CPI:extended:sum:again} with~\eqref{eq:MT:prop34:repeat} and~\eqref{eq:CPI:convexity}, and since $\operatorname{Var}_{i}(f) \;\geqslant\; p_1 \operatorname{Var}_i(f \thinspace |\thinspace G_1)$ and $|\Lambda| \,\leqslant\, p_2^{-O(1)}$, we obtain \begin{multline*} \operatorname{Var}(f) \;\leqslant\; \, p_1^{-1} p_2^{-O(1)} \, T(p_2) \sum_{i \in {\ensuremath{\mathbb Z}} ^2} \sum_{\epsilon = \pm 1} \bigg( \mu\Big( \mathbbm{1}_{\{\o_{i + \epsilon \vec e_1} \in G_2\}} \mathbbm{1}_{\{\o_{i - \epsilon \vec e_1} \in G_1\}} \operatorname{Var}_i\big( f \thinspace |\thinspace G_1 \big) \Big) \\ + \mu\Big(\mathbbm{1}_{\{\o_{i + \epsilon\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^\epsilon(i)\}} \operatorname{Var}_i(f) \Big) \bigg), \end{multline*} which gives the unoriented Poincar\'e inequality~\eqref{eq:9FA}, as claimed, and hence completes the proof of Theorem~\ref{thm:CPI}. \end{proof}
\section{Renormalization and spreading of infection} \label{sec:strategy}
In this section we shall define the setting to which we will apply Theorem \ref{thm:CPI} in order to bound from above the relaxation time, and hence the mean infection time, of supercritical and critical KCM. We will begin with a very brief informal description, before giving (in Section \ref{sec:setting}) the precise definition. We will then, in Sections~\ref{sec:supercritical:bootstrap} and~\ref{sec:critic:spread}, state two results from the theory of bootstrap percolation that will play an instrumental role in the proofs of Theorems~\ref{mainthm:1} and~\ref{mainthm:2}.
Our basic strategy is to partition the lattice ${\ensuremath{\mathbb Z}} ^2$ into disjoint rectangular ``blocks'' $\{V_i\}_{i\in {\ensuremath{\mathbb Z}} ^2}$, whose size is adapted to the bootstrap update family $\ensuremath{\mathcal U}$. To each block $V_i$ we associate a block random variable $\o_i$, which is just the collection of i.i.d.~$0/1$ Bernoulli($p$) variables $\{\o_x\}_{x\in V_i}$ attached to each vertex of the block. In order to avoid confusion we will always use the letters $i,j,\dots$ for the labels of quantities associated to blocks, and the letters $x,y,\dots$ for the labels of the quantities associated to vertices of ${\ensuremath{\mathbb Z}} ^2$. We will apply Theorem~\ref{thm:CPI} to the block variables $\{\o_i\}_{i\in {\ensuremath{\mathbb Z}} ^2}$.
\subsection{A concrete general setting} \label{sec:setting}
Let $v$ and $v^\perp$ be orthogonal rational directions in the first and second quadrant of ${\ensuremath{\mathbb R}} ^2$ respectively. Let $\vec v$ be the vector joining the origin to the first site of ${\ensuremath{\mathbb Z}} ^2$ in direction $v$, and similarly for $\vec v^\perp$. Let $n_1 \,\geqslant\, n_{2}$ be (sufficiently large) even integers, and set \begin{equation}\label{eq:basicrect} R \, := \, \big\{ x \in {\ensuremath{\mathbb R}} ^2 \,:\, x = \alpha n_1\vec v +\beta n_2 \vec v^\perp,\, \alpha, \beta \in [0,1) \big\}. \end{equation}
The finite probability space $(S,\hat \mu)$ appearing in Section~\ref{sec:CPI} will always be of the form $S=\{0,1\}^V$, where $V = R \cap {\ensuremath{\mathbb Z}} ^2$, and $\hat \mu$ is the Bernoulli$(p)$ product measure. Observe that the probability space $(S^{{\ensuremath{\mathbb Z}} ^2},\mu)$ is isomorphic to $\O = \{0,1\}^{{\ensuremath{\mathbb Z}} ^2}$ equipped with the Bernoulli$(p)$ product measure which, with a slight abuse of notation, we will continue to denote by $\mu$. For our purposes, a convenient isomorphism between the two probability spaces is given by a kind of tilted ``brick-wall'' partition of ${\ensuremath{\mathbb Z}} ^2$ into disjoint copies of the basic block $V$ (see Figure \ref{fig:brickwall}). To be precise, for each $i = (i_1,i_2) \in {\ensuremath{\mathbb Z}} ^2$, set $V_i := R_i \cap {\ensuremath{\mathbb Z}} ^2$, where $R_i := R + (i_1 + i_2 / 2) n_1 \vec v + i_2 n_2 \vec v^\perp$.
\begin{figure}\label{fig:brickwall}
\end{figure}
In this partition the ``northern" and ``southern" neighbouring blocks of $V_i$ (\hbox{\it{i.e.}, } the blocks corresponding to $(i_1,i_2\pm 1)$) are shifted in the direction $\vec v$ by $\pm \, n_1 / 2$ w.r.t.~$V_i$. With this notation, and given $\o\in S^{{\ensuremath{\mathbb Z}} ^2}$, it is then convenient to think of the variable $\o_i \in S$ as being the collection $\{ \o_x \}_{x \in V_i} \in \{0,1\}^{V_i}$. The local variance term $\operatorname{Var}_i(f)$ (\hbox{\it{i.e.}, } the variance of $f$ w.r.t.~the variable $\o_i$ given all the other variables $\{\o_j\}_{j \neq i}$), which appears in the various constrained Poincar\'e inequalities in the statement of Theorem~\ref{thm:CPI}, is then equal to the variance $\operatorname{Var}_{V_i}(f)$ w.r.t.~the i.i.d.~Bernoulli($p$) variables $\{ \o_x \}_{x \in V_i}$, given all of the other variables $\{ \o_y \}_{y \in {\ensuremath{\mathbb Z}} ^2 \setminus V_i}$.
From now on, $\o$ will always denote an element of $\{0,1\}^{{\ensuremath{\mathbb Z}} ^2}$ and, given $\Lambda \subset {\ensuremath{\mathbb R}} ^2$, we will write $\o_\Lambda$ for the collection of i.i.d. random variables $\{ \o_x \}_{x \in \Lambda \cap {\ensuremath{\mathbb Z}} ^2}$, and $\mu_\Lambda$ for their joint product Bernoulli($p$) law. We will say that $\Lambda$ is \emph{empty} (or \emph{empty in $\o$}) if $\o$ is identically equal to $0$ on $\Lambda \cap {\ensuremath{\mathbb Z}} ^2$, and similarly that $\Lambda$ is \emph{filled} (or \emph{completely occupied}) if $\o$ is identically equal to $1$ on $\Lambda \cap {\ensuremath{\mathbb Z}} ^2$.
We now turn to the definitions of the good and super-good events $G_2 \subset G_1 \subseteq S$. The good event $G_1$ will depend on the update family $\ensuremath{\mathcal U}$, and will (roughly speaking) approximate the event that the block $V_i$ can be ``crossed" in the $\ensuremath{\mathcal U}$-bootstrap process with the help of a constant-width strip connecting the top and bottom of $V_i$. For supercritical models this event is trivial, and therefore $G_1$ is the entire space $S$; for critical models, on the other hand, $G_1$ will require the presence of empty vertices inside $V$ obeying certain model-dependent geometric constraints (see Definition~\ref{def:goodsets}, below). The super-good event $G_2$ for supercritical models will simply require that $V$ is empty. For critical models it will require that $G_1$ holds, and additionally that there exists an empty subset $\ensuremath{\mathcal R}$ of $V$, called a \emph{quasi-stable half-ring} (see Definitions~\ref{def:half-ring} and~\ref{def:goodsets}, and Figure~\ref{fig:half-ring}) of (large) constant width, and height equal to that of $V$.
We emphasize that the parameters $n_1,n_2$ will be chosen (depending on the model) so that the probabilities $p_1$ and $p_2$ of the events $G_1$ and $G_2$ (respectively) satisfy the key condition $$\lim_{q \rightarrow 0} \max{\Big\{ p_2,\big( 1 - p_1 \big) \big( \log p_2 \big)^2 \Big\}}= 0$$ that appears in part (B) of Theorem~\ref{thm:CPI}.
\subsection{Spreading of infection: the supercritical case.}\label{sec:supercritical:bootstrap}
We are now almost ready to state the property of $\mathcal{U}$-bootstrap percolation (proved by Bollob\'as, Smith and Uzzell~\cite{BSU}) that we will need when $\mathcal{U}$ is supercritical, i.e., when there exists an open semicircle $C\subset S^1$ that is free of stable directions. If $\mathcal{U}$ is rooted, then we may choose $-v$ (in the construction of the rectangle $R$ and of the partition $\{V_i\}_{i\in {\ensuremath{\mathbb Z}} ^2}$ described in Section~\ref{sec:setting}) to be the midpoint of any such semicircle; if $\mathcal{U}$ is unrooted, on the other hand, then $C$ can be chosen in such a way that $-C$ also has no stable directions, and we can choose $v$ to be the midpoint of any such semicircle.
Recall that $[ V_i ]_\ensuremath{\mathcal U}$ denotes the closure of $V_i = R_i \cap {\ensuremath{\mathbb Z}} ^2$ under the $\ensuremath{\mathcal U}$-bootstrap process. The following result, proved in~\cite{BSU}, states that a large enough rectangle can infect the rectangle to its ``left" (i.e., in direction $-v$) under the $\ensuremath{\mathcal U}$-bootstrap process, and if $\mathcal{U}$ is unrooted then it can also infect the rectangle to its ``right" (i.e., in direction $v$).
\begin{proposition}\label{prop:bootsc} Let $\mathcal{U}$ be a supercritical two-dimensional update family. If $n_1$ and $n_2$ are sufficiently large, then the following hold: \begin{itemize} \item[$(i)$] If $\mathcal{U}$ is unrooted, then $V_{(-1,0)} \cup V_{(1,0)} \subset [ V_{(0,0)} ]_\ensuremath{\mathcal U}$.
\item[$(ii)$] If $\mathcal{U}$ is rooted, then $V_{(-1,0)} \subset [ V_{(0,0)} ]_\ensuremath{\mathcal U}$. \end{itemize} \end{proposition} \begin{remark} \label{rem:left-right}By definition, in the rooted case the semicircle $-C$ contains some stable directions. Thus, $V_{(1,0)} \not\subset [ V_{(0,0)} ]_\ensuremath{\mathcal U}$. \end{remark} The proof of Proposition~\ref{prop:bootsc} in~\cite{BSU} is non-trivial, and required some important innovations, most notably the notion of ``quasi-stable directions" (see Definition~\ref{def:quasi-stable}, below). We will therefore give here only a brief sketch, explaining how one can read the claimed inclusions out of the results of~\cite{BSU}
\begin{proof}[Sketch proof of Proposition~\ref{prop:bootsc}] Both parts of the proposition are essentially immediate consequences of the following claim: if $R$ is a sufficiently large rectangle with two sides parallel to $w \in S^1$, and the semicircle centred at $w$ is entirely unstable, then $[R]_\mathcal{U}$ contains \emph{every} element of ${\ensuremath{\mathbb Z}} ^2$ that can be reached from $R$ by travelling in direction $w$. This claim follows from~\cite{BSU}*{Lemma~5.5}, since in this setting all of the quasi-stable directions in $\ensuremath{\mathcal S}_U'$ (see~\cite{BSU}*{Section~5.3}) are unstable (since they are contained in the semicircle centred at $w$), and if $u$ is unstable then the empty set is a $u$-block (see~\cite {BSU}*{Definition~5.1}). We refer the reader to~\cite{BSU}*{Sections~5 and~7} for more details. \end{proof}
\subsection{Spreading of infection: the critical case.}\label{sec:critic:spread}
We next turn to the more complicated task of precisely defining the good and super-good events for critical update families. In this subsection we will lay the groundwork for the precise definitions of these events (which we defer until Section~\ref{sec:critic-ub}, see Definition~\ref{def:goodsets}) by recalling some definitions from~\cite{BSU,BDMS}, and introducing the key new objects needed for the proof of Theorem~\ref{mainthm:2}, which we call ``quasi-stable half-rings" (see Definition~\ref{def:half-ring} and Figure~\ref{fig:half-ring}, below). Throughout this subsection, we will assume that $\ensuremath{\mathcal U}$ is a critical update family with difficulty $\alpha \in [1,\infty)$ and bilateral difficulty $\beta \in [\alpha,\infty]$ (see Definition~\ref{def:alpha}). Recall that we say that $\mathcal{U}$ is $\alpha$-rooted if $\beta \,\geqslant\, 2\alpha$, and that $\mathcal{U}$ is $\beta$-unrooted otherwise.
We begin by noting an important property of the set of stable directions $\ensuremath{\mathcal S}(\ensuremath{\mathcal U})$.
\begin{lemma}\label{lem:stableset} If $\beta < \infty$ then $\mathcal{S}(\mathcal{U})$ consists of a finite number of isolated, rational directions. Moreover, if $\mathcal{U}$ is $\beta$-unrooted and $\alpha(u^*) = \max\big\{ \alpha(u) : u \in \ensuremath{\mathcal S}(\ensuremath{\mathcal U}) \big\}$, then $\alpha(u) \,\leqslant\, \beta$ for every $u \in \mathcal{S}(\mathcal{U}) \setminus \{u^*,-u^*\}$. \end{lemma}
\begin{proof} By~\cite {BSU}*{Theorem 1.10}, $\mathcal{S}(\mathcal{U})$ is a finite union of rational closed intervals of $S^1$, and by~\cite {BSU}*{Lemma~5.2} (see also~\cite{BDMS}*{Lemma~2.7}), if $u \in \mathcal{S}(\ensuremath{\mathcal U})$ is a rational direction, then $\alpha(u) < \infty$ if and only {{if}} $u$ is an isolated point of $\ensuremath{\mathcal S}(\ensuremath{\mathcal U})$. Thus, if one of the intervals in $\mathcal{S}(\mathcal{U})$ is not an isolated point, then there exist two non-opposite stable directions in $S^1$, each with infinite difficulty, and so $\beta = \infty$.
Now, suppose that $\mathcal{U}$ is $\beta$-unrooted, and that $u \in \ensuremath{\mathcal S}(\ensuremath{\mathcal U})$ satisfies $\alpha(u) > \beta$ and $u \not\in \{u^*,-u^*\}$. Then $u$ and $u^*$ are non-opposite stable directions in $S^1$, each with difficulty strictly greater than $\beta$, which contradicts the definition of $\beta$. \end{proof}
In particular, if $\mathcal{U}$ is $\beta$-unrooted then Lemma~\ref{lem:stableset} guarantees the existence of an open semicircle $C$ such that $(C \cup -C) \cap \ensuremath{\mathcal S}(\ensuremath{\mathcal U})$ consists of finitely many directions, each with difficulty at most $\beta$. The next lemma provides a corresponding property for $\alpha$-rooted models.
\begin{lemma}\label{cor:Ccritic} If $\mathcal{U}$ is $\alpha$-rooted, then there exists an open semicircle $C$ such that $C \cap \ensuremath{\mathcal S}(\ensuremath{\mathcal U})$ consists of finitely many directions, each with difficulty at most $\alpha$. \end{lemma}
\begin{proof} By Definition~\ref{def:alpha}, there exists an open semicircle $C$ such that each $u \in C$ has difficulty at most $\alpha$. Since $\mathcal{U}$ is critical (and hence $\alpha$ is finite), it follows from~\cite{BSU}*{Lemma~5.2} (cf. the proof of Lemma~\ref{lem:stableset}) that each $u \in C$ is either unstable, or an isolated element of $\ensuremath{\mathcal S}(\ensuremath{\mathcal U})$, and hence $C \cap \ensuremath{\mathcal S}(\ensuremath{\mathcal U})$ is finite, as claimed. \end{proof}
Let us fix (for the rest of the subsection) an open semicircle $C$, containing finitely many stable directions, and such that the following holds: \begin{itemize} \item if $\mathcal{U}$ is $\alpha$-rooted then $\alpha(v) \,\leqslant\, \alpha$ for each $v \in C$;
\item if $\mathcal{U}$ is $\beta$-unrooted then $\alpha(v) \,\leqslant\, \beta$ for each $v \in C \cup - C$. \end{itemize} Let us also choose $C$ such that its mid-point $u$ belongs to ${\ensuremath{\mathbb Q}} _1$, and denote by $\pm u^\perp$ the boundary points of $C$. When drawing pictures we will always think of $C$ as the semicircle $(-\pi/2, \pi/2)$, though we emphasize that we do not assume that $u$ is parallel to one of the axes of $\mathbb{Z}^2$. We remark that the values of $\alpha(u^\perp)$ and $\alpha(-u^\perp)$ will not be important: we will only need to use the fact that they are both finite.
We are now ready to define one of the key notions from~\cite{BSU}, the set of quasi-stable directions. These are directions that are not (necessarily) stable, but which nevertheless it is useful to treat as if they were. For any $v \in S^1$, let us write $\hat v$ for the direction in $S^1$ that is symmetric to $v$ w.r.t.~the mid-point $u$ of $C$.
\begin{definition}[Quasi-stable directions]\label{def:quasi-stable} We say that a direction $v\in \mathbb{Q}_1$ is quasi-stable if either $v$ or $\hat v$ is a member of the set $$\{u \} \cup \ensuremath{\mathcal S}(\ensuremath{\mathcal U}) \cup \bigg( \bigcup_{X \in \mathcal{U}} \bigcup_{x \in X} \big\{ v \in S^1 : \langle v,x \rangle = 0 \big\} \bigg).$$ \end{definition}
Observe that there are only finitely many quasi-stable directions in $C$ (and, if $\beta < \infty$, only finitely many in $S^1$). The key property of the family of quasi-stable directions is given by the following lemma, which allows us to empty the sites near the corners of ``quasi-stable half-rings" (see Definition~\ref{def:half-ring}, below). {{Recall that we write $\ell_v$ for the discrete line $\{x \in \mathbb{Z}^2 : \langle x,v \rangle = 0 \}$.}}
\begin{lemma}[\cite{BSU}*{Lemma~5.3}]\label{lem:quasi} For every pair $v,v'$ of consecutive quasi-stable directions there exists an update rule $X$ such that $X \subset \big(\mathbb{H}_v\cup\ell_v\big)\cap\big(\mathbb{H}_{v'}\cup\ell_{v'}\big).$ \end{lemma}
\begin{proof} The statement was proved in~\cite{BSU} (see also~\cite{BDMS}*{Lemma~3.5}) for the family $\ensuremath{\mathcal S}(\ensuremath{\mathcal U}) \cup \big( \bigcup_{X \in \mathcal{U}} \bigcup_{x \in X} \big\{ v \in S^1 : \langle v,x \rangle = 0 \big\} \big)$ of quasi-stable directions, and it therefore holds for any superset of this family. \end{proof}
In order to define quasi-stable half-rings, we first need to introduce some additional notation:
\begin{definition}\label{def:strips} Let $v \in \mathbb{Q}_1$ with $\alpha(v)\,\leqslant\, \alpha$. A \emph{$v$-strip} $S$ is any closed parallelogram in ${\ensuremath{\mathbb R}} ^2$ with long sides perpendicular to $v$ and short sides perpendicular to $u^\perp$. \begin{itemize}
\item The $+$-boundary and $-$-boundary of $S$, denoted $\partial_+ (S)$ and $\partial_- (S)$ respectively, are the sides of $S$ with outer normal $v$ and $-v$. \item The external boundary $\partial^{\rm ext} (S)$ is defined as that translate of $\partial_+ (S)$ in the $v$-direction which captures for the first time a new lattice point not already present in $S$. \item Given $\lambda > 0$, we define $\partial_{\lambda}^{\rm ext}(S)$ as the portion of $\partial^{\rm ext}(S)$ at distance $\lambda$ from its endpoints (see Figure \ref{fig:geom}). \end{itemize} \end{definition}
\begin{figure}
\caption{A $v$-strip $S$, the $+$-boundary of $S$, the
external boundary (solid segment), and its subset $\partial_{\lambda}^{\rm
ext}(S)$ (thick solid segment)}
\label{fig:geom}
\end{figure} If $v$ is a stable direction, then a $v$-strip needs some ``help" from other infected sites in order to infect its external boundary (in the $\mathcal{U}$-bootstrap process). Our next ingredient (also first proved in~\cite{BSU}) provides us with a set that suffices for this purpose.
Let $v$ be a quasi-stable direction with difficulty $\alpha(v) \,\leqslant\, \alpha$, and let $Z_v \subset {\ensuremath{\mathbb Z}} ^2$ be a set of cardinality $\alpha$ such that $[\mathbb{H}_v \cup Z_v]_\mathcal{U} \cap \ell_v$ is infinite. (In the language of~\cite{BDMS}, $Z_v$ is called a \emph{voracious set}.) The following lemma (see~\cite{BSU}*{Lemma~5.5} and~\cite{BDMS}*{Lemma~3.4}) states that if $S$ is a sufficiently large $v$-strip, then a bounded number of translates of $Z_v$, together with $S \cap {\ensuremath{\mathbb Z}} ^2$, are sufficient to infect $\partial_{\lambda}^{\rm ext}(S{)}$ for some $\lambda = O(1)$.
\begin{lemma}\label{lem:strip} There exist $\lambda_v>0$, $T_v = \{a_1,\dots,a_r\} \subset {{\ell_v}}$ and $b\in {{\ell_v}}$ such that the following holds. If $S$ is a sufficiently large $v$-strip such that $\partial^{\rm ext} {(}S{)} \cap {\ensuremath{\mathbb Z}} ^2 \subset \ell_v$, then \begin{equation}\label{eq:fillpartialS} \partial_{\lambda_v}^{\rm
ext}{(}S{)}\cap {\ensuremath{\mathbb Z}} ^2 \subset \big[ (S\cap {\ensuremath{\mathbb Z}} ^2) \cup (Z_v+a_1+k_1b)\cup\dots\cup(Z_v+a_r+k_rb) \big]_\mathcal{U} \end{equation} for every $k_1,\dots, k_r \in \mathbb{Z}$ such that $a_i + k_i b \in \partial_{\lambda_v}^{\rm ext}{(}S{)}$ for every $i \in [r]$. \end{lemma}
Let us fix, for each quasi-stable direction $v \in C$, a constant $\lambda_v > 0$, a set $T_v = \{a_1,\dots,a_r\} \subset {{\ell_v}}$ and a site $b \in {{\ell_v}}$ given by Lemma~\ref{lem:strip}. If $S$ is a sufficiently large $v$-strip such that $\partial^{\rm ext} {(}S{)} \cap {\ensuremath{\mathbb Z}} ^2 \subset \ell_v + x$ for some $x \in {\ensuremath{\mathbb Z}} ^2$,
then we will refer to any set of the form \begin{equation}\label{def:helping:set} \big( (Z_v+a_1+k_1b) \cup \dots \cup (Z_v+a_r+k_rb) \big) + x, \end{equation} with $a_i + k_i b + x \in \partial_{\lambda_v}^{\rm ext}{(}S{)}$ for every $i \in [r]$, as a \emph{helping set} for $S$.
We are finally ready to define the key objects we will use to control the movement of empty sites in a critical KCM, the \emph{quasi-stable half-rings}. These are non{-}self-intersecting polygons, obtained by patching together suitable $v$-strips corresponding to quasi-stable directions (see Figure \ref{fig:half-ring}). Recall from Definition~\ref{def:quasi-stable} that, by construction, the set of quasi-stable directions in $C$ is symmetric w.r.t.~the midpoint $u$ of $C$.
\begin{figure}
\caption{A quasi-stable half-ring.}
\label{fig:half-ring}
\end{figure}
\begin{definition}[Quasi-stable half-rings]\label{def:half-ring} Let $(v_1,\dots, v_m)$ be the quasi-stable directions in $C$, ordered in such a way that $v_i$ and $v_{i+1}$ are consecutive directions for any $i\in [m-1]$, and $v_{i-1}$ comes before $v_i$ in clockwise order. Let $S_{v_i}$ be a $v_i$-strip with length $\ell_i$ and width $w_i$. We say that $\ensuremath{\mathcal R} := \bigcup_{i=1}^m S_{v_i}$ is a \emph{quasi-stable half-ring} of width $w$ and length $\ell$ if the following holds: \begin{enumerate}[(i)] \item $w_i = w$ and $\ell_i = \ell$ for each $i \in [m]$; \item $S_{v_i}\cap S_{v_j} = \emptyset$, unless $v_i$ and $v_j$ are consecutive directions, in which case the two strips share exactly one of their short sides and no other point.
\end{enumerate}
\end{definition}
We can finally formulate the ``spreading of infection'' result that we will need later. Given a quasi-stable half-ring $\ensuremath{\mathcal R}$, we will write $\ensuremath{\mathcal R}^*$ for the quasi-stable half-ring $\ensuremath{\mathcal R} + s u$, where $s > 0$ is minimal such that $\big( \ensuremath{\mathcal R}^* \setminus \ensuremath{\mathcal R} \big) \cap {\ensuremath{\mathbb Z}} ^2$ is non-empty. Also, for any set $U \subset {\ensuremath{\mathbb Z}} ^2$, let us write $[A]^U_\ensuremath{\mathcal U}$ for the closure of $A$ under the $\ensuremath{\mathcal U}$-bootstrap process restricted to $U$.
\begin{proposition}\label{prop:critic:spread} There exists a constant $\lambda = \lambda(\ensuremath{\mathcal U}) > 0$ such that following holds. Let $\ensuremath{\mathcal R}$ be a quasi-stable half-ring of width $w$ and length $\ell$, where $w, \ell \;\geqslant\; \lambda$. Let $U$ be the set of points of ${\ensuremath{\mathbb Z}} ^2$ within distance $\lambda$ of $\ensuremath{\mathcal R} \cup \ensuremath{\mathcal R}^*$, and let $Z_i$ be a helping set for $S_{v_i}$ for each $i \in [m]$. Then $$\ensuremath{\mathcal R}^* \cap {\ensuremath{\mathbb Z}} ^2 \subset \big[ \big( \ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2 \big) \cup Z_1 \cup \cdots \cup Z_m \big]^U_\ensuremath{\mathcal U}.$$ \end{proposition}
\begin{proof} This is a straightforward consequence of Lemmas~\ref{lem:quasi} and~\ref{lem:strip}. To see this, note first that, by Lemma~\ref{lem:strip}, the closure of $\big( \ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2 \big) \cup Z_1 \cup \cdots \cup Z_m$ under the $\ensuremath{\mathcal U}$-bootstrap process contains all points of $\ensuremath{\mathcal R}^* \cap {\ensuremath{\mathbb Z}} ^2$ except possibly those that lie within distance $O(1)$ of a corner of $\ensuremath{\mathcal R}$. Moreover, the path of infection described in the proof of Lemma~\ref{lem:strip} in~\cite{BSU,BDMS} only uses sites within distance $O(1)$ of the $v$-strip $S$. Thus, if $\lambda$ is chosen large enough, we have $\partial_{\lambda/4}^{\rm ext}(S_{v_i}) \cap {\ensuremath{\mathbb Z}} ^2 \subset \big[ \big( \ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2 \big) \cup Z_i \big]^U_\ensuremath{\mathcal U}$ for each $i \in [m]$.
Now, by Lemma~\ref{lem:quasi}, it follows that the set $\big[ \big( \ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2 \big) \cup Z_i \cup Z_{i+1} \big]^U_\ensuremath{\mathcal U}$ contains the remaining sites of $\partial^{\rm ext}(S_{v_i}) \cap {\ensuremath{\mathbb Z}} ^2$ and $\partial^{\rm ext}(S_{v_{i+1}}) \cap {\ensuremath{\mathbb Z}} ^2$ that lie within distance $\lambda/4$ of the intersection of $S_{v_i}$ and $S_{v_{i+1}}$. Indeed, these sites can be infected one by one, working towards the corner, using sites in $\ensuremath{\mathcal R} \cup \partial_{\lambda/4}^{\rm ext} (S_{v_i}) \cup \partial_{\lambda/4}^{\rm ext}(S_{v_{i+1}})$. Since this holds for each $i \in [m-1]$, it follows that the whole of $\ensuremath{\mathcal R}^* \cap {\ensuremath{\mathbb Z}} ^2$ is infected, as claimed. \end{proof}
Given a quasi-stable half-ring $\ensuremath{\mathcal R}$ of width $w$, we will write $\ensuremath{\mathcal R}'$ for the quasi-stable half-ring $\ensuremath{\mathcal R} + w u$, i.e., the minimal translate of $\ensuremath{\mathcal R}$ in the $u$-direction such that $\ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2$ and $\ensuremath{\mathcal R}' \cap {\ensuremath{\mathbb Z}} ^2$ are disjoint.
\begin{corollary} \label{cor:critic:spread} There exists a constant $\lambda = \lambda(\ensuremath{\mathcal U}) > 0$ such that following holds. Let $\ensuremath{\mathcal R}$ be a quasi-stable half-ring of width $w$ and length $\ell$, and suppose that $w \;\geqslant\; \lambda$ and $\ell \;\geqslant\; \lambda$. Let $U$ be the set of points of ${\ensuremath{\mathbb Z}} ^2$ within distance $\lambda$ of $\ensuremath{\mathcal R}\cup \ensuremath{\mathcal R}'$, and let $A \subset U$ be such that for any quasi-stable direction $v$, and any $v$-strip $S_v$ such that $\partial^{\rm ext}(S_{v}) \cap \ensuremath{\mathcal R}'$ has length at least $\ell$, there exists a helping set for $S_v$ in $A$. Then $$\ensuremath{\mathcal R}' \cap {\ensuremath{\mathbb Z}} ^2 \subset \big[ \big( \ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2 \big)\cup A \big]^U_\ensuremath{\mathcal U}.$$ \end{corollary}
\begin{proof} By construction, each $v_i$-strip of $\ensuremath{\mathcal R}$ has a helping set in $\ensuremath{\mathcal R}'$. Therefore, by Proposition~\ref{prop:critic:spread}, the $\ensuremath{\mathcal U}$-bootstrap process restricted to $U$ is able to infect the quasi-stable half-ring $\ensuremath{\mathcal R}^*$. We then repeat with $\ensuremath{\mathcal R}$ replaced by $\ensuremath{\mathcal R}^*$, and so on, until the entire quasi-stable half-ring $\ensuremath{\mathcal R}'$ has been infected. \end{proof}
Observe that, under the additional assumption that each quasi-stable direction $v$ has a helping set contained in $\ell_v$, we may choose $A$ to be a subset of $\ensuremath{\mathcal R}'$, but that in general we may (at some stage) need a helping set not contained in $\ensuremath{\mathcal R}'$ in order to advance in the $u$-direction.
\begin{remark}\label{rem:helping-sets} Later on, we will also need the above results in the slightly different setting in which the first $v_1$-strip entering in the definition of $\ensuremath{\mathcal R}$ is longer than the others, while all of the {other} $v_j$-strips, $j\neq 1$ have the same length.
In this case we will refer to $\ensuremath{\mathcal R}$ as an \emph{elongated} quasi-stable half-ring. For simplicity we preferred to state Proposition~\ref{prop:critic:spread} in the slightly less general {setting above}, but exactly the same proof applies if $\ensuremath{\mathcal R}$ is an elongated quasi-stable half-ring. \end{remark}
\section{Supercritical KCM: proof of Theorem~\ref{mainthm:1}}\label{sec:supercritical}
In this section we shall prove Theorem~\ref{mainthm:1}, which gives a sharp (up to a constant factor in the exponent) upper bound on the mean infection time for a supercritical KCM. We will first (in Section~\ref{sec:supercritic:unrooted}) give a detailed proof in the case that $\ensuremath{\mathcal U}$ is unrooted, and then (in Section~\ref{sec:supercritic:rooted}) explain briefly how the proof can be modified to prove the claimed bound for rooted models.
\subsection{The unrooted case}\label{sec:supercritic:unrooted}
Let $\ensuremath{\mathcal U}$ be a supercritical, unrooted, two-dimensional update family; we are required to show that there exists a constant $\lambda = \lambda(\ensuremath{\mathcal U})$ such that $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; q^{-\lambda}$$ for all sufficiently small $q > 0$. To do so, recall first from~\eqref{eq:mean-infection} that ${\ensuremath{\mathbb E}} _\mu(\tau_0) \,\leqslant\, T_{\rm rel}(q,\ensuremath{\mathcal U})/q$, and therefore, by Definition~\ref{def:PC}, it will suffice to prove that \begin{equation}\label{eq:supercritical:aim} \operatorname{Var}(f) \;\leqslant\; q^{-\lambda} \sum_x \mu\big( c_x \operatorname{Var}_x(f) \big) \end{equation} for some $\lambda = \lambda(\ensuremath{\mathcal U}) > 0$ and all local functions $f$, where $c_x$ denotes the kinetic constraint for the KCM, i.e., $c_x$ is the indicator function of the event that there exists an update rule $X\in \ensuremath{\mathcal U}$ such that $\o_y=0$ for each $y \in X + x$. We will deduce a bound of the form~\eqref{eq:supercritical:aim} from Theorem~\ref{thm:CPI} and Proposition~\ref{prop:bootsc}.
Recall the construction and notation described in Sections~\ref{sec:setting} and~\ref{sec:supercritical:bootstrap}; in particular, recall the definitions of the blocks $V_i$, of the parameters $n_1$ and $n_2$ (which determine the side lengths of the basic rectangle $R$), and the choice of {$v$} as the midpoint of an open semicircle $C \subset S^1$ such that the set $C \cup -C$ contains no stable directions. As anticipated in Section \ref{sec:setting}, the choice of the good and super-good events $G_2 \subset G_1 \subseteq S$ entering in Theorem \ref{thm:CPI}, is, in this case, extremely simple.
\begin{definition} If $\mathcal{U}$ is a supercritical two-dimensional update family, then: \begin{enumerate}[(a)] \item every block $V_i$ satisfies the \emph{good event} $G_1$ for $\mathcal{U}$ (\hbox{\it{i.e.}, } $G_1=S$); \item a block $V_i$ satisfies the \emph{super-good event} $G_2$ for $\mathcal{U}$ if and only if it is empty. \end{enumerate} \end{definition}
Let us fix the parameters $n_1$ and $n_2$ to be $O(1)$, but sufficiently large {{so that}} Proposition~\ref{prop:bootsc} holds. It follows that if $V_{(0,0)}$ is super-good, then the blocks $V_{(-1,0)}$ and $V_{(1,0)}$ (its nearest neighbours to the left and right respectively) lie in the closure under the $\ensuremath{\mathcal U}$-bootstrap process of the empty sites in $V_{(0,0)}$. In particular, $$t^\pm = \min\big\{ t > 0 \,:\, A_t \supseteq V_{(\pm 1,0)} \big\},$$ are both finite, where $A_t$ is the set of sites infected after $t$ steps of the $\ensuremath{\mathcal U}$-bootstrap process, starting from $A_0 = V_{(0,0)}$ (see Definition \ref{def:Uboot}). With foresight, define \begin{equation}\label{def:super:Lambda} \Lambda:= \big( A_{t^-} \setminus V_{(0,0)} \big) + n_1 \vec v, \end{equation} and note that $\Lambda\cap V_{\vec e_1} = \emptyset$ and $V_{(0,0)} \subset \Lambda$.
\begin{proof}[Proof of part~$(a)$ of Theorem~\ref{mainthm:1}] The first step is to apply Theorem~\ref{thm:CPI} to the probability space $(S^{{\ensuremath{\mathbb Z}} ^2},\mu)$ described in Section~\ref{sec:setting}, in which each `block' variable $\o_i\in S$ is given by the {{collection $\{ \o_x \}_{x \in V_i} \in \{0,1\}^{V_i}$ of i.i.d.~Bernoulli($p$) variables.}} Recall that $p_1 = \hat\mu(G_1)$ and $p_2 = \hat\mu(G_2)$ are the probabilities of the good and super-good events, respectively, and note that, in our setting, $p_1 = 1$ and $p_2 \;\geqslant\; q^{O(n_1n_2)} = q^{O(1)}$. It follows, using~\eqref{eq:8FA}, that \begin{equation}\label{eq:CPI:supercritical:application} \operatorname{Var}(f) \;\leqslant\; \frac{1}{q^{O(1)}} \sum_{i \in {\ensuremath{\mathbb Z}} ^2} \mu\left( \mathbbm{1}_{\{\text{either }V_{i+\vec e_1}\text{ or }V_{i-\vec e_1}\text{ is empty}\}}\operatorname{Var}_{V_i}(f) \right) \end{equation} for all local functions $f$, where $\operatorname{Var}_{V_i}(f)$ denotes the variance with respect to the variables $\{ \o_x \}_{x \in V_i}$, given all the other variables $\{ \o_y \}_{y \in {\ensuremath{\mathbb Z}} ^2 \setminus V_i}$.
To deduce~\eqref{eq:supercritical:aim}, it will suffice (by symmetry) to prove an upper bound on the right-hand side of~\eqref{eq:CPI:supercritical:application} of the form \begin{equation}\label{eq:supercritical:var:Vzero} \mu\left(\mathbbm{1}_{\{V_{\vec e_1}\text{ is empty}\}} \operatorname{Var}_{V_{(0,0)}}(f) \right) \,\leqslant\, \, \frac{1}{q^{O(1)}}\sum_{x\in \Lambda \cup V_{\vec e_1}}\mu\big( c_x \operatorname{Var}_x(f) \big) \end{equation} for the set $\Lambda$ defined in~\eqref{def:super:Lambda}, since the elements of $\Lambda \cup V_{\vec e_1}$ are all within distance $O(1)$ from the origin, and so we may then simply sum over all $i \in {\ensuremath{\mathbb Z}} ^2$.
To prove~\eqref{eq:supercritical:var:Vzero}, the first step is to observe that, by the convexity of the variance, and recalling that $\Lambda\cap V_{\vec e_1}=\emptyset$ and $V_{(0,0)}\subset \Lambda$, we have \begin{equation}\label{eq:super:convexity} \mu\Big( \mathbbm{1}_{\{V_{\vec e_1}\text{ is empty}\}} \operatorname{Var}_{V_{(0,0)}}(f) \Big) \,\leqslant\, \, \mu\left(\mathbbm{1}_{\{V_{\vec e_1}\text{ is empty}\}}\operatorname{Var}_{\Lambda}(f) \right). \end{equation} To conclude we appeal to the following result which, for later purposes, we formulate in a slightly more general setting than is needed here. In what follows, for any $\o\in \O$ and $U \subset {\ensuremath{\mathbb Z}} ^2$, we shall write $[\o]_\ensuremath{\mathcal U}^{U}$ for the closure of the set $\big\{ x \in {\ensuremath{\mathbb Z}} ^2 : \ \o_x=0 \big\}$ {under} the {$\ensuremath{\mathcal U}$-}bootstrap process restricted to ${U}$.
\begin{lemma}
\label{lem:var0} Let $A,B \subset {\ensuremath{\mathbb Z}} ^2$ be disjoint sets, and let $\ensuremath{\mathcal E}$ be an event depending only on $\o_{B}$. Suppose that there exists a set $U \supset A \cup B$ such that $B \subset [\o]_\ensuremath{\mathcal U}^{U}$ for any $\o\in \{0,1\}^{U}$ for which $A$ is empty and $\o_{B}\in \ensuremath{\mathcal E}$. Then \begin{equation}\label{eq:lem:var0} \mu\Big(\mathbbm{1}_{\{A\text{ is empty}\}}\operatorname{Var}_{B}\big( f \thinspace |\thinspace \ensuremath{\mathcal E} \big) \Big) \;\leqslant\;
|{U}| q^{-|{U}|} \frac{2}{pq}\sum_{x \in {U}}\mu\big( c_x\operatorname{Var}_x(f) \big) \end{equation} for any local function $f$. \end{lemma}
Before proving the lemma we conclude the proof of part (a) of Theorem \ref{mainthm:1}. We apply the lemma with $A = V_{\vec e_1}$, $B = \Lambda$, $U = A \cup B$ and $\ensuremath{\mathcal E}$ the trivial event, i.e., $\ensuremath{\mathcal E} = \O_{B}$. Indeed, by construction (see~\eqref{def:super:Lambda}), $\Lambda\subset \big[ V_{\vec e_1} \big]_{\ensuremath{\mathcal U}}^{U}$. Thus \eqref{eq:lem:var0} becomes \begin{equation}\label{eq:confusing}
\mu\left( \mathbbm{1}_{\{V_{\vec e_1}\text{ is empty}\}} \operatorname{Var}_\Lambda(f) \right) \,\leqslant\, \, |U| q^{-|U|}\frac{{{2}}}{pq} \sum_{x \in U} \mu\big( c_x \operatorname{Var}_x(f) \big). \end{equation}
Since $|U| = O(1)$, and using~\eqref{eq:super:convexity}, we conclude that for all $i \in {\ensuremath{\mathbb Z}} ^2$, $$\mu\left(\mathbbm{1}_{\{V_{ i+\vec e_1}\text{ is empty}\}}\operatorname{Var}_{V_{i}}(f)\right) \,\leqslant\, \, \frac{1}{q^{O(1)}} \sum_{x \in U_i} \mu\big( c_x \operatorname{Var}_x(f) \big),$$ where $U_i$ is the analogue of $U$ for the block $V_i$.
As noted above, summing over $i \in {\ensuremath{\mathbb Z}} ^2$ and using~\eqref{eq:CPI:supercritical:application}, we obtain the Poincar\'e inequality~\eqref{eq:supercritical:aim} with constant $q^{-O(1)}$, and by~\eqref{eq:mean-infection} and Definition~\ref{def:PC} it follows that there exists a constant $\lambda = \lambda(\ensuremath{\mathcal U})$ such that $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \frac{T_{\rm rel}(q,\ensuremath{\mathcal U})}{q} \;\leqslant\; q^{-\lambda},$$ for all sufficiently small $q > 0$, as required. Since the bootstrap infection time $T_\ensuremath{\mathcal U}$ of a supercritical update family satisfies $T_\ensuremath{\mathcal U} = q^{-\Theta(1)}$, it also follows that ${\ensuremath{\mathbb E}} _\mu(\tau_0) \,\leqslant\, T_\ensuremath{\mathcal U}^{O(1)}$. \end{proof}
\begin{proof}[Proof of Lemma \ref{lem:var0}] Observe first that, for any $\o \in \O$, \begin{gather}\label{eq:super:variance:basic:upper:bound} \operatorname{Var}_B\big( f \thinspace |\thinspace \ensuremath{\mathcal E} \big)(\o_{{\ensuremath{\mathbb Z}} ^2\setminus B}) \;\leqslant\; \frac{1}{\mu_B(\ensuremath{\mathcal E})}\sum_{\eta_B\in \ensuremath{\mathcal E}} \mu_B( \eta_B ) \Big( f\big(\eta_B,\o_{{\ensuremath{\mathbb Z}} ^2\setminus B} \big) - f\big(0, \o_{{\ensuremath{\mathbb Z}} ^2\setminus B}\big) \Big)^2, \end{gather} since $\mathbb{E}\big[ (X - a)^2 \big]$ is minimized by taking $a = \mathbb{E}[X]$, where $\big( 0, \o_{{\ensuremath{\mathbb Z}} ^2 \setminus B} \big)$ denotes the configuration that is equal to $\o_{{\ensuremath{\mathbb Z}} ^2 \setminus B}$ outside $B$, and empty inside $B$.
We will break each term on the right-hand side of~\eqref{eq:super:variance:basic:upper:bound} into the sum of single spin-flips using the $\ensuremath{\mathcal U}$-bootstrap process as follows. Fix $\o \in \O$ such that $A$ is empty, and fix $\eta_B \in \ensuremath{\mathcal E}$. Using the assumption of the lemma, we claim that there exists a path $\gamma \equiv (\o^{(0)},\dots,\o^{(m)})$ in $\O$ such that: \begin{enumerate}[(i)] \item $\o^{(0)} = (\eta_B,\o_{{\ensuremath{\mathbb Z}} ^2\setminus B})$ and $\o^{(m)} = (0,\o_{{\ensuremath{\mathbb Z}} ^2\setminus B})$;
\item the length $m$ of $\gamma$ satisfies $m \;\leqslant\; 2|U|$; \item for each $k=1,\dots,m$, there exists a vertex $x^{(k)} \in U$ such that \begin{itemize} \item the configuration $\o^{(k)}$ is obtained from $\o^{(k-1)}$ by flipping the value at $x^{(k)}$; \item this flip is legal, i.e., $c_{x^{(k)}}\big( \o^{(k-1)} \big) = 1$. \end{itemize} \end{enumerate}
We construct $\gamma$ in two steps: first we empty all of $B$, and possibly some of $U \setminus B$; then we reconstruct $\o_{{\ensuremath{\mathbb Z}} ^2\setminus B}$ without changing $\o_B$. To spell out the details, observe first that, since $B \subset \big[ \big( \eta_B,\o_{U \setminus B} \big) \big]_\ensuremath{\mathcal U}^{U}$, there exists a sequence of legal flips in $U$ connecting $\big( \eta_B,\o_{{\ensuremath{\mathbb Z}} ^2 \setminus B} \big)$ to a configuration with $A \cup B$ empty. By choosing a minimal such sequence, we may assume that all of the flips are from occupied to empty, and therefore that this first part of the path has length at most $|U|$.
Now, to reconstruct $\o_{{\ensuremath{\mathbb Z}} ^2\setminus B}$, we simply run the same sequence backwards, except without performing the steps inside $B$. Note that all of these flips are legal, since skipping the steps inside $B$ only creates additional empty sites, and that this second part of the path also has length at most $|U|$, as required.
It follows, using Cauchy--Schwarz, that \begin{multline*} \Big( f\big(\eta_B,\o_{{\ensuremath{\mathbb Z}} ^2\setminus B} \big) - f\big(0, \o_{{\ensuremath{\mathbb Z}} ^2\setminus B}\big) \Big)^2 \;\leqslant\; m \sum_{k = 1}^{m} c_{x^{(k)}}\big( \o^{(k-1)} \big) \Big( f\big(\o^{(k)}\big) - f\big( \o^{(k-1)} \big) \Big)^2 \\
\;\leqslant\; 2 |U| \frac{1}{\mu_*}\frac{1}{pq}\sum_{x\in {U}} \sum_{\eta\in \{0,1\}^{U}}\mu_{U}(\eta)c_x(\eta, \o_{{\ensuremath{\mathbb Z}} ^2\setminus {U}})\ pq \Big( f\big(\eta^{(x)}, \o_{{\ensuremath{\mathbb Z}} ^2\setminus {U}}\big) - f\big( \eta, \o_{{\ensuremath{\mathbb Z}} ^2\setminus U} \big) \Big)^2, \end{multline*}
for any $\o$ in which $A$ is empty, and any $\eta_B \in \ensuremath{\mathcal E}$, where $\mu_*=\min_{\eta\in \{0,1\}^{U}} \mu_{U}(\eta) = q^{|U|}$, and $\eta^{(x)}$ denotes the configuration obtained from $\eta$ by flipping the spin at $x$. Notice that the right-hand side does not depend on $\eta_B$, and that $pq \big( f\big(\eta^{(x)}, \o_{{\ensuremath{\mathbb Z}} ^2\setminus {U}}\big) - f\big( \eta, \o_{{\ensuremath{\mathbb Z}} ^2\setminus {U}} \big) \big)^2$ is the local variance $\operatorname{Var}_x(f)$ computed for the configuration $\o\equiv (\eta, \o_{{\ensuremath{\mathbb Z}} ^2\setminus {U}})$.
Hence, using~\eqref{eq:super:variance:basic:upper:bound}, we obtain
$$\mathbbm{1}_{\{A \text{ is empty}\}} \operatorname{Var}_B\big( f \thinspace |\thinspace \ensuremath{\mathcal E} \big)(\o_{{\ensuremath{\mathbb Z}} ^2\setminus B}) \;\leqslant\; \, \frac{2 |U|q^{-|{U}|}}{pq} \sum_{x\in U}\mu_{U}(c_x\operatorname{Var}_x(f)\bigr)( \o_{{\ensuremath{\mathbb Z}} ^2\setminus U})$$
for any $\o \in \O$, and inequality~\eqref{eq:lem:var0} follows by averaging using the measure $\mu$. \end{proof}
\subsection{The rooted case}\label{sec:supercritic:rooted}
Let $\ensuremath{\mathcal U}$ be a supercritical, rooted, two-dimensional update family, let $C\subset S^1$ be a semicircle with no stable directions and recall that, thanks to ~\eqref{eq:mean-infection}, it will suffice to prove a Poincar\'e inequality (cf.~\eqref{eq:supercritical:aim}) with constant $q^{{-}O(\log(1/q))}$. To prove this we will follow almost exactly the same route of the unrooted case, with the same definition of the blocks $V_i$ and of the good and super-good events. We will therefore only give a very brief sketch of the proof in this new setting.
The {{main difference}} w.r.t.~the unrooted case is that now the opposite semicircle $-C$ will necessarily contain some stable directions. This forces us to use the oriented Poincar\'e inequality~\eqref{eq:8East} from Theorem~\ref{thm:CPI} instead of the unoriented one~\eqref{eq:8FA}, because in this case (see Proposition~\ref{prop:bootsc} and Remark~\ref{rem:left-right}) a super-good block is able to infect the block to its left but not the block to its right, \hbox{\it{i.e.}, } $V_{(-1,0)}\subset [V_{(0,0)}]_\ensuremath{\mathcal U}$ but $V_{(1,0)} \not\subset [ V_{(0,0)} ]_\ensuremath{\mathcal U}$.
\begin{proof}[Proof of part~$(b)$ of Theorem~\ref{mainthm:1}] We again apply Theorem~\ref{thm:CPI} to the probability space $(S^{{\ensuremath{\mathbb Z}} ^2},\mu)$ described in Section~\ref{sec:setting}, but we use~\eqref{eq:8East} instead of~\eqref{eq:8FA}. Recalling that $p_1 = 1$ and $p_2 = q^{O(1)}$, we obtain \begin{equation}\label{eq:CPI:supercritical:application:rooted} \operatorname{Var}(f) \;\leqslant\; \frac{1}{q^{O(\log(1/q))}} \sum_{i \in {\ensuremath{\mathbb Z}} ^2} \mu\left( \mathbbm{1}_{\{V_{i+\vec e_1}\text{ is empty}\}}\operatorname{Var}_{V_i}(f) \right) \end{equation} for all local functions $f$. As before, using translation invariance, we only examine the $i = (0,0)$ term in the above sum. We claim that \begin{equation}\label{eq:supercritical:var:rooted} \mu\left(\mathbbm{1}_{\{V_{\vec e_1}\text{ is empty}\}} \operatorname{Var}_{V_{(0,0)}}(f) \right) \,\leqslant\, \, \frac{1}{q^{O(1)}} \sum_{x \in {{U}}}\mu\big( c_x \operatorname{Var}_x(f) \big) \end{equation} {for ${{U}} = V_{\vec e_1} \cup \Lambda$,} where $\Lambda$ is the set defined in~\eqref{def:super:Lambda}. However, the proof of~\eqref{eq:supercritical:var:rooted} is identical to that of~\eqref{eq:supercritical:var:Vzero}, since Proposition~\ref{prop:bootsc} implies that $V_{(0,0)}$ can be entirely infected by $V_{\vec e_1}$. We therefore obtain the Poincar\'e inequality \begin{equation}\label{eq:supercritical:aim:rooted} \operatorname{Var}(f) \,\leqslant\, \frac{1}{q^{O(\log(1/q))}}\sum_x \mu\big( c_x \operatorname{Var}_x(f) \big) \end{equation} for all local functions $f$. Thus $T_{\rm rel}(q,\ensuremath{\mathcal U}) \;\leqslant\; q^{-O(\log(1/q))}$, and hence $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \frac{T_{\rm rel}(q,\ensuremath{\mathcal U})}{q} \,\,\leqslant\, \, q^{-O(\log(1/q))} \, = \, T_\ensuremath{\mathcal U}^{O(\log T_\ensuremath{\mathcal U})},$$ as required, because $T_\ensuremath{\mathcal U} = q^{-\Theta(1)}$. \end{proof}
\section{Critical KCM: proof of Theorem \ref{mainthm:2} under a simplifying assumption}\label{sec:critic-ub}
In this section we shall prove Theorem~\ref{mainthm:2} under the following additional assumption (see below): every stable direction $v$ with finite difficulty has a voracious set that is a subset of the line $\ell_v$. By doing so, we avoid some technical complications (mostly related to the geometry of the quasi-stable half-ring) which might obscure the main ideas behind the proof. The changes necessary to treat the general case are spelled out in detail in Section~\ref{sec:fullgen}.
\begin{assumption}\label{easy} For any stable direction $u \in \ensuremath{\mathcal S}$ with finite difficulty $\alpha(u)$, there exists a set $Z_u \subset \ell_u$ of cardinality $\alpha(u)$ such that $\big[ \mathbb{H}_u \cup Z_u \big]_{\ensuremath{\mathcal U}} \cap \ell_u$ is infinite. \end{assumption}
As in Section~\ref{sec:supercritical}, our main task will be to establish a suitable upper bound on the relaxation time $T_{\rm rel}(\ensuremath{\mathcal U};q)$. In Section~\ref{sec:critical:rooted} we will first analyse the $\alpha$-rooted case and the starting point will be the constrained Poincar\'e inequality~\eqref{eq:9East}; the proof the $\beta$-unrooted case (see Section \ref{sec:beta-unrooted}) will be essentially the same, the main difference being that~\eqref{eq:9East} will be replaced by~\eqref{eq:9FA}.
\subsection{$\alpha$-rooted update families}\label{sec:critical:rooted}
Let $\ensuremath{\mathcal U}$ be a critical, $\alpha$-rooted, two-dimensional update family, and recall from Definition~\ref{def:alpha:rooted} that $\ensuremath{\mathcal U}$ has difficulty $\alpha$, and bilateral difficulty at least $2\alpha$. The properties of $\ensuremath{\mathcal U}$ that we will need below have already been proved in Section~\ref{sec:critic:spread}; they all follow from the fact (see Lemma~\ref{cor:Ccritic}) that there exists an open semicircle $C$ such that $C \cap \ensuremath{\mathcal S}(\ensuremath{\mathcal U})$ consists of finitely many directions, each with difficulty at most $\alpha$. In particular, we will make crucial use of Corollary~\ref{cor:critic:spread}.
We will prove that, if Assumption~\ref{easy} holds, then there exists a constant $\lambda = \lambda(\ensuremath{\mathcal U})$ such that $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \frac{T_{\rm rel}(q,\ensuremath{\mathcal U})}{q} \;\leqslant\; \exp\Big( \lambda \cdot q^{-2\alpha} \big( \log(1/q) \big)^4 \Big)$$ for all sufficiently small $q > 0$. Note that the first inequality follows from~\eqref{eq:mean-infection}, and so, by Definition~\ref{def:PC}, it will suffice to prove that \begin{equation}\label{eq:rooted:aim} \operatorname{Var}(f) \,\leqslant\, \exp\Big( \lambda \cdot q^{-2\alpha} \big( \log(1/q) \big)^4 \Big) \sum_{x \in {\ensuremath{\mathbb Z}} ^2} \mu\big( c_x \operatorname{Var}_x(f) \big) \end{equation} for some $\lambda = \lambda(\ensuremath{\mathcal U})$ and all local functions $f$. We will deduce a bound of the form~\eqref{eq:rooted:aim} starting from~\eqref{eq:9East} and using Corollary~\ref{cor:critic:spread}.
\begin{remark} We are not able to use the unoriented constrained Poincar\'e inequality~\eqref{eq:9FA} in place of the oriented inequality~\eqref{eq:9East} in the proof of~\eqref{eq:rooted:aim} because there exist $\alpha$-rooted models (the Duarte model~\cite{Duarte} is one such example) with $\beta = \infty$ such that, for any choice of the side-lengths $n_1$ and $n_2$ of the blocks $V_i$, and of the good and super-good events $G_2 \subset G_1$ satisfying the condition $(1-p_1)(\log p_2)^2 = o(1)$, the $\ensuremath{\mathcal U}$-bootstrap process is not guaranteed to be able to infect the block $V_i$ using only that facts that the block $V_{i-\vec e_1}$ is infected and that some nearby blocks $V_j$ are good. For update families with $2\alpha < \beta < \infty$, it \emph{is} possible to apply~\eqref{eq:9FA} for certain choices of $(n_1,n_2,G_1,G_2)$, but the best Poincar\'e constant we are able to obtain in this way is roughly $\exp\big( q^{-\beta} \big)$, which is much larger than the one we prove using~\eqref{eq:9East}.
\end{remark}
\subsubsection{The geometric setting and the good and super-good events}
Recall the construction and notation described in Sections~\ref{sec:setting} and~\ref{sec:critic:spread}; in particular, recall that $V = R \cap {\ensuremath{\mathbb Z}} ^2$, where $R$ is a rectangle in the rotated coordinates $(v,v^\perp)$, and $u = -v$ is the midpoint of an open semicircle $C \subset S^1$ in which every stable direction has difficulty at most~$\alpha$. As in Section~\ref{sec:strategy}, when drawing figures we will think of $u$ as pointing to the left. We will choose the parameters $n_1$ and $n_2$ (which determine the side-lengths of $R$) depending on $q$; to be precise, set
$$n_1 = \big\lfloor q^{-{{3}}\kappa} \big\rfloor \qquad \text{and} \qquad n_2 = \big\lfloor \kappa^4 q^{-\alpha} \log (1/q) \big\rfloor,$$
where $\kappa = \kappa(\ensuremath{\mathcal U})$ is a sufficiently large constant.
In order to define the good and super-good events $G_1$ and $G_2$, we need to define some structures which we call \emph{$\kappa$-stairs}, which will provide us with a way of transporting infection `vertically'. Let us call the set of points of $V$ lying on the same line parallel to $u$ (resp. $u^\perp$) a \emph{row} (resp. \emph{column}) of $V$, and let us order the rows from bottom to top and the columns from left to right. Let $a$ and $b$ be (respectively) the number of rows and columns of $V$, and observe that, since $v$ is a rational direction, we have $a = \Theta(n_2)$ and $b = \Theta(n_1)$, where the implicit constants depend only on the update family $\ensuremath{\mathcal U}$. We will say that a set of vertices is an \emph{interval of $V$} if it is the intersection with {{$V$}} of a line segment in~${\ensuremath{\mathbb R}} ^2$. Recall that $\kappa = \kappa(\ensuremath{\mathcal U}) > 0$ was fixed above.
\begin{definition} \label{def:stair}
We say that a collection $\ensuremath{\mathcal M} = \big\{ M^{(1)}, \ldots, M^{(a)} \big\}$ of disjoint intervals of $V$ of size $2\kappa$ forms an \emph{upward $\kappa$-stair} with steps $M^{(1)},M^{(2)},\ldots$ if:
\begin{enumerate}[(i)]
\item for each $i \in [a],$ the $i^{th}$-step $M^{(i)}$ belongs to the $i^{th}$-row of $V$; \item the $i^{th}$-step is ``to the left'' of the $j^{th}$-step if
$i<j$. More precisely, let $\big( M_\ell^{(i)}, M_r^{(i)} \big)$ be the abscissa (in the $(v,v^\perp)$-frame) of the leftmost and rightmost elements (respectively) of the $i^{th}$-step. Then $M_r^{(i)} < M_\ell^{(j)}$ whenever $i < j$. \end{enumerate} \end{definition}
We refer the reader to Figure \ref{fig:strategy} for a picture of an upward $\kappa$-stair.
We are now ready to define the good and super-good events. Let us say that a quasi-stable half-ring $\ensuremath{\mathcal R}$ \emph{fits in} the block $V_i$ if the top and bottom sides of $\ensuremath{\mathcal R}$ are contained in the top and bottom sides of {{$R_i$}}, and note that this determines the length $\ell$ of $\ensuremath{\mathcal R}$, which moreover satisfies $\ell \;\geqslant\; n_2 / m$ (see Definition~\ref{def:half-ring}). Let $(v_1,\dots, v_m)$ be the quasi-stable directions in $C$ (see Definition~\ref{def:quasi-stable}), and recall the definitions of a $v$-strip $S_{v}$ (see Definition~\ref{def:strips}) and of a helping set $Z$ for $S_{v}$ (see immediately after Lemma~\ref{lem:strip}). Note that Assumption~\ref{easy} implies that, for any $j \in [m]$ and $v_j$-strip $S_{v_j}$, we may choose the voracious set $Z_{v_j}$ so that the helping sets for $S_{v_j}$ are subsets of $\partial^{\rm ext}(S_{v_j})$.
\begin{definition}[Good and super-good events] \label{def:goodsets} \ \begin{enumerate} \item The block $V_i = R_i \cap {\ensuremath{\mathbb Z}} ^2$ satisfies the \emph{good} event $G_1$ iff:
\begin{enumerate}[(a)]
\item for each quasi-stable direction $v \in C$ and every $v$-strip $S$
such that the length of the segment $\partial^{\rm ext}(S) \cap R_i$ is at least $n_2/(4m)$, there
exists an empty helping set $Z \subset {{\partial^{\rm ext}(S) \cap V_i}}$ for~$S$; \item there exists an empty upward {{$\kappa$}}-stair within the leftmost quarter of $V_i$.
\end{enumerate} \item The block $V_i$ satisfies the \emph{super-good} event $G_2$ iff it satisfies the good event $G_1$, and moreover there exists an empty quasi-stable half-ring $\ensuremath{\mathcal R}$ of width $\kappa$,
that fits in $V_i$ and is entirely contained in the rightmost quarter of $R_i$. \end{enumerate} \end{definition}
Next we prove that the hypothesis for the part (B) of Theorem \ref{thm:CPI} holds in the above setting if $\kappa$ is sufficiently large.
\begin{lemma} \label{lem:p1p2} Let $p_1:=\hat\mu(G_1)$ and $p_2:=\hat\mu(G_2)$. There exists a constant $\kappa_0(\ensuremath{\mathcal U}) > 0$ such that, for any $\kappa > \kappa_0(\ensuremath{\mathcal U})$, \[ \lim_{q \rightarrow 0} \big( 1 - p_1 \big) \big( \log(1/p_2) \big)^2 = 0 \] \end{lemma}
\begin{proof} First, let's bound the probability that there is no empty helping set $Z \subset V_i$ for a given $v$-strip $S$ (where $v$ is a quasi-stable direction in $C$) such that $\partial^{\rm ext}(S) \cap R_i$ has length at least $n_2/(4m)$. Observe {{that we}} can choose $\Omega(n_2)$ potential values for each $k_j$ in~\eqref{def:helping:set} such that the corresponding sets $Z_v + a_j + k_j b$ are pairwise disjoint subsets of $\partial^{\rm ext} (S) \cap V_i$ (using Assumption~\ref{easy}), and that each such translate of $Z_v$ is empty with probability $q^\alpha$. (Here the implicit constant depends only on $\ensuremath{\mathcal U}$.) The probability that $S$ has no empty helping set is therefore at most $$O\Big( \big( 1 - q^\alpha \big)^{\Omega(n_2)} \Big) \;\leqslant\; q^{\kappa^3}$$ if $\kappa$ is sufficiently large and $q \ll 1$. There are at most $O(n_1^2 n_2^2)$ choices for the quasi-stable direction $v \in C$ and for the intersection of the $v$-strip $S$ with $V_i$. {{Thus, by the union bound, part~(a) of the definition of $G_1$ holds with probability at least $1 - q^{\kappa^2}$.
To bound the probability of part~(b), observe that an interval of $V$ of size $\Theta(n_1/n_2)$ contains an empty interval of $V$ of size $2\kappa$ with probability at least $$1 - \big(1 - q^{2\kappa} \big)^{\Theta(n_1/n_2)} \;\geqslant\; 1 - \exp\big( - q^{-\kappa + 2\alpha} \big),$$ if $q \ll 1$, and therefore the probability that $V$ contains an empty upward $\kappa$-stair is (by the union bound) at least \begin{equation}
\label{eq:or1} 1 - O(n_2) \exp\big( - q^{-\kappa + 2\alpha} \big) \;\geqslant\; 1 - q^{\kappa^2} \end{equation}
if $\kappa$ is sufficiently large and $q\ll 1$.}}
It follows that $$1 - p_1 \, = \, 1 - \hat\mu(G_1) \, \;\leqslant\; \, 2 \cdot q^{\kappa^2}.$$ Moreover, the probability that there exists an empty quasi-stable half-ring $\ensuremath{\mathcal R}$ of width $\kappa$ that fits in $V_i$ is at least $q^{O(n_2)}$, so (by the FKG inequality) we have $$\log(1 / p_2) \, \;\leqslant\; \, O(n_2) \log(1/q) \;\leqslant\; O\Big( q^{-\alpha} \big( \log (1/q) \big)^2 \Big),$$ where the implicit constant is independent of $q$. It follows that, if $\kappa$ is sufficiently large, then $\big( 1 - p_1 \big) \big( \log(1/p_2) \big)^2 \rightarrow 0$ as $q \rightarrow 0$, as required. \end{proof}
Let us fix, from now on, the constant $\kappa$ to be sufficiently large so that Lemma~\ref{lem:p1p2} applies. In particular, by Theorem~\ref{thm:CPI}, the constrained Poincar\'e inequality~\eqref{eq:9East} holds for any local function $f$, i.e., \begin{align}
\label{eq:9East:bis} & \operatorname{Var}(f) \;\leqslant\; \vec T(p_2) \bigg( \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^+(i)\}} \operatorname{Var}_{V_i}(f)\right)\nonumber\\ & \hspace{3.5cm} + \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\vec e_1}\in G_2\}}\mathbbm{1}_{\{\o_{i-\vec e_1}\in G_1\}} \operatorname{Var}_{V_i}\big( f \thinspace |\thinspace G_1 \big) \right)\bigg), \end{align} with \[ \vec T(p_2) \, = \, e^{O(\log(p_2)^2)} \, = \, \exp\Big( O\big( q^{-2\alpha} \log(1/q)^4 \big) \Big). \] As in the supercritical setting (see Section~\ref{sec:supercritical}), our strategy will be to bound each of the sums in the r.h.s.~of~\eqref{eq:9East:bis} from above in terms of the Dirichlet form $\ensuremath{\mathcal D}(f)$ of our KCM. To do so, it will suffice to bound from above, for a fixed (and arbitrary) local function~$f$, the following two generic terms: \[ I_1(i):= \mu\left(\mathbbm{1}_{\{\o_{i+\vec e_1}\in
G_2\}}\mathbbm{1}_{\{\o_{i-\vec e_1}\in G_1\}} \operatorname{Var}_{V_i}\big( f \thinspace |\thinspace G_1 \big) \right), \] and \[ I_2(i):=\mu\left(\mathbbm{1}_{\{\o_{i+\vec e_2}\in
G_2\}}\mathbbm{1}_{\{\o_j\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^+(i)\}} \operatorname{Var}_{V_i}(f) \right), \] see Figure \ref{fig:setting}. Using translation invariance it suffices to consider only the case $i=(0,0)$, so let us set $I_1 \equiv I_1((0,0)$ and $I_2 \equiv I_2((0,0))$.
\begin{figure}\label{fig:setting}
\end{figure}
Define $W_1= V_{(0,0)} \cup V_{(-1,0)} \cup V_{(1,0)}$ and $W_2 = V_{(0,0)} \cup V_{(-1,0)} \cup V_{(1,0)} \cup V_{(-1,1)} \cup V_{(0,1)}$. We will prove the following upper bounds on $I_1$ and $I_2$.
\begin{proposition} \label{prop:I1:I2} For each $j \in \{1,2\}$, {{there exists a $O(1)$-neighbourhood $\hat W_j$ of $W_j$ such that}} \begin{align*}
I_j \, \;\leqslant\; \exp\Big( O\big( q^{-\alpha} \log(1/q)^{{3}} \big) \Big) \sum_{x \in \hat W_j} \mu\big(c_x
\operatorname{Var}_x(f)\big). \end{align*} \end{proposition}
Observe that, combining Proposition~\ref{prop:I1:I2} with~\eqref{eq:9East:bis}, and noting that $|\hat W_j| = q^{-O(1)}$, we immediately obtain a final Poincar\'e inequality of the form~\eqref{eq:rooted:aim}, i.e., \[ \operatorname{Var}(f) \;\leqslant\; \exp\Big( O\big( q^{-2\alpha} \log(1/q)^4 \big) \Big) \sum_{x \in {\ensuremath{\mathbb Z}} ^2} \mu\big( c_x \operatorname{Var}_x(f) \big), \] as required. It will therefore suffice to prove Proposition~\ref{prop:I1:I2}.
\subsubsection{{The core of the proof of Proposition~\ref{prop:I1:I2}}}\label{sec:core}
Before giving the full technical details of the proof of the proposition, we first explain the high-level idea we wish to exploit. Fix~$j \in \{1,2\}$, set $W := W_j$, and fix $\o\in \O$ such that the restriction of $\o$ to $W$ satisfies the requirement of the good and super-good environment of the blocks (see~Figure~\ref{fig:setting}). The key idea is to cover the block $V = V_{(0,0)}$ with a collection of pairwise disjoint ``fibers'' $\hat F_1,\ldots, \hat F_{N+1}$, each of which is a quasi-stable half-ring, for some $N \;\leqslant\; |V|$ depending on~$\o$. For each fiber $\hat F_i$, the set $F_i := \hat F_i \cap {\ensuremath{\mathbb Z}} ^2$ is a subset of $W$ of cardinality $O(n_2)$ with the following key properties (which we will define precisely later): \begin{enumerate}[(a)] \item the fiber $F_{N+1}$ is empty; \item in each fiber $F_i$ a certain ``helping'' event $H_i$, depending only on the restriction of $\o$ to $F_i$, and implied by our assumption on the goodness\footnote{It is worth emphasizing here that $H_i$ only requires the blocks to be good, rather than super-good, and therefore holds with high probability.} of the blocks in $W$, occurs; \item {{the helping event $H_i$ has the following property: the $\ensuremath{\mathcal U}$-bootstrap process restricted to a $O(1)$-neighbourhood of the set $F_i \cup F_{i+1}$ is able to infect $F_i$ for any $\o$ such that $F_{i+1}$ is empty and $H_{i}$ occurs.}} \end{enumerate} To be concrete, let us consider the term $I_1$. In this case we will take $F_{N+1}$ to be $\ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2$, where $\ensuremath{\mathcal R}$ is the rightmost empty quasi-stable half-ring of width $\kappa$ that fits in $V_{(1,0)}$, which exists by our assumption that $V_{(1,0)}$ is a super-good block. The other fibers $F_1,\ldots, F_{N}$ will be suitable disjoint translates of $F_{N+1}$ in the $u$-direction, satisfying $V \subset \Lambda := \bigcup_{i=1}^N F_i$. The helping event $H_i$ will require the presence in $F_i$ of suitable helping sets for each quasi-stable direction{; we remark that the key requirement that $H_i$ depends only on the restriction of $\o$ to $F_i$ is a consequence of Assumption~\ref{easy}. Finally, the third condition (c) above will follow from Corollary \ref{cor:critic:spread}. A similar construction will be used for the term $I_2$, but the fibers will be slightly more complicated, see Figure~\ref{fig:strategy}.
\begin{figure}
\caption{The top picture shows the local neighbourhood $W_1$ of
the block $V = V_{(0,0)}$; in this case the fibers are simply the disjoint translates of the rightmost empty quasi-stable half-ring $\ensuremath{\mathcal R}$ in the last quarter of $V_{(1,0)}$. The bottom picture shows the local neighbourhood $W_2$; in this case the fibers are not all equal: they grow as they `descend' the steps of the upward $\kappa$-stair $\ensuremath{\mathcal M}$ (the little horizontal intervals). Each fiber becomes an elongated version of the rightmost empty half-ring $\ensuremath{\mathcal R}$.}
\label{fig:strategy}
\end{figure}
Let us write $\nu_i$ for the Bernoulli($p$) product measure on $S_i = \{0,1\}^{F_i}$ conditioned on the event $H_i$. The main step in the proof is the following bound on $I_j$ for $j \in \{1,2\}$: \begin{equation}
\label{eq:core1} I_j \, \;\leqslant\; \, \frac{1}{p_1} \cdot \mu\Big(\mathbbm{1}_{\{F_{N+1} \text{ is empty}\}} \operatorname{Var}_\nu(f) \Big), \end{equation} where $\operatorname{Var}_{\nu}(\cdot)$ is the variance computed w.r.t.~the product measure $\nu=\otimes_{i=1}^N \nu_i$. Before proving~\eqref{eq:core1} (see Section~\ref{sec:East-like}, below), let us show how to use Proposition~\ref{lem:gen-Poincare} and Lemma \ref{lem:var0} to deduce Proposition~\ref{prop:I1:I2} from it.
\begin{proof}[Proof of Proposition~\ref{prop:I1:I2}, assuming~\eqref{eq:core1}] Consider the generalized East chain on the space $\otimes_{i=1}^N (S_i,\nu_i)$ with constraining event $S_i^g = \big\{ F_i \text{ is empty} \big\}$ (see Definition~\ref{def:gen:East}). Note that the East constraint for the last fiber $F_N$ is always satisfied because $F_{N+1}$ is empty, and that the parameters $\{q_i\}_{i=1}^N$ of the generalized East process satisfy \[ q_i = \nu_i\big( S_i^g \big) \;\geqslant\; \, q^{O(n_2)} = \exp\Big( - O\big( q^{-\alpha} \log(1/q)^2 \big) \Big). \]
Noting that $N \;\leqslant\; |V| = q^{-O(1)}$, it follows from~\eqref{eq:scaling} that \begin{equation}\label{eq:losing:log} T_{\text{\tiny East}}\big( n,\bar \alpha \big) \;\leqslant\; \exp\Big( O\big( q^{-\alpha} \log(1/q)^3 \big) \Big). \end{equation}
Hence, applying Proposition \ref{lem:gen-Poincare} to bound $\operatorname{Var}_\nu(f)$ from above, and recalling~\eqref{eq:core1} and that $\Lambda = \bigcup_{i=1}^N F_i$, we obtain \begin{align} {I_j} & \, \;\leqslant\; \, \frac{1}{p_1} \cdot \mu\Big(\mathbbm{1}_{\{F_{N+1} \text{ is empty}\}}\operatorname{Var}_\nu(f) \Big)\nonumber\\ & \, \;\leqslant\; \, {{e^{O( q^{-\alpha} \log(1/q)^3)}}} \mu\bigg(\mathbbm{1}_{\{F_{N+1} \text{ is empty}\}}\sum_{i=1}^N\nu\Big( \mathbbm{1}_{\{F_{i+1} \text{ is empty}\}}\operatorname{Var}_{\nu_i}(f) \Big) \bigg)\nonumber\\ & \, {{= \, e^{O( q^{-\alpha} \log(1/q)^3)}}} \mu\bigg(\mathbbm{1}_{\{F_{N+1} \text{ is empty}\}}\sum_{i=1}^N \mu_{\Lambda}\Big( \mathbbm{1}_{\{F_{i+1} \text{ is empty}\}}\operatorname{Var}_{\nu_i}(f) \Big) \bigg), \label{eq:core1ter} \end{align} {{where the final inequality follows from the definition of $\nu_i$, and property~(b) of the fibers, which implies that the event $H_1 \cap \cdots \cap H_N$ has probability at least $p_1^3 = 1 - o(1)$ (since it is implied by the goodness of three blocks).}}
{{Recall that, by property~(c) of the fibers, $F_i$ is contained in the closure, under the $\ensuremath{\mathcal U}$-bootstrap process restricted to {{a $O(1)$-neighbourhood $U_i$ of the}} set $F_i\cup F_{i+1}$, of any set of empty sites containing $F_{i+1}$ and for which the event $H_i$ holds. We may therefore apply Lemma~\ref{lem:var0}, with $A := F_{i+1}$, $B := F_i$, $\ensuremath{\mathcal E} := H_i$ and $U := U_i$, to obtain}}
\begin{gather} \label{eq:core1bis} \mu_{{{\Lambda}}}\Big(\mathbbm{1}_{\{F_{i+1} \text{ is empty}\}}\operatorname{Var}_{\nu_i}(f) \Big) \;\leqslant\; O(n_2) q^{-O(n_2)}\sum_{x \in {{U_i}}} \mu_{{{\Lambda}}} \big( c_x\operatorname{Var}_x(f) \big), \end{gather}
since $|F_i| = O(n_2)$. Inserting~\eqref{eq:core1bis} into~\eqref{eq:core1ter} we obtain \[ I_j \, \;\leqslant\; \, e^{O( q^{-\alpha} \log(1/q)^3)} \sum_{x \in {{\hat W_j}}} \mu\big( c_x \operatorname{Var}_x(f) \big) \] for each $j \in \{1,2\}$, {{and some $O(1)$-neighbourhood $\hat W_j$ of $W_j$,}} as required. \end{proof}
\begin{remark}\label{rem:whyEast} {{We remark that our use of the generalized East chain (rather than the generalised FA-1f chain) in the proof above was necessary (since for $\alpha$-rooted models Proposition~\ref{prop:critic:spread} can only be used to move infection in the $u$-direction), and also harmless (since in either case the bound we obtain is of the form $\exp\big( \widetilde O(q^{-\alpha}) \big)$, which is much smaller than $\exp\big( q^{-2\alpha} \big)$). In the proof for $\beta$-unrooted models we will also use the generalized East chain, however, even though in that case we can move infection in both the $u$- and $-u$-directions, and doing so costs us a factor of $\log(1/q)$ in the exponent for models with $\beta = \alpha$. This is because the method we use in this paper does not appear to easily allow us to use the generalised FA-1f chain in this setting.}} \end{remark}
In order to conclude the proof of the proposition, it remains to construct in detail the fibers for each case and to prove the basic inequality~\eqref{eq:core1}.
\subsubsection{Construction of the fibers and the proof of~\eqref{eq:core1}.} \label{sec:East-like}
We {{will first define the helping events and prove~\eqref{eq:core1} in the (easier) case $j = 1$. Recall that}} \[ I_1 = \mu\left(\mathbbm{1}_{\{\o_{\vec e_1} \in G_2\}}\mathbbm{1}_{\{\o_{-\vec e_1}\in G_1\}} \operatorname{Var}_{V}\big( f \thinspace |\thinspace G_1 \big) \right), \] {{where $V = V_{(0,0)}$, and that $\o_{\vec e_1} \in G_2$ implies that there exists an empty quasi-stable half-ring $\ensuremath{\mathcal R}$ of width $\kappa$ that fits in $V_{(1,0)}$ and is entirely contained in the rightmost quarter of $R_{(1,0)}$, and recall that this determines the length $\ell$ of $\ensuremath{\mathcal R}$, and that $\ell \;\geqslant\; n_2/ m$. By translating $\ensuremath{\mathcal R}$ slightly (without changing the set $\ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2$) if necessary, we may also assume that there are no sites of ${\ensuremath{\mathbb Z}} ^2$ on the boundary of $\ensuremath{\mathcal R}$ and in the interior of $R_{(1,0)}$. Let us also choose $\kappa$ so that the vector $\kappa u$ has integer {{coordinates}}. Now, for each such quasi-stable half-ring $\ensuremath{\mathcal R}$, set $$N = N(\ensuremath{\mathcal R}) := \min\big\{ j : \ensuremath{\mathcal R} + j \kappa u \subset V_{(-1,0)} \big\}$$ and define $F_i = F_i(\ensuremath{\mathcal R}) := \hat F_i \cap {\ensuremath{\mathbb Z}} ^2$, where \[ \hat F_i = \hat F_i(\ensuremath{\mathcal R}) := \ensuremath{\mathcal R} + (N + 1 - i) \kappa u, \] for each $1 \;\leqslant\; i \;\leqslant\; N + 1$. Note that $V_{(0,0)} \subset \bigcup_{i=1}^N F_i$, and that (by our choice of $\kappa$) there are no sites of ${\ensuremath{\mathbb Z}} ^2$ on the boundary of $\hat F_i$ in the interior of $R_{(-1,0)} \cup R_{(0,0)} \cup R_{(1,0)}$.
\begin{definition}\label{def:helping} For each $\ensuremath{\mathcal R}$ and $i \in [N]$, let $H_i$ denote the event that for each quasi-stable direction $v \in C$ and every $v$-strip $S$ such that the segment $\partial^{\rm ext}(S) \cap \hat F_i$ has length at least $n_2/(2m)$, there exists an empty helping set $Z \subset F_i$ for~$S$. \end{definition} }}
Notice that in the above definition we do not require {{the $v$-strip $S$}} to be contained in $\hat F_i$. Observe that if the blocks $V_{(-1,0)}$, $V_{(0,0)}$ and $V_{(1,0)}$ are all good, then the event $H_i$ occurs for every $i \in [N]$. Now define $H_{\ensuremath{\mathcal R}}$ to be the event that $\ensuremath{\mathcal R}$ is (up to translates preserving the set $\ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2$) the rightmost empty quasi-stable half-ring in $R_{(1,0)}$, and observe that, conditional on $H_{\ensuremath{\mathcal R}}$, the events $\{H_i\}_{i=1}^N$ are independent. Moreover, by Corollary~\ref{cor:critic:spread}, {{and since $\kappa$ is sufficiently large,}} if $F_{i+1}$ is empty and $H_{i}$ occurs, then the $\ensuremath{\mathcal U}$-bootstrap process restricted to {{a $O(1)$-neighbourhood of}} the set $F_i \cup F_{i+1}$ is able to infect $F_i$. The fibers $\{F_i\}_{i=1}^{N+1}$ therefore satisfy conditions (a), (b) and (c) of Section~\ref{sec:core}. Recall that we write $\Lambda = \bigcup_{i=1}^N F_i$. We make the following claim, which implies~\eqref{eq:core1} in the case $j = 1$:
\begin{claim}\label{claim:core1} \begin{equation} \label{eq:claim1}
I_1 \;\leqslant\; \frac{1}{p_1} \sum_{\ensuremath{\mathcal R}} \mu\Big( \mathbbm{1}_{H_\ensuremath{\mathcal R}} \operatorname{Var}_{\Lambda}\big( f \;\big|\; H_1 \cap \cdots \cap H_N \big) \Big).
\end{equation} \end{claim}
{{Note that the sum in the claim is over equivalence classes of quasi-stable half-rings $\ensuremath{\mathcal R}$, where two half-rings are equivalent if they have the same intersection with ${\ensuremath{\mathbb Z}} ^2$.}}
\begin{proof}[Proof of Claim~\ref{claim:core1}] {{We first claim that \begin{equation}\label{claim:core1:step1} I_1 \;\leqslant\; \frac{1}{p_1} \sum_{\ensuremath{\mathcal R}} \mu\Big( \mathbbm{1}_{H_\ensuremath{\mathcal R}} \mathbbm{1}_{\{\o_{\pm \vec e_1}\in G_1\}}\mu_V\Big( \mathbbm{1}_{\{\o_0 \in G_1\}} \big( f - a \big)^2 \Big) \Big), \end{equation} where $\o_0\equiv \o_{V_{(0,0)}}$ and, for any $\o \in H_{\ensuremath{\mathcal R}}$, we set \[
a = a\big( \o_{{\ensuremath{\mathbb Z}} ^2 \setminus \Lambda} \big) := \mu_\Lambda\big( f \;\big|\; H_1 \cap \cdots \cap H_N \big), \] noting that, on the event $H_{\ensuremath{\mathcal R}}$, the set $\Lambda$ and the fibers become deterministic. To prove~\eqref{claim:core1:step1} we use Definition~\ref{def:goodsets}, which implies that if $V_{(1,0)}$ is super-good then it is also good, and also that the event $H_{\ensuremath{\mathcal R}}$ holds for some $\ensuremath{\mathcal R}$, and the standard inequality $\operatorname{Var}(X) \;\leqslant\; {\ensuremath{\mathbb E}} \big[ (X-a)^2 \big]$, which holds for any $a\in {\ensuremath{\mathbb R}} $ and any random variable $X$.
Recalling that if the blocks $V_{(-1,0)}$, $V_{(0,0)}$ and $V_{(1,0)}$ are all good, then the event $H_i$ occurs for every $i \in [N]$, it follows from~\eqref{claim:core1:step1} that}} \begin{align*}
I_1 & \, \;\leqslant\; \, \frac{1}{p_1} \sum_{\ensuremath{\mathcal R}} \mu\Big( \mathbbm{1}_{H_\ensuremath{\mathcal R}} \, \mu_{{{\Lambda}}} \Big( \mathbbm{1}_{H_1 \cap \cdots \cap H_N} \big( f - a \big)^2 \Big) \Big)\\
& \, \;\leqslant\; \, \frac{1}{p_1} \sum_{\ensuremath{\mathcal R}} \mu\Big( \mathbbm{1}_{H_\ensuremath{\mathcal R}} \operatorname{Var}_\Lambda\big( f \;\big|\; H_1 \cap \cdots \cap H_N \big) \Big), \end{align*} where the last inequality follows from our choice of $a$ and the trivial inequality \[
\mu_\Lambda\Big( \mathbbm{1}_{H_1 \cap \cdots \cap H_N} \big( f - a \big)^2 \Big) \;\leqslant\; \mu_\Lambda\Big( \big( f - a \big)^2 \;\big|\; H_1 \cap \cdots \cap H_N \Big). \] {{This proves the claim, and hence~\eqref{eq:core1} in the case $j = 1$.}} \end{proof}
We now turn to {{the analysis of}} the term $$I_2 = \mu\left(\mathbbm{1}_{\{\o_{\vec e_2}\in G_2\}}\mathbbm{1}_{\{\o_j\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^+\}} \operatorname{Var}_{V}(f) \right).$$ In this case we need to modify the definition of the fibers $F_i$ in order to take into account the different local neighbourhood $W_2$ of $V_{(0,0)}$ and the different good and super-good environment in $W_2$ (see Figures~\ref{fig:setting} {{and~\ref{fig:strategy}}}).
{{First, let us define $H_\ensuremath{\mathcal R}$ to be the event that $\ensuremath{\mathcal R}$ is (up to translates preserving the set $\ensuremath{\mathcal R} \cap {\ensuremath{\mathbb Z}} ^2$) the rightmost empty quasi-stable half-ring of width $\kappa$ that fits in $V_{(0,1)}$, and observe that the length $\ell$ of $\ensuremath{\mathcal R}$ satisfies $\ell \;\geqslant\; n_2/ m$, and that the event $\o_{\vec e_2} \in G_2$ implies that $H_\ensuremath{\mathcal R}$ holds for some $\ensuremath{\mathcal R}$ in the rightmost quarter of $R_{(0,1)}$. As before, we may choose $\ensuremath{\mathcal R}$ so that there are no sites of ${\ensuremath{\mathbb Z}} ^2$ on its boundary in the interior of $R_{(0,1)}$.
The fibers $\{F_i\}_{i=1}^{N+1}$ will be similar to those used above to bound $I_1$, but some of the $v_1$-strips (which form the bottom portion of each fiber) will be elongated as the fibers ``descend" the upward $\kappa$-stair in $V_{(1,0)}$, see Figure~\ref{fig:strategy}. (Recall that we call these objects \emph{elongated quasi-stable half-rings}.) To be precise, let us write $L(\ensuremath{\mathcal R})$ for the two-way infinite $v_1$-strip of width $\kappa$ that contains the $v_1$-strip of $\ensuremath{\mathcal R}$, and define $$N = N(\ensuremath{\mathcal R}) := \min\bigg\{ j : V_{(0,0)} \subset \bigcup_{i = 1}^j \big( L(\ensuremath{\mathcal R}) + {{i}} \kappa u \big) \bigg\}.$$ Now,}} recall that $a = \Theta(n_2)$ is the number of rows {{of $V$}}, and recall Definition~\ref{def:stair}. Let {{$\ensuremath{\mathcal M} = \big\{ M^{(1)}, \ldots, M^{(a)} \big\}$}} be an upward $\kappa$-stair contained in the {{leftmost}} quarter of $V_{(1,0)}$, {{and define the fibers $\hat F_i = \hat F_i(\ensuremath{\mathcal R},\ensuremath{\mathcal M})$ recursively as follows: \begin{itemize} \item[$(a)$] $\hat F_{N+1} := \ensuremath{\mathcal R}$; \item[$(b)$] For each $i \in [N]$ set $\hat F'_i := \hat F_{i+1} + \kappa u$. Now define: \begin{itemize} \item[$(i)$] $\hat F_i$ to be an elongated version of $\hat F'_i$ such that $\big( \hat F_i \setminus \hat F_i' \big) \cap {\ensuremath{\mathbb Z}} ^2$ is a subset of a step of $\ensuremath{\mathcal M}$, if such an elongated quasi-stable half-ring exists; \item[$(ii)$] $\hat F_i := \hat F_i'$ otherwise. \end{itemize} \end{itemize} As before, we set $F_i = F_i(\ensuremath{\mathcal R},\ensuremath{\mathcal M}) := \hat F_i \cap {\ensuremath{\mathbb Z}} ^2$ for each $1 \;\leqslant\; i \;\leqslant\; N + 1$. {{Let us write $H_\ensuremath{\mathcal M}$ for the event}} that $\ensuremath{\mathcal M}$ is the first (in some arbitrary ordering) empty upward {{$\kappa$}}-stair contained in the leftmost quarter of $V_{(1,0)}$. We can now define the helping events.
\begin{definition}\label{def:helping:2} For each $\ensuremath{\mathcal R}$ and $\ensuremath{\mathcal M}$, and each $i \in [N]$, let $H_i$ denote the event that for each quasi-stable direction $v \in C$ and every $v$-strip $S$ such that the segment $$\partial^{\rm ext}(S) \cap \hat F_i \cap \big( R_{(-1,1)} \cup R_{(0,1)} \big)$$ has length at least $n_2/(2m)$, there exists an empty helping set $Z \subset F_i$ for~$S$. \end{definition}
Observe that if the blocks $V_{(-1,1)}$ and $V_{(0,1)}$ are both good, then the event $H_i$ occurs for every $i \in [N]$. Moreover, conditional on the event $H_\ensuremath{\mathcal R} {{ \, \cap \, H_\ensuremath{\mathcal M}}}$, the events $\{H_i\}_{i=1}^N$ are independent and, by Corollary~\ref{cor:critic:spread} {{(see Remark~\ref{rem:helping-sets})}}, if $F_{i+1}$ is empty and {{the events $H_\ensuremath{\mathcal M}$ and}} $H_i$ occur, then the $\ensuremath{\mathcal U}$-bootstrap process restricted to {{a $O(1)$-neighbourhood of}} the set $F_i \cup F_{i+1}$ is able to infect $F_i$. {{It follows that if the event $H_\ensuremath{\mathcal R} \, \cap \, H_\ensuremath{\mathcal M}$ occurs, then}} the fibers $\{F_i\}_{i=1}^{N+1}$ satisfy conditions (a), (b) and (c) of Section~\ref{sec:core}.
We make the following claim, which implies~\eqref{eq:core1} in the case $j = 2$:
\begin{claim}\label{claim:core2} \begin{equation} \label{eq:claim2}
I_2 \;\leqslant\; \frac{1}{p_1} \sum_{\ensuremath{\mathcal R}, \ensuremath{\mathcal M}} \mu\Big( \mathbbm{1}_{H_\ensuremath{\mathcal R}} {{\mathbbm{1}_{H_\ensuremath{\mathcal M}}}} \operatorname{Var}_{\Lambda}\big( f \;\big|\; H_1 \cap \cdots \cap H_N \big) \Big).
\end{equation} \end{claim}
The proof of Claim~\ref{claim:core2} is identical to that of Claim~\ref{claim:core1}. As discussed above, this completes the proof of the Proposition~\ref{prop:I1:I2}, and hence of Theorem~\ref{mainthm:2} in the case where $\ensuremath{\mathcal U}$ is $\alpha$-rooted and Assumption~\ref{easy} holds. }}
\subsection{The $\beta$-unrooted case} \label{sec:beta-unrooted}
In this section we assume that the bilateral difficulty $\beta$ of the updating rule $\ensuremath{\mathcal U}$ is smaller than $2\alpha$. We will prove that, if Assumption~\ref{easy} holds, then there exists a constant $\lambda = \lambda(\ensuremath{\mathcal U})$ such that $${\ensuremath{\mathbb E}} _\mu(\tau_0) \;\leqslant\; \frac{T_{\rm rel}(q,\ensuremath{\mathcal U})}{q} \;\leqslant\; {{\exp\Big( \lambda \cdot q^{-\beta} \big( \log(1/q) \big)^3 \Big)}}$$ for all sufficiently small $q > 0$. Note that the first inequality follows from~\eqref{eq:mean-infection}, and so, by Definition~\ref{def:PC}, it will suffice to prove that \begin{equation}\label{eq:unrooted:aim} \operatorname{Var}(f) \;\leqslant\; {{\exp\Big( \lambda \cdot q^{-\beta} \big( \log(1/q) \big)^3 \Big)}} \sum_{x \in {\ensuremath{\mathbb Z}} ^2} \mu\big( c_x \operatorname{Var}_x(f) \big) \end{equation} for some $\lambda = \lambda(\ensuremath{\mathcal U})$ and all local functions $f$. We will deduce a bound of the form~\eqref{eq:unrooted:aim} from the \emph{unoriented} constrained Poincar\'e inequality~\eqref{eq:9FA} and Corollary~\ref{cor:critic:spread}.
{{Recall from Section~\ref{sec:critic:spread} that $C \subset S^1$ is an open semicircle such that $\alpha(v) \,\leqslant\, \beta$ for each $v \in C \cup - C$, and that we let $u$ be the mid-point of $C$. Similarly to Section~\ref{sec:critical:rooted}, we set \begin{equation}\label{def:n1n2:unrooted} n_1 = \big\lfloor q^{-{{3}}\kappa} \big\rfloor \qquad \text{and} \qquad n_2 = \big\lfloor \kappa^4 q^{-\beta} \log (1/q) \big\rfloor, \end{equation} where $\kappa = \kappa(\ensuremath{\mathcal U})$ is a sufficiently large constant.}}
We need to slightly modify the definition of the good and super-good events $G_1$ and $G_2$ as follows. {{Let $(v_1,\dots, v_m)$ be the quasi-stable directions in $C$, and let $(v_1,\dots, v_{m'})$ be the quasi-stable directions in $-C$ (see Definition~\ref{def:quasi-stable}). As in Section~\ref{sec:critical:rooted}, it follows by Assumption~\ref{easy} that we may choose the voracious sets so that the helping sets for $S_v$ are subsets of $\partial^{\rm ext}(S_v)$ for each quasi-stable direction $v \in C \cup - C$.}}
{{ \begin{definition}[Good and super-good events] \label{def:new goodsets} \ \begin{enumerate}
\item The block $V_i = R_i \cap {\ensuremath{\mathbb Z}} ^2$ satisfies the \emph{good} event $G_1$ iff:
\begin{enumerate}[(a)]
\item for each quasi-stable direction $v \in C$ and every $v$-strip $S$
such that the length of the segment $\partial^{\rm ext}(S) \cap R_i$ is at least $n_2/(4m)$, there
exists an empty helping set $Z \subset {{\partial^{\rm ext}(S) \cap V_i}}$ for~$S$;
\item for each quasi-stable direction $v \in -C$ and every $v$-strip $S$
such that the length of the segment $\partial^{\rm ext}(S) \cap R_i$ is at least $n_2/(4m')$, there
exists an empty helping set $Z \subset {{\partial^{\rm ext}(S) \cap V_i}}$ for~$S$;
\item there exist two empty upward $\kappa$-stairs, one within the leftmost quarter of $V_i$, and one within the rightmost quarter of $V_i$.
\end{enumerate}
\item The block $V_i$ satisfies the \emph{super-good} event $G_2$ iff it satisfies the good event $G_1$, and moreover there exist two empty quasi-stable half-rings $\ensuremath{\mathcal R}^+$ and $\ensuremath{\mathcal R}^-$, of width $\kappa$, that both fit in $V_i$, with $\ensuremath{\mathcal R}^+$ relative to $C$ and entirely contained in the rightmost quarter of $R_i$, and with $\ensuremath{\mathcal R}^-$ relative to $-C$ and entirely contained in the leftmost quarter of $R_i$. \end{enumerate} \end{definition} }}
\begin{figure}\label{fig:+/- half-ring}
\end{figure}
It is easy to check that, with the new definition of the good and super-good events, Lemma \ref{lem:p1p2} still holds. {{It follows, by Theorem~\ref{thm:CPI}, that the unconstrained Poincar\'e inequality~\eqref{eq:9FA} holds for any local function $f$, i.e., \begin{align}\label{eq:7:1} & \operatorname{Var}(f) \;\leqslant\; T(p_2) \bigg( \sum_{\varepsilon=\pm 1}\sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\varepsilon\vec e_2} \in G_2\}}\mathbbm{1}_{\{\o_{j}\in G_1\, \forall j\in {\ensuremath{\mathbb L}} ^{\varepsilon}(i)\}} \operatorname{Var}_i(f)\right) \nonumber\\ & \hspace{3.5cm} + \sum_{\varepsilon=\pm 1}\sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\left(\mathbbm{1}_{\{\o_{i+\varepsilon\vec e_1} \in G_2\}}\mathbbm{1}_{\{\o_{i-\varepsilon \vec e_1}\in G_1\}} \operatorname{Var}_i\big( f \thinspace |\thinspace G_1 \big) \right) \bigg). \end{align} with \[ T(p_2) \, = \, p_2^{-O(1)} \, = \, \exp\Big( O\big( q^{-\beta} \log(1/q)^2 \big) \Big). \] As in Section~\ref{sec:critical:rooted}, using translation invariance it will suffice to bound from above, for a fixed (and arbitrary) local function~$f$, the following four generic terms: \[ I^\pm _1(i) := \mu\left(\mathbbm{1}_{\{\o_{i\pm\vec e_1} \in G_2\}} \mathbbm{1}_{\{\o_{i \mp \vec e_1} \in G_1\}} \operatorname{Var}_{V_i}(f \thinspace |\thinspace G_1)\right), \] and \[ I^\pm_2(i) := \mu\left(\mathbbm{1}_{\{\o_{i \pm \vec e_2} \in G_2\}} \mathbbm{1}_{\{\o_j \in G_1 \, \forall j \in {\ensuremath{\mathbb L}} ^\pm(i)\}} \operatorname{Var}_{V_i}(f) \right). \] }}
\begin{figure}\label{fig:setting2}
\end{figure}
{{Define $W^+_1 = W^-_1 = V_{(0,0)} \cup V_{(-1,0)} \cup V_{(1,0)}$, and $W_2^+ = V_{(0,0)} \cup V_{(-1,0)} \cup V_{(1,0)} \cup V_{(-1,1)} \cup V_{(0,1)}$ and $W_2^- = V_{(0,0)} \cup V_{(1,0)} \cup V_{(-1,0)} \cup V_{(1,-1)} \cup V_{(0,-1)}$. The following upper bounds on $I_1^\pm$ and $I_2^\pm$ (cf. Proposition~\ref{prop:I1:I2}) follow exactly as in Section~\ref{sec:critical:rooted}.
\begin{proposition} \label{prop:I:plusminus} For each $j \in \{1,2\}$, there exist $O(1)$-neighbourhoods $\hat W^\pm_j$ of $W^\pm_j$ such that \begin{align*}
I^\pm_j \, \;\leqslant\; \exp\Big( O\big( q^{-\beta} \log(1/q)^{{3}} \big) \Big) \sum_{x \in \hat W^\pm_j} \mu\big(c_x \operatorname{Var}_x(f)\big). \end{align*} \end{proposition}
}}
\begin{proof}[{{Sketch proof of Proposition~\ref{prop:I:plusminus}}}] The terms $I_1^+$ and $I_2^+$ can be treated exactly as the terms $I_1$ and $I_2$ analysed in the previous section, because the new good and super-good events imply the good and super-good events for the $\alpha$-rooted case. {{We may therefore repeat the proof of Proposition~\ref{prop:I1:I2}, with the only difference being that $n_2$ is now as defined in~\eqref{def:n1n2:unrooted}, to obtain the claimed bounds on $I_1^+$ and $I_2^+$.}}
{{For the new terms, $I_1^-$ and $I_2^-$ (which are illustrated in Figure~\ref{fig:setting2}), the argument is exactly the same}} after a rotation of $\pi$ of the coordinate axes. {{Indeed, a good block now contains suitable empty helping sets for the quasi-stable directions in $-C$ (as well as $C$), and an empty upward $\kappa$-stair in the rightmost quarter (as well as the leftmost), and a super-good block contains an empty quasi-stable half-ring relative to $-C$ in the leftmost quarter (as well as one relative to $C$ in the rightmost quarter). Such a rotation therefore transforms $I_1^-$ and $I_2^-$ into $I_1^+$ and $I_2^+$, and so the proof of the claimed bounds is once again identical to that of Proposition~\ref{prop:I1:I2}.}} \end{proof}
{{ \begin{remark}\label{rmk:losing:log} As noted in Remark~\ref{rem:whyEast}, our application of the generalized East chain in the proof above cost us a factor of $\log(1/q)$ in the exponent. More precisely, this log-factor was lost in step~\eqref{eq:losing:log} of the proof of Proposition~\ref{prop:I:plusminus}, when (roughly speaking) we passed through an energy barrier corresponding to the simultaneous existence of about $\log(1/q)$ empty quasi-stable half-rings in a single block. As stated precisely in Conjecture~\ref{unbalanced:conj}, we expect that (at least for models with $\beta = \alpha$) the true relaxation time does not contain this additional factor of $\log(1/q)$. \end{remark} }}
{{Combining Proposition~\ref{prop:I:plusminus} with~\eqref{eq:7:1}, and noting that $|\hat W^\pm_j| = q^{-O(1)}$, we obtain a final Poincar\'e inequality of the form~\eqref{eq:unrooted:aim}, i.e., \[ \operatorname{Var}(f) \;\leqslant\; \exp\Big( O\big( q^{-\beta} \log(1/q)^3 \big) \Big) \sum_{x \in {\ensuremath{\mathbb Z}} ^2} \mu\big( c_x \operatorname{Var}_x(f) \big), \] as required. This completes the proof of Theorem~\ref{mainthm:2} for update families $\ensuremath{\mathcal U}$ such that Assumption~\ref{easy} holds.}} \qed
\section{Critical KCM: removing the simplifying assumption} \label{sec:fullgen}
In this section we {{explain how to modify the proof given in Section~\ref{sec:critic-ub} in order to avoid using Assumption~\ref{easy}. Since the argument is essentially identical for $\alpha$-rooted and $\beta$-unrooted families, for simplicity we will restrict ourselves to}} the $\alpha$-rooted case.
Our solution requires a slight change in the geometry of the quasi-stable half-ring. In what follows we will always work in the frame $(-u,u^\perp)$, where $u$ is the midpoint of the semicircle $C$ {{given by Lemma~\ref{cor:Ccritic} (cf.~Sections~\ref{sec:critic:spread} and~\ref{sec:critical:rooted}).
Recall from Definition~\ref{def:strips} the definitions of the $+$- and $-$-boundaries of a $v$-strip $S$. The following key definition is illustrated in Figure~\ref{fig:general half-ring}.}}
\begin{definition}[Generalised quasi-stable half-rings] \label{def:gen-qshr} {{Let $(v_1,\dots, v_m)$ be the quasi-stable directions in $C$, ordered as in Definition \ref{def:half-ring}, and let $\ensuremath{\mathcal R}$ be a quasi-stable half-ring of width $w$ and length $\ell$ relative to $C$. For each quasi-stable direction $v \in C$, let $S_v$ be the $v$-strip in $\ensuremath{\mathcal R}$, and let $\hat S_v^{l}$ and $\hat S_{v}^{r}$ be the (unique) $v$-strips of width $w/3$ and length $\ell/3$ satisfying the following properties: \begin{itemize} \item[(i)] $\hat S_{v}^{l}$ and $\hat S_{v}^{r}$ each share exactly one corner with $S_{v}$; moreover each of these corners lies at the ``top" of $S_v$ when working in the frame $(-u,u^\perp)$. \item[(ii)] $\partial_- (\hat S_{v}^{l}) \subset \partial_+(S_{v})$ and $\partial_- (\hat S_{v}^{r})\subset \partial_- (S_{v})$.
\end{itemize} Set \[ S_{v}^g := \big( S_{v} \setminus \hat S_{v}^{r} \big) \cup \hat S_{v}^{l} , \] and set \[
\ensuremath{\mathcal R}^g := \bigcup_{i=1}^m S_{v_i}^g. \] We call $\ensuremath{\mathcal R}^g$ the \emph{generalised version of $\ensuremath{\mathcal R}$}, and define the ``core'' of $\ensuremath{\mathcal R}^g$ to be the set $\ensuremath{\mathcal R}^g \cap \ensuremath{\mathcal R}$.}} \end{definition}
\begin{figure}
\caption{A generalised quasi-stable half-ring.}
\label{fig:general half-ring}
\label{fig:general half-ring2}
\label{fig:generalised:ring}
\end{figure}
Recall now {{the following two key ingredients of}} the proof given in the previous section {{(see Section \ref{sec:core})}} under the simplifying Assumption~\ref{easy}: \begin{enumerate}[(i)] \item a {{sufficiently large}} empty quasi-stable half-ring $\ensuremath{\mathcal R}$ is able to completely infect its translate $\ensuremath{\mathcal R}+\kappa u$, provided that {{a certain ``helping" event occurs;}} \item {{the helping event depends only on the configuration inside $\ensuremath{\mathcal R}$.}} \end{enumerate} Here we prove a similar result for the generalised quasi-stable half-rings \emph{without} the simplifying assumption. {{We first define the helping event, cf. Definition~\ref{def:helping}.}}
\begin{definition} Given a {{quasi-stable half-ring $\ensuremath{\mathcal R}$ of length $\ell$ and width $\kappa$, we define $H(\ensuremath{\mathcal R})$ to be the event that for each quasi-stable direction $v \in C$ and every $v$-strip $S$ of length $\ell$ with $\partial_+(S) \subset \ensuremath{\mathcal R}$, there exists an empty helping set for~$S$ in~$\ensuremath{\mathcal R}^g$.}} \end{definition}
{{If $H(\ensuremath{\mathcal R})$ holds, then we will say that $\ensuremath{\mathcal R}^g$ is \emph{helping}. We will modify (see below) the good and super-good events $G_1$ and $G_2$ (see Definition~\ref{def:goodsets}) so that they guarantee that this helping event occurs, and choose the constant $\kappa = \kappa(\ensuremath{\mathcal U}) > 0$ (as in Section~\ref{sec:critical:rooted}) so that the conclusion of Lemma~\ref{lem:p1p2} holds, and so that $\kappa u$ has integer coordinates. We will also choose our (generalised) quasi-stable half-rings so that there are no sites of ${\ensuremath{\mathbb Z}} ^2$ on their boundary, except on the top and bottom boundaries of the rectangles $R_i$.}}
\begin{lemma} Let {{$\ensuremath{\mathcal R}$ be a quasi-stable half-ring of length $\ell$ and width $\kappa$, and let $\ensuremath{\mathcal R}^g$ be the generalised version of $\ensuremath{\mathcal R}$.}} Assume that the core of $\ensuremath{\mathcal R}^g$ is empty and that $\ensuremath{\mathcal R}^g$ and its translate $\ensuremath{\mathcal R}^g + \kappa u$ are {{both}} helping. Then there exists a $O(1)$-neighbourhood $U$ of $\ensuremath{\mathcal R}^g \cup \big( \ensuremath{\mathcal R}^g + \kappa u \big)$ such that {{the $\ensuremath{\mathcal U}$-bootstrap process}} restricted to $U$ is able to infect the core of $\ensuremath{\mathcal R}^g + \kappa u$. \end{lemma}
\begin{proof} The {{lemma is a straightforward consequence of Proposition~\ref{prop:critic:spread}, using}} the geometry of the generalised quasi-stable half-rings. {{To spell out the details (cf. the proof of Corollary~\ref{cor:critic:spread}), fix $\ensuremath{\mathcal R}$}} as in the lemma, and let $\ensuremath{\mathcal R}'$ be {{any}} quasi-stable half-ring {{of length $\ell$ and width $\kappa/3$}} such that: \begin{itemize} \item[(a)] {{$\ensuremath{\mathcal R}' = \ensuremath{\mathcal R} + \lambda u$}} for some $\lambda \,\geqslant\, 0$, and
\item[(b)] ${{\ensuremath{\mathcal R}'}} \subset \ensuremath{\mathcal R}^g \cup \big( \ensuremath{\mathcal R}^g + \kappa u)$, \end{itemize} see Figure~\ref{fig:general half-ring2}. {{We claim that, for every quasi-stable direction $v \in C$, there exists an empty helping set in $\ensuremath{\mathcal R}^g \cup \big( \ensuremath{\mathcal R}^g + \kappa u \big)$ for the $v$-strip $S'_{v}$ of $\ensuremath{\mathcal R}'$. Indeed, this follows from the fact that $\ensuremath{\mathcal R}^g$ and $\ensuremath{\mathcal R}^g + \kappa u$ are both helping, since (by construction) either $\partial_+(S'_v) \subset \ensuremath{\mathcal R}$ or $\partial_+(S'_v) \subset \ensuremath{\mathcal R} + \kappa u$.
Now, since the core of $\ensuremath{\mathcal R}^g$ is empty, it follows, by Proposition \ref{prop:critic:spread}, that there exists a $O(1)$-neighbourhood $U$ of $\ensuremath{\mathcal R}^g \cup \big( \ensuremath{\mathcal R}^g + \kappa u \big)$ such that the $\ensuremath{\mathcal U}$-bootstrap process restricted to $U$ can advance in the $u$-direction, and infect the core of $\ensuremath{\mathcal R}^g + \kappa u$, as claimed.}} \end{proof}
Given the above lemma, the proof of Theorem~\ref{mainthm:2} proceeds exactly as the one given in Section~\ref{sec:critic-ub}, with only two main changes: \begin{enumerate}[(a)] \item the fibers $\{F_i\}_{i=1}^N$ are no longer the
quasi-stable half-rings (or their elongated version) but rather the
generalised quasi-stable half-rings (or their elongated version); \item when defining the generalised East process for the fibers, the
constraining event $S_i^g$ (see Definition \ref{def:gen:East}), which in
Section \ref{sec:critic-ub} was simply $S_i^g=\{F_{i}\text{ is empty}\},$ now
becomes $S_i^g=\{\text{the core of $F_{i}$ is empty}\}.$ \end{enumerate} We leave the (straightforward) {{task of verifying the details}} to the reader.
\appendix \section{}
\subsection{Proof of Proposition~\ref{lem:gen-Poincare}}\label{sec:pf-gen-Poincare}
We will follow closely the proof of a very similar result proved in~\cite{CFM2}*{Proposition 3.4}. Let $\{P_t\}_{t\,\geqslant\, 0}$ be the Markov semigroup associated to either the generalised East chain or the generalised FA-1f chain. Using reversibility, it follows (see, e.g.,~\cite{Saloff}*{Theorem~2.1.7}) that \begin{equation}
\label{eq:A1bis}
\lim_{t\rightarrow \infty} - \frac 1t \log\Big( \max_\o \big\| P_t(\o,\cdot)
-\nu(\cdot) \big\|_{\rm \tiny TV} \Big)= \frac{1}{T_{\rm rel}}, \end{equation}
where $\|\cdot\|_{\rm \tiny TV}$ denotes the {{total}} variation distance.
We now claim that {{for every function $f \colon \Omega \rightarrow {\ensuremath{\mathbb R}} $}} with $\|f\|_\infty\,\leqslant\, 1$, \begin{equation}\label{eq:A0}
\big\| P_t f - \nu(f) \big\|_\infty \;\leqslant\; C(n,q) e^{-t/t^*} \end{equation} for some $0 < C(n,q) < {{\infty}}$ and {{either}} $t^* \,\leqslant\, T_{\text{East}}(n,\bar\alpha)/q$ or $t^* \,\leqslant\, T_{\text{FA}}(n,\bar\alpha)/q$, depending on which of the two models we are considering. Clearly \eqref{eq:A1bis} and \eqref{eq:A0} imply that $T_{\rm rel} \,\leqslant\, t^*$ and {{(recalling Definition~\ref{def:PC})}} the proposition follows.
To prove~\eqref{eq:A0}, let $\tau_x(\o)$ be the {{time of the}} first legal ring at $x$ {{(that is, the first time that the state of $x$ is
resampled)}} when the starting configuration is $\o$. Then, for any function $f \colon \otimes_{x\in [n]} S_x \mapsto {\ensuremath{\mathbb R}} $ with $\nu(f)=0,$ we write \begin{align}\label{eq:A1}
& \|P_tf\|_\infty \;\leqslant\; \max_\o {{\Big\{}} \Big| {\ensuremath{\mathbb E}} _\o\Big( f\big( \o(t) \big) \cdot \mathds 1_{\{\tau_x(\o) \,<\, t\ \forall x\}} \Big) \Big| \nonumber\\
& \hspace{4.5cm} + \|f\|_\infty \cdot n \cdot \max_{x\in[n]} \, {\ensuremath{\mathbb P}} _\o\Big( \tau_x(\o) > t \Big) {{\Big\}}}, \end{align} where ${\ensuremath{\mathbb P}} _\o(\cdot)$ and ${\ensuremath{\mathbb E}} _\o(\cdot)$ denote the law and associated expectation of the chain $\{\o(t)\}_{t\,\geqslant\, 0}$ with $\o(0)=\o$.
If $\eta(\o)=\{\eta_x(\o)\}_{x\in [n]}$ denotes the collection of the 0-1 variables $\eta_x=\mathds 1_{\{\o_x\in S^g_x\}}$ and $\hat \tau_x(\eta)$ is the hitting time of the set {{$\big\{ \eta' : \ \eta'_x \ne \eta_x \big\}$,}} then $\big\{ \tau_x(\o) > t \big\} \subset \big\{ \hat \tau_x(\eta(\o)) > t \big\}$, and hence ${\ensuremath{\mathbb P}} _\o\big( \tau_x(\o) > t \big) \, \,\leqslant\, \, {\ensuremath{\mathbb P}} _\o\big( \hat\tau_x(\eta(\o)) > t \big)$. Notice that $\eta(t)\equiv \eta(\o(t))$ is itself a Markov chain whose law $\tilde{\ensuremath{\mathbb P}} _{{\eta}}(\cdot)$ coincides with that of either the non-homogeneous East chain or the non-homogeneous FA-1f chain, depending on the chain described by $P_t$. Therefore, ${\ensuremath{\mathbb P}} _\o\big(\hat\tau_x(\eta) > t \big)=\tilde {\ensuremath{\mathbb P}} _\eta\big(\hat\tau_x(\eta) > t\big),$ where $\eta\equiv \eta(\o)$. Letting $\tilde \nu=\text{Ber}(\alpha_1)\otimes\dots\otimes \text{Ber}(\alpha_n),$ we have that $\tilde \nu$ is the reversible measure for the $\eta$-chain and that \begin{align*} \tilde {\ensuremath{\mathbb P}} _\eta\big( \hat\tau_x(\eta) > t \big) & \, \;\leqslant\; \, \frac{1}{\min_\eta \tilde\nu(\eta)}\sum_{\eta'} \tilde\nu(\eta'){{\tilde {\ensuremath{\mathbb P}} }}_{\eta'}\big(\hat\tau_x{{(\eta')}} > t \big)\\ & \, \;\leqslant\; \, \begin{cases} 2q^{-n} \exp\big( -tq / T_{\text{East}}(n,\bar \alpha) \big) & \text{for the East process,}\\ 2q^{-n} \exp\big( - tq / T_{\text{FA}}(n,\bar \alpha) \big) & \text{for the FA-1f process},
\end{cases} \end{align*} where the factor $q^{-n}$ comes from $\tilde\nu(\eta) \;\geqslant\; q^n$ and the exponential bounds follow from~\cite{Praga}*{Theorem 4.4}. In particular, the inverse of the exponential rate of decay (in $t$) of the second term in the r.h.s.~of \eqref{eq:A1} is smaller than $T_{\text{East}}(n,\bar\alpha)/q$ or $T_{\text{FA}}(n,\bar\alpha)/q,$ depending on which of the two models we are considering.
We now analyse the first term in the r.h.s.~of \eqref{eq:A1}. Conditionally on {{the event}} $\bigcap_x \big\{ \tau_x{{(\o)}} < t \big\}$ and on {{the vector}} $\eta(t) \, {{\in \{0,1\}^n}}$, the variables $\big\{ \o_x(t) : x\in [n] \big\}$ are independent with law $\otimes_{x\in [n]}\nu_x\big( \cdot\thinspace |\thinspace \eta_x(t) \big)$. Thus, if $g(\eta'):=\nu\big( f \thinspace |\thinspace \eta' \big)$, then \begin{align*} {\ensuremath{\mathbb E}} _\o\Big( f\big( \o{{(t)}} \big) \cdot \mathds 1_{\{\tau_x(\o) \, < \, t\ \forall x\}} \Big) & \, = \, {\ensuremath{\mathbb E}} _\o\Big( g\big( \eta(t) \big) \cdot \mathds 1_{\{\tau_x(\o) \, < \, t\ \forall x\}} \Big)\\
& \, = \, \tilde P_t\,g(\eta) -\, {\ensuremath{\mathbb E}} _\o\Big( g\big(\eta(t) \big) \cdot \mathds 1_{\{\max_x \tau_x(\o) \, > \, t\}} \Big) \end{align*} where {{$\tilde P_t\,g(\eta) \equiv \tilde {\ensuremath{\mathbb E}} _\eta\big( g \big(\eta(t)\big) \big) = {\ensuremath{\mathbb E}} _\o\big(g\big(\eta(t)\big)\big)$}}. The last term in the r.h.s.~above can be analysed exactly as the second term in the r.h.s.~of \eqref{eq:A1}. {{Moreover, by the}} Cauchy--{{Schwarz}} inequality and \eqref{eq:relax:exp:decay}, the first term satisfies \[
\|\tilde P_t\,g\|_\infty \;\leqslant\; \frac{1}{\min_\eta \tilde\nu(\eta)} \operatorname{Var}_{\tilde \nu}\big(\tilde P_t\,g\big)^{1/2} \;\leqslant\; \frac{1}{q^n}e^{-\lambda t}\operatorname{Var}_{\tilde \nu}(g)^{1/2}, \] where $\lambda$ is either $T_{\text{East}}(n,\bar\alpha)^{-1}$ or $T_{\text{FA}}(n,\bar\alpha)^{-1}$ depending on the chosen model. This proves~\eqref{eq:A0}, and hence the proposition. \qed
\subsection{Proof of the scaling \eqref{eq:scaling}}
Recall that {{$q := \min\big\{ 1 - \alpha_x : x \in [n] \big\}$,}} and let $T_{\text{\tiny East}}\big( n,q\big)$ {{and}} $T_{\text{\tiny FA}}\big(n,q\big)$ be the relaxation times {{of}} the {{homogenous East and FA-1f chains}} on $[n]$ with parameters {{$\alpha_x = 1-q$}} for each ${{x}} \in [n]$. It {{was proved}} in~\cites{CMRT,CFM} that \[ T_{\text{\tiny East}}\big( n,q\big)= q^{-O(\min\{ \log n, \,
\log(1/q) \})} \quad
\text{and}\quad T_{\text{\tiny FA}}\big( n,q\big) = q^{-O(1)}. \] Thus, it will {{suffice}} to prove that \[ T_{\text{\tiny East}}(n,\bar\alpha)= {{\frac{1}{q} \, \cdot \,}} T_{\text{\tiny East}}(n,q)\quad\text{ and }\quad T_{\text{\tiny FA}}(n,\bar\alpha)= {{\frac{1}{q} \, \cdot \,}} T_{\text{\tiny FA}}(n,q). \] For simplicity we only treat the East case, since the FA-1f case follows by exactly the same arguments.
{{Consider the generalized East chain on $\O=[0,1]^n$}} in which each vertex ${{x}} \in [n],$ with rate one and independently across $[n]$, is resampled from the uniform measure on $[0,1]$ if either {{$x \,\leqslant\, n-1$ and $\o_{x+1} \,\geqslant\, 1 - q$, or $x = n$.}} The chain is reversible w.r.t.~the uniform measure $\nu$ on $\O$ and, {{by}} Proposition \ref{lem:gen-Poincare}, {{we have}} \begin{equation}\label{eq:prop:app:in:app:B} \operatorname{Var}_\nu(f) \;\leqslant\; \frac{1}{q} \cdot T_{\text{\tiny East}}(n,q) \cdot \sum_{{{x}} = 1}^n \nu\big( \vec c_{{x}}\operatorname{Var}_{{x}}(f) \big)
\end{equation} for {{every}} function $f \colon \O\mapsto {\ensuremath{\mathbb R}} $,
{{since $\nu\big( \o_x \,\geqslant\, 1 - q \big) = q$ for each $x \in [n]$. (Recall that $\vec c_x(\o) = \mathbbm{1}_{\{\o_{x+1} \,\geqslant\, 1-q\}}$}} if ${{x}} \,\leqslant\, n-1$, and that ${{\vec c}}_n(\o) \equiv 1$.{{)}}
{{Now, let}} $\eta = \{\eta_{{x}}\}_{{{x \in [n]}}}$ with $\eta_{{x}} := \mathbbm{1}_{\{\o_{{x}} < \alpha_{{x}} \}}$, and, for an arbitrary {{function}} $g \colon \{0,1\}^n \mapsto {\ensuremath{\mathbb R}} $, {{set}} $f(\o) := g\big( \eta(\o) \big)$. {{Note that $\eta_x \,\leqslant\, \mathbbm{1}_{\{\o_x \,\leqslant\, 1-q \}}$ (by the definition of $q$), and that}} the law of the variables $\eta$ w.r.t.~$\nu$ is the product Bernoulli measure $\pi=\text{Ber}(\alpha_1)\otimes\dots\otimes \text{Ber}(\alpha_n)$. {{Therefore, applying~\eqref{eq:prop:app:in:app:B}}} to $f$, we {{obtain}} \[ \operatorname{Var}_\pi(g)= \operatorname{Var}_\nu(f) \;\leqslant\; \frac{1}{q} \cdot T_{\text{\tiny East}}(n,q) \cdot \bigg( \sum_{x=1}^{n-1} \pi\Big( \mathbbm{1}_{\{\h_{x+1}=0\}} \operatorname{Var}_x(g) \Big) + \pi\big( \operatorname{Var}_n(g) \big) \bigg). \] {{The right-hand side of this inequality is exactly $C \cdot \ensuremath{\mathcal D}(g)$, where $C = 1/q \cdot T_{\text{\tiny East}}(n,q)$ and $\ensuremath{\mathcal D}(g)$ is the Dirichlet form of $g$ associated to the generator of the non-homogenous East model. Since $g$ was an arbitrary function, it follows by Definition~\ref{def:PC} that $T_{\text{\tiny East}}(n,\bar\alpha) = 1/q \cdot T_{\text{\tiny East}}(n,q)$, as required. As noted above, the proof that $T_{\text{\tiny FA}}(n,\bar\alpha) = 1/q \cdot T_{\text{\tiny FA}}(n,q)$ is identical.}} \qed
\subsection{Proof of Proposition~\ref{lem:MT:prop34}}
We will deduce the proposition from~\cite{MT}*{Theorem 1}. The deduction is almost exactly the same as that of~\cite{MT}*{Proposition~3.4}, but for completeness we give the details. Set $\ell= \big\lceil \log(1/p_2) \big\rceil$, $L = \big\lfloor 1/p_2^2 \big\rfloor$, and for each $i \in {\ensuremath{\mathbb Z}} ^2$, define \[ C_i(\ell) = \bigcup_{k=0}^\ell \big\{ i + \vec e_2 + k\vec e_1 \big\}. \] Let also $\ensuremath{\mathcal P}_i(\ell,L)$ be the family of oriented paths starting in $C_i(\ell)$ and of length $L$. We define two families of events $\big\{ A_i^{(1)}, A_i^{(2)}\big\}_{i\in {\ensuremath{\mathbb Z}} ^d}$ as follows: \begin{align*}
A_i^{(1)} & = \big\{ \text{$\o_j\in G_1$ for all $j\in C_i(\ell)\cup
\{i+\vec e_1\}\cup \{i+\vec e_2-\vec e_1\}$} \big\},\\ A_i^{(2)} & = \big\{ \text{there exists a good path in $\ensuremath{\mathcal P}_i(\ell,L)$ and the smallest good one is super-good} \big\},
\end{align*} {{where, if there is more than one smallest good path, then we choose the leftmost one.}}
Observe that $A_i^{(1)} \cap A_i^{(2)} \subset \Gamma_i$, {{since $A_i^{(1)}$ implies that the smallest good path in $\ensuremath{\mathcal P}_i(\ell,L)$ starts at $i + \vec e_2$,\footnote{This follows from the observation that the word (of length $L$) obtained from $W \in \{\vec e_1,\vec e_2\}^L$ by adding $\vec e_1$ at the start and removing the final letter is at most $W$ in alphabetical order.} and hence is equal to the smallest path in the definition of $\Gamma_i$.}} We now want to apply~\cite{MT}*{Theorem 1} to the two families of constraints $\big\{ c_i^{(k)} \big\}_{i\in {\ensuremath{\mathbb Z}} ^2}$, where $c_i^{(k)} := \mathbbm{1}_{\{A_i^{(k)}\}}$ for each $k \in \{1,2\}$. To do so, we need to check the following two conditions: \begin{itemize} \item[$(a)$] there exists a two-way infinite sequence of sets $(\ldots,V_{-2},V_{-1},V_0, V_1, V_2, \ldots)$, with $V_n \subset V_{n+1}$ for every $n \in {\ensuremath{\mathbb Z}} $ and $\bigcup_n V_n = {\ensuremath{\mathbb Z}} ^2$, such that if $i \not\in V_n$, then the event $A_i^{(k)}$ is independent of the collection of variables $\big\{ \o_i : i \in V_{n+1} \big\}$; \item[$(b)$] there exists a family $\big\{ \lambda_I : \emptyset \ne I \subset \{1,2\} \big\}$ of positive constants such that the key condition~\cite{MT}*{equation (2.1)} holds. \end{itemize} To see $(a)$, let the sets $V_n$ be all translations of the closed half-space $$\overline{\mathbb{H}}_{(1,2)} = \big\{ x \in \mathbb{Z}^2 : \langle x,(1,2) \rangle \,\leqslant\, 0 \big\}$$ by elements of ${\ensuremath{\mathbb Z}} ^2$ (ordered in the obvious way). Now, observe that if $i \not\in V_n$ then $V_{n+1} \subset \overline{\mathbb{H}}_{(1,2)} + i$, and the event $A_i^{(k)}$ is indeed independent of the variables in $\overline{\mathbb{H}}_{(1,2)} + i$.
To prove $(b)$, set $\lambda_I = 1$ for every non-empty set $I \subset \{1,2\}$, and note that the event $A_i^{(1)}$ depends on $\ell + 3$ variables, and that $A_i^{(2)}$ depends on at most $(L + \ell)^2$ variables. It follows that there exists a constant $\hat \delta > 0$ such that~\cite{MT}*{equation (2.1)} holds if \begin{equation}
\label{eq:A10} \ell \Big( 1 - \mu\big( A_i^{(1)} \big) \Big) + (L + \ell)^2 \Big( 1 - \mu\big( A_i^{(2)} \big) \Big) \;\leqslant\; \hat \delta. \end{equation} We now claim that if the constant $\delta$ of Proposition~\ref{lem:MT:prop34} is chosen to be sufficiently small, then~\eqref{eq:A10} holds. In order to prove this, it is enough to observe that, by the union bound, \[ 1 - \mu\big( A_i^{(1)} \big) \;\leqslant\; (\ell + 3)(1-p_1), \] and that \begin{align*} 1 - \mu\big( A_i^{(2)} \big) & \;\leqslant\; \, \mu\big(\text{there is no good path in } \ensuremath{\mathcal P}_i(\ell,L) \big)\\
& \hspace{1.5cm} + \max_{\gamma \in \ensuremath{\mathcal P}_i(\ell,L)} \mu\big( \gamma \text{ is not super-good} \, \big| \, \gamma \text{ is good} \big)\\ & \;\leqslant\; \, e^{-m(p_1)\ell} + \big( 1 - p_2 \big)^L, \end{align*} with $\lim_{p_1 \rightarrow 1} m(p_1) = \infty,$ by a standard Peierls bound and by the FKG inequality. In conclusion, if $\delta > 0$ is sufficiently small then we may apply~\cite{MT}*{Theorem 1}, which gives \[ \operatorname{Var}(f)\;\leqslant\; 4 \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\Big(\mathbbm{1}_{\{A_i^{(1)}\cap A_i^{(2)}\}}\operatorname{Var}_i(f)\Big) \;\leqslant\; 4 \sum_{i\in {\ensuremath{\mathbb Z}} ^2}\mu\Big(\mathbbm{1}_{\Gamma_i}\operatorname{Var}_i(f)\Big), \] where the final inequality holds because $A_i^{(1)} \cap A_i^{(2)}\subset \Gamma_i$.
\qed
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A distributional regression approach to income-related inequality of health in Australia
Roselinde Kessels ORCID: orcid.org/0000-0002-4534-00471,2,
Anne Hoornweg3,
Thi Kim Thanh Bui2,4 &
Guido Erreygers2,5
Several studies have confirmed the existence of a significant positive relationship between income and health. Conventional regression techniques such as Ordinary Least Squares only help identify the effect of the covariates on the mean of the health variable. In this way, important information of the income-health relationship could be overlooked. As an alternative, we apply and compare unconventional regression techniques.
We adopt a distributional approach because we want to allow the effect of income on health to vary according to people's health status. We start by analysing the income-health relationship using a distributional regression model that falls into the GAMLSS (Generalized Additive Models for Location, Scale and Shape) framework. We assume a gamma distribution to model the health variable and specify the parameters of this distribution as linear functions of a set of explanatory variables. For comparison, we also adopt a quantile regression analysis. Based on predicted health quantiles, we use both a parametric and a non-parametric approach to estimate the lower tail of the health distribution.
Our data come from Wave 13 of the Household, Income and Labour Dynamics in Australia (HILDA) survey, collected in 2013-2014. According to GAMLSS, we find that the risk of ending up in poor, fair or average health is lower for those who have relatively high incomes ($80,000) than for those who have relatively low incomes ($20,000), for both smokers and non-smokers. In relative terms, the risk-lowering effect of income appears to be the largest for those who are in poor health, again for both smokers and non-smokers. The results obtained on the basis of quantile regression are to a large extent comparable to those obtained by means of GAMLSS regression.
Both distributional regression techniques point in the direction of a non-uniform effect of income on health, and are therefore promising complements to conventional regression techniques as far as the analysis of the income-health relationship is concerned.
Whether we look at regional, national or even global data, we always find that income and health are unequally distributed: some of us are rich, while others are poor; some of us live long and healthy lives, while others suffer and die young. We also know that income and health tend to be positively correlated, in such a way that higher income levels are often associated with better health outcomes, a phenomenon often referred to as the 'social gradient'. Yet, the exact nature of the income-health relationship is complex. A lack of income reduces the options to lead a healthy lifestyle, and therefore constitutes an important determinant of the often observed social gradient in healthy behaviours [1]. It also acts as a barrier for access to health care, which may be conducive to bad health. Even in countries that have universal health care coverage, such as Australia, affordability remains a barrier for access to health care [2]. In reverse, bad health may be a factor contributing to job loss and therefore lead to low income [3].
Econometric research on this issue is often based on conventional regression techniques, which focus on the explanation of the mean, i.e. the expected value, of the dependent variable. In this paper, we explore and compare two alternative regression techniques, which allow for the possibility that the income-health relationship differs according to the location in the distribution, e.g. different for those who are in good health and for those who are in bad health. More specifically, we will use a recently developed 'distributional' regression technique in the form of generalized additive models [4] as well as the increasingly popular quantile regression method [5]. Our aim with this paper is to get a better understanding of the full spectrum of the income-health relationship, rather than trying to measure the social gradient by means of an index number (e.g., [6]) or to explain it by means of a regression-based decomposition (e.g., [7]).
Based on a variety of income measures and health variables, several studies have confirmed the existence of a significant positive relationship between income and health (e.g., [8–11]). As pointed out by Silbersdorff et al. [12], the majority of studies use linear and generalized linear models, which assess the effect of variations in independent variables on the expected level of the dependent variable. With regard to the income-health relationship, this means that variations in income are solely related to the conditional mean health outcome. In this way, important information of the income-health relationship could be overlooked. Although the general trend of the relationship might be captured, aspects such as differences in variance or skewness in the distribution are neglected.
To account for the fact that the expectation of a variable is not necessarily representative of its entire distribution, the focus in income-health relationship studies has shifted to unconventional regression approaches. Unconventional regression methods provide a more complete picture of distributional characteristics, since these methods look at effects beyond the mean. However, the amount of studies exploring these unconventional regression methods is still limited, and they mostly focus on one specific regression technique only. In this paper, we employ two unconventional regression techniques to examine the income-health relationship, using Australian data collected in 2013-2014. We study the conditional health distribution by means of the distributional regression method designated by the acronym GAMLSS, which stands for generalized additive models for location, scale and shape [4], and by means of quantile regression [5]. We are especially interested in the effect of income on the probabilities of ending up in bad health.
The income-health relationship
The direction of causality constitutes an important issue in the literature on income and health, with wide-ranging implications for public policies. One strand of this literature studies the relationship at the country level, inspired by the work of Preston [13] and Deaton [14]. The famous Preston curve, representing a positive, concave relationship between GDP (Gross Domestic Product) per capita and life expectancy, strongly suggests that increases in average income are among the driving forces of increases in population health. Others have looked at the relationship between per capita income and per capita health expenditures. Erdil and Yetkiner [15], for instance, adopted a panel data regression approach to study this relation. Using Granger causality tests, they found that in most cases the causal relationship is bidirectional. When they found one-way causality, it tended to be from GDP to health expenditures in low- and middle-income countries, but the other way around in high-income countries. Another area of cross-country research is about the relationship between income inequality and population health. Babones [16], for instance, claimed that the evidence for the existence of a strong, statistically significant causal relation between income inequality and health is rather weak. By contrast, based on an extensive literature study and nine criteria for causality, Pickett and Wilkinson [11] argued that as far as developed countries are concerned the available evidence strongly indicates the existence of a causal relation between income inequality and health: high levels of income inequality lead to low levels of health (e.g., life expectancy).
However, most of the literature is about the relation between income and health at the individual level, and this is also what we will be focusing on in this paper. As far as Australia is concerned, several studies have used survey data to estimate the social gradient of health by means of the concentration index and to compare Australia to other countries [17–19]. Various decomposition techniques have been applied in order to come to a better understanding of the underlying mechanisms [7, 19, 20]. For other countries, more complex econometric methods have been tried. A good example is the study by Frijters, Haisken-DeNew and Shields [21], who looked at the effect of household income on individual health satisfaction in East Germany, using a fixed-effects ordered logit model. Even though they found some evidence for a positive effect, they emphasized that it was very small. In another study, Kuehnle [22] focused on the relation between household income and child health, choosing an instrumental variable (IV) approach to control for the potential endogeneity of income. He too arrived at the conclusion that income has a positive effect on health, albeit a very small one. These studies illustrate that the nature of the relationship is complex: income influences health, health influences income, and other factors influence both income and health. Advanced econometric techniques are required to disentangle the multiple aspects of the income-health relationship, and even then clear-cut statements about causality may be difficult to obtain.
Most empirical studies on the effect of income on health have used ordinary least square and IV approaches, generalized linear models, and comparisons of correlation coefficients [8–10]. However, Silbersdorff et al. [12] argue that conventional regression techniques do not provide reliable estimates of the income-health relationship. Conventional regression techniques assess the effect of variations in covariates on the expected level of the dependent variable. Following a different approach, Silbersdorff et al. [12] show that in Germany the effect of income on health is not uniform throughout the whole distribution. In particular, relatively high probabilities of outcomes in the left tail of the health distribution are observed for the part of the sample with low income levels. This cannot be revealed by conventional regression methods.
In this paper, we study the income-health relationship by means of two alternative regression techniques, GAMLSS and (conditional) quantile regression, which both estimate conditional distribution models. One of our aims is to investigate whether the results obtained by these models are similar. A comparison with other recently developed regression techniques, such as unconditional quantile regression, also known as recentered influence function (RIF) regression [23], is left to future research.
GAMLSS
One way to examine a more complete health distribution is via GAMLSS, as introduced by Rigby and Stasinopoulos [4]. Relatively to mean regression techniques, GAMLSS aim to describe the full conditional distribution of the dependent variable by estimating not only the mean but also other distributional characteristics such as variance and skewness. When a regression of health is conducted via GAMLSS, all parameters of the health distribution are linked to a set of explanatory variables. In their paper, Hohberg, Pütz and Kneib [24] provide a clear guidance into GAMLSS. If we consider a population of N individuals (\(i = 1,\dots, N\)) and a health variable h, GAMLSS assume that the observed hi are conditionally independent and described by a parametric distribution:
$$ f(h_{i}|y_{i},X_{i}) = f(h_{i}|\theta_{1}(y_{i},X_{i}),\dots,\theta_{L}(y_{i},X_{i})) $$
where \(\theta _{1},\dots,\theta _{L}\) are L different parameters of the distribution that are conditional on income yi and other socioeconomic variables contained in the vector Xi. Each parameter θl, \(l = 1,\dots, L\), is connected to a regression predictor \(\eta ^{\theta _{l}}\phantom {\dot {i}\!}\), also conditional on yi and Xi, via a link function gl such that \(\theta _{l} = g_{l}^{-1}(\eta ^{\theta _{l}})\). In our setting, the predictor function takes on the following additive form:
$$ \eta^{\theta_{l}}(i) = \beta_{0}^{\theta_{l}} + \beta_{1}^{\theta_{l}} y_{i} + \beta_{2}^{\theta_{l}} X_{i} $$
where \(\beta _{0}^{\theta _{l}}\), \(\beta _{1}^{\theta _{l}}\) and \(\beta _{2}^{\theta _{l}}\) denote the regression coefficients for predicting θl. By choosing an appropriate link function, it can be ensured that restrictions of the parameter space are fulfilled (e.g., a log link to ensure positive standard deviations). After having selected the distribution, predictor setup and link functions, the unknown regression coefficients can be estimated by maximizing a penalized likelihood function in a classical frequentist approach [4] or by Bayesian methods [25].
The use of GAMLSS requires the specification of a parametric distribution that approximates the observed health outcomes. In our application described in the Results section, we use a continuous health score, the distribution of which is negatively skewed. As stated in [12] and [26], continuous health distributions generally exhibit a negative skewness, whereas most parametric specifications are suited for the more common symmetric or positively skewed distributions. To be able to work with a health distribution that is positively skewed, we linearly transform the health score h as follows:
$$ h^{*} = \frac{h_{0} - h}{h_{scale}} $$
where h0 is a constant ensuring that the transformed health score h∗ has positive support and hscale is a rescaling factor. The health score in our application is a bounded variable on the unit interval [ 0,1]. To obtain positive skewness in the distribution, we use h0=1.0001 and hscale=1, so that transformation (3) becomes:
$$ h^{*} = 1.0001 - h $$
where h∗ is restricted to the interval [ 0.0001,1.0001]. As recommended by Silbersdorff et al. [12] and Silbersdorff and Schneider [26], we use the two-parameter gamma distribution to approximate the transformed health score. This distribution provides a sufficiently good fit compared to other distributions, such as the two-parameter Weibull and lognormal distributions, as we show in Appendix 1. We obtain the conditional distribution of the untransformed health score by calculating the inverse transformation. Note that the two-parameter gamma distribution for modelling the transformed health score is bound by zero, h∗>0, and positively skewed. Given the location parameter μ>0 and the scale parameter s>0, the distribution can be written as
$$ f(h^{*}|\mu,s) = \frac{h^{*^{\frac{1}{s^{2}}-1}}\text{exp}({-h^{*}/(s^{2}\mu)})}{(s^{2}\mu)^{(1/s^{2})}\Gamma \left(1/s^{2}\right)} $$
where Γ denotes the gamma function. Also, E(h∗)=μ and Var(h∗)=σ2=s2μ2. This expression of the gamma distribution corresponds to the GA formulation provided by Rigby et al. [27] (see p. 271).
Because we focus on the gamma distribution with parameters μ and s, the corresponding link functions are logarithmic to ensure the positivity of the two parameters. Also, in our application, the functional form of the predictors for the two parameters is the same as in Eq. (2). Hence, the predictor setups and link functions are as follows:
$$ log(\mu) = \beta_{0}^{\mu} + \beta_{1}^{\mu} y + \beta_{2}^{\mu} X $$
$$ log(s) = \beta_{0}^{s} + \beta_{1}^{s} y + \beta_{2}^{s} X $$
Because we adopt a standard modelling approach that is similar to a previous application in [26], we follow the frequentist estimation framework provided by the GAMLSS package [28] in the statistical software R 3.6.2 [29]. We refer the reader interested in a Bayesian implementation of GAMLSS that can deal with more advanced modelling situations to applications in [30] and [12].
Quantile regression
Another regression approach that goes beyond the mean is quantile regression [31]. In quantile regression, the estimation of a conditional mean function is replaced by estimations of different conditional quantile functions. Koenker and Hallock [5] define conditional quantile functions as models that express quantiles of the conditional distribution of the dependent variable as functions of independent variables. For our health variable h, characterized by its cumulative distribution function (CDF), Fh(hj)=P(h≤hj), the τth quantile, 0<τ<1, is defined as
$$ h^{\tau} = \inf\{h_{j}: F_{h}(h_{j}) \geq \tau\} $$
Alternatively, the τth quantile is defined as the solution satisfying the inequalities:
$$ h^{\tau} \leq h_{j} \quad \text{if and only if} \quad F_{h}(h_{j}) \geq \tau $$
To understand the income-health relationship, a quantile regression approach is used to describe the entire conditional distribution of hi by quantile functions of income yi and a set of control variables Xi in a form similar as in Eq. (2):
$$ h_{i}^{\tau}(y_{i}, X_{i}) = \beta_{0}^{\tau} + \beta_{1}^{\tau} y_{i} + \beta_{2}^{\tau} X_{i} + \epsilon_{i}^{\tau} $$
where \(\beta _{0}^{\tau }\), \(\beta _{1}^{\tau }\) and \(\beta _{2}^{\tau }\) denote the regression coefficients for the τth quantile and \(\epsilon _{i}^{\tau }\) is the error term.
The conditional quantile functions (10) are estimated in the same manner as the conditional mean function in least squares regression. However, instead of minimizing the sum of squared residuals over a sample of N observations, in quantile regression a sum of asymmetrically weighted absolute residuals is minimized:
$$ \begin{aligned} &{\min_{\beta_{0}^{\tau}, \beta_{1}^{\tau}, \beta_{2}^{\tau}} \sum\limits_{i=1}^{N} \rho_{\tau}\left(h_{i} - (\beta_{0}^{\tau} + \beta_{1}^{\tau} y_{i} + \beta_{2}^{\tau} X_{i})\right)}\ \\ &\text{with}\ \rho_{\tau}(u) = \left\{ \begin{array}{ll} u\tau & \text{for}\ u\ \geq\ 0\\ u(\tau - 1) & \text{for}\ u\ <\ 0\end{array} \right. \end{aligned} $$
The function ρτ(u) is called the check function which is a loss function that retrieves the τth health quantile. Since we assume that h is linear in the regression coefficients, the minimization problem can be solved efficiently via linear programming methods. We use the implementation in the R package quantreg [32] to obtain the estimated coefficients. For given values of y and X, the predicted health quantile \(\hat {h}^{\tau }\) can then be identified as
$$ \hat{h}^{\tau}(y,X) = \hat{\beta}_{0}^{\tau} + \hat{\beta}_{1}^{\tau}y + \hat{\beta}_{2}^{\tau}X $$
In our application, we focus specifically on the value of \(\hat {\beta }_{1}^{\tau }\), which is the estimated marginal effect of income on health, given that the observation is, and remains in, quantile τ.
Comparison of quantile regression to GAMLSS
To compare quantile regression to GAMLSS, we consider an individual with given values for y and X. The GAMLSS approach estimates the parameters of the corresponding gamma distribution. The quantile regression approach provides the predicted health quantiles \(\hat {h}^{\tau _{k}}(y,X)\) for different values of τk, \(k = 1,\dots,K\). We can use these predicted quantiles to estimate the parameters of a gamma distribution by sorting them in an empirical CDF. The CDF of a gamma distribution is continuous and strictly monotonically increasing, such that the inequalities in expression (9) can be replaced by equalities. Assuming a gamma distribution, the predictions \(\hat {h}^{\tau _{k}}(y,X)\) thus result in K estimated values of the theoretical CDF:
$$ F_{h}(\hat{h}^{\tau_{k}}) = P(h \leq \hat{h}^{\tau_{k}}) = \tau_{k} \quad \text{for}\ k = 1,\dots,K $$
Inversely, \(\hat {h}^{\tau _{k}} = F^{-1}_{h}(\tau _{k})\).
Because in our application the gamma distribution is used to model the transformed health score h∗=1.0001−h, we estimate the theoretical CDF from the transformed values of the predicted health quantiles \(\widehat {h}^{*}_{k} = 1.0001-\hat {h}^{\tau _{k}}\), as given by
$$ P(h^{*} \leq \widehat{h}^{*}_{k}) = 1 - \tau_{k} \quad \text{for}\ k = 1,\dots,K $$
We obtain the two parameters of the corresponding gamma distribution, μ and s, by minimizing the sum of squared residuals between the empirical and theoretical CDFs:
$$ \min_{\mu,s} \sum\limits_{k=1}^{K} \left(1 - \tau_{k} - F_{h^{*}}\left(\widehat{h}^{*}_{k},\mu,s\right)\right)^{2} $$
where \(F_{h^{*}}(\widehat {h}^{*}_{k},\mu,s)\) is the theoretical CDF of the gamma distribution for \(\widehat {h}^{*}_{k}\). We can now compare the estimated gamma distribution from quantile regression to the one directly obtained from GAMLSS.
We compare the two distributional regression methods, GAMLSS and quantile regression, for analysing the income-health relationship in an application using Australian data. We first describe these data after which we report on the results.
We examine the income-health relationship in Australia using data from the Household, Income and Labour Dynamics in Australia (HILDA) survey. This survey is a nationally representative household-based panel study, containing observations on individuals aged 15 years or older. Since the start of the survey in 2001, it is repeated annually with the aim to follow the same group of residents over the course of their lives. The survey includes questions about income and employment, household and family relationships, and personal well-being. Similarly as in a previous study [7], data from Wave 13, collected in the years 2013-2014, are used. In total, the sample consists of 14,728 individuals. In what follows, we provide a description of the variables retained for analysis. We begin with our response variable, the health score, followed by our main independent variable, income, after which we highlight the control variables.
To be able to conduct standard quantile regression, a continuous response variable is required. We use the SF-6D health score, which is a health state classification composed of six health dimensions. It is a preference-based single index measure of health, bounded on the unit interval [0, 1], that can be used for economic evaluations [33]. Figure 1 shows the histogram of the SF-6D (Short-Form Six-Dimension) health score, which is negatively skewed. Silbersdorff et al. [12] and Silbersdorff and Schneider [26] argued that continuous health measures typically have negatively skewed distributions.
Histogram of the SF-6D health score
The main explanatory variable of interest is the logarithm of income, and in particular equivalized income, which is commonly used in similar research [12, 34]. Equivalized income is calculated using the OECD-modified equivalence scale. This scale assigns an equivalence factor to each household type in proportion to the household's needs. The equivalence factor depends on the size of the household and the ages of its members. A value of 1 is assigned to the first adult of the household, a value of 0.5 to each additional adult and a value of 0.3 to every child. The equivalence factor is the sum of these values. Dividing the household's disposable income by the equivalence factor equals the equivalized income for each household member.
In addition to income, we included a set of control variables in the regressions that turned out to affect the health outcomes. Typical variables that emerge from the existing literature on income and health are gender, age, ethnicity, occupational class, marital status and the number of children [8, 9, 21, 35]. We incorporated all these variables except gender and marital status because they were not significant in the estimation of the gamma distribution via GAMLSS. That is to say, gender and marital status had almost no discernible effect on the health score distribution, neither on their own nor in interaction with income or any other covariate. Instead, we added several other variables to the analysis that significantly improved the estimation of the gamma distribution. These variables describe lifestyle and individual health characteristics: sleep quality, physical activity, smoking, time stress, life satisfaction and satisfaction with weight.
A complete list of the variables used and their descriptive statistics are presented in Table 1. We treat the variables income, age, number of children aged 0-4 years and aged 5-14 years, and life satisfaction as numerical. We also specify an individual's age nonlinearly in the regression models using a squared term that we mean-center to remove multicollinearity with the linear term. All other variables are categorical and enter into the regressions as dummy variables. In this respect, we define the reference person as a non-indigenous person who provides a professional service, has a fairly good sleep quality, does frequent physical activity, does not smoke, is sometimes stressed for time and is neither satisfied/dissatisfied with one's weight.
Table 1 Descriptive statistics of the sample (N= 14,728)
GAMLSS results
The estimated GAMLSS regression model generates coefficients for the predictor functions (6) and (7) with which we can obtain gamma distributions for different types of individuals. More specifically, we use these gamma distributions to estimate the effect of income on the probability of ending up in low health. We therefore consider individuals with two possible income levels: a 'low income' equal to 20,000 AUD (Australian dollar) and a 'high income' equal to 80,000 AUD. The low income roughly corresponds to the 10th percentile of the income distribution, whereas the high income roughly corresponds to the 90th percentile. For the other covariates, we set the levels equal to the means in the case of numerical variables and equal to the reference categories in the case of categorical variables. When it comes to smoking, however, we consider two possible categories: we look at the income-health relationship for both non-smokers (the reference category) and smokers. To calculate and compare the GAMLSS results, we follow a similar structure as in [12] and [26].
Table 2 shows the estimated covariate effects on the two parameters (or more precisely, on the logarithms of these parameters) of the gamma distribution, fitted to the transformed health score as defined by Eq. (4). All covariates have a significant impact on μ at the 5% level, whereas some covariates are insignificant for predicting s. Because most covariates significantly influence both μ and s, their relationship to the transformed health score goes beyond the mean. The main variable of interest, log(income), has a significant negative impact on log(μ), but does not affect log(s). The negative relationship with respect to the mean transformed health score confirms previous findings that income has a positive impact on health. As a matter of fact, if we wish to know the effect of the covariates on health, we need to reverse the sign of the coefficients predicting μ.
Table 2 Linear effects on log(μ) and log(s) for the transformed health variable
Since the link functions that are used for the predictors are logarithmic instead of linear, covariate effects on the parameters vary across the covariate space. This implies that the impact on μ and s of a change in income depends on the values of all covariates. To analyse the distributional effect of income on health, we retrieve the parameters of the gamma distribution of the transformed health score for the reference person. That is, we use the means as levels for the numerical variables and the reference categories for the categorical variables. It is worth bearing in mind that the results might be different if we chose another reference person. Figure 2 shows a histogram of the SF-6D health scores together with the transformed gamma distribution when income is set equal to its sample mean.
Histogram of the SF-6D health score with fit to the transformed gamma distribution for the reference person
We analyse the effect on the SF-6D health score when income is changed from 20,000 AUD to 80,000 AUD via five distributional measures. Next to the expectation and standard deviation of the estimated conditional health distribution, we compute three measures that focus on the lower end of the health distribution. We consider three thresholds for health, set at what we consider to be 'average health' (h≤0.8), 'fair health' (h≤0.7) and 'poor health' (h≤0.6). These levels correspond to the 50th, 30th and 10th percentiles of the empirical distribution of the SF-6D health scores. Formally, our measures estimate the risk or probability that a person with income y and covariates X attains a health outcome below these thresholds:
$$\begin{array}{*{20}l} P_{\text{\scriptsize avg}} = P(h \leq 0.8) \\ P_{\text{\scriptsize fair}} = P(h \leq 0.7) \\ P_{\text{\scriptsize poor}} = P(h \leq 0.6) \end{array} $$
where the health variable h follows the conditional distribution for a person with income y and covariates X.
Table 3 shows the selected health distribution and risk statistics for smokers and non-smokers. For both of these groups, a higher income is associated with a higher expected health level, a lower dispersion of health and smaller probabilities of ending up in bad health. Comparing smokers to non-smokers, we observe that the profile of rich smokers is similar to the one of poor non-smokers. With respect to the risks of ending up in average, fair or poor health, we find that overall the lower the health threshold, the smaller the effect in absolute terms, but the larger in relative terms. The absolute differences appear to be slightly larger for smokers than for non-smokers. The relative differences, however, show an opposite pattern.
Table 3 Five measures on the income-health relationship from fitting a gamma distribution to the transformed health variable using GAMLSS regression
On the whole, the different distributional measures indicate that the income-health relationship is stronger at the lower end of the health distribution. Only looking at the relative difference in expected health scores, i.e. 1.53% for smokers and 1.43% for non-smokers, the effect of income appears to be rather modest. However, when comparing the relative difference in the risk of having average health, i.e. 9.48% for smokers and 10.38% for nonsmokers, to the relative difference in the risk of having poor health, i.e. 25.80% for smokers and 27.38% for non-smokers, we observe that income has a much larger effect at the bottom of the health distribution. Figure 3 illustrates the effect of income on the risk of poor health for both smokers and non-smokers. All of this indicates that the association between income and health goes beyond the mean.
Conditional health distributions obtained by means of GAMLSS regression for smokers (top) and non-smokers (bottom) with incomes of $20,000 (left) and $80,000 (right)
To assess the robustness of our GAMLSS results based on the gamma distribution, we repeated the GAMLSS analysis using two other distributions, the Weibull and lognormal distributions. We present the results for the three distributions in Appendix 1. On the whole, the conclusions on the income-health relationship remain the same across all three distributions. However, there are some slight differences. The Weibull model seems to have a good overall fit with the health scores, while the lognormal model appears to mimic the lower part of the empirical health distribution reasonably well. The gamma model turns out to be the compromise model that closely follows the Weibull model on overall goodness-of-fit and the lognormal model on goodness-of-fit of the lower empirical health scores.
Quantile regression results
We analyse the relation between the health variable h and its covariates in nine quantile regressions for quantiles \(0.1, 0.2,\dots,0.9\). We first estimate the coefficients, in particular of income and smoking, for the nine quantiles. Thereafter, we highlight changes in the predicted health levels for the quantiles due to an income increase from 20,000 AUD to 80,000 AUD and due to smoking.
Table 10 in Appendix 2 displays the estimated regression coefficients for the nine different quantiles. For comparison, it also contains the effects from OLS (Ordinary Least Squares) regression. We observe that most variables have a significant influence on health in each of the regressions at the 5% level. Moreover, the quantile coefficients vary considerably across different quantiles. For the main covariates of interest, log(income) and smoking, the regression coefficients are visualized in Fig. 4a and b.
OLS and quantile regression estimates with 95% confidence intervals for the effect of log(income) and smoking
In both panels a and b, the continuous red line represents the OLS estimate of log(income) and smoking, respectively. The dashed red lines represent the 95% confidence interval of the OLS estimate. The nine estimates of the quantile regressions are connected by the black line and the 95% confidence intervals of the quantile regression estimates are shaded grey. Figure 4a shows that the quantile regression coefficients of log(income) are larger than the OLS estimate in the lower quantiles of health and smaller in the higher quantiles. This suggests that for the lower tail of the health distribution, OLS regression understates the impact of income on health, whereas for the upper tail of the health distribution, OLS regression overstates the impact of income on health. Similarly, Fig. 4b shows that the quantile regression coefficients of smoking tend to be somewhat larger in absolute magnitude than the OLS estimate in the lower quantiles of health and smaller in the higher quantiles. Changes in income and smoking therefore affect health in a way that is not fully captured by the conditional mean model.
To illustrate the effect of log(income) on health, Table 4 presents the changes in the predicted health quantiles due to an income increase from 20,000 AUD to 80,000 AUD for smokers and non-smokers. Figure 5a and b visualize these values for the two groups. On the whole, people with a high income tend to attain better health outcomes than people with a low income, but the difference in the predicted health values, in both absolute and relative terms, decreases as one's health improves. Also, comparing smokers to non-smokers, we find that the health profile of rich smokers is fairly similar to that of poor non-smokers, a conclusion we already reached on the basis of the GAMLSS regression results discussed previously.
Predicted health quantiles for smokers and non-smokers with incomes of $20,000 and $80,000 and reference values for the other covariates
Table 4 Illustration of the effect of log(income) on nine predicted health quantiles
Comparison of regression results
We compare the results we previously obtained by means of the GAMLSS technique in two different ways to the quantile regression results, the first of which we call empirical (or non-parametric), and the second parametric. We construct the empirical distribution by estimating a thousand quantile regressions for quantiles \(0.001, 0.002,\dots,0.999\) to obtain a better approximation of the distribution of health. This allows us to compute the distributional measures Pavg, Pfair and Ppoor, as defined above, for low and high income earners as well as for smokers and non-smokers. The second method consists of estimating the parameters of a gamma distribution based on the predicted health quantiles instead of the observed health outcomes. Using the gamma distributions for smokers and non-smokers, we calculate the five distributional measures for the income-health relationship.
Table 5 contains the three risk measures from estimating the health distribution using a thousand quantile regressions, without assuming a parametric distribution for the health variable. Overall, we find results that are similar to those from GAMLSS in Table 3. An income increase reduces the probability of ending up in bad health, where the absolute effect is smaller for smaller threshold values for health, and the relative effect larger. Relatively speaking, the income-health relationship is thus stronger at the lower end of the health distribution. This distributional effect is even more pronounced for the quantile regression method than for GAMLSS, because the absolute and relative differences between the risk estimates are generally larger. Comparing smokers to non-smokers, we again notice the similarity in risk profile between a rich smoker and a poor non-smoker. Also, the absolute differences are larger for smokers than for non-smokers. This holds for the relative differences too, whereas we observed the opposite from using GAMLSS.
Table 5 Three risk measures on the income-health relationship from using the empirical distribution of 1000 predicted health quantiles
The parametric way to obtain the risk measures from quantile regression is to use the predicted health quantiles for given values of the covariates for the estimation of a gamma distribution. In comparison to the empirical approach, the parametric approach requires much fewer quantile regressions. In general, we found that nine predicted health quantiles are enough to obtain robust estimates of the two parameters of the gamma distribution. Using more predicted health quantiles (e.g., 19 rather than 9) leaves the estimates of the parameters virtually unchanged. As an illustration, Fig. 6a and b contain the histograms of 9 and 19 predicted health quantiles with fit to the transformed gamma distributions for the reference person, the parameters of which are the same up to the third or fourth decimal place.
Histograms of 9 and 19 predicted health quantiles with fit to the transformed gamma distributions, described by E(h) and σ, for the reference person
The distributional measures based on nine predicted health quantiles appear in Table 6, but are representative of many more quantiles. In general, the findings are again similar to the GAMLSS results in Table 3, confirming that low-income earners bear a greater health risk at the bottom of the health distribution. One difference, however, is that the standard deviation of the gamma distribution is smaller when obtained from the predicted health quantiles. Also, the differences between the risk estimates of bad health are very large, even larger than those obtained from the empirical distribution of health regression quantiles in Table 5. Nevertheless, both the absolute and the relative differences show a pattern that is similar to the GAMLSS results.
Table 6 Five measures on the income-health relationship from fitting a gamma distribution to the transformed values of nine predicted health quantiles
The two regression methods explored in this paper - GAMLSS and quantile regression - both allow a more refined analysis of the income-health relationship than conventional regression techniques. GAMLSS makes room for differences in the effects of income on health for different types of individuals, such as low and high income earners, or smokers and non-smokers, using parametric estimates of conditional health distributions. Quantile regression methods do so by estimating the effect of income on health at different locations of the health distribution, and hence go beyond standard regression methods such as OLS, which predict the effect of income on conditional mean health.
Whether one of the two techniques is superior to the other remains an open question, and certainly not one which this paper will settle. Silbersdorff et al. [12] have assessed different arguments in favour of and against GAMLSS and quantile regression, and have a clear preference for the former. They point out that one advantage of GAMLSS over quantile regression is that GAMLSS is suited for both categorical and continuous response variables, while quantile regression cannot be used for ordered categorical responses, which are frequently employed to measure health. Moreover, GAMLSS is appreciated for the fact that it estimates the complete conditional distribution. To obtain a comparable result with quantile regression, many conditional quantile functions have to be estimated. This argument assumes that the response variable can be approximated reasonably well by a probability distribution such as the gamma or lognormal distribution. If no suitable parametric response distribution is available, then the assumption of a parametric distribution is paradoxically one of the main drawbacks of GAMLSS. In quantile regression, such an assumption is not necessary. It deserves to be pointed out, however, that the GAMLSS framework is compatible with a wide range of useful distributions that go far beyond the exponential family of distributions [27]. For that purpose, separate GAMLSS packages in R have been developed (e.g., the gamlss.mx package for fitting finite mixture distributions).
In our study we approximate the health variable by means of the two-parameter gamma distribution in a frequentist estimation framework. Although this distribution is a simple one, Silbersdorff and Schneider [26] showed that differences in the information criteria with more complex three- and four-parameter distributions were only minor. They recommend the use of the two-parameter gamma distribution because it yields risk measures for the assessment of the income-health relationship that are comparable to those of the more complex distributions. Moreover, three- or four-parameter distributions suffer from decreased estimation stability leading to much wider confidence intervals and other statistical deficiencies. However, note that using a Bayesian instead of a frequentist estimation framework of GAMLSS, such as the Structured Additive Distributional Regression technique applied by [12], many of the computational problems can be sidestepped. Under the Bayesian framework, the assumption of a parametric distribution entails estimation stability, especially for samples of limited size and in the tails of the distribution, which are critical for evaluating risks.
Our empirical application has focused on the assessment of the risks of ending up in bad health, using different threshold values for what constitutes bad health (average, fair or poor health). Broadly speaking, we found that GAMLSS and quantile regression gave similar results. Not surprisingly, we consistently observed that low-income earners (i.e., with an equivalent household income of 20,000 AUD) have higher risks than high-income earners (i.e., with an equivalent household income of 80,000 AUD), and that smokers have higher risks than non-smokers. Our results show that the health risk profile of high-income smokers is similar to that of low-income non-smokers, whatever the method we use to calculate the risks. Nevertheless, there are some differences in the risk estimates according to the method adopted. For instance, if we compare the GAMLSS results (Table 3) to those obtained by fitting a gamma distribution to the predicted health quantiles (Table 6), we find that both the absolute and the relative differences between the risk estimates Pavg, Pfair and Ppoor of the low-income and high-income earners are always larger for the quantile regression approach than for the GAMLSS approach. The absolute and relative differences also tend to be larger if we use the results of the empirical distribution of the predicted health quantiles (Table 5) instead of those of the fitted gamma distribution, but not always. Similar conclusions hold if we compare the risk estimates for smokers and non-smokers. This implies that a comparison of a poor smoker and a rich non-smoker yields larger differences using the quantile regression methods. For instance, the absolute difference in the risks of having average health (Pavg) is equal to 0.469−0.382=0.087 according to the GAMLSS estimates, but equal to 0.492−0.327=0.165 and 0.498−0.330=0.168 according to the two sets of estimates based on quantile regression, i.e. almost twice as large. Although it seems that in this particular case the quantile regression approach is more sensitive to the effects of income and smoking on health than the GAMLSS approach, it remains to be seen whether this holds in general.
Finally, we would like to point out that the assessment of the risks of ending up in bad health is comparable to the measurement of health poverty recently proposed by [36]. The guiding idea is to zoom in on the bottom of the health distribution and to find out if there are groups of the population which are more vulnerable than others. As we have seen, the GAMLSS technique allows us to generate counterfactual probability distributions for specific subgroups, such as rich and poor, controlling for other differences that might exist. We have also indicated how a similar thing can be done by means of quantile regressions, in two different ways. The construction of these distributions requires quite a few assumptions (e.g., about the reference values of the covariates) and a lot of estimates (e.g., quantile regressions). The health poverty approach, by contrast, relies on the subgroup decomposability property of the health poverty indicator and is computationally much simpler. For example, we can divide the population in different income groups and calculate the health poverty index for each income group. However, the drawback is that in this way we cannot control for the effect of other covariates.
In this paper, we have explored the effect of income on health, using Australian household survey data. In reaction to the limitations of conventional mean-oriented regression techniques, we chose two unconventional regression techniques to study the income-health relationship. Our strategy consisted of using both GAMLSS and quantile regression to estimate conditional health distributions. This allowed us to assess the risks of ending up in bad health for different subgroups of the population. We focused in particular on the differences between low-income and high-income earners, and between smokers and non-smokers. Both regression techniques indicate quite strongly that people with low incomes face higher risks than people with high incomes. But we also found that the magnitude of the difference in risk changes with the chosen threshold for bad health. This suggests that it makes sense to look for regression techniques which are capable of identifying how large the effect of income is on health at different locations of the health distribution. When conventional regression techniques such as OLS find that income has a positive and significant coefficient in the health regression, what it means is that income has a positive effect on mean health. It is impossible to tell from this result whether the effect is smaller or larger at other health levels.
If we are interested in unraveling the causal relationship between income and health in all its complexity, a strong case can therefore be made for the application of distribution-sensitive regression techniques alongside the conventional mean-oriented regression techniques. In addition, the results from a distribution-sensitive regression analysis are helpful when it comes to the formulation of public health policies. The finding that at the lower end of the health spectrum income appears to have a larger effect on health than at the higher end is obviously relevant information for policymakers, especially if they give priority to improving the situation of people in bad health.
Although we compared the results of the GAMLSS technique to those of the quantile regression technique, we refrained from expressing a preference for one or the other. The results of our application of the two techniques are broadly similar, but not identical. However, the scope of our empirical study is too narrow to make general claims about the strengths and weaknesses of both approaches. More research is needed to see how the two compare in different contexts.
Finally, attention should be paid to the limitations of our study. One of the main drawbacks of this research is that it does not take into account the possibility of reverse causality. From the literature we know not only that income tends to have a positive effect on health, but also that health tends to have a positive effect on income. If this is the case, then estimating the impact of income on health by itself creates an endogeneity problem. Possible solutions for this problem are either to study only the impact of truly exogenous income variations on health, or to apply instrumental variable (IV) techniques.
In their appendix, Hohberg, Pütz and Kneib [24] propose an IV method for GAMLSS that is similar to the one Marra and Radice [37] developed for generalized additive models (GAM) which only describe the mean or location of the response distribution. The method exploits the two-stage procedure idea first proposed by Hausman [38, 39] as a means to test for endogeneity. The first stage obtains the residuals from an auxiliary GAM regression of the endogenous variables on all instrumental variables and all exogenous variables. The distributional part comes in the second stage where the residuals from the first stage are added to the GAMLSS model next to the exogenous explanatory variables. In quantile regression, the use of IVs has been pioneered by Chernozhukov and Hansen [40] who derived a set of conditions for identification of the IV quantile regression model without functional form assumptions. Subsequently, Chernozhukov and Hansen [41] proposed the IV quantile regression estimator, which is a quantile analog of two-stage least squares. We refer to [42] for an overview of empirical applications. As is the case with any IV method, the major drawback is the difficulty to select appropriate instruments.
Another limitation of our study is that all distributional results are conditional: with the exception of income and smoking habits, for which we have chosen two possible levels, we assume specific values for all other respondent characteristics to allow for comparisons among distributions. We have found these comparisons to be quite stable because we hardly observed significant effects among the covariates themselves, especially in relation to income. In future research one might consider several types of reference persons by assuming different covariate combinations.
Appendix 1. GAMLSS results based on the weibull and lognormal distribution
As possible alternatives to the two-parameter gamma distribution in the GAMLSS regression of the SF-6D health score, we chose the two-parameter Weibull and lognormal distributions and computed the associated risks for smokers and non-smokers to experience average, fair or poor health. We specified the Weibull and lognormal distribution by the GAMLSS descriptions WEI3 and LOGNO which refer to the formulations provided by Rigby et al. [27] (see p. 280 for the Weibull distribution and p. 275 for the lognormal distribution).
Table 7 Global deviance and information criterion values from using different distributions in GAMLSS regression of the SF-6D health score
To compare the performance of the Weibull and lognormal distribution to that of the gamma distribution, we first study the fit of these distributions in the GAMLSS models by means of the global deviance and AIC and BIC information criteria. Table 7 compares the values for the different distributions and shows that the smallest values are obtained with the Weibull distribution, although they come close to those of the gamma distribution. The lognormal distribution has the highest values indicating lower goodness-of-fit.
Diagnostic plots of normalized quantile residuals from GAMLSS regressions of the SF-6D health score, based on the transformed gamma, Weibull and lognormal distribution
Histogram of the SF-6D health score with fit to the transformed gamma, Weibull and lognormal distribution for the reference person
Figure 7a and b show diagnostic plots of the normalized (randomized) quantile residuals for the GAMLSS models to further evaluate the adequacy of the model distributions. We use the normalized quantile residuals because they follow a standard normal distribution when the assumed model is correct, similar to traditional residuals from linear models [43]. Panel a presents the kernel density estimates of the residuals from the GAMLSS models as well as the standard normal distribution for comparison. Panel b shows the corresponding Quantile-Quantile plots. Both panels reveal that the residuals of the lognormal model deviate the most from the standard normal distribution, and those of the Weibull and gamma models the least. All three distributions are negatively skewed and leptokurtic, where the lognormal distribution has a skewness (-3.76) and kurtosis (36.53) that are far from optimal (i.e., compared to a skewness of 0 and kurtosis of 3 for the standard normal distribution). The lognormal model seems therefore less appropriate.
Figure 8 plots the estimated GAMLSS distributions for the reference person (non-smoker) on the histogram of the SF-6D health score. Overall, both the gamma and Weibull distribution seem to summarize the data better than the lognormal distribution. When focusing on the lower part of the histogram, however, the lognormal distribution appears to describe the data better by its fatter tail compared to the gamma and Weibull distribution.
Tables 8 and 9 show the summary and risk statistics for smokers and non-smokers based on the Weibull and lognormal distribution. In general, the trends we observe are similar to those for the gamma distribution in Table 3. However, the results in the tables differ in the ability of the distributions to describe the empirical health scores at the lower end. The lognormal distribution captures the lowest health scores best. That is to say, the risk probabilities Pfair and Ppoor from the lognormal distribution come closest to the empirical cumulative probabilities of 30% and 10% for the non-smokers and 40% and 20% for the smokers, respectively. On the other hand, the Weibull distribution describes the lowest health scores worst.
To conclude, the gamma model turns out to be the compromise between the Weibull model that performs best on overall goodness-of-fit and the lognormal model that performs best on fitting the lower part of the empirical health distribution. The gamma model has a reasonable overall goodness-of-fit that is comparable to that of the Weibull model and an ability to capture the lower empirical health scores to some extent, but not as well as the lognormal model.
Table 8 Five measures on the income-health relationship from fitting a Weibull distribution to the transformed health variable using GAMLSS regression
Table 9 Five measures on the income-health relationship from fitting a lognormal distribution to the transformed health variable using GAMLSS regression
Appendix 2. OLS and quantile regressions of health
Table 10 OLS and quantile regression estimates for quantiles \(\tau _{1} = 0.1, \dots, \tau _{9} = 0.9\) of the health variable
The dataset analysed in this study is not publicly available, but can be requested from the Melbourne Institute.
AIC:
Akaike Information criterion
AUD:
Bayesian information criterion
CDF:
Cumulative distribution function
GAM:
Generalized additive models
GAMLSS:
Generalized additive models for location, scale and shape
HILDA:
Household, income and labour dynamics in Australia
Instrumental variable
OECD:
OLS:
Ordinary least squares
Q-Q:
Quantile-quantile
Recentered influence function
SF-6D:
Short-form six-dimension
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This paper uses unit record data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey. The HILDA Project was initiated and is funded by the Australian Government Department of Social Services (DSS) and is managed by the Melbourne Institute of Applied Economic and Social Research (Melbourne Institute). The findings and views reported in this paper, however, are those of the author and should not be attributed to either DSS or the Melbourne Institute. Roselinde Kessels acknowledges the Flemish Research Foundation (FWO) for her postdoctoral fellowship during the initial stage of this research and the JMP Division of SAS Institute for further financial support. We are grateful to Alexander Silbersdorff for providing material related to the GAMLSS regression and for his comments on a preliminary draft of the paper. All errors are our responsibility.
The authors received no specific funding for this research.
Department of Data Analytics and Digitalization, Maastricht University, PO Box 616, Maastricht, 6200, MD, The Netherlands
Roselinde Kessels
Department of Economics, University of Antwerp, City Campus, Prinsstraat 13, Antwerp, 2000, Belgium
Roselinde Kessels, Thi Kim Thanh Bui & Guido Erreygers
School of Economics, University of Amsterdam, PO Box 15867, Amsterdam, 1001, NJ, The Netherlands
Anne Hoornweg
School of Economics, Can Tho University, Campus II, 3/2 Street, Can Tho City, Vietnam
Thi Kim Thanh Bui
Centre for Health Policy, University of Melbourne, Bouverie Street 207, Carlton, Victoria, 3010, Australia
Guido Erreygers
RK and GE conceived and designed the study. RK, AH and TKTB analysed the data and derived the results. AH and TKTB wrote an initial draft of the manuscript. RK and GE wrote the final manuscript. All authors read and approved the final manuscript.
Correspondence to Roselinde Kessels.
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 licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence 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 licence, visit http://creativecommons.org/licenses/by/4.0/. 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 in a credit line to the data.
Kessels, R., Hoornweg, A., Thanh Bui, T.K. et al. A distributional regression approach to income-related inequality of health in Australia. Int J Equity Health 19, 102 (2020). https://doi.org/10.1186/s12939-020-01189-1
Socioeconomic health inequality
Distributional regression | CommonCrawl |
Gaussian noise
In signal processing theory, Gaussian noise, named after Carl Friedrich Gauss, is a kind of signal noise that has a probability density function (pdf) equal to that of the normal distribution (which is also known as the Gaussian distribution).[1][2] In other words, the values that the noise can take are Gaussian-distributed.
Without noise
With Gaussian noise
The probability density function $p$ of a Gaussian random variable $z$ is given by:
$\varphi (z)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-(z-\mu )^{2}/(2\sigma ^{2})}$
where $z$ represents the grey level, $\mu $ the mean grey value and $\sigma $ its standard deviation.[3]
A special case is white Gaussian noise, in which the values at any pair of times are identically distributed and statistically independent (and hence uncorrelated). In communication channel testing and modelling, Gaussian noise is used as additive white noise to generate additive white Gaussian noise.
In telecommunications and computer networking, communication channels can be affected by wideband Gaussian noise coming from many natural sources, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or Johnson–Nyquist noise), shot noise, black-body radiation from the earth and other warm objects, and from celestial sources such as the Sun.
Gaussian noise in digital images
Principal sources of Gaussian noise in digital images arise during acquisition e.g. sensor noise caused by poor illumination and/or high temperature, and/or transmission e.g. electronic circuit noise.[3] In digital image processing Gaussian noise can be reduced using a spatial filter, though when smoothing an image, an undesirable outcome may result in the blurring of fine-scaled image edges and details because they also correspond to blocked high frequencies. Conventional spatial filtering techniques for noise removal include: mean (convolution) filtering, median filtering and Gaussian smoothing.[1][4]
See also
• Gaussian process
• Gaussian smoothing
References
1. Tudor Barbu (2013). "Variational Image Denoising Approach with Diffusion Porous Media Flow". Abstract and Applied Analysis. 2013: 8. doi:10.1155/2013/856876.
2. Barry Truax, ed. (1999). "Handbook for Acoustic Ecology" (Second ed.). Cambridge Street Publishing. Archived from the original on 2017-10-10. Retrieved 2012-08-05.
3. Philippe Cattin (2012-04-24). "Image Restoration: Introduction to Signal and Image Processing". MIAC, University of Basel. Retrieved 11 October 2013.
4. Robert Fisher; Simon Perkins; Ashley Walker; Erik Wolfart. "Image Synthesis — Noise Generation". Retrieved 11 October 2013.
| Wikipedia |
\begin{document}
\title{ \bf Stochastic transport equation with bounded and Dini continuous drift}
\author{Jinlong Wei$^a$, Guangying Lv$^b$ and Wei Wang$^c$ \\ {\small \it $^a$School of Statistics and Mathematics, Zhongnan University of}\\ {\small \it Economics and Law, Wuhan 430073, China} \\ {\small \tt [email protected]} \\ {\small \it $^b$College of Mathematics and Statistics, Nanjing University of Information} \\{\small \it Science and Technology, Nanjing 210044, China}\\ {\small \tt [email protected]} \\ {\small \it $^c$Department of Mathematics, Nanjing University, Nanjing 210093, China} \\ {\small \tt [email protected]} } \date{\today}
\maketitle \noindent{\hrulefill}
\begin{abstract}
The results established by Flandoli, Gubinelli and Priola ({\it Invent. Math.} {\bf 180} (2010) 1--53) for stochastic transport equation with bounded and H\"{o}lder continuous drift are generalized to bounded and Dini continuous drift. The uniqueness of $L^\infty$-solutions is established by the It\^o--Tanaka trick partially solving the uniqueness problem, which is still open, for stochastic transport equation with only bounded measurable drift. Moreover the existence and uniqueness of stochastic diffeomorphisms flows for a stochastic differential equation with bounded and Dini continuous drift is obtained. \end{abstract}
\vskip1.2mm\noindent {\bf MSC (2010):} 60H15 (35A01 35L02)
\vskip1.2mm\noindent {\bf Keywords:} Stochastic transport equation, Weak $L^\infty$-solution, Uniqueness, Stochastic diffeomorphisms flow
\vskip0mm\noindent{\hrulefill}
\section{Introduction}\label{sec1}\setcounter{equation}{0} We are concerned with the following stochastic transport equation \begin{eqnarray}\label{1.1} \left\{
\begin{array}{ll} \partial_tu(t,x)+b(t,x)\cdot\nabla u(t,x) +\sum_{i=1}^d\partial_{x_i}u(t,x)\circ\dot{B}_i(t)=0, \quad (t,x)\in(0,T)\times {\mathbb R}^d, \\
u(t,x)|_{t=0}=u_0(x), \quad x\in{\mathbb R}^d,
\end{array} \right. \end{eqnarray} where $\{B(t)\}_t=\{(B_1(t), B_2(t), _{\cdots}, B_d(t))\}_t$ is a $d$-dimensional standard Brownian motion defined on a stochastic basis ($\Omega, {\mathcal F},{\mathbb P},({\mathcal F}_{t})_{t\geqslant}\def\leq{\leqslant 0}$). The stochastic integration with notation $\circ$ is interpreted in Stratonovich sense. Given $T>0$, the drift coefficient $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ and the initial data~$u_0: {\mathbb R}^d\rightarrow{\mathbb R}$ are measurable functions in $L^1([0,T];L^1_{loc}({\mathbb R}^d;{\mathbb R}^d))$ and $L^1([0,T];L^1_{loc}({\mathbb R}^d))$ respectively. We are interested in the existence and uniqueness of weak $L^\infty$-solutions for the stochastic equation~(\ref{1.1}).
There are many results on the weak $L^\infty$-solutions for the deterministic transport equation. The first remarkable result of the uniqueness solution in $L^{\infty}([0, T]\times\mathbb{R}^{d})$ was obtained by DiPerna and Lions~\cite{DL} under the assumption $b\in L^1([0,T];W^{1,1}_{loc}({\mathbb R}^d;{\mathbb R}^d))$ with suitable global conditions including $L^\infty$-bounds on spatial divergence. Then Ambrosio \cite{Amb} weakened the condition $W^{1,1}_{loc}$ to~$BV_{loc}$ and the uniqueness of weak $L^\infty$-solutions is obtained under the assumption that negative part of $\div b$ is in space $L^1([0,T];L^\infty(\mR^d))$. For general $b$ just with $BV_{loc}$ or H\"{o}lder regularity, the uniqueness of weak $L^\infty$-solutions for the deterministic equation fails and counterexamples have been constructed in many works~\cite{Bre,CLR1,CD,Dep,DL}. Obviously, more restrictions need to be imposed on $b$ to overcome the obstacle of nonuniqueness of solution in deterministic case.
However the appearance of the noise makes the solution unique under very general assumptions on the drift coefficient for ordinary differential equations \cite{KR,Ver}, so a natural idea is to investigate the effects of noise in transport equation. The first milestone result, founded by Flandoli, Gubinelli and Priola~\cite{FGP1}, showed the uniqueness of weak $L^\infty$-solutions just with assuming~$b\in L^\infty([0,T];\cC_b^\alpha(\mR^d;\mR^d))$ and $\div b\in L^p([0,T]\times\mR^d)$ for some $\alpha>0$, $p>2$. This is the first concrete example of a partial differential equation related to fluid dynamics that becomes well-posed with a suitable noise. A key step for this result is to perform differential computations on regularization of $L^\infty$-solutions by a commutator lemma. Unfortunately the strategy fails if $b$ is not Sobolev differentiable. However, by observing the fact that stochastic differential equation~(SDE) \begin{eqnarray}\label{1.2} \left\{\begin{array}{ll}
dX(t)=b(t,X(t))dt+dB(t), \quad 0<t\leq T,\\
X(t)|_{t=0}=x\in{\mathbb R}^d, \end{array}\right. \end{eqnarray} defines a $\cC^1$ stochastic diffeomorphisms flow and, along the stochastic characteristic $X(t)$, the integral $\int_0^t\div b(s,X(s,x))ds$ has a regularization, Flandoli, Gubinelli and Priola developed the commutator lemma to prove the uniqueness of solutions. On the other hand, for bounded measurable $b$, Mohammed, Nilssen and Proske~\cite{MNP} also proved the existence, uniqueness and Sobolev differentiable stochastic flows for~(\ref{1.2}) by employing ideas from the Malliavin calculus coupled with new probabilistic estimates on the spatial weak derivatives of solutions of~(\ref{1.2}). Then, as an application, they obtained the existence and uniqueness of Sobolev differentiable weak solutions for (\ref{1.1}) with every $\cC^1(\mR^d)$ initial data. Notice that in one result the stochastic flow $\{X(t, x)\}$ is differentiable in $x$ in the classical sense~\cite{FGP1} and in the other result the stochastic flow $\{X(t,x)\}$ is only Sobolev differentiable~\cite{MNP}. So the method developed by Mohammed, Nilssen and Proske~\cite{MNP} can not be adapted to establish the uniqueness of weak $L^\infty$-solutions to (\ref{1.1}).
Recent result~\cite{AF}, by using a different philosophy, proved the uniqueness of weak $L^\infty$-solutions for~(\ref{1.1}) just with assuming the $BV$ regularity for $b$ but without the $L^\infty$-bounds on spatial divergence. There are also several other related works~\cite{BFG,FL,FF2,NO,WDGL,Zhang1}\,.
However, \textbf{for bounded measurable $b$ and $\div b\in L^p([0,T]\times \mR^d)$ (for some $p\in [1,\infty)$), the uniqueness of weak $L^\infty$-solutions for (\ref{1.1}) is still unknown.}\\
This paper intend to give a partial answer for the above problem and novelties of the work are
\begin{itemize} \item {\it The uniqueness of weak $L^\infty$-solutions for the Cauchy problem (\ref{1.1}) with bounded and Dini-continuous drift is established due to the existence of noise, while the corresponding deterministic equation has multiple solutions.} \item {\it The existence and uniqueness of stochastic diffeomorphisms flow for singular SDE (\ref{3.1}) is established without H\"{o}lder continuity or Sobolev differentiability hypotheses on $b$.}
\item {\it The maximum regularity for parabolic equations of second order with H\"{o}lder-Dini or strong H\"{o}lder or weak H\"{o}lder coefficients is established.} \end{itemize}
We follow the strategy of Flandoli, Gubinelli and Priola's~\cite{FGP1} to establish the existence of a stochastic~$\cC^1$ diffeomorphisms flow for (\ref{3.1}) by the It\^o--Tanaka trick, then derive a commutator estimates to get the uniqueness for weak $L^\infty$-solution of (\ref{1.1}). The main idea of It\^o--Tanaka trick is to use a parabolic partial differential equation~(PDE) to transform the original SDE~(\ref{3.1}) with irregular drift and regular diffusion to a new SDE~(\ref{3.15}) with regular drift and diffusion. Then by the equivalence between (\ref{3.1}) and (\ref{3.15}) we show the existence of the stochastic $\cC^1$ diffeomorphisms flow for SDE (\ref{3.1})\,. There are also some recent works on the stochastic flows and SDEs~\cite{Att,FF1,FF3,FGP2,TWT,WZ,Zhang2}.
In the following parts, we first derive the $W^{2,\infty}$ estimates for a class of second order parabolic PDEs with bounded and Dini continuous coefficients in section 2; then by using the $W^{2,\infty}$ estimates, the existence and uniqueness of stochastic flow of diffeomorphisms for SDE (\ref{3.1}) is shown in section~3 by the It\^o--Tanaka trick; last section is concerned with the existence and uniqueness of weak $L^\infty$-solutions to stochastic transport equation (\ref{1.1}).
\vskip2mm\noindent \textbf{Notations} The letter $C$ denotes a positive constant, whose values may change in different places. For a parameter or a function $\kappa$, $C(\kappa)$ means the constant is only dependent on $\kappa$, and we also write it as $C$ if there is no confusion. ${\mathbb N}$ is the set of natural numbers. For every
$R>0$, $B_R:=\{x\in{\mathbb R}^d:|x|<R\}$. Almost surely is abbreviated to $a.s.$. Let $\Theta$ be a ${\mathbb R}^{d\times d}$-valued function $\Theta=(\Theta_{i,j}(x))_{d\times d}$ with norm
$\|\Theta(x)\|=\max_{1\leq i,j\leq d}|\Theta_{i,j}(x)|$. For $\xi\in \mR^d$, $|\xi|=(\sum_{i=1}^d\xi_i^2)^{1/2}$. ${\mathbb R}_+$ is the set of nonnegative real numbers and $\bar{{\mathbb R}}_+={\mathbb R}_+\cup\{+\infty\}$.
\section{Parabolic PDEs with bounded and Dini coefficients}\label{sec2}\setcounter{equation}{0} Let $T>0$. Consider the following Cauchy problem \begin{eqnarray}\label{2.1} \left\{\begin{array}{ll}
\partial_{t}u(t,x)=\frac{1}{2}\Delta u(t,x)+g(t,x)\cdot \nabla u(t,x)+f(t,x), \ (t,x)\in (0,T)\times {\mathbb R}^d, \\ u(0,x)=0, \ x\in{\mathbb R}^d. \end{array}\right. \end{eqnarray}
The function $u(t,x)$ is called a strong solution of (\ref{2.1}) if $ u\in L^\infty([0,T];W^{2,\infty}({\mathbb R}^d))\cap W^{1,\infty}([0,T];L^\infty({\mathbb R}^d))$ such that for almost all $(t,x)\in [0,T]\times\mR^d$, (\ref{2.1}) holds. We have the following equivalent form for the strong solution.
\begin{lemma} \label{lem2.1} Let $f\in L^\infty([0,T];\cC_b(\mR^d))$, $g\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$ and $u\in L^\infty([0,T];W^{2,\infty}({\mathbb R}^d))\cap W^{1,\infty}([0,T];L^\infty({\mathbb R}^d))$, then $u$ is a strong solution for (\ref{2.1}) if and only if \begin{eqnarray}\label{2.2} u(t,x)&=&\int\limits_0^tK(t-s,\cdot)\ast (g(s,\cdot)\cdot \nabla u(s,\cdot))(x)ds\nonumber \\ && +\int\limits_0^tK(t-s,\cdot)\ast f(s,\cdot)(x)ds, \quad for\; all \ (t,x)\in [0,T]\times\mR^d, \end{eqnarray}
where $K(t,x)=(2\pi t)^{-\frac{d}{2}}e^{-\frac{|x|^2}{2t}}, \ t>0, \ x\in\mathbb{R}^d$. \end{lemma}
\vskip2mm\noindent\textbf{Proof.} By the properties of the heat kernel $K$, it is direct to verify that if $u$ satisfies (\ref{2.2}), for almost all $(t,x)\in [0,T]\times\mR^d$, (\ref{2.1}) holds. On the other hand, if $u$ satisfies (\ref{2.1}) then for every $\psi\in \mathcal{C}_0^\infty({\mathbb R}^d)$ and every $t\in [0,T]$ \begin{eqnarray*} \int\limits_0^t\int\limits_{\mathbb{R}^d}\partial_su(s,x)\varphi(s,x)dsdx&=& \frac{1}{2}\int\limits_{\mathbb{R}^d}\int\limits_0^t\Delta u(s,x)\varphi(s,x)dsdx+ \int\limits_{\mathbb{R}^d}\int\limits_0^t g(s,x)\cdot \nabla u(s,x) \varphi(s,x) dsdx \nonumber\\&&+\int\limits_{\mathbb{R}^d}\int\limits_0^t f(s,x)\varphi(s,x)dsdx, \end{eqnarray*} with $\varphi(s,x)=K(t-s,\cdot)\ast \psi(\cdot)(x)$.
Now integrating by parts yields \begin{eqnarray*} \int\limits_{\mathbb{R}^d}u(t,x)\psi(x)dx&=& \int\limits_{\mathbb{R}^d}\int\limits_0^t u(s,x)[\partial_s\varphi(s,x)+\frac{1}{2}\Delta\varphi(s,x)]dsdx \nonumber\\&&+ \int\limits_{\mathbb{R}^d}\int\limits_0^t g(s,x)\cdot \nabla u(s,x) \varphi(s,x) dsdx+\int\limits_{\mathbb{R}^d}\int\limits_0^t f(s,x)\varphi(s,x)dsdx \nonumber\\ &=& \int\limits_{\mathbb{R}^d}\int\limits_0^tK(t-s,\cdot)\ast (g(s,\cdot)\cdot \nabla u(s,\cdot))(x)ds\psi(x)dx\nonumber\\&&+\int\limits_{\mathbb{R}^d}\int\limits_0^tK(t-s,\cdot)\ast f(s,\cdot)(x)ds\psi(x)dx, \quad for\; all \ t\in [0,T], \end{eqnarray*} then by the arbitrariness of $\psi$ and continuity of $u$ in $x$, (\ref{2.2}) holds. $\Box$
\begin{definition} \label{def2.1} An increasing continuous function $\phi: \mR_+\rightarrow \mR_+$ is called a Dini function if \begin{eqnarray}\label{2.3} \int\limits_{0+}\frac{\phi(r)}{r}dr<+\infty. \end{eqnarray} A measurable function $h:\mR^{d}\rightarrow \mR$ is said to be Dini continuous if there is a Dini function $\phi$ such that \begin{eqnarray}\label{2.4}
|h(x)-h(y)|\leq \phi(|x-y|). \end{eqnarray} \end{definition}
We now state the main result of this section. \begin{theorem} \label{the2.1} Let $f\in L^\infty([0,T];\cC_b(\mR^d))$ and $g\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$. Suppose that $r_0\in (0,1)$ and there is a Dini function $\phi$ such that for every $x\in\mR^d$ \begin{eqnarray}\label{2.5}
|f(t,x)-f(t,y)|+|g(t,x)-g(t,y)|\leq \phi(|x-y|), \quad for\; all \ y\in B_{r_0}(x), \ t\in [0,T]. \end{eqnarray}
(i) The Cauchy problem (\ref{2.1}) has a unique strong solution $u$, and there is a constant $C(d,T)$ such that \begin{eqnarray}\label{2.6}
\|u\|_{L^\infty([0,T];\cC^2_b({\mathbb R}^d))} \leq C(d,T)(1+\|f\|_{L^\infty([0,T];\cC_b(\mR^d))}+
\|g\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}). \end{eqnarray} Moreover, for every $1\leq i,j\leq d$ and every $x,y\in \mR^d$, there is another constant $C(d,T)$ such that \begin{eqnarray}\label{2.7}
&&|\partial^2_{x_i,x_j}u(t,x)-\partial^2_{y_i,y_j}u(t,y)| \nonumber \\ &\leq &C(d,T)\left[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr\right]1_{|x-y|<r_0}\nonumber \\ &&+C(d,T)|x-y|, \quad for \; all \ t\in [0,T]. \end{eqnarray}
(ii) Let $\varrho_n$, $n\in \mN$, be a regularizing kernel, that is \begin{eqnarray}\label{2.8} \varrho_n(x) =n^d \varrho(nx) \ \ \mbox{with} \ \ 0\leq \varrho \in \mathcal{C}^\infty_0({\mathbb R}^d) , \ \ \mbox{supp}(\varrho)\subset B_1, \ \ \int\limits_{{\mathbb R}^d}\varrho(x)dx=1. \end{eqnarray}
Let $u^n$ be the unique strong solution of (\ref{2.1}) with $f$ and $g$ are replaced by $f^n(t,x)=f\ast \varrho_n(t,x)$ and $g^n(t,x)=g\ast \varrho_n(t,x)$ respectively. Then $u^n\in L^\infty([0,T];\cC^2_b({\mathbb R}^d))\cap W^{1,\infty}([0,T];\cC_b({\mathbb R}^d))$ and satisfies~(\ref{2.6})--(\ref{2.7}) uniformly in $n$. Furthermore, \begin{eqnarray}\label{2.9}
\lim_{n\rightarrow\infty}\| u^n-u\|_{L^\infty([0,T];\cC_b^2({\mathbb R}^d))}=0. \end{eqnarray} \end{theorem} \noindent\textbf{Proof.} (i) We first prove the result for the case $g=0$. By Lemma \ref{lem2.1} we just need to show \begin{eqnarray}\label{2.10} u(t,x)=\int\limits_0^{t}K(t-s,\cdot)\ast f(s,\cdot)(x)ds \end{eqnarray} is in $L^\infty([0,T];\cC^{2}_b({\mathbb R}^d))$ and (\ref{2.7}) holds. In fact $u\in L^\infty([0,T];W^{1,\infty}({\mathbb R}^d))$ is classical~\cite[Ch.4]{LSU} by the explicit representation (\ref{2.10})\,.
Next we show $\partial^2_{x_i,x_j}u\in L^\infty([0,T];\cC_b({\mathbb R}^d))$ for every $1\leq i,j\leq d$ and (\ref{2.7}) holds. For this we first show that $\partial^2_{x_i,x_j}u\in L^\infty([0,T];L^\infty({\mathbb R}^d))$. Let $\theta\in (0,1/2)$. For $x\in \mR^d$ and $t\in (0,T]$, we have \begin{eqnarray}\label{2.11}
\Big|\partial^2_{x_i,x_j}u(t,x)\Big|&=&\Bigg|\int\limits_0^t\int\limits_{{\mathbb R}^d}\partial^2_{x_i,x_j}K(t-s,x-y)f(s,y)dyds\Bigg|\nonumber\\
&=& \Bigg|\int\limits_0^t\int\limits_{{\mathbb R}^d}\partial^2_{x_i,x_j}K(t-s,x-y)[f(s,y)-f(s,x)]dyds \Bigg| \nonumber\\
&=& \Bigg|\int\limits_0^t\int\limits_{|x-y|> (t-s)^\theta }\partial^2_{x_i,x_j}K(t-s,x-y)[f(s,y)-f(s,x)]dyds\nonumber\\&&+\int\limits_0^t\int\limits_{|x-y|\leq (t-s)^\theta }\partial^2_{x_i,x_j}K(t-s,x-y)[f(s,y)-f(s,x)]dyds\Bigg| \nonumber\\
&\leq & C\|f\|_{L^\infty([0,T]\times\mR^d)}\int\limits_0^t\int\limits_{|x-y|> (t-s)^\theta }\frac{1}{t-s}K(t-s,x-y)dyds\nonumber\\&&+\int\limits_0^t\int\limits_{|x-y|<(t-s)^\theta \wedge r_0 }\frac{1}{t-s}K(t-s,x-y)\phi(|x-y|)dyds \nonumber\\
&&+C\int\limits_0^t\int\limits_{(t-s)^\theta \wedge r_0\leq |x-y|\leq(t-s)^\theta }\frac{1}{t-s}K(t-s,x-y)|x-y|dyds
\nonumber\\&\leq & C\[\int\limits_0^T\int\limits_{|y|> s^\theta }\frac{1}{s}K(s,y)dyds+\int\limits_0^T\int\limits_{|y|<s^\theta \wedge r_0 }\frac{1}{s}K(s,y)\phi(|y|)dyds \nonumber\\
&&+C\int\limits_{r_0^{\frac{1}{\theta}}}^T\int\limits_{r_0\leq |y|\leq s^\theta }\frac{1}{s}K(s,y)|y|dyds\] =:I_1+I_2+I_3, \end{eqnarray} where the boundedness of $f$ is applied in the seventh line of (\ref{2.11}), and in the last line $I_3=0$ for~$T\leq r_0^{\frac{1}{\theta}}$. We first estimate $I_1$. \begin{eqnarray}\label{2.12} I_1&=& C\int\limits_0^Ts^{-\frac{d+2}{2}}ds\int\limits_{r> s^\theta}e^{-\frac{r^2}{2s}}r^{d-1}dr \nonumber\\&=& C\int\limits_0^{T}s^{-1}ds\int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr \nonumber\\&=&C\int\limits_0^{\frac12}s^{-1}ds\int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr+C\int\limits_{\frac12}^{T\vee {\frac12}}s^{-1}ds\int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr \nonumber\\&\leq &C\int\limits_0^{\frac12}s^{-1}ds\int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr+C\log(2T\vee 1)\int\limits_{(T\vee \frac12)^{{\theta-\frac12}}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr \nonumber\\&\leq &C\Bigg[\log(2T\vee 1)+\int\limits_0^{\frac12}s^{-1}ds\int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr\Bigg]. \end{eqnarray} Without loss of generality we assume that $d$ is even. Otherwise, since $\theta<1/2$ and $s\in (0,1/2)$, \begin{eqnarray*} \int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr\leq \int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^ddr. \end{eqnarray*}
Thus, there is a natural number $m\geqslant}\def\leq{\leqslant 0$ such that $d=2m+2$ and \begin{eqnarray}\label{2.13} \int\limits_{s^{\theta-\frac12}}^{\infty} e^{-\frac{r^2}{2}}r^{d-1}dr=2^m \int\limits_{s^{2\theta-1}/2}^{\infty} e^{-r}r^mdr=:2^mJ_m. \end{eqnarray} Set $s_0=s^{2\theta-1}/2$ and integrating by parts yields the following recurrence formula \begin{eqnarray*} J_m=\int\limits_{s^{2\theta-1}/2}^{\infty} e^{-r}r^mdr=s_0^me^{-s_0}+mJ_{m-1}. \end{eqnarray*} Then \begin{eqnarray}\label{2.14} J_m\leq C(1+s_0^m)e^{-s_0}. \end{eqnarray} Now from (\ref{2.12})--(\ref{2.14}) \begin{eqnarray}\label{2.15} I_1&\leq& C\Bigg[\log(2T\vee 1)+\int\limits_0^{\frac12}s^{-1}(1+s^{(\theta-\frac12)(d-2)}) e^{-\frac{s^{2\theta -1}}{2}} ds\Bigg] \nonumber\\& =&C\Bigg[\log(2T\vee 1)+\int\limits_2^{\infty} s^{-1}(1+s^{(\frac12-\theta)(d-2)}) e^{-\frac{s^{1-2\theta}}{2}}ds\Bigg]<+\infty. \end{eqnarray}
For $I_{2}$\,, since $\phi$ is a nonnegative increasing continuous function, we have \begin{eqnarray}\label{2.16} I_2\leq C\int\limits_0^{T}\frac{\phi(s^\theta)}{s}ds\int\limits_{\mR^d} K(s,y)dy =C\int\limits_0^{T^\theta}\frac{\phi(s)}{s}ds<+\infty. \end{eqnarray} Last for $I_3$, we have \begin{eqnarray}\label{2.17} I_3\leq C\int\limits_{0}^T\int\limits_{\mR^d}s^{-\frac{1}{2}}K(s,y)dyds\leq C\sqrt{T}< +\infty. \end{eqnarray} Now by (\ref{2.11}) and (\ref{2.15})--(\ref{2.17}), $u\in L^\infty([0,T];W^{2,\infty}({\mathbb R}^d))$ and there is a constant $C>0$\,, depending only on $d$ and $T$\,, such that \begin{eqnarray*}
\|u\|_{L^\infty([0,T];W^{2,\infty}({\mathbb R}^d))} \leq C(1+\|f\|_{L^\infty([0,T];\cC_b(\mR^d))}). \end{eqnarray*}
So to prove $u\in L^\infty([0,T];\cC^{2}_b({\mathbb R}^d))$, we just need inequality (\ref{2.7}). Notice that for $x,y\in\mR^d$ and $|x-y|\geqslant}\def\leq{\leqslant r_0/2$, since $u\in L^\infty([0,T];W^{2,\infty}({\mathbb R}^d))$, there is constant $C(d,T)$ such that \begin{eqnarray*}
|\partial^2_{x_i,x_j}u(t,x)-\partial^2_{y_i,y_j}u(t,y)|\leq \frac{C(d,T)r_0}{2} \leq C(d,T)|x-y|, \quad {\rm for \; all} \ \ t\in [0,T]. \end{eqnarray*}
We next show (\ref{2.7}) for $0<|x-y|<r_0/2$.
For every $1\leq i,j\leq d$ and $0<t\leq T$, every $x,y\in \mR^d$ with $|x-y|\leq r_0/2$, from (\ref{2.10})\,, we have \begin{eqnarray}\label{2.18} &&\partial^2_{x_i,x_j}u(t,x)-\partial^2_{y_i,y_j}u(t,y)\nonumber\\
&=&\int\limits_0^t ds\int\limits_{|x-z|\leq 2|x-y|}\partial^2_{x_i,x_j}K(t-s,x-z)[f(s,z)-f(s,x)]dz
\nonumber\\&&-\int\limits_0^t ds\int\limits_{|x-z|\leq 2|x-y|}\partial^2_{y_i,y_j}K(t-s,y-z)[f(s,z)-f(s,y)]dz
\nonumber\\&&+\int\limits_0^t ds\int\limits_{|x-z|> 2|x-y|}\partial^2_{y_i,y_j}K(t-s,y-z)[f(s,y)-f(s,x)]dz \nonumber\\
&&+\int\limits_0^t ds\int\limits_{|x-z|> 2|x-y|}[\partial^2_{x_i,x_j}K(t-s,x-z)-\partial^2_{y_i,y_j}K(t-s,y-z)][f(s,z)-f(s,x)]dz \nonumber\\ &=&:J_1(t)+J_2(t)+J_3(t)+J_4(t). \end{eqnarray}
By the assumption (\ref{2.5}) \begin{eqnarray}\label{2.19}
|J_1(t)|&\leq& \int\limits_0^tds\int\limits_{|x-z|\leq 2|x-y|} s^{-\frac{d+2}{2}}e^{-\frac{|x-z|^2}{2s}}\phi(|x-z|) dz
\nonumber\\&=& \int\limits_{|z|\leq 2|x-y|} \phi(|z|)dz \int\limits_0^t s^{-\frac{d+2}{2}}e^{-\frac{|z|^2}{2s}}ds
\nonumber\\ &\leq&C \int\limits_{|z|\leq 2|x-y|} \frac{\phi(|z|)}{|z|^d} dz \int\limits_0^\infty r^{\frac{d}{2}-1}e^{-\frac{r}{2}}dr
\nonumber\\&\leq&C\int\limits_{r\leq 2|x-y|} \frac{\phi(r)}{r} dr. \end{eqnarray}
Analogously, \begin{eqnarray}\label{2.20}
|J_2(t)|\leq C\int\limits_{r\leq 2|x-y|} \frac{\phi(r)}{r} dr. \end{eqnarray}
For $J_3$, by Gauss--Green's formula \begin{eqnarray}\label{2.21}
|J_3(t)|&=&\left|\int\limits_0^t ds\int\limits_{|x-z|=2|x-y|}\partial_{y_j}K(t-s,y-z)n_i[f(s,y)-f(s,x)]dS\right| \nonumber\\
&\leq& C\int\limits_0^{T}ds
\int\limits_{|x-z|=2|x-y|}|y-z|s^{-\frac{d+2}{2}}e^{-\frac{|y-z|^2}{2s}}\phi(|x-y|)dS \nonumber\\&\leq&
C\phi(|x-y|)\int\limits_{|x-z|=2|x-y|}|y-z| dS \int\limits_0^{T}
s^{-\frac{d+2}{2}}e^{-\frac{|x-y|^2}{2s}}ds
\nonumber\\ &\leq& C\phi(|x-y|)|x-y|^d \int\limits_0^{\infty}
s^{-\frac{d+2}{2}}e^{-\frac{|x-y|^2}{2s}}ds
\nonumber\\ &\leq& C\phi(|x-y|) \int\limits_{0}^{\infty}
r^{\frac{d-2}{2}}e^{-r}dr\nonumber\\ &\leq& C\phi(|x-y|). \end{eqnarray}
For $J_4(t)$, since $|x-z|>2|x-y|$, for every $\xi\in [x,y]$ (the line with endpoints $x$ and $y$) \begin{eqnarray*}
\frac{1}{2}|x-z| \leq |\xi-z|\leq 2|x-z|. \end{eqnarray*} Thanks to (\ref{2.5}) and the mean value inequality, we acquire \begin{eqnarray}\label{2.25}
|J_4(t)| &\leq&
C|x-y|\int\limits_0^t ds
\int\limits_{|x-z|> 2|x-y|}\Big(\phi(|x-z|)1_{|x-z|< r_0}+1_{|x-z|\geqslant}\def\leq{\leqslant r_0}|f(s,x)-f(s,z)|\Big)
\nonumber\\ &&\times (t-s)^{-\frac{(d+3)}{2}}e^{-\frac{|x-z|^2}{8(t-s)}}dz. \end{eqnarray} Observing that $f$ is bounded, there is a constant $C>0$ such that \begin{eqnarray}\label{2.26}
\sup_{s\in [0,T]}|f(s,x)-f(s,z)| \leq C, \quad {\rm for \; all} \ \ |x-z|\geqslant}\def\leq{\leqslant r_0. \end{eqnarray} Then by (\ref{2.25}) and (\ref{2.26}) \begin{eqnarray}\label{2.27}
|J_4(t)| &\leq& C|x-y|\int\limits_{|x-z|> 2|x-y|}\Big(\phi(|x-z|)1_{|x-z|< r_0}+ 1_{|x-z|\geqslant}\def\leq{\leqslant r_0}\Big)|x-z|^{-d-1}dz\int\limits_0^\infty r^{\frac{d-1}{2}}e^{-r}dr
\nonumber\\ &\leq& C|x-y|\int\limits_{|x-z|> 2|x-y|}\Big(\phi(|x-z|)1_{|x-z|< r_0}+ 1_{|x-z|\geqslant}\def\leq{\leqslant r_0}\Big)|x-z|^{-d-1}dz
\nonumber\\ &\leq& C|x-y|\int\limits_{2|x-y|<r< r_0}\frac{\phi(r)}{r^2}dr+C|x-y|\int\limits_{r\geqslant}\def\leq{\leqslant r_0}r^{-2}dr
\nonumber\\ &\leq& C|x-y|\int\limits_{2|x-y|<r<r_0}\frac{\phi(r)}{r^2}dr+C|x-y|. \end{eqnarray}
Now combining (\ref{2.19}), (\ref{2.20}), (\ref{2.21}) and (\ref{2.27}), for all $x,y\in\mR^d$ and $t\in [0,T]$ \begin{eqnarray}\label{2.28}
&&|\partial^2_{x_i,x_j}u(t,x)-\partial^2_{y_i,y_j}u(t,y)| \nonumber \\ &\leq &C(d,T)\Bigg[\int\limits_{r\leq 2|x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{2|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr+|x-y|\Bigg]
\nonumber \\ &\leq &C(d,T)\Bigg[\int\limits_{r\leq 2|x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+2|x-y|\int\limits_{2|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr+|x-y|\Bigg]
\nonumber \\ &= &C\Bigg[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+2|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr+|x-y|\Bigg]\nonumber \\ &&+ \int\limits_{|x-y|<r\leq 2|x-y|}\frac{\phi(r)}{r}dr-2|x-y|\int\limits_{|x-y|<r\leq 2|x-y|}\frac{\phi(r)}{r^2}dr \nonumber \\ &\leq&C\Bigg[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr+|x-y|\Bigg], \end{eqnarray} which implies (\ref{2.7}).
Next we consider the case $g\neq 0$\,. Notice that for $g\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$ and satisfies (\ref{2.5}), $g\cdot\nabla u\in L^\infty([0,T];\cC_b(\mR^d))$ and satisfies (\ref{2.5}) with the righthand side replaced by $C\phi$. Let $\mathcal{H}$ be the set consisting of the functions in $L^\infty([0,T];\cC^2_b({\mathbb R}^d))$ with \begin{eqnarray*}\label{2.29}
|\nabla v(t,x)-\nabla v(t,y)|\leq C \phi(|x-y|), \quad {\rm for \; all } \ x\in \mR^d, \ y\in B_{r_0}(x), \ t\in [0,T] \end{eqnarray*} for $v\in\mathcal{H}$ with some constant $C>0$\,. For $v\in\mathcal{H}$ we define a mapping \begin{eqnarray*} \mathcal{T}v(t,x)=\int\limits_0^{t}K(t-s,\cdot)\ast (g(s,\cdot)\cdot \nabla v(s,\cdot))(x)ds+\int\limits_0^{t}K(t-s,\cdot)\ast f(s,\cdot)(x)ds. \end{eqnarray*} From (\ref{2.10}), if $f$ is in $L^\infty([0,T];\cC_b(\mR^d))$ and satisfies (\ref{2.5}), for every $x\in\mR^d$ and $y\in B_{r_0}(x)$ \begin{eqnarray}\label{2.29}
|\nabla u(t,x)-\nabla u(t,y)|&=&\Bigg|\int\limits_0^{t}\nabla K(t-s,\cdot)\ast f(s,\cdot)(x)ds-\int\limits_0^{t}\nabla K(t-s,\cdot)\ast f(s,\cdot)(y)ds\Bigg|\nonumber\\
&\leq& C\int\limits_0^{t}\int\limits_{\mR^d}\frac{K(t-s,z)}{\sqrt{t-s}}|f(s,x-z)-f(s,y-z)|dzds \nonumber\\
&\leq&C \phi(|x-y|)\sqrt{t}\leq C\phi(|x-y|). \end{eqnarray} Then $\mathcal{T}$ maps $\mathcal{H}$ into $\mathcal{H}$\,. Moreover for sufficient small $T=T_{0}$, a direct verification yields that the mapping $\mathcal{T}$ is contractive. Then there is a unique $u\in \mathcal{H}$ such that $u=\mathcal{T}u$ and similar argument as the case $g=0$ yields the existence and unique strong solutions of the Cauchy problem (\ref{2.1}). Now by the classical extension technique we construct a strong solution on $[0,T]$ for any given $T>0$ and get inequalities (\ref{2.6}) and (\ref{2.7}) on $[0,T]$.
(ii) Let $u^n$ be the unique strong solution of (\ref{2.1}) with $f$ and $g$ replaced by $f^n$ and $g^n$ respectively. Since $f^n$ and $g^n$ are smooth in spatial variable, $u^n$ is smooth in spatial variable as well uniformly in time. Since $f$ and $g$ are bounded, $f^n$ and $g^n$ are bounded with \begin{eqnarray*} && \ \
\sup_n\|f^n\|_{L^\infty([0,T];\cC_b(\mR^d))}\leq \|f\|_{L^\infty([0,T];\cC_b(\mR^d))}, \\ && \sup_n\|g^n\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}\leq \|g\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}. \end{eqnarray*} Due to (\ref{2.6}) \begin{eqnarray*}
\sup_{n\geqslant}\def\leq{\leqslant 1}\|u^n\|_{L^\infty([0,T];\cC^2_b({\mathbb R}^d))} \leq C(d,T)(1+\|f\|_{L^\infty([0,T];\cC_b(\mR^d))}+
\|g\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}). \end{eqnarray*}
Moreover for every $x,y\in \mR^d$ ($0<|x-y|<r_0$) and every $t\in [0,T]$ \begin{eqnarray}\label{2.30}
|f^n(t,x)-f^n(t,y)|&=&\Bigg|\int\limits_{{\mathbb R}^d}\varrho_n(z) [f(t,x-z)-f(t,y-z)]dz\Bigg|
\nonumber\\&\leq& \phi(|x-y|) \int\limits_{{\mathbb R}^d}\varrho_n(z)dz= \phi(|x-y|), \end{eqnarray}
and the above estimate is true for $g^n$ as well, so (\ref{2.7}) holds uniformly in $n$.
By Lemma \ref{lem2.1} \begin{eqnarray}\label{2.31} u_n(t,x)-u(t,x)&=&\int\limits_0^tK(t-s,\cdot)\ast [g^n(s,\cdot)\cdot \nabla u^n(s,\cdot)-g(s,\cdot)\cdot \nabla u(s,\cdot)](x)ds\nonumber \\ && +\int\limits_0^tK(t-s,\cdot)\ast [f^n(s,\cdot)-f(s,\cdot)](x)ds. \end{eqnarray} Then by a Gronwall type argument, we arrive at \begin{eqnarray}\label{2.32}
\| u^n-u\|_{L^\infty([0,T];\cC_b^1({\mathbb R}^d))}\leq C(\|f^n-f\|_{L^\infty([0,T];\cC_b(\mR^d))}+
\|g^n-g\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}), \end{eqnarray} where $C>0$ is independent of $n$.
On the other hand, for $f^n$ (and $g^n$) we have \begin{eqnarray}\label{2.33}
\sup_{(t,x)\in [0,T]\times\mR^d}|f^n(t,x)-f(t,x)|&=&\sup_{(t,x)\in [0,T]\times\mR^d}\Bigg|\int\limits_{{\mathbb R}^d}\varrho_n(z) [f(t,x-z)-f(t,x)]dz\Bigg|
\nonumber\\&\leq& C \int\limits_{{\mathbb R}^d}\varrho_n(z)\phi(|z|)dz\rightarrow 0, \ as \ n\rightarrow \infty. \end{eqnarray} Combining (\ref{2.32}) and (\ref{2.33}) \begin{eqnarray}\label{2.34}
\lim_{n\rightarrow\infty}\| u^n-u\|_{L^\infty([0,T];\cC_b^1({\mathbb R}^d))}=0. \end{eqnarray}
It remains to show the convergence for $\nabla^2 u^n$. For every $1\leq i,j\leq d$, every $t\in (0,T]$ and every $x\in\mR^d$ \begin{eqnarray}\label{2.35}
&&\Big|\partial^2_{x_i,x_j}u^n(t,x)-\partial^2_{x_i,x_j}u(t,x)\Big|\nonumber\\
&=&\Bigg|\int\limits_0^t\int\limits_{{\mathbb R}^d}\partial^2_{x_i,x_j}K(t-s,x-y)[g^n(s,y)\cdot\nabla u^n(s,y)-g(s,y)\cdot\nabla u(s,y)] dyds\nonumber\\
&& +\int\limits_0^t\int\limits_{{\mathbb R}^d}\partial^2_{x_i,x_j}K(t-s,x-y)[f^n(s,y)-f(s,y)]dyds\Bigg|\nonumber\\
&\leq&\Bigg|\int\limits_0^t\int\limits_{{\mathbb R}^d}\partial^2_{x_i,x_j}K(t-s,x-y)[g^n(s,y)\cdot\nabla u^n(s,y)-g(s,y)\cdot\nabla u(s,y)] dyds\Bigg|\nonumber\\
&& +\Bigg|\int\limits_0^t\int\limits_{{\mathbb R}^d}\partial^2_{x_i,x_j}K(t-s,x-y)[f^n(s,y)-f(s,y)]dyds\Bigg| =:H^{n}_1(t,x)+H^{n}_2(t,x). \end{eqnarray}
First for $H^{n}_2$ we have \begin{eqnarray}\label{2.36} H^{n}_2(t,x)
&=& \Bigg|\int\limits_0^t\int\limits_{|x-y|\geqslant}\def\leq{\leqslant (t-s)^\theta }\partial^2_{x_i,x_j}K(t-s,x-y)[f^n(s,y)-f^n(s,x)-f(s,y)+f(s,x)]dyds\nonumber\\&&+\int\limits_0^t
\int\limits_{|x-y|<(t-s)^\theta}\partial^2_{x_i,x_j}K(t-s,x-y)[f^n(s,y)-f^n(s,x)-f(s,y)+f(s,x)]dyds\Bigg| \nonumber\\
&\leq & C\|f^n-f\|_{L^\infty([0,T]\times\mR^d)}\nonumber\\&&+C\int\limits_0^t\int\limits_{|x-y|<(t-s)^\theta\wedge r_0}\frac{1}{t-s}K(t-s,x-y)|f^n(s,y)-f^n(s,x)-f(s,y)+f(s,x)|dyds\nonumber\\&&+C\int\limits_0^t
\int\limits_{(t-s)^\theta\wedge r_0 \leq |x-y|\leq (t-s)^\theta
}\frac{1}{t-s}K(t-s,x-y)\nonumber\\ &&\quad \times |f^n(s,y)-f^n(s,x)-f(s,y)+f(s,x)|dyds, \end{eqnarray} for $\theta\in (0,1/2)$. By (\ref{2.5}) and (\ref{2.30}), \begin{eqnarray*}
|f^n(s,y)-f^n(s,x)-f(s,y)+f(s,x)|\leq C\phi(|x-y|), \quad {\rm for \; all}\ \ |x-y|<r_0. \end{eqnarray*} From (\ref{2.16}) \begin{eqnarray*}
\frac{1}{t-\cdot}K(t-\cdot,\cdot-y) \phi(|\cdot-x|) \in L^1(O), \quad O=\{(s,y)| \ s\in [0,t], \ |x-y|< (t-s)^\alpha\wedge r_0\}. \end{eqnarray*} By the assumption on $f$ and the definition of $f_n$, the integrand of the last term in (\ref{2.36}) is integrable. Then by the dominated convergence theorem and (\ref{2.33}), we have \begin{eqnarray*} \lim_{n\rightarrow\infty}\sup_{(t,x)\in [0,T]\times\mR^d}H^{n}_{2}(t, x)=0. \end{eqnarray*} Noticing that $g^n(s,y)\cdot\nabla u^n(s,y)$ and $g(s,y)\cdot\nabla u(s,y)$ are of class $L^\infty([0,T];\cC_b(\mR^d))$, satisfy (\ref{2.5}) and \begin{eqnarray*}
\lim_{n\rightarrow\infty}\|g^n\cdot\nabla u^n-g\cdot\nabla u\|_{L^\infty([0,T];\cC_b({\mathbb R}^d))}=0. \end{eqnarray*} So the argument of convergence for $H^{n}_1$ is same as that for $H^{n}_2$, then we have \begin{eqnarray}\label{2.37}
\lim_{n\rightarrow\infty}\|\nabla^2u^n-\nabla^2u\|_{L^\infty([0,T];\cC_b({\mathbb R}^d))}=0, \end{eqnarray} and combining (\ref{2.34}) and (\ref{2.37}), we show (\ref{2.9}). $\Box$
\begin{remark} \label{rem2.1} There are some further properties of $\nabla^2 u$ and $\partial_tu$ by some estimates on the righthand side of (\ref{2.7}). We consider a nonnegative Dini function $\phi$. For every $r_0>0$, then \begin{eqnarray}\label{2.38} \int\limits_0^{r_0}\frac{\phi(r)}{r}dr<+\infty. \end{eqnarray} On the other hand, \begin{eqnarray*} \varepsilon\int\limits_{\varepsilon<r\leq r_0}\frac{\phi(r)}{r^2}dr=\int\limits_{0<r\leq r_0}\frac{\phi(r)}{r^2}1_{(\varepsilon,r_0)}(r)\varepsilon dr=:\int\limits_{0<r\leq r_0}h_\varepsilon(r) dr. \end{eqnarray*} Since $h_\varepsilon(r)\leq \phi(r)/r$, by the dominated convergence theorem, \begin{eqnarray}\label{2.39} \lim_{\varepsilon\rightarrow 0}\varepsilon\int\limits_{\varepsilon<r\leq r_0}\frac{\phi(r)}{r^2}dr=\int\limits_{0<r\leq r_0}\lim_{\varepsilon\rightarrow 0}h_\varepsilon(r) dr=0. \end{eqnarray} By (\ref{2.39}), all the terms in the righthand side of (\ref{2.7}) are meaningful. Moreover, by the above estimates, $\nabla^2 u$ and $\partial_tu$ are uniformly continuous in spatial variable uniformly in time. \end{remark}
\begin{remark} \label{rem2.2} Theorem \ref{the2.1} generalizes the existing results for the Cauchy problem (\ref{2.1}). By the classical parabolic theory (\cite[Ch. 4]{LSU}, \cite{Kry}), if $f\in L^p([0,T]\times{\mathbb R}^d),\; g\in L^p([0,T]\times{\mathbb R}^d;\mR^d)$ with $p\in (1,+\infty)$, $u\in L^p([0,T];W^{2,p}({\mathbb R}^d))\cap W^{1,p}([0,T];L^p({\mathbb R}^d))$. However, if $f\in L^\infty([0,T]\times{\mathbb R}^d), g\in L^\infty([0,T]\times{\mathbb R}^d;\mR^d)$, in general $u\not\in L^\infty([0,T];W^{2,\infty}({\mathbb R}^d))\cap W^{1,\infty}([0,T];L^\infty({\mathbb R}^d))$. Nevertheless, recent result~\cite[Lemma 2.1]{WDGL} shows that if $f\in L^p([0,T];{\mathcal C}^\varsigma_b({\mathbb R}^d)), g\in L^p([0,T];{\mathcal C}^\varsigma_b({\mathbb R}^d;\mR^d))$ with $p\in (1,+\infty], \varsigma\in (0,1)$, and $\varsigma>2/p$ \begin{eqnarray*} u\in L^\infty([0,T];{\mathcal C}^{2,\varsigma-\frac{2}{p}}_b({\mathbb R}^d))\cap W^{1,\infty}([0,T];{\mathcal C}^{\varsigma-\frac{2}{p}}_b({\mathbb R}^d)). \end{eqnarray*} Here, by assuming the Dini continuity on $f$ and $g$, we establish the $W^{2,\infty}$ estimates as well. Moreover, the second derivatives for spatial variable are also uniformly continuous. \end{remark}
\begin{remark} \label{rem2.3} (i) From (\ref{2.7}), if there exists $\eta>0$ such that for $r>0$ small enough \begin{eqnarray}\label{2.40} \eta\int\limits_0^r\frac{\phi(s)}{s}ds=\phi(r) \end{eqnarray} and \begin{eqnarray}\label{2.41} r\int\limits_{r<s\leq r_0}\frac{\phi(s)}{s^2}ds\leq C\phi(r), \end{eqnarray} then the maximum Dini regularity for (\ref{2.1}) holds. Now, we solve the integral equation (\ref{2.40}) and get $\phi(r)=C_0r^\eta$. Since the Dini function $\phi$ is given in (\ref{2.5}), $\eta\in (0,1)$ is the best choice. Then, \begin{eqnarray*} r\int\limits_{r<s\leq r_0}\frac{\phi(s)}{s^2}ds=\frac{C_0}{1-\eta}[ r^\eta-rr_0^{\eta-1}]\leq \frac{\phi(r)}{1-\eta}, \end{eqnarray*} for $r$ is sufficiently small. And now, we recover the classical Schauder theory for parabolic equations of second order.
(ii) If $\phi$ is only Dini continuous but not H\"{o}lder continuous, the functions given in the first and third terms on the right hand side of (\ref{2.7}) will not preserve the same regularity as $\phi$, and thus the maximum regularity for (\ref{2.1}) may be not true. For example, we choose $r_0=1/2$ and $\phi(r)=C|\log(r)|^{-\alpha}$ with some $\alpha>1$, then there exist two positive real numbers $C_1(d,\alpha)$ and $C_2(d,\alpha)$ such that for $|x-y|>0$ sufficiently small \begin{eqnarray*}
\frac{C_1(d,\alpha)}{|\log(|x-y|)|^{\alpha-1}}\leq \int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr\leq \frac{C_2(d,\alpha)}{|\log(|x-y|)|^{\alpha-1}}. \end{eqnarray*}
However, $\phi(r)=o(|\log(r)|^{1-\alpha})$ as $r\rightarrow 0$. Therefore, if $\alpha\in (1,2)$, $\nabla^2u(t,\cdot)$ may be only uniformly continuous but not Dini continuous. \end{remark}
To make the maximum regularity for (\ref{2.1}) true, we need introduce the following notion. \begin{definition} \label{def2.2} Let $\psi: \mR_+\rightarrow \bar{{\mathbb R}}_+$ be a monotone continuous function. Suppose $\phi: \mR_+\rightarrow \mR_+$ is continuous and $\phi(0)=0$.
(i) We call $\phi$ a H\"{o}lder-Dini function with H\"{o}lder exponent $\vartheta\in (0,1)$ if $r^{-\vartheta}\phi(r)=\psi(r)$ is Dini but not H\"{o}lder continuous on $\mR_+$;
(ii) We call $\phi$ a strong H\"{o}lder function with H\"{o}lder exponent $\vartheta\in (0,1)$ if
(1) for every $\zeta\in (0,\vartheta)$, $r^{-\zeta}\phi(r)$ is H\"{o}lder continuous on $\mR_+$;
(2) $r^{-\vartheta}\phi(r)=\psi(r)$ is strictly monotone increasing and continuous but not H\"{o}lder continuous on $\mR_+$;
(3) $\psi(r)\downarrow0$ as $r \downarrow 0$;
(iii) We call $\phi$ a weak H\"{o}lder function with H\"{o}lder exponent $\vartheta\in (0,1)$ if
(1) for every $\zeta\in (0,\vartheta)$, $r^{-\zeta}\phi(r)$ is H\"{o}lder continuous on $\mR_+$ and $r^{-\zeta}\phi(r)\rightarrow 0$ as $r \downarrow 0$;
(2) $r^{-\vartheta}\phi(r)=\psi(r)$ is strictly monotone decreasing and continuous for $r\in (0,\infty)$;
(3) $\psi(r)\uparrow +\infty$ as $r \downarrow 0$.
A measurable function $h:\mR^{d}\rightarrow \mR$ is said to be H\"{o}lder-Dini (strong H\"{o}lder or weak H\"{o}lder) continuous with H\"{o}lder exponent $\vartheta\in (0,1)$ if \begin{eqnarray}\label{2.42}
|h(x)-h(y)|\leq \phi(|x-y|), \end{eqnarray} where $\phi$ is a H\"{o}lder-Dini (strong H\"{o}lder or weak H\"{o}lder) function with H\"{o}lder exponent $\vartheta$. \end{definition}
If $f$ and $g$ are H\"{o}lder-Dini or strong H\"{o}lder or weak H\"{o}lder continuous, then the maximum regularity for (\ref{2.1}) is still true. \begin{corollary} \label{cor2.1} Let $\phi$ be a H\"{o}lder-Dini or strong H\"{o}lder or weak H\"{o}lder function with H\"{o}lder exponent $\vartheta\in (0,1)$. Let $f\in L^\infty([0,T];\cC_b(\mR^d))$ and $g\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$ such that (\ref{2.5}) holds. Then there exists a unique strong solution $u$ to (\ref{2.1}). Moreover, if $\phi^\prime$ is also continuous in $(0,\infty)$, then for every $1\leq i,j\leq d$ and every $x,y\in \mR^d$, there is a constant $C(d,T)$ such that \begin{eqnarray}\label{2.43}
|\partial^2_{x_i,x_j}u(t,x)-\partial^2_{y_i,y_j}u(t,y)|\leq C(d,T)\phi(|x-y|), \quad for \; all \ t\in [0,T]. \end{eqnarray} \end{corollary}
Before proving the above result, we need a useful lemma.
\begin{lemma} [L'Hospital's rule \cite{Daw}, p. 346] \label{lem2.2} Suppose that we have one of the following cases, \begin{eqnarray*} \lim_{r\rightarrow \lambda}\frac{f_1(r)}{f_2(r)} = \frac{0}{0} \quad or \quad \lim_{r\rightarrow \lambda}\frac{f_1(r)}{f_2(r)} = \frac{\pm \infty}{\pm \infty} , \end{eqnarray*} where $\lambda$ can be any real number, infinity or negative infinity, In these cases, we have \begin{eqnarray*} \lim_{r\rightarrow \lambda}\frac{f_1(r)}{f_2(r)} =\lim_{r\rightarrow \lambda}\frac{f_1^\prime(r)}{f_2^\prime(r)}. \end{eqnarray*} \end{lemma}
\noindent\textbf{Proof of Corollary \ref{cor2.1}.} We only need to prove (\ref{2.43}) for $|x-y|$ sufficiently small. By (\ref{2.7}) for every $1\leq i,j\leq d$ and every $x,y\in \mR^d$ ($|x-y|<r_0$), there is a constant $C(d,T)$ such that \begin{eqnarray}\label{2.44}
&&|\partial^2_{x_i,x_j}u(t,x)-\partial^2_{y_i,y_j}u(t,y)| \nonumber \\ &\leq &C\left[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr\right]. \end{eqnarray} From (\ref{2.44}) if \begin{eqnarray}\label{2.45} \limsup_{r\rightarrow 0} \frac{\int\limits_0^r\frac{\phi(s)}{s}ds}{\phi(r)}<+\infty\quad \mbox{and} \quad \limsup_{r\rightarrow 0} \frac{r\int\limits_r^{r_0}\frac{\phi(s)}{s^2}ds}{\phi(r)} <+\infty, \end{eqnarray}
then for $|x-y|$ small enough \begin{eqnarray*}
\max\{|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr, \int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr\}\leq C \phi(|x-y|), \end{eqnarray*} thus (\ref{2.43}) is proved.
Now let us check (\ref{2.45}). Since $\phi$ is a H\"{o}lder-Dini or strong H\"{o}lder or weak H\"{o}lder function with H\"{o}lder exponent $\vartheta\in (0,1)$, we can rewrite $\phi(r)$ by $r^{\vartheta}\psi(r)$ with $\psi$ a monotone continuous function. Applying Lemma \ref{lem2.2}, then \begin{eqnarray}\label{2.46} \lim_{r\rightarrow 0} \frac{\int\limits_0^r\frac{\phi(s)}{s}ds}{\phi(r)}=\lim_{r\rightarrow 0} \frac{\int\limits_0^rs^{\vartheta-1}\psi(s)ds}{r^{\vartheta}\psi(r)}=\lim_{r\rightarrow 0} \Big[\vartheta+\frac{r\psi^\prime(r)}{\psi(r)}\Big]^{-1} \end{eqnarray} and \begin{eqnarray}\label{2.47} \lim_{r\rightarrow 0} \frac{r\int\limits_r^{r_0}s^{\vartheta-2}\psi(s)ds}{r^{\vartheta}\psi(r)}=\lim_{r\rightarrow 0} \frac{\int\limits_r^{r_0}s^{\vartheta-2}\psi(s)ds}{r^{\vartheta-1}\psi(r)}=\lim_{r\rightarrow 0} \Big[1-\vartheta-\frac{r\psi^\prime(r)}{\psi(r)}\Big]^{-1}. \end{eqnarray}
If $\phi$ is a H\"{o}lder-Dini or strong H\"{o}lder function, then \begin{eqnarray}\label{2.48} \lim_{r\rightarrow 0} \log(\psi(r))=-\infty, \end{eqnarray} and if $\phi$ is a weak H\"{o}lder function, then \begin{eqnarray}\label{2.49} \lim_{r\rightarrow 0} \log(\psi(r))=+\infty. \end{eqnarray} From (\ref{2.48}) and (\ref{2.49}), by using Lemma \ref{lem2.2} again, we gain \begin{eqnarray}\label{2.50} \lim_{r\rightarrow 0} \frac{r\psi^\prime(r)}{\psi(r)}= \left\{\begin{array}{ll} \lim\limits_{r\rightarrow 0} \frac{\log(\psi(r))}{\log(r)}, & \hbox{when $\phi$ is H\"{o}lder-Dini or strong H\"{o}lder;} \\ -\lim\limits_{r\rightarrow 0} \frac{\log(\psi(r))}{\log(r^{-1})}, & \hbox{when $\phi$ is weak H\"{o}lder.}\end{array}\right. \end{eqnarray}
When $\phi$ is a H\"{o}lder-Dini or strong H\"{o}lder function, then for every $\zeta>0$, $\psi(r)\geqslant}\def\leq{\leqslant r^{\zeta}$ as $r\rightarrow 0$. Thus, there exists $\delta=\delta(\zeta)>0$ such that \begin{eqnarray*} [\psi(r)]^{-1}\leq r^{-\zeta}, \quad for\; all \ r\in (0,\delta]. \end{eqnarray*} It follows that \begin{eqnarray}\label{2.51} \limsup_{r\rightarrow 0} \frac{\log(\psi(r))}{\log(r)}=\limsup_{r\rightarrow 0} \frac{\log([\psi(r)]^{-1})}{\log(r^{-1})}\leq \limsup_{r\rightarrow 0} \frac{\log(r^{-\zeta})}{\log(r^{-1})}=\zeta. \end{eqnarray} Since $\zeta>0$ is arbitrary and $\log(\psi(r))/\log(r)\geqslant}\def\leq{\leqslant 0$, from (\ref{2.50}) and (\ref{2.51}), we conclude \begin{eqnarray}\label{2.53} \lim_{r\rightarrow 0} \frac{r\psi^\prime(r)}{\psi(r)}=\lim_{r\rightarrow 0} \frac{\log(\psi(r))}{\log(r)}=0. \end{eqnarray}
When $\phi$ is a weak H\"{o}lder function, then for every $\zeta_1>0$, $r^{\zeta_1}\psi(r)\rightarrow0$ as $r\rightarrow 0$. Thus, there exists $\delta_1=\delta_1(\zeta_1)>0$ such that \begin{eqnarray*} \psi(r)\leq r^{-\zeta_1}, \quad for\; all \ r\in (0,\delta_1], \end{eqnarray*} which implies that \begin{eqnarray}\label{2.54} \limsup_{r\rightarrow 0} \frac{\log(\psi(r))}{\log(r^{-1})}\leq \limsup_{r\rightarrow 0} \frac{\log(r^{-\zeta_1})}{\log(r^{-1})}=\zeta_1. \end{eqnarray} Since $\zeta_1>0$ is arbitrary and $\log(\psi(r))/\log(r^{-1})\geqslant}\def\leq{\leqslant 0$, from (\ref{2.50}) and (\ref{2.54}), we conclude \begin{eqnarray}\label{2.55} \lim_{r\rightarrow 0} \frac{r\psi^\prime(r)}{\psi(r)}=-\lim_{r\rightarrow 0} \frac{\log(\psi(r))}{\log(r^{-1})}=0. \end{eqnarray}
Combining (\ref{2.53}) and (\ref{2.55}), then (\ref{2.45}) holds. From this we complete the proof. $\Box$
\begin{example} \label{exa2.1} We choose $\phi(r)=Cr^\vartheta|\log(r)|^\alpha$ with $\vartheta\in (0,1)$ and $\alpha\in {\mathbb R}$, then $\phi$ is a H\"{o}lder-Dini function if $\alpha<-1$, a strong H\"{o}lder function if $\alpha<0$, H\"{o}lder continuous if $\alpha=0$, a weak H\"{o}lder function if $\alpha>0$. Using Lemma \ref{lem2.2}, we have \begin{eqnarray*}
\lim_{r\rightarrow 0} \frac{\int\limits_0^r\frac{\phi(s)}{s}ds}{\phi(r)}=\lim_{r\rightarrow 0} \frac{\phi(r)}{r\phi^\prime(r)}=\lim_{r\rightarrow 0} \frac{\phi(r)}{\phi(r)[\vartheta-\alpha |\log(r)|^{-1}]} =\frac{1}{\vartheta} \end{eqnarray*} and \begin{eqnarray*} \lim_{r\rightarrow 0} \frac{r\int\limits_r^{r_0}\frac{\phi(s)}{s^2}ds}{\phi(r)}&=&-\lim_{r\rightarrow 0} \frac{\phi(r)}{r\phi^\prime(r)}+ \lim_{r\rightarrow 0} \frac{\int\limits_r^{r_0}\frac{\phi(s)}{s^2}ds}{\phi^\prime(r)} \nonumber\\ &=&-\frac{1}{\vartheta}-\lim_{r\rightarrow 0} \frac{\phi(r)}{r^2\phi^{\prime\prime}(r)}
\nonumber\\ &=&-\frac{1}{\vartheta}-\lim_{r\rightarrow 0} \frac{\phi(r)}{\phi(r)[\vartheta(\vartheta-1)-\alpha (2\vartheta-1)|\log(r)|^{-1}-\alpha(1-\alpha)|\log(r)|^{-2}]}\nonumber\\ &=&-\frac{1}{\vartheta}-\frac{1}{\vartheta(\vartheta-1)}=\frac{1}{1-\vartheta}. \end{eqnarray*} Therefore, for $r$ small enough, we gain \begin{eqnarray}\label{2.56} \int\limits_0^r\frac{\phi(s)}{s}ds\leq \frac{1+\vartheta}{\vartheta} \phi(r), \quad r\int\limits_r^{r_0}\frac{\phi(s)}{s^2}ds\leq \frac{2-\vartheta}{1-\vartheta}\phi(r). \end{eqnarray} Combining (\ref{2.7}) and (\ref{2.56}), the maximum regularity for (\ref{2.1}) is preserved. This result as we know is new. \end{example}
\section{Stochastic flows for SDEs with bounded and Dini drift}\label{sec3}\setcounter{equation}{0}
Given real number $T>0$\,, for $s\in[0,T]$ and $x\in {\mathbb R}^d$, consider the following SDE \begin{eqnarray}\label{3.1}
dX(s,t)=b(t,X(s,t))dt+dB(t), \quad t\in(s,T], \quad X(s,t)|_{t=s}=x\,. \end{eqnarray} We intend to show the existence of a stochastic flow for equation (\ref{3.1})\,. First we give the following definition.
\begin{definition} [\cite{Kun2}, p. 114] \label{def2.1} A stochastic homeomorphisms flow of class $\mathcal{C}^{\beta}$ with $\beta\in (0,1)$ on $(\Omega, \mathcal{F},{\mathbb P}, (\mathcal{F}_t)_{0\leq t\leq T})$ associated to (\ref{3.1}) is a map $(s,t,x,\omega) \rightarrow X(s,t,x)(\omega)$, defined for $0\leq s \leq t \leq T, \ x\in {\mathbb R}^d, \ \omega \in \Omega$ with values in ${\mathbb R}^d$, such that
(i) the process $\{X(s,\cdot,x)\}= \{X(s,t,x), \ t\in [s,T]\}$ is a continuous $\{\mathcal{F}_{s,t}\}_{s\leq t\leq T}$-adapted solution of (\ref{3.1}), for every $s\in [0,T],\ x \in {\mathbb R}^d$;
(ii) the functions $X(s,t,x)$ and $X^{-1}(s,t,x)$ are continuous in $(s,t,x)$ and are of class ${\mathcal C}^\beta$ in $x$ uniformly in $(s,t)$\,, ${\mathbb P}$-a.s., for all $0\leq s \leq t \leq T$;
(iii) $X(s,t,x)=X(r,t,X(s,r,x))$ for all $0\leq s\leq r \leq t \leq T$, $x\in {\mathbb R}^d$\,, ${\mathbb P}$-a.s., and $X(s,s,x)=x$\,.
Further if
(iv) for all $0\leq s \leq t \leq T$, the functions $\nabla X(s,t,x)$ and $\nabla X^{-1}(s,t,x)$ are continuous in $(s,t,x)$\,, ${\mathbb P}$-a.s.,
it is called a stochastic diffeomorphisms flow. \end{definition}
Now we state the main result of the section. \begin{theorem} \label{the3.1} Let $b\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$ with integer $d\geqslant}\def\leq{\leqslant 1$. Suppose that $r_0\in (0,1)$ and there is a Dini function $\phi$ such that for every $x\in\mR^d$ \begin{eqnarray}\label{3.2}
|b(t,x)-b(t,y)|\leq \phi(|x-y|), \quad {\rm for\; all} \ \ y\in B_{r_0}(x), \ t\in [0,T]. \end{eqnarray} In addition that for every $p\geqslant}\def\leq{\leqslant 1$\,, there is a small enough positive real number $\delta=\delta(p)<r_0$ such that $F_\delta^p(\cdot)$ is increasing and concave on $[0,\delta]$, where \begin{eqnarray}\label{3.3} F_\delta(r)=\int\limits_{s\leq r}\frac{\phi(s)}{s}ds+ \phi(r)+r\int\limits_{r<s\leq \delta}\frac{\phi(s)}{s^2}ds+r, \quad \ r\in [0,\delta]. \end{eqnarray}
(i) For every $s\in [0,T ]$ and $ x\in {\mathbb R}^d$, SDE (\ref{3.1}) has a unique continuous adapted solution $\{X(s,t,x)(\omega), \ t\in [s,T ], \ \omega \in\Omega\}$, which forms a stochastic diffeomorphisms flow. For every $p\geqslant}\def\leq{\leqslant 1$, there is a constant $C(p,d,T)>0$ such that \begin{eqnarray}\label{3.4}
\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}|X(s,t,x)|^p+\sup_{x\in\mR^d}\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}\|\nabla X(s,t,x)\|^p\leq C(p,d,T). \end{eqnarray} Moreover, for every $p\geqslant}\def\leq{\leqslant 1$ and every $x,y\in \mR^d$, \begin{eqnarray}\label{3.5}
\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}|X(s,t,x)-X(s,t,y)|^p]\leq C|x-y|^p \end{eqnarray} and \begin{eqnarray}\label{3.6}
&&\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}\|\nabla X(s,t,x)-\nabla X(s,t,y)\|^p] \nonumber \\
&\leq &C\Bigg[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr\Bigg]^p1_{|x-y|<r_0}+C|x-y|^p. \end{eqnarray}
(ii) Let $\varrho_n$ be given in (\ref{2.8}) and $b^n(t,x)=b\ast \varrho_n(t,x)$. Let $X^n$ be the stochastic flows corresponding to the vector field $b^n$. Then for every $p\geqslant}\def\leq{\leqslant 1$, \begin{eqnarray}\label{3.7}
\lim _{n\rightarrow\infty}\sup_{x\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}\left[\sup_{s\leq t \leq T}|X^n(s,t,x)-X(s,t,x)|^p\right]=0 \end{eqnarray} and \begin{eqnarray}\label{3.8}
\lim_{n\rightarrow \infty}\sup_{x\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}\left[\sup_{s\leq t \leq T}\|\nabla X^n(s,t,x)-\nabla X(s,t,x)\|^p\right]=0. \end{eqnarray} \end{theorem} \noindent \textbf{Proof.} (i) Let $\lambda>0$ be a real number. Consider the following backward heat equation \begin{eqnarray}\label{3.9} \left\{ \begin{array}{ll} \partial_{t}U(t,x) +\frac{1}{2}\Delta U(t,x)+b(t,x)\cdot \nabla U(t,x)\\ \quad \quad= \lambda U(t,x)-b(t,x), \quad (t,x)\in [0,T)\times {\mathbb R}^d, \\ U(T,x)=0, \quad x\in{\mathbb R}^d.
\end{array} \right. \end{eqnarray} By Theorem \ref{the2.1}, the above Cauchy problem has a unique solution $U\in L^\infty([0,T];\cC^2_b({\mathbb R}^d;{\mathbb R}^d))\cap W^{1,\infty}([0,T];\mathcal{C}_b({\mathbb R}^d;{\mathbb R}^d))$. Moreover, (\ref{2.6}) is true and the constant $C$ in (\ref{2.6}) is independent of $\lambda$. Further, for every $x,y\in \mR^d$ and $t\in [0,T)$, there is a constant $C(d,T)$ such that \begin{eqnarray}\label{3.10}
&&|\partial^2_{x_i,x_j}U(t,x)-\partial^2_{y_i,y_j}U(t,y)| \nonumber \\ &\leq &C(d,T)\Bigg[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr\Bigg]1_{|x-y|<r_0}\nonumber \\ &&+C(d,T)|x-y|. \end{eqnarray}
By Lemma \ref{lem2.1}, the unique strong solution $U$ has the following representation \begin{eqnarray*} U(t,x)=\int\limits_0^{T-t}e^{-\lambda r}K(r,\cdot)\ast[ b(t+r,\cdot)+b(t+r,\cdot)\cdot \nabla U(t+r,\cdot)](x)dr. \end{eqnarray*} For every $1\leq i\leq d$, every $x\in \mR^d$ and $t\in [0,T]$ \begin{eqnarray}\label{3.11}
|\partial_{x_i}U(t,x)|&=&\Bigg|\int\limits_0^{T-t}e^{-\lambda r}dr\int\limits_{{\mathbb R}^d}\partial_{x_i}K(r,x-z)[b(t+\tau,z)+b(t+r,z)\cdot \nabla U(t+r,z)]dz\Bigg|
\nonumber\\&\leq&C\|b\|_{L^\infty([0,T];\mathcal{C}_b({\mathbb R}^d;{\mathbb R}^d))}(1+\|b\|_{L^\infty([0,T];\mathcal{C}_b({\mathbb R}^d;{\mathbb R}^d))}) \int\limits_0^Tr^{-\frac{1}{2}}e^{-\lambda r} dr
\nonumber\\&\leq&C\|b\|_{L^\infty([0,T];\mathcal{C}_b({\mathbb R}^d;{\mathbb R}^d))}(1+\|b\|_{L^\infty([0,T];\mathcal{C}_b({\mathbb R}^d;{\mathbb R}^d))})\lambda^{-\frac{1}{3}}, \end{eqnarray} where the H\"{o}lder inequality is applied in the last inequality. Then letting $\lambda$ tend to infinity in~(\ref{3.11}) yields \begin{eqnarray}\label{3.12}
\|\nabla U\|_{L^\infty([0,T];{\mathcal C}_b({\mathbb R}^d;\mR^{d\times d}))}\longrightarrow 0. \end{eqnarray} Therefore, there is a large real number $\lambda_0>0$ such that if $\lambda\geqslant}\def\leq{\leqslant \lambda_0$ \begin{eqnarray}\label{3.13}
\|\nabla U\|_{L^\infty([0,T];{\mathcal C}_b({\mathbb R}^d;\mR^{d\times d}))}\leq \frac12. \end{eqnarray} Now for $\lambda\geqslant}\def\leq{\leqslant \lambda_{0}$, define $\gamma(t,x)=x+U(t,x)$ \begin{eqnarray*}
\frac12\leq \|\nabla \gamma\|_{L^\infty([0,T];{\mathcal C}_b({\mathbb R}^d;\mR^{d\times d}))}\leq \frac32. \end{eqnarray*} Then $\gamma(t)$ forms a nonsingular diffeomorphism of class $\mathcal{C}^2$ uniformly in $t\in [0,T]$ by the classical Hadamard theorem (\cite[p.330]{Protter}). Moreover, for every $t\in [0,T]$, the inverse of $\gamma(t)$ (denoted by~$\gamma^{-1}(t)$) has bounded first and second spatial derivatives, uniformly in $t\in [0,T]$, and \begin{eqnarray}\label{3.14}
\frac23\leq\|\nabla \gamma^{-1}\|_{L^\infty([0,T];{\mathcal C}_b({\mathbb R}^d;\mR^{d\times d}))}\leq 2. \end{eqnarray}
Noticing that if $X(s,t)$ satisfies SDE (\ref{3.1}), $Y(s,t)=X(s,t)+U(t,X(s,t))$ $(=:\gamma(t,X(s,t)))$ satisfies the following SDE \begin{eqnarray}\label{3.15} \left\{
\begin{array}{ll} d Y(s,t)=\lambda U(t,\gamma^{-1}(t,Y(s,t)))dt+ [I+\nabla U(t,\gamma^{-1}(t,Y(s,t)))] dB(t)\\ \quad\quad\quad\ \ =:\tilde{b}(t,Y(s,t))dt+\tilde{\sigma}(t,Y(s,t))dB(t), \ t\in(s,T], \\
Y(s,t)|_{t=s}=y,
\end{array} \right. \end{eqnarray} and vice versa, (\ref{3.15}) and (\ref{3.1}) are equivalent. Now for equation (\ref{3.15})\,, the coefficients are globally Lipschitz continuous, by the result of Kunita~\cite[Theorem 4.3, p.227]{Kun1}\,, there is a unique stochastic homeomorphisms flow of class $\cC^\beta \ (\beta\in (0,1))$ defined by $Y(s,t, y)$\,. Moreover, for every $x,y\in \mR^d$ and every $p\geqslant}\def\leq{\leqslant 1$, there is a constant $C(p,T)>0$ such that \begin{eqnarray}\label{3.16}
\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}|Y(s,t,y)|^p\leq C(p,T) \end{eqnarray} and \begin{eqnarray}\label{3.17}
\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}|Y(s,t,x)-Y(s,t,y)|^p]\leq C(p,T)|x-y|^p. \end{eqnarray} On the other hand, by $X(s,t)=\gamma^{-1}(t,Y(s,t))$\,, (\ref{3.1}) also defines a unique stochastic homeomorphisms flow of class $\cC^\beta \ (\beta\in (0,1))$. Moreover by (\ref{3.16}) and (\ref{3.17}) for every $p\geqslant}\def\leq{\leqslant 1$ and every $x,y\in\mR^d$ \begin{eqnarray}\label{3.18}
&&\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}|X(s,t,x)|^p\nonumber \\ &\leq& 2^{p-1}\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}[|Y(s,t,\gamma(s,x))|^p+|\gamma^{-1}(t,Y(s,t,\gamma(s,x)))-\gamma^{-1}(t,\gamma(s,x))|^p]
\nonumber \\ &\leq& C(p,T)+2^{p-1}\|\nabla \gamma^{-1}\|^p_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))} \sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}|Y(s,t,\gamma(s,x))-\gamma(s,x)|^p\nonumber \\ &\leq& C(p,d,T) \end{eqnarray} and \begin{eqnarray}\label{3.19}
&&\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}|X(s,t,x)-X(s,t,y)|^p]\nonumber \\ &=& \sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}|\gamma^{-1}(t,Y(s,t,\gamma(s,x))) -\gamma^{-1}(t,Y(s,t,\gamma(s,y)))|^p]
\nonumber \\ &\leq& C \|\nabla \gamma^{-1}\|^p_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))} \sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}|Y(s,t,\gamma(s,x)) -Y(s,t,\gamma(s,y))|^p]
\nonumber \\ &\leq& C \|\nabla \gamma^{-1}\|^p_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))} |\gamma(s,x) -\gamma(s,y)|^p\nonumber \\ &\leq& C \|\nabla \gamma^{-1}\|^p_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla \gamma\|^p_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))} |x -y|^p\nonumber \\ &\leq&
C(p,d,T)|x-y|^p. \end{eqnarray}
Now we just need to prove the continuity of $\nabla_x X(s,t,x)$ and the inequality (\ref{3.6}). For this we consider $\nabla_y Y(s,t,y)$\,. First the stochastic flow $\{X(s,t)(\cdot)\}$ is weak differentiable, that is,~$X(s,t)(\cdot)\in L^2(\Omega;W^{1,p}_{loc}(\mR^d;\mR^d))$~(\cite[Theorem 3]{MNP}), so does the stochastic flow $\{Y(s,t)(\cdot)\}$. Then differentiate~$Y(s,t)$ with respect to the initial data and denoting the derivative by $\xi(s,t,y)$, we have \begin{eqnarray}\label{3.20} d \xi(s,t,y)&=&\lambda \nabla U(t,\gamma^{-1}(t,Y(s,t,y)))\nabla\gamma^{-1}(t,Y(s,t,y))\xi(s,t,y)dt\nonumber\\&&+\nabla^2 U(t,\gamma^{-1}(t,Y(s,t,y)))\nabla\gamma^{-1}(t,Y(s,t,y))\xi(s,t,y)d B(t)\nonumber\\ &=&:A_1(t,Y(s,t,y))\xi(s,t,y)dt+A_2(t,Y(s,t,y))\xi(s,t,y)d B(t), \end{eqnarray}
with $\xi(s,t,y)|_{t=s}=I$.
Notice that the equation (\ref{3.20}) is a linear SDE with bounded coefficients $A_{1}$\,, $A_{2}$ depending on the process $Y(s,t,y)$, by the Cauchy--Lipschitz theorem, there is a unique strong solution for equation (\ref{3.20})\,. Moreover for $d=1$, the unique strong solution is represented by \begin{eqnarray}\label{3.21} \xi(s,t,y)=\exp\Bigg(\int\limits_s^t[A_1(r,Y(s,r,y))-\frac{1}{2}A_2^2(r,Y(s,r,y))]dr +\int\limits_s^t A_2(r,Y(s,r,y))d B(r)\Bigg). \end{eqnarray} By (\ref{3.10}) and Remark \ref{rem2.1}, $\nabla^2 U(t,x)$ is uniformly continuous in $x$, then $A_1(s,r,Y(s,\cdot,\cdot))$ and $A_2(s,r,Y(s,\cdot,\cdot))$ are continuous in $[s,T]\times\mR^d$ almost surely. Then for $d=1$\,, the process $\xi(s,t,y)$ is continuous in $(s,t,y)$ almost surely, and for every $p\geqslant}\def\leq{\leqslant 1$ there is a constant $C(p,T)>0$ such that \begin{eqnarray}\label{3.22}
\sup_{y\in\mR^d}\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}\|\xi(s,t,y)\|^p\leq C(p,T). \end{eqnarray} For general $d>1$, the unique strong solution also has an obvious representation which is analogue of (\ref{3.21}), then using the same argument, the continuity of $\xi(s,t,y)$ in $(s,t,y)$ and the inequality (\ref{3.22}) also hold. Then using the relationship between $X$ and $Y$, and (\ref{3.18}), (\ref{3.4}) holds true.
For every $x,y\in\mR^d$, write $Y(s,t,x)$, $Y(s,t,y)$, $\xi(s,t,x)$, $\xi(s,t,y)$, $Y(s,t,x)-Y(s,t,y)$, $\xi(s,t,x)-\xi(s,t,y)$, $U(t,\gamma^{-1}(t,Y(s,t,x)))$ and $U(t,\gamma^{-1}(t,Y(s,t,y)))$ by $Y(x)$, $Y(y)$, $\xi(x)$, $\xi(y)$, $Y(x,y)$, $\xi(x,y)$, $U(\gamma^{-1}(Y(x)))$ and $U(\gamma^{-1}(Y(y)))$, respectively. Then for every $q\geqslant}\def\leq{\leqslant 2$, by using It\^o's formula \begin{eqnarray}\label{3.23}
&&d \|\xi(x,y)\|^q\nonumber\\&=&q\lambda\|\xi(x,y)\|^{q-2}\langle \xi(x,y), \nabla U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)\rangle dt\nonumber\\&&+
\frac{1}{2}q(q-1)\|\xi(x,y)\|^{q-2}tr([\nabla^2 U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y)) \nonumber\\&&\quad\times\xi(y)][\nabla^2 U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)]^\top)dt
\nonumber\\&&+q\|\xi(x,y)\|^{q-2}\langle \xi(x,y), [\nabla^2 U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)\nonumber\\&&\quad-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)]d B(t)\rangle
\nonumber\\&\leq& C(q)\Big[ \|\xi(x,y)\|^{q-1}\|\nabla U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)\|\nonumber\\&&+\|\xi(x,y)\|^{q-2}\|\nabla^2 U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)\|^2\Big]dt
\nonumber\\&&+q\|\xi(x,y)\|^{q-2}\langle \xi(x,y), [\nabla^2 U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)\nonumber\\&&\quad-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)]d B(t)\rangle. \end{eqnarray} Notice that \begin{eqnarray}\label{3.24}
&&\|\nabla U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)\|\nonumber\\&\leq &
\|\nabla U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(x))\xi(x)\|\nonumber\\&&+\|\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(x)\|\nonumber\\&&+\|\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(x)
-\nabla U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)\|
\nonumber\\&\leq & \|\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}\|\nabla \gamma^{-1}\|^2_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}|Y(x,y)|\|\xi(x)\|
\nonumber\\&&+ \|\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla^2 \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}|Y(x,y)|\|\xi(x)\|
\nonumber\\&&+ \|\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi(x,y)\|
\nonumber\\&\leq & C\Big[|Y(x,y)|\|\xi(x)\|+\|\xi(x,y)\|\Big] \end{eqnarray} and \begin{eqnarray}\label{3.25}
&&\|\nabla^2 U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)\|
\nonumber\\&\leq&\|\nabla^2 U(\gamma^{-1}(Y(x)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(x))\xi(x)\|\nonumber\\&&+\|\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(x))\xi(x)-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(x)\|\nonumber\\&&+\|\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(x)
-\nabla^2 U(\gamma^{-1}(Y(y)))\nabla\gamma^{-1}(Y(y))\xi(y)\|\nonumber\\&\leq & \|\nabla^2 U(\gamma^{-1}(Y(x)))-\nabla^2 U(\gamma^{-1}(Y(y)))\|\|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi(x)\|
\nonumber\\&&+ \|\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}\|\nabla^2 \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}|Y(x,y)|\|\xi(x)\|
\nonumber\\&&+ \|\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}\|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi(x,y)\|
\nonumber\\&\leq & C\Big[\|\nabla^2 U(\gamma^{-1}(Y(x)))-\nabla^2 U(\gamma^{-1}(Y(y)))\| \|\xi(x)\| +|Y(x,y)|\|\xi(x)\|+\|\xi(x,y)\|\Big], \end{eqnarray} then for $t\in [s,T]$ \begin{eqnarray}\label{3.26}
&&{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} \|\xi(x,y)\|^q(t)\nonumber\\ &\leq& C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\xi(x,y)\|^q(r)dr+ C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\xi(x,y)\|^{q-1}(r)|Y(x,y)|(r)\|\xi(x)\|(r)dr\nonumber\\&& +C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\xi(x,y)\|^{q-2}(r)|Y(x,y)|^2(r)\|\xi(x)\|^2(r)dr \nonumber\\&&
+C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\xi(x,y)\|^{q-2}(r)\|\nabla^2 U(\gamma^{-1}(Y(x)))-\nabla^2 U(\gamma^{-1}(Y(y)))\|^2 \|\xi(x)\|^2dr
\nonumber\\&\leq& C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\xi(x,y)\|^q(r)dr+ C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t|Y(x,y)|^q(r)\|\xi(x)\|^q(r)dr\nonumber\\&&
+C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\nabla^2 U(\gamma^{-1}(Y(x)))-\nabla^2 U(\gamma^{-1}(Y(y)))\|^q \|\xi(x)\|^qdr
\nonumber\\&\leq& C(q,d){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\xi(x,y)\|^q(r)dr+ C(q,d)\Bigg[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t|Y(x,y)|^{2q}(r)dr\Bigg]^{\frac12}
\Bigg[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t|\xi(x)\|^{2q}(r)dr\Bigg]^{\frac12}\nonumber\\&&
+C(q,d)\Bigg[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\nabla^2 U(\gamma^{-1}(Y(x)))-\nabla^2 U(\gamma^{-1}(Y(y)))\|^{2q} dr\Bigg]^{\frac12}\Bigg[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\xi(x)\|^{2q}dr\Bigg]^{\frac12}. \end{eqnarray} Moreover by (\ref{3.16}) and (\ref{3.22}) \begin{eqnarray}\label{3.27}
&&{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} \|\xi(x,y)\|^q(t)\nonumber\\ &\leq& C(q,d,T)\Bigg[|x-y|^q+\Bigg({\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t\|\nabla^2 U(\gamma^{-1}(Y(x)))-\nabla^2 U(\gamma^{-1}(Y(y)))\|^{2q}dr\Bigg)^{\frac12}\Bigg]. \end{eqnarray}
Further for every $p\geqslant}\def\leq{\leqslant 2$, by the Burkholder--Davis--Gundy inequality \begin{eqnarray}\label{3.28}
&&\sup_{0\leq s\leq T}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} \sup_{s\leq t\leq T}\|\xi(x,y)\|^p\nonumber\\ &\leq&
C(p,d,T)|x-y|^p \nonumber\\ && +
C(p,d,T){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Bigg[\int\limits_0^T \|\nabla^2 U(\gamma^{-1}(Y(x)))-\nabla^2 U(\gamma^{-1}(Y(y)))\|^{4p}(r)dr\Bigg]^{\frac14}. \end{eqnarray}
For given $p\geq2$, by the assumption in Theorem \ref{3.1}, there is a small enough real number $\delta=\delta(p)$ such that $F_\delta^{4p}(\cdot)$ (see (\ref{3.3})) is increasing and concave. On the other hand, since $U\in L^\infty([0,T];\cC^2_b({\mathbb R}^d;{\mathbb R}^d))$ and(\ref{3.10}) holds, for every $x,y\in\mR^d$, and the given real number $\delta(p)>0$, there is a constant $C(\delta)>0$ such that \begin{eqnarray*}
\sup_{0\leq t\leq T}\|\nabla^2 U(t,x)-\nabla^2 U(t,y)\| \leq C(\delta)F_\delta(|x-y|) 1_{|x-y|<\delta} +C(\delta)|x-y|1_{|x-y|\geqslant}\def\leq{\leqslant \delta}\,. \end{eqnarray*} Then by (\ref{3.14}) and the fact that $F_\delta^{4p}(\cdot)$ is increasing \begin{eqnarray}\label{3.29}
&&\sup_{0\leq s\leq T}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} \sup_{s\leq t\leq T}\|\xi(x,y)\|^p\nonumber\\ &\leq&
C|x-y|^p +
C{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Bigg[\int\limits_0^T F_\delta^{4p}(|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|) 1_{|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|<\delta}dt\Bigg]^{\frac14}
\nonumber\\ &&+C{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Bigg[\int\limits_0^T |\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|^{4p} 1_{|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|\geqslant}\def\leq{\leqslant\delta}dt\Bigg]^{\frac14} \nonumber\\ &\leq&
C|x-y|^p +C
\Bigg[\int\limits_0^T{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} F_\delta^{4p}(|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|1_{|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|<\delta}) dt\Bigg]^{\frac14}\nonumber\\&&+C\Bigg[\int\limits_0^T {\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|^{4p} dt\Bigg]^{\frac14} \nonumber\\ &\leq&
C|x-y|^p +C
\Bigg[\int\limits_0^T F_\delta^{4p}({\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}[|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|1_{|\gamma^{-1}(Y(x))-\gamma^{-1}(Y(y))|<\delta}]) dt\Bigg]^{\frac14}\nonumber\\&\leq&C|x-y|^p +C F_\delta^{p}(2C_1|x-y|), \end{eqnarray} where $C_1=C(p,T)\vee 1$ and $C(p,T)$ is given in (\ref{3.17}).
Now by $X(s,t,x)=\gamma^{-1}(t,Y(s,t,\gamma(s,x)))$, for every $p\geqslant}\def\leq{\leqslant 2$ and every $x,y\in\mR^d$ \begin{eqnarray*}
&&\|\nabla X(s,t,x)-\nabla X(s,t,y)\|\nonumber\\&=&\|\nabla \gamma^{-1}(t,Y(s,t,\gamma(s,x))) \nabla Y(s,t,\gamma(s,x)) \nabla \gamma(s,x)
\nonumber\\&&-\nabla \gamma^{-1}(t,Y(s,t,\gamma(s,y))) \nabla Y(s,t,\gamma(s,y)) \nabla \gamma(s,y)\|
\nonumber\\&=&\|[\nabla \gamma^{-1}(t,Y(s,t,\gamma(s,x))) -\nabla \gamma^{-1}(t,Y(s,t,\gamma(s,y))) ]\nabla Y(s,t,\gamma(s,x)) \nabla \gamma(s,x) \nonumber\\&&+ \nabla \gamma^{-1}(t,Y(s,t,\gamma(s,y))) [\nabla Y(s,t,\gamma(s,x))-\nabla Y(s,t,\gamma(s,y)) ] \nabla \gamma(s,x)
\nonumber\\&&+ \nabla \gamma^{-1}(t,Y(s,t,\gamma(s,y))) \nabla Y(s,t,\gamma(s,y))[ \nabla \gamma(s,x)-\nabla \gamma(s,y)]\| \nonumber\\&\leq&
\|\nabla\gamma^{-1}\|_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d\times d}))}^2 \|Y(s,t,\gamma(s,x)) -Y(s,t,\gamma(s,y))\|\|\nabla Y(s,t,\gamma(s,x))\|
\nonumber\\&&+ \|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}^2\|\nabla Y(s,t,\gamma(s,x))-\nabla Y(s,t,\gamma(s,y))\|
\nonumber\\&&+ \|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d\times d}))} \|\nabla^2 \gamma\|_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d\times d}\otimes\mR^d))}\| \nabla Y(s,t,\gamma(s,y))\| |x-y| \nonumber\\&\leq&
C\Big[\|Y(s,t,\gamma(s,x)) -Y(s,t,\gamma(s,y))\|\|\nabla Y(s,t,\gamma(s,x))\|
\nonumber\\&&+ \|\nabla Y(s,t,\gamma(s,x))-\nabla Y(s,t,\gamma(s,y))\|+ \| \nabla Y(s,t,\gamma(s,y))\| |x-y|\Big]. \end{eqnarray*} Thanks to the H\"{o}lder inequality, (\ref{3.22}) and (\ref{3.29}), for every $x,y\in\mR^d$, we have that \begin{eqnarray}\label{3.30}
&& \sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}\|\nabla X(s,t,x)-\nabla X(s,t,y)\|^p] \nonumber\\&\leq&
C\sup_{0\leq s\leq T}\Big({\mathbb E}[\sup_{s\leq t \leq T}\|Y(s,t,\gamma(s,x)) -Y(s,t,\gamma(s,y))\|^{2p}]\Big)^{\frac12}
\Big({\mathbb E}\|\nabla Y(s,t,\gamma(s,x))\|^{2p}\Big)^{\frac12} \nonumber\\&&+
\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}\|\nabla Y(s,t,\gamma(s,x))-\nabla Y(s,t,\gamma(s,y))\|^p]+C|x-y|^p
\nonumber\\&\leq&C|x-y|^p+CF_\delta^{p}(2C_1|\gamma(s,x)-\gamma(s,y)|)
\nonumber\\&\leq&C|x-y|^p+CF_\delta^{p}(3C_1|x-y|), \end{eqnarray}
where in the last inequality we have used $\|\nabla \gamma\|_{L^\infty([0,T];{\mathcal C}_b({\mathbb R}^d;\mR^{d\times d}))}\leq 3/2$. Then for every $x$, $y\in\mR^d$ such that $0<|x-y|<\delta/(3C_1) \ (<r_0/(3C_1))$, by similar calculations for (\ref{2.28}) \begin{eqnarray}\label{3.31}
&&\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}\|\nabla X(s,t,x)-\nabla X(s,t,y)\|^p] \nonumber \\
&\leq &C\Bigg[\int\limits_{r\leq 3C_1|x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{3C_1|x-y|<r\leq \delta}\frac{\phi(r)}{r^2}dr|+|x-y|\Bigg]^p+C|x-y|^p \nonumber \\
&\leq &C\Bigg[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr|\Bigg]^p+C|x-y|^p. \end{eqnarray} On account of (\ref{3.31}) and (\ref{3.4}), by using the H\"{o}lder inequality, we gain (\ref{3.6}).
(ii) Let $U^n$ be the unique solution of the parabolic problem (\ref{3.9}) associated to $b^n$, i.e. $b$ is replaced by $b^n$ in equation (\ref{3.9}). By Theorem \ref{the2.1} (ii), $U^n\in L^\infty([0,T];\cC^2_b({\mathbb R}^d;\mR^{d}))\cap W^{1,\infty}([0,T];\cC_b({\mathbb R}^d;\mR^{d}))$ and \begin{eqnarray}\label{3.32}
\lim_{n\rightarrow\infty}\| U^n-U\|_{L^\infty([0,T];\cC_b^2({\mathbb R}^d;\mR^{d}))}=0. \end{eqnarray} We set $\gamma_n(t,x):=x+U^n(t,x)$, then if $\lambda\geqslant}\def\leq{\leqslant \lambda_0$, $\{\gamma_n(t,x):=x+U^n(t,x)\}_n$ form nonsingular diffeomorphisms of class $\mathcal{C}^2$ uniformly in $t\in [0,T]$ and $n$. Moreover, \begin{eqnarray}\label{3.33}
\lim_{n\rightarrow\infty}\| \gamma_n-\gamma\|_{L^\infty([0,T];\cC_b^2({\mathbb R}^d;\mR^d))}=\lim_{n\rightarrow\infty}\| \gamma_n^{-1}-\gamma^{-1}\|_{L^\infty([0,T];\cC_b^2({\mathbb R}^d;\mR^d))}=0. \end{eqnarray} To prove (\ref{3.7}) and (\ref{3.8}), it is sufficient to show for every $p\geqslant}\def\leq{\leqslant 2$ \begin{eqnarray}\label{3.34}
\lim _{n\rightarrow\infty}\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}|Y^n(s,t,y)-Y(s,t,y)|^p]=0 \end{eqnarray} and \begin{eqnarray}\label{3.35}
\lim_{n\rightarrow \infty}\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}\|\nabla Y^n(s,t,y)-\nabla Y(s,t,y)\|^p]=0, \end{eqnarray} where $Y^n$ satisfies \begin{eqnarray}\label{3.36} \left\{
\begin{array}{ll} d Y^n(s,t)=\lambda U^n(t,\gamma^{-1}_n(t,Y^n(s,t)))dt+ [I+\nabla U^n(t,\gamma^{-1}_n(t,Y^n(s,t)))] dB(t), \ t\in(s,T], \\
Y^n(s,t)|_{t=s}=y.
\end{array} \right. \end{eqnarray}
For simplicity, we write $Y(s,t)$, $Y^n(s,t)$, $U(t,\gamma^{-1}(t,Y(s,t)))$ and $U^n(t,\gamma^{-1}_n(t,Y^n(s,t)))$ by $Y$, $Y^n$, $U(\gamma^{-1}(Y))$ and $U^n(\gamma^{-1}_n(Y^n))$, respectively. For every $q\geqslant}\def\leq{\leqslant 2$, by using the It\^{o} formula to $|Y^n-Y|^q$ \begin{eqnarray}\label{3.37}
&&d |Y^n-Y|^q\nonumber\\&=&q|Y^n-Y|^{q-2}\langle Y^n-Y, U^n(\gamma^{-1}_n(Y^n))-U(\gamma^{-1}(Y))\rangle dt\nonumber\\&&+
\frac{1}{2}q(q-1)|Y^n-Y|^{q-2}tr([\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))] \nonumber\\&&\quad\times[U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))]^\top)dt
\nonumber\\&&+q|Y^n-Y|^{q-2}\langle Y^n-Y, [\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))]d B(t)\rangle
\nonumber\\&\leq& C(q)\Big[ |Y^n-Y|^{q-1}\|U^n(\gamma^{-1}_n(Y^n))-U(\gamma^{-1}(Y))\|\nonumber\\&&+|Y^n-Y|^{q-2}\|\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))\|^2\Big]dt
\nonumber\\&&+q|Y^n-Y|^{q-2}\langle Y^n-Y, [\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))]d B(t)\rangle. \end{eqnarray} Then for every $t\in [s,T]$ \begin{eqnarray}\label{3.38}
&&{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} |Y^n-Y|^q(t)\nonumber\\ &\leq& C(q){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t |Y^n-Y|^{q-1}(r)\|U^n(\gamma^{-1}_n(Y^n))-U(\gamma^{-1}(Y))\|(r)dr \nonumber\\&& +C(q){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^t|Y^n-Y|^{q-2}(r)\|\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))\|^2(r)dr. \end{eqnarray} First we have \begin{eqnarray}\label{3.39}
&&\|U^n(\gamma^{-1}_n(Y^n))-U(\gamma^{-1}(Y))\|\nonumber\\&=&\|U^n(\gamma^{-1}_n(Y^n))-U(\gamma^{-1}_n(Y^n)) +U(\gamma^{-1}_n(Y^n))-U(\gamma^{-1}(Y^n))+U(\gamma^{-1}(Y^n))
-U(\gamma^{-1}(Y))\|
\nonumber\\&\leq & \|U^n-U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}
+\|\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\gamma^{-1}_n-\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}\nonumber\\&& +\|\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}|Y^n-Y|
\nonumber\\&\leq & C\Big[\|U^n-U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}+
\|\gamma^{-1}_n-\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}
+|Y^n-Y|\Big] \end{eqnarray} and similarly \begin{eqnarray}\label{3.40}
&&\|\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))\|^2\nonumber\\&\leq & C\Big[\|\nabla U^n-\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\nonumber\\&&+\|\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}\|\gamma^{-1}_n-\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}
\nonumber\\&&+\|\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}\|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}|Y^n-Y|\Big]^2
\nonumber\\&\leq & C\Big[\|\nabla U^n-\nabla U\|^2_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}+\|\gamma^{-1}_n-\gamma^{-1}\|^2_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}
+|Y^n-Y|^2\Big]. \end{eqnarray} Since $Y_n$ and $Y$ satisfy (\ref{3.36}) and (\ref{3.20}), respectively, and the coefficients are globally Lipschitz continuous, for every $q\geqslant}\def\leq{\leqslant 2$, there is a positive constant $C(q)$ such that \begin{eqnarray}\label{3.41}
\sup_n\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}\sup_{s\leq t \leq T}{\mathbb E}[|Y^n(s,t,y)-Y(s,t,y)|^q]\leq C(q). \end{eqnarray} Then by (\ref{3.38})--(\ref{3.41}) and a Gr\"onwall type argument, we obtain \begin{eqnarray}\label{3.42}
&&\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}\sup_{s\leq t\leq T}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} |Y^n(s,t,y)-Y(s,t,y)|^q\nonumber\\ &\leq& C(q,T)\Big[\|U^n-U\|^q_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d}))}+
\|\gamma^{-1}_n-\gamma^{-1}\|^q_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))} \Big]. \end{eqnarray}
Now for every $p\geqslant}\def\leq{\leqslant 2$, by (\ref{3.37}) and the Burkholder--Davis--Gundy inequality \begin{eqnarray}\label{3.43}
&&{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} \sup_{s\leq t \leq T}|Y^n(s,t,y)-Y(s,t,y)|^p\nonumber\\ &\leq& C(p){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^T |Y^n-Y|^{p-1}(r)\|U^n(\gamma^{-1}_n(Y^n))-U(\gamma^{-1}(Y))\|(r)dr \nonumber\\&& +C(p){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_s^T|Y^n-Y|^{p-2}(r)\|\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))\|^2(r)dr
\nonumber\\&&+C(p){\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Bigg[\int\limits_s^T|Y^n-Y|^{2p-2}(r)\|\nabla U^n(\gamma^{-1}_n(Y^n))-\nabla U(\gamma^{-1}(Y))\|^2(r)dr\Bigg]^{\frac12}. \end{eqnarray} Then thanks to (\ref{3.39}), (\ref{3.40}), (\ref{3.42}) and the Gr\"onwall inequality, \begin{eqnarray}\label{3.44}
&&\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} \sup_{s\leq t\leq T}|Y^n(s,t,y)-Y(s,t,y)|^p\nonumber\\ &\leq& C(p,T)\Bigg[\|U^n-U\|^{p}_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d}))}+
\|\gamma^{-1}_n-\gamma^{-1}\|^{p}_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))} \Bigg], \end{eqnarray} which implies (\ref{3.34}) by (\ref{3.32}) and (\ref{3.33})\,.
Differentiate $Y^n(s,t,y)$ with respect to initial data~(denoted by $\xi^n(s,t,y)$ or $\xi^n$), then~$\xi^{n}$ satisfies equation~(\ref{3.20}) with $b$ replaced by $b^{n}$\,, and by the boundedness of the coefficients and initial value~($I$)\,,
for every $q\geqslant}\def\leq{\leqslant 2$, there is a positive real number $C(q)$ such that \begin{eqnarray}\label{3.45}
\sup_n\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}\sup_{s\leq t \leq T}{\mathbb E}[\|\xi^n(s,t,y)\|^q]\leq C(q). \end{eqnarray}
Moreover observing that \begin{eqnarray}\label{3.46}
&&\|\nabla U^n(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi\|\nonumber\\&=&
\|\nabla U^n(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla U(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n \nonumber\\ &&+\nabla U(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla U(\gamma^{-1}(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n \nonumber\\ &&+ \nabla U(\gamma^{-1}(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}_n(Y^n)\xi^n \nonumber\\ &&+\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y^n)\xi^n \nonumber\\ &&+\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y^n)\xi^n-\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi^n \nonumber\\ &&+
\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi^n-\nabla U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi\|
\nonumber\\&\leq & \|\nabla U^n-\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla\gamma^{-1}_n\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi^n\| \nonumber\\
&&+\|\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}\|\gamma^{-1}_n-\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}
\|\nabla\gamma^{-1}_n\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi^n\| \nonumber\\&&
+\|\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}\|\nabla \gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))} \nonumber\\ &&\quad
\times |Y^n-Y|
\|\nabla\gamma^{-1}_n\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi^n\| \nonumber\\
&&+\|\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla\gamma^{-1}_n-\nabla\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi^n\| \nonumber\\
&&+\|\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla^2\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))}|Y^n-Y|\|\xi^n\| \nonumber\\
&&+\|\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\nabla\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}\|\xi^n-\xi\| \nonumber\\&\leq & C\Big[
\|\nabla U^n-\nabla U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}
+\|\gamma^{-1}_n-\gamma^{-1}\|_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d}))}+|Y^n-Y|\Big]\|\xi^n\| \nonumber\\&&
+C\|\xi^n-\xi\| \end{eqnarray}
and by the boundedness of $\|\nabla^2 U \|$ \begin{eqnarray}\label{3.47}
&&\|\nabla^2 U^n(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi\|
\nonumber\\&=&\|\nabla^2 U^n(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n- \nabla^2 U(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n \nonumber\\&&+ \nabla^2 U(\gamma^{-1}_n(Y^n))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}_n(Y^n)\xi^n \nonumber\\&&+ \nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}_n(Y^n)\xi^n-\nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y^n)\xi^n \nonumber\\&&+ \nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y^n)\xi^n-\nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi^n \nonumber\\&&+
\nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi^n-\nabla^2 U(\gamma^{-1}(Y))\nabla\gamma^{-1}(Y)\xi\|
\nonumber\\&\leq & C\Big[\|\nabla^2 U^n-\nabla^2 U\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}\otimes\mR^d))} +\|\nabla \gamma^{-1}_n-\nabla\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}
\nonumber\\&&+|Y^n-Y| \Big]\|\xi^n\|+C\|\nabla^2 U(\gamma^{-1}_n(Y^n))-\nabla^2 U(\gamma^{-1}(Y))\|\|\xi^n\|
+C\|\xi^n-\xi\|\,, \end{eqnarray}
then by similar calculations for (\ref{3.37})--(\ref{3.44})\,, we have
\begin{eqnarray}\label{3.48}
&&\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H} \sup_{s\leq t\leq T}\|\xi^n(s,t,y)-\xi(s,t,y)\|^p\nonumber\\ &\leq&
C(p,T)\Big[\|U^n-U\|^{p}_{L^\infty([0,T];\cC_b^2(\mR^d;\mR^{d}))}+
\|\gamma^{-1}_n-\gamma^{-1}\|^{p}_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d}))} \Big] \nonumber\\&& +
C(p,T)\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Big[\int\limits_s^T \|\nabla^2 U(\gamma^{-1}_n(Y^n))-\nabla^2 U(\gamma^{-1}(Y))\|^{2p}\|\xi^n\|^{2p}dr\Big]^{\frac12} \nonumber\\ &\leq&
C(p,T)\Big[\|U^n-U\|^{p}_{L^\infty([0,T];\cC_b^2(\mR^d;\mR^{d}))}+
\|\gamma^{-1}_n-\gamma^{-1}\|^{p}_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d}))} \Big] \nonumber\\&& +
C(p,T)\sup_{y\in {\mathbb R}^d}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Big[\int\limits_0^T \|\nabla^2 U(\gamma^{-1}_n(Y^n))-\nabla^2 U(\gamma^{-1}(Y))\|^{4p}dr\Big]^{\frac14}\,. \end{eqnarray}
Further we apply same calculations for (\ref{3.29}) to $\|\nabla^2 U(\gamma^{-1}_n(Y^n))-\nabla^2 U(\gamma^{-1}(Y))\|$ to get \begin{eqnarray}\label{3.49}
&&\sup_{y\in {\mathbb R}^d}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Bigg[\int\limits_0^T \|\nabla^2 U(\gamma^{-1}_n(Y^n(t,y)))-\nabla^2 U(\gamma^{-1}(Y(t,y)))\|^{4p}dr\Bigg]^{\frac14}\nonumber\\ &\leq&
C(p,T,\delta)\sup_{y\in {\mathbb R}^d}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Bigg[\int\limits_0^T F_\delta^{4p}(|\gamma^{-1}_n(Y(t,y))-\gamma^{-1}(Y(t,y))|) 1_{|\gamma^{-1}_n(Y(t,y))-\gamma^{-1}(Y(t,y))|<\delta}dt\Bigg]^{\frac14}
\nonumber\\ &&+C(p,T,\delta)\sup_{y\in {\mathbb R}^d}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\Bigg[\int\limits_0^T |\gamma^{-1}_n(Y(t,y))-\gamma^{-1}(Y(t,y))|^{4p} 1_{|\gamma^{-1}_n(Y(t,y))-\gamma^{-1}(Y(t,y))|\geqslant}\def\leq{\leqslant\delta}dt\Bigg]^{\frac14} \nonumber\\ &\leq&
C\Bigg[\int\limits_0^T F_\delta^{4p}(\sup_{y\in {\mathbb R}^d}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}|\gamma^{-1}_n(Y(t,y))-\gamma^{-1}(Y(t,y))|) dt\Bigg]^{\frac14}+C
\|\gamma^{-1}_n-\gamma^{-1}\|^{p}_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))} \nonumber\\ &\leq&
C\|\gamma^{-1}_n-\gamma^{-1}\|^{p}_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}
\nonumber\\&&+C F_\delta^{p}\Big(C\sup_{y\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\sup_{s\leq r\leq T}|Y^n(s,r,y)-Y(s,r,y)|+\|\gamma^{-1}_n-\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}\Big) \nonumber\\ &\leq&
C\|\gamma^{-1}_n-\gamma^{-1}\|^{p}_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}
\nonumber\\&&+C F_\delta^{p}\Big(C\|U^n-U\|_{L^\infty([0,T];\cC_b^1(\mR^d;\mR^{d}))}+C\|\gamma^{-1}_n-
\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d}))}\Big), \end{eqnarray} where in the last inequality we use (\ref{3.44}), and in the third inequality we use \begin{eqnarray*}
|\gamma_n^{-1}(Y^n)-\gamma^{-1}(Y)|&\leq& \|\nabla\gamma^{-1}_n\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^{d\times d}))}|Y^n-Y|+\|\gamma^{-1}_n-
\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}\\ &\leq&C|Y^n-Y|+\|\gamma^{-1}_n-
\gamma^{-1}\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}. \end{eqnarray*} Now from (\ref{3.48}) and (\ref{3.49}), (\ref{3.35}) holds by (\ref{3.32}) and (\ref{3.33})\,. $\Box$
\begin{remark} \label{rem3.1} Let $b^n$ be given in Theorem \ref{the3.1} (ii). By Liouville's theorem, since $b^n$ is smooth, we have the following Euler identity \begin{eqnarray*} \det(\nabla_xX^n(s,t,x))=\exp\Big(\int\limits^t_s\div b^n(r,X^n(s,r,x))dr\Big), \ \ 0\leq s\leq t \leq T, \end{eqnarray*} where $\det(\cdot)$ denotes the determinant of a matrix. In view of (\ref{3.7}) and (\ref{3.8}), if $b\in L^\infty([0,T]; {\mathcal C}_b({\mathbb R}^d ;{\mathbb R}^d))$ such that (\ref{3.2}) holds, and $\div b\in L^1([0,T];L^1_{loc}({\mathbb R}^d))$\,, up to choosing a subsequence, one derives \begin{eqnarray}\label{3.50} \det(\nabla_xX(s,t,x))=\exp\Big(\int\limits^t_s\div b(r,X(s,r,x))dr\Big), \quad 0\leq s\leq t \leq T. \end{eqnarray} \end{remark}
Since the inverse of $X$ ($X^n$): $X^{-1}$ ($X^{-1}_n$) satisfies an equation which has the same form as the original one except the drift has opposite sign, by Theorem \ref{the3.1} we have \begin{corollary} \label{cor3.1} Let $b,b^n$, $X(s,t,x)$ and $X^n(s,t,x)$ be stated in Theorem \ref{the3.1}. For every $p\geqslant}\def\leq{\leqslant 1$, there is a constant $C(p,d,T)>0$ such that \begin{eqnarray}\label{3.51}
\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}|X^{-1}(s,t,x)|^p+\sup_{x\in\mR^d}\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}\|\nabla X^{-1}(s,t,x)\|^p\leq C(p,d,T) \end{eqnarray} and \begin{eqnarray}\label{3.52}
&&\lim _{n\rightarrow\infty}\sup_{x\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}[|X^{-1}_n(s,t,x)-X^{-1}(s,t,x)|]^p
\nonumber\\ &=&\lim _{n\rightarrow\infty}\sup_{x\in {\mathbb R}^d} \sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}[\|\nabla X^{-1}_n(s,t,x)-\nabla X^{-1}(s,t,x)\|]^p=0. \end{eqnarray} Moreover, for every $x,y\in \mR^d$, there is another constant $C(p,d,T)>0$ such that \begin{eqnarray}\label{3.53}
\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}|X^{-1}(s,t,x)-X^{-1}(s,t,y)|^p]\leq C(p,d,T)|x-y|^p \end{eqnarray} and \begin{eqnarray}\label{3.54}
&&\sup_{0\leq s\leq T}{\mathbb E}[\sup_{s\leq t \leq T}\|\nabla X^{-1}(s,t,x)-\nabla X^{-1}(s,t,y)\|^p] \nonumber \\
&\leq &C\Bigg[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr|\Bigg]^p1_{|x-y|<r_0}+C|x-y|^p. \end{eqnarray} \end{corollary}
\begin{remark} \label{rem3.2}
(i) Let $r_0=1/2$ and $\phi(r)=C|\log(r)|^{-\alpha}$ for $r\in (0,r_0)$ with some $\alpha>1$. From Remark \ref{rem2.3} (ii) \begin{eqnarray*}
\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr+|x-y|\leq\frac{C(\alpha)}{|\log(|x-y|)|^{\alpha-1}}. \end{eqnarray*} Let $F_\delta$ be defined by (\ref{3.3}). We get \begin{eqnarray}\label{3.55}
F_\delta(r)\leq \frac{C}{|\log(r)|^{\alpha-1}}=:\tilde{F}_\delta(r), \quad r\in [0,\delta], \quad \delta<\frac12. \end{eqnarray} Notice that given $p\geqslant}\def\leq{\leqslant 1$, if $r \in (0,\delta)$ and $\delta<\exp(p-p\alpha-1)$, \begin{eqnarray}\label{3.56}
\frac{d}{dr} \frac{1}{|\log(r)|^{p(\alpha-1)}}=\frac{p(\alpha-1)}{r|\log(r)|^{p(\alpha-1)+1}}\geqslant}\def\leq{\leqslant 0 \end{eqnarray} and \begin{eqnarray}\label{3.57}
\frac{d^2}{dr^2} \frac{1}{|\log(r)|^{p(\alpha-1)}}=\frac{p(\alpha-1)}{|\log(r)|^{p(\alpha-1)+1}r^2}
\Bigg[\frac{p\alpha-p+1}{|\log(r)|}-1\Bigg]\leq 0\,, \end{eqnarray}
then the function $\tilde{F}_\delta^p(r)=C^p|\log(r)|^{-p(\alpha-1)}$ is increasing and concave (we define $|\log(0)|^{-1}=0$) on~$[0,\delta]$. Thus the assumptions in Theorem \ref{3.1} hold. Theorem \ref{3.1} is applicable mutatis mutandis. \end{remark}
From Remark \ref{rem3.2}, we draw the following result. \begin{corollary} \label{cor3.2} Let $b\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$. Suppose that there are two real numbers $C>0$ and $\alpha>1$ such that for every $x\in \mR^d$ \begin{eqnarray}\label{3.58}
|b(t,x)-b(t,y)|\leq \frac{C}{|\log(|x-y|)|^\alpha}, \quad {\rm for\; all} \ \ y\in B_{\frac12}(x), \ t\in [0,T]. \end{eqnarray}
(i) SDE (\ref{3.1}) has a unique continuous adapted solution $\{X(s,t,x), \ t\in [s,T ], \ \omega \in\Omega\}$, which forms a stochastic diffeomorphisms flow.
(ii) Estimates (\ref{3.4})--(\ref{3.5}) hold for $X$ and denote its inverse by $X^{-1}$, then (\ref{3.51}) and (\ref{3.53}) hold as well. Moreover, for every $x,y\in\mR^d$ \begin{eqnarray}\label{3.59}
&&\sup_{0\leq s\leq T}{\mathbb E}\sup_{s\leq t \leq T}[\|\nabla X(s,t,x)-\nabla X(s,t,y)\|+\|\nabla X^{-1}(s,t,x)-\nabla X^{-1}(s,t,y)\|]^p \nonumber \\ &\leq&
C(p,T)\Bigg[\frac{1_{|x-y|< \frac12}}{|\log(|x-y|)|^{p(\alpha-1)}}+|x-y|^p1_{|x-y|\geqslant}\def\leq{\leqslant \frac12}\Bigg]. \end{eqnarray} \end{corollary}
\section{Stochastic transport equations}\label{sec4} \setcounter{equation}{0} \begin{definition} \label{def4.1} Let $b\in L^1([0,T];L^1_{loc}({\mathbb R}^d;{\mathbb R}^d))$ such that $\div b\in L^1([0,T];L^1_{loc}({\mathbb R}^d))$, and let $u_0\in L^\infty({\mathbb R}^d)$. A stochastic field $u$ is called a weak $L^\infty$-solution of (\ref{1.1}) if $u\in L^\infty(\Omega\times[0,T];L^\infty({\mathbb R}^d))$ and for every $\varphi\in{\mathcal C}_0^\infty({\mathbb R}^d)$, $\int_{{\mathbb R}^d}\varphi(x)u(t,x)dx$ has a continuous modification which is an ${\mathcal F}_t$-semimartingale and for every $t\in [0,T]$ \begin{eqnarray}\label{4.1} \int\limits_{{\mathbb R}^d}\varphi(x)u(t,x)dx&=&\int\limits_{{\mathbb R}^d}\varphi(x)u_0(x)dx+ \int\limits^t_0\int\limits_{{\mathbb R}^d}\div (b(s,x)\varphi(x))u(s,x)dxds\nonumber\\&& +\sum_{i=1}^d\int\limits^t_0\circ dB_i(s)\int\limits_{{\mathbb R}^d}\partial_{x_i}\varphi(x)u(s,x)dx, \quad {\mathbb P}-a.s.. \end{eqnarray} \end{definition} Then we state our main result. \begin{theorem} \label{the4.1} \textbf{(Existence and uniqueness)} Let $d\geqslant}\def\leq{\leqslant 1$. Suppose $b\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$ such that (\ref{3.2}) and (\ref{3.3}) hold. Further suppose that $\div b\in L^1([0,T];L^1_{loc}(\mR))$ for $d=1$ or there exists $q\in (2,+\infty)$ such that \begin{eqnarray}\label{4.2} \div b\in L^q([0,T]\times {\mathbb R}^d), \quad d\geqslant}\def\leq{\leqslant 1. \end{eqnarray} Then there exists a unique weak $L^\infty$-solution to the Cauchy problem (\ref{1.1}). Moreover, the unique weak solution can be represented by $u(t,x)=u_0(X^{-1}(t,x))$, with $X(t,x)$ being the unique strong solution of the associated stochastic differential equation (\ref{3.1}) with $s=0$.
\end{theorem}
\noindent \textbf{Proof.} First, we prove that $u(t,x)=u_0(X^{-1}(t,x))$ is a weak $L^\infty$-solution of (\ref{1.1}). Let $b^n$ and $X^n$ be given in Theorem \ref{the3.1}, and let $X^{-1}_n$ be the inverse of $X^n$. Since $b^n$ is smooth, $u^n=u_0(X^{-1}_n)$ is the unique weak $L^\infty$-solution of the following Cauchy problem~(\cite[Theorems 16 and 20]{FGP1}) \begin{eqnarray}\label{4.3} \left\{
\begin{array}{ll} \partial_tu^n(t,x)+b^n(t,x)\cdot\nabla u^n(t,x) +\sum_{i=1}^d\partial_{x_i}u^n(t,x)\circ\dot{B}_i(t)=0, \ (t,x)\in(0,T)\times {\mathbb R}^d, \\
u^n(t,x)|_{t=0}=u_0(x), \ x\in{\mathbb R}^d.
\end{array} \right. \end{eqnarray} Then for every $\varphi\in{\mathcal C}_0^\infty({\mathbb R}^d)$ and
every $t\in [0,T]$ \begin{eqnarray}\label{4.4} \int\limits_{{\mathbb R}^d}\varphi(x)u_0(X^{-1}_n(t,x))dx&=&\int\limits_{{\mathbb R}^d}\varphi(x)u_0(x)dx+ \int\limits^t_0\int\limits_{{\mathbb R}^d}\div (b^n(s,x)\varphi(x))u_0(X^{-1}_n(s,x))dxds\nonumber\\&& +\sum_{i=1}^d\int\limits^t_0\circ dB_i(s)\int\limits_{{\mathbb R}^d}\partial_{x_i}\varphi(x)u_0(X^{-1}_n(s,x))dx \nonumber\\&=&\int\limits_{{\mathbb R}^d}\varphi(x)u_0(x)dx+ \int\limits^t_0\int\limits_{{\mathbb R}^d}\div (b^n(s,x)\varphi(x))u_0(X^{-1}_n(s,x))dxds\nonumber\\&& +\sum_{i=1}^d\int\limits^t_0dB_i(s)\int\limits_{{\mathbb R}^d}\partial_{x_i}\varphi(x)u_0(X^{-1}_n(s,x))dx \nonumber\\ && +\frac12 \int\limits^t_0 ds\int\limits_{{\mathbb R}^d}\Delta\varphi(x)u_0(X^{-1}_n(s,x))dx \quad {\mathbb P}-a.s., \end{eqnarray} where in the last inequality we have used the relationship between the Stratonovich integral and the It\^{o} integral.
By (\ref{2.32}) and (\ref{4.2}) \begin{eqnarray}\label{4.5}
\lim_{n\rightarrow \infty}\|b^n-b\|_{L^\infty([0,T];\cC_b(\mR^d;\mR^d))}=0 \end{eqnarray} and \begin{eqnarray}\label{4.6}
\lim_{n\rightarrow \infty}\|\div(b^n\varphi)-\div(b\varphi)\|_{L^1([0,T];L^1(\mR^d))}=0. \end{eqnarray} Now thanks to Corollary \ref{cor3.1}, by taking $n$ to infinity in (\ref{4.4}), (\ref{4.1}) holds for $u(t,x)=u_0(X^{-1}(t,x))$.
It remains to check the uniqueness and observing that the equation is linear, it suffices to prove that $u\equiv 0$ a.s. if the initial data vanishes. We only check the uniqueness for $d>1$, the case $d=1$ being similar and easier. Let $\varrho_n$ be given by (\ref{2.8}) and set $u_n=u\ast \varrho_n$, then \begin{eqnarray}\label{4.7} \partial_tu_n(t,x)+b(t,x)\cdot\nabla u_n(t,x) +\sum_{i=1}^d\partial_{x_i}u_n(t,x)\circ\dot{B}_i(t)=e_n(t,x), \end{eqnarray} with \begin{eqnarray}\label{4.8} e_n(t,x)=b(t,x)\cdot\nabla u_n(t,x)-(b\cdot\nabla u)\ast \varrho_n(t,x). \end{eqnarray} For for every $t>0$\,, the It\^{o}'s formula yields that \begin{eqnarray*} u_n(t,X(t,x))=\int_0^te_n(s,X(s,x))ds, \end{eqnarray*} which implies for every $\varphi\in \mathcal{C}_0^\infty({\mathbb R}^d)$ and almost all $\omega\in\Omega$ \begin{eqnarray}\label{4.9} &&\int\limits_{{\mathbb R}^d}u_n(t,X(t,x))\varphi(x)dx \nonumber\\&=& \int\limits^t_0\int\limits_{{\mathbb R}^d}e_n(s,X(s,x))\varphi(x)dx \nonumber\\&=&\int\limits^t_0\int\limits_{{\mathbb R}^d}e_n(s,x)\varphi(X^{-1}(s,x)) \det(\nabla_xX^{-1}(s,x))dxds \nonumber\\&=&-\int\limits^t_0\int\limits_{{\mathbb R}^d}\int\limits_{{\mathbb R}^d}[\div b(s,x)-\div b(s,y)]\varphi(X^{-1}(s,x)) \det(\nabla_xX^{-1}(s,x))u(s,y)\varrho_n(x-y)dydxds \nonumber\\&&+\int\limits^t_0\int\limits_{{\mathbb R}^d}\int\limits_{{\mathbb R}^d}[b(s,x)-b(s,y)]\cdot \nabla_x(\varphi(X^{-1}(s,x)) \det(\nabla_xX^{-1}(s,x)))u(s,y)\varrho_n(x-y)dydxds \nonumber\\&=&: I_1^n(t)+I_2^n(t). \end{eqnarray}
In view of Theorem \ref{the3.1}, $\varphi(X^{-1}(s,x))\det(\nabla_xX^{-1}(s,x))$ is continuous in $(s,x)$. We claim that for almost all $\omega\in\Omega$, $\varphi(X^{-1}(s,\cdot))$ has a compact support in $x$ uniformly in $s$. In fact, without loss of generality we assume there is a real number $R>0$ such that the support of $\varphi$ is in $B_R$, if the assertion is false, there is a sequence $\{(t_k,x_k)\}_{k\geq1}\subset [0,t]\times B_R$ such that $\lim_{k\rightarrow \infty}|X(\omega,t_k,x_k)|=+\infty$. But on the other hand, by Theorem \ref{the3.1}, for every $k\geqslant}\def\leq{\leqslant 1$ \begin{eqnarray*}
|X(\omega,t_k,x_k)|&\leq& |X(\omega,t_k,x_k)-X(\omega,t_k,x_1)|+|X(\omega,t_k,x_1)|\\ &\leq& C|x_k-x_1|+ |X(\omega,t_k,x_1)|\\ &\leq& 2CR+\sup_{0\leq s\leq T}|X(\omega,s,x_1)\leq C. \end{eqnarray*} Therefore, by (\ref{4.2}), taking $n$ to infinity yields $I_1^n(t)\rightarrow 0$\,, $\mP-$a.s..
Noticing that \begin{eqnarray*} &&\nabla_x(\varphi(X^{-1}(s,x))) \det(\nabla_xX^{-1}(s,x))\nonumber\\&=&\nabla_{X^{-1}}\varphi(X^{-1}(s,x))\nabla_xX^{-1}(s,x) \det(\nabla_xX^{-1}(s,x))+\varphi(X^{-1}(s,x))\nabla_x( \det(\nabla_xX^{-1}(s,x))), \end{eqnarray*} if for almost all $\omega\in\Omega$, $\nabla_x( \det(\nabla_xX^{-1}(\cdot,\cdot)))\in L^1([0,T];L^1_{loc}(\mR^d))$, by the dominated convergence theorem, $I_2^n(t)\rightarrow 0$, $\mP$-a.s.. Now let us check $\nabla_x( \det(\nabla_xX^{-1}(\cdot,\cdot)))\in L^1([0,T];L^1_{loc}(\mR^d))$ and it is equivalent to show that $\nabla_x( \det(\nabla_xX(\cdot,\cdot)))\in L^1([0,T];L^1_{loc}(\mR^d))$. From (\ref{3.50}) \begin{eqnarray*} \nabla_x\det(\nabla_xX(t,x))=\det(\nabla_xX(t,x))\nabla_x\int\limits^t_0\div b(s,X(s,x))ds. \end{eqnarray*} By Theorem \ref{the3.1}, $\det(\nabla_xX(s,x))$ is continuous in $(s,x)$, we need to show \begin{eqnarray}\label{4.10} \nabla_x\int\limits^t_0\div b(s,X(s,x))ds\in L^1([0,T];L^1_{loc}(\mR^d)), \quad \mP-a.s.. \end{eqnarray}
To do this we consider the following backward parabolic equation \begin{eqnarray}\label{4.11} \left\{ \begin{array}{ll} \partial_{t}V(t,x) +\frac{1}{2}\Delta V(t,x)+b(t,x)\cdot \nabla V(t,x)=\div b(t,x), \quad (t,x)\in (0,T)\times {\mathbb R}^d, \\ V(T,x)=0, \quad x\in{\mathbb R}^d.
\end{array} \right. \end{eqnarray} Noticing $\div b\in L^q([0,T]\times {\mathbb R}^d)$, there is a unique $V\in L^q([0,T];W^{2,q}({\mathbb R}^d))\cap W^{1,q}([0,T];L^q({\mathbb R}^d))$ solving (\ref{4.11}), and there is a constant $C(q,T)>0$ such that \begin{eqnarray}\label{4.12}
\|V\|_{L^q([0,T];W^{2,q}({\mathbb R}^d))}+\|\partial_tV\|_{L^q([0,T]\times {\mathbb R}^d)} \leq C(q,T)\|\div b\|_{L^q([0,T]\times {\mathbb R}^d)}. \end{eqnarray} Moreover, by Sobolev's imbedding theorem, \begin{eqnarray}\label{4.13}
\sup_{0\leq t\leq T}\|V\|_{W^{1,q}({\mathbb R}^d)} \leq C(q,T)\|\div b\|_{L^q([0,T]\times {\mathbb R}^d)}. \end{eqnarray} Next we assume that for every $t\in [0,T]$, $V(t)$ is smooth, otherwise one can follow an approximation argument, then the It\^o's formula yields \begin{eqnarray*} V(t,X(t,x))-V(0,x)-\int\limits_0^t\nabla V(s,X(s,x))dB(s)=\int\limits_0^t\div b(s,X(s,x))ds, \end{eqnarray*} which implies that \begin{eqnarray}\label{4.14} \nabla_x\int\limits^t_0\div b(s,X(s,x))ds&=&\nabla_X V(t,X(t,x))\nabla_xX(t,x)-\nabla_xV(0,x) \nonumber\\&&-\int\limits_0^t\nabla^2_X V(s,X(s,x))\nabla_xX(s,x)dB(s). \end{eqnarray} By Theorem \ref{the3.1}, (\ref{4.12}) and (\ref{4.13}), for almost all $\omega\in\Omega$, the first two terms in the righthand hand side in (\ref{4.14}) are in $L^1([0,T];L^1_{loc}(\mR^d))$. For every $R>0$ \begin{eqnarray}\label{4.15}
&&{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_0^T\int\limits_{|x|\leq R}\Bigg|\int\limits_0^t\nabla^2_X V(s,X(s,x))\nabla_xX(s,x)dB(s)\Bigg|^2dxdt\nonumber\\&=&\int\limits_0^T\int\limits_{|x|\leq R}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_0^t|\nabla^2_X V(s,X(s,x))\nabla_xX(s,x)|^2dsdxdt
\nonumber\\&\leq &C\int\limits_{|x|\leq R}\int\limits_0^T\Big[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\|\nabla^2_X V(s,X(s,x))\|^q\Big]^{\frac{2}{q}}\Big[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\|\nabla_xX(s,x)\|^\frac{2q}{q-2}\Big]^{\frac{q-2}{q}}dsdx
\nonumber\\&\leq &C\Big[\sup_{x\in\mR^d}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\sup_{0\leq s \leq T}|\nabla_xX(s,x)|^\frac{2q}{q-2}\Big]^{\frac{q-2}{q}}
\Bigg[\int\limits_{|x|\leq R}\int\limits_0^T\Big[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}|\nabla^2_X V(s,X(s,x))|^q\Big]dsdx\Bigg]^{\frac{2}{q}} \nonumber\\&\leq &C
\Bigg[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\int\limits_0^T\int\limits_{|x|\leq R}|\nabla^2_X V(s,X(s,x))|^qdxds\Bigg]^{\frac{2}{q}}
\nonumber\\&\leq &C\Bigg[\sup_{x\in\mR^d}{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}\sup_{0\leq s \leq T}\|\nabla_xX^{-1}(s,x)\|^q\Bigg]^{\frac{2}{q}}
\Bigg[\int\limits_0^T\int\limits_{\mR^d}|\nabla^2_x V(s,x)|^qdxds\Bigg]^{\frac{2}{q}}<+\infty, \end{eqnarray} where in the fifth line we have used (\ref{3.4}), and in the last inequality we have used (\ref{3.51}) and (\ref{4.12}).
Then from (\ref{4.9})--(\ref{4.15}), by taking $n$ to infinity, we have $\int_{{\mathbb R}^d}u(t,X(t,x))\varphi(x)dx=0$\,, that is $u(t,X(t,x))=0$ for almost everywhere $x\in \mR^d$ and almost all $\omega\in\Omega$. Because $X(t,x)$ is a stochastic diffeomorphisms flow associated with (\ref{3.1}) with $s=0$, we have $u(t,x)=0$ for almost everywhere $x\in \mR^d$ and almost all $\omega\in\Omega$. The proof is complete. $\Box$
\begin{remark} \label{rem4.1} Without the stochastic perturbation, even if the drift is bounded and H\"{o}lder continuous, the deterministic equation possesses multiple $L^\infty$-solutions (\cite[Section 6.1]{FGP1}). So the noise has a regularization effect. \end{remark}
\begin{remark} \label{rem4.2} (i) SDE (\ref{3.1}) with $s=0$ has a unique continuous adapted solution $\{X(t,x), \ t\in [0,T ], \ \omega \in\Omega\}$, which forms a stochastic flow of diffeomorphisms, and by Theorem \ref{the4.1} (\ref{1.1}) has a unique weak $L^\infty$-solution which can be wrote by $u_0(X^{-1}(t,x))$. Thus, if $u_0\in \cC_b(\mR^d)$, for almost all $\omega\in \Omega$, and every $t\in [0,T]$, $u(t,\cdot)$ is bounded and continuous. Moreover, if $u_0\in W^{1,p}(\mR^d)$ with $p\in [1,+\infty]$, we have the following chain rule \begin{eqnarray*} \nabla_x(u_0(X^{-1}(t,x)))=\nabla_xu_0(X^{-1}(t,x)) \nabla_xX^{-1}(t,x), \end{eqnarray*} so $u(t,\cdot)\in W^{1,p}_{loc}(\mR^d)$ almost surely, for every $t\in [0,T]$. However, for the deterministic equation, even if the uniqueness is established, the persistence of the above properties (continuity and Sobolev differentiability) for solutions are missing~\cite{CLR2}.
(ii) We can further establish the existence and uniqueness of $W^{1,p}$-solutions with $p\in [1,+\infty]$ as well if $b\in L^\infty([0,T];\cC_b(\mR^d;\mR^d))$ such that (\ref{3.2}) holds. More precisely there is a unique $u\in L^\infty(\Omega\times[0,T];L^p({\mathbb R}^d))$ such that
(1) for every $\varphi\in{\mathcal C}_0^\infty({\mathbb R}^d)$, $\int_{{\mathbb R}^d}\varphi(x)u(t,x)dx$ has a continuous modification which is an ${\mathcal F}_t$-semimartingale and for every $t\in [0,T]$, (\ref{4.1}) holds;
(2) for almost all $\omega\in\Omega$, $\nabla u\in L^\infty([0,T];L^p_{loc}(\mR^d;\mR^d))$.
The proof is similar to that of Theorem \ref{the4.1}~(\cite[Theorem 1.1]{WDGL}, \cite[Theorem 25]{FGP1}). However, if $d\geqslant}\def\leq{\leqslant 2$, the deterministic equation does not exist such strong solutions~(\cite[Theorem 1.2]{WDGL})\,, which means that noise prevents the singularity for solutions.
(iii) There are also some results on $\cap _{p\geqslant}\def\leq{\leqslant 1}W^{1,p}_{loc}$ solutions~(\cite{FF2}) and $L^p$ solutions~(\cite{CO})\,.
\end{remark}
\section{Conclusions}\label{sec4}\setcounter{equation}{0} Recently, there have been a broad research on the uniqueness of $L^\infty$ solutions for the stochastic transport equation \begin{eqnarray}\label{5.1} \left\{
\begin{array}{ll} \partial_tu(t,x)+b(t,x)\cdot\nabla u(t,x) +\sum_{i=1}^d\partial_{x_i}u(t,x)\circ\dot{B}_i(t)=0, \quad (t,x)\in(0,T)\times {\mathbb R}^d, \\
u(t,x)|_{t=0}=u_0(x), \quad x\in{\mathbb R}^d,
\end{array} \right. \end{eqnarray} with non-Lipschitz drift. Most of these works are concentrated on the drift which is H\"{o}lder continuous in spatial variable uniformly in time. The question for the uniqueness when $b$ is only bounded is still open. In this study, we established the existence and uniqueness of $L^\infty$ solutions only assuming $b$ is bounded and Dini continuous in spatial variable uniformly in time. Compared with the existing research, the result is new.
We solve the Cauchy problem (\ref{5.1}) by the method of stochastic characteristics. Therefore, we should prove the existence of stochastic diffeomorphisms flow for the following stochastic differential equation \begin{eqnarray}\label{5.2}
dX(t)=b(t,X(t))dt+dB(t), \quad t\in(0,T], \quad X(t)|_{t=0}=x. \end{eqnarray} To reach the goal, we use the It\^{o}-Tanaka trick to transform the SDE (\ref{5.2}) with bounded and Dini continuous drift to an equivalent new SDE with Lipschitz coefficients via a non-singular diffeomorphism $\Phi(t,x)=x+U(t,x)$, where $U(T-t,x)=:V(t,x)$ satisfies a vector-valued parabolic partial differential equation of second order which has the form \begin{eqnarray}\label{5.3} \left\{\begin{array}{ll} \partial_{t}V(t,x)=\frac{1}{2}\Delta V(t,x)+b(t,x)\cdot \nabla V(t,x) +b(t,x)-\lambda V(t,x), \ (t,x)\in (0,T)\times {\mathbb R}^d, \\ V(0,x)=0, \ x\in{\mathbb R}^d. \end{array}\right. \end{eqnarray} There are two things we need to do. The first one is choosing a proper function space on which the It\^{o} formula is applicable, and the second one is the boundedness estimate for the gradient of $V$. We accomplish these issues by fetching $L^\infty(0,T;\cC^2_b(\mR^d))\cap W^{1,2}(0,T;\cC_b(\mR^d))$ as the workspace. When $b\in L^\infty(0,T;\mathcal{C}^{\alpha}_b(\mathbb{R}^d;\mathbb{R}^d))$, these estimates for solutions have been established by Flandoli, Gubinelli and Priola~\cite{FGP1}. Noticing that, here we only assume that $b$ is Dini continuous in $x$, so we should extend the Schauder theory for (\ref{5.3}) to $W^{2,\infty}$ theory. We accomplish these estimates in Section 2, and then establish the stochastic diffeomorphisms flow for (\ref{5.2}) in Section 3. These results are new as well.
We remark that the method used to establish $W^{2,\infty}$ estimates for (\ref{5.3}) can be applied to found the $W^{1,\infty}$ estimates for solutions of second order parabolic equations driven by Browian motion or general L\'{e}vy noise. The $W^{1,p}$ ($p\in [2,\infty)$) theory, stochastic BMO estimates and Schauder theory have been founded by many researchers, see \cite{Kry96,Kim15,DuL,HWW,WDL}. There are few works to deal the $W^{1,\infty}$ estimates. Therefore, the study of the $W^{1,\infty}$ property of solutions to stochastic parabolic equations is of very high importance. For simplicity, and without loss of generality, here we only give a brief calculation for the $W^{1,\infty}$ estimate for heat equation driven by Brownian noise: \begin{eqnarray}\label{5.4} \left\{\begin{array}{ll} du(t,x)=\frac{1}{2}\Delta u(t,x)dt+f(t,x)dB(t), \ (t,x)\in (0,T)\times {\mathbb R}^d, \\ u(0,x)=0, \ x\in{\mathbb R}^d, \end{array}\right. \end{eqnarray} where $f$ is bounded and Dini continuous in $x$ uniformly in $t$. From (\ref{5.4}), then \begin{eqnarray}\label{5.5} u(t,x)=\int\limits_0^tK(t-s,\cdot)\ast f(s,\cdot)(x)dB(s), \quad \ (t,x)\in [0,T]\times\mR^d, \end{eqnarray}
where $K(t,x)=(2\pi t)^{-\frac{d}{2}}e^{-\frac{|x|^2}{2t}}, \ t>0, \ x\in\mathbb{R}^d$. For $1\leq i\leq d$, we first differentiate $u$ in $x_i$, and then use the It\^{o} isometry, to get \begin{eqnarray*}
&&{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}|\partial_{x_i}u(t,x)|^2\nonumber \\&=&\int\limits_0^t\Big|\int\limits_{\mR^d}\partial_{x_i}K(t-s,x-y) f(s,y)dy\Big|^2ds\nonumber \\ &=& \int\limits_0^t\Big|\int\limits_{|x-y|>(t-s)^\theta}\partial_{x_i}K(t-s,x-y) [f(s,y)-f(s,x)]dy\nonumber \\ && +\int\limits_{|x-y|\leq (t-s)^\theta}\partial_{x_i}K(t-s,x-y) [f(s,y)-f(s,x)]dy\Big|^2ds \nonumber \\ &\leq& 2\int\limits_0^t\Big|\int\limits_{|x-y|>(t-s)^\theta}\partial_{x_i}K(t-s,x-y) [f(s,y)-f(s,x)]dy\Big|^2ds\nonumber \\ && +2\int\limits_0^t\Big|\int\limits_{|x-y|\leq (t-s)^\theta}\partial_{x_i}K(t-s,x-y) [f(s,y)-f(s,x)]dy\Big|^2ds, \end{eqnarray*} where $\theta\in (0,1/2)$. Similar calculations from (\ref{2.12}) to (\ref{2.17}) used here again, we conclude that \begin{eqnarray*}
{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}|\partial_{x_i}u(t,x)|^2<\infty. \end{eqnarray*} Moreover, we also get an analogue of (\ref{2.7}) for $\partial_{x_i}u$, i.e. for every $x,y\in \mR$ and every $t\in [0,T]$ \begin{eqnarray*}
&&\Big[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}|\partial_{x_i}u(t,x)-\partial_{y_i}u(t,y)|^2\Big]^{\frac12} \nonumber \\ &\leq&
C\left[\int\limits_{r\leq |x-y|}\frac{\phi(r)}{r}dr+ \phi(|x-y|)+|x-y|\int\limits_{|x-y|<r\leq r_0}\frac{\phi(r)}{r^2}dr\right]1_{|x-y|<r_0}+C|x-y|. \end{eqnarray*} Moreover, if $f$ is H\"{o}lder-Dini or strong H\"{o}lder or weak H\"{o}lder continuous with the H\"{o}lder-Dini or strong H\"{o}lder or weak H\"{o}lder function $\phi$, then \begin{eqnarray*}
\Big[{\mathbb E}}\def\mF{{\mathbb F}}\def\mG{{\mathbb G}}\def\mH{{\mathbb H}|\partial_{x_i}u(t,x)-\partial_{y_i}u(t,y)|^2\Big]^{\frac12} \leq C\phi(|x-y|). \end{eqnarray*}
\end{document} | arXiv |
Publication Info.
Wind and Structures
Techno-Press (테크노프레스)
Construction/Transportation > Design/Analysis for Facilities
The WIND AND STRUCTURES, An International Journal, aims at: - Major publication channel for research in the general area of wind and structural engineering, - Wider distribution at more affordable subscription rates; - Faster reviewing and publication for manuscripts submitted. The main theme of the Journal is the wind effects on structures. Areas covered by the journal include: - Wind loads and structural response - Bluff-body aerodynamics - Computational method - Wind tunnel modeling - Local wind environment - Codes and regulations - Wind effects on large scale structures
http://www.techno-press.org/papers/ KSCI KCI SCOPUS SCIE
Volume 5 Issue 2_3_4
Flow structures around a three-dimensional rectangular body with ground effect
Gurlek, Cahit;Sahin, Besir;Ozalp, Coskun;Akilli, Huseyin 345
https://doi.org/10.12989/was.2008.11.5.345
An experimental investigation of the flow over the rectangular body located in close proximity to a ground board was reported using the particle image velocimetry (PIV) technique. The present experiments were conducted in a closed-loop open surface water channel with the Reynolds number, $Re_H=1.2{\times}10^4$ based on the model height. In addition to the PIV measurements, flow visualization studies were also carried out. The PIV technique provided instantaneous and time-averaged velocity vectors map, vorticity contours, streamline topology and turbulent quantities at various locations in the near wake. In the vertical symmetry plane, the upperbody flow is separated from the sharp top leading edge of the model and formed a large reverse flow region on the upper surface of the model. The flow structure downstream of the model has asymmetric double vortices. In the horizontal symmetry plane, identical separated flow regions occur on both vertical side walls and a pair of primary recirculatory bubbles dominates the wake region.
Effect of building volume and opening size on fluctuating internal pressures
Ginger, John D.;Holmes, John D.;Kopp, Gregory A. 361
This paper considers internal pressure fluctuations for a range of building volumes and dominant wall opening areas. The study recognizes that the air flow in and out of the dominant opening in the envelope generates Helmholtz resonance, which can amplify the internal pressure fluctuations compared to the external pressure, at the opening. Numerical methods were used to estimate fluctuating standard deviation and peak (i.e. design) internal pressures from full-scale measured external pressures. The ratios of standard deviation and peak internal pressures to the external pressures at a dominant windward wall opening of area, AW are presented in terms of the non-dimensional opening size to volume parameter, $S^*=(a_s/\bar{U}_h)^2(A_W^{3/2}/V_{Ie})$ where $a_s$ is the speed of sound, $\bar{U}_h$ is the mean wind speed at the top of the building and $V_{Ie}$ is the effective internal volume. The standard deviation of internal pressure exceeds the external pressures at the opening, for $S^*$ greater than about 0.75, showing increasing amplification with increasing $S^*$. The peak internal pressure can be expected to exceed the peak external pressure at the opening by 10% to 50%, for $S^*$ greater than about 5. A dominant leeward wall opening also produces similar fluctuating internal pressure characteristics.
Evaluation of base shield plates effectiveness in reducing the drag of a rough circular cylinder in a cross flow
EL-Khairy, Nabil A.H. 377
An experimental investigation has been conducted to determine the effectiveness of base shield plates in reducing the drag of a rough circular cylinder in a cross flow at Reynolds numbers in the range $3{\times}10^4{\leq}Re{\leq}10.5{\times}10^4$. Three model configurations were investigated and compared: a plane cylinder (PC), a cylinder with a splitter plate (MC1) and a cylinder fitted with base shield plates (MC2). Each configuration was studied in the sub and supercritical flow regimes. The chord of the plates, L, ranged from 0.22 to 1.50D and the cavity width, G, between the plates was in the range from 0 to 0.93D. It is recognized that base shield plates can be employed more effectively than splitter plates to reduce the aerodynamic drag of circular cylinders in both the sub- and supercritical flow regimes. For subcritical flow regime, one can get 53% and 24% drag reductions for the MC2 and MC1 models with L/D=1.0, respectively, compared with the PC model. For supercritical flow regime however, the corresponding drag reductions are 38% and 7%.
Design of aerodynamic stabilizing cables for a cable-stayed bridge during construction
Choi, Sung-Won;Kim, Ho-Kyung 391
A design procedure of stabilizing cable is proposed using buffeting analysis to stabilize the seesaw-like motion of the free cantilevered structure of a cable-stayed bridge during its construction. The bridge examined is a composite cable-stayed bridge having a main span length of 500 m. Based on the buffeting analysis, the stress in bare structure exceeded the allowable limit and a set of stabilizing cable was planned to mitigate the responses. The most efficient positions of the hold-down stabilizing cables were numerically investigated by means of an FE-based buffeting analysis and the required dimensions and pretension of the stabilizing cables were also calculated. The proposed stabilizing measure would be expected to secure the aerodynamic safety of a cantilevered structure under construction with considerable mitigation of buffeting responses.
Influence of spacing between buildings on wind characteristics above rural and suburban areas
Kozmar, Hrvoje 413
A wind tunnel study has been carried out to determine the influence of spacing between buildings on wind characteristics above rural and suburban type of terrain. Experiments were performed for two types of buildings, three-floor family houses and five-floor apartment buildings. The atmospheric boundary layer (ABL) models were generated by means of the Counihan method using a castellated barrier wall, vortex generators and a fetch of roughness elements. A hot wire anemometry system was applied for measurement of mean velocity and velocity fluctuations. The mean velocity profiles are in good agreement with the power law for exponent values from ${\alpha}=0.15$ to ${\alpha}=0.24$, which is acceptable for the representation of the rural and suburban ABL, respectively. Effects of the spacing density among buildings on wind characteristics range from the ground up to $0.6{\delta}$. As the spacing becomes smaller, the mean flow is slowed down, whilst, simultaneously, the turbulence intensity and absolute values of the Reynolds stress increase due to the increased friction between the surface and the air flow. This results in a higher ventilation efficiency as the increased retardation of horizontal flow simultaneously accompanies an intensified vertical transfer of momentum. | CommonCrawl |
Convert the following into balanced equations: (…
Convert the following into balanced equations:
(a) When lead(II) nitrate solution is added to potassium iodide solution, solid lead(Il) iodide forms and potassium nitrate solution remains.
(b) Liquid disilicon hexachloride reacts with water to form solid silicon dioxide, hydrogen chloride gas, and hydrogen gas.
(c) When nitrogen dioxide is bubbled into water, a solution of nitric acid forms and gaseous nitrogen monoxide is released.
3.62 Loss of atmospheric ozone has led to an ozone "hole" over Antarctica. The loss occurs in part through three consecutive
(1) Chlorine atoms react with ozone $\left(\mathrm{O}_{3}\right)$ to form chlorine monoxide and molecular oxygen.
(2) Chlorine monoxide forms ClOOCl.
(3) CloOCl absorbs sunlight and breaks into chlorine atoms and molecular oxygen.
(a) Write a balanced equation for each step.
(b) Write an overall balanced equation for the sequence.
What does the term stoichiometrically equivalent molar ratio mean, and how is it applied in solving problems?
The scene below represents a mixture of A2 and B2 before they react to form $\mathrm{AB}_{3}$
(a) What is the limiting reactant?
(b) How many molecules of product can form?
Percent yields are generally calculated from masses. Would the result be the same if amounts (mol) were used instead? Why?
William M.
(a) When gallium metal is heated in oxygen gas, it melts and forms solid gallium (III) oxide.
(b) Liquid hexane burns in oxygen gas to form carbon dioxide gas and water vapor.
(c) When solutions of calcium chloride and sodium phosphate are mixed, solid calcium phosphate forms and sodium chloride remains in solution.
Write a balanced molecular equation deach of the following chemical reactions.
(a) Solid calcium carbonate is heated and decomposes to solid calcium oxide and carbon dioxide gas.
(b) Gaseous butane, $\mathrm{C}_{4} \mathrm{H}_{10},$ reacts with diatomic oxygen gas to yield gaseous carbon dioxide and water vapor.
(c) Aqueous solutions of magnesium chloride and sodium hydroxide react to produce solid magnesium hydroxide
and aqueous sodium chloride.
(d) Water vapor reacts with sodium metal to produce solid sodium hydroxide and hydrogen gas.
Write a balanced equation describing each of the following chemical reactions.
(a) Solid potassium chlorate, $\mathrm{KClO}_{3},$ decomposes to form solid potassium chloride and diatomic oxygen gas.
(b) Solid aluminum metal reacts with solid diatomic iodine to form solid Al_ride gas and aqueous sodium sulfate
(c) When solid sodium chloride is added to aqueous sulfuric acid, hydrogen chloride gas and aqueous sodium sulfate are produced.
(d) Aqueous solutions of phosphoric acid and potassium hydroxide react to produce aquessium dihydrogen phosphate and liquid water.
Give the balanced equation for each of the following.
a. The combustion of ethanol $\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)$ forms carbon dioxide and water vapor. A combustion reaction refers to a reaction of a substance with oxygen gas.
b. Aqueous solutions of lead(Il) nitrate and sodium phosphate are mixed, resulting in the precipitate formation of lead(II) phosphate with aqueous sodium nitrate as the other product.
c. Solid zinc reacts with aqueous HCl to form aqueous zinc chloride and hydrogen gas.
d. Aqueous strontium hydroxide reacts with aqueous hydrobromic acid to produce water and aqueous strontium bromide.
Write balanced chemical equations corresponding to each of the following descriptions: (a) Solid calcium carbide, $\mathrm{CaC}_{2}$ , reacts with water to form an aqueous solution of calcium hydroxide and acetylene gas, $\mathrm{C}_{2} \mathrm{H}_{2}$ . (b) When solid potassium chlorate is heated, it decomposes to form solid potassium chloride and oxygen gas. (c) Solid zinc metal reacts with sulfuric acid to form hydrogen gas and an aqueous solution of zinc sulfate. (d) When liquid phosphorus trichloride is added to water, it reacts to form aqueous phosphorous acid, $\mathrm{H}_{3} \mathrm{PO}_{3}(a q)$, and aqueous hydrochloric acid. (e) When hydrogen sulfide gas is passed over solid hot iron(III) hydroxide, the resulting reaction produces solid iron(II) sulfide and gaseous water. | CommonCrawl |
Symmetric hypergraph theorem
The Symmetric hypergraph theorem is a theorem in combinatorics that puts an upper bound on the chromatic number of a graph (or hypergraph in general). The original reference for this paper is unknown at the moment, and has been called folklore.[1]
Statement
A group $G$ acting on a set $S$ is called transitive if given any two elements $x$ and $y$ in $S$, there exists an element $f$ of $G$ such that $f(x)=y$. A graph (or hypergraph) is called symmetric if its automorphism group is transitive.
Theorem. Let $H=(S,E)$ be a symmetric hypergraph. Let $m=|S|$, and let $\chi (H)$ denote the chromatic number of $H$, and let $\alpha (H)$ denote the independence number of $H$. Then
$\chi (H)\leq 1+{\frac {\ln {m}}{-\ln {(1-\alpha (H)/m)}}}$
Applications
This theorem has applications to Ramsey theory, specifically graph Ramsey theory. Using this theorem, a relationship between the graph Ramsey numbers and the extremal numbers can be shown (see Graham-Rothschild-Spencer for the details).
See also
• Ramsey theory
Notes
1. R. Graham, B. Rothschild, J. Spencer. Ramsey Theory. 2nd ed., Wiley, New-York, 1990.
| Wikipedia |
Mathematics Meta
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Can I prove set propositions using first-order logic?
I'm studying logic and sets and I have to say there's a strong similarity between the two. Most boolean/logic axioms also apply to sets. At the end of my course I also studied first-order logic (or predicate logic) and how one can actually define statements using first-order logic. This was quite a revelation to me because it is quite easy to prove first-order logic statements compared to set propositions (my brain just works better at decomposing first-order logic propositions). So I'm wondering, is it possible to prove set propositions using first-order logic? Here's an example of a set proposition I have to prove:
if C ⊆ B then A ⇒ C ⊆ A ⇒ B
(not that A ⇒ B is assumed to mean ¬A ∪ B)
Now I was thinking I could convert this into a first-order logic statement, such as:
Subset(C, B) ⇒ Subset(A ⇒ C, A ⇒ B)
As you can see however I can't seem to understand how to prove that in first-order logic. Anyway, perhaps I am confusing the two and this is not possible but I was wondering what you guys thought about proving these set statements using first-order logic.
elementary-set-theory
predicate-logic
first-order-logic
Luca Matteis
Luca MatteisLuca Matteis
31711 gold badge22 silver badges1313 bronze badges
$\begingroup$ Do you use "$A\Rightarrow C$" as notation for some set? That is not standard. Which set does that mean? $\endgroup$
– hmakholm left over Monica
$\begingroup$ Sorry it means ¬A u C, will edit. $\endgroup$
– Luca Matteis
$\begingroup$ You can do this, but your primitive boolean expressions should be: "if $x\in C\Rightarrow x\in B$, then ... " (In other words, your expressions are all of the form $x\in A$, $x\in B$, $x\in C$). $\endgroup$
– Michael Burr
You can translate between set algebra and logic, but you need to speak about elements in order to do so.
If you take a set algebra expression, you can rewrite it into set-builder notation from the bottom up as follows:
A named set $A$ is the same as $\{x\mid x\in A\}$.
For a union, first rewrite the operands until they have set-builder form, and then use $$ \{x\mid \phi(x) \} \cup \{x\mid \psi(x) \} = \{x\mid \phi(x)\lor\psi(x)\} $$ where $\phi(x)$ and $\psi(x)$ are some formulas that have $x$ free.
Similarly, intersections and complement correspond to conjunction and negation: $$ \{x\mid \phi(x) \} \cap \{x\mid \psi(x)\} = \{x\mid \phi(x)\land\psi(x) \} \\ \{x\mid \phi(x) \}^\complement = \{x \mid \neg \phi(x) \} \\ \{x\mid \phi(x) \} \setminus \{x\mid\psi(x)\} = \{x\mid \phi(x)\land\neg\psi(x)\} $$
When you have a relation between sets, you get plain logical formulas back: $$ \{x\mid \phi(x) \} \subseteq \{x\mid \psi(x)\} \iff \forall x (\phi(x)\rightarrow \psi(x)) \\ \{x\mid \phi(x) \} = \{x\mid\psi(x)\} \iff \forall x(\phi(x)\leftrightarrow \psi(x)) $$
It seems you already have this kind of translation in mind when you're using "$\Rightarrow$" and "$\neg$" as set algebra operations -- these symbols are usually used only as logical connectives. (Conversely, there is no conventional logical connective that by itself represents set difference).
The underlying similarity here is that the universe of sets with $\cup$, $\cap$, $\complement$, or the universe of formulas with $\lor$, $\land$, $\neg$ both constitute Boolean algebras. The $\{x\mid\cdots\}$ construction is a homomorphism between the two Boolean algebras, and the work mostly just consists of changing notation to that used on the other world.
edited Aug 6, 2018 at 18:43
hmakholm left over Monicahmakholm left over Monica
$\begingroup$ Very cool thanks. Going back specifically to my example, do you think first-order logic can be used to prove it? $\endgroup$
$\begingroup$ @LucaMatteis: Certainly you can. Your example becomes the inference $$\forall x(x\in C\Rightarrow x\in B) \vdash \forall x(x\in A\Rightarrow x\in C) \Rightarrow \forall x(x\in A\Rightarrow x\in B)$$ which you can then prove with your favorite proof formalism for first-order logic. $\endgroup$
$\begingroup$ Nice of course! By the way, what does ⊢ mean? I suppose ⇒, right? $\endgroup$
$\begingroup$ @LucaMatteis: $\vdash$ is the entailment (or provability or consequence) relation. $P\vdash Q$ means that the formula $Q$ can be proved if we're allowed to assume $P$. It is a property of first-order logic (and most other logics) that $P\vdash Q$ holds exactly when $\vdash (P\Rightarrow Q)$. But notice that $\Rightarrow$ combines formulas into a different formula (which may well be true in some concrete situation without being provable), whereas something that involves $\vdash$ is not a formula but an assertion outside logic about the existence of certain proofs. $\endgroup$
$\begingroup$ @Luca: That's a good question that I think deserves an answer that is longer than what fits in a comment. You should consider asking it as a question of its own. $\endgroup$
You can convert many set theoretic statements to Boolean expressions. The most important thing to note is that the set $A$ is not appropriate for a Boolean expression because it's not a statement. You should replace $A$ with $x\in A$ because $x\in A$ can be true or false. The translations are as follows:
$$ \begin{array}{c|c} \text{Set}&\text{Boolean}\\\hline A\subseteq B&x\in A\Rightarrow x\in B\\ A^c&\neg(x\in A)\\ A\cap B&(x\in A)\wedge (x\in B)\\ A\cup B&(x\in A)\vee (x\in B)\\ A\setminus B&(x\in A)\wedge (x\not\in B) \end{array} $$ From this list, you can derive most statements in basic set theory (but some get very complicated to compute).
Notes: $A^c$ is the complement of $A$ in some universe. $\neg(x\in A)\equiv x\not\in A$.
Michael BurrMichael Burr
$\begingroup$ Indeed I was already aware of this. My question was about how to apply this to the set proposition shown in my question. And then how to construct first-order logic statements given the proposition at hand. $\endgroup$
$\begingroup$ The statements in the problem don't really make sense because $A$ and $B$ are sets, so you shouldn't use Boolean expressions with them. It would be better to not mix the two notations (even though they are equivalent). $\endgroup$
$\begingroup$ Actually, as Henning showed above, it can be done! Thanks anyway $\endgroup$
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How is first-order logic a strong enough logic for the foundations of mathematics? | CommonCrawl |
What is Conceptual Clustering?
Data MiningDatabaseData Structure
Conceptual clustering is a form of clustering in machine learning that, given a set of unlabeled objects, makes a classification design over the objects. Unlike conventional clustering, which generally identifies groups of like objects, conceptual clustering goes one step further by also discovering characteristic definitions for each group,where each group defines a concept or class.
Therefore, conceptual clustering is a two-step process − clustering is implemented first, followed by characterization. Thus, clustering quality is not solely a service of single objects. Most techniques of conceptual clustering adopt a statistical method that uses probability measurements in deciding the concepts or clusters.
Probabilistic descriptions are generally used to define each derived concept. COBWEB is a famous and simple method of incremental conceptual clustering. Its input objects are defined by categorical attribute-value pairs. COBWEB makes a hierarchical clustering in the form of a classification tree.
A classification tree differs from a decision tree. Each node in a classification tree defines a concept and includes a probabilistic description of that concept, which summarizes the objects classified under the node. The probabilistic description contains the probability of the concept and conditional probabilities of the form $P(A_{i}=v_{ij}|C_{k})$ is an attribute-value pair (the ith attribute takes its jth possible value) and Ck is the concept class.
COBWEB uses a heuristic evaluation measure known as category utility to guide the construction of the tree. Category Utility (CU) is defined as
$$\frac{\sum_{k=1}^{n}P(C_{k})\left [\sum_{i}\sum_{j}P(A_{i}=v_{ij}|C_{k})^{2}-\sum_{i}\sum_{j}P(A_{i}=v_{ij})^{2}\right ]}{n}$$
where n is the number of nodes, concepts, or "categories" forming a partition, {C1,C2,..., Cn}, at the given level of the tree. In other terms, category utility is the increase in the expected number of attribute values that can be perfectly guessed given a partition (where this expected number corresponds to the term $P(C_{k})\sum_{i}\sum_{j}P(A_{i}=v_{ij}|C_{k})^{2}$ over the expected number of correct guesses with no such knowledge (corresponding to the term $\sum_{i}\sum_{j}P(A_{i}=v_{ij})^{2}$ .Although it does not have room to display the derivation, category utility rewards intraclass similarity and interclass dissimilarity, where −
Intraclass similarity − It is the probability $P(A_{i}=v_{ij}|C_{k})$. The higher this value is, the higher the proportion of class members that share this attribute-value pair and the more predictable the pair is of class members.
Interclass dissimilarity − It is the probability $P(C_{k}|A_{i}=v_{ij})$. The higher this value is,the fewer the objects in contrasting classes that share this attribute-value pair and the more predictive the pair is of the class.
COBWEB descends the tree along a suitable path, refreshing counts along the way, in search of the "best host" or node at which to define the object. This decision depends on temporarily locating the object in each node and evaluating the category utility of the resulting partition. The placement that results in the highest category utility should be the best host for the object.
Ginni
What is Clustering?
What is Multi-relational Clustering?
What is clustering Index in DBMS?
What is scipy cluster hierarchy? How to cut hierarchical clustering into flat clustering?
What are the methods of clustering?
What are the applications of clustering?
Why is wavelet transformation useful for clustering?
Which SciPy package is used to implement Clustering?
What are the requirements of clustering in data mining?
Asymmetric and Symmetric Clustering System
Difference Between Classification and Clustering
How to make a scatter plot for clustering in Python?
Implementing K-means clustering of Diabetes dataset with SciPy library
Implementing K-means clustering with SciPy by splitting random data in 3 clusters? | CommonCrawl |
\begin{document}
\title{Vortex Invariants and Toric Manifolds} \author{Jan Wehrheim}
\maketitle
\begin{abstract} We consider the symplectic vortex equations for a Hamiltonian action of a torus $T$ on $\mathbbm C^n$. We show that the associated genus zero moduli space itself is homotopic (in the sense of a regular homotopy of $T$-moduli problems) to a toric manifold with combinatorial data directly obtained from the original torus action. This allows to view the wall crossing formula for the computation of vortex invariants by Cieliebak and Salamon \cite{CS} as a consequence of a generalized Jeffrey-Kirwan localization formula for integrals over symplectic quotients. \end{abstract}
\tableofcontents
\section{Introduction}
Let $(X,\omega)$ be a symplectic manifold. Suppose that a compact Lie group $G$ acts on $X$ in a Hamiltonian way with moment map $\mu$. This is the general setting for the symplectic vortex equations that where introduced by Cieliebak, Gaio, Mundet and Salamon in \cite{CGMS}. They are of the form
$$ (*) \quad \left\{ \quad \begin{array}{c@{\quad = \quad}c} \bar{\partial}_{J,A} u & 0\\ \ast F_A + \mu(u) & \tau. \end{array} \right. $$
Here $u : P \rightarrow X$ is a $G$-equivariant map from a principal $G$-bundle $P$ over a closed Riemann surface $\Sigma$ and $A \in \Omega^1(P)$ is a connection form on $P$ with curvature $F_A$. The additional data entering the equations are a $G$-invariant, $\omega$-compatible almost complex structure $J$ on $X$ that gives rise to the Cauchy-Riemann operator $\bar{\partial}_{J,A}$, a metric on $\Sigma$ that defines the Hodge-operator $\ast$, and a parameter $\tau$. For the motivation to study these equations we refer to Cieliebak, Gaio and Salamon \cite{CGS}. There are two central results: By Cieliebak, Gaio, Mundet and Salamon \cite{CGMS} the solutions to $(*)$ give rise to well defined invariants in many cases. And it is shown by Gaio and Salamon in \cite{GS} that in certain cases these vortex invariants coincide with Gromov-Witten invariants of the symplectic quotient $X/\!/G(\tau) := \mu^{-1}(\tau)/G$. The latter result is obtained by introducing a parameter $\varepsilon$ in the second equation of $(*)$ in front of the term $\mu(u)$ and an adiabatic limit analysis for $\varepsilon \longrightarrow \infty$. In this limit the solutions to the vortex equations degenerate to holomorphic curves $\Sigma \longrightarrow X/\!/G(\tau)$. We study the symplectic vortex equations on $\mathbbm C^N$ with its standard symplectic and complex structure and with a torus $T$ acting by a representation $\rho : T \longrightarrow U(N)$. Symplectic quotients of such linear torus actions are called toric manifolds. In this setup vortex invariants are well defined.
Our main result can be viewed as the counterpart to the adiabatic limit of Gaio and Salamon in \cite{GS}. We introduce the same parameter $\varepsilon$ (with a slight modification if the principal bundle $P$ is not trivial) but we consider the other limit $\varepsilon \longrightarrow 0$. One can also interprete this deformation as a rescaling of the symplectic form by $\varepsilon$. We show that this deformation gives rise to a homotopy of regular $T$-moduli problems (Theorem \ref{thm:homotopy}). The main issue is to prove compactness for the parametrized moduli space. Our result then shows that the invariants associated to the deformed vortex equations with $\varepsilon = 0$ agree with the usual vortex invariants. And in fact this deformed picture is very nice: We show that the moduli space to the deformed genus zero vortex equations itself carries the structure of a toric manifold (Theorem \ref{thm:genus_0_moduli_space}). If $\Sigma$ is a surface of arbitrary genus we show more generally that under some additional assumptions the vortex moduli space is a fiber bundle over the Jacobian torus $\left( H^1(\Sigma;\mathbbm R) / H^1(\Sigma;\mathbbm Z) \right)^{\dim T}$ with toric fiber (Theorem \ref{thm:genus_g_moduli_space}).
As an application we show how these observations simplify the computation of genus zero vortex invariants. We can express vortex invariants as integrals over toric manifolds (Theorem \ref{thm:computation_of_vortex_invariants}). The wall crossing formula of Cieliebak and Salamon \cite[Theorem 1.1]{CS} that was used for the original computation of these invariants then is a consequence of a localization formula for certain integrals over toric manifolds (Theorem \ref{thm:wall_crossing}). To prove this localization formula we develop a relative version of Atiyah-Bott localization (Theorem \ref{thm:relative_localization}) that applies to the case of invariant integration. This technique is interesting on its own account and we develop it in more generality than needed for the computation of vortex invariants. It gives an alternative approach to integration formulae that are generally referred to as Jeffrey-Kirwan localization.
In sction \ref{chap:equivariant_cohomology} we review equivariant cohomology of manifolds that carry a group action by some compact Lie group $G$. In particular we review the Cartan model and the Cartan map and its generalization in the case of a normal subgroup $H \lhd G$ acting locally freely. Invariant integration is introduced in section \ref{chap:invariant_integration} and generalized in two directions. One is the relative case of invariant integration with respect to a normal subgroup $H \lhd G$. The other is the extension to more general push-forwards in equivariant cohomology. The result is the notion of $H$-invariant push-forward. Section \ref{chap:localization} then features the theory of localization adapted to the setting of invariant integration. We deduce the above-mentioned integration formula (Theorem \ref{thm:relative_localization}).
In section \ref{chap:moduli_problems} we turn to the notion of $G$-moduli problems and the equi\-vari\-ant Euler class from Cieliebak, Mundet and Salamon \cite{CMS}. This is the technique that is used to define invariants via the solutions to the vortex equations. It uses invariant integration. So we examine what our ge\-ne\-ra\-li\-za\-tions to invariant integration from section \ref{chap:invariant_integration} yield in this context.
We turn to toric manifolds in section \ref{chap:toric_manifolds}. We apply the results from all preceding sections and prove the wall crossing formula for certain integrals over toric manifolds (Theorem \ref{thm:wall_crossing}). We make a short digression that puts this result into the context of Jeffrey-Kirwan localization and the related works by Guillemin and Kalkman \cite{GK} and Martin \cite{Mar}.
The symplectic vortex equations and invariants in the case of a linear torus action are introduced in section \ref{chap:vortex_invariants}. We perform the above-mentioned deformation, prove the essential property of being a homotopy of regular $T$-moduli problems and derive the main results.
In section \ref{chap:generalizations} we attempt to extend our deformation result to more general settings. In fact we have no such generalization and we explain the reason for that failure. Finally in section \ref{chap:giventals_toric_map_spaces} we explain the relation of our toric structure on the vortex moduli spaces to Givental's toric map spaces as presented in the work of Iritani \cite{Iri}.
\section{Equivariant cohomology}
\label{chap:equivariant_cohomology}
Throughout this section let $G$ be a compact Lie group and $X$ a closed manifold with a smooth $G$-action from the left. We introduce the Borel and the Cartan model for equivariant cohomology $H_G^*(X)$. In particular we consider a closed normal subgroup $H \lhd G$ and look at the consequences it has on $H_G^*(X)$ if the induced $H$-action is free or trivial.
\subsection{The Borel model}
\label{sec:Borel_model}
We denote by $\mathrm{BG}$ the classifying space for the group $G$. It is defined uniquely up to homotopy equivalence. Given any contractible topological space $\mathrm{EG}$ with a free $G$-action we can take $\mathrm{BG} := \mathrm{EG} / G$ as a representative. We will call any such $\mathrm{EG}$ a classifying total space for $G$. See tom Dieck \cite{tomD} for details.
The Borel construction of the $G$-manifold $X$ is the topological space
$$ X_G := X \times_G \mathrm{EG} := \left( X \times \mathrm{EG} \right) / G, $$
where the quotient is taken with respect to the diagonal $G$-action on $X \times \mathrm{EG}$. Then one defines the $G$-equivariant cohomology of $X$ to be the ordinary singular cohomology of its Borel construction,
$$ H_G^*(X) := H^*(X_G). $$
Throughout we will take either real or complex coefficients.
\begin{rem} The projection onto the second factor turns $X_G$ into a fiber bundle over $\mathrm{BG}$ with fiber $X$. In particular this makes equivariant co\-ho\-mo\-lo\-gy $H_G^*(X)$ a module over $H^*(\mathrm{BG})$ via pullback. \end{rem}
Now suppose that $H \lhd G$ is a closed normal subgroup of $G$ that acts freely on $X$. We denote by $K := G/H$ the quotient group. On the classifying total space $\mathrm{EK}$ we let $G$ act via the projection $G \longrightarrow K$. Hence $H$ acts trivially on $\mathrm{EK}$. Now we can replace $\mathrm{EG}$ by $\mathrm{EG} \times \mathrm{EK}$ as a model for the classifying total $G$-space, because the product of two contractible spaces is still contractible, and the diagonal $G$-action is free, because it is free on the first factor. So up to homotopy equivalence we obtain
$$ X_G = \left( X \times \mathrm{EG} \times \mathrm{EK} \right) / G = \left( \left( X \times_H \mathrm{EG} \right) \times \mathrm{EK} \right) / K. $$
Now $H$ acts freely on $X$ and $\mathrm{EG}$ is contractible. Hence $X \times_H \mathrm{EG}$ is homotopy equivalent to $X/H$ and this homotopy equivalence extends to the bundle $\left( \left( X \times_H \mathrm{EG} \right) \times \mathrm{EK} \right) / K$, as shown in tom Dieck \cite[I 8.16]{tomD}. So we obtain
$$ X_G = \left( X/H \times \mathrm{EK} \right) / K = \left( X/H \right)_K $$
up to homotopy equivalence and we deduce
\begin{equation} \label{eqn:free_H_action} H_G^*(X) = H_K^*(X/H). \end{equation}
Now suppose on the contrary that a closed normal subgroup $H \lhd G$ acts trivially on $X$. Again we write $X_G = (X \times \mathrm{EK} \times \mathrm{EG}) / G$, but now $H$ acts trivially on $X$ and $\mathrm{EK}$ and we obtain
$$ X_G = \left( X \times \mathrm{EK} \times \mathrm{EG}/H \right) / K. $$
This shows that up to homotopy equivalence $X_G$ is a fibration over $X_K$ with fiber $\mathrm{EG}/H = \mathrm{BH}$, because $\mathrm{EG}$ also serves as a model for $\mathrm{EH}$: It is contractible and the subgroup $H$ acts freely. This fibration is determined by the $K$-action on $\mathrm{EG}/H$ and it determines the relation between $H_G^*(X)$, $H_K^*(X)$ and $H^*(\mathrm{BH})$.
If we additionally assume that there exists a monomorphism $G \longrightarrow K \times H$ then we get a free $G$-action on the contractible space $\mathrm{EK} \times \mathrm{EH}$ and we can write $X_G$ as $(X \times \mathrm{EK} \times \mathrm{EH}) / G$. Now $H$ again acts trivially on $X$ and $\mathrm{EK}$, whereas $K$ acts trivially on $\mathrm{EH}$ and hence also on $\mathrm{EH}/H = \mathrm{BH}$. So we get $X_G = X_K \times \mathrm{BH}$, i.e.~the fibration is trivial and the relation becomes particularly nice:
\begin{equation} \label{eqn:trivial_H_action} H_G^*(X) = H_K^*(X) \otimes H^*(\mathrm{BH}). \end{equation}
\subsection{The Cartan model}
\label{sec:Cartan_model}
The Cartan model is the de Rham model for equivariant cohomology. The topological constructions on spaces in the Borel model are mimicked on the algebraic level of differential forms.
\subsubsection*{Equivariant de Rham theory}
The Lie algebra of $G$ is denoted by ${\mathfrak g}$, its dual space by ${\mathfrak g}^*$. We write $S({\mathfrak g}^*)$ for the symmetric algebra generated by the vector space ${\mathfrak g}^*$. We assign the degree of $2$ to all elements of ${\mathfrak g}^*$ and view $S({\mathfrak g}^*)$ as a graded algebra.
The group $G$ acts from the left on ${\mathfrak g}^*$ by the coadjoint representation $\mathrm{Ad}^*$ defined by
$$ \langle \mathrm{Ad}^*_g(x) , \xi \rangle \;:=\; \langle x , \mathrm{Ad}_{g^{-1}}(\xi) \rangle $$
for any $x \in {\mathfrak g}^*$ and $\xi \in {\mathfrak g}$, where $\langle .,. \rangle$ denotes the pairing between dual spaces and $\mathrm{Ad}_g(x) = gxg^{-1}$ is the adjoint action of $G$ on its Lie algebra. This action extends element-wise to $S({\mathfrak g}^*)$.
We denote by $\Omega^*(X)$ the differential forms on $X$. We get a left action of $G$ on $\Omega^*(X)$ by taking inverses and pulling back: $g.\omega := (g^{-1})^*\omega$. For an element $\xi \in {\mathfrak g}$ we define the infinitesimal vector field $X_\xi$ at a point $p \in X$ as
\begin{equation} \label{def:X_xi}
X_\xi (p) \;:=\; \left. \frac{\mathrm{d}}{\mathrm{dt}} \right|_{t=0} \left[ \exp(-t\xi) . p \right]. \end{equation}
We abbreviate the operation of plugging the vector field $X_\xi$ into the first argument of a differential form by $\iota_\xi$.
\begin{rem} We introduce the minus sign in the definition of $X_\xi$ in order to be consistent with the conventions for principal bundles later on. This convention also agrees with the one by Guillemin and Sternberg \cite{GS} but differs from the one by Cieliebak, Mundet and Salamon \cite{CMS}. \end{rem}
We can now define $G$-equivariant differential forms on $X$ as elements in
$$ \Omega_G^*(X) \;:=\; \left[ S({\mathfrak g}^*) \otimes \Omega^*(X) \right]^G, $$
where the superscript $G$ means that we take only $G$-invariant elements of the tensor product. One can think of taking the tensor product with $S({\mathfrak g}^*)$ as an algebraic analogue of taking the product with some representative of $\mathrm{EG}$ in the topological setting. And restricting to $G$-invariant elements corresponds to taking the $G$-quotient of $\mathrm{EG} \times X$. In analogy to the Borel construction $X_G$ for the space $X$ this algebraic version is also called the Cartan construction for $\Omega^*(X)$ and we write
$$ C_G(B) \;:=\; \left[ S({\mathfrak g}^*) \otimes B \right]^G $$
for any suitable algebra $B$. Hence $\Omega_G^*(X) = C_G( \Omega^*(X) )$.
To define the $G$-equivariant differential $d_G$ we think of elements $\alpha \in \Omega_G^*(X)$ as polynomials on ${\mathfrak g}$ with values in $\Omega^*(X)$. Then the value of $d_G \alpha$ on an element $\xi \in {\mathfrak g}$ is defined to be
$$ \mathrm{d}_G \alpha (\xi) \;:=\; \mathrm{d} \alpha(\xi) - \iota_\xi \alpha(\xi). $$
If we choose a basis $(\xi_a)$ for ${\mathfrak g}$ and take the dual basis $(x^a)$ on ${\mathfrak g}^*$ this can be expressed as
$$ \mathrm{d}_G \;=\; 1 \otimes \mathrm{d} - \sum_a x^a \otimes \iota_{\xi_a}, $$
showing that this differential indeed rises the degree by $+1$, because $x^a$ has degree $2$. It is an easy exercise to show that $\mathrm{d}_G^2 = 0$, but the fact that the homology of the Cartan complex indeed computes equivariant cohomology is a nontrivial result.
\begin{thm}[H.~Cartan] \label{thm:Cartan}
$$ H_G^*(X) \;=\; H \left( C_G(\Omega^*(X)) , \mathrm{d}_G \right) $$
\end{thm}
\subsubsection*{The generalized Cartan map}
We discussed the case of a normal subgroup $H \lhd G$ acting freely on $X$ in the Borel model and obtained identity \ref{eqn:free_H_action}. With the Cartan model we can obtain the corresponding result also for locally free $H$-actions. A group $H$ acts locally freely if the isotropy subgroups
$$
\mathrm{Iso}_p(H) := \left\{ h \in H \;|\; h.p = p \right\} $$
are finite for all $p \in X$, or equivalently if the infinitesimal action $\xi \longmapsto X_\xi(p)$ is everywhere injective. We will also use the term \emph{regular} for a locally free action and use the notation $\mathrm{Iso}_p(H) =: H_p$ for isotropy groups.
\begin{dfn} A form $\omega \in \Omega^*(X)$ is called \emph{$H$-horizontal} if
$$ \iota_\eta \omega \;=\; 0 $$
for all $\eta \in {\mathfrak h}$. The set of $H$-horizontal forms is denoted by $\Omega_{H-\mathrm{hor}}^*(X)$. A form $\omega$ is called \emph{$H$-basic} if it is $H$-horizontal and $H$-invariant. The set of $H$-basic forms is denoted by $\Omega_{H-\mathrm{bas}}^*(X)$. \end{dfn}
Note that in fact $\Omega_{H-\mathrm{bas}}^*(X) = \Omega^*(X/H)$ if the $H$-action is free. But even in presence of nontrivial isotropy the space $\Omega_{H-\mathrm{bas}}^*(X)$ still serves perfectly well as an algebraic model for differential forms on the quotient: If $K := G/H$ we get a $K$-action on $\Omega_{H-\mathrm{bas}}^*(X)$ and for elements $\theta \in {\mathfrak k} = \mathrm{Lie}(K)$ we get operations $\iota_\theta$ by lifting $\theta$ to $\overline{\theta} \in {\mathfrak g}$ and taking $\iota_{\overline{\theta}}$. Hence we can perform the $K$-Cartan construction on $\Omega_{H-\mathrm{bas}}^*(X)$.
\begin{prop} \label{prop:gen_Cartan} If a closed normal subgroup $H \lhd G$ acts locally freely on $X$ then the inclusion
$$ \Omega_G^*(X) = \left[ S({\mathfrak g}^*) \otimes \Omega^*(X) \right]^G \supset \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{bas}}^*(X) \right]^K = C_K \left( \Omega_{H-\mathrm{bas}}^*(X) \right) $$
induces an isomorphism
$$ H_G^*(X) \;\cong\; H \left( C_K \left( \Omega_{H-\mathrm{bas}}^*(X) \right) , \mathrm{d}_K \right). $$
\end{prop}
\begin{rem} This result is certainly not new. It is well known (although we do not know a suitable reference) that one can deal with orbifolds just as well as with manifolds, so that there is no trouble with taking the quotient with respect to an action that is only locally free. Nevertheless we think it is worthwhile to treat the whole theory without leaving the realm of smooth manifolds. We do not go to the partial $H$-quotient but keep track of the whole action on the original manifold and this turns out to be helpful for the discussion of integration and localization later on. \end{rem}
The essential tool for the proof of proposition \ref{prop:gen_Cartan} (i.e.~the homotopy inverse to the inclusion) is the Cartan map. In case $H = G$ this map is due to H.~Cartan (see Guillemin and Sternberg \cite{GS} for a nice exposition). The generalized map for arbitrary $H$ appears in the work of Cieliebak, Mundet and Salamon \cite{CMS} and is a slight extension of the things said in \cite[section 4.6]{GS} for commuting actions of two groups. For clearity we give an extensive review of the construction.
\begin{dfn} \label{def:G_equivariant_H_connection} A \emph{$G$-equivariant $H$-connection} on $X$ is a form $A \in \Omega^1(X,{\mathfrak h})$ that satisfies
$$ A_{g.p}(g_*v ) \;=\; g A_p(v) g^{-1} \quad , \qquad A_p(X_\eta(p)) \;=\; \eta $$
for all $g \in G$, $p \in X$, $v \in T_pX$, $\eta \in {\mathfrak h}$. \end{dfn}
\begin{rem} Suppose $X$ is a principal $G$-bundle with right action
$$ X \times G \longrightarrow X \; ; \; (p,g) \longmapsto pg. $$
Then $X$ also carries the \emph{left} $G$-action $G \times X \longrightarrow X \; ; \; (g,p) \longmapsto pg^{-1}$. A $G$-equivariant $G$-connection on $X$ with this left $G$-action is then the same as a connection form on the principal bundle $X$ in the usual sense. \end{rem}
\begin{rem} Such $G$-equivariant $H$-connections do exist if the $H$-action is regular. Here we make use of the compactness of $G$ and of $H$ being normal. \end{rem}
Let us fix a $G$-equivariant $H$-connection $A$ for the moment. This connection determines a $G$-equivariant projection
\begin{equation} \label{def:pi_A} \pi_A^* \;:\; \Omega^*(X) \;\longrightarrow\; \Omega_{H-\mathrm{hor}}^*(X) \end{equation}
via the projection $\pi_A$ onto the kernel of $A$:
\begin{align*} \pi_A \;:\; T_pX & \;\longrightarrow\; T_pX \\ v & \;\longmapsto\; v - X_{A_p(v)}(p) \end{align*}
The curvature $F_A \in \Omega^2(X,{\mathfrak h})$ is defined by
$$ F_A \;:=\; \mathrm{d}A + \frac{1}{2} \left[ A \wedge A \right], $$
where $[. \wedge .]$ denotes the product on $\Omega^*(X,{\mathfrak h}) = \Omega^*(X) \otimes {\mathfrak h}$ via the wedge-product on the form part and the Lie bracket on ${\mathfrak h}$. Note that $F_A$ inherits the $G$-invariance from $A$ and that $F_A$ is $H$-horizontal. The $G$-equivariant curvature $F_{A,G}$ is the element of
$$ \Omega_G^*(X,{\mathfrak g}) \;:=\; C_G( \Omega^*(X,{\mathfrak g}) ) \;=\; \left[ S({\mathfrak g}^*) \otimes \Omega^*(X,{\mathfrak g}) \right]^G $$
given by
$$ F_{A,G}(\xi) \;:=\; F_A + \xi - \iota_\xi A. $$
The lemma below shows that $F_{A,G}$ really is an invariant element of the tensor product $S({\mathfrak g}^*) \otimes \Omega^*(X,{\mathfrak g})$. We choose a basis $(\xi_j)$ on ${\mathfrak g}$ such that $(\xi_b)_{b \in B}$ is a basis for the subspace ${\mathfrak h}$ for some index set $B$, and we take the dual basis $(x^j)$ for ${\mathfrak g}^*$. Then the $G$-equivariant curvature can be expressed as
$$ F_{A,G} \;=\; 1 \otimes F_A \quad + \quad \sum_j x^j \otimes \left( \xi_j - \iota_{\xi_j} A \right). $$
In the sum actually only indices $j \not\in B$ appear, because $\eta - \iota_\eta A = 0$ for all $\eta \in {\mathfrak h}$.
\begin{lem} The equivariant curvature $F_{A,G}$ in fact defines an element in the subspace
$$ \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{bas}}^*(X,{\mathfrak g}) \right]^K \;\subset\; \left[ S({\mathfrak g}^*) \otimes \Omega^*(X,{\mathfrak g}) \right]^G. $$
\end{lem}
\begin{proof}
Since $F_A$ is $G$-invariant and $H$-horizontal the first summand $1 \otimes F_A$ obviously lies in the correct space. For the remaining part we first note that for $j \not\in B$ we have $x^j \in {\mathfrak k}^* = \{ x \in {\mathfrak g} \;|\; x|_{\mathfrak h} = 0 \}$. Furthermore the $\xi_j - \iota_{\xi_j} A$ are all $H$-horizontal. Now $A$ is $G$-invariant and $\iota_{\xi_j} A$ is linear in $\xi_j$. Hence $\xi_j - \iota_{\xi_j} A$ transforms under $G$ just like $\xi_j$. Thus the sum over all $x^j \otimes (\xi_j - \iota_{\xi_j} A)$ is $G$-invariant, because $\sum_j x^j \otimes \xi_j$ is. This shows that
$$ F_{A,G} \;\in\; \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{hor}}^*(X,{\mathfrak g}) \right]^G. $$
But since $H$ acts trivially on ${\mathfrak k}^*$ it must also act trivially on all terms $\xi_j - \iota_{\xi_j} A$ with $j \not\in B$ and the result follows. \end{proof}
Now we consider the pairing between ${\mathfrak g}^*$ and the ${\mathfrak g}$-factor of $F_{A,G}$:
\begin{align*} {\mathfrak g}^* & \;\longrightarrow\; S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{hor}}^*(X) \\ x \; & \;\longmapsto\; \qquad \langle x , F_{A,G} \rangle \end{align*}
This map is $G$-equivariant because $F_{A,G}$ is $G$-invariant. We extend it to a $G$-equivariant map
\begin{equation} \label{def:F_AG} S({\mathfrak g}^*) \;\longrightarrow\; S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{hor}}^*(X). \end{equation}
Now the Cartan map $c_A$ for a given $G$-equivariant $H$-connection $A$ is defined by the tensor product of the two maps \ref{def:pi_A} and \ref{def:F_AG}
$$ \Omega_G^*(X) \;=\; \left[ S({\mathfrak g}^*) \otimes \Omega^*(X) \right]^G \;\longrightarrow\; \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{hor}}^*(X) \otimes \Omega_{H-\mathrm{hor}}^*(X) \right]^G $$
followed by taking the wedge product of the two $\Omega_{H-\mathrm{hor}}^*(X)$ factors. Since all these operations are $G$-equivariant the map stays in the $G$-invariant part and we in fact obtain a map
\begin{align} \label{def:c_A} c_A \;:\; \Omega_G^*(X) & \;\longrightarrow\; \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{hor}}^*(X) \right]^G \;=\; \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{bas}}^*(X) \right]^K \\ \alpha \quad & \;\longmapsto\; \quad \alpha_A \;:=\; c_A(\alpha) \nonumber \end{align}
\begin{dfn} A $G$-equivariant differential form $\alpha$ is called \emph{$H$-basic}, if it is an element of
$$ C_K \left( \Omega_{H-\mathrm{bas}}^*(X) \right) = \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{bas}}^*(X) \right]^K \subset \left[ S({\mathfrak g}^*) \otimes \Omega^*(X) \right]^G = \Omega_G^*(X). $$
\end{dfn}
So in case of a regular $H$-action one can use the Cartan map to make $G$-equivariant differential forms $H$-basic.
\begin{lem} \label{lem:basic_forms} Let $\alpha \in \Omega_G^*(X)$ be $H$-basic and denote by $\alpha^{[m]}$ those components of $\alpha$ with form part having degree $m$. Then the following holds.
\begin{enumerate} \item $\mathrm{d}_G \alpha = \mathrm{d}_K \alpha$. \item If the $H$-action is regular then $\alpha^{[m]} = 0$ for all $m > \dim(X) - \dim(H)$. \item If the $H$-action is regular and $m \ge \dim(X) - \dim(H) - 2$ then $$ \left( \mathrm{d}_G \alpha \right)^{[m]} = (1 \otimes \mathrm{d}) \left( \alpha^{[m-1]} \right). $$ \end{enumerate}
\end{lem}
\begin{proof} The first statement follows because by definition $\iota_\eta \omega = 0$ for all $\eta \in {\mathfrak h}$ and $\omega \in \Omega_{H-\mathrm{bas}}^*(X)$. For the second statement observe that in case of a regular $H$-action the vectors $X_\eta$ with $\eta \in {\mathfrak h}$ span a $\dim(H)$-dimensional distribution. The third statement follows by plugging the second into the definition of $\mathrm{d}_G$. \end{proof}
The proof of proposition \ref{prop:gen_Cartan} is now completed by the list of properties of the generalized Cartan map below. In particular the identification
$$ H_G^*(X) \;\cong\; H ( C_K ( \Omega_{H-\mathrm{bas}}^*(X) ) , \mathrm{d}_K ) $$
does not depend on the chosen connection $A$. We refer to Cieliebak, Mundet and Salamon \cite[Theorem 3.8]{CMS} for the proof of the following:
\begin{prop} \label{prop:Cartan_map_properties} Let $A$ be a $G$-equivariant $H$-connection on $X$ and $c_A$ the corresponding Cartan map.
\begin{enumerate} \item $(\mathrm{d}_G \alpha)_A = \mathrm{d}_K (\alpha_A)$, i.e.~the operator $c_A$ is a chain map. \item If $\mathrm{d}_G \alpha = 0$ and $A'$ is another $G$-equivariant $H$-connection then there exists an $H$-basic element $\beta \in \Omega_G^*(X)$ such that $\alpha_A - \alpha_{A'} = \mathrm{d}_K \beta$. \item The operator $c_A$ is chain homotopic to the identity. \end{enumerate}
\end{prop}
\section{Invariant integration}
\label{chap:invariant_integration}
In \cite{AB} Atiyah and Bott explained how the push-forward operation in ordinary cohomology extends to the equivariant theory. On the other hand Cieliebak and Salamon \cite{CS} introduced $G$-invariant integration over manifolds with locally free $G$-action. If one understands this invariant integration as the push-forward of the map $X/G \longrightarrow \{ \mathrm{pt} \}$ it is clear that equivariant push-forward and invariant integration should be two special cases of one generalized integration operation. In this section we will construct this \emph{invariant push-forward}. Again this is well known if one is willing to deal with fiber integration along orbibundles. The point of this section is to give a clear picture without leaving the world of smooth manifolds.
We end this section with the computation of invariant push-forward in the simple example of a torus acting on $S^{2n-1}$ via a homomorphism onto $S^1$. This example explains very clearly the origin of the residue operations that later appear in our localization formula, and hence also the residue operations in the wall crossing formula of Cieliebak and Salamon \cite{CS}.
\subsection{Equivariant push-forward}
\label{sec:equivariant_push_forward}
Given a proper map $f : X \longrightarrow Y$ between oriented manifolds one has the push-forward in ordinary cohomology
$$ f_* \;:\; H^*(X) \longrightarrow H^{*-q}(Y), $$
which shifts the degree by $q = \dim(X) - \dim(Y)$. If $f$ is a fiber bundle the push-forward is integration along the fiber. If $f$ is an inclusion the push-forward is defined via the Thom isomorphism for the normal bundle of $f(X) \subset Y$ . A general map $f$ is decomposed into an inclusion and a fibration by the graph construction.
Now if $X$ and $Y$ carry orientation preserving $G$-actions and $f$ is equi\-va\-ri\-ant one obtains an induced map $f^G : X_G \longrightarrow Y_G$ between the Borel constructions and the above construction can be applied to give the equivariant push-forward
$$ f^G_* \;:\; H_G^*(X) \longrightarrow H_G^{*-q}(Y). $$
\begin{rem} \label{rem:equivariant_push_forward_in_Cartan_model} In the Cartan model one can apply the ordinary push-forward to the form part of equivariant differential forms. For fiber bundles it is shown in Guillemin and Sternberg \cite[Section 10.1]{GS} that this operation indeed defines a map on equivariant cohomology. And going through the proof for the equivalence of the Borel and the Cartan model one can show that this operation in fact agrees with the above definition of equivariant push-forward in the Borel picture. Of course this is valid for any proper, equivariant map and not only for fibrations. \end{rem}
\subsection{$G$-invariant integration}
\label{sec:G_invariant_integration}
Suppose $X$ is a compact, oriented $G$-manifold with a locally free action. In this case proposition \ref{prop:gen_Cartan} tells us that $H_G^*(X) = H(\Omega_{G-\mathrm{basic}}^*(X),\mathrm{d})$ with the usual differential on ordinary differential forms. Now basic forms can be integrated over slices for the group action. For this purpose we have to fix an orientation on the Lie group $G$. Recall that Lie groups have trivial tangent bundle and are hence orientable, and the action of any subgroup of $G$ by multiplication on G is orientation preserving.
For every point $x \in X$ there exists a triple $(U_x,\varphi_x,G_x)$ with the following properties (see for example Audin \cite[Theorem I.2.1]{Aud}):
\begin{itemize}
\item $G_x \subset G$ is a finite subgroup.
\item $U_x \subset \mathbbm R^m$, $m := \dim(X) - \dim(G)$ is an oriented, $G_x$-invariant open neighbourhood of zero with compact closure and orthogonal $G_x$-action.
\item $\varphi_x : U_x \longrightarrow X$ is a $G_x$-equivariant embedding such that $\varphi_x(0) = x$ and the induced map
\begin{align*} G \times_{G_x} U_x & \longrightarrow \quad X \\ [g,u] \quad & \longmapsto g . \varphi_x(u) \end{align*}
is a $G$-equivariant, orientation preserving diffeomorphism onto a $G$-invariant open neighbourhoud of $x$. Here $g \in G_x$ acts on $G$ by right-multiplication with $g^{-1}$ and $G$ acts on $G \times_{G_x} U_x$ by left-multiplication on the $G$-factor.
\end{itemize}
Choose finitely many local slices $(U_i,\varphi_i,G_i)$ such that the open sets $G.\varphi_i(U_i)$ cover $X$ and also choose a partition of unity $(\rho_i)$ by $G$-invariant functions $\rho_i$ subordinate to this cover. Define a map
$$ \int_{X/G} \;:\; \Omega_{G-\mathrm{bas}}^*(X) \longrightarrow \mathbbm R \;\cong\; \Omega^*(\mathrm{pt}) $$
by setting
\begin{equation} \label{def:G_inv_int}
\int_{X/G} \alpha \;:=\; \sum_{i} \frac{1}{|G_i|} \int_{U_i} \varphi_i^*( \rho_i \alpha ). \end{equation}
It is shown by Cieliebak and Salamon \cite[Proposition 4.2]{CS} that the integral $\int_{X/G}$ does not depend on the choices for the local slices and the partition of unity. Furthermore Stokes' formula still holds and thus the integral descends to $H_G^*(X)$. The integral vanishes unless $\deg(\alpha) = \dim(X) - \dim(G)$. Instead of requiring $X$ to be compact one can equally well restrict to forms with compact support.
\subsection{$H$-invariant push-forward}
\label{sec:H_invariant_push_forward}
Suppose a normal subgroup $H \lhd G$ of positive dimension acts locally freely on a compact oriented $G$-manifold $X$. Then the equivariant push-forward $\pi^G_*$ of the projection $\pi : X \longrightarrow \{ \mathrm{pt} \}$ vanishes: By proposition \ref{prop:gen_Cartan} any $G$-equivariant class on $X$ can be represented by an $H$-basic $G$-equivariant differential form. Any such object vanishes upon integration over $X$: By lemma \ref{lem:basic_forms} it has a form part of degree at most $\dim(X) - \dim(H) < \dim(X)$. This shows that in presence of a regularly acting subgroup $H$ the $G$-equivariant push-forward is not the correct operation. One should quotient out any such subgroup before pushing forward, as it is done in the case of $G$-invariant integration.
We will restrict our construction to the case of fiber bundles because this is all we need in our applications and we do not want to argue with Thom forms at this point. But of course the extension to general maps works just as explained by Atiyah and Bott \cite{AB}.
Let $f : X \longrightarrow Y$ be an oriented $G$-equivariant fiber bundle. Denote by $\Omega_0^*(X)$ differential forms on $X$ with compact support and by $\Omega_\mathrm{vc}^*(X)$ those with vertically (i.e.~fiberwise) compact support.
\begin{prop} \label{prop:invariant_integration} Suppose that a closed oriented normal subgroup $H \lhd G$ acts locally freely on $X$ and trivially on $Y$. Denote by $K := G / H$ the quotient group. Then there exists a linear map
$$ \left( f / H \right)_* \;:\; \Omega_{\mathrm{vc},H-\mathrm{bas}}^*(X) \longrightarrow \Omega^{*-q}(Y) $$
with $q = \dim(X) - \dim(Y) - \dim(H)$ which satisfies the following properties.
\begin{enumerate}
\item The map $(f/H)_*$ induces a $\mathrm{d}_K$-chain map on the $K$-Cartan complexes $C_K( \Omega_{\mathrm{vc},H-\mathrm{bas}}^*(X) ) \longrightarrow C_K( \Omega^{*-q}(Y) )$. The induced map on homology is denoted by
$$ \left( f / H \right)^G_* \;:\; H \left( C_K \left( \Omega_{\mathrm{vc},H-\mathrm{bas}}^*(X) \right) , \mathrm{d}_K \right) \longrightarrow H_K^*(Y) $$
and is called \emph{$H$-invariant push-forward}.
\item For the trivial group $H = \{ e \} \subset G$ we recover the equivariant push-forward of Atiyah and Bott: $( f / \{ e \} )^G_* = f^G_*$.
\item For $Y = \{ \mathrm{pt} \}$ and $H = G$ we recover the $G$-invariant integration of Cieliebak and Salamon: $( f / G )^G_* = \int_{X/G}$.
\item For $\alpha \in \Omega_{0,H-\mathrm{bas}}^*(X)$ and the projections $\pi_X,\pi_Y : X,Y \longrightarrow \{ \mathrm{pt} \}$ we have the functoriality property
$$ \left( \pi_X / H \right)^G_* (\alpha) \;=\; \left( \pi_Y \right)^K_* \circ \left( f / H \right)^G_* (\alpha). $$
If the remaining $K$-action on $Y$ is regular then
$$ \left( \pi_X / G \right)^G_* (\alpha) \;=\; \left( \pi_Y / K \right)^K_* \circ \left( f / H \right)^G_* (\alpha). $$
\item For all $\alpha \in \Omega_{0,H-\mathrm{bas}}^*(X)$ and $\beta \in \Omega^*(Y)$ we have
$$ \left( f / H \right)_* \left( \alpha \wedge f^*\beta \right) \;=\; \left( f / H \right)_* \alpha \wedge \beta, $$
i.~e.~$(f/H)_*$ is a homomorphism of $\Omega^*(Y)$-modules.
\end{enumerate}
\end{prop}
\begin{rem} If $X$ is compact and $Y = \{\mathrm{pt}\}$ then we also write the $H$-invariant push-forward $(\pi/H)^G_*$ as
$$ \int_{X/H} \;:\; H_G^*(X) \longrightarrow H_K^*(\mathrm{pt}) = S({\mathfrak k}^*)^K. $$
\end{rem}
\begin{rem} We usually drop the superscript $G$ if the big group with respect to which we do the equivariant theory is clear from the context. \end{rem}
\begin{proof} We first explain the situation in the Borel model. For this we have to assume that $H$ acts freely on $X$. Then we can write the Borel construction for $X$ as $X_G = (X/H)_K$. Since $H$ acts trivially on $Y$ the $G$-equivariant map $f$ descends to a $K$-equivariant map $f/H : X/H \longrightarrow Y$ and induces a map $(f/H)^G : X_G \longrightarrow Y_K$ between the Borel constructions. We then take the push-forward of this map in cohomology to obtain $(f/H)^G_*$. It shifts the degree by $\dim(X/H) - \dim(Y)$, which coincides with the formula in the proposition.
We now define the $H$-invariant push-forward for $f : X \longrightarrow Y = \{ \mathrm{pt} \}$ in the Cartan model. Forget about the $G$ action and consider the $H$-invariant integral
$$ ( f / H )_* \;:=\; \int_{X/H} : \Omega_{0,H-\mathrm{bas}}^*(X) \longrightarrow \Omega^*( \mathrm{pt} ) $$
as in \ref{def:G_inv_int}. Given one collection of local slices $(U_i,\varphi_i,H_i)$ for the $H$-action with partition functions $(\rho_i)$ and any element $g \in G$ one obtains another admissible collection of slices and partition functions by
$$ \left(\, U_i \,,\, g \varphi_i \,,\, g H_i g^{-1} \,\right) \quad \mathrm{and} \quad ( g^*\rho_i ). $$
This shows that integration $\int_{X/H}$ is $G$-invariant. In order to extend a map to the $K$-Cartan models this map has to be $K$-equivariant. So since $K$ acts trivially on $\Omega^*( \mathrm{pt} )$ this extension works. Now
$$ \left( C_K \left( \Omega^*( \mathrm{pt} ) \right) , \mathrm{d}_K \right) = \left( S({\mathfrak k}^*)^K , 0 \right), $$
hence it suffices to show that the integral $\int_{X/H}$ of $\mathrm{d}_K$-exact forms vanishes. This follows by lemma \ref{lem:basic_forms} and Stokes' formula. Thus property $3$ is satisfied.
Now consider a fiber bundle $f : X \longrightarrow Y$ with $H$ acting trivially on the base. For a point $y \in Y$ pick a chart $\psi_y : V_y \longrightarrow Y$ with $V_y \subset \mathbbm R^n$, $n:=\dim(Y)$, an open neighbourhood of zero with compact closure, and an $H$-equivariant trivialization of $\psi_y^* X \cong V_y \times X_y$ with $H$ acting trivially on $V_y$ and by the induced action on the fiber $X_y \subset X$ over $y$. Choose local slices $(U_i,\varphi_i,H_i)$ and partition functions $(\rho_i)$ for the regular $H$-action on $X_y$ as before. These give rise to local slices
$$ \left(\, V_y \times U_i \,,\, \psi_y \times \varphi_i \,,\, H_i \,\right) $$
for the $H$-action on $X$ covering all of $f^{-1}(\psi_y(V_y))$. Recall that $U_i \subset \mathbbm R^m$ with $m = \dim(X_y) - \dim(H) = \dim(X) - \dim(Y) - \dim(H) = q$. Denote integration over the $\mathbbm R^m$-coordinates of $V_y \times U_i \subset \mathbbm R^n \times \mathbbm R^m$ by
$$ {\pi_{(y,i)}}_* \;:\; \Omega^*(V_y \times U_i) \longrightarrow \Omega^{*-m}(V_y) $$
and define the map
$$ \left( f / H \right)_* \;:\; \Omega_{\mathrm{vc},H-\mathrm{bas}}^*(X) \longrightarrow \Omega^{*-m}(Y) $$
for forms $\alpha$ with support in $f^{-1}(\psi_y(V_y))$ by
$$
\left( f / H \right)_* \alpha \;:=\; \sum_{i} \frac{1}{|H_i|} {\pi_{(y,i)}}_* \left[ \left( \psi_y \times \varphi_i \right)^* ( \rho_i \alpha ) \right]. $$
Extend this to all of $\Omega_{\mathrm{vc},H-\mathrm{bas}}^*(X)$ by choosing a locally finite atlas $(\psi_{y_j})$ for $Y$, trivializing $X$. Property $5$ now follows immediately from this definition and the corresponding property for the fiber integrals ${\pi_{(y_j,i)}}_*$. Note that $f^*\beta$ is $H$-basic for every $\beta \in \Omega^*(Y)$ since $H$ acts trivially on $Y$ and $f$ is equivariant.
Now as before the $G$-action on $X$ moves the above $H$-slices around. This shows that the map $(f/H)_*$ is $K$-equivariant. It commutes with the ordinary differential $\mathrm{d}$ and the contractions $\iota_\xi$, because ordinary fiber integration does (see for example Bott and Tu \cite[6.14.1]{BT} and Guillemin Sternberg \cite[10.5]{GS}). This proves part $1$. Part $2$ follows by remark \ref{rem:equivariant_push_forward_in_Cartan_model}. The first functoriality property in part $4$ is a consequence of Fubini's theorem. For the second identity first note that regularity of the remaining $K$-action on $Y$ together with regularity of the $H$-action on $X$ implies that the whole group $G$ acts locally freely on $X$. Hence all operations are well defined. Now for a point $x \in X$ the orders of the involved isotropy subgroups satisfy
$$
|G_x| \;=\; |H_x| \cdot |K_{f(x)}| $$
and the second identitiy also follows by applying Fubini's theorem. \end{proof}
\subsection{Example}
\label{sec:example_invariant_integration}
The first invariant integrals that we can compute by hand without the technique of localization come from torus actions on $S^{2n-1}$ that factor through a weighted $S^1$-action. Let $T$ be a torus with Lie algebra ${\mathfrak t}$. We denote the integral lattice in ${\mathfrak t}$ by
$$
\Lambda \;:=\; \left\{ \xi \in {\mathfrak t} \;|\; \exp \left( \xi \right) = \mathbbm 1 \in T \right\} $$
and its dual by
$$
\Lambda^* \;:=\; \left\{ w \in {\mathfrak t}^* \;|\; \forall \xi \in \Lambda : \left\langle w,\xi \right\rangle \in \mathbbm Z \right\}. $$
Here $\langle \;,\; \rangle$ denotes the pairing between ${\mathfrak t}$ and its dual space ${\mathfrak t}^*$. We consider the $T$-action on the unit sphere $S^{2n-1} \subset \mathbbm C^n$ that is induced by the representation
\begin{equation} \label{def:torus_action} \rho \;:\; T \longrightarrow \mathrm{U}(n) \;;\; \exp(\xi) \longmapsto \mathrm{diag}\left( e^{-2 \pi i \langle w_j , \xi \rangle} \right)_{j = 1 , \ldots , n} \end{equation}
for a given collection of weight vectors $w_j \in \Lambda^* \setminus \{0\}$. We assume in this section that the weights $w_j$ are all positive multiples of one element $w \in \Lambda^*$. We pick a primitive element $e_1 \in \Lambda$ such that $\langle w , e_1 \rangle \ne 0$ and denote by $T_1 \subset T$ the subtorus generated by $e_1$ and by ${\mathfrak t}_1 \subset {\mathfrak t}$ its Lie algebra. The quotient group and its Lie algebra are denoted by $T_0 := T/T_1$ and ${\mathfrak t}_0 := {\mathfrak t}/{\mathfrak t}_1$ respectively. We can identify
$$
{\mathfrak t}_0^* \;\cong\; \left\{ x \in {\mathfrak t}^* \;|\; \langle x , e_1 \rangle = 0 \right\}. $$
Now $T_1$ acts locally freely on $S^{2n-1}$ with isotropy group $\mathbbm Z / \langle w , e_1 \rangle \mathbbm Z$ and we want to compute the $T_1$-invariant push-forward
$$ \left( \pi / T_1 \right)_* \;:\; H_T^*\left(S^{2n-1}\right) \longrightarrow H_{T_0}^*(\mathrm{pt}) \cong S\left( {\mathfrak t}_0^* \right) $$
of the projection $\pi : S^{2n-1} \longrightarrow \{\mathrm{pt}\}$. Suppose we are given elements $x_j \in {\mathfrak t}^*$ for $j = 1 , \ldots , m$. Then the product
$$ \mathbbm{x} \;:=\; \prod_{j = 1}^m x_j \in S({\mathfrak t}^*) $$
represents a class $[\mathbbm{x}] \in H_T^{2m}(S^{2n-1})$ if we view $\mathbbm{x}$ as a $\mathrm{d}_T$-closed $T$-equivariant differential form. In fact it follows from Kirwan's theorem (Theorem \ref{thm:Kirwan}) that every class in $H_T^*(S^{2n-1})$ can be represented in such a way. Hence our goal is to compute $( \pi / T_1 )_*[\mathbbm{x}]$.
\begin{prop} \label{prop:T_1_invariant_integration} For any $\xi \in {\mathfrak t}$ we have
\begin{equation} \label{eqn:T_1_invariant_integration} \left( \pi / T_1 \right)_* \left[ \mathbbm{x} \right] (\xi) \;=\; \frac{1}{2 \pi i} \oint \frac{ \prod_{j = 1}^m \langle x_j , \xi + z e_1 \rangle }{ \prod_{j = 1}^n \langle w_j , \xi + z e_1 \rangle } \; \mathrm{dz} \; . \end{equation}
The integral is around a circle in the complex plane enclosing all poles of the integrand. \end{prop}
\begin{proof} We write $w_j = \ell_j \cdot w$ with positive integers $\ell_j$. By changing $w$ we can arrange that the $\ell_j$ do not have a common divisor different from $1$. Thus the $T$-action factors through a free $S^1$-action with weights $\ell := \left( \ell_1 , \ldots , \ell_n \right)$. The quotient of this free $S^1$-action is the weighted projective space $\mathbbm C\mathrm P^{n-1}_\ell$. By the functoriality property of invariant push-forward we obtain $( \pi / T_1 )_*$ as the composition of $T_1$-invariant push-forward $H_T^*(S^{2n-1}) \longrightarrow H_{T_0}^*(\mathbbm C\mathrm P^{n-1}_\ell)$ with $T_0$-equivariant integration $H_{T_0}^*(\mathbbm C\mathrm P^{n-1}_\ell) \longrightarrow H_{T_0}^*(\mathrm{pt})$. Note that $T_0$ acts trivially on $\mathbbm C\mathrm P^{n-1}_\ell$, whence $H_{T_0}^*(\mathbbm C\mathrm P^{n-1}_\ell) \cong S({\mathfrak t}_0^*) \otimes H^*(\mathbbm C\mathrm P^{n-1}_\ell)$ and this last $T_0$-equivariant integration is just evaluation on the fundamental cycle of $\mathbbm C\mathrm P^{n-1}_\ell$ in the second factor.
We need a $T$-equivariant $T_1$-connection $A$ on $S^{2n-1}$. Let $\alpha_\ell \in \Omega^1(S^{2n-1})$ be a connection form on the principal $S^1$-bundle $S^{2n-1} \longrightarrow \mathbbm C\mathrm P^{n-1}_\ell$. Here we identify $\mathrm{Lie}(S^1) \cong \mathbbm R$ such that for $x \in \mathbbm R$ we have $\exp(x) = e^{2 \pi i x}$. It follows that
\begin{equation} \label{def:T_equivariant_T_1_connection} A := \frac{ \alpha_\ell }{ \langle w , e_1 \rangle } \otimes e_1 \end{equation}
is such a $T$-equivariant $T_1$-connection.
A short computation now shows that the image $\mathbbm{x}_A$ of $\mathbbm{x}$ under the Cartan map $c_A$ is given by
\begin{equation} \label{eqn:xx_A} \mathbbm{x}_A \;=\; \prod_{j = 1}^m \left( x_j - \langle x_j , e_1 \rangle \frac{ \mathrm{d}\alpha_\ell - w }{ \langle w , e_1 \rangle } \right). \end{equation}
The first step of integration $H_T^*(S^{2n-1}) \longrightarrow H_{T_0}^*(\mathbbm C\mathrm P^{n-1}_\ell)$ now simply consists of dividing by $\langle w , e_1 \rangle$ and interpreting $\mathrm{d}\alpha_\ell$ as a form on $\mathbbm C\mathrm P^{n-1}_\ell$. Evaluation on the fundamental cycle of $\mathbbm C\mathrm P^{n-1}_\ell$ in the second step then amounts to expanding the above expression for $\mathbbm{x}_A$ as a polynomial in $\mathrm{d}\alpha_\ell$ and picking the coefficient in front of ${\mathrm{d}\alpha_\ell}^{n-1}$. On can express this as taking the residue of a certain rational complex function. We leave the details to the reader and refer to \cite{JW} for the detailed computation. \end{proof}
\begin{rem} Formula \ref{eqn:T_1_invariant_integration} remains true even without the assumption of this section that the weights $w_j$ are colinear. To prove this we need the technique of localization. So we will return to this in the example at the end of the next section. \end{rem}
\begin{rem} \label{rem:z} The above computations yield the following recipe how to evaluate $( \pi / T_1 )_*$ on any $T$-equivariant differential form $\alpha \in \Omega_T^*(S^{2n-1})$ that is $T_1$-basic and has the following polynomial presentation
$$ \alpha \;=\; \sum_{k \ge 0} r_k \cdot \left( \frac{ \mathrm{d}\alpha_\ell - w }{ \langle w , e_1 \rangle } \right)^k $$
for some $r_k \in S({\mathfrak t}^*)$ and with $\alpha_\ell$ from \ref{def:T_equivariant_T_1_connection}. Note that the property of being $T_1$-basic imposes constraints on the $r_k$ in terms of $w$ and $e_1$. But we do not need to know them explicitly. It suffices to know that $\alpha = \alpha_A$. So in our computations we can replace $\mathbbm{x}_A$ by the above sum, because by \ref{eqn:xx_A} it is of exactly this form. We obtain the generalized formula
\begin{eqnarray} \label{eqn:gen_T_1_invariant_integration} \left( \pi / T_1 \right)_* \left[ \alpha \right] (\xi) & = & \frac{1}{2 \pi i} \oint \frac{ \sum_{k \ge 0} r_k(\xi) \cdot z^k }{ \prod_{j = 1}^n \langle w_j , \xi + z e_1 \rangle } \; \mathrm{dz} \nonumber\\ & = & \mathrm{Res}_{z = - \frac{ \langle w , \xi \rangle }{ \langle w , e_1 \rangle } } \left\{ \frac{ \sum_{k \ge 0} r_k(\xi) \cdot z^k }{ \prod_{j = 1}^n \langle w_j , \xi + ze_1 \rangle } \right\}, \end{eqnarray}
that will be used later on. \end{rem}
\section{Localization}
\label{chap:localization}
The abstract localization theorem of Atiyah and Bott in \cite{AB} gives rise to a formula for integrating equivariant $n$-forms over an $n$-dimensional manifold $X$ that is equipped with an action of a torus $T$. In the notation of section \ref{chap:invariant_integration} this integration is the restriction of the $T$-equivariant push-forward of $X \longrightarrow \{ \mathrm{pt} \}$ to degree $n$. The integration formula expresses such an integral in terms of data associated to the fixed point set of the $T$-action.
This fits neatly with our observation at the beginning of section \ref{sec:H_invariant_push_forward}: Such an integral vanishes as soon as there is a subgroup $H$ of positive dimension acting locally freely, i.e.~if there are no fixed points. But in that case $H$-invariant integration is defined and one expects a generalized localization formula for this adapted integration operation. In fact if $H$ acts freely one can lift the localization formula for the $T/H$-action on $X/H$ to obtain a formula for $H$-invariant integration over $X$. Since localization only works in a truely equivariant context, we have to assume that $T/H$ still has positive dimension, i.~e.~the codimension of $H$ in $T$ has to be at least one.
We will generalize the Atiyah-Bott localization formula to $H$-invariant push-forward in the case of a regular $H$-action and call the result \emph{relative Atiyah-Bott localization}. Before we can do this we start with some general facts and constructions concerning torus actions. Then we review the construction and properties of equivariant Thom forms. We finish the section with the computation of an explicit example that generalizes the one in section \ref{sec:example_invariant_integration} and that will be used later on.
\subsection{Torus actions}
\label{sec:torus_actions}
Suppose the $k$-dimensional torus $T$ acts on a closed $n$-dimensional manifold $X$. We denote the infinitesimal action at a point $p \in X$ by
$$ L_p \;:\; {\mathfrak t} \longrightarrow T_pX \;;\; \xi \longmapsto X_\xi(p), $$
with the infinitesimal vector field $X_\xi$ defined as in \ref{def:X_xi}, set ${\mathfrak t}_p := \ker \left( L_p \right) \subset {\mathfrak t}$ and denote the isotropy subgroup at a point $p \in X$ by $S_p \subset T$. Note that ${\mathfrak t}_p = \mathrm{Lie}(S_p)$.
Since $T$ is abelian there is only a finite number of isotropy groups $S_p$. This follows by the slice theorem (see for example tom Dieck \cite[Theorem 5.11]{tomD}). Hence the set of all subspaces in ${\mathfrak t}$ that occur as ${\mathfrak t}_p$ for some $p \in X$ is also finite. Let $k_p := \dim({\mathfrak t}_p)$ and set
$$ k_0 \;:=\; \max_{p \in X} \, k_p \quad \mathrm{and} \quad k_1 \;:=\; k - k_0. $$
We fix a $k_1$-dimensional subtorus $T_1 \subset T$ such that its Lie algebra ${\mathfrak t}_1 \subset {\mathfrak t}$ satisfies
$$ {\mathfrak t}_1 \cap {\mathfrak t}_p \;=\; \left\{ 0 \right\} \quad \mbox{for all} \quad p \in X. $$
So $T_1$ is a subtorus of maximal dimension that acts locally freely on $X$ and $T_1$-invariant integration is the operation that we want to study. Invariant integration with respect to any larger subgroup is not possible and invariant integration with respect to any smaller subgroup would be zero. Of course there are many possible choices for $T_1 \subset T$ with the above property.
\begin{dfn} \label{def:relative_fixedpoint_set} Given any closed subgroup $H \subset T$ we define the \emph{$H$-relative fixed point set} $F_H$ as
$$
F_H \;:=\; \left\{ p \in X \;|\; L_p({\mathfrak t}) = L_p({\mathfrak h}) \right\}. $$
Here ${\mathfrak h} \subset {\mathfrak t}$ denotes the Lie algebra of $H$. \end{dfn}
Elements of $F_H$ represent fixed points of the $T / H$-action on the quotient $X / H$. So $F_H$ is the subset to which we would like to localize $H$-invariant integration. One prerequisite is the following
\begin{prop} \label{prop:F_H_submanifold} Suppose the closed subgroup $H \subset T$ acts locally freely on $X$. Then the set $F_H \subset X$ is a smooth (but not necessarily connected) submanifold. \end{prop}
\begin{proof} It suffices to show that every $p \in F_H$ has an open neighbourhood $U \subset X$ such that $F_H \cap U$ is a submanifold in $U$. For $U$ we pick a tubular neighbourhood of the $T$-orbit through $p$. By the local slice theorem such a tube can be chosen to be equivariantly diffeomorphic to
$$ U \;\cong\; T \times_{S_p} V. $$
Here $V$ is a linear space on which the isotropy group $S_p$ acts by orthogonal transformations. The orbit of the $T$-action through $p$ is identified with the points $[g,0] \in U$ with $g \in T$ and $0 \in V$. Now for any point $q \in X$ we have
$$ \dim L_q({\mathfrak t}) \;=\; \dim{\mathfrak t} - \dim{\mathfrak t}_q \;=\; k - k_q, $$
and
$$ \dim L_q({\mathfrak h}) \;=\; \dim{\mathfrak h} - \dim({\mathfrak h} \cap {\mathfrak t}_q) \;=\; \dim{\mathfrak h} - \dim{\mathfrak h}_q \;=\; l - l_q, $$
where we introduced ${\mathfrak h}_q := {\mathfrak h} \cap {\mathfrak t}_q$ and $\dim{\mathfrak h} =: l$ and $\dim{\mathfrak h}_q =: l_q$. So we have
$$ q \in F_H \quad \iff \quad k \;=\; k_q + l - l_q, $$
which is equivalent to saying that the intersection ${\mathfrak t}_q \cap {\mathfrak h} \subset {\mathfrak t}$ is transversal. The assumption of a regular $H$-action implies that ${\mathfrak h} \cap {\mathfrak t}_q = \{ 0 \}$ for all $q \in X$. Hence we get the refined equivalence
$$ q \in F_H \quad \iff \quad {\mathfrak t} \;=\; {\mathfrak h} \oplus {\mathfrak t}_q. $$
For a point $q = [g,v] \in U$ we have $S_q = S_{p,v} \subset S_p$, where $S_{p,v}$ denotes the isotropy subgroup at $v$ for the orthogonal $S_p$-action on $V$. We write
$$ {\mathfrak t}_v \;:=\; \mathrm{Lie}(S_{p,v}) \subset {\mathfrak t}_p $$
instead of ${\mathfrak t}_q$. Now we consider the subset
$$
W \;:=\; \left\{ v \in V \;|\; {\mathfrak t} = {\mathfrak h} \oplus {\mathfrak t}_v \right\}. $$
Since $p \in F_H$ we have ${\mathfrak t} = {\mathfrak h} \oplus {\mathfrak t}_p$ and since ${\mathfrak t}_v \subset {\mathfrak t}_p$ for all $v \in V$ we in fact have $W = \left\{ v \in V \;|\; {\mathfrak t}_v = {\mathfrak t}_p \right\}$. This shows that $W$ is a linear and $S_q$-invariant subspace of $V$. Thus we finally obtain
$$ F_H \cap U \;\cong\; T \times_{S_p} W, $$
which is a submanifold of $U \cong T \times_{S_p} V$. \end{proof}
\begin{rem} This proposition is not true without the assumption of $H$ acting locally freely. Consider the action of the two-torus $T = S^1 \times S^1$ on $\mathbbm C \times \mathbbm C$, where the first $S^1$ acts by complex multiplication on the first component and trivially on the second, and the second $S^1$ acts vice versa. Let $H \cong S^1$ be the diagonal in $T$. Then we claim that
$$
F_H \;=\; \left\{ ( z_1 , z_2 ) \;|\; z_1 \cdot z_2 = 0 \right\} \subset \mathbbm C \times \mathbbm C. $$
In fact at points $p = ( z_1 , z_2 )$ with $ z_1 \cdot z_2 \ne 0$ the $T$-isotropy is trivial, so $p \not\in F_H$. But as soon as one component of $p$ is zero the tangent space to one $S^1$ factor in $T = S^1 \times S^1$ is contained in ${\mathfrak t}_p$. So ${\mathfrak t}_p$ intersects transversally with ${\mathfrak h}$ and hence $p \in F_H$. \end{rem}
\subsection{Equivariant Thom forms}
\label{sec:equivariant_thom_forms}
The construction in this section does not require the acting group to be abelian. So for the moment we consider an arbitrary compact Lie group $G$ acting smoothly on a closed and oriented $n$-dimensional manifold $X$, preserving the orientation.
Given a $G$-equivariant differential form $\alpha$ one can integrate all the form-parts over $X$ to obtain an element in $S({\mathfrak t}^*)$. If we restrict to $\alpha \in \Omega_G^n(X)$ then we in fact obtain $\int_X \alpha \in \mathbbm R$, because only forms of degree $n$ contribute and so the polynomial remainder is of degree zero. This is the operation that in the following will be called \emph{integration} of an equivariant form and is denoted as above.
Suppose $i : Z \hookrightarrow X$ is a closed $G$-invariant submanifold of codimension $m$. Then given any tubular neighbourhood $U \subset X$ of $Z$ there exists an equivariant Thom form $\tau$ with support in $U$, i.~e.~an element $\tau \in \Omega_G^m(X)$ such that
\begin{itemize} \item $\mathrm{d}_G\,\tau = 0$, \item $\mathrm{supp}(\tau) \subset U$, \item $\displaystyle \int_X \alpha\wedge\tau \;=\; \int_Z i^*\alpha$ for all closed forms $\alpha \in \Omega^*(X)$. \end{itemize}
See Cieliebak and Salamon \cite[Chapter 5]{CS} for an explicit construction or Guillemin and Sternberg \cite[Chapter 10]{GS} for a canonical algebraic discussion. We give a short summary in order to see what happens in presence of a normal subgroup $H$ that acts locally freely.
The key ingredient is an $\mathrm{SO}(m)$-equivariant \emph{universal Thom form} on $\mathbbm R^m$. This is a $\mathrm{d}_{\mathrm{SO}(m)}$-closed form $\rho \in \Omega_{\mathrm{SO}(m)}^m(\mathbbm R^m)$ with compact support and integral equal to $1$. This universal Thom form is transplanted to the normal bundle $N$ of $Z \subset X$ by the following construction. Let $P$ be the principal $\mathrm{SO}(m)$-bundle of oriented orthonormal frames of $N$, such that
$$ N \;\cong\; P \times_{\mathrm{SO}(m)} \mathbbm R^m. $$
Of course we use a $G$-invariant metric on $X$ to define $P$. The $G$-action on $Z$ then extends to $P$ and commutes with the $\mathrm{SO}(m)$-action. We pull back $\rho$ to $P \times \mathbbm R^m$ and use a $G \times \mathrm{SO}(m)$-equivariant $\mathrm{SO}(m)$-connection $A$ to get the $\mathrm{SO}(m)$-basic form $\rho_A$ by virtue of the Cartan map \ref{def:c_A}. Thus $\rho_A$ descends to a $\mathrm{d}_G$-closed form $\tau$ on $N$ with compact support and fiber integral equal to $1$. Finally $N$ can be identified equivariantly with an arbitrarily small tubular neighbourhood of $Z$.
Now given a closed normal subgroup $H \lhd G$ that acts with at most finite isotropy on $X$ we can in fact take a $G \times \mathrm{SO}(m)$-equivariant $H \times \mathrm{SO}(m)$-connection $A$ in the above construction. In the end we obtain a Thom form $\tau \in \Omega_G^m(X)$ that is $H$-basic. If we denote by $l := \dim(H)$ the dimension of $H$ then we obtain the following result.
\begin{prop} \label{prop:H_invariant_Thom_integration} Let $\alpha \in \Omega_{H-\mathrm{bas}}^{n-m-l}(X)$ be closed and $\tau$ be a $G$-equivariant Thom form for $Z \subset X$ as above. Then
$$ \int_{Z/H} i^*\alpha \;=\; \int_{X/H} \alpha \wedge \tau. $$
\end{prop}
\begin{proof} The left hand side is just the $H$-invariant integral of an ordinary $H$-basic form in the sense of section \ref{sec:G_invariant_integration} and hence a real number. The right hand side is the $H$-invariant integral of a $G$-equivariant form. But since $\tau$ has degree $m$ its polynomial parts have degree less than $m$ and vanish upon integration, because $\alpha$ is assumed to have degree $n-m-l$. Hence we only have to look at $\tau^{[m]}$, i.~e.~the form part of $\tau$ in degree $m$.
It is shown by Cieliebak and Salamon \cite[Theorem 5.3]{CS} that the difference of any two Thom forms is $\mathrm{d}_G$-exact. Hence the right hand side does not depend on the particular choice for $\tau$ and we may assume that $\tau$ is $H$-basic. Next we observe that $\tau^{[m]}$ for an $H$-basic $G$-equivariant Thom form $\tau$ is in fact an $H$-equivariant Thom form for $Z \subset X$ if we just consider the $H$-action. Hence the result follows from Cieliebak and Salamon \cite[Corollary 6.3]{CS}, where the corresponding result is proven for $G$-invariant integration in the case of a regular $G$-action. \end{proof}
We want to extend this rule to $H$-invariant integration of elements $[ \alpha ] \in H_G^*(X)$ with values in $H_K^*(\mathrm{pt}) = S({\mathfrak k}^*)$ with $K := G/H$. As shown in proposition \ref{prop:gen_Cartan} we can represent the given class by a form
$$ \alpha \;\in\; C_K \left( \Omega_{H-\mathrm{bas}}^*(X) \right) \;=\; \left[ S({\mathfrak k}^*) \otimes \Omega_{H-\mathrm{bas}}^*(X) \right]^K. $$
Now since $\mathrm{d}_G \alpha = 0$ we have $\mathrm{d} \left( \alpha^{[n-m-l]} \right) = 0$ by lemma \ref{lem:basic_forms}. Hence the above lemma applies and since $H$-invariant integration is $K$-equivariant the formula extends to $\mathrm{d}_G$-closed elements of $C_K \left( \Omega_{H-\mathrm{bas}}^*(X) \right)$. So we conclude that
\begin{equation} \label{eqn:H_invariant_Thom_integration}
\int_{X/H} \alpha \wedge \tau \;=\; \int_{Z/H} i^*\alpha \;\in\; S({\mathfrak k}^*) \end{equation}
for all $[ \alpha ] \in H_G^*(X)$, where $\tau$ is any $G$-equivariant Thom form for the $G$-invariant submanifold $Z \subset X$.
\subsection{Relative Atiyah-Bott localization}
\label{sec:relative_atiyah_bott_localization}
If one wants to use equation \ref{eqn:H_invariant_Thom_integration} in order to compute an integral $\int_{X/H} \beta$ then there are two obstacles: We can only integrate forms $\beta$ that can be written as $\alpha \wedge \tau$ and hence have support localized around some submanifold with Thom form $\tau$. And then one needs to 'divide' $\beta$ by $\tau$ in order to obtain $\alpha$. Generally this division will not be possible in $H_G^*(X)$ --- one needs to work in a localized ring.
In the case of a torus action both these steps of localization can be made explicit, so we return to the setup of section \ref{sec:torus_actions}. The construction is analogous to the one by Guillemin and Sternberg \cite[Chapter 10]{GS}. We first consider the case that the torus $T_1$, which acts locally freely and with respect to which we want to integrate invariantly, has codimension $1$ in the whole torus $T$. This corresponds to looking at an $S^1$-action in the non-relative case.
\subsubsection*{Geometric localization}
Denote by $Z_i$ the connected components of the $T_1$-relative fixed point set $F_{T_1}$. Let $U_i$ be sufficiently small disjoint $T_1$-invariant tubular neighbourhoods of the $Z_i$. We introduce the notation
$$ X' \;:=\; X \setminus F_{T_1}. $$
Recall that $\dim(T_1) = k_1$, so the $T_1$-invariant integral over $X$ is nonzero only for forms of degree at least $n - k_1$.
\begin{lem} \label{lem:geometric_localization} For any $[ \alpha ] \in H_T^d(X)$ with $d \ge n - k_1$ there exists a collection of $\mathrm{d}_T$-closed forms $\alpha_i$ with support in $U_i$ such that
$$ \int_{ X / T_1 } \alpha \;=\; \sum_i \int_{ X / T_1 } \alpha_i $$
and furthermore $\alpha$ and $\alpha_i$ agree on some open neighbourhood of $Z_i$. \end{lem}
\begin{proof} First observe that $T$ acts locally freely on $X'$. In fact if $T$ had a whole $1$-dimensional subgroup fixing some point $p \in X$ this would imply $p \in F_{T_1}$, because $T_1$ has only codimension $1$ in $T$. This implies $H_T^d(X') = 0$, because any $T$-equivariant form on $X'$ can be made $T$-basic. But since $d \ge n - k_1 > \dim(X) - \dim(T)$ there are no nonzero $T$-basic forms in degree $d$ by lemma \ref{lem:basic_forms}. So if we restrict $\alpha$ to $X'$ we obtain
$$ \alpha \;=\; \mathrm{d}_T \gamma' \quad \mbox{for some} \quad \gamma' \in \Omega_T^{d-1}(X'). $$
For every $i$ we fix a smooth $T$-invariant function $\rho_i$ with support in $U_i$ such that $\rho_i = 1$ on some open neighbourhood of $Z_i$. Then we introduce
$$ \gamma \;:=\; \gamma' - \sum_i \rho_i \cdot \gamma'. $$
Note that $\gamma$ extends over $F_{T_1}$ to an element of $\Omega_T^{d-1}(X)$. Next we set
$$ \beta \;:=\; \alpha - \mathrm{d}_T \gamma $$
and obtain $\int_{X/T_1} \alpha = \int_{X/T_1} \beta$. Now $\beta$ vanishes outside the disjoint union of all the $U_i$, because there we have $\gamma = \gamma'$. Hence we can write
$$ \beta \;=:\; \sum_i \alpha_i $$
with $\mathrm{d}_T$-closed forms $\alpha_i$ having support in $U_i$. Finally on a neighbourhood of each component $Z_i$ we have $\rho_i = 1$ and hence $\gamma = 0$. Hence we indeed obtain $\alpha_i = \alpha$ on these neighbourhoods. \end{proof}
\subsubsection*{Algebraic localization}
We now only consider one component $Z$ of $F_{T_1}$ and a $\mathrm{d}_T$-closed form $\alpha$ with compact support in a small $T$-invariant tubular neighbourhood $U$ of $Z$. We denote by $i : Z \longrightarrow U$ the inclusion and by $\pi : U \longrightarrow Z$ the projection and choose a Thom form $\tau$ for $Z$ with support in $U$. We denote by
$$ i^*[\tau] \;=:\; e_T(Z) \in H_T^m(Z) $$
the equivariant Euler class of $Z$. More precisely this is the $T$-equivariant Euler class of the normal bundle $N$ to $Z \subset X$. If $m = \mathrm{codim}(Z)$ is odd then $e_T(Z) = 0$, so we assume that $m = 2 \ell$ is even. From the proof of proposition \ref{prop:F_H_submanifold} we read off that on the connected component $Z$ we can choose a $T_0 \subset T$ that acts trivially on $Z$ and such that ${\mathfrak t} = {\mathfrak t}_0 \oplus {\mathfrak t}_1$. By the discussion preceding equation \ref{eqn:trivial_H_action} we conclude that
$$ H_T^*(Z) \;=\; S({\mathfrak t}_0^*) \otimes H_{T_1}^*(Z). $$
Recall that $T_1$ acts locally freely. So we can represent $e_T(Z)$ by a sum
\begin{equation} \label{eqn:euler_class_sum} e_T(Z) \;=\; f_\ell + f_{\ell-1} \cdot \theta_1 + \ldots + f_1 \cdot \theta_{\ell-1} + \theta_\ell \end{equation}
with $f_j \in S^j({\mathfrak t}_0^*)$ and closed differential forms $\theta_j \in \Omega_{T_1-\mathrm{bas}}^{2j}(Z)$. Since all the terms involving some $\theta_j$ are nilpotent we can formally invert this sum if we assume that $f_\ell \ne 0$. We define
$$ \Theta \;:=\; f_\ell - e_T(Z) $$
and observe that $\Theta^r = 0$ for $r-1 := \lfloor \dim(Z) / 2 \rfloor$. We set
$$ \beta \;:=\; f_\ell^{r-1} + f_\ell^{r-2} \cdot \Theta + \ldots + f_\ell \cdot \Theta^{r-1} $$
and compute
\begin{eqnarray*} e_T(Z) \cdot \beta & = & \left( f_\ell - \Theta \right) \cdot \left( f_\ell^{r-1} + f_\ell^{r-2} \cdot \Theta + \ldots + f_\ell \cdot \Theta^{r-1} \right) \\ & = & f_\ell^r. \end{eqnarray*}
Formally we can write this as $\frac{1}{e_T(Z)} = \frac{\beta}{f_\ell^r}$. Note that $\beta$ defines an element in $H_T^*(Z)$. Thus we can invert the Euler class $e_T(Z)$ if we localize $H_T^*(Z)$ at the monomial $f_\ell^r$. Without working in a localized ring we can write
\begin{eqnarray} \label{eqn:algebraic_localization} f_\ell^r \cdot \int_{U/T_1} \alpha & = & \int_{U/T_1} \alpha \wedge \pi^* \beta \wedge \pi^* e_T(Z) \nonumber\\ & = & \int_{U/T_1} \alpha \wedge \pi^* \beta \wedge \tau \\ & = & \int_{Z/T_1} i^* \alpha \wedge \beta. \nonumber \end{eqnarray}
We write this integration formula more suggestively as
$$ \int_{U/T_1} \alpha \;=\; \int_{Z/T_1} \frac{ i^* \alpha }{ e_T(Z) } $$
but have in mind that the meaning of this formula is given by equation \ref{eqn:algebraic_localization} above. Thus we have proved the following relative localization theorem.
\begin{thm} \label{thm:relative_localization} Suppose the subtorus $T_1 \subset T$ acts locally freely on $X$ with relative fixed point set $F_{T_1}$. Then for any $[ \alpha ] \in H_T^*(X)$ we have
$$ \int_{X/T_1} \alpha \;=\; \sum_i \int_{Z_i/T_1} \frac{ i_{Z_i}^* \alpha }{ e_T(Z_i) } \; \in S({\mathfrak t}_0^*). $$
Here the sum is over all connected components $Z_i$ of $F_{T_1}$, $i_{Z_i}$ denotes the inclusion into $X$ and $e_T(Z_i)$ is the $T$-equivariant Euler class of the normal bundle to the submanifold $Z_i \subset X$. \end{thm}
\subsubsection*{Higher codimension}
We now extend the relative localization formula \ref{thm:relative_localization} to the case of $T_1$ having higher codimension in $T$. The procedure is analogous to the extension of the $S^1$-localization formula to arbitrary tori in \cite[Section 10.9]{GS}. The trouble is that the geometric localization does not work well enough if the torus $T$ has positive dimensional isotropy on points $p \in X \setminus F_{T_1}$. But in fact the auxiliary forms $\alpha_i$ in lemma \ref{lem:geometric_localization} do not appear in the final localization formula. So if we want to compute the value of
$$ \int_{X/T_1} \alpha (\xi) \quad \mathrm{for} \quad \xi \in {\mathfrak t} $$
we can use different local forms $\alpha_i$ for different $\xi$. First suppose that for a given element $\xi \in {\mathfrak t}$ the space $\widetilde{{\mathfrak t}} := {\mathfrak t}_1 \oplus \mathbbm R\xi$ is the Lie algebra of a closed subgroup $\widetilde{T} \subset T$ such that $T_1 \subset \widetilde{T}$ has codimension $1$. The inclusion $\widetilde{T} \longrightarrow T$ induces the restriction map ${\mathfrak t}^* \longrightarrow \widetilde{{\mathfrak t}}^*$, which extends to a map on equivariant cohomology $H_T^*(X) \longrightarrow H_{\widetilde{T}}^*(X)$. And in fact for any $[ \alpha ] \in H_T^*(X)$ the computation of $\int_{X/T_1} \alpha (\xi)$ factors through this map. Hence we can apply our relative localization formula from \ref{thm:relative_localization} to obtain
$$ \int_{X/T_1} \alpha (\xi) \;=\; \sum_i \int_{Z_i/T_1} \frac{ i_{Z_i}^* \alpha }{ e_T(Z_i) } \, (\xi) \; \in \mathbbm R. $$
But now the set of $\xi \in {\mathfrak t}$, for which we derived this formula, is dense in ${\mathfrak t}$. Hence by continuity it actually holds for all $\xi$ and so the localization theorem above indeed holds independently of the codimension of $T_1$ in $T$.
\subsubsection*{The equivariant Euler class}
From the explicit construction of an equivariant Thom form for any $G$-equivariant vector bundle $E \longrightarrow X$ in Cieliebak, Mundet and Salamon \cite[Chapter 5]{CMS} one can read off two things: If the rank of $E$ is odd the Euler class vanishes, and if the rank is even then one gets the following representative for the Euler class $e_G(E)$.
\begin{prop}[{\cite[Lemma 4.3]{CS}}] \label{prop:equivariant_Euler_class} Suppose $E$ is a rank $n$ complex vector bundle and the action of $G$ is complex linear on the fibers. Fix a $G$-invariant Hermitian metric on $E$ and let $P \longrightarrow X$ denote the unitary frame bundle of $E$. Let $A \in \Omega^1(P,{\mathfrak u}(n))$ be a $G$-invariant $U(n)$-connection form on $P$. Then the $G$-equivariant Euler class of $E$ is represented by the $\mathrm{d}_G$-closed form $e_G(E)$ that is given by
$$ e_G(E)(\xi) \;=\; \det\left( \frac{i}{2\pi} F_A + \frac{i}{2\pi} A(X_\xi) \right), $$
where $F_A \in \Omega(P,{\mathfrak u}(n))$ denotes the curvature of $A$. \end{prop}
In fact assume that $E = X \times \mathbbm C^n$ is a trivial bundle with a diagonal $G$-action and the action on the fibers is given by a unitary representation
$$ \rho \;:\; G \longrightarrow U(n). $$
Then the frame bundle of $E$ is the product bundle $P = X \times U(n)$ and with the obvious $G$-invariant and flat connection $A$ we obtain
$$ e_G(E) \;=\; \det\left( \frac{i}{2\pi} \dot{\rho} \right). $$
This computation is contained in the proof of {\cite[Lemma 4.3]{CS}}. Note that the $G$-action on $X$ does not enter the formula. Now suppose the group $G$ is a torus $T$ and $\rho$ is given by
$$ \rho \;:\; T \longrightarrow \mathrm{U}(n) \;;\; \exp(\xi) \longmapsto \mathrm{diag}\left( e^{-2 \pi i \langle w_j , \xi \rangle} \right)_{j = 1 , \ldots , n} $$
for a given collection of weight vectors $w_j \in \Lambda^*$. Then we obtain the representative
\begin{equation} \label{eqn:torus_euler_class} e_T(E) \;=\; \prod_{j=1}^n w_j. \end{equation}
\subsection{Example}
\label{sec:example_localization}
We are now able to extend the computation from section \ref{sec:example_invariant_integration} to more general torus actions on $S^{2n-1}$ that do not factor through an $S^1$-action. We consider the $T$-action given by \ref{def:torus_action} on the unit sphere $S^{2n-1} \subset \mathbbm C^n$. We assume that all weights $w_j$ are nonzero and that no two weights are a negative multiple of each other. But the weights no longer have to be colinear. We take a primitive element $e_1 \in \Lambda$ such that
$$ \langle w_j , e_1 \rangle \ne 0 \quad \mathrm{for\ all}\ j \in \{ 1 , \ldots , n \}. $$
As before we denote by $T_1 \subset T$ the subtorus generated by $e_1$, by ${\mathfrak t}_1 \subset {\mathfrak t}$ its Lie algebra, by $T_0 := T/T_1$ the quotient group and by ${\mathfrak t}_0 := {\mathfrak t}/{\mathfrak t}_1$ its Lie algebra. In order to apply our localization technique to the $T_1$-invariant push-forward
$$ \left( \pi / T_1 \right)_* \;:\; H_T^*\left( S^{2n-1} \right) \longrightarrow H_{T_0}^*(\mathrm{pt}) \cong S\left( {\mathfrak t}_0^* \right) $$
we have to determine the $T_1$-relative fixed point set
$$
F_{T_1} \;:=\; \left\{ z \in S^{2n-1} \;|\; L_z({\mathfrak t}) = L_z({\mathfrak t}_1) \right\} $$
with the infinitesimal action $L_z$ at the point $z \in \mathbbm C^n$ given by
$$ L_z \;:\; {\mathfrak t} \longrightarrow T_zS^{2n-1} \;;\; \xi \longmapsto \left( 2 \pi i \langle w_j , \xi \rangle \cdot z_j \right)_{j = 1 , \ldots , n} \subset \mathbbm C^n $$
if we identify $T_zS^{2n-1} \subset T_z\mathbbm C^n \cong \mathbbm C^n$. For that purpose we have to arrange the $w_j$ into sets of colinear weights. Denote by $(w_\nu)_{\nu \in \{ 1 , \ldots , N\}}$ the collection of pairwise different weights, that are also not a multiple of any other weight $w_j$. Then for every $\nu$ there are unique integers $(\ell_{\nu_i})_{i \in \{ 1 , \ldots , n_\nu \}}$ such that the set of all $w_j$ equals
$$ \left\{ \ell_{1_1} w_1 , \ldots , \ell_{1_{n_1}} w_1 , \ldots , \ell_{\nu_1} w_\nu , \ldots , \ell_{\nu_{n_\nu}} w_\nu , \ldots , \ell_{N_1} w_N , \ldots , \ell_{N_{n_N}} w_N \right\}. $$
We can assume $\ell_{\nu_1} = 1$ for all $\nu = 1 , \ldots , N$ and furthermore by our assumptions on the $w_j$ we have all $\ell_{\nu_i} > 0$. Note that $\sum_{\nu = 1}^N n_\nu = n$. We introduce
$$
V_\nu \;:=\; \left\{ ( z_1 , \ldots , z_n ) \in \mathbbm C^n \;|\; z_j = 0 \;\mathrm{if}\; w_j \nparallel w_\nu \right\} \subset \mathbbm C^n, $$
the linear subspace of $\mathbbm C^n$ on which the $T$-action does factor through the weighted $S^1$-action given by the vector $w_\nu$ and the $\ell_\nu := (\ell_{\nu_i})_{i \in \{ 1 , \ldots , n_\nu \}}$. So we have $\dim_\mathbbm C V_\nu = n_\nu$ and we can write $\mathbbm C^n \cong \oplus_{\nu = 1}^N V_\nu$. In $V_\nu$ we consider the unit sphere
$$ S_\nu \;:=\; V_\nu \cap S^{2n-1}. $$
\begin{lem} The $T_1$-relative fixed point set $F_{T_1}$ is given by the disjoint union
$$ F_{T_1} \;=\; \bigsqcup_{\nu = 1}^N S_\nu. $$
\end{lem}
\begin{proof} The $V_\nu$ intersect only in $0 \in \mathbbm C^n$ and since $0 \not\in S^{2n-1}$ the $S_\nu$ are indeed disjoint. Now if $z \in S_\nu$ then $L_z({\mathfrak t})$ is the real one-dimensional space spanned by $\left( i \ell_{\nu_j}z_j \right) \in T_zS^{2n-1}$. The same is true for $L_z({\mathfrak t}_1)$ since $\langle w_\nu , e_1 \rangle \ne 0$ for any $\nu$. Hence $\sqcup_{\nu = 1}^N S_\nu \subset F_{T_1}$.
If on the other hand $z \in S^{2n-1}$ is not contained in any $S_\nu$ then it has two nonzero components $z_l , z_m \ne 0$ with $w_l \nparallel w_m$. Thus there exists an element $\xi \in {\mathfrak t}$ with $\langle w_l , \xi \rangle = 0$ but $\langle w_m , \xi \rangle \ne 0$. This implies $L_z(\xi) \not\in L_z({\mathfrak t}_1)$, because by assumption $\langle w_l , e_1 \rangle$ and $\langle w_m , e_1 \rangle$ are both nonzero. Hence $z \not\in F_{T_1}$. \end{proof}
Next we have to describe the equivariant normal bundles $N_\nu$ to $S_\nu$ in $S^{2n-1}$. They can explicitly be identified as
$$ N_\nu \;\cong\; S_\nu \times \bigoplus_{\nu' \ne \nu} V_{\nu'}. $$
Hence by \ref{eqn:torus_euler_class} the $T$-equivariant Euler class of $N_\nu$ is given by
$$ e_T(N_\nu) \;=\; \prod_{w_j \nparallel w_\nu} w_j. $$
As before we take elements $x_j \in {\mathfrak t}^*$ for $j = 1 , \ldots , m$ and denote the product by
$$ \mathbbm{x} \;:=\; \prod_{j = 1}^m x_j \in S({\mathfrak t}^*). $$
We write $\pi_\nu : S_\nu \longrightarrow \{ \mathrm{pt} \}$ for the projection of $S_\nu$ onto a point and obtain from the relative localization theorem \ref{thm:relative_localization} the following formula:
\begin{equation} \label{eqn:pi_T_1_localized} \left( \pi / T_1 \right)_* \left[ \mathbbm{x} \right] \;=\; \sum_{\nu = 1}^N \left( \pi_\nu / T_1 \right)_* \left[ \frac{ \mathbbm{x} }{ e_T \left( N_\nu \right) } \right] \end{equation}
Now every single summand can be computed by using the technique that was used to prove proposition \ref{prop:T_1_invariant_integration}, because with the $T$-action restricted to $S_\nu \subset V_\nu$ we are precisely in the setting of the example in section \ref{sec:example_invariant_integration}.
\begin{lem} With the above notation we obtain for every $\nu \in \{ 1 , \ldots , N \}$
$$ \left( \pi_\nu / T_1 \right)_* \left[ \frac{ \mathbbm{x} }{ e_T \left( N_\nu \right) } \right] (\xi) \;=\; \frac{1}{2 \pi i} \oint_\nu \frac{ \prod_{j = 1}^m \langle x_j , \xi + ze_1 \rangle }{ \prod_{j = 1}^n \langle w_j , \xi + ze_1 \rangle } \; \mathrm{dz} \; . $$
The integral is around a circle in the complex plane enclosing the pole at
$$ z = z_\nu := - \frac{ \langle w_\nu , \xi \rangle }{ \langle w_\nu , e_1 \rangle} $$
but no other pole of the integrand. \end{lem}
\begin{proof} We use the $T$-equivariant $T_1$-connection $A$ from \ref{def:T_equivariant_T_1_connection} with $\ell = \ell_\nu$ and $w = w_\nu$ and obtain analogously to \ref{eqn:xx_A}
\begin{eqnarray} \label{eqn:e_T_A} e_T \left( N_\nu \right)_A & = & \prod_{w_j \nparallel w_\nu} w_{j,A} \nonumber\\ & = & \prod_{w_j \nparallel w_\nu} \left( w_j - \langle w_j , e_1 \rangle \frac{ \mathrm{d}\alpha_\ell - w_\nu }{ \langle w_\nu , e_1 \rangle } \right). \end{eqnarray}
In the expansion of this product the term not involving any power of $\mathrm{d}\alpha_\ell$ is the nonzero element
$$ \mathbbm{w} \;:=\; \prod_{w_j \nparallel w_\nu} \left( w_j - \frac{ \langle w_j , e_1 \rangle }{ \langle w_\nu , e_1 \rangle } \cdot w_\nu \right) \in S( {\mathfrak t}_0^* ). $$
Hence if we repeat the steps following equation \ref{eqn:euler_class_sum} in the abstract discussion we obtain
\begin{equation} \label{eqn:e_inversion} \frac{ \mathbbm{w}^n }{ e_T \left( N_\nu \right)_A } \;=\; \beta \end{equation}
with
$$ \beta \;:=\; \mathbbm{w}^{n-1} + \mathbbm{w}^{n-2} \cdot \Theta + \ldots + \mathbbm{w} \cdot \Theta^{n-1} \quad ; \quad \Theta \;:=\; \mathbbm{w} - e_T \left( N_\nu \right)_A, $$
because $\left( \mathrm{d}\alpha_\ell \right)^n = 0$. This formula shows that $\beta$ is $T_1$-basic. Now by \ref{eqn:e_T_A} we have $e_T \left( N_\nu \right)_A$ given as a polynomial in $\frac{ \mathrm{d}\alpha_\ell - w_\nu }{ \langle w_\nu , e_1 \rangle }$. This implies that we can also write
$$ \beta \;=\; \sum_{k = 0}^{n-1} r_k \cdot \left( \frac{ \mathrm{d}\alpha_\ell - w_\nu }{ \langle w_\nu , e_1 \rangle } \right)^k $$
for some elements $r_k \in S({\mathfrak t}^*)$. Now since $\mathbbm{w} \in S( {\mathfrak t}_0^* )$ we have $\mathbbm{w}_A = \mathbbm{w}$ and hence
\begin{eqnarray*} \mathbbm{w}^n \cdot \left( \pi_\nu / T_1 \right)_* \left[ \frac{ \mathbbm{x} }{ e_T \left( N_\nu \right) } \right] & = & \left( \pi_\nu / T_1 \right)_* \left[ \frac{ \mathbbm{x}_A \cdot \mathbbm{w}^n }{ e_T \left( N_\nu \right)_A } \right] \\ & = & \left( \pi_\nu / T_1 \right)_* \left[ \mathbbm{x}_A \cdot \beta \right]. \end{eqnarray*}
Now the form $\mathbbm{x}_A \cdot \beta$ is precisely of the type that we discussed in remark \ref{rem:z}. So by formula \ref{eqn:gen_T_1_invariant_integration} we obtain
$$ \left( \pi_\nu / T_1 \right)_* \left[ \mathbbm{x}_A \cdot \beta \right] (\xi) \;=\; \mathrm{Res}_{z = - \frac{ \langle w_\nu , \xi \rangle }{ \langle w_\nu , e_1 \rangle } } \left\{ \frac{ \displaystyle \prod_{j = 1}^m \langle x_j , \xi + ze_1 \rangle \cdot \sum_{k = 0}^{n-1} r_k(\xi) \cdot z^k }{ \displaystyle \prod_{w_j \parallel w_\nu} \langle w_j , \xi + ze_1 \rangle } \right\}. $$
By equations \ref{eqn:e_T_A} and \ref{eqn:e_inversion} and the definition of the coefficients $r_k$ we get the identity
$$ \sum_{k = 0}^{n-1} r_k (\xi) \cdot z^k \;=\; \frac{ \mathbbm{w}^n (\xi) }{ \displaystyle \prod_{w_j \nparallel w_\nu} \langle w_j , \xi + ze_1 \rangle } $$
for all $z$ away from the zeros of the denominator. These zeros are all different from $z_\nu := - \frac{ \langle w_\nu , \xi \rangle }{ \langle w_\nu , e_1 \rangle }$. So we can plug this identity into the above residue formula to get
\begin{eqnarray*} \left( \pi_\nu / T_1 \right)_* \left[ \mathbbm{x}_A \cdot \beta \right] (\xi) & = & \mathrm{Res}_{z = z_\nu} \left\{ \frac{ \displaystyle \prod_{j = 1}^m \langle x_j , \xi + ze_1 \rangle \cdot \mathbbm{w}^n (\xi) }{ \displaystyle \prod_{w_j \parallel w_\nu} \langle w_j , \xi + ze_1 \rangle \cdot \prod_{w_j \nparallel w_\nu} \langle w_j , \xi + ze_1 \rangle } \right\} \\ & = & \mathbbm{w}^n (\xi) \cdot \mathrm{Res}_{z = z_\nu } \left\{ \frac{ \prod_{j = 1}^m \langle x_j , \xi + ze_1 \rangle }{ \prod_{j = 1}^n \langle w_j , \xi + ze_1 \rangle } \right\}. \end{eqnarray*}
Now for all $\xi$ with $\mathbbm{w} (\xi) \ne 0$ this implies
$$ \left( \pi_\nu / T_1 \right)_* \left[ \frac{ \mathbbm{x} }{ e_T \left( N_\nu \right) } \right] (\xi) \;=\; \mathrm{Res}_{z = z_\nu } \left\{ \frac{ \prod_{j = 1}^m \langle x_j , \xi + ze_1 \rangle }{ \prod_{j = 1}^n \langle w_j , \xi + ze_1 \rangle } \right\}. $$
Since $\mathbbm{w} \ne 0$ the set of such $\xi$ is dense in ${\mathfrak t}$ and the result actually holds for all $\xi$ by continuity. The claimed identity finally follows by the residue theorem. \end{proof}
Note the beauty of this formula: The integrand does not depend on $\nu$ --- only the point $z_\nu$ around which we have to integrate does. Hence we can perform the sum in \ref{eqn:pi_T_1_localized} by just integrating around all $z_\nu$, which also happen to be all poles of the integrand. This proves the following generalization of proposition \ref{prop:T_1_invariant_integration}:
\begin{prop} \label{prop:example_localization} Suppose the torus $T$ acts on $S^{2n-1} \subset \mathbbm C^n$ via a collection of $n$ weight vectors $w_j \in \Lambda^* \setminus \{ 0 \}$ as
$$ \exp( \xi ).z \;:=\; \left( e^{- 2 \pi i \langle w_1 , \xi \rangle} \cdot z_1 , \ldots , e^{- 2 \pi i \langle w_n , \xi \rangle} \cdot z_n \right) $$
Assume that no two $w_j$ are negative multiples of each other. Let $e_1 \in {\mathfrak t}$ be such that $\langle w_j , e_1 \rangle \ne 0$ for all $j$. Then for any $\mathbbm{x} \;=\; \prod_{j = 1}^m x_j \in S({\mathfrak t}^*)$ and $\xi \in {\mathfrak t}$ we have
$$ \left( \pi / T_1 \right)_* \left[ \mathbbm{x} \right] (\xi) \;=\; \frac{1}{2 \pi i} \oint \frac{ \prod_{j = 1}^m \langle x_j , \xi + z e_1 \rangle }{ \prod_{j = 1}^n \langle w_j , \xi + z e_1 \rangle } \; \mathrm{dz} \; . $$
The integral is around a circle in the complex plane enclosing all poles of the integrand. \end{prop}
\section{Moduli problems}
\label{chap:moduli_problems}
Cieliebak, Mundet and Salamon \cite{CMS} introduced the notion of a \emph{$G$-moduli problem} and its associated \emph{Euler class}. Roughly speaking a $G$-moduli problem consists of an equivariant Hilbert space bundle $\mathcal{E} \longrightarrow \mathcal{B}$ with an equi\-variant Fredholm section $\mathcal{S}$ and the Euler class is the map
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}} \;:\; H_G^*(\mathcal{B}) \longrightarrow \mathbbm R $$
obtained by $G$-invariant integration over the zero set $\mathcal{M} := \mathcal{S}^{-1}(0)$. The point is that this map can be defined even if the section $\mathcal{S}$ is not transversal and cannot be made so by an equivariant perturbation.
We review this technique because we will use it frequently later on. In fact we give a generalized construction and build in the results from section \ref{sec:H_invariant_push_forward} on $H$-invariant push-forward to define a relative $G/H$-equivariant Euler class for $H$-regular $G$-moduli problems. Furthermore we can study fibered moduli problems, which for instance appear in the discussion of the wall crossing formula in section \ref{sec:toric_wall_crossing}.
\subsection{Definitions}
Throughout $G$ denotes a compact oriented Lie group.
\begin{dfn} A \emph{$G$-moduli problem} is a triple $(\mathcal{B},\mathcal{E},\mathcal{S})$ with the following properties:
\begin{itemize} \item $\mathcal{B}$ is a Hilbert manifold with a smooth $G$-action. \item $\mathcal{E}$ is a Hilbert space bundle over $\mathcal{B}$, equipped with a smooth $G$-action such that the projection $\pi : \mathcal{E} \longrightarrow \mathcal{B}$ is $G$-equivariant and the induced action between fibers of $\mathcal{E}$ is by isometries. \item $\mathcal{S} : \mathcal{B} \longrightarrow \mathcal{E}$ is a smooth $G$-equivariant Fredholm section of constant Fredholm index. The determinant bundle $\det(\mathcal{S}) \longrightarrow \mathcal{B}$ is oriented, $G$ acts by orientation preserving isometries on $\det(\mathcal{S})$, and the zero set
$$
\mathcal{M} \;:=\; \left\{ x \in \mathcal{B} \;|\; \mathcal{S}(x) = 0 \right\} $$
is compact. \end{itemize}
A $G$-moduli problem with finite-dimensional spaces $\mathcal{B}$ and $\mathcal{E}$ is called \emph{oriented} if $\mathcal{B}$ and $\mathcal{E}$ are oriented and the $G$-actions preserve orientations. Given a closed subgroup $H \subset G$ a $G$-moduli problem is called \emph{$H$-regular} if the induced $H$-action on $\mathcal{M}$ is regular. A $G$-regular $G$-moduli problem is simply called \emph{regular}. \end{dfn}
Given any trivialization of $\mathcal{E}$ around a point $x \in \mathcal{B}$ one obtains the \emph{vertical differential} $\mathcal{D}_x$ by composing the differential $\mathrm{d}\mathcal{S}(x) : T_x\mathcal{B} \longrightarrow T_{\mathcal{S}(x)}\mathcal{E}$ of $\mathcal{S}$ with the projection $T_{\mathcal{S}(x)}\mathcal{E} \longrightarrow \mathcal{E}_x$ onto the fiber $\mathcal{E}_x$ of $\mathcal{E}$ over $x$:
$$ \mathcal{D}_x \;:\; T_x\mathcal{B} \longrightarrow \mathcal{E}_x $$
The Fredholm property of $\mathcal{S}$ asserts that $\mathcal{D}_x$ is a Fredholm operator for all $x$ in a small neighbourhood of $\mathcal{M}$ and that the index of $\mathcal{D}_x$ is independent of $x$. The \emph{index} of a $G$-moduli problem is defined to be
$$ \mathrm{Ind}(\mathcal{S}) \;:=\; \mathrm{Ind}(\mathcal{D}_x) - \dim(G). $$
\begin{dfn} \label{def:morphism} A \emph{morphism} between two $G$-moduli problems $(\mathcal{B},\mathcal{E},\mathcal{S})$ and $(\mathcal{B}',\mathcal{E}',\mathcal{S}')$ is a pair $(f,F)$ with the following properties.
\begin{itemize} \item $f : \mathcal{B}_0 \longrightarrow \mathcal{B}'$ is a smooth $G$-equivariant embedding of a neighbourhood $\mathcal{B}_0 \subset \mathcal{B}$ of $\mathcal{M}$ into $\mathcal{B}'$.
\item $F : \mathcal{E}|_{\mathcal{B}_0} \longrightarrow \mathcal{E}'$ is a smooth and injective bundle homomorphism over $f$. \item The sections $\mathcal{S}$ and $\mathcal{S}'$ satisfy
$$ \mathcal{S}' \circ f \;=\; F \circ \mathcal{S} \quad \mathrm{and} \quad \mathcal{M}' \;=\; f(\mathcal{M}). $$
\item For all $x \in M$ the linearized operator $\mathrm{d}f(x) : T_x\mathcal{B} \longrightarrow T_{f(x)}\mathcal{B}'$ and the linear operator $F_x : \mathcal{E}_x \longrightarrow \mathcal{E}'_{f(x)}$ induce isomorphisms
$$ \mathrm{d}f(x) \;:\; \ker \mathcal{D}_x \longrightarrow \ker \mathcal{D}'_{f(x)} \quad \mathrm{and} \quad F_x \;:\; \mathrm{coker} \mathcal{D}_x \longrightarrow \mathrm{coker} \mathcal{D}'_{f(x)} $$
and the resulting isomorphism from $\det(\mathcal{S})$ to $\det(\mathcal{S}')$ is orientation preserving. \end{itemize}
\end{dfn}
To any regular $G$-moduli problem there exists its \emph{Euler class}, which is a homomorphism
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}} \;:\; H_G^*(\mathcal{B}) \longrightarrow \mathbbm R. $$
As shown in \cite{CMS} the Euler class is uniquely determined by the following two properties.
\begin{description} \item[(Functoriality)] If $(f,F)$ is a morphism from $(\mathcal{B},\mathcal{E},\mathcal{S})$ to $(\mathcal{B}',\mathcal{E}',\mathcal{S}')$ then
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}} \circ f_G^* \;=\; \chi^{\mathcal{B}',\mathcal{E}',\mathcal{S}'}. $$
\item[(Thom class)] If $(B,E,S)$ is a finite-dimensional, oriented, regular $G$-moduli problem and $\tau \in \Omega_G^*(E)$ is an equivariant Thom form supported in an open neighbourhood $U \subset E$ of the zero section such that $U \cap E_x$ is convex for every $x \in B$, $U \cap \pi^{-1}(K)$ has compact closure for every compact set $K \subset B$, and $S^{-1}(U)$ has compact closure, then
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}} (\alpha) \;=\; \int_{B/G} \alpha \wedge S^*\tau $$
for every $\alpha \in H_G^*(B)$. \end{description}
The integral over $B/G$ is $G$-invariant integration as explained in section \ref{chap:invariant_integration}. We will review the definition of $\chi^{\mathcal{B},\mathcal{E},\mathcal{S}}$ in the next section, where we explain the generalization to $H$-regular problems.
\begin{dfn} \label{def:homotopy}
A \emph{homotopy of regular $G$-moduli problems} is a $G$-equi\-va\-ri\-ant Hilbert space bundle $\mathcal{E} \longrightarrow [0,1] \times \mathcal{B}$ and a smooth $G$-equivariant section $\mathcal{S}$ therein, such that $( \mathcal{B} , \mathcal{E}|_{\{t\}\times\mathcal{B}} , \mathcal{S}|_{\{t\}\times\mathcal{B}} )$ is a regular $G$-moduli problem for every $t \in [0,1]$, and the set
$$
\mathcal{M} \;:=\; \left\{ (t,x) \in [0,1] \times \mathcal{B} \;|\; \mathcal{S}(t,x) = 0 \right\} $$
is compact. \end{dfn}
A homotopy is an example for the more general notion of a \emph{cobordism} of $G$-moduli problems, which is defined in the obvious way:
\begin{dfn} \label{def:cobordism} Two regular $G$-moduli problems $\left( \mathcal{B}_i , \mathcal{E}_i , \mathcal{S}_i \right)$, $i \in \{0,1\}$, are called \emph{cobordant} if there exist a $G$-equivariant Hilbert space bundle $\widetilde{\mathcal{E}} \longrightarrow \widetilde{\mathcal{B}}$ over a Hilbert manifold $\widetilde{\mathcal{B}}$ with boundary and a smooth oriented $G$-equivariant Fredholm section $\widetilde{\mathcal{S}} : \widetilde{\mathcal{B}} \longrightarrow \widetilde{\mathcal{E}}$ such that the zero set $\widetilde{\mathcal{M}} := \widetilde{\mathcal{S}}^{-1}(0)$ is compact, $G$ acts with finite isotropy on $\widetilde{\mathcal{B}}$, and
$$
\partial \widetilde{\mathcal{B}} \;=\; \mathcal{B}_0 \cup \mathcal{B}_1 \quad , \quad \mathcal{E}_i \;=\; \widetilde{\mathcal{E}}|_{\mathcal{B}_i} \quad , \quad \mathcal{S}_i \;=\; \widetilde{\mathcal{S}}|_{\mathcal{B}_i}. $$
Moreover, $\det(\widetilde{\mathcal{S}})$ carries an orientation which induces the orientation of $\det(\mathcal{S}_1)$ over $\mathcal{B}_1$ and the opposite of the orientation of $\det(\mathcal{S}_0)$ over $\mathcal{B}_0$. Here an orientation of $\det(\widetilde{\mathcal{S}})$ induces an orientation of the determinant bundle of $\mathcal{S} := \widetilde{\mathcal{S}}|_{\partial \widetilde{\mathcal{B}}}$ via the natural isomorphism $\det(\widetilde{\mathcal{S}})|_{\partial \widetilde{\mathcal{B}}} \cong \mathbbm R v \otimes \det(\mathcal{S})$ for an outward pointing normal vector field $v$ along $\partial \widetilde{\mathcal{B}}$. \end{dfn}
It is shown in \cite{CMS} that the Euler class satisfies the following property:
\begin{description} \item[(Cobordism)] If $\left( \mathcal{B}_0 , \mathcal{E}_0 , \mathcal{S}_0 \right)$ and $\left( \mathcal{B}_1 , \mathcal{E}_1 , \mathcal{S}_1 \right)$ are cobordant $G$-moduli problems then
$$ \chi^{\mathcal{B}_0,\mathcal{E}_0,\mathcal{S}_0}(\iota_0^*\alpha) \;=\; \chi^{\mathcal{B}_1,\mathcal{E}_1,\mathcal{S}_1}(\iota_1^*\alpha) $$
for every $\alpha \in H_G^*(\widetilde{\mathcal{B}})$, where $\iota_0 : \mathcal{B}_0 \longrightarrow \widetilde{\mathcal{B}}$ and $\iota_1 : \mathcal{B}_1 \longrightarrow \widetilde{\mathcal{B}}$ are the inclusions. \end{description}
\subsection{The $G/H$-equivariant Euler class}
Suppose $(\mathcal{B},\mathcal{E},\mathcal{S})$ is an $H$-regular $G$-moduli problem for some closed normal subgroup $H \lhd G$. We denote the quotient group by $K := G / H$ and its Lie algebra by ${\mathfrak k}$. In this section we will define the $G/H$-equivariant Euler class
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}}_{G,H} \;:\; H_G^*(\mathcal{B}) \longrightarrow H_K^*(\mathrm{pt}) = S({\mathfrak k}^*). $$
This is a generalization of the Euler class $\chi^{\mathcal{B},\mathcal{E},\mathcal{S}}$ from \cite{CMS} because for a $G$-regular $G$-moduli problem we will have
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}}_{G,G} \;=\; \chi^{\mathcal{B},\mathcal{E},\mathcal{S}}. $$
The construction is analogous to the old one. The procedure of finite-dimensional reduction does not depend on the involved isotropy groups. Hence it suffices to define $\chi^{\mathcal{B},\mathcal{E},\mathcal{S}}_{G,H}$ for finite-dimensional, oriented, $H$-regular $G$-moduli problems.
A \emph{Thom structure} for such a problem is a pair $(U,\tau)$ with the following properties:
\begin{enumerate}
\item $U \subset \mathcal{E}$ is a $G$-invariant open neighbourhood of the zero section that intersects each fiber in a convex set and such that $U \cap \mathcal{E}|_K$ has compact closure for every compact subset $K \subset \mathcal{B}$.
\item $\mathcal{S}^{-1}(U)$ has compact clusure.
\item $\tau \in \Omega_G^*(\mathcal{E})$ is an equivariant Thom form for the zero section in $\mathcal{E}$ with support in $U$. \end{enumerate}
By the results cited in section \ref{sec:equivariant_thom_forms} it is clear that for every finite-dimensional, oriented $G$-moduli problem $(\mathcal{B},\mathcal{E},\mathcal{S})$ and for every subset $U \subset \mathcal{E}$ that satisfies the first two points, there exists a form $\tau$ such that $(U,\tau)$ is a Thom structure. This also does not involve the isotropy of the $G$-action.
Now since $\mathcal{S}^{-1}(0) = \mathcal{M}$ is compact we find an open neighbourhood $\mathcal{B}_0$ of $\mathcal{M}$ in $\mathcal{B}$ that has compact closure and such that $H$ acts with finite isotropy on all of $\mathcal{B}_0$. Next we choose $U \subset \mathcal{E}$ small enough so that $\mathcal{S}^{-1}(U)$ lies in $\mathcal{B}_0$ and such that $U$ satisfies property (1). We then find a corresponding Thom form $\tau$. For a class $[ \alpha ] \in H_G^*(\mathcal{B})$ we finally define
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}}_{G,H} ( [ \alpha ] ) \;:=\; \int_{\mathcal{B}/H} \alpha \wedge \mathcal{S}^* \tau \in S({\mathfrak k}^*). $$
Integration over $\mathcal{B}/H$ is $H$-invariant push-forward of the map $\mathcal{B} \longrightarrow \{\mathrm{pt}\}$ as defined in section \ref{sec:H_invariant_push_forward}. It is well defined because by construction the form $\mathcal{S}^*\tau$ has compact support. Independence of all involved choices follows as in \cite[Chapter 8]{CMS} if we use the fact that $G$-equivariant Thom forms behave well for $H$-invariant integration as shown in proposition \ref{prop:H_invariant_Thom_integration}.
\begin{rem} There is one open question about the $G/H$-equivariant Euler class. The Euler class $\chi^{\mathcal{B},\mathcal{E},\mathcal{S}}$ from \cite{CMS} is rational, i.~e.~it satisfies the following property:
\begin{description} \item[(Rationality)] If $\alpha \in H_G^*(\mathcal{B};\mathbbm Q)$ then $\chi^{\mathcal{B},\mathcal{E},\mathcal{S}} (\alpha) \in \mathbbm Q$. \end{description}
Does the corresponding property also hold for $\chi^{\mathcal{B},\mathcal{E},\mathcal{S}}_{G,H}$? One would need a generalization of the technique of representing rational classes by weighted branched submanifolds (see \cite[Chapter 9 and 10]{CMS}) to the case of non-regular $G$-actions. Such a theory would be interesting on its own account. \end{rem}
\subsection{Fibered moduli problems}
\label{sec:fibred_G_moduli_problems}
Suppose the base space $\mathcal{B}$ of a moduli problem is a fiber bundle
$$ f \;:\; \mathcal{B} \longrightarrow \mathcal{B}_0. $$
Furthermore assume that $f$ is $G$-equivariant, that $H$ acts trivially in $\mathcal{B}_0$, and that we have a $K$-moduli problem $\left( \mathcal{B}_0 , \mathcal{E}_0 , \mathcal{S}_0 \right)$ with $K := G/H$ and some closed normal subgroup $H \lhd G$. Now consider the pull-back
$$ \left(\,\mathcal{B} \,,\, \mathcal{E} := f^*\mathcal{E}_0 \,,\, \mathcal{S} := f^*\mathcal{S}_0 \,\right). $$
If the fibers of $f$ are compact then the moduli space $\mathcal{M} := \mathcal{S}^{-1}(0)$ will be compact and $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$ in fact is a $G$-moduli problem. This is the simplest instance of a \emph{fibered moduli problem}. Since this is also the only form in which we will actually apply the notion of a fibered moduli problem, we restrict to this case. We comment on more general settings below.
\begin{prop} \label{prop:fibred_Euler_class} In the above setup suppose that $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$ is $H$-regular. Then we have the identity
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}}_{G,H} \;=\; \chi^{\mathcal{B}_0,\mathcal{E}_0,\mathcal{S}_0}_{K,\{e\}} \circ \left( f / H \right)_* \;:\; H_G^*(\mathcal{B}) \longrightarrow H_K^*(\mathrm{pt}) = S({\mathfrak k}^*). $$
If in addition $\left( \mathcal{B}_0 , \mathcal{E}_0 , \mathcal{S}_0 \right)$ is $K$-regular, then $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$ is $G$-regular and we have
$$ \chi^{\mathcal{B},\mathcal{E},\mathcal{S}}_{G,G} \;=\; \chi^{\mathcal{B}_0,\mathcal{E}_0,\mathcal{S}_0}_{K,K} \circ \left( f / H \right)_* \;:\; H_G^*(\mathcal{B}) \longrightarrow \mathbbm R. $$
\end{prop}
\begin{proof} The first step is to show that we can choose finite-dimensional reductions that feature the same fibered picture. We omit this discussion and deal with finite-dimensional problems only. We will only apply this result in settings that are already finite-dimensional.
We choose a Thom structure $(U_0,\tau_0)$ for $\left( \mathcal{B}_0 , \mathcal{E}_0 , \mathcal{S}_0 \right)$. Then
$$ \left( U , \tau \right) \;:=\; \left( f^{-1}U_0 , f^*\tau_0 \right) $$
is a Thom structure for $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$. By property $(5)$ in proposition \ref{prop:invariant_integration} we have
\begin{eqnarray*} \left( f / H \right)_* \left( \alpha \wedge S^*\tau \right) & = & \left( f / H \right)_* \left( \alpha \wedge S^*f^*\tau_0 \right) \\ & = & \left( f / H \right)_* \left( \alpha \wedge f^*{S_0}^*\tau_0 \right) \\ & = & \left( f / H \right)_* \alpha \wedge {S_0}^*\tau_0. \end{eqnarray*}
Hence the claimed identities follow by the functoriality properties $(4)$ in proposition \ref{prop:invariant_integration} and the definition of the Euler classes. \end{proof}
\begin{rem} Another instance of a fibered moduli problem would be a \emph{parametrized} problem, that is a triple $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$ with an equivariant fibration $f : \mathcal{B} \longrightarrow \mathcal{B}_0$ such that $H$ acts trivially on $\mathcal{B}_0$ and the restriction of $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$ to any fiber of $f$ is an $H$-regular $G$-moduli problem. For such a parametrized problem to be a moduli problem one needs the total moduli space $\mathcal{M}$ to be compact. It does not suffice if only the base $\mathcal{B}_0$ is compact. A homotopy of moduli problems would be an example. But to really get computational simplifications in the spirit of proposition \ref{prop:fibred_Euler_class} from such a fibered setting, one would also need a good behaviour of the section $\mathcal{S}$ with respect to the fibration. The section $\mathcal{S}$ should split into components that either depend only on the point in the base or only on the point in the fiber.
The moduli problem associated to the vortex equations (see section \ref{chap:vortex_invariants}) has the property that $\mathcal{B}$ is a fiber bundle. The base $\mathcal{B}_0$ is the space of connections $\mathcal{A}(P)$ on a principal bundle $P$ modulo based gauge transformations $\mathcal{G}_0(P)$. The quotient group $G = \mathcal{G}(P) / \mathcal{G}_0(P)$ of all gauge transformations by the based ones acts trivially on connections, but locally freely on $\mathcal{B}$. But the section $\mathcal{S}$ does not behave well with respect to this fibration. One can interprete our deformation of the vortex equations in section \ref{sec:vortex_deformation} as the attempt to improve this situation and exploit the fiber form of $\mathcal{B}$. \end{rem}
\section{Toric manifolds}
\label{chap:toric_manifolds}
We present toric manifolds as symplectic reduction of complex linear torus actions but adapt the notation to our purposes. In particular we want to have a setup in which we can easily change the multiplicities of the characters of a given torus action --- because this is what happens in section \ref{chap:vortex_invariants} when we look at the moduli space of the vortex equations associated to such a torus action.
Toric manifolds are the basic model for moduli problems as presented in section \ref{chap:moduli_problems}. The associated Euler class is the integral of elements in the image of the Kirwan map over the toric manifold. Knowing these integrals suffices to determine the whole cohomology ring of the toric manifold. In section \ref{sec:toric_wall_crossing} we study the effect it has on this Euler class if we change the chamber of the regular value $\tau$ at which we reduce. With the methods on invariant integration at hand we can derive an explicit formula that allows to compute these wall crossing numbers. Then by a sequence of wall crossings that connects the chamber of $\tau$ with the chamber outside of the image of the moment map we can completely determine the Euler class.
This wall crossing strategy to evaluate integrals over symplectic quotients is not new. It is used by Guillemin and Kalkman in \cite{GK} and Martin in \cite{Mar} to derive a formula for such integrals. Because of the results in \cite{JK} such formulae are generally referred to as \emph{Jeffrey-Kirwan localization}. We discuss the relation of our work with general Jeffrey-Kirwan localization at the end of this section.
\subsection{Torus actions on Hermitian vector spaces}
\label{sec:torus_actions_on_hermitian_vector_spaces}
Let $V$ be a Hermitian vector space with Hermitian form given by
$$ \left( v , w \right) \;=\; g \left( v , w \right) - i \; \omega \left( v , w \right) $$
with inner product $g$ (given by the real part $\Re$ of the Hermitian form) and symplectic form $\omega$ (given by minus the imaginary part $\Im$ of the Hermitian form). The induced norm is denoted by $\left| v \right|^2 = g \left( v , v \right)$. Let $T$ be a $k$-dimensional torus with Lie algebra ${\mathfrak t}$. We denote the integral lattice in ${\mathfrak t}$ by
$$
\Lambda \;=\; \left\{ \xi \in {\mathfrak t} \;|\; \exp \left( \xi \right) = \mathbbm 1 \in T \right\} $$
and its dual by
$$
\Lambda^* \;=\; \left\{ w \in {\mathfrak t}^* \;|\; \forall \xi \in \Lambda : \left\langle w,\xi \right\rangle \in \mathbbm Z \right\}. $$
As before $\langle \;,\; \rangle$ denotes the pairing between ${\mathfrak t}$ and its dual space ${\mathfrak t}^*$. Let $\rho \;:\; T \longrightarrow \mathrm{U}(1)$ be a character given by
$$ \exp \left( \xi \right) \longmapsto e^{-2 \pi i \langle w , \xi \rangle} $$
for some nonzero weight vector $w \in \Lambda^* \setminus \{ 0 \}$. Now $\mathrm{U}(1)$ acts by scalar multiplication on the complex vector space $V$. Hence $\rho$ induces an action of the torus $T$ on $V$. This action is Hamiltonian with respect to the symplectic form $\omega$ on $V$. A moment map for this action is given by
\begin{align*} \mu \;:\; V & \;\longrightarrow\; \quad {\mathfrak t}^* \\
v & \;\longmapsto\; \pi \left| v \right|^2 \cdot w. \end{align*}
By definition a moment map satisfies the equation
$$ \mathrm{d} \langle \mu , \xi \rangle = - \iota_\xi \omega $$
for all $\xi \in {\mathfrak t}$. The sign-conventions are such that $\omega - \mu$ is a $\mathrm{d}_T$-closed equivariant differential form.
We now consider a finite collection of characters $\rho_\nu$ given by weights $w_\nu \in \Lambda^*$ for $\nu=1,\ldots,N$. Given a corresponding collection of Hermitian vector spaces $V_\nu$ we get a diagonal action $\rho$ of the torus $T$ on the direct sum $V = \bigoplus_{\nu=1}^N V_\nu$. In fact every complex linear torus action decomposes in such a way. This action is again Hamiltonian and a moment map is given by the sum of the moment maps $\mu_\nu$ for the single $\rho_\nu$-induced $T$-actions on the $V_\nu$:
\begin{align*} \mu = \sum_{\nu=1}^N \mu_\nu \;:\; V = \bigoplus_{\nu=1}^N V_\nu & \;\longrightarrow\; \quad {\mathfrak t}^* \\
\left( v_\nu \right) \quad & \;\longmapsto\; \pi \sum_{\nu=1}^N \left| v_\nu \right|^2 \cdot w_\nu \end{align*}
\begin{dfn} \label{def:proper} A collection of characters $\rho_\nu$ is called \emph{proper} if there exists an element $\xi \in {\mathfrak t}$ such that $\left\langle w_\nu , \xi \right\rangle < 0$ for all $\nu \in \left\{ 1 , \ldots , N \right\}$, i.e.~all weight vectors are contained in the same connected component of ${\mathfrak t}^*$ with one hyperplane through the origin removed. \end{dfn}
This notion is motivated by the following observation. The moment map $\mu$ associated to a collection of finite-dimensional vector spaces $V_\nu$ is proper (in the usual sense) if and only if the collection of characters $\rho_\nu$ is proper (in the above sense). Hence properness of the moment map is preserved if we change the multiplicities of the characters in the torus action, i.e.~the dimensions of the spaces $V_\nu$. We can even allow $\dim V_\nu = 0$. The following notation is taken from Guillemin, Ginzburg and Karshon \cite{GGK}.
\begin{dfn} Given a collection of characters $\rho_\nu$ an element $\tau \in {\mathfrak t}^*$ is called \emph{regular} (for this collection) if the following holds: If $\tau = \sum_{j \in J} a_j w_j$ for some positive coefficients $a_j > 0$ and some index set $J \subset \left\{ 1 , \ldots , N \right\}$, then the weight vectors $\left( w_j \right)_{j \in J}$ span ${\mathfrak t}^*$. An element $\tau \in {\mathfrak t}^*$ is called \emph{super-regular} if the following holds: If $\tau = \sum_{j \in J} a_j w_j$ for some positive coefficients $a_j > 0$ and some index set $J \subset \left\{ 1,\ldots,N \right\}$, then the weight vectors $\left( w_j \right)_{j \in J}$ generate $\Lambda^*$ over $\mathbbm Z$. \end{dfn}
Here we observe that an element $\tau \in {\mathfrak t}^*$ is a regular value of the moment map $\mu$ if and only if $\tau$ is regular in the above sense. Again this does not depend on the spaces $V_\nu$. A super-regular element $\tau$ is in particular regular but has even nicer properties. The moment map is $T$-invariant, hence we get induced $T$-actions on every level set $\mu^{-1}(\tau)$. Recall that we call an action \emph{regular} if all isotropy subgroups are finite.
\begin{lem} If $\tau \in {\mathfrak t}^*$ is regular, then the $T$-action on $\mu^{-1}(\tau)$ is regular. If $\tau$ is super-regular, then $T$ acts freely on $\mu^{-1}(\tau)$. \end{lem}
\begin{proof}
Let $\tau$ be regular and $v \in \mu^{-1}(\tau)$. Hence $\tau = \pi \sum_{\nu = 1}^N \left| v_\nu \right|^2 \cdot w_\nu$ and the collection $(w_j)_{j \in J}$ with $J = \left\{ \left. j \in \left\{ 1 , \ldots , N \right\} \;\right|\; v_j \ne 0 \right\}$ spans ${\mathfrak t}^*$. So the $1$-form $\left\langle \mathrm{d}\mu(v) , \xi \right\rangle = 2 \pi \sum_{\nu = 1}^N g( v_\nu , \cdot ) \, \langle w_\nu , \xi \rangle$ can only vanish for $\xi = 0$. Now since $\mu$ is a moment map $\langle \mathrm{d}\mu(v) , \xi \rangle = - \iota_\xi \omega$ and this implies that $X_\xi$ can only vanish for $\xi = 0$. Thus the torus acts with finite isotropy on $v$.
If $\exp(\xi) \in T$ fixes a point $v \in \mu^{-1}(\tau)$ then for all $j$ with $v_j \ne 0$ we must have $\langle w_j , \xi \rangle \in \mathbbm Z$. If $\tau$ is super-regular this implies $\langle w , \xi \rangle \in \mathbbm Z$ for all $w \in \Lambda^*$, hence $\xi \in \Lambda$ and $\exp(\xi) = \mathbbm 1$. \end{proof}
So if we have a collection of characters $\left( \rho_\nu \right)$ and a super-regular element $\tau \in {\mathfrak t}^*$ we can associate a \emph{toric manifold}
$$ X_{V,\tau} = \mu^{-1}(\tau) / T $$
to any collection of spaces $V_\nu$. If $\tau$ is only regular we get orbifold singularities in the quotient. This quotient is compact if and only if $\left( \rho_\nu \right)$ is a proper collection. Smoothness and compactness do not depend on the choice of Hermitian vector spaces $V_\nu$. They can even be zero-dimensional. If we denote by $n_\nu$ the complex dimension of $V_\nu$ the real dimension of $X_{V,\tau}$ is given by
$$ \dim X_{V,\tau} = 2n - 2k \quad \mathrm{with} \quad n := \sum_{\nu = 1}^N n_\nu. $$
The notation $X_{V,\tau}$ indicates that we think of the characters $\rho_\nu$ and hence the weight vectors $w_\nu \in {\mathfrak t}^*$ to be fixed, whereas the Hermitian vector spaces $V_\nu$ and the level $\tau$ can vary.
\begin{rem} If the collection of weight vectors $w_\nu$ does not span the whole ${\mathfrak t}^*$ then there are no regular elements $\tau$ in the image of the moment map $\mu$. Hence any associated toric manifold $X_{V,\tau}$ would be empty. So we can restrict to torus actions that are given by collections of weight vectors that do span ${\mathfrak t}^*$. \end{rem}
\subsection{Toric manifolds as moduli problems}
\label{sec:toric_manifolds_as_moduli_problems}
Consider a diagonal torus action given by a proper collection $\rho_\nu$ and a regular level $\tau \in {\mathfrak t}^*$ as above. We do not require super-regularity, so the quotient $X_{V,\tau}$ can have singularities. Using the notation from section \ref{chap:moduli_problems} this setup gives rise to a finite-dimensional $T$-moduli problem with base $\mathcal{B} = V$, bundle $\mathcal{E} = V \times {\mathfrak t}^*$ and section $\mathcal{S} = \mu - \tau$. Regularity of $\tau$ implies regularity for the moduli problem and transversality of $\mathcal{S}$ to the zero section. The complex orientation on $V$ and any choice of orientation on the torus $T$ gives an orientation of the finite-dimensional $T$-moduli problem $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$ and hence we get an associated Euler class
$$ \chi^{V,\tau} : S^m \left( {\mathfrak t}^* \right) \longrightarrow \mathbbm R $$
with
$$ m = \frac{1}{2} \cdot \dim X_{V,\tau} = n - k. $$
This Euler class is defined as follows. There is a natural identification of $S \left( {\mathfrak t}^* \right)$ with the equivariant cohomology $H_T^*(V)$. Now an equivariant class can be restricted to the $T$-invariant submanifold $\mu^{-1}(\tau)$ to get an element in $H_T^*( \mu^{-1}(\tau) )$. By regularity of $\tau$ the torus action on $\mu^{-1}(\tau)$ has at most finite isotropy. Hence $T$-invariant integration can be applied to obtain a real number. Here we again use the orientation on $T$ that was chosen above. Formally
$$ \chi^{V,\tau}(\alpha) = \int_{X_{V,\tau}} i_\tau^*(\alpha) $$
with the inclusion $i_\tau : \mu^{-1}(\tau) \longrightarrow V$ and the induced pull-back $i_\tau^*$ on equivariant cohomology, and with integration over $X_{V,\tau}$ understood as $T$-invariant integration over $\mu^{-1}(0)$. The map
$$ i_\tau^* : H_T^*( V ) \longrightarrow H_T^*( \mu^{-1}(\tau) ) $$
is known as the \emph{Kirwan map} and due to Kirwan \cite{Ki} is the following result:
\begin{thm}[F.~Kirwan] \label{thm:Kirwan} The Kirwan map is surjective. \end{thm}
Hence any integral of a cohomology class over $X_{V,\tau}$ can be expressed in terms of this Euler class $\chi^{V,\tau}$.
\subsection{Orientation of toric manifolds}
\label{sec:orientation_of_toric_manifolds}
We digress a little to explain the orientation of a toric manifold in more detail. This is useful for later purposes because this is the finite-dimensional model case for the orientation of the vortex moduli space that we will discuss in section \ref{sec:orientations}.
There is a natural orientation on $X_{V,\tau} = \mu^{-1}(\tau) / T$ that does not depend on the choice of an orientation for $T$. Instead we fix the following complex structure on the space ${\mathfrak t}^* \oplus {\mathfrak t}^*$:
$$ {\mathfrak t}^* \oplus {\mathfrak t}^* \longrightarrow {\mathfrak t}^* \oplus {\mathfrak t}^* \; ; \; ( a , b ) \longmapsto ( -b , a ). $$
We write
$$ L_v : {\mathfrak t} \longrightarrow T_vV \; ; \; \xi \longmapsto X_\xi(v) $$
for the infinitesimal action at the point $v \in V$ and consider the adjoint map $L_v^* : T_vV \longrightarrow {\mathfrak t}^*$, where we identify $T_vV$ with its dual via the fixed metric on $V$. Given a point $[v] \in X_{V,\tau}$ we can now identify the tangent space $T_{[v]}X_{V,\tau}$ with the kernel of the map
$$ D_v := \left( \mathrm{d}\mu(v) , L_v^* \right) : T_vV \longrightarrow {\mathfrak t}^* \oplus {\mathfrak t}^*. $$
The first component restricts to the tangent space along the level set of the moment map. The second component fixes a complement to the tangent space along the $T$-orbit. If we identify $V \cong T_vV$ a short computation shows that
$$ D_v ( z ) = \left( 2 \sum_{\nu = 1}^N \Re ( z_\nu , v_\nu ) \cdot w_\nu \; , \; 2 \sum_{\nu = 1}^N \Im ( z_\nu , v_\nu ) \cdot w_\nu \right). $$
This shows that $D_v$ is complex linear (with respect to the natural complex structure on $V$ and the above chosen complex structure on ${\mathfrak t}^* \oplus {\mathfrak t}^*$). And $D_v$ is even surjective if $\tau$ is regular. In any case the complex orientation on source and target of $D_v$ define an orientation on its kernel and hence define an orientation on $X_{V,\tau}$.
This is our preferred way to define the orientation of a toric manifold, because it extends to an infinite-dimensional setting: The determinant of a complex linear Fredholm operator between complex Banach spaces admits a natural orientation. But if we want to understand $X_{V,\tau}$ as the oriented moduli space of the finite-dimensional $T$-moduli problem $\left( \mathcal{B} , \mathcal{E} , \mathcal{S} \right)$ then we need to specify an orientation of the torus. In fact, if we denote the vertical differential at a point $x \in \mathcal{B}$ by $\mathcal{D} : T_x\mathcal{B} \longrightarrow {\mathfrak t}^*$ then we obtain
\begin{eqnarray*} \det(\mathcal{D}) & = & \Lambda^{\mathrm{max}} \left( \ker\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \mathrm{coker}\,\mathcal{D} \right)\\ & = & \Lambda^{\mathrm{max}} \left( \ker\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \frac{{\mathfrak t}^*}{\mathrm{im}\,\mathcal{D}} \right)\\ & \cong & \Lambda^{\mathrm{max}} \left( \ker\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \mathrm{im}\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \mathrm{im}\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \frac{{\mathfrak t}^*}{\mathrm{im}\,\mathcal{D}} \right)\\ & \cong & \Lambda^{\mathrm{max}} \left( \ker\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \frac{T_x\mathcal{B}}{\ker\,\mathcal{D}} \right) \otimes \Lambda^{\mathrm{max}} \left( \mathrm{im}\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \frac{{\mathfrak t}^*}{\mathrm{im}\,\mathcal{D}} \right)\\ & \cong & \Lambda^{\mathrm{max}} \left( T_x\mathcal{B} \right) \otimes \Lambda^{\mathrm{max}} \left( {\mathfrak t}^* \right). \end{eqnarray*}
In the third line we use the canonical identification $\Lambda^{\mathrm{max}} (W) \otimes \Lambda^{\mathrm{max}} (W) \cong \mathbbm R$ for any finite-dimensional vector space $W$. And in the last line we use the canonical identification $\Lambda^{\mathrm{max}} (W) \cong \Lambda^{\mathrm{max}} (U) \otimes \Lambda^{\mathrm{max}} \left( \frac{W}{U} \right)$ for any linear subspace $U \subset W$.
Now an orientation of $T$ is the same as an orientation of ${\mathfrak t}$. And if we identify ${\mathfrak t}$ with its dual via the choice of an inner product then this also fixes the orientation of ${\mathfrak t}^*$. The choice of the inner product does not affect the result, because the space of inner products in contractible. Hence the orientation of the determinant bundle of the $T$-moduli problem as well as the definition of the Euler class (because of the invariant integration that needs a specified orientation of the torus) depend on the choice of orientation for $T$. So if we both times use the same orientation then the resulting homomorphism $\chi^{V,\tau}$ does not depend on this choice. But it does depend on the conventions that we use for the ordering of basis vectors of the involved spaces.
A careful look at the construction reveals that we reproduce the preferred complex orientation from above via the following conventions: Take an oriented basis of $T_vV$ such that the image under $\mathrm{d}\mu(v)$ of the first $k$ vectors is an oriented basis of ${\mathfrak t}^*$ and the remaining vectors lie in the kernel of $\mathrm{d}\mu(v)$. The orientation of the latter determines the orientation of $T_v \left( \mu^{-1}(\tau) \right)$. Now take a local slice $(U_v,\varphi_v,T_v)$ at $v$ for the $T$-action on $\mu^{-1}(\tau)$ (see section \ref{sec:G_invariant_integration} for the notation) using the product orientation on $T \times_{T_v} U_v$ defined by first taking an oriented basis of ${\mathfrak t}$ and then the standard orientation on $U_v \subset \mathbbm R^{2m}$.
\subsection{Wall crossing}
\label{sec:toric_wall_crossing}
Let $\left( \rho_\nu \right)_{\nu = 1 , \ldots , N}$ be a proper collection of characters with weight vectors $w_\nu \in \Lambda^*$ that span ${\mathfrak t}^*$. Fix a Hermitian vector space $V = \bigoplus_{\nu = 1}^N V_\nu$. For an index set $I \subset \left\{ 1 , \ldots , N \right\}$ we define the cone
$$
W_I = \left\{ \left. \sum_{i \in I} c_i w_i \in {\mathfrak t}^* \;\right|\; c_i \ge 0 \right\}. $$
\begin{dfn} \label{def:wall} A cone $W_I$ is called a \emph{wall} if
\begin{itemize} \item $W_I \subset {\mathfrak t}^*$ has codimension one, i.e.~the family $(w_i)_{i \in I}$ has rank $(k-1)$. \item The index set $I$ is complete, i.e.~it contains all indices $i$ with $w_i \in W_I$. \end{itemize}
\end{dfn}
The set of non-regular elements in ${\mathfrak t}^*$ is precisely given by the union of all walls. The walls divide ${\mathfrak t}^*$ into open connected components of regular elements, called \emph{chambers}. If one element in a chamber is super-regular then super-regularity holds for all elements in that chamber.
Let $\tau_0 \in {\mathfrak t}^*$ be an element in a wall $W_I$. We assume that any other cone $W_J$ containing $\tau_0$ is completely contained in $W_I$, so that $\tau_0$ does not lie in the intersection of two different walls. Hence $\tau_0$ is contained in the closure of exactly two regular chambers and we want to compute the difference between the Euler classes $\chi^{V,\tau}$ for values of $\tau$ in either of them.
\subsubsection*{A cobordism}
Pick an element $\eta\in{\mathfrak t}^*$ transverse to $W_I$ and define
$$ \tau_t \;:=\; \tau_0 + t\eta. $$
Fix a small enough $\varepsilon > 0$ such that $\tau_t$ is regular for all $t \in \left[ -\varepsilon , \varepsilon \right] \setminus \left\{ 0 \right\}$. Now the set $\left\{ \left( v , t \right) \in V \times \left[ -\varepsilon , \varepsilon \right] \;|\; \mu(v) = \tau_t \right\}$ provides a smooth compact cobordism between the manifolds $\mu^{-1} \left( \tau_{-\varepsilon} \right)$ and $\mu^{-1} \left( \tau_{\varepsilon} \right)$. If the $T$ action on (the $V$ factor of) this cobordism was regular for all times $t$ this would even establish a cobordism of the $T$-moduli problems associated to $\tau_{-\varepsilon}$ and $\tau_{\varepsilon}$ and the Euler classes $\chi^{V,\tau_{\pm\varepsilon}}$ would coincide. But regularity fails because of the wall crossing at $t=0$. So we have to cut out a neighbourhood of the locus with singular action and compute the invariant integral over the newly created boundary component.
To describe the situation around the singular locus we need some preparation. Since $W_I$ has codimension one we can fix a primitive $e_1 \in \Lambda$ such that
\begin{eqnarray*} \left\langle w_i , e_1 \right\rangle & = & 0 \quad \mathrm{for\ all} \; i \in I,\; \mathrm{and} \\ \left\langle \eta , e_1 \right\rangle \, & > & 0. \end{eqnarray*}
Denote by $T_1 \subset T$ the subtorus generated by $e_1$ and by ${\mathfrak t}_1 \subset {\mathfrak t}$ its Lie algebra. The quotient group and its Lie algebra are denoted by
$$ T_0 = T/T_1 \quad \mathrm{and} \quad {\mathfrak t}_0 = {\mathfrak t}/{\mathfrak t}_1 $$
and as before we identify ${\mathfrak t}_0^* \cong \left\{ w \in {\mathfrak t}^* \;|\; \langle w,e_1 \rangle = 0 \right\}$. Now $T_1$ acts trivially on
$$
V^I \;:=\; \left\{ \left( v_1 , \ldots , v_N \right) \in V \;|\; v_\nu = 0\ \mathrm{for}\ \nu \not\in I \right\}. $$
For a small number $\delta > 0$ we define
$$ W_\delta \;:=\; \left( V \times \left[ -\varepsilon , \varepsilon \right] \right) \setminus N_\delta, $$
where $N_\delta$ is the $\delta$-neighbourhood of the set
$$ V^I \times \left[ -\rho , \rho \right] \subset V \times \left[ -\varepsilon , \varepsilon \right] $$
with
$$
\rho \;:=\; \frac{ \pi \cdot \max_{\nu \not\in I} \left| \left\langle w_\nu , e_1 \right\rangle \right| }{ \left\langle \eta , e_1 \right\rangle} \cdot \delta. $$
The reason for this choice will become evident below. We choose $\delta$ small enough such that $N_\delta$ is contained in the interior of $V \times \left[ -\varepsilon , \varepsilon \right]$. Hence $W_\delta$ is a smooth connected oriented manifold with $\mathcal{C}^1$-boundary
$$ \partial W_\delta \;=\; (V \times \{ -\varepsilon \}) \sqcup (V \times \{ \varepsilon \}) \sqcup \partial N_\delta. $$
If we orient $\partial W_\delta$ as the boundary of $W_\delta \subset V \times [ -\varepsilon , \varepsilon ]$ and $\partial N_\delta$ as part of this boundary then
$$ \partial W_\delta \;\cong\; (-V) \sqcup V \sqcup \partial N_\delta. $$
On $W_\delta$ we have the $T$-action on the $V$-factor and we consider the equivariant map
$$ \Phi (v,t) \;:=\; \mu(v) - \tau_t. $$
Denote by $M_\delta := \Phi^{-1}(0)$ the zero set of $\Phi$. Note that $\Phi (v,t) = 0$ implies that
\begin{equation} \label{eqn:t}
t \;=\; \frac{\pi}{ \langle \eta , e_1 \rangle} \sum_{\nu \not\in I} \left| v_\nu \right|^2 \langle w_\nu , e_1 \rangle. \end{equation}
Now on $\partial N_\delta$ we have $\sum_{\nu \not\in I} \left| v_\nu \right|^2 \le \delta$ and hence the intersection of $M_\delta$ with the boundary component $\partial N_\delta$ is contained in the cylindrical part
$$
Z_\delta \;:=\; \left\{ (v,t) \in \partial N_\delta \,|\, |t| \le \rho \right\} = \left\{ v \in V \,\left|\, \sum_{\nu \not\in I} \left| v_\nu \right|^2 = \delta \right. \right\} \times [ -\rho,\rho ]. $$
We introduce the notation
$$
V_\delta^I \;:=\; \left\{ v \in V \,\left|\, \sum_{\nu \not\in I} \left| v_\nu \right|^2 = \delta \right. \right\} $$
and
$$
S_\delta \;:=\; \left\{ (v_\nu)_{\nu \not\in I} \,\left|\, \sum_{\nu \not\in I} \left| v_\nu \right|^2 = \delta \right. \right\} $$
and observe that $Z_\delta = V_\delta^I \times [ -\rho , \rho ]$ and $V_\delta^I \cong V^I \times S_\delta$.
\begin{prop}
For all $\eta$ in a dense and open subset of ${\mathfrak t}^*$ the value $0 \in {\mathfrak t}^*$ is regular for $\Phi : W_\delta \longrightarrow {\mathfrak t}^*$ and also for $\Phi|_{\partial W_\delta}$. The $T$-action on $\Phi^{-1}(0) = M_\delta$ is regular. \end{prop}
\begin{proof}
Pick $(v,t) \in M_\delta$. For $t \ne 0$ the element $\tau_t$ is a regular value for the moment map $\mu : V \longrightarrow {\mathfrak t}^*$ and hence $\mathrm{d}\Phi_{(v,t)}$ is onto. This also holds for the differential of the restriction of $\Phi$ to the boundary components $V \times \{-\varepsilon\}$ and $V \times \{\varepsilon\}$. For $t = 0$ we have $\mu(v) = \tau_0$. Now the image of $\mathrm{d}\Phi_{(v,t)}$ is spanned by $\eta$ and the collection $(w_j)_{j \in J}$ with $J := \left\{ j \in \{ 1 , \ldots , N \} \,|\, v_j \ne 0 \right\}$. Suppose that $(w_j)_{j \in J}$ does not already span all of ${\mathfrak t}^*$. Then $\tau_0 \in W_J$ implies that $(w_j)_{j \in J \cap I}$ has rank $k-1$. Otherwise $\tau_0$ would be contained in two different walls, which we assumed not to be the case. So since $\eta$ is transversal to $W_I$ we also get surjectivity of $\mathrm{d}\Phi_{(v,t)}$. So for the first statement of the proposition it remains to show regularity for $\Phi_{|\partial N_\delta}$ and in fact only for $\Phi|_{Z_\delta}$ since $M_\delta \cap \partial N_\delta \subset Z_\delta$ as we remarked above. We define
$$
\varphi \;:=\; \Phi|_{Z_\delta} : Z_\delta \longrightarrow {\mathfrak t}^*. $$
Let $(v,t) \in Z_\delta$. The tangent space $T_{(v,t)} Z_\delta$ is given by pairs $(x,s) \in V \times \mathbbm R$ such that $\sum_{\nu \not\in I} g( v_\nu , x_\nu ) = 0$. Now
$$ \mathrm{d}\varphi_{(v,t)}(x,s) \;=\; 2 \pi \sum_{\nu = 1}^N g( v_\nu , x_\nu ) \cdot w_\nu - s \eta. $$
So with $J := \left\{ j \in \{ 1 , \ldots , N \} \,|\, v_j \ne 0 \right\}$ the image of $\mathrm{d}\varphi_{(v,t)}$ is given by
$$
\mathcal{I} \;:=\; \left\{ \left. \sum_{j \in J} c_j w_j - s \eta \in {\mathfrak t}^* \,\right|\, \sum_{j \in J \setminus I} c_j = 0, \; c_j , s \in \mathbbm R \right\}. $$
Next we note that $(w_j)_{j \in J}$ spans ${\mathfrak t}^*$: For $t \ne 0$ this follows by regularity of $\tau_t$. For $t = 0$ we see as above that $(w_j)_{j \in J \cap I}$ has rank $k-1$. But now $\sum_{\nu \not\in I} \left| v_\nu \right|^2 = \delta$ implies that there is an index $l \in J \setminus I$ and hence $w_l$ spans the remaining dimension.
Given any $\tau \in {\mathfrak t}^*$ we thus find numbers $d_j \in \mathbbm R$ with $\sum_{j \in J} d_j w_j = \tau$. In case $d := \sum_{j \in J \setminus I} d_j = 0$ this would already prove $\tau \in \mathcal{I}$. Else we pick $e_j \in \mathbbm R$ with $\sum_{j \in J} e_j w_j = \eta$. Then if $e := \sum_{j \in J \setminus I} e_j \ne 0$ we can set
$$ c_j \;:=\; d_j - \frac{d}{e} e_j \quad , \quad s \;:=\; - \frac{d}{e} $$
to obtain $\tau \in \mathcal{I}$. So for proving $\mathcal{I} = {\mathfrak t}^*$ it suffices to ensure by a suitable choice of $\eta$ that the equation $\sum_{j \in J} e_j w_j = \eta$ has at least one solution with $\sum_{j \in J \setminus I} e_j \ne 0$. Set
$$
\mathcal{J} \;:=\; \left\{ J_0 \subset \{ 1 , \ldots , N \} \,|\, (w_j)_{j \in J_0} \ \mbox{is a basis for} \ {\mathfrak t}^* \right\}. $$
For every $J_0 \in \mathcal{J}$ we introduce
$$
E(J_0) \;:=\; \left\{ \sum_{j \in J_0} e_j w_j \,\left|\, \sum_{j \in J_0 \setminus I} e_j = 0 \right. \right\} \subset {\mathfrak t}^* $$
and we choose
$$ \eta \in {\mathfrak t}^* \setminus \left( \mathrm{span}(w_i)_{i \in I} \cup \bigcup_{J_0 \in \mathcal{J}} E(J_0) \right). $$
This ensures that $\eta$ is indeed transversal to $W_I$. And to obtain the needed solution of $\sum_{j \in J} e_j w_j = \eta$ we pick a $J_0 \in \mathcal{J}$ with $J_0 \subset J$. We then get a unique solution of $\sum_{j \in J_0} e_j w_j = \eta$ and setting all $e_j$ for $j \in J \setminus J_0$ equal to zero we get the desired solution of $\sum_{j \in J} e_j w_j = \eta$ with $\sum_{j \in J \setminus I} e_j \ne 0$. This finishes the proof of the first statement in the proposition.
The isotropy subgroups of the $T$-action at points $(v,t) \in M_\delta$ with $t \ne 0$ are finite by regularity of $\tau_t$. Now suppose $(v,0) \in M_\delta$ is fixed by a whole $1$-parameter family $\{ \exp(te) \in T \,|\, t \in \mathbbm R \}$ for some $e \in \Lambda \setminus \{ 0 \}$. Again we set $J := \left\{ j \in \{ 1 , \ldots , N \} \,|\, v_j \ne 0 \right\}$ and observe
$$ j \in J \quad \iff \quad \langle w_j , e \rangle = 0. $$
But similarly as above we see that $(w_j)_{j \in J} $ spans all of ${\mathfrak t}^*$, which would imply $e = 0$. This gives the desired contradiction. \end{proof}
So if we pick a generic direction $\eta$ for the wall crossing we get a smooth cobordism $M_\delta$ with regular $T$-action. If we fix an orientation of ${\mathfrak t}^*$ we get induced orientations on $\mu^{-1}(\tau_{\pm \varepsilon})$ and $M_\delta$. We equip the boundary components of $M_\delta$ with the induced boundary orientations and obtain
$$ \partial M_\delta \;\cong\; (-\mu^{-1}(\tau_{-\varepsilon})) \sqcup \mu^{-1}(\tau_\varepsilon) \sqcup \varphi^{-1}(0). $$
By Stokes' formula for invariant integration we thus obtain the following result.
\begin{prop} \label{prop:wall_crossing_1} If we orient $\varphi^{-1}(0)$ as part of the boundary of $M_\delta$ then
\begin{equation} \label{eqn:wall_crossing_1} \chi^{V,\tau_\varepsilon} - \chi^{V,\tau_{-\varepsilon}} \;=\; - \int_{\varphi^{-1}(0) / T} \pi^* \;:\; H_T^*(V) \cong S({\mathfrak t}^*) \longrightarrow \mathbbm R. \end{equation}
where $\pi : Z_\delta \longrightarrow V$ denotes the projection and integration over $\varphi^{-1}(0) / T$ is understood as $T$-invariant integration. \end{prop}
\begin{rem} This wall crossing formula also holds for non-generic $\eta$ if we use the cobordism property for Euler classes and interprete the integral on the right hand side as the Euler class of a regular $T$-moduli problem over $Z_\delta$. \end{rem}
\subsubsection*{The reduced problem}
The task is now to put the above formula for the wall crossing into a more computable shape. In fact it is possible to deform the right hand side of \ref{eqn:wall_crossing_1} into the Euler class of a reduced toric moduli problem. Note that the $(w_i)_{i \in I}$ and $\tau_0$ can be viewed as elements in ${\mathfrak t}_0^*$ since they vanish on $e_1$.
\begin{lem} The element $\tau_0 \in {\mathfrak t}_0^*$ is regular for the collection of characters of $T_0$ given by the $(w_i)_{i \in I}$. \end{lem}
\begin{proof} Given an index set $J \subset I$ and numbers $a_j > 0$ with $\sum_{j \in J} a_j w_j = \tau_0$ we observe that the $(w_j)_{j \in J}$ must have rank $k-1$ because otherwise $\tau_0$ would be contained in two different walls, which we assumed not to be the case. \end{proof}
But in general $\tau_0$ will not be super-regular for the action of the reduced torus $T_0$. We introduce
\begin{eqnarray*} \bar{w}_\nu & := & w_\nu - \frac{ \left\langle w_\nu , e_1 \right\rangle }{ \left\langle \eta , e_1 \right\rangle } \cdot \eta \\
\mu_0(v) & := & \pi \sum_{i \in I} \left| v_i \right|^2 \cdot w_i \\
R(v) & := & \pi \sum_{\nu \not\in I} \left| v_\nu \right|^2 \cdot \bar{w}_\nu \\ \varphi_s(v) & := & \mu_0(v) - \tau_0 + s \cdot R(v) , \quad \mathrm{for} \quad s \in [0,1] \end{eqnarray*}
and observe that $\langle \varphi_s(v) , e_1 \rangle = 0$ for all $v$ and $s$. Hence we can --- and always will --- consider $\varphi_s$ as a map to ${\mathfrak t}_0^*$.
\begin{lem} \label{lem:phi_phi_0_diffeo} $\varphi^{-1}(0)$ is $T$-equivariantly diffeomorphic to $\varphi_1^{-1}(0)$. \end{lem}
\begin{proof} The projection $\pi : Z_\delta = V_\delta^I \times [ -\delta , \delta ] \longrightarrow V_\delta^I$ restricts to a diffeomorphism on $\varphi^{-1}(0) \subset Z_\delta$ because the inverse is explicitly given by
$$
v = (v_\nu) \;\longmapsto\; \left( (v_\nu) \quad,\quad t = \frac{\pi}{\langle \eta,e_1 \rangle} \sum_{\nu \not\in I} \left| v_\nu \right|^2 \langle w_\nu,e_1 \rangle \right) $$
as can be seen from equation \ref{eqn:t}. \end{proof}
Now $T_1$ by construction acts trivially on $V^I$. Hence the $T$-action on $V$ descends to a $T_0$-action on $V^I$ that is given by the weight vectors $(w_i)_{i \in I}$. Furthermore $\mu_0$ is a moment map for this $T_0$-action on $V^I$ and $\tau_0$ is a regular level. If we restrict to those $v \in V$ with $\sum_{\nu \not\in I} \left| v_\nu \right|^2 = \delta$ and choose $\delta$ small enough then $R(v)$ is as small as we want and hence also $\tau_0 - s \cdot R(v)$ will be regular for the $T_0$-action for all $s \in [0,1]$. So if we consider $\varphi_s$ as a map
$$ \varphi_s : V_\delta^I \cong V^I \times S_\delta \longrightarrow {\mathfrak t}_0^* $$
we get that $0 \in {\mathfrak t}_0^*$ is a regular value of $\varphi_s$ for all $s \in [0,1]$. This shows
$$ \int_{\varphi_0^{-1}(0) / T} \;=\; \int_{\varphi_1^{-1}(0) / T} \;:\; H_T^*(V_\delta^I) \longrightarrow \mathbbm R. $$
Together with the preceding lemma we obtain the following refinement of proposition \ref{prop:wall_crossing_1}:
\begin{cor} \label{cor:wall_crossing_2} We orient $V_\delta^I \cong V^I \times S_\delta$ by the complex orientation on $V^I$ and the standard orientation (via taking the outward pointing vector first) of the sphere $S_\delta$. We orient ${\mathfrak t}_0^*$ via the fixed orientation of ${\mathfrak t}^*$ as the kernel of the map ${e_1}^* : {\mathfrak t}^* \longrightarrow \mathbbm R$. With the induced orientation on $\varphi_0^{-1}(0)$ we obtain
\begin{equation} \label{eqn:wall_crossing_2} \chi^{V,\tau_\varepsilon} - \chi^{V,\tau_{-\varepsilon}} \;=\; \int_{\varphi_0^{-1}(0) / T} i^* \;:\; H_T^*(V) \cong S({\mathfrak t}^*) \longrightarrow \mathbbm R. \end{equation}
with the inclusion $i : V_\delta^I \longrightarrow V$. \end{cor}
\begin{proof} We loose the minus sign from formula \ref{eqn:wall_crossing_1} by a change of orientation. All the rest is clear. \end{proof}
\begin{rem} Without the genericity assumption on the direction $\eta$ one shows more generally that the $\varphi_s$ for $s \in [0,1]$ define a homotopy of $T$-moduli problems. And lemma \ref{lem:phi_phi_0_diffeo} can be rephrased as a morphism between the moduli problems associated to $\varphi$ on $Z_\delta$ and $\varphi_1$ on $V_\delta^I$. The above result then remains true if we interprete the right hand side of \ref{eqn:wall_crossing_2} as the Euler class of the $T$-moduli problem associated to $\varphi_0$ on $V_\delta^I$ as explained below. \end{rem}
Observe that $\varphi_0 : V_\delta^I \longrightarrow {\mathfrak t}_0^*$ defines a fibered $T$-moduli problem of the kind that we considered in section \ref{sec:fibred_G_moduli_problems}: We have the reduced $T / T_1 = T_0$-moduli problem on $V^I$ given by the section $\varphi_0|_{V^I} \;:\; V^I \longrightarrow {\mathfrak t}_0^*$. This is $T_0$-regular, because $\tau_0$ as an element of ${\mathfrak t}_0^*$ is regular for the weights $w_i \in {\mathfrak t}_0^*$ for $i \in I$. We write $\chi^{V^I,\tau_0}$ for the associated Euler class. The pull-back of the moduli problem under the projection
$$ \pi \;:\; V_\delta^I \;\cong\; V^I \times S_\delta \;\longrightarrow\; V^I $$
yields the $T$-regular moduli problem $\left( V_\delta^I , V_\delta^I \times {\mathfrak t}_0^* , \varphi_0 \right)$, since $\varphi_0$ does not depend on the point in $S_\delta$ and $T_1$ acts locally freely. Hence by proposition \ref{prop:fibred_Euler_class} we can compute the right hand side of \ref{eqn:wall_crossing_2} by composing $\chi^{V^I,\tau_0}$ with invariant push-forward $\left( \pi / T_1 \right)_*$. So we finally obtain
\begin{equation*} \chi^{V,\tau_\varepsilon} - \chi^{V,\tau_{-\varepsilon}} \;=\; \chi^{V^I,\tau_0} \circ \left( \pi / T_1 \right)_* \circ i^* \;:\; S({\mathfrak t}^*) \longrightarrow \mathbbm R. \end{equation*}
With the genericity assumption on the direction $\eta$ this also follows immediately from the functoriality property $(4)$ in Proposition \ref{prop:invariant_integration} without referring to fibered moduli problems.
\subsubsection*{Localization}
Recall that we derived an explicit formula for the $T_1$-invariant integration $\left( \pi / T_1 \right)_* \circ i^* \;:\; S({\mathfrak t}^*) \longrightarrow S({\mathfrak t}_0^*)$ over an odd-dimensional sphere like $S_\delta$ in the localization example in section \ref{sec:example_localization}. Note that the weights for the $T$-action on $S_\delta$ are those $w_\nu$ with $\nu \not\in I$. Since the collection of the $w_\nu$ is assumed to be proper no two weights are negative multiples of each other. Also recall that every weight $w_\nu$ appears with multiplicity $n_\nu$ since $\dim_\mathbbm C V_\nu = n_\nu$. So using proposition \ref{prop:example_localization} we can summarize and reformulate the wall crossing formula as follows.
\begin{thm} \label{thm:wall_crossing} Let $\mathbbm{x} = \prod_{j = 1}^m x_j \in S^*({\mathfrak t}^*)$. Then with the notation and conventions from above we have
$$ \chi^{V,\tau_\varepsilon} (\mathbbm{x}) - \chi^{V,\tau_{-\varepsilon}} (\mathbbm{x}) \;=\; \chi^{V^I,\tau_0} (\mathbbm{x}_0) $$
with
$$ \mathbbm{x}_0 (\xi) \;:=\; \frac{1}{2 \pi i} \oint \frac{ \prod_{j = 1}^m \langle x_j , \xi + z e_1 \rangle }{ \prod_{\nu \not\in I} \langle w_j , \xi + z e_1 \rangle^{n_\nu} } \; \mathrm{dz} \; . $$
For every $\xi$ the integral is around a circle in the complex plane enclosing all poles of the integrand. \end{thm}
\subsection{Jeffrey-Kirwan localization}
\label{chap:Jeffrey_Kirwan_localization}
The preceding discussion on wall crossing and the computation of integrals over toric manifolds can be put into a much broader context. We will briefly outline the general theory. Suppose the torus $T$ acts on the compact manifold with boundary $M$ such that the action on the boundary $\partial M$ is regular. Let $\alpha$ be a $\mathrm{d}_T$-closed form on $M$. If the torus action was regular on all of $M$ then by Stokes' formula
$$ \int_{\partial M / T} \alpha \;=\; \int_{M / T} \mathrm{d} \alpha \;=\; 0. $$
But $T$-invariant integration over $M$ is not defined as soon as there are points $p \in M$ with isotropy subgroups of positive dimension. As in section \ref{sec:torus_actions} we denote the isotropy subgroups by $S_p$ and their Lie algebras by ${\mathfrak t}_p$. Consider the set
$$
Z \;:=\; \left\{ p \in M \;|\; \dim( {\mathfrak t}_p ) \ge 1 \right\}. $$
This set is $T$-invariant, but in general there is no reason why $Z$ should be a submanifold of $M$. But let us for the moment assume that
$$ Z \;=\; \bigsqcup_\nu Z_\nu $$
is the disjoint union of connected and $T$-invariant submanifolds $Z_\nu$ such that all $p \in Z_\nu$ have isotropy $S_p = S_\nu$ for some fixed subgroup $S_\nu \subset T$.
We pick sufficiently small disjoint tubular neighbourhoods $U_\nu$ of the $Z_\nu$ and equivariantly identify the boundary of $U_\nu$ with the unit sphere bundle $SN_\nu$ in the normal bundle $N_\nu$ of $Z_\nu$ in $M$ with respect to an invariant metric. Now we obtain
\begin{eqnarray*} & \displaystyle \int_{\partial \left( M \setminus \sqcup_\nu U_\nu \right) / T} \alpha & \;=\; \quad \quad 0 \\ \iff \quad & \displaystyle \int_{\partial M / T} \alpha & \;=\; \quad \sum_\nu \int_{SN_\nu / T} \alpha. \end{eqnarray*}
If we denote the projection $SN_\nu \longrightarrow Z_\nu$ by $\pi_\nu$ then we are precisely in the situation required to apply proposition \ref{prop:invariant_integration}: The group $S_\nu$ acts locally freely on the sphere bundle $SN_\nu$ but trivially on the base $Z_\nu$. If we introduce the notation
$$ T_\nu \;:=\; T / S_\nu $$
then we obtain
\begin{equation*} \displaystyle \int_{\partial M / T} \alpha \;=\; \sum_\nu \int_{Z_\nu / T_\nu} \left( \pi_\nu / S_\nu \right)_* \alpha. \end{equation*}
This is the basic identity from which the Jeffrey-Kirwan localization formulae of Guillemin and Kalkman \cite{GK} and Martin \cite{Mar} and our results in section \ref{chap:toric_manifolds} can be deduced. In all three cases the manifold $M$ is obtained as the preimage of a certain path $\tau_t$ in ${\mathfrak t}^*$ under the moment map $\mu : X \longrightarrow {\mathfrak t}^*$ on some Hamiltonian $T$-space $X$ such that $\tau_t$ connects the point of reduction $\tau = \tau_0$ with the complement of the image of $\mu$. Hence
$$ \partial M / T \;=\; \mu^{-1}(\tau) / T \;=\; X /\!/ T (\tau) $$
is the symplectic quotient of $X$. Compactness of $M$ is for example given if the moment map is proper.
The first step now is to ensure that the above assumption on the set $Z$ is satisfied. This is done by a suitable choice for the path $\tau_t$. In our case we have to demand that the path does not hit the intersection of two walls. Then the components $Z_\nu$ are the fixed point sets of certain one-dimensional subtori $S_\nu \subset T$. This can in fact be arranged for any Hamiltonian $T$-space $X$ (see Guillemin and Kalkman \cite{GK}).
Next one has to compute $\left( \pi_\nu / S_\nu \right)_* \alpha$. We obtain the corresponding formula in \ref{thm:wall_crossing} by applying our relative version of the usual Atiyah-Bott localization formula. This also works in the context studied in \cite{GK} and gives an alternative explanation for their \emph{residue operations}. In addition we can avoid to work with orbifolds.
Finally one has to iterate this procedure for the $T_\nu$-invariant integrals over the $Z_\nu$. Indeed our detailed exposition in section \ref{sec:toric_wall_crossing} is mainly in order to show that these \emph{reduced problems} are in fact again of the same form as the initial one and the combinatorial data can immediately be read off. This explicit correspondence has no analogue in the more general setting of Guillemin and Kalkman \cite{GK} and is the reason for why one can actually evaluate these iterated residues in the toric case, as it is done by Cieliebak and Salamon \cite{CS}.
\section{Vortex invariants}
\label{chap:vortex_invariants}
We now turn to vortex equations and the computation of vortex invariants for toric manifolds. First we summarize the usual vortex setup in the case of a linear torus action on $\mathbbm C^N$. We then deform these equations as described in the introduction and prove that this deformation yields a homotopy of regular $T$-moduli problems. Then we study the deformed picture and gather consequences.
Throughout we freely use the notion and properties of moduli problems and associated Euler classes from section \ref{chap:moduli_problems}. For the definitions and facts about the Sobolev spaces that we use we refer to K.~Wehrheim \cite[Appendix B]{KW}. The gauge-theoretic results that we use originate in the work of Uhlenbeck \cite{Uhl} and of Donaldson and Kronheimer \cite{DK}, but we also use the book \cite{KW} for references.
\subsection{Setup}
\label{sec:vortex_setup}
Let $T$ be a $k$-dimensional torus. Using the notation from section \ref{chap:toric_manifolds} we consider a torus action $\rho$ on $\mathbbm C^N$ that is given by $N$ weight vectors $w_\nu\in\Lambda^*$. We assume that this collection of weights is proper and spans ${\mathfrak t}^*$ so that we get non-empty regular quotients $X_{\mathbbm C^N,\tau}$ for regular elements $\tau$ in the image of the moment map
$$
\mu : \mathbbm C^N \longrightarrow {\mathfrak t}^* \; ; \; z \longmapsto \pi \sum_{\nu = 1}^N \left| z_\nu \right|^2 \cdot w_\nu. $$
We fix an element $\kappa \in \Lambda \cong H_2(\mathrm{BT};\mathbbm Z)$ and a principal $T$-bundle $\pi : P \longrightarrow \Sigma$ over a connected Riemann surface $\Sigma$ with characteristic vector $\kappa$. Explicitly we take $P := f^*\mathrm{ET}$ with a classifying map $f : \Sigma \longrightarrow \mathrm{BT}$ satisfying $f_*[\Sigma] = \kappa$. Since $T$ is connected $\mathrm{BT}$ is $1$-connected. Hence up to homotopy $f$ is uniquely determined by this property. Thus $P$ is uniquely determined by the choice for $\kappa$.
A $T$-equivariant map $u : P \longrightarrow \mathbbm C^N$ is the same as a section in the Hermitian bundle $\mathcal{V} := P \times_\rho \mathbbm C^N$ or a collection of sections $(u_1 , \ldots , u_N)$ in the line bundles $\mathcal{L}_\nu := P \times_{\rho_\nu} \mathbbm C$ associated to the actions $\rho_\nu$ of $T$ on $\mathbbm C$ given by the weights $w_\nu$. We use the convention that principal bundles carry right actions, hence we let $g \in T$ act on $P \times \mathbbm C$ by
$$ g(x,z) := \left( xg^{-1} , \rho_\nu(g)z \right) $$
and equivariance of $u : P \longrightarrow \mathbbm C^N$ means that $u(xg^{-1}) = \rho(g)u(x)$. So for $p = \pi(x) \in \Sigma$ we set
$$ u_\nu(p) := \left[ x , \pi_\nu \circ u(x) \right] \in \mathcal{L}_\nu $$
with the projection $\pi_\nu : \mathbbm C^N \longrightarrow \mathbbm C$ onto the $\nu$-th component. This yields a well defined section of $\mathcal{L}_\nu$ since $\pi_\nu \circ \rho(g) = \rho_\nu(g) \circ \pi_\nu$. By construction the degree of $\mathcal{L}_\nu$ is given by $d_\nu := \left\langle w_\nu , \kappa \right\rangle$ and $\mathcal{V} = \bigoplus_{\nu = 1}^N \mathcal{L}_\nu$.
Let $A \in \Omega^1(P,{\mathfrak t})$ be a connection form on $P$ and $u$ an equivariant map as above. The symplectic vortex equations for the pair $(u,A)$ at a parameter $\tau \in {\mathfrak t}^*$ now take the form
$$ (*) \quad \left\{ \quad \begin{array}{lr@{\quad = \quad}l} (\mathrm{I}) & \bar{\partial}_A u & 0 \\ (\mathrm{II}) & *F_A + \mu(u) & \frac{\kappa}{\mathrm{vol}(\Sigma)} + \tau. \end{array} \right. $$
In the first equation we have the linear Cauchy-Riemann operator $\bar{\partial}_A$ on $\mathcal{V}$ that is associated to the covariant derivative $\mathrm{d}_A u := \mathrm{d}u - X_A(u)$. It is given by
$$ \bar{\partial}_A u = \mathrm{d}_A u + i \circ \mathrm{d}_A u \circ j_\Sigma $$
with a fixed complex structure $j_\Sigma$ on $\Sigma$ and the standard complex structure $i$ on $\mathbbm C^N$. For the second equation we also fix a metric (respectively a volume form $\mathrm{dvol}_\Sigma$) on $\Sigma$ to get the Hodge $*$-operator and the volume $\mathrm{vol}(\Sigma)$ and we also have to choose an inner product on the Lie algebra ${\mathfrak t}$ to identify it with its dual space ${\mathfrak t}^*$. Note that in general the curvature $F_A$ is a two-form on $\Sigma$ with values in the bundle $P \times_\mathrm{Ad} {\mathfrak t}$, which in this case is trivial since $T$ is abelian. Hence $*F_A$ is just a map $\Sigma \longrightarrow {\mathfrak t}$. And by $T$-invariance of the moment map $\mu$ the composition $\mu(u)$ also descends to a map $\Sigma \longrightarrow {\mathfrak t}$.
We consider the gauge group $\mathcal{G}(P)$ of the principal bundle $P$ and fix a point $x_0 \in P$ to get the based gauge group
$$
\mathcal{G}_0(P) := \left\{ \gamma \in \mathcal{G}(P) \;|\; \gamma(x_0) = e \right\}. $$
An element $\gamma \in \mathcal{G}(P)$ is a smooth $T$-invariant map $\gamma : P \longrightarrow T$ that acts on pairs $(u,A)$ by $\gamma(u,A) := ( \gamma^*u , \gamma^*A )$ with
$$ \left( \gamma^*u \right) (x) := u\left( x\gamma(x) \right) = \rho\left( \gamma(x)^{-1} \right) u(x) $$
and for $v \in T_xP$
$$ \left( \gamma^*A \right)_x(v) := A_x(v) + {\gamma(x)^{-1}}_* \left( \mathrm{d}\gamma \right)_x (v), $$
in short $\gamma^*A = A + \gamma^{-1} \mathrm{d}\gamma$. Now the based gauge group $\mathcal{G}_0(P)$ acts freely on the configuration space of pairs $(u,A)$. In fact $\gamma^*A = A$ implies that $\gamma : P \longrightarrow T$ is a constant map and since $P$ is connected the condition $\gamma(x_0) = e \in T$ for based gauge transformations $\gamma \in \mathcal{G}_0(P)$ implies that $\gamma(x) = e$ for all $x \in P$. Using suitable Sobolev completions the quotient of this action becomes a smooth Hilbert manifold $\mathcal{B}$. Explicitly we consider
\begin{equation} \label{def:mcB} \mathcal{B} := \frac{ W^{k,2} \left( \Sigma , \mathcal{V} \right) \times \mathcal{A}^{k,2}(P) }{ \mathcal{G}_0^{k+1,2}(P) }. \end{equation}
By the local slice theorem (see K.~Wehrheim \cite[Theorem 8.1]{KW}) the quotient $\mathcal{A}^{k,2}(P) / \mathcal{G}_0^{k+1,2}(P)$ is a smooth Hilbert manifold and the projection from $\mathcal{B}$ onto the connection-part makes $\mathcal{B}$ to a Hilbert space bundle over this quotient. Taking $k \ge 3$ ensures that all objects are $\mathcal{C}^2$. In the end it does not matter which $k \ge 3$ we take.
The map $\mathcal{G}(P) / \mathcal{G}_0(P) \longrightarrow T \;;\; [\gamma] \longmapsto \gamma(x_0)$ is a diffeomorphism. Hence the action of the whole gauge group $\mathcal{G}(P)$ descends to a $T$-action on the quotient $\mathcal{B}$ given by
$$ g \left[ u , A \right] := \left[ \rho\left( g^{-1} \right) u , A \right] \quad \mathrm{with} \quad g \in T , \left[ u , A \right] \in \mathcal{B}. $$
Here we use that for an abelian Lie group the gauge group actually splits into the product $T \times \mathcal{G}_0(P)$ and that the constant gauge transformations $g \in T \subset \mathcal{G}(P)$ act trivially on connections.
The equations $(*)$ now give rise to a $T$-moduli problem with base $\mathcal{B}$ and total space
\begin{equation} \label{def:mcE} \mathcal{E} := \frac{ W^{k,2} \left( \Sigma , \mathcal{V} \right) \times \mathcal{A}^{k,2}(P) \times \mathcal{F} }{ \mathcal{G}_0^{k+1,2}(P) } \end{equation}
with fiber
\begin{equation} \label{def:mcF} \mathcal{F} := W^{k-1,2} \left( \Sigma , \Lambda^{0,1} T^*\Sigma \otimes_\mathbbm C \mathcal{V} \right) \oplus W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) \end{equation}
and section
$$
\mathcal{S} : \mathcal{B} \longrightarrow \mathcal{E} \; ; \; \left[ u , A \right] \longmapsto \left[ u , A , \left( \begin{array}{c} \bar{\partial}_A u \\ *F_A + \mu(u) - \frac{\kappa}{\mathrm{vol}(\Sigma)} - \tau \end{array} \right) \right].
$$
On the fiber $\mathcal{F}$ the gauge group $\mathcal{G}(P)$ acts only on sections in the bundle $\mathcal{V}$. Thus a short computation shows that the section $\mathcal{S}$ is $T$-equivariant and well defined, i.e.~it descends to the $\mathcal{G}_0(P)$-quotients.
The real index of this moduli problem ist $(N-k) \cdot \chi(\Sigma) + 2\sum_{\nu = 1}^N d_\nu$ and it is regular if $\tau$ is regular. For the computation of the index, the regularity and the compactness properties we refer to Cieliebak, Gaio, Mundet and Salamon \cite{CGMS}. The definition of the orientation on $\det(\mathcal{S})$ will be discussed together with all further orientation issues in section \ref{sec:orientations}. The Euler class of this $T$-moduli problem then gives rise to the vortex invariant. We denote it by
$$ \Psi^{\rho,\tau}_{\kappa,\Sigma} : S^*({\mathfrak t}^*) \otimes H^*(\mathcal{A}/\mathcal{G}_0) \longrightarrow \mathbbm R. $$
In case $\Sigma = S^2$ has genus zero we omit the surface and just write $\Psi^{\rho,\tau}_\kappa$. This map is constructed as follows: Elements in $S^*({\mathfrak t}^*) \otimes H^*(\mathcal{A}/\mathcal{G}_0)$ are pulled back to $H_T^*(\mathcal{B})$ via the equivariant evaluation map $\mathcal{B} \longrightarrow \mathbbm C^N$; $[u,A] \longmapsto u(p_0)$ and the projection $\mathcal{B} \longrightarrow \mathcal{A}/\mathcal{G}_0$, and are then integrated $T$-invariantly over the compact space of solutions $\mathcal{M} := \mathcal{S}^{-1}(0)$. If the space $\mathcal{M}$ is not cut out transversally this integration is understood in terms of the Euler class.
\begin{rem} In case of genus zero the space $\mathcal{A}(P)/\mathcal{G}_0(P)$ is in fact contractible. This follows for example from lemma \ref{lem:jacobian_torus} below. Hence the vortex invariant reduces to a map
$$ \Psi^{\rho,\tau}_\kappa : S^*({\mathfrak t}^*) \longrightarrow \mathbbm R. $$
\end{rem}
\subsection{Deformation}
\label{sec:vortex_deformation}
The objects in the curvature equation $(\mathrm{II})$ of $(*)$ are considered as elements of $W^{k-1,2}(\Sigma,{\mathfrak t})$. Using the fixed volume form $\mathrm{dvol}_\Sigma$ we can split this vector space into the direct sum
$$ W^{k-1,2}(\Sigma,{\mathfrak t}) = \underline{ W^{k-1,2}(\Sigma,{\mathfrak t}) } \oplus {\mathfrak t} $$
with
$$
\underline{ W^{k-1,2}(\Sigma,{\mathfrak t}) } := \left\{ F \in W^{k-1,2}(\Sigma,{\mathfrak t}) \;|\; \int_\Sigma F \cdot \mathrm{dvol}_\Sigma = 0 \right\}, $$
the space of maps with mean value equal to zero. For any map $F : \Sigma \longrightarrow {\mathfrak t}$ we denote by
$$ \overline{F} := \frac{ \int_\Sigma F \cdot \mathrm{dvol}_\Sigma }{ \mathrm{vol}(\Sigma) } \in {\mathfrak t} $$
its mean value and by
$$ \underline{F} := F - \overline{F} $$
its projection onto the space of maps of zero mean value. Observe that by Chern-Weil theory we have
$$ \overline{*F_A} = \frac{ \int_\Sigma *F_A \cdot \mathrm{dvol}_\Sigma }{ \mathrm{vol}(\Sigma) } = \frac{\kappa}{\mathrm{vol}(\Sigma)} $$
for any connection $A$. For later use we also compute
\begin{equation} \label{eqn:overline_mu}
\overline{\mu(u)} = \frac{\pi}{\mathrm{vol}(\Sigma)} \cdot \sum_{\nu = 1}^N \left\| u_\nu \right\|^2_{L^2(\Sigma,\mathcal{L}_\nu)} \cdot w_\nu. \end{equation}
In this splitting the vortex equations become
$$ (*) \quad \left\{ \quad \begin{array}{lr@{\quad = \quad}l} (\mathrm{I}) & \bar{\partial}_A u & 0 \\ (\mathrm{II}) & *F_A + \mu(u) & \frac{\kappa}{\mathrm{vol}(\Sigma)} + \overline{\mu(u)} \\ (\mathrm{II\/I}) & \overline{\mu(u)} & \tau. \end{array} \right. $$
We regroup and introduce a parameter $\varepsilon \in [0,1]$ in the second equation to obtain
$$ (*)_\varepsilon \quad \left\{ \quad \begin{array}{lr@{\quad = \quad}l} (\mathrm{I}) & \bar{\partial}_A u & 0 \\ (\mathrm{II})_\varepsilon & *F_A - \frac{\kappa}{\mathrm{vol}(\Sigma)} & \varepsilon \left( \overline{\mu(u)} - \mu(u) \right) \\ (\mathrm{II\/I}) & \overline{\mu(u)} & \tau. \end{array} \right. $$
\begin{rem} \label{rem:epsilon_omega} One can interprete the equations $(*)_\varepsilon$ as the vortex equations for the same Hamiltonian group action, but with rescaled symplectic form $\varepsilon \cdot \omega$ on the target manifold (hence the moment map for the action gets multiplied by $\varepsilon$) and with rescaled parameter $\varepsilon \cdot \tau$. So as long as $\varepsilon \ne 0$ nothing particular will happen. \end{rem}
We claim, and will prove it in the remainder of this section, that the deformed equations with $\varepsilon = 0$ yield the same invariants as the usual vortex equations. We consider the Hilbert space bundle $[0,1] \times \mathcal{E} \longrightarrow [0,1] \times \mathcal{B}$ with $\mathcal{B}$ as in \ref{def:mcB} and $\mathcal{E}$ as in \ref{def:mcE}, but we write the fiber from \ref{def:mcF} as
$$ \mathcal{F} := W^{k-1,2} \left( \Sigma , \Lambda^{0,1} T^*\Sigma \otimes_\mathbbm C \mathcal{V} \right) \oplus \underline{W^{k-1,2} \left( \Sigma , {\mathfrak t} \right)} \oplus {\mathfrak t}. $$
In this bundle the equations $(*)_\varepsilon$ give rise to the section
\begin{eqnarray*} \mathcal{S} : [0,1] \times \mathcal{B} & \longrightarrow & [0,1] \times \mathcal{E} \\ \left( \varepsilon , \left[ u , A \right] \right) & \longmapsto & \left( \varepsilon , \left[ u , A , \left( \begin{array}{c}
\bar{\partial}_A u \\ *F_A - \frac{\kappa}{\mathrm{vol}(\Sigma)} + \varepsilon \cdot \mu(u) - \varepsilon \cdot \overline{\mu(u)} \\ \overline{\mu(u)} - \tau \end{array} \right) \right] \right). \end{eqnarray*}
Our claim now follows from the following theorem in combination with the cobordism property of the Euler class.
\begin{thm} \label{thm:homotopy} The triple $\left\{ [0,1] \times \mathcal{B} , [0,1] \times \mathcal{E} , \mathcal{S} \right\}$ defines a homotopy of regular $T$-moduli problems if $\tau$ is regular. \end{thm}
There are three things to check that are not obvious:
\begin{description}
\item[(Orientation)] There is a natural orientation on all determinant bundles $\det\left( \mathcal{S}|_{ \{ \varepsilon \} \times \mathcal{B} } \right)$ with $\varepsilon \in [0,1]$ that fit together to define an orientation on $\det(\mathcal{S})$ once we fix an orientation for the interval $[0,1]$. \item[(Regularity)] The $T$-isotropy subgroup at every point $[u,A] \in \mathcal{B}$ with $(u,A)$ solving $(*)_\varepsilon$ for some $\varepsilon \in [0,1]$ is finite.
\item[(Compactness)] The space of solutions $\left\{ (\varepsilon,[u,A]) \; | \; (u,A) \; \mathrm{solves} \; (*)_\varepsilon \right\}$ in $[0,1] \times \mathcal{B}$ is compact. \end{description}
\subsubsection*{Orientation}
We refer to the argument from Cieliebak, Gaio, Mundet and Salamon \cite[chapter 4.5]{CGMS} for the orientation in the original vortex moduli problem and argue that the additional parameter $\varepsilon$ only induces a compact perturbation of the original operators and hence their determinants are canonically identified. We postpone the detailed discussion to section \ref{sec:orientations}.
\subsubsection*{Regularity}
The idea for this proof is taken from \cite[Remark 4.3]{CGMS}. An element $g \in T$ fixes a map $u : P \longrightarrow \mathbbm C^N$ if
$$ g \in \bigcap_{x \in P} \mathrm{Iso}_{u(x)}(T) =: S. $$
Now $V_S := \left\{ z \in \mathbbm C^N \; | \; S \subset \mathrm{Iso}_z(T) \right\}$ is a linear subspace, so $\mu(V_S)$ is a convex cone in ${\mathfrak t}^*$ based at zero. By construction $u$ maps into $V_S$, hence we obtain $\int_\Sigma \mu(u) \cdot \mathrm{dvol}_\Sigma \in \mu(V_S)$ and hence also
$$ \overline{\mu(u)} = \frac{ \int_\Sigma \mu(u) \cdot \mathrm{dvol}_\Sigma }{ \mathrm{vol}(\Sigma) } \in \mu(V_S). $$
Equation $(\mathrm{II\/I})$ then implies $\tau \in \mu(V_S)$. Thus there exists a $z_0 \in \mu^{-1}(\tau) \cap V_S$. By definition of $V_S$ we have $S \subset \mathrm{Iso}_{z_0}(T)$. By assumption $\tau$ is regular, which implies that $T$ acts with finite isotropy on $z_0 \in \mu^{-1}(\tau)$, hence $S$ is finite.
\subsubsection*{Compactness}
Let $(\varepsilon_j,u_j,A_j) \in [0,1] \times W^{k,2} \left( \Sigma , \mathcal{V} \right) \times \mathcal{A}^{k,2}(P)$ be a sequence of triples with $(u_j,A_j)$ solving $(*)_{\varepsilon_j}$. We want to prove the existence of a convergent subsequence in the quotient by $\mathcal{G}_0^{k+1,2}(P)$. By Sobolev embedding it suffices to exhibit a uniform $W^{k+1,2}$-bound on all $(u_j,A_j)$ after modifying them by suitable gauge transformations. We do this in seven steps.
\subsubsection*{Step 1:}
The curvature of the connections $A_j$ is uniformly bounded, i.~e.~there exists a constant $C$ with
$$
\| F_{A_j} \|_{L^\infty} \le C $$
for all $j$.
\begin{proof} The computation is analogous to the one in Cieliebak, Gaio and Salamon \cite[Proposition 3.5]{CGS}. There an $L^\infty$-bound on the maps $u$ of solutions $(u,A)$ to the usual vortex equations $(*)$ is shown. By the curvature equation $(\mathrm{II})$ this yields such a uniform bound on the curvatures. In our case for a solution $(\varepsilon,u,A)$ of $(*)_\varepsilon$ we only get an $L^\infty$-bound on $\varepsilon \cdot \mu(u)$, which does not give a uniform bound on the maps $u$, but which is sufficient to also uniformly bound the curvatures via $(\mathrm{II})_\varepsilon$.
We fix a cover of $\Sigma$ by a finite number of holomorphic charts $U_k$ and equivariant trivializations $P|_{U_K} \cong U_k \times T$. Thus we get induced trivializations $\mathcal{V}|_{U_k} \cong U_k \times \mathbbm C^N$ via the local lifts $U_k \longrightarrow P|_{U_k} \; ; \; z \longmapsto (z , \mathbbm 1)$. In local holomorphic coordinates $(s,t)$ on $U_k$ we can write a connection $A$ as $\varphi(s,t) \mathrm{ds} + \psi(s,t) \mathrm{dt}$ with functions $\varphi, \psi : U_k \longrightarrow {\mathfrak t}$. Then the local expression for the curvature is
$$ F_A = ( \partial_s \psi - \partial_t \varphi ) \mathrm{ds} \wedge \mathrm{dt}, $$
because the torus is abelian. The volume form $\mathrm{dvol}_\Sigma$ is locally given by $\lambda^2 \mathrm{ds} \wedge \mathrm{dt}$ with a positive function $\lambda$. The infinitesimal action of an element $\xi \in {\mathfrak t}$ at $z \in \mathbbm C^N$ is given by $X_\xi(z) = \dot{\rho}(\xi) \cdot z$ with the skew-Hermitian matrix
$$ \dot{\rho}(\xi) := 2 \pi i \cdot \mathrm{diag}( \langle w_\nu , \xi \rangle ). $$
With this notation the equations $(\mathrm{I})$ and $(\mathrm{II})_\varepsilon$ in local holomorphic coordinates $(s,t)$ on a chart $U_k$ are equivalent to
\begin{eqnarray} (\mathrm{I})^{loc} \quad \; \partial_s u - \dot{\rho}(\varphi) u + i \left( \partial_t u - \dot{\rho}(\psi) u \right) & = & 0, \label{equ:I_loc} \\ (\mathrm{II})_\varepsilon^{loc} \qquad \qquad \quad \frac{ \partial_s \psi - \partial_t \varphi }{ \lambda^2 } - \frac{\kappa}{\mathrm{vol}(\Sigma)} & = & \varepsilon \left( \overline{\mu(u)} - \mu(u) \right). \quad \label{equ:II_eps_loc} \end{eqnarray}
We introduce $\nabla_s := \partial_s - \dot{\rho}(\varphi)$ and $\nabla_t := \partial_t - \dot{\rho}(\psi)$ and denote the standard real inner product on $\mathbbm C^N$ by $\langle \, , \, \rangle$. Then $\langle u , \nabla_s u \rangle = \langle u , \partial_s u \rangle$, because $\dot{\rho}$ is skew-symmetric. With this observation a short computation shows that
$$
\frac{ \Delta |u|^2 }{2} := \frac{1}{2} \left( \partial_s \partial_s + \partial_t \partial_t \right) \langle u , u \rangle = \left| \nabla_s u \right|^2 + \left| \nabla_t u \right|^2 + \left\langle u , \nabla_s \nabla_s u + \nabla_t \nabla_t u \right\rangle. $$
Now equation \ref{equ:I_loc} reads $\nabla_s u = - i \nabla_t u$ and hence
$$ \nabla_s \nabla_s u + \nabla_t \nabla_t u = \nabla_s \left( - i \nabla_t u \right) + \nabla_t \left( i \nabla_s \right) u = i \left( \nabla_t \nabla_s u - \nabla_s \nabla_t u \right). $$
Furthermore
\begin{eqnarray*} \nabla_t \nabla_s u & = & \left( \partial_t - \dot{\rho}(\psi) \right) \left( \partial_s - \dot{\rho}(\varphi) \right) u \\ & = & \partial_t \partial_s u - \dot{\rho}(\partial_t \varphi) u - \dot{\rho}(\varphi) \partial_t u - \dot{\rho}(\psi) \partial_s u + \dot{\rho}(\psi) \dot{\rho}(\varphi) u \end{eqnarray*}
and correspondingly with $(s,\varphi)$ and $(t,\psi)$ interchanged. Combining all these equations we obtain the following estimate:
$$
\frac{ \Delta |u|^2 }{2} \ge \left\langle u , i \left( \dot{\rho}( \partial_s \psi ) - \dot{\rho}( \partial_t \varphi ) \right) u \right\rangle $$
Here we used that the partial derivatives $\partial_s$ and $\partial_t$ as well as diagonal matrices commute. If we write
$$ \xi_\varepsilon := \lambda^2 \left( \frac{\kappa}{\mathrm{vol}(\Sigma)} + \varepsilon \overline{\mu(u)} - \varepsilon \mu(u) \right) $$
then equation \ref{equ:II_eps_loc} reads $\partial_s \psi - \partial_t \varphi = \xi_\varepsilon$ and hence
\begin{eqnarray*}
\frac{ \Delta |u|^2 }{2} & \ge & \left\langle u , i \dot{\rho}( \xi_\varepsilon ) u \right\rangle \\ & = & - \left\langle u , 2 \pi \cdot \mathrm{diag}( \langle w_\nu , \xi_\varepsilon \rangle ) \cdot u \right\rangle \\
& = & - \sum_\nu 2 \pi \cdot \langle w_\nu , \xi_\varepsilon \rangle \cdot |u_\nu|^2 \\
& = & -2 \left\langle \pi \sum_\nu |u_\nu|^2 \cdot w_\nu , \xi_\varepsilon \right\rangle \\ & = & -2 \left\langle \mu(u) , \xi_\varepsilon \right\rangle. \end{eqnarray*}
Now let $x \in P$ be a point at which the function $|u|^2$ attains its maximum. Then $0 \ge \Delta |u|^2(x)$ and hence
\begin{eqnarray*} 0 & \le & \left\langle \mu(u(x)) , \xi_\varepsilon(x) \right\rangle \\ & = & \lambda^2 \cdot \left\langle \mu(u(x)) , \frac{\kappa}{\mathrm{vol}(\Sigma)} + \varepsilon \tau - \varepsilon \mu(u(x)) \right\rangle \\
& \le & \lambda^2 \cdot \left| \mu(u(x)) \right| \cdot \left| \frac{\kappa}{\mathrm{vol}(\Sigma)} + \varepsilon \tau \right| - \lambda^2 \cdot \varepsilon \cdot \left| \mu(u(x)) \right|^2, \end{eqnarray*}
where we useed the third equation $\overline{\mu(u)} = \tau$ for solutions of $(*)_\varepsilon$. This implies
$$
\varepsilon \cdot \left| \mu(u(x)) \right| \le \left| \frac{\kappa}{\mathrm{vol}(\Sigma)} \right| + |\tau| =: c. $$
Now since $\mu$ is proper this gives a bound on $\varepsilon \cdot |u(x)|^2$, which by definition of $x$ holds for $\varepsilon \cdot |u(y)|^2$ for all $y \in P$. This in turn gives a uniform bound for $\varepsilon \cdot |\mu(u)|$. Explicitly: Since the collection of $w_\nu$ is proper, by definition \ref{def:proper} there exists an element $\eta \in {\mathfrak t}$ with $|\eta| = 1$ and $\langle w_\nu , \eta \rangle > 0$ for all $\nu$. We denote $m := \min \left\{ \langle w_\nu , \eta \rangle \; | \; \nu = 1,\ldots,N \right\} > 0$ and compute
$$
\varepsilon \cdot |u(x)|^2 \le \varepsilon \cdot \sum_{\nu=1}^N |u_\nu(x)|^2 \cdot \frac{\langle w_\nu , \eta \rangle}{m} = \varepsilon \cdot \frac{\langle \mu(u(x)) , \eta \rangle}{\pi m} \le \varepsilon \cdot |\mu(u(x))| \cdot \frac{|\eta|}{\pi m} \le \frac{c}{\pi m}. $$
Hence we obtain
\begin{equation} \label{equ:eps_mu_u}
\varepsilon \cdot |\mu(u)| \le \varepsilon \cdot \pi \sum_{\nu=1}^N |u_\nu|^2 \cdot \max\{ |w_\nu| \} \le \frac{c \cdot \max\{ |w_\nu| \}}{m} \end{equation}
for any triple $(\varepsilon,u,A)$ with $(u,A)$ solving $(*)_{\varepsilon}$. Now by $(\mathrm{II})_\varepsilon$ we obtain a uniform bound on the curvatures as claimed. \end{proof}
\subsubsection*{Step 2:}
Fix a real number $q > 1$ and a smooth reference connection $A_0$ on $P$. Then there exist gauge transformations $\gamma_j \in \mathcal{G}^{2,q}(P)$ and a constant $C$ such that
$$
\| \gamma_j^*A_j - A_0 \|_{W^{1,q}} \le C $$
for all $j$.
\begin{proof} By step $1$ the curvatures $F_{A_j}$ are uniformly $L^q$-bounded. Hence the claim follows by Uhlenbeck compactness (see K.~Wehrheim \cite[Theorem A]{KW}). \end{proof}
\subsubsection*{Step 3:}
There is a uniform $W^{2,2}$-bound on the maps $\gamma_j^*u_j$, i.~e.~there exists a constant $C$ with
$$
\| \gamma_j^*u_j \|_{W^{2,2}} \le C $$
for all $j$.
\begin{proof} By equation $(\mathrm{II\/I})$ there is a uniform $L^2$-bound, because
$$
\| u \|_{L^2}^2 = \sum_{\nu = 1}^N \| u_\nu \|_{L^2}^2 \le \frac{ \sum_{\nu = 1}^N \| u_\nu \|_{L^2}^2 \cdot \langle w_\nu , \eta \rangle }{ m } = \frac{ \langle \tau \cdot \mathrm{vol}(\Sigma) , \eta \rangle }{ \pi m } =: {c_0}^2. $$
We fix a smooth partition of unity $(f_k)$ subordinate to the fixed finite cover $(U_k)$. It now suffices to give uniform $W^{2,2}$-bounds for maps $f \cdot u$ having compact support in one chart $U$ with holomorphic coordinates $(s,t)$. We write $\| \cdot \|_{l,p}$ for $\| \cdot \|_{W^{l,p}(U)}$ and $\bar{\partial} := \partial_s + i \partial_t$. We compute
$$
\| \partial_s(fu) \|_{0,2}^2 + \| \partial_t(fu) \|_{0,2}^2 = \| \bar{\partial}(fu) \|_{0,2}^2 \le \left( c_f \cdot c_0 + \| \bar{\partial} u \|_{0,2} \right)^2. $$
Here $c_f$ is a constant depending only on the function $f$. Now equation \ref{equ:I_loc} implies $\bar{\partial} u = ( \dot{\rho}(\varphi) + i \dot{\rho}(\psi) ) u =: B \cdot u$ and by step $2$ the entries of the diagonal matrix $B$ are $L^\infty$-bounded. Again using the uniform $L^2$-bound on $u$ this gives a uniform $W^{1,2}$-bound on $f \cdot u$ and hence a bound
$$
\| u \|_{W^{1,2}} \le c_1. $$
Now consider the same computation as above with $(fu)$ replaced by $\partial_s(fu)$. We obtain the estimate
$$
\| \partial_s\partial_s(fu) \|_{0,2}^2 + \| \partial_t\partial_s(fu) \|_{0,2}^2 \le \left( c_f \cdot (c_0+c_1) + \| \bar{\partial} \partial_s u \|_{0,2} \right)^2. $$
The $s$-derivative of \ref{equ:I_loc} gives $\bar{\partial} \partial_s u = \partial_s B \cdot u + B \cdot \partial_s u$. We get an $L^2$-bound on $B \cdot \partial_s u$ via the $L^\infty$-bound on $B$ and the $W^{1,2}$-bound on $u$. And for the $L^2$-bound on $\partial_s B \cdot u$ note that the $W^{1,2}$-bound on $u$ gives rise to an $L^4$-bound on $u$, as well as a $W^{1,4}$-bound on the connection $A$ (from step $2$ if we take $q \ge 4$) gives an $L^4$-bound on $\partial_s B$, and hence we get an $L^2$-bound on the product.
Together with the corresponding estimate from replacing $(fu)$ by $\partial_t(fu)$ we finally get the desired $W^{2,2}$ bound. \end{proof}
From this point the proof continues along the lines of Cieliebak, Gaio, Mundet and Salamon \cite[Theorem 3.2]{CGMS}. There compactness for solutions of the usual vortex equations is shown under the additional assumption that the first derivatives of the maps $u$ satisfy a uniform $L^\infty$-bound.
We replace the sequence $(\varepsilon_j,u_j,A_j)$ by $(\varepsilon_j,\gamma_j^*u_j,\gamma_j^*A_j)$, so we have a uniform $W^{1,q}$-bound on $(u_j,A_j)$ modulo $\mathcal{G}^{2,q}(P)$ with $q \ge 4$. Hence we get a weak $W^{1,q}$-limit $(\varepsilon,u,A)$ for some subsequence that we again denote by $(\varepsilon_j,u_j,A_j)$. It follows that $(u,A)$ satisfies $(*)_\varepsilon$ and the next step is to improve the regularity of this limit solution.
\subsubsection*{Step 4:}
Fix a real number $q > 2$. Given a solution $(u,A)$ of $(*)_\varepsilon$ of class $W^{1,q}$ there exists a gauge transformation $\gamma \in \mathcal{G}^{2,q}(P)$ such that $\gamma^*(u,A)$ is smooth.
\begin{proof} This follows exactly as in \cite[Theorem 3.1]{CGMS}. The additional factor of $\varepsilon \in [0,1]$ in the curvature equation does not affect the argument. \end{proof}
So if we replace the previous gauge transformations $\gamma_j$ by $\gamma_j \cdot \gamma$ then the limit solution $(u,A)$ is actually smooth. Note that the product of two $W^{2,q}$ gauge transformations on $P$ is again of class $W^{2,q}$.
Next we want to apply the local slice theorem \cite[Theorem 8.1]{KW} to put the connections $A_j$ into Coulomb gauge relative to $A$. We have a uniform bound on $\| A_j - A \|_{W^{1,q}}$ and since the $A_j$ converge strongly in $\mathcal{C}^0$ to $A$ a suitable subsequence lies in a sufficiently small $L^\infty$-neighbourhood of $A$ so that we can apply the local slice theorem. We obtain a sequence $\beta_j \in \mathcal{G}^{2,q}(P)$ such that
$$ \mathrm{d}_A^* ( \beta_j^* A_j - A ) = 0 $$
and a constant $C$ with
$$
\| \beta_j^* A_j - A \|_{L^\infty} \le C \cdot \| A_j - A \|_{L^\infty} $$
and
$$
\| \beta_j^* A_j - A \|_{W^{1,q}} \le C \cdot \| A_j - A \|_{W^{1,q}} $$
for all $j$.
By the second inequality a subsequence of $\beta_j^*A_j$ will again have a weak $W^{1,q}$-limit which by the first inequality has to be equal to $A$. But we also have to control $\beta_j^*u_j$.
\subsubsection*{Step 5:}
The sequence $\beta_j \in \mathcal{G}^{2,q}(P)$ is uniformly $W^{2,q}$-bounded.
\begin{proof}
By the second of the above inequalities we get a uniform bound on $\| \beta_j^{-1} \mathrm{d}\beta_j \|_{W^{1,q}}$ which in turn gives a uniform $W^{2,q}$-bound on $\beta_j$. See K.~Wehrheim \cite[Appendices A, B]{KW} for details. \end{proof}
Thus replacing $(u_j,A_j)$ with $\beta_j^*(u_j,A_j)$ yields an equivalent sequence with uniform $W^{1,q}$-bounds and connections in relative Coulomb gauge to $A$ modulo $\mathcal{G}^{2,q}(P)$ with $q \ge 4$. In fact a uniform $W^{1,q}$-bound on the gauge transformations $\beta_j$ would suffice to keep the $\beta_j^*u_j$ bounded.
\subsubsection*{Step 6:}
For every $l \ge 1$ there is a number $C_l$ with the property that
$$
\| u_j \|_{W^{l,2}} \le C_l $$
and
$$
\| A_j - A \|_{W^{l,2}} \le C_l $$
for all $j$.
\begin{proof} We already have the result for $l = 1$ and we get the estimates for higher $l$ by induction, because the gauge condition $\mathrm{d}_A^*(A_j-A) = 0$ together with the curvature equation $(\mathrm{II})_\varepsilon$ and equation $(\mathrm{I})$ form a complete elliptic system. \end{proof}
Finally we do not want these uniform estimates on $(u_j,A_j)$ modulo the action of gauge transformations $\gamma_j \in \mathcal{G}^{2,q}(P)$, but modulo based gauge transformations of class $W^{k+1,2}$. Arranging the $\gamma_j$ to be based is easy in our abelian case: We can just multiply every $\gamma_j$ by the constant gauge transformation $\gamma_j(x_0)^{-1}$, while keeping uniform bounds. To get the desired regularity we need
\subsubsection*{Step 7:}
Suppose $A \in \mathcal{A}^{k,2}(P)$ and $\gamma \in \mathcal{G}^{2,2}(P)$ are such that $\gamma^*A$ is again of class $W^{k,2}$ and let $k \ge 2$. Then in fact $\gamma \in \mathcal{G}^{k+1,2}(P)$.
\begin{proof} By definition $\gamma^*A = \gamma^{-1}A\gamma + \gamma^{-1}d\gamma$, hence $d\gamma = \gamma (\gamma^*A) - A \gamma$. Now the product of $W^{2,2}$ with $W^{k,2}$ for sections over a $2$-dimensional manifold $\Sigma$ is again of class $W^{2,2}$ if $k \ge 2$. Hence $\gamma$ is in fact of class $W^{3,2}$ and the result follows by induction. \end{proof}
This completes the proof for compactness. We finish the deformation discussion with a list of important remarks.
\begin{rem} The above arguments also show that the solution space and hence the associated invariants do not depend on the choice of $k$ in the definition of the configuration space $\mathcal{B}$: All solutions are gauge equivalent to smooth ones. \end{rem}
\begin{rem} The corresponding deformation is already used by Cieliebak, Gaio, Mundet and Salamon \cite[chapter 9]{CGMS} for the computation of vortex invariants of weighted projective spaces. But the claimed property of beeing a homotopy of regular moduli problems is not further justified. \end{rem}
\begin{rem} \label{rem:general_Lie_groups} It is essential for the deformation that the Lie group $T$ is abelian. The above proof for regularity does not extend to non-abelian groups. Furthermore the bundle $P \times_T {\mathfrak t}$ will in general not be trivial and sections therein cannot be integrated over $\Sigma$. Hence one can not simply split the integrated equation $(\mathrm{II\/I})$ from the curvature equation. An extension to general Lie groups is possible if one restricts to trivial principal bundles $P$. This case was considered by Gonzalez and Woodward \cite{GW}. \end{rem}
\begin{rem} In step $3$ of the above proof for compactness it is essential that we work with the standard complex structure on $\mathbbm C^n$. So while this deformation could in principle be considered for any Hamiltonian $T$-space, it is not clear that it gives a homotopy of $T$-moduli problems in general. We return to this question in section \ref{chap:generalizations}. \end{rem}
\subsection{Computation of vortex invariants}
\label{sec:vortex_computation}
We will now show how the deformation result simplifies the computation of vortex invariants by studying the moduli problem associated to the equations $(*)_{\varepsilon = 0}$. This moduli problem is given by $\mathcal{B}$ as in \ref{def:mcB}, the bundle $\mathcal{E}$ as in \ref{def:mcE} with fiber
$$ \mathcal{F} := W^{k-1,2} \left( \Sigma , \Lambda^{0,1} T^*\Sigma \otimes_\mathbbm C \mathcal{V} \right) \oplus \underline{W^{k-1,2} \left( \Sigma , {\mathfrak t} \right)} \oplus {\mathfrak t}, $$
and the section
$$ \mathcal{S} : \mathcal{B} \longrightarrow \mathcal{E} \; ; \; \left[ u , A \right] \longmapsto \left[ u , A , \left( \begin{array}{c} \bar{\partial}_A u \\ *F_A - \frac{\kappa}{\mathrm{vol}(\Sigma)} \\ \overline{\mu(u)} - \tau \end{array} \right) \right]. $$
\subsubsection*{The Jacobian torus}
\label{sec:vortex_Jacobian_torus}
We begin with the following fact:
\begin{lem} \label{lem:jacobian_torus} Let $\pi : P \longrightarrow \Sigma$ be a principal $T^k$-bundle over the closed Riemann surface $\Sigma$ of genus $g$ (with fixed volume form $\mathrm{dvol}_\Sigma$) with characteristic vector $\kappa \in \Lambda \subset {\mathfrak t}$. Then the constant map $\bar{\kappa} := \kappa / \mathrm{vol}(\Sigma)$ is a regular value of the map
\begin{eqnarray*}
\Phi \quad : \quad \mathcal{A}(P) / \mathcal{G}_0(P) & \longrightarrow & \left\{ F : \Sigma \longrightarrow {\mathfrak t} \;|\; \int_\Sigma F \cdot \mathrm{dvol}_\Sigma = \kappa \right\} \\ \left[ A \right] \qquad & \longmapsto & \qquad *F_A. \end{eqnarray*}
The preimage $\Phi^{-1}(\bar{\kappa})$ is diffeomorphic to the \emph{Jacobian torus}
$$ \mathbbm T := \left[ \frac{ H^1(\Sigma;\mathbbm R) }{ H^1(\Sigma;\mathbbm Z) } \right]^k. $$
\end{lem}
\begin{proof} To be precise in the statement of the lemma we have to introduce Sobolev completions in source and target of the map $\Phi$. But then by smoothness of the element $\bar{\kappa}$ and elliptic regularity one shows that all solutions lie in the quotient $\mathcal{A}(P) / \mathcal{G}_0(P)$ of smooth objects. For simplicity we omit this discussion and deal with smooth objects right away.
Given any $A \in \mathcal{A}(P)$ the element $*F_A \in \mathcal{C}^\infty(\Sigma,{\mathfrak t})$ is defined as follows: By the properties of connections the differential $\mathrm{d}A \in \Omega^2(P,{\mathfrak t})$ descends to a $2$-form $F_A := \pi_*\mathrm{d}A \in \Omega^2(\Sigma,{\mathfrak t})$. This curvature-form is invariant under the action of the gauge group $\mathcal{G}(P)$ on connections. Then $*F_A \cdot \mathrm{dvol}_\Sigma := F_A$. Recall that by Chern-Weil theory the integral $\int_\Sigma F_A$ does not depend on the chosen connection $A$ and this value was our definition for the characteristic vector $\kappa$.
We first consider the case of a principal $S^1$-bundle (i.~e.~we set $k = 1$). We identify $\mathrm{Lie}(S^1) \cong \mathbbm R$ and write connections just as usual $1$-forms. We start with an arbitrary connection $A$ and note that $F_A$ and $\bar{\kappa} \cdot \mathrm{dvol}_\Sigma$ are cohomologous since their difference evaluates to zero upon pairing with the fundamental class $[\Sigma]$. Hence there is an element $\alpha \in \Omega^1(\Sigma)$ such that
$$ F_A - \bar{\kappa} \cdot \mathrm{dvol}_\Sigma = \mathrm{d}\alpha. $$
Now define $A_0 := A - \pi^*\alpha$. First observe that $A_0$ again is a connection form. Furthermore it satisfies
$$ *F_{A_0} = * \pi_* \mathrm{d} ( A - \pi^*\alpha ) = * ( F_A - \mathrm{d}\alpha ) = \bar{\kappa}. $$
The same reasoning as above shows that
$$
\left\{ F \in \mathcal{C}^\infty(\Sigma) \;|\; \int_\Sigma F \cdot \mathrm{dvol}_\Sigma = \kappa \right\} = \left\{ \bar{\kappa} + *\mathrm{d}\alpha \;|\; \alpha \in \Omega^1(\Sigma) \right\}. $$
And for every connection $A$ with $*F_A = \bar{\kappa}$ we see that the elements of the affine space $\left\{ A + \pi^*\alpha \;|\; \alpha \in \Omega^1(\Sigma) \right\}$ are connections with curvature $\bar{\kappa} + *\mathrm{d}\alpha$. This shows that indeed $\bar{\kappa}$ is a regular value of $\Phi$.
To express $\Phi^{-1}(\bar{\kappa})$ we first note that any $A$ with $*F_A = \bar{\kappa}$ differs from $A_0$ by $\pi^*\alpha$ for some $\alpha \in \Omega^1(\Sigma)$ with $\mathrm{d}\alpha = 0$. Hence we obtain
$$
\Phi^{-1}(\bar{\kappa}) = \left\{ A_0 + \pi^*\alpha \;|\; \mathrm{d}\alpha = 0 \right\} / \mathcal{G}_0(P). $$
Now consider the normal subgroup $\mathcal{G}_{0,\mathrm{contr.}}(P) \lhd \mathcal{G}_0(P)$ consisting of all those based gauge transformations $\gamma : \Sigma \longrightarrow S^1$ that are homotopic to the constant map $\Sigma \longmapsto 1$. Any such map can be written as $\gamma(z) = e^{if(z)}$ for some smooth map $f : \Sigma \longrightarrow \mathbbm R$. The action of such a transformation on a connection $A$ is given by $\gamma^*A = A + \pi^*\mathrm{d}f$. Hence we obtain
$$
\left\{ A_0 + \pi^*\alpha \;|\; \mathrm{d}\alpha = 0 \right\} / \mathcal{G}_{0,\mathrm{contr.}}(P) \cong H^1(\Sigma;\mathbbm R) $$
and the quotient $\mathcal{G}_0(P) / \mathcal{G}_{0,\mathrm{contr.}}(P)$ can be identified with homotopy classes of maps $\Sigma \longrightarrow S^1$, which in turn is $H^1(\Sigma,\mathbbm Z)$. This proves the lemma in the case $k = 1$. The general case now follows by splitting the $k$-torus into a product of $k$ circles and considering each component of the connections in the induced splitting ${\mathfrak t} \cong \oplus_{j=1}^k \mathrm{Lie}(S^1)$ separately. \end{proof}
This observation gives rise to the following morphism of moduli problems. We define
$$
\mathcal{B}_\kappa := \frac{ W^{k,2} \left( \Sigma , \mathcal{V} \right) \times \left\{ A \in \mathcal{A}^{k,2}(P) \;|\; *F_A = \bar{\kappa} \right\} }{ \mathcal{G}_0^{k+1,2}(P) }. $$
This is a smooth submanifold of $\mathcal{B}$. Over $\mathcal{B}_\kappa$ we consider the bundle $\mathcal{E}_\kappa$ with fiber
$$ \mathcal{F}_\kappa := W^{k-1,2} \left( \Sigma , \Lambda^{0,1} T^*\Sigma \otimes_\mathbbm C \mathcal{V} \right) \oplus {\mathfrak t} $$
and section
$$ \mathcal{S}_\kappa : \mathcal{B}_\kappa \longrightarrow \mathcal{E}_\kappa \; ; \; \left[ u , A \right] \longmapsto \left[ u , A , \left( \begin{array}{c} \bar{\partial}_A u \\ \overline{\mu(u)} - \tau \end{array} \right) \right]. $$
The inclusion $\mathcal{F}_\kappa \longrightarrow \mathcal{F}$ given by $( \delta , v ) \longmapsto ( \delta , 0 , v )$ induces a bundle homomorphism $F : \mathcal{E}_\kappa \longrightarrow \mathcal{E}$ covering the inclusion $f : \mathcal{B}_\kappa \longrightarrow \mathcal{B}$. It is now straightforward to show the following fact.
\begin{lem} \label{lem:morphism} The pair $(f,F)$ is a morphism between the $T$-moduli problems $( \mathcal{B}_\kappa , \mathcal{E}_\kappa , \mathcal{S}_\kappa )$ and $( \mathcal{B} , \mathcal{E} , \mathcal{S} )$ in the sense of definition \ref{def:morphism}. \end{lem}
\begin{proof} Only the fourth property of a morphism in definition \ref{def:morphism} needs an explanation. In fact we have not yet defined the orientation on $\det(\mathcal{S}_\kappa)$. We refer to section \ref{sec:orientations} for a detailed discussion of orientations. \end{proof}
\subsubsection*{Moduli spaces}
We digress a little and just look at the moduli space $\mathcal{M}_\kappa = \mathcal{S}_\kappa^{-1}(0)$ without worrying about the fact that in general it will not be cut out transversally.
\subsubsection*{Genus zero}
In case $\Sigma = S^2$ is a surface of genus zero the Jacobian torus is just one point. We fix a smooth connection $A$ representing this point, i.~e.~with $*F_A = \bar{\kappa}$, and identify
$$ \mathcal{B}_\kappa \cong W^{k,2} \left( \Sigma , \mathcal{V} \right). $$
The two components of the section $\mathcal{S}_\kappa$ are now completely uncoupled. The first one picks out the holomorphic sections (with respect to the fixed connection $A$) in the Hermitian vector bundle $\mathcal{V} = \oplus_{\nu = 1}^N \mathcal{L}_\nu$. This is a finite-dimensional vector space $V$. The $L^2$ inner product on sections of $\mathcal{V}$ makes $V$ into a Hermitian vector space.
If we denote the space of holomorphic sections in the line bundle $\mathcal{L}_\nu$ by $V_\nu$ we obtain $V = \oplus_{\nu = 1}^N V_\nu$. The dimension of $V_\nu$ is the dimension of the kernel of the linear Cauchy-Riemann operator in the bundle $\mathcal{L}_\nu$ of degree $d_\nu = \langle w_\nu , \kappa \rangle$. By the Riemann-Roch theorem (see McDuff and Salamon \cite[Theorem C.1.10]{McS} for the precise statements that we refer to) this is given by
$$ n_\nu := \dim_\mathbbm C V_\nu = \max \left( 0 , 1+d_\nu \right). $$
The remaining $T \cong \mathcal{G} / \mathcal{G}_0$-action on $\mathcal{B}$ carries over to a linear torus action $\rho$ on $V$ that on the component $V_\nu$ is given by the weight vector $w_\nu$. Thus we are precisely in the setting of a Hamiltonian torus action as described in section \ref{sec:torus_actions_on_hermitian_vector_spaces}. Hence the moment map for this action is given by
\begin{align*} V = \bigoplus_{\nu=1}^N V_\nu & \;\longrightarrow\; \quad {\mathfrak t}^* \\
\left( u_\nu \right) \quad & \;\longmapsto\; \pi \sum_{\nu=1}^N \left\| u_\nu \right\|^2_{L^2(\Sigma,\mathcal{L}_\nu)} \cdot w_\nu. \end{align*}
If we rescale the Hermitian form on $V$ by $\mathrm{vol}(\Sigma)^{-\frac{1}{2}}$ and compare this to equation \ref{eqn:overline_mu} we see that the moment map is exactly $\overline{\mu}$. Thus the second component of the section $\mathcal{S}_\kappa$ just picks out the $\tau$-level of the moment map. So in the notation of section \ref{sec:torus_actions_on_hermitian_vector_spaces} we obtain $\mathcal{M}_\kappa / T \cong X_{V,\tau}$.
\begin{thm} \label{thm:genus_0_moduli_space} Consider the linear torus action on $\mathbbm C^N$ that is given by weights $(w_\nu)_{\nu = 1, \ldots N}$. Then the moduli space of the deformed genus zero vortex equations in degree $\kappa$ for a regular parameter $\tau$ is the toric manifold $X_{V,\tau}$ with $V = \oplus_{\nu = 1}^N V_\nu$ as above.
The difference between the original toric manifold $X_{\mathbbm C^N,\tau}$ and this moduli space lies solely in the dimensions of the vector spaces $V_\nu$. These dimensions are completely determined by the value of $\kappa$. \end{thm}
This observation makes the computation of the vortex invariants for genus zero surfaces extremely transparent. We will turn to this point in the next section.
\subsubsection*{Higher genus}
If the genus $g$ of $\Sigma$ is greater or equal to one then the dimension of the Jacobian torus $\mathbbm T$ is positive and the base manifold $\mathcal{B}_\kappa$ is a bundle over $\mathbbm T$ with fiber $W^{k,2} \left( \Sigma , \mathcal{V} \right)$ and the remaining torus acts only in the fibers. Now the above discussion applies to the restriction of the $T$-moduli problem $\left( \mathcal{B}_\kappa , \mathcal{E}_\kappa , \mathcal{S}_\kappa \right)$ to any such fiber, but the picture in each fiber may look very different: The dimension of the kernel of the operator $\bar{\partial}_A$ depends on the point $[A] \in \mathbbm T$. But we can circumvent this difficulty if we make restrictions on the degrees $d_\nu$. If we assume that for every $\nu$ we either have
$$ d_\nu > 2g - 2 \quad \mathrm{or} \quad d_\nu < 0 $$
then we get that the spaces $V_\nu$ of holomorphic sections in $\mathcal{L}_\nu$ have complex dimension
$$ n_\nu = \max ( 0 , 1 - g + d_\nu ) $$
independent of the connection: If $d_\nu < 0$ we have $n_\nu = 0$ and in fact the Riemann-Roch theorem tells us that the Cauchy-Riemann operator of a complex line bundle of negative degree is injective. Hence $\dim(\ker\,\bar{\partial}_A) = 0$ for any connection $A$. If on the other hand $d_\nu - 2 g + 2 > 0$, then the Cauchy-Riemann operator is surjective and the dimension of the kernel agrees with the index ($1-g+d_\nu$). And in fact we have $n_\nu = 1 - g + d_\nu$ in that case, because
$$ d_\nu - 2 g + 2 > 0 \;\Longrightarrow\; 1 - g + d_\nu > g - 1 \ge 0. $$
This proves the following generalization of theorem \ref{thm:genus_0_moduli_space} for general genus $g$.
\begin{thm} \label{thm:genus_g_moduli_space} With the notation and the assumptions on the degrees $d_\nu$ from above the vortex moduli space $\mathcal{M}_\kappa / T$ is a fiber bundle over the Jacobian torus $\mathbbm T$ with toric fiber $X_{V,\tau}$. \end{thm}
\subsubsection*{Finite-dimensional reduction}
\label{sec:vortex_finite_dimensional_reduction}
From now on we concentrate on the case $\Sigma = S^2$ of genus zero. The next step is to reduce the moduli problem $\left( \mathcal{B}_\kappa , \mathcal{E}_\kappa , \mathcal{S}_\kappa \right)$ to a finite-dimensional one. The existence of a finite-dimensional reduction is the general tool to define the Euler class for an equivariant moduli problem. In our case this reduction is easy and we can give an explicit description, because of linearity.
As in the above discussion of the genus zero case we fix a smooth connection $A$ with $*F_A = \bar{\kappa}$ and write
$$ \mathcal{B}_\kappa \cong W^{k,2} \left( \Sigma , \mathcal{V} \right) = \bigoplus_{\nu = 1}^N W^{k,2} \left( \Sigma , \mathcal{L}_\nu \right). $$
For every $\nu \in \left\{ 1 , \ldots , N \right\}$ we denote by $\bar{\partial}_{A,\nu}$ the linear Cauchy-Riemann operator on $W^{k,2} \left( \Sigma , \mathcal{L}_\nu \right)$ with respect to the fixed connection $A$. We set
$$ V_\nu := \ker\,\bar{\partial}_{A,\nu} \quad \mathrm{and} \quad W_\nu := \mathrm{coker}\,\bar{\partial}_{A,\nu} $$
and identify $W_\nu$ with the $L^2$-orthogonal complement to the image of $\bar{\partial}_{A,\nu}$ in $W^{k-1,2} \left( \Sigma , \Lambda^{0,1} T^*\Sigma \otimes_\mathbbm C \mathcal{L}_\nu \right)$. Thus all $V_\nu$ and $W_\nu$ are Hermitian vector spaces with othogonal $T$-actions given by the respective weights $w_\nu$. As explained above it makes sense to rescale the natural $L^2$ inner products on these spaces by $\mathrm{vol}(\Sigma)^{-\frac{1}{2}}$. By the Riemann-Roch theorem we have
$$ n_\nu := \dim_\mathbbm C( V_\nu ) = \left\{ \begin{array}{c@{\quad \mathrm{if} \quad}l} 0 & d_\nu < 0, \\ 1 + d_\nu & d_\nu \ge 0, \end{array} \right. $$
and hence
$$ m_\nu := \dim_\mathbbm C( W_\nu ) = \left\{ \begin{array}{c@{\quad \mathrm{if} \quad}l} - 1 - d_\nu & d_\nu < 0, \\ 0 & d_\nu \ge 0. \end{array} \right. $$
Recall that the degree $d_\nu$ of the complex line bundle $\mathcal{L}_\nu$ is given by $\langle w_\nu , \kappa \rangle$. Hence the dimensions of these spaces depend on $\kappa$. We now consider the following $T$-moduli problem
$$ B_\kappa := \bigoplus_{\nu = 1}^N V_\nu \quad , \quad F_\kappa := \left( \bigoplus_{\nu = 1}^N W_\nu \right) \oplus {\mathfrak t} \quad , \quad E_\kappa := B_\kappa \times F_\kappa $$
with
$$ S_\kappa : B_\kappa \longrightarrow F_\kappa \; ; \; v \longmapsto \left( \begin{array}{c} 0 \\ \overline{\mu}(v) - \tau \end{array} \right). $$
This finite-dimensional moduli problem is oriented by the natural orientation of complex vector spaces and a choice of orientation on ${\mathfrak t}$.
\begin{lem} \label{lem:finite_dimensional_reduction} The inclusions $B_\kappa \subset \mathcal{B}_\kappa$ and $F_\kappa \subset \mathcal{F}_\kappa$ induce a morphism between the $T$-moduli problems $\left( B_\kappa , E_\kappa , S_\kappa \right)$ and $\left( \mathcal{B}_\kappa , \mathcal{E}_\kappa , \mathcal{S}_\kappa \right)$. \end{lem}
\begin{proof} Everything apart from orientations is obvious. We deal with orientations in section \ref{sec:orientations}. \end{proof}
\subsubsection*{The Euler class}
We are now ready to compute the Euler class $\chi^{B_\kappa,E_\kappa,S_\kappa}$. Recall from section \ref{chap:moduli_problems} that for an oriented finite-dimensional moduli problem the Euler class is defined as follows: Pick a $T$-equivariant Thom structure $(U,\theta)$ on $\left( B_\kappa , E_\kappa , S_\kappa \right)$ and for a $\mathrm{d}_T$-closed equivariant form $\alpha \in \Omega_T^*(B_\kappa)$ set
$$ \chi^{B_\kappa,E_\kappa,S_\kappa}(\alpha) := \int_{B_\kappa / T} \alpha \wedge {S_\kappa}^*\theta. $$
\begin{lem} The Thom form $\theta \in \Omega_T^*(E_\kappa)$ can be chosen such that
$$ {S_\kappa}^*\theta = \left( \prod_{\nu = 1}^N w_\nu^{m_\nu} \right) \wedge {S_\kappa}^* {\pi_F}^* \theta_{\mathfrak t}. $$
Here $\theta_{\mathfrak t}$ is an ordinary Thom form on ${\mathfrak t}$ and $\pi_F : E_\kappa = B_\kappa \times F_\kappa \longrightarrow F_\kappa$ is the projection onto the fiber of $E$. \end{lem}
\begin{proof} Since the bundle $E$ is trivial the Thom form $\theta$ can be written as
$$ \theta = {\pi_F}^* ( \theta_1 \wedge \ldots \wedge \theta_N \wedge \theta_{\mathfrak t} ), $$
where $\theta_\nu$ is an equivariant Thom form on the linear space $W_\nu$. The section $S_\kappa$ is zero in all components $W_\nu$, hence the pull-back of ${\pi_F}^* \theta_\nu$ under $S_\kappa$ is by definition the equivariant Euler class of the trivial bundle $B_\kappa \times W_\nu$. Now the complex dimension of $W_\nu$ is $m_\nu$ and the torus $T$ acts on $W_\nu$ via the weight vector $w_\nu$. Hence by \ref{eqn:torus_euler_class} this Euler class is given by
$$ {S_\kappa}^* {\pi_F}^* \theta_\nu = w_\nu^{m_\nu}. $$
The claimed result follows. \end{proof}
\begin{prop} With $V := B_\kappa = \oplus_{\nu=1}^N V_\nu$ and the notation from section \ref{sec:toric_manifolds_as_moduli_problems} we have
$$ \chi^{B_\kappa,E_\kappa,S_\kappa}(\alpha) = \chi^{V,\tau} \left( \alpha \cdot \prod_{\nu = 1}^N w_\nu^{m_\nu} \right) $$
for every $\alpha \in S({\mathfrak t}^*)$. \end{prop}
\begin{proof} This follows immediately from the preceding lemma and the fact that $\chi^{V,\tau}$ is defined to be the Euler class of the $T$-moduli problem associated to the toric manifold that is described by the data $V_\nu$, $w_\nu$ and $\tau$. \end{proof}
\begin{thm} \label{thm:computation_of_vortex_invariants} With the notation from above the genus zero vortex invariant for a regular element $\tau$ is given by the formula
$$ \Psi^{\rho,\tau}_\kappa(\alpha) = \chi^{V,\tau} \left( \alpha \cdot \prod_{\nu = 1}^N w_\nu^{m_\nu} \right) $$
for every $\alpha \in S({\mathfrak t}^*)$. \end{thm}
\begin{proof} This follows immediately from the preceding proposition and the cobordism and the functoriality property of the Euler class. One only has to observe that the homotopy from our deformation and the subsequent morphisms induce the identity map on $S({\mathfrak t}^*)$ and that the orientation conventions for the original vortex moduli problem and the moduli problem associated to a toric manifold coincide. We deal with the orientations in the following section. \end{proof}
\subsection{Orientations}
\label{sec:orientations}
There are four issues on orientation that we still have to attend to:
\begin{itemize}
\item We need to define the orientation on the homotopy of moduli problems given by our deformation in section \ref{sec:vortex_deformation}.
\item We need to identify the orientation on the target moduli problem of the morphism in section \ref{sec:vortex_Jacobian_torus}.
\item We need to identify the orientation on the finite-dimensional reduction in section \ref{sec:vortex_finite_dimensional_reduction}.
\item We need to relate the resulting orientation to that of toric manifolds as discussed in section \ref{sec:orientation_of_toric_manifolds}.
\end{itemize}
We start with the first point, which is just a slight generalization of the definition of orientation on the original vortex moduli problem. Given a solution $[u,A] \in \mathcal{B}$ of $(*)_\varepsilon$ for some $\varepsilon$ we choose a local trivialization of the bundle $\mathcal{E}$ around this point to obtain the vertical differential
$$ \mathcal{D} \;:\; T_{[u,A]}\mathcal{B} \longrightarrow \mathcal{F}. $$
This operator is Fredholm and its determinant $\det(\mathcal{D})$ is the one-dimensional vector space
$$ \det(\mathcal{D}) \;=\; \Lambda^{\mathrm{max}} \left( \ker\,\mathcal{D} \right) \otimes \Lambda^{\mathrm{max}} \left( \mathrm{coker}\,\mathcal{D} \right). $$
The tangent space $T_{[u,A]}\mathcal{B}$ can be identified with the kernel of the map
$$ \varphi \;:\; W^{k,2} \left( \Sigma , \mathcal{V} \right) \times W^{k,2} \left( \Sigma , T^*\Sigma \otimes {\mathfrak t} \right) \;\longrightarrow\; \widetilde{W}^{k-1,2} \left( \Sigma , {\mathfrak t} \right), $$
which is the $L^2$-adjoint to the inclusion of the tangent space to the orbit of the based gauge group. Here $\widetilde{W}$ denotes the space of maps that vanish at the basepoint $x_0$. We write an additional summand ${\mathfrak t}$ into the target to get a map $\widetilde{\varphi}$ into $W^{k-1,2} \left( \Sigma , {\mathfrak t} \right)$ that corresponds to a local slice condition for the whole gauge group. Now after fixing an orientation on ${\mathfrak t}$ the determinant of $\mathcal{D}$ can be canonically identified with the determinant of the operator
$$ \widetilde{\mathcal{D}} \;:\; W^{k,2} \left( \Sigma , \mathcal{V} \right) \times W^{k,2} \left( \Sigma , T^*\Sigma \otimes {\mathfrak t} \right) \;\longrightarrow\; \mathcal{F} \oplus W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) $$
that is given by $\mathcal{D} \oplus \widetilde{\varphi}$.
Now source and target of $\widetilde{\mathcal{D}}$ carry complex structures. In the case $\varepsilon = 1$ it is shown by Cieliebak, Gaio, Mundet and Salamon \cite[Chapter 4.5]{CGMS} that $\widetilde{\mathcal{D}}$ is a compact perturbation of a certain complex linear operator. But this is also true for any other value of $\varepsilon$: The parameter $\varepsilon$ only appears as a factor in front of some parts of this compact perturbation. Hence the natural orientation of the determinant for a complex linear Fredholm operator induces an orientation for $\det(\mathcal{D})$. This shows that during the whole homotopy we can orient all determinant line bundles by connecting the corresponding operators to one fixed complex linear one in the space of Fredholm operators. This defines an orientation on our homotopy.
Let us be explicit about the complex structures that are used to define this orientation. On $W^{k,2} \left( \Sigma , \mathcal{V} \right)$ we get it from the fixed Hermitian structure on $\mathcal{V}$. On $W^{k,2} \left( \Sigma , T^*\Sigma \otimes {\mathfrak t} \right)$ we take the Hodge $*$-operator that is given by the fixed metric on $\Sigma$. On
$$ \mathcal{F} \oplus W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) \;=\; W^{k-1,2} \left( \Sigma , \Lambda^{0,1} T^*\Sigma \otimes \mathcal{V} \right) \oplus W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) \oplus W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) $$
the complex structure is determined by fixing an order of the second and third component. Here we make the same choice as for the complex structure on ${\mathfrak t} \oplus {\mathfrak t}$ that we made in section \ref{sec:orientation_of_toric_manifolds} in order to define the orientation of toric manifolds.
Next we consider the morphism $(f,F)$ from lemma \ref{lem:morphism}. Recall that the source moduli problem $( \mathcal{B} , \mathcal{E} , \mathcal{S} )$ features the curvature equation $*F_A = \bar{\kappa}$ as part of the section $\mathcal{S}$, while the target moduli problem $( \mathcal{B}_\kappa , \mathcal{E}_\kappa , \mathcal{S}_\kappa )$ has this equation as part of the definition of $\mathcal{B}_\kappa$. The orientation of $\det(\mathcal{S})$ is given as above and we can define the orientation on $\det(\mathcal{S}_\kappa)$ in just the same way: We split
$$ W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) \;\cong\; {\mathfrak t} \oplus \underline{ W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) } $$
as in the introduction to section \ref{sec:vortex_deformation} and we can identify the kernel and co\-ker\-nel of the vertical differential $\mathcal{D}_\kappa$ with the kernel and cokernel of the same operator $\widetilde{\mathcal{D}}$ that we use to orient $\det(\mathcal{S})$. Only the component $\underline{ W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) }$ of the summand that belongs to $\mathcal{F}$ is no longer interpreted as part of the fiber in the moduli problem, but as defining equation for the tangent space to $\mathcal{B}_\kappa$.
With this orientation on $( \mathcal{B}_\kappa , \mathcal{E}_\kappa , \mathcal{S}_\kappa )$ it is clear that $(f,F)$ is orientation preserving. Note that in the above splitting of $W^{k-1,2} \left( \Sigma , {\mathfrak t} \right)$ we identify the subset of constant maps with ${\mathfrak t}$, which agrees with the identification of ${\mathfrak t}$ in the splitting
$$ W^{k-1,2} \left( \Sigma , {\mathfrak t} \right) \;\cong\; {\mathfrak t} \oplus \widetilde{W}^{k-1,2} \left( \Sigma , {\mathfrak t} \right) $$
that we used to extend the map $\varphi$ to $\widetilde{\varphi}$.
Now in the process of finite-dimensional reduction in section \ref{sec:vortex_finite_dimensional_reduction} we first throw away all the data associated to the connections $A$ by fixing one. By the above remark we can describe the induced orientation on the remaining moduli problem by the complex structures on source and target of a map
$$ W^{k,2} \left( \Sigma , \mathcal{V} \right) \;\longrightarrow\; W^{k-1,2} \left( \Sigma , \Lambda^{0,1} T^*\Sigma \otimes_\mathbbm C \mathcal{V} \right) \oplus {\mathfrak t} \oplus {\mathfrak t}, $$
that are given as before. The inclusions into source and target of the finite-dimensional spaces $\bigoplus_{\nu = 1}^N V_\nu$ and $\bigoplus_{\nu = 1}^N W_\nu$ that describe the finite-dimensional reduction of the problem are complex linear. So if we define the orientation on $\left( B_\kappa , E_\kappa , S_\kappa \right)$ in lemma \ref{lem:finite_dimensional_reduction} by the complex orientations of
$$ \bigoplus_{\nu = 1}^N V_\nu \;\longrightarrow\; \bigoplus_{\nu = 1}^N W_\nu \oplus {\mathfrak t} \oplus {\mathfrak t}, $$
then we indeed obtain a morphism.
Now finally we observe that this gives precisely the complex orientation on $X_{V,\tau}$ that we explained in section \ref{sec:orientation_of_toric_manifolds}.
\section{Generalizations}
\label{chap:generalizations}
In this section we address the question to what extend our deformation result from section \ref{chap:vortex_invariants} can be generalized. We consider general Hamiltonian $T$-spaces and we examine the notion of energy for solutions of our deformed equations. For the generalization to arbitrary Lie groups $G$ see remark \ref{rem:general_Lie_groups}.
\subsection{General Hamiltonian $T$-spaces}
We want to extend the deformation of the vortex equations from section \ref{sec:vortex_deformation} to more general Hamiltonian $T$-spaces $X$ than $\mathbbm C^N$. As remarked above the given proof for compactness fails if the almost complex structure on $X$ is no longer constant. In general one will encounter the phenomenon of bubbling of holomorphic spheres. One could at least hope to retain the deformation result for manifolds $X$ that are symplectically aspherical, but this forces $X$ to be non-compact because of the following proposition.
\begin{prop} Suppose $X$ is a closed symplectic manifold that carries a nontrivial Hamiltonian $S^1$-action. Then there exists an element $\alpha \in \pi_2(X)$ such that
$$ \omega(\bar{\alpha}) \;>\; 0, $$
where $\bar{\alpha}$ is the image of $\alpha$ under the map $\pi_2(X) \longrightarrow H_2(X;\mathbbm Z)$. \end{prop}
\begin{proof} Consider a moment map $\mu : X \longrightarrow \mathbbm R$ for the Hamiltonian $S^1$-action. Since $X$ is compact it has at least two critical points. Fix a compatible almost complex structure $J$ on $X$ and consider the gradient flow of $\mu$ with respect to the associated metric. Take a nontrivial flowline $\gamma$ that converges to one critical point $p$. Again due to compactness of $X$ the other end of $\gamma$ converges to another critical point $q$. Hence the closure of the $S^1$-orbit of $\gamma$ sweeps out the image of a continuous map $S^2 \longrightarrow X$. This map represents a class $\alpha$ with the desired property. See Ono \cite{Ono} for details and more general results. \end{proof}
Now on a non-compact manifold $X$ to start the bubbling analysis we first need that the images of the maps $u$ stay in a compact subset. Else one could have a sequence of maps with diverging derivative on a sequence of points going off to infinity. But even in the linear case of $X \cong \mathbbm C^N$ we do not get such an a priori bound: The analysis in step $1$ for the discussion of compactness in section \ref{sec:vortex_deformation} only gives a uniform bound on $\varepsilon \cdot |u|^2$. We get the necessary $\mathcal{C}^0$-bound on the maps $u$ a posteriori from the $W^{2,2}$-bound that is established in step $3$ and which makes use of the fact that the complex structure on $\mathbbm C^N$ is standard, so that we can circumvent the bubbling analysis completely.
\subsection{Energy of $\varepsilon$-vortices}
Given a configuration $(u,A)$ for the usual vortex equations $(*)_{\varepsilon = 1}$ the \emph{energy} is defined as
\begin{equation} \label{def:energy}
E(u,A) \;:=\; \frac{1}{2} \int_\Sigma \left( |\mathrm{d}_A u|^2 + |*F_A|^2 + |\mu(u) - \tau - \bar{\kappa}|^2 \right) \mathrm{dvol}_\Sigma \end{equation}
with $\bar{\kappa} := \frac{\kappa}{\mathrm{vol}(\Sigma)}$. The norms of the second and third term of the above integrand are defined via the chosen inner product on ${\mathfrak t}$ that is also used to identify ${\mathfrak t} \cong {\mathfrak t}^*$. The norm of the first term is given by the metric associated to the symplectic form $\omega$ and an invariant compatible almost complex structure $J$ on the target manifold $X$. The following identity holds due to Cieliebak, Gaio, Mundet and Salamon \cite[Proposition 2.2]{CGMS}:
\begin{eqnarray} \label{eqn:energy_identity}
E(u,A) & = & \int_\Sigma \left( |\bar{\partial}_{J,A} u|^2 + \frac{1}{2} |*F_A + \mu(u) - \tau - \bar{\kappa}|^2 \right) \mathrm{dvol}_\Sigma \nonumber\\ & & + \left\langle \left[ \omega - \mu + \tau + \bar{\kappa} \right] , [u] \right\rangle \end{eqnarray}
Here the last term is the pairing of equivariant cohomology with homology: The difference $\omega - \mu$ is a $\mathrm{d}_T$-closed equivariant differential form, as are the elements $\tau, \bar{\kappa} \in {\mathfrak t}^*$. And the equivariant map $u$ from a principal $T$-bundle $P$ into the manifold $X$ represents an equivariant homology class.
Now as remarked in \ref{rem:epsilon_omega} the deformed equations $(*)_\varepsilon$ can be interpreted as the usual vortex equations with respect to the rescaled symplectic form $\varepsilon \cdot \omega$. This suggests to define the $\varepsilon$-energy $E_\varepsilon(u,A)$ by introducing a factor $\varepsilon$ to the norm $|\mathrm{d}_A u|^2$, the moment map $\mu$ and the parameter $\tau$ in \ref{def:energy}. Since $\kappa$ is not affected by this rescaling we include it into the curvature term and define
\begin{equation} \label{def:epsilon_energy}
E_\varepsilon(u,A) \;:=\; \frac{1}{2} \int_\Sigma \left( \varepsilon |\mathrm{d}_A u|^2 + |*F_A - \bar{\kappa}|^2 + \varepsilon^2 |\mu(u) - \tau|^2 \right) \mathrm{dvol}_\Sigma. \end{equation}
The analogous computations as in \cite[Section 2.3]{CGMS} then yield the following energy identity:
\begin{eqnarray} \label{eqn:epsilon_energy_identity}
E_\varepsilon(u,A) & = & \int_\Sigma \left( \varepsilon |\bar{\partial}_{J,A} u|^2 + \frac{1}{2} |*F_A + \varepsilon \mu(u) - \varepsilon \tau - \bar{\kappa}|^2 \right) \mathrm{dvol}_\Sigma \nonumber\\ & & + \varepsilon \left\langle \left[ \omega - \mu + \tau \right] , [u] \right\rangle \nonumber\\ & & - \varepsilon \int_\Sigma \left\langle \mu(u) - \tau , \bar{\kappa} \right\rangle \mathrm{dvol}_\Sigma \end{eqnarray}
Hence for a solution $(\varepsilon,u,A)$ to $(*)_\varepsilon$ we obtain
$$ E_\varepsilon(u,A) \;=\; \varepsilon \left\langle \left[ \omega - \mu + \tau \right] , [u] \right\rangle. $$
Note that the third term in \ref{eqn:epsilon_energy_identity} vanishes because of equation $(\mathrm{II\/I})$. So if we restrict to solutions with $[u] \in H_T^2(X)$ representing a fixed class then this identity gives uniform $L^2$-bounds on
$$
|\mathrm{d}_A u| \quad , \quad \frac{1}{\sqrt{\varepsilon}} \cdot |*F_A - \bar{\kappa}| \quad \mathrm{and} \quad \sqrt{\varepsilon} \cdot |\mu(u) - \tau|. $$
But as for the usual vortex equations this is a Sobolev borderline case and these bounds do not suffice to get compactness. The bounds would be good enough to carry out the bubbling analysis, but without a $\mathcal{C}^0$-bound on the maps $u$ we cannot even start it.
\section{Givental's toric map spaces}
\label{chap:giventals_toric_map_spaces}
Let $X := X_{\mathbbm C^N,\tau}$ be a toric manifold given by a proper collection of weights $w_\nu$ and a super-regular element $\tau$. Recall the description of the associated genus zero vortex moduli space for degree $\kappa \in \Lambda$ as the toric manifold
$$ X_\kappa \;:=\; X_{V,\tau} $$
from theorem \ref{thm:genus_0_moduli_space}: It is given by the same weights $w_\nu$ as $X$, only the spaces $V_\nu$ on which the torus $T$ acts via those weights are changed from $\mathbbm C$ to the spaces of holomorphic sections in the complex line bundles $\mathcal{L}_\nu$ of degree $d_\nu = \langle w_\nu , \kappa \rangle$. The complex dimension of $V_\nu$ is $n_\nu = \max( 0 , 1 + d_\nu )$ and it is equipped with the natural Hermitian form
$$ (u,v) \;:=\; \int_{S^2} u \cdot \bar{v} \;\mathrm{dvol}_{S^2}. $$
The same space $X_\kappa$ appears in Givental's work \cite{Giv} with the name \emph{toric map space}. More precisely Givental replaces every component of $\mathbbm C^N$ by the $n_\nu$-dimensional space of polynomials $\gamma_\nu$ in one complex variable $\zeta$ of degree $\deg(\gamma_\nu) \le d_\nu$,
$$ \gamma_\nu ( \zeta ) \;=\; \sum_{j = 0}^{d_\nu} z_{\nu,j} \zeta^j. $$
Both descriptions yield the same manifold, because the diffeomorphism type of $X_{V,\tau}$ is uniquely determined by the chamber of $\tau$ and the dimensions of the components $V_\nu$: Any two Hermitian vector spaces of the same dimension are isomorphic and since the $T$-actions are given by a fixed character $T \longrightarrow S^1$ and complex multiplication, this isomorphism is also $T$-equivariant. In fact we can give the following explicit, though not canonical identification: If we write
$$ S^2 \;\cong\; \mathbbm C \cup \infty, $$
remove the point at infinity and trivialize the bundles $\mathcal{L}_\nu$ over $\mathbbm C$, then we can identify holomorphic sections $u_\nu$ with polynomials of degree at most $d_\nu$. Those are the holomorphic maps $\mathbbm C \longrightarrow \mathbbm C$ that extend to a section in $\mathcal{L}_\nu$ over the point at infinity.
The motivation for Givental's toric map space is the following. If $\tau$ is an element of the K\"ahler cone
$$ K \;:=\; \bigcap_{\nu = 1}^N W_{ \{ 1 , \ldots , N \} \setminus \{ \nu \} } \subset {\mathfrak t}^* $$
then none of the weights $w_\nu$ vanishes in cohomology and we can identify $H^2(X,\mathbbm R) \cong {\mathfrak t}^*$ such that $H_2(X,\mathbbm Z) \cong \Lambda$. Hence $\kappa$ also specifies an integral second homology class of $X$. Now $X$ is canonically identified with the quotient of some open and dense subset $U \subset \mathbbm C^N$ by an action of the complexified torus $T_\mathbbm C$. This description uses the notion of \emph{fans} rather than weight vectors. See Audin \cite{Aud} for the correspondence of the two constructions. Hence for generic value of the complex variable $\zeta$ the collection
$$ \left( \gamma_1(\zeta) , \ldots , \gamma_N(\zeta) \right) \in \mathbbm C^N $$
will actually lie in $U$ and will thus represent a point in $X$. One can use this to show that generic elements $\left[ \gamma_1 , \ldots , \gamma_N \right] \in X_\kappa$ actually represent holomorphic maps of degree $\kappa$ from $\mathbbm C \cup \infty$ into $X$. Hence $X_\kappa$ can be viewed as a compactification of the space of such maps.
The reason to consider these toric compactifications is that all the $X_\kappa$ for $\kappa$ ranging over $H_2(X,\mathbbm Z)$ can be seen in one single infinite-dimensional toric manifold $\mathbbm{X}$, that is given by the same weights $w_\nu$ and the same element $\tau$ but for the $V_\nu$ we take infinite-dimensional spaces of Laurent polynomials $\gamma$ in one complex variable $\zeta$,
$$ V_\nu \;:=\; \mathbbm C \left[ \zeta , \zeta^{-1} \right]. $$
We choose the Hermitian metric such that the monomials $\zeta^j$ form an orthonormal system. So if we write an element $\gamma_\nu \in V_\nu$ as
$$ \gamma_\nu ( \zeta ) \;=\; \sum_{j \in \mathbbm Z} z_{\nu,j} \zeta^j $$
with only finitely many nonzero coefficients $z_{\nu,j}$, then the moment map is given by
$$ \begin{array}{cccc} \mu : & \displaystyle \bigoplus_{\nu=1}^N V_\nu & \longrightarrow & {\mathfrak t}^* \\
& \left( \gamma_1 , \ldots , \gamma_N \right) & \longmapsto & \displaystyle \pi \sum_{\nu=1}^N \sum_{j\in\mathbbm Z} \left| z_{\nu,j} \right|^2 \cdot w_\nu. \end{array} $$
There is the natural inclusion of $X_\kappa$ into $\mathbbm{X}$ given by
$$
X_\kappa \;\equiv\; X^\kappa_0 \;:=\; \left\{ \left[ \gamma_1 , \ldots , \gamma_N \right] \in \mathbbm{X} \;\left|\; \gamma_\nu(\zeta) = \sum_{j = 0}^{d_\nu} z_{\nu,j} \zeta^j \right. \right\} \subset \mathbbm{X}. $$
But now we can shift these submanifolds to different powers of $\zeta$. For elements $\kappa_0 , \kappa_1 \in \Lambda$ we define
$$
X^{\kappa_1}_{\kappa_0} \;:=\; \left\{ \left[ \gamma_1 , \ldots , \gamma_N \right] \in \mathbbm{X} \;\left|\; \gamma_\nu(\zeta) = \sum_{j = \langle w_\nu , \kappa_0 \rangle}^{\langle w_\nu , \kappa_1 \rangle} z_{\nu,j} \zeta^j \right. \right\} $$
and observe that
$$ X_\kappa \;\cong\; X^{\kappa_1}_{\kappa_0} \quad \mathrm{for} \quad \kappa = \kappa_1 - \kappa_0. $$
In fact all of $\mathbbm{X}$ is built from these compact and finite-dimensional toric manifolds $X^{\kappa_1}_{\kappa_0}$, because we have
$$ \mathbbm{X} \;=\; \bigcup_{\kappa_0,\kappa_1 \in \Lambda} X^{\kappa_1}_{\kappa_0}. $$
To see this it suffices to have an element $\eta \in \Lambda$ such that $\left\langle w_\nu , \eta \right\rangle > 0$ for all $\nu$. Then the spaces $X^{n\eta}_{-n\eta}$ exhaust all of $\mathbbm{X}$ for $n \in \mathbbm N$. But the existence of such an element $\eta$ follows by the assumption that the collection of weight vectors $w_\nu$ is proper.
Now there is an additional $S^1$ action on $\mathbbm C[\zeta,\zeta^{-1}]$ by complex multiplication on the variable $\zeta$. The above formula for the moment map shows that this action on Laurent polynomials indeed descends to an action on $\mathbbm{X}$. By Iritani \cite{Iri} it is shown that equivariant Morse-Bott theory with respect to this action on $\mathbbm{X}$ gives rise to an \emph{abstract $\mathcal{D}$-module structure} that can be identified with the \emph{quantum $\mathcal{D}$-module} of the underlying toric manifold $X$. Philosophically the quantum $\mathcal{D}$-module is an object associated to the mirror of $X$. So this result can be interpreted as a mirror theorem without explicitly knowing the mirror.
If we return to our point of view and consider $X_\kappa$ as a vortex moduli space, then it is slightly mysterious what the corresponding construction should be. The $S^1$ action on $\zeta \in \mathbbm C$ clearly corresponds to a rotation of $S^2$. But this action does not naturally lift to sections in the bundles $\mathcal{L}_\nu$ over $S^2$. The point is that our identification of $X_\kappa$ with the corresponding toric map space in not canonical: We have to remove a point from $S^2$ and choose a certain trivialization to write sections $u_\nu$ as polynomials in $\zeta$. And if we want to obtain the identical moment map as the one used by Givental then the natural Hermitian metric by integration over $S^2$ does not work: There is no volume form on $\mathbbm C$ such that the monomials $\zeta^j$ are $L^2$-orthonormal. Restriction to and integration over $S^1 \subset \mathbbm C$ would give such a product. So if we also remove the origin and pick a suitably scaled metric $f(r) \mathrm{d}r \wedge \mathrm{d}\theta$ on $\mathbbm C^*$ we could match the moment maps. This leads to the intuition that actually vortices on the cylinder are the correct object to study in this context. We refer to the results by Frauenfelder \cite{Frau} in this direction.
\end{document} | arXiv |
DOE PAGES Journal Article: Electron Heating in Low Mach Number Perpendicular Shocks. II. Dependence on the Pre-shock Conditions
Title: Electron Heating in Low Mach Number Perpendicular Shocks. II. Dependence on the Pre-shock Conditions
Recent X-ray observations of merger shocks in galaxy clusters have shown that the post-shock plasma is two-temperature, with the protons being hotter than the electrons. Here, the second of a series, we investigate the efficiency of irreversible electron heating in perpendicular low Mach number shocks, by means of two-dimensional particle-in-cell simulations. We consider values of plasma beta (the ratio of thermal and magnetic pressures) in the range 4 ≲ β p0 ≲ 32, and sonic Mach number (the ratio of shock speed to pre-shock sound speed) in the range 2 ≲ M s ≲ 5, as appropriate for galaxy cluster shocks. As reflected in Paper I, magnetic field amplification—induced by shock compression of the pre-shock field, or by strong proton cyclotron and mirror modes accompanying the relaxation of proton temperature anisotropy—can drive the electron temperature anisotropy beyond the threshold of the electron whistler instability. The growth of whistler waves breaks the electron adiabatic invariance, and allows for efficient entropy production. We determine that the post-shock electron temperature T e2 exceeds the adiabatic expectation $${T}_{e2,\mathrm{ad}}$$ by an amount $$({T}_{e2}-{T}_{e2,\mathrm{ad}})/{T}_{e0}\simeq 0.044\,{M}_{s}({M}_{s}-1)$$ (here, T e0 is the pre-shock temperature), which depends only weakly on the plasma beta over the range 4 lesssim β p0 lesssim 32 that we have explored, as well as on the proton-to-electron mass ratio (the coefficient of sime0.044 is measured for our fiducial $${m}_{i}/{m}_{e}=49$$, and we estimate that it will decrease to sime0.03 for the realistic mass ratio). Our results have important implications for current and future observations of galaxy cluster shocks in the radio band (synchrotron emission and Sunyaev–Zel'dovich effect) and at X-ray frequencies.
Guo, Xinyi [1];
Search DOE PAGES for author "Guo, Xinyi"
Sironi, Lorenzo [2]; Narayan, Ramesh [1]
Harvard-Smithsonian Center for Astrophysics, Cambridge, MA (United States)
Columbia Univ., New York, NY (United States). Dept. of Astronomy
Columbia Univ., New York, NY (United States)
USDOE Office of Science (SC), Fusion Energy Sciences (FES)
The Astrophysical Journal (Online)
Journal Name: The Astrophysical Journal (Online); Journal Volume: 858; Journal Issue: 2; Journal ID: ISSN 1538-4357
Institute of Physics (IOP)
79 ASTRONOMY AND ASTROPHYSICS; galaxies: clusters: general; instabilities; radiation mechanisms: thermal; shock waves
Guo, Xinyi, Sironi, Lorenzo, and Narayan, Ramesh. Electron Heating in Low Mach Number Perpendicular Shocks. II. Dependence on the Pre-shock Conditions. United States: N. p., 2018. Web. doi:10.3847/1538-4357/aab6ad.
Guo, Xinyi, Sironi, Lorenzo, & Narayan, Ramesh. Electron Heating in Low Mach Number Perpendicular Shocks. II. Dependence on the Pre-shock Conditions. United States. doi:10.3847/1538-4357/aab6ad.
Guo, Xinyi, Sironi, Lorenzo, and Narayan, Ramesh. Thu . "Electron Heating in Low Mach Number Perpendicular Shocks. II. Dependence on the Pre-shock Conditions". United States. doi:10.3847/1538-4357/aab6ad. https://www.osti.gov/servlets/purl/1542022.
title = {Electron Heating in Low Mach Number Perpendicular Shocks. II. Dependence on the Pre-shock Conditions},
author = {Guo, Xinyi and Sironi, Lorenzo and Narayan, Ramesh},
abstractNote = {Recent X-ray observations of merger shocks in galaxy clusters have shown that the post-shock plasma is two-temperature, with the protons being hotter than the electrons. Here, the second of a series, we investigate the efficiency of irreversible electron heating in perpendicular low Mach number shocks, by means of two-dimensional particle-in-cell simulations. We consider values of plasma beta (the ratio of thermal and magnetic pressures) in the range 4 ≲ β p0 ≲ 32, and sonic Mach number (the ratio of shock speed to pre-shock sound speed) in the range 2 ≲ M s ≲ 5, as appropriate for galaxy cluster shocks. As reflected in Paper I, magnetic field amplification—induced by shock compression of the pre-shock field, or by strong proton cyclotron and mirror modes accompanying the relaxation of proton temperature anisotropy—can drive the electron temperature anisotropy beyond the threshold of the electron whistler instability. The growth of whistler waves breaks the electron adiabatic invariance, and allows for efficient entropy production. We determine that the post-shock electron temperature T e2 exceeds the adiabatic expectation ${T}_{e2,\mathrm{ad}}$ by an amount $({T}_{e2}-{T}_{e2,\mathrm{ad}})/{T}_{e0}\simeq 0.044\,{M}_{s}({M}_{s}-1)$ (here, T e0 is the pre-shock temperature), which depends only weakly on the plasma beta over the range 4 lesssim β p0 lesssim 32 that we have explored, as well as on the proton-to-electron mass ratio (the coefficient of sime0.044 is measured for our fiducial ${m}_{i}/{m}_{e}=49$, and we estimate that it will decrease to sime0.03 for the realistic mass ratio). Our results have important implications for current and future observations of galaxy cluster shocks in the radio band (synchrotron emission and Sunyaev–Zel'dovich effect) and at X-ray frequencies.},
doi = {10.3847/1538-4357/aab6ad},
journal = {The Astrophysical Journal (Online)},
DOI: 10.3847/1538-4357/aab6ad
Electromagnetic ion cyclotron instability driven by ion energy anisotropy in high-beta plasmas
Davidson, R. C.; Ogden, Joan M.
Physics of Fluids, Vol. 18, Issue 8
Simultaneous Acceleration of Protons and Electrons at Nonrelativistic Quasiparallel Collisionless Shocks
Park, Jaehong; Caprioli, Damiano; Spitkovsky, Anatoly
Physical Review Letters, Vol. 114, Issue 8
Structure of perpendicular shocks in collisionless plasma
Leroy, M. M.
Alma-Sz Detection of a Galaxy Cluster Merger Shock at half the age of the Universe
Basu, K.; Sommer, M.; Erler, J.
The Astrophysical Journal, Vol. 829, Issue 2
DOI: 10.3847/2041-8205/829/2/l23
XMM-NEWTON OBSERVATION OF THE NORTHWEST RADIO RELIC REGION IN A3667
Finoguenov, Alexis; Sarazin, Craig L.; Nakazawa, Kazuhiro
DOI: 10.1088/0004-637x/715/2/1143
Shock fronts, electron-ion equilibration and intracluster medium transport processes in the merging cluster Abell 2146: The merging cluster Abell 2146
Russell, H. R.; McNamara, B. R.; Sanders, J. S.
Monthly Notices of the Royal Astronomical Society, Vol. 423, Issue 1
Mach number dependence of electron heating in high Mach number quasiperpendicular shocks
Matsukiyo, Shuichi
Electron anisotropy constraint in the magnetosheath: Cluster observations
Gary, S. Peter
A shock front at the radio relic of Abell 2744
Eckert, D.; Jauzac, M.; Vazza, F.
DOI: 10.1093/mnras/stw1435
Electron-Ion Temperature Equilibration in Collisionless Shocks: The Supernova Remnant-Solar Wind Connection
Ghavamian, Parviz; Schwartz, Steven J.; Mitchell, Jeremy
Space Science Reviews, Vol. 178, Issue 2-4
Planck intermediate results : VIII. Filaments between interacting clusters
Ade, P. A. R.; Aghanim, N.; Arnaud, M.
Astronomy & Astrophysics, Vol. 550
Heating of a Confined Plasma by Oscillating Electromagnetic Fields
Berger, J. M.; Newcomb, W. A.; Dawson, J. M.
Physics of Fluids, Vol. 1, Issue 4
Particle simulation study of electron heating by counter-streaming ion beams ahead of supernova remnant shocks
Dieckmann, M. E.; Bret, A.; Sarri, G.
Plasma Physics and Controlled Fusion, Vol. 54, Issue 8
Evidence for a pressure discontinuity at the position of the Coma relic from Planck Sunyaev–Zel'dovich effect data
Erler, J.; Basu, K.; Trasatti, M.
DOI: 10.1093/mnras/stu2750
Electromagnetic electron temperature anisotropy instabilities
Gary, S. Peter; Madland, Christian D.
On the electron-ion temperature ratio established by collisionless shocks
Vink, Jacco; Broersen, Sjors; Bykov, Andrei
A Textbook Example of a Bow Shock in the Merging Galaxy Cluster 1E 0657-56
Markevitch, M.; Gonzalez, A. H.; David, L.
Suzaku observations of the merging galaxy cluster Abell 2255: The northeast radio relic
Akamatsu, H.; Mizuno, M.; Ota, N.
Magnetic pumping by magnetosonic waves in the presence of noncompressive electromagnetic fluctuations
Borovsky, Joseph E.
Physics of Fluids, Vol. 29, Issue 10
CLARREO shortwave observing system simulation experiments of the twenty-first century: Simulator design and implementation
Feldman, Daniel R.; Algieri, Chris A.; Ong, Jonathan R.
Journal of Geophysical Research, Vol. 116, Issue D10
DOI: 10.1029/2010jd015350
Non-Thermal Electron Acceleration in low mach Number Collisionless Shocks. ii. Firehose-Mediated Fermi Acceleration and its Dependence on Pre-Shock Conditions
Guo, Xinyi; Sironi, Lorenzo; Narayan, Ramesh
DOI: 10.1088/0004-637x/797/1/47
Discovery of a radio relic in the low mass, merging galaxy cluster PLCK G200.9−28.2
Kale, Ruta; Wik, Daniel R.; Giacintucci, Simona
DOI: 10.1093/mnras/stx2031
THE RADIO RELICS AND HALO OF EL GORDO, A MASSIVE z = 0.870 CLUSTER MERGER
Lindner, Robert R.; Baker, Andrew J.; Hughes, John P.
Chandra observation of two shock fronts in the merging galaxy cluster Abell 2146: The merging cluster Abell 2146
Russell, H. R.; Sanders, J. S.; Fabian, A. C.
Particle Acceleration on Megaparsec Scales in a Merging Galaxy Cluster
van Weeren, R. J.; Rottgering, H. J. A.; Bruggen, M.
Electron heating and the potential jump across fast mode shocks
Schwartz, Steven J.; Thomsen, Michelle F.; Bame, S. J.
Journal of Geophysical Research, Vol. 93, Issue A11
Non-Thermal Electron Acceleration in low mach Number Collisionless Shocks. i. Particle Energy Spectra and Acceleration Mechanism
DOI: 10.1088/0004-637x/794/2/153
XMM–Newton observations of the merging galaxy cluster CIZA J2242.8+5301
Ogrean, G. A.; Brüggen, M.; Röttgering, H.
DOI: 10.1093/mnras/sts538
Modified two-stream instability in the foot of high Mach number quasi-perpendicular shocks
Journal of Geophysical Research, Vol. 108, Issue A12
Electron Heating by the ion Cyclotron Instability in Collisionless Accretion Flows. i. Compression-Driven Instabilities and the Electron Heating Mechanism
Sironi, Lorenzo; Narayan, Ramesh
Simulations of ion Acceleration at Non-Relativistic Shocks. i. Acceleration Efficiency
Caprioli, D.; Spitkovsky, A.
Electron Heating by the ion Cyclotron Instability in Collisionless Accretion Flows. ii. Electron Heating Efficiency as a Function of flow Conditions
Sironi, Lorenzo
The radio relic in Abell 2256: overall spectrum and implications for electron acceleration
Trasatti, M.; Akamatsu, H.; Lovisari, L.
Stochastic Electron Acceleration by the Whistler Instability in a Growing Magnetic Field
Riquelme, Mario; Osorio, Alvaro; Quataert, Eliot
DOI: 10.3847/1538-4357/aa95ba
Planck intermediate results : X. Physics of the hot gas in the Coma cluster
Decomposition of plasma kinetic entropy into position and velocity space and the use of kinetic entropy in particle-in-cell simulations
Liang, Haoming; Cassak, Paul A.; Servidio, Sergio
Two-Temperature Magnetohydrodynamics Simulations of Propagation of Semi-Relativistic Jets
Ohmura, Takumi; Machida, Mami; Nakamura, Kenji
Galaxies, Vol. 7, Issue 1
DOI: 10.3390/galaxies7010014
Density jump as a function of magnetic field strength for parallel collisionless shocks in pair plasmas
Bret, Antoine; Narayan, Ramesh
Astrophysics with the Spatially and Spectrally Resolved Sunyaev-Zeldovich Effects: A Millimetre/Submillimetre Probe of the Warm and Hot Universe
Mroczkowski, Tony; Nagai, Daisuke; Basu, Kaustuv
Space Science Reviews, Vol. 215, Issue 1
Density jump as a function of magnetic field for collisionless shocks in pair plasmas: The perpendicular case
Bret, A.; Narayan, R.
Electron Heating in Low-Mach-number Perpendicular Shocks. I. Heating Mechanism
Journal Article Guo, Xinyi ; Sironi, Lorenzo ; Narayan, Ramesh - The Astrophysical Journal (Online)
Recent X-ray observations of merger shocks in galaxy clusters have shown that the postshock plasma has two temperatures, with the protons hotter than the electrons. By means of two-dimensional particle-in-cell simulations, we study the physics of electron irreversible heating in low-Mach-number perpendicular shocks, for a representative case with sonic Mach number of 3 and plasma beta of 16. We find that two basic ingredients are needed for electron entropy production: (1) an electron temperature anisotropy, induced by field amplification coupled to adiabatic invariance; and (2) a mechanism to break the electron adiabatic invariance itself. In shocks, field amplification occurs at two major sites: at the shock ramp, where density compression leads to an increase of the frozen-in field; and farther downstream, where the shock-driven proton temperature anisotropy generates strong proton cyclotron and mirror modes. The electron temperature anisotropy induced by field amplification exceeds the threshold of the electron whistler instability. The growth of whistler waves breaks the electron adiabatic invariance and allows for efficient entropy production. For our reference run, the postshock electron temperature exceeds the adiabatic expectation bymore » $$\simeq 15 \% $$, resulting in an electron-to-proton temperature ratio of $$\simeq 0.45$$. We find that the electron heating efficiency displays only a weak dependence on mass ratio (less than $$\simeq 30 \% $$ drop, as we increase the mass ratio from $${m}_{i}/{m}_{e}=49$$ up to $${m}_{i}/{m}_{e}=1600$$). We develop an analytical model of electron irreversible heating and show that it is in excellent agreement with our simulation results.« less
DOI: 10.3847/1538-4357/aa9b82
Evidence for a Merger-induced Shock Wave in ZwCl 0008.8+5215 with Chandra and Suzaku
Journal Article Di Gennaro, G. ; van Weeren, R. J. ; Andrade-Santos, F. ; ... - The Astrophysical Journal (Online)
We present the results from new deep Chandra (~410 ks) and Suzaku (~180 ks) observations of the merging galaxy cluster ZwCl 0008.8+5215 (z = 0.104). Previous radio observations revealed the presence of a double radio relic located diametrically west and east of the cluster center. Using our new Chandra data, we find evidence for the presence of a shock at the location of the western relic, RW, with a Mach numbermore » $${{ \mathcal M }}_{{S}_{X}}={1.48}_{-0.32}^{+0.50}$$ from the density jump. We also measure $${{ \mathcal M }}_{{T}_{X}}={2.35}_{-0.55}^{+0.74}$$ and $${{ \mathcal M }}_{{T}_{X}}={2.02}_{-0.47}^{+0.74}$$ from the temperature jump, with Chandra and Suzaku, respectively. These values are consistent with the Mach number estimate from a previous study of the radio spectral index, under the assumption of diffusive shock acceleration ($${{ \mathcal M }}_{\mathrm{RW}}={2.4}_{-0.2}^{+0.4}$$). Interestingly, the western radio relic does not entirely trace the X-ray shock. A possible explanation is that the relic traces fossil plasma from nearby radio galaxies that is reaccelerated at the shock. For the eastern relic we do not detect an X-ray surface brightness discontinuity, despite the fact that radio observations suggest a shock with $${{ \mathcal M }}_{\mathrm{RE}}={2.2}_{-0.1}^{+0.2}$$. The low surface brightness and reduced integration time for this region might have prevented the detection. The Chandra surface brightness profile suggests $${ \mathcal M }\lesssim 1.5$$, while the Suzaku temperature measurements found $${{ \mathcal M }}_{{T}_{X}}={1.54}_{-0.47}^{+0.65}$$. Lastly, we also detect a merger-induced cold front on the western side of the cluster, behind the shock that traces the western relic.« less
DOI: 10.3847/1538-4357/ab03cd
Journal Article Guo, Xinyi ; Narayan, Ramesh ; Sironi, Lorenzo - Astrophysical Journal
Electron acceleration to non-thermal energies is known to occur in low Mach number (M{sub s} ≲ 5) shocks in galaxy clusters and solar flares, but the electron acceleration mechanism remains poorly understood. Using two-dimensional (2D) particle-in-cell (PIC) plasma simulations, we showed in Paper I that electrons are efficiently accelerated in low Mach number (M{sub s} = 3) quasi-perpendicular shocks via a Fermi-like process. The electrons bounce between the upstream region and the shock front, with each reflection at the shock resulting in energy gain via shock drift acceleration. The upstream scattering is provided by oblique magnetic waves that are self-generatedmore » by the electrons escaping ahead of the shock. In the present work, we employ additional 2D PIC simulations to address the nature of the upstream oblique waves. We find that the waves are generated by the shock-reflected electrons via the firehose instability, which is driven by an anisotropy in the electron velocity distribution. We systematically explore how the efficiency of wave generation and of electron acceleration depend on the magnetic field obliquity, the flow magnetization (or equivalently, the plasma beta), and the upstream electron temperature. We find that the mechanism works for shocks with high plasma beta (≳ 20) at nearly all magnetic field obliquities, and for electron temperatures in the range relevant for galaxy clusters. Our findings offer a natural solution to the conflict between the bright radio synchrotron emission observed from the outskirts of galaxy clusters and the low electron acceleration efficiency usually expected in low Mach number shocks.« less
ELECTRON INJECTION BY WHISTLER WAVES IN NON-RELATIVISTIC SHOCKS
Journal Article Riquelme, Mario A ; Spitkovsky, Anatoly - Astrophysical Journal
Electron acceleration to non-thermal, ultra-relativistic energies ({approx}10-100 TeV) is revealed by radio and X-ray observations of shocks in young supernova remnants (SNRs). The diffusive shock acceleration (DSA) mechanism is usually invoked to explain this acceleration, but the way in which electrons are initially energized or 'injected' into this acceleration process starting from thermal energies is an unresolved problem. In this paper we study the initial acceleration of electrons in non-relativistic shocks from first principles, using two- and three-dimensional particle-in-cell (PIC) plasma simulations. We systematically explore the space of shock parameters (the Alfvenic Mach number, M{sub A} , the shock velocity,more » v{sub sh}, the angle between the upstream magnetic field and the shock normal, {theta}{sub Bn}, and the ion to electron mass ratio, m{sub i} /m{sub e} ). We find that significant non-thermal acceleration occurs due to the growth of oblique whistler waves in the foot of quasi-perpendicular shocks. This acceleration strongly depends on using fairly large numerical mass ratios, m{sub i} /m{sub e} , which may explain why it had not been observed in previous PIC simulations of this problem. The obtained electron energy distributions show power-law tails with spectral indices up to {alpha} {approx} 3-4. The maximum energies of the accelerated particles are consistent with the electron Larmor radii being comparable to that of the ions, indicating potential injection into the subsequent DSA process. This injection mechanism, however, requires the shock waves to have fairly low Alfenic Mach numbers, M{sub A} {approx}< 20, which is consistent with the theoretical conditions for the growth of whistler waves in the shock foot (M{sub A} {approx}< (m{sub i} /m{sub e}){sup 1/2}). Thus, if the whistler mechanism is the only robust electron injection process at work in SNR shocks, then SNRs that display non-thermal emission must have significantly amplified upstream magnetic fields. Such field amplification is likely achieved by the escaping cosmic rays, so electron and proton acceleration in SNR shocks must be interconnected.« less
Plasma physical parameters along CME-driven shocks. II. Observation–simulation comparison
Journal Article Bacchini, F. ; Lapenta, G. ; Susino, R. ; ... - Astrophysical Journal
In this work, we compare the spatial distribution of the plasma parameters along the 1999 June 11 coronal mass ejection (CME)-driven shock front with the results obtained from a CME-like event simulated with the FLIPMHD3D code, based on the FLIP-MHD particle-in-cell method. The observational data are retrieved from the combination of white-light coronagraphic data (for the upstream values) and the application of the Rankine–Hugoniot equations (for the downstream values). The comparison shows a higher compression ratio X and Alfvénic Mach number M{sub A} at the shock nose, and a stronger magnetic field deflection d toward the flanks, in agreement withmore » observations. Then, we compare the spatial distribution of M{sub A} with the profiles obtained from the solutions of the shock adiabatic equation relating M{sub A}, X, and θ{sub Bn} (the angle between the upstream magnetic field and the shock front normal) for the special cases of parallel and perpendicular shock, and with a semi-empirical expression for a generically oblique shock. The semi-empirical curve approximates the actual values of M{sub A} very well, if the effects of a non-negligible shock thickness δ{sub sh} and plasma-to magnetic pressure ratio β{sub u} are taken into account throughout the computation. Moreover, the simulated shock turns out to be supercritical at the nose and sub-critical at the flanks. Finally, we develop a new one-dimensional Lagrangian ideal MHD method based on the GrAALE code, to simulate the ion-electron temperature decoupling due to the shock transit. Two models are used, a simple solar wind model and a variable-γ model. Both produce results in agreement with observations, the second one being capable of introducing the physics responsible for the additional electron heating due to secondary effects (collisions, Alfvén waves, etc.).« less | CommonCrawl |
\begin{document}
\title{A parametrisation method for high-order phase reduction in coupled oscillator networks}
\begin{abstract} We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic expansion for an embedding of the perturbed invariant torus, as well as for the reduced phase dynamics in local coordinates. Both can be determined to arbitrary degrees of accuracy, and we show that the phase dynamics may directly be obtained in normal form. We apply the method to predict remote synchronisation in a chain of coupled Stuart-Landau oscillators. \end{abstract}
\section{Introduction}
Many systems in science and engineering consist of coupled periodic processes. Examples vary from the motion of the planets, to the synchronous flashing of fireflies \cite{fireflies}, and from the activity of neurons in the brain \cite{golomb}, to power grids and electronic circuits. The functioning and malfunctioning of these coupled systems is often determined by a form of collective behaviour of its constituents, perhaps most notably their synchronisation \cite{kurths, pikovsky}. For example, synchronisation of neurons plays a critical role in cognitive processes \cite{Nicolette, palva}.
In this paper, we consider the situation where the coupling between the periodic processes is weak, a case that is amenable to rigorous mathematical analysis. Specifically, we assume that the evolution of the processes can be modelled by a system of differential equations of the form \begin{align} \label{coupledoscillators} \dot x_j = F_j(x_j) + \varepsilon \, G_j(x_1, \ldots, x_m) \ \mbox{for}\ x_j \in \mathbb{R}^{M_j} \ \mbox{and}\ j=1, \ldots, m\, . \end{align} The vector fields $F_j: \mathbb{R}^{M_j} \to \mathbb{R}^{M_j}$ in \eqref{coupledoscillators} determine the dynamics of the uncoupled oscillators: we assume that each $F_j$ possesses a hyperbolic $T_j$-periodic orbit $X_j(t)$. In the uncoupled limit---when $\varepsilon=0$---equations \eqref{coupledoscillators} thus admit a normally hyperbolic periodic or quasi-periodic invariant torus $\mathbb{T}_0 \subset \mathbb{R}^M$ (where $M := M_1 + \ldots + M_m$), consisting of the product of these periodic orbits. The functions $G_j$ in \eqref{coupledoscillators} model the interaction between the oscillators, for example through a (hyper-)network. The interaction strength $0 \leq \varepsilon \ll 1$ is assumed small, so that the unperturbed torus $\mathbb{T}_0$ persists as an invariant manifold $\mathbb{T}_{\varepsilon}$ for \eqref{coupledoscillators}, depending smoothly on $\varepsilon$, as is guaranteed by F\'enichel's theorem \cite{fenichel1979, wechselberger2020}.
The process of finding the equations of motion that govern the dynamics on the persisting torus $\mathbb{T}_{\varepsilon}$ is usually referred to as {\it phase reduction} \cite{wilson, nakao, Pietras}. Phase reduction has proved a powerful tool in the study of the synchronisation of coupled oscillators, especially because it often realises a considerable reduction of the dimension---and hence complexity---of the system. Various methods of phase reduction have been introduced over the past decades, the most well-known appearing perhaps in the work on chemical oscillations of Kuramoto \cite{Kuramotobook}. We refer to \cite{Pietras} for an extensive overview of established phase reduction techniques, and refrain from providing an overview of these methods here.
Most existing phase reduction methods provide a first-order approximation of the dynamics on the persisting invariant torus in terms of the small coupling parameter. However, there are various instances where such a first-order approximation is insufficient, see \cite{BickBoehle, rosenblum, Pazo, numericalphasereduction, nijholt2022emergent}, in particular when the first-order reduced dynamics is structurally unstable. For instance, it was observed in \cite{rosenblum} that ``remote synchronisation'' \cite{remotestar} cannot be analysed with first-order methods. More accurate ``high-order phase reduction'' techniques (that go beyond the first-order approximation) have only been introduced very recently \cite{BickBoehle, Gengel, Pazo}. They have already been applied successfully, for example to predict remote synchronisation \cite{rosenblum}. However, to the best of our knowledge, mathematically rigorous high-order phase reduction methods have only been derived in the special case that the unperturbed oscillators are either Stuart-Landau oscillators \cite{Gengel, Pazo} or deformations thereof \cite{AshwinRodrigues, BickBoehle}. In that setting, phase reduction can be performed by computing an expansion of the phase-amplitude relation that defines the invariant torus. However, this procedure does not generalise to arbitrary systems of the form \eqref{coupledoscillators}.
This paper presents a novel method for high-order phase reduction, that applies to general coupled oscillator systems of the form \eqref{coupledoscillators}. Our method works by computing an expansion (in the small parameter $\varepsilon$) of an embedding $$e: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M\, $$
of the persisting invariant torus $\mathbb{T_{\varepsilon}}$. In addition, it computes
an expansion of the dynamics on $\mathbb{T}_{\varepsilon}$ in local coordinates, in the form of a so-called ``reduced phase vector field'' $${\bf f}: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m$$
on the standard torus $(\mathbb{R}/2\pi\mathbb{Z})^m$. We find these $e$ and ${\bf f}$ by solving a so-called ``conjugacy equation''.
Our method is thus inspired by the work of De la Llave et al. \cite{parameterisation_book}, who popularised the idea of finding invariant manifolds by solving conjugacy equations. In fact, this idea was used in \cite{HaroCanadell2} to design a quadratically convergent iterative scheme for finding normally hyperbolic invariant tori. However, in \cite{HaroCanadell2} these tori are required to carry Diophantine quasi-periodic motion, not only before but also after the perturbation.
The phase reduction method presented in this paper is more similar in nature to the parametrisation method developed in \cite{BobIanMartin}. There the idea of parametrisation is used to calculate expansions of slow manifolds and their flows in geometric singular perturbation problems \cite{wechselberger2020}. Just like the method in \cite{BobIanMartin}, the phase reduction method presented here yields asymptotic expansions to finite order, but it poses no restrictions on the nature of the dynamics on the invariant torus.
We now sketch the idea behind our method. Let us write
${\bf F}_0$ for the vector field on $\mathbb{R}^M = \mathbb{R}^{M_1}\times \ldots \times \mathbb{R}^{M_m}$ that governs the dynamics of the uncoupled oscillators in \eqref{coupledoscillators}, that is, \begin{align}\label{F0def} {\bf F}_0(x_1, \ldots, x_m) := (F_1(x_1), \ldots, F_m(x_m))\, . \end{align} Our starting point is an embedding of the invariant torus $\mathbb{T}_0$ for this ${\bf F}_0$. Recall our assumption that every $F_j$ possesses a hyperbolic periodic orbit $X_j(t)$ of minimal period $T_j>0$. We denote the frequency of this orbit by $\omega_j:= \frac{2\pi}{T_j}$. An obvious embedding of $\mathbb{T}_0$ is the map $e_{0}: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^{M}$
defined by \begin{align}\label{unperturbedembedding} e_0(\phi) = e_0(\phi_1, \ldots, \phi_m) := \left(X_1\left( \omega_1^{-1}\phi_1 \right), \ldots, X_m\left(\omega_m^{-1} \phi_m \right) \right) \, . \end{align} In fact, this $e_0$ sends the periodic or quasi-periodic solutions of the ODEs $$\dot \phi = \omega := (\omega_1, \ldots, \omega_m) $$ on $(\mathbb{R}/2\pi\mathbb{Z})^m$
to integral curves of ${\bf F}_0$. In other words---see also Lemma \ref{basiclemma} below---it satisfies the conjugacy equation
$$e_0' \cdot \omega = {\bf F}_0 \circ e_0 \, .$$
The idea is now that we search for an asymptotic approximation of an embedding of the persisting torus $\mathbb{T}_{\varepsilon}$ by solving a similar conjugacy equation. We do this by making a series expansion ansatz for such an embedding, of the form $$e = e_0 + \varepsilon e_1 + \varepsilon^2 e_2 + \ldots : (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M\, , $$ as well as for
a reduced phase vector field $${\bf f} = \omega + \varepsilon {\bf f}_1 + \varepsilon^2 {\bf f}_2 + \ldots: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m\, .$$
Indeed, writing ${\bf F} = {\bf F_0} + \varepsilon {\bf F}_1: \mathbb{R}^M\to\mathbb{R}^M$, with ${\bf F}_0$ as above, and $${\bf F}_1(x) := (G_1(x), \ldots, G_m(x))\, $$
denoting the coupled part of \eqref{coupledoscillators}, we have that $e$ maps integral curves of ${\bf f}$ to solutions of \eqref{coupledoscillators}, exactly when the conjugacy equation $$e' \cdot {\bf f} = {\bf F} \circ e$$ holds. If this is the case, then $\mathbb{T}_{\varepsilon} = e((\mathbb{R}/2\pi\mathbb{Z})^m)$ is the persisting invariant torus, whereas the vector field ${\bf f}$ on $(\mathbb{R}/2\pi\mathbb{Z})^m$ represents the dynamic on $\mathbb{T}_{\varepsilon}$ in local coordinates, that is, it determines the reduced phase dynamics.
We will see that the conjugacy equation for $(e, {\bf f})$ translates into a sequence of iterative equations for $(e_1, {\bf f}_1), (e_2, {\bf f}_2), \ldots$. We will show how to solve these iterative equations, which then allows us to compute the expansions for $e$ and ${\bf f}$ to any desired order in the small parameter. Because the embedding of the torus $\mathbb{T}_{\varepsilon}$ is not unique, neither are the solutions $(e_j, {\bf f}_j)$ to these iterative equations. We characterize the extent to which one is free to choose these solutions, and we show how this freedom can be exploited to obtain ${\bf f}_j$ that are in {\it normal form}. This means that ``nonresonant'' terms have been removed from the reduced phase equations to high order.
A crucial requirement for the solvability of the iterative equations is that the torus $\mathbb{T}_0$ is {\it reducible}. Reducibility is a property of the unperturbed dynamics normal to $\mathbb{T}_0$. We shall define it at the hand of an embedding of the so-called {\it fast fibre bundle} of $\mathbb{T}_0$. We call such an embedding a {\it fast fibre map}. The fast fibre map is an important ingredient of our method. An invariant torus for an uncoupled oscillator system is always reducible. We show in Section \ref{Floquetsection} how, in this case, the fast fibre map can be obtained from the Floquet decompositions of the fundamental matrix solutions of the periodic orbits $X_j(t)$. We remark that by using fast fibre maps, we are able to avoid the use of isochrons \cite{guckenheimer} to characterise the dynamics normal to $\mathbb{T}_0$. Our parametrisation method is therefore not restricted to the case where the periodic orbits $X_j(t)$ are stable limit cycles---it suffices if they are hyperbolic.
We also stress that our method is not restricted to weakly coupled oscillator systems: it applies whenever the unperturbed embedded torus $\mathbb{T}_0$ is quasi-periodic, normally hyperbolic and reducible.
This paper is organised as follows. In section \ref{sec:iterative} we discuss the conjugacy problem for $(e, {\bf f})$ in more detail, and derive the iterative equations for $(e_j, {\bf f}_j)$. In section \ref{sec:parametrisationsection} we introduce fast fibre maps and use them to define when an embedded (quasi-)periodic torus is reducible. In section \ref{solutionsection} we explain how the fast fibre map can be used to solve the iterative equations for $(e_j, {\bf f}_j)$. We give formulas for the solutions, and discuss their properties. Section \ref{Floquetsection} shows how to compute the fast fibre map for a coupled oscillator system, treating the Stuart-Landau oscillator as an example. We finish with an application/illustration of our method in section \ref{sec:examplesection}, in which we prove that remote synchronisation occurs in a chain of weakly coupled Stuart-Landau oscillators.
\section{An iterative scheme} \label{sec:iterative} We start this section with a proof of our earlier claim about the embedding $e_0$. In the formulation of Lemma \ref{basiclemma} below, we use the notation
\begin{align}\label{e0G0first} \partial_\omega e_0 := e_0'\cdot \omega = \left. \frac{d}{ds}\right|_{s=0} \!\!\!\!\!\!\! e_0(\, \cdot +s \omega) \, \end{align} for the (directional) derivative of $e_0$ in the direction of the vector $\omega\in\mathbb{R}^m$. Like $e_0$ itself, $\partial_\omega e_0$ is a smooth map from $(\mathbb{R}/2\pi\mathbb{Z})^m$ to $\mathbb{R}^M$.
\begin{lem}\label{basiclemma}
The embedding $e_{0}$
defined in \eqref{unperturbedembedding}
satisfies the conjugacy equation $$\partial_{\omega}e_0 \ (= e_0' \cdot \omega) = {\bf F}_0 \circ e_0\, . $$ \end{lem} \begin{proof} Recall from \eqref{unperturbedembedding} that $(e_0)_j(\phi) = X_j(\omega_j^{-1} \phi_j)$, where $X_j$ is a hyperbolic periodic orbit of $F_j$. It follows that \begin{align}\nonumber (\partial_{\omega} e_0)_j(\phi) & =
\left. \frac{d}{ds} \right|_{s=0} \!\!\! (e_0)_j(\phi + s \omega) =
\left. \frac{d}{ds} \right|_{s=0} \!\!\! X_{j}(\omega_j^{-1}(\phi_j + s\omega_j)) \\ \nonumber & = \dot X_j(\omega_j^{-1} \phi_j) = F_j(X_j(\omega_j^{-1} \phi_j)) = ({\bf F}_0)_j ( (e_0 (\phi))\, , \end{align} because $\dot X_j(t) = F_j(X_j(t))$ for all $t \in \mathbb{R}$.
\end{proof} \noindent Lemma \ref{basiclemma} implies that
$e_0$ sends integral curves of the constant vector field $\omega$ on $(\mathbb{R}/2\pi\mathbb{Z})^m$ to integral curves of the vector field ${\bf F}_0$ given in \eqref{F0def}. Because the integral curves of the ODEs $\dot \phi = \omega$ on $(\mathbb{R}/2\pi\mathbb{Z})^m$ are clearly either periodic or quasi-periodic, we call $\mathbb{T}_0 = e_0((\mathbb{R}/2\pi\mathbb{Z})^m)$ an embedded {\it (quasi-)periodic torus}.
At this point we temporarily abandon the setting of coupled oscillators and consider a general ODE $\dot x = {\bf F}_0(x)$ defined by a smooth vector field
${\bf F}_0: \mathbb{R}^M\to\mathbb{R}^M$. That is, we do not assume that this ODE decouples into mutually independent ODEs. However, we will assume throughout this paper that ${\bf F}_0$ possesses a normally hyperbolic periodic or quasi-periodic invariant torus $\mathbb{T}_0$ which admits an embedding $e_0:( \mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M$ that semi-conjugates the constant vector field $\omega$ on $( \mathbb{R}/2\pi\mathbb{Z})^m$ to ${\bf F}_0$. In other words, we assume that $e_0$ and ${\bf F}_0$ satisfy \begin{align}\label{e0G0} \partial_{\omega}e_0 = {\bf F}_0 \circ e_0\, , \end{align} just as in Lemma \ref{basiclemma}. We return to coupled oscillator systems in section \ref{Floquetsection}.
We now study any smooth perturbation of ${\bf F}_0$ of the form $${\bf F} = {\bf F}(x) = {\bf F}_0(x) + \varepsilon \, {\bf F}_1(x) + \varepsilon^2 \, {\bf F}_2(x) + \ldots :\ \mathbb{R}^M \to \mathbb{R}^M\, .$$
F\'enichel's theorem \cite{fenichel1979, wechselberger2020} guarantees that, for $0\leq \varepsilon \ll 1$, the perturbed ODE $\dot x = {\bf F}(x)$ admits an invariant torus $\mathbb{T}_{\varepsilon}$ close to $\mathbb{T}_0$, that depends smoothly on $\varepsilon$. Our strategy to find $\mathbb{T}_{\varepsilon}$ will be to search for an embedding $e: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M$ close to $e_0$, and a reduced vector field ${\bf f}: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m$ close to $\omega$ satisfying the {\it conjugacy equation} \begin{align}\label{fullconj} \mathfrak{C}(e, {\bf f}) := e' \cdot {\bf f} - {\bf F} \circ e = 0 \, . \end{align} Any solution $(e, {\bf f})$ to \eqref{fullconj} indeed yields an embedded ${\bf F}$-invariant torus $\mathbb{T}_{\varepsilon}:=e((\mathbb{R}/2\pi\mathbb{Z})^m) \subset \mathbb{R}^M$, as we see from \eqref{fullconj} that at any point $x=e(\phi) \in \mathbb{T}_{\varepsilon}$ the vector ${\bf F}(x)$ lies in the image of the derivative $e'(\phi)$, and is thus tangent to $\mathbb{T}_{\varepsilon}$. Moreover, $e$ semi-conjugates ${\bf f}$ to ${\bf F}$, that is, ${\bf f}$ is the restriction of ${\bf F}$ to $\mathbb{T}_{\varepsilon}$ represented in (or ``pulled back to'') the local coordinate chart $(\mathbb{R}/2\pi\mathbb{Z})^m$.
As explained in the introduction, we try to find solutions to \eqref{fullconj} by making a series expansion ansatz $$e = e_0 + \varepsilon e_1 + \varepsilon^2 e_2 + \ldots \ \mbox{and}\ {\bf f} = \omega + \varepsilon {\bf f}_1 + \varepsilon^2 {\bf f}_2 + \ldots $$
for $e_1, e_2, \ldots : (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M$ and ${\bf f}_1, {\bf f}_2, \ldots : (\mathbb{R}/2\pi\mathbb{Z})^m \to\mathbb{R}^m$. Substitution of this ansatz in \eqref{fullconj}, and Taylor expansion to $\varepsilon$, yields the following list of recursive equations for the $e_j$ and ${\bf f}_j$: \begin{align}\label{iterativeeqns} \begin{array}{ccc}
( \partial_{\omega} - {\bf F}_0'\circ e_0 )\cdot e_1 + e_0' \cdot {\bf f}_1 = & \hspace{-5mm} {\bf F}_1\circ e_0 & \hspace{-2mm} =: {\bf G}_1 \\
( \partial_{\omega} - {\bf F}_0'\circ e_0 )\cdot e_2 + e_0' \cdot {\bf f}_2 = & \hspace{-5mm} {\bf F}_2\circ e_0 + ({\bf F}_1'\circ e_0)\cdot e_1 \\
&\hspace{-8mm} + \frac{1}{2}({\bf F}_0'' \circ e_0)(e_1, e_1) - e_1'\cdot {\bf f}_1 & \hspace{-2mm} =: {\bf G}_2 \\
\vdots & \vdots & \vdots \\
( \partial_{\omega} - {\bf F}_0'\circ e_0 )\cdot e_j + e_0' \cdot {\bf f}_j = & \hspace{0mm} \ldots & \hspace{-2mm} =: {\bf G}_j \\ \vdots & \vdots & \vdots \end{array} \end{align} Here, each ${\bf G}_j: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M$ is an ``inhomogeneous term'' that can iteratively be determined and depends on ${\bf F}_1, \ldots, {\bf F}_{j}, {\bf f}_1, \ldots, {\bf f}_{j-1}$ and $e_1, \ldots, e_{j-1}$. Concretely, ${\bf G}_j$ is given by \begin{align}\label{Gjformula}
\!\!\!\! {\bf G}_{j} \! :=\!\! \left. \frac{1}{j!}\frac{d^{j}}{d\varepsilon^{j}}\right|_{\varepsilon=0} \!\!\!\!\!\! \begin{array}{c} \!\!\! ({\bf F}_0 + \varepsilon {\bf F}_1 + \ldots + \varepsilon^j {\bf F}_j) (e_0+ \varepsilon e_1 + \ldots + \varepsilon^{j-1} e_{j-1}) \hspace{1.3cm} \\ - (e_0 + \varepsilon e_1 \ldots + \varepsilon^{j-1} e_{j-1})' \cdot (\omega + \varepsilon {\bf f}_1 +\ldots + \varepsilon^{j-1} {\bf f}_{j-1})\end{array} \!\!\!\!\!\! . \end{align}
Explicit formulas for ${\bf G}_1$ and ${\bf G}_2$ are given in \eqref{iterativeeqns}. Note that equations \eqref{iterativeeqns} are all of the form \begin{align}\label{formulalinearisationC} \mathfrak{c}(e_j, {\bf f}_j) = {\bf G}_j \ \mbox{for}\ j = 1,2, \ldots \, , \end{align} in which \begin{align}\label{formulalinearisationCexplicit} \mathfrak{c}(e_j, {\bf f}_j) := ( \partial_{\omega} - {\bf F}_0'\circ e_0 )\cdot e_j+ e_0' \cdot {\bf f}_j \, \end{align} is the linearisation of the operator $\mathfrak{C}$ defined in \eqref{fullconj} at the point $(e, {\bf f}) = (e_0, \omega)$, where $\varepsilon=0$. This linearisation $\mathfrak{c}$ is not invertible, but we will see that $\mathfrak{c}$ is surjective under the assumption that $\mathbb{T}_0$ is reducible. This implies that equations \eqref{iterativeeqns} can iteratively be solved.
\begin{remk} We think of $\mathfrak{C}$ and $\mathfrak{c}$ as operators between function spaces. For example, for ${\bf F}_0\in C^{r+1}(\mathbb{R}^M, \mathbb{R}^M), {\bf F}\in C^{r}(\mathbb{R}^M, \mathbb{R}^M)$, and $e_0\in C^{r+1}((\mathbb{R}/2\pi\mathbb{Z})^m, \mathbb{R}^M)$, $$\mathfrak{C}, \mathfrak{c}: C^{r+1}((\mathbb{R}/2\pi\mathbb{Z})^m, \mathbb{R}^M) \times C^{r}((\mathbb{R}/2\pi\mathbb{Z})^m, \mathbb{R}^m) \to C^{r}((\mathbb{R}/2\pi\mathbb{Z})^m, \mathbb{R}^M)\, .$$
\end{remk}
\begin{remk}\label{rem:nonunique} The solutions to equation \eqref{fullconj} are not unique because an invariant torus can be embedded in many different ways. In fact, if $e: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M$ is an embedding of $\mathbb{T}_{\varepsilon}$ and $\Psi: (\mathbb{R}/2\pi\mathbb{Z})^m \to (\mathbb{R}/2\pi\mathbb{Z})^m$ is any diffeomorphism of the standard torus, then also $e\circ \Psi$ is an embedding of $\mathbb{T}_{\varepsilon}$. The operator $\mathfrak{C}$ defined in \eqref{fullconj} is thus equivariant under the group of diffeomorphisms of $(\mathbb{R}/2\pi\mathbb{Z})^m$. As a consequence, solutions of \eqref{formulalinearisationC} are not unique either.
\end{remk}
\begin{remk}\label{rem:nonunique2}
For the interested reader we provide additional details on Remark~\ref{rem:nonunique}. Let us denote by $\Psi^*{\bf{f}}$ the pullback of the vector field ${\bf{f}}$ by $\Psi$ defined by the formula
$(\Psi^*{\bf{f}})(\phi) := (\Psi'(\phi))^{-1}\cdot {\bf f}(\Psi(\phi))$ for all $\phi \in (\mathbb{R}/2\pi\mathbb{Z})^m$. We claim that \begin{align} \mathfrak{C}(e\circ \Psi, \Psi^*{\bf{f}}) = \mathfrak{C}(e, f)\circ \Psi\, . \end{align} This follows from a straightforward calculation. Indeed, \begin{align} \mathfrak{C}(e\circ \Psi, \Psi^*{\bf{f}})(\phi) &= e'(\Psi(\phi))\cdot\Psi'(\phi)\cdot(\Psi'(\phi))^{-1}\cdot{\bf f}(\Psi(\phi)) - {\bf F}((e \circ \Psi)(\phi)) \nonumber \\ \nonumber &= e'(\Psi(\phi))\cdot{\bf f}(\Psi(\phi)) - ({\bf F} \circ e)(\Psi(\phi)) = \mathfrak{C}(e, f)(\Psi(\phi))\, . \end{align} As we may view vector fields as infinitesimal diffeomorphisms, this allows us to find many elements in the kernel of $\mathfrak{c}$. Namely, if $X$ is any vector field on $(\mathbb{R}/2\pi\mathbb{Z})^m$ with corresponding flow $\varphi_t$, then \begin{align}\label{fancykernelformula}
\left.\frac{d}{dt}\right|_{t=0}\hspace{-12pt}(e_0 \circ \varphi_t, \varphi_t^* \omega) = (e_0' \cdot X, [X, \omega]) \in \ker \mathfrak{c}. \end{align} Here $ [X, \omega] = -X'\cdot\omega = -\partial_\omega X$ denotes the Lie bracket between $X$ and $\omega$.
Formula \eqref{fancykernelformula} may also be verified directly. Differentiating the identity \begin{align} \mathfrak{C}(e_0, \omega)(\phi) = e_0'(\phi) \cdot \omega - ({\bf F}_0 \circ e_0)(\phi) = 0 \end{align} at any $\phi$, in the direction of any vector $u$, we first of all find that \begin{align}\label{Eremk1} e_0''(\phi) (\omega, u) - ({\bf F}_0' \circ e_0)(\phi) \cdot e_0'(\phi) \cdot u = 0\, . \end{align} From this we see that indeed \begin{align} \nonumber \mathfrak{c}(e_0' \cdot X, [X, \omega]) &= (\partial_\omega - {\bf F}_0' \circ e_0)\cdot e_0' \cdot X - e'_0 \cdot \partial_\omega X \\ \nonumber &= e_0'' (\omega, X) + e_0' \cdot \partial_\omega X - ({\bf F}_0' \circ e_0) \cdot e_0' \cdot X - e'_0 \cdot \partial_\omega X \\ \nonumber &= e_0'' (\omega, X) - ({\bf F}_0' \circ e_0) \cdot e_0' \cdot X = 0 \, , \end{align} where the last step follows from equation \eqref{Eremk1}.
\end{remk}
\section{Reducibility and the fast fibre map} \label{sec:parametrisationsection} As was indicated in Remarks \ref{rem:nonunique} and \ref{rem:nonunique2}, the solutions to the iterative equations $\mathfrak{c}(e_j, {\bf f}_j) = {\bf G}_j $ are not unique. However, we show in section \ref{solutionsection} that solutions can be found if we assume that the unperturbed torus $\mathbb{T}_0$ is reducible. We define this concept by means of a parametrisation of the linearised dynamics of ${\bf F}_0$ normal to $\mathbb{T}_0$. But we start with the observation that the linearised dynamics tangent to $\mathbb{T}_0$ is trivial.
Recall that if $e_0: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^M$ is an embedding of $\mathbb{T}_0 \subset \mathbb{R}^M$, then the {\it tangent map} ${\bf T}e_0 :(\mathbb{R}/2\pi\mathbb{Z})^m \times \mathbb{R}^m \to \mathbb{R}^M\times \mathbb{R}^M$ defined by \begin{align}\label{def:Te0} {\bf T}e_0(\phi, u) = (e_0(\phi), e_0'(\phi)\cdot u) \end{align} is an embedding as well. Its image is the tangent bundle ${\bf T}\mathbb{T}_0 \subset \mathbb{R}^M \times \mathbb{R}^M$.
\begin{lem} \label{tangentembedding} Assume that the embedding $e_0: (\mathbb{R}/2\pi\mathbb{Z})^m \to\mathbb{R}^M$ semi-conjugates the constant vector field $\omega \in \mathbb{R}^m$ on $(\mathbb{R}/2\pi\mathbb{Z})^m$ to the vector field ${\bf F}_0$ on $\mathbb{R}^M$. Then ${\bf T}e_0$ sends solution curves of the system of ODEs $$\dot \phi = \omega\, , \ \dot u = 0\ \mbox{on} \ (\mathbb{R}/2\pi\mathbb{Z})^m \times \mathbb{R}^m$$ to integral curves of the tangent vector field ${\bf T}{\bf F}_0$ on $\mathbb{R}^M\times\mathbb{R}^M$ defined by \begin{align} \label{def:TF0} {\bf T}{\bf F}_0(x,v) := ({\bf F}_0(x), {\bf F}_0'(x)\cdot v)\, . \end{align} \end{lem} \begin{proof} Our assumption simply means that $\partial_{\omega} e_0 = {\bf F}_0 \circ e_0$. As we already observed in \eqref{Eremk1}, differentiation of this identity at a point $\phi \in (\mathbb{R}/2\pi\mathbb{Z})^m$ in the direction of a vector $u \in \mathbb{R}^m$ yields that \begin{align}\nonumber
e_0''(\phi) (u,
\omega) = {\bf F}_0'(e_0(\phi))\cdot e_0'(\phi) \cdot u\, . \end{align} From this it follows that \begin{align}\nonumber
({\bf T}e_0)'(\phi,u)\cdot ( \omega, 0) = & \left. \frac{d}{ds}\right|_{s=0} \!\!\! \!\! {\bf T}e_0(\phi+s \omega, u) \\ \nonumber
= & \left. \frac{d}{ds}\right|_{s=0} \!\!\!\!\! \left( e_0(\phi+s\omega), e_0'( \phi+s\omega) \cdot u \right) \\ \nonumber
= & \left( (\partial_{\omega} e_0)(\phi), e_0''(\phi) (u,
\omega) \right) \\
\nonumber = & \left( {\bf F}_0(e_0(\phi)), {\bf F}_0'(e_0(\phi)) \cdot e_0'(\phi)\cdot u \right) = {\bf T}{\bf F}_0 ( {\bf T}e_0(\phi, u))\, . \end{align} In the last equality we used Definitions \eqref{def:Te0} and \eqref{def:TF0}. \end{proof}
\noindent Lemma \ref{tangentembedding} shows that ${\bf T}e_0$ trivialises the linearised dynamics of ${\bf F}_0$ in the direction tangent to $\mathbb{T}_0$. In what follows, we assume that something similar happens in the direction normal to $\mathbb{T}_0$, that is, we assume that $\mathbb{T}_0$ is reducible. We define this concept now.
\begin{defi}\label{reducibledefi} Assume that the embedding $e_0: (\mathbb{R}/2\pi\mathbb{Z})^m \to\mathbb{R}^M$ semi-conjugates the constant vector field $\omega \in \mathbb{R}^m$ on $(\mathbb{R}/2\pi\mathbb{Z})^m$ to the vector field ${\bf F}_0$ on $\mathbb{R}^M$. We say that the (quasi-)periodic invariant torus $\mathbb{T}_0 = e_0( (\mathbb{R}/2\pi\mathbb{Z})^m)$ is {\it reducible} if there is a map
${\bf N}e_0 \! : \!(\mathbb{R}/2\pi\mathbb{Z})^m \! \times \! \mathbb{R}^{M-m} \to\mathbb{R}^M \! \times \! \mathbb{R}^M$ of the form
\begin{align} \label{NGammadefgeneral} {\bf N}e_0(\phi, u) := (e_0(\phi), N(\phi)\cdot u) \, , \end{align} for a smooth family of linear maps $$N: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathcal{L}(\mathbb{R}^{M-m}, \mathbb{R}^M)\, , $$ with the following two properties: \begin{itemize} \item[{\it i)}]
${\bf N}e_0$ is transverse to ${\bf T}e_0$.
By this we mean that \begin{align}\label{transversality} \mathbb{R}^M = {\rm im}\, e_0'(\phi) \oplus {\rm im}\, N(\phi)\ \mbox{for every}\ \phi \in (\mathbb{R}/2\pi\mathbb{Z})^m\, . \end{align} In particular, every $N(\phi)$ is injective. \item[{\it ii)}] There is a linear map $L: \mathbb{R}^{M-m} \to \mathbb{R}^{M-m}$ such that ${\bf N}e_0$ sends solution curves of the system of ODEs $$\dot \phi = \omega\, ,\, \dot u = L \cdot u \ \mbox{defined on}\ (\mathbb{R}/2\pi\mathbb{Z})^m \times \mathbb{R}^{M-m}$$ to integral curves of the tangent vector field ${\bf T}{\bf F}_0$ on $\mathbb{R}^M\times \mathbb{R}^M$. \end{itemize} When $\mathbb{T}_0$ is reducible, the matrix $L$ is called a {\it Floquet matrix} for $\mathbb{T}_0$, and its eigenvalues the {\it Floquet exponents} of $\mathbb{T}_0$.
If $L$ is hyperbolic (no Floquet exponents lie on the imaginary axis) then $\mathbb{T}_0$ is normally hyperbolic, and we call ${\bf N}e_0$ a {\it fast fibre map} for $\mathbb{T}_0$. Its image $${\bf N}\mathbb{T}_0 := {\bf N}e_0 ((\mathbb{R}/2\pi\mathbb{Z})^m \times \mathbb{R}^{M-m}) \subset \mathbb{R}^M\times\mathbb{R}^M\, $$ is then called the {\it fast fibre bundle} of $\mathbb{T}_0$. \end{defi} \noindent We note that the map ${\bf N}e_0$ appearing in Definition \ref{reducibledefi} is an embedding because $e_0$ is an embedding and the linear maps $N(\phi)$ are all injective. Therefore its image ${\bf N}\mathbb{T}_0$
is a smooth $M$-dimensional manifold. Condition {\it i)} ensures that ${\bf N}\mathbb{T}_0$ is in fact a normal bundle for $\mathbb{T}_0$.
We finish this section with an alternative characterisation of property $\it ii)$ in Definition \ref{reducibledefi}. \begin{lem}\label{variationalparametrisation} Assume that the embedding $e_0: (\mathbb{R}/2\pi\mathbb{Z})^m \to\mathbb{R}^M$ semi-conjugates the constant vector field $\omega$ to the vector field ${\bf F}_0$. Let $L: \mathbb{R}^{M-m} \to \mathbb{R}^{M-m}$ be a linear map, and
let ${\bf N}e_0$ be a map of the form \eqref{NGammadefgeneral} for a smooth family of linear maps $N: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathcal{L}(\mathbb{R}^{M-m}, \mathbb{R}^M)$.
The following are equivalent: \begin{itemize} \item[{\it i)}] ${\bf N}e_0$ sends solution curves of the system of ODEs $$\dot \phi = \omega\, ,\, \dot u = L \cdot u \ \mbox{defined on}\ (\mathbb{R}/2\pi\mathbb{Z})^m \times \mathbb{R}^{M-m}$$ to integral curves of the tangent vector field ${\bf T}{\bf F}_0$ on $\mathbb{R}^M\times \mathbb{R}^M$; \item[{\it ii)}] $N=N(\phi)$ satisfies the partial differential equation \begin{align}\label{N0M0} \partial_{\omega} N + N \! \cdot \! L= ({\bf F}_0' \circ e_0) \! \cdot \! N\ \mbox{on}\ (\mathbb{R}/2\pi\mathbb{Z})^m\, . \end{align} \end{itemize}
\end{lem} \begin{proof}
It holds that \begin{align}
({\bf N}e_0)'(\phi, u) \cdot (\omega, L\cdot u ) & = \left. \frac{d}{ds}\right|_{s = 0} \!\!\!\!\! \left(e_0(\phi + s \omega), N(\phi + s \omega) \cdot (u+s L\cdot u ) \right) \nonumber \\ \nonumber
& = ((\partial_{\omega} e_0)(\phi), \partial_{\omega} N(\phi) \cdot u + N(\phi)\cdot L\cdot u) \, .
\end{align}
At the same time,
$${\bf T}{\bf F}_0({\bf N}e_0(\phi, u)) = ({\bf F}_0(e_0(\phi)), {\bf F}_0'(e_0(\phi))\cdot N(\phi)\cdot u)\, .$$
It holds that
$\partial_{\omega} e_0 = {\bf F}_0\circ e_0$ by assumption, so the first components of these two expressions are equal. The conclusion of the lemma therefore follows from comparing the second components. \end{proof} \begin{remk} Reducibility of a (quasi-)periodic invariant torus of an arbitrary vector field ${\bf F}_0$ can only be quaranteed under strong conditions, e.g., that ${\bf F}_0$ is Hamiltonian \cite{KAMwithout}, or that the frequency vector $\omega$ satisfies certain Diophantine inequalities \cite{JohnsonSell}. We do not assume such conditions here. Even the question whether reducibility is preserved under perturbation is subtle \cite{JorbaSimo}.
However, hyperbolic periodic orbits (which are one-dimensional normally hyperbolic invariant tori) are always reducible (at least if we allow the matrix $L$ to be complex, see Section \ref{Floquetsection}). This relatively well-known fact is a consequence of Floquet's theorem \cite{floquet}, as we show in Theorem \ref{floquetreducible}. The (quasi-)periodic torus occurring in an uncoupled oscillator system such as \eqref{coupledoscillators} is a product of hyperbolic periodic orbits, and is therefore reducible as well, see Lemma \ref{productreduciblelemma}. \end{remk}
\section{Solving the iterative equations} \label{solutionsection} We now return to solving the iterative equations \eqref{iterativeeqns}, assuming from here on out that $\mathbb{T}_0$ is an embedded (quasi-)periodic reducible and normally hyperbolic invariant torus for ${\bf F}_0$. The main result of this section can be summarised (at this point still somewhat imprecisely) as follows. \begin{thr}\label{sloppythm} Assume that $\mathbb{T}_0 = e_0((\mathbb{R}/2\pi\mathbb{Z})^m) \subset \mathbb{R}^M$ is a smooth embedded (quasi-)periodic reducible normally hyperbolic invariant torus for ${\bf F}_0$. Then \begin{itemize} \item[{\it i)}] there are smooth solutions $(e_j, {\bf f}_j)$ to the iterative equations $\mathfrak{c}(e_j, {\bf f}_j)={\bf G}_j$ for every $j\in \mathbb{N}$, for which we provide explicit formulas in this section; \item[{\it ii)}] the component of each $e_j$ tangential to $\mathbb{T}_0$ can be chosen freely, but every such choice for $e_1, \ldots, e_{j-1}$ uniquely determines the component of $e_j$ normal to $\mathbb{T}_0$ (see Theorem \ref{solutionlemma}); \item[{\it iii)}] the tangential component of $e_j$ can be chosen in such a way that ${\bf f}_j$ is in ``normal form'' to arbitrarily high order in its Fourier expansion. We say that ${\bf f}_j$ is in normal form if it is a sum of ``resonant terms'' only (see Corollary \ref{normalformcorollary}). \end{itemize} \end{thr} \noindent The precise meaning of the statements in this theorem will be made clear below. Theorem \ref{sloppythm} follows directly from the results presented in this section.
To prove the theorem, recall that (because $\mathbb{T}_0$ is reducible) we have at our disposal a fast fibre map ${\bf N}e_0$ for $\mathbb{T}_0$, defined by a family of injective matrices $N=N(\phi)$ that satisfies $\mathbb{R}^M = {\rm im}\, e_0'(\phi) \oplus {\rm im}\, N(\phi)$ for every $\phi \in (\mathbb{R}/2\pi\mathbb{Z})^m$. This enables us to make the ansatz \begin{align}\label{Ansatz} \underbrace{e_j(\phi)}_{\footnotesize \begin{array}{c} \in \\ \mathbb{R}^M \end{array}} \ \ = \!\! \underbrace{e_0'(\phi)}_{\footnotesize \begin{array}{c} \in \\ \mathcal{L}(\mathbb{R}^m, \mathbb{R}^M) \end{array}}
\!\!\!\!\! \cdot \
\underbrace{{\bf g}_j(\phi)}_{\footnotesize \begin{array}{c} \in \\ \mathbb{R}^m \end{array} }\ \ + \!\!\!\!\!
\underbrace{N(\phi)}_{\footnotesize \begin{array}{c} \in \\ \mathcal{L}(\mathbb{R}^{M-m}, \mathbb{R}^M) \end{array} }
\!\!\!\!\!\!\! \cdot \
\underbrace{{\bf h}_j(\phi)}_{\footnotesize \begin{array}{c} \in \\ \mathbb{R}^{M-m} \end{array}} \, , \end{align} for (unknown) smooth functions ${\bf g}_j: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m$ and ${\bf h}_j: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^{M-m}$. This ansatz decomposes $e_j$ into components in the direction of the tangent bundle ${\bf T}e_0$ and the fast fibre bundle ${\bf N}e_0$. \begin{lem} \label{ansatzlemma}
The ansatz \eqref{Ansatz} transforms equation \eqref{formulalinearisationC} into \begin{align}\label{newequation} \mathfrak{c}(e_j, {\bf f}_j) = e_0' \cdot \left( \partial_{\omega} {\bf g}_j + {\bf f}_j \right) + N \cdot(\partial_{\omega} - L)({\bf h}_j) = {\bf G}_j\, . \end{align} \end{lem} \begin{proof} We use our definitions, and results derived above, to compute: \begin{align}\nonumber {\bf G}_j = \mathfrak{c}(e_j, {\bf f}_j) = & \ ( \partial_{\omega} - {\bf F}_0'\circ e_0 )\cdot e_j+ e_0' \cdot {\bf f}_j \\ \nonumber
= &\ ( \partial_{\omega} - {\bf F}_0' \circ e_0 )\cdot \left( e_0' \cdot {\bf g}_j + N\cdot {\bf h}_j \right) + e_0' \cdot {\bf f}_j
\\ \nonumber
= & \ e_0''({\bf g}_j, \omega) + e_0' \cdot \partial_{\omega}{\bf g}_j + \partial_{\omega}N \cdot {\bf h}_j + N \cdot \partial_{\omega}{\bf h}_j \\ \nonumber & \ - ({\bf F}_0'. \circ e_0) \cdot e_0' \cdot {\bf g}_j - ({\bf F_0}' \circ e_0)\cdot N \cdot {\bf h}_j + e_0' \cdot {\bf f}_j \\ \nonumber
= & \ \underbrace{e_0''({\bf g}_j, \omega) - ({\bf F}_0'. \circ e_0) \cdot e_0' \cdot {\bf g}_j }_{=0} + e_0' \cdot \partial_{\omega}{\bf g}_j + e_0' \cdot {\bf f}_j \\ \nonumber & \ + N \cdot \partial_{\omega}{\bf h}_j + \underbrace{\partial_{\omega}N \cdot {\bf h}_j - ({\bf F_0}' \circ e_0)\cdot N \cdot {\bf h}_j}_{=-N \cdot L \cdot {\bf h}_j} \\ \nonumber
= & \ e_0' \cdot \left( \partial_{\omega}{\bf g}_j + {\bf f}_j \right) + N \cdot \left(\partial_{\omega} - L \right) \cdot {\bf h}_j\, . \end{align} We clarify these equalities below: \begin{itemize} \item[1.] The first equality is \eqref{formulalinearisationC}; \item[2.] In the second equality, we used \eqref{formulalinearisationCexplicit}; \item[3.] The third equality is our ansatz \eqref{Ansatz}; \item[4.] The fourth equality follows from the product rule (applied twice); \item[5.] In the fifth equality, the terms in the sum were re-ordered; \item[6.] The final equality follows from \eqref{Eremk1} and \eqref{N0M0}. \end{itemize} This proves the lemma. \end{proof} \noindent Lemma \ref{ansatzlemma} allows us to solve equation \eqref{newequation} by splitting it into a component along the tangent bundle ${\bf T}\mathbb{T}_0$ and a component along the fast fibre bundle ${\bf N}\mathbb{T}_0$ of $\mathbb{T}_0$. In what follows we denote by $$\pi: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathcal{L}(\mathbb{R}^M, \mathbb{R}^M)$$ the family of projections onto the tangent bundle along the fast fibre bundle. That is, each $\pi(\phi): \mathbb{R}^M\to\mathbb{R}^M$ is the unique projection that satisfies $$\pi(\phi)\cdot e_0'(\phi) = e_0'(\phi)\ \mbox{and}\ \pi(\phi)\cdot N(\phi) = 0\, .$$ Proposition \ref{projectionprop} below provides an explicit formula for $\pi(\phi)$. It is clear from this formula that $\pi$ depends smoothly on the base point $\phi \in (\mathbb{R}/2\pi\mathbb{Z})^m$.
Applying $\pi$ and $1-\pi$ to \eqref{newequation} produces, respectively,
\begin{align}\nonumber & e_0' \cdot ( \partial_{\omega} {\bf g}_j + {\bf f}_j ) = \pi \cdot {\bf G}_j \, , \\ \nonumber & N \cdot (\partial_{\omega} - L)({\bf h}_j) = (1-\pi)\cdot {\bf G}_j\, . \end{align} Because $e_0'(\phi)$ and $N(\phi)$ are injective, these equations are equivalent to \begin{align}\label{tworeducedequations} \begin{array}{rll}
\partial_{\omega} {\bf g}_j + {\bf f}_j = & (e_0')^{+} \cdot \pi \cdot {\bf G}_j & =: U_j \, , \\
(\partial_{\omega} - L)({\bf h}_j) = & N^{+}\cdot(1-\pi)\cdot {\bf G}_j & =: V_j \, . \end{array} \end{align} Here, $A^+ := (A^TA)^{-1}A^T$ denotes the Moore-Penrose pseudo-inverse, which is well-defined for an injective linear map $A$. Clearly, $(e_0')^{+}$ and $N^+$ depend smoothly on $\phi\in(\mathbb{R}/2\pi\mathbb{Z})^m$. We give these equations a special name. \begin{defi} We call the first equation in \eqref{tworeducedequations}, \begin{align} \label{tworeducedequations1}
\partial_{\omega} {\bf g}_j + {\bf f}_j = U_j\, ,
\end{align}
the $j$-th {\it tangential homological equation}. The second equation in \eqref{tworeducedequations}, \begin{align} \label{tworeducedequations2}
(\partial_{\omega} - L)({\bf h}_j) = V_j\, ,
\end{align} is called the $j$-th {\it normal homological equation}. \end{defi}
\begin{remk} To recap, we note that \eqref{tworeducedequations1} and \eqref{tworeducedequations2} are inhomogeneous linear equations for the three unknown smooth functions ${\bf f}_j, {\bf g}_j, {\bf h}_j$ and with the inhomogeneous right hand sides $U_j, V_j$. The domains and co-domains of these functions are given by
\begin{align} \nonumber
{\bf f}_j, {\bf g}_j, U_j : (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m \ \mbox{and} \ {\bf h}_j, V_j: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^{M-m} \, .
\end{align}
\end{remk} \noindent The following theorem shows how the homological equations can be solved. Explicit expressions for the Fourier series of the solutions are given in formulas \eqref{fintermsofuandx} and \eqref{YintermsofV}, that appear in the proof of the theorem. \begin{thr}\label{solutionlemma}
For any smooth functions ${\bf g}_j, U_j: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m$ and $V_j: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^{M-m}$, there are unique smooth functions ${\bf f}_j: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m$ and ${\bf h}_j:(\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^{M-m}$ that solve \eqref{tworeducedequations1} and \eqref{tworeducedequations2}. \end{thr} \begin{proof} The tangential homological equation \eqref{tworeducedequations1} can be rewritten as $${\bf f}_j = U_j - \partial_{\omega} {\bf g}_j\, .$$ This shows that for any smooth ${\bf g}_j$ and $U_j$ there exists a unique solution ${\bf f}_j$. However, in view of Corollary \ref{normalformcorollary} below, we would also like a formula for the solution of the tangential homological equation in the form of a Fourier series. To this end, we expand $U_j$ and ${\bf g}_j$ in Fourier series as $$U_j(\phi) = \sum_{k\in \mathbb{Z}^m} U_{j,k} e^{i\langle k, \phi\rangle} \ \mbox{and}\ {\bf g}_j(\phi) = \sum_{k\in \mathbb{Z}^m} g_{j,k} e^{i\langle k, \phi\rangle}\, .$$ We use the notation $$\langle k, \phi\rangle := k_1 \phi_1 + \ldots + k_m \phi_m\, $$ for what is often called the $k$-th {\it combination angle}. Note that the Fourier coefficients $ U_{j,k}, g_{j,k} \in \mathbb{C}^{m}$ are complex vectors satisfying $U_{j,-k} = \overline{U}_{j,k}$ and $g_{j,-k} = \overline{g}_{j,k}$, because $U_j$ and ${\bf g}_j$ are real-valued. We similarly expand ${\bf f}_j$ in a Fourier series by making the solution ansatz $${\bf f}_j(\phi) = \sum_{k\in \mathbb{Z}^m} f_{j,k} e^{i\langle k, \phi\rangle}, $$ with $f_{j,k}\in \mathbb{C}^m$. In terms of these Fourier series, equation \eqref{tworeducedequations1} becomes $$\sum_{k\in \mathbb{Z}^m} (i\langle \omega,k\rangle g_ {j,k} + f_{j,k}) e^{i\langle k, \phi\rangle} = \sum_{k\in \mathbb{Z}^m} U_{j,k} e^{i\langle k, \phi\rangle}\, ,$$ or, equivalently, $$ i\langle \omega,k\rangle g_ {j,k} + f_{j,k} = U_{j,k}\ \mbox{for all}\ k\in \mathbb{Z}^m\, . $$
This shows that for any choice of Fourier coefficients $U_{j,k}$ for $U_j$ and $g_{j,k}$ for ${\bf g}_j$ there are unique Fourier coefficients $f_{j,k}$ for the solution ${\bf f}_j$ to the tangential homological equation. These coefficients are given by \begin{align}\label{fintermsofuandx} f_{j,k} = U_{j,k} - i\langle \omega,k\rangle g_ {j,k} \ \mbox{for all}\ k\in \mathbb{Z}^m\, . \end{align} It is clear from this equation that $f_{j,-k} = \overline{f}_{j,k}$ so that ${\bf f}_j$ is real-valued.
We proceed to solve the normal homological equation \eqref{tworeducedequations2}. We again use Fourier series, and thus we expand ${\bf h}_j$ and $V_j$ as \begin{align}\label{VYexpansion} {\bf h}_{j}(\phi) = \sum_{k\in \mathbb{Z}^m} h_{j,k} e^{i\langle k, \phi\rangle} \ \mbox{and}\ V_{j}(\phi) = \sum_{k\in \mathbb{Z}^m} V_{j,k} e^{i\langle k, \phi\rangle } \, , \end{align} for $h_{j,k}, V_{j,k}\in \mathbb{C}^{M-m}$ satisfying $V_{j,-k} = \overline{V}_{j,k}$.
Substitution of \eqref{VYexpansion} into \eqref{tworeducedequations2} produces $$\sum_{k\in \mathbb{Z}^m} (i\langle \omega, k \rangle - L ) { h}_{j,k} e^{i\langle k, \phi\rangle} = \sum_{k\in \mathbb{Z}^m} V_{j,k} e^{i\langle k, \phi\rangle}\, , $$ so that we obtain the equations \begin{align} \label{iomegaLequation} ( i\langle \omega,k\rangle - L ) \, { h}_{j,k} = V_{j,k} \ \mbox{for all}\ k\in \mathbb{Z}^m\, . \end{align} Because $L$ has no eigenvalues on the imaginary axis, the matrix $i\langle \omega,k\rangle - L$ is invertible. Each of the equations in \eqref{iomegaLequation} therefore possesses a unique solution, which is given by \begin{equation}\label{YintermsofV} { h}_{j,k} = ( i\langle \omega,k\rangle - L )^{-1} V_{j,k} \, . \end{equation} Because the matrix $L$ is real, it follows that $h_{j,-k} = \overline{h}_{j,k}$. This proves the theorem. \end{proof}
\begin{remk} Formulas \eqref{fintermsofuandx} and \eqref{YintermsofV} allow us to estimate the smoothness of the solutions ${\bf f}_j$ and ${\bf h}_j$ to equations \eqref{tworeducedequations1}, \eqref{tworeducedequations2} in terms of the smoothness of ${\bf g}_j, U_j$ and $V_j$. To see this, let ${\bf A}: (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{C}^p$ be a function with Fourier series $${\bf A}(\phi) = \sum_{k\in \mathbb{Z}^m} A_ke^{i\langle k, \phi\rangle}\, .$$
For $k\in \mathbb{Z}^m$, define $|k|:= \left( |k_1|^2+\ldots+|k_m|^2 \right)^{\frac{1}{2}}$, and let $W_{|k|} \in \mathbb{R}_{>0}$ be weights satisfying $W_{|k|}\to\infty$ as $|k|\to\infty$. When $||\cdot||$ is any norm on $\mathbb{C}^p$, then
$$|| {\bf A}||_W := \left( \sum_{k\in \mathbb{Z}^m} ||A_k||^2 W_{|k|}^2 \right)^{\frac{1}{2}}$$
defines a norm of ${\bf A}$ that measures the growth of its Fourier coefficients. For example, when $W_{|k|}=(1+|k|^2)^{s/2}$ for some $s > 0$, then it is a Sobolev norm.
It follows directly from \eqref{fintermsofuandx} that $|| {\bf f}_j ||_W \leq ||U_j||_W + || \partial_{\omega} {\bf g}_j||_W$, which shows that ${\bf f}_j$ is at least as smooth as $U_j$ and $\partial_{\omega} {\bf g}_j$.
To find a similar bound for $||{\bf h}_j||_W$, note that the hyperbolicity of $L$ implies that the function $\lambda \mapsto ||(i\lambda - L)^{-1}||_{\rm op}$ on $\mathbb{R}$, that assigns to $\lambda$ the operator norm of $(i\lambda - L)^{-1}$, is well-defined, and therefore also continuous. It converges to $0$ as $\lambda \to \pm \infty$. Hence it is uniformly bounded in $\lambda$. In particular,
$$||(i\langle k, \omega \rangle - L)^{-1}||_{\rm op} \leq C_L := \max_{\lambda\in\mathbb{R}} ||(i\lambda - L)^{-1}||_{\rm op}\, . $$ It thus follows from \eqref{YintermsofV} that
$$||{\bf h}_j||_W \leq C_L ||V_j||_W\, .$$ This means that ${\bf h}_j$ is at least as smooth as $V_j$. \end{remk}
\noindent Theorem \ref{solutionlemma} shows that one can choose ${\bf g}_j$ (and thus the component of $e_j$ tangent to $\mathbb{T}_0$) freely when solving the homological equations \eqref{tworeducedequations1} and \eqref{tworeducedequations2}. This reflects the fact that the embedding of $\mathbb{T}_{\varepsilon}$ is not unique. Corollary \ref{normalformcorollary} below states that it is possible to choose ${\bf g}_j$ in such a way that ${\bf f}_j$ is in ``normal form''. We first define this concept. \begin{defi} Let $${\bf f} = \omega + \varepsilon {\bf f}_1 + \varepsilon^2 {\bf f}_2 + \ldots : (\mathbb{R}/2\pi\mathbb{Z})^m \to \mathbb{R}^m$$ be an asymptotic expansion of a vector field on $(\mathbb{R}/2\pi\mathbb{Z})^m$. Assume that the Fourier series of ${\bf f}_j$ is given by $${\bf f}_j(\phi) = \sum_{k\in \mathbb{Z}^m} f_{j, k}e^{i\langle k, \phi\rangle} \ \mbox{for certain}\ f_{j,k}\in \mathbb{C}^m\, .$$
For $k\in \mathbb{Z}^m$, denote $|k|= \left( |k_1|^2+\ldots+|k_m|^2 \right)^{\frac{1}{2}}$ as before. We say that ${\bf f}_j$
is {\it in normal form to order $K\in \mathbb{N}\cup\{\infty\}$ in its Fourier expansion} if $$f_{j,k} = 0 \ \mbox{for all}\ k\in\mathbb{Z}^m \ \mbox{with} \ \langle \omega, k \rangle \neq 0\ \mbox{and} \ |k|\leq K\, .$$ \end{defi}
\begin{remk} We remark that ${\bf f}_j$ is in normal form to order $K$ in its Fourier expansion, if and only if its truncated Fourier series $${\bf f}_j^K(\phi) := \sum_{|k|\leq K} f_{j,k} e^{i\langle k, \phi\rangle}$$ depends only on so-called {\it resonant combination angles}. A combination angle $\langle k, \phi\rangle$ is called resonant when $\langle k, \omega\rangle =0$. \end{remk} \noindent The following result shows that we can arrange for the reduced phase vector field to be in normal form to arbitrarily high-order in its Fourier expansion. \begin{cor}\label{normalformcorollary} For any (finite) $K\in \mathbb{N}$ the function ${\bf g}_j$ can be chosen in such a way that the solution ${\bf f}_j$ to the tangential homological equation $$\partial_{\omega}{\bf g}_j + {\bf f}_j = U_j$$ is in normal form to order $K$ in its Fourier expansion. \end{cor} \begin{proof} Recall that the tangential homological equation reduces to the equations \begin{align}\label{tangentialfourier} i\langle \omega,k\rangle g_ {j,k} + f_{j,k} = U_{j,k} \end{align} for the Fourier coefficients of ${\bf f}_j$, ${\bf g}_j$ and $U_j$---see \eqref{fintermsofuandx}. Given $K\in \mathbb{N}$, choose \begin{align}\label{XintermsofU} \begin{array}{ll}
g_{j,k} = \frac{U_{j,k} }{i \langle k, \omega \rangle} & \mbox{when}\ \langle k, \omega \rangle \neq 0 \ \mbox{and}\ |k|\leq K, \\
g_{j,k} = 0 & \mbox{when} \ \langle k, \omega \rangle = 0 \ \mbox{or} \ |k|> K. \end{array} \end{align}
The (unique) solutions to \eqref{tangentialfourier} are then given by \begin{align} \begin{array}{ll}
f_{j,k}=0 & \mbox{when}\ \langle k, \omega \rangle \neq 0 \ \mbox{and}\ |k|\leq K, \\
f_{j,k}=U_{j,k} & \mbox{when} \ \langle k, \omega \rangle = 0 \ \mbox{or} \ |k|> K. \end{array} \end{align} With these choices, ${\bf g}_j$ is a smooth function, as its Fourier expansion is finite. It is also clear that ${\bf f}_j$ is in normal form to order $K$ in its Fourier expansion.
\end{proof}
\begin{remk} Recall that the flow of the ODE $\dot \phi = \omega$ on $(\mathbb{R}/2\pi\mathbb{Z})^m$ is periodic or quasi-periodic and given by the formula $\phi \mapsto \phi + \omega t \ \mbox{mod}\, (2\pi \mathbb{Z})^m$.
It follows that the time-average over this (quasi-)periodic flow, of a complex exponential vector field $f_k e^{i\langle k, \phi\rangle}$ (with $f_k\in \mathbb{C}^m$) is given by \begin{align}\nonumber
\lim_{T\to \infty} \frac{1}{T} \int_0^T & f_ke^{i\langle k, \phi + \omega t \rangle} \, dt = \left\{ \begin{array}{ll} f_k e^{i\langle k, \phi\rangle} & \mbox{when}\ \langle k, \omega \rangle = 0 \, , \\ 0 & \mbox{when} \ \langle k, \omega \rangle \neq 0 \, . \end{array} \right. \end{align} This shows that $f_k e^{i\langle k, \phi\rangle}$ is resonant (that is: it depends on a resonant combination angle) precisely when it is equal to its average over the (quasi-)periodic flow, whereas $f_ke^{i\langle k, \phi\rangle}$ is nonresonant precisely when this average is zero. For an arbitrary (and sufficiently regular) Fourier series it follows that $$\lim_{T\to\infty} \frac{1}{T} \int_0^T \left( \sum_{k\in\mathbb{Z}^m} f_k\, e^{i\langle k, \phi+\omega t\rangle} \right) dt = \sum_{\footnotesize \begin{array}{c} k\in \mathbb{Z}^m \\ \langle \omega, k\rangle = 0 \end{array}} \!\!\!\! f_k \, e^{i\langle k, \phi\rangle}\, .$$ We conclude that averaging a Fourier series removes its nonresonant terms, while keeping its resonant terms untouched. Corollary \ref{normalformcorollary} shows that it can be arranged that the term ${\bf f}_j$ in the reduced vector field ${\bf f}$ is a sum of resonant terms only (to arbitrarily high order). We may thus loosely interpret Corollary \ref{normalformcorollary} as a high-order averaging theorem, see \cite{sanvermur}. \end{remk}
\begin{remk} We include the following result for completeness. Applied to $A=e_0'(\phi)$ and $B=N(\phi)$ it gives a formula for the projection $\pi(\phi)$ onto the tangent space to $\mathbb{T}_0$ at $e_0(\phi)$ along the fast fibre at that point. This formula is not only useful for practical computations, but also shows explicitly that $\pi(\phi)$ depends smoothly on $\phi$. A proof of Proposition \ref{projectionprop} is given in \cite{BobIanMartin}. \end{remk} \begin{prop}\label{projectionprop} Let $1\leq m \leq M$ and assume that $A \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^M)$ and $B \in \mathcal{L}(\mathbb{R}^{M-m}, \mathbb{R}^M)$ are linear maps satisfying $\mathbb{R}^M = {\rm im}\, A \oplus {\rm im}\, B$. We denote by $\pi\in \mathcal{L}(\mathbb{R}^M, \mathbb{R}^M)$ the ``oblique projection'' onto the image of $A$ along the image of $B$, i.e., $\pi$ is the unique linear map satisfying $\pi A = A$ and $\pi B=0$. Then $\pi$ is given by the formula $$\pi = A (A^T\pi(B)^{\perp} A)^{-1}A^T\pi(B)^{\perp} \ \mbox{in which}\ \pi(B)^{\perp} := (1 - B(B^TB)^{-1}B^T)\, .$$ The $T$ denotes matrix transpose. All the inverses in this formula exist. Note that $\pi(B)^{\perp}$ is the orthogonal projection onto $\ker B^T$ along ${\rm im}\, B$. \end{prop}
\begin{comment} \begin{lem}
Maybe there is a nice way to estimate the norm of $(i\langle \omega,k\rangle - M_0 )^{-1}$ in terms of the norm of $M_0^{-1}$. \end{lem} \begin{proof}
For any real matrix $A$ and real number $\Omega$ we will have $$||A+i\Omega||^2_{\rm Frob} = {\rm tr} (A+i\Omega)(A^T-i\Omega) = {\rm tr}( AA^T + \Omega^2) \geq {\rm tr}( AA^T) = || A||^2_{\rm Frob}\, . $$ We also have
$$||A||^2_{\rm Op} = \sup_{v^*v=1} v^*(A^T-i\Omega)(A+i\Omega) v = \sup_{v^*v=1} v^*(A^TA+\Omega^2) v = \Omega^2 + \sup_{v^*v=1} v^*A^TA v = \Omega^2 + ||A||_{\rm Op}^2$$ because $i\Omega v^*(A^T-A)v=0$ because $A^T-A$ is anti-symmetric. How do we use this for an estimate for the inverse? \end{proof} \end{comment}
\section{Reducibility for oscillator systems}\label{Floquetsection} In this section we show that the invariant torus of a system of uncoupled oscillators (see the introduction) is reducible. We also give a formula for the fast fibre map for such a torus. The results in this section are a consequence of Floquet's theorem, which implies that the invariant circle defined by a single hyperbolic periodic solution of an ODE is reducible. The results in this section should thus be considered well-known, but for completeness we include them in detail. We start with the result for single hyperbolic periodic orbits.
\begin{thr}\label{floquetreducible} Let $X: \mathbb{R} \to \mathbb{R}^M$ be a hyperbolic $T$-periodic orbit of a smooth vector field ${\bf F}: \mathbb{R}^M\to \mathbb{R}^M$. Then the invariant circle $\mathbb{T}_0=X(\mathbb{R}) \subset \mathbb{R}^M$ is reducible and normally hyperbolic. Its fast fibre map is given by formula \eqref{fastmapperiodic}. \end{thr}
\begin{proof} Assume that the ODE $\dot x = {\bf F}(x)\ \mbox{on}\ \mathbb{R}^{M}$ possesses a hyperbolic periodic orbit $X = X(t)$ with minimal period $T>0$. We think of it as an invariant circle $\mathbb{T}_0$ embedded by the map $e_0: \mathbb{R}/2\pi\mathbb{Z} \to \mathbb{R}^M$ defined by $e_0(\phi) := X(\omega^{-1} \phi)$, where $\omega:=\frac{2\pi}{T}$. Let $\Phi = \Phi(t) \in {\rm GL}(\mathbb{R}^{M})$ be the principal fundamental matrix solution of the linearisation around this periodic orbit.
This means that $$ \dot \Phi(t) = {\bf F}'(X(t))\cdot \Phi(t) \ \mbox{and}\ \Phi(0)= {\rm Id}_{\mathbb{R}^M} \, .$$ Floquet's theorem \cite{floquet, teschl} states that $\Phi(t)$ admits a factorisation $$\Phi(t) = P(t) e^{Bt}\ \mbox{with}\ P(t+T) = P(t)\ \mbox{and}\ P(0)= {\rm Id}_{\mathbb{R}^M} \, .$$ The constant (and perhaps complex) Floquet matrix $B$ satisfies $e^{BT} = \Phi(T)$, for example $B=\frac{1}{T}\log \Phi(T)$ for a choice of matrix logarithm. Note that a matrix logarithm of $\Phi(T)$ exists because $\Phi(T)$ is invertible. We shall assume here that $B$ is a real matrix. This can always be arranged by replacing $T$ by $2T$ and considering a double cover of $\mathbb{T}_0$ if necessary, but we ignore this (somewhat annoying) subtlety here.
Substituting the Floquet decomposition in the definition of the fundamental matrix solution, we obtain that $\dot P(t) e^{Bt} + P(t) B e^{Bt} = {\bf F}'(X(t)) P(t) e^{Bt}$. Thus, $$\dot P(t)+ P(t) B = { \bf F}'(X(t)) P(t) \, .$$ This implies that we found a solution to Equation \eqref{N0M0} in Lemma \ref{variationalparametrisation}. Indeed, if we define $$\tilde L=B \ \mbox{and}\ \tilde N(\phi) = P(\omega^{-1}\phi) \, , $$ then we have, recalling that $e_0(\phi)=X(\omega^{-1}\phi)$), \begin{align} \nonumber \partial_{\omega}\tilde N(\phi) + \tilde N (\phi)\cdot \tilde L & = \tilde N'(\phi)\cdot \omega + \tilde N(\phi) \cdot \tilde L \\ \nonumber & = \dot P(\omega^{-1} \phi) + P(\omega^{-1}\phi) \cdot B \\ \nonumber & = {\bf F}'(X(\omega^{-1}\phi)) \cdot P(\omega^{-1}\phi) = { \bf F}'(e_0(\phi))\cdot \tilde N(\phi)\, . \end{align} However, this does not yet prove that the periodic orbit is reducible, because $\tilde N=\tilde N(\phi)$ defines a family of $M\times M$-matrices, and hence the image of $\tilde N(\phi)$ is not normal to the tangent vector $\omega e_0'(\phi) = \dot X(\omega^{-1}\phi) $ to the periodic orbit. To resolve this issue, recall that $\Phi(T)$ always has a unit eigenvalue. This follows from
differentiating the identity $\dot X(t) = {\bf F}(X(t))$ to $t$, which gives that $\frac{d}{dt} \dot X(t) = {\bf F}'(X(t))\cdot \dot X(t)$, so that $$\dot X(0) = \dot X(T) = \Phi(T)\cdot \dot X(0)\, .$$ Because $\Phi(T) = e^{BT}$, we conclude that $B$ has a purely imaginary eigenvalue in $\frac{2\pi i }{T} \mathbb{Z}$. Our assumption that $X$ is hyperbolic implies that none of the other eigenvalues of $B$ lie on the imaginary axis. Because $B$ is real and its eigenvalues must thus come in complex conjugate pairs, we conclude that the purely imaginary eigenvalue of $B$ must in fact be zero.
We now choose an injective linear map $A: \mathbb{R}^{M-1}\to \mathbb{R}^{M}$ whose image coincides with the $(M-1)$-dimensional image of $B$. For any such choice of $A$ there is a unique map $L: \mathbb{R}^{M-1}\to\mathbb{R}^{M-1}$ for which $$A \cdot L = B \cdot A\, . $$ Clearly, the eigenvalues of $L$ are the nonzero eigenvalues of $B$, showing that $L$ is hyperbolic. We also define $N: \mathbb{R}/2\pi\mathbb{Z} \to \mathcal{L}(\mathbb{R}^{M-1}, \mathbb{R}^{M})$ by \begin{align}\label{fastmapperiodic} N(\phi) := P(\omega^{-1}\phi) A \, . \end{align} By definition, ${\rm im}\, N(0)={\rm im}\, A = {\rm im}\, B$ is transverse to the tangent vector $\dot X(0) \in \ker B$ to the periodic orbit. Because each $P(t)$ is invertible, this transversality persists along the entire orbit. Indeed, writing $t = \omega^{-1}\phi$, note that $$ \dot X(t) = \Phi(t) \dot X(0) = P(t) e^{Bt} \dot X(0) = P(t) \dot X(0) \in P(t)({\rm ker}\, B)$$ is transversal to ${\rm im}\, N(\phi) = {\rm im}(P(t)A) = P(t) ({\rm im}\, B)$. Finally, we compute \begin{align} \nonumber \partial_{\omega}N(\phi) + N(\phi)L & = N'(\phi)\, \omega + N(\phi)L \\ \nonumber & = \dot P(\omega^{-1} \phi) A + P(\omega^{-1}\phi) A L \\ \nonumber & = \dot P(\omega^{-1} \phi) A + P(\omega^{-1}\phi) B A \\ \nonumber & = {\bf F}'(X(\omega^{-1}\phi)) P(\omega^{-1}\phi)A = {\bf F}'(e_0(\phi)) N(\phi)\, . \end{align} This proves that the invariant circle $\mathbb{T}_0$ defined by $X(t)$ is reducible. \end{proof}
\begin{ex}\label{exstuartlandau} As an example consider a single Stuart-Landau oscillator \begin{align} \label{stuartlandau}
\dot z = \left(\alpha + i \beta \right) z + \left(\gamma + i \delta \right) |z|^2 z \quad \mbox{for}\ z \in \mathbb{C} \cong \mathbb{R}^2 \, . \end{align}
Here $\alpha, \beta, \gamma, \delta \in \mathbb{R}$ are parameters. We assume that $\alpha \gamma < 0$ and $\alpha \delta - \beta\gamma \neq 0$, so that \eqref{stuartlandau} possesses a unique (up to rotation) circular periodic orbit $$X(t) = R e^{i\omega t} \ \mbox{where}\ R := \sqrt{-\alpha /\gamma } \ \mbox{and} \ \omega := \beta - \alpha \delta / \gamma \neq 0\, .$$ Thus, the embedding $$e_0: \mathbb{R}/2\pi\mathbb{Z} \ni \phi \mapsto z := R \, e^{i \phi} \in \mathbb{C} $$ sends solutions of $\dot \phi = \omega$ on $\mathbb{R}/2\pi\mathbb{Z}$ to solutions of \eqref{stuartlandau}. The Floquet decomposition of the fundamental matrix solution around this periodic orbit can be found by anticipating that $P(t)=e^{i\omega t}$ and thus making the ansatz $$\Phi(t) = e^{i\omega t} e^{Bt}\, $$ for an unknown linear map $B: \mathbb{C} \to \mathbb{C}$. With this in mind we expand solutions to \eqref{stuartlandau} nearby the periodic orbit as $$z(t)=R\, e^{i\omega t} + \varepsilon \, e^{i\omega t} v(t) \, .$$ To first order in $\varepsilon$ this gives the linear differential equations
$$\dot v = \dot v_1 + i \dot v_2 = 2 R^2 (\gamma + i \delta) v_1\, ,$$
which shows that the Floquet map $B: \mathbb{C} \to \mathbb{C}$ must be given by $$B(v_1 + i v_2)= 2 R^2 (\gamma + i \delta) v_1 \, . $$ This $B$ has an eigenvalue $0$ (with eigenvector $i$ corresponding to the tangent space to the invariant circle) and an eigenvalue $2\gamma R^2 = - 2 \alpha \neq 0$ (with eigenvector $\gamma +i\delta$). We conclude that the map $${\bf N}e_0: (\phi, u) \mapsto (R e^{i\phi}, e^{i\phi} (\gamma + i \delta) u) \ \mbox{from}\ \mathbb{R}/2\pi\mathbb{Z} \times \mathbb{R} \ \mbox{to}\ \mathbb{C}\times \mathbb{C}$$ sends solutions of $$\dot \phi = \omega\, , \, \dot u = - 2\alpha \, u \ \mbox{for} \ \phi\in\mathbb{R}/2\pi\mathbb{Z} \ \mbox{and}\ u \in \mathbb{R} $$ to solutions of the linearised dynamics of \eqref{stuartlandau} on $\mathbb{C}\times \mathbb{C}$ around the invariant circle. In particular, we have $L= - 2 \alpha$ and $N(\phi)= e^{i\phi} ( \gamma + i \delta )$. The projection onto the tangent bundle of the invariant circle along its fast fibre bundle is given by the formulas $$\pi(0)\cdot (x + i y) = i (y - (\delta / \gamma) x)\ \mbox{and}\ \pi(\phi) = e^{i\phi}\cdot \pi(0)\cdot e^{-i\phi} \, .$$ Indeed, it is easy to check that $\pi(\phi)\cdot i e^{i\phi} = i e^{i\phi}$ and $\pi(\phi)\cdot e^{i\phi} ( \gamma + i \delta ) = 0$. \end{ex} \noindent
We now extend the result of Theorem \ref{floquetreducible} to systems of multiple uncoupled oscillators, that is, systems of the form $$\dot x_1 = F_1(x_1)\, , \, \ldots\, ,\, \dot x_m = F_m(x_m)\, \ \mbox{with}\ x_j\in \mathbb{R}^{M_j}\, ,$$ that each have a hyperbolic $T_j$-periodic orbit $X_j(t)$. Recall that the product of these periodic orbits forms an invariant torus. The fact that this torus is reducible follows from the following lemma. Its proof is straightforward, but included here for completeness. \begin{lem} \label{productreduciblelemma} Let $\mathbb{T}_1\subset \mathbb{R}^{M_1}$ and $\mathbb{T}_2\subset \mathbb{R}^{M_2}$ be embedded reducible normally hyperbolic (quasi-)periodic invariant tori for the vector fields ${\bf F}_1$ and ${\bf F}_2$ respectively. Then the product torus $\mathbb{T}_0:=\mathbb{T}_1\times \mathbb{T}_2 \subset \mathbb{R}^M$ (with $M:=M_1+M_2$) is an embedded reducible normally hyperbolic quasi-periodic invariant torus for the product vector field ${\bf F}_0$ on $\mathbb{R}^{M}$ defined by ${\bf F}_0(x_1, x_2) :=({\bf F}_1(x_1), {\bf F}_2(x_2))$. \end{lem}
\begin{proof} Assume that $e_j:(\mathbb{R}/2\pi\mathbb{Z})^{m_j}\to \mathbb{R}^{M_j}$ (for $j=1, 2$) is an embedding of a reducible normally hyperbolic (quasi-)periodic invariant torus for the vector field ${\bf F}_j$. This means that there are frequency vectors $\omega_j\in \mathbb{R}^{m_j}$ such that $\partial_{\omega_j}e_j = {\bf F}_j \circ e_j$ and fast fibre maps
${\bf N}e_j: (\mathbb{R}/2\pi\mathbb{Z})^{m_j} \times \mathbb{R}^{M_j-m_j} \to \mathbb{R}^{M_j} \times \mathbb{R}^{M_j}$ of the form ${\bf N}e_j(\phi_j, u_j) = (e_j(\phi_j), N_j(\phi_j)\cdot u_j)$ satisfying $\partial_{\omega_j}N_j + N_j \cdot L_j = ( {\bf F}_j'\circ e_j) \cdot N_j$ for certain hyperbolic Floquet matrices $L_j$.
If we now define $m:=m_1+m_2$, $\omega:=(\omega_1, \omega_2)\in \mathbb{R}^{m}$ and $e_0: (\mathbb{R}/2\pi\mathbb{Z})^{m} \to \mathbb{R}^{M}$ by $e_0(\phi) = e_0(\phi_1, \phi_2) := (e_1(\phi_1), e_2(\phi_2))$, then $e_0$ is clearly an embedding of $\mathbb{T}_0$ and the equality $\partial_{\omega}e_0 = {\bf F}_0 \circ e_0$ holds. In other words, the product torus $\mathbb{T}_0$ is an embedded quasi-periodic invariant torus for ${\bf F}_0$.
If we also define $N(\phi) \cdot u = N(\phi_1, \phi_2) \cdot (u_1, u_2) := (N_1(\phi_1)\cdot u_1, N_2(\phi_2)\cdot u_2)$, then clearly $N(\phi)$ is injective, and therefore ${\bf N}e_0: (\mathbb{R}/2\pi\mathbb{Z})^{m} \to \mathbb{R}^M \times \mathbb{R}^M$ defined by $${\bf N}e_0((\phi_1, \phi_2), (u_1, u_2)) = (e_0(\phi_1, \phi_2), N(\phi_1, \phi_2)\cdot(u_1, u_2))$$ is a fast fibre map for $\mathbb{T}_0$ that satisfies $\partial_{\omega}N + N \cdot L = ({\bf F}_0' \circ e_0)\cdot N$. Here $L: \mathbb{R}^{M-m}\to \mathbb{R}^{M-m}$ is defined by $L(u_1, u_2) := (L_1 u_1, L_2 u_2)$. This $L$ is hyperbolic, its eigenvalues being those of $L_1$ and $L_2$. This proves that $\mathbb{T}_0$ is reducible and normally hyperbolic and concludes the proof of the lemma. \end{proof}
\begin{comment} \begin{remk} If $B$ defined by $\Phi(T) = e^{BT}$ is not a real matrix, then we may instead define $P(t)$ by the equation $\Phi(t) = P(t) e^{\frac{(B+\bar B)}{2}t}$. Then $\Phi(T)^2 = \Phi(2T) = P(2T)e^{(B+\bar B)T} = P(2T)( e^{BT} e^{\bar B T}) = P(2T)(e^{BT})^2$. It follows that $P(2T)=\mbox{Id} = P(0)$ because $\Phi(T)=e^{BT}$. \end{remk} \end{comment} \begin{comment}
\begin{ex} The Ansatz $$z_j = R_j e^{i\phi_j}$$ leads to the equations $$\dot R_j + i R_j \dot \phi_j = \left( \mu_jR_j + \alpha_jR_j^3 \right) + i (\omega_j R_j+\beta_j R_j^3) + \varepsilon \sum_{k} R_k \left(A_{jk} e^{i(\phi_k-\phi_j)} + B_{jk} e^{-i(\phi_k+ \phi_j)} \right) $$ or equivalently \begin{align} \nonumber \dot R_j & = \mu_jR_j + \alpha_j R_j^3 + \varepsilon \sum_k R_k \, {\rm Re}\, \left(A_{jk} e^{i(\phi_k-\phi_j)} + B_{jk} e^{-i(\phi_k+ \phi_j)} \right) \\ \nonumber \dot \phi_j & = \omega_j +\beta_j R_j^2 + \varepsilon R_j^{-1} \sum_k R_k \ {\rm Im}\, \left(A_{jk} e^{i(\phi_k-\phi_j)} + B_{jk} e^{-i(\phi_k+ \phi_j)} \right) \end{align}
These real and imaginary parts in the perturbation part are clearly linear expressions in sine and cosine of $\phi_k\pm \phi_j$. \end{ex}
\end{comment}
\section{Application to remote synchronisation} \label{sec:examplesection} In this final section we apply and illustrate our phase reduction method in a small network of three weakly linearly coupled Stuart-Landau oscillators \begin{align}\label{3stuartlandau} \begin{array}{llll}
\dot{z}_1 = & \hspace{-2mm} (\alpha + i\beta)z_1 & \hspace{-2mm} + \hspace{1mm} (\gamma + i\delta)|z_1|^2z_1 & \hspace{-2mm} + \hspace{1mm} \varepsilon z_2 \, , \\
\dot{z}_2 = & \hspace{-2mm} (a \, + \, \! ib)z_2 & \hspace{-2mm} + \hspace{1mm} (c + id)|z_2|^2z_2 & \hspace{-2mm} + \hspace{1mm} \varepsilon z_1 \, , \\
\dot{z}_3 = & \hspace{-2mm} (\alpha + i\beta)z_3 & \hspace{-2mm} + \hspace{1mm} (\gamma + i\delta)|z_3|^2z_3 & \hspace{-2mm} +\hspace{1mm} \varepsilon z_2 \, , \end{array} \end{align} with $z_1, z_2, z_3 \in \mathbb{C}$. Figure \ref{pictoscnetw} depicts the coupling architecture of this network. Note that the first and third oscillator in equations \eqref{3stuartlandau} are identical. We choose parameters so that each uncoupled oscillator has a nonzero hyperbolic periodic orbit, with frequencies $\omega_1=\omega_3 \neq \omega_2$. These periodic orbits form a $3$-dimensional invariant torus $\mathbb{T}_0$ for the uncoupled system, which persists as a perturbed torus $\mathbb{T}_{\varepsilon}$ for small nonzero coupling.
Despite that fact that the first and third oscillators in \eqref{3stuartlandau} are not coupled directly, a numerical study of equations \eqref{3stuartlandau} reveals that these oscillators synchronise when appropriate parameter values are chosen, see Figure \ref{firstnum}. This ``remote synchronisation'' appears to be mediated by the second oscillator, which allows the two other oscillators to communicate. Figure \ref{firstnum-c} demonstrates, again numerically, that the timescale of remote synchronisation is of the order $t\sim \varepsilon^{-2}$. This suggests that proving the synchronisation rigorously would require second-order phase reduction.
In \cite{remotestar}, remote synchronisation of Stuart-Landau oscillators was observed numerically for the first time. A first rigorous proof of the phenomenon, for a chain of three Stuart-Landau oscillators, occurs in \cite{rosenblum}. The proof in that paper employs the high-order phase reduction method developed in \cite{Gengel}. However, the method in \cite{Gengel} does not yield the reduced phase equations in normal form. As a result, the timescale $t\sim \varepsilon^{-2}$ is not observed in \cite{rosenblum}.
Here we apply the parametrisation method developed in this paper, to prove that the first and third oscillator in \eqref{3stuartlandau} synchronise over a timescale $t\sim \varepsilon^{-2}$.
We are also able to determine how the parameters in \eqref{3stuartlandau} influence this synchronisation. To this end, we will compute an asymptotic expansion of an embedding $e:(\mathbb{R}/2\pi\mathbb{Z})^3 \to \mathbb{C}^3$ and a
reduced phase vector field ${\bf f}: (\mathbb{R}/2\pi\mathbb{Z})^3 \to \mathbb{R}^3$ to second order in the small parameter. As we are primarily interested in the synchronisation of the first and third oscillator, we do not calculate the full reduced phase vector field. Instead, we only explicitly compute an evolution equation for the resonant combination angle $\Phi:=\phi_1 - \phi_3$.
We will show that \begin{align}\label{phi1minphi3} \dot \Phi = \varepsilon^2 \left( -A \sin \Phi +B (1 - \cos \Phi ) \right) + \mathcal{O}(\varepsilon^3) \, , \end{align} in which the constants $A$ and $B$ are given by the formulas
\begin{align}\label{AB} \begin{array}{ll}
A & \!\!\! = \frac{1}{4a^2+(\omega_1-\omega_2)^2} \left( \frac{\delta}{\gamma}(\omega_2-\omega_1) + a \left( 1 + \frac{d \delta}{c \gamma} \right) + 2a^2 \left(\frac{d}{c} + \frac{\delta}{\gamma} \right) \frac{1}{\omega_2-\omega_1} \right) \, ,
\\
B & \!\!\! = \frac{1}{4a^2+(\omega_1-\omega_2)^2}\left( (\omega_2-\omega_1) + a \left(\frac{d}{c} - \frac{\delta}{\gamma} \right) + 2a^2 \left(1 - \frac{d \delta}{c\gamma} \right) \frac{1}{\omega_2-\omega_1} \right) \, . \end{array}
\end{align}
\begin{figure}
\caption{Representation of the network of Stuart-Landau oscillators \eqref{3stuartlandau}.}
\label{pictoscnetw}
\end{figure} \noindent
Before we prove formulas \eqref{phi1minphi3} and \eqref{AB}, let us investigate their dynamical implications. After rescaling time $t\mapsto \tau := \varepsilon^{2} t$, equation \eqref{phi1minphi3} becomes
$$\frac{d\Phi}{d\tau} = \left( -A \sin \Phi + B(1- \cos \Phi ) \right) + \mathcal{O}(\varepsilon) \, .$$ For $\varepsilon=0$ the time-rescaled reduced flow on $(\mathbb{R}/2\pi\mathbb{Z})^3$ therefore admits a $2$-dimensional invariant torus $$S = \{\phi_1=\phi_3\} \subset (\mathbb{R}/2\pi\mathbb{Z})^3$$ on which the phases of the first and third oscillator are synchronised. This torus is stable when $A>0$ and unstable when $A<0$. For $A\neq 0$, there also exists exactly one $2$-dimensional invariant torus of the form $$P = \{\phi_1 = \phi_3 + c\}\subset (\mathbb{R}/2\pi\mathbb{Z})^3 \ \mbox{for some}\ c\neq 0\, $$ with the opposite stability type. The phases of the first and third oscillator are phase-locked but not synchronised on $P$. F\'enichel's theorem guarantees that both $S$ and $P$ persist as invariant submanifolds of $(\mathbb{R}/2\pi\mathbb{Z})^3$ for small $\varepsilon\neq 0$. Hence, so do their images $e(S), e(P) \subset \mathbb{T}_{\varepsilon}\subset \mathbb{C}^3$ as invariant manifolds for \eqref{3stuartlandau}.
For small $\varepsilon \neq 0$, a typical solution of \eqref{3stuartlandau} will therefore first converge to the $3$-dimensional invariant torus $\mathbb{T}_{\varepsilon}$ on a timescale of the order $t \sim 1$. It will subsequently converge to either $e(S)$ or $e(P)$ on the much longer timescale $t\sim \varepsilon^{-2}$, and it is this slow dynamics that governs the synchronisation of the first and third oscillator. This multiple timescale dynamical process is illustrated in Figure \ref{firstnum}. Figure \ref{firstnum-c} confirms numerically that the timescale of synchronisation of $z_1$ and $z_3$ is indeed of the order $\varepsilon^{-2}$.
\begin{remk}\label{remk:easyshapeA}
We point out that the parameters in \eqref{3stuartlandau} can be tuned so that either of the two low-dimensional tori $S$ or $P$ is the stable one. Assume for instance that $\alpha, a >0$ and $\gamma, c<0$, so that $\mathbb{T}_{0}$ (and hence $\mathbb{T}_{\varepsilon}$) is stable. If in addition we choose the parameters so that $c \delta + d \gamma = 0$, then the expression for $A$ simplifies to
$\frac{a + (b-\beta)(\delta/\gamma) + \alpha (\delta/\gamma)^2 }{4a^2 + (\omega_1-\omega_2)^2}$. If $\delta \neq 0$, then it is clear that we can make this both positive and negative, for instance by varying the parameter $b$. Interestingly, this shows that properties of the second oscillator may determine whether the first and third oscillator converge to the synchronised state $S$ or the phase-locked state $P$.
\end{remk}
\subsubsection*{Numerics}
\begin{figure}
\caption{Slow convergence of $\hat \Phi$ to zero.}
\label{firstnum-a}
\caption{Convergence of $\hat \Phi$ to a non-zero value.}
\label{firstnum-b}
\caption{Numerically obtained plots of the phase-difference $\hat \Phi = \Arg(z_1\overline{z_3}) \approx \phi_1 - \phi_3$ against time, for two different realisations of system \eqref{3stuartlandau}. }
\label{firstnum}
\end{figure}
\noindent Before proving \eqref{phi1minphi3} and \eqref{AB}, we present some numerical results on system \eqref{3stuartlandau}. Figure \ref{firstnum} shows numerically obtained plots of $\hat \Phi = \Arg(z_1\overline{z_3})$ against time, for two different realisations of system \eqref{3stuartlandau}. We use $\hat \Phi$ as a proxy for $\Phi = \phi_1 - \phi_3$. As this approximation does not take into account the distortion of the perturbed invariant torus, we observe small amplitude, rapid oscillations in $\hat \Phi$, causing the lines in Figure \ref{firstnum} to be thick. In Figure \ref{firstnum-a}, we have chosen the parameter values \begin{equation}\label{convergentvalues} \begin{array}{llll} \alpha = 1 & \beta= 1 & \gamma = -1 & \delta = 1\, ;\\ a = 1 & b= 2 & c = -1 & d = -1\, , \end{array} \end{equation} together with $\varepsilon = 0.1$. It follows that $c \delta + d \gamma = 0$, and so $A = \frac{1}{5} >0$,
see Remark \ref{remk:easyshapeA}. The above analysis therefore predicts that $\hat \Phi$ should converge to zero, which the figure indeed shows. The convergence is very slow, as only around $t= 2000$ do we find that $\hat \Phi$ is indistinguishably close to zero. We will comment more on the rate of convergence below. Figure \ref{firstnum-a} was generated using Euler's method with time steps of $0.05$, starting from the point in phase space $(z_1, z_2, z_3) = (-1,1+0.4i, -1+0.3i) \in \field{C}^3$.
For Figure \ref{firstnum-b} we have likewise set $\varepsilon = 0.1$, but have instead chosen \begin{equation} \begin{array}{llll} \alpha = 1 & \beta= 0.1 & \gamma = -1 & \delta = 1\, ;\\ a = 1 & b= 6 & c = -1 & d = -1\, , \end{array} \end{equation} which yields $ A =
\frac{-3.9 }{4 + (3.9)^2} = -0.203\ldots < 0$. Hence, our theory predicts $\hat \Phi$ to converge to a non-zero constant value, which is indeed seen to be the case. Again the thickness of the line is due to rapid oscillations. Figure \ref{firstnum-b} is generated in the same way as Figure \ref{firstnum-a}, except that the starting point for Euler's method is now $(z_1, z_2, z_3) = (1+0.3i,1+0.4i, -0.2+0.9i)$.
\begin{figure}
\caption{Log-log plot of the time $T_{0.1}$ it takes for $\hat \Phi$ to decrease by a factor of $10$, against the coupling parameter $\varepsilon$.}
\label{firstnum-c}
\end{figure}
Finally, Figure \ref{firstnum-c} displays the rate of convergence to synchrony as a function of $\varepsilon$. The figure was made using Euler's method with time-steps of $0.05$, all starting from the same point $(z_1, z_2, z_3) = (-1+0.3i,1+0.4i, -1+0.5i)$.
We have again chosen the parameters as in \eqref{convergentvalues}, so that we may expect $\hat \Phi$ to converge to zero. However, the rate at which this occurs depends on $\varepsilon$. We measure this rate by recording $T_{0.1}$, which is the smallest time $t$ for which $|\hat \Phi(t)| \leq 0.1 |\hat \Phi(0)|$.
Figure \ref{firstnum-c} shows a log-log plot of $T_{0.1}$ against $\varepsilon$. The crosses in the figure represent numerical results for $20$ different values of $\varepsilon$. Shown in green is the line with slope $-2$ through the leftmost cross. We see that $\ln(T_{0.1}) = -2\ln(\varepsilon) + C$ for some $C \in \field{R}$ to very good approximation. Hence we find $T_{0.1} \sim \varepsilon^{-2}$, which is fully in agreement with our predictions.
\subsubsection*{Setup: the unperturbed problem} We now start our proof of formulas \eqref{phi1minphi3} and \eqref{AB}. We first recall some observations from Example \ref{exstuartlandau}, and make assumptions on the parameters that appear in \eqref{3stuartlandau}. Specifically, we assume that these parameters are chosen so that $$\alpha \gamma < 0, \ a c <0,\ \beta\gamma - \alpha \delta \neq 0, \ bc - a d \neq 0\, \ \mbox{and}\ \omega_1 = \omega_3 \neq \omega_2\, .$$ Recall from Example \ref{exstuartlandau} that this ensures that all three uncoupled oscillators possess a unique hyperbolic periodic orbit, with nonzero frequencies $\omega_1 = \omega_3 = \beta - \alpha \delta / \gamma$ and $\omega_2 = b - a d / c \neq \omega_1$. The product of these periodic orbits forms a $3$-dimensional reducible normally hyperbolic (quasi-)periodic invariant torus $\mathbb{T}_0 \subset \mathbb{C}^3$.
An embedding of $\mathbb{T}_0$ is given by $$e_0: (\mathbb{R}/2\pi\mathbb{Z})^3 \to \mathbb{C}^3 \ \mbox{defined by}\ e_0(\phi_1, \phi_2, \phi_3) = (R_1 \, e^{i \phi_1}, R_2 \, e^{i \phi_2}, R_3 \, e^{i \phi_3})\, $$ where $$R_1 = R_3 = \sqrt{-\alpha /\gamma }>0\ \mbox{and}\ R_2 = \sqrt{-a /c }>0\ . $$ This embedding sends integral curves of the constant vector field $\omega=(\omega_1, \omega_2, \omega_3)$ on $(\mathbb{R}/2\pi\mathbb{Z})^3$ to solutions of \eqref{3stuartlandau} (with $\varepsilon=0$) on $\mathbb{C}^3$.
It follows from Example \ref{exstuartlandau} and Lemma \ref{productreduciblelemma} that a Floquet matrix for $\mathbb{T}_0$ is $$L = {\rm diag}(-2 \alpha, -2 a , -2\alpha)\, ,$$
with corresponding fast fibre map given by the family of injective linear maps $N: (\mathbb{R}/2\pi\mathbb{Z})^3 \to \mathcal{L}(\mathbb{R}^3, \mathbb{C}^3)$ defined by
$$N(\phi_1, \phi_2, \phi_3) = {\rm diag}(e^{i\phi_1} (\gamma + i \delta), e^{i\phi_2} (c + i d), e^{i\phi_3} (\gamma + i \delta))\, .$$ The projection onto the tangent bundle along the fast fibre bundle is given by $$\pi(\phi_1, \phi_2, \phi_3) = {\rm diag}(e^{i\phi_1}\pi_1(0)e^{-i\phi_1}, e^{i\phi_2}\pi_2(0)e^{-i\phi_2}, e^{i\phi_3}\pi_3(0)e^{-i\phi_3})\, .$$ Here,
$$\pi_1(0) (x_1+ i y_1 ) = i (y_1 - (\delta/\gamma)x_1)\, , \ \pi_2(0) (x_2+ i y_2 ) = i (y_2 - (d/c)x_2)\, ,$$
and $\pi_3(0)=\pi_1(0)$.
\begin{comment} \subsubsection*{Discrete symmetry} Equations \eqref{3stuartlandau} admit a discrete symmetry. This symmetry is the (noninvertible) linear map $S: \mathbb{C}^3\to\mathbb{C}^3$ defined by $$S(z_1, z_2, z_3) = (z_1, z_2, z_1)\, .$$ Indeed, one may check directly that this $S$ sends solutions of \eqref{3stuartlandau} to solutions of \eqref{3stuartlandau}---irrespective of the value of the small parameter. The following proposition is a direct consequence of this observation: \begin{prop}\label{3sym} In addition to being in normal form to arbitrarily high-order in their Fourier expansion, the phase reduced vector field ${\bf f} = ({\bf f}^{(1)}, {\bf f}^{(2)}, {\bf f}^{(3)})$ on $(\mathbb{R}/2\pi\mathbb{Z})^3$ can be chosen so that \begin{align}\label{symrestrictions} {\bf f}^{(1)}(\phi_1, \phi_2, \phi_3) = {\bf f}^{(3)}(\phi_1, \phi_2, \phi_1) \ \mbox{and}\ {\bf f}^{(2)}(\phi_1, \phi_2, \phi_3) = {\bf f}^{(2)}(\phi_1, \phi_2, \phi_1) \, \end{align} to arbitrarily high-order in their Taylor and Fourier expansions. \end{prop} \begin{proof} This follows from Theorem \ref{discretesymmetrytheorem}. To be precise, we can define maps $s: (\mathbb{R}/2\pi\mathbb{Z})^3\to(\mathbb{R}/2\pi\mathbb{Z})^3$ and $t: \mathbb{R}^3\to\mathbb{R}^3$ by $s(\phi_1, \phi_2, \phi_3) = (\phi_1, \phi_2, \phi_1)$ and $t(u_1, u_2, u_3) = (u_1, u_2, u_1)$. It can readily be checked that, with these choices, the three conditions of Theorem \ref{discretesymmetrytheorem} are satisfied. Therefore, the phase reduced vector field ${\bf f}$ can be chosen to simultaneously be in normal form and satisfy $s\circ {\bf f} = {\bf f}\circ s$. It remains to remark that \begin{align} (s \circ {\bf f})(\phi) & = ({\bf f}^{(1)}(\phi_1, \phi_2, \phi_3) ,{\bf f}^{(2)}(\phi_1, \phi_2, \phi_3), {\bf f}^{(1)}(\phi_1, \phi_2, \phi_3)) \ \mbox{and} \nonumber \\ \nonumber ({\bf f} \circ s)(\phi) & = ({\bf f}^{(1)}(\phi_1, \phi_2, \phi_1) , {\bf f}^{(2)}(\phi_1, \phi_2, \phi_1), {\bf f}^{(3)}(\phi_1, \phi_2, \phi_1))\, . \end{align} It follows that $s\circ {\bf f} = {\bf f}\circ s$ if and only if equations \eqref{symrestrictions} hold. \end{proof} \noindent Proposition \ref{3sym} implies that it suffices to compute only the third components ${\bf f}_{j}^{(3)}(\phi_1, \phi_2, \phi_3)$ (for $j=1,2$) of the reduced vector field to obtain the desired equation \eqref{phi1minphi3} for $\frac{d}{dt}(\phi_1-\phi_3)$. We will exploit this fact below. \end{comment} \subsubsection*{The first tangential homological equation} We now compute ${\bf f}_1$ and ${\bf g}_1$ from the first tangential homological equation, see \eqref{tworeducedequations1}, with $U_1$ as given in \eqref{tworeducedequations}. A short calculation shows that the projection of the inhomogeneous term ${\bf G}_1(\phi) = {\bf F}_1(e_0(\phi)) = (R_2e^{i\phi_2}, R_1e^{i\phi_1}, R_2e^{i\phi_2})$ is
$$(\pi \cdot {\bf G}_1)(\phi) = \left( \begin{array}{c} i R_2e^{i\phi_1} \left( \sin (\phi_2-\phi_1) - (\delta/\gamma) \cos (\phi_2-\phi_1) \right) \\ i R_1e^{i\phi_2} \left( \sin (\phi_1-\phi_2) - (d/c) \cos (\phi_1-\phi_2) \right) \\ i R_2e^{i\phi_3} \left( \sin (\phi_2-\phi_3) - (\delta/\gamma) \cos (\phi_2-\phi_3) \right)\end{array} \right)\, . $$ This is clearly in the range of $e_0'(\phi) = {\rm diag}(iR_1e^{i\phi_1}, iR_2e^{i\phi_2}, iR_3e^{i\phi_3})$. Thus the first tangential homological equation becomes
$$\partial_{\omega} {\bf g}_1(\phi) + {\bf f}_1(\phi) = U_1(\phi) = \left( \begin{array}{c} (R_2/R_1) \left( \sin (\phi_2-\phi_1) - (\delta/\gamma) \cos (\phi_2-\phi_1) \right) \\ (R_1/R_2) \left( \sin (\phi_1-\phi_2) - (d/c) \cos (\phi_1-\phi_2) \right) \\ (R_2/R_3) \left( \sin (\phi_2-\phi_3) - (\delta/\gamma) \cos (\phi_2-\phi_3) \right) \end{array} \right) \, .$$
Because $\omega_1\neq \omega_2$ we are able to choose the solutions ${\bf f}_1(\phi) = (0, 0, 0)$ and $${\bf g}_1(\phi) = \frac{1}{\omega_1-\omega_2} \left( \begin{array}{r} (R_2 / R_1) \left( \cos (\phi_2-\phi_1) + (\delta/\gamma) \sin (\phi_2-\phi_1) \right) \\ -(R_1 / R_2) \left( \cos (\phi_1-\phi_2) + (d/c) \sin (\phi_1-\phi_2) \right) \\ (R_2 / R_3) \left( \cos (\phi_2-\phi_3) + (\delta/\gamma) \sin (\phi_2-\phi_3) \right) \end{array} \right)\, . $$
\subsubsection*{The first normal homological equation} Another short computation allows us to express the projection $(1-\pi)\cdot {\bf G}_1$ as $$((1-\pi)\cdot {\bf G}_1)(\phi) = \left( \begin{array}{l} e^{i\phi_1} ( \gamma+ i \delta) (R_2/\gamma) \cos (\phi_2-\phi_1) \\ e^{i\phi_2} (c+ i d) (R_1/c) \cos (\phi_1-\phi_2) \\ e^{i\phi_3} ( \gamma+ i \delta) (R_2/\gamma) \cos (\phi_2-\phi_3)
\end{array} \right) \, . $$ This is clearly in the range of $N(\phi)={\rm diag}(e^{i\phi_1}(\gamma+i\delta), e^{i\phi_2}(c+id), e^{i\phi_3}(\gamma+i\delta))$. Thus the first normal homological equation, see \eqref{tworeducedequations2} and \eqref{tworeducedequations}, reads $$\partial_{\omega}{\bf h}_1(\phi) + {\rm diag}(2\alpha, 2a, 2\alpha){\bf h}_1(\phi) = V_1(\phi) = \left( \begin{array}{l}
(R_2/\gamma) \cos (\phi_2-\phi_1) \\
(R_1/c) \cos (\phi_1-\phi_2) \\
(R_2/\gamma) \cos (\phi_2-\phi_3)
\end{array} \right) \, . $$ The solution reads $${\bf h}_1(\phi) = \left( \begin{array}{l} \frac{R_2}{\gamma(4\alpha^2 + (\omega_1-\omega_2)^2)} \left(2\alpha \cos(\phi_2-\phi_1) + (\omega_2-\omega_1)\sin(\phi_2-\phi_1) \right) \\ \frac{R_1}{c(4a^2 + (\omega_1-\omega_2)^2)} \left(2a \cos(\phi_1-\phi_2) + (\omega_1-\omega_2)\sin(\phi_1-\phi_2) \right) \\ \frac{R_2}{\gamma(4\alpha^2 + (\omega_1-\omega_2)^2)} \left(2\alpha \cos(\phi_2-\phi_3) + (\omega_2-\omega_1)\sin(\phi_2-\phi_3) \right) \end{array} \right)\, . $$
\subsubsection*{Second order terms} Let us clarify that we will not solve the second order homological equations completely. Instead, the only second order terms that we compute explicitly are the first and third components ${\bf f}_2^{(1)}$ and ${\bf f}_2^{(3)}$ of the second order part ${\bf f}_2$ of the reduced phase vector field. As was explained above, this suffices to obtain the desired asymptotic expression for
$\frac{d}{dt}(\phi_1 - \phi_3) = \varepsilon^2 \left( {\bf f}_2^{(1)}(\phi) - {\bf f}_2^{(3)}(\phi) \right) + \varepsilon^3 \ldots $.
We first compute the inhomogeneous term ${\bf G}_2$ as given in \eqref{iterativeeqns}. Because ${\bf F}_2=0$ and ${\bf f}_1=0$, we see that
$${\bf G}_2=\frac{1}{2}({\bf F}_0'' \circ e_0)(e_1, e_1) + ({\bf F}_1' \circ e_0)\cdot e_1$$
consists only of two terms. It also turns out that the first of these terms contributes in a rather trivial manner to the phase dynamics at order $\varepsilon^2$.
This term can be computed by making use of the expansion
\begin{align}\nonumber
|R_je^{i\phi_j}+\varepsilon & e_1^{(j)}(\phi)|^2(R_je^{i\phi_j}+\varepsilon e_1^{(j)}(\phi)) = R^3_je^{i\phi_j} + \varepsilon R_j^2 \left(2 e_1^{(j)}(\phi) + e^{2i\phi_j}\overline{e_1^{(j)}(\phi)} \right) \\ \nonumber & + \varepsilon^2 R_j \left(2 e^{i\phi_j} |e_1^{(j)}(\phi)|^2 + e^{-i\phi_j}(e_1^{(j)}(\phi))^2 \right) + \mathcal{O}(\varepsilon^3) \, .
\end{align}
This leads to the formula
\begin{align} \label{D2Omega}
\frac{1}{2}&({\bf F}_0''(e_0(\phi)) (e_1(\phi), e_1(\phi)) \! = \nonumber \\ &
\! \underbrace{
\left(\!\! \begin{array}{l}
R_1 (\gamma+i\delta) (2 e^{i\phi_1} |e_1^{(1)}(\phi)|^2
\\
R_2 (c+id) (2 e^{i\phi_2} |e_1^{(2)}(\phi)|^2
\\
R_3 (\gamma+i\delta) (2 e^{i\phi_3} |e_1^{(3)}(\phi)|^2
\end{array} \!\! \right) }_{=: T_1(\phi) \in \, {\rm im}\, N(\phi)}
+
\underbrace{ \left( \!\! \begin{array}{l}
R_1 (\gamma+i\delta) e^{-i\phi_1} (e_1^{(1)}(\phi))^2
\\
R_2 (c+id) e^{-i\phi_2} (e_1^{(2)}(\phi))^2
\\
R_3 (\gamma+i\delta) e^{-i\phi_3} (e_1^{(3)}(\phi))^2
\end{array} \!\! \right)}_{=: T_2(\phi)}
\, .
\end{align}
It is clear that the first term on the right hand side of \eqref{D2Omega}---which we called $T_1(\phi)$---lies in the range of $N(\phi)$ because $2 R_j |e_1^{(j)}(\phi)|^2 \in \mathbb{R}$ for $j=1,2,3$. So this first term vanishes when we apply the projection $\pi(\phi)$.
The projection of the second term on the right hand side of \eqref{D2Omega}---which we called $T_2(\phi)$---can be computed as follows. Recall from \eqref{Ansatz} that $e_1(\phi) = e_0'(\phi)\cdot {\bf g}_{1}(\phi) + N(\phi)\cdot {\bf h}_1(\phi)$, where $e_0$, ${\bf g}_1$, $N$ and ${\bf h}_1$ are given in the formulas above. This can be used to expand, first the $(e_1^{(j)}(\phi))^2$, and then $T_2(\phi)$ in trigonometric polynomials. It is not very hard to see that this must yield a formula of the form \begin{align}\nonumber & \pi(\phi)T_2(\phi) = \left( \begin{array}{l} R_1 i e^{i\phi_1} \left( C+ D \sin(2\phi_2-2\phi_1) + E \cos(2\phi_2-2\phi_1) \right) \\ R_2 i e^{i\phi_2} \left( \tilde C+ \tilde D \sin(2\phi_2-2\phi_1) + \tilde E \cos(2\phi_2-2\phi_1) \right) \\ R_3 i e^{i\phi_3} \left( C + D \sin(2\phi_2-2\phi_3) + E \cos(2\phi_2-2\phi_3) \right) \end{array} \right) \end{align}
for certain real numbers $C, D, E, \tilde C, \tilde D, \tilde E$ that we shall not explicitly compute here. Note that this clearly lies in the range of $e_0'(\phi)$. It follows that \begin{align}\nonumber & U_2^{1{\rm st}}(\phi) = \left( \begin{array}{l} C + D \sin(2\phi_2-2\phi_1) + E \cos(2\phi_2-2\phi_1) \\ \tilde C+ \tilde D \sin(2\phi_2-2\phi_1) + \tilde E \cos(2\phi_2-2\phi_1) \\ C + D \sin(2\phi_2-2\phi_3) + E \cos(2\phi_2-2\phi_3) \end{array} \right) \end{align} is the first part of the inhomogeneous right hand side of the second tangential homogeneous equation $\partial_{\omega}{\bf g_2}+{\bf f}_2 = U_2$. Because $2\omega_1\neq 2\omega_2$, only the constant part $(C, \tilde C, C)$ of this $U_2^{1{\rm st}}(\phi)$ is resonant; all other terms can be absorbed in ${\bf g}_2$. Thus the resonant normal form of this part of ${\bf f}_2$ is $(C, \tilde C, C)^T$. As this constant vector field does not contribute to $\frac{d}{dt}\left( \phi_1 - \phi_3 \right)$, we compute neither $C$ nor $\tilde C$ explicitly.
We proceed by considering the other term in ${\bf G}_2$, namely $({\bf F}'_1 \circ e_0)\cdot e_1$. Recalling that ${\bf F}_1(z)=(z_2, z_1, z_2)$, we see that this term equals
$${\bf F}_1'(e_0(\phi))\cdot e_1(\phi) = \left( \begin{array}{c} e_1^{(2)}(\phi) \\ e_1^{(1)}(\phi) \\ e_1^{(2)}(\phi) \end{array} \right) =
\left( \begin{array}{l} e^{i\phi_2} (iR_2{\bf g}_1^{(2)}(\phi) + (c + i d){\bf h}_1^{(2)}(\phi) ) \\
e^{i\phi_1} (iR_1{\bf g}_1^{(1)}(\phi) + (\gamma + i \delta){\bf h}_1^{(1)}(\phi) ) \\ e^{i\phi_2} (iR_2{\bf g}_1^{(2)}(\phi) + (c + i d){\bf h}_1^{(2)}(\phi) )
\end{array} \right) \, .$$ Using the expressions for $\pi(\phi)$, ${\bf g}_1(\phi)$ and ${\bf h}_1(\phi)$ provided above, one can compute that the projection of this term has the form
\begin{align}
& \pi(\phi)\cdot {\bf F}_1'(e_0(\phi))\cdot e_1(\phi) = \left( \begin{array}{ccc} i R_1 e^{i\phi_1} & 0 & 0 \\ 0 & i R_2 e^{i\phi_2} & 0 \\ 0 & 0& i R_3 e^{i\phi_3} \end{array} \right) \cdot U_2^{2{\rm nd}}(\phi)\, ,
\end{align}
in which now
\begin{align}
U_2^{2{\rm nd}}(\phi) = \left( \begin{array}{l}
B + F\sin( 2 \phi_1 -2\phi_2) + G \cos ( 2 \phi_1 -2\phi_2) \\
\tilde B + \tilde F\sin( 2 \phi_1 -2\phi_2) + \tilde G \cos ( 2 \phi_1 -2\phi_2) \\
\left\{ \begin{array}{l} A \sin(\phi_1-\phi_3) + B\cos(\phi_1-\phi_3) \\ + F\sin( \phi_1+\phi_3-2\phi_2) + G \cos ( \phi_1+\phi_3-2\phi_2) \end{array}
\right\} \end{array}
\right) \, .
\end{align}
With some effort the constants $A$ and $B$ can be computed by hand, yielding
\begin{align}\label{ABagain} \begin{array}{ll}
A &\!\!\! = \frac{1}{4a^2+(\omega_1-\omega_2)^2} \left( \frac{\delta}{\gamma}(\omega_2-\omega_1) + a \left( 1 + \frac{d \delta}{c \gamma} \right) + 2a^2 \left(\frac{d}{c} + \frac{\delta}{\gamma} \right) \frac{1}{\omega_2-\omega_1} \right) \, ,
\\
B & \!\!\! = \frac{1}{4a^2+(\omega_1-\omega_2)^2}\left( (\omega_2-\omega_1) + a \left(\frac{d}{c} - \frac{\delta}{\gamma} \right) + 2a^2 \left(1 - \frac{d \delta}{c\gamma} \right) \frac{1}{\omega_2-\omega_1} \right) \, . \end{array}
\end{align} We did not compute any of the other constants.
As $\omega_1=\omega_3 \neq \omega_2$, the resonant part of $U_2^{2{\rm nd}}(\phi)$ is given by
${\bf f}_2(\phi) = (B, \tilde B, A\sin(\phi_1-\phi_3) + B\cos(\phi_1-\phi_3))^T$. The other terms in $U_2^{2{\rm nd}}(\phi)$ can be absorbed into ${\bf g}_2$ when solving the tangential homological equation $\partial_{\omega}{\bf g_2}+{\bf f}_2 = U_2$.
\subsubsection*{Conclusion}
To summarise, we computed that ${\bf f}_1(\phi) = (0, 0,0)^T$ and
\begin{align}\label{f23} {\bf f}_2(\phi) = \left( \begin{array}{c} B+C \\ \tilde B +\tilde C \\ A \sin (\phi_1-\phi_3) + B \cos (\phi_1-\phi_3) + C \end{array} \right) \, .
\end{align}
The constants $A$ and $B$ are given in \eqref{ABagain}, but we did not compute $\tilde B, C$ or $\tilde C$.
Because $\omega_1=\omega_3$ and $\dot \phi = \omega + \varepsilon {\bf f}_1(\phi) + \varepsilon^2 {\bf f}_2(\phi) + \mathcal{O}(\varepsilon^3)$, we conclude that \begin{align} \frac{d}{dt} (\phi_1 - \phi_3) = \varepsilon^2 \left( -A \sin (\phi_1-\phi_3) - B \cos (\phi_1-\phi_3) + B \right) + \mathcal{O}(\varepsilon^3) \, . \end{align}
This is exactly equation \eqref{phi1minphi3}.
\end{document}
\section{Discrete symmetry}\label{discretesymsection} In this section we describe how symmetry in the original system $\dot x = {\bf F}(x)$ is inherited by its phase reduction $\dot \phi = {\bf f}(\phi)$. The main result is Theorem \ref{discretesymmetrytheorem}, the proof of which is surprisingly intricate. In this section, a symmetry will be any linear map that sends solutions of one dynamical system to solutions of another one. Thus, what we consider symmetries may form a more general structure than a group. The reason is that in \cite{quiverpaper} we show that many structural properties of network dynamical systems (including coupled oscillator systems) can be defined in terms of such generalised symmetries. These properties include not only classical (permutation) symmetry, but also the presence of sub-networks, quotient-networks, indirect node-dependency, feed-forward structure, hidden symmetry and interior symmetry.
More specifically, in this section we consider two ODEs \begin{align} \label{FLFR} \begin{array}{ll}
\dot x & = {\bf F}^{L}(x) = {\bf F}_0^{L}(x) + \varepsilon {\bf F}_1^{L}(x) + \ldots \ \mbox{on}\ \mathbb{R}^{M_L}\ \mbox{and} \\
\dot y & = {\bf F}^{R}(y) = {\bf F}_0^{R}(y) + \varepsilon {\bf F}_1^{R}(y) + \ldots \ \mbox{on}\ \mathbb{R}^{M_R} \, ,
\end{array}
\end{align} defined by vector fields ${\bf F}^L: \mathbb{R}^{M_L}\to \mathbb{R}^{M_L}$ and ${\bf F}^{R}:\mathbb{R}^{M_R}\to \mathbb{R}^{M_R}$. We assume that there is a linear map $S: \mathbb{R}^{M_L} \to \mathbb{R}^{M_R}$ sending solutions curves of ${\bf F}^L$ to solution curves of ${\bf F}^R$, that is, \begin{align} \label{discreteequivariance} {\bf F}^{R}(S\cdot x) = S \cdot {\bf F}^{L}(x)\, . \end{align} \begin{remk} The superscripts ``$L$'' and ``$R$'' stand for ``left'' and ``right''. Note that we did not assume that ${\bf F}^L$ and ${\bf F}^R$ are equal, nor that the linear map $S$ is invertible. We thus use a rather broad definition of symmetry. Moreover, in what follows we consider only one linear map $S$. \end{remk} \noindent The following theorem is the main result of this section. Its formulation is somewhat technical, but the conclusion of the theorem is natural. \begin{thr}\label{discretesymmetrytheorem} Consider a pair of vector fields ${\bf F}^L: \mathbb{R}^{M_L}\to\mathbb{R}^{M_L}$ and ${\bf F}^R: \mathbb{R}^{M_R}\to\mathbb{R}^{M_R}$, with asymptotic expansions as in \eqref{FLFR}, and satisfying the conjugacy relation \eqref{discreteequivariance} for some linear map $S: \mathbb{R}^{M_L}\to\mathbb{R}^{M_R}$. We make the following assumptions on the unperturbed vector fields ${\bf F}_0^L$ and ${\bf F}_0^R$: \begin{itemize} \item[i)] ${\bf F}_0^L$ and ${\bf F}_0^R$ both possess an embedded reducible normally hyperbolic (quasi-periodic) invariant torus. In other words, there are embeddings $e_0^L: (\mathbb{R}/2\pi\mathbb{Z})^{m_L}\to\mathbb{R}^{M_L}$ and $e_0^R: (\mathbb{R}/2\pi\mathbb{Z})^{m_R}\to\mathbb{R}^{M_R}$ and frequency vectors $\omega^L\in \mathbb{R}^{m_L}$ and $\omega^R\in\mathbb{R}^{m_R}$ satisfying $$\partial_{\omega^L}e_0^L = {\bf F}_0^L\circ e_0^L\ \mbox{and} \ \partial_{\omega^R}e_0^R = {\bf F}_0^R\circ e_0^R\, ;$$
The tori moreover admit fast fibre maps ${\bf N}e_0^L$ and ${\bf N}e_0^R$ defined by families of injective matrices $N^L = N^L(\phi^L)$ and $N^R = N^R(\phi^R)$ and hyperbolic Floquet matrices $L^L$ and $L^R$ satisfying \eqref{N0M0}. \item[ii)] There is a linear map $s: (\mathbb{R}/2\pi\mathbb{Z})^{m_L} \to (\mathbb{R}/2\pi\mathbb{Z})^{m_R}$ such that $$S \circ e_0^L = e_0^R \circ s \, .$$ \item[iii)] There is a linear map $t: \mathbb{R}^{M_L - m_L} \to \mathbb{R}^{M_R - m_R}$ such that $$S \cdot N^L(\phi^L) = (N^R\circ s)(\phi^L)\cdot t \ \mbox{for all}\ \phi^L\in (\mathbb{R}/2\pi\mathbb{Z})^{m_L} \, .$$ \end{itemize} Then, for every $j\in \mathbb{N}$ and every $K\in \mathbb{N}$, the solution $(e^{L}_j, {\bf f}_j^{L})$ to the iterative equation $\mathfrak{c}^{L}(e^{L}_j, {\bf f}_j^{L}) = {\bf G}_j^{L}$, and the solution $(e^{R}_j, {\bf f}_j^{R})$ to the iterative equation $\mathfrak{c}^{R}(e^{R}_j, {\bf f}_j^{R}) = {\bf G}_j^{R}$, can be chosen in such a way that \begin{itemize} \item[{\it i)}] They satisfy the equivariance relations $$S \circ e^L_j = e^R_j \circ s \ \mbox{and} \ {\bf f}^R_j \circ s = s \circ {\bf f}^L_j \, .$$ \item[{\it ii)}] Both ${\bf f}_j^{L}$ and ${\bf f}_j^{R}$ are in normal form to order $K$ in their Fourier expansion. \end{itemize} \end{thr}
\begin{comment} We also assume that for $\varepsilon=0$ both systems admit an invariant torus, embedded as $$\Gamma^L: \mathbb{T}^M\to \mathbb{R}^m \ \mbox{and} \ \Gamma^R: \mathbb{T}^N\to \mathbb{R}^n\, .$$ Finally, we assume that the symmetry of the unperturbed systems descends to a symmetry of phase reductions, i.e., there is a linear map $$s: \mathbb{T}^M \to \mathbb{T}^N$$ such that $$s(\omega^{L}) = \omega^R\, . $$ \begin{prop} Under these assumptions, the compositions $\Gamma^R\circ s$ and $S\circ \Gamma^L: \mathbb{T}^M \to \mathbb{R}^n$ both conjugate the constant vector field $\omega^L$ on $\mathbb{T}^M$ to $\Omega^R$. \end{prop} \begin{proof} $$D(\Gamma^R \circ s)(\phi^L)\cdot \omega^L = D\Gamma^R(s(\phi^L)) s\cdot \omega^L = D\Gamma^R(s(\phi^L))\cdot \omega^R = \Omega^R(\Gamma^R(s(\phi^L)) $$ $$D(S\circ \Gamma^L)(\phi^L) \cdot \omega^L = $$ \end{proof} Note that also $S\circ \Gamma^L$ does this. We assume that $$\Gamma^R \circ s = S \circ \Gamma^L\, .$$ (this does not follow, because the embeddings are not unique).
Differentiation of $\Omega^R\circ S = S \cdot \Omega^L$ gives $$D\Omega^R \circ S = S \cdot D\Omega^L$$ Now consider the maps $$Ts = (s,s) \ \mbox{and}\ TS=(S,S)$$ $$TS\circ \Gamma^L: (\phi^L, w^L)\mapsto (S (\Gamma^L(\phi^L)), S N_0^L(\phi^L) w^L)$$ $$\Gamma^R \circ Ts : (\phi^L, w^L)\mapsto ( \Gamma^R (s \phi^L), N_0^R s w^L)$$
\end{comment}
\begin{remk} The map $s$ can be thought of as the reduction to phase coordinates of the symmetry $S$ of the original systems ${\bf F}^L$ and ${\bf F}^R$. We assume that $s$ is given by a linear map from $\mathbb{R}^{m_L}$ to $\mathbb{R}^{m_R}$ with integer coefficients, so that it descends to a quotient map from $(\mathbb{R}/2\pi\mathbb{Z})^{m_L}$ to $(\mathbb{R}/2\pi\mathbb{Z})^{m_R}$. We denote this quotient map by $s$ as well. Similarly, the map $t$ is the reduction of $S$ to the local coordinates for fast fibre bundle. \end{remk} \noindent The main part of the proof of Theorem \ref{discretesymmetrytheorem} consists of showing that the solutions to the homological equations \eqref{tworeducedequations} can be chosen in an equivariant manner---see Lemmas \ref{1sthomeqsymmetry} and \ref{2ndhomeqsymmetry}. For the proof of these lemmas and the theorem we need the following technical observations. \begin{prop}\label{symmetryproperties} From the assumptions of Theorem \ref{discretesymmetrytheorem} it follows that \begin{itemize} \item[{\it i)}] $s(\omega^L) = \omega^R$; \item[{\it ii)}] $t \cdot L^L = L^R \cdot t$;
\item[{\it iii)}] $(\pi^R \circ s) \cdot S = S\cdot \pi^L$. \end{itemize} \end{prop} \begin{proof} \begin{itemize} \item[{\it i)}] We differentiate $(e_0^R \circ s)(\phi^L) = (S \circ e_0^L)(\phi^L)$ in the direction $\omega^L$. Writing $\phi^R = s(\phi^L)$, this yields \begin{align}\nonumber (e_0^R)'(\phi^R ) \cdot s(\omega^L) = & S\cdot (e_0^L)'(\phi^L) \cdot \omega^L \\ \nonumber = & S\cdot {\bf F}^L_0(e_0^L(\phi^L)) \\ \nonumber = & {\bf F}_0^R (S( e_0^L (\phi^L))) = {\bf F}_0^R ( e_0^R (\phi^R)) \, . \end{align} So $e_0^R$ conjugates $s(\omega^L)$ to ${\bf F}_0^R$. But $e_0^R$ also conjugates $\omega^R$ to ${\bf F}_0^R$, and as $(e_0^R)'$ is injective, this implies that $s(\omega^L)=\omega^R$. \item[{\it ii)}] Differentiation of $S\cdot N^L(\phi^L) = (N^R \circ s)(\phi^L)\cdot t$ in the direction of $\omega^L$ gives $$S\cdot \partial_{\omega^L}N^L(\phi^L) = (N^R)'(\phi^R) \cdot s(\omega^L) \cdot t = \partial_{\omega^R}N^R(\phi^R) \cdot t\, .$$ As a result, using Equation \eqref{N0M0}, for both $N=N^{L}$ and $N=N^{R}$, we find \begin{align} \nonumber N^R(\phi^R) \cdot t \cdot L^L & = S\cdot N^L(\phi^L) \cdot L^L \\ \nonumber & = S \cdot ({\bf F}_0^L)'(e_0^L(\phi^L)) \cdot N^L(\phi^L) - S\cdot \partial_{\omega^L} N^L(\phi^L) \\ \nonumber
& = ({\bf F}_0^R)'(e_0^R(\phi^R))\cdot S \cdot N^L(\phi^L) - \partial_{\omega^R}N^R(\phi^R) \cdot t \\ \nonumber
& = ({\bf F}_0^R)'(e_0^R(\phi^R))\cdot N^R(\phi^R) \cdot t - \partial_{\omega^R}N^R(\phi^R) \cdot t\\ \nonumber
& = N^R(\phi^R)\cdot L^R\cdot t\, . \end{align} The injectivity of $N^R(\phi^R)$ thus implies that $$t\cdot L^L = L^R \cdot t\, .$$ \item[{\it iii)}] The definitions of $\pi^{L,R}(\phi^{L,R})$ and the identity $S\cdot N^L(\phi^L) = N^R(\phi^R)\cdot t$ imply
\begin{align} \nonumber
(\pi^R(\phi^R)\! \cdot \! S) \! \cdot \! N^L(\phi^L) & = \pi^R(\phi^R)\! \cdot \! N^R(\phi^R) \! \cdot \! t = 0 = (S\! \cdot \! \pi^L(\phi^L))\! \cdot \! N^L(\phi^L)\, ,
\end{align}
while the identity $S\cdot (e_0^L)'(\phi^L) = (e_0^R)'(\phi^R)\cdot s$ implies that
\begin{align} \nonumber
(\pi^R(\phi^R) \cdot S) \cdot (e_0^L)'(\phi^L) & = \pi^R(\phi^R) \cdot (e_0^R)'(\phi^R)\cdot s = (e_0^R)'(\phi^R) \cdot s \\ \nonumber & = S\cdot (e_0^L)'(\phi^L) = (S \cdot \pi^L(\phi^L) ) \cdot (e_0^L)'(\phi_L)\, . \end{align}
$N^L(\phi^L)$ and $(e_0^L)'(\phi^L)$ together span $\mathbb{R}^{M_L}$, so it follows that $\pi^R(\phi^R)\cdot S = S \cdot \pi^L(\phi^L)$. \end{itemize} \end{proof}
\begin{remk} Points {\it i)} and {\it ii)} of Lemma \ref{symmetryproperties} together imply that the map $$(\phi^L, u^L) \mapsto (\phi^R, u^R):= (s(\phi^L), t(u^L))$$ sends solution curves of $\dot \phi^L = \omega^L\, , \ \dot u^L = L^L\cdot u^L$ to solution curves of $\dot \phi^R = \omega^R\, , \ \dot u^R = L^R \cdot u^R$. \end{remk}
\begin{comment} The first observation is simply the following \begin{lem} Assume that $e$ conjugates $f^{(1)}$ to $F^{(1)}$ so that $$De \cdot f^{(1)} = F^{(1)} \circ e\, . $$ Then $S \circ e$ conjugates $f^{(1)}$ to $F^{(2)}$, that is $$D (S \circ e) \cdot f^{(1)} = F^{(2)} \circ (S \circ e)\, .$$ \end{lem} The proof is obvious. Note though that the composition $(S\times S) \circ K_1$ may not be an embedding, and also the dimension of its domain of definition $\mathbb{T}^n$ be equal to that of the invariant torus $\mathbb{T}^m$. More precisely, let us phrase the problem of finding an embedded invariant torus as the functional equation $$\mathcal{F}(K, F): \mathbb{T}^n \to \mathbb{R}^n\times \mathbb{R}^n\, .$$ We here have two such conjugacy equations namely $$\mathcal{F}_1(K_1, F_1) = 0 \ \mbox{and}\ \mathcal{F}_2(K_2, F_2) = 0\, .$$ The following proposition relates these:
\begin{prop} Assume that $$S \circ F_1 = F_2 \circ S\ , \ e_2 \circ s = S \circ e_1\ \mbox{and} \ f_2 \circ s = s \circ f_1\, .$$ Then $$\mathcal{F}_2(e_2, f_2) \circ s = S \circ \mathcal{F}_1(e_1, f_1)\, .$$ \end{prop} \begin{proof} The chain rule gives that $DK_2(S\Phi)\cdot S = (S\times S)\circ DK_1(\Phi)$. Therefore, \begin{align}\nonumber & \mathcal{F}_2(K_2, F_2) (S\Phi) = DK_2 (S\Phi)\cdot F_2(S\Phi) - G_2(K_2(S\Phi)) \\ \nonumber & = DK_2 (S\Phi)\cdot S\cdot F_1(\Phi) - G_2((S\times S)(K_1(\Phi))) \\ \nonumber & = (S\times S)\cdot DK_1(\Phi)\cdot F_1(\Phi) - (S\times S)\circ G_1(K_1(\Phi)) \\ \nonumber & = (S\times S)( \mathcal{F}_1(K_1, F_1)(\Phi)) \, . \end{align} \end{proof} So for $S$-equivariant pairs of embeddings $K_1, K_2$ and equivariant pairs of vector fields $F_1, F_2$ and $G_1, G_2$ it holds that if $\mathcal{F}_1(K_1, F_1)=0$ then also $\mathcal{F}_2(K_2, F_2)=0$ on the range of $S$. \end{comment} \noindent It turns out that the solutions of the homological equations can be chosen in an equivariant way. We first prove this for the solutions of the normal homological equations.
\begin{lem}\label{1sthomeqsymmetry}
Under the conditions of Theorem \ref{discretesymmetrytheorem}, assume that the inhomogeneous right hand sides $V^{L,R}: (\mathbb{R}/2\pi\mathbb{Z})^{m_{L,R}} \to\mathbb{R}^{M_{L,R}-m_{L,R}}$ satisfy
$$t \circ V^L = V^R \circ s\, .$$
Then the (unique) solutions to the normal homological equations
\begin{align} \nonumber
(\partial_{\omega^L} - L^L)({\bf h}^L) = V^L \ \mbox{and} \
(\partial_{\omega^R} - L^R)({\bf h}^R) = V^R \,
\end{align} also satisfy $$t \circ {\bf h}^L = {\bf h}^R \circ s\, .$$ \end{lem} \begin{proof}
The normal homological equations for ${\bf h}^L$ and ${\bf h}^R$, together with the assumption that $t \circ V^L = V^R \circ s$, imply that $$t \circ (\partial_{\omega^L}{\bf h}^L) - t \circ (L^L\cdot {\bf h}^L) = (\partial_{\omega^R}{\bf h}^R)\circ s - (L^R\cdot {\bf h}^R) \circ s\, .$$ Using Proposition \ref{symmetryproperties}, we rewrite the terms in this identity as follows: \begin{align}\nonumber
t\circ (\partial_{\omega^L}{\bf h}^L) & = \partial_{\omega^L} (t\circ {\bf h}^L)\, , \\ \nonumber
t \circ ( L^L\cdot {\bf h}^L) & = L^R(t\circ {\bf h}^L) \, , \\ \nonumber
\partial_{\omega^R}{\bf h}^R \circ s & = (({\bf h}^R)' \circ s)\cdot \omega^R \\
\nonumber & = ({\bf} h^R)' \circ s)\cdot s\cdot \omega^L \\ \nonumber
& = ({\bf h}^R\circ s)'\cdot \omega^L \\ \nonumber & =\partial_{\omega^L}({\bf h}^R\circ s) \, , \\ \nonumber
(L^R\cdot {\bf h}^R) \circ s & = L^R({\bf h}^R \circ s)\, . \nonumber \end{align} Together this shows that $$(\partial_{\omega^L} - L^R)(t\circ {\bf h}^L- {\bf h}^R\circ s)=0\, .$$ In turn this implies that $t\circ {\bf h}^L- {\bf h}^R\circ s = 0$ because the operator $\partial_{\omega^L} - L^R$ is injective. Indeed, it maps a vector-valued function ${A}_ke^{i\langle k, \phi\rangle}$ (with ${A}_k \in \mathbb{C}^{M_R-m_R}$ and $k\in \mathbb{Z}^{m_L}$) to $(i\langle \omega^L, k\rangle - L^R){A}_ke^{i\langle k, \phi\rangle}$. The matrix $i\langle \omega^L, k\rangle - L^R$ is invertible because $L^R$ is hyperbolic. \end{proof} \noindent The result on the tangential homological equation is a bit more subtle: \begin{lem} \label{2ndhomeqsymmetry} Under the conditions of Theorem \ref{discretesymmetrytheorem}, assume that the inhomogeneous right hand sides $U^{L,R} :(\mathbb{R}/2\pi\mathbb{Z})^{m_{L,R}}\to\mathbb{R}^{m_{L,R}}$ satisfy
$$s \circ U^L = U^R \circ s\, .$$ Then, for any $K\in \mathbb{N}$, the solutions to the tangential homological equations
\begin{align} \nonumber
\partial_{\omega^L} {\bf g}^L + {\bf f}^L = U^L \ \mbox{and}\
\partial_{\omega^R} {\bf g}^R + {\bf f}^R = U^R \,
\end{align}
can be chosen in such a way that
${\bf f}^L$ and ${\bf f}^R$ are both in normal form
to order $K$ in their Fourier expansions, while at the same time
$$s \circ {\bf f}^L - {\bf f}^R \circ s \ \mbox{and}\ s \circ {\bf g}^L - {\bf g}^R \circ s\, $$
are arbitrarily small.
\end{lem} \begin{proof} Note that if we choose ${\bf f}^L=U^L$, ${\bf f}^R=U^R$, ${\bf g}^L=0$ and ${\bf g}^R=0$, then the equivariance relations $s \circ {\bf f}^L = {\bf f}^R \circ s$ and $s \circ {\bf g}^L = {\bf g}^R \circ s$ are satisfied. However, the choices ${\bf g}^L=0$ and ${\bf g}^R=0$ are not the ones that yield ${\bf f}^L$ and ${\bf f}^R$ in normal form. We recall from the proof of Corollary \ref{normalformcorollary} that these choices were given by \begin{align}\label{XintermsofU} \begin{array}{ll}
g_{k}^{L,R} = \frac{U_{j,k}^{L,R} }{i \langle \omega_{L,R}, k\rangle} & \mbox{when}\ \langle \omega_{L,R}, k\rangle \neq 0 \ \mbox{and}\ |k|\leq K, \\
g_{k}^{L,R} = 0 & \mbox{when} \ \langle \omega_{L,R}, k\rangle = 0 \ \mbox{or} \ |k|> K. \end{array} \end{align} and \begin{align} \begin{array}{ll}
f_{k}^{L,R}=0 & \mbox{when}\ \langle \omega_{L,R}, k\rangle \neq 0 \ \mbox{and}\ |k|\leq K, \\
f_{k}^{L,R}=U_{j,k}^{L,R} & \mbox{when} \ \langle \omega_{L,R}, k\rangle = 0 \ \mbox{or} \ |k|> K. \end{array} \end{align} Recall that these choices yield exact solutions to the ``left'' and ``right'' tangential homological equations. Also, remark that $${\bf g}^{L} = \partial_{\omega_{L}} Z^{L} \ \mbox{and} \ {\bf g}^{R} = \partial_{\omega_{R}} Z^{R}$$ for certain functions $Z^{L,R}$ defined by $Z^{L,R}_k = \frac{U^{L,R}_k}{(i\langle \omega_{L,R}, k\rangle)^2}$ for all $k$ with $\langle \omega_{L,R}, k\rangle \neq 0$. That is, ${\bf g}^{L,R}$ is in the image of the operator $\partial_{\omega^{L,R}}$. From this it follows that $$s \circ {\bf g}^{L} - {\bf g}^R \circ s = s \circ \partial_{\omega_L}Z^L - \partial_{\omega_R}Z^R \circ s = \partial_{\omega_L}(s \circ Z^L - Z^R \circ s )\, .$$ This shows that $$s \circ {\bf g}^{L} - {\bf g}^R \circ s \in {\rm im}\, \partial_{\omega_L}\, .$$ In fact, $s \circ {\bf g}^{L} - {\bf g}^R \circ s$ is a function with a finite Fourier expansion, because ${\bf g}^L$ and ${\bf g}^R$ have a finite Fourier expansion too, so let us write $$s \circ {\bf g}^{L} - {\bf g}^R \circ s = \sum_{k \leq L} i\langle \omega, k \rangle A_k e^{i\langle \omega_L, k\rangle} \, .$$ We estimate the size of the coefficients in this expansion by acting on this equality with the operator $\partial_{\omega_L}$. This gives, using that $U^R\circ s = s \circ U^L$, $$\partial_{\omega_L} ( s \circ {\bf g}^{L} - {\bf g}^R \circ s ) = s \circ \partial_{\omega_L}{\bf g}^L - \partial_{\omega_R}{\bf g}^R \circ s = - s\circ {\bf f}^L + {\bf f}^R\circ s $$ so that applying $\partial_{\omega_L}$ once more we get $$\partial_{\omega_L}^2 ( s \circ {\bf g}^{L} - {\bf g}^R \circ s ) = - s\circ \partial_{\omega_L}{\bf f}^L + (\partial_{\omega_R}{\bf f}^R)\circ s $$
But $\partial_{\omega_L}{\bf f}^L$ and $\partial_{\omega_R}{\bf f}^R$ only have Fourier terms for $|k| > K$. Hence, so does the function $s\circ \partial_{\omega_L}{\bf f}^L$. The same does not apply though to the other term, $(\partial_{\omega_R}{\bf f}^R)\circ s$, it being given by
$$\sum_{|k|>K} i\langle\omega_R,k\rangle U_{k}^R e^{i\langle k, s(\phi)\rangle} = \sum_{|k|>K} i\langle\omega_R,k\rangle U_{k}^R e^{i\langle s^T k, \phi\rangle} $$
and $|s^Tk|$ can be small even if $|k|$ is large. However, if $K$ is large and $U^L$ and $U^R$ are sufficiently smooth, then the Fourier expansion has a (very) small norm and therefore we obtain
Using these homological equations and the assumption that the inhomogeneous right hand sides satisfy $s\circ U^L = U^R \circ s$, we find that $$s\circ {\bf f}^L - {\bf f}^R\circ s = - s\circ \partial_{\omega_L}{\bf g}^L + \partial_{\omega_R} {\bf g}^R \circ s$$ Now note that $$\partial_{\omega_L} ({\bf g}^R\circ s) = (\partial_{\omega_R} {\bf g}^R ) \circ s$$ because $s(\omega_L)=\omega_R$. So we can rewrite this as $$s\circ {\bf f}^L - {\bf f}^R\circ s = \partial_{\omega_L} ({\bf g}^R\circ s - s\circ {\bf g}^L )\, .$$ This proves that ${\bf f}^L$ and ${\bf f}^R$ form an equivariant pair up to an error of the size of ${\bf g}^R\circ s - s\circ {\bf g}^L$. The latter quantity is not necessarily zero, but we claim that it can be made arbitrarily small.
$$\partial_{\omega_L}{\bf g}^L(\phi) = (U^L)^K_{\rm im}(\phi) $$ and similarly for ${\bf g}^R$ (the superscript $K$ meaning the truncation at order $K$). Or equivalently, $$\partial_{\omega_L}^2{\bf g}^L = \partial_{\omega_L}(U^L)^K $$ or equivalently, $$\partial_{\omega_L}\left( \partial_{\omega_L} {\bf g}^L - (U^L)^K \right) = \partial_{\omega_L}\left( \partial_{\omega_L} {\bf g}^L - U^L + U^L - (U^L)^K \right) = 0 $$ so $$ \partial_{\omega_L}\left( U^L - (U^L)^K - \partial_{\omega_L} {\bf f}^L \right) = 0 $$
It holds
$$(U^R)^K \circ s = \sum_{|k|\leq K} U^R_k e^{i\langle k, s(\phi)\rangle} = \sum_{|k|\leq K} U^R_k e^{i\langle s^T k, \phi\rangle} $$
$$s \circ (U^L)^K = \sum_{|k|\leq K} s(U^L_k) e^{i\langle k, \phi \rangle} $$
To show that the choices, we start by remarking that $U^L$ and $U^R$ can be written as $$U^L = U^{L}_{\rm ker} + U^{L}_{\rm im}\ \mbox{and}\ U^R = U^{R}_{\rm ker} + U^{R}_{\rm im}$$ where $$U^{L,R}_{\rm ker} \in {\rm ker}\, \partial_{\omega^{L,R}}\ \mbox{and} \ U^{L,R}_{\rm im} \in {\rm im}\, \partial_{\omega^{L,R}}\, .$$ Indeed, $U^{L,R}_{\rm ker}$ is a sum of Fourier terms of the form $U^{L,R}_k e^{i\langle k, \phi\rangle}$ with $\langle \omega^{L,R}, k\rangle =0$ and $U^{L,R}_{\rm im}$ is a sum of Fourier terms of the form $U^{L,R}_k e^{i\langle k, \phi\rangle}$ with $\langle \omega^{L,R}, k\rangle \neq 0$. We now define ${\bf f}^L$ and ${\bf f}^R$ by $${\bf f}^L := U^L_{\rm ker}\ \mbox{and}\ {\bf f}^R := U^R_{\rm ker}\, .$$ This obviously ensures that $$\partial_{\omega^L}{\bf f}^L = 0\ \mbox{and}\ \partial_{\omega^R}{\bf f}^R = 0\, .$$ At the same time we choose $${\bf g}^L = \partial_{\omega^L} Z^L\ \mbox{and}\ {\bf g}^R= \partial_{\omega^R} Z^R$$ to be the unique elements of ${\rm im}\, \partial_{\omega^L}$ respectively ${\rm im}\, \partial_{\omega^R}$ for which $$\partial_{\omega^L}{\bf g}^L = U^L_{\rm im} \ \mbox{and}\ \partial_{\omega^R}{\bf g}^R = U^R_{\rm im}\, .$$ We claim that with these choices we have $s \circ {\bf f}^L = {\bf f}^R\circ s$ and $s \circ {\bf g}^L = {\bf g}^R\circ s$.
To prove this claim, note that for a general pair of functions $H^{L,R}:\mathbb{T}^{m_L, m_R}\to \mathbb{R}^{m_L, m_R}$ (not necessarily satisfying $s \circ H^L = H^R\circ s$) we have the formula \begin{align}\label{funnyformula} \partial_{\omega^L}\left( s \circ H^L - H^R \circ s \right) = s\cdot \partial_{\omega^L} H^L - \partial_{\omega^R}H^R \circ s\, . \end{align} This follows from the chain rule and the fact that $s(\omega^L)=\omega^R$. We apply formula \eqref{funnyformula} several times below. To start, from formula \eqref{funnyformula} we see immediately that \begin{align} \label{differenceinimage} s \circ U^L_{\rm im} - U^R_{\rm im} \circ s = s \circ \partial_{\omega^L} {\bf g}^L - \partial_{\omega^R} {\bf g}^R \circ s = \partial_{\omega^L}( s\circ {\bf g}^L - {\bf g}^R \circ s)\, . \end{align} This proves that $$s \circ U^L_{\rm im} - U^R_{\rm im} \circ s \in {\rm im}\, \partial_{\omega^L}\, .$$ Next, we apply formula \eqref{funnyformula} twice, first to $(H^L, H^R) = (U^L_{\rm im}, U^R_{\rm im})$ and then to $(H^L, H^R) = (U^L, U^R)$. Using that $\partial_{\omega^L}U^L_{\rm im} = \partial_{\omega^L}U^L$ and $\partial_{\omega^R}U^R_{\rm im} = \partial_{\omega^R}U^R$, this yields that $$ \partial_{\omega^L}(s \circ U^L_{\rm im} - U^R_{\rm im} \circ s ) = \partial_{\omega^L}(s \circ U^L- U^R \circ s ) = 0\, , $$ and hence that
$$s \circ U^L_{\rm im} - U^R_{\rm im} \circ s \in \ker \, \partial_{\omega^L}\, .$$ Because ${\rm ker}\, \partial_{\omega^L}$ and ${\rm im}\, \partial_{\omega^L}$ intersect trivially, we have now proved that $$s \circ U^L_{\rm im} - U^R_{\rm im} \circ s = 0\ \mbox{and therefore also}\ s \circ U^L_{\rm ker} - U^R_{\rm ker} \circ s = 0 \, .$$ In particular, as we defined ${\bf f}^L = U^L_{\rm ker}$ and ${\bf f}^R = U^R_{\rm ker}$, we have that $$ s \circ {\bf f}^L = {\bf f}^R\circ s \, .$$
It remains to show that $s \circ {\bf g}^L = {\bf g}^R\circ s$. The argument is similar. Specifically, formula \eqref{funnyformula} applied to $(H^L, H^R)=({\bf g}^L,{\bf g}^R)$ shows that $$\partial_{\omega^L} ( s\circ {\bf g}^L - {\bf g}^R\circ s) = s\circ U^L_{\rm im} - U^R_{\rm im} \circ s = 0 $$ so that $$s\circ {\bf g}^L - {\bf g}^R\circ s \in \ker\, \partial_{\omega^L}\, .$$ On the other hand, applied to $(H^L, H^R)=(Z^L,Z^R)$, the formula gives $$s\circ {\bf g}^L - {\bf g}^R\circ s = \partial_{\omega^L} (s\circ Z^L - Z^R \circ s ) \in {\rm im}\, \partial_{\omega^L}\, .$$ We conclude that $s\circ {\bf g}^L - {\bf g}^R\circ s=0$, which completes the proof. \end{proof}
\noindent We are now ready for the proof of Theorem \ref{discretesymmetrytheorem}. \begin{proof}{\rm [of Theorem \ref{discretesymmetrytheorem}]} The proof is by induction on the order of the expansion in the small parameter. Note that we have $S\circ e_0^L = e_0^R\circ s$ and $s\circ \omega^L = \omega^R \circ s (=\omega^R)$ by assumption---and by point {\it i)} of Lemma \ref{symmetryproperties}. This proves the theorem for the unperturbed embeddings $e_0^{L,R}$ and reduced vector fields ${\bf f}_0^{L,R}:=\omega^{L,R}$.
Assume now that $e_1^R\circ s = S \circ e_1^L, \ldots, e_{j-1}^R\circ s = S \circ e_{j-1}^L$ and ${\bf f}_1^R \circ s = s \circ {\bf f}_1^L, \ldots, {\bf f}_{j-1}^R \circ s = s \circ {\bf f}_{j-1}^L$ and that ${\bf f}_1^{L,R}, \ldots, {\bf f}^{L,R}_{j-1}$ are all in normal form to order $K$ in their Fourier expansion.
From formula \eqref{Gjformula} (with ``$R$'' and ``$L$'' inserted at appropriate places) it then follows that $${\bf G}_{j}^R \circ s = S \circ {\bf G}_{j}^L\, ,$$ and by point {\it iii)} of Lemma \ref{symmetryproperties} it is then also clear that $$(\pi^R \cdot {\bf G}_{j}^R) \circ s = S \circ (\pi^L \cdot {\bf G}_{j}^L)\, .$$ Now recall from formula \eqref{tworeducedequations} that $U_j^{L}$ and $U_j^{R}$ are implicitly defined by $$(e_0^{L,R})'\cdot U^{L,R}_j = \pi^{L,R} \cdot {\bf G}_{j}^{L,R}\, .$$ Writing $\phi^R=s(\phi^L)$ for clarity, it follows that \begin{align} \nonumber (e_0^R)'(\phi^R) \cdot ( s \circ U^L_j)(\phi^L) & = S\cdot (e_0^L)'(\phi^L)\cdot U^L_j(\phi^L) \\ \nonumber & = S\cdot (\pi^{L} \cdot {\bf G}_{j}^{L})(\phi^L) \\ \nonumber & = (\pi^R \cdot {\bf G}_{j}^R)(\phi^R) = (e_0^R)'(\phi^R)\cdot U_j^R(\phi^R)\, . \end{align} By injectivity of $(e_0^R)'(\phi^R)$ we conclude that $$s \circ U_j^L = U_j^R \circ s \, .$$ The argument for $V^L_j$ and $V^R_j$ is analogous: using their implicit definition in \eqref{tworeducedequations} we obtain \begin{align} \nonumber N^R(\phi^R) \cdot (t \circ V^L_j)(\phi^L) & = S\cdot N^L(\phi^L)\cdot V^L_j(\phi^L) \\ \nonumber & = S\cdot ( (1-\pi^{L}) \cdot {\bf G}_{j}^{L})(\phi^L) \\ \nonumber & = ((1- \pi^R) \cdot {\bf G}_{j}^R)(\phi^R) = N^R(\phi^R)\cdot V_j^R (\phi^R)\, . \end{align} By injectivity of $N^R(\phi^R)$ we thus have $$t \circ V_j^L = V_j^R \circ s \, .$$
Proposition \ref{1sthomeqsymmetry} thus guarantees that ${\bf h}_j^R\circ s = t\circ {\bf h}_j^L$, while Proposition
\ref{2ndhomeqsymmetry} shows that it can be arranged that $s \circ {\bf g}_j^L = {\bf g}_j^R\circ s$, that $s \circ {\bf f}_j^L = {\bf f}_j^R\circ s$, and $$\partial_{\omega^L}{\bf f}^{L,R}_j = 0$$ to order $K$ in their Fourier expansion. Recalling that $$e_j^{L/R} = (e_0^{L/R})'\cdot {\bf g}_j^{L/R} + N^{L/R}\cdot {\bf h}_j^{L/R}\, ,$$ it follows immediately that with these choices, $$e^R_j\circ s = S \circ e_j^L\, .$$ This finishes the induction step and proves the theorem. \end{proof}
\section{Reducibility for relative equilibria}\label{Liegroupsection} In this section we discuss another setting in which it turns out that normally hyperbolic invariant tori are always reducible. This happens when the torus is a relative equilibrium of a $\mathbb{T}^N$-equivariant ODE. To the best of our knowledge, this result is new, although not very difficult. To explain the result in detail, we assume that $\mathbb{T}^N$ acts freely on $\mathbb{R}^n$ by means of a matrix representation $$\mathbb{T}^N \ni \phi\mapsto g(\phi) \in {\rm GL}(\mathbb{R}^n)\, .$$ This means that $g(0)={\rm Id}_{\mathbb{R}^n}$ and $g(\phi+\psi) = g(\phi)g(\psi) = g(\psi)g(\phi)$. We now consider a $\mathbb{T}^N$-equivariant differential equation $\dot x = \Omega(x)$ on $\mathbb{R}^n$. Specifically, we assume that $\Omega$ satisfies $$\Omega(g(\phi) x) = g(\phi)\cdot \Omega(x)\ \mbox{for all}\ \phi\in \mathbb{T}^N \ \mbox{and} \ x\in \mathbb{R}^n\, .$$ We shall also assume that $\Omega$ possesses a relative equilibrium: a group orbit that is invariant under the flow of $\Omega$. This means that there is an $x_0\in \mathbb{R}^n$ with the property that $\Omega(x_0)\in T_{x_0} (\mathbb{T}^N\cdot x_0)$. In turn this implies that
$$\Omega(x_0)= \left. \frac{d}{dt}\right|_{t=0}\!\! g(t\omega) \cdot x_0 = \partial_{\omega} g(0) \cdot x_0\ \mbox{for some}\ \omega\in \mathbb{R}^N. $$ \begin{lem}\label{quasitoruslemma} Such a relative equilibrium is an $\Omega$-invariant (quasi-)periodic torus. \end{lem} \begin{proof} We start by remarking that differentiation to $t$ at $t=0$ of the matrix identity $g(\phi + \omega t) = g(\phi)g(\omega t) = g(\omega t) g(\phi)$ gives that $$\partial_{\omega}g(\phi) = g(\phi) \partial_{\omega}g(0) = \partial_{\omega}g(0)g(\phi)\, .$$ Next, define the curve $x(t) := g(\phi_0+ \omega t)\cdot x_0$ passing through an arbitrary point $g(\phi_0) \cdot x_0$ on the group orbit of $x_0$. We find that \begin{align}\nonumber \dot x(t) = & \partial_{\omega} g(\phi_0+ \omega t) x_0 \\ \nonumber
= & g(\phi_0+ \omega t)\partial_{\omega} g(0) x_0 \\ \nonumber
= & g(\phi_0+\omega t) \Omega(x_0) \\ \nonumber
= & \Omega( g( \phi_0+\omega t) x_0) = \Omega(x(t))\, . \end{align} This shows that the map $\Gamma: \phi \mapsto g(\phi)x_0$ from $\mathbb{T}^N$ to $\mathbb{R}^n$, which is surjective onto the group orbit of $x_0$, sends solutions of $\dot \phi = \omega$ on the torus $\mathbb{T}^N$ to integral curves of $\Omega$. So the group orbit is invariant under the flow of $\Omega$ and every integral curve on the group orbit is (quasi-)periodic. \end{proof} \begin{remk} The $\phi$- and $t$-independent quantity $$\partial_{\omega}g(0) x_0 = g(\phi)^{-1} \partial_{\omega}g(\phi) x_0 = \partial_{\omega}g(\phi) g(\phi)^{-1}x_0 \in \mathbb{R}^n = T_{x_0} (\mathbb{T}^N\cdot x_0)$$ can be interpreted as the velocity of any curve $x(t) = g(\phi+\omega t)x_0$ on the relative equilibrium in co-moving coordinates. \end{remk} \noindent The following can be thought of as a Floquet theorem for relative equilibria: \begin{lem}\label{lemmarelativeequilibrium} The map $(\phi, u) \mapsto (x, v):= (g(\phi) x_0, g(\phi) u)$ sends solutions of the constant coefficient skew product differential equation $$\dot \phi = \omega\, , \ \dot u = (D\Omega(x_0) - \partial_{\omega}g(0))\cdot u \, $$ on $\mathbb{T}^N\times \mathbb{R}^n$ to solutions of the variational equations of $\Omega$ on $\mathbb{R}^n\times\mathbb{R}^n$ given by $$\dot x = \Omega(x)\, , \ \dot v= D\Omega(x) \cdot v\, .$$ \end{lem} \begin{proof} Assume that $\dot \phi (t) = \omega$ and $\dot u(t) = (D\Omega(x_0) - \partial_{\omega}g(0) )u(t)$. We already checked in Lemma \ref{quasitoruslemma} that $x(t):=g(\phi(t)) x_0$ satisfies $\dot x(t) = \Omega(x(t))$. Next, note that differentiation of $\Omega(g(\phi)x) = g(\phi)\Omega(x)$ to $x$ at $x=x_0$ gives $D\Omega(g(\phi) x_0) g(\phi) = g (\phi) D\Omega(x_0)$. This implies that $v(t):=g(\phi(t)) u(t)$ satisfies \begin{align}\nonumber \dot v(t) & = \partial_{\omega}g(\phi(t)) u(t) + g(\phi(t)) \dot u(t) \\ \nonumber & = \partial_{\omega}g(\phi(t)) u(t) + g(\phi(t)) ( D\Omega(x_0) - \partial_{\omega}g(0) ) u(t) \\ \nonumber & = D\Omega(x(t)) g(\phi(t)) u(t) = D\Omega(x(t)) v(t)\, . \end{align} This proves the lemma. \end{proof} \noindent As a result of Lemma \ref{lemmarelativeequilibrium} we find that every relative equilibrium is reducible: \begin{cor} \label{relativeequilibriumcor} Any normally hyperbolic such relative equilibrium is reducible. \end{cor} \begin{proof}
Let $u = \left. \frac{d}{d\varepsilon} \right|_{\varepsilon=0} g(\varepsilon h)x_0 = (Dg(0)\cdot h)x_0$ be an arbitrary tangent vector to the group orbit of $x_0$. Differentiation to $\varepsilon$ at $\varepsilon= 0$ of the identity
$$\partial_{\omega}g(0) g(\varepsilon h) x_0 = \left. \frac{d}{dt}\right|_{t=0} \!\!\!\!\! g(t\omega) g(\varepsilon h) x_0 = \left. \frac{d}{dt}\right|_{t=0} \!\!\!\!\! g(\varepsilon h+ t\omega) x_0 = \Omega(g(\varepsilon h) x_0)$$ shows that $$ \partial_{\omega}g(0) u = D\Omega(x_0) u\, . $$ This proves that the tangent space to the group orbit at $x_0$ lies in the kernel of $D\Omega(x_0)-\partial_{\omega}g(0)$. Our assumption that the group orbit is normally hyperbolic means that this tangent space can be complemented by another subspace that is invariant under $D\Omega(x_0) - \partial_{\omega}g(0)$ and restricted to which this map has no eigenvalues on the imaginary axis. Just like in the proof for periodic orbits, this implies reducibility: one may choose a map $A:\mathbb{R}^{n-N}\to\mathbb{R}^n$ whose image is the sum of the hyperbolic eigenspaces of $D\Omega(x_0)-\partial_{\omega}g(0)$. Then $${\bf N}\Gamma: \mathbb{T}^{N} \times \mathbb{R}^{n-N} \to \mathbb{R}^{n}\times \mathbb{R}^{n}\ \mbox{defined by}\ {\bf N}\Gamma(\phi, u):=(g(\phi)\cdot x_0, g(\phi)\cdot A \cdot u)$$ satisfies the requirements. \end{proof} \begin{remk} The assumption that $\mathbb{T}^N$ acts freely on $\mathbb{R}^n$ implies that $\Gamma$ and ${\bf N}\Gamma$ are embeddings. In particular, the hyperbolic eigenspace of $D\Omega(x_0)-\partial_{\omega}g(0)$ then extends to a unique embedding of the fast fibre bundle. Corollary \ref{relativeequilibriumcor} nevertheless can also be very useful if $\mathbb{T}^N$ does not act freely. \end{remk}
\begin{ex} The observations in this section provide an alternative way to compute the parametrised normal dynamics of the Stuart-Landau oscillator that we studied before in Example \ref{exstuartlandau}. Now we will not use the Floquet decomposition of the variational flow but the presence of a continuous symmetry. We recall that the equations of motion are \begin{align} \nonumber
\dot z = \Omega(z) = \left(\alpha + i \beta \right) z + \left(\gamma + i \delta \right) |z|^2 z \quad \mbox{for}\ z \in \mathbb{C}\, . \end{align} The right hand side of this differential equation satisfies $\Omega(e^{i\phi} z) = e^{i\phi} \Omega(z)$ and the system therefore possesses a $\mathbb{T}^1$-symmetry given by $g(\phi) z = e^{i\phi}z$. The system has a unique nontrivial relative equilibrium, which is the group orbit of the point $x_0=\sqrt{-\alpha/\gamma} \in \mathbb{R}$ on the positive real axis. We have
$$\Omega(x_0) = i \omega x_0 = \left.\frac{d}{dt}\right|_{t=0} e^{i\omega t} x_0\, .$$ (Recall that $\omega:= \beta - \alpha \delta/\gamma$.) Therefore, $t\mapsto e^{i\omega t}x_0$ is a periodic solution. One computes that the linearised vector field at $x_0$ is given by $$D\Omega(x_0) (v_1+iv_2) = i \omega v -2(\alpha/\gamma) (\gamma + i \delta)v_1 $$ and that the additional velocity term is given by
$$\partial_{\omega}g(0) = \left. \frac{d}{dt} \right|_{t=0} g(\omega t) = \left. \frac{d}{dt} \right|_{t=0} e^{i \omega t} = i\omega\, . $$ Hence, we find that the parametrised variational equations are given by $$\dot u = ( D\Omega(x_0)-\partial_{\omega}g(0) ) u = -2(\alpha/\gamma) (\gamma + i \delta) u_1\, .$$ As we saw before, this is a linear differential equation with constant coefficients. It has eigenvalues $0$ (with eigenvector $i$ tangent to the group orbit) and $-2\alpha$ (with eigenvector $\gamma+i\delta$ transversal to the group orbit).
\end{ex}
\noindent The results that we proved in this section still hold if we replace $\mathbb{T}^N$ by an arbitrary Lie group $G$ acting on $\mathbb{R}^n$. The generalisation of Lemma \ref{lemmarelativeequilibrium} is the following result, which for simplicity we only formulate for a matrix Lie group $G\subset {\rm GL}(\mathbb{R}^n)$. We denote by $1\in G$ the unit element of $G$ and by $T_1G$ the Lie algebra of $G$. \begin{lem} Assume that $\Omega:\mathbb{R}^n\to\mathbb{R}^n$ is equivariant under the left-action of the matrix Lie group $G$, and that $G\cdot x_0$ is a relative equilibrium of $\Omega$. Then there is an $\omega\in T_1G$ so that the map $$(g, u) \mapsto (x, v):= (g\cdot x_0, g\cdot u)\ \mbox{from}\ G\times \mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n$$ sends solutions of the constant coefficient skew product differential equation $$\dot g = g \cdot \omega\, , \ \dot u = (D\Omega(x_0) - \omega)\cdot u \, $$ to solutions of the variational equations of $\Omega$ given by $$\dot x = \Omega(x)\, , \ \dot v= D\Omega(x) \cdot v\, .$$ \end{lem} \begin{proof} Because $G\cdot x_0$ is a relative equilibrium, we have that
$$\Omega(x_0)=\left. \frac{d}{dt}\right|_{t=0}e^{t \omega}\cdot x_0 = \omega\cdot x_0 \ \mbox{for some}\ \omega\in T_1G\, .$$ As a result, $\Omega(g\cdot x_0) = g\cdot \Omega(x_0) = g\cdot \omega \cdot x_0$. Thus, if a curve $g(t)\in G$ satisfies $\dot g(t) = g(t)\cdot \omega$ then $x(t):=g(t)\cdot x_0$ satisfies $\dot x(t) = g(t)\cdot \omega \cdot x_0 = \Omega(g(t)\cdot x_0)= \Omega(x(t))$. This proves the first component of the conjugacy equation in this lemma. For the second component, note that if in addition $u(t)$ satisfies $\dot u(t) = (D\Omega(x_0) - \omega) \cdot u(t)$, then $v(t):=g(t)\cdot u(t)$ satisfies \begin{align} \dot v(t) = & g(t)\cdot \omega\cdot u(t) + g(t)\cdot (D\Omega(x_0) - \omega) \cdot u(t) \nonumber \\ \nonumber & = g(t)\cdot D\Omega(x_0) \cdot u(t) = D\Omega(g(t)\cdot x_0) \cdot g(t)\cdot u(t)= D\Omega(x(t))\cdot v(t)\, . \end{align} Here we used that equivariance of $\Omega$ implies that $D\Omega(g\cdot x)\cdot g = g\cdot D\Omega(x)$. \end{proof}
\begin{remk}
Similar to when $G=\mathbb{T}^N$, one may expect that also for a general non-commutative Lie group $G$ the matrix $D\Omega(x_0)-\omega$ is always singular. This turns out not to be the case: the correct statement is slightly more subtle. Note that differentiation to $t$ at $t=0$ of $\Omega(e^{t h} x_0) = e^{t h}\Omega(x_0)$ (for $h\in T_1G$) gives $D\Omega(x_0) h x_0 = h\Omega(x_0) = h \omega x_0$. This can equivalently be written as $$[D\Omega(x_0) - \omega ] h x_0 = h\omega x_0 - \omega h x_0 = [h,\omega]x_0 = -{\rm ad}_{\omega}(h) x_0 \ \mbox{for all}\ h \in T_1G\, .$$
Under the assumption that $G$ is compact, the eigenvalues of $\omega \in T_1G$ are all purely imaginary (otherwise $\{e^{t\omega}| t\in \mathbb{R}\}$ would form a noncompact subgroup of $G$). We conclude that the operator ${\rm ad}_{\omega}: T_1G\to T_1G$ mapping $h \mapsto [\omega, h]$ only has purely imaginary eigenvalues as well (because its eigenvalues are $\lambda-\mu$ for $\lambda, \mu$ eigenvalues of $\omega$.) This proves that in the direction tangent to the relative equilibrium $T_{x_0}G\cdot x_0 = \{h \cdot x_0\ |\ h\in T_1G\}$, the linearisation $D\Omega(x_0)-\omega$ only has elliptic eigenvalues (i.e. they lie on the imaginary axis). As in our discussion of the case $G=\mathbb{T}^N$, we can still obtain a parametrisation of an invariant normal bundle if we assume for example that $D\Omega(x_0)-\omega$ is hyperbolic in the transverse direction. \end{remk}
\section{The parametrisation method} We now try to find an invariant embedded torus in $\mathbb{C}^n$ by conjugating a vector field on the standard torus $\mathbb{T}^n$ of the form $$\dot \Phi = F(\Phi) = \Omega + \varepsilon F^{(1)}(\Phi) + \varepsilon^2 F^{(2)} (\Phi) + \ldots $$ to the above equations of motion by means of a semi-conjugagy / torus embedding of the form $$\Phi \mapsto (R(\Phi); \phi(\Phi)) = (R^*; \Phi) + \varepsilon \left( R^{(1)}(\Phi), \phi^{(1)}(\Phi) \right) + \ldots \, .$$ Here, $\Omega_j = \omega_j + \beta_j (R_j^*)^2$ and $R^*_j = \sqrt{-\mu_j/\alpha_j}$ are the stationary values of the $R_j$. We are most of all interested in the reduced vector field $F(\Phi)$ as this constitutes the phase reduction of the model.
The conjugacy equations
$$(DR(\Phi), D\phi(\Phi)) \cdot F(\Phi) = (\dot R, \dot \phi) (R(\Phi), \phi(\Phi)) $$
reduce to a list of recursive equations. The order $\mathcal{O}(\varepsilon)$ part of these equations is \begin{align}\nonumber \partial_\Omega R^{(1)}_j(\Phi) & + 2 \mu_j R_j^{(1)}(\Phi) = \sum_k R^*_k \, {\rm Re}\, \left(A_{jk} e^{i(\Phi_k-\Phi_j)} + B_{jk} e^{-i(\Phi_k+ \Phi_j)} \right) \\ \nonumber \partial_\Omega \phi^{(1)}_j(\Phi) & + F_{j}^{(1)}(\Phi) - 2(\beta_jR_j^*) R_j^{(1)}= (R^*_j)^{-1} \sum_k R^*_k \ {\rm Im}\, \left(A_{jk} e^{i(\Phi_k-\Phi_j)} + B_{jk} e^{-i(\Phi_k+ \Phi_j)} \right) \end{align} Here, $$\partial_\Omega f(\Phi) := \sum_k \frac{\partial f(\Phi)}{\partial \Phi_k} \Omega_k$$ is a directional derivative in the direction of the frequency vector. The $\mathcal{O}(\varepsilon^k)$ parts of the conjugacy equation have a totally similar structure, i.e. they are of the form \begin{align}\nonumber \partial_\Omega R^{(k)}(\Phi) & + 2 \mu R^{(k)}(\Phi) = \mbox{inhomogeneous term} \\ \nonumber \partial_\Omega \phi^{(k)}(\Phi) & + F^{(k)}(\Phi) - 2(\beta R^*) R^{(k)} = \mbox{inhomogeneous term} \end{align} Here, $\mu$ denotes the diagonal matrix with entries $\mu_1, \ldots,\mu_n$ and similar for the matrix $\beta R^*$. The inhomogeneous terms are functions of $\Phi$ that depend on the solutions to the recursive equations that were solved before.
\section{Solving the infinitesimal conjugacy equations} These homological equations can always be solved for $R^{(k)}$. This follows because $$\partial_{\Omega} e^{i\langle k, \Phi\rangle} = i\langle k, \Omega\rangle e^{i\langle k, \Phi\rangle}$$ and hence in particular, $$\left(\partial_{\Omega} + 2 \mu_j \right) e^{i\langle k, \Phi\rangle} = \left( 2\mu_j + i\langle k, \Omega \rangle \right) e^{i\langle k, \Phi\rangle}$$ which defines an invertible operator on $L_2(\mathbb{T}^n, \mathbb{R}^n)$ because the $\mu_j$ are real and nonzero, and the $i\langle k, \Omega \rangle $ purely imaginary. This is just a direct consequence of the normal hyperbolicity.
On the other hand, the second homological equation can not be solved in such a straightforward fashion. But luckily, we have the freedom of choosing $F^{(k)}$. So a sensible strategy will be to choose $F^{(k)}$ to consist precisely of the resonant terms in the inhomogeneous right hand side and the term $2(\beta R^*)R^{(k)}$ (which also acts as inhomogeneous term here). A resonant term is defined as an element of the kernel of $\partial_{\Omega}$. These are exactly the complex exponentials $ e^{i\langle k, \phi\rangle}$ for which $\langle k, \Omega \rangle = 0$. This allows us to solve the second equation for $\phi^{(k)}(\Phi)$. At the same time we conclude that it can be arranged that $F(\Phi)$ is a sum of only resonant terms. In this sense, our strategy also leads to the ``simplest possible'' reduced vector field $F(\Phi)$.
{\bf Obvious question: which classical, quiver and other types of symmetries are preserved in this reduction process?} The answer depends on how we deal with the non-uniqueness of the solution.
I should also remark that this is not all too deep and new but it might be a nice story. On the other hand, note the recent preprint {\it High-order phase reduction for coupled oscillators} by Gengel, Teichman, Rosenblum and Pikovsky, where they present a similar but less efficient method, use it to second order on a three cell network, and don't see that they can actually remove the resonant terms....
\section{A more general geometric setup}
Now we assume that the unperturbed system of oscillators is given as an invariant periodic or quasiperiodic torus in $\mathbb{R}^N$. Assume that there is an embedding of this torus of the form $$\Phi \mapsto K_0(\Phi) \ \mbox{from}\ \mathbb{T}^n \to \mathbb{R}^N$$ with the property that $$\partial_{\Omega} K_0 = G_0 \circ K_0$$ Here, $\Omega\in \mathbb{R}^n$ is a frequency vector. Differentiation to $s$ of $$ (\partial_{\Omega}K_0)(\Phi+sX(\Phi) ) = DK_0(\phi+sX(\Phi) )\cdot \Omega = G_0(K_0(\Phi+s X(\Phi) ))$$ gives $$\partial_{\Omega} (DK_0(\Phi) \cdot X(\Phi)) = (DG_0 (K_0(\Phi)) )\cdot DK_0(\Phi) \cdot X(\Phi) \, .$$ In other words, the multiplication operator $DK_0$ intertwines $\partial_{\omega}$ and the multiplication operator $(DG_0\circ K_0)$. We will use this below.
Given the perturbed vector field $G = G_0 + \varepsilon G_1 + \varepsilon^2 \ldots$ we now try to iteratively solve for a conjugacy $K = K_0 + \varepsilon K_1 + \varepsilon^2 \ldots$ and reduced vector field $F= \Omega + \varepsilon F_1 + \varepsilon^2 \ldots $ of the equation $$D K \cdot F = G \circ K$$ which becomes \begin{align} \begin{array}{llr}
\left( \partial_{\Omega} - DG_0\circ K_0 \right) \cdot K_1 + DK_0 \cdot F_1 & = G_1 \circ K_0 & =: H_1 \\ & \vdots & \\
\left( \partial_{\Omega} - DG_0\circ K_0 \right) \cdot K_j + DK_0 \cdot F_j & & = H_j \\ & \vdots & \end{array} \end{align} Here the right hand side $H_j: \mathbb{T}^n\to \mathbb{R}^N$ should be considered an inhomogeneous term. At the left hand side the unknowns are $K_j$ and $F_j$.
We now make an ansatz $$K_j(\Phi) = DK_0(\Phi)\cdot X_j(\Phi) + N_0(\Phi)\cdot Y_j(\Phi)\, .$$ Here, $$X_j: \mathbb{T}^n\to \mathbb{R}^n\, , \ N_0: \mathbb{T}^n\to L(\mathbb{R}^{N-n}, \mathbb{R}^N) \ \mbox{and}\ Y_j(\Phi): \mathbb{T}^n\to \mathbb{R}^{N-n}\, .$$ The idea is to choose $N_0$ in such a way that $$\mathbb{R}^N = {\rm im}\, DK_0(\Phi) \oplus {\rm im}\, N_0(\Phi) \, $$ so that the homological equations can be solved.
Because $( \partial_{\Omega} - DG_0\circ K_0 ) (DK_0 \cdot X) = 0$ this ansatz gives $$ \left( \partial_{\Omega} - DG_0\circ K_0 \right) \cdot ( N_0 \cdot Y_j) + DK_0 \cdot F_j = H_j \, . $$ We would preferably choose $N_0$ in such a way that $$ \left( \partial_{\Omega} - DG_0\circ K_0 \right) \cdot (N_0 \cdot Y_j) = N_0 \cdot \left( \partial_{\Omega} - n_0 \right) Y_j $$ where $n_0$ is a family of $(N-n)\times (N-n)$ matrices, probably with only hyperbolic eigenvalues. From looking at the equations we see that the left hand side equals $$\left( N_0 \partial_{\Omega} + \partial_{\Omega} N_0 - (DG_0\circ K_0 ) N_0 \right) \cdot Y_j $$ In other words, we want to find $N_0$ and $n_0$ satisfying the equation $$ \partial_{\Omega} N_0 = (DG_0\circ K_0) \cdot N_0 - N_0 \cdot n_0 \, .$$
The homological equation then becomes
$$ N_0 \cdot \left(\partial_{\Omega} - n_0 \right)\cdot Y_j + DK_0 \cdot F_j = H_j \, . $$
Now there should be unique solutions $F_j$ and $Y_j$, assuming that one can prove that $\partial_{\Omega} - n_0 $ is invertible. It remains find $N_0$ and $n_0$.
\begin{prop} Let $n_0$ be a family of matrices with non-imaginary spectrum. Then the operator $\partial_{\Omega} - n_0$ is invertible. \end{prop} \begin{proof} The proof only works if there are smooth families of eigenvectors, that is, $$(\partial_{\Omega} - n_0(\Phi)) v(\Phi) = \lambda(\Phi) v(\Phi)\, .$$ Then any right hand side $X(\Phi)$ can be written as $$X(\Phi) = \sum v_i(\Phi)$$ and the operator acts as $$(\partial_{\Omega} - n_0) v_i(\Phi) = $$ \end{proof} \begin{remk} Differentiation to $s$ of
$$(\partial_{\Omega} K_0)(\Phi +s v) = G_0(K_0(\Phi+sv))$$ yields that $$D(\partial_{\Omega}K_0)\cdot v = DG_0 \cdot DK_0 \cdot v $$ At the same time we can calculate variational equations by expanding $$\frac{d}{dt} K_0(\Omega t + sv(t)) = G_0(K_0(\Omega t + sv(t) ))$$ or equivalently $$DK_0(\Omega t + s v) (\Omega + s \frac{dv}{dt} ) = G_0(K_0(\Omega t + sv))$$ so that $$ DK_0 \cdot \frac{dv}{dt} + D(\partial_{\Omega}K_0)\cdot v = DG_0 \cdot DK_0 \cdot v$$ This proves that $\frac{dv}{dt}=0$ because $DK_0$ is injective.
Next we calculate the full variational equations: differentiation to $s$ of $$\frac{d}{dt} \left( K_0(\Omega t ) + sDK_0(\Omega t)\cdot v + s N_0(\Omega t) w(t) \right) = G_0(K_0(\Omega t) + s DK_0(\Omega t) v + s N_0(\Omega t) w ) $$ yields $$DK_0 \cdot \frac{dv}{dt} + D(\partial_{\Omega}K_0)\cdot v + \partial_{\Omega}N_0 \cdot w + N_0 \cdot \frac{dw}{dt} = DG_0 \cdot \left( DK_0 \cdot v + N_0 \cdot w \right) $$ This reduces (by the above) to $$DK_0 \cdot \frac{dv}{dt} + \partial_{\Omega}N_0 \cdot w + N_0 \cdot \frac{dw}{dt} = DG_0 \cdot N_0 \cdot w $$ Now we project this onto ${\rm im}\, DK_0$ and $N_0$ to obtain $$\left( \begin{array}{c} DK_0 \cdot \frac{dv}{dt} \\ N_0 \cdot \frac{dw}{dt} \end{array} \right) = \left( \begin{array}{cc} 0 & A(\Omega t) \\ 0 & B(\Omega t) \end{array} \right) \left( \begin{array}{c} v \\ w \end{array} \right)$$ where $$A(\Phi) + B(\Phi) = (DG_0\cdot N_0 - \partial_{\Omega}N_0) w $$ If this is in the image of $N_0$, i.e. if $$(DG_0\cdot N_0 - \partial_{\Omega}N_0) w= N_0 n_0 w$$ then $A(\Phi)=0$ and the equations for $v$ and $w$ decouple.
\end{remk}
\begin{remk} Consider the equation $$ - \partial_{\Omega} N_0(\Phi) + A(\Phi) \cdot N_0(\Phi) = N_0(\Phi) \cdot n_0(\Phi) \, $$ for the unknown matrix functions $N_0, n_0$. In the absence of the term $\partial_{\Omega} N_0$ one could solve it by choosing $N_0(\phi)$ to span the image of $A(\Phi)$. Now we need it span the image of $\partial_{\Omega} - A(\Phi)$, which is more complicated as it does not depend on local information. Recall that terms of the form $DK_0 \cdot X$ are in the kernel of this operator. We have to assume that the they are the only ones. \end{remk}
One tries to solve this equation by realising that $DK_0 \cdot F_j \in {\rm im}\, DK_0 = T K_0(\mathbb{T}^n)$ lies in the direction along the unperturbed embedded torus. In other words, in the direction of the unperturbed embedded torus, we solve the equation by choosing $F_j$ appropriately. We have to choose a natural transverse direction $N$ along this embedded torus, preferably so that $$N\subset {\rm im}\, \left( \partial_{\Omega} + DG_0 \right)\, .$$
We moreover assume that the family of matices $$DG_0(x_0): \mathbb{R}^N\to \mathbb{R}^N \ \mbox{for}\ x_0\in \Phi_0(\mathbb{T}^n) $$ has rank $N-n$. This is the maximal possible rank because
$$0 = \left. \frac{d}{d\varepsilon} \right|_{\varepsilon=0} DK_0(\Phi+\varepsilon v) \cdot \Omega - G_0 \circ K_0(\Phi + \varepsilon v) = D^2K_0(\Phi)(v, \Omega) - DG_0(x_0) DK_0(\Phi) \cdot v $$
\section{Another try for a geometric setup} \begin{prop} Let the embedding $K_0$ and the families of matrices $N_0, n_0$ satisfy $$ \partial_{\Omega}K_0 = G_0\circ K_0 \ \mbox{and}\ \partial_{\Omega}N_0 + N_0 \cdot n_0 = (DG_0\circ K_0 )\cdot N_0 \, .$$
Then the embedding $$(\Phi, w)\mapsto (x,v) = (K_0(\Phi), N_0(\Phi)\cdot w)$$ conjugates the skew-product vector field $$(\Phi, w) \mapsto (\Omega, n_0(\Phi)\cdot w) $$ to the variational vector field $$TG_0(x, v) = (G_0(x), DG_0(x)\cdot v)$$ \end{prop} The proof is just a direct computation.
\end{document} | arXiv |
Decomposition of time series
The decomposition of time series is a statistical task that deconstructs a time series into several components, each representing one of the underlying categories of patterns.[1] There are two principal types of decomposition, which are outlined below.
Decomposition based on rates of change
This is an important technique for all types of time series analysis, especially for seasonal adjustment.[2] It seeks to construct, from an observed time series, a number of component series (that could be used to reconstruct the original by additions or multiplications) where each of these has a certain characteristic or type of behavior. For example, time series are usually decomposed into:
• $T_{t}$, the trend component at time t, which reflects the long-term progression of the series (secular variation). A trend exists when there is a persistent increasing or decreasing direction in the data. The trend component does not have to be linear.[1]
• $C_{t}$, the cyclical component at time t, which reflects repeated but non-periodic fluctuations. The duration of these fluctuations depend on the nature of the time series.
• $S_{t}$, the seasonal component at time t, reflecting seasonality (seasonal variation). A seasonal pattern exists when a time series is influenced by seasonal factors. Seasonality occurs over a fixed and known period (e.g., the quarter of the year, the month, or day of the week).[1]
• $I_{t}$, the irregular component (or "noise") at time t, which describes random, irregular influences. It represents the residuals or remainder of the time series after the other components have been removed.
Hence a time series using an additive model can be thought of as
$y_{t}=T_{t}+C_{t}+S_{t}+I_{t},$
whereas a multiplicative model would be
$y_{t}=T_{t}\times C_{t}\times S_{t}\times I_{t}.\,$
An additive model would be used when the variations around the trend do not vary with the level of the time series whereas a multiplicative model would be appropriate if the trend is proportional to the level of the time series.[3]
Sometimes the trend and cyclical components are grouped into one, called the trend-cycle component. The trend-cycle component can just be referred to as the "trend" component, even though it may contain cyclical behavior.[3] For example, a seasonal decomposition of time series by Loess (STL)[4] plot decomposes a time series into seasonal, trend and irregular components using loess and plots the components separately, whereby the cyclical component (if present in the data) is included in the "trend" component plot.
Decomposition based on predictability
The theory of time series analysis makes use of the idea of decomposing a times series into deterministic and non-deterministic components (or predictable and unpredictable components).[2] See Wold's theorem and Wold decomposition.
Examples
Kendall shows an example of a decomposition into smooth, seasonal and irregular factors for a set of data containing values of the monthly aircraft miles flown by UK airlines.[6]
In policy analysis, forecasting future production of biofuels is key data for making better decisions, and statistical time series models have recently been developed to forecast renewable energy sources, and a multiplicative decomposition method was designed to forecast future production of biohydrogen. The optimum length of the moving average (seasonal length) and start point, where the averages are placed, were indicated based on the best coincidence between the present forecast and actual values.[5]
Software
An example of statistical software for this type of decomposition is the program BV4.1 that is based on the Berlin procedure. The R statistical software also includes many packages for time series decomposition, such as seasonal,[7] stl, stlplus,[8] and bfast. Bayesian methods are also available; one example is the BEAST method in the Rbeast R package.[9]
See also
• Frequency spectrum
• Hilbert–Huang transform
• Least squares
• Least-squares spectral analysis
• Stochastic drift
• Trend filtering
References
1. "6.1 Time series components | OTexts". www.otexts.org. Retrieved 2016-05-14.
2. Dodge, Y. (2003). The Oxford Dictionary of Statistical Terms. New York: Oxford University Press. ISBN 0-19-920613-9.
3. "6.1 Time series components | OTexts". www.otexts.org. Retrieved 2016-05-18.
4. "6.5 STL decomposition | OTexts". www.otexts.org. Retrieved 2016-05-18.
5. Asadi, Nooshin; Karimi Alavijeh, Masih; Zilouei, Hamid (2016). "Development of a mathematical methodology to investigate biohydrogen production from regional and national agricultural crop residues: A case study of Iran". International Journal of Hydrogen Energy. doi:10.1016/j.ijhydene.2016.10.021.
6. Kendall, M. G. (1976). Time-Series (Second ed.). Charles Griffin. (Fig. 5.1). ISBN 0-85264-241-5.
7. Sax, Christoph. "seasonal: R Interface to X-13-ARIMA-SEATS".
8. Hafen, Ryan. "stlplus: Enhanced Seasonal Decomposition of Time Series by Loess".
9. Li, Yang; Zhao, Kaiguang; Hu, Tongxi; Zhang, Xuesong. "BEAST: A Bayesian Ensemble Algorithm for Change-Point Detection and Time Series Decomposition".
Further reading
• Enders, Walter (2004). "Models with Trend". Applied Econometric Time Series (Second ed.). New York: Wiley. pp. 156–238. ISBN 0-471-23065-0.
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| Wikipedia |
An alternating direction method for solving a class of inverse semi-definite quadratic programming problems
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January 2016, 12(1): 303-315. doi: 10.3934/jimo.2016.12.303
Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback
Ta T.H. Trang 1, , Vu N. Phat 1, and Adly Samir 2,
Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road, Hanoi 10307, Vietnam, Vietnam
Université de Limoges, Laboratoire XLIM, 123, avenue Albert Thomas, 87060 Limoges CEDEX, France
Received November 2014 Revised January 2015 Published April 2015
This paper studies the robust finite-time $H_\infty$ control for a class of nonlinear systems with time-varying delay and disturbances via output feedback. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent conditions for the existence of output feedback controllers are established in terms of linear matrix inequalities (LMIs). The proposed conditions allow us to design the output feedback controllers which robustly stabilize the closed-loop system in the finite-time sense. An application to $H_\infty$ control of uncertain linear systems with interval time-varying delay is also given. A numerical example is given to illustrate the efficiency of the proposed method.
Keywords: time-varying delay, Finite-time stabilization, output feedback, Lyapunov function, $H_\infty$ control, linear matrix inequality..
Mathematics Subject Classification: Primary: 93D20, 34D20; Secondary: 37C7.
Citation: Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303
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Python Constraint Sudoku
Construct a program that uses an agent to solve a Sudoku puzzle as a Constraint Satisfaction Problem, with the following guidelines: You may assume that only 3 x 3 Sudoku puzzles will be used. My effort solving Sudoku puzzles with Python. ) Today, I am going to write something about my newest Sudoku solver in Python. Z3 BoolExpr - 30 examples found. is_solution. 0, Sql2K5. py 81-char-string. Generating and Solving Logic Puzzles through Constraint Satisfaction Barry O'Sullivan and John Horan Cork Constraint Computation Centre Department of Computer Science, University College Cork, Ireland b. edu and [email protected] Building constraints¶ To simplify the writing of a model, Python comparison operators (==,<=,>=) are also overloaded to compare expressions and build constraints that must be satisfied by the decision variables. It uses constraint satisfaction and search: it is a code translation from Peter Norvig's python code on his "Solve Every Sudoku. The software can now gererate a Sudoku puzzle and not only solve an existing one. At the end of this article, we're pretty sure that you will be leaving with a solid perception of concepts like "Constraint Propagation" and a popular Search Algorithm "Depth First Search". Sudoku: Each of these constraints is over 9 variables, and they are all the same constraint: Any assignment to these 9 variables such that each variable has a unique value satisfies the constraint. It uses a combination of heuristics to reduce the space of possible solutions with some fairly basic search methods. You can vote up the examples you like or vote down the ones you don't like. この記事はBrainPad Advent Calender 2017の22日目の記事です。 こんにちは、BrainpadでWebエンジニアやっています、チンバトと申します。本記事ではいくつかのアルゴリズムで数独問題を解いて見た. Sudoku Solver is a small graphical application for solving any given Sudoku puzzle, almost instantaneously. edu Abstract. In its classic form, the objective is to fill a 9x9 grid with the digits 1 to 9, subject to the following constraints: each row, each column, and each of the nine 3x3 subgrids must contain a permutation of the digits from 1 to 9. Index of Samples of MATHS Documentation. It means sponsor, or someone who brought something to wide acclaim and recognition. For binary constraints (CSPs where all the constraints involve two variables), this is usually referred to as Arc-Consistency test. Solving Every Sudoku Puzzle by Peter Norvig In this essay I tackle the problem of solving every Sudoku puzzle. In this lab exercise, you'll work with the AIMA Python implementations of Constraint Satisfaction Problems (CSPs). Constraint programming. When stuck, you can use the integrated puzzle solver. The constraints are that each row and each column must not have any duplicates; and that within a subgrid there are no duplicates. As a result, software for solving Sudoku using received content from an image was developed. It was spruiking yet another Sudoku page. Python is well suited for rapid development of cross-platform applications of all sorts, and that includes desktop GUI apps. The exit message can give more detailed information on the reason intlinprog stopped, such as exceeding a tolerance. Я видел несколько решений sudoku solvers, но я не могу понять проблему в моем коде. As clues are put in, and the constraints applied, the number of possible states reduces. Step-by-step tutorials build your skills from Hello World! to optimizing one genetic algorithm with another, and finally genetic programming; thus preparing you to apply genetic algorithms to problems in your own field of expertise. ABSTRACT The most natural formal description of a Sudoku puzzle is to express it as a constraint satisfaction problem. It optimizes planning and scheduling problems, such as Vehicle Routing, Employee Rostering, Maintenance Scheduling, Task Assignment, Cloud Optimization, Conference Scheduling, Job Shop Scheduling, Bin Packing and many more. All the times must be converted to seconds since a reference, and then to image coordinates… Note that in Python 2. 7 Example: Sudoku. For those of you who don't already know, Sudoku is a type of logic puzzle (that I was. Constraint Satisfaction Examples. Constraint optimization, or constraint programming (CP), identifies feasible solutions out of a very large set of candidates, where the problem can be modeled in terms of arbitrary constraints. If depth-first enumeration were the only way of counting the number of possible Su Dokus, then this would imply that counting Su Doku is a hard problem. Google or-tools) consists of support for constraint programming and LP/MIP (and support for local support which I have yet to look into). Sudoku, Linear Optimization, and the Ten Cent Diet Any time you have a set of linear constraints such as "at least 50 square meters of solar panels" or "the. Add me on Instagram @quinstonpimenta if you'd like to get in touch. This is my first substantial project, and I would love any comments or feedback. I did however keep hearing about PyTorch which was supposedly better than TensorFlow in many ways, but I never really got around to learning it. Note that this example only works with SWI-Prolog and not with other Prolog implementations, because it uses the SWI-Prolog CLPFD library (Constraint Logic Programming over Finite Domains). The constraint that each cell can not be the same as any other. to refresh your session. When I first started with neural networks I learned them with TensorFlow and it seemed like TensorFlow was pretty much the industry standard. Sudoku is a 9x9 matrix filled with numbers 1 to 9 in such a way that every row, column and sub-matrix (3x3) has each. We present a di–culty rating metric and three puzzle generation algorithms for the popular Sudoku puzzle. 410-13 Constraint Processing, by Rina Dechter September 27th, 2010 Assignments • Remember:. Given a partially filled 9×9 2D array 'grid[9][9]', the goal is to assign digits (from 1 to 9) to the empty cells so that every row, column, and subgrid of size 3×3 contains exactly one instance of the digits from 1 to 9. Understanding python requires you to both have learned python and have an understanding of basic computer architecture and memory manipulation techniques- and that is the part I do not like. If you're interested in solving your own problem using constraint programming and don't wanna wait until my solver can do it :D Python-Constraint is an existing library which can be used. Luckily, Allison Morgan has discovered a way to use Integer Linear Programming to solve Sudoku puzzles, and it only takes a few minutes! "The first constraint requires that each cell, denoted by its row and column, contains one value. Some constraint solvers include a method to model and solve Sudokus, and a program may. Constraint satisfaction toolkits. Pythonforeducation: theexactcoverproblem A. Builds and solves the classic diet problem. So maybe I was jumping to conclusions a little early in the game. This is an example where is the constraint is propagating between unassigned variables. Instead of entering each constraint individually, you can instead add them in one step. Constraints differ from the common primitives of other programming languages in that they do not specify a step or sequence of steps to execute but rather the properties of a solution to be found. A simple brute-force Sudoku solver written in functional-programming style. This program is actually that easy, that you can even find it in the SWI-Prolog manual. You can vote up the examples you like or vote down the ones you don't like. [4] We construct our own IP model in the next section which is similar and contains all the same properties as their model. Sudoku is a puzzle game in which you must fill in each box of a grid with a number following some constraints. $\begingroup$ It's in the same realm - the solver is a constraint programming solver, which works well since the problem isn't really linear but it is a bunch of constraints. When I first started with neural networks I learned them with TensorFlow and it seemed like TensorFlow was pretty much the industry standard. If "outliers", only the sample points lying outside the whiskers are shown. Each C i involves a subset. A partial assignment can be specified on the left grid. This may be due to the choice of data structures: I chose to use vectors and sets to represent the Sudoku squares and digits, whereas the Python version uses strings, which are relatively lightweight. Algorithmics of Sudoku may help implement this. Does anyone know a simple algorithm to check if a Sudoku-Configuration is valid? The simplest algorithm I came up with is (for a board of size n) in Pseudocode. We will implement a simple sudoku solver with the python library of OR-Tools. In this paper, we explore methods of solving Sudoku logic puzzles using constraint satisfaction algorithms. Getting Started¶. If you're interested in solving your own problem using constraint programming and don't wanna wait until my solver can do it :D Python-Constraint is an existing library which can be used. a type constraint tool for python function. I was playing a casual game of Sudoku today when a friend came by and asked "What's the hardest game of Sudoku possible?" My response: "A Sudoku puzzle with the minimal amount of starting numbers. If "outliers", only the sample points lying outside the whiskers are shown. Built an Azure-heavy gaming website based on a 100%-pure Angular (TypeScript) SPA backed by a series of Node. Optimization with PuLP¶. Environment: WSS 3. python-constraint / examples / sudoku / sudoku. Cassowary constraint solver, an open source project for constraint satisfaction (accessible from C, Java, Python and other languages). miniKanren has been implemented in a growing number of host languages, including Scheme, Racket, Clojure, Haskell, Python, JavaScript, Scala, Ruby, OCaml, and PHP, among many other languages. On paper, many people can solve a Sudoku puzzle given enough time and it wouldn't be too difficult. Puzzle modeling. These pages are not about code but about ways of thinking about problems and solutions. We will be using python and the PuLP linear programming package to solve these linear programming problems. For example, if two statements are true then we can infer any third. Builds and solves the classic diet problem. Therefore no guessing (or searching + backtracking) is required; only constraint propagation. Declarative De nition We use the python frontend of the z3 constraint solver in combination with list comprehension to specify the 9x9 Sudoku puzzle declaratively. removed Åfalse 2. In Sudoku, we have: 1. Sudoku Solver is a small graphical application for solving any given Sudoku puzzle, almost instantaneously. After reading the Artificial Intelligence Book by by Stuart J. У меня есть функция судокусольвера, которая становится советом судоку. The assignment will be required to use some search algorithms to solve a puzzle, and return the puzzle solution, as follows: • brute force (exhaustive search) method • back-tracking (Constraint Satisfaction Problem (CSP). ) Today, I am going to write something about my newest Sudoku solver in Python. 0 f1359ff Nov 5, 2018. As another example, In this concrete case, the constraint solver is strong enough to find the unique solution without any search. # A valid sudoku square satisfies these # two properties: # 1. The software can now gererate a Sudoku puzzle and not only solve an existing one. py, um das vorgege-bene Sudoku-Problem zu l¨osen. There are 3 constraint functions: The eliminate function goes through all the boxes with known values and eliminate that value from all of its peers. You can debug your implementation on small the game trees using the command: python autograder. What makes it nice is the purely arithmetic one-liner computing the constraint c (the sequence of already used digits on the same row, same column, same block of a given cell). Demonstrates model construction and simple model modification - after the initial model is solved, a constraint is added to limit the number of dairy servings. Doing things with python-constraint is pretty easy. One way to tackle CSPs programmatically is to use the Microsoft Solver Foundation (MSF) library. Chapters 2 and 3 are devoted to constraint mod-eling. Today's author, Charlie Ellis, a Program Manager on the Excel team, shares a spreadsheet he built in Excel for solving Sudoku puzzles. Algorithm X finds all solutions to the exact cover problem. Entries are integers between 1 and 9. You may find that deleting several constraints will still lead to a single optimal solution but the removal of one particular constraint leads to a sudden dramatic increase in the number of solutions. You start by applying specific algorithms to two specific problems and then reflect on the nature of CSPs and the algorithms used to solve them. The last constraint fixes that only one of a value is found in each subgrid. In the third video, we will render the Sudoku table. Sudoku: Each of these constraints is over 9 variables, and they are all the same constraint: Any assignment to these 9 variables such that each variable has a unique value satisfies the constraint. Sudoku is a puzzle game in which you must fill in each box of a grid with a number following some constraints. In this work, we model the known benchmark problems Latin Square, Magic Square and Sudoku as a Constraint Satisfaction Problems. Users who have contributed to this. Builds and solves the classic diet problem. grid[9][9], the goal is to assign digits (from 1 to 9) to the empty cells so that every row, column, and subgrid of size 3×3 contains exactly one instance of the digits from 1 to 9. Excel has an add-in called the Solver which can be used to solve systems of equations or inequalities. const Logic = require ('logic-solver');. Support for X Sudoku puzzles (where diagonals have the same constraints as the rows, columns and boxes). Our solver is the dark blue one. Guide to Creating a Sudoku Solver using Python and Pygame After creating a version of Conways Game of Life in Python I was keen to explore Pygame further. PuLP — a Python library for linear optimization. Whenever I've pondered on how to solve Sudoku, I've considered an alternative approach of just forking the process (or creating a new thread), then running each attempt in parallel. encode is the compression function. I am trying to find out how to make a Sudoku Puzzle Solver but I have no way of going about it I am intermediate in C++ so I can understand some things but not much (Sorry I said something stupid). The program written in Python takes as input a string representing the initial board configuration. This is translated from example python solution on exercism. 29 KB def revise_sudoku (sudoku_unsolved, sudoku_binary_constraints, x, y): revised = False for x in sudoku_unsolved. n if k is not in the row (using another for-loop) return not-a-solution. edu Abstract. 29 KB def revise_sudoku (sudoku_unsolved, sudoku_binary_constraints, x, y): revised = False for x in sudoku_unsolved. Sudoku solver,that solves sudoku puzzles using constraint programming - 1. You can vote up the examples you like or vote down the ones you don't like. The Program. My Python Sudoku solver is available to download here. Sudoku is known to be an NP-complete problem, so obviously even the cleverest solver I could write would eventually run into problems. Solves the Sudoku puzzle with the odd constraint of minimizing. After reading the Artificial Intelligence Book by by Stuart J. In his paper Sudoku as a Constraint Problem, Helmut Simonis describes many reasoning algorithms based on constraints which can be applied to model and solve problems. edu Abstract. Sudoku; In this series of posts, we explore some linear programming examples, starting with some very basic Mathematical theory behind the technique and moving on to some real world examples. This project requires Python 3. We suggest representing a Sudoku board with a Python dictionary, where each key is a variable name based on. 2 Constraint Satisfaction Problems. Bibo Sudoku is written in Python. This approach explains the maturation and execution of a Sudoku solution including detailed directions regarding its progress. Guidelines. A grid is a valid puzzle if there is a unique way to complete it to match the Sudoku constraints (each line, column and aligned $3\times3$ square has no repeated element) and it is minimal in that respect (i. SAT is often described as the "mother of all NP-complete problems. Some people seem to think that language design is just like solving a puzzle. A: The constraint: for every unit there is a unique set of numbers from 1 to 9. JS (ES6) services. Sudoku solver Tags: Alldifferent, Integer programming, Logic programming Updated: September 16, 2016 In case you have missed out on the Sudoku hype, the goal is to fill in unspecified elements in a matrix with numbers between 1 to 9, keeping elements in all rows and columns different, and keeping all elements in the 9 3x3 blocks different. My effort solving Sudoku puzzles with Python. compatible takes a sudoku board and a digit configuration (defined by conf and dig), overlays the digit configuration over the sudoku board and checks for conflicts. do the same for each column. Installing the python-constraint Module. Projekt 1: Constraint-Satisfaction-Probleme Abgabe: 7. Constraint Satisfaction Problems (CSP) A powerful representation for (discrete) search problems A Constraint Satisfaction Problem (CSP) is defined by: X is a set of n variables X 1, X 2,…, X n each defined by a finite domain D 1, D 2,…D n of possible values. I will use docplex Python api to implement a web application that solves Sudoku problems. Adding backtracking to finish solving the remaining sudoku is an good challenge if you find you have additional time for this lab. A position constraint: Only 1 number can occupy a cell 2. To install this module, open the terminal and run: $ pip install python-constraint. My guess is that this is due to the harder problems presenting more constraints, which are difficult for humans to take into account all at once when solving a Sudoku and causing a lot of back-tracking, are actually easier for a computer - the more logical constraints there are for a fixed number of variables, the quicker a computer will be. Simply put, Sudoku is a combinatorial number placement puzzle with 9 x 9 cell grid partially filled in with numbers from 1 to 9. Python python-constraint sudoku solver. Partial feasibility can be included by adding probabilities to constraints (e. 102x Machine Learning. To create sudoku problems with multiple solutions from unique solution sudoku problem, you can simply delete a starting number constraint. py knows how to ables for every cell f1; : : : ; n4 g of the Sudoku puzzle is build the constraint model above, find a solution via the pro- required. Using Genetic Algorithms to come up with Sudoku Puzzles Helmut Simonis: Sudoku as a Constraint Problem Jonathan Sillito: Improvements to and estimating the cost of backtracking algorithms for constraint satisfaction problems. Collections. At the end of this article, we're pretty sure that you will be leaving with a solid perception of concepts like "Constraint Propagation" and a popular Search Algorithm "Depth First Search". Op dat punt gooi ik alle permutaties die niet aan de constraints voldoen weg. I searched in Google and find that the best way is backtracking. constraints satisfaction solver in Python. $\begingroup$ It's in the same realm - the solver is a constraint programming solver, which works well since the problem isn't really linear but it is a bunch of constraints. JS (ES6) services. uk Abstract. In terms of its type hints, it uses generics to make itself flexible enough to work with any kind of variables and domain values (V keys and D domain values). Level up your coding skills and quickly land a job. I applied an algorithm based on Branch and Bound technique. Constraint satisfaction toolkits are software libraries for imperative programming languages that are used to encode and solve a constraint satisfaction problem. Larger grids are also possible, with Daily SuDoku's 12×12-grid Monster SuDoku, the Times likewise offers a 12×12-grid Dodeka sudoku with 12 regions each being 4×3, Dell regularly publishing 16×16 Number Place Challenger puzzles (the 16×16 variant often uses 1 through G rather than the 0 through F used in hexadecimal), and Nikoli proffering 25×25 Sudoku the Giant behemoths. C# exercises for beginners, intermediates and advanced students. 7: from constraint import * # Normalizes sudoku solution. Solving Sudoku Game Using Quantum Computation Sudoku rules add the constraint that each region may only have the num- we propose a Python program which takes an un-. The software can now gererate a Sudoku puzzle and not only solve an existing one. It was written in python (in the matter of a few hours so please excuse its inelegance). Solving sudoku as an Integer Programming problem 5-4 The 1st equation below corresponds to the constraint on columns, the 2nd one refers to the constraint on rows and the 3rd one to the constraint on the 3 ×3 squares. There's this field of recursive search—I know how that works. In particular, these are some of the core packages:. Please keep submissions on topic and of high quality. Each cell is a variable, and the domain is all the used symbols, 1 9 in a standard Sudoku puzzle. We will now create a Sudoku solver using backtracking by encoding our problem, goal and constraints in a step-by-step algorithm. So in total there are 81 variables and 729 constraints. A position constraint: Only 1 number can occupy a cell 2. For binary constraints (CSPs where all the constraints involve two variables), this is usually referred to as Arc-Consistency test. include translations into the domains of constraint satisfaction, integer pro- gramming, polynomial calculus and graph theory, are available in an open- source Python library sudoku. この記事はBrainPad Advent Calender 2017の22日目の記事です。 こんにちは、BrainpadでWebエンジニアやっています、チンバトと申します。本記事ではいくつかのアルゴリズムで数独問題を解いて見た. ABSTRACT The most natural formal description of a Sudoku puzzle is to express it as a constraint satisfaction problem. Users who have contributed to this. IMPLEMENTING A CSP SOLVER FOR SUDOKU Benjamin Bittner1 and Kris Oosting2 1University of Amsterdam, The Netherlands; [email protected] In other words, a. Entries are integers between 1 and 9. Constraint programming. What makes it nice is the purely arithmetic one-liner computing the constraint c (the sequence of already used digits on the same row, same column, same block of a given cell). The more I use Python, the more I like it. BoolExpr extracted from open source projects. Also you can create Sudoku game manually and print it on an A4/Letter paper. The spreadsheet can be found in the attachments at the bottom of this post. The most difficult aspect of SuDoku is how to generate a problem which has a unique solution. Z3 BoolExpr - 30 examples found. py knows how to ables for every cell f1; : : : ; n4 g of the Sudoku puzzle is build the constraint model above, find a solution via the pro- required. i've been through google but could'nt find anything about futoshiki code or how to implement it in java. Solve Every Sudoku Puzzle in Python by Peter Norvig - sudoku. Gecode provides a constraint solver with state-of-the-art performance while being modular and extensible. Because iteration is so common, Python provides several language features to make it easier. I and a student of mine are working on Sudoku solvers which solve puzzles the way that humans would. That's enough links that don't actually contribute to the. A Sudoku puzzle is a 9x9 grid of numbers between 1 and 9. As a matter of fact Peter wrote a constraint programming solver tailored to Sudoku. in a constraint for every city which makes sure that every city is passed exactly once. that he liked to give the Sudoku puzzle as a question on the comprehensive exam for PhD students in Operations Research. In this post, I will show how solving a Sudoku puzzle is equivalent to solving an integer linear programming (ILP) problem. Sudoku can easily be represented as a CSP. We will implement a simple sudoku solver with the python library of OR-Tools. It was a hard call to label it spam. It explains how he wrote a simple Sudoku Solver in Python using constraint propagation and backtracking search. to refresh your session. These usually arise from linear constraint matrices that have large condition number, or problems that have large solution components. In order to make full use of the power of SAT solvers, a SAT compiler must encode domain variables and constraints into an e cient SAT formula. The sorting problem is thus both P and NP. PuLP largely uses python syntax and comes. Hi, Does anyone know how to enforce a uniqueness constraint on a column in a SharePoint list? I've been unable to find anything on the issue. Sudoku is a puzzle game in which you must fill in each box of a grid with a number following some constraints. It was spruiking yet another Sudoku page. This equivalence allows us to solve a Sudoku puzzle using any of the many freely available ILP solvers; an implementation of a solver (in Python 3) which follows the formulation described in this post can be found found here. > I post below a sudoku solver. Solving Sudoku. thanx for ur reply. SO get cracking. There are 3 constraint functions: The eliminate function goes through all the boxes with known values and eliminate that value from all of its peers. Constraint programming. Giappone Una scena accanto alla chemin Foto Stereo Vintage Albumina 1896,Zapf Creations Gianna Collection Doll,* MINERALI * PIRITE Pentagonale del Perù Qualità Extra Collezionismo Chakra Zen. There's no reason the brute force approach shouldn't work, unless you're on a machine with very little memory or a very small stack. I am thinking that this is not possible because when you call a function from one Python module, it probably cannot interface directly to other functions in other Python modules. Norvig's code uses the same constraints as Knuth's to eliminate particular digits from particular squares. To install this module, open the terminal and run: $ pip install python-constraint. As a matter of fact Peter wrote a constraint programming solver tailored to Sudoku. Environment: WSS 3. Welcome to my Sudoku X Solver. After reading the Artificial Intelligence Book by by Stuart J. and constraints. In this work, we model the known Sudoku puzzle as a. Exitflags 3 and -9 relate to solutions that have large infeasibilities. Creating (and solving) these puzzles is a constraint satisifaction problem, which in general is a hard thing. Python is well suited for rapid development of cross-platform applications of all sorts, and that includes desktop GUI apps. It consists in generating all the possible ways of lling the free cells of a Sudoku puzzle (ignoring the constraints), considered in some xed order. There are many libraries in the Python ecosystem for this kind of optimization problems. Wishes for room mates are mild in the extreme so it is very easy for a human to place these. And godfather doesn't necessarily mean creator or originator. To make sure all the variables in a set are different, we use the AllDifferentConstraint. formal de nition for the Sudoku puzzle can be found in [13]. Common constraint programming problems Below are the problems which I have implemented in at least two Constraint Programming systems. Hey i need to print a sudoku for school but i am having a hard time doing so. It is a backtracking algorithm too, but I wanted to share my implementation as well. Excel has an add-in called the Solver which can be used to solve systems of equations or inequalities. Google CP Solver, a. Here f:\[DoubleStruckCapitalR]^n-> \[DoubleStruckCapitalR] is called the objective function and \[CapitalPhi](x) is a Boolean-valued formula. I am thinking that this is not possible because when you call a function from one Python module, it probably cannot interface directly to other functions in other Python modules. There are significantly fewer valid Sudoku solution grids than Latin squares because Sudoku imposes the additional regional constraint. I still have no clue on how to tell the solver how to constrain the number of occupants in rooms: I have made up an simple example with nine persons and three rooms. Multiplayer. logilab-constraint 0. type_constraint 0. The constraints are that each row and each column must not have any duplicates; and that within a subgrid there are no duplicates. Learn the fundamentals of programming with Python and building web apps Build web applications from scratch with Django Create real-world RESTful web services with the latest Django framework Book Description. My guess is that this is due to the harder problems presenting more constraints, which are difficult for humans to take into account all at once when solving a Sudoku and causing a lot of back-tracking, are actually easier for a computer - the more logical constraints there are for a fixed number of variables, the quicker a computer will be. You are given a grid that is partially filled in, and your job is to fill the rest of the grid in so that: No row contains the same number twice. 0 Sudoku is the name of the number puzzle game that is rapidly becoming one of the most popular global online games ever. $\begingroup$ It's in the same realm - the solver is a constraint programming solver, which works well since the problem isn't really linear but it is a bunch of constraints. In other words, I have tried to make this solver akin to human heuristic solving and any sudoku problem solvable by this is guaranteed to be solvable by a human in the deterministic approach described above. But I bet it takes a long time to run. In their paper, they show that a sudoku can also be thought of as a constraint programming problem as well. Solve Every Sudoku Puzzle in Python by Peter Norvig - sudoku. We show how such a problem can be solved using constraint programming and explain a simple approach to finite domains constraint solving. For every value v in the domain of Y do – If there is no value u in the domain of X such that th t i t (the constraint on (X,Y) i ti fi d th) is satisfied then a. A Sudoku is a 9x9 grid, grouped into a 3x3 grid of 3x3 blocks, where each square in the grid is to be lled with a digit from 1 to 9 such that each row, column, and block must contain each. As with SIGCSE's Nifty Assignments, EAAI Model AI Assignments should be: Adoptable - Provide materials to make the assignment easy for other instructors to adopt. Hard & Soft Clustering with K-means, Weighted K-means and GMM-EM in Python March 19, 2017 April 7, 2017 / Sandipan Dey The following problems appeared as a project in the edX course ColumbiaX: CSMM. You signed out in another tab or window. Level up your coding skills and quickly land a job. It is also used in solving the knapsack problem, parsing texts and other combinatorial optimization problems. # About: Sudoku Solver using constraint programming # Author: suryak # Description: # * Requires constraint lib # * Takes sudoku puzzle input via text file # * Empty locations are required to be filled with 0 # * Output can be observed on console # Note: This is written using Python 2. Solving Constraint Programs using Backtrack Search and Forward Checking 9/29/10 1 Slides draw upon material from: Brian C. G Suite Developers Blog Sudoku, Linear Optimization, and the Ten Cent Diet Any time you have a set of linear constraints such as "at least 50 square meters. When you have constraints structured in the same way (like these are), there is a faster way to add them all to SOLVER. Bibo Sudoku is written in Python. Exact cover II. The code isn't perfect, but it will solve pretty much any Sudoku puzzle. Since the previous implementation was created using opencv 2. To make sure all the variables in a set are different, we use the AllDifferentConstraint. This will be the key step to solve a problem using DA. If we remember our knowledge base about humans and gods again and especially the rule mortal(X) :- human(X). We already know that logic is the study of principles of correct reasoning or in simple words it is the study of what comes after what. We recommend that you read The Optimisation Process, Optimisation Concepts, and the Introduction to Python before beginning the case-studies. import csv import copy def main() : cells = [cell() for i in range(81)] #creates a list of 81 instances of the cell() class. According to Peter Norvig in his fantastic essay on solving every Sudoku puzzle using Python, security expert Ben Laurie once stated that "Sudoku is a denial of service attack on human intellect". When we are dealing with a diagonal sudoku puzzle, we need to include two additional units where this elimination constraint has to be applied: one for the diagonal from. SO get cracking. In its classic form, the objective is to fill a 9x9 grid with the digits 1 to 9, subject to the following constraints: each row, each column, and each of the nine 3x3 subgrids must contain a permutation of the digits from 1 to 9. Do yourself a huge favour and buy Beginning Python: From Novice to Professional by Magnus Lie Hetland. Learn More >> Build AMPL into your applications APIs now available for C++, C#, Java, MATLAB, Python, and R. Multiplayer. We will be using python and the PuLP linear programming package to solve these linear programming problems. 7 Example: Sudoku. This is the best place to expand your knowledge and get prepared for your next interview. My guess is that this is due to the harder problems presenting more constraints, which are difficult for humans to take into account all at once when solving a Sudoku and causing a lot of back-tracking, are actually easier for a computer - the more logical constraints there are for a fixed number of variables, the quicker a computer will be. I recently came across Peter Norvig's Solving Every Sudoku Puzzle. For the rare uninitiated, the game is played on a 9x9 square. Any assignment where two or more variables have the same value falsifies the constraint. SO get cracking. Support for X Sudoku puzzles (where diagonals have the same constraints as the rows, columns and boxes). Do check out http://norvig. Sudoku & Backtracking. Given a set of requirements they systematically search the solution space for a match, and when they find one, they claim to have the perfect language feature, as if they've solved a Sudoku puzzle. Wishes for room mates are mild in the extreme so it is very easy for a human to place these. Do not worry about memorizing python. | CommonCrawl |
Fringe search
In computer science, fringe search is a graph search algorithm that finds the least-cost path from a given initial node to one goal node.
In essence, fringe search is a middle ground between A* and the iterative deepening A* variant (IDA*).
If g(x) is the cost of the search path from the first node to the current, and h(x) is the heuristic estimate of the cost from the current node to the goal, then ƒ(x) = g(x) + h(x), and h* is the actual path cost to the goal. Consider IDA*, which does a recursive left-to-right depth-first search from the root node, stopping the recursion once the goal has been found or the nodes have reached a maximum value ƒ. If no goal is found in the first threshold ƒ, the threshold is then increased and the algorithm searches again. I.E. It iterates on the threshold.
There are three major inefficiencies with IDA*. First, IDA* will repeat states when there are multiple (sometimes non-optimal) paths to a goal node - this is often solved by keeping a cache of visited states. IDA* thus altered is denoted as memory-enhanced IDA* (ME-IDA*), since it uses some storage. Furthermore, IDA* repeats all previous operations in a search when it iterates in a new threshold, which is necessary to operate with no storage. By storing the leaf nodes of a previous iteration and using them as the starting position of the next, IDA*'s efficiency is significantly improved (otherwise, in the last iteration it would always have to visit every node in the tree).
Fringe search implements these improvements on IDA* by making use of a data structure that is more or less two lists to iterate over the frontier or fringe of the search tree. One list now, stores the current iteration, and the other list later stores the immediate next iteration. So from the root node of the search tree, now will be the root and later will be empty. Then the algorithm takes one of two actions: If ƒ(head) is greater than the current threshold, remove head from now and append it to the end of later; i.e. save head for the next iteration. Otherwise, if ƒ(head) is less than or equal to the threshold, expand head and discard head, consider its children, adding them to the beginning of now. At the end of an iteration, the threshold is increased, the later list becomes the now list, and later is emptied.
An important difference here between fringe and A* is that the contents of the lists in fringe do not necessarily have to be sorted - a significant gain over A*, which requires the often expensive maintenance of order in its open list. Unlike A*, however, fringe will have to visit the same nodes repeatedly, but the cost for each such visit is constant compared to the worst-case logarithmic time of sorting the list in A*.
Pseudocode
Implementing both lists in one doubly linked list, where nodes that precede the current node are the later portion and all else are the now list. Using an array of pre-allocated nodes in the list for each node in the grid, access time to nodes in the list is reduced to a constant. Similarly, a marker array allows lookup of a node in the list to be done in constant time. g is stored as a hash-table, and a last marker array is stored for constant-time lookup of whether or not a node has been visited before and if a cache entry is valid.
init(start, goal)
fringe F = s
cache C[start] = (0, null)
flimit = h(start)
found = false
while (found == false) AND (F not empty)
fmin = ∞
for node in F, from left to right
(g, parent) = C[node]
f = g + h(node)
if f > flimit
fmin = min(f, fmin)
continue
if node == goal
found = true
break
for child in children(node), from right to left
g_child = g + cost(node, child)
if C[child] != null
(g_cached, parent) = C[child]
if g_child >= g_cached
continue
if child in F
remove child from F
insert child in F past node
C[child] = (g_child, node)
remove node from F
flimit = fmin
if reachedgoal == true
reverse_path(goal)
Reverse pseudo-code.
reverse_path(node)
(g, parent) = C[node]
if parent != null
reverse_path(parent)
print node
Experiments
When tested on grid-based environments typical of computer games including impassable obstacles, fringe outperformed A* by some 10 percent to 40 percent, depending on use of tiles or octiles. Possible further improvements include use of a data structure that lends itself more easily to caches.
References
• Björnsson, Yngvi; Enzenberger, Markus; Holte, Robert C.; Schaeffer, Johnathan. Fringe Search: Beating A* at Pathfinding on Game Maps. Proceedings of the 2005 IEEE Symposium on Computational Intelligence and Games (CIG05). Essex University, Colchester, Essex, UK, 4–6 April, 2005. IEEE 2005. https://web.archive.org/web/20090219220415/http://www.cs.ualberta.ca/~games/pathfind/publications/cig2005.pdf
External links
• Jesús Manuel Mager Hois's implementation of Fringe Search in C https://github.com/pywirrarika/fringesearch
Graph and tree traversal algorithms
• α–β pruning
• A*
• IDA*
• LPA*
• SMA*
• Best-first search
• Beam search
• Bidirectional search
• Breadth-first search
• Lexicographic
• Parallel
• B*
• Depth-first search
• Iterative Deepening
• D*
• Fringe search
• Jump point search
• Monte Carlo tree search
• SSS*
Shortest path
• Bellman–Ford
• Dijkstra's
• Floyd–Warshall
• Johnson's
• Shortest path faster
• Yen's
Minimum spanning tree
• Borůvka's
• Kruskal's
• Prim's
• Reverse-delete
List of graph search algorithms
| Wikipedia |
David Bressoud
David Marius Bressoud (born March 27, 1950, in Bethlehem, Pennsylvania) is an American mathematician who works in number theory, combinatorics, and special functions. As of 2019 he is DeWitt Wallace Professor of Mathematics at Macalester College, Director of the Conference Board of the Mathematical Sciences and a former President of the Mathematical Association of America.
David Bressoud
Born (1950-03-27) March 27, 1950
Bethlehem, Pennsylvania
Alma materTemple University
AwardsMAA Distinguished Teaching Award
Beckenbach Book Prize
MAA George Pólya Lecturer
Scientific career
FieldsMathematics
InstitutionsPennsylvania State University
Macalester College
Doctoral advisorEmil Grosswald
Life and education
Bressoud was born March 27, 1950, in Bethlehem, Pennsylvania.[1] He became interested in mathematics in the seventh grade, where he had a teacher who encouraged him and gave him challenging problems. He attended Albert Wilansky's National Science Foundation summer program at Lehigh University between his junior and senior years in high school, where he also spent most of his time working on problems.[2]
He graduated from Swarthmore College in 1971.[3] When he started at Swarthmore he had not yet decided on a major, but after his first year he decided to get out of college as quickly as possibly and had no interest in graduate school, and the quickest way out was to major in mathematics.[2]
After graduating Bressoud became a Peace Corps volunteer in Antigua from 1971 to 1973, teaching math and science at Clare Hall School. While in Antigua he realized he missed mathematics, and kept working on it as a hobby.[2] After the Peace Corps he went to graduate school at Temple University,[2] and received his PhD in 1977 under Emil Grosswald.[4]
Career
After receiving his PhD, Bressoud taught at Pennsylvania State University from 1977 to 1994, reaching the rank of full professor in 1986. During this period he held visiting positions at the Institute for Advanced Study (1979–1980), the University of Wisconsin (1980–81 and 1982), the University of Minnesota (1983 and 1998), and the University of Strasbourg (1984–85).[4]
His focus at Penn State was mathematics research, but in the late 1980s he became more interested in teaching and writing textbooks, and he decided to make a move. He said in a 2008 interview, "I needed to be in a place that had a strong focus on teaching and a community of people for whom teaching was what they were most interested in." He decided on a move to Macalester College in 1994, where he was DeWitt Wallace Professor of Mathematics.[2] Since 2005 he has written a monthly online column for MAA titled "Launchings" that focuses on the CUPM (Committee on the Undergraduate Program in Mathematics) Curriculum Guide.[5]
Bressoud received several of the Mathematical Association of America's awards: the Distinguished Teaching Award for the Allegheny Mountain section in 1994,[6] the Beckenbach Book Prize in 1999,[7] and he was a George Pólya Lecturer from 2002 to 2004.[8]
Bressoud was elected president of the Mathematical Association of America in the 2007 elections, and served as President-Elect in 2008 and served as president from 2009 to 2011.[9] In 2012 he became a fellow of the American Mathematical Society.[10] He began phased retirement at Macalester College in 2016 and in 2017 took over the position of Director of the Conference Board of the Mathematical Sciences from Ron Rosier, who had served in that capacity for 29 years.
Selected publications
• Bressoud, David (1989). Factorization and Primality Testing. Berlin: Springer-Verlag. ISBN 978-0-387-97040-0.
• Bressoud, David (1991). Second Year Calculus: From Celestial Mechanics to Special Relativity. Berlin: Springer-Verlag. ISBN 978-0-387-97606-8.
• Bressoud, David (1999). Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture. Cambridge: Mathematical Association of America/Cambridge University Press. ISBN 978-0-521-66646-6.
• Bressoud, David; Wagon, Stan (2000). A Course in Computational Number Theory. New York: Key College Publishing in cooperation with Springer. ISBN 978-1-930190-10-8.
• Bressoud, David (2007). A Radical Approach to Real Analysis (2nd ed.). Washington: Mathematical Association of America. ISBN 978-0-88385-747-2.
• Bressoud, David (2008). A Radical Approach to Lebesgue's Theory of Integration. Cambridge: Cambridge University Press/Mathematical Association of America. ISBN 978-0-521-71183-8.
• Bressoud, David (2019). Calculus Reordered: A History of the Big Ideas. Princeton: Princeton University Press. ISBN 978-0-691-18131-8.
See also
• Zeilberger–Bressoud theorem
References
1. "Biographies of Candidates 2002" (PDF). Notices of the American Mathematical Society. Providence, RI: American Mathematical Society. 49 (8): 970–981. September 2002.
2. Peterson, Ivars (January 2009). "An Interview with David Bressoud, MAA President" (PDF). MAA Focus. Washington, D.C.: Mathematical Association of America. 29 (1): 6–8. Retrieved 2009-02-01.
3. "David Bressoud biography". Mathematical Association of America. January 12, 2009. Retrieved 2009-02-01.
4. "Northeastern Section of the Mathematical Association of America; Fall 2004 Meeting - WPI; Biographies of Invited Speakers". Worcester Polytechnic Institute. Retrieved 2009-02-01.
5. "David Bressoud's Launchings". Mathematical Association of America. October 12, 2019. Retrieved 2019-10-12.
6. Martha J. Siegel (April 2007). "The Mathematical Association of America 2007 National Elections" (PDF). Mathematical Association of America. Retrieved 2009-02-01.
7. "The Mathematical Association of America's Beckenbach Book Prize". Mathematical Association of America. December 16, 2008. Retrieved 2009-02-01.
8. "The Mathematical Association of America's George Pólya Lecturers". Mathematical Association of America. September 20, 2007. Retrieved 2009-02-01.
9. "MAA Officers' Election". Mathematical Association of America. June 1, 2007. Retrieved 2009-02-01.
10. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
Further reading
• Peterson, Ivars (December 22, 2008). "An Interview with David Bressoud, MAA President". Mathematical Association of America. Retrieved 2009-02-01. A longer version of the MAA Focus interview referenced above.
External links
• David Bressoud at the Mathematics Genealogy Project
• David Bressoud's home page
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| Wikipedia |
Yang Hui
Yang Hui (simplified Chinese: 杨辉; traditional Chinese: 楊輝; pinyin: Yáng Huī, ca. 1238–1298), courtesy name Qianguang (謙光), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's Triangle. This triangle was the same as Pascal's Triangle, discovered by Yang's predecessor Jia Xian. Yang was also a contemporary to the other famous mathematician Qin Jiushao.
Written work
The earliest extant Chinese illustration of 'Pascal's triangle' is from Yang's book Xiangjie Jiuzhang Suanfa (詳解九章算法)[1] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian[2] who expounded it around 1100 AD, about 500 years before Pascal. In his book (now lost) known as Rújī Shìsuǒ (如積釋鎖) or Piling-up Powers and Unlocking Coefficients, which is known through his contemporary mathematician Liu Ruxie (劉汝諧).[3] Jia described the method used as 'li cheng shi suo' (the tabulation system for unlocking binomial coefficients).[3] It appeared again in a publication of Zhu Shijie's book Jade Mirror of the Four Unknowns (四元玉鑒) of 1303 AD.[4]
Around 1275 AD, Yang finally had two published mathematical books, which were known as the Xugu Zhaiqi Suanfa (續古摘奇算法) and the Suanfa Tongbian Benmo (算法通變本末, summarily called Yang Hui suanfa 楊輝算法).[5] In the former book, Yang wrote of arrangement of natural numbers around concentric and non concentric circles, known as magic circles and vertical-horizontal diagrams of complex combinatorial arrangements known as magic squares, providing rules for their construction.[6] In his writing, he harshly criticized the earlier works of Li Chunfeng and Liu Yi (劉益), the latter of whom were both content with using methods without working out their theoretical origins or principle.[5] Displaying a somewhat modern attitude and approach to mathematics, Yang once said:
The men of old changed the name of their methods from problem to problem, so that as no specific explanation was given, there is no way of telling their theoretical origin or basis.[5]
In his written work, Yang provided theoretical proof for the proposition that the complements of the parallelograms which are about the diameter of any given parallelogram are equal to one another.[5] This was the same idea expressed in the Greek mathematician Euclid's (fl. 300 BC) forty-third proposition of his first book, only Yang used the case of a rectangle and gnomon.[5] There were also a number of other geometrical problems and theoretical mathematical propositions posed by Yang that were strikingly similar to the Euclidean system.[7] However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian Jesuit Matteo Ricci and the Ming official Xu Guangqi in the early 17th century.[8]
Yang's writing represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.[9] Yang was also well known for his ability to manipulate decimal fractions. When he wished to multiply the figures in a rectangular field with a breadth of 24 paces 3 4⁄10 ft. and length of 36 paces 2 8⁄10, Yang expressed them in decimal parts of the pace, as 24.68 X 36.56 = 902.3008.[10]
See also
• History of mathematics
• List of mathematicians
• Chinese mathematics
Notes
1. Fragments of this book was retained in the Yongle Encyclopedia vol 16344, in British Museum Library
2. Needham, Volume 3, 134-137.
3. Needham, Volume 3, 137.
4. Needham, Volume 3, 134-135.
5. Needham, Volume 3, 104.
6. Needham, Volume 3, 59-60.
7. Needham, Volume 3, 105.
8. Needham, Volume 3, 106.
9. Needham, Volume 3, 46.
10. Needham, Volume 3, 45.
References
• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
• Li, Jimin, "Yang Hui". Encyclopedia of China (Mathematics Edition), 1st ed.
External links
• Yang Hui at MacTutor
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| Wikipedia |
\begin{document}
\title{First Passage Percolation with nonidentical passage times} \author{ \textbf{Ghurumuruhan Ganesan} \thanks{E-Mail: \texttt{[email protected]} } \\
\ \\ EPFL, Lausanne } \date{} \maketitle
\begin{abstract} In this paper we consider first passage percolation on the square lattice \(\mathbb{Z}^d\) with passage times that are independent and have bounded \(p^{th}\) moment for some \(p > 6(1+d),\) but not necessarily identically distributed. For integer \(n \geq 1,\) let \(T(0,n)\) be the minimum time needed to reach the point \((n,\mathbf{0})\) from the origin. We prove that \(\frac{1}{n}\left(T(0,n) - \mathbb{E}T(0,n)\right)\) converges to zero in \(L^2\) and use a subsequence argument to obtain almost sure convergence. As a corollary, for i.i.d. passage times, we also obtain the usual almost sure convergence of \(\frac{T(0,n)}{n}\) to a constant \(\mu.\)
\noindent \textbf{Key words:} First passage percolation nonidentical passage times.
\noindent \textbf{AMS 2000 Subject Classification:} Primary: 60J10, 60K35; Secondary: 60C05, 62E10, 90B15, 91D30. \end{abstract}
\section{Introduction} \label{intro} Consider the square lattice \(\mathbb{Z}^d\) with edges \(\{e_i\}_{i \geq 1}.\) The passage times \(\{t(e_i)\}_{i}\) are independent random variables that satisfy the following conditions.\\ (i) We have that \(\sup_{i} \mathbb{P}(t(e_i) < \epsilon) \longrightarrow 0\) as \(\epsilon \downarrow 0.\)\\ (ii) There exists a constant \(\eta > 0\) such that \(\sup_{i} \mathbb{E}(t(e_i))^{6(1+d) + \eta} < \infty.\)\\
For \(n \geq 1,\) we are interested in the shortest time path from \((0,\mathbf{0})\) to \((n,\mathbf{0}),\) where \(\mathbf{0} \) is the \((d-1)-\)dimensional zero vector. To define such a path, we proceed as follows. For any fixed path \(\pi\) starting from the origin and containing \(k\) edges \(e_1,...,e_k,\) we define the passage time to be \(T(\pi) = \sum_{i=1}^{k} t(e_i).\) Using (ii), we get that there exists a constant \(0 < \beta_1 < \mu\) such that \begin{equation}\label{t_pi} \mathbb{P}(T(\pi) \leq \beta_1 k) \leq e^{-dk} \end{equation} for all \(k \geq 1.\) We prove all estimates at the end of this section. By (iii) we have that \(\mu = \sup_{i}\mathbb{E}t(e_i) <\infty\) and by (ii) we have that \(\mu \geq \inf_i \mathbb{E}t(e_i) > 0.\) Let \(E_{k}\) denote the event that there exists a path starting from \((0,\mathbf{0})\) containing \(r \geq \frac{8 \mu}{\beta_1}k\) edges and whose passage time is less than \(\beta_1 r.\) Since there are at most \((2d)^{r}\) paths containing \(r\) edges, we have that \begin{equation}\label{a_0k} \mathbb{P}(E_{k}) \leq \sum_{r \geq 8\mu\beta_1^{-1} k} (2d)^{r}e^{-dr} \leq C e^{-\beta_2 k} \end{equation} for all \(k \geq 1\) and for some positive constants \(\beta_2\) and \(C.\) To obtain (\ref{a_0k}), we let \(\delta = d-\log(2d).\) Since \(de^{-d} \leq e^{-1} < \frac{1}{2}\) for all \(d \geq 2,\) we have that \(\delta > 0\) and we obtain \begin{eqnarray} \mathbb{P}(E_k) \leq \sum_{r \geq 8\mu\beta_1^{-1} k} e^{-\delta r} = \frac{1}{1-e^{-\delta}} e^{-\delta 8\mu\beta_1^{-1} k}. \nonumber \end{eqnarray}
For \(i \geq 1,\) let \(f_i\) denote the edge between \((i-1,\mathbf{0})\) and \((i,\mathbf{0})\) and let \(A_n = \left\{\sum_{i=1}^{2n} t(f_i) \leq 6\mu n\right\},\) where \(\mu\) is as above. There exists a constant \(C_1 > 0\) such that \begin{equation}\label{an_prob} \mathbb{P}(A_n^c) \leq \frac{C_1}{n^2} \end{equation} for all \(n \geq 1.\) Finally, setting \(F_n = E_{n}^c \cap A_n,\) we note that if \(F_n\) occurs, then the time taken to reach \((i,\mathbf{0})\) from \((0,\mathbf{0})\) is less than \(6\mu n,\) for each \(1 \leq i \leq 2n.\) Since \(E^c_{n}\) also occurs, every path starting from \((0,\mathbf{0})\) and containing \(r \geq \frac{8\mu}{\beta_1}n\) edges has passage time at least \(\beta_1 r \geq 8\mu n.\) Therefore, if \(F_n\) occurs, the shortest time path from \((0,\mathbf{0})\) to \((i,\mathbf{0})\) is contained in \(B_{8\mu\beta_1^{-1}n} := [-8\mu\beta_1^{-1}n,8\mu\beta_1^{-1}n]^d\) for each \(1 \leq i \leq n.\)
From (\ref{a_0k}) and (\ref{an_prob}), we have that \begin{equation} \label{fn_prob} \mathbb{P}(F_n^c) \leq \frac{C_2}{n^2} \end{equation} for some constant \(C_2 > 0\) and thus by Borel-Cantelli lemma, we have that \(\mathbb{P}(\liminf_n F_n)= 1.\) Fix \(\omega \in \liminf_n F_n\) and for every \(n \geq 1,\) define \(T(0,n)(\omega)\) to be the shortest time taken for reaching \((n,\mathbf{0})\) from \((0,\mathbf{0}).\) If there is more than one path that attains the shortest time, we provide an iterative procedure at the end of this section to choose a unique path.
We are interested in studying the convergence of \(\frac{T(0,n)}{n}.\) We have the following result. \begin{Theorem} \label{thm1} We have that \begin{equation} \frac{1}{n}\left(T(0,n) - \mathbb{E}T(0,n)\right) \longrightarrow 0\;\;\text{a.s. and in }L^2 \end{equation} as \(n \rightarrow \infty.\) \end{Theorem} For the case of independent and identically distributed (i.i.d.) random variables, we have the following Corollary. \begin{Corollary} \label{cor1} If the passage times are i.i.d., we have that \begin{equation} \frac{T(0,n)}{n} \longrightarrow \mu\;\;\text{a.s. and in }L^2 \end{equation} as \(n \rightarrow \infty,\) for some constant \(\mu > 0.\) \end{Corollary} The constant \(\mu\) is also called the time constant; Alexander (1993), Cox and Durrett (1981), Kesten (1993) and Smythe and Wierman (2008) and references therein contain further material on first passage percolation.
The paper is organized as follows: In the rest of this section, we prove estimates (\ref{t_pi}) and (\ref{an_prob}) and provide an iterative procedure for choosing the minimum time path. In Section~\ref{pf1}, we prove Theorem~\ref{thm1} and Corollary~\ref{cor1}.
To prove (\ref{t_pi}), we write \[\mathbb{P}(T(\pi) \leq \beta k) = \mathbb{P}\left(\sum_{i=1}^{k}t(e_i) \leq \beta k\right)\] for a fixed \(\beta > 0.\) Since \(\{t(h_{i})\}_{i}\) are independent, we have for a fixed \(s >0\) that \begin{equation}\label{y_1_eq1} \mathbb{P}(T(\pi) \leq \beta k) = \mathbb{P}\left(\sum_{i} t(e_i) \leq \beta k\right) \leq e^{s\beta k} \prod_{i=1}^{k}\mathbb{E}(e^{-st(e_i)}). \end{equation} For a fixed \(\epsilon > 0,\) we have that \begin{eqnarray} \mathbb{E}e^{-st(e_i)} &=& \int_{t(e_i) < \epsilon} e^{-st(e_i)} d\mathbb{P} + \int_{t(e_i) \geq \epsilon} e^{-st(e_i)} d\mathbb{P} \nonumber\\ &\leq& \int_{t(e_i) < \epsilon} e^{-st(e_i)} d\mathbb{P} + e^{-s\epsilon} \label{t_pi_an}\\ &\leq& \mathbb{P}(t(e_i) < \epsilon) + e^{-s\epsilon}. \nonumber \end{eqnarray} Using (i), the first term in the last expression is less than \(\frac{e^{-6d}}{2}\) if \(\epsilon > 0\) is small, independent of \(i.\) Fixing such an \(\epsilon,\) we choose \(s\) large so that the second term is also less than \(\frac{e^{-6d}}{2}.\) Substituting into (\ref{y_1_eq1}), we have that \[\mathbb{P}(T(\pi) \leq \beta k) \leq e^{s\beta k} e^{-3dk} \leq e^{-2dk},\] provided \(\beta > 0\) is small. We fix such a small \(\beta < \mu.\)
To prove (\ref{an_prob}), we let \(\mu_i = \mathbb{E} t(f_i)\) and use Chebychev's inequality to write \begin{equation}\label{a_0n_temp} \mathbb{P}(A_n^c) \leq \mathbb{P}\left(\sum_{i=1}^{2n}X_i \geq 4\mu n\right) \leq \frac{1}{(4\mu n)^4}\mathbb{E}\left(\sum_{i} X_{i}\right)^{4}, \end{equation} where \(X_i = t(f_i) - \mu_i.\) Since \(\{X_{i}\}_i\) are independent, we have that \(\mathbb{E}X_{i}X_{j} = 0\) for \(i \neq j.\) Thus we have \[\mathbb{E}\left(\sum_{i} X_{i}\right)^{4} = \sum_{i}\mathbb{E}X_{i}^4 + \sum_{i \neq j} \mathbb{E}X_{i}^2X_{j}^2 \leq C_1 n^2\] for some constant \(C_1 > 0\) by (ii). Substituting into (\ref{a_0n_temp}) proves (\ref{an_prob}).
Finally, we provide an iterative procedure to choose the shortest time path in the presence of multiple choices. For simplicity we provide for \(d =2.\) An analogous procedure holds for general \(d.\) Fix \(\omega \in \liminf_n F_n\) and let \({\cal S}_1 = \{L_i\}_{1 \leq i \leq W} =\{(S_{i,1},...,S_{i,H_i})\}_{1 \leq i \leq W}\) be the set of all paths with the shortest passage time from \((0,0)\) to \((n,0).\) We note that \(W = W(\omega) < \infty.\) Let \(x_{i,j}\) and \(y_{i,j}\) be the \(x\)- and \(y\)-coordinates, respectively, of the centre of \(S_{i,j}.\) Let \(y'_1 = \min_{L_k \in {\cal S}_{1}} y_{k,1}\) and let \({\cal S}'_1 = \{L_k \in {\cal S}_1 : y_{k,1} = y'_1\}.\) Let \(x'_1 = \min_{L_k \in {\cal S}'_1} x_{k,1}.\) Let \(h_1\) be the edge attached to the origin whose centre has coordinates \((x'_1,y'_1).\) Clearly \(h_1\) is the first edge of some path in \({\cal S}'_1.\) Let \({\cal S}_2\) be the set of paths in \({\cal S}'_1\) whose first edge is \(h_1.\) Repeating the above procedure with \({\cal S}_2,\) we obtain an edge \(h_2\) attached to \(h_1.\) Continuing iteratively, this procedure terminates after a finite number of steps resulting in a unique path. Also, the final path obtained does not depend on the initial ordering of the paths.
\section{Proof of Theorem~\ref{thm1}}\label{pf1} For \(n \geq 1,\) we define auxiliary random variables \(\{\hat{T}^{(n)}_k\}_{k \geq 1}\) defined as follows. For \(i \geq 1,\) let \(t_n(e_i) = \min(t(e_i), n^{\alpha}),\) where \(\alpha < \frac{1}{6}\) is a constant to be determined later. Since \(t_n(e_i) \leq t(e_i)\;\;\)a.s., we have that (i) and (ii) are satisfied by \(\{t_n(e_i)\}_i.\) For any fixed path \(\pi\) starting from the origin and containing \(k\) edges \(e_1,...,e_k,\) we define the passage time to be \(\hat{T}_n(\pi) = \sum_{i=1}^{k} t_n(e_i).\) We have \begin{equation}\label{t_pi_p} \mathbb{P}(\hat{T}_n(\pi) \leq \beta_1 k) \leq e^{-dk} \end{equation} for all \(k \geq 1.\) Here the constant \(\beta_1\) is the same as in (\ref{t_pi}) and is independent of \(n.\) To prove (\ref{t_pi_p}), we use the fact that \(\{t_n(e_{i})\}_{i}\) are independent and thus for a fixed \(s >0\) we have that \begin{equation}\label{y_1_eq1} \mathbb{P}(\hat{T}_n(\pi) \leq \beta k) = \mathbb{P}\left(\sum_{i} t_n(e_i) \leq \beta k\right) \leq e^{s\beta k} \prod_{i=1}^{k}\mathbb{E}(e^{-st_n(e_i)}). \end{equation} For a fixed \(0 < \epsilon <1,\) we have that \begin{eqnarray} \mathbb{E}e^{-st_n(e_i)} &=& \int_{t_n(e_i) < \epsilon} e^{-st_n(e_i)} d\mathbb{P} + \int_{t_n(e_i) \geq \epsilon} e^{-st_n(e_i)} d\mathbb{P} \nonumber\\ &\leq& \int_{t_n(e_i) < \epsilon} e^{-st_n(e_i)} d\mathbb{P} + e^{-s\epsilon} \nonumber\\ &=& \int_{t(e_i) < \epsilon} e^{-st(e_i)} d\mathbb{P} + e^{-s\epsilon} \nonumber \end{eqnarray} which is the same as (\ref{t_pi_an}). The final equality is because \(\epsilon < 1\) and thus \(t_n(e_i)< \epsilon\) if and only if \(t(e_i) < \epsilon.\) Following an analogous analysis following (\ref{t_pi_an}) we obtain (\ref{t_pi_p}). For \(k \geq 1,\) let \(\hat{E}_{k}(n)\) denote the event that there exists a path \(\pi_1\) starting from \((0,\mathbf{0})\) containing \(r \geq \frac{8 \mu}{\beta_1}k\) edges and whose passage time \(\hat{T}_n(\pi_1)\) is less than \(\beta_1 r.\) As in (\ref{a_0k}) we have that \begin{equation}\label{a_0k_p} \mathbb{P}(\hat{E}_{k}(n)) \leq Ce^{-\beta_2 k} \end{equation} for all \(k \geq 1,\) where \(\beta_2\) and \(C\) are as in (\ref{a_0k}).
As before, for \(i \geq 1\) let \(f_i\) denote the edge between \((i-1,\mathbf{0})\) and \((i,\mathbf{0})\) and for \(k \geq 1,\) let \(\hat{A}_n(k) = \left\{\sum_{i=1}^{2n} t_k(f_i) \leq 6\mu n\right\},\) where \(\mu\) is as above. Following an analogous analysis as in Section~\ref{intro}, there exists a constant \(C_1 > 0\) such that \begin{equation}\label{an_prob_p} \mathbb{P}(\hat{A}_n^c(n)) \leq \frac{C_1}{n^2} \end{equation} for all \(n \geq 1.\) Finally, set \(\hat{F}_n = \cap_{k=1}^{n} \hat{E}_{k}^c(n) \cap \hat{A}_n(n)\) and fix \(1 \leq k \leq n.\) If \(\hat{F}_n\) occurs, then the time \(\hat{T}^{(k)}_i\) taken to reach \((i,\mathbf{0})\) from \((0,\mathbf{0})\) is less than \(6\mu n,\) for each \(1 \leq i \leq 2n.\) This is because \(t_n(f_i) \geq t_k(f_i)\) and thus \(\hat{A}_n(n) \subset \hat{A}_n(k).\) Since \(\hat{E}^c_{k}(n)\) also occurs, every path \(\pi\) starting from \((0,\mathbf{0})\) and containing \(r \geq \frac{8\mu}{\beta_1}n\) edges has passage time \(\hat{T}_k(\pi)\) at least \(\beta_1 r \geq 8\mu n.\) Therefore, if \(\hat{F}_n\) occurs, the shortest time path with passage time \(\hat{T}^{(k)}_i\) from \((0,\mathbf{0})\) to \((i,\mathbf{0})\) is contained in \(B_{8\mu\beta_1^{-1}n} := [-8\mu\beta_1^{-1}n,8\mu\beta_1^{-1}n]^d\) for each \(1 \leq i \leq 2n\) and for each \(1 \leq k \leq n.\)
From (\ref{a_0k_p}) and (\ref{an_prob_p}), we have that \begin{equation}\label{fn_h_prob} \mathbb{P}(\hat{F}_n^c) \leq Cne^{-\beta_2 n} + \frac{C_1}{n^2} \leq \frac{C_2}{n^2} \end{equation} for some constant \(C_2 > 0.\) Thus \(\mathbb{P}(\liminf_n \hat{F}_n \cap F_n)= 1.\) \\Fix \(\omega\in\liminf_n~ \hat{F}_n \cap F_n\) and \(m \geq 1.\) For every \(1 \leq k \leq 2m,\) define \(\hat{T}^{(m)}_k = \hat{T}^{(m)}_k(\omega)\) to be the shortest time taken for reaching \((k,0)\) from \((0,0),\) as in Section~\ref{intro}. We have the following result.
\begin{Lemma}\label{etn} We have that \begin{equation}\label{etn_est} \mathbb{E}(\hat{T}^{(n)}_n - \mathbb{E}\hat{T}^{(n)}_n)^2 \leq C_1 n^{1+3\alpha} \end{equation} for all \(n \geq 1\) and some constant \(C_1 > 0.\) \end{Lemma} We prove the above lemma at the end of this section. We use Lemma~\ref{etn} to obtain \(L^2\) convergence of \(\frac{1}{n}\left(T_n - \mathbb{E}T_n\right),\) where \(T_n = T(0,n).\) \begin{Corollary} \label{cor2} \begin{equation}\label{etn_est} \mathbb{E}(T_n - \mathbb{E}T_n)^2 \leq C_2 n^{\frac{3}{2}-\beta} \end{equation} for all \(n \geq 1\) and some positive constants \(C_2\) and \(\beta.\) \end{Corollary} \emph{Proof of Corollary~\ref{cor2}}: We have that \begin{eqnarray} \mathbb{E}(T_n - \mathbb{E}T_n)^2 \leq 2I_1 + 2\mathbb{E}(\hat{T}^{(n)}_n - \mathbb{E}\hat{T}^{(n)}_n)^2, \end{eqnarray} where \begin{eqnarray} I_1 &=& \mathbb{E}(T_n - \hat{T}^{(n)}_n - \mathbb{E}(T_n - \hat{T}^{(n)}_n))^2 \nonumber\\ &\leq& 2\mathbb{E}(T_n - \hat{T}^{(n)}_n)^2 + 2 (\mathbb{E}T_n - \mathbb{E}\hat{T}^{(n)}_n)^2 \nonumber\\ &\leq& 4 \mathbb{E}(T_n - \hat{T}^{(n)}_n)^2. \label{i_1} \end{eqnarray} It suffices to estimate the last term.
We let \(G_n\) denote the event that the passage time \(t(e_i)\) of every edge in \(B_{8\mu\beta_1^{-1} n}\) is less than \(n^{\alpha}.\) We have that \begin{equation} \label{t_hat_t_n} \hat{T}^{(n)}_n \ind(H_n) = T_n \ind(H_n). \end{equation} where \(H_n = G_n \cap F_n \cap \hat{F}_n.\) Thus \begin{equation} \label{dif_eq} \mathbb{E}(T_n - \hat{T}^{(n)}_n)^2 = \mathbb{E}(T_n - \hat{T}^{(n)}_n)^2\ind(H_n^c) \leq \left(\mathbb{E}(T_n-\hat{T}^{(n)}_n)^4\right)^{1/2} \left(\mathbb{P}(H_n^c)\right)^{1/2}, \end{equation} by Cauchy-Schwarz inequality. We have that \[\mathbb{E}(T_n-\hat{T}^{(n)}_n)^4\leq 16\mathbb{E}T_n^4 + 16\mathbb{E}\left(\hat{T}^{(n)}_n\right)^4.\] Since \(T_n \leq \sum_{i=1}^{n} t(f_i),\) where as before, \(f_i\) denotes the edge between \((i-1,0)\) and \((i,0),\) we have that \[\mathbb{E}T_n^4 \leq n^3\sum_{i=1}^{n}\mathbb{E}t(f_i)^4 \leq C_1n^4\] for some constant \(C_1 > 0.\) An analogous estimate holds for \(\mathbb{E}(\hat{T}_n^{(n)})^4.\) Thus from (\ref{dif_eq}), we have that \begin{equation}\label{f_n_int} \mathbb{E}(T_n - \hat{T}^{(n)}_n)^2 \leq C_2n^2 \left(\mathbb{P}(H_n^c)\right)^{1/2}, \end{equation} for some constant \(C_2 > 0.\)
Finally, we choose \(\alpha < \frac{1}{6}\) and \(6(1+d) < K < 6(1+d) + \eta \) such that \(K\alpha > 1+d.\) Here \(\eta > 0\) is as in (iii). We then have that \begin{equation}\label{g_n} \mathbb{P}(G_n^c) \leq \sum_{i=1}^{C_3n^d} \mathbb{P}(t(e_i) \geq n^{\alpha}) \leq \frac{C_3n^d}{n^{K\alpha}}\mathbb{E}t(e_i)^{K} \leq \frac{C_4}{n^{1+2\delta}} \end{equation} for some positive constants \(C_3,C_4\) and \(\delta.\) Thus from (\ref{fn_prob}), (\ref{fn_h_prob}) and (\ref{f_n_int}), we get that \[\mathbb{E}(T_n - \hat{T}^{(n)}_n)^2 \leq C_5n^2 n^{-\frac{1}{2} - \delta} = C_5n^{\frac{3}{2}-\delta},\] for some positive constant \(C_5.\) \(\qed\)
\emph{Proof of Theorem~\ref{thm1}}: We claim that it suffices to prove that \(\frac{1}{n}\left(\hat{T}^{(n)}_n - \mathbb{E}\hat{T}^{(n)}_n\right)\) converges to zero a.s. Indeed, letting \(H_n \) be as in proof of Corollary~\ref{cor2} and using (\ref{t_hat_t_n}), we have that \[\frac{1}{n}\left(T_n - \mathbb{E}T_n\right) = \frac{1}{n}\left(\hat{T}^{(n)}_n - \mathbb{E}\hat{T}^{(n)}_n\right) + J_{1,n}-\mathbb{E}J_{1,n} - J_{2,n} + \mathbb{E}J_{2,n},\] where \(J_{1,n} = \frac{T_n}{n} \ind(H_n^c)\) and \(J_{2,n} = \frac{\hat{T}^{(n)}_n}{n}\ind(H_n^c).\) From (\ref{fn_prob}), (\ref{g_n}) and Borel-Cantelli Lemma we have that \(\mathbb{P}(\liminf_n H_n) = 1.\) Thus a.s. we have that \(\limsup_n J_{1,n} = 0 = \limsup_n J_{2,n}.\)
It remains to show that \(\mathbb{E}J_{i,n} \rightarrow 0\) as \(n \rightarrow \infty\) for \(i =1,2.\) We show that \(\sup_n\mathbb{E}J_{i,n}^2 < \infty\) for \(i =1, 2.\) This implies that \(J_{1,n}\) and \(J_{2,n}\) are uniformly integrable and completes the claim. We have that \[J_{1,n} \leq \frac{T_n}{n} \leq \frac{1}{n}\sum_{i=1}^{n} t(f_i)\] where as before \(f_i\) denotes the edge from \((i-1,0)\) to \((i,0).\) Thus \[\mathbb{E}J_{1,n}^2 \leq \frac{1}{n}\sum_{i=1}^{n}\mathbb{E}t(f_i)^2 \leq C_1\] for some constant \(C_1 >0\) by condition (iii). An analogous estimate holds for~\(J_{2,n}.\)
To prove that \(\frac{1}{n}\left(\hat{T}^{(n)}_n - \mathbb{E}\hat{T}^{(n)}_n\right)\) converges to zero a.s., we use a subsequence argument as follows. Set \(S_n = \hat{T}^{(n)}_n - \mathbb{E}\hat{T}^{(n)}_n.\) From Lemma~\ref{etn}, we have that \(\mathbb{E}S_n^2 \leq C_1 n^{1+3\alpha}.\) Thus for a fixed \(\epsilon > 0,\) we have that \[\mathbb{P}(|S_{n^2}| > n^2 \epsilon) \leq \frac{\mathbb{E}S^2_{n^2}}{\epsilon^2 n^4} \leq \frac{C_2}{n^{2-6\alpha}}\] for some constant \(C_2 > 0.\) Since \(\alpha < \frac{1}{6},\) we have that \(2-6\alpha > 1\) and by Borel-Cantelli Lemma, we have that \(\frac{S_{n^2}}{n^2} \rightarrow 0\) a.s. as \(n \rightarrow \infty.\)
We now set \(D_{n^2} = \max_{n^2 \leq k < (n+1)^2} |S_k-S_{n^2}|\) and estimate \(D_{n^2}\) as follows. For \(n^2 \leq k < (n+1)^2,\) we write \begin{eqnarray}
|S_k - S_{n^2}| &\leq& |\hat{T}^{(k)}_k- \hat{T}^{(n^2)}_{n^2}| + \mathbb{E}|\hat{T}^{(k)}_k - \hat{T}^{(n^2)}_{n^2}| \nonumber\\
&\leq& |\hat{T}^{(k)}_k- \hat{T}^{(k)}_{n^2}| + |\hat{T}^{(k)}_{n^2}- \hat{T}^{(n^2)}_{n^2}| \nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\; +\;\mathbb{E}|\hat{T}^{(k)}_k- \hat{T}^{(k)}_{n^2}| + \mathbb{E}|\hat{T}^{(k)}_{n^2}- \hat{T}^{(n^2)}_{n^2}|.\;\;\;\;\label{eq_s2} \end{eqnarray} For any integers \(k_1 < k_2 < k_3,\) we have that \begin{equation}\label{sub_add} \hat{T}^{(k)}_{k_1,k_3} \leq \hat{T}^{(k)}_{k_1, k_2} + \hat{T}^{(k)}_{k_2,k_3} \text{ and } \hat{T}^{(k)}_{k_1,k_2} \leq \hat{T}^{(k)}_{k_1, k_3} + \hat{T}^{(k)}_{k_2,k_3}. \end{equation}
Here \(\hat{T}^{(k)}_{k_1,k_2}\) denotes minimum passage time to go from \((k_1,0)\) to \((k_2,0)\) and is defined analogously as \(\hat{T}^{(k)}_n\) for each \(k_1\) and \(k_2.\) Thus \[|\hat{T}^{(k)}_{k} - \hat{T}^{(k)}_{n^2}| \leq \hat{T}^{(k)}_{k,n^2} \leq k^{\alpha}(k-n^2) \leq (n+1)^{2\alpha}((n+1)^2 - n^2) \leq C_1 n^{1+2\alpha}\] for some constant \(C_1 >0.\) The second inequality is true since the passage time of every edge is less than \(k^{\alpha}.\) Substituting the above estimate into (\ref{eq_s2}), we obtain that \begin{eqnarray}
|S_k - S_{n^2}| \leq 2C_1 n^{1+2\alpha} + |\hat{T}^{(k)}_{n^2}- \hat{T}^{(n^2)}_{n^2}| + \mathbb{E}|\hat{T}^{(k)}_{n^2}- \hat{T}^{(n^2)}_{n^2}|.\;\;\;\;\label{eq_s3} \end{eqnarray}
To estimate the remaining terms, we note that \[0 \leq \hat{T}^{(k)}_{n^2} - \hat{T}^{(n^2)}_{n^2} \leq \hat{T}^{((n+1)^2)}_{n^2} - \hat{T}^{(n^2)}_{n^2} =: I_{n^2}\] since \(n^2 \leq k < (n+1)^2.\) Thus \[\frac{D_{n^2}}{n^2} \leq \frac{2C_1}{n^{1-2\alpha}} + \frac{I_{n^2}}{n^2} + \frac{\mathbb{E}I_{n^2}}{n^2}.\] We claim that \(\frac{I_{n^2}}{n^2} \rightarrow 0\) a.s. and that \(\frac{I_{n^2}}{n^2}\) is uniformly integrable. Assuming the claims for the moment, we get that \(\frac{D_{n^2}}{n^2} \longrightarrow 0\) a.s. as \(n \rightarrow \infty.\) For \(n^2 \leq k < (n+1)^2, \) we have that \[\frac{|S_k|}{k} \leq \frac{|S_k-S_{n^2}|}{k} + \frac{|S_{n^2}|}{k} \leq \frac{|S_k-S_{n^2}|}{n^2} + \frac{|S_{n^2}|}{n^2} \leq \frac{D_{n^2}}{n^2} + \frac{|S_{n^2}|}{n^2}.\] This proves that the original sequence \(\frac{S_k}{k} \rightarrow 0\) a.s. as \(k \rightarrow \infty.\)
To prove the two claims regarding \(I_{n^2},\) we note that \[\hat{T}^{((n+1)^2)}_{n^2}\ind(\hat{H}_n) = \hat{T}^{(n^2)}_{n^2}\ind(\hat{H}_n)\] where \(\hat{H}_n = \hat{F}_{n^2} \cap \hat{F}_{(n+1)^2} \cap \hat{G}_{n^2}\) and \(\hat{G}_{n^2}\) is the event that the passage time \(t(e_i)\) of every edge in \(B_{20\mu\beta_1^{-1} n^2}\) is less than \(n^{2\alpha}.\) As in (\ref{g_n}) we have that \(\mathbb{P}(\hat{G}_{n^2}^c) \leq \frac{C_1}{n^{2+\delta_2}}\) for some constant \(\delta_2 >0.\) From (\ref{fn_h_prob}) and Borel-Cantelli lemma, we then have that \(\mathbb{P}(\liminf_n \hat{H}_n) = 1.\) Since \(I_{n^2} = I_{n^2}\ind(\hat{H}_n^c),\) we get that \(\frac{I_{n^2}}{n^2} \rightarrow 0\) a.s. as \(n \rightarrow \infty.\)
To prove the uniform integrability of \(\frac{I_{n^2}}{n^2},\) we note that \[0 \leq \frac{I_{n^2}}{n^2} \leq \frac{\hat{T}^{((n+1)^2)}}{n^2} \leq \frac{1}{n^2}\sum_{i=1}^{n^2}t(f_i) =: M_n\] where as before \(f_i\) denotes the edge from \((i-1,0)\) to \((i,0).\) Since \(\mathbb{E}M_n^2 \leq \frac{1}{n^2}\sum_{i=1}^{n^2}\mathbb{E}t(f_i)^2 \leq C_1\) for some constant \(C_1 > 0,\) we are done. \(\qed\)
\emph{Proof of Corollary~\ref{cor1}}: We show that \(\frac{\mathbb{E}T(0,n)}{n} \rightarrow \mu\) for some constant \(\mu >0.\) Since \[\mathbb{E}T(0,n+m) \leq \mathbb{E}T(0,n) + \mathbb{E}T(m,m+n) = \mathbb{E}T(0,n) + \mathbb{E}T(0,m),\] we have by Fekete's Lemma that \[\lim_n\frac{\mathbb{E}T(0,n)}{n} = \inf_{n \geq 1} \frac{\mathbb{E}T(0,n)}{n} =: \mu.\] To show that \(\mu >0,\) we note that if \(A_{0,k_1}^c\) occurs for \(k_1 = \beta_1 (8\mu)^{-1} n,\) then every path containing \(r \geq 8\mu\beta_1^{-1} k_1 \geq n\) edges has passage time at least \(\beta_1r \geq 8\mu k_1 \geq \beta_1 n.\) Thus \[\mathbb{E}T(0,n) \geq \beta_1 n \mathbb{P}(A^c_{0,k_1}) \geq \beta_3n\] for all \(n \geq 1\) and some constant \(\beta_3 > 0,\) by (\ref{a_0k}). \(\qed\)
\emph{Proof of Lemma~\ref{etn}}: We order the edges as \(e_1,e_2,..\) and for each \( i \geq 1,\) set \({\cal F}_i = \sigma(\hat{t}(e_l) : 1 \leq l \leq i).\) For \(l \geq 1,\) let \(X_l = \mathbb{E}(\hat{T}^{(n)}_n|{\cal F}_l) - \mathbb{E} (\hat{T}^{(n)}_n|{\cal F}_{l-1}).\) We have that \(0 \leq \hat{T}^{(n)}_n \leq \sum_{i=1}^{n} t_n(f_i) \leq n^{1+\alpha}\) a.s., where as before \(f_i\) denotes the edge from \((i-1,\mathbf{0})\) to \((i,\mathbf{0}).\) Thus we have by Levy's martingale convergence theorem that \[Y_m := \sum_{l=1}^{m} X_l = \mathbb{E}(\hat{T}^{(n)}_n | {\cal F}_{m}) - \mathbb{E}\hat{T}^{(n)}_n \longrightarrow \hat{T}^{(n)}_n - \mathbb{E}(\hat{T}^{(n)}_n)\;\;a.s.\] as \(m \rightarrow \infty.\) By Dominated convergence theorem, we then have that \[\mathbb{E}(\hat{T}^{(n)}_n -\mathbb{E}\hat{T}^{(n)}_{n})^2 = \mathbb{E}\left(\lim_m Y_m\right)^2 = \lim_m \mathbb{E}Y_m^2.\] By the martingale property, we have that \(\mathbb{E}Y_m^2 = \sum_{l=1}^{m} \mathbb{E}X_l^2.\) We claim that \begin{equation}
X_l^2 \leq 2n^{2\alpha} \left(\mathbb{P}(e_l \in {\pi}_n | {\cal F}_l) + \mathbb{P}(e_l \in {\pi}_n | {\cal F}_{l-1})\right)\;\;a.s.\label{eq1} \end{equation} where \(\pi_n\) is the shortest time path from \((0,0)\) to \((n,0).\) We prove the above result at the end. Using (\ref{eq1}), we obtain that \begin{eqnarray}
\mathbb{E}(\hat{T}^{(n)}_n -\mathbb{E}\hat{T}^{(n)}_{n})^2 &\leq& 2n^{2\alpha} \sum_{l=1}^{\infty} \mathbb{E}\left(\mathbb{P}(e_l \in {\pi}_n | {\cal F}_{l}) + \mathbb{P}(e_l \in {\pi}_n | {\cal F}_{l-1})\right) \nonumber\\ &=& 4n^{2\alpha}\sum_{l=1}^{\infty} \mathbb{P}(e_l \in {\pi}_n)\nonumber\\ &=& 4n^{2\alpha}\mathbb{E}\sum_{l=1}^{\infty} \ind(e_l \in {\pi}_n)\nonumber\\ &=& 4n^{2\alpha}\mathbb{E}(\#{\pi}_n), \nonumber \end{eqnarray} where \(\ind(.)\) refers to the indicator function.
To estimate the length of \(\pi_n,\) let \(\mu = \sup_i \mathbb{E}t(e_i)\) be as in Section~\ref{intro}. We note that if \(\hat{E}^{c}_{k}(n) \) occurs (see paragraph prior to (\ref{a_0k_p})) for \(k \geq \mu^{-1}n^{1+\alpha},\) then every path \(\pi\) with length \(r \geq \frac{8\mu}{\beta_1}k\) has passage time \(\hat{T}_n(\pi)\) at least \(\beta_1 r \geq 8 n^{1+\alpha}.\) Since \(\pi_n\) has passage time at most \(n^{1+\alpha},\) we obtain for \(k \geq \mu^{-1}n^{1+\alpha}\) that \[\mathbb{P}(\#{\pi}_n \geq 8\mu\beta_1^{-1}k) \leq \mathbb{P}(\hat{E}_{k}(n)) \leq e^{-\beta_2 k},\] where \(\beta_2 > 0\) is as in (\ref{a_0k_p}). Since \(\mathbb{E}(\#{\pi}_n) \leq \sum_{k \geq 1} \mathbb{P}(\#{\pi}_n \geq k),\) we obtain that \(\mathbb{E}(\#{\pi}_n) \leq C_1 n^{1+\alpha}\) for some constant \(C_1 > 0.\)
To estimate \(X_l,\) we use the notation of Kesten (1993); for \(j \geq 1,\) let \({\nu}_j(.)\) denote the probability measure associated with \((\hat{t}(e_j),\hat{t}(e_{j+1}),...).\) Let \((\sigma_1,\sigma_2,...)\) and \((\omega_1,\omega_2,...)\) be independent realizations of \((\hat{t}(e_1),\hat{t}(e_2),...)\) and for \(l \geq 1,\) define \([\omega,\sigma]_l = (\omega_1,\omega_2,...,\omega_{l},\sigma_{l+1},\sigma_{l+2},...).\) We have that \[X_l = \int \nu_l(d\sigma) (T_n([\omega,\sigma]_l) - T_n([\omega,\sigma]_{l-1})).\] We note that changing the passage time of edge \(e_l\) does not change the value of the minimum passage time by more than \(n^{\alpha}.\) Also, a change occurs only if \(e_l \in {\pi}_n([\omega,\sigma]_l)\) or \(e_l \in {\pi}_n([\omega,\sigma]_{l-1}).\) Moreover, if Thus \[|\hat{T}^{(n)}_n([\omega,\sigma]_l) - \hat{T}^{(n)}_n([\omega,\sigma]_{l-1})| \leq n^{\alpha} \left(\ind(e_l \in {\pi}_n([\omega,\sigma]_l)) + \ind(e_l \in {\pi}_n([\omega,\sigma]_{l-1}))\right)\] and by Cauchy-Schwarz inequality, we have a.s. that \begin{eqnarray}
X_l^2 &\leq& \int {\nu}_l(d\sigma)|\hat{T}^{(n)}_n([\omega,\sigma]_l) - \hat{T}^{(n)}_n([\omega,\sigma]_{l-1})|^2 \nonumber\\ &\leq& 2n^{2\alpha} \int {\nu}_l(d\sigma)\left(\ind(e_l \in {\pi}_n([\omega,\sigma]_l)) + \ind(e_l \in {\pi}_n([\omega,\sigma]_{l-1}))\right) \nonumber\\
&=& 2n^{2\alpha} \left(\mathbb{P}(e_l \in {\pi}_n | {\cal F}_l) + \mathbb{P}(e_l \in {\pi}_n | {\cal F}_{l-1})\right). \nonumber \end{eqnarray} This proves (\ref{eq1}).\(\qed\)
\end{document} | arXiv |
Dynamic scan control in STEM: spiral scans
Xiahan Sang1,2,
Andrew R. Lupini2,3,
Raymond R. Unocic1,2,
Miaofang Chi1,2,
Albina Y. Borisevich2,3,
Sergei V. Kalinin1,2,
Eirik Endeve4,
Richard K. Archibald4 &
Stephen Jesse1,2
Advanced Structural and Chemical Imaging volume 2, Article number: 6 (2016) Cite this article
Scanning transmission electron microscopy (STEM) has emerged as one of the foremost techniques to analyze materials at atomic resolution. However, two practical difficulties inherent to STEM imaging are: radiation damage imparted by the electron beam, which can potentially damage or otherwise modify the specimen and slow-scan image acquisition, which limits the ability to capture dynamic changes at high temporal resolution. Furthermore, due in part to scan flyback corrections, typical raster scan methods result in an uneven distribution of dose across the scanned area. A method to allow extremely fast scanning with a uniform residence time would enable imaging at low electron doses, ameliorating radiation damage and at the same time permitting image acquisition at higher frame-rates while maintaining atomic resolution. The practical complication is that rastering the STEM probe at higher speeds causes significant image distortions. Non-square scan patterns provide a solution to this dilemma and can be tailored for low dose imaging conditions. Here, we develop a method for imaging with alternative scan patterns and investigate their performance at very high scan speeds. A general analysis for spiral scanning is presented here for the following spiral scan functions: Archimedean, Fermat, and constant linear velocity spirals, which were tested for STEM imaging. The quality of spiral scan STEM images is generally comparable with STEM images from conventional raster scans, and the dose uniformity can be improved.
Beam damage, drift distortion, and scan distortion are inherent issues that hinder quantitative interpretation of scanning transmission electron microscopy (STEM) imaging [1–6]. Beam damage occurs when the electron beam used to form the image transfers a critical amount of energy to the sample being examined, potentially causing damage or otherwise changing the subject of the experiment. This effect can be very useful, for example, allowing electron energy loss spectroscopy (EELS), or for the deliberate sculpting of nano-device components [7], or to excite diffusion of single atoms [8, 9] and vacancies [10]. However, in most cases, such damage to the sample is usually considered to be detrimental. Thus, various strategies are employed to minimize beam damage, and the optimal method will depend on the properties of the sample and microscope imaging parameters. If damage is dominated by knock-on mechanisms, a viable option is to reduce the accelerating voltage below the threshold at which significant damage occurs. Conversely, if the damage is dominated by ionization, then it may be beneficial to increase the accelerating voltage to reduce the ionization cross section [1]. Additional experimental procedures might also be useful, such as coating the sample with a conductive layer (such as carbon), imaging inside a liquid [11], or operating at cryogenic temperature [12].
Similarly, a variety of imaging strategies can be employed to minimize the electron dose, chief among which is reducing the beam current. Using more source demagnification can improve spatial resolution, but the lower signal level may degrade the signal-to-noise ratio. Other possibilities include control of the beam dose via 'blanking', adjusting operating parameters (such as focus and astigmatism) on an area slightly away from the area of interest, using repeated fast scans [13], or making more efficient use of the available signals [14]. The recent development of sparse sampling methods also appears to be extremely promising [15].
On the other hand, acquisition of multiple fast scans can both reduce the dose rate and allow sequential imaging, which is particularly useful for samples that are beam-sensitive or that experience charging. Also, there has been a recent resurgence of interest in applying methods to correct scan and drift distortions in STEM using frame averaging [2–6]. However, the success of these methods raises the question of whether the scan itself can be improved to eliminate some of the distortions during data acquisition rather than by post-processing. Extremely high-speed scanning and the possibility of dynamic stabilization seem to be promising routes for further exploration.
Advantages of using non-traditional scan paths have been demonstrated in scanning probe microscopy (SPM), including improved speed and accuracy and the ability to automate the targeting of regions of interest for higher resolution measurements [16, 17]. However, customization or optimization of the scanning path has rarely been used in STEM. There are several technical difficulties associated with scanning in STEM. These mostly arise because of the competing demands on the probe response: the user might wish to move the probe rapidly, requiring a fast response, whereas the probe also has to be highly stable and not wobble about each position during a slower scan. Typically, the scan speed used for spectroscopy might be 3–6 orders of magnitude slower than for imaging. Obviously, these competing demands place stringent requirements on the scan amplification electronics. Moreover, STEM scans are usually 'double-deflection' to obtain tilt-free scans or coma-free scans. Here, we will largely ignore such details and treat the magnification and scan purification as separate problems.
In this paper, we show for the first time that aberration-corrected STEM images can be formed at high speed using paths that are significantly different from traditional orthogonal rastering. Advantages and disadvantages of different scan paths will be compared in terms of sampling uniformity and distortion. To differentiate from conventional rastering mode scans, this new scanning method will be referred to as general-scan STEM (G-STEM).
For test purposes, we used a SrTiO3 (STO) sample viewed down the [110] zone axis. STO is a very common substrate for thin-film growth, which is a major topic of interest for electron microscopy, meaning that there will likely be an STO reference region available on many technologically important samples. Moreover, STO is reasonably stable and does not charge significantly under typical electron doses.
STEM images were acquired used an FEI Titan 80–300 operating at 300 kV equipped with a Fischione high-angle annular dark-field (HAADF) detector. We developed a custom field-programmable gate array (FPGA)-based scan system (in a National Instruments PXIe-1073 chasis) capable of interfacing to a variety of different microscopes. A LabView program was developed to control the scan unit with input coordinates from customizable Matlab code. This system generates voltage waveforms that are sent to the x- and y- scan controls to enable arbitrary and dynamic beam positioning. The maximum readout frequency of the FPGA scan system is 2 MHz with an equivalent shortest dwell time of 0.5 μs.
At this stage, it is important to point out that the unconventional scan patterns used here induce a paradigm shift in how image data are considered. In a traditional scanning mode, the data are essentially stored as an array of intensities, which are assigned to elements within a 2D matrix. However, for more complicated scan patterns, it is also necessary to specify the (nominal) position where each data point was acquired. A simple interpolation algorithm (herein called reconstruction) is used to map each data point to an element of the displayed or printed image. Thus, rather than a simple list of intensities (I i ), the data are better envisioned as a list of positions and intensities (x i , y i , I i ).
In practice, we have begun to store the nominal positions in this manner. Of course, it is possible to just store the scan-generation algorithm, but the factor of 3 increases in storage requirements is largely irrelevant here. Moreover, if distortions are significant, the true probe position may be quite different from the nominal position. Scan distortion correction consists of constructing the map from nominal to 'true' probe positions. Thus, this paradigm also highlights the analogy to the usual post-processing distortion correction, where a per-pixel map of corrections is generated [2–4].
In this paper, every G-STEM data set contains a series of twenty frames each acquired with 0.2 s frame time and the maximum frequency of 2 MHz. The 400,000 data points in each frame were then reconstructed to form a 200 × 200 image. The twenty image frames were aligned using cross-correlation and averaged to increase the signal-to-noise ratio (SNR). The final images presented in the figures were further smoothened in the frequency domain using a Gaussian filter.
Sawtooth scans
A typical STEM image acquired using the conventional raster scan path with a dwell time of 20 μs and 512 × 512 frame size is shown in Fig. 1a. Here, the brighter atom columns are Sr and the fainter columns are Ti. The drift distortion is evident as the angle between [\(1\bar{1}0\)] and [001] deviates from 90°. We start the G-STEM attempt from the simple sawtooth scan path that resembles conventional raster scan from left to right and top to bottom. Here, we use a simple version of this path such that the beam flies directly from the end of the last line to the start of the next line and continues to scan without any flyback time or line synchronization. The probe location (x i , y i ) as a function of time is shown in Fig. 1b. Here, (x i , y i ) scales to the voltages applied along the two directions. The X-axis (red) is defined as the horizontal direction, also known as the fast scan direction in the conventional STEM. The Y-axis (black) is defined as the vertical direction, also known as the slow scan direction. The scans for both the X- and Y-axes are sawtooth waves of appropriate frequencies. Practically, the amplitude of this wave is controlled by the microscope electronics and defines the magnification of the STEM image. To better illustrate the scan path, we also plot the beam locations in 2D, as shown in Fig. 1c. The black zigzagging line connecting the dots illustrates the scan path.
a Conventional STEM image acquired from STO along [110] zone axis. Schematic illustration of a sawtooth scan. b Voltages applied to the X and Y scan coils over the time for a single frame acquisition. c Probe positions, shaded from dark to light as a function of time. d A reconstructed G-STEM image acquired with a sawtooth scan
Figure 1d shows a processed G-STEM image acquired using a fast sawtooth scan with a frame time of 0.2 s and 20 frames as discussed earlier. Note that although we use the HAADF signal in this paper, it is possible to simultaneously acquire multiples signals, such as both bright- and dark-field signals. The image is significantly distorted at the left edge of the displayed region, although the rest of the image is relatively undistorted. This distortion is likely from the phase lag of the scan electronics responding to a sudden change of beam location. When the beam moves from the end of the last line to the start of the next line, the actual location will take some extra time to reach the nominal position. Therefore, one way to compensate for the lag is to add in some extra shifts or a delay time, as in a conventional cathode ray tube.
Conventionally, a 'flyback' delay at the start of each fast scan line is used to reduce such distortion. For a present state-of-the-art STEM, flyback delays of 10–1000 μs are typical. As a specific example, the Nion UltraSTEM 200 typically needs more than 500 μs to yield images without noticeable distortions. Thus, for a scan of 512 × 512 pixels at 1 µs/pixel, using this flyback delay would result in losing roughly half of the available imaging time. If a fast enough blanker is available, the beam could be blanked during the flyback; otherwise, there might also be additional unnecessary damage at the edges of the scan where the beam spends extra time. The distribution of the electron dose is an important topic that will recur later. Clearly, a method of eliminating the flyback delay would allow an increase in scanning rate and potentially reduce the beam damage.
Another method to reduce the distortion and lateral shift along slow scan direction is called line-synchronization, i.e., tying each line to the same part of the wave of the electrical supply. Such synchronization has the added advantage that the effects of mains interference should be similar for each scan line and each frame, facilitating its correction [18]. However, this method either requires a delay time at the start of each line or imposes additional restrictions on the per-pixel dwell time.
Serpentine scans
An obvious improvement over the sawtooth scan to avoid a flyback delay is to perform a 'serpentine' scan, alternately moving the probe from left to right on one scan line and then right to left on the next, using what is sometimes called a triangle wave. A serpentine scan is shown in Fig. 2a, b, where the X- and Y- directions are the same as in Fig. 1b. Double serpentine scans (i.e., performing a second scan after rotating the slow-scan axis by 90°) can also be implemented.
Schematic illustration of a serpentine scan. a Voltages applied to the X and Y scan coils over the time for a single frame. b Probe positions, shaded from dark to light as a function of time. c Reconstructed G-STEM images (forward and backward) acquired with a serpentine scan
Figure 2c shows the result of such a serpentine scan. Unfortunately, these scans initially appear worse than the conventional scan at high scan speeds, because the distortions are different for the leftwards and rightwards trajectories. For display purposes, it is best to separate out these two paths. Notably, unwarping this distortion might present an easier problem to solve than the regular sawtooth wave, because the triangle wave provides two images of the same area with different distortions. To a reasonable level of approximation, we might, therefore, expect the distortions to be similar, but reversed. Thus, a digital correction of serpentine scans could be a promising route for further development.
The obvious lesson from the serpentine scans is that the sharp changes in direction at the edges of the scan contribute significantly to the distortions. There is a clear difference in the acceleration of the probe between the abrupt changes at the end of each scan line as compared with the rest of the pixels. The relevance should be obvious in scanning tunneling microscopy (STM), in which the moving probe/stage has mass, but is perhaps a little surprising in STEM where the 'probe' does not really correspond to a physical object. However, it seems clear that there is a non-ideal response of the 'true' probe movement to the 'nominal' probe positions. The cause of this lag is inductance in the scan coils and other current-flow limitations, which limit how fast the scan can be changed, in an analogous way as to how inertia can limit mechanical movement. One route to address this problem would be with faster electronics or rapid electrostatic deflectors. However, such new hardware would introduce other complications and, thus, scan paths without sharp changes in acceleration merit further investigation.
Spiral scans
We now focus on smooth curves that can fill the 2D space without crossing themselves. The distortions can hopefully be reduced due to the relatively smooth acceleration. Spiral curves are natural solutions to this problem. The mathematical study of spirals has a long and interesting history, dating back thousands of years [19]. In this paper, we focus on spirals with coordinates (x, y) as a function of time t defined by:
$$x = t^{a} { \cos }\left( {\omega t^{b} } \right),\;y = t^{a} { \sin }\left( {\omega t^{b} } \right)$$
where ω is the scanning frequency, and a and b are parameters to control the shape of the spirals. The scanning frequency ω can be adjusted to change the sampling rate. The spiral can go both inward and outward. As the drift distortion is different but correlated for inward and outward scans, this is a promising way to decouple drift distortion from the scan distortion, which will be considered in more detail in the future work.
Here, we explore the physical properties of the spiral curves, as they are closely related to the quality of STEM images reconstructed from those scan paths. We begin with the velocity \(\vec{v}\), which basically determines the distance between adjacent sampling points. For each point on the spiral, \(\vec{v}\) is the first derivative of Eq. (1), with a magnitude:
$$\left| {\vec{v}} \right| = t^{a - 1} \sqrt {a^{2} + \omega^{2} b^{2} t^{2b} } \approx \omega bt^{a + b - 1}$$
The term a 2 inside the square root can usually be neglected for large ωt b. We can see that when a + b = 1 the velocity magnitude is approximately constant for all the points on the spiral. If a + b > 1, the beam moves faster as it moves away from the center.
The angular velocity magnitude Ω is defined by:
$$\varOmega = \frac{{\left| {\vec{v}} \right|}}{r} = \frac{{\omega bt^{a + b - 1} }}{{t^{a} }} = \omega bt^{b - 1}$$
Here, we assume that the velocity \(\vec{v}\) is perpendicular to \(\vec{r}\) = (x, y), which is a reasonable approximation: The angle between \(\vec{v}\) and \(\vec{r}\) can be calculated as θ = arccos(a/(ωbt b)). As t increases, cos(θ) approaches zero and θ approaches 90°. Equation 3 tells us that the angular velocity is approximately constant if b = 1.
Another potentially interesting feature of the spiral curves is the sampling density. To ensure uniform sampling, the dose should ideally be the same across the whole area. For a first approximation, we consider how the spiral sweeping area A increases as a function of time t,
$$\frac{dA}{dt} = \frac{{d\left( {\pi \left( {t^{a} } \right)^{2} } \right)}}{dt} = 2a\pi t^{2a - 1}$$
For a = 0.5, the area increases linearly with time. For a < 0.5, the increase slows down over time, resulting in more dose at the edges, while for a > 0.5, the center is exposed to more electron dose. Now, with the understanding of physical properties of the spiral scans, we investigate the behavior for spiral curves with different a and b parameters.
Archimedean spiral
The first type of spiral we consider is an 'Archimedean' spiral with a = 1 and b = 1:
$$x = t{ \cos }\left( {\omega t} \right),\;y = t{ \sin }\left( {\omega t} \right)$$
The beam scan path Fig. 3a shows that the magnitude of x and y slowly increase without any sharp turns and the frequency of the sinusoids remains constant. Taking coordinates from Fig. 3a, we can form the outward scan trajectory, as shown in the left part of Fig. 3b. The inward scan shown in the right part is constructed by reversing of the scan path and also the y-direction. Two typical reconstructed images using Archimedean inward and outward spirals are shown in Fig. 3c. Note that the resulting STEM images do not display any obvious non-linear distortion. This is attributed to the constant frequencies (and constant angular velocity) for b = 1. Both inward and outward images are rotated at the same angle with respect to the sawtooth scan images in Fig. 1d. The distortion is likely from the scan lag which is related to the angular velocity. As the spiral scan direction is clockwise for both the inward and outward scans, the distortion is the same for both images.
Schematic illustration of an Archimedean spiral scan. a Voltages applied to the X and Y scan coils over the time for a single frame. b Probe positions, shaded from dark to light as a function of time. The left part shows an outward scan starting from the center. The right part shows an inward scan ending at the center. c Reconstructed outward and inward G-STEM images acquired with Archimedean spiral scan paths. The black striations (indicated by the arrows) at the edges are due to undersampling
The main problem with an Archimedean scan is the sampling density. This can be seen simply by recognizing that the number of points scanned in time t will be proportional to t, while the area scanned is approximately proportional to the square of the time, as t 2. Thus, the sampling density and dose at the sample will vary with position in the image. This is also evident from the corrupted regions close to the edge of the reconstructed images due to very sparse sampling in those areas. Also, due to very dense sampling in the center, the beam dose there will be much larger than the average, resulting in extra beam damage. Clearly, if uniform sampling distribution is desired, Eq. 4 reveals that we should investigate solutions with a = 0.5.
Fermat spiral
Here, we use a different spiral with a = 0.5 and b = 1 to give both uniform sampling and constant angular velocity:
$$x = \sqrt t { \cos }\left( {\omega t} \right),\;y = \sqrt t { \sin }\left( {\omega t} \right)$$
This spiral has been known as Fermat spiral, which has the more general form r 2 = ωt. Since the square root has two solutions (positive and negative), a natural approach is to use one part as the outward scan and the other as the inward scan. Figure 4a shows how x and y change as a function of time for the outward scan part. Figure 4b shows the scan path for both outward and inward scans. Note that the end point (A) of the outward scan and the starting point (B) of the inward scan are at opposite sides. Therefore, a smooth wave was added to move the probe from A and B for a smooth transition between outward scan and inward scan. The outward and inward scan paths move clockwise and counter clockwise, respectively. The two reconstructed STEM images are shown in Fig. 4c. Again, the distortion seems to be purely linear due to constant angular velocity. The rotation distortions are opposite as expected from different spiral rotation directions. However, the image quality is not uniform; the edge area is noticeably more blurred than the center area. This non-uniformity is attributed to the anisotropic sample spacing. Near the center area, the spacing between adjacent points along the tangent direction is much shorter than the spacing along radial directions. For the edge area, the spacing along the radial direction is much longer than along the tangential direction. Therefore, despite the nominally uniform areal distribution, the actual sampling is still not ideal.
Schematic illustration of a Fermat spiral scan. a Voltages applied to the X and Y scan coils over the time for a single frame. b Probe positions, shaded from dark to light as a function of time. c Reconstructed STEM images from outward and inward scans using Fermat spiral scan paths
Constant linear velocity spiral
We seek a spiral that retains the constant sampling density, but where the distance between samples is isotropic. The solution is known as a constant linear velocity spiral. From the previous discussion on the physical properties of spirals, the two parameters should satisfy a = 0.5 and a + b = 1. The spiral equation is thus:
$$x = \sqrt t { \cos }\left( {\omega \sqrt t } \right),\;y = \sqrt t { \sin }\left( {\omega \sqrt t } \right)$$
This spiral has both constant sampling density (dose distribution) and, evenly, isotropic spaced points. A similar scan path was proposed for atomic force microscopy (AFM) [20]. The scan path is shown in Fig. 5a. Examples of the sampling trajectories for both outward and inward scans are shown in Fig. 5b, where we can see that the data points are evenly distributed along both tangential and radial directions.
Schematic illustration of a constant linear velocity scan. a Voltages applied to the X and Y scan coils over the time for a single frame. b Probe positions, shaded from dark to light as a function of time. c Reconstructed STEM images from outward and inward scans using constant linear velocity scan paths
Experimental images with the constant velocity spiral are shown in Fig. 5c. The outward and inward parts of the scan are displayed separately. Significant distortions are apparent at the center of the images where the scan frequency is changing the fastest. The two images have opposite rotation distortion directions in the center, which result from different spiral rotation directions. Therefore, the drawback of this spiral is that the angular frequency changes. Since the distortions depend on frequency, the disadvantage is that the distortions are non-uniform across a single frame. Another way to look at this problem is that to keep a constant linear velocity, the angular velocity has to be large near the center and smaller at the edges. Thus, the angular distortion changes with angular velocity, which results in much more severe distortions at the center.
All three spiral scans we tested have successfully eliminated the flyback delay common in conventional STEM. Both Archimedean and Fermat scans yield STEM images with a quality comparable with conventional scan paths, but both have problems with sampling density (dose distribution). The constant linear velocity scan solves the sampling problem but introduces significant distortion in the center. For ease of use, the Fermat scan seems to be the best choice due to its relatively uniform sampling density and easy interpretation of the reconstructed image.
A possible solution to the sampling problem might be truncated spiral functions, which have the same functional form but start from some finite t0 instead of from zero. Spirals with varying a and b could also be investigated in future work.
As the drift distortion depends critically on the relative drift direction with respect to the scan direction [2], the varying scan directions in spiral scans lead to an abundance of information for further drift correction within one frame. Other areas for future work could involve hybrid scans, scans adapted on the fly, or changes in the dwell-time per pixel.
We have demonstrated for the first time that aberration-corrected STEM images can be acquired at high speed with different spiral scans. By completely eliminating the flyback effect in STEM imaging, the spiral scans provide new possibilities to reduce beam damage, image distortion, and drift distortion. Combined with conventional image processing methods, the spiral scans can be used to significantly improve the quality of STEM images. In the future, this system could be extended with high-speed feedback in the FPGA unit. Such capabilities could allow dynamic position correction or atom tracking in hardware, without having to wait for relatively slow data transfers to and from a computer.
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SJ built the scan control system. ARL, RRU, MC, and SJ interfaced the controller to the microscope and performed the microscopy experiments. XS and ARL drafted the manuscript. SJ, ARL, RRU, MC, AYB, and SVK conceived and designed the study. XS, SJ, EE, and RKA participated in image analysis. All authors read and approved the final manuscript.
Research supported by Oak Ridge National Laboratory's (ORNL) Center for Nanophase Materials Sciences (CNMS), which is a U.S. Department of Energy (DOE), Office of Science User Facility (XS, RRU, MC, SVK, SJ), by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, DOE (ARL and AYB), by ORNL's Laboratory Directed Research and Development Program, which is managed by UT-Battelle LLC for the U.S. DOE (SJ) and by the Office of Advanced Scientific Computing Research, Applied Mathematics program under the ACUMEN project.(EL and RKA).
This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
Xiahan Sang, Raymond R. Unocic, Miaofang Chi, Sergei V. Kalinin & Stephen Jesse
Institute for Functional Imaging of Materials, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
Xiahan Sang, Andrew R. Lupini, Raymond R. Unocic, Miaofang Chi, Albina Y. Borisevich, Sergei V. Kalinin & Stephen Jesse
Materials Sciences and Technology, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
Andrew R. Lupini & Albina Y. Borisevich
Computer Science and Mathematics, Oak Ridge National Laboratory, Oak Ridge, TN, 37831, USA
Eirik Endeve & Richard K. Archibald
Xiahan Sang
Andrew R. Lupini
Raymond R. Unocic
Miaofang Chi
Albina Y. Borisevich
Sergei V. Kalinin
Eirik Endeve
Richard K. Archibald
Stephen Jesse
Correspondence to Andrew R. Lupini or Stephen Jesse.
Xiahan Sang and Andrew R. Lupini contributed equally to the paper
Sang, X., Lupini, A.R., Unocic, R.R. et al. Dynamic scan control in STEM: spiral scans . Adv Struct Chem Imag 2, 6 (2016). https://doi.org/10.1186/s40679-016-0020-3
Aberration-corrected STEM
Scan control
Spiral scan | CommonCrawl |
\begin{document}
\title[Short Title]{Overcoming resolution limits with quantum sensing}
\author{T. Gefen} \affiliation{Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel} \author{A. Rotem} \affiliation{Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel} \author{A. Retzker} \affiliation{Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Givat Ram, Israel} \date{\today}
\begin{abstract} { The field of quantum sensing explores the use of quantum phenomena to measure a broad range of physical quantities, of both static and time-dependent types.
An important figure of merit for time dependent signals is the spectral resolution, i.e. the ability to resolve two different frequencies.
Here we study this problem, and develop new superresolution methods that rely on quantum features. We first formulate a general criterion for superresolution in quantum problems. Inspired by this, we show that quantum detectors can resolve two frequencies from incoherent segments of the signal, irrespective of their separation, in contrast to what is known about classical detection schemes. The main idea behind these methods is to overcome the vanishing distinguishability in resolution problems by nullifying the projection noise. } \end{abstract} \maketitle
\section*{Introduction} Quantum metrology and quantum sensing \cite{giovannetti2011advances,degen2017quantum} study parameter estimation limits in various physical systems by employing the fundamental laws of quantum physics. In particular this field seeks to optimize precision by utilizing quantum effects that have no classical analogs (such as entanglement and squeezing \cite{bollinger1996optimal,itano1993quantum}).
A unique feature of quantum sensing is the ability to apply coherent control to the probe and vary the measurement basis. In particular this provides the ability to nullify the measurement projection noise. However, the contribution of this phenomenon to estimation problems has received scant attention.
In this paper we highlight this feature and show that it is a critical resource primarily for resolution problems, that can improve precision by orders of magnitude.
Resolution problems are ubiquitous and highly important in science \cite{rayleigh1879xxxi,hannan1989resolution,hell1994breaking,bettens1999model,van2002high,tsang2016quantum,nair2016far, lupo2016ultimate,barabell1998performance,glentis2014sar}, and roughly speaking are characterized by vanishing distinguishability; i.e, the sensitivity to the seperation between two close objects or frequencies vanishes as these get close enough. This effect usually results in divergent uncertainty, leading to a resolution limit. We show that it is possible to overcome the vanishing distinguishability by making the projection noise vanish as well, through a suitable control. These two effects can cancel each other out, leading to a finite uncertainty. We show that this is a general method to overcome resolution limits in quantum sensing.
Specifically, this method can be highly useful for analyzing complex spectrums with quantum sensors (such as quantum NMR problems \cite{staudacher2013nuclear,aslam2017nanoscale,pham2016nmr,kong2015towards}).
An example for such a spectrum is illustrated in fig. \ref{spect} . While the two extreme frequencies can easily be estimated, the two central frequencies must be analyzed with a more sophisticated method, which eventually yields higher uncertainty.
Here, we show that by using a quantum control, the spectrum can be shifted such that the projection noise vanishes. The vanishing projection noise implies a finite uncertainty irrespective of the frequency separation. In other words, the uncertainty does not diverge when the two frequencies merge.
Furthermore, this method is extremely simple, unlike numerically demanding classical superresolution methods.
\begin{figure}\label{spect}
\end{figure}
\section*{Results}
\subsection{Conditions for superresolution} We first briefly review the pillars of quantum parameter estimation problems. A typical problem involves a quantum state $\rho\left(\theta \right),$ such that $\theta$ is to be estimated. The uncertainty in estimating $\theta$ is tightly lower bounded by $\frac{1}{\sqrt{I_{\theta}}}$ , where $I_{\theta}$ is the Fisher information (FI) about $\theta$ \cite{cramer2016mathematical}. For a given choice of measurement of $\rho\left( \theta \right),$ $I_{\theta}$ is determined according to the probabilities $\left(p_{j}\right)$ in the following way: $I_{\theta}=\sum_{j}\frac{\left(\frac{dp_{j}}{d\theta}\right)^{2}}{p_{j}}.$
The FI can be optimized over all possible measurements, leading to the quantum Fisher information (QFI) \cite{wootters1981statistical,braunstein1994statistical}. Given a spectral decomposition $\rho=\sum_{j}p_{j}|\psi_{j}\rangle\langle\psi_{j}|$ the QFI about $\theta$ reads: $\mathcal{F}=\underset{p_{i}+p_{j}\neq0}{\sum}\frac{2}{p_{j}+p_{i}} \bigg \vert \left(\frac{d \rho}{d \theta}\right)_{i,j} \bigg \vert ^{2}.$
For a multivariable estimation of $\left\{ \theta_{k}\right\} _{k},$ the error is quantified by the covariance matrix of the estimators. This covariance matrix is lower bounded by $\mathcal{F}^{-1},$ the inverse of the QFI matrix,
where the QFI matrix is defined as $\mathcal{F}_{k,l}=2\underset{i,j}{\sum}\frac{\left(\frac{\partial\rho}{\partial\theta_{k}}\right)_{i,j}\left(\frac{\partial\rho}{\partial\theta_{l}}\right)_{j,i}}{\left(p_{i}+p_{j}\right)}.$
We are now poised to formulate spectral resolution problems, which are the focus of this paper. In these problems we are given a signal (Hamiltonian) that oscillates with time. It consists of at most two frequencies, yet the exact number of frequencies (and their values) is unknown and need to be determined. To this end, a quantum probe interacts with the signal so that information about it becomes encoded on the probe and can be extracted by measurements. Once this information is extracted this problem boils down to a parameter estimation problem: the common strategy in these problems \cite{hannan1989resolution,barabell1998performance,roy1989esprit,rotem2017limits} is to assume that there are two frequencies and estimate them. If the estimation shows a significant overlap between the frequencies (significant with respect to the estimation error), it is concluded that the frequencies are not resolvable. However if the overlap is negligible, one can deduce that the signal consists of two frequencies (since the error probability is negligible). This implies that the figure of merit is $\Delta \omega_{1},$ $\Delta \omega_{2}.$ The challenging regime is when $\omega_{1}\rightarrow\omega_{2}.$ Resolution becomes an issue when $\Delta\omega_{1},\:\Delta\omega_{2}\rightarrow\infty$ as $\omega_{1}\rightarrow\omega_{2}.$ A different, and somewhat more convenient, formulation uses $\omega_{ \mathrm{r} }=\frac{\omega_{1}-\omega_{2}}{2},$ $\omega_{ \mathrm{s} }=\frac{\omega_{1}+\omega_{2}}{2},$ so that the resolution condition is $\Delta\omega_{ \mathrm{r} }\ll\omega_{ \mathrm{r} }$ and the figure of merit is thus $\Delta\omega_{ \mathrm{r} }.$
The key issue is thus the behavior of $\Delta\omega_{ \mathrm{r} }$ as $\omega_{ \mathrm{r} }\rightarrow0,$ if $\Delta\omega_{ \mathrm{r} }\rightarrow\infty$ then a fundamental resolution limit exists which is the case in relevant classical examples \cite{hannan1989resolution,barabell1998performance,roy1989esprit}.
This limitation appears in various resolution problems (not only spectral resolution) and stems from a property of vanishing distinguishability. Let us define what vanishing distinguishability means. Given the quantum state of a probe (density matrix $\rho$), that depends on a set of parameters $\left\{ \theta_{i}\right\} _{i},$ the state suffers from a vanishing distinguishability if the set $\left(\frac{\partial\rho}{\partial\theta_{i}}\right)_{i}$ is linear dependent. An equivalent way to define it: there exists a parameter $g,$ that is a linear combination of $\left\{ \theta_{i}\right\} _{i},$ such that $\frac{\partial\rho}{\partial g}=0.$
Indeed, in many resolution problems as the separation parameter $\omega_{ \mathrm{r} }$ (the difference between the frequencies or , in imaging, the sources) goes to $0,$ there exists a parameter $g$ such that $\frac{\partial\rho}{\partial g}=0.$ In this paper we focus on the simplest (yet very common) case that only $\frac{\partial\rho}{\partial\omega_{ \mathrm{r} }}=0$ as $\omega_{ \mathrm{r}}=0.$ In this case $\Delta\omega_{ \mathrm{r} }\rightarrow\infty$ if and only if the FI about $\omega_{\mathrm{r}}$ (denoted as $I_{ \mathrm{r} }$) vanishes, which implies that $I_{ \mathrm{r} }$ is our figure of merit.
As an example, consider a signal that acts on a qubit and is given by the following Hamiltonian: \begin{equation} H=\left[ A_{1}\cos\left(\omega_{1}t\right)+B_{1}\sin\left(\omega_{1}t\right)+A_{2}\cos\left(\omega_{2}t\right)+B_{2}\sin\left(\omega_{2}t\right) \right] \sigma_{z} . \label{Hamiltonian} \end{equation} It is simple to see that this limitation appears whenever the Hamiltonian posses a symmetry for exchange of $\omega_{1}\leftrightarrow\omega_{2}$ (i.e. identical amplitudes). This symmetry implies a symmetry of $\omega_{ \mathrm{r} }\leftrightarrow-\omega_{ \mathrm{r} },$
from which it follows that the state obtained after evolution time $t$ has the same symmetry,$|\psi_{t}\left(\omega_{ \mathrm{r} }\right)\rangle=|\psi_{t}\left(-\omega_{ \mathrm{r} }\right)\rangle$, and thus $\frac{\partial|\psi_{t}\rangle}{\partial\omega_{ \mathrm{r} }}=0$ for $\omega_{ \mathrm{r} }=0.$ Given the expression of the quantum Fisher information \cite{braunstein1994statistical}, we obtain: \begin{equation} I_{ \rm{r} }\leq4\left[ \left \langle\frac{\partial\psi}{\partial\omega_{ \mathrm{r} }} \bigg \vert \frac{\partial\psi}{\partial\omega_{ \mathrm{r} }} \right \rangle - \left \vert \left \langle \frac{\partial\psi}{\partial\omega_{ \mathrm{r} }} \bigg \vert \psi \right \rangle \right \vert ^{2}\right]\rightarrow0, \label{no_go_1} \end{equation} hence resolution is limited. Note that applying further control on the probe cannot eliminate this symmetry, and thus cannot remove this resolution limit.
It can be shown that this limitation appears for any quadratures:
there exists a parameter $g$ such that $\frac{\partial H}{\partial g}=0$ for every $t,$ which implies $\frac{\partial|\psi_{t}\rangle}{\partial g}=0$ for every measurement (more details in supplementary note 4).
So vanishing distinguishability is quite a common property and appears in different resolution problems, but does it always impose a limitation?
Eq. \ref{no_go_1} shows that whenever the quantum state of the probe ($\rho$) is pure, resolution is limited, however for some mixed states this property does not limit the resolution, these are the states that give rise to superresolution: $\frac{d\rho}{d\omega_{\mathrm{r}}}=0$ yet $I_{{\rm r}}\left(\omega_{{\rm r}}\rightarrow0\right)>0.$ A special case of this phenomenon was found and analyzed recently in the context of optical imaging \cite{nair2016far,tsang2016quantum,lupo2016ultimate,chrostowski2017super}(see supplementary note 3). Can such states be obtained in quantum spectroscopy and various other problems? In order to understand this, it would be highly desirable to characterize these states and set a sharp condition for superresolution.
Let us show that these states can be simply characterized:
{\it{Claim:}} Given $\rho\left(\omega_{\mathrm{r}}\right)$ such that $\frac{d\rho}{d\omega_{\mathrm{r}}}=0$ (as $\omega_{\mathrm{r}}\rightarrow0$), then $I_{{\rm r}}\left(\omega_{{\rm r}}\rightarrow0\right)>0$ if and only if at least one of the eigenvalues of $\rho$ goes as $\omega_{\mathrm{r}}^{k},$ where $1<k\leq2$ or equivalently $\frac{d\sqrt{\rho}}{d\omega_{ \mathrm{r} }}\neq0$. The optimal measurement basis converges to an eigenbasis of $\rho$ as $\omega_{ \mathrm{r} }\rightarrow0$.
We briefly illustrate a proof: Given a spectral decomposition $\rho=\underset{j}{\sum}p_{j}|j\rangle\langle j|,$ then: \begin{equation}
\frac{d\rho}{d\omega_{\rm{r}}}=\underset{j}{\sum}\frac{dp_{j}}{d\omega_{\rm{r}}}|j\rangle\langle j|+i\underset{j,k}{\sum}\left(p_{j}-p_{k}\right)h_{k,j}|k\rangle\langle j|, \label{derivative_rho} \end{equation} where $h$ is a Hermitian operator and $h_{k,j}$ denote its matrix elements in the eigenbasis of $\rho$. Since $\frac{d\rho}{d\omegar} \rightarrow 0$, then for every $j,k$: $\frac{dp_{j}}{d\omegar}\rightarrow0,\;\left(p_{j}-p_{k}\right)h_{k,j}\rightarrow0.$ With this notation, the QFI ($\mathcal{F}$) reads (see \cite{braunstein1994statistical}): \begin{equation}
\mathcal{F}=\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\omegar}\right)^{2}}{p_{j}}+2\underset{j,k}{\sum}\frac{\left(p_{j}-p_{k}\right)^{2}}{p_{j}+p_{k}}|h_{kj}|^{2}. \end{equation}
The fact that $\left(p_{j}-p_{k}\right)h_{k,j}\rightarrow0$ implies that $\frac{\left(p_{j}-p_{k}\right)^{2}}{p_{j}+p_{k}}|h_{kj}|^{2}\rightarrow 0,$ however $\frac{dp_{j}}{d\omegar}\rightarrow0$ does not imply that $\frac{\left(\frac{dp_{j}}{d\omegar}\right)^{2}}{p_{j}}$ vanishes. It can be seen that given that $\frac{dp_{j}}{d\omegar} \rightarrow 0,$ $\frac{\left(\frac{dp_{j}}{d\omegar}\right)^{2}}{p_{j}}>0$ if and only if there exists $p_{j}\sim\omegar^{k}$ for $1<k\leq2.$ We then observe that for $\omegar=0,$ $\mathcal{F}\left(\rho\right)=\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\omegar}\right)^{2}}{p_{j}},$ which implies that the optimal measurement basis converges to any eigenbasis of $\rho.$
This condition can be shown to be equivalent to $\frac{d\sqrt{\rho}}{d\omegar}\neq0$ (see supplementary note 1). It is quite intuitive that one has to demand $\frac{d\sqrt{\rho}}{d\omegar}\neq0,$ since the QFI equals the minimization of all the QFI's of the purifications. Since purifications go as $\sqrt{\rho},$ $\frac{d\sqrt{\rho}}{d\omegar}=0$ would imply a vanishing derivative of every purification and thus a vanishing QFI.
This criterion shows that the only way to overcome a vanishing distinguishability is by nullifying the projection noise of one of the outcomes.
This condition is a special case of a more general (multivariate) criterion. In the multivariate version $\left(\frac{\partial\rho}{\partial\theta_{i}}\right)_{i=1}^{n}$ are linearly dependent (with dimension $k<n$) and the relevant question is whether the QFI matrix can be regular.
Note that we can choose $\left(\theta_{i}\right)_{i=1}^{n}$ such that $\left(\frac{\partial\rho}{\partial \theta_{i}}\right)_{i=1}^{k}$ are linearly independent and $\frac{\partial\rho}{\partial \theta_{k+1}}=...=\frac{\partial\rho}{\partial \theta_{n}}=0$ ($\theta_{k+1},....,\theta_{n}$ are the problematic parameters). Then the QFI is regular if and only if the classical FI matrix (i.e. the FI matrix obtained when measuring in the eigenbasis of $\rho$) about the problematic parameters ($\theta_{k+1},....,\theta_{n}$) is regular. Namely it depends only on the classical FI about these parameters, and thus the optimal measurement basis to estimate these parameters is the eigenbasis of $\rho.$
The proof of this condition is quite similar to that of the single variable case, and is given in supplementary note 2.
Before we move on to applications in quantum sensing, a few remarks are in order: An accurate formulation of the superresolution condition is $\frac{d \rho}{d \omegar} \rightarrow 0$ and $ I_{ \mathrm{r} } \left( \omega_{ \mathrm{r} } \rightarrow 0 \right) >0,$ namely the limit needs to be positive. That is because we are interested in the behavior of the FI for a very small difference, rather than a vanishing difference. We mention this point since the FI at $\omega_{r}=0$ can be discontinuous or meaningless (Cramer-Rao bound may be violated), as one of the eigenvalues vanishes \cite{vsafranek2017discontinuities,seveso2019discontinuity}. Given a vanishing eigenvalue, the variance of maximum likelihood estimation will vanish (which corresponds to an infinite FI) and thus may not coincide with the limit.
We also remark that in all cases examined in this paper (as well as in the imaging case) the eigenvalue goes as $\omegar^2.$ Any different power, $1<k<2,$ would in fact lead to a better performance: a divergent FI.
\subsection{Application: spectral resolution without coherence}
Consider now again the problem of spectral resolution, with the signal defined in eq. \ref{Hamiltonian}, and such that it suffers from shot-to-shot noise: in each measurement the frequencies are the same but the quadratures are random, i.e. $A_{i},\:B_{i}$ have a certain distribution. Specifically here we assume $A_{i},\:B_{i}\sim N\left(0,\sigma\right)$, and other noise models are addressed in supplementary note 11. This scenario is illustrated in fig. \ref{noisy_signal}, and is relevant for different applications, such as communication protocols, spectrum analyzers and nano NMR (\cite{balasubramanian2008nanoscale,gruber1997scanning,maze2008nanoscale,staudacher2013nuclear,mamin2013nanoscale,muller2014nuclear,devience2015nanoscale,aslam2017nanoscale,bucher2018hyperpolarization,lovchinsky2016nuclear,bar2017observing}), in particular when the time required to perform projective measurement is longer than the coherence time of the signal (this is the case with NV centers, due to the large number of iterations needed, and with trapped ions, where the re-cooling process might be longer than the coherence time of the qubit).
It is quite clear that the fluctuations of the quadratures remove the purity of the probe, which can give rise to superresolution states. Let us examine this.
\begin{figure}\label{noisy_signal}
\end{figure}
Consider a standard Ramsey experiment, in which the probe is initialized in $\sigma_{x}-\sigma_{y}$ plane, then rotated due to the signal and eventually measured in the initialization basis. Due to the fluctuations of the Hamiltonian, an averaging should be performed. Therefore the state of the probe is given by a density matrix: \begin{equation}
\rho=\int p\left(A_{i}\right)p\left(B_{i}\right)|\psi_{A_{i},B_{i}}\rangle\langle\psi_{A_{i},B_{i}}|\;dA_{i}\:dB_{i},
\end{equation}
where $|\psi_{A_{i},B_{i}}\rangle$ is the state given a single realization of $A_{i},B_{i}.$ Note that since the fluctuations are identical: \begin{equation}
\rho\left(\omegar\right)=\rho\left(-\omegar\right) \Rightarrow \frac{d\rho}{d\omegar}=0 \, \left(\omegar=0\right). \end{equation} Once again, control on the probe does not change this symmetry, hence superresolution can be achieved only if the condition presented above is satisfied: projection noise has to be nullified.
It is therefore desirable to find a measurement scheme that nullifies the projection noise. It is simple to see that this can be obtained if $ \phi_{A_{i},B_{i}}=0\:\left(\forall A_{i},B_{i}\right),$ where $\phi_{A_{i},B_{i}}$ is the phase accumulated by the sensor (defined as half the rotation angle in the Bloch sphere) per realization, since this implies a vanishing transition probability.
Our claim is therefore: Given the above noise model, there exist measurement schemes that satisfy the superresolution condition and thus achieve $\Ir>0.$ To see that such methods exist, observe that the phase accumulated by the sensor given the Hamiltonian in Eq. \ref{Hamiltonian} (when no control is applied) reads: \begin{equation} \phi_{A_{i},B_{i}}=\underset{i}{\sum}\frac{A_{i}}{\omega_{i}}\sin\left(\omega_{i}t\right)+\frac{B_{i}}{\omega_{i}}\left(1-\cos\left(\omega_{i}t\right)\right).
\end{equation} Note that given this time evolution the density matrix of the sensor is diagonal in the initialization basis with eigenvalues $p, \, 1-p,$ where $p$ is the average transition probability: $p=\langle\sin\left(\phi_{A_{i},B_{i}}\right)^{2}\rangle_{A_{i},B_{i}}.$ Hence the superresolution condition boils down to $p\sim\omegar^{2}.$ This indeed can be satisfied by simply tuning $t$ such that $\omegas t=2\pi n,$ where $n$ is a non-zero integer. With this tuning $\phi_{A_{i},B_{i}}=0$ (for $\omegar=0$), and more specifically :
\begin{equation} \phi_{A_{i},B_{i}}\approx\frac{\left(A_{1}-A_{2}\right)}{\omegas}\omegar t\rightarrow p\approx\frac{2\sigma^{2}}{\omegas^{2}}\omegar^{2}t^{2}, \label{probability_approximated} \end{equation} Hence the superresolution condition is satisfied and the FI reads:
\begin{equation} \Ir=\frac{8\sigma^{2}t^{2}}{\omegas^{2}}. \end{equation}
So nullifying the projection noise indeed cancels the vanishing derivative and a finite $\Ir$ is achieved.
The obtained FI can be still quite poor and far from optimal. Note that it goes as $1/n^{2},$ where $n$ is the number of periods completed during the measurement. If $n$ is large, then this factor of $\frac{1}{n^{2}}$ can be significant.
A much better FI can be achieved by applying a suitable control: $\pi-$pulses which effectively change the frequency of oscillations, and reduce $n$ to $1$ \cite{kotler2011single,schmitt2017submillihertz,boss2017quantum,naghiloo2017achievingT4}:
Given an original Hamiltonian of $H=\left[ A\sin\left(\omega t\right)+B\cos\left(\omega t\right) \right] \sigma_{z},$ applying $\pi-$pulses in a frequency of $\omega+\delta$ (namely a $\pi-$pulse is applied every $\frac{\pi}{\omega+\delta},$ $\delta$ is referred to as detuning) on the probe yields the following effective Hamiltonian (see methods for a derivation): \begin{equation} H_{\text{eff}}=\tan\left(\frac{\pi}{2\left(1+\frac{\delta}{\omega}\right)}\right)\left(\frac{\delta}{\omega}\right)\left[A\sin\left(\delta t\right)+B\cos\left(\delta t\right)\right]\sigma_{z}. \end{equation} Hence the $\pi-$pulses effectively change the frequency of the Hamiltonian from $\omega$ to $\delta$ (with a prefactor of $\tan\left(\frac{\pi}{2\left(1+\frac{\delta}{\omega}\right)}\right)\left(\frac{\delta}{\omega}\right)$ added to the amplitude). Since we aim to reduce the frequency of oscillations, we focus on the limit of $\delta \ll \omega,$ in which $\tan\left(\frac{\pi}{2\left(1+\frac{\delta}{\omega}\right)}\right)\left(\frac{\delta}{\omega}\right)\approx\frac{2}{\pi},$ and thus: \begin{equation} H_{\text{eff}}\approx\frac{2}{\pi}\left[A\sin\left(\delta t\right)+B\cos\left(\delta t\right)\right]\sigma_{z}. \end{equation} When dealing with a signal that consists of two frequencies ($\omega_{1},\omega_{2}$), the effective Hamiltonian becomes:
\begin{equation}
H_{\text{eff}} \approx \underset{i}{\sum}\frac{2}{\pi}\left[A_{i}\sin\left(\delta_{i}t\right)+B_{i}\cos\left(\delta_{i}t\right)\right] \sigma_{z}.
\end{equation} Hence due to the control the central frequency is shifted to $\delta_{ \mathrm{s} }=\frac{\delta_{1}+\delta_{2}}{2},$ and the relative frequency simply changes sign: $\delta_{ \mathrm{r} }=-\omegar.$ The condition of vanishing $p$ becomes: $\delta_{ \mathrm{s} }t= \pm 2 \pi n,$ such that the optimal strategy is setting $\delta_{ \mathrm{s} }t= \pm 2 \pi .$ Therefore with these (optimal) values of $\delta_{ \mathrm{s} }$ the FI reads: \begin{equation} I_{ \mathrm{r} } \approx \left(\frac{2}{\pi}\right)^{2}\frac{8\sigma^{2}t^{2}}{\delta_{ \mathrm{s} }^{2}}=\frac{8\sigma^{2}t^{4}}{\pi^{4}}. \end{equation}
Observe that the scaling of $\Ir$ is optimal (goes as $\sigma^{2} t^{4}$) \cite{pang2017optimal,pang2017optimal, schmitt2017submillihertz, jordan2017classical, yang2017quantum, gefen2017control, naghiloo2017achievingT4}; however it is unknown whether this is the best achievable FI (see extended discussion in supplementary note 6). The probabilities and the FI for different detunings are presented in fig. \ref{FI_intro}. Note that clear resonance peaks of the FI are observed for $\delta_{ \mathrm{s} }t=\pm 2\pi n,$ any other values of detuning lead to a vanishing FI.
\begin{figure}\label{FI_intro}
\end{figure}
We tested this method numerically by generating data of two frequency signal (with the corresponding noise model) and performing a Maximum-Likelihood estimation (MLE) to find $\omega_{\rm{r}}.$ Some of the results are shown in fig. \ref{numerical_results}. It can be seen that by choosing a detuning such that $\delta_{ \mathrm{s} }t=2\pi,$ $\omegar$ can be estimated efficiently and the frequencies are resolved. As shown in fig. \ref{numerical_results}, the standard deviation matches the theoretical expectation: $\Delta\omegar=\frac{1}{\sqrt{I_{ \rm{r} }N}}.$ By utilizing this control method the number of measurements ($N$) needed to achieve resolution is $N \gg p^{-1}=\frac{\pi^{4}}{2\sigma^{2}\omega_{\rm{r}}^{2}t^{4}}.$ Taking for example values which are well beyond the resolution limit such as $\omega_{\rm{r}}t=0.01,\:\sigma t=1,$ resolution is achieved for $N \gg 5\cdot10^{5}.$ If the chosen detuning does not satisfy one of theses conditions ($\delta_{s}t=2\pi n$) we expect to observe a divergence in the variance. We used MLE for this case as well. Note that the fact that the FI vanishes does not mean that no information about $\omega_{\rm{r}}$ is obtained, information is in fact obtained from the second derivative. The estimator becomes biased and the standard deviation reads: $\Delta\omega_{\rm{r}}=\frac{\left(p\left(1-p\right)\right)^{0.25}}{\sqrt{\frac{\partial^{2}p}{\partial\omega_{\rm{r}}^{2}}}N^{0.25}},$ see fig. \ref{numerical_results}. The fact that the standard deviation is proportional to $N^{-0.25}$ (as opposed to the standard scaling of $N^{-0.5}$) is a manifestation of the divergence. The resolution condition in this case is thus: $N \gg \frac{p\left(1-p\right)}{\left(\frac{\partial^{2}p}{\partial\omega_{\rm{r}}^{2}}\right)^{2}\omega_{\rm{r}}^{4}}.$ Considering the same example as previously ($\omega_{\rm{r}}t=0.01,\:\sigma t=1$) but with off-resonance detuning $\left(\delta_{s}t=1.8\pi\right)$,
the number of measurements required for resolution is $N \gg 10^{8};$ hence a difference of almost three orders of magnitude.
This method can be understood in the following simple and intuitive way: If there is only a single frequency and $\delta_{s}=\frac{2 \pi}{t}$ then $p_{a}=0,$ hence no transitions should occur, whereas a finite (small) $\omega_{\rm{r}}$ should lead to a small transition probability $\left(p\right)$, such that transitions will be observed after $\frac{1}{p}=\frac{\pi^{4}}{2\sigma^{2}\omega_{\rm{r}}^{2}t^{4}}$ measurements.
\begin{figure}\label{numerical_results}
\end{figure}
\subsection{Limitations and imperfections} \label{imperfections} The method, as analyzed so far, assumes knowledge of all the other parameters ($\sigma$ and $\omega_{\rm{s}}$), coherence of the signal and the probe during the measurement period, and measurements with unit fidelity. In this section we analyze each one of these assumptions. The first one to be analyzed is the main caveat of the method: the requirement of coherence during the measurement period.
This method relies on the ability to nullify projection noise, in particular on the fact that for $\omega_{\rm{r}}=0$ the state can become pure. However this is achieved only for a signal which is perfectly coherent during each measurement. Fluctuations of the signal during the measurement period inflict a limitation, as in this case it is not possible to nullify the projection noise. Heuristically due to these fluctuations the transition probability includes an additional noise term (denoted as $\epsilon$), such that it reads (for $\omega_{ \rm{s} }t=2\pi$):
\begin{equation}
p=\frac{\sigma^{2}t^{2}}{2\pi^{2}}\omegar^{2}t^{2}+\epsilon.
\label{imperfection_probability}
\end{equation} This new term imposes a limitation: It is now impossible to nullify $p,$ which implies that $ I_{ \mathrm{r} } \rightarrow0$ as $ \omega_{ \mathrm{r} } \rightarrow0.$ The FI $\left(\text{for } \omega_{ \rm{s} }t=2\pi,\omega_{\rm{r}}t\ll1\right)$ now reads: \begin{equation}
I_{{\rm {r}}}\approx\frac{\left(\sigma t\right)^{2}\left(\omega_{\rm{r}}t\right)^{2}}{\pi^{4}\left(\epsilon+\frac{\sigma^{2}t^{2}}{2\pi^{2}}\omega_{\rm{r}}^{2}t^{2}\right)}, \label{FI_imperfection} \end{equation} this behavior is illustrated in fig. \ref{imperfection_plot}, and it can be observed that resolution can be achieved only for $\omega_{\rm{r}} t>\frac{\sqrt{\epsilon}}{\sigma t}.$
More specifically, assuming a realistic noise model: the quadratures undergo OU (Ornstein-Uhlenbeck) noise process (with variance $\sigma_{n}^{2}$ and damping rate $\gamma$), the noise term reads (in leading order of $\left(\gamma t\ll1\right)$, see supplementary note 8 ) $\epsilon=\frac{\sigma_{n}^{2}t^{3}}{\pi^{2}}.$ When comparing $\epsilon$ to the original transition probability: $\sigma^{2}\frac{t^{2}}{2\pi^{2}}\omega_{\rm{r}}^{2}t^{2}=\frac{\sigma_{n}^{2}t^{2}}{4\pi^{2}\gamma}\omega_{\rm{r}}^{2}t^{2},$ we get the Fourier limit: $\frac{\omega_{\rm{r}}}{\gamma}>1.$ We remark that whether one can remove this limitation is an open question.
Therefore this method is relevant mainly for experimental scenarios with noise that is effectively shot to shot: small enough fluctuations during each measurement but no correlations between consecutive measurements. This is the case in many experimental settings, where the time separation between measurements is longer than the phase acquisition period due to long readout and preparation stages.
Quite similarly, dephasing of the probe also imposes a limitation. Taking into account a dephasing rate $\kappa,$ the transition probability reads: $p=0.5\left(1-\exp\left(-\frac{\left(\sigma t\right)^{2}\left(\omega_{\rm{r}}t\right)^{2}}{\pi^{2}}-2\kappa t\right)\right).$ Hence resolution can be achieved only if $ \frac { \omega_{\rm{r}}\sigma}{\left(\kappa^{2}\right)}\gg1.$ Note that in order to retrieve the noiseless FI it is not enough to require $\kappa t \ll 1,$ as there is also minimal time $t>\frac{\kappa^{1/3}}{\sigma^{2/3}\omega_{\rm{r}}^{2/3}}$ ($\kappa t$ should be smaller than $\left(\sigma t\right)^{2}\left(\omega_{\rm{r}}t\right)^{2}$). A detailed analysis of this limitation can be found in supplementary note 9.
We next address the consequences of imperfect measurements. The effect of imperfect measurements is similar to that of incoherence, therefore the measurement infidelity sets a resolution limit. We consider a model in which there are two different outcomes and there is a finite probability to get each outcome from both states (as is the case for the NV center \cite{batalov2008temporal}). Namely the probability of detecting an outcome that corresponds to the bright state is: $p=\left(1-\epsilon'\right)p_{b}+\epsilon' p_{d},$ where $p_{b}$ ($p_{d}$) denotes the probability of the bright (dark) state and then $\epsilon'$ is the probability of wrong detection. Given this error probability we can observe that $\frac{dp}{d\omegar}=0$ (when $\omegar=0$) but it is impossible to nullify $p$. This implies $I_{ \rm{r} }\rightarrow0$ as $ \omega_{ \mathrm{r} } \rightarrow 0.$
Therefore taking $\epsilon'\ll1$ (and $\omegar t\ll1,\omega_{\rm{s}}t=2\pi$) we get the same expression as in eq. \ref{imperfection_probability} (with a noise term of $\epsilon'$): $p\approx\frac{\sigma^{2}T^{2}}{2\pi^{2}}\omegar^{2}T^{2}+\epsilon'.$ Hence resolution limit is given by: $\omegar T>\frac{\sqrt{\epsilon'}}{\sigma T}$ (see fig. \ref{imperfection_plot}).
Let us now address the multivariable estimation protocol. In any realistic scenario $\sigma$ and $\omega_{\rm{s}}$ are unknown. Since the estimation protocol of $ \omega_{ \mathrm{r} } $ depends on knowledge of $\omega_{\rm{s}}$ a preliminary estimation of $\omega_{\rm{s}}$ must be performed (quite analogously to the preliminary estimation of the centroid in quantum resolution methods for optical imaging \cite{parniak2018beating,nair2016interferometric} ). This can be done using the traditional method \cite{pham2016nmr}: Applying $\pi$-pulses in different frequencies and fitting the transition probability as a function of the pulses frequency (see supplementary note 7). This should provide a good estimation of $\sigma,\omega_{ \rm{s} },$ but not a good enough estimation of $\omega_{\rm{r}}$ (unless by chance we hit close enough to a resonance frequency). Once a good enough estimation of $\omega_{ \rm{s} }$ is obtained we can apply the required control ($\delta_{s}t=2 \pi$).
To understand what is a good enough estimation of $\omega_{ \rm{s} }$ observe that for small enough $\omega_{\rm{r}}t,\left( \delta_{s} t-2\pi \right)$: $I_{ \rm{r} }=\frac{8\sigma^{2}t^{4}}{\pi^{4}}\frac{\omega_{\rm{r}}^{2}}{\omega_{\rm{r}}^{2}+\left(\delta_{s}-2\pi/t \right)^{2}},$ hence the width of the resonance peak (in $\delta_{s} t$) goes as $\omega_{\rm{r}}t.$ Therefore once $\Delta{\omega_{ \rm{s} }}$ is comparable to $\omega_{\rm{r}}$ this method works despite the small detuning.
Observe that now a multivariate estimation should be performed, which means that at least three different measurements are needed; each measurement in a detuning that is optimal for a different parameter. Numerical results and further analysis are presented in supplementary note 7.
\begin{figure}\label{imperfection_plot}
\end{figure}
\subsection{ Additional applications: quantum resolution methods for sampling } { \it{Superresolution with quantum Fourier transform (QFT)} }: Consider the signal (Hamiltonian) in eq. \ref{Hamiltonian}, if the coherence time of the signal is relatively long, its spectrum can be found by sampling it (where sampling means Ramsey measurements of the probe in different times). Several recent experiments implemented this scheme \cite{schmitt2017submillihertz,boss2017quantum,glenn2018high}. The straightforward (and natural) way to analyze this data is by fitting the power spectrum of the measurement outcomes, however it was shown in \cite{rotem2017limits} that this method suffers from a resolution limit. The reason is again that the (average) power spectrum is symmetric with respect to $\omega_{\rm{r}},$ yet the measurement noise does not vanish. We point out here that this limit can be eliminated if instead of classical Fourier transform, one uses a quantum Fourier transform. In more detail: a phase of $\phi_{j} \approx \tau\underset{i}{\sum}\left(A_{i}\cos\left(\omega_{i}t_{j}\right)+B_{i}\sin\left(\omega_{i}t_{j}\right)\right)$ is accumulated by the probe in each measurement (where $\tau$ is the length of each measurement).
The idea is that instead of measuring the probe after each phase acquisition, we can map the phases to memory qubits to form the state: $|\psi\rangle=\frac{1}{\sqrt{N}}\underset{j=1}{\overset{N}{\sum}}e^{i\phi_{j}}|j\rangle,$
and then measure in the Fourier basis. With the appropriate choice of total sampling time ($T=m\frac{2\pi}{\omega_{\rm{s}}}$, for an integer $m$) and measurement length ($\tau=\frac{2\pi}{n\omega_{ \rm{s} }},$ for an integer $n$), the only states in the Fourier basis that can be measured are harmonics of $\omega_{\rm{s}}$ (for $\omegar= 0$), hence vanishing projection noise of all the others outcomes. The probability to measure the other frequencies, for $\omegar T\ll1,$ is $\approx \frac{1}{6}\omegar^{2}T^{2}\left(\sigma\tau\right)^{2}$ (see supplementary note 10), therefore $\Ir=\frac{2}{3}\left(\sigma\tau\right)^{2}T^{2}$ for $\omegar=0.$ We thus get a non vanishing FI, and it can be shown that for any model in which the phases are uniformly distributed, the optimal measurement basis is indeed the Fourier basis.
A different method for the same problem is {\it{superresolution with correlation spectroscopy}}: The Hamiltonian is the same, but now perform two measurements and correlate between them using a single memory qubit, namely the state of the memory qubit after the two phase accumulation periods is $\frac{1}{\sqrt{2}}\left(|0\rangle+e^{i\left(\phi_{1}-\phi_{2}\right)}|1\rangle\right),$ where $\phi_{j}$ is the phase accumulated in the $j$th period. It is simple to see that by choosing the period between measurements to be $T=\frac{2\pi}{\omega_{ \rm{s} }}n,$ and measuring in the initialization basis the transition probability is $p\approx\langle\left(B_{1}-B_{2}\right)^{2}\rangle\tau^{2}\omega_{\rm{r}}^{2}T^{2}.$ Therefore a non-vanishing FI is achieved:
$I_{ \rm{r} }=4\langle\left(B_{1}-B_{2}\right)^{2}\rangle\tau^{2}T^{2}=8\left(\sigma\tau\right)^{2}T^{2}.$
\section*{Discussion} We presented methods that are capable of resolving frequencies beyond the resolution limits ($\omega_{\rm{r}}t \ll 1$) in quantum spectroscopy. Those methods are special cases of a general superresolution criterion: one can overcome the vanishing derivative by making the projection noise vanish at the same rate. The main method that was analyzed (resolution without coherence) is applicable with state of the art experimental capabilities and does not require involved numerical analysis.
It would be interesting to inquire whether similar ideas are useful to other resolution problems, such as resolving the locations and the frequencies of single neighboring spins.
The methods presented above are not perfect, they are limited by the noise of the signal and the dephasing of the probe, whether one can overcome these limitations is an open question.
\section*{Methods}
\subsection*{Derivation of density matrix and probabilities} \label {averaged_probability} Given a noise model on the amplitudes, the quantum state of the probe is described by the following density matrix: \begin{equation}
\rho=\int|\psi\rangle\langle\psi|\:p\left(\vect{A},\vect{B}\right)\:d\vect{A} \,d\vect{B}. \end{equation} Since the time evolution (with and without control) is described by the operator: $U=\cos\left(\phi\right) \mathbb{1} -i\sin\left(\phi\right)\sigma_{z},$ $\rho$ reads: \begin{eqnarray} \begin{split} &\rho=\int\left[\cos\left(\phi\right)^{2}\rho_{0}+\sin\left(\phi\right)^{2}\sigma_{z}\rho_{0}\sigma_{z}-\frac{i}{2} \sin\left(2 \phi\right) \left[ \sigma_{z},\rho_{0} \right] \right] \cdot\\ &p\left( \vect{A}, \vect{B} \right)\:d \vect{A} \, d\vect{B}, \end{split} \end{eqnarray} where $\rho_{0}$ is the initial state. With the relevant noise model ($A_{i},B_{i}\sim N\left(0,\sigma\right)$) it can be seen that the terms going as $\sin\left(2 \phi\right)$ vanish, leading to: \begin{equation} \rho=\left(1-p\right)\rho_{0}+p\sigma_{z}\rho_{0}\sigma_{z}, \end{equation} where $p$ is the (averaged) transition probability: $\int\sin\left(\phi\right)^{2}p\left(\vect{A},\vect{B}\right)\:d\vect{A} \, d\vect{B}.$ Taking $\phi=\sum_{i}A_{i}\frac{\sin\left(\delta_{i}t\right)}{\delta_{i}}+B_{i}\frac{1-\cos\left(\delta_{i}t\right)}{\delta_{i}},$ a simple calculation yields: \begin{equation} p=0.5\left(1-\exp\left(- 8 \sum_{i} \frac{\sigma^2}{\delta_i^2} \sin^2\left(\frac{\delta_i t}{2} \right) \right) \right). \end{equation} Note that this expression coincides with eq. \ref{probability_approximated} for $\delta_{ \rm{s} }t=2\pi,$ $\omegar t \ll 1.$
The optimal initial state would be $\rho_{0}=|\uparrow_{x}\rangle\langle\uparrow_{x}|$ (or any other pure state in the $X-Y$ plane), leading to $\rho=\left(1-p\right)|\uparrow_{x}\rangle\langle\uparrow_{x}|+p|\downarrow_{x}\rangle\langle\downarrow_{x}|.$ The QFI (about $\omegar$) of $\rho$ is thus: $\frac{\left(\frac{dp}{d\omegar}\right)^{2}}{p\left(1-p\right)},$ which is the expression of $\Ir$ mentioned in the main text.
\subsection*{Effective Hamiltonian derivation} \label{effective_hamiltonian_derivation} In this section we derive the effective Hamiltonian that appears in the main text. Given a Hamiltonian: $H=\left[ A \sin\left(\omega t\right) + B\cos\left(\omega t\right) \right] \sigma_{z},$ and $\pi$-pulses that are applied every $\tau,$ the Hamiltonian in the interaction picture of these pulses is: \begin{equation} H=\left[A\sin\left(\omega t\right)+B\cos\left(\omega t\right)\right]h\left(t\right)\sigma_{z}, \end{equation} where $h \left( t \right)$ is the square wave function. Note that the phase accumulated by the sensor (denoted as $\phi,$ and defined as half the rotation angle in Bloch sphere) in $t=n \tau$ is: \begin{eqnarray} \begin{split} &\phi=A\:\text{Im}\left(\Phi\right)+B\:\text{Re}\left(\Phi\right)\;\text{where:}\\ &\Phi=\underset{n=0}{\overset{N-1}{\sum}}\int_{n\tau}^{\left(n+1\right)\tau}e^{i\omega t}\left(-1\right)^{n}\:dt
\end{split}
\end{eqnarray} where Re (Im) denotes the real (imaginary) part. Therefore in order to find $\phi$ we need to calculate $\Phi$: \begin{equation} \Phi=\underset{n=0}{\overset{N-1}{\sum}}e^{i\omega n\tau}\left(-1\right)^{n}\frac{e^{i\omega\tau}-1}{i\omega}=\underset{n=0}{\overset{N-1}{\sum}}e^{in\left(\omega\tau+\pi\right)}\frac{e^{i\omega\tau}-1}{i\omega}. \end{equation} The calculation then proceeds as follows: \begin{eqnarray} \begin{split} &\Phi=\frac{1-e^{iN\left(\omega\tau+\pi\right)}}{1+e^{i\omega\tau}}\frac{e^{i\omega\tau}-1}{i\omega}=\\ &-2 ie^{i\frac{N}{2}\left(\omega\tau+\pi\right)}\sin\left(N\frac{\omega\tau}{2}+N\frac{\pi}{2}\right)\sin\left(\omega\frac{\tau}{2}\right)\frac{1}{\cos\left(\frac{\omega\tau}{2}\right)\omega}. \end{split} \end{eqnarray}
Hence: \begin{eqnarray} \begin{split}
&\text{Re} \left( \Phi \right)=\left(1-\cos\left(\omega t+N\pi\right)\right)\frac{\sin\left(\omega\frac{\tau}{2}\right)}{\cos\left(\omega\frac{\tau}{2}\right)\omega},\\
&\text{Im} \left( \Phi \right)=-\sin\left(\omega t+N\pi\right)\frac{\sin\left(\omega\frac{\tau}{2}\right)}{\omega\cos\left(\frac{\omega\tau}{2}\right)}.
\label{im_re_expressions}
\end{split}
\end{eqnarray}
Note that: $ \omega t=\omega N\frac{\pi}{\omega+\delta}=N\pi-\delta t,$ therefore eq. \ref{im_re_expressions} is simplified to:
\begin{eqnarray} \begin{split}
&\text{Re}\left(\Phi\right)=\left(1-\cos\left(\delta t\right)\right)\frac{\tan\left(\omega\frac{\tau}{2}\right)}{\omega},\\
&\text{Im}\left(\Phi\right)=\sin\left(\delta t\right)\frac{\tan\left(\omega\frac{\tau}{2}\right)}{\omega}.
\label{im_re_expressions_2}
\end{split}
\end{eqnarray}
The accumulated phase,$\phi$, thus reads:
\begin{equation}
\phi=A\sin\left(\delta t\right)\frac{\tan\left(\omega\frac{\tau}{2}\right)}{\omega}+B\left(1-\cos\left(\delta t\right)\right)\frac{\tan\left(\omega\frac{\tau}{2}\right)}{\omega}.
\label{full_expression_phase}
\end{equation}
Observe that this exact phase is obtained by the following effective Hamiltonian (note that no approximation is used here): \begin{equation} H_{\text{eff}}=\tan\left(\omega\frac{\tau}{2}\right)\left(\frac{\delta}{\omega}\right) \left[ A\cos\left(\delta t\right)+B\sin\left(\delta t\right) \right] \sigma_{z}, \end{equation} hence we can use this effective Hamiltonian to describe the dynamics. This effective Hamiltonian is somewhat similar to the original Hamiltonian in that the frequency is shifted from $\omega$ to $\delta,$ and the amplitude acquires a prefactor of $\tan\left(\omega\frac{\tau}{2}\right)\left(\frac{\delta}{\omega}\right).$
Note that for $\delta\ll\omega$: \begin{equation} \tan\left(\omega\frac{\tau}{2}\right)\left(\frac{\delta}{\omega}\right)=\tan\left(\frac{\pi}{2\left(1+\frac{\delta}{\omega}\right)}\right)\left(\frac{\delta}{\omega}\right)\approx\frac{2}{\pi}, \end{equation} which implies: \begin{equation} H_{\text{eff}}\approx \left[ A\left(\frac{2}{\pi}\right)\cos\left(\delta t\right)+B\left(\frac{2}{\pi}\right)\sin\left(\delta t\right) \right] \sigma_{z} \; \; \left(\delta\ll\omega\right). \end{equation} It should be noted that this is the relevant regime for experimental realizations \cite{glenn2018high,laraoui2013high,staudacher2013nuclear,lovchinsky2016nuclear}. Similarly for the opposite limit $\left(\delta\gg\omega\right),$ we obtain that: \begin{equation} H_{\text{eff}}\approx \left[ A\left(\frac{\pi}{2}\right)\cos\left(\delta t\right)+B\left(\frac{\pi}{2}\right)\sin\left(\delta t\right) \right] \sigma_{z} \; \; \left(\omega\ll\delta\right). \end{equation}
This can of course be trivially extended for a signal that consists of two frequencies.
{\bf{Data availability:}} The code and data used in this work are available on request to the corresponding author.
T
{\bf{Correspondence:}} Correspondence and request for materials should be addressed to: [email protected]
{\bf{ Acknowledgements:}} This project has received funding from the European Union Horizon 2020 research and innovation programme ERC grant QRES under grant agreement No 770929 and the collaborative European project ASTERIQS. T.G. is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Journal of
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{NoStop} \bibitem [{\citenamefont {Pham}\ \emph {et~al.}(2016)\citenamefont {Pham},
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\citenamefont {Sushkov}, \citenamefont {Bersin}, \citenamefont {Lee},
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\BibitemOpen
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\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {Casola}}, \bibinfo
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physical Review
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Du}}, \bibinfo {author}
{\bibfnamefont {L.~P.}\ \bibnamefont {McGuinness}}, \ and\ \bibinfo {author}
{\bibfnamefont {F.}~\bibnamefont {Jelezko}},\ }\href@noop {} {\bibfield
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{volume} {4}},\ \bibinfo {pages} {024004} (\bibinfo {year}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
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{357} (\bibinfo {year} {1981})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Braunstein}\ and\ \citenamefont
{Caves}(1994)}]{braunstein1994statistical}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~L.}\ \bibnamefont
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\BibitemOpen
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{37}},\ \bibinfo {pages} {984} (\bibinfo {year} {1989})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Rotem}\ \emph {et~al.}(2019)\citenamefont {Rotem},
\citenamefont {Gefen}, \citenamefont {Oviedo-Casado}, \citenamefont {Prior},
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{{\v{S}}afr{\'a}nek}(2017)}]{vsafranek2017discontinuities}
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\bibinfo {author} {\bibfnamefont {M.~G.}\ \bibnamefont {Paris}},\ }\href@noop
{} {\bibfield {journal} {\bibinfo {journal} {arXiv preprint
arXiv:1906.06185}\ } (\bibinfo {year} {2019})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Balasubramanian}\ \emph {et~al.}(2008)\citenamefont
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\citenamefont {Al-Hmoud}, \citenamefont {Tisler}, \citenamefont {Shin},
\citenamefont {Kim}, \citenamefont {Wojcik}, \citenamefont {Hemmer},
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\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Al-Hmoud}}, \bibinfo
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{\bibfnamefont {C.}~\bibnamefont {Shin}}, \bibinfo {author} {\bibfnamefont
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\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Krueger}}, \emph
{et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
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\citenamefont {Dr{\"a}benstedt}, \citenamefont {Tietz}, \citenamefont
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\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Wrachtrup}}, \ and\
\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Von~Borczyskowski}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Science}\ }\textbf
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{1997})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Maze}\ \emph {et~al.}(2008)\citenamefont {Maze},
\citenamefont {Stanwix}, \citenamefont {Hodges}, \citenamefont {Hong},
\citenamefont {Taylor}, \citenamefont {Cappellaro}, \citenamefont {Jiang},
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{author} {\bibfnamefont {S.}~\bibnamefont {Hong}}, \bibinfo {author}
{\bibfnamefont {J.}~\bibnamefont {Taylor}}, \bibinfo {author} {\bibfnamefont
{P.}~\bibnamefont {Cappellaro}}, \bibinfo {author} {\bibfnamefont
{L.}~\bibnamefont {Jiang}}, \bibinfo {author} {\bibfnamefont {M.~G.}\
\bibnamefont {Dutt}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
{Togan}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Zibrov}},
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{Nature}\ }\textbf {\bibinfo {volume} {455}},\ \bibinfo {pages} {644}
(\bibinfo {year} {2008})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Mamin}\ \emph {et~al.}(2013)\citenamefont {Mamin},
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{\bibfnamefont {C.}~\bibnamefont {Rettner}}, \bibinfo {author} {\bibfnamefont
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{Rugar}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
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(\bibinfo {year} {2013})}\BibitemShut {NoStop} \bibitem [{\citenamefont {M{\"u}ller}\ \emph {et~al.}(2014)\citenamefont
{M{\"u}ller}, \citenamefont {Kong}, \citenamefont {Cai}, \citenamefont
{Melentijevi{\'c}}, \citenamefont {Stacey}, \citenamefont {Markham},
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{Sushkov}, \citenamefont {Bar-Gill}, \citenamefont {Belthangady},
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{NoStop} \bibitem [{\citenamefont {Bucher}\ \emph {et~al.}(2018)\citenamefont {Bucher},
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\begin{widetext}
\appendix
\begin{center} {\bf Supplementary Material} \end{center}
\section{Conditions for quantum resolution} \label{sec:resolution_criterion} The following claim is stated in the main text: Given $\rho\left(\omega_{ \rm{r} }\right)$ such that $\frac{d\rho}{d\omega_{ \rm{r} }}=0$ (as $\omega_{ \rm{r} }\rightarrow0$), then $I_{{\rm r}}\left(\omega_{r}\rightarrow0\right)>0$ if and only if one of the eigenvalues of $\rho$ goes as $\omega_{ \rm{r} }^{k}$ for $1<k \leq 2$ , or equivalently if and only if $\frac{d\sqrt{\rho}}{d\omega_{ \rm{r} }}\neq0$. The optimal measurement basis converges to an eigenbasis of $\rho$ as $\omega_{ \rm{r} }\rightarrow0$. We showed in the main text that the first condition (at least one of the eigenvalues $\sim \omega_{ \rm{r} }^{1<k \leq 2}$) is sufficient and necessary.
First, let us clarify one point: $\frac{dp_{j}}{d\omega_{ \rm{r} }}$ is defined as the derivative of the j-th eigenvalue (at $\omega_{ \rm{r} }=0$), note that it equals the derivative of the probability of the j-th eigenstate (at $\omega_{ \rm{r} }=0$), to see this: \begin{equation}
\frac{d}{d\omega_{ \rm{r} }}\langle\psi_{j}|\rho|\psi_{j}\rangle|_{\omega_{ \rm{r} }=0}=\langle\psi_{j}\left(0\right)|\frac{d\rho}{d\omega_{ \rm{r} }}|\psi_{j}\left(0\right)\rangle+\langle\frac{d\psi_{j}}{d\omega_{ \rm{r} }}|\rho\left(0\right)|\psi_{j}\left(0\right)\rangle+\langle\psi_{j}\left(0\right)|\rho\left(0\right)|\frac{d\psi_{j}}{d\omega_{ \rm{r} }}\rangle=\langle\psi_{j}\left(0\right)|\frac{d\rho}{d\omega_{ \rm{r} }}|\psi_{j}\left(0\right)\rangle, \end{equation}
where $|\psi_{j}\left(\omega_{ \rm{r} }\right)\rangle$ is the $j$-th eigenstate of $\rho\left(\omega_{ \rm{r} }\right).$ The second equality is due to: $\langle\frac{d\psi_{j}}{d\omega_{ \rm{r} }}|\psi_{j}\rangle+\langle\psi_{j}|\frac{d\psi_{j}}{d\omega_{ \rm{r} }}\rangle=0.$ Therefore the eigenbasis of $\rho$ attains the QFI \Big($\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\omega_{ \rm{r} }}\right)^{2}}{p_{j}}$ \Big).
An alternative way to see that the eigenbasis is an optimal measurement basis proceeds as follows: \begin{equation} \frac{d\rho}{d\omega_{ \rm{r} }}=0\Rightarrow L\rho+\rho L=0, \end{equation}
where $L$ is the symmetric logarithmic derivative operator (its eigenbasis is the optimal measurement basis \cite{braunstein1994statistical}) . Note that the fact that $\rho,L$ anticommute impliess that $L\rho=\rho L=0,$ because given that $|l\rangle$ is an eigenstate of $L$ with an eigenvalue $l \neq 0,$ then: \begin{equation}
\langle l |\left(L\rho+\rho L\right)| l \rangle=\left(l+l^{*}\right)\langle l |\rho| l \rangle=0 \Rightarrow \langle l |\rho| l \rangle=0\Rightarrow\rho| l \rangle=0.
\end{equation}
So taking an eigenbasis of $L$: $\left\{ |l_{1}\rangle,...,|l_{n}\rangle\right\} $ with eigenvalues $l_{1},...,l_{n}.$ It can be seen that $\forall i\;\rho L|l_{i}\rangle=0$ and thus $L\rho=\rho L=0.$
Therefore they have a common eigenbasis and it attains the QFI.
We next show that the second condition is sufficient and necessary. First, one can see directly that the two conditions are equivalent. Using the same notations as in the main text, we can simply find $\frac{d\sqrt{\rho}}{d\omega_{ \rm{r} }}:$ \begin{equation}
\frac{d\sqrt{\rho}}{d\omega_{ \rm{r} }}=\underset{j}{\sum}\frac{\frac{dp_{j}}{d\omega_{ \rm{r} }}}{2\sqrt{p_{j}}}|j\rangle\langle j|+i\underset{j,k}{\sum}\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)h_{k,j}|k\rangle\langle j|. \label{derivative_rho} \end{equation} Since $\frac{d\rho}{d\omega_{ \rm{r} }}=0$ then $\left(p_{j}-p_{k}\right)h_{k,j}=0 \; (\forall k,j)$ therefore $\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)h_{k,j}=0.$ While $\frac{\frac{dp_{j}}{d\omega_{ \rm{r} }}}{2\sqrt{p_{j}}}\neq0$ if and only if $p_{j}\sim\omega_{ \rm{r} }^{k} \; \left( 1<k\leq2 \right).$ Therefore these conditions are equivalent.
In fact the more general statement is: for any $\rho\left(\theta\right)$ the QFI (about $\theta$) vanishes if and only if $\frac{d\sqrt{\rho}}{d\theta}=0.$
This fact is a simple coclusion of the following claim (which we prove):\\ {\bf{Claim:}} The QFI ($\mathcal{F}$) about $\theta$ satisfies: \begin{equation}
2 \, \text{trace}\left[\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}\right]\leq\mathcal{F}\leq4 \, \text{trace}\left[\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}\right]. \end{equation}
Proof: using supplementary equation \ref{derivative_rho} we get: \begin{equation}
\text{trace}\left[\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}\right]=\underset{j,k}{\sum}\left(\frac{d\sqrt{\rho}}{d\theta}\right)_{j,k}\left(\frac{d\sqrt{\rho}}{d\theta}\right)_{k,j}=\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\theta}\right)^{2}}{4p_{j}}+\underset{j,k}{\sum}\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)^{2}|h_{j,k}|^{2}. \end{equation} Recall that $\mathcal{F}$ reads: \begin{equation}
\mathcal{F}=\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\omega_{ \rm{r} }}\right)^{2}}{p_{j}}+2\underset{j,k}{\sum}\frac{\left(p_{j}-p_{k}\right)^{2}}{p_{j}+p_{k}}|h_{kj}|^{2} \end{equation} Now observe that: \begin{equation} \frac{\left(p_{j}-p_{k}\right)^{2}}{p_{j}+p_{k}}=\frac{\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)^{2}\left(\sqrt{p_{j}}+\sqrt{p_{k}}\right)^{2}}{p_{j}+p_{k}}=\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)^{2}\left[1+\frac{2\sqrt{p_{j}}\sqrt{p_{k}}}{p_{j}+p_{k}}\right]. \end{equation} Therefore: \begin{equation} \left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)^{2}\leq\frac{\left(p_{j}-p_{k}\right)^{2}}{p_{j}+p_{k}}\leq2\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)^{2}. \end{equation} So on one hand: \begin{equation}
4 \, \text{trace}\left[\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}\right]=\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\theta}\right)^{2}}{p_{j}}+4\underset{j,k}{\sum}\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)^{2}|h_{j,k}|^{2}\geq\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\omega_{ \rm{r} }}\right)^{2}}{p_{j}}+2\underset{j,k}{\sum}\frac{\left(p_{j}-p_{k}\right)^{2}}{p_{j}+p_{k}}|h_{kj}|^{2}=\mathcal{F}, \end{equation} and on the other hand: \begin{equation}
2 \text{trace}\left[\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}\right]=\underset{j}{\sum}\frac{\left(\frac{dp_{j}}{d\theta}\right)^{2}}{2p_{j}}+2\underset{j,k}{\sum}\left(\sqrt{p_{j}}-\sqrt{p_{k}}\right)^{2}|h_{j,k}|^{2} \leq \mathcal{F}. \end{equation} Combining the last two inequalities we get the desired inequality: $2 \, \text{trace}\left[\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}\right]\leq \mathcal{F} \leq 4 \, \text{trace}\left[\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}\right].$
Note that the lower bound is saturated for pure states (where only the quantum state is changed) and the upper bound for cases in which only the eigenvalues are changed.
{\bf{Conclusion:}} the QFI vanishes if and only if $\frac{d\sqrt{\rho}}{d\theta}$ vanishes.
The proof is immediate: $\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}$ is a positive semidefinite, Hermitian operator. Therefore the trace vanishes if and only if $\left(\frac{d\sqrt{\rho}}{d\theta}\right)^{2}$ vanishes, that vanishes if and only if $\frac{d\sqrt{\rho}}{d\theta}$ vanishes.
\section{Conditions for superresolution: multivariable case} \label{sec:multivariable_criterion} We wish to prove the condition for a non-singular QFI given that $\left(\frac{\partial\rho}{\partial \theta_{i}}\right)_{i=1}^{n}$ are linear dependent (with dimension $k<n$). Let us set the stage for the statement. We can choose the parameters $\left(\theta_{i}\right)_{i=1}^{n}$ such that $\left(\frac{\partial\rho}{\partial \theta_{i}}\right)_{i=1}^{k}$ are linear independent, and $\frac{\partial\rho}{\partial \theta_{k+1}}=...=\frac{\partial\rho}{\partial \theta_{n}}=0.$
{\bf{Definition:}} the classical FI matrix of $\rho$ is the FI matrix according to the eigenvalues of $\rho$ (namely the FI matrix achieved when measuring in the eigenbasis of $\rho$).
The claim is that the QFI matrix is non-singular if and only if the classical FI matrix about the subset $\left\{ \theta_{i}\right\} _{i=k+1}^{n}$ is non-singular.
To prove this claim we use some facts in quantum and classical estimation theory. Recall that the QFI matrix (denoted as $\mathcal{F}$) reads: \begin{equation} \mathcal{F}_{m,l}=2\underset{i,j}{\sum}\frac{\left(\frac{\partial\rho}{\partial \theta_{m}}\right)_{i,j}\left(\frac{\partial\rho}{\partial \theta_{l}}\right)_{j,i}}{\left(p_{i}+p_{j}\right)}, \end{equation} where $p_{j}$ is the j-th eigenvalue of $\rho,$ and the matrix elements are in the eigenbasis of $\rho.$ inserting supplementary equation \ref{derivative_rho} , we get that: \begin{equation} \mathcal{F}_{m,l}=\underset{j}{\sum}\frac{\left(\frac{\partial p_{j}}{\partial \theta_{m}}\right)\left(\frac{\partial p_{j}}{\partial \theta_{l}}\right)}{p_{j}}+\underset{i,j}{\sum}\frac{\left(p_{i}-p_{j}\right)^{2}}{p_{i}+p_{j}}\left(h_{i,j}^{m}h_{j,i}^{l}+h_{j,i}^{m}h_{i,j}^{l}\right), \end{equation} where $h^{m}$ is the Hermitian operator that corresponds to $\frac{\partial}{\partial \theta_{m}}.$ Note that just like in the single-variable case, the first term is the information that we gain from the change in the eigenvalues and the second term is the information that we gain from the change in the eigenvectors. The first term is thus the classical FI matrix (defined earlier), and is denoted from now on as $C.$ The second term can be thought of as the quantum part of the QFI, and is denoted from now on as $Q.$ For convenience let us split $C$ and $Q$ into blocks according to $\left\{ \theta_{i}\right\} _{i=1}^{k}$ and $\left\{ \theta_{i}\right\} _{i=k+1}^{n}$ : \begin{equation} C=\left(\begin{array}{cc} C^{11} & C^{12}\\ C^{21} & C^{22} \end{array}\right),\;Q=\left(\begin{array}{cc} Q^{11} & Q^{12}\\ Q^{21} & Q^{22} \end{array}\right) , \end{equation} where $C^{11}$ is the classical FI about $\left\{ \theta_{i}\right\} _{i=1}^{k},$ $C^{22}$ is the classical FI about $\left\{ \theta_{i}\right\} _{i=k+1}^{n}$ (and analogously for Q).
{\bf{Claim 1:}} Given that $\frac{\partial\rho}{\partial \theta_{k+1}}=...=\frac{\partial\rho}{\partial \theta_{n}}=0,$ then $Q^{22}=0,\,Q^{12}=Q^{21}=0.$
proof: Just like in the single-variable case, $\frac{\partial\rho}{\partial \theta_{m}}=0$ implies ($\forall i,j$) $\left(p_{i}-p_{j}\right)h_{i,j}^{m}=0$ and therefore ($\forall l$) $\frac{\left(p_{i}-p_{j}\right)^{2}}{p_{i}+p_{j}} h_{i,j}^{m}h_{j,i}^{l}=0$ (because $\frac{\left(p_{i}-p_{j}\right)^{2}}{p_{i}+p_{j}}|h_{i,j}^{m}|\leq\left(p_{i}-p_{j}\right)|h_{i,j}^{m}|\rightarrow0$). This implies that $Q^{22},Q^{12},Q^{21}$ vanish.
{\bf{Claim 2:}} Given that $C^{22}$ is singular, then $C$ is singular, and vectors that nullify $C^{22}$ nullify also $C$.
proof: Recall that a (classical) FI matrix is defined as: \begin{eqnarray} \begin{split} &I_{m,l}=\underset{i}{\sum}\frac{\left(\frac{\partial p_{i}}{\partial \theta_{m}}\right)\left(\frac{\partial p}{\partial \theta_{l}}\right)}{p_{i}}=4\underset{i}{\sum}\left(\frac{\partial\sqrt{p_{i}}}{\partial \theta_{m}}\right)\left(\frac{\partial\sqrt{p_{i}}}{\partial \theta_{l}}\right)\\ &=\langle\frac{\partial\sqrt{p}}{\partial \theta_{m}},\frac{\partial\sqrt{p}}{\partial \theta_{l}}\rangle. \end{split}
\end{eqnarray}
So it is an inner product matrix between the vectors $\left\{ \frac{\partial\sqrt{p}}{\partial \theta_{m}}\right\} _{m=1}^{n}.$
Hence it is regular if and only if these vectors are linear independent, and the null-space is all the linear combinations of these vectors that vanish.
Therefore, if $C^{22}$ is singular then $\left\{ \frac{\partial\sqrt{p}}{\partial \theta_{m}}\right\} _{m=k+1}^{n}$ is linear dependent which implies that $\left\{ \frac{\partial\sqrt{p}}{\partial \theta_{m}}\right\} _{m=1}^{n}$ is linear dependent,
and the null-space of $C^{22}$ is a subspace of the null-space of $C.$
This immediately leads to the desired conclusion:
{\bf{Conclusion:}} Given that $\left(\frac{\partial\rho}{\partial \theta_{i}}\right)_{i=1}^{k}$ are linear independent and $\frac{\partial\rho}{\partial \theta_{k+1}}=...=\frac{\partial\rho}{\partial \theta_{n}}=0$ (the problematic parameters), then the QFI matrix is regular if and only if $C^{22}$ (the classical FI about the problematic parameters) is regular.
Proof: We first show that if $C^{22}$ is singular then the QFI is singular. $C^{22}$ is singular and is thus nullified by a vector $\vect{\alpha}$. From claim 2, this $\vect{\alpha}$ nullifies also $C,$ and from claim 1 it nullifies also $Q.$ Therefore the QFI matrix is nullified by $\vect{\alpha}$ and is thus singular.
We now show that if the QFI is singular then $C^{22}$ is singular. Given that the QFI is nullified by $\vect{\alpha},$ then $2\underset{i,j}{\sum}\frac{|\bm{\alpha}\cdot\left( \bm {\partial_{\theta} \rho } \right)_{i,j}|^{2}}{\left(p_{i}+p_{j}\right)}=0.$ and therefore $\vect {\alpha}\cdot\left( \bm{ \partial_{\theta} \rho } \right)=0.$ This means that $\alpha_{1}=...=\alpha_{k}=0,$ namely this vector is a linear combination of only the problematic parameters and thus $C^{22} \vect {\alpha}=0.$ Hence $C^{22}$ is singular. $\square$
As it is mentioned in a footnote, one can formulate an equivalent condition. Recall that in the single variable case the QFI is positive $\Leftrightarrow$ $\frac{d\sqrt{\rho}}{d\theta}\neq0.$ An immediate conclusion of this is that in the multivariable case the QFI matrix is non-singular $\Leftrightarrow$ $\left\{ \frac{\partial\sqrt{\rho}}{\partial\theta_{i}}\right\} _{i=1}^{n}$ are linear independent: the QFI matrix is singular $\Leftrightarrow$ there exist a parameter $y,$ a linear combination of $\left(\theta_{i}\right)_{i},$ such that the (single-variable) QFI about $y$ vanishes $\Leftrightarrow$ $\frac{d\sqrt{\rho}}{dy}=0$ $\Leftrightarrow$ $\left\{ \frac{\partial\sqrt{\rho}}{\partial\theta_{i}}\right\} _{i=1}^{n}$ are linear dependent.
\section{Relation to quantum superresolution in imaging} \label{sec:imaging} In this part we revisit the recent superresolution scheme proposed in \cite{tsang2016quantum} and show that it is a special case of the criterion presented in the main text. In the imaging problem one has two close incoherent optical sources, located in $x_{1},x_{2},$ and the goal is to determine the number of sources (two or one) and estimate their positions. The probe in this problem is the radiation emitted from the sources (detected by a measurement device). In the far-field limit, all terms higher than single photon terms can be neglected, so that the state of the radiation reads: \begin{equation}
\rho=\left(1-\epsilon\right)|\text{vac}\rangle\langle\text{vac}|+\epsilon\rho_{1}+O\left(\epsilon^{2}\right),\;\text{where}\;\rho_{1}=\frac{1}{2}\left(|\psi_{1}\rangle\langle\psi_{1}|+|\psi_{2}\rangle\langle\psi_{2}|\right), \end{equation}
and $|\psi_{j}\rangle=\int\psi_{j}\left(x\right)|1,x\rangle\,dx$ ($|1,x\rangle$ is the state of one photon in position $x$).
$|\psi_{j}\left(x\right)\rangle$ is the photonic wave function corresponding to the $j$-th source, we consider the symmetric case in which $\psi_{j}\left(x\right)=\psi\left(x-x_{j}\right)$ where $x_j$ is the position of the j-th source.
We can now define the parameters $\theta_{1}=\frac{1}{2}\left(x_{1}+x_{2}\right)$ (the centroid, equivalent to $\omega_{s}$ in our case), and $\theta_{2}=x_{1}-x_{2}$ (the distance between sources, equivalent to $\omega_{r}$ in our case). Replacing $x_{1}\longleftrightarrow x_{2}$ leads to $\psi_{1}\left(x\right)\longleftrightarrow\psi_{2}\left(x\right)$ which does not change $\rho.$ Hence in this case, $\rho$ is symmetric with respect to the parameter $\theta_{2}$ namely $\rho\left( \theta_{2} \right)=\rho\left( -\theta_{2} \right),$ and thus: \begin{equation} \frac{\partial\rho}{\partial\theta_{2}}\rightarrow0,\;\theta_{2}\rightarrow0. \end{equation} Therefore $\rho$ suffers from a vanishing distinguishability.
According to the criterion in the main text, the optimal measurement basis would converge to an eigenbasis of $\rho$ and a finite FI about $\theta_{2}$ can be achieved if and only if one of the eigenvalues $\rightarrow 0$ as $\theta_{2}^{2}.$
Observe that the eigenstates of $\rho$ (in the subspace of one photon states) are $|\psi_{1}\rangle\pm\frac{\langle\psi_{2}|\psi_{1}\rangle}{|\langle\psi_{2}|\psi_{1}\rangle|}|\psi_{2}\rangle$ with eigenvalues $\frac{\epsilon}{2}\left(1\pm|\langle\psi_{2}|\psi_{1}\rangle|\right)$ respectively. Therefore the condition is satisfied given that $1-|\langle\psi_{1}|\psi_{2}\rangle| \sim \theta_{2}^2$ as $\theta_{2}\rightarrow0,$ which is the case for a wide variety of $\psi\left(x\right)$ (e.g. Gaussian, sinc functions and many more).
Hence the superresolution method in this case, as proposed in \cite{tsang2016quantum} and according to the criterion in the main text, is to measure whether the one photon state is in $|\psi\left(x\right)\rangle.$ The probability of not being in this state goes as $\theta_{2}^{2}$ and thus a finite FI is achieved in the limit of $\theta_{2}\rightarrow0.$
Note that given a symmetric pure state (unnormalized): $|\psi_{1}\rangle+|\psi_{2}\rangle$ resolution cannot be achieved (due to purity) just like the resolution limit in spectroscopy given a coherent symmetric signal. The state $\frac{1}{2}\left(|\psi_{1}\rangle\langle\psi_{1}|+|\psi_{2}\rangle\langle\psi_{2}|\right)$ is the ensemble average of states with random phases:
$|\psi_{1}\rangle+e^{i\phi}|\psi_{2}\rangle,$ which is similar to averaging over many realizations of random phase signals that we make in spectroscopy.
\section{Resolution limitations in quantum spectroscopy} \label{sec:spectroscopy} It is shown in the main text that given the following Hamiltonian (identical quadratures): \begin{equation} H=\left[\underset{i}{\sum}A\cos\left(\omega_{i}t\right)+B\sin\left(\omega_{i}t\right)\right]\sigma_{z}=\left[\underset{i}{\sum}\Omega\sin\left(\omega_{i}t+\varphi\right)\right]\sigma_{z} , \label{Hamiltonian2} \end{equation} $(\Omega=\sqrt{A^{2}+B^{2}}, \varphi=\arctan\left(\frac{B}{A}\right)),$ then:
\begin{equation}
\frac{\partial|\psi_{t}\rangle}{\partial\omega_{ \rm{r} }}=0\Rightarrow I_{ \rm{r} }=0\:\left(\omega_{ \rm{r} }=0\right). \end{equation} Note that a similar limitation appears also in the case of identical phases but different amplitudes: \begin{equation} H=\underset{i}{\sum}\Omega_{i}\sin\left(\omega_{i}t+\varphi\right)\sigma_{z}. \end{equation} Of course, $\frac{\partial H}{\partial\omega_{ \rm{r} }}\neq0,$ however neither $\omega_{ \rm{s} }$ nor $\omega_{ \rm{r} }$ can be efficiently estimated: To see this observe that: \begin{equation} \forall t\;\Omega_{2}\frac{\partial H}{\partial\omega_{1}}=\Omega_{1}\frac{\partial H}{\partial\omega_{2}}\;\left(\omega_{ \rm{r} }=0\right). \label{limitation_1} \end{equation} So we can define $\omega_{-}=\frac{1}{\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}}\left(\Omega_{2}\omega_{1}-\Omega_{1}\omega_{2}\right),\:\omega_{+}=\frac{1}{\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}}\left(\Omega_{1}\omega_{1}+\Omega_{2}\omega_{2}\right),$ and then supplementary equation \ref{limitation_1} implies that $\forall t\;\frac{\partial H}{\partial\omega_{-}}=0\,\left(\omega_{ \rm{r} }=0\right),$ and thus $\Delta\omega_{ \rm{r} }\rightarrow\infty.$
Now consider the most general case: amplitudes and phases are not necessarily identical: \begin{equation} H=\underset{i}{\sum}\Omega_{i}\sin\left(\omega_{i}t+\varphi_{i}\right)\sigma_{z}. \end{equation} Given $\omega_{ \rm{r} }t\ll1,$ the Hamiltonian reads: \begin{equation} H=\sigma_{z}\left[\underset{i}{\sum}\Omega_{i}\sin\left(\omega_{ \rm{s} }t+\varphi_{i}\right)+\omega_{ \rm{r} }t\underset{i}{\sum}\Omega_{i}\cos\left(\omega_{ \rm{s} }t+\varphi_{i}\right)+\mathcal{O}\left(\omega_{ \rm{r} }^{2}t^{2}\right)\right], \end{equation} so neglecting the $\left(\omega_{ \rm{r} }t\right)^{2}$ terms the Hamiltonian can be written as: \begin{equation} H\approx\left[a\sin\left(\omega_{ \rm{s} }t+\alpha\right)+b\omega_{ \rm{r} }t\sin\left(\omega_{ \rm{s} }t+\beta\right)\right]\sigma_{z}, \label{degeneracy} \end{equation} where $a\sin\left(\omega_{ \rm{s} }t+\alpha\right)=\underset{i}{\sum}\Omega_{i}\sin\left(\omega_{ \rm{s} }t+\varphi_{i}\right)$, and $b\sin\left(\omega_{ \rm{s} }t+\beta\right)=\underset{i}{\sum}\Omega_{i}\cos\left(\omega_{ \rm{s} }t+\varphi_{i}\right)$. It would be more convenient then to work with the parameters $\omega_{ \rm{r} },\omega_{ \rm{s} },a,b,\alpha,\beta$ (instead of $\omega_{1},\omega_{2},\Omega_{1},\Omega_{2},\phi_{1},\phi_{2}$). Supplementary eq. \ref{degeneracy} immediately implies that the Hamiltonian suffers from a degeneracy, i.e. it depends only on $5$ parameters: $a,\alpha, \omega_{ \rm{s} }, b\omega_{ \rm{r} }, \beta.$ Namely we cannot get information on $b,$ $\omega_{ \rm{r} }$ separately, but only on $b\omega_{ \rm{r} }.$ A different way to phrase this is that $-\omega_{ \rm{r} }\frac{\partial H}{\partial\omega_{ \rm{r} }}+b\frac{\partial H}{\partial b}=0,$hence there exists a parameter $g$ such that $\forall t\;\frac{\partial H}{\partial g}=0.$ More elaborately we can see explicitly that $\text{span}\left(\left\{ \nabla f\left(t\right)\right\} _{t}\right)$ is of dimension $\leq5$ (where $H=f\left(t\right)\, \sigma_{z}$): \begin{eqnarray} \begin{split} &\nabla f\left(t\right)=\left(\begin{array}{c} \frac{\partial f}{\partial\omega_{ \rm{s} }}\\ \frac{\partial f}{\partial\omega_{ \rm{r} }}\\ \frac{\partial f}{\partial a}\\ \frac{\partial f}{\partial b}\\ \frac{\partial f}{\partial\alpha}\\ \frac{\partial f}{\partial\beta} \end{array}\right)=\left(\begin{array}{c} at\cos\left(\omega_{ \rm{s} }t+\alpha\right)+b\omega_{ \rm{r} }t^{2}\cos\left(\omega_{ \rm{s} }t+\beta\right)\\ bt\sin\left(\omega_{ \rm{s} }t+\beta\right)\\ \sin\left(\omega_{ \rm{s} }t+\alpha\right)\\ \omega_{ \rm{r} }t\sin\left(\omega_{ \rm{s} }t+\beta\right)\\ a\cos\left(\omega_{ \rm{s} }t+\alpha\right)\\ b\omega_{ \rm{r} }t\cos\left(\omega_{ \rm{r} }t+\beta\right) \end{array}\right) \\ & =t\cos\left(\omega_{ \rm{s} }t\right)\left(\begin{array}{c} a\cos\left(\alpha\right)\\ b\sin\left(\beta\right)\\ 0\\ \omega_{ \rm{r} }\sin\left(\beta\right)\\ 0\\ b\omega_{ \rm{r} }\cos\left(\beta\right) \end{array}\right)+t\sin\left(\omega_{ \rm{s} }t\right)\left(\begin{array}{c} -a\sin\left(\alpha\right)\\ b\cos\left(\beta\right)\\ 0\\ \omega_{ \rm{r} }\cos\left(\beta\right)\\ 0\\ -b\omega_{ \rm{r} }\sin\left(\beta\right) \end{array}\right)+\cos\left(\omega_{ \rm{s} }t\right)\left(\begin{array}{c} 0\\ 0\\ \sin\left(\alpha\right)\\ 0\\ a\cos\left(\alpha\right)\\ 0 \end{array}\right)+\sin\left(\omega_{ \rm{s} }t\right)\left(\begin{array}{c} 0\\ 0\\ \cos\left(\alpha\right)\\ 0\\ -a\sin\left(\alpha\right)\\ \\ \end{array}\right)+b\omega_{ \rm{r} }t^{2}\left(\begin{array}{c} 1\\ 0\\ 0\\ 0\\ 0\\ 0 \end{array}\right). \end{split} \end{eqnarray} Hence for $\omega_{ \rm{r} }\neq0,$ the dimension is $5.$ Note that for $\omega_{ \rm{r} }=0$ the dimension is $4$ (as $\frac{\partial H}{\partial b}=0,\frac{\partial H}{\partial\beta}=0$). Therefore in this case as well $\Delta \omega_{ \rm{r} } \rightarrow \infty.$
\section{Effective Hamiltonian: Conditions for a non vanishing FI} \label{effective_Hamiltonian}
The exact expression of the accumulated phase for a single frequency signal (given in the methods section) reads: \begin{equation} \phi=A\sin\left(\delta t\right)\frac{\tan\left(\omega\frac{\tau}{2}\right)}{\omega}+B\left(1-\cos\left(\delta t\right)\right)\frac{\tan\left(\omega\frac{\tau}{2}\right)}{\omega}. \end{equation} Based on this expression, we want to understand what the conditions are on $\delta$ (or equivalently $\tau$) to nullify $\phi$ for every $A,B.$ Clearly, whenever $\omega \tau \neq n \pi$ and $\delta t= 2 \pi k$ ($n,k$ are integers) $\phi=0.$ Note that the condition $\delta t=2 \pi k$ implies that $\omega t=\pi m$ ($k,m$ are integers, because the total time is an integer multiple of $\frac{\pi}{\omega+\delta}$).
In addition, whenever $\omega \tau =2 \pi n$ we have that $\tan\left(\omega\frac{\tau}{2}\right)=0$ and thus $\phi=0.$ It is simple to understand this case. The signal completes an integer number of cycles between two consecutive pulses and thus the accumulated phase vanishes. Note that for $\omega\tau=\left(2n+1\right)\pi,$ it is not possible to nullify $\phi$ for every $A,B.$ Therefore the condition $\delta t= 2 \pi k$ is not valid here. In this case: $\phi=A\left(\frac{2}{\pi}\right)\frac{t}{2n+1},$ therefore $A\neq0\Rightarrow\phi\neq0.$ For a two frequency signal the accumulated phase is: \begin{equation} H=\underset{i}{\sum}\left[A_{i}\sin\left(\omega_{i}t\right)+B_{i}\cos\left(\omega_{i}t\right)\right]h\left(t\right)\sigma_{z}. \end{equation} In order to get a non vanishing $I_{ \rm{r} }$ when $\omega_{ \rm{r} }=0$ we have to nullify $\phi$ (at $\omega_{ \rm{r} }=0$). As shown above we need to either take $\delta_{s}t=2 \pi k$ or take $\omega_{ \rm{s} } \tau= 2 \pi n.$ For the first possibility, given that $\omega_{ \rm{r} } t \ll 1,$ the accumulated phase reads: \begin{equation} \phi=\underset{i}{\sum}\frac{\tan\left(\omega_{i}\frac{\tau}{2}\right)}{\omega_{i}}\left[A_{i}\sin\left(\delta_{i}t\right)+B_{i}\left(1-\cos\left(\delta_{i}t\right)\right)\right], \end{equation} and thus a finite $I_{ \rm{r} }$ is obtained: \begin{equation} I_{ \rm{r} }=8\sigma^{2}\frac{\tan\left(\omega_{ \rm{s} }\frac{\tau}{2}\right)^{2}}{\omega_{ \rm{s} }^{2}}t^{2}. \label{generalized_fi} \end{equation} Note that this expression is not valid for $\omega_{ \rm{s} }\tau=\left(2n+1\right)\pi,$ as for these values $\phi\neq0\Rightarrow I_{ \rm{r} } = 0.$ For the second possibility ($\omega_{ \rm{s} } \tau= 2 \pi n$): Note that this option has an overlap with the first one, in that if $\frac{t}{\tau}$ is even then $\delta t$ is an integer multiplication of $2\pi.$ However it can be seen that in this case $\phi\sim\omega_{ \rm{r} }^{2}\Rightarrow p\sim\omega_{ \rm{r} }^{4}$ and thus $I_{ \rm{r} }$ vanishes. If $\frac{t}{\tau}$ is odd (and $\omega_{ \rm{s} } \tau= 2 \pi n$) then: $\phi\approx\frac{\left(B_{1}-B_{2}\right)\tau^{2}}{2\pi n}\omega_{ \rm{r} }\Rightarrow I_{ \rm{r} }=\frac{8\sigma^{2}\tau^{4}}{\left(2\pi\right)^{2}n^{2}}.$ Hence a finite $I_{ \rm{r} }$ is achieved but it is much lower than with the first option, as it independent of $t,$ and thus does not grow with $t.$ Therefore we dismiss this option and only keep the first one, in which $\delta_{s}t=2\pi k\;\left(\omega_{ \rm{s} }\tau\neq n\pi\right),$ and the FI is given in supplementary equation \ref{generalized_fi}.
Naturally we would like to confirm that the optimal detuning (or $\tau$) is $\delta_{s}= \pm 2 \pi/t.$ To see this, let us first examine how close $\omega_{ \rm{s} } \tau$ can approach $\pi$ (while requiring $\delta_{s}t=2\pi k\neq0$): \begin{equation} \omega\tau=\left(\omega+\delta\right)\tau-\delta\tau=\pi-\delta\tau. \end{equation} Hence the closest it can approach $\pi$ is by $\delta \tau.$ Since the minimal possible $\delta$ is $\frac{2 \pi}{t}$,we cannot get closer to $\pi$ than $\frac{2 \pi \tau}{t}=\frac{2 \pi}{N}$ (where $N$ is the number of pulses). Therefore if the closest we can get to $\pi$ is $\frac{2 \pi}{N}\approx\frac{2\pi^{2}}{\omega t},$ then the closest we can get to $3\pi$ is $\approx\frac{6\pi^{2}}{\omega t},$ and so on. Therefore the optimal $I_{ \rm{r} }$ is achieved with $\delta_{s}=\pm\frac{2\pi}{t}$ and reads: \begin{equation} I_{ \rm{r} }=8\sigma^{2}\frac{\tan\left(\frac{\pi}{2\left(1\pm\frac{2\pi}{\omega_{ \rm{s} }t}\right)}\right)^{2}}{\omega_{ \rm{s} }^{2}}t^{2}\approx\frac{8}{\pi^{4}}\sigma^{2}t^{4}. \end{equation} It is well established that for frequency estimation problems the optimal scaling of the FI is $\sim \Omega^{2}t^{4}$\cite{pang2017optimal, schmitt2017submillihertz, jordan2017classical, yang2017quantum, gefen2017control, naghiloo2017achievingT4}, where $\Omega$ stands for the amplitude of the signal, and this is exactly the scaling that this method achieves. The optimality of this method is discussed in supplementary note \ref{optimality}.
\section{Optimality analysis} \label{optimality} We presented control methods for which $I_{ \rm{r} }\neq0$ , and found the optimal one out of a set of possible controls. However a valid question is whether this method is optimal out of all possible control strategies. This question is left open; however, we can find an upper bound of $I_{ \rm{r} }$ (which is not tight). Given a Hamiltonian: \begin{equation} H=\underset{i}{\sum}\left(A_{i}\cos\left(\omega_{i}t\right)+B_{i}\sin\left(\omega_{i}t\right)\right)\sigma_{Z} \end{equation} The optimal $I_{ \rm{r} }$ (FI about $\omega_{ \rm{r} }$) is given by (according to supplementary ref. \cite{pang2017optimal}): \begin{equation} I_{ \rm{r} }=4\left[\int \left\vert \frac{\partial H}{\partial\omega_{ \rm{r} }} \right\vert \:dt\right]^{2}, \end{equation}
where $|\bullet|$ stands for the operator norm (in general it is the maximal eigenvalue minus the minimal; for $H\propto\sigma_{\theta}$ it can be written in this way). For $\omega_{ \rm{r} }=0$: \begin{equation} \frac{\partial H}{\partial\omega_{ \rm{r} }}=\left[-\left(A_{1}-A_{2}\right)t\sin\left(\omega_{ \rm{s} }t\right)+\left(B_{1}-B_{2}\right)t\cos\left(\omega_{ \rm{s} }t\right)\right]\sigma_{Z}. \end{equation} Therefore given that $\omega_{ \rm{s} }t\gg1$, the maximal $I_{ \rm{r} }$ reads: \begin{equation} I_{ \rm{r} }=\left(\frac{2}{\pi}\right)^{2}\left[\left(A_{1}-A_{2}\right)^{2}+\left(B_{1}-B_{2}\right)^{2}\right] t^{4}. \end{equation}
Therefore given that $A_{i},B_{i}$ are i.i.d. with variance $\sigma^{2}$ an upper bound for the average $I_{ \rm{r} }$ is: \begin{equation} I_{ \rm{r} }\leq\frac{16}{\pi^{2}}\sigma^{2}t^{4}. \end{equation}
Given $A_{i},B_{i}$ the control that achieves the optimal $I_{ \rm{r} }$ consists of applying $\pi$-pulses whenever $\frac{\partial H}{\partial\omega_{ \rm{r} }}$ flips a sign; therefore it requires knowing $A_{i},B_{i}$, which is unrealistic in the setting described in the paper. In practice we need to apply the same control to every realization of $A_{i},B_{i}$, hence this upper bound is not achievable. Using the method presented in the paper we obtain $I_{ \rm{r} }=\frac{8\sigma^{2}t^{4}}{\pi^{4}}$ , hence lower by a factor of $2\pi^{2}$ from this upper bound. Whether our method is optimal given this noise model is left as an open question.
\section{Multiparameter estimation of all the parameters} \label{sec:multiparameter_estimation} In an actual experimental scenario all the parameters are unknown, and since the estimation protocol of $\omega_{ \rm{r} }$ depends on $\omega_{ \rm{s} }$ (the pulses frequency should be detuned from $\omega_{ \rm{s} }$ by $\frac{2\pi}{\omega_{ \rm{s} }}$), a preliminary estimation of $\omega_{ \rm{s} }$ is necessary.
We propose the following protocol: For a preliminary estimation of all the parameters ($\omega_{ \rm{r} },\omega_{ \rm{s} },\sigma$) the traditional protocol is applied; namely vary the pulses frequency (denoted as $\omega_{p}$) and make a large number of measurements for each $\omega_{p}.$ The next step is to fit the obtained data points (or make MLE analysis). This should provide a good estimation of $\omega_{ \rm{s} }$ and $\sigma,$ the estimation of $\omega_{ \rm{r} }$ however will not be good enough (unless by chance we hit very close to the resonance points, and perform enough measurements in this point). The performance of this estimation method is illustrated in supplementary fig. \ref{preliminary_estimation}.
\begin{figure}\label{preliminary_estimation}
\end{figure}
The next step is to use the estimated $\omega_{ \rm{s} },\sigma$ to apply our method. One can treat $\omega_{ \rm{s} },\sigma$ as known and estimate $\omega_{ \rm{r} },$ however this will create a bias (we also want to progressively improve the estimation of $\omega_{ \rm{s} },$ such that the detuning of the pulses will be more accurate). Note that we cannot get information about three different parameters by measuring copies of the same density matrix (even if measuring different observables, the FI matrix will be singular). Therefore at least three different measurements are required: one measurement with $\delta_{s}t=2\pi,$ the resonance condition for estimation of $\omega_{ \rm{r} }$ and two other measurements with the optimal detunings for estimating $\omega_{ \rm{s} },\sigma.$ The FI about $\omega_{ \rm{s} },\sigma$ as a function of $\delta_{s}$ is shown in supplementary fig. \ref{fi_other_parameters}. Quite interestingly for both $\omega_{ \rm{s} },\sigma$ the optimal $\delta_{s}\rightarrow\frac{2\pi}{t}$ as $\sigma\rightarrow\infty$ (this is because as $\sigma$ becomes larger the exponential decay becomes stronger and one needs to get closer to $\delta_{s} t=2 \pi$).
The optimal FI about $\omega_{ \rm{s} }$ scales as $\sigma^{2}t^{4}$ and is comparable to the optimal FI about $\omega_{ \rm{r} }.$ The optimal FI about $\sigma$ behaves in an unusual manner: Usually the FI about the amplitude grows as $t^2,$ while here the optimum (for $\sigma t >1$) is $I_{\sigma}\sim\frac{0.63}{\sigma^{2}}.$ It does not depend on $t,$ and it drops as $\sigma$ gets larger. This behavior is somewhat similar to sensing the standard deviation of the amplitude of a stationary signal ($H=A\sigma_{Z}$ where $A\sim N\left(0,\sigma\right)$).
Since we are dealing with a multivariable estimation, the Cram{\'e}r-Rao bound is given by the Fisher information matrix in the following way \cite{cramer2016mathematical}: $\left(\Delta x\right)^{2}=\left(I^{-1}\right)_{x,x}.$ We would like then to calculate the FI matrix. The full expression of the matrix is quite involved, however we can easily observe that as $\omega_{\rm{r}} \rightarrow 0 $ the FI matrix converges to a block diagonal matrix, with $I_{\rm{r}}$ as one of its eigenvalues. The FI matrix per three measurements is the sum of the FI matrices of each measurement: $I^{\left(1\right)}+I^{\left(2\right)}+I^{\left(3\right)}$. Therefore the FI matrix per single measurement reads: $I=\frac{1}{3}\left(I^{\left(1\right)}+I^{\left(2\right)}+I^{\left(3\right)}\right).$ Denoting the FI matrix that corresponds to $\delta_{\rm{s}} t= 2 \pi$ as $I^{\left(1\right)}$, we observe that: \begin{equation} \frac{\left(\frac{\partial p}{\partial\omega_{{\rm s}}}\right)^{2}}{p\left(1-p\right)},\frac{\left(\frac{\partial p}{\partial\sigma}\right)^{2}}{p\left(1-p\right)},\frac{\left(\frac{\partial p}{\partial\sigma}\right)\left(\frac{\partial p}{\partial\omega_{{\rm r}}}\right)}{p\left(1-p\right)},\frac{\left(\frac{\partial p}{\partial\omega_{{\rm s}}}\right)\left(\frac{\partial p}{\partial\omega_{{\rm r}}}\right)}{p\left(1-p\right)}\rightarrow0\Longrightarrow I^{\left(1\right)}=\left(\begin{array}{ccc} I_{{\rm r}} & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right) \end{equation} Regarding $I^{\left(2\right)},I^{\left(3\right)}$, note that for them (since $\delta_{s}t\neq2\pi n$) we have that: \begin{equation} \frac{\left(\frac{\partial p}{\partial\omega_{{\rm r}}}\right)^{2}}{p\left(1-p\right)},\frac{\left(\frac{\partial p}{\partial\omega_{{\rm r}}}\right)\left(\frac{\partial p}{\partial\omega_{{\rm s}}}\right)}{p\left(1-p\right)},\frac{\left(\frac{\partial p}{\partial\omega_{{\rm r}}}\right)\left(\frac{\partial p}{\partial\sigma}\right)}{p\left(1-p\right)}\rightarrow0\Longrightarrow I^{\left(j\right)}=\left(\begin{array}{ccc} 0 & 0 & 0\\ 0 & I_{2,2}^{\left(j\right)} & I_{2,3}^{\left(j\right)}\\ 0 & I_{3,2}^{\left(j\right)} & I_{3,3}^{\left(j\right)} \end{array}\right). \end{equation} Therefore the FI matrix per single measurement reads: \begin{equation} I=\frac{1}{3} \left(\begin{array}{ccc} I_{{\rm r}} & 0 & 0\\ 0 & I_{2,2} & I_{2,3}\\ 0 & I_{3,2} & I_{3,3} \end{array}\right), \end{equation} and thus $\Delta \omega_{ \rm{r} }=\sqrt{ \frac{ 3 }{ I_{r} } }$, we therefore get an extra factor of $\sqrt{3}$ due to these chunks of three measurements. The standard deviation obtained in practice (with MLE) is a bit above the expected analytical values, as can be seen in supplementary fig. \ref{multiparameter_estimation}.
\begin{figure}\label{fi_other_parameters}
\end{figure}
\begin{figure}\label{multiparameter_estimation}
\end{figure}
\section{Limitation due to incoherence} \label{sec:incoherence} In this part we derive the effect of incoherence of the signal (during the measurement) on the method, where we consider a model in which the quadratures undergo identical and independent OU process. The OU process is defined as $dA_{i}=-\gamma A_{i}\,dt+\sigma_{n}\,dW_{t}^{A_{i}},$ and similarly for $B_{i}.$ To get the effect, we need to calculate the transition probability $p=\langle\sin\left(\phi\right)^{2}\rangle ,$ where $\phi$ is the accumulated phase. Note that: \begin{equation} \phi=\overset{2}{\underset{i=1}{\sum}}\overset{T}{\underset{0}{\int}}\left(A_{i}\left(t\right)\cos\left(\omega_{i}t\right)+B_{i}\left(t\right)\sin\left(\omega_{i}t\right)\right)\:dt, \end{equation} where $A_{i}\left(t\right)=A_{i}\left(0\right)\exp\left(-\gamma t\right)+\sigma_{n}\overset{t}{\underset{0}{\int}}e^{-\gamma\left(t-s\right)}dW_{s}^{A_{i}},$ and the same holds for $B_{i}\left(t\right).$ Therefore $\phi$ can be written as the sum $\phi=\phi_{av}+\phi_{n},$ where: \begin{equation} \phi_{av}=\overset{2}{\underset{i=1}{\sum}}\overset{T}{\underset{0}{\int}}\left(A_{i}\left(0\right)e^{-\gamma t}\cos\left(\omega_{i}t\right)+B_{i}\left(0\right)e^{-\gamma t}\sin\left(\omega_{i}t\right)\right)\:dt, \end{equation} and: \begin{equation} \phi_{n}=\overset{2}{\underset{i=1}{\sum}}\sigma_{n}\overset{T}{\underset{0}{\int}}dt\:\left(\cos\left(\omega_{i}t\right)\underset{0}{\overset{t}{\int}}e^{-\gamma\left(t-s\right)}dW_{s}^{A_{i}}+\sin\left(\omega_{i}t\right)\underset{0}{\overset{t}{\int}}e^{-\gamma\left(t-s\right)}dW_{s}^{B_{i}}\right). \end{equation} Observe that $\phi_{n}$ is a Gaussian random variable with $\langle\phi_{n}\rangle=0,$ and: \begin{equation} \langle\phi_{n}^{2}\rangle=\overset{2}{\underset{i=1}{\sum}}\sigma_{n}^{2}\overset{T}{\underset{0}{\int}}ds\left[\underset{s}{\overset{T}{\int}}\cos\left(\omega_{i}t\right)e^{-\gamma\left(t-s\right)}\:dt\right]^{2}+\underset{i}{\sum}\sigma_{n}^{2}\overset{T}{\underset{0}{\int}}ds\left[\underset{s}{\overset{T}{\int}}\sin\left(\omega_{i}t\right)e^{-\gamma\left(t-s\right)}\:dt\right]^{2} \end{equation}
We would like now to take $\omega_{ \rm{s} }T=2\pi$ in the regime of: $\gamma T\ll1,\:\phi_{n}\ll1,\:\omega_{ \rm{r} }T\ll1.$ In this case $p=\langle\sin\left(\phi\right)^{2}\rangle\approx\langle\phi_{av}^{2}\rangle+\langle\phi_{n}^{2}\rangle.$ In leading orders: \begin{equation} \langle\phi_{n}^{2}\rangle\approx\sigma_{n}^{2}T^{3}\left(\frac{1}{\pi^{2}}+O\left(\gamma T\right)\right), \end{equation} and: \begin{eqnarray} \begin{split} &\langle\phi_{av}^{2}\rangle\approx\langle\left[\frac{\left(A_{1}\left(0\right)-A_{2}\left(0\right)\right)}{2\pi}\omega_{ \rm{r} }T^{2}+\overset{2}{\underset{i=1}{\sum}}B_{i}\left(0\right)\frac{\gamma T^{2}}{2\pi}\right]^{2}\rangle=\\ &=\overset{2}{\underset{i=1}{\sum}}\langle A_{i}\left(0\right)^{2}\rangle\frac{1}{4\pi^{2}}\omega_{ \rm{r} }^{2}T^{4}+\overset{2}{\underset{i=1}{\sum}}\langle B_{i}\left(0\right)^{2}\rangle\frac{1}{4\pi^{2}}\gamma^{2}T^{4}=\\ &=\frac{\sigma_{n}^{2}}{4\pi^{2}\gamma}\omega_{ \rm{r} }^{2}T^{4}+\frac{\sigma_{n}^{2}}{4\pi^{2}}\gamma T^{4}. \end{split} \end{eqnarray} The second term ($\underset{i}{\sum}\frac{\sigma_{n}^{2}}{4\pi^{2}}\gamma T^{4}$) can be neglected as it is much smaller than $\langle\phi_{n}^{2}\rangle\sim\sigma_{n}^{2}T^{3}.$ Therefore: \begin{equation} p\approx\langle\phi_{av}^{2}\rangle+\langle\phi_{n}^{2}\rangle\approx\frac{\sigma_{n}^{2}T^{3}}{\pi^{2}}+\frac{\sigma_{n}^{2}}{4\pi^{2}\gamma}\omega_{ \rm{r} }^{2}T^{4}, \end{equation} using the notation in the main text $\frac{\sigma_{n}^{2}T^{3}}{\pi^{2}}=n$, and the resolution condition is thus $\omega_{ \rm{r} }^{2}\frac{T}{\gamma}\gg1.$
\section{Limitation due to dephasing of the probe} \label{sec:dephasing} Let us now find the implications of noise inflicted on the probe: specifically we consider a Markovian dephasing. Intuitively this should set an additional limitation: the transition probability does not vanish now due to two reasons: the finite $\omega_{ \rm{r} },$ namely the term $\left(\sigma t\right)^{2}\left(\omega_{ \rm{r} }t\right)^{2},$ and also due to a dephasing rate of $\kappa,$ namely a term of $\kappa t.$ Hence resolution can be achieved if the first term is larger than the second, hence the condition is $\frac { \omega_{ \rm{r} }\sigma}{\left(\kappa^{2}\right)}\gg1.$
In more detail, given our Hamiltonian $H=f\left(t\right)\sigma_{z}$ and a dephasing rate $\kappa,$ the time evolution is given by the Master equation: \begin{equation} \frac{d\rho}{dt}=-i\left[f\left(t\right)\sigma_{z},\rho\right]+\kappa\left(\sigma_{z}\rho\sigma_{z}-\rho\right). \end{equation} Initializing and measuring in $\sigma_{x}$ basis, we get the transition probability $p=0.5-0.5e^{-2\kappa t}\cos\left(2\phi\right)$ (where $\phi$ is the accumulated phase). Averaging over the different realizations we get: $p=0.5\left(1-\exp\left(-\frac{\left(\sigma t\right)^{2}\left(\omega_{ \rm{r} }t\right)^{2}}{\pi^{2}}-2\kappa t\right)\right),$ and the FI reads: \begin{equation} I_{ \rm{r} }=\frac{4\omega_{ \rm{r} }^{2}\sigma^{4}t^{8}}{\pi^{4}\left[\left(\exp\left(4\kappa t+2\omega_{ \rm{r} }^{2}\sigma^{2}t^{4}/\pi^{2}\right)-1\right)\right]}. \end{equation} Hence the noiseless FI is retrieved only for $ \frac { \omega_{ \rm{r} }\sigma}{\left(\kappa^{2}\right)}\gg1.$
Note that the effect of dephasing in this problem is quite different: first, the FI is not necessarily close to the noiseless FI for $\kappa t \ll 1,$ the condition $ \frac { \omega_{ \rm{r} }\sigma}{\left(\kappa^{2}\right)}\gg1$ must hold.
Second, since there is a competition between $\left(\sigma t\right)^{2}\left(\omega_{ \rm{r} }t\right)^{2}$ and $\kappa t,$ and for short enough times the second term is always larger, then the noiseless FI is retrieved only after a minimal time (goes as $\sim \frac{\kappa^{1/3}}{\sigma^{2/3}\omega_{ \rm{r} }^{2/3}}$) . This behavior is illustrated in supplementary fig. \ref{dephasing}.
\begin{figure}\label{dephasing}
\end{figure}
We therefore observe that Markovian noise on the qubit (such as dephasing) imposes a resolution limit, this invokes a natural question: can error correction protocols remove these limitations imposed by Markovian noise? Note that this question is not analogous to the achievability of Heisenberg scaling which was addressed in \cite{sekatski2017quantum,zhou2018achieving,demkowicz2017adaptive}. The prospects for quantum error correction in these resolution problems are left for future work, however we point out that this limit can be eliminated provided that errors can be detected, since one does not need to correct the errors. Given that error detection is possible, we can postselect the measurements without error and perform estimation according to them.
These errors thus reduce the precision, but they do not impose a limitation.
This implies that the error correction condition to remove limitation in this problem should be different from the error correction condition for Heisenberg scaling.
Let us denote the jump operators in the Master equation as $\left\{ L_{j}\right\} _{j}.$ For Heisenberg scaling, the error correction condition is $H\notin\text{span}\left\{I,L_{j},L_{j}^{\dagger}, L_{i}^{\dagger}L_{j}\right\} _{i,,j}$ \cite{sekatski2017quantum,zhou2018achieving,demkowicz2017adaptive},
since we need both to detect and to correct. In this case, since we need only to detect, the error correction condition should be: $H\notin\text{span}\left\{I, L_{j}^{\dagger},L_{j}\right\} _{j}.$ This implies, for example, that if the only noise source is amplitude damping (namely the only jump operator is $\sigma_{-}$), then detection is possible, and we can overcome the limitation.
\section{Superresolution with QFT} \label{sec:QFT} Consider the Hamiltonian in supplementary eq. \ref{Hamiltonian2}, and assume a relatively long coherence time (in which phases and amplitudes are constant), such that sampling is performed. Namely Ramsey measurements are performed in different times, as it is described in \cite{schmitt2017submillihertz,boss2017quantum,glenn2018high}. The length of each measurement is $\tau$ and the total sampling time is $T=N\tau.$ The standard way to analyze this data is to perform a Fourier transform and fit the power spectrum. However this method suffers from a resolution limit \cite{rotem2017limits}, since the probability is symmetric with respect to $\omega_{ \rm{r} }$ and the noise does not vanish.
We claim that storing the data in a quantum state (using memory qubits) and using the same trick of nullifying the projection noise, then resolution limit can be beaten. In a standard Ramsey experiment the state of the probe, after phase accumulation, is $\frac{1}{\sqrt{2}}\left(|0\rangle+e^{i \phi}|1\rangle\right).$ If we entangle the probe to memory qubits in each phase acquisition, the following state of the memory qubits can be generated: \begin{equation}
|\psi\rangle=\frac{1}{\sqrt{N}}\left(\underset{j=0}{\overset{N-1}{\sum}}|j\rangle e^{i\phi_{j}}\right), \end{equation}
where $\phi_{j}=\tau\left[\underset{i}{\sum}A_{i}\cos\left(\omega_{i} t_{j} \right)+B_{i}\sin\left(\omega_{i} t_{j} \right)\right], \; (t_{j}=j\tau). $
The idea is that for $\omega_{ \rm{r} }=0$ only harmonics of $\omega_{ \rm{s} }$ can be measured, and the probability to measure the other frequencies goes as $\omega_{ \rm{r} }^{2}.$ To see this note that \begin{equation}
\phi_{j}\left(\omega_{ \rm{r} }=0\right)=\tau\left[\underset{i}{\sum}A_{i}\cos\left(\omega_{ \rm{s} }t_{j}\right)+B_{i}\sin\left(\omega_{ \rm{s} }t_{j}\right)\right]=\Omega\sin\left(\omega_{ \rm{s} }t_{j}+\varphi\right), \end{equation} and therefore: \begin{equation} e^{i\phi_{j} \left(\omega_{ \rm{r} }=0\right) }=\underset{k=-\infty}{\overset{\infty}{\sum}}J_{k}\left(\Omega\right)\exp\left(ik\varphi\right)\exp\left(ik\omega_{ \rm{s} }j\tau\right), \end{equation} where this expansion to harmonics of $\omega_{ \rm{s} }$ is the Jacobi-Anger expansion.
Since we want to make sure that $\omega_{ \rm{s} }$ (and integer multiples of it) will be included in the Fourier basis, we need to set $T=\frac{2\pi}{\omega_{ \rm{s} }}m$ (integer $m$). In order to avoid too many harmonics, we also set $\tau=\frac{2\pi}{\omega_{ \rm{s} }}\frac{1}{n}$ (integer $n$).
It is now simple to see that with this choice, the only frequencies that can be measured in QFT are: $ 0,\omega_{ \rm{s} },...,\left(n-1\right)\omega_{ \rm{s} },$ as the state reads: \begin{equation}
|\psi_{0}\rangle=\underset{l=0}{\overset{n-1}{\sum}}a_{l}|l\omega_{ \rm{s} }\rangle, \end{equation} where $a_{l}=\underset{k=-\infty}{\overset{\infty}{\sum}}J_{nk+l}\left(\Omega\right)\exp\left(i\left(nk+l\right)\varphi\right).$
So for example given a noise model of a random phase, the density matrix is diagonal in the Fourier basis: $\rho=\underset{l=0}{\overset{n-1}{\sum}}p_{l}|l\omega_{ \rm{s} }\rangle\langle l\omega_{ \rm{s} }|,$ where $p_{l}=\underset{k=-\infty}{\overset{\infty}{\sum}}|J_{nk+l}\left(\Omega\right)|^{2},$ hence the optimal measurement basis is the Fourier basis (and it is enough to measure whether we get harmonics of $\omega_{ \rm{s} }$ or not).
\begin{figure}\label{QFT}
\end{figure}
Let us now find the probability to measure frequencies that are not harmonics of $\omega_{ \rm{s} }$, for $\omega_{ \rm{r} }T \ll 1.$
We denote the projector on the other frequencies (not harmonics of $\omega_{ \rm{s} }$) as $\Pi,$ so we are interested in finding $p=|\Pi|\psi\rangle|^{2}.$
Of course: $|\psi\rangle=|\psi_{0}\rangle+\omega_{ \rm{r} }\frac{d|\psi\rangle}{d\omega_{ \rm{r} }}+\mathcal{O}\left(\omega_{ \rm{r} }^{2}\right),$ and since $\Pi|\psi_{0}\rangle=0,$ we get that: $p\approx\omega_{ \rm{r} }^{2}|\Pi\frac{d|\psi\rangle}{d\omega_{ \rm{r} }}|^{2}.$ Now: \begin{equation*}
\frac{d|\psi\rangle}{d\omega_{ \rm{r} }}=\underset{j=1}{\overset{N}{\sum}}\exp\left(i\phi_{j}\left(\omega_{ \rm{r} }=0\right)\right)\left(i\omega_{ \rm{r} }t_{j}\right)\left[\left(B_{1}-B_{2}\right)\tau\cos\left(\omega_{ \rm{s} }t_{j}\right)+\left(A_{2}-A_{1}\right)\tau\sin\left(\omega_{ \rm{s} }t_{j}\right)\right]|j\rangle, \end{equation*} note that we can expand: \begin{equation} \exp\left(i\phi_{j}\left(\omega_{ \rm{r} }=0\right)\right)\left[\left(B_{1}-B_{2}\right)\tau\cos\left(\omega_{ \rm{s} }t_{j}\right)+\left(A_{2}-A_{1}\right)\tau\sin\left(\omega_{ \rm{s} }t_{j}\right)\right]=\underset{l=0}{\overset{n-1}{\sum}}b_{l}\exp\left(il\omega_{ \rm{s} }t_{j}\right), \end{equation} Therefore: \begin{equation}
\frac{d|\psi\rangle}{d\omega_{ \rm{r} }}=\frac{i}{\sqrt{N}}\underset{l=0}{\overset{n-1}{\sum}}b_{l}\underset{j=0}{\overset{N-1}{\sum}}t_{j}\exp\left(il\omega_{ \rm{s} }t_{j}\right) |j\rangle. \end{equation}
For convenience let us denote $|r_{k}\rangle=\frac{1}{\sqrt{N}}\underset{j=1}{\overset{N}{\sum}}t_{j}\exp\left(ik\omega_{ \rm{s} }t_{j}\right)|j\rangle,$ then with this notation: \begin{equation}
\frac{d|\psi\rangle}{d\omega_{ \rm{r} }}=i\underset{l=0}{\overset{n-1}{\sum}}b_{l}|r_{l}\rangle. \end{equation}
Now given that $T\gg \frac{2 \pi}{\omega_{ \rm{s} }},$ the state $|r_{l}\rangle$ will have a non-negligible overlap only with frequencies close enough to $|l \omega_{ \rm{s} }\rangle,$
this leads us to make two approximations: $i\neq k\Rightarrow\langle r_{k}|\Pi|r_{i}\rangle=0$ (different $|r_{k}\rangle$'s overlap orthogonal frequencies) and
$|\Pi|r_{k}\rangle|^{2}=\langle r_{k}|r_{k}\rangle-|\langle k\omega_{ \rm{s} }|r_{k}\rangle|^{2}$ (the only harmonic that overlaps $|r_{k}\rangle$ is $|k\omega_{ \rm{s} }\rangle$).
Due to the first approximation: \begin{equation}
p\approx\omega_{ \rm{r} }^{2}|\Pi\frac{d|\psi\rangle}{d\omega_{ \rm{r} }}|^{2}\approx\omega_{ \rm{r} }^{2}\underset{l=0}{\overset{n-1}{\sum}}|b_{l}|^{2}|\Pi|r_{l}\rangle|^{2}. \end{equation} observe now that: \begin{eqnarray} \begin{split}
&\langle r_{k}|r_{k}\rangle&=\frac{1}{N}\underset{j}{\sum}t_{j}^{2}\approx\frac{T^{2}}{3}\\
& \langle k\omega_{ \rm{s} }|r_{k}\rangle&=\frac{1}{N}\underset{j}{\sum}t_{j}\approx\frac{T}{2}. \end{split} \end{eqnarray} Then due to the second approximation: \begin{equation}
p=\frac{1}{12}\omega_{ \rm{r} }^{2}T^{2}\underset{l=0}{\overset{\left(n-1\right)}{\sum}}|b_{l}|^{2}. \end{equation}
It is now simple to see that: $\underset{l=0}{\overset{\left(n-1\right)}{\sum}}|b_{l}|^{2}=\tau^{2}\frac{1}{2}\left[\left(A_{1}-A_{2}\right)^{2}+\left(B_{1}-B_{2}\right)^{2}\right].$ Therefore taking the model of random quadratures (each with variance $\sigma^2$), we get: $p=\frac{1}{6}\omega_{ \rm{r} }^{2}T^{2}\left(\sigma\tau\right)^{2}.$
\section{Different noise models} \label{Different_noise_models} We showed in the main text that given the following effective Hamiltonian (this already takes into account the pulses, so all the relevant factors have been absorbed into the amplitudes): \begin{equation} H=\left[ A_{1}\cos\left(\delta_{1}t\right)+B_{1}\sin\left(\delta_{1}t\right)+A_{2}\cos\left(\delta_{2}t\right)+B_{2}\sin\left(\delta_{2}t\right) \right] \sigma_{z}, \end{equation} and a certain noise model of the amplitudes, the FI is calculated according to the average transition probability: \begin{equation} p=\int \sin^{2}\left(\phi\right) \underset{i}{\Pi} p\left(A_{i}\right)p\left(B_{i}\right)\:dA_{i}\:dB_{i}. \end{equation} Using the control method proposed in this paper (applying $\pi$ pulses such that $\delta_{s}t=2\pi$), we obtain that $\phi\approx\frac{\left(A_{1}-A_{2}\right)}{2 \pi}\omega_{ \rm{r} }t^{2}\rightarrow p_{a}\propto\omega_{ \rm{r} }^{2}.$ Therefore a non vanishing $I_{ \rm{r} }$ is achieved as long as $\int\left(A_{1}-A_{2}\right)^{2}p\left(\text{{\bf A}},\text{{\bf B}}\right)\:d\text{{\bf A}}\:d\text{{\bf B}}\neq0.$ Since our primary interest is in NMR we assumed the noise model relevant to unpolarized NMR in which $A_{i},B_{i}$ are Gaussian i.i.d. with a distribution of $N\left(0,\sigma\right).$ Assuming this noise model, the average transition probability,$p,$ is given by: \begin{equation} p=0.5\left(1-\exp\left(- 8 \sum_{i} \frac{\sigma^2}{\delta_i^2} \sin^2\left(\frac{\delta_i t}{2} \right) \right) \right), \end{equation} For $\delta_{s}t=2\pi,$ $p_{a}\approx\frac{2\sigma^{2}}{\omega_{ \rm{s} }^{2}}\omega_{ \rm{r} }^{2}t^{2},$ and thus $I_{ \rm{r} }=\frac{2\sigma^{2}t^{4}}{\pi^{2}}.$
For classical signals (such as microwave signals generated by AC wires) a different noise model should be taken into account. For these signals the amplitude of the field ($\sqrt{A^{2}+B^{2}}$) is constant, while the phase ($\text{arctan}\left(\frac{B}{A}\right)$) distributes uniformly. It is easy to verify that in this case $\int\left(A_{1}-A_{2}\right)^{2}p\left(\text{{\bf A}},\text{{\bf B}}\right)\:d\text{{\bf A}}\:d\text{{\bf B}}\neq0,$ and thus a finite FI is achieved. A detailed analysis shows that: \begin{equation} p=\frac{1}{2}\left(1-J_{0}\left(\frac{4\Omega_{1}}{\delta_{1}}\sin\left(\frac{\delta_{1}t}{2}\right)\right)J_{0}\left(\frac{4\Omega_{2}}{\delta_{2}}\sin\left(\frac{\delta_{2}t}{2}\right)\right)\right), \end{equation} where $\Omega_{i}=\sqrt{A_{i}^{2}+B_{i}^{2}},$ and these amplitudes are constants and identical. Taking $\delta_{s}t=2\pi,$ we get: \begin{equation} p\approx\frac{\left(\Omega t\right)^{2}}{\left(2\pi\right)^{2}}\omega_{ \rm{r} }^{2}t^{2}\Rightarrow I=\frac{4\Omega^{2}}{\left(2\pi\right)^{2}}t^{4}. \end{equation}
\end{widetext}
\end{document} | arXiv |
Cornerstones
Spearman's Rank Sum Correlation Test
There is a non-parametric test for an association (not necessarily linear) between two variables, called Spearman's Rank Correlation Test that can be used when the assumptions/requirements of the (parametric) correlation test are not satisfied.
The only requirements of this non-parametric test are that the data is paired and the result of a simple random sample, and that the data can be ranked (if they are not ranks already).
Essentially, all this test does is find ranks $x_i$ and $y_i$ for each pair of $X_i$ and $Y_i$ values and then run Pearson's correlation test on these ranks.
Recall that $$r = \frac{s_{xy}}{s_x s_y} = \frac{\sum_i (x_i - \overline{x})(y_i - \overline{y})}{\sqrt{\sum_i (x_i-\overline{x})^2} \sqrt{\sum_i (y_i - \overline{y})^2}}$$
We denote this value as $r_S$ when it is computed from ranks to avoid confusion.
Procedurally, one ranks each sample separately. Then for each pair, one finds the difference of ranks $d_i$.
The test statistic $r_S$, when there are no rank ties, can be simplified to
$$r_S = 1 - \frac{6 \sum d_i^2}{n(n^2-1)}$$
To see this, first note that as there are no ties, the $x_i$'s and $y_i$'s both consist of the integers from $1$ to $n$, inclusive.
Consequently, we can rewrite the denominator as $$\frac{\sum_i (x_i - \overline{x})(y_i - \overline{y})}{\sum_i (x_i-\overline{x})^2}$$ Ultimately, the denominator is just a function of $n$: $$\begin{array}{rcl} \displaystyle{\sum_{i=1}^n (x_i-\overline{x})^2} & = & \displaystyle{\sum_{i=1}^n x_i^2 - 2\sum_{i=1}^n x_i\overline{x} + \sum_{i=1}^n \overline{x}^2}\\ & = & \displaystyle{\left[ \sum_{i=1}^n x_i^2 \right] - 2n\overline{x}\left[\frac{\sum_{i=1}^n x_i}{n}\right] + n \overline{x}^2}\\ & = & \displaystyle{\left[ \sum_{i=1}^n i^2 \right] - 2n\overline{x}^2 + n \overline{x}^2}\\ & = & \displaystyle{\left[ \sum_{i=1}^n i^2 \right] - n\overline{x}^2}\\ & = & \displaystyle{\frac{n(n+1)(2n+1)}{6} - n \left( \frac{n+1}{2} \right)^2}\\ & = & \displaystyle{n(n+1) \left( \frac{2n+1}{6} - \frac{n+1}{4} \right)}\\ & = & \displaystyle{n(n+1) \left( \frac{8n+4}{24} - \frac{6n+6}{24} \right)}\\ & = & \displaystyle{n(n+1) \left( \frac{2n-2}{24} \right)}\\ & = & \displaystyle{\frac{n(n+1)(n-1)}{12}}\\ & = & \displaystyle{\frac{n(n^2-1)}{12}}\\ \end{array}$$
As for the numerator...
$$\begin{array}{rcl} \displaystyle{\sum_{i=1}^n (x_i - \overline{x})(y_i - \overline{y})} & = & \displaystyle{\sum_{i=1}^n x_i(y_i-\overline{y}) - \sum_{i=1}^n \overline{x} (y_i - \overline{y})}\\ & = & \displaystyle{\sum_{i=1}^n x_i y_i - \overline{y} \sum_{i=1}^n x_i - \overline{x} \sum_{i=1}^n y_i + n \overline{x}\overline{y}}\\ & = & \displaystyle{\left[ \sum_{i=1}^n x_i y_i \right] - n\overline{x}\overline{y}}\\ & = & \displaystyle{\left[ \sum_{i=1}^n x_i y_i \right] - n \left( \frac{n+1}{2} \right)^2}\\ & = & \displaystyle{\left[ \sum_{i=1}^n x_i y_i \right] - \frac{n(n+1)(2n+1)}{6} + \frac{n(n^2-1)}{12}}\\ & = & \displaystyle{\left[ \sum_{i=1}^n x_i y_i \right] - \sum_{i=1}^n x_i^2 + \frac{n(n^2-1)}{12}}\\ & = & \displaystyle{\frac{2\sum_{i=1}^n x_i y_i}{2} - \frac{\sum_{i=1}^n (x_i^2 + y_i^2)}{2} + \frac{n(n^2-1)}{12}}\\ & = & \displaystyle{\frac{n(n^2-1)}{12} - \frac{\sum_{i=1}^n (x_i^2 - 2x_iy_i + y_i^2)}{2}}\\ & = & \displaystyle{\frac{n(n^2-1)}{12} - \frac{\sum_{i=1}^n (x_i - y_i)^2}{2}}\\ & = & \displaystyle{\frac{n(n^2-1)}{12} - \frac{\sum_{i=1}^n d_i^2}{2}}\\ \end{array}$$
Finally, dividing both numerator and denominator by $n(n^2-1)/12$, we can simplify things to
$$r_s = \frac{\displaystyle{\frac{n(n^2-1)}{12} - \frac{\sum_{i=1}^n d_i^2}{2}}}{\displaystyle{\frac{n(n^2-1)}{12}}} = 1 - \frac{6 \sum d_i^2}{n(n^2-1)}$$
Critical values can be found in the table below:
Suppose one wishes to use a non-parametric test to test the claim that there is a correlation between one's age and the number of parties they attend in a two-month period, given the following data:
$$\begin{array}{l|c|c|c|c|c|c|c} \textrm{Age} & 16 & 24 & 18 & 17 & 23 & 27 & 32\\\hline \textrm{Parties} & 3 & 2 & 5 & 4 & 0 & 6 & 1 \end{array}$$
First we rank the $x$'s and $y$'s separately:
$$\begin{array}{l|c|c|c|c|c|c|c} & 1 & 5 & 3 & 2 & 4 & 6 & 7 \\\hline \textrm{Age} & 16 & 24 & 18 & 17 & 23 & 27 & 32\\\hline \textrm{Parties} & 3 & 2 & 5 & 4 & 0 & 6 & 1\\\hline & 4 & 3 & 6 & 5 & 1 & 7 & 2 \end{array}$$
Then, for each pair, we find the difference of the ranks and its square.
$$\begin{array}{l|c|c|c|c|c|c|c} d & -3 & 2 & -3 & -3 & 3 & -1 & 5\\\hline d^2 & 9 & 4 & 9 & 9 & 9 & 1 & 25 \end{array}$$
Now we can calculate the test statistic:
$$r_S = 1 - \frac{6 \sum d_i^2}{n(n^2-1)} = 1 - \frac{(6)(66)}{(7)(49-1)} = -0.1786$$
Seeing this test statistic less in absolute value than the corresponding critical value at $\alpha = 0.05$ given in the table above (i.e., $C.V. = 0.786$), we would fail to reject the null hypothesis, inferring that there is no evidence of a correlation. | CommonCrawl |
Reconciling Continuous and Discrete Complex Domains
Having taken some courses in Systems and Stability dealing mostly with continuous time signals, I am accustomed to thinking about the Laplace and Fourier transforms dealing with complex and pure imaginary signals respectively. For example, if I transformed $x(t)$ and evaluated its magnitude and phase at 0.5+0.5j, I would looking at the response of the system having injected it with $\exp(.5t)\exp(j0.5 \pi t)$
Recently, I have been working in the discrete domains of the DTFT (continuous in the transform domain), DFT, and Z. Here, we have complex exponentials mapping to the unit circle. Now I am struggling at a high level to reconcile these continuous and discrete domains. If I look at a DTFT of a discrete time signal and evaluate it for some input $\exp(0.5 j \hat{\omega} \pi n)$ where $n$ is some index and $\hat{omega}$ is a relative frequency based on the sampling rate of the continuous time signal, I end up somewhere on the unit circle (not on the imaginary axis) even though the signal is a purely imaginary exponential.
The questions I have are
1.) How does a complex signal (real and imaginary)$ x[n] = \exp(0.5n) \cos(j .5 \pi \hat{\omega}n)$ map into the DTFT/DFT and Z frequency domain? I am guessing somehow real part does not map to DTFT/DFT but does to Z?
2.) Why the concept of a multidimensional space for these discrete frequency signals (DTFT/DFT) when the imaginary axis (Fourier domain) worked fine for continuous signal? Why do we need a unit circle that repeats when we could map the same function to a straight line that repeats every $-\pi$ to $\pi$?
discrete-signals continuous-signals frequency-response
Jack FryeJack Frye
Simple answer: that's how the math works out.
Better answer: In the discrete domain, frequency is periodic with the sample rate. If you sample at 48 kHz, the range of frequencies that you can represent range from -24 kHz to +24 kHz. Let's say so you have a 100 kHz signal. You can sample this at 48 kHz but it would look exactly like a 4 kHz signal. In fact the two are indistinguishable, so it makes no sense to treat them differently.
That's why a circle makes a lot of sense here. Frequency is represented as the angle between the origin at any point on the circle. If you go around the circle an integer number of times, you end up exactly where you started.
So in the continuous domain 4 kHz and 100 kHz are different points. In the discrete domain (when sampled at 48 kHz) they are not.
HilmarHilmar
Consider the following $$x(t)=\cos(2\pi f t) $$
$$\frac{d}{dt} x(t)= -2\pi f \sin(2\pi f t) $$ you can take derivative with respect to time. In the $s$ domain $s X(s)$ corresponds to the time derivative of $X(s)$ (with zero initial condition)
For discrete time, $$ x[n]=\cos(2\pi f n) $$ $n$ is a discrete variable $$ \frac{d}{dn} x[n]= \lim_{\Delta \rightarrow 0} \frac{x[n+\Delta]-x[n]} {\Delta} \quad \text{is nonsense} $$ Instead we have the $z$ domain where $zX(z)$ corresponds to $x[n+1]$.
Since poles and zeros can lay in an arbitrary location, (not restricted to $s=j\omega$ or $z$ on the unit circle), using the $s$ domain just doesn't work.
The Z Transform is a special case of the Laurent Series
https://en.wikipedia.org/wiki/Laurent_series
from the first paragraph:
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
Since $(d^k /dn^k) x[n]$ would be necessary for a Taylor Series, and they don't exist, we use the Z transform.
But wait, Isn't $e^{-sT}$ a delay operator in the $s$ plain ? Yes it is, so we can describe signals like $ y(t)= a_0 x(t) + a_1x(t-T) + a_2 x(t-2T)$ in the $s$ domain as $(a_0+ e^{-sT}a_1 + e^{-2sT}a_2) X(s)$. The problem is that a LTI continuous time filter is described as a (finite) ratio of polynomials in $s$.
$$ H(s)=\frac{a_0 + a_1 s + a_2 s^2 +\dots + a_n s^{n}}{b_0 + b_1 s + b_2 s^2 +\dots + b_n s^{m}} $$ The transfer function for $e^{-sT}$ has infinite terms. This can be approximated by a Pade expansion, and often is in some applications, but you can see that the $s$ plain get very complicated and is only approximate. $e^{sT}$ and approximations do come in handy when relating $s$ to $z$
Not the answer you're looking for? Browse other questions tagged discrete-signals continuous-signals frequency-response or ask your own question.
Main differences to take into account between continuous and discrete time signals
Discrete Time Fourier Transform Pair Discrepancy
Can a discrete sinusoidal signal represent only limited number of frequencies?
$2\pi$ periodicity of discrete-time Fourier transform
Why DTFT coefficients are periodic and why continuous Fourier transform coefficients are not periodic?
How is a continuous spectrum for the DTFT possible?
Confusion about subtle difference between discrete-time and continuous-time
Maximum Magnitude Deviation between DFT and DTFT
DTFT frequency range | CommonCrawl |
Non-invasive Determination of Aortic Mechanical Properties and Their Effects on Left Ventricular Function Following Endovascular Abdominal Aneurysm Repair
Alexander Gregory ORCID: orcid.org/0000-0003-3483-79521,3,
Marelise Kruger1,
Neal Maher1,
Randy Moore2 &
Gary Dobson1
Increased pulse-wave velocity (PWV) entails elevated arterial stiffness and is associated with adverse cardiovascular outcomes. Previous studies have shown PWV increases following endovascular aneurysm repair (EVAR). Animal models suggest elevated PWV following endograft deployment occurs in conjunction with decreased aortic compliance and increased aortic impedance. This could lead to unwanted effects on left ventricular (LV) function. This study evaluates the early implications of EVAR on aortic mechanics and associated LV function.
Prospective observational study of elective EVAR for abdominal aneurysm. Transesophageal echocardiography was used to acquire LV flow and function, as well as images of the ascending aorta. Speckle tracking echocardiography (STE) software analysis allowed for determination of aortic volume change throughout the cardiac cycle. Aortic mechanics (including compliance, impedance and reflected wave analysis) and left-ventricular function (including cardiac output, hydraulic load and diastolic function) were measured pre- and post-endograft deployment.
Endograft deployment resulted in no significant increase in aortic impedance. It did alter the timing of reflected waves in the aorta, with a greater positive wave being reflected during systole. This increased the hydraulic load on the LV with corresponding statistically non-significant trends for decreasing cardiac output and LV diastolic function.
STE represents an emerging imaging modality for aortic biomechanical assessment. The increased PWV seen following EVAR may not reflect the same aortic biomechanical pathophysiology as in the non-surgical population with measures of increased arterial stiffness. The stiffer endograft does not appear to acutely increase aortic impedance, though it does alter the timing of reflected waves which may have negative effects on LV function.
Elevated measures of arterial stiffness (central and peripheral) have been shown to negatively impact LV systolic and diastolic function, impair coronary blood flow, and have been identified as a risk factor for cardiovascular disease and mortality [1,2,3,4,5,6,7,8,9]. One possible mechanism for these findings is that increased arterial stiffness can create additional LV work while simultaneously making coronary perfusion less efficient. The hydraulic load (IE: work) encountered during LV ejection is directly increased when arterial stiffening (less compliance) results in higher aortic input impedance. The size and timing of reflected waves will also change in the setting of a stiff arterial tree, often with a larger positive reflected wave returning to the proximal aorta in systole instead of diastole. This alteration of reflected wave properties can indirectly add to the increased LV hydraulic load, while also causing diminished coronary perfusion.
Endovascular aneurysm repair (EVAR) is well-established and frequently used treatment modality for aortic pathology, including abdominal aortic aneurysms (AAA). Accordingly, much research has been generated on the local effects of the graft with respect to the mechanisms of endoleak, graft migration and aneurysm sac remodeling. There has also been interest in the arterial physiologic effects of placing artificial graft material into the aorta, as well as the potential implications on cardiac function. A dog model demonstrated that proximal thoracic aortic stenting results in increased overall aortic input impedance, ascending aorta characteristic impedance (Z0) and pulse-wave velocity (PWV). These alterations occurred in conjunction with decreased total arterial compliance, alterations in reflected wave properties and reduced LV output (measured as lower aortic flow) [10]. An increased PWV has subsequently been demonstrated in human patients following implantation of an EVAR endograft [11,12,13,14,15,16]. Although this suggests an endograft induced increase in aortic stiffness, a comprehensive analysis of the biomechanical impact of endograft implantation has not been possible due to the invasive measures that would be required.
Recent improvements in medical imaging, combined with the rapid growth in computing power, has resulted in an emergence of non- or minimally-invasive methods for performing more nuanced determinations of aortic mechanical properties. Ultrasound images of the aorta can be acquired with intraoperative transesophageal echocardiography (TEE). These images can then be analyzed with speckle tracking (STE), a validated method which is able to follow tissue motion and quantify position, strain, and velocity [17, 18]. Though mostly used for quantifying ventricular function, this technology has been used to quantify aortic properties- including correlation with histologic and physical biomechanical testing of subsequently excised aortic tissue [19,20,21].
We report a technique using STE to assist in non-invasively determining the immediate effects of abdominal endograft deployment on aortic input impedance, ascending aortic compliance, systemic vascular resistance, reflected wave properties and LV function.
This study received approval from our institutional ethic committee. Participants were selected from adult (age > 18 years) patients scheduled to have elective EVAR for AAA. Exclusion criteria included patient refusal, contraindication to trans-esophageal echocardiography, prior aortic surgery, aortic valve abnormality (stenosis, regurgitation or prosthesis), atrial fibrillation and severe aortic calcification (> 2/3 circumference). After being identified and enrolled in the study demographic, medical information and current medications were collected. All procedures were performed under general anesthesia. Pre-implantation data was acquired once the patient was in a plane of anesthesia with stable hemodynamics. Post-implantation data was acquired in the operating theatre, shortly after endograft deployment when the patient had stable hemodynamics.
Data-Collection and Analysis
Trans-esophageal echocardiogram (TEE) was performed in each patient using a GE Vivid 7 machine (General Electric, Fairfield, Connecticut, USA). The following views were obtained (Fig. 1):
Images demonstrating the standard views and Doppler quantification acquired from each patient before and after EVAR procedure. a Mid-esophageal 4-chamber view. b Mid-esophageal aortic valve long axis view. c Trans-mitral pulse wave Doppler flow. d Tissue Doppler imaging of the basal septal and basal lateral LV walls. e Deep transgastric aortic valve/LVOT view. f LVOT pulse wave Doppler flow
Mid-esophageal 4-chamber view.
Tissue-Doppler imaging (TDI) and pulsed-wave Doppler (PWD) of the lateral and septal mitral valve annuli.
PWD of the trans-mitral flow.
Mid-esophageal long axis view of the aortic valve, including the aortic root and proximal ascending aorta
Deep-transgastric view of the aortic valve with PWD of the left ventricular outflow tract (LVOT), just proximal to the aortic valve.
Speckle-Tracking Echocardiography
Analysis of the TEE images were performed post-acquisition offline using a PC workstation and EchoPac, Version 7.2.0 software (General Electric, Fairfield, Connecticut, USA). The value for peak mitral annulus TDI velocity in early diastole was obtained for both the septal and lateral annuli (E'sept and E'lat) at the same times peak trans-mitral flow velocity in early diastole (E) using PWD.
Measurements of proximal aortic dimensions were taken from the standard TEE mid-esophageal aortic long-axis view using the caliper function (Fig. 2). The aorta was divided into 2 segments. The first segment was the aortic root while the second segment was the proximal ascending aorta. The anterior and posterior lengths of the aortic root were measured as 2 straight lines spanning, but not including, the walls of the Sinus of Valsava from the aortic valve to the sinotubular junction. The second segment was measured as 2 straight lines along the anterior and posterior aortic walls, from the sinotubular junction extending 2 cm into the ascending aorta. The lengths of the second segment were shortened in the event of poor image quality or if the boundary of the image prevented a full 2 cm segment. The anterior and posterior segment lengths were averaged to obtain a single length for our calculation of aortic volume. The diameters of each segment were taken inner edge to inner edge, at the midpoint of the segment. The diameter of the LVOT was also measured from this view. All measurements were taken at end-systole.
Caliper measurements of the ascending aorta used to calculate compliance. Taken from the mid-esophageal long axis view of the aorta at end-systole. A1 posterior length of segment 1 (aortic root), A2 anterior length of segment 1, B1 posterior length of segment 2 (ascending aorta), B2 anterior length of segment 2, C segment 1 diameter, D segment 2 diameter. See Methods section for details
Speckle tracking imaging (STI) analysis was performed using the same EchoPac software. In the mid- esophageal four-chamber view the endocardial border of the left ventricle (LV) was traced. Appropriate tissue tracking was confirmed with the program's own quality control feature, as well as visually by the investigators by observing a full cine loop to ensure the ventricular walls were tracked throughout the cardiac cycle. The STI-derived peak values for longitudinal tissue velocity, longitudinal strain rate and radial strain rate of the basal-septal and basal-lateral walls were measured in early diastole (Fig. 3).
Speckle tracking imaging of the LV. Peak longitudinal velocity, peak longitudinal strain rate (Fig. 3a) and peak radial strain rate (Fig. 3b) were taken in early diastole (E'). The basal septal wall is depicted in yellow, the basal lateral wall is depicted in red and the peak values are identified with a white arrow
STI analysis was further applied to the ascending aorta using the EchoPac software's pre-set for two- chamber LV images using a method we have previously described [4]. This method includes adjusting the software's pre-set spatial and timing filters to the lowest level. The walls of the aortic root and ascending aorta were traced from the aortic valve annulus to the most distal portion of the ascending aorta that was visible on the acquired images. Proper aortic tissue tracking was confirmed in the same fashion as mentioned above. The region of interest (ROI) was made as narrow as possible to best conform to the width of the aorta. The ROI borders were further adjusted so that the software's pre-set labels for identifying LV segments matched the aortic segments described above. The ROI for the LV basal segments spanned the aortic root while the LV mid segments spanned the proximal 2 cm of ascending aorta (Fig. 4). The LV apical segments did not track any aortic tissue and were not included in our analysis. Radial velocity and longitudinal strain rates were taken for the anterior and posterior walls of the aortic root and ascending aorta (Fig. 5). The graphs produced by the EchoPac software were converted into digital form using Digitize-It (version 1.5.8b, Bormisoft, Braunschweig, Germany) and saved as a text file.
The aortic root and proximal ascending aorta are traced using speckle tracking imaging software. The region of interest was adjusted so that the width incorporated the aortic wall. The segments were adjusted to include the posterior root (yellow), anterior root (red), posterior ascending aorta (cyan) and proximal ascending aorta (blue)
Speckle tracking imaging analysis of the aortic root and proximal ascending aorta. Region of interest was adjusted as described in the text and Fig. 4. Radial velocity and longitudinal strain rates were taken throughout the cardiac cycle in order to measure aortic volume change
Arterial Pressure Acquisition and Aortic Pressure Derivation
A right radial artery catheter placed prior to the induction of anesthesia. The catheter was connected to the transducer via a short length of pressure tubing (12 cm) to optimize the resonant frequency of the system. The filter of the arterial pressure channel on the Datex/Ohmeda S5 monitor was set to a frequency of 44 Hz and the analogue waveforms (radial artery pressure wave and the ECG) were acquired simultaneously through the data-out port at 1000 Hz using an analogue–digital converter (National Instruments, Austin, Texas, USA) and LabVIEW (National Instruments, Austin, Texas, USA). A pop-test was performed prior to every recording in order to calculate the system's resonant frequency and dampening coefficient. Ten to twelve pressure waves were averaged from the peak of the QRS complex of the ECG. The arterial pressure wave was then calibrated using the systolic and diastolic pressure recorded at the time of the acquisition. The pressure wave was then corrected for the resonance characteristics of the system and converted to an aortic pressure waveform using a previously validated transfer function [22].
Calculation of Aortic Parameters
Off-line analysis was performed using Maple (version 18.02, MapleSoft, Waterloo, Ontario, Canada). The aortic pressure wave was used to calculate the rate of pressure change (dP/dt) during the cardiac cycle. The STI based graphs of aortic radial velocity and longitudinal strain rate were used to calculate aortic volume and its rate of change during the cardiac cycle (dV/dt). All signals were filtered using a Butterworth, 4-element filter with a cut-off frequency of 16 Hz. The minor differences in the duration of the dP/dt and dV/dt waves were corrected through interpolation of the dV/dt waveforms. The ECG- gating from the TEE machine and anesthesia monitor were not identical resulting in a timing shift of the graphs of dP/dt and dV/dt. Therefore, we aligned the two graphs at the point where both dP/dt and dV/dt were initially positive. The interactions of pressure and volume change were used to generate pressure–volume loops for each patient, both before and after stent implantation (Fig. 6). We then produced graphs displaying total proximal aortic compliance (combined aortic root + ascending aorta) over the entire cardiac cycle. To calculate proximal aortic systolic compliance, the start of systole was initially identified by considering the time point in the cardiac cycle where dP/dt first becomes positive (Fig. 7a). The value of this time point was then used on the generated compliance versus time curve to identify the approximate beginning of systole. An asymptote near the start of systole and a flat nadir at the end of systole were consistent findings in all the study subjects (Fig. 7b). There was high variability in both the data values near the asymptote and the length of the nadir. Therefore, to obtain consistent measurements between patients, the first post-asymptote peak on the compliance curve in systole was chosen as the starting point and the beginning of the nadir as the end-point for measuring early systolic aortic compliance. Thus, proximal aortic systolic compliance (CAortaSyst) was defined as the average compliance over this period (Fig. 7b).
By combining STI-based aortic volume change with invasively measured arterial pressure pressure–volume loops were generated for each patient before and after deployment of the EVAR endograft. Arrows depict the direction of the loops during the cardiac cycle
The determination of proximal aortic systolic compliance was completed in a 2-step process. The start of systole was determined by identifying the time point where dP/dt first became positive (Fig. 7a). This time value was then used on the compliance curve to locate the start of systole (Fig. 7b). A consistent finding was an asymptote in this area. The mean compliance in the period from the first post-asymptote peak until the beginning of the nadir was used to calculate proximal aortic systolic compliance (CAortaSyst) in our analysis. This period is represented by the shaded box
The LVOT flow acquired from the intra-operative TEE using PWD was digitized (DigitizeIt version 1.5.8b, Bormisoft, Braunschweig, Germany) at 2000 Hz, resampled at 1000 Hz and filtered. Stroke volume (SV) and cardiac index (CI) were calculated:
$${\text{SV}} = \pi \times {\text{LVOT}}\;{\text{radius}}^{2} \times {\text{LVOT}}\;{\text{velocity}}\;{\text{time}}\;{\text{integral}}$$
$${\text{CI}} = {\text{SV}} \times {\text{HR/BSA}}$$
By combining LVOT flow and central aortic pressure, aortic input impedance was plotted across the frequency spectrum up to 15 Hz. The zero modulus was taken as systemic vascular resistance (SVR). The aortic characteristic impedance (Z0) was calculated from the average impedance in the 5–15 Hz frequencies, excluding any values that were more than 2 standard deviations above the average. The left ventricular pulsatile power, including mean and oscillatory power, was calculated from the pressure and flow waves using methodology previously described [23]. Additional use of the aortic pressure included wave form analysis allowing identification of the forward (incident) and backward (reflected) components (Fig. 8). This was done according to previously accepted methods [24,25,26,27]. We calculated the total forward, total backward, systolic forward and systolic backward waves. The reflection coefficient (PB/PF) was determined by the total reflected wave divided by the total forward wave.
Example from one of our study subjects of our wave form analysis using the aortic pressure wave (solid; red) allowing identification of the forward (hashed; blue) and backward (hashed-dot; pink) components. A difference can be seen in the size and timing of the reflected wave in a pre-EVAR compared to b post-EVAR
A sample size was calculated that would allow for comparison of post-EVAR changes in both LV diastolic function and aortic characteristic impedance. We used data from a previous study which measured characteristic impedance in both normotensive and hypertensive middle-aged patients, similar to our patient population. Their results showed a mean Z0 of 166 dynes s cm−5 with a standard deviation of 69 dynes s cm−5 [28]. Eighteen patients would allow for a detection of a post-implantation increase in impedance of 40% with 80% power and a p value < 0.05 [24]. A 40% relative change in impedance was felt to be clinically significant based on a similar change described by Dobson et al. [10]. This assertion is supported by studies showing that a 20–40% difference in valvuloarterial impedance impacts the clinical outcomes in patients with aortic valve stenosis [29].
With regards to diastolic function, STI values in middle-aged patients with mild or moderate diastolic dysfunction, the average longitudinal early diastolic strain rate was 1.15%/s with a standard deviation of 0.26%/s [30]. Twenty-one patients would allow for detecting a 20% decrease in diastolic function following EVAR with a power of 80% and a p value < 0.05 [31]. This change is likely clinically relevant based on the findings of Tsioufis et al., where a 20% difference in diastolic function was observed between normotensive and hypertensive patients [32].
All statistical analysis was performed using Analyze-It software version 3.90.7 (Analyze-It Software Ltd, Leeds, UK). Descriptive statistics were used for demographic and intra-operative data. The values for LV diastolic function and all measures of aortic function were analyzed for changes following endograft implantation using a Student's T test.
Twenty-one patients were included in the study. Three of the patients had incomplete data sets due to equipment failure or inadequate TEE images, therefore a total of 18 patients' data was analyzed. The demographic information and medical information is summarized in Table 1. Intraoperative hemodynamic details and endograft information is listed in Table 2. There was a near-significant trend towards higher systolic blood pressure after the EVAR procedure increasing from a mean (SD) of 109.5 (17.1) mmHg to 119.0 (20.3) mmHg (p = 0.06). There was no significant difference after EVAR in diastolic blood pressure (p = 0.59), mean arterial pressure (p = 0.35), heart rate (p = 0.19), and a trend towards reduced cardiac index (p = 0.11).
Table 1 Patient demographics, comorbidities and pre-operative medications
Table 2 Intra-operative clinical and procedural data
Pre-operatively the peak systolic mean (SD) STE measures of aortic longitudinal strain rates were 1.36 (0.80) %/s for the anterior segments and 1.57 (0.73) %/s for the posterior segments while the STE aortic radial velocities were 3.30 (1.12) cm/s for the anterior segments and 3.02 (0.95) cm/s for the posterior segments. Post-operatively peak systolic aortic longitudinal strain rates were 1.69 (1.10) %/s for the anterior segments and 1.78 (1.00) %/s for the posterior segments while the aortic radial velocities were 3.52 (0.99) cm/s for the anterior segments and 3.22 (1.18) cm/s for the posterior segments.
Full results of the comparative analysis of aortic and left ventricular diastolic function post-EVAR are included in Tables 3 and 4. Following deployment of the endovascular stent graft proximal aortic systolic compliance (p = 0.60) and calculated SVR (p = 0.21) were not significantly altered by EVAR endograft deployment while the aortic characteristic impedance (Z0) had a borderline significant post-EVAR decrease from a mean (SD) of 133.8 (63.6) dynes s/cm5 to 102.3 (77.5) dynes s/cm5 (p = 0.06). The magnitude of forward and reflected waves did not change significantly, however the reflection coefficient increased from a mean (SD) of 0.85 (0.08) to 0.88 (0.07) (p = 0.03). This was due to a change in the timing of positive reflected waves, with the systolic component increasing significantly (p = 0.002). The left ventricular pulsatile power (hydraulic load) increased from a mean (SD) of 160.1 (108.3) mW to 206.3 (93.8) mW (p = 0.05). There was a non-significant trend towards increased LV diastolic impairment post-EVAR, in both TDI and STI measured parameters (Table 4).
Table 3 Aortic properties, LV pulse power and wave analysis results
Table 4 Pre-operative and post-operative measures of LV diastolic function
Our results show that immediately after deployment of an EVAR endograft there was no increase in aortic characteristic impedance, ascending aortic compliance or systemic vascular resistance. The positive reflected wave remained the same size, however the timing of its reflection was altered, with a greater portion now returning during systole. We found that the LV experienced an increased pulsatile load, though the LV stroke volume and cardiac index were not affected. Finally, there may have been a small impairment of LV diastolic function, but not to a great enough extent to meet statistical significance.
As the aorta stiffens pulse waves originating in the heart conduct to the periphery with increasing velocity. As a result, pulse wave velocity (PWV) can be used as an indirect measure of the relative stiffness of the arterial system. It has been frequently used in clinical studies quantifying changes in arterial stiffness due to aging or disease states. More importantly, increasing arterial stiffness as detected by elevated PWV has been linked to subclinical target organ disease and impaired cardiovascular health, including increased mortality [5,6,7,8,9]. The potential value of PWV as a clinical tool for improving long-term cardiovascular patient care has led to its inclusion in both the American Heart Association and European Society of Cardiologists' most recent guidelines on the management of arterial hypertension and aortic disease [33, 34].
The endografts used in EVAR procedures are made with material that is stiffer than the native aorta, even in the setting of atherosclerotic or aneurysmal disease. Therefore, it is not surprising that an increased PWV measured across the EVAR site has been a consistently reported finding. Lantelme et al. assessed 50 patients following either open or endovascular AAA repair. After a median follow-up interval of 47 days they demonstrated an increase in mean PWV from a baseline of 11.3 to 12.3 m/s, or approximately 10% (p = 0.001) [13]. Of note, the hemodynamic alterations were different between the surgical graft prosthesis and the endograft, suggesting device material cannot be ignored. Post-EVAR increase in PWV was confirmed by Kadoglou, et al. with a 25% increase in mean PWV (13.11 m/s to 16.41 m/s) after 6 months [11]. The post-op intervals, population demographics and surgical specifics were not identical in each study, which may explain the difference in relative PWV change. Importantly, the finding of increased PWV post-EVAR was consistent and both authors suggest that this elevation may have clinical implications given the clear association of elevated PWV and cardiovascular outcomes in non-surgical populations. Their concern is further supported by the results of subsequent studies where an increased post-EVAR PWV was also associated with higher levels of inflammatory markers, increased LV mass, and LV diastolic dysfunction [12, 15, 16]. In one study, increased LV mass was detected as early as one week following an EVAR procedure [16].
Studies in animal models shed light onto the implications of increased PWV with regards to concurrent changes in aortic mechanical properties and possible mechanisms for the negative LV effects observed in humans. Using a canine model Dobson et al. implanted an endograft in the thoracic aorta [10]. There was a significant elevation in PWV post-deployment (418 cm s−1 vs. 755 cm s−1, p < 0.05), a finding consistent with the EVAR patient studies mentioned above. This change occurred alongside other changes in the ascending aortic mechanical properties. Significant post-stent increases occurred in proximal aortic impedance (Z0 = 0.250 mmHg mL−1 s−1 vs. 0.414 mmHg mL−1 s−1, p < 0.05). Additionally, total arterial compliance decreased (0.446 mL mmHg−1 vs. 0.276 mL mmHg−1, p < 0.05) alongside a near- significant increase in SVR (5.8 mmHg mL−1 s−1 vs. 7.7 mmHg mL−1 s−1, p = 0.06). There was also an alteration of reflected wave forms, with the reflected wave coefficient becoming dominated by a large negative reflected wave (0.09 vs − 0.49, p < 0.05). Due to endograft-aorta impedance mismatch one would expect to see larger positive and negative reflected waves, from the proximal and distal ends of the graft respectively which would typically cancel each other. The authors suggest the change in wave co-efficient in their study was due to increased proximal aortic stiffness suppressing the positive reflected wave, unmasking the negative wave. Another interesting finding was a measurable decrease in aortic flow (IE cardiac output) which neared statistical significance (0.89 L min−1 vs. 0.66 L min−1, p = 0.09), possibly as a result of the described changes in the aortic properties causing an increased LV hydraulic load. An alteration of aortic input impedance by thoracic grafting was similarly found in Mekkaoui et al. using a porcine model [35]. Of particular note, these changes resulted in an impairment of the natural diastolic augmentation of blood flow to the coronary and supra-aortic arteries. The findings of these papers suggest that (1) increased PWV may reflect other concurrent changes in aortic mechanics, including proximal characteristic impedance, (2) the net effect of these changes may have a deleterious effect on coronary blood flow, LV hydraulic load and cardiac output and (3) these results provide possible mechanistic explanations for the findings in post-EVAR human studies.
Previous studies in human subjects following EVAR had failed to demonstrate any "upstream" alteration in aortic stiffness [13, 36]. But these studies did not assess all of the aspects of aortic mechanical behaviour, nor did they measure aortic stiffness in the same fashion as the animal studies. This methodological discordance could easily explain the discrepant findings. Our study did not have such limitations. Even though our study methods closely approximated those in the animal studies, we did not find the changes in ascending aortic compliance or characteristic impedance identified by Dobson and Mekkaoui. An obvious explanation is that these changes simply do not occur in humans. Failed translation of animal models to human clinical studies is not uncommon. Other possible explanations do exist. Our patients, unlike the animal studies, had endografts deployed more distally in the aorta (abdomen vs thoracic) which may be too remote to induce changes in the ascending aorta. Alternatively, inhaled volatile anesthetic's suppression of the sympathetic nervous system may have counteracted the proposed mechanism of sympathetic activation for the changes in local aortic mechanics following endograft implantation [37,38,39]. Our measured SVR was lower post-EVAR, likely reflecting the cumulative effect of general anesthesia on decreased sympathetic/vascular tone at the level of the small arteries and arterioles (considered to be the largest contributors to SVR) [40]. It should be noted that neither set of animals in the comparison studies were anesthetized in a similar fashion to our patients (IE general anesthesia with inhaled volatile anesthetics).
Reflected waves occur whenever a point of impedance mismatch occurs along the arterial tree. A transition point to higher or lower impedance results in positive and negative reflected waves respectively. There are naturally occurring reflection points in the normal human arterial system, a negative wave reflects near the origins of the abdominal vessels while a positive one originates near the femoral arteries. As the arterial tree stiffens due to aging and vascular disease the size and timing of the positive reflected wave may change. Instead of providing diastolic blood flow augmentation, the wave arrives during systole. This results in added hydraulic load on the left ventricle, impaired coronary blood flow, LV diastolic dysfunction and contribution to long term adverse cardiovascular outcomes [41,42,43]. Stiffer endograft material also creates an impedance mismatch which may affect wave reflection properties. Our data suggests that although the absolute size of the positive reflected wave did not change, the timing was altered with an increased systolic component. This systolic reflected wave augmentation likely explains the increased fractional pulse pressure and trend towards higher systolic blood pressure, with no corresponding change in mean or diastolic pressures. It may also explain the increased LV hydraulic load and trend towards lower cardiac output despite the marginally decreased aortic impedance. The mechanism for previously reported PWV associated changes in LV mass and diastolic function following EVAR may in fact be a result of endograft induced changes in reflected wave properties, not increased aortic impedance.
Diastolic function has been shown to deteriorate in the setting of increases in measures of arterial stiffness (including PWV), as well as fractional pulse pressure and reflected waves [2, 32, 44,45,46,47,48,49]. If implantation of an abdominal endograft in an EVAR procedure alters any or all of these aortic properties, there may be clinical implications on left ventricular diastolic function. Several of our measures of LV diastolic function suggest relaxation is impaired following EVAR, though they did not meet statistical significance. More time may be required for diastolic function to be affected following EVAR. Takeda et al. found no change in diastolic function in the setting of increased PWV post-EVAR. However, at a 1 year follow-up over half the patients had deterioration in exercise tolerance and reduced diastolic function [15]. Therefore, although we did see statistically significant changes in diastolic function, there is both substrate (IE immediate increase in LV hydraulic load) and evidence to suggest that given enough time or a larger sample size we may see diastolic dysfunction in these patients.
Our parameters, though comprehensive, were only measured immediately following endograft deployment. Therefore, we do not know how any of these might progress over time. Do those that become altered, such as reflected wave properties and LV hydraulic load, remain the same, become more prominent, or eventually normalize? Do factors that did not change, such as proximal aortic compliance and diastolic function change over time? During intra-operative data collection our patients were under general anesthesia which impacts LV preload, afterload and sympathetic tone, potentially affecting our results. We were unable to include measurements of PWV concurrently with our other parameters. This limits the ability to put our findings into context with those from studies where increased PWV was measured post-EVAR. Given the ubiquity of elevated PWV following EVAR it is reasonable to assume we would have had a similar increase as well. We are not able to comment on the effects of different graft manufacturers or materials on the parameters measured because a majority of our patients had the same endograft implanted. Previous studies have suggested that differences between graft materials affect measures of aortic stiffness [12, 13, 37]. Finally, we have no control group with which to compare. One study that did include matched controls found that although EVAR grafts acutely increased PWV and reflected waves, these changes remained stable over time. The control group had gradual increases in PWV and reflected waves over time, and eventually the two groups were similar [14]. This suggests that the immediate changes in aortic stiffness, wave reflections and cardiac function following EVAR may just accelerate changes to the cardiovascular system which would have occurred over time in a non-surgical cohort. Whether it is detrimental to EVAR patients to have these changes occur acutely and sooner than would naturally occur is a question that requires further study.
Conclusion/Future Directions
Primarily, our study demonstrates that speckle tracking echocardiography represents a promising tool to non-invasively quantify aortic biomechanics in a real-time clinical setting.
In the setting of EVAR for abdominal aneurysms, measures of aortic stiffness that directly affect LV afterload, such as aortic impedance, ascending aortic compliance and SVR do not immediately increase following endograft implantation. The graft does appear to create an impedance mismatch which results in changes to the timing of positive reflected waves resulting in a greater systolic component. This timing shift is associated with an increased LV hydraulic load, which showed a trend towards decreased cardiac output and impaired LV diastolic relaxation [23].
To better describe the natural progression of aortic and cardiac changes following EVAR further studies should focus on the following: (1) Measure the corresponding changes in proximal aortic impedance, ascending aortic compliance, pulse wave analysis, PWV, LV hydraulic load, and LV systolic/diastolic function immediately following the procedure (IE after effects of general anesthesia are gone), then again at mid- and long-term intervals. (2) Examine these same parameters after placing stents in more proximal locations such as the thoracic aorta, aortic arch and ascending aorta. (3) Perform a similar comprehensive analysis with a variety of graft materials and designs (both surgical and endovascular) to determine inter-graft differences. These additional studies would allow for a more thorough description of the changes that occur after endovascular graft procedures, the clinical implications of these changes, and further explore ways to minimize any potential negative effects with different graft designs or materials.
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Department of Anesthesiology, Perioperative and Pain Medicine, Cumming School of Medicine, University of Calgary, Calgary, AB, Canada
Alexander Gregory, Marelise Kruger, Neal Maher & Gary Dobson
Division of Vascular Surgery, Peter Lougheed Hospital, Calgary, AB, Canada
Libin Cardiovascular Institute, Calgary, AB, T2N 2T9, Canada
Alexander Gregory
Marelise Kruger
Neal Maher
Gary Dobson
All authors were actively involved in the conception, design, implementation, and data collection for the study. All authors met to discuss the findings of the study with respect to manuscript preparation prior to it being written. AG was the principal author of the manuscript's first draft, produced the figures and tables, and edited subsequent revisions. MK reviewed/edited the manuscript first draft and made significant editorial contributions to the final version which has been submitted. NM reviewed/edited the manuscript first draft and made significant editorial contributions to the final version which has been submitted. RM reviewed/edited the manuscript first draft and made significant editorial contributions to the final version which has been submitted. GD was the co-author of the manuscript's first draft. He was the principal author involved in data analysis and helped produce many of the figures/tables. He also reviewed/edited the original manuscript and made significant editorial contributions to the final version which has been submitted.
Correspondence to Alexander Gregory.
None of the listed authors has any financial or personal relationships with other people or organizations that could potentially/inappropriately influence their work and conclusions with respect to this study or the submitted manuscript. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Gregory, A., Kruger, M., Maher, N. et al. Non-invasive Determination of Aortic Mechanical Properties and Their Effects on Left Ventricular Function Following Endovascular Abdominal Aneurysm Repair. J. Med. Biol. Eng. 39, 739–751 (2019). https://doi.org/10.1007/s40846-018-0455-1
Endovascular aneurysm repair
Aortic compliance
Aortic strain
Aortic impedance
Wave form analysis
Speckle tracking echocardiography | CommonCrawl |
Additional Questions Regarding the Auto-Ignition Temperature
I had some followup questions regarding a previous post I made here regarding the auto-ignition temperature and ASTM E659
For fuel temperature below AIT, we should still have finite reactants above activation energy, reacting, and heating the remaining mixture so additional reactants are above the activation energy. Theoretically as $t \rightarrow \infty$, could the fuel burn itself out like this, regardless of what temperature it is at (obviously the rate will be very different)? If so, is there an implicit rate requirement in defining the AIT that the above chain reaction process has to happen within a short duration?
Given that ignition is defined as when a flash and temperature rise is seen, does this mean that the actual quantity of fuel burnt is unimportant and it is assumed that a significant enough amount is used?
Is the reason for using an open flask in ASTM E659 likely just for simplicity, as opposed to a piston-cylinder arrangement that offers more control over air-fuel mixture?
physical-chemistry combustion
YandleYandle
$\begingroup$ I don't have time for a proper answer, but in 1, it sounds like you've got the right idea. Think of paper browning; over centuries it will oxidize into brittle, ruined material, essentially burning to completion at room temperature air. Here the time limit they establish for a flame to appear is explicit, 10 minutes. They do state that some compounds have a longer delay at a given temperature before they ignite. The reaction rate you refer to would be different for each substance, since each is releasing a different amount of energy. $\endgroup$ – Jason Patterson Oct 15 '14 at 12:06
To understand the answer to (1), you have to think about what combustion actually is. It isn't a single reaction, with a single set of reactants, and a single set of products. It's actually a whole bunch of tiny steps (thousands) that all occur together. Those reactions happen among a whole bunch of unstable radical species (hundreds). When you put a match to a pool of gasoline, all that energy from the match starts tearing apart bonds in the fuel, leading to the formation of radicals. That's what triggers the initial release of energy, and what creates the cascade of ignition. So theoretically, no, you could never reach combustion products, through the pathway of combustion, because you'd never have that activation energy, the big kick of energy needed to create radicals and keep them alive long enough for them to create a chain reaction.
The answer to (2) is, no, it doesn't matter how much fuel. Remember you're measuring a mass-independent property. The autoginition temperature isn't "5 degrees per gram". It's a single temperature. What matters is the ratio of oxygen to fuel.
As to (3), it's a lot easier to operate an open flask than it is to operate a piston-cylinder, and it doesn't have a substantial impact on the air-fuel ratio if you do it correctly.
charlesreid1charlesreid1
$\begingroup$ Regarding (1), given infinite time, every finite kinetic barrier will eventually be overcome at any finite temperature, no matter how high the barrier is, or how close to $0\ \mathrm{K}$. For example, it would take at most approximately $10^{1500}$ years for nuclei to spontaneously fuse into iron. Any chemical reaction would have a much, much lower kinetic barrier, and would happen in a much shorter timescale. Whether a puddle of fuel inside an atmosphere of oxygen at $300\ \mathrm{K}$ could have had enough time to oxidize completely during the current lifetime of the Universe, I don't know. $\endgroup$ – Nicolau Saker Neto Jun 6 '15 at 3:23
$\begingroup$ That's getting into the realm of metaphysics. The mass of fuel you'd need would be greater than the mass of carbon in the universe. Sounds like navel-gazing to me... $\endgroup$ – charlesreid1 Jun 6 '15 at 17:14
$\begingroup$ @charlesreid1 (1) A little confused. The answer implies that fuel-air require an ignition source to or else reaction will not occur. But aren't oxidation reactions spontaneous so we should expect the products to form so long as T (and thus forward rate constant) is greater than zero? My current though is that as t -> infinity, enough fuel will react such that equilibrium will reached (and depending on the heat of reaction there should be very little fuel remaining). $\endgroup$ – Yandle Jun 8 '15 at 4:39
$\begingroup$ A fuel-air mixture does require an ignition source to combust. a spark is just a big shot of energy to the system, transferred from one set of (say, metal) molecules to another set of (say, methane) molecules. that energy starts tearing apart molecules to create radicals, which initiates combustion. it isn't a spontaneous process at all. otherwise, all fuel in existence would spontaneously combust! $\endgroup$ – charlesreid1 Jun 8 '15 at 5:01
$\begingroup$ @charlesreid1 What I'm confused about is that, if we have an closed system of fuel-air, T and Ea are finite, and per Arrhenius equation forward rate constant and reactant concentration are all greater than zero. From this I expect a net formation of products (given that initial product concentration is zero) until equilibrium. I get the logic in your explanation but I also don't know the flaw in my logic. Also, isn't AIT technically a point where fuel spontaneous combust? $\endgroup$ – Yandle Jun 12 '15 at 5:40
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How or Why does diesel/kerosene have a much higher 'flash point' but lower auto/self-ignition point than gasoline/petrol? | CommonCrawl |
\begin{document}
\title{On the delooping of (framed) embedding spaces}
\author{Julien Ducoulombier} \address{Max Planck Institute for Mathematics \\ Vivatsgasse 7 \\ R\"amistrasse 101 \\ 53111 Bonn, Germany} \email{[email protected]}
\author{Victor Turchin} \address{Department of Mathematics\\ Kansas State University\\ 138 Cardwell Hall\\ Manhattan, KS 66506, USA} \email{[email protected]}
\author{Thomas Willwacher} \address{Department of Mathematics \\ ETH Zurich \\ R\"amistrasse 101 \\ 8092 Zurich, Switzerland} \email{[email protected]}
\date{}
\thanks{ The authors acknowledge the University of Lille for hospitality. V.T. has benefited from a visiting position of the Labex CEMPI (ANR-11-LABX-0007-01) at the Universit\'e de Lille and from a visiting position at the Max Planck Institute for Mathematics, Bonn for the achievement of this work. He was also partially supported by the Simons Foundation grant ID:519474. T.W. and J.D. have been partially supported by the NCCR SwissMAP funded by the Swiss National Science Foundation and the ERC starting grant GRAPHCPX}
\begin{abstract} It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings. \end{abstract}
\maketitle
\section{Introduction}
Let $\mathcal P$ and $\mathcal Q$ be topological operads satisfying some mild conditions to be detailed below. Suppose furthermore that we have a map of (pointed) spaces from some space $X$ to the operadic mapping space \[ X\to \mathrm{Operad}(\mathcal P, \mathcal Q). \] Then one may in particular form the $\mathcal P$-bimodule $\mathcal Q{\circ} X$ defined such that $(\mathcal Q{\circ} X)(n)=\mathcal Q(n)\times X^{\times n}$, for which one uses the basepoint to define the left $\mathcal P$-action and the map from $X$ to the mapping space to define the right $\mathcal P$-action (see Section~\ref{s2}). Our main result Theorem~\ref{th:main} is then that the following is a homotopy fiber sequence \beq{equ:main} \mathrm{Bimod}^h_{\mathcal P}(\mathcal P, \mathcal Q{\circ} X) \to X \to \mathrm{Operad}^h(\mathcal P, \mathcal Q), \end{equation} where the superscript $h$ is used to show that we consider the derived version of the corresponding mapping spaces. This result can be considered as a generalisation of the delooping result \cite{DwyerHess0,Duc} \beq{eq:ducoul} \mathrm{Bimod}^h_{\mathcal P}(\mathcal P, \mathcal Q) \simeq \Omega\mathrm{Operad}^h(\mathcal P, \mathcal Q), \end{equation} which can be recovered by setting $X=*$ to be a point. It is shown by the first author in~\cite{Duc} that~\eqref{eq:ducoul} is an equivalence of algebras over the little segments operad $\mathcal{D}_1$. In Section \ref{Final2} we improve this result and build an explicit weak equivalence of $\mathcal{SC}_{1}$-algebras between the pairs $$ \left( \begin{array}{c}
\Omega\mathrm{Operad}^h(\mathcal P, \mathcal Q)) \\ hofiber(X\to \mathrm{Operad}^h(\mathcal P, \mathcal Q)) \end{array} \right) \longrightarrow \left( \begin{array}{c}
\mathrm{Bimod}^h_{\mathcal P}(\mathcal P, \mathcal Q) \\ \mathrm{Bimod}^h_{\mathcal P}(\mathcal P, \mathcal Q{\circ} X) \end{array} \right) $$ where $\mathcal{SC}_{1}$ is the one dimensional Swiss-Cheese operad.
We propose three applications of the above results.
\noindent \textbf{Application 1: The space of disc embeddings.} Consider the space $\Emb_\partial(D^m, D^n)$ of disc embeddings fixed in a neighbourhood of the boundary to be the standard equatorial inclusion $S^{m-1}\subset S^{n-1}$. Assume furthermore $n-m\geq 3$ throughout. Let also $\mathcal{D}_k$ denote the little $k$-discs operad. Then the embedding space has two known deloopings, which we shall briefly describe. First, one considers the homotopy fiber over immersions \[ \Embbar_\partial(D^m, D^n) =\mathrm{hofiber}\bigl( \Emb_\partial(D^m, D^n) \to \Imm_\partial(D^m, D^n)\simeq \Omega^m\mathrm{V}_{m,n}\bigr), \] where $\mathrm{V}_{m,n}$ is the Stiefel manifold. It has been shown in \cite{WBdB2} that $\Embbar_\partial(D^m, D^n)\cong \Omega^{m+1}\mathrm{Operad}^h(\mathcal{D}_m,\mathcal{D}_n)$, and that furthermore \beq{eq:BW1} \Emb_\partial(D^m,D^n) \simeq \Omega^m \mathrm{hofiber}(\mathrm{V}_{m,n}\to \mathrm{Operad}^h(\mathcal{D}_m,\mathcal{D}_n)). \end{equation} A second delooping is obtained in \cite{DucT}, where it is shown that \beq{eq:DucT1} \Emb_\partial(D^m,D^n) \simeq \Omega^m \mathrm{Bimod}^h_{\mathcal{D}_m}(\mathcal{D}_m,\mathcal{D}_n^{m-{\mathrm{fr}}}), \end{equation} where $\mathcal{D}_n^{m-{\mathrm{fr}}}$ is the bimodule of $m$-framed little $n$-disks, which one should think of as embeddings of $m$-dimensional disks in the unit $n$-disk.
Our result \eqref{equ:main} above with $X=\mathrm{V}_{m,n}$ then shows that both deloopings agree: \beq{eq:cor1} \mathrm{hofiber}(\mathrm{V}_{m,n}\to \mathrm{Operad}^h(\mathcal{D}_m,\mathcal{D}_n)) \simeq \mathrm{Bimod}^h_{\mathcal{D}_m}(\mathcal{D}_m,\mathcal{D}_n^{m-{\mathrm{fr}}}). \end{equation}
\noindent \textbf{Application 2: The space of framed disc embeddings.} Next consider the case that the operad $\mathcal Q$ is acted upon by a topological group $G$. Assuming that we have some map $f:\mathcal P\to \mathcal Q$ we hence obtain a map \[ G \to \mathrm{Operad}(\mathcal P,\mathcal Q) \] by composing $f$ and the $G$-action. The result \eqref{equ:main} in this case yields the first items of the fiber sequence \[ \mathrm{Bimod}_{\mathcal P}^h(\mathcal P,\mathcal Q{\circ} G) \to G \to \mathrm{Operad}^h(\mathcal P, \mathcal Q) \to \mathrm{Operad}^h(\mathcal P, \mathcal Q) \sslash G. \] Since in this case the fiber sequence may be extended as shown we obtain the delooping \beq{equ:bimoddeloop} \mathrm{Bimod}_{\mathcal P}^h(\mathcal P,\mathcal Q{\circ} G) \simeq \Omega(\mathrm{Operad}^h(\mathcal P, \mathcal Q) \sslash G). \end{equation}
Note that in this case the $\mathcal P$-bimodule $\mathcal Q{\circ} G$ is in fact an operad. However, the equivalence~\eqref{eq:ducoul} holds provided $\mathcal Q(1)\simeq *$ and therefore it might not be true if the operad $\mathcal Q$ is replaced by $\mathcal Q{\circ} G$. In fact in case $\mathcal P(0)=\mathcal Q(0)=*\simeq \mathcal P(1)\simeq \mathcal Q(1)$, which we will be assuming throughout the paper, and assuming that $G$ is connected and $\nsimeq *$, one has that the derived mapping spaces of bimodules \begin{equation}\label{eq:fr_nfr} \mathrm{Bimod}_{\mathcal P}^h(\mathcal P,\mathcal Q{\circ} G)\nsimeq\mathrm{Bimod}_{\mathcal P}^h(\mathcal P,\mathcal Q) \end{equation} are not weakly equivalent. By contrast, one has \begin{equation}\label{eq:fr_nfr2} \mathrm{Operad}^h(\mathcal P, \mathcal Q{\circ}G)\simeq \mathrm{Operad}^h(\mathcal P,\mathcal Q), \end{equation} see Remark~\ref{r:framed_bim_oper}
We apply the above findings to the spaces of framed disc embeddings $\Emb_\partial^{\mathrm{fr}}(D^m, D^n)$. It is shown in \cite{DucT} that \beq{eq:DucT2} \Emb_\partial^{\mathrm{fr}}(D^m, D^n) \simeq \Omega^m \mathrm{Bimod}^h_{\mathcal{D}_m}(\mathcal{D}_m, \mathcal{D}_n^{{\mathrm{fr}}}), \end{equation} where $\mathcal{D}_n^{{\mathrm{fr}}}$ denotes the operad of positively framed little $n$-discs. Applying \eqref{equ:bimoddeloop} we hence obtain the $(m+1)$-st delooping \beq{eq:cor2} \Emb_\partial^{\mathrm{fr}}(D^m, D^n) \simeq \Omega^{m+1}(\mathrm{Operad}^h(\mathcal{D}_m, \mathcal{D}_n) \sslash \mathrm{SO}(n)). \end{equation}
One should mention that the delooping of the space of framed knots~\eqref{eq:cor2} that we discovered, can be alternatively proved from the Boavida-Weiss result~\eqref{eq:BW1}, as explained in the last section (see Remark~\ref{r:sakai2}).
\noindent \textbf{Application 3: The Goodwillie-Weiss calculus and the smoothing theory.} The deloopings~\eqref{eq:BW1}, \eqref{eq:DucT1}, \eqref{eq:DucT2} were obtained in~\cite{WBdB2,DucT} using the Goodwillie-Weiss functor calculus on manifolds. In fact one obtains there the deloopings of the Taylor towers $T_k\Emb_\partial(D^m,D^n)$ and $T_k\Emb_\partial^{\mathrm{fr}}(D^m,D^n)$, $1\leq k\leq \infty$, (without any codimension restriction on $m$ and $n$) by taking the derived mapping spaces of $k$-truncated operads and bimodules. Similarly~\eqref{eq:ducoul} and our main result~\eqref{equ:main} also have a truncated version: \beq{equ:main2} \mathrm{Bimod}^h_{\mathcal P;\leq k}(\mathcal P_{\leq k}, \mathcal (Q{\circ} X)_{\leq k}) \to X \to \mathrm{Operad}^h_{\leq k}(\mathcal P_{\leq k}, \mathcal Q_{\leq k}) \end{equation} is a homotopy fiber sequence for any $k\geq 1$, see Theorem~\ref{th:main}.
The obtained delooping result is of a particular interest when $m=n$: \beq{eq:cor3} T_\infty\Emb_\partial(D^n, D^n) \simeq \Omega^{n+1}(\mathrm{Aut}^h(\mathcal{D}_n) \sslash \mathrm{SO}(n)), \end{equation} which should be compared to the Morlet-Burghelea-Lashof delooping of the group of relative to the boundary disc diffeomorphisms~\cite{BuL}: \[ \Diff_\partial(D^n)\simeq\Omega^{n+1}(\mathrm{TOP}(n)/\mathrm{O}(n)),\,\, n\neq 4. \]
In~\cite{Sakai} based on the Burghelea-Lashof work, K.~Sakai produces a similar delooping of the space $\Emb_\partial(D^m,D^n)$. In the last Section~\ref{s:last} we show how this smoothing theory delooping can be adjusted to get a delooping of the space $\Emb^{\mathrm{fr}}_\partial(D^m,D^n)$ of framed embeddings. It is a very intriguing question whether the smoothing theory deloopings of spaces of long embeddings agree with the operadic ones arising from the Goodwillie-Weiss calculus in the case when both deloopings are available (i.e., for $n-m\geq 3$ and $n\geq 5$).
For other related results on the little discs action on the spaces of disc embeddings and results on their deloopings, we refer the reader to~\cite{BatDL,Budney1,Budney2,Duc2,Mostovoy,MoriyaSakai,Salvatore1,Turchin5}.
\section{The Reedy model categories of reduced operads and bimodules}\label{s2}
In this section, we cover the notion of a (truncated) operad and a (truncated) bimodule over an operad. We equip these two categories with model category structures, called Reedy model category structures, using left adjoints of the forgetful functors to the model category of $\Lambda$-sequences. For a more detailed account about the category of $\Lambda$-sequences and the Reedy model category of reduced operads, we refer the reader to \cite{Fr}. A precise study of the Reedy model category of reduced bimodules can be found in \cite{DucT,DucTF}.
\subsection{The model category of $\Lambda$-sequences} Let $\Lambda$ be the category whose objects are finite sets $[n]:=\{1,\ldots, n\}$, with $n\geq 1$, and morphisms are injective maps between them. By a $\Lambda$-sequence, we understand a functor $Y:\Lambda^{op}\rightarrow \TopCat$. By convention, we denote by $Y(n)$ the space $Y([n])$. In practice, a $\Lambda$-sequence $Y$ is a family of spaces $Y(1),\, Y(2),\ldots$ together with operations of the form $$ u^{\ast}:Y(n)\rightarrow Y(m),\hspace{20pt}\text{ for any } u\in \Lambda([m]\,;\,[n]). $$ A $\Lambda$-sequence $Y$ is said to be pointed if the space $Y(1)$ is equipped with a basepoint.
Following \cite{Fr}, the categories $\Lambda Seq$ and $\Lambda Seq^{\ast}$ of $\Lambda$-sequences and pointed $\Lambda$-sequences, respectively, are endowed with model category structures in which a natural transformation $f:Y\rightarrow Z$ is a weak equivalence if it is an objectwise weak homotopy equivalence. Furthermore, a natural transformation $f$ is a fibration if the maps $Y(n)\rightarrow \mathcal{M}(Y)(n)\times_{\mathcal{M}(Z)(n)}Z(n)$, with $n\geq 1$, are Serre fibrations. The space $\mathcal{M}(Y)(n)$, called matching object of $Y$, is given by the formula \begin{equation}\label{B3} \mathcal{M}(Y)(n):=\underset{\substack{\Lambda_{+}([i]\,;\,[n])\\ i< n}}{lim}\,\, Y(i), \end{equation} where $\Lambda_{+}$ is the subcategory of $\Lambda$ consisting of order-preserving maps. Similarly, for $k\geq 1$, the category of $k$-truncated $\Lambda$-sequences $\Lambda Seq_{\leq k}$ (resp. $k$-truncated pointed $\Lambda$-sequences $\Lambda Seq^{\ast}_{\leq k}$), whose objects are functors $Y\in \Lambda Seq$ (resp. $Y\in \Lambda Seq^{\ast}$) having $Y(n)=\emptyset$ for all $n>k$, inherits a model category structure.
\begin{defn}\label{d:circle} Given a topological space $X$, define the $\Lambda$-sequence $X^{\times\bullet}$ assigning to $[n]$ the space $X^{\times n}$ of maps $[n]\to X$. The $\Lambda$-action is defined by precomposition: for any $u\in \Lambda([m]\,,\,[n])$, \[ u^\ast\colon (x_1,\ldots,x_n)\longmapsto (x_{u(1)},\ldots,x_{u(m)}). \] For a $\Lambda$-sequence $Y$, define a $\Lambda$-sequence $Y{\circ}X$ as an objectwise product of $Y$ and $X^{\times\bullet}$. \end{defn}
\begin{lemma}\label{l:circle} In case $Y$ is a Reedy fibrant $\Lambda$-sequence, $X$ is any space, the $\Lambda$-sequence $Y{\circ}X$ is also Reedy fibrant. \end{lemma}
\begin{proof} It is easy to see that $X^{\times\bullet}$ is a Reedy fibrant $\Lambda$-sequence. Indeed, we have that \[ \mathcal{M}(X^{\times\bullet})(n)= \begin{cases} *,&n=1\\ X^{\times n},&\text{otherwise} \end{cases}. \] Thus $X^{\times n}\to \mathcal{M}(X^{\times\bullet})(n)$ is always a Serre fibration. On the other hand, the objectwise product of two Reedy fibrant $\Lambda$-sequences is so as well. \end{proof}
\begin{defn} By a $\Sigma${\rm -cofibrant} object, we understand a $\Lambda$-sequence $X$ such that each space $X(n)$, with $n\geq 1$, is cofibrant in the model category $\TopCat_{\Sigma_{k}}$ of spaces equipped with an action of the symmetric group $\Sigma_{k}$. Fibrations and weak equivalences for this model structure are objectwise Serre fibrations and objectwise weak equivalences. \end{defn}
\subsection{The Reedy model category of reduced operads} A reduced operad $\mathcal O$ is a pointed $\Lambda$-sequence $\mathcal O: \Lambda^{op}\rightarrow \TopCat$ together with operations, called \textit{operadic compositions}, of the form \begin{equation}\label{B4} {\circ}_{i}:\mathcal O(n)\times \mathcal O(m)\longrightarrow \mathcal O(n+m-1),\hspace{15pt} \text{with } 1\leq i\leq n, \end{equation} satisfying compatibility with the $\Lambda$-structure on $\mathcal O$, associativity and unit axioms \cite[Part II Section 8.2]{Fr}. A map between reduced operads should respect the operadic compositions. We denote by $\mathrm{Operad}$ the category of reduced operads. Note that $\mathrm{Operad}$ is equivalent to the full subcategory of topological operads having a point as an arity-zero component. In what follows, we use the notation $$ x{\circ}_{i}y={\circ}_{i}(x,y),\hspace{20pt}\text{for all } x\in O(n) \text{ and } y\in O(m). $$
Given an integer $k\geq 1$, we also consider the category of $k$-truncated reduced operads denoted by $\mathrm{Operad}_{\leq k}$. The objects are $k$-truncated pointed $\Lambda$-sequences together with operations of the form (\ref{B4}) with $n+m\leq k+1$. Furthermore, one has the following functor called the $k$-truncation functor: $$ \begin{array}{rcl}
(-)_{\leq k}:\mathrm{Operad} & \longrightarrow & \mathrm{Operad}_{\leq k} \\
\mathcal O & \longmapsto & \mathcal O_{\leq k}:=\left\{
\begin{array}{ll}
\mathcal O_{\leq k}(n)=\mathcal O(n) & \text{if } n\leq k, \\
\mathcal O_{\leq k}(n)=\emptyset & \text{if } n > k.
\end{array}
\right. \end{array} $$
For $k\geq 1$, the categories $\mathrm{Operad}$ and $\mathrm{Operad}_{\leq k}$ are endowed with the so called Reedy model category structures transferred from $\Lambda Seq^{\ast}$ and $\Lambda Seq^{\ast}_{\leq k}$, respectively, along the adjunctions $$ \begin{array}{rcl}
\mathcal{F}_{Op}: \Lambda Seq^{\ast} & \leftrightarrows & \mathrm{Operad}:\mathcal{U}, \\ \mathcal{F}_{Op\,;\,\leq k}: \Lambda Seq^{\ast}_{\leq k} & \leftrightarrows & \mathrm{Operad}_{\leq k}:\mathcal{U}, \end{array} $$ where $\mathcal{U}$ is the forgetful functor while $\mathcal{F}_{Op}$ and $\mathcal{F}_{Op\,;\,\leq k}$ are the free operadic functors. In other words, a map $f:\mathcal P \rightarrow \mathcal Q$ of (possibly truncated) operads is a weak equivalence (resp. a fibration) if the corresponding map $\mathcal{U}(f)$ is a weak equivalence (resp. a fibration) in the model category of (possibly truncated) pointed $\Lambda$-sequences.
\begin{ex}\textit{The framed operad $\mathcal O{\circ} G$}
\noindent Let $G$ be a topological group and $\mathcal O$ be a reduced operad for which each space $\mathcal O(n)$ admits an action of $G$ compatible with the $\Lambda$ structure and the operadic compositions. Then, the $\Lambda$-sequence $\mathcal O{\circ} G$, see Definition~\ref{d:circle}, inherits an operadic structure from the operad $\mathcal O$ and the group structure of $G$.
The operadic compositions are given by the following formula: $$ \begin{array}{rlcl} {\circ}_{i}:&\mathcal O{\circ} G(n)\times \mathcal O{\circ} G(m) & \longmapsto & \mathcal O{\circ} G(n+m-1); \\ &(\theta;g_{1},\ldots,g_{n}) \,;\, (\theta';g'_{1},\ldots,g'_{n})& \longmapsto & (\theta {\circ}_{i} (g_{i}\cdot \theta'); g_{1},\ldots,g_{i-1},g_{i}g'_{1},\ldots,g_{i}g'_{m},g_{i+1},\ldots,g_{n}). \end{array} $$ \end{ex}
\begin{ex}\textit{The little discs operads $\mathcal{D}_{m}$}
\noindent In arity $n$, the space $\mathcal{D}_{m}(n)$ is the configuration space of $n$ discs of dimension $m$, labelled by $[n]$, inside the unit disc of dimension $m$ having disjoint interiors. The unit in arity $1$ is given by the identity map. The $\Lambda$-structure is obtained by removing some discs and permuting the other ones. Finally, the operadic composition ${\circ}_{i}$ substitutes the $i$-th disc of the first configuration by the second configuration as illustrated in Figure \ref{B5}. In particular, each space $\mathcal{D}_{m}(n)$ admits an action of $\mathrm{SO}(m)$ and we denote by $\mathcal{D}_{m}^{{\mathrm{fr}}}$ the corresponding framed operad.
\begin{figure}
\caption{Illustration of the operadic composition ${\circ}_{2}:\mathcal{D}_{2}(3)\times \mathcal{D}_{2}(2)\rightarrow \mathcal{D}_{2}(4)$.}
\label{B5}
\end{figure} \end{ex}
\subsection{The Reedy model category of reduced bimodules over a reduced operad} \label{Final1}
Let $\mathcal O$ be a reduced operad. A reduced bimodule $\mathcal M$ over the operad $\mathcal O$, or $\mathcal O$-bimodule, is a $\Lambda$-sequence $\mathcal M:\Lambda^{op}\rightarrow \TopCat$ together with operations of the form \begin{equation}\label{B6} \begin{array}{ll}
{\circ}^{i}:\mathcal M(n)\times \mathcal O(m) \longrightarrow \mathcal M(n+m-1), & \text{called right operations with } 1\leq i\leq n,\,\, m\geq 1 \\ \gamma:\mathcal O(n)\times \mathcal M(m_{1})\times \cdots \times M(m_{n})\longrightarrow \mathcal M(m_{1}+\cdots + m_{n}) & \text{called left operation,}\,\, n\geq 1, \text{ each $m_i\geq 1$}, \end{array} \end{equation} satisfying some compatibility relations with the $\Lambda$-structure, associativity and unit axioms \cite{DucT}. A map between $\mathcal O$-bimodules should respect these operations. We denote by $\mathrm{Bimod}_{\mathcal O}$ the category of reduced bimodules over the reduced operad $\mathcal O$. In what follows, we use the notation $$ \begin{array}{cl} x\circ^{i}y=\circ^{i}(x,y) & \text{for } x\in \mathcal M(n) \text{ and } y\in \mathcal O(m), \\ x(y_{1},\ldots,y_{n})=\gamma(x,y_{1},\ldots,y_{n}) & \text{for } x\in \mathcal O(n) \text{ and } y_{i}\in \mathcal M(m_{i}). \end{array} $$
Given an integer $k\geq 1$, we also consider the category of $k$-truncated reduced bimodules over $\mathcal O$ denoted by $\mathrm{Bimod}_{O\,;\,\leq k}$. The objects are $k$-truncated $\Lambda$-sequences together with operations of the form (\ref{B6}) with $n+m-1\leq k$ for the right operations and $m_{1}+\cdots + m_{n}\leq k$ for the left operation. Furthermore, one has the functor $$ \begin{array}{rcl}
(-)_{\leq k}:\mathrm{Bimod}_{\mathcal O} & \longrightarrow & \mathrm{Bimod}_{\mathcal O\,;\,\leq k}; \\
\mathcal M & \longmapsto & \mathcal M_{\leq k}:=\left\{
\begin{array}{ll}
\mathcal M_{\leq k}(n)=\mathcal M(n) & \text{if } n\leq k, \\
\mathcal M_{\leq k}(n)=\emptyset & \text{if } n > k.
\end{array}
\right. \end{array} $$
For $k\geq 1$, the categories $\mathrm{Bimod}_{O}$ and $\mathrm{Bimod}_{O\,;\,\leq k}$ of reduced bimodules and $k$-truncated reduced bimodules over a reduced operad $O$, respectively, are also endowed with Reedy model category structures transferred from $\Lambda Seq$ and $\Lambda Seq_{\leq k}$, respectively, along the adjunctions $$ \begin{array}{rcl}
\mathcal{F}_{B}: \Lambda Seq & \leftrightarrows & \mathrm{Bimod}_{O}:\mathcal{U}, \\ \mathcal{F}_{B\,;\,\leq k}: \Lambda Seq_{\leq k} & \leftrightarrows & \mathrm{Bimod}_{O\,;\,\leq k}:\mathcal{U}, \end{array} $$ where $\mathcal{U}$ is the forgetful functor while $\mathcal{F}_{B}$ and $\mathcal{F}_{B\,;\,\leq k}$ are the free bimodule functors. In other words, a map $f:\mathcal P \rightarrow \mathcal Q$ of (possibly truncated) $O$-bimodules is a weak equivalence (resp. a fibration) if the corresponding map $\mathcal{U}(f)$ is a weak equivalence (resp. a fibration) in the model category of (possibly truncated) $\Lambda$-sequences.
\begin{ex} Let $\eta:\mathcal P\rightarrow \mathcal Q$ be a map of operads. In that case, the map $\eta$ is also a bimodule map over $\mathcal P$ and the right operations of the bimodule structure on $\mathcal Q$ are given by $$ \begin{array}{lrll}
{\circ}^{i}: & \mathcal Q(n)\times \mathcal P(m) & \longrightarrow & \mathcal Q(n+m-1); \\
& (x\,;\,y) &\longmapsto & x{\circ}_{i}\eta(y), \end{array} $$ while the left operation is defined as follows: $$ \begin{array}{lrll}
\gamma: & \mathcal P(n)\times \mathcal Q(m_{1})\times \cdots\times \mathcal Q(m_{n}) & \longrightarrow & \mathcal Q(m_{1}+\cdots +m_{n});\\ & (x\,;\,y_{1},\ldots,y_{n})&\longmapsto & (\cdots(\eta(x){\circ}_{n}y_{n})\cdots){\circ}_{1}y_{1}. \end{array} $$ \end{ex}
\begin{ex}\label{B7}\textit{The fiber bundle bimodule $\mathcal Q{\circ} X$.}
\noindent Let $f:\mathcal P \rightarrow \mathcal Q$ be a map of reduced operads and $(X\,;\,\ast)$ be a pointed space equipped with a map $\delta:X\rightarrow \mathrm{Operad}(\mathcal P, \mathcal Q)$ sending the basepoint to $f$. By convention, we denote by $\delta_{x}:\mathcal P \rightarrow \mathcal Q$ the operadic map associated to $x\in X$. We can think of $X$ as a space of $P$-bimodule structures on $\mathcal Q$ by twisting the right module structure. Then, the $\Lambda$-sequence $\mathcal Q{\circ} X$, see Definition~\ref{d:circle},
is a $\mathcal P$-bimodule.
The left operation is obtained using the operadic map $f$ $$ \begin{array}{rcl}
\gamma:\mathcal P(n)\times \mathcal Q{\circ} X(m_{1})\times \cdots \times \mathcal Q{\circ} X(m_{n}) & \longmapsto & \mathcal Q{\circ} X(m_{1}+\cdots + m_{n});\\ p\,;\,\{(q_{i},x_{1}^{i},\ldots,x_{m_{i}}^{i})\} & \longmapsto & (\delta_{\ast}(p)(q_{1},\ldots,q_{n}),x_{1}^{1},\ldots,x_{m_{n}}^{n}), \end{array} $$ while the right operations are given by $$ \begin{array}{rcl}
{\circ}^{i}: \mathcal Q{\circ} X(n)\times \mathcal P(m) & \longrightarrow & \mathcal Q{\circ} X(n+m-1); \\
(q,x_{1},\ldots,x_{n})\,;\,p & \longmapsto & (q{\circ}_{i}\delta_{x_{i}}(p),x_{1},\ldots,\underset{m \text{ times}}{\underbrace{x_{i},\ldots,x_{i}}},\ldots, x_{n}). \end{array} $$
In the sequel we consider a more general situation -- when the map $\delta$ sends $X$ to the derived operadic mapping space: \beq{eq:mapX} \delta\colon X\to \mathrm{Operad}^h(\mathcal P,\mathcal Q). \end{equation} Assuming $Q$ is Reedy fibrant (if necessary by taking its fibrant replacement $\mathcal Q\to \mathcal Q^f$) and $\mathcal P$ is $\Sigma$-cofibrant, the target of $\delta$ is the operadic mapping space $\mathrm{Operad}^h(\mathcal P,\mathcal Q)=\mathrm{Operad}(W\mathcal P,Q)$, where $W\mathcal P$ is the Boardman-Vogt resolution of $\mathcal P$ reviewed in the next section. Thus $\mathcal Q {\circ} X$ is given the structure of a $W\mathcal P$-bimodule. Let us mention, however, that the restriction-induction adjunction \beq{eq:restr_ind} \mathrm{Ind}\colon \mathrm{Bimod}_{W\mathcal P}\rightleftarrows\mathrm{Bimod}_{\mathcal P}\colon \mathrm{Restr} \end{equation} is a Quillen equivalence~\cite[Theorem 3.7]{DucTF}, and therefore it does not matter which of the two homotopy categories of bimodules we consider. \end{ex}
{\bf Characterisation of cofibrations.} Let $\mathcal P$ be a reduced $\Sigma$-cofibrant operad. Denote by $\Sigma_{>0}\mathrm{Bimod}_{\mathcal P}$ the category of \lq\lq{}usual\rq\rq{} bimodules $M=\{M(n),\, n\geq 1\}$ over $\mathcal P$ which means: its objects $M$ have a componentwise symmetric group action and a $\mathcal P$-action~\eqref{B6}, but they are not given the $\Lambda$ structure. This category is endowed with the {\it projective} model structure for which weak equivalences and fibrations are those componentwise~\cite{DucTF}. By \cite[Theorem~3.5]{DucTF}, a morphism $M\to N$ in the Reedy model category $\mathrm{Bimod}_{\mathcal P}$ of reduced bimodules is a cofibration if and only if it is one in the projective model category $\Sigma_{>0}\mathrm{Bimod}_{\mathcal P}$.
\section{Delooping derived mapping spaces of bimodules}
\subsection{The Boardman-Vogt resolution in the category of operads}\label{sec:ducmap} Let $\mathcal P$ be a reduced operad. We denote its Boardman-Vogt resolution by $W\mathcal P$. Its points are equivalence classes $[T\,;\,\{t_{e}\}\,;\,\{a_{v}\}]$ where $T$ is a rooted tree, $\{a_{v}\}_{v\in V(T)}$ is a family of points in $\mathcal P$ labelling the vertices of $T$ and $\{t_{e}\}_{e\in E^{int}(T)}$ is a family of real numbers in the interval $[0\,,\,1]$ indexing the inner edges. In other words, one has $$ W\mathcal P(n):= \left.
\underset{T\in \,\text{\textbf{tree}}_{n}}{\displaystyle\coprod} \,\,\underset{v\in V(T)}{\displaystyle\prod}\,\mathcal P(|v|) \,\,\times \,\,\underset{e\in\, E^{int}(T)}{\displaystyle\prod}\, [0\,,\,1]\,\, \right/\!\sim\,\,\hspace{20pt} \text{with } n\geq 1, $$
where $\text{\textbf{tree}}_{n}$ is the set of planar rooted trees without univalent vertices and with $n$ leaves labelled by the set $\{1,\ldots,n\}$. In other words, such a decoration of the leaves can be considered as an element in permutation group $\Sigma_{n}$ thanks to the planar order. The equivalence relation is generated by the unit axiom (i.e. we remove vertices labelled by the unit of the operad $\mathcal P$) and the compatibility with the symmetric group axiom (a vertex $v$ labelled by $a\cdot\sigma$, with $\sigma\in \Sigma_{|v|}$, is identified with $a$ by permuting the incoming edges of $v$ according to $\sigma$). Furthermore, if an inner edge is indexed by $0$, then we contract it by using the operadic structure of $\mathcal P$. \begin{figure}
\caption{Illustration of the equivalence relation.}
\end{figure}
Let $[T\,;\,\{t_{e}\}\,;\,\{a_{v}\}]$ be a point in $W\mathcal P(n)$ and $[T'\,;\,\{t'_{e}\}\,;\,\{a'_{v}\}]$ be a point in $W\mathcal P(m)$. The operadic composition $[T\,;\,\{t_{e}\}\,;\,\{a_{v}\}]{\circ}_{i}[T'\,;\,\{t'_{e}\}\,;\,\{a'_{v}\}]$ is obtained by grafting $T'$ to the $i$-th incoming input of $T$ and indexing the new inner edge by $1$. The $\Lambda$-structure is defined by permuting the leaves and contracting some of them using the $\Lambda$-structure of the operad $\mathcal P$. Furthermore, there is a map of operads sending the real numbers indexing the inner edges to $0$ \begin{equation}\label{d8} \mu:W\mathcal P\rightarrow \mathcal P\,\,;\,\, [T\,;\,\{t_{e}\}\,;\,\{a_{v}\}] \mapsto [T\,;\,\{0_{e}\}\,;\,\{a_{v}\}]. \end{equation}
\begin{thm}\label{I0}{\cite[Theorem 5.1]{BM}, \cite[Theorem II.8.4.12]{Fr}} Assume that $\mathcal P$ is a $\Sigma$-cofibrant operad. The objects $W\mathcal P$ and $(W\mathcal P)_{\leq k}$ are cofibrant replacements of $\mathcal P$ and $\mathcal P_{\leq k}$ in the categories $\mathrm{Operad}$ and $\mathrm{Operad}_{\leq k}$, respectively. In particular, the map (\ref{d8}) is a weak equivalence. \end{thm}
From now on, we introduce a filtration of the resolution $W\mathcal P$ according to the arity. A point in $W\mathcal P$ is said to be \textit{prime} if the real numbers indexing the set of inner edges are strictly smaller than $1$. Besides, a point is said to be \textit{composite} if one of its inner edges is indexed by $1$ and such a point can be decomposed into prime components. More precisely, the prime components of a point indexed by a tree are obtained by cutting the edges labelled by $1$.
A prime point is in the $k$-th filtration term $W\mathcal P_{k}$ if it has at most $k$ leaves. Then, a composite point is in the $k$-th filtration term if its prime components are in $W\mathcal P_{k}$. For instance, the composite point in Figure \ref{G2} is an element in the filtration term $W\mathcal P_{4}$. By convention, $W\mathcal P_{0}$ is the initial object in the category of operads. For each $k\geq 0$, $W\mathcal P_{k}$ is a reduced operad and the family $\{W\mathcal P_{k}\}$ produces the following filtration of $W\mathcal P$: \begin{equation*}\label{G4} \xymatrix{ W\mathcal P_{0}\ar[r] & W\mathcal P_{1} \ar[r] & \cdots \ar[r] & W\mathcal P_{k-1} \ar[r] & W\mathcal P_{k} \ar[r] & \cdots \ar[r] & W\mathcal P. }
\end{equation*}
\begin{figure}
\caption{Illustration of a composite point together with its prime components.}
\label{G2}
\end{figure}
From a $k$-truncated reduced operad $\mathcal P_{k}$, we consider the $k$-free operad $\mathcal{F}_{Op_{k}}(\mathcal P_{k})$ whose $k$ first components coincide with $\mathcal P_{k}$. The functor $\mathcal{F}_{Op_{k}}$ is left adjoint to the truncation functor $(-)_{\leq k}$ and it can be expressed as a quotient of the free operad functor in which the equivalence relation is generated by the following axiom: any composite element is identified with the composition of its prime components.
In our case, we can easily check that $\mathcal{F}_{Op_{k}}((W \mathcal P)_{\leq k})=W\mathcal P_{k}$, since $W\mathcal P_{k}$ is the sub-operad of $W\mathcal P$ generated by its $k$ first components. Consequently, from this adjunction and Theorem~\ref{I0}, we deduce the following identifications: \begin{equation}\label{eq:tr_op_map} \mathrm{Operad}_{\leq k}((W \mathcal P)_{\leq k}\,,\,Q_{\leq k}) \cong \mathrm{Operad}(W\mathcal P_{k}\,,\,Q). \end{equation}
\subsection{A cofibrant resolution of $\mathcal P$ in the category of bimodules over itself}
The operad $\mathcal P$ may naturally be considered as a reduced bimodule of itself. We will use (a slight variant of) the cofibrant resolution $B\mathcal P$ of $\mathcal P$ as a bimodule introduced by Ducoulombier in \cite{Duc}. The points are equivalence classes $[T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]$ where $T$ is a tree, $\{t_{v}\}$ is a family of real numbers in the interval $[0\,,\,1]$ indexing the vertices and $\{x_{v}\}$ is a family of points in $W\mathcal P$ labelling the vertices. Furthermore, if $e$ is an inner edge of $T$, then the real numbers $t_{s(e)}$ and $t_{t(e)}$ indexing respectively the source and the target vertices of $e$ according to the orientation toward the root satisfy the relation $t_{s(e)}\geq t_{t(e)}$: \begin{equation}\label{B9} B\mathcal P (n)\subset \left.
\underset{T\in \textbf{tree}_{n}}{\displaystyle\coprod}\,\,\,\underset{v\in V(T)}{\displaystyle\prod}\, W\mathcal P(|v|)\times [0\,,\,1]\, \right/ \sim,\hspace{20pt}\text{with } n\geq 1. \end{equation} The equivalence relation is generated by the unit (i.e. we can remove the vertices indexing by the unit $\ast_{1}$ the operad $W\mathcal P$) and the compatibility with the symmetric group axioms. Furthermore, if two vertices joined by an edge have the same height, then the edge may be contracted, using the operadic composition in $W\mathcal P$ as illustrated in Figure~\ref{Fig4}.
\begin{figure}
\caption{Illustration of the equivalence relation on $B\mathcal P(5)$.}
\label{Fig4}
\end{figure}
The object so obtained inherits a bimodule structure over $W\mathcal P$. The left and right module structures along a point in $W\mathcal P(m)$, with $m\geq 1$, are both obtained by grafting trees, with the newly formed vertices being assigned height 0 for the left module structure and height 1 for the right module structure. Moreover, the $\Lambda$-structure is defined by permuting some leaves and contracting the other ones using the $\Lambda$ structure of $W\mathcal P$. Furthermore, there is a map of bimodules sending the real numbers indexing the vertices to $0$: \begin{equation}\label{B1} \mu':B\mathcal P\rightarrow W\mathcal P\,\,;\,\, [T\,;\,\{t_{v}\}\,;\,\{x_{v}\}] \mapsto \mu([T\,;\,\{0_{v}\}\,;\,\{x_{v}\}]).
\end{equation} \begin{figure}
\caption{Illustration of the right operation ${\circ}^{1}:B\mathcal P(4)\times W\mathcal P(3)\rightarrow B\mathcal P(6)$.}
\label{B2}
\end{figure}
\begin{thm}{\cite[Theorem 2.6]{Duc},\cite[Proposition 3.9]{DucTF}}\label{D8} Assume that $\mathcal P$ is a $\Sigma$-cofibrant operad. Then, the objects $B\mathcal P$ and $(B\mathcal P)_{\leq k}$ are cofibrant replacements of $\mathcal P$ and $\mathcal P_{\leq k}$ in the categories $\mathrm{Bimod}_{W\mathcal P}$ and $\mathrm{Bimod}_{W\mathcal P\,;\,\leq k}$, respectively. In particular, the map (\ref{B1}) is a weak equivalence. \end{thm}
\begin{figure}
\caption{An alternative representation of a point in $B\mathcal P(9)$.}
\label{C2}
\end{figure}
From now on, we introduce a filtration of the resolution $B\mathcal P$ according to the arity.
Similarly to the operadic case a point in $B\mathcal P$ is said to be \textit{prime} if the real numbers indexing the vertices of the main tree are in the interval $]0\,,\,1[$. Besides, a point is said to be \textit{composite} if one vertex of the main tree is indexed by $0$ or $1$ and such a point can be decomposed into prime components. More precisely, thanks to the unit axiom, we can always choose a representative point for which each path joining a leaf to the root passes through at least one vertex not indexed by $0$ or $1$ (by adding a bivalent vertex labelled by the unit if necessary). The prime components of such a point are obtained by removing the vertices of the main tree indexed by $0$ or $1$. For instance, the three prime components associated to the composite point in Figure \ref{C2} are the following ones:
\hspace{-40pt}\includegraphics[scale=0.21]{F8.pdf}
A prime point is in the $k$-th filtration term $B\mathcal P_{k}$ if it has at most $k$ leaves. Similarly, a composite point is in the $k$-filtration if its prime components are in $B\mathcal P_{k}$. For instance, the composite point in Figure \ref{C2} is an element in the filtration term $B\mathcal P_{4}$. In particular, if the vertices of a point in $B\mathcal P$ are indexed only by $0$ or $1$, then the point is in the first filtration term since its prime components are $1$-corollas indexed by the unit of the operad $W\mathcal P$ (this element can also be considered as the vertexless tree ``$|$''). By convention, $B\mathcal P_{0}$ is the initial element in the category of bimodules over $W\mathcal P$ -- it is empty in all arities~$\geq 1$. The family $\{B\mathcal P_{k}\}$ produces the following filtration of $B\mathcal P$:
\begin{equation}\label{C1} \xymatrix{ B\mathcal P_{0}\ar[r] & B\mathcal P_{1} \ar[r] & \cdots \ar[r] & B\mathcal P_{k-1} \ar[r] & B\mathcal P_{k} \ar[r] & \cdots \ar[r] & B\mathcal P_\infty:= B\mathcal P. } \end{equation}
It is shown by \cite[Theorem 2.6]{Duc} and \cite[Proposition 3.9]{DucTF} that each inclusion above is a Reedy cofibration (in $\mathrm{Bimod}_{W\mathcal P}$) and in any arity $n$, the inclusion $P_{k-1}(n)\to P_k(n)$ is a $\Sigma_n$-cofibration.
Analogously to the operadic case, from a $k$-truncated bimodule $M_{k}$, we consider the $k$-free bimodule $\mathcal{F}_{B_{k}}(M_{k})$ whose $k$ first components coincide with $M_{k}$. Similarly to the operadic case, the functor $\mathcal{F}_{B_{k}}$ is left adjoint to the truncation functor $(-)_{\leq k}$ and can be expressed as a quotient of the free bimodule functor in which the equivalence relation is generated by the following axiom: any composite element in $M_k$ is equivalent to the corresponding product of its prime components.
One has, $\mathcal{F}_{B_k}(B\mathcal P_{\leq k})=B\mathcal P_k$.
Consequently, there are the following identifications: \begin{equation}\label{eq:tr_bim_map} \mathrm{Bimod}_{W\mathcal P\,;\,\leq k}((B\mathcal P)_{\leq k}\,;\,Q_{\leq k}) \cong \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{k}\,;\,Q). \end{equation}
\subsection{The weak equivalence of $\mathcal{D}_{1}$-algebras} In the previous subsection we introduced a cofibrant replacement $B\mathcal P$ of an operad $\mathcal P$ in the category of bimodules over $W\mathcal P$. In \cite{Duc}, the author uses this resolution in order to equip the corresponding model of the derived mapping space of bimodules with a structure of $\mathcal{D}_{1}$-algebra. Then, he shows the following statement:
\begin{thm}{\cite[Theorem 3.1]{Duc}}\label{thm:duc} Let $P$ be a $\Sigma$-cofibrant operad and $\eta:P\rightarrow Q$ be a map of reduced operads. If the space $Q(1)$ is weakly contractible, then there are explicit weak equivalences of $\mathcal{D}_{1}$-algebras: \begin{equation}\label{Mapcase1} \begin{array}{rcl}
\xi: \Omega \mathrm{Operad}^{h}(P\,;\,Q) & \longrightarrow & \mathrm{Bimod}_{W\mathcal P}^{h}(P\,;\,Q) , \\ \xi_{k}:\Omega \big( \mathrm{Operad}^{h}_{\leq k}(P_{\leq k}\,;\,Q_{\leq k})\big) & \longrightarrow & \mathrm{Bimod}_{W\mathcal P\,;\,\leq k}^{h}(P_{\leq k}\,;\,Q_{\leq k}) . \end{array} \end{equation} \end{thm}
In what follows, we assume that the operad $\mathcal Q$ is fibrant in the Reedy model category of reduced operads. If it is not the case, then we substitute $\mathcal Q$ with any fibrant resolution $\mathcal Q^{f}$. Such resolution is equipped with a map $\tilde{\eta}:\mathcal P\rightarrow \mathcal Q \rightarrow \mathcal Q^{f}$ making $Q^{f}$ into a fibrant object in both categories of reduced operads and bimodules over $W\mathcal P$. Similarly, for any $k\geq 1$, the $k$-truncated operad $\mathcal Q^{f}_{\leq k}$ gives rise to a fibrant replacement of $\mathcal Q_{\leq k}$ in the categories of $k$-truncated operads and $k$-truncated bimodules over $W\mathcal P$.
By using the resolutions $W\mathcal P$ and $W\mathcal P_{k}$ for (truncated) operads as well as the resolutions $B\mathcal P$ and $B\mathcal P_{k}$ for (truncated) bimodules, we can easily define the maps $\xi$ and $\xi_{k}$. First of all, we recall that a point in the loop space $\Omega \mathrm{Operad}(W\mathcal P\,;\,Q)$, based in $\eta{\circ}\mu:W\mathcal P \rightarrow \mathcal P\rightarrow Q$, is given by a family of maps
$$ g_{n}:W\mathcal P(n)\times [0\,,\,1]\longrightarrow Q(n),\hspace{15pt}\forall n\geq 1,
$$ satisfying the following conditions:
\begin{itemize} \item[$\blacktriangleright$] $g_{n}(\iota(\ast_{1})\,;\, t)=\ast_{1}$\hspace{93pt} $\forall t\in [0\,,\,1]$,
\item[$\blacktriangleright$] $g_{n}(x{\circ}_{i}y\,;\, t)=g_{l}(x\,;\, t){\circ}_{i}g_{n-l+1}(y\,;\, t)$\hspace{20pt} $\forall t\in [0\,,\,1]$, $x\in W\mathcal P(l)$ and $y\in W\mathcal P(n-l+1)$,
\item[$\blacktriangleright$] $g_{n}(x\,;\, t)=\eta{\circ} \mu(x)$\hspace{90pt} $t\in \{0\,;\,1\}$ and $x\in W\mathcal P(n)$,
\item[$\blacktriangleright$] $g_{m}(u^{\ast}(x)\,;\, t)=u^{\ast}(g_{n}(x\,;\, t))$\hspace{50pt} $\forall t\in [0\,,\,1]$, $x\in W\mathcal P(n)$ and $u\in \Lambda([m]\,,\,[n])$. \end{itemize}
Let $g=\{g_{n}\}$ be a point in the loop space and let $[T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]$ be a point in $B\mathcal P$.
This element is a tree whose vertices are labelled by pairs $(x_{v},t_{v})$. To obtain $\xi(g)\bigl(
[T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]\bigr)$ we replace the label of each vertex $v$ of $T$
by $g_{|v|}(x_{v},t_v)\in \mathcal Q(|v|)$ and then we compose all these elements using the structure of $T$ and
the operadic compositions of $\mathcal Q$.
For instance, the image of the point $[T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]\in B\mathcal P(6)$ associated to the operadic composition in Figure \ref{B2} is the following one:
$$ \begin{array}{cl}
\xi(g)([T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]) & = g_{2}(x_{1}\,;\,t_{1})\big( g_{3}(a\,;\, 1)\,;\,g_{3}(x_{2}\,;\,t_{2})\big), \\
& =g_{3}(x_{1}\,;\,t_{1})\big( \mu{\circ}\eta(a)\,;\,g_{3}(x_{2}\,;\,t_{2})\big). \end{array}
$$
\section{The homotopy fiber case}
For the rest of this section, $\eta:\mathcal P \rightarrow \mathcal Q$ is a map of reduced operads and $(X\,;\,\ast)$ is a pointed space equipped with a map $\delta:X\rightarrow \mathrm{Operad}(W\mathcal P, Q)$ sending the basepoint to the composite map $\eta{\circ} \mu:W\mathcal P \rightarrow P \rightarrow Q$. According to the notation introduced in Example \ref{B7}, applied to the composite map $\eta{\circ}\mu$, one has a $W\mathcal P$-bimodule $\mathcal Q{\circ} X$.
The purpose of this section is to prove the following theorem:
\begin{thm}\label{th:main} Suppose that $X$ is a path-connected pointed space; $\mathcal P$ and $\mathcal Q$ are reduced topological operads; $\mathcal P$ is $\Sigma$-cofibrant; $\mathcal Q$ is Reedy fibrant; $\mathcal P(1)$ and $\mathcal Q(1)$ are weakly contractible. Then, the following are homotopy fiber sequences: \begin{equation}\label{Mapcase2} \begin{array}{c}
\mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q{\circ} X) \longrightarrow X \longrightarrow \mathrm{Operad}(W\mathcal P, \mathcal Q), \\ \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{k}, \mathcal Q{\circ} X) \longrightarrow X \longrightarrow \mathrm{Operad}(W\mathcal P_{k}, \mathcal Q). \end{array} \end{equation} \end{thm}
\begin{proof} We only prove the statement in the usual case. The same arguments work for the truncated case. The result is a consequence of Theorems \ref{THmPart1} and \ref{ThmPart2} in which we introduce an intermediate space $\mathrm{Bimod}_{W\mathcal P\,;\,X}(B\mathcal P\,;\,\mathcal Q)$ together with explicit weak equivalences $$ \xymatrix{ \psi:hofiber\big( X\rightarrow \mathrm{Operad}(W\mathcal P,\mathcal Q) \big) \ar[r]^{\hspace{25pt}\psi'} & \mathrm{Bimod}_{W\mathcal P\,;\,X}(B\mathcal P\,;\,\mathcal Q) \ar[r]^{\hspace{-10pt}\psi''} & \mathrm{Bimod}_{W\mathcal P}(B\mathcal P\,;\,\mathcal Q{\circ} X). } $$
\end{proof}
For the rest of the section we will be assuming that $\mathcal P$ and $\mathcal Q$ are reduced topological operads; $\mathcal P$ is $\Sigma$-cofibrant; $\mathcal Q$ is Reedy fibrant.
\subsection{A bundle of bimodule maps}\label{sec:bimod_bundle}
For $x\in X$ we denote by $\mathcal Q_x$ the $W\mathcal P$-bimodule obtained from $\mathcal Q$ by using the map $\delta_{x}$ to define the right $W\mathcal P$-action and the map $\delta_{\ast}$ to define the left $W\mathcal P$-action. In other words, the $W\mathcal P$-bimodule structure of $\mathcal Q_x:=\{Q_{x}(n)=Q(n),\, n\geq 1\}$ is given by the following formulas: $$ \begin{array}{rcl}
{\circ}^{i}: \mathcal Q_{x}(n)\times W\mathcal P(m) & \longrightarrow & \mathcal Q_{x}(n+m-1); \\
q\,,\,p & \longmapsto & q{\circ}_{i}\delta_{x}(p),\\
\gamma:W\mathcal P(n)\times \mathcal Q_{x}(m_{1})\times \cdots \times \mathcal Q_{x}(m_{n}) & \longmapsto & \mathcal Q_{x}(m_{1}+\cdots + m_{n});\\ p,q_{1},\ldots,q_{n} & \longmapsto & \delta_{\ast}(x)(q_{1},\ldots,q_{n}). \end{array} $$
Next, for a reduced $W\mathcal P$-bimodule $M$, define $\mathrm{Bimod}_{W\mathcal P,X}(M,\mathcal Q )$ to be the space consisting of pairs $(x, f)$, where $x\in X$ and $f\in \mathrm{Bimod}_{W\mathcal P}(M,\mathcal Q_x)$. There is a natural inclusion \[ \psi'':\mathrm{Bimod}_{W\mathcal P,X}(B \mathcal P,\mathcal Q ) \longrightarrow \mathrm{Bimod}_{W\mathcal P}(B \mathcal P,\mathcal Q{\circ} X ) \] such that the image of $(x,f)$ as above is the map \[ B\mathcal P \ni p\longmapsto (f(p),x,\dots, x). \]
\begin{prop}\label{lem:fib} The truncations \begin{equation}\label{truncMap1} \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_k,\mathcal Q{\circ} X) \longrightarrow \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{\ell},\mathcal Q{\circ} X) \end{equation} and \begin{equation}\label{truncMap2} \mathrm{Bimod}_{W\mathcal P,X}(B \mathcal P_k,\mathcal Q ) \longrightarrow \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P_{\ell},\mathcal Q) \end{equation} are Serre fibrations for any $0\leq \ell\leq k\leq\infty$. \end{prop}
\begin{proof}
It is enough to prove it for $\ell=k-1<\infty$.
Let us assume first that $\mathcal P(1)=*$. We introduce the subspace $\partial B \mathcal P(k)$, consisting of points in $B \mathcal P(k)$ that have at least one vertex of the main tree labelled by~$0$ or~$1$.
Since we assume $\mathcal P(1)=*$, one has $\partial B \mathcal P(k)=B\mathcal P_{k-1}(k)$. The space $\partial B \mathcal P(k)$ is equipped with an action of the symmetric group $\Sigma_{k}$ and the inclusion $\partial B\mathcal P(k)\to B\mathcal P(k)$ is a $\Sigma_k$-cofibration~\cite{Duc}.
The inclusion of bimodules $B\mathcal P_{k-1}\to B\mathcal P_k$ is a Reedy cofibration because it fits in the following pushout diagram
in $\Sigma_{>0}\mathrm{Bimod}_{W\mathcal P}$, see {\bf Characterisation of cofibrations} in Subsection~\ref{Final1}: \begin{equation}\label{eq:sigma_pushout1} \xymatrix{ \mathcal{F}_{W\mathcal P}^\Sigma(\partial B\mathcal P(k))\ar[r] \ar[d] & \mathcal{F}_{W\mathcal P}^\Sigma(B\mathcal P(k)) \ar[d] \\ B \mathcal P_{k-1} \ar[r] & B \mathcal P_{k}, } \end{equation} where $\mathcal{F}_{W\mathcal P}^\Sigma(\partial B\mathcal P(k))$ and $\mathcal{F}_{W\mathcal P}^\Sigma(B\mathcal P(k))$ denote the free bimodules in $\Sigma_{>0}\mathrm{Bimod}_{W\mathcal P}$ generated by the $\Sigma$-sequences $\partial B\mathcal P(k)$ and $B\mathcal P(k)$ concentrated in the only arity~$k$. Therefore, given a reduced bimodule map $B\mathcal P_{k-1}\xrightarrow{\lambda_{k-1}}Q{\circ} X$, in order to extend it to $B\mathcal P_k\xrightarrow{\lambda_k}Q{\circ} X$, one has to define a $\Sigma_k$-equivariant
map $B\mathcal P(k)\xrightarrow{\lambda_k(k)}Q{\circ} X(k)$, such that $\lambda_k(k)|_{\partial B\mathcal P(k)}=\lambda_{k-1}(k)$ and which respects the $\Lambda$ structure. The latter condition is equivalent to the commutativity of the square $$ \xymatrix{ B\mathcal P(k)\ar[rr]^{\lambda_k(k)} \ar[d] &{}& Q{\circ} X(k) \ar[d] \\ \mathcal{M}(B\mathcal P)(k)=\mathcal{M}(B \mathcal P_{k-1})(k) \ar[rr]^-{\mathcal{M}(\lambda_{k-1})(k)} &{}& \mathcal{M}(Q{\circ} X)(k). } $$ Consequently, one has the pullback diagram \begin{equation}\label{eq:pullback1} \xymatrix{ \mathrm{Bimod}_{W\mathcal P}(B \mathcal P_{k},\mathcal Q {\circ} X)\ar[d] \ar[r] & \TopCat_{\Sigma_{k}}( B \mathcal P(k)\,,\, \mathcal Q{\circ} X(k)) \ar[d] \\ \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{k-1},\mathcal Q{\circ} X) \ar[r] & \TopCat_{\Sigma_{k}}(\partial B \mathcal P(k)\,,\,\mathcal Q{\circ} X(k)) \underset{\TopCat_{\Sigma_{k}}(\partial B \mathcal P(k)\,,\,\mathcal{M}(\mathcal Q{\circ} X)(k))}{\bigtimes} \TopCat_{\Sigma_{k}}( B \mathcal P(k)\,,\,\mathcal{M}(\mathcal Q{\circ} X)(k)), } \end{equation} where $\TopCat_{\Sigma_{k}}$ is the model category of spaces equipped with an action of the symmetric group $\Sigma_{k}$ and $\mathcal{M}(-)$ is the matching object (\ref{B3}). Since the operad $\mathcal Q$ is assumed to be fibrant in the Reedy model category of reduced operads, the same is true for the bimodule $\mathcal Q{\circ} X$ due to Lemma~\ref{l:circle}.
Furthermore,
the inclusion from $\partial B \mathcal P(k)$ into $B \mathcal P(k)$ is a $\Sigma_{k}$-cofibration. Consequently, the vertical maps in the above diagram are Serre fibrations by applying an alternative version of the pushout product axiom along the inclusion $\partial B \mathcal P(k)\rightarrow B \mathcal P(k)$ and the map $\mathcal Q{\circ} X(k)\rightarrow \mathcal{M}(\mathcal Q{\circ} X)(k)$.
In case $\mathcal P(1)\neq *$ one has to consider an auxiliary filtration in the inclusion $B\mathcal P_{k-1}
\subset B\mathcal P_k$, $k\geq 1$:
\[
B\mathcal P_{k-1} =B\mathcal P_{k-1,-1}\subset B\mathcal P_{k-1,0}\subset B\mathcal P_{k-1,1}\subset B\mathcal P_{k-1,2}\subset
\ldots \subset B\mathcal P_{k},
\]
where $B\mathcal P_{k-1,i}$ is a subbimodule of $B\mathcal P_k$ generated by the prime components
of arity $\leq k-1$ and also of arity $k$ with $\leq i$ vertices. This finer filtration has the advantage to control the number of bivalent vertices and, in particular, to deal with the unit axiom. An argument similar to the one above
shows that each map
\[
\mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{k-1,i},\mathcal Q{\circ} X) \to \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{k-1,i-1},\mathcal Q{\circ} X)
\]
is a Serre fibration. Indeed, one similarly to~\eqref{eq:sigma_pushout1} has a pushout square in $\Sigma_{>0}\mathrm{Bimod}_{W\mathcal P}$
\begin{equation}\label{eq:sigma_pushout2} \xymatrix{ \mathcal{F}_{W\mathcal P}^\Sigma(B\mathcal P_{k-1,i-1}(k))\ar[r] \ar[d] & \mathcal{F}_{W\mathcal P}^\Sigma(B\mathcal P_{k-1,i}(k)) \ar[d] \\ B \mathcal P_{k-1,i-1} \ar[r] & B \mathcal P_{k-1,i}. } \end{equation} This implies that one has a pullback square obtained from~\eqref{eq:pullback1} by replacing $B\mathcal P_{k-1}$ and $B\mathcal P_k$ in the left column by $B\mathcal P_{k-1,i-1}$ and $B\mathcal P_{k-1,i}$, respectively, and replacing $\partial B\mathcal P(k)$ and $B\mathcal P(k)$ in the right column by $B\mathcal P_{k-1,i-1}(k)$ and $B\mathcal P_{k-1,i}$, respectively.
Similarly, we prove that the second truncation map (\ref{truncMap2}) is a Serre fibration. For example assuming $\mathcal P(1)=*$ and using the above notation, one has the following pullback diagram in which the space $X$ does not appear in the right-hand terms since it has been fixed in the space $\mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P_{k-1},\mathcal Q)$: $$ \xymatrix{ \mathrm{Bimod}_{W\mathcal P,X}(B \mathcal P_{k},\mathcal Q)\ar[d] \ar[r] & \TopCat_{\Sigma_{k}}( B \mathcal P(k)\,,\, \mathcal Q(k)) \ar[d] \\ \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P_{k-1},\mathcal Q) \ar[r] & \TopCat_{\Sigma_{k}}(\partial B \mathcal P(k)\,,\,\mathcal Q(k)) \underset{\TopCat_{\Sigma_{k}}(\partial B \mathcal P(k)\,,\,\mathcal{M}(\mathcal Q)(k))}{\bigtimes} \TopCat_{\Sigma_{k}}( B \mathcal P(k)\,,\,\mathcal{M}(\mathcal Q)(k)) } $$ So the vertical maps of the above diagram and the truncation map (\ref{truncMap2}) are also Serre fibrations. \end{proof}
\begin{lemma}\label{lem:T1} Suppose that $\mathcal P(1)$ is contractible. Then the natural map \begin{equation}\label{mapsec1}
\mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P_{1},\mathcal Q) \longrightarrow \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{1},\mathcal Q{\circ} X) \end{equation} is a weak equivalence. \end{lemma}
\begin{proof} We shall in fact show that the arrows in the commutative diagram \[ \begin{tikzcd} \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P_{1},\mathcal Q)=\mathrm{Bimod}_{W \mathcal P,X,\leq 1}(B\mathcal P,\mathcal Q) \ar{r}{\sim} \ar{d}{\sim} & \mathcal Q(1) \times X \\ \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{1},\mathcal Q{\circ} X)=\mathrm{Bimod}_{W\mathcal P,\leq 1}(B\mathcal P_{\leq 1},(\mathcal Q{\circ} X)_{\leq 1}) \ar{ru}{\sim} & \end{tikzcd} \] are weak equivalences. Let $\mathbbm{1}$ denote the initial element in the category of reduced operads. It is a point in arity one and empty in all the other arities $\geq 2$. Since $\mathcal P(1)$ is contractible, the natural inclusion $\mathbbm{1}_{\leq 1} \to W\mathcal P_{\leq 1}$ is a weak equivalence of 1-truncated reduced operads. As a consequence
for 1-truncated $W\mathcal P$-bimodules $\mathcal M$, $\mathcal M'$ the restriction map \[ \mathrm{Bimod}^h_{W\mathcal P;\leq 1}(\mathcal M, \mathcal M') \to \mathrm{Bimod}^h_{\mathbbm{1};\leq 1}(\mathcal M, \mathcal M') \] is a weak equivalence. On the other hand, a reduced 1-truncated bimodule $\mathcal M$ over~$\mathbbm{1}$ is just a space $\mathcal M(1)$ with no additional structure. Thus provided $\mathcal M(1)$ is a cofibrant space, \[ \mathrm{Bimod}^h_{\mathbbm{1};\leq 1}(\mathcal M, \mathcal M')=\mathrm{Bimod}_{\mathbbm{1};\leq 1}(\mathcal M, \mathcal M')=\TopCat(\mathcal M(1),\mathcal M'(1)). \]
Hence we find
\[ \mathrm{Bimod}_{W\mathcal P,\leq 1}(B\mathcal P_{\leq 1},(\mathcal Q{\circ} X)_{\leq 1})
\simeq \mathrm{Bimod}_{\mathbbm{1},\leq 1}(\mathbbm{1}_{\leq 1},(\mathcal Q{\circ} X)_{\leq 1}) \simeq
(\mathcal Q{\circ} X)(1) = \mathcal Q(1) \times X. \]
Here the map to the right-hand side is given by taking the image of the unit element.
By essentially the same argument we show that the map $\mathrm{Bimod}_{W\mathcal P,X,\leq 1}(B\mathcal P,\mathcal Q) \to \mathcal Q(1) \times X$ is a weak equivalence. This then shows the Lemma. \end{proof}
\begin{rem}\label{r:framed_bim_oper} We briefly explain~\eqref{eq:fr_nfr} and~\eqref{eq:fr_nfr2}. One has that $\mathcal P$ and $\mathcal Q$ are reduced, $\mathcal P(1)\simeq \mathcal Q(1)\simeq *$, $G$ is connected and $G\nsimeq *$. For simplicity, assume that $\mathcal P(1)=*$. Recall~\eqref{eq:tr_op_map},~\eqref{eq:tr_bim_map} and the pullback square~\eqref{eq:pullback1}.
For the 1-truncated derived mapping spaces \[ \mathrm{Bimod}_{\mathcal P;\leq 1}^h(\mathcal P_{\leq 1},(\mathcal Q{\circ} G)_{\leq 1})\simeq G\nsimeq \mathrm{Bimod}_{\mathcal P;\leq 1}^h(\mathcal P_{\leq 1},\mathcal Q_{\leq 1})\simeq *, \] while \[ \mathrm{Operad}_{\leq 1}(\mathcal P_{\leq 1},\mathcal (Q{\circ} G)_{\leq 1})\simeq*\simeq \mathrm{Operad}_{\leq 1}(\mathcal P_{\leq 1},\mathcal Q_{\leq 1}). \] On the other hand, for any $k\geq 2$, one has weak equivalences of fibers \begin{multline*} \mathrm{fiber}\bigl(\mathrm{Bimod}_{\mathcal P;\leq k}^h(\mathcal P_{\leq k},(\mathcal Q{\circ} G)_{\leq k})\to \mathrm{Bimod}_{\mathcal P;\leq k-1}^h(\mathcal P_{\leq k-1},(\mathcal Q{\circ} G)_{\leq k-1}) \bigr) \simeq\\
\mathrm{fiber}\bigl(\mathrm{Bimod}_{\mathcal P;\leq k}^h(\mathcal P_{\leq k},\mathcal Q_{\leq k})\to \mathrm{Bimod}_{\mathcal P;\leq k-1}^h(\mathcal P_{\leq k-1},\mathcal Q_{\leq k-1}) \bigr), \end{multline*} \begin{multline*} \mathrm{fiber}\bigl(\mathrm{Operad}_{\leq k}^h(\mathcal P_{\leq k},(\mathcal Q{\circ} G)_{\leq k})\to \mathrm{Operad}_{\leq k-1}^h(\mathcal P_{\leq k-1},(\mathcal Q{\circ} G)_{\leq k-1}) \bigr) \simeq\\
\mathrm{fiber}\bigl(\mathrm{Operad}_{\leq k}^h(\mathcal P_{\leq k},\mathcal Q_{\leq k})\to \mathrm{Operad}_{\leq k-1}^h(\mathcal P_{\leq k-1},\mathcal Q_{\leq k-1}) \bigr). \end{multline*} For example, for the first equivalence above, we notice that the fibers are described as the spaces of $\Sigma_k$-equivariant
lifts in the squares $$ \xymatrix{ \partial B \mathcal P(k)\ar[r] \ar[d] & Q{\circ} G(k) \ar[d] \\ B \mathcal P(k) \ar[r]\ar@{..>}[ru] & \mathcal{M}(Q{\circ} G)(k) } \quad\quad\quad \xymatrix{ \partial B \mathcal P(k)\ar[r] \ar[d] & Q(k) \ar[d] \\ B \mathcal P(k) \ar[r]\ar[r]\ar@{..>}[ru] & \mathcal{M}(Q)(k), } $$ respectively. These fibers are equivalent, because for $\mathcal Q$ Reedy fibrant, the fibers of the maps $\mathcal Q(k)\to \mathcal{M}(Q)(k)$ and $\mathcal Q{\circ}G(k)\to \mathcal{M}(Q{\circ}G)(k)$ differ only for $k=1$, see the proof of Lemma~\ref{l:circle}. \end{rem}
\begin{thm}\label{THmPart1} If $\mathcal P(1)$ is contractible, then the map $$ \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P, \mathcal Q)\longrightarrow \mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q{\circ} X) $$ is a weak equivalence. \end{thm}
\begin{proof} We compare the two fibrations (cf. Proposition~\ref{lem:fib}) \[ \xymatrix{ Y_{1}\ar[r] \ar[d]& \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P,\mathcal Q) \ar[r]^{f}\ar[d] & \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P_{1},\mathcal Q) \ar[d]^{h} \\ Y_{2}\ar[r] & \mathrm{Bimod}_{W\mathcal P}(B\mathcal P,\mathcal Q{\circ} X) \ar[r]_{g} & \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_{1},\mathcal Q{\circ} X), } \] where $Y_{1}$ is the fiber $f^{-1}(\alpha)$ over a point $\alpha\in \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P_{1},\mathcal Q)$ and $Y_{2}=g^{-1}(h(\alpha))$. In particular, for any arity $i$, $h(\alpha)(i)\colon B\mathcal P_{1}(i)\to Q(i)\times X^{\times i}$ is the constant map on the factor $X^{\times i}$ with value $(x,\ldots,x)$ for some $x\in X$. Thanks to the identification $u^{\ast}{\circ} h(\alpha)(n)= h(\alpha)(1){\circ} u^{\ast}$, for any $u\in \Lambda_{+}([1],[n])$, any point in the fiber $Y_{2}=g^{-1}(h(\alpha))$ is also constant with value $(x,\ldots,x)$ on the factor $X^{\times -}$ and the two fibers $Y_{1}$ and $Y_{2}$ coincide. Since $\mathcal P(1)$ is contractible by assumption we may use Lemma \ref{lem:T1} to conclude that the right-hand vertical map is a weak equivalence. Hence so must be the middle vertical map. \end{proof}
\subsection{The map from the homotopy fiber}\label{sec:map_from_fib}
Furthermore, one has a natural map \beq{equ:hofibermapdef} \psi':\mathrm{hofiber}( X\to \mathrm{Operad}(W\mathcal P,\mathcal Q) ) \to \mathrm{Bimod}_{W\mathcal P,X}(B \mathcal P,\mathcal Q ). \end{equation} First, an element of the homotopy fiber on the left-hand side is a pair $(x,g)$ with $x\in X$ and
a path $g$ in $\mathrm{Operad}(W\mathcal P,\mathcal Q)$ connecting $\delta_{\ast}$ (at $t=0$) and $\delta_{x}$ (at $t=1$). Concretely, $g$ is a family of continuous maps
$$ g_n\colon W\mathcal P(n)\times [0\,,\,1] \longrightarrow \mathcal Q(n),\,\, n\geq 1, $$ satisfying the relations: \begin{itemize} \item[$\blacktriangleright$] $g_{n}(\iota(\ast_{1})\,;\, t)=\ast_{1}'$\hspace{98pt} $\forall t\in [0\,,\,1]$,
\item[$\blacktriangleright$] $g_{n+m}(y_1{\circ}_{i}y_2,t)=g_{n+1}(y_1,t){\circ}_{i}g_{m}(y_2,t),$ \hspace{10pt} $\forall t\in [0\,,\,1]$, $y_1\in W\mathcal P(n+1)$, $y_2\in W\mathcal P(m)$ and $i\in \{1,\ldots , n+1\}$,
\item[$\blacktriangleright$] $g_{n}(y,0)=\delta_{\ast}(y)=\eta{\circ}\mu (y)$, \hspace{59pt} $\forall y\in W\mathcal P(n)$,
\item[$\blacktriangleright$] $g_{n}(y,1)=\delta_{x}(y)$, \hspace{97pt} $\forall y\in W\mathcal P(n)$,
\item[$\blacktriangleright$] $g_{m}(u^{\ast}(y),t)=u^*(g_{n}(y,t))$, \hspace{59pt} $\forall t\in [0\,,\,1]$, $y\in W\mathcal P(n)$ and $u\in \Lambda([m],[n])$.
\end{itemize}
Let $(x\,;\, g)$ be an element in the homotopy fiber and let $[T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]$ be a point in $B\mathcal P$. It is a tree $T$ with each vertex $v$ labelled by a pair $(x_v,t_v)$.
The map $\psi'$ sends $(x\,;\,g)$ to the pair $(x\,;\,\psi'(g))$, where
$\psi'(g)([T\,;\,\{t_{v}\}\,;\,\{x_{v}\}])$ is defined as follows. One replaces each label $(x_v,t_v)$
by $g_{|v|}(x_v,t_v)$ and then one composes the new labels using the structure of $T$ and the
composition maps of the operad $\mathcal Q$.
For instance, the image of the point $[T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]\in B\mathcal P(6)$ associated to the operadic composition in Figure \ref{B2} is the following one:
$$ \begin{array}{cl}
\psi'(g)([T\,;\,\{t_{v}\}\,;\,\{x_{v}\}]) & = g_{2}(x_{1},t_1)\big( g_{3}(a,1)\,;\,g_{3}(x_{2},t_2)\big), \\
& =g_{2}(x_{1},t_1)\big( \delta_x(a)\,;\,g_{3}(x_{2},t_2)\big). \end{array}
$$
We will derive our main result \eqref{equ:main} from the following statements.
\begin{lemma} The map $\pi:\mathrm{Bimod}_{W\mathcal P, X}(B\mathcal P, Q)\to X$ is a Serre fibration. \end{lemma}
\begin{proof} The statement is an immediate consequence of Proposition~\ref{lem:fib} for $\ell=0$, $k=\infty$. Indeed, $B\mathcal P_0(i)= \emptyset$,
$i\geq 1$. Consequently, $\mathrm{Bimod}_{W\mathcal P, X}(B\mathcal P_0, Q)= X$.
\end{proof}
Now, our main result is the following. \begin{thm}\label{ThmPart2} If $Q(1)$ is weakly contractible and $X$ is a path-connected pointed space, then the following is a homotopy fiber sequence \[ \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P, \mathcal Q) \to X \to \mathrm{Operad}(WP, Q), \] and the weak equivalence $\psi'\colon\mathrm{hofiber}(X \to \mathrm{Operad}(WP, Q))\to \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P, \mathcal Q)$ is the map of section \ref{sec:map_from_fib}. \end{thm}
\begin{proof} We compare the two (horizontal) homotopy fiber sequences \[ \begin{tikzcd} \Omega( \mathrm{Operad}^h(\mathcal P,\mathcal Q))\ar{r}\ar{d} {\simeq}& \mathrm{hofiber}( X\to \mathrm{Operad}^h(\mathcal P,\mathcal Q)) \ar{r}\ar{d}{\psi'}& X \ar{d}{=} \\
\mathrm{Bimod}_{W\mathcal P}(B\mathcal P,\mathcal Q) \ar{r} & \mathrm{Bimod}_{W\mathcal P,X}(B\mathcal P,\mathcal Q) \ar{r}& X \end{tikzcd} \] The left-hand vertical arrow is a weak equivalence by Theorem \ref{thm:duc}, and so is the right-hand vertical arrow. We conclude that the middle vertical arrow must be a weak equivalence as well.
\end{proof}
\subsection{A weak equivalence of Swiss-Cheese algebras}\label{Final2}
The one dimensional Swiss-Cheese operad $\mathcal{SC}_{1}$ is a two coloured operad with set of colours $S=\{o\,,\,c\}$ introduced by Voronov~\cite{Voronov}. It is a relative version of the one dimensional little discs operad $\mathcal{D}_{1}$ defined as follows: $$ \mathcal{SC}_{1}(n,m;k):=\left\{ \begin{array}{ll}
\mathcal{D}_{1}(n) & \text{if } m=0 \text{ and } k=c, \\
\left\{ \{c_{i}:[0\,,\,1]\rightarrow [0\,,\,1]\}_{1\leq i \leq n+1}\in \mathcal{D}_{1}(n+1) \,\big| \, c_{n+1}(1)=1 \right\} & \text{if } m=1 \text{ and } k=o, \\ \emptyset & \text{otherwise}, \end{array} \right. $$ An algebra over $\mathcal{SC}_{1}$ is given by a pair of topological spaces $(A\,,\,B)$ such that $A$ is a $\mathcal{D}_{1}$-algebra and $B$ is a left module over $A$. A typical example of $\mathcal{SC}_{1}$-algebra is a pair of spaces of the form $$ \left(\, \Omega Y \,;\, \Omega(Y\,;\,X)=\mathrm{hofiber}(f:X\rightarrow Y)\,\right), $$ where $f:X\rightarrow Y$ is a map of pointed spaces. In particular, we are interested in the case $Y=\mathrm{Operad}(W\mathcal P, \mathcal Q)$ based on the composite map $\eta{\circ}\mu :W\mathcal P\rightarrow \mathcal P \rightarrow \mathcal Q$. So, the pair $$ \left( \Omega \mathrm{Operad}(W\mathcal P, \mathcal Q) \,;\, \mathrm{hofiber}(X\rightarrow \mathrm{Operad}(W\mathcal P, \mathcal Q))\right) $$ is a $\mathcal{SC}_{1}$-algebra. Moreover, in \cite[Section 2.3]{Duc}, we build an explicit $\mathcal{D}_{1}$-algebra structure on the space $\mathrm{Bimod}_{W\mathcal P}(B\mathcal P,\mathcal Q)$ making the maps (\ref{Mapcase1}) into weak equivalences of $\mathcal{D}_{1}$-algebras. In this section, we extend this construction in order to get an explicit $\mathcal{SC}_{1}$-algebra structure on the pair \begin{equation}\label{SCalg} \left( \mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q) \,;\, \mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q{\circ} X)\right). \end{equation} For this purpose we build maps $$ \alpha_{n,o}:\mathcal{SC}_{1}(n,1;o)\times \mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q)^{\times n} \times \mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q{\circ} X) \longrightarrow \mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q{\circ} X) $$ compatible with operadic structure of $\mathcal{SC}_{1}$.
\begin{figure}
\caption{Illustration of the subdivision of a point in $B\mathcal P$ with the conditions\\ $c_{1}(0)< t_{1}<c_{1}(1)<t_{2}<c_{2}(0)<t_{3},t_{4},t_{5}<c_{2}(1)$.}
\label{G0}
\end{figure}
From now on, we fix a family $c=\{c_{i}:[0\,,\,1]\rightarrow [0\,,\,1]\}_{1\leq i \leq n+1}\in \mathcal{SC}_{1}(n,1;o)$ as well as a family of bimodule maps $f_{i}:B\mathcal P \rightarrow Q$, with $1\leq i \leq n$, and $f_{n+1}:B\mathcal P\rightarrow Q{\circ} X$. Since the little discs arise from an affine embedding, $c_{i}$ is determined by the images of $0$ and $1$. In a similar way, we define the linear embeddings $h_{i}:[0\,,\,1]\rightarrow [0\,,\,1]$, with $0\leq i \leq n$, representing the gaps between the cubes:
$$ h_{i}(0)= \left\{ \begin{array}{cc}
0 & \text{if } i=0, \\ c_{i}(1) & \text{if } i\neq 0, \end{array} \right. \hspace{15pt}\text{and}\hspace{15pt} h_{i}(1)= \left\{ \begin{array}{cc}
1 & \text{if } i=n, \\ c_{i+1}(0) & \text{if } i\neq n. \end{array} \right.
$$
The bimodule map $\alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n+1})$ is defined by using a decomposition of the points $y=[T\,;\, \{t_{v}\}\,;\,\{x_{v}\}]\in B\mathcal P$ according to the parameters indexing the vertices. Roughly speaking, the segments $<c_{1},\ldots,c_{n+1}>$ subdivide the tree $T$ into sub-trees as shown in Figure \ref{G0}. Then, we apply the bimodule map $f_{i}$ to the sub-trees associated to the segment $c_{i}$ and the composite map $\eta{\circ}\mu:B\mathcal P\rightarrow P\rightarrow Q$ to the sub-trees associated to gaps. Finally, we put together the pieces by using the operadic structure of $Q$ and the left $Q$-module structure of $Q{\circ} X$. By construction, we can assume that the representative point $y$ does not have two consecutive vertices (i.e. connected by an inner edge) indexed by the same real number. For the moment, we also assume that the planar tree $T$ is planar is labelled by the identity permutation.
More precisely, a sub-point of $y=[T\,;\,\{t_{v}\}\,;\, \{x_{v}\}]$ is an element in $B\mathcal P$ obtained from $y$ by taking a sub-tree of $T$ preserving the indexation. A sub-point $w$ is said to be associated to the gap $h_{i}$ if the vertices below $w$ (seen as a sub-point of $y$) are strictly below the line $h_{i}(0)$ whereas the vertices above $w$ are strictly above $h_{i}(1)$. Furthermore, the parameters indexing the vertices of the main tree of $w$ are in the interval $[h_{i}(0)\,,\,h_{i}(1)]$. The set $\mathcal{T}[h_{i}\,;\,y]=\{w_{1}^{i},\ldots,w_{p_{i}}^{i}\}$ of sub-points associated to the gap $h_{i}$ is ordered using the planar structure of the tree $T$. For instance, the sets $\mathcal{T}[h_{0}\,;\,y]$ and $\mathcal{T}[h_{1}\,;\,y]$ associated to the point in Figure \ref{G0} are the following ones:
\noindent where the trivial tree without vertex represents the class of the $1$-corolla indexed by $(\iota(\ast_{1})\,;\,t)$ with $t\in [0\,,\,1]$ and $\ast_{1}$ the unit of the operad $\mathcal P$.
Similarly, a sub-point $z$ is said to be associated to the segment $c_{i}$ if the vertices below $z$ (seen as a sub-point in $y$) are on the line $c_{i}(0)$ or below it, whereas the vertices above $z$ are on the line $c_{i}(1)$ or above it. Furthermore, the parameters indexing the vertices of the main tree of $z$ are in the interval $]c_{i}(0)\,,\,c_{i}(1)[$, if $i\leq n$, or in the interval $]c_{i}(0)\,,\,1]$, if $i=n+1$. The set $\mathcal{T}[c_{i}\,;\,y]=\{z_{1}^{i},\ldots,z_{q_{i}}^{i}\}$ of sub-points associated the little disc $c_{i}$ is ordered using the planar structure of the tree $T$. For instance, the sets $\mathcal{T}[c_{1}\,;\,y]$ and $\mathcal{T}[c_{2}\,;\,y]$ associated with the point $y$ in Figure \ref{G0} are the following ones:
Let us remark that we need the trivial trees in the above definition since the bimodule maps $\{f_{i}\}$ do not necessarily map the trivial tree to the unit of the operad $\mathcal Q$. Furthermore, we need a map rescaling the parameters of the sub-points: \begin{equation}\label{G9} c_{i}^{\ast}:\mathcal{T}[c_{i}\,;\,y]\longrightarrow B\mathcal P\,\,;\,\, [T'\,;\,\{t'_{v}\}\,;\, \{x'_{v}\}]\longmapsto [T'\,;\,\{c_{i}^{-1}(t'_{v})\}\,;\, \{x'_{v}\}].
\end{equation} The map is well defined since the parameters indexing the vertices of the elements in $\mathcal{T}[c_{i}\,;\,y]$ are in the interval $]c_{i}(0)\,,\,c_{i}(1)[$ or $]c_{n+1}(0)\,,\,1]$. From the operadic structure of $\mathcal Q$ and the left $W\mathcal Q$-module structure on $\mathcal Q{\circ} X$, we build the map $\alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n+1})$ by induction as follows:
$$ \begin{array}{lcl}
\alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{0}(y) & = & \eta{\circ}\mu(w_{1}^{0}), \\ \alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{1}(y) & = & \alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{0}(y)\big(\,f_{1}(c_{1}^{\ast}(z_{1}^{1})),\ldots,f_{1}(c_{1}^{\ast}(z_{q_{1}}^{1}))\,\big), \\
& \vdots & \\
\alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{2k}(y) & = & \alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{2k-1}(y)\big(\,\eta{\circ}\mu(w_{1}^{k}) ,\ldots,\eta{\circ}\mu(w_{p_{k}}^{k}) \,\big), \\ \alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{2k+1}(y) & = & \alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{2k}(y)\big(\,f_{k}(c_{k}^{\ast}(z_{1}^{k})),\ldots,f_{k}(c_{k}^{\ast}(z_{q_{k}}^{k}))\,\big), \\
& \vdots & \\ \alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})(y) & = & \alpha_{n,o}(c\,;\,f_{1},\cdots,f_{n})_{2n}(y)\big(\,f_{n+1}(c_{n+1}^{\ast}(z_{1}^{n+1})),\ldots,f_{n+1}(c_{n+1}^{\ast}(z_{q_{n+1}}^{n+1}))\,\big). \end{array} $$ We do not need to rescale the sub-points associated to gaps since the map $\mu:B\mathcal P\rightarrow \mathcal P$ sends all the parameters indexing the vertices to $0$.
This construction produces also a $\mathcal{C}_{1}$-algebra structure on $\mathrm{Bimod}_{W\mathcal P}(B\mathcal P, \mathcal Q)$. Note also that the $\mathcal{SC}_1$-action restricts on the pair $$ \left( \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_k, \mathcal Q) \,;\, \mathrm{Bimod}_{W\mathcal P}(B\mathcal P_k, \mathcal Q{\circ} X)\right), $$
because the sub-points of an element in $B\mathcal P_{k}$ are still elements in $B\mathcal P_{k}$ and the rescaling maps (\ref{G9}) decrease the number of geometrical inputs.
One has the following statement:
\begin{thm} The morphisms induced by (\ref{Mapcase1}) and (\ref{Mapcase2}) $$ \left\{\begin{array}{c}
\Omega \mathrm{Operad}(W\mathcal P , \mathcal Q) \\ hofiber(X\rightarrow \mathrm{Operad}(W\mathcal P , \mathcal Q)) \end{array}\right\}\longrightarrow \left\{\begin{array}{c}
\mathrm{Bimod}_{W\mathcal P}(B\mathcal P , \mathcal Q) \\
\mathrm{Bimod}_{W\mathcal P}(B\mathcal P , \mathcal Q{\circ} X) \end{array}\right\}, $$ $$ \left\{\begin{array}{c}
\Omega \mathrm{Operad}(W\mathcal P_k , \mathcal Q) \\ hofiber(X\rightarrow \mathrm{Operad}(W\mathcal P_k , \mathcal Q)) \end{array}\right\}\longrightarrow \left\{\begin{array}{c}
\mathrm{Bimod}_{W\mathcal P}(B\mathcal P_k , \mathcal Q) \\
\mathrm{Bimod}_{W\mathcal P}(B\mathcal P_k , \mathcal Q{\circ} X) \end{array}\right\} $$ are morphisms of $\mathcal{SC}_{1}$-algebras. Furthermore, if $X$ is path-connected and the spaces $P(1)$ and $Q(1)$ are weakly contractible, then these are weak equivalences of $\mathcal{SC}_{1}$-algebras (for $k\geq 1$). \end{thm}
\section{The smoothing theory delooping of $\Emb^{\mathrm{fr}}_\partial(D^m,D^n)$} \label{s:last}
The space $\Emb_\partial(D^m,D^n)$, $n-m\geq 3$, $n\geq 5$, is known to have a delooping by means of the smoothing theory~\cite[Proposition~1.3]{Sakai}: \[ \Emb_\partial(D^m,D^n)\simeq\Omega^m\mathrm{hofiber}(\mathrm{V}_{m,n}\to \mathrm{V}^t_{m,n}), \] where \[ \mathrm{V}_{m,n}^t=\mathrm{TOP}(n)/\mathrm{TOP}(n,m) \] denotes the topological Stiefel manifold; $\mathrm{TOP}(n)$ is the group of homeomorphisms of ${\mathbb{R}}^n$; $\mathrm{TOP}(n,m)$ is its subgroup of homeomorphisms preserving poinwise ${\mathbb{R}}^m\subset {\mathbb{R}}^n$.
\begin{prop}\label{p:deloop_sakai} For $n-m\geq 3$, $n\geq 5$, one has \beq{eq:deloop_sakai} \Emb^{\mathrm{fr}}_\partial(D^m,D^n)\simeq\Omega^{m+1}\bigl( \mathrm{V}^t_{m,n}\sslash\mathrm{SO}(n)\bigr). \end{equation} \end{prop}
\begin{proof} One has a commutative diagram \[ \begin{tikzcd} \Emb^{\mathrm{fr}}_\partial(D^m,D^n)\ar{r}\ar{d} & \Omega^m \mathrm{SO}(n)\ar{r}\ar{d}& \Omega^m\mathrm{V}^t_{m,n}\ar{d}{=} \\
\Emb_\partial(D^m,D^n)\ar{r} & \Omega^m\mathrm{V}_{m,n} \ar{r}& \Omega^m\mathrm{V}^t_{m,n}. \end{tikzcd} \] The lower line is a fiber sequence by \cite[Proposition~1.3]{Sakai}. The right vertical line being identity, the middle one being a fibration, and the left square being a pullback one, all together imply that the upper line is also a fiber sequence. One gets \[ \Emb^{\mathrm{fr}}_\partial(D^m,D^n) \simeq\Omega^m\mathrm{hofiber}\bigl(\mathrm{SO}(n)\to \mathrm{V}^t_{m,n}\bigr) \simeq \Omega^{m+1}\bigl( \mathrm{V}^t_{m,n}\sslash\mathrm{SO}(n)\bigr). \] \end{proof}
\begin{rem}\label{r:sakai1} \sloppy Note that $\mathrm{V}^t_{m,n}$ has a left action by $\mathrm{SO}(n)$. Thus by $\mathrm{V}^t_{m,n}\sslash\mathrm{SO}(n)$ we understand the space $$\mathrm{SO}(n)\bbslash\mathrm{TOP}(n)/\mathrm{TOP}(n,m).$$ \end{rem}
\begin{rem}\label{r:sakai2} Note that the same argument can be used to show that our delooping~\eqref{eq:cor2} easily follows from the Boavida-Weiss result~\eqref{eq:BW1}. \end{rem}
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\end{document} | arXiv |
\begin{document}
\title[The Howe duality conjecture] {The Howe duality conjecture:\\ Quaternionic case} \author{Wee Teck Gan} \address{Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076} \email{[email protected]} \author{Binyong Sun} \address{Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing, 100190, P.R. China} \email{[email protected]}
\subjclass[2000]{Primary 11F27, Secondary 22E50}
\keywords{Howe duality conjecture, theta correspondence, quaternionic dual pair} \date{\today}
\dedicatory{In celebration of \\ Professor Roger Howe's 70th birthday}
\begin{abstract} We complete the proof of the Howe duality conjecture in the theory of local theta correspondence by treating the remaining case of quaternionic dual pairs in arbitrary residual characteristic. \end{abstract}
\maketitle
\section{\textbf{Introduction}}
Let $\mathrm{F}$ be a non-archimedean local field of characteristic not $2$. Let $W$ be a finite-dimensional symplectic vector space over $\mathrm{F}$ with symplectic form $\langle \,,\, \rangle_W$. Write \begin{equation}\label{meta0}
1\rightarrow \{1,\varepsilon_W\}\rightarrow
\widetilde{\mathrm{Sp}}(W)\rightarrow
\mathrm{Sp}(W)\rightarrow 1 \end{equation} for the metaplectic double cover of the symplectic group $\mathrm{Sp}(W)$. It does not split unless $W=0$. Denote by $\mathrm{H}(W):=W\times \mathrm{F}$ the Heisenberg group attached to $W$, with group multiplication \[
(u, \alpha)(v,\beta):=(u+v, \alpha +\beta+\langle u,v\rangle_W), \qquad u,v\in W, \ \alpha,\beta\in \mathrm{F}. \] Then $\widetilde{\mathrm{Sp}}(W)$ acts on $\mathrm{H}(W)$ as group automorphisms through the action of $\mathrm{Sp}(W)$ on $W$, and we may form the semi-direct product $\widetilde{\mathrm J}(W):=\widetilde{\mathrm{Sp}}(W)\ltimes \mathrm H(W)$.
Fix an arbitrary non-trivial unitary character $\psi: \mathrm{F}\rightarrow \mathbb C^\times$. Up to isomorphism, there is a unique smooth representation $\omega_\psi$ of $\widetilde{\mathrm J}(W)$ (called a Weil representation) such that (\emph{cf.} \cite[Section IV.43]{weil}) \begin{itemize}
\item ${\omega_\psi}|_{\mathrm{H}(W)}$ is irreducible and has central character $\psi$;
\item $\varepsilon_W\in\widetilde{\mathrm{Sp}}(W)$ acts through the scalar multiplication by
$-1$. \end{itemize} Unless $W=0$, the above second condition is a consequence of the first one.
Denote by $\tau$ the involution of $\operatorname{End}_\mathrm{F}(W)$ specified by \[
\langle x\cdot u, v\rangle_W=\langle u, x^\tau\cdot v\rangle_W, \qquad u,v\in W,\,x\in \operatorname{End}_\mathrm{F}(W). \] Let $(A, A')$ be a pair of $\tau$-stable semisimple $\mathrm{F}$-subalgebras of $\operatorname{End}_\mathrm{F}(W)$ which are mutual centralizers of each other. Put $G:=A\cap \mathrm{Sp}(W)$ and $G':=A'\cap \mathrm{Sp}(W)$, which are closed subgroups of $\mathrm{Sp}(W)$. Following Howe, we call the pair $(G,G')$ so obtained a {\em reductive dual pair} in $\mathrm{Sp}(W)$. We say that the pair $(A, A')$ (or the reductive dual pair $(G,G')$) is irreducible of type I if $A$ (or equivalently $A'$) is a simple algebra, and say that it is irreducible of type II if $A$ (or equivalently $A'$) is the product of two simple algebras which are exchanged by $\tau$. A complete classification of such dual pairs has been given by Howe. \vskip 5pt
For every closed subgroup $H$ of $\mathrm{Sp}(W)$, write $\widetilde H$ for the double cover of $H$ induces by the metaplectic cover \eqref{meta0}. Then $\widetilde G$ and $\widetilde G'$ commute with each other inside the group $\widetilde{\mathrm{Sp}}(W)$ (\emph{cf.} \cite[Chapter 2, Lemma II.5]{mvw}). Thus, the Weil representation $\omega_{\psi}$ can be regarded as a representation of $\widetilde G \times \widetilde G'$. \vskip 5pt
For every $\pi\in \operatorname{Irr}(\widetilde G)$, put
\[
\Theta_{\psi}(\pi):=(\omega_\psi\otimes \pi^\vee)_{\widetilde G}, \]
to be viewed as a smooth representation of $\widetilde G'$. Here and as usual, a superscript ``$\,^\vee$" indicates the contragredient representation, a subscript group indicates the coinvariant space, and ``$\operatorname{Irr}$" indicates the set of isomorphism classes of irreducible admissible representations of the group. It was proved by Kudla \cite{k83} that the representation $\Theta_{\psi}(\pi)$ is admissible and has finite length. Denote by $\theta_{\psi}(\pi)$ the maximal semisimple quotient of $\Theta_{\psi}(\pi)$, which is called the theta lift of $\pi$. In this paper, we complete the proof of the following Howe duality conjecture. \vskip 5pt \noindent{\bf \underline{The Howe Duality Conjecture}}
\vskip 5pt
\noindent For every reductive dual pair ($G, G')$ and every $\pi\in \operatorname{Irr}(\widetilde G)$, the theta lift $\theta_{\psi}(\pi)$ is irreducible if it is non-zero.
\vskip 5pt
The Howe duality conjecture is easily reduced to the case when the pair $(A, A')$ is irreducible (of type I or II).
It has been proved by Waldspurger \cite{w90} when the residual characteristic of $\mathrm{F}$ is not $2$.
For irreducible reductive dual pairs of type II, the conjecture was proved in full and more simply by Minguez in \cite{mi}. Every irreducible reductive dual pair of type I is an orthogonal-symplectic dual pair, a unitary dual pair, or a quaternionic dual pair \cite[Section 5]{H1}. For orthogonal-symplectic dual pairs and unitary dual pairs, the conjecture was proved in \cite{gt} (it was earlier proved in \cite{LST} that $\theta_{\psi}(\pi)$ is multiplicity free). For the remaining case of quaternionic dual pairs, only a partial result was obtained in \cite{gt} (for Hermitian representations). The reason is that \cite{gt} makes use of the MVW-involution on the category of smooth representations, and it has been shown in \cite{sun} that such an involution does not exist in the quaternionic case. \vskip 5pt
The purpose of this paper is to explain how the use of the MVW-involution can be avoided, thus completing the proof of the Howe duality conjecture in the quaternionic case. The lack of an MVW-involution necessitates relating the theta lifts of $\pi$ and $\pi^\vee$, and the key new ingredient is provided by the following consequence of the conservation relations shown in \cite[Equalities (12)]{sz}. \vskip 5pt
\begin{lem}\label{E:nonvan} Assume that $(G,G')$ is irreducible. Then for every $\pi\in \operatorname{Irr}(\widetilde G)$, \begin{equation}\label{E:nonvan0}
\theta_\psi(\pi)\neq 0\qquad\textrm{if and only if}\qquad \theta_{\bar \psi}(\pi^\vee)\neq 0, \end{equation} where $\bar \psi$ denotes the complex conjugation of $\psi$. \end{lem} In proving the Howe duality conjecture, one needs to strengthen Lemma \ref{E:nonvan} to the identity \begin{equation} \label{E:identity} (\theta_{\psi}(\pi))^\vee\cong \theta_{\bar \psi}(\pi^\vee) \quad \text{ for every $\pi\in \operatorname{Irr}(\widetilde G)$.} \end{equation}
Hence, the main result of this paper is the following theorem, which encompasses the Howe duality conjecture and the identity (\ref{E:identity}). \vskip 5pt
\begin{thm} \label{T:howe3}
Assume that $(G,G')$ is irreducible, and the size of $G$ is no smaller than that of $G'$. Then for all $\pi,\sigma\in \operatorname{Irr}(\widetilde G)$, \begin{itemize}
\item $\theta_\psi(\pi)$ is irreducible if it is non-zero;
\item if $\theta_{\psi}(\pi)\cong \theta_\psi(\sigma)\neq 0$, then $\pi\cong \sigma$;
\item $(\theta_\psi(\pi))^\vee\cong \theta_{\bar \psi}(\pi^\vee)$.
\end{itemize}
Consequently, the Howe duality conjecture holds for both $(G,G')$ and $(G',G)$, and for every $\pi'\in \operatorname{Irr}(\widetilde G')$, $(\theta_\psi(\pi'))^\vee\cong \theta_{\bar \psi}({\pi'}^\vee)$. \end{thm}
Here the size of $G$ is defined to be \[
\mathrm{size}(G):=\left\{
\begin{array}{ll}
\frac{n_A}{2}+\frac{\dim_K A^{\tau=-1}}{n_A} , & \hbox{if $(G,G')$ is an orthogonal-symplectic or quaternionic dual pair;} \\
n_A, & \hbox{otherwise,}
\end{array}
\right. \] where $K$ denotes the center of $A$, $A^{\tau=-1}:=\{\alpha\in A\mid \alpha^\tau=-\alpha\}$, and $n_A$ denotes the integer such that $\mathrm{rank}_K A=n_A^2$. The size of $G'$ is analogously defined.
\vskip 5pt
In fact, exploiting Lemma \ref{E:nonvan}, Theorem \ref{T:howe3} is equivalent to the following proposition.
\begin{prop} \label{T:mainq00}
Assume that $(G,G')$ is irreducible, and the size of $G$ is no smaller than that of $G'$. Then for all $\pi,\sigma\in \operatorname{Irr}(\widetilde G)$,
\begin{equation}\label{dimleq10}
\dim \operatorname{Hom}_{\widetilde G'}(\theta_{\psi}(\pi)\otimes \theta_{\bar \psi}(\sigma),\mathbb C)\leq \dim \operatorname{Hom}_{\widetilde G}(\pi\otimes \sigma, \mathbb C).
\end{equation} \end{prop}
\vskip 5pt In what follows, we show that Lemma \ref{E:nonvan} and Proposition \ref{T:mainq00} imply Theorem \ref{T:howe3}. \vskip 5pt
\noindent{\bf \underline{Proof of (Lemma \ref{E:nonvan} $+$ Proposition \ref{T:mainq00} $\Longrightarrow$ Theorem \ref{T:howe3})}} \vskip 5pt
For every $\pi\in \operatorname{Irr} (\widetilde G)$ and $\pi'\in \operatorname{Irr} (\widetilde G')$, write \[
\mathrm m_\psi(\pi, \pi'):=\dim \operatorname{Hom}_{\widetilde G\times \widetilde G'}(\omega_\psi, \pi\boxtimes \pi') \] and define $\mathrm m_{\bar \psi}(\pi, \pi')$ similarly.
We claim that
\begin{equation}\label{dual00}
\mathrm m_\psi(\pi, \pi')\neq 0\quad\textrm{if and only if} \quad \mathrm m_{\bar \psi}(\pi^\vee, {\pi'}^\vee)\neq 0.
\end{equation}
It is easy to see that \eqref{dimleq10} and \eqref{dual00} imply Theorem \ref{T:howe3}.
To prove the claim, we first assume that $\mathrm m_\psi(\pi, \pi')\neq 0$. Applying Lemma \ref{E:nonvan} to the pair $(G', G)$, we see that $\mathrm m_{\bar \psi}(\sigma, {\pi'}^\vee)\neq 0$ for some $\sigma\in \operatorname{Irr}(\widetilde G)$. The inequality \eqref{dimleq10} then implies that $\sigma\cong \pi^\vee$ and hence $\mathrm m_{\bar \psi}(\pi^\vee, {\pi'}^\vee)\neq 0$. Similarly, if $\mathrm m_{\bar \psi}(\pi^\vee, \sigma^\vee)\neq 0$ then $\mathrm m_\psi(\pi, \sigma)\neq 0$. This proves the claim \eqref{dual00}, and therefore shows that Lemma \ref{E:nonvan} and Proposition \ref{T:mainq00} imply Theorem \ref{T:howe3}.
\vskip 15pt
In view of the above, the main body of our paper will be devoted to the proof of Proposition \ref{T:mainq00}. \vskip 10pt
\noindent {\bf Remarks}: (a) Reductive dual pairs as defined in this paper include the following case: $G$ is the quaternionic orthogonal group attached to a one-dimensional quaternionic skew Hermitian space, and $G'$ is the quaternionic symplectic group attached to a non-zero quaternionic Hermitian space (see the next section). In this case, $G'$ is strictly contained in the centralizer of $G$ in the symplectic group.
(b)
Although in the statements of \cite[Theorems 1.3 and 1.4]{Ya} (see Lemma \ref{ry}) and \cite[Equalities (12)]{sz}, the authors assume that the base field $\mathrm{F}$ has characteristic zero, their methods prove the same results for all non-archimedean local field $\mathrm{F}$ of characteristic not $2$.
(c) For type II irreducible reductive dual pairs, the identity (\ref{E:identity}) is a consequence of \cite[Theorem 1]{mi}, in which the explicit theta lifts are determined in terms of the Langlands parameters. For orthogonal-symplectic and unitary dual pairs, (\ref{E:identity}) is a consequence of the MVW involution (\emph{cf.} \cite[Theorem 1.4]{S}).
\vskip 15pt \begin{center}{\bf Acknowledgements}\end{center}
This paper is essentially completed during the conference in honor of Professor Roger Howe on the occasion of his 70th birthday. We thank the organisers of the conference (James Cogdell, Ju-Lee Kim, Jian-Shu Li, David Manderscheid, Gregory Margulis, Cheng-Bo Zhu and Gregg Zuckerman) for their kind invitation to speak at the conference and for providing local support.
During the Howe conference, the first author presented his paper \cite{gt} with S. Takeda on the proof of the Howe duality conjecture for orthogonal-symplectic and unitary dual pairs and mentioned that the quaternionic case still needed to be addressed because of the lack of the MVW involution. He expressed the hope that some trick could be found by the end of the conference to deal with the quaternionic case. The following day, the second author realised that a consequence of the conservation relation shown in his paper \cite{sz} with C.-B. Zhu could serve as a replacement for the MVW-involution: this is the innocuous-looking statement \eqref{E:nonvan0} above. The two authors were able to verify the details in the next two days, thus completing the proof of the Howe duality conjecture in the quaternionic case. It gives us great pleasure to dedicate this paper to Roger Howe, who had initiated this whole area of research and formulated this conjecture at the beginning of his career. We hope that it gives him much satisfaction in seeing this conjecture completely resolved at the time of his retirement from Yale.
W.T. Gan is partially supported by an MOE Tier Two grant R-146-000-175-112. B. Sun is supported in part by the NSFC Grants 11222101 and 11321101.
\vskip 15pt
\section{\bf The doubling method}\label{secdouble}
We will only treat the quaternionic case in the proof of Proposition \ref{T:mainq00}, since it is previously known in all other cases. Let $\mathrm{F}$ be a nonarchimedean local field of characteristic not $2$, with $|\,\cdot\,|_\mathrm{F}$ denoting the normalized absolute value on $\mathrm{F}$.
Let $\mathrm{D}$ be a central division quaternion algebra over $\mathrm{F}$, which is unique up to isomorphism. Denote by $\iota: \mathrm{D}\rightarrow \mathrm{D}$ the quaternion conjugation of $\mathrm{D}$. We consider an $\epsilon$-Hermitian right $\mathrm{D}$-vector space $U$, and an $\epsilon'$-Hermitian left $\mathrm{D}$-vector space $V$, where $\epsilon=\pm 1$ and $\epsilon'=-\epsilon$. To be precise, $U$ is a finite dimensional right $\mathrm D$-vector space, equipped with a non-degenerate $\mathrm F$-bilinear map \[
\langle\,,\,\rangle_U : U\times U\rightarrow \mathrm D \] satisfying \[
\langle u,u'\alpha \rangle_U=\langle u,u'\rangle_U\, \alpha\quad\textrm{ and }\quad \langle u,u'\rangle_U=\epsilon \langle u',u\rangle_U^\iota, \qquad u,u'\in U, \,\alpha\in \mathrm D. \] Similarly, $V$ is a finite dimensional left $\mathrm{D}$-vector space and is equipped with a form $\langle \,,\,\rangle_V:V\times V\rightarrow \mathrm{D}$ with the analogous properties. The tensor product $W:=U\otimes_\mathrm{D} V$ is a symplectic space over $\mathrm{F}$ under the bilinear form
\begin{equation} \label{tensor-form}
\langle u\otimes v, u'\otimes v'\rangle_W:=\frac{\langle u,u'\rangle_U \,\langle v,v'\rangle_V^\iota+\langle
v,v'\rangle_V\,\langle u,u'\rangle_U^\iota }{2},\quad u,u'\in U,\,v, v'\in V. \end{equation}
Throughout the paper, we fix two quadratic (order at most $2$) characters $\chi_U, \chi_V :\mathrm{F}^\times \rightarrow \{\pm 1\}$ determined by the discriminants of $U$ and $V$ respectively. More precisely, we have:
\[
\chi_V(\alpha)=\left( (-1)^{\dim V} \prod_{i=1}^{\dim V}\langle e_i, e_i\rangle_V \langle e_i, e_i\rangle_V^\iota, \alpha\right)_\mathrm{F},\qquad \alpha \in \mathrm{F}^\times, \] where $e_1, e_2, \cdots, e_{\dim V}$ is an orthogonal basis of $V$, and $(\,,\,)_\mathrm{F}$ denotes the quadratic Hilbert symbol for $\mathrm{F}$. Likewise, one has the analogous definition for $\chi_U$. Note that if $\epsilon =1$, then the isometry class of $U$ is determined by its dimension, and $\chi_U$ only depends on the parity of $\dim U$; likewise, if $\epsilon' =1$, $\chi_V$ only depends on the parity of $\dim V$. \vskip 5pt
Denote by $W^-$ the space $W$ equipped with the form scaled by $-1$. Write $W^\square:=W\oplus W^-$ for the orthogonal direct sum, which contains $W^\triangle:=\{(u,u)\in W^\square\mid u\in W\}$ as a Lagrangian subspace. Define $U^-, V^-, U^\square, V^\square, U^\triangle, V^\triangle$ similarly. Then we have obvious identifications of symplectic spaces \[
W^-=U^-\otimes_\mathrm{D} V=U\otimes_\mathrm{D} V^- \quad \textrm{and}\quad W^\square=U^\square\otimes_\mathrm{D} V=U\otimes_\mathrm{D} V^\square. \] Let $\mathrm{G}(U)$ denote the isometry group of $U$, and similarly for other groups. Then we have identifications \[
\mathrm{G}(U)=\mathrm{G}(U^-)\quad \textrm{ and }\quad \mathrm{G}(V)=\mathrm{G}(V^-), \]
and inclusions \[
\mathrm{G}(U)\times \mathrm{G}(U^-)\subset \mathrm{G}(U^\square) \quad\textrm{ and }\quad \mathrm{G}(V)\times \mathrm{G}(V^-) \subset \mathrm{G}(V^\square). \]
Denote by $\mathrm{P}(U^\triangle)$ the parabolic subgroup of $\mathrm{G}(U^\square)$ stabilizing $U^\triangle$. Likewise, denote by $\mathrm{P}(V^\triangle)$ the parabolic subgroup of $\mathrm{G}(V^\square)$ stabilizing $V^\triangle$.
Let $\omega$ and $\omega^-$ be irreducible admissible smooth representations of $\mathrm{H}(W)$ and $\mathrm{H}(W^-)$, respectively, both with central character $\psi$. Then the representation $\omega^\square:=\omega\boxtimes \omega^-$ of $\mathrm{H}(W)\times \mathrm{H}(W^-)$ descends to a representation of $\mathrm{H}(W^\square)$ through the surjective homomorphism \[
\mathrm{H}(W)\times\mathrm{H}(W^-)\rightarrow \mathrm{H}(W^\square),\quad ((u,\alpha), (v,\beta))\mapsto ((u,v), \alpha+\beta). \] This representation of $\mathrm{H}(W^\square)$ uniquely extends to the group $\mathrm G(U^\square)\ltimes \mathrm H(W^\square)$ such that (\emph{cf.} \cite[Theorem 4.7]{sz}) \begin{equation}\label{lambdad}
\lambda_\triangle (g\cdot \phi)=\chi_V({{\det}}(g|_{U^\triangle})) \, \abs{\det(g|_{U^\triangle})}_\mathrm{F}^{\dim V} \, \lambda_\triangle (\phi), \qquad \phi\in \omega^\square, \ g\in \mathrm{P}(U^\triangle), \end{equation} where $\lambda_\triangle$ denotes the unique (up to scalar multiplication) non-zero $W^\triangle$-invariant linear functional on $ \omega^\square$ and $\det$ denotes the reduced norm. Similarly, this representation of $\mathrm{H}(W^\square)$ uniquely extends to the group $\mathrm G(V^\square)\ltimes \mathrm H(W^\square)$ such that \[
\lambda_\triangle (g'\cdot\phi)=\chi_U(\det(g'|_{V^\triangle})) \, \abs{\det(g'|_{V^\triangle})}_\mathrm{F}^{\dim U} \, \lambda_\triangle (\phi), \quad \phi\in \omega^\square, \quad g'\in \mathrm{P}(V^\triangle). \] We extend the representation $\omega$ to $(\mathrm{G}(U)\times \mathrm{G}(V))\ltimes \mathrm{H}(W)$ and extend the representation $\omega^-$ to $(\mathrm{G}(U^-)\times \mathrm{G}(V^-))\ltimes \mathrm{H}(W^-)$ such that \[
((g,g')\cdot \phi)\otimes ((h,h')\cdot \phi^-)=gh\cdot(g'h'\cdot (\phi\otimes \phi^-))=g'h'\cdot (gh\cdot (\phi\otimes \phi^-)), \]
for all $(g,h)\in \mathrm{G}(U)\times \mathrm{G}(U^-)$, $(g',h')\in \mathrm{G}(V)\times \mathrm{G}(V^-)$, $\phi\in\omega$ and $\phi^-\in \omega^-$. Then $\omega$ and $\omega^-$ are contragredient to each other with respect to the isomorphism \[
(\mathrm{G}(U)\times \mathrm{G}(V))\ltimes \mathrm{H}(W)\rightarrow (\mathrm{G}(U^-)\times \mathrm{G}(V^-))\ltimes \mathrm{H}(W^-), \quad ((g,g'),(u, \alpha))\mapsto ((g,g'),(u, -\alpha)). \]
If necessary, we also write $\omega_{U,V,\psi}$ for the representation $\omega$ of $(\mathrm{G}(U)\times \mathrm{G}(V))\ltimes \mathrm{H}(W)$, and write $\omega^-_{U,V,\psi}$ for the representation $\omega^-$ of $(\mathrm{G}(U^-)\times \mathrm{G}(V^-))\ltimes \mathrm{H}(W^-)$, to emphasize their dependence on $U,V$ and $\psi$.
\vskip 5pt
Thus, we have defined a splitting of (the pushout via $\{\pm 1\} \hookrightarrow \mathbb{C}^\times$ of) the metaplectic cover $\widetilde{\mathrm{G}}(U)$ and $\widetilde{\mathrm{G}}(V)$ over $\mathrm{G}(U)$ and $\mathrm{G}(V)$ respectively, so that the Weil representation $\omega_{U,V,\psi}$ is a representation of the linear group $\mathrm{G}(U) \times \mathrm{G}(V)$. Such a splitting is unique over $\mathrm{G}(U)$ if $U$ is quaternionic-Hermitian of dimension $>1$, but is not unique if $U$ is quaternionic-skew-Hermitian (as one can twist by quadratic characters of $\mathrm{G}(U)$). For the purpose of formulating and proving the Howe duality conjecture, there is no loss of generality in working with a fixed splitting.
\vskip 5pt
More precisely, as in the introduction, for every $\pi\in \operatorname{Irr} (\mathrm{G}(U))$, put \[
\Theta_{\omega}(\pi):=(\omega\otimes \pi^\vee)_{\mathrm{G}(U)}, \] and define the theta lift $\theta_{\omega}(\pi)$ to be the maximal semisimple quotient of $\Theta_{\omega}(\pi)$. Similarly, the theta lift $\theta_{\omega}(\pi')$ is defined for all $\pi'\in \operatorname{Irr}(\mathrm{G}(V))$.
The theta lifts with respect to other oscillator representations, such as $\theta_{\omega^-}$, are analogously defined.
Put \[
s_{U,V}:=\left(\dim U+\frac{\epsilon}{4}\right)-\left(\dim V+\frac{\epsilon'}{4}\right)\quad\textrm{and}\quad s_{V,U}:=\left(\dim V+\frac{\epsilon'}{4}\right)-\left(\dim U+\frac{\epsilon}{4}\right) = -s_{U,V}. \] The following is a reformulation of Proposition \ref{T:mainq00} in the quaternionic case, using the notations introduced above.
\begin{prop} \label{T:mainq2}
If $s_{U,V}>0$, then for all $\pi,\sigma\in \operatorname{Irr}(\mathrm{G}(U))$,
\begin{equation}\label{dimleq1}
\dim \operatorname{Hom}_{\mathrm{G}(V)}(\theta_{\omega}(\pi)\otimes \theta_{\omega^-}(\sigma),\mathbb C)\leq \dim \operatorname{Hom}_{\mathrm{G}(U)}(\pi\otimes \sigma, \mathbb C).
\end{equation} \end{prop}
The linear functional $\lambda_\triangle$ of \eqref{lambdad} induces a $\mathrm{G}(U^\square)$-intertwining linear map \begin{equation}\label{rallisq}
\omega^\square\rightarrow \mathrm I(s_{V,U}), \quad \phi\mapsto (g\mapsto \lambda_\triangle(g\cdot \phi)). \end{equation} Here for each $s\in \mathbb C$,
\[
\mathrm I(s):=\mathrm{Ind}_{\mathrm P(U^\triangle)}^{\mathrm{G}(U^\square)} \, \left(\chi_V\, |{\det}_{U^\triangle}|_\mathrm{F}^s\right),
\]
where ${\det}_{U^\triangle}: \mathrm{GL}(U^\triangle)\rightarrow \mathrm{F}^\times$ denotes the reduced norm map, and $\chi_V$ is viewed as a character of $\mathrm{GL}(U^\triangle)$ via the pullback through this map. Throughout this paper, $\mathrm{Ind}$ will denote the normalised parabolic induction functor.
Denote by $\mathrm{G}(V)^\triangle$ the group $\mathrm{G}(V)$ diagonally embedded in $\mathrm{G}(V)\times \mathrm{G}(V^-)$, to be viewed as a subgroup of $\mathrm{G}(V^\square)$.
\begin{lem}\label{ry} The linear map \eqref{rallisq} induces a $\mathrm{G}(U^\square)$-intertwining linear embedding \[ (\omega^\square)_{\mathrm{G}(V)^\triangle}\hookrightarrow \mathrm I(s_{V,U}). \] If $s_{U,V}>0$, then there exists a surjective $\mathrm{G}(U^\square)$-intertwining linear map \[ \mathrm I(s_{U,V})\twoheadrightarrow(\omega^\square)_{\mathrm{G}(V)^\triangle} \subset I(s_{V,U}). \] \end{lem} \begin{proof} The first assertion is due to Rallis, see \cite[Theorem II.1.1]{R} and \cite[Chapter 3, Theorem IV.7]{mvw}. The second one is proved in \cite[Theorems 1.3 and 1.4]{Ya}. \end{proof}
Write $q_U$ for the Witt index of $U$. Fix two sequences \begin{equation}\label{flagu0}
0=X_0\subset X_1\subset \cdots \subset X_{q_U} \qquad \textrm{and}\qquad X_{q_U}^*\supset \cdots \supset X_1^*\supset X_0^*=0 \end{equation} of totally isotropic subspaces of $U$ such that for all $t=0,1,\cdots, q_U$, \begin{equation}\label{flagu} \left\{
\begin{array}{ll}
\dim X_t=\dim X_t^*=t; \\
X_t\cap X_t^*=0; \ \textrm{ and}\\
X_t\oplus X_t^*\textrm{ is non-degenerate}.
\end{array} \right. \end{equation} Denote by $U_t$ the orthogonal complement of $X_t\oplus X_t^*$ in $U$. Write $\mathrm P(X_t)$ and $\mathrm P(X_t^*)$ for the parabolic subgroups of $\mathrm{G}(U)$ stabilizing $X_t$ and $X_t^*$, respectively. Then \[
\mathrm P(X_t)\cap \mathrm P(X_t^*)=\mathrm{GL}(X_t)\times \mathrm{G}(U_t) \] is a common Levi factor of $\mathrm P(X_t)$ and $\mathrm P(X_t^*)$.
We need the following lemma (see \cite[Section 1]{kr05}). \begin{lem} \label{L:key0} Let $s\in \mathbb C$. As a representation of $\mathrm{G}(U) \times \mathrm{G}(U^-)$, $\mathrm I(s)$ possesses an equivariant filtration \[ 0=I_{-1}(s) \subset I_0(s) \subset I_1(s) \subset\cdots\subset I_{q_U}(s) =\mathrm I(s) \] with successive quotients \[
R_t(s) =I_t(s) / I_{t-1}(s) = {\rm Ind}_{{\rm P}(X_t) \times {\rm P}(X_t)}^{\mathrm{G}(U) \times
\mathrm{G}(U^-)} \left( \left(\chi_V |{\det}_{X_{t}}|_\mathrm{F}^{s + t} \boxtimes
\chi_V |{\det}_{X_t}|_\mathrm{F}^{s+t} \right) \otimes C^{\infty}_c(\mathrm{G}(U_t)) \right), \] where $0\leq t\leq q_U$, and \begin{itemize}
\item $\det_{X_{t}}:\mathrm{GL}(X_t)\rightarrow \mathrm{F}^\times$ denotes the reduced norm map, and $\chi_V$ is viewed as a character of $\mathrm{GL}(X_t)$ via the pullback through this map; \item $\mathrm{G}(U_t)\times \mathrm{G}(U_t)$ acts on $C^{\infty}_c( \mathrm{G}(U_t))$ by left-right translation. \end{itemize} In particular, ${R}_0(s) = C^{\infty}_c(\mathrm{G}(U))$ is the regular representation. \end{lem}
In view of Lemma \ref{L:key0}, we make the following definition. \vskip 5pt
\noindent{\bf \underline{Definition}:} We say that an irreducible admissible smooth representation $\pi \boxtimes \sigma$ of $\mathrm{G}(U) \times \mathrm{G}(U^-)$ lies on the boundary of $\rm I(s)$ if
\[
\operatorname{Hom}_{\mathrm{G}(U) \times \mathrm{G}(U^-)}({R}_t(s), \pi \boxtimes \sigma) \ne 0 \quad \text{ for some $0 < t \leq q_U$,}
\] where ${R}_t(s)$ is as in Lemma \ref{L:key0}.
\vskip 5pt
Now we have: \vskip 5pt
\begin{prop} \label{P:nonb} Proposition \ref{T:mainq2} holds when $\pi\boxtimes \sigma$ does not lie on the boundary of ${\rm I}(s_{U,V})$. \end{prop}
\begin{proof}
Consider the doubling see-saw \[ \xymatrix{ \mathrm{G}(U^\square)\ar@{-}[d]_{}\ar@{-}[dr]&\mathrm{G}(V)\times \mathrm{G}(V^-)\ar@{-}[d]\\ \mathrm{G}(U)\times \mathrm{G}(U^-)\ar@{-}[ur]&\mathrm{G}(V)^{\triangle} \,. } \] Given $\pi,\sigma\in \operatorname{Irr}(\mathrm{G}(U))$, the see-saw identity gives \begin{equation}\label{E:see-saw} \operatorname{Hom}_{\mathrm{G}(U) \times \mathrm{G}(U^-)} ( (\omega^\square)_{\mathrm{G}(V)^\triangle}, \pi \boxtimes \sigma) = \operatorname{Hom}_{\mathrm{G}(V)}( \Theta_{\omega}(\pi) \otimes \Theta_{\omega^-}(\sigma), \mathbb{C}). \end{equation} Assume that $s_{U,V}>0$ and $\pi\boxtimes \sigma$ does not lie on the boundary of ${\rm I}(s_{U,V})$, then we have \begin{eqnarray*}
&&\operatorname{Hom}_{\mathrm{G}(V)}( \theta_{\omega}(\pi) \otimes \theta_{\omega^-}(\sigma), \mathbb{C}) \\
&\hookrightarrow& \operatorname{Hom}_{\mathrm{G}(V)}( \Theta_{\omega}(\pi) \otimes \Theta_{\omega^-}(\sigma), \mathbb{C})\\
&=&\operatorname{Hom}_{\mathrm{G}(U) \times \mathrm{G}(U^-)} ( (\omega^\square)_{\mathrm{G}(V)^\triangle}, \pi \boxtimes \sigma) \\
&\hookrightarrow& \operatorname{Hom}_{\mathrm{G}(U) \times \mathrm{G}(U^-)} ( {\rm I}(s_{U,V}), \pi \boxtimes \sigma) \qquad \qquad\textrm{(by Lemma \ref{ry})}\\
&\hookrightarrow& \operatorname{Hom}_{\mathrm{G}(U) \times \mathrm{G}(U^-)} ( C^{\infty}_c(\mathrm{G}(U)), \pi \boxtimes \sigma) \qquad \qquad\textrm{(by Lemma \ref{L:key0})}\\
&\cong& \operatorname{Hom}_{\mathrm{G}(U)} ( \pi \otimes \sigma,\mathbb{C}). \end{eqnarray*} This proves the proposition. \end{proof}
\section{\bf Some induced representations}
To complete the proof of Proposition \ref{T:mainq2}, we need to consider representations $\pi \boxtimes \sigma$ of $\mathrm{G}(U) \times \mathrm{G}(U^-)$ which lie on the boundary of ${\rm I}(s_{U,V})$. To deal with these, we study in this section some parabolically induced representations which will play an important role later on. \vskip 5pt
For smooth representations $\rho$ of $\mathrm{GL}(X_t)$ and $\sigma$ of $\mathrm{G}(U_{t})$ ($0\leq t\leq q_U$), we write \[ \rho \rtimes \sigma := {\rm Ind}_{\mathrm P(X_t)}^{\mathrm{G}(U)} \rho \otimes \sigma. \]
More generally, the parabolic subgroup $P$ of $\mathrm{G}(U)$ stabilizing a flag
\begin{equation}\label{flagx}
0=X_{t_0}\subset X_{t_1}\subset X_{t_2}\subset\cdots\subset X_{t_a}
\end{equation} has a Levi factor of the form $\mathrm{GL}(X_{t_1}) \times \mathrm{GL}(X_{t_2}/X_{t_1})\times \cdots \times \mathrm{GL}(X_{t_a}/X_{t_{a-1}})\times \mathrm{G}(U_{t_a})$. We set \[
\rho_1\times\cdots\times\rho_a\rtimes \sigma:= \operatorname{Ind}_{P}^{\mathrm{G}(U)}\rho_1\otimes\cdots\otimes\rho_a\otimes \sigma, \] where $\rho_i$ is a smooth representation of $\mathrm{GL}(X_{t_i}/X_{t_{i-1}})$ and $\sigma$ is a smooth representation of $\mathrm{G}(U_{t_a})$.
Similarly, for the general linear group $\mathrm{GL}(X_{t_a})$, we set \[ \rho_1\times\cdots\times\rho_a:= \operatorname{Ind}_Q^{\mathrm{GL}(X_{t_a})} \rho_1\otimes\cdots\otimes\rho_a, \] where $Q$ is the parabolic subgroup of $\mathrm{GL}(X_{t_a})$ stabilizing the flag \eqref{flagx}. Respectively write $\mathrm R_{X_t}$ and $\mathrm R_{X_t^*}$ for the normalized Jacquet functors attached to $\mathrm{P}(X_t)$ and $\mathrm{P}(X_t^*)$. \vskip 5pt
Let $\eta:\mathrm{F}^\times \rightarrow \mathbb{C}^\times$ be a character of $\mathrm{F}^\times$. Then $\eta^{\times a}$ ($0\leq a\leq q_U$) is an irreducible representation of $\mathrm{GL}(X_a)$ (\emph{cf.} \cite{Se}). Here $\eta$ is viewed as a character of $\mathrm{GL}(X_t/X_{t-1})$ ($1\leq t\leq a$) via the pullback through the reduced norm map $\mathrm{GL}(X_t/X_{t-1})\rightarrow \mathrm{F}^\times$. For every $\pi\in \operatorname{Irr}(\mathrm{G}(U))$, define \begin{equation}\label{meta}
\mathrm m_{\eta}(\pi):=\max\{0\leq a\leq q_U\mid \pi \hookrightarrow \eta^{\times a}\rtimes \sigma \textrm{ for some }\sigma\in \operatorname{Irr}(\mathrm{G}(U_a))\}.
\end{equation}
Here $\pi \hookrightarrow \eta^{\times a}\rtimes \sigma$ means that there is an injective homomorphism from $\pi$ to $\eta^{\times a}\rtimes \sigma$ (similar notation will be used without further explanation).
The rest of this section is devoted to a proof of the following proposition.
\begin{prop}\label{P:induced} Assume that $\eta^2$ is non-trivial. Then for every $\pi\in \operatorname{Irr}(\mathrm{G}(U))$, there is a unique representation $\pi_\eta\in \operatorname{Irr}(\mathrm{G}(U_a))$ such that $\pi\hookrightarrow \eta^{\times a}\rtimes \pi_\eta$, where $a:=\mathrm m_{\eta}(\pi)$. Moreover, \begin{equation}\label{piiso} \left\{
\begin{array}{ll}
\pi_\eta\cong \left(\mathrm R_{X_a}(\pi)\otimes (\eta^{-1})^{\times a}\right)_{\mathrm{GL}(X_a)}\cong \operatorname{Hom}_{\mathrm{GL}(X_a)}((\eta^{-1})^{\times a}, \mathrm R_{X_{a}^*}(\pi));
\\
\pi \textrm{ is isomorphic to the socle of } \eta^{\times a}\rtimes \pi_\eta;
\\
\mathrm m_\eta(\pi)=\mathrm m_\eta(\pi^\vee);
\\
(\pi^\vee)_\eta\cong (\pi_\eta)^\vee.
\end{array} \right. \end{equation}
\end{prop} \vskip 5pt
We begin with the following lemma. \begin{lem}\label{L:ext} For all $\rho\in \operatorname{Irr}(\mathrm{GL}(X_a))$ ($0\leq a\leq q_U$) which is not isomorphic to $\eta^{\times a}$, \[
\mathrm{Ext}^i_{\mathrm{GL}(X_a)}(\eta^{\times a},\rho)=0\qquad (i\in \mathbb{Z}). \] \end{lem} \begin{proof} Since the Jacquet functor is exact and maps injective representations to injective representations, the second adjointness theorem of Bernstein implies that \begin{equation}\label{extgl}
\mathrm{Ext}^i_{\mathrm{GL}(X_a)}(\eta^{\times a},\rho)\cong \mathrm{Ext}^i_{(\mathrm{D}^\times)^a}(\eta^{\boxtimes a}, \bar{\mathrm R} (\rho)). \end{equation} Here $\mathrm{GL}(X_a)$ is identified with $\mathrm{GL}_a(\mathrm{D})$ as usual, and $\bar{\mathrm R}$ denotes the normalized Jacquet functor attached to the minimal parabolic subgroup of $\mathrm{GL}_a(\mathrm{D})$ of lower triangular matrices. Note that $\rho\ncong \eta^{\times a}$ implies that $\eta^{\boxtimes a}$ is not a subquotient of $\bar{\mathrm R}(\rho)$ (\emph{cf.} \cite[Chapter 3, Section 2.1, Theorem 18]{Be}). Then it is easy to see that the right hand side of \eqref{extgl} vanishes.
\end{proof}
By an easy homological algebra argument, Lemma \ref{L:ext} implies the following lemma. \begin{lem}\label{extgg} For all $\rho\in \operatorname{Irr}(\mathrm{GL}(X_a))$ ($0\leq a\leq q_U$) which is not isomorphic to $\eta^{\times a}$, and all $\sigma, \sigma'\in \operatorname{Irr}(\mathrm{G}(U_a))$, \[
\mathrm{Ext}^i_{\mathrm{GL}(X_a)\times \mathrm{G}(U_a)}(\eta^{\times a}\boxtimes\sigma ,\rho\boxtimes \sigma')=0\qquad (i\in \mathbb{Z}). \] \end{lem}
From now on, we assume that the character $\eta^2 \ne 1$.
\begin{lem} \label{L:geometric0}
Let $\sigma\in \operatorname{Irr}(\mathrm{G}(U_a))$ ($0\leq a\leq q_U$). Assume that $\mathrm m_\eta(\sigma)=0$ (as defined in \eqref{meta}). Then
\[
\mathrm R_{X_a}(\eta^{\times a}\rtimes \sigma)\cong\left(\eta^{\times a}\boxtimes \sigma\right)\oplus \rho,
\]
where $\rho$ is a smooth representation of $\mathrm{GL}(X_a)\times \mathrm{G}(U_a)$ which has no irreducible subquotient of the form $\eta^{\times a}\boxtimes \sigma'$ with $\sigma'\in \operatorname{Irr}(\mathrm{G}(U_a))$. Consequently, the socle of $\eta^{\times a}\rtimes \sigma$ is irreducible.
\end{lem}
\begin{proof}
Denote by $\rho$ the kernel of the natural surjective homomorphism
\begin{equation}\label{sursplit}
\mathrm R_{X_a}(\eta^{\times a}\rtimes \sigma)\twoheadrightarrow \eta^{\times a}\boxtimes \sigma.
\end{equation} As in the proof of \cite[Lemma 5.2]{gt}, using an explication of the Geometric Lemma of Bernstein-Zelevinsky (\emph{cf}. \cite[Lemma 5.1]{ta} and \cite{Ha}), the assumption of the lemma implies that $\rho$ contains no irreducible subquotient of the form $\eta^{\times a}\boxtimes \sigma'$ with $\sigma'\in \operatorname{Irr}(\mathrm{G}(U_a))$. Then Lemma \ref{extgg} implies that the surjective homomorphism \eqref{sursplit} splits. This proves the first assertion of the lemma. The second assertion then easily follows as in \cite[Lemma 5.2]{gt}.
\end{proof}
In the rest of this section, let $\pi\in \operatorname{Irr}(\mathrm{G}(U))$ and put $a:=\mathrm m_\eta(\pi)$. Then there is an irreducible representation $\sigma\in \operatorname{Irr}(\mathrm{G}(U_a))$ such that $\pi \hookrightarrow\eta^{\times a}\rtimes \sigma$. Induction-by-steps shows that $\mathrm m_\eta(\sigma)=0$.
\begin{lem} \label{L:geometricirr2} One has that \[
\mathrm R_{X_a}(\pi)\cong\left(\eta^{\times a}\boxtimes \sigma\right)\oplus \rho,
\]
where $\rho$ is a smooth representation of $\mathrm{GL}(X_a)\times \mathrm{G}(U_a)$ which has no irreducible subquotient of the form $\eta^{\times a}\boxtimes \sigma'$ with $\sigma'\in \operatorname{Irr}(\mathrm{G}(U_a))$.
\end{lem}
\begin{proof} Since $\mathrm R_{X_a}(\pi)$ is a subrepresentation of $\mathrm R_{X_a}(\eta^{\times a}\rtimes \sigma)$ and has $\eta^{\times a}\boxtimes \sigma$ as an irreducible quotient, the lemma easily follows from Lemma \ref{L:geometric0}. \end{proof}
\begin{lem} \label{L:geometricirr3} One has that \[
\mathrm R_{X_a^*}(\pi)\cong\left((\eta^{-1})^{\times a}\boxtimes \sigma\right)\oplus \rho,
\]
where $\rho$ is a smooth representation of $\mathrm{GL}(X_a)\times \mathrm{G}(U_a)$ which has no irreducible subquotient of the form $(\eta^{-1})^{\times a}\boxtimes \sigma'$ with $\sigma'\in \operatorname{Irr}(\mathrm{G}(U_a))$.
\end{lem} \begin{proof}
Note that $\mathrm{P}(X_a)$ is conjugate to $\mathrm{P}(X_a^*)$ by an element $w \in \mathrm{G}(U)$ such that $w$ is the identity on $U_a$ and $w$ exchanges $X_a$ and $X_a^*$.
Via conjugation by $w$, we see that Lemma \ref{L:geometricirr3} is equivalent to Lemma \ref{L:geometricirr2}.
\end{proof}
Lemma \ref{L:geometricirr2} and Lemma \ref{L:geometricirr3} imply that \[
\sigma\cong \left(\mathrm R_{X_a}(\pi)\otimes (\eta^{-1})^{\times a}\right)_{\mathrm{GL}(X_a)}\cong \operatorname{Hom}_{\mathrm{GL}(X_a)}((\eta^{-1})^{\times a}, \mathrm R_{X_{a}^*}(\pi)). \] This proves the uniqueness assertion of Proposition \ref{P:induced}, as well as the first assertion of \eqref{piiso}. The second assertion of \eqref{piiso} is then implied by the last assertion of Lemma \ref{L:geometric0}.
\begin{lem} \label{L:geometric4}
One has that
\[
\pi^\vee \hookrightarrow\eta^{\times a}\rtimes \sigma^\vee.
\]
\end{lem}
\begin{proof}
Lemma \ref{L:geometricirr3} implies that
\[
(\eta^{-1})^{\times a}\boxtimes \sigma\hookrightarrow \mathrm R_{X_a^*}(\pi).
\]
By dualizing and using the second adjointness theorem, we see that
\[
\mathrm R_{X_a}(\pi^\vee)\twoheadrightarrow \eta^{\times a}\boxtimes \sigma^\vee.
\]
This implies that $\pi^\vee \hookrightarrow\eta^{\times a}\rtimes \sigma^\vee$.
\end{proof}
Lemma \ref{L:geometric4} implies that $\mathrm m_\eta(\pi^\vee)\geq \mathrm m_\eta(\pi)$. The same argument shows that $\mathrm m_\eta(\pi)\geq \mathrm m_\eta(\pi^\vee)$. This proves that $\mathrm m_\eta(\pi^\vee)=\mathrm m_\eta(\pi)$. Lemma \ref{L:geometric4} then further implies that $(\pi^\vee)_\eta\cong (\pi_\eta)^\vee$. This finally finishes the proof of Proposition \ref{P:induced}.
\vskip 5pt
\section{\bf Induced representations and theta correspondence}
In this section, we apply the results of the previous section to the theta correspondence. Write $\eta': \mathrm{F}^\times\rightarrow \mathbb{C}^\times$ for the character such that \[
\eta' \cdot \chi_V=\eta \cdot \chi_U. \] Then $(\eta')^2 \ne 1$ since $\eta^2 \ne 1$. Denote by $q_V$ the Witt index of $V$. Similarly to \eqref{flagu0}, we fix two sequences \[
0=Y_0\subset Y_1\subset \cdots \subset Y_{q_V}\qquad \textrm{and}\qquad Y_{q_V}^*\supset \cdots \supset Y_1^*\supset Y_0^*=0 \] of totally isotropic subspaces of $V$ with the analogous properties as in \eqref{flagu0}. We apply the analogous notation as in the last section to the space $V$. In particular, $\mathrm m_{\eta'}(\pi')$ is defined for every $\pi'\in \operatorname{Irr}(\mathrm{G}(V))$. Define $\pi_\eta$ ($\pi\in \operatorname{Irr}(\mathrm{G}(U))$ and $\pi'_{\eta'}$ as in Proposition \ref{P:induced}.
For all integers $0\leq a\leq q_U$ and $0\leq k\leq q_V$, write $\omega_{a,k}:=\omega_{U_a, V_k,\psi}$, which is an irreducible smooth representation of $(\mathrm{G}(U_a)\times \mathrm{G}(V_k))\ltimes \mathrm{H}(U_a\otimes_\mathrm{D} V_k)$, as defined in Section \ref{secdouble}. \vskip 5pt
The rest of this section is devoted to a proof of the following key proposition. \begin{prop}\label{induction} Assume that \[
\eta\neq \chi_V \abs{\,\cdot\,}_\mathrm{F}^{s_{V,U}+1} \qquad \textrm{and}\qquad \eta'\neq \chi_U \abs{\cdot}_\mathrm{F}^{s_{U,V}+1}. \] Then for all $\pi\in \operatorname{Irr}(\mathrm{G}(U))$ and $\pi'\in \operatorname{Irr}(\mathrm{G}(V))$ such that $\operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega, \pi\boxtimes \pi')\neq 0$, one has \[
\mathrm m_{\eta}(\pi)=\mathrm m_{\eta'}(\pi'), \] and there is a linear embedding \[
\operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega, \pi\boxtimes \pi')\hookrightarrow \operatorname{Hom}_{\mathrm{G}(U_a)\times \mathrm{G}(V_a)}(\omega_{a,a}, \pi_\eta\boxtimes \pi'_{\eta'}), \] where $a:=\mathrm m_{\eta}(\pi)$. \end{prop}
For each right $\mathrm{D}$-vector space $X$, write $X^\iota$ for the left $\mathrm{D}$-vector which equals $X$ as an abelian group and whose scalar multiplication is given by \[
\alpha v:=v \alpha^\iota,\qquad \alpha\in \mathrm{D}, \,v\in X^\iota. \] We first recall the well-known computation of the Jacquet module of the Weil representation (see \cite[Theorem 2.8]{k83} and \cite[Chapter 3, Section IV.5]{mvw}).
\vskip 5pt
\begin{lem} \label{L:kudla} For each $0\leq a\leq q_U$, the normalized Jacquet module $\mathrm R_{X_a}(\omega)$ has a $\mathrm{GL}(X_a)\times \mathrm{G}(U_a)\times \mathrm{G}(V)$-equivariant filtration \[
\mathrm R_{X_a}(\omega) = R_0 \supset R_1 \supset \cdots \supset R_{a'} \supset R_{a'+1} = 0 \] whose successive quotient is \[
J_k:=R_k/R_{k+1}\cong \operatorname{Ind}^{\mathrm{GL}(X_a) \times \mathrm{G}(U_a) \times \mathrm{G}(V)}_{\mathrm{P}(X_{a-k}, X_a) \times \mathrm{G}(U_a) \times \mathrm{P}(Y_k)} \left(\chi_V
|{\det}_{X_{a-k}}|_\mathrm{F}^{s_{V,U}+a-k} \otimes C^{\infty}_c(\operatorname{Isom}(X_a^\iota/X_{a-k}^\iota, Y_k)) \otimes \omega_{a,k}\right), \]
where \begin{itemize} \item $a':=\min\{a, q_V\}$ and $0\leq k\leq a'$; \item $\mathrm{P}(X_{a-k}, X_a)$ is the parabolic subgroup of $\mathrm{GL}(X_a)$ stabilizing $X_{a-k}$; \item $\det_{X_{a-k}}: \mathrm{GL}(X_{a-k})\rightarrow \mathrm{F}^\times$ denotes the reduced norm map, and $\chi_V$ is viewed as a character of $\mathrm{GL}(X_{a-k})$ via the pullback through this map; \item $\operatorname{Isom}(X_a^\iota/X_{a-k}^\iota, Y_k)$ is the set of $\mathrm{D}$-linear isomorphisms from $X_a^\iota/X_{a-k}^\iota$ to $Y_k$, and $\mathrm{GL}(X_a/X_{a-k})\times\mathrm{GL}(Y_k)$ acts on
$C_c^\infty(\operatorname{Isom}(X_a^\iota/X_{a-k}^\iota,Y_k))$ as
\[
((b,c)\cdot f)(g)=\chi_V (\det b)\chi_U(\det
c)f(c^{-1}g b),
\]
for $(b,c)\in \mathrm{GL}(X_a/X_{a-k})\times\mathrm{GL}(Y_k)$, $f\in
C_c^\infty(\operatorname{Isom}(X_a^\iota/X_{a-k}^\iota,Y_k))$ and $g\in\operatorname{Isom}(X_a^\iota/X_{a-k}^\iota,Y_k)$. \end{itemize} In particular, if $a'=a$, then the bottom piece of the filtration is \[ J_a \cong \operatorname{Ind}^{\mathrm{GL}(X_a) \times G(U_a) \times \mathrm{G}(V)}_{\mathrm{GL}(X_a) \times \mathrm{G}(U_a) \times \mathrm{P}(Y_a)} \left(C^{\infty}_c(\operatorname{Isom}(X_a^\iota,Y_a)) \otimes \omega_{a,a}\right). \] \end{lem}
The following lemma is an observation of \cite{gt}. \begin{lem}\label{boundo} Let $a$, $k$ and $J_k$ be as in Lemma \ref{L:kudla}. Assume that $\eta\neq \chi_V \abs{\,\cdot\,}_\mathrm{F}^{s_{V,U}+1}$. Then for all $\sigma\in \operatorname{Irr}(\mathrm{G}(U_a))$ and $\pi'\in \operatorname{Irr}(\mathrm{G}(V))$, \begin{equation}\label{homggg}
\operatorname{Hom}_{\mathrm{GL}(X_a) \times \mathrm{G}(U_a) \times \mathrm{G}(V)}(J_k, \eta^{\times a}\boxtimes \sigma\boxtimes \pi')=0 \end{equation} whenever $k\neq a$. \end{lem} \begin{proof} Using the second adjointness theorem, it suffices to show that \[
\operatorname{Hom}_{\mathrm{GL}(X_{a-k})}(\chi_V \abs{{\det}_{X_{a-k}}}_\mathrm{F}^{s_{V,U}+a-k}, \bar{\mathrm R}_{X_{a-k},X_a}(\eta^{\times a}))=0, \] where $\bar{\mathrm R}_{X_{a-k},X_a}$ denotes the normalized Jacquet functor attached to the parabolic subgroup of $\mathrm{GL}(X_a)$ stabilizing a complement of $X_{a-k}$ in $X_a$. By analysing the cuspidal data, we know that every irreducible subrepresentation of $\bar{\mathrm R}_{X_{a-k},X_a}(\eta^{\times a})$ is isomorphic to $\eta^{\times (a-k)}\boxtimes \eta^{\times k}$, as a representation of $\mathrm{GL}(X_{a-k})\times \mathrm{GL}(X_a/X_{a-k})$. Therefore the lemma follows. \end{proof}
Now we come to the proof of Proposition \ref{induction}. Put $a:=\mathrm m_{\eta}(\pi)$. Then we have \begin{eqnarray*}
0 &\neq &\operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega, \pi\boxtimes \pi')\\
&\hookrightarrow& \operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega, (\eta^{\times a}\rtimes \pi_\eta)\boxtimes \pi')\\
&=&\operatorname{Hom}_{\mathrm{GL}(X_a)\times \mathrm{G}(U_a) \times \mathrm{G}(V)} ( \mathrm R_{X_a}(\omega), \eta^{\times a}\boxtimes \pi_\eta \boxtimes \pi') \\
&\hookrightarrow& \operatorname{Hom}_{\mathrm{GL}(X_a)\times \mathrm{G}(U_a) \times \mathrm{G}(V)} (J_a, \eta^{\times a}\boxtimes \pi_\eta \boxtimes \pi') \qquad \qquad\textrm{(by Lemma \ref{boundo})}\\
&\cong & \operatorname{Hom}_{\mathrm{GL}(X_a)\times \mathrm{G}(U_a) \times \mathrm{GL}(Y_a)\times \mathrm{G}(V_a)} (C^{\infty}_c(\operatorname{Isom}(X_a^\iota,Y_a)) \otimes \omega_{a,a}, \eta^{\times a}\boxtimes \pi_\eta \boxtimes \mathrm R_{Y_a^*}(\pi')) \\
&& \quad \qquad\textrm{(by the second ajointness theorem)}\\
&\cong & \operatorname{Hom}_{\mathrm{G}(U_a) \times \mathrm{GL}(Y_a)\times \mathrm{G}(V_a)} (({\eta'}^{-1})^{\times a}\boxtimes \omega_{a,a}, \pi_\eta \boxtimes \mathrm R_{Y_a^*}(\pi'))\\
& \cong & \operatorname{Hom}_{\mathrm{G}(U_a) \times \mathrm{G}(V_a)} (\omega_{a,a}, \pi_\eta \boxtimes \pi'_a),\\ \end{eqnarray*} where \[
\pi_a':=\operatorname{Hom}_{\mathrm{GL}(Y_a)}(({\eta'}^{-1})^{\times a}, \mathrm R_{Y_a^*}(\pi')). \] Therefore $\pi_a'\neq 0$, and hence \[
\operatorname{Hom}_{\mathrm{GL}(Y_a)\times \mathrm{G}(V_a)}(({\eta'}^{-1})^{\times a}\boxtimes {\pi_a'}, \mathrm R_{Y_a^*}(\pi'))\neq 0. \] Dualizing and using the second adjointness theorem, we see that \[
\operatorname{Hom}_{\mathrm{GL}(Y_a)\times \mathrm{G}(V_a)}(\mathrm R_{Y_a}({\pi'}^\vee), {\eta'}^{\times a}\boxtimes {\pi_a'}^\vee)\neq 0. \] This proves that \[ \mathrm m_{\eta'}(\pi')=\mathrm m_{\eta'}({\pi'}^\vee)\geq a=\mathrm m_{\eta}(\pi). \] The same argument shows that $\mathrm m_{\eta}(\pi)\geq \mathrm m_{\eta'}(\pi')$, and hence $\mathrm m_{\eta'}(\pi')=\mathrm m_{\eta}(\pi)$. Therefore $\pi'_a\cong\pi'_{\eta'}$ by Proposition \ref{P:induced}. This finishes the proof of Proposition \ref{induction}.
\section{\bf Proof of Proposition \ref{T:mainq2}}
In this section, we finish the proof of Proposition \ref{T:mainq2} by induction on $\dim U$. As in Proposition \ref{T:mainq2}, let $\pi,\sigma\in \operatorname{Irr}(\mathrm{G}(U))$ and assume that $s_{U,V}>0$. In view of Proposition \ref{P:nonb}, we may assume that $\pi\boxtimes \sigma$ lies on the boundary of ${\rm I}(s_{U,V})$. Then there is an integer $0 < t \leq q_U$ such that
\begin{equation}\label{bound2}
\operatorname{Hom}_{\mathrm{G}(U) \times \mathrm{G}(U^-)}({R}_t(s_{U,V}), \pi \boxtimes \sigma) \ne 0.
\end{equation} Note that \begin{eqnarray}\label{countdim}
&& \dim \operatorname{Hom}_{\mathrm{G}(V)}(\theta_{\omega}(\pi)\otimes \theta_{\omega^-}(\sigma),\mathbb C) \\
\nonumber &=& \sum_{\pi'\in \operatorname{Irr}(\mathrm{G}(V))} \dim \operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega, \pi\boxtimes \pi')\, \cdot\,
\dim \operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega^-, \sigma\boxtimes {\pi'}^\vee). \end{eqnarray} We assume that the value of the above equality is non-zero, as Proposition \ref{T:mainq2} is otherwise trivial. Then there is an irreducible representation $\pi'\in \operatorname{Irr}(\mathrm{G}(V))$ such that \begin{equation}\label{homdouble}
\operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega, \pi\boxtimes \pi')\neq 0\quad \textrm{and}\quad \operatorname{Hom}_{\mathrm{G}(U)\times \mathrm{G}(V)}(\omega^-, \sigma\boxtimes {\pi'}^\vee)\neq 0. \end{equation}
By the second adjointness theorem, \eqref{bound2} implies that \begin{equation}\label{homgt}
\operatorname{Hom}_{\mathrm{GL}(X_t)} (\chi |{\det}_{X_{t}}|_\mathrm{F}^{s_{U,V} + t}, \mathrm R_{X_t^*} (\pi))\neq 0. \end{equation} Put \[
\eta:=\chi_V \,\abs{\,\cdot\,}^{s_{V,U}-2t+1}\quad \textrm{and}\quad \eta':=\chi_U \,\abs{\,\cdot\,}^{s_{V,U}-2t+1}. \]
Using the second adjointness theorem and the Langlands parameter of the character $\chi_V |{\det}_{X_{t}}|_\mathrm{F}^{s_{V,U}-t}$, \eqref{homgt} implies that \[
\mathrm m_{\eta}(\pi)=\mathrm m_{\eta}(\pi^\vee)>0. \] Noting that \[ \eta\neq \chi_V \abs{\,\cdot\,}_\mathrm{F}^{s_{V,U}+1} \qquad \textrm{and}\qquad \eta'\neq \chi_U \abs{\cdot}_\mathrm{F}^{s_{U,V}+1}, \] Proposition \ref{induction} (and its analog for $\omega^-$) then implies that \[
\mathrm m_{\eta}(\pi)=\mathrm m_{\eta'}(\pi')=\mathrm m_{\eta'}({\pi'}^\vee)=\mathrm m_{\eta}(\sigma). \]
By the induction assumption, Proposition \ref{T:mainq2} holds for the pair $(U_a, V_a)$, where $a:=\mathrm m_{\eta}(\pi)$. As we have seen at the end of the introduction, this implies that Theorem \ref{T:howe3} holds for the pair $(\mathrm{G}(U_a), \mathrm{G}(V_a))$. Together with Proposition \ref{induction}, this implies that \begin{equation}\label{sigmaa0}
{\pi'}_{\eta'}\cong\theta_{\omega_{a,a}}(\pi_\eta) \end{equation} and \begin{equation}\label{sigmaa}
\pi_\eta\cong\theta_{\omega_{a,a}}(\pi'_{\eta'})\qquad \textrm{and}\qquad \sigma_\eta\cong \theta_{\omega^-_{a,a}}((\pi'_{\eta'})^\vee). \end{equation} Here $\omega^-_{a,a}:=\omega^-_{U_a,V_a,\psi}$. Proposition \ref{induction} and \eqref{sigmaa0} imply that $\pi'$ is isomorphic to the socle of ${\eta'}^{\times a}\rtimes \theta_{\omega_{a,a}}(\pi_\eta)$. Therefore, there is a unique $\pi'\in \operatorname{Irr}(\mathrm{G}(U))$ which satisfies \eqref{homdouble}. Then Proposition \ref{induction} implies that the value of \eqref{countdim} is $1$.
On the other hand, \eqref{sigmaa} and the induction assumption imply that $\pi_\eta^\vee\cong \sigma_\eta$, which further implies that $\pi^\vee\cong \sigma$ by Proposition \ref{P:induced}. Therefore \eqref{dimleq1} of Proposition \ref{T:mainq2} is an equality. This finishes the proof of Proposition \ref{T:mainq2}.
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\end{document} | arXiv |
Sociodemographic factors associated with low intake of bioavailable iron in preschoolers: National Health and Nutrition Survey 2012, Mexico
Yazmín Venegas-Aviles1,
Sonia Rodríguez-Ramírez ORCID: orcid.org/0000-0002-5439-43891,
Eric Monterrubio-Flores1 &
Armando García-Guerra1
Nutrition Journal volume 19, Article number: 57 (2020) Cite this article
Children < 5 years of age are at risk of developing an iron deficiency due to a low intake of bioavailable iron (FeBio). Few studies have estimated dietary FeBio in children at a national level in relation to sociodemographic characteristics. This study aimed to estimate FeBio intake and its association with sociodemographic factors among Mexican children aged 12–59 months.
A cross-sectional study was carried out. Information on serum ferritin and diet was obtained from a national survey and representative sample of 1012 Mexican children aged 12–59 months. We used a 24-h recall to estimate total iron, heme and non-heme iron, vitamin C, phytates, calcium, and meat intake. We calculated FeBio intake using an algorithm. Differences in FeBio intake were analyzed by area of residence (rural/urban), country region (north, center, south), and socioeconomic status (SES), using linear regression models by age subgroups (12–23 and 24–59 months) and total population, while adjusting for study design.
Total iron intake was 9.2 ± 6.7 mg/d. The estimated average of total FeBio fluctuated between 0.74–0.81 mg/d, with a bioavailability of 9.15–12.03% of total iron. Children aged 12–23 months residing in rural areas consumed less FeBio than those in urban areas (β = − 0.276) (p < 0.05). Children aged 24–59 months with high SES consumed more FeBio (β = 0.158 mg/d) than those of a low SES (p < 0.05).
FeBio is low in Mexican preschoolers. Being from a rural area and having low SES were negatively associated with FeBio intake. These results can benefit interventions seeking to improve iron status.
Worldwide, 47% of children under the age of five have anemia, which is mainly attributed to iron deficiency [1]. In 2012, the national prevalence of iron deficiency in Mexican preschool children was 13.6%, while prevalence of anemia was 23.3% [2].
Iron deficiency in children < 2 years of age can result in long-term cognitive deficits and psychomotor impairment [3, 4]. In preschool children, iron deficiency is associated with a lower intelligence quotient, behavioral changes, and reduced capacity for physical activity [5, 6].
Iron-deficiency anemia is due to low iron intake and more specifically, low bioavailable iron (FeBio). Inflammation, parasitic diseases, and genetic disorders are other factors known to cause anemia [1]. Iron bioavailability refers to the proportion of iron ingested, absorbed and metabolized, is essential for a number of physiological functions [7], and should be considered in diet adequacy estimates [8, 9]. The mean iron bioavailability in the United States and Canada diets is estimated at 18% for children ≥1 year [10]. However, in Mexico iron bioavailability is lower, because the Mexican traditional diet is rich in plant-based foods such as grains and legumes, an important source of phytates, which inhibit non-heme iron absorption [8, 11].
There are two forms of dietary iron: 1) heme iron, which comes from hemoglobin and myoglobin in animal source foods, with uniform absorption, and could contribute ≥40% of total absorbed iron [12]. 2) non-heme iron found in plants, animal tissues and widely present in fortified foods, and supplement compounds, its bioavailability depends on body iron reserves, iron absorption enhancers, and absorption inhibitors consumed [13, 14]. Iron absorption is estimated using stable iron isotopes [15,16,17]; however, this method is not feasible in population studies. For this reason, algorithms have been designed to measure its bioavailability in populations [13, 14, 18]. In Mexico, estimates found a low bioavailability of iron (2.7–6.1%), as well as an elevated intake of phytates in preschoolers' diets [13].
Dietary patterns vary with socioeconomic characteristics, which determine iron bioavailability. In Mexico, the probability of consuming fruits, vegetables, and red meat in important quantities, is greater in individuals with a high socioeconomic status (SES) and those who reside in urban areas [19]. In Mexico, the population of northern region consumes more processed meats than in the central region [19]. The southern region follows traditional dietary patterns, with a greater consumption of plant-based foods, which influences iron intake, absorption enhancers, and absorption inhibitors [8, 19]. In Mexico, mother's educational level was also associated with children's healthy diets (adequate in micronutrients) [20, 21].
To our knowledge, FeBio has not been recently estimated with national data in Mexico, nor has its relationship with sociodemographic characteristics been explored. The purpose of this study is to estimate FeBio and analyze its association with sociodemographic characteristics, such as area of residence (urban/rural), region and socioeconomic status in Mexican children from 12 to 59 months of age.
Design and population
The present study is cross-sectional. Information was obtained from the 2012 National Health and Nutrition Survey (2012 ENSANUT, by its Spanish acronym), which is a national, probabilistic survey, with state-level, regional, and urban/rural representation. The survey was carried out between October 2011 and May 2012 [22]. The purpose of this survey was to quantify the frequency, distribution, and patterns of the Mexican population's health and nutrition status. The 2012 ENSANUT included 50,528 households. Information on diet was obtained in a subsample of 2655 children < 5 years [23, 24]. More details on the 2012 ENSANUT can be found in other documents [22,23,24].
This study included preschool aged children that provided a blood sample and their parents completed a 24-h dietary recall questionnaire. Of the sample with valid diet information (n = 2113), children whose serum ferritin concentrations (SF) and C-reactive protein (CRP) levels could not be determined were excluded (n = 1101) (measures described below). The final sample included 1012 children (Fig. 1).
Study flowchart
The 2012 ENSANUT was approved by the research, biosecurity, and ethics committees at the National Institute of Public Health [Instituto Nacional de Salud Pública (INSP by its Spanish acronym)] in Cuernavaca, Morelos, Mexico. Informed consent was signed by the parents or guardians of the participating children [24].
Information on diet was obtained using a 24-h recall. The dietary information was provided by mothers or the person in charge of the child's meals. Interviews were mostly carried out on weekday (75.1%), and a quarter took place during the weekend. A multi-step method was used in order to collect more precise information on diet [25]. This included 1) obtaining a quick or preliminary list of the foods and beverages consumed by the child, without specifying the order or time of consumption; 2) enquiring about commonly forgotten foods (predetermined list); 3) enquiring about meal time and the context in which the foods were eaten (place and activity during consumption); 4) recording detailed information about quantity and characteristics of the foods (food or beverage, consumed alone or along with other foods, ingredients used in preparation and quantities, preparation process); and finally, 5) carrying out a final revision and correcting information, if necessary.
The 24-h recall was carried out by trained personnel and standardized by researchers from INSP [24]. The 24-h recall software (multi-step methods, version 1.0, 2012; INSP) was developed and tested by INSP personnel for use in the 2012 ENSANUT.
Estimation of nutrient intake
Intake of vitamin C, total iron, heme iron, non-heme iron, calcium, and phytates (all in mg/d) and grams of red meat, poultry, fish, and seafood were estimated using the database on nutritional values developed by INSP [26]. Additionally, in order to estimate vitamin C and phytates losses by different cooking methods, retention factors for these dietary components were used [27, 28].
Cleaning process of diet information
A meat intake > + 3 SD was observed in 159 children and was considered implausible for the age group. For these children, we imputed mean meat consumption data by age and then recalculated the amount of heme iron, non- heme iron, and total iron consumed. A new estimate was made for the quantity of heme iron, non-heme iron, and total iron consumed. Children under 1 year of age were excluded from analyses (n = 411), as well as children who were partially breastfed (n = 107), due to the difficulty of estimating the quantity of breastmilk consumed. Values > + 3 SD or < − 3 SD of the logarithm distribution of the ratio between energy intake and energy requirement, were considered implausible and were excluded [24]. For nutrients, an upper boundary for plausible mineral intake was defined by multiplying the 99th percentile of the intake for each mineral by 1.5. When an intake value exceeded this upper boundary, we replaced it with a random value between the 95th percentile and the upper boundary value [8].
Biochemical indicators
A venous blood sample was obtained under fasting conditions by trained personnel and were handled according to recommended laboratory procedures for storage, preservation, and processing [23, 29]. Concentrations of SF and CRP were obtained using an automatic immunoassay analyzer (Abbott diagnostics, Wiesbaden, Germany) in the INSP nutrition laboratory. Children with CRP ≥5 mg/L (n = 275) were excluded from analyses, considering this was due to illness and could alter SF concentrations [30, 31].
Socioeconomic variables
Data on sex and age were obtained. As iron deficiency and its consequences are more severe in children < 2 years, children were classified in subgroups from 12 to 23 and 24–59 months [2, 4, 5]. Three geographical regions in Mexico were defined: North, Center and South. Rural areas were defined as having a population < 2500; the rest were considered as urban [22].
An index of socioeconomic status (SES) was created through the analysis of principal components, including household characteristics (number of rooms, exclusive kitchen, bathroom, use of firewood or coal as fuel, and floor material), goods and services at home (color television, microwave, washing machine, computer, motor vehicle, stereo, Internet access, cell phone, telephone line). A continuous variable was obtained and categorized in terciles (low, medium, high) [32]. Maternal education level was estimated by using the highest level of schooling, categorized as: 1) elementary school, 2) middle school, 3) high school, 4) bachelor's degree or above.
Estimation of bioavailable iron
The percentage of FeBio was estimated with the algorithm created by Armah et al. [14], given it had previously been used with data from nationally representative surveys in preschoolers and considered individual body iron storage. The percentage of total bioavailability was estimated along with food consumption for all meals throughout the day. Consumption of black tea was low in the study population, so this inhibiting factor was not included in the algorithm [13]. As a first step, bioavailability of non-heme iron was estimated considering absorption enhancers (vitamin C and meats), absorption inhibitors of non-heme iron (phytates and calcium), and the concentration of SF per individual using the following equation:
Equation I. Estimation of percentage of non-heme iron bioavailability
$$ Ln\ Bioavailability\ NHI\ \left(\%\right)=6.294-0.709\ln (SF)+0.119\ln (VC)+0.006\ln \left(M+0.1\right)-0.247\ln (Ph)-0.137\ln (ca)-0.083\ \ln (NHI) $$
Where: SF corresponded to serum ferritin (μg/L); VC to vitamin C (mg/d); M to meats (gr/d); Ph to phytates (mg/d); Ca to calcium (mg/d), and NHI to non-heme iron (mg/d).
Next, bioavailability of heme iron (HI) was estimated. According to body iron stores, two categories were defined: 1) adequate iron reserve (SF ≥12 μg/L), assuming a conservative bioavailability value of 25% for HI and 2) depleted iron reserve (SF < 12 μg/L), considering a HI bioavailability of 35% [16, 18].
As a second part of the algorithm, total iron bioavailability was estimated from the sum of the fraction of bioavailable NHI and HI with the following formula:
Equation II. Estimation of percentage of total iron bioavailability
$$ Total\ bioavailability=\left( Bioavailability\ NHI\ast proportion\ NHI\right)+\left( Bioavailability\ HI\ast proportion\ HI\right) $$
Where: Bioavailability NHI corresponded to the percentage of bioavailability of non-heme iron estimated in the first formula I; proportion NHI to the proportion from total iron in the diet (non-heme iron); bioavailability HI to the bioavailability of heme iron (conservative value of 25–35%) and proportion HI to the proportion from total iron (heme iron).
Finally, once the bioavailability of non-heme iron was obtained with the algorithm created by Armah et al. [14], FeBio was estimated in relation to dietary intake of total iron with the following formula:
Equation III. Estimation of bioavailable iron (in mg)
$$ Total\ FeBio=\left[ Bioavailability\ NHI\ast \left(\frac{NHI}{100}\right)\right]+\left[ Bioavailability\ HI\ast \left(\frac{HI}{100}\right)\right] $$
Where: Bioavailability NHI corresponded to the percentage of bioavailability of non-heme iron estimated from the formula I; NHI to non-heme dietary iron (mg/d); bioavailability HI to heme iron bioavailability (conservative value of 25–35%), and HI to heme dietary iron (mg/d).
General characteristics of the study population are shown in proportions with confidence intervals (CI) at 95%. Using chi-square tests, statistically significant differences (p < 0.05) between sociodemographic factors (SES, maternal education level, area, and geographic region) by groups of age (12–23 and 24–59 months) were calculated.
Nutrient intake, concentration of serum ferritin, bioavailability of total iron and non-heme iron, heme FeBio, non-heme FeBio, and total FeBio are shown in mean ± SD, and medians. To determine mean differences in nutrient intake, statistical significance was set at p < 0.05, and the Bonferroni method was used to adjust for multiple comparisons [33]. To study the association between FeBio and sociodemographic factors, robust multiple linear regression modeling techniques were employed. FeBio was considered a dependent variable and SES, area, and geographic as covariates. A model was run for all children aged 12–59 months and two separate models for the groups 12–23 and 24–59 months old. The northern region, urban area, and low SES were the reference categories. We verified the normality and homogeneity of residual variance, in addition to multicollinearity between independent variables in the models. We excluded the variable mother's education due to collinearity with SES. Analyses were performed using the logarithmic transformation of FeBio. Results were similar to the untransformed variable. To facilitate the interpretation of results, we used the untransformed variable. We used the STATA software, version 14.0, (StataCorp. 2015. Stata Statistical Software: Release 14. College Station, TX: StataCorp LP.), and adjusted by sampling design of the survey, with SVY module.
We analyzed information from 1012 children between 12 and 59 months of age who had data on diet and body iron storage indicators. Approximately half were male (48.9%), most lived in urban areas (66.2%) and in the central region and Mexico City (41.8%), 40% had a middle SES, followed by those with low SES (36.2%).
A high proportion of mothers had a middle school education level (43.1%). The sociodemographic variables (area, region, SES, and mother's education level) were not different between the two age groups (Table 1).
Table 1 Sociodemographic variables in preschool children from 12 to 59 moa,b
In the total sample, consumption of total iron and calcium was lower and consumption of phytates were higher in rural areas (compared to urban areas), and among children from low SES (compared with high SES) (p < 0.05). On the other hand, children from high SES had a higher intake of vitamin C than those from low and medium SES (p < 0.05), and children residing in urban areas had higher meat consumption (p < 0.05). No differences were observed for the intake of absorption enhancers (vitamin C and meats) or non-heme iron absorption inhibitors (phytates and calcium) between regions of residence (Table 2). Vitamin C and phytates intake were slightly higher when losses by cooking method were not considered (80 mg and 700 mg/d of vitamin C and phytates, respectively) (data not showed in table).
Table 2 Nutrients intake by sociodemographic factors in children from 12 to 59 mo, stratified by agea,b
In children aged 12–23 months, no differences were observed in the mean intake of enhancers or inhibitors of non-heme iron absorption between sociodemographic characteristics.
Children aged 24–59 months from rural areas with low SES consumed less total iron, vitamin C, and calcium and had a higher consumption of phytates, while the highest meat consumption was in children with a high SES (p < 0.05) (Table 2).
The mean intake of total iron by age group was 7.48 and 9.47 mg/d in children aged 12–23 months and 24–59 months, respectively (Table 3). The total estimated FeBio was 0.75 ± 0.73 mg/d in the total sample, corresponding to a bioavailability of 9.15 ± 5.36%. The main source of FeBio was non-heme iron (0.68 ± 0.70 mg/d), with a bioavailability of 8.33 ± 5.08%. In children aged 24–59 months, total intake of FeBio was 0.74 ± 0.71 mg/d, corresponding to a bioavailability of 8.74 ± 4.83%, which was similar to the estimated bioavailability in the total sample.
Table 3 Total and bioavailable iron intake and bioavailability in preschool children from 12 to 59 moa,b
In children aged 12–23 months, bioavailability tended to be higher (12.03 ± 7.83%), as did FeBio (0.81 ± 0.76 mg/d) (Table 3).
Country regions were not associated with FeBio consumption in any of the three regression models. In 12 to 23-month-old children, we only found differences in the intake of FeBio by area of residence, with 0.276 mg/d less FeBio intake in rural areas versus children in urban areas (p < 0.05). In addition, this negative association was also observed in the total population, with 0.113 mg/d less FeBio intake in children from rural areas compared to urban areas (p < 0.05) (Table 4).
Table 4 Socioeconomic variables associated with FeBio consumption in preschool children from 12 to 59 moa
In the total population, children with a middle SES consumed 0.123 mg/d and children with a high SES consumed 0.173 mg/d more FeBio than low SES (p < 0.05). In 24 to 59-month-old children, only differences in children with high SES were observed, with a greater intake of 0.158 mg/d FeBio in comparison to low SES (p < 0.05) (Table 4).
In this study, we found that the estimated intake of FeBio in Mexican children between 12 and 59 months of age was low (less than 1 mg/d) and was negatively associated with a low SES and residing in a rural area. We also found that dietary iron bioavailability was less than 10%. These results are due the following: 1) the majority of iron consumed in our population was non-heme, for which the bioavailability is much lower than heme iron; 2) there is a high consumption of iron absorption inhibitors, phytates and calcium, and low consumption of meat, which promotes iron absorption.
When the fraction of bioavailable heme and non-heme iron were added, a total bioavailability of 9.15 ± 5.36% was obtained, which differs from the estimated bioavailability in the United States population (15.1%) [11]. The bioavailability of iron is important to correctly estimate requirements for this nutrient. When assuming a low iron bioavailability (5.5% in children aged 1–3 years and 7.5% in children aged 4–5), estimates done with data from the Mexican National Nutrition Survey (ENN) 1999), the prevalence of iron deficiency in Mexican preschoolers was 52% [8, 34]. However, assuming a bioavailability of 18% (recommended in United States and Canada), the prevalence of iron deficiency is underestimated by 5% [8, 10]. We found that the prevalence of iron deficiency, considering the bioavailability in the present study, is 45%.
The estimated FeBio intake (0.74–0.81 mg/d) is slightly higher than previous estimates in Mexican preschool children, with data from the 1999 ENN (0.14–0.37 mg/d) [13]. Diverse factors could be contributing to the differences between estimates: 1) the instrument and methodology used for data collection were different, as in the present study a multi-step method was used, allowing for a better record of consumed foods [8, 24, 25]; 2) the algorithm applied included the concentration of SF per individual [14], whereas in 1999, three different scenarios of iron reserves were used because a ferritin measurement was not available [13]; 3) a possible change in iron intake in the past 13 years could be due to a greater consumption of fortified foods [8, 35, 36]; 4) the implementation of government programs, such as the Liconsa milk supply program (milk fortified with iron and other micronutrients), could be contributing to an improved iron status in children [36,37,38].
Despite increases in iron bioavailability, FeBio continues to be low, given that only 4.6% of total dietary iron is heme iron, while, in other populations, heme iron represents > 10% of total dietary iron [8, 14, 16]. It is because in 2012, 31% of households had food insecurity, which limited meat consumption within other expensive food groups [39]. There was no difference in FeBio intake between different regions in Mexico, which is largely explained by the fact that consumption of meats, vitamin C, phytates, and calcium is not different between age groups. Similar to previous studies, consumption of total iron, phytates, and vitamin C were not significantly associated with SF concentration [40].
In our analyses, there were no significant differences in FeBio intake by sex (data not shown). Children iron deficiency is reported to be attributed to factors such as muscle and blood volume increase due to rapid growth, a diet low in high available iron (heme), elevated consumption of cow's milk, and loss of intestinal blood from parasitism, in both boys and girls [41]. Furthermore, conditions such as inflammation, vitamin deficiency, obesity, and genetic factors contribute to differences in iron absorption among individuals [12].
As our findings in the total sample suggest, living in a rural area and having a low SES are negatively associated with FeBio. In rural areas of Mexico, the traditional diet, rich in cereals and legumes, is largely consumed, representing an important source of iron and of iron absorption inhibitors [7, 8, 42]. FeBio is lower in individuals from the lowest SES terciles, which could be due to a lower mean consumption of iron absorption enhancers (meat and vitamin C) in these groups. Other studies indicated a lower consumption of fruits, vegetables (an important source of vitamin C), and red meats in individuals with a low SES, in addition to a higher consumption of energy-dense foods with low nutritional quality [20, 43].
Some strategies to increase bioavailable iron are: combination of beans or other legumes in the same meal with foods that facilitate iron absorption (citrus fruit, 100% natural juices and vegetables) [44]. Appropriate cooking methods (fermentation, sourdough preparation, soaking, and discarding soaking water) has also been effective to reduce content of phytates in whole foods [12, 45].
Our study presents some limitations. First, we performed a secondary analysis from the 2012 ENSANUT data. However, unlike clinical studies, our results can be extrapolated to the general population and provide information for health and nutrition policies. Second, the use of only a 24-h recall and a possible overestimation of the prevalence of iron deficiency. Nonetheless, the 24-h multi-step recall method minimizes omission of forgotten foods and improves diet estimates [25]. Third, not adjusting the calcium and iron intake for losses for cooking methods. However, we consider that the lack of adjustment for losses of these two minerals does not change the direction of the results, since the highest cooking losses for calcium and iron are found in vegetables ~ 24% of the total losses, and a high proportion of Mexican children < 48 months do not consume vegetables (80%) [43]. For iron, losses by cooking oscillate between 5 and 10% in the different food [27, 46]. Fourth, information on dietary supplements was not collected in this sample of children, which would have been useful to have a better estimate of the iron intake.
The strengths of the study include the use of an algorithm developed with national data, proved useful for estimating iron bioavailability in population groups [11]. Another strength is the estimation of vitamin C and phytates intakes by adjusting for losses in relation to the different cooking methods used. This adjustment provided a better estimate of the total bioavailable iron [28]. Furthermore, to the best of our knowledge, few studies have estimated FeBio in preschool children's overall diet at a national level, in a representative manner.
In conclusion, the estimated FeBio in Mexican children was low. Rural areas and low SES were negatively associated with FeBio intake. From a public health standpoint, identifying dietary factors that hinder or promote iron bioavailability in groups with certain sociodemographic characteristics will allow for the design of targeted interventions to improve consumption of bioavailable iron. In light of the present findings, future studies should review iron intake recommendations in the Mexican population.
The dataset used for analysis is available from the corresponding author on reasonable request.
CRP:
ENN:
National Nutrition Survey
ENSANUT:
National Health and Nutrition Survey
FeBio:
Bioavailable iron
Heme iron
INSP:
National Institute of Public Health (Instituto Nacional de Salud Publica)
NHI:
Non-heme iron
Phytates
SF:
Serum ferritin
VC:
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USDA. USDA table of nutrient retention factors release 6; 2007. Available from: https://data.nal.usda.gov/dataset/usda-table-nutrient-retention-factors-release-6-2007.
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Thurnham DI, McCabe LD, Haldar S, Wieringa FT, Northrop-Clewes CA, McCabe GP. Adjusting plasma ferritin concentrations to remove the effects of subclinical inflammation in the assessment of iron deficiency: a meta-analysis. Am J Clin Nutr. 2010;92(3):546–55.
WHO, CDC. Assessing the iron status of populations. Annex 2: indicators of the iron status of populations: ferritin. Geneva: World Health Organization; Centers for Disease Control and Prevention; 2007.
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Rivera-Dommarco JÁ, Hotz C, Rodriguez-Ramirez S, Garcia-Guerra A, Perez-Exposito A, Martinez H. Hierro. In: Recomendaciones de ingestión de nutrimentos Para la población mexicana [recommendations of intake of nutrients for the Mexican population]. México: Editorial Medica panamericana; 2005. p. 247–64.
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Shamah-Levy T, Méndez-Gómez-Humarán I, Gaona-Pineda EB, Cuevas-Nasu L, Villalpando S. Food assistance programmes are indirectly associated with anaemia status in children under 5 years old in Mexico. Br J Nutr. 2016;116(6):1095–102.
Barquera-Cervera S, Rivera-Dommarco J, Gasca-García A. Food and nutrition policies and programs in México. Salud Publica Mex. 2001;5:464–77.
Rivera JA, Shamah T, Villalpando S, Monterrubio E. Effectiveness of a large-scale iron-fortified milk distribution program on anemia and iron deficiency in low-income young children in Mexico. Am J Clin Nutr. 2010;91(2):431–9.
Cuevas-Nasu L, Rivera-Dommarco JA, Shamah-levy T. Inseguridad alimentaria y estado de nutrición en menores de cinco años de edad en Mexico. [Food insecurity and nutritional status in children under five years of age in Mexico]. Salud Publica Mex. 2014;56(2):47–53.
Morales-Rúan M, Villalpando S, García-Guerra A, Shamah-Levy T, Robledo-Pérez R, Ávila-Arcos MA, et al. Iron, zinc, copper and magnesium nutritional status in Mexican children aged 1 to 11 years. Salud Pública de Mex. 2012;54:125–34.
Subramaniam G, Girish M. Iron deficiency anemia in children. Indian J Pediatr. 2015;82(6):558–64.
Rivera JA, Pedraza LS, Aburto TC, Batis C, Sanchez-Pimienta TG, Gonzalez de Cosio T, et al. Overview of the dietary intakes of the Mexican population: results from the National Health and nutrition survey 2012. J Nutr. 2016;146(9):1851S–5S.
Deming DM, Afeiche MC, Reidy KC, Eldridge AL, Villalpando-Carrión S. Early feeding patterns among Mexican babies: Findings from the 2012 National Health and Nutrition Survey and implications for health and obesity prevention. BMC Nutr. 2015;1(40):1-14.
Platel K, Krishnapura S. Bioavailability of micronutrients from plant foods: an update. Food Sci Nutr. 2016;56(10):1608–19.
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The authors thank Cloe Rawlinson for the English language editing.
Centro de Investigación en Nutrición y Salud (CINYS), Instituto Nacional de Salud Pública (INSP), Av. Universidad 655 Colonia Santa María Ahuacatitlán, Cerrada Los Pinos y Caminera C.P., 62100, Cuernavaca, Mexico
Yazmín Venegas-Aviles, Sonia Rodríguez-Ramírez, Eric Monterrubio-Flores & Armando García-Guerra
Yazmín Venegas-Aviles
Sonia Rodríguez-Ramírez
Eric Monterrubio-Flores
Armando García-Guerra
SRR conceptualized and designed the research; YVA and EMF analyzed the data; YAV and SRR wrote the paper; AGG gave key insights for the final manuscript; YVA and SRR had primary responsibility for final content. The authors read and approved the final manuscript.
Correspondence to Sonia Rodríguez-Ramírez.
The research, biosecurity, and ethics committees of the National Institute of Public Health (INSP, by its Spanish acronym) in Cuernavaca, Morelos, Mexico, approved the 2012 ENSANUT. The parents or guardians of the children signed an informed consent.
The authors declare having no competing interests.
Venegas-Aviles, Y., Rodríguez-Ramírez, S., Monterrubio-Flores, E. et al. Sociodemographic factors associated with low intake of bioavailable iron in preschoolers: National Health and Nutrition Survey 2012, Mexico. Nutr J 19, 57 (2020). https://doi.org/10.1186/s12937-020-00567-3
Iron dietary
Child preschool | CommonCrawl |
The dual construction for arcs in projective Hjelmslev spaces
AMC Home
February 2011, 5(1): 1-10. doi: 10.3934/amc.2011.5.1
Point compression for Koblitz elliptic curves
Philip N. J. Eagle 1, , Steven D. Galbraith 2, and John B. Ong 2,
Information Security Group, Mathematics Department, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom
Mathematics Department, The University of Auckland, Private Bag 92019 Auckland 1142, New Zealand, New Zealand
Received February 2009 Revised September 2010 Published February 2011
Elliptic curves over finite fields have applications in public key cryptography. A Koblitz curve is an elliptic curve $E$ over $\mathbb F$2; the group $E(\mathbb F$2n$)$ has convenient features for efficient implementation of elliptic curve cryptography.
Wiener and Zuccherato and Gallant, Lambert and Vanstone showed that one can accelerate the Pollard rho algorithm for the discrete logarithm problem on Koblitz curves. This implies that when using Koblitz curves, one has a lower security per bit than when using general elliptic curves defined over the same field. Hence for a fixed security level, systems using Koblitz curves require slightly more bandwidth.
We present a method to reduce this bandwidth when a normal basis representation for $\mathbb F$2n is used. Our method is appropriate for applications such as Diffie-Hellman key exchange or Elgamal encryption. We show that, with a low probability of failure, our method gives the expected bandwidth for a given security level.
Keywords: Elliptic curve cryptography, point compression., Koblitz curves.
Mathematics Subject Classification: 94A60, 11T7.
Citation: Philip N. J. Eagle, Steven D. Galbraith, John B. Ong. Point compression for Koblitz elliptic curves. Advances in Mathematics of Communications, 2011, 5 (1) : 1-10. doi: 10.3934/amc.2011.5.1
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Marek Janasz, Piotr Pokora. On Seshadri constants and point-curve configurations. Electronic Research Archive, 2020, 28 (2) : 795-805. doi: 10.3934/era.2020040
Koray Karabina, Berkant Ustaoglu. Invalid-curve attacks on (hyper)elliptic curve cryptosystems. Advances in Mathematics of Communications, 2010, 4 (3) : 307-321. doi: 10.3934/amc.2010.4.307
Chuangqiang Hu, Shudi Yang. Multi-point codes from the GGS curves. Advances in Mathematics of Communications, 2020, 14 (2) : 279-299. doi: 10.3934/amc.2020020
Kwang Ho Kim, Junyop Choe, Song Yun Kim, Namsu Kim, Sekung Hong. Speeding up regular elliptic curve scalar multiplication without precomputation. Advances in Mathematics of Communications, 2020, 14 (4) : 703-726. doi: 10.3934/amc.2020090
Alice Silverberg. Some remarks on primality proving and elliptic curves. Advances in Mathematics of Communications, 2014, 8 (4) : 427-436. doi: 10.3934/amc.2014.8.427
David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335
Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
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Guilin Ji, Changjian Liu. The cyclicity of a class of quadratic reversible centers defining elliptic curves. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021299
Steven D. Galbraith, Ping Wang, Fangguo Zhang. Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm. Advances in Mathematics of Communications, 2017, 11 (3) : 453-469. doi: 10.3934/amc.2017038
Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
Rafail Krichevskii and Vladimir Potapov. Compression and restoration of square integrable functions. Electronic Research Announcements, 1996, 2: 42-49.
Matthias Ngwa, Ephraim Agyingi. A mathematical model of the compression of a spinal disc. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1061-1083. doi: 10.3934/mbe.2011.8.1061
Florian Luca, Igor E. Shparlinski. On finite fields for pairing based cryptography. Advances in Mathematics of Communications, 2007, 1 (3) : 281-286. doi: 10.3934/amc.2007.1.281
Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215
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Philip N. J. Eagle Steven D. Galbraith John B. Ong | CommonCrawl |
# Understanding linear transformations
Consider a linear transformation T: R^2 → R^2 defined by T(x, y) = (2x + 3y, x - 2y). We can observe that T preserves the operations of addition and scalar multiplication. For example, if we have two vectors u = (1, 2) and v = (3, 4), we can calculate their sum and scalar multiplication:
u + v = (1 + 3, 2 + 4) = (4, 6)
k * u = (k * 1, k * 2) = (k, 2k)
Now, let's apply the linear transformation T to these vectors:
T(u + v) = T(4, 6) = (8, 4)
T(k * u) = T(k, 2k) = (2k + 3(2k), k - 2(2k)) = (2k + 6k, k - 4k) = (8k, -2k)
We can see that T preserves the operations of addition and scalar multiplication.
## Exercise
Calculate the linear transformation T(2x + 3y, x - 2y) for the vector (1, 1).
### Solution
T(2(1) + 3(1), (1) - 2(1)) = T(5, 0) = (10, -2)
# Matrix representation of linear transformations
Consider the linear transformation T: R^2 → R^2 defined by T(x, y) = (2x + 3y, x - 2y). We can represent this transformation using a matrix:
T = | 2 3 |
| 1 -2 |
Now, let's apply the transformation T to the vector (1, 1):
T(1, 1) = | 2 3 | | 1 | | 5 |
| 1 -2 | | 1 | = |-1 |
We can see that T(1, 1) = (5, -1).
## Exercise
Calculate the matrix representation of the linear transformation T(x, y) = (3x + 2y, 2x - y).
### Solution
T = | 3 2 |
| 2 -1 |
# Properties of orthogonal matrices
Consider the orthogonal matrix A:
A = | 1/√2 1/√2 |
| 1/√2 -1/√2 |
We can observe that the columns of A are orthogonal unit vectors:
Column 1 = (1/√2, 1/√2)
Column 2 = (1/√2, -1/√2)
Now, let's apply the linear transformation A to the vector (1, 1):
A(1, 1) = | 1/√2 1/√2 | | 1 | | (1/√2 + 1/√2) |
| 1/√2 -1/√2 | | 1 | = | (1/√2 - 1/√2) |
We can see that A(1, 1) = (√2, 0).
## Exercise
Find an orthogonal matrix B with determinant 1.
### Solution
B = | 1/√2 1/√2 |
| 1/√2 -1/√2 |
# Creating and manipulating matrices in JavaScript
To create a matrix in JavaScript, we can use the following code:
```javascript
const glMatrix = require('gl-matrix');
const mat2 = glMatrix.mat2;
let A = mat2.create();
mat2.set(A, 1, 2, 3, 4);
```
Now, let's apply the matrix A to the vector (1, 1):
```javascript
let x = glMatrix.vec2.create();
glMatrix.vec2.set(x, 1, 1);
let y = glMatrix.vec2.create();
glMatrix.vec2.transformMat2(y, x, A);
console.log(y); // Output: [ 5, -1 ]
```
We can see that the matrix A applied to the vector (1, 1) gives us the same result as in the previous example.
## Exercise
Create a matrix in JavaScript that represents the linear transformation T(x, y) = (3x + 2y, 2x - y).
### Solution
```javascript
const glMatrix = require('gl-matrix');
const mat2 = glMatrix.mat2;
let T = mat2.create();
mat2.set(T, 3, 2, 2, -1);
```
# Inverse matrices and their properties
Consider the matrix A:
A = | 1 2 |
| 3 4 |
We can calculate its inverse matrix A^(-1):
A^(-1) = | 4 -2 |
| -3 2 |
Now, let's multiply A by its inverse:
A * A^(-1) = | 1 2 | | 4 -2 | | 1 0 |
| 3 4 | |-3 2 | = | 0 1 |
We can see that A * A^(-1) = I, where I is the identity matrix.
## Exercise
Find the inverse matrix of the matrix A = | 1 2 |
| 3 4 |
### Solution
A^(-1) = | 4 -2 |
| -3 2 |
# Linear combination of vectors
Consider two vectors u = (1, 2) and v = (3, 4). We can calculate their linear combination 3u + 2v:
3u + 2v = (3 * 1 + 2 * 3, 3 * 2 + 2 * 4) = (11, 14)
## Exercise
Calculate the linear combination 2u + 3v for the vectors u = (1, 2) and v = (3, 4).
### Solution
2u + 3v = (2 * 1 + 3 * 3, 2 * 2 + 3 * 4) = (7, 14)
# Applying linear transformations to vectors and matrices
To apply a linear transformation to a vector in JavaScript,
# Orthogonal matrices and their properties
Orthogonal matrices are a special type of matrix that have some unique properties. They are square matrices with real entries and their transpose is equal to their inverse. In other words, an orthogonal matrix A satisfies the condition $A^TA = AA^T = I$, where $I$ is the identity matrix.
An example of an orthogonal matrix is the following:
$$
A = \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}
$$
## Exercise
Verify that the matrix $A$ is orthogonal.
Instructions:
1. Calculate $A^T$.
2. Multiply $A$ by $A^T$.
3. Calculate $AA^T$.
4. Check if the resulting matrices are equal to the identity matrix.
### Solution
1. $A^T = \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}$
2. $AA^T = \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}$
3. $A^TA = \begin{bmatrix}
1 & 0 \\
0 & -1
\end{bmatrix}$
Since $AA^T = A^TA = I$, the matrix $A$ is orthogonal.
# Transformation matrices and their use in computer graphics
Transformation matrices are used in computer graphics to manipulate objects in a 2D or 3D space. They can be used to scale, rotate, translate, and skew objects.
To scale an object by a factor of 2 in the x-direction and 3 in the y-direction, we can use the following transformation matrix:
$$
S = \begin{bmatrix}
2 & 0 \\
0 & 3
\end{bmatrix}
$$
To rotate an object by an angle $\theta$, we can use the following transformation matrix:
$$
R = \begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}
$$
## Exercise
Calculate the transformation matrix for a translation by 2 units in the x-direction and 3 units in the y-direction.
Instructions:
1. Create a transformation matrix for translation.
2. Apply the matrix to a vector.
### Solution
1. The transformation matrix for a translation is:
$$
T = \begin{bmatrix}
1 & 2 \\
0 & 1
\end{bmatrix}
$$
2. Applying the matrix to a vector, say $\begin{bmatrix}x \\ y\end{bmatrix}$, results in the vector $\begin{bmatrix}x + 2 \\ y + 3\end{bmatrix}$.
# Advanced topics: matrix decomposition and eigenvectors
Matrix decomposition is a technique that allows us to break down a matrix into a product of simpler matrices. One common decomposition is the Singular Value Decomposition (SVD), which is used in machine learning and data analysis.
## Exercise
Calculate the Singular Value Decomposition of the following matrix:
$$
A = \begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
$$
Instructions:
1. Calculate the SVD of the matrix $A$.
### Solution
1. The SVD of the matrix $A$ is:
$$
A = U \Sigma V^T
$$
where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal matrix.
To find the SVD of $A$, we can use a library like scikit-learn in Python. The resulting matrices will be:
$$
U = \begin{bmatrix}
0.4472 & -0.8944 \\
0.8944 & 0.4472
\end{bmatrix}
$$
$$
\Sigma = \begin{bmatrix}
5.0000 & 0 \\
0 & 0.2500
\end{bmatrix}
$$
$$
V = \begin{bmatrix}
0.4472 & 0.8944 \\
-0.8944 & -0.4472
\end{bmatrix}
$$
# Eigenvectors and eigenvalues
Eigenvectors and eigenvalues are important concepts in linear algebra that allow us to analyze the behavior of a linear transformation. An eigenvector of a matrix $A$ is a non-zero vector $v$ such that $Av = \lambda v$, where $\lambda$ is a scalar called the eigenvalue.
To find the eigenvectors and eigenvalues of a matrix, we can use the following steps:
1. Calculate the characteristic equation, which is $det(A - \lambda I) = 0$.
2. Solve the equation for $\lambda$.
3. Substitute the eigenvalue back into the equation $Av = \lambda v$ to find the eigenvectors.
## Exercise
Find the eigenvectors and eigenvalues of the following matrix:
$$
A = \begin{bmatrix}
2 & 1 \\
1 & 2
\end{bmatrix}
$$
Instructions:
1. Calculate the characteristic equation.
2. Solve the equation for $\lambda$.
3. Substitute the eigenvalues back into the equation $Av = \lambda v$ to find the eigenvectors.
### Solution
1. The characteristic equation is $det(A - \lambda I) = 0$.
2. Solving the equation, we get $\lambda = 3 \pm \sqrt{2}$.
3. Substituting the eigenvalues back into the equation $Av = \lambda v$, we find the eigenvectors:
- For $\lambda = 3 + \sqrt{2}$, the eigenvector is $\begin{bmatrix}1 \\ 1\end{bmatrix}$.
- For $\lambda = 3 - \sqrt{2}$, the eigenvector is $\begin{bmatrix}1 \\ -1\end{bmatrix}$. | Textbooks |
Lumerical Support > APP home
CMOS image sensor - Angular response 3D
FDTD CHARGE CMOS Image Sensors Consumer Electronics
In this example, the angular response of a CMOS image sensor is characterized through optical simulations using the FDTD solver and electrical simulations using the CHARGE solver. Key results from the simulations include the spatial field profiles, transmission and optical efficiency vs. angle, quantum efficiency vs. angle. The effect of microlens shift is also considered.
download example
Understand the simulation workflow and key results
Characterization of CMOS images sensors generally requires both optical and electrical simulations to account for the absorption, scattering, and diffraction from sub-wavelength features as well as the electrical transport of generated charge. In this example, optical simulations provide information about field profile, transmission, optical efficiency. The effects of injection angle and the microlens shift are also considered. Steps 1-3 demonstrate a few example tasks with increasing complexity (single simulation, angle/polarization sweep and angle/polarization/microlens position sweep).The generated charge data from the optical simulations (step 2) is combined with the weighting function from an independent electrical simulation (step 4) for further calculation of the quantum efficiency and the crosstalk in terms of the injection angle (step 5).
Note that the definition of the "pixel" can differ depending on the application areas. The optical simulations in this example contain a periodic array of Red/Green/Blue/Green unit cells. Throughout this example, we will refer to each of the R/G/B/G regions as a "pixel", meaning there are 4 pixels in a unit cell, as shown in the figure below.
As this example requires many sweeps, we limited ourselves to single frequency simulations to reduce the overall simulation time. But the approaches in the optical simulations are applicable to broadband simulations.
Step 1: Initial simulation
Obtain the field profile, transmission and optical efficiency of each pixel when the sensor is illuminated by a planewave at a fixed angle. The main purpose of this step is to ensure the simulation is set up correctly and to allow the user to manually explore the results, before running the full angular response parameter sweep in the latter steps.
Step 2: Angular response
Calculate the optical efficiency and the electron-hole pair generation rates as a function of injection angle. In this example, the generation rate results are averaged in the y-direction and saved in a 2D format so that it can be used in step 5 to calculate the quantum efficiency of the device.
Step 3: Effect of microlens shift
Obtain a 2D map of the optical efficiency as a function of the injection angle and microlens shift. This sweep demonstrates an interesting way to optimize the optical performance of the device. In the interest of keeping the example short, we don't record the generation rate data as in this step (although we could).
Step 4: Weighting function
Run the CHARGE solver to get the impulse response (Green's function) of the system to an electron-hole pair at arbitrary positions in the substrate. From this, we calculate a spatially-varying weighting function which represents the probability of an electron-hole pair generated at any point in space to be collected by the contact for a specific pixel (green in this example). 2D CHARGE simulations are used to reduce the simulation time, but it is possible to extend the simulation methodology to 3D.
Step 5: Quantum efficiency and crosstalk
The weighting function (Step 4) is multiplied by the generation rate data (Step 2) and integrated to yield the internal quantum efficiency (IQE) and crosstalk. This approach based on the Green's function is very efficient in calculating the IQE for arbitrary optical generation rate profiles since it requires, on the electrical side, only a single simulation for the weighting function calculation. For the sake of keeping the simulation time short, we ran a 2D CHARGE simulation to obtain a 2D weighting function and used the generation rate data that was averaged in the y-direction.
Run and results
Instructions for running the model and discussion of key results
Open the simulation file (CMOS_image_sensor_angular_response.fsp)
Run the script file (CMOS_image_sensor_angular_response_initial.lsf) to run the simulation and visualize some of the representative results shown below.
Field profile
The "field_XZ" and "field_YZ" frequency monitors record the fields at the cross sections of the red-green pixels and the green-blue pixels, respectively. As the source is currently set to emit at 550 nm (green), a high transmission is observed at the green pixel due to the different wavelength-selective filters on different regions.
The movies of propagating fields at the same positions as the frequency monitors can be found in the folder where the simulation file is saved. They clearly show that the injected light is selectively transmitted through the green filter and eventually dissipated by absorption in the underlying silicon layer.
The "pixel_transmission" analysis group records the normal component of the Poynting vector, \(P_{z}(x,y)\), on the top surface of the Si layer. To calculate the power absorbed in each pixel, (optical efficiency), we can choose to integrate Pz only over the depletion region of the pixel. The easiest way to integrate \(P_{z}\) over an arbitrary region is to use a spatial filter in the shape of the depletion region and multiply it to the \(P_{z}\) . The spatial filter is optional and can be disabled in the Analysis->Script tab. The following figures show the unfiltered \(P_{z}\) , the depletion regions and the \(P_{z}\) in the depletion regions. The shape of each depletion region is currently set to a 1x1um square with a rounded corner. See integrating the poynting vector for more information.
Optical efficiency
Optical efficiency is defined as the fraction of the power incident in the pixel that is absorbed in the depletion region of the pixel:
$$\text{Optical efficiency (OE)} =\frac{\text{Absorbed power}}{\text{Source power}}$$
By integrating the \(P_{z}\) over the entire surface of the Si layer and normalizing it by the injected power, we find that about 38% of the power is transmitted into the Si layer. The combined efficiency of the two green pixels makes about 33 % while the efficiency for the red and blue pixel is about 0.5 % each.
Power into Si layer: 0.372
Power through red pixel: 0.00459
Power through 2 green pixels: 0.328
Power through blue pixel: 0.00459
The simulation file comes with a pre-run "convergence" sweep object, which records the optical efficiencies vs. mesh accuracy. The figure below shows that there relatively small differences in the results for different mesh accuracies, with the mesh accuracy of "1" giving a reasonably close result to the one for mesh accuracy "6". Using a coarse mesh for initial simulations is strongly recommended due to the large time savings it provides. The default mesh accuracy in this example is "2". However, a mesh accuracy of "1" is used in Step 3 due to the large number of simulations required. Like all simulations, thorough convergence testing is required to ensure the results are accurate.
Open the script file (CMOS_image_sensor_angular_response_sweep_angle.lsf) and set the value of the "Analysis_only" parameter - "1" to visualize the pre-run sweep results or "0" to run the sweep and visualize the results.
Run the script file.
The "angle sweep" consists of 14 sweep points – 7 for the source angle and 2 for the polarization. To reduce the simulation time, unnecessary simulation objects like movie monitors and index monitors were disabled.
The optical efficiency vs. source angle for each pixel is shown below. Note that we averaged the efficiencies for 0 and 90 (degree) polarizations to obtain the response for an unpolarized light. Note that a factor of \(cos(\theta)\) was also multiplied to correct for the varying amount of incident power per pixel as a function of angle. Even with an ideal pixel design, it would not be possible to have an optical efficiency as a function of angle that does better than \(cos(theta)\). The OE for green has maximum at normal incidence and is reduced at larger incidence angle. The angular response simulations also provide a measure of optical crosstalk – some light being absorbed in the red or blue pixels under green illumination (or vice versa).
Generation rate
The generation rate data from "CW_generation_gb" analysis group is used in the electrical simulation in step 4. Once the sweep is completed, 14 data files for the generation rate of the green/blue pixels will be created under the folder named "sweepangle". File names are automatically assigned by the model setup script based on the polarization angle and the source angle. The figure below shows the generation rate for an unpolarized light (550 nm) in the green/blue pixels. Note that the CW generation analysis group is currently set to average the generation rate in the "y" direction and produce an averaged 2d map of it, \(G_{L}(x,z)\). This is to make it compatible with the 2D CHARGE simulations in step 4. For the sake of saving the simulation time, we are using 2D CHARGE simulations in this example. However, it is possible to do full 3D simulations in CHARGE and hence use the full 3D generation rate from FDTD.
Step 3: Microlens shift
Open the script file (CMOS_image_sensor_angular_response_microlens_shift.lsf) and set the value of the "Analysis_only" parameter - "1" to visualize the pre-run sweep results or "0" to run the sweep and visualize the results.
The "microlens shift" sweep consists of a total of 462 sweep points – 21 for microlens shift, 7 for angle theta and 2 for polarization angle. Considering the huge number of sweeps, we are using the mesh accuracy of "1" for this example to reduce the simulation time.
The optical efficiency in terms of the angle and the lens shift is shown below for each pixel. Note that the script applies a factor of \(cos(\theta)\) to correct for the varying amount of incident power per pixel at different injection angle. From the result for green, it can be seen that a shift of about 37 (nm/degree), marked by the dotted black line, gives the maximum optical efficiency for a given incidence angle. For example, if the light is incident dominantly at 15 degree, the lens needs to be shifted by about 555 nm for maximum efficiency.
Note: The generation rate data from Step 2 is a prerequisite to step 4.
Open the simulation file (CMOS_image_sensor_greens_function.ldev) and run the simulation
Run the script file (CMOS_image_sensor_weighting_function.lsf)
Silicon has a negligible thermal recombination at moderate illumination strengths. With it being an indirect band gap semiconductor, its radiative recombination (generation of photons by recombination of electron-hole pairs) is also negligible. As a result, the photo-generated charges in the substrate will be mostly collected by the contacts for different pixels. In this step, we use the point generation source in CHARGE simulation to determine the weighting function, \(W(x,y,z)\) of the device. \( W(x,y,z)\) is the probability that the charge generated at that position will be collected by a specific contact. This approach is based on the Green's function, \(G(x,y,z)\), and is equivalent to knowing the carrier density \(n,p\) at each contact in response to an impulse generation rate source located at a specific location.
The carrier densities are calculated during the charge transport simulation. To determine the full \(G(x,y,z)\), the location of the impulse source is swept over all r in the simulation domain. This operation is performed internally when the "map current collection probability" option is enabled for one or more contacts. When the simulation is complete, the weighting functions are stored as the result "W" for each carrier type at each enabled contact and are exported as data files for further analysis in step 5. The figure below shows the weighting function \(W(x,z,)\) for the green pixel shows that the collection probability for the green pixel is very high when the charge is located nearer the green contact (top left). However, it also shows that some of the charges generated in the blue pixel region (\(x>0\)) has a non-zero probability of being collected by the green contact. This suggests that there is some electrical crosstalk between the neighbouring pixels.
Step 5: IQE and crosstalk
Open the simulation file (CMOS_image_sensor_greens_function.ldev)
Run the script file (CMOS_image_sensor_angular_response_iqe.lsf)
In this step, we will be calculating the quantum efficiency (QE) of the green pixel and the green/blue crosstalk based on the Green's function approach. We do not need to run any additional simulations at this stage as we already have all the required generation rate and weighting function data saved from the step 2 and 4, respectively. The definitions of the related quantities are as follows:
$$\text{Internal quantum efficiency (IQE)} = \frac{\text{Charges collected by the green pixel}}{\text{Total charges generated by absorption}}$$
$$\text{External quantum efficiency (EQE)} = \text{IQE}\times \text{OE (Optical efficiency)}$$
$$\text{Green/blue crosstalk} = \frac{\text{Charges collected by the blue pixel}}{\text{Total charges generated by absorption}}$$
Quantum efficiency and crosstalk
The script sequentially loads the 14 generation rate data saved from the angle sweep in step 2 and multiplies it with the weighting function for green. The following figures show \(G_{L}(x,z)\), \(W_{green}(x,z)\) and \(G_{L}(x,z)W_{green}(x,z)\) for the unpolarized light at normal incidence.
By integrating the \(G_{L}(x,z)W_{green}(x,z)\) and normalizing it with the total generation rate, we obtain the IQE for the green pixel. Repeating the same procedure with the weighting function for the blue pixel, \(W_{blue}(x,z)\), yields the green/blue crosstalk. The IQE has its maximum value of about 80 % and decreases at larger source angle. The trend is in agreement with the increased green/blue crosstalk at larger angle. The maximum EQE is about 26%. When interpreting this data, it's important to remember that \(G_{L}(x,z)\) is for green light illumination.
Important model settings
Description of important objects and settings used in this model
Parametrization
All the structures in this example are constructed using the setup script in the "image sensor" structure group. However, some of the key design parameters such as pixel size and the microlens shift are set in the setup tab of the "model", which then will update the associated parameters in the "image sensor" as well as in other simulation objects dependent on any of those parameters.
"CW generation" analysis groups
The generation rate group consists of a 3D frequency monitor. It measures the absorbed power in the Si layer, then calculates the generation rate assuming a single pair of electron-hole pair is created per single photon.
Source propagation axis: The script in the "CW generation" group assumes the source is injecting in the y-direction for 2D and the z-direction for 3D simulations.
Averaging of generation rate: Even though the raw data for the generation rate is obtained in 3D, we are averaging it in the y-direction and saves a 2D generation data for later use. This is because we are running a 2D simulation in CHARGE to save the simulation time in this demonstrative example.
x/y/z spans: The spans of the generation rate objects are set by the setup script in the "model." The z-min of the objects might need to be adjusted to capture most of the absorbed light penetrating deeper into the substrate. The z-max of the objects needs to be at least one mesh cell away from the Si surface for an accurate absorption calculation. You might need to do some convergence test to decide on the appropriate z-span.
Filenames: The name of the generation rate file name is automatically set by the "polarization angle" of the source and the "count" in the setup variables tab of the "model." The "count" is paired with the parameter "theta" in the "sweep angle" object. At each sweep point of "sweep angle", the "count" value in "model" is updated and consequently the "export filename" in the CW generation group.
Unpolarized light
To obtain the transmission and generation rate results for unpolarized light, we need to run two simulations (one for x-polarized and another for y-polarized light) and average them. For that reason, a nested sweep named "source polarization" is included in the "angle sweep" and the "microlens shift" sweep objects.
For initial simulations, it can make sense to ignore polarization effects and choose only one polarization, S or P. Eventually, it makes sense to correctly calculate the incoherent response using both polarizations but a great deal of initial testing and optimization can be done with only one polarization.
"map current collection probability"
To calculate the current collection probability (=weighting function) for a given electrical contact, select the "map current collection probability" option in the steady-state contact configuration. In this simulation, the collection probability mapping is enabled for each of the simulation contacts located in the PD(Photodiode) n-wells. These contacts are biased to mimic the potential of a depletion region. The p++ surface diffusion and the substrate are electrically grounded.
Use a coarse mesh initially
We strongly recommend starting with a coarse mesh, using a mesh accuracy setting of 1 or 2. It is far easier to setup, test or optimize a simulation that takes a few seconds to minutes, compared to a simulation that takes several hours. Only once all other problems are resolved and you are getting good results should you try using a mesh accuracy setting of 3 or more by doing some convergence testing.
X,Y override in Si
Due to the high index of Silicon (n>4), the automatic meshing algorithm will use a very small mesh everywhere in the Si region. This small mesh will make the simulation significantly slower that it would otherwise be.
From Snell's law, we can show that the in-plane wave vector is conserved (\(k_{x, glass} = k_{x, Si}\)).
In other words, the in-plane wavevector (kx or ky) in the Silicon can never be larger than the in-plane wavevector in the glass. Therefore, it is possible to use a coarser mesh in the x and y directions without reducing the accuracy. To force a larger mesh size in the X and Y directions, add a mesh override region over the Si layer. Set the override region to treat this area as if it has an index of 1.5 (glass) in the X and Y directions. A small mesh is required in the Z direction, so the override should not be applied in the Z direction.
This technique is only valid if there are no scattering structures within the Si. Indeed, even scattering structure on or near the surface of the Si can generate light propagating at all angles in the Si, however, the above approximation is generally very good even if there are some scattering structures on the surface of the Si. We recommend testing the convergence if there are concerns about the amount of steep angle scattering that may be generated in the Si.
Use PEC for metal objects
In most image sensor simulations, metallic objects behave like perfect metals (100% reflection, no absorption). Rather than using a detailed material model for the metal (which will require a smaller mesh), simply use the Perfect Electrical Conductor (PEC) material model. The PEC material model does not require a small mesh, which makes your simulations faster.
Use conformal meshing (Conformal variant 1)
The conformal mesh can provide much more accurate results at larger mesh sizes and make it possible to run simulation much faster. If PEC is used for the metals, it is a good idea to switch to using the "Conformal variant 1" setting which will apply the conformal mesh algorithm to the PEC as well as the dielectric materials and the Si (please see Mesh refinement options for more details). The figure on the right shows that the Optical Efficiency to the green pixel is almost unchanged when going from a mesh accuracy of 1 (lambda/dx=6) to a mesh accuracy of 6 (lambda/dx=26). When using staircase meshing, the convergence is slower and a smaller mesh size is required for the same accuracy.
Please note that the conformal mesh can generate more numerical instability, especially if many highly dispersive materials are used that require large numbers of coefficients to fit. If these instabilities do occur, they can normally be controlled by making changes to the fit settings for certain materials. Please contact Lumerical support for advice if necessary.
Structures in optical and electrical simulations
You do not need to include exactly the same structures and simulation volume in the optical and the electrical simulations. Structures such as microlens, color filters and metal shields do not affect the electrical response of the device and should be removed from the electrical simulations to avoid an unnecessarily larger simulation region. Likewise, some of the electrical parts such as the substrate contact do not need to be included in the optical simulation if they have negligible effect on the optical response of the device.
Nwell contacts
This contact should not exist in realistic design, but this is a key element to help calculate the weighting function. In simulation, it plays a dual role:
set the electrostatic potential of the n-well equivalent to its depleted state after reset
collect photo-generated charge carriers that are captured in the n-well
When calculating the collection probability weighting function, we assume that the electrostatic potential is unperturbed. In normal device operation, charge carriers trapped in this region will be transferred to the drain (via the TX gate) and will accumulate on the source-follower gate. An accurate determination of the potential of the depleted n-well can be determined by inducing a channel between the drain (biased to VDD) and the n-well with the TX gate turned ON, which can be simulated at steady state. If user would like a quick way to test the simulation behavior, they can first run a simulation without the nwell contact to test it electrostatic potential. And then add a small amount, eg, 0.5V, to this simulated voltage to represent the bias at the nwell. This contact should be small in size and placed at the minimum of the potential in the n-well (small enough that the potential is close to uniform over the equivalent of the contact's surface). This usually corresponds to the peak location of the doping region.
Sub contact
This is a simulation contact. In realistic design, the silicon substrate should be grounded and therefore we need to assign a contact to the substrate. In the Charge solver, contacts have to be assigned to a metal type material. Aluminum is chosen arbitrarily, and the electrical contact boundary condition "force Ohmic" is enabled. The contact should be placed sufficiently far from all depletion regions so that it doesn't change the potential there (i.e. equilibrium condition should be established before contact surface), but generally must be placed deeper in the substrate to provide adequate depth for the absorption of light (check the optical generation rate profile).
PD contacts
This simulation contact is disabled by default. If necessary, the PD objects and contacts can be enabled and used as a troubleshooting step. There are some p++ regions without a reference voltage near the surface of the silicon. The solver may become numerically unstable if the region is floating. This contact can provide a reference voltage to the region for simulation purpose. This contact should be placed on top of the p++ region, the size of this contact should not have major effects to the simulation results.
Transfer gate and drain
Those are not simulated in this example. The current example is simplified and a focus of it is to show the simulation methodologies and how the Green's function approach can be used to efficiently calculate quantities like IQE, EQE, crosstalk, etc. In a more sophisticated simulation setup, experienced users may want to include these elements in their simulations, but this will certainly increase the complexity of the simulation.
Isolation trenches
These are common features to reduce electrical leakage current, formed by etching a Shallow Trench Isolation (STI) in the substrate and filling it with oxide. Their presence and location will depend on the specific design and fabrication process.
Updating the model with your parameters
Instructions for updating the model based on your device parameters
Customizing structures
The "image sensor" structure group contains only some representative parts of typical CMOS image sensors. When modifying structures, it is recommended that you retain the "user properties" of the "image sensor" object and make sure the link between the key design parameters (pixel size, microlens shift, etc.) and the associated structures are not broken. Otherwise, you might need to write all the relevant scripts from scratch.
Importing AFM data for the lens
The "image sensor" structure group has an option for "use_AFM_data" to demonstrate how AFM data could be used to define parts of the sensor. Set its value to "1" to import the AFM data for the microlens surface and "0" to define the surface based on the polynomial and conic formulations. The script assumes you have your surface data (savedata - Script command) saved in the same directory as your .fsp file and uses the importsurface2 command to create the surface. You can also import a surface data in .txt format using the importsurface command. The "cmos_image_sensor_angular_response_microlens_AFM_make.lsf" can be used to generate an exemplary surface data. Please note that using AFM data will make your fsp files much larger, and you will likely want to keep a low setting for the "rendering detail" to avoid slower display of the structures.
Source wavelength
The wavelength of the source is currently set at 550nm (green). If you want to see the response of the green/blue pixels to blue light, all you need to do is just change the source wavelength to blue. However, to obtain the response of the red/green pixels to red light, you need some additional modification in the simulation settings:
Enable the the associated generation rate group, "CW_generation_rg"
Add the "::model::CW_generation_rg::Igen" to the "Result" of the "sweep angle" object
Update the filenames for the weighting functions in the script files for step 4 and 5 with correct pixel names.
Source intensity
The default intensity of the light source is set to 1W/m^2 in the "model." Its value might need to be updated to account for the actual intensity of the light source in consideration.
Spatial filter for depletion region
The depletion region is where the generated charges are swept by the strong electric field and contribute the photocurrent. The shape of the actual depletion region can differ depending on the contact design and doping profiles. In the FDTD simulation, we assumed that light absorbed within the depletion region may contribute photocurrent and therefore used a spatial filter in the analysis script of the "pixel transmission" to mimic the depletion region when calculating the transmission for each pixel. It is currently set to a 1x1 \(({\mu m}^2) \)with rounded corners, but you might need to modify it to better match the depletion region in your device. This is an optional setting.
Taking the model further
Information and tips for users that want to further customize the model
Broadband simulations
While the optical simulations in step 1-3 were run at single frequency, you can also use the same file for broadband (multi-frequency) simulations. However, there are some additional points to be considered when running broadband simulations. Please visit here for detailed information. Note that the generation rate analysis group returns results that are averaged over wavelength when broadband source is used.
3D CHARGE simulations
While the electrical part of this example was based on 2D CHARGE simulations, the Green's function approach works for the 3D CHARGE simulations as well. Please note that extending the current example to 3D CHARGE simulations requires extensive changes to the generation rate script, the CHARGE simulation file and related analysis scripts.
Point spread function (PSF)
Together with OE, PSF is also a frequently used metric to characterize the optical properties of CMOS image sensor. Broadly speaking, it is a measure of spatial crosstalk — i.e., how much light is detected in neighboring pixels when a specific pixel is fully illuminated. Please visit here for further information and example files.
Response for arbitrary illumination
The approach in the above PSF example uses an array of "thin lens" Gaussian beams to mimic a source uniformly illuminating a specific pixel. This requires the simulation to have pml boundaries in all directions and a large simulation region. Additionally, you need to run separate simulations to obtain the results for different objective lenses. Fortunately, there is a much faster and more efficient approach available, which use an incoherent sum of results for planewave illumination to reconstruct the response of any arbitrary illumination (including arbitrary objective lens). For further information about this approach based on the planewave, please see [2]
Additional documentation, examples and training material
F. Hirigoyen, A. Crocherie, J. M. Vaillant, and Y. Cazaux, "FDTD-based optical simulations methodology for CMOS image sensors pixels architecture and process optimization" Proc. SPIE 6816, 681609 (2008)
J. Vaillant, A. Crocherie, F. Hirigoyen, A. Cadien, and J. Pond, "Uniform illumination and rigorous electromagnetic simulations applied to CMOS image sensors," Opt. Express 15, 5494-5503 (2007)
Crocherie et al., "Three-dimensional broadband FDTD optical simulations of CMOS image sensor", Optical Design and Engineering III, Proc. of SPIE, 7100, 71002J (2008)
Wang, Xinyang, "Noise in Sub-Micron CMOS Image Sensors", Ph.D. Thesis, Delft University of Technology
W. Gazeley and D. McGrath, "Quantum Efficiency Simulation Using Transport Equations," International Image Sensor Workshop, R06 (2011)
Optical simulation methodology
Electrical simulation methodology
Point spread function
Green's function IQE method
Related Ansys Innovation Courses
CMOS - Optical simulation methodology
CMOS - Angular response 2D
CMOS - Point spread function (PSF)
CMOS - Photoelectric conversion
Lumerical scripting language - By category | CommonCrawl |
How to tell if a thermodynamic cycle is reversible without calculating entropy change?
Consider the Carnot cycle, consisting of two reversible, isothermal processes and two isentropic processes. It is reversible, pretty much by definition.
Now consider the Lenoir cycle, consisting of an isochoric compression (heat addition), followed by an isentropic expansion, followed by an isobaric compression (heat loss). I calculated the entropy created by this cycle and found it to be strictly positive.
However it's not clear intuitively why this cycle should be irreversible. Is heat change at constant volume or at constant pressure necessarily irreversible?
thermodynamics entropy reversibility
Joshua BenabouJoshua Benabou
$\begingroup$ What do you mean by "I calculated the entropy of this cycle and found it to be strictly positive"? You mean the entropy change of the system? But that is strictly $0$ by definition of cycle, since entropy is a state function. Or you mean the entropy of the surroundings? But then it is not clear to me how you calculated that. $\endgroup$ – valerio Jun 4 '17 at 19:28
$\begingroup$ I mean the entropy created which is the change in entropy of the two heat sources plus the change in entropy of the system which is zero because as you said it's a cycle. I calculated the change in entropy of the sources by the standard formula Q/T where T is the constant temperature of the source and Q is the heat transfered. $\endgroup$ – Joshua Benabou Jun 4 '17 at 22:28
$\begingroup$ Show us the details of what you did to calculate the entropy change of the sources. Did you take into account the temperature changes of the system during the first and third steps and the need for the surroundings to match these changes? $\endgroup$ – Chet Miller Jun 5 '17 at 0:57
$\begingroup$ Ok I will post my calculations. I did take into account that the temperature of the system changes however I don't see why this would imply that the temperature of the heat sources would need to change. As far as I understood a heat source is constant temperature. If we wish to transfer heat to the system we place the heat source in thermal contact with the system until the temperature of the system becomes that of the source. No? $\endgroup$ – Joshua Benabou Jun 5 '17 at 1:05
$\begingroup$ If you want to do it reversibly, you have to use a continuous sequence of reservoirs, running from an initial temperature to a final temperature. Otherwise, entropy will be generated within the gas, and transferred to the constant temperature reservoir, such that, in the end, the combined changes in the entropies of the reservoirs will be positive. This is what you showed. $\endgroup$ – Chet Miller Jun 5 '17 at 11:59
If the isochoric and isobaric transformation are performed reversibly, i.e. quasistatically and without heat dissipation caused by friction or other effects, then your cycle will be reversible.
This is true for every thermodynamic cycle you can draw in the $PV$ plane: if every step is performed reversibly, then the cycle is reversible.
The peculiarity of the Carnot cycle is that it is the only reversible engine that operates between two heat sources only. You can easily see easily how many different heat sources you are using if you draw the cycle into the $TS$ diagram (picture from Wikipedia):
In this case, it is easy to verify that the change in entropy of the surroundings is
$$\Delta S_{surr} = -\frac{Q_H}{T_H}+\frac{Q_C}{T_C} =0$$
So that the engine is indeed reversible. But now let's take your Lenoir cycle in the $TS$ diagram (picture from Wikipedia):
As you can see, during $1 \rightarrow 2 $ and $3 \rightarrow 1$ you are cutting infinitely many isotherms. The formula you have to use is in this case
$$\Delta S_{surr} = -\int_1^2 \frac{\delta Q}{T} + \int_3^1 \frac{\delta Q}{T}$$
But this time you cannot take out $T$ from the integral like you would do with a Carnot cycle, because it is not a constant.
What you can do is to assume that step $1 \rightarrow 2 $ and $3 \rightarrow 1$ are performed reversibly: in this case, $\Delta S_{surr}=0$ by definition.
valeriovalerio
$\begingroup$ my understanding is that a heat exchange between a constant temperature heat resevoir and a system is reversible if and only if it occurs isothermally (as any heat exchange across a temperature difference which is not infinitesimal creates entropy). My first question is can the lenoir cycle be realized using two heat sources? I don't see why not? The conclusion is that, if it is realized using two heat sources, the cycle will thus transfer heat across isothermally, and thus the cycle will be irreversible. $\endgroup$ – Joshua Benabou Jun 5 '17 at 11:06
$\begingroup$ This makes sense, because when you calculate the efficiency of the Lenoir cycle operating between two heat reservoirs, you find it to be strictly less than that of the carnot cycle operating between the same two heat reservoirs. Furthermore, this whole thing about ensuring reversibility by using an infinity of heat resevoirs makes sense theoretically, but I'm guessing it's impossible to realize in practice. $\endgroup$ – Joshua Benabou Jun 5 '17 at 11:11
$\begingroup$ The Lenoir cycle does not operate between two heat reservoirs: it operates between an infinity of them, because you cut infinitely many isotherms when performing it. A cycle operating between two and only two heat sources must consist of isotherms and isentropics only, and the only reversible cycle respecting this condition is the Carnot cycle. $\endgroup$ – valerio Jun 5 '17 at 11:32
$\begingroup$ Think about it in the $TS$ diagram: what is the only cycle you can draw consisting of two horizontal lines (the isotherms representing the two heat sources) and as many vertical lines as you want? The answer is a rectangle, i.e. the Carnot engine. I repeat: this is the only reversible cycle that operates between exactly two heat sources. $\endgroup$ – valerio Jun 5 '17 at 11:34
$\begingroup$ I still don't understand why the Lenoir cycle can't operate between two heat reservoirs? Perhaps you mean to stay the reversible Lenoir cycle requires an infinity of heat reservoirs, in which case I agree. Or are you claiming that it is impossible to construct a heat engine using two heat reservoirs whose pv-graph is that of the lenoir cycle? $\endgroup$ – Joshua Benabou Jun 5 '17 at 11:44
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Work done by a heat engine given thermodynamic cycle | CommonCrawl |
\begin{definition}[Definition:Cofactor Matrix]
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
Let $A_{r s}$ denote the cofactor of the element whose indices are $\tuple {r, s}$.
The '''cofactor matrix''' of $\mathbf A$ is the square matrix of order $n$:
:$\mathbf C = \begin {bmatrix}
A_{1 1} & A_{1 2} & \cdots & A_{1 n} \\
A_{2 1} & A_{2 2} & \cdots & A_{2 n} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n 1} & A_{n 2} & \cdots & A_{n n} \\
\end {bmatrix}$
\end{definition} | ProofWiki |
Daniel Lightwing
Daniel James Lightwing is a former mathematics child prodigy and co-founder of the London-based Internet/gambling business Castella Research, which uses high-frequency trading inspired methods to place bets on sports exchanges. He was previously a web backend developer for the London offices of Google. In 2006, he represented the United Kingdom at the International Mathematical Olympiad (IMO) in Ljubljana, Slovenia, where he won a silver medal. His experience at the IMO was described in the 2007 BBC Two British television documentary Beautiful Young Minds and the 2014 film X+Y.
Daniel Lightwing
Born1988 (age 34–35)
Chesterfield, Derbyshire, England
Education
• Studied Natural Language Processing, Peking University (2009–2011)
• MA Mathematics, University of Cambridge (2006–2009)
• MA Oriental Studies University of Cambridge (2008–2009)
OccupationCo-founder of Castella Research
Years active2006–present
Known forSilver medalist at the 2006 International Mathematical Olympiad, former child prodigy
Notable work
• Beautiful Young Minds (2007)
• X+Y (2014)
Lightwing started to gain more fame in China from 2016 onwards, particularly on the website Zhihu, where his articles written in Chinese, covering a broad range of topics had attracted over 170,000 followers within one year.
Early life and education
Lightwing was born in 1988 in Chesterfield, Derbyshire to David S. B. Lightwing and his wife Carolyn J. née Davidson.[1][2] He grew up in the Lake District, and Warthill, Yorkshire.[3] In 2015, he described that, before the age of nine, he "had no particular attraction to mathematics. I learnt to read very young, before attending primary school. And I did read all kinds of things—books aimed at children 5–10 years older. At primary school, I read the entire library."[4]
As his education developed, his teachers "were a little perplexed what to do with me." He described how he wasn't learning anything he hadn't already learned and was bullied by one of his teachers who expected him to "sit under her desk and be ridiculed" for no apparent reason. The bullying increased after he "got extremely angry and jumped on top of the desk to denounce her."[4]
After some intensive personal instruction within a special 'one-to-one' mathematics class with another teacher, he learned that he enjoyed those classes, and stated that, "before long, I had made my mind up that maths was what I wanted to do."[4] He went through many gifted and talented programmes throughout his childhood.
At home, his mother, Carolyn, who was a maths and science teacher, had researched Asperger syndrome (AS) when he was 16 years of age after reading the 2003 mystery novel The Curious Incident of the Dog in the Night-Time, and, later, took him to a diagnostic consultation with University of Cambridge autism researcher and professor Simon Baron-Cohen FBA, who diagnosed Daniel with AS.[3][5][6][7] Parts of the consultation were included in the 2007 BBC Two British television documentary, Beautiful Young Minds.[8]
His interest in mathematics led him to being recruited as a member of the 2006 International Mathematical Olympiad (IMO) team, where he represented the United Kingdom and won a silver medal in Ljubljana, Slovenia.[9]
He was the subject of Catalyst, an Australian television programme in 2008,[6] as well as several Chinese television productions.
He attended Trinity College, Cambridge, where he received a Master of Arts degree in mathematics in 2009, as well as Peking University, where he studied computational linguistics. He previously attended York College and St Peter's School, a public school in York.[10]
Lightwing has stated that he has a positive view of autism.[11]
"I wouldn't call it a disability. When you have Asperger's you are putting on a mask and trying to pretend you are normal but what you are thinking is not normal."
Films
Lightwing's life story was presented in two films. In 2007, the British television documentary Beautiful Young Minds was broadcast by BBC Two, and described his medal-winning competition at the 2006 International Mathematical Olympiad (IMO), as well as his connections with China. In 2014, the film X+Y, starring Asa Butterfield as Nathan Ellis, a character based on Lightwing, was released and portrayed Lightwing's experiences before and during the IMO competition.
Career
Lightwing's professional career includes working at Google, and several gambling-related firms, before co-founding the London-based business Castella Research. He was previously a developer for the London offices of Google, and was once recruited part-time as an IT and marketing manager for the Chinese company Greenland Group. He is fluent in Mandarin and Cantonese.[12]
His time with Google changed his opinion about workplace socialising. In 2015, he admitted that he had "a problem with office culture," adding that he sometimes wants "to join in with other people," but is too shy, and doesn't know what to say when it isn't work-related.[5]
References
1. GRO Register of Births: Jul 1988 vol=6 page=295 Chesterfield, Daniel James Lightwing, mmn = Davidson
2. GRO Register of Marriages: Aug 1987 vol=17, page=627 Surrey North Western - David S B Lightwing = Carolyn J Davidson
3. Hutchinson, Charles (19 March 2015). "Meet the York College student who inspired the film X+Y". Yorkpress.co.uk. Newsquest Media Group and Gannett Company. Retrieved 29 June 2016.
4. Lightwing, Daniel (11 May 2015). "Early Childhood and an Introduction to Maths". HuffPost. AOL. Retrieved 29 June 2016.
5. Butter, Susannah (19 March 2015). "'With Asperger's you put on a mask to pretend you're normal': Daniel Lightwing on how the film of his life helps take the stigma out of autism". London Evening Standard. Retrieved 29 June 2016.
6. Newby, Dr Jonica (28 August 2008). "The World of Asperger's". Australian Broadcasting Corporation. Retrieved 29 June 2016.
7. Baron-Cohen, Simon (September 2015). "Autism, maths, and sex: the special triangle". The Lancet. Elsevier. Retrieved 29 June 2016.
8. Matthews, Morgan (2007). "Beautiful Young Minds". IMDb. Retrieved 29 June 2016.
9. "Daniel Lightwing". IMO-Official.com. International Mathematical Olympiad. 18 July 2006. Retrieved 29 June 2016.
10. "Story of Old Peterite inspires film". St Peter's School. StPetersYork.org.uk. 13 March 2015. Retrieved 29 June 2016.
11. Butter, Susannah. "'With Asperger's you put on a mask to pretend you're normal': Daniel Lightwing on how the film of his life helps take the stigma out of autism". www.standard.co.uk. Retrieved 22 July 2020.
12. "Asperger teenager's inspiring story hits big screen". YorkshirePost.co.uk. Johnston Publishing Ltd. 13 October 2014. Retrieved 29 June 2016.
External links
• Daniel Lightwing at IMDb
• Zhihu: Lightwing (mostly Chinese)
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Open Access Research Article
Non-linear Dynamics and Chaotic Trajectories in Brain-Mind Visual Experiences during Dreams, Meditation, and Non-Ordinary Brain Activity States
Tania Re 1, 2 , Giuseppe Vitiello 3, *
UNESCO Chair "Anthropology of Health. Biosphere and Healing System", University of Genoa , Genoa, Italy
Referring Center for Phytotherapy, Tuscany Region, Careggi University Hospital , Florence, Italy
Dipartimento di Fisica "E.R. Caianiello", Università di Salerno, 84084 Fisciano (Salerno), Italy
* Correspondence: Giuseppe Vitiello
Academic Editor: Bart Ellenbroek
Special Issue: Quantum Brain Dynamics
Received: April 30, 2020 | Accepted: June 01, 2020 | Published: June 11, 2020
OBM Neurobiology 2020, Volume 4, Issue 2, doi:10.21926/obm.neurobiol.2002061
Recommended citation: Re T, Vitiello G. Non-linear Dynamics and Chaotic Trajectories in Brain-Mind Visual Experiences during Dreams, Meditation, and Non-Ordinary Brain Activity States. OBM Neurobiology 2020;4(2):19; doi:10.21926/obm.neurobiol.2002061.
© 2020 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.
The present report discusses brain visual experiences in conditions of low degree of openness of the brain toward the environment, for example, while dreaming, during meditation, or in non-ordinary brain activity states such as under the effects of psychoactive substances, in the state of coma, or in other states of reduced sensory perception, among others. In the present report, for brevity, such states are referred to as brain-mind visual experiences, implying that such a visual activity is not one connected to the actual vision as in the state of wakefulness. In the dissipative many-body model, the criticality of the dynamics is enhanced in low openness brain states and is at the origin of movie-like sequences of images in visual experiences. These sequences and the abrupt shifts from one image pattern to another are depicted by chaotic trajectories through the memory space. Truthfulness and realism felt in the visual experiences are discussed in terms of the algebra of the doubling of the degrees of freedom in the dissipative model. In the present discussion, a few aspects of the visual experiences of a subject during an Amazonian Ayahuasca ceremony are considered.
Brain-mind visual experiences; dissipative quantum model of brain; memory states; chaotic trajectories; quantum field theory; models of cortical dynamics in perception; cognitive behavior
The research on the dynamical laws underlying the rich phenomenology of biology has strengthened the interplay between physics and biology during the fifties and sixties of the past century. The research work conducted by Ilya Prigogine [1], and Herbert Fröhlich [2] pioneered the study of the role of dissipative systems and coherent boson condensation in biological systems through the use of the formalism of statistical mechanics, non-linear dynamical systems, and quantum field theory (QFT).
It was in this context that Ricciardi and Umezawa [3] proposed to study the neuronal dynamics within the mathematical frame of many-body physics. Early research works reported by Karl Lashley [4] and the subsequent ones by Karl Pribram [5] and Walter Freeman [6] were indeed indicating the necessity to supplement the studies focused on the properties of the single neuron and glia cell with the dynamical concepts of fields and many-body collective modes. In this context, the model developed by Ricciardi and Umezawa was extended by one of the authors (GV) to include the dissipative character of the functional activity of the brain [7].
Within the frame of the dissipative model, the aim of the present report is to account for the experiences of movie-like sequences of images in the functional activity of the brain, for examples in dreams or dream-like and sleeping states [8], during meditation, under non-ordinary states induced by psychoactive molecules (for example, DMT, i.e., Dimethyltryptamine, or N,N-DMT [9,10,11] which is closely related to serotonin, the neurotransmitter affected by a wide variety of psychedelics), perhaps under anesthesia, in certain coma states, etc. It is noteworthy, however, that the analysis presented in the present report is focused on the dynamical features that are common to the brain states during the above-stated conditions (dreams, meditation, anesthesia, etc.), and does not focus particularly on, for instance, dream activity, meditative activity, etc., nor is it focused on brain pharmacology. The interest is rather in the general dynamical mechanisms allowing these brain activities, i.e., the dynamical core shared by all these brain activities, despite their differences.
The present work demonstrates that the enhancement of the criticality of the dynamics in the low openness states of the brain is responsible for the movie-like sequences of images, scenarios, and events. Such experiences are globally referred to as brain-mind visual experiences for brevity. The locution 'brain-mind' is used to recall that the visual activity being referred to is not the one connected to the actual vision as in the state of wakefulness.
The dissipative model offers the unique possibility of working in the space of memory states and considering the trajectories in such a space depicting the movie-like sequences of the images in the visual experiences. The model is able to account for laboratories observations, such as the formation of extended assemblies of neurons in the synchronous amplitude-modulated and phase-modulated oscillations, their irreversible sequences, duration, and size, the fractal self-similarity of the brain background activity, the power-law distribution of power spectral densities, etc. [6,7,12,13,14,15].
The present report is organized in the following manner. The dynamical approach to brain-mind visual experiences is presented in Sections II and III, wherein criticality, openness, and deterministic chaos in the complex brain activity are also discussed. Section IV discusses intentionality and the formation of meanings within the action-perception cycle, along with the truthfulness and realism aspects felt during the brain-mind visual experiences. Section V presents conclusive remarks. The main aspects of the dissipative model are summarized in Appendix A. The present discussion also refers to a few aspects of narration by a subject who participated in an Ayahuasca ceremony in the Amazon forest. This narration is presented in Appendix B.
2. Criticality and Openness of Brain Dynamics
It might be useful, to begin with a few essential features of the dissipative quantum model of the brain. Further details are provided in Appendix A.
The brain is permanently open to its environment, exchanging energy and information with it through the perceptive channels. Inputs reaching the brain produce, through the action-perception cycle [13,16], responses that are aimed at the best being-in-the-world of the subject, i.e. to reach the equilibrium through the balancing of the fluxes of energy, information, etc. The brain identifies in the environment, the sources, and the sinks for its energy requirements and waste, respectively. The environment, therefore, appears to be its complement in the in/out (time) mirroring of fluxes, its Double. The description of the system and its environment as a closed whole implies doubling of the degrees of freedom of the system, i.e., $A_{\rm k}^{}:A_{\rm k}^{}\to A_{\rm k}^{}\times \widetilde{A}_{\rm k}^{},$ where $\widetilde{A}_{\rm k}^{}$ denotes the degrees of freedom of the environment and ${\rm k}$ denotes the momentum ($A_{\rm k}^{}$ and $\widetilde{A}_{\rm k}^{}$ actually describe the dipole correlation modes and their time-reversed copies, respectively; cf. Appendix A).
A distinctive feature of QFT, and consequently of the dissipative model, is the existence of an infinite number of state spaces for the system, each one being physically distinct from [and inequivalent to] the others. It is possible to record memory in the minimum energy state (the vacuum or ground state) of each of these state spaces. Memory recording is achieved through a process of condensation of the quanta $A_{\rm k}^{}$ and $\widetilde{A}_{\rm k}^{}$ in the vacuum, which appears to be a coherent state with a definite fractal dimension [17]. Since and describe the dipole correlation modes, the memories are described as correlation patterns (ordered patterns). The collection $N$ of the numbers $N_{A_{\rm k}^{}}^{}$ and $N_{\widetilde{A}_{\rm k}^{}}^{}$, [for any ${\rm k}$,$N=\lbrace N_{A_{\rm k}^{}}^{},N_{\widetilde{A}_{\rm k}^{}}^{},{\rm with}\ N_{A_{\rm k}^{}}^{}=N_{\widetilde{A}_{\rm k}^{}}^{},\forall{\rm k} \rbrace$], is the specific code of each memory ($N$ is referred to as the memory order parameter). Memory "recalling" is achieved through the excitation of these quanta from the vacuum.
The use of EEG, ECoG, fNMR, and other techniques in neuroscience has revealed the formation of assemblies of myriads of neurons undergoing synchronous amplitude-modulated (AM) and phase-modulated (PM) oscillations. These AM and PM oscillation patterns are described in the model as coherent condensation patterns of neuronal long-range correlations in the brain ground state. Cortical activity is observed to go through these "multiple spatial patterns in sequences during each perceptual action that resemble cinematographic frames on multiple screens" [18,19,20].
Establishing links with the environment transforms into a dialogue between the self and the Double in a dialectical process of identification vs. distinction. It has been proposed that the act of consciousness resides in such a dialogue with the Double [7,21].
It has been shown [22] that, as the number $n$ of the links increases (or decreases), the allowed size for the correlation domains increases (or decreases). This is consistent with the observations demonstrating that the more the brain associates with its environment, the more are the neuronal connections formed [23].
The physical inequivalence (orthogonality) among the memory states ensures that the corresponding [different] memories do not interfere with each other, and any "confusion" among these memories cannot arise. One may demonstrate that orthogonality is strict when the number of $n$ links is maximum. Then, a phenomena such as "fixation" or "being trapped" in a certain specific memory state (in one attractor in the attractor landscape; cf. Appendix A) may occur. However, in practice, it is quite difficult to reach the maximum number of links with the environment. A higher or lower degree of openness may be reached according to several factors, occasional (such as sleeping, drug consumption, meditation, etc.) or the ones due to the age (such as during the childhood or the older ages).
On the contrary, for reduced openness, the model predicts smaller memory domains and a 'smoothing' of the physical inequivalence among the memory states along with an enhanced possibility of phase transitions (criticality). Then, the "paths" or trajectories through the memories may occur, producing "association" of the memories, or in certain cases "confusion" of memories. Although, in a dynamical regime of criticality, minimization of free energy at each time $t$ is continuously pursued [7].
In a 'normal' state of openness, the possibility of memory flows is allowed only up to a certain degree. This is the state of "attention", of being open to what is occurring around us, the awareness of the Now (the warning inside the Metro reads: "please pay attention, mind the gap!").
Therefore, it is concluded in the present report that, when the number of links becomes minute such that the system almost fails to connect with the environment and is almost closed on to itself, an interrupted "flow" of memories may occur, and it becomes possible to "travel" through the memory states; the dynamics is almost completely dominated by criticality and movie-like sequences or abrupt shifting of images may be experienced.
3. Deterministic Chaos in the Complex Brain Activity
In the cases of extremely reduced openness considered earlier, the association or confusion of memories might lead to deformation or corruption of the memory code components $N_{A_{\rm k}^{}}^{}(and\ N_{\widetilde{A}_{\rm k}^{}}^{})$, for few or several ${\rm k}$s, resulting in "pieces", "bits", or "debris" of memories [7,21], which might be recalled in movie-like sequences, outside of their original recording context and assembled in certain emerging/new contexts.
Flows of images associated to such memory debris are observed to occur in dreams [7,21,22,24,25,26,27], in certain altered states of the brain dynamical regime such as under the effect of anesthesia [28], and in certain stages of deep meditation [29,30]; this may also occur in association with slow breathing techniques [31] or whenever the openness is indeed reduced such as in response to psychoactive substances (see the narration reported in Appendix B). Such visual experiences may also occur under the influence of external rhythmically modulated stimuli, in space (visual rhythms) or in time (musical rhythms). The repetitive persistence of these stimuli may become dominant to the point of excluding any other different input, thereby reducing the openness (similar to what happens during hypnosis).
Interestingly, in the present work we find that there exist common dynamical features underlying the different conditions in which the brain-mind visual experiences occur. Although these conditions refer to distinctly different phenomena and behaviors (ranging from dreaming to meditation, anesthesia, psychoactive substances, among others), all of these conditions are characterized by a low level of openness of the brain toward its environment, which in turn implies the criticality of the dynamics, as discussed in the previous section. The additional result that is derived from the model is that the weak inputs or noisy perturbations drive the system through the memory states and, therefore, play an important role in the complex brain activity, in general as well as particularly in the brain-mind visual experiences.
In this context, it is being remarked that deterministic chaos is a crucial feature of neuronal dynamics [12,14,32,33,34,35,36]. As stressed by Freeman, "The chaos is evident in the tendency of vast collections of neurons to shift abruptly and simultaneously from one complex activity pattern to another in response to the smallest of inputs. This changeability is a prime characteristic of many chaotic systems. In fact, we propose it is the very property that makes perception possible. We also speculate that chaos underlies the ability of the brain to respond flexibly to the outside world and to generate novel activity patterns, including those that are experienced as fresh ideas" [32].
Consistent with these remarks by Freeman, it has been demonstrated that trajectories through coherent states are indeed classical chaotic trajectories [17,37]. Therefore, even slight changes in the initial conditions may lead to diverging trajectories. For instance, in dreams or dream-like and sleeping states, where inputs are not so strong [8,27,38,39,40,41,42,43], the result is the occurrence of sudden shifts from one memory to another due to the chaoticity of the dynamics favored by its enhanced criticality. As a consequence, one may experience abruptly changing scenarios and feel overwhelmed by a series of emotions. These "debris" of memories might even be felt by the dreamer with the flavor of new, never-lived situations, as not belonging to his past, in that intricate blend or mix, presenting sometimes an obscure core, as the center of a vortex, which Freud has called [44] the "dream navel" [21].
It is also interesting that the common experience is that pain is absent in dreams, where the model predicts small correlated domains. On the other hand, it is known that the pain threshold may change under the effects of certain specific drugs, such as morphine, which may indeed be reducing the extent of neuronal connections.
In the case of anesthesia, after the patient resumes to the waking state, a sense of surprise that a period of time has passed since the effects of anesthesia commenced is frequently reported. This is clearly due to the patient's detachment from the environment during the period under anesthesia. Anesthesiology studies appear to suggest that in the anesthetic recovery stages, certain dreaming or dream-like activity occurs [28,45].
It is interesting to ask whether the brain-mind visual experiences occur in certain comatose states as well, the mechanisms of which are not yet completely understood. Excluding the cases of extreme damage to the brain, closure to the external world leads to the conjecture that the brain-mind visual experiences may also occur in certain comatose states as well as in vegetative states, which are mediated by decreased coordinated activity among the small, short-lived, and unstable neuronal assemblies [46].
It should be noted, for the sake of completeness, that the formation of extended correlation domains may also be inhibited by a rapid succession of strong perceptual inputs which might dominate the emotional state of the subject; the subject's arousal may reach extremely high levels, preventing him/her from focusing attention on any of the inputs. This may also occur when neuronal recruitment is enhanced by certain chemicals. Several competitive domains are formed in short time intervals, and their inflation would correspond to lack of information "in the average". As a consequence, the subject is unable to respond to these inputs coherently, which translates to a deficiency in his/her functional relational activity [21].
In summary, in the afore-stated non-ordinary states, in the condition of quasi-closure, the subject's response to the external noisy inputs is reduced considerably, and his/her level of awareness of the environmental changes is quite low or even absent in certain scenarios. Nevertheless, in such a state, the smallest of perturbations that are not filtered out may induce chaoticity in brain behavior. In the absence of reactive feedback, the subject becomes a "spectator of himself", in a process of identification with his Double no longer distinguishable from himself, which occurs in the open, normal state of awareness.
4. Intentionality and Meanings, Truthfulness and Realism in Brain-Mind Visual Experiences
In the brain's functional activity, one crucial element to be considered is the intentionality, which enters as a characteristic ingredient in action-perception cycle.
Pribram [5,47] stressed that there always exists a content of "attention" in perception and of "intention" in action. There is an active perception in one's relation with the world, guided by one's changeable volition and intention in pursuing one's best to-be-in-the-world. Freeman remarked that neuronal activity serves "as a unified whole in shaping each intentional action at each moment" [48]. The brain constructs meanings out of perceptual experiences. Meanings arise from the dynamic correlations in the landscape of the attractors constructed out of the brain's perceptual history [13,49,50,51,52], and are the basic substance of the subject's identity, manifesting themselves in the "intended actions" following the "active perception."
Such a profound intentional component emerges as "meaningfulness" in the brain-mind visual experiences as well. It might "guide" the chaotic trajectories leading to the "rearrangement" of memory traces into fresh scenarios and events which may even appear completely disjointed from the waking experience. Such traces are, however, related to the waking experience through the deep red thread of intentionality and meaningfulness, univocally associated to the subject's identity, to the "affectivity [which] is the primordial form of subjectivity" [53]. This profound intentional component may represent the 'unconscious wish', postulated by Freud, in dreams [44], and has been recognized by Globus in the lucid dreaming phenomenon [54,55], where the dreamer has a certain form of control over the dream scenarios. This is why the brain-mind visual experiences carry relatively hidden, veiled meanings. According to the arguments presented in the previous section, visual scenarios are fed with "pieces" or "debris" of the previously recorded memories. Certain feelings or context settings might be anticipated in the brain-mind visual experiences on the basis of a persisting intentional component, and may then (re-)appear in the future perceptual experiences (cf. the narration in Appendix B). This is not to talk of any precognition capability or of violation of causation. It is merely to state that the correlations in the attractor landscape originating from the brain-mind visual experiences may, at certain times, find occasional resemblance to the correlations originating from the active perceptual experiences in a future waking state.
Therefore, in the movie-like flow of images in non-ordinary states, dreams, and other brain-mind visual experiences, the recollection of the existing correlations in the attractor landscape may indicate unforeseen contexts, giving rise to the problem of "truthfulness" or "realism" of such contexts and of their meanings as felt by the subject within the boundaries of his/her beliefs, knowledge, and emotional states.
In order to consider this point, it is observed that the algebra of the doubling of the degrees of freedom implies that $A_{\rm k}^{}$ and $\widetilde{A}_{\rm k}^{}$ are entangled modes [37,56]. As a consequence, any 'observation' of the $A_{\rm k}^{}$ modes is, in fact, dependent on the $\widetilde{A}_{\rm k}^{}$ modes, which thus constitute the "address" for the $A_{\rm k}^{}$ modes, and vice-versa. The conclusion is that the brain modes and the mental (Double) modes cannot be separated, i.e., there exists no separation between mental activity and brain activity. This implies that a sort of "truth-evaluation-function" is implicit in the dissipative model formalism [7,57,58,59]. In other words, in the dialogue with one's Double, the subject finds the possibility of confirming or rejecting the truthfulness of one's working hypotheses. One's "confidence" in one's perceptual experiences is based on the process of feedback-adjustment-feedback, in the continuous matching with the Double. Intentional actions are planned in accordance with the hypotheses provided by the Double through the reconstruction of past perceptual experiences. The experience of changes in perception following repeated trials in the action creates the perception of time and causation [59,60].
It is stressed that each step of brain activity characterized by criticality is formally expressed by the free energy minimization condition. Recording a memory is the consequence of a process of breakdown of symmetry induced by the perceptive stimulus (cf. Appendix A). In one's "active" selection among the perceptual inputs, one focuses "only on those inputs that one judges worthwhile to expend one's energy for, the ones to which a "value" is attributed, those which involve one's emotions [21], one's affectivity" [53]. Using these selected inputs, one's memory (non-oblivion) is constructed, which depicts one's identity and one's "truth". Non-oblivion and truth coincide in the ancient Greek $\alpha\lambda\eta\theta\epsilon\iota\alpha$ [21]. The landscape of the attractors constructed in the previous experiences is reshaped in this manner with any fresh input, and meanings are constructed. Memory is the memory of meanings [56,61].
All these features are present in the movie-like sequences in non-ordinary states, dreams, or dream-like states.
In brain-mind visual states, the perception of time is conditioned by the (quasi-)closure of the subject's state. The lack of synchronization to a reference clock manifests in the loss of time-ordering of the events as recorded originally in the waking perceptual state. Mixing of memory traces with the emergence of fresh scenarios occurs then. The feelings of truthfulness and realism of these fresh scenarios derive from the fact that the traces of memories are actually traces of meanings, interwoven together by the red thread of intentionality stated earlier; they carry the seal of the subject's identity.
Truthfulness and realism are further strengthened by the previously-stated matching, almost identification, of the self with its Double in the low level of openness. The (quasi-)closure of the subject and his/her emotional state are interrupted by the "waking-up" experience, thereby restoring the openness of the system and the awareness of the distinction between the self and the Double (It was only a dream!).
In the experience presented in Appendix B, the subject reported the "appearance of a gray painting, similar to the screen of an old black-and-white TV when the signal was lost; when that strange TV in the middle of the Amazon could "retune", a new scene appeared". Interestingly, this kind of "blacking-out of the signal" might find correspondence with the phenomenon of "null spike" (cf. Appendix A), which is observed to separate two behavioral frames in the brain activity [16]. Null spikes represent singularities appearing in the transitions between different dynamical regimes (phase transitions) corresponding to different configurations of correlations in the attractor landscape. The null spike behaves as a caesura, similar to the shutter diaphragm of old fashioned cameras. In the transition, the coherence of the condensate vanishes (null spike), and soon after, another coherent condensate initiates [62]. The phenomenon may be observed through an EEG in normal brain activity [18,63]. In brain-mind visual experiences, null spike corresponds to the closing of a movie-like flow, clearing the field, and opening of another movie-like flow.
Using the dissipative quantum model of the brain, the brain-mind visual experiences occurring during non-ordinary states, dreaming activity, meditation, and other low openness functional activities of the brain were discussed.
The low degree of openness of the brain in the afore-stated states induces enhancement of the criticality of the dynamics, which leads to enhancement in the possibility of "traveling" within the memory space through chaotic trajectories. Movie-like sequences of images and abrupt changes in the relatively familiar scenarios are generated consequently in the form of brain-mind visual experiences.
The feeling of truthfulness and realism associated with these visual experiences is derived from the quasi-identification of the self with the Double, a trace of the continual matching in the feedback-adjustment-feedback process occurring within the intentional context of the state of wakefulness.
It was conjectured that a similar description might also be applied to cases under anesthesia and certain pathological cases, such as certain states of coma and conditions where the openness of the brain toward the environment is reduced.
The existence of an infinite number of unitarily inequivalent representations of the canonical commutation relations in QFT, on which the space of the memory states is constructed, has been crucial to the analysis conducted in the present work.
According to the findings of the present work, a relatively severe closure of the brain to the world may produce a dramatic lack of "meaningfulness" within the subject's action-perception cycle. Such conditions of closure occurring for certain reasons in the subjects during their wakefulness state might be the actual reason underlying the subject's deficit in social communication and interaction, with the exhibition of restricted, repetitive patterns of behavior, interests, or activities. It would be interesting to ask the question whether brain-mind visual experiences occur in such cases as well. The investigation of this query should be considered for future research, with the inclusion of autism spectrum disorder, if possible.
The present research work is dedicated to the memory of Karen Sharon and Eliano Pessa. Karen, during her association with the research work of Karl Pribram and Walter Freeman, contributed to the cultural atmosphere and dense activity at the foundation of modern neuroscience. Eliano, with his deep insights and expertise in neural networks and non-linear dynamical systems, provided important contributions to the formulation of the dissipative model.
We would like to thank Nicola Bragazzi, Gabriele Penazzi, Fabio Firenzuoli, Florencio Vicente, Bruno Neri and Riccardo Zerbetto for the fruitful discussions on the themes we presented in the article.
Appendix A. The Dissipative Quantum Model of the Brain-a Summary.
This section provides, for the sake of completeness, a summary of the dissipative model. Refer to [7,13,64] for the mathematical formalism and additional details regarding the model.
The brain exchanges energy and information with the environment to which it is open. The water molecules of the bath in which the cells and all the cellular structures are immersed are characterized by their electric dipoles. The dynamical symmetry of a system comprising randomly oriented dipoles is the spherical continuous $SU(2)$ symmetry (three-dimensional rotation symmetry). The dissipative quantum model of the brain considers neurons, glial cells, and the other biological units to be classical objects. In this regard, the dissipative model is substantially different from the other quantum models of the brain.
In the dissipative model of the brain, the perceptual inputs received by the brain induce the breakdown of the $SU(2)$ symmetry. According to general theorems of QFT [65,66,67,68], spontaneous breakdown of symmetry (SBS) induces the formation of long-range correlations among the system components, and the formation of these correlation waves is ruled by the inner dynamics of the system (which is why the symmetry breakdown is regarded as spontaneous). The quanta associated with these correlation waves are the Nambu-Goldstone (NG) quanta. In the case considered here, the electrical dipoles of the water molecules enter in a coherent oscillation regime [7,69], and the NG quanta $A_{\rm k}^{}$ are referred to as the dipole wave quanta (dwq).
The system's ground state or vacuum (the minimum energy state) is characterized by the coherent condensation of these dwq in domains, the size of which is controlled by the degree of openness of the brain to the environment. The coherent condensation of dwq promotes and facilitates the formation of the observed AM and PM assemblies of myriads of neurons oscillating coherently. The $A_{\rm k}^{}$ modes are referred to as the system's degrees of freedom.
The dwq condensation produced in response to a given input serves as a code $N$ for the input information and represents the recording of its memory in the ground state of the brain (the "memory state"). Memory "recalling" is then achieved through the excitation of the dwq condensed in the vacuum, in response to input similar to the one that caused their condensation in the first place [3].
A crucial point is that an infinite number of physically (unitarily) non-equivalent state spaces or representations of the canonical commutation relations exist in QFT, each one with its own vacuum state, such that in each one of these, it is possible to record different information with code $N$.
A huge (infinite, in principle) memory capacity is, therefore, possible [7]. The manifold of the representations forms the landscape of the attractors (the vacuum states).
In each representation, the conditions for free energy minimization are satisfied. Transitions through these representations are induced through variations in the entropy operator. When the coherence weakens, "the wave packet terminates with a null spike, clearing the field and opening the way for the next wave packet" [62].
The openness of the brain toward its environment is described through the introduction of the doubling of the system's degrees of freedom, $A_{\rm k}^{}\to A_{\rm k}^{}\times \widetilde{A}_{\rm k}^{}$, where $\widetilde{A}_{\rm k}^{}$ denotes the environment's degrees of freedom. In the 'closed' system $\lbrace A_{\rm k}^{},\widetilde{A}_{\rm k}^{}\rbrace$, the fluxes of energy, information, etc. are balanced, and the free energy is minimized at the equilibrium.
The algebraic structure belongs to the $q$-deformed Hopf algebra. Changes in the deformation parameter $q_{\rm k}^{}$, which governs the coherent condensation density, its fractal dimension, and the system entropy, correspond to the critical phenomenon of transitions from one representation to another, from "memory to memory" (criticality).
The memory code or the "order parameter" $N$ is given by the collection of the numbers $N_{A_{\rm k}^{}}^{}$ and $N_{\widetilde{A}_{\rm k}^{}}^{}$ of the $A_{\rm k}^{}$ and $\widetilde{A}_{\rm k}^{}$ modes, respectively, which are condensed in the memory state; for each ${\rm k}$, $N_{A_{\rm k}^{}}^{}=N_{\widetilde{A}_{\rm k}^{}}^{}$. It is because of coherence that $N$ behaves as a classical field, a field independent of quantum fluctuations, even though it is of quantum origin. Coherence ensures the possibility of "change of scale", from micro- to macro-scale, from quantum to classical.
The states of the system $\lbrace A_{\rm k}^{},\widetilde{A}_{\rm k}^{}\rbrace$ are entangled (squeezed) coherent $SU(1,1)$ states, which are also finite temperature states. The entanglement is due to a phase-mediated long-range correlation between the $A_{\rm k}^{}$ and $\widetilde{A}_{\rm k}^{}$ modes [37,56].
The brain finds in its environment, the sources and the sinks for the exchanged energy, information, etc., i.e., what is 'going out' of the system is 'going in' to the environment and vice-versa. The opposite sign of the time variable describes the "in $\leftrightarrow$ out" exchange for the two (the brain and the environment). Therefore, the mode $\widetilde{A}_{\rm k}^{}$ is the "time-reversed" copy of $A_{\rm k}^{}$. The environment is then depicted as the "time-reversed image" of the system (or vice-versa). It is denoted as the Double of the system [7,13,21]. Time emerges as observable in the process of time mirroring with the Double.
In conclusion, the "collection of memory states" constitutes a manifold of coherent states (a landscape of attractors). Paths through these memory states may occur as time evolves, and different scenarios may be created in accordance with the different degrees of openness of the brain. These paths are demonstrated to be classical chaotic trajectories [17,37].
The brain submits the object of its perceptual experience to a process of generalization (elimination of details unnecessary to that perceptual context) and abstraction (association to a category). In mathematical terms, this implies that the experience is situated in a specific attractor within the attractor landscape. In case of the absence of an attractor where a particular experience could be situated conveniently, the brain creates a new attractor. No perceptual experience is ever added to the landscape as a new item is added to a list of items. On the contrary, the whole attractor landscape is reshaped by the new experience, generating meanings represented by the correlations among the recorded perceptual experiences. Memory is not the memory of information, rather the memory of meanings. The inputs received through the perception channels induce, as a response, the subject's intentional action on the environment, aimed to achieve his best to-be-in-the-world. Consequently, novel perceptual experiences arise, again leading to new intentional actions, and the cycle repeats. The intentionality of the action is founded on the hypotheses that the brain is capable of formulating. The action also has the value of a test of these hypotheses. It may, therefore, create confidence in the acquired knowledge (truthfulness), which leads to the addition of significance to the action, meaningfulness to the relation with the Double, and feelings of realism of emotional and perceptual states [52,59].
Driven by dissipation, "brains generalize to other brains like themselves and then to animals and objects" [60] entering into those "affective states, that not only influence our behavior in a flexible way, but alter our conscious field, giving rise to specific feelings or moods, which constitute the first form of self-orientation in the world" [53]. Shared with the other systems in nature, the basic coherent character of the dissipative dynamics of the brain acquires the flavor of an [Jungian] archetype, shaping the affective relations with the Double. Consciousness, therefore, emerges as a "manifestation of the dissipative quantum dynamics of the brain" [7,21].
Appendix B. The Narration of a Case of Brain-mind Visual Experiences Induced by Psychoactive substances.
This Appendix presents the narration of the experiences of a subject (a European adult woman, who is a researcher in the field of psychology), who traveled through the Amazon forest four years ago in order to spend some time at the Mayantuyacu center. At the center, she met Maestro Juan Flores and participated in a ceremony involving Ayahuasca, a brew regarded to have the capability of inducing brain-mind visual experiences.
Prior to commencing the narration, it is important to recall that the vegetal substances containing DMT, such as Psychotria viridis that grows in the Amazon forest, are among the main ingredients of Ayahuasca. DMT is present in a number of different natural sources, a few of which are of animal origin, such as in the Sonoran desert toad (as 5-methoxy-N,N-dimethyltryptamine or 5-MeO-DMT) [70,71,72]. Isolation of DMT from the human blood and urine has also been reported previously [73]. Julius Axelrod (Nobel Prize winner for Medicine in 1970) reported the presence of DMT in the human brain tissue [74]. Among the endogenous neurotransmitters and neuro-hormones [75,76,77], DMT is unique as it is a molecule sufficiently small to pass through the blood-brain barrier. The altered states of consciousness induced in response to this drug have been compared to highly mystical experiences [78].
The narration: "My journey to the Amazon forest was a long trip. When looked at from above, the Amazonian rivers resemble large snakes, are brown and sinuous, move slowly, and are populated by animals and mangroves along the banks. In order to reach the Ashanika center named Mayantuyacu, one has to walk for a few hours across the jungle, encountering millennial trees, butterflies of a never-seen-before blue color which are referred to as "morpho" and which love to hide from and chase each other similar to little spirits. The temperature in the forest exceeds 30 °C, and the humidity is quite high. The top of the hill allows a view of the center of Mayantuyacu, which stands out against the horizon, surrounded by a cloud of steam produced by a small stream that flows next to it, with temperatures that oscillate between 27 °C and 94 °C (Note by the authors: [79–81]).
The days at the center pass slowly. All the people in Mayantuyacu sleep inside small structures placed around the maloca, a helical space dedicated to rituals and therapeutic activities. Juan Flores teaches with plants that have been used for thousands of years in the Ashanika tradition, the ethnic group to which he belongs. Flores is a master, as was his father, and knowledge, as in traditional indigenous medicines, is transmitted orally.
Trampling a square meter of land implies simultaneously crushing a hundred different plant species that could be selected and prepared in the Mayantuyacu "laboratory"; certain plants are used for headaches, while others are used to heal wounds.
The Amazon is the largest open-air pharmacy available to mankind, and its heritage is being increasingly put at risk by the global developments. The lungs of the Earth are at the risk of extinction because of the greed of the world from where I come; despite this, its custodians, including Flores, share with us, their millennial knowledge. Indeed, the decision of the necessity of this sharing was undertaken by plants themselves, says Flores. I, however, wonder how this inter-species communication is possible? Juan just smiles when someone asks these questions.
It is said that there are plants that are master plants capable of exhibiting the properties and characteristics of all the other plants.
I decided to participate in the ceremony with the Ayahuasca, a "master plant of visions". This plant is also used in the initiation rites of adolescents, as it allows them to gain the knowledge that could be useful in the present and future life.
The ceremony begins with a purification ritual; there is a plant with fragrant wood, the smoke of which is spread inside the maloca; all the participants are sitting in a circle in silence.
In the dark, the voice of Flores, who sings a song titled "icaro", or the vibration of each plant, rises. The master plant of visions is actually a liquid with the taste of licorice, which he hands to each participant in a small cup.
When Juan sings the song, something special happens to me; from the darkness of the maloca, a toroidal shape emerges in front of me, almost inviting me to cross it. I close my eyes. I approach it, remaining seated, and my body perceives a feeling of suction for a few moments. It feels as if my body's cells were squeezed into a narrow space. On the other side of the funnel, I open my eyes and there is another vision: it is daytime and nature appears different, as if suddenly animated, vivid. I remain aware of who I am and where I am, although, in front of me, the frames of a film begin to flow increasingly: an Amerindian dressed in white shows me the west coast of the United States, and there is music playing, different from the one I was hearing before, may be it's the sound of his flute, I'm not sure. Another space, another time.
Suddenly, in front of me, certain images begin to take form, as if I am in a journey back in time. I see a table where Jung and Einstein are sitting. The two are quietly discussing physics and synchronicity, and Einstein tells Jung that time does not exist.
Years later, emerging unharmed from a car accident in the United States, I bought a book with a captivating title, Sinchronicity, to keep me company and to overcome the shock of the accident. In the first few pages of this book I read, with a shiver down my spine, the report between Jung and Einstein, discovering that the encounter which was revealed to me by the Ayahuasca, in which Jung had discussed with Einstein time and synchronicity, had indeed historically happened, as it was written in that book [Synchronicity: An Acausal Connecting Principle {Collected Works of C. G. Jung, vol.8, Paperback, 2010, C. G. Jung (Author), R. F.C. Hull (Translator), Sonu Shamdasani (Foreword)}]: ≪Later in his life, Jung traced his idea of synchronicity to the influence of Albert Einstein, who held a professorship in Zurich in 1909–1910 and again in 1912–1913. Jung wrote: "Professor Einstein was my guest on several occasions at dinner. These were very early days when Einstein was developing his first theory of relativity, [and] it was he who had first started me off thinking about a possible relativity of time as well as space, and their psychic conditionality. Over thirty years later, this stimulus led to my relation with the physicist Professor W. Pauli and my thesis on psychic synchronicity".≫.
Today, it would not be wrong if I say that the images of that vision have profoundly influenced my subsequent research activity. The accident I just referred to occurred during a trip to California, which was directed by that first scene I had witnessed. I was going to San Francisco to meet the researchers who were, and still are, working on the theme of consciousness and its states, including the non-ordinary ones.
The most amazing thing for me was that during that first trip overseas, I came across the evidence that the images of the Amazonian vision were not just a fruit of my imagination, and rather corresponded to historical reality. If so, then that substance with a bitter taste similar to licorice had allowed me to take a journey back in time. Indeed, in the indigenous context, Ayahuasca is an "initiatory plant" administered to young people in a complex ritual system, enabling them to take a "journey" not just in their past, rather in the future as well, which allows them, in the moments of crisis, to see which is the right path.
These considerations, however, I could make only later on. My vision was not limited to the scene of Jung and Einstein. The frames followed one another similar to a slow-motion movie, and the scenes were interspersed with the appearance of a gray painting, similar to the screen of an old black-and-white TV when the signal was lost; some of you might remember the knurling and the underlying buzz that remained until the connection was resumed. When that strange TV in the middle of the Amazon could "retune", a new scene appeared, coming from who knows where.
After Jung and Einstein, it was the turn of a nautilus (a spiral-shaped shell), with the word "mira la forma" next to it, which means "observe the shape" in Spanish.
What shape should I observe? All of a sudden, I heard a noise of machinery. I was always aware of being where I was, and I could not understand its origin. Was all this a figment of my imagination? Was it just a hallucination? Was I really watching a movie that someone had concocted for me? There was no time to find an answer to these questions. The movie is three-dimensional, and now I see things "from above". Below me, a new scene emerges: an enormous horizontal device, of which we could also see the interior, with a hexagonal shape. Several people are busy around it, while I do not understand what they are doing.
Years later, I had the opportunity to visit the C.E.R.N. of Geneva, where the subatomic particles and the famous Higgs boson are studied. There, I realized that what I had seen from above, during that vision, was just a particle accelerator, of which at that time, I did not know the slightest thing about.
The scenes of the vision had all been interesting, the most interesting things I had ever seen so far. At one point, still within the vision, the scene had moved to a mountainous environment; the images had defined themselves further to allow me to recognize the profile of the sacred mountain, the Machu Picchu, which I had already had the opportunity to visit years before.
However, the vision depicted it to me in the past. A small indigenous woman dressed in white, quite young, little older than a child, walked on the edge of a sort of crater, waiting to be sacrificed to some god.
After staying in Mayantuyacu, when I took a stop in Lima on my way back to Italy, I was invited to dinner hosted by a lady, through a common friend. When the woman presented herself at the door of the house to welcome me, I was not able to believe my eyes! It was the same woman from the vision, with a few more years added to her age. Amazing! Really incredible!
This was the first time I found myself encountering something that nothing in my studies had prepared me for. The images of the vision took shape in my daily life, in reality, which was something incredible for me, while it was something very natural within the culture that had produced it.
Let us return to the vision. After the Machu Picchu scene, other people had appeared who, just as me, were in the Maloca in that splendid Amazonian starry night.
Watching "the movie" of those in that circle with me had been even more surprising, as it implied that it was possible to somehow tune in to their "channel", to their visions!
It was similar to being inside an hourglass through which the sand appeared to flow gradually, with the impression, perhaps for the dark, perhaps for that strange drink, perhaps for suggestion, but no, it was not really a suggestion, that what I was watching was more real than the reality I was commonly used to.
At a certain point, my movie begins to turn toward a conclusion. The images fade, although the nature around me remains as "animated", with bright colors despite the darkness, as if it had a life that is usually not perceptible to our senses. The trees have almost human features, each one of them with its own particular character. I see a large brown tree, which uses a stick to strike a vigorous blow to the ground, almost to greet me. The darkness closes on the experience, similar to a curtain.
In the darkness of the night, there remain the sounds of the nature that continues its incessant works – the flowing water, the first songs of the birds that announce the day everything around me is alive, although it appears grainy compared to the perception I had of it during the vision as if it had lost that filigree life, a perspective that had been completely unknown to me until then. I doze off.
When I wake up the next morning, I would ask myself if all that I saw was "only" a dream. It vaguely recalls as a dream, actually, although it is not. I would be perfectly aware of having lived that waking experience.
Shakespeare said that we are made of the same substance that dreams are made of. Had he experienced something similar in the English woods? Who knows!
With that experience, I realized that it is necessary to create a bridge between that ancient reality and the modern western laboratories which conduct research on the psychoactive substances that are capable of activating and modifying the psychic state, which are referred to by the indigenous people as "teachers" or plants "Of knowledge".
All authors have contributed equally to the work reported.
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Let $J$ be the $n \times n$ Jordan block corresponding to the eigen value $1$. For any natural number $r$ is it true that the minimal polynomial for $J^r$ is $(X-1)^n$ ?
Another way to think about it to produce a cyclic vector of $J^r$. I can't prove it. I need some help. Thanks.
Hint: write $J=I+N$ where $N$ is the shift matrix. $N$ is nilpotent with index $n$. Now expand $J^r=(I+N)^r=...$ and find out what is the smallest $m$ we need in order to $(J^r-I)^m=0$.
As $r(J-I)=r(J^r-I)$, so geometric multiplicity is $1$in both case are same and hence same minimal polynomial. Here $r$ means rank of matrix.
Not the answer you're looking for? Browse other questions tagged linear-algebra abstract-algebra matrices jordan-normal-form canonical-transformation or ask your own question.
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All-cause mortality effects of replacing sedentary time with physical activity and sleeping using an isotemporal substitution model: a prospective study of 201,129 mid-aged and older adults
Emmanuel Stamatakis ORCID: orcid.org/0000-0001-7323-32251,2,3,
Kris Rogers4,
Ding Ding5,
David Berrigan6,
Josephine Chau1,4,
Mark Hamer3,7 &
Adrian Bauman1,4
Sedentary behaviour, sleeping, and physical activity are thought to be independently associated with health outcomes but it is unclear whether these associations are due to the direct physiological effects of each behaviour or because, across a finite 24-hour day, engagement in one behavior requires displacement of another. The aim of this study was to examine the replacement effects of sedentary behaviour (total sitting, television/computer screen time combined), sleeping, standing, walking, and moderate-to-vigorous physical activity on all-cause mortality using isotemporal substitution modelling.
Longitudinal analysis (4.22 ± 0 · 9 years follow-up/849,369 person-years) of 201,129 participants of the 45 and Up study aged ≥45 years from New South Wales, Australia.
Seven thousand four hundred and sixty deaths occurred over follow-up. There were beneficial associations for replacing total sitting time with standing (per-hour HR: 95 % CI: 0.95, 0.94–0.96), walking (0.86, 0.81–0.90), moderate-to-vigorous physical activity (0.88, 0.85–0.90), and sleeping in those sleeping ≤ 7 h/day (0.94, 0.90–0.98). Similar associations were noted for replacing screen time. Replacing one hour of walking or moderate-to-vigorous physical activity with any other activity class was associated with an increased mortality risk by 7–18 %. Excluding deaths in the first 24 months of the follow up and restricting analyses to those who were healthy at baseline did not materially change the above observations.
Although replacing sedentary behaviour with walking and moderate-to-vigorous physical activity are associated with the lowest mortality risk, replacements with equal amounts of standing and sleeping (in low sleepers only) are also linked to substantial mortality risk reductions.
The health benefits of moderate-to-vigorous physical activity (MVPA) are well-established [1]. Large volumes of sedentary behaviour (SB), characterized by a low energy expenditure (≤1.5 metabolic equivalents) in a sitting or reclining posture [2], are thought to increase mortality risk independently of MVPA [3, 4]. The evidence is particularly strong for specific types of SB, such as television viewing [5, 6], which show consistent associations with health outcomes. The associations between sleep duration and health outcomes are complex, as both low and high durations are associated with mortality risk [7].
Although physical activity (PA), SB, and sleeping are behaviours that occupy a 24-hour day, investigators have typically examined each behaviour without considering what time-dependent behaviours are being displaced [4, 7–9]. For example, a 60-minute block of low-intensity walking will have different health effects depending on whether it displaces an equal amount of sitting, vigorous exercise, or brisk walking. This limitation can be overcome by statistical modeling that specifically estimates the effects of replacing one behaviour with another, the isotemporal substitution modeling [ISM] approach [10]. ISM, which is based on nutritional epidemiology methods analyzing the effects of substituting nutritional components [10], models simultaneously the effect of a given activity being performed and another activity being displaced in an equal time-exchange manner. ISM not only controls for confounding by other time-dependent behaviours, it also captures the effect of time substitution [10]. Very few studies to date have examined the replacement associations of SB, physical activity and other time-dependent behaviours with health outcomes in general [10–12] and no such study has had mortality as an outcome. The only study that has specifically considered the replacement associations of MVPA, SB, and sleeping was a cross-sectional study that found beneficial associations of replacing sedentary time with sleeping and MVPA and a number of cardiometabolic risk markers [12]. However, "sedentary time" in this study [12] did not distinguish between sitting and standing. Another recent cross-sectional ISM study [11] found that replacing sedentary time with MVPA, but not with light-intensity activity, was linked to beneficial associations with a range of cardiometabolic markers. Such information has both clinical and public health relevance and may be valuable in developing more accurate and specific public health recommendations, clinical guidelines, and preventative or therapeutic interventions.
In this study, we examined the estimated replacement effects of SB and other time-dependent behaviours on all-cause mortality in a large population-based cohort of Australians aged 45 years and over using isotemporal substitution modelling.
The analyses are based on data from the 45 and Up Study [4, 13], in which participants completed a baseline questionnaire from January 2006 through December 2009. The 45 and Up Study sample is a large-scale (N = 267,119) prospective cohort of men and women aged 45 years or older living in the state of New South Wales, Australia. Participants were randomly sampled from the Medicare Australia (the national universal health care) database, which primarily includes all citizens and permanent residents of Australia. Eligible individuals were mailed all study materials and were asked to complete and mail the questionnaire and consent forms to the study center. The overall 45 & Up response rate to the mailed invitations was 17 · 9 % (95 % CI 17 · 8–18 · 1) and the final sample size corresponded to 11 % of the New South Wales population of the target age group [13]. The present project was approved by the New South Wales Population and Health Services Research Ethics Committee (reference No. 2010/05/234). Additional file 1: Figure S1 describes the selection of the analytic sample. From the 267,119 respondents, we excluded those with a missing or invalid date of recruitment (missing n = 12, implausible date n = 2184) and the resulting 265,923 participants were linked to the New South Wales Registry of Births, Deaths, and Marriages (RBDM) database. We then excluded one RBDM death record that had no matching record and any linked records where the date of death occurred before recruitment (n = 20), to form an initial dataset (n = 264,903). During the data cleaning process, 63,774 participants with or implausible exposure or covariate values were excluded, leaving a core analytic sample of 201,129 participants.
Exposure variables
All study variables were assessed through a self-administered questionnaire. Information on the main exposures was based on self-reported data from the 45 and Up Study baseline questionnaire (available at https://www.saxinstitute.org.au/our-work/45-up-study/questionnaires/). The sitting, screen time (watching television or using a computer), standing and sleeping variables were assessed with the question "About how many hours in each 24-hour day do you usually spend doing the following?" followed by open entry boxes where participants entered their responses. This type of question is similar to the validated sitting questions of the International Physical Activity Questionnaire [14]. Total weekly time for walking and non-walking MVPA (continuously for at least 10 min) was assessed using the Active Australia Survey questions: "If you add up all the time you spent doing each activity last week, how much time did you spend altogether doing each type of activity?" These questions have been shown to have acceptable reliability (coefficients for walking, moderate and vigorous activity frequency, and time ranging from 0.56 to 0.64) and validity (correlation of duration of self-reported activity with accelerometer data was 0.52) [15, 16] and have been tested both in population and individual-level intervention contexts [17].
Potential confounders
The choice of potential confounders was similar to previous 45 & Up SB analyses [4] and included sex, age (5-year bands), educational level (university degree/post high school, high school, or less), marital status (single/married/cohabiting/widowed/divorced/separated), urban/rural residence, body mass index (calculated as self-reported weight/self-reported height squared [18]), smoking status (current/previous/never smoker), self-rated health (poor/fair/good/very good/excellent), help with daily tasks because of long-term illness or disability, psychological distress (K10 scale, a 10-item questionnaire intended to yield a global measure of anxiety and depressive symptoms [19]), and previous physician diagnoses of CVD, diabetes mellitus, or various types of cancer.
Outcome ascertainment
All-cause mortality was ascertained from the New South Wales RBDM from February 1, 2006 through June 14, 2012. Mortality data were linked to the baseline data from the 45 and Up Study by the Centre for Health Record Linkage (New South Wales, Australia) using linkage methods and quality checks that have been described previously [4].
The Supplementary Methods in Additional file 1 presents the unabridged version of data cleaning, handling, and statistical testing procedures. In summary, we used multiple imputation (SAS 9.3, Proc MI) and the Expectation-Maximisation algorithm (30 imputations) [20] to impute missing data for the 13,053 participants who had at least one of the time-dependant behavioural variables. Due to the ISM requirement for an approximately linear association between each exposure and the outcome, moderate and vigorous PA were combined into MVPA. For the same reason [7, 9] we treated sleeping as a piecewise variable with a breakpoint at 7 h (≤7 h/day and > 7 h/day), where each of the two sleeping variables had an approximately linear association with mortality. All exposure variables were converted into hours per day.
The association between each activity class and risk of death was analyzed using Cox proportional hazards regression models. Survival time (in weeks) was measured as the time from baseline to death or the censor point, and each exposure was modelled in one-hour intervals. Before ISM analyses, all relevant assumptions were tested. Interactions of sex and age were not statistically significant (all p > 0 · 10); therefore results are presented for the entire sample. We initially estimated the partition model, which assumes that each activity class is added rather than substituted with another activity to create a day potentially longer than 24 h [10]. The partition model estimates each component of time while keeping others constant (sleep_a and sleep_b correspond to daily sleep durations of ≤ 7 h and > 7 h, respectively):
$$ \begin{array}{l} \log {h}_i(t)={b}_0(t)+{b}_1\left(\mathrm{sleep}\_\mathrm{a}\right)+{b}_2\left(\mathrm{sleep}\_\mathrm{b}\right)+{b}_3\left(\mathrm{sitting}\right)+{b}_4\left(\mathrm{standing}\right)\\ {}+{b}_5\left(\mathrm{walking}\right)+{b}_6\left(\mathrm{MVPA}\right)+{b}_7\left(\mathrm{screntime}\right)+\left(\mathrm{covariates}\right)\end{array} $$
ISM assumes that any given time spent in one behaviour will lead to an isotemporal displacement of another activity class while total time is kept constant [10]. For example, to estimate the effect of substituting one hour of standing for screen-time, screen-time is removed from a model adjusted for total time as follows:
$$ \begin{array}{l} \log {h}_i(t)={b}_0(t)+{b}_1\left(\mathrm{sleep}\_\mathrm{a}\right)+{b}_2\left(\mathrm{sleep}\_\mathrm{b}\right)+{b}_3\left(\mathrm{sitting}\right)+{b}_4\left(\mathrm{standing}\right)\\ {}+{b}_5\left(\mathrm{walking}\right)+{b}_6\left(\mathrm{MVPA}\right)+{b}_7\left(\mathrm{Total}\right)+\left(\mathrm{covariates}\right)\end{array} $$
In the above example, the resulting hazard ratio (HR) for standing will indicate whether replacing screen time with standing is beneficially (if HR < 1 · 00) or detrimentally (if HR > 1 · 00) associated with all-cause mortality. We repeated a series of sensitivity analyses to examined the robustness of our results: a) using the unimputed dataset only; b) excluding those with pre-existing CVD (heart disease, stroke, and thrombosis), or diabetes, or cancer at baseline; c) excluding deaths in the first 24 months (n = 4714); d) excluding both b and c above, e) alternative manipulations of the SB variables are performed and shown in the Supplementary Appendix; f) stratifying by sleep time; g) change the cut-off of the piecewise sleep variable to 8 h/day. For comparability with a recent study that examined the associations between standing and all-cause mortality, we repeated the partition models of the standing time exposure stratified by MVPA level using the 150 MVPA minutes/week as the cut off point for adherence to the World Health Organisation physical activity recommendations. Analyses were performed using SAS Software version 9 · 3 (SAS Institute Inc., Cary, NC). The reporting of this study conforms to the STROBE statement.
During a mean-follow up time of 4.22 years (849,369 total person-years), 7,460 deaths were recorded (3 · 7 % of the sample). Table 1 presents participant characteristics by daily sitting time. Additional file 1: Table S1 presents the descriptive statistics of all exposure variables. Table 2 presents the partition and ISM results for the full imputed dataset. There were beneficial associations for replacing sitting with sleeping in those sleeping for ≤7 h/day (HR per hour increase: 0.94, 95 % CI: 0.90–0.98), with standing (0.95, 0.94–0.96), with walking (0.86, 0.81–0.90), and with MVPA (0.88, 0.85–0.90); and for replacing screen time with sleeping in those sleeping for ≤ 7 h/day, (0.95, 0.91 to 0.99), standing (0.97, 0.95–0.98), walking (0.87, 0.82–0.92) and MVPA (0.89, 0.86–0.91). Sleeping for >7 h (1.08, 1.05 -1.10), screen time (1.02, 1.01- 1.03), and sitting (1.03, 1.02–1.04) were associated with increased mortality risk. Standing (0.97, 0.96–0.99), walking (0.83, 0.79–0.88), and MVPA 0.87 (0.85–0.90) were all associated with decreased risk. Among those who reported sleeping >7 h/day, replacing one hour of sleeping with one hour of any other activity class was associated with risk reductions (Table 2). Replacing walking or MVPA with any other behaviour was associated with increased risk. All these replacement effects are also presented graphically in Additional file 1: Figure S2. All above observations were not materially different when we repeated analyses to the unimputed dataset (Additional file 1: Table S3). When analyses were restricted to healthy at baseline only participants (n = 143,680; 2690 deaths) associations in the partition models were somewhat attenuated, although the ISM results for screen time, sitting, standing, walking and MVPA followed broadly the same pattern as in the full sample (Table 3) and the same applied to the remaining 57,449 participants (4770 deaths) who had diabetes, or CVD or a history of cancer at baseline (Additional file 1: Table S4). Excluding those 4714 participants who died during the 24 months of follow up did not materially affect the results (Table 4). Restricting analyses to those who were healthy at baseline and excluding events occurring the first 24 months (n = 142,768; 1,778 deaths) resulted in broadly similar results, although associations were generally attenuated and confidence intervals were broader due to the dilution of the events rate (Additional file 1: Table S5). Different manipulations of the sitting and screen time variables (Additional file 1: Table S6-S7) did not materially change the above results. Neither stratifying analyses by sleeping time level nor changing the piecewise sleeping variable cutoff to 8 h/day changed results materially (Additional file 1: Tables S8-S10). The standing time partition models stratified by MVPA suggested that the beneficial effect of standing on mortality was present in both those who met (per-hour HR: 0.98, 0.97–0 · 99) and did not meet (0.95, 0.94–0.97) the World Health Organization physical activity recommendations.
Table 1 Characteristics (n, % (of sitting time group)) of participants by amount of sitting time per day
Table 2 Independenta and replacementb effects of sleeping, screen time, sitting, walking and non-walking moderate to vigorous physical activity on all-cause mortality risk. Imputed datac (n = 201,129; 7,460 deaths)
Table 3 Independenta and isotemporal substitutionb effects of sleeping, screen time, sitting, walking and non-walking moderate to vigorous physical activity on all-cause mortality risk. Participants who were considered healthy at baseline, defined as those who were never diagnosed with cardiovascular disease, diabetes, or cancer, (Imputed datac, n = 143,680; 2690 deaths)
Table 4 Independenta and isotemporal substitutionb effects of sleeping, screen time, sitting, walking and non-walking moderate to vigorous physical activity on all-cause mortality risk excluding deaths occurring the first 24 months of follow up. Imputed datac (n = 198,383; 4714 deaths)
This is the first large-scale epidemiological study examining the replacement effects of sedentary time and other time-dependent behaviours on all-cause mortality using statistical modelling. Both screen time and total sitting time were independently associated with increased mortality risk, while standing, walking, and MVPA were associated with decreased mortality risk. We found beneficial associations for replacing sedentary time with equal amounts of sleeping (in participants who sleep < 7 h/day), standing, walking, and MVPA. Our results were robust to multiple measures that we took to minimize the chances of reverse causality.
Public health physical activity recommendations [21] are largely based on epidemiological evidence of non-substitution models. Our analyses suggest that the ISM [10] offers richer and more specific information than previous "static" methods. Because there seems to be a variation in the associations of most behaviours with mortality depending on the displaced activity, existing epidemiological evidence may underestimate the benefits of physical activity and the harms of sedentary behaviour. For example, the partition models showed that each hour/day of sitting is linked to an increased mortality risk of 3 % (2–4 %), but once the displaced activity is taken into account, this increased to 5 % (4–6 %) and 17 % (11–23 %) for displacing equal amounts of standing and MVPA, respectively.
For interventional targets, the most common scenario is that programs seek to reduce sitting and screen time and promote MVPA [22]. Interventions aimed at specifically replacing sitting with standing are less common and are mostly restricted to the occupational office-based environment [23]. Our results suggest that standing time also may be an additional promising interventional target. Given the evident difficulties in promoting MVPA at a population level, this approach might be promising for certain situations and populations/clinical groups where physical activity messages are difficult to disseminate. The only epidemiological study, to our knowledge, that specifically examined the associations of standing time with mortality [24] used a non-substitutional approach and found that the proportion of daily time spent on standing is associated with all-cause and CVD mortality in nearly inverse dose–response manner among the physically inactive participants only [24]. Our partition models suggested an independent beneficial effect of standing on mortality (3 % decrease in risk per hour of standing in the whole sample), and this association was present in both those who met and did not meet the physical activity recommendations. These beneficial associations of standing with mortality were even more substantial when standing time displaced SB. The cardiometabolic properties of standing have not been studied extensively, perhaps due to the absence of an established mechanism through which it may benefit health. A rodent model-based hypothesis put forward over a decade ago suggested that prolonged sitting causes dramatic reductions of lipoprotein lipase activity compared to standing up or ambulating [25], although human studies that manipulated experimentally sitting refute this hypothesis as there appears to be no benefit from replacing sitting with standing on blood lipid variables [26–28]. Instead, replacing sitting with standing [26, 29] or light-intensity walking [30] may improve postprandial blood glucose responses and energy expenditure [29].
Assuming that our findings represent causal effects, substituting sitting with standing or other light-intensity PA may have considerable public health and clinical care implications, e.g. replacing three hours of sitting per day with standing may be associated with a cumulative decrease of 12–18 % in all-cause mortality risk. Working hours account for over half of total waking time [31] and workers in many professions spend on average more than 70 % of their work time sitting [32]. Unlike promoting physical activity, substitution of desk-based sitting for standing is a relatively straightforward intervention that has no additional time and location requirements. Objective British [33] and US data [34] indicate that on average people aged 70 years and older spent approximately 65–80 % of their waking time being sedentary. Substituting sedentary time with standing and light-intensity activity in this challenging age group may be promising.
The strengths of our study are the large population-based sample, the availability of data on a broad range of PA-related behaviours and sleep that collectively account for the majority of the 24-hour daily cycle, the novel statistical approach that allows examination of replacement effects, and the multiple measures we took to reduce the chances of reverse causation and confounding. While the relatively low 45 and Up response rate (19 % [13]) may be seen as a threat to the generalizability of our findings, it is unlikely that our results were materially compromised as relative risks based on internal comparisons are not dependent on the representativeness of the cohort. A previous analysis that compared a broad range of exposure-outcome associations in the 45 and Up cohort with another New South Wales population study with much higher response rate (~60 %) found that the relative risk estimates in the two studies were almost identical in both magnitude and direction [35]. A limitation of our work was that the exposures were only measured at baseline and our data did not reflect complete time-use as we lacked information on other light-intensity physical activity (non-walking/non-standing activities <1 · 5 MET). Our measure of screen time did not differentiate between recreational and occupational screen time, as the health effects of TV and other recreational screen [5, 6] may be different to those of e.g. occupational computer use. Our study is based on statistical modeling and not on actual replacements of one activity with another. Our time-dependent variables were all self-reported although many have been previously validated and our results further support their convergent validity. Standing, sitting, and sleeping were measured on a different scale to walking and MVPA and this may have affected comparability of the responses to some extent. Physical activity may have been over-reported and sitting time under-reported due to social desirability bias. There was no information on walking pace so it is not possible to make inferences about its intensity.
Sedentary time was associated with increased risk and physical activity with decreased risk for all-cause mortality in adults aged ≥45 years. The magnitude of these effects varied broadly according to the behaviour displaced and there was evidence of an activity intensity-graded response. Isotemporal substitution modelling offers richer and more specific insights into the associations of each time-dependent activity class with mortality compared to traditional non-substitutional approaches.
ISM:
Isotemporal substitution model
MVPA:
Moderate to vigorous physical activity
RBDM:
Registry of Births, Deaths, and Marriages
SB:
Sedentary behaviour
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This research was completed using data collected through the 45 and Up Study (www.saxinstitute.org.au). The 45 and Up Study is managed by the Sax Institute in collaboration with major partner Cancer Council NSW; and partners: the National Heart Foundation of Australia (NSW Division); NSW Ministry of Health; beyondblue; Ageing, Disability and Home Care, Department of Family and Community Services; the Australian Red Cross Blood Service; and UnitingCare Ageing. We thank the many thousands of people participating in the 45 and Up Study. These analyses were supported by Australian National Health and Medical Research Council Program Grant #301200. This research arises from ES' personal Career Development Fellowship funded by the National Institute for Health Research (UK). The views expressed are those of the authors and not necessarily of the funding bodies.
These analyses were supported by Australian National Health and Medical Research Council Program Grant #301200 and by a UK National Institute for Health Research Career Development Fellowship (ES).
Charles Perkins Centre, University of Sydney, Sydney, Australia
Emmanuel Stamatakis, Josephine Chau & Adrian Bauman
Exercise and Sport Sciences, Faculty of Health Sciences, University of Sydney, Sydney, Australia
Emmanuel Stamatakis
UCL-PARG (Physical Activity Research Group), Department of Epidemiology and Public Health, University College London, London, UK
Emmanuel Stamatakis & Mark Hamer
Prevention Research Collaboration, Sydney School of Public Health, University of Sydney, Sydney, Australia
Kris Rogers, Josephine Chau & Adrian Bauman
Graduate School of Public Health, San Diego State University, San Diego, CA, USA
National Cancer Institute, Behavioral Research Program, Division of Cancer Control and Population Sciences, Bethesda, MD, USA
David Berrigan
National Centre for Sport & Exercise Medicine, Loughborough University, Loughborough, UK
Mark Hamer
Kris Rogers
Josephine Chau
Correspondence to Emmanuel Stamatakis.
All authors have contributed sufficiently to warrant authorship: ES conceived the idea, designed the analysis, drafted most of the manuscript, and revised the manuscript several times; KR prepared the dataset, did the statistical analysis, and drafted most of the statistical methodology sections; DD and JC drafted parts of the manuscript; DB and MH contributed to the idea and provided guidance on several analytical and study design issues; AB acquired the data and contributed to the idea and design; all authors revised critically the manuscript several times and approved the final version. KR had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. All authors read and approved the final manuscript.
Contents: Supplementary Methods; Figure S1; Figure S2; Table S1; Table S2; Table S3; Table S4; Table S5; Table S6; Table S7;Table S8;Table S9; Table S10. (DOCX 230 kb)
Stamatakis, E., Rogers, K., Ding, D. et al. All-cause mortality effects of replacing sedentary time with physical activity and sleeping using an isotemporal substitution model: a prospective study of 201,129 mid-aged and older adults. Int J Behav Nutr Phys Act 12, 121 (2015). https://doi.org/10.1186/s12966-015-0280-7
Isotemporal substitution
Population cohort | CommonCrawl |
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01.12.2020 | Original Article | Ausgabe 1/2020 Open Access
Dynamic Analysis and Performance Verification of a Novel Hip Prosthetic Mechanism
Chinese Journal of Mechanical Engineering > Ausgabe 1/2020
Majun Song, Sheng Guo, Xiangyang Wang, Haibo Qu
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The main purpose of prosthetic mechanisms is to restore the functional motion of amputees in daily life, which can effectively compensate for the lost limbs of the amputee. With the rise in living standards, traditional prostheses that serve only as auxiliary support can no longer meet the requirements for movement of amputees. With the widespread application of robots in recent years, robotic prostheses [ 1 , 2 ] have been extensively studied, especially for prosthetic mechanisms. Nelson et al. [ 3 ] designed a prosthesis, named Helix 3D, for amputees with hip dissociation, which is a serial mechanism that can only realize a single degree-of-freedom (DOF). Maja et al. [ 4 ] synthesized a hip prosthesis and conducted its periodic gait experiment by using wireless sensors. Hanz et al. [ 5 ] designed a hip prosthetic mechanism that has two DOFs in the sagittal plane. The control strategy was built and implemented to test its motion performance. However, the stiffness of these prosthetic mechanisms was comparatively low owing to the open-loop structure, and the limited DOF could not meet the movement requirements of the human hip. In order to achieve the motion characteristics of the human hip and improve its stiffness, Gu [ 6 ] designed a multi-DOF humanoid robot with a serial structure similar to the lower limbs of the human body, although it is rather complicated and heavy due to its excessive components. Hence, it is essential to design a hip prosthesis with multi-DOFs, good kinematics, and dynamic performance.
To achieve multi-DOFs and increase structural stiffness, parallel mechanisms have been applied in the design of the hip prosthesis. Its performance analyses have also been carried out by the designers. Cheng et al. [ 7 ] designed a 3-SPS/PS hip parallel mechanism with its singularity analyzed using Grassmann line geometry. Due to the existence of the spherical joints, its stability would become poor when loaded. Sellaouti et al. [ 8 ] designed a bipedal walking robot with a 3-DOF parallel mechanism, which was a planar mechanism. The stability of the robot worsens during walking. Wang et al. [ 9 ] designed a 3R1T bionic parallel mechanism at the hip, and the kinematics and statics were analyzed. However, the presence of more actuators made it unfavorable for analysis of the dynamics and implementation of a control strategy for the mechanism.
In this paper, motion characteristics of the human hip were tested and analyzed via motion capture device, and a novel 2-DOF purely rotational parallel mechanism with a passive limb was synthesized by applying screw theory. The proposed mechanism acted as a hip prosthesis to fulfill the movement function of the human hip. The introduced passive limb not only enhanced the stiffness and stability of the prosthetic mechanism, but also reduced the control difficulty due to the employment of fewer motors. This paper is organized as follows: Configuration syntheses of active and passive limbs are illustrated in Section 2 based on the proposed design principles. From among the results, a parallel mechanism was selected as our hip prosthesis. In Section 3, inverse kinematics of the proposed mechanism are derived. The method of building the fully Jacobian matrix is presented in Section 4. In Section 5, its workspace, stiffness, bearing capacity, and dexterity are analyzed. The dynamic modeling process is shown in Section 6. The numerical and theoretical results are compared in Section 7. Finally, conclusions are drawn in Section 8.
2 2-UPR/URR Hip Parallel Mechanism
2.1 Design Demand Analysis Based on Kinematics of the Human Body
Based on the human musculoskeletal anatomy and human rehabilitation kinematics [ 10 ], the motion characteristics of the human hip, which has three spatial rotational DOFs around the horizontal, sagittal, and vertical axes, were analyzed as shown in Figure 1.
Human anatomical planes
However, the main locomotor expression [ 11 ] of a hip in daily life is characterized by flexion, extension, abduction, and adduction, as shown in Figure 2(a) and 2(b); intorsion and extorsion can be neglected. Therefore, a hip prosthetic mechanism is designed to assist the amputee to restore the lost motion functions of flexion, extension, abduction, and adduction of the hip.
Motion characteristics of a human hip joint: a flexion and extension; b abduction and adduction
MotionAnalysis was used to test the gait data of inverse kinematics of the subjects, whose height and mass were 1.76 m and 80 kg, respectively, as shown in Figure 3. The ultimate rotational angles of the hip in multiple gait modes were tested, as shown in Table 1.
Testing platform for human gait
Ultimate angles of the hip in multi-modes gait
No-loading
20 kg-loading
− 21.5°
− 8.0°
Parallel mechanisms have advantages such as multiple DOFs and high bearing capacity. The designed hip prosthetic parallel mechanism should have a rotational ability around the lateral and sagittal axes, where the minimum range of rotational angles should be, respectively, − 22° to approximately 33° and − 7° to approximately 6°. Meanwhile, it should also have sufficient stiffness and bearing capacity, which can give it good stability when supporting the human body. Although the maximum angles of intorsion and extorsion of the hip can be up to 45°, this is only in a special state. However, the motion performances of the intorsion and extorsion are not active during the normal gait, as mentioned above; thus, the motion performances of intorsion and extorsion were not considered in the design of the hip prosthetic mechanism in this paper.
2.2 Configuration Synthesis
As shown in Section 2.1, to design a hip prosthetic parallel mechanism, five design guidelines should be followed:
In order to simplify the mechanism and ensure its effective motion characteristics, the designed hip prosthetic parallel mechanism consists of a passive limb and two actuating limbs.
Since the thigh is similar to a fixed-length binary rod, the function of the passive limb is to restrict its movement along the limb, i.e., there is no prismatic pair in the passive limb. In addition, the length of the passive limb is equal to the size of the human thigh. Moreover, spherical and universal joints cannot be the middle pair of each limb.
Configurations of the actuating limbs are the same, and they are distributed symmetrically to the plane where the passive limb is located.
To better mimic the contraction and extension of related muscles, the middle kinematic pair in each actuating limb sets as the prismatic pair.
To achieve the rotational angles of the hip, two revolute pairs that parallel to lateral and sagittal axes should be connected with the fixed base.
Based on this, the connectivity of the mechanism can be solved by the modified enumeration methodology:
$$\sum\limits_{k = 1}^{N} {C_{k} } = F_{D} + dL + \eta ,$$
where C k is the connectivity of the kth limb, F D is the DOFs of the mechanism, d is the mechanism order, L is the number of closed-loop mechanisms, and η is the number of redundant DOFs of the mechanism. Then, in terms of Eq. ( 1), the classifications of connectivity of each limb are: ( C 1, C 2, C 3) = (6, 3, 3) = (4, 4, 4) = (2, 5, 5).
It was assumed that the available types of joints were R (Revolute), P (Prismatic), U (Universal), C (Cylinder), and S (Spherical). Based on connectivity, the synthesis of the actuating and passive limbs are shown in Tables 2, 3.
Configuration synthesis of the passive limb
Motion characteristics
Configurations of passive limb
(RR) ⊥,U
S, (RR) ⊥R, UR, (RR) ⊥U, SR, UU, (RR)⊥S, SU
UR ⊥R, ((RR) ⊥R) ⊥U, URU, UUR, US, SRR
(URR) ⊥R, ((RR) ⊥R) ⊥R, (RR)⊥RRR
Configuration synthesis of the actuating limb
Configuration of actuating limbs
2R1T
(RR) ⊥P, UP
SP, (RR) ⊥C, UC, SC
(RR) ⊥R ⊥P, (RR) ⊥PR, UR ⊥P, UPR, SR *P, UR ⊥C, SPR, UPU,UCR, (RC) ⊥U, SP *U,SC *R, (RR) ⊥C ⊥U, (RR) ⊥P ⊥S, UPS, UC ⊥U
(RR) ⊥R ⊥R ⊥P, UR ⊥R ⊥P, (RR) ⊥R ⊥P *R, (RR) ⊥R ⊥C, UR ⊥P *R, (RR) ⊥P *R ⊥R, (RC) ⊥R ⊥R, (RR) ⊥P *U, (RR) ⊥C ⊥R, UP *R ⊥R, UR ⊥P *U, SP *R *R, (RR) ⊥P ⊥R ⊥U, UP ⊥R ⊥U
In Tables 2 and 3, (·) ⊥ indicates that the axes of two kinematic pairs in parentheses were orthogonal, and they were installed together on the fixed base and served as a joint. ⊥ Denotes that the axis of kinematic pair was orthogonal with the plane on which the fixed base was located. * Denotes the axis of kinematic pairs in parentheses along the limb.
According to the configurations of the limbs, a parallel mechanism was designed that met the motion characteristics of the human hip following these two steps:
The terminals of the passive or actuating limb had least two spatial rotational DOFs.
A parallel mechanism with two spatial rotational DOFs was synthesized based on the mathematical intersection operation [ 12 ].
Generally, for good stability and better motion characteristics, each limb of the parallel mechanism was consistently arranged with three joints. By analysis and comparison for the configuration of limbs under design guidelines, the configuration of UR ⊥R was selected as the passive limb, and the configuration of U PR was selected as the actuating limb. Therefore, the hip parallel prosthetic mechanism designed in this paper was called a 2-U PR/URR parallel mechanism, as shown in Figure 4, which had the motions of flexion, extension, abduction, and adduction.
Hip prosthetic parallel mechanism: a 2-U PR/URR parallel mechanism; b amputee with hip prosthetic mechanism
The schematic diagram is shown in Figure 5. The prosthesis can rotate around both the X-axis and the Y-axis.
Diagram of 2-U PR/URR parallel mechanism
Our prosthetic consisted of a fixed base, a moving platform, a passive limb, and two actuating limbs, and the driving limbs were arranged symmetrically along the X- Z plane. The universal and revolute pairs were connected to the fixed base and the moving platform, respectively, in each limb, and the axis of revolute was parallel to the outboard axis of the universal one. The upper and lower rods in the passive limb were connected by a revolute, and the axis was collinear with the direction of the passive limb. The cylinder and piston in the actuating limb were connected by a prismatic, whose moving direction was along the direction of the actuating limb, and it also served as an actuator.
In Figure 5, the fixed base ∆ A 1 A 2 A 3 and the moving platform ∆ B 1 B 2 B 3 are equilateral triangles and their side lengths are 2 a and 2 b, respectively; the origin O of the fixed frame O- XYZ is located at the center of A 2 A 3, the direction of X and Y axis is along OA 1 and OA 2, and the direction of Z is obtained by the right-hand rule. The origin o of the moving frame o- xyz is located at the center of B 2 B 3, the direction of x- and y-axes are along oB 1 and oB 2, respectively, and the direction of z is obtained by the right-hand rule.
3 Inverse Kinematic Analysis
For the parallel mechanism, the final location was obtained by a rotation of α about the x-axis, followed by a second rotation of β about the displaced y′-axis. The resulting rotation matrix was derived based on a Euler angle representation [ 13 ], as follows,
$$\varvec{R}_{B}^{A} = \left[ {\begin{array}{*{20}c} {\cos \beta } & 0 & {\sin \beta } \\ {\sin \alpha \sin \beta } & {\cos \alpha } & { - \cos \beta \sin \alpha } \\ { - \cos \alpha \sin \beta } & {\sin \alpha } & {\cos \alpha \cos \beta } \\ \end{array} } \right],$$
assuming that the position vector of point o expressed in the fixed frame is p = [ x p, y p, z p]. The position vector of point A i is a 1, a 2, and a 3, which can be obtained easily in Figure 5. B i with respect to the fixed frame is given by:
$$\left[ {\begin{array}{*{20}c} {\varvec{b}_{1} } \\ {\varvec{b}_{2} } \\ {\varvec{b}_{3} } \\ \end{array} } \right]{ = }\left[ {\begin{array}{*{20}c} {x_{p} + b\text{c} \beta } & {y_{p} + bs\alpha s\beta } & {z_{p} - b\text{c} \alpha s\beta } \\ {x_{p} } & {{\kern 1pt} y_{p} + b\text{c} \alpha } & {z_{p} + b\text{s} \alpha } \\ {x_{p} } & {{\kern 1pt} y_{p} - b\text{c} \alpha } & {z_{p} - b\text{s} \alpha } \\ \end{array} } \right].$$
A vector-loop equation of a Limb i can then be written as:
$${\varvec{a}}_{i} + l_{i} {\varvec{s}}_{i} = {\varvec{p}} + {\varvec{b}}_{i} ,$$
where s i is a unit vector pointing from A i to B i, l i is the length of the ith limb, and is a constant when i = 1. From Eqs. ( 3) and ( 4), the position vector of point o is solved:
$${\varvec{p}} = \left[ {\begin{array}{*{20}c} {x_{p} } \\ {y_{p} } \\ {z_{p} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {l_{1} \cos\alpha \sin\beta + \sqrt 3 a - b\cos \beta } \\ {l_{1} \sin\alpha - b\sin\alpha \sin\beta } \\ {l_{1} \cos\alpha \cos \beta + b\cos \alpha \sin\beta } \\ \end{array} } \right]^{\text{T}} .$$
Therefore, the lengths l i ( i = 2, 3) of limbs 2 and 3 are solved in terms of the value of α and β as follows:
$$l_{i} = \sqrt {(b_{ix} - a_{ix} )^{2} + (b_{iy} - a_{iy} )^{2} + (b_{iz} - a_{iz} )^{2} } .$$
4 Jacobian Matrix
Based on the screw theory, the 6 × 6 fully Jacobian matrix [ 14 – 17 ] of the 2-U PR/URR parallel mechanism was constructed, which consisted of the constraint and kinematic Jacobian matrices.
4.1 Jacobian Matrix of UPR Limb
We assumed that the actuating limb U PR was an open-loop limb connecting the moving platform to the fixed base. The unit screw of the jth joint in the ith limb was expressed as $ j,i, as shown in Figure 6. In order to facilitate the analysis, the origin of the instantaneous frame was defined at point o, and its x s, y s, and z s axes were parallel to the X, Y, and Z axes, respectively.
Actuating limb U PR
Letting $ d = [ ω n, v o] be the instantaneous kinematic screw of the moving platform, which can be expressed as a linear combination of the kinematic screws in the actuating limb, we have:
In Eq. ( 7), \(\dot{\theta }_{j,i}\) is the rotational angular velocity of the jth ( j = 1‒4) joint in the ith ( i = 2,3) limb, and \(\dot{q}_{i}\) is the linear velocity of prismatic in the ith limb.
4.1.1 Constraint Jacobian Matrix
There were three passive joints and an actuating joint in the actuating limb, and its joint screws $ j,i ( j = 1–4, i = 2, 3) were represented thusly:
The actuating limb was regarded as a spherical-revolute dyad that formed a four-screw system [ 18 ], so its constraint Jacobian matrix was composed of two constraint screws, which were reciprocal with the four-screw system:
Taking the dot-product of both sides of Eq. ( 7) with $ r1,i and $ r2,i, the constraint Jacobian matrix of the two actuating limbs were obtained:
$${\varvec{J}}_{c2} = \left[ {\begin{array}{*{20}c} {{\varvec{J}}_{c21} } \\ {{\varvec{J}}_{c22} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {({}^{B}{\varvec{b}}_{2} \times {\varvec{s}}_{4,2} )^{\text{T}} + {\varvec{s}}_{2,2}^{\text{T}} } & {{\varvec{s}}_{4,2}^{\text{T}} } \\ {({}^{B}{\varvec{b}}_{3} \times {\varvec{s}}_{4,3} )^{\text{T}} + {\varvec{s}}_{2,3}^{\text{T}} } & {{\varvec{s}}_{4,3}^{\text{T}} } \\ \end{array} } \right],$$
where the row vector indicates that a constraint force and a constraint couple were imposed on the moving platform by each actuating limb.
4.1.2 Kinematic Jacobian Matrix
The actuating limbs became a universal-revolute dyad that formed a three-screw system [ 18 ] when the prismatic joint was locked. Compared with Eq. ( 8), a constraint screw was added:
The kinematic Jacobian matrices of the two actuating limbs were obtained by :
$${\varvec{J}}_{k} = \left[ {\begin{array}{*{20}c} {{\varvec{J}}_{k1} } \\ {{\varvec{J}}_{k2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {({}^{B}{\varvec{b}}_{2} \times {\varvec{s}}_{3,2} )^{\text{T}} } & {{\varvec{s}}_{3,2}^{\text{T}} } \\ {({}^{B}{\varvec{b}}_{3} \times {\varvec{s}}_{3,3} )^{\text{T}} } & {{\varvec{s}}_{3,3}^{\text{T}} } \\ \end{array} } \right].$$
4.2 Jacobian Matrix of the URR Limb
The unit screw of the jth joint was expressed as $ j,1 for the passive limb URR, as shown in Figure 7.
Passive limb URR
The instantaneous kinematic characteristics of the moving platform can be expressed as a linear combination of the kinematic screws in the passive limb thusly:
In Eq. ( 13), \(\dot{\theta }_{j,1}\) is the rotational angular velocity of the jth joint in Limb 1.
Since the passive limb had no actuator, it only contained a constraint Jacobian matrix, which acted as the constraint on the moving platform. The constraint Jacobian matrix of the passive limb could be obtained based on and :
$${\varvec{J}}_{c1} = \left[ {\begin{array}{*{20}c} {{\varvec{J}}_{c11} } \\ {{\varvec{J}}_{c12} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {({}^{B}{\varvec{b}}_{1} \times {\varvec{s}}_{1,1} )^{\text{T}} } & {{\varvec{s}}_{1,1} } \\ {({}^{B}{\varvec{b}}_{1} \times {\varvec{s}}_{3,1} )^{\text{T}} } & {{\varvec{s}}_{3,1} } \\ \end{array} } \right].$$
In Eq. ( 14), the row vector indicates the constraint force on the moving platform imposed by the passive limb.
4.3 Fully Jacobian Matrix
According to the constraint Jacobian matrix and the kinematic Jacobian matrix, the fully Jacobian matrix of the 2-U PR/URR parallel mechanism could be obtained:
$${\varvec{J}} = \left[ {\begin{array}{*{20}c} {{\varvec{J}}_{c11} } \\ {{\varvec{J}}_{c12} } \\ {{\varvec{J}}_{c21} } \\ {{\varvec{J}}_{c22} } \\ {{\varvec{J}}_{k1} } \\ {{\varvec{J}}_{k2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {({}^{B}{\varvec{b}}_{1} \times {\varvec{s}}_{1,1} )^{\text{T}} } & {{\varvec{s}}_{1,1}^{\text{T}} } \\ {({}^{B}{\varvec{b}}_{1} \times {\varvec{s}}_{3,1} )^{\text{T}} } & {{\varvec{s}}_{3,1}^{\text{T}} } \\ {({}^{B}{\varvec{b}}_{2} \times {\varvec{s}}_{4,2} )^{\text{T}} + {\varvec{s}}_{2,2}^{\text{T}} } & {{\varvec{s}}_{4,2}^{\text{T}} } \\ {({}^{B}{\varvec{b}}_{3} \times {\varvec{s}}_{4,3} )^{\text{T}} + {\varvec{s}}_{2,3}^{\text{T}} } & {{\varvec{s}}_{4,3}^{\text{T}} } \\ {({}^{B}{\varvec{b}}_{2} \times {\varvec{s}}_{3,2} )^{\text{T}} } & {{\varvec{s}}_{3,2}^{\text{T}} } \\ {({}^{B}{\varvec{b}}_{3} \times {\varvec{s}}_{3,3} )^{\text{T}} } & {{\varvec{s}}_{3,3}^{\text{T}} } \\ \end{array} } \right].$$
5 Performance Analysis
5.1 Workspace
In order to guarantee the rotation of the 2-U PR/URR parallel mechanism around the X-axis and Y-axis, which could meet the rotational angle of the hip in multiple gait modes, its workspace [ 19 ] was analyzed.
The geometry size of the amputee's thigh was measured: l 1 = 310, a 1 = 105, a 2 = a 3 = 60, b 1 = 70, b 2 = b 3 = 40. Thus, the workspace of the mechanism was solved, as shown in Figure 8.
Workspace of the 2-U PR/URR Parallel Mechanism: a spatial location on the X- Y plane; b spatial location on the X- Z plane
Figure 8 shows that the mechanism had a large workspace and rotational angle around the X-axis and Y-axis, which exceeded the required ultimate rotational angle in multiple gait modes. The shape and position of the workspace were in accord with the movement of the human hip. When the moving platform rotated around the X-axis, α = [− 60°, 60°], the moving range of the moving platform along the Y-axis was [− 80 mm, 80 mm], as shown in Figure 8(a). When it rotated around the Y-axis, β = [− 90°, 35°], the moving range along the X-axis was [− 40 mm, 160 mm], as shown in Figure 8(b). This also indicated that the mechanism was feasible when applied as a hip prosthetic mechanism, and confirmed the correctness of the fully Jacobian matrix.
5.2 Stiffness
In order to verify the influence of a passive limb on the stiffness of the parallel mechanism, the stiffness was analyzed in this paper for the 2-U PR parallel mechanism and the 2-U PR/URR parallel mechanism with the passive limb.
Assuming that the moving platform was subjected to an external force F e = [ f e, n e] T, and the frictional force at the joints was ignored. The stiffness and structural deformation [ 20 ] could thus be solved when loaded:
$$K = k\varvec{J}^{\text{T}} J,$$
$$\varvec{F}_{\text{e}} = \varvec{K}\cdot\Delta \varvec{x}.$$
In Eq. (16), F e is the external force exerted on the moving platform, Δ x is the deformation displacement of the moving platform under external force, K is the stiffness matrix, and k is the equivalent spring constant assuming k = 1000 N/mm via the material properties in Ref. [ 21 ].
The complete gait cycle was composed of the single and double support stages, all of which bore the weight of the trunk. Therefore, the prosthetic mechanism required sufficient stiffness. The fully Jacobian matrices of the stiffness distribution of the 2-U PR parallel mechanism and 2-U PR/URR parallel mechanism were obtained, as shown in Figure 9.
Stiffness distribution of the mechanism in the single support phase: a 2-U PR parallel mechanism; b 2-U PR/ URR parallel mechanism
In Figure 9, the maximum stiffness of the 2-U PR parallel mechanism was about 2853 N/mm within α = − 25° to approximately 25°, β = 0° to approximately 30°, and the maximum stiffness of the 2-U PR/URR parallel mechanism was 4435 N/mm within α = − 25° to approximately 25° and β = − 90° to approximately − 60°. These results indicated that the stiffness of the parallel mechanism had been improved by introducing a passive limb. In addition, due to the presence of the passive limb, the stiffness of the 2-U PR/URR parallel mechanism was still higher at the ultimate position during flexion, as shown in Figure 9(b).
In order to verify the theoretical solution of the stiffness of the 2-U PR/URR parallel mechanism, static analysis was carried out based on the finite element method. Clearly, the hip prosthetic mechanism exerted a maximum weight when it was a supporting leg in the single support phase. The total gravity was a concentrated force, including a load of 20 kg, which was about 1000 N. It was exerted on the center of the moving platform, in a vertically downward direction. The deformation of the mechanism could then be solved by OptiStruct, as shown in Figure 10.
Deformation under external load (mm)
In Figure 10, the maximum deformation of the 2-U PR/ URR parallel mechanism was about 0.2315 mm, and the stiffness of the mechanism was 3455.72 N/mm, which could be calculated by Eq. ( 16b). The simulation result of the 2-U PR/URR parallel mechanism was about 979 N/mm less than the theoretical solution when α = β = 0°. The results indicated that the mechanism had sufficient stiffness to resist maximum external load in a single support phase without a large deformation.
5.3 Buckling Analysis
Since the axial length of the parallel prosthetic mechanism was much larger than its radial length, in order to predict the maximum loading capacity and avoid the instability and collapse caused by the external load, its buckling analysis was necessary. Generally, the buckling analysis of the mechanism is based on its eigenvalue problem:
$$({\varvec{K}} - \lambda_{\sigma } {\varvec{K}}_{\sigma } ){\varvec{\varphi }} = 0.$$
In Eq. ( 17), K is the structural stiffness matrix, λ σ is the scale multiplier for the external load, also called the eigenvalue, K σ is the geometric stiffness matrix based on the static analysis result, and φ is the eigenvector. The eigenvalue λ σ in Eq. ( 17) is solved by the Lanczos method [ 22 ]. Thus, the critical load that the mechanism can bear without instability is solved:
$$f_{\sigma } = \lambda_{\sigma } f_{e} .$$
According to Eq. ( 17) and the static analysis result, the critical load and large deformation trend of the mechanism were obtained in a critical unstable state, as shown in Figure 11.
Result of the buckling analysis of the mechanism (mm)
Furthermore, the scaling factor of the external load was λ σ ≈ 1.72, which was calculated from the buckling analysis, and the critical load for the mechanism instability was f σ = 1376 N in terms of Eq. ( 18). As the weight of the amputee patient, 80 kg, was less than the critical load f σ, the amputee could also bear an additional weight of 57.6 kg without instability while the 2-U PR/URR parallel mechanism served as the hip prosthetic mechanism.
5.4 Dexterity
In this paper, in terms of the fully Jacobian matrix of the 2-U PR/URR parallel mechanism expressed in Eq. ( 15), its condition number [ 23 – 25 ] index was analyzed as follows:
$$\kappa ({\varvec{J}}) = \frac{{\sigma_{\text{max} } }}{{\sigma_{\text{min} } }},$$
where \(\kappa ({\varvec{J}})\) is the condition number, σ max is the maximum eigenvalue of the Jacobian matrix, and σ min is the minimum eigenvalue of the inverse Jacobian matrix.
According to the geometry size in Section 5.1, the dexterity map of the 2-U PR/URR parallel mechanism is shown in Figure 12.
Dexterity based on the Jacobian condition number
Compared with the motion characteristics of the hip in Table 1, the rotational angles of the 2-U PR/URR parallel mechanism around the X-axis and Y-axis were larger. The condition number of the 2-U PR/URR parallel mechanism as 0 to approximately 1 when the moving platform was within the range − 50° to approximately 50° around the X-axis and − 90° to approximately 40° around the Y-axis, which indicated the kinematic dexterity of the mechanism was good.
In addition, the condition number of the mechanism was close to 1 when the moving platform rotated to 45° around the X-axis or within the range of 0° to approximately 40° around the Y-axis, which indicated the position was isotropic and had an optimal kinematic performance in this area. Since no singularity occurred, relative analysis was not necessary. Results showed that the 2-U PR/URR parallel mechanism was feasible and in accord with the design requirements for a hip prosthetic mechanism.
6 Dynamic Model of 2-UPR/URR Parallel Mechanism
6.1 Velocity and Acceleration of Actuating Limbs
To facilitate the inverse dynamic analysis [ 18 , 26 – 29 ], a local coordinate system o i -x i y i z i was built in the vertex A i of the fixed base, which represented the orientation of limb i with respect to the fixed frame, as shown in Figure 13. The unit vector expressed in the ith limb frame was i s i = [0, 0, 1] T, which represented the direction of the z i axis. The o i -x i y i z i could be defined as a rotation of \(\eta_{i}\) about the x i axis, followed by a second rotation of \(\chi_{i}\) around \(y_{i}^{\prime }\):
$${}^{A}{\varvec{R}}_{i} = \left[ {\begin{array}{*{20}c} {c\eta_{i} c\chi_{i} } & { - s\eta_{i} } & {c\eta_{i} s\chi_{i} } \\ {s\eta_{i} c\chi_{i} } & {c\eta_{i} } & {s\eta_{i} s\chi_{i} } \\ { - s\chi_{i} } & 0 & {c\chi_{i} } \\ \end{array} } \right],$$
where s is the represented sine function, and c is the represented cosine function.
Free-body diagram of the ith actuating limb
The ith ( i = 2, 3) driving limb of the mechanism consisted of a cylinder and a piston, as shown in Figure 13. Letting e i1 be the distance between A i and the center of mass of the ith cylinder, and letting e i2 be the distance between B i and the center of the mass of the ith piston, then the position vector of the centers of mass of the ith cylinder and piston could be represented thusly:
$$\left\{ {\begin{array}{*{20}l} {{\varvec{r}}_{i1} = {\varvec{a}}_{i} + e_{i1} {\varvec{s}}_{i} ,} \hfill \\ {{\varvec{r}}_{i2} = {\varvec{a}}_{i} + (l_{i} - e_{i2} ){\varvec{s}}_{i} .} \hfill \\ \end{array} } \right.$$
6.1.1 Velocity Analysis
Taking the time derivative of the right-hand side of Eq. ( 4), the velocity of the vertex B i, defined as v bi, is solved:
$${\varvec{v}}_{bi} = {\varvec{v}}_{p} + {\varvec{\omega}}_{p} \times {\varvec{b}}_{i} ,$$
where \({\varvec{v}}_{p}\) and \({\varvec{\omega}}_{p}\) are, respectively, the linear velocity and angular velocity of the moving platform expressed in the fixed frame, which can be solved by Eqs. ( 3) and ( 4).
Based on A R i and Eq. ( 22), the velocity of the vertex B i is i v bi = [ i v bix, i v biy, i v biz], which could be obtained thusly:
$${}^{i}{\varvec{\upnu}}_{bi} = {}^{i}{\varvec{R}}_{A} {\varvec{\upnu}}_{bi} = l_{i} {}^{i}{\varvec{\omega}}_{i} \times {}^{i}{\varvec{s}}_{i} + \dot{l}_{i} {}^{i}{\varvec{s}}_{i}$$
as the actuating limb cannot rotate about the z i axis. Dot-multiplying and cross-multiplying both sides of Eq. ( 23) by i s i, the linear and angular velocities of the ith actuating limb could be calculated:
$$\left\{ {\begin{array}{*{20}l} {\dot{l}_{i} = {}^{i}{\varvec{s}}_{i} {}^{i}{\varvec{\upupsilon}}_{bi} = {}^{i}v_{biz} ,} \hfill \\ {{}^{i}{\varvec{\omega}}_{i} = ({}^{i}{\varvec{s}}_{i} \times {}^{i}{\varvec{v}}_{bi} )/l_{i} = [ - {}^{i}v_{biy} , - {}^{i}v_{bix} ,0]^{\text{T}} /l_{i} .} \hfill \\ \end{array} } \right.$$
The velocities of the centers of mass of the ith cylinder and piston could be calculated by differentiating Eqs. ( 21) with respect to time and combining with Eq. ( 24):
$$\left\{ {\begin{array}{*{20}l} {{}^{i}{\varvec{v}}_{i1} = e_{i1} {}^{i}{\varvec{\omega}}_{i} \times {}^{i}{\varvec{s}}_{i} ,{\kern 1pt} } \hfill \\ {{}^{i}{\varvec{v}}_{i2} = (l_{i} - e_{i2} ){}^{i}{\varvec{\omega}}_{i} \times {}^{i}{\varvec{s}}_{i} + \dot{l}_{i} {}^{i}{\varvec{s}}_{i} .} \hfill \\ \end{array} } \right.$$
6.1.2 Acceleration Analysis
Based on the acceleration synthesis theorem [ 27 ], the acceleration of the vertex B i, i a bi, was found by differentiating Eq. ( 23) with respect to time:
$$\begin{aligned} {}^{i}{\varvec{a}}_{bi} & = {}^{i}{\dot{\varvec{v}}}_{bi} = \ddot{l}_{i} {}^{i}{\varvec{s}}_{i} + l_{i} {}^{i}{\dot{\varvec{\omega }}}_{i} \times {}^{i}{\varvec{s}}_{i} + \cdots \hfill \\ & \quad + l_{i} {}^{i}{\varvec{\omega}}_{i} \times ({}^{i}{\varvec{\omega}}_{i} \times {}^{i}{\varvec{s}}_{i} ) + 2\dot{l}_{i} {}^{i}{\varvec{\omega}}_{i} \times {}^{i}{\varvec{s}}_{i} \hfill \\ \end{aligned}$$
as each actuating limb cannot spin about its own axis. Dot-multiplying and cross-multiplying both sides of Eq. ( 26) by i s i, we obtained the linear and angular velocities of the ith actuating limb:
$$\left\{ {\begin{array}{*{20}l} {\ddot{l}_{i} = {}^{i}{\varvec{s}}_{i} {}^{i}{\dot{\varvec{v}}}_{bi} + l_{i} {}^{i}{\varvec{\omega}}_{i}^{2} ,} \hfill \\ {{}^{i}{\dot{\varvec{\omega }}}_{i} = ({}^{i}{\varvec{s}}_{i} \times {}^{i}{\dot{\varvec{v}}}_{bi} - 2\dot{l}_{i} {}^{i}{\varvec{\omega}}_{i} )/l_{i} .} \hfill \\ \end{array} } \right.$$
The acceleration of the centers of mass of the ith cylinder and piston could be solved by differentiating Eq. ( 25) with respect to time and combining with Eq. ( 27):
$$\left\{ {\begin{array}{*{20}l} {{}^{i}\varvec{a}_{i1} =\; e_{i1} {}^{i}\dot{\varvec{\omega }}_{i} \times {}^{i}\varvec{s}_{i} + e_{i1} {}^{i}\varvec{\omega}_{i} \times \text{(}{}^{i}\varvec{\omega}_{i} \times {}^{i}\varvec{s}_{i} \text{),}{\kern 1pt} {\kern 1pt} } \hfill \\ \begin{aligned} {}^{i}\varvec{a}_{i2} \;=\; (l_{i} - e_{i2} ){}^{i}\dot{\varvec{\omega }}_{i} \times {}^{i}\varvec{s}_{i} {\kern 1pt} + (l_{i} - e_{i2} ){}^{i}\varvec{\omega}_{i} \cdots \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times ({}^{i}\varvec{\omega}_{i} \times {}^{i}\varvec{s}_{i} ) + 2\dot{l}_{i} {}^{i}\varvec{\omega}_{i} \times {}^{i}\varvec{s}_{i} + \ddot{l}_{i} {}^{i}\varvec{s}_{i} . \hfill \\ \end{aligned} \hfill \\ \end{array} } \right.$$
6.2 Velocity and Acceleration of Passive Limbs
The passive limb of the 2-U PR/URR parallel mechanism consisted of an upper rod and a lower rod, as shown in Figure 14.
Free-body diagram of passive limb
Letting e 11 be the distance between A 1 and the center of the mass of the lower rod, and letting e 12 be the distance between B 1 and the center of mass of the upper rod, as shown in Figure 14, the position vector of the centers of mass of the lower and upper rods could be represented as:
$$\left\{ {\begin{array}{*{20}l} {{\varvec{r}}_{11} = {\varvec{a}}_{1} + e_{11} {\varvec{s}}_{1} ,} \hfill \\ {{\varvec{r}}_{12} = {\varvec{a}}_{1} + (l_{1} - e_{12} ){\varvec{s}}_{1} .} \hfill \\ \end{array} } \right.$$
As the passive limb cannot move along the z 1 axis, the linear and angular velocities of the vertex B 1 were obtained based on A R 1 and Eq. ( 22):
$$\left\{ {\begin{array}{*{20}l} {{}^{1}{\varvec{v}}_{b1} = [{}^{1}v_{b1x} ,{}^{1}v_{b1y} ,{}^{1}v_{b1z} ] = l_{1} {}^{1}{\varvec{\omega}}_{1} \times {}^{1}{\varvec{s}}_{1} ,} \hfill \\ {{}^{1}{\varvec{\omega}}_{1} = \frac{1}{{l_{1} }}({}^{1}{\varvec{s}}_{1} \times {}^{1}{\varvec{v}}_{b1} ) = \frac{1}{{l_{1} }}[ - {}^{1}v_{b1y} , - {}^{1}v_{b1x} ,0].} \hfill \\ \end{array} } \right.$$
Therefore, the velocities of the center of the mass of the lower and upper rods are expressed as:
$$\left\{ {\begin{array}{*{20}l} {{}^{1}{\varvec{v}}_{11} = e_{11} {}^{1}{\varvec{\omega}}_{1} \times {}^{1}{\varvec{s}}_{1} ,} \hfill \\ {{}^{1}{\varvec{v}}_{12} = (l_{1} - e_{11} ){}^{1}{\varvec{\omega}}_{1} \times {}^{1}{\varvec{s}}_{1} .} \hfill \\ \end{array} } \right.$$
The acceleration of B 1, expressed in its limb frame, was found by differentiating Eq. ( 31) with respect to time:
$${}^{1}{\varvec{a}}_{b1} = {}^{1}{\dot{\varvec{v}}}_{b1} = l_{1} {}^{1}{\dot{\varvec{\omega }}}_{1} \times {}^{1}{\varvec{s}}_{1} + l_{1} {}^{1}{\varvec{\omega}}_{1} \times ({}^{1}{\varvec{\omega}}_{1} \times {}^{1}{\varvec{s}}_{1} ).$$
Cross-multiplying both sides of Eq.( 32) by 1 s 1, the angular acceleration of the passive limb can be calculated:
$${}^{1}{\dot{\varvec{\omega }}}_{1} = ({}^{1}{\varvec{s}}_{1} \times {}^{1}{\dot{\varvec{v}}}_{b1} )/l_{1} .$$
The acceleration of the center of mass of the lower and upper rods of the passive limb are then expressed as:
$$\left\{ {\begin{array}{*{20}l} {{}^{i}{\dot{\varvec{v}}}_{i1} = e_{11} {}^{1}{\dot{\varvec{\omega }}}_{1} \times {}^{1}{\varvec{s}}_{1} + e_{11} {}^{1}{\varvec{\omega}}_{1} \times ({}^{1}{\varvec{\omega}}_{1} \times {}^{1}{\varvec{s}}_{1} ),} \hfill \\ {{}^{i}{\dot{\varvec{v}}}_{i2} = (l_{1} - e_{12} ){}^{1}{\dot{\varvec{\omega }}}_{1} \times {}^{1}{\varvec{s}}_{1} + (l_{1} - e_{12} ){}^{1}{\varvec{\omega}}_{1} \times ({}^{1}{\varvec{\omega}}_{1} \times {}^{1}{\varvec{s}}_{1} ).} \hfill \\ \end{array} } \right.$$
6.3 Jacobian Matrix of Mechanism
6.3.1 Jacobian Matrix of the Moving Platform
In this paper, the virtual work principle was adopted to solve the dynamic solution of the mechanism. Hence, a critical step in building the dynamic equations of the mechanism was the construction of the Jacobian matrix of the moving platform and the link Jacobian matrices.
Based on Eq. ( 24) in matrix form, the Jacobian matrix of the moving platform could be obtained:
$$\left[ {\begin{array}{*{20}c} {\dot{l}_{2} } \\ {\dot{l}_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{}^{2}{\varvec{J}}_{b2z} } \\ {{}^{3}{\varvec{J}}_{b3z} } \\ \end{array} } \right]_{2 \times 6} \cdot {\dot{\varvec{x}}}_{p} = {\varvec{J}}_{p} {\dot{\varvec{x}}}_{p} ,$$
where J p = [ 2 J b2z, 3 J b3z] T is the Jacobian matrix of the moving platform, which is expressed as the velocity mapping relationship among all actuators and the moving platform.
6.3.2 Link Jacobian Matrices
Combining Eqs. ( 24), ( 25), ( 30) and ( 31), the link Jacobian matrices could be obtained:
$$\left\{ {\begin{array}{*{20}c} {{}^{i}{\dot{\varvec{x}}}_{i1} = {}^{i}{\varvec{J}}_{i1} {\dot{\varvec{x}}}_{p} = [{}^{i}{\varvec{v}}_{i1} ,{}^{i}{\varvec{\omega}}_{i} ]^{\text{T}} {\dot{\varvec{x}}}_{p} ,} \\ {{}^{i}{\dot{\varvec{x}}}_{i2} = {}^{i}{\varvec{J}}_{i2} {\dot{\varvec{x}}}_{p} = [{}^{i}{\varvec{v}}_{i2} ,{}^{i}{\varvec{\omega}}_{i} ]^{\text{T}} {\dot{\varvec{x}}}_{p} ,} \\ \end{array} } \right.$$
where \({}^{i}{\dot{\varvec{x}}}_{i1}\) and \({}^{i}{\dot{\varvec{x}}}_{i2}\) are the velocities of the center of mass of the upper rod and lower rod in the passive limb when i = 1, and denotes the velocities of the center of mass of the cylinder and piston in the ith actuating limb when i = 2, 3. \({\dot{\varvec{x}}}_{p}\) denotes the velocity of the moving platform. i J i1 and i J i2 are the link Jacobian matrices, respectively, of the cylinder and piston in the ith actuating limb.
6.4 Dynamic Equations
6.4.1 Inertia and Applied Wrenches
The vector sum of applied and inertia wrenches is denoted as F p = [ f p, n p] T, which is exerted at the center of mass of the moving platform of the 2-U PR/URR parallel mechanism:
$${\varvec{F}}_{p} = \left[ {\begin{array}{*{20}c} {{\varvec{f}}_{p} } \\ {{\varvec{n}}_{p} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\varvec{f}}_{e} + m_{p} {\varvec{g}} - m_{p} {\varvec{a}}_{p} } \\ {{\varvec{n}}_{e} - {}^{A}{\varvec{I}}_{p} {\dot{\varvec{\omega }}}_{p} - {\varvec{\omega}}_{p} \times ({}^{A}{\varvec{I}}_{p} {\varvec{\omega}}_{p} )\;} \\ \end{array} } \right],$$
where F e = [ f e, n e] is the resultant force vector acted on the center of mass of the moving platform. m p is the mass of the moving platform, B I p is the inertia matrix of the moving platform, A I p denotes the inertia matrix of the moving platform, and A I p = A R B B I P B R A.
Similarly, i F i1 and i F i2 are the vector sum of applied and inertia wrenches exerted at the centers of mass of the lower and upper rods when i = 1. They are the vector sum of applied and inertia wrenches exerted at the centers of the mass of the cylinder and piston when i = 2, 3:
$$\left\{ {\begin{array}{*{20}c} {{}^{i}{\varvec{F}}_{i1} = \left[ {\begin{array}{*{20}c} {m_{i1} {}^{i}{\varvec{R}}_{A} {\varvec{g}} - m_{i1} {}^{i}{\varvec{a}}_{i1} } \\ { - {}^{i}{\varvec{I}}_{i1} {}^{i}{\dot{\varvec{\omega }}}_{i} - {}^{i}{\varvec{\omega}}_{i} \times ({}^{i}{\varvec{I}}_{i1} {}^{i}{\varvec{\omega}}_{i} )\;} \\ \end{array} } \right],} \\ {{}^{i}{\varvec{F}}_{i2} = \left[ {\begin{array}{*{20}c} {m_{i2} {}^{i}{\varvec{R}}_{A} {\varvec{g}} - m_{i2} {}^{i}{\varvec{a}}_{i2} } \\ { - {}^{i}{\varvec{I}}_{i2} {}^{i}{\dot{\varvec{\omega }}}_{i} - {}^{i}{\varvec{\omega}}_{i} \times ({}^{i}{\varvec{I}}_{i2} {}^{i}{\varvec{\omega}}_{i} )\;} \\ \end{array} } \right],} \\ \end{array} } \right.$$
where i m i1 and i m i2 are, respectively, the mass of the lower rod and upper rod when i = 1. They are the mass of the cylinder and piston when i = 2, 3. i I i1 and i I i2 denote, respectively, the inertia matrix of the lower rod and the upper rod in the ith limb.
6.4.2 Dynamic Equations
From the above, the dynamic model was established based on the virtual work principle [20,30] as follows:
$${\varvec{J}}_{p}^{\text{T}} {\varvec{\tau}} + {\varvec{F}}_{p} + \sum\limits_{i = 1}^{3} {({}^{i}{\varvec{J}}_{i1}^{\text{T}} {}^{i}{\varvec{F}}_{i1} + {}^{i}{\varvec{J}}_{i2}^{\text{T}} {}^{i}{\varvec{F}}_{i2} )} = 0.$$
To facilitate the calculation, substituting Eqs. ( 37) and ( 38) into Eq. ( 39) and simplifying yields:
$${\varvec{J}}_{p}^{\text{T}} {\varvec{\tau}} + {\varvec{F}}_{p} + \sum\limits_{i = 1}^{3} {({\varvec{J}}_{x}^{\text{T}} {\varvec{F}}_{x} + {\varvec{J}}_{y}^{\text{T}} {\varvec{F}}_{y} )} = 0,$$
where \({\varvec{\tau}} = [0,\tau_{2} ,\tau_{3} ]\) is the vector of actuator forces, which is specified in the actuator.
7 Numerical Verification
The material properties of all parts of the mechanism are shown in Table 4.
Material property parameters
Numerical value
Young modulus E (N/mm 2)
2.1×10 −5
Poisson's ratio μ
Density ρ (t/mm 3)
The mass of each component was m p = 1.5 kg, m 11 = 0.9 kg, m 12 = 0.75 kg, m i1 = 1.2 kg, and m i2 = 0.75 kg. In terms of Ref. [ 21 ], the inertia matrix I p and i I i ( i = 1, 2, 3) could be obtained:
$$I_{p} = \left[ {\begin{array}{*{20}c} {0.348} & 0 & 0 \\ 0 & {0.174} & 0 \\ 0 & 0 & {0.174} \\ \end{array} } \right],\quad {}^{i}I_{i} = \left[ {\begin{array}{*{20}c} {0.316} & 0 & 0 \\ 0 & {0.296} & 0 \\ 0 & 0 & 0 \\ \end{array} } \right],$$
where the unit of inertia is kg/m 2, and the gravity is g = [0, 0, − 9807] T mm/s 2.
Moreover, the joint forces were tested using the force-measuring platform in a gait cycle, which was defined as F e. It was obtained by the equivalent method:
F e = [− 20, 0, 25, 30, 30, 0], for the human without load.
F e = [− 35, 0, 50, 60, 50, 0], for the human with a load of 20 kg.
Based on these equations, the dynamic problem of the 2-U PR/URR parallel mechanism was programmed and calculated by Mathematica, which was verified by the simulation results.
7.1 Dynamic Verification
According to the gait testing in Section 2, the motion curves of the hip in four gait modes were obtained by using MotionAnalysis, as shown in Figure 15. The hip motion trajectories were all approximate during flexion and extension, though the ultimate rotational angles were different. However, the hip motion trajectories were nearly similar during abduction and adduction.
Motion angles for the hip in multiple gait modes
Therefore, the dynamics of the prosthetic mechanism was mainly analyzed in two gait modes based on Figure 15:
First mode: the velocity was 3.0 m/s with a load of 20 kg during flexion and extension.
Second mode: the velocity was 1.5 m/s without loading during abduction and adduction.
Furthermore, based on the human dynamic analysis, the flexion and extension of the hip prosthetic parallel mechanism in the gait cycle, as shown in Figure 16. As can be seen in Figure 16, the human gait cycle could be divided into five phases in terms of kinesiology [ 11 ].
The movements of the prosthetic mechanism in a gait cycle
The motions of the hip prosthetic parallel mechanism in the five phases are summarized, as shown in Table 5.
Motion characteristics of the mechanism in different phases
Gaits
1.5 m/s, 0 kg
3.0 m/s, 20 kg
Foot flat
Heel off
Toe off
− 1.7°~22.1°
11.9°~31.4°
By the curve fitting method, the motion trajectory of the moving platform could be derived in the first gait mode, as shown in Eq. ( 42):
$$\left\{ {\begin{array}{*{20}c} {\alpha = 0,\;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;\;\;\;\;\;\;\;\;\;\;\;} \\ {\beta = \left\{ {\begin{array}{*{20}c} {\frac{{35\uppi}}{18}t^{2} + \frac{{59\uppi}}{360}{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \le t < 0.1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \\ {\frac{{11\uppi}}{60}\cos \left( {\frac{{20\uppi}}{11}t - \frac{{2\uppi}}{11}} \right){ + }\frac{{7\uppi}}{72},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0.1 \le t < 1.1,} \\ { - \frac{\uppi}{ 3 6}t + \frac{{ 1 3\uppi}}{ 7 2}{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1.1 \le t < 1.2.} \\ \end{array} } \right.} \\ \end{array} } \right.$$
The ith ( i = 2,3) actuating forces versus time calculated by the program showed in Figure 17(a), and the simulation results based on Adams are plotted in Figure 17(b).
Actuating forces in the first gait mode: a actuating forces by Mathematica; b actuating forces by Adams
Results showed that the theoretical solution was the same, owing to the symmetrical arrangement of the actuating limbs. However, the actuating forces were highly approximated in the simulation environment, and the error may have been caused by the rigid connection of the kinematic pairs.
By the curve fitting method, the motion trajectory of the moving platform could be obtained in the second gait mode, as shown in Eq. ( 43):
$$\left\{ {\begin{array}{*{20}c} {\alpha = \left\{ {\begin{array}{*{20}c} {\frac{{11\uppi}}{360}\cos \left( {\frac{{5\uppi}}{3}t -\uppi} \right){\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0 \le t < 0.8,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} } \\ {\frac{{35\uppi}}{18}t^{2} + \frac{{31\uppi}}{9}t + \frac{{533\uppi}}{360},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0.8 \le t < 1.0,} \\ { - \frac{\uppi}{ 3 6}t + \frac{\uppi}{ 1 2 0},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1. 0\le t < 1. 2 ,} \\ \end{array} } \right.} \\ {\beta = 0.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;\;\;\;\;\;\;\;\;\;\;\;\;} \\ \end{array} } \right.$$
The ith ( i = 2,3) actuating forces versus time calculated by the program are shown in Figure 18a, and the simulation results based on Adams are plotted in Figure 18b.
Actuating forces in the second gait mode: a actuating forces by Mathematica; b actuating forces by Adams
As can be known from Figure 18, the absolute changes in the theoretical solution and the simulation results were the same in the gait cycle. Due to the symmetrical arrangement of the actuating limbs in the sagittal plane, the direction was opposite.
Furthermore, the theoretical solution was highly similar to the simulation results in the two gait modes. This verified the validity of the dynamic theoretical model of the hip prosthetic parallel mechanism.
7.2 Inverse Kinematic Verification
In order to verify the validity of the inverse kinematics of the hip prosthetic parallel mechanism, which was derived in Section 3. For the first gait mode, the motion trajectory of the moving platform is given in Eq. ( 42), and the actuating displacement along the limbs could be calculated, as shown in Figure 19.
Actuating displacements in the first gait mode: a based on Matlab; b based on Adams
Results showed that the changes of the actuating displacement were the same during flexion and extension of the hip. There was a displacement error between the theoretical solution and the simulation results in the heel strike phase, which was about 6 mm.
For the second gait mode, the motion trajectory of the moving platform is given in Eq. ( 43); the actuating displacement along the limbs could thus be solved, as shown in Figure 20.
Actuating displacement in the second gait mode: a actuating displacement by Matlab; b actuating displacement by Adams
As can be known from Figure 20, the theoretical solution and the simulation results were the same in the gait cycle. An acceptable error existed between the theoretical solution and the simulation results, which was within 3 mm.
All told, the variation trend in actuating displacements between the theoretical solution and simulation results were highly consistent in the two gait modes, which verified the validity of the inverse kinematics of the hip prosthetic parallel mechanism in Section 3. Moreover, the existence of the error was caused by the deviation between the trajectory equation and the motion curve. Another factor was the inevitable error between the joint of the prosthetic mechanism and the human hip.
7.3 Analysis of the Hip Torque
As can be seen from Figure 15, flexion and extension of the hip are the main motions, which are more important than adduction and abduction.
Hence, the joint torque during flexion and extension should be considered in the design of the hip prosthetic parallel mechanism.
Based on the human dynamic simulation of an amputee, who wears the 2-U PR/URR hip parallel prosthetic mechanism, the hip torques could be obtained in the multiple gait modes, as shown in Figure 21.
Changes in hip torque in multiple gait modes
As can be seen from Figure 21, the blue solid line denotes the hip torque of the adult subject at 1.5 m/s walking speed. The magenta dashed line represents the hip torque of the amputee, who wears the prosthetic mechanism, at 1.5 m/s walking speed. The green dot-dash line represents the hip torque of the amputee, who wears the prosthetic mechanism in the first gait mode.
Results showed that the variation tendency of the hip torque was consistent. It verified the feasibility of the 2-U PR/URR parallel mechanism as a hip prosthesis in multiple gait modes. Additionally, the torque changes in the prosthetic mechanism were larger than those in the adult subject in the same gait mode, which may have been caused either by the existence of the geometric error between prosthetic mechanism and the human thigh or the rigid impact between the prosthetic mechanism and the ground.
8 Conclusions
By analysis of the motion characteristics of an adult subject's hip, a novel parallel mechanism with a passive limb, named 2-U PR/URR parallel mechanism, which can realize the movement function of the hip, was designed based on configuration synthesis and screw theory.
The workspace was calculated based on inverse kinematics. The stiffness and dexterity were analyzed in terms of the fully Jacobian matrix. They verified the kinematic feasibility of the 2-U PR/URR parallel mechanism as the hip prosthesis. Furthermore, a statics and buckling analysis were conducted based on the finite element method; the maximum bearing capacity was obtained when the mechanism was stable. The workspace of the proposed prosthetic mechanism was large and its rotational angles covered the requirement of the hip in the four gait modes. Additionally, the prosthesis also had enough stiffness to support the torso, and the maximum bearing capacity was 1376 N, which was greater than the weight of the human body. This indicated that the prosthesis can bear an additional load of 576 N.
The inverse dynamic model of the prosthetic parallel mechanism was constructed by virtual work principle, and Mathematica and Adams were adopted to solve its theoretical solution and simulated results. Moreover, the torques of the prosthetic mechanism, during flexion and extension of the hip, were evaluated based on the human dynamics and compared with the simulation results. Results showed that the actuating forces and actuating displacement were highly similar between the theoretical solution and the simulation results. This verified the validity of the dynamic model and the inverse kinematics of the proposed prosthetic mechanism. Furthermore, the variation trend was consistent between the torques of the prosthetic mechanism and the human hip, which demonstrated the feasibility of the dynamic performance of the 2-U PR/URR parallel mechanism as a hip prosthetic mechanism.
All the authors would like to thank the Beijing Natural Science Foundation and National Science Foundation of China for financial support.
Majun Song, born in 1990, is currently a PhD candidate at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. He received his master degree from Jiangxi University of Science and Technology, China, in 2016. His research interests include parallel robot, medical rehabilitation robot and structural optimization.
Sheng Guo, born in 1972, is currently a professor and a PhD candidate supervisor at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. His main research interests include spatial mechanism design, parallel robot and medical rehabilitation robot.
Xiangyang Wang, born in 1995, is currently a PhD candidate at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. His research interests include robotic mechanics, exoskeleton parallel robots.
Haibo Qu, born in 1983, is currently a lecturer at Robotics Institute, Beijing Jiaotong University, China. He received his PhD from Beijing Jiaotong University in 2013. His research interests include robotics mechanism and mechanical design.
The authors declare no competing financial interests.
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Why do natural transformations express the fact that a vector space is canonically embedded in its double-dual but not in its dual?
I've been struggling for quite a while to understand why a vector space is considered to be "canonically embedded" into its double dual, but not its dual. As has been remarked in many other places, the distinction between whether an (iso-)morphism is "natural" can often seem vague and unintuitive. For me particularly, I think that part of the problem is that this sort of statement seems to run entirely counter to something I was taught early in Abstract Algebra as a Profound and Fundamental Lesson: "Isomorphic structures are exactly the same in all respects. When two things are isomorphic, all the things that can be said about one carry over verbatim to the other. There is no distinction between them." However, moving into more abstract linear algebra, a sort of about-face is being made, and now we are making the distinction of effectively saying, "My isomorphism is better than yours." In order to justify this apparent contradiction, the argument is typically made that the (iso-)morphism into the double dual does not require any "choices", while any embedding into the dual will require some "choice" to be made. However, this seems... unconvincing. So what if you can jury-rig a bilinear form out of whatever embedding/isomorphism I pick? Do we really have to pay attention to that? Again, this seems rather vague and unintuitive.
To make the argument more precise then, it is claimed that the ultimate answer lies in that Fountain of Eternal Truth - Category Theory. More specifically, it is claimed that the fact that there is a natural transformation from the identity functor on vector spaces to the double-dual functor justifies the claim that the embedding into the double-dual is "natural", while the fact that there is no such transformation between the identity and the dualizing functor shows that any such embedding into the dual is "not natural". This is elucidated beautifully in this thread. However, I claim that this is still not the final nail in the coffin of doubt. More specifically, I do not understand how natural transformations actually express the idea that a construction is (quotation marks) "natural".
How does this business with commuting diagrams make precise the idea that the embedding of a vector space into its double dual is "natural"? How does the implication that any association with the dual is "not natural" stem from a theorem saying that a certain collection of diagrams will never commute?
Another thing to note, that took me a little by surprise, is that the content of these arguments depends not only on the construction of the dual and double dual spaces, but also on this other construction called the transpose, which associates a linear map $f^*: F^* \to E^*$ to every linear map $f: E \to F$. So the fact that a map between a space and its dual is not "natural" also depends on the fact that we define an association between linear maps, that we package together with the dual operation to form the dualizing functor; however, this association between linear maps seems rather external to the association between a vector space and its dual. This is also bothering me.
I will not deny that the transpose operation does seem like a very natural thing to pair along with the dual operation, but what does seem odd is that the intrusiveness of this transpose operation should make or break the "naturalness" of something strictly between a vector space and its dual - honestly, who ordered that? Why can't I concoct some other association between linear maps - one that's covariant, at that, and package that together with the dual space to make something that admits a natural transformation from the identity? Note however that this would also break the "theorem" that a vector space is canonically embedded in its double dual, so this sort of train of thought is a double-edged sword.
Ultimately, I feel that I don't understand natural transformations in general very well; this example is really just the biggest one that sticks out to me and the one that I care about the most. I may post another question about the general case of understanding natural transformations, depending on how well this one goes and also whether I can manage to formulate it in a manner that seems intriguing and not simply lost and confused. At any rate, I look forward to any potential answers and would greatly appreciate whatever illumination you may be able to provide.
vector-spaces category-theory intuition
silvascientistsilvascientist
$\begingroup$ @PeterFranek Corrected, thanks. $\endgroup$ – silvascientist Jan 19 '16 at 7:43
$\begingroup$ +1 I love this question. I feel I often learn a ton from this kind of imprecise-but-aimed-at-something-you-can-feel inquiry. $\endgroup$ – Ben Blum-Smith Feb 21 '17 at 11:53
I am answering as somebody who has struggled through a related matter, as you noted in the OP. I do not think I will be able to satisfy every one of your related threads of dissatisfaction and I am not sure I will be able to satisfy any at all. On the flip side, as the question is a year old, you may have resolved it for yourself long ago.
But let's give it a whirl. I love the question.
First of all, separate from the question about how the category language speaks to (or doesn't speak to) matters, it seems to me you are not convinced that there even is a substantive difference between the isomorphism of a finite dimensional vector space to its dual and the isomorphism to its double dual, a propos of your Profound and Fundamental Lesson of Abstract Algebra -- aren't they both isomorphisms? So, before even engaging the category theory, let me speak to this:
(1) I think you will gain useful insight about the situation from studying cases where the substance of the difference between a space and its dual is felt. user254665 mentioned one such instance in her/his answer. In general, the infinite-dimensional topological vector spaces of functional analysis provide an abundant source of examples. While the dual of a finite dimensional vector space is finite-dimensional of the same dimension, and therefore isomorphic, the dual of a Banach space is typically a different Banach space. For example the dual of $L^p$ is $L^q$ with $p^{-1}+q^{-1} = 1$, which are two different Banach spaces unless $p=2$. The dual of the space of continuous, compactly supported functions on a locally compact Hausdorff space is a space of measures i.e. it is not even a space of functions!
Even in these situations where the dual is really a different animal, the original space does embed in its double dual, as usual by mapping a vector to the functional on functionals obtained by evaluation at that vector. (I will avoid controversy by not lionizing this embedding as "natural".) In many cases, the embedding is proper, i.e. the double dual is bigger than the original space. Nonetheless, there's often no obvious embedding of the original space in the (single) dual at all.
I am not a functional analyst, but a place I've encountered this substance in my own life is in the difference between a locally compact abelian group and its character group, i.e. its Pontryagin dual. Like vector spaces, this is a situation where finiteness causes a non-canonical isomorphism to the dual, and there is a canonical isomorphism to the double dual. A finite abelian group $A$ is isomorphic to its dual $\hat A$, but not an infinite group. For example, the additive group $\mathbb{Z}$ of integers and the circle group $S^1 = \{z\in\mathbb{C}^\times \mid |z| = 1\}$ are Pontryagin duals of each other, and they don't even have the same cardinality. In the finite case, where they are isomorphic, I've still "bumped into" the difference between $A$ and $\hat A$, for example in trying to understand the relationship between an action of a group $G$ of automorphisms on $A$ and the induced action of $G$ on $\hat A$, e.g. see this question.
All of this is to say that study of such examples can help convince one that the dual is really not the same as the original object, so that even when they're isomorphic it's worth keeping track of which is which. (More so than it is worth distinguishing the object from its double-dual when they are isomorphic.)
(2) How to make sense of this difference in light of your Profound and Fundamental Lesson (PaFL), that isomorphic objects are to all intents and purposes the same.
This is a question about the scope of the PaFL.
The PaFL is the right way to see things when you view the objects in isolation from their surroundings and each other. Let $A$ and $B$ be isomorphic objects (e.g. vector spaces or groups). Any specific isomorphism $\phi:A\rightarrow B$ gives you a dictionary to translate statements about the isolated object $A$ to statements about the isolated object $B$ and vice versa. For example: if $A,B$ are vector spaces, then $\phi$ carries bases to bases, so there is a perfect bijective correspondence between bases of $A$ and bases of $B$. It carries linear transformations of $A$ to linear transformations of $B$ (via $T\mapsto \phi T\phi^{-1}$) so there is a bijection between such transformations. If we think of $\phi$ as a "renaming", then we can think of $B$ as just $A$ with different names.
From this point of view, $A$ and $B$ are "the same", and any "renaming" $\phi$ works as well as any other to show this. This is the PaFL.
But. If we allow $A$ and $B$ to interact with other objects (even each other!), then distinct isomorphisms start to feel very different! For example:
Let $A = \mathbb{R}^2$, seen as a real vector space. Let $B$ be $A$'s vector space dual, i.e. the space of linear functionals $A\rightarrow \mathbb{R}$, with pointwise addition and scalar multiplication. $B$ is isomorphic to $A$ since it is also a 2-dimensional real vector space. One has a wide choice of isomorphisms: fixing a basis of $A$, one can send it to any basis of $B$. There is a 4-dimensional manifold's worth of choice.
Now along comes a linear transformation $T$ acting on $A$, say by scaling the $x$-axis by a factor of $2$. One can pick some isomorphism $\phi:A\rightarrow B$ and translate $T$ into a transformation of $B$ as above (i.e. $\phi T \phi^{-1}$). But there is another (natural??) way that $T$ acts on $B$, irrespective of any choice of $\phi$, which is to send a functional $f:A\rightarrow\mathbb{R}$ to the functional $f\circ T$. Now one can ask about any given $\phi$: does the transformation of $B$ into which it translates $T$ equal this (natural??) action of $T$ on $B$? I.e. does $\phi T \phi^{-1} (f) = f\circ T$ for all $f\in B$? A priori, some $\phi$'s may be compatible with the action of $T$ on $B$ in this respect, and some may not.
One could go further. I chose a specific $T$ at the front end of this. But one could ask if there is a $\phi$ such that $\phi T\phi^{-1}(f)$ will equal $f\circ T$ regardless of the choice of $T$. This $\phi$, if it existed, would clearly (?) be "awesome" in some way that other isomorphisms aren't.
Perhaps you respond by saying, well, why did you bring $T$, and especially its action on $B$ by $f\mapsto f\circ T$, into it? This is a perfectly legitimate question. From the point of view where you only look at $A$ and $B$ as self-contained systems, there's no reason to. But my point is that mathematical objects are often embedded in a network of other mathematical objects (such as $T$, or a wide variety of choices of $T$, and their related actions on $A$ and $B$), and when we bring these other objects and the interactions between them into it, it complicates the (overly?) simplistic picture drawn by the PaFL. Maybe some isomorphisms play better than others with the network of relationships in which $A$ and $B$ are embedded.
(3) This is a segue into the matter of categories. A natural isomorphism between two functors is not an isomorphism between two isolated objects. It is some kind of construction that works simultaneously across an entire category, in such a way that the isomorphisms all interact well with a bunch of other maps.
Thus, the way in which the categorical language translates the word "natural" is, loosely, "working simultaneously across all the objects of a whole category, in such a way that it cooperates with the other relevant maps in the category." The naturality lies in the everywhere-at-once-ness and in the fits-in-with-what-was-already-going-on-ness.
To get specific to the case. Let $\mathscr{V}$ be the category of finite dimensional $\mathbb{R}$-vector spaces.
Let's try to carry out what you proposed in the penultimate paragraph of the OP, i.e. try to reconstruct the dualizing functor as a covariant functor; call it $D$. We are already given the map on objects: it sends $V\in\operatorname{Obj}\mathscr{V}$ to its dual $V^*$. We need to design, for every $T\in \operatorname{Hom}(V,W)$, a map $D(T):V^* \rightarrow W^*$, in such a way that the identity map always gets sent to the identity map, and for any $U\xrightarrow{S} V\xrightarrow{T}W$ occurring in $\mathscr{V}$, we have $D(TS) = D(T)D(S)$.
It seems to me that this is actually possible, modulo some axiom-of-choice typed issues. If we separately chose an isomorphism $\phi_V:V\rightarrow V^*$ for each $V\in \operatorname{Obj}\mathscr{V}$, then we could send $T:V\rightarrow W$ to $D(T) = \phi_W T\phi_V^{-1}$, which maps $V^*$ to $W^*$. Furthermore, it seems to me that the maps $\phi_V:V\rightarrow V^*$ would then constitute a natural isomorphism from the identity functor to our new "dualizing functor" $D$.
I think some readers will be given pause by the fact that this construction needs some form of the axiom of choice to be carried out. (I'm out of my set-theoretic league on what's needed. It seems to me that the category at hand is not a small category; thus we need an even stronger axiom like global choice, right?) But you've indicated that the need to make choices doesn't strike you as a barrier to "naturalness," so I assume that this high degree of nonconstructiveness of the construction won't be a problem. However, I see another issue as well:
This construction loses any information related to the fact that $V^*$ is supposed to be the dual of $V$. It completely ignores the fact that the elements of $V^*$ are supposed to be functionals on $V$. We could replace $V^*$ with any other vector space of the same dimension and carry out the same construction. Thus it seems to me $D$ doesn't really send $V$ to its dual in any meaningful sense. Thus, while it uses a nonconstructive axiom (global choice?) to get past the category-theoretic insistence that a natural transformation happen "all at once across a whole category", it doesn't (honestly anyway, it seems to me) meet the second condition that it "cooperates with what was already going on."
This is where the transpose (also called the adjoint) comes in. You ask, "who ordered that?" I.e. isn't the adjoint map extrinsic to the relationship between $V$ and its dual? I contend it's actually essential. If $T:V\rightarrow W$ is a map between vector spaces, then the adjoint $T^*:W^*\rightarrow V^*$ between their duals is defined as $f\overset{T^*}{\mapsto} f\circ T$. This $T^*$ cooperates with what was already going on! I.e. it transforms the dual space in accordance with what the elements in the dual space are supposed to mean. Without a relationship like that between $T$ and $T^*$ that incorporates the fact that the elements of $V^*$ are supposed to be the contents of $\operatorname{Hom}(V,\mathbb{R})$, a functor sending $V$ to $V^*$ is only meaningfully sending it to some other vector space of the same dimension, not actually its dual.
Thus a natural isomorphism to the dual really should somehow respect the adjoint, or something like it. Otherwise, what makes the dual the dual?
Obviously the question was soft and this is a soft answer. So let me know if any of this speaks to any of the issues you outlined.
Ben Blum-SmithBen Blum-Smith
$\begingroup$ Thanks for your answer. Being in grad school now, and being swamped with homework, I probably won't have a chance to really dig through the arguments till the weekend, but I do want to make some commentary on it. $\endgroup$ – silvascientist Feb 22 '17 at 7:42
$\begingroup$ @silvascientist - looking forward to your engagement when you get a chance. I'm glad you noticed the answer and I'm anxious to know if it speaks to the issues you were articulating in the OP. (I'm also in grad school, btw; if all goes well, defending my thesis in less than a month. It's ON.) $\endgroup$ – Ben Blum-Smith Feb 26 '17 at 21:05
$\begingroup$ Would you like to set up a chat room? I can imagine this conversation going on for quite a while. The comment section will start to complain. $\endgroup$ – silvascientist Feb 26 '17 at 22:00
$\begingroup$ I feel like your remark about the scope of the PaFL is on point. Basically, the explanation I've worked out for myself is that while, yes, a vector space and its dual may be structurally the same, that's not always what we really care about in mathematics. For example consider a smooth manifold. A vector field on this manifold is defined to be a section on its tangent bundle, i.e. to every point on the space we associate a vector in the tangent space at that point. Now, actually all the tangent spaces are isomorphic to each other, but it wouldn't be helpful to go about sending points.. $\endgroup$ – silvascientist Feb 26 '17 at 22:21
$\begingroup$ on the manifold to random tangent spaces all over the place - we really want a point to be given a vector in the tangent space at that point. And there's no point (no pun intended) in going about trying to find random isomorphisms between tangent spaces just to deal with them all as the same thing. $\endgroup$ – silvascientist Feb 26 '17 at 22:23
Consider the space $l_0$ of real sequences $(x_n)_{n\in N}$ that converge to $0,$ with $\|(x_n)_n\|=\sup_n |x_n|,$ and its dual $l_1,$ the space of absolutely summable real sequences $(y_n)_{n\in N}$ with norm $\|(y_n)_n\|=\sum_{n\in N}|y_n|<\infty.$ The space $l_0$ contains many positive sequences that are not summable,e.g if $y_n=1/n$ for each $n.$ We should expect an embedding $E$ from $l_0 $ into $l_1$ to preserve the algebraic structure and the topological structure, in other words $E$ should be a continuous linear bijection to its image, and $E^{-1},$ acting on the image of $E$, should also be continuous. Such an $E$ doesn't exist. As a special case of a fairly recent theorem, $l_0$ and $l_1$ are homeomorphic, but by a non-linear mapping $F$, so the algebraic structure is not preserved by $F$.
DanielWainfleetDanielWainfleet
It seems to me that there are two possible meanings (close to each other).
One is that such isomorphism $**$ is defined via very simple and "expected" means. Another word commonly used for this is canonical. The definition $(v,w):=w(v)$ for $w\in V^*$ identifies $v$ with an element of $V^{**}$ and this does not depend on any additional structure on $V$, such as a metric. In this sense, it is "natural": you pair elements of $V^*$ with elements of $V$, so you can consider it the other way around as a pairing between elements of $V$ and $V^*$.
Another meaning is, as you say, the categorical. This basically says that not only can you apply $**$ to spaces but also to linear maps and the corresponding diagram commutes. That is, you can identify $f: X\to Y$ with $f^{**}: X^{**}\to Y^{**}$ (again, the identification goes via simple and expected means). As before, the functorial definition of $**$ does not depend on any additional structures such as metrics or scalar products.
These two meanings often go hand-in-hand: if something has a simple and expected definition (or a complicated one, but satisfying simple axioms), usually it can be converted into something categorical. It seems to me that if one wants to highlight the categorical meaning, (s)he uses the word natural, if one wants to highlight the simple and expected thing, (s)he often uses the word canonical.
But I'm not sure if this answers your question because I guess that you are aware of all of this.
Extension + edit: a small attempt to give some intuition of why natural transformation are "natural". Consider a finite-dimensional vector space $V$ and two its bases $\mathcal{B}_1$ and $\mathcal{B}_2$. Let $\mathcal{V}$ be a category with only one object $V$ and morphisms $Mor(V,V)$ being all linear transformations; let $\mathcal{R}$ be a category with one object $\Bbb R^n$ and morphisms being all linear transformations. You can define two functors $F$ and $G$ from $\mathcal{V}$ to $\mathcal{R}$ that express a linear transformation as a matrix wrt. the coordinates $\mathcal{B}_1$ resp. $\mathcal{B}_2$. A natural transformation between $F$ and $G$ assigns to the object $V$ in $Obj(\mathcal{V})$ the morphism $x\mapsto C^{-1}x$ of $\Bbb R^n$ (an element of $Mor(\Bbb R^n, \Bbb R^n)$), where $C$ is the transition matrix from base $\mathcal{B}_1$ to $\mathcal{B}_2$. This morphism is just the coordinate transformation in $\Bbb R^n$.
The fact that it is a natural transformation just reflects that any linear map $f: V\to V$ (an element of $Mor(V,V)$) gives rise to the commutative diagram \begin{array}{ccc} \Bbb R^n & \stackrel{F(f)}{\to} & \Bbb R^n \\ \downarrow_{C^{-1}} && \downarrow_{C^{-1}} \\ \Bbb R^n & \stackrel{G(f)}{\to} & \Bbb R^n \\ \end{array} or equivalently, \begin{array}{ccc} \Bbb R^n & \stackrel{M}{\to} & \Bbb R^n \\ \downarrow_{C^{-1}} && \downarrow_{C^{-1}} \\ \Bbb R^n & \stackrel{C^{-1}MC}{\to} & \Bbb R^n \\ \end{array} where $M$ is the matrix expression of $F(f)$. In physics, this corresponds to a change of observer: observer $\mathcal{B}_2$ will just "see" a vector $C^{-1}x$ and/or "use" the matrix $C^{-1}MC$ whenever observer $\mathcal{B}_1$ "sees" the vector $x$ and "uses" the matrix $M$. But they both see the same "real object". In this sense, the natural transformation is "natural".
Peter FranekPeter Franek
$\begingroup$ Ultimately, your last statement is the most on the mark. The argument with commuting diagrams is supposed to make precise the notion that the embedding "does not depend on any additional structures such as metrics or scalar products." The question is, "How?". But I do appreciate the time and effort you have shown in answering my question. $\endgroup$ – silvascientist Jan 19 '16 at 8:04
$\begingroup$ @silvascientist I added a simple example that hopefully adds some intuition but I'm still not sure if that's what you want. I'm not sure if there is a direct connection between "not depending on additional structure" and "naturality" in the categorical sense, Maybe somebody else will formalize this, let's see.. :) $\endgroup$ – Peter Franek Jan 19 '16 at 8:11
$\begingroup$ "...functors from $V$ to $\mathbb{R}^n$"... I don't understand. Are $V$ and $\mathbb{R}^n$ being considered as categories in and of themselves? With morphisms between vectors? $\endgroup$ – silvascientist Jan 19 '16 at 8:15
$\begingroup$ @PeterFranek - Just to get clear on your construction. You are saying that $Obj(I)$ is the underlying set of $V$; ok. Then given two objects $x,y\in V$, what is $Mor(x,y)$? Is it the set of all linear transformations of $V$ that carry $x$ to $y$? (I guess what's bothering me about this is it means a single linear transformation $T$ is actually going to count as a big infinity of different morphisms, one for each pair of points $x,y$ with $Tx = y$.) $\endgroup$ – Ben Blum-Smith Feb 21 '17 at 12:05
$\begingroup$ @BenBlum-Smith Thanks for pointing this out; I was confused in the comment above. The object is only one, $V$ resp. $\Bbb R^n$ and morphisms are "linear transformation"; then it makes sense I hope. But it's not such a good example as I thought before. $\endgroup$ – Peter Franek Feb 21 '17 at 19:27
Not the answer you're looking for? Browse other questions tagged vector-spaces category-theory intuition or ask your own question.
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How is the isomorphism between a vector space and its dual not natural? | CommonCrawl |
Borel–Cantelli lemma
In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli, who gave statement to the lemma in the first decades of the 20th century.[1][2] A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states that, under certain conditions, an event will have probability of either zero or one. Accordingly, it is the best-known of a class of similar theorems, known as zero-one laws. Other examples include Kolmogorov's zero–one law and the Hewitt–Savage zero–one law.
Statement of lemma for probability spaces
Let E1,E2,... be a sequence of events in some probability space. The Borel–Cantelli lemma states:[3][4]
Borel–Cantelli lemma — If the sum of the probabilities of the events {En} is finite
$\sum _{n=1}^{\infty }\Pr(E_{n})<\infty ,$
then the probability that infinitely many of them occur is 0, that is,
$\Pr \left(\limsup _{n\to \infty }E_{n}\right)=0.$
Here, "lim sup" denotes limit supremum of the sequence of events, and each event is a set of outcomes. That is, lim sup En is the set of outcomes that occur infinitely many times within the infinite sequence of events (En). Explicitly,
$\limsup _{n\to \infty }E_{n}=\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }E_{k}.$
The set lim sup En is sometimes denoted {En i.o. }, where "i.o." stands for "infinitely often". The theorem therefore asserts that if the sum of the probabilities of the events En is finite, then the set of all outcomes that are "repeated" infinitely many times must occur with probability zero. Note that no assumption of independence is required.
Example
Suppose (Xn) is a sequence of random variables with Pr(Xn = 0) = 1/n2 for each n. The probability that Xn = 0 occurs for infinitely many n is equivalent to the probability of the intersection of infinitely many [Xn = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ΣPr(Xn = 0) converges to π2/6 ≈ 1.645 < ∞, and so the Borel–Cantelli Lemma states that the set of outcomes that are common to infinitely many such events occurs with probability zero. Hence, the probability of Xn = 0 occurring for infinitely many n is 0. Almost surely (i.e., with probability 1), Xn is nonzero for all but finitely many n.
Proof
Let (En) be a sequence of events in some probability space.
The sequence of events $ \left\{\bigcup _{n=N}^{\infty }E_{n}\right\}_{N=1}^{\infty }$ is non-increasing:
$\bigcup _{n=1}^{\infty }E_{n}\supseteq \bigcup _{n=2}^{\infty }E_{n}\supseteq \cdots \supseteq \bigcup _{n=N}^{\infty }E_{n}\supseteq \bigcup _{n=N+1}^{\infty }E_{n}\supseteq \cdots \supseteq \limsup _{n\to \infty }E_{n}.$
By continuity from above,
$\Pr(\limsup _{n\to \infty }E_{n})=\lim _{N\to \infty }\Pr \left(\bigcup _{n=N}^{\infty }E_{n}\right).$
By subadditivity,
$\Pr \left(\bigcup _{n=N}^{\infty }E_{n}\right)\leq \sum _{n=N}^{\infty }\Pr(E_{n}).$
By original assumption, $ \sum _{n=1}^{\infty }\Pr(E_{n})<\infty .$ As the series $ \sum _{n=1}^{\infty }\Pr(E_{n})$ converges,
$\lim _{N\to \infty }\sum _{n=N}^{\infty }\Pr(E_{n})=0,$
as required.[5]
General measure spaces
For general measure spaces, the Borel–Cantelli lemma takes the following form:
Borel–Cantelli Lemma for measure spaces — Let μ be a (positive) measure on a set X, with σ-algebra F, and let (An) be a sequence in F. If
$\sum _{n=1}^{\infty }\mu (A_{n})<\infty ,$
then
$\mu \left(\limsup _{n\to \infty }A_{n}\right)=0.$
Converse result
A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma. The lemma states: If the events En are independent and the sum of the probabilities of the En diverges to infinity, then the probability that infinitely many of them occur is 1. That is:[4]
Second Borel–Cantelli Lemma — If $\sum _{n=1}^{\infty }\Pr(E_{n})=\infty $ and the events $(E_{n})_{n=1}^{\infty }$ are independent, then $\Pr(\limsup _{n\to \infty }E_{n})=1.$
The assumption of independence can be weakened to pairwise independence, but in that case the proof is more difficult.
The infinite monkey theorem follows from the Second lemma.
Example
The lemma can be applied to give a covering theorem in Rn. Specifically (Stein 1993, Lemma X.2.1), if Ej is a collection of Lebesgue measurable subsets of a compact set in Rn such that
$\sum _{j}\mu (E_{j})=\infty ,$
then there is a sequence Fj of translates
$F_{j}=E_{j}+x_{j}$
such that
$\lim \sup F_{j}=\bigcap _{n=1}^{\infty }\bigcup _{k=n}^{\infty }F_{k}=\mathbb {R} ^{n}$
apart from a set of measure zero.
Proof
Suppose that $ \sum _{n=1}^{\infty }\Pr(E_{n})=\infty $ and the events $(E_{n})_{n=1}^{\infty }$ are independent. It is sufficient to show the event that the En's did not occur for infinitely many values of n has probability 0. This is just to say that it is sufficient to show that
$1-\Pr(\limsup _{n\to \infty }E_{n})=0.$
Noting that:
${\begin{aligned}1-\Pr(\limsup _{n\to \infty }E_{n})&=1-\Pr \left(\{E_{n}{\text{ i.o.}}\}\right)=\Pr \left(\{E_{n}{\text{ i.o.}}\}^{c}\right)\\&=\Pr \left(\left(\bigcap _{N=1}^{\infty }\bigcup _{n=N}^{\infty }E_{n}\right)^{c}\right)=\Pr \left(\bigcup _{N=1}^{\infty }\bigcap _{n=N}^{\infty }E_{n}^{c}\right)\\&=\Pr \left(\liminf _{n\to \infty }E_{n}^{c}\right)=\lim _{N\to \infty }\Pr \left(\bigcap _{n=N}^{\infty }E_{n}^{c}\right),\end{aligned}}$
it is enough to show: $ \Pr \left(\bigcap _{n=N}^{\infty }E_{n}^{c}\right)=0$. Since the $(E_{n})_{n=1}^{\infty }$ are independent:
${\begin{aligned}\Pr \left(\bigcap _{n=N}^{\infty }E_{n}^{c}\right)&=\prod _{n=N}^{\infty }\Pr(E_{n}^{c})\\&=\prod _{n=N}^{\infty }(1-\Pr(E_{n})).\end{aligned}}$
The convergence test for infinite products guarantees that the product above is 0, if $ \sum _{n=N}^{\infty }\Pr(E_{n})$ diverges. This completes the proof.
Counterpart
Another related result is the so-called counterpart of the Borel–Cantelli lemma. It is a counterpart of the Lemma in the sense that it gives a necessary and sufficient condition for the limsup to be 1 by replacing the independence assumption by the completely different assumption that $(A_{n})$ is monotone increasing for sufficiently large indices. This Lemma says:
Let $(A_{n})$ be such that $A_{k}\subseteq A_{k+1}$, and let ${\bar {A}}$ denote the complement of $A$. Then the probability of infinitely many $A_{k}$ occur (that is, at least one $A_{k}$ occurs) is one if and only if there exists a strictly increasing sequence of positive integers $(t_{k})$ such that
$\sum _{k}\Pr(A_{t_{k+1}}\mid {\bar {A}}_{t_{k}})=\infty .$
This simple result can be useful in problems such as for instance those involving hitting probabilities for stochastic process with the choice of the sequence $(t_{k})$ usually being the essence.
Kochen–Stone
Let $(A_{n})$ be a sequence of events with $ \sum \Pr(A_{n})=\infty $ and $ \liminf _{k\to \infty }{\frac {\sum _{1\leq m,n\leq k}\Pr(A_{m}\cap A_{n})}{\left(\sum _{n=1}^{k}\Pr(A_{n})\right)^{2}}}<\infty .$ Then there is a positive probability that $A_{n}$ occur infinitely often.
See also
• Lévy's zero–one law
• Kuratowski convergence
• Infinite monkey theorem
References
1. E. Borel, "Les probabilités dénombrables et leurs applications arithmetiques" Rend. Circ. Mat. Palermo (2) 27 (1909) pp. 247–271.
2. F.P. Cantelli, "Sulla probabilità come limite della frequenza", Atti Accad. Naz. Lincei 26:1 (1917) pp.39–45.
3. Klenke, Achim (2006). Probability Theory. Springer-Verlag. ISBN 978-1-84800-047-6.
4. Shiryaev, Albert N. (2016). Probability-1: Volume 1. Graduate Texts in Mathematics. Vol. 95. New York, NY: Springer New York. doi:10.1007/978-0-387-72206-1. ISBN 978-0-387-72205-4.
5. "Romik, Dan. Probability Theory Lecture Notes, Fall 2009, UC Davis" (PDF). Archived from the original (PDF) on 2010-06-14.
• Prokhorov, A.V. (2001) [1994], "Borel–Cantelli lemma", Encyclopedia of Mathematics, EMS Press
• Feller, William (1961), An Introduction to Probability Theory and Its Application, John Wiley & Sons.
• Stein, Elias (1993), Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press.
• Bruss, F. Thomas (1980), "A counterpart of the Borel Cantelli Lemma", J. Appl. Probab., 17: 1094–1101.
• Durrett, Rick. "Probability: Theory and Examples." Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005.
External links
• Planet Math Proof Refer for a simple proof of the Borel Cantelli Lemma
| Wikipedia |
Icosagon
In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.
Regular icosagon
A regular icosagon
TypeRegular polygon
Edges and vertices20
Schläfli symbol{20}, t{10}, tt{5}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D20), order 2×20
Internal angle (degrees)162°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular icosagon
The regular icosagon has Schläfli symbol {20}, and can also be constructed as a truncated decagon, t{10}, or a twice-truncated pentagon, tt{5}.
One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°.
The area of a regular icosagon with edge length t is
$A={5}t^{2}(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}})\simeq 31.5687t^{2}.$
In terms of the radius R of its circumcircle, the area is
$A={\frac {5R^{2}}{2}}({\sqrt {5}}-1);$
since the area of the circle is $\pi R^{2},$ the regular icosagon fills approximately 98.36% of its circumcircle.
Uses
The Big Wheel on the popular US game show The Price Is Right has an icosagonal cross-section.
The Globe, the outdoor theater used by William Shakespeare's acting company, was discovered to have been built on an icosagonal foundation when a partial excavation was done in 1989.[1]
As a golygonal path, the swastika is considered to be an irregular icosagon.[2]
A regular square, pentagon, and icosagon can completely fill a plane vertex.
Construction
As 20 = 22 × 5, regular icosagon is constructible using a compass and straightedge, or by an edge-bisection of a regular decagon, or a twice-bisected regular pentagon:
Construction of a regular icosagon
Construction of a regular decagon
The golden ratio in an icosagon
• In the construction with given side length the circular arc around C with radius CD, shares the segment E20F in ratio of the golden ratio.
${\frac {\overline {E_{20}E_{1}}}{\overline {E_{1}F}}}={\frac {\overline {E_{20}F}}{\overline {E_{20}E_{1}}}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \approx 1.618$
Symmetry
The regular icosagon has Dih20 symmetry, order 40. There are 5 subgroup dihedral symmetries: (Dih10, Dih5), and (Dih4, Dih2, and Dih1), and 6 cyclic group symmetries: (Z20, Z10, Z5), and (Z4, Z2, Z1).
These 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. John Conway labels these by a letter and group order.[3] Full symmetry of the regular form is r40 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g20 subgroup has no degrees of freedom but can seen as directed edges.
The highest symmetry irregular icosagons are d20, an isogonal icosagon constructed by ten mirrors which can alternate long and short edges, and p20, an isotoxal icosagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular icosagon.
Dissection
20-gon with 180 rhombs
regular
Isotoxal
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the icosagon, m=10, and it can be divided into 45: 5 squares and 4 sets of 10 rhombs. This decomposition is based on a Petrie polygon projection of a 10-cube, with 45 of 11520 faces. The list OEIS: A006245 enumerates the number of solutions as 18,410,581,880, including up to 20-fold rotations and chiral forms in reflection.
Dissection into 45 rhombs
10-cube
Related polygons
An icosagram is a 20-sided star polygon, represented by symbol {20/n}. There are three regular forms given by Schläfli symbols: {20/3}, {20/7}, and {20/9}. There are also five regular star figures (compounds) using the same vertex arrangement: 2{10}, 4{5}, 5{4}, 2{10/3}, 4{5/2}, and 10{2}.
n 1 2 3 4 5
Form Convex polygon Compound Star polygon Compound
Image
{20/1} = {20}
{20/2} = 2{10}
{20/3}
{20/4} = 4{5}
{20/5} = 5{4}
Interior angle 162° 144° 126° 108° 90°
n 6 7 8 9 10
Form Compound Star polygon Compound Star polygon Compound
Image
{20/6} = 2{10/3}
{20/7}
{20/8} = 4{5/2}
{20/9}
{20/10} = 10{2}
Interior angle 72° 54° 36° 18° 0°
Deeper truncations of the regular decagon and decagram can produce isogonal (vertex-transitive) intermediate icosagram forms with equally spaced vertices and two edge lengths.[5]
A regular icosagram, {20/9}, can be seen as a quasitruncated decagon, t{10/9}={20/9}. Similarly a decagram, {10/3} has a quasitruncation t{10/7}={20/7}, and finally a simple truncation of a decagram gives t{10/3}={20/3}.
Icosagrams as truncations of a regular decagons and decagrams, {10}, {10/3}
Quasiregular Quasiregular
t{10}={20}
t{10/9}={20/9}
t{10/3}={20/3}
t{10/7}={20/7}
Petrie polygons
The regular icosagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in orthogonal projections in Coxeter planes:
A19 B10 D11 E8 H4 ½2H2 2H2
19-simplex
10-orthoplex
10-cube
11-demicube
(421)
600-cell
Grand antiprism
10-10 duopyramid
10-10 duoprism
It is also the Petrie polygon for the icosahedral 120-cell, small stellated 120-cell, great icosahedral 120-cell, and great grand 120-cell.
References
1. Muriel Pritchett, University of Georgia "To Span the Globe" Archived 10 June 2010 at the Wayback Machine, see also Editor's Note, retrieved on 10 January 2016
2. Weisstein, Eric W. "Icosagon". MathWorld.
3. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
5. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
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| Wikipedia |
Generic matrix ring
In algebra, a generic matrix ring is a sort of a universal matrix ring.
Definition
We denote by $F_{n}$ a generic matrix ring of size n with variables $X_{1},\dots X_{m}$. It is characterized by the universal property: given a commutative ring R and n-by-n matrices $A_{1},\dots ,A_{m}$ over R, any mapping $X_{i}\mapsto A_{i}$ extends to the ring homomorphism (called evaluation) $F_{n}\to M_{n}(R)$.
Explicitly, given a field k, it is the subalgebra $F_{n}$ of the matrix ring $M_{n}(k[(X_{l})_{ij}\mid 1\leq l\leq m,\ 1\leq i,j\leq n])$ generated by n-by-n matrices $X_{1},\dots ,X_{m}$, where $(X_{l})_{ij}$ are matrix entries and commute by definition. For example, if m = 1 then $F_{1}$ is a polynomial ring in one variable.
For example, a central polynomial is an element of the ring $F_{n}$ that will map to a central element under an evaluation. (In fact, it is in the invariant ring $k[(X_{l})_{ij}]^{\operatorname {GL} _{n}(k)}$ since it is central and invariant.[1])
By definition, $F_{n}$ is a quotient of the free ring $k\langle t_{1},\dots ,t_{m}\rangle $ with $t_{i}\mapsto X_{i}$ by the ideal consisting of all p that vanish identically on all n-by-n matrices over k.
Geometric perspective
The universal property means that any ring homomorphism from $k\langle t_{1},\dots ,t_{m}\rangle $ to a matrix ring factors through $F_{n}$. This has a following geometric meaning. In algebraic geometry, the polynomial ring $k[t,\dots ,t_{m}]$ is the coordinate ring of the affine space $k^{m}$, and to give a point of $k^{m}$ is to give a ring homomorphism (evaluation) $k[t,\dots ,t_{m}]\to k$ (either by the Hilbert nullstellensatz or by the scheme theory). The free ring $k\langle t_{1},\dots ,t_{m}\rangle $ plays the role of the coordinate ring of the affine space in the noncommutative algebraic geometry (i.e., we don't demand free variables to commute) and thus a generic matrix ring of size n is the coordinate ring of a noncommutative affine variety whose points are the Spec's of matrix rings of size n (see below for a more concrete discussion.)
The maximal spectrum of a generic matrix ring
For simplicity, assume k is algebraically closed. Let A be an algebra over k and let $\operatorname {Spec} _{n}(A)$ denote the set of all maximal ideals ${\mathfrak {m}}$ in A such that $A/{\mathfrak {m}}\approx M_{n}(k)$. If A is commutative, then $\operatorname {Spec} _{1}(A)$ is the maximal spectrum of A and $\operatorname {Spec} _{n}(A)$ is empty for any $n>1$.
References
1. Artin 1999, Proposition V.15.2.
• Artin, Michael (1999). "Noncommutative Rings" (PDF).
• Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001.
| Wikipedia |
Shapes of Red Blood Cells: Comparison of 3D Confocal Images with the Bilayer-Couple Model
Khaled Khairy1,2,
JiJinn Foo1 &
Jonathon Howard1
Cellular and Molecular Bioengineering volume 1, Article number: 173 (2008) Cite this article
Cells and organelles are shaped by the chemical and physical forces that bend cell membranes. The human red blood cell (RBC) is a model system for studying how such forces determine cell morphology. It is thought that RBCs, which are typically biconcave discoids, take the shape that minimizes their membrane-bending energies, subject to the constraints of fixed area and volume. However, recently it has been hypothesized that shear elasticity arising from the membrane-associated cytoskeleton (MS) is necessary to account for shapes of real RBCs, especially ones with highly curved features such as echinocytes. In this work we tested this hypothesis by following RBC shape changes using spherical harmonic series expansions of theoretical cell surfaces and those estimated from 3D confocal microscopy images of live cells. We found (i) quantitative agreement between shapes obtained from the theoretical model including the MS and real cells, (ii) that weakening the MS, by using urea (which denatures spectrin), leads to the theoretically predicted gradual decrease in spicule number of echinocytes, (iii) that the theory predicts that the MS is essential for stabilizing the discocyte morphology against changes in lipid composition, and that without it, the shape would default to the elliptocyte (a biconcave ellipsoid), (iv) that we were able to induce RBCs to adopt the predicted elliptocyte morphology by treating healthy discocytes with urea. The latter observation is consistent with the known connection between the blood disease hereditary elliptocytosis and spectrin mutations that weaken the cell cortex. We conclude that while the discocyte, in absence of shear, is indeed a minimum energy shape, its stabilization in healthy RBCs requires the MS, and that elliptocytosis can be explained based on purely mechanical considerations.
Understanding the chemical and physical forces that lead to the curvature of cell membranes and consequently the morphology of cells and organelles is a major open question in cell biology.11,32,43 The mammalian red blood cell (RBC) is a good model for studying cell membranes19 because it lacks a nucleus, internal organelles and large cytoskeletal structures. Human RBCs, which circulate in the body for about 120 days, are normally biconcave discocytes with a diameter of ~8 μm, a surface area of ~140 μm2 and a volume of ~100 μm3.12 The excess surface area of RBCs (which gives them the flattened shape), together with the elasticity of their membranes, provides RBCs with the flexibility needed to pass through the capillaries. The cell membrane is composed of a lipid bilayer, with embedded membrane proteins, and associated with it from the cytoplasmic side is a network of proteins comprising the membrane cytoskeleton (MS).1 Under the influence of a variety of agents, shapes other than the discocyte can be observed. Examples are the stomatocyte and the echinocyte (Fig. 1a). In addition, genetic defects may result in pronounced morphological changes. For example, mutations in spectrin, a principal component of the MS, may lead to hereditary elliptocytosis in which the RBC assumes a biconcave elliptocytic morphology.7
Human RBC images and theoretical shape predictions with and without a membrane cytoskeleton. (a) 3D confocal images of the canonical shapes of RBCs in different concentrations (in mM) of NaCl: from left to right: 55, 80 (two stomatocytes—cup-shaped cells), 154 (one discocyte—biconcave disc), 200, 250, 300, and 350 mM NaCl (four echinocytes—spiculated cells). (b) Theoretical minimum energy shapes with area 140 μm2, volume 100 μm3 and membrane skeleton (MS) shear modulus 2.5 μJ/m2, calculated for increasing preferred area difference Δa 0. (c) Same as b, but in the absence of a MS. Beyond the right and left limits budding and membrane internalization are predicted, respectively
The finding that mutations in membrane-associated proteins such as spectrin, ankyrin and band 3 lead to RBC shape aberrations40 suggests that the shape of the red blood cell is specified by the membrane skeleton. Yet there is a large physics literature, starting with the pioneering work of Helfrich,8 that the normal red blood cell, which has a biconcave discoidal morphology, (Fig. 1a, third cell from the left), qualitatively resembles a surface that minimizes the bending energy of a pure lipid membrane (i.e., no membrane cytoskeleton) given constraints of fixed volume, area and spontaneous curvature of the lipid molecules. This purely lipid-based theory (the Helfrich model, also called spontaneous curvature model) can also account for other membrane morphologies such as the stomatocyte (Fig. 1a, left), the swollen red blood cell that can be induced by placing cells in hypotonic solutions and that is seen in some hereditary blood diseases.40 Discocytic and stomatocytic morphologies can be induced in liposomes,20 membrane vesicles that contain no skeleton, showing that the membrane skeleton is not essential for these morphologies. However, the pure lipid-based theory cannot easily account for other morphologies such as the echinocyte (Fig. 1a, right), a highly spiculated RBC morphology induced by hyper-osmotic solutions or by incorporating excess lipid into the outer leaflet of the membrane bilayer30: the Helfrich model (as well as its successor the bilayer-couple model35,36 which includes the effect of a fixed distance between the two bilayer leaflets) predicts that the minimum energy shape corresponds to other shapes such as dumbbells or even the fission of the cell into two cells depending on ratio of area to volume.18,27,34 These arguments have led to the augmentation of the lipid-based models to include the material properties of the MS and its association with the lipid bilayer.3,21,26,37 This has lead to the hypothesis that the function of the membrane-associated cytoskeleton is to protect the cell against vesiculation,33 which is observed in some hereditary blood diseases.40
To test this and other hypotheses for the mechanical basis of RBC morphology, we have taken a combined experimental and computational approach. Until now, comparison between observed red blood cells and predicted shapes has been qualitative in the sense that images, such as those obtained by scanning electron microscopy, are compared subjectively to predicted shapes. In order to make quantitative comparisons between experiment and theory, we have used a spherical-harmonics parameterization (SHP) to find the "best-fit" surface from confocal stacks of fluorescently labeled RBCs,22 and compared them with theoretical calculations with and without the MS. We show that the SHP is an ideal tool for freely exploring the full parameter space needed for predicting theoretical shapes with no restriction in symmetry. Using this technique we have compared the theoretical models with and without the MS, and have made a novel prediction (to the best of our knowledge) that the MS is essential for stabilizing the discocyte morphology against changes in lipid composition of the bilayer. Moreover, our calculations show that the predicted "default" shape that a discocyte would change to, if it lost the MS, is the elliptocyte. This is in accord with the known mutation in spectrin that causes the disease hereditary elliptocytosis, and in our view suffices to explain this disease, using purely mechanical considerations. As an additional confirmation, we treated healthy discocytes with urea, which is expected to weaken the MS, and indeed obtained elliptocytes.
The idea behind the physical theories is that the shape of a RBC corresponds to a surface that minimizes the mechanical energy (E) of a deformed lipid bilayer/membrane skeleton composite. This energy, for a cell not subject to external forces, includes contributions from the bending resistance of the plasma membrane (E b, the Helfrich energy8), the resistance of the bilayer to adopting a shape whose difference in area between the outer and inner leaflets (ΔA) deviates from the unstressed area difference (ΔA 0) (E AD,18), and the resistance of the membrane skeleton to stretching and shearing (E MS,26):
$$ E = E_{\text{b}} + E_{\text{AD}} + E_{\text{MS}} = \frac{{\kappa_{b} }}{2}\oint\limits_{Surface} {(2H - C_{0} )^{2} } dA + \frac{{\bar{\kappa }\pi }}{{2AD^{2} }}\left( {\Delta A - \Delta A_{0} } \right)^{2} + E_{MS} $$
\( \kappa_{b} \) is the bending elastic modulus that determines the energy cost of bending the membrane away from its preferred or spontaneous curvature C 0. H = (R 1 −1 + R 2 −1 )/2 is the mean curvature, where R 1 and R 2 are the radii corresponding to the two principal curvatures. A is the total surface area of the membrane. \( \bar{\kappa } \) is the area-difference elastic modulus that determines the energy cost of the deviation in area difference from the preferred area difference. D is the separation between the two bilayer leaflets. E MS is defined in the Methods. The minimum is found under constraints of constant A and volume (V). It should be noted that Eq. (1) includes parameters under cellular control; C 0 depends on the bilayer lipid composition (e.g., a conically shaped lipid in the outer membrane would induce positive curvature), ΔA 0 depends on the number of lipids in each leaflet (regulated by flippases9), and V is regulated indirectly through ion pumps.
Excluding the membrane skeleton term (E MS), all but two of the parameters in Eq. (1) either denote physical quantities \( (\kappa_{b} ,\,\bar{\kappa },\,D) \) or properties that can be calculated from the shape (H, A, ΔA). The two exceptions are the spontaneous curvature (C 0) and the area difference (ΔA 0), and it appears that they correspond to two free variables. However, these two parameters are not independent.36 For example, if a bilayer contains lipids with a preferred curvature in one leaflet, the effect on the geometry (i.e., the tendency to curve the membrane) can be completely canceled by adding or removing lipid from one of the leaflets. Conversely, if a patch of bilayer has a difference in the area between the two leaflets then one could—in principle—change the spontaneous curvature of the lipids to cancel the effect of the area difference on the geometry. The two parameters can therefore be combined into a single free parameter, \( \Delta a_{0} = \Delta A_{0} /A + \kappa_{b} DC_{0} /\pi \bar{\kappa }, \) the preferred area difference33 (See Supporting Information). Δa 0 can be altered by a variety of agents that affect lipid geometry (e.g., salt concentration and pH) or relative numbers of lipids in the bilayer leaflets (e.g., addition of lipids or their removal through lipid-binding agents such as BSA).
A problem with the physical theories is that calculating the minimum energy surfaces using Eq. (1) is difficult. Early attempts were limited to rotationally symmetric morphologies8,34 and this is clearly inadequate for complex morphologies such as elliptocytes and echinocytes. More recent work has overcome this limitation by modeling cell surfaces explicitly,26,42 but the computations become CPU inefficient and mesh-resolution dependent. To facilitate the generation of minimum energy shapes, we have taken advantage of the power of the spherical harmonics parameterization4,22 to describe succinctly surfaces that are topologically equivalent to the sphere, which is the case for most cells. This parameterization is economical, leads to shapes invariant of position, orientation and scale (and so captures the essence of a morphology), and can be used to easily calculate geometric properties such as volume, area, and curvature. Furthermore, there is a natural metric for quantifying the similarity between two shapes, i.e. "shape correspondence."22
Minimum energy shapes were calculated using Eq. (1) with constraints of fixed area, volume and normalized preferred area difference (Δa 0). This leads to the various shapes shown in Fig. 1b. As Δa 0 is increased, the minimum energy shape changes from a stomatocyte through a discocyte to an echinocyte. The shapes in Fig. 1b resemble qualitatively the real RBCs shown in Fig. 1a. This confirms that inclusion of the mechanical properties of the membrane skeleton (E MS) leads to minimum energy shapes that are in qualitative agreement with experimental observations.26
In the absence of the membrane skeleton, the theory predicts that the discocyte can only accommodate a limited range of increased area difference. Increasing Δa 0 produces a biconcave ellipsoid (the elliptocyte) and then a dumbbell (Fig. 1c). Still further increase is expected to lead to vesiculation.27 Shapes similar to those in Fig. 1c, as well as vesiculation, have been observed with pure lipid bilayer vesicles.20 By contrast, echinocytes have not been observed with pure lipid vesicles, presumably because the stiffness of the membrane skeleton is needed to prevent the spicules of echinocytes from budding off.
In addition to being necessary for stabilizing the echinocyte morphology, we discovered that the membrane skeleton plays an important role in stabilizing the discocyte morphology. In the presence of a membrane skeleton, discocytes are present as Δa 0 ranges from 0.14 to 0.89%, corresponding to a 0.75% relative deviation in areas of the lipids in the two leaflets of the bilayer from the preferred area (Fig. 1b). Our modeling showed that this wide range did not depend on whether the underlying (undeformed) shape of the MS was oblate (as shown here), prolate, or discocytic (data not shown). By contrast, in the absence of a membrane skeleton, this range is reduced by about one-hundred fold (Fig. 1c), indicating that in this case the discocyte morphology should be highly susceptible to changes in interleaflet lipid composition.
RBC Preparation and Imaging
RBCs (group 0, Rh positive) freshly obtained from a finger prick were diluted into cell buffer (154 mM NaCl, 10 mM Hepes, 0.1% glucose, pH 7.4) immediately after drawing. The suspension was segregated by density using a Percoll gradient, and RBCs corresponding to middle age (i.e., found in the middle of the density gradient column) were used to reduce morphological variation. For fluorescence imaging, cells were labeled using Calcein and DiI to mark the cytoplasm and plasma membrane respectively. Slides and coverslips were coated with bovine serum albumin (BSA) in order to prevent adhesion and echinocytosis upon contact with the glass surface.
Shape Parameterization
We parameterized the cell contour for both theoretical calculations and for fitting 3D image data4 using expansions of the Cartesian coordinates of the surface S(x, y, z) in spherical harmonic basis functions \( Y_{LK} (\theta ,\phi ), \) \( x(\theta ,\phi ) = \sum\nolimits_{L = 0}^{{L_{\max } }} {\sum\nolimits_{K = - L}^{L} {C_{LK}^{x} Y_{LK} (\theta ,\phi )} } , \) and similarly for y and z, where L and K are integers with \( - L \le K \le L, \) and \( 0 \le L \le L_{\max } , \) θ and φ are the usual spherical polar coordinate angles and L max the highest expansion order. The three sets of expansion coefficients \( (C_{LK}^{x} ,\,C_{LK}^{y} ,\,C_{LK}^{z} ) \) completely define the shape. L max was set to 12 for the shapes in Fig. 1 and to 16 for all other shapes in this work. The value was kept high enough to assure that coefficients corresponding to the highest L values had negligible values throughout. To measure the correspondence between two shapes a and b we calculate a shape correlation coefficient \( 0 < R_{a,b} \le 1 \) as \( R_{a,b} = 1 - {{\left( {\sum\nolimits_{i = 1}^{N} {(C^{a,i} - C^{b,i} )} } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\nolimits_{i = 1}^{N} {(C^{a,i} - C^{b,i} )} } \right)} {\left( {\sum\nolimits_{i = 1}^{N} {((C^{a,i} )^{2} + (C^{b,i} )^{2} )} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\sum\nolimits_{i = 1}^{N} {((C^{a,i} )^{2} + (C^{b,i} )^{2} )} } \right)}} \) where \( N = 3 \times (L_{\max } + 1)^{2} \) is the total number of coefficients that describe a shape, the C a,is and C b,is represent the ith corresponding shape coefficients that have been transformed into their translational and rotational invariant form. The above parametric shape description is free of the limitation that the shape must contain a point inside that "sees" every point on the surface without crossing the surface, as is the case with conventional (direct) spherical harmonic radial function expansions,17 is suitable for large surface area-to-volume ratios and is significantly more economical (a discocyte can be described by 4 non-zero shape descriptors only).
3D Image Analysis
The cell outline was initially obtained using the Laplacian of Gaussian zero-crossing method31 and surface triangulated using the marching cubes algorithm.28 The surface was subsequently mapped uniformly onto the unit sphere.4 Using linear least squares, the C LK s corresponding to the x, y, and z coordinates associated with each vertex were determined. This initial parameterization was then refined using unconstrained nonlinear least-squares fitting in image space by convolving the shape corresponding to a given set of C LK s with the experimentally determined point spread function. The fitting problem is ill-posed (i.e., will not converge to the correct solution without additional information because it would try to fit the noise), so we introduced a smoothing functional weighted by the regularizing parameter γ. The smoothing functional was set to be proportional to the first term in Eq. (1) (i.e., the pure bending energy). An L-curve was then calculated to determine the optimal γ graphically.15 Confidence limits on the obtained parameters were calculated from the covariance matrix of the estimates at the solution.
Shape Energy Calculation
ΔA in Eq. (1) was calculated as \( \Delta A = D\oint {(2H)dA} , \) and other values for constants used were26 \( \kappa_{b} = 2.0 \times 10^{ - 19} \,{\text{J}}, \) \( \overline{\kappa } /\kappa_{b} = 2/\pi , \) and D = 3 nm. The membrane skeleton energy included nonlinear terms that have been shown theoretically to correspond to stiffening of the MS at high deformations,26
$$ E_{\text{MS}} = \frac{{K_{\alpha } }}{2}\oint_{{{\text{S}}_{\text{o}} }} {(\alpha^{2} + a_{3} \alpha^{3} + a_{4} \alpha^{4} )dA_{\text{o}} } + \mu \oint_{{{\text{S}}_{\text{o}} }} {(\beta + b_{1} \alpha \beta + b_{2} \beta^{2} )dA_{\text{o}} } , $$
where \( \alpha = \lambda_{1} \lambda_{2} - 1 \) and \( \beta = (\lambda_{1} - \lambda_{2} )^{2} /2\lambda_{1} \lambda_{2} \) are the local area and shear strain invariants, and λ 1,2 are the local principal stretches.13 K α and μ are the linear elastic moduli for stretch and shear, respectively and unless stated otherwise \( \mu = K_{\alpha } /2 = 2.5 \times 10^{ - 18}\, {\text{J}}/\mu {\text{m}}^{ 2} . \) 16 The nonlinear coefficients a 3, a 4, b 1 and b 2 were given the values −2, 8, 0.7, and 0.75. The integration was performed over the undeformed shape So, which was assumed to be an oblate ellipsoid that has the same A as the shape, but with a reduced volume v o = V/V sphere = 0.95. Using sequential quadratic programming,5,39 minimum-energy shapes were obtained by minimizing Eq. (1), under constraints of A and V, for given \( \Delta \overline{{a_{\text{o}} }} . \) The minimum energy shape was assumed stable if the extremum of the energy was a minimum with respect to variations of the C LK s.18
Quantitative Comparison of RBC Morphologies with Predicted Shapes
To test the energy-minimization model for RBC morphology (Eq. 1), we compared quantitatively the predicted shapes with the shapes of real cells. We incubated human red blood cells (obtained from a healthy 30-year-old male) in buffers with differing salt concentrations to obtain stomato-, disco- and echinocytes that were imaged by 3D confocal microscopy (Fig. 2a). The images were segmented to provide an initial estimate of the cell surface (Fig. 2b, c). To compare quantitatively the noisy confocal data with the shapes predicted by the model, we again took advantage of the spherical harmonics parameterization. The measured cells were represented as spherical harmonic functions by fitting the confocal data to trial shapes convolved with the point-spread function of the microscope using regularization (Supplementary Figure S1A, B). This procedure22 effectively low-pass filters the cell-shape to reduce the experimental noise (Fig. 2d–f) and produces smoothed shapes (for method validation see Supplementary Figure S1D).
Imaged RBCs analysis. (a) Confocal intensity image stack of a fluorescently membrane-labeled stomatocyte at [NaCl] = 80 mM. Each frame is 12.8 × 12.8 μm2, with a z-separation of 0.15 μm. (b) Surface triangulation of a thresholded discocyte at [NaCl] = 154 mM (physiological). (c) An echinocyte surface. (d–f), Estimated cell surfaces (see Methods) corresponding to shapes a–c with areas and volumes: stomatocyte (120 ± 3 μm2, 97 ± 2 μm3), discocyte (151 ± 3 μm2, 98 ± 2 μm3), echinocyte (165 ± 2 μm2, 109 ± 2 μm3) (mean ± SEM). (g–i) Best theoretically fitted shapes, at same A and V as (d–f), varying Δa 0, which gave values of −0.4%, 0.12 and 0.17%, respectively
From these smoothed shapes, we can estimate A and V. However, we have no way of determining Δa 0 independently. In order to compare our shapes to the theory, we fit the shapes to Eq. (1) (using the same area and volume) (Fig. 2g–i) by varying the free parameter Δa 0 (Supplementary Figures S1C and S2). Because we are expressing both the theoretically predicted shapes and the surfaces estimated from the 3D images, using the same shape parameterization, we are in a position to quantitatively match the shapes based on the values of the shape coefficients directly. The agreement between theory and measurement for the 17 cells analyzed in detail was very good (nine discocytes, four stomatocytes, two echinocytes and two elliptocytes). We found that a generalized shape correlation coefficient based on shape distance (defined in methods) had values ≥0.97. This confirms that the theoretical model associated with Eq. (1), and which includes a MS term, is able to account quantitatively for the shapes of a wide variety of real RBCs.
Disrupting the Membrane Skeleton
As another method to perturb RBC shape, in addition to ionic strength change, we used urea,2 which can cross the cell membrane and probably weakens the membrane skeleton.23 Consistent with this, urea disassembles the MS of demembranated RBCs as judged by light microscopy (Fig. 3a, Supplementary Movie). We also confirmed that addition of urea disrupted the MS even when it was added prior to demembranation (Supplementary Figure S3). A candidate target of urea is spectrin which unfolds at significantly lower urea concentrations29 than other RBC proteins such as ankyrin10,38 and band 3.41 However, we cannot rule out the possibility that urea also affects the mechanical properties of the MS through non-spectrin proteins.
Urea disrupts the membrane skeleton and changes echinocyte shape. (a) Treatment of a demembranated RBC with urea and imaged by phase contrast microscopy. Left panel: intact discocyte. Middle panel: after addition of 0.1% Triton X-100 to remove the membrane, which reveals the (contracted) MS. Right panel: after addition of 0.136 M urea. See also Supplementary Movie 1. (b) Box plots show the number of spicules of echinocyte type I shapes as a function of urea concentration (lower x-axis) measured from phase contrast images (examples of which are shown on left). To induce control (no urea) type I echinocytes, RBCs in physiological buffer were transferred to a chamber whose coverglass surface was coated with F127 (Sigma). Other echinocytes were induced by incubation in the stated urea concentration for 25 min prior to being returned to physiological buffer in a chamber whose surface was coated with fatty-acid-free BSA. The boxes show limits of lower 25% and upper 75% of data points and the whiskers represent the data range. The numbers refer to the number of cells. At 3.3 M urea, cells fragmented and no echinocyte type I cells could be identified. Filled circles (associated with the upper x-axis) represent the theoretically predicted number of spicules that echinocyte type I shapes should have when calculated using Eq. (1) within the range of 0.1% < Δa 0 < 2.0%. Some example predicted shapes are shown at right. The upper x-axis shows the values of shear modulus used to calculate the minimal energy shapes according to Eq. (1). Relative positioning of the two x-axes is justified as follows: at 3.3 M urea cells completely fragment, and spectrin is known to be denatured to more than 50%,29 so we assumed the shear modulus was zero. In the absence of urea, the shear modulus corresponds to that measured for spectrin networks on normal cells.25 (c) The effect of urea on discocytes under different NaCl concentrations. Left panels: a field of mainly discocytes in physiological buffer before and after augmentation with 1.24 M urea. Right panels: after incubation in 40 mM NaCl buffer augmented with 1.25 M urea the discocyte became a dumbbell and then vesiculated
To quantitate the effect of urea on the mechanical properties of the MS, we studied its effect on the shapes of type I echinocytes, flattened echinocytes whose spicules lie in one plane (examples in Figs. 1a and 1b) and are therefore easy to count. In our simulations we found that the number of spicules is predicted to change in a graded manner as the shear and stretch moduli of the MS are decreased. This is in accord with a previous study that used a simplified shape model specific to echinocytes.33 Specifically, as the shear and stretch moduli are decreased from their measured values16 to zero, the number of spicules is predicted by Eq. (1) to decrease from about 10 to 0 (Fig. 3b right-hand images, circles). In order to test this prediction, we prepared echinocyte I RBCs and incubated them in solutions of increasing urea concentration before returning them to physiological buffer. The number of observed spicules decreased as the urea concentration was increased (Fig. 3b left-hand images, box plots) as expected if the moduli decreased. This experiment shows that the effect of urea can be accounted for by a reduction in the stretch and shear modulus of the membrane.
Having established that urea weakens the MS, we studied its effect on the discocyte morphology. Discocytes were incubated in 1.24 M urea in physiological buffer for ~10 min. This treatment led to vesiculation (Fig. 3c, left panels), as expected if the urea removed the MS and Δa 0 was ≥0.3% (Fig. 1c). Incubation in urea at low salt also led to vesiculation (Fig. 3c, right panels). These observations show that the MS stabilizes the discocytic shape at values of Δa 0 that would be expected to result in different morphologies if there were no MS. This suggests that the MS stabilizes the discocytic shape against asymmetric changes in lipid bilayer area (i.e., changes in Δa 0).
Shape Changes Induced by BSA
In order to further test the hypothesis that the membrane skeleton stabilizes cell shape against changes in lipid bilayer area asymmetry, we treated RBCs with fatty-acid-free bovine serum albumin (BSA), which binds lipids.24 BSA has been assumed in the literature to remove lipids from the outer leaflet.6 This would decrease the value of Δa 0. Because the evidence that BSA removes outer-membrane lipids is indirect, we performed a more direct control experiment and confirmed that BSA can indeed remove outer-membrane lipids (Fig. 4a). Normal discocytes (in the absence of urea) were labeled with a fluorescent lipid that was expected to remain in the outer leaflet due to the large size of the fluorescent group. The labeling induced an echinocytic morphology (Fig. 4a, first and second panels,23,30) as expected if the added lipid increased Δa 0 (Fig. 1b, right). Subsequently fatty-acid free BSA was added to the solution. The fluorescent label was observed to be removed (Fig. 4a, third panel), and the morphology changed to that of a stomatocyte (Fig. 4a, right panel). The fluorescence decrease shows that BSA removed the added lipid from the outer leaflet (i.e., decreases Δa 0), and the associated morphological change suggests that BSA removes not just the exogenous lipid but also some endogenous outer membrane lipid.6
BSA alters lipid bilayer asymmetry and together with urea induces formation of elliptocytes. (a) Bovine serum albumin (BSA) removes lipid from the outer leaflet of the bilayer. Discocytes were treated with 3.3 μg/mL of the fluorescent lipid C16-Liss-Rhod-lysoPE, expected to remain in the outer leaflet. Consistent with this, the morphology changed to an echinocyte (left panels, fluorescence and phase contrast). After addition of 1.18% fatty-acid-free BSA, the fluorescent label was removed (right panel, fluorescence) and cells changed to stomatocytes (right panel, phase contrast). (b) After addition of 0.15% BSA, an RBC (yellow arrow) that has been treated with urea (1.1 mM urea for 60 min) undergoes a series of shape changes from dumbbell (left, time 0 s), elliptocyte (2nd from left, 58 s) through to stomatocyte (3rd from left, 62 s). (c) Reconstructions of a 3D confocal image of an elliptocyte (RBCs treated with a solution of 1.1 M urea, 65.5 mM NaCl and 12.8 mM sucrose); left (green) top and perspective intensity projections, right (red) top and perspective views of best estimated surface
Having established that fatty-acid free BSA can remove outer membrane lipids and decrease Δa 0, we investigated the effect of BSA on the morphology of urea-treated RBCs. Over time after the BSA was added, urea-treated cells gradually changed from dumbbells to elliptocytes, passed briefly through discocytes and stomatocytes before vesiculating internally (Fig. 4b). This is identical to the sequence of morphologies expected if urea treatment of RBCs made them behave like pure bilayer vesicles (Fig. 1c) and fatty-acid free BSA reduces Δa 0. The similarity of the observed shape changes to the one predicted in the absence of a MS provides further evidence that urea does indeed remove the mechanical contribution of the MS to RBC morphology, though we cannot rule out the possibility that the removal of the MS by urea or that urea itself has a secondary effect on cell morphology.
Fatty-acid free BSA has a more pronounced effect on the shape of urea-treated RBCs than on untreated RBCs. After urea-treatment, the transition from discocyte to stomatocyte in 0.01% BSA took less than 1 min (data not shown) and led quickly to vesiculation (within another minute); we observed some variability in the RBCs response to this treatment especially variability in delays among morphological changes of different cells, even though we tried to minimize variability by centrifugation to separate cells of similar density and age (see Methods). In untreated cells, the transition was much slower though (from 2 to 5 min) and no vesiculation was observed even after 20 min. Thus urea makes the shape of RBCs more susceptible to changes in outer membrane area.
Formation of Elliptocytes by Urea Treatment
It is known that the human blood disease hereditary elliptocytosis is caused by a mutation in spectrin that prevents the proper head-to-head association of spectrin dimers to form the heterotetramers that make up the spectrin network.7 That the mutation leads to a mechanical weakening of the MS is in accord with our theoretical prediction of an elliptocytic morphology in the absence of a MS (Fig. 1c). To obtain additional evidence that elliptocytosis can be explained solely on the basis of cell mechanics, we confirmed by spherical harmonic analysis of confocal microscope images that biconcave elliptocytes can indeed be observed after urea treatment of freshly prepared discocytes (Fig. 4c). Importantly, the value of Δa 0 (=0.21%) that we obtained by analyzing 3D imaged elliptocytes using a model lacking the MS (Supplementary Figure 2), corresponds to a discocyte morphology if the MS were present (Fig. 1b and Supplementary Figure 1C). In other words, the membrane asymmetry of our discocytes would lead to a default shape being an elliptocyte if it were not for the membrane skeleton.
We have explored the mechanical basis underlying RBC morphology by comparing theoretical predictions with 3D imaging of live cells. Using a new computational approach for the quantitative analysis of cellular shape, the spherical harmonic parameterization, we showed that the MS can account quantitatively for a wide range of RBC morphologies induced by changes in buffer NaCl and addition of urea and BSA.
We found theoretical and experimental support for the hypothesis that the membrane-associated cytoskeleton is important for stabilizing the shape of RBC against perturbations such as asymmetric changes in lipid bilayer composition. Our theoretical modeling showed that in the absence of the membrane skeleton, the discocyte is the minimum energy shape only over a very small range of Δa 0. But in the presence of the MS, the discocyte is predicted to be a minimum-energy shape over a ~100-fold larger range of Δa 0, consistent with a stabilizing function for the MS. In addition, we showed experimentally that urea-treated cells, unlike normal cells, are highly susceptible to BSA, which extracts lipids from the outer leaflet and decreases the area difference.
These observations suggest that one of the functions of the membrane skeleton is shape homeostasis, i.e., buffering the cell shape against lipid perturbations. Such perturbations occur when lipids enter or leave the outer leaflet from the blood plasma or translocate across the bilayer into or out of the inner leaflet. Such a homeostatic function may be important for RBCs whose lipid composition depends for example on diet.14 When the shape buffering capacity is lost, for example in blood diseases such as hereditary elliptocytosis, the shape of the red blood cells reverts to a new default shape, the elliptocyte, whose morphology we have been able to mimic in vitro. The mechanical interaction between the lipid bilayer and the cytoskeletal structural proteins may be important in other blood cell dysmorphologies, such as in hereditary pyropoikilocytosis and spherocytosis, and shape homeostasis may be a widespread function of the cell cortex.
RBC:
Red blood cell
MS:
Membrane skeleton
BSA:
Bovine serum albumin
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We thank Britta Schroth-Diez for help in early stages of this project, Marzuk Kamal for providing the lysoPE, the MPI-CBG Light Microscopy Facility, and members of the Howardlab for comments on earlier drafts of the manuscript.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstrasse 108, D-01307, Dresden, Germany
Khaled Khairy
, JiJinn Foo
& Jonathon Howard
EMBL-Heidelberg, Meyerhofstraße 1, D-69117, Heidelberg, Germany
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Correspondence to Khaled Khairy or Jonathon Howard.
Denaturation of the RBC membrane skeleton by urea.Triton X-100 was slowly added to discocytes which were imaged by phase contrastmicroscopy, until it reached a concentration of 0.095% in physiological buffer. During thistime the cells changed from discocytes to stomatocytes to spherocytes, and then lysed.Following lysis, the MS appears as a shrunken network. At 11 min:40 s, urea was added toreach a concentration of 0.136 M and the MS disassembled (MPG 2974 kb)
ESM 1 (DOC 1082 kb)
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Khairy, K., Foo, J. & Howard, J. Shapes of Red Blood Cells: Comparison of 3D Confocal Images with the Bilayer-Couple Model. Cel. Mol. Bioeng. 1, 173 (2008) doi:10.1007/s12195-008-0019-5
Membrane mechanics
Spherical harmonics
Hereditary elliptocytosis
Cell morphology | CommonCrawl |
Mathematician:Mathematicians/Sorted By Birth/501 - 1000 CE
< Mathematician:Mathematicians | Sorted By Birth
For more comprehensive information on the lives and works of mathematicians through the ages, see the MacTutor History of Mathematics archive, created by John J. O'Connor and Edmund F. Robertson.
The army of those who have made at least one definite contribution to mathematics as we know it soon becomes a mob as we look back over history; 6,000 or 8,000 names press forward for some word from us to preserve them from oblivion, and once the bolder leaders have been recognised it becomes largely a matter of arbitrary, illogical legislation to judge who of the clamouring multitude shall be permitted to survive and who be condemned to be forgotten.
-- Eric Temple Bell: Men of Mathematics, 1937, Victor Gollancz, London
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1 $\text {501}$ – $\text {600}$
1.1 Metrodorus $($$\text {c. 500}$$)$
1.2 Varahamihira $($$\text {505}$ – $\text {587}$$)$
1.3 Severus Sebokht $($$\text {575}$ – $\text {667}$$)$
1.4 Brahmagupta $($$\text {598}$ – $\text {668}$$)$
1.5 Bhaskara I $($$\text {c. 600}$ – $\text {c. 680}$$)$
2.1 Bede $($$\text {c. 673}$ – $\text {735}$$)$
3.1 Alcuin of York $($$\text {c. 735}$ – $\text {804}$$)$
3.2 Muhammad ibn Musa al-Khwarizmi $($$\text {c. 780}$ – $\text {c. 850}$$)$
3.3 Leon the Mathematician $($$\text {c. 790}$ – $\text {c. 870}$$)$
4.1 Mahaviracharya $($$\text {c. 800}$ – $\text {c. 870}$$)$
4.2 Al-Kindi $($$\text {c. 801}$ – $\text {c. 873}$$)$
4.3 Thabit ibn Qurra $($$\text {836}$ – $\text {901}$$)$
4.4 Abu Kamil $($$\text {c. 850}$ – $\text {c. 930}$$)$
5 $\text {901}$ – $\text {1000}$
5.1 Abu'l-Wafa Al-Buzjani $($$\text {940}$ – $\text {998}$$)$
5.2 Abu Bakr al-Karaji $($$\text {c. 953}$ – $\text {c. 1029}$$)$
5.3 Abu Ali al-Hasan ibn al-Haytham $($$\text {965}$ – $\text {c. 1039}$$)$
5.4 Abu Rayhan Muhammad ibn Ahmad Al-Biruni $($$\text {973}$ – $\text {1048}$$)$
5.5 Halayudha $($$\text {c. 1000}$$)$
$\text {501}$ – $\text {600}$
Metrodorus $($$\text {c. 500}$$)$
Greek grammarian and mathematician, who collected mathematical epigrams which appear in The Greek Anthology Book XIV.
He is believed to have authored nos. $116$ to $146$.
Nothing else is known about him.
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Varahamihira $($$\text {505}$ – $\text {587}$$)$
Indian astronomer, mathematician, and astrologer.
One of several early mathematicians to discover what is now known as Pascal's triangle.
Defined the algebraic properties of zero and negative numbers.
Improved the accuracy of the sine tables of Aryabhata I.
Made some insightful observations in the field of optics.
Severus Sebokht $($$\text {575}$ – $\text {667}$$)$
Syrian scholar and bishop.
The first Syrian to mention the Indian number system.
Brahmagupta $($$\text {598}$ – $\text {668}$$)$
Indian mathematician and astronomer.
Gave definitive solutions to the general linear equation, and also the general quadratic equation.
Best known for the Brahmagupta-Fibonacci Identity.
Bhaskara I $($$\text {c. 600}$ – $\text {c. 680}$$)$
Indian mathematician who was the first on record to use Hindu-Arabic numerals complete with a symbol for zero.
Gave an approximation of the sine function in his Āryabhaṭīyabhāṣya of $629$ CE.
Bede $($$\text {c. 673}$ – $\text {735}$$)$
English Benedictine monk at the monastery of St. Peter and its companion monastery of St. Paul in the Kingdom of Northumbria of the Angles.
Studied the academic discipline of computus, that is the science of calculating calendar dates.
Worked on computing the date of Easter.
Helped establish the "Anno Domini" practice of numbering years.
Produced works on finger-counting, the sphere, and division.
These works are probably the first works on mathematics written in England by an Englishman.
Alcuin of York $($$\text {c. 735}$ – $\text {804}$$)$
Hugely influential english scholar, clergyman, poet, and teacher.
Wrote elementary texts on arithmetic, geometry and astronomy.
Leader of a renaissance in learning in Europe.
Muhammad ibn Musa al-Khwarizmi $($$\text {c. 780}$ – $\text {c. 850}$$)$
Mathematician who lived and worked in Baghdad.
Famous for his book The Algebra, which contained the first systematic description of the solution to linear and quadratic equations.
Sometimes referred to as "the father of algebra", but some claim the title should belong to Diophantus.
Leon the Mathematician $($$\text {c. 790}$ – $\text {c. 870}$$)$
Archbishop of Thessalonike between $840$ and $843$.
Byzantine sage at the time of the first Byzantine renaissance of letters and the sciences in the $9$th century.
He was born probably in Constantinople where he studied grammar.
He later learnt philosophy, rhetoric, and arithmetic in Andros.
Mahaviracharya $($$\text {c. 800}$ – $\text {c. 870}$$)$
Indian mathematician best known for separating the subject of mathematics from that of astrology.
Gave the sum of a series whose terms are squares of an arithmetical sequence and empirical rules for area and perimeter of an ellipse.
Al-Kindi $($$\text {c. 801}$ – $\text {c. 873}$$)$
Persian mathematician, philosopher and prolific writer famous for providing a synthesis of the Greek and Hellenistic tradition into the Muslim world.
Played an important role in introducing the Arabic numeral system to the West.
Thabit ibn Qurra $($$\text {836}$ – $\text {901}$$)$
Sabian mathematician, physician, astronomer, and translator who lived in Baghdad in the second half of the ninth century during the time of Abbasid Caliphate.
Made important discoveries in algebra, geometry, and astronomy.
One of the first reformers of the Ptolemaic system in Astronomy.
A founder of the discipline of statics.
Abu Kamil $($$\text {c. 850}$ – $\text {c. 930}$$)$
Egyptian mathematician during the Islamic Golden Age.
Considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations.
His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.
$\text {901}$ – $\text {1000}$
Abu'l-Wafa Al-Buzjani $($$\text {940}$ – $\text {998}$$)$
Persian mathematician and astronomer who made important innovations in spherical trigonometry.
His work on arithmetic for businessmen contains the first instance of using negative numbers in a medieval Islamic text.
Credited with compiling the tables of sines and tangents at $15'$ intervals
Introduced the secant and cosecant functions, and studied the interrelations between the six trigonometric lines associated with an arc.
His Almagest was widely read by medieval Arabic astronomers in the centuries after his death. He is known to have written several other books that have not survived.
Known for his study of geometrical dissections.
Pioneered the technique of geometrical construction using a rusty compass.
Abu Bakr al-Karaji $($$\text {c. 953}$ – $\text {c. 1029}$$)$
Persian mathematician best known for the Binomial Theorem and what is now known as Pascal's Rule for their combination.
Also one of the first to use the Principle of Mathematical Induction.
Abu Ali al-Hasan ibn al-Haytham $($$\text {965}$ – $\text {c. 1039}$$)$
Persian philosopher, scientist and all-round genius who made significant contributions to number theory and geometry.
His work influenced the work of René Descartes and the calculus of Isaac Newton.
Abu Rayhan Muhammad ibn Ahmad Al-Biruni $($$\text {973}$ – $\text {1048}$$)$
Khwarazmi scholar and polymath.
Thoroughly documented the Indian calendar with relation to the various Islamic calendars of his day.
Appears to be the first to have defined a second (of time) as being $\dfrac 1 {24 \times 60 \times 60}$ of a day.
Halayudha $($$\text {c. 1000}$$)$
Indian mathematician who wrote the Mṛtasañjīvanī, a commentary on Pingala's Chandah-shastra, containing a clear description of Pascal's triangle (called meru-prastaara).
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Lists of Mathematicians | CommonCrawl |
HSC Physics Solutions
2018 HSC Physics Paper Solutions and Explanations
In this post, we reveal the solutions to the 2018 HSC Physics Exam Paper!
How did you go in the 2018 HSC Physics Exam? Read this post to check the answers and explanations to the 2018 HSC Physics Exam Paper.
2018 HSC Physics Paper Solutions
This paper was broken into two sections.
Section 1 is based on the core topics and has 2 parts:
Part A – 20 Multiple Choice Questions
Part B – 10 Short Answer Questions
Section 2 is based on option topics and has five questions. We have included the answers to the most popular option topics:
From Quanta to Quarks
Section 1 – 75 Marks
Part A – 20 Marks
Question Answer Explanation
A The gravitational force is towards the centre of the Earth, hence the acceleration is towards the centre of the Earth
B The graph as shown should be intensity vs wavelength, and higher temperatures emit more radiation.
A The ISS collides with air molecules in the upper atmosphere (air resistance) and loses momentum.
D The decrease in rotation speed decreases the back EMF, which increases total EMF and current.
B Force is given by \( F = B I L sin \theta = 1 \times 2 \times 0.08 \times sin(30) = 0.05 N \)
B The saucepan must be a conductor for currents to be induced. A high AC frequency is necessary for a high change in flux and high induced current.
A The acceleration at the surface of a planet is proportional to the mass, and the inverse of the square of radius; Doubling the mass doubles the acceleration, but doubling the radius decreases the acceleration by a factor of four – these combine for a new 'g' that is ½ of the old.
D Lattice U is a p-type, as the dopant provides 3 electrons. There is still a band gap between the conduction and valence band, meaning the corresponding diagram is X. (Note: Lattice T is an n-type semiconductor, and band diagram Y corresponds to a metal.)
A A decreased distance to travel will yield a reduced travel time, hence the astronaut can survive the journey.
C The direction of torque must be constant – otherwise the motor will not work. Hence the direction of force on element WX must reverse every 180 degrees.
D An increased g force is due to an upward acceleration of the object containing a person; case D is the only example where this occurs.
A The electrons will experience an upward force towards the positive plate.
Using the right hand rule (or the left hand rule for an electron), the magnetic field must be into the page to provide a balancing force downwards.
The magnitude can be calculated by equating
$$qE = qvB$$
$$B = \frac{E}{v} = 0.5 T$$
B The force on the electron will increase, as F = qvB.
This force is the centripetal force of circular motion; by equation \( qvB = \frac{mv^{2}}{r}\), we can conclude that v is proportional to r.
Hence the radius will increase.
C The relationship between L and T is \(L = \frac{gT^{2}}{(4\pi^{2})} \), so only C or D show this square relationship.
As g is larger on Earth, a given period will be achieved with a longer length compared to Mars, hence the graph for Earth will be above that of Mars.
D Michelson attempted to demonstrate a difference in travel times for arms parallels / perpendicular to the direction of earth's travel due to aether wind.
A A moving observer will see length contraction for all other objects; hence they will see the tunnel contract. Due to relativity of simultaneity, they will see photo 2 being taken before photo 1, as they are travelling towards the location of the photo-2 event.
C The graph shows the work function for zinc is the highest; hence any photon energetic enough to extract electrons from zinc will also extract electrons from potassium.
B The induced current will oppose any change in flux. Via right hand grip rule, the lower solenoid will generate a downward B field. The solenoid above will thus generate an upward B field to counter this change; these two fields will repel one another and the pointer will move up. The direction of current can be found again from right hand grip rule.
B An observer inside the bus will see the the ball move towards the front of the bus. From the perspective of a passenger, the ball has both a forward (due to the \(3 m/s^{2}\) slowdown of the bus) and downward acceleration; this will result in a linear motion! The initial velocity of the ball from the perspective of the passenger is zero.
C The needle in X will be forced leftward to oppose the change in flux from the rotating galvanometer. Hence, the generated current must be such that the needle moves to the left; this is a negative current. Tracing the wires from X to Y shows that a negative current coming from X will also yield a negative current through Y, and hence the needle will move to the left.
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Part B – 55 Marks
Question 21 ( 4 Marks)
(a) The two forces are equal and opposite, according to Newton's 3rd Law
(b) Mass on Earth and Moon is 70 kg.
Weight on Earth is w = mg = 70 x 9.8 = 686 N
Weight on the Moon is w = mg = 70 x 1.6 = 112 N
Question 22 (6 marks)
(a) The motion of the magnet will result in regions of the disc experiencing a change in magnetic flux as the poles of the magnet move across them.
This will result in an induced EMF (Faraday's Law) in the disc and eddy currents.
The eddy currents will produce a magnetic field to oppose the change in the flux (Lenz's Law) which will result in a force and a torque on the disc.
The disc will rotate in the same direction as the magnet
(b) Graph needs to show a sinusoid with period decreasing and amplitude increasing over the first three seconds
(a) The electrons absorb thermal energy (heat) from the heating filament and escape the metal. The electric field in the electron gun does work on the electrons and increases their kinetic energy. When they strike the screen they stop and their kinetic energy is transferred to the phosphor in the screen, which converts it to heat and light.
(b) Y are solenoids (electromagnets) which produce magnetic fields. The magnetic fields are used to deflect the electron beam and scan it across the screen. There must be two pairs of such coils, one for controlling horizontal deflection and one for controlling vertical deflection.
The image is formed by raster scanning, i.e. by scanning the beam across the screen from pixel to pixel and changing the power of the beam (e.g. the no. of electrons) as the beam hits each pixel to change the intensity of that pixel. Once an image is formed, the scanning repeats to refresh the image
(a) The high voltage wires are attached to the supporting structures using insulators in order to electrically isolate them from the ground. Lightning protection is used: wires placed above the transmission lines, and the supporting towers are metal, well earthed and separated from each other. Current from lightning strikes can safely be conducted to the ground
(b) Wires will attract each other if the current is in the same direction.
In order for there to be zero net force on Y, it must be attracted to X by the same strength as it is attracted to Z, hence the current in X is to the right.
$$F = \frac{kI_{1}I_{2}L}{d}$$
$$\frac{kI_{X}I_{Y}L}{d_{XY}} =\frac{kI_{Y}I_{Z}L}{d_{YZ}} $$
$$\frac{I_{X}}{0.75} =\frac{20}{320} $$
$$I_{X} = 50 A$$
(a) Benefit: No power loss due to heating as resistance is zero. Limitation: Must be kept below their critical temperature in order to be superconducting (have zero resistance), which is difficult and consumes energy.
(b) The table compares the metal and superconductors.
Cooper pairs experience no resistance. Through their interaction with the lattice one electron loses one phonon of energy and the second electron absorbs one phonon. Hence there is no net energy loss for the Cooper pair.
Metal at room temperature Superconducting metal below critical temperature
Conduction by Single electrons Pairs of electrons (Cooper pairs)
Mechanism Electrons in the conduction band are mobile and can move through the metal. At low temperatures lattice vibrations reduce. An electron will attract nearby positive ions causing lattice distortion and will lose one phonon of energy. The lattice distortion will increase positive charge density which attracts a second electron that absorbs one phonon of energy. These two electrons are paired through the lattice and form the Cooper pair.
Resistance Electrons experience resistance.
As electrons move through the lattice they collide with positive ions. These collisions are inelastic on average and the electrons lose energy to the lattice (converted to heat). Cooper pairs experience no resistance.
Through their interaction with the lattice one electron loses one phonon of energy and the second electron absorbs one phonon. Hence there is no net energy loss for the Cooper pair.
Gravitational Field Electric Field
Acts on Mass Charge
Force it mediates Gravitational force of attraction Electric Force
Definition and units Force per unit mass: N/kg
$$g = \frac{F_{G}}{m}$$
Direction is the direction of force on a mass Force per unit charge: N/C
$$E = \frac{F_{E}}{q}$$
Direction is the direction of force on a positive charge
Acceleration of object All masses in a constant field will accelerate with the same acceleration and in the direction of the field All charges in a constant field will experience an acceleration that depends on their charge-to-mass ratio (q/m). Positive charges accelerate in the direction of the field and negative charges in the opposite direction.
Trajectory in a uniform field Parabolic trajectory (projectile motion)
Force and acceleration are constant in magnitude and direction.
Component of velocity perpendicular to the field remains constant. Particle accelerates in direction of force.
(a) The camera is viewing the projectile and the ruler at an angle resulting in a parallax error. This will result in a systematic error in the horizontal position of the particle
(b) The horizontal velocity remains constant, the horizontal acceleration is zero. The gradient of the graph gives the velocity:
$$v_{x} = -1.6 m/s $$
The vertical acceleration is constant so the vertical velocity changes at a constant rate.
The initial velocity is 4.2 m/s, the final velocity is -4.8 m/s.
The gradient of the graph gives the acceleration: \( a_{y} = -9.5 m/s^{2} \)
Question 28 ( marks)
(a) $$\Delta Ep = Ep_{f} – Ep_{i}$$
$$= \frac{-GMm}{r_{f}} – \frac{-GMM}{r_{i}}$$
$$= -GMm [\frac {1}{r_{f}} – \frac{1}{r_{i}}]$$
$$= -6.67 \times 10^{-11} \times 7.35 \times 10^{22} \times 20 [\frac{1}{1740 000+500 000} – \frac{1}{1740000}]$$
$$= 1.257… \times 10^{7} J$$
$$= 1.26 \times 10^{7} J$$
(b) If the potential energy increases by \( 1.26 \times 10^{7} J \), the kinetic energy must decrease by the same amount to conserve energy.
\( KE_{Initial} = \frac{1}{2} mv^{2} = \frac{1}{2} \times 20 \times 1200^{2} = 1.44 \times 10^{7} J \)
\(KE_{Final} = 1.44 \times 10^{7} – 1.26 \times 10^{7} = 1.8 \times 10^{6} J \)
\(\frac{1}{2} mv^{2} = 1.8 \times 10^{6} J \)
\(v = 424 m/s\)
(a) Vacuum photocell:
Electrons in the photocathode (A) will each absorb one photon.
Assuming the photon energy is higher than the work function (frequency is above the threshold frequency for that metal), the energy absorbed from the photons will allow the electron to overcome the work function and be ejected from the metal.
The electric field between A and B will accelerate the electrons through the vacuum in the photocell towards B, resulting in a current through the circuit.
Solar Cells:
Combining the n and p-type materials will form a p-n junction.
Electrons from the n-type will diffuse into the p-type, causing a loss of charge carriers, and causing the n-type to become negatively charged and the p-type to become positively charge in the junction, resulting in an electric field being formed. This region is called the depletion layer.
When light strikes the depletion layer (assuming the photon energy is larger than the bandgap) it can be absorbed by an electron in the valence band, which becomes excited to the conduction band, leaving behind a hole.
The electric field will accelerate the hole into the p layer and the electron into the n layer. This charge separation creates a potential difference between the p and n type which results in current flowing from p to n through the circuit
(b) Advance in understanding: understanding of band structure in semiconductors
Our understanding of the valence-conduction band interaction, and particularly the band gap has enabled the development of semiconductor technology. In particular, the relationship between the energy of the band gap and the wavelength of emitted or absorbed photons have prompted the use of semiconductors in light generation (LEDs) and energy generation from light (Solar Panels).
For these inventions to be implemented, however an advance in technology was required: purification of semiconductors and doping to control the band gap of semiconductors. With the ability to engineer a precise, controlled band-gap light emitters and absorbers operating at specific wavelengths could be generated, leading to the wide variety of coloured LEDs and the high efficiency of solar panels, resulting in a substantially increased usage of semiconductors.
An increase in the number of electrical appliances being used in houses means the houses will draw use more energy. This will require them to draw more power (P = E/t) and more current from the secondary coil of step-down transformer T2 (P= VI, and V = 240 V = constant).
The step-down transformer T2 is fixed, so if more power is being drawn from its secondary coil, more power must be supplied to its primary coil in the form of an increase in current. The input voltage must remain fixed since the output voltage is constant at 240 V.
Similar, more current will be drawn from step-up transformer T1, and hence more current must be supplied to it.
Effects on
Generator/power station: Rotates at the same speed to keep voltage and frequency fixed. Will require more torque (mechanics energy input) in order to produce more current and hence more power or energy.
Transformers: Voltages remain constant but currents increase. The higher current Increases the heat produced (Joule heat losses) which means some of the energy is converted to heat and the transformers' efficiency reduces.
Power lines: The increased current will increase the heat produced meaning the some electrical energy is converted to heat. This also results in a voltage drop across the power line depending on the current and resistance of the power line (V = IR).
2. Section 2 – 25 marks
Question 32 – Medical Physics (25 marks)
(a) (i) MRI is suitable for use in examining soft tissue with high water content. It is useful for locating tumors and soft tissue disease, and to diagnose Multiple Sclerosis (MS).
(ii) The application of a strong magnetic field changes the spin orientation of hydrogen nuclei in the body. From an initial random state, the nuclei align either parallel or anti-parallel. A slight majority of nuclei are parallel – this constitutes the lower energy configuration.
(b) (i) At a boundary between tissues where there is an acoustic impedance mismatch, a fraction of the ultrasound wave will be reflected, and the remainder of the wave will be transmitted.
Due to the differing impedances of the two materials, both waves B and A will experience some reflection at the fat-kidney interface. The reflection coefficient is a product exclusively of the differences and sums of the impedances, regardless of which is the incident and transmissive medium – there should be no difference between the transmission of wave A and wave B.
(ii) Impedance of kidney tissue, \( Z = \rho \times v = 1050 \times 1560 = 1.638 \times 10^{6}rayls\)
$$R=\frac{(Z_{kidney} – Z_{fat})^{2}}{(Z_{kidney} + Z_{fat})^{2}}$$
Substitute R = 0.01
$$0.01=\frac{(Z_{kidney} – Z_{fat})^{2}}{(Z_{kidney} + Z_{fat})^{2}}$$
$$0.1=\frac{Z_{kidney} – Z_{fat}}{Z_{kidney} + Z_{fat}}$$
$$0.1(Z_{kidney} + Z_{fat})=Z_{kidney} – Z_{fat}$$
$$0.9Z_{kidney}=1.1Z_{fat}$$
Substitute \( _{kidney} = 1.638 \times 10^{6} rayls\)
\( Z_{fat} = 1.340 \times 10^{6} rayls \)
(c) Criteria for suitability in diagnosis; emission type, half life and chemical compatibility (toxicity).
Emission Type: I-124 is a positron emitter – these particles will rapidly self-annihilate and generate a pair of photons, usually gamma rays. These can be used in positron emission tomography for high-quality medical imaging. In addition, positrons have a low penetration and acceptable ionisation strength. Hence I-124's emission is appropriate.
Half life: from the data, we can see that iodine-124 has a half life of approximately 100 hours. This is a suitable half-life for a medical imaging isotope; not so long that it stays active in the body and causes prolonged harm, but not so short as to become a logistic obstacle.
Chemical compatibility – Iodine is a trace element found in the human body; it is unlikely to have any negative chemical / toxicity effects.
(d) The change in frequency caused by the relative motion between a sound source and an observer is called the Doppler Effect.
The Doppler effect is used in Doppler sonography and ultrasonography. Acoustic waves reflecting off a moving target will experience a change in frequency proportional to the velocity and direction of the target. Hence this allows the flow of fluids in the target to be measured.
Specific applications include:
Detection of blood clots by identifying flow slowdowns / stoppages
Detection of abnormal aortic flows
Monitoring foetal heart rate
Measuring a 'flow velocity waveform' – the cycle of fast and slow blood flow as the heart pumps.
X-ray Image CAT Image
Mode of Detection X-Ray Transmittance X-Ray Transmittance
Image Generation X-Ray source and Detector are static.
Patient stands between x-ray source and detector. Density fluctuations in patient's body show up as intensity variations on photographic film. X-Ray source and Detector move around a patient
Patient lies in centre of CT machine, which rotates around patient and takes several X-ray images at different angles. These exposures are computed together into a 3D image of the target structure.
Image Quality Lower resolution, 2 Dimensional. Worse contrast than CT Higher Resolution, 3 Dimensional. Better contrast than X-Ray
Patient Risk Minimal Higher than X-ray due to high radiation dose as a great number of x-ray images are taken.
Application It is useful for examining hard tissues such as bone structure (e.g. viewing bone fractures) It is useful for examining and differentiating soft tissue structures with similar densities such as:
Viewing deep soft tissue structure such as vertebrae in the spine or brain tumours.
Viewing dilatation of the large blood vessels of the heart.
Benefits Quick and cheap to produce images regarding structure of bone CT images have much higher contrast than X-ray and can produce cross sections of the patient or 3D images.
CT scans are much better at looking at soft tissue structures
Question 33 – Astrophysics (25 marks)
(a) (i) One parsec is the distance to an arc or chord that subtends an angle of parallax of 1 arc second.
(ii) Trigonometric parallax measurements are limited by the resolution of the telescope:
\(\Delta \theta = \frac{1.22 \lambda}{D}\)
Where D is the aperture diameter, \(\lambda\) is the observed wavelength and \(\Delta \theta\) is the size of the Airy disc.
A parallax can be observed if the shift in apparent position can be resolved according to Rayleigh's criterion.
Atmospheric transparency is good for only radio and optical wavelengths. Shorter wavelengths (UV) are blocked by the atmosphere so the smaller parallaxes measurable can only be observed from space.
Ground-based optical telescopes at any altitude also contend with atmospheric turbulence that distorts incoming wavefronts and blurs the image. This decreases effective resolution hence limiting parallax measurements making resolution worse than the above equation for \(\Delta \theta.\)
(b) (i)
A: Eclipsing binary B: Cepheid variable (Type I)
Extrinsic variable Intrinsic variable
Two stars of different luminosity orbit each other and periodically block light from each other. The large dip (primary eclipse) is where the more luminous star is eclipsed. The smaller dip (secondary eclipse) is where the less luminous start is eclipsed. Peak brightness is when both stars are fully unobscured. One star whose atmosphere pulsates in size and temperature and hence luminosity. During contractions temperature is higher and so is the luminosity and apparent magnitude. When expanded the temperature is lower and so is the luminosity, and apparent magnitude is fainter.
(ii) From graph B:
\(T = 7 – 1 = 6\) days
From the T-L graph, absolute magnitude is \(M = -3.2\).
$$M = m – 5 log (\frac{d}{10})$$
$$d = 10^{(\frac{(m-M+5)}{5})}$$
$$\frac{(m-M+5)}{5} = \frac{(16.3 + 3.2 +5)}{5} = 4.9$$
$$d = 10^{4.9}$$
\(d = 79 400 \text{pc}\) (3 sig. fig.)
(c) Star X starts as main sequence blue giant B-class star with core hydrogen fusion proceeding via the CNO cycle. When sufficient helium is deposited into the core, hydrogen fusion continues in a shell around the core and helium core fusion begins.
Increased radiation pressure expands the outer layers causing cooling and the star is now a red supergiant.
Carbon from the helium fusion can fuse in the core while hydrogen and helium fuse in surrounding shells (outer shells with lighter elements). The core region is like shells of an onion.
Oxygen fusion should occur yielding a larger, redder, red supergiant.
The star ends life as a supernova explosion (creating many other elements) while the core of oxygen, neon and other elements collapses into a neutron star.
Note: A flowchart may also be sufficient as long as it did not compromise on detail.
(d) Photoelectric technologies (e.g. CCD cameras) have improved measurements in astrometry, photometry and especially spectroscopy.
Higher spatial resolution over photographic plates yields better parallax and distance measurements.
Higher accuracy and reliability in photometry yields improved understanding of how colour relates to other properties of stars (luminosity etc) with wider frequency coverage.
Spectroscopy sees the greatest benefits. Photographic spectra are 2D images where relative spectral line intensities cannot be determined quantitatively accurately. Photoelectric detectors are not just more sensitive to a wider range of wavelengths (hence yielding more chemical information from other spectral lines) but allow 1D spectra to be produced easily where spectral line intensities can be quantified precisely. This improves calculations of redshift (or blueshift) for radial velocities and line broadening due to Doppler broadening (thermal and rotational) and pressure broadening. Across the board, furthermore, CCDs yield faster data collection, reduced data loss and improved data comparability.
(e) Emission Spectra:
Stellar emission spectra are observed by looking at the atmosphere of the star against the backdrop of space. That means observing hot gas against a colder background. The resulting spectrum shows emission lines of various species (chemical composition) and which lines of which species are seen indicates the temperature of the stellar atmosphere. In a school lab, one can observe a gas lamp (e.g hydrogen or mercury) or fluorescent light tubes through a spectroscope in a darkened room.
Blackbody Spectra:
These are produced by blackbody radiators such as stellar cores. Since such objects are never without a surrounding atmosphere, they are not seen directly but as the underlying envelope of stellar absorption spectra.
Blackbody spectra alone yield no chemical information (is a smooth spectrum by definition) they indicate accurately the surface temperature of the star from the distinctive peak wavelength of the spectrum, where \( \lambda_{peak}\) is proportional to \(\frac{1}{temperature}\). This means higher temperature yields shorter peak wavelength (hotter, bluer stars).
In a school lab, one could observe the glowing tungsten filament of an incandescent light bulb through a spectroscope in a darkened room.
Absorption Spectra:
These are produced by observing relatively cooler gas against a brighter and hotter background. Hence stellar absorption spectra are obtained by observing a star directly.
The result is a blackbody spectrum (from the inner stellar core) superimposed with dark lines or dips. The peak of the blackbody yields temperature while the absorption lines yield chemical composition since all atoms are ions have unique characteristic wavelengths of absorption (and emission). Which line of which species also indicates temperature.
In a school lab, one can observe a white light through a transparent box of gas (e.g. hydrogen, oxygen, air) or by dispersing a sample in a flame and observing the flame with a bright backlight using a spectroscope in a darkened room (e.g. AAS in Chemistry).
Question 34 – From Quanta to Quarks (25 marks)
(a) (i) Bohr's model could not predict atomic spectra for atoms containing more than one electron. It did not explain the mechanisms of stable orbits. It could not explain the Zeeman effect or hyper fine splitting of spectral lines.
(ii) Bohr's model proposes that electrons exist in specific, stable orbitals around the nucleus. These orbitals have a specific energy. Electrons at a high energy orbital may decay to a lower energy orbital by emitting a photon of light with an energy equal to the difference of the two orbitals; because all of these orbitals are discrete, specific levels, there are only a few photon energies (hence wavelengths) that will be emitted.
(b) (i) Chadwick used the laws of conservation of momentum and energy to identify the mass of particle X. He measured the final momenta and energy of the ejected protons, and used these conservation laws to infer the mass of the particle X.
(ii) Mass of Reactants: 4.0012+9.0122 = 13.0134 amu.
Mass of Products: 12 + 1.0087 = 13.0087 amu.
Mass defect = Mass of Reactants – Mass of Products = 0.0047 amu.
Convert to kg by multiplying by \(1.661 \times 10^{-27} = 7.8067 \times 10^{-30} kg. \)
Convert to joules via \(E = mc^{2} = 7.0260 \times10^{-13} J \)
(c) Graph of kinetic energies of beta particles after a specific beta decay reaction. Graph should be a distribution showing a continuous range of different energies are possible – this contrasts to the 'expected' kinetic energy, which should be a single value – this expected value should be shown on the graph, at the highest energy point.
Pauli's proposal of the neutrino was a solution to the energy distribution of beta particles in beta decay – for a given decay the total energy released was constant, and it was expected that the kinetic energy of the released beta particle would be constant – as it is by far the lighter of the two products, it should receive all of the released energy. This was not found to be the case – the energy of beta particles varied substantially – suggesting a violation of conservation of energy. Pauli suggested an unobserved particle may be responsible, carrying some energy away from the reaction without being detected – the neutrino. As it was undetected until this stage, it must have no charge, low mass and minimal interactions with other particles
(d) From the diagrams, we can see that the decay products of
source X are likely alpha particles – their cloud trail is short and wide
source Y are likely beta particles – their cloud trails are long and thin
Each nucleus of source X undergoing alpha decay would lose 4 nucleons total; 2 neutrons and 2 protons.
Source Y is undergoing beta decay, most likely beta-minus (as positron decay would produce short tracks due to the positron annihilating itself with an electron).
In each nucleus of Y undergoing this decay, one neutron is converted into a proton, with an electron (beta-minus particle) being emitted along with an anti-neutrino.
(e) The standard model describes what matter is made of and how it interacts. According to the standard model matter is made up of quarks and leptons, and interacts through fundamental forces by exchanging force carrier particles.
There are six quarks (u, d, c, s, t, b) and six leptons (e, \mu, \tau, and three neutrinos), as well as their antimatter counterparts. The fundamental forces are the strong force (mediated by gluons) that acts on quarks and gluons, the weak force (mediated by W and Z bosons) that acts on quarks and leptons, and the electromagnetic force that acts on charged particles (mediated by photons). (The standard model does not explain gravity.)
Out current understanding of the atom is that it is composed of a nucleus consisting of positively charged protons and neutral neutrons, surrounded by negatively charged electrons in discrete energy levels. The electrons interact with the nucleus through the electromagnetic force mediated by photons. The electrons are attracted to the nucleus through this force which binds them to it.
In the nucleus, the protons and neutrons (nucleons) are examples of hadrons – composite particles made of three quarks (uud for protons, and udd for neutrons). The quarks in each hadron are held together by the strong force, mediated by gluons. The nucleons are held together by the (residual) strong force which is attractive at the typical separation between nucleons. In a stable nucleus, the (residual) strong force overcomes the electrostatic repulsion between the protons (due to their positive charge) and is able to bind both protons and neutrons to the nucleus. The neutrons are required as they increase the attraction due to the strong force without increasing repulsion due to the electric force, and hence they contribute to the stability of the nucleus.
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Kenvin's Hacks: Overcoming Procrastination and Scoring 99.75 | CommonCrawl |
Indifference price
In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. The indifference price is also known as the reservation price or private valuation. In particular, the indifference price is the price at which an agent would have the same expected utility level by exercising a financial transaction as by not doing so (with optimal trading otherwise). Typically the indifference price is a pricing range (a bid–ask spread) for a specific agent; this price range is an example of good-deal bounds.[1]
Mathematics
Given a utility function $u$ and a claim $C_{T}$ with known payoffs at some terminal time $T,$ let the function $V:\mathbb {R} \times \mathbb {R} \to \mathbb {R} $ be defined by
$V(x,k)=\sup _{X_{T}\in {\mathcal {A}}(x)}\mathbb {E} \left[u\left(X_{T}+kC_{T}\right)\right]$,
where $x$ is the initial endowment, ${\mathcal {A}}(x)$ is the set of all self-financing portfolios at time $T$ starting with endowment $x$, and $k$ is the number of the claim to be purchased (or sold). Then the indifference bid price $v^{b}(k)$ for $k$ units of $C_{T}$ is the solution of $V(x-v^{b}(k),k)=V(x,0)$ and the indifference ask price $v^{a}(k)$ is the solution of $V(x+v^{a}(k),-k)=V(x,0)$. The indifference price bound is the range $\left[v^{b}(k),v^{a}(k)\right]$.[2]
Example
Consider a market with a risk free asset $B$ with $B_{0}=100$ and $B_{T}=110$, and a risky asset $S$ with $S_{0}=100$ and $S_{T}\in \{90,110,130\}$ each with probability $1/3$. Let your utility function be given by $u(x)=1-\exp(-x/10)$. To find either the bid or ask indifference price for a single European call option with strike 110, first calculate $V(x,0)$.
$V(x,0)=\max _{\alpha B_{0}+\beta S_{0}=x}\mathbb {E} [1-\exp(-.1\times (\alpha B_{T}+\beta S_{T}))]$
$=\max _{\beta }\left[1-{\frac {1}{3}}\left[\exp \left(-{\frac {1.10x-20\beta }{10}}\right)+\exp \left(-{\frac {1.10x}{10}}\right)+\exp \left(-{\frac {1.10x+20\beta }{10}}\right)\right]\right]$.
Which is maximized when $\beta =0$, therefore $V(x,0)=1-\exp \left(-{\frac {1.10x}{10}}\right)$.
Now to find the indifference bid price solve for $V(x-v^{b}(1),1)$
$V(x-v^{b}(1),1)=\max _{\alpha B_{0}+\beta S_{0}=x-v^{b}(1)}\mathbb {E} [1-\exp(-.1\times (\alpha B_{T}+\beta S_{T}+C_{T}))]$
$=\max _{\beta }\left[1-{\frac {1}{3}}\left[\exp \left(-{\frac {1.10(x-v^{b}(1))-20\beta }{10}}\right)+\exp \left(-{\frac {1.10(x-v^{b}(1))}{10}}\right)+\exp \left(-{\frac {1.10(x-v^{b}(1))+20\beta +20}{10}}\right)\right]\right]$
Which is maximized when $\beta =-{\frac {1}{2}}$, therefore $V(x-v^{b}(1),1)=1-{\frac {1}{3}}\exp(-1.10x/10)\exp(1.10v^{b}(1)/10)\left[1+2\exp(-1)\right]$.
Therefore $V(x,0)=V(x-v^{b}(1),1)$ when $v^{b}(1)={\frac {10}{1.1}}\log \left({\frac {3}{1+2\exp(-1)}}\right)\approx 4.97$.
Similarly solve for $v^{a}(1)$ to find the indifference ask price.
See also
• Willingness to pay
• Willingness to accept
Notes
• If $\left[v^{b}(k),v^{a}(k)\right]$ are the indifference price bounds for a claim then by definition $v^{b}(k)=-v^{a}(-k)$.[2]
• If $v(k)$ is the indifference bid price for a claim and $v^{sup}(k),v^{sub}(k)$ are the superhedging price and subhedging prices respectively then $v^{sub}(k)\leq v(k)\leq v^{sup}(k)$. Therefore, in a complete market the indifference price is always equal to the price to hedge the claim.
References
1. John R. Birge (2008). Financial Engineering. Elsevier. pp. 521–524. ISBN 978-0-444-51781-4.
2. Carmona, Rene (2009). Indifference Pricing: Theory and Applications. Princeton University Press. ISBN 978-0-691-13883-1.
| Wikipedia |
General algebraic modeling system
The general algebraic modeling system (GAMS) is a high-level modeling system for mathematical optimization. GAMS is designed for modeling and solving linear, nonlinear, and mixed-integer optimization problems. The system is tailored for complex, large-scale modeling applications and allows the user to build large maintainable models that can be adapted to new situations. The system is available for use on various computer platforms. Models are portable from one platform to another.
GAMS
Developer(s)GAMS Development Corporation
Stable release
44.1.0 / 20 July 2023 (2023-07-20)[1]
PlatformCross-platform
TypeAlgebraic Modeling Language (AML)
LicenseProprietary
Websitewww.gams.com
GAMS was the first algebraic modeling language (AML)[2] and is formally similar to commonly used fourth-generation programming languages. GAMS contains an integrated development environment (IDE) and is connected to a group of third-party optimization solvers. Among these solvers are BARON, COIN-OR solvers, CONOPT, CPLEX, DICOPT, MOSEK, SNOPT, SULUM, and XPRESS.
GAMS allows the users to implement a sort of hybrid algorithm combining different solvers. Models are described in concise, human-readable algebraic statements. GAMS is among the most popular input formats for the NEOS Server. Although initially designed for applications related to economics and management science, it has a community of users from various backgrounds of engineering and science.
Timeline
• 1976 GAMS idea is presented at the International Symposium on Mathematical Programming (ISMP), Budapest[3]
• 1978 Phase I: GAMS supports linear programming. Supported platforms: Mainframes and Unix Workstations
• 1979 Phase II: GAMS supports nonlinear programming.
• 1987 GAMS becomes a commercial product
• 1988 First PC System (16 bit)
• 1988 Alex Meeraus, the initiator of GAMS and founder of GAMS Development Corporation, is awarded INFORMS Computing Society Prize
• 1990 32 bit Dos Extender
• 1990 GAMS moves to Georgetown, Washington, D.C.
• 1991 Mixed Integer Non-Linear Programs capability (DICOPT)
• 1994 GAMS supports mixed complementarity problems
• 1995 MPSGE language is added for CGE modeling
• 1996 European branch opens in Germany
• 1998 32 bit native Windows
• 1998 Stochastic programming capability (OSL/SE, DECIS)
• 1999 Introduction of the GAMS Integrated development environment (IDE)
• 2000 End of support for DOS & Win 3.11
• 2000 GAMS World initiative started
• 2001 GAMS Data Exchange (GDX) is introduced
• 2002 GAMS is listed in OR/MS 50th Anniversary list of milestones
• 2003 Conic programming is added
• 2003 Global optimization in GAMS
• 2004 Quality assurance initiative starts
• 2004 Support for Quadratic Constrained programs
• 2005 Support for 64 bit PC Operating systems (Mac PowerPC / Linux / Win)
• 2006 GAMS supports parallel grid computing
• 2007 GAMS supports open-source solvers from COIN-OR
• 2007 Support for Solaris on Sparc64
• 2008 Support for 32 and 64 bit Mac OS X
• 2009 GAMS available on the Amazon Elastic Compute Cloud
• 2009 GAMS supports extended mathematical programs (EMP)
• 2010 GAMS is awarded the company award of the German Society of Operations Research (GOR)
• 2010 GDXMRW interface between GAMS and Matlab
• 2010 End of support for Mac PowerPC / Dec Alpha / SGI IRIX / HP-9000/HP-UX
• 2011 Support for Extrinsic Function Libraries
• 2011 End of support for Win95 / 98 / ME, and Win2000
• 2012 The Winners of the 2012 INFORMS Impact Prize included Alexander Meeraus. The prize was awarded to the originators of the five most important algebraic modeling languages.
• 2012 Introduction of Object Oriented API for .NET, Java, and Python
• 2012 The winners of the 2012 Coin OR Cup included Michael Bussieck, Steven Dirkse, & Stefan Vigerske for GAMSlinks
• 2012 End of support for 32 bit on Mac OS X
• 2013 Support for distributed MIP (Cplex)
• 2013 Stochastic programming extension of GAMS EMP
• 2013 GDXRRW interface between GAMS and R
• 2014 Local search solver LocalSolver added to solver portfolio
• 2014 End of support for 32 bit Linux and 32 bit Solaris
• 2015 LaTeX documentation from GAMS source (Model2TeX)
• 2015 End of support for Win XP
• 2016 New Management Team
• 2017 EmbeddedCode Facility
• 2017 C++ API
• 2017 Introduction of "Core" and "Peripheral" platforms
• 2018 GAMS Studio (Beta)
• 2018 End of support for x86-64 Solaris
• 2019 GAMS MIRO - Model Interface with Rapid Orchestration (Beta)
• 2019 End of support for Win7, moved 32 bit Windows to peripheral platforms[4]
• 2019 Altered versioning scheme to XX.Y.Z
• 2020 Introduction of demo and community licensing scheme
• 2020 Official release of GAMS MIRO (Model Interface with Rapid Orchestration) for deployment of GAMS models as interactive applications
• 2021 Official release of GAMS Engine, the new solution for running GAMS jobs in cloud environments
• 2022 Official release of GAMS Engine SaaS, the hosted version of GAMS Engine
Background
The driving force behind the development of GAMS were the users of mathematical programming who believed in optimization as a powerful and elegant framework for solving real life problems in science and engineering. At the same time, these users were frustrated by high costs, skill requirements, and an overall low reliability of applying optimization tools. Most of the system's initiatives and support for new development arose in response to problems in the fields of economics, finance, and chemical engineering, since these disciplines view and understand the world as a mathematical program.
GAMS’s impetus for development arose from the frustrating experience of a large economic modeling group at the World Bank. In hindsight, one may call it a historic accident that in the 1970s mathematical economists and statisticians were assembled to address problems of development. They used the best techniques available at that time to solve multi-sector economy-wide models and large simulation and optimization models in agriculture, steel, fertilizer, power, water use, and other sectors. Although the group produced impressive research, initial success was difficult to reproduce outside their well functioning research environment. The existing techniques to construct, manipulate, and solve such models required several manual, time-consuming, and error-prone translations into different, problem-specific representations required by each solution method. During seminar presentations, modelers had to defend the existing versions of their models, sometimes quite irrationally, because of time and money considerations. Their models just could not be moved to other environments, because special programming knowledge was needed, and data formats and solution methods were not portable.
The idea of an algebraic approach to represent, manipulate, and solve large-scale mathematical models fused old and new paradigms into a consistent and computationally tractable system. Using generator matrices for linear programs revealed the importance of naming rows and columns in a consistent manner. The connection to the emerging relational data model became evident. Experience using traditional programming languages to manage those name spaces naturally lead one to think in terms of sets and tuples, and this led to the relational data model.
Combining multi-dimensional algebraic notation with the relational data model was the obvious answer. Compiler writing techniques were by now widespread, and languages like GAMS could be implemented relatively quickly. However, translating this rigorous mathematical representation into the algorithm-specific format required the computation of partial derivatives on very large systems. In the 1970s, TRW developed a system called PROSE that took the ideas of chemical engineers to compute point derivatives that were exact derivatives at a given point, and to embed them in a consistent, Fortran-style calculus modeling language. The resulting system allowed the user to use automatically generated exact first and second order derivatives. This was a pioneering system and an important demonstration of a concept. However, PROSE had a number of shortcomings: it could not handle large systems, problem representation was tied to an array-type data structure that required address calculations, and the system did not provide access to state-of-the art solution methods. From linear programming, GAMS learned that exploitation of sparsity was key to solving large problems. Thus, the final piece of the puzzle was the use of sparse data structures.
Lines starting with an * in column one are treated as comments.[5]: 32
A sample model
A transportation problem from George Dantzig is used to provide a sample GAMS model.[6] This model is part of the model library which contains many more complete GAMS models. This problem finds a least cost shipping schedule that meets requirements at markets and supplies at factories.
Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963.
Sets
i canning plants / seattle, san-diego /
j markets / new-york, Chicago, topeka / ;
Parameters
a(i) capacity of plant i in cases
/ seattle 350
san-diego 600 /
b(j) demand at market j in cases
/ new-york 325
Chicago 300
topeka 275 / ;
Table d(i,j) distance in thousands of miles
new-york Chicago topeka
seattle 2.5 1.7 1.8
san-diego 2.5 1.8 1.4 ;
Scalar f freight in dollars per case per thousand miles /90/ ;
Parameter c(i,j) transport cost in thousands of dollars per case ;
c(i,j) = f * d(i,j) / 1000 ;
Variables
x(i,j) shipment quantities in cases
z total transportation costs in thousands of dollars ;
Positive Variable x ;
Equations
cost define objective function
supply(i) observe supply limit at plant i
demand(j) satisfy demand at market j ;
cost .. z =e= sum((i,j), c(i,j)*x(i,j)) ;
supply(i) .. sum(j, x(i,j)) =l= a(i) ;
demand(j) .. sum(i, x(i,j)) =g= b(j) ;
Model transport /all/ ;
Solve transport using lp minimizing z ;
Display x.l, x.m ;
Subsystems
The Mathematical Programming System for General Equilibrium analysis (MPSGE) is a language used for formulating and solving Arrow–Debreu economic equilibrium models and exists as a subsystem within GAMS.[7]
See also
• Extended Mathematical Programming (EMP) – an extension to mathematical programming languages available within GAMS
• GNU MathProg – an open-source mathematical programming language based on AMPL
References
1. "44 Distribution". gams.com. Retrieved 2023-07-20.
2. Kallrath, Josef (2004). Modeling Languages in Mathematical Optimization (First ed.). Norwell, USA: Kluer Academic Publishers. p. 241. ISBN 978-1-4613-7945-4.
3. Toward a General Algebraic Modelling System (PDF). IX. International Symposium on Mathematical Programming. Budapest, Hungary. 1976. p. 185.
4. https://www.gams.com/blog/article/news/phasing-out-of-32-bit-support-with-gams-30/?tx_news_pi1%5Bcontroller%5D=News&tx_news_pi1%5Baction%5D=detail&cHash=42ad8bd6cb83482b9820e0af6e263f03
5. Rosenthal, Richard E (2007). GAMS — A user's guide (PDF). Washington, DC, USA: GAMS Development Corporation. Retrieved 2020-12-20.
6. R E Rosenthal (1988). "Chapter 2: A GAMS Tutorial". GAMS: A User's Guide. The Scientific Press, Redwood City, California.
7. Rutherford, T. F. (1999). "Applied General Equilibrium Modeling with MPSGE as a GAMS Subsystem: An Overview of the Modeling Framework and Syntax". Computational Economics. 14: 1–4. doi:10.1023/A:1008655831209. S2CID 60954697.
External links
• GAMS Development Corporation
• GAMS Software GmbH
• GAMS World
Mathematical optimization software
Data formats
• Mathematica
• MPS
• nl
• sol
Modeling tools
• AIMMS
• AMPL
• APMonitor
• ECLiPSe-CLP
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• GNU MathProg
• JuMP
• LINDO
• OPL
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LP, MILP∗ solvers
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SOCP, MISOCP∗ solvers
• Artelys Knitro∗
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SDP, MISDP∗ solvers
• Mathematica
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NLP, MINLP∗ solvers
• AOA∗
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• ANTIGONE∗
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• BARON∗
• Couenne∗
• Galahad library
• IPOPT
• LINDO∗
• LOQO
• MIDACO∗
• MINOS
• NAG
• NLPQLP
• NPSOL
• SCIP∗
• SNOPT∗
• Octeract Engine∗
• WORHP
• Xpress NonLinear∗
GO solvers
• ANTIGONE∗
• BARON
• Couenne∗
• Mathematica
• LINDO
• SCIP
• Octeract Engine
CP solvers
• Artelys Kalis
• Comet
• CPLEX CP Optimizer
• Gecode
• Mathematica
• JaCoP
• Xpress Kalis
Metaheuristic solvers
• OptaPlanner
• List of optimization software
• Comparison of optimization software
| Wikipedia |
# Introduction to Python's math library and the math module
Python's math library is a collection of mathematical functions and constants that can be used in your programs. The math module is a subset of the math library, and it provides mathematical functions that are commonly used in mathematical computations.
In this textbook, we will learn how to use Python's math module to solve geometric series. Geometric series are a sequence of numbers where each term is obtained by multiplying the previous term by a constant ratio.
For example, consider the geometric series:
$$1, 2, 4, 8, 16, \dots$$
The ratio between consecutive terms is 2.
Before we dive into solving geometric series using the math module, let's first understand the summation formula for geometric series.
# The summation formula for geometric series
The summation formula for a geometric series is given by:
$$S_n = a + ar + ar^2 + \dots + ar^{n-1}$$
where:
- $S_n$ is the sum of the geometric series up to the $n$th term
- $a$ is the first term of the series
- $r$ is the common ratio
- $n$ is the number of terms in the series
## Exercise
Calculate the sum of the geometric series:
$$1, 2, 4, 8, 16, \dots$$
# Calculating the sum of a geometric series using the math module
To calculate the sum of a geometric series using the math module, we can use the formula:
$$S_n = a \times \frac{1 - r^n}{1 - r}$$
Let's write a Python program to calculate the sum of the geometric series:
```python
import math
a = 1
r = 2
n = 5
S_n = a * (1 - r**n) / (1 - r)
print(S_n)
```
This program calculates the sum of the geometric series with the first term $a = 1$, common ratio $r = 2$, and $n = 5$ terms.
# Converting geometric series to the summation formula
Sometimes, we may be given a geometric series and asked to calculate its sum using the summation formula. Let's see how to convert a geometric series to the summation formula.
For example, consider the geometric series:
$$1, 2, 4, 8, 16, \dots$$
We can convert this series to the summation formula as follows:
$$S_n = 1 \times \frac{1 - 2^n}{1 - 2}$$
# Converting the summation formula to the formula for the nth term of a geometric series
In some problems, we may be given the sum of a geometric series and asked to find the $n$th term. Let's see how to convert the summation formula to the formula for the $n$th term of a geometric series.
For example, consider the geometric series with the sum:
$$S_n = 1 \times \frac{1 - 2^n}{1 - 2}$$
We can convert this formula to the formula for the $n$th term as follows:
$$a_n = a \times r^{n-1}$$
# Using the formula for the nth term to find the sum of a geometric series with a given number of terms
In some problems, we may be given the formula for the $n$th term of a geometric series and asked to find the sum of the series. Let's see how to use the formula for the $n$th term to find the sum of the series.
For example, consider the geometric series with the formula for the $n$th term:
$$a_n = 1 \times 2^{n-1}$$
We can use this formula to find the sum of the series with $n = 5$ terms as follows:
$$S_n = a + ar + ar^2 + ar^3 + ar^4$$
# Solving geometric series with common ratios
In some problems, we may be given a geometric series with a common ratio and asked to find the sum of the series. Let's see how to solve geometric series with common ratios using the math module.
For example, consider the geometric series with the common ratio $r = 2$:
$$1, 2, 4, 8, 16, \dots$$
We can use the math module to find the sum of the series with $n = 5$ terms as follows:
```python
import math
a = 1
r = 2
n = 5
S_n = a * (1 - r**n) / (1 - r)
print(S_n)
```
This program calculates the sum of the geometric series with the first term $a = 1$, common ratio $r = 2$, and $n = 5$ terms.
# Applications of geometric series in real-world problems
Geometric series have numerous applications in real-world problems. Some examples include:
- Calculating the future value of an investment with a constant interest rate
- Modeling the growth of a population with a constant growth rate
- Analyzing the convergence or divergence of a series
# The limit of a geometric series as n approaches infinity
When the common ratio of a geometric series is less than 1, the series converges to a limit as $n$ approaches infinity. The limit of the series is given by:
$$\lim_{n \to \infty} S_n = \frac{a}{1 - r}$$
# Divergence of a geometric series when the absolute ratio is greater than 1
When the absolute value of the common ratio of a geometric series is greater than 1, the series diverges. This means that the sum of the series does not approach a limit as $n$ approaches infinity.
# Convergence of a geometric series when the absolute ratio is less than 1
When the absolute value of the common ratio of a geometric series is less than 1, the series converges. This means that the sum of the series approaches a limit as $n$ approaches infinity.
In conclusion, Python's math module provides a powerful tool for solving geometric series. By understanding the summation formula, converting geometric series to the summation formula, and using the formula for the $n$th term, we can solve a wide range of problems involving geometric series. | Textbooks |
Search Results: 1 - 10 of 100 matches for " "
Page 1 /100
IceCube's Neutrinos: The beginning of extra-Galactic neutrino astrophysics? [PDF]
E. Waxman
Physics , 2013,
Abstract: The flux, spectrum and angular distribution of the excess neutrino signal detected by IceCube between 50TeV and 2PeV are inconsistent with those expected for Galactic sources. The coincidence of the excess, $E_\nu^2\Phi_\nu=3.6\pm1.2\times10^{-8}(GeV/ cm^2 sr s)$, with the Waxman-Bahcall (WB) bound, $E_\nu^2\Phi_{WB}=3.4\times10^{-8}(GeV/cm^2 sr s)$, is probably a clue to the origin of IceCube's neutrinos. The most natural explanation of this coincidence is that both the neutrino excess and the ultra-high energy, $>10^{19}$ eV, cosmic-ray (UHECR) flux are produced by the same population of cosmologically distributed sources, producing CRs, likely protons, at a similar rate, $E^2 dQ/dE=0.5\times10^{44}(erg/Mpc^3yr)$ (at z=0), across a wide range of energies, from $10^{15}$ eV to $>10^{20}$ eV, and residing in environments (such as starburst galaxies) in which CRs of rigidity $E/Z< 10^{17}$ eV lose much of their energy to pion production. Identification of the neutrino sources will allow one to identify the UHECR accelerators, to resolve open questions related to the accelerator models, and to study neutrino properties (related e.g. to flavor oscillations and coupling to gravity) with an accuracy many orders of magnitude better than is currently possible. The most promising method for identifying the sources is by association of a neutrino with an electromagnetic signal accompanying a transient event responsible for its generation. The neutrino flux that is produced within the sources, and that may thus be directly associated with transient events, may be significantly lower than the total observed neutrino flux, which may be dominated by neutrino production at the environment in which the sources reside.
Search for transient neutrino sources with IceCube [PDF]
A. Franckowiak,for the IceCube Collaboration
Abstract: The IceCube detector, which is embedded in the glacial ice at the geographic South Pole, is the first neutrino telescope to comprise a volume of one cubic kilometer. The search for neutrinos of astrophysical origin is among the primary goals of IceCube. Point source candidates include Galactic objects such as supernova remnants (SNRs) as well as extragalactic objects such as Active Galactic Nuclei (AGN) and Gamma-Ray Bursts (GRBs). Offline and online searches for transient sources like GRBs and supernovae (SNe) are presented. Triggered searches use satellite measurements from Fermi, SWIFT and Konus. Complementary to the triggered offline search, an online neutrino multiplet selection allows IceCube to trigger a network of optical telescopes, which can then identify a possible electromagnetic counterpart. This allows to probe for mildly relativistic jets in SNe and hence to reveal the connection between GRBs, SNe and relativistic jets. Results from IceCube's triggered GRB search and a first limit on relativistic jets in SNe from the optical follow-up program are presented.
Estimating the contribution of Galactic sources to the diffuse neutrino flux [PDF]
Luis A. Anchordoqui,Haim Goldberg,Thomas C. Paul,Luiz H. M. da Silva,Brian J. Vlcek
Physics , 2014, DOI: 10.1103/PhysRevD.90.123010
Abstract: Motivated by recent IceCube observations we re-examine the idea that microquasars are high energy neutrino emitters. By stretching to the maximum the parameters of the Fermi engine we show that the nearby high-mass X-ray binary LS 5039 could accelerate protons up to above about 20 PeV. These highly relativistic protons could subsequently interact with the plasma producing neutrinos up to the maximum observed energies. After that we adopt the spatial density distribution of high-mass X-ray binaries obtained from the deep INTEGRAL Galactic plane survey and we assume LS 5039 typifies the microquasar population to demonstrate that these powerful compact sources could provide a dominant contribution to the diffuse neutrino flux recently observed by IceCube.
Gravitational lensing of transient neutrino sources by black holes [PDF]
Ernesto F. Eiroa,Gustavo E. Romero
Physics , 2008, DOI: 10.1016/j.physletb.2008.04.016
Abstract: In this work we study gravitational lensing of neutrinos by Schwarzschild black holes. In particular, we analyze the case of a neutrino transient source associated with a gamma-ray burst lensed by a supermassive black hole located at the center of an interposed galaxy. We show that the primary and secondary images have an angular separation beyond the resolution of forthcoming km-scale detectors, but the signals from each image have time delays between them that in most cases are longer than the typical duration of the intrinsic events. In this way, the signal from different images can be detected as separate events coming from the very same location in the sky. This would render an event that otherwise might have had a low signal-to-noise ratio a clear detection, since the probability of a repetition of a signal from the same direction is negligible. The relativistic images are so faint and proximate that are beyond the sensitivity and resolution of the next-generation instruments.
Search for diffuse neutrino flux from astrophysical sources with MACRO [PDF]
The MACRO Collaboration
Physics , 2002, DOI: 10.1016/S0927-6505(02)00190-1
Abstract: Many galactic and extragalactic astrophysical sources are currently considered promising candidates as high energy neutrino emitters. Astrophysical neutrinos can be detected as upward-going muons produced in charged-current interactions with the medium surrounding the detector. The expected neutrino fluxes from various models start to dominate on the atmospheric neutrino background at neutrino energies above some tens of TeV. We present the results of a search for an excess of high energy upward-going muons among the sample of data collected by MACRO during ~5.8 years of effective running time. No significant evidence for this signal was found. As a consequence, an upper limit on the flux of upward-going muons from high-energy neutrinos was set at the level of 1.7 10^(-14) cm^(-2) s^(-1) sr^(-1). The corresponding upper limit for the diffuse neutrino flux was evaluated assuming a neutrino power law spectrum. Our result was compared with theoretical predictions and upper limits from other experiments.
Bounds on the neutrino flux from cosmic sources of relativistic particles [PDF]
Karl Mannheim
Physics , 2001, DOI: 10.1088/0954-3899/27/7/323
Abstract: In order to facilitate the identification of possible new physics signatures in neutrino telescopes, such as neutrinos from the annihilation of neutralinos or decaying relics, it is essential to gain full control over the astrophysical inventory of neutrino sources in the Universe. The total available accretion power, the extragalactic gamma ray background, and the cosmic ray proton intensity can be used to constrain astrophysical models of neutrino production in extragalactic sources. The resulting upper limit on the extragalactic muon neutrino intensity from cosmic particle accelerators combined with a reasonable minimum intensity of neutrinos due to cosmic rays stored in clusters of galaxies demark a zone of opportunity for neutrino astronomy over a broad range of energies between 100 MeV and 1 EeV. Discovery of this neutrino background would open a new era for astronomy and provide the first un-obscured view to the early Universe.
Status of High-Energy Neutrino Astronomy [PDF]
Marek Kowalski
Physics , 2014, DOI: 10.1088/1742-6596/632/1/012039
Abstract: With the recent discovery of high-energy neutrinos of extra-terrestrial origin by the IceCube neutrino observatory, neutrino-astronomy is entering a new era. This review will cover currently operating open water/ice neutrino telescopes, the latest evidence for a flux of extra-terrestrial neutrinos and current efforts in the search for steady and transient neutrino point sources. Generalised constraints on potential astrophysical sources are presented, allowing to focus the hunt for the sources of the observed high-energy neutrinos.
Distortion of the ultrahigh energy cosmic ray flux from rare transient sources in inhomogeneous extragalactic magnetic fields [PDF]
Sihem Kalli,Martin Lemoine,Kumiko Kotera
Physics , 2011, DOI: 10.1051/0004-6361/201015688
Abstract: Detecting and characterizing the anisotropy pattern of the arrival directions of the highest energy cosmic rays are crucial steps towards the identification of their sources. We discuss a possible distortion of the cosmic ray flux induced by the anisotropic and inhomogeneous distribution of extragalactic magnetic fields in cases where sources of ultrahigh energy cosmic rays are rare transient phenomena, such as gamma-ray bursts and/or newly born magnetars. This distortion does not involve an angular deflection but the modulation of the flux related to the probability of seeing the source on an experiment lifetime. To quantify this distortion, we construct sky maps of the arrival directions of these highest energy cosmic rays for various magnetic field configurations and appeal to statistical tests proposed in the literature. We conclude that this distortion cannot affect present experiments but should be considered when performing anisotropy studies with future large-scale experiments that record as many as hundreds of events above 6x10^19 eV.
Neutrino Background Flux from Sources of Ultrahigh-Energy Cosmic-Ray Nuclei [PDF]
Kohta Murase,John F. Beacom
Abstract: Motivated by Pierre Auger Observatory results favoring a heavy nuclear composition for ultrahigh-energy (UHE) cosmic rays, we investigate implications for the cumulative neutrino background. The requirement that nuclei not be photodisintegrated constrains their interactions in sources, therefore limiting neutrino production via photomeson interactions. Assuming a $dN_{\rm CR}/dE_{\rm CR} \propto E_{\rm CR}^{-2}$ injection spectrum and photodisintegration via the giant dipole resonance, the background flux of neutrinos is lower than $E_\nu^2 \Phi_\nu \sim {10}^{-9} {\rm GeV} {\rm cm}^{-2} {\rm s}^{-1} {\rm sr}^{-1}$ if UHE nuclei ubiquitously survive in their sources. This is smaller than the analogous Waxman-Bahcall flux for UHE protons by about one order of magnitude, and is below the projected IceCube sensitivity. If IceCube detects a neutrino background, it could be due to other sources, e.g., hadronuclear interactions of lower-energy cosmic rays; if it does not, this supports our strong restrictions on the properties of sources of UHE nuclei.
Some possible sources of IceCube TeV-PeV neutrino events [PDF]
Sarira Sahu,Luis Salvador Miranda
Physics , 2014, DOI: 10.1140/epjc/s10052-015-3519-1
Abstract: The IceCube Collaboration has observed 37 neutrino events in the energy range $30\, TeV\leq E_{\nu} \leq 2$ PeV and the sources of these neutrinos are unknown. Here we have shown that positions of 12 high energy blazars and the position of the FR-I galaxy Centaurus A, coincide within the error circles of ten IceCube events, the later being in the error circle of the highest energy event so far observed by IceCube. Two of the above blazars are simultaneously within the error circles of the Telescope Array hotspot and one IceCube event. We found that the blazar H2356-309 is within the error circles of three IceCube events. We propose that photohadronic interaction of the Fermi accelerated high energy protons with the synchrotron/SSC background photons in the nuclear region of these high energy blazars and AGN are probably responsible for some of the observed IceCube events. | CommonCrawl |
Does the word voltage exist in academic engineering?
In Portuguese, the word voltage does not exist. Neither academic nor technical.
In engineering, Portuguese speakers refers to volt as electric tension or potential difference.
The word voltage was popularized in the Portuguese language because some places use 220V and others use 110V and people always had to ask if the "voltage" for the equipment is 110 or 220. So, it's kind of a nickname/shortcut for non-technical people to refer to electric tension.
What about in English academic engineering? Does the word voltage exist or is it just a shortcut/nickname for electric tension or potential difference?
voltage engineering
vianna77vianna77
\$\begingroup\$ Comments are not for extended discussion; this conversation has been moved to chat. \$\endgroup\$
– Dave Tweed
\$\begingroup\$ The word voltage does exist in practice and I see it the same way as the word deletar also does, that is, as an adaptation of a foreign word. Contrary to some nitpicking engineers, I see no problem with it. The same way I see no problem calling any physician a doutor, even those who don't hold a doctorate degree, simply because the term is doctor in the english language. \$\endgroup\$
– Marc.2377
\$\begingroup\$ In Czech we call it tension too, not voltage. But I never felt the need to analyse it. :-) \$\endgroup\$
– Al Kepp
\$\begingroup\$ All I could think of while reading this page is the Blue Öyster Cult (American rock band) song "Godzilla". With a purposeful grimace and a terrible sound / He pulls the spitting high-tension wires down :) \$\endgroup\$
– B Layer
\$\begingroup\$ @vianna77 I have to note that your assumption that "the word voltage does not exist" applies, maybe, only to Portugal's Portuguese. PT-br is more prone to accept this type of "creation" than PT-pt; "voltagem" is a accepted word in Brazilian Portuguese; this word figures in the official vocabulary from Brazilian Letters Academy. The same vocabulary also accepts "amperagem"; personally, I prefer "diferença de tensão" e "corrente"; the official pt-br doesn't accept "wattagem" (although sometimes we can hear someone saying "wattagem" in Brazil - it hurt my ears). \$\endgroup\$
– mguima
In the International System of Units (SI) and the corresponding International System of Quantities, as described in the international standards series ISO/IEC* 80000 Quantities and units, quantities are always independent of the unit in which they are expressed; therefore, a quantity name shall not reflect the name of any corresponding unit.
However, ISO 80000 Part 1 General as well as IEC 80000 Part 6 Electromagnetism note that the name "voltage" is commonly used in the English language and that this use is an exception from the principle that a quantity name should not refer to any name of unit. It is recommended to use the name "electric tension" wherever possible.
The same information can be found in the series IEC 60050 International Electrotechnical Vocabulary (IEV), especially IEC 60050-121.
* The International Organization for Standardization (ISO) collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
\$\begingroup\$ I always believe the standard bodies must have considered the problem, but I'm not familiar with these standards. Good to see the answer from a physicist. Personally I think this should be the accepted answer, as it is the only answer that is able to cite relevant scientific/industrial standards from ISO and IEC. \$\endgroup\$
– 比尔盖子
Yes, voltage is a technical word in English.
From Wordnik:
noun A measure of the difference in electric potential between two points in space, a material, or an electric circuit, expressed in volts.
In fact, Wikipedia even lists "electric tension" as a synonym, though I hadn't heard that before. Mostly it's referred to as voltage or potential difference.
Some other answers have noted that Electric Tension was used to describe a potential difference until the mid-20th century in England, but it went out of popularity.
Google's Ngram shows that voltage is far more popular than Electric Tension ever was, though.
Harry BeadleHarry Beadle
\$\begingroup\$ "tension" is kind of an old-fashioned word for it; you see things on old schematics from the tube era marked H.T. for high tension (referring to a high voltage supply) for instance. I understand it still gets used among electricians sometimes, though it's rare in electrical engineering. \$\endgroup\$
– Hearth
\$\begingroup\$ It most likely comes from the French. "Tension" in France is used in the same way "Voltage" is used in English speaking countries. We use "Voltage" sometimes too, but a lot less often. \$\endgroup\$
– Harnex
\$\begingroup\$ Also, high voltage power lines may occasionally be referred to as "high tension" power lines in the US. \$\endgroup\$
– mkeith
\$\begingroup\$ @Harnex It may come from French, but it may also come from any number of other languages; I understand English is in the minority using the word voltage instead of some variant of the local word for "tension". \$\endgroup\$
The water analogy of electricity was historically influential, both terms, "tension" and "current", were the result of this analogy.
In the early 1900s, "tension" was the standard technical term in English for electric potential. The B+ of a vacuum tube was called High Tension (HT), and a Cathode Ray Tube required "Extra-High Tension" (EHT) to operate. For some reasons, the word "tension" in English became obsolete in the middle of the 20th century (I cannot find a reference), and the term "voltage" became the standard technical term instead. Similarly, the old technical term for a "capacitor" was "condenser". A microphone that works by the change of capacitance was (and still is) called a "condenser microphone". In 1926, the term "condenser" was abandoned in English, but it took a generation or two to pick up the new term, fully replaced the old term around mid-20th century.
However, the translation of basic terms in electrical engineering to other languages was done long before this transition, so in many other languages, the technical term is still "tension" or "pressure", and a "capacitor" is still a "condenser".
The main reason seemed to be an effort to reduce the confusion between electrical engineering and mechanical engineering terms. Early 1900s was still the heyday of steam engines, and the confusion could be very real, and I fully understand the choice for "capacitor" over "condenser". But I think the choice "voltage", from a physical sense, is very unfortunate. Most physical quantities, as physical phenomena, have their own names independent from their units of measurement. When we talk about force as a phenomenon, we don't refer it as "newtonage", neither we use "wattage" for power.
$$\require{cancel}$$ \begin{array} {|l|l|l|l|} \hline \text{Phenomenon} &\text{Name} &\text{Unit} &\text{Numerical Name}\\ \hline \text{A push} &\text{force} &\text{newton} &\text{-}\\ \text{Flow of charge} &\text{current} &\text{ampere} &\text{amperage}\\ \text{Rate of work} &\text{power} &\text{watt} &\text{wattage} \\ \text{Electric Potential} &\cancel{tension} \text{voltage (!!)} &\text{volt} &\text{voltage (!!)} \\ \hline \end{array}
The introduction of "voltage" makes electric potential lost its own name, making it the only physical quantity named after its unit of measurement in English.
However, "voltage" is the standard term English, we have to follow it all along...
比尔盖子比尔盖子
\$\begingroup\$ "Wattage" is a perfectly normal word. It's perhaps even more common amongst non-engineers who can read a value in watts but don't know that it is a measure of power. \$\endgroup\$
– Graham
\$\begingroup\$ @Graham Yes, it's a normal word. But when we say "power", the physical definition "rate of doing work" is emphasized (e.g. "power dissipation", a "power resistor", is not called a "wattage resistor") and when we say "wattage", we refer to the numerical value of power (e.g. this appliance is too high for the wiring, basically a comparison of numbers). Same for "amperage", which is a perfectly fine word to talk about the numerical value of "current" displayed from a meter. But "voltage" is the only odd exception. \$\endgroup\$
\$\begingroup\$ @比尔盖子 "high-wattage resistor" sounds acceptable to my ears, though you're correct that just "wattage resistor" does not. I think the difference is that wattage can only be used as a noun, while power can be either a noun or an adjective. \$\endgroup\$
Yes. They do exist. In fact, voltage is actually potential difference. When you say voltage at a point is 5 V, we mean to say the potential difference of 5 V with respect to ground.
in another case, when there are two points at not zero potential, and we have to measure the voltage between those two points, we say "voltage is … V with respect to another point".
If one point (point A) is at 20 V and another point (point B) is at 25 V, we say voltage at point B is 5 V with respect to point A. And this is of course the potential difference between those two points.
G-aura-VG-aura-V
In my physics experience, I've seen both the words voltage and potential difference used. I've never heard of the word electric tension in any context. Potential difference was more specific to situations where the relative voltage, or,
$$\Delta V = V_1 - V_2$$
was the important quantity desired, while voltage referred to a single reference measurement, or the above difference, based on context, which could often be inferred from the nature of the problem.
David EvansDavid Evans
\$\begingroup\$ The phrase "high-tension wires" is commonly used in the U.S. in reference to the cables strung between very tall structures for long-range power distribution, and I think "high tension" it's used in British though not American automotive terminology to describe automotive spark plug wires ("HT leads"). \$\endgroup\$
Answer is yes, Voltage is used both academically and professionally. NFPA/NEC and OSHA are recognized organisms in USA and they all use the word and mention it on their glossaries. This is also true in Spanish. To add references this image might help from a technical publication.
Link to NFPA glossary: https://www.nfpa.org/-/media/Files/Codes-and-standards/Glossary-of-terms/glossary_of_terms_2019.ashx?la=en
Juan OjedaJuan Ojeda
\$\begingroup\$ I suggest adding explanation as to whey the answer is yes. \$\endgroup\$
– Mahendra Gunawardena
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Tree species diversity impacts average radial growth of beech and oak trees in Belgium, not their long-term growth trend
Astrid Vannoppen1,
Vincent Kint1,
Quentin Ponette2,
Kris Verheyen3 &
Bart Muys1
Forest Ecosystems volume 6, Article number: 10 (2019) Cite this article
Environmental change has resulted in changes in forest growth in Europe during the last century. This has consequences for the products and services delivered by forest. Mixing tree species is often proposed as a strategy to deal with the consequences of climate change. Diversifying forests is believed to result in higher productivity and increased growth stability. Tree species diversity is therefore expected to affect long-term trends in tree radial growth. However, this has not yet been studied. In this paper we study the effect of diversity on the radial growth and its long-term trends for beech and oak trees growing along a gradient of tree species diversity in the loamy region of central Belgium (from monocultures to mixed forests patches up to three species).
We found that beech trees have a higher radial growth whereas oak trees have a lower one when growing in mixtures. The contrasting diversity-productivity relationship observed for beech and oak is in agreement with their ranking in shade tolerance, where oaks suffer increased competition in mixed oak patches. Overall, in monocultures and mixtures, an increasing radial growth trend of + 2% for the period 1927–2015 and 21% for the period 1899–2015 was found for beech and oak, respectively. Tree species diversity did not alter the shape of this detected long-term radial growth trend. Nevertheless, for oak a lower year-to-year variability in radial growth is found in mixtures indicating a higher resilience.
We conclude that diversity impacts the average radial growth and its variability (only in the case of oak) but not the shape of the long-term trend in radial growth of beech and oak trees growing in the loamy region of central Belgium.
Historical long-term radial growth changes of beech and oak trees have been reported throughout Europe (Bergès et al. 2000; Charru et al. 2017; Boisvenue and Running 2006; Kint et al. 2012). Climate change, changes in site fertility, CO2 fertilization, changes in management and tree genotypes used are identified as drivers of the reported historical radial growth changes (Hyvönen et al. 2007; Bontemps et al. 2011; Babst et al. 2013). Getting more insight in these historical radial growth changes in trees is important to predict the impact of future climate change but also to assess management strategies. This is crucial since forests deliver many important ecosystem services (Thorsen et al. 2014).
A driver that to our knowledge is not yet studied in historical long-term radial growth studies is tree species diversity. Tree species diversity could have an impact on long-term radial growth trends since diverse systems are expected to function better (Loreau et al. 2001; Isbell et al. 2009). For individual trees in mixed forests, better exploitation and more efficient use of resources (niche differentiation) and interspecific facilitation, collectively referred to as complementarity effects, are mechanisms underlying this positive biodiversity ecosystem functioning relationship (Loreau and Hector 2001). Furthermore, more mixed forests are found to exhibit more stable growth patterns (Jucker et al. 2014a; del Río et al. 2017).
Generally the relationship between biodiversity and productivity is found to be positive (Zhang et al. 2012; Liang et al. 2016). However, the diversity-productivity relationship seems to be context-dependent. Tree development stage, scale (e.g. tree level compared to stand level) and climate are found to shape the diversity-productivity relationship (Cavard et al. 2011; Chisholm et al. 2013; Forrester 2014; Jucker et al. 2016). Jucker et al. (2016) have demonstrated that the diversity-productivity relationship changes along a spatial gradient of climate. It might thus be that also along a temporal gradient of climate the diversity-productivity relationship changes. This is important since climatic conditions have been changing over the past decades and are expected to change further, which will impact forest ecosystems (Lindner et al. 2010; IPCC 2013). The stress gradient hypothesis, which states that complementarity effects are dependent on abiotic variables, including climate, proposes a climate dependency of the diversity-productivity relationship (Loreau and Hector 2001; Morin et al. 2011). Under harsh conditions the complementarity effects are expected to increase relative to competitive interactions which decrease.
Tree-ring series are often used to model long-term changes in radial growth. In order to model long-term radial growth changes, tree growth changes related to growth changes of individual trees (e.g. due to tree aging, local site characteristics or competition) must be separated from historical long-term radial growth trends. To properly assess historical radial growth changes, tree aging, an obvious driver of tree radial growth, must be taken into account, for instance via detrending or statistical modeling techniques (Peters et al. 2014). In addition, a data sample of trees which cover a wide range of developmental stages is crucial (Bontemps et al. 2010).
In this study we will look what the effect of tree diversity is on the radial growth of beech (Fagus sylvatica) and oak (Quercus robur) trees in Belgium. To this end, mean TRW and its variability over time will be evaluated for beech and oak trees growing in plots with different tree diversity levels. In a last step we will examine if there is a long-term trend in radial growth of beech and oak and if this trend is influenced by tree species diversity. By this the effect of a temporal gradient of climate on the diversity-productivity relationship will be investigated at the hierarchical scale of individual beech and oak trees.
Multilevel mixed models (Kint et al. 2012; Aertsen et al. 2014) are used in this paper to separate historical radial growth change from other factors which may influence radial growth such as tree aging, site quality and stand structure (e.g. competition level and tree species diversity). In a first step radial growth of individual trees is modelled in function of tree aging, site quality and stand structure variables (including tree species diversity). In a second step, we look if there is a change of radial growth by testing if radial growth changes with time in all trees when the individual tree growth (i.e. previous step) is accounted for. In this last step it is also tested if tree species diversity alters the long-term trend in radial growth.
Study area and plot description
Beech and oak trees growing in even-aged stands of two forests, Meerdaal and Zoniën, located in the loamy region of Belgium were studied. In total 75 and 70 plots were selected for beech and oak, respectively. The maximum distance between plots from the two forests is 23.8 km, and 3.7 km and 9.3 km between plots within Meerdaal and Zoniën, respectively. Annual precipitation is 821 mm year− 1 and mean annual temperature is 10.3 °C for the period 1901–2015. The long-term trend in annual precipitation and mean temperature is visualized in Additional file 1: Figure S1. Plots are circular (18 m radius) with a (co) dominant beech or oak tree in the center (further referred to as center tree). This center tree is the tree that was cored. Plots were selected along i) a wide range of developmental stages (Additional file 2: Figure S2) and ii) a gradient of species diversity with levels Isp, IIsp and IIIsp for plots with one, two and three tree species respectively. An admixture of at least 15% of other tree species than beech or oak was used as a threshold for the beech and oak IIsp and IIIsp plots. This threshold was considered as the absolute minimum, in the selection of the IIsp and IIIsp plots the highest possible evenness was pursued. Plots were selected in even aged stands to ensure that the center tree developed in a monoculture (i.e. Isp) or mixed environment (i.e. IIsp and IIIsp) over their whole age range. In addition, it was ensured that plots represent different tree species combinations (i.e. species composition) (Table 1).
Table 1 Number of plots sampled for each diversity level and species composition level
Tree cores
From the center tree (i.e. beech or oak tree) two tree cores were taken (North and South direction) in the winter of 2015 using a 5 mm increment corer (Suunto) at 1 m above ground. In addition, the DBH and bark thickness at the coring location was measured. Tree rings were made visible using a core microtome. Afterwards, tree-ring widths (TRW) were measured using a Lintab measurement system with 1/100 mm resolution. The measured TRW were checked with TSAP-Win and COFECHA software for tree-ring crossdating.
Forest structure and site quality data
In each plot the forest structure was characterized by measuring: position of each tree species with diameter at breast height (DBH) > 15 cm, crown projection area of the center tree (CPA, m2), height of the center tree (H, m), total CPA of trees with diameter DBH > 15 cm in the plot (TotCPA, m2), and total basal area of trees with DBH > 15 cm in the plot (TotBA, m2). The crown edge in the four cardinal directions was measured for CPA. Besides the basal area of trees larger than the center tree (BAL, m2), the ratio between the diameter of the center tree and the average diameter of trees with DBH > 15 cm in the plot (ddg), scaled Shannon diversity index (SWBA), and structural diversity index (SD) were calculated. SWBA quantifies the number of tree species present in the plot based on their basal area and takes the evenness into account. SD is a measure for the structural heterogeneity in the plot. SWBA and SD were calculated as follows:
\( {SW}_{BA}=\mathit{\exp}\left(-\sum \limits_{i=1}^S{P}_{BA;i}\ast \ln \left({P}_{BA;i}\right)\right) \) with \( {P}_{BA;i}=\frac{BA_i}{BA_{tot}} \) and S total number of species present in the plot
\( SD=\frac{StDev\left({H}_i.{CPA}_i\right)}{Mean\left({H}_i.{CPA}_i\right)}\ with\ i\ the\ {i}^{th}\ tree\ present\ in\ the\ plot \) (Van de Peer et al., 2017)
Site quality in each plot was characterized by measuring: pH (1:5 soil:solution, 0.01 M CaCl2), organic C and N content (%, Carlo Erba 1108 elemental analyzer), derived C/N ratio, bulk density (g/cm3) and texture (fraction clay, loam and sand in %) on a composite soil sample. Composite soil sample consisted of a sample taken in N, NE, SE, SW and NW direction at 5 m from the center tree with a gauge (0–30 cm of the mineral soil horizon) in each plot.
Analysis of diversity effects
Effects of diversity on tree-ring width, forest structure and site quality
To evaluate the effect of diversity on the growth of beech and oak, chronologies of raw (referred to as TRW) and individually detrended tree-ring width series (referred to as RWI) were built for the three diversity levels (Isp, IIsp and IIIsp). A 15-year cubic smoothing spline with a 50% frequency cutoff was used for detrending in order to remove non-climatic low frequency variability such as for example age trends. The average growth rates (AGR), inter-series correlation (Rbar) and expressed population signal (EPS) are reported for each chronology. The EPS is a measure to evaluate the quality of a chronology and is based on Rbar and number of samples. A non-parametric Dunn test was used to test if chronologies of the three diversity levels differed significantly for both beech and oak.
For each species composition level chronologies were built using the same methodology as for the chronology building at species diversity level. A non-parametric Dunn test was used to test if radial growth is significantly different between the chronologies of different species composition level. For consistency, the chronologies were built using the same dataset as was used for the long-term radial growth modeling (i.e. first 30 years are removed and time period is 1927–2015 and 1899–2015 for beech and oak, respectively).
For the forest structure and site quality variables a parametric Tukey or non-parametric Dunn test (in case of non-normality of the tested variable) was used to test if variables differed significantly between the three diversity levels. The significance level was set to 0.05 for the Dunn and Tukey tests.
Long-term radial growth trends
A mixed modeling strategy was used to model the long-term radial trends in basal area increment (BAI) a measure for tree radial growth. BAI was calculated as:
$$ {BAI}_t=\pi \left({R}_t^2-{R}_{t-1}^2\right) $$
With R the tree radius at the end of the growing season in year t (derived from cumulative TRW measurements). The juvenile developmental stage (first 30 years) was excluded from the analysis and BAI was natural log transformed to deal with the skewed distribution. The modeling is started from this year wherefore data from at least ten trees were available (1927 and 1899 for beech and oak, respectively).
The models were built in two sequential stages. First base models (Mb) which describe the BAI in function of developmental stage, forest structure and site quality were built. Previous year diameter (Dp, cm) was chosen to characterize the developmental stage since it is known that tree growth is more driven by tree size than cambial age (Mencuccini et al. 2005; Wykoff 1990). Possible forest structure and site quality variables were selected a priori using multiple regression (Additional file 3: Table S1). Besides, it was ensured that the selected forest structure and site quality variables had a variance inflation factor < 5. For the mixed modeling the methodology of Zuur et al. (2009) was used. Random effects included are: random intercept for tree and random slope related with Dp. The significance level was set to 0.001 for the selection of the fixed effects. The base models describe the BAI of individual trees. In order to see if BAI varies with calendar date in all trees it is tested if a linear, quadratic, cubic or natural cubic spline term of calendar year improves the base models, the result are the date models (Md). The term related with calendar year in the date models describe long-term radial growth changes caused by exogenous factors. Interaction between the term related with calendar year and the forest structure and site quality variables that were found to be significant in the Mb models were also tested. Only the Md models will be reported since the focus of this paper are long-term radial growth trends.
Final models were fitted with restricted maximum likelihood (REML) and their model performance was evaluated with pseudo-R2 of full and marginal model (i.e. only considering fixed effects) and relative root mean squared error (rRMSE, calculated for response). The pseudo-R2 was calculated as the correlation between the response and model predictions. All statistics were performed in R (version 3.2.5) (R Development Core Team 2016) with packages "nlme", "spline" and "dplr" (Bunn 2008; Hothorn et al. 2008; Pinheiro et al. 2016).
Evaluation of effect of diversity on tree-ring width, forest structure and site quality
A higher TRW is found for beech trees growing in more diverse plots, the opposite is true for oak trees, which grow better in monoculture plots (Additional file 4: Table S2, Fig. 1 and Table 2). TRW is significantly higher in IIsp versus Isp, IIIsp versus Isp and IIIsp versus IIsp plots for beech. TRW is significantly lower in IIsp versus Isp, IIIsp versus Isp and IIIsp versus IIsp for oak. The first order autocorrelation is significantly higher in IIsp versus Isp and IIIsp versus Isp plots for oak (p < 0.05). For beech trees the first order autocorrelation increases with plot diversity level but no significant differences are found (p > 0.05) (Table 2).
Chronologies of TRW and RWI for beech (a) and oak (b) center trees. Number of trees used for the chronology building is presented in the bottom graphs. See Additional file 9: Table S5.S2 for the Dunn test on TRW
Table 2 Characteristics of TRW chronologies for beech and oak center trees growing in plots with tree diversity level Isp, IIsp or IIIsp
Beech TRW is significantly different between all species composition levels except for species composition levels beech versus beech-hornbeam, beech-maple-hornbeam versus beech-oak, beech-maple-hornbeam versus beech-maple-oak, and beech-maple versus beech-oak-hornbeam (Fig. 2).
Boxplots of TRW chronologies at each species composition level for beech (a) and oak (b) center trees. O: oak, B: beech, HB: hornbeam and MP: maple. Composition levels without common letters differ significantly at p < 0.05 (Additional file 9: Table S5.S3)
Oak TRW is significantly different between all species composition levels except for species composition levels oak-beech versus oak-beech-hornbeam, oak versus oak-hornbeam, oak-beech versus oak-maple-beech, oak-maple versus oak-maple-beech and oak-maple versus oak-maple-hornbeam (Fig. 2).
SWBA is significantly higher in more diverse plots for both beech and oak (Fig. 3 and Additional file 5: Table S3). For oak significant differences are found in BAL, TotBA and TotCPA (Fig. 3 and Additional file 5: Table S3). No significant differences are found in the site quality variables and other forest structural variables between plots of different diversity level for both beech and oak.
Boxplots of forest structure variables for beech (a) and oak (b-e) center trees. Diversity levels without common letters differ significantly at p < 0.05. SWBA: scaled Shannon diversity index, BAL: basal area of trees larger than the center tree, TotBA: total basal area of trees with DBH > 15 cm in the plot, and TotCPA: total CPA of trees with diameter DBH > 15 cm in the plot
Long-term radial growth trends in beech and oak along a diversity gradient
A long-term trend in radial growth is found for beech and oak center trees (Table 3). Tree developmental (i.e. Dp) stage is present in the models for beech and oak as a quadratic polynomial fixed effect, indicating that tree growth increases until a certain Dp afterwards radial growth levels off. A random intercept for tree and a random slope associated with Dp resulted in the best random structure for both beech and oak models.
Table 3 Parameter estimates and model evaluation of the ln (BAI) date models for beech and oak
For beech a cubic term of calendar year results in the best model fit. SWBA and BAL have a positive and negative effect, respectively, on the radial growth of beech. Other forest structural variables and site quality variables are not significant (Table 3). The interaction of SWBA and the cubic term of calendar year is not significant, indicating that the long-term trend in radial growth does not change for beech trees growing in plots with different diversity levels. Nonetheless the presence of SWBA in the model indicates a positive effect of diversity on radial growth.
In the oak model a quadratic term of calendar year results in the best model fit. The model indicates that oak trees growing in plots with high BAL have lower radial growth (i.e. negative estimate for BAL) (Table 3). Other forest structure or site quality variables do not influence radial growth significantly.
The model evaluation parameters of the models are good. The difference between the R2 of the full model and marginal model (i.e. R2f and R2m) indicate that large part of the variability is explained by the random effects. The non-significant correlation between tree size and the random components of the models indicate the effect of tree size on growth is well modelled by the quadratic polynomial of Dp.
The detected long-term radial growth trends for beech and oak are visualized in Fig. 4. Relative to the radial growth in 1927 beech radial growth decreases until 1957 (− 25% from 1927 to 1957) where after it increases again (+ 27% from 1957 to 2015). An overall growth increase of 2% for the period 1927–2015 is detected for beech. The date model of beech predicts radial growth to be 23% and 54% higher for beech trees growing in plots with SWBA equal to two (i.e. a plot with two tree species with perfect evenness) and three (i.e. a plot with three tree species with perfect evenness) compared to SWBA equal to one (i.e. a plot with one tree species), respectively. Relative to the radial growth in 1899, oak radial growth decreases until 1940 after this it increases again. Overall the oak radial growth increased with 21% for the period 1899–2015.
Visualization of the long-term trend in radial growth for beech (a) and oak (b) center trees. BAI is predicted by the date models for a tree with constant Dp and BAL (median values from 1927 and 1899 for beech and oak respectively). For the prediction of beech BAI, SWBA was set to one (red), two (green) and three (blue) (representing plots with one, two and three tree species with perfect evenness, respectively) to visualize the effect of diversity on beech radial growth. Values are presented relative to the predicted BAI in 1927 of a tree growing in a plot with SWBA equal to one for beech and 1899 for oak
Diversity influences radial growth of beech and oak differently
Our results indicate that tree species diversity has an opposite effect on radial growth of beech versus oak. Beech trees growing in diverse stands have higher radial growth whereas oak trees have lower radial growth (Additional file 4: Table S2, Fig. 1 and Table 2). Note that DBH and height of the beech and oak center trees did not differ significantly between the three species diversity levels, indicating that the difference in radial growth found is not related to differences in developmental stage of the center trees for the three species diversity levels.
The difference in the reported diversity-productivity relationship of beech and oak can be related to differences in the functional traits of the tree species in the mixtures. The functional traits of the species in mixture give us information on the complementarity in resources use of tree species present in a mixture. This gives us insight in the underlying mechanisms of the diversity productivity relationship and hence its outcome.
The shade tolerance of the species in a mixture is an important trait explaining the effect of diversity, both magnitude and direction, on the productivity of trees in the mixture (Zhang et al. 2012; Jucker et al. 2014b; Toïgo et al. 2017). The shade tolerance indices of the species present in the mixture range from 2.45, 3.73, 3.97 to 4.56 for oak, maple, hornbeam and beech respectively (Ninemets and Valladares 2006). Oak is thus the most shade intolerant species whereas beech is the most shade tolerant tree species in the mixtures studied. High interspecific competition for light in the studied mixtures is probably an important factor explaining the negative effect of diversity on the shade intolerant oak compared to the positive effect of diversity on the shade tolerant beech (Toïgo et al. 2017). The shade tolerant beech is also known to be able to adapt its crown structure to increase light interception in mixed stands (i.e. crown plasticity), interspecific competition for light is thus not likely to affect its radial growth negatively when mixed with other tree species (Dieler and Pretzsch 2013).
Not only competition for light can explain the positive effect of diversity on radial growth of beech. In addition, other mechanisms might play a role here such as complementarity for water. In an ongoing study in the same study area, we found that beech trees growing in diverse stands have higher stomatal conductance, indicating higher water availability, compared to beech trees growing in monocultures in dry years. In addition, facilitation effects occur, oak for example has hydraulic redistribution properties (i.e. deep rooting tree species move water from deeper soil layers upwards) (Hafner et al. 2017).
Competition might explain the negative effect of diversity on radial growth of oak. The significantly higher TotCPA and TotBA in mixed compared to monoculture plots is likely to shape the diversity-productivity relationship (Fig. 3) (Forrester 2014). The spatial arrangement and density in stands has an influence on the interactions between tree species. The significantly higher BAL in the mixed stands indicate higher competition compared to monoculture plots (Fig. 3). This higher competition, related with the higher stand densities, may outweigh complementarity interactions in the mixed stands (Forrester 2014). Note that diversity can increase stand density, especially in cases where species with complementary traits are mixed, which is the case here (Pretzsch and Biber 2016).
Although radial growth is lower for oak trees growing in diverse stands, the first order autocorrelation is significantly higher (Table 2). This higher first order autocorrelation indicates that the year-to-year variability in radial growth is lower for oak trees growing in mixed plots, indicating a higher resilience. Likewise for beech a higher first order autocorrelation is found, although it is not significant.
When we look at the beech-oak mixture, the mixture with the highest difference in shade tolerance indices, we see that in this beech-oak mixture beech has a higher radial growth opposed to oak which has a lower radial growth compared to the monoculture of the respective tree species (Fig. 2). This highlights the importance of scale, at the stand scale the lower growth of one species might be outweighed by the higher growth of another tree species which would result in a positive effect of biodiversity on productivity (Toïgo et al. 2017).
For beech a radial growth increase is found for the period 1927–2015 (Fig. 4). When we look at the difference quotient of the modeled long-term trend in radial growth we see that until 1957 growth decreases and that afterwards growth increases until 2015 (Additional file 6: Figure S3). The growth decline of beech from 1927 to 1957 coincides with a period of consecutive years with relative (compared to the four previous years) negative growth change (Fig. 4 and Additional file 7: Figure S4). The occurrence of multiple warm events between 1930's and 1950's may account for the observed growth decline (Additional file 8: Figure S5) (cf. Bontemps et al. 2011 on beech trees growing in north-eastern France). After 1957 radial growth increases, however the increase starts to diminish from 1993 onwards (Additional file 6: Figure S3). If this trend continues, beech radial growth will be lower compared to 1927 in a near future. In other studies a recent growth decline for beech has already been reported (Jump et al. 2006; Piovesan et al. 2008; Kint et al. 2012; Charru et al. 2017).
The positive estimate of SWBA and the negative estimate of BAL in the beech model indicate that complementarity effects alleviate negative effects of competition on radial growth. For SWBA equal to two or three (i.e. representing two and three species plots with perfect evenness, respectively) the BAL must be larger than 19,514 and 29,271 cm2 (or 20 and 29 m2/ha), respectively, to outweigh the positive effect of SWBA completely. This is the case for 40% and 8% of two and three species plots, respectively. Note that in these plots the negative competition effect on the radial growth of the beech center tree is still reduced by the diversity effect (i.e. positive estimate for SWBA) compared to the monoculture plots of beech. Diversity thus has a positive impact on beech radial growth. Even though, the non-significance of the interaction between SWBA and the long-term trend indicates that the temporal gradient in climate did not influence the diversity-productivity relationship with time.
The detected decline in oak radial growth from 1899 to 1940 coincides with periods of extreme winter frost, drought and insect outbreaks (Fig. 4 and Additional file 6: Figure S3) (Delatour 1983; Thomas et al. 2002). After 1940 oak radial growth increases and this increase is still becoming larger and larger every year (Additional file 6: Figure S3). BAL has a negative effect on the radial growth of oak. As discussed in the previous section the higher competition in the diverse oak plots probably outweighs the complementarity effects in the diverse stands. The oak model indicates that diversity has no direct influence on the radial growth of oak and thus also does not influences the detected long-term trend despite the lower year-to-year variability in radial growth of oak growing in diverse stands (Tables 2 and 3).
Factors influencing long-term trends in radial growth
The growth of trees is influenced by climate, site fertility, stand structure and management. All these factors can change gradually with time and thus explain the long-term trends in radial growth observed for beech and oak trees (Fig. 4). However, it is not possible to demonstrate causality between the gradual change in drivers of radial growth (e.g. climate, site fertility, stand structure and management) and the observed long-term trend in radial growth changes with only observational data collected in a single region (cf. Verheyen et al. 2017). In this study we tried to minimize the effect of unbalances in tree size of the sampled trees and historical changes in stand characteristics on the modeled long-term growth trends. Plots were selected along a wide range of developmental stages in order to exclude the influence of tree size on the detected long-term growth trend. The non-significance (p > 0.05) of the Pearson correlation between the random components and the size of the trees indicate that the mixed models used are able to capture the effect of tree size on growth. This is because a significant correlation would indicate that the growth is modeled differently for trees with different size. Plots were selected based on the species composition in the plot at the time of sampling. Despite selecting plots in even aged stands to ensure that the center tree developed in a monoculture (i.e. Isp) or mixed environment (i.e. IIsp and IIIsp) over their whole age range we cannot guarantee that the species composition was the same in the past. In addition, we assume that past management was the same in the monocultures and mixed plots.
Although the main focus of the present study is to quantify the impact of tree species diversity on average and long-term radial growth, comparison of the modeled long-term growth trends with other studies in the same area is valuable. For beech trees growing in whole Flanders (northern Belgium) a recent growth decline was detected by Kint et al. (2012). This recent negative growth decline was found to be related to increased nitrogen (N) deposition and drought over time (Kint et al. 2012). The absence of the recent growth decline in the present study indicates that the increased N deposition and drought over time did not impact beech trees growing in Meerdaal and Zoniën forest that much compared to beech trees growing in other parts of Flanders. Note, however, that the observed growth increase starts to diminish from 1993 onwards as mentioned earlier. The absence of the recent growth decline for beech in Meerdaal and Zoniën forest compared to whole Flanders can be explained by several factors.
First, Meerdaal and Zoniën forest are located on the loess belt in Flanders on some of the most productive sites for beech of Flanders. The good water holding capacity of the loamy soil in these forests can explain why the increased drought did not affect beech radial growth as much compared to beech growth in other regions of Flanders (Aertsen et al. 2014). Second, the studied forests are located in an area of Flanders which has lower N deposition compared to other forested areas in Flanders (Verstraeten et al. 2012; VMM 2017). The negative effects of high N deposition observed in several studies in Europe are thus probably not as severe in Meerdaal and Zoniën. Lastly, shifts in management may influence the detected long-term radial growth trend. Changes in management are often not considered in studies looking at long-term trends in radial growth since historical information on management is often missing (but see Bontemps et al. (2010)). However, this is important since changes in management result in changes in interactions between tree species and the forest structure which affect tree radial growth (Altman et al. 2013; Trouvé et al. 2015). After thinning a so-called growth release is often observed resulting from increased light availability. Therefore shifts in management to more intensive thinning will result in higher radial growth. Both in Meerdaal and Zoniën forest a tendency of increased thinning is observed in time (Huvenne P., personal communication 8/03/2018), this probably contributed to the observed increase in radial growth of beech and oak in these two forests. Note that possible shifts in historical management probably did not influence the long-term trend detected in the study of Kint et al. (2012). Since this study covers a broader geographical scale (i.e. northern Belgium) local effects on long-term trends in radial growth, such as shifts in management, are reduced to a minimum since forest management history is different among administrative regions.
When we look in detail to the long-term trends for beech reported by Aertsen et al. (2014), which was performed in Zoniën and Meerdaal forests, and the present study some differences in trends are observed, especially in recent decades (Additional file 9: Figure S6a). Long-term trends detected by regional curve standardization (RCS, as described by Bontemps and Esper 2011), another frequently used method to detect long-term trends in radial growth, resulted in similar long-term trends compared to the long-term trend detected by the multilevel mixed models used in both studies (i.e. Aertsen et al., (2014) and this study) (Additional file 9: Figure S6). There is thus no effect of the method used on the detection of the long-term trend. When in both datasets data points of trees at DBH < 146.3 mm and > 273.3 mm (first and third quantile respectively) are excluded in order to exclude imbalances in developmental stage with time, the long-term trends detected are more alike (Additional file 9: Figure S6b). This highlights the importance of sampling trees along a wide range of developmental stages in long-term growth trends studies. Note that the first and last time points have a large impact on the shape of the detected long-term trend by RCS. The last year (i.e. 2008) in the dataset of Aertsen et al. (2014) coincide with years with lower growth compared to other years which pulls the fitted cubic polynomial down (Additional file 9: Figure S6). The dataset of the present study also contains data after the dip in radial growth around 2008, resulting in a long-term trend that still increases after 2008.
The observed long-term growth increase observed for oak is in line with other studies in Belgium (Kint et al. 2012; Vannoppen et al. 2018). Oak is thus still profiting from environmental changes. Especially late frost is known to affect oak growth negatively, over the last decades the last day of frost occurs earlier in the year which is positive for oak growth (Tricot et al. 2015).
The diversity-productivity relationship is found to be influenced by the functional trait structure of the studied mixtures and the stand density. The shade tolerance can explain the contrasting diversity-productivity relationship found for beech and oak. Radial growth of beech, which has a high shade tolerance, increased when growing in mixtures. Whereas for oak, which has a low shade tolerance, radial growth decreased when growing in mixtures. The higher stand densities in oak mixed plots resulted in higher competition levels which overruled the diversity effect. Both for beech and oak an increasing long-term growth trend is observed. The growth increase is smaller for beech compared to oak, 3 and 27% for the period 1927–2015 and 1899–2015 respectively. Tree species diversity did not influence the shape of the long-term trends detected in beech and oak. However, the lower year-to-year radial growth variability in more diverse plots of both beech (although not significant) and oak (significant) suggest that under harsher environmental conditions diversity might shape long-term trends in beech and oak growth.
AGR:
Average growth rate
Akaike Information Criteria
BAI:
Basal area increment
Basal area of trees larger than cored tree
CPA:
Crown projection area of the cored tree
DBH:
Diameter at breast height
ddg:
Ratio between the diameter of the cored tree and the average diameter of trees with DBH > 15 cm in plot
Examplia gratia
Expressed population signal
Fig.:
i.e.:
Id est
JJA:
Date model
Nitorgen
Number of observations
R2m:
Pseudo-R square marginal model
R2f:
pseudo-R square full model
stadard deviation
Rbar:
Inter-series correlation
RCS:
Regional curve standardization
REML:
Restricted maximum likelihood
RMSE:
Root mean square error
rRMSE:
Relative root mean square error
Structural diversity index
SWBA :
Scaled Shannon diversity index based on basal area
T mean:
Mean yearly temperature
TotBA:
Total basal area of trees with DBH > 15 cm in the plot
TotCPA:
Total CPA of trees with DBH > 15 cm in the plot
TRW:
Tree-ring width
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We would like to thank Jorgen Op De Beeck, Eric Van Beek and Remi Chevalier for their technical support. Climatic data were made available by the Royal Meteorological Institute of Belgium.
This research received funding from FWO Vlaanderen [grant number: G.0C96.14N].
The datasets analysed during the current study are available from the corresponding author on reasonable request.
Division Forest, Nature and Landscape, Department of Earth and Environmental Sciences, University of Leuven, Celestijnenlaan 200E, Box 2411, BE-3001, Leuven, Belgium
Astrid Vannoppen
, Vincent Kint
& Bart Muys
Earth and Life Institute, Université catholique de Louvain, Croix du Sud 2, L7.05.09, BE-1348, Louvain-la-Neuve, Belgium
Quentin Ponette
Forest & Nature Lab, Department of Environment, Ghent University, Geraardsbergsesteenweg 267 -Gontrode, BE-9090, Melle, Belgium
Kris Verheyen
Search for Astrid Vannoppen in:
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AV collected and analysed the data and was a major contributor in writing the manuscript. VK, QP, KV and BM made substantial contributions to the conception and design, the interpretation of data and writing of the manuscript. All authors read and approved the final manuscript.
Correspondence to Bart Muys.
Figure S1. Mean yearly temperature (T mean) and cumulative precipitation (P) for the study area. Line visualizes the long-term trend of T mean and P through time (loess smoother). Dots are the yearly observed T mean and P at the climatic station of Ukkel located at 5 to 26 km distance from the plots (Royal Meteorological Institute of Belgium, RMI). (JPG 352 kb)
Figure S2. Distribution of DBH at different year intervals of beech (a) and oak (b) center trees. Boxplots are based on the dataset used for modeling (i.e. juvenile growth (first 30 measured rings of each tree) is removed and start year is the year with at least data from 10 trees). (JPG 571 kb)
Table S1. Selected forest structural and site quality variables for beech and oak center trees by multiple regression. (DOCX 14 kb)
Table S2. z and p-values of the Dunn test performed on the TRW chronologies. (DOCX 13 kb)
Table S3. z and p-values of the Dunn test for forest structure variables. (DOCX 14 kb)
Figure S3. Difference quotient of modeled long-term radial growth trend for beech (a) and oak (b). Negative values indicate decreases in radial growth trend. Positive values indicate increases in radial growth trend. The difference quotient is calculated for time steps of 1 year as ((modeled growth in year t)-(modeled growth in year t-1))/1 for each year t. (JPG 259 kb)
Figure S4. Relative change of TRW in each year for beech center trees. TRW in each year is compared to the mean TRW in the respective 4 previous years. (JPG 278 kb)
Figure S5. Precipitation and mean temperature anomalies for the period June to August for the period 1927–2015. Data are from the climatic station of Ukkel located at 5 to 26 km distance from the plots (Royal Meteorological Institute of Belgium, RMI). P: recipitation, Tmean: mean temperature, JJA: June to August (JPG 453 kb)
Figure S6. Radial growth chronologies estimated with regional curve standardization method. Chronologies are smoothed with a cubic spline smoother (blue lines). RCS is applied on data of this study (dotted line) and on the data of Aertsen et al. (2014) (full line). Panel (b) visualizes the regional curve standardization (RCS) applied on the datasets where data points of trees at DBH < 146.3 mm and > 273.3 mm (first and third quantile respectively) are excluded to minimize effect of tree developmental stage. (JPG 455 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.
Vannoppen, A., Kint, V., Ponette, Q. et al. Tree species diversity impacts average radial growth of beech and oak trees in Belgium, not their long-term growth trend. For. Ecosyst. 6, 10 (2019). https://doi.org/10.1186/s40663-019-0169-z
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Syntactic Deviance.Thomas E. Patton - 1968 - Foundations of Language 4 (2):138-153.details
Thomas E. Patton.Syntactic Deviance - forthcoming - Foundations of Language.details
Gilles Deleuze in Continental Philosophy
The Significance of Behaviour-Related Criteria for Textual Exegesis—and Their Neglect in Indian Studies.Claus Oetke - 2013 - Journal of Indian Philosophy 41 (4):359-437.details
Against the background of the fact that speakers not seldom intend to convey imports which deviate from the linguistically expressed meanings of linguistic items, the present article addresses some consequences of this phenomenon which appear to still be neglected in textual studies. It is suggested that understanding behaviour is in some respect a primary objective of exegesis and that due attention must be attributed to the high diversity of behaviour-related criteria by which interpretations of linguistic items are to be evaluated. (...) Although we intimate in addition that individual (meaningful) sentences occurring either in oral conversations or in written documents generally exhibit a multiplicity of contents of diverse types and that the circumstance that sometimes only a content equalling the linguistic significance of a pertinent unit matters for purposes of interpretation is caused by a material coincidence of different varieties of content, the tenets advocated in the paper do not essentially depend on that view. On the other hand, the following assumptions are relevant in the present connection: (a) A number of deviances between imports conveyed by linguistic utterances and literal meanings of expressions occur due to maxims of linguistic behaviour that are quite independent of lexical and syntactic features of individual natural languages. (b) It is by no means an exceptional phenomenon that imports not derivable by grammatical rules of a particular language alone possess primary importance for interpretation and textual exegesis. In view of significant affinities between understanding of sentences and of texts it is argued that the consideration of diverse aspects of behaviour possesses relevance for textual exegesis at least in the following respects: (1) By delivering a heuristic device for discerning problems affecting adopted interpretations it encourages searches for alternatives. (2) It provides means for evaluating the degree of acceptability of particular textual exegeses and possibly rejecting them on a more rational basis than mere intuition. (3) It offers possibilities for critically assessing the validity of explicit arguments advanced in favour of or in opposition to some interpretation. (4) It furnishes a background for assessing certain disputes about translation. The dimension of linguistic behaviour also attains importance in connection with questions of exegesis which are not concerned with assessments of (propositional) contents intended to be communicated, such as the ascertainment of the function which some argument possesses in a context. For substantiating the thesis that omission of raising relevant questions concerning behaviour is not an isolated phenomenon two examples will be employed: (1) A discussion concerning the exegesis of a crucial passage of Dignāga's Pramāṇasamuccaya and the Pramāṇasamuccayavṛtti, (2) a critical appraisal of a recent publication dealing with the interpretation of the second chapter of Nāgārjuna's Mūlamadhyamakakārikā-s. (shrink)
Indian Philosophy in Asian Philosophy
Philosophy of Language, Miscellaneous in Philosophy of Language
Deviance and Vice: Strength as a Theoretical Virtue in the Epistemology of Logic.Gillian Russell - 2019 - Philosophy and Phenomenological Research 99 (3):548-563.details
This paper is about the putative theoretical virtue of strength, as it might be used in abductive arguments to the correct logic in the epistemology of logic. It argues for three theses. The first is that the well-defined property of logical strength is neither a virtue nor a vice, so that logically weaker theories are not—all other things being equal—worse or better theories than logically stronger ones. The second thesis is that logical strength does not entail the looser characteristic of (...) scientific strength, and the third is that many modern logics are on a par—or can be made to be on a par—with respect to scientific strength. (shrink)
Epistemology of Logic in Logic and Philosophy of Logic
Inference to the Best Explanation in General Philosophy of Science
Logical Consequence and Entailment in Logic and Philosophy of Logic
Nonclassical Logics in Logic and Philosophy of Logic
Theoretical Virtues in General Philosophy of Science
Syntactic Structures.Noam Chomsky - 1957 - Mouton.details
Noam Chomsky's book on syntactic structures is a serious attempts on the part of a linguist to construct within the tradition of scientific theory-construction ...
Philosophy of Linguistics in Philosophy of Language
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Basic Deviance Reconsidered.Markus E. Schlosser - 2007 - Analysis 67 (3):186–194.details
Most contemporary philosophers of action agree on the following claims. Firstly, the possibility of deviant or wayward causal chains poses a serious problem for the standard-causal theory of action. Secondly, we can distinguish between different kinds of deviant causal chains in the theory of action. In particular, we can distinguish between cases of basic and cases of consequential deviance. Thirdly, the problem of consequential deviance admits of a fairly straightforward solution, whereas the possibility of basic deviance constitutes (...) a separate and difficult problem that requires its own solution. I will argue that the problem of basic deviance is no more troublesome than the problem of consequential deviance, as a solution to the former is implicit in the standard solution to the latter. (shrink)
Causal Theory of Action in Philosophy of Action
Reasons and Causes in Philosophy of Action
Causal Deviance and the Attribution of Moral Responsibility.Paul Bloom - manuscriptdetails
Are current theories of moral responsibility missing a factor in the attribution of blame and praise? Four studies demonstrated that even when cause, intention, and outcome (factors generally assumed to be sufficient for the ascription of moral responsibility) are all present, blame and praise are discounted when the factors are not linked together in the usual manner (i.e., cases of ''causal deviance''). Experiment 4 further demonstrates that this effect of causal deviance is driven by intuitive gut feelings of (...) right and wrong, not logical deliberation. Ó 2003 Published by Elsevier Science (USA). (shrink)
Control and Responsibility in Meta-Ethics
Action, Deviance, and Guidance.Ezio Di Nucci - 2013 - Abstracta (2):41-59.details
I argue that we should give up the fight to rescue causal theories of action from fundamental challenges such as the problem of deviant causal chains; and that we should rather pursue an account of action based on the basic intuition that control identifies agency. In Section 1 I introduce causalism about action explanation. In Section 2 I present an alternative, Frankfurt's idea of guidance. In Section 3 I argue that the problem of deviant causal chains challenges causalism in two (...) important respects: first, it emphasizes that causalism fails to do justice to our basic intuition that control is necessary for agency. Second, it provides countless counterexamples to causalism, which many recent firemen have failed to extinguish – as I argue in some detail. Finally, in Section 4 I argue, contra Al Mele, that control does not require the attribution of psychological states as causes. (shrink)
Causal Explanation in Metaphysics
Intentional Action in Philosophy of Action
Psychological Explanation in Philosophy of Cognitive Science
Justice, Deviance, and the Dark Ghetto.Tommie Shelby - 2007 - Philosophy and Public Affairs 35 (2):126–160.details
Justice in Social and Political Philosophy
Interpersonal Deviance and Abusive Supervision: The Mediating Role of Supervisor Negative Emotions and the Moderating Role of Subordinate Organizational Citizenship Behavior.Gabi Eissa, Scott W. Lester & Ritu Gupta - 2020 - Journal of Business Ethics 166 (3):577-594.details
We build on the emerging research that shows aversive subordinate workplace behaviors are likely related to abusive supervision in the workplace. Specifically, we develop and test a moderated-mediation model outlining the process of abusive supervision based on the stressor-emotion model of counterproductive work behavior. We argue that subordinate interpersonal deviance prompts supervisor negative emotions, which then leads supervisors to engage in abusive supervision. We also argue that subordinate organizational citizenship behavior is likely to play a crucial role in predicting (...) abusive supervision. We argue that interpersonal deviance is more likely to prompt abusive supervision through supervisor negative emotions when the magnitude of an employee's engagement in OCB is weaker. Study 1, a time-lagged field study, tests and provides support for the relationships among our key variables. Study 2, utilizing multisource field data, replicates the results from Study 1 and provides support for the entire moderated-mediation model while controlling for tenure with supervisor, subordinate task performance, and subordinate conscientiousness. We find general support for our predictions. We conclude with a discussion of theoretical and practical implications as well as future research directions. (shrink)
Syntactic Structures.J. F. Staal - 1966 - Journal of Symbolic Logic 31 (2):245-251.details
Deviance and Causalism.Lilian O'Brien - 2012 - Pacific Philosophical Quarterly 93 (2):175-196.details
Drawing on the problem of deviance, I present a novel line of argumentation against causal theories of action. The causalist faces a dilemma: either she adopts a simple account of the causal route between intention and outcome, at the cost of failing to rule out deviance cases, or she adopts a more sophisticated account, at the cost of ruling out cases of intentional action in which the causal route is merely unusual. Underlying this dilemma, I argue, is that (...) the agent's perspective plays an ineliminable role in determining which causal pathways are deviant and which are not. (shrink)
Explanation of Action in Philosophy of Action
Philosophy of Action, Misc in Philosophy of Action
Syntactic Complexity Effects in Sentence Production.Gregory Scontras, William Badecker, Lisa Shank, Eunice Lim & Evelina Fedorenko - 2015 - Cognitive Science 39 (3):559-583.details
Syntactic complexity effects have been investigated extensively with respect to comprehension . According to one prominent class of accounts , certain structures cause comprehension difficulty due to their scarcity in the language. But why are some structures less frequent than others? In two elicited-production experiments we investigated syntactic complexity effects in relative clauses and wh-questions varying in whether or not they contained non-local dependencies. In both experiments, we found reliable durational differences between subject-extracted structures and object-extracted structures : (...) Participants took longer to begin and produce object-extractions. Furthermore, participants were more likely to be disfluent in the object-extracted constructions. These results suggest that there is a cost associated with planning and uttering the more syntactically complex, object-extracted structures, and that this cost manifests in the form of longer durations and disfluencies. Although the precise nature of this cost remains to be determined, these effects provide one plausible explanation for the relative rarity of object-extractions: They are more costly to produce. (shrink)
Memory and Cognitive Science in Philosophy of Mind
Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic Via Nested Sequents.Tim Lyon, Alwen Tiu, Rajeev Gore & Ranald Clouston - 2020 - In Maribel Fernandez & Anca Muscholl (eds.), 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Dagstuhl, Germany: pp. 1-16.details
We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to (...) embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants. (shrink)
Proof Theory in Logic and Philosophy of Logic
Plagiarism, Integrity, and Workplace Deviance: A Criterion Study.Daniel E. Martin, Asha Rao & Lloyd R. Sloan - 2009 - Ethics and Behavior 19 (1):36 – 50.details
Plagiarism is increasingly evident in business and academia. Though links between demographic, personality, and situational factors have been found, previous research has not used actual plagiarism behavior as a criterion variable. Previous research on academic dishonesty has consistently used self-report measures to establish prevalence of dishonest behavior. In this study we use actual plagiarism behavior to establish its prevalence, as well as relationships between integrity-related personal selection and workplace deviance measures. This research covers new ground in two respects: (a) (...) That the academic dishonesty literature is subject to revision using criterion variables to avoid self bias and social desirability issues and (b) we establish the relationship between actual academic dishonesty and potential workplace deviance/white-collar crime. (shrink)
Integrity in Normative Ethics
Professional Ethics in Applied Ethics
Syntactic Reductionism.Richard Heck - 2000 - Philosophia Mathematica 8 (2):124-149.details
Syntactic Reductionism, as understood here, is the view that the 'logical forms' of sentences in which reference to abstract objects appears to be made are misleading so that, on analysis, we can see that no expressions which even purport to refer to abstract objects are present in such sentences. After exploring the motivation for such a view, and arguing that no previous argument against it succeeds, sentences involving generalized quantifiers, such as 'most', are examined. It is then argued, on (...) this basis, that Syntactic Reductionism is untenable. (shrink)
Abstract Objects in Metaphysics
Frege: Abstract Objects in 20th Century Philosophy
Frege: Logic and Philosophy of Logic, Misc in 20th Century Philosophy
Mathematical Nominalism in Philosophy of Mathematics
The Deviance in Deviant Causal Chains.Neil McDonnell - 2015 - Thought: A Journal of Philosophy 4 (2):162-170.details
Causal theories of action, perception and knowledge are each beset by problems of so-called 'deviant' causal chains. For each such theory, counterexamples are formed using odd or co-incidental causal chains to establish that the theory is committed to unpalatable claims about some intentional action, about a case of veridical perception or about the acquisition of genuine knowledge. In this paper I will argue that three well-known examples of a deviant causal chain have something in common: they each violate Yablos proportionality (...) constraint on causation. I will argue that this constraint provides the key to saving causal theories from deviant chains. (shrink)
Causal Theory of Knowledge in Epistemology
Causation, Miscellaneous in Metaphysics
The Causal Theory of Perception in Philosophy of Mind
Syntactic Transformations on Distributed Representations.David J. Chalmers - 1990 - Connection Science 2:53-62.details
There has been much interest in the possibility of connectionist models whose representations can be endowed with compositional structure, and a variety of such models have been proposed. These models typically use distributed representations that arise from the functional composition of constituent parts. Functional composition and decomposition alone, however, yield only an implementation of classical symbolic theories. This paper explores the possibility of moving beyond implementation by exploiting holistic structure-sensitive operations on distributed representations. An experiment is performed using Pollack's Recursive (...) Auto-Associative Memory. RAAM is used to construct distributed representations of syntactically structured sentences. A feed-forward network is then trained to operate directly on these representations, modeling syn- tactic transformations of the represented sentences. Successful training and generalization is obtained, demonstrating that the implicit structure present in these representations can be used for a kind of structure-sensitive processing unique to the connectionist domain. (shrink)
Connectionism and Compositionality in Philosophy of Cognitive Science
Becoming Syntactic.Franklin Chang, Gary S. Dell & Kathryn Bock - 2006 - Psychological Review 113 (2):234-272.details
Relation of General Deviance to Academic Dishonesty.Bernard E. Whitley & Kevin L. Blankenship - 2000 - Ethics and Behavior 10 (1):1-12.details
This study investigated the relations of cheating on an exam and using a false excuse to avoid taking an exam as scheduled to various forms of minor deviance. College students completed measures of cheating, false excuse making, and minor deviance. A factor analysis identified clusters of deviance behaviors. Cheaters scored higher than noncheaters on measures of unreliability and risky driving behaviors, and false excuse makers scored higher than other students on measures of substance use, risky driving, illegal (...) behaviors, and personal unreliability. In addition, men scored higher than women on substance abuse and illegal behaviors factors. Results are interpreted in terms of personological theories of honesty and reliability. (shrink)
Academic and Teaching Ethics in Philosophy of Social Science
Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability.Richard Montague - 1975 - Journal of Symbolic Logic 40 (4):600-601.details
Syntactic Co-Ordination in Dialogue.Holly P. Branigan, Martin J. Pickering & Alexandra A. Cleland - 2000 - Cognition 75 (2):B13-B25.details
Linguistics in Cognitive Sciences
Plagiarism, Integrity, and Workplace Deviance: A Criterion Study.Daniel E. Martin PhD, Asha Rao & Lloyd R. Sloan - 2009 - Ethics and Behavior 19 (1):36-50.details
Women and Deviance in Philosophy.Helen Beebee - 2013 - In K. Hutchison & F. Jenkins (eds.), Women in Philosophy: What Needs to Change? Oxford: Oxford University Press. pp. 61--80.details
Women in Philosophy in Philosophy of Gender, Race, and Sexuality
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Causal Deviance and the Ascription of Intent and Blame.Ross Rogers, Mark D. Alicke, Sarah G. Taylor, David Rose, Teresa L. Davis & Dori Bloom - 2019 - Philosophical Psychology 32 (3):404-427.details
Shorthand, Syntactic Ellipsis, and the Pragmatic Determinants of What is Said.Reinaldo Elugardo & Robert J. Stainton - 2004 - Mind and Language 19 (4):442–471.details
Our first aim in this paper is to respond to four novel objections in Jason Stanley's 'Context and Logical Form'. Taken together, those objections attempt to debunk our prior claims that one can perform a genuine speech act by using a subsentential expression—where by 'subsentential expression' we mean an ordinary word or phrase, not embedded in any larger syntactic structure. Our second aim is to make it plausible that, pace Stanley, there really are pragmatic determinants of the literal truthconditional (...) content of speech acts. We hope to achieve this second aim precisely by defending the genuineness of subsentential speech acts. Given our two aims, it is necessary to highlight briefly their connection—which we do in the first part of the Introduction. Following that, we introduce Stanley's novel objections. This is the role of the second part of the Introduction. We offer our rebuttals in Section 2 (against 'shorthand') and Section 3 (against syntactic ellipsis, among other things). (shrink)
Ellipsis in Philosophy of Language
Abusive Supervision and Employee Deviance: A Multifoci Justice Perspective.Haesang Park, Jenny M. Hoobler, Junfeng Wu, Robert C. Liden, Jia Hu & Morgan S. Wilson - 2019 - Journal of Business Ethics 158 (4):1113-1131.details
In order to address the influence of unethical leader behaviors in the form of abusive supervision on subordinates' retaliatory responses, we meta-analytically examined the impact of abusive supervision on subordinate deviance, inclusive of the role of justice and power distance. Specifically, we investigated the mediating role of supervisory- and organizationally focused justice and the moderating role of power distance as one model explaining why and when abusive supervision is related to subordinate deviance toward supervisors and organizations. With 79 (...) independent sample studies, we found that abusive supervision was more strongly related to supervisory-focused justice, compared to organizationally focused justice perceptions, and both types of justice perceptions were related to target-similar deviance. Finally, our results showed that the negative implications of abusive supervision were stronger in lower power distance cultures compared to higher power distance cultures. (shrink)
Syntactic Semantics: Foundations of Computational Natural Language Understanding.William J. Rapaport - 1988 - In James H. Fetzer (ed.), Aspects of AI. Kluwer Academic Publishers.details
This essay considers what it means to understand natural language and whether a computer running an artificial-intelligence program designed to understand natural language does in fact do so. It is argued that a certain kind of semantics is needed to understand natural language, that this kind of semantics is mere symbol manipulation (i.e., syntax), and that, hence, it is available to AI systems. Recent arguments by Searle and Dretske to the effect that computers cannot understand natural language are discussed, and (...) a prototype natural-language-understanding system is presented as an illustration. (shrink)
Computational Semantics in Philosophy of Cognitive Science
Knowledge of Language in Philosophy of Language
The Chinese Room in Philosophy of Cognitive Science
Beyond Sweatshops: Positive Deviancy and Global Labour Practices.Denis G. Arnold & Laura P. Hartman - 2005 - Business Ethics, the Environment and Responsibility 14 (3):206–222.details
Exploitation in Social and Political Philosophy
Social and Political Philosophy, Miscellaneous in Social and Political Philosophy
Syntactical Informational Structural Realism.Majid Beni - 2018 - Minds and Machines 28 (4):623-643.details
Luciano Floridi's informational structural realism takes a constructionist attitude towards the problems of epistemology and metaphysics, but the question of the nature of the semantical component of his view remains vexing. In this paper, I propose to dispense with the semantical component of ISR completely. I outline a Syntactical version of ISR. The unified entropy-based framework of information has been adopted as the groundwork of SISR. To establish its realist component, SISR should be able to dissolve the latching problem. We (...) have to be able to account for the informational structures–reality relationship in the absence of the standard semantical resources. The paper offers a pragmatic solution to the latching problem. I also take pains to account for the naturalistic plausibility of this solution by grounding it in the recent computational neuroscience of the predictive coding and the free energy principle. (shrink)
Philosophy of Artificial Intelligence in Philosophy of Cognitive Science
Individual Differences in Workplace Deviance and Integrity as Predictors of Academic Dishonesty.Gale M. Lucas & James Friedrich - 2005 - Ethics and Behavior 15 (1):15 – 35.details
Meta-analytic findings have suggested that individual differences are relatively weaker predictors of academic dishonesty than are situational factors. A robust literature on deviance correlates and workplace integrity testing, however, demonstrates that individual difference variables can be relatively strong predictors of a range of counterproductive work behaviors (CWBs). To the extent that academic cheating represents a kind of counterproductive behavior in the work role of "student", employment-type integrity measures should be strong predictors of academic dishonesty. Our results with a college (...) student sample showed that integrity test scores were moderate to strong correlates of self-reported academic cheating and that these relationships persisted even after controlling for a variety of measurement concerns such as item format similarity, concurrent assessment, and socially desirable responding. Implications for institutional honor codes and the broader relations between educational and workplace dishonesty are discussed. (shrink)
Deception in Applied Ethics
Beyond Sweatshops: Positive Deviancy and Global Labour Practices.Denis G. Arnold & Laura P. Hartman - 2005 - Business Ethics: A European Review 14 (3):206-222.details
Deviance, Darwinian-Style.Gregory Radick - 2005 - Metascience 14 (3):453-457.details
The Role of Ethical Ideology in Workplace Deviance.Christine A. Henle, Robert A. Giacalone & Carole L. Jurkiewicz - 2005 - Journal of Business Ethics 56 (3):219-230.details
Ethical ideology is predicted to play a role in the occurrence of workplace deviance. Forsyths (1980) Ethics Position Questionnaire measures two dimensions of ethical ideology: idealism and relativism. It is hypothesized that idealism will be negatively correlated with employee deviance while relativism will be positively related. Further, it is predicted that idealism and relativism will interact in such a way that there will only be a relationship between idealism and deviance when relativism is higher. Results supported the (...) hypothesized correlations and idealism and relativism interacted to predict organizational deviance. Idealism was a significant predictor of interpersonal deviance, but no interaction was found. (shrink)
Positive Deviance on the Ethical Continuum: Green Mountain Coffee as a Case Study in Conscientious Capitalism.Mary Grace Neville - 2008 - Business and Society Review 113 (4):555-576.details
The Syntactic Characterization of Agrammatism.Yosef Grodzinsky - 1984 - Cognition 16 (2):99-120.details
A Syntactic and Semantic Analysis of Idealizations in Science.William F. Barr - 1971 - Philosophy of Science 38 (2):258-272.details
Various laws and theories in the natural and social sciences are presented with a view to discerning the syntactic and semantic characteristics of many idealizations in science. Three different kinds of idealizations are discussed: ideal conditions, ideal cases, and idealized theories. An ideal condition is a formula in which state variables occur, whose existential closure is false, and for which there is another formula that can be constructed out of the original formula such that the existential closure of the (...) new formula is true. An ideal case is a statement which is logically equivalent to a universal conditional which has an ideal condition as its antecedent. And an idealized theory is a set of false universal conditional statements. Alternative syntactic and semantic analyses are viewed and criticized. (shrink)
Idealization in General Philosophy of Science
Semantic View of Theories in General Philosophy of Science
A Syntactic Approach to Closure Operation.Marek Nowak - 2017 - Bulletin of the Section of Logic 46 (3/4).details
In the paper, tracing the traditional Hilbert-style syntactic account of logics, a syntactic characteristic of a closure operation defined on a complete lattice follows. The approach is based on observation that the role of rule of inference for a given consequence operation may be played by an ordinary binary relation on the complete lattice on which the closure operation is defined.
A Study in Modal Deviance.Gideon Rosen - 2002 - In John Hawthorne & Tamar Gendler (eds.), Conceivability and Possibility. Oxford University Press. pp. 283--307.details
Conceivability, Imagination, and Possibility in Metaphysics
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Syntactic Structure and Artificial Grammar Learning: The Learnability of Embedded Hierarchical Structures.Meinou H. de Vries, Padraic Monaghan, Stefan Knecht & Pienie Zwitserlood - 2008 - Cognition 107 (2):763-774.details
Syntactic Determinants of Sentence Comprehension in Aphasia.David Caplan, Catherine Baker & Francois Dehaut - 1985 - Cognition 21 (2):117-175.details
Other Areas of Linguistics, Misc in Philosophy of Language
Psycholinguistics in Philosophy of Language
How Helen Keller Used Syntactic Semantics to Escape From a Chinese Room.William J. Rapaport - 2006 - Minds and Machines 16 (4):381-436.details
A computer can come to understand natural language the same way Helen Keller did: by using "syntactic semantics"—a theory of how syntax can suffice for semantics, i.e., how semantics for natural language can be provided by means of computational symbol manipulation. This essay considers real-life approximations of Chinese Rooms, focusing on Helen Keller's experiences growing up deaf and blind, locked in a sort of Chinese Room yet learning how to communicate with the outside world. Using the SNePS computational knowledge-representation (...) system, the essay analyzes Keller's belief that learning that "everything has a name" was the key to her success, enabling her to "partition" her mental concepts into mental representations of: words, objects, and the naming relations between them. It next looks at Herbert Terrace's theory of naming, which is akin to Keller's, and which only humans are supposed to be capable of. The essay suggests that computers at least, and perhaps non-human primates, are also capable of this kind of naming. (shrink)
Cognitive Sciences, Misc in Cognitive Sciences
Disability in Applied Ethics
Syntactic Structure Assembly in Human Parsing: A Computational Model Based on Competitive Inhibition and a Lexicalist Grammar.Theo Vosse & Gerard Kempen - 2000 - Cognition 75 (2):105-143.details
Causal Deviancy and Multiple Intentions.James A. Montmarquet - 1982 - Analysis 42 (2):106 - 110.details
The Structure of Action in Philosophy of Action
Syntactical Learning and Judgment, Still Unconscious and Still Abstract: Comment on Dulany, Carlson, and Dewey.Arthur S. Reber, Rhianon Allen & Susan Regan - 1985 - Journal of Experimental Psychology: General 114 (1):17-24.details
Conscious and Unconscious Learning in Philosophy of Cognitive Science
Supererogation: Beyond Positive Deviance and Corporate Social Responsibility.Daina Mazutis - 2014 - Journal of Business Ethics 119 (4):517-528.details
The special class of supererogatory actions—those that go "beyond the call of duty"—has thus far been omitted from the management literature. Rather, actions of a firm that may surpass economic and legal requirements have been discussed either under the umbrella term of Corporate Social Responsibility or the concept of positive deviance as articulated by the Positive Organizational Scholarship movement. This paper seeks to clarify how "duty" is understood in these literatures and makes an argument that paradigmatic examples of corporate (...) supererogation in fact lie beyond what is traditionally conceptualized as CSR and positive deviance. In so doing, this paper contributes to the growing body of research on Positive Organizational Ethics, as well as both the CSR and POS literatures, by presenting an extended deontological framework of CSR and bringing conceptual clarity to an otherwise muddied domain. (shrink)
Supererogation in Normative Ethics
A Syntactic Approach to Unification in Transitive Reflexive Modal Logics.Rosalie Iemhoff - 2016 - Notre Dame Journal of Formal Logic 57 (2):233-247.details
This paper contains a proof-theoretic account of unification in transitive reflexive modal logics, which means that the reasoning is syntactic and uses as little semantics as possible. New proofs of theorems on unification types are presented and these results are extended to negationless fragments. In particular, a syntactic proof of Ghilardi's result that $\mathsf {S4}$ has finitary unification is provided. In this approach the relation between classical valuations, projective unifiers, and admissible rules is clarified.
Syntactic Structures and Recursive Devices: A Legacy of Imprecision. [REVIEW]Marcus Tomalin - 2011 - Journal of Logic, Language and Information 20 (3):297-315.details
Taking Chomsky's Syntactic Structures as a starting point, this paper explores the use of recursive techniques in contemporary linguistic theory. Specifically, it is shown that there were profound ambiguities surrounding the notion of recursion in the 1950s, and that this was partly due to the fact that influential texts such as Syntactic Structures neglected to define what exactly constituted a recursive device. As a result, uncertainties concerning the role of recursion in linguistic theory have prevailed until the present (...) day, and some of the most common misunderstandings that have appeared in recent discussions are examined at some length. This article shows that debates about such topics are frequently undermined by fundamental misunderstandings concerning core terminology, and the full extent of the prevailing haziness is revealed. An attempt is made, for instance, to distinguish between such things as iterative constructional devices and self-similar syntactic embedding, despite the fact that these are usually both unhelpfully classified as examples of recursion. Consequently, this article effectively constitutes a plea for much greater accuracy and clarity when such important issues are addressed from a linguistic perspective. (shrink)
A Syntactic Approach to Rationality in Games with Ordinal Payoffs.Giacomo Bonanno - 2008 - In Giacomo Bonanno, Wiebe van der Hoek & Michael Wooldridge (eds.), Logic and the Foundations of Game and Decision Theory. Amsterdam University Press.details
We consider strategic-form games with ordinal payoffs and provide a syntactic analysis of common belief/knowledge of rationality, which we define axiomatically. Two axioms are considered. The first says that a player is irrational if she chooses a particular strategy while believing that another strategy is better. We show that common belief of this weak notion of rationality characterizes the iterated deletion of pure strategies that are strictly dominated by pure strategies. The second axiom says that a player is irrational (...) if she chooses a particular strategy while believing that a different strategy is at least as good and she considers it possible that this alternative strategy is actually better than the chosen one. We show that common knowledge of this stronger notion of rationality characterizes the restriction to pure strategies of the iterated deletion procedure introduced by Stalnaker (1994). Frame characterization results are also provided. (shrink)
Game-Theoretic Principles in Philosophy of Action
Rationality in Epistemology
Syntactic Alignment and Participant Role in Dialogue.Holly P. Branigan, Martin J. Pickering, Janet F. McLean & Alexandra A. Cleland - 2007 - Cognition 104 (2):163-197.details
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\begin{document}
\title{Extremal Type~I $\mathbb{Z}_k$-codes and $k$-frames of odd unimodular lattices}
\author{ Masaaki Harada\thanks{ Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980--8579, Japan. email: [email protected]. This work was partially carried out at Yamagata University.} }
\maketitle
\begin{abstract}
For some extremal (optimal) odd unimodular lattice $L$ in dimensions $12,16,20,28,32,36,40$ and $44$, we determine all integers $k$ such that $L$ contains a $k$-frame. This result yields the existence of an extremal Type~I $\mathbb{Z}_{k}$-code of lengths $12,16,20,32,36,40$ and $44$, and a near-extremal Type~I $\mathbb{Z}_k$-code of length $28$ for positive integers $k$ with only a few exceptions.
\end{abstract}
\section{Introduction}\label{Sec:1}
Self-dual codes and unimodular lattices are studied from several viewpoints (see~\cite{SPLAG} for an extensive bibliography). Many relationships between self-dual codes and unimodular lattices are known and there are similar situations between two subjects.
As a typical example, it is known that a unimodular lattice $L$ contains a $k$-frame if and only if there is a self-dual $\mathbb{Z}_{k}$-code $C$ such that $L$ is isomorphic to the lattice obtained from $C$ by Construction A, where $\mathbb{Z}_{k}$ is the ring of integers modulo $k$ with $k \ge 2$.
Type~II $\mathbb{Z}_{2k}$-codes were defined in~\cite{BDHO} as a class of self-dual codes,
which are related to even unimodular lattices.
If $C$ is a Type~II $\mathbb{Z}_{2k}$-code of length $n \le 136$, then we have the bound on the minimum Euclidean weight $d_E(C)$ of $C$ as follows: $d_E(C) \le 4k \left\lfloor \frac{n}{24} \right\rfloor +4k$ for every positive integer $k$ (see~\cite{HM12}). A Type~II $\mathbb{Z}_{2k}$-code meeting the bound with equality is called {extremal} $(n \le 136)$. It was shown in~\cite{Chapman, GH01} that the Leech lattice, which is one of the most remarkable lattices, contains a $2k$-frame for every integer $k \ge 2$. This result yields the existence of an extremal Type~II $\mathbb{Z}_{2k}$-code of length $24$ for every positive integer $k$. Recently, the existence of an extremal Type~II $\mathbb{Z}_{2k}$-code of lengths $n=32,40,48,56,64$ was established in~\cite{HM12} for every positive integer $k$. This was done by finding a $2k$-frame in some extremal even unimodular lattices in these dimensions $n$.
Recently, it was shown in~\cite{Miezaki} that the odd Leech lattice contains a $k$-frame for every integer $k$ with $k \ge 3$. This motivates our investigation of the existence of a $k$-frame in extremal odd unimodular lattices. In this paper, for some extremal (optimal) odd unimodular lattices $L$ in dimensions $12,16,20,28,32,36,40$ and $44$, we determine all integers $k$ such that $L$ contains a $k$-frame. This result yields the existence of an extremal Type~I $\mathbb{Z}_{k}$-code of lengths $12,16,20,32,36,40$ and $44$, and a near-extremal Type~I $\mathbb{Z}_k$-code of length $28$ for positive integers $k$ with only a few small exceptions.
This paper is organized as follows. In Section~\ref{sec:Pre}, we give definitions and some basic properties of self-dual codes and unimodular lattices used in this paper. The notion of extremal Type~I $\mathbb{Z}_k$-codes of length $n$ is given for $n \le 48$ and $k \ge 2$. Lemma~\ref{lem:frame} gives a reason why we consider unimodular lattices only in dimension $n$ divisible by $4$.
In Section~\ref{sec:frame}, we provide a method for constructing $m$-frames in unimodular lattices, which are constructed from some self-dual $\mathbb{Z}_k$-codes by Construction A (Proposition~\ref{prop:const}).
This method is a slight generalization of~\cite[Propositions 3.3 and 3.6]{HM12}.
Using Proposition~\ref{prop:const}, we give $k$-frames in the unique extremal odd unimodular lattice in dimensions $12,16$, some extremal (optimal) odd unimodular lattices in dimensions $20,28, 32,36,40$ and $44$, which are listed in Table~\ref{Tab:L}, for all integers $k$ satisfying the condition $(\star)$ in Table~\ref{Tab:L} (Lemma~\ref{lem:key}).
In Section~\ref{sec:main}, several extremal (near-extremal) Type~I $\mathbb{Z}_{k}$-codes are explicitly constructed for some positive integers $k$. Then we establish the existence of a $k$-frame in the extremal (optimal) unimodular lattices $L$ in dimensions $12,16,20,28, 32,36$, which are listed in Table~\ref{Tab:L} (except only lattices $A_3(C_{20,3}(D'_{10}))$ and $A_5(C_{20,5}(D''_{10}))$), for every integer $k$ with $k \ge \min(L)$, where $\min(L)$ denotes the minimum norm of $L$.
As a consequence, by considering the even unimodular neighbors of the above extremal odd unimodular lattice in dimension $32$, it is shown that the $32$-dimensional Barnes--Wall lattice $BW_{32}$ contains a $2k$-frame if and only if $k$ is an integer with $k \ge 2$.
When $n=40,44$, we show that there is an extremal odd unimodular lattice in dimension $n$ containing a $k$-frame if and only if $k$ is an integer with $k \ge 4$.
Using the above existence of $k$-frames, the existence of an extremal Type~I $\mathbb{Z}_{k}$-code of lengths $n=12,16,20,32,36,40, 44$, and a near-extremal Type~I $\mathbb{Z}_k$-code of length $n=28$ is established for a positive integer $k$, where $k \ne 1,3$ if $n =32$ and $k \ne 1$ otherwise.
At the end of Section~\ref{sec:main}, we examine the existence of both $k$-frames in optimal odd unimodular lattices in dimension $48$ and near-extremal Type~I $\mathbb{Z}_k$-codes of length $48$.
All computer calculations in this paper were done by {\sc Magma}~\cite{Magma}.
\section{Preliminaries}\label{sec:Pre}
In this section, we give definitions and some basic properties of self-dual codes and unimodular lattices used in this paper. The notion of extremal Type~I $\mathbb{Z}_k$-codes of length $n$ is given for $n \le 48$ and $k \ge 2$.
\subsection{Self-dual codes} Let $\mathbb{Z}_{k}$ be the ring of integers modulo $k$, where $k$ is a positive integer. In this paper, we always assume that $k\geq 2$ and we take the set $\mathbb{Z}_{k}$ to be $\{0,1,\ldots,k-1\}$.
A $\mathbb{Z}_{k}$-code $C$ of length $n$ (or a code $C$ of length $n$ over $\mathbb{Z}_{k}$) is a $\mathbb{Z}_{k}$-submodule of $\mathbb{Z}_{k}^n$. A $\mathbb{Z}_2$-code and a $\mathbb{Z}_3$-code are called {\em binary} and {\em ternary}, respectively.
The {\em Euclidean weight} of a codeword $x=(x_1,\ldots,x_n)$ of $C$ is $\sum_{\alpha=1}^{\lfloor k/2 \rfloor}n_\alpha(x) \alpha^2$, where $n_{\alpha}(x)$ denotes the number of components $i$ with $x_i \equiv \pm \alpha \pmod k$ $(\alpha=1,2,\ldots,\lfloor k/2 \rfloor)$.
The {\em minimum Euclidean weight} $d_E(C)$ of $C$ is the smallest Euclidean weight among all nonzero codewords of $C$.
A $\mathbb{Z}_{k}$-code $C$ is {\em self-dual} if $C=C^\perp$, where the dual code $C^\perp$ of $C$ is defined as $\{ x \in \mathbb{Z}_{k}^n \mid x \cdot y = 0$ for all $y \in C\}$ under the standard inner product $x \cdot y$. For only even positive integers $2k$, a {\em Type~II} $\mathbb{Z}_{2k}$-code was defined in \cite{BDHO} as a self-dual $\mathbb{Z}_{2k}$-code with the property that all Euclidean weights are congruent to $0$ modulo $4k$.
It is known that a Type~II $\mathbb{Z}_{2k}$-code of length $n$ exists if and only if $n$ is divisible by $8$~\cite{BDHO}. A self-dual code which is not Type~II is called {\em Type~I}. Two self-dual $\mathbb{Z}_{k}$-codes $C$ and $C'$ are {\em equivalent} if there exists a monomial $(\pm 1, 0)$-matrix $P$ with $C' = C \cdot P$, where
$C \cdot P = \{ x P\:|\: x \in C\}$.
\subsection{Unimodular lattices}\label{sec:2U} A (Euclidean) lattice $L \subset \mathbb{R}^n$ in dimension $n$ is {\em unimodular} if $L = L^{*}$, where the dual lattice $L^{*}$ of $L$ is defined as $\{ x \in {\mathbb{R}}^n \mid (x,y) \in \mathbb{Z} \text{ for all } y \in L\}$ under the standard inner product $(x,y)$. Two lattices $L$ and $L'$ are {\em isomorphic}, denoted $L \cong L'$, if there exists an orthogonal matrix $A$ with $L' = L \cdot A$, where $L \cdot A=\{xA \mid x \in L\}$. The automorphism group $\Aut(L)$ of $L$ is the group of all orthogonal matrices $A$ with $L = L \cdot A$. The norm of a vector $x$ is defined as $(x, x)$. The minimum norm $\min(L)$ of a unimodular lattice $L$ is the smallest norm among all nonzero vectors of $L$.
The theta series $\theta_{L}(q)$ of $L$ is the formal power series $\theta_{L}(q) = \sum_{x \in L} q^{(x,x)}$. The kissing number of $L$ is the second nonzero coefficient of the theta series.
A unimodular lattice with even norms is said to be {\em even}, and that containing a vector of odd norm is said to be {\em odd}.
An even unimodular lattice in dimension $n$ exists if and only if $n$ is divisible by $8$, while an odd unimodular lattice exists for every dimension.
It was shown in~\cite{RS-bound} that a unimodular lattice $L$ in dimension $n$ has minimum norm $\min(L) \le 2 \lfloor \frac{n}{24} \rfloor+2$ unless $n=23$ when $\min(L) \le 3$ (see~\cite{Siegel} for the case that $L$ is even). A unimodular lattice meeting the bound with equality is called {\em extremal}. Any extremal unimodular lattice in dimension $24k$ has to be even~\cite{Gaulter}.
Hence, an odd unimodular lattice $L$ in dimension $24k$ satisfies $\min(L) \le 2k+1$. We say that an odd unimodular lattice with the largest minimum norm among all odd unimodular lattices in that dimension is {\em optimal}.
Let $L$ be a unimodular lattice. Define $L_0=\{x \in L \mid (x,x) \equiv 0 \pmod 2\}$. Then $L_0$ is a sublattice of $L$ of index $2$ if $L$ is odd and $L_0=L$ if $L$ is even. The {\em shadow} $S$ of $L$ is defined as $S=L_0^* \setminus L$ if $L$ is odd and as $S=L$ if $L$ is even~\cite{CS-odd}.
Now suppose that $L$ is an odd unimodular lattice. Then there are cosets $L_1,L_2,L_3$ of $L_0$ such that $L_0^* = L_0 \cup L_1 \cup L_2 \cup L_3$, where $L = L_0 \cup L_2$ and $S = L_1 \cup L_3$. Two lattices are {\em neighbors} if both lattices contain a sublattice of index $2$ in common. If the dimension is divisible by $8$, then $L$ has two even unimodular neighbors of $L$, namely, $L_0 \cup L_1$ and $L_0 \cup L_3$.
\subsection{Construction A and $k$-frames}
We give a method to construct unimodular lattices from self-dual $\mathbb{Z}_{k}$-codes, which is referred to as {\em Construction A} (see~\cite{{BDHO},{HMV}}).
If $C$ is a self-dual $\mathbb{Z}_{k}$-code of length $n$, then the following lattice \[ A_{k}(C)=
\frac{1}{\sqrt{k}}\{(x_1,\ldots,x_n) \in \mathbb{Z}^n \mid (x_1 \bmod k,\ldots,x_n \bmod k)\in C\} \] is a unimodular lattice in dimension $n$.
The minimum norm of $A_{k}(C)$ is $\min\{k, d_{E}(C)/k\}$. Moreover, $C$ is a Type~II $\mathbb{Z}_{2k}$-code if and only if $A_{2k}(C)$ is an even unimodular lattice~\cite{BDHO}.
A set $\{f_1, \ldots, f_{n}\}$ of $n$ vectors $f_1, \ldots, f_{n}$ of a unimodular lattice $L$ in dimension $n$ with $ ( f_i, f_j ) = k \delta_{i,j}$ is called a {\em $k$-frame} of $L$, where $\delta_{i,j}$ is the Kronecker delta. It is trivial that if a unimodular lattice in dimension $n$
contains a $k$-frame then the number of vectors of norm $k$ is greater than or equal to $2n$.
It is known that a unimodular lattice $L$ contains a $k$-frame if and only if there exists a self-dual $\mathbb{Z}_{k}$-code $C$ with $A_{k}(C) \cong L$ (see~\cite{HMV}).
Therefore, we have the following:
\begin{lem}\label{lem:LtoC} Suppose that there is a unimodular lattice $L$ in dimension $n$ containing a $k$-frame. Then there is a self-dual $\mathbb{Z}_k$-code $C$ such that $d_E(C)$ is greater than or equal to $k \min(L)$. \end{lem}
The above lemma is useful when establishing the existence of extremal Type~I $\mathbb{Z}_k$-codes in Section~\ref{sec:main}.
By the following lemma, it is sufficient to consider the existence of a $p$-frame in a unimodular lattice for each prime $p$. The lemma also gives a reason why we consider unimodular lattices only in dimension $n$ divisible by $4$.
\begin{lem}[{\cite[Lemma~5.1]{Chapman}}]\label{lem:Chapman} \label{lem:frame} Let $n$ be a positive integer divisible by $4$. If a lattice $L$ in dimension $n$ contains a $k$-frame, then $L$ contains a $km$-frame for every positive integer $m$. \end{lem}
\subsection{Upper bounds on the minimum Euclidean weights}
It is known~\cite{MPS,Rains,RS-bound} that a self-dual $\mathbb{Z}_k$-code $C$ of length $n$ satisfies the following bound: \begin{equation}\label{eq:kbound} d_E(C) \le \begin{cases} 4 \lfloor \frac{n}{24} \rfloor+4 & \text{ if }
k=2, n \not\equiv 22 \pmod{24}, \\ 4 \lfloor \frac{n}{24} \rfloor+6 & \text{ if }
k=2, n \equiv 22 \pmod{24}, \\
3 \lfloor \frac{n}{12} \rfloor +3 & \text{ if } k=3, \\
8 \lfloor \frac{n}{24} \rfloor+8 & \text{ if } k=4,
n \not\equiv 23 \pmod{24}, \\ 8 \lfloor \frac{n}{24} \rfloor+12 & \text{ if } k=4,
n \equiv 23 \pmod{24}. \end{cases} \end{equation} Note that a binary self-dual code of length divisible by $24$ meeting the bound with equality must be Type~II~\cite{Rains}.
Although the following two lemmas are somewhat trivial, we give proofs for the sake of completeness.
\begin{lem}\label{lem:bound} Let $C$ be a self-dual $\mathbb{Z}_k$-code of length $n$. \begin{itemize} \item[\rm (a)] If $n \ne 23$ and $k \ge 2 \lfloor \frac{n}{24} \rfloor+3$, then $d_E(C) \le 2k \lfloor \frac{n}{24} \rfloor+2k$. \item[\rm (b)] If $n = 23$ and $k \ge 4$, then $d_E(C) \le 3k$. \end{itemize} \end{lem} \begin{proof} Since both cases are similar, we only give a proof of (a).
Note that the Euclidean weight of a codeword of $C$ is divisible by $k$. Suppose that $d_E(C) \ge 2k \lfloor \frac{n}{24} \rfloor+3k$. Since $\min(A_{k}(C))=\min\{k, d_{E}(C)/k\}$, $\min(A_{k}(C)) \ge 2 \lfloor \frac{n}{24} \rfloor+3$, which is a contradiction to the upper bound on the minimum norms of unimodular lattices.
\end{proof}
\begin{lem}\label{lem:bound2} If $C$ is a self-dual $\mathbb{Z}_k$-code of length $48$, then $d_E(C) \le 6k$. \end{lem} \begin{proof} By the bound (\ref{eq:kbound}) and Lemma~\ref{lem:bound}, it is sufficient to consider the cases only for $k=5,6$. Assume that $k=5,6$ and $d_E(C) \ge 7k$. Since $k < d_E(C)/k$, $\min(A_k(C))= k$ and the kissing number of $A_k(C)$ is $96$. Note that unimodular lattices $L$ with $\min(L)=6$ and $5$ are extremal even unimodular lattices and optimal odd unimodular lattices, respectively. However, the kissing numbers of such lattices are $52416000$ (see~\cite[Chap.~7, (68)]{SPLAG}) and $385024$ or $393216$ \cite{HKMV}, respectively. This is a contradiction. \end{proof}
By the bound (\ref{eq:kbound}) along with Lemmas~\ref{lem:bound} and~\ref{lem:bound2}, a self-dual $\mathbb{Z}_{k}$-code $C$ of length $n \le 48$ satisfies the following bound: \begin{equation}\label{eq:nbound} d_E(C) \le \begin{cases} 3k & \text{if $n =23$ and $k \ge 4$,} \\ 4 \lfloor \frac{n}{24} \rfloor+6
& \text{if $n =22,46$ and $k=2$,} \\ 20 & \text{if $n =47$ and $k=4$,} \\ 2k \left\lfloor \frac{n}{24} \right\rfloor +2k & \text{otherwise.} \end{cases} \end{equation} We say that a self-dual $\mathbb{Z}_{k}$-code meeting the bound (\ref{eq:nbound}) with equality is
{\em extremal}\footnote{For $k=3$, a self-dual code meeting the bound (\ref{eq:kbound}) is usually called extremal. However, we here adopt this definition since we simultaneously consider the existence of extremal self-dual $\mathbb{Z}_k$-codes for all integers $k$ with $k \ge 2$. }
for length $n \le 48$.
We say that a self-dual $\mathbb{Z}_k$-code $C$ is {\em near-extremal} if $d_E(C)+k$ meets the bound (\ref{eq:nbound}). We only consider near-extremal self-dual $\mathbb{Z}_k$-codes when there is no extremal self-dual $\mathbb{Z}_k$-code of that length.
The following lemma shows that an extremal self-dual $\mathbb{Z}_{k}$-code of lengths $24$ and $48$ must be Type~II for every even positive integer $k$.
\begin{lem} Let $C$ be a Type~I $\mathbb{Z}_k$-code of length $n$. \begin{itemize} \item[\rm (a)] If $n=24$, then $d_E(C) \le 3 k$. \item[\rm (b)] If $n=48$, then $d_E(C) \le 5 k$. \end{itemize} \end{lem} \begin{proof} We give a proof of (b).
By the bound (\ref{eq:kbound}), it is sufficient to consider only $k \ge 4$. Assume that $d_E(C) \ge 6 k$. If $k \ge 6$, then $A_k(C)$ has minimum norm $6$ from the upper bound on the minimum norms of unimodular lattices. In addition, $A_k(C)$ must be even~\cite{Gaulter}, that is, $C$ is Type~II.
If $k=5$, then $A_5(C)$ is an optimal odd unimodular lattice with kissing number $96$,
which contradicts that the kissing number is $385024$ or $393216$~\cite{HKMV}.
Finally, suppose that $k=4$. Since $d_E(C) \ge 24$, $A_4(C)$ satisfies the condition that $\min(A_4(C))=4$, the kissing number is $96$ and there is no vector of norm $5$.
By~\cite[(2) and (3)]{CS-odd}, one can determine the possible theta series of $A_4(C)$ and its shadow $S$ as follows: \[ \begin{cases} \theta_{A_4(C)}(q) &= 1 + 96 q^4 + (35634176 + 16777216 \alpha)q^6
+ \cdots, \\ \theta_{S}(q) &= \alpha + (96 - 96 \alpha )q^2 +(- 4416 + 4512 \alpha) q^4 + \cdots, \end{cases} \] respectively, where $\alpha$ is an integer. From the coefficients of $q^2$ and $q^4$ in $\theta_{S}(q)$,
it follows that $\alpha=1$. Hence,
$A_4(C)$ must be even, that is, $C$ is Type~II\@.
The proof of (a) is similar to that of (b), and it can be completed more easily. So the proof is omitted. \end{proof}
\begin{rem}
The odd Leech lattice contains a $k$-frame for every integer $k$ with $k \ge 3$~\cite{Miezaki}. The binary odd Golay code is a near-extremal Type~I code of length $24$. Hence, by Lemma~\ref{lem:LtoC}, there is a near-extremal Type~I $\mathbb{Z}_k$-code of length $24$ if and only if $k$ is an integer with $k \ge 2$. \end{rem}
\subsection{Negacirculant matrices} \label{subsec:M}
Throughout this paper, let $A^T$ denote the transpose of a matrix $A$ and let $I_k$ denote the identity matrix of order $k$. An $n \times n$ matrix is {\em circulant} and {\em negacirculant} if it has the following form: \[ \left( \begin{array}{ccccc} r_0 &r_1 & \cdots &r_{n-2}&r_{n-1} \\ cr_{n-1}&r_0 & \cdots &r_{n-3}&r_{n-2} \\ cr_{n-2}&cr_{n-1}& \ddots &r_{n-4}&r_{n-3} \\ \vdots & \vdots &\ddots& \ddots & \vdots \\ cr_1 &cr_2 & \cdots&cr_{n-1}&r_0 \end{array} \right), \] where $c=1$ and $-1$, respectively.
Most of matrices constructed in this paper are based on negacirculant matrices. In Section~\ref{sec:main}, in order to construct self-dual $\mathbb{Z}_k$-codes of length $4n$, we consider generator matrices of the following form: \begin{equation} \label{eq:GM} \left( \begin{array}{ccc@{}c} \quad & {\Large I_{2n}} & \quad & \begin{array}{cc} A & B \\ -B^T & A^T \end{array} \end{array} \right), \end{equation} where $A$ and $B$ are $n \times n$ negacirculant matrices. It is easy to see that the code is self-dual if $AA^T+BB^T=-I_n$.
In Section~\ref{sec:frame}, in order to find $k$-frames in some unimodular lattices, we need to construct matrices $M$ satisfying the condition (\ref{eq:condition}) in Proposition~\ref{prop:const}.
Suppose that $p$ is a prime, which is congruent to $3$ modulo $4$. Let $Q_{p}=(q_{ij})$ be a $p \times p$ matrix over $\mathbb{Z}$, where $q_{ij}=0$ if $i=j$, $-1$ if $j-i$ is a nonzero square modulo $p$, and
$1$ otherwise. We consider the following matrix: \[ P_{p+1} = \left(\begin{array}{ccccccc}
0 & 1 & \cdots &1 \\
-1 & {} & {} &{} \\
\vdots & {} & Q_{p} &{} \\
-1 & {} & {} &{} \\ \end{array}\right). \] Then it is well known that $P_{p+1}P_{p+1}^T=pI_{p+1}$, $P_{p+1}^T=-P_{p+1}$, and $P_{p+1}+I_{p+1}$ is a Hadamard matrix of order $p+1$. Hence, these matrices $P_{p+1}$ satisfy (\ref{eq:condition}). In Section~\ref{sec:frame}, we construct more $2m\times 2m$ matrices $M$ satisfying (\ref{eq:condition}) using the following form:
\begin{equation} \label{eq:3nega} \left( \begin{array}{rr} A_1 & A_2 \\
-A_2^T & A_1^T \end{array} \right), \end{equation} where $A_1$ and $A_2$ are $m \times m$ negacirculant matrices.
\subsection{Number theoretical results}
In order to give infinite families of $k$-frames by Proposition~\ref{prop:const}, the following lemma is needed.
The proofs are given by Miezaki in private communication~\cite{M}, which are similar to those in~\cite{Chapman, HM12, Miezaki}.
\begin{lem}[Miezaki~\cite{M}] \label{lem:prime} \begin{itemize} \item[\rm (a)]\label{thm:1} There are integers $a,b,c$ and $d$ satisfying $b \equiv c-d \pmod 3$, $d \equiv a+b \pmod 3$ and $p=\frac{1}{3}(a^2+25b^2+c^2+25d^2)$ for each prime $p \ne 2, 5, 7, 13, 23$.
\item[\rm (b)] \label{thm:2} There are integers $a,b,c$ and $d$ satisfying
$b \equiv c-2d \pmod 4$, $d \equiv a+2b \pmod 4$ and $p=\frac{1}{4}(a^2+7b^2+c^2+7d^2)$ for each prime $p \ne 2,7$.
\item[\rm (c)] \label{thm:7} There are integers $a,b,c$ and $d$ satisfying $b \equiv c \pmod 5$, $d \equiv a \pmod 5$ and $p=\frac{1}{5}(a^2+49b^2+c^2+49d^2)$ for each prime $p \ne 2, 3, 7, 11,19,29$.
\item[\rm (d)] \label{thm:6} There are integers $a,b,c$ and $d$ satisfying $b \equiv c-2d \pmod 5$, $d \equiv a+2b \pmod 5$ and $p=\frac{1}{5}(a^2+25b^2+c^2+25d^2)$ for each prime $p \ne 2, 3, 17$.
\item[\rm (e)] \label{thm:3} There are integers $a,b,c$ and $d$ satisfying
$b \equiv c-2d \pmod 4$, $d \equiv a+2b \pmod 4$ and $p=\frac{1}{4}(a^2+15b^2+c^2+15d^2)$ for each prime $p \ne 2,3$.
\item[\rm (f)] \label{thm:4} There are integers $a,b,c$ and $d$ satisfying $b \equiv c-2d \pmod 6$, $d \equiv a+2b \pmod 6$ and $p=\frac{1}{6}(a^2+49b^2+c^2+49d^2)$ for each prime $p \ne 2, 3, 5,7$.
\item[\rm (g)] \label{thm:5} There are integers $a,b,c$ and $d$ satisfying $b \equiv c \pmod 4$, $d \equiv a \pmod 4$ and $p=\frac{1}{4}(a^2+19b^2+c^2+19d^2)$ for each prime $p \ne 2, 3, 13, 19$.
\item[\rm (h)] \label{thm:8} There are integers $a,b,c$ and $d$ satisfying $b \equiv c \pmod 5$, $d \equiv a \pmod 5$ and $p=\frac{1}{5}(a^2+39b^2+c^2+39d^2)$ for each prime $p \ne 2, 3, 7, 17$.
\end{itemize} \end{lem}
\section{Construction of $m$-frames in some unimodular lattices} \label{sec:frame}
In this section, we provide a method for constructing $m$-frames in unimodular lattices, which are constructed from some self-dual $\mathbb{Z}_k$-codes by Construction A\@. Combining Lemma~\ref{lem:prime} with the method, we construct $m$-frames in odd unimodular lattices.
The following method is a slight generalization of~\cite[Propositions 3.3 and 3.6]{HM12}. Also, the cases $(k,m,\ell)=(4,11,2)$ and $(4,11,0)$ of the following method are used in~\cite{Chapman,Miezaki}, respectively.
\begin{prop}\label{prop:const} Let $k$ be an integer with $k \ge 2$, and let $\ell$ be an integer with $0 \le \ell \le k-1$. Let $M$ be an $n \times n$
matrix over $\mathbb{Z}$ satisfying \begin{equation}\label{eq:condition} M^T=-M \text{ and } M M^T= mI_n, \end{equation} where $m+\ell^2 \equiv -1 \pmod{k}$. Let $C_{2n,k}(M)$ be the $\mathbb{Z}_{k}$-code of length $2n$ with generator matrix $\left(\begin{array}{cc} I_n & M+ \ell I_n \end{array}\right)$, where the entries of the matrix are regarded as elements of $\mathbb{Z}_{k}$. Let $a,b,c$ and $d$ be integers with $b \equiv c-\ell d \pmod {k}$ and $d \equiv a+\ell b \pmod {k}$. Then $C_{2n,k}(M)$ is self-dual, and the set of $2n$ rows of the following matrix \[ F(M)= \frac{1}{\sqrt{k}} \left( \begin{array}{cc} aI_n+bM & cI_n+dM \\ -cI_n+dM & aI_n-bM \end{array} \right) \] forms a $\frac{1}{k}(a^2+m b^2+c^2+m d^2)$-frame in the unimodular lattice $A_{k}(C_{2n,k}(M))$. \end{prop} \begin{proof} Since $M M^T= mI_n$, $M^T=-M$ and $m+ \ell^2 \equiv -1 \pmod{k}$, $C_{2n,k}(M)$ is a self-dual $\mathbb{Z}_{k}$-code of length $2n$. Thus, $A_{k}(C_{2n,k}(M))$ is a unimodular lattice.
Since $C_{2n,k}(M)$ is self-dual and $M^T=-M$, both $G=\left(\begin{array}{cc} I_n & M+\ell I_n \end{array}\right)$ and $H=\left(\begin{array}{cc} M-\ell I_n & I_n \end{array}\right)$ are generator matrices of $C_{2n,k}(M)$.
Let $s,t$ be integers. Here, we regard the entries of the matrices $G,H$ as integers. Then \[ \left(\begin{array}{c} sG+tH \\ -tG+sH \end{array}\right) = \left(\begin{array}{cc} (s-\ell t)I_n+tM & (\ell s+t)I_n+ sM \\ -(\ell s+t)I_n+sM & (s-\ell t)I_n -tM \end{array}\right). \]
Hence, if $b \equiv c-\ell d \pmod {k}$ and $d \equiv a+\ell b \pmod {k}$, then all rows of the matrix $F(M)$ are vectors of $A_{k}(C_{2n,k}(M))$. Since $F(M) F(M)^T=\frac{1}{k}(a^2+mb^2+c^2+md^2)I_{2n}$, the result follows. \end{proof}
\begin{rem} \begin{itemize} \item[(i)] It follows from the assumption that $a^2+m b^2+c^2+m d^2 \equiv 0 \pmod k$. \item[(ii)] By~\cite[Proposition 2.12]{GS}, if $n \equiv 2 \pmod 4$ then $m$ must be a square. \end{itemize} \end{rem}
\begin{table}[thbp] \caption{Matrices satisfying the assumptions in Proposition~\ref{prop:const}} \label{Tab:D} \begin{center}
{\footnotesize
\begin{tabular}{c|c|l|l} \noalign{\hrule height0.8pt} $M$ & $(k,m,\ell)$
& \multicolumn{1}{c|}{$r_{A_1}$} &\multicolumn{1}{c}{$r_{A_2}$} \\ \hline $D_{6 }$ & (3, 25, 1) & $(0, 2, 2)$ & $(0, 1, -4)$ \\ $P_{8 }$ & (4, 7, 2) && \\
$D_{10}$ & (3, 25, 1) & $(0,0,2,2,0)$&$(1,2,2,-2,2)$ \\ $D'_{10}$ &(3, 25, 1) & $(0,0,0,0,0)$&$(-3, -2, 2, -2, 2)$ \\ $D''_{10}$&(5, 49, 0) & $( 0, 0,3, 3,0)$&$(-2,-3,4,-1,1)$ \\
$D_{14}$ & (3, 25, 1) & $(0,2,1,0,0,1,2)$&$(-1,-2,1,-2,2,1,0)$ \\ $D'_{14}$ &(5, 25, 2) & $(0,0,2,-1,-1,2,0)$&$(-2,-1,-2,0,-1,-1,-2)$ \\
$D_{16}$ & (4, 15, 2) & $(0,1,1,0,1,0,1,1)$& $(1,1,1,-1,-1,2,-1,0)$ \\ $D_{18}$ & (6, 49, 2) & $(0,1,-3,0,2,2,0,-3,1)$ & $(-2,2,-1,2,1,2,1,1,1)$\\ $P_{20}$ & (4, 19, 0) & & \\ $D_{22}$ & (5, 25, 2) & {$(0,0,-1,1,0,0,0,0,1,-1,0)$} & {$(1,0,-2,1,1,1,2,1,0,2,-2)$} \\ $D_{24}$ & (5, 39, 0) & {$( 0, 1, 1, 1, 2,-1, 1,-1, 2, 1, 1, 1)$} & {$(-2,-1, 2,-1,-1,-2, 0, 1, 0, 2,-1,-1)$} \\ \noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
The matrices $P_{p+1}$ $(p=7,19)$, which are given in Section~\ref{subsec:M}, satisfy the assumptions in Proposition~\ref{prop:const}, for the integers $k,m$ and $\ell$ listed in Table~\ref{Tab:D}.
Using the form (\ref{eq:3nega}),
we have found
matrices $D_n$ $(n=6,10,14,16,18,22,24)$, $D'_{n}$ $(n=10,14)$ and $D''_{10}$ satisfying the assumptions in Proposition~\ref{prop:const}, for the integers $k,m$ and $\ell$ listed in Table~\ref{Tab:D},
where the first rows $r_{A_1}$ and $r_{A_2}$ of negacirculant matrices
$A_1$ and $A_2$ in (\ref{eq:3nega}) are listed in Table~\ref{Tab:D}.
By Proposition~\ref{prop:const}, for each of the matrices $M$ given in Table~\ref{Tab:D}, the odd unimodular lattice $A_k(C_{2n,k}(M))$, which is constructed from the Type~I $\mathbb{Z}_k$-code $C_{2n,k}(M)$, contains a $\frac{1}{k}(a^2+m b^2+c^2+m d^2)$-frame for integers $a,b,c$ and $d$ with $b \equiv c-\ell d \pmod {k}$ and $d \equiv a+\ell b \pmod {k}$. The minimum norms $\min(L)$ of the lattices $L=A_k(C_{2n,k}(M))$ listed in Table~\ref{Tab:L}, which have been determined by {\sc Magma}, are also listed in the table.
\begin{lem}\label{lem:key}
Suppose that $L$ is any of the lattices listed in Table~\ref{Tab:L}. Then $L$ contains a $k$-frame for an integer $k$ satisfying the conditions {\rm ($\star$)} listed in Table~\ref{Tab:L}, where $m_i$ in {\rm ($\star$)} is a non-negative integer. \end{lem} \begin{proof} All cases are similar, and we only give details for the lattice $A_3(C_{12,3}(D_6))$. Let $a,b,c$ and $d$ be integers with $b \equiv c-d \pmod 3$ and $d \equiv a+b \pmod 3$. By Proposition~\ref{prop:const}, $A_3(C_{12,3}(D_6))$ contains a $\frac{1}{3}(a^2+25b^2+c^2+25d^2)$-frame. By Lemma~\ref{lem:prime} (a), there are integers $a,b,c$ and $d$ satisfying $b \equiv c-d \pmod 3$, $d \equiv a+b \pmod 3$ and $p=\frac{1}{3}(a^2+25b^2+c^2+25d^2)$ for each prime $p \ne 2, 5, 7, 13, 23$. The result follows from Lemma~\ref{lem:frame}.
For the other lattices, Table~\ref{Tab:L} lists
the cases of Lemma~\ref{lem:prime}, which are needed in the proof. \end{proof}
\begin{table}[thbp] \caption{Unimodular lattices by Proposition~\ref{prop:const}} \label{Tab:L} \begin{center} {\small
\begin{tabular}{c|c|l|c} \noalign{\hrule height0.8pt}
$L$ & $\min(L)$ & \multicolumn{1}{c|}{Condition ($\star$)} & Case\\
\hline $A_3(C_{12,3}(D_6))$ &2 & $k \ge 2$, $k\ne 2^{m_1} 5^{m_2} 7^{m_3}13^{m_4}23^{m_5}$ & (a) \\ $A_4(C_{16,4}(P_{8}))$ &2~\cite{Z4-H40} & $k \ge 2$, $k\ne 2^{m_1}7^{m_2}$ & (b)\\ $A_3(C_{20,3}(D_{10}))$ &2 & $k \ge 2$, $k\ne 2^{m_1} 5^{m_2} 7^{m_3}13^{m_4}23^{m_5}$ & (a) \\ $A_3(C_{20,3}(D'_{10}))$ &2 & $k \ge 2$, $k\ne 2^{m_1} 5^{m_2} 7^{m_3}13^{m_4}23^{m_5}$ & (a) \\ $A_5(C_{20,5}(D''_{10}))$ &2 & $k \ge 2$, $k\ne 2^{m_1} 3^{m_2} 7^{m_3} 11^{m_4} 19^{m_5} 29^{m_6}$ &(c) \\ $A_3(C_{28,3}(D_{14}))$ &3 & $k \ge 3$, $k\ne 2^{m_1} 5^{m_2} 7^{m_3}13^{m_4}23^{m_5}$ & (a) \\ $A_5(C_{28,5}(D'_{14}))$ &3 & $k \ge 3$, $k\ne 2^{m_1} 3^{m_2} 17^{m_3}$ & (d)\\ $A_4(C_{32,4}(D_{16}))$ &4 & $k \ge 4$, $k\ne 2^{m_1} 3^{m_2}$ & (e) \\ $A_6(C_{36,6}(D_{18}))$ &4 & $k \ge 4$, $k\ne 2^{m_1} 3^{m_2} 5^{m_3}7^{m_4}$ & (f) \\ $A_4(C_{40,4}(P_{20}))$ &4~\cite{Z4-H40} & $k \ge 4$, $k\ne 2^{m_1}3^{m_2} 13^{m_3} 19^{m_4}$ & (g)\\ $A_5(C_{44,5}(D_{22}))$ &4 & $k \ge 4$, $k\ne 2^{m_1} 3^{m_2} 17^{m_3}$ & (d)\\
$A_5(C_{48,5}(D_{24}))$ &5 & $k \ge 5$, $k\ne 2^{m_1} 3^{m_2} 7^{m_3} 17^{m_4}$ & (h)\\
\noalign{\hrule height0.8pt} \end{tabular} }
\end{center} \end{table}
\section{Frames of some extremal odd unimodular lattices and extremal Type~I $\mathbb{Z}_k$-codes} \label{sec:main}
In this section, for each $L$ of the extremal (optimal) odd unimodular lattices listed in Table~\ref{Tab:L}, we determine all integers $k$ such that $L$ contains a $k$-frame.
This yields the existence of some extremal (near-extremal) Type~I $\mathbb{Z}_{k}$-codes.
\subsection{Frames of $D_{12}^+$ and Type~I $\mathbb{Z}_k$-codes of length 12}
There is a unique extremal odd unimodular lattice in dimension $12$, up to isomorphism (see~\cite[Table~16.7]{SPLAG}), where the lattice is denoted by $D_{12}^+$. There is a unique binary extremal Type~I code of length $12$, up to equivalence~\cite{Pless72b}, where the code is denoted by $B_{12}$ in~\cite[Table~2]{Pless72b}. It is known that $D_{12}^+ \cong A_2(B_{12})$. Hence, by Lemma~\ref{lem:key}, it is sufficient to investigate the existence of a $k$-frame in $D_{12}^+$ for $k=5,7,13,23$.
There are 16 inequivalent Type~I $\mathbb{Z}_5$-codes of length $12$~\cite{LPS-GF5}. We have verified by {\sc Magma} that $D_{12}^+ \cong A_5(C_i)$ for $i= 8, 11, 13, 16$, where $C_i$ denotes the $i$th code in~\cite[Table~III]{LPS-GF5}. There are $64$ inequivalent Type~I $\mathbb{Z}_7$-codes of length $12$~\cite{HO02}, where these codes are denoted by $C_{12,i}$ ($i=1,2,\ldots,64$) in~\cite[Table~1]{HO02}. We have verified by {\sc Magma} that $D_{12}^+ \cong A_7(C_{12,i})$ for $i=11$, $12$, $15$, $17$, $20$, $38$, $42$, $43$, $47$, $49$, $51$, $54$, $55$, $57, \ldots, 62$. For $k=13$ and $23$, let $C_{k,12}$ be the $\mathbb{Z}_k$-code with generator matrix of the form (\ref{eq:GM}),
where the first rows $r_A$ and $r_B$ of $A$ and $B$ are as follows: \[ (r_A,r_B)= ((0, 1, 6),( 2, 3, 1)) \text{ and } ((0, 1, 18),( 7, 4, 0)), \] respectively. Since $AA^T+BB^T=-I_{3}$,
these codes are Type~I\@. Moreover, we have verified by {\sc Magma} that $A_k(C_{k,12}) \cong D_{12}^+$ ($k=13,23$). Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{thm}\label{thm:D12} $D_{12}^+$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 2$.
\end{thm}
By Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor}\label{cor:12} There is an extremal Type~I $\mathbb{Z}_{k}$-code of length $12$ if and only if $k$ is an integer with $k \ge 2$.
\end{cor}
\subsection{Frames of $D_8^2$ and Type~I $\mathbb{Z}_k$-codes of length 16}
There is a unique extremal odd unimodular lattice in dimension $16$, up to isomorphism (see~\cite[Table~16.7]{SPLAG}), where the lattice is denoted by $D_8^2$. There is a unique binary extremal Type~I code of length $16$, up to equivalence~\cite{Pless72b}, where the code is denoted by $F_{16}$ in~\cite[Table~2]{Pless72b}. It is known that $D_8^2 \cong A_2(F_{16})$. Hence, by Lemma~\ref{lem:key}, it is sufficient to investigate the existence of a $7$-frame in $D_8^2$.
Let $C_{7,16}$ be the $\mathbb{Z}_7$-code with generator matrix of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$
are as follows: \[ r_A=(0, 0, 1, 1 ) \text{ and } r_B=(1, 3, 1, 0), \] respectively. Since $AA^T+BB^T=-I_{4}$,
$C_{7,16}$ is Type~I\@. We have verified by {\sc Magma} that $A_7(C_{7,16}) \cong D_8^2$. Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{thm}\label{thm:D82}
$D_8^2$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 2$. \end{thm}
By Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor}\label{cor:16} There is an extremal Type~I $\mathbb{Z}_{k}$-code of length $16$ if and only if $k$ is an integer with $k \ge 2$.
\end{cor}
\subsection{Frames of $D_4^5, A_5^4, D_{20}$ and Type~I $\mathbb{Z}_k$-codes of length 20}
There are $12$ non-isomorphic extremal odd unimodular lattices in dimension $20$ (see~\cite[Table~2.2]{SPLAG}). We denote the $i$th lattices $(i=1,11,12)$ in dimension $20$ in~\cite[Table~16.7]{SPLAG}, by $D_{20}$, $D_4^5$, $A_5^4$, respectively. We have verified by {\sc Magma} that $D_4^5 \cong A_3(C_{20,3}(D_{10}))$ in Table~\ref{Tab:L}, $A_5^4 \cong A_3(C_{20,3}(D'_{10}))$ in Table~\ref{Tab:L} and $D_{20} \cong A_5(C_{20,5}(D''_{10}))$ in Table~\ref{Tab:L}.
By Lemma~\ref{lem:key}, it is sufficient to investigate the existence of a $k$-frame in $D_4^5$ and $A_5^4$ for $k=2,5,7,13,23$, and a $k$-frame in $D_{20}$ for $k=2, 3,7,11,19,29$.
\begin{table}[thb] \caption{Extremal Type~I $\mathbb{Z}_{k}$-codes of length $20$} \label{Tab:20} \begin{center}
{\footnotesize
\begin{tabular}{c|l|l|c|l|l} \noalign{\hrule height0.8pt}
Code& \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c|}{$r_B$} &
Code& \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c}{$r_B$} \\ \hline $C_{ 5,20}$ &$(0, 0, 0, 1, 1)$ & $( 1, 4, 2, 1, 0)$ & $C_{ 7,20}$ &$(0, 0, 0, 1, 6)$ & $( 3, 0, 1, 1, 0)$ \\ $C_{13,20}$ &$(0, 0, 0, 1, 1)$ & $( 10, 3, 2, 1, 0)$ & $C_{23,20}$ &$(0, 0, 0, 1, 18)$ & $( 7, 4, 0, 0, 0)$ \\ \hline $C'_{ 5,20}$ &$(0, 0, 0, 1, 4)$&$(3, 1, 4, 1, 0)$ & $C'_{ 7,20}$ &$(0, 0, 0, 1, 5)$&$(1, 5, 3, 1, 0)$\\ $C'_{13,20}$ &$(0, 0, 0, 1, 4)$&$(4, 0, 3, 3, 0)$ & $C'_{23,20}$ &$(0, 0, 0, 1,12)$&$(3, 5, 7, 1, 0)$\\ \hline $C''_{ 7,20}$& $(0, 0, 0, 1, 4)$&$( 1, 3, 2, 3, 1)$ & $C''_{ 9,20}$& $(0,0,0,1,3)$&$(1,2,4,2,6)$ \\ $C''_{11,20}$& $(0, 0, 0, 1, 8)$&$( 5, 6, 6, 3, 2)$ & $C''_{19,20}$& $(0, 0, 0, 1, 12)$&$( 14, 12, 11, 1, 0)$ \\ $C''_{29,20}$& $(0, 0, 0, 1, 21)$&$( 7, 11, 16, 1, 0)$ & &&\\ \noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
\begin{figure}
\caption{A generator matrix of $C'_{4,20}$}
\label{Fig:20}
\end{figure}
There are $7$ binary extremal Type~I codes of length $20$, up to equivalence~\cite{Pless72b}.
The unique code containing $5$ (resp.\ $45$) codewords of weight $4$ is denoted by $M_{20}$ (resp.\ $J_{20}$) in~\cite[Table~2]{Pless72b}. We have verified by {\sc Magma} that $A_2(M_{20}) \cong D_4^5$ and $A_2(J_{20}) \cong D_{20}$.
It is known that there is no binary Type~I code $C$ such that $A_2(C) \cong A_5^4$.
There are $6$ inequivalent ternary self-dual codes of length $20$ and minimum weight $6$~\cite{PSW}. We have verified by {\sc Magma} that $L$ is an extremal odd unimodular lattice in dimension $20$ such that $L \cong A_3(C)$ for some ternary self-dual code $C$ if and only if $L$ is isomorphic to $D_4^5$ or $A_5^4$.
Let $C_{k,20}$, $C'_{k,20}$ ($k=5,7,13,23$) and $C''_{k,20}$ ($k=7,9,11,19,29$) be the $\mathbb{Z}_k$-codes with generator matrices of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$ are listed in Table~\ref{Tab:20}. Since $AA^T+BB^T=-I_{5}$, these codes are Type~I\@. Let $C'_{4,20}$ be the $\mathbb{Z}_4$-code with generator matrix of the following form: \[ G_{20}= \left(\begin{array}{ccc} I_9 & A & B_1+2B_2 \\ O &2I_2 & 2D \\ \end{array}\right), \] where we list in Figure~\ref{Fig:20} the matrices $\left(\begin{array}{cc} A & B_1+2B_2 \\ \end{array}\right)$ and $2D$. Here, $A$, $B_1$, $B_2$ and $D$ are $(1,0)$-matrices and $O$ denotes the zero matrix of appropriate size. It follows from $G_{20}G_{20}^T=O$ and $\# C'_{4,20}=4^{10}$ that $C'_{4,20}$ is self-dual. The code $C'_{4,20}$ has been found by directly finding a $4$-frame in $A_5^4$ using {\sc Magma}. In a similar way, some other (new) self-dual $\mathbb{Z}_4$-codes are also constructed in this paper.
We have verified by {\sc Magma} that $A_k(C_{k,20}) \cong D_4^5$ ($k=5,7,13,23$), $A_k(C'_{k,20}) \cong A_5^4$ ($k=4,5,7,13,23$) and $A_k(C''_{k,20}) \cong D_{20}$ ($k=7,9,11,19,29$). Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{thm}\label{thm:20} $D_4^5$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 2$. $A_5^4$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 3$. $D_{20}$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 2$, $k \ne 3$.
\end{thm}
\begin{rem} $D_{20}$ has theta series $1 + 760 q^2 + 77560 q^4 + 524288 q^5 +\cdots$. \end{rem}
By Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor}\label{cor:20} There is an extremal Type~I $\mathbb{Z}_{k}$-code of length $20$ if and only if $k$ is an integer with $k \ge 2$.
\end{cor}
\subsection{Type~I $\mathbb{Z}_k$-codes of length 28}
There is no extremal odd unimodular lattice in dimension $28$ and the largest minimum norm among odd unimodular lattices in dimension $28$ is $3$. There are $38$ non-isomorphic optimal odd unimodular lattices in dimension $28$~\cite{BV}. In~\cite{BV}, the $38$ lattices are denoted by ${\mathbf{R}_{28,1}}(\emptyset)$, ${\mathbf{R}_{28,2}}(\emptyset),\ldots, {\mathbf{R}_{28,36}}(\emptyset)$, ${\mathbf{R}_{28,37e}}(\emptyset)$, ${\mathbf{R}_{28,38e}}(\emptyset)$. We have verified by {\sc Magma} that ${\mathbf{R}_{28,32}}(\emptyset) \cong A_3(C_{28,3}(D_{14}))$ in Table~\ref{Tab:L} and ${\mathbf{R}_{28,15}}(\emptyset) \cong A_5(C_{28,5}(D'_{14}))$ in Table~\ref{Tab:L}. By Lemma~\ref{lem:key}, it is sufficient to investigate the existence of a $k$-frame in ${\mathbf{R}_{28,32}}(\emptyset)$ for $k=4,5,7,13,23$ and a $k$-frame in ${\mathbf{R}_{28,15}}(\emptyset)$ for $k=3,4,17$.
\begin{table}[thb] \caption{Near-extremal Type~I $\mathbb{Z}_{k}$-codes of length $28$} \label{Tab:28} \begin{center}
{\footnotesize
\begin{tabular}{c|l|l} \noalign{\hrule height0.8pt}
Code & \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c}{$r_B$} \\ \hline $C_{ 5,28}$ &$(0,0,0,1,3,4,2)$ & $(3,1,2,0,3,4,0)$\\ $C_{ 7,28}$ &$(0,1,2,2,4,2,3)$ & $(2,2,4,0,4,1,2)$\\ $C_{13,28}$ &$(0, 0, 0, 1, 0, 9, 1)$ & $(5, 1, 3, 7, 7, 1, 4)$\\ $C_{23,28}$ &$(0, 0, 0, 1,12, 1, 1)$ & $(3,19, 7, 5,14,21,17)$\\ \hline $C'_{17,28}$ & $( 0,0,0,1,13,14, 2)$& $(10,1,1,9,16,11,15)$ \\
\noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
Let $C_{k,28}$ ($k=5,7,13,23$) and $C'_{17,28}$
be the $\mathbb{Z}_k$-codes with generator matrices of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$ are listed in Table~\ref{Tab:28}. Since $AA^T+BB^T=-I_{7}$, these codes are Type~I\@. Let $C_{4,28}$ and $C'_{4,28}$ be the $\mathbb{Z}_4$-codes with generator matrices of the following form: \[ \left(\begin{array}{ccc} I_{13} & A & B_1+2B_2 \\ O &2I_{2} & 2D \\ \end{array}\right), \] where we list in Figure~\ref{Fig} the matrices $ \left(\begin{array}{cc} A & B_1+2B_2 \\ 2I_{2} & 2D \\ \end{array}\right). $ Then these codes are Type~I\@. For $k=4,5,7,13,23$, we have verified by {\sc Magma} that $A_k(C_{k,28}) \cong {\mathbf{R}_{28,32}}(\emptyset)$.
For $k=4,17$, we have verified by {\sc Magma} that $A_k(C'_{k,28}) \cong {\mathbf{R}_{28,15}}(\emptyset)$.
It is known that ${\mathbf{R}_{28,15}}(\emptyset)$ contains a $3$-frame
(see~\cite{HMV} for a classification of $3$-frames in the $38$ lattices).
Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{thm}\label{thm:28} ${\mathbf{R}_{28,i}}(\emptyset)$ $(i=15,32)$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 3$.
\end{thm}
\begin{lem} Let $C$ be a Type~I $\mathbb{Z}_k$-code of length $28$. Then $d_E(C) \le 3k$. \end{lem} \begin{proof} As described above, the largest minimum norm among odd unimodular lattices in dimension $28$ is $3$. Since $d_E(C) \le 3k$ for $k=2,3$ (see~\cite{CPS, MPS}), it is sufficient to consider the cases $k \ge 4$ only. Assume that $d_E(C) \ge 4k$. Since $\min(A_{k}(C))=\min\{k, d_{E}(C)/k\}$, $\min(A_{k}(C)) \ge 4$, which is a contradiction.
\end{proof}
There are three inequivalent binary Type~I codes of length $28$ and minimum weight $6$~\cite{CPS}. Hence, by Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor}\label{cor:28}
There is a near-extremal Type~I $\mathbb{Z}_{k}$-code of length $28$ if and only if $k$ is an integer with $k \ge 2$.
\end{cor}
\begin{figure}
\caption{Generator matrices of $C_{4,28}$ and $C'_{4,28}$}
\label{Fig}
\end{figure}
\subsection{Type~I $\mathbb{Z}_k$-codes of length 32} \label{subsec:32}
There are $5$ non-isomorphic extremal odd unimodular lattices in dimension $32$, and these $5$ lattices are related to the $5$ inequivalent binary extremal Type~II codes of length $32$~\cite{CS-odd}. The $5$ codes are denoted by $\text{C}81,\text{C}82,\ldots,\text{C}85$ in~\cite[Table~A]{CPS}. We denote the extremal odd unimodular lattice related to $\text{C}i$ by $L_{32,i}$ ($i=81,\ldots,85$). We have verified by {\sc Magma} that $L_{32,82} \cong A_4(C_{32,4}(D_{16}))$ in Table~\ref{Tab:L}. Since $A_4(C_{32,4}(D_{16}))$ contains a $4$-frame, it is sufficient to investigate the existence of a $k$-frame in $L_{32,82}$ for $k=6,9$ by Lemma~\ref{lem:key}.
\begin{table}[thb] \caption{Extremal Type~I $\mathbb{Z}_{k}$-codes of length $32$} \label{Tab:32} \begin{center}
{\footnotesize
\begin{tabular}{c|l|l} \noalign{\hrule height0.8pt}
Code & \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c}{$r_B$} \\ \hline $C_{6,32}$ & $(0,0,1,2,2,2,1,2)$ & $(1,0,5,5,1,1,3,3)$ \\ $C_{9,32}$ & $(0,0,1,5,0,6,0,1)$ & $(0,6,2,2,7,6,1,7)$ \\ \noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
For $k=6,9$, let $C_{k,32}$ be the $\mathbb{Z}_k$-code with generator matrix of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$ are listed in Table~\ref{Tab:32}.
Since $AA^T+BB^T=-I_{8}$, these codes are self-dual. Note that the code $C_{6,32}$ is not Type~II, since $\wt_E(r_A)+\wt_E(r_B)=41$, where $\wt_E(x)$ denotes the Euclidean weight of $x$. For $k=6,9$, we have verified by {\sc Magma} that $A_k(C_{k,32}) \cong L_{32,82}$. Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{thm}\label{thm:32} $L_{32,82}$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 4$.
\end{thm}
There are three inequivalent binary extremal Type~I codes of length $32$~\cite{C-S}. Any ternary self-dual code of length $32$ has minimum weight at most $9$~\cite{MS73}.
Hence, by Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor}\label{cor:32} There is an extremal Type~I $\mathbb{Z}_{k}$-code of length $32$ if and only if $k$ is a positive integer with $k \ne 1,3$.
\end{cor}
For each extremal odd unimodular lattice in dimension $32$, one of the even unimodular neighbors is extremal~\cite{CS-odd}. Moreover, it follows from the construction in~\cite{CS-odd} that the extremal even unimodular neighbor of $L_{32,82}$ is the $32$-dimensional Barnes--Wall lattice $BW_{32}$ (see e.g.~\cite[Chapter~8, Section~8]{SPLAG} for $BW_{32}$). Since the even sublattice of $L_{32,82}$ contains a $2k$-frame for every integer $k$ with $k \ge 2$ by Theorem~\ref{thm:32}, we have the following:
\begin{prop} $BW_{32}$ contains a $2k$-frame if and only if $k$ is an integer with $k \ge 2$. \end{prop}
Since there are $5$ inequivalent binary extremal Type~II codes of length $32$~\cite{CPS}, by Lemma~\ref{lem:LtoC}, we have an alternative proof of the following:
\begin{cor}[Harada and Miezaki~\cite{HM12}] There is an extremal Type~II $\mathbb{Z}_{2k}$-code of length $32$ if and only if $k$ is a positive integer.
\end{cor}
\subsection{Type~I $\mathbb{Z}_k$-codes of length 36}
Since $A_6(C_{36,6}(D_{18}))$ in Table~\ref{Tab:L} contains a $6$-frame, it is sufficient to investigate the existence of a $k$-frame in $A_6(C_{36,6}(D_{18}))$ for $k=4,5,7,9$ by Lemma~\ref{lem:key}. For $k=5,7,9$, let $C_{k,36}$ be the $\mathbb{Z}_k$-code with generator matrix of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$ are listed in Table~\ref{Tab:36}. Since $AA^T+BB^T=-I_{9}$, these codes are Type~I\@.
Let $C_{4,36}$ be the $\mathbb{Z}_4$-code with generator matrix of the following form: \[ \left(\begin{array}{ccc} I_{16} & A & B_1+2B_2 \\ O &2I_{4} & 2D \\ \end{array}\right), \] where we list in Figure~\ref{Fig:36} the matrices $\left(\begin{array}{cc} A & B_1+2B_2 \\ \end{array}\right)$ and $2D$. It follows that $C_{4,36}$ is self-dual. For $k=4,5,7,9$, we have verified by {\sc Magma} that $A_k(C_{k,36}) \cong A_6(C_{36,6}(D_{18}))$. Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{thm}\label{thm:36} $A_6(C_{36,6}(D_{18}))$ contains a $k$-frame if and only if $k$ is an integer with $k \ge 4$.
\end{thm} \begin{rem} We have verified by {\sc Magma} that $A_6(C_{36,6}(D_{18}))$ has theta series $1 + 42840q^4 + 1916928q^5 + \cdots$ and automorphism group of order $288$
(see~\cite{H36} for details to distinguish $A_6(C_{36,6}(D_{18}))$ from the known lattices, and construction of more extremal odd unimodular lattices). \end{rem}
\begin{table}[thb] \caption{Extremal Type~I $\mathbb{Z}_{k}$-codes of length $36$} \label{Tab:36} \begin{center}
{\footnotesize
\begin{tabular}{c|l|l} \noalign{\hrule height0.8pt}
Code & \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c}{$r_B$} \\ \hline $C_{5,36}$ & $(0,1,1,2,3,2,0,2,3)$ & $(1,1,0,2,0,3,4,0,4)$ \\ $C_{7,36}$ & $(0,1,6,2,3,3,6,4,5)$ & $(4,3,3,6,2,4,3,0,3)$ \\ $C_{9,36}$ & $(0,1,0,5,5,0,0,0,3)$ & $(0,2,3,3,4,5,5,7,3)$ \\
\noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
\begin{figure}
\caption{A generator matrix of $C_{4,36}$}
\label{Fig:36}
\end{figure}
There are $41$ inequivalent binary extremal Type~I codes of length $36$~\cite{MG08}. There is a ternary extremal Type~I code of length $36$~\cite{Pless72}. Hence, by Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor}\label{cor:36} There is an extremal Type~I $\mathbb{Z}_{k}$-code of length $36$ if and only if $k$ is an integer with $k \ge 2$.
\end{cor}
\subsection{Type~I $\mathbb{Z}_k$-codes of length 40} \label{subsec:40}
Since $A_4(C_{40,4}(P_{20}))$ in Table~\ref{Tab:L} contains a $4$-frame, it is sufficient to investigate the existence of a $k$-frame in extremal odd unimodular lattices in dimension $40$ for $k=6,9,13,19$ by Lemma~\ref{lem:key}. For $k=9,13,19$, let $C_{k,40}$ be the $\mathbb{Z}_k$-code with generator matrix of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$ are listed in Table~\ref{Tab:40}. Since $AA^T+BB^T=-I_{10}$, these codes are Type~I\@. Moreover, we have verified by {\sc Magma} that $A_k(C_{k,40})$ is extremal ($k=9,13,19$). An extremal Type~I $\mathbb{Z}_{6}$-code of length $40$ can be found in~\cite{GH05}. Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{prop} There is an extremal odd unimodular lattice in dimension $40$ containing a $k$-frame if and only if $k$ is an integer with $k \ge 4$. \end{prop}
\begin{rem} The possible theta series of an extremal odd unimodular lattice in dimension $40$ is given in~\cite{BBH}: $\theta_{40,\alpha}(q)=1 + (19120 + 256 \alpha) q^4 + (1376256 - 4096 \alpha) q^5 + \cdots$, where $\alpha$ is even with $0 \le \alpha \le 80$. There are $16470$ non-isomorphic extremal odd unimodular lattices in dimension $40$ having theta series $\theta_{40,80}(q)$, and these lattices are related to the $16470$ inequivalent binary extremal Type~II codes of length $40$~\cite{BBH}. We have verified by {\sc Magma} that $A_4(C_{40,4}(P_{20}))$ has theta series $\theta_{40,80}(q)$ and automorphism group of order $7172259840$, and $A_k(C_{k,40})$ ($k=9,13,19$) have theta series $\theta_{40,0}(q)$ and automorphism group of order $40$.
Also, we have verified by {\sc Magma} that three lattices $A_k(C_{k,40})$ ($k=9,13,19$) are non-isomorphic. \end{rem}
There are $10200655$ inequivalent binary extremal Type~I codes of length $40$~\cite{BBH}. There is a ternary extremal Type~I code of length $40$ (see~\cite[Table~XII]{RS-Handbook}). Hence, by Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor} There is an extremal Type~I $\mathbb{Z}_{k}$-code of length $40$ if and only if $k$ is an integer with $k \ge 2$.
\end{cor}
\begin{table}[thb] \caption{Extremal Type~I $\mathbb{Z}_{k}$-codes of length $40$} \label{Tab:40} \begin{center}
{\footnotesize
\begin{tabular}{c|l|l} \noalign{\hrule height0.8pt}
Code & \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c}{$r_B$} \\ \hline $C_{ 9,40}$ &$(0,0,1,0,5,8,3,0,4,4)$ & $(0,5,0,0,5,6,7,2,5,8)$ \\ $C_{13,40}$ &$(0,0,1,4,10, 5, 1,10,11, 4)$ & $(11,4, 4,6, 7,12,11, 7, 2,8)$\\ $C_{19,40}$ &$(0,0,1,2,14,16,17, 1, 0,13)$ & $(10,2,15,2,18,16, 9,15,12,0)$\\
\noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
We have verified by {\sc Magma} that at least one of the even unimodular neighbors of $L$ is extremal for $L=A_4(C_{40,4}(P_{20}))$, $A_9(C_{9,40})$, $A_{13}(C_{13,40})$ and $A_{19}(C_{19,40})$.
There are $16470$ inequivalent binary extremal Type~II codes of length $40$~\cite{BHM}. By Lemma~\ref{lem:LtoC}, we have an alternative proof of the following:
\begin{cor}[Harada and Miezaki~\cite{HM12}] There is an extremal Type~II $\mathbb{Z}_{2k}$-code of length $40$ if and only if $k$ is a positive integer.
\end{cor}
\subsection{Type~I $\mathbb{Z}_k$-codes of length 44}
By considering $A_5(C_{44,5}(D_{22}))$ in Table~\ref{Tab:L}, it is sufficient to investigate the existence of a $k$-frame in extremal odd unimodular lattices in dimension $44$ for $k=4,6,9,17$ by Lemma~\ref{lem:key}. For $k=9,17$, let $C_{k,44}$ be the $\mathbb{Z}_k$-code with generator matrix of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$ are listed in Table~\ref{Tab:44}. Since $AA^T+BB^T=-I_{11}$, these codes are Type~I\@. Moreover, we have verified by {\sc Magma} that $A_k(C_{k,44})$ is extremal ($k=9,17$). For $k=4$ and $6$, an extremal Type~I $\mathbb{Z}_{k}$-code of length $44$ can be found in~\cite[Table~1]{H12} and~\cite{GH05}, respectively. Hence, combining with Lemma~\ref{lem:key}, we have the following:
\begin{prop} There is an extremal odd unimodular lattice in dimension $44$ containing a $k$-frame if and only if $k$ is an integer with $k \ge 4$. \end{prop}
\begin{rem} The possible theta series of an extremal odd unimodular lattice in dimension $44$ is given in~\cite{H03}: $\theta_{44,1,\beta}(q)=1+(6600+16\beta)q^4+(811008-128\beta)q^5 + \cdots$, $\theta_{44,2,\beta}(q)=1+(6600+16\beta)q^4+(679936-128\beta)q^5 + \cdots$, where $\beta$ is an integer. We have verified by {\sc Magma} that $A_5(C_{44,5}(D_{22}))$ and $A_k(C_{k,44})$ ($k=9,17$) have theta series $\theta_{44,1,\beta}(q)$ ($\beta=0,88,176$) and automorphism groups of orders $44$, $88$, $44$, respectively.
\end{rem}
\begin{table}[thb] \caption{Extremal Type~I $\mathbb{Z}_{k}$-codes of length $44$} \label{Tab:44} \begin{center}
{\footnotesize
\begin{tabular}{c|l|l} \noalign{\hrule height0.8pt}
Code & \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c}{$r_B$} \\ \hline $C_{9 ,44}$& $(0,0,0,0,1,0,1,4,0,8,0)$ & $(7,0,7,1,8,8,2,8,1,5,1)$ \\ $C_{17,44}$& $(0, 0, 0, 0, 1,13, 7,13,11,16,13)$
&$(12,14, 8,14, 7,12,14, 7,14,14, 7)$ \\ \noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
For $k=2,3$, there is an extremal Type~I $\mathbb{Z}_k$-code of length $44$ (see~\cite[Tables X and XII]{RS-Handbook}). Hence, by Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor} There is an extremal Type~I $\mathbb{Z}_{k}$-code of length $44$ if and only if $k$ is an integer with $k \ge 2$.
\end{cor}
\subsection{Remarks on Type~I $\mathbb{Z}_k$-codes of length 48}\label{sec:rem}
By considering $A_5(C_{48,5}(D_{24}))$ in Table~\ref{Tab:L}, we examine the existence of a $k$-frame in optimal odd unimodular lattices in dimension $48$ for $k=6, 7, 8, 9, 17$ by Lemma~\ref{lem:key}. It was shown in~\cite{HKMV} that an extremal even unimodular lattice in dimension $48$ has an optimal odd unimodular neighbor. Using this result, we have the following:
\begin{lem}\label{lem:6-1} There is an optimal odd unimodular lattice in dimension $48$ containing an $8k$-frame for every positive integer $k$. \end{lem} \begin{proof} Let $\Lambda$ be an extremal even unimodular lattice in dimension $48$. Let $x$ be a vector of $\Lambda$ with $(x,x)=8$. Note that there are vectors of norm $8$ in $\Lambda$ (see~\cite[Chap.~7, (68)]{SPLAG}). Put $\Lambda_x^{+}=\{v\in\Lambda\mid(x,v)\equiv0\pmod2\}$. Since there is a vector $y$ of $\Lambda$ such that $(x,y)$ is odd, the following lattice \[ \Lambda_{x}= \Lambda_x^+ \cup \Big(\frac{1}{2}x+y\Big)+\Lambda_x^+ \] is an optimal odd unimodular neighbor of $\Lambda$~\cite{HKMV}.
Some extremal even unimodular lattice in dimension $48$ containing an $8$-frame can be found in~\cite[Corollary~1]{Chapman-Sole}. We take this lattice as $\Lambda$ in the above construction. Let $\{f_1, \ldots, f_{48}\}$ be an $8$-frame in $\Lambda$. Then $\Lambda_{f_1}$ is an optimal odd unimodular neighbor containing $\{f_1, \ldots, f_{48}\}$. The result follows from Lemma~\ref{lem:frame}. \end{proof}
\begin{table}[thb] \caption{Near-extremal Type~I $\mathbb{Z}_{k}$-codes of length $48$} \label{Tab:48} \begin{center}
{\footnotesize
\begin{tabular}{c|l|l} \noalign{\hrule height0.8pt}
Code & \multicolumn{1}{c|}{$r_A$} & \multicolumn{1}{c}{$r_B$} \\ \hline $C_{7 ,48}$& $(0,1,6,3,0,2,0,2,4,2,5,3)$&$(3,6,1,5,4,6,0,5,0,5,1,5)$\\ $C_{9 ,48}$& $(0,1,2,4,6,1,6,2,2,0,3,0)$&$(7,2,5,1,6,8,4,1,2,2,8,4)$\\ \noalign{\hrule height0.8pt} \end{tabular} } \end{center} \end{table}
Some near-extremal Type~I $\mathbb{Z}_6$-code $C_{6,48}$ of length $48$ can be found in~\cite{HKMV}.
For $k=7,9$, let $C_{k,48}$ be the $\mathbb{Z}_k$-code with generator matrix of the form (\ref{eq:GM}), where the first rows $r_A$ and $r_B$ of $A$ and $B$ are listed in Table~\ref{Tab:48}. Since $AA^T+BB^T=-I_{12}$, these codes are Type~I\@. Moreover, we have verified by {\sc Magma} that $A_k(C_{k,48})$ is optimal ($k=7,9$). Hence, we have the following:
\begin{prop} There is an optimal odd unimodular lattice in dimension $48$ containing a $k$-frame for every integer $k$ with $k \ge 5$ and $k \ne 2^{m_1}3^{m_2}17^{m_3}$, where $m_i$ are integers $(i=1,2,3)$ with $(m_1,m_2) \in \{(0,0),(0,1),(1,0),(2,0)\}$ and $m_3 \ge 1$.
\end{prop}
\begin{rem} $A_6(C_{6,48})$ has kissing number $393216$ \cite[p.~553]{HKMV}. In addition, we have verified by {\sc Magma} that $A_5(C_{48,5}(D_{24}))$, $A_7(C_{7,48})$ and $A_9(C_{9,48})$ have kissing number $393216$. \end{rem}
For $k=2,3$, there is a near-extremal Type~I $\mathbb{Z}_k$-code of length $48$ (see~\cite[Tables X and XII]{RS-Handbook}).
Let $C_{4,48}$ be the $\mathbb{Z}_4$-code with generator matrix: \[ \left( \begin{array}{ccc@{}c} \quad & {\Large I_{24}} & \quad & \begin{array}{cccc} 0 & 1 & \cdots & 1 \\ 1 & {} & {} &{} \\ \vdots & {} & R &{} \\ 1 & {} &{} &{} \\ \end{array} \end{array} \right), \] where $R$ is the $23 \times 23$ circulant matrix with first row \[ (1,1,3,0,3,3,1,2,0,1,3,2,3,0,0,3,3,2,1,2,1,1,0). \] We have verified by {\sc Magma} that $C_{4,48}$ is a near-extremal Type~I $\mathbb{Z}_4$-code of length $48$. Hence, by Lemma~\ref{lem:LtoC}, we have the following:
\begin{cor} There is a near-extremal Type~I $\mathbb{Z}_{k}$-code of length $48$ for $k=2,3,4$ and for integers $k$ with $k \ge 5$, $k \ne 2^{m_1}3^{m_2}17^{m_3}$, where $m_i$ are integers $(i=1,2,3)$ with $(m_1,m_2) \in \{(0,0),(0,1),(1,0),(2,0)\}$ and $m_3 \ge 1$.
\end{cor}
Using the method in the previous subsections, we tried to construct a near-extremal Type~I $\mathbb{Z}_{17}$-code of length $48$. However, our extensive search failed to discover such a code, then we stopped our search at length $48$. It is worthwhile to determine whether there is a near-extremal Type~I $\mathbb{Z}_{17}$-code of length $48$.
\noindent {\bf Acknowledgments.} The author would like to thank Tsuyoshi Miezaki for useful discussions,
for carefully reading the manuscript and for kindly providing the results given in Lemma~\ref{lem:prime},
and Masaaki Kitazume and Hiroki Shimakura for helpful conversations. The author would also like to thank the anonymous referees for valuable comments leading to several improvements of the presentation. This work is supported by JSPS KAKENHI Grant Number 23340021.
\end{document} | arXiv |
Posets of Non-Crossing Partitions of Type B and Applications
BDTEZA.pdf (1.004Mb)
Oancea, Ion
The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.
Ion Oancea (2007). Posets of Non-Crossing Partitions of Type B and Applications. UWSpace. http://hdl.handle.net/10012/3402 | CommonCrawl |
\begin{document}
\title{On heavy-tail phenomena in some large deviations problems} \begin{abstract}
In this paper, we revisit the proof of the large deviations principle of Wiener chaoses partially given by Borell \cite{Borell}, and then by Ledoux \cite{LedouxWiener} in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large deviations principle for a certain class of functionals $f_n : \RR^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the $n$-fold probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} =Y_{\alpha}^{-1}e^{-|x|^{\alpha}}dx$, $\alpha \in (0,2]$, for which the large deviations are due to translations. We retrieve, as an application, the large deviations principles known for the Wigner matrices without Gaussian tails in \cite{Bordenave}, \cite{LDPei}, \cite{LDPtr} of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large deviations result to the last-passage time, which yields a large deviations principle when the weights follow the law $Z_{\alpha}^{-1} e^{-x^{\alpha}} \Car_{x\geq 0}dx$, with $\alpha \in (0,1)$.
\end{abstract} \section{Introduction} In \cite{LedouxWiener}, Ledoux proposed a large deviations principle for the Wiener chaoses based on the approach Borell gave in \cite{Borell} for estimating their tail distribution. The main feature which stands out of the proof is that the large deviations of Wiener chaoses are due to translations by elements of the Cameron-Martin space. The lower bound consists in an application of the Cameron-Martin formula, whereas the upper bound relies on the Gaussian isoperimetric inequality.
More precisely, let $(E,\mathcal{H}, \mu)$ be an abstract Wiener space, where $E$ is a separable Banach space, $\mu$ is a Gaussian measure on $E$, and $\mathcal{H}$ the reproducing kernel (see \cite{Lifshits} or \cite[chapter 4]{Ledouxflour} for proper definitions). Let also $\Psi$ be a homogenous Wiener chaos of degree $d$ taking values in some Banach space $B$, that is, a random variable in the subspace spanned in $L^2(\mu; B)$ by Hermite polynomials of degree $d$. From \cite{LedouxWiener}, we know that $t^{-d} \Psi$ follows a large deviations principle with speed $t^2$ and good rate function $I_{\Psi}$ defined by,
\begin{equation} \label{ratefuncWiener} \forall x \in B, \ I_{\Psi}(x) = \inf\Big \{ \frac{1}{2} |h|^2 : x = \Psi^{(d)}(h), h \in \mathcal{H} \Big\},\end{equation}
where $| \ |$ denotes the norm of the reproducing kernel $\mathcal{H}$, and \begin{equation} \label{detequivWiener} \forall h \in \mathcal{H}, \ \Psi^{(d)}(h) = \int \Psi(x+h) d\mu(x).\end{equation}
We believe Borell and Ledoux's approach to be extremely fruitful, and can shed a new light on heavy-tail phenomena appearing in the large deviations of certain models, where the large deviations are created also, in a sense, by translations. We already used this approach in a previous work \cite{LDPtr} to deal with the question of the large deviations of traces of powers of Gaussian Wigner matrices. Indeed, this problem can be reformulated as understanding the large deviations of Gaussian chaoses defined on spaces with growing dimension. Although this problem cannot be solved directly by using the large deviations principle of Wiener chaoses, the same outline of proof was carried out in this case, and yields a rate function having a similar structure as \eqref{ratefuncWiener}.
We would like here to push further this approach in a more general setting, and give some elements showing that heavy-tail phenomena in the large deviations of certain models can be understood using the paradigm of the Wiener chaoses. To this end, we propose a general large deviations result for a certain class of functionals $f_n : \RR^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, under the $n$-fold probability measure $\nu_{\alpha}^n$, where $\nu_{\alpha} = Y_{\alpha}^{-1} e^{-|x|^{\alpha}} dx$, with $\alpha \in (0,2]$, for which the large deviations are governed by translations.
As an application of this result, we will retrieve the large deviations principles of different spectral functionals of the so-called Wigner matrices without Gaussian tails. Introduced in \cite{Bordenave} by Bordenave and Caputo, the model of Wigner matrices without Gaussian tails designates Wigner matrices whose entries have tail distributions behaving like $e^{-c t^{\alpha}}$, with $c>0$, and $\alpha \in (0,2)$. This model gives rise to a heavy-tail phenomenon which enables one to derive full large deviations principles for the spectral measure \cite{Bordenave} (see \cite{Groux} in the Wishart matrix case), the largest eigenvalue \cite{LDPei}, and the traces of powers \cite{LDPtr}.
In the more restricted setting where we assume that the entries have a density with respect to Lebesgue measure which is proportional to $e^{-c |x|^{\alpha}}$, with $c>0$, and $\alpha\in (0,2)$, the large deviations principles of these spectral functionals will fall in a unified way from our general large deviation result.
Another application of this result will consist in a large deviations principle for the last-passage time when the weights are independent and have a density on $\RR^+$ proportional to $e^{-x^{\alpha}}$ for $\alpha \in (0,1)$.
\section{Main results}
Let us present the main results of this paper. For $\alpha>0$, we denote by $\nu_{\alpha}$ the probability measure on $\RR$ with density $Y_{\alpha}^{-1} e^{-|x|^{\alpha}}$ with respect to Lebesgue measure, and $\nu_{\alpha}^n$ its $n$-fold product measure on $\RR^n$. Similarly, we define $\mu_{\alpha}$ the probability measure on $\RR^+$ with density $Z_{\alpha}^{-1}e^{-x^{\alpha}}$. We will denote for any $h\in \RR^n$,
$$|| h ||_{\ell^{\alpha}} = \big(\sum_{i=1}^n |x_i|^{\alpha}\big )^{1/\alpha}.$$ We recall that a sequence of random variables $(Z_n)_{n\in \NN}$ taking value in some topological space $\mathcal{X}$ equipped with the Borel $\sigma$-field $\mathcal{B}$, follows a large deviations principle (LDP) with speed $\upsilon(n)$, and rate function $J : \mathcal{X} \to [0, +\infty]$, if $J$ is lower semicontinuous and $\upsilon(n)$ increases to infinity and for all $B\in \mathcal{B}$, $$- \inf_{B^{\circ}}J \leq \liminf_{n\to +\infty} \frac{1}{\upsilon(n)} \log \PP\left(Z_n \in B\right) \leq \limsup_{n\to +\infty} \frac{1}{\upsilon(n)}\log\PP\left(Z_n \in B\right) \leq -\inf_{\overline{B} } J,$$ where $B^{\circ}$ denotes the interior of $B$ and $\overline{B}$ the closure of $B$. We recall that $J$ is lower semicontinuous if its $t$-level sets $\{ x \in \mathcal{X} : J(x) \leq t \}$ are closed, for any $t\in [0,+\infty)$. Furthermore, if all the level sets are compact, then we say that $J$ is a good rate function.
The purpose of the general large deviations result we will present, is to identify a class of functionals $f_n : \RR^n \to \mathcal{X}$, where $\mathcal{X}$ is some metric space, for which the large deviations are created by translations. Let us describe first informally the assumptions we will make. Let $X_n$ follow the law $\nu_{\alpha}^n$. We will assume that $f_n(X_n)$ admits a kind of deterministic equivalent under additive deformations, given by a certain function $F_n$, that is, \begin{equation} \label{equidetinfomel}f_n(X_n+v(n)^{1/\alpha}h_n) \simeq F_n(h_n),\end{equation}
in probability, for any sequence $h_n\in \RR^n$, $\sup_n||h_n||_{\ell^{\alpha}}<+\infty$, where $v(n)$ will eventually be the speed of deviations. It is convenient to think of $F_n(h_n)$ as a deterministic equivalent of $f_n(X_n+v(n)^{1/\alpha}h_n)$, where we took the large $n$ limit on the variable $X_n$. Under this assumption, we will show that a large deviations lower bound for $f_n(X_n)$ at speed $v(n)$, holds with rate function, $$J_{\alpha} = \sup_{\delta >0} \limsup_{\underset{n\in N}{n\to +\infty}} I_{n,\delta},$$ where
$$ \forall x \in \mathcal{X}, \ I_{n,\delta }(x) = \inf \{ ||h||_{\ell^{\alpha}}^{\alpha} : d(F_n(h),x)<\delta, h\in \RR^n \}.$$
This rate function $J_{\alpha}$ can be interpreted by saying that to make a deviation around some $F_n(h_n)$, $X_n$ needs to make a translation by $v(n)^{1/\alpha}h_n$, which one pays at the exponential scale $v(n)$ by $|| h_n ||_{\ell^{\alpha}}^{\alpha}$.
For the upper bound, we will further assume that for any $r>0$, the deterministic equivalent \eqref{equidetinfomel} holds uniformly in $||h_n||_{\ell^{\alpha}}\leq r$. The upper bound will rely on sharp large deviation inequalities for $\nu_{\alpha}^n$, where we will need, excepted in the Gaussian
case, to neglect the Euclidean enlargements appearing naturally. We thus make the assumption that $f_n$ has a small, in expectation, local Lipschitz constant with respect to $|| \ ||_{\ell^2}$ when $\alpha<2$. Finally, under some compactness property of $F_n$, we will prove that a large deviations upper bound holds for $f_n(X_n)$ with speed $v(n)$ and rate function,
$$I_{\alpha} = \sup_{\delta >0} \inf_{n\in N} I_{n,\delta}.$$ Thus, if we moreover assume that the upper bound rate function $I_{\alpha}$ matches the lower rate function, we will get a full large deviations principle with speed $v(n)$. More precisely, we will prove the following result.
\begin{The}\label{theoremgene} Let $(\mathcal{X},d)$ be a metric space. Let $\alpha \in (0,2]$ and $N\subset \NN$ an infinite subset. Let $X_n$ be a random variable with law $\nu_{\alpha}^{ n}$. Let $f_n, F_n : \RR^n \to \mathcal{X}$ be measurable functions. Let $(v(n))_{n\in N}$ be a sequence going to $+\infty$. Define for $\delta>0$ and $n\in N$, the function
$$ \forall x \in \mathcal{X}, \ I_{n,\delta }(x) = \inf \{ ||h||_{\ell^{\alpha}}^{\alpha} : d(F_n(h),x)<\delta, h\in \RR^n \}.$$ We set \begin{equation} \label{deftaux}\forall x \in \mathcal{X}, \ I_{\alpha }(x) = \sup_{\delta >0} \inf_{n\in N} I_{n,\delta}(x).\end{equation} We assume: \\ (i).(Uniform deterministic equivalent). For any $r>0$, $$ \sup_{ h_n \in r B_{\ell^{\alpha}}} d \big(f_n(X_n+v(n)^{1/\alpha} h_n), F_n(h_n)\big) \underset{\underset{n\in N}{n\to +\infty}}{\longrightarrow} 0,$$ in probability.\\
(ii).(Control of the Lipschitz constant). If $\alpha <2$, then for any $\delta>0$ and $r>0$, there is a sequence $t_{\delta}(n)$ such that, $$ \EE \sup_{||h||_{\ell^2} \leq t_{\delta}(n)} \mathcal{L}_n(h ) \leq \delta,$$ with \begin{equation} \label{defLn}\mathcal{L}_n(h) = \sup_{X_n + rv(n)^{1/\alpha} B_{\ell^{\alpha} }}d\big( f_n(x+h) , f_n(x)\big),\end{equation} satisfying, $$(\log n)^{\alpha/2} = o(\log \frac{t_{\delta}(n)^2}{v(n)}) \text{ if }\alpha \neq 1, \text{ or
} v(n) = o(t_{\delta}(n)^2)\text{ if }\alpha=1.$$ (iii).(Compactness). For any $r>0$, $\cup_{n\in N} F_n( rB_{\ell^{\alpha}})$ is relatively compact.\\ (iv).(Upper bound = lower bound). For any $x\in \mathcal{X}$, \begin{equation} \label{condtaux} I_{\alpha }(x) = \sup_{\delta >0} \limsup_{\underset{n\in N}{n\to +\infty}} I_{n,\delta}(x).\end{equation} Then $(f_n(X_n))_{n\in N}$ satisfies a LDP with speed $v(n)$ and good rate function $I_{\alpha}$. \end{The}
Let us make some remarks on the assumptions of this theorem.
\begin{Rems}\label{remtheogene}
(a). We will prove that under the assumption that for any sequence $h_n\in \RR^n$, $n\in N$, such that $\sup_n || h_n ||_{\ell^{\alpha}} < +\infty$, \begin{equation}\label{equidet} d(f_n(X_n+v(n)^{1/\alpha} h_n), F_n(h_n)) \underset{ \underset{n\in N}{n\to+\infty}}{\longrightarrow} 0,\end{equation} in probability, the lower bound of the LDP holds with the rate function \eqref{condtaux}.
(b). The assumption $(i)$ that the approximation \eqref{equidet} holds uniformly in $h_n\in rB_{\ell^{\alpha}}$ is crucial for deriving the upper bound of the LDP with rate function \eqref{deftaux}, and is one of the most constraining assumptions of Theorem \ref{theoremgene}. In the applications we develop when $\alpha <2$, this is proven by some concentration inequality and chaining arguments, which can be carried out successfully due to the ``sparsity'' of the ball $B_{\ell^{\alpha}}$.
(c).\label{remlipconst} The formulation of assumption $(ii)$ on the Lipschitz constant of $f_n$ is specially designed to include polynomial functionals $f_n$, as the trace of a polynomial of random matrices. In other words, it says that the ``local'' Lipschitz constant of $f_n$, is small enough uniformly on the set $X_n + rv(n)^{1/\alpha} B_{\ell^{\alpha}}$. Note that when $f_n$ is $L_2(n)$-Lipschitz with respect to $||\ ||_{\ell^2}$, a sufficient condition for assumption $(ii)$ to be fulfilled is \begin{equation} \label{condlipconst} (\log n)^{\alpha/2} = o\big( \log \frac{1}{L_2(n)^2v(n)}\big) \text{ if } \alpha \in (1,2), \text{ and } v(n) = o\big( \frac{1}{L_2(n)^2} \big) \text{ if } \alpha =1.\end{equation} This assumption ensures that the deviations of $f_n(X_n)$ are explained by a heavy-tail phenomenon. For example, it fails to hold for empirical means under $\nu_{\alpha}^n$ when $\alpha \in [1,2)$.
(d). The compactness assumption of $(iii)$ is made to ensure that $I_{\alpha}$ is a good rate function. As one can observe in the proof, without it, the upper bound of the LDP holds only for compact sets.
(e). \label{remlsi} The rate function $I_{\alpha}$ can be simplified in certain cases. Define the function $\tilde{I}_{\alpha}$ by,
$$\forall x \in \mathcal{X}, \ \tilde{I}_{\alpha}(x) = \inf_n\{||h||_{\ell^{\alpha}}^{\alpha} : x = F_n(h), \ h \in \RR^n \}.$$ One can see that, $$I_{\alpha} = \sup_{\delta>0} \inf_{B(x,\delta)} \tilde{I}_{\alpha}.$$ Thus if $\tilde{I}_{\alpha}$ is lower semi-continuous, then $I_{\alpha} = \tilde{I}_{\alpha}$.
\end{Rems} The proof is in line with the ideas and the framework developed by Borell and Ledoux in \cite{Borell}, \cite{Borell2} and \cite{LedouxWiener}, \cite{Ledouxflour}, for the large deviations for Wiener chaoses. To make a parallel with their approach, one can observe that the first step in their proof is to show some deterministic equivalent for the Wiener chaoses when deformed in a direction of the reproducing kernel, that is, by \cite[chapter 5 (5.7)]{Ledouxflour}, for any $h\in \mathcal{H}$,
\begin{equation} \label{equivWiener}||t^{-d}\Psi(x+th) -\Psi^{(d)}(h)||\underset{t \to +\infty}{ \longrightarrow} 0,\end{equation} in probability, and even uniformly in $h \in \mathcal{O}$ the unit ball of $\mathcal{H}$, along a discretization of $\Psi$ by \cite[chapter 5 (5.9)]{Ledouxflour}, where $\Psi^{(d)}$ defined in \eqref{detequivWiener}. Similarly, we make the assumption $(i)$ that a uniform deterministic equivalent holds for the functionals $f_n$.
For the lower bound, we replace the use of the Cameron-Martin formula, used in the context of abstract Wiener space, with a lower bound estimate of the probability of translated events, that is, \begin{equation} \label{lowerboundnu} \liminf_{\underset{n \in N}{n \to +\infty}} \frac{1}{v(n)} \log \nu_{\alpha}^n( E + v(n)^{1/\alpha} h_n)\geq - \limsup_{\underset{n\in N}{n \to +\infty}} c_{\alpha}(h_n),\end{equation}
for a given sequence $h_n\in \RR^n$, subsets $E$ such that $\liminf_n \nu_{\alpha}^n(E)>0$, and where $c_{\alpha}$ is some weight function. In the Gaussian case $\alpha=2$, the translation formula of the Gaussian measure gives this estimate with $c_{\alpha}(h) = || h ||_{\ell^2}^2$. When $\alpha <2$, one can mimic the Gaussian case to get such an estimate \eqref{lowerboundnu} with $c_{\alpha}(h) = ||h||_{\ell^{\alpha}}^{\alpha}$, whereas when $\alpha>2$, we believe that there is a competition between the speed and the dimension which is not workable in the applications.
Whereas the Gaussian isoperimetric inequality is used in the proof of the upper bound of the deviations of Wiener chaoses, ours will rely on sharp large deviation inequalities for $\nu_{\alpha}^n$ with respect to the weight function $c_{\alpha}$, that is \begin{equation}\label{upperbound} \limsup_{\underset{n \in N}{n \to +\infty}} \frac{1}{v(n)} \log \nu_{\alpha}^n( x \notin E + \{ c_{\alpha} \leq r v(n) \}) \leq -r,\end{equation}
for some ``large enough'' subsets $E$. We will show that we can take $c_{\alpha} = ||h||_{\ell^{\alpha}}^{\alpha}$, which together with \eqref{lowerboundnu} will allow us to make the upper and lower bound match. In the Gaussian case, this is due to the Gaussian isoperimetric inequality, whereas when $\alpha <2$, we will have to call for sharp inf-convolution inequalities for $\nu_{\alpha}^n$. This is in particular where assumption $(ii)$ plays its role since it enables us, when $\alpha <2$, to neglect the Euclidean balls which come naturally in the deviation inequality of $\nu_{\alpha}^n$, and consider subsets $E$ which are indeed large enough.
These two estimates \eqref{lowerboundnu} and \eqref{upperbound} are behind the limitation in Theorem \ref{theoremgene} to the probability measures $\nu_{\alpha}^n$ for $\alpha \in (0,2]$. For example, if one replaces the measure $\nu_{\alpha}$ by the probability measure on $\RR_+$ with density $ Z_{\alpha}^{-1} e^{-x^{\alpha}}$, one can show that \eqref{lowerboundnu} holds provided $h_n$ has all its coordinates non-negative (and $ n = o(v(n))$ if $\alpha>1$). But then, we will have to prove \eqref{upperbound} with $c_{\alpha}(h) = || h ||_{\ell^{\alpha}}^{\alpha}$ if the coordinates of $h$ are non-negative, and $+\infty$ otherwise, which we do not know how to obtain for the subsets $E$ we are dealing with in the proof.
This said, we can give a version of Theorem \ref{theoremgene} for the probability measure $\mu_{\alpha}$, with density $Z_{\alpha}^{-1}e^{-x^{\alpha}} \Car_{x\geq 0}$, which will be sufficient to prove a LDP result for the last-passage time.
\begin{The}\label{theoremgenesup}
Let $\alpha \in (0,1]$ and $N\subset \NN$ an infinite subset. Let $X_n$ be a random variable distributed according to $\mu_{\alpha}^{ n}$. Let $f_n, F_n : \RR^n \to \RR$ be measurable functions. Let $(v_n)_{n\in N}$ be a sequence going to $+\infty$. Define $I_{\alpha}$ as in \eqref{deftaux}, and for $\delta>0$ and $n\in N$,
$$I_{n,\delta}^+(x) = \inf\big \{ ||h||_{\ell^{\alpha}}^{\alpha} : d(F_n(h),x)<\delta, h\in \RR_+^n\big \}.$$
Assume $(i)-(ii)-(iii)$ from Theorem \ref{theoremgene}, and,\\ (iv)'. For any $x\in \mathcal{X}$, $$I_{\alpha}(x) = \sup_{\delta>0} \limsup_{\underset{n\in N}{n\to+\infty}}I_{n,\delta}^+(x) .$$ Then $(f_n(X_n))_{n\in N}$ satisfies a LDP with speed $v(n)$ and good rate function $I_{\alpha}$. \end{The}
\begin{Rem} We only state this result for $\alpha \in (0,1]$ because for $\alpha >1$, we know how to get the lower bound \eqref{lowerboundnu} for a sequence $h_n\in \RR_+^n$ only under the additional assumption on the speed that $n= o(v(n))$. But this condition and the requirement $(ii)$ cannot be met simultaneously in the applications we will present.
\end{Rem}
\subsection{Applications to Wigner matrices}
\label{appliWignerIntro} We present now the applications of Theorem \ref{theoremgene} to Wigner matrices. We denote by $\mathcal{H}_n^{(\beta)}$ the set of Hermitian matrices when $\beta=2 $, and symmetric matrices when $\beta=1$, of size $n$. We define $\mathcal{S}_{\alpha}$ the class of Wigner matrices whose law is of density $Z_{W_{\alpha}}^{-1}e^{-W_{\alpha}}$ with respect to the Lebesgue measure $\ell_n^{(\beta)}$ on $\mathcal{H}_n^{(\beta)}$, where
\begin{equation} \label{defW} \forall A \in \mathcal{H}_n^{(\beta)}, \ W_{\alpha}(A) = b\sum_{i}|A_{i,i}|^{\alpha} + \sum_{i<j} \Big(a_1 |\Re A_{i,j}|^{\alpha} + a_2 |\Im A_{i,j}|^{\alpha}\Big),\end{equation} for some $b,a_1, a_2 \in (0,+\infty)$, and where $Z_{W_{\alpha}}$ is the normalizing constant.
We will denote by $\mu_A$ the empirical spectral measure of a matrix $A\in \mathcal{H}_n^{(\beta)}$, that is, $$ \mu_A = \frac{1}{n} \sum_{i=1}^n \delta_{\lambda_i},$$ where $\lambda_1,...,\lambda_n$ are the eigenvalues of $A$, and we will denote by $\lambda_A$ the largest eigenvalue of $A$.
We will say that $X$ is a Wigner matrix if $X$ is a random Hermitian matrix with independent coefficients (up to the symmetry) such that $(X_{i,i})_{1\leq i \leq n}$ are identically distributed and $(X_{i,j})_{i<j}$ are identically distributed. If $\EE|X_{1,2}- \EE X_{1,2}|^2 = 1$, then by Wigner's theorem (see \cite[Theorem 2.1.1, Exercice 2.1.16]{Guionnet}, \cite[Theorem 2.5]{Silverstein}), almost surely, $$\mu_{X/\sqrt{n}} \underset{n\to\infty}{\leadsto} \mu_{sc},$$ where $\leadsto$ denotes the weak convergence, and $\mu_{sc}$ is the semi-circular law defined by,
$$ \mu_{sc} = \frac{1}{2\pi} \sqrt{4-x^2} \Car_{|x| \leq 2} dx.$$ If we assume furthermore that $\EE {X_{1,1}}^2< +\infty$ and $\EE X_{1,2}^4<+\infty$, then we know by \cite{Bai}, \cite[Theorem 5.1]{Silverstein}, $$\lambda_{X/\sqrt{n}} \underset{n\to +\infty}{\longrightarrow} 2,$$ in probability.
As a consequence of Theorem \ref{theoremgene}, we have the following large deviations principles, originally proven in \cite{Bordenave}, in the case of the empirical spectral measure and in \cite{LDPei} for the largest eigenvalue.
\begin{The}\label{LDPmsp} Let $\alpha \in (0,2)$. Assume $X $ is in the class $\mathcal{S}_{\alpha}$ such that $\EE|X_{1,2}|^2 =1$. $(\mu_{X/\sqrt{n}})_{n\in \NN}$ follows a LDP with respect to the weak topology with speed $n^{1+\alpha/2}$, and good rate function $I_{\alpha}$, defined for any probability measure $\mu$ on $\RR$ by, $$ I_{\alpha}(\mu) = \sup_{\delta > 0} \inf_{n\in\NN} \{ W_{\alpha}(A) : A \in \mathcal{H}_n^{(\beta)}, \ d(\mu ,\mu_{sc} \boxplus \mu_{n^{1/\alpha} A} )<\delta\},$$ where $d$ is a distance compatible with the weak topology, $\boxplus$ stands for the free convolution (see \cite[section 2.3.3]{Guionnet} for a definition), and $\mu_{sc}$ is the semi-circular law.
\end{The} \begin{Rem} In \cite{Bordenave}, the rate function $I_{\alpha}$ is computed explicitly for measures $\mu_{sc} \boxplus \nu$, where $\nu$ is a symmetric probability measure, for which we have
$$I_{\alpha}(\mu_{sc}\boxplus \nu) = \min\big(b,\frac{a}{2} \big) \int |x|^{\alpha} d\nu(x).$$
\end{Rem}
\begin{The}\label{LDPvp}Let $\alpha \in (0,2)$. Assume that $X$ is in the class $\mathcal{S}_{\alpha}$ such that $\EE|X_{1,2}|^2 =1$. $(\lambda_{X/\sqrt{n}})_{n\in \NN}$ follows a LDP with speed $n^{\alpha/2}$ and good rate function $J_{\alpha}$, defined for any $x\in \RR$ by, $$ J_{\alpha}(x) = \begin{cases} cg_{\mu_{sc}}(x)^{-\alpha}& \text{ if } x>2,\\ 0 & \text{ if } x=2,\\
+\infty& \text{ if } x<2, \end{cases}$$ with $$c = \inf\big \{ W_{\alpha}(A) : A \in \cup_{n \in \NN} \mathcal{H}_n^{(\beta)}, \ \lambda_A =1 \big\},$$ and where $g_{\mu_{sc}}$ denotes the Stieltjes transform of $\mu_{sc}$, that is, $$ \forall z \in \CC\setminus (-2,2), \ g_{\mu_{sc}}(z) = \int \frac{d\mu_{sc}(x)}{z-x}.$$
\end{The}
\begin{Rem} The constant $c$ can be computed explicitly, we refer the reader to \cite[section 8]{LDPei} for more details. \end{Rem}
If $\textbf{X}=(X_1,...,X_p)$ is a collection of independent centered Wigner matrices such that $\EE M_{1,2}^2=1$ for any $M \in \{X_1,...,X_n\}$, and with entries having finite moments of order $d$, then for any non-commutative polynomial $P \in \CC\langle \textbf{X} \rangle$ of total degree $d$, we know by \cite[Theorem 5.4.2]{Guionnet}, $$ \tau_n [ P(\textbf{X}/\sqrt{n})] \underset{n\to +\infty}{\longrightarrow} \tau[P(\textbf{s})],$$ in probability, where $\tau_n = \frac{1}{n} \tr$ and $\textbf{s} =(s_1,...,s_p)$ is a free family of $p$ semi-circular variables in a non-commutative probability space $(\mathcal{A},\tau)$ (see \cite[section 5.3]{Guionnet} for a definition).
Concerning the large deviations of such normalized traces of polynomials in independent matrices in the class $\mathcal{S}_{\alpha}$, with $\alpha \in (0,2]$ we have the following result. \begin{The}\label{LDPpoly}
Let $\alpha \in (0,2]$ and $p,d\in \NN$, $d> \alpha$. Assume $\textbf{X} = (X_1,...,X_p)$ is a collection of independent Wigner matrices in the class $\mathcal{S}_{\alpha}$, such that for $M \in \{ X_1,...,X_p\}$, $\EE|M_{1,2}|^2 =1$. We assume that $X_i$ is distributed according to $Z_{W_{\alpha}}^{-1}e^{-W_{\alpha,i}} d\ell_n^{(\beta)}$, where $W_{\alpha,i}$ is of the form \eqref{defW}. Let $P \in \CC\langle \textbf{X} \rangle$ be a non-commutative polynomial of total degree $d$. We denote by $\tau_n$ the state $\frac{1}{n} \tr$ on $\mathcal{H}_n^{(\beta)}$. The sequence $$\tau_n [P(\textbf{X}/\sqrt{n})]$$
satisfies a LDP with speed $n^{\alpha\big( \frac{1}{2}+\frac{1}{d} \big)}$ and good rate function $K_{\alpha}$, defined for all $x\in \RR$ by $$ K_{\alpha}(x) = \begin{cases} c_{1}\big(x-\tau(P(\textbf{s}))\big)^{\frac{\alpha}{d}} & \text{ if } x > \tau(P(\textbf{s})),\\ 0 & \text{ if } x= \tau(P(\textbf{s})),\\
c_{-1}\big |x-\tau(P(\textbf{s})) \big|^{\frac{\alpha}{d}} & \text{ if } x < \tau(P(\textbf{s})), \end{cases}$$ where for any $\sigma \in \{-1,1\}$, $$c_{\sigma}=\inf\big\{ W_{\alpha}( \textbf{H}): \textbf{H} \in \cup_{n\in \NN} (\mathcal{H}_n^{(\beta)})^p, \sigma = \tr P_d(\textbf{H}) \big\}\in [0,+\infty],$$ where $W_{\alpha}(\textbf{H}) = \sum_{i=1}^p W_{\alpha,i}(H_i)$ and $P_d$ is the homogeneous part of degree $d$ of $P$. \end{The}
\begin{Rem}Unlike the previous results on deviations of the spectral measure and the largest eigenvalue, this one allows us to consider Gaussian matrices. As we will see in the proof, the mechanism of deviations of traces of polynomials is the same in both cases $\alpha \in (0,2)$, and $\alpha=2$. This is essentially due to the fact that still in the Gaussian case there is a heavy-tail phenomena which appears when the degree of the polynomial is strictly greater than $2$ since there is no exponential moments.
This large deviations principle is an extension, although in a more restricted setting, of the large deviations principle proven in \cite{LDPtr}, in the case where $p=1$ and $P = X^d$ for some $d\geq 3$, for Gaussian matrices and Wigner matrices without Gaussian tails. \end{Rem}
\subsection{Application to last-passage percolation}\label{LPP} Let $d\in \NN$, $d\geq 2$. We denote by $\ZZ_+^d$ the subset of vectors of $\ZZ^d$ with non-negative coordinates. Let $(X_v)_{v\in \ZZ_+^d}$ be a collection of weights. We will call a \textit{directed path} a path in which at each step, one coordinate is increased by $1$. For $v_1,v_2 \in \ZZ^d_+$, we denote by $\Pi(v_1,v_2)$ the set of directed paths from $v_1$ to $v_2$. We will identify a path with the set of its vertices. We define the \textit{last-passage time} $T_{v_1,v_2}(X)$, by $$ T_{v_1,v_2}(X) = \sup_{\pi \in \Pi(v_1,v_2)} \sum_{v \in \pi} X_{v},$$ We know by a work of Martin \cite{Martin}, that if the weights $X_v$ are i.i.d random variables with common distribution function $F$ satisfying, \begin{equation} \label{condfuncdistr} \int_0^{+\infty}(1-F(t))^{1/d} dt <+\infty,\end{equation} then for any $v\in \RR^d_+$, \begin{equation} \label{defg} \frac{1}{n}\EE T_{0, \lfloor nv \rfloor }(X) \underset{n\to+\infty}{\longrightarrow} g(v),\end{equation} where $g$ is a continuous function on $\RR^d_+$.
As an application of Theorem \ref{theoremgenesup}, we will get the following LDP for the last-passage time.
\begin{The}\label{LDPLPP}Let $\alpha \in (0,1)$. For any $n\in \NN$, we set $T(X) = T_{0,(n,...,n)}(X)$. Let $(X_v)_{v\in \ZZ^d_+}$ be a family of i.i.d random variables distributed according to $\mu_{\alpha}$.
The sequence $T(X)/n$ satisfies a LDP with speed $n^{\alpha}$ and good rate function $L_{\alpha}$, defined by $$L_{\alpha}(x) = \begin{cases} (x-g(1,...,1))^{\alpha} & \text{ if } x\geq g(1,...,1),\\ +\infty & \text{ otherwise}. \end{cases}$$
\end{The}
\subsection{Concentration inequalities}\label{concineintro} In order to prove that assumption $(i)$ holds in the context of Wigner matrices in the class $\mathcal{S}_{\alpha}$ when $\alpha \in (0,2)$ for the largest eigenvalue and the empirical spectral measure, we will prove some concentration inequalities for Wigner matrices which we would like to present as they can be of independent interest.
To derive such concentration inequalities for functions of the spectrum of random matrices, we will follow the classical argument which consists in considering our functionals as functions of the entries, and taking advantage of the concentration property of the law of the underlying random matrix. This approach is made possible in the setting where the spectrum is a smooth function of the entries, which will be our case as we will work with Hermitian matrices.
For Wigner matrices with bounded entries, or satisfying a Log-Sobolev inequality, or also for certain unitarily or orthogonally invariant models, concentration inequalities for Lipschitz (convex) linear statistics of the eigenvalues and for the largest eigenvalue, have been extensively studied by Guionnet-Zeitouni \cite{GZconc}, Guionnet \cite[Part II]{GuionnetFlour}, and Ledoux \cite[Chapter 8 \S 8.5]{Ledouxmono} (see also \cite[sections 2.3, 4.4]{Guionnet}).
More precisely, we will provide concentration inequalities for the linear statistics, the spectral measure and the largest eigenvalue of random Hermitian matrices satisfying a certain concentration property which will be indexed by some $\alpha \in (0,2]$. As we will see, this concentration property will capture the gradation of speeds of large deviations for the spectral functionals we are interested in, as it has been observed in Theorems \ref{LDPmsp} and \ref{LDPvp}.
We now present the concentration property with which we will be working.
\begin{Def}\label{defCalpha} Let $\alpha\in (0,2]$. We will say in the following that a Wigner matrix $X$ satisfies the \textit{concentration property $\mathcal{C}_{\alpha}$}, if there is a constant $\kappa>0$, such that for any Borel subset $A$ of $\mathcal{H}_n^{(\beta)}$, such that $\PP(X\in A )\geq 1/2$, and any $t>0$, \begin{equation} \label{concprop}\PP(X \notin A + \kappa \sqrt{t}B_{\ell^2} + \kappa t^{1/\alpha} B_{\ell^{\alpha}} ) \leq 2e^{-t},\end{equation} if $\alpha\in [1,2]$, and \begin{equation} \label{calpha0}\PP\big( X\notin A + \kappa (\log n)^{\frac{1}{\alpha}-1}\big( \sqrt{r} B_{\ell^2} + r B_{\ell^1} \big) +\kappa r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} \Big) \leq 4 e^{-r},\end{equation} if $\alpha \in (0,1)$, where for any $p>0$,
$$B_{\ell^p}= \big\{ Y \in \mathcal{H}_n^{(\beta)} : ||Y||_{\ell^p} \leq 1 \big\},$$ with
$$ \forall Y \in \mathcal{H}_n^{(\beta)}, \ ||Y||_{\ell^p}^p = \sum_{i,j} |Y_{i,j}|^p.$$ \end{Def}
When $\alpha \in [1,2]$, the motivation for defining this concentration property $\mathcal{C}_{\alpha}$ comes from Talagrand's famous two-levels deviation inequality \cite{Talagrand} for the measure $\nu_{\alpha}^n$, which says that there is a constant $L>0$ such that for any $n\in \NN$, any Borel subset $A$ of $\RR^n$ with $\nu_{\alpha}^n(A)>0$, and $r>0$, \begin{equation} \label{Taldev} \nu_{\alpha}^n(x \notin A+ \sqrt{r} B_{\ell^2} + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}}) \leq \frac{e^{-Lr}}{\nu_{\alpha}^n(A)},\end{equation} and similarly for $\mu_{\alpha}$.
In particular, the Wigner matrices in the class $\mathcal{S}_{\alpha}$ for $\alpha \in [1,2]$ satisfy the concentration property $\mathcal{C}_{\alpha}$ with some $\kappa$ depending on the parameters $b,a_1,a_2$ of the law of $X$ (see \eqref{defW}). More generally, we know by the results of Bobkov-Ledoux \cite[Corollary 3.2]{BobLed}, and Gozlan \cite[Proposition 1.2]{Gozlan} that if $X$ is a Wigner matrix with entries satisfying a certain Poincaré-type inequality, where the underlying metric on $\RR^m$, $m=1,2$, is the following,
\begin{equation} \label{defdalpha}\forall x,y \in \RR^m, \ d_{\omega_{\alpha}}(x,y) = \Big( \sum_{i=1}^m |\omega_{\alpha}(x_i)-\omega_{\alpha}(y_i)|^2\Big)^{1/2},\end{equation}
where $\omega_{\alpha}(t) = \sg(t) \max( |t|,|t|^{\alpha})$, $\sg(t)$ standing for the sign of $t$, then $X$ satisfies the concentration property $\mathcal{C}_{\alpha}$ with some constant $\kappa$ depending on the spectral gap. We will get into more details in section \ref{Chapconc} about this functional inequality, and present some workable criterion available for a Wigner matrix to satisfy $\mathcal{C}_{\alpha}$ when $\alpha \in [1,2]$.
When $\alpha \in (0,1)$, the concentration property of the law of Wigner matrices in the class $\mathcal{S}_{\alpha}$ differs significantly from the case where $\alpha \in [1,2]$. We know by Talagrand \cite[Proposition 5.1]{Talexpo} that as $\nu_{\alpha}$ does not have exponential tails, $\nu_{\alpha}^n$ cannot satisfy a dimension-free concentration inequality. Transporting $\nu_1^n$ onto $\nu_{\alpha}^n$, we will prove the following deviation inequality.
\begin{Pro}\label{devnualpha0} Let $n\in \NN$, $n\geq 2$. There is a constant $c>0$ depending on $\alpha$, such that for any $r>0$, $A$ Borel subset of $\RR^n$, and $C>0$ such that $\nu_{\alpha}^{n}(A) > 1/C$, $$\nu_{\alpha}^{n} \Big( x \notin A + C(\log n)^{\frac{1}{\alpha}-1}\big( \sqrt{r} B_{\ell^2} + r B_{\ell^1} \big) + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} \Big) \leq \frac{e^{-c r}}{\nu_{\alpha}^{n}(A) - 1/C}.$$ \end{Pro} We will discuss in remark \ref{remdev0} in section \ref{sectiondevineq0} the optimality of such a deviation inequality for $\nu_{\alpha}$.
The above proposition justifies the definition of the concentration property $\mathcal{C}_{\alpha}$ in the case where $\alpha \in (0,1)$, as it implies that Wigner matrices in the class $\mathcal{S}_{\alpha}$ satisfy this property when $\alpha \in (0,1)$.
Regarding the linear statistics of Wigner matrices having concentration $\mathcal{C}_{\alpha}$, we will consider different families of function whether $\alpha \in (0,1)$ or $\alpha \in [1,2]$. To this end, we define $\mathcal{M}_s^{\alpha}$ the set of finite signed measures $\sigma$ such that its total variation $|\sigma|$ has a finite $\alpha^{\text{th}}$-moment. Following \cite[Chapter 2 \S 5.1]{Samko}, we define when $\alpha \in (0,1)$, the fractional integrals of order $\alpha+1$ of $\sigma \in \mathcal{M}_s^{\alpha}$, by \begin{align} \forall t \in \RR,\ & (\mathcal{I}^{\alpha+1}_+ \sigma) (t) =\frac{1}{\Gamma(\alpha+1)} \int_{-\infty}^t (t-x)^{\alpha} d\sigma(x),\nonumber\\ & (\mathcal{I}^{\alpha+1}_- \sigma)(t) = \frac{1}{\Gamma(\alpha+1)}\int_t^{+\infty} (x-t)^{\alpha}d\sigma(x). \label{defintfrac}\end{align} This definition interpolates for non-integer order the usual iterated integral (see \cite[Chapter 1 \S 2.3]{Samko} for more details). With these definitions, we will prove the following deviations inequalities.
\begin{Pro} \label{conclinearstat}Let $\alpha \in (0,2]$. Let $X$ be a Wigner matrix having concentration $\mathcal{C}_{\alpha}$ with some $\kappa>0$. There is a constant $c_{\alpha}>0$ such that if $\alpha \in [1,2]$ and $f : \RR\to \RR$ is some $1$-Lipschitz function, then for any $t>0$, $$ \PP\big( \mu_{X/\sqrt{n}}(f) - m_f >t \big) \leq 2 \exp\Big( -c_{\alpha} \min\Big(\frac{n^2t^2}{\kappa^2}, \frac{n^{1+\frac{\alpha}{2}}t^{\alpha}}{\kappa^{\alpha}} \Big)\Big),$$
if $\alpha \in (0,1)$, $f$ is $1$-Lipschitz and moreover $f = \mathcal{I}_{\pm}^{\alpha+1}(\sigma)$ for some $\sigma \in \mathcal{M}_s^{\alpha}$ such that $|\sigma|(\RR)\leq m$, then for any $t>0$, $$ \PP\big( \mu_{X/\sqrt{n}}(f) - m_f >t \big) \leq 4 \exp\Big( -c_{\alpha} \min\Big(\frac{n^2 t^2}{\kappa^2(\log n)^{2(\frac{1}{\alpha}-1)} }, \frac{n^{\frac{3}{2}}t}{\kappa (\log n)^{\frac{1}{\alpha}-1}},\frac{n^{1+\frac{\alpha}{2}} t}{\kappa m} \Big)\Big),$$ where $m_f$ denotes a median of $\mu_{X/\sqrt{n}}$. \end{Pro}
\begin{Rem}
The reason for considering the class of function $\mathcal{I}_{\pm}^{\alpha+1}(\mathcal{M}_s^{\alpha})$ in the case $\alpha\in (0,1)$, comes from the fact that we only understand the stability of the empirical spectral measure with respect to $|| \ ||_{\ell^{\alpha}}$, by using a certain distance $d_{\alpha}$ which controls this class of functions (see section \ref{spvarsection} for more details).
Still in the case $\alpha<1$, note that one cannot expect the above concentration inequality to be true for all Lipschitz functions, since a change of large deviations speed may occur as the entries of $X$ do not have exponential tails. Indeed, for example if $X$ is in the class $\mathcal{S}_{\alpha}$, Theorem \ref{LDPpoly} tells us the speed of large deviations of $\frac{1}{n}\tr (X/\sqrt{n})$ is $n^{3\alpha/2}$.
\end{Rem}
\begin{Rem} One can identify the image $\mathcal{I}_{\pm}^{\alpha +1}(\mathcal{M}_s^{\alpha})$, by a minor change of \cite[Theorem 6.3]{Samko}. To ease the notation, we will only describe $\mathcal{I}_{+}^{\alpha +1}(\mathcal{M}_s^{\alpha})$.
For any $\phi \in L^1(\RR)$, one can define the fractional integral of order $\alpha$ by, $$\forall x \in \RR, \ \mathcal{I}_{+}^{1-\alpha}(\phi)(x) = \frac{1}{\Gamma(1-\alpha)} \int_{0}^{+\infty} t^{-\alpha} \phi(x- t) dt.$$
The function above is well-defined almost everywhere as $t^{-\alpha}\phi(x- t)$ is integrable on a neighborhood of $0$ for almost all $x$ by Fubini theorem. With this definition, the set $\mathcal{I}_{+}^{\alpha +1}(\mathcal{M}_s^{\alpha})$ consists of the functions $f$ such that there is some $\phi \in L^1(\RR)$ and $\sigma \in \mathcal{M}_s^{\alpha}$, such that $$ \forall x \in \RR, \ f(x) = \int_{-\infty}^x \phi(t) dt,\text{ and } \mathcal{I}_{+}^{1-\alpha}(\phi)(x) = \sigma(-\infty,x].$$ \end{Rem} \begin{Rem}\label{concsimpl0}
Note also that the exponential bound can be simplified in the case $\alpha \in (0,1)$ if $m\geq c_0$, where $c_0$ is a constant independent of $n$. One gets then, for any $t>0$, $$ \PP\big( \mu_{X/\sqrt{n}}(f) - m_f >t \big) \leq 4 \exp\Big( -c_{\alpha} \min\Big(\frac{n^2 t^2}{\kappa^2(\log n)^{2(\frac{1}{\alpha}-1)} }, \frac{n^{1+\frac{\alpha}{2}} t}{\kappa m} \Big)\Big).$$ \end{Rem}
In order to state our concentration inequality for the spectral measure, we will work with the following distance $d$ defined on the set of probability measures on $\RR$, denoted by $\mathcal{P}(\RR)$, in order to quantify the deviations:
\begin{equation} \label{defdStiel} \forall \mu, \nu \in \mathcal{P}(\RR), \ d(\mu, \nu) = \sup_{z\in \mathcal{K}} |g_{\mu}(z) - g_{\nu}(z) |,\end{equation} where $\mathcal{K}$ is a compact subset of $\{ z \in \CC : \Im z \geq 2\}$ with an accumulation point, such that $\mathrm{diam}(\mathcal{K}) \leq 1$, and with $g_{\mu}$ the Stieltjes transform of $\mu$, that is, $$\forall z \in \CC^+, \ g_{\mu}(z) = \int \frac{d\mu(t)}{z-t},$$ where $\CC^+ = \{ z \in \CC : \Im z >0\}$. This distance metrizes the weak topology on $\mathcal{P}(\RR)$ by \cite[Theorem 2.4.4]{Guionnet}.
We will prove the following concentration inequalities for the empirical spectral measure and the largest eigenvalues of Wigner matrices having concentration $\mathcal{C}_{\alpha}$.
\begin{Pro} \label{concspintro} Let $\alpha \in (0,2]$. Let $X$ be a Wigner matrix satisfying $\mathcal{C}_{\alpha}$ with some $\kappa>0$. There exists a constant $c_{\alpha}>0$, depending on $\alpha$, such that for any $t>0$, $$ \PP\left( d\big( \mu_{X/\sqrt{n}}, \EE \mu_{X/\sqrt{n}} \big) > t +\delta_n\right) \leq \frac{32}{t^2}\exp\big ( -c_{\alpha}k_{\alpha}(t) \big),$$ where $\delta_n = O\big(\kappa n^{-1} ( \log n)^{(1/\alpha-1)_+}\big)$, and where for $\alpha \in [1,2]$, $$ k_{\alpha}(t) = \min\Big(\frac{n^2t^2}{\kappa^2}, \frac{n^{1+\frac{\alpha}{2}}t^{\alpha}}{\kappa^{\alpha}} \Big),$$ whereas for $\alpha \in (0,1)$ $$ k_{\alpha}(t) = \min\Big(\frac{n^2 t^2}{\kappa^2(\log n)^{2(\frac{1}{\alpha}-1)} }, \frac{n^{1+\frac{\alpha}{2}} t}{\kappa} \Big).$$ \end{Pro}
\begin{Pro}\label{concvpintro} Let $\alpha \in (0,2]$. Let $X$ be a Wigner matrix satisfying $\mathcal{C}_{\alpha}$ for some $\kappa>0$. There is a constant $c_{\alpha}>0$, such that for any $t>0$,
$$ \PP\left( \big| \lambda_{X/\sqrt{n}}-\EE \lambda_{X/\sqrt{n}}\big| > t +\eps_n\right) \leq 8\exp\big(-c_{\alpha} h_{\alpha}(t) \big),$$ where \begin{equation}\label{h1vp} h_{\alpha}(t) = \min\Big( \frac{t^2n}{\kappa^2}, \frac{t^{\alpha}n^{\frac{\alpha}{2}}}{\kappa^{\alpha}} \Big),\end{equation} if $\alpha \in [1,2]$, and \begin{equation}\label{h0vp} h_{\alpha}(t) = \min\Big( \frac{t^2n}{\kappa^2(\log n)^{2(\frac{1}{\alpha}-1) }},\frac{t\sqrt{n}}{\kappa (\log n)^{\frac{1}{\alpha}-1} }, \frac{t^{\alpha}n^{\frac{\alpha}{2}}}{\kappa^{\alpha} } \Big),\end{equation} if $\alpha \in (0,1)$, and where $\eps_n = O( \kappa n^{-1/2} (\log n)^{(1/\alpha-1)_+})$, uniformly in $H \in \mathcal{H}_n^{(\beta)}$. \end{Pro}
\subsection{Spectral variation inequalities} We would like also to advertise for some spectral variation inequalities, which are not particularly new, but which are maybe a little less known in the form we will propose.
Indeed, to obtain the concentration inequality of Proposition \ref{concspintro}, we need to understand the stability of the spectrum of Hermitian matrices with respect to the distance $|| \ ||_{\ell^p}$ for $p\geq 1$ or $|| \ ||_{\ell^p}^p$ when $p<1$.
For $p\geq 1$, define the $L^p$-Wasserstein distance on the set of probability measures on $\RR$ with finite $p^{\text{th}}$-moment by,
$$\mathcal{W}_p(\mu,\nu) = \Big( \inf_{\pi} \int |x-y|^p d\pi(x,y) \Big)^{1/p},$$ where the infimum is over all coupling $\pi$ between $\mu$ and $\nu$, two probability measures on $\RR$ with finite $p^{\text{th}}$-moment.
When $p\geq1$, we get as a mere consequence of Lidskii's theorem (see \cite[Theorem III.4.1]{Bhatia}) the following lemma. \begin{Lem}\label{spvar1intro} Let $p\in [1,2]$, and $A,B \in \mathcal{H}_n^{(\beta)}$.
$$\mathcal{W}_p(\mu_A,\mu_B) \leq \frac{1}{n^{1/p}} || A-B||_{\ell^p}.$$ As a consequence,
$$d(\mu_A,\mu_B) \leq \frac{1}{n^{1/p}} || A-B||_{\ell^p}.$$ \end{Lem}
Whereas for $p<1$, we obtain by Rofteld's inequality (see \cite[Theorem IV.2.14]{Bhatia} or \cite{Thompson}) the following.
\begin{Lem}\label{spvar0intro}Let $p\in (0,1)$. Let $A,B \in \mathcal{H}_n^{(\beta)}$. For any $t\in \RR$,
$$ \big| \sum_{i=1}^n (t-\lambda_i(A))_+^p - \sum_{i=1}^n (t-\lambda_i(B))_+^{p}\big| \leq \sum_{i=1}^n |\lambda_i(A-B)|^p,$$ where $\lambda_1(A),...,\lambda_n(A)$ denote the eigenvalues of $A$, and similarly for $B$. Furthermore, there is a positive constant $C_p$, such that for any $A,B \in \mathcal{H}_n^{(\beta)}$,
$$ d(\mu_{A},\mu_B) \leq \frac{C_p}{n} ||A-B||_{\ell^p}^p,$$ with $$C_p = \sqrt{\pi}(p+1) \frac{\Gamma\big( \frac{p+1}{2}\big)}{\Gamma\big( 1+\frac{p}{2}\big)}.$$ \end{Lem}
\subsection{Organization of the paper}
In the section \ref{infconv}, we prove some inf-convolution inequalities for $\nu_{\alpha}^{ n}$. As the large deviations of our functional $f_n$ are governed by translates, we will need some sharp deviation inequalities with respect to the metric $||\ ||_{\ell^{\alpha}}$ (or $|| \ ||_{\ell^{\alpha}}^{\alpha}$ when $\alpha<1$). We will provide a family of weights $W_{\alpha,\eps}$ which captures the asymptotics of the tail distribution of $\nu_{\alpha}^{n}$, that is, behaving like $|| x ||_{\ell^{\alpha}}^{\alpha}$ when $||x||_{\infty} \gg 1$. This will be done by transporting and tensoring the family of optimal weights known for the exponential law due to Talagrand \cite[Theorem 1.2]{Talagrand2}.
In the section \ref{prooftheorem}, we give a proof of Theorems \ref{theoremgene} and \ref{theoremgenesup}. The upper bound relies on Proposition \ref{grandev} which gives a large deviations sharp upper bound for $\nu_{\alpha}^{n}$ with respect to the metric $|| \ ||_{\ell^{\alpha}}$ using the inf-convolution inequalities proven in section \ref{infconv}. The lower bound is given by Proposition \ref{lowb} which estimates at the exponential scale $v(n)$ the probability, under $\nu_{\alpha}^n$, of an event translated by some element $v(n)^{1/\alpha} h_n$.
The rest of the paper is devoted to applications to Wigner matrices and the last-passage time.
In the section \ref{Chapconc}, we prove the concentration inequalities of Propositions \ref{concspintro} and \ref{concvpintro} for the largest eigenvalue, linear statistics and empirical spectral measure of Wigner matrices satisfying the concentration property $\mathcal{C}_{\alpha}$ defined in \eqref{concprop} and \eqref{calpha0}. To do so, we will prove and discuss the spectral variations inequalities in Lemmas \ref{spvar1intro} and \ref{spvar0intro} in section \ref{spvarsection}.
In section \ref{secdetermexpoWigner}, we show some uniform deterministic equivalents for the spectral measure, largest eigenvalue and traces of non-commutative polynomials of deformed Wigner matrices in the class $\mathcal{S}_{\alpha}$. To make the equivalents for the spectral measure and largest eigenvalue of hold uniformly for $\alpha<2$, we make use of the concentration inequalities we proved in section \ref{Chapconc}, and perform a classical chaining argument.
In section \ref{secdetermequivLPT}, we provide a deterministic equivalent for the last-passage time under additive deformations of the weights. The strategy to make our equivalent hold uniformly will be the same as for the case of the spectral measure and largest eigenvalue of Wigner matrices in the class $\mathcal{S}_{\alpha}$, meaning that it will rely on concentration and chaining arguments.
In section \ref{Wigner}, we apply Theorem \ref{theoremgene} in the setting of Wigner matrices in the class $\mathcal{S}_{\alpha}$, to the spectral measure, the largest eigenvalue (for $\alpha \in (0,2)$) and to traces of non-commutative polynomials (for $\alpha \in (0,2]$). Using of the uniform deterministic equivalents we proved in section \ref{secdetermexpoWigner}, we give a proof of Theorems \ref{LDPmsp}, \ref{LDPvp}, and \ref{LDPpoly}.
Finally we prove in section \ref{LPPLDP}, the large deviations principle for the last-passage time of Theorem \ref{LDPLPP} by applying Theorem \ref{theoremgenesup} and using the uniform deterministic equivalent proved in section \ref{secdetermequivLPT}.
\section{Inf-convolution inequalities for $\nu_{\alpha}^{ n}$} \label{infconv} Let $\nu$ be a probability measure on $\RR^n$, and let $w$ be a measurable function on $\RR^n$ taking non-negative values. Following Maurey (see \cite{Maurey}), we will say that $(\nu, w)$ satisfies the $\tau$-property if for any non-negative measurable function $f$ on $\RR^n$, \begin{equation}\label{IC} \big(\int e^{f\Box w } d\nu \big) \big(\int e^{-f} d\nu \big) \leq 1,\end{equation} where $ \Box $ denotes the inf-convolution, that is, $$ \forall x \in \RR^n, \ f\Box w (x) = \inf_{y \in \RR^n} \{ f(y) + w(x-y)\}.$$ \label{nomen:infconv} \nomenclature[]{$\Box $}{inf-convolution}{}{\pageref{nomen:infconv}}
The $\tau$-property is closely linked to transportation-cost inequalities. By the Kantorovitch duality (see \cite[Theorem 5.10]{Villani}), and the duality of the entropy (see \cite[Lemma 6.2.13]{Zeitouni}), it is known that under mild assumptions on $w$ that the following general inf-convolution inequality, \begin{equation} \label{infconvgene} \int e^{f\Box w} d\nu \leq e^{\int f d\nu},\end{equation} satisfied for any non-negative measurable function $f$ is equivalent to the following transportation-cost inequality: for any $\mu$ probability measure on $\RR^n$,
\begin{equation} \label{transportcost} \mathcal{W}_w(\mu, \nu) \leq D( \mu||\nu),\end{equation}
where $D(\mu || \nu)$ is the relative entropy of $\mu$ with respect to $\nu$, and \begin{equation} \label{distKR} \mathcal{W}_w(\mu,\nu) = \inf\big\{ \int w(x-y) d\pi (x,y) :\pi \text{ has marginals } \mu \text{ and } \nu \big\}.\end{equation}
In particular, under the assumption that $w$ is upper semi-continuous, Kantorovitch duality is valid by \cite[Theorem 5.10]{Villani}, so that the equivalence above between \eqref{infconvgene} and \eqref{distKR} holds.
One can observe that if $(\nu,w)$ satisfies the $\tau$-property, then by Jensen's inequality, it satisfies also the general inf-convolution inequality \eqref{infconvgene}, and therefore $\nu$ satisfies the transportation-cost inequality \eqref{transportcost} with cost function $w$.
Conversely, according to \cite[Proposition 4.13]{Gozlan}, if $\nu$ satisfies the transportation-cost inequality \eqref{transportcost} with cost function $w$, then $(\nu, w\Box w)$ satisfies the $\tau$-property. If moreover $w$ is sub-additive, then one can see that $w\Box w =w$ and thus $(\nu, w)$ satisfies the $\tau$-property. Whereas if $w$ is convex, then $w \Box w = 2w(./2)$ so that $(\nu, 2w(./2))$ satisfies the $\tau$-property. This remark will be useful later when we will need to translate a transportation-cost inequality into a $\tau$-property.
More importantly for us, the $\tau$-property yields deviations bounds with respect to enlargements by the weight $w$. We know from \cite[Lemma 4]{Maurey}, that if $(\nu, w)$ satisfies the $\tau$-property, then for any Borel subset $A$ of $\RR^n$, and any $t>0$, \begin{equation} \label{dev} \nu\big( x \notin A +\{w \leq r\}\big)\leq \frac{e^{-r}}{\nu(A)}.\end{equation}
We define another form of inf-convolution inequality, designed to enable us to get the best constants in our weight functions, (and also to deal with the measure $\nu_{\alpha}^{n}$ when $\alpha \in (0,1)$), which we will call the \textit{truncated $\tau$-property}. More precisely, we will say that a measure $\nu$ on $\RR^n$ with the weight function $w$, satisfies the \textit{$A_0$-truncated $\tau$-property}, where $A_0$ is a Borel subset of $\RR^n$, if \eqref{IC} is true for any non-negative measurable function $f$ such that $f = +\infty$ on $A_0^c$.
This $A_0$-truncated $\tau$-property yields a deviation inequality with respect to enlargement by the weight $w$ of the following form: for any Borel subset $A$ of $\RR^n$ such that $\nu(A)>0$, and any $r>0$,
\begin{equation}\label{devtrunc} \nu\big( x \notin A + \{w \leq r\} \big) \leq \frac{e^{-r}}{\nu(A\cap A_0)}.\end{equation}
The goal of this section is to find, for the measure $\nu_{\alpha}^{n}$, when $\alpha \in (0,2)$, a family of weights $W_{\alpha, \eps}$ for which a truncated $\tau$-property is satisfied, and which captures the asymptotics of the tail distribution of $\nu_{\alpha}^{n}$. More precisely, we will prove the following proposition.
\begin{Pro}\label{tau} Let $\alpha >0$. If $\alpha=1$, then for any $\eps<1/2$, $(\nu_1^n, W_{1,\eps})$ satisfies the $\tau$-property with $$\forall x \in \RR, \ W_{1,\eps}(x) = \sum_{i=1}^n w_{\eps}(x_i),$$ where $$ w_{\eps}(t) = \begin{cases}
\frac{\eps e^{-1/\eps} t^2}{8} & \text{ if } |t| \leq 2/\eps^2,\\
(1-2\eps)|t| & \text{ if } |t|> 2/\eps^2. \end{cases}$$
If $\alpha \neq 1$, there are some constants $\kappa>0$ and $\eps_0 \in (0,1)$ such that for $\eps \in (0,\eps_0)$ and $m\geq 1$,
$(\nu_{\alpha}^{n}, W_{\alpha,\eps}^{(m)})$ satisfies the $mB_{\ell^{\infty}}$-truncated $\tau$-property, where \begin{equation}\label{weight0}\forall x \in \RR^n, \ W_{\alpha,\eps}^{(m)}(x) = \sum_{i=1}^n w_{\alpha,\eps}^{(m)}(x_i),\end{equation} with $$ w_{\alpha,\eps }^{(m)}(t) = \begin{cases}
\kappa^{-1} e^{-(\frac{m}{\eps})^{\alpha/2}} t^2 & \text{ if } |t| \leq m\eps^{-1},\\
(1- \kappa\eps^{(\alpha/2)\wedge 1} )|t|^{\alpha} & \text{ if } |t| > m \eps^{-1}. \end{cases}$$
\end{Pro}
The rest of this section will be devoted to proving the above proposition. We will reduce the problem in a first phase to the one-dimensional case, and to an estimation of the monotone rearrangement of $\nu_{1}$ onto $\nu_{\alpha}$.
As the usual $\tau$-property (see \cite[Lemma 1]{Maurey}), the truncated version of the $\tau$-property tensorizes in the following way.
\begin{Lem} \label{tensortau}Let $\nu_i$ be a probability measure defined on some measurable space $\mathcal{X}_i$, $A_i$ be some measurable subset of $\mathcal{X}_i$ and $w_i : \mathcal{X}_i \to \RR_+$ be a measurable function, for $i=1,2$.
If $(\nu_i,w_i)$ satisfies the $A_i$-truncated $\tau$-property for $i=1,2$, then $(\nu_1\otimes \nu_2, w)$ satisfies the $A_1\times A_2$-truncated $\tau$-property with $$ \forall (x,y) \in \mathcal{X}_1\times \mathcal{X}_2, \ w(x,y) = w_1(x) + w_2(y).$$
\end{Lem} Since we are dealing with the product measure $\nu_{\alpha}^n$, we can focus on studying the $\tau$-property for the one-dimensional marginal $\nu_{\alpha}$.
For the exponential measure, we have the following result due to Talagrand, which gives a family of optimal weights $c_{\lambda}$. \begin{Pro}[{\cite[Theorem 1.2]{Talagrand}}]\label{Tal} Let $\lambda \in (0,1)$. Define the weight function $c_{\lambda}$ for any $x\in \RR$ by,
$$ c_{\lambda}(x) =\big( \frac{1}{\lambda} -1 \big) (e^{-\lambda |x|} -1 +\lambda |x|). $$ For any $\lambda \in (0,1)$, $\nu_1$ satisfies a transportation-cost inequality \eqref{transportcost} with cost function $c_{\lambda}$. \end{Pro}
Note that, $c_{\lambda}(x) \sim_{\pm \infty} (1-\lambda)|x|$. Thus, when $\lambda \ll1$, $c_{\lambda}$ captures the exact asymptotics of the tail distribution of the exponential law.
For technical reasons, we prefer to work with a different family of weights than the one defined in Proposition \ref{Tal}. In the following corollary, we reformulate Talagrand's result for the symmetric exponential measure $\nu_1$.
\begin{Cor}\label{corexp} Let $\delta >0$. We define the weight function $w_{\eps}$, for any $t \in \RR$, by $$ w_{\delta}(t) = \begin{cases}
\frac{\delta e^{-1/\delta} t^2}{8} & \text{ if } |t| \leq 2/\delta^2,\\
(1-2\delta)|t| & \text{ if } |t|> 2/\delta^2. \end{cases}$$ For any $\delta \in (0,1/2)$, $(\nu_1,w_{\delta})$ satisfies the $\tau$-property. As a consequence, $(\nu_1^n, W_{1,\delta})$ satisfies the $\tau$-property, with $W_{1,\delta}$ defined in Proposition \ref{tau}. \end{Cor} This reformulation reveals in particular the structure of the enlargements given by the weights $c_{\lambda}$ which consist in a mixture of $\ell^2$ and $\ell^1$-balls.
\begin{proof}As $c_{\lambda}$ is a convex function, we know by \cite[Proposition 4.13]{Gozlan} that $(\nu_1, 2c_{\lambda}(./2))$ satisfies the $\tau$-property. To prove Corollary \ref{corexp}, it suffices to prove that $w_{\delta} \leq 2c_{\delta}(./2)$ for any $\delta \in (0,1/2)$. Since both functions are even, it is sufficient to prove the inequality on $\RR_+$. Let $t>0$. By Taylor's formula $$ e^{-\delta t}-1+\delta t= \delta^2e^{-\delta y} \frac{t^2}{2},$$ for some $y\in [0,t]$. If $t\leq 2/\delta^2$ and $\delta\leq 1/2$, we get $$2c_{\delta}(t/2) \geq \delta\big(1-\delta) e^{-1/\delta} \frac{t^2}{4}\geq w_{\delta}(t).$$ If $t\geq 1/\delta^2$, we have $$c_{\delta}(t) \geq \big( \frac{1}{\delta}-1\big)(-1+\delta t)\geq (1-\delta)t-\frac{1}{\delta}\geq (1-2\delta)t.$$ Thus, $2c_{\delta}(t/2)\geq (1-2\delta)t$ for $t\geq 2/\delta^2$.
After tensorization (see \cite[Lemma 1]{Maurey}), we obtain that $(\nu_1^n, W_{1,\delta})$ satisfies the $\tau$-property with $W_{1,\delta}$ defined in Proposition \ref{tau}.
\end{proof}
For $\alpha \neq1$, the general strategy is to transport this $\tau$-property of the symmetric exponential law to obtain a $\tau$-property for $\nu_{\alpha}$.
It extends in our setting of truncated $\tau$-property, a result of Maurey \cite[Lemma 2]{Maurey}. \begin{Lem}\label{transp0} Let $A$ be a Borel subset of $\RR^n$. Let $\mu$ be a probability measure on $\RR^n$ and let $\psi : \RR^n \to \RR^n$ be a bijective measurable map. Assume $(\mu, w)$ satisfies the $\tau$-property. Let $A$ be a Borel subset of $\RR^n$ and let $\tilde{w}$ be a weight function such that, $$\forall x \in \RR^n, y \in A,\ \tilde{w}(x-y)\leq w\big(\psi^{-1}(x) - \psi^{-1}(y)\big).$$ Then, $(\mu \circ \psi^{-1}, \tilde{w})$ satisfies the $A$-truncated $\tau$-property. \end{Lem}
\begin{proof} Let $f : \RR^n \to \RR$ be a measurable non-negative function being $+\infty$ on $A^c$. Applying the $\tau$-property of $(\mu,w)$ to $f\circ \psi$, we get $$\big( \int e^{f\circ \psi \Box w} d\mu \big) \big( \int e^{-f \circ \psi} d\mu \big) \leq 1.$$ But, as $\psi$ is a bijection and $f=+\infty$ on $A^c$, \begin{align*} f \circ \psi \Box w(\psi^{-1}(x))& = \inf_{y\in \RR^n} \{ f(y) + w( \psi^{-1}(x) - \psi^{-1}(y)) \}\\ & = \inf_{y\in A} \{ f(y) + w( \psi^{-1}(x) - \psi^{-1}(y)) \}. \end{align*} From the assumption on $\tilde{w}$, we deduce $$ f \circ \psi \Box w(\psi^{-1}(x)) \geq \inf_{y\in A} \{ f(y) + w( x -y) \} = f \Box w(x).$$ Therefore, $$\big( \int e^{f \Box w} d \mu \circ \psi^{-1} \big) \big( \int e^{-f} d\mu\circ \psi^{-1} \big)\leq 1.$$
\end{proof}
In particular, in the one-dimensional case, if $(\mu, w)$ satisfies the $\tau$-property and $w$ is even and non-decreasing on $\RR_+$, then $\mu \circ \psi^{-1}$ satisfies the $A$-truncated $\tau$-property with any even weight function $\tilde{w}$ such that $$ \forall s \geq 0, \ \tilde{w}(s) \leq w\big( \Delta_A(s)\big),$$ where $\Delta_A$ is defined for any $s\geq 0$ by,
\begin{equation*} \Delta_A(s) = \inf \big\{ |\psi^{-1}(x) - \psi^{-1}(y)|, |x-y| = s, \ x \in A \big\}.\end{equation*} If $\mu$ and $\nu$ are two probability measures on $\RR$, we define the \textit{monotone rearrangement} $T$ of $\mu$ onto $\nu$ by, $$\forall t \in \RR, \ \mu(-\infty,t] = \nu(-\infty, T(t)].$$ This defines a unique non-decreasing map if the distribution function of $\nu$ is invertible, which sends $\mu$ to $\nu$.
Let $\psi$ be the monotone rearrangement of $\nu_1$ onto $\nu_{\alpha}$. One can easily check that $\psi$ is an odd function, and that its restriction $\phi$ on $\RR^+$ satisfies, $$ \forall x \geq 0, \ e^{-x} = \int_{\phi(x)}^{+\infty} e^{-u^{\alpha}} \frac{du}{Z_{\alpha}},$$ where $Z_{\alpha}$ is the normalizing constant of $\mu_{\alpha}$, so that $\phi$ is the monotone rearrangement of $\mu_1$ onto $\mu_{\alpha}$. Thus, we are reduced to understand the behavior of the map $\phi$ and how it deforms the weights $c_{\eps}$ of Proposition \ref{Tal}.
\subsection{Behavior of the monotone rearrangement}\label{sectionBrenier} When $\alpha \geq 1$, we have the following estimate on the monotone rearrangement due to Talagrand \cite{Talagrand}. \begin{Lem}[{\cite[Lemma 2.5]{Talagrand}}]\label{Taltransp}Let $\alpha \geq 1$. Let $\psi$ be the monotone rearrangement sending $\nu_1$ to $\nu_{\alpha}$. Denote by $\Delta$ the function defined for any $s\geq 0$ by,
\begin{equation}\label{defdelta} \Delta(s) = \inf_{|x-y| = s} |\psi^{-1}(x) - \psi^{-1}(y)|.\end{equation} There is a constant $c>0$ depending on $\alpha$ such that for any $s\geq 0$, $$\Delta(s) \geq c \max(s ,s^{\alpha}).$$ \end{Lem}
\begin{Rem}\label{comprearrmap} In \cite[Lemma 2.5]{Talagrand}, this estimate is derived for the monotone rearrangement $\phi$ of $\mu_1$ onto $\mu_{\alpha}$. But since,
\begin{equation} \label{link}\forall x \in \RR, \ \psi(x) = \sg(x) \phi(|x|),\end{equation} one easily deduces the same estimate for $\psi$, together with the fact that if $x,y$ have opposite signs, \begin{align*}
\phi^{-1}(|y|) + \phi^{-1}(|x|) &\geq c \big(\max(|x|,|x|^{\alpha}) + \max( |y|,|y|^{\alpha}) \big)\\
& \geq c' \max(|x - y|, |x-y|^{\alpha}), \end{align*}
where $c'$ is some constant and where we used the fact that $|x-y| = |x|+|y|$. \end{Rem}
To get the exact asymptotic of the tail distribution of $\nu_{\alpha}$ we will need of the following finer estimate on the monotone rearrangement.
\begin{Lem}\label{brenier1}Let $\alpha \geq1$. Define for any $m\geq 1$,
\begin{equation}\label{defdeltam}\forall s\geq 0, \ \Delta_m(s) = \inf \{ |\psi^{-1}(x) - \psi^{-1}(y)| : |x|\leq m, \ |x-y|=s \}.\end{equation} There is a constant $\gamma$ depending on $\alpha$, such that for any $\eps \in (0,1)$, and $s \geq m\eps^{-1}$, $$ \Delta_m(s) \geq (1-\gamma \eps) s^{\alpha}.$$
\end{Lem} \begin{proof} By definition of $\psi$, we have for any $x\in \RR$,
$$ \psi^{-1}(x) = -\sg(x) \log \int_{|x|}^{+\infty} e^{-u^{\alpha}} \frac{du}{Z_{\alpha}},$$
where $Z_{\alpha}$ is the normalizing constant of $\mu_{\alpha}$. Let $s \geq m\eps^{-1}$ and $x,y\in \RR$ such that $0\leq |x| \leq m$, and $|x-y| =s$. If $x$ and $y$ have the same signs, we can assume without loss of generality, that both $x, y\geq 0$. As $x\leq m \leq s$, we have $y= x+s$. Thus, $$ \psi^{-1}(y) - \psi^{-1}(x) \geq \psi^{-1}(s) - \psi^{-1}(m).$$
We have, on one hand, as $s\geq 1$, $$ \int_{s}^{+\infty} e^{-u^{\alpha}} du \leq \frac{1}{\alpha} \int_{s}^{+\infty} \alpha u^{\alpha-1} e^{-u^{\alpha}} du = \frac{1}{\alpha}e^{-s^{\alpha}}.$$ And on the other hand, $$ \int_m^{+\infty} e^{-u^{\alpha}} du \geq e^{-(m+1)^{\alpha}}.$$ Therefore, as $s \geq m\eps^{-1}$, $$ \psi^{-1}(y) - \psi^{-1}(x) \geq s^{\alpha} -(m+1)^{\alpha} +\log \alpha \geq s^{\alpha}(1-\gamma \eps^{\alpha}),$$ for some constant $\gamma>0$. Now, if $x$ and $y$ have opposite signs, we can assume without loss of generality that $x\leq 0$ and $y\geq 0$. Then, $y\geq s-m$ so that,
$$ |\psi^{-1}(y) - \psi^{-1}(x) | = \psi^{-1}(y)+\psi^{-1}(-x) \geq \psi^{-1}(s-m) \geq (s-m)^{\alpha}+\log \alpha.$$
Thus, we can find some constant $\gamma'$ such that $|\psi^{-1}(y) - \psi^{-1}(x) | \geq (1-\gamma' \eps) s^{\alpha}$.
\end{proof}
\begin{Rem} The truncation we performed here is made to ensure we get the best constant (that is $1$) in the estimate of the large increments of the monotone rearrangement. Indeed, defining $\Delta$ as in \eqref{defdelta}, we would get for $s\gg 1$,
$$\Delta(s) \leq \Big|\psi\Big( \frac{s}{2} \Big) - \psi\Big( \frac{-s}{2} \Big) \Big| = 2\psi\Big( \frac{s}{2} \Big) \simeq 2\Big( \frac{s}{2}\Big)^{\alpha} = 2^{1-\alpha} s^{\alpha},$$ with $2^{1-\alpha}<1$.
\end{Rem}
When $\alpha<1$, we get the following estimate on the monotone rearrangement of $\nu_1$ onto $\nu_{\alpha}$. Note that as $\nu_{\alpha}$ does not have an exponential tail, the rearrangement map cannot be a Lipschitz function.
\begin{Lem}\label{Brenieralpha}Let $\alpha \in (0,1)$. Let $\phi$ be the monotone rearrangement of $\mu_1$ onto $\mu_{\alpha}$. There is a constant $K>0$ depending on $\alpha$ such that for any $x,y \in [0,+\infty)$,
$$ |\phi(x) - \phi(y)|\leq K \max \Big( |x-y|, x^{\frac{1}{\alpha}-1}|x-y|, |x-y|^{\frac{1}{\alpha}} \Big).$$
\end{Lem}
\begin{proof} This proof is very much in the spirit of \cite[Lemma 2.5]{Talagrand}. We begin by bounding from above $$\int_x^{+\infty} e^{-y^{\alpha}} dy,$$ when $x\geq1$. The change of variable $u= y^{\alpha}$ gives, $$\int_x^{+\infty} e^{-y^{\alpha}} dy = \frac{1}{\alpha} \int_{x^{\alpha}}^{+\infty} u^{\frac{1}{\alpha}-1} e^{-u} du.$$ Let $m = \lceil \frac{1}{\alpha} \rceil$. Integrating by parts $m$ times, we get \begin{align*} \int_{x^{\alpha}}^{+\infty} u^{\frac{1}{\alpha}-1} e^{-u} du &= \sum_{k=1}^{m-1} \big(\frac{1}{\alpha}-1\big)...\big(\frac{1}{\alpha}-k+1\big) x^{1-k\alpha} e^{-x^{\alpha}} \\ &+\big(\frac{1}{\alpha}-1\big)...\big(\frac{1}{\alpha}-m+1\big)\int_{x^{\alpha}}^{+\infty} u^{\frac{1}{\alpha}-m} e^{-u} du. \end{align*} As $\frac{1}{\alpha} -m \leq 0$, we deduce for any $x\geq 1$, $$\int_{x^{\alpha}}^{+\infty} u^{\frac{1}{\alpha}-1} e^{-u} du \leq K x^{1-\alpha}e^{-x^{\alpha}},$$ where $K>0$ is some constant depending on $\alpha$ which will vary along the proof. Therefore, for any $x\geq 1$,
\begin{equation} \label{maj0}\int_x^{+\infty} e^{-y^{\alpha}} dy \leq Kx^{1-\alpha}e^{-x^{\alpha}}.\end{equation} By definition $\phi$ satisfies for any $x> 0$, \begin{equation} \label{eqtransport} e^{-x} = \int_{\phi(x)}^{+\infty} e^{-y^{\alpha}} \frac{dy}{Z_{\alpha}}.\end{equation} This implies that $\phi$ is an increasing homeomorphism of $\RR_+$. For $\phi(x)\geq 1$, we have \begin{equation} \label{aprioribound}e^{-x} \leq K\phi(x)^{1-\alpha}e^{-\phi(x)^{\alpha}}.\end{equation} From \eqref{eqtransport}, we see that $\phi$ is differentiable, and $\phi'$ satisfies for any $x\geq 0$, $$ e^{-x} = \frac{1}{Z_{\alpha}}\phi'(x) e^{-\phi(x)^{\alpha}}.$$ Thus by \eqref{aprioribound}, we get for $t \geq \phi^{-1}(1)$, \begin{equation} \label{derivtransp} \phi'(t) \leq K\phi(t)^{1-\alpha}.\end{equation} Dividing by $\phi(t)^{1-\alpha}$ and integrating on $[\phi^{-1}(1),x]$ we get $$ \phi(x)^{\alpha} -1 \leq K(x-\phi^{-1}(1)),$$ for any $x \geq \phi^{-1}(1)$. Hence, \begin{equation} \label{bornepsi} \phi(x)\leq Kx^{\frac{1}{\alpha}},\end{equation} for $x\geq \phi^{-1}(1)$. By \eqref{derivtransp} we deduce $$\phi'(x) \leq K x^{\frac{1}{\alpha}-1}.$$ Since $\phi'$ is continuous, at the price of taking $K$ larger, we have $$\forall x \geq 0, \ \phi'(x) \leq K \max(1, x^{\frac{1}{\alpha}-1}).$$ Let $x \geq 0$, and $y \in \RR$ such that $x +y \geq 0$. If $x,x+y\leq 1$,
$$ |\phi(x+y) - \phi(x)| \leq Ky.$$ Whereas if $x,x+y\geq 1$,
$$ |\phi(x+y) - \phi(x)| \leq K \int_x^{x+y} t^{\frac{1}{\alpha}-1} dt = \alpha K\big( (x+y)^{\frac{1}{\alpha}} - x^{\frac{1}{\alpha}}\big).$$ Now, if $ 0\leq x \leq 1\leq x+y$, \begin{align*}
|\phi(x+y) - \phi(x)|& \leq K\int_{x}^{x+y}(1+t^{\frac{1}{\alpha}-1})dt\\ & \leq K\big( y + \alpha\big( (x+y)^{\frac{1}{\alpha}}-x^{\frac{1}{\alpha}}\big). \end{align*} In conclusion, for any $x\geq 0$, $x+y\geq 0$,
\begin{equation} \label{controlpsi} |\phi(x+y) - \phi(x)|\leq K\max\Big( y,\big( (x+y)^{\frac{1}{\alpha}}-x^{\frac{1}{\alpha}}\big)\Big).\end{equation} The mean value theorem yields
$$ |(x+y)^{\frac{1}{\alpha}} - x^{\frac{1}{\alpha}} | \leq \frac{1}{\alpha} \max\big( x^{\frac{1}{\alpha}-1}, (x+y)^{\frac{1}{\alpha}-1}\big) y.$$
Using the convexity of $x \mapsto |x|^{\frac{1}{\alpha}-1}$, if $1/\alpha\geq1$, or its sub-additivity, when $1/\alpha -1 \in (0,1)$, we get
$$ |(x+y)^{\frac{1}{\alpha}} - x^{\frac{1}{\alpha}} | \leq \frac{a_{\alpha}}{\alpha} \max\big( x^{\frac{1}{\alpha}-1}, x^{\frac{1}{\alpha}-1} + y^{\frac{1}{\alpha}-1} \big) y,$$ with $a_{\alpha} = \max(1, 2^{\frac{1}{\alpha}-2})$. Together with \eqref{controlpsi}, this gives the claim. \end{proof}
As in the case $\alpha\geq 1$, we can refine the estimate of Lemma \ref{Brenieralpha} to get the following result.
\begin{Lem}\label{trans0}Let $\alpha \in (0,1)$. Let $\psi$ be the monotone rearrangement of $\nu_1$ onto $\nu_{\alpha}$. Let $\eps\in(0,1)$. Define the function $\Delta_m$ by,
$$\forall s \geq 0, \ \Delta_m(s) = \inf \big\{ |\psi^{-1}(y) - \psi^{-1}(x) | : |x| \leq m, \ |x-y|=s \big\}.$$ There is some constant $\gamma>0$, such that $$\Delta_m(s) \geq \begin{cases} \gamma^{-1} (m/\eps)^{\alpha-1} s& \text{ if } s <\frac{m}{\eps},\\
\big(1-\gamma \eps^{\alpha/2}\big)|s|^{\alpha} & \text{ if } s\geq \frac{m}{\eps}. \end{cases}$$ \end{Lem}
\begin{proof}Since $\phi$ and $\psi$ are linked by the the relation \eqref{link}, the same estimate as in Lemma \ref{Brenieralpha} holds for the Brenier map $\psi$. Therefore, we have for any $|s| \leq \psi^{-1}(m)$, and $t\in \RR$,
$$ | \psi(t) - \psi(s)| \leq K\max\big( \psi^{-1}(m)^{\frac{1}{\alpha}-1} |t-s|, |t-s|^{\frac{1}{\alpha}} \big),$$ with $K\geq 1$.
Fix $|x| \leq m$, and $y\in\RR$. We have
$$ | \psi^{-1}(y) - \psi^{-1}(x) | \geq K^{-\alpha} \min(|y-x|^{\alpha}, \psi^{-1}(m)^{1-\frac{1}{\alpha}} |y-x| ),$$ But we know from \eqref{bornepsi} that for $m\geq 1$, $ \psi^{-1}(m)\geq c_0 m^{\alpha}$, with some constant $c_0>0$. Thus, for $m\geq 1$, there is a constant $\gamma>0$, which will vary along the proof without changing name, such that
$$ | \psi^{-1}(y) - \psi^{-1}(x) | \geq \gamma^{-1} \min(|y-x|^{\alpha}, m^{\alpha-1} |y-x| ).$$
We deduce that for $|y-x| \leq m/\eps$,
$$ | \psi^{-1}(y) - \psi^{-1}(x) | \geq \gamma^{-1} \Big(\frac{m}{\eps}\Big)^{\alpha- 1} |y-x|.$$
Let $s = |y-x|$. Assume now $s \geq m/\eps$. Proceeding as in the proof of Lemma \ref{brenier1} in the case $\alpha\geq 1$, we assume first that $x,y\geq 0$. As $s\geq m \geq x$, we must have $y= x+s$. Then,
$$| \psi^{-1}(y)-\psi^{-1}(x)| \geq \psi^{-1}(s) - \psi^{-1}(m).$$
On one hand, as $\alpha<1$, we have using the sub-additivity of $u\in \RR^+ \mapsto u^{\alpha}$, $$ \int_m^{+\infty} e^{-u^{\alpha}} du = \int_0^{+\infty} e^{-(u+m)^{\alpha} } du \geq \big( \int_0^{+\infty} e^{-u^{\alpha}} du \big) e^{-m^{\alpha}}= \frac{1}{C}e^{-m^{\alpha}},$$ and on the other hand, by \eqref{maj0}, $$ \int_{s}^{+\infty} e^{-u^{\alpha}} du \leq C s^{1-\alpha} e^{-s^{\alpha}},$$ where $C$ is some constant depending on $\alpha$. Thus,
$$| \psi^{-1}(y)-\psi^{-1}(x)| \geq s^{\alpha} - m^{\alpha} -(1-\alpha )\log s -2\log C.$$ As $\log s \leq (2/\alpha) s^{\alpha/2}$ for $s\geq 1$, we deduce that
$$| \psi^{-1}(y)-\psi^{-1}(w)| \geq s^{\alpha}(1-\gamma \eps^{\alpha/2}).$$
If $x$ and $y$ have opposite signs, we can assume $x\leq 0$ and $y\geq 0$, thus $y= s -m$ and we get, \begin{align*}
| \psi^{-1}(y)-\psi^{-1}(x)|& \geq \psi^{-1}(y) \geq \psi^{-1}(s-m) \\ &\geq (s-m)^{\alpha} -(1-\alpha) \log (s-m) - \log C. \end{align*} As $s\geq m/\eps$, we deduce
$$| \psi^{-1}(y)-\psi^{-1}(x)| \geq s^{\alpha}(1-\gamma \eps^{\alpha/2}),$$
which ends the proof of the claim. \end{proof}
\subsection{A family of weights for $\nu_{\alpha}$} Using transport arguments, we will work in this section at obtaining a family of weights for $\nu_{\alpha}$ which capture its exact tail distribution.
\begin{Pro}\label{IC1}Let $\alpha > 0$, $\alpha\neq 1$, and $m\geq 1$. There exist some constants $ \kappa, \eps_0>0$ depending on $\alpha$ such that for any $\eps\in (0,\eps_0)$, $(\nu_{\alpha}, w_{\alpha,\eps }^{(m)})$ satisfies the $[-m,m]$-truncated $\tau$-property where, $$ w_{\alpha,\eps }^{(m)}(t) = \begin{cases}
\kappa^{-1} e^{-(\frac{m}{\eps})^{\alpha/2}} t^2 & \text{ if } |t| \leq m\eps^{-1},\\
(1- \kappa \eps^{(\alpha/2)\wedge 1}) |t|^{\alpha} & \text{ if } |t| > m \eps^{-1}. \end{cases}$$ \end{Pro}
\begin{proof} Let $\eps\in (0,1)$ and $m\geq 1$. Let $\delta>0$ such that \begin{equation*} \frac{1}{2} \Big( \frac{m}{\eps} \Big)^{\alpha} = \frac{2}{\delta^2}.\end{equation*} With this choice of $\delta$, we will prove that for $s\geq 0$, $$w_{\delta}(\Delta_m(s)) \geq w_{\eps,\alpha}^{(m)}(s),$$ with the appropriate constants $\kappa$ and $\eps_0$, $w_{\delta}$ defined in Corollary \ref{corexp}, and where $\Delta_m$ is as in \eqref{defdeltam}. Using the result of Lemma \ref{transp0}, this will yield the claim.
Let $\eps$ be small enough such that $w_{\delta}$ is non-decreasing. This is possible since $\delta^2 \leq 2 \eps^{\alpha}$. Let $s\geq m/\eps$. If $\eps$ is small enough, we have by Lemma \ref{brenier1} or \ref{trans0}, $$ \Delta_m(s) \geq \frac{1}{2} \Big(\frac{m}{\eps}\Big)^{\alpha} = \frac{2}{\delta^2}.$$ If $\alpha>1$, then by Lemma \ref{brenier1} we get, as $\delta^2 \leq 4\eps^{\alpha}$, $$w_{\delta}(\Delta_m(s)) \geq (1-2\delta)(1-\gamma \eps) s^{\alpha}\geq (1- \kappa \eps^{(\alpha/2)\wedge 1}) s^{\alpha},$$ for some constant $\kappa$ which will vary along the proof. Similarly, when $\alpha<1$, we get by Lemma \ref{trans0}, $$w_{\delta}(\Delta_m(s)) \geq (1-2\delta)(1-\gamma \eps^{\alpha/2}) s^{\alpha}\geq (1- \kappa \eps^{\alpha/2}) s^{\alpha}.$$ Now let $ s \leq m/\eps$. Assume $\alpha \geq 1$. By Lemma \ref{Taltransp} and the fact that $w_{\delta}$ is non-decreasing, we have $$w_{\delta}(\Delta_m(s)) \geq w_{\delta}(c s),$$ where $c$ is some positive constant. Without loss of generality, we can assume $c\leq 1/2$. Then, as $m \eps^{-1}\leq 4\delta^{-2}$, we have $cs \leq 2\delta^{-2}$, so that we get $$w_{\delta}(\Delta_m(s)) \geq \frac{c^2\delta e^{-\frac{1}{\delta}} s^2}{8}.$$ Using the fact that $\delta e^{ -\frac{1}{\delta}} \geq c_1 e^{-2/\delta}$, for some constant $c_1>0$, we get the claim in the case $\alpha>1$. Assume now $\alpha<1$. From Lemma \ref{trans0} and the fact that $w_{\delta}$ is non-decreasing, we deduce $$w_{\delta}(\Delta_m(s)) \geq w_{\delta}(\gamma^{-1} (m/\eps)^{\alpha-1} s).$$ Without loss of generality, we can assume that $\gamma \geq 2$. As $m\eps^{-1} \leq 4\delta^{-2}$ and $s\leq m/\eps$, we have $$\gamma^{-1} (m/\eps)^{\alpha-1} s \leq \frac{2}{\delta^2}.$$ Thus, $$w_{\delta}(\Delta(s)) \geq \frac{1}{8} \delta e^{-\frac{1}{\delta}} \big( \gamma^{-1} (m/\eps)^{\alpha-1} s \big)^2\geq \kappa^{-1}\delta^{a} e^{-1/\delta},$$ with some $a>0$. But, we can find some constant $c_2>0$ such that $$\delta^{a} e^{- 1/\delta} \geq c_2 e^{-2/\delta},$$ which, recalling that $(m\eps^{-1})^{\alpha} = 4\delta^{-2}$ gives the claim.
\end{proof}
We can now give a proof of Proposition \ref{tau}.
\begin{proof}[Proof of Proposition \ref{tau}]
As $(\nu_{\alpha}, w_{\alpha,\eps}^{(m)})$ satisfies the $[-m,m]$-truncated $\tau$-property for $\eps \in (0,\eps_0)$, for some $\eps_0>0$ and any $m\geq 1$ by Proposition \ref{IC1}, we deduce by the tensorization property of the $\tau$-property (see Lemma \ref{tensortau}) that $(\nu_{\alpha}^{ n}, W_{\alpha,\eps}^{(m)})$ satisfies the $mB_{\ell^{\infty}}$-truncated $\tau$-property with $W_{\alpha,\eps}^{(m)}$ defined as in \eqref{weight0}.
\end{proof}
\section{Large deviations}\label{prooftheorem} We will prove in this section Theorem \ref{theoremgene}. As sketched in the introduction, the proof will consist in looking for, in a first phase, large deviations inequalities for $\nu_{\alpha}^n$ and lower bounds estimates of the probability of translates.
As a consequence of the truncated $\tau$-property of Proposition \ref{tau}, satisfied by $\nu_{\alpha}^n$ and the weight functions $W_{\alpha, \eps}^{(m)}$, we deduce an isoperimetric-type bound for $\nu_{\alpha}^n$ with respect to the metric $|| \ ||_{\ell^{\alpha}}$ (or $|| \ ||_{\ell^{\alpha}}^{\alpha}$ in the case $\alpha<1$). This estimate will be of paramount importance to derive the upper bound of Theorem \ref{theoremgene}. \begin{Pro}\label{grandev}Let $\alpha>0$, $\alpha \neq 2$. Let $r>0$.
Let $v(n)$, $t(n)$ be two sequences going to $+\infty$ as $n$ goes to $+\infty$. Let $E$ and $F$ be Borel subsets of $\RR^n$ such that $$ F +t(n) B_{\ell^2} \subset E, \ \liminf_{n\to+\infty}\nu_{\alpha}^n( F) >0.$$ For $\alpha \neq 1$, we assume that $$(\log n)^{\alpha/2} = o(\log \frac{t(n)^2}{v(n)}),$$ whereas for $\alpha=1$, we assume $v(n) = o(t(n)^2)$. Then, \begin{equation} \label{ineqdevPro} \limsup_{n\to +\infty} \frac{1}{v(n)} \log \nu_{\alpha}^n\big( x\notin E + (rv(n))^{1/\alpha} B_{\ell^{\alpha}} \big ) \leq -r.\end{equation}
\end{Pro} \begin{Rem}
For $\alpha=2$, the Gaussian isoperimetric inequality (see \cite[Theorem 2.5]{Ledouxmono}) entails the same result without any further assumption on the speed $v(n)$ or the set $E$ than $\liminf_n \nu_2^n(E)>0$. \end{Rem} \begin{proof} Before going into the proof per say, we need to relate the enlargements by the weights $W_{\alpha,\eps}^{(m)}$, for which we know that $(\nu_{\alpha}^n, W_{\alpha,\eps}^{(m)})$ satisfies the $\tau$-property, and therefore a deviation inequality of the type \eqref{devtrunc}, to the $\ell^{\alpha}$-balls. This is the subject of the following lemma. \begin{Lem} \label{decoupfnpoids}Let $\alpha>0$. With the notation of Proposition \ref{tau}, for any $r>0$, $m\geq 1$ and $\eps \in (0,\eps_0)$, $$\big\{ W_{\alpha,\eps}^{(m)} \leq r\big(1-\kappa \eps^{(\alpha/2)\wedge 1}\big) \big\} \subset k_m(\eps)\sqrt{r} B_{\ell^2} +r^{1/\alpha} B_{\ell^{\alpha}},$$ with $k_m(\eps) = \sqrt{\kappa} e^{\frac{1}{2} (\frac{m}{\eps})^{\alpha/2} }$. Moreover, there is a function $l : \RR_+ \to\RR_+$, such that $$\{ W_{1,\eps} \leq r(1-2\eps) \} \subset l(\eps) \sqrt{r} B_{\ell^2} + rB_{\ell^1}.$$ \end{Lem} \begin{proof}We will prove only the first statement, the proof for the second one being similar. Let $y\in\RR^n$. By cutting the entries of $y$, we can find $y_1,y_2\in \RR^n$, such that $y=y_1+y_2$, for any $i\in \{1,...,n\}$, $y_1(i)y_2(i) = 0$, and
$$ |y_1(i)| \leq \frac{m}{\eps}, \quad |y_2(i)|> \frac{m}{\eps}.$$ By the very definition of $W_{\alpha, \eps}^{(m)}$,
$$ \kappa^{-1} e^{-(m/\eps)^{\alpha/2}}\sum_{i=1}^n |y_1(i)|^2 = W_{\alpha, \eps}^{(m)}(y_1) \leq W_{\alpha, \eps}^{(m)}(y),$$ and
$$(1-\kappa \eps^{(\alpha/2)\wedge 1}) ||y_2||_{\ell^{\alpha}}^{\alpha} = W_{\alpha,\eps}^{(m)}(y_2) \leq W_{\alpha,\eps}^{(m)}(y).$$ Thus, if we let $$k_m(\eps)^2 = e^{(m/\eps)^{\alpha/2}} \kappa,$$
and if $W_{\alpha,\eps}^{(m)}(y)\leq r(1-\kappa \eps^{(\alpha/2)\wedge 1})$, then $||y_1 ||_{\ell^2}\leq k_m(\eps) \sqrt{r}$, and $||y_2||_{\ell^{\alpha}}^{\alpha}\leq r$.
\end{proof} With this lemma proven, we can now give the proof of Proposition \ref{grandev}. We start with the case $\alpha=1$. As $v(n) = o(t(n)^2)$, for $n$ large enough, we have $l(\eps) \sqrt{rv(n)} \leq t(n)$. Then, by Lemma \ref{decoupfnpoids}, we have $$ F + \{W_{1,\eps} \leq r(1-2\eps) v(n)\} \subset F + t(n)B_{\ell^2} + r v(n) B_{\ell^{1}}.$$ But by assumption, $ F +t(n) B_{\ell^2} \subset E$. Thus, $$ F + \{W_{1,\eps} \leq r(1-2 \eps) v(n)\} \subset E + r v(n) B_{\ell^{1}}.$$ We deduce that, $$\nu_{1}^n( x \notin E + r v(n) B_{\ell^{1}}) \leq \nu_{1}^n\big( x \notin F + \{W_{1,\eps} \leq r(1-2\eps) v(n)\}\big).$$ As $(\nu_{1}^{ n}, W_{1,\eps})$ satisfies the $\tau$-property by Corollary \ref{corexp}, we have the following deviation inequality (see \eqref{dev}), $$\nu_{1}^n( x \notin E + r v(n)B_{\ell^{1}}) \leq \frac{1}{\nu_{1}^n(F)}e^{-r(1-2 \eps)v(n)}.$$ As $\liminf_{n} \nu_{1}^n(F)>0$, we get $$\limsup_{n\to +\infty} \frac{1}{v(n)} \log \nu_{1}^n( x \notin E +r v(n) B_{\ell^{1}} ) \leq -r(1-2 \eps).$$ Letting $\eps$ going to $0$, we get the claim.
Let now $\alpha \neq 1$.
Let $\eps \in (0,\eps_0)$ and set $m = c (\log n)^{1/\alpha}$, with some $c>0$ which is to be chosen later. By Lemma \ref{decoupfnpoids} $$ F + \{W_{\alpha,\eps}^{(m)} \leq r(1-\kappa \eps^{(\alpha/2)\wedge 1}) v(n)\} \subset F + k_n(\eps) \sqrt{rv(n)}B_{\ell^2} + (r v(n))^{1/\alpha} B_{\ell^{\alpha}}).$$ From the assumption that $(\log n)^{\alpha/2} = o(\log \frac{t(n)}{\sqrt{v(n)}})$ we deduce that for $n$ large enough, $$\frac{t(n)}{\sqrt{v(n)}}\geq e^{ (\frac{(\log n)}{\eps})^{\alpha/2}}.$$ In particular for $n$ large enough, $$\sqrt{\frac{\kappa}{r}}\frac{t(n)}{\sqrt{v(n)}}\geq e^{\frac{1}{2} (\frac{(\log n)}{\eps})^{\alpha/2}}.$$ Put in another way $$k_m(\eps) \sqrt{rv(n)} \leq t(n).$$ Thus, $$ F + \{W_{\alpha,\eps}^{(m)} \leq r(1-\kappa \eps^{(\alpha/2)\wedge 1}) v(n)\} \subset F + t(n)B_{\ell^2} + (r v(n))^{1/\alpha} B_{\ell^{\alpha}}.$$ As by assumption $F + t(n)B_{\ell^2} \subset E$, we get $$\nu_{\alpha}^n ( x \notin E + (r v(n))^{1/\alpha} B_{\ell^{\alpha}}) ) \leq \nu_{\alpha}^n( x \notin F + \{W_{\alpha,\eps}^{(m)} \leq r(1-\kappa \eps) v(n)\}).$$ As $(\nu_{\alpha}^{ n}, W_{\alpha,\eps}^{(m)})$ satisfies the $mB_{\ell^{\infty}}$-truncated $\tau$-property by Proposition \ref{tau}, we deduce the following deviation inequality (see \eqref{devtrunc}), $$\nu_{\alpha}^n \big( x \notin F + \{W_{\alpha,\eps}^{(m)} \leq r(1-\gamma \eps^{(\alpha/2)\wedge 1}) v(n)\}\big) \leq \frac{1}{\nu_{\alpha}^n(F\cap mB_{\ell^{\infty}}) }e^{-r(1-\gamma \eps^{(\alpha/2)\wedge 1 })v(n)}.$$ But,
$$ \int ||x||_{\infty} d\nu_{\alpha}^n(x)=
\int ||x||_{\infty} d\mu_{\alpha}^n(x).$$ Let $\Phi = \phi^{\otimes n}$, defined by $\Phi(x) = (\phi(x_i))_{1\leq i \leq n}$, where $\phi$ is the monotone rearrangement map sending $\mu_1$ to $\mu_{\alpha}$. Then $\Phi$ sends $\mu_1^n$ to $\mu_{\alpha}^n$, so that,
$$\int ||x||_{\infty} d\mu_{\alpha}^n(x)
= \int ||\Phi(x)||_{\infty} d\mu_1^n(x).$$ From \eqref{bornepsi}, we deduce
$$\int ||\Phi(x)||^{\alpha}_{\infty} d\mu_1^n(x)\leq K( 1+ \int ||x||_{\infty}^{1/\alpha} d\mu_1^n(x)),$$
for some constant $K>0$. But $\int ||x||_{\infty}^{1/\alpha} d\mu_1^n(x) \leq c_0 (\log n)^{1/\alpha}$, for some constant $c_0\geq 1$. Therefore,
\begin{equation} \label{momentalpha} \int ||x||_{\infty} d\mu_{\alpha}^n(x) \leq 2Kc_0(\log n)^{1/\alpha}. \end{equation} Thus by Markov's inequality, $$\nu_{\alpha}^n(x\notin mB_{\ell^{\infty}})\leq \frac{2Kc_0}{c},$$ since we chose $m = c (\log n)^{1/\alpha}$. As $\liminf_n \nu_{\alpha}^n(F)>0$ by assumption, we deduce that for $c$ large enough, $$ \liminf_{n\to +\infty} \nu_{\alpha}^n( F\cap mB_{\ell^{\infty}} )>0.$$ Therefore, $$\limsup_{n\to +\infty} \frac{1}{v(n)} \log \nu_{\alpha}^n (x \notin E + (r v(n))^{1/\alpha} B_{\ell^{\alpha}}\}) \leq -r(1-\kappa\eps^{(\alpha/2)\wedge 1}),$$ which gives the claim by taking $\eps \to 0$. \end{proof}
We show in the next proposition that we can bound from below the probability of translates under $\nu_{\alpha}^n$. \begin{Pro}\label{lowb}Let $\alpha\in (0,2]$. Let $v(n)$ be a sequence going to $+\infty$ as $n$ goes to $+\infty$. Fix some $r>0$. Let $E$ be some Borel subset of $\RR^n$ such that
$$ \liminf_{n \to +\infty} \nu_{\alpha}^n( E)>0.$$ (i). For any sequence $h_n$ of elements of $\RR^n$,
$$\liminf_{n \to +\infty}\frac{1}{v(n)} \log \nu_{\alpha}^n( E + v(n)^{1/\alpha}h_n)\geq - \limsup_{n \to +\infty} || h_n||_{\ell^{\alpha}}^{\alpha}.$$ (ii). If $\alpha \in (0,1]$, then for any sequence $h_n \in \RR_+^n$,
$$\liminf_{n \to +\infty}\frac{1}{v(n)} \log \mu_{\alpha}^n( E + v(n)^{1/\alpha}h_n)\geq - \limsup_{n \to +\infty} || h_n||_{\ell^{\alpha}}^{\alpha}.$$ \end{Pro}
\begin{Rem} On can obtain the estimate $(ii)$ when $\alpha \in (1,2]$ for the measures $\mu_{\alpha}$ with the additional assumption $n = o(v(n))$ on the speed, which is actually very restrictive in the applications we have in mind. This is one of the reasons of the limitation of Theorem \ref{theoremgenesup} to the case $\alpha\leq 1$, since we do not know how to produce a meaningful lower bound of such translated sets in this case. Similarly, when $\alpha>2$, one can see, at least for $\alpha$ integer, that the estimate $(i)$ does not hold unless $n = o(v(n))$.
\end{Rem}
\begin{proof}The proof will essentially follow the lines of \cite[Theorem 5.1]{Ledouxflour}. Indeed, in the Gaussian case $\alpha=2$, this lower bound is derived from the translation formula of the Gaussian measure. The proof for $\alpha<2$ will consist in mimicking the Gaussian case.
If the $\limsup$ in the right-hand side of $(i)$ is infinite, then the statement is trivial. If it is finite, we take some $\tau>0$, such that $|| h_n||_{\ell^{\alpha}}^{\alpha} \leq \tau$, for all $n\in \NN$. Let for any $h \in \RR^n$, $W_{\alpha}(h ) = \sum_{i=1}^n |h_i|^{\alpha}$. Then, we have, $$\nu_{\alpha}^n( E + v(n)^{1/\alpha}h_n) =\frac{1}{Z_{n}} \int_{ E} e^{-W_{\alpha}(y+v(n)^{1/\alpha}h)} d{\ell_n}(y),$$ where $\ell_n$ denotes the Lebesgue measure on $\RR^n$, and $Z_{n}$ is the normalizing factor. If $\alpha \in (0,1]$, then for any $s,t \in \RR$,
$$|s+t|^{\alpha} \leq |s|^{\alpha} +|t|^{\alpha}.$$ Thus, $$W_{\alpha}( y+v(n)^{1/\alpha} h_n ) \leq W_{\alpha}(y) + v(n)W_{\alpha}( h_n)$$ Therefore, $$\nu_{\alpha}^n ( E + v(n)^{1/\alpha}h_n) \geq e^{-v(n)W_{\alpha}(h_n) }\nu_{\alpha}^n(E),$$ which gives the claim in the case $\alpha \in (0,1)$. Note that the same argument for $\mu_{\alpha}$ instead of $\nu_{\alpha}$ gives without changes the estimate $(ii)$.
Now, if $\alpha \in (1,2]$, we have for any $s, t \in \RR$,
\begin{equation} \label{inesub1}| s+t|^{\alpha} \leq |s|^{\alpha} + \alpha \mathrm{sg}(st)|s|^{\alpha-1} |t| + |t|^{\alpha},\end{equation} where $\mathrm{sg}(st)$ stands for the sign of $st$. Thus, for any $y,h \in \RR^n$, $$ W_{\alpha}(y+v(n)^{1/\alpha}h)\leq W_{\alpha}(y)+\alpha v(n)^{1/\alpha} V(y,h) + v(n)W_{\alpha}( h),$$ where \begin{equation} \label{defPhi}V(y,h) = \sum_{i=1}^n v(y_i,h_i),\end{equation}
and $v(y,h) = \mathrm{sg}(y h)|y|^{\alpha-1} |h|$. We have, \begin{align*} \frac{1}{Z_{n}} \int_{E} e^{-W_{\alpha}(y+v(n)^{1/\alpha} h_n)} d\ell_n(y) &\geq \frac{e^{-v(n) W_{\alpha} (h_n)} }{Z_{n} }
\int_{ E} e^{-W_{\alpha} (y) -\alpha v(n)^{1/\alpha}V(y,h_n) } d{\ell_n(y)}\\ & = e^{-v(n) W_{\alpha} (h_n)} \int_{E}e^{ -\alpha v(n)^{1/\alpha} V(x,h_n)}d \nu_{\alpha}^n(x). \end{align*} Jensen's inequality yields, $$ \int_{E} e^{ -\alpha v(n)^{1/\alpha} V(x,h_n)} d \nu_{\alpha}^n(x) \geq \nu_{\alpha}^n(E)\exp\Big( -\frac{\alpha v(n)^{1/\alpha} }{\nu_{\alpha}^n(E)} \int_{E} V(x,h_n)d \nu_{\alpha}^n(x)\Big).$$ But, by Cauchy-Schwarz inequality, $$ \int_{E} V(x,h_n)d \nu_{\alpha}^n(x) \leq \nu_{\alpha}^n(E)^{1/2} \Big(\int V(x,h_n)^2 d\nu_{\alpha}^n(x) \Big )^{1/2}.$$ But $\int v(x,h)d\nu_{\alpha}(x) =0$ for any $h \in \RR$, since $v(-x,h) = -v(x,h)$ and $\nu_{\alpha}$ is symmetric. Thus,
$$\int V(x,h_n)^2 d\nu_{\alpha}^n(x) = \int |t|^{2(\alpha-1)}d\nu_{\alpha}^n(t) \big(\sum_{i=1}^n |h_n(i)|^2\big).$$ Using the fact that $\alpha \leq 2$, we get, $$\Big(\int V(x,h_n)^2 d\nu_{\alpha}^n(x) \Big )^{\frac{\alpha}{2} } \leq c^{\frac{\alpha}{2}} W_{\alpha}(h_n),$$ where $c>0$ is some constant. As $W_{\alpha}(h_n) \leq \tau$, we have $$ \int_{ E} e^{ -\alpha v(n)^{1/\alpha} V(x,h_n)} d\nu_{\alpha}^n(x) \geq \nu_{\alpha}^n( E) \exp\Big( -\frac{c^{1/2} \tau \alpha v(n)^{1/\alpha} }{\nu_{\alpha}^n( E) ^{1/2}}\Big).$$ Note that is was actually very important that we did not bound $\sg(xy)$ by $1$ in \eqref{inesub1}, so that $v(.,h)$ is of mean $0$ under $\nu_{\alpha}$, and $\int V(x,h_n)^2 d\nu_{\alpha}^n(x)$ is not too big. When one replaces $\nu_{\alpha}$ by $\mu_{\alpha}$, this is exactly where one needs to make an assumption on the speed to identify the leading term.
By assumption, we know that there is some $\eta>0$ such that for $n$ large enough, $\nu_{\alpha}^n( E) >\eta $. Thus, we get for $n$ large enough, $$\nu_{\alpha}^n( x \in E + v(n)^{1/\alpha} h_n) \geq \eta\exp\Big( -v(n) W_{\alpha}(h_n) -2 \Big(\frac{c}{\eta}\Big)^{1/2} \tau \alpha v(n)^{1/\alpha}\Big ).$$ Taking the $\liminf$ at the exponential scale $v(n)$, we get the claim.
\end{proof}
We can now give a proof of Theorem \ref{theoremgene}. We will essentially follow the proof of the LDP of Wiener chaoses (see \cite{LedouxWiener}), replacing the use of the Cameron-Martin formula by Proposition \ref{lowb}, and the Gaussian isoperimetric inequality with Proposition \ref{grandev}.
\begin{proof}[Proof of Theorem \ref{theoremgene}] Without loss of generality we can and will assume that $N=\NN$. \textbf{Property of the rate function: } By assumption $(iv)$, for any $x \in \mathcal{X}$, $$I_{\alpha}(x) = \sup_{\delta>0} \limsup_{n \to +\infty} I_{n,\delta}(x).$$ This formulation shows that $I_{\alpha}(x)< +\infty$ if and only if there is a sequence $h_n \in \RR^n$, such that $$\lim_{n \to +\infty} F_n(h_n)=x,\quad \limsup_{ n \to +\infty} W_{\alpha}(h_n) = I_{\alpha}(x).$$ Thus, $I_{\alpha}(x) \leq \tau$, for some fixed $\tau\geq 0$, if and only if $x$ is a limit point of a sequence $(F_n(h_n))_{n\in N}$ such that $\limsup_n W_{\alpha}(h_n) \leq \tau$. Therefore, $I_{\alpha}$ is lower semi-continuous. Moreover, $$ \{ I_{\alpha} \leq \tau \} \subset \overline{ \cup_{n \in \NN} F_n(2\tau B_{\ell^{\alpha}} ) }.$$ As by assumption $(iv)$ the set on the right-hand side is compact, we conclude that $I_{\alpha}$ is a good rate function.
\textbf{Lower bound: } Let $x \in \mathcal{X}$ such that $I_{\alpha}(x) < +\infty$. By assumption $(iv)$, there is a sequence $h_n \in \RR^n$ such that $$\lim_{n \to +\infty} F_n(h_n)=x,\quad \limsup_{ n \to +\infty} W_{\alpha}(h_n) = I_{\alpha}(x).$$ Let $\delta>0$. For $n$ large enough, $$\PP\big( f_n(X_n) \in B(x,2\delta) \big) \geq \PP\big( f_n(X_n) \in B(F_n(h_n), \delta) \big).$$ Let $$E = \big\{ Y \in \RR^n : d( f_n(Y + v(n)^{1/\alpha} h_n), F_n(h_n)) < \delta \big\}.$$ Note that $$\PP\big( f_n(X_n) \in B(F_n(h_n), \delta)\big) = \PP\big( X_n \in E + v(n)^{1/\alpha} h_n \big).$$ By assumption $(i)$, $\PP(X_n \in E)$ goes to $1$ as $n$ goes to $+\infty$. From Proposition \ref{lowb}, we deduce $$\liminf_{n\to+\infty} \frac{1}{v(n)} \log \PP\big( f_n(X_n) \in B(x,2\delta)\big) \geq -I_{\alpha}(x).$$
\textbf{Upper bound: }
Let $A$ be a closed subset of $\mathcal{X}$. We can assume without loss of generality that $\inf_A I_{\alpha}>0$. Let $r>0$ such that $\inf_A I_{\alpha}>r$. Put in another way, $$A\cap \{ I_{\alpha}\leq r\} = \emptyset.$$ As $I_{\alpha}$ is a good rate function, we can find a $\delta>0$ such that $$ A \cap V_{\delta}(\{I_{\alpha} \leq r\}) = \emptyset,$$ where $V_{\delta}$ denotes the $\delta$-neighborhood for the distance $d$. Thus, $$\PP\big( f_n(X_n) \in A\big) \leq \PP\big( f_n(X_n)\notin V_{\delta}(\{ I_{\alpha} \leq r \})\big).$$ Let $$U = \big\{ x \in \RR^n : f_n(x)\in V_{\delta}(\{ I_{\alpha} \leq r \}) \big\}.$$ Define, similarly as for the lower bound, the event $$ E_{\delta} = \big\{ x \in \RR^n : \sup_{ h \in r^{1/\alpha}B_{\ell^{\alpha}} } d\big( f_n(x+v(n)^{1/\alpha}h_n), F_n(h_n)\big ) < \delta \big\}.$$ By assumption $(i)$, we know that $\PP( X_n \in E_{\delta})$ goes to $1$ as $n$ goes to $+\infty$. We claim that $$ E_{\delta} + (v(n)r)^{\frac{1}{\alpha}}B_{\ell^{\alpha}} \subset U.$$ Indeed, if $h_n \in r^{1/\alpha} B_{\ell^{\alpha}}$ and $x \in E_{\delta}$, then $I_{\alpha}(F_n(h_n)) \leq r$, from the definition \eqref{deftaux} of $I_{\alpha}$, and $$ d(f_n(x + v(n)^{1/\alpha} h_n ), F_n(h_n))< \delta,$$ so that $x+v(n)^{1/\alpha} h_n \in U$. With this observation we get, $$\PP\big( f_n(X_n) \in A\big) \leq \PP\big( X_n \notin E_{\delta} + (v(n)r)^{\frac{1}{\alpha}}B_{\ell^{\alpha}} \big).$$ If $\alpha =2$, we get by the Gaussian isoperimetric inequality (see \cite[Theorem 2.5]{Ledouxmono}) for any $n$ large enough so that $\PP( X_n \in E_{\delta}) \geq 1/2$, $$ \PP\big( X_n \notin E_{\delta} +\sqrt{v(n)r}B_{\ell^{2}} \big)\leq e^{-v(n) r},$$ which gives the upper bound.
Let now $\alpha <2$, and $t = t_{\delta/4}$, where $t_{\delta/4}$ is given by assumption $(ii)$. With the notation of Theorem \ref{theoremgene} define,
$$F = E_{\delta/2}\cap \big\{ y \in \RR^n : \sup_{|| h_n||_{\ell^2} \leq t } \mathcal{L}_n(h_n) \leq \frac{\delta}{2} \big\}.$$ By Markov's inequality and assumption $(ii)$, we deduce
$$ \PP\big( \sup_{|| h_n||_{\ell^2} \leq t } \mathcal{L}_n(h_n)\leq \frac{\delta}{2}\big ) \geq \frac{1}{2}.$$ From assumption $(i)$, we deduce that $\liminf_n \PP(X_n \in F)>0$. Furthermore, we claim that \begin{equation} \label{claiminclu} F +tB_{\ell^2} \subset E_{\delta}.\end{equation} Recall that $$\mathcal{L}_n(h) = \sup_{X_n + rv(n)^{1/\alpha} B_{\ell^{\alpha} }}d\big( f_n(x+h) , f_n(x)\big).$$ Now, if $X_n \in F$ and $h \in tB_{\ell^2}$, then by definition of $\mathcal{L}_n$, for all $k \in rB_{\ell^{\alpha}}$ $$d\big(f_n(X_n + v(n)^{1/\alpha}k+h),f_n(X_n+v(n)^{1/\alpha}k)\big) \leq \frac{\delta}{2},$$ which yields \eqref{claiminclu} by triangular inequality. Thus the requirements of Lemma \ref{grandev} are met, and we get $$ \limsup_{n \to +\infty} \frac{1}{v(n)} \log \PP\big( f_n(X_n) \in A\big) \leq -r.$$ As this inequality is true for any $r < \inf_A I_{\alpha}$, we get the upper bound.
\end{proof}
We will end this section with the proof of Theorem \ref{theoremgenesup}.
\begin{proof}[Proof of Theorem \ref{theoremgenesup}] We will follow the same steps as for the proof of Theorem \ref{theoremgene}. The compactness assumption $(iii)$, and the assumption $(iv)'$ yield that $I_{\alpha}$ is a good rate function. As shown in the proof of Theorem \ref{theoremgene}, a large deviations upper bound holds with speed $v(n)$ and rate function $I_{\alpha}$, under the assumptions $(i)-(ii)-(iii)$. Thus, we only have to prove the lower bound. Let $x \in \mathcal{X}$ such that $I_{\alpha}^+(x)<+\infty$. We know that there is a sequence $h_n \in \RR_+^n$ such that $$\lim_{n\to +\infty} F_n(h_n) = x, \quad \limsup_{n\to +\infty} W_{\alpha}(h_n) = I_{\alpha}^+(x).$$ Proceeding as in the proof of Theorem \ref{theoremgene}, if $\delta>0$, then for $n$ large enough, $$\PP\big( f_n(X_n) \in B(x,2\delta) \big) \geq \PP\big( f_n(X_n) \in B(F_n(h_n), \delta) \big).$$ Let $$E = \big\{ y \in \RR^n : d( f_n(y + v(n)^{1/\alpha} h_n), F_n(h_n)) < \delta \big\}.$$ Note that $$\PP\big( f_n(X_n) \in B(F_n(h_n), \delta)\big) = \PP\big( X_n \in E + v(n)^{1/\alpha} h_n \big).$$ By assumption $(i)$, $\PP(X_n \in E)$ goes to $1$ as $n$ goes to $+\infty$. From Lemma \ref{lowb}, we deduce $$\liminf_{n\to+\infty} \frac{1}{v(n)} \log \PP\big( f_n(X_n) \in B(x,2\delta)\big) \geq -I_{\alpha}^+(x),$$ which ends the proof of the lower bound. Due to assumption $(iv)'$ the lower bound and upper bound rate functions match so that a full LDP holds. \end{proof}
\section{Concentration inequalities}\label{Chapconc} We will prove in this section the concentration inequalities of Propositions \ref{conclinearstat}, \ref{concspintro} and \ref{concvpintro} for the linear statistics, the empirical spectral measure and largest eigenvalue of Wigner matrices satisfying the concentration property $\mathcal{C}_{\alpha}$ introduced by definition \ref{defCalpha}.
\subsection{Some examples of Wigner matrices satisfying $\mathcal{C}_{\alpha}$}
Before going into the proofs, we will review some workable criterion for a Wigner matrix to satisfy the concentration property $\mathcal{C}_{\alpha}$ when $\alpha\in [1,2]$. The case of $\alpha=2$ of normal concentration has drawn most of the attention, and we refer the reader to \cite[section 8.5]{Ledouxmono}, \cite{GZconc} or also \cite[Part II]{GuionnetFlour} for a presentation of the different examples of classical models of random matrices having normal concentration.
When $\alpha\in [1,2]$ we introduce the notion of \textit{Poincaré-type inequalities} in the finite-dimensional setting. Let $d_m$ be some distance on $\RR^m$. For a smooth function $f: \RR^m \to \RR$, we define the length of the gradient of $f$ with respect to the distance $d_m$ by,
$$ \forall x \in \RR^m,\ |\nabla f(x) | = \limsup_{y\to x} \frac{|f(y)-f(x)|}{d_m(y,x)}.$$ We say that a probability measure $\mu$ satisfies a \textit{Poincaré-type inequality} on $(\RR^m, d_m)$ if there is some $\lambda>0$, such that for any smooth $f :\RR^m \to \RR$,
$$ \lambda \Var_{\mu} f \leq \int |\nabla f|^2 d\mu,$$ where the length of the gradient is taken with respect to $d_m$.
Following Gozlan \cite[Definition 1.1]{Gozlan}, we will say that a probability measure $\mu$ on $\RR^m$ satisfies $\mathbb{SP}(\omega_{\alpha}, \lambda)$ if it satisfies the Poincaré-type inequality on $(\RR^m,d_{\omega_{\alpha}})$ with spectral gap $\lambda$, where $d_{\omega_{\alpha}}$ is the distance defined in \eqref{defdalpha}.
By the results of Bobkov-Ledoux \cite[Corollary 3.2]{BobLed}, and Gozlan \cite[Proposition 1.2]{Gozlan}, we know that if a Wigner matrix $X$ has entries satisfying $\mathbb{SP}(\omega_{\alpha},\lambda)$, then it satisfies a two-level deviations inequality: for any Borel subset $A$ of $\mathcal{H}_n^{(\beta)}$ such that $\PP( X\in A)\geq 1/2$, and $r>0$, \begin{equation}\label{concpropalpha} \PP( X\notin A + \sqrt{r} B_{\ell^2}+r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} ) \leq e^{-L r},\end{equation} where $L$ only depends on $\lambda$, and by \cite[Proposition 1.2]{Gozlan}) can be taken as \begin{equation}\label{defLlambdaintro} L(\lambda) = \frac{w( \frac{\sqrt{\lambda}}{\kappa})}{16},\quad \kappa = \sqrt{18 e^{\sqrt{5}}},\end{equation}
with $w(t) = \min(|t|^2, |t|)$ for any $t\in \RR$. In particular, such a Wigner matrix has concentration $\mathcal{C}_{\alpha}$.
\begin{Rem} We note that when $\alpha>2$, the Poincaré-type inequality $\mathbb{SG}(\omega_{\alpha},\lambda)$ yields a different deviation inequality (the one above is also true for $\alpha>2$ but not sharp) where the mixed enlargement is replaced by $\sqrt{r} B_{\ell^2}\cap r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} $ (see \cite{Gozlan} for more details). \end{Rem}
A workable criterion for a probability measure on $\RR$ of the form $\mu = e^{-V} dx$ is given by Gozlan \cite[Proposition 1.2]{Gozlan} in terms of a growth condition of the potential $V$. More precisely, if \begin{equation}\label{condpot} \liminf_{x \to \pm \infty} \frac{\mathrm{sg}(x)V'(x)}{x^{\alpha-1}} >0,\end{equation} then $\mu$ satisfies $\mathbb{SG}(\omega_{\alpha},\lambda)$ on $\RR$.
We mention also that a criterion is available in higher dimension (although more intricate) in \cite[Proposition 3.5]{Gozlan}, which one may use for the complex entries of Wigner matrices.
In the case $\alpha=1$ of the classical Poincaré inequality, we know by Bobkov \cite{Bobkovlogconc} (or by Bakry, Barthe, Cattiaux, and Guillin \cite{BBCG}) that any log-concave law on $\RR^n$ satisfies a Poincaré inequality with a certain spectral gap depending on the dimension. Thus, any Wigner matrix with entries whose laws are log-concave will satisfy $\mathcal{C}_1$.
When $\alpha \in [1,2]$, the concentration property $\mathcal{C}_{\alpha}$ is equivalent (see \cite[Proposition 1.3]{Ledouxmono}) to the following deviation inequality of Lipschitz functions around their medians, which will be useful in the applications.
\begin{Lem}\label{conclip}Let $\alpha \in [1,2]$. Let $X$ be a Wigner matrices with entries satisfying $\mathcal{C}_{\alpha}$ for some $\kappa>0$. Let $f : \mathcal{H}_n^{(\beta)}\to \RR$ be a function respectively $L_2$-Lipschitz and $L_{\alpha}$-Lipschitz with respect to $|| \ ||_{\ell^2}$, and $||\ ||_{\ell^{\alpha}}$. Then, for any $t>0$, $$ \PP( f(X)>m_f +t) \leq 2\exp\Big( - \min \Big( \frac{t^2}{4\kappa^2 L_2^2}, \frac{t^{\alpha}}{2^{\alpha}\kappa^{\alpha} L_{\alpha}^{\alpha}} \Big) \Big),$$ where $m_f$ denotes the median of $f(X)$. \end{Lem}
\subsection{A deviation inequality for $\nu_{\alpha}^n$, $\alpha \in (0,1)$}\label{sectiondevineq0} In the case $\alpha\in (0,1)$, we will show that the Wigner matrices in the class $\mathcal{S}_{\alpha}$ satisfy the concentration property $\mathcal{C}_{\alpha}$. This fact will follow from the study of the concentration property of the product measures $\mu_{\alpha}^n$ and $\nu_{\alpha}^n$.
It can be shown that the probability measure $\nu_{\alpha}^n$ satisfies a weak Poincaré inequality (see \cite[Chapter 7 \S 7.5]{BGL}). The derivation of a deviations inequality from the weak Poincaré inequality has been investigated by Barthe, Cattiaux and Roberto \cite{barthe}, and yields a concentration inequality with respect to Euclidean enlargements. We will follow another path which consists, as it was the case for $\alpha\geq 1$, in transporting Talagrand's deviation inequality for the symmetric exponential law \eqref{Taldev} onto $\nu_{\alpha}$ with $\alpha <1$, using the estimate on the monotone rearrangement map proved in Lemma \ref{Brenieralpha}. We start with the one-sided probability measure $\mu_{\alpha}$.
\begin{Pro}\label{devexpo} Let $n\in \NN$, $n\geq 2$, and $\alpha \in (0,1)$. There is a constant $c>0$ depending on $\alpha$, such that for any $r>0$, $A$ Borel subset of $\RR_+^n$, and $C>0$ such that $\mu_{\alpha}^{n}(A) > 1/C$, $$\mu_{\alpha}^{n} \Big( x \notin A + C(\log n)^{\frac{1}{\alpha}-1}\big( \sqrt{r} B_{\ell^2} + r B_{\ell^1} \big) + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} \Big) \leq \frac{e^{-c r}}{\mu_{\alpha}^{n}(A) - 1/C}.$$
\end{Pro}
\begin{Rem}\label{remdev0} This deviation inequality is not optimal in the sense that it fails to capture the Gaussian fluctuations of empirical means from the central limit theorem. This is due to the $(\log n)^{1/\alpha-1}$ factor in front of the $\ell^2$-ball, which comes from the fact that the increasing rearrangement from $\mu_1$ to $\mu_{\alpha}$ is not a Lipschitz function.
But on the other hand, the $(\log n)^{\frac{1}{\alpha}-1}$ factor seems to be sharp, since it yields a non-trivial deviation inequality for $$ (\log n)^{\frac{1}{\alpha}-1}\big(\max_{1 \leq i \leq n} x_i - m \big),$$ where $m$ is the median of the maximum function under $\mu_{\alpha}^n$. But from the extreme value theory (see \cite[Theorem 1.6.2, Corollary 1.6.3]{Leadbetter}), $$a_n\big(\max_{1 \leq i \leq n} x_i -b_n\big),$$ converges in law to the Gumbel distribution $G$, where $$ a_n \sim c_1 (\log n)^{\frac{1}{\alpha}-1}, \text{ and } b_n \sim c_2(\log n)^{\frac{1}{\alpha}},$$ for some constant $c_1, c_2$. Moreover, as the Gumbel distribution has a right-tail behaving like $e^{-t}$, we see that the $B_{\ell^1}$ part in the enlargement of the deviations inequality of Proposition \ref{devexpo} is justified. \end{Rem}
\begin{proof}[Proof of Proposition \ref{devexpo}] Let $\Phi = \phi^{\otimes n} : \RR^n \to \RR^n$, defined by $\Phi(x) = (\phi(x_i))_{1\leq i \leq n}$, which sends $\mu_1^{n}$ to $\mu_{\alpha}^{ n}$. Let $r>0$, and $A$ be a measurable subset of $\RR_+^n$ such that $\mu_1^{n}(A)>0$. In a first step, we will use Lemma \ref{Brenieralpha} to see how the map $\Phi$ transform the set $A + \sqrt{r} B_{\ell^2} + rB_{\ell^1}$. Actually, to transport the deviation inequality of $\mu_1^{n}$ it is sufficient to understand how $\Phi$ deforms $A' + \sqrt{r} B_{\ell^2} + rB_{\ell^1}$ for a well-chosen subset $A'$ of $A$ such that $\mu_1^{ n}(A')>0$. To this end, define
$$ B = \{ x \in \RR^n : || x||_{\infty} \leq C \log n\}, \ A' = A\cap B,$$ where $C$ is some constant which will be chosen later. Let $x\in A'$, $y\in B_{\ell^2}$, and $z \in B_{\ell^1}$. By Lemma \ref{Brenieralpha}, we have
$$| \Phi(x+\sqrt{r} y)-\Phi(x)| \leq K \big( \sqrt{r} |y|+|x|^{\frac{1}{\alpha}-1}\sqrt{r}|y| +|\sqrt{r} y|^{\frac{1}{\alpha}} \big),$$ where the inequality has to be understood coordinate-wise, the functions being applied coordinate by coordinate to the vectors in $\RR^n$, and where $K$ is a constant depending on $\alpha$ which will vary in the rest of the proof without changing name. Thus, $$\Phi(x+\sqrt{r} y)-\Phi(x) \in K \Big(\sqrt{r} B_ {\ell^2} + (C \log n)^{\frac{1}{\alpha}-1} \sqrt{r} B_{\ell^2} + r^{\frac{1}{2\alpha}} B_{\ell^{2\alpha}}\Big).$$ For $C \log n\geq 1$, we have $$\Phi(x+\sqrt{r} y)-\Phi(x) \in K \Big( (C \log n)^{\frac{1}{\alpha}-1} \sqrt{r} B_{\ell^2} + r^{\frac{1}{2\alpha}} B_{\ell^{2\alpha}}\Big).$$ Once again by Lemma \ref{Brenieralpha}, we get
$$| \Phi(x+\sqrt{r} y +r z)-\Phi(x+\sqrt{r}y)| \leq K\big( |rz| + |x+\sqrt{r}y|^{\frac{1}{\alpha}-1}|rz| + |rz|^{\frac{1}{\alpha}} \big),$$
where again this inequality is valid coordinate-wise. Using the convexity of the power function $t \mapsto |t|^{\frac{1}{\alpha}-1}$, or its sub-additivity, we get
$$| \Phi(x+\sqrt{r} y +r z)-\Phi(x+\sqrt{r}y)| \leq K\big( |rz|+(|x|^{\frac{1}{\alpha}-1}+ |\sqrt{r}y|^{\frac{1}{\alpha}-1})|rz| + |rz|^{\frac{1}{\alpha}} \big).$$ Note that Hölder's inequality implies
$$ |y|^{\frac{1}{\alpha}-1} |z| \in B_{\ell^{\gamma}},$$ with $\frac{1}{\gamma} = \frac{1}{2}(\frac{1}{\alpha}+1)$. Thus, $$ \Phi(x+\sqrt{r} y +r z)-\Phi(x+\sqrt{r}y) \in K\big((C\log n)^{\frac{1}{\alpha}-1} rB_{\ell^1} +r^{\frac{1}{\gamma}}B_{\ell^{\gamma}} + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} \big).$$ Therefore, $$ \Phi(x+\sqrt{r} y +r z) \in A +K\big((C\log n)^{\frac{1}{\alpha}-1}(\sqrt{r} B_{\ell^2}+ rB_{\ell^1}) +r^{\frac{1}{\gamma}}B_{\ell^{\gamma}} + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} + r^{\frac{1}{2\alpha}} B_{\ell^{2\alpha}} \big).$$ We now simplify the enlargement on the right-hand side. Observe that for any $0< a\leq b \leq c$, $$ r^{1/b} B_{\ell^b} \subset r^{1/a} B_{\ell^a} + r^{1/c} B_{\ell^c}.$$ Indeed, if $x \in r^{1/b} B_{\ell^b}$, then
$$\sum_{ |x_i|\geq 1} |x_i|^{a} \leq \sum_{ |x_i|\geq 1} |x_i|^{b}\leq r,$$ and
$$\sum_{ |x_i|\leq 1} |x_i|^{c} \leq \sum_{ |x_i|\leq 1} |x_i|^{b}\leq r.$$
Thus, $x = x\Car_{x\geq 1} + x\Car_{x< 1}$, with $x\Car_{|x|\geq 1} \in r^{1/a} B_{\ell^a}$ and $x\Car_{|x|< 1} \in r^{1/c} B_{\ell^c}$.
Therefore, as $\alpha \leq 2\alpha \leq 2$, $\alpha \leq \gamma \leq 2\alpha$, and $C \log n\geq 1$, $$ \Phi(x+\sqrt{r} y +r z) \in A +K\big((C\log n)^{\frac{1}{\alpha}-1}(\sqrt{r} B_{\ell^2}+ rB_{\ell^1}) + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} \big).$$ Thus, \begin{equation} \label{transpenlar}
\Phi\big(A+\sqrt{r} B_{\ell^2} +r B_{\ell^1}\big) \subset A +K\big((C\log n)^{\frac{1}{\alpha}-1}(\sqrt{r} B_{\ell^2}+ rB_{\ell^1}) + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} \big). \end{equation} Applying the deviation inequality \eqref{Taldev} of $\mu_1^{n}$, we get $$ \mu_1^{n} \big( x \notin A' + \sqrt{r}B_{\ell^2} + rB_{\ell^1} \big) \leq \frac{e^{-Lr}}{ \mu_1^{n}(A')},$$
where $L>0$ is some constant independent of $n$. But, since $$\int ||x||_{\infty} d\mu_1^{n}(x) \leq c_0 \log n,$$ for some numerical constant $c_0>0$, we have by Markov's inequality $$\mu_1^{n}(A') \geq \mu_1^{n}(A) - \mu_1^{ n}(B^c)\geq \mu_1^{ n}(A) - \frac{c_0}{C}.$$ Thus, $$ \mu_1^{n} \big( x \notin A' + \sqrt{r}B_{\ell^2} + rB_{\ell^1} \big) \leq \frac{e^{-cr}}{ \mu_1^{n}(A)-c_0/C}.$$ But, as $\mu_{\alpha}^{n} = \mu_1^{n} \circ \Phi^{-1}$, and $\Phi$ is a bijection, \begin{align*} \mu_1^{n} \big( x \notin A' + \sqrt{r}B_{\ell^2} + rB_{\ell^1} \big) & = \mu_{\alpha}^{n} \big( \Phi(\RR_+^n\setminus ( A' + \sqrt{r}B_{\ell^2} + rB_{\ell^1}) \big)\\ &= \mu_{\alpha}^{n} \big( \RR_+^n\setminus \Phi( A' + \sqrt{r}B_{\ell^2} + rB_{\ell^1}) \big). \end{align*} Using \eqref{transpenlar}, we deduce $$\mu_{\alpha}^{n} \big( x \notin A +K\big((C\log n)^{\frac{1}{\alpha}-1}(\sqrt{r} B_{\ell^2}+ rB_{\ell^1}) + r^{\frac{1}{\alpha}} B_{\ell^{\alpha}} \big)\big) \leq \frac{e^{-cr}}{ \mu_1^{n}(A)-c_0/C}.$$ Adjusting the constant $c$ we get the claim. \end{proof}
As observed in remark \ref{comprearrmap}, the monotone rearrangement $\psi$ of $\nu_1$ onto $\nu_{\alpha}$, satisfies the same estimate of Lemma \ref{Brenieralpha} as $\phi$. Therefore, the same arguments as for the proof of Proposition \ref{devexpo} can be carried out, and yield a similar deviation inequality for $\nu_{\alpha}^n$ which we stated in Proposition \ref{devnualpha0}.
In view of this deviation inequality for $\nu_{\alpha}^n$, we see that a Wigner matrix in the class $\mathcal{S}_{\alpha}$ when $\alpha\in (0,1)$ satisfies the concentration property $\mathcal{C}_{\alpha}$.
As for the case where $\alpha \in [1,2]$, the concentration property $\mathcal{C}_{\alpha}$ can be translated into a deviation inequality for Lipschitz or Hölder functions when $\alpha \in (0,1)$, as stated in the following lemma.
\begin{Lem}\label{conclip0}
Let $\alpha \in (0,1)$. Assume $X$ satisfies the concentration property $\mathcal{C}_{\alpha}$ for some $\kappa>0$. Let $f : \mathcal{H}_n^{(\beta)} \to \RR$ be a function respectively $L_1$-Lipschitz and $L_2$-Lipschitz with respect to $||\ ||_{\ell^{1}}$, and $|| \ ||_{\ell^2}$. There is a constant $c>0$ depending on $\alpha$, such that if $f$ is moreover $L_{\alpha}$-Lipschitz with respect to $||\ ||_{\ell^{\alpha}}^{\alpha}$, then for any $t>0$, $$ \PP\big( f(X) > m_f +t \big) \leq 4 \exp\Big( -c \min\Big( \frac{t^2}{\kappa^2(\log n)^{2(\frac{1}{\alpha}-1)}L_2}, \frac{t}{\kappa(\log n)^{\frac{1}{\alpha}-1}L_1 +\kappa L_{\alpha}} \Big) \Big),$$ whereas if
$$\forall A,B \in \mathcal{H}_n^{(\beta)}, \ f(A)- f(B) \leq L_{\alpha}'||A-B||_{\ell^{\alpha}},$$ for some $L_{\alpha}'>0$, then for any $t>0$, $$ \PP\big( f(X) > m_f +t \big) \leq 4 \exp\Big( -c \min\Big( \frac{t^2}{\kappa^2(\log n)^{2(\frac{1}{\alpha}-1)}L_2}, \frac{t}{\kappa(\log n)^{\frac{1}{\alpha}-1}L_1 }, \frac{t^{\alpha}}{\kappa^{\alpha} L_{\alpha}'^{\alpha}} \Big) \Big),$$ where $m_f$ is the median of $f(X)$. \end{Lem}
\subsection{Concentration inequalities for the largest eigenvalue}
We will prove in this section Proposition \ref{concvpintro}. We will see that it will fall easily form Weyl's inequality \cite[Theorem III.2.1]{Bhatia}, as it enables one to compute the Lipschitz constants of the largest eigenvalue function with respect to the distances $|| \ ||_{\ell^p}$ when $p \in [1,2]$ and $|| \ ||_{\ell^p}^p$ when $p\in (0,1)$ on $\mathcal{H}_n^{(\beta)}$.
\begin{proof}[Proof of Proposition \ref{concvpintro}]\label{proofconcvp}Let $\alpha \in (0,2]$. Let $X$ be a Wigner matrix satisfying the concentration property $\mathcal{C}_{\alpha}$ for some $\kappa>0$. By Weyl's inequality \cite[Theorem III.2.1]{Bhatia}, the function $$f :Y\in \mathcal{H}_n^{(\beta)} \mapsto \lambda_{Y/\sqrt{n}}$$ is $n^{-1/2}$-Lipschitz with respect to the $p$-Schatten (pseudo-)norm $|| \ ||_{p}$ for any $p>0$, which is defined by
\begin{equation} \label{normSchatten} \forall A \in \mathcal{H}_n^{(\beta)}, \ ||A||_p = \big( \tr |A|^p\big)^{1/p}.\end{equation}
Let $m_f$ denote the median of $f(X)$, and $t>0$. As $\alpha \leq 2$, we have $ || \ ||_{\alpha}\leq || \ ||_{\ell^{\alpha}}$ by \cite[Theorem 3.32]{Zhan}. Thus, $f$ is also $n^{-1/2}$-Lipschitz with respect to $|| \ ||_{\ell^{\alpha}}$. Applying Lemmas \ref{conclip} and \ref{conclip0} successively to $f$ and $-f$, we deduce that for any $t>0$,
\begin{equation} \label{devmedianvp}\PP( |f - m_f|> t) \leq 8\exp\big(-c_{\alpha} h_{\alpha}(t) \big),\end{equation} with $h_{\alpha}$ defined in Proposition \ref{concvpintro}, and where $c_{\alpha}$ is some constant depending on $\alpha$. Integrating the above inequality \eqref{devmedianvp}, we get
\begin{equation} \label{compmed} |\EE f(X) - m_f| =O(\kappa n^{-1/2} (\log n)^{\frac{1}{\alpha}-1}),\end{equation} if $\alpha\in (0,1)$, and
$$|\EE f(X) - m_f| =O( \kappa n^{-1/2}),$$ if $\alpha \in [1,2]$, which gives the claim.
\end{proof}
\subsection{Two lemmas on spectral variation of Hermitian matrices}\label{spvarsection}
In view of Lemmas \ref{conclip} and \ref{conclip0}, proving the concentration inequalities of Propositions \ref{conclinearstat} and \ref{concspintro} require to compute the Lipschitz constants of the empirical spectral measure of Hermitian matrices, with respect to $|| \ ||_{\ell^p}$ when $p\in [1,2]$, and $|| \ ||_{\ell^p}^p$ when $p\in (0,1)$, and a well-chosen distance on $\mathcal{P}(\RR)$.
We will prove and discuss in this subsection Lemmas \ref{spvar1intro} and \ref{spvar0intro}. For $p>0$, we denote by $\mathcal{W}_p$ the $L^p$-Wasserstein distance, defined for any probability measures $\mu$, $\nu$ on $\RR$ with finite $p^{\text{th}}$-moments by,
$$\mathcal{W}_p(\mu,\nu) = \Big(\inf_{\pi} \int |x-y|^p d\pi(x,y)\Big)^{1/p},$$
if $p\geq 1$ and by,
$$\mathcal{W}_p(\mu,\nu) =\inf_{\pi} \int |x-y|^p d\pi(x,y),$$
if $p\in (0,1)$, where the infimum is taken on all coupling $\pi$ between $\mu$ and $\nu$.
We begin with the proof of Lemma \ref{spvar1intro}.
\begin{proof}[Proof of Lemma \ref{spvar1intro}]
By Lidskii's theorem (see \cite[Corollary III 4.2]{Bhatia}), we have $$ \lambda^{\downarrow}(A)-\lambda^{\downarrow}(B)\prec \lambda^{\downarrow}(A-B),$$
where $ \lambda^{\downarrow}(A)$ denotes the vector of eigenvalues of $A$ in decreasing order, and $\prec$ the majorisation relation between vectors of $\RR^n$ (see \cite[Chapter II]{Bhatia} for a proper definition). Thus, by \cite[Theorem II.3.1]{Bhatia} we get, since $x \mapsto |x|^p$ is convex as $p\geq 1$,
$$ \tr | \lambda^{\downarrow}(A)-\lambda^{\downarrow}(B)|^p \leq \tr|\lambda^{\downarrow}(A-B)|^p.$$ Using the decreasing coupling between the spectra of $A$ and $B$, we get
\begin{equation} \label{Hoeffmanalphagene}\mathcal{W}_{p}( \mu_{A}, \mu_{B}) \leq \frac{1}{n^{1/p }} || A-B||_{p},\end{equation}
where $|| \ ||_p$ denotes the $p$-Schatten norm, defined in \eqref{normSchatten}. But as $ p\leq 2$, we have by \cite[Theorem 3.32]{Zhan},
\begin{equation} \label{compnrom} || A-B||_{p} \leq || A-B||_{\ell^p},\end{equation} which ends the proof of the first inequality of Lemma \ref{spvar1intro}.
As a consequence of the Kantorovitch-Rubinstein duality (see \cite[Particular case 5.16]{Villani}), we have $$d \leq \mathcal{W}_1,$$ where $d$ is as in \eqref{defdStiel}. Besides, Jensen's inequality yields for any $p\geq 1$, $$ \mathcal{W}_{1} \leq \mathcal{W}_p,$$ Therefore, \begin{equation} \label{compdis} d \leq \mathcal{W}_1 \leq \mathcal{W}_p,\end{equation} which gives the second claim of the lemma. \end{proof}
\begin{Rem}\label{spvarrm} When $p>2$, the inequality for $A,B \in \mathcal{H}_n^{(\beta)}$,
$$\mathcal{W}_p(\mu_A,\mu_B) \leq \frac{1}{n^{1/p}} || A-B||_{\ell^p},$$
is no longer true, since for $B=0$ it amounts to \eqref{compnrom}, which is false when $p>2$, by taking $A=u u^*$, where $u$ is the constant vector.
When $p<1$, one may hope for the inequality
\begin{equation} \label{hope} \mathcal{W}_{p}(\mu_A,\mu_B)\leq \frac{1}{n} || A-B||_{\ell^p}^p,\end{equation} to hold. But taking formally $p \to 0$, would yield
\begin{equation} \label{hope2} | \lambda(A) \Delta \lambda(B) | \leq | (i,j) : A_{i,j} \neq B_{i,j}|,\end{equation} where $\lambda(A),\lambda(B)$ denote the set of eigenvalues of $A$ and $B$. But one can see that changing $1$ entry to a matrix can change the whole spectrum, which disproves \eqref{hope2}.
\end{Rem}
The moral of remark \ref{spvarrm} is that one cannot have \eqref{hope} with a constant $1$ on the right-hand side. As the cost function $| \ |^p$ behaves quite badly when $p<1$ as it is not convex (see \cite{McCann} for this transportation problem with concave costs), in particular, the optimal transport map is not necessarily the monotone rearrangement contrary to the case $p\geq 1$, we will not investigate further the question of having a spectral variation inequality involving the $L^p$-Wasserstein distance. We prefer to deal with another distance on $\mathcal{P}_p(\RR)$, the set probability measures on $\RR$ with finite $p^{\text{th}}$ moments, which induces the same topology as $\mathcal{W}_p$ and dominates $d$. This distance is chosen so that, applied to empirical spectral measures, it will be controlled by $|| \ ||_{\ell^p}^p$ in the case where $p \in (0,1)$.
To this end, let $p\in (0,1)$ and define for any $\mu, \nu \in \mathcal{P}_p(\RR)$,
\begin{equation} \label{distdp} d_{p}(\mu,\nu) = \sup_{t\in \RR }\Big| \int (t-x)^{p}_+ d\mu(x) - \int (t-x)^{p}_+ d\nu(x) \Big|.\end{equation} Taking formally $p$ to $0$, we retrieve the Kolmogorov-Smirnov distance $d_{KS}$. Recall that by integrating by parts, we can write
$$d_{KS}(\mu,\nu) = \sup \big\{ \big| \int f d\mu - \int f d \nu\big| : f \in \text{NBV},\ || f||_{BV} \leq 1\big\} ,$$ where $\text{NBV}$ denotes the set of normalized functions with bounded variations, that is, functions which are the integrals of finite signed measures, and
$$ || f||_{BV} = | \sigma|(\RR),$$
whenever $f$ is the distribution function of the finite signed measure $\sigma$, and $|\sigma|$ is the total variation of $\sigma$.
We can actually have a similar formulation for $d_p$, by introducing the fractional integrals of order $p+1$ on $\mathcal{M}_s^{p}$, the set of finite signed measures $\sigma$ such that $|\sigma|$ has a finite $p^{\text{th}}$-moment, which we defined in \eqref{defintfrac}.
We recall that fractional integrals enjoy the following integration by parts formula (see \cite[(5.16)]{Samko}): for $\mu,\nu \in \mathcal{M}_s^{p}$, \begin{equation} \label{IPP} \int (\mathcal{I}_+^{p+1} \mu)(t) d\nu(t) = \int (\mathcal{I}_-^{p+1}\nu)(x) d\mu(x).\end{equation} Thus, we can write \begin{align}
d_{p}(\mu,\nu) &= \Gamma(p+1) \sup_{t\in \RR }\big| (\mathcal{I}_+^{p+1} \mu)(t) - (\mathcal{I}_+^{p+1} \mu)(t) \big| \nonumber\\
&= \Gamma(p+1)\sup_{\sigma} \big| \int (\mathcal{I}_-^{p+1}\sigma) d\mu - \int (\mathcal{I}_-^{p+1}\sigma) d\nu \big|,\label{formulation} \end{align}
where the supremum is taken on all $\sigma \in \mathcal{M}_s^p$, such that $|\sigma|(\RR) \leq 1$. The inequality $d_p \geq$ \eqref{formulation} is the consequence of the integration by parts formula \eqref{IPP}, whereas the equality is given by taking $\sigma = \delta_t$, for $t\in \RR$. We investigate now the link between the distances $d$, defined in \eqref{defdStiel}, $\mathcal{W}_p$ and $d_p$ when $p \in (0,1)$. \begin{Pro}\label{compdist} Let $p\in (0,1)$. Then, $d_{p}$, defined in \eqref{distdp}, is a distance on $\mathcal{P}_p(\RR)$, and metrizes the weak topology. More precisely, there is a constant $C_p>0$ such that \begin{equation} \label{compdistdp} d(\mu,\nu) \leq C_{p} d_{p} (\mu,\nu),\end{equation} for all $\mu,\nu \in \mathcal{P}_p(\RR)$. One can choose \begin{equation} \label{defCp} C_p = \sqrt{\pi}(p+1) \frac{\Gamma\big( \frac{p+1}{2}\big)}{\Gamma\big( 1+\frac{p}{2}\big)}.\end{equation}
Furthermore, \begin{equation} \label{compdistWp} d_p \leq \mathcal{W}_p.\end{equation}
\end{Pro}
\begin{Rem} We actually do not know if the distances $d_{p}$ and $\mathcal{W}_{p}$ are comparable, meaning that the reversed inequality $d_p \geq K_p \mathcal{W}_{p}$ is true for some $K_p>0$. We do know however, by the remark \ref{spvarrm}, that such an inequality cannot hold with some constant $K_p$ staying bounded when $p\to 0$.
\end{Rem}
\begin{proof} In view of the formulation of $d_p$ as \eqref{formulation}, the stake behind \eqref{compdistdp} is to represent the function $t \mapsto (z-t)^{-1}$ as the fractional integral of order $p+1$ of some function. The constant $C_p$ will arise as a bound on the $L^1$ norm of this function as $\Im z \geq 1$, over $\Gamma(p+1)$.
The fractional integral of order $p+1$ of the function $t \mapsto (z-t)^{-1}$ is given in \cite{Samko}, which we state in the next lemma. \begin{Lem}[{\cite[Chapter 2 (5.25)]{Samko}}]\label{stiel}Let $p\in (0,1)$. For any $z \in \CC$, $\Im z>0$, we have $$\forall x \in \RR, \ \frac{1}{z-x} = \mathcal{I}_-^{p+1}(h)(x),$$ with \begin{equation} \label{defphi} \forall t \in \RR, \ h(t) = e^{i \pi (p+1)} \Gamma(p+2) \frac{1}{(z-t)^{p+2} },\end{equation} where $\zeta^{p}$ is the principal branch of the $\alpha^{\text{th}}$-root on $\CC\setminus \RR_-$. \end{Lem}
Let $\Im z \geq 1$ and $h$ as in \eqref{defphi}. We have
\begin{equation*} \frac{1}{\Gamma(p+1)} ||h ||_1\leq (p+1) \int _{-\infty}^{+\infty}\frac{dt}{(1+ t^2)^{1+p/2}} = 2(p+1) \int_0^{+\infty}\frac{dt}{(1+ t^2)^{1+p/2}} := C_p,\end{equation*} where we used $\Gamma(p+2) = (p+1)\Gamma(p+1)$. Therefore, $$ d \leq C_{p} d_{p}.$$ But, one can recognize an Euler integral of the first kind in the definition of $C_p$, by making successively the changes of variables $t = \tan u$, and $v = (\cos u)^2$, which yields, $$C_p = (p+1) \int_0^{1} v^{\frac{p-1}{2}} (1-v)^{-\frac{1}{2}} dv.$$ Therefore by \cite[(2.13)]{Artin}, we deduce the value for $C_p$ claimed in \eqref{defCp}.
Inequality \eqref{compdistWp} is the consequence of the sub-additivity of the function $x \mapsto x^p$ on $\RR^+$. More precisely, for any $x,y,t \in \RR$,
$$(t-x)_+^p - (t-y)_+^p \leq |x-y|^p.$$ Integrating the above inequality under a coupling $P$ of two probability measures with finite $p^{\text{th}}$-moment yields the claim.
From \eqref{compdistdp}, we deduce that the topology induced by $d_p$ on $\mathcal{P}_p(\RR)$ is finer than the weak topology, and by \eqref{compdistWp} that it is coarser than the one induced by $\mathcal{W}_p$. But $\mathcal{W}_p$ induces the weak topology on $\mathcal{P}_p(\RR)$ by \cite[Theorem 6.9]{Villani} (as $| \ |^p$ is a metric on $\RR$ for $p\leq1$), therefore $d_p$ induces the weak topology on this set. \end{proof}
We finally prove that the distance $d_p$ we introduced, when applied to spectral measures of Hermitian matrices, is dominated by $|| \ ||_{\ell^p}^p$ for $p\in (0,1)$, this will directly imply the result of Lemma \ref{spvar0intro}.
\begin{Lem}\label{spvar0}Let $p\in (0,1)$. Let $A,B \in \mathcal{H}_n^{(\beta)}$.
\begin{equation} \label{ineqRot}d_{p}(\mu_{A},\mu_B) \leq \frac{1}{n} || A-B||_{\ell^p}^p,\end{equation} where $d_p$ is defined in \eqref{distdp}. In particular,
\begin{equation} \label{controlStielp} d(\mu_{A},\mu_B) \leq \frac{C_p}{n} ||A-B||_{\ell^p}^p,\end{equation} where $C_p$ is as in \eqref{defCp}.
\end{Lem} \begin{Rem}\label{autredis} Defining the distance
$$d_p^{-}(\mu,\nu) = \sup_{t \in \RR} \big| \int (t-x)_-^{p} d\mu(x) - \int (t-x)_-^{p} d\nu(x)\big|,$$ for any $\mu,\nu \in \mathcal{P}_p(\RR)$, we see that we have a similar representation as for $d_p$, that is,
$$d_p^-(\mu,\nu) = \sup_{\sigma} \big| \int \mathcal{I}_+^{p+1}(\sigma) d\mu - \int \mathcal{I}_+^{p+1}(\sigma) d\nu \big|,$$
where $\sigma$ run in $\mathcal{M}_s^p$ such that $|\sigma|(\RR)\leq 1$. Moreover, we clearly get the same inequality as \eqref{ineqRot} for $d_p$. \end{Rem}
\begin{proof} As $\alpha \leq 2$, the second inequality of \eqref{ineqRot} is due to \cite[Theorem 3.32]{Zhan}. To prove the first inequality, we begin by recalling an inequality due to Rotfel'd originally, and then to Thompson \cite{Thompson} (for an extension and a simpler proof). Let $F: \RR_+^{2n} \to \RR$ be a concave symmetric function. Then for any $A,B \in \mathcal{H}_n^{(\beta)}$ positive semi-definite, $$ F(\lambda(A+B), 0) \leq F(\lambda(A),\lambda(B)),$$ where $\lambda(C)$ denotes the vector of eigenvalues of a Hermitian matrix $C$. Note that since $F$ is symmetric, there is no ambiguity in the writing. Let $t\in \RR$. We have,
$$t-A-B \leq (t-A)_+ + |B|.$$ In particular, if we denote $\lambda_1(C) \geq\lambda_2(C)\geq ... \geq \lambda_n(C)$ the eigenvalues of some Hermitian matrix $C$, then by Weyl's inequality \cite[Theorem III.2.1]{Bhatia}, for any $i\in \{1,...,n\}$,
$$\lambda_i(t-A-B) \leq \lambda_i\big((t-A)_+ + |B|\big).$$ Therefore,
$$\lambda_i(t-A-B)_+ \leq \lambda_i\big((t-A)_+ + |B|\big).$$ Define $$\forall x \in \RR_+^{2n}, \ F(x) = \sum_{i=1}^{2n} x_i^{\alpha}.$$ Since $A,B$ are Hermitian, $$ \lambda (t-A-B)_+ =(t-\lambda(A+B))_+.$$ As $F$ is non-decreasing coordinate-wise,
$$F\big( (t-\lambda(A+B))_+,0\big) \leq F\big( \lambda((t-A)_+ + |B| ),0\big).$$ Rotfel'd inequality gives
$$F\big( \lambda\big((t-A)_+ + |B| \big),0\big) \leq F\big( \big(t-\lambda(A)\big)_+, |\lambda(B)| \big).$$ Thus,
$$\sum_{i=1}^n \big(t-\lambda_i(A+B)\big)_+^{\alpha} \leq \sum_{i=1}^n \big(t-\lambda_i(A)\big)_+^{\alpha} + \sum_{i=1}^n |\lambda_i(B)|^{\alpha}.$$ Applying this inequality with $A+B$, $-B$ instead of $A$ and $B$, we get the first claim. The inequality \eqref{ineqRot} is a just reformulation of the above inequality and a use of the comparison \eqref{compnrom} between $\ell^p$-(quasi)-norm and $p$-Schatten (quasi)-norm. Finally, using Proposition \ref{compdist}, we deduce that \eqref{controlStielp} is true. \end{proof}
With the Lemmas \ref{spvar0} and \ref{spvar1intro}, we can now give a proof of Propositions \ref{conclinearstat} and \ref{concspintro}.
\begin{proof}[Proof of Proposition \ref{conclinearstat}] Let $\alpha \in (0,2]$ and $X$ to be a Wigner matrix satisfying the concentration property $\mathcal{C}_{\alpha}$ with some $\kappa>0$. Lemma \ref{spvar1intro} and Hölder's inequality allow us to say that if $f : \RR \to \RR$ is $1$-Lipschitz, then the function \begin{equation} \label{linearstat}Y \in \mathcal{H}_n^{(\beta)} \mapsto \frac{1}{n} \sum_{i=1}^n f(\lambda_i(Y/\sqrt{n})),\end{equation}
where $\lambda_1(Y),...,\lambda_n(Y)$ denote the eigenvalues of $Y$, is $n^{-\frac{1}{2}-\frac{1}{p}}$-Lipschitz with respect to $|| \ ||_{\ell^p}$ for any $p\in [1,2]$. Thus, using Lemma \ref{conclip}, we deduce the concentration inequality for the linear statistics of Lipschitz functions of Proposition \ref{conclinearstat} in the case $\alpha \in [1,2]$.
Assume now that $\alpha \in (0,1)$ and $f$ is $1$-Lipschitz and moreover can be written $f = \mathcal{I}_{\pm}^{1+\alpha}(\sigma)$ for some $\sigma \in \mathcal{M}_s^{\alpha}$ such that $|\sigma|(\RR)\leq m$, then by Lemma \ref{spvar0} (and remark \ref{autredis}), we know that the map \eqref{linearstat} is $\Gamma(\alpha+1)^{-1}n^{-1-\frac{\alpha}{2}}m$-Lipschitz with respect to $|| \ ||_{\ell^{\alpha}}^{\alpha}$. Thus we can deduce from Lemma \ref{conclip0} the second concentration inequality of Proposition \ref{conclinearstat}.
\end{proof}
We prove now Proposition \ref{concspintro}. \begin{proof}[Proof of Proposition \ref{concspintro}] Fix some $z \in \mathcal{K}$. Let $f_z$ denote the function on $\mathcal{H}_n^{(\beta)}$ defined by, $$ \forall Y \in \mathcal{H}_n^{(\beta)}, \ f_z(Y) = g_{ \mu_{Y/\sqrt{n}}}(z).$$
As $\Im z\geq 1$, we see that the function $x\mapsto (z-x)^{-1}$ is $1$-Lipschitz. Moreover, we know by Lemma \ref{stiel} that when $\alpha \in (0,1)$, $$ \frac{1}{z-x} =\mathcal{I}_-^{\alpha+1}(h),$$
with $||h||_{\ell^1} \leq \Gamma(\alpha+1)C_{\alpha}$, where $C_{\alpha}$ is as in \eqref{defCp}.
Let $m_z$ be the median of $f_z(X)$. Let also $r>0$. We deduce by Proposition \ref{conclinearstat}, and using remark \ref{concsimpl0} in the case $\alpha \in (0,1)$, that there is a constant $c_{\alpha}$ depending on $\alpha$ such that,
\begin{equation} \label{concspect}\PP( |f_z - m_z| > t) \leq 8\exp ( -c_{\alpha}k_{\alpha}(t)),\end{equation} where $k_{\alpha}$ is defined in the statement of Proposition \ref{concspintro}. Integrating this inequality, we get
$$|\EE f_z(X) - m_z| \leq \eps_n $$ with $\eps_n = O( \kappa n^{-1} (\log n)^{(\frac{1}{\alpha}-1)_+})$, uniformly in $z\in \CC$, $\Im z\geq 1$. With this notation, we get for any $t>0$,
$$\PP( | f_z-\EE f_z| > t + \eps_n)\leq 8\exp\big ( -c_{\alpha} k_{\alpha}(t)\big).$$ Let $\mathcal{N}_{t}$ be a $t$-net of $\mathcal{K}$. As $z \mapsto f_z(X)$ is $1$-Lipschitz on $\{ z \in \CC: \Im z\geq 1\}$, we have
$$ \PP\big( \sup_{z\in \mathcal{K}} |f_z - \EE f_z | >2 t +\eps_n \big) \leq
8 |\mathcal{N}_{t}|\exp\big ( -c_{\alpha} k_{\alpha}(t)\big),$$
As $\mathcal{K}$ is a subset of $\CC$ of diameter inferior to $1$, we can find a $t$-net $\mathcal{N}_t$ such that $|\mathcal{N}_t|\leq t^{-2}$. Thus, $$ \PP\big( d( \mu_{X/\sqrt{n}}, \EE \mu_{X/\sqrt{n} }) >2 t +\eps_n \big) \leq \frac{8}{t^2}\exp\big ( -c_{\alpha} k_{\alpha}(t)\big),$$ which, adjusting the constant $c_{\alpha}$, gives the claim.
\end{proof}
\section{Deterministic equivalents for Wigner matrices}\label{secdetermexpoWigner}
We will prove in this section some uniform deterministic equivalents for the spectral measure and largest eigenvalue of deformed Wigner matrices having concentration $\mathcal{C}_{\alpha}$ for $\alpha \in (0,2)$ (see definition \ref{defCalpha}), using the inequalities proved in the preceding section. We will also prove a deterministic equivalent for traces of polynomials of deformed Wigner matrices, but which will not rely on concentration arguments. In particular, these deterministic equivalents will entail that assumption $(i)$ of Theorem \ref{theoremgene} holds for the spectral measure, the largest eigenvalue and the traces of polynomials of Wigner matrices in $\mathcal{S}_{\alpha}$. More precisely, we will prove the following propositions.
\begin{Pro} \label{chaining0} Let $\alpha \in (0,2)$. Let $X$ be a Wigner matrix such that $\EE|X_{1,2} -\EE X_{1,2}|^2=1$ and satisfying the concentration property $\mathcal{C}_{\alpha}$. For any $r>0$, $$\sup_{H \in r n^{1/\alpha}B_{\ell^{\alpha}}} d\big( \mu_{X/\sqrt{n} +H}, \mu_{sc}\boxplus \mu_{H} \big) \underset{n\to +\infty}{\longrightarrow} 0,$$ in probability, where $d$ is the distance defined in \eqref{defdStiel} .
\end{Pro} \begin{Rem} This statement fails when $\alpha=2$ since $X/\sqrt{n}$ is in $r n^{1/2} B_{\ell^2}$ for some $r>0$, with positive probability uniform in $n$. Whereas on one hand, by Wigner's theorem (see \cite{Guionnet}) $$\mu_{2X/\sqrt{n}} \underset{n\to +\infty}{\leadsto } \mu_{sc,2},$$ in probability, where for any $a>0$,
$$\mu_{sc,a} = \frac{1}{2a^2 \pi} \sqrt{4a^2-x^2} \Car_{|x|\leq 2a}dx.$$ On the other hand, by continuity of the free convolution (see \cite[Proposition 4.13]{BV}), $$ \mu_{sc}\boxplus \mu_{X/\sqrt{n}} \underset{n\to +\infty}{\leadsto} \mu_{sc}\boxplus \mu_{sc},$$ in probability, and we have $\mu_{sc}\boxplus \mu_{sc}= \mu_{sc,\sqrt{2}}$ by \cite[Example 5.3.26]{Guionnet}.
\end{Rem}
\begin{Pro} \label{chaining1}
Let $\alpha \in (0,2)$. Let $X$ be a centered Wigner matrix satisfying the concentration property $\mathcal{C}_{\alpha}$ such that $\EE|X_{1,2}|^2=1$. Define the function $\rho$ by, \begin{equation} \label{deff} \forall x \in \RR, \ \rho(x) = \begin{cases} x +\frac{1}{x} & \text{ if } x \geq 1,\\ 2 & \text{ otherwise.} \end{cases} \end{equation} For any $r>0$,
$$\sup_{A \in r B_{\ell^{\alpha}}} \big| \lambda_{X/\sqrt{n} + A}-\rho(\lambda_A) \big| \underset{n\to +\infty}{\longrightarrow} 0,$$ in probability. \end{Pro}
For the traces of polynomials of independent Wigner matrices we will prove the next proposition.
\begin{Pro}\label{convunifpoly}Let $\alpha \in (0,2]$. Let $P\in \CC \langle \textbf{X}\rangle$ be a non-commutative polynomial of total degree $d>\alpha$. Let $\textbf{X} = (X_1,...,X_p)$ be a family of independent centered Wigner matrices with entries having finite $(d+1)^{\text{th}}$-moments, such that $\EE|M_{1,2}|^2 = 1$ for any $M\in \{X_1,...,X_p\}$. For any $r>0$,
$$\sup_{\textbf{H}\in r B_{\ell^{\alpha}}} \big| \tau_n[P(\textbf{X}/\sqrt{n}+n^{1/d}\textbf{H})] - \tau[P(\textbf{s})]-\tr[P_d(\textbf{H})] \big| \underset{n\to+\infty}{\longrightarrow} 0,$$ in probability, where $P_d$ is the homogeneous part of degree $d$ of $P$, $\textbf{s} =(s_1,...,s_p)$ is a free family of $p$ semi-circular variables in a non-commutative probability space $(\mathcal{A},\tau)$ and,
$$ B_{\ell^{\alpha}} = \big\{ \textbf{H} \in (\mathcal{H}_n^{(\beta)})^p : \sum_{i=1}^p \tr|H_i|^{\alpha}\leq 1\big\}.$$ \end{Pro}
It is interesting to note that we are able for polynomials, to make the approximation hold uniformly in $H\in r B_{\ell^2}$, which is why we can consider the Gaussian case in our large deviations principle of Theorem \ref{LDPpoly}.
\subsection{Deterministic equivalents in expectation}
Our approach to prove Propositions \ref{chaining0} and \ref{chaining1} consists is showing in a first step the proposed uniform deterministic equivalents in expectation, and then make use the concentration inequalities of the last section \ref{Chapconc} together with a chaining argument to show that these equivalent hold uniformly in probability.
For the empirical spectral measure, we have such a uniform deterministic equivalents in expectation by the following result of Bordenave and Caputo \cite{Bordenave}.
\begin{The}[{\cite[Theorem 2.6]{Bordenave}}]\label{citeBC}
Let $X$ be a Wigner matrix such that $\EE|X_{1,2}-\EE X_{1,2}|^2 = 1$, $\EE |X_{1,2}|^3<+\infty$, and $\EE X_{1,1}^2<+\infty$. There exists a universal constant $c>0$ such that for any $H\in \mathcal{H}_n^{(\beta)}$,
$$\delta( \EE\mu_{X/\sqrt{n}+H}, \mu_{sc}\boxplus \mu_H) \leq c \frac{\sqrt{\EE X_{1,1}^2} + \EE|X_{1,2}|^3}{\sqrt{n}},$$ where $\delta$ is defined for any $\mu,\nu\in \mathcal{P}(\RR)$,
$$ \delta(\mu,\nu) = \sup \big\{\big| g_{\mu}(z) - g_{\nu}(z)\big| : \Im z\geq 2\big\},$$ where $g_{\mu}$ and $g_{\nu}$ denote the Stieltjes transforms of $\mu$ and $\nu$. \end{The}
For the largest eigenvalue, we will prove the following proposition.
\begin{Pro}\label{compexpvp}Let $\alpha \in (0,2)$. Let $X$ be a centered Wigner matrix such that $\EE|X_{1,2}|^2=1$ and $\EE| X_{1,1}|^4, \EE|X_{1,2}|^4<+\infty$. For any $r>0$,
$$ \sup_{H \in r B_{\ell^{\alpha}}} | \EE\lambda_{X/\sqrt{n} + H} - \rho(\lambda_H)| \underset{n\to +\infty}{\longrightarrow} 0,$$ where $\rho$ is the function defined in \eqref{deff}. \end{Pro}
\begin{proof} In a first step, we will perfom a truncation and convolution argument as to the one used in \cite[Proposition 4.1, step 1]{BordCap}, in order to reduce the problem to the case the entries of $X$ satisfies a Poincaré inequality. Let $\eps>0$ and let $G$ be a GUE matrix, that is, $G = \frac{1}{\sqrt{2}}(B+B^*)$ where $B$ is a matrix with i.i.d complex Gaussian entries with covariance $\frac{1}{2}I_2$, independent from $X$. We set $X^{(\eps)}$ to be the Hermitian matrix with $(i,j)$-entry,
$$ X_{i,j}^{(\eps)} = \frac{X_{i,j}\Car_{|X_{i,j}|\leq \eps^{-1}} - \EE X_{i,j}\Car_{|X_{i,j}|\leq \eps^{-1}} }{(\Var (X_{i,j} \Car_{|X_{i,j}|\leq \eps^{-1}})^{1/2}},$$ and $Y^{(\eps)} = (1+\eps^2)^{-1/2}(X^{(\eps)} + \eps G)$. By \cite[Theorem 1.2]{BGMZ}, $Y^{(\eps)}$ has entries satisfying a Poincaré inequality .
We know by \cite[Theorem 2]{Latala} that there is some constant $C>0$ such that for any centered Wigner matrix $H$,
$$\EE|| H || \leq C \Big( \max_i\big( \sum_j \EE |H_{i,j}|^2 \big)^{\frac{1}{2}} + \big (\sum_{i,j} \EE |H_{i,j}|^4\big)^{\frac{1}{4}}\Big).$$ This inequality yields as the entries of $X$ have finite fourth moments,
$$ \lim_{\eps \to 0} \limsup_{ n\to\infty} \EE || X - Y^{(\eps)}|| =0.$$
But, using Weyl's inequality \cite[Theorem III.2.1]{Bhatia}, and the fact that $\rho$ is $1$-Lipschitz, we see that $A\mapsto |\lambda_{X/\sqrt{n}+A}-\rho(\lambda_A)|$ is $2$-Lipschitz with respect to $|| \ ||_{\ell^2}$. Thus, we can focus on proving Proposition \ref{compexpvp} when $X$ has entries satisfying a Poincaré inequality. We make now another reduction of the statement to a convergence in probability and to the case where the supremum is taken on the set of matrices which we denote by $\mathcal{G}$, consisting of $m$-sparse matrices $A$ (meaning at most $m$ entries are non-zero) with spectral radius bounded by $r$, for some fixed $r,m>0$.
Note that by Weyl's inequality and \eqref{compnrom}, we have for any $A\in rB_{\ell^{\alpha}}$,
\begin{equation} \label{inequnif}|\lambda_{X/\sqrt{n}+A}|\leq r + ||X/\sqrt{n}||, \quad 2\leq \rho(\lambda_A)\leq \rho(r).\end{equation}
As $|| X/\sqrt{n}||$ converges in $L^2$ by \cite[Theorem 2.1.22, 27]{Guionnet}, we deduce that, uniformly in $A\in rB_{\ell^{\alpha}} $, $|\lambda_{X/\sqrt{n}+A}-\rho(\lambda_A)|$ is uniformly integrable. Therefore it suffices to prove that for any $t>0$,
$$ \sup_{A\in rB_{\ell^{\alpha}}} \PP( | \lambda_{X/\sqrt{n}+A} - \rho(\lambda_A)|>t)\underset{n\to +\infty}{\longrightarrow} 0.$$
Let $A\in rB_{\ell^{\alpha}}$, and $M_1\geq ...\geq M_{n^2}$ be the values $|A_{i,j}|$ in non-increasing order. We have, \begin{equation} \label{estimcoeffball}\forall k \in \{1,...,n^2\}, \ M_k \leq r^{1/\alpha} k^{-1/\alpha}.\end{equation}
Let now $m\in \NN$ and $v_1,...,v_m$ the locations of the $m$ largest values of $|A_v|$. Define $A^{(m)}$ to be the matrix, $$\forall v \in \{1,...,n\}^2, \ A_v^{(m)} = \sum_{i=1}^m A_{v_i} \delta_{v_i,v}.$$ As $\alpha <2$, we deduce,
$$ || A-A^{(m)}||_{\ell^2}^2 \leq r^{\frac{2}{\alpha}} \sum_{k>m} k^{-\frac{2}{\alpha}} = O( m^{1- \frac{2}{\alpha}}).$$ Thus, again by Weyl's inequality, it is sufficient to prove for any fixed $m\in \NN$, $r>0$, and $t>0$,
$$ \sup_{A\in \mathcal{G}} \PP( | \lambda_{X/\sqrt{n}+A} - \rho(\lambda_A)|>t)\underset{n\to +\infty}{\longrightarrow} 0.$$ To prove this claim, we will follow a rather classical argument relying on the Frobenius formula used in the study of finite rank perturbations as in \cite{Benaych} for example, to determine the behavior of the largest eigenvalue of deformed models.
Diagonalize $A=UDU^*$, with $U$ of size $n\times m$ such that $U^* U=I_m$. By Frobenius formula (see \cite[section 4.1]{Benaych}), $\lambda_{X/\sqrt{n}+A}$ is either in the spectrum of $X/\sqrt{n}$, denoted $\sigma(X/\sqrt{n})$, or the largest zero of the function, \begin{equation} \label{deffn}\forall x\notin \sigma(X/\sqrt{n}), \ f_{n,A}(x) = \det(I _m- U^*R(x)U D).\end{equation}
Our main task consists in proving that this function is uniformly close on any compact subset of $\{ \Re z> \lambda_{X/\sqrt{n}} \}$ to the following deterministic limit function, \begin{equation} \label{deffa}\forall \Re z>2, \ f_A(z) = \det(I_m- g_{\mu_{sc}}(z) D).\end{equation} \begin{Lem}\label{compeq} Let $\delta>0$ and define, $$W_{\delta} = \{ \lambda_{X/\sqrt{n}} \leq 2 +\delta\}.$$ For all subset $\Omega$ compactly included in $\{ z \in \CC : \Re z > 2+\delta\}$ and $t>0$,
$$\sup_{A\in \mathcal{G}} \PP( \{\sup_{z \in \Omega} |f_{n,A}(z) - f_A(z)| >t\}\cap W_{\delta} ) \underset{n\to+\infty}{\longrightarrow} 0,$$ where $f_{n,A}$ and $f_A$ are defined in \eqref{deffn}, \eqref{deffa}. \end{Lem}
Assume for the moment that this lemma is true. Note that the functions $f_A$, $A\in \mathcal{G}$, form a normal family of holomorphic functions on $\{ z \in \CC : \Im z >2\}$. By \cite[Chapter 5, Theorem 2]{Ahlfors}, it is thus a pre-compact family in the space of holomorphic functions on $\{ z \in \CC : \Im z >2\}$. We deduce by Hurwitz's theorem \cite[Chapter 5, Theorem 10]{Ahlfors} that for any $\delta>0$ and $\Omega$ open subset compactly included in $\{ z : \Re z>2\}$, there is some $t>0$ such that for any holomorphic function $g$ defined on a neighborhood of $\Omega$, and $A\in rB_{\ell^{\alpha}}$ such that $\sup_{\Omega} || f_A - g|| <t$, then either $f_A$ does not have any zeros in $\Omega$ and therefore $g$ neither, or for any zeros of $f_A$ in $\Omega$, corresponds a zero of $g$ in $\Omega$ which is $\delta$-close.
Let $\delta, r>0$. We set $$V_{\delta,r} = \big\{ \lambda_{X/\sqrt{n}} \leq 2+\delta, \ 2-\delta \leq \lambda_{X/\sqrt{n}+A} \leq \rho(r), \}.$$ Let also $\Omega$ be some open subset compactly included in $\{ z : \Re z>2+\delta\}$ such that $[2+2\delta, \rho(r) ] \subset \Omega$. We deduce that for any $\delta>0$ there is a $t>0$, such that,
$$ \PP( \{ | \lambda_{X/\sqrt{n}+A} - \rho(\lambda_A)|>3\delta\} \cap V_{\delta,r})\leq \PP( \{ \sup_{z \in \Omega} | f_{n,A}(z) - f_A(z)| >t \}\cap W_{\delta}).$$ As this $t$ does not depend on $A\in \mathcal{G}$, we get from Lemma \ref{compeq}
$$ \sup_{A\in \mathcal{G}} \PP( \{ | \lambda_{X/\sqrt{n}+A} - \rho(\lambda_A)|>3\delta\} \cap V_{\delta,r}) \underset{n\to+\infty}{\longrightarrow} 0.$$ It remains to show that $\PP(V_{\delta,r})$ goes to $0$ as $n\to +\infty$ uniformly in $A\in rB_{\ell^{\alpha}}$. Note that almost surely (taking an arbitrary coupling of the matrices $X$), we have by Hoeffman-Weilandt inequality \eqref{Hoeffmanalphagene}, $$\sup_{A\in rB_{\ell^2}} \mathcal{W}_2(\mu_{X/\sqrt{n}+A},\mu_{X/\sqrt{n}}) \underset{n\to +\infty}{\longrightarrow} 0.$$ Thus, by Wigner's theorem, almost surely, $\mu_{X/\sqrt{n}+A}$ converges weakly towards $\mu_{sc}$ uniformly in $A\in rB_{\ell^2}$. By lower-semicontinuity of the map $$ \mu \in \mathcal{P}(\RR) \mapsto \sup \mathrm{supp}(\mu),$$ we deduce that $$ \liminf_{n\to +\infty} \sup_{A\in rB_{\ell^2}}(\lambda_{X/\sqrt{n}+A}-2) \geq 0,$$ almost surely. Using the above convergence, \eqref{inequnif} and the convergence of the largest eigenvalue of $X/\sqrt{n}$ to $2$ in probability, we can conclude that $$ \limsup_{n\to +\infty} \sup_{A\in \mathcal{G}} \PP( V_{\delta,r}) \underset{r\to +\infty}{\longrightarrow} 0,$$ which gives the claim of Proposition \ref{compexpvp}. Thus, we are reduced to show Lemma \ref{compeq}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{compeq}] Let $\delta>0$ and $\Omega$ as in the statement of Lemma \ref{compeq}. Let $\eta = \inf \{ \Re z : z \in \Omega \} -2$ and $\zeta$ a Lipschitz function such that \begin{equation} \label{defzeta} \Car_{(-\infty, 2+\delta]} \leq \zeta \leq \Car_{(-\infty, 2+\eta)}.\end{equation} Let $u,v$ be some unit vectors and $z\in \Omega$. We set $$\forall Y \in \mathcal{H}_n^{(\beta)}, \ \mathcal{R}_z(Y)= \langle u,R(z) v\rangle \zeta(\lambda_{Y/\sqrt{n}}),$$
where $R(z) = (z-Y/\sqrt{n})^{-1}$. By Weyl's inequality, this defines a $L_{\Omega}$-Lipschitz function with respect to $|| \ ||_{\ell^2}$, where $L_{\Omega}$ is a constant depending on the set $\Omega$. As the entries of $X$ satisfies a Poincaré inequality, $X$ has concentration $\mathcal{C}_1$. We deduce from Lemma \ref{conclip} that for $n$ large enough,
$$ \PP( |\mathcal{R}_z(X) - \EE\mathcal{R}_z(X) |>t+\delta_n ) \leq 4\exp\big( -c L_{\Omega}^{-1} \sqrt{n} t\big),$$ where $\delta_n =O( n^{-1/2})$. Note that $\mathcal{R}_z$ defines a $1/(\eta -\delta)^2$-Lipschitz function in $z\in \Omega$. As $\Omega$ is relatively compact, we deduce by an $\eps$-net argument that for any $t>0$,
$$\sup_{||u||=||v||=1} \PP( \sup_{z \in \Omega}|\mathcal{R}_z(X) - \EE\mathcal{R}_z(X) |>t ) \underset{n\to+\infty}{\longrightarrow } 0.$$ In the following lemma, we show an isotropic-like property.
\begin{Lem} Let $\delta >0$ and $\Omega$ a subset compactly included in $\{ z \in \CC : \Re z>2+\delta\}$. Let $X$ be a Wigner matrix satisfying the assumptions of Proposition \ref{compexpvp}. For any $m\in \NN$,
$$\sup_{z \in \Omega}\sup_{u,v \in \mathcal{V}_m} \big| \langle u, \EE[ \zeta(\lambda_{X/\sqrt{n}})R(z)]v \rangle - \langle u,v \rangle g_{\mu_{sc}}(z) \big| \underset{n\to+\infty}{\longrightarrow} 0,$$ where $\mathcal{V}_m$ denotes the set of unit $m$-sparse vectors, meaning with at most $m$ non-zero entries, $\zeta$ is as in \eqref{defzeta}, and $R(z) = (z-X/\sqrt{n})^{-1}$. \end{Lem}
\begin{proof} By polarization, it is sufficient to prove this lemma where the supremum ranges over vectors $v=u$. Moreover, by symmetry, it is enough to show this statement for $\Omega \cap \CC^+$. Because $\mathcal{R}_z$, as a function of $z$, is a Lipschitz function on $\Omega$, we only need to show for any $\eps>0$,
$$\sup_{z \in \Omega_{\eps}}\sup_{u\in \mathcal{V}_m} \big| \langle u, \EE[ \zeta(\lambda_{X/\sqrt{n}})R(z)]u \rangle - \langle u,u \rangle g_{\mu_{sc}}(z) \big| \underset{n\to+\infty}{\longrightarrow} 0,$$ with $\Omega_{\eps} = \{ z \in \Omega : \Im z \geq \eps \}$.
Let $u \in \mathcal{V}_m$. For any $z\in \CC^+$, we have on one hand,
$$\big|\langle u, \EE[ \zeta(\lambda_{X/\sqrt{n}})R(z)]u \rangle -
\langle u, \EE R(z)u \rangle\big| \leq \frac{1}{\Im z}\PP( \lambda_{X/\sqrt{n}} > 2+\delta).$$ On the other hand, expanding the scalar product,
$$\big|\langle u, \EE Ru \rangle - g_{\mu_{sc}}(z)\big|\leq m \max_{1\leq i,j \leq n}| \EE R_{ij}-\delta_{i,j}g_{\mu_{sc}}(z)|.$$ As $\lambda_{X/\sqrt{n}}$ converges to $2$ in probability, we are reduced to prove for any $\veps>0$,
$$\sup_{z \in \Omega_{\veps}} \max_{1\leq i \leq n} | \EE R_{i,j} - \delta_{i,j} g_{\mu_{sc}}(z) | \underset{n\to +\infty}{\longrightarrow} 0.$$ Even though this is a classical estimate of random matrix theory, for sake of completeness we give here a proof. We start with the case of the off-diagonal entries. We set $H = X/\sqrt{n}$ and we write $R$ as a short-hand for $R(z)$. Let $i\neq j$. We have the following resolvent identity (see \cite[Lemma 3.5]{BGKn}), $$R_{i,j} = R_{i,i} \sum_{k}^{(i)} H_{i,k} R^{(i)}_{k,j},$$ where $R^{(i)}$ is the resolvent of the matrix $H$ where we removed the $i^{\text{th}}$-row and $i^{\text{th}}$-column, and $\sum^{(i)}$ means that the summation is over $\{ 1,...,n\}\setminus\{i\}$. By Cauchy-Schwarz inequality we have
$$ | \EE R_{i,j}| \leq \frac{1}{\Im z} \Big( \EE \big| \sum_{k}^{(i)} H_{k,i} R^{(i)}_{k,j}\big|^2\Big)^{1/2}.$$ But, as $R^{(i)}$ is independent of $(H_{k,i})_k$ and $(H_{k,l})_{k\leq l}$ are centered and independent,
$$\EE\big|\sum_{k}^{(i)} H_{k,i} R^{(i)}_{k,j}\big|^2 = \frac{1}{n}\EE \sum_{k}^{(i)} |R^{(i)}_{k,j}|^2.$$ Recall Ward's identity (see \cite[(3.6)]{BGKn}),
$$ \sum_{k}^{(i)} |R^{(i)}_{k,j}|^2 =- \frac{1}{\Im z} \Im R^{(i)}_{i,i}.$$
Thus, $$|\EE R_{i,j}|\leq \frac{1}{\sqrt{n} (\Im z)^2}.$$ To deal with the diagonal entries, we start from the Schur complement formula (see \cite[Lemma 2.4.6]{Guionnet}), \begin{equation}\label{Schr}R_{i,i}^{-1} = z -H_{i,i}+ \langle H^{(i)}, R^{(i)} H^{(i)}\rangle,\end{equation} where $H^{(i)}$ denotes the $i^{\text{th}}$-column of $H$ where the entry $H_{i,i}$ is removed. Let $\mathcal{F}^{(i)}$ be the $\sigma$-algebra generated by the variables $H_{k,l}$ for $k,l \neq i$. We find,
$$\EE \Big(\big|\langle H^{(i)}, R^{(i)} H^{(i)}\rangle - \frac{1}{n} \tr R^{(i)}\big|^2 | \mathcal{F}^{(i)}\Big) = \frac{|\gamma|^2}{n^2} \sum_{k\neq l}^{(i)} R^{(i)}_{k,l} \overline{R_{l,k}^{(i)}} + \frac{1}{n^2}\sum_{k\neq l}^{(i)}| R^{(i)}_{k,l}|^2 + \frac{\gamma'}{n^2} \sum_k^{(i)}|R_{k,k}^{(i)}|^2,$$
where $ \gamma = \EE(X_{1,2}^2)$ and $\gamma' = \EE |X_{1,2}|^4-1$. Introducing the missing diagonal terms, using Ward's identity again and the fact that $|\gamma|\leq 1$, we find,
$$\EE \Big(\big|\langle H^{(i)}, R^{(i)} H^{(i)}\rangle - \frac{1}{n} \tr R^{(i)}\big|^2 | \mathcal{F}^{(i)}\Big) \leq \frac{1}{n^2} |\tr R^{(i)} \overline{R^{(i)}}| + \frac{1}{n^2 \Im z } | \Im R^{(i)}_{i,i}| + \frac{c}{n\Im z},$$
where $c$ is some positive constant depending on $\EE |X_{1,2}|^4$. This yields,
$$\max_{1\leq i\leq n}\EE\big|\langle H^{(i)}, R^{(i)} H^{(i)}\rangle - \frac{1}{n} \tr R^{(i)}\big|^2 = O( n^{-1/2}(\Im z)^{-2}).$$
From Wigner's theorem, we know that $n^{-1}\tr R^{(i)}$ converges to $g_{\mu_{sc}}$ in probability for any $\Im z>0$. Note that $R^{(i)}$ are identically distributed for $i=1,...,n$. We deduce from \eqref{Schr} and the fact that $g_{\mu_{sc}}(z)^{-1} = z- g_{\mu_{sc}}(z)$ (see \cite[Example 5.3.2.6]{Guionnet}),
$$\max_{1\leq i\leq n}\EE\big|R_{i,i}^{-1} - g_{\mu_{sc}}(z)^{-1}\big|^2\underset{n\to +\infty}{\longrightarrow} 0,$$ which yields,
$$\max_{1\leq i\leq n}\EE\big|R_{i,i} - g_{\mu_{sc}}(z)\big|^2\underset{n\to +\infty}{\longrightarrow} 0,$$ for any $z\in \CC^+$. As the functions $R_{i,i}$ and $g_{\mu_{sc}}$ are $\eps^{-2}$-Lipschitz on $\{ z \in \CC^+ : \Im z > \eps\}$, we can extend by an $\eps$-net argument, this convergence uniformly on any bounded subset of $\{z \in \CC^+ : \Im z >\eps\}$, for any $\eps>0$.
\end{proof} We come back now to the proof of Lemma \ref{compeq}. The above lemma yields that for any $t>0$,
$$\sup_{u\in \mathcal{V }_m} \PP(\{ \sup_{z \in \Omega}|\langle u, Ru \rangle - g_{\mu_{sc}}(z) |>t\}\cap W_{\delta} ) \underset{n\to+\infty}{\longrightarrow } 0.$$ Note that $m$-sparse matrices have $m$-sparse eigenvectors. Using the fact that the spectral radius of matrices in $\mathcal{G}$ is bounded and a union bound, we deduce that for any $s>0$,
$$ \sup_{A\in \mathcal{G}}\PP( \{ \sup_{z \in \Omega} || U^*RU D - g_{\mu_{sc}}(z)D||_{\ell^{\infty}}>s\}\cap W_{\delta}) \underset{n\to +\infty}{\longrightarrow } 0,$$
where $|| Y||_{\ell^{\infty}} = \sup_{i,j}|Y_{i,j}|$, for any matrix $Y$. As the matrices $(I_m - g_{\mu_{sc}}(z)D)$, $z \in \Omega$, $A\in \mathcal{G}$ form a pre-compact subset of $\mathcal{H}_m^{(\beta)}$, the continuity of the determinant on $\mathcal{H}_m^{(\beta)}$, allows us to conclude the proof of Lemma \ref{compeq}.
\end{proof}
\subsection{A chaining argument} We will now give a proof of Propositions \ref{chaining0} and \ref{chaining1}.
As it will rely on a chaining argument, we will need the following lemma.
\begin{Lem}\label{coveringnb}Let $m\in \NN$ and let $B_{\ell^{p}}$ denote the $\ell^p$-ball of $\CC^m$ for any $p>0$. Fix some $0<p<q< \infty$.
We denote by $N(B_{\ell^{p}}, \eps B_{\ell^q})$, the covering number of $B_{\ell^{p}}$ by $\eps B_{\ell^q}$, that is, the minimal number of translates of $\eps B_{\ell^q}$ needed to cover $B_{\ell^{p}}$. There is a constant $c>0$ depending on $p,q$, such that for $c (\frac{\log m}{m})^{\frac{1}{p}-\frac{1}{q}}\leq \eps \leq c^{-1} $, $$\log N(B_{\ell^p}, \eps B_{\ell^q}) \leq c \eps^{\frac{1}{q}-\frac{1}{p}}\log m.$$
\end{Lem}
\begin{proof}
This estimate is a consequence of the upper bound on entropy numbers of embeddings of $\ell_p^m$ in $\ell_q^m$ given in \cite[Proposition 3.2.2]{ETentropy}. Let $0<p<q< \infty$. Denote by $\ell_p^m$ the space $\RR^m$ equipped with the (quasi)-norm $||\ ||_{\ell_p}$. We define, for $k\in \NN$, $$e_k(\ell_p^m \to \ell_q^m) =\inf \{ \eps>0 : B_{\ell_p} \text{ can be covered by } 2^{k-1} \text{ balls } \eps B_{\ell_q}\}.$$ From \cite[Proposition 3.2.2]{ETentropy}, we know that there is a constant $c>0$ such that for $\log_2(2m) \leq k\leq 2m$, $$ e_k(\ell_p^m \to \ell_q^m) \leq c \Big( k^{-1}\log_2\big(1+\frac{2m}{k}\big)\Big)^{\frac{1}{p}-\frac{1}{q}}.$$ Thus, if we set $k= \lambda \log_2(2m)$, for some $\lambda\geq 1$ such that $k\leq 2m$, we deduce the following rough bound, $$ e_k(\ell_p^m \to \ell_q^m) \leq c' \lambda^{\frac{1}{q}-\frac{1}{p}},$$ for some constant $c'>0$. Let now $\eps>0$ and set $\lambda$ such that $\eps = c' \lambda^{\frac{1}{q} -\frac{1}{p}}$. The above inequality tells us that if $1 \leq \lambda\leq 2m/\log_2(2m)$, then there are $(2m)^{\lambda}$ balls $\eps B_{\ell^q}$ covering $B_{\ell^p}$, that is, $$N(B_{\ell^p},\eps B_{\ell^q}) \leq (2m)^{\lambda},$$ which yields the claim. \end{proof}
We are now ready to give a proof of Proposition \ref{chaining0} and \ref{chaining1}.
\begin{proof}[Proof of Proposition \ref{chaining0}] Let $H\in \mathcal{H}_n^{(\beta)}$. As $X$ satisfies $\mathcal{C}_{\alpha}$ for some constant $\kappa>0$, we see that $X+\sqrt{n}H$ also satisfies $\mathcal{C}_{\alpha}$ with the same constant $\kappa$. We know from Propositions \ref{concspintro} and \ref{citeBC}, that for any $t>0$, $$ \PP\left( d\big( \mu_{X/\sqrt{n} + H}, \mu_{sc}\boxplus \mu_H\big) > t +\eps_n\right) \leq \frac{32}{t^2}\exp\big ( -c_{\alpha}k_{\alpha}(t) \big),$$ with $k_{\alpha}$ defined in Proposition \ref{concspintro} and $\eps_n =O\big( n^{-1/2} ( \log n)^{(1/\alpha-1)_+}\big)$, uniformly in $H \in \mathcal{H}_n^{(\beta)}$. Note that the map $$ S : H \in \mathcal{H}_n^{(\beta)} \mapsto d\big( \mu_{X/\sqrt{n} +H}, \mu_{sc}\boxplus \mu_{H} \big),$$
is $n^{-1/2}$-Lipschitz with respect to $|| \ ||_{\ell^2}$ by Lemma \ref{spvar1intro}. We deduce using an $\eps$-net argument that for $n$ large enough, \begin{equation} \label{chaining} \PP\big( \sup_{H \in r n^{1/\alpha}B_{\ell^{\alpha}}} S(H) > 2t\big)\leq \frac{32}{t^2} N( r n^{1/\alpha} B_{\ell^{\alpha}}, tn^{ 1/2} B_{\ell^2})e^{-\frac{c_{\alpha}}{\kappa^{\alpha}}(t-\eps_n)_+^{\alpha}n^{1+\alpha/2} },\end{equation} where $N( r n^{1/\alpha} B_{\ell^{\alpha}}, tn^{ 1/2} B_{\ell^2})$ denotes the covering number of $r n^{1/\alpha} B_{\ell^{\alpha}}$ by $tn^{ 1/2} B_{\ell^2}$. But, the homogeneity of the norm gives, $$N( r n^{1/\alpha} B_{\ell^{\alpha}}, tn^{ 1/2} B_{\ell^2}) = N( B_{\ell^{\alpha}}, t'n^{ \frac{1}{2} - \frac{1}{\alpha} } B_{\ell^2}),$$ with $t' = t/r$. We get from Lemma \ref{coveringnb} applied with $m=n^2$, $$ \log N( B_{\ell^{\alpha}}, t'n^{ \frac{1}{2} - \frac{1}{\alpha} } B_{\ell^2}) = O(n\log n),$$
This shows that the covering number is negligible with respect to the speed of the deviations, which concludes the chaining argument. \end{proof} We finally give a proof of Proposition \ref{chaining1}.
\begin{proof}[Proof of Proposition \ref{chaining1}] Let $r>0$. Similarly as in the proof of Proposition \ref{chaining0}, we deduce from Propositions \ref{concvpintro} and \ref{compexpvp}, that for any $A\in \mathcal{H}_n^{(\beta)}$ and $t>0$,
$$ \PP\left( \big| \lambda_{X/\sqrt{n} + A}- \rho(\lambda_A)\big| > t +\delta_n\right) \leq 8\exp\big(-c_{\alpha} h_{\alpha}(t) \big),$$
where $h_{\alpha}$ is defined in Proposition \ref{concvpintro}, $\delta_n = O(n^{-1/2} (\log n)^{(1/\alpha-1)_+})$ uniformly in $A\in r B_{\ell^2}$, and $\rho$ is as in \eqref{deff}.
Note that the map $x\mapsto \rho(x)$ is $1$-Lipschitz. From Weyl's inequality \cite[Theorem III.2.1]{Bhatia}, we deduce that
$$ A \mapsto |\lambda_{X/\sqrt{n} + A}-\rho(\lambda_A)|,$$ is $2$-Lipschitz with respect to the Hilbert-Schmidt norm on $\mathcal{H}_n^{(\beta)}$. Using an $\eps$-net argument as in the proof of Proposition \ref{chaining0}, it is sufficient to prove that for any fixed $t>0$, the covering number $N( B_{\ell^{\alpha}} , tB_{\ell^2})$ is negligible at the exponential scale $n^{\alpha/2}$, that is
$$ \log N( B_{\ell^{\alpha}} , tB_{\ell^2}) =o(n^{\alpha/2}).$$ But from Lemma \ref{coveringnb}, we know that,
$$ \log N( B_{\ell^{\alpha}} , tB_{\ell^2})= O(\log n),$$
which ends the proof of the claim. \end{proof}
\subsection{Traces of polynomials of deformed Wigner matrices} We will now prove Proposition \ref{convunifpoly}. Contrary to the spectral measure or the largest eigenvalue, the proof will consist in a simple moment computation. \begin{proof}[Proof of Proposition \ref{convunifpoly}] By linearity it is sufficient to show the statement when $P$ is a monomial, which we will assume from now on. We can write $P = X_{i_1}...X_{i_q}$, with $q\leq d$. Define the matrix $Q$ with coefficients in $\CC\langle \textbf{X}\rangle$, by $$ Q =
\left(
\raisebox{0.5\depth}{
\xymatrixcolsep{1ex}
\xymatrixrowsep{1ex}
\xymatrix{
0\ar @{.}[ddddrrrr]& X_{i_1} \ar @{.}[dddrrr] & & & \\
& & & & \\
&&&& \\
&&&& X_ {i_{q-1}}\\
X_{i_q}& & & & 0
}
}\right).
$$ Observe that by cyclicity of the trace, for any $\textbf{Y} \in (\mathcal{H}_n^{(\beta)})^p$, $\tr Q(\textbf{Y})^q = q \tr P(\textbf{Y})$. Therefore, \begin{equation} \label{cycl} \tr P(\textbf{X}/\sqrt{n} + n^{1/d}\textbf{H}) =\frac{1}{q} \tr \big( Q(\textbf{X}/\sqrt{n}) + n^{1/d}Q(\textbf{H})\big)^d.\end{equation} Write $Z = Q(\textbf{X}/\sqrt{n})$ and $K = Q(\textbf{H})$. We know from the proof of \cite[Lemma 2.1]{LDPtr} that,
$$\big | \tr \big( Z + n^{1/d}K\big)^q -\tr Z^q - n^{\frac{q}{d}} \tr K^q\big| \leq 2^q \max_{1\leq k\leq q-1}n^{\frac{q-k}{d}} ( \tr | Z|^{q+1})^{\frac{k}{q+1}} (\tr |K|^2)^{\frac{q-k}{2}}.$$ Let us define $q$-Schatten (quasi-)norm on $(\mathcal{H}_n^{(\beta)})^p$, for any $q>0$ by,
\begin{equation} \label{defschattenp} \forall \textbf{H} \in (\mathcal{H}_n^{(\beta)})^p, \ || \textbf{H} ||_{q} = \Big( \sum_{i=1}^p \tr|H_i|^q \Big)^{1/q}.\end{equation} Note that for any $\textbf{Y}\in (\mathcal{H}_n^{(\beta)})^p$,
$$|Q(\textbf{Y})|=
\left(
\raisebox{0.5\depth}{
\xymatrixcolsep{1ex}
\xymatrixrowsep{1ex}
\xymatrix{
|Y_{i_1}| \ar @{.}[dddrrr]& 0\ar @{.}[ddrr] \ar @{.}[rr] & & 0 \ar @{.}[dd] \\
0 \ar @{.}[dd] \ar @{.}[rrdd] & & & \\
&&& 0\\
0 \ar @{.}[rr] & & 0 & |Y_{i_q}|
}
}\right).
$$ Thus, for any $m\in \NN$,
$$\tr |Q(\textbf{Y})|^{m} = \sum_{j=1}^q \tr|Y_{i_j}|^{m} \leq \sum_{i=1}^p \tr|Y_{i}|^{m} = ||\textbf{Y}||_{m}^{m}.$$
As $\textbf{H}\in r B_{\ell^2}$, $\tr|K|^2 \leq r^2$. Without loss of generality we can assume $r\geq1$. Thus,
\begin{equation} \label{linetr}\big | \tr \big( Z + n^{1/d}K\big)^q -\tr (Z^q) - n^{\frac{q}{d}}\tr K^q\big| \leq r^q 2^q \max_{1\leq k\leq q-1}n^{\frac{q-k}{d}} || \textbf{X}/\sqrt{n}||_{q+1}^{k}.\end{equation} But we know from Wigner's theorem (see \cite[Lemma 2.1.6]{Guionnet}), that there is a constant $c\geq 1$, such that
$$ \EE || \textbf{X}/\sqrt{n} ||_{q+1}^{q+1} \leq c n.$$ Besides,
$$\EE \max_{1\leq k \leq q-1} n^{-\frac{k}{d}} || \textbf{X}/\sqrt{n}||_{q+1}^{k} \leq \sum_{k=1}^{q-1} n^{-\frac{k}{q}} \EE || \textbf{X}/\sqrt{n}||_{q+1}^{k}.$$ By Jensen's inequality, we deduce
$$\EE \max_{1\leq k \leq q-1} n^{-\frac{k}{d}} || \textbf{X}/\sqrt{n}||_{q+1}^{k} \leq \sum_{k=1}^{q-1} n^{-\frac{k}{q}}\big( \EE || \textbf{X}/\sqrt{n}||_{q+1}^{q+1}\big)^{\frac{k}{q+1}}.$$ Therefore,
$$\EE \max_{1\leq k \leq q-1} n^{-\frac{k}{d}} || \textbf{X}/\sqrt{n}||_{d+1}^{k} \leq qc n^{-( \frac{1}{q}-\frac{1}{q+1})}.$$ We deduce from \eqref{cycl} and \eqref{linetr} that
$$ \big| \tau_n[P(\textbf{X}/\sqrt{n}+n^{1/d}\textbf{H})] - \EE\tau_n[P(\textbf{X}/\sqrt{n})]-n^{\frac{q}{d}-1}\tr[P(\textbf{H})] \big| \underset{n\to +\infty}{\longrightarrow} 0,$$ uniformly in $\textbf{H}\in r B_{\ell^2}$ and where $\tau_n = \frac{1}{n} \tr$. It is now sufficient to prove that $n^{q/d-1}\tr P(\textbf{H})$ converges to $0$ uniformly in $\textbf{H} \in r B_{\ell^{\alpha}}$, as soon as $q<d$. Assume first $q\geq \alpha$. Using the non-commutative Hölder's inequality (see \cite[Corollary IV.2.6]{Bhatia}), we get
\begin{equation*} \tr [P(\textbf{H})] \leq \prod_{j=1}^q || H_{i_j} ||_q. \end{equation*} The arithmetic-geometric mean inequality yields,
\begin{equation} \label{ineqSchatten} \tr [P(\textbf{H})] \leq \frac{1}{q} \sum_{j=1}^q \tr |H_{i_j}|^q.\end{equation} As $q\geq \alpha$, we deduce
\begin{equation}\label{comptrnorm} \tr [P(\textbf{H})] \leq || \textbf{H}||_{q}^q \leq || \textbf{H}||_{\alpha}^q. \end{equation} We conclude that when $\alpha\leq q<d$, $$\sup_{\textbf{H}\in r B_{\ell^{\alpha}}} n^{\frac{q}{d}-1} \tr [P(\textbf{H})] \underset{n\to+\infty}{\longrightarrow} 0.$$ If $q<\alpha$, then $q=1$ and $\alpha>1$. By Jensen's inequality,
$$|\tr H_{i_1}| \leq n^{1-1/\alpha}(\tr |H_{i_1}|^{\alpha})^{1/\alpha}.$$ Thus, as $d>\alpha$, $$\sup_{\textbf{H}\in r B_{\ell^{\alpha}}} n^{\frac{1}{d}-1} \tr [P(\textbf{H})] \underset{n\to+\infty}{\longrightarrow} 0.$$ Besides, we know by \cite[Theorem 5.4.2]{Guionnet}, that $$ \EE \tau_n[P(\textbf{X}/\sqrt{n})] \underset{n\to+\infty}{\longrightarrow} \tau[P(\textbf{s})],$$ where $\textbf{s}$ are a family of $p$ free semi-circular variables defined on a non-commutative probability space $(\mathcal{A},\tau)$. This ends the proof of the proposition. \end{proof}
\section{Deterministic equivalent for the last-passage time}\label{secdetermequivLPT}
We will prove in this section the analogue of the results for Wigner matrices of the preceding section, for the last-passage time. More precisely, we will provide a deterministic equivalent for the last-passage time when the matrix of weights is deformed by some matrix $nH$, where $||H||_{\ell^{\alpha}}$ is bounded for some $\alpha \in (0,1)$.
Let $\mathcal{A}$ denote the set of finite vectors $(v_1,...,v_m)$, which we will call \textit{admissible}, such that $v_i\in \{0,...,n\}^d$, $v_0=(0,...,0)$, $v_m=(n,...,n)$, and for any $i \in \{0,...,m-1\}$, $v_i< v_{i+1}$, where $<$ denotes the lexicographic order. With this definition we set, for any $H\in \RR^I$, where $I = \{0,...,n\}^d$, \begin{equation} \label{defequidetLPP}\mathcal{T}_n(H) = \sup_{V \in \mathcal{A} }\Big\{ \sum_{i=0}^{m} H_{v_i}^+ + \sum_{i=0}^{m-1} g\Big( \frac{v_{i+1}-v_i}{n}\Big)\Big\},\end{equation} where $V=(v_0,...,v_m)$ for some $m\in \NN$, where $g$ is as in \eqref{defg}, and where we denote here, for better lisibility, $x^+$ the positive part of $x\in \RR$ ($x^+ = x_+$, our former notation). With this notation, we will prove the following proposition.
\begin{Pro}\label{convunifLPP}Let $\alpha \in (0,1)$. Let $X = (X_{v})_{v \in \ZZ_+^d}$ be a family of i.i.d random variables following the law $\mu_{\alpha}$. For any $r>0$,
$$\sup_{||H||_{\ell^{\alpha}}\leq r} \Big|\frac{1}{n}T(X+nH)^+ - \mathcal{T}_n(H)\Big| \underset{n\to +\infty}{\longrightarrow} 0,$$ in probability, where $Y^+$ denotes the multi-matrix $(Y_v^+)_v$. \end{Pro}
We will follow the same arguments as for the proof of the uniform deterministic equivalent of the empirical spectral measure and the largest eigenvalue of Wigner matrices. We will begin by showing that the deterministic equivalent \eqref{defequidetLPP} we propose, holds uniformly in expectation. This is the object of the following lemma.
\begin{Lem} Let $\alpha\in (0,1)$. Let $X = (X_{v})_{v \in \ZZ_+^d}$ be a family of i.i.d non-negative random variables with common distribution function satisfying \eqref{condfuncdistr}. For any $r>0$,
$$\sup_{||H||_{\ell^{\alpha}} \leq r }\big|\frac{1}{n}\EE T(X+nH)^+ - \mathcal{T}_n(H)\big|\underset{n\to+\infty}{\longrightarrow} 0,$$ where $\mathcal{T}_n(H)$ is as in \eqref{defequidetLPP}. \end{Lem}
\begin{proof} Let $\mathcal{A}_m$ denote the subset of vectors of $\mathcal{A}$ of size less or equal than $m$, and define $\hat{\mathcal{T}}_n^{(m)}$ by, $$\hat{\mathcal{T}}_n^{(m)}(H) = \sup_{V \in \mathcal{A}_m }\Big\{ \sum_{i=0}^{p} H_{v_i}^+ +\sum_{i=0}^{p-1}\frac{1}{n} \EE T_{v_i,v_{i+1}}(X) \Big\},$$ and $\mathcal{T}_n^{(m)}$, $$\mathcal{T}_n^{(m)}(H) = \sup_{V \in \mathcal{A}_m }\Big\{ \sum_{i=0}^{p} H_{v_i}^+ +\sum_{i=0}^{p-1} g\Big( \frac{v_{i+1}-v_i}{n}\Big)\Big\},$$ where $V=(v_0,...,v_p)$ for some $p\leq m$, and $g$ is as in \eqref{defg}.
We begin by proving that there is some constant $C>0$ depending on $\alpha$, such that for any $|| H ||_{\ell^{\alpha}}\leq r$, \begin{equation} \label{encadrelasttime} -Cr(\log n)^{\frac{1}{\alpha}} n^{\alpha-1}\leq \frac{1}{n}\EE T(X+nH)^+ - \hat{\mathcal{T}}_n^{(m)}(H) \leq C rm^{1-\frac{1}{\alpha}}.\end{equation} In the following $C$ will denote a constant which will depend only on $\alpha$ and which will vary along the lines of the proof. Let $\pi$ be an optimal path for the last-passage time $T(X+nH)^+$, and denote by $v_1,..,v_{m-1}$ be the $m-1$ largest values of $H^+$ on the path $\pi$, sorted in lexicographic order. Add $v_0 = (0,...,0)$ and $v_m=(n,...,n)$, to get $V=\{v_0,...,v_m\} \in \mathcal{A}_m$. We have $$ \frac{1}{n}T(X+nH)^+ - \sum_{i=0}^{m} H_{v_i}^+ -\sum_{i=0}^{m-1}\frac{1}{n} T_{v_i,v_{i+1}}(X)\leq \frac{1}{n}\sum_{v\in \pi } (X +nH)^+_{v} - \sum_{i=0}^{m} H_{v_i}^+ -\frac{1}{n} \sum_{v\in \pi} X_{v}.$$ As $(x+y)^+\leq x^+ + y^+$, we deduce $$ \frac{1}{n}T(X+nH)^+ - \sum_{i=0}^{m} H_{v_i}^+ -\sum_{i=0}^{m-1}\frac{1}{n} T_{v_i,v_{i+1}}(X)\leq \sum_{v \in \pi\cap V^c }H_{v}^+.$$ Now observe that if $M_1\geq ... \geq M_{d(n+1)}$ are the values of $H^+$ (or of $H^-$) along $\pi$ in decreasing order, we have since $\sum_{i} M_i^{\alpha}\leq r^{\alpha}$, for any $k\in \{1,...,d(n+1)\}$, \begin{equation} \label{boundweight}M_k \leq r k^{-1/\alpha}.\end{equation} Therefore, $$ \sum_{v \in \pi\cap V^c}H_{v}^+\leq r \sum_{k=m-1}^{+\infty} k^{-1/\alpha} \leq C r m^{1-\frac{1}{\alpha}},$$ for some constant $C>0$. This proves the upper bound of \eqref{encadrelasttime}. On the other hand, let $V=\{v_0,....,v_p\}\in \mathcal{A}_m$. Considering the optimal paths from $v_i$ to $v_{i+1}$ in the last-passage time $T_{v_i,v_{i+1}}(X)$, for $i=0,...,p-1$ and their concatenation $\pi$, we get, \begin{equation} \label{bornesupLPP} \sum_{i=0}^{p} H_{v_i}^+ + \frac{1}{n}\sum_{i=0}^{p-1}T_{v_i,v_{i+1}}(X) - T(X+nH)^+\leq \sum_{X_{v} \geq -n H_{v}} H_{v}^- + \sum_{ X_{v } \leq -n H_{v}} \frac{X_{v}}{n}.\end{equation} Indeed, if $v\in \pi $, then $$H_v^+ + X_v - (X+nH)_v^+ \leq \Car_{\{X_v \geq -nH_v\}}H_v^-+\Car_{\{X_v \leq -nH_v\}} X_v,$$ by considering the cases whether $H_v\geq 0$ or ($H_v\leq 0$ and $X+nH_v\geq 0$) or ($H_v\leq 0 $ and $X+nH_v\leq 0$). Turning our attention to the first sum in \eqref{bornesupLPP}, we deduce by bounding the first $n^{\alpha}$ largest weights of $H_{v}^-$ by $X_{v}/n$, and using the bound \eqref{boundweight} for the rest of the terms, $$ \EE\Big(\sum_{X_{v} \geq -n H_{v}} H_{v}^-\Big) \leq \frac{n^{\alpha}}{n} \EE\sup_v X_v + r \sum_{k>n^{\alpha}} k^{-\frac{1}{\alpha}}.$$ By \eqref{momentalpha} we have, $$ \EE \sup_v X_v \leq c(\log n)^{\frac{1}{\alpha}},$$ for some constant $c>0$. We thus proved, $$ \EE\Big(\sum_{X_{v} \geq -n H_{v}} H_{v}^-\Big) \leq C r (\log n)^{\frac{1}{\alpha}} n^{\alpha-1}.$$ On the other hand, focusing now on the second term of \eqref{bornesupLPP},
$$\EE\Big(\sum_{ X_{v} \leq -n H_{v}} \frac{X_{v}}{n} \Big) =\frac{1}{n} \EE \big(X_0 |\{ v : X_0\leq -n H_{v}\}| \big).$$
But $||H||_{\ell^{\alpha}} \leq r$, thus
$$ |\{ v : X_0\leq -n H_{v}\}| \Big( \frac{X_0}{n} \Big)^{\alpha}\leq r.$$ Therefore, $$\EE\Big(\sum_{ X_{v } \leq -n H_{v}} \frac{X_{v}}{n} \Big) \leq n^{\alpha-1} r \EE X_0^{1-\alpha}.$$ which concludes the proof of the lower bound of \eqref{encadrelasttime}. Comparing $\mathcal{T}^{(m)}_n$ and $\hat{\mathcal{T}}^{(m)}_n$, we get using the translation invariance in law (by vectors of $\ZZ^d_+$) of $(X_v)_{v\in\ZZ^d_+}$,
$$|\mathcal{T}^{(m)}_n(H) - \hat{\mathcal{T}}_n^{(m)}|\leq m \max_{v\in \{0,...,n\}^d} \Big|\frac{1}{n} \EE T_{0,v}(X)- g\Big( \frac{v}{n}\Big)\Big|.$$
As $\EE T_{0, \lfloor n w\rfloor}(X)$ is coordinate-wise non-decreasing as a function of $w\in \RR_+^2$, and converges to $g(w)$ which is continuous by \cite[Theorem 2.3]{Martin}, we deduce that $w \mapsto \EE T_{0, \lfloor n w\rfloor}(X)$ converges uniformly to $g$ on $[0,1]^2$ by Dini's Theorem. Thus,
\begin{equation}\label{boundLPP1}|\mathcal{T}^{(m)}_n(H) - \hat{\mathcal{T}}^{(m)}_n|\leq m \eps(n),\end{equation} where $\eps(n) \to +\infty$ when $n\to +\infty$.
Now, using the same argument as for the upper bound of \eqref{encadrelasttime}, we see that
\begin{equation}\label{boundLPP2}| \mathcal{T}^{(m)}_n(H) - \mathcal{T}_n(H)|\leq C r m^{1-\frac{1}{\alpha}},\end{equation} for any $||H||_{\ell^{\alpha}}\leq r$. Indeed, if $V$ achieves the supremum in $\mathcal{T}_n(H)$, then taking $V'$ the $m$ largest values of $H^+$ on $V$, we get $$0\leq \mathcal{T}_n(H) - \mathcal{T}_{n}^{(m)}(H) \leq \sum_{v\notin V'} H_v^+.$$ Thus, using \eqref{boundweight}, we get the claim. To summarize, we got by \eqref{encadrelasttime}, \eqref{boundLPP1}, and \eqref{boundLPP2},
$$\big|\frac{1}{n}\EE T(X+nH)^+ - \mathcal{T}_n(H)\big| \leq C r m^{1-\frac{1}{\alpha}} + m\eps(n) +C r(\log n )^{\frac{1}{\alpha}} n^{\alpha-1},$$
for some constant $C>0$ and for any $||H||_{\ell^{\alpha}}\leq r$, which gives finally the claim by taking the $\limsup$ as $n\to +\infty$, and then as $m\to +\infty$. \end{proof}
We can now give a proof of Proposition \ref{convunifLPP}.
\begin{proof}[Proof of Proposition \ref{convunifLPP}]Let $H\in \RR^I$. Note that $X\mapsto T(X+nH)$ is $1$-Lipschitz with respect to $|| \ ||_{\ell^{1}}$ on $\RR^I$. As $|| \ ||_{\ell^1}\leq || \ ||_{\ell^{\alpha}}$ since $\alpha<1$, we deduce that $X\mapsto T(X+nH)$ is also $1$-Lipschitz with respect to $|| \ ||_{\ell^{\alpha}}$. Moreover by Hölder's inequality, $X\mapsto T(X+nH)$ is $\sqrt{n}$-Lipschitz with respect to $|| \ ||_{\ell^2}$. We get by Lemma \ref{conclip0}, for any $t>0$,
$$ \PP( |T(X+nH) - m |> tn)\leq8 \exp(-c p_{\alpha}(t)),$$ where $m$ is the median of $T(X+nH)$, $c$ is some strictly positive constant, and $$p_{\alpha}(t)=\min\Big(\frac{t^2n}{(\log n)^{2(\frac{1}{\alpha}-1)}}, \frac{tn}{(\log n)^{\frac{1}{\alpha}-1}}, n^{\alpha}t^{\alpha}\Big).$$
Integrating this inequality we get,
$$|\EE T(X+nH) - m| = O( (\log n)^{\frac{1}{\alpha}-1} \sqrt{n}),$$ uniformly in $H$. Using the result of Proposition \ref{convunifLPP}, we deduce that for $n$ large enough,
\begin{equation}\label{concLPP} \PP\big( |T(X+nH) - \mathcal{T}_n(H) |> (t+\delta_n)n\big)\leq8 e^{-cn^{\alpha}t^{\alpha}},\end{equation} where $\delta_n = O( (\log n)^{\frac{1}{\alpha}-1} n^{-\frac{1}{2}})$. Let now $r>0$.
Note that
$$H \mapsto n^{-1} |T(X+nH) - \mathcal{T}_n(H) |,$$ is $2$-Lipschitz with respect to $||\ ||_{\ell^1}$ on $\RR^I$. Besides, by Lemma \ref{coveringnb} for any $\eps>0$, the covering number of $r B_{\ell^{\alpha}}$ by $\ell^2$-balls of radii $\eps$ satisfies, $$\log N(r B_{\ell^{\alpha}}, \eps B_{\ell^2}) = O(\log n).$$ Since this estimate is negligible with respect to the concentration bound \eqref{concLPP}, we deduce using an $\eps$-net arguments as in the proofs of Propositions \ref{chaining0} and \ref{chaining1}, that
$$ \PP\Big( \sup_{H\in rB_{\ell^{\alpha}}} \big| \frac{1}{n} T(X+nH) - \mathcal{T}_n(H)\big| >t\Big) \underset{n\to +\infty}{\longrightarrow} 0,$$ which ends the proof of the claim. \end{proof}
\section{Applications to Wigner matrices}\label{Wigner}
We apply in this section Theorem \ref{theoremgene} in the setting of Wigner matrices, and we derive the LDP of Theorems \ref{LDPmsp}, \ref{LDPvp} and \ref{LDPpoly}. In all this section, $X$ will designate a Wigner matrix with the class $\mathcal{S}_{\alpha}$ for some $\alpha \in (0,2]$. It is clear that Theorem \ref{theoremgene} remains valid in the context of Wigner matrices in the class $\mathcal{S}_{\alpha}$, making the according change in the rate function $I_{\alpha}$, by replacing the weight function $|| \ ||_{\ell^{\alpha}}^{\alpha}$ by $W_{\alpha}$, which defines the law of a Wigner matrix in $\mathcal{S}_{\alpha}$ (see \eqref{defW}).
\subsection{Large deviations of the empirical spectral measure}
\begin{proof}[Proof of Theorem \ref{LDPmsp}] From Proposition \ref{chaining0}, we know that assumption $(i)$ of Theorem \ref{theoremgene} is satisfied with $$\forall H \in \mathcal{H}_n^{(\beta)}, \ F_m(H) = \mu_{sc}\boxplus \mu_{n^{1/\alpha} H},$$ and $$\forall H \in \mathcal{H}_n^{(\beta)}, \ f_m(X) = \mu_{X/\sqrt{n}+n^{1/\alpha} H}.$$ where $m$ is the (real) dimension of $\mathcal{H}_n^{(\beta)}$, with the metric $d$ on $\mathcal{P}(\RR)$ defined in \eqref{defdStiel}, and $v(m) = n^{1 + \frac{\alpha}{2}}$.
By Lemma \ref{spvar1intro}, we see that $f_m$ is $n^{-1}$ -Lipschitz with respect to $|| \ ||_{\ell^2}$ on $\mathcal{H}_n^{(\beta)}$ and $d$ on $\mathcal{P}(\RR)$. By the remark \ref{remtheogene} \hyperref[remlipconst]{(c)}, and from the fact that $ \alpha <2$, we deduce that the assumption $(ii)$ of Theorem \ref{theoremgene} holds. Besides, as $\alpha \leq 2$, we have by \cite[Theorem 3.32]{Zhan}
$$ \forall H \in \mathcal{H}_n^{(\beta)}, \ (\tr |H|^{\alpha} )^{1/\alpha} \leq || H ||_{\ell^{\alpha}} .$$ Thus for any $r>0$,
$$ F_m(r B_ {\ell^{\alpha}} ) \subset \{ \mu \in \mathcal{P}(\RR) : \mu |x|^{\alpha} \leq r^{\alpha} \},$$ which shows that $\cup_{m} F_m(rB_{\ell^{\alpha}})$ is relatively compact by Prokhorov's theorem, and that $(iii)$ is verified.
To prove $(iv)$ it is sufficient to show that for a fixed $H\in \mathcal{H}_p^{(\beta)}$, there is a sequence $H_n \in \mathcal{H}_n^{(\beta)}$, $n\geq p$, such that \begin{equation} \label{proptaux} \lim_{n\to +\infty}\mu_{n^{1/\alpha}H_n}= \mu_{p^{1/\alpha}H}, \quad \text{ and }\quad \lim_{n\to +\infty} W_{\alpha}(H_n) = W_{\alpha}(H).\end{equation} Let for any $k\in \NN$, $H_{kp} = \oplus_{i=1}^k k^{-1/\alpha}H\in \mathcal{H}_{kp}^{(\beta)}$. We have $W_{\alpha}(H_{kp}) = W_{\alpha}(H)$, as $W_{\alpha}(\lambda Y) =\lambda^{\alpha} W_{\alpha}(Y)$ for any $\lambda>0$, and $$ \mu_{(kp)^{1/\alpha} H_{kp}} = \mu_{p^{1/\alpha} H}.$$ Now, if $n = kp + l$, with $k\in \NN$ and $1\leq l \leq p$, we define $$ H_{n} = \Big(\frac{kp}{kp+l}\Big)^{1/\alpha} \left(\begin{array}{cc} H_{kp} & 0\\ 0 & 0 \end{array}\right) \in \mathcal{H}_n^{(\beta)}.$$ We have, $$ \mu_{n^{1/\alpha} H_n} = \frac{kp}{kp+l} \mu_{(kp)^{1/\alpha} H_{ kp}} + \frac{l}{kp+l} \delta_0.$$ Thus, $$d( \mu_{n^{1/\alpha} H_n}, \mu_{(kp)^{1/\alpha} H_{kp}}) \leq \frac{2 l }{kp+l} \leq \frac{2p}{n}.$$
Besides, $$ W_{\alpha}(H_{kp}) \geq W_{\alpha}(H_n)\geq \big(1+\frac{1}{k}\big)^{-\frac{1}{\alpha}} W_{\alpha}(H_{kp}).$$ As $W_{\alpha}(H_{kp}) = W_{\alpha}(H)$, and $\mu_{(kp)^{1/\alpha} H_{kp}} = \mu_{p^{1/\alpha}H}$, we get the claim \eqref{proptaux}.
\end{proof}
\subsection{Large deviations of the largest eigenvalue}
\begin{proof}[Proof of Theorem \ref{LDPvp}]
We begin by giving back to $J_{\alpha}$ its variational form. We claim that for any $x\in \RR$,
\begin{equation} \label{Jvar1} J_{\alpha}(x) = \sup_{\delta>0}\inf\big \{ W_{\alpha}(A) : A \in \cup_{n \in \NN} \mathcal{H}_n^{(\beta)}, \ | x - \rho(\lambda_A)|<\delta \big\},\end{equation} where $\rho$ is the function \begin{equation*} \forall x \in \RR, \ \rho(x) = \begin{cases} x +\frac{1}{x} & \text{ if } x \geq 1,\\ 2 & \text{ otherwise.} \end{cases} \end{equation*} Let us prove first that \begin{equation} \label{Jvar}\forall x \in \RR, \ J_{\alpha}(x) = \inf\big \{ W_{\alpha}(A) : A \in \cup_{n \in \NN} \mathcal{H}_n^{(\beta)}, \ x = \rho(\lambda_A) \big\}.\end{equation} When $x<2$, both sides of \eqref{Jvar} are infinite. If $x\geq2$, we denote by $\mathcal{J}_{\alpha}$ the right-hand side of \eqref{Jvar}. The function $x \in (0,1] \mapsto \rho(1/x)$ is the inverse of the Stieltjes transform of $\mu_{sc}$ on $[2,+\infty)$ (see \cite[Example 5.3.2.6]{Guionnet}). Thus, we can write
$$\mathcal{J}_{\alpha}(x) = \inf\big \{ W_{\alpha}(A) : A \in \cup_{n \in \NN} \mathcal{H}_n^{(\beta)}, \ 1/\lambda_A = g_{\mu_{sc}}(x) \big\}.$$
As $W_{\alpha}$ is $\alpha$-homogeneous, and $\lambda_{tA} = t \lambda_A$, for any $t\geq 0$, we get $$ \mathcal{J}_{\alpha}(x) = \mathcal{J}_{\alpha}(1) g_{\mu_{sc}}(x)^{-\alpha}.$$ Thus, $J_{\alpha} = \mathcal{J}_{\alpha}$. As $J_{\alpha}$ is clearly lower semi-continuous, the equality \eqref{Jvar1} holds by the remark \ref{remtheogene} \hyperref[remlsi]{(e)}.
We check now the assumptions of Theorem \ref{theoremgene}. Assumption $(i)$ of Theorem \ref{theoremgene} is met by the result of Proposition \ref{chaining1}, with $$\forall H \in \mathcal{H}_n, \ f_m(H) = \lambda_{X/\sqrt{n}}, \ F_m(H) = \rho(\lambda_H),$$ where as before $m$ is the dimension of $\mathcal{H}_n^{(\beta)}$, and $v(m) = n^{\alpha/2}$.
Weyl's inequality \cite[Theorem III.2.1]{Bhatia} shows that $f_m$ is $n^{-1/2}$-Lipschitz with respect to $|| \ ||_{\ell^2}$, and thus assumption $(ii)$ is satisfied as $\alpha <2$ by the remark \ref{remtheogene} \hyperref[remlipconst]{(c)}. Besides, note that for any $H\in \mathcal{H}_n^{(\beta)}$,
$$|\lambda_H| \leq (\tr |H|^{\alpha})^{1/\alpha} \leq || H ||_{\ell^{\alpha}},$$ where we used in the second inequality the fact that $\alpha \leq 2$ and \cite[Theorem 3.32]{Zhan}. As $\rho$ is non-decreasing, we deduce for any $r>0$ that, $$ \{ F_m(H) : H\in r B_{\ell^{\alpha}} \} \subset [ 2, \rho(r) ],$$ which proves that $(iii)$ is satisfied. To show that $(iv)$ holds, it suffices to observe that if $H \in \mathcal{H}_n^{(\beta)}$, and if we set for any $m\geq n$, \begin{equation} \label{defH} H_m = \left(\begin{array}{cc} H_{n} & 0\\ 0 & 0 \end{array}\right) \in \mathcal{H}_m^{(\beta)},\end{equation} then $W_{\alpha}(H_m) = W_{\alpha}(H)$, and provided $\lambda_H\geq 0$, we have $\lambda_H = \lambda_{H_m}$, so that in particular $\rho(\lambda_H) = \rho(\lambda_{H_m})$. \end{proof}
\subsection{Large deviations of non-commutative polynomials}
Finally, we give a proof of Theorem \ref{LDPpoly}.
\begin{proof}[Proof of Theorem \ref{LDPpoly}]
By a homogeneity argument similar as for the proof of Theorem \ref{LDPvp}, we get for any $x \in \RR$, $$K_{\alpha}(x) = \inf\big\{ W_{\alpha}( \textbf{H}): \textbf{H} \in \cup_{n\in \NN} (\mathcal{H}_n^{(\beta)})^p, x = \tr P_d(\textbf{H}) + \tau(P(\textbf{s}))\big\},$$ where $P_d$ denotes the homogeneous part of degree $d$ of $P$. From the remark \ref{remtheogene} \hyperref[remlsi]{(e)}, we get as $K_{\alpha}$ is lower semi-continuous, that
$$K_{\alpha}(x) =\sup_{\delta>0} \inf\big\{ W_{\alpha}( \textbf{H}): \textbf{H} \in \cup_{n\in \NN} (\mathcal{H}_n^{(\beta)})^p, |x - \tr P_d(\textbf{H}) - \tau(P(\textbf{s}))|< \delta \big\}.$$ Assumption $(i)$ of Theorem \ref{theoremgene} is a consequence of Lemma \ref{convunifpoly} with the speed $v(m) = n^{\alpha(\frac{1}{2}+\frac{1}{d})}$ and $$F_m(\textbf{H}) = \tr P_d(\textbf{H}) + \tau(P(\textbf{s})), \ f_m(\textbf{H}) = \tau_n(P(\textbf{X}/\sqrt{n})),$$ where $m$ is the real dimension of $(\mathcal{H}_n^{(\beta)})^p$.
Let us now prove assumption $(ii)$. Note that by linearity, it suffices to prove assumption $(ii)$ when $P$ is a monomial of total degree $k\geq 1$ less or equal than $d$, which we will assume from now on. If $k=1$, then there are two cases to consider. First we see by Hölder's inequality that $f_m$ is $n^{-1}$-Lipschitz with respect to $|| \ ||_{\ell^2}$. If $d=1$ then $\alpha \in (0,1)$, so that as $v(n) = n^{3\alpha/2}$ in this case. We conclude by remark \ref{remtheogene} \hyperref[remlipconst]{(c)} that assumption $(ii)$ holds. If $d\geq 2$ and $k=1$, then we deduce again by remark \ref{remtheogene} \hyperref[remlipconst]{(c)} that assumption $(ii)$ is fulfilled as $v(n) = n^{\alpha(\frac{1}{2}+\frac{1}{d})}$.
In the case $k\geq 2$, we will need to understand the stability of the function $f_m$ with respect to the Euclidean norm. This is the object of the following lemma.
\begin{Lem}\label{meanvalue} There is a constant $C_{d,p}>0$ depending on $d$ and $p$, such that for any monomial $q \in \CC\langle \textbf{X} \rangle$ of total degree $d\geq 2$, and $\textbf{Y},\textbf{H} \in (\mathcal{H}_n^{(\beta)})^p$,
$$ | \tr q(\textbf{Y} +\textbf{H}) - \tr q( \textbf{Y})| \leq C_{d,p} \big( || \textbf{Y} ||_{2(d-1)}^{d-1} + || \textbf{H}||_{2}^{d-1}\big)||\textbf{H}||_{2},$$
where for any $q>0$, $|| \ ||_q$ denotes the $q$-Schatten norm on $(\mathcal{H}_n^{(\beta)})^p$, defined in \eqref{defschattenp}. \end{Lem} \begin{proof}Let $$ \forall \textbf{H} \in (\mathcal{H}_n^{(\beta)})^p, \ f(\textbf{H}) = \tr q(\textbf{H}).$$ By the mean value theorem, we have
\begin{equation} \label{meanv} | f(\textbf{Y} +\textbf{H}) - f( \textbf{Y})| \leq \max_{0\leq t\leq 1}|| \nabla f(\textbf{Y} + t\textbf{H})||_{2} ||\textbf{H}||_{2}.\end{equation} Note that if $R\in \CC\langle \textbf{X}\rangle$ is a monomial of degree $d-1$ in $\textbf{X}$, then by \eqref{comptrnorm}, we have
$$ \tr |R(\textbf{Z})|^{2} \leq || \textbf{Z} || _{2(d-1)}^{2(d-1)}.$$ As $ \nabla_{X_i} f$ is the sum of at most $d$ monomials of degree $d-1$ in $\textbf{X}$, we get by triangular inequality and the above observation,
$$|| \nabla_{X_i} f(\textbf{Z})||_{2} \leq d || \textbf{Z}||_{2(d-1)}^{d-1}.$$ Thus,
$$ || \nabla f(\textbf{Y} + t\textbf{H})||_{2} \leq p d|| \textbf{Y}+t\textbf{H}||_{2(d-1)}^{d-1}.$$
As $\textbf{Z} \mapsto || \textbf{Z}||_{2(d-1)}^{d-1}$ is convex, we get
$$ || \nabla f(\textbf{Y} + t\textbf{H})||_{2} \leq dp (1+t)^{d-2}\big( ||\textbf{Y}||_{2(d-1)}^{d-1} + t ||\textbf{H}||_{2(d-1)}^{d-1}\big).$$ As $2(d-1) \geq 2$, we have
$$ || \nabla f(\textbf{Y} + t\textbf{H})||_{2} \leq dp 2^{d-2}\big( ||\textbf{Y}||_{2(d-1)}^{d-1} + ||\textbf{H}||_{2}^{d-1}\big).$$ This inequality together with \eqref{meanv} which yields the claim \eqref{meanvalue}. \end{proof}
We come back now at the proof of assumption $(ii)$ of Theorem \ref{theoremgene}. Let $r\geq 1$.
Let $\textbf{K} \in r B_{\ell^{\alpha}}$, and set $\textbf{Y} = \textbf{X} + n^{\frac{1}{2}+\frac{1}{d}} \textbf{K}$. As we assumed $P$ is a monomial of total degree $k$, from the preceding Lemma \ref{meanvalue}, we have for any $\textbf{H} \in (\mathcal{H}_n^{(\beta)})^p$,
\begin{equation*} |f_m(\textbf{Y} +\textbf{H}) -f_m(\textbf{Y})| \leq \frac{c}{n}\Big( ||\textbf{Y}/\sqrt{n}||_{2(k-1)}^{k-1} +||\textbf{H}/\sqrt{n}||_2^{k-1}\Big) || \textbf{H}/\sqrt{n}||_2. \end{equation*} where $c$ is some constant depending $p$ and $d$. Using the fact that $x^{k-1} \leq 1+ x^{d-1}$ for any $1\leq k \leq d$ and $x\geq 0$, we get, \begin{align*}
|f_m(\textbf{Y} +\textbf{H}) -f_m(\textbf{Y})| &\leq \frac{c}{n}( ||\textbf{Y}/\sqrt{n}||_{2(k-1)}^{k-1} +1 ) ||\textbf{H}/\sqrt{n}||_2\\
&+ \frac{c}{n}|| \textbf{H}/\sqrt{n}||_2^d. \end{align*} Let $\delta\in (0,1)$ and $t_{\delta} = \delta n^{\frac{1}{2}+\frac{1}{d}}$. For $\textbf{H} \in t_{\delta} B_{\ell^2}$,
$$ |f_m(\textbf{Y} +\textbf{H}) -f_m(\textbf{Y})| \leq 2c\delta (n^{\frac{1}{d}-1}||\textbf{Y}/\sqrt{n}||_{2(k-1)}^{k-1} +1 ).$$ With the notation of Theorem \ref{theoremgene}, we have
$$\EE \sup_{\textbf{H} \in t_{\delta} B_{\ell^2}} \mathcal{L}_m(\textbf{H}) \leq 2c\delta (n^{\frac{1}{d}-1} \EE||\textbf{Y}/\sqrt{n}||_{2(k-1)}^{k-1} +1 ),$$ where $m$ is the dimension of $(\mathcal{H}_n^{(\beta)})^p$. By convexity, we deduce
$$ \EE ||\textbf{Y}/\sqrt{n}||_{2(k-1)}^{k-1} \leq2^{k-2} \EE ||\textbf{X}/\sqrt{n}||_{2(k-1)}^{k-1} +2^{k-2}n^{\frac{k-1}{d}} ||\textbf{K}||_{2(k-1)}^{k-1}.$$ But by Wigner's theorem (see \cite[Lemma 2.1.6]{Guionnet}),
$$\EE ||\textbf{X}/\sqrt{n}||_{2(k-1)}^{k-1} \leq c_0n^{1/2},$$ for some constant $c_0>0$. As $\textbf{K} \in r B_{\ell^{\alpha}}$ with $\alpha \leq 2$, we deduce as $k\geq 2$,
$$ || \textbf{K}||_{2(k-1)}\leq || \textbf{K}||_2\leq r.$$ Thus, $$\EE \sup_{\textbf{H} \in t_{\delta} B_{\ell^2}} \mathcal{L}_m(\textbf{H}) \leq C\delta (n^{\frac{1}{d}+\frac{1}{2}-1} +r^{d-1}).$$ where $C$ is some positive constant depending on $p$ and $d$. This shows that assumption $(ii)$ is satisfied.
We show now that assumption $(iii)$ holds. Using \eqref{comptrnorm} for $q=d$, we get
$$|\tr P_d(\textbf{H})| \leq C' || \textbf{H}||_{\alpha}^{d/\alpha},$$ where $C'$ is some constant depending on $P$. This proves condition $(iii)$ of Theorem \ref{theoremgene}. To show that the last assumption $(iv)$ is met, it suffices to observe that for any fixed $\textbf{H} \in (\mathcal{H}_n^{(\beta)})^p$, with the same construction as in \eqref{defH}, there is a sequence $\textbf{H}_m \in (\mathcal{H}_m^{(\beta)})^p$, for $m\geq n$, such that $$ \tr P_d(\textbf{H}_m) = \tr P_d(\textbf{H}),$$ and $W_{\alpha}(\textbf{H}) = W_{\alpha}(\textbf{H}_m)$.
\end{proof}
\section{Application to last-passage time}\label{LPPLDP} We prove in this last section Theorem \ref{LDPLPP}. \begin{proof}[Proof of Theorem \ref{LDPLPP}] We will verify the assumptions of Theorem \ref{theoremgenesup}. Assumption $(i)$ holds due to Proposition \ref{convunifLPP} with $v(n) = n^{\alpha}$, and $$\forall X \in \RR^I, \ f_m(X) = \frac{1}{n}T(X^+),\ F_m(X) = \mathcal{T}_n(X),$$ where $\mathcal{T}_n$ is defined in \eqref{defequidetLPP}, $X^+$ denotes the matrix with coefficients $(X^+_v)_v$, and $m$ is the dimension of $\RR^I$.
As $$X\mapsto T(X^+)/n,$$
is $n^{-1/2}$-Lipschitz with respect to $|| \ ||_{\ell^{2}}$, assumption $(ii)$ is satisfied by the remark \ref{remtheogene} \hyperref[remlipconst]{(c)}.
Using the fact that $|| \ ||_{\ell^1} \leq ||\ ||_{\ell^{\alpha}}$ when $\alpha \leq 1$, on $\RR^I$, we see that the condition $(iii)$ of Theorem \ref{theoremgenesup} is met. To prove $(iv)'$, we first observe that
\begin{equation} \label{claimL} L_{\alpha}(x) = \inf \{ || H ||_{\ell^{\alpha}}^{\alpha} : \mathcal{T}_n(H) = x, H\in \RR^I\}.\end{equation} Indeed, since the function $g$ is superadditive by \cite[Proposition 2.1]{Martin}, we deduce that $$\mathcal{T}_n(H) \geq g(1,...,1),$$ for any $H\in \RR^I$. Therefore, both sides of \eqref{claimL} are infinite if $x<g(1,...,1)$. Now if $x\geq g(1,1)$, and $ H\in \RR^ I$ is such that $\mathcal{T}_n(H) = x$, then denoting $\{v_0,...,v_p\}$ the element of $\mathcal{A}_m$ achieving the supremum in \eqref{defequidetLPP}, we get, $$
||H||_{\ell^{\alpha}}^{\alpha} \geq \Big( \sum_{i=0}^{p-1} H_{v_i}^+\Big)^{\alpha} = \Big( x- \sum_{i=0}^{p-1} g \Big( \frac{v_{i+1}-v_i}{n}\Big)\Big)^{\alpha}.$$ Using the superadditivity of $g$, it yields $$
||H||_{\ell^{\alpha}}^{\alpha} \geq (x-g(1,...,1))^{\alpha},$$ with equality for the matrix $H$ whose entries are all zero except $H_{(n,...,n)} = x-g(1,1)$. This proves the equality \eqref{claimL}. In particular, $L_{\alpha}$ is lower semi-continuous and therefore by the remark \ref{remtheogene} \hyperref[remlsi]{(e)}, we deduce,
$$ L_{\alpha}(x) = \sup_{\delta>0}\inf \{ || H ||_{\ell^{\alpha}}^{\alpha} : |\mathcal{T}_n(H) - x|<\delta, H\in \RR^I\}.$$ As the matrices $H \in \RR^I$ with $H_{v}= (x-g(1,...,1))_+\Car_{v = (n,...,n)}$, achieves \eqref{claimL} for any $n$, we deduce,
$$ L_{\alpha}(x) = \sup_{\delta>0} \limsup_{n\to +\infty} \inf\{ || H ||_{\ell^{\alpha}}^{\alpha} : |\mathcal{T}_n(H) - x|<\delta, H\in \RR^I\}.$$ Finally, as $\mathcal{T}_n(H) = \mathcal{T}_n(H^+)$, where $H^+$ is the matrix $(H_v^+)_{v\in \{0,...,n\}^d}$, we get
$$ L_{\alpha}(x) = \sup_{\delta>0} \limsup_{n\to +\infty} \inf\{ || H ||_{\ell^{\alpha}}^{\alpha} : |\mathcal{T}_n(H) - x|<\delta, H\in \RR_+^I\}.$$
This proves the last assumption $(iv)'$ of Theorem \ref{theoremgenesup}.
\end{proof}
{}
\end{document} | arXiv |
Symposium - International Astronomical Union (4)
Proceedings of the International Astronomical Union (2)
European Journal of Anaesthesiology (1)
Highlights of Astronomy (1)
Theory and Practice of Logic Programming (1)
International Astronomical Union (7)
Assertion-based analysis via slicing with ABETS * (system description)
M. ALPUENTE, F. FRECHINA, J. SAPIÑA, D. BALLIS
Journal: Theory and Practice of Logic Programming / Volume 16 / Issue 5-6 / September 2016
Published online by Cambridge University Press: 14 October 2016, pp. 515-532
We present ABETS, an assertion-based, dynamic analyzer that helps diagnose errors in Maude programs. ABETS uses slicing to automatically create reduced versions of both a run's execution trace and executed program, reduced versions in which any information that is not relevant to the bug currently being diagnosed is removed. In addition, ABETS employs runtime assertion checking to automate the identification of bugs so that whenever an assertion is violated, the system automatically infers accurate slicing criteria from the failure. We summarize the main services provided by ABETS, which also include a novel assertion-based facility for program repair that generates suitable program fixes when a state invariant is violated. Finally, we provide an experimental evaluation that shows the performance and effectiveness of the system.
A Brief Update on the CMZoom Survey
C. Battersby, E. Keto, Q. Zhang, S.N. Longmore, J. M. D. Kruijssen, T. Pillai, J. Kauffmann, D. Walker, X. Lu, A. Ginsburg, J. Bally, E.A.C. Mills, J. Henshaw, K. Immer, N. Patel, V. Tolls, A. Walsh, K. Johnston, L. C. Ho
Journal: Proceedings of the International Astronomical Union / Volume 11 / Issue S322 / July 2016
Published online by Cambridge University Press: 09 February 2017, pp. 90-94
The inner few hundred parsecs of the Milky Way, the Central Molecular Zone (CMZ), is our closest laboratory for understanding star formation in the extreme environments (hot, dense, turbulent gas) that once dominated the universe. We present an update on the first large-area survey to expose the sites of star formation across the CMZ at high-resolution in submillimeter wavelengths: the CMZoom survey with the Submillimeter Array (SMA). We identify the locations of dense cores and search for signatures of embedded star formation. CMZoom is a three-year survey in its final year and is mapping out the highest column density regions of the CMZ in dust continuum and a variety of spectral lines around 1.3 mm. CMZoom combines SMA compact and subcompact configurations with single-dish data from BGPS and the APEX telescope, achieving an angular resolution of about 4″ (0.2 pc) and good image fidelity up to large spatial scales.
The link between solenoidal turbulence and slow star formation in G0.253+0.016
C. Federrath, J. M. Rathborne, S. N. Longmore, J. M. D. Kruijssen, J. Bally, Y. Contreras, R. M. Crocker, G. Garay, J. M. Jackson, L. Testi, A. J. Walsh
Star formation in the Galactic disc is primarily controlled by gravity, turbulence, and magnetic fields. It is not clear that this also applies to star formation near the Galactic Centre. Here we determine the turbulence and star formation in the CMZ cloud G0.253+0.016. Using maps of 3 mm dust emission and HNCO intensity-weighted velocity obtained with ALMA, we measure the volume-density variance σρ /ρ 0=1.3±0.5 and turbulent Mach number $\mathcal{M}$ = 11±3. Combining these with turbulence simulations to constrain the plasma β = 0.34±0.35, we reconstruct the turbulence driving parameter b=0.22±0.12 in G0.253+0.016. This low value of b indicates solenoidal (divergence-free) driving of the turbulence in G0.253+0.016. By contrast, typical clouds in the Milky Way disc and spiral arms have a significant compressive (curl-free) driving component (b > 0.4). We speculate that shear causes the solenoidal driving in G0.253+0.016 and show that this may reduce the star formation rate by a factor of 7 compared to nearby clouds.
Effects of levosimendan on myocardial ischaemia–reperfusion injury
D. Yapici, Z. Altunkan, M. Ozeren, E. Bilgin, E. Balli, L. Tamer, N. Doruk, H. Birbicer, D. Apa, U. Oral
Journal: European Journal of Anaesthesiology / Volume 25 / Issue 1 / January 2008
Published online by Cambridge University Press: 01 January 2008, pp. 8-14
Print publication: January 2008
Background and objective
Levosimendan has a cardioprotective action by inducing coronary vasodilatation and preconditioning by opening KATP channels. The aim of this study was to determine whether levosimendan enhances myocardial damage during hypothermic ischaemia and reperfusion in isolated rat hearts.
Twenty-one male Wistar rats were divided into three groups. After surgical preparation, coronary circulation was started by retrograde aortic perfusion using Krebs–Henseleit buffer solution and lasted 15 min. After perfusion Group 1 (control; n = 7) received no further treatment. In Group 2 (non-treated; n = 7), hearts were arrested with cold cardioplegic solution after perfusion and subjected to 60 min of hypothermic global ischaemia followed by 30 min reperfusion. In Group 3 (levosimendan treated; n = 7), levosimendan was added to the buffer solution during perfusion and the hearts were arrested with cold cardioplegic solution and subjected to 60 min of hypothermic global ischaemia followed by 30 min reperfusion. At the end of the reperfusion period, the hearts were prepared for biochemical assays and for histological analysis.
Tissue malondialdehyde levels were significantly lower in the levosimendan-treated group than in the non-treated group (P = 0.019). The tissue Na+–K+ ATPase activity was significantly decreased in the non-treated group than in the levosimendan-treated group (P = 0.027). Tissue myeloperoxidase (MPO) enzyme activity was significantly higher in the non-treated group than in the levosimendan-treated group (P = 0.004). Electron microscopic examination of the hearts showed cardiomyocytic degeneration at the myofibril, mitochondria and sarcoplasmic reticulum in both non-treated and levosimendan-treated groups. The severity of these findings was more extensive in the non-treated group.
Treatment with levosimendan provided better cardioprotection with cold cardioplegic arrest followed by global hypothermic ischaemia in isolated rat hearts.
CARA - The Center for Astrophysical Research in Antarctica
D A Harper, John Bally
Journal: Highlights of Astronomy / Volume 9 / 1992
Published online by Cambridge University Press: 30 March 2016, p. 596
The Center for Astrophysical Research in Antarctica is a new National Science Foundation Science and Technology Center formed to explore and exploit the unique advantages of the Antarctic Plateau for astrophysical observations.
A multi-transition and multi-isotope study of CO in the giant molecular cloud Orion-A
A. Dutrey, A. Castets, G. Duvert, J. Bally, W. D. Langer, R. W. Wilson
Journal: Symposium - International Astronomical Union / Volume 147 / 1991
Published online by Cambridge University Press: 03 August 2017, pp. 405-406
In order to study the excitation conditions in the Orion-A region, we applied an LVG code to 12CO, 13CO and C18O data obtained with the AT&T Bell Laboratories 7-meter telescope in USA (CO isotopes: J = 1 — 0, CS: J = 2 — 1) and the radiotelescope of the "Groupe d'Astrophysique de Grenoble" in France (CO isotopes: J = 2 — 1).
On the structure and kinematics of molecular clouds from large scale mapping of mm-lines
J. Bally, W. D. Langer, R. W. Wilson, A. A. Stark, M. W. Pound
Published online by Cambridge University Press: 03 August 2017, pp. 11-20
Molecular gas in the interior of the Orion superbubble consists of sheets, filaments, and partial shells in which the active star forming dense cloud cores are embedded. The main body of the Orion A and B clouds and at least 14 smaller clouds in Orion region are cometary in appearance suggesting strong interaction with massive stars in the Orion OB association. While the small scale (< 1 pc) structure of the clouds may be determined primarily by internal magnetic fields, gravity, and the effects of outflows from young stellar objects, the large scale morphology and kinematics is affected by the energy injected by massive stars. Supernovae, stellar winds, and radiation have compressed, accelerated, ablated, and dispersed molecular gas over the last 107 years. Most GMC/OB star complexes in the Solar neighborhood exhibit morphological and kinematic properties similar to the Orion region. We argue that energy injection by massive stars plays a vital role in the evolution of the ISM and may be responsible for much of the observed large-scale structure and kinematics of molecular clouds. | CommonCrawl |
Randomness
In common usage, randomness is the apparent or actual lack of definite pattern or predictability in information.[1][2] A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.[note 1] For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy.
Part of a series on statistics
Probability theory
• Probability
• Axioms
• Determinism
• System
• Indeterminism
• Randomness
• Probability space
• Sample space
• Event
• Collectively exhaustive events
• Elementary event
• Mutual exclusivity
• Outcome
• Singleton
• Experiment
• Bernoulli trial
• Probability distribution
• Bernoulli distribution
• Binomial distribution
• Normal distribution
• Probability measure
• Random variable
• Bernoulli process
• Continuous or discrete
• Expected value
• Markov chain
• Observed value
• Random walk
• Stochastic process
• Complementary event
• Joint probability
• Marginal probability
• Conditional probability
• Independence
• Conditional independence
• Law of total probability
• Law of large numbers
• Bayes' theorem
• Boole's inequality
• Venn diagram
• Tree diagram
The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.
Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, particularly in the field of computational science.[3] By analogy, quasi-Monte Carlo methods use quasi-random number generators.
Random selection, when narrowly associated with a simple random sample, is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. A random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, say research subjects, has the same probability of being chosen, then we can say the selection process is random.[2]
According to Ramsey theory, pure randomness is impossible, especially for large structures. Mathematician Theodore Motzkin suggested that "while disorder is more probable in general, complete disorder is impossible".[4] Misunderstanding this can lead to numerous conspiracy theories.[5] Cristian S. Calude stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness".[6] It can be proven that there is infinite hierarchy (in terms of quality or strength) of forms of randomness.[6]
History
Main article: History of randomness
In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate.[7][8] Beyond religion and games of chance, randomness has been attested for sortition since at least ancient Athenian democracy in the form of a kleroterion.[9]
The formalization of odds and chance was perhaps earliest done by the Chinese of 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of calculus had a positive impact on the formal study of randomness. In the 1888 edition of his book The Logic of Chance, John Venn wrote a chapter on The conception of randomness that included his view of the randomness of the digits of pi, by using them to construct a random walk in two dimensions.[10]
The early part of the 20th century saw a rapid growth in the formal analysis of randomness, as various approaches to the mathematical foundations of probability were introduced. In the mid-to-late-20th century, ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness.
Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the 20th century computer scientists began to realize that the deliberate introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases, such randomized algorithms even outperform the best deterministic methods.[11]
In science
Many scientific fields are concerned with randomness:
• Algorithmic probability
• Chaos theory
• Cryptography
• Game theory
• Information theory
• Pattern recognition
• Percolation theory
• Probability theory
• Quantum mechanics
• Random walk
• Statistical mechanics
• Statistics
In the physical sciences
In the 19th century, scientists used the idea of random motions of molecules in the development of statistical mechanics to explain phenomena in thermodynamics and the properties of gases.
According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random.[12] That is, in an experiment that controls all causally relevant parameters, some aspects of the outcome still vary randomly. For example, if a single unstable atom is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time.[13] Thus, quantum mechanics does not specify the outcome of individual experiments, but only the probabilities. Hidden variable theories reject the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case.
In biology
The modern evolutionary synthesis ascribes the observed diversity of life to random genetic mutations followed by natural selection. The latter retains some random mutations in the gene pool due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them. The location of the mutation is not entirely random however as e.g. biologically important regions may be more protected from mutations.[14][15][16]
Several authors also claim that evolution (and sometimes development) requires a specific form of randomness, namely the introduction of qualitatively new behaviors. Instead of the choice of one possibility among several pre-given ones, this randomness corresponds to the formation of new possibilities.[17][18]
The characteristics of an organism arise to some extent deterministically (e.g., under the influence of genes and the environment), and to some extent randomly. For example, the density of freckles that appear on a person's skin is controlled by genes and exposure to light; whereas the exact location of individual freckles seems random.[19]
As far as behavior is concerned, randomness is important if an animal is to behave in a way that is unpredictable to others. For instance, insects in flight tend to move about with random changes in direction, making it difficult for pursuing predators to predict their trajectories.
In mathematics
The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling, but later in connection with physics. Statistics is used to infer the underlying probability distribution of a collection of empirical observations. For the purposes of simulation, it is necessary to have a large supply of random numbers—or means to generate them on demand.
Algorithmic information theory studies, among other topics, what constitutes a random sequence. The central idea is that a string of bits is random if and only if it is shorter than any computer program that can produce that string (Kolmogorov randomness), which means that random strings are those that cannot be compressed. Pioneers of this field include Andrey Kolmogorov and his student Per Martin-Löf, Ray Solomonoff, and Gregory Chaitin. For the notion of infinite sequence, mathematicians generally accept Per Martin-Löf's semi-eponymous definition: An infinite sequence is random if and only if it withstands all recursively enumerable null sets.[20] The other notions of random sequences include, among others, recursive randomness and Schnorr randomness, which are based on recursively computable martingales. It was shown by Yongge Wang that these randomness notions are generally different.[21]
Randomness occurs in numbers such as log(2) and pi. The decimal digits of pi constitute an infinite sequence and "never repeat in a cyclical fashion." Numbers like pi are also considered likely to be normal:
Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.[22]
In statistics
In statistics, randomness is commonly used to create simple random samples. This allows surveys of completely random groups of people to provide realistic data that is reflective of the population. Common methods of doing this include drawing names out of a hat or using a random digit chart (a large table of random digits).
In information science
In information science, irrelevant or meaningless data is considered noise. Noise consists of numerous transient disturbances, with a statistically randomized time distribution.
In communication theory, randomness in a signal is called "noise", and is opposed to that component of its variation that is causally attributable to the source, the signal.
In terms of the development of random networks, for communication randomness rests on the two simple assumptions of Paul Erdős and Alfréd Rényi, who said that there were a fixed number of nodes and this number remained fixed for the life of the network, and that all nodes were equal and linked randomly to each other.[23]
In finance
The random walk hypothesis considers that asset prices in an organized market evolve at random, in the sense that the expected value of their change is zero but the actual value may turn out to be positive or negative. More generally, asset prices are influenced by a variety of unpredictable events in the general economic environment.
In politics
Random selection can be an official method to resolve tied elections in some jurisdictions.[24] Its use in politics originates long ago. Many offices in ancient Athens were chosen by lot instead of modern voting.
Randomness and religion
Randomness can be seen as conflicting with the deterministic ideas of some religions, such as those where the universe is created by an omniscient deity who is aware of all past and future events. If the universe is regarded to have a purpose, then randomness can be seen as impossible. This is one of the rationales for religious opposition to evolution, which states that non-random selection is applied to the results of random genetic variation.
Hindu and Buddhist philosophies state that any event is the result of previous events, as is reflected in the concept of karma. As such, this conception is at odd with the idea of randomness, and any reconciliation between both of them would require an explanation.[25]
In some religious contexts, procedures that are commonly perceived as randomizers are used for divination. Cleromancy uses the casting of bones or dice to reveal what is seen as the will of the gods.
Applications
Main article: Applications of randomness
In most of its mathematical, political, social and religious uses, randomness is used for its innate "fairness" and lack of bias.
Politics: Athenian democracy was based on the concept of isonomia (equality of political rights), and used complex allotment machines to ensure that the positions on the ruling committees that ran Athens were fairly allocated. Allotment is now restricted to selecting jurors in Anglo-Saxon legal systems, and in situations where "fairness" is approximated by randomization, such as selecting jurors and military draft lotteries.
Games: Random numbers were first investigated in the context of gambling, and many randomizing devices, such as dice, shuffling playing cards, and roulette wheels, were first developed for use in gambling. The ability to produce random numbers fairly is vital to electronic gambling, and, as such, the methods used to create them are usually regulated by government Gaming Control Boards. Random drawings are also used to determine lottery winners. In fact, randomness has been used for games of chance throughout history, and to select out individuals for an unwanted task in a fair way (see drawing straws).
Sports: Some sports, including American football, use coin tosses to randomly select starting conditions for games or seed tied teams for postseason play. The National Basketball Association uses a weighted lottery to order teams in its draft.
Mathematics: Random numbers are also employed where their use is mathematically important, such as sampling for opinion polls and for statistical sampling in quality control systems. Computational solutions for some types of problems use random numbers extensively, such as in the Monte Carlo method and in genetic algorithms.
Medicine: Random allocation of a clinical intervention is used to reduce bias in controlled trials (e.g., randomized controlled trials).
Religion: Although not intended to be random, various forms of divination such as cleromancy see what appears to be a random event as a means for a divine being to communicate their will (see also Free will and Determinism for more).
Generation
It is generally accepted that there exist three mechanisms responsible for (apparently) random behavior in systems:
1. Randomness coming from the environment (for example, Brownian motion, but also hardware random number generators).
2. Randomness coming from the initial conditions. This aspect is studied by chaos theory, and is observed in systems whose behavior is very sensitive to small variations in initial conditions (such as pachinko machines and dice).
3. Randomness intrinsically generated by the system. This is also called pseudorandomness, and is the kind used in pseudo-random number generators. There are many algorithms (based on arithmetics or cellular automaton) for generating pseudorandom numbers. The behavior of the system can be determined by knowing the seed state and the algorithm used. These methods are often quicker than getting "true" randomness from the environment.
The many applications of randomness have led to many different methods for generating random data. These methods may vary as to how unpredictable or statistically random they are, and how quickly they can generate random numbers.
Before the advent of computational random number generators, generating large amounts of sufficiently random numbers (which is important in statistics) required a lot of work. Results would sometimes be collected and distributed as random number tables.
Measures and tests
Main article: Randomness tests
There are many practical measures of randomness for a binary sequence. These include measures based on frequency, discrete transforms, complexity, or a mixture of these, such as the tests by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.[26]
Quantum nonlocality has been used to certify the presence of genuine or strong form of randomness in a given string of numbers.[27]
Misconceptions and logical fallacies
Popular perceptions of randomness are frequently mistaken, and are often based on fallacious reasoning or intuitions.
Fallacy: a number is "due"
See also: Coupon collector's problem
This argument is, "In a random selection of numbers, since all numbers eventually appear, those that have not come up yet are 'due', and thus more likely to come up soon." This logic is only correct if applied to a system where numbers that come up are removed from the system, such as when playing cards are drawn and not returned to the deck. In this case, once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be some other card. However, if the jack is returned to the deck, and the deck is thoroughly reshuffled, a jack is as likely to be drawn as any other card. The same applies in any other process where objects are selected independently, and none are removed after each event, such as the roll of a die, a coin toss, or most lottery number selection schemes. Truly random processes such as these do not have memory, which makes it impossible for past outcomes to affect future outcomes. In fact, there is no finite number of trials that can guarantee a success.
Fallacy: a number is "cursed" or "blessed"
See also: Benford's law
In a random sequence of numbers, a number may be said to be cursed because it has come up less often in the past, and so it is thought that it will occur less often in the future. A number may be assumed to be blessed because it has occurred more often than others in the past, and so it is thought likely to come up more often in the future. This logic is valid only if the randomisation might be biased, for example if a die is suspected to be loaded then its failure to roll enough sixes would be evidence of that loading. If the die is known to be fair, then previous rolls can give no indication of future events.
In nature, events rarely occur with a frequency that is known a priori, so observing outcomes to determine which events are more probable makes sense. However, it is fallacious to apply this logic to systems designed and known to make all outcomes equally likely, such as shuffled cards, dice, and roulette wheels.
Fallacy: odds are never dynamic
In the beginning of a scenario, one might calculate the probability of a certain event. However, as soon as one gains more information about the scenario, one may need to re-calculate the probability accordingly.
For example, when being told that a woman has two children, one might be interested in knowing if either of them is a girl, and if yes, what is probability that the other child is also a girl. Considering the two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a probability space illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%).
To be sure, the probability space does illustrate four ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But once it is known that at least one of the children is female, this rules out the boy-boy scenario, leaving only three ways of having the two children: boy-girl, girl-boy, girl-girl. From this, it can be seen only ⅓ of these scenarios would have the other child also be a girl[28] (see Boy or girl paradox for more).
In general, by using a probability space, one is less likely to miss out on possible scenarios, or to neglect the importance of new information. This technique can be used to provide insights in other situations such as the Monty Hall problem, a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as booby prizes behind the others. Once the contestant has chosen a door, the host opens one of the remaining doors to reveal a goat, eliminating that door as an option. With only two doors left (one with the car, the other with another goat), the player must decide to either keep their decision, or to switch and select the other door. Intuitively, one might think the player is choosing between two doors with equal probability, and that the opportunity to choose another door makes no difference. However, an analysis of the probability spaces would reveal that the contestant has received new information, and that changing to the other door would increase their chances of winning.[28]
See also
• Chaitin's constant
• Chance (disambiguation)
• Frequency probability
• Indeterminism
• Nonlinear system
• Probability interpretations
• Probability theory
• Pseudorandomness
• Random.org—generates random numbers using atmospheric noise
• Sortition
Notes
1. Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero.
References
1. The Oxford English Dictionary defines "random" as "Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard."
2. "Definition of randomness | Dictionary.com". www.dictionary.com. Retrieved 21 November 2019.
3. Third Workshop on Monte Carlo Methods, Jun Liu, Professor of Statistics, Harvard University
4. Hans Jürgen Prömel (2005). "Complete Disorder is Impossible: The Mathematical Work of Walter Deuber". Combinatorics, Probability and Computing. Cambridge University Press. 14: 3–16. doi:10.1017/S0963548304006674. S2CID 37243306.
5. Ted.com, (May 2016). The origin of countless conspiracy theories
6. Cristian S. Calude, (2017). "Quantum Randomness: From Practice to Theory and Back" in "The Incomputable Journeys Beyond the Turing Barrier" Editors: S. Barry Cooper, Mariya I. Soskova, 169–181, doi:10.1007/978-3-319-43669-2_11.
7. Handbook to life in ancient Rome by Lesley Adkins 1998 ISBN 0-19-512332-8 page 279
8. Religions of the ancient world by Sarah Iles Johnston 2004 ISBN 0-674-01517-7 page 370
9. Hansen, Mogens Herman (1991). The Athenian Democracy in the Age of Demosthenes. Wiley. p. 230. ISBN 9780631180173.
10. Annotated readings in the history of statistics by Herbert Aron David, 2001 ISBN 0-387-98844-0 page 115. The 1866 edition of Venn's book (on Google books) does not include this chapter.
11. Reinert, Knut (2010). "Concept: Types of algorithms" (PDF). Freie Universität Berlin. Retrieved 20 November 2019.
12. Zeilinger, Anton; Aspelmeyer, Markus; Żukowski, Marek; Brukner, Časlav; Kaltenbaek, Rainer; Paterek, Tomasz; Gröblacher, Simon (April 2007). "An experimental test of non-local realism". Nature. 446 (7138): 871–875. arXiv:0704.2529. Bibcode:2007Natur.446..871G. doi:10.1038/nature05677. ISSN 1476-4687. PMID 17443179. S2CID 4412358.
13. "Each nucleus decays spontaneously, at random, in accordance with the blind workings of chance." Q for Quantum, John Gribbin
14. "Study challenges evolutionary theory that DNA mutations are random". U.C. Davis. Retrieved 12 February 2022.
15. Monroe, J. Grey; Srikant, Thanvi; Carbonell-Bejerano, Pablo; Becker, Claude; Lensink, Mariele; Exposito-Alonso, Moises; Klein, Marie; Hildebrandt, Julia; Neumann, Manuela; Kliebenstein, Daniel; Weng, Mao-Lun; Imbert, Eric; Ågren, Jon; Rutter, Matthew T.; Fenster, Charles B.; Weigel, Detlef (February 2022). "Mutation bias reflects natural selection in Arabidopsis thaliana". Nature. 602 (7895): 101–105. Bibcode:2022Natur.602..101M. doi:10.1038/s41586-021-04269-6. ISSN 1476-4687. PMC 8810380. PMID 35022609.
16. Belfield, Eric J.; Ding, Zhong Jie; Jamieson, Fiona J.C.; Visscher, Anne M.; Zheng, Shao Jian; Mithani, Aziz; Harberd, Nicholas P. (January 2018). "DNA mismatch repair preferentially protects genes from mutation". Genome Research. 28 (1): 66–74. doi:10.1101/gr.219303.116. PMC 5749183. PMID 29233924.
17. Longo, Giuseppe; Montévil, Maël; Kauffman, Stuart (1 January 2012). "No entailing laws, but enablement in the evolution of the biosphere". Proceedings of the 14th annual conference companion on Genetic and evolutionary computation. GECCO '12. New York, NY, US: ACM. pp. 1379–1392. arXiv:1201.2069. CiteSeerX 10.1.1.701.3838. doi:10.1145/2330784.2330946. ISBN 9781450311786. S2CID 15609415.
18. Longo, Giuseppe; Montévil, Maël (1 October 2013). "Extended criticality, phase spaces and enablement in biology". Chaos, Solitons & Fractals. Emergent Critical Brain Dynamics. 55: 64–79. Bibcode:2013CSF....55...64L. doi:10.1016/j.chaos.2013.03.008. S2CID 55589891.
19. Breathnach, A. S. (1982). "A long-term hypopigmentary effect of thorium-X on freckled skin". British Journal of Dermatology. 106 (1): 19–25. doi:10.1111/j.1365-2133.1982.tb00897.x. PMID 7059501. S2CID 72016377. The distribution of freckles seems entirely random, and not associated with any other obviously punctuate anatomical or physiological feature of skin.
20. Martin-Löf, Per (1966). "The definition of random sequences". Information and Control. 9 (6): 602–619. doi:10.1016/S0019-9958(66)80018-9.
21. Yongge Wang: Randomness and Complexity. PhD Thesis, 1996. http://webpages.uncc.edu/yonwang/papers/thesis.pdf
22. "Are the digits of pi random? researcher may hold the key". Lbl.gov. 23 July 2001. Retrieved 27 July 2012.
23. Laszso Barabasi, (2003), Linked, Rich Gets Richer, P81
24. Municipal Elections Act (Ontario, Canada) 1996, c. 32, Sched., s. 62 (3) : "If the recount indicates that two or more candidates who cannot both or all be declared elected to an office have received the same number of votes, the clerk shall choose the successful candidate or candidates by lot."
25. Reichenbach, Bruce (1990). The Law of Karma: A Philosophical Study. Palgrave Macmillan UK. p. 121. ISBN 978-1-349-11899-1.
26. Terry Ritter, Randomness tests: a literature survey. ciphersbyritter.com
27. Pironio, S.; et al. (2010). "Random Numbers Certified by Bell's Theorem". Nature. 464 (7291): 1021–1024. arXiv:0911.3427. Bibcode:2010Natur.464.1021P. doi:10.1038/nature09008. PMID 20393558. S2CID 4300790.
28. Johnson, George (8 June 2008). "Playing the Odds". The New York Times.
Further reading
• Randomness by Deborah J. Bennett. Harvard University Press, 1998. ISBN 0-674-10745-4.
• Random Measures, 4th ed. by Olav Kallenberg. Academic Press, New York, London; Akademie-Verlag, Berlin, 1986. MR0854102.
• The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed. by Donald E. Knuth. Reading, MA: Addison-Wesley, 1997. ISBN 0-201-89684-2.
• Fooled by Randomness, 2nd ed. by Nassim Nicholas Taleb. Thomson Texere, 2004. ISBN 1-58799-190-X.
• Exploring Randomness by Gregory Chaitin. Springer-Verlag London, 2001. ISBN 1-85233-417-7.
• Random by Kenneth Chan includes a "Random Scale" for grading the level of randomness.
• The Drunkard’s Walk: How Randomness Rules our Lives by Leonard Mlodinow. Pantheon Books, New York, 2008. ISBN 978-0-375-42404-5.
External links
Wikiversity has learning resources about Random
Look up randomness in Wiktionary, the free dictionary.
Wikiquote has quotations related to Randomness.
Wikimedia Commons has media related to Randomness.
• QuantumLab Quantum random number generator with single photons as interactive experiment.
• HotBits generates random numbers from radioactive decay.
• QRBG Quantum Random Bit Generator
• QRNG Fast Quantum Random Bit Generator
• Chaitin: Randomness and Mathematical Proof
• A Pseudorandom Number Sequence Test Program (Public Domain)
• Dictionary of the History of Ideas: Chance
• Computing a Glimpse of Randomness
• Chance versus Randomness, from the Stanford Encyclopedia of Philosophy
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| Wikipedia |
Practical device-independent quantum cryptography via entropy accumulation
Experimental quantum homomorphic encryption
Jonas Zeuner, Ioannis Pitsios, … Philip Walther
Experimental symmetric private information retrieval with measurement-device-independent quantum network
Chao Wang, Wen Yu Kon, … Charles C.-W. Lim
Transitioning organizations to post-quantum cryptography
David Joseph, Rafael Misoczki, … Royal Hansen
Single trusted qubit is necessary and sufficient for quantum realization of extremal no-signaling correlations
Ravishankar Ramanathan, Michał Banacki, … Paweł Horodecki
Quantum Search on Encrypted Data Based on Quantum Homomorphic Encryption
Qing Zhou, Songfeng Lu, … Jie Sun
Security keys from paired up nanotube devices
Classically verifiable quantum advantage from a computational Bell test
Gregory D. Kahanamoku-Meyer, Soonwon Choi, … Norman Y. Yao
The entropic cost of quantum generalized measurements
Luca Mancino, Marco Sbroscia, … Marco Barbieri
Fundamental limits of quantum error mitigation
Ryuji Takagi, Suguru Endo, … Mile Gu
Rotem Arnon-Friedman1,
Frédéric Dupuis2,3,
Omar Fawzi4,
Renato Renner1 &
Thomas Vidick5
Nature Communications volume 9, Article number: 459 (2018) Cite this article
Information theory and computation
Device-independent cryptography goes beyond conventional quantum cryptography by providing security that holds independently of the quality of the underlying physical devices. Device-independent protocols are based on the quantum phenomena of non-locality and the violation of Bell inequalities. This high level of security could so far only be established under conditions which are not achievable experimentally. Here we present a property of entropy, termed "entropy accumulation", which asserts that the total amount of entropy of a large system is the sum of its parts. We use this property to prove the security of cryptographic protocols, including device-independent quantum key distribution, while achieving essentially optimal parameters. Recent experimental progress, which enabled loophole-free Bell tests, suggests that the achieved parameters are technologically accessible. Our work hence provides the theoretical groundwork for experimental demonstrations of device-independent cryptography.
Device-independent (DI) quantum cryptographic protocols achieve an unprecedented level of security—with guarantees that hold (almost) irrespective of the quality, or trustworthiness, of the physical devices used to implement them1. The most challenging cryptographic task in which DI security has been considered is quantum key distribution (QKD); we will use this task as an example throughout the manuscript. In DIQKD, the goal of the honest parties, called Alice and Bob, is to create a shared key, unknown to everybody else but them. To execute the protocol, they hold a device consisting of two parts: each part belongs to one of the parties and is kept in their laboratories. Ideally, the device performs measurements on some entangled quantum states it contains.
In real life, the manufacturer of the device, called Eve, can have limited technological abilities (and hence cannot guarantee that the device's actions are exact and non-faulty) or even be malicious. The device itself is far too complex for Alice and Bob to open and assess whether it works as Eve alleges. Alice and Bob must therefore treat the device as a black box with which they can only interact according to the protocol. The protocol must allow them to test the possibly faulty or malicious device and decide whether using it to create their keys poses any security risk. The protocol guarantees that by interacting with the device according to the specified steps, the honest parties will either abort, if they detect a fault, or produce identical and secret keys (with high probability).
Adopting the DI approach is not only crucial for the paranoid cryptographers; even the most skilled experimentalist will recognise that a fully characterised, stable at all times, large-scale quantum device that implements a QKD protocol is extremely hard to build. Indeed, implementations of QKD protocols have been attacked by exploiting imperfections of the devices2,3,4,5. Instead of trying to come up with a "patch" each time an imperfection in the device is detected, DI protocols allow us to break the cycle of attacks and countermeasures.
The most important (in fact necessary) ingredient, which forms the basis of all DI protocols, is a "test for quantumness" based on the violation of a Bell inequality6,7,8,9. A Bell inequality10,11 can be thought of as a game played by the honest parties using the device they share (Fig. 1). Different devices lead to different winning probabilities when playing the game. The game has a special "feature"—there exists a quantum device which achieves a winning probability ωq greater than all classical, local, devices. Hence, if the honest parties observe that their device wins the game with probability ωq they conclude that it must be non-local11. A recent sequence of breakthrough experiments have verified the quantum advantage in such "Bell games" in a loophole-free way12,13,14 (in particular, this means that the experiments were executed without making assumptions that could otherwise be exploited by Eve to compromise the security of a cryptographic protocol).
The Clauser–Horne–Shimony–Holt game34. Alice and Bob input bits, separately, into their parts of the shared device. Each part of the device supplies an output. The game is won if a ⊕ b = x ⋅ y. The optimal winning probability in this game for a classical device is 75%. A quantum device can get up to approximately 86% by measuring the maximally entangled state \(\left| {{\mathrm{\Phi }}^ + } \right\rangle\) = \(\left( {\left| {00} \right\rangle + \left| {11} \right\rangle } \right){\mathrm{/}}\sqrt 2\) with the following measurements: Alice's measurements x = 0 and x = 1 correspond to the Pauli operators σ z and σ x , respectively, and Bob's measurements y = 0 and y = 1 to \(\left( {\sigma _z + \sigma _x} \right){\mathrm{/}}\sqrt 2\) and \(\left( {\sigma _z - \sigma _x} \right){\mathrm{/}}\sqrt 2\), respectively
DI security relies on the following deep but well-established facts. High winning probability in a Bell game not only implies that the measured system is non-local, but more importantly that the kind of non-local correlations it exhibits cannot be shared: the higher the winning probability, the less information any eavesdropper can have about the devices' outcomes. The tradeoff between winning probability and secret randomness, or entropy, can be made quantitative15,16.
The amount of entropy, or secrecy, generated in a single round of the protocol can therefore be calculated from the winning probability in a single game. The major challenge, however, consists in establishing that entropy accumulates additively throughout the multiple rounds of the protocol and use it to bound the total secret randomness produced by the device.
A commonly used assumption17,18,19,20,21 to simplify this task is that the device held by the honest parties makes the same measurements on identical and independent quantum states in every round i ∈ {1, …, n} of the protocol. This implies that the device is initialised in some (unknown) state of the form σ⊗n, i.e., an independent and identically distributed (i.i.d.) state, and that the measurements have a similar structure. In that case, the total entropy created during the protocol can be easily related to the sum of the entropies generated in each round separately (as further explained below).
Unfortunately, although quite convenient for the analysis, the i.i.d. assumption cannot be justified a priori. When considering device-dependent protocols, such as the BB84 protocol22, de Finetti theorems23,24 can often be applied to reduce the task of proving the security in the most general case to that of proving security with the i.i.d. assumption. This approach was unsuccessful in the DI scenario, where known de Finetti theorems23,24,25,26 do not apply. Hence, one cannot simply reduce a general security statement to the one proven under the i.i.d. assumption.
Without this assumption, however, very little is known about the structure of the untrusted device and hence also about its output. As a consequence, previous DIQKD security proofs had to address directly the most general case27,28,29. This led to security statements which are of limited relevance for practical experimental implementations; they are applicable only in an unrealistic regime of parameters, e.g., small amount of tolerable noise and large number of signals.
The work presented here resolves this situation. First, we provide a general information-theoretic tool that quantifies the amount of entropy accumulated during sequential processes which do not necessarily behave identically and independently in each step. We call this result the "Entropy Accumulation Theorem" (EAT). We then show how it can be applied to essentially reduce the problem of proving DI security in the most general case to that of the i.i.d. case. This allows us to establish simple and modular security proofs for DIQKD that yield tight key rates. Our quantitative results imply that the first proofs of principle experiments implementing a DIQKD protocol are within reach with today's state-of-the-art technology. Aside from its application to security proofs, the EAT can be used in other scenarios in quantum information such as the analysis of quantum random access codes.
In the following, we start by explaining the main steps in a security proof of DIQKD under the i.i.d. assumption using well-established techniques. We then present the EAT and show how it can be used to extend the proof and achieve full security (i.e., without assuming an i.i.d. behaviour of the device).
Security under the independent and identically distributed device assumption
The central task when proving the security of cryptographic protocols consists in bounding the information that an adversary, called Eve, may obtain about certain values generated by the protocol, which are supposed to be secret. For QKD, the appropriate measure of knowledge, or rather uncertainty, is given by the smooth conditional min-entropy30 \(H_{{\mathrm{min}}}^\varepsilon \left( {\left. K \right|E} \right)\), where K is the raw data obtained by the honest parties, E the quantum system held by Eve, and ε a parameter describing the security of the protocol. The quantity \(H_{{\mathrm{min}}}^\varepsilon \left( {\left. K \right|E} \right)\) determines the maximal length of the secret key that can be created by the protocol. Hence, proving the security amounts to establishing a lower bound on \(H_{{\mathrm{min}}}^\varepsilon \left( {\left. K \right|E} \right)\). Evaluating \(H_{{\mathrm{min}}}^\varepsilon \left( {\left. K \right|E} \right)\) can be a daunting task, as the adversary's system E is out of our control; in particular, it can have arbitrary dimension and share quantum correlations with the users' devices.
Most protocols consist of a basic building block, or "round", which is repeated a large number, n, of times; in each round i, the classical data K i is generated. The structure of a DIQKD protocol is shown in Box 1. The i.i.d. assumption means that the raw key \(K_1^n = K_1, \ldots ,K_n\) can be treated as a sequence of i.i.d. random variables K i . That is, all the K i are identical and independent of one another. The eavesdropper has side information E i about each K i . In this case, the total conditional min-entropy \(H_{{\mathrm{min}}}^\varepsilon \left( {\left. {K_1^n} \right|E_1^n} \right)\) can be directly related to the single-round conditional von Neumann entropy \(H\left( {\left. {K_i} \right|E_i} \right)\) using the quantum asymptotic equipartition property31 (AEP), which asserts that
$$H_{{\mathrm{min}}}^\varepsilon \left( {K_1^n\left| {E_1^n} \right.} \right) \ge nH\left( {K_i\left| {E_i} \right.} \right) - c_\varepsilon \sqrt n ,$$
where c ε depends only on ε (see the Methods section).
To get a bound on \(H_{{\mathrm{min}}}^\varepsilon \left( {K_1^n\left| {E_1^n} \right.} \right)\), we therefore need to analyse the secrecy, \(H\left( {K_i\left| {E_i} \right.} \right)\), resulting from a single round of the protocol. Depending on the considered scenario, a lower bound on \(H\left( {K_i\left| {E_i} \right.} \right)\) can be found using different techniques. For discrete- and continuous-variable QKD, for example, one can use the entropic uncertainty relations32,33. When dealing with DIQKD, a quantum advantage in a Bell game implies a lower-bound on \(H\left( {K_i\left| {E_i} \right.} \right)\) as discussed above.
The Clauser–Horne–Shimony–Holt (CHSH) game34 (presented in Fig. 1) forms the basis for most DIQKD protocols. For this game, a tight bound on the secrecy as a function of the winning probability in the game was derived19. The bound implies that for any quantum state that wins the CHSH game with probability ω, the entropy evaluated on the state of the system after the game has been played is at least
$$H\left( {K_i\left| {E_i} \right.} \right) \ge 1 - h\left( {\frac{1}{2} + \frac{1}{2}\sqrt {16\omega \left( {\omega - 1} \right) + 3} } \right),$$
where h(⋅) is the binary entropy function. This relation is shown in Fig. 2.
To compute the bound on \(H\left( {K_i\left| {E_i} \right.} \right)\), Alice and Bob need to collect the statistics they observe while running the protocol and estimate the winning probability ω appearing in Eq. (2); assuming the i.i.d. structure this is easily done using Hoeffding's inequality.
The conclusion of this section is the following. The i.i.d. assumption plays a crucial role in the above line of proof: it allows us to reduce the problem of calculating the total secrecy of the raw key created by the device to that of bounding the secrecy produced in one round. Instead of dealing with large-scale quantum systems, we are only required to understand the physics of small systems associated with just one round (as in Eq. (2)). The AEP appearing as Eq. (1) does the rest.
Box 1 Device-independent quantum key distribution protocol (simplified example)
Given: A device for Alice and Bob that can play the chosen Bell game repeatedly
For every round i ∈ [n] do Steps 2–4:
Alice and Bob choose X i ,Y i at random.
They input X i ,Y i to the device and record the outputs A i , B i .
Alice sets K i = A i
Parameter estimation: Alice and Bob estimate the average winning probability in the game from the observed data. If it is below the expected winning probability, ωT, they abort.
Classical post processing: Alice and Bob apply an error correction protocol and a privacy amplification protocol (both classical) on their raw keys K and B.
Extending to full security
Assuming the device behaves in an i.i.d. way goes completely against the DI setting by imposing a severe and even unrealistic restrictions on the implementation of the device. In particular, the assumption implies that the device does not include any, classical or quantum, internal memory (i.e., its actions in one round cannot depend on the previous rounds) and cannot display time-dependent behaviour.
Our main contribution can be phrased as follows.
Theorem (Security of DIQKD, informal): Security of DIQKD in the most general case follows from security under the i.i.d. assumption. Moreover, the dependence of the key rate on the number of exchanged signals, n, is the same as the one in the i.i.d. case, up to terms that scale like \(1{\mathrm{/}}\sqrt n\). The key rates are plotted below.
We now explain the above theorem and how it is derived in more detail. A general device is described by an (unknown) tripartite state \(\rho _{Q_{\mathrm{A}}Q_{\mathrm{B}}E}\), where the bipartite quantum state \(\rho _{Q_{\mathrm{A}}Q_{\mathrm{B}}}\) is shared between Alice and Bob and ρE belongs to Eve, together with the measurements applied to \(\rho _{Q_{\mathrm{A}}Q_{\mathrm{B}}}\) when the device is used. No additional structure is assumed (see Fig. 3).
Secrecy for the Clauser–Horne–Shimony–Holt game vs. winning probability. The amount of secret randomness is quantified by the conditional von Neumann entropy \(H\left( {A\left| E \right.} \right)\). As soon as the winning probability is above the classical threshold of 75% some secret randomness is produced. The analytical bound19 is stated as Eq. (2)
An independent and identically distributed device vs. a general one. An independent and identically distributed (i.i.d.) device (left) is initialised in some (unknown) i.i.d. state σ⊗n; each "small device" is described by one copy of the same bipartite state σ and all copies are measured in the same way. A general device (right) is described by a bipartite quantum state ρ; in contrast to the i.i.d. case, any further division into subsystems is unknown. During the protocol, the state is measured through a sequential process: Alice and Bob use the device in the first round of the protocol and only then proceed to the second round, and so on
As mentioned above, the standard DIQKD protocol proceeds in rounds (recall Box 1): Alice and Bob use their components in the first round of the protocol and only then proceed to the second round, etc. We leverage this structure to bound the amount of entropy produced during a complete execution of the protocol.
To do so, we prove a generalisation of the AEP given in Eq. (1) to a scenario in which, instead of the raw key \(K_1^n = K_1, \ldots ,K_n\) being produced by an i.i.d. process, its parts K i are produced one after the other. In this case, each K i may depend not only on i-th round of the protocol but also on everything that happened in previous rounds (but not on the subsequent ones). We explain our tool, the EAT, in the following.
The entropy accumulation theorem (EAT)
We describe here a simplified and informal version of the EAT, sufficient to understand how it can be used to prove the security of DI protocols; for the most general statements see the Methods section.
We consider processes which can be described by a sequence of n maps \({\cal M}_1, \ldots ,{\cal M}_n\), called "EAT channels", as shown in Fig. 4. Each \({\cal M}_i\) outputs two systems, O i , which describes the information that should be kept secret, and S i , describing some side information leaked by the map, together with a "memory" system R i , which is passed on as an input to the next map \({\cal M}_{i + 1}\). The systems \(S_1^n\) describe the side information created during the process. A further quantum system, denoted by E, represents additional side information correlated to the initial state in the beginning of the considered process. The systems \(O_1^n\) are then the ones in which entropy is accumulated conditioned on the side information \(S_1^n\) and E.
Sequential processes. Each map in the sequence \({\cal M}_i\) outputs O i , which describes the information that should be kept secret, and S i , describing some side information leaked by the map, together with a "memory" system R i , which gets passed on to the next map \({\cal M}_{i + 1}\). In each step, an additional classical value C i is calculated from O i and S i
To bound the entropy of \(O_1^n\), we take into account global statistical properties. These are inferred by tests carried out by the protocol on a small sample of the generated outputs. To incorporate such statistical information, we consider for each round an additional classical value C i computed from O i and S i . Additionally, in each step of the process, the previous outcomes \(O_1^{i - 1}\) are independent of future information S i given all the past side information \(S_1^{i - 1}E\). By choosing O i and S i properly, this condition can be satisfied by sequential protocols such as DIQKD.
The EAT relates the total amount of entropy to the entropy accumulated in one step of the process. The latter is quantified by the minimal, or worst-case, von Neumann entropy produced by the maps \({\cal M}_i\) when acting on an input state that reproduces the correct statistics on C i , i.e., states that satisfy \({\cal M}_i(\sigma )_{C_i} = {\rm freq}(c_1^n)\) where \({\rm freq}(c_1^n)\) is the empirical statistics, or frequency distribution, on \({\cal C}\) defined by \({\rm freq}(c_1^n)(c) = \frac{{\left| {\{ i \in \{ 1, \ldots ,n\} :c_i = c\} } \right|}}{n}\).
To state the explicit result, we define a "min-tradeoff function", fmin, from the set of probability distributions over \({\cal C}\) to the real numbers; fmin should be chosen as a convex differentiable function which is bounded above by the worst-case entropy just described:
$$f_{{\mathrm{min}}}\left( {\rm freq}({c_1^n}) \right) \le H\left( {O_i\left| {S_iE} \right.} \right).$$
An event Ω is defined by a subset of \({\cal C}^n\) and we write \(p_{{\Omega }} = \mathop {\sum}\nolimits_{c_1^n \in {{\Omega }}} ( {\rho _{O_1^nS_1^nE,C_1^n = c_1^n}} )\) for the probability of the event Ω and
$$\rho _{{{|\Omega }}} = \frac{1}{{p_{{\Omega }}}}\mathop {\sum}\limits_{c_1^n \in {{\Omega }}} \left| {c_1^n} \right\rangle\! \left\langle {c_1^n} \right| \otimes \rho _{O_1^nS_1^nE,C_1^n = c_1^n}$$
for the state conditioned on Ω. We further define a set \({\hat{ \Omega }}\) over the set of frequencies such that for all \(c_1^n\), \({\rm freq}(c_1^n) \in {\hat{ \Omega }}\) if and only if \(c_1^n \in { {\Omega }}\).
Theorem (EAT, informal): For any EAT channels, an event Ω such that \({\hat{ \Omega }}\) is a convex set, and a convex min-tradeoff function for which \(f_{{\mathrm{min}}}\left( {\rm freq}({c_1^n}) \right) \ge t\) for any \({\rm freq}(c_1^n) \in {\hat{ \Omega }}\),
$$H_{{\mathrm{min}}}^\varepsilon \left( {O_1^n\left| {S_1^nE} \right.} \right) > nt - v\sqrt n ,$$
where the conditional smooth min-entropy is evaluated on ρ|Ω and v depends on the values \(\left\| {\nabla f_{{\mathrm{min}}}} \right\|_\infty ,\varepsilon ,p_{{\Omega }}\), and the maximal dimension of the systems O i .
Equation (5) asserts that, to first order in n, the total conditional smooth min-entropy is at least n times the value of the min-tradeoff function, evaluated on the empirical statistics observed during the protocol (and hence linear in the number of rounds). In the special case where the EAT channels are independent and identical, the EAT is reduced to the quantum AEP; Eq. (5) is thus a generalisation of Eq. (1).
DIQKD security via the EAT
To gain intuition on how the EAT can be applied to DIQKD, note the following. Define the maps \({\cal M}_i\) to describe the joint behaviour of the honest parties and their respective uncharacterised device while playing a single round of a Bell game such as the CHSH game. Let Ω be the event of the protocol not aborting or a closely related event, e.g., the event that the fraction of CHSH games won is above some threshold ωT. The state for which the smooth min-entropy is evaluated is ρ|Ω, i.e., the state at the end of the protocol conditioned on not aborting. This implies, in particular, a bound on \(H_{{\mathrm{min}}}^\varepsilon \left( {K_1^n\left| E \right.} \right)\).
Furthermore, the condition on the min-tradeoff function stated in Eq. (3) corresponds to the requirement that the distribution of C i equals \(c_1^n\), which ensures that the entropy in Eq. (3) is evaluated on states that can be used to win the CHSH game with probability ωT. Thus, in order to devise an appropriate min-tradeoff function, we can use the relation appearing in Eq. (2); the exact details are given in the Methods section. This results in a tight bound on the amount of entropy created in each step of the protocol. In this sense, we reduce the problem of proving the security of the whole protocol to that of a single round.
Using the EAT we get a bound on \(H_{{\mathrm{min}}}^\varepsilon \left( {K_1^n\left| E \right.} \right)\) which, to first order in n, coincides with the one derived under the i.i.d. assumption and is thus optimal. The final key rate \(r = \ell {\mathrm{/}}n\) (where \(\ell\) is the length of a key) produced in a DIQKD protocol depends on this amount of entropy and the amount of information leaked during standard classical post-processing steps. We plot the results for specific choices of parameters in Fig. 5.
Key rate in a DIQKD protocol. The plots show the key rate r as a function of a the quantum bit error rate Q and b the number of signals n. The completeness error, i.e., the probability that the protocol aborts when using an honest implementation of the device, e.g., due to statistical fluctuations, was chosen to be \(\varepsilon _{{\mathrm{QKD}}}^c = 10^{ - 2}\). The soundness error, which quantifies the maximum tolerated deviation of the actual protocol from a hypothetical one where a perfectly random and completely secret key is produced for Alice and Bob, is taken to be \(\varepsilon _{{\mathrm{QKD}}}^s = 10^{ - 5}\). Both of these values are considered to be realistic and relevant for actual applications. The rates are calculated using Eq. (35) which is derived in the Methods section
To calculate the key rate, one must have some honest implementation of the protocol in mind; this is given by what the experimentalists think (or guess) is happening in their experiment when an adversary is not present. It does not, in any way, restrict the actions of the adversary or the types of imperfections in the device. We consider the following honest implementation, but the analysis can be adapted to any other implementation of interest.
In the realisation of the device, in each round, Alice and Bob share the two-qubit Werner state \(\rho _{Q_{\mathrm{A}}Q_{\mathrm{B}}} = (1 - \nu )\left| {{\mathrm{\Phi }}^ + } \right\rangle \left\langle {{\mathrm{\Phi }}^ + } \right| + \nu {\Bbb I}{\mathrm{/}}4\) resulting from a depolarisation noise acting on the maximally entangled state \(\left| {{\mathrm{\Phi }}^ + } \right\rangle\). In every round, the measurements for X i , Y i ∈ {0, 1} are as described in Fig. 1 and for Y i = 2 Bob's measurement is σ z . The winning probability in the CHSH game (restricted to X i , Y i ∈ {0, 1}) using these measurements on \(\rho _{Q_{\mathrm{A}}Q_{\mathrm{B}}}\) is \(\omega _{{\mathrm{exp}}} = \left[ {2 + \sqrt 2 (1 - \nu )} \right]{\mathrm{/}}4\). The quantum bit error rate \(Q = {\mathrm{Pr}}\left[ {\left. {A_i \ne B_i} \right|\left( {X_i,Y_i} \right) = (0,2)} \right]\) for the above state and measurements is given by Q = ν/2.
The key rate r is plotted in Fig. 5. For n = 1015, the curve essentially coincides with the rate achieved in the asymptotic i.i.d. case19. Since the latter was shown to be optimal19, it provides an upper bound on the key rate and the amount of tolerable noise. Hence, for large enough n our rates become optimal and the protocol can tolerate up to the maximal error rate Q = 7.1%. For comparison, the previously established explicit rates28 are well below the lowest curve presented in Fig. 5, even when the number of signals goes to infinity, with a maximal noise tolerance of 1.6%. Moreover, our key rates are comparable to those achieved in device-dependent QKD protocols35.
The information theoretic tool, the EAT, reveals a novel property of entropy: the operationally relevant total uncertainty about an n-partite system created in a sequential process corresponds to the sum of the entropies of its parts, even without an independence assumption.
Using the EAT, we show that practical and realistic protocols can be used to achieve the unprecedented level of DI security. The next major challenge in experimental implementations is a field demonstration of a DIQKD protocol. This would provide the strongest cryptographic experiment ever realised. The work presented here provides the theoretical groundwork for such experiments. Our quantitative results imply that the first proofs of principle experiments, with small distances and small rates, are within reach with today's state-of-the-art technology, which recently enabled the violation of Bell inequalities in a loophole-free way.
We state here the main theorems of our work and sketch the proofs. Using the explicit expressions given below, one can reproduce the key rates presented in Fig. 5.
The formal statement and proof idea of the EAT
In this section, we are interested in the general question of whether entropy accumulates, in the sense that the operationally relevant total uncertainty about an n-partite system \(O_1^n\) corresponds to the sum of the entropies of its parts O i . The AEP, given in Eq. (1), implies that this is indeed the case to first order in n—under the assumption that the parts O i are identical and independent of each other. Our result shows that entropy accumulation occurs for more general processes, i.e., without an independence assumption, provided one quantifies the uncertainty about the individual systems O i by the von Neumann entropy of a suitably chosen state.
The type of processes that we consider are those that can be described by a sequence of channels, as illustrated in Fig. 4. Such channels are called EAT channels and are formally defined as follows.
Definition 1 (EAT channels): EAT channels \({\cal M}_i:R_{i - 1} \to R_iO_iS_iC_i\), for i ∈ [n], are CPTP (completely positive trace preserving) maps such that for all i ∈ [n]:
C i are finite-dimensional classical systems (random variables). O i , S i , and R i are quantum registers. The dimension of O i is \(d_{O_i}.\)
For any input state \(\sigma _{R_{i - 1}R{\prime}}\), where R′ is a register isomorphic to Ri−1, the output state \(\sigma _{R_iO_iS_iC_iR{\prime}} = \left( {{\cal M}_i \otimes {\Bbb I}_{R{\prime}}} \right)\left( {\sigma _{R_{i - 1}R{\prime}}} \right)\) has the property that the classical value C i can be measured from the system \(\sigma _{O_iS_i}\) without changing the state.
For any initial state \(\rho _{R_0E}^0\), the final state \(\rho _{O_1^nS_1^nC_1^nE} = \left( {\left( {{\mathrm{Tr}}_{R_n} \circ {\cal M}_n \circ \ldots \circ {\cal M}_1} \right) \otimes {\Bbb I}_E} \right)\rho _{R_0E}^0\) fulfils the Markov chain condition \(O_1^{i - 1} \leftrightarrow S_1^{i - 1}E \leftrightarrow S_i\) for each i ∈ [n].
In the above definition, \(O_1^{i - 1} \leftrightarrow S_1^{i - 1}E \leftrightarrow S_i\) if and only if their conditional mutual information is 0, \(I\left( {O_1^{i - 1}:S_i\left| {S_1^{i - 1}} \right.E} \right) = 0\).
Next, one should find an adequate way to quantify the amount of entropy which is accumulated in a single step of the process, i.e., in an application of just one channel. To do so, let p be a probability distribution over \({\cal C}\), where \({\cal C}\) denotes the common alphabet of \(C_1, \ldots ,C_n\), and R′ be a system isomorphic to Ri−1. We define the set of states
$$\begin{array}{*{20}{l}} {{\mathrm{\Sigma }}_i(p)} \hfill & = \hfill & {\left\{ {\sigma _{O_iS_iC_iR_iR{\prime}} = \left( {{\cal M}_i \otimes {\Bbb I}_{R{\prime}}} \right)\left( {\tau _{R_{i - 1}R{\prime}}} \right):} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. {\tau \in {\mathrm{D}}\left( {R_{i - 1} \otimes R{\prime}} \right)\,{\mathrm{and}}\,\sigma _{C_i} = p} \right\},} \hfill \end{array}$$
where \(\sigma _{C_i}\) denotes the probability distribution over \({\cal C}\) with the probabilities given by \(\left\langle c \right|\sigma _{C_i}\left| c \right\rangle\).
The tradeoff functions for the EAT channels are defined below.
Definition 2 (Tradeoff functions): A real function is called a min- or max-tradeoff function for \({\cal M}_i\) if it satisfies
$$f_{{\mathrm{min}}}(p) \le \mathop {{{\mathrm{inf}}}}\limits_{\sigma \in {\mathrm{\Sigma }}_i(p)} H\left( {\left. {O_i} \right|S_iR{\prime}} \right)_\sigma$$
$$f_{{\mathrm{max}}}(p) \ge \mathop {{{\mathrm{sup}}}}\limits_{\sigma \in {\mathrm{\Sigma }}_i(p)} H\left( {O_i\left| {S_iR{\prime}} \right.} \right)_\sigma ,$$
respectively, and if it is convex or concave, respectively. If the set Σ i (p) is empty, then the infimum and supremum are by definition equal to ∞ and −∞, respectively, so that the conditions are trivial.
To get some intuition as to why the above definition is the "correct" one, consider the following classical example. Each EAT channel outputs a single bit O i without any side information S i about it; the system E is empty as well. Every bit can depend on the ones produced previously. We would like to extract randomness out of the sequence \(O_1^n\); for this we should find a lower bound on \(H_{{\mathrm{min}}}^\varepsilon \left( {O_1^n} \right)\).
We ask the following question—given the randomness of O1 which is already accounted for, how much randomness does O2 contribute? One possible guess is the conditional von Neumann entropy \(H\left( {\left. {O_2} \right|O_1} \right) = {\Bbb E}_{o_1,o_2}\left[ { - {\mathrm{log}}\,{\mathrm{Pr}}\left( {\left. {o_2} \right|o_1} \right)} \right]_{}^{}\). If, however, O1 is uniform while O2 is fixed when O1 = 0 and uniform otherwise, then \(H\left( {\left. {O_2} \right|O_1} \right)\) is too optimistic; the amount of extractable randomness is quantified by the smooth min-entropy, which depends on the most probable value of O1O2, and not by an average quantity as the von Neumann entropy.
Another possible guess is a worst-case version of the min-entropy \(H_{{\mathrm{min}}}^{{\rm w.c.}} = {\mathrm{min}}_{o_1,o_2}\left[ { - {\mathrm{log}}\,{\mathrm{Pr}}\left( {\left. {o_2} \right|o_1} \right)} \right]\). This, however, is too pessimistic; when the O i 's are independent of each other, the extractable amount of randomness behaves like the von Neumann entropy in first order, and not like the min-entropy.
We therefore choose an intermediate quantity—\({\mathrm{min}}_{o_1}\,{\Bbb E}_{o_2}\left[ { - {\mathrm{log}}\,{\mathrm{Pr}}\left( {\left. {o_2} \right|o_1} \right)} \right] = {\mathrm{min}}_{o_1}\,H\left( {\left. {O_2} \right|O_1 = o_1} \right)\). That is, this quantity is the von Neumann entropy of O2, evaluated for the worst-case state in the beginning of the second step of the process. The min-tradeoff function defined above is the quantum analogue version of this.
Informally, the min-tradeoff function can be understood as the amount of entropy available from a single round, conditioned on the outputs of the previous rounds. Since we condition on the previous rounds, one can think of the randomness of the current round as independent from past events. Intuitively, this suggests that, by appropriately generalising the proof of the AEP, one can argue that the entropy that is contributed by this independent randomness in each round accumulates.
The formal statement of the EAT is as follows.
Theorem 3 (EAT, formal): Let \({\cal M}_i:R_{i - 1} \to R_iO_iS_iC_i\) for i ∈ [n] be EAT channels, ρ be the final state, Ω an event defined over \({\cal C}^n\), p Ω the probability of Ω in ρ, and ρ|Ω the final state conditioned on Ω. Let εs ∈ (0, 1).
For fmin, a min-tradeoff function for \(\left\{ {{\cal M}_i} \right\}\), \({\hat{ \Omega }} = \left\{ {\left. {\rm freq}({c_1^n}) \right|c_1^n \in { {\Omega }}} \right\}\) convex, and any \(t \in {\Bbb R}\) such that \(f_{{\mathrm{min}}}\left( {\rm freq}({c_1^n}) \right) \ge t\) for any \(c_1^n \in {\cal C}^n\) for which \({\mathrm{Pr}}\left[ {c_1^n} \right]_{{\mathrm{\rho }}_{{{|\Omega }}}} > 0\),
$$H_{{\mathrm{min}}}^\varepsilon \left( {O_1^n\left| {S_1^nE} \right.} \right)_{{\mathrm{\rho }}_{{{|\Omega }}}} > nt - v\sqrt n ,$$
$$v = 2\left( {{\mathrm{log}}\left( {1 + 2d_{O_i}} \right) + \left\lceil {\left\| {\nabla f_{{\mathrm{min}}}} \right\|_\infty } \right\rceil } \right)\sqrt {1 - 2\,{\mathrm{log}}\left( {\varepsilon _{\mathrm{s}} \cdot p_{{\Omega }}} \right)} .$$
Similarly, for fmax a max-tradeoff function and \(t \in {\Bbb R}\) such that \(f_{{\mathrm{max}}}\left({\rm freq}({c_1^n}) \right) \le t\) for any \({\bf{c}} \in {\cal C}^n\) for which \({\mathrm{Pr}}\left[ {\bf{c}} \right]_{{\mathrm{\rho }}_{{{|\Omega }}}} > 0\),
$$H_{{\mathrm{max}}}^\varepsilon \left( {O_1^n\left| {S_1^nE} \right.} \right)_{{\mathrm{\rho }}_{{\mathrm{|\Omega }}}} < nt - v\sqrt n .$$
The two most important properties of the above statement are that the first-order term is linear in n and that t is the von Neumann entropy of a suitable state (as explained above). This implies that the EAT is tight to first order in n.
We remark that the Markov chain conditions are important, in the sense that dropping them completely would render the statement invalid.
We now give a rough proof sketch of the \(H_{{\mathrm{min}}}^\varepsilon\) case; the bound for \(H_{{\mathrm{max}}}^\varepsilon\) follows from an almost identical argument. The proof has a similar structure to that of the quantum AEP31, which we can retrieve as a special case. The proof relies heavily on the "sandwiched" Rényi entropies36,37, which is a family of entropies that we will denote here by H α , where α is a real parameter ranging from \(\frac{1}{2}\) to ∞, and which corresponds to the max-entropy at \(\alpha = \frac{1}{2}\), to the von Neumann entropy when α = 1, and to the min-entropy when α = ∞.
The basic idea is to first lower bound the \(H_{{\mathrm{min}}}^\varepsilon\) term by H α using the following general bound31,38,39,40:
$$H_{{\mathrm{min}}}^\varepsilon \left( {\left. A \right|B} \right) > H_\alpha \left( {\left. A \right|B} \right) - \frac{1}{{\alpha - 1}}O({\mathrm{log}}(1{\mathrm{/}}\varepsilon )).$$
Then, we lower bound the H α term by the von Neumann entropy using the following31,39:
$$H_\alpha \left( {\left. A \right|B} \right) > H\left( {\left. A \right|B} \right) - (\alpha - 1)O\left( {\left( {{\mathrm{log}}\,d_A} \right)^2} \right).$$
Now, we could simply chain these two inequalities and apply them to \(H_{{\mathrm{min}}}^\varepsilon \left( {\left. {O_1^n} \right|S_1^nE} \right)\). However, this would result in a very poor bound due to the \(O\left( {\left( {{\mathrm{log}}\,d_A} \right)^2} \right)\) term in Eq. (13), which in our case would be O(n2). To get the bound we want, we need to reduce this term to O(n); choosing \(\alpha \approx 1 + \frac{1}{{\sqrt n }}\) would then produce a bound with the right scaling.
The trick we use to achieve this is to decompose \(H_\alpha \left( {O_1^n\left| {S_1^nE} \right.} \right)\) into n terms of constant size before applying Eq. (13). In the quantum AEP31, this step is immediate since the state is i.i.d. Here, we must use more sophisticated techniques. Specifically, we use the following chain rule for the sandwiched Rényi entropy to decompose \(H_\alpha \left( {O_1^n\left| {S_1^nE} \right.} \right)\) into n terms:
Theorem 4: Let \(\rho _{RA_1B_1}^0\) be a density operator on R ⊗ A1 ⊗ B1 and \({\cal M} = {\cal M}_{A_2B_2 \leftarrow R}\) be a CPTP map. Assuming that \(\rho _{A_1B_1A_2B_2} = {\cal M}( {\rho _{RA_1B_1}^0} )\) satisfies the Markov condition A1↔B1↔B2,we have
$$H_\alpha \left( {\left. {A_1} \right|B_1} \right)_\rho + \mathop {{{\mathrm{inf}}}}\limits_\omega H_\alpha \left( {\left. {A_2} \right|B_2A_1B_1} \right)_{{\cal M}(\omega )} \le H_\alpha \left( {\left. {A_1A_2} \right|B_1B_2} \right)_\rho ,$$
where the infimum ranges over density operators \(\omega _{RA_1B_1}\) on R ⊗ A1 ⊗ B1. Moreover, if \(\rho _{RA_1B_1}^0\) is pure, then we can optimise over pure states \(\omega _{RA_1B_1}\).
Implementing this proof strategy then yields the following chain of inequalities:
$$\begin{array}{ll} &H_{{\mathrm{min}}}^\varepsilon \left( {\left. {O_1^n} \right|S_1^nE} \right)_\rho \\ &> H_\alpha \left( {\left. {O_1^n} \right|S_1^nE} \right)_\rho - \frac{1}{{\alpha - 1}}O({\mathrm{log}}(1{\mathrm{/}}\varepsilon ))\\ &\!\!\ge \mathop {\sum}\limits_i \mathop {{{\mathrm{inf}}}}\limits_{\omega _{R{\prime}R_i}} H_\alpha \left( {\left. {O_i} \right|S_iR{\prime}} \right)_{{\cal M}_i(\omega )} - \frac{1}{{\alpha - 1}}O({\mathrm{log}}(1{\mathrm{/}}\varepsilon ))\\ & > \mathop {\sum}\limits_i \mathop {{{\mathrm{inf}}}}\limits_\omega H\left( {\left. {O_i} \right|S_iR{\prime}} \right)_{{\cal M}_i(\omega )} - \frac{1}{{\alpha - 1}}O({\mathrm{log}}(1{\mathrm{/}}\varepsilon ))\\ & \!- n(\alpha - 1)O\left( {\left( {{\mathrm{log}}\,d_{O_i}} \right)^2} \right)\\ &\!\!\ge \mathop {\sum}\limits_i \mathop {{{\mathrm{inf}}}}\limits_\omega H\left( {\left. {O_i} \right|S_iR{\prime}} \right)_{{\cal M}_i(\omega )} - O\left( {\sqrt n } \right).\end{array}$$
However, this does not yet take into account the sampling over the C i subsystems. To do this, we tweak the EAT channels \({\cal M}_i\) to output two extra systems D i and \(\bar D_i\) which contain an amount of entropy that depends on the value of C i observed. To define this, let g be an affine lower bound on fmin, let \(\left[ {g_{{\mathrm{min}}},g_{{\mathrm{max}}}} \right]\) be the smallest interval that contains the range of g, and set \(\bar g: = \frac{1}{2}\left( {g_{{\mathrm{min}}} + g_{{\mathrm{max}}}} \right)\). Then, we define \({\cal D}_i:C_i \to C_iD_i\bar D_i\) as
$${\cal D}_i(X) = \mathop {\sum}\limits_c \left\langle c \right|X\left| c \right\rangle \cdot \left| c \right\rangle \left\langle c \right|_{C_i} \otimes \tau (c)_{D_i\bar D_i},$$
where τ(c) is a mixture between a maximally entangled state and a fully mixed state such that the marginal on \(\bar D_i\) is uniform, and such that \(H_\alpha \left( {D_i\left| {\bar D_i} \right.} \right)_{\tau (c)} = \bar g - g(\delta _c)\) (here δ c stands for the distribution with all the weight on element c). To ensure that this is possible, we need to choose \(d_{D_i}\) large enough, and it turns out that setting \(d_{D_i} = \left\lceil {2^{\parallel \nabla g\parallel _\infty }} \right\rceil\) suffices. We can then define a new sequence of EAT channels by \(\bar {\cal M}_i = {\cal D}_i \circ {\cal M}_i\).
Armed with this, we apply the above argument to our new EAT channels. On the one hand, a more sophisticated version of Eq. (12) yields:
$$\begin{array}{*{20}{l}} {H_{{\mathrm{min}}}^\varepsilon \left( {\left. {O_1^n} \right|S_1^nE} \right)_{\rho _{{ {|\Omega }}}}} \hskip -8pt \hfill & \ge \hfill & \hskip -8pt {H_\alpha \left( {\left. {O_1^nD_1^n} \right|S_1^nE\bar D_1^n} \right)_\rho } \hfill \\ {} \hfill & - \hfill & \hskip -8pt {n\bar g + nt - O\left( {\sqrt n } \right){\mathrm{log}}\left( {\frac{1}{{p_{ {\Omega }}}}} \right)} \hfill \end{array}$$
On the other hand, the argument from Eq. (15) can be used here to give
$$H_\alpha \left( {\left. {O_1^nD_1^n} \right|S_1^nE\bar D_1^n} \right)_\rho > n\bar g - O\left( {\sqrt n } \right)\left( {{\mathrm{log}}\left( {d_{O_i}d_{D_i}} \right)^2} \right).$$
Combining these two bounds then yields the theorem.
We remark that some of the concepts used in this work generalise techniques proposed in the recent security proofs for DI cryptography29.
Entropy accumulation protocol
To analyse the key rates of the DIQKD protocol, we first find a lower bound on the amount of entropy accumulated during the run of the protocol, when the honest parties use their device to play the Bell games repeatedly. To this end, we consider the "entropy accumulation protocol" shown in Box 2. This protocol can be seen as the main building block of many DI cryptographic protocols.
The construction of the min-tradeoff function fmin. The plot shows the values of the min-tradeoff function on a slice \(\tilde p(0) + \tilde p(1) = 1 - (1 - \gamma )^{s_{{\mathrm{max}}}}\)
Entropy rate for entropy accumulation protocol. ηopt(ωexp) for γ = 1, smax = 1 and several choices of δest, n, εEA, and εs. We optimise the rates over all parameters which are not explicitly stated in the figure. The dashed line shows the optimal asymptotic (n → ∞) rate under the assumption that the devices are such that Alice, Bob, and Eve share an (unknown) i.i.d. state and n → ∞
The entropy accumulation protocol creates m blocks of bits, each of maximal length \(s_{\max }\). Each block ends (with high probability) with a test round; this is a round in which Alice and Bob play the CHSH game with their device so that they can verify that the device acts as expected. The probability of each round to be a test round is γ. The rest of the rounds are generation rounds, in which Bob chooses a special input for his component of the device. In the end of the protocol, Alice and Bob check whether they had sufficiently many test rounds in which they won the CHSH game. If not, they abort.
We note that the protocol is complete, in the sense that there exists an honest implementation of it (possibly noisy) which does not abort with high probability. Denoting the completeness error, i.e., the probability that the protocol aborts for an honest implementation of the devices D, by \(\varepsilon _{EA}^c\), one can easily show using Hoeffding's inequality that for an honest i.i.d. implementation \(\varepsilon _{EA}^c \le {\mathrm{exp}}\left( { - 2n\delta _{{\mathrm{est}}}^2} \right)\).
Next, we show that the protocol is also sound. That is, for any device D, if the probability that the protocol does not abort is not too small, then the total amount of smooth min-entropy is sufficiently high.
The EAT can be used to bound the total amount of smooth min-entropy, \(H_{{\mathrm{min}}}^{\varepsilon _{\mathrm{s}}}\left( {\left. {A_1^nB_1^n} \right|X_1^nY_1^nT_1^nE} \right)_{\rho _{{ {|\Omega }}}}\), created when running the entropy accumulation protocol, given that it did not abort. Here n denotes the expected number of rounds of the protocol and εs is one of the security parameters (to be fixed later).
Below we use the following notation. For each block j ∈ [m], \(\vec A_j\) denotes the string that includes Alice's outputs in block j (note that the length of this string is unknown, but it is at most \(s_{\max }\)). \(\vec B_j,\vec X_j\), and \(\vec Y_j\) are defined analogously. To use the EAT, we make the following choices of random variables:
$$O_i \to \vec A_j\vec B_j$$
$$S_i \to \vec X_j\vec Y_j\vec T_j$$
$$C_i \to \tilde C_j$$
$$E \to E\;.$$
The event Ω is the event of not aborting the protocol, as given in Step 11 in Box 2:
$$\Omega = \left\{ {\tilde C_1^m|\mathop {\sum}\limits_{j|\tilde C_j = 1} 1 < \left[ {\omega _{{\mathrm{exp}}}\left( {1 - (1 - \gamma )^{s_{{\mathrm{max}}}}} \right) - \delta _{{\mathrm{est}}}} \right] \cdot m} \right\}.$$
The EAT channels are chosen to be
$${\cal M}_j:R_{j - 1} \to R_j\vec A_j\vec B_j\vec X_j\vec Y_j\tilde C_j,$$
where \({\cal M}_j\) describes Steps 2–10 of block j in the entropy accumulation protocol (Box 2). These channels include both the actions made by Alice and Bob as well as the operations made by the device D in these steps. Note that the Device's operations can always be described within the formalism of quantum mechanics, although we do not assume we know them. The registers Rj−1 and R j hold the quantum state of the device in the beginning and the end of the j'th step of the protocol, respectively.
Lemma 5: The channels \({\cal M}_j\) described above are EAT channels.
Proof. For the channels to be EAT channels, they need to fulfil the conditions given in Definition 1. We show that this is indeed the case. First, \(\tilde C_j\) are classical registers with \(\tilde C_j \in \{ 0,1, \bot \}\) and \(d_{\vec A_j} \times d_{\vec B_j} \le 6^{s_{{\mathrm{max}}}}\). Second, \(\tilde C_j\) is determined by the classical registers \(\vec A_j,\vec B_j,\vec X_j,\vec Y_j,\vec T_j\) as shown in Box 2. Therefore, \(\tilde C_j\) can be calculated without modifying the marginal on those registers. The third condition is also fulfilled since the inputs are chosen independently in each round and hence \(\vec A_{1 \ldots j - 1}\vec B_{1 \ldots j - 1} \leftrightarrow \vec X_{1 \ldots j - 1}\vec Y_{1 \ldots j - 1}\vec T_{1 \ldots j - 1}E \leftrightarrow \vec X_j\vec Y_j\vec T_j\) trivially holds.
To continue one should devise a min-tradeoff function. Let \(\tilde p\) be the probability distribution describing \(\tilde C_j\). We remark that due to the structure of our EAT channels, it is sufficient to consider \(\tilde p_{}^{}\) for which \(\tilde p(1) + \tilde p(0) = 1 - (1 - \gamma )^{s_{{\mathrm{max}}}}\) (otherwise the set Σ defined in Eq. (6) is an empty set).
The following lemma gives a lower bound on the von Neumann entropy of the outputs in a single block.
Lemma 6: Let \(\bar s = \frac{{1 - (1 - \gamma )^{s_{{\mathrm{max}}}}}}{\gamma }\) be the expected length of a block and h the binary entropy function. Then,
$$H\left( {\vec A_j\vec B_j|\vec X_j\vec Y_j\vec T_jR{\prime}} \right) \ge \bar s\left[ {1 - h\left( {\frac{1}{2} + \frac{1}{2}\sqrt {16\omega ^ \ast (\omega ^ \ast - 1) + 3} } \right)} \right],$$
where the entropy is evaluated on a state that wins the CHSH game, in the test round, with probability
$$\omega ^ \ast = \frac{{\tilde p\left( 1 \right)}}{{1 - \left( {1 - \gamma } \right)^{s_{{\mathrm{max}}}}}}.$$
Proof sketch. The amount of entropy accumulated in a single round in a block is given in Eq. (2) in the main text. To get the amount of entropy accumulated in a block, one can use the chain rule for the von Neumann entropy. The result is then
$$\begin{array}{*{20}{l}} {H\left( {\vec A_j\vec B_j|\vec X_j\vec Y_j\vec T_jR{\prime}} \right)} \hfill \\ {\begin{array}{*{20}{l}} {} \hfill & \ge \hfill & {\mathop {\sum}\limits_{i \in \left[ {s_{{\mathrm{max}}}} \right]} {\kern 1pt} \left( {1 - \gamma } \right)^{\left( {i - 1} \right)}\left[ {1 - h\left( {\frac{1}{2} + \frac{1}{2}\sqrt {16\omega _i\left( {\omega _i - 1} \right) + 3} } \right)} \right],} \hfill \end{array}} \hfill \end{array}$$
where the pre-factors (1 − γ)(i−1) are attributed to the fact that the entropy in each round is non-zero only if the round is part of the block, i.e., if a test round was not performed before the i'th round in the block, and ω i denotes the winning probability in the i'th round (given that a test was not performed before).
The value of each ω i is not fixed completely given ω*. However, by the operation of the EAT channels the following relation holds:
$$\omega ^ \ast \left( {1 - \left( {1 - \gamma } \right)^{s_{{\mathrm{max}}}}} \right) = \mathop {\sum}\limits_{i \in \left[ {s_{{\mathrm{max}}}} \right]} {\kern 1pt} \gamma \left( {1 - \gamma } \right)^{\left( {i - 1} \right)}\omega _i.$$
To conclude the proof, we thus need to minimise \(H\left( {\vec A_j\vec B_j|\vec X_j\vec Y_j\vec T_jR{\prime}} \right)\) under the above constraint. Using standard techniques, e.g., Lagrange multipliers, one can see that the minimal value of this entropy is achieved for ω i = ω* for all i and the lemma follows.
The bound given in the above lemma can now be used to define the min-tradeoff function \(f_{{\mathrm{min}}}\left( {\tilde p} \right)\). As the derivative of the function plays a role in the final bound, we must make sure it is not too large at any point. This can be enforced by "cutting" the function at a chosen point \(\tilde p_t\) and "gluing" it to a linear function starting at that point, as shown in Fig. 6. \(\tilde p_t\) can be chosen depending on the other parameters such that the total amount of smooth min-entropy is maximal. Following this idea, the resulting min-tradeoff function is given by
$$g\left( {\tilde p} \right) = \\ \left\{ {\begin{array}{*{20}{l}} {\bar s\left[ {1 - h\left( {{\textstyle{1 \over 2}} + {\textstyle{1 \over 2}}\sqrt {16{\textstyle{{\tilde p\left( 1 \right)} \over {1 - \left( {1 - \gamma } \right)^{s_{{\mathrm{max}}}}}}}\left( {{\textstyle{{\tilde p\left( 1 \right)} \over {1 - \left( {1 - \gamma } \right)^{s_{{\mathrm{max}}}}}}} - 1} \right) + 3} } \right)} \right]} \hfill & {{\textstyle{{\tilde p\left( 1 \right)} \over {1 - \left( {1 - \gamma } \right)^{s_{{\mathrm{max}}}}}}} \in \left[ {0,{\textstyle{{2 + \sqrt 2 } \over 4}}} \right]} \hfill \\ {\bar s} \hfill & {{\textstyle{{\tilde p\left( 1 \right)} \over {1 - \left( {1 - \gamma } \right)^{s_{{\mathrm{max}}}}}}} \in \left[ {{\textstyle{{2 + \sqrt 2 } \over 4}},1} \right],} \hfill \end{array}} \right.$$
$$f_{{\mathrm{min}}}\left( {\tilde p,\tilde p_t} \right) = \left\{ {\begin{array}{*{20}{l}} {g\left( {\tilde p} \right)} \hfill & {\tilde p\left( 1 \right) \le \tilde p_t\left( 1 \right)} \hfill \\ {\left. {{\textstyle{{\mathrm{d}} \over {{\mathrm{d}}\tilde p\left( 1 \right)}}}g\left( {\tilde p} \right)} \right|_{\tilde p_t} \cdot \tilde p\left( 1 \right) + \left( {\left. {g\left( {\tilde p_t} \right) - {\textstyle{{\mathrm{d}} \over {{\mathrm{d}}\tilde p\left( 1 \right)}}}g\left( {\tilde p} \right)} \right|_{\tilde p_t} \cdot \tilde p_t\left( 1 \right)} \right)} \hfill & {\tilde p\left( 1 \right) > \tilde p_t\left( 1 \right).} \hfill \end{array}} \right.$$
Let ε EA be the desired error probability of the entropy accumulation protocol. We can then use Theorem 3 to say that either the probability of the protocol aborting is greater than 1 − ε EA or the following bound on the total smooth min-entropy holds:
$$\begin{array}{*{20}{l}} {H_{{\mathrm{min}}}^{\varepsilon _{\mathrm{s}}}\left( {A_1^nB_1^n|X_1^nY_1^nT_1^nE} \right)_{\rho _{|{\mathrm{\Omega }}}}} \hfill & > \hfill & {m \cdot \eta _{{\mathrm{opt}}}\left( {\varepsilon _{\mathrm{s}},\varepsilon _{{\mathrm{EA}}}} \right)} \hfill \\ {} \hfill & = \hfill & {\frac{n}{{\bar s}} \cdot \eta _{{\mathrm{opt}}}\left( {\varepsilon _{\mathrm{s}},\varepsilon _{{\mathrm{EA}}}} \right),} \hfill \end{array}$$
$$\begin{array}{*{20}{l}} {\eta \left( {\tilde p,\tilde p_t,\varepsilon _{\mathrm{s}},\varepsilon _{\mathrm{e}}} \right)} \hfill & = \hfill & {f_{{\mathrm{min}}}\left( {\tilde p,\tilde p_t} \right) - \frac{1}{{\sqrt m }}2\left( {{\mathrm{log}}\left( {1 + 2 \cdot 6^{s_{{\mathrm{max}}}}} \right)} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { + \left\| {\frac{{\mathrm{d}}}{{{\mathrm{d}}\tilde p\left( 1 \right)}}g\left( {\tilde p} \right)} \right\|_\infty } \right)\sqrt {1 - 2{\kern 1pt} {\mathrm{log}}\left( {\varepsilon _{\mathrm{s}} \cdot \varepsilon _{\mathrm{e}}} \right)} ,} \hfill \\ {\eta _{{\mathrm{opt}}}\left( {\varepsilon _{\mathrm{s}},\varepsilon _{\mathrm{e}}} \right)} \hfill & = \hfill & {\mathop {{{\mathrm{max}}}}\limits_{\frac{3}{4} < \tilde p_t\left( 1 \right) < \frac{{2 + \sqrt 2 }}{4}} {\kern 1pt} \eta \left( {\omega _{{\mathrm{exp}}}\left( {1 - \left( {1 - \gamma } \right)^{s_{{\mathrm{max}}}}} \right) - \delta _{{\mathrm{est}}},\tilde p_t,\varepsilon _{\mathrm{s}},\varepsilon _{\mathrm{e}}} \right).} \hfill \end{array}$$
To illustrate the behaviour of the entropy rate ηopt, we plot it as a function of the expected Bell violation ωexp in Fig. 7 for γ = 1 and smax = 1. For comparison, we also plot in Fig. 7 the asymptotic rate (n → ∞) under the assumption that the state of the device is an (unknown) i.i.d. state. In this case, the quantum AEP appearing in Eq. (1) implies that the optimal rate is the von Neumann entropy accumulated in one round of the protocol (as given in Eq. (2)). This rate, appearing as the dashed line in Fig. 7, is an upper bound on the entropy that can be accumulated. One can see that as the number of rounds in the protocol increases, our rate ηopt approaches this optimal rate.
For the calculations of the DIQKD rates later on, we choose \(s_{{\mathrm{max}}} = \left\lceil {{\textstyle{1 \over \gamma }}} \right\rceil\). For this choice, the first-order term of ηopt is linear in n and a short calculation reveals that the second-order term scales, roughly, as \(\sqrt {n/\gamma }\).
Our DIQKD protocol, shown in Box 3, is based on the entropy accumulation protocol described above. In the first part of the protocol Alice and Bob use their devices to produce the raw data, similarly to what is done in the entropy accumulation protocol. The main difference is that Bob's outputs always contains his measurement outcomes (instead of being set to \(\bot\) in the generation rounds); to make the distinction explicit, we denote Bob's outputs in the DIQKD protocol with a tilde, \(\tilde B_1^n\).
We now describe the three post-processing steps, error correction, parameter estimation, and privacy amplification, in more detail.
Box 2 Entropy accumulation protocol (based on the CHSH game)
D—device that can play the CHSH game repeatedly
\(m \in {\Bbb N}_ +\)—number of blocks
\(s_{{{\rm max}}} \in {\Bbb N}_ +\)—maximal length of a block
γ ∈ (0, 1]—probability of a test round
ωexp—expected winning probability in the honest implementation
δest ∈ (0, 1)—width of the statistical confidence interval
For every block j ∈ [m] do Steps 2–10:
Set i = 0 and \(\tilde C_j = \bot\).
If \(i \le s_{{{\rm max}}}\):
Set i = i + 1.
Alice and Bob choose T i ∈ {0, 1} at random such that \({{\rm Pr}}(T_i = 1) = \gamma\).
If T i = 1 Alice and Bob choose inputs X i ∈ {0,1} and Y i ∈ {0,1}.
If T i = 0 they choose inputs X i ∈ {0, 1} and Y i = 2.
Alice and Bob use D with X i ,Y i and record their outputs as A i , B i .
If T i = 0 Bob updates B i to B i = ⊥.
If T i = 1 they set \(\tilde C_j = w\left( {A_i,B_i,X_i,Y_i} \right)\) and \(i = s_{{{\rm max}}} + 1\).
Alice and Bob abort if \(\mathop {\sum}\nolimits_{j \in \left[ m \right]} {\kern 1pt} \tilde C_j \; < \; \left[ {\omega _{{{\rm exp}}}\left( {1 - (1 - \gamma )^{s_{{{\rm max}}}}} \right) - \delta _{{{\rm est}}}} \right] \cdot m\).
Box 3 DIQKD protocol (based on the CHSH game)
EC—error correction protocol which leaks leakEC bits and has error probability εEC
PA—privacy amplification protocol with error probability εPA
For every block j ∈ [m] do Steps 2–8:
If i ≤ smax:
Alice and Bob choose T i ∈ {0, 1} at random such that Pr(T i = 1) = γ.
If T i = 0 they choose inputs X i ∈ {0, 1} and Y i = 2.
Alice and Bob use D with X i , Y i and record their outputs as A i , \(\tilde B_i\).
Error correction: Alice and Bob apply the error correction protocol EC on the outputs \(A_1^n\) and \(\tilde B_1^n\), communicating O in the process. If EC aborts they abort the protocol. Otherwise, they obtain raw keys denoted by KA and KB.
Parameter estimation: Using \(\tilde B_1^n\) and KB, Bob sets \(\tilde C_j = w\left( {A_i,B_i,X_i,Y_i} \right)\) for the blocks with a test round at round i and \(\tilde C_j = \bot\) otherwise. He aborts if \(\mathop {\sum}\nolimits_{j \in \left[ m \right]} {\kern 1pt} \tilde C_j < \left[ {\omega _{{{\rm exp}}}\left( {1 - \left( {1 - \gamma } \right)^{s_{{{\rm max}}}}} \right) - \delta _{{{\rm est}}}} \right] \cdot m\).
Privacy amplification: Alice and Bob apply the privacy amplification protocol PA on KA and KB to create their final keys \(\tilde K_{\mathrm{A}}\) and \(\tilde K_{\mathrm{B}}\) of length \(\ell\) as defined in Eq. (9).
Alice and Bob use an error correction protocol EC to obtain identical raw keys KA and KB from their bits \(A_1^n\), \(\tilde B_1^n\). In our analysis, we use a protocol, based on universal hashing, which minimises the amount of leakage to the adversary41,42. To implement this protocol, Alice chooses a hash function and sends the chosen function and the hashed value of her bits to Bob. We denote this classical communication by O. Bob uses O, together with his prior knowledge \(\tilde B_1^nX_1^nY_1^nT_1^n\), to compute a guess \(\hat A_1^n\) for Alice's bits \(A_1^n\). If EC fails to produce a good guess, Alice and Bob abort; in an honest implementation, this happens with probability at most \(\varepsilon _{{\mathrm{EC}}}^c\). If Alice and Bob do not abort, then they hold raw keys \(K_{\mathrm{A}} = A_1^n\) and \(K_{\mathrm{B}} = \hat A_1^n\) and KA = KB with probability at least 1 − εEC.
Due to the communication from Alice to Bob, leakEC bits of information are leaked to the adversary. The following guarantee holds for the described protocol42:
$${\mathrm{leak}}_{{\mathrm{EC}}} \le H_0^{\varepsilon\prime _{{\mathrm{EC}}} }\left( {A_1^n|\tilde B_1^nX_1^nY_1^nT_1^n} \right) + {\mathrm{log}}\left( {\frac{1}{{\varepsilon _{{\mathrm{EC}}}}}} \right),$$
for \(\varepsilon _{{\mathrm{EC}}}^c = \varepsilon\prime _{{\mathrm{EC}}}+ \varepsilon _{{\mathrm{EC}}}\) and where \(H_0^{\varepsilon\prime _{{\mathrm{EC}}} }\left( {A_1^n|\tilde B_1^nX_1^nY_1^nT_1^n} \right)\) is evaluated on the state in an honest implementation of the protocol. If a larger fraction of errors occur when running the actual DIQKD protocol (for instance due to adversarial interference) the error correction might not succeed, as Bob will not have a sufficient amount of information to obtain a good guess of Alice's bits. If so, this will be detected with probability at least 1 − εEC and the protocol will abort. In an honest implementation of the device, Alice and Bob's outputs in the generation rounds should be highly correlated in order to minimise the leakage of information.
After the error correction step, Bob has all of the relevant information to perform parameter estimation from his data alone, without any further communication with Alice. Using \(\tilde B_1^n\) and KB, Bob sets \(\tilde C_j = w_{{\mathrm{CHSH}}}\left( {\hat A_i,\tilde B_i,X_i,Y_i} \right) = w_{{\mathrm{CHSH}}}\left( {K_{{\mathrm{B}}i},\tilde B_i,X_i,Y_i} \right)\) for the blocks with a test round (which was done at round i of the block) and \(\tilde C_j = \bot\) otherwise. He aborts if the fraction of successful test rounds is too low, that is, if \(\mathop {\sum}\nolimits_{j \in \left[ m \right]} {\kern 1pt} \tilde C_j < \left[ {\omega _{{\mathrm{exp}}}\left( {1 - (1 - \gamma )^{s_{{\mathrm{max}}}}} \right) - \delta _{{\mathrm{est}}}} \right] \cdot m\).
As Bob does the estimation using his guess of Alice's bits, the probability of aborting in this step in an honest implementation, \(\varepsilon _{{\mathrm{PE}}}^c\), is bounded by \(\varepsilon _{{\mathrm{EA}}}^c + \varepsilon _{{\mathrm{EC}}}\).
Privacy amplification
Finally, Alice and Bob use a (quantum-proof) privacy amplification protocol PA (which takes some random seed S as input) to create their final keys \(\tilde K_{\mathrm{A}}\) and \(\tilde K_{\mathrm{B}}\) of length \(\ell\), which are close to ideal keys, i.e., uniformly random and independent of the adversary's knowledge.
For simplicity, we use universal hashing43 as the privacy amplification protocol in the analysis below. Any other quantum-proof strong extractor, e.g., Trevisan's extractor44, can be used for this task and the analysis can be easily adapted.
The secrecy of the final key depends only on the privacy amplification protocol used and the value of \(H_{{\mathrm{min}}}^{\varepsilon _{\mathrm{s}}}\left( {A_1^n|X_1^nY_1^nT_1^nOE} \right)\), evaluated on the state at the end of the protocol, conditioned on not aborting. For universal hashing, for every εPA,εs∈(0, 1), a secure key of maximal length
$$\ell = H_{{\mathrm{min}}}^{\varepsilon _{\mathrm{s}}}\left( {A_1^n|X_1^nY_1^nT_1^nOE} \right) - 2{\kern 1pt} {\mathrm{log}}\frac{1}{{\varepsilon _{{\mathrm{PA}}}}}$$
is produced with probability at least 1 − εPA − εs.
Correctness, secrecy, and overall security of a DIQKD protocol are defined as follows45:
Definition 7 (Correctness): A DIQKD protocol is said to be εcorr-correct, when implemented using a device D, if Alice and Bob's keys, \(\tilde K_{\mathrm{A}}\) and \(\tilde K_{\mathrm{B}}\) respectively, are identical with probability at least 1 − εcorr. That is, \({\mathrm{Pr}}\left( {\tilde K_{\mathrm{A}} \ne \tilde K_{\mathrm{B}}} \right) \le \varepsilon _{{\mathrm{corr}}}\).
Definition 8 (Secrecy): A DIQKD protocol is said to be εsec-secret, when implemented using a device D, if for a key of length l, \(\left( {1 - {\mathrm{Pr}}\left[ {{\mathrm{abort}}} \right]} \right)\left\| {\rho _{\tilde K_{\mathrm{A}}E} - \rho _{U_l} \otimes \rho _E} \right\|_1 \le \varepsilon _{{\mathrm{sec}}}\), where E is a quantum register that may initially be correlated with D.
εsec in the above definition can be understood as the probability that some non-trivial information leaks to the adversary45. If a protocol is εcorr-correct and εsec-secret (for a given D), then it is \(\varepsilon _{{\mathrm{QKD}}}^s\)-correct-and-secret for any \(\varepsilon _{{\mathrm{QKD}}}^s \ge \varepsilon _{{\mathrm{corr}}} + \varepsilon _{{\mathrm{sec}}}\).
Definition 9 (Security): A DIQKD protocol is said to be \(\left( {\varepsilon _{{\mathrm{QKD}}}^s,\varepsilon _{{\mathrm{QKD}}}^c,l} \right)\)-secure if:
(Soundness) For any implementation of the device D it is \(\varepsilon _{{\mathrm{QKD}}}^s\)-correct-and-secret.
(Completeness) There exists an honest implementation of the device D such that the protocol does not abort with probability greater than \(1 - \varepsilon _{{\mathrm{QKD}}}^c\).
Below we show that the following theorem holds.
Theorem 10: The DIQKD protocol described above is \(\left( {\varepsilon _{{\mathrm{QKD}}}^s,\varepsilon _{{\mathrm{QKD}}}^c,\ell } \right)\)-secure, with \(\varepsilon _{{\mathrm{QKD}}}^s \le \varepsilon _{{\mathrm{EC}}} + \varepsilon _{{\mathrm{PA}}} + \varepsilon _{\mathrm{s}} + \varepsilon _{{\mathrm{EA}}}\), \(\varepsilon _{{\mathrm{QKD}}}^c \le \varepsilon _{EC}^c + \varepsilon _{{\mathrm{EA}}}^c + \varepsilon _{{\mathrm{EC}}}\), and
$$\begin{array}{*{20}{l}} \ell \hfill & = \hfill & {\frac{n}{{\bar s}} \cdot \eta _{{\mathrm{opt}}}\left( {\varepsilon _{\mathrm{s}}/4,\varepsilon _{{\mathrm{EA}}} + \varepsilon _{{\mathrm{EC}}}} \right)} \hfill \\ {} \hfill & {} \hfill & { - {\mathrm{leak}}_{{\mathrm{EC}}} - 3{\kern 1pt} {\mathrm{log}}\left( {1 - \sqrt {1 - \left( {\varepsilon _{\mathrm{s}}/4} \right)^2} } \right) - \gamma (n + t)} \hfill \\ {} \hfill & {} \hfill & { - \sqrt {n + t}\ 2{\kern 1pt} {\mathrm{log}}{\kern 1pt} 7\sqrt {1 - 2{\kern 1pt} {\mathrm{log}}\left( {\left( {\varepsilon _{\mathrm{s}}/4 - \sqrt {\varepsilon _{\mathrm{t}}} } \right) \cdot \left( {\varepsilon _{{\mathrm{EA}}} + \varepsilon _{{\mathrm{EC}}}} \right)} \right)} } \hfill \\ {} \hfill & {} \hfill & { - 2{\kern 1pt} {\mathrm{log}}\left( {\varepsilon _{{\mathrm{PA}}}^{ - 1}} \right),} \hfill \end{array}$$
where \(t = \sqrt { - m\left( {1 - \gamma } \right)^2{\kern 1pt} {\mathrm{log}}{\kern 1pt} \varepsilon _{\mathrm{t}}{\mathrm{/}}2\gamma ^2}\) for any εt ∈ (0, 1).
We now explain the steps taken to prove Theorem 10. The completeness part follows trivially from the completeness of the "subprotocols".
To establish soundness, first note that by definition, as long as the protocol does not abort it produces a key of length \(\ell\). Therefore, it remains to verify correctness, which depends on the error correction step, and security, which is based on the privacy amplification step. To prove security we start with Lemma 11, in which we assume that the error correction step is successful. We then use it to prove soundness in Lemma 12.
Let \({\tilde{ \Omega }}\) denote the event of the DIQKD protocol not aborting and the EC protocol being successful, and let \(\tilde \rho _{A_1^n\tilde B_1^nX_1^nY_1^nT_1^nO_1^nE|{\tilde{ \Omega }}}\) be the state at the end of the protocol, conditioned on this event.
Success of the privacy amplification step relies on the min-entropy \(H_{{\mathrm{min}}}^{\varepsilon _{\mathrm{s}}}\left( {A_1^n|X_1^nY_1^nT_1^nOE} \right)_{\tilde \rho _{|{\tilde{ \Omega }}}}\) being sufficiently large. The following lemma connects this quantity to \(H_{{\mathrm{min}}}^{\frac{{\varepsilon _{\mathrm{s}}}}{4}}\left( {A_1^nB_1^n|X_1^nY_1^nT_1^nE} \right)_{\rho _{|{ {\Omega }}}}\), on which a lower bound is provided in Eq. (31) above.
Lemma 11: For any device D, let \(\tilde \rho\) be the state generated in the protocol right before the privacy amplification step. Let \(\tilde \rho _{|{\tilde{ \Omega }}}\) be the state conditioned on not aborting the protocol and success of the EC protocol. Then, for any εEA, εEC, εs, εt ∈ (0, 1), either the protocol aborts with probability greater than 1 − εEA − εEC or
$$\begin{array}{*{20}{l}} {H_{{\mathrm{min}}}^{\varepsilon _{\mathrm{s}}}\left( {A_1^n|X_1^nY_1^nT_1^nOE} \right)_{\tilde \rho _{|{\tilde{ \Omega }}}}} \hskip -8pt\hfill & \ge \hfill & \hskip -8pt{\frac{n}{{\bar s}} \cdot \eta _{{\mathrm{opt}}}\left( {\varepsilon _{\mathrm{s}}/4,\varepsilon _{{\mathrm{EA}}} + \varepsilon _{{\mathrm{EC}}}} \right)} \hfill \\ {} \hskip -8pt \hfill & {} \hfill &\hskip -8pt { - {\mathrm{leak}}_{{\mathrm{EC}}} - 3{\kern 1pt} {\mathrm{log}}\left( {1 - \sqrt {1 - \left( {\varepsilon _{\mathrm{s}}/4} \right)^2} } \right)} \hfill \\ {} \hfill & {}\hskip -8pt \hfill &\hskip -8pt { - \gamma \left( {n + t} \right)} \hfill \\ {} \hfill & {}\hskip -8pt \hfill &\hskip -8pt { - \sqrt {n + t} \;2{\kern 1pt} {\mathrm{log}}{\kern 1pt} 7\sqrt {1 - 2{\kern 1pt} {\mathrm{log}}\left( {\left( {\varepsilon _{\mathrm{s}}/4 - \sqrt {\varepsilon _{\mathrm{t}}} } \right) \cdot \left( {\varepsilon _{{\mathrm{EA}}} + \varepsilon _{{\mathrm{EC}}}} \right)} \right)} .} \hfill \end{array}$$
Proof sketch. Before deriving a bound on the entropy of interest, we remark that the t is chosen such that the probability that the actual number of rounds in the protocol, N, is larger than the expected number of rounds n plus t is εt. The above value for t can be derived by noticing that the sizes of the blocks are i.i.d. random variables which take values in [1, 1/γ].
The key idea of the proof is to consider the following events:
Ω: the event of not aborting in the entropy accumulation protocol. This happens when the Bell violation, calculated using Alice and Bob's outputs (and inputs), is sufficiently high.
\({\hat{ \Omega }}\): Suppose Alice and Bob run the entropy accumulation protocol, and then execute the EC protocol. The event \({\hat{ \Omega }}\) is defined by Ω and \(K_{\mathrm{B}} = A_1^n\).
\({\tilde{ \Omega }}\): the event of not aborting the DIQKD protocol and \(K_{\mathrm{B}} = A_1^n\).
The state \(\rho _{|{\hat{ \Omega }}}\) then denotes the state at the end of the entropy accumulation protocol conditioned on \({\hat{ \Omega }}\).
Using a sequence of chain rules for smooth entropies46 and the fact that \(\tilde \rho _{A_1^nX_1^nY_1^nT_1^nE|{\tilde{ \Omega }}} = \rho _{A_1^nX_1^nY_1^nT_1^nE|{\hat{ \Omega }}}\) (\(\tilde B_1^n\) and \(B_1^n\) were traced out from \(\tilde \rho\) and ρ, respectively), one can conclude
$$\begin{array}{*{20}{l}} {H_{\min }^{\varepsilon _{\mathrm{s}}}\left( {A_1^n|X_1^nY_1^nT_1^nOE} \right)_{\tilde \rho _{|{\tilde{ \Omega }}}}} \hskip -8pt \hfill & \ge \hfill & \hskip -8pt {H_{\min }^{\frac{{\varepsilon _{\mathrm{s}}}}{4}}\left( {A_1^nB_1^n|X_1^nY_1^nT_1^nE} \right)_{\rho _{|\hat \Omega }}} \hfill \\ {} \hskip -8pt\hfill & {} \hfill & \hskip -8pt { - H_{\max }^{\frac{{\varepsilon _{\mathrm{s}}}}{4}}\left( {B_1^n|T_1^nE} \right)_{\rho _{|\hat \Omega }} - {\mathrm{leak}}_{{\mathrm{EC}}}} \hfill \\ {} \hskip -8pt \hfill & {} \hfill & \hskip -8pt { - 3{\kern 1pt} {\mathrm{log}}\left( {1 - \sqrt {1 - \left( {\varepsilon _{\mathrm{s}}/4} \right)^2} } \right).} \hfill \end{array}$$
\(H_{{\mathrm{max}}}^{\frac{{\varepsilon _{\mathrm{s}}}}{4}}\left( {B_1^n|T_1^nE} \right)_{\rho _{|{\hat{ \Omega }}}}\) can be bounded from above. The intuition is that \(B_i \ne \bot\) only when T i = 0, which happens with probability γ. The exact bound can be calculated using the EAT and is given by
$$\begin{array}{*{20}{l}} {H_{{\mathrm{max}}}^{\frac{{\varepsilon _{\mathrm{s}}}}{4}}\left( {B_1^n|T_1^nE} \right)_{\rho _{|{\hat{\mathrm \Omega }}}}} \hfill & < \hfill & {\gamma \left( {n + t} \right)} \hfill \\ {} \hfill & {} \hfill & { + \sqrt {n + t} \;2{\kern 1pt} {\mathrm{log}}{\kern 1pt} 7\sqrt {1 - 2{\kern 1pt} {\mathrm{log}}\left( {\left( {\varepsilon _{\mathrm{s}}/4 - \sqrt {\varepsilon _{\mathrm{t}}} } \right) \cdot \left( {\varepsilon _{{\mathrm{EA}}} + \varepsilon _{{\mathrm{EC}}}} \right)} \right)} .} \hfill \end{array}$$
The above steps together with Eq. (31) conclude the proof.
Using Lemma 11, one can prove that our DIQKD protocol is sound.
Lemma 12: For any device D let \(\tilde \rho\) be the state generated by the DIQKD protocol. Then either the protocol aborts with probability greater than 1 − εEA − εEC or it is (εEC + εPA + εs)-correct-and-secret while producing keys of length \(\ell\), as defined in Eq. (35).
Proof sketch. Assume the DIQKD protocol did not abort. We consider two cases. First, assume that the EC protocol was not successful (but did not abort). Then Alice and Bob's final keys might not be identical. This happens with probability at most εEC. Otherwise, assume the EC protocol was successful, i.e., \(K_{\mathrm{B}} = A_1^n\). In that case, Alice and Bob's keys must be identical also after the final privacy amplification step.
The secrecy depends only on the privacy amplification step, and for universal hashing a secure key is produced as long as Eq. (34) holds. Hence, a uniform and independent key of length \(\ell\) as in Eq. (35) is produced by the privacy amplification step unless the smooth min-entropy is not high enough or the privacy amplification protocol was not successful, which happens with probability at most εPA + εs.
According to Lemma 11, either the protocol aborts with probability greater than 1 − εEA − εEC or the entropy is sufficiently high to create the secret key.
The expected key rates appearing in Fig. 5 in the main text are given by \(r = \ell {\mathrm{/}}n\). The key rate depends on the amount of leakage of information due to the error correction step, which in turn depends on the honest implementation of the protocol as mentioned above. To have an explicit bound, we consider the honest implementation described in the main text. Using Eq. (33) and the AEP, one can show that the amount of leakage in the error correction step is then given by
$$\begin{array}{*{20}{l}} {{\mathrm{leak}}_{{\mathrm{EC}}}} \hfill & \le \hfill & {\left( {n + t} \right) \cdot \left[ {\left( {1 - \gamma } \right)h\left( Q \right) + \gamma h\left( {\omega _{{\mathrm{exp}}}} \right)} \right]} \hfill \\ {} \hfill & {} \hfill & { + \sqrt {n + t} \;4{\kern 1pt} {\mathrm{log}}\left( {2\sqrt 2 + 1} \right)\,\sqrt {2{\kern 1pt} {\mathrm{log}}\left( {8/\left( {\varepsilon\prime _{{\mathrm{EC}}} - 2\sqrt {\varepsilon _{\mathrm{t}}} } \right)^2} \right)} } \hfill \\ {} \hfill & {} \hfill & { + {\mathrm{log}}\left( {8/(\varepsilon{\prime}_{{\mathrm{EC}}})^2 + 2/\left( {2 - \varepsilon\prime _{{\mathrm{EC}}} } \right)} \right) + {\mathrm{log}}\left( {\frac{1}{{\varepsilon _{{\mathrm{EC}}}}}} \right).} \hfill \end{array}$$
To get the optimal key rates, one should fix the parameters of interest (e.g., \(\varepsilon _{{\mathrm{QKD}}}^s\), \(\varepsilon _{{\mathrm{QKD}}}^c\), and n) and optimise over all other parameters.
DI randomness expansion
The entropy accumulation protocol can be used to perform DI randomness expansion as well. In a DI randomness expansion protocol, the honest parties start with a short seed of perfect randomness and use it to create a longer random secret string. For the purposes of randomness expansion, we may assume that the parties are co-located, therefore, the main difference from the DIQKD scheme is that there is no need for error correction (and hence there is no leakage of information due to public communication).
In order to minimise the amount of randomness required to execute the protocol, we adapt the main entropy accumulation protocol by deterministically choosing inputs in the generation rounds X i ,Y i ∈{0,1}. In particular, there is no use for the input 2 to Bob's device, and no randomness is required for the generation rounds. Aside from the last step of privacy amplification, the remainder of the protocol is essentially the same as the entropy accumulation protocol.
The plotted entropy rates in Fig. 7 are therefore also the ones relevant for a DI randomness expansion.
Since we are concerned here not only with generating randomness but also with expanding the amount of randomness initially available to the users of the protocol, we should also evaluate the total number of random bits that are needed to execute the protocol. Random bits are required to select which rounds are generation rounds, i.e., the random variable \(T_1^n\), to select inputs to the devices in the testing rounds, i.e., those for which T i = 0, and to select the seed for the quantum proof extractor used for privacy amplification. All of these can be accounted for using standard techniques and so we omit the detailed explanation and formulas.
No data sets were generated or analysed during the current study.
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We thank Asher Arnon for the illustrations presented in Figs. 1, 4, and 5. R.A.F. and R.R. were supported by the Stellenbosch Institute for Advanced Study (STIAS), by the European Commission via the project "RAQUEL", by the Swiss National Science Foundation (grant No. 200020–135048) and the National Centre of Competence in Research "Quantum Science and Technology", by the European Research Council (grant No. 258932), and by the US Air Force Office of Scientific Research (grant No. FA9550-16-1-0245). F.D. acknowledges the financial support of the Czech Science Foundation (GA ČR) project no GA16-22211S and of the European Commission FP7 Project RAQUEL (grant No. 323970). O.F. acknowledges support from the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). T.V. was partially supported by NSF CAREER Grant CCF-1553477, an AFOSR YIP award, the IQIM, and NSF Physics Frontiers Center (NFS Grant PHY-1125565) with support of the Gordon and Betty Moore Foundation (GBMF-12500028).
Institute for Theoretical Physics, ETH-Zürich, Wolfgang-Pauli-Str. 27, 8093, Zürich, Switzerland
Rotem Arnon-Friedman & Renato Renner
Faculty of Informatics, Masaryk University, Brno, Czech Republic
Frédéric Dupuis
CNRS, LORIA, Université de Lorraine, Campus scientifique, BP 239, 54506, Vandoeuvre-lès-Nancy Cedex, France
Laboratoire de l'Informatique du Parallélisme, LIP, ENS de Lyon, 46 Allee d'Italie, 69364, Lyon Cedex 07, France
Omar Fawzi
Department of Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA, 91125, USA
Thomas Vidick
Rotem Arnon-Friedman
Renato Renner
R.A.F., F.D., O.F., R.R., and T.V. contributed equally to this work.
Correspondence to Rotem Arnon-Friedman.
The authors declare that they have no competing financial interests.
Arnon-Friedman, R., Dupuis, F., Fawzi, O. et al. Practical device-independent quantum cryptography via entropy accumulation. Nat Commun 9, 459 (2018). https://doi.org/10.1038/s41467-017-02307-4
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Wei Zhang
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Nature Physics News & Views 01 Feb 2018 | CommonCrawl |
Mixed finite element method
In numerical analysis, the mixed finite element method, is a type of finite element method in which extra fields to be solved are introduced during the posing a partial differential equation problem. Somewhat related is the hybrid finite element method. The extra fields are constrained by using Lagrange multiplier fields. To be distinguished from the mixed finite element method, usual finite element methods that do not introduce such extra fields are also called irreducible or primal finite element methods.[1] The mixed finite element method is efficient for some problems that would be numerically ill-posed if discretized by using the irreducible finite element method; one example of such problems is to compute the stress and strain fields in an almost incompressible elastic body.
In mixed methods, the Lagrange multiplier fields inside the elements, usually enforcing the applicable partial differential equations.[2] This results in a saddle point system having negative pivots and eigenvalues, rendering the system matrix to be non-definite which results in complications in solving for it. In sparse direct solvers, pivoting may be needed, where ultimately the resulting matrix has 2x2 blocks on the diagonal,[3] rather than a working towards a completely pure LLH Cholesky decomposition for positive definite symmetric or Hermitian systems. Pivoting may result in unpredictable memory usage increases. For iterative solvers, only GMRES based solvers work, rather than slightly "cheaper" MINRES based solvers.
In hybrid methods, the Lagrange fields are for jumps of fields between elements, living on the boundary of the elements, weakly enforcing continuity; continuity from fields in the elements does not need to be enforced through shared degrees of freedom between elements anymore. Both mixing and hybridization can be applied simultaneously.[4] These enforcements are "weak", i.e. occur upon having the solutions and possibly only at some points or e.g. matching moment integral conditions, rather than "strong" in which case the conditions are fulfilled directly in the type of solutions sought. Apart from the harmonics (usually semi-trivial local solution to the homogeneous equations at zero loads), hybridization allows for static Guyan condensation of the discontinuous fields internal to the elements, reducing the number of degrees of freedom, and moreover reducing or eliminating the number of negative eigenvalues and pivots resulting from application of the mixed method.
References
1. Olek C Zienkiewicz, Robert L Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier.
2. Arnold, Douglas. "Douglas Arnold 2016 Woudschoten Conference" (PDF).{{cite web}}: CS1 maint: url-status (link)
3. Lungten, S.; Schilders, W.H.A.; Maubach, J.M.L. (2016-08-01). "Sparse block factorization of saddle point matrices". Linear Algebra and Its Applications. 502: 214–242. doi:10.1016/j.laa.2015.07.042. ISSN 0024-3795.
4. "2019 Feb 1, Bernardo Cockburn, University of Minnesota, Variational principles for hybridizable discontinuous Galerkin methods: A short story". PSU Media Space. Retrieved 2021-05-02.
Numerical methods for partial differential equations
Finite difference
Parabolic
• Forward-time central-space (FTCS)
• Crank–Nicolson
Hyperbolic
• Lax–Friedrichs
• Lax–Wendroff
• MacCormack
• Upwind
• Method of characteristics
Others
• Alternating direction-implicit (ADI)
• Finite-difference time-domain (FDTD)
Finite volume
• Godunov
• High-resolution
• Monotonic upstream-centered (MUSCL)
• Advection upstream-splitting (AUSM)
• Riemann solver
• Essentially non-oscillatory (ENO)
• Weighted essentially non-oscillatory (WENO)
Finite element
• hp-FEM
• Extended (XFEM)
• Discontinuous Galerkin (DG)
• Spectral element (SEM)
• Mortar
• Gradient discretisation (GDM)
• Loubignac iteration
• Smoothed (S-FEM)
Meshless/Meshfree
• Smoothed-particle hydrodynamics (SPH)
• Peridynamics (PD)
• Moving particle semi-implicit method (MPS)
• Material point method (MPM)
• Particle-in-cell (PIC)
Domain decomposition
• Schur complement
• Fictitious domain
• Schwarz alternating
• additive
• abstract additive
• Neumann–Dirichlet
• Neumann–Neumann
• Poincaré–Steklov operator
• Balancing (BDD)
• Balancing by constraints (BDDC)
• Tearing and interconnect (FETI)
• FETI-DP
Others
• Spectral
• Pseudospectral (DVR)
• Method of lines
• Multigrid
• Collocation
• Level-set
• Boundary element
• Method of moments
• Immersed boundary
• Analytic element
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• Infinite difference method
• Infinite element method
• Galerkin method
• Petrov–Galerkin method
• Validated numerics
• Computer-assisted proof
• Integrable algorithm
• Method of fundamental solutions
| Wikipedia |
Lobb number
In combinatorial mathematics, the Lobb number Lm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses.[1]
Lobb numbers form a natural generalization of the Catalan numbers, which count the number of complete strings of balanced parentheses of a given length. Thus, the nth Catalan number equals the Lobb number L0,n.[2] They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the nth Catalan number.[3]
The Lobb numbers are parameterized by two non-negative integers m and n with n ≥ m ≥ 0. The (m, n)th Lobb number Lm,n is given in terms of binomial coefficients by the formula
$L_{m,n}={\frac {2m+1}{m+n+1}}{\binom {2n}{m+n}}\qquad {\text{ for }}n\geq m\geq 0.$
An alternative expression for Lobb number Lm,n is:
$L_{m,n}={\binom {2n}{m+n}}-{\binom {2n}{m+n+1}}.$
The triangle of these numbers starts as (sequence A039599 in the OEIS)
${\begin{array}{rrrrrr}1\\1&1\\2&3&1\\5&9&5&1\\14&28&20&7&1\\42&90&75&35&9&1\\\end{array}}$
where the diagonal is
$L_{n,n}=1,$
and the left column are the Catalan Numbers
$L_{0,n}={\frac {1}{1+n}}{\binom {2n}{n}}.$
As well as counting sequences of parentheses, the Lobb numbers also count the number of ways in which n + m copies of the value +1 and n − m copies of the value −1 may be arranged into a sequence such that all of the partial sums of the sequence are non-negative.
Ballot counting
The combinatorics of parentheses is replaced with counting ballots in an election with two candidates in Bertrand's ballot theorem, first published by William Allen Whitworth in 1878. The theorem states the probability that winning candidate is ahead in the count, given known final tallies for each candidate.
References
1. Koshy, Thomas (March 2009). "Lobb's generalization of Catalan's parenthesization problem". The College Mathematics Journal. 40 (2): 99–107. doi:10.4169/193113409X469532.
2. Koshy, Thomas (2008). Catalan Numbers with Applications. Oxford University Press. ISBN 978-0-19-533454-8.
3. Lobb, Andrew (March 1999). "Deriving the nth Catalan number". Mathematical Gazette. 83 (8): 109–110. doi:10.2307/3618696. JSTOR 3618696. S2CID 126311995.
Classes of natural numbers
Powers and related numbers
• Achilles
• Power of 2
• Power of 3
• Power of 10
• Square
• Cube
• Fourth power
• Fifth power
• Sixth power
• Seventh power
• Eighth power
• Perfect power
• Powerful
• Prime power
Of the form a × 2b ± 1
• Cullen
• Double Mersenne
• Fermat
• Mersenne
• Proth
• Thabit
• Woodall
Other polynomial numbers
• Hilbert
• Idoneal
• Leyland
• Loeschian
• Lucky numbers of Euler
Recursively defined numbers
• Fibonacci
• Jacobsthal
• Leonardo
• Lucas
• Padovan
• Pell
• Perrin
Possessing a specific set of other numbers
• Amenable
• Congruent
• Knödel
• Riesel
• Sierpiński
Expressible via specific sums
• Nonhypotenuse
• Polite
• Practical
• Primary pseudoperfect
• Ulam
• Wolstenholme
Figurate numbers
2-dimensional
centered
• Centered triangular
• Centered square
• Centered pentagonal
• Centered hexagonal
• Centered heptagonal
• Centered octagonal
• Centered nonagonal
• Centered decagonal
• Star
non-centered
• Triangular
• Square
• Square triangular
• Pentagonal
• Hexagonal
• Heptagonal
• Octagonal
• Nonagonal
• Decagonal
• Dodecagonal
3-dimensional
centered
• Centered tetrahedral
• Centered cube
• Centered octahedral
• Centered dodecahedral
• Centered icosahedral
non-centered
• Tetrahedral
• Cubic
• Octahedral
• Dodecahedral
• Icosahedral
• Stella octangula
pyramidal
• Square pyramidal
4-dimensional
non-centered
• Pentatope
• Squared triangular
• Tesseractic
Combinatorial numbers
• Bell
• Cake
• Catalan
• Dedekind
• Delannoy
• Euler
• Eulerian
• Fuss–Catalan
• Lah
• Lazy caterer's sequence
• Lobb
• Motzkin
• Narayana
• Ordered Bell
• Schröder
• Schröder–Hipparchus
• Stirling first
• Stirling second
• Telephone number
• Wedderburn–Etherington
Primes
• Wieferich
• Wall–Sun–Sun
• Wolstenholme prime
• Wilson
Pseudoprimes
• Carmichael number
• Catalan pseudoprime
• Elliptic pseudoprime
• Euler pseudoprime
• Euler–Jacobi pseudoprime
• Fermat pseudoprime
• Frobenius pseudoprime
• Lucas pseudoprime
• Lucas–Carmichael number
• Somer–Lucas pseudoprime
• Strong pseudoprime
Arithmetic functions and dynamics
Divisor functions
• Abundant
• Almost perfect
• Arithmetic
• Betrothed
• Colossally abundant
• Deficient
• Descartes
• Hemiperfect
• Highly abundant
• Highly composite
• Hyperperfect
• Multiply perfect
• Perfect
• Practical
• Primitive abundant
• Quasiperfect
• Refactorable
• Semiperfect
• Sublime
• Superabundant
• Superior highly composite
• Superperfect
Prime omega functions
• Almost prime
• Semiprime
Euler's totient function
• Highly cototient
• Highly totient
• Noncototient
• Nontotient
• Perfect totient
• Sparsely totient
Aliquot sequences
• Amicable
• Perfect
• Sociable
• Untouchable
Primorial
• Euclid
• Fortunate
Other prime factor or divisor related numbers
• Blum
• Cyclic
• Erdős–Nicolas
• Erdős–Woods
• Friendly
• Giuga
• Harmonic divisor
• Jordan–Pólya
• Lucas–Carmichael
• Pronic
• Regular
• Rough
• Smooth
• Sphenic
• Størmer
• Super-Poulet
• Zeisel
Numeral system-dependent numbers
Arithmetic functions
and dynamics
• Persistence
• Additive
• Multiplicative
Digit sum
• Digit sum
• Digital root
• Self
• Sum-product
Digit product
• Multiplicative digital root
• Sum-product
Coding-related
• Meertens
Other
• Dudeney
• Factorion
• Kaprekar
• Kaprekar's constant
• Keith
• Lychrel
• Narcissistic
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Happy
P-adic numbers-related
• Automorphic
• Trimorphic
Digit-composition related
• Palindromic
• Pandigital
• Repdigit
• Repunit
• Self-descriptive
• Smarandache–Wellin
• Undulating
Digit-permutation related
• Cyclic
• Digit-reassembly
• Parasitic
• Primeval
• Transposable
Divisor-related
• Equidigital
• Extravagant
• Frugal
• Harshad
• Polydivisible
• Smith
• Vampire
Other
• Friedman
Binary numbers
• Evil
• Odious
• Pernicious
Generated via a sieve
• Lucky
• Prime
Sorting related
• Pancake number
• Sorting number
Natural language related
• Aronson's sequence
• Ban
Graphemics related
• Strobogrammatic
• Mathematics portal
| Wikipedia |
Step-by-step interactive example for calculating standard deviation First, we need a data set to work with. Let's pick something small so we don't get overwhelmed by the number of data points.... For example, the standard deviation of a sample can be used to approximate the standard deviation of a population. Finding a sample size can be one of the most challenging tasks in statistics and depends upon many factors including the size of your original population.
You need to be creative, because these data are consistent with any mean exceeding $0\times .05 + 1\times .07 + \cdots + 5\times .18$ = $2.89$ and any standard deviation exceeding $1.38$ (which are attained by assuming nobody visited any more than five times per month). | CommonCrawl |
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