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Faster quantum mixing for slowly evolving sequences of Markov chains Davide Orsucci1, Hans J. Briegel1,2, and Vedran Dunjko1,3,4 1Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, 6020 Innsbruck, Austria 2Department of Philosophy, University of Konstanz, Fach 17, 78457 Konstanz, Germany 3Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany 4LIACS, Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands Markov chain methods are remarkably successful in computational physics, machine learning, and combinatorial optimization. The cost of such methods often reduces to the mixing time, i.e., the time required to reach the steady state of the Markov chain, which scales as $δ^{-1}$, the inverse of the spectral gap. It has long been conjectured that quantum computers offer nearly generic quadratic improvements for mixing problems. However, except in special cases, quantum algorithms achieve a run-time of $\mathcal{O}(\sqrt{δ^{-1}} \sqrt{N})$, which introduces a costly dependence on the Markov chain size $N,$ not present in the classical case. Here, we re-address the problem of mixing of Markov chains when these form a slowly evolving sequence. This setting is akin to the simulated annealing setting and is commonly encountered in physics, material sciences and machine learning. We provide a quantum memory-efficient algorithm with a run-time of $\mathcal{O}(\sqrt{δ^{-1}} \sqrt[4]{N})$, neglecting logarithmic terms, which is an important improvement for large state spaces. Moreover, our algorithms output quantum encodings of distributions, which has advantages over classical outputs. Finally, we discuss the run-time bounds of mixing algorithms and show that, under certain assumptions, our algorithms are optimal. @article{Orsucci2018fasterquantummixing, doi = {10.22331/q-2018-11-09-105}, url = {https://doi.org/10.22331/q-2018-11-09-105}, title = {Faster quantum mixing for slowly evolving sequences of {M}arkov chains}, author = {Orsucci, Davide and Briegel, Hans J. and Dunjko, Vedran}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {2}, pages = {105}, month = nov, year = {2018} } [1] Newman, M. E. J. and Barkema, G. T., Monte Carlo Methods in Statistical Physics. Oxford University Press (1999). [2] Sinclair, A., Algorithms for random generation and counting: a Markov chain approach. Springer (1993). [3] Bellman, R., A Markovian decision process. 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Physics Articles Physics Tutorials Physics Guides Physics FAQs Math Articles Math Guides Math FAQs Bio/Chem/Tech Bio/Chem Articles Computer Science Tutorials Learn Lagrangians in Mathematical Quantum Field Theory November 14, 2017 /42 Comments/in Physics Articles /by Urs Schreiber This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 4. Field Variations. The next chapter is 6. Symmetries. 5. Lagrangians Given any type of fields (def. 3.1), those field histories that are to be regarded as "physically realizable" (if we think of the field theory as a description of the observable universe) should satisfy some differential equation — the equation of motion — meaning that realizability of any field histories may be checked upon restricting the configuration to the infinitesimal neighbourhoods (example 3.30) of each spacetime point. This expresses the physical absence of "action at a distance" and is one aspect of what it means to have a local field theory. By remark 4.3 this means that equations of motion of a field theory are equations among the coordinates of the jet bundle of the field bundle. For many field theories of interest, their differential equation of motion is not a random partial differential equations, but is of the special kind that exhibits the "principle of extremal action" (prop. 7.36 below) determined by a local Lagrangian density (def. 5.1 below). These are called Lagrangian field theories, and this is what we consider here. Namely among all the variational differential forms (def. 4.11) two kinds stand out, namley the 0-forms in ##\Omega^{0,0}_\Sigma(E)## — the smooth functions — and the horizontal ##p+1##-forms ##\Omega^{p+1,0}_\Sigma(E)## — to be called the Lagrangian densities ##\mathbf{L}## (def. 5.1 below) — since these occupy the two "corners" of the variational bicomplex (38). There is not much to say about the 0-forms, but the Lagrangian densities ##\mathbf{L}## do inherit special structure from their special position in the variational bicomplex: Their variational derivative ##\delta \mathbf{L}## uniquely decomposes as the Euler-Lagrange derivative ##\delta_{EL} \mathbf{L}## which is proportional to the variation of the fields (instead of their derivatives) the total spacetime derivative ##d \Theta_{BFV}## of a potential ##\Theta_{BFV}## for a presymplectic current ##\Omega_{BFV} := \delta \Theta_{BFV}##. This is prop. 5.12 below: \delta \mathbf{L} \;=\; \underset{ \text{Euler-Lagrange variation} }{\underbrace{\delta_{EL}\mathbf{L}}} – d \underset{\text{presymplectic current}}{\underbrace{\Theta_{BFV}}} \,. These two terms play a pivotal role in the theory: The condition that the first term vanishes on field histories is a differential equation on field histories, called the Euler-Lagrange equation of motion (def. 5.24 below). The space of solutions to this differential equation, called the on-shell space of field histories \label{InclusionOfOnShellSpaceOfFieldHistories} \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} \overset{\phantom{AAA}}{\rightarrow} \Gamma_\Sigma(E) has the interpretation of the space of "physically realizable field histories". This is the key object of study in the following chapters. Often this is referred to as the space of classical field histories, indicating that this does not yet reflect the full quantum field theory. Indeed, there is also the second term in the variational derivative of the Lagrangian density, the presymplectic current ##\Theta_{BFV}##, and this implies a presymplectic structure on the on-shell space of field histories (def. 8.2 below) which encodes deformations of the algebra of smooth functions on ##\Gamma_\Sigma(E)##. This deformation is the quantization of the field theory to an actual quantum field theory, which we discuss below. \array{ &&& \delta \mathbf{L} &&& = & & \delta_{EL}\mathbf{L} &- & d \Theta_{BFV} & & \swarrow && && \searrow \text{classical} \text{field theory} && && && \text{deformation to} \text{quantum} Definition 5.1. (local Lagrangian density) Given a field bundle ##E## over a ##(p+1)##-dimensional Minkowski spacetime ##\Sigma## as in example 3.4, then a local Lagrangian density ##\mathbf{L}## (for the type of field thus defined) is a horizontal differential form of degree ##(p+1)## (def. 4.11) on the corresponding jet bundle (def. 4.1): \mathbf{L} \;\in \; \Omega^{p+1,0}_{\Sigma}(E) By example 4.12 in terms of the given volume form on spacetimes, any such Lagrangian density may uniquely be written as \label{LagrangianFunctionViaVolumeForm} \mathbf{L} = L \, dvol_\Sigma where the coefficient function (the Lagrangian function) is a smooth function on the spacetime and field coordinates: L = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots ) where by prop. 4.6 ##L((x^\mu), \cdots)## depends locally on an arbitrary but finite order of derivatives ##\phi^a_{,\mu_1 \cdots \mu_k}##. We say that a field bundle ##E \overset{fb}{\to} \Sigma## (def. 3.1) equipped with a local Lagrangian density ##\mathbf{L}## is (or defines) a prequantum Lagrangian field theory on the spacetime ##\Sigma##. Remark 5.2. (parameterized and physical unit-less Lagrangian densities) More generally we may consider parameterized collections of Lagrangian densities, i.e. functions \mathbf{L}_{(-)} \;\colon\; U \longrightarrow \Omega^{p+1,0}_\Sigma(E) for ##U## some Cartesian space or generally some super Cartesian space. For example all Lagrangian densities considered in relativistic field theory are naturally smooth functions of the scale of the metric ##\eta## (def. 2.15) \mathbb{R}_{\gt 0} &\overset{}{\longrightarrow}& \Omega^{p+1,0}_\Sigma(E) r &\mapsto& \mathbf{L}_{r^2\eta} But by the discussion in remark 2.16, in physics a rescaling of the metric is interpreted as reflecting but a change of physical units of length/distance. Hence if a Lagrangian density is supposed to express intrinsic content of a physical theory, it should remain unchanged under such a change of physical units. This is achieved by having the Lagrangian be parameterized by further parameters, whose corresponding physical units compensate that of the metric such as to make the Lagrangian density "physical unit-less". This means to consider parameter spaces ##U## equipped with an action of the multiplicative group ##\mathbb{R}_{\gt 0}## of positive real numbers, and parameterized Lagrangians \mathbf{L}_{(-)} \;\colon\; U \longrightarrow \Omega^{p+1,0}_\Sigma(E) which are invariant under this action. Remark 5.3. (locally variational field theory and Lagrangian p-gerbe connection) If the field bundle (def. 3.1) is not just a trivial vector bundle over Minkowski spacetime (example 3.4) then a Lagrangian density for a given equation of motion may not exist as a globally defined differential ##(p+1)##-form, but only as a p-gerbe connection. This is the case for locally variational field theories such as the charged particle, the WZW model and generally theories involving higher WZW terms. For more on this see the exposition at Higher Structures in Physics. Example 5.4. (local Lagrangian density for free real scalar field on Minkowski spacetime) Consider the field bundle for the real scalar field from example 3.5, i.e. the trivial line bundle over Minkowski spacetime. According to def. 4.1 its jet bundle ##J^\infty_\Sigma(E)## has canonical coordinates \left\{ \{x^\mu\}, \phi, \{\phi_{,\mu}\}, \{\phi_{,\mu_1 \mu_2}\}, \cdots \right\} In these coordinates, the local Lagrangian density ##L \in \Omega^{p+1,0}(\Sigma)## (def. 5.1) defining the free real scalar field of mass ##m \in \mathbb{R}## on ##\Sigma## is := \tfrac{1}{2} \left( \eta^{\mu \nu} \phi_{,\mu} \phi_{,\nu} m^2 \phi^2 \right) \mathrm{dvol}_\Sigma This is naturally thought of as a collection of Lagrangians smoothly parameterized by the metric ##\eta## and the mass ##m##. For this to be physical unit-free in the sense of remark 5.2 the physical unit of the parameter ##m## must be that of the inverse metric, hence must be an inverse length according to remark 2.16 This is the inverse Compton wavelength ##\ell_m = \hbar / m c## (9) and hence the physical unit-free version of the Lagrangian density for the free scalar particle is \mathbf{L}_{\eta,\ell_m} \::=\; \tfrac{\ell_m^2}{2} \left( \tfrac{m c}{\hbar} \right)^2 \phi^2 Example 5.5. (phi^n theory) Consider the field bundle for the real scalar field from example 3.5, i.e. the trivial line bundle over Minkowski spacetime. More generally we may consider adding to the free field Lagrangian density from example 5.4 some power of the field coordinate \mathbf{L}_{int} \;:=\; g \phi^n \, dvol_\Sigma \,, for ##g \in \mathbb{R}## some number, here called the coupling constant. The interacting Lagrangian field theory defined by the resulting Lagrangian density \mathbf{L} \;0\; g \phi^n is usually called just phi^n theory. Example 5.6. (local Lagrangian density for free electromagnetic field) Consider the field bundle ##T^\ast \Sigma \to \Sigma## for the electromagnetic field on Minkowski spacetime from example 3.6, i.e. the cotangent bundle, which over Minkowski spacetime happens to be a trivial vector bundle of rank ##p+1##. With fiber coordinates taken to be ##(a_\mu)_{\mu = 0}^p##, the induced fiber coordinates on the corresponding jet bundle ##J^\infty_\Sigma(T^\ast \Sigma)## (def. 4.1) are ##( (x^\mu), (a_\mu), (a_{\mu,\nu}), (a_{\mu,\nu_1 \nu_2}), \cdots )##. Consider then the local Lagrangian density (def. 5.1) given by \label{ElectromagnetismLagrangian} f_{\mu \nu} f^{\mu \nu} dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(T^\ast \Sigma) where ##f_{\mu \nu} := \tfrac{1}{2}(a_{\nu,\mu} – a_{\mu,\nu})## are the components of the universal Faraday tensor on the jet bundle from example 4.4. This is the Lagrangian density that defines the Lagrangian field theory of free electromagnetism. Here for ##A \in \Gamma_\Sigma(T^\ast \Sigma)## an electromagnetic field history (vector potential), then the pullback of ##f_{\mu \nu}## along its jet prolongation (def. 4.2) is the corresponding component of the Faraday tensor (20): j^\infty_\Sigma(A) \right)^\ast(f_{\mu \nu}) & = (d A)_{\mu \nu} & = F_{\mu \nu} It follows that the pullback of the Lagrangian (43) along the jet prologation of the electromagnetic field is \right)^\ast \mathbf{L} \tfrac{1}{2} F \wedge \star_\eta F Here ##\star_\eta## denotes the Hodge star operator of Minkowski spacetime. More generally: Example 5.7. (Lagrangian density for Yang-Mills theory on Minkowski spacetime) Let ##\mathfrak{g}## be a finite dimensional Lie algebra which is semisimple. This means that the Killing form invariant polynomial k \colon \mathfrak{g} \otimes \mathfrak{g} \longrightarrow \mathbb{R} is a non-degenerate bilinear form. Examples include the special unitary Lie algebras ##\mathfrak{so}(n)##. Then for ##E = T^\ast \Sigma \otimes \mathfrak{g}## the field bundle for Yang-Mills theory as in example 3.7, the Lagrangian density (def. 5.1) ##\mathfrak{g}##-Yang-Mills theory on Minkowski spacetime is k_{\alpha \beta} f^\alpha_{\mu \nu} f^{\beta \mu \nu} dvol_\Sigma f^\alpha_{\mu \nu} a^\alpha_{\nu,\mu} a^\alpha_{\mu,\nu} \gamma^{\alpha}{}_{\beta \gamma} a^\beta_{\mu} a^\gamma_{\nu} \Omega^{0,0}_\Sigma(E) is the universal Yang-Mills field strength (31). Example 5.8. (local Lagrangian density for free B-field) Consider the field bundle ##\wedge^2_\Sigma T^\ast \Sigma \to \Sigma## for the B-field on Minkowski spacetime from example 3.9. With fiber coordinates taken to be ##(b_{\mu \nu})## with b_{\mu \nu} = – b_{\nu \mu} the induced fiber coordinates on the corresponding jet bundle ##J^\infty_\Sigma(T^\ast \Sigma)## (def. 4.1) are ##( (x^\mu), (b_{\mu \nu}), (b_{\mu \nu, \mu_1}), (b_{\mu \nu, \mu_1 \mu_2}), \cdots )##. \label{LagrangianForBField} h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} \, dvol_\Sigma \Omega^{p+1,0}_\Sigma(\wedge^2_\Sigma T^\ast \Sigma) where ##h_{\mu_1 \mu_2 \mu_3}## are the components of the universal B-field strength on the jet bundle from example 4.5. Example 5.9. (Lagrangian density for free Dirac field on Minkowski spacetime) For ##\Sigma## Minkowski spacetime of dimension ##p + 1 \in \{3,4,6,10\}## (def. 2.17), consider the field bundle ##\Sigma \times S_{odd} \to \Sigma## for the Dirac field from example 3.50. With the two-component spinor field fiber coordinates from remark 2.32, the jet bundle has induced fiber coordinates as follows: \left(\psi^\alpha\right) \psi^\alpha_{,\mu} \cdots (\chi_a), (\chi_{a,\mu}), \cdots \right), ( \xi^{\dagger \dot a}), (\xi^{\dagger \dot a}_{,\mu}), \cdots All of these are odd-graded elements (def. 3.35) in a Grassmann algebra (example 3.36), hence anti-commute with each other, in generalization of (28): \label{DiracFieldJetCoordinatesAnticommute} \psi^\alpha_{,\mu_1 \cdots \mu_r} \psi^\beta_{,\mu_1 \cdots \mu_s} The Lagrangian density (def. 5.1) of the massless free Dirac field on Minkowski spacetime is \label{DiracFieldLagrangianMassless} \overline{\psi} \, \gamma^\mu \psi_{,\mu}\, dvol_\Sigma given by the bilinear pairing ##\overline{(-)}\Gamma(-)## from prop. 2.31 of the field coordinate with its first spacetime derivative and expressed here in two-component spinor field coordinates as in (15), hence with the Dirac conjugate ##\overline{\psi}## (14) on the left. Specifically in spacetime dimension ##p + 1 = 4##, the Lagrangian function for the massive Dirac field of mass ##m \in \mathbb{R}## is & := \underset{ \text{kinetic term} \underbrace{ i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} \text{mass term} m \overline{\psi} \psi This is the inverse Compton wavelength ##\ell_m = \hbar / m c## (9) and hence the physical unit-free version of the Lagrangian density for the free Dirac field is \ell_m i \overline{\psi} \gamma^\mu \psi_{,\mu} + \left( \tfrac{m c}{\hbar} \right) \overline{\psi} \psi \right) dvol_\Sigma Remark 5.10. (reality of the Lagrangian density of the Dirac field) The kinetic term of the Lagrangian density for the Dirac field form def. 5.9 is a sum of two contributions, one for each chiral spinor component in the full Dirac spinor (remark 2.32): i \overline{\psi} \gamma^\mu \psi_{,\mu} -(\partial_\mu \xi^a ) \sigma^\mu_{a \dot c} \xi^{\dagger \dot c} + \partial_\mu(\chi^a \sigma^\mu_{a \dot c} \chi^{\dagger \dot c}) \xi^a \sigma^\mu_{a \dot c} \partial_\mu \xi^{\dagger \dot c} \xi^\dagger_{\dot a} \tilde \sigma^{\mu \dot a c} \partial_\mu \xi_c \xi^\dagger \tilde \sigma^\mu \partial_\mu \xi \chi^\dagger \tilde \sigma^\mu \partial_\mu \chi \partial_\mu(\xi \sigma^\mu \xi^\dagger) Here the computation shown under the brace crucially uses that all these jet coordinates for the Dirac field are anti-commuting, due to their supergeometric nature (45). Notice that a priori this is a function on the jet bundle with values in ##\mathbb{K}##. But in fact for ##\mathbb{K} = \mathbb{C}## it is real up to a total spacetime derivative:, because i \chi^\dagger \tilde \sigma^\mu \partial_\mu\chi \right)^\dagger -i \left( \partial_\mu \chi\right)^\dagger \sigma^\mu \chi i \chi^\dagger \sigma^\mu \partial_\mu \chi + i \partial_\mu\left( \chi^\dagger \sigma^\mu \chi \right) and similarly for ##i \xi^\dagger \tilde \sigma^\mu \partial_\mu\xi## (e.g. Dermisek I-9) Example 5.11. (Lagrangian density for quantum electrodynamics) Consider the fiber product of the field bundles for the electromagnetic field (example 3.6) and the Dirac field (example 3.50) over 4-dimensional Minkowski spacetime ##\Sigma := \mathbb{R}^{3,1}## (def. 2.17): \underset{ \array{ \text{electromagnetic} \\ \text{field} } }{\underbrace{T^\ast \Sigma}} \times \text{Dirac} \\ \text{field} S_{odd} This means that now a field history is a pair ##(A,\Psi)##, with ##A## a field history of the electromagnetic field and ##\Psi## a field history of the Dirac field. On the resulting jet bundle consider the Lagrangian density L_{int} i g \, \overline{\psi} \gamma^\mu \psi a_\mu for ##g \in \mathbb{R}## some number, called the coupling constant. This is called the electron-photon interaction. Then the sum of the Lagrangian densities for the free electromagnetic field (example 5.6); the free Dirac field (example 5.9) the above electron-photon interaction \mathbf{L}_{EM} + \mathbf{L}_{Dir} + \mathbf{L}_{int} \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} \;+\; i \, \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} + m \overline{\psi} \psi \, dvol_\Sigma defines the interacting field theory Lagrangian field theory whose perturbative quantization is called quantum electrodynamics. In this context the square of the coupling constant \alpha := \frac{g^2}{4 \pi} is called the fine structure constant. The beauty of Lagrangian field theory (def. 5.1) is that a choice of Lagrangian density determines both the equations of motion of the fields as well as a presymplectic structure on the space of solutions to this equation (the "shell"), making it the "covariant phase space" of the theory. All this we discuss below. But in fact all this key structure of the field theory is nothing but the shadow (under "transgression of variational differential forms", def. 7.31 below) of the following simple relation in the variational bicomplex: Proposition 5.12. (Euler-Lagrange form and presymplectic current) Given a Lagrangian density ##\mathbf{L} \in \Omega^{p+1,0}_\Sigma(E)## as in def. 5.1, then its de Rham differential ##\mathbf{d}\mathbf{L}##, which by degree reasons equals ##\delta \mathbf{L}##, has a unique decomposition as a sum of two terms \label{dLDecomposition} \mathbf{d} \mathbf{L} \delta_{EL} \mathbf{L} d \Theta_{BFV} such that ##\delta_{EL}\mathbf{L}## is proportional to the variational derivative of the fields (but not their derivatives, called a "source form"): \Omega^{p+1,0}_{\Sigma}(E) \wedge \delta C^\infty(E) \;\subset\; \Omega^{p+1,1}_{\Sigma}(E) \delta_{EL} \;\colon\; \Omega^{p+1,0}_{\Sigma}(E) \longrightarrow \Omega^{p+1,0}_{\Sigma}(E) \wedge \delta \Omega^{0,0}_{\Sigma}(E) thus defined is called the Euler-Lagrange operator and is explicitly given by the Euler-Lagrange derivative: \label{EulerLagrangeEquationGeneral} \delta_{EL} L \, dvol_\Sigma \frac{\delta_{EL} L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma \frac{\partial L}{\partial \phi^a} \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} \frac{d^2}{d x^{\mu_1} d x^{\mu_2}} \frac{\partial L}{\partial \phi^a_{\mu_1, \mu_2}} \delta \phi^a \wedge dvol_\Sigma The smooth subspace of the jet bundle on which the Euler-Lagrange form vanishes \label{ShellInJetBundle} \mathcal{E} x \in J^\infty_\Sigma(E) \;\vert\; \delta_{EL}\mathbf{L}(x) = 0 \;\overset{i_{\mathcal{E}}}{\rightarrow}\; J^\infty_\Sigma(E) is called the shell. The smaller subspace on which also all total spacetime derivatives vanish (the "formally integrable prolongation") is the prolonged shell \label{ProlongedShellInJetBundle} \mathcal{E}^\infty \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \delta_{EL}\mathbf{L} \right)(x) = 0 \overset{i_{\mathcal{E}^\infty}}{\rightarrow} Saying something holds "on-shell" is to mean that it holds after restriction to this subspace. For example a variational differential form ##\alpha \in \Omega^{\bullet,\bullet}_\Sigma(E)## is said to vanish on shell if ##\alpha\vert_{\mathcal{E}^\infty} = 0##. The remaining term ##d \Theta_{BFV}## in (47) is unique, while the presymplectic potential \label{PresymplecticPotential} \Theta_{BFV} \in \Omega^{p,1}_{\Sigma}(E) is not unique. (For a field bundle which is a trivial vector bundle (example 3.4 over Minkowski spacetime (def. 2.17), prop. 4.14 says that ##\Theta_{BFV}## is unique up to addition of total spacetime derivatives ##d \kappa##, for ##\kappa \in \Omega^{p-1,1}_\Sigma(E)##.) One possible choice for the presymplectic current ##\Theta_{BFV}## is \label{StandardThetaForTrivialVectorFieldBundleOnMinkowskiSpacetime} \Theta_{BFV} & := \phantom{+} \; \wedge \iota_{\partial_\mu} dvol_\Sigma & \phantom{=} \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu} \wedge \iota_{\partial_\mu} dvol_\Sigma & \phantom{=} + \cdots \iota_{\partial_{\mu}} dvol_\Sigma (-1)^{\mu} d x^0 \wedge \cdots d x^{\mu-1} \wedge d x^{\mu+1} \wedge \cdots \wedge d x^p denotes the contraction (def. 1.20) of the volume form with the vector field ##\partial_\mu##. The vertical derivative of a chosen presymplectic potential ##\Theta_{BFV}## is called a pre-symplectic current for ##\mathbf{L}##: \label{PresymplecticCurrent} \Omega_{BFV} \delta \Theta_{BFV} \;\;\; \in \Omega^{p,2}_{\Sigma}(E) Given a choice of ##\Theta_{BFV}## then the sum \label{TheLepage} \mathbf{L} + \Theta_{BFV} \;\in\; \Omega^{p+1,0}_\Sigma(E) \oplus \Omega^{p,1}_\Sigma(E) is called the corresponding Lepage form. Its de Rham derivative is the sum of the Euler-Lagrange variation and the presymplectic current: \label{DerivativeOfLepageForm} \mathbf{d}( \mathbf{L} + \Theta_{BFV} ) \delta_{EL} \mathbf{L} + \Omega_{BFV} (Its conceptual nature will be elucidated after the introduction of the local BV-complex in example 8.11 below.) Proof. Using ##\mathbf{L} = L dvol_\Sigma## and that ##d \mathbf{L} = 0## by degree reasons (example 4.12), we find \mathbf{d}\mathbf{L} \frac{\partial L}{\partial \phi^a} \delta \phi^a \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \frac{\partial L}{\partial \phi^a_{,\mu_1 \mu_2}} \delta \phi^a_{,\mu_1 \mu_2} \wedge dvol_{\Sigma} The idea now is to have ##d \Theta_{BFV}## pick up those terms that would appear as boundary terms under the integral ##\int_\Sigma j^\infty_\Sigma(\Phi)^\ast \mathbf{d}L## if we were to consider integration by parts to remove spacetime derivatives of ##\delta \phi^a##. We compute, using example 4.12, the total horizontal derivative of ##\Theta_{BFV}## from (52) as follows: \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{\mu \nu}} d \frac{\partial L}{\partial \phi^a_{,\mu}} \right) \wedge \delta \phi^a \frac{\partial L}{\partial \phi^a_{,\mu}} \delta d \phi^a \left(d \frac{\partial L}{\partial \phi^a_{,\nu \mu}}\right) \wedge \delta \phi^a_{,\nu} \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta d \phi^a_{,\nu} \left( d \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \right) \wedge \delta d \phi^a \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\nu \mu} \frac{d^2}{ d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \frac{d}{d x^\nu} + \cdots \wedge dvol_\Sigma where in the last line we used that d x^{\mu_1} \wedge \iota_{\partial_{\mu_2}} dvol_\Sigma dvol_\Sigma &\vert& \text{if}\, \mu_1 = \mu_2 0 &\vert& \text{otherwise} \right. Here the two terms proportional to ##\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu}## cancel out, and we are left with Hence ##-d \Theta_{BFV}## shares with ##\mathbf{d} \mathbf{L}## the terms that are proportional to ##\delta \phi^a_{,\mu_1 \cdots \mu_k}## for ##k \geq 1##, and so the remaining terms are proportional to ##\delta \phi^a##, as claimed: \mathbf{d}L + d \Theta_{BFV} = \delta_{EL}\mathbf{L} \frac{d}{d x^\mu}\frac{\partial L}{\partial \phi^a_{,\mu}} \frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}} The following fact is immediate from prop. 5.12, but of central importance, we futher amplify this in remark 5.16 below: Proposition 5.13. (total spacetime derivative of presymplectic current vanishes on-shell) Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1). Then the Euler-Lagrange form ##\delta_{EL} \mathbf{L}## and the presymplectic current (prop. 5.12) are related by d \Omega_{BFV} = – \delta(\delta_{EL}\mathbf{L}) In particular this means that restricted to the prolonged shell ##\mathcal{E}^\infty \rightarrow J^\infty_\Sigma(E)## (50) the total spacetime derivative of the presymplectic current vanishes: \label{HorizontalDerivativeOfPresymplecticCurrentVanishesOnShell} d \Omega_{BFV} \vert_{\mathcal{E}^\infty} \;=\; 0 Proof. By prop. 5.12 we have \delta \mathbf{L} = \delta_{EL} \mathbf{L} – d \Theta_{BFV} The claim follows from applying the variational derivative ##\delta## to both sides, using (37): ##\delta^2 = 0## and ##\delta \circ d = – d \circ \delta##. Many examples of interest fall into the following two special cases of prop. 5.12: Example 5.14. (Euler-Lagrange form for spacetime-independent Lagrangian densities) Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) whose field bundle ##E## is a trivial vector bundle ##E \simeq \Sigma \times F## over Minkowski spacetime ##\Sigma## (example 3.4). In general the Lagrangian density ##\mathbf{L}## is a function of all the spacetime and field coordinates \mathbf{L} = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots) dvol_\Sigma Consider the special case that ##\mathbf{L}## is spacetime-independent in that the Lagrangian funtion ##L## is independent of the spacetime coordinate ##(x^\mu)##. Then the same evidently holds for the Euler-Lagrange form ##\delta_{EL}\mathbf{L}## (prop. 5.12). Therefore in this case the shell (50) is itself a trivial bundle over spacetime. In this situation every point ##\varphi## in the jet fiber defines a constant section of the shell: \label{ConstantSectionOfTrivialShellBundle} \Sigma \times \{\varphi\} \subset \mathcal{E}^\infty Example 5.15. (canonical momentum) Consider a Lagrangian field theory ##(E, \mathbf{L})## (def. 5.1) whose Lagrangian density ##\mathbf{L}## does not depend on the spacetime-coordinates (example 5.14); depends on spacetime derivatives of field coordinates (hence on jet bundle coordinates) at most to first order. Hence if the field bundle ##E \overset{fb}{\to} \Sigma## is a trivial vector bundle over Minkowski spacetime (example 3.4) this means to consider the case that L\left( (\phi^a), (\phi^a_{,\mu}) \right) \wedge dvol_\Sigma Then the presymplectic current (def. 5.12) is (up to possibly a horizontally exact part) of the form \label{CanonicalMomentumPresymplecticCurrent} \delta p_a^\mu \iota_{\partial_\mu} dvol_\Sigma \label{CanonicalMomentumInCoordinates} p_a^\mu \frac{\partial L}{ \partial \phi^a_{,\mu}} denotes the partial derivative of the Lagrangian function with respect to the spacetime-derivatives of the field coordinates. p_a p_a^0 \frac{\partial L}{\partial \phi^a_{,0}} is called the canonical momentum corresponding to the "canonical field coordinate" ##\phi^a##. In the language of multisymplectic geometry the full expression p_a^\mu \wedge \iota_{\partial_\mu} dvol_\Sigma \Omega^{p,1}_\Sigma(E) is also called the "canonical multi-momentum", or similar. Proof. We compute: \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \delta \Theta_{BFV} \delta \delta p_a^\mu \wedge \delta \phi^a \wedge \iota_{\partial_\mu} dvol_\Sigma Remark 5.16. (presymplectic current is local version of (pre-)symplectic form of Hamiltonian mechanics) In the simple but very common situation of example 5.15 the presymplectic current (def. 5.12) takes the form (59) with ##\phi^a## the field coordinates ("canonical coordinates") and ##p_a^\mu## the "canonical momentum" (59). Notice that this is of the schematic form "##(\delta p_a \wedge \delta q^a) \wedge dvol_{\Sigma_p}##", which is reminiscent of the wedge product of a symplectic form expressed in Darboux coordinates with a volume form for a ##p##-dimensional manifold. Indeed, below in Phase space we discuss that this presymplectic current "transgresses" (def. 7.31 below) to a presymplectic form of the schematic form "##d P_a \wedge d Q^a##" on the on-shell space of field histories (def. 5.24) by integrating it over a Cauchy surface of dimension ##p##. In good situations this presymplectic form is in fact a symplectic form on the on-shell space of field histories (theorem 8.7 below). This shows that the presymplectic current ##\Omega_{BFV}## is the local (i.e. jet level) avatar of the symplectic form that governs the formulation of Hamiltonian mechanics in terms of symplectic geometry. In fact prop. (56) may be read as saying that the presymplectic current is a conserved current (def. 6.6 below), only that it takes values not in smooth functions of the field coordinates and jets, but in variational 2-forms on fields. There is a conserved charge associated with every conserved current (prop. 8.13 below) and the conserved charge associated with the presymplectic current is the (pre-)symplectic form on thephase space of the field theory (def. 8.2 below). Example 5.17. (Euler-Lagrange form and presymplectic current for free real scalar field) Consider the Lagrangian field theory of the free real scalar field from example 5.4. Then the Euler-Lagrange form and presymplectic current (prop. 5.12) are \label{RealScalarFieldLEForm} \left(\eta^{\mu \nu} \phi_{,\mu \nu} – m^2 \right) \delta \phi \wedge dvol_\sigma \left(\eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \right) \wedge \iota_{\partial_\nu} dvol_{\Sigma} \Omega^{p,2}_{\Sigma}(E) Proof. This is a special case of example 5.15, but we spell it out in detail again: We need to show that Euler-Lagrange operator ##\delta_{EL} \colon \Omega^{p+1,0}(\Sigma) \to \Omega^{p+1,1}_S(\Sigma)## takes the local Lagrangian density for the free scalar field to \delta_EL L \eta^{\mu \nu} \phi_{,\mu \nu} – m^2 \phi \delta \phi \wedge \mathrm{dvol}_\Sigma First of all, using just the variational derivative (vertical derivative) ##\delta## is a graded derivation, the result of applying it to the local Lagrangian density is \delta L \eta^{\mu \nu} \phi_{,\mu} \delta \phi_{,\nu} m^2 \phi \delta \phi \wedge \mathrm{dvol}_\Sigma By definition of the Euler-Lagrange operator, in order to find ##\delta_{EL}\mathbf{L}## and ##\Theta_{BFV}##, we need to exhibit this as the sum of the form ##(-) \wedge \delta \phi – d \Theta_{BFV}##. The key to find ##\Theta_{BFV}## is to realize ##\delta \phi_{,\nu}\wedge \mathrm{dvol}_\Sigma## as a total spacetime derivative (horizontal derivative). Since ##d \phi = \phi_{,\mu} d x^\mu## this is accomplished by \delta \phi_{,\nu} \wedge \mathrm{dvol}_\Sigma \delta d \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma where on the right we have the contraction (def. 1.20) of the tangent vector field along ##x^\nu## into the volume form. Hence we may take the presymplectic potential (51) of the free scalar field to be \label{PresymplecticPotentialOfFreeScalarField} \eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu} because with this we have \eta^{\mu \nu} \phi_{,\mu \nu} \delta \phi \right) \wedge \mathrm{dvol}_\Sigma In conclusion this yields the decomposition of the vertical differential of the Lagrangian density = \delta_{EL} \mathcal{L} which shows that ##\delta_{EL} L## is as claimed, and that ##\Theta_{BFV}## is a presymplectic potential current (51). Hence the presymplectic current itself is \Omega_{BFV} &:= \delta \Theta_{BFV} \delta \left( \eta^{\mu \nu} \phi_{,\mu} \delta \phi \wedge \iota_{\partial_\nu} \mathrm{dvol}_\Sigma \right) Example 5.18. (Euler-Lagrange form for free electromagnetic field) Consider the Lagrangian field theory of free electromagnetism from example 5.6. The Euler-Lagrange variational derivative is \label{ElectromagneticFieldEulerLagrangeForm} – \frac{d}{d x^\mu} f^{\mu \nu} \delta a_\nu Hence the shell (49) in this case is \Sigma \times (a_\mu) , (a_{\mu,\mu_1}), (a_{\mu,\mu_1 \mu_2}), \cdots \;\vert\; f^{\mu \nu}{}_{,\mu} = 0 \right\} J^\infty_\Sigma(T^\ast \Sigma) Proof. By (48) we have \frac{\delta_{EL} L}{\delta a_\mu} \delta a_\mu \frac{\partial}{\partial a_\mu} \tfrac{1}{2} a_{[\mu,\nu]} a^{[\mu,\nu]} \frac{d}{d x^\rho} \frac{\partial}{\partial a_{\alpha,\rho}} \delta a_\alpha a_{\mu,\nu} a^{[\mu,\nu]} a^{[\alpha,\rho]} \delta a_{\alpha} – f^{\mu \nu}{}_{,\mu} \delta a_{\nu} Example 5.19. (Euler-Lagrange form for Yang-Mills theory on Minkowski spacetime) Let ##\mathfrak{g}## be a semisimple Lie algebra and consider the Lagrangian field theory ##(E,\mathbf{L})## of ##\mathfrak{g}##-Yang-Mills theory from example 5.7. Its Euler-Lagrange form (prop. 5.12) is f^{\mu \nu \alpha}_{,\mu} \gamma^\alpha{}_{\beta' \gamma} a_\mu^{\beta'} f^{\mu \nu \gamma} k_{\alpha \beta} \,\delta a_\mu^\beta Proof. With the explicit form (48) for the Euler-Lagrange derivative we compute as follows: \delta_{EL} k_{\alpha \beta} f^\alpha_{\mu\nu} f^{\beta \mu \nu} \frac{\partial}{\partial a_{\mu'}^{\alpha'}} a_{\nu,\mu}^\alpha \gamma^{\alpha}{}_{\alpha_2 \alpha_3} a_{\mu}^{\alpha_2} a_\nu^{\alpha_3} f^{\beta \mu \nu} \frac{d}{d x^{\nu'}} \frac{\partial}{\partial a_{\mu',\nu'}^{\alpha'}} \delta a_{\mu'}^{\alpha'} \gamma^{\alpha}{}_{\alpha' \alpha_3} a_\nu^{\alpha_3} \frac{d}{d x^{\mu}} f^{\beta \mu \nu} \delta a_{\nu}^{\alpha} &= f^{\alpha \mu \nu}_{,\mu} \gamma^\alpha{}_{\beta \gamma} a_\mu^\beta f^{\gamma \mu \nu} \delta a_\nu^\beta In the last step we used that for a semisimple Lie algebra ##\gamma_{\alpha \beta \gamma} := k_{\alpha \alpha'} \gamma^{\alpha'}{}_{\beta \gamma}## is totally skew-symmetric in its indices (this being the coefficients of the Lie algebra cocycle) which is in transgression with the Killing form invariant polynomial ##k##. Example 5.20. (Euler-Lagrange form of free B-field) Consider the Lagrangian field theory of the free B-field from example 3.9. h^{\mu \nu \rho}{}_{,\rho} \delta b_{\mu \nu} where ##h_{\mu_1 \mu_2 \mu_3}## is the universal B-field strength from example 4.5. \frac{\delta_{EL} L}{\delta b_{\mu \nu}} \delta b_{\mu \nu} \frac{\partial}{\partial b_{\mu \nu}} \tfrac{1}{2} b_{[\mu_1 \mu_2, \mu_3]} b^{[\mu_1 \mu_2, \mu_3]} \frac{\partial}{\partial b_{\mu \nu, \rho}} \delta b_{\mu \nu} \tfrac{1}{2} b_{\mu_1 \mu_2, \mu_3} b^{[\mu_1 \mu_2, \mu_3]} b^{[\mu \nu, \rho]} Example 5.21. (Euler-Lagrange form and presymplectic current of Dirac field) Consider the Lagrangian field theory of the Dirac field on Minkowski spacetime of dimension ##p + 1 \in \{3,4,6,10\}## (example 5.9). the Euler-Lagrange variational derivative (def. 5.12) in the case of vanishing mass ##m## is$$ 2 i\, \overline{\delta \psi} \,\gamma^\mu\, \psi_{,\mu} $$and in the case that spacetime dimension is ##p +1 = 4## and arbitrary mass ##m\in \mathbb{R}##, it is$$ i \gamma^\mu \psi_{,\mu} + m \psi – i \gamma^\mu\overline{\psi_{,\mu}} + m \overline{\psi} (\delta \psi) its presymplectic current (def. 5.12) is$$ \overline{\delta \psi}\,\gamma^\mu \,\delta \psi \, \iota_{\partial_\mu} dvol_\Sigma Proof. In any case the canonical momentum of the Dirac field according to example 5.15 is p^\alpha_\mu \frac{\partial }{\partial \psi^\alpha_{,\mu}} i \overline {\psi} \, \gamma^\nu \, \psi_{,\nu} \overline{\psi}^\beta (\gamma^\mu)_\beta{}^\alpha This yields the presymplectic current as claimed, by example 5.15. Now regarding the Euler-Lagrange form, first consider the massless case in spacetime dimension ##p+1 \in \{3,4,6,10\}##, where i \overline{\psi} \, \gamma^\mu \, \psi_{,\mu} Then we compute as follows: \delta_{EL} L i \,\overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} i \overline{\psi_{,\mu}} \, \gamma^\mu \, \delta \psi 2 i \, \overline{\delta \psi} \, \gamma^\mu \, \psi_{,\mu} Here the first equation is the general formula (48) for the Euler-Lagrange variation, while the identity under the braces combines two facts (as in remark 5.23 above): the symmetry (12) of the spinor pairing ##\overline{(-)}\gamma^\mu(-)## (prop. 2.31); the anti-commutativity (45) of the Dirac field and jet coordinates, due to their supergeometric nature (remark 3.52). Finally in the special case of the massive Dirac field in spacetime dimension ##p+1 = 4## the Lagrangian function is i \, \overline{\psi} \gamma^\mu \psi_{,\mu} + m \overline{\psi}\psi where now ##\psi_\alpha## takes values in the complex numbers ##\mathbb{C}## (as opposed to in ##\mathbb{R}##, ##\mathbb{H}## or ##\mathbb{O}##). Therefore we may now form the derivative equivalently by treeating ##\psi## and ##\overline{\psi}## as independent components of the field. This immediately yields the claim. Example 5.22. (trivial Lagrangian densities and the Euler-Lagrange complex) If a Lagrangian density ##\mathbf{L}## (def. 5.4) is in the image of the total spacetime derivative, hence horizontally exact (def. 4.11) \mathbf{L} \;=\; d \mathbf{\ell} for any ##\mathbf{\ell} \in \Omega^{p,0}_\Sigma(E)##, then both its Euler-Lagrange form as well as its presymplectic current (def. 5.12) vanish: \delta_{EL}\mathbf{L} = 0 \phantom{AA} \Omega_{BFV} = 0 This is because with ##\delta \circ d = – d \circ \delta## (37) the defining unique decomposition (47) of ##\delta \mathbf{L}## is given by \delta d \mathbf{\ell} \underset{= \delta_{EL}\mathbf{L}}{\underbrace{0}} d \underset{\Theta_{BFV}}{\underbrace{\delta \mathbf{l}}} which then implies with (53) that \delta \delta \mathbf{\ell} Therefore the Lagrangian densities which are total spacetime derivatives are also called trivial Lagrangian densities. If the field bundle ##E \overset{fb}{\to} \Sigma## is a trivial vector bundle (example 3.4) over Minkowski spacetime (def. 2.17) then also the converse is true: Every Lagrangian density whose Euler-Lagrange form vanishes is a total spacetime derivative. Stated more abstractly, this means that the exact sequence of the total spacetime from prop. 4.14 extends to the right via the Euler-Lagrange variational derivative ##\delta_{EL}## to an exact sequence of the form \mathbb{R} \overset{}{\rightarrow} \overset{d}{\longrightarrow} \overset{\delta_{EL}}{\longrightarrow} \Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E)) \overset{\delta_{H}}{\longrightarrow} In fact, as shown, this exact sequence keeps going to the right; this is also called the Euler-Lagrange complex. (Anderson 89, theorem 5.1) The next differential ##\delta_{H}## after the Euler-Lagrange variational derivative ##\delta_{EL}## is known as the Helmholtz operator. By definition of exact sequence, the Helmholtz operator detects whether a partial differential equation on field histories, induced by a variational differential form ##P \in \Omega^{p+1,0}_\Sigma(E) \wedge \delta(C^\infty(E))## as in (63) comes from varying a Lagrangian density, hence whether it is the equation of motion of a Lagrangian field theory via def. 5.24. This way homological algebra is brought to bear on core questions of field theory. For more on this see the exposition at Higher Structures in Physics. Remark 5.23. (supergeometric nature of Lagrangian density of the Dirac field) Observe that the Lagrangian density for the Dirac field (def. 5.9) makes sense (only) due to the supergeometric nature of the Dirac field (remark 3.52): If the field jet coordinates ##\psi_{,\mu_1 \cdots \mu_k}## were not anti-commuting (45) then the Dirac's field Lagrangian density (def. 5.9) would be a total spacetime derivative and hence be trivial according to example 5.22. This is because d \left( \overline{\psi} \,\gamma^\mu\, \psi \tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma \underset{ = (-1) \tfrac{1}{2} \overline{\psi_{,\mu}} \,\gamma^\mu\, \psi \, dvol_\Sigma }{ \tfrac{1}{2}\overline{\psi} \,\gamma^\mu\, \psi_{,\mu} \, dvol_\Sigma Here the identification under the brace uses two facts: the symmetry (12) of the spinor bilinear pairing ##\overline{(-)}\Gamma (-)##; The second fact gives the minus sign under the brace, which makes the total expression vanish, if the Dirac field and jet coordinates indeed are anti-commuting (which, incidentally, means that we found an "off-shell conserved current" for the Dirac field, see example 6.9 below). If however the Dirac field and jet coordinates did commute with each other, we would instead have a plus sign under the brace, in which case the total horizontal derivative expression above would equal the massless Dirac field Lagrangian (46), thus rendering it trivial in the sense of example 5.22. The same supergeometric nature of the Dirac field will be necessary for its intended equation of motion, the Dirac equation (example 5.30) to derive from a Lagrangian density; see the proof of example 5.21 below, and see remark 5.31 below. The key implication of the Euler-Lagrange form on the jet bundle is that it induces the equation of motion on the space of field histories: Definition 5.24. (Euler-Lagrange equation of motion) Given a Lagrangian field theory ##(E,\mathbf{L})## (def. 5.1 then the corresponding Euler-Lagrange equations of motion is the condition on field histories (def. 3.46) \Phi_{(-)} \;\colon\; U \longrightarrow \Gamma_\Sigma(E) to have a jet prolongation (def. 4.2) j^\infty_\Sigma(\Phi_{(-)}(-) ) \;\colon\; U \times \Sigma \longrightarrow J^\infty_\Sigma(E) that factors through the shell inclusion ##\mathcal{E} \overset{i_{\mathcal{E}}}{\rightarrow} J^\infty_\Sigma(E)## (49) defined by vanishing of the Euler-Lagrange form (prop. 5.12) \label{EquationOfMotionEL} j^\infty_\Sigma(\Phi_{(-)}(-)) U \times \Sigma \longrightarrow \overset{i_{\mathcal{E}}}{\rightarrow} (This implies that ##j^\infty_\Sigma(\Phi_{(-)})## factors even through the prolonged shell ##\mathcal{E}^\infty \overset{i_{\mathcal{E}^\infty}}{\rightarrow} J^\infty_\Sigma(E)## (50).) In the case that the field bundle is a trivial vector bundle over Minkowski spacetime as in example 3.4 this is the condition that ##\Phi_{(-)}## satisfies the following differential equation (again using prop. 5.12): (x^\mu), (\Phi^a), \left( \frac{\partial \Phi^a_{(-)}}{\partial x^\mu}\right), \left( \frac{\partial^2 \Phi^a_{(-)}}{\partial x^\mu \partial x^\nu} \right), where the differential operator (def. 4.7) \label{DifferentialOperatorEulerLagrangeDerivative} j^\infty_\Sigma(-)^\ast \frac{\delta_{EL}L}{\delta \phi^{(-)}} \Gamma_\Sigma(T^\ast_\Sigma E) from the field bundle (def. 3.1) to its vertical cotangent bundle (def. 1.13) is given by the Euler-Lagrange derivative (48). The on-shell space of field histories is the space of solutions to this condition, namely the the sub-super smooth set (def. 3.40) of the full space of field histories (22) (def. 3.46) \label{OnShellFieldHistories} \Gamma_\Sigma(E)_{\delta_{EL} L = 0} whose plots are those ##\Phi_{(-)} \colon U \to \Gamma_\Sigma(E)## that factor through the shell (63). More generally for ##\Sigma_r \rightarrow \Sigma## a submanifold of spacetime, we write \label{OnShellFieldHistoriesInHigherCodimension} \Gamma_{\Sigma_r}(E)_{\delta_{EL} L = 0} \Gamma_{\Sigma_r}(E) for the sub-super smooth ste of on-shell field histories restricted to the infinitesimal neighbourhood of ##\Sigma_r## in ##\Sigma## (25). Definition 5.25. (free field theory) A Lagrangian field theory ##(E, \mathbf{L})## (def. 5.1) with field bundle ##E \overset{fb}{\to} \Sigma## a vector bundle (e.g. a trivial vector bundle as in example 3.4) is called a free field theory if its Euler-Lagrange equations of motion (def. 5.24) is a differential equation that is linear differential equation, in that with \Phi_1, \Phi_2 \;\in\; \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0} any two on-shell field histories (65) and ##c_1, c_2 \in \mathbb{R}## any two real numbers, also the linear combination c_1 \Phi_1 + c_2 \Phi_2 \;\in\; \Gamma_\Sigma(E) which a priori exists only as an element in the off-shell space of field histories, is again a solution to the equations of motion and hence an element of ##\Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}##. A Lagrangian field theory which is not a free field theory is called an interacting field theory. Remark 5.26. (relevance of free field theory) In perturbative quantum field theory one considers interacting field theories in the infinitesimal neighbourhood (example 3.30) of free field theories (def. 5.25) inside some super smooth set of general Lagrangian field theories. While free field theories are typically of limited interest in themselves, this perturbation theory around them exhausts much of what is known about quantum field theory in general, and therefore free field theories are of paramount importance for the general theory. We discuss the covariant phase space of free field theories below in Propagators and their quantization below in Free quantum fields . Example 5.27. (equation of motion of free real scalar field is Klein-Gordon equation) By example 5.17 its Euler-Lagrange form is Hence for ##\Phi \in \Gamma_\Sigma(E) = C^\infty(X)## a field history, its Euler-Lagrange equation of motion according to def. 5.24 is \eta^{\mu \nu} \frac{\partial^2 }{\partial x^\mu \partial x^\nu} \Phi – m^2 \Phi \;=\; 0 often abbreviated as \label{KleinGordonEquation} (\Box – m^2) \Phi \;=\; 0 This PDE is called the Klein-Gordon equation on Minowski spacetime. If the mass ##m## vanishes, ##m = 0##, then this is the relativistic wave equation. Hence this is indeed a free field theory according to def. 5.25. The corresponding linear differential operator (def. 4.7) \label{KleinGordonOperator} (\Box – m^2) \Gamma_\Sigma(\Sigma \times \mathbb{R}) is called the Klein-Gordon operator. For later use we record the following basic fact about the Klein-Gordon equation: Example 5.28. (Klein-Gordon operator is formally self-adjoint ) The Klein-Gordon operator (68) is its own formal adjoint (def. 4.9) witnessed by the bilinear differential operator (33) given by \label{WitnessForFormalSelfadjointnessOfKleinGordonEquation} K(\Phi_1, \Phi_2) \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2 \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu} \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma Proof. $$ d K(\Phi_1, \Phi_2) \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2 \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu} \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu} \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu} \Box(\Phi_1) \Phi_2 – \Phi_1 \Box (\Phi_2) Example 5.29. (equations of motion of vacuum electromagnetism are vacuum Maxwell's equations) Consider the Lagrangian field theory of free electromagnetism on Minkowski spacetime from example 5.6. \frac{d}{d x^\mu}f^{\mu \nu} \delta a_\nu Hence for ##A \in \Gamma_{\Sigma}(T^\ast \Sigma) = \Omega^1(\Sigma)## a field history ("vector potential"), its Euler-Lagrange equation of motion according to def. 5.24 is & \frac{\partial}{\partial x^\mu} F^{\mu \nu} = 0 \Leftrightarrow\;\; & d \star_\eta F = 0 where ##F = d A## is the Faraday tensor (20). (In the coordinate-free formulation in the second line "##\star_\eta##" denotes the Hodge star operator induced by the pseudo-Riemannian metric ##\eta## on Minkowski spacetime.) These PDEs are called the vacuum Maxwell's equations. This, too, is a free field theory according to def. 5.25. Example 5.30. (equation of motion of Dirac field is Dirac equation) Consider the Lagrangian field theory of the Dirac field on Minkowski spacetime from example 5.9, with field fiber the spin representation ##S## regarded as a superpoint ##S_{odd}## and Lagrangian density given by the spinor bilinear pairing i \overline{\psi} \gamma^\mu \partial_\mu \psi + m \overline{\psi}\psi (in spacetime dimension ##p+1 \in \{3,4,6,10\}## with ##m = 0## unless ##p+1 = 4##). By example 5.21 the Euler-Lagrange differential operator (64) for the Dirac field is of the form \label{DiracOperatorAsELOperator} \Gamma_\Sigma(\Sigma \times S) &\overset{ }{\longrightarrow}& \Gamma_\Sigma(\Sigma \times S^\ast) \Psi &\mapsto& \overline{(-)} D \psi so that the corresponding Euler-Lagrange equation of motion (def. 5.24) is equivalently \label{DiracEquation} \underset{D}{ \left(-i \gamma^\mu \partial_\mu + m\right) \psi \;=\; 0 This is the Dirac equation and ##D## is called a Dirac operator. In terms of the Feynman slash notation from (16) the corresponding differential operator, the Dirac operator reads – i \partial\!\!\!/\, + m Hence this is a free field theory according to def. 5.25. Observe that the "square" of the Dirac operator is the Klein-Gordon operator ##\Box – m^2## (67) +i \gamma^\mu \partial_\mu + m \left(-i \gamma^\mu \partial_\mu + m\right)\psi \left(\partial_\mu \partial^\mu – m^2\right) \psi \left(\Box – m^2\right) \psi This means that a Dirac field which solves the Dirac equations is in particular (on Minkowski spacetime) componentwise a solution to the Klein-Gordon equation. Remark 5.31. (supergeometric nature of the Dirac equation as an Euler-Lagrange equation) While the Dirac equation (71) of example 5.30 would make sense in itself also if the field coordinates ##\psi## and jet coordinates ##\psi_{,\mu}## of the Dirac field were not anti-commuting (45), due to their supergeometric nature (remark 3.52), it would, by remark 5.23, then no longer be the Euler-Lagrange equation of a Lagrangian density, hence then Dirac field theory would not be a Lagrangian field theory. Example 5.32. (Dirac operator on Dirac spinors is formally self-adjoint differential operator) The _Dirac operator, hence the differential operator corresponding to the Dirac equation of example 5.30 via def. 4.7 is a formally anti-self adjoint (def. 4.9): D^\ast = – D Proof. By (70) we are to regard the Dirac operator as taking values in the dual spin bundle by using the Dirac conjugate ##\overline{(-)}## (14): Then we need to show that there is ##K(-,-)## such that for all pairs of spinor sections ##\Psi_1, \Psi_2## we have \overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1) \overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2) d K(\psi_1, \psi_2) But the spinor-to-vector pairing is symmetric (12), hence this is equivalent to \overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2 By the product law of differentiation>, this is solved, for all ##\Psi_1, \Psi_2##, by K(\Psi_1, \Psi_2) \left( \overline{\Psi_1} \gamma^\mu \Psi_2\right) \iota_{\partial_\mu} dvol This concludes our discussion of Lagrangian densities and their variational calculus. In the next chapter we consider the infinitesimal symmetries of Lagrangians. Urs Schreiber I am a researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague. Presently I am on leave at the Max Planck Institute for Mathematics in Bonn. Tags: Lagrangians, Quantum Field Theory https://www.physicsforums.com/insights/wp-content/uploads/2017/10/qft_lang.png 135 240 Urs Schreiber https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png Urs Schreiber2017-11-14 01:34:392021-04-21 11:01:09Learn Lagrangians in Mathematical Quantum Field Theory Learn the Geometry of Mathematical Quantum Field Theory Interview with Mathematician and Physicist Arnold Neumaier Struggles with the Continuum: Quantum Electrodynamics Learn Symmetries in Mathematical Quantum Field Theory Learn Spacetime in Mathematical Quantum Field Theory Introduction to Perturbative Quantum Field Theory strangerep says: ##beta## is dimensionless (regardless of which system of units you choose to use). I might have misunderstood you, but the only thing I thought was "wrong" is the statement that a physical quantity ("##S##", say) with dimensions of "action" can be made "dimensionless" by setting ##hbar=1##. However, if one works instead with ##S/hbar##, that quantity is now indeed dimensionless, and by choosing units s.t. ##hbar=1## one can reduce some of the mess. vanhees71 says: Yes, so what's wrong with what I wrote before? In natural units velocities are measured with dimensionless numbers, because you set ##c=1##. In the SI it's just ##beta=v/c## (with ##v## and ##c## measured in metre over second, m/s). vanhees71 I don't know, how you call a quantity which is just a number like, e.g., the Sommerfeld fine-structure constant ##alpha=frac{e^2}{4 pi} simeq 1/137## (where ##e## is again a pure number in the Heaviside-Lorentz units with ##hbar=c=1##), which conventionally is defined as (##alpha=frac{e^2}{4 pi epsilon_0 hbar c}## in SI units) or radians for angles. […] I call ##alpha## dimensionless because, in full gory detail, it is: $$ alpha ~:=~ frac{1}{4pivarepsilon_0} , frac{e^2}{hbar c} ~.$$ Urs Schreiber says: strangerep I did read those examples, of course, but failed to get a grip on what the presymplectic current means physically. I've been able to (more-or-less) follow your other stuff, since I could relate it to ordinary classical mechanics and pedestrian Lagrangian field theory. But the presymplectic current didn't ring a bell with anything I'd learned before. Probably I'm just missing something important.Oh, I see. i should maybe add some outlook explanation in that section. The role of the presymplectic current will be elucidated in the chapter "Phase Space": it is the jet bundle avatar of the symplectic form on the phase space. In other words, it is the current whose conserved charge with respect to some spacelike Cauchy surface is the symplectic form for the phase space corresponding to that Cauchy surface. You can see this well in the example of the scalar field: if you write "q" for the field "phi" and "p" for the field derivative, then the presymplectic current has the form ##delta p wedge delta q wedge volume ##. After transgression this becomes the familiar ##d P wedge d Q## on the phase space of the scalar field. I don't have time today to add an outlook remark on this in the present chapter. But please remind me to do so. I don't think "dimensionless" is the right word here. Just because you give something a value of 1 doesn't cause it to lose its physical type. E.g., if both ##hbar## and ##c## were dimensionless, I should be able to add them together. But that could never make sense because they're different types. You can't sensibly add a velocity to an action.I don't know, how you call a quantity which is just a number like, e.g., the Sommerfeld fine-structure constant ##alpha=frac{e^2}{4 pi} simeq 1/137## (where ##e## is again a pure number in the Heaviside-Lorentz units with ##hbar=c=1##), which conventionally is defined as (##alpha=frac{e^2}{4 pi epsilon_0 hbar c}## in SI units) or radians for angles. Of course, it doesn't make physical sense to add ##hbar## and ##c##. Also in conventional units you have examples for this: E.g., it doesn't make sense to add an energy (dimension Force times Distance) to a torque (also dimension Force times Distance). The usual and most convenient convention in HEP is to set ##hbar=c=1##. These are just conversion constants depending on the used system of units. It is expected that next year the entire SI will be redefined by giving these constants a fixed value (as is already the case for the speed of light, coupling the definition of the metre to that of the second, which is defined by a hyperfine transition of Cs). The only thing you need to know to convert from these natural units of the theoreticians to SI units is that ##hbar c simeq 0.197 text{GeV},text{fm}##. In the natural system you have the choice of only one unit left. Usually one works with GeV for masses, energies, momenta (in the conventional system multiplied with the appropriate powers of ##c##) and fm for times (conventionally ##text{fm}/c##) and distances. This, together with the value of ##hbar c##, given in these units, is all you need in HEP, and the natural units make dimensional analysis, as exemplified in the previous posting, easier. If you carry that thought to the end, you arrive at th rescaling of the metric: Because what does it mean to "rescale the (co)tangent bundle"? This can only mean to 1) have a scale on it and then 2) change that. But a "scale" on the (co)tangent bundle, that's precisely a (pseudo-)Riemannian metric. Actually, when I carry that thought further, I think of $$ds^2 ~=~ g_{munu} dx^mu dx^nu ~,$$and I want ##ds## to have the same dimensions as ##dx##. Also, if I re-scale ##dx^mu to k dx^mu## I want that to induce ##ds to kds##. That seems to force ##g## to be dimensionless. OTOH,… I suppose that if one wanted ##ds## to be invariant under ##dx^mu to k dx^mu##, then ##g## must scale (inversely) also. Usually we use natural units in HEP physics. Then ##hbar=c=1##. So actions are dimensionless. I don't think "dimensionless" is the right word here. Just because you give something a value of 1 doesn't cause it to lose its physical type. E.g., if both ##hbar## and ##c## were dimensionless, I should be able to add them together. But that could never make sense because they're different types. You can't sensibly add a velocity to an action. [##Theta_{BFV}##] spelled out earlier, in examples 5.11 and 5.13! I did read those examples, of course, but failed to get a grip on what the presymplectic current means physically. I've been able to (more-or-less) follow your other stuff, since I could relate it to ordinary classical mechanics and pedestrian Lagrangian field theory. But the presymplectic current didn't ring a bell with anything I'd learned before. Probably I'm just missing something important. Of course the fundamental form ##eta_{mu nu}=mathrm{diag}(1,-1,-1,-1)## (or the opposite sign, depending on your convention) is dimensionless.A detailed study of QFT with respect to "physical scaling" of the metric tensor is in https://arxiv.org/abs/1710.01937, following https://arxiv.org/abs/1411.1302. This is well adapted to QFT on curved spacetimes. Maybe I find time to come back to this later. In Example 5.4, I would have thought this is because of the 2 derivatives of ##phi## by the coordinate ##x^mu##. Normally the metric ##eta_{munu}## would be dimensionless. And btw, what are your dimensions of ##phi##? I'm a bit confused about how you're apparently dispensing with physical units here. In ordinary Lagrangian mechanics, we'd have $$S ~=~ int L ,dt$$where ##S## has dimensions of action (##ML^2/T##), hence ##L## must have dimensions ##ML^2/T^2##, i.e., energy. If we express ##L## in terms of a density ##{mathcal L}## via $$L ~=~ int {mathcal L} ; d^3x$$then ##{mathcal L}## must have dimensions ##M/LT^2##. But perhaps you mean to make ##S## dimensionless by replacing it by ##S/hbar## ? :oldconfused:Usually we use natural units in HEP physics. Then ##hbar=c=1##. So actions are dimensionless. In (1+3)-Minkowski space (physical case) thus the Lagrangian (to be precise the Lagrangian density) must be of dimension ##text{Length}^{-4}##. Of course the fundamental form ##eta_{mu nu}=mathrm{diag}(1,-1,-1,-1)## (or the opposite sign, depending on your convention) is dimensionless. The kinetic term for a KG field is $$mathcal{L}=(partial_{mu} phi)^*(partial^{mu} phi).$$ Each derivative has dimension ##1/text{Length}##. Consequently the KG field must have also dimension ##1/text{length}## such that ##mathcal{L}## has dimension ##1/text{Length}^4##, such that the action is dimensionless. In natural units energies, momenta, and masses have of course the dimension ##1/text{Length}## or the other way around length has dimension ##1/text{Energy}##. Usually one counts in terms of energ-momenta-mass dimensions. Then the counting rules for the superficial degree of divergence in renormalization theory becomes more natural, and you can simply state that a Lagrangian leads to a superficially renormalizable theory (in (1+3) spacetime dimensions) if the energy-mopmetnum-mass dimensions of the coefficients in the lagrangian (derivatives, masses, and coupling constants) are not negative. Re: "shell": I'm glad you're using this term in a unified way in both the classical and quantum cases. I once used the term "on-shell" in a classical context to mean "in the space of trajectories", but was scolded for doing so. o_OHm, saying "on-shell" for "when the equations of motion hold" is widely adopted standard. You should feel relaxed about saying it!:-) Prop 5.8, 1st para: Is ##{mathbf {LL}}## a typo ? Also, in defn 5.18: typo: mathfbThanks! Fixed now. In the examples from 5.20 onwards: it wouldn't hurt to illustrate the pre-symplectic current explicitly in each case. (I must say I had great difficulty getting my head around ##Theta_{BFV}## in the abstract.) :oldfrown:That's spelled out earlier, in examples 5.11 and 5.13! Pardon my ignorance, but,… why not? I would have thought one could at least perform a position-dependent rescaling of the (co-)tangent space at a point, hence rescaling ##dx^mu## and ##partial/partial x^mu##.If you carry that thought to the end, you arrive at th rescaling of the metric: Because what does it mean to "rescale the (co)tangent bundle"? This can only mean to 1) have a scale on it and then 2) change that. But a "scale" on the (co)tangent bundle, that's precisely a (pseudo-)Riemannian metric. In the examples from 5.20 onwards: it wouldn't hurt to illustrate the pre-symplectic current explicitly in each case. (I must say I had great difficulty getting my head around ##Theta_{BFV}## in the abstract.) :oldfrown: Re: "shell": I'm glad you're using this term in a unified way in both the classical and quantum cases. I once used the term "on-shell" in a classical context to mean "in the space of trajectories", but was scolded for doing so. o_O On curved spacetime it makes no sense to "rescale coordinates", […] Pardon my ignorance, but,… why not? I would have thought one could at least perform a position-dependent rescaling of the (co-)tangent space at a point, hence rescaling ##dx^mu## and ##partial/partial x^mu##. I would have thought this is because of the 2 derivatives of ##phi## by the coordinate ##x^mu##. Normally the metric ##eta_{munu}## would be dimensionless.Right, these are two different ways to see the dimension. These two ways lead to equivalent dimensions over Minkowski spacetime, but only the other one, which i use in the series, generalizes well to curved spacetime: On curved spacetime it makes no sense to "rescale coordinates", but it always makes sense to rescale the metric. Even though I don't discuss this generalization in the series, my aim is to present everything in a form such that this generalization will be straightforward. The alternative perspective that you have in mind I have spelled out on the nLab here For this to be physical unit-free in the sense of remark 5.2 the physical unit of the parameter ##m## must be that of the inverse metric, hence must be an inverse length I would have thought this is because of the 2 derivatives of ##phi## by the coordinate ##x^mu##. Normally the metric ##eta_{munu}## would be dimensionless. But perhaps you mean to make ##S## dimensionless by replacing it by ##S/hbar## ? :oldconfused: High Energy, Nuclear, Particle Physics Receive Insights Articles to Your Inbox Blog Information Blog Author List astronomy (16) black holes (16) classical physics (28) cosmology (15) education (21) electromagnetism (14) general relativity (16) gravity (22) interview (22) mathematics (25) mathematics self-study (16) Physicist (26) Physics Career (13) programming (13) Quantum Field Theory (31) quantum mechanics (32) quantum physics (19) relativity (31) Special Relativity (13) universe (19) 2021 © PHYSICS FORUMS, ALL RIGHTS RESERVED - Contact Us - Privacy Policy - About PF Insights Learn Field Variations in Mathematical Quantum Field Theory	CommonCrawl
\begin{document} \leftline{ \scriptsize \it International Journal of Group Theory Vol. {\bf\rm XX} No. X {\rm(}201X{\rm)}, pp XX-XX.} \title{Groups of order 2048 with three generators and three relations} \author{S. FOULADI and R. ORFI$^$} \thanks{{\scriptsize \hskip -0.4 true cm MSC(2010): Primary: 20F05 ; Secondary: 20D15. \newline Keywords: Schur multiplier, presentation, deficiency zero, finite $p$-group, lower exponent-$p$ central series.\\ Received: 26 July 2011, Accepted: 21 June 2010.\\ $$Corresponding author \newline\indent{\scriptsize $\copyright$ 2011 University of Isfahan}}} \maketitle \begin{center} Communicated by\; \end{center} \begin{abstract} It is shown that there are exactly seventy-eight 3-generator 2-groups of order $2^{11}$ with trivial Schur multiplier. We then give 3-generator, 3-relation presentations for forty-eight of them proving that these groups have deficiency zero. \end{abstract} \vskip 0.2 true cm \pagestyle{myheadings} \markboth{\rightline {\scriptsize FOULADI and ORFI}} {\leftline{\scriptsize Groups of order 2048 with three generators and three relations }} \section{\bf Introduction} \vskip 0.4 true cm A finite group is said to have deficiency zero if it has a deficiency zero presentation, namely a presentation with an equal number of generators and relations. A classical fact is that finite groups of deficiency zero have trivial Schur multiplier, for example see [\ref{J}, p.87]. So the Schur multiplier provides a useful criterion in the search for finite groups of deficiency zero. But the converse is not true since there are many examples of finite groups with trivial Schur multiplier and non-zero deficiency. These groups are all non-nilpotent. In fact, it is a long-standing question about finite $p$-groups with trivial Schur multiplier whether they have deficiency zero, see [\ref{W}, Question 12]. In [\ref{G}], the authors prove a number of $p$-groups have deficiency zero and give explicit presentations for them with an equal number of generators and relations. It is noted in [\ref{G}] that there are no $3$-generator $2$-groups of order less than $2^9$ having trivial Schur multiplier and there exist exactly two such groups of order $2^9$. Moreover in [\ref{F}] we see that there are exactly eighteen $3$-generator $2$-groups of order $2^{10}$ with trivial Schur multiplier all having deficiency zero. Many finite $d$-generator, $d$-relation groups are known for $d=1,2,3$. Trivial examples are the finite cyclic groups with $d=1$ and the symmetric group of degree 3 with $d=2$. In fact many examples with $d=2$ have been given by several authors. Examples of finite groups with $d=3$ are infrequent, see [\ref{Ja}] and the references therein. It might be worth noting that there are no known examples of finite groups with $d=4$ and finite nilpotent $4$-generator groups require at least $5$ defining relations by a celebrated theorem of Golod-Shafarevich. Such groups have been constructed in [\ref{GN}, \ref{HN}], of orders $2^{14}, 2^{16}, 2^{17}, 2^{18}$ and $2^{19}$. In this paper, using computational methods we show that there are exactly seventy-eight $3$-generator $2$-groups of order $2^{11}$ with trivial Schur multiplier. We then give $3$-generator, $3$-relation presentations for forty-eight of them proving that these groups all have deficiency zero. Our notation is standard. $\mathbb{Z}_{n}$ is the cyclic group of order $n$. The direct product of $\ell$ copies of $\mathbb{Z}_{n}$ is denoted by $ \mathbb{Z}_{n}^\ell$. The Schur multiplier of the group $G$ is denoted by $M(G)$. We write SmallGroup$(n,m)$ for the $m$th group of order $n$ as quoted in the "Small Groups" library in \textsf{GAP} [\ref{GA}]. \section{\textbf{Method}} \vspace{0.4cm} In this section our first step is to determine all $3$-generator groups of order $2^{11}$ with trivial Schur multiplier. Then our second step is to show that some of these groups have deficiency zero.\\ We describe below a method that enables one to determine $3$-generator groups of order $2^{11}$ having trivial Schur multiplier. We use the computer algebra systems \textsf{GAP} [\ref{GA}] and {\sc Magma} [\ref{MA}] which contain a data library \lq \lq Small Groups\rq \rq providing access to the descriptions of the groups of order at most $2000$ except $2^{10}$, prepared by Besche {\it et al} [\ref{BEO}]. Following [\ref{G}], our main strategy is to determine some particular extensions, called descendants, of specified $3$-generator $2$-groups $G$, $\|G\|\leq 2^9$, in the hope of finding $2$-groups of order $2^{11}$ with trivial Schur multiplier. To do this we will use the following theorems. \begin{thm}\label{2.1} \textup{[\ref{K}, Theorem 3.2.1]} Suppose that $N$ is a normal subgroup of a finite group $G$. If $F$ is a free group of finite rank, $R$ is a normal subgroup of $F$ for which $G\cong F/R$ and $S$ is a normal subgroup of $F$ for which $SR/R$ corresponds to $N$, then there is an exact sequence $$1\rightarrow(\frac{R\cap[F,S])}{([F,R]\cap[F,S])}\rightarrow M(G)\rightarrow M(\frac{G}{N})\rightarrow \frac{(N\cap G')}{[N,G]}\rightarrow1.$$ \end{thm} \begin{thm}\label{2.2} \textup{[\ref{K}, Corollary 3.2.2]} Suppose that $N$ is a normal subgroup of a finite group $E$. If $M(E)=1$, then $M(\frac{E}{N})\cong \frac{(N\cap E')}{[N, E]}.$ \end{thm} Recall that the lower exponent-$p$ central series of $G$ is a descending series of subgroups defined recursively by $P_{0}(G)=G$, $P_{i+1}(G)=[P_{i}(G), G] P_{i}(G)^{p}$ for $i\geq 0$. If $c$ is the smallest integer such that $P_{c}(G)=1$, then $G$ has exponent-$p$ class $c$. A group $E$ is said to be a descendant of a finite $d$-generator $p$-group $G$ with exponent-$p$ class $c$ if the quotient $E/P_{c}(E)$ is isomorphic to $G$. A group is called an immediate descendant of $G$ if it is a descendant of $G$ and has exponent-$p$ class $c+1$. Both \textsf{GAP} and {\sc Magma} compute the lower exponent-$p$ central series of a finite group using the $p$-quotient algorithm described in [\ref{MB}] and are able to construct all immediate descendants of a given $p$-group by the $p$-group generation algorithm [\ref{O}]. Now we determine all $3$-generator $2$-groups $E$ of order $2^{11}$ with trivial Schur multiplier. \begin{lem}\label{2.3} Let $E$ be a $3$-generator $2$-group of order $2^{11}$ with trivial Schur multiplier. Then $E$ is an immediate descendant of a $3$-generator group $G$ of order $2^n$ $(n\leq 10)$ which satisfies $M(G)\cong \mathbb{Z}_{2}^\ell$, where \break $0\leq \ell \leq 11-n$. \end{lem} \begin{proof} Suppose that $E$ has exponent-$p$ class $c+1.$ Using Theorem \ref{2.2}, with $N=P_{c}(E)$, we have $M(E/P_{c}(E))\cong (P_{c}(E)\cap E')/[ P_{c}(E), E].$ By our hypothesis on the class of $E$, we observe that $P_{c}(E)^2=1$, from which we conclude that $P_{c}(E)$ is an elementary abelian $2$-group and that $M(E/P_{c}(E))\cong P_{c}(E)\cap E'$. Now the group $G:=E/P_{c}( E)$ is a $3$-generator group with $M(G)\hookrightarrow P_{c}(E)$ and so $\|M(G)\|\leq \|E\|/\|G\|$. \end{proof} The above lemma reduces the number of groups that need to be considered dramatically. We use {\sc Magma} and \textsf{GAP} to construct all immediate descendants $E$ of such groups $G$ and rule out those having non-trivial Schur multiplier. Since all groups $G$ of order $2^n$ $(n\leq 9)$ are available in \textsf{GAP}, first we determine all immediate descendants of groups $G$ which satisfy $M(G)\cong \mathbb{Z}_{2}^\ell$, where $0\leq \ell \leq 11-n$. In the list below there are forty $3$-generator groups of order $2^{11}$ with trivial Schur multiplier with the above property. We use the notation $[n, m, k]$ for the group $E$, where $E$ is the $k$th immediate descendant of the group $G$=SmallGroup$(n, m)$. \vspace{0.2cm} \noindent [512, 6489, 2], [512, 6489, 3], [512, 6490, 2], [512, 6490, 3], [512, 9113, 4],\break [512, 9113, 5], [512, 9114, 4], [512, 9114, 5], [512, 9121, 4], [512, 9121, 5],\break [512, 9122, 4], [512, 9122, 5], [512, 9137, 4], [512, 9137, 5], [512, 9146, 4], \break [512, 9146, 5] , [512, 12397, 4], [512, 12397, 5], [512, 12398, 4],\break [512, 12398, 5], [512, 12399, 4], [512, 12399, 5], [512, 12400, 4],\break [512, 12400, 5], [512, 12401, 4], [512, 12401, 5], [512, 12402, 4],\break [512, 12402, 5], [512, 12403, 2], [512, 12403, 3], [512, 12404, 2],\break [512, 12404, 3], [512, 12413, 4], [512, 12413, 5], [512, 12414, 4],\break [512, 12414, 5], \hspace{.2 cm}[512, 12423, 4], [512, 12423, 5], [512, 12424, 4], \break [512, 12424, 5].\\ Now by Lemma \ref{2.3}, we have to consider 3-generator groups $G$ of order $2^{10}$ with $M(G)\cong \mathbb{Z}_{2}^\ell$, where $0\leq \ell \leq 1$. All 3-generator groups of order $2^{10}$ with trivial Schur multiplier are classified in [\ref{F}]. By using \textsf{GAP} we see that there is no immediate descendant of order $2^{11}$ of these eighteen groups of order $2^{10}$. Since groups of order $2^{10}$ are not available in \textsf{GAP}, we state the following theorem to construct groups of order $2^{10}$ with Schur multiplier of order 2. \begin{thm}\label{2.4} Let $E$ be a $3$-generator $2$-group of order $2^{10}$ with $M(E)\cong \mathbb{Z}_{2}$. Then $E$ is an immediate descendant of a $3$-generator group $G$ of order $2^n$ with $n\leq 9$ which satisfies either $M(G)\cong \mathbb{Z}_{2}^\ell$, $0\leq \ell \leq 11-n$ or $M(G)\cong \mathbb{Z}_{4}\times \mathbb{Z}_{2}^\ell$, $0\leq \ell \leq 9-n$. \end{thm} \begin{proof} Suppose that $E$ has exponent-$p$ class $c+1.$ By Theorem \ref{2.1}, we have the following exact sequence: $1\rightarrow (R\cap[F,S])/([F,R]\cap[F,S])\rightarrow M(E)\xrightarrow{\alpha} M(E/P_{c}(E))\xrightarrow{\beta} (P_{c}(E)\cap E')/[P_{c}(E),E]\rightarrow 1,$ where $F$ is a free group of finite rank, $R$ is a normal subgroup of $F$ for which $E\cong F/R$ and $S$ is a normal subgroup of $F$ for which $SR/R$ corresponds to $P_{c}(E)$. On setting $G=E/P_{c}(E)$ we see that $M(G)/Ker{\beta}\cong P_{c}(E)\cap E'$ and $M(E)/Ker{\alpha}\cong Ker{\beta}$. Therefore $Ker{\beta}=1$ or $Ker{\beta}\cong \mathbb{Z}_{2}$ since $M(E)\cong \mathbb{Z}_{2}$. Now since $M(G)/Ker{\beta}\hookrightarrow P_{c}(E)$ and $P_{c}(E)$ is elementary abelian, we deduce that $\|M(G)\|\leq 2\|P_{c}(E)\|$ and $M(G)$ is either elementary abelian or $M(G)\cong \mathbb{Z}_{4}\times \mathbb{Z}_{2}^\ell$. \end{proof} Now it only remains to determine all immediate descendants of groups of order $2^{10}$ with Schur multiplier of order 2. In the list below there are thirty-eight groups of order $2^{11}$ with trivial Schur multiplier with the above property. We use the notation $[n, m, k, t]$ for the group $E$, where $E$ is the $t$th immediate descendant of the group $L$ such that $L$ is the $k$th immediate descendant of $G$=SmallGroup$(n, m)$, in fact $L$ is a 3-generator group of order $2^{10}$ with Schur multiplier of order 2. \vspace{0.2cm} \noindent [512, 53479, 3, 1], [512, 53479, 3, 2], [512, 53480, 1, 1], [512, 53480, 1, 2],\break [512, 53480, 2, 1], [512, 53480, 2, 2], [256, 2525, 8, 1], [256, 2525, 8, 2],\break [256, 2525, 9, 1], [256, 2525, 9, 2], [256, 2528, 5, 1], [256, 2528, 5, 2],\break [256, 2528, 6, 1], [256, 2528, 6, 2], [256, 3638, 8, 1], [256, 3638, 8, 2],\break [256, 3639, 8, 1], [256, 3639, 8, 2], [256, 3640, 5, 1], [256, 3640, 5, 2],\break [256, 3640, 6, 1], [256, 3640, 6, 2], [256, 3641, 8, 1], [256, 3641, 8, 2],\break [256, 3641, 9, 1], [256, 3641, 9, 2], [256, 3641, 10, 1], [256, 3641, 10, 2],\break [256, 3643, 8, 1], [256, 3643, 8, 2], [256, 3643, 9, 1], [256, 3643, 9, 2],\break [256, 3643, 10, 1], [256, 3643, 10, 2], [256, 2522, 6, 1], [256, 2522, 6, 2], \break [256, 2523, 6, 1], [256, 2523, 6, 2]. \vspace{0.2cm} The second step is to give $3$-generator, $3$-relation presentations for the groups obtained in the first step. We used mainly the method described in [\ref{HN}] to find such a presentation for each group $G$ under consideration. Our first attempt towards obtaining such presentations for $G$ was to find several triples of generators for each group. On each generating triple, we computed a presentation $\la X \| R \ra$ using the relation finding algorithm of Cannon [\ref{C}] which is available in \textsf{GAP} and {\sc Magma}. Then an attempt was made to find a subset $S$ of $R$ having three elements such that $\la X \| S\ra $ defines $G$. In searching for generating triples for each group a small set of group elements was chosen by a knowledge of conjugacy classes and checked for generating triples. This technique was also used in [\ref{F}] to determine deficiency zero presentations for all 3-generator, 2-groups of order $2^{10}$ with trivial Schur multiplier. The authors obtained seventeen deficiency zero presentations from eighteen groups in [\ref{F}] by this method. It seems that this method is useful to find deficiency zero presentations. Moreover in this paper to find the order of the groups defined by the presentations $\la X \| S\ra $ as above, we use Knuth-Bendix algorithm in {\sc KBMAG} package, which was written by Derek Holt [\ref{GA}]. By the above observation we show that forty-eight groups from seventy-eight 3-generator groups of order $2^{11}$ with trivial Schur multiplier, have deficiency zero. \section{\textbf{Results}} \vspace*{0.2cm} In three tables below we list all $3$-generator $2$-groups of order $2^{11}$ with trivial Schur multiplier. In tables 1 and 2 we list forty-eight groups with deficiency zero. Also table 3 give a presentation for the remaining thirty groups with more than three relations in which we show that the above method failed to find a balanced presentation for these groups. Entries of the form $[n, m, k]$ and $[n, m, k, t]$ were described in the previous section. An attempt was made to choose a presentation for each group with a reasonably small length. \hspace{4.5cm} \noindent \textbf{ Table 1 } \noindent \begin{tabular}{ l l l l } \hline Group No. & Relators & $[n, m, k]$ \\ \hline $\#1$ & $b^{-1}acabc^{-1}$, $ c^2ab^2a $, $ab^{-1}cacb^{-1}$ & $[ 512, 6489, 2 ]$ \\ $\#2$ & $bca^{-1}c^{-1}ba$, $bac^{-1}b^{-3}ca$, $ cbca^{-1}c^2a^{-1}b$ & $[ 512, 6489, 3 ]$ \\ $\#3$ & $ba^{-1}c^{-2}ba$, $ a^2cbcb^{-1}$, $ bcbaca^{-1}$ & $[ 512, 6490, 2 ]$\\ $\#4$ & $bac^{-1}b^{-1}ca$, $ b^2cac^{-1}a$, $b^{-1}cbc^3a^{-2}$ & $[ 512, 6490, 3 ]$\\ $\#5$ & $b^2ca^{-1}ca$, $c^{-1}a^2b^{-1}cb$, $ ba^{-1}b^{-1}c^4a$ & $[512, 9113, 4 ]$\\ $\#6$ & $ba^{-1}b^3a$, $b^{-1}c^3bc^{-1}$, $a^3b^{-1}cba^{-1}c^{-1}$ & $[ 512, 9113, 5 ]$\\ $\#7$ & $ bca^{-1}b^{-1}ca$, $b^3cbc$, $a^3b^{-1}a^{-1}cbc^{-1}$ & $[512, 9114, 4 ]$\\ $\#8$ & $ cbcb^{-1}$, $ba^{-1}b^3a$, $a^3c^3a^{-1}c^{-1}$ & $[ 512, 9114, 5 ]$\\ $\#9$ & $ ba^{-1}c^{-1}bc^{-1}a$, $ab^{-2}cac^{-1}$, $a^2c^3bc^{-1}b^{-1}$ & $[ 512, 9122, 4 ]$\\ $\#10$ & $ ba^{-1}c^{-1}bc^{-1}a$, $ca^{-1}c^{-1}b^2a^{-1}$, $a^2c^{-1}bc^3b^{-1}$ & $[ 512, 9122, 5 ]$\\ $\#11$ & $ a^{-1}cbcab^{-1}$, $b^3a^{-1}ba$, $ cac^3a^{-3}$ & $[ 512, 9137, 4 ]$\\ \end{tabular}\\ \hspace{4.5cm} \noindent \textbf{ Table 1 } \noindent \begin{tabular}{ l l l l } \hline Group No. & Relators & $[n, m, k]$ \\ \hline $\#12$ & $ a^{-1}cbcab^{-1}$, $b^3a^{-1}ba$, $ a^3ca^{-1}c^3$ & $[ 512, 9137, 5 ]$\\ $\#13$ & $ cbcb^{-1}$, $a^{-1}ba^{-2}b^2a^{-1}b$, $c^3aca^{-3}$ & $[512, 9146, 4 ]$\\ $\#14$ & $ cbcb^{-1}$, $ a^{-1}ba^{-2}b^2a^{-1}b$, $a^3c^3a^{-1}c$ & $[ 512, 9146, 5 ]$\\ $\#15$ & $a^3bab, acab^{-1}c^{-1}b, b^2cac^{-3}a$ & $[ 512, 12397, 4]$\\ $\#16$ & $ a^3bab, acab^{-1}c^{-1}b, b^2c^{-1}ac^3a$ & $[ 512,12397, 5 ]$\\ $\#17$ & $ baba^{-1}, a^2ca^{-1}cbab^{-1}, b^3c^3b^{-1}c^{-1}$ & $[ 512, 12398, 4 ]$ \\ $\#18$ & $cac^{-1}a, b^{-1}a^{-1}ba^2b^2a, bac^{-3}bca$ & $[ 512, 12398, 5 ]$\\ $\#19$ & $ bab^{-1}a, a^{-1}c^3ac^{-1}, a^3c^{-1}abc^{-1}b^{-3}$ & $[512, 12400, 4 ]$\\ $\#20$ & $bab^{-1}a, a^{-1}c^3ac^{-1}, a^2b^{-1}a^{-1}ca^{-1}bcb^{-2}$ & $[ 512, 12400, 5 ]$\\ $\#21$ & $cbc^{-1}b, ac^3a^{-1}c^{-1}, ba^{-1}ba^2b^2a$ & $[512, 12401, 4 ]$\\ $\#22$ & $cbc^{-1}b, a^{-1}c^3ac^{-1}, ba^{-1}ba^2b^2a$ & $[ 512, 12401, 5 ]$\\ $\#23$ & $cbc^{-1}b, ba^{-1}b^3a^{-1}, a^3cac^{-3}$ & $[ 512, 12402, 4]$\\ $\#24$ & $ab^{-3}ab^{-1}, bc^{-3}bc^{-1}, a^2bca^{-1}ca^{-1}b^{-1}$ & $[512, 12404, 2 ]$\\ $\#25$ & $bab^3a, cbc^3b, bacacb^{-1}a^{-2}$ & $[ 512, 12404, 3 ]$\\ $\#26$ & $ba^{-1}bc^2a, ac^2bab^{-1}, (ca)^2cbcb^{-1}$ & $[ 512, 12413, 5 ]$\\ $\#27$ & $b^{-1}a^{-1}b^3a, ca^{-1}cbab^{-1}, a^2c^{-2}(bc^{-1})^2$ & $[ 512, 12414, 4 ]$\\ $\#28$ & $cbac^{-1}ab^{-1}, acbca^{-1}b^{-1}, a^2c^{-1}a^{-1}bc^{-1}ab$ & $[ 512, 12424,4 ]$\\ \end{tabular}\\ \hspace{4.5cm} \noindent \textbf{ Table 2 } \noindent \begin{tabular}{ l l l l } \hline Group No. & Relators & $ [n, m, k, t]$ \\ \hline $\#29$ & $cbc^{-1}b, a^3b^{-1}c^{-1}ba^{-1}c, a^3bc^{-1}a^{-1}c^{-1}b $ & $[ 512, 53480, 2, 1 ]$\\ $\#30$ & $cbc^{-1}b, a^{-3}bcacb, a^3c^{-1}a^{-1}cb^2 $ & $[ 512, 53480, 2, 2 ]$ \\ $\#31$ & $ cbcb^{-1}, ba^{-1}c^{-1}bca, a^2b^{-1}acbac $ & $ [ 256, 2528, 5, 1 ]$\\ $\#32$ & $a^2c^2, bcb^{-1}ac^{-1}a, bacb^3ca^{-1} $ & $[ 256, 2528, 5, 2 ]$ \\ $\#33$ & $ a^2(bc)^2, bcb^{-1}aca^{-1}, a^3c^{-2}bab^{-1} $ & $[256, 2528, 6, 1 ]$ \\ $\#34$ & $ ba^{-1}cb^{-1}ca, cb^2ca^{-2}, a^3bcba^{-1}c^{-1} $ & $[ 256, 2528, 6, 2 ]$\\ $\#35$ & $c^{-1}bcb, ac^{-1}ac^{-1}b^2, a^2ca^{-1}bc^{-1}ab$ & $ [ 256, 3640, 5, 1 ]$ \\ $\#36$ & $ ba^{-1}cbc^{-1}a, ac^3a^{-1}c^{-1}, b^{-3}c^{-1}bca^2 $ & $[ 256, 3640, 5, 2 ]$ \\ $\#37$ & $ a^{-1}b^3ab^{-1}, b^{-1}a^{-1}cabc, c^{-1}ac^{-3}bab$ & $ [ 256, 3640, 6, 1 ]$ \\ \end{tabular}\\ \hspace{4.5cm} \noindent \textbf{ Table 2 } \noindent \begin{tabular}{ l l l l } \hline Group No. & Relators & $ [n, m, k, t]$ \\ \hline $\#38$ & $ a^{-1}b^3ab^{-1}, b^{-1}a^{-1}cabc, c^{-2}ac^{-1}bcab $ & $[ 256, 3640, 6, 2 ]$ \\ $\#39$ & $ a^{-1}cba^{-1}c^{-1}b, b^3cb^{-1}c^{-1}, a^3b^{-1}c^2ab^{-1} $ & $[ 256, 3641, 10, 1 ]$ \\ $\#40$ & $ caca^{-1}, a^{-1}b^3ab^{-1}, acabc^{-3}b $ & $[ 256, 3641, 10, 2 ]$\\ $\#41$ & $ cac^3a, bcba^{-1}b^{-2}ca, babca^{-1}c^{-1}a^{-2} $ & $[ 256, 3643, 8, 1 ]$ \\ $\#42$ & $ ac^{-3}ac^{-1}, bcba^{-1}b^{-2}ca, babca^{-1}c^{-1}a^{-2}$ & $ [ 256, 3643, 8, 2 ]$ \\ $\#43$ & $ ac^{-3}ac^{-1}, a^{-1}cbcb^2a^{-1}b^{-1}, babca^{-1}c^{-1}a^{-2} $ & $[ 256, 3643, 10, 1 ]$ \\ $\#44$ & $ cac^3a, b^3cb^{-1}a^{-1}ca, babca^{-1}c^{-1}a^{-2} $ & $ [ 256, 3643, 10, 2 ]$\\ $\#45$ & $a^2(ab^{-1})^2, bc^3b^{-1}c^{-1}, a^3b^2cac^{-1} $ & $[ 256, 2522, 6, 1 ]$\\ $\#46$ & $ a^2(ab^{-1})^2, bc^3b^{-1}c^{-1}, a^3c^{-1}ab^2c $ & $[ 256, 2522, 6, 2 ]$ \\ $\#47$ & $ ca^{-1}bca^{-1}b^{-1}, cab^{-2}ca^{-1}, a^2c^2(ba)^2 $ & $ [ 256, 2523, 6, 1 ]$\\ $\#48$ & $a^2(ab^{-1})^2, b^{-1}c^3bc^{-1}, a^3cabcb $ & $[ 256, 2523, 6, 2 ]$\\ \end{tabular}\\ \hspace{4.5cm} \noindent \textbf{ Table 3 } \noindent \begin{tabular}{ l l l l } \hline Group No. & Relators & $ [n, m, k, t]$ \\ \hline $\#49$ & $ bac^{-1}b^{-1}ca, ba^{-1}b^3a, ab^5a^{-1}b^{-1}, a^2bcb^3c^{-1}, $ & $ [ 512, 9121, 4 ]$ \\ & $cabc^{-3}ba$ & \\ $\#50$ & $ bac^{-1}b^{-1}ca, ba^{-1}b^3a, ab^5a^{-1}b^{-1}, a^2bcb^3c^{-1}, $ & $[ 512, 9121, 5 ] $ \\ & $ c^{-1}bc^{-2}acab$ & \\ $\#51$ & $a^3b^{-1}a^{-1}b, ac^3a^{-1}c^{-1}, a^5bab^{-1}, a^2c^2a^{-2}c^{-2}, $ & $ [ 512, 12399, 4 ]$\\ & $ aca^{-1}bcb^{-3}$ & \\ $\#52$ & $bc^3b^{-1}c^{-1}, b^3a^3b^{-1}a^{-1}, aca^{-1}c^{-1}a^{-1}cac^{-1}, $ & $[ 512, 12399, 5 ]$\\ & $ ab^{-1}c^{-1}a^{-1}cb^3, a^2cac^{-2}ac $ & \\ $\#53$ & $ bab^3a, b^2cb^2c^{-1}, a^{-1}c^3ac^{-1}, bc^{-1}bca^{-4},$ & $ [ 512, 12402, 5 ]$\\ & $(bc)^2(bc^{-1})^2 $ & \\ $\#54$ & $ a^3b^{-1}a^{-1}b, a^5bab^{-1}, a^3c^{-1}ac^3, $ & $[ 512, 12403, 2 ] $ \\ & $a^2c^2a^{-2}c^{-2}, babc^{-1}b^{-1}cba$ & \\ $\#55$ & $ a^3b^{-1}a^{-1}b, a^5bab^{-1}, a^3c^{-3}ac, $ & $ [ 512, 12403, 3 ]$ \\ & $a^2c^2a^{-2}c^{-2}, bab^2cb^{-1}c^{-1}a $ & \\ \end{tabular} \hspace{4.5cm} \noindent \textbf{ Table 3 } \noindent \begin{tabular}{ l l l l } \hline Group No. & Relators & $ [n, m, k, t]$ \\ \hline $\#56$ & $ ba^{-1}bc^2a, ac^2bab^{-1}, b^2cb^2c^{-1}, $ & $[ 512, 12413, 4 ] $\\ & $ cacb^{-1}c^{-1}bc^{-1}a$ & \\ $\#57$ & $ bcb^{-1}ca^{-2}, a^2b^{-1}cbc, a^{-1}c^3ac^{-1},$ & $[ 512, 12414, 5 ]$\\ & $ a^3b^{-3}a^{-1}b $ & \\ $\#58$ & $ ac^2ab^2, c^2a^2b^{-2}, a^{-1}cbcab, $ & $[ 512, 12423, 4 ] $\\ & $ c^{-2}ac^{-1}bca^{-1}b, cbaca^{-1}bc^{-1}a^{-1}c^{-1}ba^{-1}b^{-1}$ & \\ $\#59$ & $ac^{-1}abc^{-1}b^{-1}, b^{-1}cbaca, cbc^{-3}b, $ & $[ 512, 12423, 5 ] $\\ & $ a^3b^{-1}cac^{-1}b^{-1} $ & \\ $\#60$ & $bcbc^{-1}a^{-2}, b^2c^{-1}aca^{-1}, b^{-1}ca^{-1}cba, $ & $ [ 512, 12424, 5 ]$\\ & $ a^2c^3bcb $ & \\ $\#61$ & $ b^2ab^2a^{-1}, c^2bc^2b^{-1}, (bc)^2ac^{-1}a^{-1}c, $ & $[ 512, 53479, 3, 1 ] $\\ & $ cbcaba^{-3}, a^3cb^{-1}a^{-1}bc^{-1}, a^2c^7b^{-1}c^{-1}b$ & \\ $\#62$ & $ a^2ca^{-2}c^{-1}, bacbc^{-1}a, bca^{-1}c^{-1}ba,$ & $ [ 512, 53479, 3, 2 ]$\\ & $acabc^{-1}b^{-1}, c^2bc^2b^{-1}, b^{16}ca^{-2}c $ & \\ $\#63$ & $ cab^2c^{-1}a^{-3}, cba^{-1}cb^{-1}a^3, a^2c^{-1}b^{-1}a^{-1}cab, $ & $[ 512, 53480, 1, 1 ] $\\ & $ b^{-2}cbaba^{-1}c, a^{-1}(bc)^2ac^2$ & \\ $\#64$ & $ c^{-1}a^{-1}c^{-1}bab, b^{-1}c^2aba^{-1}, a^2ba^2b^{-1}, $ & $ [ 512, 53480, 1, 2 ]$\\ & $ a^2ca^2c^{-1}, bc^3bca^2c^2$ & \\ $\#65$ & $ c^4, a^2ba^2b^{-1}, b^{-1}cacba, acab^{-1}c^{-1}b, $ & $[ 256, 2525, 8, 1 ] $\\ & $ b^5c^{-1}a^{-1}cb^{-1}a$ & \\ $\#66$ & $ c^4, a^2ba^2b^{-1}, b^{-1}cacba, acab^{-1}c^{-1}b, $ & $ [ 256, 2525, 8, 2 ]$\\ & $b^5cb^{-1}aca^{-1} $ & \\ $\#67$ & $ bcac^{-1}ba, b^2cb^{-2}c^{-1}, a^3bc^{-1}bac^{-1}, $ & $[ 256, 2525, 9, 1 ] $\\ & $ a^2c^{-1}bc^3b^{-1}, a^3cab^{-1}cb^{-1}$ & \\ $\#68$ & $ c^{-1}b^{-2}cb^2, bc^{-1}acba, b^{-1}acbca^{-3}, $ & $ [ 256, 2525, 9, 2 ]$\\ & $c^2b^{-1}a^{-1}bc^2a, a^3b^{-1}cb^{-1}ac $ & \\ $\#69$ & $ ac^{-2}a^{-1}c^2, ac^{-1}b^{-1}abc, acbac^{-1}b^{-3},$ & $ [ 256, 3638, 8, 1 ]$\\ & $ c^{-1}a^2cba^2b^{-1}, a^3ca^{-1}b^{-2}c$ & \\ \end{tabular} \hspace{4.5cm} \noindent \textbf{ Table 3 } \noindent \begin{tabular}{ l l l l } \hline Group No. & Relators & $ [n, m, k, t]$ \\ \hline $\#70$ & $ b^{-1}c^{-1}acba, bc^2b^{-1}c^{-2}, a^4c^2b^{-2},$ & $[ 256, 3638, 8, 2 ] $\\ & $a^3cb^{-1}cab $ & \\ $\#71$ & $a^2ca^2c^{-1}, ba^{-1}cb^{-1}ca, b^2cb^{-2}c^{-1}, $ & $ [ 256, 3639, 8, 1 ]$\\ & $ bc^2b^{-1}c^{-2}, a^3b^{-1}ca^{-1}cb^{-1}, b^6c^2$ & \\ $\#72$ & $a^2ca^2c^{-1}, bacb^{-1}ca^{-1}, b^2cb^{-2}c^{-1}, $ & $ [ 256, 3639, 8, 2 ]$\\ & $bc^2b^{-1}c^{-2}, a^{-1}bc^{-1}ba^3c, b^2c^6 $ & \\ $\#73$ & $ a^2ca^2c^{-1}, b^{-1}cbaca^{-3}, cab^2ca^{-3},$ & $[ 256, 3641, 8, 1 ] $\\ & $ a^3b^{-1}c^2ab^{-1}$ & \\ $\#74$ & $ a^2ca^2c^{-1}, b^{-1}cbaca^{-3}, cab^2ca^{-3},$ & $[ 256, 3641, 8, 2 ] $\\ & $ba^{-1}c^2ba^{-3} $ & \\ $\#75$ & $a^2b^{-2}, a^2cb^2c^{-1}, a^{-1}c^3ac^{-1}, $ & $[ 256, 3641, 9, 1 ] $\\ & $ (bc)^2(b^{-1}c^{-1})^2, acbc^{-1}a(b^{-1}a^{-1})^2b^{-1}$ & \\ $\#76$ & $ a^2b^2, a^2cb^{-2}c^{-1}, a^{-1}c^3ac^{-1},$ & $[ 256, 3641, 9, 2 ] $\\ & $ (bc)^2(b^{-1}c^{-1})^2, babc^{-1}aba^{-1}cb^{-1}a^{-1}$ & \\ $\#77$ & $ a^2ca^2c^{-1}, ab^2a^{-1}b^{-2}, b^3cb^{-1}ca^{-2}, $ & $, [ 256, 3643, 9, 1 ] $\\ & $ bcacb^{-1}a^{-3}, bacbc^{-3}a^{-1}$ & \\ $\#78$ & $ a^2ca^2c^{-1}, ab^2a^{-1}b^{-2}, a^2cb^{-1}cb^3, $ & $[ 256, 3643, 9, 2 ] $\\ & $ bcacb^{-1}a^{-3}, bac^{-1}bc^3a^{-1}$ & \\ \end{tabular}\\ \begin{center}{\textbf{Acknowledgments}} \end{center} The authors are grateful to the referees for their valuable suggestions. The work of authors was in part supported by Arak University. \\ {\footnotesize \par\noindent{\bf Authors:}\; \\ {Department of Mathematics}, {University of Arak, P.O.Box 38156-88349,} {Arak, Iran}\\ {\tt Email: [email protected]}\\ {\tt Email: [email protected]}\\ \end{document}	arXiv
\begin{document} \title[Multicomponent Transport]{On the homogenization of multicomponent transport} \author{Gr\'egoire Allaire, Harsha Hutridurga} \date{\today} \maketitle \begin{abstract} This paper is devoted to the homogenization of weakly coupled cooperative parabolic systems in strong convection regime with purely periodic coefficients. Our approach is to factor out oscillations from the solution via principal eigenfunctions of an associated spectral problem and to cancel any exponential decay in time of the solution using the principal eigenvalue of the same spectral problem. We employ the notion of two-scale convergence with drift in the asymptotic analysis of the factorized model as the lengthscale of the oscillations tends to zero. This combination of the factorization method and the method of two-scale convergence is applied to upscale an adsorption model for multicomponent flow in an heterogeneous porous medium. \end{abstract} \section{Introduction} \label{sec:int} Upscaling reactive transport models in porous media is a problem of great practical importance and homogenization theory is a method of choice for achieving this goal (see \cite{HOR} and references therein). In this paper we focus on a model problem of reactive multicomponent transport for $N$ diluted chemical species in a saturated periodically varying media. The fluid velocity is assumed to be known. On top of usual convective and diffusive effects we consider linear reaction terms which satisfy a specific condition, namely that the reaction matrix is cooperative (see the precise definition in Section \ref{sec:model}). This assumption is quite natural for a linear system, as we consider here, since it ensures a maximum (or positivity) principle for solutions which, being concentrations, should indeed be non-negative for obvious physical reasons. As usual the ratio between the period of the coefficients and a characteristic lengthscale of the porous domain is denoted by a small parameter $0<\varepsilon\ll1$. Denoting the unknown concentrations by $u^\varepsilon_\alpha$, for $1\le\alpha\le N$, we study in the entire space $\R^d$ the following weakly coupled (i.e., no coupling in the derivatives) system of $N$ parabolic equations with periodic bounded coefficients: \begin{equation} \label{eq:intro} \rho_\alpha\Big(\frac{x}{\varepsilon}\Big)\frac{\partial u^\varepsilon_\alpha}{\partial t} + \frac{1}{\varepsilon}b_\alpha\Big(\frac{x}{\varepsilon}\Big)\cdot\nabla u^\varepsilon_\alpha - {\rm div}\Big(D_\alpha\Big(\frac{x}{\varepsilon}\Big)\nabla u^\varepsilon_\alpha\Big) + \frac{1}{\varepsilon^2}\sum_{\beta=1}^N\Pi_{\alpha\beta}\Big(\frac{x}{\varepsilon}\Big)u^\varepsilon_\beta = 0, \end{equation} for $1\le\alpha\le N$, where $b_\alpha$ are velocity fields, $D_\alpha$ are symmetric and coercive diffusion tensors and $\Pi$ is the reaction (or coupling) matrix, assumed to be cooperative (see (\ref{eq:ass:irr}) for a precise definition). All coefficients are $Y$-periodic, where $Y:=]0,1[^d$ is the unit cell in $\R^d$. Our main result, Theorem \ref{thm:hom}, states that a solution to the Cauchy problem for (\ref{eq:intro}) admits the following asymptotic representation (for every $1\le\alpha\le N$): $$ u^\varepsilon_\alpha(t,x)=\varphi_\alpha\Big(\frac{x}{\varepsilon}\Big)\exp{(-\lambda t/\varepsilon^2)}\Big(v\Big(t, x-\frac{b^t}{\varepsilon}\Big)+\mathcal{O}(\varepsilon)\Big), $$ where $\{\lambda,(\varphi_\alpha)_{1\le\alpha\le N}\}$ is the first eigenpair for a periodic system posed in the unit cell $Y:=]0,1[^d$, $b^$ is a constant drift vector and $v(t,x)$ solves a scalar parabolic homogenized problem with constant coefficients. Our result generalizes the works \cite{AR:07} and \cite{DP:05}, which were restricted to a single (scalar) parabolic equation. In \cite{Ca:00}, \cite{Ca:02} a similar result was obtained for a cooperative elliptic system without convective terms. Our present work is thus the first to combine large convective terms and multiple equations. Let us explain the specific $\varepsilon$-scaling of the coefficients in (\ref{eq:intro}), which is not new and is well explained, e.g., in \cite{AMP}. Before adimensionalization, the physical system of equations, in original time-space coordinates $(\tau,y)$, is, for $1 \le \alpha \le N$, $$ \rho_\alpha\frac{\partial u_\alpha}{\partial\tau} + b_\alpha\cdot\nabla u_\alpha - {\rm div}(D_\alpha\nabla u_\alpha) + \sum_{\beta=1}^N \Pi_{\alpha\beta}u_\beta = 0 . $$ Interested by a macroscopic view and long time behaviour of this parabolic system, we perform a ``parabolic'' scaling of the time-space variables, i.e., $(\tau,y)\to(\varepsilon^{-2}t,\varepsilon^{-1}x)$, which yields the scaled model (\ref{eq:intro}). \begin{remark} Another scaling that one could consider is the ``hyperbolic'' scaling, i.e., $(\tau,y)\to(\varepsilon^{-1}t,\varepsilon^{-1}x)$. This has been addressed in \cite{MS:13} (for $N=2$) where the scaled system is: $$ \rho_\alpha\Big(\frac{x}{\varepsilon}\Big)\frac{\partial u^\varepsilon_\alpha}{\partial t} + b_\alpha\Big(\frac{x}{\varepsilon}\Big) \cdot\nabla u^\varepsilon_\alpha - \varepsilon{\rm div}\Big(D_\alpha\Big(\frac{x}{\varepsilon}\Big)\nabla u^\varepsilon_\alpha\Big) + \frac1{\varepsilon}\sum_{\beta=1}^N \Pi_{\alpha\beta}\Big(\frac{x}{\varepsilon}\Big)u^\varepsilon_\beta = 0 , $$ for $1\le\alpha\le N$. The main result of \cite{MS:13} is that the solution to the Cauchy problem for the above system admits the asymptotic representation: $$ u^\varepsilon_\alpha(t,x) \approx \phi_\alpha\Big(\frac{x}{\varepsilon}\Big)\delta(x-b^t) $$ where $\phi_\alpha$ is the first eigenfunction and there is no time exponential because $\lambda=0$ happens to be the first eigenvalue for the specific choice of cooperative matrix $\Pi_{\alpha\beta}$ made in \cite{MS:13}. In the above equation $\delta$ is the Dirac mass which appears because of a concentration assumption on the initial data. The main difference with the parabolic scaling in our work is that there is no diffusion homogenized problem. The drift velocity can be interpreted as $b^=\nabla H(0)$ with $H$ being some effective Hamiltonian. \end{remark} The organization of this paper is as follows. In Section \ref{sec:model}, we describe the mathematical model of cooperative parabolic systems and the precise hypotheses made on the coefficients. Section \ref{sec:qa} briefly recalls the existence and uniqueness theory for system (\ref{eq:intro}). Since no uniform a priori estimates can be obtained for (\ref{eq:intro}), a factorization principle (or change of unknowns) is performed in Section \ref{sec:fp}. Then, uniform a priori bounds are deduced for the solution of this factorized problem. The definition of two-scale convergence with drift is recalled in Section \ref{sec:2scl}. Then, based on the uniform a priori estimates of Section \ref{sec:fp}, we obtain a two-scale compactness result for the sequence of solutions (see Theorem \ref{thm:3:2scl}). Our main homogenization result is Theorem \ref{thm:hom} which is proved in Section \ref{sec:hom}. Eventually, Section \ref{sec:apm} generalizes our analysis to a similar, but more involved, system which is meaningful from a physical point of view. The differences are that $(i)$ the convection-diffusion takes place in a perforated porous medium and $(ii)$ the chemical reactions are localized on the holes' boundaries rather than in the fluid bulk. This is a frequent case for adsorption or deposition of the chemical on the solid surface (cf. the discussion and references in \cite{AMP}). \section{The model} \label{sec:model} Before we present our model, let us introduce the following shorthands: $$ \rho^\varepsilon_\alpha(x):=\rho_\alpha\Big(\frac{x}{\varepsilon}\Big);\hspace{0.5 cm}b^\varepsilon_\alpha(x):=b_\alpha\Big(\frac{x}{\varepsilon}\Big);\hspace{0.5 cm}D^\varepsilon_\alpha(x):=D_\alpha\Big(\frac{x}{\varepsilon}\Big);\hspace{0.5 cm}\Pi_{\alpha\beta}^\varepsilon(x):=\Pi_{\alpha\beta}\Big(\frac{x}{\varepsilon}\Big), $$ where the small positive parameter $\varepsilon\ll1$ represents the lengthscale of oscillations. We consider the following Cauchy problem: \begin{equation} \label{eq:cd} \displaystyle\rho^\varepsilon_\alpha \frac{\partial u^\varepsilon_\alpha}{\partial t} + \frac1\varepsilon b^\varepsilon_\alpha\cdot\nabla u^\varepsilon_\alpha - {\rm div}(D^\varepsilon_\alpha\nabla u^\varepsilon_\alpha) + \frac1{\varepsilon^2}\sum_{\beta=1}^N \Pi^\varepsilon_{\alpha\beta}u^\varepsilon_\beta = 0\hspace{0.1 cm}\mbox{ in }\hspace{0.1 cm}(0,T)\times\R^d, \end{equation} \begin{equation} \label{eq:in} u^\varepsilon_\alpha(0,x)=u^{in}_\alpha(x)\hspace{0.1 cm}\mbox{ for }x\in\R^d. \end{equation} For a normed vector space $\mathcal H$, we use the following standard notation for $Y$-periodic function spaces: $$ L^p_\#(\R^d;\mathcal H):=\Big\{f:\R^d\to\mathcal H \mbox{ s.t. }f\mbox{ is }Y\mbox{-periodic} \mbox{ and } \\|\\|f\\|_{\mathcal H}\\|_{L^p(Y)}<\infty \Big\}. $$ The assumptions made on the coefficients of (\ref{eq:cd}) are the following: \begin{equation} \label{eq:ass:rho} \rho_\alpha\in L^\infty_\#(\R^d;\R)\mbox{ and }\exists c_\alpha>0\mbox{ s.t. } \rho_\alpha(y)\geq c_\alpha , \end{equation} \begin{equation} \label{eq:ass:vel} b_\alpha\in L^\infty_\#(\R^d;\R^d) \mbox{ and } {\rm div} b_\alpha\in L^\infty_\#(\R^d;\R), \end{equation} \begin{equation} \label{eq:ass:diff} D_\alpha=(D_\alpha)^\in L^\infty_\#(\R^d;\R^{d\times d})\mbox{ and }\exists c_\alpha>0\mbox{ s.t. }c_\alpha\|\xi\|^2\le D_\alpha(y)\xi\cdot\xi \end{equation} for all $\xi\in\R^d$ and for almost every $y\in\R^d$ (where $(D_\alpha)^$ is the adjoint or transposed matrix of $D_\alpha$), \begin{equation} \label{eq:ass:cple} \Pi\in L^\infty_\#(\R^d;\R^{d\times d})\mbox{ and }\Pi_{\alpha\beta}\le0\mbox{ for }\alpha\not=\beta, \end{equation} we also assume that the coupling matrix $\Pi$ is irreducible, i.e., there exists no partition $\mathcal{B}\not=\emptyset$,$\mathcal{B}'\not=\emptyset$ of $\{1,\cdots,N\}$ such that \begin{equation} \label{eq:ass:irr} \{1,\cdots,N\}=\mathcal{B}\cup\mathcal{B}'\mbox{ with } \mathcal{B}\cap\mathcal{B}'=\emptyset \mbox{ and }\Pi_{\alpha\beta}=0\mbox{ for all }\alpha\in\mathcal{B}, \beta\in\mathcal{B}'. \end{equation} This irreducibility assumption ensures that the system (\ref{eq:cd}) cannot be decoupled in two disjoint subsystems (see Remark \ref{rem:irr} below). \begin{remark} The only assumption made on the convective fields $b_\alpha$ in (\ref{eq:ass:vel}) is that they are bounded as well as their divergences. No divergence-free assumption is made on these vector fields. The hypotheses (\ref{eq:ass:cple})-(\ref{eq:ass:irr}) are borrowed from \cite{Sw:92, MS:95, AC:00, Ca:02}. A matrix satisfying (\ref{eq:ass:cple}) is sometimes referred to as ``cooperative matrix'' (up to the addition of a multiple of the identity it is also an $M$-matrix). Hence the system (\ref{eq:cd}) gets the name ``cooperative parabolic system''. \end{remark} Finally, we assume that the initial data in (\ref{eq:in}) has following regularity: $u^{in}_\alpha\in L^2(\R^d)$ for each $1 \le \alpha \le N$. \section{Qualitative Analysis} \label{sec:qa} Results of existence and uniqueness of solutions to (\ref{eq:cd}) are classical. The ``cooperative'' hypothesis (\ref{eq:ass:cple}) is actually not necessary to obtain well-posedness. Standard approach is to derive a priori estimates on the solution. Classical technique is to multiply (\ref{eq:cd}) by $u^\varepsilon_\alpha$ and integrate over $\R^d$: $$ \frac12\frac{d}{dt}\int_{\R^d}\rho_\alpha^\varepsilon\|u^\varepsilon_\alpha\|^2{\rm d}x+\int_{\R^d} D^\varepsilon_\alpha\nabla u^\varepsilon_\alpha\cdot\nabla u^\varepsilon_\alpha{\rm d}x $$ $$ =\frac{1}{2\varepsilon}\int_{\R^d}{\rm div}(b^\varepsilon_\alpha)\|u^\varepsilon_\alpha\|^2{\rm d}x - \frac1{\varepsilon^2}\sum_{\beta=1}^N\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}u^\varepsilon_\beta u^\varepsilon_\alpha {\rm d}x. $$ Since the divergences of the convective fields are bounded, summing the above expression over $1\le\alpha\le N$ followed by the application of Cauchy-Schwarz inequality, Young's inequality, Gronwall's lemma and an integration over $(0,T)$ leads to the following a priori estimates: \begin{equation} \label{eq:apr:ve} \displaystyle\sum_{\alpha=1}^N\\|u^\varepsilon_\alpha\\|_{L^\infty((0,T);L^2(\R^d))} + \sum_{\alpha=1}^N\\|\nabla u^\varepsilon_\alpha\\|_{L^2((0,T)\times\R^d)}\le C_\varepsilon\sum_{\alpha=1}^N\\|u^{in}_\alpha\\|_{L^2(\R^d)}, \end{equation} where the constant $C_\varepsilon$ depends on the small parameter $\varepsilon$. For any fixed $0<\varepsilon$, we can use the a priori estimates (\ref{eq:apr:ve}) and Galerkin method to establish existence and uniqueness of the solution $u^\varepsilon_\alpha\in L^2((0,T);H^1(\R^d))\cap C((0,T);L^2(\R^d))$, $1\le\alpha\le N$. Maximum principles are a different story altogether. In general we have no maximum principles for systems. However, the hypotheses (\ref{eq:ass:cple})-(\ref{eq:ass:irr}) guarantee a maximum principle. In \cite{Sw:92, MS:95}, weakly coupled cooperative elliptic systems with coupling matrices satisfying (\ref{eq:ass:cple})-(\ref{eq:ass:irr}) are studied with emphasis on maximum principles and on the well-posedness of associated spectral problems. The results of \cite{Sw:92} on the cooperative elliptic systems have their parabolic counterpart. We state the result from \cite{Sw:92} adapted to cooperative parabolic systems: \begin{lemma}[see \cite{Sw:92, MS:95} for a proof] Let the conditions (\ref{eq:ass:rho})-(\ref{eq:ass:irr}) on the coefficients of (\ref{eq:cd}) be satisfied. Then, for any fixed $\varepsilon>0$, the following holds: (i) There is a unique solution $u^\varepsilon_\alpha\in L^2((0,T);H^1(\R^d))\cap C((0,T);L^2(\R^d))$ for $1 \le \alpha \le N$. (ii) If $u^{in}_\alpha\ge0$ for all $1 \le \alpha \le N$, then $u^{\varepsilon}_\alpha\ge0$ for all $1 \le \alpha \le N$. \end{lemma} \begin{remark} In order to make an asymptotic analysis on (\ref{eq:cd}), as $\varepsilon\to0$, one demands uniform (with respect to $\varepsilon$) estimates on the solution $u^\varepsilon_\alpha$. But the estimates in (\ref{eq:apr:ve}) are not uniform in $\varepsilon$. This renders the application of standard compactness theorems from homogenization theory useless for (\ref{eq:cd}). \end{remark} \section{Factorization Principle} \label{sec:fp} The difficulty with the derivation of a priori estimates in presence of large lower order terms has long been recognized \cite{Va:81, Ji:84, Ko:84, AC:00, ACPSV:04, DP:05}. The idea is to use information from an associated spectral cell problem. The basic principle is to factor out principal eigenfunction from the solution to arrive at a new ``factorized system'', amenable to uniform a priori estimates. This idea of factoring our oscillations from the solution was first introduced in \cite{Va:81} in the context of elliptic eigenvalue problems. In case of scalar parabolic equations it is shown in \cite{Ji:84, Ko:84, DP:05, AR:07} that the factorized equations have no zero order terms and that the first order terms are divergence free. In case of cooperative elliptic systems with large lower order terms studied in \cite{AC:00, Ca:02}, however, it is shown that the factorized systems still have zero order terms and are transformed as ``difference terms''. We adopt the ``factorization principle'', extensively used in the above mentioned references, to remedy the difficulty we have with the derivation of uniform a priori estimates for (\ref{eq:cd}). We first define the following spectral problem associated with (\ref{eq:cd}) and posed in the unit cell with periodic boundary conditions: \begin{equation} \label{eq:scp} \left\{ \begin{array}{ll} \displaystyle b_\alpha \cdot \nabla_y \varphi_\alpha - {\rm div}_y \Big(D_\alpha \nabla_y \varphi_\alpha\Big) + \sum_{\beta=1}^N \Pi_{\alpha\beta} \varphi_\beta = \lambda\rho_\alpha \varphi_\alpha & \mbox{ in }Y, \\[0.3cm] y \to \varphi_\alpha(y) \hspace{1 cm} Y\mbox{-periodic.} \end{array} \right. \end{equation} The above spectral cell problem is not self-adjoint. The associated adjoint problem is: \begin{equation} \label{eq:ascp} \left\{ \begin{array}{ll} \displaystyle -{\rm div}_y(b_\alpha \varphi^_\alpha) - {\rm div}_y \Big(D_\alpha \nabla_y \varphi^_\alpha\Big) + \sum_{\beta=1}^N \Pi^_{\alpha\beta} \varphi^_\beta = \lambda\rho_\alpha \varphi^_\alpha & \mbox{ in }Y,\\[0.3cm] y \to \varphi^_\alpha(y) \hspace{1 cm} Y\mbox{-periodic,} \end{array} \right. \end{equation} where $\Pi^$ is the transpose of $\Pi$. The well-posedness of the above spectral problems is a delicate issue which is addressed in \cite{Sw:92, MS:95}. The following proposition is an adaptation to our periodic setting of the main result of \cite{Sw:92, MS:95}. \begin{proposition}[see \cite{Sw:92} for a proof] \label{prop:spec} Under the assumptions (\ref{eq:ass:rho})-(\ref{eq:ass:irr}) on the coefficients, the spectral problems (\ref{eq:scp}) and (\ref{eq:ascp}) admit a common first eigenvalue (i.e., smallest in modulus) which satisfies: (i) the first eigenvalue $\lambda$ is real and simple, (ii) the corresponding first eigenfunctions $(\varphi_\alpha)_{1\le\alpha\le N}\in (H^1_\#(Y))^N$ for (\ref{eq:scp}), $(\varphi^_\alpha)_{1\le\alpha\le N}\in (H^1_\#(Y))^N$ for (\ref{eq:ascp}) are positive, $\varphi_\alpha,\varphi^_\alpha>0$ for $1\le\alpha\le N$, and unique up to normalization. \end{proposition} \begin{remark} \label{rem:spec} The first eigenvalue $\lambda$ in Proposition \ref{prop:spec} measures the balance between convection-diffusion and reaction. Also, the uniqueness of first eigenfunctions in Proposition \ref{prop:spec} is up to a chosen normalization. The normalization that we consider is the following: \begin{equation} \label{eq:norm} \sum_{\alpha=1}^N\:\:\int_Y \rho_\alpha \varphi_\alpha\varphi^_\alpha{\rm d}y = 1. \end{equation} \end{remark} In the proof of our a priori estimates it will be convenient to scale the spectral problems (\ref{eq:scp})-(\ref{eq:ascp}) to the entire domain $\R^d$ via the change of variables $y\to\varepsilon^{-1}x$. More precisely, (\ref{eq:scp})-(\ref{eq:ascp}) are equivalent to \begin{equation} \label{eq:scp:scl} \left\{ \begin{array}{ll} \displaystyle \varepsilon b^\varepsilon_\alpha \cdot \nabla \varphi^\varepsilon_\alpha - \varepsilon^2{\rm div} \Big(D^\varepsilon_\alpha \nabla \varphi^\varepsilon_\alpha\Big) + \sum_{\beta=1}^N \Pi^\varepsilon_{\alpha\beta} \varphi^\varepsilon_\beta = \lambda\rho^\varepsilon_\alpha \varphi^\varepsilon_\alpha & \mbox{ in }\R^d, \\[0.3cm] x \to \varphi^\varepsilon_\alpha(x)\equiv \varphi_\alpha(x/\varepsilon) \hspace{1 cm} \varepsilon Y\mbox{-periodic,} \end{array} \right. \end{equation} \begin{equation} \label{eq:ascp:scl} \left\{ \begin{array}{ll} \displaystyle -\varepsilon{\rm div}(b^\varepsilon_\alpha \varphi^{\varepsilon}_\alpha) - \varepsilon^2{\rm div} \Big(D^\varepsilon_\alpha \nabla \varphi^{\varepsilon}_\alpha\Big) + \sum_{\beta=1}^N \Pi^{\varepsilon}_{\alpha\beta} \varphi^{\varepsilon}_\beta = \lambda\rho^\varepsilon_\alpha \varphi^{\varepsilon}_\alpha & \mbox{ in }\R^d,\\[0.3cm] x \to \varphi^{\varepsilon}_\alpha(x)\equiv\varphi^_\alpha(x/\varepsilon) \hspace{1 cm} \varepsilon Y\mbox{-periodic.} \end{array} \right. \end{equation} Now, we get down to the task of reducing (\ref{eq:cd}) to a ``factorized system''. As explained in \cite{AC:00, ACPSV:04, DP:05, AR:07} the first eigenvalue $\lambda$ governs the time decay or growth of the solution $u^\varepsilon_\alpha$. So, as is done in the references cited, we perform time renormalization in the spirit of the factorization principle. Also the first eigenfunction $\varphi^\varepsilon_\alpha$ is factored out of $u^\varepsilon_\alpha$. In other words we make the following change of unknowns: \begin{equation} \label{eq:fcp} \displaystyle v^\varepsilon_\alpha(t,x)=\exp{(\lambda t/\varepsilon^2)}\frac{u^\varepsilon_\alpha(t,x)}{\varphi^\varepsilon_\alpha(x)}. \end{equation} The above change of unknowns is valid, thanks to the positivity result in Proposition \ref{prop:spec}. Now we state a result that gives the factorized system satisfied by the new unknown $(v^\varepsilon_\alpha)_{1\leq\alpha\leq N}$. \begin{lemma} \label{lem:equiv:cd} The system (\ref{eq:cd})-(\ref{eq:in}) is equivalent to \begin{equation} \label{eq:cd1} \varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha\frac{\partial v^\varepsilon_\alpha}{\partial t} + \frac{1}{\varepsilon}\tilde b^\varepsilon_\alpha \cdot \nabla v^\varepsilon_\alpha - {\rm div}\left(\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\right)+ \frac{1}{\varepsilon^2}\displaystyle\sum_{\beta=1}^N\:\:\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha) = 0 \end{equation} in $(0,T)\times\R^d$ for each $1\le\alpha\le N$ complemented with the initial data: \begin{equation} \label{eq:in1} \displaystyle v^\varepsilon_\alpha(0,x) = \frac{u^{in}_\alpha(x)}{\varphi^\varepsilon_\alpha(x)}\hspace{4 cm}x\in\R^d, \end{equation} for each $1\le\alpha\le N$, where the components of $(v^\varepsilon_\alpha)_{1\le\alpha\le N}$ are defined by (\ref{eq:fcp}). The convective velocities, $\tilde b^\varepsilon_\alpha(x) = \tilde b_\alpha\Big(\frac{x}{\varepsilon}\Big)$, in (\ref{eq:cd1}) are given by \begin{equation} \label{eq:b1} \tilde b_\alpha(y) = \varphi_\alpha\varphi^_\alpha b_\alpha + \varphi_\alpha D_\alpha\nabla_y\varphi^_\alpha -\varphi^_\alpha D_\alpha \nabla_y\varphi_\alpha\hspace{0.5 cm}\textrm{for every}\:\:1\le\alpha\le N \end{equation} and the diffusion matrices, $\tilde D^\varepsilon_\alpha(x) = \tilde D_\alpha\Big(\frac{x}{\varepsilon}\Big)$, in (\ref{eq:cd1}) are given by \begin{equation} \label{eq:D1} \tilde D_\alpha(y) = \varphi_\alpha\varphi^_\alpha D_\alpha\hspace{0.5 cm}\textrm{for every}\:\:1\le\alpha\le N. \end{equation} \end{lemma} The proof of Lemma \ref{lem:equiv:cd} is just a matter of simple algebra, using (\ref{eq:scp:scl}), and we refer to \cite{Ca:02}, \cite{Hu:13} for more details, keeping in mind the following chain rule formulae: $$ \left\{ \begin{array}{l} \displaystyle\frac{\partial u^\varepsilon_\alpha}{\partial t}(t,x) = \exp{(-\lambda t/\varepsilon^2)} \left( \frac{-\lambda}{\varepsilon^2}\varphi_\alpha\Big(\frac{x}{\varepsilon}\Big)v^\varepsilon_\alpha(t,x)+\varphi_\alpha\Big(\frac{x}{\varepsilon}\Big)\frac{\partial v^\varepsilon_\alpha}{\partial t}(t,x)\right),\\[0.5 cm] \displaystyle\nabla\Big(u^\varepsilon_\alpha(t,x)\Big) = \exp{(-\lambda t/\varepsilon^2)} \left( \frac{1}{\varepsilon}v^\varepsilon_\alpha(t,x)\Big(\nabla_y\varphi_\alpha\Big)\Big(\frac{x}{\varepsilon}\Big)+\varphi_\alpha\Big(\frac{x}{\varepsilon}\Big)\nabla_x v^\varepsilon_\alpha(t,x) \right) . \end{array}\right. $$ \begin{remark} \label{rem:div} The divergence of the convective fields $\tilde b_\alpha$ satisfy \begin{equation} \label{eq:divb} {\rm div}_y\tilde b_\alpha = \displaystyle\sum_{\beta=1}^N\:\:\Pi^_{\alpha\beta}\varphi_\alpha\varphi^_\beta-\displaystyle\sum_{\beta=1}^N\:\:\Pi_{\alpha\beta}\varphi^_\alpha\varphi_\beta. \end{equation} It follows that $$ \sum_{\alpha=1}^N\:\:{\rm div}_y\tilde b_\alpha=0. $$ \end{remark} \begin{remark} \label{rem:diff} The factorized system (\ref{eq:cd1}) still has large lower order terms. But, as noticed in \cite{AC:00, Ca:02}, the terms are transformed as ``difference terms''. This factorization is the key for getting a priori estimate on the differences $(v^\varepsilon_\alpha-v^\varepsilon_\beta)$. \end{remark} The following lemma gives the a priori estimates on the new unknown. \begin{lemma} \label{lem:apriori} Let $(v^\varepsilon_\alpha)_{1\le\alpha\le N}$ be a weak solution of (\ref{eq:cd1})-(\ref{eq:in1}). There exists a constant $C$, independent of $\varepsilon$, such that \begin{equation} \label{eq:apriori} \begin{array}{ll} \displaystyle\sum_{\alpha=1}^N\:\:\Big\\|v^\varepsilon_\alpha\Big\\|_{L^\infty((0,T);L^2(\R^d))}+\sum_{\alpha=1}^N\:\:\Big\\|\nabla v^\varepsilon_\alpha\Big\\|_{L^2((0,T)\times\R^d)}\\[0.4 cm] +\displaystyle\frac1\varepsilon\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\Big\\|v^\varepsilon_\alpha-v^\varepsilon_\beta\Big\\|_{L^2((0,T)\times\R^d)}\leq C\sum_{\alpha=1}^N\:\:\\|v^{in}_\alpha\\|_{L^2(\R^d)}. \end{array} \end{equation} \end{lemma} \begin{proof} To derive the a priori estimates, we multiply (\ref{eq:cd1}) by $v^\varepsilon_\alpha$ followed by integrating over $\R^d$ and sum the obtained expressions over $1\le\alpha\le N$: \begin{equation} \label{eq:1:nrj} \begin{array}{cc} \displaystyle\frac12\frac{d}{dt}\displaystyle\sum_{\alpha=1}^N\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha\|v^\varepsilon_\alpha\|^2{\rm d}x -\frac{1}{2\varepsilon}\displaystyle\sum_{\alpha=1}^N\int_{\R^d}{\rm div}(\tilde b^\varepsilon_\alpha)\|v^\varepsilon_\alpha\|^2{\rm d}x\\[0.3 cm] +\displaystyle\sum_{\alpha=1}^N\int_{\R^d}\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot\nabla v^\varepsilon_\alpha{\rm d}x +\frac{1}{\varepsilon^2}\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\int_{\R^d}\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha)v^\varepsilon_\alpha=0. \end{array} \end{equation} To simplify the above expressions, we now use the scaled spectral problems (\ref{eq:scp:scl})-(\ref{eq:ascp:scl}). Multiply (\ref{eq:scp:scl}) by $\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2$ followed by integration over the space domain $\R^d$: $$ \frac1\varepsilon\int_{\R^d}(b^\varepsilon_\alpha \cdot \nabla \varphi^\varepsilon_\alpha)\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2{\rm d}x - \int_{\R^d}{\rm div} \Big(D^\varepsilon_\alpha \nabla \varphi^\varepsilon_\alpha\Big)\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ +\frac{1}{\varepsilon^2} \sum_{\beta=1}^N \int_{\R^d}\Pi^\varepsilon_{\alpha\beta} \varphi^\varepsilon_\beta\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2{\rm d}x - \frac{1}{\varepsilon^2}\int_{\R^d}\lambda\rho^\varepsilon_\alpha \varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ = - \frac1\varepsilon\int_{\R^d}{\rm div}(b^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha)\varphi^\varepsilon_\alpha(v^\varepsilon_\alpha)^2{\rm d}x - \frac1\varepsilon\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha b^\varepsilon_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ + \int_{\R^d} \varphi^{\varepsilon}_\alpha D^\varepsilon_\alpha \nabla \varphi^\varepsilon_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x + \int_{\R^d}(v^\varepsilon_\alpha)^2 D^\varepsilon_\alpha \nabla \varphi^\varepsilon_\alpha\cdot\nabla\varphi^{\varepsilon}_\alpha{\rm d}x $$ $$ +\displaystyle\frac{1}{\varepsilon^2}\sum_{\beta=1}^N\:\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\alpha)^2{\rm d}x - \frac{1}{\varepsilon^2}\int_{\R^d} \lambda\rho^\varepsilon_\alpha\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ = - \frac1\varepsilon\int_{\R^d}{\rm div}(b^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha)\varphi^\varepsilon_\alpha(v^\varepsilon_\alpha)^2{\rm d}x - \frac1\varepsilon\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha b^\varepsilon_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ + \int_{\R^d} \varphi^{\varepsilon}_\alpha D^\varepsilon_\alpha \nabla \varphi^\varepsilon_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x + \int_{\R^d}(v^\varepsilon_\alpha)^2 D^\varepsilon_\alpha \nabla \varphi^\varepsilon_\alpha\cdot\nabla\varphi^{\varepsilon}_\alpha{\rm d}x $$ $$ - \int_{\R^d} \varphi^\varepsilon_\alpha D^\varepsilon_\alpha \nabla \varphi^{\varepsilon}_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x + \int_{\R^d} \varphi^\varepsilon_\alpha D^\varepsilon_\alpha \nabla \varphi^{\varepsilon}_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ + \displaystyle\frac{1}{\varepsilon^2}\sum_{\beta=1}^N\:\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\alpha)^2{\rm d}x - \frac{1}{\varepsilon^2}\int_{\R^d} \lambda\rho^\varepsilon_\alpha\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ = - \frac1\varepsilon\int_{\R^d}{\rm div}(b^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha)\varphi^\varepsilon_\alpha(v^\varepsilon_\alpha)^2{\rm d}x - \int_{\R^d} {\rm div}(D^\varepsilon_\alpha \nabla \varphi^{\varepsilon}_\alpha)\varphi^\varepsilon_\alpha (v^\varepsilon_\alpha)^2{\rm d}x $$ $$ - \frac{1}{\varepsilon^2}\int_{\R^d} \lambda\rho^\varepsilon_\alpha\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha(v^\varepsilon_\alpha)^2{\rm d}x - \frac1\varepsilon\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha b^\varepsilon_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ + \int_{\R^d} \varphi^{\varepsilon}_\alpha D^\varepsilon_\alpha \nabla \varphi^\varepsilon_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x - \int_{\R^d} \varphi^\varepsilon_\alpha D^\varepsilon_\alpha \nabla \varphi^{\varepsilon}_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x $$ $$ + \displaystyle\frac{1}{\varepsilon^2}\sum_{\beta=1}^N\:\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\alpha)^2{\rm d}x = 0. $$ In the above expression, we recognize the scaled adjoint cell problem (\ref{eq:ascp:scl}). We also recognize the scaled expression of (\ref{eq:b1}) for the convective field $\tilde b_\alpha$. Taking all these into consideration, we have the following: $$ - \frac1\varepsilon\int_{\R^d}\tilde b^\varepsilon_\alpha\cdot\nabla(v^\varepsilon_\alpha)^2{\rm d}x + \displaystyle\frac{1}{\varepsilon^2}\sum_{\beta=1}^N\:\int_{\R^d} \Big(\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta - \Pi^{\varepsilon}_{\alpha\beta}\varphi^{\varepsilon}_\beta\varphi^\varepsilon_\alpha\Big)(v^\varepsilon_\alpha)^2{\rm d}x=0. $$ Summing over $\alpha$, we have: \begin{equation} \label{eq:divb:nrj} \displaystyle-\frac1{2\varepsilon}\sum_{\alpha=1}^N\,\,\int_{\R^d}{\rm div}(\tilde b^\varepsilon_\alpha) (v^\varepsilon_\alpha)^2{\rm d}x = \displaystyle\frac{1}{2\varepsilon^2}\sum_{\alpha=1}^N\sum_{\beta=1}^N\:\int_{\R^d} \Big(\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta - \Pi^{\varepsilon}_{\alpha\beta}\varphi^{\varepsilon}_\beta\varphi^\varepsilon_\alpha\Big)(v^\varepsilon_\alpha)^2{\rm d}x. \end{equation} Now, let us employ (\ref{eq:divb:nrj}) in the estimate (\ref{eq:1:nrj}) which leads to: $$ \displaystyle\frac12\frac{d}{dt}\displaystyle\sum_{\alpha=1}^N\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha\|v^\varepsilon_\alpha\|^2{\rm d}x +\displaystyle\sum_{\alpha=1}^N\int_{\R^d}\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot\nabla v^\varepsilon_\alpha{\rm d}x $$ $$ +\displaystyle\frac{1}{\varepsilon^2}\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\int_{\R^d}\Big\{\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta\Big(v^\varepsilon_\beta v^\varepsilon_\alpha - \frac12 (v^\varepsilon_\alpha)^2\Big) - \frac12\Pi^{\varepsilon}_{\alpha\beta}\varphi^{\varepsilon}_\beta\varphi^\varepsilon_\alpha (v^\varepsilon_\alpha)^2 \Big\}{\rm d}x=0. $$ The above expression is nothing but the following energy estimate: \begin{equation} \label{eq:nrj} \begin{array}{ll} \displaystyle\frac12\frac{d}{dt}\displaystyle\sum_{\alpha=1}^N\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha\|v^\varepsilon_\alpha\|^2{\rm d}x +\displaystyle\sum_{\alpha=1}^N\int_{\R^d}\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot\nabla v^\varepsilon_\alpha{\rm d}x\\[0.3 cm] -\displaystyle\frac{1}{2\varepsilon^2}\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\int_{\R^d}\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta\|v^\varepsilon_\alpha-v^\varepsilon_\beta\|^2{\rm d}x=0. \end{array} \end{equation} Each one of the integrands in the above estimate is positive because of the positivity assumption (\ref{eq:ass:rho}), coercivity assumption (\ref{eq:ass:diff}) and the cooperative assumption (\ref{eq:ass:cple}). Integrating the energy estimate (\ref{eq:nrj}) over $(0,T)$ yields the a priori estimates (\ref{eq:apriori}). \end{proof} \section{Two-scale Compactness} \label{sec:2scl} The homogenization procedure is to consider the weak formulation of (\ref{eq:cd1})-(\ref{eq:in1}) with appropriately chosen test functions and passing to the limit as $\varepsilon\to0$. The usual approach is to obtain two-scale limits using a priori estimates of Lemma \ref{lem:apriori} by employing some compactness theorems. As it has been noticed in \cite{MP:05, DP:05, AR:07}, the classical notion of two-scale convergence from \cite{Al:92b, nguetseng} needs to be modified in order to address the homogenization of parabolic problems in strong convection regime. We recall this modified notion of two-scale convergence with drift, as first defined in \cite{MP:05}. \begin{definition} \label{def:2scl} Let $b^\in\R^d$ be a constant vector. A sequence of functions $u_\varepsilon(t,x)$ in $L^2((0,T)\times\R^d)$ is said to two-scale converge with drift $b^$, or equivalently in moving coordinates $\displaystyle(t,x)\rightarrow\Big(t,x-\frac{b^* t}{\varepsilon}\Big)$, to a limit $u_0(t,x,y)\in L^2((0,T)\times\R^d\times Y)$ if, for any function $\phi(t,x,y)\in C^\infty_c((0,T)\times\R^d;C^\infty_\#(Y))$, we have \begin{equation} \label{eq:hom:defn2d} \lim_{\varepsilon\to0}\int_0^T\int_{\R^d} u_\varepsilon(t,x)\phi\Big(t,x-\frac{b^}{\varepsilon}t,\frac{x}{\varepsilon}\Big){\rm d}x{\rm d}t = \int_0^T\int_{\R^d}\int_Y u_0(t,x,y)\phi(t,x,y){\rm d}y{\rm d}x{\rm d}t. \end{equation} We denote this convergence by $u_\varepsilon \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} u_0$. \end{definition} Now we state a compactness theorem, again borrowed from \cite{MP:05}, which guarantees the existence of two-scale limits with drift for certain sequences. \begin{proposition} \label{prop:2scl1}\cite{MP:05, Al:08} Let $b^$ be a constant vector in $\mathbb{R}^d$ and let the sequence $u_\varepsilon$ be uniformly bounded in $L^2((0,T)\times\R^d)$. Then, there exist a subsequence, still denoted by $\varepsilon$, and a function $u_0(t,x,y)\in L^2((0,T)\times\R^d;L^2_\#(Y))$such that $$ u_\varepsilon \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} u_0. $$ \end{proposition} \begin{remark} \label{rem:2scl} Note that the case $b^=0$ coincides with the classical notion of two-scale convergence from \cite{Al:92b, nguetseng}. It should also be noted that the two-scale limits obtained according to Proposition \ref{prop:2scl1} depend on the chosen drift velocity $b^\in\R^d$. These issues are addressed in \cite{Hu:13}. Unfortunately, the notion of convergence in Definition \ref{def:2scl} does not carry over to the case when the drift velocity $b^$ varies in space. \end{remark} If the sequence $\{u_\varepsilon\}$ has additional bounds, then the result of Proposition \ref{prop:2scl1} can be improved. The following result addresses this issue when the sequence has uniform $H^1$ bounds in space. \begin{proposition} \label{prop:2scl}\cite{MP:05, Al:08} Let $b^$ be a constant vector in $\mathbb{R}^d$ and let the sequence $u_\varepsilon$ be uniformly bounded in $L^2((0,T);H^1(\mathbb{R}^d))$. Then, there exist a subsequence, still denoted by $\varepsilon$, and functions $u_0(t,x) \in L^2((0,T);H^1(\mathbb{R}^d))$ and $u_1(t,x,y) \in L^2((0,T)\times\mathbb{R}^d;H^1_\#(Y))$ such that $$ u_\varepsilon \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} u_0 $$ and $$ \nabla u_\varepsilon \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} \nabla_x u_0 + \nabla_y u_1. $$ \end{proposition} Having given the notion of convergence, we shall state a result that gives the two-scale limits corresponding to solution sequences for (\ref{eq:cd1})-(\ref{eq:in1}). \begin{theorem} \label{thm:3:2scl} Let $b^\in\R^d$ be a constant vector. There exist $v\in L^2((0,T);H^1(\R^d))$ and $v_{1,\alpha}\in L^2((0,T)\times\R^d;H^1_\#(Y))$, for each $1\le\alpha\le N$, such that a subsequence of solutions $(v^\varepsilon_\alpha)_{1\le\alpha\le N}\in L^2((0,T);H^1(\R^d))^N$ of the system (\ref{eq:cd1})-(\ref{eq:in1}), two-scale converge with drift $b^$, as $\varepsilon \to 0$, in the following sense: \begin{equation} \label{eq:2scl} \begin{array}{cc} \displaystyle v^\varepsilon_\alpha \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} v, \hspace{1 cm} \nabla v^\varepsilon_\alpha \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} \nabla_x v + \nabla_y v_{1,\alpha},\\[0.2 cm] \displaystyle\frac{1}{\varepsilon}\Big(v^\varepsilon_\alpha-v^\varepsilon_\beta\Big) \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} v_{1,\alpha} - v_{1,\beta}, \end{array} \end{equation} for every $1\le\alpha,\beta\le N$. \end{theorem} \begin{proof} Consider the a priori bounds (\ref{eq:apriori}) on $v^\varepsilon_\alpha$ obtained in Lemma \ref{lem:apriori}. It follows from Proposition \ref{prop:2scl} that there exist a subsequence (still indexed by $\varepsilon$) and two-scale limits, say $v_\alpha\in L^2((0,T);H^1(\mathbb{R}^d))$ and $v_{1,\alpha}\in L^2((0,T)\times\R^d;H^1_\#(Y))$ such that \begin{equation} \label{eq:2scl:1} \begin{array}{cc} v^\varepsilon_\alpha\stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} v_\alpha\\[0.2 cm] \nabla v^\varepsilon_\alpha \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} \nabla_x v_\alpha + \nabla_y v_{1,\alpha} \end{array} \end{equation} for every $1\le\alpha\le N$. Also from the a priori estimates (\ref{eq:apriori}) we have: \begin{equation} \label{eq:apri:mod} \displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d} \Big(v^\varepsilon_\alpha-v^\varepsilon_\beta\Big)^2{\rm d}x{\rm d}t\leq C\varepsilon^2. \end{equation} The estimate (\ref{eq:apri:mod}) implies that the two-scale limits obtained in the first line of (\ref{eq:2scl:1}) do match i.e., $v_\alpha=v$ for every $1\le\alpha\le N$. However, the limit of the coupled term isn't straightforward. Since $\displaystyle\frac1\varepsilon(v^\varepsilon_\alpha - v^\varepsilon_\beta)$ is bounded in $L^2((0,T)\times\R^d)$, we have the existence of a subsequence and a function $q(t,x,y)\in L^2((0,T)\times\R^d;L^2_\#(Y))$ from Proposition \ref{prop:2scl1} such that \begin{equation} \label{eq:2scl:2} \frac1\varepsilon(v^\varepsilon_\alpha - v^\varepsilon_\beta) \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} q(t,x,y) . \end{equation} Taking $\Psi\in L^2((0,T)\times\R^d\times Y)^d$, let us consider \begin{equation} \label{eq:2scl:3} \begin{array}{cc} \displaystyle\int_0^T\int_{\R^d}\Big(\nabla v^\varepsilon_\alpha - \nabla v^\varepsilon_\beta\Big)\cdot\Psi\Big(t,x-\frac{b^}{\varepsilon}t,\frac{x}{\varepsilon}\Big){\rm d}x{\rm d}t=\\[0.3 cm] \displaystyle-\int_0^T\int_{\R^d}\Big(v^\varepsilon_\alpha - v^\varepsilon_\beta\Big){\rm div}_x\Psi\Big(t,x-\frac{b^}{\varepsilon}t,\frac{x}{\varepsilon}\Big){\rm d}x{\rm d}t\\[0.3 cm] \displaystyle-\int_0^T\int_{\R^d}\frac{1}{\varepsilon}\Big(v^\varepsilon_\alpha - v^\varepsilon_\beta\Big){\rm div}_y\Psi\Big(t,x-\frac{b^}{\varepsilon}t,\frac{x}{\varepsilon}\Big){\rm d}x{\rm d}t. \end{array} \end{equation} Let us pass to the limit in (\ref{eq:2scl:3}) as $\varepsilon\to0$. The first term on the right hand side vanishes as the limits of $v^\varepsilon_\alpha$ and $v^\varepsilon_\beta$ match. To pass to the limit in the second term of the right hand side, we shall use (\ref{eq:2scl:2}). Considering the two-scale limit in the second line of (\ref{eq:2scl:1}), upon passing to the limit as $\varepsilon\to0$ in (\ref{eq:2scl:3}) we have: \begin{equation} \label{eq:2scl:4} \begin{array}{cc} \displaystyle\int_0^T\int_{\R^d}\int_Y\nabla_y\Big(v_{1,\alpha} - v_{1,\beta}\Big)\cdot\Psi(t,x,y){\rm d}y{\rm d}x{\rm d}t =\\[0.3 cm] -\displaystyle\int_0^T\int_{\R^d}\int_Y q(t,x,y){\rm div}_y\Psi(t,x,y){\rm d}y{\rm d}x{\rm d}t. \end{array} \end{equation} >From (\ref{eq:2scl:4}) we deduce that $(v_{1,\alpha} - v_{1,\beta})$ and $q(t,x,y)$ differ by a function of $(t,x)$, say $l(t,x)$. As $v_{1,\alpha}$ and $v_{1,\beta}$ are also defined up to the addition of a function solely dependent on $(t,x)$, we can get rid of $l(t,x)$ and we recover indeed the following limit $q(t,x,y) = v_{1,\alpha} - v_{1,\beta}$. \end{proof} \section{Homogenization Result} \label{sec:hom} This section deals with the homogenization of the coupled system (\ref{eq:cd1})-(\ref{eq:in1}). To begin with, we state a Fredholm alternative for solving the cell problem, which is a key ingredient in the homogenization result. \begin{lemma} \label{lem:fh} Let $(f_\alpha)_{1\le\alpha\le N}\in(L^2_\#(Y))^N$. Consider the following cooperative system: \begin{equation} \label{eq:fh} \left\{ \begin{array}{l} \tilde b_\alpha\cdot \nabla_y\zeta_\alpha - {\rm div}_y \Big(\tilde D_\alpha \nabla_y\zeta_\alpha\Big) + \displaystyle\sum_{\beta=1}^N\: \Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta\Big(\zeta_\beta-\zeta_\alpha\Big)= f_\alpha \mbox{ in }Y,\\[0.3cm] y \to \zeta_\alpha(y) \quad Y\mbox{-periodic}, \end{array} \right. \end{equation} for every $1\le\alpha\le N$, where the coefficients $(\tilde b_\alpha, \tilde D_\alpha)$ are as in (\ref{eq:b1})-(\ref{eq:D1}) and the hypotheses (\ref{eq:ass:vel})-(\ref{eq:ass:irr}) hold. Then there exists a unique solution $(\zeta_\alpha)_{1\le\alpha\le N}\in(H^1_\#(Y))^N/(\R\times\mathds{1})$ to (\ref{eq:fh}), where $\mathds{1}=(1,\cdots,1)\in\R^N$, if and only if the following compatibility condition holds true: \begin{equation} \label{eq:fhc} \sum_{\alpha=1}^N\int_Y f_\alpha{\rm d}y=0. \end{equation} \end{lemma} \begin{proof} To prove that condition (\ref{eq:fhc}) is necessary, let us integrate the left hand side of (\ref{eq:fh}) over the unit cell. Exploiting the periodic boundary conditions, we will be left with: $$ -\int_Y{\rm div}_y(\tilde b_\alpha)\zeta_\alpha{\rm d}y + \displaystyle\sum_{\beta=1}^N\int_Y \Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta\Big(\zeta_\beta-\zeta_\alpha\Big){\rm d}y. $$ Substituting for the divergence term in the above expression from (\ref{eq:divb}) and summing over $\alpha$ indeed guarantees that the condition (\ref{eq:fhc}) on the source term is necessary. To prove sufficiency, let us assume that (\ref{eq:fhc}) is satisfied. Consider the following norm on the quotient space $\mathscr{H}(Y) :=(H^1_\#(Y))^N/(\R\times\mathds{1})$: \begin{equation} \label{eq:quo:norm} \\|(z_\alpha)_{1\le\alpha\le N}\\|^2_{\mathscr{H}(Y)}=\sum_{\alpha=1}^N\\|\nabla_y z_\alpha\\|^2_{L^2(Y)} + \sum_{\alpha=1}^N\sum_{\beta=1}^N\\|z_\alpha-z_\beta\\|^2_{L^2(Y)}. \end{equation} (It is easy to show that (\ref{eq:quo:norm}) is a norm on $\mathscr{H}(Y)$ since the zero set of (\ref{eq:quo:norm}) is the subspace spanned by $\mathds{1}$.) The variational formulation of (\ref{eq:fh}) in $\mathscr{H}(Y)$ is: find $\zeta=(\zeta_a)_{1\le\alpha\le N} \in \mathscr{H}(Y)$ such that \begin{equation} \label{eq:fh:vf} \int_Y Q(\zeta)\cdot\eta\:{\rm d}y = L(\eta) \mbox{ for any }\eta=(\eta_a)_{1\le\alpha\le N} \in \mathscr{H}(Y), \end{equation} with $$ \begin{array}{cc} \displaystyle \int_Y Q(\zeta)\cdot\eta{\rm d}y := \sum_{\alpha=1}^N\int_Y\Big(\tilde b_\alpha(y)\cdot \nabla_y\zeta_\alpha\Big)\eta_\alpha{\rm d}y + \sum_{\alpha=1}^N\int_Y\tilde D_\alpha(y)\nabla_y\zeta_\alpha\cdot\nabla_y\eta_\alpha{\rm d}y\\[0.2 cm] \displaystyle+\sum_{\alpha=1}^N\sum_{\beta=1}^N\int_Y\Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta\Big(\zeta_\beta-\zeta_\alpha\Big)\eta_\alpha{\rm d}y \end{array} $$ and $$ L(\eta):=\sum_{\alpha=1}^N\int_Y f_\alpha\eta_\alpha{\rm d}y. $$ The compatibility condition (\ref{eq:fhc}) implies that $(f_\alpha)_{1\le\alpha\le N}$ is orthogonal to $\mathds{1}$ in $L^2$ and consequently that the linear form $L(\eta)$ in (\ref{eq:fh:vf}) is continuous. By performing similar computations as in the proof of Lemma \ref{lem:apriori}, we can show that the bilinear form in (\ref{eq:fh:vf}) is coercive in $\mathscr{H}(Y)$ i.e., $$ \int_Y Q(\zeta)\cdot\zeta\:{\rm d}y \ge C\sum_{\alpha=1}^N\int_Y \|\nabla_y \zeta_\alpha\|^2{\rm d}y + \sum_{\alpha=1}^N\sum_{\beta=1}^N\int_Y \|\zeta_a - \zeta_b\|^2{\rm d}y. $$ To show that the bilinear form in (\ref{eq:fh:vf}) is continuous on $\mathscr{H}(Y)\times\mathscr{H}(Y)$ we remark that, first, $\int_Y Q(\eta)\cdot\mathds{1}\,{\rm d}y=0$ for any $\eta\in \mathscr{H}(Y)$ (this is precisely the computation which yields the compatibility condition (\ref{eq:fhc})) and, second, $Q(\eta - c \mathds{1}) = 0$ for any $\eta\in \mathscr{H}(Y)$ and any $c\in\R$. Therefore, for any $\zeta,\eta \in \mathscr{H}(Y)$, we have the following: $$ \int_Y Q(\zeta)\cdot\eta\:{\rm d}y = \int_Y Q\Big(\zeta - \mathds{1}c_\zeta\Big) \cdot \Big(\eta - \mathds{1}c_\eta\Big)\:{\rm d}y\hspace{0.5 cm}\mbox{ for any constants } c_\zeta, c_\eta \in\R, $$ which implies $$ \Big\|\int_Y Q(\zeta)\cdot\eta\:{\rm d}y\Big\|\le C\Big\\|\Big(\zeta - \mathds{1}c_\zeta\Big)\Big\\|_{(H^1_\#(Y))^N}\Big\\|\Big(\eta - \mathds{1}c_\eta\Big)\Big\\|_{(H^1_\#(Y))^N}=C\\|\zeta\\|_{\mathscr{H}(Y)}\\|\eta\\|_{\mathscr{H}(Y)}. $$ We can thus apply the Lax-Milgram lemma in $\mathscr{H}(Y)$ to obtain the existence and uniqueness of a solution to (\ref{eq:fh}). \end{proof} \begin{remark} \label{rem:quot} The well-posedness result of the Lemma \ref{lem:fh} is given in the quotient space $(H^1_\#(Y))^N/(\R\times\mathds{1})$ i.e., the solutions are unique up to the addition of a constant. The constant being the same for each component of the solution. \end{remark} In the previous section, using the a priori estimates, we have obtained two-scale limits with drift for the solution sequence. Now, by choosing an appropriate drift constant $b^$, we shall characterize the two scale limits. Contrary to the compactness result of Theorem \ref{thm:3:2scl} which gives the convergence up to a subsequence, the next result guarantees that the entire sequence $v^\varepsilon_\alpha$ converges to $v$ for every $1\le\alpha\le N$. The main result of this article is the following. \begin{theorem} \label{thm:hom} Let $(v^\varepsilon_\alpha)_{1\le\alpha\le N}$ be the sequence of solutions to the system (\ref{eq:cd1})-(\ref{eq:in1}). The entire sequence $v^\varepsilon_\alpha$ converges, in the sense of Theorem \ref{thm:3:2scl}, to the limits $v\in L^2((0,T);H^1(\R^d))$ and $v_{1,\alpha}\in L^2((0,T)\times\R^d;H^1_\#(Y))$ for every $1\le\alpha\le N$ (see (\ref{eq:2scl}) for details). The two-scale limits $v_{1,\alpha}$ are explicitly given by \begin{equation} \label{eq:2scl:corr} v_{1,\alpha}(t,x,y) = \sum_{i=1}^d \frac{\partial v}{\partial x_i}(t,x)\omega_{i,\alpha}(y)\hspace{0.5 cm}\mbox{ for every }1\le\alpha\le N, \end{equation} where $(\omega_{i,\alpha})_{1\le\alpha\le N}\in(H^1_\#(Y))^N/(\R\times\mathds{1})$ satisfy the cell problem: \begin{equation} \label{eq:cpb} \left\{ \begin{array}{lll} \tilde b_\alpha(y)\cdot \Big(\nabla_y\omega_{i,\alpha} + e_i\Big) - {\rm div}_y \Big(\tilde D_\alpha \Big(\nabla_y\omega_{i,\alpha}+e_i\Big)\Big)\\[0.3cm] + \displaystyle\sum_{\beta=1}^N\: \Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta\Big(\omega_{i,\beta}-\omega_{i,\alpha}\Big)= \varphi_\alpha\varphi^_\alpha\rho_\alpha b^\cdot e_i & \mbox{ in }Y,\\[0.3cm] y \to \omega_{i,\alpha}(y) & Y\mbox{-periodic}, \end{array} \right. \end{equation} for every $1\le i\le d$, where the drift velocity $b^$ is given by \begin{equation} \label{eq:drift} \displaystyle b^ = \displaystyle\sum_{\alpha=1}^N\: \int_Y \tilde b_\alpha(y){\rm d}y. \end{equation} Further, the two-scale limit $v(t,x)$ is the unique solution of the scalar diffusion equation: \begin{equation} \label{eq:hom} \left\{ \begin{array}{ll} \displaystyle\frac{\partial v}{\partial t} - {\rm div}({\mathcal D}\nabla v) = 0 & \mbox{ in }(0,T)\times\R^d,\\[0.3 cm] v(0,x) = \displaystyle\sum_{\alpha=1}^N u^{in}_\alpha(x) \displaystyle\int_Y\rho_\alpha(y) \varphi^_\alpha(y){\rm d}y & \mbox{ in }\R^d, \end{array} \right. \end{equation} with the elements of the dispersion matrix $\mathcal D$ given by \begin{equation} \label{eq:disp} \begin{array}{ll} {\mathcal D}_{ij}=\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\tilde D_\alpha\Big(\nabla_y\omega_{i,\alpha} + e_i\Big)\cdot\Big(\nabla_y\omega_{j,\alpha} + e_j\Big){\rm d}y\\[0.5 cm] -\displaystyle\frac12\sum_{\alpha,\beta=1}^N\:\:\int_Y\varphi^_\alpha\varphi_\beta\Pi_{\alpha\beta}\Big(\omega_{i,\alpha} - \omega_{i,\beta}\Big)\Big(\omega_{j,\alpha} - \omega_{j,\beta}\Big){\rm d}y. \end{array} \end{equation} \end{theorem} \begin{remark} \label{rem:irr} The irreducibility assumption (\ref{eq:ass:irr}) on the coupling matrix $\Pi$ ensures microscopic equilibrium among all $v^\varepsilon_\alpha$ resulting in a single homogenized limit $v(t,x)$ i.e., if the coupling matrix $\Pi\equiv0$ (say), we get $N$ different homogenized limits. \end{remark} \begin{remark} Our main homogenization result (Theorem \ref{thm:hom}) holds only for weakly coupled cooperative parabolic systems. Our approach does not answer the homogenization of general weakly coupled parabolic systems, not to mention fully coupled systems. We heavily rely upon the cooperative assumption on the coupling matrix as the positivity and spectral theorems are known only in the cooperative case. \end{remark} \begin{remark} \label{rem:lim} The homogenized limit $v(t,x)$ is proven to satisfy a scalar diffusion equation (\ref{eq:hom}), which is a bit deceptive by its simplicity. However, if we make the following change of functions: $$ \tilde v(t,x) = \exp{(-\lambda t/\varepsilon^2)}v\Big(t,x-\frac{b^}{\varepsilon}t\Big), $$ we remark that $\tilde v(t,x)$ indeed satisfies the following scalar convection-diffusion-reaction equation: $$ \displaystyle\frac{\partial \tilde v}{\partial t} + \frac{b^}{\varepsilon}\cdot\nabla \tilde v - {\rm div}({\mathcal D}\nabla \tilde v) + \frac{\lambda}{\varepsilon^2}\tilde v = 0\mbox{ in }(0,T)\times\R^d. $$ Therefore, $b^/\varepsilon$ is precisely the effective drift while $\lambda/\varepsilon^2$ is the effective reaction rate. Remark that because of the large drift $\varepsilon^{-1}b^$, we cannot work in bounded domains. \end{remark} \begin{remark} The assumption of pure periodicity on the coefficients of (\ref{eq:cd}) is crucial for the results obtained in this article. The natural thought for generalizing the results of this article is to explore the possibility of considering ``locally periodic'' coefficients i.e., coefficients of the type $b(x,x/\varepsilon)$, where the function is $Y$-periodic in the second variable. If the convective fields $b^\varepsilon_\alpha$ were locally periodic, then it is clear that the drift vector $b^(x)$ should depend on $x$. However, in such a case, we have no idea on how to extend the method of two-scale asymptotic expansion, not to mention the even greater difficulties in generalizing the notion of two-scale convergence with non-constant drift (as already mentioned in Remark \ref{rem:2scl}). Such a generalization still remains as an outstanding open problem in the theory of Taylor dispersion. \end{remark} \begin{remark} This article only addresses the homogenization of linear systems. We have also considered only diagonal diffusion models. Cross diffusion phenomena occurs naturally in the physics of multicomponent gaseous mixtures, population dynamics and porous media (cf. \cite{BGS:12} and references therein). The natural nonlinear transport model to consider is the Maxwell-Stefan's equations. A complete mathematical study of the Maxwell-Stefan laws is still missing. There have been some recent studies in this direction (cf. \cite{BGS:12, DLM:14, MT:14, DT:p} for example). One approach would be to consider the ``parabolically'' scaled Maxwell-Stefan's equations and arrive at an homogenization result. The obvious questions to ask is the following: Is there a scalar diffusion limit even in case of nonlinear Maxwell-Stefan's equations? This problem might involve mathematical techniques quite different from the ones used here as the spectral problems (which is the crux of the Factorization method) in the nonlinear counterpart have not been well understood. We hope to return to this question in subsequent publications. \end{remark} Before we present the proof of Theorem \ref{thm:hom}, we state a lemma that gives some qualitative information on the dispersion matrix. \begin{lemma} \label{lem:disp} The dispersion matrix $\mathcal D$ given by (\ref{eq:disp}) is symmetric positive definite. \end{lemma} \begin{proof} The symmetric part is obvious. By the hypothesis on the coupling matrix $\Pi$ and the positivity of the first eigenvector functions, the factor $\Pi_{\alpha\beta}\varphi^_\alpha\varphi_\beta$ is always non-positive for $\alpha\neq\beta$. By the hypothesis (\ref{eq:ass:diff}), we know that the diffusion matrices $D_\alpha$ are coercive with coercivity constants $c_\alpha>0$. For $\xi\in\R^d$, we define $$ \omega_{\alpha\xi} := \sum_{i=1}^d\omega_{i,\alpha}\xi_i. $$ Then, $$ {\mathcal D}\xi\cdot\xi \geq\displaystyle\sum_{\alpha=1}^N\:\:c_\alpha\displaystyle\int_Y\Big\|\nabla_y\omega_{\alpha\xi}+\xi\Big\|^2{\rm d}y \geq0. $$ Now, we need to show that ${\mathcal D}\xi\cdot\xi>0$ for all $\xi\not=0$. Suppose that ${\mathcal D}\xi\cdot\xi=0$ which in turn implies that $\omega_{\alpha\xi}+\xi\cdot y\equiv C_\alpha$ for some constant $C_\alpha$. As the cell solutions $(\omega_{i,\alpha})_{1\le\alpha\le N}$ are $Y$-periodic, they cannot be affine. Thus the above equalities are possible only when $\xi=0$ which implies the positive definiteness of $\mathcal D$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:hom}] In the sequel we use the notations $$ \begin{array}{cc} \displaystyle\phi\equiv\phi(t,x) \, , \quad \phi^\varepsilon\equiv\phi\left(t,x-\frac{b^* t}{\varepsilon}\right) \, ,\\[0.3 cm] \displaystyle\phi_{1,\alpha}\equiv\phi_{1,\alpha}(t,x,y) \, , \quad \phi^\varepsilon_{1,\alpha}\equiv\phi_{1,\alpha}\left(t,x-\frac{b^* t}{\varepsilon},\frac{x}{\varepsilon}\right). \end{array} $$ The idea is to test the factorized equation (\ref{eq:cd1}) with $$ \phi^\varepsilon_\alpha=\phi^\varepsilon+\varepsilon\,\phi^\varepsilon_{1,\alpha}, $$ where $\phi(t,x)$ and $\phi_{1,\alpha}(t,x,y)$ are smooth functions with compact support in $x$, which vanish at the final time $T$ and are $Y$-periodic with respect to $y$. We get $$ \begin{array}{cc} \displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha\frac{\partial v^\varepsilon_\alpha}{\partial t}\phi^\varepsilon_\alpha{\rm d}x{\rm d}t + \frac{1}{\varepsilon}\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\tilde b^\varepsilon_\alpha \cdot \nabla v^\varepsilon_\alpha\phi^\varepsilon_\alpha{\rm d}x{\rm d}t\\[0.3 cm] +\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} \tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot \nabla \phi^\varepsilon_\alpha{\rm d}x{\rm d}t +\frac{1}{\varepsilon^2}\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d}\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha)\phi^\varepsilon_\alpha = 0. \end{array} $$ Substituting for $\phi^\varepsilon_\alpha$ in the above variational formulation and integrating by parts leads to \begin{equation} \label{eq:vf:1} \begin{array}{cc} -\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha v^\varepsilon_\alpha\frac{\partial \phi^\varepsilon}{\partial t}{\rm d}x{\rm d}t -\displaystyle\sum_{\alpha=1}^N\:\:\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha v^\varepsilon_\alpha(0,x)\phi^\varepsilon(0,x){\rm d}x\\[0.3 cm] \displaystyle+\frac{1}{\varepsilon}\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} v^\varepsilon_\alpha\Big(\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha b^ - \tilde b^\varepsilon_\alpha\Big)\cdot \nabla_x \phi^\varepsilon{\rm d}x{\rm d}t\\[0.3 cm] \displaystyle-\frac{1}{\varepsilon}\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} {\rm div}\Big(\tilde b^\varepsilon_\alpha\Big) v^\varepsilon_\alpha\phi^\varepsilon{\rm d}x{\rm d}t +\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot\nabla_x\phi^\varepsilon{\rm d}x{\rm d}t\\[0.3 cm] +\displaystyle\frac{1}{\varepsilon^2}\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha)\phi^\varepsilon{\rm d}x{\rm d}t + \sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} \Big(\tilde b^\varepsilon_\alpha \cdot \nabla v^\varepsilon_\alpha\Big)\phi_{1,\alpha}^\varepsilon{\rm d}x{\rm d}t\\[0.3 cm] +\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha v^\varepsilon_\alpha b^\cdot \nabla \phi_{1,\alpha}^\varepsilon{\rm d}x{\rm d}t + \displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} \tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot \nabla_y \phi_{1,\alpha}^\varepsilon{\rm d}x{\rm d}t\\[0.3 cm] + \displaystyle\frac{1}{\varepsilon}\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha)\phi_{1,\alpha}^\varepsilon{\rm d}x{\rm d}t + {\mathcal O}(\varepsilon) = 0. \end{array} \end{equation} In a first step we choose $\phi^\varepsilon\equiv0$ in (\ref{eq:vf:1}) and pass to the limit as $\varepsilon\to0$ which yields: \begin{equation} \label{eq:vf:cpb} \begin{array}{cc} -\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\int_Y\varphi_\alpha\varphi^_\alpha\rho_\alpha b^\cdot\nabla_x v\phi_{1,\alpha}{\rm d}y{\rm d}x{\rm d}t\\[0.3 cm] + \displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\int_Y\tilde b_\alpha\cdot\Big(\nabla_x v + \nabla_y v_{1,\alpha}\Big) \phi_{1,\alpha}{\rm d}y{\rm d}x{\rm d}t\\[0.3 cm] -\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\int_Y {\rm div}_y\Big(\tilde D_\alpha\Big(\nabla_x v + \nabla_y v_{1,\alpha}\Big)\Big)\phi_{1,\alpha}{\rm d}y{\rm d}x{\rm d}t\\[0.3 cm] + \displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d}\int_Y \Pi_{\alpha\beta}\varphi^_\alpha\varphi_\beta\Big(v_{1,\beta}-v_{1,\alpha}\Big)\phi_{1,\alpha}(t,x,y){\rm d}y{\rm d}x{\rm d}t = 0. \end{array} \end{equation} The above expression is the variational formulation for the following PDE: \begin{equation} \left\{ \begin{array}{cc} \displaystyle\tilde b_\alpha\cdot \Big(\nabla_y v_{1,\alpha} + \nabla_x v\Big) - {\rm div}_y \Big(\tilde D_\alpha \Big(\nabla_y v_{1,\alpha}+\nabla_x v\Big)\Big)\\[0.3 cm] + \displaystyle\sum_{\beta=1}^N \Pi_{\alpha\beta}\varphi^_\alpha\varphi_\beta(v_{1,\beta}-v_{1,\alpha})= \varphi_\alpha\varphi^_\alpha\rho_\alpha b^\cdot \nabla_x v & \mbox{ in }Y, \\[0.2 cm] y \to v_{1,\alpha}(y) & Y\mbox{-periodic,} \end{array} \right. \label{eq:vv1} \end{equation} for every $1\le\alpha\le N$. By the Fredholm result of Lemma \ref{lem:fh}, we have the existence and uniqueness of $(v_{1,\alpha})_{1\le\alpha\le N}\in L^2((0,T)\times\R^d;\mathscr{H}(Y))$ if and only if the compatibility condition (\ref{eq:fhc}) is satisfied. Writing down the compatibility condition for (\ref{eq:vv1}) yields the expression (\ref{eq:drift}) for the drift velocity $b^$. Also by linearity of (\ref{eq:vv1}), we deduce that we can separate the slow and fast variables in $v_{1,\alpha}$ as in (\ref{eq:2scl:corr}) with $(\omega_{i,\alpha})_{1\le\alpha\le N}$ satisfying the coupled cell problem (\ref{eq:cpb}). In a second step we choose $\phi^\varepsilon_{1,\alpha}\equiv0$ in (\ref{eq:vf:1}) and substitute (\ref{eq:in1}) for the initial data $v^\varepsilon_\alpha(0,x)$, which yields \begin{equation} \label{eq:vf:2} \begin{array}{ll} -\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho_\alpha^\varepsilon v^\varepsilon_\alpha\frac{\partial \phi^\varepsilon}{\partial t}{\rm d}x{\rm d}t -\displaystyle\sum_{\alpha=1}^N\:\:\int_{\R^d}\varphi^{\varepsilon}_\alpha\rho_\alpha^\varepsilon u^{in}_\alpha(x)\phi^\varepsilon(0,x){\rm d}x\\[0.2 cm] +\displaystyle\frac{1}{\varepsilon}\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha v^\varepsilon_\alpha\rho_\alpha^\varepsilon b^\cdot \nabla_x \phi^\varepsilon{\rm d}x{\rm d}t -\frac{1}{\varepsilon}\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} v^\varepsilon_\alpha\tilde b^\varepsilon_\alpha \cdot\nabla_x\phi^\varepsilon{\rm d}x{\rm d}t\\[0.2 cm] -\displaystyle\frac{1}{\varepsilon}\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} {\rm div}\Big(\tilde b^\varepsilon_\alpha\Big) v^\varepsilon_\alpha\phi^\varepsilon{\rm d}x{\rm d}t +\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot\nabla_x\phi^\varepsilon{\rm d}x{\rm d}t\\[0.2 cm] +\displaystyle\frac{1}{\varepsilon^2}\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha)\phi^\varepsilon{\rm d}x{\rm d}t = 0. \end{array} \end{equation} Using the expression (\ref{eq:divb}) for the divergence of $\tilde b_\alpha$ allows us to obtain $$ \displaystyle-\frac{1}{\varepsilon}\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} {\rm div}\Big(\tilde b^\varepsilon_\alpha\Big) v^\varepsilon_\alpha\phi^\varepsilon{\rm d}x{\rm d}t +\displaystyle\frac{1}{\varepsilon^2}\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d} \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha)\phi^\varepsilon{\rm d}x{\rm d}t $$ \begin{equation} \label{eq:vf:div} \begin{array}{ll} = \displaystyle\frac{1}{\varepsilon^2}\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d} \Big(\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta v^\varepsilon_\alpha - \Pi_{\alpha\beta}^{\varepsilon }\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\beta v^\varepsilon_\alpha\Big)\phi^\varepsilon{\rm d}x{\rm d}t\\[0.2 cm] +\displaystyle\frac{1}{\varepsilon^2}\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_0^T\int_{\R^d} \Big(\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta v^\varepsilon_\beta - \Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta v^\varepsilon_\alpha \Big)\phi^\varepsilon{\rm d}x{\rm d}t=0. \end{array} \end{equation} Thanks to (\ref{eq:vf:div}) all terms of order $\mathcal O (\varepsilon^{-2})$ in (\ref{eq:vf:2}) cancel each other. There are, however, terms of $\mathcal O (\varepsilon^{-1})$ in (\ref{eq:vf:2}) which still prevent us to pass to the limit as $\varepsilon\to0$. In order to remedy the situation, we introduce the following auxiliary problem posed in the unit cell: \begin{equation} \label{eq:aux} \left\{ \begin{array}{ll} \displaystyle-\Delta \Xi = \displaystyle\sum_{\alpha=1}^N\:\Big(\varphi_\alpha\varphi^_\alpha\rho_\alpha b^ - \tilde b_\alpha\Big) & \mbox{ in }Y, \\[0.2 cm] y \to \Xi(y) & Y\mbox{-periodic.} \end{array} \right. \end{equation} The above auxiliary problem is well-posed, thanks to our choice (\ref{eq:drift}) of the drift velocity and the chosen normalization (\ref{eq:norm}). We scale (\ref{eq:aux}) to the entire domain via the change of variables $y\to\varepsilon^{-1}x$. The vector-valued function $\Xi^\varepsilon(x) =\Xi(x/\varepsilon)$ satisfies \begin{equation} \label{eq:aux:scl} \left\{ \begin{array}{ll} \displaystyle-\varepsilon^2 \Delta \Xi^\varepsilon = \displaystyle\sum_{\alpha=1}^N\:\Big(\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho_\alpha^\varepsilon b^ - \tilde b^\varepsilon_\alpha\Big) & \mbox{ in }\R^d,\\[0.2 cm] x \to \Xi^\varepsilon & \varepsilon Y\mbox{-periodic.} \end{array} \right. \end{equation} Getting back to the variational formulation (\ref{eq:vf:2}), let us regroup the problematic terms of order $\mathcal O (\varepsilon^{-1})$: $$ \displaystyle\frac{1}{\varepsilon}\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d} v^\varepsilon_\alpha\Big(\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha v^\varepsilon_\alpha\rho_\alpha^\varepsilon b^ - b^\varepsilon_\alpha\Big)\cdot \nabla_x \phi^\varepsilon{\rm d}x{\rm d}t $$ \begin{equation} \label{eq:vf:rerng} \begin{array}{ll} \displaystyle=-\frac{\varepsilon^2}{\varepsilon}\sum_{i=1}^d\int_0^T\int_{\R^d} \Delta \Xi_i^\varepsilon \frac{\partial \phi^\varepsilon}{\partial x_i}v^\varepsilon_\alpha{\rm d}x{\rm d}t +\frac{1}{\varepsilon}\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}(v^\varepsilon_\alpha-v^\varepsilon_\beta)\tilde b^\varepsilon_\beta \cdot\nabla_x\phi^\varepsilon{\rm d}x{\rm d}t\\[0.3 cm] +\displaystyle\frac{1}{\varepsilon}\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\beta\varphi^{\varepsilon}_\beta\rho_\beta^\varepsilon(v^\varepsilon_\beta-v^\varepsilon_\alpha) b^\cdot \nabla_x \phi^\varepsilon{\rm d}x{\rm d}t \end{array} \end{equation} \begin{equation} \label{eq:vf:sing} \begin{array}{ll} \displaystyle= \varepsilon\sum_{i=1}^d\int_0^T\int_{\R^d}\nabla \Xi_i^\varepsilon\cdot\nabla\Big(\frac{\partial \phi^\varepsilon}{\partial x_i}\Big)v^\varepsilon_\alpha{\rm d}x{\rm d}t + \varepsilon\sum_{i=1}^d\int_0^T\int_{\R^d}\nabla \Xi_i^\varepsilon\cdot\nabla v^\varepsilon_\alpha \Big(\frac{\partial \phi^\varepsilon}{\partial x_i}\Big){\rm d}x{\rm d}t\\[0.3 cm] \displaystyle+\frac{1}{\varepsilon}\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\beta\varphi^{\varepsilon}_\beta\rho_\beta^\varepsilon(v^\varepsilon_\beta-v^\varepsilon_\alpha) b^\cdot \nabla_x \phi^\varepsilon{\rm d}x{\rm d}t\\[0.3 cm] \displaystyle+\frac{1}{\varepsilon}\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}(v^\varepsilon_\alpha-v^\varepsilon_\beta)\tilde b^\varepsilon_\beta \cdot\nabla_x\phi^\varepsilon{\rm d}x{\rm d}t , \end{array} \end{equation} where we have used the scaled auxiliary problem (\ref{eq:aux:scl}). We can now pass to the limit in (\ref{eq:vf:sing}) since the sequences $(v^\varepsilon_\beta-v^\varepsilon_\alpha)/\varepsilon$ are bounded. Taking into consideration (\ref{eq:vf:div}) and (\ref{eq:vf:sing}), the variational formulation (\ref{eq:vf:2}) rewrites as \begin{equation} \label{eq:vf:3} \begin{array}{ll} -\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho_\alpha^\varepsilon v^\varepsilon_\alpha\frac{\partial \phi^\varepsilon}{\partial t}{\rm d}x{\rm d}t -\displaystyle\sum_{\alpha=1}^N\:\:\int_{\R^d}\varphi^{\varepsilon}_\alpha\rho_\alpha^\varepsilon u^{in}_\alpha(x)\phi^\varepsilon(0,x){\rm d}x\\[0.2 cm] \displaystyle+\varepsilon\sum_{i=1}^d\int_0^T\int_{\R^d}\nabla \Xi_i^\varepsilon\cdot\nabla\Big(\frac{\partial \phi^\varepsilon}{\partial x_i}\Big)v^\varepsilon_\alpha{\rm d}x{\rm d}t + \varepsilon\sum_{i=1}^d\int_0^T\int_{\R^d}\nabla \Xi_i^\varepsilon\cdot\nabla v^\varepsilon_\alpha \Big(\frac{\partial \phi^\varepsilon}{\partial x_i}\Big){\rm d}x{\rm d}t\\[0.2 cm] \displaystyle+\frac{1}{\varepsilon}\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}\varphi^\varepsilon_\beta\varphi^{\varepsilon}_\beta\rho_\beta^\varepsilon(v^\varepsilon_\beta-v^\varepsilon_\alpha) b^\cdot \nabla_x \phi^\varepsilon{\rm d}x{\rm d}t +\displaystyle\sum_{\alpha=1}^N\:\:\int_0^T\int_{\R^d}\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\cdot\nabla_x\phi^\varepsilon{\rm d}x{\rm d}t\\[0.2 cm] \displaystyle+\frac{1}{\varepsilon}\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}(v^\varepsilon_\alpha-v^\varepsilon_\beta)\tilde b^\varepsilon_\beta \cdot\nabla_x\phi^\varepsilon{\rm d}x{\rm d}t=0. \end{array} \end{equation} Using the compactness results from Theorem \ref{thm:3:2scl}, we pass to the limit as $\varepsilon\to0$ in the above variational formulation leading to: \begin{equation} \label{eq:vf:4} \begin{array}{ll} -\displaystyle\int_0^T\int_{\R^d} v\frac{\partial \phi}{\partial t}{\rm d}x{\rm d}t -\displaystyle\sum_{\alpha=1}^N\:\int_{\R^d}\int_Y u^{in}_\alpha(x)\phi(0,x) \varphi^{}_\alpha\rho_\alpha{\rm d}y{\rm d}x\\[0.2 cm] +\displaystyle\sum_{\alpha=1}^N\:\int_0^T\int_{\R^d}\int_Y\tilde D_\alpha \Big(\nabla v + \nabla_y v_{1,\alpha}\Big)\cdot\nabla_x\phi{\rm d}y{\rm d}x{\rm d}t\\[0.2 cm] \displaystyle+\sum_{i=1}^d\int_0^T\int_{\R^d}\int_Y\nabla_y\Xi_i\cdot\nabla_y v_{1,\alpha} \frac{\partial \phi}{\partial x_i}{\rm d}y{\rm d}x{\rm d}t\\[0.2 cm] +\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}\int_Y\varphi_\beta\varphi^_\beta\rho_\beta\Big(v_{1,\beta}-v_{1,\alpha}\Big) b^\cdot \nabla_x \phi{\rm d}y{\rm d}x{\rm d}t\\[0.2 cm] +\displaystyle\sum_{\beta=1}^N\:\int_0^T\int_{\R^d}\int_Y\Big(v_{1,\alpha}-v_{1,\beta}\Big)\tilde b_\beta \cdot\nabla_x\phi{\rm d}y{\rm d}x{\rm d}t = 0. \end{array} \end{equation} Substituting (\ref{eq:2scl:corr}) for $v_{1,\alpha}$ in (\ref{eq:vf:4}), we obtain \begin{equation} \label{eq:vf:lim} \begin{array}{ll} -\displaystyle\int_0^T\int_{\R^d} v\frac{\partial \phi}{\partial t}{\rm d}x{\rm d}t -\displaystyle\sum_{\alpha=1}^N\:\int_{\R^d} u^{in}_\alpha(x)\phi(0,x)\int_Y \varphi^{}_\alpha(y)\rho_\alpha(y){\rm d}y{\rm d}x\\[0.2 cm] +\displaystyle\sum_{\alpha=1}^N\:\sum_{i,j=1}^d\:\int_0^T\int_{\R^d}\frac{\partial v}{\partial x_j}\frac{\partial \phi}{\partial x_i}\int_Y \tilde D_\alpha\Big(\nabla y_j + \nabla_y\omega_{j,\alpha}\Big)\cdot\nabla y_i{\rm d}y{\rm d}x{\rm d}t\\[0.2 cm] -\displaystyle\sum_{i,j=1}^d\int_0^T\int_{\R^d}\frac{\partial v}{\partial x_j}\frac{\partial \phi}{\partial x_i}\int_Y\Big(\Delta_y\Xi_i\Big) \omega_{j,\alpha}{\rm d}y{\rm d}x{\rm d}t\\[0.2 cm] +\displaystyle\sum_{\beta=1}^N\:\sum_{i,j=1}^d\:\int_0^T\int_{\R^d}\frac{\partial v}{\partial x_j}\frac{\partial \phi}{\partial x_i}\int_Y\varphi_\beta\varphi^_\beta\rho_\beta \Big(\omega_{j,\beta} - \omega_{j,\alpha}\Big) b^\cdot\nabla y_i{\rm d}y{\rm d}x{\rm d}t\\[0.2 cm] +\displaystyle\sum_{\beta=1}^N\:\sum_{i,j=1}^d\:\int_0^T\int_{\R^d}\frac{\partial v}{\partial x_j}\frac{\partial \phi}{\partial x_i}\int_Y\Big(\omega_{j,\alpha} - \omega_{j,\beta}\Big) \tilde b_\beta\cdot\nabla y_i{\rm d}y{\rm d}x{\rm d}t = 0. \end{array} \end{equation} Using the information from the auxiliary cell problem (\ref{eq:aux}) in (\ref{eq:vf:lim}) and making a rearrangement similar to that of (\ref{eq:vf:rerng}), we deduce that (\ref{eq:vf:lim}) is nothing but the variational formulation for a scalar diffusion equation (\ref{eq:hom}) for $v(t,x)$ with the entries of the diffusion matrix given by $$ {\mathcal D}_{ij} = \displaystyle\sum_{\alpha=1}^N\:\int_Y\tilde D_\alpha\Big(\nabla y_j + \nabla_y\omega_{j,\alpha}\Big)\cdot\nabla y_i{\rm d}y +\displaystyle\sum_{\alpha=1}^N\:\:\int_Y \omega_{j,\alpha}\Big(\varphi_\alpha\varphi^_\alpha v^\varepsilon_\alpha\rho_\alpha^\varepsilon b^ - b^\varepsilon_\alpha\Big)\cdot e_i{\rm d}y. $$ By integration by parts, it is clear that the diffusion matrix $\mathcal D$ is contracted with the Hessian matrix $\nabla\nabla v$, which is symmetric. Thus the non-symmetric part of $\mathcal D$ does not contribute to the homogenized equation (\ref{eq:hom}). So, the above expression for the diffusion matrix is symmetrized: \begin{equation} \label{eq:disp:sym} \begin{array}{ll} {\mathcal D}_{ij} = \displaystyle\sum_{\alpha=1}^N\:\:\int_Y\tilde D_\alpha e_j\cdot e_i{\rm d}y +\frac12 \Big\{\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\Big(\tilde D_\alpha\nabla_y\omega_{i,\alpha}\cdot e_j + \tilde D_\alpha\nabla_y\omega_{j,\alpha}\cdot e_i\Big){\rm d}y\Big\}\\[0.2 cm] +\displaystyle \frac12 \Big\{\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\Big( \omega_{i,\alpha}(\varphi_\alpha\varphi^_\alpha\rho_\alpha b^ -\tilde b_\alpha)\cdot e_j + \omega_{j,\alpha}(\varphi_\alpha\varphi^_\alpha\rho_\alpha b^ -\tilde b_\alpha)\cdot e_i\Big){\rm d}y\Big\}. \end{array} \end{equation} To obtain the desired expression (\ref{eq:disp}) for the diffusion matrix, we consider the variational formulation for the cell problem (\ref{eq:cpb}) with test functions $(\psi_\alpha)_{1\le\alpha\le N}$ \begin{equation} \label{eq:vf:cpb1} \begin{array}{ll} \displaystyle\sum_{\alpha=1}^N\:\:\int_Y\Big(\tilde b_\alpha\cdot\nabla_y\omega_{i,\alpha}\Big)\psi_\alpha{\rm d}y+\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_Y\Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta\Big(\omega_{i,\beta}-\omega_{i,\alpha}\Big)\psi_\alpha{\rm d}y\\[0.2 cm] +\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\tilde D_\alpha \Big(\nabla_y\omega_{i,\alpha}+e_i\Big)\cdot\nabla_y\psi_\alpha{\rm d}y=\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\Big(\varphi_\alpha\varphi^_\alpha\rho_\alpha b^* - \tilde b_\alpha\Big)\cdot e_i\psi_\alpha{\rm d}y. \end{array} \end{equation} In (\ref{eq:vf:cpb1}) we first choose the test function $(\psi_\alpha)=(\omega_{j,\alpha})$. Similarly, in (\ref{eq:vf:cpb1}) for $j$ instead of $i$, we choose the test function $(\psi_\alpha)=(\omega_{i,\alpha})$. This leads to \begin{equation} \label{eq:vf:cpb2} \begin{array}{ll} \displaystyle\frac12 \Big\{\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\Big( \omega_{i,\alpha}(\varphi_\alpha\varphi^_\alpha\rho_\alpha b^ -\tilde b_\alpha)\cdot e_j + \omega_{j,\alpha}(\varphi_\alpha\varphi^_\alpha\rho_\alpha b^ -\tilde b_\alpha)\cdot e_i\Big){\rm d}y\Big\}\\[0.2 cm] =\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\tilde D_\alpha\nabla_y\omega_{i,\alpha}\cdot\nabla_y\omega_{j,\alpha}{\rm d}y+\frac12 \Big\{\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\Big(\tilde D_\alpha\nabla_y\omega_{i,\alpha}\cdot e_j + \tilde D_\alpha\nabla_y\omega_{j,\alpha}\cdot e_i\Big){\rm d}y\Big\}\\[0.2 cm] \displaystyle-\frac12\Big\{\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\omega_{i,\alpha}\omega_{j,\alpha}{\rm div}_y\tilde b_\alpha{\rm d}y\Big\}\\[0.2 cm] \displaystyle+\frac12 \Big\{\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_Y\Big(\Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta(\omega_{i,\beta}-\omega_{i,\alpha})\omega_{j,\alpha} + \Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta(\omega_{j,\beta}-\omega_{j,\alpha})\omega_{i,\alpha}\Big){\rm d}y\Big\}. \end{array} \end{equation} Using formula (\ref{eq:divb}) for the divergence of $\tilde b_\alpha$ in (\ref{eq:vf:cpb2}), its right hand side simplifies as \begin{equation} \label{eq:vf:cpb3} \begin{array}{ll} \displaystyle\sum_{\alpha=1}^N\:\:\int_Y\tilde D_\alpha\nabla_y\omega_{i,\alpha}\cdot\nabla_y\omega_{j,\alpha}{\rm d}y+\frac12 \Big\{\displaystyle\sum_{\alpha=1}^N\:\:\int_Y\Big(\tilde D_\alpha\nabla_y\omega_{i,\alpha}\cdot e_j + \tilde D_\alpha\nabla_y\omega_{j,\alpha}\cdot e_i\Big){\rm d}y\Big\}\\[0.2 cm] \displaystyle-\frac12\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\int_Y \varphi^_\alpha\varphi_\beta\Pi_{\alpha\beta}\Big(\omega_{i,\alpha} - \omega_{i,\beta}\Big)\Big(\omega_{j,\alpha} - \omega_{j,\beta}\Big){\rm d}y. \end{array} \end{equation} Plugging (\ref{eq:vf:cpb3}) in the symmetrized formula (\ref{eq:disp:sym}) leads to the desired equation (\ref{eq:disp}). Eventually the scalar homogenized equation (\ref{eq:hom}) has a unique solution since, by virtue of Lemma \ref{lem:disp}, the dispersion matrix is positive definite. This guarantees that the entire sequence $v^\varepsilon_\alpha$ converges to $v$, for $1\le\alpha\le N$, and not merely a subsequence as in Theorem \ref{thm:3:2scl}. \end{proof} \section{Adsorption in Porous Media} \label{sec:apm} In this section, we give a generalization of our previous result in a more applied context. Our goal is to upscale a model of multicomponent transport in an highly heterogeneous porous medium in presence of adsorption reaction at the fluid-pore interface. In \cite{AR:07}, the authors study the homogenization of one single scalar convection-diffusion-reaction equation posed in an $\varepsilon$-periodic infinite porous medium: \begin{equation} \label{eq:ar07} \left\{ \begin{array}{ll} \displaystyle\rho^\varepsilon\frac{\partial u^\varepsilon}{\partial t} + \frac{1}{\varepsilon}b^\varepsilon\cdot\nabla u^\varepsilon - {\rm div}(D^\varepsilon\nabla u^\varepsilon) + \frac{1}{\varepsilon^2} c^\varepsilon u^\varepsilon = 0 & \mbox{ in }(0,T)\times\Omega_\varepsilon,\\[0.2 cm] \displaystyle-D^\varepsilon\nabla u^\varepsilon\cdot n = \frac{1}{\varepsilon}\kappa u^\varepsilon & \mbox{ on }(0,T)\times\partial\Omega_\varepsilon. \end{array} \right. \end{equation} Typically, an $\varepsilon$-periodic infinite porous medium is built out of $\R^d$ ($d=2$ or $3$, being the space dimension) by removing a periodic distribution of solid obstacles which, after rescaling, are all similar to the unit obstacle $\Sigma^0$. More precisely, let $Y = ]0,1[^d$ be the unit periodicity cell. Let us consider a smooth partition $Y = \Sigma^0 \cup Y^0$ where $\Sigma^0$ is the solid part and $Y^0$ is the fluid part. The fluid part (extended by periodicity) is assumed to be a smooth connected open subset whereas no particular assumptions are made on the solid part. For each multi-index $j\in\mathbb{Z}^d$, we define $Y^j_\varepsilon := \varepsilon(Y^0+j)$, $\Sigma^j_\varepsilon := \varepsilon(\Sigma^0+j)$, $S^j_\varepsilon := \varepsilon(\partial\Sigma^0+j)$, the periodic porous medium $\Omega_\varepsilon := \displaystyle \cup_{j\in\mathbb{Z}^d} Y^j_\varepsilon$ and the $(d-1)-$dimensional surface $\partial\Omega_\varepsilon := \cup_{j\in\mathbb{Z}^d} S^j_\varepsilon$. In this section, we generalize the results of \cite{AR:07} to the multicomponent case. We consider the following weakly coupled cooperative parabolic system with Neumann boundary condition at the fluid-pore interface. \begin{equation} \label{eq:cdb} \left\{ \begin{array}{cc} \displaystyle\rho^\varepsilon_\alpha \frac{\partial u^\varepsilon_\alpha}{\partial t} + \frac1\varepsilon b^\varepsilon_\alpha\cdot\nabla u^\varepsilon_\alpha - {\rm div}(D^\varepsilon_\alpha\nabla u^\varepsilon_\alpha) = 0&\mbox{ in }(0,T)\times\Omega_\varepsilon,\\[0.2 cm] \displaystyle- D^\varepsilon_\alpha\nabla u^\varepsilon_\alpha\cdot n = \frac1{\varepsilon}\sum_{\beta=1}^N \Pi^\varepsilon_{\alpha\beta}u^\varepsilon_\beta &\mbox{ on }(0,T)\times\partial\Omega_\varepsilon,\\[0.2 cm] \displaystyle u^\varepsilon_\alpha(0,x) = u^{in}_\alpha(x)& \mbox{ in }\Omega_\varepsilon . \end{array} \right. \end{equation} \begin{remark} Note the different scaling in front of the surface reaction terms. It is of order $\varepsilon^{-1}$ because it balances a flux rather than a diffusive term, as in the previous model of Section \ref{sec:model}. As usual, by the change of variable $(\tau,y)\to(\varepsilon^{-2}t,\varepsilon^{-1}x)$ all singular powers of $\varepsilon$ disappears in (\ref{eq:cdb}) written in the $(\tau,y)$ variables. \end{remark} The hypotheses on the coefficients in (\ref{eq:cdb}) are exactly the same as in Section \ref{sec:model}. As before it is impossible to obtain uniform (in $\varepsilon$) estimates on the solutions $u^\varepsilon_\alpha$ of (\ref{eq:cdb}). As was done in Section \ref{sec:fp}, we employ the method of factorization by introducing a new unknown: $$ v^\varepsilon_\alpha(t,x)=\exp{(\lambda t/\varepsilon^2)}\frac{u^\varepsilon_\alpha(t,x)}{\varphi_\alpha\Big(\frac{x}{\varepsilon}\Big)}, $$ where $(\lambda,\varphi_\alpha)$ and $(\lambda,\varphi^_\alpha)$ are the principal eigenpairs associated with the (new) following spectral problems respectively: \begin{equation} \label{eq:scp:b} \left\{ \begin{array}{ll} \displaystyle b_\alpha(y) \cdot \nabla_y \varphi_\alpha - {\rm div}_y \Big(D_\alpha \nabla_y \varphi_\alpha\Big) = \lambda\rho_\alpha \varphi_\alpha & \mbox{ in } Y^0, \\[0.2 cm] \displaystyle- D_\alpha\nabla_y\varphi_\alpha\cdot n = \sum_{\beta=1}^N \Pi_{\alpha\beta} \varphi_\beta & \mbox{ on }\partial\Sigma^0,\\[0.2 cm] y \to \varphi_\alpha(y) & Y\mbox{-periodic.} \end{array} \right. \end{equation} \begin{equation} \label{eq:ascp:b} \left\{ \begin{array}{ll} \displaystyle -{\rm div}_y(b_\alpha \varphi^_\alpha) - {\rm div}_y \Big(D_\alpha \nabla_y \varphi^_\alpha\Big) = \lambda\rho_\alpha \varphi^_\alpha & \mbox{ in } Y^0, \\[0.2 cm] \displaystyle- D_\alpha\nabla_y\varphi^_\alpha\cdot n - b_\alpha(y)\cdot n \varphi^_\alpha = \sum_{\beta=1}^N \Pi^_{\alpha\beta} \varphi^_\beta & \mbox{ on }\partial\Sigma^0,\\[0.2 cm] y \to \varphi^_\alpha(y) & Y\mbox{-periodic.} \end{array} \right. \end{equation} Proposition \ref{prop:spec}, which guarantees the existence of principal eigenpairs for the spectral problems (\ref{eq:scp})-(\ref{eq:ascp}), carries over to the above spectral problems (\ref{eq:scp:b})-(\ref{eq:ascp:b}) as well. This is apparent from the proofs in \cite{Sw:92, MS:95}. The normalization (ensuring uniqueness of the eigenfunctions) that we choose is: $$ \displaystyle \sum_{\alpha=1}^N\: \int_{Y^0} \varphi_\alpha\varphi^_\alpha \rho_\alpha \,{\rm d}y = 1. $$ As in Section \ref{sec:fp} it is a matter of simple algebra to obtain the factorized system for (\ref{eq:cdb}) with the new unknown which is, for each $1 \le \alpha \le N$, \begin{equation} \label{eq:cdb1} \left\{ \begin{array}{ll} \displaystyle\varphi^\varepsilon_\alpha\varphi^{\varepsilon}_\alpha\rho^\varepsilon_\alpha\frac{\partial v^\varepsilon_\alpha}{\partial t} + \frac{1}{\varepsilon}\tilde b^\varepsilon_\alpha \cdot \nabla v^\varepsilon_\alpha - {\rm div}\left(\tilde D^\varepsilon_\alpha \nabla v^\varepsilon_\alpha\right) = 0 & \mbox{ in }(0,T)\times\Omega_\varepsilon,\\[0.2 cm] \displaystyle-\tilde D^\varepsilon_\alpha\nabla v^\varepsilon_\alpha\cdot n = \frac{1}{\varepsilon}\displaystyle\sum_{\beta=1}^N\:\:\Pi^\varepsilon_{\alpha\beta}\varphi^{\varepsilon}_\alpha\varphi^\varepsilon_\beta(v^\varepsilon_\beta-v^\varepsilon_\alpha) & \mbox{ on }(0,T)\times\partial\Omega_\varepsilon,\\[0.2 cm] \displaystyle v^\varepsilon_\alpha(0,x) = \frac{u^{in}_\alpha(x)}{\varphi_\alpha\Big(\frac{x}{\varepsilon}\Big)}& \mbox{ in } \Omega_\varepsilon, \end{array} \right. \end{equation} where the convective fields $\tilde b_\alpha$ and diffusion matrices $\tilde D_\alpha$ are given by the same formulae (\ref{eq:b1}) and (\ref{eq:D1}). A proof, completely similar to that of Lemma \ref{lem:apriori}, yields the following a priori estimates (\ref{eq:cdb1}): \begin{equation} \label{eq:cdb1:ap} \begin{array}{ll} \displaystyle\sum_{\alpha=1}^N\:\:\Big\\|v^\varepsilon_\alpha\Big\\|_{L^\infty((0,T);L^2(\Omega_\varepsilon))}+\sum_{\alpha=1}^N\:\:\Big\\|\nabla v^\varepsilon_\alpha\Big\\|_{L^2((0,T)\times\Omega_\varepsilon)}\\[0.4 cm] +\displaystyle\sum_{\alpha=1}^N\:\:\sum_{\beta=1}^N\:\:\sqrt{\varepsilon}\Big\\|\frac1\varepsilon(v^\varepsilon_\alpha-v^\varepsilon_\beta)\Big\\|_{L^2((0,T)\times\partial\Omega_\varepsilon)}\leq C\sum_{\alpha=1}^N\:\:\\|v^{in}_\alpha\\|_{L^2(\R^d)}. \end{array} \end{equation} \begin{remark} \label{rem:surf} Since the $(d-1)$ dimensional measure of the periodic surface $\partial\Omega_\varepsilon$ is of order $\mathcal O(\varepsilon^{-1})$, a bound of the type $\sqrt{\varepsilon}\\|z_\varepsilon\\|_{L^2(\partial\Omega_\varepsilon)}\le C$ means that the sequence $z_\varepsilon$ is bounded on the surface $\partial\Omega_\varepsilon$. \end{remark} In the a priori estimates (\ref{eq:cdb1:ap}), we have bounds in function spaces defined on the periodic surface $\partial\Omega_\varepsilon$. In order to speak of the convergence of sequences in such function spaces, we need to generalize the Definition \ref{def:2scl} of two-scale convergence with drift for periodic surfaces. This generalization was introduced in \cite{Hu:13}. We state this definition together with the corresponding compactness result (the proof of which is similar to that of Theorem 9.1 in \cite{Al:08}). \begin{lemma} \label{lem:2ds} Let $b^\in\R^d$ be a constant vector. Suppose that $u_\varepsilon(t,x)$ is a sequence of functions uniformly bounded in $L^2((0,T)\times\partial\Omega_\varepsilon)$ in the sense that $$ \sqrt{\varepsilon}\\|u_\varepsilon\\|_{L^2((0,T)\times\partial\Omega_\varepsilon)}\le C . $$ Then, there exists a subsequence, still denoted by $u_\varepsilon(t,x)$, and a function $u_0(t,x,y)\in L^2((0,T)\times\R^d\times \partial\Sigma^0)$ such that \begin{equation} \label{eq:defn2ds} \begin{array}{cc} \displaystyle\lim_{\varepsilon\to0}\varepsilon\int_0^T\int_{\partial\Omega_\varepsilon} u_\varepsilon(t,x)\phi\Big(t,x-\frac{b^}{\varepsilon}t,\frac{x}{\varepsilon}\Big){\rm d}\sigma_\varepsilon(x){\rm d}t \\[0.2 cm] \displaystyle= \int_0^T\int_{\R^d}\int_{\partial\Sigma^0} u_0(t,x,y)\phi(t,x,y){\rm d}\sigma(y){\rm d}x{\rm d}t , \end{array} \end{equation} for any function $\phi(t,x,y)\in C^\infty_c((0,T)\times\R^d;C^\infty_\#(Y))$. \end{lemma} In (\ref{eq:defn2ds}), ${\rm d}\sigma_\varepsilon(x)$ and ${\rm d}\sigma(y)$ denote the standard surface measures on $\partial\Omega_\varepsilon$ and $\partial\Sigma^0$ respectively. We denote this convergence on periodic surfaces in moving coordinates by $u_\varepsilon \stackrel{2s-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} u_0$. \begin{remark} \label{rem:2ds:1} Let $u_\varepsilon(t,x)$ be a sequence of functions defined on $(0,T)\times\Omega_\varepsilon$. Let $\gamma$ be the trace operator, i.e., $\gamma u = u\|_{\partial\Omega_\varepsilon}$. Suppose that we have a well-defined sequence of associated trace functions $\gamma u_\varepsilon(t,x)$ on $(0,T)\times\partial\Omega_\varepsilon$. If $u_\varepsilon\stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} u_0$ and $\gamma u_\varepsilon\stackrel{2s-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} v_0$ with the same drift velocity for both convergences, then $\gamma u_0 = v_0$ i.e., $\gamma u_0 = u_0\|_{\partial\Sigma^0}=v_0$ (see \cite{ADH} for details). In the sequel we systematically identify the ``bulk'' and ``surface'' two-scale limits. \end{remark} We now define the homogenized velocity which is chosen as the constant drift in the definition of two-scale convergence with drift: \begin{equation} \label{eq:cdb:drift} \displaystyle b^ = \sum_{\alpha=1}^N\,\int_{Y^0} \tilde b_\alpha(y){\rm d}y. \end{equation} \begin{theorem} \label{thm:cdb:hom} Let $(v^\varepsilon_\alpha)_{1\le\alpha\le N}\in L^2((0,T);H^1(\Omega_\varepsilon))^N$ be the sequence of solutions of (\ref{eq:cdb1}). Let $b^\in\R^d$ be given by (\ref{eq:cdb:drift}). There exist $v\in L^2((0,T);H^1(\R^d))$ and $\omega_{i,\alpha}\in H_\#^1(Y^0)$, for $1\le\alpha\le N$ and $1\le i\le d$, such that $v^\varepsilon_\alpha$ two-scale converges with drift $b^$, as $\varepsilon \to 0$, in the following sense: \begin{equation} \label{eq:3:brxn:2scl} \left\{ \begin{array}{ll} v^\varepsilon_\alpha \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} v,\\[0.2 cm] \displaystyle\nabla v^\varepsilon_\alpha \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} \nabla_x v + \nabla_y\Big(\sum_{i=1}^d \omega_{i,\alpha}\frac{\partial v}{\partial x_i}\Big),\\[0.3 cm] \displaystyle\frac{1}{\varepsilon}\Big(v^\varepsilon_\alpha-v^\varepsilon_\beta\Big) \stackrel{2s-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} \sum_{i=1}^d \Big(\omega_{i,\alpha} - \omega_{i,\beta}\Big)\frac{\partial v}{\partial x_i}, \end{array}\right. \end{equation} for every $1\le\alpha,\beta\le N$. The two-scale limit $v(t,x)$ in (\ref{eq:3:brxn:2scl}) satisfies the following homogenized equation: \begin{equation} \label{eq:3:brxn:hom} \left\{ \begin{array}{ll} \displaystyle\frac{\partial v}{\partial t} - {\rm div}({\mathcal D}\nabla v) = 0 & \textrm{in}\:\: (0,T)\times\R^d, \\[0.3cm] v(0,x) = \displaystyle\sum_{\alpha=1}^N u^{in}_\alpha(x) \displaystyle\int_{Y^0}\rho_\alpha(y) \varphi^_\alpha(y){\rm d}y & \textrm{in}\:\: \R^d, \end{array} \right. \end{equation} where the dispersion tensor $\mathcal D$ is given by \begin{equation} \label{eq:3:brxn:disp} \begin{array}{ll} {\mathcal D}_{ij}=\displaystyle\sum_{\alpha=1}^N\:\:\int_{Y^0}\tilde D_\alpha\Big(\nabla_y\omega_{i,\alpha} + e_i\Big)\cdot\Big(\nabla_y\omega_{j,\alpha} + e_j\Big){\rm d}y\\[0.5 cm] -\displaystyle\frac12\sum_{\alpha,\beta=1}^N\:\:\int_{\partial\Sigma^0} \varphi^_\alpha\varphi_\beta\Pi_{\alpha\beta}\Big(\omega_{i,\alpha} - \omega_{i,\beta}\Big)\Big(\omega_{j,\alpha} - \omega_{j,\beta}\Big){\rm d}\sigma(y) \end{array} \end{equation} and the components $(\omega_{i,\alpha})_{1\le\alpha\le N}$, for every $1\leq i\leq d$, are the solutions of the cell problems \begin{equation} \label{eq:3:brxn:cellpb} \left\{ \begin{array}{lll} \tilde b_\alpha(y)\cdot \Big(\nabla_y\omega_{i,\alpha} + e_i\Big) - {\rm div}_y \Big(\tilde D_\alpha \Big(\nabla_y\omega_{i,\alpha}+e_i\Big)\Big)= \varphi_\alpha\varphi^_\alpha\rho_\alpha b^\cdot e_i& \textrm{in}\:\: Y^0, \\[0.3cm] -\tilde D_\alpha(\nabla_y \omega_{i,\alpha}+e_i)\cdot n = \displaystyle\sum_{\beta=1}^N\: \Pi_{\alpha\beta} \varphi^_\alpha\varphi_\beta\Big(\omega_{i,\beta}-\omega_{i,\alpha}\Big) & \textrm{on} \: \: \partial\Sigma^0, \\[0.3cm] y \to \omega_{i,\alpha} & Y\textrm{-periodic.} \end{array} \right. \end{equation} \end{theorem} \begin{proof} As we have $L^2$ bounds on the solution sequence, we have the existence of a subsequence and a two-scale limit, say $(v_\alpha)_{1\le\alpha\le N}\in L^2((0,T);L^2(\R^d))^N$ such that \begin{equation} \label{eq:cpb:2scl} v^\varepsilon_\alpha\stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} v_\alpha \end{equation} for every $1\le\alpha\le N$. For $w\in H^1(\Omega_\varepsilon)$, consider the following Poincar\'e type inequality derived in \cite{Co:85}: \begin{equation} \label{eq:Co:poin} \\|w\\|^2_{L^2(\Omega_\varepsilon)}\le C\Big(\varepsilon^2\\|\nabla w\\|^2_{L^2(\Omega_\varepsilon)} + \varepsilon\\|w\\|^2_{L^2(\partial\Omega_\varepsilon)}\Big). \end{equation} Taking $\displaystyle w = \frac{1}{\varepsilon}\Big(v^\varepsilon_\alpha-v^\varepsilon_\beta\Big)$, we deduce from (\ref{eq:Co:poin}) and a priori estimates (\ref{eq:cdb1:ap}) that \begin{equation} \label{eq:cpb:diff} \displaystyle\sum_{\alpha=1}^d\sum_{\beta=1}^d\\|v^\varepsilon_\alpha-v^\varepsilon_\beta\\|_{L^2((0,T)\times\Omega_\varepsilon)}\le C\,\varepsilon. \end{equation} The above estimate (\ref{eq:cpb:diff}) implies that the limits obtained in (\ref{eq:cpb:2scl}) do match i.e., $v_\alpha=v$ for every $1\le\alpha\le N$. The $H^1$ a priori estimate in space as in (\ref{eq:cdb1:ap}) does imply that $v\in L^2((0,T);H^1(\R^d))$ and that there exist limits $v_{1,\alpha}\in L^2((0,T)\times\R^d;H^1_\#(Y^0))$ such that \begin{equation} \label{eq:cdb:corr} \nabla v^\varepsilon_\alpha \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} \nabla_x v + \nabla_y v_{1,\alpha} \end{equation} for every $1\le\alpha\le N$. In order to arrive at the two-scale limit of the coupled term on the boundary, we use Lemma \ref{lem:tech} below. Taking $\phi$ from (\ref{eq:tech:pb}) as the test function, consider the following expression with the coupled term: $$ \varepsilon \int_0^T\int_{\partial\Omega_\varepsilon} \frac{1}{\varepsilon}\Big( v^\varepsilon_\alpha - v^\varepsilon_\beta\Big) \phi\left(t,x-\frac{b^ t}{\varepsilon},\frac{x}{\varepsilon}\right) {\rm d}\sigma_\varepsilon(x) {\rm d}t $$ $$ = \int_0^T\int_{\Omega_\varepsilon} {\rm div}\left((v^\varepsilon_\alpha - v^\varepsilon_\beta) \theta\left(t,x-\frac{b^* t}{\varepsilon},\frac{x}{\varepsilon}\right)\right) {\rm d}x {\rm d}t $$ $$ = \int_0^T\int_{\Omega_\varepsilon}\Big(\nabla v^\varepsilon_\alpha - \nabla v^\varepsilon_\beta\Big) \cdot \theta\left(t,x-\frac{b^* t}{\varepsilon},\frac{x}{\varepsilon}\right){\rm d}x {\rm d}t $$ $$ + \int_0^T\int_{\Omega_\varepsilon}\Big(v^\varepsilon_\alpha - v^\varepsilon_\beta\Big) \left({\rm div}_x \theta\right)\left(t,x-\frac{b^*t}{\varepsilon},\frac{x}{\varepsilon}\right) {\rm d}x {\rm d}t $$ $$ \stackrel{2-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} \int_0^T\int_{\mathbb{R}^d}\int_{Y^0} \Big(\nabla_y v_{1,\alpha} - \nabla_y v_{1,\beta}\Big) \cdot \theta{\rm d}y {\rm d}x {\rm d}t $$ $$ = \int_0^T\int_{\R^d}\int_{\partial\Sigma^0} \Big( v_{1,\alpha} - v_{1,\beta}\Big) \theta \cdot n {\rm d}\sigma(y) {\rm d}x {\rm d}t = \displaystyle\int_0^T\int_{\R^d}\int_{\partial\Sigma^0} \Big( v_{1,\alpha} - v_{1,\beta}\Big) \phi {\rm d}\sigma(y) {\rm d}x {\rm d}t, $$ which implies that $$ \displaystyle\frac{1}{\varepsilon}\Big(v^\varepsilon_\alpha-v^\varepsilon_\beta\Big) \stackrel{2s-drift}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightharpoonup} v_{1,\alpha} - v_{1,\beta}\:\:\mbox{ for every }1\le\alpha,\beta\le N. $$ The rest of the proof is completely similar to the proof of Theorem \ref{thm:hom}. We safely leave it to the reader. \end{proof} We finish by stating a technical lemma which was useful in the proof of Theorem \ref{thm:cdb:hom}. \begin{lemma} \label{lem:tech} For a function $\phi(t,x,y) \in L^2((0,T)\times\mathbb{R}^d\times\partial\Sigma^0)$ such that \begin{equation} \label{eq:tech:comp} \int\limits_{\partial\Sigma^0} \phi(t,x,y) {\rm d}\sigma(y) = 0 \hspace{1 cm}\forall\:(t,x)\in(0,T)\times \mathbb{R}^d, \end{equation} there exists a vector field $\theta(t,x,y) \in L^2((0,T)\times\mathbb{R}^d;L^2_\#(Y^0))^d$ such that \begin{equation} \label{eq:tech:pb} \left\{ \begin{array}{ll} {\rm div}_y \theta = 0 & \textrm{in} \: \: Y^0,\\[0.2cm] \theta \cdot n = \phi & \textrm{on} \: \: \partial\Sigma^0,\\[0.2 cm] y \to \theta(t,x,y) & Y\mbox{-periodic.} \end{array}\right. \end{equation} \end{lemma} \begin{proof} Consider the following stationary diffusion problem posed in the unit cell: $$ \begin{array}{ll} \Delta_y\xi(y)=0 & \mbox{ in }Y^0,\\[0.2 cm] \nabla_y\xi\cdot n = \phi& \mbox{ on }\partial\Sigma^0, \end{array} $$ with $Y$-periodic boundary conditions and the Neumann data $\phi$ satisfying (\ref{eq:tech:comp}). The existence and uniqueness of $\xi\in H^1_{\#}(Y^0)/\R$ is guaranteed for the above problem as (\ref{eq:tech:comp}) is indeed the compatibility condition from the Fredholm alternative. Choosing $\theta=\nabla_y\xi$ gives one possible solution for (\ref{eq:tech:pb}). \end{proof} {\bf Acknowledgements.} This work was initiated during the PhD thesis of H.H. This work was partially supported by the GdR MOMAS from the CNRS. G.A. is a member of the DEFI project at INRIA Saclay Ile-de-France. H.H. acknowledges the support of the ERC grant MATKIT. \signga \signhh \end{document}	arXiv
Ergodic Theory and Dynamical Systems (2) ESAIM: Probability and Statistics (1) Highlights of Astronomy (1) Cambridge Planetary Science (1) Cambridge Studies in Nineteenth-Century Literature and Culture (1) 22 - Spectroscopy of Pluto and Its Satellites from Part IV - Applications to Planetary Surfaces By Dale P. Cruikshank, William M. Grundy, Donald E. Jennings, Catherine B. Olkin, Silvia Protopapa, Dennis C. Reuter, Bernard Schmitt, S. Alan Stern Edited by Janice L. Bishop, James F. Bell III, Arizona State University, Jeffrey E. Moersch, University of Tennessee, Knoxville Book: Remote Compositional Analysis The near-infrared reflectance spectra of Pluto and its satellites are rich with diagnostic absorption bands of ices of CH4, N2, CO, H2O, and an incompletely identified ammonia-bearing molecule. Following years of investigation of the spectra of Pluto and Charon with ground-based telescopes, NASA's New Horizons spacecraft obtained spectral maps of these bodies and three small satellites on its passage through the system on July 14, 2015, showing the distribution of these ices, as well as a colored, non-ice component. Spectral modeling mapped the distribution of the various ices and showed their abundance and mixing details in relationship to regions of differing surface elevation, albedo, and geologic structure. Additionally, owing to their greatly different degrees of volatility, the ices of Pluto are distributed in patterns responsive to Pluto's climatic changes on both short and long terms. The surface of Charon is dominated spectrally by H2O ice with one or more ammoniated compounds, and three of the four very small satellites show both H2O ice and the ammonia signature. By Theresa Jill Buckland, Elizabeth Edwards, Oliver Gaycken, John Holmes, Barbara Larson, Bernard Lightman, Sadiah Qureshi, Cannon Schmitt, Kirsten E. Shepherd-Barr, Carla Yanni, Bennett Zon Edited by Bernard V. Lightman, York University, Toronto, Bennett Zon, University of Durham Book: Evolution and Victorian Culture Print publication: 29 May 2014, pp xiii-xvi Genetic poverty of an extremely specialized wetland species, Nehalennia speciosa: implications for conservation (Odonata: Coenagrionidae) R. Bernard, T. Schmitt Journal: Bulletin of Entomological Research / Volume 100 / Issue 4 / August 2010 Oligo- and mesotrophic wetlands, such as bogs, fens and swamps, have become more and more restricted in Europe, and wetland species related to them have increasingly been threatened. Due to increasing habitat fragmentation, the exchange of individuals of these species among sites and, as a consequence, gene flow has been reduced or even eliminated. Therefore, we analysed the genetic structure of 11 populations of an endangered stenotopic damselfly, Nehalennia speciosa (Odonata: Coenagrionidae), in Poland and Lithuania by means of allozyme electrophoresis of 14 gene loci. The overall genetic diversity of all populations was low (A: 1.32; H: 2.6%; Ptot: 29.2%), and no significant differences were observed among the different groupings of populations (degree of fragmentation, habitat type and size, population size). The genetic differentiation among populations was also low (FST: 2.0%) and no regional groups were detected. A low degree of isolation by distance was observed for genetic distances. Taking into account these results, the conservation effort for this species should be focused on large local populations and not necessarily on metapopulation structures. Furthermore, N. speciosa could be (re-)introduced in extinct patches and seemingly suitable localities. Genetically, such relocations should be feasible due to the generally high genetic homogeneity of populations. Exponential inequalities for VLMC empirical trees Antonio Galves, Véronique Maume-Deschamps, Bernard Schmitt Journal: ESAIM: Probability and Statistics / Volume 12 / 2008 A seminal paper by Rissanen, published in 1983, introduced the class of Variable Length Markov Chains and the algorithm Context which estimates the probabilistic tree generating the chain. Even if the subject was recently considered in several papers, the central question of the rate of convergence of the algorithm remained open. This is the question we address here. We provide an exponential upper bound for the probability of incorrect estimation of the probabilistic tree, as a function of the size of the sample. As a consequence we prove the almost sure consistency of the algorithm Context. We also derive exponential upper bounds for type I errors and for the probability of underestimation of the context tree. The constants appearing in the bounds are all explicit and obtained in a constructive way. Infrared Spectroscopy of the Centaur Asbolus with the ESO-Very Large Telescope Catherine de Bergh, Maria-Antonietta Barucci, Aurélie Le Bras, Jennifer Romon, Bernard Schmitt, Jean-Gabriel Cuby Journal: Highlights of Astronomy / Volume 12 / 2002 As part of an observing programme with the ISAAC infrared spectrometer at the 8-m Antu telescope of the ESO-Very Large Telescope (Chile) devoted to the study of the surface composition of Kuiper Belt Objects and Centaurs, we obtained, in May 1999, spectra of the red Centaur Asbolus in the J, H and K bands. The spectra appear featureless. Conformal measures for multidimensional piecewise invertible maps JÉRÔME BUZZI, FRÉDÉRIC PACCAUT, BERNARD SCHMITT Journal: Ergodic Theory and Dynamical Systems / Volume 21 / Issue 4 / August 2001 Given a piecewise invertible map T:X\to X and a weight g:X\rightarrow\ ]0,\infty[, a conformal measure \nu is a probability measure on X such that, for all measurable A\subset X with T:A\to TA invertible, \nu(TA)= \lambda \int_{A}\frac{1}{g}\ d\nu with a constant \lambda>0. Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that \log g has bounded distortion and an irreducibility condition, we build such a conformal measure. Abnormal escape rates from nonuniformly hyperbolic sets VIVIANE BALADI, CHRISTIAN BONATTI, BERNARD SCHMITT Journal: Ergodic Theory and Dynamical Systems / Volume 19 / Issue 5 / October 1999 Consider a $C^{1+\epsilon}$ diffeomorphism $f$ having a uniformly hyperbolic compact invariant set $\Omega$, maximal invariant in some small neighbourhood of itself. The asymptotic exponential rate of escape from any small enough neighbourhood of $\Omega$ is given by the topological pressure of $-\log \|J^u f\|$ on $\Omega$ (Bowen–Ruelle in 1975). It has been conjectured (Eckmann–Ruelle in 1985) that this property, formulated in terms of escape from the support $\Omega$ of a (generalized Sinai–Ruelle–Bowen (SRB)) measure, using its entropy and positive Lyapunov exponents, holds more generally. We present a simple $C^\infty$ two-dimensional counterexample, constructed by a surgery using a Bowen-type hyperbolic saddle attractor as the basic plug.	CommonCrawl
\begin{document} \begin{comment} \pagestyle{empty} \thispagestyle{empty} \begin{center} \Huge \textsc{Friedrich-Schiller University}\\ \large \textsc{Faculty for Mathematics and Computer Science}\\ \textsc{Department of Analysis}\\ \Huge \textsc{Li-Yau Inequalities on \\ Finite Graphs \\[0.5\baselineskip]} \large \textsc{Master article for attainment of the academic degree of Master of Science (M. Sc.)}\\ \large \textsc{presented by}\\ \Large \textsc{ Florentin Münch}\\ {\normalsize \textsc{born in Gera, January 31, 1990}}\\ \large \textsc{primary evaluator: Dr. M. Keller \\ secondary evaluator: Prof. Dr. D. Lenz}\\ \textsc{Jena, \today}\\ \end{center} \end{comment} \maketitle \begin{abstract} \Abstract \end{abstract} \tableofcontents \pagestyle{plain} \section{Introduction} In 1986, Li and Yau proved a gradient estimate for manifolds, later known as Li-Yau inequality (cf. \cite{Li1986}). This states that for positive solutions $u$ to the heat equation $\mathcal L(u) = 0$ (with $\mathcal L = \Delta - \partial_t$) on a $d$-dimensional compact manifold without boundary, the following implication holds. \begin{equation} Ricc \geq 0 \Longrightarrow - \Delta \log u (x,t) \leq \frac d {2t} \label{LY86}. \end{equation} The term $Ricc$ denotes the Ricci curvature of the manifold. An important application of this inequality is the Harnack inequality (diffusion) which can be seen as an integrated form of the Li-Yau inequality. The different forms of the Harnack inequality (logarithmic, harmonic, diffusion) are closely related. In 2006, Bakry and Ledoux generalized the Li-Yau inequality result to more general Laplacians that satisfy the chain rule and improved the result by using a curvature-dimension inequality ($CD$-inequality) instead of the non-negative Ricci curvature and by giving a characterization of the $CD$-inequality via a logarithmic Sobolev inequality containing the Li-Yau inequality (cf. \cite{Bakry2006}). The $CD$-inequality will be introduced in the next subsection. After Li's and Yau's breakthrough in 1986, great effort was made to establish an analog result on graphs. This turned out to be very complicated since all known proofs of the Li-Yau inequality had made an extensive use of the chain rule, but this chain rule does not hold on graphs. As a first step to consider graphs with non-negative curvature, Chung and Yau introduced the concept of Ricci-flat graphs in 1996 (cf. \cite{Chung1996}) and obtained the following results for graphs. \begin{equation} d \mbox{-Ricci-flat} \Longrightarrow \mbox{Harnack inequality (logarithmic)}. \nonumber \end{equation} Ricci-flat graphs are a slight generalization of Abelian Cayley graphs (cf. Subsection \ref{SRF}). It remains unclear, whether there are Ricci-flat graphs which are not Abelian Cayley graphs. In 2012, Chung, Lin, and Yau found a first Li-Yau type result using the $CD(d,0)$ inequality on graphs (cf. \cite{Chung2012}). They showed for graphs \begin{eqnarray} d\mbox{-Ricci-flat} &\Longrightarrow& CD(d,0) \\ &\Longrightarrow& \mbox{ Harnack inequality (harmonic). } \nonumber \end{eqnarray} One year later, in 2013, Bauer, Horn, Lin, Lippner, Mangoubi, and Yau proved a result on graphs which is very similar to the original Li-Yau inequality from 1986 (cf. \cite{Bauer2013}). This result is the following. If $G$ is a graph and $u \in C^1( V \times \nnegR)$ is a positive solution to the heat equation on $G$, then \begin{eqnarray} d\mbox{-Ricci-flat} &\Longrightarrow& CDE(d,0) \\ &\Longrightarrow& \frac {\Gamma(\sqrt u)}{u} - \frac {\partial_t \sqrt u}{\sqrt u} \leq \frac d {2t} \label{BLY} \\ &\Longrightarrow& \mbox{Harnack inequality (diffusion)} . \nonumber \end{eqnarray} The $CDE(d,0)$ condition is the exponential curvature-dimension inequality, a substitute of $CD(d,0)$. The gradient form $\Gamma$ has been introduced by Bakry and Émery (cf. \cite{Bakry1985}). We will define this in the next subsection. There are many generalizations of this statement. In \cite{Bauer2013}, general curvature bounds and potentials have been discussed. In \cite{Qian2013}, the gradient estimate have been proven with time dependent coefficients. Obviously, the gradient estimate (\ref{BLY}) which we will call the $\sqrt{\cdot}$ Li-Yau inequality, has a different form than the logarithmic Li-Yau inequality (\ref{LY86}). Additionally, we can see in the following that in the $CDE$ inequality, there is a break of analogy to the original result on manifolds. In the examples section (Section \ref{CExamples}), we will explain this break of analogy and why it occurs. They overcome the missing chain rule in a remarkable way by observing that a version of the chain rule for the square root surprisingly also holds on graphs. In this article, we prove an inequality on graphs which has the same form as the logarithmic Li-Yau inequality (\ref{LY86}) from 1986. To do so, we introduce a new version of the $CD$ inequality to avoid the use of the chain rule (cf. Subsection \ref{SCD}). We will show \begin{eqnarray} d C_{\log} \mbox{-Ricci-flat} &\Longrightarrow& CD \log(d,0) \\ &\Longrightarrow& - \Delta \log u (x,t) \leq \frac d {2t} \label{logLY} \\ &\Longrightarrow& \mbox{Harnack inequality (diffusion)}, \nonumber \end{eqnarray} where $u \in C^1( V \times \nnegR)$ is a positive solution to the heat equation, and $C_{\log}$ is a positive constant. The $CD \log(d,0)$ inequality is a new curvature-dimension condition on graphs. If there is a chain rule for the Laplacian as in the case of manifolds, the $CD \log(d,0)$ inequality is equivalent to the $CD (d,0)$ inequality (cf. Subsection \ref{SMF}). In the case of graphs, the $CD \log(d,0)$ inequality implies the $CD (d,0)$ inequality (cf. Subsection \ref{SLT}). When establishing our results, we even introduce a more general concept by replacing the logarithm by a concave function $\psi \in C^1(\mathbb{R}^+)$. This leads us to the non-linear Laplacian $\Delta^\psi$ (cf. Subsection \ref{SDpsi}) as a substitution for the expression $\Delta \log (\cdot)$. With the choice $\psi=\sqrt{\cdot}$ respectively $\psi=\log$, we obtain the logarithmic respectively the $\sqrt{\cdot}$ Li-Yau inequality (cf. Example \ref{Elog}). As discussed above, one goal of this article is to use these Li-Yau estimates to deduce Harnack-inequalities which have the form \[ \frac{u(x,T_1)}{u(y,T_2)} \leq C(d(x,y),T_1,T_2) \] with a positive solution $u \in C^1( V \times \nnegR)$ to the heat equation, positive real numbers $T_1 < T_2$, and a constant $C(d(x,y),T_1,T_2)$ depending only on the distance of $(x,T_1)$ and $(y,T_2)$ in space-time. It turns out that the Harnack-estimates presented here are stronger than the ones of Bauer, Horn, Lin, Lippner, Mangoubi, and Yau in \cite{Bauer2013} (cf. Corollary \ref{CHR}). In this article, we use finite graphs as the basis of our discrete setting. In future work, we plan to also consider infinite graphs or even Dirichlet forms as a more general setting. {\bf Acknowledgements} I wish to thank Matthias Keller and Daniel Lenz, who have been the supervisors of my master thesis which is presented here in this article, for many useful discussions and for providing a very enjoyable and constructive atmosphere. Moreover, I wish to acknowledge Matthias Keller for proposing the topic of my master thesis. \section{Basics} We are interested in giving discrete analoga of the so called Li-Yau inequality on manifolds. \begin{defn}[Graph] A pair $G=(V,E)$ with a finite set $V$ and a relation $E \subset V\times V$ is called a \emph{finite graph} if $(v,v) \notin E$ for all $v \in V$ and if $(v,w) \in E$ implies $(w,v) \in E$ for $v,w \in V$. For $v,w \in V$, we write $v \sim w$ if $(v,w) \in E$. In this case, we say that the vertices $v$ and $w$ are \emph{adjacent}. For $v\in V$, we denote $\deg v := \#\{w \in V : w \sim v\}$. \end{defn} \begin{defn}[Path metric on graphs] Let $G=(V,E)$ be a finite graph. A sequence $(v_0,\ldots,v_n) \in V^{n+1}$ is called a \emph{path} of the length $n$ from $v_0$ to $v_n$ if all $v_i$ are pairwise distinct and $v_i \sim v_{i-1}$ for all $i \in \{1,\ldots,n\}$. We call $d:V \times V \to [0,\infty]$, \[ d(x,y):= \inf \{n \in {\mathbb{N}}: \mbox{there is a path of the length $n$ from $x$ to $y$} \} \] the \emph{distance} of $x$ and $y$. The finite graph $G$ is called \emph{connected} if there is a path from $x$ to $y$ for all $x,y \in V$ . \end{defn} A well studied Laplacian on manifolds is the Laplace-Beltrami operator. On graphs, there is an analogon of this operator. \begin{defn}[Laplacian $\Delta$] Let $G=(V,E)$ be a finite graph. The domain of the Laplacian $\Delta$ is \[ C(V):= {\mathbb{R}}^V := \{f: V \to {\mathbb{R}}\ \}. \] The \emph{Laplacian} $\Delta : C(V) \to C(V)$ is defined for $f \in C(V)$ and $v \in V$ as \[ \Delta f (v) := \sum_{w \sim v} (f(w) - f(v)). \] \end{defn} \begin{rem} Following the definition of Li and Yau in \cite{Li1986}, we deal with the Laplacian with a negative sign. \end{rem} We introduce the $\Gamma$-calculus by Bakry and Émery (cf. \cite{Bakry1985}). Note that such a calculus can be defined whenever a sufficiently nice Laplacian is given. Especially, it is useful on manifolds and on graphs. On manifolds, this calculus has been studied e.g. in \cite{Bakry1985, Bakry1992, Bakry2006}, and on graphs, it has been studied in \cite{Bauer2012,Bauer2013,Chung2012,Jost2014,Lin2010,Lin2011}. \begin{defn}[$\Gamma$-calculus] \label{defGamma} Let $G=(V,E)$ be a finite graph. Then, the \emph{gradient form} or \emph{carré du champ} operator $\Gamma : C(V) \times C(V) \to C(V)$ is defined by \[ 2 \Gamma (f,g) := \Delta(fg) - f\Delta g - g\Delta f. \] Similarly, the \emph{second gradient form} $\Gamma_2 : C(V) \times C(V) \to C(V)$ is defined by \[ 2 \Gamma_2 (f,g) := \Delta \Gamma (f, g) - \Gamma(f, \Delta g) - \Gamma (g, \Delta f). \] We write $\Gamma (f):= \Gamma (f,f)$ and $\Gamma_2 (f):= \Gamma_2 (f,f)$. \end{defn} On manifolds, there is a nice relation between its Ricci curvature and the second gradient form which is a consequence of Bochner's formula (cf. \cite{Bochner1953}). This relation is \[ \Gamma_2(f) \geq \frac 1 d (\Delta f)^2 + Ricc(\nabla f) \] for all $f$, where $d$ is the dimension of the manifold and $Ricc$ is the Ricci curvature. Especially, if $Ricc \geq 0$, then \[ \Gamma_2 (f) \geq \frac 1 d (\Delta f)^2. \] This motivates to introduce a curvature-dimension inequality on graphs, where no suitable explicit definition of the Ricci curvature is known yet. \begin{defn} We write $\mathbb{R}^+ := (0,\infty)$ and $\mathbb{R}^+_0 := [0,\infty)$. Let $G=(V,E)$ be a finite graph. Then, we write \[ C^+(V) := \{f:V \to \mathbb{R}^+\}. \] \end{defn} \begin{defn}[$CD(d,0)$ condition] \label{DCD} Let $G=(V,E)$ be a finite graph and $d \in \mathbb{R}^+$. We say $G$ satisfies the \emph{curvature-dimension inequality} $CD(d,0)$ if for all $f \in C(V)$, \[ \Gamma_2(f) \geq \frac 1 d (\Delta f)^2. \] We can interpret this as meaning that that the graph $G$ has a dimension (at most) $d$ and a non-negative Ricci curvature. \end{defn} \begin{rem} This curvature-dimension inequality has been studied on manifolds e.g. by Bakry and Émery in \cite{Bakry1985, Bakry1992, Bakry2006}. On graphs, it has been studied in \cite{Bauer2012,Bauer2013,Chung2012,Jost2014,Lin2010,Lin2011}. There is a great interest in giving generalizations of the Ricci curvature. Apart from the Bakry Émery approach, there is a well studied concept of Ricci curvature on metric measure spaces via optimal transport (cf. \cite{Lott2009,Ollivier2009,Ollivier2010,Sturm2006}). Connections between these different approaches are given in \cite{Ambrosio2012, Jost2014}. A notion of Ricci curvature on cell complexes by counting neighboring cells is presented in \cite{Forman2003}. \end{rem} \begin{defn}[$C^1(I)$ for intervals $I$] Let $I \subseteq {\mathbb{R}}$ be a (not necessarily open) interval and let $\phi : I \to {\mathbb{R}}$ be a function. We call $\phi$ (continuously) differentiable if $\phi$ can be extended to a (continuously) differentiable function on an open interval $J$ with $I \subset J \subset {\mathbb{R}}$. The derivative of $\phi$ is the derivative of that extension of $\phi$ restricted to $I$. We write \begin{eqnarray} C^1(I) &:=& \{\phi: I \to {\mathbb{R}} \; \| \; \phi \mbox { is continuously differentiable } \}, \\ C(I) &:=& \{\phi: I \to {\mathbb{R}} \; \| \; \phi \mbox { is continuous } \}. \end{eqnarray} \end{defn} We will be mostly interested in the case $I \in \{{\mathbb{R}}, \mathbb{R}^+, \mathbb{R}^+_0 \}$. Since the Li-Yau inequality deals with solutions to the heat equation, we define the heat operator. \begin{defn}[Heat operator $\mathcal L$] Let $G=(V,E)$ be a graph. The domain of the heat operator $\mathcal L$ is \[ C^1(V \times \posR) := \{u: V \times \posR \to {\mathbb{R}} \; \| \; u \mbox { is continuously differentiable in the second variable} \}. \] For $u \in C^1(V \times \posR)$ we write \[ u_t(v):=u(v,t) \] for all $v \in V$ and $t \in \posR$. We call $t \in \posR$ the \emph{time}, and we call $v \in V$ the \emph{location} respectively \emph{position}. The range of the heat operator is \[ C(V \times \posR) := \{u: V \times \posR \to {\mathbb{R}} \; \| \; u \mbox { is continuous in the second variable} \}. \] The \emph{heat operator} is defined by $\mathcal L(u) := \Delta u - \partial_t u$ for all $u \in C^1(V \times \posR)$. We call a function $u \in C^1( V \times \nnegR)$ a \emph{solution to the heat equation} on $G$ if $\mathcal L(u) = 0$. \end{defn} Later we will also use the heat operator on $C^1(V \times I)$ for an interval $I \subset {\mathbb{R}}$. The definition above can be extended to this case in a natural way. Most proofs of the Li-Yau inequality on manifolds make an extensive use of the chain rule for the Laplace-Beltrami operator given in the next proposition. \begin{proposition}[Chain rule on manifolds] Let $\Delta$ be the Laplace-Beltrami operator on a complete, connected manifold $M$ (with its Riemannian measure) and $\Phi \in C^\infty( \mathbb{R}^+)$ and $f,g \in C^\infty (M)$ with $f>0$. Then, the chain rules \begin{eqnarray} \Delta \Phi(f) &=& \Phi'(f) \Delta f + \Phi''(f) \Gamma (f) \label{CR}, \\ \Gamma(\Phi(f),g) &=& \Phi'(f) \Gamma (f,g) \label{CRG},\\ \Gamma(f,gh) &=& h \Gamma (f,g) + g \Gamma (f,h) \label{CRP} \end{eqnarray} are valid. \end{proposition} \begin{proof} A proof of this claim is given in \cite{Bakry1992}. \end{proof} \begin{rem} There is an established theory of more general Laplace operators which satisfy the three equations above. These operators are said to be diffusion operators (cf. \cite{Bakry1985, Bakry1992, Bakry2006}). \end{rem} Unfortunately, there is no chain rule for the Laplacian on graphs. So, we have to find a way to bypass the chain rule, as we want to prove the Li-Yau inequality. We will use a key identity for $\Gamma_2$ in the case of manifolds to introduce a new second gradient by which we can avoid the use of the chain rule. This identity states that for all solutions $u \in C^1( V \times \nnegR)$ to the heat equation, one has \[ \mathcal L(u \Delta (\log u)) = -2 u\Gamma_2 (\log u). \] Note that this statement is not true in general if there is no chain rule. But in the next section, we will define a scaling invariant Laplacian $\Delta^\psi$ and a second gradient $\Gamma_2^\psi$, such that \[ \mathcal L(u \Delta^\psi u)) = -2 u\Gamma_2^\psi (u) \] holds for all solutions $u \in C^1( V \times \nnegR)$ to the heat equation and no chain rule is needed. \section{The $\Gamma^\psi$-calculus} We start this section with a short summary of its contents: As the basis for our new calculus, we will introduce the operator $\Delta^\psi$ as a scaling invariant and non-linear generalization of the Laplacian with the parameter $$\psi \in C^1(\mathbb{R}^+) =\{\phi: \mathbb{R}^+ \to {\mathbb{R}} \mbox{ \| } \phi' \mbox{ is continuous} \}.$$ Using this, we introduce new gradients $\Gamma^\psi$ and $\Gamma_2^\psi$. Let $G=(V,E)$ be a finite graph. Our goal is to generalize the Li-Yau inequality to a $\psi$-Li-Yau inequality with the form \begin{equation} -\Delta^\psi u \leq \frac d {2t} \label{psiLY} \end{equation} for all positive solutions $u \in C^1( V \times \nnegR)$ to the heat equation which satisfy the $CD\psi(d,0)$ inequality which states \[ \Gamma_2^\psi (f) \geq \frac 1 d (\Delta^\psi f)^2 \] for all positive $f \in C(V)$. The second $\psi$-gradient $\Gamma_2^\psi$ will be defined in the $CD\psi$ subsection (Subsection \ref{SCD}). With appropriate choices of $\psi$, we obtain $\Delta^{\log} = \Delta {\log}$ and $\Delta^{\sqrt{\cdot}}(u)=\frac {\Gamma(\sqrt u)} u - \frac {\Delta u}{2u}$ for all positive functions $u$ (cf. Example \ref{Elog}). Especially with these two choices, the $\psi$-Li-inequality (\ref{psiLY}) turns into the logarithmic Li-Yau inequality (\ref{logLY}), respectively into the $\sqrt{\cdot}$ Li-Yau inequality (\ref{BLY}). We will introduce a first $\psi$-gradient in Subsection \ref{SGF}. We will need this to prove Harnack inequalities. In the manifolds subsection (Subsection \ref{SMF}) of this section, we will show among others that, in the case of manifolds for suitable $\psi$, one has \begin{eqnarray} \Delta^\psi &=& \Delta \log (\cdot),\\ \Gamma^\psi &=& \Gamma (\log (\cdot)),\\ \Gamma_2^\psi &=& \Gamma_2 (\log (\cdot)). \end{eqnarray} In the limit theorem subsection (Subsection \ref{SLT}), we will describe the classical operators as a limit of the $\psi$-operators. I.e. for all $f \in C(V)$ and all concave $\psi \in C^\infty(\mathbb{R}^+)$, one has \begin{eqnarray} \lim_{\varepsilon \to 0} \frac 1 {\varepsilon} \Delta^\psi (1+ \varepsilon f) &=& \psi'(1) \Delta f, \\ \lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \Gamma^\psi (1+ \varepsilon f) &=& -\psi''(1) \Gamma(f), \\ \lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \Gamma_2^\psi (1+ \varepsilon f) &=& -\psi''(1) \Gamma_2(f). \end{eqnarray} By this, we show that the $CD\psi$ inequality implies the $CD$ inequality. We will prove below our key identity \[ \mathcal L(u \Delta^\psi u)) = -2u\Gamma_2^\psi (u) \] which holds for all solutions $u \in C^1( V \times \nnegR)$ to the heat equation. In the examples section (Section \ref{CExamples}), we will show that Ricci-flat graphs satisfy the $CD\psi(d,0)$ inequality with appropriate $\psi \in C^1(\mathbb{R}^+)$ and $d \in \mathbb{R}^+$. We will see that concavity of the function $\psi$ is crucial for deriving Harnack inequalities (cf. Section \ref{CHarnack}) and for proving $CD\psi$ inequalities on Ricci-flat graphs (cf. Subsection \ref{SRF}). \subsection{The operator $\Delta^\psi$} \label{SDpsi} In this subsection, we will introduce the scaling invariant Laplacian $\Delta^\psi$ as replacement of $\Delta \circ \log$ which appeared in the manifold case \ref{LY86}. We will show in Subsection \ref{SMF} that both coincide in the case of manifolds if $\psi'(1)=1=-\psi''(1)$, as holds e.g. for $\psi = \log$. \begin{defn}[$\psi$-Laplacian $\Delta^\psi$] \label{DLP} Let $\psi \in C^1(\mathbb{R}^+)$ and let $G=(V,E)$ be a finite graph. Then, we call $\Delta^\psi : C^+(V) \to C(V)$, defined as \[ (\Delta^\psi f ) (v) := \left( \Delta \left[ \psi \left( \frac f {f(v)} \right) \right] \right) (v), \] the $\psi$\emph{-Laplacian}. \end{defn} We show that the operator $\Delta^\psi$ is scaling invariant in the argument and linear in the parameter $\psi$. In particular, $\Delta^\psi$ is not a linear operator. \begin{lemma} \label{Lcalc} Let $G=(V,E)$ be a graph, $\phi, \psi \in C^1(\mathbb{R}^+)$, $a,b \in {\mathbb{R}}$, $r \in \mathbb{R}^+$, and $f \in C^+(V)$. Then, \begin{enumerate}[(1)] \item $\Delta^{a \phi + b \psi} (rf) = a \Delta^\phi f + b \Delta^\psi f$, \item $\Delta^\psi f = 0$ if $f$ is constant or $\psi$ is constant. \end{enumerate} \end{lemma} \begin{proof} The first claim is a direct consequence of the definition of $\Delta^\psi$ and the linearity of $\Delta$. The second claim follows from the fact that the Laplacian vanishes on constant functions. \end{proof} With appropriate choices of $\psi$, the $\psi$-Li-Yau inequality (\ref{psiLY}) turns into the logarithmic Li-Yau inequality (\ref{logLY}) respectively into the $\sqrt{\cdot}$ Li-Yau inequality (\ref{BLY}). This is discussed next. \begin{example} \label{Elog} Let $G=(V,E)$ be a finite graph. (1) If $\psi = \log$, and $f \in C(V)$, and $v \in V$, then we get \begin{eqnarray} \Delta^{\log}f(v) &=& \Delta \left[ \log \left(\frac f {f(v)}\right)\right](v) = (\Delta \log f) (v) - \Delta (\log f(v)) \\ &=& (\Delta \log f) (v) \end{eqnarray} since $f(v)$ is a constant and the Laplacian vanishes on constants. Hence, we have \[ -\Delta^{\log}f = -\Delta \log f. \] This means, the $\psi$-Li-Yau inequality (\ref{psiLY}) turns into the logarithmic Li-Yau inequality (\ref{logLY}). (2) If $\psi = \sqrt{\cdot}$ and $u \in C^1( V \times \nnegR)$ is a positive solution to the heat equation on $G$ and $v \in V$, then \[ \Delta^{\psi}u(v) = \Delta \sqrt{\frac u {u(v)}} (v) = \frac { (\Delta \sqrt{u})(v)} {\sqrt{u(v)}} \] since the Laplacian is linear. Thus, \begin{eqnarray} -\Delta^{\psi}u &=& -\frac {\Delta \sqrt u} {\sqrt u} \\ &=& \frac {\Delta \left[{\sqrt u}^2\right] - 2 \sqrt u \Delta \sqrt{u} } {2u} - \frac {\Delta u} {2 u} \\ &=& \frac {\Gamma(\sqrt u)} u - \frac {\Delta u} {2 u} \\ &\stackrel{\mathcal L(u) =0}{=}& \frac {\Gamma(\sqrt u)} u - \frac {\partial_t u}{2u} \\ &=& \frac {\Gamma(\sqrt u)} u - \frac {\partial_t \sqrt {u}}{\sqrt{u}}. \end{eqnarray} This means, the $\psi$-Li-Yau inequality (\ref{psiLY}) turns into the $\sqrt{\cdot}$ Li-Yau inequality (\ref{BLY}). \end{example} \begin{rem} In the Example \ref{Elog}.(2), we have seen the chain rule for the square root on graphs which is a key identity in \cite{Bauer2013}. This chain rule states that on finite graphs $G=(V,E)$ for all $f \in C^+(V)$ \[ 2 \sqrt{f}\Delta \sqrt{f} = \Delta f - 2 \Gamma (\sqrt{f}). \] \end{rem} There is a useful representation of $\Delta^\psi$. \begin{lemma} [Representation of $\Delta^\psi$] \label{LRL} Let $\psi \in C^1(\mathbb{R}^+)$. Let $G=(V,E)$ be a finite graph, let $f \in C^+(V)$ and let $v \in V$. If $\psi(1)=0$, then \begin{equation} \Delta^\psi f (v) = \sum_{w \sim v} \psi \left( \frac {f(w)}{f(v)} \right). \label{repG} \end{equation} \end{lemma} \begin{proof} This claim is obvious, since $0 = \psi(1) = \psi \left( \frac {f(v)}{f(v)} \right)$ . \end{proof} \subsection{The $CD\psi$ condition and $\Gamma_2^\psi$} \label{SCD} In this subsection, we introduce the non bilinear operator $\Gamma_2^\psi$ and formulate the $CD\psi$ condition. Furthermore, we will prove our key identity $\mathcal L(u \Delta^\psi u)) = -2 u\Gamma_2^\psi (u)$ whenever $\mathcal L u = 0$. We will use this key identity to characterize the $CD\psi$ condition. \begin{defn}[Second $\psi$-gradient $\Gamma_2^\psi$] \label{DefG2} Let $\psi \in C^1(\mathbb{R}^+)$, and let $G=(V,E)$ be a finite graph. Then, we define $\Omega^\psi : C^+(V) \to C(V)$ by \[ (\Omega^\psi f ) (v) := \left( \Delta \left[ \psi' \left( \frac f {f(v)} \right) \cdot \frac f {f(v)} \left[ \frac{\Delta f} {f} - \frac{(\Delta f)(v)} {f(v)} \right] \right] \right) (v). \] Furthermore, we define the \emph{second} $\psi$-\emph{gradient} $\Gamma_2^\psi : C^+(V) \to C(V)$ by \[ 2 \Gamma_2^\psi (f) := \Omega^\psi f + \frac {\Delta f \Delta^\psi f} f - \frac {\Delta \left(f \Delta^\psi f\right)} f. \] \end{defn} In the next subsection, we will also define the first $\psi$-gradient $\Gamma^\psi$. But there is no obvious derivation of $\Gamma_2^\psi$ from $\Gamma^\psi$ in a similar way as $\Gamma_2$ derives from $\Gamma$ (see Definition \ref{defGamma}). Instead, one has a variant of our key identity, namely \[ \mathcal L\left(u \Gamma^\psi u \right) = 2 u \Gamma_2^\psi u , \quad \mbox{if } \mathcal L u =0. \] Moreover, in the case of manifolds, one has $\Gamma_2^\psi = \Gamma_2 (\log (\cdot))$ for suitable $\psi$. This result will be discussed in the manifolds subsection (Subsection \ref{SMF}) With the previous definition, we have all ingredients for defining the $CD\psi$ condition. \begin{defn}[$CD\psi$ condition] Let $G=(V,E)$ be a finite graph and $d \in \mathbb{R}^+$. We say $G$ satisfies the $CD\psi(d,0)$ \emph{inequality} if for all $f \in C^+(V)$, one has \begin{equation} \Gamma_2^\psi (f) \geq \frac 1 d (\Delta^\psi f)^2. \label{CDpsi} \end{equation} \end{defn} Now, we will give the key identity of the second $\psi$-gradient $\Gamma_2^\psi$. It will be used to show the analogy between the classical Bakry-Emery calculus and the $\Gamma^\psi$-calculus, and to characterize the validity of the $CD\psi$ inequality. \begin{lemma} [Representation of $\Gamma_2^\psi$] Let $\psi \in C^1(\mathbb{R}^+)$, let $I \subset {\mathbb{R}}$ be an interval, and let $t_0 \in I$. Let $G=(V,E)$ be a finite graph, $u : C^1(V \times I)$ a positive function with $\mathcal L(u) (\cdot,t_0)=0$ . Then \begin{equation} \mathcal L \left( - u \Delta^\psi u \right) (\cdot,t_0) = 2 u \Gamma_2^\psi (u) (\cdot,t_0). \label{repGG} \end{equation} \end{lemma} \begin{proof} All computations of the proof of the first claim are taking place at the time $t=t_0$. \begin{eqnarray} \mathcal L \left(-u \Delta^\psi u\right) &=& - \Delta \left(u \Delta^\psi u \right) + \partial_t \left(u \Delta^\psi u \right) \\ &=& - \Delta \left(u \Delta^\psi u \right) + \left( \Delta u \right) \Delta^\psi u + u \partial_t \Delta^\psi u .\\ \end{eqnarray} For all $v \in V$ we compute \begin{eqnarray} \partial_t \Delta^\psi u (v) &=& \Delta \partial_t \left[ \psi\left(\frac u {u(v)}\right)\right](v) \\ &=& \Delta \left[ \psi'\left(\frac u {u(v)}\right) \cdot \partial_t \left(\frac u {u(v)}\right) \right](v) \\ &=& \Delta \left[ \psi'\left(\frac u {u(v)}\right) \cdot \frac{u(v)\partial_t u - u \partial_t u(v)}{u(v)^2} \right](v) \\ &=& \Delta \left[ \psi'\left(\frac u {u(v)}\right) \cdot \frac u {u(v)} \cdot \left[\frac{\Delta u} {u} - \frac{\Delta u(v)} {u(v)} \right] \right](v) \\ &=& \Omega^\psi u (v). \end{eqnarray} Putting the two equalities above together results for all $v \in V$ in \[ \mathcal L \left(-u \Delta^\psi u\right) = - \Delta \left(u \Delta^\psi u \right) + \left(\Delta u \right) \Delta^\psi u + u \Omega^\psi u = 2u\Gamma_2^\psi (u). \] This finishes the proof. \end{proof} Next, we present a characterization of the $CD\psi$ condition which is the key to prove the $\psi$-Li-Yau inequality (\ref{psiLY}). \begin{theorem} [Characterization $CD\psi$] \label{tchar} Let $G=(V,E)$ be a finite graph, $\psi \in C^1(\mathbb{R}^+)$, and $d \in \mathbb{R}^+$. Then, the following statements are equivalent. \begin{enumerate}[(i)] \item \label{1} $G$ satisfies the $CD\psi(d,0)$ inequality. \item \label{2} For all positive solutions $u \in C^1( V \times \nnegR)$ to the heat equation on $G$, one has \begin{equation} \mathcal L \left( -u \Delta^\psi u \right) \geq \frac 2 d u \left( \Delta^\psi u \right)^2. \label{CCD} \end{equation} \end{enumerate} \end{theorem} This characterization shows the connection between the time-independent $CD\psi$ condition and its consequences for solutions to the heat equation. \begin{proof} The implication $\emph{(\ref{1})} \Rightarrow \emph{(\ref{2})}$ is a direct consequence of the identity (\ref{repGG}) from the representation lemma of $\Gamma_2^\psi$, i.e. \[ \mathcal L \left( -u \Delta^\psi u \right) \stackrel{(\ref{repGG})}{=} 2 u \Gamma_2^\psi (u) \stackrel{\emph{(\ref{1})}}{\geq} \frac 2 d u \left( \Delta^\psi u \right)^2. \] Next, we show $\emph{(\ref{2})} \Rightarrow \emph{(\ref{1})}$. Given $f \in C^+(V)$, there is a positive solution $u \in C^1( V \times \nnegR)$ to the heat equation on $G$, such that $u(\cdot,0) = f$. Such a solution $u$ can be obtained by \[ u_t := e^{\Delta t} f := \sum_{k=0}^{\infty} \frac {t^k \Delta^k f}{k!} \] for $x \in V$ and $t \geq 0$. This statement will be proved in Proposition \ref{SG}. We use the continuity of $u$ and $\Delta^\psi$ and $\Gamma_2^\psi$, and we apply (\ref{repGG}) to get \begin{eqnarray} 2f \Gamma_2^\psi (f) &=& \lim_{t {\searrow} 0} 2u \Gamma_2^\psi (u)(\cdot,t) \stackrel{(\ref{repGG})}{=} \lim_{t {\searrow} 0} \mathcal L \left( -u \Delta^\psi u \right) (\cdot,t) \\ &\stackrel{\emph{(\ref{2})}}{\geq}& \lim_{t {\searrow} 0} \frac 2 d u \left( \Delta^\psi u \right)^2 (\cdot,t) = \frac 2 d f \left( \Delta^\psi f \right)^2. \end{eqnarray} This finishes the proof. \end{proof} \subsection{The gradient form $\Gamma^\psi$} \label{SGF} In this subsection, we introduce the gradient form $\Gamma^\psi$. This is not necessary to understand the $CD \psi$ condition and the $\psi$-Li-Yau inequality. But it enables us to formulate the $\psi$-Li-Yau inequality as a gradient estimate. This formulation is crucial to derive Harnack inequalities. \begin{defn} For $\psi \in C^1(\mathbb{R}^+)$, we define \[ \overline{\psi}(x):= \psi'(1)\cdot(x-1) - (\psi(x) - \psi(1)). \] \end{defn} If $\psi$ is a concave function, then obviously, one has $\overline{\psi}(x) \geq 0$ for all $x>0$. We can see $\overline{\psi}(1)=0$. Hence, if $\psi$ is concave and if $G=(V,E)$ is a finite graph, then for every $v_0 \in V$ and every positive $f \in C^+(V)$, the function $V \to {\mathbb{R}}$ with \[ v \mapsto \overline{\psi} \left( \frac {f(v)}{f(v_0)} \right) \] has a minimum in $v=v_0$. Consequently \[ \Delta^{\overline{\psi}} f(v_0) = \Delta \overline{\psi} \left( \frac {f}{f(v_0)} \right) \geq 0. \] This motivates the following definition. \begin{defn}[$\psi$-gradient $\Gamma^\psi$] Let $\psi \in C^1(\mathbb{R}^+)$ be a concave function and let $G=(V,E)$ be a finite graph. We define the $\psi$-\emph{gradient} as $\Gamma^\psi : C^+(V) \to C(V)$, \[ \Gamma^\psi := \Delta^{\overline{\psi}}. \] \end{defn} \begin{rem} By using this, we can also introduce a $CD\psi$ inequality for a dimension $d>0$ with a non-zero curvature bound $K \in {\mathbb{R}}$ denoted by $CD\psi(d,K)$ via the inequality \[ \Gamma_2^\psi( f) \geq \frac 1 d \left(\Delta^\psi f \right)^2 + K\Gamma^\psi( f) \quad \mbox{for all } f \in C^+(V). \] In this paper, we focus on the case $K=0$. \end{rem} In the following lemma, we give a representation of $\Delta^\psi$ which allows us to understand the $\psi$-Li-Yau inequality as a gradient estimate. This lemma will be used in Section \ref{CHarnack} to derive Harnack inequalities. \begin{lemma}[Gradient representation of $\Delta^\psi$] \label{LGR} Let $G=(V,E)$ be a finite graph and let $\psi \in C^1({\mathbb{R}})$ be a concave function. Then for all $f \in C^+(V)$, one has \begin{equation} -\Delta^\psi f = \Gamma^\psi (f) - \psi'(1) \frac {\Delta f} f. \label{EGamma} \end{equation} \end{lemma} \begin{proof} The claim follows immediately from Lemma \ref{Lcalc} and from the definition of $\Delta^\psi$ (Definition \ref{DLP}). \end{proof} \subsection{The $\Gamma^\psi$-calculus on manifolds} \label{SMF} In this subsection, we will see that the new $\Gamma^\psi$ calculus can be transferred to the setting of manifolds. Our goal is to show that in this setting, the $\Gamma^\psi$ calculus coincides with the common $\Gamma$-calculus by Bakry and Émery (cf. \cite{Bakry1985,Bakry1992,Bakry2006}). For basics on Riemannian manifolds, we refer the reader to \cite{Spivak1965}. \begin{defn}[$\psi$-Laplacian on manifolds] Let $\psi \in C^1(\mathbb{R}^+)$ and let $\Delta$ be the Laplace-Beltrami operator on a complete, connected Riemannian manifold $M$. Then, we call $\Delta^\psi : C_+^\infty(M) \to C^\infty(M)$ with \[ (\Delta^\psi f ) (v) := \left( \Delta \left[ \psi \left( \frac f {f(v)} \right) \right] \right) (v) \] the $\psi$\emph{-Laplacian}. \end{defn} Analogously, we can transfer the definitions from the previous three subsections to the setting of manifolds. By doing this transfer, the equations (\ref{repGG}) and (\ref{EGamma}) remain valid also on manifolds. The proofs are analogous to the ones in the graph case. But in contrast to the graph case, the $\Gamma^\psi$-calculus on manifolds only depends on $\psi'(1)$ and $\psi''(1)$. We will discuss this in the following representation theorem. \begin{theorem} [Representation of the $\psi$-operators] \label{TPO} Let $\psi \in C^1(\mathbb{R}^+)$. Let $\Delta$ be the Laplace-Beltrami operator on a complete, connected Riemannian manifold $M$ and $\psi \in C^\infty(\mathbb{R}^+)$. Then for all positive $f \in C^\infty(M)$, one has \begin{eqnarray} \Delta^\psi f &=& \psi'(1) \frac{\Delta f}{f} + \psi''(1) \frac{\Gamma (f)}{f^2}, \label{TMP}\\ \Gamma^\psi &=& -\psi''(1) \Gamma (\log (\cdot)), \label{clGa} \\ \Gamma_2^\psi &=& -\psi''(1) \Gamma_2 (\log (\cdot)). \label{TM2} \end{eqnarray} \end{theorem} \begin{proof} Let $f \in C^\infty(M)$ and $x \in M$. First, we prove identity (\ref{TMP}). We use the chain rule (\ref{CR}) for \[ \Phi(s) = \psi \left( \frac s {f(x)} \right), \qquad s>0 \] to get \begin{eqnarray} \Delta^\psi f (x) &=& (\Delta \Phi(f))(x) \stackrel{(\ref{CR})}{=} \Phi'(f(x)) (\Delta f) (x) + \Phi''(f(x)) (\Gamma (f)) (x) \\ &=& \frac {\psi'(1)}{f(x)} (\Delta f) (x) + \frac{\psi''(1)}{(f(x))^2} (\Gamma (f)) (x) \\ &=& \left[ \psi'(1) \frac {\Delta f}{f} + \psi''(1) \frac{\Gamma (f)}{f^2} \right] (x). \end{eqnarray} Since $f$ and $x$ are arbitrary, the claim follows immediately. Next, we show identity (\ref{clGa}). Since equation (\ref{EGamma}) remains valid on manifolds, we can calculate \[ \Gamma^\psi (f) \stackrel{(\ref{EGamma})}{=} \psi'(1) \frac {\Delta f} f -\Delta^\psi f = - \psi''(1) \frac{\Gamma (f)}{f^2} \stackrel{(\ref{CRG})}{=} -\psi''(1) \Gamma (\log f) \] by using the already proven first statement of the present theorem and the chain rule (\ref{CRG}). Finally, we prove identity (\ref{TM2}) in two steps. In the first step, we show $\Gamma_2^{\log} (f) = \Gamma_2 (\log f)$, and in the second step, we show $\Gamma_2^\psi = -\psi''(1)\Gamma_2^{\log}$. We start with $\Gamma_2^{\log} (f) = \Gamma_2 (\log f)$. For all $x \in M$, we obtain \begin{eqnarray} (\Omega^{\log} f ) (x) &=& \left( \Delta \left[ {\left( \frac f {f(x)} \right)}^{-1} \cdot \frac f {f(x)} \left[ \frac{\Delta f} {f} - \frac{(\Delta f)(x)} {f(x)} \right] \right] \right) (x) \\ &=& \left( \Delta \left[ \frac{\Delta f} {f} - \frac{(\Delta f)(x)} {f(x)} \right] \right) (x) \\ &=& \left( \Delta \left[ \frac{\Delta f} {f} \right] \right) (x). \\ \end{eqnarray} This implies \[ \Omega^{\log} f = \Delta \left(\frac{\Delta f}{f} \right). \] Now, we use the chain rules \begin{eqnarray} \Gamma(\log f,g) &=& \frac {\Gamma(f,g)} f, \label{crl1} \\ \Delta \log f &=& \frac {\Delta f} f - \frac {\Gamma(f)} {f^2} \label{crl2} \end{eqnarray} to obtain \begin{eqnarray} 2\Gamma_2(\log f) &=& \Delta \Gamma (\log f) - 2 \Gamma (\log f, \Delta \log f) \\ &\stackrel{(\ref{crl1})}{=}& \Delta \left( \frac{\Gamma(f)}{f^2} \right) - \frac 2 f \Gamma(f,\Delta \log f) \\ &=& \Delta \left( \frac{\Gamma(f)}{f^2} \right) - \frac 1 f [\Delta(f \Delta \log f) - f \Delta \Delta \log f - \Delta f \Delta \log f] \\ &=& \Delta \left[ \Delta \log f + \frac{\Gamma(f)}{f^2} \right]- \frac{\Delta(f \Delta \log f)} f + \frac {\Delta f \Delta \log f} f \\ &\stackrel{(\ref{crl2})}{=}& \Delta \left( \frac{\Delta f} {f} \right)- \frac{\Delta(f \Delta \log f)} f + \frac {\Delta f \Delta \log f} f \\ &=& \Omega^{\log} f - \frac{\Delta(f \Delta \log f)} f + \frac {\Delta f \Delta \log f} f \\ &=& 2 \Gamma_2^{\log} (f). \end{eqnarray} Before we prove the second step, we establish the equation \begin{equation} \mathcal L\left(u \Gamma^\psi u \right)=\mathcal L\left( -u \Delta^\psi u \right) = 2u\Gamma_2^\psi u \label{ErepG} \end{equation} for all positive solutions $u$ to the heat equation on $M \times \mathbb{R}^+_0$. The proof of the second identity is analogous to the proof of the key identity (\ref{repGG}) on graphs. For showing the first identity, we write \begin{eqnarray} \mathcal L \left( -u \Delta^\psi u \right) &=& \mathcal L\left[ u \left( \Gamma^\psi (u) - \psi'(1) \frac {\Delta u} u \right) \right] \\ &=& \mathcal L\left(u \Gamma^\psi u \right)- \psi'(1) \mathcal L(\Delta u) \\ &=& \mathcal L\left(u \Gamma^\psi u \right). \end{eqnarray} We prove $\Gamma_2^\psi (f) = \Gamma_2^{\log} (f)$. We will use the already proven second claim (\ref{clGa}) of the present theorem, and we will use the identity (\ref{ErepG}) from above. Therefore, we extend the positive function $f \in C^\infty(M)$ to a positive function $u \in C^\infty(M \times {\mathbb{R}})$, such that $u(\cdot,0) = f$ and $\mathcal L(u) (\cdot,0) = 0$. We can do this by \[ u(\cdot,t) := f \cdot \exp \left( \frac{\Delta f}{f} t \right). \] At the time $t=0$, we calculate \begin{eqnarray} 2 f \Gamma_2^\psi (f) &= & 2 u \Gamma_2^\psi (u) \stackrel{(\ref{ErepG})} {=} \mathcal L \left( u \Gamma^\psi u \right) \stackrel{(\ref{clGa})} {=} -\psi''(1) \cdot \mathcal L \left( u \Gamma^{\log} u \right) \\ &\stackrel{(\ref{ErepG})} {=}& -\psi''(1) \cdot 2 u \Gamma_2^{\log} (u) = -\psi''(1)\cdot 2 f \Gamma_2^{\log} (f). \end{eqnarray} This finishes the proof, since $f$ is an arbitrary positive function. \end{proof} \begin{rem} In the proof of \cite[Lemma 3.12]{Bauer2013}, similar computations can be found to show the connection between their $CDE$ condition and the $CD$ condition. \end{rem} With appropriate $\psi$, the representation of the $\psi$-operators simplifies to the following corollary. \begin{corollary}[Coincidence of the $\psi$-operators with the $\Gamma$ calculus] Let $\psi \in C^\infty$ with $\psi'(1) = 1 = - \psi''(1)$, and let $\Delta$ be the Laplace-Beltrami operator on a complete, connected Riemannian manifold $M$. Then for all positive $f \in C^\infty(M)$, one has \begin{eqnarray} \Delta^\psi f \;=& \Delta^{\log} f &=\; \Delta \log f, \label{Apsi} \\ \Gamma^\psi (f) \;=& \Gamma^{\log} (f) &=\; \Gamma (\log f), \\ \Gamma_2^\psi (f) \;=& \Gamma_2^{\log} (f) &=\; \Gamma_2 (\log f). \end{eqnarray} \end{corollary} This corollary shows that our replacement $\Delta \log (\cdot) \leadsto \Delta^\psi$ is consistent. Moreover, we can see the equivalence of the $CD$ condition and the $CD \psi$ condition on manifolds. \begin{proof} If we choose $\psi=\log$, then we obtain $\psi'(1) = 1 = - \psi''(1)$. Thus, the claim is an easy consequence of the representation of the $\psi$-operators (Theorem \ref{TPO}). \end{proof} \subsection{A limit theorem on graphs} \label{SLT} In this subsection, we will interpret the classical operators $\Delta, \Gamma, \Gamma_2$ as directional derivatives of the corresponding $\psi$-operators. By this, we can show that the $CD\psi$ condition implies the $CD$ condition on graphs. \begin{theorem} [Limit of the $\psi$-operators] Let $G=(V,E)$ be a finite graph. Then for all $f \in C(V)$, one has the pointwise limits \begin{eqnarray} \lim_{\varepsilon \to 0} \frac 1 {\varepsilon} \Delta^\psi (1+ \varepsilon f) \; =& \psi'(1) \Delta f &\quad \mbox{for } \psi \in C^1(\mathbb{R}^+), \label{ELL}\\ \lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \Gamma^\psi (1+ \varepsilon f) \; =& -\psi''(1) \Gamma(f) &\quad \mbox{for } \psi \in C^2(\mathbb{R}^+), \label{EL1}\\ \lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \Gamma_2^\psi (1+ \varepsilon f) \; =& -\psi''(1) \Gamma_2(f) &\quad \mbox{for } \psi \in C^2(\mathbb{R}^+). \label{EL2} \end{eqnarray} Since all $f \in C(V)$ are bounded, one obviously has $1+ \varepsilon f> 0$ for small enough $\varepsilon>0$. \end{theorem} \begin{rem} The above limits can be understood as directional derivatives of the $\psi$-operators to the direction $f$ at the constant function $1$. For a function $H : C(V) \to C(V)$ and for $f,g \in C(V)$ and for $x \in V$, we can define $h : {\mathbb{R}} \to {\mathbb{R}}, \; t \mapsto H(g + tf)(x)$ and the directional derivatives $\partial_f H, \partial^{2}_f H: C(V) \to C(V)$ via $ \big[\partial_f H (g)\big](x) = h'(0)$ and $ \big[\partial_f^2 H (g)\big](x) = h''(0)$. By using this notation, for $\psi \in C^2(\mathbb{R}^+)$ with $\psi'(1)\neq 0 \neq \psi''(1)$, the above theorem can be written as \begin{eqnarray} \Delta f &=& \left[ \partial_f \Delta^\psi \right] (1) / \psi'(1) ,\\ \Gamma(f) &=& -\frac 1 2 \left[ \partial^2_f \Gamma^\psi \right] (1) / \psi''(1) ,\\ \Gamma_2(f) &=& -\frac 1 2 \left[ \partial^2_f \Gamma_2^\psi \right] (1) / \psi''(1) . \end{eqnarray} \end{rem} \begin{proof} First, we prove (\ref{EL1}). Let $x \in V$, let $\psi \in C^2(\mathbb{R}^+)$ and let $f \in C(V)$. By the definition of $\Gamma^\psi$, we can write \[ \frac 1 {\varepsilon^2} \Gamma^{\psi}(1+ \varepsilon f)(x) = \Delta \frac 1 {\varepsilon^2} \overline{\psi} \left( \varepsilon \cdot \frac {f-f(x)}{1+\varepsilon f(x)} + 1 \right)(x). \] We recall $\overline{\psi}(t) = \psi'(1) (t-1) - \psi(t) $. W.l.o.g., $\psi(1)=0$ and, hence, $\overline{\psi}(1)=0=\overline{\psi}'(1)$ and $\overline{\psi}''(1)=-\psi''(1)$. We apply Taylor's theorem to $\overline{\psi}$ at point $1$ to obtain \begin{equation} \overline{\psi} \left( \varepsilon \cdot \frac {f-f(x)}{1+\varepsilon f(x)} + 1 \right) = \left(\frac 1 2 \overline{\psi}''(1)+ h(t_\varepsilon) \right) t_\varepsilon^2 \end{equation} with $t_\varepsilon = \varepsilon \cdot \frac {f-f(x)}{1+\varepsilon f(x)}$ and a function $h: {\mathbb{R}} \to {\mathbb{R}}$ with $\lim_{s\to 0} h(s) = 0$. Thus, we can calculate the pointwise limit \[ \lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \overline{\psi} \left( \varepsilon \cdot \frac {f-f(x)}{1+\varepsilon f(x)} + 1 \right) = \frac 1 2 \overline{\psi}''(1) (f-f(x))^2. \] Since we are on finite graphs, we can do the following computation at the point $x \in V$: \begin{eqnarray} \lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \Gamma^\psi (1+ \varepsilon f) &=& \Delta \lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \overline{\psi} \left( \varepsilon \cdot \frac {f-f(x)}{1+\varepsilon f(x)} + 1 \right)\\ &=& \Delta \frac 1 2 \overline{\psi}''(1) (f-f(x))^2\\ &=& -\psi''(1) \Gamma(f). \end{eqnarray} Next, we prove (\ref{ELL}) similarly. Let $f \in C(V)$ and let $\psi \in C^1(\mathbb{R}^+)$. Since $\overline{\psi}(1)=0$, we observe that \[ \frac 1 \varepsilon \Gamma^\psi (1+ \varepsilon f) (x) = \Delta \frac 1 \varepsilon \overline{\psi} \left(\frac{1+ \varepsilon f}{1+ \varepsilon f(x)}\right) (x) = \Delta \frac 1 \varepsilon \left[ \overline{\psi} \left( \varepsilon \cdot \frac {f-f(x)}{1+\varepsilon f(x)} + 1 \right) -\overline{\psi}(1) \right](x) \] for all $x \in V$. Due to Taylor's theorem of the first order and since $\overline{\psi}'(1)$=0, we can compute the pointwise limit \[ \lim_{\varepsilon \to 0} \frac 1 {\varepsilon} \overline{\psi} \left( \varepsilon \cdot \frac {f-f(x)}{1+\varepsilon f(x)} + 1 \right) = \overline{\psi}'(1) (f-f(x)) = 0. \] Hence, we have $\frac 1 \varepsilon \Gamma^\psi (1+ \varepsilon f) \stackrel{\varepsilon \to 0}{\longrightarrow} 0$. We use the gradient representation of $\Delta^\psi$ (cf. Lemma~\ref{LGR}) to obtain \begin{eqnarray} \frac 1 \varepsilon \Delta^\psi (1+ \varepsilon f) &=& \frac 1 \varepsilon \psi'(1) \frac {\Delta (1+ \varepsilon f)}{1+ \varepsilon f} - \frac 1 \varepsilon \Gamma^\psi (1+ \varepsilon f) \\ &=& \psi'(1) \frac {\Delta f}{1+ \varepsilon f} - \frac 1 \varepsilon \Gamma^\psi (1+ \varepsilon f)\\ &\stackrel{\varepsilon \to 0}{\longrightarrow}& \psi'(1) {\Delta f}. \end{eqnarray} Finally, we prove (\ref{EL2}). Let $x \in V$, let $\psi \in C^2(\mathbb{R}^+)$ and let $g \in C^+(V)$. Again, we use the gradient representation of $\Delta^\psi$ (cf. Lemma \ref{LGR}) and calculate \begin{equation} \Delta(g \Delta^\psi g) = \Delta (\psi'(1) \Delta g) - \Delta \left( g \Gamma^\psi (g) \right). \label{LgGg} \end{equation} We recall the non-linear operator $\Omega^\psi$ from the definition of $\Gamma_2^\psi$ (cf. Definition \ref{DefG2}). We can write \begin{eqnarray} \left[g\Omega^\psi(g)\right](x) &=& \left[ g \Delta \left[ \psi'\left(\frac g {g(x)}\right) \frac g {g(x)} \cdot \left( \frac {\Delta g} g - \frac {\Delta g (x)}{g(x)} \right) \right] \right](x) \nonumber\\ &=& \left[ \Delta \left[ \psi'\left(\frac g {g(x)}\right) \Delta g \right] - \Delta g \cdot \Delta \left[ \psi'\left(\frac g {g(x)}\right) \frac g {g(x)} \right] \right](x). \label{gOg} \end{eqnarray} We define $\nu(t)= \psi'(t)-\psi'(1)$ and $\omega(t)= \psi(t)-t\psi'(t)$ for $t \in \mathbb{R}^+$. By using the linearity of $\Delta^\psi$ in the parameter and by resolving $g\Omega^\psi(g)$ with (\ref{gOg}) and $\Delta(g \Delta^\psi g)$ with (\ref{LgGg}), we obtain \begin{eqnarray} \left[ 2g \Gamma_2^\psi(g) \right](x) &=& \left[ g\Omega^\psi(g) + \Delta g \Delta^\psi g - \Delta(g \Delta^\psi g) \right](x) \\ &=& \left[ \Delta\left[\nu \left( \frac g {g(x)} \right) \Delta g \right] + \Delta g \Delta^\omega g + \Delta \left( g \Gamma^\psi (g) \right) \right](x). \end{eqnarray} Let $f \in C(V)$. We set $g = g_\varepsilon = 1+\varepsilon f$ and get \[ \left[ \frac {2g_\varepsilon} {\varepsilon^2} \Gamma_2^\psi(1+ \varepsilon f) \right](x) = \left[ \frac 1 {\varepsilon^2} \Delta\left[\nu \left( \frac {g_\varepsilon} {g_\varepsilon(x)} \right) \Delta g_\varepsilon \right] + \frac 1 {\varepsilon^2} \Delta g_\varepsilon \Delta^\omega g_\varepsilon + \frac 1 {\varepsilon^2} \Delta \left( g_\varepsilon \Gamma^\psi (g_\varepsilon) \right) \right](x) . \] Now, we compute the limits of the three summands on the right hand side. We start with the last one and proceed backwards. Since $g_\varepsilon \stackrel{\varepsilon \to 0}{\longrightarrow} 1$ and $\frac 1 {\varepsilon^2} \Gamma^\psi (g_\varepsilon) \stackrel{\varepsilon \to 0}{\longrightarrow} -\psi''(1) \Gamma(f)$ which was proven above, one has \[ \frac 1 {\varepsilon^2} \Delta \left( g_\varepsilon \Gamma^\psi (g_\varepsilon) \right) \stackrel{\varepsilon \to 0}{\longrightarrow} -\psi''(1) \Delta \Gamma (f). \] Since $\frac 1{\varepsilon} \Delta g_\varepsilon = \Delta f$ and $\frac 1 {\varepsilon} \Delta^\omega (g_\varepsilon) \stackrel{\varepsilon \to 0}{\longrightarrow} \omega'(1) \Delta f = -\psi''(1) \Delta f$ which was proven above, one has \[ \frac 1 {\varepsilon^2} \Delta g_\varepsilon \Delta^\omega g_\varepsilon \stackrel{\varepsilon \to 0}{\longrightarrow} -\psi''(1) (\Delta f)^2. \] The function $\nu$ is differentiable since $\psi \in C^2(\mathbb{R}^+)$, and obviously, $\nu(1)=0$. By defining $\delta := \varepsilon \cdot \frac {f-f(x)}{g_\varepsilon}$, we obtain \[ \frac 1 \varepsilon \nu \left( \frac {g_\varepsilon}{g_\varepsilon(x)}\right) = \frac{1}{\delta} \big[ \nu(1+\delta)-\nu(1) \big] \cdot \frac {f-f(x)}{g_\varepsilon} \stackrel{\varepsilon \to 0}{\longrightarrow} \nu'(1)(f-f(x)) = \psi''(1)(f-f(x)) \] and hence, \begin{eqnarray} \frac 1 {\varepsilon^2} \Delta\left[\nu \left( \frac {g_\varepsilon} {g_\varepsilon(x)} \right) \Delta g_\varepsilon \right](x) &\stackrel{\varepsilon \to 0}{\longrightarrow}& \Delta \left[ \psi''(1)(f-f(x)) \Delta f \right] (x) \\ &=& -\psi''(1)\left[ f \Delta \Delta f - \Delta (f \Delta f)\right](x). \end{eqnarray} Putting together the three limits calculated above yields \begin{eqnarray} \frac {2g_\varepsilon} {\varepsilon^2} \Gamma_2^\psi(1+ \varepsilon f) &\stackrel{\varepsilon \to 0}{\longrightarrow} & -\psi''(1) \left[ \Delta \Gamma (f) + (\Delta f)^2 + f \Delta \Delta f - \Delta (f \Delta f) \right] \\ &=& -\psi''(1) \left[ \Delta \Gamma (f) - 2 \Gamma(f,\Delta f) \right]\\ &=& -\psi''(1) \cdot 2 \Gamma_2 (f). \end{eqnarray} Since $g_\varepsilon \stackrel{\varepsilon \to 0}{\longrightarrow} 1$, we conclude $\frac {1} {\varepsilon^2} \Gamma_2^\psi(1+ \varepsilon f) \stackrel{\varepsilon \to 0}{\longrightarrow} -\psi''(1) \Gamma_2 (f) $ which finishes the proof. \end{proof} \begin{corollary} Let $\psi \in C^2(\mathbb{R}^+)$ be concave with $\psi''(1)\neq 0 \neq \psi'(1)$ and let $d \in \mathbb{R}^+$. Let $G=(V,E)$ be a graph satisfying the $CD\psi(d,0)$ condition. Then, $G$ also satisfies the $CD\left(\frac{ -\psi''(1)}{\psi'(1)^2}d,0 \right)$ condition. \end{corollary} \begin{proof} Let $f \in C(V)$. Since $G$ satisfies the $CD\psi(d,0)$ condition, we have \[ -\psi''(1)\Gamma_2(f)=\lim_{\varepsilon \to 0} \frac 1 {\varepsilon^2} \Gamma_2^\psi(1+\varepsilon f) \geq \lim_{\varepsilon \to 0} \frac 1 {d \varepsilon^2}\left[ \Delta^\psi(1+\varepsilon f)\right]^2 = \frac {\psi'(1)^2 }{d} (\Delta f)^2. \] Since $\psi$ is concave and $\psi''(1) \neq 0$, one has $-\psi''(1)>0$. Thus, we obtain that $G$ satisfies the $CD\left(\frac{ -\psi''(1)}{\psi'(1)^2}d,0 \right)$ condition. \end{proof} \begin{example} On finite graphs, the $CD\log(d,0)$ condition implies the $CD(d,0)$ condition. \end{example} The concept of the $\psi$-Laplacian and the $CD\psi$-condition can be extended to infinite graphs. In \cite{Hua2014}, Hua and Lin show that the $CD(d,0)$ condition, together with a geometric completeness property, implies stochastic completeness. Thus, we obtain that the $CD\psi$ condition on infinite graphs also implies stochastic completeness. \section{Li-Yau inequalities} \label{CLiYau} In this section, we will give two proofs of the $\psi$-Li-Yau inequality. First, we will prove the $\psi$-Li-Yau inequality via a maximum principle which was used in \cite{Li1986, Bauer2013}. After that, we will give another characterization of the $CD \psi$ condition by using semigroup methods introduced in \cite{Bakry2006}. This characterization turns out to be a stronger version of the $\psi$-Li-Yau inequality. \subsection{A proof via the maximum principle} First, we present a monotonicity lemma which can be found in \cite[Lemma 4.1]{Bauer2013}. This gives an important maximum property of the heat operator. \begin{lemma} [Monotonicity of $\mathcal L$] Let $G = (V,E)$ be a graph and let $g,F : V \times (0,T] \to {\mathbb{R}}$ be differentiable functions, such that $g\geq 0$ and such that $F$ attains a local maximum in some $(x_0,t_0)$. Then, one has \begin{equation} \mathcal L(g)F (x_0,t_0) \geq \mathcal L (gF)(x_0,t_0). \label{MON} \end{equation} \end{lemma} \begin{proof} A short calculation gives \begin{eqnarray} \Delta(g)F (x_0,t_0) &=& \sum_{y \sim x_0} g(y,t_0)F(x_0,t_0) - g(x_0,t_0)F(x_0,t_0)\\ &\geq& \sum_{y \sim x_0} g(y,t_0)F(y,t_0) - g(x_0,t_0)F(x_0,t_0) \\ &=& \Delta (gF)(x_0,t_0). \end{eqnarray} Similarly, \[ F \partial_t g (x_0,t_0) \leq F \partial_t g (x_0,t_0) + g \partial_t F (x_0,t_0) = \partial_t (gF) (x_0,t_0) \] since $\partial_t F =0$ if $t_0 \in (0,T)$ and $\partial_t F \geq0$ if $t_0=T$. The difference of these estimates yields the claim. \end{proof} \begin{theorem}[$\psi$-Li-Yau inequality] \label{TPLY} Let $G = (V,E)$ be a finite graph satisfying $CD\psi(D,0)$ and $u \in C^1( V \times \nnegR)$ a positive solution to the heat equation on $G$ . Then for all $x\in V$ and $t \in \mathbb{R}^+$, one has \[ -\Delta^\psi u(x,t) \leq \frac {d} {2 t}. \] \end{theorem} \begin{proof} We define for $x \in V$ and $t \in \mathbb{R}^+_0$ \[F(x,t) := -t \Delta^\psi u(x,t).\] It is sufficient to show that for all $T>0$, one has \[ \sup_{x\in V, 0\leq t\leq T} F(x,t) \leq \frac {d} {2}. \] Since $V \times [0,T]$ is compact and $F$ is continuous, the restriction $F\big\|_{V\times [0,T]}$ attains its maximum in some $(x_0,t_0)$. We assume without loss of generality that $F(x_0,t_0)$ is positive. Since $F(\cdot,0)=0$, we can deduce $t_0 > 0$ and thus, the maximum is attained on $V \times (0,T]$. Hence, we can use the estimate (\ref{CCD}) from the characterization of $CD\psi$ and the estimate (\ref{MON}) from the monotonicity lemma with $g(\cdot,t) := \frac u t$ for all $t \in(0,T]$. The following computation is understood to take place at the point $(x_0,t_0)$. We obtain \begin{eqnarray} \frac u {t_0^2} F &=& \mathcal L \left(g\right) F \stackrel{(\ref{MON})}{\geq} \mathcal L \left( gF \right) = \mathcal L \left(-u\Delta^\psi u \right) \stackrel{(\ref{CCD})}{\geq} \frac {2}{d} u (\Delta^\psi u)^2 = \frac {2}{d} \frac u {t_0^2} F^2. \end{eqnarray} Since $F(x_0,t_0) >0$, we can conclude $F \leq \frac {d} {2}$. \end{proof} \subsection{A proof via semigroup methods} Next, we use semigroup methods to give another proof of the $\psi$-Li-Yau inequality. Moreover, we give another characterization of the $CD\psi$ condition which is inspired by a similar result that Bakry and Ledoux showed on diffusion semigroups (cf. \cite{Bakry2006}). \begin{defn}[Operator semigroup] Let $G=(V,E)$ be a finite graph. The Laplacian $\Delta$ generates an \emph{operator semigroup} $\left(P_t\right)_{t\geq 0}: C(V) \to C(V)$ with \[ P_t f := e^{\Delta t} f := \sum_{k=0}^{\infty} \frac {t^k \Delta^k f}{k!} \mbox{ for all } t \in \mathbb{R}^+_0 \mbox{ and all } f \in C(V). \] \end{defn} \begin{proposition}[Basic properties of operator semigroups] {\label{SG}} Let $G=(V,E)$ be a finite graph with Laplacian $\Delta$ and the generated semigroup $\left(P_t\right)_{t\geq 0}$. \begin{enumerate}[(1)] \item The semigroup satisfies the property ${P_t ( P_s f ) = P_{t+s} f}$ for all $f \in C(V)$ and all $s,t \geq 0$. \item The semigroup gives a solution to the heat equation. I.e. $P_t f$ is continuous in $t$ for all $t \geq 0$ and $f \in C(V)$, and furthermore $\mathcal L ( P_t f) = 0$ for all $t \geq 0$ and all $f \in C(V)$. \item The semigroup satisfies the initial condition $P_0 f = f$ for all $f \in C(V)$. \item If the initial condition $f \in C(V)$ is positive, then $P_t f \in C(V)$ is also positive for all $t \geq 0$. \end{enumerate} \end{proposition} This proposition is a standard one and a proof can be found e.g. in \cite{Chung2000}. \begin{theorem}[Semigroup form of the $\psi$-Li-Yau inequality] Let $\psi \in C^1(\mathbb{R}^+)$, and let $G=(V,E)$ be a finite graph with Laplacian $\Delta$ which generates the semigroup $\left(P_t\right)_{t\geq 0}$. Then, the following statements are equivalent. \begin{enumerate}[(i)] \item $G$ satisfies $CD\psi({d},0)$. \item For all positive functions $f \in C^+(V)$ and all $t\geq 0$, one has \[ P_t f \Delta^\psi P_t f \geq P_t(f \Delta^\psi f)\left(1 + \frac {{2} t} n \Delta^\psi P_t f \right). \] \end{enumerate} If one of these statements is true, then the $\psi$-Li-Yau inequality \[ -\Delta^\psi P_t f \leq \frac n {2 t} \] holds for all $f \in C^+(V)$ and all $t>0$. \end{theorem} The proof of this theorem is in the spirit of Bakry and Ledoux (cf. \cite[Theorem 1]{Bakry2006}). In contrast to their proof, we are able to bypass the chain rule by using Theorem \ref{tchar} which is the characterization of the $CD\psi$ condition. It seems to be quite a challenge to obtain logarithmic Sobolev inequalities (cf. \cite[inequalities (1.10), (1.11)]{Bakry2006}) on graphs by using our methods. Nevertheless in \cite{Chung2000}, Chung, Grigor'yan and Yau have shown another version of the Sobolev inequality on manifolds and on graphs. \begin{proof} First, we show $\emph{(\ref{2})} \Rightarrow \emph{(\ref{1})}$. We will use identity (\ref{repGG}) from the representation lemma of $\Gamma_2^\psi$. For all $f \in C^+(V)$, the assumption $\emph{(\ref{2})}$ implies \begin{eqnarray} 0 &\leq& \lim_{t \to 0} \frac 1 {2t} \left[ P_tf \Delta^\psi P_t f - P_t (f\Delta^\psi f) - \frac {{2} t} n P_t(f \Delta^\psi f) \Delta^\psi P_t f \right] \\ &=& \lim_{t \to 0} \frac 1 {2t} \left[ (P_tf \Delta^\psi P_t f - P_0f \Delta^\psi P_0 f) - (P_t (f\Delta^\psi f - P_0 (f\Delta^\psi f))\right] \\&& - \frac {{1}} {n} f (\Delta^\psi f)^2\\ &=& \frac 1 2 \partial_t\left[ P_tf \Delta^\psi P_t f - P_t (f\Delta^\psi f) \right]_{t=0} - \frac {{1}} {n} f (\Delta^\psi f)^2 \\ &=& \frac 1 2 \left[ \partial_t( P_tf \Delta^\psi P_t f) - \Delta P_0 (f\Delta^\psi f) \right]_{t=0} - \frac {{1}} {n} f (\Delta^\psi f)^2 \\ &=& \frac 1 2 \left[ \partial_t( P_tf \Delta^\psi P_t f) - \Delta (P_t f\Delta^\psi P_t f) \right]_{t=0} - \frac {{1}} {n} f (\Delta^\psi f)^2 \\ &=& \frac 1 2 [\mathcal L (-P_tf \Delta^\psi P_t f)]_{t=0} - \frac {{1}} {n} f (\Delta^\psi f)^2 \\ &\stackrel{(\ref{repGG})}{=}& \left[P_tf {\Gamma_2^\psi} (P_tf) \right]_{t=0} - \frac {{1}} {n} f (\Delta^\psi f)^2 \\ &=& f {\Gamma_2^\psi} (f) - \frac {{1}} {n} f (\Delta^\psi f)^2. \end{eqnarray} Thus, \[ {\Gamma_2^\psi} (f) \geq \frac {{1}} {n} (\Delta^\psi f)^2. \] Since $f$ is arbitrary, we obtain the $CD\psi(n,0)$ inequality. Next, we show $\emph{(\ref{1})} \Rightarrow \emph{(\ref{2})}$. Let $f \in C(V)$ and $t>0$. Following the notation of Bakry and Ledoux, for $s \in [0,t]$, we denote \begin{eqnarray} g &:=& P_s f,\\ A &:=& g\Delta^\psi g,\\ \phi(s) &:=& P_{t-s}(P_s f \Delta^\psi P_s f) = P_{t-s}(A). \end{eqnarray} We take the derivative of $\phi$. By using identity (\ref{repGG}) from the representation lemma of $\Gamma_2^\psi$, we obtain \begin{eqnarray} \phi'(s) &=& \partial_s P_{t-s}(A) = -\Delta P_{t-s}(A) + P_{t-s}(\partial_s A) = P_{t-s}\mathcal L (-A) = P_{t-s}\mathcal L (-g\Delta^\psi g)\\ &\stackrel{(\ref{repGG})}{=}& P_{t-s}(2 g {\Gamma_2^\psi} (g)) \stackrel{CD\psi}{\geq}\frac {{2}} n P_{t-s}\left( g \left(\Delta^\psi g\right)^2 \right) = \frac {{2}} n P_{t-s}\left(\frac {A^2} g \right)\\ &\geq& \frac {{2}} n \frac {(P_{t-s}A)^2}{P_{t-s} g} = \frac {{2}} n \frac {\phi(s)^2}{P_t f} \geq 0. \end{eqnarray} In the first inequality, we also used the positivity of $P_{t-s}$. This calculation implies \begin{equation} P_tf \Delta^\psi P_t f = \phi(t)\geq \phi(0) = P_t(f \Delta^\psi f).\label{ptpo} \end{equation} Suppose $\Delta^\psi P_t f \geq 0$ and $P_t(f \Delta^\psi P_t f)\leq 0$. Then, the claim is obvious. Suppose not. Since $\phi$ is monotonically non-decreasing, we can conclude $\phi(s) \neq 0$ for all $s \in [0,t]$ . Thus, we obtain \[ - \left( \frac 1 {\phi} \right)' = \frac {\phi'}{\phi^2} \geq \frac {{2}} {n P_t f}. \] By integrating this identity from $0$ to $t$, we get \[ \frac 1 {\phi(0)} - \frac 1 {\phi(t)} \geq \frac {{{2}} t} {n P_t f}. \] Since $\phi(0)$ and $\phi(t)$ have the same sign, we see \[ {\phi(t)} - {\phi(0)} \geq \phi(t) \phi(0) \frac {{{2}}t} {n P_t f}. \] Substituting $\phi(t)$ and $\phi(0)$ into this estimation yields the result. Now, we will deduce the $\psi$-Li-Yau inequality from the equivalent statements $\emph{(\ref{1})}$ and $\emph{(\ref{2})}$. Suppose $\Delta^\psi P_t f > 0$. Then, the claim is obvious. Suppose not. We have already seen in inequality (\ref{ptpo}) that $\emph{(\ref{1})}$ implies $P_tf \Delta^\psi P_t f \geq P_t(f \Delta^\psi f)$. Thus, \[ 0 \geq P_tf \Delta^\psi P_t f \geq P_t(f \Delta^\psi f). \] Consequently, $\emph{(\ref{2})}$ implies \[ 1 + \frac {{2} t} n \Delta^\psi P_t f \geq 0 \] which is equivalent to the Li-Yau inequality \[ -\Delta^\psi P_t f \leq \frac n {{2} t}. \] This finishes the proof. \end{proof} \section{Harnack inequalities} \label{CHarnack} The Harnack inequality states that if $u$ is a solution to the heat equation and if $(x_i,t_i)_{i=1,2}$ are two points in space-time, then $\frac{u(x_1,t_1)}{u(x_2,t_2)}$ can be estimated by a function only depending on a certain distance of the points $(x_1,t_1)$ and $(x_2,t_2)$. In this section, we show in which sense the $\psi$-Li-Yau inequality can be understood as a gradient estimate and how to use this property to derive Harnack type inequalities. To do so, we will use the methods introduced in \cite{Bauer2013} which turn out to be applicable here with minor changes. We will note in this section that the concavity of $\psi$ is crucial to derive Harnack inequalities from the $\psi$-Li-Yau inequality. Furthermore, we will see in Subsection \ref{SRF} that we need concavity of $\psi$ to prove $CD\psi$ inequalities on Ricci-flat graphs. \subsection{Preliminaries} To understand the $\psi$-Li-Yau inequality as a gradient estimate, we use the $\psi$-gradient $\Gamma^\psi$ (see Subsection \ref{SGF}). This is defined as $\Gamma^\psi = \Delta^{\overline{\psi}}$ with $$\overline{\psi}(x)= \psi'(1)\cdot(x-1) - (\psi(x) - \psi(1))$$ for all $x>0$ and all concave $\psi \in C(\mathbb{R}^+)$. We recall the gradient representation of $\Delta^\psi$ (cf. Lemma \ref{LGR}). Let $G=(V,E)$ be a finite graph, and let $\psi \in C^1({\mathbb{R}})$ be a concave function. Then, the gradient representation of $\Delta^\psi$ states that for all $f \in C^+(V)$, one has \[ -\Delta^\psi f = \Gamma^\psi (f) - \psi'(1) \frac {\Delta f} f. \] This formulation will turn out to be a convenient basis for deducing Harnack inequalities from the Li-Yau inequality. We only need to introduce one more constant dependent on $\psi$. \begin{defn}[Harnack constant] \label{DHC} Let $\psi \in C^1(\mathbb{R}^+)$ be a concave function. We define the \emph{Harnack-constant} $H_\psi$ of $\psi$ as \begin{equation} H_\psi := \sup_{x>1} \frac{(\log x)^2}{\overline{\psi}(x)} \in [0,\infty]. \end{equation} \end{defn} This constant is defined to give the connection between $\psi$-Li-Yau type gradient estimates and Harnack inequalities. The case $H_\psi = \infty$ is allowed, but this only occurs if $\psi''(1)=0$ (cf. Lemma \ref{LDH}). The following lemma is the key to prove Harnack inequalities by using the methods introduced in \cite[Section 5]{Bauer2013} and gives the link between the $\Gamma^\psi$-calculus and these methods. \begin{lemma} [Estimate of ${\Gamma^\psi}$] Let $\psi \in C^1(\mathbb{R}^+)$ be a concave function, let $G=(V,E)$ be a graph, let $f \in C^+(V)$ and let $v,w \in V$ with $v \sim w$. Then, \begin{equation} \log {\frac {f(w)}{f(v)}} \leq \sqrt{H_\psi} \sqrt{\Gamma^\psi (f)}(v) \label {EG}. \end{equation} \end{lemma} \begin{proof} First, we show the following claim. For all $x \in \mathbb{R}^+$, one has \begin{equation} \log x \leq \sqrt{H_\psi}\sqrt{\overline{\psi}(x)}. \label{claimG} \end{equation} This claim is obvious for $x \leq 1$, since we obtain $\log x \leq 0$ in this case. If $x > 1$, the claim follows from the definition of $H_\psi$. For the assertion of the lemma, we use $\overline{\psi} \geq 0$ and apply inequality (\ref{claimG}) proven above with $x = \frac {f(w)}{f(v)}$ to compute \begin{eqnarray} \sqrt{H_\psi} \sqrt{\Gamma^\psi (f)}(v) &=& \sqrt{H_\psi} \sqrt{\Delta^{\overline{\psi}} f}(v) \\ &\stackrel{(\ref{repG})}{=}& \sqrt{H_\psi} \sqrt{\sum_{\widetilde v \sim v} \overline{\psi}\left(\frac{f(\widetilde v)}{f(v)} \right)}\\ &{\geq}& \sqrt{H_\psi} \sqrt{ \overline{\psi}\left(\frac{f(w)}{f(v)} \right) }\\ &\geq& \log {\frac {f(w)}{f(v)}}. \end{eqnarray} This finishes the proof. \end{proof} Next, we give a lemma which is a special case of \cite[Lemma 5.3]{Bauer2013}. \begin{lemma}[Minimal integral estimate] \label{MIL} Let $T_1,T_2 \in {\mathbb{R}}$ with $T_2 > T_1$. Let $\gamma: [T_1,T_2] \to \mathbb{R}^+_0$ be a continuous function and let $C_1, C_2 \in \mathbb{R}^+$ be positive constants. Then, one has \begin{equation} \frac {C_2^2}{C_1(T_2-T_1)} \geq \inf_{s\in [T_1,T_2]} \left( C_2 \sqrt{\gamma(s)} - C_1 \int_s^{T_2} \gamma(t) dt \right) . \label{HL} \end{equation} \end{lemma} We include the proof for convenience of the reader. \begin{proof} We estimate the infimum by an average integral with the weight function $\phi:[T_1,T_2] \to \mathbb{R}^+, s \mapsto s-T_1$. \begin{eqnarray} && \inf_{s\in [T_1,T_2]} \left( -C_1 \int_s^{T_2} \gamma(t) dt + C_2 \sqrt{\gamma(s)} \right) \\ &\leq& \frac{ \int_{T_1}^{T_2} \phi(s) \left(-C_1 \int_s^{T_2} \gamma(t) dt + C_2 \sqrt{\gamma(s)} \right) ds }{\int_{T_1}^{T_2} \phi(s)ds} \\ &=& \frac 1 {(T_2-T_1)^2} \int_{T_1}^{T_2} \left( - 2 C_1 \gamma(s) \int_{T_1}^s \phi(t) dt + 2 C_2(s-T_1)\sqrt{\gamma(s)} \right) ds\\ &=& \frac 1 {(T_2-T_1)^2} \int_{T_1}^{T_2} \left( - C_1 \gamma(s) (s-T_1)^2 + 2 C_2(s-T_1)\sqrt{\gamma(s)} \right) ds \\ &\leq& \frac 1 {(T_2-T_1)^2} \int_{T_1}^{T_2} \frac{C_2^2}{ C_1} ds \\ &=& \frac {C_2^2}{C_1(T_2-T_1)}. \end{eqnarray} In the last estimate, we used $-C_1 x^2 + 2C_2 x \leq \frac{C_2^2}{ C_1}$ with $x = (s-T_1)\sqrt{\gamma(s)}$. \end{proof} \subsection{Main theorem} The following theorem is in the spirit of Bauer, Horn, Lin, Lippner, Mangoubi, and Yau (cf. \cite[Theorem 5.1]{Bauer2013}). In contrast to their version and in order to focus on our new methods, we do not consider the effect of potentials but instead, we consider more general gradient forms. To prove the Harnack inequality, we require the gradient estimate \[ D_1 \Gamma^\psi (u) (x,t) - \partial_t \log u (x,t) \leq \frac{D_2}t + D_3 \] for all positive solutions $u \in C^1( V \times \nnegR)$ to the heat equation and some positive constants $D_1,D_2,D_3 \in \mathbb{R}^+$. By using the gradient representation (Lemma \ref{LGR}) \[ -\Delta^\psi f = \Gamma^\psi (f) - \psi'(1) \frac {\Delta f} f, \] we will be able to guarantee the required gradient estimate due to the $\psi$-Li-Yau inequality (Theorem \ref{TPLY}) \[ -\Delta^\psi u(x,t) \leq \frac {d} {2 t} \] for graphs satisfying the $CD\psi(d,0)$ condition. \begin{theorem}[Harnack inequality as a consequence of a $\psi$-gradient estimate] \label{THarnack} Let G=(V,E) be a finite and connected graph, $D_1,D_2,D_3 \in \mathbb{R}^+$ positive constants, and let $u \in C^1( V \times \nnegR)$ be a function satisfying \begin{equation} D_1 \Gamma^\psi (u) (x,t) - \partial_t \log u (x,t) \leq \frac{D_2}t + D_3 \label{Hreq} \end{equation} for all $x\in V$ and $t \in \mathbb{R}^+_0$. Then, \[ \frac{u(x_1,T_1)}{u(x_2,T_2)} \leq \left( \frac{T_2}{T_1} \right)^{D_2} \exp(D_3(T_2-T_1)) \exp\left(\frac {H_\psi d(x_1,x_2)}{ D_1 (T_2-T_1)}\right) \] holds for all $x_1,x_2 \in V$ and all positive $T_1<T_2$. \end{theorem} The proof of this theorem is very similar to the one given in \cite[Theorem 5.1]{Bauer2013}. \begin{proof} First, we consider the case $x_1 \sim x_2$. Let $T_1 < T_2$ and $s \in \left[ T_1,T_2 \right]$. We use the assumption (\ref{Hreq}) of the theorem and the estimate (\ref{EG}) of $\Gamma^\psi$ to estimate \begin{eqnarray} \log \frac {u(x_1,T_1)} {u(x_2,T_2)} &=& \log \frac {u(x_1,T_1)} {u(x_1,s)} + \log \frac {u(x_1,s)} {u(x_2,s)} + \log \frac {u(x_2,s)} {u(x_2,T_2)} \\ &=& \int_{T_1}^s -\partial_t \log u(x_1,t) dt + \log \frac {u(x_1,s)} {u(x_2,s)} -\int_{s}^{T_2} \partial_t \log u(x_2,t) dt \\ &\stackrel{(\ref{Hreq})}{\leq}& \int_{T_1}^s \left( \frac {D_2}t + D_3 - D_1 \Gamma^\psi (u)(x_1,t) \right) dt + \log \frac {u(x_1,s)} {u(x_2,s)} \\&& + \int_{s}^{T_2} \left( \frac {D_2}t + D_3 - D_1 \Gamma^\psi (u)(x_2,t) \right) dt \\ &\stackrel{D_1 \Gamma^\psi \geq 0}{\leq}& \int_{T_1}^{T_2} \left( \frac {D_2}t + D_3 \right) dt + \log \frac {u(x_1,s)} {u(x_2,s)} - \int_{s}^{T_2} D_1 \Gamma^\psi (u)(x_2,t) dt \\ &\stackrel{(\ref{EG})}{\leq}& D_2\log\ \frac {T_2}{T_1} + D_3(T_2-T_1) \\&& + \sqrt{H_\psi}\sqrt{\Gamma^\psi (u)(x_2,s)} - \int_{s}^{T_2} D_1 \Gamma^\psi (u)(x_2,t) dt. \\ \end{eqnarray} Now, we take the infimum over all $s \in \left[ T_1,T_2 \right]$ and, by using the minimal integral estimate (\ref{HL}), we obtain \begin{eqnarray} \log \frac {u(x_1,T_1)} {u(x_2,T_2)} &\leq& D_2\log\ \frac {T_2}{T_1} + D_3(T_2-T_1) \\&& + \inf_{s\in[T_1,T_2]} \left( \sqrt{H_\psi}\sqrt{\Gamma^\psi (u)(x_2,s)} - \int_{s}^{T_2} D_1 \Gamma^\psi (u)(x_2,t) dt \right) \\ &\stackrel{(\ref{HL})}{\leq}& D_2\log\ \frac {T_2}{T_1} + D_3(T_2-T_1) + \frac {H_\psi} {D_1(T_2-T_1)}. \\ \end{eqnarray} The minimal integral estimate (\ref{HL}) was applied with $\gamma(t) = \log u(x_2,t)$, and $C_2 = \sqrt{H_\psi}$, and $C_1 = D_1$. Now, we consider the general case (i.e. the vertices $x_1$ and $x_2$ are not necessarily adjacent). Denote $n:= d(x_1,x_2)$. Since $G$ is connected, there is a path $x_1=v_0 \sim \ldots \sim v_{n} = x_2$ of length $n$, and there are positive numbers $T_1 = t_0 < \ldots < t_{n} = T_2$ with $t_i - t_{i-1} = \frac {T_2 - T_1}{n}$ for $i\in \{1,\ldots,n\}$. Using the estimation from above yields \begin{eqnarray} \log \frac {u(x_1,T_1)} {u(x_2,T_2)} &=& \sum_{i=1}^{n} \log \frac {u(v_{i-1},t_{i-1})} {u(v_i,t_i)} \\ &\leq& \sum_{i=1}^{n} D_2\log\ \frac {t_i}{t_{i-1}} + D_3(t_i-t_{i-1}) + \frac {H_\psi} {D_1(t_i-t_{i-1})} \\ &=& D_2 \log \frac {T_2}{T_1} + D_3(T_2-T_1) + \frac {H_\psi d(x_1,x_2)^2} {D_1(T_2-T_1)}. \\ \end{eqnarray} Hence, \[ \frac{u(x_1,T_1)}{u(x_2,T_2)} \leq \left( \frac{T_2}{T_1} \right)^{D_2} \exp(D_3(T_2-T_1)) \exp\left(\frac {H_\psi d(x_1,x_2)}{ D_1 (T_2-T_1)}\right) \] which is the claim of the theorem. \end{proof} \begin{corollary}[Harnack inequalities as a consequence of the $CD\psi$ condition] Let $\psi \in C^1(\mathbb{R}^+)$ be a concave function with $\psi'(1)=1$, and let $G=(V,E)$ be a graph satisfying the $CD\psi(d,0)$ inequality for some $d \in \mathbb{R}^+$. Then for all positive solutions $u \in C^1( V \times \nnegR)$ to the heat equation on $G$, all $x_1,x_2 \in V$, and all positive $T_1<T_2$, one has \[ \log \frac{u(x_1,T_1)}{u(x_2,T_2)} \leq \frac d {2} \log \frac{T_2}{T_1} + \frac{H_\psi d(x_1,x_2)^2}{(T_2-T_1)}. \] \end{corollary} \begin{proof} Since $G$ satisfies the $CD\psi(d,0)$ inequality, the $\psi$-Li-Yau inequality holds. Thus, \[ \Gamma^\psi (u) - \frac {\Delta u} u = -\Delta^\psi u \leq \frac d {2t}. \] Hence, we can apply the Harnack inequality with $D_1=1$, $D_2 = \frac d {2}$, and $D_3 = 0$, and we obtain the claim. \end{proof} \section{Examples} \label{CExamples} To show that the Harnack inequality presented here is stronger than the one given in \cite[Corollary 5.2]{Bauer2013}, we have to assure a unified context for both. Unfortunately, the different curvature-dimension conditions seem to be not unifiable. Instead, so called Ricci-flat graphs will turn out to be an appropriate basis to compare the Harnack inequalities. To do this comparison, we need some preliminary considerations. In the following subsection, we will show that Ricci-flat graphs satisfy the $CD\psi(d,0)$ condition for suitable $\psi$. For a given Ricci-flat graph, the dimension bound $d$ depends on the parameter $\psi$. This dependence can be described by a constant $C_\psi$. In the second subsection of this section, we compute this constant $C_\psi$ and the Harnack constant $H_\psi$ (cf. Definition \ref{DHC}) for several $\psi$. We use these computations to discuss the announced break of analogy between the $CDE$ condition introduced in \cite[Definition 3.9]{Bauer2013} and the $CD$ condition (cf. Definition \ref{DCD}). In the third subsection, we establish a Harnack inequality on Ricci-flat graphs. Finally, we will compare this inequality with the one given in \cite[Corollary 5.2]{Bauer2013}. \subsection{Ricci-flat graphs}\label{SRF} Ricci-flat graphs were introduced by Chung and Yau \cite{Chung1996} as a generalization of Abelian Cayley graphs to prove Harnack inequalities and log-Sobolev inequalities. These graphs have been the basis to establish new notions of Ricci curvature on graphs (cf. \cite{Bauer2013, Lin2010}). The goal of this subsection is to prove that Ricci-flat graphs satisfy the $CD \psi$ inequality with curvature bound zero. This subsection is in the spirit of \cite[Subsection 6.3]{Bauer2013}, where the $CDE$ condition is proved for Ricci-flat graphs. \begin{defn}[Ricci-flat graphs] Let $D \in {\mathbb{N}}$. A finite graph $G = (V,E)$ is called $D$\emph{-Ricci-flat} in $v \in V$ if all $w \in N(v):=\{v\} \cup \{w \in V: w \sim v\}$ have the degree $D$, and if there are maps $\eta_1,\ldots,\eta_D : N(v) \to V $, such that for all $w \in N(v)$ and all $i, j \in \{1,\ldots,D\}$ with $i \neq j$, one has \begin{eqnarray} \eta_i(w) &\sim& w, \label{r1} \\ \eta_i(w) &\neq& \eta_j(w), \label{r2}\\ \bigcup_k \eta_k(\eta_i(v)) &=& \bigcup_k \eta_i(\eta_k(v)) \label{r3}. \end{eqnarray} The graph $G$ is called $D$\emph{-Ricci-flat} if it is $D$-Ricci-flat in all $v \in V$. \end{defn} \begin{example} All Abelian Cayley graphs with degree $D$ are $D$-Ricci-flat as mentioned already in \cite{Chung1996}. \end{example} In the next lemma, we collect some facts which are already used in \cite{Bauer2013}. \begin{lemma} [Basic properties of Ricci-flat graphs] \label{Lj} Let $G = (V,E)$ be Ricci-flat in $v \in V$ with according maps $\eta_1,\ldots,\eta_D : N(v) \to V $. \begin{enumerate}[(1)] \item Let $f \in C(V)$ be a function. Then for all $i \in \{1,\ldots,D\}$, one has \begin{equation} \sum_k f(\eta_k \eta_i(v)) = \sum_k f(\eta_i \eta_k(v)). \label{komm} \end{equation} \item For all $i \in \{1,\ldots,D\}$, there is a unique $j = j(i)$, such that $\eta_{i} (\eta_j(v)) = v$. Additionally, the map $i \mapsto j(i)$ is a permutation of $\{1,\ldots,D\}$. \end{enumerate} \end{lemma} \begin{proof} First, we prove \emph{(1)}. Since $\bigcup_k \eta_k(\eta_i(v)) = \bigcup_k \eta_i(\eta_k(v))$, it is sufficient to show that no vertex in (\ref{komm}) is summed up twice. This is clear if $\bigcup_k \eta_k(\eta_i(v))$ and if $\bigcup_k \eta_i(\eta_k(v))$ are disjoint unions. We know, $\bigcup_k \eta_k(\eta_i(v))$ is a disjoint union by (\ref{r2}). Since $D < \infty$, identity (\ref{r3}) implies that $\bigcup_k \eta_i(\eta_k(v))$ is a disjoint union. Next, we prove \emph{(2)}. The uniqueness of $j(i)$ is obvious. Suppose $i \mapsto j(i)$ is not a permutation. Then, there are $i,k \in \{1,\ldots,D\}$ with $i \neq k$ and $j = j(i) = j(k)$. This means $\eta_i\eta_j(v) = \eta_k\eta_j(v)$. Hence, \[ \# \bigcup_l \eta_l \eta_j (v) < D. \] This is a contradiction. \end{proof} To prove a $CD\psi$ inequality on Ricci-flat graphs, we need to introduce the constant $C_\psi$. \begin{defn}\label{DCP} Let $\psi \in C^1(\mathbb{R}^+)$. Then for all $x,y > 0$, we write \[ \widetilde{\psi}(x,y):= \left[\psi'(x) + \psi'(y) \right](1-xy) + x [\psi(y) - \psi(1/x)] + y [\psi(x) - \psi(1/y)] \] and \[ C_\psi := \inf_{x,y>0} \frac{\widetilde{\psi}(x,y)}{(\psi(x) + \psi(y) - 2\psi(1))^2} \in [-\infty, \infty]. \] \end{defn} To obtain useful results, we need $C_\psi>0$. We will discuss this condition in the next subsection. \begin{rem} For $\psi=\log$, it suffices to consider the infimum of the function in $C_\psi$ for the diagonal, i.e., $x=y$. This behavior is also indicated by numerical computations for various concave $\psi$ and it would be interesting to know whether this behavior can be established for arbitrary concave $\psi$. \end{rem} \begin{theorem}[$CD\psi$ for Ricci-flat graphs] \label{TRicci} Let $D \in {\mathbb{N}}$, let $G=(V,E)$ be a $D$-Ricci-flat graph, and let $\psi \in C^1(\mathbb{R}^+)$ be a concave function, such that $C_\psi>0$. Then, $G$ satisfies the $CD \psi (d,0)$ inequality with $d = D / C_\psi$. \end{theorem} The proof and the notation are inspired by the proof of Theorem 6.7 in \cite{Bauer2013}. \begin{proof} We have to show for all $f \in C(V)$ and all $v \in V$ that \[ 2 \Gamma_2^\psi (f) (v) \geq \frac {2 C_\psi}{D} \left[\Delta^\psi f (v)\right]^2. \] First, we recall the definitions \begin{eqnarray} \Delta^\psi f(v) &=& \sum_{w\sim v} \psi\left( \frac{f(w)}{f(v)} \right) - \psi (1),\\ \Omega^\psi f(v) &=& \sum_{w\sim v} \psi'\left( \frac{f(w)}{f(v)} \right) \cdot \frac{f(w)}{f(v)} \cdot \left[ \frac{\Delta f(w)}{f(w)} - \frac{\Delta f(v)}{f(v)} \right], \\ 2 \Gamma_2^\psi (f) &=& \Omega^\psi f + \frac {\Delta f \Delta^\psi f} f - \frac {\Delta \left(f \Delta^\psi f\right)} f. \end{eqnarray} We can assume $$\psi(1)=0$$ without loss of generality since $\Gamma_2^\psi$, $\Delta^\psi$ and $C_\psi$ are invariant under adding constants to $\psi$. Let $v \in V$ and $f \in C(V)$. Since $G$ is Ricci-flat, there are maps $\eta_1,\ldots,\eta_D : N(v) \to V$ as demanded in the definition. For all $i,j \in \{1,\ldots,D\}$, we denote \begin{eqnarray} y &:=& f(v),\\ y_i &:=& f(\eta_i(v)),\\ y_{ij} &:=& f(\eta_j(\eta_i(v))),\\ z_i &:=& y_i / y, \\ z_{ij} &:=& y_{ij}/y_{i}. \end{eqnarray} We use the representation of $\Delta^\psi$ (Lemma \ref{LRL}) to obtain the following two identities \begin{eqnarray} \Delta^\psi f (v) &=& \sum_i \psi(z_i), \\ \Delta^\psi f (\eta_i(v)) &=& \sum_j \psi(z_{ij}). \\ \end{eqnarray} Thus, we can compute \begin{eqnarray} &&\frac {\Delta \left(f \Delta^\psi f\right)} {f} (v) - \frac {\left(\Delta f \right) \Delta^\psi f} {f} (v)\\ &=& \frac {\sum_{w \sim v}- f(v) \Delta^\psi f(v) + f(w) \Delta^\psi f(w) } {f (v)} - \frac {\left(\sum_{w\sim v} - f(v) + f(w) \right) \sum_i \psi(z_i)} {f(v)} \\ &=& \left[ \sum_{i,j} z_j \psi(z_{ji}) - \psi(z_i) \right] - \left[ \sum_{i,j} \left( z_j - 1 \right) \psi (z_i) \right]\\ &=& \sum_{i,j} z_j [\psi(z_{ji}) - \psi(z_i)] \end{eqnarray} and \begin{eqnarray} \Omega^\psi f (v) &=& \sum_i \psi'(z_i)z_i \left[ \frac {(\Delta f)(\eta_i(v))}{y_i} - \frac {(\Delta f)(v)}{y} \right] \\ &=& \sum_{i,j} \psi'(z_i)z_i (z_{ij}-z_j) \\ &\stackrel{(\ref{komm})}{=}& \sum_{i,j} \psi'(z_i)z_j (z_{ji}-z_i). \end{eqnarray} We summarize \begin{equation} 2 \Gamma_2^\psi (f) (v) = \sum_{i,j} z_j \left( \psi'(z_i)(z_{ji}-z_i) - [\psi(z_{ji}) - \psi(z_i)] \right) .\label{GRF} \end{equation} Since $\psi$ is concave, every summand is positive. As we showed in the second claim of Lemma \ref{Lj}, for each $i$, there is a unique $j=j(i)$ with $\eta_i(\eta_j(v)) = v$. Now, we disregard all other summands of (\ref{GRF}) and use $z_ {ji} = 1/z_{j(i)}$ if $j=j(i)$ to estimate \begin{eqnarray} 2 \Gamma_2^\psi (f) (v)&\geq& \sum_i z_{j(i)}\left( \psi'(z_i)\left( \frac 1 {z_{j(i)}} - z_i \right) -\left[ \psi\left( \frac 1 {z_{j(i)}} \right) - \psi(z_i) \right] \right) \nonumber \\ &=& \sum_i \psi'(z_i) - \sum_i z_{j(i)} \psi\left( \frac 1 {z_{j(i)}} \right) + \sum_i z_{j(i)}\left( \psi(z_i) - z_i \psi'(z_i) \right). \end{eqnarray} The next step is to symmetrize the sum. Unfortunately, we do not have $j(j(i))=i$ in general. But instead, we can use the rearrangement inequality. This states that for all permutations $\sigma$ on $\{1,\ldots,D\}$ and all $a_1 \leq \ldots \leq a_D$ and all $b_1 \leq \ldots \leq b_D$, one has \[ \sum_{i=1}^D a_{\sigma(i)} b_i \geq \sum_{i=1}^D a_{D+1-i} b_i. \] Since $\psi$ is concave, we observe that the map $z \mapsto \psi(z) - z \psi'(z)$ is monotonically non-decreasing. Without loss of generality, we have $0 < z_1 \leq \ldots \leq z_D$. Furthermore, by the second claim of Lemma \ref{Lj}, the map $i \mapsto j(i)$ is a permutation. Thus, we can apply the rearrangement inequality to obtain \[ \sum_i z_{j(i)}\left( \psi(z_i) - z_i \psi'(z_i) \right) \geq \sum_i z_{i'}\left( \psi(z_i) - z_i \psi'(z_i) \right) \] with $i':=D+1-i$. Especially, we have $i''=i$. Furthermore, the map $i \mapsto i'$ is a permutation. Hence, \begin{eqnarray} 2 \Gamma_2^\psi (f) (v) &\geq& \sum_i \psi'(z_i) - \sum_i z_{j(i)} \psi\left( \frac 1 {z_{j(i)}} \right) + \sum_i z_{j(i)}\left( \psi(z_i) - z_i \psi'(z_i) \right)\\ &\geq& \sum_i \psi'(z_i) - \sum_i z_{i'} \psi\left( \frac 1 {z_{i'}} \right) + \sum_i z_{i'}\left( \psi(z_i) - z_i \psi'(z_i) \right)\\ &=& \frac 1 2 \left( \sum_i + \sum_{i'}\right)\left[\psi'(z_i) - z_{i'} \psi\left( \frac 1 {z_{i'}}\right) + z_{i'}\left( \psi(z_i) - z_i \psi'(z_i) \right) \right] \\ &=& \frac 1 2 \sum_i \left[\psi'(z_i) - z_{i'} \psi\left( \frac 1 {z_{i'}}\right) + z_{i'}\left( \psi(z_i) - z_i \psi'(z_i) \right) \right]\\ && + \frac 1 2 \sum_i \left[\psi'(z_{i'}) - z_i \psi\left( \frac 1 {z_i}\right) + z_i\left( \psi(z_{i'}) - z_{i'} \psi'(z_{i'}) \right) \right].\\ \end{eqnarray} In the first identity, we used the permutation property of the map $i \mapsto i'$ and its consequence $\sum_i = \sum_{i'}$. In the second identity, we used $i''=i$. Now, we employ the definitions $ \widetilde{\psi}(x,y)= \left[\psi'(x) + \psi'(y) \right](1-xy) + x [\psi(y) - \psi(1/x)] + y [\psi(x) - \psi(1/y)] $ and $ C_\psi = \inf_{x,y>0} {\widetilde{\psi}(x,y)}/{(\psi(x) + \psi(y) - 2\psi(1))^2} $ from Definition \ref{DCP} to obtain \begin{eqnarray} \ldots&=& \frac 1 2 \sum_i \widetilde{\psi}(z_i, z_{i'}) \geq \frac 1 2 \sum_i C_\psi \left[ \psi(z_i) + \psi(z_{i'}) \right]^2 \\ &\geq& \frac {C_\psi}{2D} \left[ \sum_i \psi(z_i) + \psi(z_{i'}) \right]^2 = \frac {C_\psi}{2D} \left[ 2 \Delta^\psi f (v) \right]^2\\ &=& \frac { 2 C_\psi}{D} \left[ \Delta^\psi f (v) \right]^2.\\ \end{eqnarray} This finishes the proof since $v\in V$ and $f\in C^+(V)$ are arbitrary. \end{proof} \subsection{Special cases for the function $\psi$} \label{SSC} One objective of this subsection is to show that the Harnack inequality on Ricci-flat graphs established in this article is stronger than the one given in \cite[Corollary 5.2]{Bauer2013}. Moreover, we will discuss that there is a break of analogy in the curvature-dimension condition introduced in \cite{Bauer2013} compared to the $CD$ condition on manifolds. From the perspective of our approach, the authors of \cite{Bauer2013} consider the instance $\psi=\sqrt{\cdot}$. We will give examples for the constants $C_\psi$ and $H_\psi$. Especially, we are interested in the cases $\psi = \log$ and $\psi = \sqrt{\cdot}$. Furthermore, we will give useful criteria, whether $C_\psi=0$ respectively $H_\psi=\infty$. These cases should be avoided since then, the $CD\psi$ condition for Ricci-flat graphs, respectively the Harnack inequality, degenerates. We will see that $C_\psi = 0$ if $\psi$ does not satisfy a certain symmetry property. Moreover, we will see that $0 < H_\psi < \infty$ if $\psi$ is concave and $\psi''(1)<0$. First, we discuss the constant $H_\psi$ and next, the constant $C_\psi$. Then, we discuss the break of analogy in \cite{Bauer2013} and finally, we give the comparison between the Harnack inequalities. \begin{lemma}[Degeneration of $H_\psi$] \label{LDH} Let $\psi \in C^1(\mathbb{R}^+)$. If $\psi$ is concave and $\psi''(1)<0$, then $0< H_\psi < \infty$. \end{lemma} \begin{proof} To prove the lemma, we recall the definition of $H_\psi$, \[ H_\psi = \sup_{x>1} \frac{(\log x)^2}{\overline{\psi}(x)} \] with \[ \overline{\psi}(x)= \psi'(1)\cdot(x-1) - (\psi(x) - \psi(1)). \] By assumption, $\psi$ is concave and $\psi''(1)<0$. Thus, we also see that $\overline{\psi}$ is concave and $\overline{\psi}''(1)>0$. Additionally, we have $\overline{\psi}'(1)=0$. Hence, we see $\overline{\psi}(x) > 0$ for $x>1$ and $\overline{\psi}(x) \geq C x$ for some $C>0$ and large $x$. Consequently, \[ \lim_{x\to \infty} \frac{(\log x)^2}{\overline{\psi}(x)} = 0. \] By l'Hopital's rule, we obtain \[ \lim_{x\to 1} \frac{(\log x)^2}{\overline{\psi}(x)} = \frac 2 {\overline{\psi}''(1)} > 0. \] Thus, $H_\psi>0$. The previous observations guarantee that the function $[1,\infty] \to {\mathbb{R}}$, $x \mapsto \frac{(\log x)^2}{\overline{\psi}(x)}$ is continuous. Thus, it attains its maximum and hence, $H_\psi < \infty$. \end{proof} \begin{example}[$H_{\log}$ and $H_{\sqrt{\cdot}}$] We will prove the identities \begin{eqnarray} H_{\log} &=& 2, \\ H_{\sqrt{\cdot}} &=& 8. \\ \end{eqnarray} \begin{proof} We start with $H_{\log} = 2$. The function $(1,\infty) \to {\mathbb{R}}$ with $x \mapsto \frac{(\log x)^2}{\overline{\log}(x)}$ is monotonically non-increasing. Thus, \[ H_{\log} = \sup_{x>1} \frac{(\log x)^2}{\overline{\log}(x)} = \lim_{x\to 1} \frac{(\log x)^2}{\overline{\log}(x)} = \frac 2 {\overline{\log}''(1)} = 2. \] We prove $H_{\sqrt{\cdot}} = 8$. The function $(1,\infty) \to {\mathbb{R}}$ with $x \mapsto \frac{(\log x)^2}{\overline{\sqrt{\cdot}}(x)}$ is monotonically non-increasing. Thus, \[ H_{\sqrt{\cdot}} = \sup_{x>1} \frac{(\log x)^2}{\overline{\sqrt{\cdot}}(x)} = \lim_{x\to 1} \frac{(\log x)^2}{\overline{\sqrt{\cdot}}(x)} = \frac 2 {\overline{\sqrt{\cdot}}''(1)} = 8. \] This finishes the proof. \end{proof} \end{example} \begin{lemma}[Degeneration of $C_\psi$] Let $\psi \in C^1(\mathbb{R}^+)$. \begin{enumerate}[(1)] \item If $\psi$ is concave, then $C_\psi \geq 0$. \item If $\psi(x)+\psi(1/x) \neq 2 \psi(1)$ for some $x>0$, then $C_\psi \leq 0$. \end{enumerate} \end{lemma} \begin{proof} We recall the definition of $C_\psi$, \[ C_\psi = \inf_{x,y>0} \frac{\widetilde{\psi}(x,y)}{(\psi(x) + \psi(y) - 2\psi(1))^2} \] with \[ \widetilde{\psi}(x,y)= \left[\psi'(x) + \psi'(y) \right](1-xy) + x [\psi(y) - \psi(1/x)] + y [\psi(x) - \psi(1/y)]. \] First, we prove \emph{(1)}. It is sufficient to show $\widetilde{\psi}(x,y) \geq 0$ for all $x,y>0$. For $x,y>0$, we can write \[ \widetilde{\psi}(x,y) = y \left[\psi'(x) \left(\frac 1 y - x \right) + \psi(x) - \psi\left( \frac 1 y \right) \right] + x \left[\psi'(y) \left(\frac 1 x - y \right) + \psi(y) - \psi\left( \frac 1 x \right) \right]. \] Since $\psi$ is concave by assumption, we obtain $\psi'(x) \left(\frac 1 y - x \right) + \psi(x) - \psi\left( \frac 1 y \right) \geq 0$ for all $x,y>0$. Consequently, $\widetilde{\psi}(x,y) \geq 0$ for all $x,y>0$. Next, we show \emph{(2)}. We observe $\widetilde{\psi}(x,1/x)=0$ for all $x>0$. By the assumption, there exists $x>0$, such that $\psi(x)+\psi(1/x) \neq 2 \psi(1)$. Hence, \[ C_\psi \leq \frac{\widetilde{\psi}(x,1/x)}{(\psi(x) + \psi(1/x) - 2\psi(1))^2} =0. \] This finishes the proof. \end{proof} \begin{rem} It would be interesting to know whether the properties concavity of $\psi$ and $\psi(x) + \psi(1/x)= 2\psi(1)$ for all $x>0$, already characterize the case $C_\psi>0$. \end{rem} \begin{example}[$C_{\log}$ and $C_{\sqrt{\cdot}}$] We will prove \begin{eqnarray} \frac 1 2 \leq C_{\log} &\leq& 1, \\%&\approx& 0.7951229668476556380644481118711831948532790323170393603186\\ C_{\sqrt{\cdot}}&=& 0. \\ \end{eqnarray} \begin{rem} Numerical computations via Mathematica \cite{Wolfram2007} have shown that $$C_{\log} \approx 0.795.$$ There seems to be no analytic expression for $C_{\log}$. \end{rem} \begin{proof} First, we show $C_{\sqrt{\cdot}}= 0$. By the degeneration lemma of $C_\psi$, we obtain $C_{\sqrt{\cdot}}= 0$, since the square root does not satisfy the symmetry condition $\psi(x)+\psi(1/x)=2\psi(1)$ for $x>0$. Next, we show $\frac 1 2 \leq C_{\log} \leq 1$. For $x,y>0$ and $\psi = \log$, we observe \[ \frac{\widetilde{\psi}(x,y)}{(\psi(x) + \psi(y) - 2\psi(1))^2} = (x+y)\frac{ \frac 1 {xy} - 1 + \log xy}{(\log xy)^2} \geq 2 \sqrt{xy} \frac{ \frac 1 {xy} - 1 + \log xy}{(\log xy)^2}. \] The inequality is sharp if $x=y$. Hence, we have \[ C_{\log} = \inf_{x,y>0} \frac{\widetilde{\psi}(x,y)}{(\psi(x) + \psi(y) - 2\psi(1))^2} = \inf_{z>0} 2 \sqrt{z} \frac{ \frac 1 {z} - 1 + \log z}{(\log z)^2}. \] Denote $\varphi(\log x) := \frac {\sqrt{x} ( 1/x - 1 + \log x)} {(\log x)^2}$ for $x>0$ and $x\neq 1$. Then, we obtain \[ \varphi(x) = \frac {e^{x/2}\left( e^{-x} - 1 + x \right)}{x^2} \] and $C_{\log} = 2 \inf_{x\neq 0} \varphi(x)$. By l'Hopital's rule, we see the upper bound \[ \lim_{x\to 0} \varphi(x) = \lim_{x\to 0} \frac {\ddx \left( e^{-x} - 1 + x \right)} {\ddx \left( x^2\right)} = \lim_{x\to 0} \frac {-e^{-x} + 1 } {2x} = \frac 1 2 . \] Consequently, $ C_{\log} \leq 1$. To prove the lower bound, we write \begin{eqnarray} \varphi(x) &=& \frac {e^{x/2}\left( e^{-x} - 1 + x \right)}{x^2}\\ &=& \frac {e^{x/2}\left( e^{-x} - 1 + x \right)}{(e^{x/4} - e^{-x/4})^2} \cdot \frac {(e^{x/4} - e^{-x/4})^2} {x^2} \\ &=& \frac { e^{-x} - 1 + x }{(1 - e^{-x/2})^2} \cdot \left( \frac {e^{x/4} - e^{-x/4}} {x} \right)^2. \end{eqnarray} Since $e^{-x/2}\geq 1 - \frac x 2$ and consequently \begin{eqnarray} e^{-x} - 1 + x &=& (e^{-x} + 1)- 2\left(1 - \frac x 2\right) \\ &\geq& (e^{-x} + 1) - 2e^{-x/2} \\ &=& (1 - e^{-x/2})^2 \end{eqnarray} we get \[ \frac { e^{-x} - 1 + x }{(1 - e^{-x/2})^2} \geq 1. \] On the other hand, we can compute \begin{eqnarray} \frac {e^{x/4} - e^{-x/4}} {x} &=& \frac 1 x \left[ \sum_{k\geq 0} \frac {\left( x / 4\right)^k}{ k!} - \frac {\left(- x / 4\right)^k} { k!} \right] \\ &=& \frac 2 x \sum_{j \geq 0} \frac {\left( x / 4\right)^{2j+1}}{ (2j+1)!} \\ &=& \sum_{j \geq 0} \frac {2x^{2j} }{ (2j+1)! \cdot 4^{2j+1}} \\ &\geq& \left[ \frac {2x^{2j} }{ (2j+1)! \cdot 4^{2j+1}} \right]_{j=0} \\ &=& \frac 1 2 . \end{eqnarray} Putting these estimates together yields the lower bound of $C_{\log}$ \begin{eqnarray} \varphi(x)&=& \frac { e^{-x} - 1 + x }{(1 - e^{-x/2})^2} \cdot \left( \frac {e^{x/4} - e^{-x/4}} {x} \right)^2 \\ &\geq& 1 \cdot \left(\frac 1 2 \right)^2 = \frac 1 4 \end{eqnarray} for all $x \neq 0$ and hence, $C_{\log} \geq \frac 1 2$. \end{proof} \end{example} Since $C_{\sqrt{\cdot}}=0$, the $CD\sqrt{\cdot} $ condition degenerates for Ricci-flat graphs. In \cite{Bauer2013}, this problem has been solved by breaking the analogy to the manifolds case. More specifically, they introduced a weaker form of the $CD\sqrt{\cdot} $ inequality which requires $\Gamma_2^{\sqrt{\cdot}} (f)(x) \geq \frac 1 d \Delta^{\sqrt{\cdot}} f (x)$ only if $\Delta^{\sqrt{\cdot}}f(x) \leq 0$ for all $f \in C^+(V)$ and $x \in V$ with a graph $G=(V,E)$. This additional condition $\Delta^{\sqrt{\cdot}}f(x) \leq 0$ is the break of analogy. There seems to be no possibility to use the semigroup methods from \cite{Bakry2006} to derive a $\psi$-Li Yau inequality from a weak $CD \psi$ condition. But nevertheless, this weak $CD \psi$ condition is sufficient to prove Li-Yau type gradient estimates via the maximum principle. \subsection{Harnack inequalities on Ricci-flat graphs} A remarkable result of \cite{Bauer2013} is the Harnack inequality on Ricci-flat graphs. This states the following. If $G = (V,E)$ is a $D$-Ricci-flat graph, then one has \[ \log \frac{u(x,T_1)}{u(y,T_2)} \leq D \log \frac{T_2}{T_1} + \frac{4d(x,y)^2}{T_2-T_1} \] for all positive solutions $u \in C^1( V \times \nnegR)$ to the heat equation, for all $x,y \in V$ and for all positive $T_1<T_2$. As claimed in the introduction, we achieve an improvement for Ricci-flat graphs. \begin{corollary}[Harnack inequality for Ricci-flat graphs] \label{CHR} Let $\psi \in C^1(\mathbb{R}^+)$ be concave and let $D \in {\mathbb{N}}$. If $G = (V,E)$ is a $D$-Ricci-flat graph, then we have \[ \log \frac{u(x,T_1)}{u(y,T_2)} \leq \frac D {2 C_{\psi}} \log \frac{T_2}{T_1} + \frac{H_\psi d(x,y)^2}{T_2-T_1} \] for all positive solutions $u \in C^1( V \times \nnegR)$ to the heat equation, for all $x,y \in V$ and for all positive $T_1<T_2$. \end{corollary} \begin{proof} This is an easy consequence of the Harnack inequality (Theorem \ref{THarnack}) and the $CD\psi$ condition for Ricci-flat graphs (Theorem \ref{TRicci}). \end{proof} If we choose $\psi = \log$, then by using $C_{\log} \geq \frac 1 2$ and $H_{\log}=2$ (cf. Subsection \ref{SSC}), we obtain \[ \log \frac{u(x,T_1)}{u(y,T_2)} \leq D \log \frac{T_2}{T_1} + \frac{2 d(x,y)^2}{T_2-T_1} . \] Using the numerical result $C_{\log} \approx 0.795$, the previous corollary improves to \[ \log \frac{u(x,T_1)}{u(y,T_2)} \leq 0.629 D \log \frac{T_2}{T_1} + \frac{2 d(x,y)^2}{T_2-T_1} . \] This means, our upper bound is by a factor of 1.59 smaller than the one obtained in \cite{Bauer2013}. Florentin Münch, \\ Mathematisches Institut, Friedrich Schiller Universität Jena, \\ D-07745 Jena, Germany\\ \texttt{[email protected]} \end{document}	arXiv
\begin{document} \title{Approximate performance analysis of generalized join the shortest queue routing} \renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \footnotetext[1]{Department of Mechanical Engineering, Eindhoven University of Technology} \footnotetext[2]{Department of Mathematics and Computer Science, Eindhoven University of Technology} \footnotetext[3]{Department of Industrial Engineering \& Innovation Sciences, Eindhoven University of Technology} \blfootnote{E-mail address: {\tt [email protected]}} \renewcommand{\fnsymbol{footnote}}{\arabic{footnote}} \setcounter{footnote}{0} \begin{abstract} In this paper we propose a highly accurate approximate performance analysis of a heterogeneous server system with a processor sharing (PS) service discipline and a general job-size distribution under a \textit{generalized join the shortest queue} (GJSQ) routing protocol. The GJSQ routing protocol is a natural extension of the well-known join the shortest queue (JSQ) routing policy that takes into account the non-identical service rates in addition to the number of jobs at each server. The performance metrics that are of interest here are the equilibrium distribution and the mean and standard deviation of the number of jobs at each server. We show that the latter metrics are near-insensitive to the job-size distribution using simulation experiments. By applying a \textit{single queue approximation} (SQA) we model each server as a single server queue with a state-dependent arrival process, independent of other servers in the system, and derive the distribution of the number of jobs at the server. These state-dependent arrival rates are intended to capture the inherent correlation between servers in the original system and behave in a rather atypical way. \end{abstract} \section{Introduction} \label{sec:introduction} \subsection{Motivation} \label{subsec:motivation} This work is motivated by web server farms. Server farms have gained popularity for providing scalable and reliable computing and web services. Most commonly the objective in analyzing such a system lies in the determination of an optimal or near-optimal load balancing routing protocol so as to maximize the performance of the system, see, e.g., \cite{HeterogeneousPSFarms_LoadBalancing_Altman2011,Harchol-Balter2013,HeterogeneousServerFarms_SizeStateAwareRouting_Hyytia2012}, where the performance of interest is usually the mean response time for an arbitrary job. In this paper the objective is to report some interesting properties of the arrival flow to each server and suggest an approximation approach for the GJSQ routing protocol. We consider farms with heterogeneous servers, which is motivated by the different hardware and the wide variety of computing capacities regarding processing power and memory access performance seen in practice in server farms \cite{Ortiz}. We assume that service requests arrive to the system according to a Poisson process. Upon arrival, a front-end dispatcher routes the request to one of the servers. After the request has been routed to the server, we assume that it cannot balk or jokey. All requests routed to a server are sharing the provided service (think of bandwidth, CPU, or RAM). We assume a PS service discipline at each server since it closely approximates the scheduling policies \cite{HB2003,SLA} employed by most commodity operating systems (e.g., Linux CPU time-sharing) and is a popular policy in computing centers (e.g., Cisco Local Director, IBM Network Dispatcher and Microsoft Sharepoint, see \cite{Cardellini2002} for a survey). In \cite{JSQ_WebServerFarms_Gupta2007} the authors consider a server farm consisting of homogeneous servers, where upon arrival jobs are routed according to the JSQ routing protocol. This protocol in case of homogeneous servers, due to the PS service discipline, is performing near-optimal in terms of the mean response time. However, as indicated by Whitt in \cite{Whitt1986}, the JSQ policy is far from optimal in case of heterogeneous servers. In \cite{Banawan1989} the authors comment on the performance of various systems under different routing protocols and conclude that the shortest expected delay (SED) routing protocol is near-optimal in terms of mean response time. The SED policy is a policy that routes jobs upon arrival to the queue promising the minimum expected delay (which also includes the processing time). In case of exponential job-size distributions, the GJSQ and SED routing protocols are identical and in case of homogeneous servers GJSQ and JSQ are the same. However, in case of general job-size distributions and heterogeneous servers we assume that the only available information are the service rates and the number of jobs at each server, i.e.~we do not keep track of residual processing times. Due to the complexity and the various challenges that the model at hand presents, we restrict our analysis to the case of two heterogeneous servers with a general job-size distribution under the GJSQ routing protocol. From here onwards we refer to this model as the $M/G(1,s)/2/GJSQ/PS$ system, abbreviated as the GJSQ model, where $G$ is the job-size distribution and 1 and $s$ are the service rates at servers 1 and 2, respectively. The approach described in this paper can be seen as a first stepping stone towards the analysis of heterogeneous server farms with PS servers; a very broad area, full of interesting problems. Moreover, the ideas presented here extend the work of Gupta et al.~\cite{JSQ_WebServerFarms_Gupta2007} on the analysis of the JSQ routing for homogeneous web server farms. \subsection{Related work} \label{subsec:related_work} To the best of our knowledge there is no previous mathematical analysis of the GJSQ system. In \cite{SED_Selen2015}, Selen et al. derive the joint equilibrium distribution of the number of jobs at each server in the $M/M(1,s)/2/SED/FCFS$ model. They prove that this distribution can be expressed as an infinite series of product forms using the compensation approach. The benefit of that approach is that it produces, by truncating the series expression, numerical results with an a priori set accuracy level. Unfortunately, the compensation approach is not appropriate (in its current setting) for multiple servers, nor for general job-size distributions. Before \cite{SED_Selen2015}, very little was known regarding the mathematical analysis of the SED policy. In \cite{Lui1995}, the authors suggest two models that act as upper and lower bounds to the SED system. However, they do not provide closed form expressions for the equilibrium distribution of these two bounding models, but only an algorithmic approach based on matrix analytic methods. Furthermore, in \cite{Foschini1978,Laws1992}, the authors show that the SED routing policy is asymptotically optimal in terms of the mean response time and results in complete resource pooling in the heavy traffic limit. This heavy traffic limit result may be used in a similar manner as in \cite{Nelson1989}. However, after a few numerical experiments, we concluded that this approximation in our case results in poor estimates and for this reason we did not proceed in this direction. On the contrary, the approach developed by Gupta et al. \cite{JSQ_WebServerFarms_Gupta2007} on approximating the distribution of the number of jobs at each server, as we show in this paper, is appropriate for the GJSQ setting with heterogeneous servers. More concretely, in \cite{JSQ_WebServerFarms_Gupta2007}, the authors develop the SQA method that accurately determines the distribution of the number of jobs at each server by modeling each queue as an $M_n/M/1/PS$ system with state-dependent arrival rates. These state-dependent arrival rates are referred to as the \textit{conditional arrival rates} and are constructed in such a way that they capture the inherent correlated behavior of the complete server farm. \subsection{Contributions} \label{subsec:contributions} We believe that we provide the first approximate analysis of the equilibrium distribution and moments of the number of jobs at each server in the GJSQ system (and by Little's law also the mean response time for an arbitrary job). Moreover, the approximation is highly accurate: we encounter a maximum relative difference between the approximation and simulations of 2.2\%. In deriving these approximations, we provide three key contributions: \begin{enumerate}[label = \arabic.] \item The mean and standard deviation of the number of jobs at each server and the conditional arrival rates are near-insensitive to the job-size distribution. This allows us to study the more tractable model with an exponential job-size distribution. \item In case of an exponential job-size distribution, the SQA method yields the same equilibrium distribution for the number of jobs at each server as in the original GJSQ model. \item For the application of the SQA method we present an approach for the derivation of the conditional arrival rates. In particular, we show that the conditional arrival rates, say $\lambda_i(n), ~ i = 1,2, ~ n \in \mathbb{N}_0$, to server 1 satisfy \begin{equation} \lambda_1(n) \to \rho^{1 + s} \textup{ as } n \to \infty, \end{equation} where $\rho$ is the load on the system, see Section~\ref{sec:model_description}, and the conditional rates to server 2 for large $n$ oscillate between $s$ different points. Note that the former result is similar to the result obtained in \cite{JSQ_WebServerFarms_Gupta2007} for the case $s = 1$, however the latter result is very atypical and is discussed in greater detail in Section~\ref{subsec:arrival_rates}. \end{enumerate} \subsection{Outline} \label{subsec:outline} The rest of the paper is organized as follows. In Section~\ref{sec:model_description} we give a detailed model description and formally define and investigate the time-average and conditional arrival rates. Section~\ref{sec:insensitivity_results} is devoted to showing that the performance metrics of interest are near-insensitive to the job-size distribution. We describe the SQA and determine the conditional arrival rates in Section~\ref{sec:single_queue_approximation}. The approximations are evaluated in Section~\ref{sec:evaluating_approximation}. In Section~\ref{sec:conclusion} we present some conclusions. \section{Model description} \label{sec:model_description} \subsection{Heterogeneous servers} \label{subsec:heterogeneous_servers} We consider a system of two heterogeneous servers and a single dispatcher. The servers employ a PS service discipline and can have different service rates, i.e.~server 1 has service rate 1 and server 2 has service rate $s$. Jobs arrive to the dispatcher according to a Poisson process with rate $\lambda$ and are routed immediately to one of the servers. Jobs cannot switch servers after being routed. We detail the routing policy in Section~\ref{subsec:routing_policy}. The size of a job is drawn from a general distribution $G$. Without loss of generality we assume that the mean job size is 1. Note that, for example, the (residual) processing time of a (residual) size $G$ job that runs on server 2 that is currently serving $q_2$ jobs is given by $G q_2/s$. In what follows we assume that $s$ is a positive integer number. In the general case $s \in \mathbb{R}_+$ we can bound the corresponding system by two systems with service rates given by the closest two integers to $s$. \subsection{Routing policy} \label{subsec:routing_policy} The routing policy employed by the dispatcher is a state-aware policy, i.e.~the dispatcher is aware of the number of jobs at each server just before an arrival instant, $q_1$ and $q_2$, and the service rates. The GJSQ routing policy routes an arriving job to the server with the smallest index $(q_i + 1) / s_i$, where $s_i$ is the service rate at server $i$. In case of a tie, the job is randomly routed to one of the servers. These indexes may be interpreted as an estimate of the expected processing time for the arriving job, made by the dispatcher who is unaware of the job-size distribution and the remaining processing times of the jobs currently in service, and furthermore ignores future arrivals. Under this routing policy, we define the load on this system as \begin{equation} \rho := \lambda / (1 + s). \end{equation} Throughout the rest of this paper we assume that $\rho < 1$. Although not necessarily optimal, GJSQ routing outperforms JSQ routing when servers are non-identical. GJSQ routing attempts to balance the load on the servers by taking into account the different service rates in addition to the information on the current number of jobs at each server. In Figure~\ref{fig:time-average_arrival_rate} we show that the long-term fraction of jobs routed to the two servers is a function of the load $\rho$. In light traffic GJSQ assigns all jobs to the fast server and in heavy traffic the load is divided proportionally according to the service rates. This is in contrast with JSQ routing, which assigns a long-term fraction of the jobs to server 1 that decreases from 1/2 to $1/(1 + s)$ for increasing load $\rho$ (verified through simulation). \begin{figure} \caption{Simulated long-term fraction of jobs routed to server $i$ as a function of the load $\rho$, where $s = 4$ and the job-size distribution is exponential. Dashed lines represent expected behavior.} \label{fig:time-average_arrival_rate} \end{figure} \subsection{Arrival rates} \label{subsec:arrival_rates} We briefly introduce two important concepts related to the (time-average) arrival rates to each server. These concepts will be used throughout the paper. \begin{definition}\label{def:time-average_arrival_rates} In the GJSQ model, the \textit{time-average arrival rate} to server $i$ is defined as \begin{equation} \overline{\la}_i := \lim_{t \to \infty} \frac{A_i(t)}{t}, \end{equation} where $A_i(t)$ is the number of arrivals at server $i$ during the time interval $[0,t]$. \end{definition} \begin{definition}\label{def:conditional_arrival_rates} In the GJSQ model, the \textit{conditional arrival rate} to server $i$, given that server $i$ has $n$ jobs, is defined as \begin{equation} \lambda_i(n) := \lim_{t \to \infty} \frac{A_{i,n}(t)}{T_{i,n}(t)}, \label{eqn:definition_conditional_arrival_rates} \end{equation} where $A_{i,n}(t)$ is the number of arrivals at server $i$ during the time interval $[0,t]$ that see $n$ jobs at server $i$ on arrival (excluding themselves), and $T_{i,n}(t)$ is the total time spent by server $i$ with $n$ jobs during the time interval $[0,t]$. \end{definition} The two definitions above are related. Assuming it exists, let $\pi_i(n)$ be the equilibrium probability that there are $n$ jobs at server $i$, then $\overline{\la}_i = \sum_{n = 0}^\infty \lambda_i(n) \pi_i(n)$. Figure~\ref{fig:interesting_conditional_arrival_rates_server_2} depicts the conditional arrival rates to server 2 for varying $s$. Intuitively it makes sense that if a server has many jobs, the other server will probably have few jobs and thus it is less likely that the dispatcher routes the job to that server. However, what we see here is a peculiar repeating pattern that has $s$ different points and does not align with this intuition. We see that if server 2 has a multiple of $s$ jobs (or one less), fewer jobs are routed to server 2. This pattern is difficult to explain, but it is definitely related to the probability that server 1 has a lower index than server 2, given that server 2 currently has $n$ jobs. We expect and indeed verify that this probability also follows such a repeating pattern. Additionally, states in server 2 are somewhat similar if they differ by a multiple of $s$ jobs, which can be derived from the equilibrium distribution in \cite{SED_Selen2015}. \begin{figure} \caption{The conditional arrival rates to server 2 oscillate between $s$ points.} \label{fig:interesting_conditional_arrival_rates_server_2} \end{figure} \section{Near-insensitivity} \label{sec:insensitivity_results} In \cite{JSQ_WebServerFarms_Gupta2007} the authors establish a near-sensitivity property in the setting of a homogeneous server farm with JSQ routing. In particular, the first and second moment of the number of jobs at server $i$, $Q_i$, and the conditional arrival rates are near-insensitive to the job-size distribution. The near-insensitivity of these two metrics seems related to the insensitivity of the equilibrium distribution to the job-size distribution in PS servers, see, e.g., \cite[Theorem~4.2]{JSQ_WebServerFarms_Gupta2007}; and the fact that the routing policy only uses the number of jobs at each server when making a decision, as opposed to, e.g., using residual processing times. The GJSQ routing decisions are based on the dynamically changing number of jobs at each server as well as the service rates. Indeed, one expects the near-insensitivity properties to extend also to the case of heterogeneous servers and GJSQ routing. Establishing this near-insensitivity property is important, since it allows us to limit our attention to the more tractable GJSQ system with an exponential job-size distribution. \subsection{Simulation settings} \label{subsec:simulation_settings} To support our claims, we simulate the GJSQ model. A simulation consists of $2 \cdot 10^6$ job departures and each simulation is repeated 50 times. Statistics are only computed for departed jobs, i.e.~data of jobs that are still in service at the end of the simulation are discarded. In Table~\ref{tbl:job-size_distributions} we list the four job-size distributions considered in this paper. \begin{table} \centering \begin{tabular}{{5}{l}} Name & Distribution & Support & Variance \\ \hline \hline uni & Uniform & $[0,2]$& $1/3$ \\ exp & Exponential & $[0,\infty)$ & 1 \\ weib & Weibull & $(0,\infty)$ & 5 \\ logn & Log-normal & $(0,\infty)$ & 10 \end{tabular} \caption{Job-size distributions used in simulations.} \label{tbl:job-size_distributions} \end{table} \subsection{Near-insensitivity results} \label{subsec:near-insensitivity_results} In Table~\ref{tbl:simulation_2servers_mean_queue_length} we show simulated statistics on the mean and standard deviation $\sigma(\cdot)$ of $Q_i$ for the GJSQ model with various job-size distributions. For the settings considered in Table~\ref{tbl:simulation_2servers_mean_queue_length}, the mean number of jobs at server $i$ deviates by no more than 3.6\% from the exponential case, while the standard deviation deviates by at most 4.4\%. The largest deviations from the exponential case occur for the log-normal job-size distribution. This is as expected, since this job-size distribution has a variance that is 10 times higher than the variance of the exponential job-size distribution. Although the results are not as strong as those shown in \cite[Figure~3]{JSQ_WebServerFarms_Gupta2007}, we conclude that the more volatile environment of heterogeneous servers and GJSQ routing also has the near-insensitivity property for $\E{Q_i}$ and $\sigma(Q_i)$. Moreover, the performance in terms of the mean response time is also near-insensitive to the job-size distributions by Little's law. \begin{table} \centering \footnotesize \begin{tabular}{ccc\|cccc\|cr} & & & \multicolumn{4}{c}{Job-size distribution} & \multicolumn{2}{c}{SQA} \\ $s$ & $\rho$ & Metric & uni & exp & weib & logn & Value & Diff. \\ \hline \hline 2 & 0.7 & $\E{Q_1}$ & 0.9139 (0.0030) & 0.9232 (0.0030) & 0.9223 (0.0029) & 0.9361 (0.0038) & 0.9077 & 1.7\% \\ & & $\sigma(Q_1)$ & 1.0404 (0.0049) & 1.0505 (0.0050) & 1.0560 (0.0056) & 1.0704 (0.0067) & 1.0462 & 0.4\% \\ & & $\E{Q_2}$ & 2.0111 (0.0059) & 2.0289 (0.0061) & 2.0222 (0.0057) & 2.0519 (0.0074) & 2.0329 & $-$0.2\% \\ & & $\sigma(Q_2)$ & 2.0302 (0.0099) & 2.0465 (0.0106) & 2.0506 (0.0114) & 2.0813 (0.0141) & 2.0484 & $-$0.1\% \\ \cline{2-9} & 0.9 & $\E{Q_1}$ & 3.2244 (0.0336) & 3.2797 (0.0336) & 3.2316 (0.0298) & 3.2396 (0.0266) & 3.2188 & 1.9\% \\ & & $\sigma(Q_1)$ & 3.2208 (0.0590) & 3.2716 (0.0723) & 3.2186 (0.0575) & 3.2002 (0.0505) & 3.2161 & 1.7\% \\ & & $\E{Q_2}$ & 6.6841 (0.0676) & 6.7915 (0.0674) & 6.6834 (0.0587) & 6.6988 (0.0524) & 6.6424 & 2.2\% \\ & & $\sigma(Q_2)$ & 6.4289 (0.1185) & 6.5288 (0.1453) & 6.4186 (0.1141) & 6.3828 (0.1016) & 6.4091 & 1.9\% \\ \hline \hline 4 & 0.7 & $\E{Q_1}$ & 0.4688 (0.0016) & 0.4747 (0.0017) & 0.4705 (0.0017) & 0.4667 (0.0022) & 0.4741 & 0.1\% \\ & & $\sigma(Q_1)$ & 0.6685 (0.0024) & 0.6730 (0.0026) & 0.6700 (0.0029) & 0.6652 (0.0031) & 0.6655 & 1.1\% \\ & & $\E{Q_2}$ & 2.5386 (0.0063) & 2.5507 (0.0069) & 2.5177 (0.0070) & 2.4997 (0.0067) & 2.5866 & $-$1.4\% \\ & & $\sigma(Q_2)$ & 2.5082 (0.0102) & 2.5179 (0.0115) & 2.4936 (0.0133) & 2.4744 (0.0122) & 2.5457 & $-$1.1\% \\ \cline{2-9} & 0.9 & $\E{Q_1}$ & 1.8662 (0.0191) & 1.8793 (0.0145) & 1.8830 (0.0196) & 1.9400 (0.0223) & 1.8813 & $-$0.1\% \\ & & $\sigma(Q_1)$ & 1.9404 (0.0338) & 1.9539 (0.0314) & 1.9801 (0.0444) & 2.0394 (0.0408) & 1.9566 & $-$0.1\% \\ & & $\E{Q_2}$ & 8.2405 (0.0769) & 8.2773 (0.0597) & 8.2631 (0.0783) & 8.4863 (0.0861) & 8.3642 & $-$1.0\% \\ & & $\sigma(Q_2)$ & 7.6982 (0.1374) & 7.7507 (0.1264) & 7.8485 (0.1815) & 8.0912 (0.1652) & 7.7692 & $-$0.2\% \\ \end{tabular} \caption{Simulated mean and standard deviation of $Q_i$, for the GJSQ system with various $s$, $\rho$ and job-size distributions. Sample standard deviation is shown in parentheses. Last two columns show the value obtained by the SQA and the relative difference with respect to the exponential case.} \label{tbl:simulation_2servers_mean_queue_length} \end{table} Concerning the conditional arrival rates, we see in Figure~\ref{fig:conditional_arrival_rate} that the simulated values for the job-size distributions of Table~\ref{tbl:job-size_distributions} match the results of the algorithm for the exponential case \cite{SED_Selen2015}. Simulated values for states where the sample standard deviation is not too high differ by at most 5\% from the results for the exponential case. So, also the conditional arrival rates are near-insensitive to the job-size distribution. \begin{figure} \caption{Simulated conditional arrival rates in the GJSQ system with various job-size distributions. The dotted curves represent values determined by the algorithm in \cite{SED_Selen2015} for the exponential job-size distribution.} \label{fig:conditional_arrival_rate} \end{figure} \section{Single queue approximation} \label{sec:single_queue_approximation} We have established near-insensitivity of $\E{Q_i}$, $\sigma(Q_i)$ and the conditional arrival rates to the job-size distributions. Thus, we may limit our attention to systems with an exponential job-size distribution. In this section we derive an approximation for the distribution of the number of jobs at each server using the SQA, which models server $i$ as an $M_n/M_i/1/PS$ queue with state-dependent arrival rates $\lambda_i(n)$, see also \cite[Section~3]{JSQ_WebServerFarms_Gupta2007}. SQA is exact when the job-size distribution is exponential and the routing belongs to a specific class of routing policies; the following theorem is a version of \cite[Theorem~3.1]{JSQ_WebServerFarms_Gupta2007} that is applicable to the GJSQ model. \begin{definition}\label{def:stationary_state-aware_routing_policy} A \textit{stationary state-aware routing policy} is a time-stationary routing policy that only uses information about the number of jobs at the servers and the service rates at the instant of an arrival. The decisions may be made probabilistically, possibly biased in favor of certain servers. \end{definition} \begin{theorem}\label{thm:SQA_exact_Markovian} Consider the $M/M(1,s)/2/\mathcal{R}/\mathcal{S}$ queueing model, where $\mathcal{R}$ is any stationary state-aware routing policy, e.g.~GJSQ, and $\mathcal{S}$ is any stationary, size-independent, work-conserving service discipline, e.g.~PS. Consider server $i$ in this model. Then SQA with the exact conditional arrival rates $\lambda_i(\cdot)$ yields the same equilibrium distribution for the number of jobs at each server as in the original model. \end{theorem} It remains to specify the conditional arrival rates $\lambda_i(n)$ for both servers. We combine exact limiting results for $n \ge N_i$ and approximation results for $n < N_i$, where $N_1 = 3$ and $N_2 = 2s$. These choices for $N_i$ result in accurate approximations. We note that Theorem~\ref{thm:SQA_exact_Markovian} implies that in order to determine the conditional arrival rates, we may assume a FCFS service discipline. In Figure~\ref{fig:interesting_conditional_arrival_rates_server_2} we have seen that the conditional arrival rates $\lambda_i(n)$ exhibit a repeating pattern from some $n$ and onwards. We rigorously characterize this limiting repeating pattern in the next theorem. \begin{theorem}\label{thm:limiting_conditional_arrival_rates} For the $M/M(1,s)/2/GJSQ/PS$ queueing model with $s \in \mathbb{N}$, \begin{align} \lambda_1^{\textup{lim}} \coloneqq \lim_{n \to \infty} \lambda_1(n) &= \rho^{1 + s}, \label{eqn:limiting_conditional_arrival_rate_server_1}\\ \lambda_2^{\textup{lim}}(r) \coloneqq \lim_{n \to \infty} \lambda_2(sn + r) &= \begin{cases} s \frac{A(r + 1)}{A(r)}, & r = 0,1,\ldots,s - 2, \\ s \rho^{1 + s} \frac{A(0)}{A(s - 1)}, & r = s - 1, \end{cases}\label{eqn:limiting_conditional_arrival_rate_server_2} \end{align} where \begin{equation} A(r) = \sum_{i = 1}^s \eta_i \frac{\beta_i}{\rho^{1 + s} - \beta_i} \ipos{\rho^{1 + s},\beta_i,r} + h(r) + \frac{\beta_{s + 1}}{1 - \beta_{s + 1}} \ineg{\rho^{1 + s},\beta_{s + 1},r}, \label{eqn:A(r)} \end{equation} and the variables $\beta_1,\beta_2,\ldots,\beta_{s+1}$, $\eta_1,\eta_2,\ldots,\eta_{s}$, and the functions $h(\cdot)$, $\ipos{\cdot}$, $\ineg{\cdot}$ are defined in Appendix~\ref{app_sec:definitions}. \end{theorem} \begin{proof} See Appendix~\ref{app_sec:proof_limiting_conditional_arrival_rates}. \end{proof} For the rates $\lambda_1(n), ~ n < 3$ and $\lambda_2(n), ~ n < 2s$ we provide approximations that are functions of $s$ and $\rho$. For server 1 we use a multiple linear regression model to fit an approximate function for the conditional arrival rates on data obtained from the algorithm in \cite{SED_Selen2015} for $s = 1,2,3,4$ and $\rho$ from 0.3 to 0.99. Obviously, one can also use conditional arrival rates obtained by simulation for these fitting purposes. We carefully select a set of 5 independent variables for each conditional arrival rate. This leads to the following approximate conditional arrival rates for server 1: \begin{align} \frac{\lambda_1(0)}{\rho^{1 + s}} &\approx \begin{bmatrix} s\rho & s & \frac{s}{\rho} & 1 & \frac{\rho^2}{s^2} \end{bmatrix} \begin{bmatrix} 0.669 & -1.90 & 1.23 & 1.86 & -0.192 \end{bmatrix}^T, \label{eqn:conditional_arrival_rate_server_1_n_0} \\ \frac{\lambda_1(1)}{\rho^{1 + s}} &\approx \begin{bmatrix} s\rho^2 & 1 & \frac{1}{\rho} & \frac{1}{s\rho} & \rho^{1/s} \end{bmatrix} \begin{bmatrix} -0.00856 & 1.37 & -0.0578 & 0.123 & -0.254 \end{bmatrix}^T, \label{eqn:conditional_arrival_rate_server_1_n_1} \\ \frac{\lambda_1(2)}{\rho^{1 + s}} &\approx 1 + \begin{bmatrix} s\rho & \frac{1}{s\rho} & \frac{\rho}{s^2} & \frac{1}{s^2} & \rho^{1/s} \end{bmatrix} \frac{1}{100} \begin{bmatrix} -0.131 & -0.820 & -6.48 & 10.4 & 0.893 \end{bmatrix}^T \label{eqn:conditional_arrival_rate_server_1_n_2} \end{align} with $\bld{x}^T$ the transpose of a vector $\bld{x}$. For $s = 1$, one should consider $\lambda_1(\cdot) = \lambda_2(\cdot)$ and use the approximations presented in \eqref{eqn:conditional_arrival_rate_server_1_n_0}-\eqref{eqn:conditional_arrival_rate_server_1_n_2}. For server 2, let us note that $\lambda_2(n) = \lambda, ~ n = 0,\ldots,s - 2$ due to the GJSQ routing. Using a multiple regression model in this case is more difficult, since the number of states for which we need to obtain a fit increases with $s$. To circumvent a possibly complex fitting procedure, we establish a relation between the conditional arrival rates for the states $n = s - 1,s,\ldots,2s - 1$ and the limiting conditional arrival rates determined in Theorem~2. Namely, \begin{align} \lambda_2(n) &\approx \Bigl( 1 + \bigl( \frac{1}{s} - \frac{\rho}{2s - 1} \bigr) \frac{1}{2^{n - (s - 1)}} \Bigr) \lambda_2^{\mathrm{lim}}(n - s), \label{eqn:conditional_arrival_rate_server_2_n_s-1_until_2s-1} \end{align} where for convenience $\lambda_2^\textup{lim}(-1) = \lambda_2^\textup{lim}(s - 1)$. The approximations \eqref{eqn:conditional_arrival_rate_server_1_n_0}-\eqref{eqn:conditional_arrival_rate_server_2_n_s-1_until_2s-1} behave in various limiting regimes as expected: \begin{proposition}\label{prop:conditional_arrival_rates_limiting_regimes}\hspace{1em} \begin{enumerate}[label = \textup{\arabic.}] \item For $s \to \infty$, we have that $\lambda_1(n) \downarrow 0$ and $\lambda_2(n) = \lambda$ for all $n \in \mathbb{N}_0$. No job will join server 1, since the processing times in server 2 are instantaneous. \item In the light-traffic regime, i.e.~$\rho \downarrow 0$, we find that $\lambda_1(n) \downarrow 0, ~ n \in \mathbb{N}_0$ and $\lambda_2(n) \downarrow 0, ~ n \ge s - 1$. \item In the heavy-traffic regime, i.e.~$\rho \uparrow 1$, we establish that $\overline{\la}_1/\lambda = 1/(1 + s)$ and $\overline{\la}_2/\lambda = s/(1 + s)$ which is consistent with the findings in Figure~\ref{fig:time-average_arrival_rate}. \end{enumerate} \end{proposition} \begin{proof} 1. Follows straightforwardly by taking the limit $s \to \infty$ in \eqref{eqn:conditional_arrival_rate_server_1_n_0}-\eqref{eqn:conditional_arrival_rate_server_1_n_2} while taking into account that $\rho = \lambda/(1 + s)$. Furthermore, observe that $\lambda_2(n) = \lambda, ~ n = 0,\ldots,s - 2$, so that $\lim_{s \to \infty} \lambda_2(n) = \lambda, ~ n \in \mathbb{N}_0$. \noindent 2. See Appendix~\ref{app_sec:proof_limiting_regimes}. \noindent 3. From the approximate conditional arrival rates $\lambda_1(\cdot)$ one can derive (approximate) equilibrium probabilities $\pi_1(\cdot)$. Then, $\overline{\la}_1 = \sum_{n = 0}^\infty \lambda_1(n) \pi_1(n) = \sum_{n = 0}^\infty \pi_1(n+1) = 1 - \pi_1(0)$ by exploiting the balance equations. For $\rho \uparrow 1$ it can be verified that $\pi_1(0) \downarrow 0$, so that $\lim_{\rho \uparrow 1} \overline{\la}_1/\lambda = 1/(1 + s)$. The result for server 2 follows analogously. \end{proof} \section{Evaluating the approximation} \label{sec:evaluating_approximation} We are now in a position to evaluate the proposed approximations. First, we show that the approximations for the conditional arrival rates follow closely the exact values, which were determined using the algorithm \cite{SED_Selen2015}. Second, we establish that the mean and standard deviation of the number of jobs at each server is also well approximated. Figure~\ref{fig:comparison_conditional_arrival_rates} compares the conditional arrival rates obtained from the algorithm in \cite{SED_Selen2015} and the approximations derived in the previous section. For the cases considered in the figure, the maximum relative difference of the approximation with respect to the values determined by the algorithm is 1.5\% for $\lambda_1(\cdot)$ and 4.1\% for $\lambda_2(\cdot)$. Since both methods consider exponential job-size distributions, the difference is due to the fitting error introduced in the approximations of the conditional arrival rates in Section~\ref{sec:single_queue_approximation} and the truncation error in the algorithm in \cite{SED_Selen2015}. However, since the truncation error has been chosen to be of the order $10^{-5}$, it has little influence. \begin{figure} \caption{Comparison of conditional arrival rates determined by the algorithm in \cite{SED_Selen2015} (lines) and our approximations (marks) for $\rho = 0.4$ (\linesolid,\marktriangle), $\rho = 0.7$ (\linedashed,\markcircle), and $\rho = 0.9$ (\linedotted,\marksquare).} \label{fig:comparison_conditional_arrival_rates} \end{figure} In the two rightmost columns of Table~\ref{tbl:simulation_2servers_mean_queue_length} we provide the mean and standard deviation of the number of jobs at both servers determined using the SQA. We report highly accurate approximations that deviate less than 2.2\% from the case with an exponential job-size distribution for the listed values of $s$ and $\rho$. Although our approximations are not aimed at the case $s = 1$, we report accurate approximations also in this setting with maximum relative differences of the same order as in \cite[Section~6.1]{JSQ_WebServerFarms_Gupta2007}. \section{Conclusion} \label{sec:conclusion} In this paper, we provide an approximate performance analysis of a queueing system consisting of two heterogeneous PS servers with service rates 1 and $s \in \mathbb{N}$, respectively, a general job-size distribution and GJSQ routing. More concretely, we derived the approximate equilibrium distribution of the number of jobs at each server using the SQA method. In order to apply SQA, we established that the GJSQ system is near-insensitive to the job-size distribution and thus we approximated the system at hand with exponentially distributed job-sizes. We then approximated the conditional arrival rates for the exponential case, by combining exact limiting results for large number of jobs and approximation results, which were obtained using a multiple linear regression model, for small number of jobs. Ultimately, the aforementioned approach resulted in approximations that are highly accurate; we reported a maximum relative difference with respect to exact or simulation results of 4.1\% for the conditional arrival rates and 2.2\% for the mean and standard deviation of the number of jobs at each server. In this paper we set the groundwork for the analysis of server farms with heterogeneous servers under the GJSQ routing policy by analyzing the case of two servers. Of course, server farms consist of multiple servers so it is in our future plans to extend the analysis presented in this paper to more than two servers. The most difficult aspect of this task would be the derivation of the conditional arrival rates, which possibly has to rely on simulation data, since the approach in \cite{SED_Selen2015} is in its current setting restricted to two servers. In Figure~\ref{fig:conditional_arrival_rates_three_servers} we present an example of the simulated conditional arrival rates in case of three servers with service rates 1, 2 and 5. Note that the structure of the conditional arrival rates is as expected, i.e.~the number of points in the repeating pattern is directly related to the rate of service, but the exact values of these points differ from the values obtained by formulas \eqref{eqn:limiting_conditional_arrival_rate_server_1} and \eqref{eqn:limiting_conditional_arrival_rate_server_2}. \begin{figure} \caption{Simulated conditional arrival rates for a system with three servers with service rates 1 (\linedotted), 2 (\linedashed), and 5 (\linesolid), with $\rho = 0.7$.} \label{fig:conditional_arrival_rates_three_servers} \end{figure} \subsection{Acknowledgments} \label{subsec:acknowledgements} This work was supported by an NWO free competition grant and the NWO Gravitation Project NETWORKS. \appendix \section{Definition of the variables and functions in Theorem~\ref{thm:limiting_conditional_arrival_rates}} \label{app_sec:definitions} The definitions can be found in \cite[Lemma~5.11]{SED_Selen2015}, but we summarize them here. We denote $\alpha = \rho^{1 + s}$. The functions $\ipos{\cdot}$ and $\ineg{\cdot}$ are vectors of size $s$ and have their first element equal to 1, i.e.~$\ipos{\alpha,\beta,0} = \ineg{\alpha,\beta,0} = 1$. Furthermore, the vectors satisfy \begin{equation} \frac{\ipos{\alpha,\beta,r}}{\ipos{\alpha,\beta,0}} = \Bigl( \frac{\alpha \beta (1 + s)(\rho + 1) - \beta^2 (1 + s) \rho - \alpha^2}{\alpha \beta s} \Bigr)^r, \quad r = 0,1,\ldots,s - 1 \label{eqn:inner_pos_eigenvector} \end{equation} and \begin{equation} \frac{\ineg{\alpha,\beta,r}}{\ineg{\alpha,\beta,0}} = \frac{\Func{\alpha,\beta,\func{-}{\beta}} \func{+}{\beta}^r - \Func{\alpha,\beta,\func{+}{\beta}} \func{-}{\beta}^r}{\Func{\alpha,\beta,\func{-}{\beta}} - \Func{\alpha,\beta,\func{+}{\beta}}}, \quad r = 0,1,\ldots,s - 1 \end{equation} with \begin{equation} \func{\pm}{\beta} = \frac{(1 + s)(\rho + 1) - \beta \pm \sqrt{(\beta - (1 + s)(\rho + 1))^2 - 4s(1 + s)\rho}}{2s} \end{equation} and \begin{equation} \Func{\alpha,\beta,\psi} = \beta - (1 + s)(\rho + 1) + s \psi + \frac{\beta}{\alpha}(1 + s)\rho \psi^{s-1} = (1 + s) \rho \psi^{-1} \bigl( \frac{\beta}{\alpha} \psi^s - 1 \bigr). \end{equation} The variables $\beta_1,\beta_2,\ldots,\beta_s$ are the $s$ roots that satisfy $\|\beta_i\| < \|\alpha\|, ~ i = 1,2,\ldots,s$ of the equation \begin{equation} \bigl( \alpha\beta(1 + s)(\rho + 1) - \beta^2(1 + s)\rho - \alpha^2 \bigr)^s - \beta (\alpha\beta s)^s = 0, \label{eqn:determinant_inner_pos} \end{equation} and $\beta_{s + 1}$ with $\|\beta_{s + 1}\| < \|\alpha\|$ is the single root of \begin{equation} \alpha^2 s^s + \beta^2 ((1 + s)\rho)^s - \alpha \beta s^s ( \func{+}{\beta}^s + \func{-}{\beta}^s ) = 0. \label{eqn:determinant_inner_neg} \end{equation} The vector $\bld{h} = (\h{0},\h{1},\ldots,\h{s - 1})$ is given by \begin{equation} \bld{h} = \alpha \bigl( \frac{1}{2}(1 + s)\rho \M{0,s - 1} + \alpha I \bigr)^{-1} \sum_{i = 1}^s \eta_i \ibpos{\alpha,\beta_i}, \end{equation} where the coefficients $\eta_1,\eta_2,\ldots,\eta_s$ satisfy \begin{align} &\sum_{i = 1}^s \eta_i \Bigl( \beta_i \bigl( (1 + s)\rho I + \alpha s\M{s - 1,0} \bigr) \notag \\ &\hspace{1.2cm} + \alpha^2 \bigl( -(1 + s)(\rho + 1)I + sL^T + (1 + s)\rho L \bigr) \bigl( \frac{1}{2}(1 + s)\rho \M{0,s - 1} + \alpha I \bigr)^{-1} \Bigr) \ibpos{\alpha,\beta_i} \notag \\ &= - \beta_{s + 1} \bigl( (1 + s)\rho\M{0,s - 1} + \alpha I \bigr) \ibneg{\alpha,\beta_{s + 1}}, \end{align} where $I$ is the $s \times s$ identity matrix, $\M{x,y}$ is the $s \times s$ binary matrix with element $(x,y)$ equal to one and zeros elsewhere, and $L$ is an $s \times s$ subdiagonal matrix with elements $(x,x - 1), ~ x = 1,2,\ldots,s - 1$ equal to one and zeros elsewhere. For consistency with indexing of all vectors, the indexing of a matrix starts at 0. \section{Proof of Theorem~\ref{thm:limiting_conditional_arrival_rates}} \label{app_sec:proof_limiting_conditional_arrival_rates} The proof is based on the exact results of the related $M/M(1,s)/2/SED/FCFS$ system, with $s \in \mathbb{N}$, presented in \cite{SED_Selen2015}. Although we obtain similar results for the limiting conditional arrival rates for server 1 as in \cite{JSQ_WebServerFarms_Gupta2007}, we use here a completely different approach in deriving the limits. In \cite{SED_Selen2015}, the state space $\{ (q_1,q_2) \mid (q_1,q_2) \in \mathbb{N}_0^2\}$ of the Markov process is transformed to the state space $\{ (m,n,r) \mid m \in \mathbb{N}_0, ~ n \in \mathbb{Z}, ~ r = 0,1,\ldots,s - 1 \}$ where $m = \min(q_1,\lfloor \frac{q_2}{s} \rfloor)$, $n = \lfloor \frac{q_2}{s} \rfloor - q_1$ and $r = \modu{q_2,s}$. Let us denote the equilibrium probabilities for the three-dimensional state space as $p(m,n,r)$. The equilibrium probability $p(m,n,r)$ has a series expression, i.e.~$p(m,n,r) = \sum_{l = 0}^\infty x(l,m,n,r)$, namely, for $m \ge 0, ~ n \ge 1$, \begin{align} p(m,n,r) &= C \sum_{l = 0}^\infty \sum_{i = 1}^{(s + 1)^l} \sum_{j = 1}^s \beta_{l,d(i) + j}^n \bigl( \eta_{l,d(i) + j} \alpha_{l,i}^m \notag \\ &\hspace{4.5cm}+ \nu_{l + 1,d(i) + j} \alpha_{l + 1,d(i) + j}^m \bigr) \ipos{\alpha_{l,i},\beta_{l,d(i) + j},r}. \label{eqn:equilibrium_distribution_3d_pos} \intertext{For $m \ge 0$,} p(m,0,r) &= C \sum_{l = 0}^\infty \sum_{i = 1}^{(s + 1)^l} \alpha_{l,i}^m h_{l,i}(r). \label{eqn:equilibrium_distribution_3d_hor} \intertext{For $m \ge 0, ~ n \le -1$,} p(m,n,r) &= C \sum_{l = 0}^\infty \sum_{i = 1}^{(s + 1)^l} \eta_{l,i(s + 1)} \alpha_{l,i}^m \beta_{l,i(s + 1)}^{-n} \ineg{\alpha_{l,i},\beta_{l,i(s + 1)},r} \notag \\ &\quad + C \sum_{l = 0}^\infty \sum_{i = 1}^{(s + 1)^l} \nu_{l+1,i(s + 1)} \alpha_{l + 1,i(s + 1)}^m \beta_{l,i(s + 1)}^{-n} \ineg{\alpha_{l + 1,i(s + 1)},\beta_{l,i(s + 1)},r}. \label{eqn:equilibrium_distribution_3d_neg} \end{align} For the exact interpretation of each variable we refer the reader to \cite{SED_Selen2015}. In \cite{SED_Selen2015} the authors establish the following properties: \begin{enumerate}[label = \arabic.] \item There exists a positive integer $N$ such that the series in \eqref{eqn:equilibrium_distribution_3d_pos} and \eqref{eqn:equilibrium_distribution_3d_neg} converge absolutely for all $m \ge 0, ~ \|n\| \ge 1$ with $m + \|n\| > N$ and the series \eqref{eqn:equilibrium_distribution_3d_hor} converges absolutely for all $m \ge N$. \item For $m + \|n\| > N$, we have $\|x(l,m,n,r)\| < u(l)$ and $\sum_{l = 0}^\infty u(l) < \infty$. \item The series $\sum_{m + \|n\| > N} p(m,n,r), ~ r = 0,1,\ldots,s - 1$ converges absolutely. \item $\|\alpha_{l,i}\| > \|\beta_{l,d(i) + j}\|$ and $\|\beta_{l,i}\| > \|\alpha_{l + 1,i}\|$ with $\alpha_{0,1} = \rho^{1 + s} < 1$. \end{enumerate} In this proof we make use of the dominated convergence theorem for complex-valued functions. \subsection{Server 1} \label{app_subsec:limiting_conditional_arrival_rate_server_1} The limiting conditional arrival rate to server 1 can be determined from \begin{equation} \lim_{n \to \infty} \lambda_1(n) = \lim_{n \to \infty} \frac{\pi_1(n + 1)/\alpha_{0,1}^{n + 1}}{\pi_1(n)/\alpha_{0,1}^{n}} \alpha_{0,1}. \label{eqn:app_limiting_conditional_arrival_rate_server_1} \end{equation} The marginal distribution for server 1 is given by, where $m = \lfloor \frac{q_2}{s} \rfloor$ and $r = \modu{q_2,s}$, \begin{align} \pi_1(n) &= \sum_{m = 0}^\infty \sum_{r = 0}^{s - 1} p(\min(n,m),m - n,r) \notag \\ &= \sum_{m = 0}^{n - 1} \sum_{r = 0}^{s - 1} p(m,m - n,r) + \sum_{r = 0}^{s - 1} p(n,0,r) + \sum_{m = 1}^\infty \sum_{r = 0}^{s - 1} p(n,m,r). \label{eqn:app_marginal_server_1} \end{align} Furthermore, \begin{align} \lim_{n \to \infty} \frac{\pi_1(n)}{\alpha_{0,1}^{n}} &= \lim_{n \to \infty} \sum_{m = 0}^{n - 1} \sum_{r = 0}^{s - 1} \frac{p(m,m - n,r)}{\alpha_{0,1}^{n}} \notag \\ &\quad + \sum_{r = 0}^{s - 1} \lim_{n \to \infty} \frac{p(n,0,r)}{\alpha_{0,1}^{n}} + \sum_{m = 1}^\infty \sum_{r = 0}^{s - 1} \lim_{n \to \infty} \frac{p(n,m,r)}{\alpha_{0,1}^{n}}, \label{eqn:app_marginal_server_1_scaled} \end{align} where the interchange of the limit and the series for the third term on the right-hand side of \eqref{eqn:app_marginal_server_1_scaled} is allowed by the dominated convergence theorem, because one can bound $p(n,m,r)$ from above by $p(0,m,r)$ and $\sum_{m = 0}^\infty p(0,m,r) < \infty$ since it is a subseries of $\sum_{m + \|n\| > N} p(m,n,r)$, which converges absolutely by property 3. Furthermore, we know that $\lim_{m \to \infty} p(m,n,r) = \lim_{m \to \infty} \sum_{l = 0}^\infty x(l,m,n,r)$ which is equal to $\sum_{l = 0}^\infty \lim_{m \to \infty} x(l,m,n,r)$ by the dominated convergence theorem for complex-valued functions in combination with property 2. This allows us to compute the second and third term on the right-hand side of \eqref{eqn:app_marginal_server_1_scaled}. The first term on the right-hand side of \eqref{eqn:app_marginal_server_1_scaled} can be determined as follows \begin{align} &\lim_{n \to \infty} \sum_{m = 0}^{n - 1} \sum_{r = 0}^{s - 1} \frac{p(m,m - n,r)}{\alpha_{0,1}^{n}} \notag \\ &= C \Bigl(\lim_{n \to \infty} \sum_{l = 0}^\infty \sum_{i = 1}^{(s + 1)^l} \eta_{l,i(s + 1)} \frac{\left( \frac{\alpha_{l,i}}{\alpha_{0,1}} \right)^{n} - \left( \frac{\beta_{l,i(s + 1)}}{\alpha_{0,1}} \right)^{n}}{\frac{\alpha_{l,i}}{\beta_{l,i(s + 1)}} - 1} \sum_{r = 0}^{s - 1} \ineg{\alpha_{l,i},\beta_{l,i(s + 1)},r} \notag \\ &\quad+ \lim_{n \to \infty} \sum_{l = 0}^\infty \sum_{i = 1}^{(s + 1)^l} \nu_{l + 1,i(s + 1)} \frac{\left( \frac{\beta_{l,i(s + 1)}}{\alpha_{0,1}} \right)^{n} - \left( \frac{\alpha_{l + 1,i(s + 1)}}{\alpha_{0,1}} \right)^{n}}{1 - \frac{\alpha_{l + 1,i(s + 1)}}{\beta_{l,i(s + 1)}}} \sum_{r = 0}^{s - 1}\ineg{\alpha_{l + 1,i(s + 1)},\beta_{l,i(s + 1)},r} \Bigr) \notag \\ &= C \frac{\eta_{0,s + 1}}{\frac{\alpha_{0,1}}{\beta_{0,s + 1}} - 1} \sum_{r = 0}^{s - 1} \ineg{\alpha_{0,1},\beta_{0,s + 1},r}. \end{align} Interchange of the limit and series is again allowed here since one can bound the absolute value of the summands from above by $v(l)$ for sufficiently large $n$ and $\sum_{l = 0}^\infty v(l) < \infty$. One can finally establish that \begin{align} \lim_{n \to \infty} \frac{\pi_1(n)}{\alpha_{0,1}^{n}} = C \Bigl(& \frac{\eta_{0,s + 1}}{\frac{\alpha_{0,1}}{\beta_{0,s + 1}} - 1} \sum_{r = 0}^{s - 1} \ineg{\alpha_{0,1},\beta_{0,s + 1},r} \notag \\ &+ \sum_{r = 0}^{s - 1} h_{0,1}(r) + \sum_{j = 1}^s \frac{\eta_{0,j} \beta_{0,j}}{1 - \beta_{0,j}} \sum_{r = 0}^{s - 1} \ipos{\alpha_{0,1},\beta_{0,j},r} \Bigr). \label{eqn:app_marginal_server_1_scaled_computed} \end{align} Thus, by \eqref{eqn:app_limiting_conditional_arrival_rate_server_1} and \eqref{eqn:app_marginal_server_1_scaled_computed}, $\lim_{n \to \infty} \lambda_1(n) = \alpha_{0,1} = \rho^{1 + s}$. \subsection{Server 2} \label{app_subsec:limiting_conditional_arrival_rate_server_2} The limiting conditional arrival rate to server 2 can be determined from \begin{equation} \lim_{n \to \infty} \lambda_2(sn + r) = \begin{cases} \lim_{n \to \infty} s \frac{\pi_2(sn + r + 1)/\alpha_{0,1}^{n}}{\pi_2(sn + r)/\alpha_{0,1}^{n}}, & r = 0,1,\ldots,s - 2, \\ \lim_{n \to \infty} s \frac{\pi_2(sn + r + 1)/\alpha_{0,1}^{n + 1}}{\pi_2(sn + r)/\alpha_{0,1}^{n}} \alpha_{0,1}, & r = s - 1. \end{cases} \label{eqn:app_limiting_conditional_arrival_rate_server_2} \end{equation} The marginal distribution for server 2 is given by, for $r = 0,1,\ldots,s - 1$, \begin{align} \pi_2(sn + r) &= \sum_{q_1 = 0}^\infty p(\min(q_1,\lfloor \frac{sn + r}{s} \rfloor),\lfloor \frac{sn + r}{s} \rfloor - q_1,\modu{sn + r,s}) \notag \\ &= \sum_{q_1 = 0}^\infty p(\min(q_1,n),n - q_1,r) \notag \\ &= \sum_{q_1 = 0}^{n - 1} p(q_1,n - q_1,r) + p(n,0,r) + \sum_{q_1 = 1}^\infty p(n,-q_1,r). \label{eqn:app_marginal_server_2} \end{align} For $\pi_2(sn + r + 1), ~ r = s - 1$ we should replace $n$ by $n + 1$ and $r$ by 0 in \eqref{eqn:app_marginal_server_2}. Furthermore, for $r = 0,1,\ldots,s - 1$, \begin{equation} \lim_{n \to \infty} \frac{\pi_2(sn + r)}{\alpha_{0,1}^{n}} = \lim_{n \to \infty} \sum_{q_1 = 0}^{n - 1} \frac{p(q_1,n - q_1,r)}{\alpha_{0,1}^n} + \lim_{n \to \infty} \frac{p(n,0,r)}{\alpha_{0,1}^n} + \lim_{n \to \infty} \sum_{q_1 = 1}^\infty \frac{p(n,- q_1,r)}{\alpha_{0,1}^n}. \label{eqn:app_marginal_server_2_scaled} \end{equation} Using identical arguments as for the limiting conditional arrival rate for server 1, we establish \begin{equation} \lim_{n \to \infty} \frac{\pi_2(sn + r)}{\alpha_{0,1}^{n}} = A(r), \quad r = 0,1,\ldots,s - 1, \label{eqn:app_marginal_server_2_scaled_computed} \end{equation} where $A(r)$ is given in \eqref{eqn:A(r)}. Finally, combining \eqref{eqn:app_limiting_conditional_arrival_rate_server_2} and \eqref{eqn:app_marginal_server_2_scaled_computed} proves \eqref{eqn:limiting_conditional_arrival_rate_server_2}. \section{Proof of Proposition~\ref{prop:conditional_arrival_rates_limiting_regimes}, point 2} \label{app_sec:proof_limiting_regimes} By letting $\rho \downarrow 0$ in \eqref{eqn:conditional_arrival_rate_server_1_n_0}-\eqref{eqn:conditional_arrival_rate_server_1_n_2} and $\lambda_1(n) \approx \rho^{1 + s}, ~ n \ge 3$ we immediately find that $\lambda_1(n) \downarrow 0, ~ n \in \mathbb{N}_0$. We note that in \eqref{eqn:conditional_arrival_rate_server_2_n_s-1_until_2s-1} the factors on the right-hand side in front of $\lambda_2^\textup{lim}(n-s)$ go to a constant for $\rho \downarrow 0$. So, what remains is that we establish that $\lim_{\rho \downarrow 0} \lambda_2^\textup{lim}(r) = 0, ~ r = 0,1,\ldots,s-1$. This part of the proof relies heavily on the asymptotic results of \cite{SED_Selen2015}. We denote $\alpha = \rho^{1 + s}$ and investigate for $r = 0,1,\ldots,s - 1$, \begin{equation} \frac{A(\alpha,r)}{\alpha^{r/s}} = \sum_{i = 1}^s \eta_i \frac{\beta_i/\alpha}{1 - \beta_i/\alpha} u_i \bigl( \frac{\beta_i}{\alpha} \bigr)^{r/s} + \alpha^{1/s} \frac{h(r)}{\alpha^{(r + 1)/s}} + \alpha^{1 - r/s} \frac{\beta_{s + 1}/\alpha}{1 - \beta_{s + 1}} \ineg{\alpha,\beta_{s + 1},r}, \end{equation} where we used that $\ipos{\alpha,\beta_i,r} = u_i \beta_i^{r/s}$ with $u_i$ the $i$-th unit root of $u^s = 1$, which is established in \cite[Lemma~5.6]{SED_Selen2015}. Now, \begin{equation} \lim_{\alpha \downarrow 0} \frac{A(\alpha,r)}{\alpha^{r/s}} = c(r), \label{eqn:app_limiting_scaled_A(r)} \end{equation} where $c(r)$ is some constant. In the following we denote $c_i$ as some constant that can be a function of $r$. Equation \eqref{eqn:app_limiting_scaled_A(r)} follows from the fact that for $\alpha \downarrow 0$ we have that $\beta_i/\alpha \to c_1 < 1, ~ i = 1,2,\ldots,s$, $\beta_{s + 1}/\alpha \to c_2$ \cite[Lemma~5.15(i)(a) and (i)(c)]{SED_Selen2015}; $h(r)/\alpha^{(r + 1)/s} \to c_3(r)$ \cite[Appendix~B, part (c)]{SED_Selen2015}; $\ineg{\alpha,\beta_{s + 1},r} \to c_4(r)$ \cite[Lemma~5.15(i)(d)]{SED_Selen2015}; $\beta_{s + 1} \to 0$ \cite[Corollary~5.14]{SED_Selen2015}; and $\sum_{i = 1}^s \eta_i u_i \to c_5$ ($\alpha \downarrow 0$ in (5.46) of \cite{SED_Selen2015}). Finally, for $r = 0,1,\ldots,s - 2$, \begin{equation} \lim_{\alpha \downarrow 0} \frac{A(\alpha,r + 1)}{A(\alpha,r)} = \lim_{\alpha \downarrow 0} \frac{\alpha^{(r + 1)/s}}{\alpha^{r/s}} \frac{A(\alpha,r + 1)/\alpha^{(r + 1)/s}}{A(\alpha,r)/\alpha^{r/s}} = \lim_{\alpha \downarrow 0} \alpha^{1/s} \frac{c(r + 1)}{c(r)} = 0 \end{equation} and for $r = s - 1$, \begin{equation} \lim_{\alpha \downarrow 0} \alpha \frac{A(\alpha,0)}{A(\alpha,s - 1)} = \lim_{\alpha \downarrow 0} \frac{\alpha}{\alpha^{(s - 1)/s}} \frac{A(\alpha,0)}{A(\alpha,r)/\alpha^{(s - 1)/s}} = \lim_{\alpha \downarrow 0} \alpha^{1/s} \frac{c(0)}{c(s - 1)} = 0. \end{equation} This concludes the proof. \end{document}	arXiv
\begin{document} \begin{frontmatter} \title{Regularity of languages generated by non context-free grammars over a~singleton terminal alphabet} \author[address1,address2]{Alberto Pettorossi\corref{1}} \cortext[1]{Corresponding author.} \ead{[email protected]} \author[address2]{Maurizio Proietti} \ead{[email protected]} \address[address1]{DICII, University of Rome Tor Vergata, Rome, Italy} \address[address2]{CNR-IASI, Rome, Italy} \begin{abstract} It is well-known that: (i)~every context-free language over a singleton terminal alphabet is regular~\cite{Har78}, and (ii)~the class of languages that satisfy the Pumping Lemma (for context-free languages) is a proper super-class of the context-free languages. We show that any language in this super-class over a singleton terminal alphabet is regular. Our proof is based on an elementary transformational approach and does not rely on Parikh's Theorem~\cite{Par66}. Our result extends previously known results because there are languages that are not context-free, do satisfy the Pumping Lemma, and do not satisfy the hypotheses of Parikh's Theorem~\cite{Ra&98}. \end{abstract} \begin{keyword} Context-free languages, pumping lemma (for context-free languages), Parikh's Theorem, regular languages. \end{keyword} \end{frontmatter} Let us begin by introducing our terminology and notations. The set of the natural numbers is denoted by $N$. The set of the \mbox{$n$-tuples} of natural numbers is denoted by $N^{n}$. We say that a language $L$ is over the terminal alphabet $\Sigma$ iff $L\!\subseteq\!\Sigma^{}$. Given a word $w\!\in\!\Sigma^{}$, $w^{0}$ is the empty word~$\varepsilon$, and, for any $i\!\geq\!0$, $w^{i+1}$ is $w^{i}\,w$, that is, the concatenation of $w^{i}$ and $w$. The length of a word $w$ is denoted by $\|w\|$. Given a symbol $a\!\in\!\Sigma$, the number of occurrences of~$a$ in $w$ is denoted by $\|w\|_{\displaystyle a}$. The cardinality of a set $A$ is denoted by $\|A\|$. Given an alphabet $\mathrm{\Sigma}$ such that $\|\Sigma\|\!=\!1$, the concatenation of any two words $w_{1}, w_{2}$ in $\mathrm{\Sigma}^{}$ is commutative, that is, $w_{1}\,w_{2}=w_{2}\,w_{1}$. In Theorem~\ref{thm:PL-CF-1-Regular} below we extend the well known result stating that any context-free language over a singleton terminal alphabet is a regular language~\cite{Har78}. An early proof of this result appears in a paper by Ginsburg and Rice~\cite{GiR62}. That proof is based on Tarski's fixpoint theorem and it is not based on the Pumping Lemma (contrary to what has been stated in a paper by Andrei et al.~\cite{An&03}). Our extension is due to the facts that: (i)~our proof does not rely on Parikh's Theorem~\cite{Par66}, like the proof in Harrison's book~\cite{Har78}, and (ii)~there are non context-free languages that do satisfy the Pumping Lemma (see Definition~\ref{def:pumpinglemma-cf}) and do not satisfy Parikh's Condition (see Definition~\ref{def:parikh-condition}) (and thus Parikh's Theorem cannot be applied)~\cite{Ra&98}. Our proof is very much related to one presented in a book by~Shallit~\cite{Sha08}, but we believe that ours is more elementary. \begin{definition}[Pumping Lemma {\rm{\cite{Ba&61}}}]\label{def:pumpinglemma-cf} {\rm{We}} {\rm{say that a language $L\!\subseteq\! \Sigma^{}$ satisfies the Pumping Lemma (for context-free languages) iff the following property, denoted $\mathit{PL}(L)$, holds:\\ $\exists\, n \!>\!0$, $\forall\, z\in L$, if $\|z\|\geq n$, then $\exists\, u,v,w,x,y\in \mathrm{\Sigma}^$, such that {\rm{(1)}}$~$ {\textup{$z=u\,v\,w\,x\,y$,}} {\rm{(2)}}$~$ {\textup{$v\,x\not=\varepsilon$,}} {\rm{(3)}}$~$ {\textup{$\|v\,w\,x\|\leq n$, ~and }} {\rm{(4)}}$~$ {\textup{$\forall\, i\geq0$, $u\,v^i\,w\,x^i\,y\in L$.}} }} $\Box$ \end{definition} \begin{definition}[Parikh's Condition~{\rm{\cite{Par66}}}]\label{def:parikh-condition} {\rm{(i)~A subset $S$ of $N^{n}$ is said to be a {\em linear} set iff there exist $v_{0},\ldots,v_{k}\!\in\!N^{n}$ such that $S\!=\!\{v_{0}+n_{1}\,v_{1}+\ldots+n_{k}\,v_{k}\mid n_{1},\ldots,n_{k}\!\in\!N\}$}}, where, for any given $u\!=\!\langle u_{1},\ldots,u_{n}\rangle$ and $v\!=\!\langle v_{1},\ldots,v_{n}\rangle$ in $N^{n}$, $u\!+\!v$ denotes $\langle u_{1}\!+\!v_{1},\ldots,u_{n}\!+\!v_{n}\rangle$ and, for any \mbox{$m\!\geq\!0$}, $m\, u$ denotes $\langle m \,u_{1},\ldots,m \, u_{n}\rangle$. (ii)~Given the alphabet $\Sigma\!=\!\{a_{1},\ldots,a_{n}\}$, we say that a language $L\!\subseteq\! \Sigma^{}$ satisfies Parikh's Condition iff $\{\langle \|w\|_{\displaystyle a_{1}},\ldots, \|w\|_{\displaystyle a_{n}}\rangle \mid w\!\in\!L\}$ is a finite union of linear subsets of~$N^{n}$. $\Box$ \end{definition} Let us first state and prove the following lemma whose proof is by transformation from Definition~\ref{def:pumpinglemma-cf}. \begin{lem}\label{lem:PumpingLemma-CFL-Card1} {\rm{For any language $L$ over a terminal alphabet $\mathrm{\Sigma}$ such that $\|\Sigma\|\!=\!1$, $\mathit{PL}(L)$ holds iff the following property, denoted $\mathit{PL}1(L)$, holds:\\ $\exists\, n \!>\!0$, $\forall\, z\!\in\! L$, if $\|z\|\!\geq\! n$, then $\exists\,p\!\geq\!0,$ $q\!\geq\!0,$ $m\!\geq\!0$, such that ${\rm{(1.1)}}~$ $\|z\|=p+q$, ${\rm{(2.1)}}~$ $0\!<\!q\!\leq \!n$, ${\rm{(3.1)}}~$ $0\!<\!m\!+\!q \!\leq\! n$, ~and ${\rm{(4.1)}}~$ $\forall\,s\in \mathrm{\Sigma}^,$ $\forall\,i\!\geq\!0$, if $\|s\|=p+i\,q$, then $s\in L$.}} \end{lem} \begin{proof} If $\|\mathrm{\Sigma}\|\!=\!1$, then commutativity of concatenation implies that in~$\mathit{PL}(L)$ we can replace $v\,w\,x$ by $w\,v\,x$, and $u\,v^{i}\,w\,x^{i}\,y$ by $u\,y\,w\,(v\,x)^{i}$. Then, we can replace: $u\,y$ by $\widetilde u$, $v\,x$ by $\widetilde{v}$, and $(\exists\, u,\,v,\,w,\,x,\, y)$ by $(\exists\, \widetilde u,\,\widetilde v,\,w)$. Thus, from $\mathit{PL}(L)$, we get: \noindent {\textup{$\exists\, n \!>\!0$, $\forall\, z\in L$, if $\|z\|\!\geq\! n$, then $\exists\, \widetilde u,\widetilde v,w\in \mathrm{\Sigma}^$, such that}} ${\rm{(1')}}~$ {\textup{$z=\widetilde u\, w\, \widetilde v$,}} ${\rm{(2')}}~$ {\textup{$\widetilde v\not=\varepsilon$,}} ${\rm{(3')}}~$ {\textup{$\|w\,\widetilde v\|\leq n$, and}} ${\rm{(4')}}~$ {\textup{$\forall\, i\!\geq\!0$, $\widetilde u\,w\,\widetilde v\hspace{.7pt} ^i\in L$.}} \noindent Now if we take the lengths of the words and we denote $\|\widetilde u\,w\|$ by $p$, $\|\widetilde v\|$ by $q$, and $\|w\|$ by $m$, we get: \noindent {\textup{$\exists\, n \!>\!0$, $\forall\, z\in L$, if $\|z\|\!\geq\! n$, then $\exists\,p\!\geq\!0,$ $q\!\geq\!0,$ $m\!\geq\!0$, such that}}\nopagebreak ${\rm{(1'')}}~$ {\textup{$\|z\|=p+q$,}} ${\rm{(2'')}}~$ {\textup{$q\!>\!0$,}} ${\rm{(3'')}}~$ {\textup{$m\!+\!q \leq n$, ~and}} ${\rm{(4'')}}~$ {\textup{$\forall\,s\!\in\! \mathrm{\Sigma}^,$ $\forall\,i\!\geq\!0$, if $\|s\|=p+i\,q$, then $s\in L$.}} \noindent For all $n\!>\!0$, $q\!>\!0$, and $m\!\geq\!0$, we have that $(q\!>\!0 \wedge m\!+\!q \!\leq\! n)$ iff $(0\!<\!q\!\leq \!n ~\wedge~ 0\!<\! m\!+\!q \!\leq\! n)$. Thus, we get $\mathit{PL}1(L)$. $\Box$ \end{proof} We say that $\mathit{PL}1(L)$ {\it holds for} $b$ if $b$ is a witness of the quantification `$\exists\, n \!>\!0$' in $\mathit{PL}1(L)$. The following theorem states our main result. \begin{thm}\label{thm:PL-CF-1-Regular} {\textup{Let $L$ be any language over a terminal alphabet $\Sigma$ such that $\|\Sigma\|\!=\!1$. If $\mathit{PL}(L)$ holds, then $L$ is a regular language.}} \end{thm} \noindent \begin{proof} Without loss of generality, let us consider a language~$L$ over the terminal alphabet~$\{a\}$, such that $\mathit{PL}(L)$ holds. By Lemma~\ref{lem:PumpingLemma-CFL-Card1}, we have that $\mathit{PL}1(L)$ holds for some positive integer $b$. Let us consider the following two disjoint languages whose union is $L$: \noindent (i)~$L_{\displaystyle{<\!b}} = \{ a^{k} \mid a^{k}\!\in\! L \wedge k\!<\!b\}$ ~~and~~ (ii)~$L_{\displaystyle{\geq\!b}} = \{ a^{k} \mid a^{k}\!\in\! L \wedge k\!\geq\! b\}$. \noindent Now, $L_{\displaystyle{<\!b}}$ is a regular language, because it is finite.\rule{0mm}{3.3mm} Since regular languages are closed\rule{0mm}{3.7mm} under finite union and intersection~\cite{HoU79}, in order to prove that $L$ is regular, it is enough to prove, as we now do, that $L_{\displaystyle \geq\! b} = \bigcup \mathcal{S} ~\cap~ \{a^{i}\mid i\!\geq\!b\} $ $(\dagger 1)$~ \noindent where: (i)~$\mathcal S$ is a set of languages which is\rule{0mm}{3.7mm} a subset of the following {\rm finite} set $\mathcal L$ of languages ($k, p_{h}, q_{0}, \ldots, q_{k}$ are integers): $\mathcal{L} = \{L^{\displaystyle{\langle p_h,q_0,\ldots,q_k\rangle}} \mid ~ (0\!\leq \!k\!<\!b) ~\wedge~ (0\!\leq\! p_{h}\! <\!b) ~\wedge~ $\vspace{-1mm} $(\dagger 2)$~\nopagebreak \hspace{44mm}$ (0\!<\! q_{0}\! \leq\!b) ~\wedge \ldots \wedge~ (0\!<\! q_{k}\! \leq\!b) ~\wedge~ $\nopagebreak \hspace{45mm}$q_{0},\ldots,q_{k}$ are all distinct\,$\}$ \noindent and (ii)~for all $k, p_{h}, q_{0}, \ldots, q_{k}$, the language: $L^{\displaystyle{\langle p_h,q_0,\ldots,q_k\rangle}}\! =\! \{a^{\displaystyle{\hspace{.5mm}p_h+i_0 \, q_0+\ldots+i_k\, q_k}} \mid i_0\!>\!0 ~\wedge \ldots \wedge~ i_k\!>\!0\}$ $(\dagger 3)$~ \noindent is {\rm regular}. Indeed, (i)~$\{a^{i}\mid i\!\geq\!b\}$ is regular, \noindent {(ii)}~$\mathcal{L}$ is a {\rm{finite}} set of languages because, for any $b$, there exists only a finite number of tuples $\langle p_h,q_0,\ldots,q_k\rangle$ satisfying all the conditions stated inside the set expression $(\dagger 2)$, and (iii)~the language $L^{\displaystyle{\langle p_h,q_0,\ldots,q_k\rangle}}$ is {\rm regular} because it is recognized by the following nondeterministic\rule{0mm}{3.5mm} finite automaton with initial state~$A$ and final state~$B$: \hangindent=0mm \begin{center} \VCDraw{ \begin{VCPicture}{(0,-1.5)(10,2.5)} \normalsize \FixStateDiameter{12mm} \FixStateLineDouble{0.4}{1.3} \ChgStateLineWidth{1.2} \SetEdgeArrowWidth{6pt} \SetEdgeArrowLengthCoef{1.8} \State[A]{(0,0)}{1} \FinalState[B]{(7,0)}{2} \SetStateLineColor{white} \FixStateDiameter{1mm} \State[.]{(9.0,1.2)}{dot1} \State[.]{(9.2,1.0)}{dot2} \State[.]{(9.4,0.8)}{dot3} \Initial{1} \Edge{1}{2}\LabelL[0.47]{a^{\displaystyle{\hspace{.5mm}p_h\!+\!q_0\!+\!\ldots\!+\!q_k}}} \LoopN{2}{}\LabelL[0.74]{\,a^{\displaystyle{\,q_0}}} \LoopE{2}{}\LabelL[0.5]{a^{\displaystyle{\,q_k}}} \end{VCPicture}} \end{center} \noindent In order to prove Equality~$(\dagger 1)$ it remains to prove that, for any $z\!\in\!L_{\displaystyle{\geq\!b}}$, there exists a tuple of the form $\langle p_{h}, q_{0}, q_{1},\ldots, q_{k}\rangle$ such\rule{0mm}{3.7mm} that $z\!\in\!L^{\displaystyle {\langle p_{h}, q_{0}, q_{1},\ldots, q_{k}\rangle}}$. Given any word $z\!\in\!L_{\displaystyle{\geq\!b}}$, the following algorithm constructs a tuple of the form $\langle p_{h}, q_{0}, q_{1},\ldots, q_{h}\rangle$, for\rule{0mm}{3.7mm} some $h\!\geq\!0$. \begin{framed} \hspace{8mm}$\{\,z\!\in\!L_{\displaystyle{\geq\!b}}\,\}$ {\it Tuple Generation Algorithm} \indent $\ell\! :=\! \|z\|;~~~ i\!:=\!0;~~~ \langle p_{0},q_{0}\rangle\!:=\!\pi(\ell);$ \hspace{8mm}$\{\,\|z\|= p_{i} + \sum_{j=0}^{i} q_{j} ~~\wedge~~ \bigwedge_{j=0}^{i} ~0\!<\!q_{j}\!\leq\!b ~~\wedge~~ 0\!\leq\!p_{i}\,\} $ {\bf while}~ $p_{i}\!\geq\!b$ ~{\bf do}~ $\ell \!:=\! p_{i}; ~~ i \!:=\! i \!+\!1; ~~ \langle p_{i},q_{i}\rangle\!:=\!\pi(\ell)$ ~{\bf od}; $h \!:=\! i;$ \hspace{8mm}$\{\,\|z\|= p_{h} + \sum_{j=0}^{h} q_{j} ~~\wedge~~ \bigwedge_{j=0}^{h} ~0\!<\!q_{j}\!\leq\!b ~~\wedge~~ 0\!\leq\! p_{h}\!<\!b\,\} \vspace{-2mm}$ \end{framed} \vspace{-2mm} \noindent In this algorithm $\pi$ is a function from $N$ to $N\!\times\!N$, whose existence follows from the validity of~$\mathit{PL}1(L)$, satisfying the following condition: for every $\ell\!\geq\!b$, $\pi(\ell)\!=\!\langle p,q\rangle$ such that $\ell\! =\! p\!+\!q$ and $0\!<\!q\!\leq\!b$ (take $i\!=\!1$ in Condition~(4.1) of $\mathit{PL}1(L)$ in Lemma~\ref{lem:PumpingLemma-CFL-Card1}). The termination of the Tuple Generation Algorithm is a consequence of the fact that, for every~$z\!\in\! L_{\displaystyle{\geq\!b}}$, for every~$i\!\geq\!0$, \mbox{$p_{i}\!=\!p_{i+1}+q_{i+1}$} and $q_{i}\!>\!0$. This implies that $p_{0},p_{1},\ldots$ is a\rule{0mm}{3.5mm} strictly decreasing sequence of integers, and eventually in that sequence we will get an element smaller than $b$, and the while-loop terminates. Thus, for every $z\!\in\! L_{\displaystyle{\geq\!b}}$, there exist $h\!\geq\!0,$ $ p_0,$ $q_0,$ $p_1,$ $q_1,$ $p_2,$ $q_2,$ $\ldots,$ $p_h, q_h$ such that: \makebox[6mm]{$z$}$=a^{\displaystyle{\,(p_h+q_h)+q_{h-1}+\ldots+q_2+q_1+q_0}}$ $(\dagger 4)$~ \noindent \makebox[12mm][l]{where:} \noindent $0\!\leq\! p_h \!<\! b$ ~and~ for every $i$, if $0\!\leq\!i\!<\!h$, then ($p_{i}\!\geq \!b$ and $0 \!<\! q_i\! \leq\! b$). \noindent In general, in Equality~$(\dagger 4)$ the $q_i$'s are {\it{not\/}} all distinct. Thus, by rearranging the summands, and writing $i_{j}\, q_{j}$, instead of $({q_{j}+\ldots+q_{j}})$ with $i_{j}$ occurrences of $q_{j}$, we have that, for every word $z\!\in\! L_{\displaystyle{\geq\!b}}$, there exist some integers $k,p_h,i_{0},q_0,\ldots,i_{k},q_k$ such that $z=a^{\displaystyle{\,p_h+i_0\, q_0+\ldots+i_k \,q_k}}$, ~~~ where: \noindent ~{\makebox[9mm][l]{($\ell$\hspace{.7pt} 0)}$0\!\leq\!k$,}\hspace{15mm} \noindent ~\makebox[9mm][l]{($\ell$\hspace{.7pt} 1)}$0\!\leq\!p_h\!<\!b$,\hspace{15mm} \noindent \noindent ~\makebox[9mm][l]{($\ell$\hspace{.7pt} 2)}\makebox[51mm][l]{$0\!<q_0\!\leq\!b, \ldots, 0\!<q_k\!\leq\!b$,} \noindent ~\makebox[9mm][l]{($\ell$\hspace{.7pt} 3)}$q_0,\ldots,q_k$ are {all distinct}, ~~and~ ~\makebox[9mm][l]{($\ell$\hspace{.7pt} 4)}$i_0\!>\!0,\ldots, i_k\!>\!0$. \ \noindent From ($\ell$\hspace{.7pt} 2) and ($\ell$\hspace{.7pt} 3), we have that $k\!<\!b$. Hence, Condition~($\ell$\hspace{.7pt} 0) can be strengthened to: \makebox[10mm][l]{($\ell\hspace{.7pt} {0}^{})$}$0\!\leq\!k\!<\!b$. We also have that $k\!\leq\!h$, and $k\!=\!h$ when in Equality~$(\dagger 4)$ the values of $q_{0},\ldots,q_{h}$ are all distinct. Since Conditions~$(\ell\hspace{.7pt} {0}^{}$), $(\ell\hspace{.7pt} {1})$, $(\ell\hspace{.7pt} {2})$, and $(\ell\hspace{.7pt} {3})$ are those occurring in the set expressions~$(\dagger 2)$, and Condition~$(\ell\hspace{.7pt} {4})$ is the one occurring in the set expressions~$(\dagger 3)$, we have concluded the proof of Equality~$(\dagger 1)$ and that of Theorem~\ref{thm:PL-CF-1-Regular}.~ $\Box$ \end{proof} Let us make a few of remarks on the proof of Theorem~\ref{thm:PL-CF-1-Regular}. \noindent (i)~The validity of~$\mathit{PL}1(L)$ tells us that the function~$\pi$ exists, but it does not tell us how to compute $\pi(\ell)$, for any given $\ell\!\geq\!b$. \noindent (ii)~Since summation is commutative, it may be the case that a language in~$\mathcal L$ corresponds to more than one tuple $\langle p_h,q_0,\ldots,q_k\rangle$. In particular, we have that $L^{\displaystyle{\langle p_h,q_0,\ldots,q_k\rangle}}$ $=$ $L^{\displaystyle{\langle p_h,q'_0,\ldots,q'_k\rangle}}$, whenever $\langle q_0,\ldots,q_k\rangle$ is a permutation of $\langle q'_0,\ldots,q'_k\rangle$.\rule{0mm}{3.5mm} \noindent (iii)~If $b\!=\!1$, then $k\!=\!h\!=\!0$. Thus, from \mbox{Conditions~($\ell$\hspace{.7pt} 1)} \mbox{and~($\ell$\hspace{.7pt} 3)} we have: $\langle p_{h},q_{h}\rangle \!=\!\langle p_{0},q_{0}\rangle \!=\!\langle 0,1\rangle$. We also have that $\mathcal{L}$ is the singleton $\{L^{\textstyle{\langle 0,1\rangle}}\}$, where $L^{\textstyle{\langle 0,1\rangle}}$ is the language $\{a^{i}\mid i\!>\!0\}$. \noindent (iv)~In Equality~$(\dagger 1)$ the set $\mathcal S$ of languages may be a {\it proper} subset of $\mathcal L$. Indeed, let us consider the language $L\!=\!\{a\,(a\,a)^{n}\mid n\!\geq\!0\}$ generated by the context-free grammar $S\rightarrow a\,S\,a \mid a$. Since $\mathit{PL}1(L)$ holds for $3$, we can take the constant $b$ occurring in Equality~$(\dagger 1)$ to be $3$. If we consider the word $z\!=\!a\,a\,a$, then the set~$\mathcal L$ of languages includes, among others, the languages $L^{\langle 0,3\rangle}\!=\! \{a^{0\, +\, i \cdot 3}\mid i\!>\!0\}$, $L^{\langle 1,2\rangle}\!=\! \{a^{1\, +\, i \cdot 2}\mid i\!>\!0\}$, and $L^{\langle 2,1\rangle}\!=\! \{a^{2\, +\, i \cdot 1}\mid i\!>\!0\}$ (these three languages are obtained for $k\!=\!h\!=\!0$). Now, $L_{\displaystyle \geq 3}= L^{\langle 1,2 \rangle} \cap \{a^{i} \mid i\!\geq\!3\}=L^{\langle 1,2 \rangle}$, while $L^{\langle 0,3 \rangle}\!\not\subseteq \!L_{\displaystyle \geq 3}$ and $L^{\langle 2,1 \rangle}\!\not\subseteq \!L_{\displaystyle \geq 3}$. \noindent (v)~It may be the case that the\rule{0mm}{3.7mm} length ${p_{h}\!+\!q_{0}\!+\!\ldots\!+\!q_{k}}$ of the word labeling the arc from state $A$ to state $B$ of the finite automaton depicted above, is smaller than $b$. Thus, in the definition of $L_{\displaystyle{\geq\!b}}$ the intersection of $\bigcup \mathcal S$ with $\{a^{i} \mid i\!\geq\!b\}$ ensures that only words whose length is\rule{0mm}{3.7mm} at least~$b$ are considered. \section*{References} \end{document}	arXiv
\begin{definition}[Definition:Open Set (Neighborhood Space)] Let $\struct {S, \NN}$ be a neighborhood space. Let $U \subseteq S$ be a neighborhood of each of its elements. Then $U$ is an '''open set''' of $S$. \end{definition}	ProofWiki
Observables and dimensional analysis Here is a simple question about physical units that I hope has a simple satisfying answer. In mathematically sophisticated treatments of both quantum and classical physics one often speaks of an algebra of observables and this leads to thinking about Jordan algebras, C* algebras and so on. As a mathematician I tend to not think about physical units too much but isn't there something funny about, for example, considering arbitrary functions of p and q when one considers units (to take classical phase space for example). For example, what sense is there in adding p and q which have different dimensions? Or even something like p+p^2. Supposedly, every smooth function on phase space, that is every smooth function of p's and q's, is supposed to be an "observable". But most such functions will be dimensionally incongruent as long as there are any units at all. I can think of some rather as hoc or otherwise unsatisfying excuses but I don't think I have come up with the best way to explain this to myself. In any given algebraic expression one can throw in various "constants" to bring things into line but nothing about the mathematical notion of an algebra of functions involves such distinctions. The whole thing seems awkward. What is the right way to think about this? By the way, it is occasionally strikes me as odd that we graphically represent in the same space the vectors of velocity, acceleration and angular momentum as if all lived in the same vector space. The very definition of a vector implies one may add any two vectors but again there is this issue of dimension. mp.mathematical-physics oa.operator-algebras cyberkatrucyberkatru $\begingroup$ It is not $p+p^2$ but rather $ap+bp^2$ where $a,b$ are physical constants that have units as well. You are not surprised that the usual free fall equation reads $h(t)=h_0-v_0t-gt^2/2$ despite if you only think of it as a function of time, you can get an impression that you add a dimensionless constant, seconds, and seconds squared. $\endgroup$ – fedja $\begingroup$ fedja, that is the answer I would have given but it isn't satisfying to me secretly. It gives the impression that nothing can go wrong when forming function, but every function is an observable in the algebra, which is also vector space. I use scalars a and b to form the linear combination $ap+bp^2$. Of course, a and b are real numbers and both could be 1. But they can both be the "same" 1 since they have different units in your view. What is the mathematical structure here? It still seems odd. $\endgroup$ – cyberkatru $\begingroup$ @cyberkatru: $a$ and $b$ are not real numbers. Real numbers are dimensionless. They are unital quantities. $\endgroup$ – Qiaochu Yuan $\begingroup$ Related question: mathoverflow.net/questions/63749 $\endgroup$ – Michael Bächtold A dimensionful real-valued quantity takes values in a $1$-dimensional real vector space (a 'line') rather than in the space of real numbers as such. Given two such quantities taking values in the line $L$, their sum also takes values in $L$; given a quantity taking values in $L$ and a quantity taking values in $L'$, their product takes values in $L \otimes L'$ (which is also a line). Dimensionless quantities are included in this, since $\mathbb{R}$ is itself a line, in fact the unit of the tensor product. So, instead of the Jordan (or whatever) algebra $A$, we really have an algebra $A^\bullet$ (I am making up this notation here), which is a completion of the direct sum over all lines $L$ of $A \otimes L$. It is worth checking that whatever functional calculus you rely on in $A$ also extends to $A^\bullet$, which is true if you include enough lines; for example, if $q$ takes values in $L$ (and never takes the value $0$), then $1/q$ takes values in the dual space $L^$ (which is also a line). Formally, we can add quantities with different dimensions, obtaining a result in $L \oplus L' \subseteq A^\bullet$ (and even do things like $\sin q$ if $A^\bullet$ is completed appropriately), but what's important is that each expression in which the units balance has a $1$-dimensional subspace of $A^\bullet$ in which all of its values lie. To relate this to Qiaochu's answer, Qiaochu's $A$ is my $A^\bullet$, and my $A$ is the subspace of Qiaochu's $A$ that is fixed by the action of $G$. Toby BartelsToby Bartels $\begingroup$ Toby, do you have a reference to a discussion in the literature about this? By the way, I recall that Hassler Whitney had a series of two articles about abstracting the notion of physical units. It was disappointingly complicated. Given how naturally everyone works with units in practice, it seems surprising that a precise mathematical abstraction of the notion should turn out to be so bulky. $\endgroup$ $\begingroup$ @cyberkatru: does terrytao.wordpress.com/2012/12/29/… count as part of the literature? $\endgroup$ $\begingroup$ I've also read something about this by John Baez in his proto-blog This Week's Finds in Mathematical Physics, but I can't track it down now. (I'm also adding a paragraph to my answer relating it to Qiaochu's.) $\endgroup$ – Toby Bartels What we mean when we say that two quantities have different units is that if we change something about how we measure quantities, the two quantities will behave differently. For example, if one quantity has a unit of length and another has a unit of time, then when we change how we measure lengths we'll modify the first quantity but not the second. Formally, we can think of thinking about units as introducing the action of a group, let's say $G = (\mathbb{R}^{+})^n$, where $n$ is the number of units you're considering. Each factor of this group rescales a different unit. Formally we can think of unital quantities in this setup as an algebra $A$ equipped with an action of $G$. That action equips it with a grading with respect to the characters of $G$, and this grading is what we mean by units. In particular, once we remember this group action there's no reason to restrict our attention to homogeneous elements of $A$. If we want to do this anyway, we can always multiply by unital constants (which are also elements of $A$). Qiaochu YuanQiaochu Yuan $\begingroup$ Just to once again underscore my discomfort, imagine I look at an element of the space of observables on a classical phase space. Does it have units? If so, can I or can I not add it to another observable of different units? If not, then we aren't in an algebra or even a vector space. Also, does the function f(q,p)=p have units? If so, can I exponentiate it without qualification? exp(f(p,q)) $\endgroup$ $\begingroup$ @cyber: yes, yes, and you need to distinguish the function $f$ from its output when evaluated on $p$ and $q$. $\endgroup$ $\begingroup$ I'm probably missing something here, and y'all will enlighten me: $f(q,p)=p$ should read $f(q,p)=ap$, with $a$ a parameter having dimension of length, so this function has no units. The appearance of the parameter $a$ exemplifies what Qiaochu Yuan says in his answer: when you change something (read $a$) the two quantities $q$ (read $q/a$) and $p$ (read $ap$) will behave differently. --- Is there more? $\endgroup$ – Carlo Beenakker The way most physicists would think about this, is to choose units where Planck's constant is 1, so $p$ has dimension of 1/length, and then assign a characteristic length $a$ to the system. If the problem is formulated on a lattice, $a$ could be the lattice constant, but any other length will do. Then whenever you see $p$ you think $ap$ and whenever you see $q$ you think $q/a$. Carlo BeenakkerCarlo Beenakker $\begingroup$ Actually, this answer is pretty good and sort of obvious in retrospect. I think this allows me to continue thinking about C algebras and such without the nagging discomfort that entered my mind the day I asked the question. On the other hand the comments of Toby and Qiaochu's are interesting and food for thought. I am looking forward to Toby's promised additional paragraph. $\endgroup$ $\begingroup$ I already added that (see the edit on my shorter answer); it's very brief (at the very end). $\endgroup$ Here is another, completely different answer. As Carlo indicated, one can use units in which Planck's constant is $1$. This is no arbitrary choice, but one dictated by fundamental physics. Similarly, one may set the vacuum speed of light to $1$. It is a historical accident (mainly due to our being massive creatures bound to a planet) that we think of length and time as being of different dimensions (in the sense of nonstandard analysis), so that we write $E = m c^2$, but really it's just $E = m$. Conversely, it's a historical accident (this time a lucky one) that we don't think of force as having its own dimension, so that we have $f = m a$ instead of $f = m a/g$ (where $g$ is Galileo's constant, about $32 \operatorname{lb}_{[M]} \operatorname{ft} \operatorname{s}^{-2} \operatorname{lb}_{[F]}^{-1}$, with a dimension of $[M] [L] [T]^{-2} [F]^{-1}$). This probably only happened because a clear distinction between the pound-force and the pound-mass came after Newton's laws (but before such other units such as the slug, the poundal, the gram, the dyne, and of course the newton). In electromagnetism, people often make do with only the dimensions $[L]$, $[T]$, and $[M]$, because they set Coulomb's constant to $1$. (Depending on where you put the $c$s, this is either the electrostatic, electromagnetic, or Gaussian system of dimensions; combined with $\operatorname{cm}$, $\operatorname{s}$, and $\operatorname{g}$ as the respective units of $[L]$, $[T]$, and $[M]$, this is called the electrostatic, electromagnetic, or Gaussian system of units.) Only the fuddy-duddies at the BIPM insist on making electric current an independent dimension $[A]$. (They also use $\operatorname{m}$, $\operatorname{s}$, and $\operatorname{kg}$ as the base units, so people write about this as 'cgs vs mks', when that's not what it's about at all.) Similarly, set Boltzmann's constant to $1$ to show that energy and temperature have the same dimension, and set Newton's gravitational constant to $1$ as well. Since of course $1 \operatorname{mol} \approx 6.02 \times 10^{23}$, all $6$ of the physical dimensions implicitly endorsed by the BIPM in the SI system of units (the candela depends on human biology) can now be seen to be utterly dimensionless! (The $6$ constants are Planck's, Maxwell's, Coulomb's, Boltzmann's, Newton's, and Avogadro's, and they are log-linearly independent, giving a unique solution to the system of $6$ homogeneous log-linear equations made by setting them all to $1$.) The point is: Every quantity is dimensionless, and every unit is simply some real number, so we may calculate with them as if they were real numbers because they are! (In fact, they are all positive real numbers, justifying our use of them in division and inequalities.) The $A^\bullet$ in my first answer is just $A$, and the group in Qiaochu's answer is trivial. Here is a problem: Although the constants we set to $1$ do come from fundamental physics, there is still some choice (even controversy) about how we do this. First, Planck's constant $h = 2 \pi \hbar$ derives from work on cyclic wave phenomena, and the really basic quantity is Dirac's constant $\hbar = h/(2 \pi)$ (which is also often called Planck's, so Carlo may have meant this all along). Similarly, Coulomb's constant $1/(4 \pi \epsilon_0)$ derives from work on spherically symmetric charge distributions, and the really basic quantity is $1/\epsilon_0$ itself (which, following the Gaussian system on placement of $c$, gives us the Heaviside–Lorentz system of dimensions when we set it to $1$). A similar remark applies to Newton's constant $G = c^2 \kappa/(8 \pi)$; Einstein's constant $\kappa$ is the more basic one. Planck himself, who first came up with all of this, not only used $h$ and $G$ instead of $\hbar$ and $\kappa$, but also used the charge of the proton instead of $\epsilon_0$ or Coulomb's constant, clearly a great error. So while every unit is a real number, different people disagree over which real numbers they are! (And not just because of experimental uncertainty, which is also an issue somewhat.) All of the possible different conventions to eliminate a given set of dimensions are mediated by a group of symmetries, the group in Qiaochu's answer, so keeping track of them all brings us back to the sophisticated answers that he and I gave. But the point is: You don't have to choose a convention. Since some convention is possible (and you already knew this when you saw your first system of units, however arbitrary it may have been), it is valid to say that every unit is a real number (even though which real number depends on the convention chosen), and so we may calculate with them as if they were (positive) real numbers. To me, an "observable" is just a device---often an actual, physical device, like a magnetic field meter or a radar gun---that outputs a number. For example, if I have a device that measures the airspeed of a plane in knots, measures the altitude of the plane in fathoms, and then outputs the sum of those two numbers, that's an observable. From this perspective, it makes perfect sense that an observable should be modeled in classical mechanics by a function on phase space, which precisely captures the idea of a number that may depend on physical circumstances. Confusingly, there's another type of thing in classical mechanics which is usually represented by a function on phase space: a Hamiltonian. Although Hamiltonians are sometimes called observables, I think there's an important sense in which they're not observables: they have units! I'll try to explain what I mean by this, but I'm really shaky on the details, so corrections are welcome. Like Toby Bartels said here, a dimensionful quantity is a physical quantity that lives in a one-dimensional real vector space, and a choice of units is a choice of isomorphism between that vector space and $\mathbb{R}$. In symplectic mechanics, we model phase space as a manifold $M$ with a symplectic form $\omega$. This doesn't involve any "unnatural identifications with $\mathbb{R}$," so no units show up. The story is different, however, when it comes to time. In classical mechanics, time is naturally thought of as a one-dimensional affine space, but in symplectic mechanics we typically identify it with the vector space $\mathbb{R}$. That's a choice of units! (And a choice of origin, too.) Once we identify time with $\mathbb{R}$, each function $H \colon M \to \mathbb{R}$ can be interpreted as a prediction about the motion of the system, in the following way. Let $\beta$ be the 2-form $\omega - dH \wedge dt$, where $t$ is the natural coordinate function on $\mathbb{R}$. A tangent vector $v \in T(\mathbb{R} \times M)$ is predicted to lie along a possible worldline of the system if and only if the cotangent vector $\beta(v, \cdot)$ is zero. Now, what if we want to avoid choosing units, and just think of time as an affine space $C$ modeled on a one-dimensional vector space $D$? Then a Hamiltonian ought to be a function $H \colon M \to D^$, so its exterior derivative will be a $D^$-valued 1-form on $M$, which can be interpreted as a 2-form on $C \times M$. Since $D$ is the tangent space of the time line, an element of $D$ is a dimensionful quantity with units of time. An element of $D^$ eats elements of $D$ and turns them into numbers, so it must have units of inverse time. That means a Hamiltonian has units of inverse time—which is exactly what you'd expect, because energy is the conjugate variable of time! 122 silver badges33 bronze badges VectornautVectornaut Not the answer you're looking for? Browse other questions tagged mp.mathematical-physics oa.operator-algebras or ask your own question. Dimensional Analysis in Mathematics W-completion of a C-algebra? What is the analog of a topos in quantum logic? 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\begin{document} \title{The Casimir force between parallel plates separated by anisotropic media } \author{Gang \surname{Deng}}\email{[email protected].} \author{Bao-Hua \surname{Tan}} \email{[email protected].} \author{Ling \surname{Pei}} \author{Ni \surname{Hu}} \author{Jin-Rong \surname{Zhu}} \affiliation{Hubei Collaborative Innovation Center for High-efficiency Utilization of Solar Energy, Hubei University of technology, Wuhan, 430068, P. R. China} \affiliation{School of Science, Hubei University of Technology, Wuhan, 430068, P. R. China} \begin{abstract} The Casimir force between two parallel plates separated by anisotropic media is investigated. We theoretically calculate the Casimir force between two parallel plates when the interspace between the plates is filled with anisotropic media. Our result shows that the anisotropy of the material between the plates can significantly affect the Casimir force, especially the direction of the force. If ignoring the anisotropy of the in-between material makes the force to be repulsive (attractive), by contrast, taking the anisotropy into account may produce an extra attractive (repulsive) force. The physical explanation for this phenomenon is also discussed. \end{abstract} \pacs{12.20.Ds, 77.22.-d, 78.20.Fm, 03.70.+k} \maketitle \section{Introduction} The Casimir effect originating form the quantum fluctuations of the electromagnetic field is one of the most remarkable macroscopic effects of quantum physics \cite{1,2,3,4,5}. Great effort has been put into both theoretical and experimental studies on the Casimir effect\cite{4,5}, as it plays an important role in various fields of physics \cite{4}. As we know, Lifshitz's original theory for the Casimir force is only applicable to isotropic dielectrics \cite{2,3}. However, anisotropy can bring us new features about the Casimir effect, like the Casimir torque. After the early works on the van der Waals interaction between anisotropic bodies in 1970s \cite{6,7}, Munday \emph{et al} numerically calculated the Casimir torque between parallel birefringent plates immersed in liquid, and proposed an experiment to observe this torque \cite{8}. As supplemental works, Shao \emph{et al} gave the analytical expressions of the Casimir torque and repulsive Casimir force between two birefringent plates with constant permittivity and permeability \cite{9}. In one of our previous works, we calculated the Casimir torque between two parallel anisotropic plates with nontrivial permeabilities, and discussed the impact of magnetic properties of the plates on the torque \cite{10}. On the other hand, anisotropy can also affect the Casimir force directly. Romanowsky show that the orientation of the optical axis could have significant influence on Casimir force between highly anisotropic plates\cite{11}. In one of our previous works, we calculated the Casimir force between anisotropic metamaterial plates \cite{12}. The results show that the direction of Casimir force could change with both the separation and the anisotropy. Ran Zeng and his collaborators investigated the Casimir force between anisotropic single-negative metamaterial slabs and show that the electromagnetic responses of the metamaterial parallel and perpendicular to the optical axis affected the Casimir force differently \cite{13}. Recent years, the research works have extended to the interaction between anisotropic particles and a surface \cite{14,15}. Most of the previous works \cite{7,8,9,10,11,12,13,14,15} were based on the assumption that the region between the two boundaries was vacuum or filled with isotropic media. However, in fact, this region can also be anisotropic. Some liquid, such as nitrobenzene and liquid crystal, can be anisotropic under special circumstances. One might naturally ask: what new phenomena can be seen if the region between the two slabs is filled with anisotropic media? Parsegian first calculated the van der Waals energy between two anisotropic bodies acting across a planar slab filled with a third anisotropic material in the non-retarded case \cite{6}. Kornilovitch investigated the Van der Waals interactions between flat surfaces in uniaxial anisotropic media in the non-retarded limit and discussed the effect of nonzero tilt between the optical axis and the surface normal on the interaction \cite{16}. On the other hand, repulsive Casimir force is always of great interests to the researchers \cite{2,3,4,5,8,9,12,13}. According to Lifshitz's theory, when the space between the two slabs is filled with isotropic media, the Casimir force can be repulsive if the permittivities of the plates ($\epsilon_{1}$ and $\epsilon_{2}$ ) and the interspatial media ($\epsilon_{3}$ ) satisfy the relation $\epsilon_{1}<\epsilon_{3}<\epsilon_{2}$ (or $\epsilon_{1}>\epsilon_{3}>\epsilon_{2}$)\cite{2,3}. However, if the interspatial media is birefringent, $\epsilon_{3}$ will have different values in different directions. Naturally, another question might be raised: when will the Casimir force be repulsive (or attractive) in this situation? Unfortunately, no clear clarification to this has been reported. Without doubt, the two problems above are of great importance. Therefore, it is necessary to study the possible new features of the Casimir effect between parallel plates separated by anisotropic media. In this work we use the quantized surface mode technique \cite{9,10,12,17} to calculate the Casimir force between two isotropic plates when the interspace between them is filled with anisotropic media. Our major concern is focused on the impact of the anisotropy on the Casimir force, especially the direction of the force. The result shows that the direction of the Casimir force can change with the anisotropy of the interspatial media. The physical understanding of this phenomenon is also investigated. The detailed discussion will be presented in the following sections. \section{The Casimir force between parallel plates separated by anisotropic media} The system considered is shown in Fig. 1. Two isotropic plates (with the diameter $D$ and the thickness $d$) made of different materials are kept parallel to each other and separated with a distance $a$. The region between the plates is filled with uniaxial media. The plates and the media between them are considered to be nonmagnetic. The \emph{x-y} plane is chosen to be parallel to the surfaces of the plates. It should be noted that if we only consider the anisotropy of the media between the plates, but consider the plates to be isotropic, there will be no Casimir torque. In this work, the optical axis of the anisotropic media is chosen to be in $z$ direction, which is often referred as the out-of-plane case. \begin{figure} \caption{Two isotropic plates are separated by anisotropic media with a distance \emph{a}. The surfaces of the plates are parallel to the \emph{x}-\emph{y} plane. The optic axis of the interspatial anisotropic media is in \emph{z} direction.} \label{fig:epsart} \end{figure} Assuming $D$ and $d$ are much greater than $a$, it is reasonable to disregard the edge effect and finite thickness effect. And the space can be approximately considered to be divided into three regions with corresponding permittivities, as shown in Fig.2. The relative permittivities can be expressed as diagonal matrixes, respectively, as following. \begin{eqnarray} \epsilon_{1}=\left( \begin{array}{ccc} \epsilon_{1} & 0 & 0 \\ 0 & \epsilon_{1} & 0 \\ 0 & 0 & \epsilon_{1} \\ \end{array} \right),\label{eqn1}\\ \epsilon_{2}=\left( \begin{array}{ccc} \epsilon_{2} & 0 & 0 \\ 0 & \epsilon_{2} & 0 \\ 0 & 0 & \epsilon_{2} \\ \end{array} \right),\label{eqn2}\\ \epsilon_{3}=\left( \begin{array}{ccc} \epsilon_{3x} & 0 & 0 \\ 0 & \epsilon_{3x} & 0 \\ 0 & 0 & \epsilon_{3z} \\ \end{array} \right),\label{eqn3} \end{eqnarray} where the subscripts $z$ and $x$ indicate the components parallel and perpendicular to the optical axis, respectively. \begin{figure} \caption{The schematic configuration. The space can be approximately considered to be divided into three regions. The optical property of each region is described by corresponding relative permittivity.} \label{fig:epsart} \end{figure} According to the quantized surface mode technique \cite{9,10,12,17}, we only need to consider the zero point energy associated with the surface modes \emph{q} which are exponentially decaying when \emph{z}$>$\emph{a} and \emph{z}$<$0. Because each region in Fig.2 is considered as homogenous, the electric and magnetic fields of the surface mode \emph{q} can be expressed as \cite{9,10,12,17,18} \begin{eqnarray} \textbf{E}_{q}={\rm i} N [a_{q}\textbf{e}_{q}(\textbf{k})-a^{\dag}_{q}{\textbf{e}}^{}_{q}(\textbf{k})],\label{eqn4}\\ \textbf{H}_{q}=N[a_{q}\textbf{h}_{q}(\textbf{k})+a^{\dag}_{q}\textbf{h}^{}_{q}(\textbf{k})],\label{eqn5} \end{eqnarray} where \emph{N} is the normalization factor, and the parameters $a_{q}$ and $a^{\dag}_{q}$ are the usual creation and annihilation operators, respectively. The parameters $\textbf{e}_{q}$ and $\textbf{h}_{q}$ are the electric and magnetic field polarization vectors. $\textbf{k}=(k_{x}, k_{y}, k_{z})$ is the wave vector. We can choose the wave vector in \emph{x-y} plane to be parallel to the \emph{x} direction, and then the wave vector can be written as $\textbf{k}=(k_{x}, 0,k_{z})=K_{0}(\alpha, 0, \gamma)$, with $K_{0}=\omega c^{-1}$. We introduce $M_{1}=\epsilon_{1}/\epsilon_{3x}$, $M_{2}=\epsilon_{2}/\epsilon_{3x}$ and $M_{3}=\epsilon_{3z}/\epsilon_{3x}$ to describe the relative values of $\epsilon_1$, $\epsilon_2$ and $\epsilon_{3z}$ to $\epsilon_{3x}$. $M_{3}$ can describe the anisotropy of the material filled between the plates. $M_{3}=1$ refers to isotropic. The Casimir energy per unit area at zero temperature can be expressed as ($\xi=-{\rm i}\omega$) \begin{equation} E(\emph{a})=\frac{\hbar}{4\pi^{2}c^{2}}\int_{1}^{\infty}p{\rm d}p \int_{0}^{\infty}\epsilon_{3,\emph{x}} \xi^{2}{\rm d}\xi[\ln G_1({\rm i} \xi)+\ln G_2({\rm i} \xi)] \label{eqn6} \end{equation} where \emph{c} is the speed of light in vacuum, and $\hbar$ is Plank constant divided by 2$\pi$. The variable $p$ is introduced as $p^{2}=1-\alpha^{2}/\epsilon_{3x}$. The detailed representations for the functions $G_1({\rm i} \xi)$ and $G_2({\rm i} \xi)$ in Eq.(6) are expressed as (The detailed derivation of Eqs.(\ref{eqn6})-(\ref{eqn8}) is presented in Appendix A.) \begin{eqnarray} G_1({\rm i} \xi)=1-\frac{(s_{1}-p)(s_{2}-p)}{(s_{1}+p)(s_{2}+p)}{\rm exp}\left(-\frac{2pa\xi}{c}\sqrt{\epsilon_{3x}}\right), \label{eqn7}\\ G_2({\rm i} \xi)=1-\frac{(s_{1}-M_{1}P)(s_{2}-M_{2}P)}{(s_{1}+M_{1}P)(s_{2}+M_{2}P)}{\rm exp}\left(-\frac{2Pa\xi}{c}\sqrt{\epsilon_{3x}}\right), \label{eqn8} \end{eqnarray} where $s_{1,2}=\sqrt{M_{1,2}-1+p^2}$ and $P=\sqrt{(M_{3}-1+p^2)/M_{3}}$. The Casimir force on the plates per unit area is \begin{align} &F=\frac{\partial E(a)}{\partial a}=\frac{\hbar}{2\pi^{2}c^{3}}\int_{1}^{\infty}p{\rm d}p \int_{0}^{\infty}\epsilon_{3\emph{x}}^{3/2} \xi^{3}{\rm d}\xi\nonumber\\ &\left[p\frac{1-G_{1}}{G_{1}}+P\frac{1-G_{2}}{G_{2}}\right]\nonumber\\ \label{eqn9} \end{align} As $G_2$ is a function of $M_3$, the Casimir force $F$ will also depend on $M_3$ which refers to the anisotropy of the material between the plates. And it can be found that $G_1$ does not depend on $M_3$, which means the anisotropy affects the force mainly through the second term in the brackets in Eq. (\ref{eqn9}). For the case that the material between the plates is isotropic, $\epsilon_{3x}=\epsilon_{3z}=\epsilon_{3}$ ($M_{3}=1$ and $P=p$), Eq.(\ref{eqn9}) becomes \begin{align} &F^{\rm isotropic}=\frac{\hbar}{2\pi^{2}c^{3}}\int_{1}^{\infty}p^{2}{\rm d}p \int_{0}^{\infty}\epsilon_{3}^{3/2} \xi^{3}{\rm d}\xi\nonumber\\ &\left\{\left[\frac{s_{1}+p}{s_{1}-p}\frac{s_{2}+p}{s_{2}-p}{\rm exp}\left(\frac{2pa\xi}{c}\sqrt{\epsilon_{3}}\right)-1\right]^{-1}\right. \nonumber\\ &\left.+\left[\frac{s_{1}+M_{1}p}{s_{1}-M_{1}p}\frac{s_{2}+M_{2}p}{s_{2}-M_{2}p}{\rm exp}\left(\frac{2pa\xi}{c}\sqrt{\epsilon_{3}}\right)-1\right]^{-1}\right\}, \label{eqn10} \end{align} which recovers Lifshitz's result about force on two bodies separated by a gap filled with a third isotropic media (equation 4.14 in Ref.\cite{3}). Let's turn to the limiting case that the separation $a$ is larger than the characteristic absorption wavelength of the material. In this case, we can replace $\epsilon_{1}$, $\epsilon_{2}$, $\epsilon_{3x}$ and $\epsilon_{3z}$ by their values at $\xi=0$, i.e. the static dielectric constants \cite{3}. The approximate Casimir force can be expressed as following \begin{equation} F\approx\frac{3\hbar c}{16\pi^2a^4\sqrt{\epsilon_{3x}}}\Psi \label{eqn11} \end{equation} with \begin{equation} \Psi (M_1,M_2,M_3)=\int_{1}^{\infty}p{\rm d}p\left[\frac{1}{p^3}\left(\frac{s_{10}-p}{s_{10}+p}\right )\left(\frac{s_{20}-p}{s_{20}+p}\right )+\frac{1}{P^3}\left(\frac{s_{10}-M_1P}{s_{10}+M_1P}\right )\left(\frac{s_{20}-M_2P}{s_{20}+M_2P}\right )\right] \label{eqn12} \end{equation} where $s_{10}$ and $s_{20}$ are the values of $s_1$ and $s_2$ at $\xi=0$. (The detail of the approximate calculation of the integration in the Casimir force function is presented in Appendix B) \section{The impact of the anisotropy of the media between the plates on the direction of the Casimir force} The positive value of $F$ (or $\Psi$) in Eq. (\ref{eqn11}) corresponds to the attractive force, while the negative value corresponds to the repulsive force. As we are interested in the impact of the anisotropy of the interspatial media on the direction of the Casimir force, we can just discuss the sign of the function $\Psi$. From Eq.(\ref{eqn11}) and Eq.(\ref{eqn12}), it is clear that if the permittivities satisfy the relations $\epsilon_{1}<\epsilon_{3x}<\epsilon_{2}$ and $\epsilon_{1}<\epsilon_{3z}<\epsilon_{2}$ (or $\epsilon_{1}>\epsilon_{3x}>\epsilon_{2}$ and $\epsilon_{1}>\epsilon_{3z}>\epsilon_{2}$) at the same time, $\Psi$ will be minus and the force will be repulsive. This is because although $\epsilon_{3}$ has different values in different directions, all the possible values of $\epsilon_{3}$ are still in the range form $\epsilon_{2}$ to $\epsilon_{1}$. This can make the force repulsive, which is similar to the Lifshitz's result \cite{3}. However, what we mainly concern is not this case but the case when $\epsilon_{3x}$ or $\epsilon_{3z}$ is out of the range $(\epsilon_{2}, \epsilon_{1})$ (Here, $(\epsilon_{2}, \epsilon_{1})$ means the range form $\epsilon_{2}$ to $\epsilon_{1}$). First, we come to the case that only $\epsilon_{3x}$ is in the range $(\epsilon_{2}, \epsilon_{1})$. We let $\epsilon_{1}>\epsilon_{3x}>\epsilon_{2}$ ($M_1>1$ and $M_2<1$). Fig.3 shows the value of $\Psi$ for different $M_3$. The positive value of $\Psi$ corresponds to the attractive force, while the negative value corresponds to the repulsive force. Two vertical dash-dot lines, corresponding to $M_3=M_2$ and $M_3=M_1$, divide Fig.3 into 3 regions. In the middle region, where $M_2<M_3<M_1$ ($\epsilon_{2}<\epsilon_{3z}<\epsilon_{1}$), the force is always repulsive, which has been discussed previously. The other two regions (left and right) are the regions that we are interested in. In the left region, where $M_3<M_2$ ($\epsilon_{3z}<\epsilon_{2}$), the force is not always in the same direction and it can be either repulsive or attractive depending on the value of $M_3$ which refers to the anisotropy. In the region on the right, where $M_3>M_1$ ($\epsilon_{3z}>\epsilon_{1}$), the result is similar and the force can also be either repulsive or attractive depending on the value of $M_3$. \begin{figure} \caption{$\Psi$, $\Psi_1$ and $\Psi_2$ vs. $M_3$. $M_1=1.5$ and $M_2=0.8$. The two vertical dash-dot lines divide the range of $M_3$ into 3 parts corresponding to $\epsilon_{3z}<\epsilon_{2}$, $\epsilon_{2}<\epsilon_{3z}<\epsilon_{1}$, and $\epsilon_{3z}>\epsilon_{1}$ form left to right, respectively. The positive value corresponds to the attractive force, while the negative value corresponds to the repulsive force.} \label{fig:epsart} \end{figure} In order to see this more clearly, we can resolve $\Psi$ in Eq.(\ref{eqn12}) into two parts as flowing \begin{equation} \Psi(M_1,M_2,M_3)=\Psi_1+\Psi_2 \label{eqn13} \end{equation} with \begin{eqnarray} \Psi_1(M_1,M_2)=\int_{1}^{\infty}p{\rm d}p\left[\frac{1}{p^3}\frac{s_{10}-p}{s_{10}+p}\frac{s_{20}-p}{s_{20}+p}\right] \label{eqn14}\\ \Psi_2(M_1,M_2,M_3)=\int_{1}^{\infty}p{\rm d}p\left[\frac{1}{P^3}\frac{s_{10}-M_{10}P}{s_{10}+M_{10}P}\frac{s_{20}-M_{20}P}{s_{20}+M_{20}P}\right] \label{eqn15} \end{eqnarray} It is clear that $\Psi_1$ is independent of $\epsilon_{3z}$ and $M_3$, which means anisotropy of the in-between media does not affect $\Psi_1$. If $\epsilon_{1}>\epsilon_{3x}>\epsilon_{2}$ ($M_1>1$ and $M_2<1$), as we assumed previously, $\Psi_1$ will always be negative and contribute to a repulsive force, no matter how much the anisotropy ($M_3$) is. And $\Psi_2$ is the one that the anisotropy is mainly associated with. From Fig.3 we can see that $\Psi_2$ takes the dominant place in most of the range. It can be either positive or negative depending on the anisotropy. Although $\epsilon_{1}>\epsilon_{3x}>\epsilon_{2}$ always produces a repulsive force, the total force will not necessarily to be repulsive, if the anisotropy makes $\epsilon_{3z}$ to be out of this range, as shown in Fig.3. This is a result of the competition between $\Psi_1$ and $\Psi_2$. If the anisotropy makes $\Psi_2$ to be positive and have a greater amplitude than $\Psi_1$, the direction of the force will switch to attractive from repulsive. But without anisotropy, $\Psi_2$ will be surely negative and the force will be always repulsive. To achieve an attractive force, anisotropy is a must in this case. Therefore, we can conclude that it is the anisotropy of the media between the plates that produces the attractive force in this case. This can also be understood in an intelligible and vivid manner as flowing. According to Lifshitz's theory\cite{3}, the attractive force arises when $\epsilon_{3}$ is out of the range $(\epsilon_{2}, \epsilon_{1})$. However, in above case, $\epsilon_{3x}$ is assumed to be just in $(\epsilon_{2}, \epsilon_{1})$, which may always contribute to repulsive force. To produce an attractive force, one must let $\epsilon_{3z}$ to be a little far form this range. So that the "average value" (the quotation marks here mean that it is not the real mathematical average value \footnote{The quotation marks here mean that it is not the real mathematical average value. In Fact, it might be some value between $\epsilon_{3x}$ and $\epsilon_{3z}$. This is just a vivid way to make the problem easier to be understood. }) of the $\epsilon_{3}$ can be out of this range, and the attractive force can arise. And to make $\epsilon_{3z}$ out of the range $(\epsilon_{2}, \epsilon_{1})$, anisotropy is a must in this situation. This is why the attractive force ($\Psi>0$) happens only on the left and right edges of Fig.3. \begin{figure} \caption{$\Psi$, $\Psi_1$ and $\Psi_2$ vs. $M_3$. $M_1=1.5$ and $M_2=1.1$. The two vertical dash-dot lines divide the range of $M_3$ into 3 parts corresponding to $\epsilon_{3z}<\epsilon_{2}$, $\epsilon_{2}<\epsilon_{3z}<\epsilon_{1}$, and $\epsilon_{3z}>\epsilon_{1}$ form left to right, respectively. The positive value corresponds to the attractive force, while the negative value corresponds to the repulsive force.} \label{fig:epsart} \end{figure} Now let's turn to the case that $\epsilon_{3x}$ is not in the range $(\epsilon_{2}, \epsilon_{1})$. If we let $\epsilon_{3x}<\epsilon_{2}$ ($M_1>M_2>1$), $\Psi_1$ will be always positive and produce an attractive force. As shown in Fig. 4, $\Psi_2$ which is associated with anisotropy is still dominant in most of the range. Without anisotropy, $\Psi_2$ will be positive and the force will be always attractive. To achieve a repulsive force, $M_3$ must not be 0 in this case. We can also conclude that it is the anisotropy of the media between the plates that produces the repulsive force in this case. According to Lifshitz's theory \cite{3}, the force is repulsive only when $\epsilon_{1}>\epsilon_{3}>\epsilon_{2}$ (or $\epsilon_{1}<\epsilon_{3}<\epsilon_{2}$) is satisfied, and it is attractive in all the other cases. However, it can be found in Fig.4 that the total force can be repulsive, even when both $\epsilon_{3x}$ and $\epsilon_{3z}$ are out of the range $(\epsilon_{2}, \epsilon_{1})$ ! The explanation is similar to the previous case. As shown in Fig.4, if $\epsilon_{3x}<\epsilon_{2}$, to achieve a repulsive force, $M_3$ ($\epsilon_{3z}$) must be larger than $M_2$ ($\epsilon_{2}$) to bring the "average value" of $\epsilon_{3}$ in $(\epsilon_{2}, \epsilon_{1})$. However, it should not be too much greater than $M_1$ ($\epsilon_{1}$), as this may make the "average value" of $\epsilon_{3}$ beyond $\epsilon_{1}$. That is why the repulsive force only appears in the middle region of Fig.4. From above discussion, we can find that the direction of the force is affected by $M_3$ which refers to the anisotropy of the media between the plates. If anisotropy of the in-between material makes the "average value" of $\epsilon_{3}$ to be in the range $(\epsilon_{2}, \epsilon_{1})$, the force will be repulsive. Otherwise, the force will be attractive. The border curve defined by $\Psi=0$ is shown in Fig.5 (for convenient, $M_1$ is fixed to be 1.5). The regions with $\Psi>0$ are the attractive regions, while the regions with $\Psi<0$ are the repulsive regions. \begin{figure} \caption{The repulsive and attractive regions in $M_2-M_3$ plane. The solid curve corresponds to $\Psi=0$ ($M_1=1.5$)} \label{fig:epsart} \end{figure} \section{Conclusion} We have calculated the Casimir force between two plates separated by a gap filled with anisotropic media. The result shows that the anisotropy of the media between the plates plays an important, or sometimes dominant, role in the total force. The more important thing is that it can affect the direction of the force. If ignoring the anisotropy of the in-between material makes the force to be repulsive, by contrast, taking the anisotropy into account can produce an extra attractive force; and if ignoring the anisotropy of the in-between material makes the force to be attractive, taking the anisotropy into account can produce an extra repulsive force. This can be explained as the result of the competition of the attractive term and repulsive term in the force function. And it can also be understood, in an intelligible and vivid way, as the anisotropy makes the "average value" of the permittivity of the intermediate material to be in a certain range to produce the force in the corresponding direction. Finally, we should mention that it is still not easy to observe the above effects experimentally, since it is difficult to achieve high anisotropy in liquid. \section{Acknowledgment} This work was partially supported by the Open Foundation of Hubei Collaborative Innovation Center for High-efficient Utilization of Solar Energy (HBSKFMS2014042 and HBSKFZD2014015), and the National Natural Science Foundation of China (11304091). The authors would like to show special thanks to Hubei University of Technology for providing early stage financial support form the Doctoral research program (BSQD12068) \appendix \section{The detailed derivation of the Casimir energy} In this section we will present the detailed derivation of Eqs.(\ref{eqn6})-(\ref{eqn8}). The electric and magnetic field polarization vectors $\textbf{e}_{q}$ and $\textbf{h}_{q}$ in Eqs.(\ref{eqn4}) and (\ref{eqn5}) with frequency $\omega$ can be determined according to the classical Maxwell equations (In SI Units) \begin{eqnarray} \nabla \times \textbf{E}=-\frac{\partial}{\partial t} \textbf{B}\label{eqna1}\\ \nabla \times \textbf{B}=\frac{1}{c^2}\frac{\partial}{\partial t} \epsilon_i \textbf{E}\label{eqna2} \end{eqnarray} $\epsilon_i$ (\emph{i}=1,2,3) is the relative permittivity in region \emph{i}, and its detailed representation can be found in Eqs.(\ref{eqn1})-(\ref{eqn3}), respectively. The solution of the electric and magnetic fields can be expressed in the form of plane wave as \begin{equation} \left( \begin{array}{c} \textbf{E} \\ \textbf{B} \end{array} \right)=\left( \begin{array}{c} e_x \\ e_y \\ e_z \\ b_x \\ b_y \\ b_z \end{array} \right){\rm e}^{{\rm i} K_{0}(\alpha x+\gamma z)-{\rm i}\omega t} \label{eqna3a} \end{equation} $e_x$, $e_y$, and $e_z$ are the elements of the electric field (E) polarization vectors, while $b_x$, $b_y$, and $b_z$ are the elements of the magnetic field (B) polarization vectors. As we have assumed that all the materials considered in our work are nonmagnetic, we can have $b_x=\mu_0 h_x$, $b_y=\mu_0 h_y$, and $b_z=\mu_0 h_z$. In Region I (as shown in Fig.2), substituting Eq.(\ref{eqn1}) and Eq.(\ref{eqna3a}) into Eqs.(\ref{eqna1}) and (\ref{eqna2}), we can have the eigenequations of the transverse elements of the electromagnetic field polarization vectors $e_x$, $e_y$, $b_x$, and $b_y$. \begin{equation} \left( \begin{array}{cccc} 0 & 0 & 1-\frac{\alpha ^2}{\epsilon_1} & 0 \\ 0 & 0 & 0 & -1 \\ \epsilon_1 & 0 & 0 & 0 \\ 0 & -\epsilon_1+ \alpha ^2 & 0 & 0 \end{array} \right)\left( \begin{array}{c} e_x \\ e_y \\ cb_y \\ cb_x \end{array} \right)=\gamma \left(\begin{array}{c} e_x \\ e_y \\ cb_y \\ cb_x \end{array} \right) \label{eqna3} \end{equation} The eigenvectors and the corresponding eigenvalues can be written in the matrix form as following: \begin{equation} W^{\rm I}=\left( \begin{array}{cccc} 0 & -\frac{{\rm i} t_1}{\epsilon_1} & 0 & \frac{{\rm i} t_1}{\epsilon_1} \\ -\frac{{\rm i}}{ t_1} & 0 & \frac{{\rm i}}{ t_1} & 0 \\ 0& 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{array} \right) \label{eqna4} \end{equation} \begin{equation} \gamma ^{\rm I}=( \begin{array}{cccc} \gamma _1^{\rm I} & \gamma _2^{\rm I} & \gamma _3^{\rm I} & \gamma _4^{\rm I} \end{array} ) =( \begin{array}{cccc} -{\rm i}t_1 & -{\rm i}t_1 & {\rm i}t_1 & {\rm i}t_1 \end{array} ) \label{eqna5} \end{equation} with $t^2_1=\alpha^2-\epsilon_1$. Each column of the matrix $W^{\rm I}$ represents an eigenvector. Four eigenvalues correspond to four independent mode solutions. The general solution of the electromagnetic field should be the linear superposition of the four mode solutions, and the superposition coefficients are the amplitudes of each mode. Then the transverse elements of the electromagnetic field in region I can be written as \begin{equation} \left(\begin{array}{c} E_x \\ E_y \\ cB_y \\ cB_x \end{array}\right) =W^{\rm I} \left(\begin{array}{c} A^{\rm I}_1{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _1^{\rm I}z)} \\ A^{\rm I}_2{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _2^{\rm I}z)} \\ A^{\rm I}_3{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _3^{\rm I}z)} \\ A^{\rm I}_4{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _4^{\rm I}z)} \end{array}\right){\rm e}^{-{\rm i}\omega t} \label{eqna6} \end{equation} where $A^{\rm I}_j$ (\emph{j}=1,2,3,4) is the amplitude of the \emph{j}th mode solution. Similarly, in region II, where it is also isotropic, the transverse elements of the electromagnetic field can be written as \begin{equation} \left(\begin{array}{c} E_x \\ E_y \\ cB_y \\ cB_x \end{array}\right) =W^{\rm II} \left(\begin{array}{c} A^{\rm II}_1{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _1^{\rm II}z)} \\ A^{\rm II}_2{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _2^{\rm II}z)} \\ A^{\rm II}_3{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _3^{\rm II}z)} \\ A^{\rm II}_4{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _4^{\rm II}z)} \end{array}\right){\rm e}^{-{\rm i}\omega t} \label{eqna7} \end{equation} with \begin{equation} W^{\rm II}=\left( \begin{array}{cccc} 0 & -\frac{{\rm i} t_2}{\epsilon_2} & 0 & \frac{{\rm i} t_2}{\epsilon_2} \\ -\frac{{\rm i}}{ t_2} & 0 & \frac{{\rm i}}{ t_2} & 0 \\ 0& 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{array} \right) \label{eqna8} \end{equation} and \begin{equation} \gamma ^{\rm I}=( \begin{array}{cccc} \gamma _1^{\rm II} & \gamma _2^{\rm II} & \gamma _3^{\rm II} & \gamma _4^{\rm II} \end{array} ) =( \begin{array}{cccc} -{\rm i}t_2 & -{\rm i}t_2 & {\rm i}t_2 & {\rm i}t_2 \end{array} ) \label{eqna9} \end{equation} where $t^2_2=\alpha^2-\epsilon_2$, and $A^{\rm II}_j$ (\emph{j}=1,2,3,4) is the amplitude of the \emph{j}th mode solution. In region III, where it is anisotropic, the case is different, as the permittivity has different form as shown in Eq.(\ref{eqn3}). Substitute Eq.(\ref{eqn3}) and Eq.(\ref{eqna3a}) into Eqs.(\ref{eqna1}) and (\ref{eqna2}), and we can have the eigenequations of the transverse elements of the electromagnetic field polarization vectors $ e_x$, $e_y$, $b_x$, and $b_y$. \begin{equation} \left( \begin{array}{cccc} 0 & 0 & 1-\frac{\alpha ^2}{\epsilon_{3z}} & 0 \\ 0 & 0 & 0 & -1 \\ \epsilon_{3x} & 0 & 0 & 0 \\ 0 & -\epsilon_{3x}+ \alpha ^2 & 0 & 0 \end{array} \right)\left( \begin{array}{c} e_x \\ e_y \\ cb_y \\ cb_x \end{array} \right)=\gamma \left(\begin{array}{c} e_x \\ e_y \\ cb_y \\ cb_x \end{array} \right) \label{eqna10} \end{equation} The eigenvectors and the corresponding eigenvalues can be written in the matrix form as following: \begin{equation} W^{\rm III}=\left( \begin{array}{cccc} 0 & -\frac{{\rm i} t_{3z}}{\epsilon_{3x}} & 0 & \frac{{\rm i} t_{3z}}{\epsilon_{3x}} \\ -\frac{{\rm i}}{ t_{3x}} & 0 & \frac{{\rm i}}{ t_{3x}} & 0 \\ 0& 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{array} \right) \label{eqna11} \end{equation} \begin{equation} \gamma ^{\rm III}=( \begin{array}{cccc} \gamma _1^{\rm III} & \gamma _2^{\rm III} & \gamma _3^{\rm III} & \gamma _4^{\rm III} \end{array} ) =( \begin{array}{cccc} -{\rm i}t_{3x} & -{\rm i}t_{3z} & {\rm i}t_{3x} & {\rm i}t_{3z} \end{array} ) \label{eqna12} \end{equation} with $t^2_{3x}=\alpha^2-\epsilon_{3x}$ and $t^2_{3z}=(\alpha^2-\epsilon_{3z})(\epsilon_{3x} / \epsilon_{3z})$ . Similarly, in region III, the transverse elements of the electromagnetic field can be written as \begin{equation} \left(\begin{array}{c} E_x \\ E_y \\ cB_y \\ cB_x \end{array}\right) =W^{\rm III} \left(\begin{array}{c} A^{\rm III}_1{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _1^{\rm III}z)} \\ A^{\rm III}_2{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _2^{\rm III}z)} \\ A^{\rm III}_3{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _3^{\rm III}z)} \\ A^{\rm III}_4{\rm e}^{{\rm i} K_{0}(\alpha x+\gamma _4^{\rm III}z)} \end{array}\right){\rm e}^{-{\rm i}\omega t} \label{eqna13} \end{equation} where $A^{\rm III}_j$ (\emph{j}=1,2,3,4) is the amplitude of the \emph{j}th mode solution. As the surface modes should be exponentially decaying for $z>0$ and $z<a$, we have $A^{\rm I}_3=A^{\rm I}_4 =A^{\rm II}_1=A^{\rm II}_2=0$ \cite{9,17}. As the transverse elements of the electromagnetic field are continuous at $z=0$, we have \begin{equation} \left(\begin{array}{cc} 0 & -\frac{{\rm i} t_1}{\epsilon_1} \\ -\frac{{\rm i}}{t_1} & 0\\ 0 & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{c} A^{\rm I}_1 \\ A^{\rm I}_2 \end{array}\right) =W^{\rm III} \left(\begin{array}{c} A^{\rm III}_1 \\ A^{\rm III}_2 \\ A^{\rm III}_3 \\ A^{\rm III}_4 \end{array}\right) \label{eqna14} \end{equation} As the transverse elements of the electromagnetic field are continuous at $z=a$, we have, \begin{equation} \left(\begin{array}{cc} 0 & \frac{{\rm i} t_2}{\epsilon_2} \\ \frac{{\rm i}}{t_2} & 0\\ 0 & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{c} A^{\rm II}_3 {\rm e}^{-K_0 t_2 a}\\ A^{\rm II}_4 {\rm e}^{-K_0 t_2 a} \end{array}\right) =W^{\rm III} \left(\begin{array}{c} A^{\rm III}_1 {\rm e}^{K_0 t_{3x} a}\\ A^{\rm III}_2 {\rm e}^{K_0 t_{3z} a}\\ A^{\rm III}_3 {\rm e}^{-K_0 t_{3x} a} \\ A^{\rm III}_4 {\rm e}^{-K_0 t_{3z} a} \end{array}\right) \label{eqna15} \end{equation} Eqs.(\ref{eqna14})and (\ref{eqna15}) include eight linear homogeneous equations relating the unknown parameters $A^{\rm I}_1$, $A^{\rm I}_2$, $A^{\rm II}_3$, $A^{\rm II}_4$, $A^{\rm III}_1$, $A^{\rm III}_2$, $A^{\rm III}_3$, and $A^{\rm III}_4$. It has the nontrivial solutions if the determinant of its coefficients is equal to zero, which leads to the equation for the determination of the proper frequency $\omega$ . And the equation of determinant of coefficients \footnote{In fact, we cancel $A^{\rm III}_1$, $A^{\rm III}_2$, $A^{\rm III}_3$, and $A^{\rm III}_4$, and reduce the problem to be 4 linear equations with 4 unknown parameters $A^{\rm I}_1$, $A^{\rm I}_2$, $A^{\rm II}_3$ and $A^{\rm II}_4$. This makes the determinant of coefficients to be $4\times 4$ rather than $8\times 8$.} equalling zero can be transformed into following form \begin{equation} Y[G_1(\omega _{\perp})*G_2(\omega _{\parallel})]=0 \label{eqna16} \end{equation} where \emph{Y} is a function that is not always equal to zero. The subscripts $\perp$ and $\parallel$ indicate the mode with the polarization of the electric field perpendicular and parallel to the plane formed by $\vec{k}_{\parallel}=(k_{x}, k_{y})$ and \emph{z} (As we have assumed $\vec{k}_{\parallel}$ to be parallel to the \emph{x} direction, this plane is also \emph{x-z} plane). Introducing $\omega = {\rm i} \xi$, the functions $G_1(\omega _{\perp})$ and $G_2(\omega _{\parallel})$ can be expressed as following (Using the relation $p^{2}=1-\alpha^{2}/\epsilon_{3x}$, we can transform $t^2_1$, $t^2_2$, $t^2_{3x}$ and $t^2_{3z}$ into the functions of \emph{p} or $s_{1,2}$ \footnote {With the relation $p^{2}=1-\alpha^{2}/\epsilon_{3x}$, we have $t^2_1=-s^2_1\epsilon_{3x}$, $t^2_2=-s^2_2\epsilon_{3x}$, $t^2_{3x}=-p^2\epsilon_{3x}$, and $t^2_{3z}=[\epsilon_{3x}(1-p^2)-\epsilon_{3z}]\frac{\epsilon_{3x}}{\epsilon_{3z}}$. This transformation might make the variables similar to the notations used in Lifshitz's theory.}) \begin{eqnarray} G_1(\omega _{\perp})=G_1({\rm i} \xi)=1-\frac{(s_{1}-p)(s_{2}-p)}{(s_{1}+p)(s_{2}+p)}{\rm exp}\left(-\frac{2pa\xi}{c}\sqrt{\epsilon_{3x}}\right), \label{eqna17}\\ G_2(\omega _{\parallel})=G_2({\rm i} \xi)=1-\frac{(\epsilon_{3x} s_{1}-\epsilon_1 \sqrt{\frac{\epsilon_{3x}(p^2-1)+\epsilon_{3z}}{\epsilon_{3z}}})(\epsilon_{3x} s_{2}-\epsilon_2 \sqrt{\frac{\epsilon_{3x}(p^2-1)+\epsilon_{3z}}{\epsilon_{3z}}})}{(\epsilon_{3x} s_{1}+\epsilon_1 \sqrt{\frac{\epsilon_{3x}(p^2-1)+\epsilon_{3z}}{\epsilon_{3z}}})(\epsilon_{3x} s_{2}+\epsilon_2 \sqrt{\frac{\epsilon_{3x}(p^2-1)+\epsilon_{3z}}{\epsilon_{3z}}})}{\rm e}^{-\frac{2a\xi\sqrt{\epsilon_{3x}^2(p^2-1)+\epsilon_{3z} \epsilon_{3x}}}{c \sqrt{\epsilon_{3z}}}}, \label{eqna18} \end{eqnarray} Eq.(\ref{eqna17}) is the same as Eq.(\ref{eqn7}). Substituting $M_{1}=\epsilon_{1}/\epsilon_{3x}$, $M_{2}=\epsilon_{2}/\epsilon_{3x}$, $M_{3}=\epsilon_{3z}/\epsilon_{3x}$ and $P=\sqrt{(M_{3}-1+p^2)/M_{3}}$ into Eq.(\ref{eqna18}), we will get Eq.(\ref{eqn8}) The Casimir energy can be expressed as \cite{9,17} \begin{equation} E(\emph{a})=\frac{\hbar}{8\pi^{2}}\int k{\rm d}k \int_{0}^{2\pi}{\rm d}\theta \left(\sum\limits_{n}\omega _{n, \perp}+\sum\limits_{n}\omega _{n, \parallel} \right) \label{eqna19} \end{equation} The summations over \emph{n} can be performed with the help of the argument theorem which has been applied in Refs.\cite{4,9,17}. And then we have \begin{equation} E(\emph{a})=\frac{\hbar}{8\pi^{3}}\int k{\rm d}k \int_{0}^{2\pi}{\rm d}\theta \int_{0}^{\infty} (\ln G_1+\ln G_2){\rm d}\xi \label{eqna20} \end{equation} Substituting $\alpha ^2 = \epsilon _{3x}(1-p^2)$ and $k=\frac{\omega}{c}\alpha$ into Eq.(\ref{eqna20}), we can arrive at Eq.(\ref{eqn6}). \section{The approximate calculation of the integration in the Casimir force} In this part we will introduce how we approximately calculate the integration over $\xi$ in the Casimir force in the limiting case. The Casimir force in Eq.(\ref{eqn9}) can be rewritten as following \begin{equation} F=F_1+F_2 \label{eqnb1} \end{equation} with \begin{eqnarray} F_{1}=\frac{\hbar}{2\pi^{2}c^{3}}\int_{1}^{\infty}p^2{\rm d}p\int_{0}^{\infty}\epsilon_{3x}^{3/2} \xi^{3}{\rm d}\xi\left[\frac{1-G_{1}}{G_{1}}\right] \label{eqnb2}\\ F_{2}=\frac{\hbar}{2\pi^{2}c^{3}}\int_{1}^{\infty}pP{\rm d}p\int_{0}^{\infty}\epsilon_{3x}^{3/2} \xi^{3}{\rm d}\xi\left[\frac{1-G_{2}}{G_{2}}\right] \label{eqnb3} \end{eqnarray} The detailed expressions of $G_1$ and $G_2$ can be found in Eq. (\ref{eqn7}) and Eq. (\ref{eqn8}). For the limiting case that the separation $a$ is larger than the characteristic absorption wavelength of the material, we can replace $\epsilon_{1}$, $\epsilon_{2}$, $\epsilon_{3x}$ and $\epsilon_{3z}$ by their values at $\xi=0$, i.e. the statistic dielectric constants \cite{3}. We introduce Eq.(\ref{eqnb2}) a new variable of integration $X=2pa\xi\sqrt{\epsilon_{3x}}/c$, and the integration should be taken over $X$ and $p$. Eq.(\ref{eqnb2}) becomes \begin{align} F_{1}=&\frac{\hbar c}{32\pi^2a^4\sqrt{\epsilon_{3x}}}\int_{1}^{\infty}\frac{{\rm d}p}{p^2}\int_{0}^{\infty}X^{3}{\rm d}X\nonumber\\ &\left[\frac{s_{10}+p}{s_{10}-p}\frac{s_{20}+p}{s_{20}-p}{\rm exp}(X)-1\right]^{-1} \label{eqnb4} \end{align} The integration over $X$ can be taken by using approximate equation $\frac{m}{n!}\int_{0}^{\infty}\frac{x^{n}{\rm d}x}{m{\rm exp}(x)-1}\approx 1$\cite{3}. \begin{equation} F_{1}\approx\frac{3\hbar c}{16\pi^2a^4\sqrt{\epsilon_{3x}}}\int_{1}^{\infty}\frac{{\rm d}p}{p^2}\left[\frac{s_{10}-p}{s_{10}+p}\frac{s_{20}-p}{s_{20}+p}\right] \label{eqnb5} \end{equation} similarly, we can have the approximate result of $F_{2}$ \begin{equation} F_{2}\approx\frac{3\hbar c}{16\pi^2a^4\sqrt{\epsilon_{3x}}}\int_{1}^{\infty}\frac{p{\rm d}p}{P^3}\left[\frac{s_{10}-M_{1}P}{s_{10}+M_{1}P}\frac{s_{20}-M_{2}P}{s_{20}+M_{2}P}\right] \label{eqnb6} \end{equation} Then we can have the result in Eq. (\ref{eqn11}). And from Eqs. (\ref{eqn14}) and (\ref{eqn15}) we can find that \begin{eqnarray} F_{1}=\frac{3\hbar c}{16\pi^2a^4\sqrt{\epsilon_{3x}}}\Psi_1 \label{eqnb7}\\ F_{2}=\frac{3\hbar c}{16\pi^2a^4\sqrt{\epsilon_{3x}}}\Psi_2 \label{eqnb8} \end{eqnarray} \end{document}	arXiv
\begin{document} \title{Tamagawa Numbers of Elliptic Curves with Prescribed Torsion Subgroup or Isogeny} \begin{abstract} We study Tamagawa numbers of elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over cubic fields and of elliptic curves with an $n-$isogeny over $\mathbb{Q}$, for $n\in\{6,8,10,12,14,16,17,18,19,37,43,67,163\}$. Bruin and Najman \cite{BN} proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}.$ We find that Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field are always divisible by $14^2$, with each factor $14$ coming from a rational prime with split multiplicative reduction of type $I_{14k},$ one of which is always $p=2.$ The only exception is the curve \lmfdbec{1922c1}{1922.e2}, with $c_E=c_2=14.$ The same curves defined over cubic fields over which they have torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ turn out to have the Tamagawa number divisible by $14^3$. As for $n-$isogenies, Tamagawa numbers of elliptic curves with an $18-$isogeny must be divisible by 4, while elliptic curves with an $n-$isogeny for the remaining $n$ from the mentioned set must have Tamagawa numbers divisible by 2, except for finite sets of specified curves. \end{abstract} \section{Introduction} Let $E$ be an elliptic curve over a number field $K$ and denote by $\Sigma$ the set of all finite primes of $K$. For each $v\in\Sigma,$ the completion of $K$ at $v$ will be denoted by $K_v$ and the residue field of $v$ by $k_v=\mathcal{O}_{K_v}/(\pi)$, where $\mathcal{O}_{K_v}$ is the ring of integers of $K_v$ and $\pi$ is a uniformizer of $\mathcal{O}_{K_v}.$ The subgroup $E_0(K_v)$ of $E(K_v)$ consists of all the points that reduce modulo $\pi$ to a non-singular point of $E(k_v)$. It is known that this group has finite index in $E(K_v)$ so we can define the Tamagawa number $c_v$ of $E$ at $v$ to be that index, i.e., $$c_v:=\left[E(K_v) : E_0(K_v)\right].$$ In light of this, we define the Tamagawa number of $E$ over $K$ to be the product $c_{E/K}:=\prod_{v\in\Sigma}c_v.$ We will write $c_E$ instead of $c_{E/K}$ wherever it does not cause confusion. It makes sense to study how the value $c_E$ depends on $E(K)_{tors},$ since $c_E/\#E(K)_{tors}$ appears as a factor in the leading term of the $L-$function of $E/K$ in the conjecture of Birch and Swinnerton-Dyer (see, for example, \cite[Conj. F.4.1.6]{HS}). We start with some known results about Tamagawa numbers, first of which is given by Lorenzini in \cite{DL} on Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with a specific torsion subgroup. \begin{proposition}{\cite[Proposition 1.1]{DL}}\label{lor} Let $E/\mathbb{Q}$ be an elliptic curve with a $\mathbb{Q}-$rational point of order $N$. The following statements hold with at most five explicit exceptions for a given $N$. \begin{itemize} \item[(a)] If $N=4$, then $(N/2)\mid c_E$, except for $E=\lmfdbec{15a8}{X_1(15)}, \lmfdbec{15a7}{15.a4},$ and $ \lmfdbec{17a4}{17.a4}.$ \item[(b)] If $N=5,6$ or $12$, then $N\mid c_E$, except for $E=\lmfdbec{11a3}{X_1(11)}, \lmfdbec{14a4}{X_1(14)}, \lmfdbec{14a6}{14.a4},$ and $ \lmfdbec{20a2}{20.a3}.$ \item[(c)] If $N=10$, then $(N^2/2)\mid c_E$. \item[(d)] If $N=7,8$ or $9$, then $N^2\mid c_E$, except for $E=\lmfdbec{15a4}{15.a8}, \lmfdbec{21a3}{21.a3}, \lmfdbec{26b1}{26.b2}, \lmfdbec{42a1}{42.a5}, \lmfdbec{48a6}{48.a6}, \lmfdbec{54b3}{54.b2}$ and $ \lmfdbec{102b1}{102.c5}.$ \end{itemize} Without exception, $N\mid c$ if $N=7,8,9,10$ or $12$. \end{proposition} He also proved that the smallest possible ratio $c_E/\#E(K)_{tors}$ for elliptic curves over $\mathbb{Q}$ is $1/5$, achieved only by the modular curve \lmfdbec{11a3}{$X_1(11)$}. He gave as well some results about Tamagawa numbers of elliptic curves over quadratic extentions. Some of his results mentioned in Proposition \ref{lor} were later expanded upon by Krumm in \cite{K} which are presented with the following result. \begin{proposition}{\cite[Propositions 5.2.2, 5.2.3]{K}}\label{kru} Let $E/\mathbb{Q}$ be an elliptic curve with a $\mathbb{Q}-$rational point of order $N$. \begin{itemize} \item[(a)] If $N=7,$ then $7\mid c_2.$ \item[(b)] If $N=9,$ then $9\mid c_2$ and $3\mid c_3.$ \end{itemize} \end{proposition} Furthermore, Krumm proved some results on Tamagawa numbers of elliptic curves with prescribed torsion over number fields of degree up to 5. He also conjectured that $\text{ord}_{13}(c_E)$ is even for all elliptic curves defined over quadratic fields with a point of order 13 and the same conjecture was later proved by Najman in \cite{N}. In their recent work \cite{BR}, Barrios and Roy explicitly classified Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with non-trivial torsion subgroups at primes of additive reduction. The results mentioned in Prosopitions \ref{lor} and \ref{kru} were the motivation to explore further the problem of finding Tamagawa numbers of elliptic curves with a certain torsion subgroup and at a certain prime. Bruin and Najman \cite{BN} proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}.$ Using that fact, we prove in Section \ref{tors} that the Tamagawa numbers of all elliptic curves defined over $\mathbb{Q}$ that have torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field are always divisible by $14^2$, except in the case of the curve \lmfdbec{1922c1}{1922.e2}, where $c_E=c_2=14.$ For each such curve we prove that at $p=2$ the reduction is split multiplicative, so $c_2=14k,$ and there always exists one more prime, distinct from 2, at which the reduction is also split multiplicative of type $I_{14t}.$ As a consequence of this result we get that elliptic curves defined over a cubic field with torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ have Tamagawa numbers divisible by $14^3.$ We mention the explicit results in the following theorem. \begin{theorem} \begin{itemize} \item Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field. \begin{itemize} \item[(a)] The reduction at $2$ is split multiplicative of type $I_{14k}$ and $c_2=14k,$ where $k\in\mathbb{Z}, \: k\geq 1.$ \item[(b)] There exist at least 2 rational primes with split multiplicative reduction of type $I_{14k},$ where $k\in\mathbb{Z}, \: k\geq 1,$ one of which is always the prime $2$, so $14^2 \mid c_E,$ except for the curve $\lmfdbec{1922c1}{1922.e2}$, where $c_E=c_2=14.$ \end{itemize} \item Let $E$ be an elliptic curve defined over a cubic field $K$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$. Let $\mathfrak{P}$ be a prime of $K$ over $2$. Then the reduction at $\mathfrak{P}$ is split multiplicative of type $I_{14k}$ and $c_\mathfrak{P}=14k,$ where $k\in\mathbb{Z}, \: k\geq 1.$ Furthermore, $14^3 \mid c_E.$ \end{itemize} \end{theorem} The proof of this theorem is given by the proofs of Propositions \ref{prop:p=2}, \ref{main} and Corollary \ref{kor} in Section \ref{tors}. The question which naturally appears next is how does the Tamagawa number of an elliptic curve depend on the isogenies of that elliptic curve. In Section \ref{iso} we give a series of propositions which gives us first results about Tamagawa numbers of elliptic curves with prescribed isogeny. For elliptic curves defined over $\mathbb{Q}$, we were able to prove that if an elliptic curve has an $18-$isogeny, then its Tamagawa number is always divisible by $4$, and if it has an $n-$isogeny, for $n\in \{6,8,10,12,14,16,17,18,19,37,43,67,163\}$, then it has to be divisible by 2. There are finitely many exceptions for some of these results, all of which we list and give their Tamagawa numbers. \begin{theorem} Let $E$ be an elliptic curve over $\mathbb{Q}$ with an $N-$isogeny. \begin{itemize} \item If $N=18,$ then $4\|c_E,$ except for the curves $\lmfdbec{14a3}{14.a2}, \lmfdbec{14a4}{14.a5}, \lmfdbec{14a5}{14.a1}, \lmfdbec{14a6}{14.a4}$, where $ c_E=2$. \item If $N=10,$ then $2\|c_E.$ \item If $N=8,$ then $2\|c_E$, except for the curves $\lmfdbec{15a7}{15.a4}$, $\lmfdbec{15a8}{15.a7}$, $\lmfdbec{48a4}{48.a5}$, where $c_E=1.$ \item If $N=6,$ then $2\|c_E,$ except for the curve $\lmfdbec{20a2}{20.a3}$, where $c_E=3,$ and also the curves $\lmfdbec{80b2}{80.b3}$, $ \lmfdbec{80b4}{80.b1},$ $\lmfdbec{20a4}{20.a1}$, $\lmfdbec{27a3}{27.a4}$ and infinitely many twists of $\lmfdbec{27a3}{27.a4}$, for which $c_E=1$. \item If $n\in\{14,17,19,37,43,67,163\}$, then $2\|c_E.$ \end{itemize} \end{theorem} The proof of this theorem is given by the proofs of Propositions \ref{prop1}, \ref{10izo}, \ref{8izo}, \ref{6izo} and \ref{ostalo} in Section \ref{iso}. Now let $E$ be an elliptic curve defined over $K_v$, given by a Weierstrass equation $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ with discriminant $\Delta$, invariants $c_4$ and $c_6$, and $j-$invariant $j_E=\frac{c_4^3}{\Delta}.$ It will be important for us to distinguish between different types of reductions at finite primes, especially to know when the reduction is multiplicative. For that, we will often use the following well known result. \begin{proposition}\label{prop:mult} \textup{(see \cite[Proposition VII.5.1.b]{S})} With the above notation, the curve $E$ in its minimal model has multiplicative reduction at $v$ of type $I_k$ if and only if $k:=\text{ord}_{v}(\Delta)>0$ and $\text{ord}_{v}(c_4)=0$. \end{proposition} As most Tamagawa numbers that we will consider in this paper are coming from primes of multiplicative reduction, it will be important to also distinguish between split and non-split multiplicative reductions and their Tamagawa numbers. One way to do that is by using the algorithm of Tate \cite[Sections 7,8]{T} which works in any characteristic of $k_v.$ Going through the algorithm with a specific elliptic curve and a prime $p$, we get the reduction type at $p$, its Kodaira symbol and the Tamagawa number $c_p.$ It turns out that in the case of split multiplicative reduction $I_k$ we have $c_v=k$ and in the case of non-split multiplicative reduction $I_k$ we have $c_v=1$ or $c_v=2$, depending on the parity of $k$, as indicated in Table \ref{table1}, where we can find all the Tamagawa numbers associated to different reduction types. For distinguishing reduction types in $char(k_v)\neq 2,3$ one can also use the tables in \cite[Table 15.1]{S} or \cite[Section 6]{T}. \renewcommand{1.2}{1.2} \begin{table}[h] \begin{tabular}{\|c\|c\|c\|} \hline reduction type at $v$ & Kodaira symbol, $k\geq 1$ & Tamagawa number at $v$ \\ \hline good & $I_0$ & 1 \\ split multiplicative & $I_k$ & $k$ \\ non-split multiplicative& $I_{2k}$ & $2$ \\ non-split multiplicative & $I_{2k-1}$ & $1$ \\ additive& $II,II^$ & 1 \\ additive& $III,III^$ & 2 \\ additive& $IV,IV^$ & 1,3 \\ additive & $I_0^$ & $1,2,4$ \\ potentially multiplicative & $I_{2k}^$ & $2,4$ \\ potentially multiplicative & $I_{2k-1}^$ & $2,4$ \\ \hline \end{tabular} \caption{types of reduction and their Tamagawa numbers}\label{table1} \end{table} The computations in this paper were executed in the computer algebra system Magma \cite{MAGMA}. The code used in this paper can be found at \url{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3.htm}. Many of the proofs in this paper omit the information used in them, such as polynomials of very high degree or with large coefficients, but those can be computed with the given code. For the reader who wants to verify the calculations, we recommend that they go through the proofs and the code simultaneously. \noindent All of the specific curves will be mentioned using their LMFDB labels, with a clickable link to the corresponding webpage in \cite{LMFDB}. \section{Tamagawa numbers of elliptic curves with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$}\label{tors} As already mentioned, Bruin and Najman \cite{BN} proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}.$ Filip Najman and the author have examined the reduction types at primes with multiplicative reduction of such elliptic curves defined over $\mathbb{Q}$ in \cite[Prop. 3.1]{NT}. We will examine those primes further, as we want to be able to say more about their Tamagawa numbers. It was proved in \cite[Prop. 3.2]{NT} that those elliptic curves always have multiplicative reduction of type $I_{14k}$ at the rational prime 2. In this section we are going to prove that the mentioned multiplicative reduction at 2 always has to be split multiplicative, giving the Tamagawa number $c_2=14k$, as shown in Table \ref{table1}. We are also going to prove that there always exists one more prime $p$, with the exception of the curve \lmfdbec{1922c1}{1922.e2}, at which we have split multiplicative reduction of type $I_{14t}$ and $c_p=14t$, which means that the Tamagawa number of the elliptic curve contains the factor $14^2.$ For the base change of every elliptic curve to a field $K$ over which they have torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ it turns out that their Tamagawa number is always divisible by $14^3.$ Bruin and Najman in \cite{BN} also showed that elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ are parameterized with $\mathbb{P}^1(\mathbb{Q})$, so we can write each such curve as $E_u$, for some $u\in\mathbb{Q}.$ They also provided a model, which was used for obtaining the results of \cite[\S 3]{NT}. We used a different model here, specifically, the one given by Jeon and Schweizer in \cite[\S 2.4]{JS}, since the one in \cite{NT} was dependant on 2 parameters. It did not impose a problem there, since we did not have the need to work with the coefficients of the curve. Even though Jeon and Schweizer do not state that their family consists of all elliptic curves over cubic fields with torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$, it turns out that it is the case and the reasoning behind it can be found in the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/2families.txt}{Magma code}. Briefly, we compute the isomorphism between different fields of definition of elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$, those are $F$ and $L$ given in \cite{BN} and \cite[\S 2.4]{JS}, respectively. With that isomorphism we map every curve from the family in \cite{BN} and we see that it is isomorphic to one of the curves from the family in \cite[\S 2.4]{JS}. Since \cite{BN} gives us all of the elliptic curves with needed properties, we see that it suffices to only look at the family from \cite[\S 2.4]{JS}. Jeon and Schweizer provided two models for $E_u$, one of which is $$y^2+xy=x^3+A_2(u)x^2+A_4(u)x+A_6(u),$$ and its short Weierstrass model $$y^2=x^3+A(u)x+B(u),$$ where we omit $A_2(u),A_4(u), A_6(u), A(u), B(u)$, since they are very large, but thay can be found in the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3.htm}{Magma code} or in \cite[\S 2.4]{JS}. We will be working with the long Weierstrass model when considering the reduction at the prime 2, but generally we will be using the short Weierstrass model, since it is easier to work with. In \Cref{prop:mult} we mentioned a way of confirming whether the curve has multiplicative reduction at a finite prime. As already stated, it will be very important to distinguish between split and non-split multiplicative reduction, since the associated Tamagawa numbers are different (see \Cref{table1}). The following lemma will be useful in differentiating between those, and it is taken directly from a step in Tate's algorithm. \begin{lemma}\label{lema1} \textup{(\cite[\S 7. Case 2)]{T})} Let $E$ be an elliptic curve and let $p$ be a prime of multiplicative reduction of type $I_{t}$ for $E$. Let $\text{ord}_p(a_i)>0,$ for $i=3,4,6$, and $\text{ord}_p(b_2)=0.$ If $T^2+a_1T-a_2$ splits over $k_p$, then $E$ has split multiplicative reduction at $p$ and $c_p=t.$ \end{lemma} As a part of the proof of the following proposition we will show that the reduction at the prime 2 is multiplicative of type $I_{14k}$, which is already proved in \cite[Proposition 3.2]{NT}. We had to include it here again and could not continue from there because of the already mentioned differences in the models we used. \begin{proposition}\label{prop:p=2} Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field. Then the reduction at $2$ is split multiplicative of type $I_{14k}$ and $c_2=14k,$ where $k\in\mathbb{Z}, \: k\geq 1.$ \end{proposition} \begin{proof} From the long Weierstrass model of $E_u$ from \cite[\S 2.4]{JS} we get the associated discriminant $$\Delta(u)=\dfrac{2^{14}(u-1)^{14}(u+1)^{14}f_1(u)}{f_2(u)}$$ and the $c_4-$invariant $c_4(u)$, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop2.2.txt}{Magma code}. The polynomials $f_{i}(u), \: i=1,2,$ are monic polynomials in $\mathbb{Z}[u]$. We will go through all of the possibilities of the prime 2 dividing $u$ and see that the reduction at 2 in all of those cases is split multiplicative and $14\mid c_2.$ \begin{itemize} \item[(1)] If $\text{ord}_2(u)>0,$ we compute $\text{ord}_2\left(\frac{\Delta(u)}{2^{14}} \right)=\text{ord}_2(c_4(u))=0$ and from \Cref{prop:mult} we conclude that the reduction at 2 is multiplicative of type $I_{14}.$ We compute $a_1=1$ and $\text{ord}_2(a_2)>0$ and since our model satisfies the conditions of Lemma \ref{lema1}, we get that $c_2=14.$ \item[(2)] If $\ell:=\text{ord}_2(u)<0,$ then we make the substitution $u\mapsto \frac{1}{m}$ so $\text{ord}_2(m)>0,$ and in the new model we get the discriminant $$\Delta(m)=\dfrac{2^{14}(m-1)^{14}m^{14}(m+1)^{14}g_1(m)}{g_2(m)}$$ and the $c_4-$invariant $c_4(m)$, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop2.2.txt}{Magma code}. The polynomials $g_{i}(m), \: i=1,2,$ are monic polynomials in $\mathbb{Z}[m]$. Using the fact that $\text{ord}_2(m)>0,$ we compute $\text{ord}_2\left(\frac{\Delta(m)}{2^{14}m^{14}} \right)=\text{ord}_2(c_4(m))=0$ and as in the previous case, using \Cref{prop:mult} and \Cref{lema1} we get that the reduction at 2 is split multiplicative of type $I_{14(\ell+1)}$ and $c_2=14(\ell+1).$ \item[(3)] If $\text{ord}_2(u)=0,$ then $\ell:=\text{ord}_2(u-1)>0.$ After the substitution $u-1\mapsto m$ we have $\ell=\text{ord}_2(m)>0$, the discriminant $$\Delta(m)=\dfrac{2^{14}m^{14}(m+2)^{14}h_1(m)}{h_2(m)}$$ and the $c_4-$invariant $c_4(m)$, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop2.2.txt}{Magma code}. The polynomials $h_{i}(m), \: i=1,2,$ are monic polynomials in $\mathbb{Z}[m]$. \noindent We can divide both the numerator and the denominator of $\Delta(m)$ by $2^{48}$ and we get that $\text{ord}_2(\Delta (m))=14(\ell-1)$ and if we divide the numerator and the denominator of $c_4(m)$ by $2^{24}$ we get that $\text{ord}_2(c_4(m))=0.$ So if $\ell>1,$ by \Cref{prop:mult} we have that the reduction at 2 is multiplicative of type $I_{14(\ell-1)}$. We compute $a_1=1$ and $\text{ord}_2(a_2)>0$ (after dividing both numerator and the denominator by $2^{12}$) and since our model satisfies the conditions of Lemma \ref{lema1}, we get that $c_2=14(\ell-1).$ \noindent Obviously we have to look at the case $\ell=1$ separately. This means that $u=2n+1,$ where $\text{ord}_2(n)=0.$ After the substitution $u\mapsto 2n+1$ we have the discriminant $$\Delta(n)=\dfrac{n^{14}(n+1)^{14}p_1(n)}{p_2(n)}$$ and the $c_4-$invariant $c_4(n)$, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop2.2.txt}{Magma code}. The polynomials $p_{i}(n), \: i=1,2,$ are monic polynomials in $\mathbb{Z}[n]$. Since $\text{ord}_2(n)=0,$ we have $t:=\text{ord}_2(n+1)>0$ and $\text{ord}_2(p_i(n))=0,$ for each $i$, so $\text{ord}_2(\Delta (n))=14t$ and $\text{ord}_2(c_4(n))=0$. By \Cref{prop:mult} we see that the reduction at 2 is multiplicative of type $I_{14t}$ and similarly as in previous cases, Lemma \ref{lema1} gives that the reduction is split multiplicative with $c_2=14t.$ \end{itemize} \end{proof} \begin{example} As it was verified in part (1) of the proof of Proposition \ref{prop:p=2}, if $\text{ord}_2(u)>0,$ then the reduction at 2 is multiplicative of type $I_{14}$ with $c_2=14.$ This allows us to generate an infinite family of elliptic curves that have torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field and Tamagawa number exactly $14$ at the prime $2$, i.e., $c_2=14.$ Namely, if we put $u=2^k,$ for any $k\in\mathbb{Z}, \: k\geq 1$, in the long Weierstrass model of $E_u$ from \cite[\S 2.4]{JS} we will get an elliptic curve with $c_2=14.$ For example, with $k=1$ (which gives $u=2$) we get a curve whose minimal model is defined by $$y^2 + xy = x^3 - 31714388875x + 2132064170125553,$$ with $c_2=14$ and torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over the field $\mathbb{Q}(\alpha),$ where $\alpha$ is a root of the polynomial $3x^3-4x^2-27x+4.$ In a similar manner, part (2) of the proof of Proposition \ref{prop:p=2} allows us to generate an infinite family of examples of elliptic curves that have torsion $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field and Tamagawa number $c_2=14t,$ where $t>1.$ When $k:=\text{ord}_2(u)<0,$ the long Weierstrass model of $E_u$ from \cite[\S 2.4]{JS} gives us an elliptic curve with $c_2=14(k+1)$ and torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field. Namely, if we specify $u=\frac{1}{2^k}, \: k\geq 1,$ we get a family of elliptic curves with Tamagawa number $c_2=14(k+1)$. For example, with $k=1$ (which gives $u=\frac{1}{2}$) we get a curve whose minimal model is defined by $$y^2 + xy = x^3 - 35365397163613670x + 2559848051274532647229668,$$ with $c_2=28$ and torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over the field $\mathbb{Q}(\alpha),$ where $\alpha$ is a root of the polynomial $-6x^3-47x^2+54x+47.$ All of the statements regarding specific elliptic curves in this example can be verified using the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/ex2.3.txt}{Magma code}. \end{example} \begin{corollary}\label{kor} Let $E$ be an elliptic curve defined over a cubic field $K$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$. Let $\mathfrak{P}$ be a prime of $K$ over $2$. Then the reduction at $\mathfrak{P}$ is split multiplicative of type $I_{14k}$ and $c_\mathfrak{P}=14k,$ where $k\in\mathbb{Z}, \: k\geq 1.$ Furthermore, $14^3 \mid c_E.$ \end{corollary} \begin{proof} Recall that $E$ is an elliptic curve defined over $\mathbb{Q}$. We will denote by $E_K$ the base change of $E$ to $K$. From \cite[Proposition 3.6]{NT} we know that $2$ splits completely in $K$, i.e. $2\mathcal{O}_K=\mathfrak{P}_1\cdot \mathfrak{P}_2\cdot \mathfrak{P}_3.$ This means that the residue field $k_{\mathfrak{P}_i}=\mathcal{O}_K/\mathfrak{P}_i,$ where $\mathfrak{P}_i$ is a prime lying over $2$, $i=1,2,3$, is isomorphic to $k_p$. For each $\mathfrak{P}_i$ we have that $E_K \text{ mod } \mathfrak{P}_i=E \text{ mod } 2$ and hence $c_{\mathfrak{P}_i}=c_2=14k,$ for $i=1,2,3,$ where $k\in\mathbb{Z}, \: k\geq 1.$ \end{proof} In the following proposition we will deal with primes distinct from 2, for which we have a simpler way of determining split multiplicative reduction than going through Tate's algorithm as we did in Proposition \ref{prop:p=2}. \begin{lemma}\label{lema2} \textup{(\cite[Lemma 2.2]{CCH})} Let $p\neq 2$ be a prime and let $E$ be an elliptic curve defined over $\mathbb{Q}_p$ with multiplicative reduction at $p$. The reduction is split multiplicative if and only if $-c_6$ is a square in $\mathbb{F}_p^{\times}.$ \end{lemma} \begin{proposition}\label{main} Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field. Then there exist at least 2 rational primes with split multiplicative reduction of type $I_{14k},$ where $k\in\mathbb{Z}, \: k\geq 1,$ one of which is always the prime $2$, so $14^2 \mid c_E,$ except for the curve $\lmfdbec{1922c1}{1922.e2}$, where $c_E=c_2=14.$ \end{proposition} \begin{proof} In \Cref{prop:p=2} we have already seen that the reduction at 2 is split multiplicative of type $I_{14k}$ and therefore $c_2=14k.$ It remains to prove that there exists one more prime with the same property for each of those curves. From the short Weierstrass model of $E_u$ from \cite[\S 2.4]{JS} we get the associated discriminant $$\Delta(u)=2^{14}(u - 1)^{14}(u + 1)^{14}f(u)$$ and the $c_4-$invariant $c_4(u)$, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop2.6.txt}{Magma code}. The polynomial $f(u)$ is a monic polynomial in $\mathbb{Z}[u]$. \begin{itemize} \item Assume that there exists a prime $p$ such that $k:=\text{ord}_p(u-1)>0.$ Let $\res(q, r)$ denote the resultant of two arbitrary polynomials $q$ and $r$. We compute $$\res\left(u-1, \frac{\Delta(u)}{(u-1)^{14}}\right)=2^{82}, $$ $$\res\left(u-1, c_4(u)\right)=2^{32}. $$ For $p\neq 2$ this means that $p^{14k}\mid\Delta(u)$ and $p\nmid c_4(u)$ and from \Cref{prop:mult} we find that the reduction of $E$ at $p$ is multiplicative of type $I_{14k}.$ We want to see that $c_p=14k,$ i.e., that the reduction at $p$ is split multiplicative. According to \Cref{lema2}, it will suffice to check the value of $-c_6$ modulo $p$. Having in mind that $u\equiv 1 \pmod p$, we compute that $-c_6\equiv 2^{48} \pmod p,$ which is a square mod $p$. \item Assume now that there exists a prime $p$ such that $k:=\text{ord} _p(u-1)<0.$ We put $m:=\frac{1}{u-1}$ so $\text{ord}_p(m)=k>0$ and we get an elliptic curve with the discriminant $$\Delta(m)=\frac{1}{2^{14}}m^{14}(m + 1/2)^{14}g(m)$$ and the $c_4-$invariant $c_4(m)$, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop2.6.txt}{Magma code}. The polynomial $g(m), \: i=1,2,$ is a monic polynomial in $\mathbb{Z}[u]$. \noindent We compute $$\res\left(m, \frac{\Delta(m)}{m^{14}}\right)=2^{-82}, $$ $$\res\left(m, c_4(m)\right)=2^{-32}. $$ For $p\neq 2$ this means that $p^{14k}\mid\Delta(m)$ and $p\nmid c_4(m)$ and from \Cref{prop:mult} we find that the reduction of $E$ at $p$ is multiplicative of type $I_{14k}.$ Having in mind that $m\equiv 0 \pmod p$, we get that $-c_6\equiv 2^{-48} \pmod p,$ which is a square mod $p$, so by \Cref{lema2} we have $c_p=14k.$ \end{itemize} So far we have proved that if we have a prime $p\neq 2$ and $k:=\text{ord}_p(u-1)\neq 0$, then we have split multiplicative reduction at $p$ with $c_p=14\|k\|.$ We have several possibilities when $\text{ord}_p(u-1)= 0$ and those are $u-1=0$ or $u-1=\pm 2^k,\: k\in\mathbb{Z}.$ \noindent When $u-1=\pm 2^k,\: k\neq 0,1,$ then $\text{ord}_p(u+1)>0,$ for some prime $p\neq 2.$ In the cases $u-1=\pm 2^k,\: k=0,1$, or $u-1=0$ we get that $u\in \{0,\pm 1, 3\}.$ For $u=\pm 1$ we get a singular curve and for $u\in \{0, 3\}$ we get the same curve, $\lmfdbec{1922c1}{1922.e2}$, with $c_E=c_2=14.$ \noindent Therefore, if we have a curve distinct from $\lmfdbec{1922c1}{1922.e2}$, it certainly has a prime $p$ such that $\text{ord}_p(u-1)\neq 0$ or $\text{ord}_p(u+1)> 0.$ It remains to see what happens in the case $\text{ord}_p(u+1)> 0.$ \begin{itemize} \item Assume that there exists a prime $p$ such that $k:=\text{ord}_p(u+1)>0.$ We compute $$\res\left(u+1, \frac{\Delta(u)}{(u+1)^{14}}\right)=2^{82}, $$ $$\res\left(u+1, c_4(u)\right)=2^{32}. $$ For $p\neq 2$ this means that $p^{14k}\mid\Delta(u)$ and $p\nmid c_4(u)$ and from \Cref{prop:mult} we find that the reduction of $E$ at $p$ is multiplicative of type $I_{14k}.$ Having in mind that $u\equiv -1 \pmod p$, we get that $-c_6\equiv 2^{48} \pmod p,$ which is a square mod $p$, so by \Cref{lema2} we have $c_p=14k.$ \end{itemize} \end{proof} \begin{remark} In Proposition \ref{main} we proved that $c_E=14$ is the least possible value of the Tamagawa number of an elliptic curve defined over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over some cubic field. This is true only for elliptic curve $E=\lmfdbec{1922c1}{1922.e2}$. Consequently, for the same curve we get the least possible value amongst those curves of the ratio $c_E/\#E(\mathbb{Q})_{tors}$ which appears as a factor in the leading term of the $L-$function of $E/\mathbb{Q}$ in the conjecture of Birch and Swinnerton-Dyer, in this case it is $c_E/\#E(\mathbb{Q})_{tors}=\frac{14}{1}=14,$ since the curves defined over $\mathbb{Q}$ with torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over some cubic field have trivial torsion over $\mathbb{Q}.$ Similarly, using Corollary \ref{kor} we see that the value of the Tamagawa number of an elliptic curve defined over a cubic field with torsion subgroup $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ is always divisible by $14^3.$ The value $c_E=14^3$ is actually a possible value, and it is achieved for the curve $E=\lmfdbec{1922c1}{1922.e2}$, which has the mentioned torsion over the cubic field $\mathbb{Q}(\alpha),$ where $\alpha$ is a root of the polynomial $x^3 + 2x^2 - 9x - 2$. This gives the smallest possibe ratio of $c_E/\#E(K)_{tors}=98$ for all such curves. \end{remark} \section{Tamagawa numbers of elliptic curves with prescribed isogeny}\label{iso} In \cite[Table 3]{ALR} we can find the $j-$invariants of elliptic curves parameterized by points on modular curves $X_0(n)$ defined over $\mathbb{Q},$ for $X_0(n)$ of genus $0$, and in \cite[Table 4]{ALR} there are $j-$invariants of elliptic curves parameterized by points on modular curves $X_0(n)$ defined over $\mathbb{Q},$ with genus of $X_0(n)$ larger than $0$. In this section we will examine the properties of Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with an $n-$isogeny, i.e., the properties of Tamagawa numbers of elliptic curves obtained from the mentioned $j-$invariants. In Section \ref{tors} we worked with a specific model for the curve $X_1(2,14)$. However, the points on $X_0(n)$ give us $j-$invariants of curves with an $n-$isogeny, which give us elliptic curves up to a twist, so now, as opposed to the situation in Section \ref{tors}, we also have to take into consideration the twists of the curves we get from those $j-$invariants. Therefore, we will be interested in how the reduction types at primes $p\in\mathbb{Q}$ change under the twisting of the curve. Let $E$ be an elliptic curve, which will always be defined over $\mathbb{Q}$ in this section. Denote by $E^d$ its quadratic twist by $d$, where $d$ is a squarefree integer. When $p\neq 2$, the reduction type change is quite straightforward, and is presented in \Cref{tab2}. In essence, if $p\nmid d,$ the reduction type does not change, and when $p\mid d,$ reduction types change as indicated in the third column. \begin{table}[h] \begin{tabular}{\|c\|c\|c\|} \hline \begin{tabular}[c]{@{}c@{}}reduction type \\ of $E$ at $p$\end{tabular} & \begin{tabular}[c]{@{}c@{}}reduction type\\ of $E^d$ at $p\nmid d$\end{tabular} & \begin{tabular}[c]{@{}c@{}}reduction type\\ of $E^d$ at $p\mid d$\end{tabular}\\ \hline $I_0$ & $I_0$ & $\:\:\:I_0^\:$ \\ $\:I_m$ & $\:I_m$ & $\:\:\:\:I_m^\:$ \\ $II$ & $II$ & $\:\:\:\:\:IV^$ \\ $\:\:III$ & $\:\:III$ & $\:\:\:\:\:\:III^$ \\ $\:IV$ & $\:IV$ & $\:\:\:II^$ \\ $I_0^\:$ & $I_0^\:$ & $I_0$ \\ $\:I_m^\:$ & $\:I_m^\:$ & $\:I_m$ \\ $\:\:IV^$ & $\:\:IV^$ & $II$ \\ $\:\:III^$ & $\:\:III^$ & $\:\:III$ \\ $\:\:II^\:\:$ & $\:\:II^\:\:$ & $\:IV$ \\ \hline \end{tabular} \caption{change of reduction types at $p\neq 2$ under twisting \cite[Prop.1]{SC}}\label{tab2} \end{table} When $p=2,$ the situation gets more complicated. As most of the relevant Tamagawa numbers we will have in the following proofs come from primes of multiplicative reduction, we give a lemma that will be especially useful for dealing with quadratic twists of a large family of elliptic curves with multiplicative reduction at $p=2.$ \begin{lemma}\label{lemma1} \textup{(\cite[Thm.A.5]{D2}, \cite[Thm.2.8]{DL2})} Let $E$ be an elliptic curve with multiplicative reduction of type $I_n$ at $p=2.$ Denote by $E^d$ the twist of $E$ by $d,$ where $d$ is a squarefree integer. \begin{itemize} \item[(a)] If $d\equiv 2,3\pmod{4},$ then the reduction of $E^d$ at $p$ is of type $I_n^.$ \item[(b)] If $d\equiv 1\pmod{4},$ then the reduction of $E^d$ at $p$ is of type $I_n.$ \end{itemize} \end{lemma} For other types of reduction, some results can also be found in \cite[Section 2]{SC}. Since we will deal here with only finitely many explicitly known elliptic curves with non-multiplicative reduction at $p=2,$ for those curves we can simply check all of the possibilities for reduction type at $p=2$ of quadratic twists, since $\mathbb{Q}_2^\times/\left( \mathbb{Q}_2^\times\right)^2=\langle -1,2,5 \rangle$. \begin{proposition}\label{prop1} Let $E$ be an elliptic curve over $\mathbb{Q}$ with an $18-$isogeny. Then $4\|c_E,$ except for the curves $\lmfdbec{14a3}{14.a2}, \lmfdbec{14a4}{14.a5}, \lmfdbec{14a5}{14.a1}, \lmfdbec{14a6}{14.a4}$, where $ c_E=2$. \end{proposition} \begin{proof} From \cite[Table 3]{ALR} we take the parameterization of the $j-$invariants of the curves that are non-cuspidal points on $X_0(18),$ $$j(h)=\dfrac{(h^3-2)^3(h^9-6h^6-12h^3-8)^3}{h^9(h^3-8)(h^3+1)^2}, h\in\mathbb{Q}.$$ From it we can acquire the discriminant $$\Delta(h)=(h-2)h^9(h+1)^2(h^2 - h + 1)^2(h^2 + 2h + 4)f(h)$$ and the $c_4-$invariant $c_4(h)$ of the minimal model up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.2.txt}{Magma code}. The polynomial $f(h)$ is a monic polynomial in $\mathbb{Z}[h].$ Assume that there exists a prime $p$ such that $k:=\text{ord} _p(h+1)>0.$ We compute $$\res\left(h+1, \frac{\Delta(h)}{(h+1)^2}\right)=3^{34}, $$ $$\res\left(h+1, c_4(h)\right)=3^{12}. $$ If $p\neq 3,$ this means that $p^{2k}\mid\Delta(h)$ and $p\nmid c_4(h)$, so from \Cref{prop:mult} we find that the reduction of $E$ at $p$ is multiplicative of type $I_{2k},$ and therefore $c_p$ is even (see \Cref{table1}). If there exists a second prime $p'$ distinct from $p$ and 3 with $k':=\text{ord} _{p'}(h+1)>0,$ then we have another prime with multiplicative reduction of type $I_{2k'}$ and therefore with even $c_{p'}.$ Assume now that there exists a prime $p$ such that $k:=\text{ord} _p(h+1)<0.$ We put $m:=\frac{1}{h+1}$ and after the substitution $x\mapsto x\cdot 3^{-6}, y\mapsto y\cdot 3^{-9}$ we get an elliptic curve with the discriminant $$ \Delta(m)=(m - 1)^9(3m-1)m^{18}(3m^2 -3 m + 1)^2(3m^2 + 1)g(m)$$ and the $c_4-$invariant $c_4(h)$ up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.2.txt}{Magma code}. The polynomial $g(m)$ is a monic polynomial in $\mathbb{Z}[m].$ \noindent We compute $$\res\left(m, \frac{\Delta(m)}{m^{18}}\right)=1, $$ $$\res\left(m, c_4(m)\right)=1. $$ We see from \Cref{prop:mult} that the reduction at $p$ is multiplicative of type $I_{18k},$ with even $c_p$ (see \Cref{table1}). If we have another prime $p'\neq p$ with $k':=\text{ord} _{p'}(h+1)<0,$ then the reduction at $p'$ is also multiplicative of type $I_{18k'}$ with even $c_{p'}.$ Assume now that there exists at most one prime $p$ such that $\text{ord}_p(h+1)\neq 0.$ That means that either $h+1=\pm p^k,$ where $k\in \mathbb{Z}, \: k\geq 0,$ or $m=\frac{1}{h+1}=\pm p^k,$ where $k\in \mathbb{Z}, \: k> 0.$ We consider the following cases: \begin{itemize} \item[(1)] If $h+1=\pm p^k, p\neq 3,$ we have $\text{ord} _{p}(h^2-h+1)=0,$ since $\res(h+1, h^2-h+1)=3$, where $h^2-h+1$ is one of the factors in the discriminant. Then there exists $p'\neq p,3$ such that $\text{ord}_{p'}(h^2-h+1)>0$, unless we have $h^2-h+1\in\{\pm 1, \pm 3\},$ i.e., $h\in\{0,\pm 1, 2\}.$ For $h=1$ we get a twist of the curve \lmfdbec{14a4}{14.a5} which has $c_E=2,$ while for $h=0, -1,2$ we do not get an elliptic curve (look at the j-invariant). \item[(2)] Suppose $h+1=\pm 3^k.$ If $k=0$ we have $h+1=\pm 1,$ i.e., $h\in \{0,-2\}.$ We already know that $h$ cannot be 0, but for $h=-2$ we get a twist of the curve \lmfdbec{14a6}{14.a4}, for which we have $c_E=2.$ When $k=1$ we have $h+1=\pm 3,$ i.e., $h\in\{-4,2\}.$ For $h=-4$ we get a twist of \lmfdbec{14a5}{14.a1}, with $c_E=2,$ and $h=2$ cannot happen. Assume now that $h+1=\pm 3^k, k>1.$ Counting the multiplicities of 3 in $\Delta(h)$ and $c_4(h)$ we get that the factor $3^{2-2k}$ appears in the j-invariant. Furthermore, if we write $\pm 3^k-1$ instead of $h$ in the equation for $E$ and make the substitution $x\mapsto x\cdot 3^{6}, y\mapsto y\cdot 3^{9},$ we get a model where $\text{ord}_3(c_4)=0,$ and it follows from \Cref{prop:mult} that for $k>1$ we have multiplicative reduction $I_{2k-2}$ at 3, with $c_3$ being even (see \Cref{table1}). Note that in any case we also have a prime $p\neq 3$ dividing $h^2-h+1$ in $\Delta(h)$ with multiplicative reduction $I_{2n},$ which makes $c_E$ divisible by 4. \item[(3)] If $m=\frac{1}{h+1}=\pm p^k,$ for some prime $p,$ and clearly $\text{ord} _{p}(3m^2-3m+1)=0,$ since $\res(m, 3m^2-3m+1)=1$. Then there exists $p'\neq p$ such that $\text{ord}_{p'}(3m^2-3m+1)>0.$ Otherwise, we have $3m^2-3m+1\in\{\pm 1\},$ i.e., $m\in\{0,1\}$ which only makes sense for $h=0$ but, as we noted earlier, $h$ cannot be 0. \end{itemize} The only thing left to consider is when we have only 2 primes with $\text{ord}_p(h+1)\neq 0,$ one of which is 3 and divides the numerator; in other words the cases $h+1=\pm 3^kp^l$ and $h+1=\pm \frac{3^k}{p^l}, p\neq 3, k, l>0.$ From the reasoning in (2) above, it is clear that if $k>1,$ we have multiplicative reduction at 3 and from the part of the proof where we had $\text{ord}_p(h+1)<0$ we see that the reduction is multiplicative at $p$ as well, which gives us $c_E$ that is divisible by 4. For $k=1,$ we have $h+1=\pm 3p^l$ or $h+1=\pm \frac{3}{p^l}.$ \begin{itemize} \item If $h+1=\pm 3p^l,$ we have another prime $p'\neq p,3$ dividing $h^2-h+1$ in the discriminant (similarly as in (1)) with multiplicative reduction. \item If $h+1=\frac{1}{m}=\pm \frac{3}{p^l},$ we also have another prime $p'\neq p$ dividing the numerator of $3m^2-3m+1$ in the discriminant (as in (3)) with multiplicative reduction, except possibly when $3m^2-3m+1=\frac{1}{a^n}, a\in\mathbb{Z}, n>0$ (this situation couldn't have happened in (3), because we had $m\in\mathbb{Z}$). By putting $ \pm\frac{p^l}{3} $ instead of $m$, we get $$ \pm\dfrac{p^{2l}}{3}\mp p^l+1=\dfrac{1}{a^n}, $$ which only has solutions for $a=3, n=1, p=2, l=1,$ i.e., if $h\in\left\{-\frac{5}{2},\frac{1}{2}\right\}.$ For $h=\frac{1}{2}$ we get a twist of the elliptic curve \lmfdbec{14a3}{14.a2}, with $c_E=2,$ and for $h=-\frac{5}{2}$ we get a curve that already has 2 primes of reduction type $I_{2k}$, namely 2 and 13. \end{itemize} To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawa numbers would change under the twisting of the curves. All even Tamagawa numbers mentioned in the proof above come from multiplicative reductions $I_{2n}$ at primes $p,$ so by using Table \ref{table1}, Table 2 and Lemma \ref{lemma1}, we conclude that all reduction types of twists at $p$ are either $I_{2n}$ or $I_{2n}^,$ so the Tamagawa numbers stay even. As for the curves \lmfdbec{14a3}{14.a2}, \lmfdbec{14a4}{14.a5}, \lmfdbec{14a5}{14.a1} and \lmfdbec{14a6}{14.a4}, they have $c_E=c_2=2.$ By using the fact that $\mathbb{Q}_p^\times/\left( \mathbb{Q}_p^\times\right)^2=\langle -1,2,5 \rangle$, we explicitly compute all possible reduction types of quadratic twists at $p=2$ and conclude that for every twist of those curves $4\mid c_E.$ \end{proof} \begin{comment} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 16-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 3]{ALR} we take the parameterization of the j-invariants of the curves that are non-cuspidal points on $X_0(12),$ $$j(h)=\dfrac{(h^8-16h^4+16)^3}{h^4(h^4-16)}, h\in\mathbb{Q}.$$ From it we can acquire the discriminant and the $c_4-$invariant up to a twist, $$\Delta(h)=(h - 2)h^4(h + 2)(h^2 + 4)(h^4 - 8)^6(h^8 - 16h^4 - 8)^6(h^8 - 16h^4 + 16)^6,$$ $$c_4(h)=(h^4 - 8)^2(h^8 - 16h^4 - 8)^2(h^8 - 16h^4 + 16)^3.$$ Assume that there exists a prime $p$ such that $k:=ord _p(h)>0.$ We compute $$res\left(h, \frac{\Delta(h)}{h^4}\right)=-2^{64}, $$ $$res\left(h, c_4(h)\right)=2^{24}. $$ If $p\neq 2,$ this means that $p^4\mid\Delta(h)$ and $p\nmid c_4(h),$ from which we find that the reduction of $E$ at $p$ is multiplicative of type $I_{4k},$ and $c_p$ is even. For the case $h=\pm 2^k,$ after the change of variables $x\mapsto x\cdot 2^{12}, y\mapsto y\cdot 2^{18},$ counting the multiplicities of 2 in $\Delta(h)$ and $c_4(h)$ we get that the factor $2^{2k-8}$ appears in the j-invariant, with $2\nmid c_4(h).$ Therefore, when $k>4,$ we have multiplicative reduction at 2 of type $I_{2k-8}$ with even $c_p.$ For $k=0$ we have $h=\pm 1$ and for the both values we get a twist of the curve 15a8 with $c_E=1.$ For $k=1$ we have $h=\pm 2,$ i.e. a curve 48a4 up to a twist, with $c_E=1.$ When $k=2$ we do not get a curve and, for $k=3$ and $h=\pm 4$ we have a twist of 24a3, where $c_E=2,$ and finally for $k=3$ and $h=\pm 16$ we have a twist of 15a7, where $c_E=1.$ Assume now that there exists a prime $p$ such that $k:=ord _p(h)<0.$ We put $m:=\frac{1}{h}$ and after the substitution $x\mapsto x\cdot 2^{-12}, y\mapsto y\cdot 2^{-18}$ we get an elliptic curve with \begin{equation} \Delta(m) = -(4m - 1)m^8(4m + 1)(8m^2 - 1)^6(16m^4 - 16m^2 + 1)^6(8m^4 + 16m^2 - 1)^6 \end{equation} up to a twist, and \begin{equation} c_4(m) =(8m^2 - 1)^2(16m^4 - 16m^2 + 1)^3(8m^4 + 16m^2 - 1)^2 \end{equation} up to a twist, where $$res\left(m, \frac{\Delta(m)}{m^{8}}\right)=1, $$ $$res\left(m, c_4(m)\right)=1. $$ We see that the reduction at $p$ is multiplicative of type $I_{8k},$ with even $c_p.$ ... \end{proof} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 12-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 3]{ALR} we take the parameterization of the j-invariants of the curves that are non-cuspidal points on $X_0(12),$ $$j(h)=\dfrac{(h^2-3)^3(h^6-9h^4+3h^2-3)^3}{(h^2-1)^3(h^2-9)h^4}, h\in\mathbb{Q}.$$ From it we can acquire the discriminant and the $c_4-$invariant up to a twist, $$\Delta(h)=(h - 3)(h - 1)^3h^4(h + 1)^3(h + 3)(h^2 - 3)^6(h^4 - 6h^2 - 3)^6(h^6 - 9h^4 + 3h^2 - 3)^6(h^8 - 12h^6 + 30h^4 - 36h^2 + 9)^6,$$ $$c_4(h)=(h^2 - 3)^3(h^4 - 6h^2 - 3)^2(h^6 - 9h^4 + 3h^2 - 3)^3(h^8 - 12h^6 + 30h^4 - 36h^2 + 9)^2.$$ Assume that there exists a prime $p$ such that $k:=ord _p(h)>0.$ We compute $$res\left(h, \frac{\Delta(h)}{h^4}\right)=3^{32}, $$ $$res\left(h, c_4(h)\right)=3^{12}. $$ If $p\neq 3,$ this means that $p^4\mid\Delta(h)$ and $p\nmid c_4(h),$ from which we find that the reduction of $E$ at $p$ is multiplicative of type $I_{4k},$ and $c_p$ is even. For the case $h=\pm 3^k,$ we have that $k>1,$ since for $k\in\{0,1\}$ we get $h\in\{ \pm 1, \pm 3 \}$ and those $h$ do not give elliptic curves (look at the $j-$invariant). After the change of variables $x\mapsto x\cdot 3^{6}, y\mapsto y\cdot 3^{9},$ counting the multiplicities of 3 in $\Delta(h)$ and $c_4(h)$ we get that the factor $3^{4k-4}$ appears in the j-invariant. Furthermore, $ord_3(c_4)=0.$ For $k>1$ that means that the reduction at 3 is of type $I_{4k-4},$ with even $c_3.$ Assume now that there exists a prime $p$ such that $k:=ord _p(h)<0.$ We put $m:=\frac{1}{h}$ and after the substitution $x\mapsto x\cdot 3^{-6}, y\mapsto y\cdot 3^{-9}$ we get an elliptic curve with \begin{equation} \begin{split} \Delta(m) =& \left(m - 1\right)^3m^{12}(m + 1)^3\left(3m - 1\right)\left(3m +1\right)\left(3m^2 - 1\right)^6\left(3m^4 + 6m^2 -1\right)^6\\ & \left(3m^6 - 3m^4 + 9m^2 - 1\right)^6\left(9m^8 - 36m^6 + 30m^4 - 12m^2 + 1\right)^6 \end{split} \end{equation} up to a twist, and \begin{equation} c_4(m)= \left(3m^2 - 1\right)^3 \left(3m^4 + 6m^2 - 1\right)^2 \left(3m^6 - 3m^4 + 9m^2 - 1\right)^3\left(9m^8 - 36m^6 + 30m^4 - 12m^2 + 1\right)^2 \end{equation} up to a twist, where $$res\left(m, \frac{\Delta(m)}{m^{12}}\right)=1, $$ $$res\left(m, c_4(m)\right)=1. $$ We see that the reduction at $p$ is multiplicative of type $I_{12k},$ with even $c_p.$ The same reasoning as at the end of Proposition \ref{prop1} (using Lemma \ref{lemma1}) shows that the parity of the Tamagawa number does not change with the twisting of the curves. \end{proof} \end{comment} \begin{proposition}\label{10izo} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a $10-$isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 3]{ALR} we take the parameterization of the $j-$invariants of the curves that are non-cuspidal points on $X_0(10),$ $$j(h)=\dfrac{(h^6-4h^5+16h+16)^3}{(h+1)^2(h-4)h^5}, h\in\mathbb{Q}.$$ From it we can acquire the discriminant $$\Delta(h)=(h - 4)h^5(h + 1)^2(h^2 - 2h - 4)^6(h^2 - 2h + 2)^6f(h)$$ and the $c_4-$invariant $c_4(h)$ up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.3.txt}{Magma code}. The polynomial $f(h)$ is a monic polynomial in $\mathbb{Z}[h].$ Assume that there exists a prime $p$ such that $k:=\text{ord} _p(h+1)>0.$ We compute $$\res\left(h+1, \frac{\Delta(h)}{(h+1)^2}\right)=5^{22}, $$ $$\res\left(h+1, c_4(h)\right)=5^{8}. $$ If $p\neq 5,$ this means that $p^{2k}\mid\Delta(h)$ and $p\nmid c_4(h),$ and we find from \Cref{prop:mult} that the reduction of $E$ at $p$ is multiplicative of type $I_{2k},$ and therefore $c_p$ is even (see \Cref{table1}). For the case $h+1=\pm 5^k,$ after the change of variables $x\mapsto x\cdot 5^{4}, y\mapsto y\cdot 5^{6},$ counting the multiplicities of 5 in $\Delta(h)$ and $c_4(h)$ we get that the factor $5^{2-2k}$ appears in the $j-$invariant, with $5\nmid c_4(h).$ Therefore, when $k>1,$ by \Cref{prop:mult} we have multiplicative reduction at 5 of type $I_{2k-2}$ with even $c_p$ (see \Cref{table1}). For $k\in\{0,1\}$ we have $h\in\{-6,-2,0,4\}.$ For the values $h\in\{0,4\}$ we do not have an elliptic curve, and for the values $h\in\{-6,-2\}$ we get twists of curves \lmfdbec{768d3}{768.h1} and \lmfdbec{768d1}{768.h3}, which have $c_E=2,$ both with bad prime 2 with reduction type $III,$ so $c_2=2.$ Assume now that there exists a prime $p$ such that $k:=\text{ord}_p(h+1)<0.$ We put $m:=\frac{1}{h+1}$ and after the substitution $x\mapsto x\cdot 5^{-4}, y\mapsto y\cdot 5^{-6}$ we get an elliptic curve with the discriminant $$\Delta(m) =(m - 1)^5(5m - 1)m^{10}(5m^2 - 4m + 1)^6(5m^2 - 2m + 1)^9g(m)$$ and the $c_4-$invariant $c_4(m)$ up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.3.txt}{Magma code}. The polynomial $g(m)$ is a monic polynomial in $\mathbb{Z}[m].$ \noindent We compute $$\res\left(m, \frac{\Delta(m)}{m^{10}}\right)=1, $$ $$\res\left(m, c_4(m)\right)=1. $$ We see by \Cref{prop:mult} that the reduction at $p$ is multiplicative of type $I_{10k},$ with even $c_p$ (see \Cref{table1}). To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawa numbers would change under the twisting of the curves. All even Tamagawa numbers mentioned in the proof above come from multiplicative reductions $I_{2n}$ at primes $p,$ so by using Table \ref{table1}, Table 2 and Lemma \ref{lemma1}, we conclude that all reduction types of twists at $p$ are either $I_{2n}$ or $I_{2n}^,$ so the Tamagawa numbers stay even. As for the curves \lmfdbec{768d3}{768.h1} and \lmfdbec{768d1}{768.h3}, they have $c_E=c_2=2.$ By using the fact that $\mathbb{Q}_p^\times/\left( \mathbb{Q}_p^\times\right)^2=\langle -1,2,5 \rangle$, we explicitly compute all possible reduction types of quadratic twists at $p=2$ and conclude that for every twist of those curves $2\mid c_E.$ \end{proof} \begin{proposition}\label{8izo} Let $E$ be an elliptic curve over $\mathbb{Q}$ with an $8-$isogeny. Then $2\|c_E,$ except for the curves $\lmfdbec{15a7}{15.a4}$, $\lmfdbec{15a8}{15.a7}$, $\lmfdbec{48a4}{48.a5}$, where $c_E=1.$ \end{proposition} \begin{proof} From \cite[Table 3]{ALR} we take the parameterization of the $j-$invariants of the curves that are non-cuspidal points on $X_0(8),$ $$j(h)=\dfrac{(h^4-16h^2+16)^3}{(h^2-16)h^2}, h\in\mathbb{Q}.$$ From it we can acquire the discriminant $$\Delta(h)=(h - 4)h^2(h + 4)f(h)$$ and the $c_4-$invariant $c_4(h)$ up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.4.txt}{Magma code}. The polynomial $f(h)$ is a monic polynomial in $\mathbb{Z}[h].$ Assume that there exists a prime $p$ such that $k:=\text{ord}_p(h)>0.$ We compute $$\res\left(h, \frac{\Delta(h)}{h^2}\right)=2^{64}, $$ $$\res\left(h, c_4(h)\right)=2^{24}. $$ If $p\neq 2,$ then this means that $p^{2k}\mid\Delta(h)$ and $p\nmid c_4(h),$ and from \Cref{prop:mult} we find that the reduction of $E$ at $p$ is multiplicative of type $I_{2k},$ and therefore $c_p$ is even (see \Cref{table1}). For the case $h=\pm 2^k,$ after the change of variables $x\mapsto x\cdot 2^{12}, y\mapsto y\cdot 2^{18},$ counting the multiplicities of 2 in $\Delta(h)$ and $c_4(h)$ we get that the factor $2^{8-2k}$ appears in the $j-$invariant, with $2\nmid c_4(h).$ Therefore, when $k>4,$ we have multiplicative reduction at 2 of type $I_{2k-8}$ with even $c_p,$ by \Cref{prop:mult} and \Cref{table1}. For $k=0$ we have $h=\pm 1$ and for the both values we get a twist of the curve \lmfdbec{15a8}{15.a7} with $c_E=1.$ For $k=1$ we have $h=\pm 2,$ i.e., a curve \lmfdbec{48a4}{48.a5} up to a twist, with $c_E=1.$ When $k=2$ we do not get a curve and, for $k=3$ and $h=\pm 8$ we have a twist of \lmfdbec{24a3}{24.a2}, where $c_E=2,$ and finally for $k=4$ and $h=\pm 16$ we have a twist of \lmfdbec{15a7}{15.a4}, where $c_E=1.$ Assume now that there exists a prime $p$ such that $k:=\text{ord} _p(h)<0.$ We put $m:=\frac{1}{h}$ and after the substitution $x\mapsto x\cdot 2^{-12}, y\mapsto y\cdot 2^{-18}$ we get an elliptic curve with the discriminant \begin{equation} \Delta(m) = -(4m - 1)m^8(4m + 1)g(m) \end{equation} and the $c_4-$invariant $c_4(m)$ up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.4.txt}{Magma code}. The polynomial $g(m)$ is a monic polynomial in $\mathbb{Z}[m].$ \noindent We compute $$\res\left(m, \frac{\Delta(m)}{m^{8}}\right)=1, $$ $$\res\left(m, c_4(m)\right)=1. $$ We see from \Cref{prop:mult} and \Cref{table1} that the reduction at $p$ is multiplicative of type $I_{8k},$ with even $c_p.$ To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawa numbers would change under the twisting of the curves. All even Tamagawa numbers mentioned in the proof above come from multiplicative reductions $I_{2n}$ at primes $p,$ so by using Table \ref{table1}, Table 2 and Lemma \ref{lemma1}, we conclude that all reduction types of twists at $p$ are either $I_{2n}$ or $I_{2n}^,$ so the Tamagawa numbers stay even. As for the curves \lmfdbec{15a7}{15.a4}, \lmfdbec{15a8}{15.a7} and \lmfdbec{48a4}{48.a5}, they have $c_E=1.$ By using the fact that $\mathbb{Q}_p^\times/\left( \mathbb{Q}_p^\times\right)^2=\langle -1,2,5 \rangle$, we explicitly compute all possible reduction types of quadratic twists at $p=2$ and conclude that for every twist of those curves $2\mid c_E.$ Lastly, for every twist of the curve \lmfdbec{24a3}{24.a2} we have $2\mid c_E.$ \end{proof} \begin{proposition}\label{6izo} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a $6-$isogeny. Then $2\|c_E,$ except for the curve $\lmfdbec{20a2}{20.a3}$, where $c_E=3,$ and also the curves $\lmfdbec{80b2}{80.b3}$, $ \lmfdbec{80b4}{80.b1},$ $\lmfdbec{20a4}{20.a1}$, $\lmfdbec{27a3}{27.a4}$ and infinitely many twists of $\lmfdbec{27a3}{27.a4}$, for which $c_E=1$. \end{proposition} \begin{proof} From \cite[Table 3]{ALR} we take the parameterization of the $j-$invariants of the curves that are non-cuspidal points on $X_0(6),$ $$j(h)=\dfrac{(h+6)^3 (h^3+18h^2+84h+24)^3}{h(h+8)^3 (h+9)^2}, h\in\mathbb{Q}.$$ From it we can acquire the discriminant $$\Delta(h)=h(h + 6)^6(h + 8)^3(h + 9)^2f(h),$$and the $c_4-$invariant $c_4(h)$ up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.5.txt}{Magma code}. The polynomial $f(h)$ is a monic polynomial in $\mathbb{Z}[h].$ Assume that there exists a prime $p$ such that $k:=\text{ord}_p(h+9)>0.$ We compute $$\res\left(h+9, \frac{\Delta(h)}{(h+9)^2}\right)=3^{32}, $$ $$\res\left(h+9, c_4(h)\right)=3^{12}. $$ If $p\neq 3,$ this means that $p^{2k}\mid\Delta(h)$ and $p\nmid c_4(h),$ and from \Cref{prop:mult} we find that the reduction of $E$ at $p$ is multiplicative of type $I_{2k},$ and therefore $c_p$ is even (see \Cref{table1}). For the case $h+9=\pm 3^k,$ after the change of variables $x\mapsto x\cdot 3^{6}, y\mapsto y\cdot 3^{9},$ counting the multiplicities of 3 in $\Delta(h)$ and $c_4(h)$ we get that the factor $3^{4-2k}$ appears in the $j-$invariant, with $3\nmid c_4(h).$ Therefore, when $k>2,$ we have multiplicative reduction at 3 of type $I_{2k-4}$ with even $c_p,$ by \Cref{prop:mult} and \Cref{table1}. For $k=0$ we have $h\in\{-10,-8\}.$ With $h=-10$ we have a twist of the elliptic curve \lmfdbec{20a2}{20.a3} which has $c_E=3,$ coming from the reduction at 2 of type $IV,$ and $h=-8$ does not give us an elliptic curve. For $k=1$ we have $h\in\{-12,-6\}.$ For $h=-12$ we get the curve \lmfdbec{36a2}{36.a2} with $c_E=6$ and for $h=-6$ we have \lmfdbec{27a3}{27.a4}, where $c_E=1.$ Lastly, if $k=2,$ then $h\in\{-18,0\}.$ For $h=0$ we do not get an elliptic curve, but for $h=-18$ we get a twist of the curve \lmfdbec{80b4}{80.b1}, with $c_E=1.$ Assume now that there exists a prime $p$ such that $k:=\text{ord} _p(h+9)<0.$ We put $m:=\frac{1}{h+9}$ and after the substitution $x\mapsto x\cdot 3^{-6}, y\mapsto y\cdot 3^{-9}$ we get an elliptic curve with the discriminant \begin{equation} \Delta(m) = (m - 1)^3(3m - 1)^6(9m - 1)m^6g(m) \end{equation} and the $c_4-$invariant $c_4(m)$ up to a twist, which can be computed with the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/prop3.5.txt}{Magma code}. The polynomial $g(m)$ is a monic polynomial in $\mathbb{Z}[m].$ \noindent We compute $$\res\left(m, \frac{\Delta(m)}{m^{6}}\right)=1, $$ $$\res\left(m, c_4(m)\right)=1. $$ We see that the reduction at $p$ is multiplicative of type $I_{6k},$ with even $c_p,$ by \Cref{prop:mult} and \Cref{table1}. To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawa numbers would change under the twisting of the curves. All even Tamagawa numbers mentioned in the proof above come from multiplicative reductions $I_{2n}$ at primes $p,$ so by using Table \ref{table1}, Table 2 and Lemma \ref{lemma1}, we conclude that all reduction types of twists at $p$ are either $I_{2n}$ or $I_{2n}^,$ so the Tamagawa numbers stay even. The curve \lmfdbec{36a2}{36.a2} already has reduction type $III$ at $3$, which stays the same under twisting or changes to $III^$, as stated in Table \ref{tab2}. In any case we get $c_3=2$ (see Table \ref{table1}). As for the curve \lmfdbec{20a2}{20.a3}, we examine all twists by $d$ such that there exists a prime $p\mid d$ such that \lmfdbec{20a2}{20.a3} has good reduction at $p$, i.e., $p\neq 2,5.$ At such $p$ we have reduction type $I_0$, and after twisting by $p$ we get that the reduction type of the twist at $p$ is $I_0^$, as stated in Table \ref{tab2}. By Tate's algorithm \cite[\S 8. Case 6)]{T} we get that $c_p=1+\text{number of roots of $P(T)$ in $k_p$}$, where $P(T)=T^3+T^2-T=T(T^2+T-1).$ Polynomial $T^2+T-1$ has roots modulo $p$ if and only if $5$ is a quadratic residue modulo $p.$ Therefore, we get $$c_p=\begin{cases} 2, \text{ if $p\equiv 2,3\pmod 5$}\\4, \text{ if $p\equiv 1,4\pmod 5$} \end{cases}.$$ It remains to see what happens with the twists by $d$, where $d$ has no divisors of good reduction for the curve \lmfdbec{20a2}{20.a3}. Since the only primes of bad reduction are $2$ and $5$, by explicitly computing all possible twists we get that \lmfdbec{20a2}{20.a3}, where $c_E=3$ and \lmfdbec{80b2}{80.b3}, where $c_E=1,$ are the only possible twists for which $2\nmid c_E.$ We use the same approach with the curve \lmfdbec{80b4}{80.b1}, which has $c_E=1.$ We examine all twists by $d$ such that there exists a prime $p\mid d$ such that \lmfdbec{80b4}{80.b1} has good reduction at $p$, i.e., $p\neq 2,5.$ At such $p$ we have reduction type $I_0$, and after twisting by $p$ we get that the reduction type of the twist at $p$ is $I_0^$, as stated in Table \ref{tab2}. By Tate's algorithm \cite[\S 8. Case 6)]{T} we get that $c_p=1+\text{number of roots of $P(T)$ in $k_p$}$, where $P(T)=T^3 - T^2 - 41T + 116=(T-4)(T^2+3T-29).$ Polynomial $T^2+3T-29$ has roots modulo $p$ if and only if $5$ is a quadratic residue modulo $p.$ Therefore, we get $$c_p=\begin{cases} 2, \text{ if $p\equiv 2,3\pmod 5$}\\4, \text{ if $p\equiv 1,4\pmod 5$} \end{cases}.$$ It remains to see what happens with the twists by $d$, where $d$ has no divisors of good reduction for the curve \lmfdbec{80b4}{80.b1}. Since the only primes of bad reduction are $2$ and $5$, by explicitly computing all possible twists we get that \lmfdbec{80b4}{80.b1} and \lmfdbec{20a4}{20.a1}, where $c_E=1,$ are the only possible twists for which $2\nmid c_E.$ For the curve \lmfdbec{27a3}{27.a4}, which has $c_E=1$, the situation is more complicated, and we will prove that the curve has infinitely many twists with Tamagawa number 1. For \lmfdbec{27a3}{27.a4} we have that $c_{E}=c_3=1$, where $3$ is the only prime of bad reduction. Similarly as for the curves \lmfdbec{20a2}{20.a3} and \lmfdbec{80b4}{80.b1}, we are interested in what happens with the Tamagawa numbers of twists $E^d$ when $p\mid d$, for some prime $p$ of good reduction for $E$, in this case $p\neq 3$. At such $p$ we have reduction type $I_0$, and after twisting by $d$ we get that the reduction type of the twist at $p$ is $I_0^$, as stated in Table \ref{tab2}. By Tate's algorithm \cite[\S 8. Case 6)]{T} we get that $c_p=1+\text{number of roots of $P(T)$ in $k_p$}$, where $P(T)=T^3+11664.$ The Tamagawa number at $p$ when the reduction type is $I_0^$ can be $1$, $2$ or $4$ (see Table \ref{table1}). As opposed to the aforementioned two curves, all of those cases are possible here, as noted in Table \ref{table4}. Furthermore, the Galois group of the polynomial $P$ is $S_3$, hence non-abelian, which means that we have no straightforward description in terms of congruences of the primes $p$ for which there is a root modulo $p$ \cite[pp.576]{W}. However, using Frobenius' density theorem \cite[pp.32]{LS} we know that the density of all primes $p$ such that $P$ remains irreducible modulo $p$ is $\frac{1}{3}.$ That means that we have infinitely many primes $p$ such that $c_p=1.$ It remains to see that $c_E=1$ as well. The prime $3$ is the only prime of bad reduction for $E$ and the reduction type of $E$ at $3$ is $II.$ The reduction type at $3$ of $E^d$ when $3\nmid d$ stays the same (see Table \ref{tab2}) and hence the Tamagawa number at $3$ stays $1$. As for the prime $p=2$, by using the fact that $\mathbb{Q}_p^\times/\left( \mathbb{Q}_p^\times\right)^2=\langle -1,2,5 \rangle$, we explicitly compute all possible reduction types of quadratic twists at $p=2$ and conclude that for every twist of those curves $c_2=1.$ Therefore, we have that $c_{E^d}=c_d=1$ for $d$ in the set of primes for which $P \text{ mod } p$ stays irreducible. \end{proof} \begin{example} Denote by $E$ the curve \lmfdbec{27a3}{27.a4}. We have that $c_{E}=c_3=1$, where $3$ is the only prime of bad reduction. As noted in the proof of Proposition \ref{6izo}, if $p$ is a prime of good reduction for $E$, then by Tate's algorithm \cite[\S 8. Case 6)]{T} we get that the Tamagawa number of $E^d$ at $p$ such that $p\mid d$ is $c_p=1+\text{number of roots of $P(T)$ in $k_p$}$, where $P(T)=T^3+11664.$ The curve $E^d$ has reduction type $I_0^$ at $p$ and the Tamagawa number at $p$ can be $1$, $2$ or $4$ (see Table \ref{table1}), depending on the number of roots of $P \text{ mod } p$. All of those cases are possible here, and they are presented in Table \ref{table4}. \begin{table}[h] \begin{tabular}{\|c\|c\|c\|c\|c\|} \hline twist of \lmfdbec{27a3}{27.a4} by $d$ & curve & reduction type at $d$ & $c_d$ & $c_E$ \\ \hline $d=7$ & \lmfdbec{21168cx1}{21168.bv4} & $I_0^$ & $1$&$1$ \\ $d=5$ & \lmfdbec{675a1}{675.e4} & $I_0^$ & $2$ & $2$ \\ $d=31$ & \lmfdbec{415152ci1}{415152.ci4} & $I_0^$ & $4$ & $4$ \\ \hline \end{tabular} \caption{some twists of \lmfdbec{27a3}{27.a4} and their Tamagawa numbers}\label{table4} \end{table} All computations made in this example can be verified using the accompanying \href{https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3/ex3.6.txt}{Magma code}. \end{example} \begin{proposition}\label{ostalo} Let $E$ be an elliptic curve over $\mathbb{Q}$ with an $n-$isogeny, $n\in\{14,17,19,37,43,67,163\}$. Then $2\|c_E.$ \end{proposition} \begin{proof} For each value of $n$, from \cite[Table 4]{ALR} we took all the possible $j-$invariants. They can be found in \Cref{tab1} in the second column. In the third column we have a Cremona label of one of the curves in the class of twists represented by each $j-$invariant. For each of those curves in the fourth column we have a prime of bad reduction of type $III.$ That reduction can only change to $III^,$ and vice versa, after twisting, as we see in \Cref{tab2}. \Cref{table1} tells us that the Tamagawa number at primes of reduction type $III$ and $III^$ is always 2, so the claim follows. \end{proof} \begin{table}[h!] \centering \begin{tabular}{\|r\|ccc\|}\hline \multicolumn{1}{\|c\|}{$\mathit{n}$} & $j-$invariant & Cremona label & \begin{tabular}[c]{@{}c@{}}bad prime with \\ reduction type $III$\end{tabular} \\ \hline 14 & $-3^3\cdot 5^3$ & \lmfdbec{49a1}{49.a4} & 7 \\ & $3^3\cdot 5^3\cdot 17^3$ & \lmfdbec{49a2}{49.a3} & 7 \\ \hline 17 & $-\frac{17^2\cdot 101^3}{2}$ & \lmfdbec{14450p1}{14450.b2} & 5 \\ & $-\frac{17\cdot 373^3}{2^{17}}$ & \lmfdbec{14450p2}{14450.b1} & 5 \\ \hline 19 & $-2^{15}\cdot 3^3$ & \lmfdbec{361a1}{361.a2} & 19 \\ \hline 37 & $-7\cdot 11^3$ & \lmfdbec{1225h1}{1225.b2} & 5 \\ & $-7\cdot 137^3\cdot 2083^3$ & \lmfdbec{1225h2}{1225.b1} & 5 \\ \hline 43 & $-2^{18}\cdot 3^3\cdot 5^3$ & \lmfdbec{1849a1}{1849.b2} & 43 \\ \hline 67 & $-2^{15}\cdot 3^3\cdot 5^3\cdot 11^3$ & \lmfdbec{4489a1}{4489.b2} & 67 \\ \hline 163 & $-2^{18}\cdot 3^3\cdot 5^3\cdot 23^3\cdot 29^3$ & \lmfdbec{26569a1}{26569.a2} & 163 \\ \hline \end{tabular} \caption{$j-$invariants of the curves $X_0(n)$; their Cremona labels are representatives in the class of twists of least conductor with reduction type $III$ at some prime}\label{tab1} \end{table} \subsection{Acknowledgments} The author would like to thank Filip Najman for comments and helpful discussions. \begin{comment} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 14-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 4]{ALR} we see that the only possible j-invariants of curves with a 14-isogeny are $j_1=-3^3\cdot 5^3$ and $j_2=3^3\cdot 5^3\cdot 17^3.$ For $j_1$ we get the curve 49a1 up to a twist, and for $j_2$ we get the curve 49a2 up to a twist. They both have bad reduction only at 7. That reduction is of type $III$ which can only change to $III^,$ and vice versa, after twisting \cite[Proposition 1]{SC}. \end{proof} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 17-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 4]{ALR} we see that the only possible j-invariants of curves with a 17-isogeny are $j_1=-\frac{17^2\cdot 101^3}{2}$ and $j_2=-\frac{17\cdot 373^3}{2^{17}}.$ For $j_1$ we get the curve 14450p1 up to a twist, and for $j_2$ we get the curve 14450p2 up to a twist. They both have bad reduction at 5 of type $III$ which can only change to $III^,$ and vice versa, after twisting \cite[Proposition 1]{SC}. \end{proof} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 19-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 4]{ALR} we see that the only possible j-invariant of curves with a 19-isogeny is $j=-2^{15}\cdot 3^3,$ which is the curve 361a1 up to a twist. The curve has bad reduction at 19 of type $III$ which can only change to $III^,$ and vice versa, after twisting \cite[Proposition 1]{SC}. \end{proof} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 37-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 4]{ALR} we see that the only possible j-invariants of curves with a 17-isogeny are $j_1=-7\cdot 11^3$ and $j_2=-7\cdot 137^3\cdot 2083^3.$ For $j_1$ we get the curve 1225h1 up to a twist, and for $j_2$ we get the curve 1225h2 up to a twist. They both have bad reduction at 5 of type $III$ which can only change to $III^,$ and vice versa, after twisting \cite[Proposition 1]{SC}. \end{proof} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 43-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 4]{ALR} we see that the only possible j-invariant of curves with a 19-isogeny is $j=-2^{18}\cdot 3^3\cdot 5^3,$ which is the curve 1849a1 up to a twist. The curve has bad reduction at 43 of type $III$ which can only change to $III^,$ and vice versa, after twisting \cite[Proposition 1]{SC}. \end{proof} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 67-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 4]{ALR} we see that the only possible j-invariant of curves with a 19-isogeny is $j=-2^{15}\cdot 3^3\cdot 5^3\cdot 11^3,$ which is the curve 4489a1 up to a twist. The curve has bad reduction at 67 of type $III$ which can only change to $III^,$ and vice versa, after twisting \cite[Proposition 1]{SC}. \end{proof} \begin{proposition} Let $E$ be an elliptic curve over $\mathbb{Q}$ with a 163-isogeny. Then $2\|c_E.$ \end{proposition} \begin{proof} From \cite[Table 4]{ALR} we see that the only possible j-invariant of curves with a 19-isogeny is $j=-2^{18}\cdot 3^3\cdot 5^3\cdot 23^3\cdot 29^3,$ which is the curve 1849a1 up to a twist. The curve has bad reduction at 163 of type $III$ which can only change to $III^,$ and vice versa, after twisting \cite[Proposition 1]{SC}. \end{proof} \end{comment} \begin{comment} Let $\alpha'$ be the restriction of $\alpha$ to $X_1(14)$ and similarly $\beta'$. They act freely on the cusps we need (??) of $X_1(14)$ (TODO: prove this - the map $\phi$ ramifies in these cusps?). Let $\sigma$ be the generator of $\operatorname{Gal}(K/\mathbb{Q})$. Then either $y^\sigma=\alpha'(y)$ or $y^\sigma=\beta'(y)$, suppose without loss the former is true. Let $\overline y=\overline {C_0}$, $\overline y^{\sigma}=\overline {C_1}$, $\overline y^{\sigma^2}=\overline {C_2}$. By \Cref{lem:fd}, all $\overline C_i$ are defined over $\mathbb{F}_q$. We can conclude that $D=0$ (napisati argument za 2, svugdje drugdje je redukcija modulo $p$ injektivna) and hence $\phi^(D)=0$ (TODO:prove). It follows that $D$ is the divisor of a function $f\in \mathbb{Q}(X_1(14))$, in particular, quotienting by $\alpha'$. It follows that $C_0$ is a fixed point of $\alpha'$, which is a contradiction. The curve $\overline X$ has 3 components (cite: Gross-Zagier or Katz-Mazur) $\mathcal{F}_{0,2}$, $\mathcal{F}_{1,1}$ and $\mathcal{F}_{2,0}$ which is each isomorphic to $\overline{X}_1(7)$ and which intersect at the supersingular points. From this we can see that our point reduces to the smooth part of $\overline X$. \begin{proof} From \cite[Chapter 3]{BN2} we know the equation for $X_1(2,14),$ \begin{equation}\label{jedn} \left(u^3+u^2-2u-1\right)v\left(v+1\right)+\left(v^3+v^2-2v-1\right)u\left(u+1\right)=0, \end{equation} and that there exist two maps of degree 3 from $X_1(2,14)$ to $\mathbb P^1$. These maps are projections onto $u$ and $v.$ Taking into account the automorphism of $X_1(2,14)$ that interchanges $u$ and $v,$ we actually only have one map of degree 3, up to automorphisms of $X_1(2,14)$. Hence all cubic fields over which there $X_1(2,14)$ has a non-cuspidal point are roots of specializations of the polynomial from \eqref{jedn} for some $u \in \mathbb{Q}$. For Divide now equation \eqref{jedn} by $u(u+1)$ and we get \begin{equation}\label{jedn2} v^3+v^2-2v-1 + A v\left(v+1\right)=0, \end{equation} where $A=\frac{\left(u^3+u^2-2u-1\right)}{u\left(u+1\right)}.$ Write $A=m\cdot2^{-k}$, where $m$ is unit in $\mathbb{Z}_2$. Observe that we always have $k>0$. Making the change of variables $x:=v2^{-k}$ and multiplying everything by $2^{-3k}$ we obtain $$x^3+2^kx^2-2^{2k+1}x-2^{3k}+m(x^2+2^kx).$$ Let $f_u(v)\in \mathbb{Q}[v]$ be the specialization of the polynomial in \eqref{jedn} for some $u \in \mathbb{Q}$ and let $\theta$ be a root of this polynomial, and let $K=\mathbb{Q}(\theta)$. We have $$\Delta(f_u(v))=u^{12} + 2u^{11} + 5u^{10} + 22u^9 + 46u^8 + 70u^7 + 141u^6 + 238u^5 + 238u^4 + 138u^3 + 49u^2 + 10u + 1.$$ Suppose the opposite, so 2 does not split. As 2 cannot ramify in any cubic cyclic extension, this implies that 2 is inert. Suppose first that the numerator and denominator of $u$ are both odd. Then there exists an odd integer $a$ such that $\alpha=a\theta \in O_K$. Let $f_\alpha \in \mathbb{Z}[v]$ be the minimal polynomial of $\alpha$. We have $$\Delta (f_\alpha)=a^6 \Delta (f_u) \in \mathbb{Z}.$$ It is easy to see that $\Delta (f_\alpha)$ is odd, and hence the conductor of $\mathbb{Z}[\alpha]$ is odd. We now apply \cite[Proposition 8.3]{N}, which tells us that $2$ splits in the same way as the $\overline{f_\alpha}$, the reduction of $f_\alpha$ Hence, every cubic number field $K$ over which $X_1(2,14)$ has a point is generated by a root $\theta$ of the polynomial $$ f(v)=\left(u^3+u^2-2u-1\right)v\left(v+1\right)+\left(v^3+v^2-2v-1\right)u\left(u+1\right) $$ in $\mathbb{Q}[v],$ for some fixed $u\in\mathbb{Q}.$ The discriminant of $f$ is odd, so the conductor of the order $\mathbb{Z}[\theta]$ is relatively prime to (2). Since the polynomial $f$ completely factorizes in $\mathbb{F}_2[v],$ we conclude that 2 must completely split in $K$ \cite[Proposition 8.3]{N}. \end{proof} \end{comment} \end{document}	arXiv
Stanley's reciprocity theorem In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior. Definitions A rational cone is the set of all d-tuples (a1, ..., ad) of nonnegative integers satisfying a system of inequalities $M\left[{\begin{matrix}a_{1}\\\vdots \\a_{d}\end{matrix}}\right]\geq \left[{\begin{matrix}0\\\vdots \\0\end{matrix}}\right]$ where M is a matrix of integers. A d-tuple satisfying the corresponding strict inequalities, i.e., with ">" rather than "≥", is in the interior of the cone. The generating function of such a cone is $F(x_{1},\dots ,x_{d})=\sum _{(a_{1},\dots ,a_{d})\in {\rm {cone}}}x_{1}^{a_{1}}\cdots x_{d}^{a_{d}}.$ The generating function Fint(x1, ..., xd) of the interior of the cone is defined in the same way, but one sums over d-tuples in the interior rather than in the whole cone. It can be shown that these are rational functions. Formulation Stanley's reciprocity theorem states that for a rational cone as above, we have $F(1/x_{1},\dots ,1/x_{d})=(-1)^{d}F_{\rm {int}}(x_{1},\dots ,x_{d}).$ Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues. Develin has said that this amounts to proving the result "without doing any work". Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes. See also • Ehrhart polynomial References • Stanley, Richard P. (1974). "Combinatorial reciprocity theorems" (PDF). Advances in Mathematics. 14 (2): 194–253. doi:10.1016/0001-8708(74)90030-9. • Beck, M.; Develin, M. (2004). "On Stanley's reciprocity theorem for rational cones". arXiv:math.CO/0409562.	Wikipedia
# Evaluation metrics for search algorithms Precision measures the proportion of relevant items retrieved out of the total items retrieved. It is defined as the number of relevant items divided by the total number of items retrieved. $$\text{Precision} = \frac{\text{Number of relevant items}}{\text{Total number of items retrieved}}$$ Recall measures the proportion of relevant items retrieved out of the total relevant items. It is defined as the number of relevant items divided by the total number of relevant items. $$\text{Recall} = \frac{\text{Number of relevant items}}{\text{Total number of relevant items}}$$ The F1-score is a single metric that combines precision and recall. It is the harmonic mean of precision and recall. $$\text{F1-score} = 2 \cdot \frac{\text{Precision} \cdot \text{Recall}}{\text{Precision} + \text{Recall}}$$ Mean average precision (MAP) is another metric for evaluating search algorithms. It calculates the average precision for each query and then takes the mean of these values. $$\text{MAP} = \frac{1}{N} \sum_{i=1}^{N} \text{Precision}_i$$ ## Exercise Calculate the precision, recall, F1-score, and MAP for a search algorithm with the following values: - Number of relevant items: 100 - Total number of items retrieved: 200 - Total number of relevant items: 500 # Introduction to Python for implementing neural networks TensorFlow is an open-source machine learning library developed by Google. It provides a flexible platform for defining and running machine learning models. Keras is a high-level neural networks API that is written in Python. It allows for easy implementation of neural networks and is compatible with TensorFlow. ## Exercise Install TensorFlow and Keras in your Python environment. # Implementing neural networks for search in Python A neural network consists of an input layer, one or more hidden layers, and an output layer. The input layer receives the input data, the hidden layers perform the computations, and the output layer produces the final output. Activation functions are used to introduce non-linearity in the neural network. Common activation functions include the sigmoid function, ReLU (Rectified Linear Unit), and tanh. Loss functions are used to measure the difference between the predicted output and the true output. Common loss functions include mean squared error (MSE) and cross-entropy loss. ## Exercise Implement a simple neural network using TensorFlow for a search problem. Use a single hidden layer with 10 neurons and a sigmoid activation function. # Optimization techniques for neural network-based search algorithms Gradient descent is a first-order optimization algorithm that uses the gradient of the loss function to update the weights of the neural network. It is a simple and widely used optimization technique. Stochastic gradient descent (SGD) is a variation of gradient descent that updates the weights using a single training example. It is faster than gradient descent but may have a higher variance. Mini-batch gradient descent is a compromise between gradient descent and stochastic gradient descent. It updates the weights using a small subset of the training data, called a mini-batch. ## Exercise Implement the optimization techniques discussed in this section using TensorFlow. Compare the performance of gradient descent, stochastic gradient descent, and mini-batch gradient descent for a neural network implemented for a search problem. # Case studies of neural network-based search methods in Python Neural networks have been successfully applied to information retrieval tasks, such as document classification and ranking. They can capture complex interactions between words and improve search relevance. Neural networks have also been used for image search tasks, such as object detection and image classification. They can leverage the spatial and semantic information in images to improve search results. Recommendation systems have been revolutionized by the use of neural networks. These systems can learn user preferences and item features to provide personalized recommendations. ## Exercise Research and implement a neural network-based search method in Python for a specific problem or domain. Evaluate its performance using the evaluation metrics discussed in previous sections. # Evaluating the performance of neural network-based search methods Cross-validation is a resampling technique used to evaluate the performance of a model on different subsets of the training data. It helps to estimate the generalization performance of the model. Hyperparameter tuning is the process of selecting the best values for the hyperparameters of a neural network. It is crucial for achieving optimal performance. Validation curves are a visualization technique that helps to identify the best hyperparameter values for a neural network. They plot the performance of the model as a function of the hyperparameter values. ## Exercise Implement cross-validation and hyperparameter tuning for a neural network-based search method in Python. Use the validation curves to visualize the performance of the model. # Comparing neural network-based search methods to traditional search methods Neural network-based search methods have several advantages over traditional search methods. They can capture complex interactions and patterns in data, leading to improved search relevance and accuracy. However, neural network-based search methods can also have disadvantages. They require large amounts of data for training and may be computationally expensive. Additionally, they may be less interpretable and harder to understand compared to traditional search methods. ## Exercise Compare the performance of a neural network-based search method to a traditional search method, such as keyword search or information retrieval. Discuss the advantages and disadvantages of each approach. # Future directions for neural network-based search methods Transformers and attention mechanisms have revolutionized natural language processing and other domains. They can capture long-range dependencies and improve the performance of neural network-based search methods. Reinforcement learning and deep learning are other promising areas for future research. Reinforcement learning can be used to optimize search algorithms by learning from user feedback. Deep learning can be used to improve the performance of neural network-based search methods by leveraging large amounts of data. ## Exercise Research and discuss future directions for neural network-based search methods. Discuss potential advancements in neural network architectures and the integration of other machine learning techniques. # Conclusion and summary of key findings In conclusion, neural network-based search methods have shown great promise in improving the performance of search algorithms. They have the potential to capture complex interactions and patterns in data, leading to improved search relevance and accuracy. However, neural network-based search methods also have limitations. They require large amounts of data for training and may be computationally expensive. Additionally, they may be less interpretable and harder to understand compared to traditional search methods. Future research directions include advancements in neural network architectures, such as transformers and attention mechanisms, as well as the integration of other machine learning techniques, such as reinforcement learning and deep learning. ## Exercise Summarize the key findings of this textbook and discuss potential future research directions for neural network-based search methods.	Textbooks
« Kan Extension Seminar Talks at CT2014 \| Main \| The Categorical Origins of Lebesgue Integration » The Linearity of Traces Posted by Mike Shulman At long last, the following two papers are up: Kate Ponto and Mike Shulman, The linearity of traces in monoidal categories and bicategories Kate Ponto and Mike Shulman, The linearity of fixed-point invariants I'm super excited about these, and not just because I like the results. Firstly, these papers are sort of a culmination of a project that began around 2006 and formed a large part of my thesis. Secondly, this project is an excellent "success story" for a methodology of "applied category theory": taking seriously the structure that we see in another branch of mathematics, but studying it using honest category-theoretic tools and principles. For these reasons, I want to tell you about these papers by way of their history. (I've mentioned some of their ingredients before when I blogged about previous papers in this series, but I won't assume here you know any of it.) To begin with, recall that an object XX of a symmetric monoidal category is dualizable if, when regarded as a 1-cell in the associated one-object bicategory, it has an adjoint DXD X. Then any endomorphism f:X→Xf:X\to X has a trace defined by I→ηX⊗DX→f⊗1X⊗DX→≅DX⊗X→ϵI. I \xrightarrow{\eta} X \otimes D X \xrightarrow{f\otimes 1} X\otimes D X \xrightarrow{\cong} D X \otimes X \xrightarrow{\epsilon} I. In VectVect, the dualizable objects are the finite-dimensional ones, traces reproduce the usual trace of a matrix (incarnated as a 1×11\times 1 matrix), and in particular tr(1 X)=dim(X)tr(1_X) = dim(X). In the stable homotopy category, this is Spanier-Whitehead duality, traces produce the Lefschetz number L(f)L(f) (incarnated as the degree of a self-map of a sphere), and we have L(1 X)=χ(X)L(1_X) = \chi(X), the Euler characteristic. The Lefschetz fixed point theorem follows by abstract nonsense. The recent part of the story began in 2001, when Peter May wrote "The additivity of traces in triangulated categories". The Euler characteristic (and Lefschetz number) are additive: if XX is a cell complex and A⊆XA\subseteq X a subcomplex, then χ(X)=χ(A)+χ(X/A)\chi(X) = \chi(A) + \chi(X/A) and L(f)=L(f\| A)+L(f/A)L(f) = L(f\|_A) + L(f/A). Peter showed an abstract version of this: if a symmetric monoidal category is compatibly triangulated, then for any distinguished triangle X→Y→Z→ΣXX\to Y\to Z\to \Sigma X, we have χ(Y)=χ(X)+χ(Z)\chi(Y) = \chi(X) + \chi(Z). A few years later, Peter and Johann Sigurdsson realized that "Costenoble-Waner duality" for parametrized spaces was naturally about adjunctions in a bicategory whose objects were topological spaces and whose 1-cells from AA to BB are spectra "parametrized over A×BA\times B". (The 2-cells are fiberwise stable maps; note the conspicuous absence of continuous maps of base spaces.) Peter thus wondered whether additivity generalized to bicategories. In the book that he and Johann wrote, they generalized some of his axioms for triangulated monoidal categories to "locally triangulated" bicategories, but the final axiom (TC5) used the symmetry, which doesn't make sense in a bicategory. It was also not clear how to generalize "traces", since the definition of trace also uses symmetry. Enter Kate Ponto, who was studying topological fixed-point theory. This subject "begins" with the Lefschetz fixed point theorem, but continues with more refined invariants such as the Reidemeister trace, which supports a converse to this theorem (under suitable hypotheses). One definition of the Reidemeister trace uses the Hattori-Stallings trace, which is a sort of trace for matrices over a noncommutative ring: you'd like to sum along the diagonal, but the result is basis-dependent until you map it from RR into the quotient abelian group ⟨⟨R⟩⟩=R/(rs∼sr)\langle\langle R \rangle\rangle = R / (r s \sim s r). Kate realized that the Hattori-Stallings trace, and hence also the Reidemeister trace, was a sort of "bicategorical trace" that she was able to define for endo-2-cells of dualizable 1-cells in any bicategory equipped with some extra structure that she named a shadow. Pleasingly to fans of the microcosm principle, a shadow on a bicategory B\mathbf{B} is a "categorified trace", consisting of functors ⟨⟨−⟩⟩:B(A,A)→T\langle\langle-\rangle\rangle:\mathbf{B}(A,A) \to \mathbf{T} that are "cyclic up to isomorphism": ⟨⟨X⊙Y⟩⟩≅⟨⟨Y⊙X⟩⟩\langle\langle X \odot Y \rangle\rangle \cong \langle\langle Y \odot X \rangle\rangle plus some coherence axioms. Given this, if X:A→BX:A\to B has an adjoint DXD X and f:X→Xf:X\to X, Kate defined its trace tr(f)\tr(f) to be ⟨⟨U A⟩⟩→η⟨⟨X⊙DX⟩→f⊙1⟨⟨X⊙DX⟩⟩→≅⟨⟨DX⊙X⟩⟩→ϵ⟨⟨U B⟩⟩ \langle\langle U_A \rangle\rangle \xrightarrow{\eta} \langle\langle X \odot D X\rangle \xrightarrow{f\odot 1} \langle\langle X\odot D X \rangle\rangle \xrightarrow{\cong} \langle\langle D X \odot X \rangle\rangle \xrightarrow{\epsilon} \langle\langle U_B \rangle\rangle where U AU_A, U BU_B are the unit 1-cells. I blogged about this here. So Kate had solved half of the problem of generalizing additivity to bicategories. At about the same time, I was intrigued by a different aspect of Peter and Johann's bicategory. Parametrized spaces and spectra can be "pulled back" and "pushed forward" along maps of base spaces. Moreover, pushforward and "copushforward" generalize homology and cohomology, hence should preserve duality. But how can we show this abstractly, since the maps between base spaces are missing from the bicategory of parametrized spectra? Peter and Johann solved this with "base change objects": for any continuous map f:A→Bf:A\to B they defined spectra S fS_f and fS{}_f S over B×AB\times A and A×BA\times B such that composing with them was equivalent to pulling back and pushing forward. Moreover, S fS_f and fS{}_f S are dual; thus, since adjunctions compose, if XX is Costenoble-Waner dualizable, so is its pushforward to a point (π A) !X(\pi_A)_! X. This clean and easy argument, when they noticed it, replaced a long and messy calculation. I, however, was unsatisfied with the fact that the maps of base spaces were not actually present in the bicategory, leading me to invent framed bicategories, which are actually double categories with extra properties. The horizontal arrows give it an underlying bicategory, while the vertical arrows supply the missing morphisms, and the additional 2-cells let us characterize the base change objects with a universal property. Soon, I realized that a "framing" on a bicategory was equivalent to giving pseudofunctorial "base change objects" with adjoints, a structure which had been defined by Richard Wood under the name proarrow equipment. However, the double-categorical viewpoint has certain advantages: e.g. it looks a little less ad hoc, it makes it easier to define functors and transformations between such structures (this had already been observed by Dominic Verity), and it generalizes to situations where the horizontal 1-cells can't be composed. Another thing that bothered me about Peter and Johann's bicategory was that, to be honest, they hadn't finished constructing it. They defined the composition and units and constructed associativity and unit isomorphisms, but didn't prove the coherence axioms. In order to remedy this cleanly and abstractly, I isolated the properties of parametrized spectra that were necessary for the construction, leading to the notion of monoidal fibration or indexed monoidal category: a pseudofunctor C:S op→MonCat\mathbf{C}:S^{op} \to MonCat. The only assumptions needed beyond this are that SS is cartesian monoidal and that the "pullback" functors f :C(B)→C(A)f^\ast:\mathbf{C}(B) \to \mathbf{C}(A) have "pushforward" Hopf left adjoints f !f_! satisfying the Beck-Chevalley condition for pullback squares (or homotopy pullback squares). Thus, from any such C\mathbf{C} we can construct a (framed) bicategory Fr(C)Fr(\mathbf{C}), whose objects are those of SS and with Fr(C)(A,B)=C(A×B)Fr(\mathbf{C})(A,B)=\mathbf{C}(A\times B). This was the main result of Framed bicategories and monoidal fibrations. Now since Kate and I were both graduate students of Peter's at the time, it was natural to put our work together. The mass of material that we produced eventually got sorted into three papers: Traces in symmetric monoidal categories, an expository paper containing the background we wanted to assume in the other papers, plus some fun unusual examples. Shadows and traces in bicategories. Kate originally defined shadows and traces in her thesis, but here we took a more systematic category-theoretic perspective. We described a string diagram calculus for shadows, and generalized the basic properties of symmetric monoidal trace that had been axiomatized by Joyal, Street, and Verity in Traced monoidal categories. For example, we showed that if XX and YY are right dualizable and f:X→Xf:X\to X and g:Y→Yg:Y\to Y, then tr(f⊙g)=tr(g)∘tr(f)\tr(f\odot g) = \tr(g) \circ \tr(f). You might say the intent was to make bicategorical traces "category-theoretically respectable". Duality and traces for indexed monoidal categories, in which we finally combined our theses. Using another string diagram calculus, we showed that Fr(C)Fr(\mathbf{C}) has a shadow and related its bicategorical traces to symmetric monoidal traces in the C(A)\mathbf{C}(A)s. To elaborate on this last one, any X∈C(A)X\in \mathbf{C}(A) can be regarded as a 1-cell in Fr(C)Fr(\mathbf{C}) in two ways: from AA to 11 or from 11 to AA. We denote these by X^\hat{X} and Xˇ\check{X} respectively. Then X^\hat{X} has a (right) adjoint just when XX is dualizable in C(A)\mathbf{C}(A). If we think of XX as an "AA-indexed family" (X a) a∈A(X_a)_{a\in A}, or as a map X→AX\to A with fiber X aX_a over a∈Aa\in A, then this generally means just that each X aX_a is dualizable. However, the trace of tr(f^)tr(\hat{f}) contains more information than tr(f)tr(f), and sometimes strictly more. The former has domain ⟨⟨U A⟩⟩\langle\langle U_A \rangle\rangle, which is generally like the free loop space of AA, and tr(f^)tr(\hat{f}) maps a loop α\alpha to the trace of f a∘X αf_a\circ X_\alpha, where X αX_\alpha is the monodromy around α\alpha and f af_a is the action of ff over some point a∈αa\in \alpha. By contrast, the trace of ff in C(A)\mathbf{C}(A) only knows about these traces for constant loops. (Right) dualizability of Xˇ\check{X} is a stronger condition; in parametrized spectra, for I Aˇ\check{I_A} (with I AI_A the unit of C(A)\mathbf{C}(A)) it is Costenoble-Waner duality. The composing-adjunctions argument mentioned above shows that if Xˇ\check{X} is right dualizable, then (π A) !X(\pi_A)_! X is dualizable in C(1)\mathbf{C}(1). In particular, a Costenoble-Waner dualizable space is also Spanier-Whitehead dualizable. Now Kate and I showed that tr(fˇ)tr(\check{f}) also contains more information than tr((π A) !(f))tr((\pi_A)_!(f)): the latter is the composite I 1→tr(fˇ)⟨⟨A⟩⟩→I 1I_1 \xrightarrow{tr(\check{f})} \langle\langle A \rangle\rangle \to I_1. This also follows completely formally, from the basic property of bicategorical traces that I mentioned above: if you compose two dualizable 1-cells, then the trace of an induced endomorphism is the composite of the original two traces. (The map ⟨⟨A⟩⟩→I 1\langle\langle A \rangle\rangle \to I_1 is the trace of the identity map of the base change object for π A\pi_A.) In particular, this explains how the Reidemeister trace refines the Lefschetz number. As we worked on these papers, Kate and I were also trying to generalize additivity to bicategories. This was harder than we expected, mainly because triangulated categories are no good. Since their axioms are about nonunique existence, when you add more axioms like Peter's, you get "there exists an X as in axiom A, and also a Y as in axiom B, together satisfying axiom C, and also …". Peter's axioms were manageable, but the bicategorical generalization was too much for us. If we had believed triangulated categories were a "correct thing", we might have pushed through; but clearly the "correct thing" is a stable (∞,1)-category. However, we weren't really enthusiastic about using those either. This led us to derivators; which may not really be a "correct thing" either, but their structure is categorically sensible and characterizes objects by universal properties, so they are much nicer to work with than triangulated categories. The obvious place to start was to prove that symmetric monoidal derivators satisfy Peter's axioms. In May 2011 I visited Kate in Kentucky, and we spent an intense week filling blackboards with string diagrams and checking that squares were homotopy exact. I even wrote a little computer program to do the latter for us. Eventually we joined forces with Moritz Groth, who contributed (among other things) the right definition of "closed monoidal derivator". But we stayed stuck on things like Peter's axiom (TC3). Then in November 2011 we discovered a totally different approach to additivity. Consider the bicategory Prof(V)Prof(\mathbf{V}) of categories and profunctors enriched in a symmetric monoidal V\mathbf{V}. We have embeddings like (−)^\hat{(-)} and (−)ˇ\check{(-)}, but with variance: a functor X:A→VX:A\to \mathbf{V} becomes a profunctor X^:A⇸1\hat{X}:A ⇸ 1, while a functor Φ:A op→V\Phi:A^{op}\to \mathbf{V} becomes a profunctor Φˇ:1⇸A\check{\Phi}:1 ⇸ A. As before, X^\hat{X} is right dualizable when each X aX_a is dualizable, and tr(f^)tr(\hat{f}) records the traces of f a∘X αf_a\circ X_\alpha as α\alpha ranges over endomorphisms in AA. And right dualizability of Φˇ\check{\Phi} says that Φ\Phi is a weight for absolute colimits in V\mathbf{V}; thus the composing-adjoints argument implies Theorem: If X:A→VX:A\to \mathbf{V} is such that each X aX_a is dualizable, while Φ\Phi is a weight for absolute colimits, then the weighted colimit colim Φ(X)\colim^{\Phi}(X) is dualizable. I would be surprised if no one had noticed this before, but I don't recall seeing it written down. Even more interestingly, however, the "composition of traces" property now implies: Theorem: In the above situation, given f:X→Xf:X\to X, the trace of colim Φ(f)\colim^\Phi(f) is the composite I→tr(1 Φ)⟨⟨U A⟩⟩→tr(f^)II \xrightarrow{\tr(1_\Phi)} \langle\langle U_A \rangle\rangle \xrightarrow{\tr(\hat{f})} I . If V\mathbf{V} is additive and AA is finite, ⟨⟨U A⟩⟩\langle\langle U_A \rangle\rangle is a direct sum of copies of II over "conjugacy classes" of endomorphisms in AA. Thus, tr(colim Φ(f))\tr(\colim^\Phi(f)) is a linear combination of the traces of f a∘X αf_a\circ X_\alpha, with coefficients determined by Φ\Phi. So for completely formal reasons, we have a very general "linearity formula" (hence the paper titles) for traces of absolute colimits. We obtain Peter's original additivity theorem by generalizing V\mathbf{V} to be a symmetric monoidal derivator, with Φ\Phi the weight for cofibers. Absoluteness of this weight is equivalent to stability of V\mathbf{V}, and its coefficients are 11 and −1-1, yielding the original formula in a rewritten form: L(f/A)=L(f)−L(f\| A).L(f/A) = L(f) - L(f\|_A). Finally, this argument can be entirely straightforwardly generalized to bicategories, since we know how to define "categories and profunctors enriched in a bicategory". Before going on, I want to emphasize why I consider this a success story for applied category theory. We started out by looking at something that arose naturally in another branch of mathematics; in this case, the Reidemeister trace in topological fixed-point theory. Its definition looked somewhat ad hoc, but it was a generalization of something that did have a nice category-theoretic description (the Lefschetz number), so we (and here I mean Kate) trusted that it probably had one too. So we (i.e. Kate) wrote down a categorical description of the structure being used, and then abstracted away the particulars to arrive at a general definition: shadows and bicategorical traces. This general definition might have looked a bit peculiar to a category theorist, but we took it seriously and went on to study it using category-theoretic tools. We proved a coherence theorem (the string diagram calculus), ensuring that the definition was not missing any axioms. We investigated its abstract properties, not because we had any particular reason to need them at the moment, but because past experience suggested that they would eventually be necessary to know, and useful to have collected in one place. It then turned out that one of these abstract properties — the composition theorem for traces — enabled a clean and essentially completely formal proof, and generalization, of a result (additivity) that used to require long calculations and lots of commutative diagrams. It took us a while to notice this. But I dare say it would have taken much longer if we hadn't previously written down the composition theorem. That's why I say it was a success story for applying category theory seriously. In fact, there are a couple more similar success stories hiding inside this larger story. The first involves shadows on framed bicategories, which were slated for inclusion in Shadows and traces in bicategories but got omitted out of consideration for the intended readership. Such a shadow is easiest to define using the double-categorical perspective: it's a single functor whose domain is the category whose objects are all the endo-horizontal-1-cells and whose morphisms are the squares with equal horizontal sources and targets: A →X A f↓ ⇓ ↓ f B →Y B. \array{ A & \xrightarrow{X} & A \\ ^f\downarrow & \Downarrow & \downarrow^f \\ B & \xrightarrow{Y} & B. } Such a shadow can be defined on any double category, but in the framed case, a shadow on the horizontal bicategory extends uniquely to one on the framed bicategory — by the construction of twisted traces! When we first noticed it, this seemed like just a cute bit of trivia. But in the linearity paper, it turned out to be crucial in identifying the components of tr(f^)\tr(\hat{f}), which we did by applying the composition theorem again using a base change object, whose trace we identified using this characterization of framed shadows. I'll omit the details; you can find them in the paper. The point is that just as before, having previously found and studied abstractly the correct categorical structure gave us the tools we needed later on for a concrete result. The second additional success story has to do with derivator bicategories: bicategories enriched over the monoidal bicategory of derivators. We needed these to get linearity for the Reidemeister trace, which is a bicategorical trace and also requries "stable" additivity. In particular, we needed to extend Peter and Johann's bicategory to a derivator bicategory. This might have been a lot of work, except that in Framed bicategories and monoidal fibrations I had already shown that FrFr was 2-functorial. My motivation for this was pure category-theoretic principle: every construction should be a functor. But now, since a monoidal derivator is a 2-functor Cat op→MONCATCat^{op}\to MONCAT (with extra properties), we can essentially just apply the 2-functor FrFr to an "indexed monoidal derivator" to obtain a derivator bicategory. And the indexed monoidal derivator is essentially right there in Peter and Johann's book. (When we shared these papers with Peter, he remarkede "so that is what we were doing way back then!") I'll finish this long post by mentioning a story that has yet to be told, relating to the construction of Prof(V)Prof(\mathbf{V}) for a derivator V\mathbf{V}. Kate and I needed this bicategory for the linearity story, so we joined forces with Moritz Groth (who had the first idea of how to construct it) to do it in a separate paper. However, the three of us then discovered that Prof(V)Prof(\mathbf{V}) would also solve the original problem of proving that Peter's axioms hold in a stable monoidal derivator. This seemed a good way to make the bicategory paper stand on its own, so we retitled it The additivity of traces in monoidal derivators (and eventually split it in two as well). (We still don't know whether Peter's proof generalizes directly to bicategorical trace. Even using derivators, there seems to be another roadblock or two. I'd be happy to elaborate if anyone is interested; it's possible they could be circumvented with a little thought.) Now unfortunately, the objects of Prof(V)Prof(\mathbf{V}) are not actually categories enriched in V\mathbf{V}, but ordinary unenriched categories. (No one knows how to define "categories (coherently) enriched in a monoidal derivator"; it may be impossible with the current definition of derivator.) Now given a monoidal derivator V:Cat op→MONCAT\mathbf{V}:Cat^{op}\to MONCAT, the hom-category Prof(V)(A,B)Prof(\mathbf{V})(A,B) should be V(A×B op)\mathbf{V}(A\times B^{op}). This should look familiar! Indeed, a monoidal derivator is a CatCat-indexed monoidal category, and the construction of Prof(V)Prof(\mathbf{V}) is very similar to that of Fr(C)Fr(\mathbf{C}) (recall Fr(C)(A,B)=C(A×B)Fr(\mathbf{C})(A,B) = \mathbf{C}(A\times B)). However, the pushforward functors in a derivator don't satisfy the Beck-Chevalley condition for (homotopy) pullback squares, which we required for FrFr; instead, they satisfy it for comma squares, or more generally homotopy exact squares. The unit and composition in Fr(C)Fr(\mathbf{C}) and Prof(V)Prof(\mathbf{V}) also look very similar. For instance, in Fr(C)Fr(\mathbf{C}) we compose X∈C(A×B)X\in \mathbf{C}(A\times B) and Y∈C(B×C)Y\in \mathbf{C}(B\times C) by pulling them both back to A×B×B×CA\times B\times B\times C and tensoring them there, pulling back again along the diagonal to A×B×CA\times B\times C, then pushing forward to A×CA\times C. In Prof(V)Prof(\mathbf{V}), we compose X∈V(A×B op)X\in \mathbf{V}(A\times B^{op}) and Y∈V(B×C op)Y\in \mathbf{V}(B\times C^{op}) by pulling them both back to A×B op×B×C opA\times B^{op}\times B\times C^{op} and tensoring them there, pulling back again along the projection of the twisted arrow category to A×tw(B) op×C opA\times tw(B)^{op}\times C^{op}, then pushing forward to A×C opA\times C^{op}. Note that if AA, BB, and CC are groupoids, then B op≅BB^{op} \cong B and tw(B)≃Btw(B)\simeq B, and the two constructions do agree. This leads to a natural Question: Is there an abstract construction producing a (framed) bicategory from some input data, which reduces in one case to FrFr and in another case to ProfProf? If such a thing existed, maybe we could apply it to "derivators" with CatCat replaced by something else, such as the 2-category of internal categories in a topos, or a 2-category of (∞,1)-categories. The latter would include in particular the (∞,0)-categories, i.e. spaces; thus when V=Spectra\mathbf{V}=Spectra it should reproduce Peter and Johann's bicategory (c.f. also Ando-Blumberg-Gepner). In fact, Kate and I had already used a version of the linearity story with Peter and Johann's bicategory replacing Prof(V)Prof(\mathbf{V}) to prove the multiplicativity of the Lefschetz number and Reidemeister trace. Roughly, multiplicativity means that given a fibration E→BE\to B with fiber FF, and compatible endomorphisms f:E→Ef:E\to E and f¯:B→B\bar{f}:B\to B, we have L(f)=L(f\| F)⋅L(f¯)L(f) = L(f\|_F) \cdot L(\bar{f}). However, if BB is not simply-connected, then L(f\| F)L(f\|_F) can differ between fibers; thus instead of a simple product we need a sum of fiberwise traces over loops in BB — whose coefficients turn out to be none other than the Reidemeister trace of f¯\bar{f}. In other words, it is another linearity formula, with BB acting like the weight Φ\Phi and the Reidemeister trace acting like its coefficient vector. And we proved it in the same way: composing the Costenoble-Waner dualizable I Bˇ:1⇸B\check{I_B}:1⇸ B with the fiberwise dualizable E^:B⇸1\hat{E}:B⇸1 yields the ordinary space E:1⇸1E:1⇸1, and we can apply the composition-of-traces theorem. Now a fibration E→BE\to B is equivalently an (∞,1)-functor B→∞GpdB \to \infty Gpd, and the total space EE is its (homotopy) colimit. Thus, additivity and multiplicativity are really two special cases of a single theorem about absolute colimits of (∞,1)-diagrams; the only thing missing is a construction of the appropriate bicategory. Posted at July 1, 2014 4:35 AM UTC Kan Extension Seminar Talks at CT2014 — Jun 28, 2014 Enriched Indexed Categories, Again — Jun 27, 2014 Categorical Homotopy Theory — Jun 07, 2014 Codescent Objects and Coherence — Jun 02, 2014 Classifying by Generalizing: The Theory of Accessibility Relative to a Limit Doctrine — May 20, 2014 On Two-Dimensional Monad Theory — Apr 28, 2014 Finite Products Theories — Apr 20, 2014 Elementary Observations on 2-Categorical Limits — Apr 18, 2014 Re: The Linearity of Traces Bravo! And thanks for explaining the story behind how all this came to be; it's not something that goes in a paper, but certainly helps others to understand the flow of ideas. Posted by: David Roberts on July 2, 2014 4:16 PM \| Permalink \| Reply to this About the question you ask: have you considered the extension of derivators to (∞,1)(\infty,1)-categories? I don't know it this reaches the level of generality you are after, but the construction of Prof(V)Prof(\mathbf{V}) for a derivator defined on (∞,1)(\infty,1)-categories can be restricted to ∞\infty-groupoids and thus gives a theory of parametrized objects in V\mathbf{V}, generalizing May and Sigurdsson's parametrized spectra. Moreover, any derivator in the usual sense gives rise to such a thing. Indeed, given a derivator V\mathbf{V}, one can extend it to (∞,1)(\infty,1)-categories in two (equivalent) ways. One can associate to a small category CC together with a subcategory W⊂CW\subset C, the full subcategory V(C,W)\mathbf{V}(C,W) of V(C)\mathbf{V}(C) which consists of objects whose corresponding functor C→V(𝟙)C\to\mathbf{V}(\mathbb{1}) sends arrows of WW to isomorphisms. This category is equivalent to the category of cocontinuous morphisms of derivators from P(C,W)\mathbf{P}(C,W) to V\mathbf{V}, where P(C,W)\mathbf{P}(C,W) is the derivator associated to the left Bousfield localization of the projective model structure on the category of simplicial presheaves on CC by WW; in particular, it only depends on the (∞,1)(\infty,1)-category obtained by inverting the maps of WW in CC. Equivalently, one can also define directly, for any simplicial set XX, V(X)\mathbf{V}(X) as the category of cocontinuous functors from P(X)\mathbf{P}(X) to V\mathbf{V}, where P(X)\mathbf{P}(X) is the derivator associated to the contravariant model structure on SSet/XSSet/X. The assignment X↦V(X)X\mapsto\mathbf{V}(X) is a 22-functor from the (opposite of the) 22-category of (∞,1)(\infty,1)-categories (as considered by Riehl And Verity for instance) to the 22-category of categories (because X↦P(X)X\mapsto\mathbf{P}(X) has this property). Furthermore, one can see that this extension of V\mathbf{V} satifies all the axioms of the theory of derivators, replacing small categories by simplicial sets and pullbacks of small categories by homotopy pullbacks in the Joyal model structure. In this sense, there is no difference between the theory of derivators defined on CatCat or on (∞,1)-Cat(\infty,1)\text{-}Cat. Finally, let me add that you can consider the restriction of (the extension of) Prof(V)Prof(\mathbf{V}) to ∞\infty-groupoids, and that, if we stick to the language of pairs (C,W)(C,W) as above, but with C=WC=W, this essentially is the content of my paper "Locally constant functors", the last section of which being closely related to the sructure of a framed bicategory constructed functorially from any derivator, whose objects are ∞\infty-groupoids. Posted by: Denis-Charles Cisinski on July 3, 2014 1:38 PM \| Permalink \| Reply to this Indeed, I have thought about that, and about the first of the ways you mention of extending an ordinary derivator to (∞,1)(\infty,1)-categories. However, I have not yet managed to verify the axioms for the extension, or to extend the construction of Prof(V)Prof(V) to "(∞,1)(\infty,1)-derivators". Are you saying you've done both of those? Are they written down anywhere? Posted by: Mike Shulman on July 3, 2014 4:57 PM \| Permalink \| Reply to this The case of the extension to ∞\infty-groupoids is essentially done in the paper "Locally constant functors" (which you may find on my web page). I don't know a reference where the fact that the extension to (∞,1)(\infty,1)-categories satisfies all the axioms of derivators is written down explicitely, but this follows right away from results which can be found in Lurie's `Higher topos theory'. If we define V(X)\mathbf{V}(X) as the category of cocontinuous functors from P(X)\mathbf{P}(X) to V\mathbf{V}, everything follows from known properties of the functorialities on the P(X)\mathbf{P}(X)'s. Indeed, for any map of simplicial sets u:X→Yu:X\to Y, we have an adjunction in the 22-category of derivators (where 11-cells are cocontinuous morphisms of derivators) u !:P(Y)⇄P(X):u .u_{!}:\mathbf{P}(Y)\rightleftarrows\mathbf{P}(X): u^{}. Moreover, the inverse image functor u :V(Y)→V(X)u^:\mathbf{V}(Y)\to\mathbf{V}(X) is obtained by composition with u !u_!. In other words, if you already know that presheaves of ∞\infty-groupoids on (∞,1)(\infty,1)-categories satisfy the axioms of derivators (which is already in Lurie's book), we get the axioms of derivator over (∞,1)(\infty,1)-categories for an arbitrary V\mathbf{V} by pure 22-functoriality (note that we can play the game of replacing V\mathbf{V} by V op\mathbf{V}^{op} whenever we want). I did not check that one gets framed bicategories or proarrow equipements from there, but I am confident that the proofs you already know in the standard case of derivators over ordinary categories extend to this `higher level of generality'. Posted by: Denis-Charles Cisinski on July 6, 2014 10:36 PM \| Permalink \| Reply to this Can you point to where these facts are in HTT? I expect they are there, but I remember looking for them and failing to find them, specifically (Der4). Assuming they are there, this is the second way that you suggested to extend a derivator to (∞,1)(\infty,1)-categories. Does that mean you haven't found a way to make the first one work directly? There is then, as I mentioned, the additional problem of generalizing the construction of a bicategory from a derivator to have (∞,1)(\infty,1)-categories as objects. Do you know how to do that? To check Der4 for presheaves of ∞\infty-groupoids (in the language of u u^* and f !f_!), you just reproduce the proof of Prop. 1.5.9 in these notes of Maltsiniotis), then apply Prop. 4.1.2.11 in HTT to the smooth maps of the form x\X→Xx\backslash X\to X (here, for readability with respect to HTT, I refer to Lurie's notion of smoothness, which corresponds to what Joyal and Grothendieck (in the case of ordinary categories) call properness). This will give you Der4 (in the laguage of u u^ and f !f_!) for any derivator V\mathbf{V}, and thus the dual version as well (replacing V\mathbf{V} by V op\mathbf{V}^{op}). I have never tried to check the axioms of derivators using the first approach directly (the main difficulty is then to get a good notion of comma categories (or of `Grothendieck fibration with ∞\infty-groupoidal fibers') at hand in order to formulate Der4 in a nice way). Once we work with quasi-categories with the second construction, I don't see any particular difficulty to get nice bicategories. Note that we have (again by 22-functoriality) the analogue of Prop. 4.1.2.11 in HTT for any derivator, as well as nice (co)finality properties (obtained from Prop. 4.1.2.8 in HTT). We have a bicategory QcatQcat whose objects are quasi-categories (or even simplicial sets!) and whose category of morhisms from XX to YY is given by the full subcategory of simplicial sets M→X×YM\to X\times Y such that the projection p:M→Xp:M\to X is a coCartesian fibration while q:M→Yq:M\to Y is a Cartesian fibration. If DerDer denotes the 22-category of derivators (with cocontinuous functors as 11-cells), we get a bifunctor Qcat→DerQcat\to Der defined by X↦P(X)X\mapsto\mathbf{P}(X) on objects and by the functors M↦q!p M\mapsto q!p^ (here, as coCartesian fibrations are smooth (see Prop. 4.1.2.15 in HTT), we have the adequate Beck-Chevalley properties to see the compatibility conditions for composition of such things in DerDer). This bifunctor is symmetric monoidal in a suitable sense with respect to the cartesian product of simplicial sets, which follows from the fact that, given simplicial sets X 1,…,X nX_1,\ldots,X_n and a derivator V\mathbf{V}, there is a canonical equivalence of categories between morphisms of prederivators $$\mathbf{P}(X_1)\times\dots\times\mathbf{P}(X_n)\to\mathbf{V}whichpreservecolimitsineachvariablesandcocontinuousmorphismsofderivators which preserve colimits in each variables and cocontinuous morphisms of derivators P(X 1×…×X n)→V\mathbf{P}(X_1\times\dots\times X_n)\to\mathbf{V}(toseethis,wemayassumethateach (to see this, we may assume that each X_iisthelocalisationofanordinarycategory,andreducetheproblemtothecasewhereeaxh is the localisation of an ordinary category, and reduce the problem to the case where eaxh X_iis(thenerveof)asmall is (the nerve of) a small 1−category).Notethatthefunctors-category). Note that the functors Qcat(X,Y)→Der(P(X),P(Y))Qcat(X,Y)\to Der(\mathbf{P}(X),\mathbf{P}(Y))sendJoyalweakequivalencesover send Joyal weak equivalences over X\times Ytoisomorphisms,andthusinducefunctors to isomorphisms, and thus induce functors Ho(Qcat(X,Y))→Der(P(X),P(Y))=P(X op×Y)(pt)Ho(Qcat(X,Y))\to Der(\mathbf{P}(X),\mathbf{P}(Y))=\mathbf{P}(X^{op}\times Y)(pt)Thesefunctorshavefullyfaithfulrightadjointswhoseessentialimagescanbedescribedintermsoftheobjects These functors have fully faithful right adjoints whose essential images can be described in terms of the objects (p,q):M\to X\times Ywith with paleftfibrationand a left fibration and qarightfibration.Unknown character/pUnknown characterUnknown characterpUnknown characterFromthere,itlooksstraightforwardtomethat,if a right fibration. </p> <p>From there, it looks straightforward to me that, if \mathbf{V}isamonoidalderivator,thencomposingthefunctor is a monoidal derivator, then composing the functor Der(-,\mathbf{V})withthemonoidalbifunctorabovegivesabicategorywhoseobjectsarequasi−categorieswithcategoriesofmorphismsgivenby with the monoidal bifunctor above gives a bicategory whose objects are quasi-categories with categories of morphisms given by \mathbf{V}(X\times Y^{op})$ (although writing down all this with care might require some time!). Posted by: Denis-Charles Cisinski on July 8, 2014 1:06 AM \| Permalink \| Reply to this With a good dependent linear type theory in place, would it be possible to couch some/many of your ideas in its terms? Posted by: David Corfield on July 7, 2014 11:30 AM \| Permalink \| Reply to this Thanks for asking that! The theory of indexed monoidal traces, and hence the multiplicativity theorem, can indeed be described in dependent linear type theory. For additivity, however, it seems that we would need an answer to the question I asked at the end (which is another motivation for asking it). Ordinary traces in symmetric monoidal categories can be described in ordinary linear type theory. Recall (for bystanders) that in linear type theory every variable present in a context must be used exactly once. Just as ordinary "nonlinear" type theory corresponds to categories with (among other things) finite products, linear type theory corresponds to (usually symmetric) monoidal categories, in which there are no diagonals X→X×XX\to X\times X or projections X→1X\to 1; thus we cannot "duplicate" or "discard" information. In particular, instead of a cartesian product type X×YX\times Y, we have a "tensor product type" X⊗YX\otimes Y. It has the same introduction rule: given x:Xx:X and y:Yy:Y we can form (x,y):X⊗Y(x,y):X\otimes Y. But the linearity restriction means that the same variable can't occur in both xx and yy, so that in particular we don't have (x,x):X⊗X(x,x):X\otimes X. The elimination rule is best phrased in terms of pattern-matching rather than projections: given p:X⊗Yp:X\otimes Y and a term z:Zz:Z involving two variables x:Xx:X and y:Yy:Y (exactly once each!) we have a term (let(x,y)≔pinz):Z. (let\;(x,y)\coloneqq p\;in\; z) : Z. Note that pp appears only once in this expression, as required. Technically, xx and yy appear twice, once in (x,y)(x,y) and once in zz, but the first is a binding occurrence which cancels out the other, so overall the term does not contain them as free variables. Similarly, the "unit type" II has the usual introduction rule tt:Itt:I, and an elimination rule that essentially says any variable of type II can be ignored: given a variable u:Iu:I and a term z:Zz:Z not involving uu, we have a term (discarduinz):Z (discard\;u\;in\;z) : Z which does involve uu. In linear type theory, we can say that a type XX is dualizable if there is a type DXD X, a term η:X⊗DX\eta : X\otimes D X (the domain II can be left out), and a term (p:DX⊗X)⊢ϵ(p):I(p:D X\otimes X) \vdash \epsilon(p):I, such that (let(x′,ξ)≔ηin(discardϵ(ξ,x)inx′))=x (let\;(x',\xi)\coloneqq \eta\;in\;(discard\;\epsilon(\xi,x)\;in\;x')) = x for any x:Xx:X, and (let(x,ξ′)≔ηin(discardϵ(ξ,x)inξ′))=ξ (let\;(x,\xi')\coloneqq \eta\;in\;(discard\;\epsilon(\xi,x)\;in\;\xi')) = \xi for any ξ:DX\xi:D X. Then given f:X→Xf:X\to X, its trace is let(x,ξ)≔ηinϵ(ξ,f(x)). let\;(x,\xi)\coloneqq\eta\;in\;\epsilon(\xi,f(x)). Note that the order of xx and ξ\xi gets reversed between their binding and their use. This corresponds to the use of symmetry in the categorical definition of trace. One way to think of this is that when we break η\eta into xx and ξ\xi, we can consider ξ\xi to be a sort of "equals-xx predicate", which is applied by ϵ\epsilon. Thus, the trace measures the extent to which f(x)f(x) equals xx. The linear type theory I just described doesn't have any dependent types. It's tricky to say what it would mean for one linear type to be dependent on another, but if we allow both "linear" and "nonlinear" types, we can easily have the linear types dependent on the nonlinear ones (and the nonlinear ones dependent on each other). This is the sort of theory that David is asking about, and it corresponds to an indexed monoidal category, where the base category is cartesian monoidal (the nonlinear types) but the fibers are only symmetric monoidal (the linear types). One of the type constructors we can have in this type theory is the dependent sum of a family of linear types dependent on a nonlinear one. That is, if we have A:NLTypeA:NLType and (a:A)⊢X(a):LType(a:A)\vdash X(a)\;:LType, then we have ∑ (a:A)X(a):LType\sum_{(a:A)} X(a) : LType, with the usual pairing constructor (a,x):∑ (a′:A)X(a′)(a,x):\sum_{(a':A)} X(a') and a matching eliminator like the above "letlet", except that the first variable matched is nonlinear and can be used multiple times (or none). In particular, we can "linearize" a nonlinear type AA by summing up the unit linear type, writing ΣA\Sigma A for ∑ (a:A)I\sum_{(a:A)} I. In Peter and Johann's world, the nonlinear types are spaces, the linear ones are spectra, and ΣA\Sigma A is the suspension spectrum of a space. Now suppose we have (a:A),(b:B)⊢X(a,b):LType(a:A),(b:B)\vdash X(a,b)\;:LType and (b:B),(c:C)⊢Y(b,c):LType(b:B),(c:C)\vdash Y(b,c)\;:LType. Their bicategorical composite is defined by (X⊙Y)(a,c)=∑ (b:B)X(a,b)⊗Y(b,c). (X\odot Y)(a,c) = \sum_{(b:B)} X(a,b)\otimes Y(b,c). And the bicategorical unit of a nonlinear type AA is (U A)(a,a′)=Σ(a=a′) (U_A)(a,a') = \Sigma(a=a') (assuming the nonlinear type theory has an identity type). Finally, a nonlinear type family (a:A)⊢X(a):LType(a:A)\vdash X(a):LType is Costenoble-Waner dualizable if there is a family of linear types (a:A)⊢DA(a):LType(a:A)\vdash D A(a) : LType, a term η:∑ (a:A)A(a)⊗DA(a)\eta:\sum_{(a:A)} A(a) \otimes D A(a), and a term (a,a′:A),(ξ:DA(a)),(x:A)⊢ϵ(a,a′,ξ,x):Σ(a=a′)(a,a':A),(\xi:D A(a)),(x:A) \vdash \epsilon(a,a',\xi,x):\Sigma(a=a'), such that the triangle identities hold in an appropriate way. However, to deal with linearity, we need a bicategory whose objects are categories. To formulate that in a similar way to the above, we'd need the "nonlinear types" to be categories, so that their type theory would be some sort of directed type theory. But without a general context in which to perform the derivator-to-bicategory construction, it's not obvious to me exactly what this directed type theory should look like, or how it should interact with the dependent linear types. Posted by: Mike Shulman on July 7, 2014 11:49 PM \| Permalink \| Reply to this In the first paper, …derivators, which were inverted by Grothendieck… should be 'invented'. Thanks, will fix. So that example in 4.16 of the first paper is what is called the Frobenius formula here. Can you approach things like the Selberg trace formula with your resources? I don't have any ideas about how to approach the Selberg trace formula, but I suppose it might be possible.	CommonCrawl
Distribution of ratio of 2 points drawn from normal distribution? Let's say we have a known normal distribution $N(\mu,\sigma^2)$. I now draw 2 points $p1$ and $p2$ randomly from this Gaussian distribution for every observation, and repeat this process large number of times. What will the distribution of $\frac{p1}{p2}$ look like? Will it be normal? Can we say something about it's mean and standard deviation? What will the distribution of $\operatorname{max} (\frac{p1}{p2},\frac{p2}{p1})$ look like? Will it be normal? Can we say something about it's mean and standard deviation? What will the distribution of $\frac{e^p_1}{ e^p_2}$ and the distribution of $\operatorname{max} (\frac{e^p_1}{ e^p_2}, \frac{e^p_2}{e^p_1})$ look like? Will it be normal? Can we say something about it's mean and standard deviation? distributions normal-distribution mathematical-statistics standard-deviation expected-value kjetil b halvorsen vigs1990vigs1990 $\begingroup$ Just about all conceivable variations of (1) are addressed in other threads: search our site. Those answers will also reply to question (2). Question (3), presumably about $\exp(p_1)/\exp(p_2)$ and its reciprocal, is about the lognormal distribution with parameters $\mu-\mu=0$ and $\sigma^2+\sigma^2=2\sigma^2$, which you can find answered in many places. The second part of (3) may be new (albeit straightforward). I would therefore encourage respondents to focus on that. $\endgroup$ – whuber♦ Oct 22 '15 at 5:33 $\begingroup$ Gaussian Ratio distribution $\endgroup$ – Glen_b -Reinstate Monica Oct 22 '15 at 7:04 For Q 1 and Q 2: This is a gaussian ratio distribution, see https://en.wikipedia.org/wiki/Ratio_distribution#Gaussian_ratio_distribution and https://www.amazon.com/Probability-Distributions-Involving-Gaussian-Variables/dp/0387346570 and search this site. Si I will concentrate an Q3. First part is straightforward, as noted in whuber's coooent, it is lognormal with parameters $\mu=0, 2\sigma^2$. The second part is the a maximum of two related lognormals: $$ \max\left( e^{p_1-p_2}, e^{p_2-p_1} \right) $$ Note that this maximum is given by $e^{\|p_1-p_2\|}$ so we could call it a log-halfnormal distribution, in this case a half-normal (or absolute value of normal with zero-mean) based on normal with parameters $\mu=0, 2\sigma^2$ with density function $$ f(x) =\frac{2}{\sqrt{2\pi}\sqrt{2\sigma^2}}e^{-\frac12 \frac{x^2}{2\sigma^2}} = \frac1{\sqrt{\pi}\sigma} e^{-\frac14(\frac{x}{\sigma})^2}, \quad x\ge 0 $$ The density of such a half-normal variate exponentiated can be found from first principles. The result becomes $$ f_Y(y) = \frac1{\sqrt{\pi}\sigma} e^{-\frac14 (\frac{\log y}{\sigma})^2}, \quad y> 1 $$ expectation and variance can be found to be $$ 2 e^{\sigma^2} \Phi(\sqrt{2}\sigma), \\ 2e^{4\sigma^2} \Phi(2\sqrt{2}\sigma) - 4e^{2\sigma^2} \Phi(\sqrt{2}\sigma)^2 $$ (calculated with help of maple). kjetil b halvorsenkjetil b halvorsen Not the answer you're looking for? Browse other questions tagged distributions normal-distribution mathematical-statistics standard-deviation expected-value or ask your own question. Is there any paper about the distribution of difference of log-normal variable? How can I find the standard deviation of the sample standard deviation from a normal distribution? Generate distribution based on descriptive statistics Comparing two probabilities from the same normal distribution The standard normal distribution vs the t-distribution Samples from a multivariate t distribution How to calculate mean, median, mode, std dev from distribution Standard deviation of (assumed) normal distribution Proof of the standard error of the distribution between two normal distributions in A/B testing	CommonCrawl
Problems in Mathematics Problems by Topics Gauss-Jordan Elimination Inverse Matrix Linear Transformation Vector Space Eigen Value Cayley-Hamilton Theorem Diagonalization Exam Problems Abelian Group Group Homomorphism Sylow's Theorem Module Theory Ring Theory LaTex/MathJax Login/Join us Solve later Problems My Solved Problems You solved 0 problems!! Solved Problems / Solve later Problems by Yu · Published 09/18/2017 An Example of a Matrix that Cannot Be a Commutator Let $I$ be the $2\times 2$ identity matrix. Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$. Read solution Click here if solved 20 Add to solve later by Yu · Published 09/15/2017 · Last modified 01/16/2018 7 Problems on Skew-Symmetric Matrices Let $A$ and $B$ be $n\times n$ skew-symmetric matrices. Namely $A^{\trans}=-A$ and $B^{\trans}=-B$. (a) Prove that $A+B$ is skew-symmetric. (b) Prove that $cA$ is skew-symmetric for any scalar $c$. (c) Let $P$ be an $m\times n$ matrix. Prove that $P^{\trans}AP$ is skew-symmetric. (d) Suppose that $A$ is real skew-symmetric. Prove that $iA$ is an Hermitian matrix. (e) Prove that if $AB=-BA$, then $AB$ is a skew-symmetric matrix. (f) Let $\mathbf{v}$ be an $n$-dimensional column vecotor. Prove that $\mathbf{v}^{\trans}A\mathbf{v}=0$. (g) Suppose that $A$ is a real skew-symmetric matrix and $A^2\mathbf{v}=\mathbf{0}$ for some vector $\mathbf{v}\in \R^n$. Then prove that $A\mathbf{v}=\mathbf{0}$. Determine a Condition on $a, b$ so that Vectors are Linearly Dependent \[\mathbf{v}_1=\begin{bmatrix} 1 \\ \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} a \\ \end{bmatrix}\] be vectors in $\R^3$. Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent. Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular. (b) Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then: The matrix $B$ is nonsingular. The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.) A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors Let $A$ be a singular $n\times n$ matrix. \[\mathbf{e}_1=\begin{bmatrix} \vdots \\ \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} \end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix} \end{bmatrix}\] be unit vectors in $\R^n$. Prove that at least one of the following matrix equations \[A\mathbf{x}=\mathbf{e}_i\] for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$. Click here if solved 9 The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$. Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector. Then the product $A\mathbf{b}$ is an $n$-dimensional vector. Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$. Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$. Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$ For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$. (a) $A=\begin{bmatrix} 4 & 2\\ 2& 1 \end{bmatrix}$. (b) $A=\begin{bmatrix} The Transpose of a Nonsingular Matrix is Nonsingular Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular. If the Quotient is an Infinite Cyclic Group, then Exists a Normal Subgroup of Index $n$ Let $N$ be a normal subgroup of a group $G$. Suppose that $G/N$ is an infinite cyclic group. Then prove that for each positive integer $n$, there exists a normal subgroup $H$ of $G$ of index $n$. Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by \[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where $I$ is the $n\times n$ identity matrix. Prove that $A$ is a symmetric matrix and $AA=I$. Conclude that the inverse matrix is $A^{-1}=A$. The Range and Null Space of the Zero Transformation of Vector Spaces Let $U$ and $V$ be vector spaces over a scalar field $\F$. Define the map $T:U\to V$ by $T(\mathbf{u})=\mathbf{0}_V$ for each vector $\mathbf{u}\in U$. (a) Prove that $T:U\to V$ is a linear transformation. (Hence, $T$ is called the zero transformation.) (b) Determine the null space $\calN(T)$ and the range $\calR(T)$ of $T$. If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial Let $x, y$ be generators of a group $G$ with relation xy^2=y^3x,\tag{1}\\ yx^2=x^3y.\tag{2} Prove that $G$ is the trivial group. Find the Inverse Linear Transformation if the Linear Transformation is an Isomorphism Let $T:\R^3 \to \R^3$ be the linear transformation defined by the formula \[T\left(\, \begin{bmatrix} x_1 \\ x_3 \end{bmatrix} \,\right)=\begin{bmatrix} x_1+3x_2-2x_3 \\ 2x_1+3x_2 \\ x_2-x_3 \end{bmatrix}.\] Determine whether $T$ is an isomorphism and if so find the formula for the inverse linear transformation $T^{-1}$. Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A\|I]$, where $I$ is the $3\times 3$ identity matrix. 1 & 3 & -2 \\ 2 &3 &0 \\ 0 & 1 & -1 \end{bmatrix}$ 1 & 0 & 2 \\ -1 &-3 &2 \\ Click here if solved 112 The Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos ^2\theta_1+\cdots+\cos^2 \theta_n=1.\] Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors Consider the $2\times 2$ matrix \[A=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix},\] where $\theta$ is a real number $0\leq \theta < 2\pi$. (a) Find the characteristic polynomial of the matrix $A$. (b) Find the eigenvalues of the matrix $A$. (c) Determine the eigenvectors corresponding to each of the eigenvalues of $A$. Using the Wronskian for Exponential Functions, Determine Whether the Set is Linearly Independent By calculating the Wronskian, determine whether the set of exponential functions \[\{e^x, e^{2x}, e^{3x}\}\] is linearly independent on the interval $[-1, 1]$. A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$ An $n\times n$ matrix $A$ is said to be invertible if there exists an $n\times n$ matrix $B$ such that $AB=I$, and $BA=I$, where $I$ is the $n\times n$ identity matrix. If such a matrix $B$ exists, then it is known to be unique and called the inverse matrix of $A$, denoted by $A^{-1}$. In this problem, we prove that if $B$ satisfies the first condition, then it automatically satisfies the second condition. So if we know $AB=I$, then we can conclude that $B=A^{-1}$. Let $A$ and $B$ be $n\times n$ matrices. Suppose that we have $AB=I$, where $I$ is the $n \times n$ identity matrix. Prove that $BA=I$, and hence $A^{-1}=B$. Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$. Find Inverse Matrices Using Adjoint Matrices Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$. The matrix $\Adj(A)$ is called the adjoint matrix of $A$. When $A$ is invertible, then its inverse can be obtained by the formula \[A^{-1}=\frac{1}{\det(A)}\Adj(A).\] For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula. 0 &-1 &2 \\ 0 & 0 & 1 (b) $B=\begin{bmatrix} Page 10 of 38« First«...7891011121314...2030...»Last » This website's goal is to encourage people to enjoy Mathematics! This website is no longer maintained by Yu. ST is the new administrator. 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Determine Whether Each Set is a Basis for $\R^3$ Prove Vector Space Properties Using Vector Space Axioms Express a Vector as a Linear Combination of Other Vectors How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix Matrix of Linear Transformation with respect to a Basis Consisting of Eigenvectors Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis 12 Examples of Subsets that Are Not Subspaces of Vector Spaces The Intersection of Two Subspaces is also a Subspace Site Map & Index abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam field theory finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear algebra linear combination linearly independent linear transformation matrix matrix representation nonsingular matrix normal subgroup null space Ohio State Ohio State.LA rank ring ring theory subgroup subspace symmetric matrix system of linear equations transpose vector vector space Search More Problems Membership Level Free If you are a member, Login here. 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Journal of Wood Science Official Journal of the Japan Wood Research Society Analysis of drying stresses in green-glued plywood of Bete (Mansonia Altissima) specie Valentin Makomra ORCID: orcid.org/0000-0002-0215-36981,3, René Oum Lissouck2,3, Régis Pommier1, Anh Phan Ngoc1, Denys Breysse1, Louis Denaud4 & Louis Max Ayina Ohandja2 Journal of Wood Science volume 66, Article number: 70 (2020) Cite this article The wood veneer dried artificially before gluing for the manufacture of plywood generally suffers several damages (cracks, deformations), decreasing the competitiveness of the process and limiting the use of nervous species of wood. The proposed solution is to glue them at the green state. However, during the drying process, superficial and internal cracks are observed. These cracks appear when stresses due to the anisotropic shrinkage of the plywood exceed the mechanical strength of the wood material. In the present study, the response of a 3-ply plywood of Bete (Mansonia altissima) glued at the green state is simulated in order to evaluate these stresses and deformations during drying using a finite element method program. The numerical results and associated experimental data make it possible to better analyze and understand the mechanical behavior of the plywood during drying in order to look for reliable plywood through new process. Plywood factories have been using wood veneer obtained by unrolling some species for a long time. The artificial drying of these wood veneers, the only industrial process, leads to a certain number of incidents that may occur during the process. Some defects are visible (superficial cracks and deformations) while others may be non-visible (internal cracks). Nevertheless, they are susceptible to induce future failure [1]. To provide solutions, several works have developed products glued at the green state [2, 3], and in particular plywood panels using different technologies [4,5,6,7]. At the industrial level, this technique may allow some energy savings and reduce the degradation of the material integrity of veneers during handling. Lavalette [8] has shown that the product from the vacuum drying of green-glued plywood processing, meets the standard requirements for shear stress according to EN 314–1 [9]. But, she showed some superficial cracks probably due to the stresses induced by the gradient of moisture and anisotropic shrinkage. To forecast these internal stresses and strains, which is a prerequisite to design the products, it is difficult to build an experimental design without the use of numerical simulations. The numerical model of the hydro-mechanical behavior during the drying of solid wood products has been the subject of several studies on sawn timber products [10, 11] and crossed lamellae panels [12]. These questions are still to be answered for plywood, especially as the thickness of the wood veneer and the method of their production are different from those of the products mentioned above. Thus, finite element simulations can be performed to study how the internal structure and properties of the material affect the plywood shape stability. The objective of the study is to determine the drying stresses in green-glued plywood by FEM analysis (Cast3M Finite Element Software). To perform an accurate simulation, it is essential to have a sufficiently detailed description of the properties of the wood. The mechano-sorption, the elastic, visco-elastic behaviors are implemented in Cast3M finite element software during drying. For this purpose, experimental drying tests of plywood consisting of three plies of size 100 × 100 × 6 mm3 (Fig. 1) permit to follow their deformation. The hydro-mechanical test setup for the veneers have been carried out and the results used in the numerical drying tests. They have made it possible to reproduce the experimental behavior (deformations, drying kinetics, etc.) and to provide estimates of the internal stresses in plywood. Geometry and mesh of the plywood sample Methods and experiments Numerical model Several models of wood behaviors were offered during those last years. The models of stress/deformation were developed, and most of them assumed that mechanical properties change during the drying. They provided a better knowledge of the mechanical behavior and may aid to improve the quality of the wood during drying [13, 14]. The rheological model implemented in the numerical approach is briefly described below. The constitutive equation given by Eq. (1) expresses the total strain $\Delta \varepsilon$ as a sum of three separate strains, describing the fundamental behaviors of wood. It is illustrated by the Maxwell figure of r branches (Fig. 2). Illustration of the rheological Maxwell model for a direction i of the wood. With i ∈ {L, T, R}, the orthotropic direction of wood; ${E}_{i}^{\mu }$, the elastic modulus of the branch μ ∈ {1, 2, …, r} along the direction i; ηi, the viscosity in direction i; ${E}_{i}^{0}$, the real modulus of wood in direction i; ${E}^{\varnothing}_{i}$, the mechano-sorptive modulus in the direction i $$\Delta \varepsilon = \Delta {\varepsilon _{eve}} + \Delta {\varepsilon _{ms}} + \Delta {\varepsilon _w},$$ where ${\Delta \varepsilon}_{\text{w}}$ is the free shrinkage/swelling that occurs when the water content is below the fiber saturation point (Wfsp). It is defined as: $$\Delta {\varepsilon _w} = \alpha \Delta w,$$ with α as the shrinkage/swelling coefficient (independent of moisture) and $\Delta w$ is the variation of moisture content; ${\Delta \varepsilon}_{\text{eve}}$ is the elastic and visco-elastic strain defined by using the chain model of Maxwell as follows: for $$\Delta t,\Delta {\varepsilon _{eve}} = {\rm{ }}{\left[ {\tilde K} \right]^{ - 1}}\Delta \sigma - {\left[ {\tilde K} \right]^{ - 1}}{\sigma^{his}}\left( t \right),$$ with $\tilde K = {E^0}({\rm{ }}w)\left[ {1 + \sum\limits_{\mu = 1}^r {{\gamma _\mu }} {\mkern 1mu} \left( {\frac{{1 - {e^{ - {\alpha _\mu }\Delta t}}}}{{{\alpha _\mu }\Delta t}}} \right)} \right],$ where ${\gamma _\mu } = \frac{{{E^0}(w)}}{{{E^\mu }}}{\rm{ }}and{\rm{ }}{\alpha _\mu } = \frac{{{E^\mu }}}{{{\eta ^\mu }}}(unit {\rm{ }} in {\rm{ }}{s^{ - 1}})$, and ${\sigma ^{his}}\left( t \right) = \sum\limits_{\mu = 1}^r {\left( {1 - {e^{ - {\kern 1pt} {\alpha _\mu }\Delta t}}} \right)} {\mkern 1mu} {\sigma ^\mu }(t)$, and $\stackrel{\sim }{K}$ is the fictious rigidity. It depends on the length of the time step, the parameters of Maxwell; ${E}^{0}(w)$ is the elastic modulus at the beginning of the increment. It depends on moisture content w; ${E}^{\mu } and {\eta}^{\mu }$ are, respectively, the elastic modulus and the viscosity of the branch ${\mu }$. ${\sigma }^{his}\left(t\right)$ is the term of the history. Its depends on the length of the step time Δt, the state of the deformation ε(t) and the recent values gained by the internal stress ${\sigma }^{\mu }(t)$ at the beginning of the increment. ${\Delta \varepsilon}_{\text{ms}}$ is the mechano-sorptive deformation, and depends linearly on the stress and the variation of the water content. This definition leads to the formulation of the following expression of deformation which does not depend on the previous ones [10]: $$\Delta {\varepsilon_{ms}} = m{\rm{ }}\sigma{\mkern 1mu}\Delta w$$ where m is the compliance of mechano-sorptive creep. From Eq. (1), the total formulation then can be written: $$\Delta \varepsilon = \alpha \Delta w + {[\tilde K]^{ - 1}}\Delta \sigma - {[\tilde K]^{ - 1}}{\sigma ^{his}}\left( t \right){\rm{ }} + m{\rm{ }}\sigma {\mkern 1mu} \Delta w$$ Identification and experimental determination of input parameters of the model In order to validate the numerical simulation by experiments, we choose the Bete specie as a reference. It permits to describe and identify the input parameters of the model like the isothermal desorption, coefficients of diffusion and exchange, the test boundary condition and material data. Description of Bete specie Bete (Mansonia altissima) is one of the most abundant hardwood species from the Congo Basin forest [15]. The color of the wood is brown and the grain is straight. The wood can be used in frame, parquet and paneling. According to the CIRAD (Agricultural Research Centre for International Development, in French) technological database [16], the density at 12% moisture content varies from 0.59 to 0.72. The average elastic modulus in the longitudinal direction (MOE) is 13,600 MPa, with a standard deviation of 1124 MPa. The mean value of the modulus of rupture in bending (MOR) is 110 MPa, with a standard deviation of 10 MPa. The compressive rupture stress is 60 MPa, with a variation coefficient of 10%. The average Wfsp value is 28% [15]. The tangential and radial shrinkage coefficients vary, respectively, from 0.241% to 0.286% and from 0.15% to 0.178%. Determination of the isothermal desorption of Bete specie Ten samples in green state (saturated) of size 20 × 20 × 2 mm3 were placed in a climatic chamber at constant temperature of 40 °C. Then the relative humidity (RH) inside the device was varied from 95 to 10%. For each RH value, the samples remained in the oven until their masses stabilized. A given mass was considered as stabilized in this study when its relative variation was equal or less than 0,001. Knowing the saturated and anhydrous mass values (obtained at 105 °C), we deduced the water content Wc, and the couple (RH, Wc) provided each point of the isotherm desorption curve. Determination of the shrinkage coefficient of Bete specie The full shrinkage radial and tangential of Bete specie were determined experimentally by using 5 samples in green state (saturated) of size 20 × 20 × 2 mm3. We assumed that the coefficients are constants. They were placed in an oven at constant temperature of 40 °C and humidity of 90% until their mass stabilized in order to have the Wfsp value. The tangential and radial dimensions were measured. The same sizes were also measured for the same specimens in anhydrous state. From those dimensions, we deduced the radial and tangential shrinkage coefficients. Determination of the density of the Bete species To determine the density, the same specimens for the determination of shrinkage coefficients were used. We assume that the longitudinal shrinkage of the specimen is neglected. Knowing the dimension and the mass, we deduced the density ρw at 0% of Wc and by the relation ρw(W%) = ρw(0%) (1 + W/100) [22], the density versus the Wc. Determination of elastic properties of the Bete species For the determination of elastic properties of Bete plywood, the tensile tests were carried out on specimen obtained after unrolling and conditioning at a temperature of 40 °C and a relative humidity of 70% (which corresponds to an average moisture content of about 12% at the hygroscopic equilibrium). The tensile tests are inspired by the standard NF EN 326–1 [17]. However, a different specimen geometry was used because preliminary tests showed that the rupture occurred in the jaws of the standard specimen. More precisely, we used a "dumbbell" specimen [8], reinforced by wooden heels at the jaws, in order to ensure the rupture in the useful part of the sample. It was cut, respectively, in the parallel and perpendicular directions of the fibers in order to obtain the elasticity modulus in those two principal directions. Fabrication of 3-ply plywood About 2-mm-thick Mansonia altissima rotary cut veneer (from the production forest of Cameroon) was used for plywood manufacture. Wood plies were selected defect free, with a regular slope of grain in order to avoid their effect on test results. They were cut to the panel dimensions (600 × 600 × 2 mm3) and initially stored in a conditioning chamber at 4 °C to keep their moisture content beyond the Wfsp. Then the wood plies were taken off the chamber and glued. The adhesive used was a one-component polyurethane (ref: Collano RP 2554) with a viscosity of 1000 mPa/s at 20 °C, developed from the adhesive patented for green plywood gluing [18]. It was spread on the plies by using a notched squeegee so that the glue was evenly distributed. 3-ply plywood panels were manufactured by using the vacuum process technique [7, 8]. The plies were oriented according to two arrangements options. The first one was antisymmetric ($-$ 15°/0°/ + 15°). It was considered in order to validate the numerical model of drying plywood. The second one was symmetric (0°/90°/0°) which represents the conventional plywood. The panels were placed in a vacuum dryer set to 150 mbar to ensure the bonding and equipped with a device (microprocessor) indicating their average moisture content (about 55%). The average moisture content of the panels when they were removed from the drier was 35%. The panels were cut according to the dimensions of the samples (section of 100 × 100 mm2) and placed in an oven where the relative humidity and the temperature were, respectively, 95% and 10 °C. Thus, their moisture content could be close to the Wfsp. We remind that the Wfsp average value, according to the literature is 28%. Determination of coefficients of diffusion and exchange The relationship between the diffusion coefficient KD and the water content in the wood is far from being definitively established. The KD values found in the literature depend on wood species and widely vary. The transfer of water in the wood is hindered by two resistances: an internal one which can be described by the diffusion coefficient KD, and a second one which is developed at the interface between the specimen and the external environment, which can be described by the surface exchange coefficient KC [19,20,21]. The boundary conditions associated with the mass diffusion equation are literally written in the unidirectional framework by the following system of equations as (Eq. 6): $$\left\{ {\begin{array}{*{20}{l}} {\frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial z}}\left( {{K_D}(w)\frac{{\partial w}}{{\partial z}}} \right)}&{with\;0 < z < a}\\ {w = {w_{ini}}}&{for\;z \in [0,{\rm{ }}a]{\rm{ }}\;at\;t = 0}\\ {{K_D}(w)\frac{{\partial w}}{{\partial z}} = {K_C}\left( {{w_{surf}} - {w_{eq}}} \right) = {Q_m};}&{\left( {z = a,\;t > 0} \right)} \end{array}} \right.$$ with KD as the coefficient of diffusion of the material; W, the moisture content in the material; Wsurf, the moisture content on the surface of the material; Wini, the initial moisture content in the material (at the beginning of drying); Weq, the equivalent moisture content to the relative humidity of the study environment; KC, the exchange coefficient between the material and the ambient environment; a, the specimen thickness; t, the time; and Qm, the flux of surface water content. For experimental determination, 10 green-glued plywood specimens were placed in a climatic chamber in which the air relative humidity and the temperature were, respectively, 50% and 40 °C. In such conditions, the moisture content at the hygroscopic equilibrium was 10%. These parameters correspond to a desorption of the specimens. The four lateral faces (RT and RL planes) of each sample were insulated with an EPI (emulsion polymer isocyanate) adhesive of a Kleiberit brand in order to impose the diffusion in the radial direction of the sample (Fig. 3). Sample of sealing specimen At each hour, the samples were weighed. The anhydrous masses were obtained after setting the samples in the oven at 105° C till their mass stopped varying. Then, we deduced the evolution of the moisture content with the time in order to determine the diffusion and exchange coefficients. Validation of the numerical model In order to validate the numerical results, the configuration of plywood chosen was arranged with asymmetrical ring orientations (longitudinal direction). The first and third plies are inclined by ± 15° about the direction of the fiber as indicated in Fig. 4. The process was conducted by following to steps. Configuration of plywood specimen (plies arrangement -15°/0°/ + 15°) In the first step, five green-glued plywood specimens with an initial average moisture content of approximately 32% were placed in a drying oven with the diffusion and theoretical mechanical boundary conditions mentioned in Fig. 5. These mechanical boundary conditions corresponded to simple supports experimentally. During the experiment, the specimens were placed on a grilling of the dryer, without holding, allowing them to perform free flexural deformation and to avoid any generation of external stress. The displacements Uz of the point P1 of the plywood (Fig. 5) of each sample were measured each hour using a digital caliper. Boundary conditions on the plywood In the second step, the fitting between experimental and numerical values was realized by minimizing the quadratic gap between the numerical and experimental displacements of the point P1 according to two scenarios. The first one did not take into account the visco-elastic behavior of the plywood. In the second one, that behavior was considered. The material parameters used for the numerical simulation are listed in Table 1 and are representative of the wood veneers of the Bete species. The following assumptions were considered: The influence of the moisture content W (%) on the elastic properties specified in Table 1 and the density of the woods was taken into account by using linear corrections given by Guitard [22] on the modulus of elasticity (Eq. 7) and density (Eq. 8): $${\text{S}}_{\text{ij}}^{-1}\left({\text{W}}\right)\text{=}{ \, {\text{S}}}_{\text{ij}}^{-1}\left(\text{12\%}\right)\left[{1}- \text{ } {\text{C}}_{\text{ij}}\text{(W-}{12}\text{)}\right]$$ with Cij constant coefficients and ${\text{S}}_{\text{ij}}^{-1}\left({\text{W}}\right)$ the elastic properties. $${\rho }_{\mathrm{W }\left(\mathrm{W \%}\right)}={\rho }_{\mathrm{W }\left(0\mathrm{ \%}\right)}\left(1+W/100\right)$$ ${\text{S}}_{\text{ij}}^{-1}\left({\text{W}}\right)$ are defined as follows (Eq. 9): $$S_{11}^{ - 1} = {E_R};{\rm{ }}S_{22}^{ - 1} = {E_T};{\rm{ }}S_{33}^{ - 1} ={E_L};{\rm{ }}S_{44}^{ - 1} = {G_{LT}};{\rm{ }}S_{55}^{ - 1} = {G_{LR}};{\rm{ }}S_{66}^{ - 1} = {G_{RT}}$$ The shrinkage coefficients were constant during drying. We assumed that the viscosity parameters of the Maxwell model with three branches (n = 3) are the same for the direction of the plywood. The validated numerical model permits to quantify and predict the fields of deformations and stresses according to two configurations: the first one was asymmetrical and the second was conventional (Fig. 6). The main interest of the asymmetrical configuration was the comparison of experimental and numerical results. In the second configuration, veneers are arranged orthogonally (Fig. 7) and the main purpose was the comparison of drying stresses and the ultimate strength of the veneers. Configuration of plywood Isothermal desorption of Bete Isothermal desorption of the Bete Figure 7 presents the plot of the equilibrium moisture content versus the relative humidity of the Bete species. Experimental results were fitted according to the Henderson model [24] (Eq. 10): $$HR=1-\mathrm{exp}\left[-A\left(T+B\right){\left({W}_{eq}\right)}^{C}\right] .$$ The estimated values of A, B and C at the end of the fitting process were, respectively, 0.14, 100 and 1.9. T is the temperature in Kelvin degree. The Wfsp value, corresponding to the limit of non-hygroscopic domain, was 32% (Fig. 7). Shrinkage coefficient of Bete specie The experimental result of tangential and radial shrinkage is given in Table 2. We notice that the value of the tangential shrinkage is bigger than the one found in the literature. On the other hand, we have almost the same result of radial shrinkage as in the literature. Volumic mass and density of specie of Bete The average density value obtained experimentally in anhydrous conditions is 0.59, with a standard deviation of 0.05. By using Eq. 8, the value at 12% practically showed no variation (from 0.655 to 0.666). Such results correspond to the literature mean value of 0.66 (volumic mass of 660 kg/m3). Elastic properties of plywood The results of tensile tests of the Bete plies are shown in Table 3. Table 1 Material parameters used The results show that the elastic modulus and the failure strength of the samples in the longitudinal direction are close to those of the solid wood obtained from the reference [16]. In the direction perpendicular to the fibers, the average elastic modulus is much lower than the module found in the literature. This can be explained by the presence of peeling slots. Diffusion and exchange coefficients Figure 8 shows an exponential decrease of the moisture content with the drying time. It was therefore possible to determine the expression of diffusion coefficient KD as an exponential function of the moisture content. Similar trends are available in the literature [19,20,21]. By fitting the experimental curve of the evolution of the moisture content (Fig. 8), we found the analytical expression of the diffusion coefficient (Eq. 11). We remind that the relative humidity and the temperature conditions are, respectively, 50% and 40 °C: Evolution of the moisture content during drying $$K_{D} = K_{0} \exp (K_{0w}w),$$ with K0, as the constant diffusion (5,84 10–10 m2/s); K0w, as the constant coefficient (1,97), and w, the moisture content (%) (Table 4). Table 2 Result of tangential and radial shrinkage of Bete The value of the coefficient KC and the expression of KD are presented in Table 4. In the literature, several authors conducted investigations concerning the mass transfer properties of wood [25, 26]. Concerning central Africa tropical woods, some results on species like sapelli (Entandrophragma cylindricum) and sipo (Entandrophragma utile) are available [27,28,29]. Such timber species present similar technological properties with Bete [30]. For instance, their mean density varies from 0.62 to 0.66. Explicit results concerning the diffusion coefficients were determined by Nsouandelé [29]. At a temperature of 40 °C and a RH of 85%, the mean values of diffusion coefficients of sapelli and sipo, in the radial direction, are, respectively, 6,45 10–12 m2/s and 3,82 10−12m2/s when the moisture content varies from 22 to 27%. The thickness of the samples (solid wood) was 21 mm, for a length of 100 mm and a width of 30 mm. Therefore, significant comparisons between the diffusion coefficients of green-glued plywood and solid wood may not be easily established because the green-glued plywood is a highly heterogeneous material, compared to the solid wood. Therefore, diffusion coefficients determined in this study should be considered as reference values for further studies concerning plywood. Experimental results and numerical validation of the model Concerning the experimental results, Fig. 9 presents the shape of the asymmetrical green-glued plywood after 50 h of drying. Visible and considerable displacements of the plywood shape were observed. Displacement of P1 obtained experimentally after 50 h The displacement Uz of the point P1 of the plywood (Fig. 9) of each sample was measured at each hour using a digital caliper and is shown in Fig. 10. Evolution of the displacement of the point P1 of the plywood Concerning the numerical validation of these results, we remind that two scenarios were considered. The first one, we considered just the elastic and the mechano-sorptive model without taking into account the visco-elastic behavior of the plywood. In the second one, that behavior was considered. Scenario 1: the visco-elastic behavior of the plywood is not considered Simulations were carried out. Only the change in tangential shrinkage coefficient greatly influences the displacement of the plywood sample. Variation of the tangential shrinkage coefficient (αT), starting from the data obtained in the literature for the specie Bete [15] and the experimental result, provides the evolution of the displacement at the point P1 of the plywood presented in Fig. 11 for four cases of tangential shrinkage coefficient values retained (αT1 = 0.246, αT2 = 0.339, αT3 = 0.634, αT4 = 0.780). Evolution of the displacement at the point P1 of the plywood according to the variation of the tangential shrinkage (scenario 1) During the first 5 h of drying, the displacement as a function of time is practically linear and in this range, all the coefficients of tangential shrinkage used predict well the elastic behavior of drying of the plywood. The likelihood of this scenario is limited by its ability to predict displacements in the post-elastic phase. The value of the tangential shrinkage coefficient of the veneer is retained by minimizing the quadratic criterion between the experimental and numerical results of the first 10 h, assuming that at this time the visco-elastic effect is negligible. This scenario's value (αT3 = 0.634) differs considerably from that of solid wood. This can be justified by the presence of cracks resulting from the unwinding operation. It is therefore chosen for the rest of the numerical model. Scenario 2: the visco-elastic behavior of the plywood is considered The three-branched Maxwell model was considered and the parameters of viscosity ${\gamma }_{1}^{i}$ and ${\left({\alpha }_{1}^{i}\right)}^{-1}$ case studied and retained are presented in Table 5. Table 3 Results of tensile tests (of wood plies) longitudinal and perpendicular to fibers Table 4 The data used in the diffusion model Table 5 Evolution and selection of the visco-elastic parameters The displacements illustrating the evolution of the plywood at the point P1 according to the variation of the viscosity parameters are shown in Fig. 12. We remind that the various displacements at the point P1 were measured on asymmetrical plywood samples. Evolution of the displacement at the point P1 of the plywood according to the variation of viscosity parameters (scenario 2) The visco-elastic model parameters chosen in this scenario are those corresponding to the curve Uz_Numerical_αT3 _v2 obtained by minimizing the quadratic criterion between the experimental results to those obtained numerically. The numerical displacement field on asymmetrical samples after 50 h of drying is shown in Fig. 13. Displacement field of plywood obtained numerically after 50 h of drying (scenario 2) Since the scenario 2 presented a more interesting ability to predict displacements in the post-elastic phase, it has been considered in the next parts of the study. The validation of the model attached to that scenario will subsequently permit to determine the deformations and the stress in the conventional plywood configuration during drying. Experimental and numerical results of the conventional configuration The conventional configuration corresponded to a plies' arrangement of 0°/90°/0°. Experimental result The experimental results in the same condition of drying shows that there was some surface cracking at the both sides of the plywood (Fig. 14). Experimental results of the faces of plywood Numerical results of stress fields and evolution of the stress at a plywood surface point The numerical results of the stress fields presented in Fig. 15 show that: along the X direction (aligned with the longitudinal direction of wood on the outer plies), the maximum surface stress of the plywood is in compression, corresponding to a value of -4.3 MPa; the inner part of the plywood is in tension with a maximum value of 6 MPa. along the Y direction (the transverse direction of the outer plies), the surface in under tension with a maximum value of 4.4 MPa; the inner part of the plywood is under compression with a maximum stress value of -8.6 MPa. Tensile stress in Y direction which is due to the orientation of the central ply (longitudinal in Y direction) that prevents the deformation in y direction (transversal) of the external plies, and put them in tension. Stress fields obtained numerically after 50 h In order to understand the behavior of the stress at the surface of the plywood, the stress evolution curve of a point at the center of the plywood is plotted in the longitudinal and tangential directions. The results are illustrated by Fig. 16. The figure shows that, after five hours of drying, any stress at the surface of the plywood almost reaches its maximum value. Evolution of stresses in X and in Y at a surface point of plywood The numerical model provides valuable information concerning the importance of an accurate description of the green-glued plywood during drying, regarding stress, deformation and the drying kinetics. In the conventional configuration where the plies are orthogonally arranged, the numerical results show that the stresses of the plywood, in the direction tangential to the fibers at the surface at 12% moisture content (4.4 MPa) are greater than the average rupture stress obtained experimentally (4.05 MPa). The accuracy of these results should make possible for plywood manufacturers to correctly plan drying programs before they are actually put into practice. In other words, the model can aid in predicting the quality (levels of shape stability and material integrity) of green-glued plywood before starting a drying program. Concerning the drying kinetics (the evolution of the stress during drying in Fig. 16), we noted that the threshold of the maximum stress is practically reached after the first 5 h. In other words, from 32 to 25% of water content, the stresses in the plywood are practically at their maximum. As a result, it is therefore possible for plywood manufacturers to quantify correctly the energy savings for two drying situations. In the first one, the automatic drying program is managed until one reaches the desired moisture content. In the second one, the automatic drying program is interrupted and substituted by a natural drying of the products. In addition, some early stage measures concerning the drying program could therefore be taken if one wants to reduce the drying stress. In this paper, the behavior of green-glued plywood during the drying process was simulated. A constant relative humidity (50%) and temperature (40 °C) were considered during drying. Small plywood specimens, manufactured by using Bete (Mansonia altissima), an abundant species from the Congo Basin, were used. The first step of the study focused on the experimental determination of some important material properties of the plywood and veneers, namely diffusion and exchange coefficients, tensile strength and modulus of elasticity, shrinkage coefficients and density. In the second step, the drying behavior of the plywood was quantified thanks to FEM model in which elastic, visco-elastic and mechano-sorptive aspects of the wood veneer were considered. The surface cracks in the plywood during drying are justified by the fact that the drying stress is higher than the tensile ultimate strength of the veneer. In addition, the stress at the surface of plywood reaches practically its maximum value during the first 5 h of drying. The simulation yields information about unfavorable deformations and stresses during the drying process. Such information presents an interesting potential to be valuable in plywood factories, especially the quality control and the planning program of the green-glued plywood. Further investigations will be needed in order to improve the FEM model, in the perspective to be applied on practically sized plywood according to the length and width. 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Tremblay C, Cloutier A, Fortin Y (2000) Experimental determination of the convective heat and mass transfer coefficients for wood drying. Wood Sci Techno 34(3):253–276 Perré P, Agoua E (2010) Mass transfer on wood: identification of structural parameters from diffusivity and permeability measurements. J Por Med 13(11):1017–1024 Simo-Tagne M, Rémond R, Rogaune Y, Zoulalian A, Bonoma B (2016) Sorption behavior of four tropical woods using a dynamic vapor standard analysis system. Maderas : Ciencia y Tecnologia 18(3):2006. https://doi.org/10.4067/S0718-221X2016005000036 Nsouandele JL, Tamba JG, Bonoma B (2010) Desorption isotherms of heavy (AZOBE, EBONY) and light heavyweight tropical woods (IROKO, SAPELLI) of Cameroon. Heat Mass Transf 54:3089–3096. https://doi.org/10.1007/s00231-018-2350-2 Simo Tagne M (2019) Modeling, numerical simulation and validation of a convective dryer in steady conditions: case study of tropical woods. Int J Modelling Simulation. https://doi.org/10.1080/02286203.2019.1575111 Oum Lissouck R, Pommier R, Breysse D, Ayina Ohandja LM, Mansié DA, R, (2016) Clustering for preservation of endangered timber species from the Congo Basin. J of Trop For Sci 28:4–20 Mukudai J, Yata S (1986) Modeling and simulation of viscoelastic behavior (tensile strain) of wood under moisture change. Wood Sci Technol 20:335–348 All sources of funding for the research are from the University of Bordeaux (Grant No. +237677321544). Univ. Bordeaux, I2M, UMR 5295, 351 cours de la Libération, 33400, Talence, France Valentin Makomra, Régis Pommier, Anh Phan Ngoc & Denys Breysse The University Institute of Wood Technology, The University of Yaounde I, P.O. Box 306, Mbalmayo, Cameroon René Oum Lissouck & Louis Max Ayina Ohandja Laboratory of Civil Engineering and Mechanics, National Advanced School of Engineering, The University of Yaounde I, P.O. Box 8392, Yaounde, Cameroon Valentin Makomra & René Oum Lissouck LaBoMaP, Laboratory of Material and Process of Bourgogne, ENSAM Cluny, 1 rue Porte de Paris, 71250, Cluny, France Louis Denaud Valentin Makomra René Oum Lissouck Régis Pommier Anh Phan Ngoc Denys Breysse Louis Max Ayina Ohandja MV, PR and LD had elaborated the experimental drying of plywood. MV, PR, PA, OR, and BD had simulated numerically the plywood glued at the green state. MV, PR and OR and DB analyzed and interpreted the drying curve. All authors read and approved the final manuscript. Correspondence to Valentin Makomra. 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/. Makomra, V., Oum Lissouck, R., Pommier, R. et al. Analysis of drying stresses in green-glued plywood of Bete (Mansonia Altissima) specie. J Wood Sci 66, 70 (2020). https://doi.org/10.1186/s10086-020-01911-1 Green-glued plywood Drying stresses Finite element	CommonCrawl
The base-10 numbers 217 and 45 are multiplied. The product is then written in base-6. What is the units digit of the base-6 representation? The units digit of a positive integer when expressed in base 6 is the same as the remainder when the integer is divided by 6. For example, the number $1502_6$ is equal to $1\cdot 6^3+5\cdot 6^2+0\cdot 6+2$, and 6 divides every term except the units digit, 2. When 217 is divided by 6, the remainder is 1. When 45 is divided by 6, the remainder is 3. Therefore, the product of 217 and 45 has a remainder of $1\cdot 3=\boxed{3}$ when divided by 6.	Math Dataset
\begin{document} \title[Second order analysis of partially-affine control problems]{ Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems} \footnotetext{This article has been accepted for publication in Discrete Contin. Dyn. Syst. Ser. S.} \author[M.S. Aronna]{M. Soledad Aronna} \address{M.S. Aronna\\ Escola de Matem\'atica Aplicada, Funda\c c\~ ao Getulio Vargas, Praia de Botafogo 190, 22250-900 Rio de Janeiro - RJ, Brazil} \email{[email protected]} \maketitle \begin{abstract} In this article we study optimal control problems for systems that are affine with respect to some of the control variables and nonlinear in relation to the others. We consider finitely many equality and inequality constraints on the initial and final values of the state. We investigate singular optimal solutions for this class of problems, for which we obtain second order necessary and sufficient conditions for weak optimality in integral form. We also derive Goh pointwise necessary optimality conditions. We show an example to illustrate the results. \end{abstract} \section{Introduction}\label{Introduction} The purpose of this paper is to investigate optimal control problems governed by systems of ordinary differential equations of the form \begin{equation} \dot{x}=f_0(x,u)+ \sum_{i=1}^m v_{i} f_i(x,u),\quad {\rm a.e.}\ {\rm on}\ [0,T]. \end{equation} Here $x:[0,T]\to \mathbb{R}^n$ is the state variable, $v_i:[0,T]\to \mathbb{R}$ are the {\em affine} controls for $i=1,\dots m,$ while $u:[0,T]\to \mathbb{R}^l$ is the vector of {\em nonlinear} controls and $f_i:\mathbb{R}^{n+l}\to \mathbb{R}^n$ is a vector field, for each $i=0,\dots m.$ Many models that enter into this framework can be found in practice and, in particular, in the existing literature. Among these we can mention: the Goddard's problem in three dimensions \cite{Goddard} analyzed in Bonnans et al. \cite{BLMT09}, several models concerning the motion of rockets as the ones treated in Lawden \cite{Law63}, Bell and Jacobson \cite{BelJac}, Goh \cite{GohThesis,Goh08}, Oberle \cite{Obe77}, Azimov \cite{Azi05} and Hull \cite{Hul11}; an hydrothermal electricity production problem studied in Bortolossi et al. \cite{BPT02}, the problem of atmospheric flight considered by Oberle in \cite{Obe90}, and the optimal production processes studied in Cho et al. \cite{ChoAbadParlar93} and Maurer at al. \cite{MauKimVos05}. All the systems investigated in these cited articles are {\em partially-affine} in the sense that they have at least one affine and at least one nonlinear control. The subject of second order optimality conditions for these partially-affine problems has been studied by Goh in \cite{GohThesis,Goh66a,Goh67,Goh08}, Dmitruk in \cite{Dmi11}, Dmitruk and Shishov in \cite{DmiShi10}, Bernstein and Zeidan \cite{BerZei90}, Frankowska and Tonon \cite{FraTon13}, and Maurer and Osmolovskii \cite{MauOsm09}. The first works were by Goh, who introduced a change of variables in \cite{Goh66a} and used it to obtain necessary optimality conditions in \cite{Goh66a,GohThesis,Goh66}, always assuming {\em normality} of the optimal solution. The necessary conditions we present imply those by Goh \cite{Goh66}, when there is only one multiplier (see Corollary \ref{CoroCBsym}). Recently, Dmitruk and Shishov \cite{DmiShi10} analyzed the quadratic functional associated with the second variation of the Lagrangian function, and provided a set of necessary conditions for the nonnegativity of this quadratic functional. Their results are consequence of a second order necessary condition that we present (see Theorem \ref{NCP2}). In \cite{Dmi11}, Dmitruk proposed, without proof, necessary and sufficient conditions for a problem having a particular structure: the affine control variable applies to a term depending only on the state variable, i.e. the affine and nonlinear controls are {\em uncoupled} or, equivalently $H_{uv}$ is identically zero, where $H$ denotes the {\em unmaximized Hamiltonian.} This hypothesis is not used in our work. Nevertheless, the conditions established here coincide with those suggested in Dmitruk \cite{Dmi11}, when the latter are applicable. In \cite{BerZei90}, Bernstein and Zeidan derived the Riccati equation for the {\em singular linear-quadratic regulator,} which is a modification of the classical linear-quadratic regulator where only some components of the control enter quadratically in the cost function. Frankowska and Tonon proved in \cite{FraTon13} second order necessary conditions for problems with {\em closed} control constraints and optimal controls containing arcs along which the second order derivative $H_{uu}$ of the unmaximized Hamiltonian vanishes. The necessary conditions given in \cite{FraTon13} hold for problems either with no endpoint constraints, or with smooth endpoint constraints and additional hypotheses as {\em calmness} and the {\em abnormality} of Pontryagin's Maximum Principle. All the articles mentioned in this paragraph use {\em Goh's transformation} to derive their optimality conditions, as it is done in the current paper, while none of them proved sufficient conditions of second order which is the main contribution of this article. It is worth mentioning that sufficient conditions were shown by Maurer and Osmolovskii in \cite{MauOsm09}, but for the case of a scalar control subject to bounds and {\em bang-bang} optimal solutions (i.e. no singular arc). This structure is not studied here since no closed control constraints are considered and thus our optimal control is supposed to be {\em singular} along the whole interval. The contributions of this article are as follows. We provide a pair of necessary and sufficient conditions in integral form for weak optimality of singular solutions of partially-affine problems (Theorems \ref{NCP2}-\ref{SC}). These conditions are {\em `no gap'} in the sense that the sufficient condition is obtained from the necessary one by strengthening an inequality. We consider fairly general endpoint constraints and we do not assume uniqueness of multiplier. The main result is the sufficient condition of Theorem \ref{SC}, which, up to our knowledge, cannot been found in the existing literature, and has important practical applications. As a product of the necessary condition \ref{NCP2} we get the {\em pointwise Goh conditions} in Corollary \ref{CoroCBsym}, extending this way previous results (see \cite{Goh66,FraTon13}) to problems with general endpoint constraints, and removing the hypothesis of vanishing $H_{uu}$ imposed in \cite{FraTon13}. In order to obtain the sufficient condition we impose a regularity assumption on the optimal controls, that in some practical situations is a consequence of the {\em generalized Legendre-Clebsch condition} (see Remark \ref{RemarkLC}). We provide a simple example to illustrate our results. As a main application of the sufficient condition provided in this article we can mention the proof of convergence of an associated {\em shooting algorithm} as stated in Aronna \cite{Aro13} and shown in detail in the technical report Aronna \cite{Aro11}. It is worth mentioning that, for practical interest, this shooting algorithm and its proof of convergence can be also used to solve partially-affine problems with bounds on the control and associated bang-singular solutions. The article is organized as follows. In Section \ref{SectionPb} we present the problem, the basic definitions and first order optimality conditions. In Section \ref{SectionSOC} we give the tools for second order analysis and establish a second order necessary condition. We introduce Goh's transformation in Section \ref{GohT}. In Section \ref{SectionNC} we show a new second order necessary condition. In Section \ref{SectionSC} we present the main result of this article that is a second order sufficient condition. We show an example to illustrate our results in Section \ref{SectionExample}, while Section \ref{SectionConclusion} is devoted to the conclusions and possible extensions. Finally, we include an Appendix containing some proofs of technical results that are omitted throughout the article. \noindent\textbf{Notations.} Given a function $h$ of variable $(t,x)$, we write $D_th$ or $\dot{h}$ for its derivative in time, and $D_xh$ or $h_x$ for the differentiations with respect to space variables. The same convention is extended to higher order derivatives. We let $\mathbb{R}^k$ denote the $k$-dimensional real space, i.e. the space of column real vectors of dimension $k;$ and by $\mathbb{R}^{k,}$ its corresponding dual space, which consists of $k-$dimensional real row vectors. By $L^p(0,T;\mathbb{R}^k)$ we mean the Lebesgue space with domain equal to the interval $[0,T]\subset \mathbb{R}$ and with values in $\mathbb{R}^k.$ The notation $W^{q,s}(0,T;\mathbb{R}^k)$ refers to the Sobolev spaces (see e.g. Adams \cite{Ada75}). Given $A$ and $B$ two $k\times k$ symmetric real matrices, we write $A\succeq B$ to indicate that $A-B$ is positive semidefinite. Given two functions $k_1:\mathbb{R}^N \rightarrow \mathbb{R}^{M}$ and $k_2: \mathbb{R}^N \rightarrow \mathbb{R}^L,$ we say that $k_1$ is a {\it big-O} of $k_2$ around 0 and write \begin{equation} k_1(x) = \mathcal{O} (k_2(x)), \end{equation} if there exists positive constants $\delta$ and $M$ such that $\|k_1(x)\| \leq M\|k_2(x)\|$ for $\|x\|<\delta.$ It is a {\it small-o} if $M$ goes to 0 as $\|x\|$ goes to 0, and in this case we write \begin{equation} k_1(x) = o(k_2(x)). \end{equation} \section{Statement of the problem and assumptions}\label{SectionPb} \subsection{Statement of the problem.} We study the optimal control problem (P) given by \begin{align} \min\,\,&\label{cost} \varphi_0(x(0),x(T)),\\ &\label{stateeq}\dot{x}=F(x,u,v),\quad {\rm a.e.}\ {\rm on}\ [0,T],\\ & \label{finaleq} \eta_j(x(0),x(T))=0,\quad \mathrm{for}\ j=1\hdots,d_{\eta},\\ &\label{finalineq} \varphi_i(x(0),x(T))\leq 0,\quad \mathrm{for}\ i=1,\hdots,d_{\varphi},\\ &\label{UV} u(t)\in U ,\,\, v(t)\in V,\quad {\rm a.e.}\ {\rm on}\ [0,T], \end{align} where the function $F\colon \mathbb{R}^{n+l+m}\to\mathbb{R}^n$ can be written as \begin{equation} F(x,u,v):=f_0(x,u) + \sum_{i=1}^m v_{i} f_i(x,u). \end{equation} Here $f_i \colon \mathbb{R}^{n+l}\rightarrow \mathbb{R}^n$ for $i=0,\hdots,m,$ $\varphi_i \colon \mathbb{R}^{2n}\rightarrow \mathbb{R}$ for $i=0,\hdots,d_{\varphi},$ $\eta_j \colon \mathbb{R}^{2n}\rightarrow \mathbb{R}$ for $j=1,\hdots,d_{\eta}.$ The sets $U$ and $V$ are open domains of $\mathbb{R}^l$ and $\mathbb{R}^m,$ respectively. The control $u(\cdot)$ is called {\em nonlinear,} while $v(\cdot)$ is named {\em affine control.} We consider the function spaces $\mathcal{U}:=L^{\infty}(0,T;\mathbb{R}^l)$ and $\mathcal{V}:=L^{\infty}(0,T;\mathbb{R}^m)$ for the controls, and $\mathcal{X}:=W^{1,\infty}(0,T;\mathbb{R}^n)$ for the state. When needed, we use $w(\cdot):=(x,u,v)(\cdot)$ to refer to a point in $\mathcal{W}:=\mathcal{X}\times \mathcal{U} \times \mathcal{V}.$ We call {\em trajectory} an element $w(\cdot)\in\mathcal{W}$ that satisfies the state equation \eqref{stateeq}. If in addition, the endpoint constraints \eqref{finaleq} and \eqref{finalineq} and the control constraint \eqref{UV} hold for $w(\cdot),$ then we say that it is a \textit{feasible trajectory} of problem (P). We consider the following regularity hypothesis throughout the article. \begin{assumption} \label{regular} All data functions have Lipschitz-continuous second order derivatives. \end{assumption} In this paper we study optimality conditions for {\em weak minima} of problem (P). A feasible trajectory $\hat{w}(\cdot)=(\hat{x},\hat{u},\hat{v})(\cdot)$ is said to be a {\em weak minimum} if there exists $\varepsilon>0$ such that the cost function attains at $\hat{w}(\cdot)$ its minimum in the set of feasible trajectories $w(\cdot)=(x,u,v)(\cdot)$ satisfying \begin{equation} \\|x-{\hat{x}}\\|_{\infty}<\varepsilon,\quad \\|u-\hat{u}\\|_{\infty}<\varepsilon,\quad \\|v-\hat{v}\\|_{\infty}< \varepsilon. \end{equation} For the remainder of the article, we fix a nominal feasible trajectory $\hat{w}(\cdot):=(\hat{x},\hat{u},\hat{v})(\cdot)$ for which we provide optimality conditions. We assume that the controls $\hat{u}(\cdot)$ and $\hat{v}(\cdot)$ do not accumulate at the boundaries of $U$ and $V,$ respectively. This is, letting $\mathbb{B}$ denote the closed unit ball of $\mathbb{R}^{l+m},$ we impose: \begin{assumption} \label{uvboundary} There exists $\delta>0$ such that $(\hat{u},\hat{v})(t)+\delta \mathbb{B} \subset U\times V,$ for almost all $t\in [0,T].$ \end{assumption} An element $\delta w(\cdot)\in \mathcal{W}$ is termed \textit{feasible variation for $\hat{w}(\cdot)$} if $\hat{w}(\cdot)+\delta w(\cdot)$ is a feasible trajectory for (P). For $\lambda=(\alpha,\beta,p(\cdot))$ in the space $\mathbb{R}^{d_{\varphi}+1,}\times \mathbb{R}^{d_{\eta},}\times W^{1,\infty}(0,T;\mathbb{R}^{n,}),$ we define the following functions: \begin{itemize} \item the \textit{pre-Hamiltonian (or unmaximized Hamiltonian)} function $H[\lambda]\colon \mathbb{R}^n\times \mathbb{R}^m\times \mathbb{R}^l\times [0,T]\to \mathbb{R}$ given by \begin{equation} H[\lambda](x,u,v,t):=p(t)\left(f_0(x,u)+\sum_{i=1}^m v_i f_i(x,u)\right), \end{equation} \item the \textit{endpoint Lagrangian} function $\ell[\lambda]\colon \mathbb{R}^{2n}\to \mathbb{R},$ \begin{equation} \ell[\lambda](x_0,x_T):=\sum_{i=0}^{d_{\varphi}} \alpha_i\varphi_i(x_0,x_T)+\sum_{j=1}^{d_{\eta}}\beta_j \eta_j(x_0,x_T), \end{equation} \item and the \textit{Lagrangian function} $\mathcal{L}[\lambda]\colon \mathcal{W}\to \mathbb{R},$ \be\label{lagrangian} \mathcal{L}[\lambda](w):= \ell[\lambda](x(0),x(T)) + \int_0^{T} p \left(f_0(x,u)+\sum_{i=1}^{m}v_{i}f_i(x,u)-\dot x\right)\mathrm{d}t. \ee \end{itemize} We assume, in sake of simplicity of notation that, whenever some argument of $F,$ $f_i,$ $H,$ $\ell,$ $\mathcal{L}$ or their derivatives is omitted, they are evaluated at $\hat{w}(\cdot).$ If we further want to explicit that they are evaluated at time $t,$ we write $F[t],$ $f_i[t],$ etc. The same convention notations hold for other functions of the state, control and multiplier that we define throughout the article. We assume, without any loss of generality, that $$ \varphi_i(\hat{x}(0),\hat{x}(T))=0,\ \mathrm{for}\ \mathrm{all}\ i=1,\hdots,d_{\varphi}. $$ \subsection{Lagrange multipliers} We introduce here the concept of {\em multiplier.} The second order conditions that we prove in this article are expressed in terms of the second variation of the Lagrangian function $\mathcal{L}$ given in \eqref{lagrangian} and the {set of Lagrange multipliers} associated with $\hat{w}(\cdot)$ that we define below. \begin{definition} \label{DefMul} An element $\lambda=(\alpha,\beta,p(\cdot))\in \mathbb{R}^{d_{\varphi}+1,}\times \mathbb{R}^{d_{\eta},}\times W^{1,\infty}(0,T;\mathbb{R}^{n,})$ is a \textit{Lagrange multiplier} associated with $\hat{w}(\cdot)$ if it satisfies the following conditions: \begin{align} \label{nontriv}&\|\alpha\|+\|\beta\|=1,\\ &\label{alphapos}\alpha=(\alpha_0,\alpha_1,\hdots,\alpha_{d_{\varphi}})\geq0, \end{align} the function $p(\cdot)$ is solution of the \textit{costate equation} \be \label{costateeq} -\dot{p}(t)=H_x[\lambda](\hat{x}(t),\hat{u}(t),\hat{v}(t),t), \ee it satisfies the \textit{transversality conditions} \be \label{transvcond} \begin{split} p(0)&=-D_{x_0}\ell[\lambda](\hat{x}(0),\hat{x}(T)),\\ p(T)&=D_{x_T}\ell[\lambda](\hat{x}(0),\hat{x}(T)), \end{split} \ee and the \textit{stationarity conditions} \be \label{stationarity} \left\{ \ba{l} \displaystyle H_u[\lambda](\hat{x}(t),\hat{u}(t),\hat{v}(t),t)=0,\\ H_v[\lambda](\hat{x}(t),\hat{u}(t),\hat{v}(t),t)=0, \ea \right. \quad {\rm a.e.}\ {\rm on}\ [0,T]. \ee We let $\Lambda$ denote the {\it set of Lagrange multipliers} associated with $\hat{w}(\cdot).$ \end{definition} The following result constitutes a {\em first order necessary condition} and yields the existence of Lagrange multipliers. \begin{theorem} \label{LambdaCompact} If $\hat{w}(\cdot)$ is a weak minimum for (P), then the set $\Lambda$ is non empty and compact. \end{theorem} \begin{proof} The existence of a Lagrange multiplier follows from Milyutin-Osmolovskii \cite[Thm. 2.1]{MilOsm98} or equivalent results proved in Alekseev et al. \cite{AleTikFom79} and Kurcyusz-Zowe \cite{KurZow}. In order to prove the compactness, observe that $\Lambda$ is closed and that $p(\cdot)$ may be expressed as a linear continuous mapping of $(\alpha,\beta).$ Thus, since the normalization \eqref{nontriv} holds, $\Lambda$ is necessarily a finite-dimensional compact set. \end{proof} In view of previous Theorem \ref{LambdaCompact}, note that $\Lambda$ can be identified with a compact subset of $\mathbb{R}^s,$ where $s:=d_{\varphi}+d_{\eta}+1.$ The main results of this article are stated on a restricted subset of $\Lambda$ for which the matrix $D^2_{(u,v)^2} H[\lambda] (\hat{w},t)$ is singular and, consequently, the pairs $(\hat{w},\lambda)$ result to be {\em singular extremals}. We comment again on this fact in Remark \ref{RemSing} below. Given $(\bar{x}_0,\bar{u}(\cdot),\bar{v}(\cdot))\in \mathbb{R}^n\times \mathcal{U}\times \mathcal{V},$ consider the \textit{linearized state equation} \begin{align} \label{lineareq} \dot{\bar{x}} &= F_{x}\,\bar{x} + F_{u}\,\bar{u} + F_{v}\,\bar{v},\quad {\rm a.e.}\ {\rm on}\ [0,T],\\ \label{lineareq0} \bar{x}(0) &= \bar{x}_0. \end{align} The solution $\bar{x}(\cdot)$ of \eqref{lineareq}-\eqref{lineareq0} is called \textit{linearized state variable.} \if{ \begin{remark} For the interest of the reader, we explicit the expression of the matrices involved in \eqref{lineareq}. For each $t\in [0,T],$ $F_{x}$ is an $n\times n-$matrix given by $\frac{\partial f_0}{\partial x}(\hat{x},\hat{u})+\sum_{i=1}^m \hat{v}_{i} \frac{\partial f_i}{\partial x}(\hat{x},\hat{u}),$ $F_{u}$ is $n\times l$ and is equal to $\frac{\partial f_0}{\partial u}(\hat{x},\hat{u})+\sum_{i=1}^m \hat{v}_{i} \frac{\partial f_i}{\partial u}(\hat{x},\hat{u})$ and, finally, $F_{v}$ is an $n\times m-$matrix whose $i$th column is $f_i(\hat{x},\hat{u}),$ for $i=1,\dots,m.$ \end{remark} }\fi \subsection{Critical cones}\label{ParCritical} We define here the sets of critical directions associated with $\hat{w}(\cdot),$ both in the $L^{\infty}$- and the $L^2$-norms. Even if we are working with control variables in $L^{\infty}$ and hence the control perturbations are naturally taken in $L^{\infty},$ the second order analysis involves quadratic mappings that require to continuously extend the cones to $L^2.$ Set $\mathcal{X}_2:=W^{1,2}(0,T;\mathbb{R}^n),$ $\mathcal{U}_2:=L^2(0,T;\mathbb{R}^l)$ and $\mathcal{V}_2:=L^2(0,T;\mathbb{R}^m),$ and write $\mathcal{W}_2:=\mathcal{X}_2\times \mathcal{U}_2\times \mathcal{V}_2$ to refer to the corresponding product space. Given $\bar{w}(\cdot)\in\mathcal{W}_2$ satisfying the linearized state equation \eqref{lineareq}-\eqref{lineareq0}, consider the \textit{linearization of the endpoint constraints and cost function,} \begin{gather} \label{linearconseq} D\eta_j(\hat{x}(0),\hat{x}(T))(\bar{x}(0),\bar{x}(T))=0,\quad {\rm for}\ j=1,\hdots,d_{\eta}, \\ \label{linearconsineq} D\varphi_i(\hat{x}(0),\hat{x}(T))(\bar{x}(0),\bar{x}(T))\leq 0,\quad {\rm for}\ i=0,\hdots,d_{\varphi}. \end{gather} The \textit{critical cones} in $\mathcal{W}_2$ and $\mathcal{W}$ are given, respectively, by \begin{gather} \label{C2}\mathcal{C}_2:=\{\bar{w}(\cdot)\in\mathcal{W}_2:\text{\eqref{lineareq}-\eqref{lineareq0}}\ \text{and}\ \text{\eqref{linearconseq}-\eqref{linearconsineq}}\ \text{hold}\},\\ \label{C} \mathcal{C}:= \mathcal{C}_2 \cap \mathcal{W}. \end{gather} The following density result holds. \begin{lemma} \label{conedense} The critical cone $\mathcal{C}$ is dense in $\mathcal{C}_2$ with respect to the $\mathcal{W}_2$-topology. \end{lemma} The proof of previous lemma follows from the following technical result (due to Dmitruk \cite[Lemma 1]{Dmi08}). \begin{lemma}[on density of cones] \label{lemmadense} Consider a locally convex topological space $X,$ a finite-faced cone $Z\subset X,$ and a linear space $Y$ dense in $X.$ Then the cone $Z\cap Y$ is dense in $Z.$ \end{lemma} \if{ Let the cone $C$ be given by \be C=\{x\in X:(p_i,x)=0,\ \mathrm{for}\ i= 1,\hdots,\mu,\ (q_j,x)\leq 0,\ \mathrm{for}\ j=1,\hdots,\nu\}. \ee Let us show first, without lost of generality, that the equality constraints can be removed from the formulation. It suffices to consider the case where $C$ is given by only one equality $(p,x)=0.$ Take any point $x(0)\in C$ and a convex neighborhood$\mathcal{O}(x(0)).$ We have to show that there exists $x$ in $C\cap L\cap \mathcal{O}(x(0)).$ Since the set $(p,x)<0$ is open, its intersection with $\mathcal{O}(x(0))$ is open too and obviously nonempty, hence it contains a point $x_1$ from the set $L,$ because the last one is dense in $X.$ Similarly, the intersection of the sets $(p,x)<0$ and $\mathcal{O}(x(0))$ contains a point $x_2\in L.$ Since $\mathcal{O}(x(0))$ is convex, it contains a point $x$ such that $(p,x)=0,$ which belongs to $C$ and to $L\cap \mathcal{O}(x(0)).$ We can then consider only the case where $C$ is given by a finite number of inequalities. Suppose first that there exists $\hat{x}\in C$ such that $(q_j,\hat{x})<0$ for all $j,$ hence $\hat{x} \in \mathrm{int}{C}.$ Take any $x(0)\in C$ and any convex neighborhood $\mathcal{O}(x(0)).$ We have to find a point $x\in C\cap \mathcal{O}(x(0))\cap L.$ We know that, for any positive $\varepsilon,$ the point $x_{\varepsilon}:=x(0)+\varepsilon \hat{x}$ lies in $\mathrm{int}(C),$ and then there exists a positive $\varepsilon$ such that this point lies also in $\mathcal{O}(x(0)).$ Thus, the open set $\mathrm{int}{C}\cap \mathcal{O}(x(0))$ is nonempty, and then contains a point $x$ from the dense set $L.$ Suppose now that the above point $\hat{x}\in\mathrm{int}{C}$ does not exist, and, without lost of generality, that $q_j\neq 0$ for every $j.$ In this case, by the Dubovitskii-Milyutin theorem, there exist multipliers $\alpha_j\geq0,\ j=1,\hdots,\nu,$ not all zero, such that Euler-Lagrange equation holds: $\alpha_1 q_1+\hdots+\alpha_{\nu}q_{\nu}=0.$ Suppose, without lost of generality, that $\alpha_{\nu}>0.$ Then, for all $x \in C$ we actually have $(q_{\nu},x)=0,$ not just $\leq0.$ This means that the cone $C$ can be given by the constraints $(q_j,x)\leq 0,\ j=1,\hdots,\nu-1,\ (q_{\nu},x)=0.$ But, as was already shown, the last equality can be removed, so the cone can be given by a smaller number of inequalities. Applying induction arguments, we arrive at a situation when either all the inequalities are changed into equalities and then removed, or the strict inequalities have a nonempty intersection. Since both cases are already considered, the proof is complete. }\fi \noindent{\em Proof of Lemma \ref{conedense}.} Set $X:=\{\bar{w}(\cdot)\in\mathcal{W}_2:\text{\eqref{lineareq}-\eqref{lineareq0}}\, \text{hold}\},$ $Y:=\{\bar{w}(\cdot)\in\mathcal{W}:\text{\eqref{lineareq}-\eqref{lineareq0}}\, \text{hold}\},$ and $Z:=\mathcal{C}_2$ and apply Lemma \ref{lemmadense}. The desired density follows. \findem \section{Second order analysis}\label{SectionSOC} We begin this section by giving an expression of the {\em second order derivative of the Lagrangian function $\mathcal{L},$} in terms of derivatives of $\ell$ and $H.$ We let $\Omega$ denote this second variation. All the second order conditions we present are established in terms of either $\Omega$ or some transformed form of $\Omega.$ The main result of the current section is the necessary condition in Theorem \ref{strengthNC}, which is applied in Section \ref{SectionNC} to get the stronger condition given in Theorem \ref{NCP2}. \subsection{Second variation} Let us consider the quadratic mapping \be \label{Omega} \begin{split} \Omega [\lambda] &(\bar{x},\bar{u},\bar{v}):= \, \mbox{$\frac{1}{2}$} D^2\ell[\lambda](\hat{x}(0),\hat{x}(T))(\bar{x}(0),\bar{x}(T))^2 + \int_0^T \Big(\mbox{$\frac{1}{2}$}\bar{x}^\top H_{xx}[\lambda] \bar{x} \ \\ & + \bar{u}^\top H_{ux}[\lambda]\bar{x} + \bar{v}^\top H_{vx}[\lambda]\bar{x} + \mbox{$\frac{1}{2}$}\bar{u}^\top H_{uu}[\lambda]\bar{u} + \bar{v}^\top H_{vu}[\lambda]\bar{u}\Big) \mathrm{d}t. \end{split} \ee \if{ \begin{remark} Note that $H_{xx}[\lambda] = p\left(\frac{\partial^2 f_0}{\partial x^2} + \sum_{k=1}^m \hat{v}_k \frac{\partial^2 f_k}{\partial x^2}\right)$ is an $n\times n$-matrix, $H_{ux}[\lambda]$ is equal to $p\left(\frac{\partial^2 f_0}{\partial u\partial x}+\sum_{k=1}^m \hat{v}_k \frac{\partial^2 f_k}{\partial u\partial x}\right)$ and it is $l\times n,$ $H_{vx}[\lambda]$ is an $m\times n-$matrix whose $i$th row is given by $H_{v_ix}[\lambda]= p \frac{\partial f_i}{\partial x}.$ The matrix $H_{uu}[\lambda]$ is equal to $p\left(\frac{\partial^2 f_0}{\partial u^2}+\sum_{k=1}^m \hat{v}_k \frac{\partial^2 f_k}{\partial u^2}\right)$ and it is $l\times l,$ and the $i-$th. row of the $m\times l-$matrix $H_{vu}[\lambda]$ is $H_{v_i u}[\lambda]= p \frac{\partial f_i}{\partial u}.$ \end{remark} }\fi The result that follows gives an expression of the Lagrangian $\mathcal{L}$ at the nominal trajectory $\hat{w}(\cdot).$ For the sake of simplicity, the time variable is omitted in the statement. \begin{lemma}[Lagrangian expansion] \label{expansionlagrangian} Let $w(\cdot)=(x,u,v)(\cdot)\in\mathcal{W}$ be a trajectory and set $\delta w(\cdot)=(\delta x,\delta u,\delta v)(\cdot):=w(\cdot)-\hat{w}(\cdot).$ Then, for every multiplier $\lambda\in\Lambda,$ the following expansion of the Lagrangian holds \be \label{expansionLag} \mathcal{L}[\lambda](w) =\mathcal{L}[\lambda](\hat{w}) + \Omega[\lambda](\delta x,\delta u,\delta v) +\omega[\lambda](\delta x,\delta u,\delta v) +\mathcal{R}(\delta x,\delta u,\delta v), \ee where $\omega$ is a cubic mapping given by \begin{align} \omega[\lambda]&(\delta x,\delta u,\delta v) := \\ &\int_0^T \left[ H_{vxx}[\lambda](\delta x, \delta x,\delta v) + 2H_{vux}[\lambda](\delta x,\delta u,\delta v) + H_{vuu}[\lambda](\delta u,\delta u,\delta v) \right] \mathrm{d}t, \end{align} and $\mathcal{R}$ satisfies the estimate \begin{equation} \mathcal{R}(\delta x,\delta u,\delta v) = L_\ell \|(\delta x(0),\delta x(T))\|^3 +L K (1+\\|v\\|_\infty)\, \\|(\delta x, \delta u)\\|_{\infty}\\|(\delta x,\delta u) \\|_2^2. \end{equation} Here $L_\ell$ is a Lipschitz constant for $D^2\ell[\lambda]$ uniformly with respect to $\lambda\in \Lambda,$ $L$ is a Lipschitz constant for $D^2f_i$ uniformly in $i=0,\dots,m,$ and $K:=\displaystyle\sup_{\lambda\in \Lambda} \\|p(\cdot)\\|_\infty.$ \end{lemma} \begin{proof} See Appendix \ref{proofexpansionlagrangian}. \end{proof} \begin{remark} From previous lemma one gets the identity \begin{equation} \Omega[\lambda] (\bar{w}) = \mbox{$\frac{1}{2}$} D^2\mathcal{L}[\lambda] (\hat{w})\, \bar{w}^2. \end{equation} \end{remark} \subsection{Second order necessary condition} The following result is a classical second order condition for weak minima. \begin{theorem}[Second order necessary condition] \label{classicalNC} If $\hat{w}(\cdot)$ is a weak minimum of problem (P), then \be \label{classicalNCeq} \max_{\lambda\in \Lambda} \Omega [\lambda] (\bar{x},\bar{u},\bar{v}) \geq 0,\ \mathrm{for}\ \mathrm{all}\ (\bar{x},\bar{u},\bar{v})\in\mathcal{C}. \ee \end{theorem} A proof of Theorem \ref{classicalNC} can be found in Levitin, Milyutin and Osmolovskii \cite{LevMilOsm1985}. Nevertheless, for the sake of completeness, we give a proof in the Appendix \ref{AppendixclassicalNC} that uses techniques of optimization in abstract spaces. An extension of the condition \eqref{classicalNCeq} to the cone $\mathcal{C}_2$ can be easily proved and gives the following, stronger, second order condition. \begin{theorem} \label{NCC2} If $\hat{w}(\cdot)$ is a weak minimum of problem (P), then \be \label{classicalNCeqC2} \max_{\lambda\in \Lambda} \Omega [\lambda] (\bar{x},\bar{u},\bar{v}) \geq 0,\quad \mathrm{for}\ \mathrm{all}\ (\bar{x},\bar{u},\bar{v})\in \mathcal{C}_2. \ee \end{theorem} \begin{proof} Observe first that $\Omega[\lambda]$ can be extended to the space $\mathcal{W}_2$ since all the coefficients are essentially bounded. The result follows by the density property of Lemma \ref{conedense} and the compactness of the Lagrange multipliers set $\Lambda$ proved in Theorem \ref{LambdaCompact}. \end{proof} \subsection{Strengthened second order necessary condition} In the sequel we aim at strengthening the necessary condition of Theorem \ref{NCC2} by proving that the maximum in \eqref{classicalNCeqC2} remains nonnegative when taken in a {\em possibly} smaller set of multipliers, whenever $\Lambda$ is convex. Let ${\rm co}\, \Lambda$ denote the {\em convex hull} of $\Lambda.$ Observe that if $\lambda=(\alpha,\beta,p(\cdot))$ is in $ {\rm co}\, \Lambda$ then it verifies \eqref{alphapos}-\eqref{stationarity} and, if $\hat{w}(\cdot)$ is a weak minimum, also the second order condition \eqref{classicalNCeqC2} is fulfilled for $\lambda.$ However, $\lambda$ may not verify the nontriviality condition \eqref{nontriv}, thus ${\rm co}\, \Lambda$ may content the trivial (i.e. identically zero) multiplier. Set \be\label{H2} \mathcal{H}_2:=\{(\bar{x},\bar{u},\bar{v})(\cdot)\in\mathcal{W}_2:\eqref{lineareq}\ \text{holds}\}, \ee and consider the subset of ${\rm co}\, \Lambda$ given by \begin{equation} ({\rm co}\, \Lambda)^{\#}:=\{ \lambda\in {\rm co}\, \Lambda: \Omega[\lambda]\ \text{is weakly-l.s.c. on }\mathcal{H}_2\}. \end{equation} Next we prove that $({\rm co}\, \Lambda)^{\#}$ can be characterized in a quite simple way (see Lemma \ref{Lambdawlsc} below). Theorem \ref{strengthNC} stated afterwards yields a {\em new} necessary optimality condition. \begin{lemma} \label{Lambdawlsc} \be ({\rm co}\, \Lambda)^{\#} =\{ \lambda\in {\rm co}\, \Lambda: H_{uu} [\lambda]\succeq0\ {\rm and}\ H_{vu}[\lambda] = 0,\,\, {\rm a.e.}\, {\rm on}\,[0,T]\}. \ee \end{lemma} \begin{remark}[About {\em singular} solutions] \label{RemSing} From now on we restrict the set $({\rm co}\, \Lambda)^{\#}$ or some subset of it and, therefore, $H_{uv}[\lambda]\equiv 0$ along the nominal trajectory $\hat{w}(\cdot).$ Consequently, \begin{equation} D^2_{(u,v)^2}H[\lambda] (\hat{w},t)\ \mathrm{is}\ \mathrm{a}\ \mathrm{singular}\ \mathrm{matrix}\ \mathrm{a.e.\,\, on}\ [0,T]. \end{equation} The latter assertion together with the stationarity condition \eqref{stationarity} imply that $(\hat{w},\lambda)$ is a {\em singular extremal} (as defined in Bryson-Ho \cite[Page 246]{BryHo}). That is, if we write $\nu:=(u,v)$ for the control, we say that $(\hat{w}, \lambda)$ is a singular extremal if $H_\nu[\lambda]=0$ and $H_{\nu\nu}[\lambda]$ is singular a.e. on $[0,T]$. Let us comment on the terminology used in the literature for the class of problems where $H_{\nu\nu}$ is a singular matrix. In Bell-Jacobson \cite[Definition 1.2]{BelJac} and Ruxton-Bell \cite{RuxtonBell1995} they refer to singular extremals (as defined above) as {\em totally singular}, while they use the term {\em partially singular} to refer to controls for which $H_{\nu}=0$ only on some subintervals of $[0,T],$ which is not the class of controls studied here. The same definition is adopted in Poggiolini and Stefani \cite{PogioliniStefani2007}. On the other hand, O'Malley in \cite{OMalley1977} calls {\em partially singular} the linear-quadratic problems in which the matrix $H_{\nu\nu}$ is (singular but) not of constant non-zero rank, that is a framework included in our class of problems. \end{remark} In order to prove Lemma \ref{Lambdawlsc} we shall notice that $\Omega[\lambda]$ can be written as the sum of two maps: the first one being a weakly-continuous function on the space $\mathcal{H}_2$ given by \be\label{Omega1} (\bar{x},\bar{u},\bar{v})\mapsto\mbox{$\frac{1}{2}$} D^2\ell[\lambda](\bar{x}(0),\bar{x}(T))^2 + \int_0^T \Big( \mbox{$\frac{1}{2}$}\bar{x}^\top H_{xx}[\lambda] \bar{x} + \bar{u}^\top H_{ux}[\lambda]\bar{x} + \bar{v}^\top H_{vx}[\lambda]\bar{x} \Big) \mathrm{d}t, \ee and the second one being the quadratic operator \be\label{Omega2} (\bar{u},\bar{v})\mapsto \int_0^T \Big( \mbox{$\frac{1}{2}$}\bar{u}^\top H_{uu} [\lambda]\bar{u} + \bar{v}^\top H_{vu}[\lambda]\bar{u} \Big) \mathrm{d}t. \ee The weak-continuity of the mapping in \eqref{Omega1} follows easily. Additionally, in view of Hestenes \cite[Theorem 3.2]{Hes51}, the following characterization holds. \begin{lemma} \label{Lemmawlsc} The mapping in \eqref{Omega2} is weakly-lower semicontinuous on $\mathcal{U}\times \mathcal{V}$ if and only if the matrix \be \label{Huvuv} D^2_{(u,v)^2}H[\lambda]=\begin{pmatrix} H_{uu}[\lambda] & H_{vu}[\lambda]^\top \\ H_{vu}[\lambda] & 0\\ \end{pmatrix}, \ee is positive semidefinite almost everywhere on $[0,T].$ \end{lemma} \begin{remark} \label{remarkLC} The fact that the matrix in \eqref{Huvuv} is positive semidefinite is known as the {\it Legendre-Clebsch necessary optimality condition} for the extremal $(\hat{w},\lambda)$ (see e.g. Bliss \cite{Bliss1946} in the framework of Calculus of Variations, and Bryson-Ho \cite{BryHo}, Agrachev-Sachkov \cite{AgrSac} or Corollary \ref{NCunique} below for Optimal Control). \end{remark} We can now prove Lemma \ref{Lambdawlsc}. \noindent{\em Proof of Lemma \ref{Lambdawlsc}.} It follows from the decomposition given in \eqref{Omega1}-\eqref{Omega2} and the characterization of weak-lower semicontinuity stated in previous Lemma \ref{Lemmawlsc}. \findem \begin{theorem}[Strengthened second order necessary condition] \label{strengthNC} If $\hat{w}(\cdot)$ is a weak minimum of problem (P), then \be \label{strengthNCeq} \max_{\lambda\in ({\rm co}\, \Lambda)^{\#}} \Omega [\lambda] (\bar{x},\bar{u},\bar{v}) \geq 0,\quad \mathrm{on}\ \mathcal{C}_2. \ee \end{theorem} \begin{remark}[On {\em unqualified} solutions] Notice that it may occur that $0\in ({\rm co}\, \Lambda)^{\#}$ and, in this case, the second order condition in Theorem \ref{strengthNC} above does not provide any information. This situation may arise when the endpoint constraints are {\it not qualified,} in the sense of the {\em constraint qualification condition \eqref{QC}} introduced in the Appendix, which is a natural generalization of the {\em Mangasarian-Fromovitz} condition \cite{ManFro67} to the infinite-dimensional framework. \end{remark} In order to achieve Theorem \ref{strengthNC}, let us recall the following result on quadratic forms (taken from Dmitruk \cite[Theorem 5]{Dmi84}). \begin{lemma} \label{quadform} Given a Hilbert space $H,$ and $a_1,a_2,\hdots,a_p$ in $H,$ set \be K:=\{x\in H:(a_i,x)\leq 0,\ \mathrm{for}\ i=1,\hdots,p\}. \ee Let $M$ be a convex and compact subset of $\mathbb{R}^s,$ and let $\{Q^{\psi}:\psi\in M\}$ be a family of continuous quadratic forms over $H,$ the mapping $\psi \rightarrow Q^{\psi}$ being affine. Set $M^{\#}:=\{ \psi \in M:\ Q^{\psi}\ \text{is weakly-l.s.c.}\text{ on } H\}$ and assume that \be \max_{\psi\in M} Q^{\psi}(x)\geq 0,\ \mathrm{for}\ \mathrm{all}\ x\in K. \ee Then \be \max_{\psi\in M^{\#}} Q^{\psi}(x)\geq 0,\ \mathrm{for}\ \mathrm{all}\ x\in K. \ee \end{lemma} We are now able to show Theorem \ref{strengthNC} as desired. \noindent{\em Proof of Theorem \ref{strengthNC}.} It is a consequence of Theorem \ref{NCC2}, Lemmas \ref{Lambdawlsc} and \ref{quadform}. \findem We finish this section with the following extension of the classical second order pointwise Legendre-Clebsch condition, which follows as a corollary of Theorem \ref{strengthNC}. \begin{corollary}[Legendre-Clebsch condition] \label{NCunique} If $\hat{w}(\cdot)$ is a weak minimum of (P) with a unique associated Lagrange multiplier $\hat\lambda,$ then $(\hat{w},\hat\lambda)$ satisfies the {\em Legendre-Clebsch condition}, this is, the matrix in \eqref{Huvuv} is positive semidefinite and, consequently, \be \label{R0pos} H_{uu} [\hat\lambda]\succeq0\ {\rm and}\ H_{vu}[\hat\lambda] \equiv 0. \ee \end{corollary} \begin{proof} It follows easily from Theorem \ref{strengthNC}. In fact, as the Lagrange multiplier is unique, ${\rm co}\, \Lambda = \Lambda= \{\hat\lambda\},$ and the inequality in \eqref{strengthNCeq} implies that $({\rm co}\, \Lambda)^{\#}\neq \emptyset.$ Therefore, $({\rm co}\, \Lambda)^{\#}=\Lambda^{\#} = \{\hat\lambda\}$ and \eqref{R0pos} necessarily holds. \end{proof} \section{Goh Transformation}\label{GohT} In this section we introduce the {\em Goh trasformation} which is a linear change of variables applied usually to a linear differential equation, and that is motivated by the facts explained in the sequel. In the previous section we were able to provide a necessary condition involving the nonnegativity on $\mathcal{C}_2$ of the maximum of $\Omega[\lambda]$ over the set $({\rm co}\, \Lambda)^{\#}$ (Theorem \ref{strengthNC}). Our next step is finding a sufficient condition. To achieve this one would naturally try to strengthen the inequality \eqref{strengthNCeq} to convert it into a condition of strong positivity. However, since no quadratic term on $\bar{v}(\cdot)$ appears in $\Omega,$ the latter cannot be strongly positive with respect to the norm of the controls. Thus, what we do here to find the desired sufficient condition is transforming $\Omega$ into a new quadratic mapping that may result strongly positive on an appropriate transformed critical cone. For historical interest, we recall that Goh introduced this change of variables in \cite{Goh66a} and employed it to derive necessary conditions in \cite{Goh66a,Goh66}. Since then, many optimality conditions were obtained by using that transformation as already mentioned in the Introduction. For the remainder of the article, we consider the following regularity hypothesis on the controls. \begin{assumption} \label{SmoothControls} The controls $\hat{u}(\cdot)$ and $\hat{v}(\cdot)$ are smooth. \end{assumption} This hypothesis is not restrictive since it is a consequence of the {\em strengthened generalized Legendre-Clebsch condition} as explained in Aronna \cite{Aro11,Aro13}, where it is shown that, whenever this generalized condition holds, one can write the controls as smooth functions of the state and costate variable. See also Remark \ref{RemarkLC} below. Consider hence the linearized state equation \eqref{lineareq} and the {\em Goh transformation} defined by \be \label{Goht} \left\{ \ba{l} \bar{y}(t):= \displaystyle\int_0^t \bar{v}(s) {\rm d}s, \\ \bar{\xi}(t) := \bar{x}(t)- F_{v}[t]\,\bar{y}(t), \ea \right. \quad {\rm for}\ t\in [0,T]. \ee Observe that $\bar{\xi}(\cdot)$ defined in that way satisfies the linear equation \be \label{xieq} \dot\bar{\xi} = F_{x}\,\bar{\xi} + F_{u}\,\bar{u} +B\,\bar{y},\quad \bar{\xi}(0)=\bar{x}(0), \ee where \be \label{B1} B:= F_{x} F_{v}-\ddt F_{v}. \ee Here $B$ is an $n\times m$-matrix whose $i$th column is given by \begin{equation} -[f_i,f_0]^x-\sum_{j=1}^m \hat{v}_j[f_i,f_j]^x + D_u f_i\, \dot{\hat{u}}, \end{equation} where $[f_i,f_j]^x:=({\rm D}_xf_i)f_j-(D_xf_j)f_i$ and it is referred as the {\it Lie bracket with respect to $x$} of the vector fields $f_i$ and $f_j.$ \subsection{Tranformed critical cones} In this paragraph we present the critical cones obtained after Goh's transformation. We shall recall the linearized endpoint constraints \eqref{linearconseq}-\eqref{linearconsineq} and the critical cones \eqref{C2}-\eqref{C}. Let $(\bar{x},\bar{u},\bar{v})(\cdot)\in \mathcal{C}$ be a critical direction. Define $(\bar{\xi},\bar{y})(\cdot)$ by Goh's transformation \eqref{Goht} and set $\bar{h}:=\bar{y}(T).$ From \eqref{linearconseq}-\eqref{linearconsineq} we get \begin{gather} \label{tlinearconseq} D\eta_j (\hat{x}(0),\hat{x}(T))\big(\bar{\xi}(0),\bar{\xi}(T)+F_{v}[T]\bar{h}\big)=0,\quad \mathrm{for}\,\, j=1,\hdots,d_{\eta}, \\ \label{tlinearconsineq} D\varphi_i(\hat{x}(0),\hat{x}(T))\big(\bar{\xi}(0),\bar{\xi}(T)+F_{v}[T]\bar{h}\big)\leq 0,\quad \mathrm{for}\,\, i=0,\hdots,d_{\varphi}. \end{gather} Remind the definition of the linear space $\mathcal{W}_2$ given in paragraph \ref{ParCritical}. Let $\mathcal{Y}$ denote the Sobolev space $W^{1,\infty}(0,T;\mathbb{R}^m),$ and consider the cones \be \label{P} \mathcal{P}:= \{(\bar{\xi}(\cdot), \bar{u}(\cdot),\bar{y}(\cdot),\bar{h})\in \mathcal{W} \times \mathbb{R}^m:\,\bar{y}(0)=0,\,\bar{y}(T)=\bar{h},\,\text{\eqref{xieq}, \eqref{tlinearconseq}-\eqref{tlinearconsineq} hold} \}, \ee \be \label{P2} \mathcal{P}_2:= \{(\bar{\xi}(\cdot), \bar{u}(\cdot),\bar{y}(\cdot),\bar{h})\in \mathcal{W}_2 \times \mathbb{R}^m:\,\text{\eqref{xieq}, \eqref{tlinearconseq}-\eqref{tlinearconsineq} hold} \}. \ee \begin{remark} \label{PandC} Observe that $\mathcal{P}$ is the cone obtained from $\mathcal{C}$ via Goh's transformation \eqref{Goht}. \end{remark} The next result shows the density of $\mathcal{P}$ in $\mathcal{P}_2.$ This fact is used afterwards when we extend a necessary condition stated in $\mathcal{P}$ to the bigger cone $\mathcal{P}_2$ by continuity arguments, as it was done for $\mathcal{C}$ and $\mathcal{C}_2$ in Section \ref{SectionSOC}. \begin{lemma} \label{PdenseP2} $\mathcal{P}$ is a dense subspace of $\mathcal{P}_2$ in the $ \mathcal{W}_2 \times \mathbb{R}^m$-topology. \end{lemma} \begin{proof} Notice that the inclusion $\mathcal{P}\subset \mathcal{P}_2$ is immediate. In order to prove the density, consider the linear spaces \begin{gather} X:= \{ (\bar{\xi}(\cdot),\bar{u}(\cdot),\bar{y}(\cdot),\bar{h})\in \mathcal{W}_2\times \mathbb{R}^m:\ \eqref{xieq}\ {\rm holds}\},\\ Y:=\{ (\bar{\xi}(\cdot),\bar{u}(\cdot),\bar{y}(\cdot),\bar{h})\in \mathcal{W}\times \mathbb{R}^m:\ \bar{y}(0)=0,\, \bar{y}(T)=\bar{h}\ {\rm and}\ \eqref{xieq}\ {\rm holds}\}, \end{gather} and the cone \begin{equation} Z:= \{ (\bar{\xi}(\cdot),\bar{u}(\cdot),\bar{y}(\cdot),\bar{h})\in X:\ \text{\eqref{tlinearconseq}-\eqref{tlinearconsineq}}\ {\rm holds}\}. \end{equation} Notice that $Y$ is a dense linear subspace of $X$ (Dmitruk-Shishov \cite[Lemma 6]{DmiShi10} or Aronna et al. \cite[Lemma 8.1]{ABDL11}), and $Z$ is a finite-faced cone of $X. $ The desired density follows by Lemma \ref{lemmadense}. \end{proof} \subsection{Transformed second variation} Next we write the quadratic mapping $\Omega$ in the variables $(\bar{\xi}(\cdot),\bar{u}(\cdot),\bar{y}(\cdot),\bar{v}(\cdot),\bar{h}).$ Set, for $\lambda\in ({\rm co}\, \Lambda)^{\#},$ \be \label{OmegaP} \begin{split} \Omega_{\mathcal{P}} &[\lambda] (\bar{\xi},\bar{u},\bar{y},\bar{v},\bar{h}) := g[\lambda] (\bar{\xi}(0),\bar{\xi}(T),\bar{h}) + \displaystyle \int_0^T \left( \mbox{$\frac{1}{2}$}\bar{\xi}\,^\top H_{xx}[\lambda] \bar{\xi} + \bar{u}^\top H_{ux}[\lambda]\bar{\xi} \right.\\ &\left. +\, \bar{y}^\top M[\lambda] \bar{\xi} + \mbox{$\frac{1}{2}$}\bar{u}^\top H_{uu}[\lambda] \bar{u} + \bar{y}^\top E[\lambda] \bar{u} + \mbox{$\frac{1}{2}$}\bar{y}^\top R[\lambda] \bar{y} + \bar{v}^\top G[\lambda] \bar{y} \right) \mathrm{d}t, \end{split} \ee where \begin{gather} \label{M} M:= F_v^\top H_{xx}-\dot H_{vx}-H_{vx}F_x,\quad E:= F_v^\top H_{ux}^\top - H_{vx}F_u, \\ \label{SV} S:=\mbox{$\frac{1}{2}$} (H_{vx}F_v + (H_{vx}F_v)^\top),\quad G:= \mbox{$\frac{1}{2}$} (H_{vx}F_v - (H_{vx}F_v)^\top), \\ \label{R1} R := F_v^\top H_{xx}F_v - (H_{vx}B+(H_{vx}B)^\top) - \dot S, \\ \label{g} g[\lambda] (\xi_0,\xi_T,h):= \mbox{$\frac{1}{2}$}\ell''(\xi_0,\xi_T+F_{v}[T]\,h)^2 +h^\top(H_{vx}[T]\, \xi_T+\mbox{$\frac{1}{2}$} S[T] h). \end{gather} Observe that, in view of Assumptions \ref{regular} and \ref{SmoothControls}, all the functions defined above are continuous in time. \begin{remark} We can see that $M$ is an $m\times n$-matrix whose $i$th row is given by the formula \begin{equation} M_i = p\sum_{j=0}^m \hat{v}_j \left( \frac{\partial^2 f_j}{\partial x^2} f_i - \frac{\partial^2 f_i}{\partial x^2} f_j + \frac{\partial f_j}{\partial x} \frac{\partial f_i}{\partial x} - \frac{\partial f_i}{\partial x} \frac{\partial f_j}{\partial x} \right)- p \frac{\partial^2 f_i}{\partial x\partial u} \dot{\hat{u}}, \end{equation} $E$ is $m\times l$ with $ E_{ij}=p\displaystyle \frac{\partial^2 F}{\partial u_j \partial x} f_i - p \frac{\partial f_i}{\partial x}\frac{\partial F}{\partial u_j}, $ the $m\times m-$matrices $S$ and $G$ have entries $ S_{ij} = \displaystyle\mbox{$\frac{1}{2}$} p \left( \frac{\partial f_i}{\partial x}f_j + \frac{\partial f_j}{\partial x}f_i \right), $ and \be \label{Vcrochet} G_{ij}=p[f_i,f_j]^x, \ee respectively. The components of the matrix $R$ have a quite long expression, that is simplified for some multipliers as it is detailed in equation \eqref{Rij} in the next section. \end{remark} The identity between $\Omega$ and $\Omega_\mathcal{P}$ stated in the following lemma holds. \begin{lemma} \label{Omegat} Let $\lambda \in ({\rm co}\, \Lambda)^{\#},$ $(\bar{x},\bar{u},\bar{v})(\cdot) \in \mathcal{H}_2$ (given in \eqref{H2}) and $(\bar{\xi},\bar{y})(\cdot)$ be defined by Goh's transformation \eqref{Goht}. Then \begin{equation} \Omega[\lambda] (\bar{x},\bar{u},\bar{v}) = \Omega_{\mathcal{P}} [\lambda] (\bar{\xi},\bar{u},\bar{y},\bar{v},\bar{y}(T)). \end{equation} \end{lemma} The proof of this lemma is merely technical and we leave it to the Appendix \ref{AppendixOmegat}. Finally let us remind the strengthened necessary condition of Theorem \ref{strengthNC}. Observe that by Goh's transformation \eqref{strengthNCeq} and in view of Remark \ref{PandC}, we obtain the following form of the second order necessary condition. \begin{corollary} \label{transNC} If $\hat{w}(\cdot)$ is a weak minimum of problem (P), then \be \label{maxOmegaP} \max_{\lambda\in ({\rm co}\, \Lambda)^{\#}} \Omega_{\mathcal{P}} [\lambda] (\bar{\xi},\bar{u},\bar{y},\dot\bar{y},\bar{h}) \geq 0,\quad \mathrm{on}\ \mathcal{P}. \ee \end{corollary} \section{New second order necessary condition}\label{SectionNC} We aim at removing the dependence on $\bar{v}$ in the formulation of the second order necessary condition of Corollary \ref{transNC} above. Note that in the inequality \eqref{maxOmegaP}, $\bar{v}=\dot \bar{y}$ appears only in the term $\bar{v}^\top G[\lambda] \bar{y}.$ We prove in the sequel that we can restrict the maximum in \eqref{maxOmegaP} to the subset of $({\rm co}\, \Lambda)^{\#}$ consisting of the multipliers for which $G[\lambda]$ vanishes. Let $G({\rm co}\, \Lambda)^{\#}$ refer to the subset of $({\rm co}\, \Lambda)^{\#}$ for which $G[\lambda]$ vanishes, i.e. \be G({\rm co}\, \Lambda)^{\#}:=\{\lambda\in ({\rm co}\, \Lambda)^{\#}: G[\lambda] \equiv 0\}. \ee Hence, the following optimality condition holds. \begin{theorem}[New necessary condition]\label{newNC} If $\hat{w}(\cdot)$ is a weak minimum of problem (P), then \be \max_{\lambda\in G({\rm co}\, \Lambda)^{\#}} \Omega_{\mathcal{P}} [\lambda] (\bar{\xi},\bar{u},\bar{y},\dot\bar{y},\bar{y}(T)) \geq 0,\quad \text{on}\ \mathcal{P}. \ee \end{theorem} Theorem \ref{newNC} is an extension of similar results given in Dmitruk \cite{Dmi77}, Milyutin \cite{Mil81} and recently in Aronna et al. \cite{ABDL11}. The proof given in Aronna et al. \cite[Theorem 4.6]{ABDL11} holds for Theorem \ref{newNC} with minor modifications and hence we do not include it in the present article. Notice that when $\hat{w}(\cdot)$ has a unique associated multiplier, from Theorem \ref{newNC} one can deduce that $G({\rm co}\, \Lambda)^{\#}$ is not empty, and since the latter is a singleton, the corollary below follows. This result gives an extension of the necessary conditions stated by Goh in \cite{Goh66} to the present framework. \begin{corollary}[Goh conditions] \label{CoroCBsym} Assume that $\hat{w}(\cdot)$ is a weak minimum having a unique associated multiplier. Then the following conditions holds. \begin{itemize} \item[(i)] $G\equiv 0 $ or, equivalently, the matrix $H_{vx}F_v$ is symmetric, which, in view of \eqref{Vcrochet}, can be written as \begin{equation} p[f_i,f_j]^x(\cdot)\equiv0,\quad \text{for}\ i,j=1,\dots,m, \end{equation} where $p(\cdot)$ is the unique associated adjoint state. \item[(ii)] The matrix \be \label{R2} \begin{pmatrix} H_{uu} & E^\top \\ E& R \end{pmatrix} \ee is positive semidefinite. \end{itemize} \end{corollary} We aim now at stating a necessary condition that does not depend on $\bar{v}(\cdot).$ Let us note that, for $\lambda\in G({\rm co}\, \Lambda)^{\#},$ the quadratic form $\Omega[\lambda]$ does not depend on $\bar{v}(\cdot)$ since its coefficients vanish. We can then consider its continuous extension to $\mathcal{P}_2$ for multipliers $\lambda\in G({\rm co}\, \Lambda)^{\#},$ given by \be \label{OmegaP2} \begin{split} \Omega_{\mathcal{P}_2}[\lambda](\bar{\xi},\bar{u},\bar{y},\bar{h}):= &\,g[\lambda] (\bar{\xi}(0),\bar{\xi}(T),\bar{h}) + \displaystyle \int_0^T \left( \mbox{$\frac{1}{2}$}\bar{\xi}\,^\top H_{xx}[\lambda] \bar{\xi} + \bar{u}^\top H_{ux}[\lambda]\bar{\xi} \right.\\ &\left. +\, \bar{y}^\top M[\lambda] \bar{\xi} + \mbox{$\frac{1}{2}$}\bar{u}^\top H_{uu}[\lambda] \bar{u} + \bar{y}^\top E[\lambda] \bar{u} + \mbox{$\frac{1}{2}$}\bar{y}^\top R[\lambda] \bar{y} \right) \mathrm{d}t, \end{split} \ee where the involved matrices and the function $g$ were defined in \eqref{M}-\eqref{g}. Observe that, since $G[\lambda] \equiv 0,$ one has that $H_{vx}[\lambda] F_v$ is symmetric and, therefore, the $ij$ entry of $R[\lambda]$ can be written as \be \label{Rij} \begin{split} R_{ij}[\lambda]=&-p\left\{ [f_j,[f_0,f_i]^x]^x + \sum_{k=1}^m \hat{v}_k [f_j,[f_k,f_i]^x]^x \right.\\ &+ \left.\left( 2\frac{\partial f_i}{\partial x} \frac{\partial f_j}{\partial u} +\frac{\partial f_j}{\partial x} \frac{\partial f_i}{\partial u} +\frac{\partial^2 f_i}{\partial u\partial x} f_j \right) \dot{\hat{u}} \right\}, \end{split} \ee for each $i,j=1,\dots,m.$ From Theorem \ref{newNC}, it follows: \begin{theorem}[Second order necessary condition in new variables] \label{NCP2} If $\hat{w}(\cdot)$ is a weak minimum of problem (P), then \be \label{Omegapos} \max_{\lambda\in G({\rm co}\, \Lambda)^{\#}} \Omega_{\mathcal{P}_2} [\lambda] (\bar{\xi},\bar{u},\bar{y},\bar{h}) \geq 0,\quad \text{on}\ \mathcal{P}_2. \ee \end{theorem} \section[Second order sufficient condition]{Second order sufficient condition for weak minimum}\label{SectionSC} In this section we present the main contribution of the article: a second order sufficient condition for strict weak optimality. The optimality to be investigated here is with respect to the following \textit{$\gamma$-order:} \be \gamma_\mathcal{P}\big(\bar{x}(0),\bar{u}(\cdot),\bar{y}(\cdot),\bar{h}\big):=\|\bar{x}(0)\|^2+ \|\bar{h}\|^2+ \int_0^T (\|\bar{u}(t)\|^2+\|\bar{y}(t)\|^2)\mathrm{d}t, \ee defined for $(\bar{x}(0),\bar{u}(\cdot),\bar{y}(\cdot),\bar{h})\in \mathbb{R}^n\times\mathcal{U}_2\times \mathcal{V}_2 \times \mathbb{R}^{m}.$ Let us note that $\gamma_\mathcal{P}$ can also be considered as a function of $(\bar{x}(0),\bar{u}(\cdot),\bar{v}(\cdot))\in \mathbb{R}^n\times \mathcal{U}_2\times \mathcal{V}_2$ by setting \be \gamma(\bar{x}(0),\bar{u}(\cdot),\bar{v}(\cdot)):= \gamma_\mathcal{P}(\bar{x}(0),\bar{u}(\cdot),\bar{y}(\cdot),\bar{y}(T)), \ee with $\bar{y}(\cdot)$ being the primitive of $\bar{v}(\cdot)$ defined as in Goh transform \eqref{Goht}. This $\gamma$-order was proposed in Dmitruk \cite{Dmi11} for a simpler {\em partially-affine} problem and it is a natural extension of the order suggested (for control-affine problems) in Dmitruk \cite{Dmi77}. \begin{definition}\label{qgdef}[$\gamma$-growth] We say that $\hat{w}(\cdot)$ satisfies the $\gamma$-\textit{growth condition in the weak sense} if there exist $\varepsilon,\rho> 0$ such that \be \label{qg} \varphi_0(x(0),x(T)) \geq \varphi_0(\hat{x}(0),\hat{x}(T)) + \rho \gamma(x(0)-\hat{x}(0),u(\cdot)-\hat{u}(\cdot),v(\cdot)-\hat{v}(\cdot)), \ee for every feasible trajectory $w(\cdot)$ with $\\|w(\cdot)-\hat{w}(\cdot)\\|_{\infty} < \varepsilon.$ \end{definition} \begin{theorem}[Sufficient condition for weak optimality]\label{SC} \begin{itemize} \item[(i)] Assume that there exists $\rho > 0$ such that \be \label{unifpos} \max_{\lambda\in G(\mathrm{co}\,\Lambda)^{\#}} \Omega_{\mathcal{P}_2} [\lambda] (\bar{\xi},\bar{u},\bar{y},\bar{h}) \geq \rho\gamma_\mathcal{P}(\bar{\xi}(0),\bar{u},\bar{y},\bar{h}),\quad \text{on}\ \mathcal{P}_2. \ee Then $\hat{w}(\cdot)$ is a weak minimum satisfying $\gamma$-growth in the weak sense. \item[(ii)] Conversely, if $\hat{w}(\cdot)$ is a weak solution satisfying $\gamma$-growth in the weak sense and such that $\alpha_0>0$ for every $\lambda\in G(\mathrm{co}\,\Lambda)^{\#},$ then \eqref{unifpos} holds for some positive $\rho.$ \end{itemize} \end{theorem} In the absence of the {\em nonlinear control} $u,$ Theorem \ref{SC} was proved in Dmitruk \cite{Dmi77}. In Aronna et al. \cite{ABDL11} the same result was shown for the case of scalar control subject to bounds. As a consequence of Theorem \ref{SC} and standard results on positive quadratic mappings due to Hestenes \cite{Hes51} we get the following pointwise condition. \begin{corollary} \label{CoroSC} If $\hat{w}(\cdot)$ satisfies the uniform positivity in \eqref{unifpos} and it has a unique associated multiplier, then the matrix in \eqref{R2} is uniformly positive definite, i.e. $$ \begin{pmatrix} H_{uu} & E^\top \\ E & R \end{pmatrix} \succeq \rho I,\quad \text{on} \ [0,T], $$ where $I$ refers to the identity matrix. \end{corollary} \begin{remark} \label{RemarkLC} Under suitable hypotheses, Goh in \cite{GohThesis} proved that the {\it strengthened generalized Legendre-Clebsch condition} is a consequence of the uniform positivity in \eqref{unifpos} (see Goh \cite[Section 4.8]{GohThesis} and Aronna \cite[Remark 8.2]{Aro11}). Thus, in that situation, the controls can be expressed as smooth functions of the state and costate variable, as was assumed here. \end{remark} The remainder of this section is devoted to the proof of Theorem \ref{SC}. Several technical lemmas that are used in the following proof were stated and proved in the Appendix \ref{AppendixSC}. \noindent{\em Proof of Theorem \ref{SC}.} {\em (i)} We shall prove that if \eqref{unifpos} holds for some $\rho>0,$ then $\hat{w}(\cdot)$ satisfies $\gamma$-growth in the weak sense. By the contrary, let us assume that the $\gamma$-growth condition \eqref{qg} is not satisfied. Consequently, there exists a sequence of feasible trajectories $\{w_k(\cdot)=(x_k(\cdot),u_k(\cdot),v_k(\cdot))\}$ converging to $\hat{w}(\cdot)$ in the weak sense, such that \begin{equation} \label{qgrowth} \varphi_0(x_k(0),x_k(T))\leq \varphi_0(\hat{x}(0),\hat{x}(T))+o(\gamma_k), \end{equation} with \begin{equation} (\delta x_k(\cdot),\bar{u}_k(\cdot),\bar{v}_k(\cdot)):= w_k(\cdot)-\hat{w}(\cdot) \,\,\, \text{and} \,\,\, \gamma_k:= \gamma (\delta x_{k}(0),\bar{u}_k(\cdot),\bar{v}_k(\cdot)). \end{equation} Let $(\bar{\xi}_k(\cdot),\bar{u}_k(\cdot),\bar{y}_k(\cdot))$ be the transformed directions defined by Goh transformation \eqref{Goht}. We divide the remainder of the proof of item {\em (i)} in the following two steps: \begin{itemize} \item[(A)] First we prove that the sequence given by \begin{equation} (\mathring{\xi}_k(\cdot),\mathring{u}_k(\cdot),\mathring{y}_k(\cdot),\mathring{h}_k):= (\bar{\xi}_k(\cdot),\bar{u}_k(\cdot),\bar{y}_k(\cdot),\bar{h}_k)/{\sqrt{\gamma_k}} \end{equation} where $\bar{h}_k:=\bar{y}_k(T),$ contains a weak converging subsequence whose weak limit is an element\\ $(\mathring\xi(\cdot),\mathring u(\cdot),\mathring y(\cdot),\mathring h)$ of $\mathcal{P}_2.$ \item[(B)] Afterwards, making use of the latter sequence and its weak limit, we show that the uniform positivity hypothesis \eqref{unifpos} together with \eqref{qgrowth} lead to a contradiction. \end{itemize} We shall begin by {Part (A).} For this we take an arbitrary Lagrange multiplier $\lambda$ in $(\mathrm{co}\,\Lambda)^{\#}.$ By multiplying the inequality \eqref{qgrowth} by $\alpha_0,$ and adding the nonpositive term \be \sum_{i=1}^{d_{\varphi}}\alpha_i\varphi_i(x_{k}(0),x_{k}(T))+\sum_{j=1}^{d_{\eta}}\beta_j\eta_j(x_{k}(0),x_k(T)), \ee to its left-hand side, we get \begin{equation} \label{quadlag} \mathcal{L}[\lambda](w_k)\leq\mathcal{L}[\lambda](\hat{w})+o(\gamma_k). \end{equation} Note that the elements of the sequence $(\mathring\xi_{k}(0),\mathring u_k(\cdot),\mathring y_k(\cdot),\mathring h_k)$ have unit $\mathbb{R}^n\times \mathcal{U}_2\times \mathcal{V}_2\times \mathbb{R}^m$-norm. The Banach-Alaoglu Theorem (see e.g. Br\'ezis \cite[Theorem III.15]{Bre83}) implies that, extracting if necessary a subsequence, there exists $(\mathring\xi(0),\mathring u(\cdot),\mathring y(\cdot),\mathring h)\in \mathbb{R}^n\times \mathcal{U}_2\times \mathcal{V}_2\times \mathbb{R}^m$ such that \begin{equation} \label{limityk} \mathring\xi_{k}(0)\rightarrow \mathring\xi(0),\quad \mathring u_k\rightharpoonup \mathring u,\quad \mathring y_k\rightharpoonup \mathring y,\quad \mathring h_k\rightarrow \mathring h, \end{equation} where the two limits indicated with $\rightharpoonup$ are considered in the weak topology of $\mathcal{U}_2$ and $\mathcal{V}_2,$ respectively. Let $\mathring\xi(\cdot)$ denote the solution of the equation \eqref{xieq} associated with $(\mathring \xi(0),\mathring u(\cdot),\mathring y(\cdot)).$ Hence, it follows easily that $\mathring\xi(\cdot)$ is the limit of $\mathring\xi_k(\cdot)$ in (the strong topology of) $\mathcal{X}_2.$ With the aim of proving that $(\mathring\xi(\cdot),\mathring u(\cdot),\mathring v(\cdot),\mathring h)$ belongs to $\mathcal{P}_2,$ it remains to check that the linearized endpoint constraints \eqref{tlinearconseq}-\eqref{tlinearconsineq} are verified. Observe that, for each index $0\leq i\leq d_{\varphi},$ one has \be \label{phineg'} D\varphi_i(\hat{x}(0),\hat{x}(T)) (\mathring\xi(0),\mathring\xi(T)+B[T]\mathring h) = \lim_{k\rightarrow \infty} D\varphi_i(\hat{x}(0),\hat{x}(T))\left(\frac{\bar{x}_{k}(0),\bar{x}_{k}(T)}{\sqrt{\gamma_k}}\right). \ee In order to prove that the right hand-side of \eqref{phineg'} is nonpositive, we consider the following first order Taylor expansion of $\varphi_i$ around $(\hat{x}(0),\hat{x}(T)):$ \begin{equation} \begin{split} \varphi_i(x_{k}&(0),x_{k}(T))\\ &= \varphi_i(\hat{x}(0),\hat{x}(T)) + D \varphi_i(\hat{x}(0),\hat{x}(T)) (\delta x_{k}(0), \delta x_{k}(T)) +o(\|(\delta x_{k}(0), \delta x_{k}(T))\|). \end{split} \end{equation} Previous equation and Lemmas \ref{lemmaxbar} and \ref{lemmaeta} imply \begin{equation} \label{taylor_phi} \varphi_i(x_{k}(0),x_{k}(T))=\varphi_i(\hat{x}(0),\hat{x}(T))+D\varphi_i(\hat{x}(0),\hat{x}(T))(\bar{x}_{k}(0),\bar{x}_{k}(T))+o(\sqrt{\gamma_k}). \end{equation} Thus, the following approximation for the right hand-side of \eqref{phineg'} holds, \be \label{difphi} D\varphi_i(\hat{x}(0),\hat{x}(T)) \left( \frac{\bar{x}_{k}(0),\bar{x}_{k}(T)}{\sqrt{\gamma_k}} \right) =\frac{\varphi_i(x_{k}(0),x_{k}(T))-\varphi_i(\hat{x}(0),\hat{x}(T))}{\sqrt{\gamma_k}}+o(1). \ee Since $w_k(\cdot)$ is a feasible trajectory, it satisfies the final inequality constraint \eqref{finalineq} and, therefore, equations \eqref{phineg'} and \eqref{difphi} yield, for $1\leq i\leq d_{\varphi},$ $$ D\varphi_i(\hat{x}(0),\hat{x}(T))(\mathring\xi(0),\mathring\xi(T)+B[T]\mathring h)\leq 0. $$ Now, for $i=0,$ use \eqref{qgrowth} to get the corresponding inequality. Analogously, one has \be \label{eta0} D\eta_j(\hat{x}(0),\hat{x}(T))(\mathring\xi(0),\mathring\xi(T)+B[T]\mathring h) = 0,\quad \mathrm{for}\ j=1,\hdots,d_{\eta}. \ee Thus $(\mathring\xi(\cdot),\mathring u(\cdot),\mathring y(\cdot),\mathring h)$ satisfies \eqref{tlinearconseq}-\eqref{tlinearconsineq}, and hence it belongs to $\mathcal{P}_2.$ Let us now pass to {Part (B).} Notice that from the expansion of $\mathcal{L}$ given in \eqref{taylor0} of Lemma \ref{lemmasc2}, and the inequality \eqref{quadlag} we get \be \Omega_{\mathcal{P}_2} [\lambda] (\mathring\xi_k,\mathring u_k,\mathring y_k,\mathring h_k) \leq o(1), \ee and thus \be \label{lims} \liminf_{k\rightarrow \infty}\, \Omega_{\mathcal{P}_2} [\lambda] (\mathring\xi_k,\mathring u_k,\mathring y_k,\mathring h_k) \leq 0. \ee Let us consider the subset of $G({\rm co}\,\Lambda)^{\#}$ defined by \be \Lambda^{\#,\rho}:= \{\lambda\in G({\rm co}\,\Lambda)^{\#}: \Omega_{\mathcal{P}_2}[\lambda] - \rho\gamma_\mathcal{P}\ {\rm is}\ {\rm weakly}\,{\rm l.s.c.}\,{\rm on}\ \mathcal{H}_2\times\mathbb{R}^m \}. \ee By applying Lemma \ref{quadform} to the inequality of uniform positivity \eqref{unifpos} one gets \be \label{maxLamrho} \max_{\lambda\in \Lambda^{\#,\rho}} \Omega_{\mathcal{P}_2} [\lambda] (\bar{\xi},\bar{u},\bar{y},\bar{h}) - \rho\gamma_\mathcal{P}(\bar{\xi}(0),\bar{u},\bar{y},\bar{h}) \geq 0,\quad \text{on}\ \mathcal{P}_2. \ee Let us take the multiplier $\mathring\lambda\in \Lambda^{\#,\rho}$ that attains the maximum in \eqref{maxLamrho} for the direction $(\mathring \xi(\cdot),\mathring u(\cdot),\mathring y(\cdot),\mathring h)$ of $\mathcal{P}_2.$ We get \be \begin{split} 0 &\leq \Omega_{\mathcal{P}_2}[\mathring\lambda](\mathring \xi,\mathring u,\mathring y,\mathring h) - \rho\gamma_\mathcal{P}(\mathring \xi(0),\mathring u,\mathring y,\mathring h)\\ &\leq \liminf_{k\rightarrow \infty} \Omega_{\mathcal{P}_2}[\mathring\lambda](\mathring\xi_k,\mathring u_k,\mathring y_k,\mathring h_k) - \rho\gamma_\mathcal{P}(\mathring\xi_{k}(0),\mathring u_k,\mathring y_k,\mathring h_k) \leq -\rho, \end{split} \ee since $\Omega_{\mathcal{P}_2}[\mathring\lambda] - \rho\gamma_\mathcal{P}$ is weakly-l.s.c., $\gamma_\mathcal{P}(\mathring\xi_{k}(0),\mathring u_k,\mathring y_k,\mathring h_k)=1$ for every $k$ and inequality \eqref{lims} holds. This leads us to a contradiction since $\rho\gr0.$ Therefore, the desired result follows, this is, the uniform positivity \eqref{unifpos} implies strict weak optimality with $\gamma$-growth. {\em (ii)} Let us now prove the second statement of the theorem. Assume that $\hat{w}(\cdot)$ is a weak solution satisfying $\gamma$-growth in the weak sense for some constant $\rho'>0,$ and such that $\alpha_0>0$ for every multiplier $\lambda \in G({\rm co}\,\Lambda)^{\#}.$ Let us consider the modified problem \be\label{tildeP}\tag{$\tilde P$} \min \{ \varphi_0(x(0),x(T))-\rho' \gamma(x(0)-\hat{x}(0),u(\cdot)-\hat{u}(\cdot),v(\cdot)-\hat{v}(\cdot)) : \text{\eqref{stateeq}-\eqref{finalineq}} \}, \ee and rewrite it in the Mayer form \be\label{2tildeP}\tag{${\breve P}$} \begin{split} \min\,\, &\varphi_0(x(0),x(T))-\rho' \big(\|x(0)-\hat{x}(0)\|^2 + \|y(T)-\hat{y}(T)\|^2 +\pi_{1}(T)+\pi_{2}(T)\big),\\ &\text{\eqref{stateeq}-\eqref{finalineq}},\\ & \dot y=v,\\ & \dot\pi_1 = (u-\hat{u})^2,\\ & \dot \pi_2 = (y-\hat{y})^2,\\ & y(0)=0,\ \pi_{1}(0)=0,\ \pi_{2}(0)=0. \end{split} \ee We will next apply the second order necessary condition of Theorem \ref{NCP2} to \eqref{2tildeP} at the point $(w(\cdot)=\hat{w}(\cdot),y(\cdot)=\hat{y}(\cdot),\pi_1(\cdot)\equiv0,\pi_2(\cdot)\equiv0).$ Simple computations show that at this solution each critical cone (see \eqref{P}) is the projection of the corresponding critical cone of \eqref{2tildeP}, and that the same holds for the set of multipliers. Furthermore, the second variation of \eqref{2tildeP} evaluated at a multiplier ${\breve \lambda} \in G ({\rm co}\, {\breve \Lambda}^{\#})$ is given by \be \Omega_{\mathcal{P}_2}[\lambda] (\bar{\xi},\bar{u},\bar{y},\bar{y}(T)) - \alpha_0 \rho'\gamma_\mathcal{P}(\bar{x}(0),\bar{u},\bar{y},\bar{y}(T)), \ee where $\lambda \in G ({\rm co}\, {\Lambda})^{\#}$ is the corresponding multiplier for problem \eqref{P}. Hence, the necessary condition in Theorem \ref{NCP2} (see Remark \ref{NCt} below) implies that for every $(\bar{\xi}(\cdot),\bar{u}(\cdot),\bar{v}(\cdot),\bar{h})\in \mathcal{P}_2,$ there exists $\lambda \in G ({\rm co}\, {\Lambda})^{\#}$ such that \begin{equation} \Omega_{\mathcal{P}_2}[\lambda] (\bar{\xi},\bar{u},\bar{y},\bar{y}(T)) - \alpha_0 \rho'\gamma_\mathcal{P}(\bar{x}(0),\bar{u},\bar{y},\bar{y}(T)) \geq 0. \end{equation} Setting $\displaystyle\rho:= \min_{G ({\rm co}\, {\Lambda})^{\#}} \alpha_0 \rho' >0$ the desired result follows. This completes the proof of the theorem. \findem \begin{remark}\label{NCt} Since the dynamics of \eqref{2tildeP} are not autonomous, what we applied above is an extension of Theorem \ref{NCP2} to time-dependent dynamics. The latter follows easily by adding a state variable $\kappa$ with dynamics $\dot\kappa = 1$ and $\kappa(0)=0.$ \end{remark} \section{Example}\label{SectionExample} We consider the following example from Dmitruk-Shishov \cite{DmiShi10}: \be \label{Pexample}\tag{PE} \begin{split} \min\,\, & -2x_1(T)x_2(T)+x_3(T),\\ & \dot x_1 = x_2+u,\\ & \dot x_2 = v,\\ & \dot x_3 = x_1^2 + x_2^2 + x_2 v+u^2\\ & x_1(0) = x_2(0) = x_3(0) =0. \end{split} \ee Let us use $p_1,p_2,p_3$ to denote the costate variables associated to \eqref{Pexample}. Observe that $\dot p_3(\cdot)\equiv 0$ and $p_3(T)=1,$ thus $p_3(\cdot)\equiv 1.$ Note as well that the linearized state equation implies $\dot\bar{x}_2=\bar{v},\, \bar{x}_1(0)=\bar{x}_2(0)=\bar{x}_3(0)=0.$ Consequently, $\bar{y}(\cdot)=\bar{x}_2(\cdot),$ $\bar{\xi}_1(0)=\bar{\xi}_2(0)=\bar{\xi}_3(0)=0,$ and $$ \bar{\xi}_2(\cdot)=\bar{x}_2(\cdot)-\bar{y}(\cdot) \equiv 0, $$ where the first equality follows from Goh's transformation \eqref{Goht}. Recalling the definitions given in \eqref{M}-\eqref{g}, the second variation $\Omega_{\mathcal{P}_2}$ (defined in \eqref{OmegaP2}) on the critical cone $\mathcal{P}_2$ of \eqref{Pexample} gives: \be \Omega_{\mathcal{P}_2}(\bar{x},\bar{u},\bar{y},\bar{h}) = \mbox{$\frac{1}{2}$} \bar{h}^2+\int_0^T (\bar{x}_1^2+\bar{u}^2+\bar{y}^2)\mathrm{d}t. \ee We see that $\Omega_{\mathcal{P}_2}$ verifies the sufficient condition \eqref{unifpos}. We should now look for a feasible solution that verifies the first order optimality conditions. In Aronna \cite{Aro13} we used the {\em shooting algorithm} to solve problem \eqref{Pexample} numerically. The numerical tests converged to the optimal solution $(\hat{u},\hat{v})(\cdot)\equiv 0$ for arbitrary guesses of the initial values of the costate variables. It is inmediate to check that $\hat{w}(\cdot)\equiv 0$ is a feasible trajectory that verifies the first order optimality conditions. Since the second variation at this $\hat{w}$ verifies the sufficient condition of Theorem \ref{SC}, we conclude that $\hat{w}(\cdot)$ is a strict weak optimal trajectory that satisfies $\gamma$-growth. \section{Conclusion and possible extensions}\label{SectionConclusion} We studied optimal control problems in the Mayer form governed by systems that are affine in some components of the control variable. A set of `no gap' necessary and sufficient second order optimality conditions was provided. These conditions apply to a weak minimum, consider fairly general endpoint constraints and do not assume uniqueness of multiplier. We further derived the Goh conditions when we assume uniqueness of multiplier. The main result of the article is Theorem \ref{SC}. The interest of this result is that it can be applied either to prove optimality of some candidate solution of a given problem, or to show convergence of an associated shooting algorithm as stated in Aronna \cite{Aro13} and proved in the detail in the technical report Aronna \cite{Aro11}. This algorithm and its proof of convergence apply also to partially-affine problems with bounds on the control and bang-singular solutions, and hence its convergence has strong practical interest. The results here presented can be pursued by many interesting extensions. One of the most important extensions are the optimality conditions for bang-singular solutions for problems containing closed control constraints. \section{Acknowledgments} Part of this work was done during my Ph.D. under the supervision of Fr\'ed\'eric Bonnans, who I thank for the great guidance. I also acknowledge the anonymous referee for his careful reading and useful remarks. \appendix \section{Proofs of technical results} We include in this part the proofs that were omitted throughout the article. \subsection{}\label{proofexpansionlagrangian} \noindent{\em Proof of Lemma \ref{expansionlagrangian}.} We shall omit the dependence on $\lambda$ for the sake of simplicity of notation. Let us consider the following second order Taylor expansions, written in a compact form, \begin{gather} \label{expell} \ell (x(0),x(T)) = \ell + {D\ell} (\delta x(0),\delta x(T)) + \mbox{$\frac{1}{2}$} {D^2\ell} (\delta x(0),\delta x(T))^2 + L_\ell\|(\delta x(0),\delta x(T))\|^3, \\ \label{expfi} f_i(x,u) = f_{i} + {D f_{i}} (\delta x,\delta u) + \mbox{$\frac{1}{2}$} {D^2 f_{i}} (\delta x,\delta u)^2 +L \|(\delta x,\delta u) \|^3. \end{gather} Observe that, in view of the transversality conditions \eqref{transvcond} and the costate equation \eqref{costateeq}, one has \be \label{Dell} \begin{split} {D\ell}\,&(\delta x(0),\delta x(T)) = -p(0)\, \delta x(0) + p(T) \, \delta x(T) \\ &= \int_0^T \left[ \dot{p}\, \delta x + p \dot{\delta x} \right] \mathrm{d}t = \int_0^T p\,\Big[-\big({D_xf_i}+\sum_{i=1}^m\hat{v}_i {D_xf_i}\big) \delta x +\dot{\delta x}\Big] \mathrm{d}t. \end{split} \ee In the definition of $\mathcal{L}$ given in \eqref{lagrangian}, replace $\ell(x(0),x(T))$ and $f_i(x,u)$ by their Taylor expansions \eqref{expell}-\eqref{expfi} and use the identity \eqref{Dell}. This yields \begin{align} \mathcal{L}(w) =&\,\,\mathcal{L}(\hat{w}) + \displaystyle \int_0^T \big[ H_u\delta u + H_v\delta v \big]\mathrm{d}t + \Omega(\delta x,\delta u,\delta v) \\ &+ \displaystyle \int_0^T \big[ H_{vxx}(\delta x, \delta x,\delta v) + 2H_{vux}(\delta x,\delta u,\delta v) + H_{vuu}(\delta u,\delta u,\delta v) \big] \mathrm{d}t \\ &+ \, L_\ell \|(\delta x(0),\delta x(T))\|^3 + L(1+\\|v\\|_\infty)\,\\|(\delta x,\delta u) \\|_{\infty} \displaystyle\int_0^T p\,\|(\delta x,\delta u) \|^2 \mathrm{d}t. \end{align} Finally, to obtain \eqref{expansionLag}, remove the first order terms by the stationarity conditions \eqref{stationarity}, and use the Cauchy-Schwarz inequality in the last integral. This completes the proof. \findem \subsection{Proof of Theorem \ref{classicalNC}}\label{AppendixclassicalNC} Let us write problem (P) in an {\em abstract} form defining, for $j=1,\dots,d_\eta$ and $i=0,\dots,d_\varphi,$ \begin{gather} \bar{\eta}_j:\mathbb{R}^n\times \mathcal{U} \times \mathcal{V} \rightarrow \mathbb{R},\quad (x(0),u(\cdot),v(\cdot))\mapsto \bar{\eta}_j(x(0),u(\cdot),v(\cdot)):=\eta_j(x(0),x(T)),\\ \bar{\varphi}_i:\mathbb{R}^n\times \mathcal{U} \times \mathcal{V} \rightarrow \mathbb{R},\quad (x(0),u(\cdot),v(\cdot))\mapsto \bar{\varphi}_i(x(0),u(\cdot),v(\cdot)):=\varphi_i(x(0),x(T)), \end{gather} where $x(\cdot)\in\mathcal{X}$ is the solution of \eqref{stateeq} associated with $(x(0),u(\cdot),v(\cdot)).$ Hence, (P) can be written as the following problem in the space $\mathbb{R}^n\times \mathcal{U} \times \mathcal{V},$ \be \label{AP}\tag{AP} \begin{split} \min\,\,&\bar\varphi_0(x(0),u,v);\\ \,\,\text{s.t.}\,\, &\bar\eta_j(x(0),u,v)=0,\ \text{for}\ j=1,\dots,d_{\eta},\\ &\bar\varphi_i(x(0),u,v)\leq 0,\ \text{for}\ j=1,\dots,d_{\varphi}. \end{split} \ee Notice that if $\hat{w}(\cdot)$ is a weak solution of (P) then $(\hat{x}(0),\hat{u}(\cdot),\hat{v}(\cdot))$ is a local solution of (AP). \begin{definition} \label{DefCQ} We say that the endpoint equality constraints are {\it qualified} if \be\label{QC} D\bar\eta(\hat{x}(0),\hat{u},\hat{v})\ \text{is onto from}\ \mathbb{R}^n\times \mathcal{U}\times\mathcal{V}\ \text{to}\ \mathbb{R}^{d_{\eta}}. \ee When \eqref{QC} does not hold, the constraints are {\em not qualified} or {\em unqualified}. \end{definition} The proof of Theorem \ref{classicalNC} is divided in two cases: qualified and not qualified endpoint equality constraints. In the latter case the condition \eqref{classicalNCeq} follows easily and it is shown in Lemma \ref{degNC} below. The proof for the qualified case is done by means of an auxiliary linear problem and duality arguments. \begin{lemma}\label{degNC} If the equality constraints are not qualified then \eqref{classicalNCeq} holds. \end{lemma} \begin{proof} Observe that since $D\bar\eta(\hat{x}(0),\hat{u},\hat{v})$ is not onto there exists $\beta\in\mathbb{R}^{d_{\eta},}$ with $\|\beta\|=1$ such that $\sum_{j=1}^{d_{\eta}} \beta_j D\bar\eta_j(\hat{x}(0),\hat{u},\hat{v})=0$ and consequently, \begin{equation} \sum_{j=1}^{d_{\eta}} \beta_j D\eta_j(\hat{x}(0),\hat{x}(T))=0. \end{equation} Set $\lambda:=(p(\cdot),\alpha,\beta)$ with $p(\cdot)\equiv 0$ and $\alpha=0.$ Then both $\lambda$ and $-\lambda$ are in $\Lambda.$ Observe that \begin{equation} \Omega[\lambda](\bar{x},\bar{u},\bar{v})= \mbox{$\frac{1}{2}$} \sum_{j=1}^{d_{\eta}} \beta_j D^2\eta_j(\hat{x}(0),\hat{x}(T))(\bar{x}(0),\bar{x}(T))^2. \end{equation} Thus, either $\Omega[\lambda](\bar{x},\bar{u},\bar{v})$ or $\Omega[-\lambda](\bar{x},\bar{u},\bar{v})$ is necessarily nonnegative. The desired result follows. \end{proof} Let us now deal with the qualified case. Take a critical direction $\bar{w}(\cdot)=(\bar{x},\bar{u},\bar{v})(\cdot)\in \mathcal{C}$ and consider the problem in the variables $\tau\in\mathbb{R}$ and $r=(r_{x_0},r_u,r_v)\in\mathbb{R}^n\times \mathcal{U}\times \mathcal{V}$ given by \be\label{QPw}\tag{QP$_{\bar{w}}$} \begin{split} \min\,\, &\tau \\ \text{s.t.}\,\,\, & D\bar\eta(\hat{x}(0),\hat{u},\hat{v}) r +D^2\bar\eta(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 =0,\\ & D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v}) r +D^2\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 \leq \tau,\ \ i=0,\dots,d_{\varphi}. \end{split} \ee \begin{proposition}\label{dualpb} Assume that $\hat{w}(\cdot)$ is a weak solution of \eqref{AP} for which the endpoint equality constraints are qualified. Let $\bar{w}(\cdot)\in \mathcal{C}$ be a critical direction. Then the problem \eqref{QPw} is feasible and has nonnegative value. \end{proposition} \noindent{\em Proof of Proposition \ref{dualpb}.} Step I. Let us first show feasibility. Since $D\bar\eta(\hat{x}(0),\hat{u},\hat{v})$ is onto, there exists $r\in \mathbb{R}^n\times \mathcal{U}\times \mathcal{V}$ for which the equality constraint in \eqref{QPw} is satisfied. Set \be \label{zeta1} \tau:=\max_{0\leq i \leq d_{\varphi}}\{D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v}) r +D^2\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 \}. \ee Then $(\tau,r)$ is feasible for \eqref{QPw}. Step II. Let us now prove that \eqref{QPw} has nonnegative value. Suppose on the contrary that there is $(\tau,r)\in \mathbb{R}\times\mathbb{R}^n\times \mathcal{U}\times \mathcal{V}$ feasible for \eqref{QPw} with $\tau<0.$ We shall look for a family of feasible solutions of \eqref{AP} referred as $\{r(\sigma)\}_{\sigma}$ with the following properties: it is defined for small positive values of $\sigma$ and it satisfies \be\label{estrsigma} r(\sigma)\underset{\sigma\to0}{\longrightarrow} (\hat{x}(0),\hat{u},\hat{v})\ \text{in} \ \mathbb{R}^n\times \mathcal{U}\times \mathcal{V},\ \text{and}\ \bar\varphi_0(r(\sigma) )< \bar\varphi_0(\hat{x}(0),\hat{u},\hat{v}). \ee The existence of such family $\{r(\sigma)\}_{\sigma}$ will contradict the local optimality of $(\hat{x}(0),\hat{u},\hat{v}).$ Consider hence \begin{equation} \tilde{r}(\sigma):= (\hat{x}(0),\hat{u},\hat{v})+\sigma(\bar{x}(0),\bar{u},\bar{v})+\mbox{$\frac{1}{2}$}\sigma^2 r. \end{equation} Let $0\leq i\leq d_{\varphi}$ and observe that \be\label{estvarphi} \begin{split} \bar\varphi_i(\tilde{r}(\sigma)) =& \,\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})+\sigma D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})\\ &+\mbox{$\frac{1}{2}$}\sigma^2\left[ D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})r+D^2 \bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 \right] + o(\sigma^2)\\ \leq&\,\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})+\mbox{$\frac{1}{2}$}\sigma^2\tau+o(\sigma^2), \end{split} \ee where last inequality holds since $(\bar{x},\bar{u},\bar{v})(\cdot)$ is a critical direction and in view of the definition of $\tau$ in \eqref{zeta1}. Analogously, one has \begin{equation} \bar\eta(\tilde{r}(\sigma))=o(\sigma^2). \end{equation} Since $D\bar\eta(\hat{x}(0),\hat{u},\hat{v})$ is onto, there exists $r(\sigma) \in \mathbb{R}\times \mathcal{U}\times \mathcal{V}$ such that $\\| r(\sigma)-\tilde{r}(\sigma)\\|_{\infty}=o(\sigma^2)$ and $\bar\eta(r(\sigma))=0.$ This follows by applying the Implicit Function Theorem to the mapping \begin{equation} (r,\sigma)\mapsto \bar\eta\left((\hat{x}(0),\hat{u},\hat{v})+\sigma(\bar{x}(0),\bar{u},\bar{v})+\mbox{$\frac{1}{2}$}\sigma^2 r\right)=\bar\eta(\tilde{r}(\sigma)). \end{equation} On the other hand, by taking $\sigma$ sufficiently small in estimate \eqref{estvarphi}, we obtain \begin{equation} \bar\varphi_i({r}(\sigma))<\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v}), \end{equation} since $\tau<0.$ Hence $r(\sigma)$ is feasible for \eqref{AP} and verifies \eqref{estrsigma}. This contradicts the optimality of $(\hat{x}(0),\hat{u},\hat{v}).$ We conclude then that all the feasible solutions of \eqref{QPw} have $\tau\geq 0$ and, therefore, its value is nonnegative. \findem We shall now proceed to prove Theorem \ref{classicalNC}. \noindent{\em Proof of Theorem \ref{classicalNC}.} The unqualified case is covered by Lemma \ref{degNC} above. Hence, for this proof, assume that \eqref{QC} holds. Given $\bar{w}(\cdot)\in \mathcal{C}, $ note that \eqref{QPw} can be regarded as a linear problem in the variables $(\zeta, r),$ whose associated dual is given by \begin{align} \label{dualQP1} \max_{(\alpha,\beta)}\,\,\, &\sum_{i=0}^{d_{\varphi}} \alpha_i D^2\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 + \sum_{j=1}^{d_{\eta}} \beta_j D^2\bar\eta_j(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 \\ \label{dualQP2} {\rm s.t.}\,\,\, & \sum_{i=0}^{d_{\varphi}} \alpha_i D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})+\sum_{j=1}^{d_{\eta}} \beta_j D\bar\eta_j(\hat{x}(0),\hat{u},\hat{v})=0,\\ \label{dualQP3} &\sum_{i=0}^{d_{\varphi}} \alpha_i=1,\quad \alpha \geq 0. \end{align} The Proposition \ref{dualpb} above and the linear duality result Bonnans \cite[Theorem 3.43]{BonOC} imply that \eqref{dualQP1}-\eqref{dualQP3} has finite nonnegative value (the reader is referred to Shapiro \cite{Sha01} and references therein for a general theory on linear duality). Consequently, there exists a feasible solution $(\bar\alpha,\bar\beta)\in \mathbb{R}^{d_\varphi+d_\eta+1}$ to \eqref{dualQP1}-\eqref{dualQP3}, with associated nonnegative and finite value. Set $(\alpha,\beta) := (\bar\alpha,\bar\beta)/(\sum_{i=0}^{d_{\varphi}} \|\bar\alpha_i\|+\sum_{j=1}^{d_{\eta}} \|\bar\beta_j\|),$ where the denominator is not zero in view of \eqref{dualQP3}. We get that $(\alpha,\beta)\in \mathbb{R}^{d_\varphi+d_\eta+1}$ verifies \eqref{nontriv}-\eqref{alphapos}, \eqref{dualQP2} and \be \label{DM2} \sum_{j=1}^{d_{\eta}} \beta_j D^2\bar\eta_j(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2+ \sum_{i=0}^{d_{\varphi}} \alpha_i D^2\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 \geq0. \ee \if{ implies that there cannot exist $(\tau,r)\in \mathbb{R}\times \mathbb{R}^n\times \mathcal{U}\times \mathcal{V}$ such that \begin{equation} \left\{ \begin{split} &D\bar\eta(\hat{x}(0),\hat{u},\hat{v}) r +D^2\bar\eta(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 =0,\\ &D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v}) r +D^2\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 \leq \tau,\ \text{for}\ i=0,\dots,d_{\varphi},\\ &\tau <0. \end{split} \right. \end{equation} Therefore, the Dubovitskii-Milyutin Theorem \cite{DubMil} guarantees the existence of $(\alpha,\beta)\in\mathbb{R}^{1+d_{\varphi}+d_{\eta}}$ such that $(\alpha,\beta)\neq0,$ $\alpha\geq 0$ and \begin{gather} \label{DM1} \displaystyle\sum_{i=0}^{d_{\varphi}} \alpha_i D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v}) + \sum_{j=1}^{d_{\eta}} \beta_j D\bar\eta_j(\hat{x}(0),\hat{u},\hat{v}) =0,\\ \label{DM2} \displaystyle\sum_{i=0}^{d_{\varphi}}\alpha_i D^2\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 + \sum_{j=1}^{d_{\eta}} \beta_j D^2\bar\eta_j(\hat{x}(0),\hat{u},\hat{v})(\bar{x}(0),\bar{u},\bar{v})^2 \geq 0 \end{gather} }\fi For this $(\alpha,\beta),$ let $p(\cdot)$ be the solution of \eqref{costateeq} with final condition \be \label{DM5} p(T)=\sum_{i=0}^{d_{\varphi}} \alpha_i D_{x_T}\varphi_i(\hat{x}(0),\hat{x}(T)) + \sum_{j=1}^{d_{\eta}} \beta_j D_{x_T}\eta_j(\hat{x}(0),\hat{x}(T)).\ee We shall prove that $\lambda := (\alpha,\beta,p(\cdot))$ is in $\Lambda,$ i.e. that also the first line in \eqref{transvcond} and the stationarity conditions \eqref{stationarity} hold. Let $(\tilde{x},\tilde{u},\tilde{v})(\cdot)\in\mathcal{W}$ be the solution of the linearized state equation \eqref{lineareq}. In view of \eqref{dualQP2}, \be \label{DM4} \sum_{i=0}^{d_{\varphi}} \alpha_i D\bar\varphi_i(\hat{x}(0),\hat{u},\hat{v})(\tilde{x}(0),\tilde{u},\tilde{v}) + \sum_{j=1}^{d_{\eta}} \beta_j D\bar\eta_j(\hat{x}(0),\hat{u},\hat{v})(\tilde{x}(0),\tilde{u},\tilde{v}) =0, \ee Hence, rewriting in terms of the endpoint Lagrangian $\ell$ and using \eqref{DM5}-\eqref{DM4}, one has \begin{equation} \label{DM3} 0 = D\ell [\lambda] (\hat{x}(0),\hat{x}(T))(\tilde{x}(0),\tilde{x}(T)) = D_{x_0}\ell [\lambda] (\hat{x}(0),\hat{x}(T))\tilde{x}(0) + p(T) \tilde{x}(T) \pm p(0) \tilde{x}(0). \end{equation} By regrouping terms in the previous equation, we get \be \label{DM6} \begin{split} 0&= \Big( D_{x_0}\ell[\lambda](\hat{x}(0),\hat{x}(T))+p(0)\Big) \tilde{x}(0) + \int_0^T (\dot{p}\tilde{x} +p \dot{\tilde{x}} ) \mathrm{d}t \\ &= \Big( D_{x_0}\ell[\lambda](\hat{x}(0),\hat{x}(T))+p(0)\Big) \tilde{x}(0) + \int_0^T \big( H_u[\lambda] \tilde{u} + H_v[\lambda] \tilde{v} \big) \mathrm{d}t, \end{split} \ee where we used \eqref{costateeq} and \eqref{lineareq} in the last equality. Since \eqref{DM6} holds for all $(\tilde{x}(0),\tilde{u}(\cdot),\tilde{v}(\cdot))$ in $\mathbb{R}^n \times \mathcal{U}\times \mathcal{V},$ the first line in \eqref{transvcond} and the stationarity conditions in \eqref{stationarity} are necessarily verified. Thus, $\lambda$ is an element of $\Lambda.$ On the other hand, simple computations yield that \eqref{DM2} is equivalent to \begin{equation} \Omega[\lambda] (\bar{x},\bar{u},\bar{v})\geq0, \end{equation} and, therefore, the result follows. \findem \subsection{} {\em Proof of Lemma \ref{Omegat}.}\label{AppendixOmegat} First recall that the term $\bar{v}^\top H_{vu}[\lambda] \bar{u}$ in $\Omega[\lambda]$ vanishes since we are taking $\lambda\in \Lambda^{\#}$ and, in view of Lemma \ref{Lambdawlsc}, $H_{vu}[\lambda] \equiv 0.$ In the remainder of the proof we omit the dependence on $\lambda$ for the sake of simplicity. Replacing $\bar{x}$ in the definition of $\Omega$ in equation \eqref{Omega} by its expression in \eqref{Goht} yields \be \label{J2} \begin{split} \Omega & (\bar{x},\bar{u},\bar{v}) =\\ &\, \mbox{$\frac{1}{2}$}\ell''(\hat{x}(0),\hat{x}(T))\big(\bar{\xi}(0),\bar{\xi}(T)+F_{v}[T]\,\bar{y}(T)\big)^2 + \displaystyle \int_0^T \left[ \mbox{$\frac{1}{2}$}(\bar{\xi}+F_v\,\bar{y})^\top H_{xx}(\bar{\xi}+F_v\,\bar{y}) \right. \\ & \left. +\,\bar{u}^\top H_{ux} (\bar{\xi}+F_v\,\bar{y}) + \bar{v}^\top H_{vx} (\bar{\xi}+F_v\,\bar{y}) + \mbox{$\frac{1}{2}$}\bar{u}^\top H_{uu}\,\bar{u} \right] \mathrm{d}t. \end{split} \ee In view of \eqref{xieq} one gets \be \label{J2.2.1} \int_0^T \bar{v}^\top H_{vx}\, \bar{\xi} \mathrm{d}t =[\bar{y}^\top H_{vx}\, \bar{\xi}]_0^T -\int_0^T \bar{y}^\top \{\dot{H}_{vx}\,\bar{\xi}+H_{vx}(F_x\,\bar{\xi}+F_u\,\bar{u}+B\,\bar{y})\} \mathrm{d}t. \ee The decomposition of $H_{vx}\,F_v$ introduced in \eqref{SV} followed by an integration by parts leads to \be\label{J2.2.2} \begin{split} \int_0^T \bar{v}^\top H_{vx}\,F_v \bar{y} \mathrm{d}t &=\int_0^T \bar{v}^\top (S+G) \bar{y} \mathrm{d}t\\ &=\mbox{$\frac{1}{2}$}[\bar{y}^\top S\bar{y}]_0^T+\int_0^T(-\mbox{$\frac{1}{2}$} \bar{y}^\top \dot S\bar{y} + \bar{v}^\top G\bar{y} )\mathrm{d}t. \end{split} \ee The result follows by replacing using \eqref{J2.2.1} and \eqref{J2.2.2} in \eqref{J2}. \findem \section{Technical lemmas used in the proof of the main Theorem \ref{SC}}\label{AppendixSC} Recall first the following classical result for ordinary differential equations. \begin{lemma}[Gronwall's Lemma] \label{GronLem} Let $a(\cdot)\in W^{1,1}(0,T;\mathbb{R}^n),$ $b(\cdot)\in L^1(0,T)$ and $c(\cdot)\in L^1(0,T)$ be such that $\|\dot{a}(t)\|\leq b(t) + c(t) \|a(t)\|$ for a.a. $t\in (0,T).$ Then \begin{equation} \\|a(\cdot)\\|_\infty \leq e^{\\|c(\cdot)\\|_1}\big(\|a(0)\|+\\|b(\cdot)\\|_1 \big). \end{equation} \end{lemma} For the lemma below recall the definition of the space $\mathcal{H}_2$ given in \eqref{H2}. \begin{lemma} \label{lemmaxbar} There exists $\rho> 0$ such that \be \label{xbargamma} \|\bar{x}(0)\|^2+\\|\bar{x}(\cdot)\\|_2^2+\|\bar{x}(T)\|^2\leq \rho \gamma(\bar{x}(0),\bar{u}(\cdot),\bar{v}(\cdot)), \ee for every linearized trajectory $(\bar{x},\bar{u},\bar{v})(\cdot)\in\mathcal{H}_2.$ The constant $\rho$ depends on $\\|A\\|_{\infty},$ $\\|F_v\\|_{\infty},$ $\\|E\\|_{\infty}$ and $\\|B\\|_{\infty}.$ \end{lemma} \begin{proof} Throughout this proof, whenever we put $\rho_i$ we refer to a positive constant depending on $\\|A\\|_{\infty},$ $\\|F_v\\|_{\infty},$ $\\|E\\|_{\infty},$ and/or $\\|B\\|_{\infty}.$ Let $(\bar{x},\bar{u},\bar{v})(\cdot)\in\mathcal{H}_2$ and $(\bar{\xi},\bar{y})(\cdot)$ be defined by Goh's Transformation \eqref{Goht}. Thus $(\bar{\xi},\bar{u},\bar{y})(\cdot)$ is solution of \eqref{xieq}. Gronwall's Lemma \ref{GronLem} and Cauchy-Schwarz inequality yield \be\label{lemmazxit} \\|\bar{\xi}\\|_{\infty}\leq \rho_1 (\|\bar{\xi}(0)\|^2+\\|\bar{u}\\|_2^2+\\|\bar{y}\\|_2^2)^{1/2}\leq \rho_1 \gamma_\mathcal{P}(\bar{x}(0),\bar{u},\bar{y},\bar{y}(T))^{1/2}, \ee with $\rho_1=\rho_1(\\|A\\|_1,\\|E\\|_{\infty},\\|B\\|_{\infty}).$ This last inequality together with the relation between $\bar{\xi}(\cdot)$ and $\bar{x}(\cdot)$ provided by \eqref{Goht} imply \be\label{lemmazz} \\|\bar{x}\\|_2\leq \\|\bar{\xi}\\|_2+\\|F_v\\|_{\infty}\\|\bar{y}\\|_2\leq\rho_2 \gamma_\mathcal{P}(\bar{x}(0),\bar{u},\bar{y},\bar{y}(T))^{1/2}, \ee for $\rho_2=\rho_2(\rho_1,\\|F_v\\|_{\infty}).$ On the other hand, \eqref{Goht} and estimate \eqref{lemmazxit} lead to \begin{equation} \|\bar{x}(T)\|\leq \|\bar{\xi}(T)\|+\\|F_v\\|_{\infty}\|\bar{y}(T)\|\leq \rho_1 \gamma_\mathcal{P}(\bar{x}(0),\bar{u},\bar{y},\bar{y}(T))^{1/2} +\\|F_v\\|_{\infty}\|\bar{y}(T)\|. \end{equation} Then, in view of Young's inequality `$2{ab}\leq {a^2+b^2} $' for real numbers $a,b,$ one gets \be\label{lemmazzT} \|\bar{x}(T)\|^2\leq \rho_3 \gamma_\mathcal{P}(\bar{x}(0),\bar{u},\bar{y},\bar{y}(T)), \ee for some $\rho_3=\rho_3(\rho_1,\\|F_v\\|_{\infty}).$ The desired estimate follows from \eqref{lemmazz} and \eqref{lemmazzT}. \end{proof} Notice that Lemma \ref{lemmaxbar} above gives an estimate of the linearized state in the order $\gamma.$ The following result shows that the analogous property holds for the variation of the state variable as well and it is a natural extension of a similar result given in Dmitruk \cite{Dmi87} for control-affine systems. \begin{lemma} \label{lemmadeltax} Given $C> 0,$ there exists $\rho> 0$ such that \be \|\delta x(0)\|^2+\\|\delta x(\cdot)\\|^2_2+\|\delta x(T)\|^2\leq \rho \gamma(\delta x(0),\delta u(\cdot),\delta v(\cdot)), \ee for every $(x,u,v)(\cdot)$ solution of the state equation \eqref{stateeq} having $\\|v(\cdot)\\|_2\leq C,$ and where $\delta w(\cdot):=w(\cdot)-\hat{w}(\cdot).$ The constant $\rho$ depends on $C,$ $\\|B\\|_{\infty},$ $\\|\dot B\\|_{\infty}$ and the Lipschitz constants of $f_i.$ \end{lemma} \begin{proof} In order to simplify the notation we omit the dependence on $t.$ Consider $(x,u,v)(\cdot)$ solution of \eqref{stateeq} with $\\|v(\cdot)\\|_2\leq C.$ Let $\delta w(\cdot):=w(\cdot)-\hat{w}(\cdot),$ $\delta y(t):=\int_0^t \delta v(s) {\rm d}s,$ and $\xi(\cdot):=\delta x(\cdot)-B[\cdot]\delta y(\cdot),$ with $y(t):=\int_0^t v(s)\mathrm{d}s.$ Note that \be \label{xidot} \begin{split} \dot\xi= & f_0(x,u)-f_0(\hat{x},\hat{u})+\sum_{i=1}^m \left[v_if_i(x,u)-\hat{v}_if_i(\hat{x},\hat{u})\right]-\dot B\delta y- \sum_{i=1}^m \delta v_i\, f_i(\hat{x},\hat{u})\\ =&f_0(x,u)-f_0(\hat{x},\hat{u})+\sum_{i=1}^m v_i[f_i(x,u)-f_i(\hat{x},\hat{u})]-\dot{B}\delta y. \end{split} \ee In view of the Lipschitz-continuity of $f_i,$ \be \|f_i(x,u)-f_i(\hat{x},\hat{u})\|\leq L (\|\delta x\|+\|\delta u\|) \leq L(\|\xi\|+\\|B\\|_{\infty}\|\delta y\|+\|\delta u\|), \ee for some $L> 0.$ Thus, from \eqref{xidot} it follows \begin{equation} \|\dot\xi\| \leq L(\|\xi\|+\\|B\\|_{\infty}\|\delta y\|+\|\delta u\|)(1+\|v\|)+\\|\dot B\\|_{\infty}\|\delta y\|. \end{equation} Applying Gronwall's Lemma \ref{GronLem} one gets \begin{equation} \\| \xi\\|_\infty \leq e^{L\\|1+\|v\|\,\\|_1}\Big[ \|\xi(0)\| + \left\\|L(1+\|v\|)(\\|F_v\\|_\infty \|\delta y\| + \|\delta u\|)+\\|\dot{F}_v\\|_\infty \|\delta y\|\,\right\\|_1 \Big], \end{equation} and Cauchy-Schwarz inequality applied to previous estimate yields \be \\|\xi\\|_{\infty} \leq \rho_1\big(\|\xi(0)\|+\\|\delta y\\|_1+\\|\delta u\\|_1 + \\|\delta y\\|_2\\|v\\|_2 + \\|\delta u\\|_2\\|v\\|_2\big), \ee for $\rho_1=\rho_1(L,C,\\|F_v\\|_{\infty},\\|\dot{F}_v\\|_{\infty}).$ Since $\\|\delta x\\|_2\leq \\|\xi\\|_2+\\|F_v\\|_{\infty}\\|\delta y\\|_2,$ by previous estimate and Cauchy-Schwarz inequality, the result follows. \end{proof} Finally, the following lemma gives an estimate for the difference between the variation of the state variable and the linearized state. \begin{lemma} \label{lemmaeta} Consider $C> 0$ and $w(\cdot)=(x,u,v)(\cdot)\in\mathcal{W}$ a trajectory with $\\|w(\cdot)-\hat{w}(\cdot) \\|_{\infty}\leq C.$ Set $(\delta x,\delta u,\delta v)(\cdot):= w(\cdot)-\hat{w}(\cdot)$ and let $\bar{x}(\cdot)$ be the linearization of $\hat{x}(\cdot)$ associated with $(\delta x,\delta u,\delta v)(\cdot).$ Define \be \vartheta(\cdot):=\delta x(\cdot)-\bar{x}(\cdot). \ee Then, $\vartheta(\cdot)$ is solution of the differential equation \be \label{doteta} \begin{split} \dot\vartheta &= D_xf_{0} (\hat{x},\hat{u})\vartheta +\sum_{i=1}^m \hat{v}_i D_xf_{i} (\hat{x},\hat{u})\vartheta + \sum_{i=1}^m \delta v_i Df_{i} (\hat{x},\hat{u})(\delta x,\bar{u}) + \zeta,\\ \vartheta(0) &= 0, \end{split} \ee where the remainder $\zeta(\cdot)$ is given by \be \label{zeta} \zeta:= \mbox{$\frac{1}{2}$} D^2f_0(\hat{x},\hat{u})(\delta x,\bar{u})^2+\sum_{i=1}^m \mbox{$\frac{1}{2}$} v_i D^2f_i(\hat{x},\hat{u})(\delta x,\bar{u})^2+L \left(1+\sum_{i=1}^m v_i\right)\|(\delta x,\bar{u})\|^3, \ee and $L$ is a Lipschitz constant for $D^2f_i,$ uniformly in $i=0,\dots,m.$ Furthermore, $\zeta(\cdot)$ satisfies the estimates \be \label{estzeta} \\|\zeta(\cdot)\\|_{\infty} < \rho_1C,\quad \\|\zeta(\cdot)\\|_2 < \rho_1C\sqrt\gamma, \ee where $\rho_1= \rho_1(C,\\|D^2 f\\|_{\infty},L,\\|v\\|_\infty+1).$ If in addition, $C\rightarrow 0,$ the following estimates for $\vartheta(\cdot)$ hold \be \label{esteta} \\|\vartheta(\cdot)\\|_{\infty} = o(\sqrt\gamma), \quad \\|\dot\vartheta(\cdot)\\|_2 = o(\sqrt\gamma). \ee \end{lemma} \begin{proof} We shall note first that \be \label{dotdeltax1} \dot{\delta x} = f_0(x,u)-f_0(\hat{x},\hat{u})+\sum_{i=1}^m v_i \big[f_i(x,u)-f_i(\hat{x},\hat{u}) \big] + \sum_{i=1}^m \delta v_{i}\,f_i(\hat{x},\hat{u}). \ee Consider the following second order Taylor expansions for $f_i,$ \be \label{Taylorfi} f_i(x,u)= f_i(\hat{x},\hat{u}) + D f_i(\hat{x},\hat{u})(\delta x,\delta u) + \mbox{$\frac{1}{2}$} D^2f_i(\hat{x},\hat{u})(\delta x,\delta u)^2 + {L}\|(\delta x,\delta u)\|^3. \ee Combining \eqref{dotdeltax1} and \eqref{Taylorfi} yields \be \label{dotdeltax} \dot{\delta x} = D f_0(\hat{x},\hat{u})(\delta x,\delta u)+\sum_{i=1}^m v_i D f_i(\hat{x},\hat{u})(\delta x,\delta u)+\sum_{i=1}^m \delta v_{i}f_i(\hat{x},\hat{u})+\zeta, \ee with the remainder being given by \eqref{zeta}. The linearized equation \eqref{lineareq} together with \eqref{dotdeltax} lead to \eqref{doteta}. In view of \eqref{zeta} and Lemma \ref{lemmadeltax}, it can be seen that the estimates in \eqref{estzeta} hold. On the other hand, by applying Gronwall's Lemma \ref{GronLem} to \eqref{doteta}, and using Cauchy-Schwarz inequality afterwards lead to \begin{equation} \\|\vartheta\\|_{\infty} \leq \rho_3\left\\|\sum_{i=1}^m \delta v_i D f_{i} (\hat{x},\hat{u}) (\delta x,\delta u) + \zeta\right\\|_1 \leq \rho_4 \Big[ \\|\delta v\\|_2(\\|\delta x\\|_2 + \\|\delta u\\|_2) + \\|\zeta\\|_2 \Big], \end{equation} for some positive $\rho_3,\rho_4$ depending on $\\|\hat{v}\\|_{\infty}$ and $\\|Df\\|_{\infty}.$ Finally, using the estimate in Lemma \ref{lemmadeltax} and \eqref{estzeta} just obtained, the inequalities in \eqref{esteta} follow. \end{proof} In view of Lemmas \ref{expansionlagrangian}, \ref{lemmaxbar}, \ref{lemmadeltax} and \ref{lemmaeta} we can justify the following technical result that is an essential point in the proof of the sufficient condition of Theorem \ref{SC}. \begin{lemma} \label{lemmasc2} Let $w(\cdot)\in\mathcal{W}$ be a trajectory. Set $(\delta x,\delta u,\delta v)(\cdot):= w(\cdot)-\hat{w}(\cdot),$ and $\bar{x}(\cdot)$ its corresponding linearized state, i.e. the solution of \eqref{lineareq}-\eqref{lineareq0} associated with $(\delta x(0),\delta u(\cdot),\delta v(\cdot)).$ Assume that $\\|w(\cdot)-\hat{w}(\cdot)\\|_{\infty} \rightarrow 0.$ Then \be \label{taylor0} \mathcal{L}[\lambda](w) = \mathcal{L}[\lambda](\hat{w}) + \Omega[\lambda](\bar{x},\delta u,\delta v)+o(\gamma), \ee for every $\lambda \in {\rm co}\, \Lambda.$ \end{lemma} \begin{proof} For the sake of simplicity of notation, we shall omit the dependence on $\lambda.$ Let us recall the expansion of the Lagrangian function given in Lemma \ref{expansionlagrangian}, and observe that it also holds for any $\lambda$ in ${\rm co}\, \Lambda.$ Next, notice that, by Lemma \ref{lemmadeltax}, $ \mathcal{L}(w) = \mathcal{L}(\hat{w})+ \Omega(\delta x,\delta u,\delta v)+o(\gamma). $ Hence, \be \label{taylorlemma} \mathcal{L}(w) = \mathcal{L}(\hat{w})+ \Omega(\bar{x},\delta u,\delta v)+\Delta\Omega + o(\gamma), \ee with $ \Delta\Omega:= \Omega(\delta x,\delta u,\delta v)-\Omega(\bar{x},\delta u,\delta v). $ The next step is using Lemmas \ref{lemmaxbar}, \ref{lemmadeltax} and \ref{lemmaeta} to prove that \begin{equation} \label{rest} \Delta \Omega=o(\gamma). \end{equation} Note that $\mathcal{Q}(a,a)-\mathcal{Q}(b,b)=\mathcal{Q}(a+b,a-b),$ for any bilinear mapping $\mathcal{Q},$ and any pair $a,b$ of elements in its domain. Set $\vartheta(\cdot):=\delta x(\cdot)- \bar{x}(\cdot)$ as it is done in Lemma \ref{lemmaeta}. Hence, \begin{equation} \begin{split} \Delta \Omega = &\, \mbox{$\frac{1}{2}$}\ell'' \Big( (\delta x(0)+\bar{x}(0),\delta x(T)+\bar{x}(T)) , (0,\vartheta(T)) \Big) \\ & +\displaystyle\int_0^T [\mbox{$\frac{1}{2}$}(\delta x+\bar{x})^\top Q\vartheta + \delta u^\top E\vartheta + \delta v^\top C\vartheta ] \mathrm{d}t. \end{split} \end{equation} The estimates in Lemmas \ref{lemmaxbar}, \ref{lemmadeltax} and \ref{lemmaeta} yield $\Delta \Omega = \int_0^T \delta v^\top C\vartheta \mathrm{d}t + o(\gamma).$ Integrating by parts in the latter expression and using \eqref{esteta} leads to \begin{equation} \int_0^T \delta v^\top C\vartheta \mathrm{d}t = [\bar{y}^\top C\vartheta]_0^T - \int_0^T \bar{y}^\top(\dot {C}\vartheta + C \dot\vartheta )\mathrm{d}t = o(\gamma), \end{equation*} and hence the desired result follows. \end{proof} \end{document}	arXiv
Accepted Manuscript Publisher's Accepted Manuscript DOE PAGES Journal Article: Continuum of quantum fluctuations in a three-dimensional S=1 Heisenberg magnet Title: Continuum of quantum fluctuations in a three-dimensional S=1 Heisenberg magnet Conventional crystalline magnets are characterized by symmetry breaking and normal modes of excitation called magnons, with quantized angular momentum $$\hbar$$. Neutron scattering correspondingly features extra magnetic Bragg diffraction at low temperatures and dispersive inelastic scattering associated with single magnon creation and annihilation. Exceptions are anticipated in so-called quantum spin liquids, as exemplified by the one-dimensional spin-1/2 chain, which has no magnetic order and where magnons accordingly fractionalize into spinons with angular momentum $$\hbar$$/2. This is spectacularly revealed by a continuum of inelastic neutron scattering associated with two-spinon processes. Here, we report evidence for these key features of a quantum spin liquid in the three-dimensional antiferromagnet NaCaNi2F7. We show that despite the complication of random Na1+–Ca2+ charge disorder, NaCaNi2F7 is an almost ideal realization of the spin-1 antiferromagnetic Heisenberg model on a pyrochlore lattice. Magnetic Bragg diffraction is absent and 90% of the neutron spectral weight forms a continuum of magnetic scattering with low-energy pinch points, indicating NaCaNi2F7 is in a Coulomb-like phase. Our results demonstrate that disorder can act to freeze only the lowest-energy magnetic degrees of freedom; at higher energies, a magnetic excitation continuum characteristic of fractionalized excitations persists. Plumb, K. W. [1]; Search DOE PAGES for author "Plumb, K. W." Search DOE PAGES for ORCID "0000-0002-2372-8139" Search orcid.org for ORCID "0000-0002-2372-8139" Changlani, Hitesh J. [1]; Scheie, A. [1]; Zhang, Shu [1]; Krizan, J. W. [2]; Rodriguez-Rivera, J. A. [3]; Qiu, Yiming [4]; Winn, B. [5]; Cava, R. J. [2]; Broholm, C. L. [6] Johns Hopkins Univ., Baltimore, MD (United States) Princeton Univ., NJ (United States). Dept. of Chemistry National Inst. of Standards and Technology (NIST), Gaithersburg, MD (United States). Center for Neutron Research; Univ. of Maryland, College Park, MD (United States). Dept. of Materials Science and Engineering National Inst. of Standards and Technology (NIST), Gaithersburg, MD (United States). Center for Neutron Research Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States) Johns Hopkins Univ., Baltimore, MD (United States); National Inst. of Standards and Technology (NIST), Gaithersburg, MD (United States). Center for Neutron Research; Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States) USDOE Grant/Contract Number: AC05-00OR22725 Nature Physics Journal Volume: 15; Journal Issue: 1; Journal ID: ISSN 1745-2473 Nature Publishing Group (NPG) 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS Plumb, K. W., Changlani, Hitesh J., Scheie, A., Zhang, Shu, Krizan, J. W., Rodriguez-Rivera, J. A., Qiu, Yiming, Winn, B., Cava, R. J., and Broholm, C. L.. Continuum of quantum fluctuations in a three-dimensional S=1 Heisenberg magnet. United States: N. p., 2018. Web. https://doi.org/10.1038/s41567-018-0317-3. Plumb, K. W., Changlani, Hitesh J., Scheie, A., Zhang, Shu, Krizan, J. W., Rodriguez-Rivera, J. A., Qiu, Yiming, Winn, B., Cava, R. J., & Broholm, C. L.. Continuum of quantum fluctuations in a three-dimensional S=1 Heisenberg magnet. United States. https://doi.org/10.1038/s41567-018-0317-3 Plumb, K. W., Changlani, Hitesh J., Scheie, A., Zhang, Shu, Krizan, J. W., Rodriguez-Rivera, J. A., Qiu, Yiming, Winn, B., Cava, R. J., and Broholm, C. L.. Mon . "Continuum of quantum fluctuations in a three-dimensional S=1 Heisenberg magnet". United States. https://doi.org/10.1038/s41567-018-0317-3. https://www.osti.gov/servlets/purl/1559715. title = {Continuum of quantum fluctuations in a three-dimensional S=1 Heisenberg magnet}, author = {Plumb, K. W. and Changlani, Hitesh J. and Scheie, A. and Zhang, Shu and Krizan, J. W. and Rodriguez-Rivera, J. A. and Qiu, Yiming and Winn, B. and Cava, R. J. and Broholm, C. L.}, abstractNote = {Conventional crystalline magnets are characterized by symmetry breaking and normal modes of excitation called magnons, with quantized angular momentum $\hbar$. Neutron scattering correspondingly features extra magnetic Bragg diffraction at low temperatures and dispersive inelastic scattering associated with single magnon creation and annihilation. Exceptions are anticipated in so-called quantum spin liquids, as exemplified by the one-dimensional spin-1/2 chain, which has no magnetic order and where magnons accordingly fractionalize into spinons with angular momentum $\hbar$/2. This is spectacularly revealed by a continuum of inelastic neutron scattering associated with two-spinon processes. Here, we report evidence for these key features of a quantum spin liquid in the three-dimensional antiferromagnet NaCaNi2F7. We show that despite the complication of random Na1+–Ca2+ charge disorder, NaCaNi2F7 is an almost ideal realization of the spin-1 antiferromagnetic Heisenberg model on a pyrochlore lattice. Magnetic Bragg diffraction is absent and 90% of the neutron spectral weight forms a continuum of magnetic scattering with low-energy pinch points, indicating NaCaNi2F7 is in a Coulomb-like phase. Our results demonstrate that disorder can act to freeze only the lowest-energy magnetic degrees of freedom; at higher energies, a magnetic excitation continuum characteristic of fractionalized excitations persists.}, doi = {10.1038/s41567-018-0317-3}, journal = {Nature Physics}, month = {10} Free Publicly Available Full Text Accepted Manuscript (DOE) Publisher's Version of Record Citation Metrics: Cited by: 17 works Citation information provided by Works referenced in this record: Insulating spin glasses journal, March 1979 Villain, Jacques Zeitschrift f�r Physik B Condensed Matter and Quanta, Vol. 33, Issue 1 DOI: 10.1007/BF01325811 Molecular Spin Resonance in the Geometrically Frustrated Magnet MgCr 2 O 4 by Inelastic Neutron Scattering journal, October 2008 Tomiyasu, K.; Suzuki, H.; Toki, M. 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Nature Physics, Vol. 15, Issue 1 Experimental signatures of a three-dimensional quantum spin liquid in effective spin-1/2 Ce2Zr2O7 pyrochlore Gao, Bin; Chen, Tong; Tam, David W. Nature Physics, Vol. 15, Issue 10 A quantum liquid of magnetic octupoles on the pyrochlore lattice Sibille, Romain; Gauthier, Nicolas; Lhotel, Elsa Thermodynamics of the pyrochlore-lattice quantum Heisenberg antiferromagnet Müller, Patrick; Lohmann, Andre; Richter, Johannes Physical Review B, Vol. 100, Issue 2 DOI: 10.1103/physrevb.100.024424 Goldstone modes in the emergent gauge fields of a frustrated magnet Garratt, S. J.; Chalker, J. T. Dynamical Structure Factor of the Three-Dimensional Quantum Spin Liquid Candidate NaCaNi 2 F 7 Zhang, Shu; Changlani, Hitesh J.; Plumb, Kemp W. text, January 2019 DOI: 10.17169/refubium-26440 All Cited By Similar Records in DOE PAGES and OSTI.GOV collections: Extended Scattering Continua Characteristic of Spin Fractionalization in the Two-dimensional Frustrated Quantum Magnet Cs2CuCl4Observed by Neutron Scattering Journal Article Coldea, Radu ; Tennant, D. A. ; Tyleczynski, Z. - Physical Review B The magnetic excitations of the quasi-2D spin-1/2 frustrated Heisenberg antiferromagnet Cs{sub 2}CuCl{sub 4} are explored throughout the 2D Brillouin zone using high-resolution time-of-flight inelastic neutron scattering. Measurements are made both in the magnetically ordered phase, stabilized at low temperatures by the weak interlayer couplings, as well as in the spin liquid phase above the ordering temperature T{sub N}, when the 2D magnetic layers are decoupled. In the spin liquid phase the dynamical correlations are dominated by highly dispersive excitation continua, a characteristic signature of fractionalization of S = 1 spin waves into pairs of deconfined S = 1/2 spinons andmore » the hallmark of a resonating-valence-bond (RVB) state. The boundaries of the excitation continua have strong 2D-modulated incommensurate dispersion relations. Upon cooling below T{sub N} magnetic order in an incommensurate spiral forms due to the 2D frustrated couplings. In this phase sharp magnons carrying a small part of the total scattering weight are observed at low energies, but the dominant continuum scattering which occurs at medium to high energies is essentially unchanged compared to the spin liquid phase. Linear spin-wave theory including one- and two-magnon processes can describe the sharp magnon excitation, but not the dominant continuum scattering, which instead is well described by a parametrized two-spinon cross section. Those results suggest a crossover in the nature of the excitations from S = 1 spin waves at low energies to deconfined S = 1/2 spinons at medium to high energies, which could be understood if Cs{sub 2}CuCl{sub 4} was in the close proximity of transition between a fractional RVB spin liquid and a magnetically ordered state. A large renormalization factor of the excitation energies [R = 1.63(5)], indicating strong quantum fluctuations in the ground state, is obtained using the exchange couplings determined from saturation-field measurements. We provide an independent consistency check of this quantum renormalization factor using measurements of the second moment of the paramagnetic scattering.« less Identifying spinon excitations from dynamic structure factor of spin-1/2 Heisenberg antiferromagnet on the Kagome lattice Journal Article Zhu, W. ; Gong, Shou-shu ; Sheng, D. N. - Proceedings of the National Academy of Sciences of the United States of America A spin-more » 1 / 2 lattice Heisenberg Kagome antiferromagnet (KAFM) is a prototypical frustrated quantum magnet, which exhibits exotic quantum spin liquids that evade long-range magnetic order due to the interplay between quantum fluctuation and geometric frustration. So far, the main focus has remained on the ground-state properties; however, the theoretical consensus regarding the magnetic excitations is limited. Here, we study the dynamic spin structure factor (DSSF) of the KAFM by means of the density matrix renormalization group. By comparison with the well-defined magnetically ordered state and the chiral spin liquid sitting nearby in the phase diagram, the KAFM with nearest neighbor interactions shows distinct dynamical responses. The DSSF displays important spectral intensity predominantly in the low-frequency region around the Q = M point in momentum space and shows a broad spectral distribution in the high-frequency region for momenta along the boundary of the extended Brillouin zone. The excitation continuum identified from momentum- and energy-resolved DSSF signals emergent spinons carrying fractional quantum numbers. These results capture the main observations in the inelastic neutron scattering measurements of herbertsmithite and indicate the spin liquid nature of the ground state. By tracking the DSSF across quantum-phase transition between the chiral spin liquid and the magnetically ordered phase, we identify the condensation of two-spinon bound state driving the quantum-phase transition.« less Low-energy spin dynamics in rare-earth perovskite oxides Journal Article Podlesnyak, Andrey A. ; Nikitin, Stanislav ; Ehlers, Georg - Journal of Physics. Condensed Matter We review recent studies of spin dynamics in rare-earth orthorhombic perovskite oxides of the type RMO3, where R is a rare-earth ion and M is a transition-metal ion, using single-crystal inelastic neutron scattering (INS). After a short introduction to the magnetic INS technique in general, the results of INS experiments on both transition-metal and rare-earth subsystems for four selected compounds (YbFeO3, TmFeO3, YFeO3, YbAlO3) are presented. We show that the spectrum of magnetic excitations consists of two types of collective modes that are well separated in energy: gapped magnons with a typical bandwidth of <70 meV, associated with the antiferromagneticallymore » (AFM) ordered transition-metal subsystem, and AFM fluctuations of <5 meV within the rare-earth subsystem, with no hybridization of those modes. We discuss the high-energy conventional magnon excitations of the 3d subsystem only briefly, and focus in more detail on the spectacular dynamics of the rare-earth sublattice in these materials. We observe that the nature of the ground state and the low-energy excitation strongly depends on the identity of the rare-earth ion. In the case of non-Kramers ions, the low-symmetry crystal field completely eliminates the degeneracy of the multiplet state, creating a rich magnetic field-temperature phase diagram. In the case of Kramers ions, the resulting ground state is at least a doublet, which can be viewed as an effective quantum spin-1/2. Equally important is the fact that in Yb-based materials the nearest-neighbor exchange interaction dominates in one direction, despite the three-dimensional nature of the orthoperovskite crystal structure. The observation of a fractional spinon continuum and quantum criticality in YbAlO3 demonstrates that Kramers rare-earth based magnets can provide realizations of various aspects of quantum low-dimensional physics.« less https://doi.org/10.1088/1361-648x/ac1367 Magnons and continua in a magnetized and dimerized spin - 1 2 chain Journal Article Stone, M. B. ; Chen, Y. ; Reich, D. H. ; ... - Physical Review. B, Condensed Matter and Materials Physics We examine the magnetic field dependent excitations of the dimerized spin -1/2 chain, copper nitrate, with antiferromagnetic intra-dimer exchangemore » $$J_1=0.44$$ (1) meV and exchange alternation $$\alpha=J_2/J_1=0.26$$ (2). Magnetic excitations in three distinct regimes of magnetization are probed through inelastic neutron scattering at low temperatures. At low and high fields there are three and two long-lived magnon-like modes, respectively. The number of modes and the anti-phase relationship between the wave-vector dependent energy and intensity of magnon scattering reflect the distinct ground states: A singlet ground state at low fields $$\mu_0H < \mu_0H_{c1} = 2.8$$ T and an $$S_z=1/2$$ product state at high fields $$\mu_0H > \mu_0H_{c2} = 4.2$$ T. Lastly, in the intermediate field regime, a continuum of scattering for $$\hbar\omega\approx J_1$$ is indicative of a strongly correlated gapless quantum state without coherent magnons.« less Coexistence and Interaction of Spinons and Magnons in an Antiferromagnet with Alternating Antiferromagnetic and Ferromagnetic Quantum Spin Chains Journal Article Zhang, H. ; Zhao, Z. ; Gautreau, D. ; ... - Physical Review Letters In conventional quasi-one-dimensional antiferromagnets with quantum spins, magnetic excitations are carried by either magnons or spinons in different energy regimes: they do not coexist independently, nor could they interact with each other. In this Letter, by combining inelastic neutron scattering, quantum Monte Carlo simulations and Random Phase Approximation calculations, we report the discovery and discuss the physics of the coexistence of magnons and spinons and their interactions in Botallackite-Cu2(OH)3Br. This is a unique quantum antiferromagnet consisting of alternating ferromagnetic and antiferromagnetic Spin-1/2 chains with weak interchain couplings. Furthermore, our study presents a new paradigm where one can study the interactionmore » between two different types of magnetic quasiparticles: magnons and spinons.« less	CommonCrawl
\begin{document} \large \title{On the computation of coefficients of modular forms: the reduction modulo p approach} \author{Jinxiang Zeng and Linsheng Yin} \address{Department of Mathematical Science, Tsinghua University, Beijing 100084, P. R. China} \email{[email protected]} \subjclass[2012]{Primary 11F30, 11G20, 11Y16, 14Q05, 14H05} \keywords{modular forms, Hecke algebra, modular curves, elliptic curves, Jacobian} \begin{abstract} In this paper, we present a probabilistic algorithm to compute the coefficients of modular forms of level one. Focusing on the Ramanujan's tau function, we give the explicit complexity of the algorithm. From a practical viewpoint, the algorithm is particularly well suited for implementations. \end{abstract} \maketitle \section{Introduction and Main Results} In the book \cite{Edixhoven}, Couveignes, Edixhoven et el. described an algorithm for computing coefficients of modular forms for the group SL$_2(\mathbb{Z})$, and Bruin \cite{Bruin} generalized the method to modular forms for the congruence subgroups of the form $\Gamma_1(n)$. Their methods lead to polynomial time algorithms for computing coefficients of modular forms. However, efficient ways to implement the algorithms and explicit complexity analysis are still being studied. Working with complex number field, Bosman's explicit computations show the power of these new methods, see \cite{Edixhoven}. As one of the applications, he largely improved the known result on Lehmer's nonvanishing conjecture for Ramanujan's tau function. For the recent progress in this direction see \cite{Mascot}. Following Couveignes's idea \cite{Couveignes}, we give a probabilistic algorithm, which seems to be more suitable to deal with complexity analysis. Instead of using Brill-Noether's algorithm, we work with the function field of the modular curve, using He\ss's algorithm to make computations in the Jacobian of the modular curve. We illustrate our method on the discriminant modular form, which is defined as $$\Delta(q)=q\prod_{n=1}^{\infty}(1-q^n)^{24}=\sum_{n=1}^{\infty}\tau(n)q^n,$$ where $z \in \mathcal{H},q=e^{2\pi i z}$. Let $p$ be a prime, using Deligne's bound we have $\|\tau(p)\|\le2p^{11/2}$, therefore to compute $\tau(p)$ it suffices to compute $\tau(p)\mod\ell$ for all primes $\ell$ bounded above by a constant in $\textrm{O}(\log p)$. Let $\ell$ be a prime, the mod-$\ell$ Galois representation associated to $\Delta(q)$ is denoted as $$\rho_\ell:\Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to \mbox{\rm GL}_2(\mathbb{F}_\ell),$$ which satisfying that for any prime $p\not=\ell$, the characteristic polynomial of the Frobenius endomorphism $\textrm{Frob}_p$ is $x^2-\tau(p)x+p^{11}\mod \ell$. Therefore, to compute $\tau(p)\mod\ell$ it suffices to compute the Galois representation $\rho_\ell$. It is well known that $\rho_\ell$ can be realized by the subspace $V_\ell$ in the $\ell$-torsions $J_1(\ell)(\overline{\mathbb{Q}})[\ell]$ of the Jacobian variety $J_1(\ell)$ of the modular curve $X_1(\ell)$, which can be written as a finite intersection $$V_\ell=\bigcap_{1\le i\le (\ell^2-1)/6} \ker(T_i-\tau(i),J_1(\ell)(\overline{\mathbb{Q}})[\ell]),$$ where $T_i,i=1 ,\ldots,\frac{\ell^2-1}{6}$ are Hecke operators. Indeed $V_\ell$ is a group scheme over $\mathbb{Q}$ of order $\ell^2$, which is called the Ramanujan subspace \cite{Couveignes}. As showed in \cite{Edixhoven}, the heights of the elements of $V_\ell$ are well bounded, which enables us to know $V_\ell$ explicitly. More precisely, we have function $\iota:V_\ell\to \mathbb{A}_{\mathbb{Q}}^1$, such that the heights of the coefficients of $P(X):=\prod_{\alpha\in V_\ell\setminus\textrm{O}}(X-\iota(\alpha))$ are bounded above by $\textrm{O}(\ell^\delta)$, where the constant $\delta$ is independent of $\ell$ and the function $\iota$ is constructed explicitly in \cite{Bruin}. Our approach is to compute $V_\ell\mod p$ for sufficiently many auxiliary small primes $p$ as in Schoof's algorithm, and then reconstruct $V_\ell$ by the Chinese Remainder Theorem. The main results of the paper are as follows. \begin{theorem}\label{theorem:complexity}Let $\ell\ge 13$ be a prime and $p$ an $s$-good prime. Given the Zeta function of the modular curve $X_1(\ell)_{\mathbb{F}_p}$, then $V_\ell \mod p$ can be computed in time $\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{1+\epsilon}p\cdot(\ell+\log p))$. \end{theorem} \begin{cor}\label{cor:complexity}(1)The Ramanujan subspace $V_\ell$ can be computed in time $\textrm{O}(\ell^{5+2\omega+\delta+\epsilon})$. (2) For prime $p$, $\tau(p)$ can be computed in time $\textrm{O}(\log^{6+2\omega+\delta+\epsilon} p)$. \end{cor} \begin{remark}See \ref{s-goodprime} for the definition of $s$-good prime. The constant $\omega$ refers to that, the complexity of a single group operation in the Jacobian variety $J_1(\ell)$ is in $\textrm{O}(g^\omega)$, where $g$ is the genus of the modular curve $X_1(\ell)$. Using Khuri-Makdisi's algorithm, the constant $\omega$ can be $2.376$. Using He\ss's algorithm, $\omega$ is known to be in $[2,4]$. The constant $\delta$ is bigger than $\mbox{\rm dim} V_\ell=2$. \end{remark} \begin{theorem}\label{theorem:main2} The nonvanishing of $\tau(n)$ holds for all $n$ such that $$n<982149821766199295999\approx9\cdot10^{20}.$$ \end{theorem} \begin{remark}In \cite{Bosman} the nonvanishing of $\tau(n)$ was verified for all $n$ such that $$n < 22798241520242687999 \approx 2 \cdot10^{19}.$$ \end{remark} {\bf Notation}: The running time will always be measured in bit operations. Using FFT, multiplication of two $n$-bit length integers can be done in $\textrm{O}(n^{1+\epsilon})$ time. Multiplication in finite field $\mathbb{F}_q$ can be done in $\textrm{O}(\log^{1+\epsilon} q)$. The paper is organized as follows. In Section 2 we provide some necessary background on computing a convenient plane model and the function field of the modular curve, results for computing isogenies of elliptic curve over finite fields are also recalled. A better bound on the generators of the maximal ideal of Hecke algebra is proved, which is used to reduce the complexity of the algorithm. Section 3 contains the application of He\ss's algorithm to the computation in the Jacobian of the modular curve over finite fields. Here, we introduce methods to find the correspondence between the places of the function field and the cusps of the modular curve, and compute the action of Hecke operators on places of the function field. The main algorithm is given in Section 4, including complexity analysis. Section 5 is concerned with some real computations of the Ramanujan's tau function. All of our computations are based on Magma computational algebra system \cite{Bosma}. \section{Function field of modular curves} In this section we study the plane model and the function field of the modular curve $X_1(\ell)$. Let $\Gamma_1(\ell)$ be a congruence subgroup of SL$_2(\mathbb{Z})$, defined as $$\Gamma_1(\ell)=\left\{ \left[\begin{matrix} a & b\\ c & d \end{matrix}\right]\in \textrm{SL}_2(\mathbb{Z}): \left[\begin{matrix} a & b\\ c & d \end{matrix}\right]\equiv \left[\begin{matrix} 1 & \\ 0 & 1 \end{matrix}\right](\mbox{\rm mod~} \ell) \right\},$$ (where `` '' means `` unspecified '') and $\mathcal{H}$ the upper half complex plane. The modular curve $Y_1(\ell)$ is defined as the quotient space of orbits under $\Gamma_1(\ell)$, $$Y_1(\ell)=\Gamma_1(\ell)\backslash \mathcal{H}.$$ We can add cusps $\mathbb{P}^1(\mathbb{Q})$ to $Y_1(\ell)$ to compactify it and obtain the modular curve $$X_1(\ell)=\Gamma_1(\ell)\backslash \mathcal{H}\cup \mathbb{P}^1(\mathbb{Q}).$$ This complex algebraic curve is defined over $\mathbb{Q}$, denoted by $X_1(\ell)_{\mathbb{Q}}$. Moreover, for $\ell\ge 5$, $X_1(\ell)$ has natural model over $\mathbb{Z}[1/\ell]$. Let $K$ be a number field, then $K$-valued points of $Y_1(\ell)_{\mathbb{Q}}$ can be interpreted as $$Y_1(\ell)_{\mathbb{Q}}(K)=\{(E,P):E/K, P\in E[\ell](K)\setminus O\}/{\sim},$$ where $E$ is an elliptic curve over $K$, $P$ is a $K$-rational point of order $\ell$, and $(E_1,P_1)\sim(E_2,P_2)$ means that, there exists a $\overline{K}$-isomorphism $\phi:E_1\to E_2$, such that $\phi(P_1)=P_2$. Such a moduli interpretation implies a way to obtain a plane model of the modular curve, as the following proposition (see \cite{BAAZIZ}), \begin{prop}Suppose that $\ell \ge 4$. Then every $K$-isomorphism class of pairs $(E,P)$ with $E$ an elliptic curve over $K$ and $P \in E(K)$ a torsion point of order $\ell$ contains a unique model of the Tate normal form \begin{equation}\label{TateNormalForm} E_{(b,c)}:y^2+(1-c)xy-by =x^3-bx^2,P=(0,0), \end{equation} with $c\in K,b \in K^$. \end{prop} Thus, points of $X_1(\ell)$ can be represented as pairs $(b,c)$ in a unique way. The $\ell$-th division polynomial gives a polynomial in $b$ and $c$, which defines a plane curve birationally equivalent to $X_1(\ell)$. The defining equation becomes much simpler, through a carefully chosen sequence of rational transformations. We use the table listed in \cite{Sutherland12}, for example a plane model of $X_1(19)$ is \begin{equation} \begin{split} f(x,y)=&y^5 - (x^2 + 2)y^4 - (2x^3 + 2x^2 + 2x - 1)y^3+ (x^5 + 3x^4 + 7x^3 + 6x^2 + 2x)y^2\\ &-(x^5 + 2x^4 + 4x^3 + 3x^2)y + x^3 + x^2, \end{split} \end{equation} where \begin{equation}\label{equation:r(xy)} r=1+\frac{x(x+y)(y-1)}{(x+1)(x^2-xy+2x-y^2+y)},s=1+\frac{x(y-1)}{(x+1)(x-y+1)}, \end{equation} and \begin{equation} c=s(r-1),b=rc. \end{equation} So the function field of $X_1(19)$ over $\mathbb{Q}$ is $$\mathbb{Q}(X_1(19))=\mathbb{Q}(b,c)=\mathbb{Q}(x)[y]/(f(x,y)).$$ Cusps of $X_1(\ell)$ correspond to those pairs $(b,c)$ such that the $j$-invariant $j(E_{(b,c)})=\infty$. For $\ell$ an odd prime, the modular curve $X_1(\ell)$ has $\ell-1$ cusps, half of which are in $X_1(\ell)(\mathbb{Q})$ and the rest are defined over the maximal real subfield of $\mathbb{Q}(\zeta_\ell)$. Accordingly, every $\mathbb{Q}$-rational cusp corresponds to a degree one place of $\mathbb{Q}(X_1(\ell))$, denote the rational cusps as $O_1,\ldots,O_{(\ell-1)/2}$, the $\mathbb{Q}(\zeta_\ell)$-cusps corresponds to a degree $\frac{\ell-1}{2}$ place of $\mathbb{Q}(X_1(\ell))$. It is easy to get these places after writing down the exact expression of $j(E_{(b,c)})$ in variables $x,y$. As in the above example, one of the $\mathbb{Q}$-rational cusps looks like \begin{displaymath} \begin{split} O_1=&\left(x,\frac{y^4}{x^4+x^3}+\frac{y^3(-x^3-x^2+x-1)}{x^4+x^3}+ \frac{y^2(-x^3-2x^2-2x-2)}{x^3+x^2}+\frac{2y}{x}+\frac{2x-1}{x}\right), \end{split} \end{displaymath} where the place $O_1$ is represented by a prime ideal of the maximal orders of the function field $\mathbb{Q}(X_1(\ell))$. It's well known that, the modular curve $X_1(\ell)$ has good reduction at prime $p\nmid \ell$, see \cite{Diamond}. The reduction curve is denoted by $X_1(\ell)_{\mathbb{F}_p}$. Having a nonsingular affine model of $X_1(\ell)$, we can easily get an affine model for $X_1(\ell)_{\mathbb{F}_p}$ and then have the function field of $X_1(\ell)_{\mathbb{F}_p}$. For simplicity, the plane model of $X_1(\ell)_{\mathbb{F}_p}$ and the $\mathbb{F}_p$-rational cusps of $X_1(\ell)_{\mathbb{F}_p}$, which are the reductions of the $\mathbb{Q}$-rational cusps of $X_1(\ell)$, are also denoted by $f(x,y)$ and $O_i,i \in\{1,\ldots,\frac{\ell-1}{2}\}$, respectively. The Ramanujan subspace $V_\ell \mod p$ is a subgroup scheme of the Jacobian variety $J_1(\ell)_{\mathbb{F}_p}$ of $X_1(\ell)_{\mathbb{F}_p}$. Similarly, it can be written as a finite intersection $$V_\ell \mod p=\bigcap_{1\le i \le \frac{\ell^2-1}{6}}\ker(T_i-\tau(i),J_1(\ell)_{\mathbb{F}_p}[\ell]),$$ where $T_i,~1\le i\le \frac{\ell^2-1}{6}$ are Hecke operators, the number $\frac{\ell^2-1}{6}$ follows from \cite{Stu87}. In fact, the Hecke algebra $\mathbb{T}=\mathbb{Z}[T_n:n\in \mathbb{Z}^+]$ $\subset$ End$(J_1(\ell))$ is a free $\mathbb{Z}$-module of rank $g=\frac{(\ell-5)(\ell-7)}{24}$. After representing each Hecke operator as a matrix, see \cite{Edixhoven}, the generators can be extracted from $T_1,\ldots,T_{(\ell^2-1)/6}$ by solving linear equations. For example, when the level $\ell=17$, the Hecke algebra $\mathbb{T}$ is equal to $\mathbb{Z}T_1+\ldots+\mathbb{Z}T_{48}$ as an $\mathbb{Z}$-module, which can be replaced by $\mathbb{Z}T_1+\mathbb{Z}T_2+\mathbb{Z}T_3+\mathbb{Z}T_4+\mathbb{Z}T_6$ as a free $\mathbb{Z}$-module, so there are fewer Hecke operators and isogenies of lower degrees needed to take into account. But, in practice, we can do much better, notice that our goal is to find nonzero elements in $J_1(\ell)_{\mathbb{F}_p}[\ell]$, which are canceled by $T_k-\tau(k), k\ge 1$. Assume there is an element $D\in J_1(\ell)_{\mathbb{F}_p}[\ell]$ satisfying $(T_2-\tau(2))(D)=0$, then if we have the relations $T_k-\tau(k)=\phi_k\cdot(T_2-\tau(2))$, for some endomorphism $\phi_k\in$ End($J_1(\ell)_{\mathbb{F}_p}$) in advance, then $(T_k-\tau(k))(D)$ is equal to zero automatically, which implies that $D\in V_\ell$. The action of $T_k-\tau(k)$ on $J_1(\ell)_{\mathbb{F}_p}[\ell]$ can be represented by a matrix over finite field $\mathbb{F}_\ell$, and the existence of $\phi_k$ is equivalent to the existence of some matrix $M_k$ over $\mathbb{F}_\ell$ such that $T_k-\tau(k)=M_k\cdot(T_2-\tau(2))$. For example, when the level $\ell\in \{13,17,19,23,29,37,41,43\}$, for any $k\ge 3$, we have $T_k-\tau(k)=M_k\cdot( T_2-\tau(2))$, for some matrix $M_k$ over finite field $\mathbb{F}_\ell$, while $\ell=31$, $T_3-\tau(3)$ also satisfies this property. In fact, we proved the following proposition. \begin{prop}\label{propositon:linearlyinl}Let $\ell$ be a prime bigger than 3 and $S_2(\Gamma_1(\ell))$ the space of cusp modular forms of weight 2 level $\ell$ over $\mathbb{C}$. Let $\mathbb{T}=\mathbb{Z}[T_n:n\ge1]$ $\subset$ $\textrm{End}(S_2(\Gamma_1(\ell)))$ be the Hecke algebra and $\mathfrak{m}$ the maximal ideal generated by $\ell$ and the $T_n-\tau(n)$ with $n\ge1$. Then $\mathfrak{m}$ can be generated by $\ell$ and the $T_n-\tau(n)$ with $1\le n\le \lceil\frac{2\ell+1}{12}\rceil$. \end{prop} \begin{proof}Let $b_\ell:=\lceil\frac{2\ell+1}{12}\rceil$ and $R:=\mathbb{T}\otimes_{\mathbb{Z}}\mathbb{F}_\ell $, then $R$ is an Artin ring, which can be decomposed as $$R=\prod_{\wp}R_{\wp},$$ where $\wp$ runs through all maximal ideals of $R$, and $R_{\wp}$ is the localization of $R$ at $\wp$. Let $\tilde{\mathfrak{m}}$ be the image of $\mathfrak{m}$ in $R$. Then, it is enough to prove $\tilde{\mathfrak{m}}$ can be generated by the $T_n-\tau(n)$ with $n\le b_\ell$, which is equivalent to show that any $T_k-\tau(k),k>b_\ell$ can be represented as \begin{equation}\label{equation:prop1} T_k-\tau(k)=\sum_{i=1}^{b_\ell}A_i\cdot(T_i-\tau(i)), \end{equation} where $A_i$ are operators in $R$. Let $S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell):=S_2(\Gamma_1(\ell);\mathbb{Z})\otimes \overline{\mathbb{F}}_\ell$, which is an $R$-module, decomposed as $$S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell)=\prod_{\wp}S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell)_{\wp},$$ where $\wp$ runs through all maximal ideals of $R$, and $S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell)_{\wp}$ is the localization of $S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell)$ at $\wp$, which is an $R_{\wp}$-module. To show (\ref{equation:prop1}), it's enough to show for each $\wp$, the action $T_k-\tau(k),k>b_\ell$ on $S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell)_{\wp}$ can be represented as \begin{equation}\label{equation:prop2} T_k-\tau(k)=\sum_{i=1}^{b_\ell}B_i\cdot(T_i-\tau(i)), \end{equation} where $B_i$ are operators in $R_{\wp}$. The maximal ideal $\wp$ of $R$ corresponds to a $\textrm{Gal}(\overline{\mathbb{F}}_\ell/\mathbb{F}_\ell)$-conjugate class $[f]$ of normalized eigenforms in $S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell)$, they are newforms of level $\ell$. Thus the localization $S_2(\Gamma_1(\ell);\overline{\mathbb{F}}_\ell)_{\wp}$ is a vector space spanned by these newforms, so $R_{\wp}$ is a field isomorphic to the field $\mathbb{F}_{\ell}(f)$, which is generated by the coefficients of $f$ over $\mathbb{F}_\ell$. Let $a_i(f)$ be the $i$-th coefficient of the Fourier expansion of $f$, to show (\ref{equation:prop2}) it is enough to show $a_k(f)-\tau(k)$ is always equal to zero, or there exists at least one nonzero element among the $a_i(f)-\tau(i)$ with $1\le i\le b_\ell$. If $f$ is congruent to $\Delta(q)\mod\ell$, then $a_k(f)-\tau(k)$ is always equal to zero. Now suppose $f$ is not congruent to $\Delta(q)$ modulo $\ell$ and $a_i(f)-\tau(i)=0$ for $1\le i\le b_\ell$. The Proposition 4.10(b) of \cite{Gross} together with Theorem 3.5(a) of \cite{Ash-Ste} imply that $f$ comes from a level one newform of weight $k_1$ with $k_1\le2\ell$. Since for any prime $p$, $\tau(p^2)=\tau(p)^2-p^{11}$ and $a_{p^2}(f)=a_p(f)^2-p^{k_1-1}$, we have $p^{k_1-12}\equiv 1\mod \ell$ for all primes $p$ satisfying $p^2\le b_\ell$. Since the least primitive root modulo $\ell$ is in $\textrm{O}(\ell^{\frac{1}{4}+\epsilon})$, we have $k_1-12\equiv 0\mod(\ell-1)$. Let $A(q)=1$ be the Hass invariant, then $f_1:=A(q)^{(k_1-12)/(\ell-1)}\Delta(q)$ is a newform of level one weight $k_1$, and $a_k(f_1)=a_k(f)$ for all $k\le b_\ell$. Since $b_\ell=\lceil\frac{2\ell+1}{12}\rceil\ge\frac{k_1+1}{12}$, the theorem of Sturm \cite{Stu87} (or see \cite{CG11}) tells that $f_1$ is congruent to $f$, that is $\Delta(q)$ is congruent to $f$ modulo $\ell$, which is a contradiction. So the proposition is proved.. \end{proof} Since $\ell$ and a subset of $\{T_k-\tau(k):1\le k\le \lceil\frac{2\ell+1}{12}\rceil\}$ may also suffice to generate $\mathfrak{m}$, we introduce the optimal subset as follows. \begin{definition}\label{definition:optimalset}Let $\mathcal{S}$ be a set of positive integers, such that $\ell$ and $T_n-\tau(n),n\in\mathcal{S}$ generate $\mathfrak{m}$. The set $\mathcal{S}$ is called optimal if $\prod_{n\in \mathcal{S}}n$ is minimal among all the $\mathcal{S}$. \end{definition} In practice only those operators in the optimal subset need to be considered, which will accelerate the algorithm. Let $n\ge 2$ be a prime not equal to $\ell$, and $Q$ a point of $X_1(\ell)_{\mathbb{F}_p}(\overline{\mathbb{F}}_p)$ represented by $(E_{(b,c)},(0,0))$, the computation of $T_n(Q)$ comes down to computing isogenies of elliptic curves over some finite extension fields of $\mathbb{F}_p$ $$T_n(E_{(b,c)},(0,0))=\sum_{C}(E/C,(0,0)+C),$$ where $C$ runs over all the order $n$ subgroups of $E_{(b,c)}$. The following result about the complexity of computing $n$-isogeny is in \cite{BOSTAN}, when $n$ is small compared to the characteristic of the field, \begin{prop}Let $\mathbb{F}_q$ be a finite field of characteristic $p$, $n$ a prime not equal to $p$, $E$ and $\tilde{E}$ two elliptic curves over $\mathbb{F}_q$ in Weierstrass form. Assume there is a normalized isogeny $\phi:E\to \tilde{E}$ of degree $n$, then $\phi$ can be computed in $O(n^{1+\epsilon})$ multiplications in the field $\mathbb{F}_q$. \end{prop} We give two notices here: The first is that, the original elliptic curve $E_{(b,c)}$ is defined over some finite field $\mathbb{F}$, but the isogeny may lie in the extension field of $\mathbb{F}$. The exact definition field of the isogeny can be obtained by computing the $n$-th classical modular polynomial $\Phi_n(X,Y)$, and solving the equation $\Phi_n(X,j(E_{(b,c)}))=0$, so the extension degree is less or equal to the degree of $\Phi_n(X,j(E_{(b,c)}))$. The second is that, in general the isogenous curve $\tilde{E}$ is not in Tate normal form (\ref{TateNormalForm}). Using the map $\phi$, or rather the point $\phi((0,0))\in \tilde{E}$, $\tilde{E}$ can be transformed into Tate normal form after some coordinate changes, this gives a new point on the modular curve $X_1(\ell)_{\mathbb{F}_p}$. \section{Computing in the jacobian of modular curves} Let $\mathbb{F}_q$ be a finite extension of the finite field $\mathbb{F}_p$ and $X_1(\ell)_{\mathbb{F}_q}$ the base change of $X_1(\ell)_{\mathbb{F}_p}$ to $\mathbb{F}_q$. One of the most important tasks of our algorithm is computing in the Jacobian $J_1(\ell)_{\mathbb{F}_q}$ of the modular curve $X_1(\ell)_{\mathbb{F}_q}$. For general curves, we already have polynomial time algorithms to perform operation (addition and subtraction) in their Jacobians \cite{Huang} \cite{Volcheck} \cite{Hess} \cite{KM}. For the Jacobian of modular curve, Couveignes uses Brill-Noether algorithm to do the computation \cite{Couveignes}, while Bruin uses Khuri-Makdisi's algorithm \cite{Bruin}. We choose He\ss's algorithm, with the advantage that it's easy know the correspondences between points of the modular curve and places of its function field. Let's recall the main idea of He\ss's algorithm, for the detail see \cite{Hess}. Let $\mathbb{K}=\mathbb{F}_q(x)[y]/(f(x,y))$ be the function field of the modular curve $X_1(\ell)_{\mathbb{F}_q}$. There are isomorphisms $$J_1(\ell)(\mathbb{F}_q)\cong \textrm{Pic}^0(X_1(\ell)_{\mathbb{F}_q})\cong \textrm{Cl}^0(\mathbb{K}).$$ Notice that, there are some calculations behind the second isomorphism, i.e. computing the change of representations. He\ss's algorithm is based on the arithmetic of the function field $\mathbb{K}$. Let $\mathbb{P}$ be the set of all places of $\mathbb{K}$ and $S$ the set of the places of $\mathbb{K}$ over the infinite place $\infty$ of $\mathbb{F}_q(x)$. The ring of elements of $\mathbb{K}$ being integral at all places of $S$ and $\mathbb{P}\setminus S$ are denoted by $\mathcal{O}_S$ and $\mathcal{O}^S$, respectively. They are Dedekind domains, called infinite and finite order of $\mathbb{K}$, whose divisor groups are denoted by $\textrm{Div}(\mathcal{O}_S)$ and $\textrm{Div}(\mathcal{O}^S)$, respectively. Place of $\mathbb{K}$ corresponds to prime ideal of $\mathcal{O}_S$ or $\mathcal{O}^S$. In fact, let $\textrm{Div}(\mathbb{K})$ be the divisor group of $\mathbb{K}$, then $\textrm{Div}(\mathbb{K})$ can be decomposed as \cite{Hess} $$\textrm{Div}(\mathbb{K})\xrightarrow{\sim}\textrm{Div}(\mathcal{O}_S)\times \textrm{Div}(\mathcal{O}^S).$$ Notice that the plane model of $X_1(\ell)$ given in \cite{Sutherland12} has singularities above $x=0$ and $x=-1$, we can check that places of $\mathbb{K}$ over the places $(x)$,$(x+1)$ and $(\frac{1}{x})$ of $\mathbb{F}_q(x)$ are cusps of $X_1(\ell)$. We will discuss how to compute the action of Hecke operators on these places in Section 3.2. Now, let's focus on how to compute the action of Hecke operators on the place corresponding to a set of smooth points of $f(x,y)=0$. Such a place $\wp$ can be represented by two elements of $\mathbb{K}^{\times}$, which can be normalized as $f_1(x)=x^d+a_{d-1}x^{d-1}+\ldots+a_1x+1$ and $f_2(x,y)=y^m+b_{m-1}(x)y^{m-1}+\ldots+b_1(x)y+b_0(x)$, where $a_i,0\le i\le d-1$, belong to the constant field $\mathbb{F}_q$, and $b_i(x),~0\le i\le m-1$ are elements of $\mathbb{F}_q[x]$, with degrees less than $d$. So the point set corresponding to $\wp$ can be computed as follows. Let $f_1(x)$ and $f_2(x,y)$ be the normalized generators of $\wp$. We first compute the roots of $f_1(x)=0$, denoted by $x_i,1\le i\le d$. Then, for each root $x_i$, compute the roots of $f_2(x_i,y)=0$, denoted by $y_{ij},1 \le j\le m$. The point set corresponding to $\wp$ is $\{(x_i,y_{ij}):{1\le i\le d,1\le j\le m} \}$. Remind that every point $(x_i,y_{ij})$ satisfies $f(x_i,y_{ij})=0$. Conversely, given a point set $\{(x_i,y_{ij}):{1\le i\le d,1\le j\le m} \}$, the two generators for the corresponding prime ideal $\wp$ of $\mathcal{O}^S$ can be computed as follows. The first generator is clear, which is $f_1(x)=\prod_{i=1}^d(x-x_i)$. The second one can be recovered as follow: Let $b_i(x)=\sum_{k=0}^{d-1}c_{ik}x^k,0\le i\le m-1$, where $c_{ik}$ are parameters belong to $\mathbb{F}_q$, after interpolating the points $(x_i,y_{ij})$, we have linear equations of $md$ variables, the second generator can be known by solving these equations. So, a degree $d$ place $\wp$ as in above corresponds to a point set, denoted by $\{(x_i,y_i):1\le i\le d\}$, which forms a complete $\mbox{\rm Gal}(\mathbb{F}_{q^d}/\mathbb{F}_q)$-conjugate set. Using the coordinate transformation formulae (\ref{equation:r(xy)}), for each point $(x_i,y_i)$, the corresponding point on $X_1(\ell)_{\mathbb{F}_p}$ of the form $(E_{(b_i,c_i)},(0,0))$ is clear, where $E_{(b_i,c_i)}$ is an elliptic curve in Tate normal form, and $(0,0)$ is a point of order $\ell$. As discussed in Section 2, the action of Hecke operator $T_n$ on each point $(E_{(b_i,c_i)},(0,0))$ leads to a sequence of elliptic curves. Using the inverse transformation formulae these curves give the point sets on the affine curve $f(x,y)=0$. Further, we have the corresponding places of the function field $\mathbb{K}$. For $P$ a place of $\mathbb{K}=\mathbb{F}_q(X_1(\ell))$, which is not equal to any of the cusps of $X_1(\ell)$, and $D$ a divisor of $\mathbb{K}$ consists of such places, we define \begin{definition} Let $P$, $D$ as above and $T_n$ a Hecke operator. $T_n(P)$ is defined to be the divisor of $\mathbb{K}$, corresponding to the point set $\sum_{i=1}^d T_n(E_{(b_i,c_i)},(0,0))$, which is effective of degree $\Psi(n)d$, where $\Psi(n)=n\prod_{p\|n}(1+\frac{1}{p})$. Decompose $D $ as $\sum_{i=1}^m a_i P_i$, where $P_i$ are places, then $T_n(D)$ is defined to be $\sum_{i=1}^m a_iT_n(P_i)$. \end{definition} Let $D_0$ be a fixed degree one place of $\mathbb{K}$, which will be served as an origin. Every element of $\textrm{Cl}^0(\mathbb{K})$ can be represented by $D-gD_0$, where $g$ is the genus of $\mathbb{K}$, and $D$ is an effective divisor of degree $g$. Addition in $\textrm{Cl}^0(\mathbb{K})$ means that, given effective divisors $A$ and $B$ of degree $g$, find an effective divisor $D$ of degree $g$, such that $D-gD_0$ is linearly equivalent to $A-gD_0+B-gD_0$. The complexity of doing this operation can be found in \cite{Hess}, \begin{prop}Notation is as above. There exists a constant $\omega\in[2,4]$ such that, the divisor $D$ can be computed in $\textrm{O}(g^{\omega})$ multiplications in the field $\mathbb{F}_q$, i.e. $\textrm{O}(g^{\omega}\log^{1+\epsilon}q)$ bit operations. \end{prop} \begin{remark}We have not yet seen the precise value of $\omega$. However, using Khuri-Makdisi's algorithm the complexity of a single group operation is known, i.e. $\omega=2.376$, when fast algorithms for the linear algebra are used. \end{remark} The following definition in \cite{Hess} is very useful in our algorithm. \begin{definition}Let $A$ be a divisor with $\deg(A)\ge1$. The divisor $\tilde{D}$ is called maximally reduced along $A$ if $\tilde{D}\ge 0$ and $\textrm{dim}(\tilde{D}-rA)=0 $ holds for all $r\ge1$. Let now $D$ be any divisor and $\tilde{D}$ is a divisor maximally reduced along $A$ such that $D$ is linearly equivalent to $\tilde{D}+rA$ for some $r\in\mathbb{Z}$. Then $\tilde{D}$ is called a reduction of $D$ along $A$. \end{definition} If $\deg(A)=1$, then the reduction divisor $\tilde{D}$ is effective and unique. \subsection{Searching an $\ell$-torsion point} One of the main steps in computing the Ramanujan subspace $V_\ell\mod p$ is to find an $\ell$-torsion point in $J_1(\ell)$. In order to do that we have to work with some large enough extension field $\mathbb{F}_q/\mathbb{F}_p$, such that $J_1(\ell)(\mathbb{F}_q)$ contains $\ell$-torsion points. A direct way to get such a point is, pick a random point $Q_0$ in $J_1(\ell)(\mathbb{F}_q)$ and then compute $Q_1:=N_\ell Q_0$, where $N_\ell$ is the prime-to-$\ell$ part of $\#J_1(\ell)(\mathbb{F}_q)$. If $Q_1$ is nonzero and $\ell$-torsion, then we succeed. Otherwise, try $Q_2:=\ell Q_1$, check again, after several steps, we can obtain a nonzero $\ell$-torsion point. As $\#J_1(\ell)(\mathbb{F}_q)$ is bounded above by $q^g$, using fast exponentiation, the running time of getting an $\ell$-torsion point is about $\log (q^g)\cdot \textrm{O}( g^{\omega})=\textrm{O}(g^{1+\omega}\log q)$ multiplications in the field $\mathbb{F}_q$ or $\textrm{O}(g^{1+\omega}\log^{2+\epsilon} q )$ bit operations. The computation is costly, due to the huge factor $N_\ell$, which is nearly $q^g$. However, we can accelerate the calculation by introducing some tricks. For each newform $f$ in $S_2(\Gamma_1(\ell))$, there is an associated abelian variety $A_f$, and $\prod_f A_f$ is an isogeny decomposition of $J_1(\ell)$, where $f$ runs through a set of representatives for the Galois conjugacy classes of newforms in $S_2(\Gamma_1(\ell))$. Let $f_\ell$ be the newform, which is congruent to $\Delta(q)$ modulo $\ell$. Then the Ramanujan subspace $V_\ell$ lands inside $A_{f_\ell}$. Let $A':=\prod_f'A_f$ be the product of the abelian varieties, except $A_{f_\ell}$. The minimal polynomial of the Hecke operator $T_2$ acting on $A'$ is denoted by $P_2[X]\in\mathbb{Z}[X]$. Now for a random point $Q_0$ in $J_1(\ell)(\mathbb{F}_q)$, $Q_1:=P_2(T_2)(Q_0)$ is a point in $A_{f_\ell}(\mathbb{F}_q)$. Similarly, let $n_\ell$ be the prime-to-$\ell$ part of $\#A_{f_\ell}(\mathbb{F}_q)$, by computing $n_\ell Q_1$, we can obtain an $\ell$-torsion point of $J_1(\ell)(\mathbb{F}_q)$. Since the calculation of $P_2(T_2)(Q_0)$ is easy and $n_\ell$ is smaller than $N_\ell$, which is bounded above by $q$ to the dimension of $A_{f_\ell}$, we can accelerate the algorithm, especially when the dimension of $A_{f_\ell}$ is small compared to $g$. However, it seems hard to get a theoretical bound of the dimension of $A_{f_\ell}$. We remark here several small examples \begin{center} \renewcommand\arraystretch{1.5} \begin{tabular}{ccccccccccccc} \hline Level $\ell$ & 13 & 17 & 19 & 29 & 31 & 37 & 41 &43 &47 &53 &59 &61 \\ \hline $\mbox{\rm dim} J_1(\ell)$ & 2 & 5 & 7 & 22 & 26 & 40 & 51 &57 &70 &92 &117 &126\\ \hline $\mbox{\rm dim} A_{f_\ell}$& 2 & 4 & 6 & 12 & 4 & 18 & 6 &36 &66 &48 &112 &8\\ \hline $\mbox{\rm dim} J_H(\ell)$ & & & & & 6 & &11 & & & & &26\\ \hline \end{tabular} \end{center} Another method, suggested by Maarten Derickx, works perfectly when the level $\ell$ satisfying $\ell\equiv 1\mod10$. More precisely, let $\chi$ the Dirichlet character associated to the newform $f_\ell$, then we have for any prime $p\not=\ell$, $\chi(p)\cdot p\equiv p^{11}\mod \ell$, hence $\chi(p^{\frac{\ell-1}{10}})\equiv p^{\ell-1}\equiv 1\mod\ell$. Which means that $f_\ell$ is invariant under the action of diamond operators in the cyclic subgroup $H=\{p^{\frac{\ell-1}{10}}:p\not=\ell~ \textrm{prime}\}$ of $G=(\mathbb{Z}/\ell\mathbb{Z})^\times$. The Ramanujan subspace $V_\ell$ lands inside the Jacobian variety $J_H(\ell)$, where $J_H(\ell)$ is isogenous to the Jacobian of the modular curve $X_H(\ell)$ associated to the subgroup of $\mbox{\rm SL}_2(\mathbb{Z})$ of matrices $[a,b;c,d]$ with $c$ divisible by $\ell$ and $a$ in $H$ modulo $\ell$. We can apply the algorithm to the modular curve $X_H(\ell)$ and the Jacobian variety $J_H(\ell)$ instead of $X_1(\ell)$ and $J_1(\ell)$, respectively. This useful observation enables us to carry out the calculation of the level $\ell=31$ case. We now explain how to compute $\#J_1(\ell)(\mathbb{F}_q)$. \begin{lemma}(Manin, Shokurov, Merel, Cremona). For $\ell$ a prime and $p\not\in\{5,\ell\}$ another prime, the Zeta function of $X_1(\ell)_{\mathbb{F}_p}$ can be computed in deterministic polynomial time in $\ell$ and $p$. \end{lemma} \begin{remark}Given this Zeta function we can easily compute $\#J_1(\ell)(\mathbb{F}_q)$, for $q$ is a power of $p$, see \cite{Couveignes}. Similarly, we can compute $\#A_f(\mathbb{F}_q)$ in deterministic polynomial time in $\ell$ and $p$, by applying the algorithm to newforms in $[f]$, where $[f]=\{f^\sigma~\|~\sigma\in\Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \}$. \end{remark} \subsection{Distinguishing the rational cusps} In our algorithm, we will compute the action of Hecke operator $T_n$ on points of $J_1(\ell)$. Especially, we need to know the action of $T_n$ on the $\mathbb{Q}$-rational cusps of $X_1(\ell)$. Notice that, $T_n,n\in\mathbb{Z}^+$ is defined over $\mathbb{Q}$, so it maps $\mathbb{Q}$-rational point to $\mathbb{Q}$-rational point. As far as we know, there is no an easy way to know the action directly from only the places $O_i,i\in \{1,\ldots,\frac{\ell-1}{2}\}$. But, a coset representatives of the $\mathbb{Q}$-rational cusps of $X_1(\ell)$ is clear, i.e. \{$\frac{1}{1},\frac{1}{2},\ldots,\frac{1}{(\ell-1)/2}$\}. And for prime $n\ne \ell$, we have $T_n(\frac{1}{m})=\frac{1}{m}+n\frac{1}{\overline{nm}}$, where $\overline{nm}$ is the class of integer $nm$ in $(\mathbb{Z}/\ell\mathbb{Z})^/\{\pm1 \}$. So knowing the 1-1 correspondence between $\{\frac{1}{i},1\le i\le \frac{\ell-1}{2}\}$ and $\{O_i,1\le i\le \frac{\ell-1}{2} \}$ leads to knowing the action of $T_n$ on $O_i$. In general, it's not easy to know the correspondence \cite{HEON}. Our strategy is to reduce the problem to finite field, as follows. Choose a prime $p$, such that $\#J_1(\ell)(\mathbb{F}_p)$ has a small factor $d$, let $g$ be the genus of the function field $\mathbb{F}_p(X_1(\ell))$. Fix a cusp $O_i$ served as origin. As discussed above, let $D-gO_i$ be a degree zero random divisor of order $d$, in general $D$ doesn't contain cusps, if it does, try a new one. Now, assume $O_i$ corresponds to $\frac{1}{m}$ for some $m\in\{1,\ldots,\frac{\ell-1}{2}\}$ and $O_j$ corresponds to $\frac{1}{\overline{nm}}$ for some $j\in\{1,\dots,\frac{\ell-1}{2}\}$. Then, for prime $n$, compute $D_n:=T_n(D-gO_i)$ by the assumption as follows $$D_n=T_n(D)-gT_n(O_i)=T_n(D)-g(O_i+nO_j).$$ If $D_n$ is not of order $d$, then the assumption is wrong. Replace $O_i$ by another $\frac{1}{m}$, or $\frac{1}{\overline{nm}}$ by another $O_j$, try again. The complete correspondence can be detected after several tries\footnote{We can use the diamond operators instead, which is faster, suggested by Maarten Derickx}. In fact, the correspondence is known up to cyclic permutation, but it is enough for our algorithm. There is one more thing should be noticed, the degree 0 divisors of the form $O_i-O_j,1\le i,j\le \frac{\ell-1}{2}$ generate a subgroup of $J_1(\ell)(\mathbb{F}_p)$, which is called cuspidal subgroup, so the chosen factor $d$ should not be a multiple of the order of the cuspidal subgroup. As the example in Section 2, the correspondence between $\{O_1,\ldots,O_9\}$ and $\{\frac{1}{1},\ldots,\frac{1}{9}\}$ is \begin{displaymath} \begin{array}{ccccccccc} \frac{1}{1} &\frac{1}{5} &\frac{1}{6} &\frac{1}{7} &\frac{1}{4} &\frac{1}{3} &\frac{1}{2} &\frac{1}{8} &\frac{1}{9}\\ \updownarrow & \updownarrow &\updownarrow &\updownarrow &\updownarrow &\updownarrow &\updownarrow & \updownarrow &\updownarrow \\ O_1 & O_2 & O_3 & O_4 & O_5 &O_6 &O_7&O_8&O_9. \end{array} \end{displaymath} After the above preparation, we can now estimate the complexity of computing $T_n(Q)$, where $Q$ is a degree zero $\mathbb{F}_q$-divisor of the form $\sum_{i=1}^m a_i P_i-gO$, $O$ is fixed to be the cusp $O_1$ and $n$ is a prime not equal to $\ell$. As mentioned above, we first compute the point set of the degree $d_i$ place $P_i$ and pick one of them forms a point $(E_{(b_i,c_i)},(0,0))$ on the modular curve, which is defined over $\mathbb{F}_{q^{d_i}}$. Denote the factorization of $\Phi_n(X,j(E_{(b_i,c_i)}))$ over $\mathbb{F}_{q^{d_i}}[X]$ as $\prod_{j=1}^h F_j(X)$, with $F_j(X)$ irreducible of degree $f_j$. Now for each root of $F_j(X)=0$, there is an isogeny of degree $n$ defined over $\mathbb{F}_{q^{d_if_j}}$. Computing this isogeny takes $$\textrm{O}(n^{1+\epsilon} \cdot (\log q^{d_if_j})^{1+\epsilon} )=\textrm{O}((nd_if_j\log q)^{1+\epsilon})$$ bit operations. Notice that, isogenous curves corresponding to the roots of $F_j(X)$ forms a $\textrm{Gal}(\mathbb{F}_{q^{d_if_j}}/\mathbb{F}_{q^{d_i}})$-conjugate set. So it suffices to compute any one of them. So the complexity of computing $T_n((E_{(b_i,c_i)},(0,0)))$ is about $$C_i:=\sum_{j=1}^h \textrm{O}((nd_if_j\log q)^{1+\epsilon}).$$ Since $\sum_{j=1}^h f_j=n+1$, $C_i$ is bounded above by $\textrm{O}((nd_in\log q)^{1+\epsilon})$. The complexity of computing $T_n(Q)$ is $C:=\sum_{i=1}^m C_i$. Since $\sum_{i=1}^m a_i d_i=g$, $C$ is bounded above by $\textrm{O}((n^2g\log q)^{1+\epsilon} )$ bit operations. Notice that after the action of $T_n$, the divisor $T_n(Q)$ becomes complicated. We would like to simplify it before going into further calculation. In general, the reduction of $T_n(Q)$ along the degree one divisor $O$ comes down to performing at most $n$ additions in the Jacobian, with a complexity of $\textrm{O}(ng^\omega\log^{1+\epsilon} q)$ bit operations. \section{Computing the coefficients of modular forms} After some modification, the simplified algorithm proposed in \cite{Couveignes}, can be used to compute the Ramanujan subspace $V_\ell\mod p$ efficiently. We can even give a complexity analysis of the algorithm. Roughly speaking, the algorithm works as follows: Step one, pick some random points in $J_1(\ell)(\mathbb{F}_q)$. Step two, construct $\ell$-torsions points by multiplying these points by some suitable factors of $\#J_1(\ell)(\mathbb{F}_q)$. Step three, project the $\ell$-torsions points into the space $V_\ell\mod p$ by the Hecke operators. Step four, reconstruct $V_\ell/\mathbb{Q}$ from sufficiently many $V_\ell\mod p$, by the Chinese Remainder Theorem. \subsection{Computing the Ramanujan subspace modulo $p$} The characteristic polynomial of the Frobenius endomorphism $\mbox{\rm Frob}_p$ acting on the Ramanujan subspace $V_\ell\mod p$ is $X^2-\tau(p)X+p^{11}\mod \ell$ and the field of definition of each point of $V_\ell\mod p$ is an extension of $\mathbb{F}_{p}$ with degree $\le d_p$, where $$d_p=\min_t\{t:X^t\equiv1\mod (X^2-\tau(p)X+p^{11},\ell),t\ge1\}.$$ For convenience, we define \begin{definition}\label{s-goodprime}Notations as above, a prime $p$ is called $s$-good if $d_p<\ell$. \end{definition} The following proposition shows that nearly half of the primes are $s$-good. \begin{prop}Let $d_p$ defined as above, then we have $$\lim_{x\to\infty}\frac{\|\{p:p~ \textrm{prime},d_p<\ell,p<x\}\|}{\|\{p:p~\textrm{prime},p<x\}\|}\ge\frac{\ell^3-\ell^2-2\ell+2}{2(\ell^3-\ell)}.$$ \end{prop} \begin{proof}Let $\rho_\ell:\Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to\mbox{\rm GL}_2(\mathbb{F}_\ell)$ be the mod-$\ell$ representation associated to the discriminant modular form $\Delta(q)$. Denote $K_\ell$ the fixed field of $\ker\rho_\ell$. Then $\rho_\ell$ factors through $\rho_\ell:\textrm{Gal}(K_\ell/\mathbb{Q})\to\mbox{\rm GL}_2(\mathbb{F}_\ell)$, which is unramified outside $\ell$. Now, for a prime $p$ not equal to $\ell$, we have $\rho_\ell(\mbox{\rm Frob}_p)\in \mbox{\rm GL}_2(\mathbb{F}_\ell)$ and $d_p$ is the order of the matrix $\rho_\ell(\mbox{\rm Frob}_p)$. By the Chebotarev Density Theorem, we have for any conjugacy class $C$ of $G:=\textrm{Gal}(K_\ell/\mathbb{Q})$ the set $\{p:p~\textrm{a prime},p\not=\ell,\mbox{\rm Frob}_p\in C\}$ has density $\|C\|/\|G\|$. Define $\mathcal{C}=\cup_{\mbox{\rm ord}(C)<\ell} C$, where $\mbox{\rm ord}(C)$ represents the order of any element in $C$, then we have $$\lim_{x\to\infty}\frac{\|\{p:p~ \textrm{prime},d_p<\ell,p<x\}\|}{\|\{p:p~\textrm{prime},p<x\}\|}=\frac{\|\mathcal{C}\|}{\|G\|}.$$ Hence we would like to know the image of the representation $\rho_\ell$. If $\ell$ is an exceptional prime for $\Delta(q)$, i.e. $\ell\in\{2,3,5,7,23,691\}$, then the representation $\rho_\ell$ is reducible or $\textrm{Im}(\rho_\ell)$ in $\mbox{\rm GL}_2(\mathbb{F}_{23})$ is dihedral. We can check that $\frac{\|\mathcal{C}\|}{\|G\|}\ge\frac{1}{2}$. If $\ell$ is not exceptional, we have $\textrm{Im}(\rho_\ell)=\{g\in \mbox{\rm GL}_2(\mathbb{F}_\ell): \det(g)\in(\mathbb{F}_\ell^\times)^{11}\}$. The representatives of conjugacy classes in $\mbox{\rm GL}_2(\mathbb{F}_\ell)$ are as follow $c_1(x):=\left[\begin{smallmatrix} x & 0 \\ 0 & x \end{smallmatrix}\right], x\in\mathbb{F}_\ell^\times,$ $c_2(x):=\left[\begin{smallmatrix} x & 1 \\ 0 & x \end{smallmatrix}\right], x\in\mathbb{F}_\ell^\times,$ $c_3(x,y):=\left[\begin{smallmatrix} x & 0 \\ 0 & y \end{smallmatrix}\right], x\not=y\in\mathbb{F}_\ell^\times,c_3(x,y)=c_3(y,x),$ $c_4(z):=\left[\begin{smallmatrix} x & Dy \\ y & x \end{smallmatrix}\right], z=x+\sqrt{D}y\in\mathbb{F}_{\ell^2} \setminus\mathbb{F}_{\ell},c_4(z)=c_4(\bar z)$, where $\bar z:=x-\sqrt{D}y.$ Here $c_3(x,y)=c_3(y,x)$ means that the conjugacy classes of these two elements agree. Let $C_1(x)$ be the conjugacy class with representative $c_1(x)$, then we have $\|C_1(x)\|=1$, $\mbox{\rm ord}(C_1(x))\|(\ell-1)$ and there are $N_1=\ell-1$ such classes. Similarly, we have $\|C_2(x)\|=\ell^2-1$, $\mbox{\rm ord}(C_2(x))\|\ell(\ell-1)$, $N_2=\ell-1$; $\|C_3(x,y)\|=\ell(\ell+1)$, $\mbox{\rm ord}(C_3(x,y))\|(\ell-1)$, $N_3=\frac{1}{2}(\ell-1)(\ell-2)$; $\|C_4(z)\|=\ell(\ell-1)$, $\mbox{\rm ord}(C_4(z))\|(\ell^2-1)$, $N_4=\frac{1}{2}\ell(\ell-1)$, we can see $\|C_1(x)\|N_1+\|C_2(x)\|N_2+\|C_3(x,y)\|N_4+\|C_4(z)\|N_4=\|\mbox{\rm GL}_2(\mathbb{F}_\ell)\|=(\ell-1)^2\ell(\ell+1)$. The subgroup $\textrm{Im}(\rho_\ell)$ consists of those conjugacy classes whose representative matrices have determinant in $(\mathbb{F}_\ell^\times)^{11}$. For example $\textrm{Im}(\rho_\ell)$ contains conjuagcy classes with representatives $c_1(x)$ satisfying $x^2\in(\mathbb{F}_\ell^\times)^{11}$. Denote the number of such classes by $M_1$ and set $L:=\|(\mathbb{F}_\ell^\times)^{11}\|$. Then we have $M_1=L$. Similarly $M_2=L$, $M_3=\frac{1}{2}L(\ell-2)$ and $M_4=\frac{1}{2}\ell L$. So we have $$\frac{\|\mathcal{C}\|}{\|G\|}\ge\frac{M_1+M_3\cdot \ell(\ell+1)}{M_1+M_2\cdot(\ell^2-1)+M_3\cdot\ell(\ell+1)+M_4\cdot\ell(\ell-1)}=\frac{\ell^3-\ell^2-2\ell+2}{2(\ell^3-\ell)}.$$ \end{proof} Now let $p$ be an $s$-good prime and $\mathbb{F}_q:=\mathbb{F}_{p^{d_p}}$, then $V_\ell\mod p$ is a subgroup of $J_1(\ell)(\mathbb{F}_q)[\ell]$. For every integer $n\ge 2$, the characteristic polynomial of $T_n$ acting on $S_2(\Gamma_1(\ell))$ is a degree $g$ monic polynomial belonging to $\mathbb{Z}[X]$. We denote it by $A_n(X)$, which can be factored as $$A_n(X)\equiv B_n(X)(X-\tau(n))^{e_n}~\mod \ell,$$ with $B_n(X)$ monic and $B_n(\tau(n))\not=0 \in \mathbb{F}_\ell$. The exponent $e_n$ is $\ge 1$ due to the theorem of congruence of modular forms (Thm2.5.7 of \cite{Edixhoven}). We call $\pi_n:J_1(\ell)(\mathbb{F}_q)[\ell]\to J_1(\ell)(\mathbb{F}_q)[\ell] $ the projection map. Which maps an $\ell$-torsion point $Q\in J_1(\ell)(\mathbb{F}_q)[\ell] $ to an $\ell$-torsion point $B_n(T_n)(Q)$ of $J_1(\ell)(\mathbb{F}_q)[\ell]$, and maps bijectively $V_\ell\mod p$ onto itself. Assume $E:=\pi_n(Q)\not=0$, define the exponent $d_n$ as the nonnegative integer satisfying $$(T_n-\tau(n))^{{d_n}}(E)\not=0~\textrm{and} ~(T_n-\tau(n))^{{d_n}+1}(E)=0,$$ then $d_n$ is in $[0,e_n)$, since $(T_n-\tau(n))^{e_n}(E)=0$. Let $\tilde{\pi}_n$ be the composition map of $\pi_n$ and $(T_n-\tau(n))^{d_n}$ and $\pi_{\mathcal{S}}:=\prod_{n\in\mathcal{S}}\tilde{\pi}_n$, where $\mathcal{S}$ is an optimal set defined in Definition \ref{definition:optimalset}. Then we have $\pi_{\mathcal{S}}(Q)\in V_\ell\mod p$ . So, the complexity of finding a nonzero point in $V_\ell\mod p$ can be determined as following. As described in Section 3, it takes $\textrm{O}(g^{1+\omega}\log^{2+\epsilon} q )=\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{2+\epsilon}p)$ bit operations to get an $\ell$-torsion point of $J_1(\ell)(\mathbb{F}_q)$. Denote the $\ell$-torsion point as $Q_0=D-gO$, where $D$ is an effective divisor of degree $g$. For $n\in \mathcal{S}$, the map $\tilde{\pi}_n$ can be written as $T_n^d+a_{d-1}T_n^{d-1}+\ldots+a_1T_n+a_0$, where $a_i\in\mathbb{F}_\ell$ and $d<g$. The divisor $\tilde{\pi}_n(Q_0)$ can be computed recursively, i.e. compute and simplify(as mentioned in Section 3.2) $Q_{i+1}:=T_n(Q_i)$ for $i=0,\ldots,d-1$. The complexity of each step is $\textrm{O}((n^2g\log q)^{1+\epsilon})+\textrm{O}(ng^\omega\log^{1+\epsilon} q)$, which is $\textrm{O}(\ell^{2+2\omega+\epsilon}\log^{1+\epsilon}p)$, since $n\in \textrm{O}(\ell)$, $g\in\textrm{O}(\ell^2)$ and $\omega\in[2,4]$. As $d$ is in $\textrm{O}(g)$, computing and simplifying $Q_0,\ldots,Q_d$ takes $d\cdot\textrm{O}(\ell^{2+2\omega+\epsilon}\log^{1+\epsilon}p)=\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{1+\epsilon}p)$. Given $Q_i$, since $a_i\in\mathbb{F}_\ell$ the complexity of computing $Q_d+a_{d-1}Q_{d-1}+\ldots+a_1Q_1+a_0Q_0$ is bounded above by $d\ell\cdot\textrm{O}( g^{\omega}\log^{1+\epsilon} q )=\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{1+\epsilon}p)$. In summary, the complexity of computing $\tilde{\pi}_n(Q_0)$ is still in $\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{1+\epsilon}p)$. So the complexity of computing $\pi_{\mathcal{S}}(Q_0)=\prod_{n\in \mathcal{S}} \tilde{\pi}_n(Q_0)$ is bounded above by $\textrm{O}(\ell^{5+2\omega+\epsilon}\log^{1+\epsilon}p)$, since $\|\mathcal{S}\|<\ell$. So it takes \begin{equation}\label{complexityofVlmodp} \textrm{O}(\ell^{4+2\omega+\epsilon}\log^{2+\epsilon}p)+\textrm{O}(\ell^{5+2\omega+\epsilon}\log^{1+\epsilon}p) =\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{1+\epsilon}p\cdot(\ell+\log p)) \end{equation} bit operations to get a point in $V_\ell \mod p$. For two nonzero random points $Q_1,Q_2\in J_1(\ell)(\mathbb{F}_q)[\ell]$, the probability that $\pi_S(Q_1)$ and $\pi_S(Q_2)$ are linearly independent is close to 1-$\frac{1}{\ell}$. So after several attempts we will get a base of $V_\ell\mod p$. The complexity of checking linearly independence of two elements in $V_\ell\mod p$ is $\textrm{O}(\ell g^\omega\log^{1+\epsilon} q)=\textrm{O}(\ell^{2+2\omega+\epsilon}\log^{1+\epsilon}p)$. Hence the complexity of getting a base of $V_\ell \mod p$ is the same as given in (\ref{complexityofVlmodp}). The Frobenius endomorphism $\mbox{\rm Frob}_p$ may help us to get a base of $V_\ell\mod p$ with lower cost in some cases. Namely, if the characteristic polynomial $X^2-\tau(p)X+p^{11}\mod \ell$ is irreducible, then for any nonzero point $Q\in V_\ell\mod p$, $\mbox{\rm Frob}_p(Q)$ and $Q$ are linearly independent. So $V_\ell \mod p=\mathbb{F}_\ell Q+\mathbb{F}_\ell(\mbox{\rm Frob}_p(Q))$. If $X^2-\tau(p)X+p^{11}\mod\ell$ has two different roots in $\mathbb{F}_\ell$, then for any nonzero point $Q\in V_\ell\mod p$ the probability that $Q$ and $\mbox{\rm Frob}_p(Q)$ are linearly independent is $1-\frac{1}{\ell}$. \begin{remark}(1)In practice, the optimal set $\mathcal{S}$ contains only small primes, for example, $\mathcal{S}=\{2\}$ for $\ell\in\{13,17,19\}$. The algorithm takes the main effort to get an $\ell$-torsion point. (2)If without Proposition \ref{propositon:linearlyinl}, the complexity comes up to \begin{equation} \textrm{O}(\ell^{4+2\omega+\epsilon}\log^{2+\epsilon}p)+\textrm{O}(\ell^{7+2\omega+\epsilon}\log^{1+\epsilon}p) =\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{1+\epsilon}p\cdot(\ell^3+\log p)). \end{equation} \end{remark} \subsection{Computing the Ramanujan subspace} Fix a $\mathbb{Q}$-rational cusp $O$ of $X_1(\ell)$, which will be served as the origin of the Jacobi map. For every point $x\in V_\ell$, let $D$ be the reduction of $x$ along $O$, i.e. $D=x+dO$. Then $D$ is an effective divisor of degree $d$, which can be decomposed as $D=Q_1+\ldots+Q_d$, here $d$ is called the stability of $x$, denoted by $\theta(x)$. Choose a rational function $\psi(x)\in\mathbb{Q}(X_1(\ell))$, which has no pole except at $O$, and define a function $\iota:V_\ell\to \overline{\mathbb{Q}}$ as $\iota(x)=\psi(Q_1)+\ldots+\psi(Q_d)\in \overline{\mathbb{Q}}$. From the uniqueness of $D$, we have $\iota(\sigma(x))=\sigma(\iota(x))$ for any $\sigma\in\textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, which is very important. Let $\wp$ be a prime ideal of $\mathbb{Q}(x)$ over $p$, we hope that the uniqueness property still holds in the reduction world, which means the following: let $\tilde{x},\tilde{O},\tilde{D}$ and $\tilde{Q}_i$ be the reductions of $x,O,D$ and $Q_i$ modulo $\wp$ respectively. The reduction of $\tilde{x}$ along $\tilde{O}$ is denoted by $\tilde{E}=\tilde{x}+\theta(\tilde{x})\tilde{O}$, where $\theta(\tilde{x})\le \theta(x)$. If $\theta(\tilde{x})=\theta(x)$, then by the uniqueness property, we have $\tilde{E}=\tilde{D}=\tilde{Q}_1+\ldots+\tilde{Q}_d$. As we don't know the value $\theta(x)$ in advance, an algorithm to determine when $\theta(\tilde{x})=\theta(x),~\forall x\in V_\ell$ is needed. In \cite{Bruin}, Bruin gave such an algorithm and proved that, for at least half of the primes smaller than $\ell^\textrm{O(1)}$, the following holds: $\theta(\tilde{x})=\theta(x),~ \forall x \in V_\ell$ , such primes are called $\mathfrak{m}$-good primes. In practice, since $\theta(x)$ is less or equal to the genus of $X_1(\ell)$, if $\theta(\tilde{x})$ is equal to the genus for every $\tilde{x}\in V_\ell\mod p$, then $\theta(\tilde{x})=\theta(x)$ automatically for all $x\in V_\ell$. We remark here that our computation suggests that most of the primes are $\mathfrak{m}$-good. A prime $p$ is called good if it simultaneously satisfies: $\mathfrak{m}$-good and $s$-good. It is reasonable to assume that there exists an absolute constant $c$ such that the density of good primes is bigger than $c$. Here we make no attempt to prove this. Now for any good prime $p$ $$P(X):=\prod_{x\in V_\ell\setminus\textrm{O}}(X-\iota(x))$$ is a polynomial in $\mathbb{Q}[X]$ of degree $\ell^2-1$, whose reduction modulo $p$ is exactly the polynomial $$\tilde{P}(X):=\prod_{\tilde{x}\in V_\ell~\textrm{mod}~p\setminus\textrm{O}}(X-\tilde{\iota}(\tilde{x})),$$ where $\tilde{\iota}$ is the reduction map of $\iota$. Given $V_\ell \mod p=\mathbb{F}_\ell e_1+\mathbb{F}_\ell e_2$, the computation of $\tilde{P}(X)$ comes down to performing $\ell^2$ additions in the Jacobian, with a complexity $\ell^2\cdot\textrm{O}( \ell^{1+2\omega}\log^{1+\epsilon} p)=\textrm{O}( \ell^{3+2\omega}\log^{1+\epsilon} p)$. Since the heights of coefficients of $P(X)$ are expected to be in $\textrm{O}(\ell^\delta)$, for some absolute constant $\delta$. Using the fact, there exists a constant $c$ such that $$\prod_{p\le L,\textrm{prime}}p>c\cdot\exp(L).$$ To recover $P(X)$ from $\tilde{P}(X)$'s, it suffices to take the upper bound $L$ of good primes to be $\textrm{O}(\ell^\delta)$. So the complexity of computing $P(X)$ will be $$\sum_{p \le L,\textrm{prime}}\textrm{O}(\ell^{4+2\omega+\epsilon}\log^{1+\epsilon}p\cdot(\ell+\log p))=\ell^{5+2\omega+\delta+\epsilon}.$$ In practice, it would be better to choose the function $\psi(x)\in\mathbb{Q}(X_1(\ell))$, such that the degree is equal to the gonality of the curve $X_1(\ell)$. For $\ell\le 40$, $\psi(x)$ have been computed by Derickx and Hoeij, see \cite{hoeij} and \cite{Derickx}. \subsection{Finding the Frobenius endomorphism } The Galois representation associated to $\Delta(q)$ is denoted by $\rho_\ell:\textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \textrm{GL}_2(\mathbb{F}_\ell)$. Let $K_\ell:=\overline{\mathbb{Q}}^{\ker{\rho_\ell}}$ be the field cut out by the representation, then $K_\ell$ is the splitting field of the polynomial $P(X)$. For any prime $p\not=\ell$, the trace of Frobenius $\textrm{Tr}(\rho_\ell(\textrm{Frob}_p))$ is equal to $\tau(p)\mod\ell$. So we would like to identify the conjugacy class of the Galois group $\textrm{Gal}(K_\ell/\mathbb{Q})$, where the Frobenius $\textrm{Frob}_p$ lands inside. The algorithm described in \cite{Dokchitser} can be used perfectly to do the computation. Let $(a_i)_{1\le i\le\ell^2-1}$ be the roots of $P(X)$ in $K_\ell$ and $h(X)$ some polynomial in $\mathbb{Q}[X]$. Then for each conjugacy class $C\subset \textrm{Gal}(K_\ell/\mathbb{Q})$, \begin{equation}\label{traceformula} \textrm{Frob}_p\in C\Leftrightarrow \Gamma_C(\textrm{Tr}_{\frac{\mathbb{F}_p[x]}{P(x)}/\mathbb{F}_p}(h(x)x^p))\equiv0\mod p, \end{equation} where the polynomial $\Gamma_C(X)$ is given by \begin{equation}\label{GammaCX} \Gamma_C(X)=\prod_{\sigma\in C}\left(X-\sum_{i=1}^{\ell^2-1}h(a_i)\sigma(a_i) \right). \end{equation} Our strategy for computing $\Gamma_C(X)$ is using Hensel lifting. \begin{prop}Let $p$ be a prime such that the extension degree $d_p$ is in $\textrm{O}(1)$. Given the polynomial $P(X)$ as in Section 4.2 and the Ramanujan subspace $V_\ell\mod p$. Then for any conjugacy class $C\subset \textrm{Gal}(K_\ell/\mathbb{Q})$, the polynomial $\Gamma_C(X)\in\mathbb{Q}[X]$ can be computed in $\textrm{O}(\ell^{6+\delta+\epsilon})$ bit operations. \end{prop} \begin{proof} Set $\mathbb{F}_{q}:=\mathbb{F}_{p^{d_p}}$. We can lift each root $\tilde{\iota}(x)\in \mathbb{F}_q$ of $\tilde{P}(X)\in \mathbb{F}_p[X]$ to the root $\iota(x)\in \mathbb{Q}_q$ of $P(X)\in\mathbb{Q}_p[X]$ by Hensel's lemma, where $\mathbb{Q}_q$ is the unramified extension of $\mathbb{Q}_p$ with extension degree $d_p$. Since the heights of coefficients of $\Gamma_C(X)$ are bounded above by $N:=\|C\|(1+\deg h)\ell^\delta$, to recover $\Gamma_C(X)$, it suffices to lift each $\tilde{\iota}(x)$ to $\iota(x)$ with precision $N$. The complexity of a single lifting is about $\textrm{O}((\|C\|(d_pN))^{1+\epsilon})$ bit operations, see \cite{Avanzi-Cohen}. The length of the conjugacy class $C$ is in $\textrm{O}(\ell^2)$, the extension degree $d_p$ is in $\textrm{O}(1)$ and $h(X)$ can be chosen to be a polynomial with small degree (e.g. $\deg h(X)=2$ ) showed in \cite{Dokchitser}, so the complexity of lifting a single root is in $\textrm{O}(\ell^{4+\delta+\epsilon})$. There are $\ell^2-1$ roots need to be lifted, so the complexity of computing $\Gamma_C(X)$ is $\textrm{O}(\ell^{6+\delta+\epsilon})$. \end{proof} \begin{remark}The Galois group $\textrm{Gal}(K_\ell/\mathbb{Q})$ is a subgroup of $\textrm{GL}_2(\mathbb{F}_\ell)$, and the action of $\textrm{Gal}(K_\ell/\mathbb{Q})$ on the roots $a_i,1\le i\le \ell^2-1$ can be computed from the action of $\textrm{GL}_2(\mathbb{F}_\ell)$ on $V_\ell \mod p$. In practice, we would like to choose the prime $p$ with large size and small extension degree $d_p$ at the same time. \end{remark} \subsection{Complexity analysis} Now we can prove Theorem \ref{theorem:complexity} and Corollary \ref{cor:complexity}. As shown in above, the complexity of computing $P(X)$ is $\textrm{O}(\ell^{5+2\omega+\delta+\epsilon})$. There are $\ell^2-1$ conjugacy classes in $\textrm{GL}_2(\mathbb{F}_\ell)$. So the complexity of computing $\Gamma_C(X)$ for all the conjugacy classes $C\subset \textrm{GL}_2(\mathbb{F}_\ell)$ is $\textrm{O}(\ell^{8+\delta+\epsilon})$. Let $p$ be a prime not equal to $\ell$. Denote the polynomial $P(X)$ as $$P(X)=X^{\ell^2-1}+c_{\ell^2-2}X^{\ell^2-2}+\ldots+c_0.$$ The trace in (\ref{traceformula}) can be interpreted as a trace of a matrix $$\textrm{Tr}_{\frac{\mathbb{F}_p[x]}{P(x)}/\mathbb{F}_p}(x^d)=\textrm{Tr}\left( \begin{array}{cccc} 0 & & & -c_0 \\ 1 & & & -c_1 \\ & \ddots & & \vdots \\ & & 1&-c_{\ell^2-1} \\ \end{array} \right)^d \mod p. $$ Using Coppersmith-Winograd algorithm, the complexity of multiplying two $\ell^2\times\ell^2$ matrices over $\mathbb{F}_p$ is in $\textrm{O}(\ell^{4.752}\log^{1+\epsilon}p)$ bit operations. So $t:=\textrm{Tr}_{\frac{\mathbb{F}_p[x]}{P(x)}/\mathbb{F}_p}(h(x)x^p)$ can be computed in $\textrm{O}(\ell^{4.752}\log^{2+\epsilon}p)$. Given $t$, the complexity of checking whether $\Gamma_C(t)\mod p$ is equal to zero for all the conjugacy classes $C$ is bounded above by $\textrm{O}(\ell^4\log^{1+\epsilon}p)$. So, for any prime $p\not=\ell$, the complexity of computing $\tau(p)\mod\ell$ consists of the complexity of computing $P(X)$, $\Gamma_C(X)$, $\textrm{Tr}(h(x)x^p)$ and $\Gamma_C(\textrm{Tr}(h(x)x^p))$ for all conjugacy classes $C\subset\mbox{\rm GL}_2(\mathbb{F}_\ell)$, which sums up to \begin{equation} \textrm{O}(\ell^{5+2\omega+\delta+\epsilon})+\textrm{O}(\ell^{4.752}\log^{2+\epsilon}p)+\textrm{O}(\ell^4\log^{1+\epsilon}p). \end{equation} For prime $p$, we have $\tau(p)\in \textrm{O}(p^6)$. Therefore it suffices to recover $\tau(p)$ from $\tau(p)\mod \ell$ with $\ell\le L$, where $L$ is in $\textrm{O}(\log p)$. So the total complexity of computing $\tau(p)$ is about \begin{equation} \sum_{\ell<L,\textrm{prime}}\textrm{O}(\ell^{5+2\omega+\delta+\epsilon})+\textrm{O}(\ell^{4.752}\log^{2+\epsilon}p)+ \textrm{O}(\ell^4\log^{1+\epsilon}p) =\textrm{O}(\log^{6+2\omega+\delta+\epsilon}p). \end{equation} \begin{remark} If the constant $\omega$ reaches $2.376$, the complexity of the algorithm is $\textrm{O}(\log^ {10.752+\delta+\epsilon}p )$. Moreover, if $\delta$ is bounded above by $3$, the complexity is $\textrm{O}(\log^{13.752+\epsilon}p )$. \end{remark} \section{Implementation and results} The algorithm has been implemented in MAGMA. One big advantage of the algorithm is that, it is rather straightforward to implement, where the major work is dealing with the action of Hecke operators on divisors of the function field. The following computation was done on a personal computer AMD FX(tm)-6200 Six-Core Processor 3.8GHz. Let $Q_\ell(x)$ be the polynomial corresponding to the projective representation, defined as $$Q_\ell(X):=\prod_{L\in\mathbb{P}(V_\ell)}(X-\sum_{\alpha\in L\setminus\textrm{O}}{\iota}(\alpha)),$$ which can be used to check whether $\tau(p)\equiv 0 \mod\ell$. More precisely, we have the following lemma, see \cite{Bosman} \begin{lemma}Let $Q_\ell(X)$ be the polynomial defined as above and $p\nmid\textrm{Disc}(Q_\ell(X))$ a prime. Then $\tau(p)\equiv 0 \mod\ell$ if and only if $Q_\ell(X)\mod p$ has an irreducible factor of degree 2 over $\mathbb{F}_p$. \end{lemma} \begin{example}$\ell=13$. To recover $Q_{13}(X)$, it suffices to take a good prime set as $$\{ 19, 23, 29, 43, 53, 61, 67, 71, 79, 83, 89, 109, 127, 149, 157, 163, 179, 193, 211, 223, 229, 233, 239, 241 \},$$ with a total of 24 primes, whose product is a 52 digits number. We have \begin{displaymath} \begin{split} 2535853\cdot Q_{13}(X)=&2535853X^{14} + 760835865X^{13} + 96570870461X^{12} + 7083218145770X^{11} +\\ &341554192651282X^{10} + 11596551892957577X^9 + 288394789072144586X^8 +\\ &5369247990154339694X^7 + 75509842125272520446X^6 +\\ &800346109631330635243X^5 + 6303044886777591079517X^4 +\\ &35793920471135235999031X^3 + 138667955645963961606844X^2 +\\ &328650624808255716476451X + 361128579432826593902125. \end{split} \end{displaymath} The computation took several minutes. The product of the primes needed to recover $P_{13}(X)$ is about 130 digits and the computation took nearly one hour. In order to recover the polynomials $\Gamma_C(X)$ for $C\subset\textrm{GL}_2(\mathbb{F}_{13})$, we first choose a good prime $p=34939$ with $d_p=2$, and then compute the roots of $P_{13}(X) \mod p$ by computing $V_{\ell}\mod p$. Notice that all of the roots are in $\mathbb{F}_{p^2}$. Using Hensel lemma, we lift each root to the $p$-adic field $\mathbb{Q}_{p^2}$ with a precision nearly 5000 digits and then reconstruct $\Gamma_X(X)$ by the formula (\ref{GammaCX}). The computation took several hours. \end{example} We summarize the results in the following table, where the matrices in the last column are the representatives of the conjugacy classes where the Frobenius endomorphism $\mbox{\rm Frob}_p$ land inside and prime $p$ is set to be $10^{1000}+1357$. \begin{center} \renewcommand\arraystretch{1.5} \begin{tabular}{ccccc} \hline level & \begin{tabular}{cc} \multicolumn{2}{c}{$Q_\ell(X)$} \\ good primes & time\\ \end{tabular} & \begin{tabular}{cc} \multicolumn{2}{c}{$P_\ell(X)$} \\ good primes & time \\ \end{tabular} & \begin{tabular}{c} $\Gamma_C(X)$ \\ time \\ \end{tabular} & $\mbox{\rm Frob}_p$ \\ \hline 13& \begin{tabular}{cc} 52 digits & few minutes\\ \end{tabular} & \begin{tabular}{cc} 130 digits & few minutes \\ \end{tabular}& few hours & $\left[\begin{smallmatrix} 10 & 0 \\ 0 & 7 \end{smallmatrix}\right]$\\ \hline 17& \begin{tabular}{cc} 467 digits & few hours \\ \end{tabular} & \begin{tabular}{cc} 740 digits & few hours \\ \end{tabular}& one day & $\left[\begin{smallmatrix} 15 & 1\\ 0 & 15 \end{smallmatrix}\right]$\\ \hline 19& \begin{tabular}{cc} 832 digits & few days \\ \end{tabular} & \begin{tabular}{cc} 1681 digits & few days \\ \end{tabular}& few days & $\left[\begin{smallmatrix} 17 & 1 \\ 0 & 17 \end{smallmatrix}\right]$\\ \hline \end{tabular} \end{center} Using a plane model for $X_H(31)$, we also finished the level 31 case. It took several days to recover the polynomial $Q_{31}(X)$. The coefficients of $Q_{31}(X)$ are very large, where the biggest one reaches 2426 digits. Similar to the $\ell\in\{13,17,19\}$ cases, $Q_{31}(X)$ can be reduced to a polynomial with small coefficients. One of the reduced polynomials is \begin{displaymath} \begin{split} f_{31}= &X^{32} - 4X^{31} - 155X^{28} + 713X^{27} - 2480X^{26} + 9300X^{25} - 5921X^{24}\\ &+ 24707X^{23} + 127410X^{22} - 646195X^{21} + 747906X^{20} - 7527575X^{19} +\\ &4369791X^{18} - 28954961X^{17} - 40645681X^{16} + 66421685X^{15} - 448568729X^{14}\\ &+ 751001257X^{13} - 1820871490X^{12} + 2531110165X^{11} - 4120267319X^{10} +\\ &4554764528X^9 - 5462615927X^8 + 4607500922X^7 - 4062352344X^6 + 2380573824X^5\\ &- 1492309000X^4 + 521018178X^3 - 201167463X^2 + 20505628X - 1261963. \end{split} \end{displaymath} Set $K=\mathbb{Q}[X]/(f_{31})$ and $\mathcal{O}_K$ the maximal order of $K$, then we can check that the discriminant of $\mathcal{O}_K$ is equal to $-31^{41}$, and the Galois group of the polynomial $f_{31}$ is isomorphic to $\textrm{PGL}_2(\mathbb{F}_{31})$. These facts prove $f_{31}$ is the polynomial corresponding to the mod-31 projective representation associated to $\Delta(q)$. An easy calculation shows that the first few primes satisfying Serre's criteria as well as $\tau(p)\equiv0\mod 11\cdot 13\cdot 17\cdot 19\cdot 31$ are $$982149821766199295999,3748991773540147199999,$$ $$3825907566871689215999,3903375187595059199999.$$ So we proved Theorem \ref{theorem:main2}. From the table above, we have $\tau(10^{1000}+1357)\equiv 15\mod19$. So the missing sign in the table of \cite{Edixhoven} is found. Moreover, we have $$\tau(10^{1000}+1357)\equiv\pm 18\mod 31.$$ The Magma code of our algorithm can be downloaded from the web at the address \begin{center}\url{http://faculty.math.tsinghua.edu.cn/~lsyin/publication.htm}\end{center} \section{acknowledgments} Our interest in computing coefficients of modular forms is motivated by the wonderful courses given by Bas Edixhoven and Jean-Marc Couveignes at Tsinghua University. Many thanks to them for their encouragement. The first author wishes to thank Jean-Marc Couveignes for his continuous assistance and many helpful suggestions. Thanks to Ye Tian for his helpful comments. Many thanks to Maarten Derickx for fruitful discussions, helpful comments and suggestions. Thanks to Mark van Hoeij for providing us a plane model for $X_H(31)$ and helping us reduce the polynomial $Q_{31}(X)$, these make the computation of level 31 a reality. \end{document}	arXiv
Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier Julián López-Gómez1 & Juan Carlos Sampedro ORCID: orcid.org/0000-0002-0791-39081 Journal of Fixed Point Theory and Applications volume 24, Article number: 8 (2022) Cite this article In this paper, we prove an analogue of the uniqueness theorems of Führer [15] and Amann and Weiss [1] to cover the degree of Fredholm operators of index zero constructed by Fitzpatrick, Pejsachowicz and Rabier [13], whose range of applicability is substantially wider than for the most classical degrees of Brouwer [5] and Leray–Schauder [22]. A crucial step towards the axiomatization of this degree is provided by the generalized algebraic multiplicity of Esquinas and López-Gómez [8, 9, 25], $\chi $, and the axiomatization theorem of Mora-Corral [28, 32]. The latest result facilitates the axiomatization of the parity of Fitzpatrick and Pejsachowicz [12], $\sigma (\cdot ,[a,b])$, which provides the key step for establishing the uniqueness of the degree for Fredholm maps. The Leray–Schauder degree was introduced in [22] to get some rather pioneering existence results on Nonlinear Partial Differential Equations. It refines, very substantially, the finite-dimensional degree introduced by Brouwer [5] to prove his celebrated fixed point theorem. The Leray–Schauder degree is a generalized topological counter of the number of zeros that a continuous map, f, can have on an open bounded subset, $\Omega $, of a real Banach space, X. To be defined, f must be a compact perturbation of the identity map. Although this always occurs in finite-dimensional settings, it fails to be true in many important applications where the involved operators are not compact perturbations of the identity map but Fredholm operators of index zero between two real Banach spaces X and Y. For Fredholm maps, it is available the degree of Fredholm maps of Fitzpatrick, Pejsachowicz and Rabier [13], a refinement of the Elworthy and Tromba degree, [7], based on the topological concepts of parity and orientation discussed by Fitzpatrick and Pejsachowicz in [12]. Very recently, the authors of this article established in [29] the hidden relationships between the degree for Fredholm maps of [13] and the concept of generalized algebraic multiplicity of Esquinas and López-Gómez in [8, 9, 25], in a similar manner as the Leray–Schauder formula relates the Leray–Schauder degree to the classic algebraic multiplicity. The main goal of this paper is axiomatizing the Fitzpatrick–Pejsachowicz–Rabier degree in the same vein as the Brouwer and Leray–Schauder degrees were axiomatized by Führer [15] and Amann and Weiss [1], respectively. In other words, we will give a minimal set of properties that characterize the topological degree of Fitzpatrick, Pejsachowicz and Rabier. Throughout this paper, for any given pair of real Banach spaces X, Y with $X\subset Y$, we denote by $\mathcal {L}_c(X,Y)$ the set of linear and continuous operators, $L \in \mathcal {L}(X,Y)$, that are a compact perturbation of the identity map, $L=I_X-K$. Then, the linear group, GL(X, Y) is defined as the set of linear isomorphisms $L\in \mathcal {L}(X,Y)$. Similarly, the compact linear group, $GL_c(X,Y)$, is defined as $GL(X,Y)\cap \mathcal {L}_c(X,Y)$. For any $L \in \mathcal {L}(X,Y)$, the sets N[L] and R[L] stand for the null space (kernel) and the range (image) of L, respectively. An operator $L \in \mathcal {L}(X,Y)$ is said to be a Fredholm operator if $$\begin{aligned} \mathrm {dim\,}N[L]<\infty \quad \hbox {and}\quad \mathrm {codim\,}R[T]<\infty . \end{aligned}$$ In such case, R[L] must be closed and the index of L is defined by $$\begin{aligned} \mathrm {ind\,} L:= \mathrm {dim\,}N[L]-\mathrm {codim\,}R[L]. \end{aligned}$$ In this paper, the set of Fredholm operators of index zero, $L \in \mathcal {L}(X,Y)$, is denoted by $\Phi _0(X,Y)$, and we set $\Phi _0(X):=\Phi _0(X,X)$. Moreover, a map $f:\overline{\Omega }\subset X\rightarrow Y$ is said to be compact if it sends bounded subsets of $\overline{\Omega }$ into relatively compact sets of Y. In the context of the Leray–Schauder degree, for any real Banach space X, any open and bounded domain $\Omega \subset X$ and any map $f:\overline{\Omega }\subset X \rightarrow X$, it is said that $(f,\Omega )$ is an admissible pair if $f\in \mathcal {C}(\overline{\Omega },X)$; f is a compact perturbation of the identity map $I_X$; $0\notin f(\partial \Omega )$. The class of admissible pairs will be denoted by $\mathscr {A}_{LS}$. Note that $(I_X,\Omega )\in \mathscr {A}_{LS}$ for every open and bounded subset $\Omega \subset X$, such that $0\notin \partial \Omega $. Actually, $(I_X,\Omega )\in \mathscr {A}_{LS,GL_c}$, where $\mathscr {A}_{LS,GL_c}$ stands for the set of admissible pairs $(L,\Omega )\in \mathscr {A}_{LS}$, such that $L\in GL_c(X)$. A homotopy $H\in \mathcal {C}([0,1]\times \overline{\Omega }, X)$ is said to be admissible if $0\notin H([0,1]\times \partial \Omega )$ and $H(t,x)=x-C(t,x)$, where $C:[0,1]\times \Omega \rightarrow X$ is a compact map. The class of admissible homotopies $(H,\Omega )$ will be denoted by $\mathscr {H}_{LS}$. The next fundamental theorem establishes the existence and the uniqueness of the Leray–Schauder degree. The existence goes back to Leray and Schauder [22] and the uniqueness is attributable to Amann and Weiss [1], though Führer [15] had already proven the uniqueness of the Brouwer degree when [1] was published. Theorem 1.1 For any real Banach space X, there exists a unique integer-valued map, $\deg _{LS}:\mathscr {A}_{LS} \rightarrow \mathbb {Z}$, satisfying the following properties: Normalization: $\deg _{LS}(I_X,\Omega )=1$ if $0\in \Omega $. Additivity: For every $(f,\Omega )\in \mathscr {A}_{LS}$ and any pair of open disjoint subsets, $\Omega _{1}$ and $\Omega _{2}$, of $\Omega $, such that $0\notin f(\overline{\Omega }\backslash (\Omega _{1}\uplus \Omega _{2}))$ $$\begin{aligned} \deg _{LS}(f,\Omega )=\deg _{LS}(f,\Omega _{1})+\deg _{LS}(f,\Omega _{2}). \end{aligned}$$ Homotopy Invariance: For every admissible homotopy $(H,\Omega )\in \mathscr {H}_{LS}$ $$\begin{aligned} \deg _{LS}(H(0,\cdot ),\Omega )=\deg _{LS}(H(1,\cdot ),\Omega ). \end{aligned}$$ Moreover, for every $(L,\Omega )\in \mathscr {A}_{LS,GL_c}$ with $0\in \Omega $ $$\begin{aligned} \deg _{LS}(L,\Omega )=(-1)^{\sum _{i=1}^{q}\mathfrak {m}_\mathrm {alg}[I_{X}-L;\mu _{i}]} \end{aligned}$$ $$\begin{aligned} {{\,\mathrm{Spec}\,}}(I_{X}-L)\cap (1,\infty )=\{\mu _{1},\mu _{2},...,\mu _{q}\}, \qquad \mu _i\ne \mu _j\quad \hbox {if}\;\; i\ne j. \end{aligned}$$ The map $\deg _{LS}$ is refereed to as the Leray–Schauder degree. In (1.2), setting $K:= I_X-L$, for any eigenvalue $\mu \in {{\,\mathrm{Spec}\,}}(K)$, we have denoted by $\mathfrak {m}_\mathrm {alg}[K;\mu ]$ the classical algebraic multiplicity of $\mu $, that is $$\begin{aligned} \mathfrak {m}_{\mathrm {alg}}[K;\mu ]= \mathrm {dim\,}\mathrm {Ker}[(\mu I_{X}-K)^{\nu (\mu )}], \end{aligned}$$ where $\nu (\mu )$ is the algebraic ascent of $\mu $, i.e., the minimal integer, $\nu \ge 1$, such that $$\begin{aligned} \mathrm {Ker}[(\mu I_{X}-K)^{\nu }]=\mathrm {Ker}[(\mu I_{X}-K)^{\nu +1}]. \end{aligned}$$ In Theorem 1.1, the axiom (N) is called the normalization property, because, for every $n\in \mathbb {Z}$, the map $n\, \mathrm {deg}_{LS}$ also satisfies the axioms (A) and (H), though not (N). Thus, the axiom (N) normalizes the degree so that, for the identity map, it provides us with its exact number of zeroes. The axiom (A) packages three basic properties of the Leray–Schauder degree. Indeed, by choosing $\Omega =\Omega _1=\Omega _2=\emptyset $, it becomes apparent that $$\begin{aligned} \mathrm {deg}_{LS}(f,\emptyset )=0, \end{aligned}$$ so establishing that no continuous map can admit a zero in the empty set. Moreover, in the special case, when $\Omega = \Omega _{1}\uplus \Omega _{2}$, (1.2) establishes the additivity property of the degree. Finally, in the special case, when $\Omega _2=\emptyset $, it follows from (1.2) and (1.3) that: $$\begin{aligned} \mathrm {deg}_{LS}(f,\Omega )= \mathrm {deg}_{LS}(f,\Omega _1), \end{aligned}$$ which is usually refereed to as the excision property of the degree. If, in addition, also $\Omega _1=\emptyset $, then $$\begin{aligned} \mathrm {deg}_{LS}(f,\Omega )= 0 \quad \hbox {if}\;\; f^{-1}(0)\cap \overline{\Omega } =\emptyset . \end{aligned}$$ Therefore, for every $(f,\Omega )\in \mathscr {A}_{LS}$, such that $\deg _{LS}(f,\Omega )\ne 0$, the equation $f(x)=0$ admits, at least, a solution in $\Omega $. This key property is refereed to as the fundamental, or solution, property of the degree. The axiom (H) establishes the invariance by homotopy of the degree. It allows to calculate the degree in the practical situations of interest from the point of view of the applications. Not surprisingly, when dealing with analytic maps, f, in $\mathbb {C}$, it provides us with the exact number of zeroes of f, counting orders, in $\Omega $ (see, e.g., Chapter 11 of [24]). From a geometrical point of view, the construction of the Leray–Schauder degree relies on the concept of orientation, that is, on the fact that $GL_{c}(X)$ consists of two path connected components. Let $\mathfrak {L}\in \mathcal {C}([0,1], GL_{c}(X))$ be a continuous operator curve on $GL_{c}(X)$. Since $\mathfrak {L}$ can be regarded as the admissible homotopy $H\in \mathcal {C}([0,1]\times \overline{\Omega },X)$ with $(H(t,\cdot ),\Omega )\in \mathscr {A}_{LS,GL_c}$ for each $t\in [0,1]$ defined by $H(t,\cdot ):=\mathfrak {L}(t)$, $t\in [0,1]$, by the axiom (H), the integer $\deg _{LS}(\mathfrak {L}(t),\Omega )$ is constant for all $t\in [0,1]$. This introduces an equivalence relation between the operators of $GL_c(X)$. Indeed, for every pair of operators $L_0$, $L_1\in GL_c(X)$, it is said that $L_0 \sim L_1$ if $L_0$ and $L_1$ are homotopic in $\mathscr {A}_{LS,GL_c}$ in the sense that $L_0=\mathfrak {L}(0)$ and $L_1= \mathfrak {L}(1)$ for some curve $\mathfrak {L}\in \mathcal {C}([0,1], GL_{c}(X))$. This equivalence relation divides $GL_{c}(X)$ into two path-connected components, $GL^{+}_{c}(X)$ and $GL^{-}_{c}(X)$, separated away by $\mathcal {S}(X)\cap GL_c(X)$, where $$\begin{aligned} \mathcal {S}(X):=\mathcal {L}(X)\setminus GL(X). \end{aligned}$$ This allows us to define a map $$\begin{aligned} \deg _{LS}(L,\Omega ):=\left\{ \begin{array}{ll} 1 &{} \text { if } L\in GL^{+}_{c}(X)\;\;\hbox {and}\;\; 0\in \Omega , \\ -1 &{} \text { if } L\in GL^{-}_{c}(X)\;\;\hbox {and}\;\; 0\in \Omega ,\\ 0 &{} \text { if } L\in GL_{c}(X)\;\;\hbox {and}\;\; 0\notin \Omega , \end{array}\right. \end{aligned}$$ verifying the three axioms of the Leray–Schauder degree in the class $\mathscr {A}_{LS,GL_c}$ and, in particular, the homotopy invariance. Once defined the degree in $\mathscr {A}_{LS,GL_c}$, one can extend this restricted concept of degree to the regular pairs $(f,\Omega )$ through the identity $$\begin{aligned} \deg _{LS}(f,\Omega )=\sum _{x\in f^{-1}(0)\cap \Omega }\deg _{LS}(Df(x),\Omega ). \end{aligned}$$ A pair $(f,\Omega )$ is said to be regular if 0 is a regular value of $f:\overline{\Omega }\subset X\rightarrow X$, i.e., if $Df(x)\in GL_{c}(X)$ for each $x\in f^{-1}(0)\cap \Omega $. Finally, according to the Sard–Smale theorem and the homotopy invariance property, it can be extended to be defined for general admissible pairs, $(f,\Omega )\in \mathscr {A}_{LS}$. A crucial feature that facilitates this construction of the degree is the fact that the space $GL_{c}(X)$ consists of two path-connected components. This fails to be true in the general context of Fredholm operators of index zero, which makes the mathematical analysis of this paper much more sophisticated technically. The main goal of this paper is establishing an analogous of Theorem 1.1 for Fredholm Operators of index zero within the context of the degree for Fredholm maps of Fitzpatrick, Pejsachowicz and Rabier [13]. Let $\Omega $ be an open and bounded subset of a real Banach space X. Then, an operator $f:\overline{\Omega }\subset X \rightarrow Y$ is said to be $\mathcal {C}^{1}$-Fredholm of index zero if $$\begin{aligned} f\in \mathcal {C}^{1}(\overline{\Omega },Y)\quad \hbox {and}\quad Df\in \mathcal {C}(\Omega ,\Phi _{0}(X,Y)). \end{aligned}$$ In this paper, the set of all these operators is denoted by $\mathscr {F}^{1}_{0}(\Omega ,Y)$. A given operator $f\in \mathscr {F}^{1}_{0}(\Omega ,Y)$ is said to be orientable when $Df:\Omega \rightarrow \Phi _{0}(X,Y)$ is an orientable map (see Sect. 3 for the concept of orientability of maps). Moreover, for any open and bounded subset, $\Omega $, of X and any map $f:\overline{\Omega }\subset X \rightarrow Y$ satisfying $f\in \mathscr {F}^{1}_{0}(\Omega ,Y)$ is orientable with orientation $\varepsilon $, f is proper in $\overline{\Omega }$, i.e., $f^{-1}(K)$ is compact for every compact subset $K\subset Y$, $0\notin f(\partial \Omega )$, it will be said that $(f,\Omega ,\varepsilon )$ is a Fredholm admissible triple. The set of all Fredholm admissible triples in the context of Fitzpatrick, Pejsachowicz and Rabier [13] is denoted by $\mathscr {A}$. Given $(f,\Omega ,\varepsilon )\in \mathscr {A}$, it is said that $(f,\Omega ,\varepsilon )$ is a regular triple if 0 is a regular value of f, i.e., $Df(x)\in GL(X,Y)$ for all $x \in f^{-1}(0)$. The set of regular triples is denoted by $\mathscr {R}$. Finally, a map $H\in \mathcal {C}^{1}([0,1]\times \overline{\Omega },Y)$ is said to be a $\mathcal {C}^{1}$-Fredholm homotopy if $D_{x}H(t,x)\in \Phi _{0}(X,Y)$ for each $(t,x)\in [0,1]\times \Omega $, and it is called orientable if $D_{x}H:[0,1]\times \Omega \rightarrow \Phi _{0}(X,Y)$ is an orientable map. The main theorem of this paper reads as follows. There exists a unique integer-valued map $\deg : \mathscr {A}\rightarrow \mathbb {Z}$ satisfying the next properties: Normalization: $\deg (L,\Omega ,\varepsilon )=\varepsilon (0)$ for all $L\in GL(X,Y)$ if $0\in \Omega $. Additivity: For every $(f,\Omega ,\varepsilon )\in \mathscr {A}$ and any pair of disjoint open subsets $\Omega _{1}$ and $\Omega _{2}$ of $\Omega $ with $0\notin f(\Omega \backslash (\Omega _{1}\uplus \Omega _{2}))$ $$\begin{aligned} \deg (f,\Omega ,\varepsilon )=\deg (f,\Omega _{1},\varepsilon )+\deg (f,\Omega _{2},\varepsilon ). \end{aligned}$$ Homotopy Invariance: For each proper $\mathcal {C}^{1}$-Fredholm homotopy $H\in \mathcal {C}^{1}([0,1]\times \overline{\Omega }, Y)$ with orientation $\varepsilon $ and $(H(t,\cdot ),\Omega ,\varepsilon _{t})\in \mathscr {A}$ for each $t\in [0,1]$ $$\begin{aligned} \deg (H(0,\cdot ),\Omega ,\varepsilon _{0})=\deg (H(1,\cdot ),\Omega ,\varepsilon _{1}). \end{aligned}$$ Moreover, given $(f,\Omega ,\varepsilon )\in \mathscr {R}$ with $\Omega $ connected and $\mathcal {R}_{Df}\ne \emptyset $, for each $p\in \mathcal {R}_{Df}$ $$\begin{aligned} \deg (f,\Omega ,\varepsilon )=\varepsilon (p)\cdot \sum _{x\in f^{-1}(0)\cap \Omega } (-1)^{\chi [\mathfrak {L}_{\omega ,x},[a,b]]}, \end{aligned}$$ where $\mathfrak {L}_{\omega ,x}\in \mathscr {C}^\omega ([a,b],\Phi _{0}(X,Y))$ is any analytic curve $\mathcal {A}$-homotopic to $Df\circ \gamma $ (see Sect. 3 for the precise meaning), for some $\gamma \in \mathcal {C}([a,b],\Omega )$, such that $\gamma (a)=p$, $\gamma (b)=x$, and $$\begin{aligned} \chi [\mathfrak {L}_{\omega ,x},[a,b]]:=\sum _{\lambda _{x}\in \Sigma (\mathfrak {L}_{\omega ,x})\cap [a,b]}\chi [\mathfrak {L}_{\omega ,x},\lambda _{x}], \end{aligned}$$ where $\chi $ is the generalized algebraic multiplicity introduced by Esquinas and López-Gómez in [8, 9, 25] (see Sect. 2 for its definition and main properties). As in the context of the Leray–Schauder degree, the axiom (A) packages three fundamental properties of the degree. Namely, the additivity and excision properties, as well as the existence property, that is, whenever $(f,\Omega ,\varepsilon )\in \mathscr {A}$ satisfies $\deg (f,\Omega ,\varepsilon )\ne 0$, there exists $x\in \Omega $, such that $f(x)=0$. The existence of the map $\deg $ was established by Fitzpatrick, Pejsachowicz and Rabier in [13] in the $\mathcal {C}^{2}$ case based on the concept of orientability introduced by Fitzpatrick and Pejsachowicz [12] and later generalized to cover the $\mathcal {C}^{1}$ setting by Pejsachowicz and Rabier in [35]. The identity (1.5) is a substantial sharpening of the classical Leray–Schauder formula in the context of the degree for Fredholm maps; it was proven by the authors in [29] and, by the sake of completeness, it will be proved in this article again in Sect. 5. Thus, the main novelty of Theorem 1.2 is establishing the uniqueness of $\deg $ as a direct consequence of (1.5); so, establishing an analogue of Theorem 1.1. Benevieri and Furi [4] have established the uniqueness of another formulation of the topological degree for Fredholm operators [2, 3]. In particular, using different techniques, they introduced another concept of orientability for continuous maps $h:\Lambda \rightarrow \Phi _{0}(X,Y)$ on a topological space $\Lambda $. When h has a regular point, that is $$\begin{aligned} \mathcal {R}_{h}:=\{p\in \Lambda : h(p)\in GL(X,Y)\}\ne \emptyset , \end{aligned}$$ the two notions coincide in the sense that $h:\Lambda \rightarrow \Phi _{0}(X,Y)$ is orientable in the Benivieri–Furi sense (BF-orientable for short) if and only if it is Fitzpatrick–Pejsachowicz orientable (FP-orientable for short). However, when $\mathcal {R}_{h}=\emptyset $, these two concepts are different. Although the singular maps (with $\mathcal {R}_{h}=\emptyset $) are orientable adopting the FP-orientation, there are examples of singular h's that are not BF-orientable. More precisely, given a Banach space X of Kuiper type, i.e., such that GL(X) is contractible, consider the map defined by $$\begin{aligned} \mathfrak {S}:\mathbb {S}^{1}\longrightarrow \Phi _{0}(X\times \mathbb {R}), \quad \mathfrak {S}(t):=\left( \begin{array}{c@{\quad }c} \mathfrak {L}(t) &{} 0 \\[1ex] 0 &{} 0 \end{array}\right) , \end{aligned}$$ where $\mathbb {S}^{1}$ stands for the unit circle, the matrix decomposition is given through the canonical projections $$\begin{aligned} P_{1}&: X\times \mathbb {R}\rightarrow X, \quad P_{1}(x,\lambda )=x,\\ P_{2}&:X\times \mathbb {R}\rightarrow \mathbb {R}, \quad P_{2}(x,\lambda )=\lambda , \end{aligned}$$ and $\mathfrak {L}:\mathbb {S}^{1}\rightarrow \Phi _{0}(X)$ is some BF-nonorientable map, whose existence is guaranteed by [4, Th. 3.15]. Then, clearly, $\mathfrak {S}$ is singular, i.e., $\mathcal {R}_{\mathfrak {S}}=\emptyset $, and hence, it is FP-orientable, thought, owing to [4, Pr. 3.8], $\mathfrak {S}$ is not BF-orientable. Based on this fact, the degree constructed by Benevieri and Furi does not coincide with the degree of Fitzpatrick, Pejsachowicz and Rabier, because there are admissible triples $(f,\Omega ,\varepsilon )$, such that $Df:\Omega \rightarrow \Phi _{0}(X,Y)$ is not BF-orientable. Thus, although Benevieri and Furi proved in [4] an uniqueness result for their degree, our Theorem 1.2 here is independent of their main uniqueness result. Actually, both uniqueness results are independent in the sense that no one implies the other, though in some important applications, both degrees coincide. However, since the algebraic multiplicity $\chi $ is defined for Fredholm operator curves $\mathfrak {L}:[a,b]\rightarrow \Phi _{0}(X,Y)$ and the orientability notion of Fitzpatrick and Pejsachowicz is defined through the use of this type of curves by means of their notion of parity, we see far more natural the degree of Fitzpatrick, Pejsachowicz and Rabier for delivering an analogue of the uniqueness theorem of Amann and Weiss through (1.5), within the same vein as in the classical context of the Leray–Schauder degree. This paper is organized as follows. Section 2 contains some necessary preliminaries on the Leray–Schauder degree and the generalized algebraic multiplicity, $\chi $, used in the generalized Leray–Schauder formula (1.5). Section 3 introduces the concepts of parity and orientation of Fitzpatrick and Pejsachowicz [12] and collects some of the findings of the authors in [29], where the Fitzpatrick–Pejsachowicz parity, $\sigma $, was calculated through the generalized algebraic multiplicity $\chi $. These results are needed for axiomatizing the parity $\sigma $ in Sect. 4. The main result of Sect. 4 is Theorem 4.2, which characterizes $\sigma $ through a normalization property, a product formula, and its invariance by homotopy, by means of the algebraic multiplicity $\chi $. This result is reminiscent of the uniqueness theorem of Mora-Corral [32] for the multiplicity $\chi $ (see also Chapter 6 of [28]). Finally, based on these results, the proof of Theorem 1.2 is delivered in Sect. 5 after revisiting, very shortly, the main concepts of the Fitzpatrick–Pejsachowicz–Rabier degree. Generalized algebraic multiplicity As the generalized algebraic multiplicity introduced by Esquinas and López-Gómez in [8, 9, 25] is a pivotal technical device in the proof of Theorem 1.2 through the formula (1.5), we will collect some of its most fundamental properties, among them the uniqueness theorem of Mora-Corral [28, 32]. Given two Banach spaces, X and Y, by a Fredholm path, or curve, it is meant any map $\mathfrak {L}\in \mathcal {C}([a,b],\Phi _{0}(X,Y))$. Given a Fredholm path, $\mathfrak {L}$, it is said that $\lambda \in [a,b]$ is an eigenvalue of $\mathfrak {L}$ if $\mathfrak {L}(\lambda )\notin GL(X,Y)$. Then, the spectrum of $\mathfrak {L}$, $\Sigma (\mathfrak {L})$, consists of the set of all these eigenvalues, that is $$\begin{aligned} \Sigma (\mathfrak {L}):=\{\lambda \in [a,b]: \mathfrak {L}(\lambda )\notin GL(X,Y)\}. \end{aligned}$$ According to Lemma 6.1.1 of [25], $\Sigma (\mathfrak {L})$ is a compact subset of [a, b], though, in general, one cannot say anything more about it, because for any given compact subset of [a, b], J, there exists a continuous function $\mathfrak {L}:[a,b]\rightarrow \mathbb {R}$, such that $J=\mathfrak {L}^{-1}(0)$. Next, we will deliver a concept introduced in [25] to characterize whether, or not, the algebraic multiplicity of Esquinas and López-Gómez [8, 9, 25] is well defined. Let $\mathfrak {L}\in \mathcal {C}([a,b], \Phi _{0}(X,Y))$ and $k\in \mathbb {N}$. An eigenvalue $\lambda _{0}\in \Sigma (\mathfrak {L})$ is said to be a k-algebraic eigenvalue if there exits $\varepsilon >0$, such that $\mathfrak {L}(\lambda )\in GL(X,Y)$ if $0<\|\lambda -\lambda _0\|<\varepsilon $; There exits $C>0$, such that $$\begin{aligned} \Vert \mathfrak {L}^{-1}(\lambda )\Vert<\frac{C}{\|\lambda -\lambda _{0}\|^{k}}\quad \hbox {if}\;\; 0<\|\lambda -\lambda _0\|<\varepsilon ; \end{aligned}$$ k is the least positive integer for which (2.1) holds. The set of algebraic eigenvalues of $\mathfrak {L}$ or order k will be denoted by ${{\,\mathrm{Alg}\,}}_k(\mathfrak {L})$. Thus, the set of algebraic eigenvalues can be defined by $$\begin{aligned} {{\,\mathrm{Alg}\,}}(\mathfrak {L}):=\biguplus _{k\in \mathbb {N}}{{\,\mathrm{Alg}\,}}_k(\mathfrak {L}). \end{aligned}$$ By Theorems 4.4.1 and 4.4.4 of [25], when $\mathfrak {L}(\lambda )$ is real analytic in [a, b], i.e., $\mathfrak {L}\in \mathcal {C}^{\omega }([a,b], \Phi _{0}(X,Y))$, then either $\Sigma (\mathfrak {L})=[a,b]$, or $\Sigma (\mathfrak {L})$ is finite and $\Sigma (\mathfrak {L})\subset {{\,\mathrm{Alg}\,}}(\mathfrak {L})$. According to [28, Ch. 7], $\lambda _0\in {{\,\mathrm{Alg}\,}}(\mathfrak {L})$ if, and only if, the lengths of all Jordan chains of $\mathfrak {L}$ at $\lambda _0$ are uniformly bounded above, which allows to characterize whether, or not, $\mathfrak {L}(\lambda )$ admits a local Smith form at $\lambda _0$ (see Rabier [28, 36]). The next concept allows to introduce a generalized algebraic multiplicity, $\chi [\mathfrak {L},\lambda _0]$, in a rather natural manner. It goes back to [9]. Subsequently, we will denote $$\begin{aligned} \mathfrak {L}_{j}:=\frac{1}{j!}\mathfrak {L}^{(j)}(\lambda _{0}), \quad 1\le j\le r \end{aligned}$$ if these derivatives exist. Given a path $\mathfrak {L}\in \mathcal {C}^{r}([a,b],\Phi _{0}(X,Y))$ and an integer $1\le k \le r$, a given eigenvalue $\lambda _{0}\in \Sigma (\mathfrak {L})$ is said to be a k-transversal eigenvalue of $\mathfrak {L}$ if $$\begin{aligned} \bigoplus _{j=1}^{k}\mathfrak {L}_{j}\left( \bigcap _{i=0}^{j-1}{{\,\mathrm{Ker}\,}}(\mathfrak {L}_{i})\right) \oplus R(\mathfrak {L}_{0})=Y\;\; \hbox {with}\;\; \mathfrak {L}_{k}\left( \bigcap _{i=0}^{k-1}{{\,\mathrm{Ker}\,}}(\mathfrak {L}_{i})\right) \ne \{0\}. \end{aligned}$$ For these eigenvalues, the algebraic multiplicity of $\mathfrak {L}$ at $\lambda _{0}$, $\chi [\mathfrak {L},\lambda _0]$, is defined through $$\begin{aligned} \chi [\mathfrak {L}; \lambda _{0}]:=\sum _{j=1}^{k}j\cdot \dim \mathfrak {L}_{j}\left( \bigcap _{i=0}^{j-1}{{\,\mathrm{Ker}\,}}(\mathfrak {L}_{i})\right) . \end{aligned}$$ By Theorems 4.3.2 and 5.3.3 of [25], for every $\mathfrak {L}\in \mathcal {C}^{r}([a,b], \Phi _{0}(X,Y))$, $k\in \{1,2,...,r\}$ and $\lambda _{0}\in {{\,\mathrm{Alg}\,}}_{k}(\mathfrak {L})$, there exists a polynomial $\Phi : \mathbb {R}\rightarrow \mathcal {L}(X)$ with $\Phi (\lambda _{0})=I_{X}$, such that $\lambda _{0}$ is a k-transversal eigenvalue of the path $$\begin{aligned} \mathfrak {L}^{\Phi }:=\mathfrak {L}\circ \Phi \in \mathcal {C}^{r}([a,b], \Phi _{0}(X,Y)). \end{aligned}$$ Moreover, $\chi [\mathfrak {L}^{\Phi };\lambda _{0}]$ is independent of the curve of trasversalizing local isomorphisms $\Phi $ chosen to transversalize $\mathfrak {L}$ at $\lambda _0$ through (2.3), regardless $\Phi $ is a polynomial or not. Therefore, the next generalized concept of algebraic multiplicity is consistent $$\begin{aligned} \chi [\mathfrak {L};\lambda _0]:= \chi [\mathfrak {L}^\Phi ;\lambda _0]. \end{aligned}$$ This concept of algebraic multiplicity can be easily extended by setting $$\begin{aligned} \chi [\mathfrak {L};\lambda _0] =0 \quad \hbox {if}\;\; \lambda _0\notin \Sigma (\mathfrak {L}) \end{aligned}$$ $$\begin{aligned} \chi [\mathfrak {L};\lambda _0] =+\infty \quad \hbox {if}\;\; \lambda _0\in \Sigma (\mathfrak {L}) \setminus {{\,\mathrm{Alg}\,}}(\mathfrak {L}) \;\; \hbox {and}\;\; r=+\infty . \end{aligned}$$ Thus, $\chi [\mathfrak {L};\lambda ]$ is well defined for all $\lambda \in (a,b)$ of any smooth path $\mathfrak {L}\in \mathcal {C}^{\infty }([a,b],\Phi _{0}(X,Y))$ and, in particular, for any analytical curve $\mathfrak {L}\in \mathcal {C}^{\omega }([a,b],\Phi _{0}(X,Y))$. In other words, $\chi $ can be viewed for each $\lambda \in (a,b)$ as a map $$\begin{aligned} \chi [\cdot ,\lambda ]: \mathcal {C}^{\infty }([a,b],\Phi _{0}(X,Y))\longrightarrow [0,\infty ]. \end{aligned}$$ The next uniqueness result goes back to Mora-Corral [32] and [28, Ch. 6]. Let U be a non-zero real Banach space, $\lambda _{0}\in \mathbb {R}$, and let $\mathfrak {I}(U)$ be a set of Banach spaces isomorphic to U, such that $U\in \mathfrak {I}(U)$. Then, for every $\varepsilon >0$, the algebraic multiplicity $\chi $ is the unique map $$\begin{aligned} \chi [\cdot ; \lambda _{0}]:\bigcup _{X,Y\in \mathfrak {I}(U)} \mathcal {C}^{\infty }((\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon ), \Phi _{0}(X,Y))\longrightarrow [0,\infty ] \end{aligned}$$ satisfying the next two axioms If $X, Y, Z\in \mathfrak {I}(U)$ with $$\begin{aligned} \mathfrak {L}\in \mathcal {C}^{\infty }((\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon ), \Phi _{0}(X,Y)), \quad \mathfrak {M}\in \mathcal {C}^{\infty }((\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon ); \Phi _{0}(Y,Z)) \end{aligned}$$ then, the next product formula holds $$\begin{aligned} \chi [\mathfrak {M}\circ \mathfrak {L};\lambda _{0}] =\chi [\mathfrak {L};\lambda _{0}]+\chi [\mathfrak {M};\lambda _{0}]. \end{aligned}$$ There exits a rank one projection $P_{0}\in \mathcal {L}(U)$, such that $$\begin{aligned} \chi [(\lambda -\lambda _{0})P_{0}+I_{U}-P_{0};\lambda _{0}]=1. \end{aligned}$$ The axiom (P) is the product formula and the axiom (N) is a normalization property for establishing the uniqueness of the algebraic multiplicity. From these axioms, one can derive all the remaining properties of the generalized algebraic multiplicity $\chi $. Among them, that it equals the classical algebraic multiplicity when $$\begin{aligned} \mathfrak {L}(\lambda )= \lambda I_{X} - K \end{aligned}$$ for some compact operator K. Indeed, according to [25] and [28], for every smooth path $\mathfrak {L}\in \mathcal {C}^{\infty }((\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon ),\Phi _{0}(X,Y))$, the following properties hold: $\chi [\mathfrak {L};\lambda _{0}]\in \mathbb {N}\uplus \{+\infty \}$; $\chi [\mathfrak {L};\lambda _{0}]=0$ if, and only if, $\mathfrak {L}(\lambda _0) \in GL(X,Y)$; $\chi [\mathfrak {L};\lambda _{0}]<\infty $ if, and only if, $\lambda _0 \in {{\,\mathrm{Alg}\,}}(\mathfrak {L})$. If $X=Y =\mathbb {R}^N$, then, in any basis $$\begin{aligned} \chi [\mathfrak {L};\lambda _{0}]= \mathrm {ord}_{\lambda _{0}}\det \mathfrak {L}(\lambda ). \end{aligned}$$ Let $L\in \mathcal {L}(X)$ be such that $\lambda I_X-L\in \Phi _0(X)$. Then, for every $\lambda _0\in {{\,\mathrm{Spec}\,}}(L)$, there exists $k\ge 1$, such that $$\begin{aligned} \begin{aligned} \chi [\lambda I_X-L;\lambda _{0}]&=\underset{n\in \mathbb {N}}{\sup } \dim {{\,\mathrm{Ker}\,}}[(\lambda _{0}I_{X}-L)^{n}] \\&= \dim {{\,\mathrm{Ker}\,}}[(\lambda _{0}I_{X}-L)^{k}]=\mathfrak {m}_\mathrm {alg}[L;\lambda _0]. \end{aligned} \end{aligned}$$ Therefore, $\chi $ extends, very substantially, the classical concept of algebraic multiplicity. Parity and orientability This section collects some very recent findings of the authors in [29] in connection with the concepts of parity and orientability introduced by Fitzpatrick and Pejsachowicz in [12]. We begin by recalling some important features concerning the structure of the space of linear Fredholm operators of index zero, $\Phi _{0}(X,Y)$, which is an open path-connected subset of $\mathcal {L}(X,Y)$; in general, $\Phi _{0}(X,Y)$ is not linear. Subsequently, for every $n\in \mathbb {N}$, we denote by $\mathcal {S}_{n}(X,Y)$ the set of singular operators of order n $$\begin{aligned} \mathcal {S}_{n}(X,Y):=\{L\in \Phi _{0}(X,Y):\;\; \dim N[L] =n\}. \end{aligned}$$ Thus, the set of singular operators is given through $$\begin{aligned} \mathcal {S}(X,Y):=\Phi _{0}(X,Y)\backslash GL(X,Y)=\biguplus _{n\in \mathbb {N}}\mathcal {S}_{n}(X,Y). \end{aligned}$$ According to [11], for every $n\in \mathbb {N}$, $\mathcal {S}_{n}(X,Y)$ is a Banach submanifold of $\Phi _{0}(X,Y)$ of codimension $n^{2}$. This feature allows us to view $\mathcal {S}(X,Y)$ as a stratified analytic set of codimension 1 of $\Phi _{0}(X,Y)$. By Theorem 1 of Kuiper [19], the space of isomorphisms, GL(H), of any separable infinite-dimensional Hilbert space, H, is path connected. Thus, it is not possible to introduce an orientation in GL(X, Y) for general Banach spaces X, Y, since, in general, GL(X, Y) is path connected. This fact reveals a fundamental difference between finite- and infinite-dimensional normed spaces, because, for every $N\in \mathbb {N}$, the space $ GL(\mathbb {R}^{N}) $ is divided into two path connected components, $GL^\pm (\mathbb {R}^N)$. A key technical tool to overcome this difficulty to define a degree in $\Phi _0(X,Y)$ is provided by the concept of parity introduced by Fitzpatrick and Pejsachowicz [12]. The parity is a generalized local detector of the change of orientability of a given admissible path. Subsequently, a Fredholm path $\mathfrak {L}\in \mathcal {C}([a,b],\Phi _{0}(X,Y))$ is said to be admissible if $\mathfrak {L}(a), \mathfrak {L}(b)\in GL(X,Y)$, and we denote by $\mathscr {C}([a,b],\Phi _{0}(X,Y))$ the set of admissible paths. Moreover, for every $r\in \mathbb {N}\uplus \{+\infty ,\omega \}$, we set $$\begin{aligned} \mathscr {C}^r([a,b],\Phi _{0}(X,Y)):= \mathcal {C}^{r}([a,b],\Phi _{0}(X,Y))\cap \mathscr {C}([a,b],\Phi _{0}(X,Y)). \end{aligned}$$ The fastest way to introduce the notion of parity consists in defining it for $\mathscr {C}$-transversal paths and then for general admissible curves through the density of $\mathscr {C}$-transversal paths in $\mathscr {C}([a,b],\Phi _{0}(X,Y))$, already established in [11]. A Fredholm path, $\mathfrak {L}\in \mathcal {C}([a,b],\Phi _{0}(X,Y))$, is said to be $\mathscr {C}$-transversal if $\mathfrak {L}\in \mathscr {C}^{1}([a,b],\Phi _{0}(X,Y))$; $\mathfrak {L}([a,b])\cap \mathcal {S}(X,Y)\subset \mathcal {S}_{1}(X,Y)$ and it is finite; $\mathfrak {L}$ is transversal to $\mathcal {S}_{1}(X,Y)$ at each point of $\mathfrak {L}([a,b])\cap \mathcal {S}(X,Y)$. The path $\mathfrak {L}\in \mathcal {C}^{1}([a,b],\Phi _{0}(X,Y))$ is said to be transversal to $\mathcal {S}_{1}(X,Y)$ at $\lambda _{0}$ if $$\begin{aligned} \mathfrak {L}'(\lambda _{0})+T_{\mathfrak {L}(\lambda _{0})}\mathcal {S}_{1}(X,Y)=\mathcal {L}(X,Y), \end{aligned}$$ where $T_{\mathfrak {L}(\lambda _{0})}\mathcal {S}_{1}(X,Y)$ stands for the tangent space to the manifold $\mathcal {S}_{1}(X,Y)$ at $\mathfrak {L}(\lambda _{0})$. When $\mathfrak {L}$ is $\mathscr {C}$-transversal, the parity of $\mathfrak {L}$ in [a, b] is defined by $$\begin{aligned} \sigma (\mathfrak {L},[a,b]):=(-1)^{k}, \end{aligned}$$ where $k\in \mathbb {N}$ equals the cardinal of $\mathfrak {L}([a,b])\cap \mathcal {S}(X,Y)$. Thus, the parity of a $\mathscr {C}$-transversal path, $\mathfrak {L}(\lambda )$, is the number of times, mod 2, that $\mathfrak {L}(\lambda )$ intersects transversally the stratified analytic set $\mathcal {S}(X,Y)$. The fact that the $\mathscr {C}$-transversal paths are dense in the set of all admissible paths, together with the next stability property: for any $\mathscr {C}$-transversal path $\mathfrak {L}\in \mathcal {C}([a,b],\Phi _{0}(X,Y))$, there exists $\varepsilon >0$, such that $$\begin{aligned} \sigma (\mathfrak {L},[a,b])=\sigma (\tilde{\mathfrak {L}},[a,b]) \end{aligned}$$ for all $\mathscr {C}$-transversal path $\tilde{\mathfrak {L}}\in \mathcal {C}([a,b],\Phi _{0}(X,Y))$ with $\Vert \mathfrak {L}-\tilde{\mathfrak {L}}\Vert _{\infty }<\varepsilon $ (see [11]); allows us to define the parity for a general admissible path $\mathfrak {L}\in \mathscr {C}([a,b],\Phi _{0}(X,Y))$ through $$\begin{aligned} \sigma (\mathfrak {L},[a,b]):=\sigma (\tilde{\mathfrak {L}},[a,b]), \end{aligned}$$ where $\tilde{\mathfrak {L}}$ is any $\mathscr {C}$-transversal curve satisfying $\Vert \mathfrak {L}-\tilde{\mathfrak {L}}\Vert _{\infty }<\varepsilon $ for sufficiently small $\varepsilon >0$. Subsequently, a given homotopy $H\in \mathcal {C}([0,1]\times [a,b],\Phi _{0}(X,Y))$ is said to be admissible if $H([0,1]\times \{a,b\})\subset GL(X,Y)$. Moreover, two given paths, $\mathfrak {L}_{1}$ and $\mathfrak {L}_{2}$, are said to be $\mathcal {A}$-homotopic if they are homotopic through an admissible homotopy. A fundamental property of the parity is its invariance under admissible homotopies, which was established in [12]. The next result, proven by the authors in [29], establishes that, as soon as the Fredholm path $\mathfrak {L}(\lambda )$ is defined in $\mathcal {L}_{c}(X)$, every transversal intersection with $\mathcal {S}(X)$ induces a change of orientation, i.e., a change of path-connected component. Let $\mathfrak {L}\in \mathscr {C}([a,b],\mathcal {L}_{c}(X))$ be an admissible curve with values in $\mathcal {L}_{c}(X)$. Then, $\sigma (\mathfrak {L},[a,b])=-1$ if, and only if, $\mathfrak {L}(a)$ and $\mathfrak {L}(b)$ lie in different path-connected components of $GL_{c}(X)$. Theorem 3.1 motivates the geometrical interpretation of the parity as a local detector of the change of orientation of the operators of a Fredholm path. As illustrated by Fig. 1, each transversal intersection of the path $\mathfrak {L}(\lambda )$ with $\mathcal {S}(X)$ can be viewed as a change of path-connected component. Geometrical interpretation of the parity on $\mathcal {L}_{c}(X)$ The next result, proven by the authors in [29], shows how the parity of any admissible Fredholm path $\mathfrak {L}\in \mathscr {C}([a,b], \Phi _{0}(X,Y))$ can be computed through the algebraic multiplicity $\chi $. This result is important from the point of view of the applications. Any continuous path $\mathfrak {L}\in \mathscr {C}([a,b],\Phi _{0}(X,Y))$ is $\mathcal {A}$-homotopic to an analytic Fredholm curve $\mathfrak {L}_{\omega }\in \mathscr {C}^{\omega }([a,b],\Phi _{0}(X,Y))$. Moreover, for any of these analytic paths $$\begin{aligned} \sigma (\mathfrak {L},[a,b])=(-1)^{\sum _{i=1}^{n}\chi [\mathfrak {L}_{\omega };\lambda _{i}]}, \end{aligned}$$ $$\begin{aligned} \Sigma (\mathfrak {L}_{\omega })=\{\lambda _{1},\lambda _{2},...,\lambda _{n}\}. \end{aligned}$$ Subsequently, we consider $\mathfrak {L}\in \mathcal {C}([a,b],\Phi _{0}(X,Y))$ and an isolated eigenvalue $\lambda _{0}\in \Sigma (\mathfrak {L})$. Then, the localized parity of $\mathfrak {L}$ at $\lambda _{0}$ is defined through $$\begin{aligned} \sigma (\mathfrak {L},\lambda _{0}):=\lim _{\eta \downarrow 0}\sigma (\mathfrak {L},[\lambda _{0}-\eta ,\lambda _{0}+\eta ]). \end{aligned}$$ As a consequence of Theorem 3.2, the next result, going back to [29], holds. Corollary 3.3 Let $\mathfrak {L}\in \mathcal {C}^{r}([a,b],\Phi _{0}(X,Y))$ with $r\in \mathbb {N}\uplus \{\infty ,\omega \}$ and $\lambda _{0}\in {{\,\mathrm{Alg}\,}}_{k}(\mathfrak {L})$ for some $1\le k \le r$. Then $$\begin{aligned} \sigma (\mathfrak {L},\lambda _{0})=(-1)^{\chi [\mathfrak {L};\lambda _{0}]}. \end{aligned}$$ The identity (3.1) establishes the precise relationship between the topological notion of parity and the algebraic concept of multiplicity. The importance of Corollary 3.3 relies on the fact that, since the localized parity detects any change of orientation, (3.1) makes intrinsic to the concept of algebraic multiplicity any change of the local degree. As the principal difficulty to introduce a topological degree for Fredholm operators of index zero is the absence of orientation in the space of linear isomorphisms $GL(X,Y)\subset \Phi _{0}(X,Y)$, the notion introduced in the next definition, going back to Fitzpatrick, Pejsachowicz and Rabier [13], restricts the admissible maps to the ones where is possible to introduce a notion of orientability. In the sequel, the notation $\Lambda $ stands for a fixed topological space, and $\mathcal {R}_{h}$ is defined through (1.6). Definition 3.4 A continuous map $h:\Lambda \rightarrow \Phi _{0}(X,Y)$ is said to be orientable if there exists a map $\varepsilon :\mathcal {R}_{h}\rightarrow \mathbb {Z}_{2}$, called orientation, such that $$\begin{aligned} \sigma (h\circ \gamma ,[a,b])=\varepsilon (\gamma (a))\cdot \varepsilon (\gamma (b)) \end{aligned}$$ for each curve $\gamma \in \mathcal {C}([a,b],\Lambda )$ with $\gamma (a),\gamma (b)\in \mathcal {R}_{h}$. A subset $\mathcal {O}\subset \Phi _{0}(X,Y)$ is said to be orientable if the inclusion map $i:\mathcal {O}\hookrightarrow \Phi _{0}(X,Y)$ is orientable, i.e., if there exists a map $\varepsilon : \mathcal {O}\cap GL(X,Y) \rightarrow \mathbb {Z}_{2}$, such that $$\begin{aligned} \sigma (\mathfrak {L},[a,b])=\varepsilon (\mathfrak {L}(a))\cdot \varepsilon (\mathfrak {L}(b)) \quad \hbox {for all} \;\; \mathfrak {L}\in \mathscr {C}([a,b],\mathcal {O}). \end{aligned}$$ Observe that if $\mathcal {R}_{h}=\emptyset $, then h is trivially orientable. Since the parity of a Fredholm curve $\mathfrak {L}$ can be regarded as a generalized local detector of any change of orientation, it is natural to define an orientation $\varepsilon $ of a subset $\mathcal {O}$ of $\Phi _{0}(X,Y)$ as a map satisfying (3.3). Indeed, owing to (3.3), $\sigma (\mathfrak {L},[a,b])=-1$ if $\varepsilon (\mathfrak {L}(a))$ and $\varepsilon (\mathfrak {L}(b))$ have contrary sign. Also, note that if $\mathcal {O}$ is an orientable subset of $\Phi _{0}(X,Y)$ with orientation $\varepsilon $, then $\varepsilon $ is locally constant, i.e., $\varepsilon $ is constant on each path connected component of $\mathcal {O} \cap GL(X,Y)$. This is a rather natural property of an orientation. The same is true for maps h; the map $\varepsilon :\mathcal {R}_{h}\rightarrow \mathbb {Z}_{2}$ is constant in each path-connected component of $\mathcal {R}_{h}$. An orientable map $h:\Lambda \rightarrow \Phi _{0}(X,Y)$ with $\Lambda $ path connected and $\mathcal {R}_{h}\ne \emptyset $, admits, exactly, two different orientations. Precisely, given $p\in \mathcal {R}_{h}$, the two orientations of h are defined through $$\begin{aligned} \varepsilon ^{\pm }: \mathcal {R}_{h} \longrightarrow \mathbb {Z}_{2}, \quad q \mapsto \pm \sigma (h\circ \gamma ,[a,b]), \end{aligned}$$ where $\gamma \in \mathcal {C}([a,b],\Lambda )$ is an arbitrary path linking p with q, and the sign ± determines the orientation of the path-connected component of p in $\mathcal {R}_{h}$, i.e., if we choose $\varepsilon ^+$, then the orientation of the path connected component of p is 1, whereas it is $-1$ if $\varepsilon ^-$ is chosen. Finally, note that if $\Lambda '$ is any subspace of $\Lambda $, then the restriction of an orientation to $\mathcal {R}_{h}\cap \Lambda '$ gives an orientation for $h\|_{\Lambda '}$. According to [13], if $\Lambda $ is simply connected, any $h:\Lambda \rightarrow \Phi _{0}(X,Y)$ is orientable. Therefore, the set of orientable maps is really large. More generally, $h:\Lambda \rightarrow \Phi _{0}(X,Y)$ is orientable if the $\mathbb {Z}_{2}$-cohomology group $H^{1}(\Lambda ,\mathbb {Z}_{2})$ is trivial. The next result, going back to [29], justifies the geometrical interpretation of the parity as a local detector of change of orientation for the operators of a Fredholm path. Proposition 3.5 Let $\mathcal {O}$ be an orientable subset of $\Phi _{0}(X,Y)$ and $\mathfrak {L}\in \mathscr {C}([a,b],\mathcal {O})$. Then, $\sigma (\mathfrak {L},[a,b])=-1$ if, and only if, $\mathfrak {L}(a)$ and $\mathfrak {L}(b)$ lye in different path connected components of $\mathcal {O}\cap GL(X,Y)$ with opposite orientations. Finally, the next result, going back as well to [29], reduces the problem of detecting any change of orientation to the problem of the computation of the local multiplicity. It allows to interpret the algebraic multiplicity as a local detector of change of orientation for the operators of a Fredholm path. Given $\mathfrak {L}\in \mathcal {C}([a,b],\Phi _{0}(X,Y))$ and $\delta >0$, an isolated eigenvalue $\lambda _{0}\in \Sigma (\mathfrak {L})$ is said to be $\delta $-isolated if $$\begin{aligned} \Sigma (\mathfrak {L})\cap [\lambda _{0}-\delta ,\lambda _{0}+\delta ]=\{\lambda _{0}\}. \end{aligned}$$ Suppose that $\mathcal {O}\subset \Phi _{0}(X,Y)$ is an orientable subset, $\mathfrak {L}\in \mathcal {C}([a,b],\mathcal {O})$ is a Fredholm curve and $\lambda _{0}\in \Sigma (\mathfrak {L})$ a $\delta $-isolated eigenvalue. Then, the next assertions are equivalent: $\sum _{\lambda \in \Sigma (\mathfrak {L}_{\omega })}\chi [\mathfrak {L}_{\omega };\lambda _{0}]$ is odd for any analytical path $\mathfrak {L}_{\omega }\in \mathscr {C}^{\omega }([\lambda _{0}-\delta ,\lambda _{0}+\delta ],\Phi _{0}(X,Y))$, such that $\mathfrak {L}\vert _{[\lambda _{0}-\delta ,\lambda _{0}+\delta ]}$ and $\mathfrak {L}_\omega $ are $\mathcal {A}$-homotopic. $\mathfrak {L}(\lambda _{0}-\delta )$ and $\mathfrak {L}(\lambda _{0}+\delta )$ live in different path-connected components of $\mathcal {O}\cap GL(X,Y)$ with opposite orientations. Axiomatization and uniqueness of the parity The aim of this section is axiomatizing the concept of parity. Therefore, establishing its uniqueness. Our axiomatization is based on Theorem 2.1 and Corollary 3.3. Thanks to this axiomatization, we are establishing the uniqueness of a local detector of change of orientability. We will begin by axiomatizing the parity as a local object. Then, we will do it in a global setting. Subsequently, for any interval $\mathcal {I}\subset \mathbb {R}$ and $\lambda _{0}\in \text {Int\,}\mathcal {I}$, we will denote by $\mathcal {C}^{\omega }_{\lambda _{0}}(\mathcal {I},\Phi _{0}(X,Y))$ the space of all the analytic paths $\mathfrak {L}\in \mathcal {C}^{\omega }(\mathcal {I},\Phi _{0}(X,Y))$, such that $\mathfrak {L}(\lambda )\in GL(X,Y)$ for all $\lambda \in \mathcal {I}\setminus \{\lambda _0\}$. For every $\varepsilon >0$ and $\lambda _{0}\in \mathbb {R}$, there exists a unique $\mathbb {Z}_{2}$-valued map $$\begin{aligned} \sigma (\cdot ,\lambda _{0}):\;\; \mathcal {C}^\omega _{\lambda _0}\equiv \mathcal {C}^{\omega }_{\lambda _0}((\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon ),\Phi _{0}(X)) \longrightarrow \mathbb {Z}_{2}, \end{aligned}$$ such that Normalization: $\sigma (\mathfrak {L},\lambda _{0})=1$ if $\mathfrak {L}(\lambda _{0})\in GL(X)$, and there exists a rank one projection $P_{0}\in \mathcal {L}(X)$, such that $\sigma (\mathfrak {E},\lambda _{0})=-1$ where $$\begin{aligned} \mathfrak {E}(\lambda ):=(\lambda -\lambda _{0})P_{0}+I_{X}-P_{0}. \end{aligned}$$ Product Formula: For every $\mathfrak {L}, \mathfrak {M}\in \mathcal {C}^{\omega }_{\lambda _{0}}$, $$\begin{aligned} \sigma (\mathfrak {L}\circ \mathfrak {M},\lambda _{0})=\sigma (\mathfrak {L},\lambda _{0}) \cdot \sigma (\mathfrak {M},\lambda _{0}). \end{aligned}$$ Moreover, for every $\mathfrak {L}\in \mathcal {C}^{\omega }_{\lambda _{0}}$, the parity map is given by First, we will prove that, for every rank one projection $P\in \mathcal {L}(X)$, setting $$\begin{aligned} \mathfrak {F}(\lambda )=(\lambda -\lambda _{0})P+I_{X}-P, \end{aligned}$$ one has that $$\begin{aligned} \sigma (\mathfrak {F},\lambda _{0})=-1. \end{aligned}$$ Indeed, by Lemma 6.1.1 of [28], there exists $T\in GL(X)$, such that $P=T^{-1}P_{0}T$. Thus $$\begin{aligned} \mathfrak {F}(\lambda )=T^{-1}[(\lambda -\lambda _{0})P_{0}+I_{X}-P_{0}]T=T^{-1}\mathfrak {E}(\lambda )T, \end{aligned}$$ and hence, by axioms (P) and (N) $$\begin{aligned} \sigma (\mathfrak {L},\lambda _{0})=\sigma (T^{-1},\lambda _{0})\cdot \sigma (\mathfrak {E}(\lambda ),\lambda _{0})\cdot \sigma (T,\lambda _{0})=-1. \end{aligned}$$ On the other hand, for any given $\mathfrak {L}\in \mathcal {C}^{\omega }_{\lambda _{0}}$, by Corollary 5.3.2(b) of [28], which goes back to the proof of Theorem 5.3.1 of [25], there exist k finite-rank projections $\Pi _{0},\Pi _{2},\dots ,\Pi _{k-1}\in \mathcal {L}(X)$ and a (globally invertible) path $\mathfrak {I}\in \mathcal {C}^{\omega }((\lambda _{0}-\varepsilon ,\lambda _{0}+\varepsilon ),GL(X))$, such that setting $$\begin{aligned} \mathfrak {C}_{\Pi _{i}}(t):=t\, \Pi _{i}+I_{X}-\Pi _{i},\qquad i\in \{0,1,...,k-1\}, \;\; t\in \mathbb {R}, \end{aligned}$$ $$\begin{aligned} \mathfrak {L}(\lambda )=\mathfrak {I}(\lambda )\circ \mathfrak {C}_{\Pi _{0}} (\lambda -\lambda _{0})\circ \mathfrak {C}_{\Pi _{1}}(\lambda -\lambda _{0})\circ \cdots \circ \mathfrak {C}_{\Pi _{k-1}}(\lambda -\lambda _{0}). \end{aligned}$$ Moreover, for every $i\in \{0,1,...,k-1\}$, there are $r_{i}=\text {rank}\,\Pi _{i}$ projections of rank one, $P_{j,i}$, $1\le j \le r_{i}$, such that $$\begin{aligned} \mathfrak {C}_{\Pi _{i}}=\mathfrak {C}_{P_{1,i}}\circ \mathfrak {C}_{P_{2,i}}\circ \cdots \circ \mathfrak {C}_{P_{r_{i},i}}. \end{aligned}$$ Consequently, we find from the axiom (P) and (4.1) that $$\begin{aligned} \sigma (\mathfrak {C}_{\Pi _{i}},\lambda _{0})= \sigma (\mathfrak {C}_{P_{1,i}},\lambda _{0})\cdots \sigma (\mathfrak {C}_{P_{r_{i},i}},\lambda _{0}) =(-1)^{r_{i}}=(-1)^{\mathrm {rank}\,\Pi _{i}} \end{aligned}$$ and, therefore, applying again the axiom (P) yields $$\begin{aligned} \sigma (\mathfrak {L},\lambda _{0})=\sigma (\mathfrak {J},\lambda _{0}) \cdot \sigma (\mathfrak {C}_{\Pi _{0}},\lambda _{0})\cdots \sigma (\mathfrak {C}_{\Pi _{k-1}},\lambda _{0}) =(-1)^{\sum _{i=0}^{k-1}{\mathrm {rank}\,\Pi _{i}}}. \end{aligned}$$ Finally, since owing to Corollary 5.3.2 of [28], we have that $$\begin{aligned} \chi [\mathfrak {L};\lambda _{0}]=\sum _{i=0}^{k-1}{\mathrm {rank}\,\Pi _{i}}, \end{aligned}$$ it becomes apparent that This concludes the proof. $\square $ Once the local parity is determined, we will give the global axiomatization. A pair $(\mathfrak {L},[a,b])$ is said to be admissible if $\mathfrak {L}\in \mathscr {C}([a,b],\Phi _{0}(X))$. The set of admissible pairs will be denoted by $\mathscr {A}$. There exists a unique $\mathbb {Z}_{2}$-valued map $\sigma :\mathscr {A}\rightarrow \mathbb {Z}_{2}$, such that Normalization: For every $\mathfrak {L}\in \mathcal {C}^{\omega }_{\lambda _0}([\lambda _{0}-\eta ,\lambda _{0}+\eta ],\Phi _{0}(X))$, $$\begin{aligned} \sigma (\mathfrak {L},[\lambda _{0}-\eta ,\lambda _{0}+\eta ])=(-1)^{\chi [\mathfrak {L};\lambda _{0}]}. \end{aligned}$$ Product Formula: For every $(\mathfrak {L},[a,b])\in \mathscr {A}$ and $c\in (a,b)$, such that $c\notin \Sigma (\mathfrak {L})$ $$\begin{aligned} \sigma (\mathfrak {L},[a,b])=\sigma (\mathfrak {L},[a,c])\cdot \sigma (\mathfrak {L},[c,b]). \end{aligned}$$ Homotopy Invariance: For every homotopy $H\in \mathcal {C}([0,1]\times [a,b],\Phi _{0}(X))$ such that $(H(t,\cdot ),[a,b])\in \mathscr {A}$ for all $t\in [0,1]$ $$\begin{aligned} \sigma (H(0,\cdot ),[a,b])=\sigma (H(1,\cdot ),[a,b]). \end{aligned}$$ Moreover, $\sigma (\mathfrak {L},[a,b])$ equals the parity map of Fitzpatrick and Pejsachowitz [12]. Pick $(\mathfrak {L},[a,b])\in \mathscr {A}$. By Theorem 3.2, we already know that there exists an analytic curve $\mathfrak {L}_{\omega }\in \mathscr {C}^{\omega }([a,b],\Phi _{0}(X))$ $\mathcal {A}$-homotopic to $\mathfrak {L}$, i.e., there exists $H\in \mathcal {C}([0,1]\times [a,b],\Phi _{0}(X))$, such that $H([0,1]\times \{a,b\})\subset GL(X)$ $$\begin{aligned} H(0,\lambda )=\mathfrak {L}(\lambda ) \quad \hbox {and}\quad H(1,\lambda )=\mathfrak {L}_\omega (\lambda ) \quad \hbox {for all}\;\;\lambda \in [a,b]. \end{aligned}$$ Then, by the axiom (H) $$\begin{aligned} \sigma (\mathfrak {L},[a,b])=\sigma (\mathfrak {L}_{\omega },[a,b]). \end{aligned}$$ Suppose that $\Sigma (\mathfrak {L}_{\omega })\cap [a,b]=\emptyset $ and pick any $\lambda _0\in (a,b)$. Then, since $\chi [\mathfrak {L}_\omega ;\lambda _0]=0$, it follows from (N) that: $$\begin{aligned} \sigma (\mathfrak {L}_{\omega },[a,b])=(-1)^{\chi [\mathfrak {L}_\omega ;\lambda _0]}=1. \end{aligned}$$ Therefore, (4.2) implies that $$\begin{aligned} \sigma (\mathfrak {L},[a,b])=1. \end{aligned}$$ Now, suppose that $\Sigma (\mathfrak {L}_{\omega })\ne \emptyset $. Since $\mathfrak {L}_\omega (a)\in GL(X)$, it follows from Theorems 4.4.1 and 4.4.4 of [25] that $\Sigma (\mathfrak {L}_{\omega })$ is discrete. Thus $$\begin{aligned} \Sigma (\mathfrak {L}_\omega )=\{\lambda _{1},\lambda _{2},....,\lambda _{n}\} \end{aligned}$$ for some $$\begin{aligned} a<\lambda _1<\lambda _2<\cdots<\lambda _n<b. \end{aligned}$$ Let $\varepsilon >0$ be sufficiently small, so that $\lambda _i$ is $\varepsilon $-isolated for all $i\in \{1,...,n\}$. Then, by the axioms (P) and (N), we find that $$\begin{aligned} \sigma (\mathfrak {L}_{\omega },[a,b])&= \prod _{i=1}^{n}\sigma (\mathfrak {L}_{\omega },[\lambda _{i}-\varepsilon ,\lambda _{i}+\varepsilon ])\\&= \prod _{i=1}^n (-1)^{\chi [\mathfrak {L}_{\omega },\lambda _{i}]} =(-1)^{\sum _{i=1}^{n}\chi [\mathfrak {L}_{\omega };\lambda _{i}]}. \end{aligned}$$ Therefore, by (4.2), it becomes apparent that $$\begin{aligned} \sigma (\mathfrak {L},[a,b])=(-1)^{\sum _{i=1}^{n}\chi [\mathfrak {L}_{\omega };\lambda _{i}]}. \end{aligned}$$ Consequently, owing to Theorem 3.2, the map $\sigma :\mathscr {A}\rightarrow \mathbb {Z}_{2}$ is the parity defined by Fitzpatrick and Pejsachowitz in [12]. This concludes the proof. $\square $ Note that the normalization property (N) in Theorem 4.2 is determined by the local uniqueness of the parity provided by Theorem 4.1. Axiomatization and uniqueness of the topological degree The aim of this section is delivering the proof of Theorem 1.2. We begin by recalling our main theorem. As already discussed in Sect. 1, for any open and bounded subset, $\Omega $, of a Banach space X, an operator $f:\overline{\Omega }\subset X \rightarrow Y$ is said to be $\mathcal {C}^{1}$-Fredholm of index zero if $f\in \mathcal {C}^{1}(\overline{\Omega },Y)$ and $Df\in \mathcal {C}(\Omega ,\Phi _{0}(X,Y))$, and the set of all these operators is denoted by $\mathscr {F}^{1}_{0}(\Omega ,Y)$. An operator $f\in \mathscr {F}^{1}_{0}(\Omega ,Y)$ is said to be orientable when $Df:\Omega \rightarrow \Phi _{0}(X,Y)$ is an orientable map. Moreover, for any open and bounded subset, $\Omega $, of a Banach space X and any operator $f:\overline{\Omega }\subset X \rightarrow Y$ satisfying $f\in \mathscr {F}^{1}_{0}(\Omega ,Y)$ is orientable with orientation $\varepsilon :\mathcal {R}_{Df}\rightarrow \mathbb {Z}_{2}$, it is said that $(f,\Omega ,\varepsilon )$ is a Fredholm admissible triple. The set of all Fredholm admissible triples is denoted by $\mathscr {A}$. Given $(f,\Omega ,\varepsilon )\in \mathscr {A}$, it is said that $(f,\Omega ,\varepsilon )$ is a regular triple if 0 is a regular value of f, i.e., $Df(x)\in GL(X,Y)$ for all $x \in f^{-1}(0)$. The set of regular triples is denoted by $\mathscr {R}$. Finally, a map $H\in \mathcal {C}^{1}([0,1]\times \overline{\Omega },Y)$ is said to be $\mathcal {C}^{1}$-Fredholm homotopy if $D_{x}H(t,x)\in \Phi _{0}(X,Y)$ for each $(t,x)\in [0,1]\times \Omega $ and it is called orientable if $D_{x}H:[0,1]\times \Omega \rightarrow \Phi _{0}(X,Y)$ is an orientable map. Henceforth, the notation $\varepsilon _{t}$ stands for the restriction $$\begin{aligned} \varepsilon _{t}: \mathcal {R}_{H_{t}}\longrightarrow \mathbb {Z}_{2}, \quad \varepsilon _{t}(x):=\varepsilon (t,x) \end{aligned}$$ for each $t\in [0,1]$, where $H_{t}(\cdot )=H(t,\cdot )$. Theorem 1.2 reads as follows. There exists a unique integer-valued map $\deg : \mathscr {A}\rightarrow \mathbb {Z}$ satisfying the next properties Additivity: For every $(f,\Omega ,\varepsilon )\in \mathscr {A}$ and any pair of disjoint open subsets $\Omega _{1}$ and $\Omega _{2}$ of $\Omega $ with $0\notin f(\Omega \backslash (\Omega _{1}\uplus \Omega _{2}))$, where $\mathfrak {L}_{\omega ,x}\in \mathscr {C}^\omega ([a,b],\Phi _{0}(X,Y))$ is any analytic curve $\mathcal {A}$-homotopic to $Df\circ \gamma $, for some $\gamma \in \mathcal {C}([a,b],\Omega )$, such that $\gamma (a)=p$ and $\gamma (b)=x$, and $$\begin{aligned} \chi [\mathfrak {L}_{\omega ,x},[a,b]]:=\sum _{\lambda _{x}\in \Sigma (\mathfrak {L}_{\omega ,x})\cap [a,b]}\chi [\mathfrak {L}_{\omega ,x},\lambda _{x}]. \end{aligned}$$ Observe that the right-hand side of (5.1) is well defined, because every open and connected set, $\Omega $, in a locally convex topological space, X, is path connected. Thus, it always exists a path $\mathfrak {L}_{x}\in \mathcal {C}([a,b],\Omega )$ joining p and x. The existence of the analytic $\mathcal {A}$-homotopic curve was established in [29]. The right-hand side of (5.1) is taken as zero if $f^{-1}(0)\cap \Omega =\emptyset $. Applying axiom (A) with $\Omega _1=\Omega _2=\Omega =\emptyset $, it becomes apparent that $\mathrm {deg}(f,\emptyset ,\varepsilon )=0$. If $(f,\Omega ,\varepsilon )\in \mathscr {A}$ and $f^{-1}(0)\cap \Omega =\emptyset $. Applying again (A) with $\Omega _1=\Omega _2=\emptyset $, we find that $\mathrm {deg}(f,\Omega ,\varepsilon )=0$. Equivalently, f admits a zero in $\Omega $ if $\mathrm {deg}(f,\Omega ,\varepsilon )\ne 0$. As already commented in Sect. 1, since the existence of $\deg $ goes back to Fitzpatrick, Pejsachowicz and Rabier [13], and the formula (5.1) was proven by the authors in [29], Theorem 5.1 actually establishes the uniqueness of $\deg $. To prove the uniqueness, it is appropriate to sketch briefly the construction of $\deg $ carried over in [13] and [35]. Let $(f,\Omega ,\varepsilon )\in \mathscr {A}$. By definition, $f\in \mathscr {F}^{1}_{0}(\Omega ,Y)$ is $\mathcal {C}^{1}$-Fredhom of index zero and it is $\varepsilon $-orientable, i.e., $Df:\Omega \rightarrow \Phi _{0}(X,Y)$ is an orientable map with orientation $$\begin{aligned} \varepsilon :\mathcal {R}_{Df} \longrightarrow \mathbb {Z}_{2}. \end{aligned}$$ Once an orientation has been introduced, the degree $\deg $ can be defined as the Leray–Schauder degree $\deg _{LS}$ as soon as 0 is a regular value of f, because since in such case, $f^{-1}(0)\cap \Omega $ is finite, possibly empty, one can define, in complete agreement with the axioms (N), (A), and (H) $$\begin{aligned} \deg (f,\Omega ,\varepsilon ):=\sum _{x\in f^{-1}(0)\cap \Omega } \varepsilon (x). \end{aligned}$$ If $f^{-1}(0)\cap \Omega =\emptyset $, as we already mentioned, $\mathrm {deg}(f,\Omega ,\varepsilon )=0$. When 0 is not a regular value, then, by definition $$\begin{aligned} \deg (f,\Omega ,\varepsilon ):=\deg (f-x_{0},\Omega ,\varepsilon ), \end{aligned}$$ where $x_{0}$ is any regular value of f belonging to a sufficiently small neighborhood of 0. The existence of such regular values is guaranteed by a theorem of Quinn and Sard [34], a version of the Sard–Smale Theorem, [40], not requiring the separability of the involved Banach spaces. Once introduced the Leray–Schauder degree, many experts generalized it to cover more general operators than compact perturbations of the identity. It is worth mentioning that the degree of Fitzpatrick, Pejsachowicz and Rabier covers most of them under the notion of the degree for $\mathcal {F}$-maps, where $\mathcal {F}$ is a fixed orientable subset of $\Phi _{0}(X,Y)$. A map $f\in \mathscr {F}^{1}_{0}(\Omega ,Y)$ is called an $\mathcal {F}$-map if $Df(\Omega )\subset \mathcal {F}$. Clearly, an $\mathcal {F}$-map $f:\Omega \subset X\rightarrow Y$ inherits a unique orientation induced by the given orientation on $\mathcal {F}$. Indeed, if $\varepsilon :\mathcal {F}\cap GL(X,Y)\rightarrow \mathbb {Z}_{2}$ denotes the orientation of $\mathcal {F}$, then $$\begin{aligned} \varepsilon _{f}:\mathcal {R}_{Df}\longrightarrow \mathbb {Z}_{2}, \quad \varepsilon _{f}(x)=\varepsilon (Df(x)), \end{aligned}$$ defines an orientation for f. Many of the existing degrees can be viewed as special cases of the degree for $\mathcal {F}$-maps. For instance, $\deg $ extends $\deg _{LS}$ to this more general setting if we restrict ourselves to consider Leray–Schauder admissible pairs of class $\mathcal {C}^{1}$. Indeed, $\mathcal {L}_{c}(X)$ is simply connected and, hence, according to [13], orientable. Choose $\mathcal {F}=\mathcal {L}_{c}(X)$ and the orientation $\varepsilon : GL_{c}(X)\rightarrow \mathbb {Z}_{2}$ defined by $$\begin{aligned} \varepsilon (L)=\deg _{LS}(L,\Omega ), \end{aligned}$$ where the right-hand side of (5.3) is defined by (1.4). Then, for every $\mathcal {C}^{1}$ Leray–Schauder regular pair $(f,\Omega )$, one has that $(f,\Omega ,\varepsilon _{f})\in \mathscr {R}$ and, thanks to (5.2) and (5.3) $$\begin{aligned} \mathrm {deg}_{LS}(f,\Omega )&=\sum _{x\in f^{-1}(0)\cap \Omega } \mathrm {deg}_{LS}(Df(x),\Omega )\\&=\sum _{x\in f^{-1}(0)\cap \Omega }\varepsilon (Df(x))=\sum _{x\in f^{-1}(0)\cap \Omega }\varepsilon _{f}(x)=\deg (f,\Omega ,\varepsilon _{f}). \end{aligned}$$ $$\begin{aligned} \deg (f,\Omega ,\varepsilon _{f})=\deg _{LS}(f,\Omega ). \end{aligned}$$ Many others, like the Nussbaum–Sadovkii degree, [33, 39], the Laloux–Mawhin coincidence degree [20, 21, 31], the Tromba degree for Röthe maps [41], the Isnard degree [17], and the Fenske degree [10], can be also regarded as special cases of the degree for $\mathcal {F}$-maps for a suitable choice of $\mathcal {F}$. The interested reader is sent to Sect. 2 of Fitzpatrick, Pejsachowicz and Rabier [13] for any further details. Before proving the uniqueness, it is convenient to illustrate the theory by establishing the generalized Schauder formula (5.1), as it was done in [29]. Proof of (5.1) Take $(f,\Omega ,\varepsilon )\in \mathscr {R}$ with $\Omega $ connected and $\mathcal {R}_{Df}\ne \emptyset $ and choose $p\in \mathcal {R}_{Df}$. By (5.2), it follows that: Fix $x\in f^{-1}(0)\cap \Omega $. According to (3.2) $$\begin{aligned} \varepsilon (x)=\varepsilon (p)\cdot \sigma (Df\circ \gamma ,[a,b]), \end{aligned}$$ where $\gamma \in \mathcal {C}([a,b],\Omega )$ is a path linking x with p. By Theorem 3.2, for any analytic curve $\mathfrak {L}_{\omega ,x}\in \mathscr {C}^{\omega }([a,b],\Phi _{0}(X,Y))$ $\mathcal {A}$-homotopic to $Df\circ \gamma $, we have that $$\begin{aligned} \varepsilon (x)=\varepsilon (p)\cdot \sigma (Df\circ \gamma ,[a,b])=\varepsilon (p)\cdot (-1)^{\chi [\mathfrak {L}_{\omega ,x},[a,b]]}, \end{aligned}$$ $$\begin{aligned} \chi [\mathfrak {L}_{\omega ,x},[a,b]]=\sum _{\lambda _{x}\in \Sigma (\mathfrak {L}_{\omega ,x})\cap [a,b]}\chi [\mathfrak {L}_{\omega ,x},\lambda _{x}]. \end{aligned}$$ Therefore, by (5.4) which ends the proof. $\square $ We have all necessary ingredients to prove Theorem 5.1. Naturally, it suffices to prove the uniqueness. Proof of the uniqueness We first prove that, for every $(f,\Omega ,\varepsilon )\in \mathscr {R}$, the topological degree is given by (5.1) in each connected component of $\Omega $, if $\mathcal {R}_{Df}\ne \emptyset $ and $\deg (f,\Omega ,\varepsilon )=0$ if $\mathcal {R}_{Df}=\emptyset $. Pick $(f,\Omega ,\varepsilon )\in \mathscr {R}$. Then, $f^{-1}(0)\cap \Omega $ is finite, possibly empty. If it is empty, then, applying axiom (A) with $\Omega _1=\Omega _2=\Omega =\emptyset $, it becomes apparent that $\mathrm {deg}(f,\emptyset ,\varepsilon )=0$. Thus, applying again (A) with $\Omega _1=\Omega _2=\emptyset $, we find that $$\begin{aligned} \mathrm {deg}(f,\Omega ,\varepsilon )=0. \end{aligned}$$ If $\mathcal {R}_{Df}=\emptyset $, necessarily $f^{-1}(0)\cap \Omega =\emptyset $ and, therefore, $\deg (f,\Omega ,\varepsilon )=0$ as required. Now, suppose that, for some $n\ge 1$ and $x_i\in \Omega $, $i\in \{1,...,n\}$ $$\begin{aligned} f^{-1}(0)\cap \Omega =\{x_{1},x_{2},...,x_{n}\}. \end{aligned}$$ By the finiteness of $f^{-1}(0)\cap \Omega $ and the additivity property (A), we can suppose that $\Omega $ is connected. Indeed, let us denote by $\mathscr {O}:=\{D_{j}\}_{j=1}^{m}$, $m\le n$, the connected components of $\Omega $ with $f^{-1}(0)\cap D_{j}\ne \emptyset $. Since $0\notin f(\Omega \backslash \uplus _{j=1}^{m} D_{j})$, where $\uplus $ stands for the disjoint union, applying (A) with $\Omega _{1}=\uplus _{j=1}^{m} D_{j}$ and $\Omega _{2}=\emptyset $, we can infer that $$\begin{aligned} \deg (f,\Omega ,\varepsilon )=\deg (f,\uplus _{j=1}^{m} D_{j},\varepsilon )= \sum _{j=1}^{m}\deg (f,D_{j},\varepsilon ), \end{aligned}$$ where the second equality follows by an inductive application of the additivity property (A). Thus, without loss of generality, we can assume that $\Omega $ is connected. Since 0 is a regular value of f, by the inverse function theorem, $f\vert _{B_{\eta _{i}}(x_{i})}$ is a diffeomorphism for each $i\in \{1,2,...,n\}$ for sufficiently small $\eta _{i}>0$ and therefore by axiom (A) $$\begin{aligned} \deg (f,\Omega ,\varepsilon )=\sum _{i=1}^{n}\deg (f,B_{\eta _{i}}(x_{i}),\varepsilon ). \end{aligned}$$ Subsequently, we fix $i\in \{1,2,...,n\}$ and consider the homotopy $H_{i}$ defined by $$\begin{aligned} \begin{array}{lccl} H_{i}: &{} [0,1]\times B_{\eta _{i}}(x_{i}) &{} \longrightarrow &{} Y \\ &{} (t,x) &{} \mapsto &{} tf(x)+(1-t)Df(x_{i})(x-x_{i}). \end{array} \end{aligned}$$ The next result of technical nature holds. $\square $ Lemma 5.2 $H_{i}\in \mathcal {C}^{1}([0,1]\times \overline{B_{\tau _i}(x_i)},Y)$ and it is proper for sufficiently small $\tau _i>0$. Obviously, $H_{i}\in \mathcal {C}^{1}([0,1]\times B_{\eta _{i}}(x_{i}),Y)$ and $$\begin{aligned} D_{x}H_{i}(t,\cdot )=tDf(\cdot )+(1-t)Df(x_{i})=Df(x_{i})+t(Df(\cdot )-Df(x_{i})). \end{aligned}$$ Since $Df\in \mathcal {C}(\Omega ,\Phi _{0}(X,Y))$, $Df(x_{i})\in GL(X,Y)$, and GL(X, Y) is open, we have that, for sufficiently small $\eta _i>0$ $$\begin{aligned} D_{x}H_{i}(t,B_{\eta _{i}}(x_{i}))\subset GL(X,Y) \quad \hbox {for all} \;\, t \in [0,1]. \end{aligned}$$ In particular, $D_{x}H_{i}(t,x)\in \Phi _0(X,Y)$ for all $(t,x)\in [0,1]\times B_{\eta _{i}}(x_{i})$. Thus, by definition, $H_{i}(t,\cdot )\in \mathscr {F}^{1}_{0}(\Omega ,Y)$ for all $t\in [0,1]$. This also entails that $DH_{i}$ is a Fredholm operator of index one from $\mathbb {R}\times X$ to Y. Thus, by Theorem 1.6 of Smale [40], $H_{i}$ is locally proper, i.e., for every $t\in [0,1]$, there exists an open interval containing t, $\mathcal {I}(t)\subset [0,1]$, and an open ball centered in $x_{i}$ with radius $\delta _{t}$, $B_{\delta _t}(x_i)$, such that $H_{i}$ is proper in $\overline{\mathcal {I}(t)}\times \overline{B_{\delta _t}(x_i)}$. In particular $$\begin{aligned}{}[0,1]\times \{x_{i}\}\subset \bigcup _{t\in [0,1]}\mathcal {I}(t)\times B_{\delta _t}(x_i). \end{aligned}$$ Since $[0,1]\times \{x_{i}\}$ is compact, there exists a finite subset $\{t_{1},t_{2},...,t_{n}\}\subset [0,1]$, such that $$\begin{aligned}{}[0,1]\times \{x_{i}\}\subset \bigcup _{j=1}^{n}\mathcal {I}(t_{j})\times B_{\delta _{_{t_j}}}(x_{i}), \end{aligned}$$ as illustrated in Fig. 2. Let $$\begin{aligned} \delta _{i}:=\min \{\delta _{t_{1}},\delta _{t_{2}},...,\delta _{t_{n}}\}. \end{aligned}$$ $$\begin{aligned}{}[0,1]\times \{x_{i}\}\subset [0,1]\times B_{\delta _{i}}(x_{i}). \end{aligned}$$ Let $\tau _{i}<\min \{\eta _{i},\delta _{i}\}$. Then, $H_{i}$ is proper in $\overline{\mathcal {I}(t)}\times \overline{B_{\tau _{i}}(x_{i})}$, since the restriction of a proper map to a closed subset is also proper. On the other hand, since $$\begin{aligned}{}[0,1]\times \overline{B_{\tau _{i}}(x_{i})}=\bigcup _{i=1}^{n}\overline{\mathcal {I}(t_{i})}\times \overline{B_{\tau _{i}}(x_{i})} \end{aligned}$$ and each $\overline{\mathcal {I}(t_{i})}\times \overline{B_{\tau _{i}}(x_{i})}$ is closed, necessarily $H_{i}$ is proper on $[0,1]\times \overline{B_{\tau _{i}}(x_{i})}$. Therefore $$\begin{aligned} H_{i}\in \mathcal {C}^{1}([0,1]\times \overline{B_{\tau _{i}}(x_{i})},Y), \end{aligned}$$ and it is proper. The proof is complete. $\square $ Scheme of the construction $0\notin H_{i}(t,\partial B_{\tau _{i}}(x_i))$ for each $t\in [0,1]$ and sufficiently small $\tau _i>0$. On the contrary, assume that $0\in H_{i}(t,\partial B_{\tau _{i}}(x_i))$ for some $t\in [0,1]$ and $\tau _i<\eta _i$, i.e., there exists $x\in \partial B_{\tau _{i}}(x_i)$, such that $H_{i}(t,x)=0$. Thus $$\begin{aligned} t[f(x)-Df(x_{i})(x-x_{i})]+Df(x_{i})(x-x_{i})=0. \end{aligned}$$ Necessarily, $t>0$, because $t=0$ implies $Df(x_{i})(x-x_{i})=0$, and in such case, $Df(x_i)$ cannot be an isomorphism. Therefore, dividing (5.7) by t yields $$\begin{aligned} f(x)-Df(x_{i})(x-x_{i})=-\frac{1}{t}Df(x_{i})(x-x_{i}). \end{aligned}$$ Taking norms and dividing by $\Vert x-x_{i}\Vert >0$, we obtain that $$\begin{aligned} \frac{\Vert f(x)-Df(x_{i})(x-x_{i})\Vert }{\Vert x-x_{i}\Vert }=\frac{1}{t}\Big \Vert Df(x_{i})\Big (\frac{x-x_{i}}{\Vert x-x_{i}\Vert } \Big )\Big \Vert \ge \frac{1}{t}\inf _{\Vert x\Vert =1}\Vert Df(x_{i})(x)\Vert .\nonumber \\ \end{aligned}$$ Since $Df(x_{i})\in GL(X,Y)$ and $\partial B_{1}(x_i)$ is closed, $Df(x_{i})(\partial B_{1}(x_i))$ is closed. Hence $$\begin{aligned} m\equiv \inf _{\Vert x\Vert =1}\Vert Df(x_{i})(x)\Vert \end{aligned}$$ is attained and, since $0\notin Df(x_{i})(\partial B_{1}(x_i))$, we find that $m>0$, and therefore, it follows from (5.8) that: $$\begin{aligned} \frac{\Vert f(x)-Df(x_{i})(x-x_{i})\Vert }{\Vert x-x_{i}\Vert }\ge \frac{m}{t}>0. \end{aligned}$$ On the other hand, since f is differentiable at $x_i$ and $f(x_{i})=0$ $$\begin{aligned} \frac{\Vert f(x)-Df(x_{i})(x-x_{i})\Vert }{\Vert x-x_{i}\Vert }\xrightarrow [x\rightarrow x_{i}]{}0. \end{aligned}$$ Thus, for sufficiently small $\tau _{i}>0$, we have that $$\begin{aligned} \frac{\Vert f(x)-Df(x_{i})(x-x_{i})\Vert }{\Vert x-x_{i}\Vert }<\frac{m}{t}, \end{aligned}$$ which contradicts (5.9) and ends the proof. $\square $ By construction, $f\vert _{B_{\eta _{i}}(x_{i})}$ is a diffeomorphism. Thus, $Df(B_{\eta _{i}}(x_{i}))$ is a path connected subset of GL(X, Y), and hence, $B_{\eta _{i}}(x_{i})\subset \mathcal {R}_{Df}$. Since the orientation $$\begin{aligned} \varepsilon \|_{B_{\eta _{i}}(x_{i})}: \; B_{\eta _{i}}(x_{i})\longrightarrow \mathbb {Z}_{2} \end{aligned}$$ is always constant in each path-connected component of its domain, it is actually constant. Denote its constant value by $\varepsilon _{0}$. Subsequently, we consider the map $$\begin{aligned} \begin{array}{lccl} \varepsilon ^{H_{i}}: &{} [0,1]\times B_{\eta _{i}}(x_{i})&{} \longrightarrow &{} \mathbb {Z}_{2} \\ &{} (t,x) &{} \mapsto &{} \varepsilon _{0}. \end{array} \end{aligned}$$ Note that, thanks to (5.6), for each $t\in [0,1]$ $$\begin{aligned} D_xH_i(t,B_{\eta _{i}}(x_{i}))\subset GL(X,Y). \end{aligned}$$ Consequently, $D_xH_i([0,1]\times B_{\eta _{i}}(x_{i}))$ is a path connected subset of GL(X, Y) and $[0,1]\times B_{\eta _{i}}(x_{i})=\mathcal {R}_{D_{x}H_{i}}$. Hence, $\varepsilon ^{H_{i}}$ provides us with an orientation of $H_{i}$. Therefore, thanks to Lemmas 5.2 and 5.3, it becomes apparent that $H_{i}$ is a proper $\mathcal {C}^{1}$-Fredholm homotopy with orientation $\varepsilon ^{H_{i}}$ and $0\notin H_{i}([0,1]\times \partial B_{\tau _{i}}(x_{i}))$ for sufficiently small $\tau _i>0$. Moreover, $\varepsilon ^{H_{i}}_{t}(\equiv \varepsilon _{0})$ provides us with an orientation of the section $H_{i}(t,\cdot )$ and, therefore, $(H_{i}(t,\cdot ),\Omega ,\varepsilon ^{H_{i}}_{t})\in \mathscr {A}$ for each $t\in [0,1]$. By the axiom (H), and taking into account that $\varepsilon ^{H_{i}}_{j}= \varepsilon \|_{B_{\eta _{i}}(x_{i})}(\equiv \varepsilon _{0})$ for each $j\in \{0,1\}$ $$\begin{aligned} \deg (f,B_{\tau _{i}}(x_{i}),\varepsilon )=\deg (Df(x_{i})(\cdot -x_{i}),B_{\tau _{i}}(x_{i}),\varepsilon ). \end{aligned}$$ To remove the affine term in (5.11), we consider the homotopy $$\begin{aligned} \begin{array}{lccl} G_{i}: &{} [0,1]\times \overline{\Pi } &{} \longrightarrow &{} Y \\ &{} (t,x) &{} \mapsto &{} Df(x_{i})(x-tx_{i}), \end{array} \end{aligned}$$ where $\Pi =\bigcup _{t\in [0,1]}B_{\tau _{i}}(tx_{i})$. Obviously, $G_{i}\in \mathcal {C}^{1}([0,1]\times \overline{\Pi },Y)$. Moreover, since, for every $t\in [0,1]$, $G_{i}(t,\cdot )=Df(x_{i})(\cdot -t x_{i})$ is a diffeomorphism, we have that $G_{i}(t,\cdot )$ is proper for each $t\in [0,1]$. Therefore, since $G_{i}$ is uniformly continuous in t, it follows from Theorem 3.9.2 of [42] that $G_{i}$ is proper. As $G_{i}(t,\cdot )=Df(x_{i})(\cdot -tx_{i})$ is a diffeomorphism for each $t\in [0,1]$ and $G_{i}(t,tx_{i})=0$, it is obvious that $$\begin{aligned} 0\notin G_{i}(t,\partial \Pi )\quad \hbox {for all} \;\; t\in [0,1]. \end{aligned}$$ Moreover, since, for every $t\in [0,1]$ $$\begin{aligned} D_x G_{i}(t,\cdot )=Df(x_{i})\in GL(X,Y), \end{aligned}$$ it is apparent that $$\begin{aligned} D_{x}G_{i}([0,1]\times \Pi )=\{Df(x_{i})\}\subset GL(X,Y) \end{aligned}$$ and if we choose the orientation $$\begin{aligned} \varepsilon ^{G_{i}}: [0,1]\times \Pi \longrightarrow \mathbb {Z}_{2} \end{aligned}$$ to be $\varepsilon ^{G_{i}}\equiv \varepsilon _{0}$, $G_{i}$ is a proper $\mathcal {C}^{1}$-Fredholm homotopy with orientation $\varepsilon ^{G_{i}}$, such that $0\notin G_{i}([0,1]\times \partial \Pi )$. Moreover, since $D_{x}G_{i}(\{t\}\times \Pi )\cap GL(X,Y)=\{Df(x_{i})\}\ne \emptyset $ for each $t\in [0,1]$, $\varepsilon ^{G_{i}}_{t}(\equiv \varepsilon _{0})$ provides us with an orientation of the section $G_{i}(t,\cdot )$ and, therefore, $(G_{i}(t,\cdot ),\Pi ,\varepsilon ^{G_{i}}_{t})\in \mathscr {A}$ for each $t\in [0,1]$. Thanks to the axiom (H), we find that $$\begin{aligned} \deg (Df(x_{i}),\Pi ,\varepsilon ^{{G}_{i}}_{0})&= \deg (G_{i}(0,\cdot ),\Pi ,\varepsilon ^{{G}_{i}}_{0})=\deg (G_{i}(1,\cdot ),\Pi ,\varepsilon ^{G_{i}}_{1}) \nonumber \\&= \deg (Df(x_{i})(\cdot -x_{i}),\Pi ,\varepsilon ^{G_{i}}_{1}). \end{aligned}$$ $$\begin{aligned} F_{t}(x)=Df(x_{i})(x-tx_{i}), \quad x\in \Pi , \end{aligned}$$ is a diffeomorphism for each $t\in \{0,1\}$ and $F_{t}(tx_{i})=0$, we have that $0\notin F_{t}(\Pi \backslash B_{\tau _{i}}(tx_{i}))$. Thus, applying the axiom (A) with $\Omega =\Pi $, $\Omega _1=B_{\tau _i}(tx_i)$ and $\Omega _2=\emptyset $, it becomes apparent that $$\begin{aligned} \deg (Df(x_{i}),\Pi ,\varepsilon ^{G_{i}}_{0})=\deg (Df(x_{i}),B_{\tau _{i}}(0),\varepsilon ^{G_{i}}_{0}\|_{B_{\tau _{i}}(0)}) \end{aligned}$$ $$\begin{aligned} \deg (Df(x_{i})(\cdot -x_{i}),\Pi ,\varepsilon ^{G_{i}}_{1})=\deg (Df(x_{i})(\cdot -x_{i}),B_{\tau _{i}}(x_{i}),\varepsilon ^{G_{i}}_{1}\|_{B_{\tau _{i}}(x_{i})}). \end{aligned}$$ Therefore, by (5.12), we infer that $$\begin{aligned} \deg (Df(x_{i})(\cdot -x_{i}),B_{\tau _{i}}(x_{i}),\varepsilon ^{G_{i}}_{0}\|_{B_{\tau _{i}}(0)})= \deg (Df(x_{i}),B_{\tau _{i}}(0),\varepsilon ^{G_{i}}_{1}\|_{B_{\tau _{i}}(x_{i})}). \end{aligned}$$ Observe that $\varepsilon ^{G_{i}}_{1}\|_{B_{\tau _{i}}(x_{i})}=\varepsilon \|_{B_{\tau _{i}}(x_{i})}$ and that $$\begin{aligned} \varepsilon ^{G_{i}}_{0}\|_{B_{\tau _{i}}(0)}=\varepsilon \|_{B_{\tau _{i}}(x_{i})}\circ T, \end{aligned}$$ where $T:B_{\tau _{i}}(0)\rightarrow B_{\tau _{i}}(x_{i})$ is the translation given by $T(x)=x+x_{i}$. Consequently, by the axiom (N) $$\begin{aligned} \deg (Df(x_{i})(\cdot -x_{i}),B_{\tau _{i}}(x_{i}),\varepsilon )= \deg (Df(x_{i}),B_{\tau _{i}}(0),\varepsilon \|_{B_{\tau _{i}}(x_{i})}\circ T)=\varepsilon (T(0))=\varepsilon (x_{i}). \end{aligned}$$ Thus, combining the last identity with (5.11) yields $$\begin{aligned} \deg (f,B_{\tau _{i}}(x_{i}),\varepsilon )=\varepsilon (x_{i}), \end{aligned}$$ and therefore, by (5.5) $$\begin{aligned} \deg (f,\Omega ,\varepsilon )=\sum _{i=1}^{n}\varepsilon (x_{i}). \end{aligned}$$ Now, since $f^{-1}(0)\cap \Omega \ne \emptyset $ and 0 is a regular value, necessarily $\mathcal {R}_{Df}\ne \emptyset $. Take $p\in \mathcal {R}_{Df}$. Then, according to (3.2), we have $$\begin{aligned} \varepsilon (x_{i})=\varepsilon (p)\cdot \sigma (Df\circ \gamma _{x_{i}},[a,b]), \end{aligned}$$ where $\gamma _{x_{i}}\in \mathcal {C}([a,b],\Omega )$ is a path linking $x_{i}$ with p. By Theorem 3.2, for any analytical curve $\mathfrak {L}_{\omega ,x_{i}}\in \mathscr {C}^{\omega }([a,b],\Phi _{0}(X,Y))$ $\mathcal {A}$-homotopic to $Df\circ \gamma _{x_{i}}$, we have $$\begin{aligned} \varepsilon (x_{i})=\varepsilon (p)\cdot \sigma (Df\circ \gamma _{x_{i}},[a,b])=\varepsilon (p)\cdot (-1)^{\chi [\mathfrak {L}_{\omega ,x_{i}},[a,b]]}, \end{aligned}$$ $$\begin{aligned} \chi [\mathfrak {L}_{\omega ,x_{i}},[a,b]]=\sum _{\lambda _{x_{i}}\in \Sigma (\mathfrak {L}_{\omega ,x_{i}})\cap [a,b]}\chi [\mathfrak {L}_{\omega ,x_{i}},\lambda _{{x}_{i}}]. \end{aligned}$$ Therefore, by (5.13) $$\begin{aligned} \deg (f,\Omega ,\varepsilon )=\varepsilon (p)\cdot \sum _{i=1}^{n} (-1)^{\chi [\mathfrak {L}_{\omega ,x_{i}},[a,b]]}, \end{aligned}$$ which ends the proof of the theorem in the regular case. We have actually proven that in the regular case, any map satisfying the axioms (N), (A), and (N) must coincide with the degree of Fitzpatrick, Pejsachowicz and Rabier [13]. If the general case, when $(f,\Omega ,\varepsilon )\notin \mathscr {R}$, for every $\eta >0$, by a theorem of Quinn and Sard, [34], there exists a regular value $x_{0}$, such that $\Vert x_{0}\Vert <\eta $. Let H be the homotopy defined by $$\begin{aligned} \begin{array}{lccl} H: &{} [0,1]\times \overline{\Omega } &{} \longrightarrow &{} Y \\ &{} (t,x) &{} \mapsto &{} f(x)-tx_{0}. \end{array} \end{aligned}$$ Then, $H\in \mathcal {C}^{1}([0,1]\times \overline{\Omega },Y)$ and it is proper. Obviously $H\in \mathcal {C}^{1}([0,1]\times \overline{\Omega },Y)$ and $$\begin{aligned} D_xH(t,\cdot )=Df(\cdot )\in \Phi _{0}(X,Y). \end{aligned}$$ First, we will prove that $H(t,\cdot )$ is proper for each $t\in [0,1]$. By the definition of H, for any compact subset, K, of Y $$\begin{aligned} H(t,\cdot )^{-1}(K)=f^{-1}(K+tx_{0}) \end{aligned}$$ is compact, because $K+tx_{0}$ is compact and f proper. Therefore, as the map H is uniformly continuous in t, as above, it follows from Theorem 3.9.2 of [42] that H is proper. Now, we will show that $0\notin H(t,\partial \Omega )$ for each $t\in [0,1]$. On the contrary, suppose that $0\in H(t,\partial \Omega )$ for some $t\in [0,1]$. Then, there exists $x\in \partial \Omega $, such that $f(x)=tx_{0}$. In particular, $tx_{0}\in f(\partial \Omega )$. Since f is proper, by Lemma 3.9.1 of [42], f is a closed map, and since $\partial \Omega $ is closed, $f(\partial \Omega )$ is closed. Since $0\notin f(\partial \Omega )$ and $f(\partial \Omega )$ is closed, there exists $\eta >0$, such that $B_{\eta }(0)\cap f(\partial \Omega )=\emptyset $. As we have already taken $\Vert x_{0}\Vert <\eta $, we also have that $tx_{0}\in B_{\eta }(0)$ and, therefore, $tx_{0}\notin f(\partial \Omega )$. This contradicts $tx_{0}\in f(\partial \Omega )$. Since $D_xH(\{t\}\times \Omega )=Df(\Omega )$ for each $t\in [0,1]$, $\mathcal {R}_{D_{x}H}=[0,1]\times \mathcal {R}_{Df}$ and if we define $$\begin{aligned} \varepsilon ^{H}: [0,1]\times \mathcal {R}_{Df}\longrightarrow \mathbb {Z}_{2}, \quad \varepsilon ^{H}(t,x)=\varepsilon (x), \end{aligned}$$ where $\varepsilon $ is the orientation of Df, then H is a proper $\mathcal {C}^{1}$-Fredholm homotopy with orientation $\varepsilon ^{H}$ and $0\notin H([0,1]\times \partial \Omega )$. Observe that in this case, the domain is the whole $\Omega $ and, therefore, the orientation $\varepsilon $ might not be, in general, constant. Moreover, since $D_xH(t,\cdot )=Df(\cdot )$ and Df is orientable with orientation $\varepsilon (=\varepsilon ^{H}_{t})$ for each $t\in [0,1]$, necessarily $(H(t,\cdot ),\Omega ,\varepsilon ^{H}_{t})\in \mathscr {A}$ for each $t\in [0,1]$. 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Sci. 204, 543–714 (2015) The authors express their deepest gratitude to the two (anonymous) reviewers of this paper for their truly professional work; very specially to the second one, whose technical comments have greatly helped them to improve, in a truly substantial way, the presentation of this paper. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Institute of Interdisciplinary Mathematics (IMI) Department of Analysis and Applied Mathematics, Complutense University of Madrid, 28040, Madrid, Spain Julián López-Gómez & Juan Carlos Sampedro Julián López-Gómez Juan Carlos Sampedro Correspondence to Julián López-Gómez. The authors have been supported by the Research Grant PGC2018-097104-B-I00 of the Spanish Ministry of Science, Technology and Universities and by the Institute of Interdisciplinar Mathematics of Complutense University. The second author has been also supported by PhD Grant PRE2019_1_0220 of the Basque Country Government. 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/. López-Gómez, J., Sampedro, J.C. Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier. J. Fixed Point Theory Appl. 24, 8 (2022). https://doi.org/10.1007/s11784-021-00916-7 Mathematics Subject Classification 55M25 Degree for Fredholm maps Axiomatization generalized additivity Homotopy invariance Orientability	CommonCrawl
Thermal production of charmonia in Pb-Pb collisions at ${ \sqrt{{ s}_{\bf{ {\rm NN}}}}{\bf {=5.02}}}$ TeV Baoyi Chen 1,2, Department of Physics, Tianjin University, Tianjin 300350, China Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany This work uses the Boltzmann transport model to study the thermal production of $J/\psi$ and $\psi(2S)$ in the quark gluon plasma (QGP) produced by $\sqrt{s_{\rm NN}}=5.02$ TeV Pb-Pb collisions. The $J/\psi$ nuclear modification factors are studied in detail alongside the mechanisms of primordial production and the recombination of charm and anti-charm quarks in the thermal medium. The $\psi(2S)$ binding energy is much smaller in the hot medium compared to the ground state; thus, $\psi(2S)$ with middle to low $p_{\rm T}$ can be thermally regenerated in the later stages of QGP expansions, enabling $\psi(2S)$ to inherit larger collective flows from the bulk medium. 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Department of Physics, Tianjin University, Tianjin 300350, China 2. Institut für Theoretische Physik, Goethe-Universität Frankfurt, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Abstract: This work uses the Boltzmann transport model to study the thermal production of $J/\psi$ and $\psi(2S)$ in the quark gluon plasma (QGP) produced by $\sqrt{s_{\rm NN}}=5.02$ TeV Pb-Pb collisions. The $J/\psi$ nuclear modification factors are studied in detail alongside the mechanisms of primordial production and the recombination of charm and anti-charm quarks in the thermal medium. The $\psi(2S)$ binding energy is much smaller in the hot medium compared to the ground state; thus, $\psi(2S)$ with middle to low $p_{\rm T}$ can be thermally regenerated in the later stages of QGP expansions, enabling $\psi(2S)$ to inherit larger collective flows from the bulk medium. We quantitatively study the nuclear modification factors of both $J/\psi$ and $\psi(2S)$ in different centralities and transverse momentum bins for $\sqrt{s_{\rm NN}}=5.02$ TeV Pb-Pb collisions. Due to their large masses, heavy flavors have unique advantages in both experimental and theoretical studies of quantum chromodynamics (QCD). Since $ J/\psi $ was initially proposed as a probe of the deconfined matter known as "quark-gluon plasma" (QGP) [1], its anomalous yield suppression through parton scattering and enhancement from the recombination of charm and anti-charm quarks in QGP have been widely studied in experiments [2–5] and theoretical models [6–12]. Charmonium is produced by an initial hard-scattering process referred to as "primordial production", which occurs in hadronic collisions with spectator nucleons [13]. Charmonium states are assumed to form before QGP reaches local equilibrium, they then undergo inelastic scatterings and color screening effects when moving through QGP, which results in dissociations and also transitions between different eigenstates ($ J/\psi $, $ \psi^{\prime} $, $ \chi_c $) [14–20]. These eigenstates are finally detected in experiments through their decay into dileptons. At LHC energies, abundant charm pairs are produced in nuclear collisions, this significantly enhances the probability of c and $ \bar c $ combining to generate new $ J/\psi $s in the QGP [21–23], in a process called "regeneration". As most charm quarks are distributed in low and middle $ p_{\rm T} $ regions, the regeneration process dominates the nuclear modification factor and the collective flows of $ J/\psi $ in the low and middle $ p_{\rm T} $ bins [24]. In the high $ p_{\rm T} $ bin, charmonia are mainly produced from the initial hadronic collisions [10]. More experimental data about the charmonium excited state $ \psi(2S) $ has been measured in $ \sqrt{s_{\rm NN}} = 2.76 $ TeV [25] and 5.02 TeV [26] Pb-Pb collisions, across different centralities and transverse momentum bins. The ratio $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ has been measured for these two collision energies, and there is considerable discrepancy between the values reported. At 2.76 TeV, $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ becomes larger than unity in the most central collisions when $ 3<p_{\rm T}<30 $ GeV/c. At 5.02 TeV, the ratio is $ \sim 0.5 $ in a similar centrality and momentum bin. Both sets of experimental data feature large error bars, which prevents any solid conclusions being drawn. In contrast to $ J/\psi $, $ \psi(2S) $ is a loosely bound state. Its wavefunction is significantly modified by the high-temperature medium, which can obscure its dissociation and regeneration rates. Having a smaller binding energy, $ \psi(2S) $ is thermally produced more frequently in the lower temperature region than $ J/\psi $, and inherits larger collective flows from the bulk medium [17, 27, 28]. This sequential regeneration can affect the $ p_{\rm T} $ dependence of the ratio $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $. In this paper, a two-component transport model is employed to study both $ J/\psi $ and $ \psi(2S) $ production in different centralities and momentum bins for $ \sqrt{s_{\rm NN}} = 5.02 $ TeV Pb-Pb collisions. The decay rates of excited states are updated with a more realistic formula instead of employing a survival temperature $ T_d $ above which no charmonia can survive. This improvement can explain effectively the ratio of $ \psi(2S)/J/\psi $ in 5.02 TeV Pb-Pb collisions. The rest of this paper proceeds as follows. In Sec. 2, the details of an improved Boltzmann transport model for charmonium evolution, and the hydrodynamic equations for QGP expansion, are introduced. In Sec. 3, realistic calculations for $ J/\psi $ and $ \psi(2S) $ in $ \sqrt{s_{\rm NN}} = 5.02 $ TeV Pb-Pb collisions are performed and the results are compared with experimental data. A final summary is given in Sec. 4. 2. Transport model and hydrodynamics Heavy quarkonium evolution in phase-space has, though the use of Boltzmann transport models, been well-studied for hot deconfined mediums in the SPS [9] and LHC [29, 30], for both p-Pb and Pb-Pb collisions. Focusing on hot-medium effects, one can start quarkonium evolutions after their production in hard-scattering processes. The three-dimensional transport equation for charmonium evolution is simplified as, $ \left[\cosh(y-\eta){\frac{\partial}{\partial\tau}}+{\frac{\sinh(y-\eta)}{\tau}}{\frac{\partial}{\partial \eta}}+{\bf v}_{\rm T}\cdot\nabla_{\rm T}\right]f_{\Psi} = -\alpha_{\Psi} f_{\Psi}+\beta_{\Psi}, $ $ f_{\Psi} $ is the Ψ phase-space density. y and η are the rapidities in momentum and coordinate space, respectively. $ { v}_{\rm T} = { p}_{\rm T}/E_{\rm T} = { p}_{\rm T}/\sqrt{m_{\Psi}^2+p_{\rm T}^2} $ is the transverse velocity of charmonium, which represents the leakage effects for a cooling system with finite size, i.e., charmonia with large velocities tend to escape from the thermal medium instead of dissociating. Primordially-produced charmonia in the initial hadronic collisions suffer color screening effects and parton inelastic scatterings, which are both included in the decay rate $ \alpha_{\Psi} $, $ \alpha_{\Psi} = {1\over E_{\rm T}} \int {{\rm d}^3{ k}\over {(2\pi)^3E_{\rm g}}}\sigma_{{\rm g}\Psi}({ p},{ k},T)F_{{\rm g}\Psi}({ p},{ k})f_{\rm g}({ k},T), $ where $ E_{\rm g} $ and $ f_{\rm g} $ are gluon energy and density in the thermal medium, respectively. $ F_{{\rm g}\Psi} $ is the flux factor. In the expanding QGP, $ u^\mu $ represents the four-velocity of the fluid. The gluonic Ψ cross-section in a vacuum is extracted from the perturbative calculation with a Coulomb potential approximation. For the thermal medium, this paper takes a similar approach to Ref. [9], using a reduced charmonium binding energy of the form, $ \sigma_{{\rm g}\Psi}(w) = A_0 {(w/\epsilon_{\Psi}-1)^{3/2}\over (w/\epsilon_{\Psi})^5} $ with $ A_0 = (2^{11}\pi/27)(m_c^3\epsilon_{\Psi})^{-1/2} $ and $ \epsilon_{\Psi} $ representing the binding energy of Ψ. The charm quark mass is taken as the mass of a D meson, to fit the binding energy of charmonium in a vacuum. $ w = p_{\Psi}^\mu p_{{\rm g}\mu}/m_{\Psi} $ is the gluon energy in the Ψ rest-frame. In Fig.1, the $ J/\psi $ decay rate $ \alpha_{\Psi} $ is compared with the quasi-free dissociation rate [27]. Most $ J/\psi $s can survive in the relatively low temperature region, and two different transport models of the decay rate present similar final results for the region $ T<300 $ MeV, where most QGP and charmonia are located [7, 30]. Figure 1. $ J/\psi $ decay rate in the thermal medium as a function of temperature T. The decay rate from quasi-free dissociation is shown for comparison. The solid line is from the improved version of the transport model, developed by TSINGHUA Group [9, 10, 31], and the dotted line is from the calculations of TAMU Group [6, 7, 27]. The heavy quark potential $ V(r,T) $ can be partially screened by the thermal medium, especially at the large distances and high temperatures suggested by lattice QCD calculations [32]. Charmonium bound states may disappear sequentially in the static medium. The maximum survival temperature of a certain bound state is called the "dissociation temperature" $ T_{\rm d} $, above this the bound state disappears. In nuclear collisions, the assumption that no bound states survive at $ T>T_{\rm d} $ strongly suppresses the $ \psi(2S) $ production, where no excited states can survive inside the QGP at $ T>T_{\rm d}^{\chi_c,\psi(2S)}\approx 1.1T_{\rm c} $. Similarly to fast cooling systems, charmonium states might survive in the region $ T>T_{\rm d} $ if the medium quickly cools down to below $ T_{\rm d} $. In this work, the approximation of $ \alpha_{\Psi}(T>T_{\rm d}) = +\infty $ employed in [33] is replaced by a large but finite value, shown in Fig.1. The new decay rate increases the survival probability of excited states, and weakly affects the production of $ J/\psi $s due to their large $ T_{\rm d} $. The $ \psi(2S) $ decay rate is extracted from the nuclear geometry scale, $ \alpha_{\psi(2S)} = \alpha_{J/\psi}\times \langle r\rangle^2_{\psi(2S)}/\langle r\rangle^2_{J/\psi} $, and $ \chi_c $ is found in a similar way. The mean radius of charmonia in a vacuum is calculated with the potential model $ \langle r\rangle_{J/\psi, \chi_c,\psi(2S)} = (0.5, 0.72, 0.9) $ fm [14]. At LHC energies, many charm pairs are also produced in Pb-Pb collisions, which can significantly increase the recombination of uncorrelated charm and anti-charm quarks in the QGP. This process is included in Eq.(1) by the term $ \beta_{\Psi} $. The regeneration rate depends on both charm and anti-charm quark densities in the QGP, and also their recombination probability. At high temperatures, charmonium binding energies are reduced significantly, which suppresses the regeneration probability of charmonium. As $ \psi(2S) $s are loosely bound, they are thermally produced in the hadronization of QGP. Charm quarks with color charges strongly couple with the QGP and lose energy. In relativistic heavy ion collisions, large quench factors and collective flows for charmed mesons have been observed [34–36]. Therefore, one can approximately take the kinetically thermalized phase-space distribution for charm quarks as being at $ \tau\geqslant \tau_0 $, where $ \tau_0 $ is the time-scale of the QGP local equilibrium [37]. As heavy quarks are rarely produced in the thermal medium due to their large masses, the total number of charm pairs is conserved with spatial diffusions inside the QGP [38]. The spatial density is controlled by the conservation equation. $ \partial_{\mu}(\rho_{\rm c}u^\mu) = 0. $ The initial charm quark density at $ \tau_0 $ is obtained from nuclear geometry, $ \rho_c({ x}_{\rm T}, \eta,\tau_0) = {{{\rm d}\sigma_{\rm pp}^{c{\bar c}}} \over {\rm d}\eta}{ T_A({ x}_{\rm T})T_B({ x}_{\rm T}-{ b})\cosh(\eta)\over \tau_0}, $ where $ T_A $ and $ T_B $ are the thickness functions of two colliding nuclei, with the definition $ T_{A(B)}({ x}_{\rm T}) = $ $ \int_{-\infty}^{\infty} {\rm d}z\rho_{A(B)}({ x}_{\rm T},z) $. $ \rho_{A(B)}({ x}_{\rm T},z) $ is understood to be the Woods-Saxon nuclear density. The rapidity distribution of charm pairs in $ \sqrt{s_{\rm NN}} = 5.02 $ TeV pp collisions is obtained by interpolation of the experimental data for 2.76 TeV and 7 TeV collisions, $ {\rm d}\sigma_{\rm pp}^{c\bar c}/{\rm d}y = 0.86 $ mb in the central rapidity region $ \|y\|<0.9 $, and 0.56 mb in the forward rapidity region $ 2.5<\|y\|<4 $ [39]. The momentum distribution of charmonium primordially produced in A-A collisions is scaled from its distribution in pp collisions. The parametrization of the charmonium initial distribution at 5.02 TeV is similar to the form it takes at 2.76 TeV, $ {{\rm d}^2\sigma_{{\rm pp}}^{J/\psi}\over {\rm d}y2\pi p_{\rm T}{\rm d}p_{\rm T}} = f_{J/\psi}^{\rm{Norm}}(p_{\rm T}\|y) \cdot {{\rm d}\sigma_{{\rm pp}}^{J/\psi}\over {\rm d}y}, $ $ f_{J/\psi}^{\rm{Norm}}(p_{\rm T}\|y) = {(n-1)\over {\pi(n-2)\langle p_{\rm T}^2\rangle_{\rm pp}}} \left[1+{p_T^2\over {(n-2)\langle p_{\rm T}^2\rangle_{\rm pp}}}\right]^{-n}. $ The charmonium rapidity differential cross-section at 5.02 TeV is $ {\rm d}\sigma_{\rm pp}^{J/\psi}/{\rm d}y = $ 5.0 µb in the central rapidity $ \|y\|<1 $, and 3.25 µb in the forward rapidity $ 2.5<\|y\|<4 $, these were found through an interpolation between the experimental results for 2.76 TeV [40] and 7 TeV [41] pp collisions. $ f_{J/\psi}^{\rm{Norm}}(p_{\rm T}\|y) $ is the normalized transverse momentum distribution of charmonium with rapidity y. The mean squared transverse momentum $ \langle p_{\rm T}^2\rangle $ and the parameter n are calculated as $ \langle p_{\rm T}^2\rangle_{\rm pp}\|_{y = 0} = 12.5\ {\rm{(GeV/c)^2}} $ and $ n = 3.2 $. For the charmonium momentum distribution in other rapidities, which have less constraints, $ \langle p_{\rm T}^2\rangle_{\rm pp}(y) $ is determined by the relation, $ \langle p_{\rm T}^2\rangle_{\rm pp}^{J/\psi}(y) = \langle p_{\rm T}^2\rangle_{\rm pp}^{J/\psi}\|_{y = 0}\times \left[1-\left({y\over y_{\rm{max}}}\right)^2\right], $ where $ y_{\rm{max}} = \cosh^{-1}(\sqrt{s_{\rm NN}}/(2E_{\rm T})) $, and is the maximum rapidity of charmonium in pp collisions with zero transverse momentum. As the masses of charmonium excited states ($ \chi_c,\psi(2S) $) are similar to $ J/\psi $s, their initial momentum distributions are approximated to be the same as those in Eqs.(6,7). In nuclear collisions, the charmonium initial distribution is also modified by shadowing effects in the nucleus [42, 43]. This paper employs the EPS09 NLO model [44] to generate the modification factors for primordially-produced charmonium in $ \sqrt{s_{\rm NN}} = 5.02 $ TeV Pb-Pb collisions. This suppression factor is $ \sim 0.8 $ depending on the impact parameter. For the regeneration, shadowing effects reduce the number of charm pairs by around 20%, and suppress the regeneration by a factor of $ \sim 0.8^2 $. The expanding QGP background of charmonium evolution is simulated with the (2+1)-dimensional ideal hydrodynamic equations in the transverse plane, under the assumption of Bjorken expansion in the longitudinal direction. $ \partial_{\mu} T^{\mu\nu} = 0 , $ $ T^{\mu\nu} = (e+p)u^\mu u^\nu -g^{\mu\nu}p $, and is the energy-momentum tensor. e and p are the energy density and pressure, respectively. $ u^\mu $ is the four-velocity of the QGP fluid, which can affect charm quark spatial diffusions through Eq.(4) as well as charmonium regeneration. It also determines the collective flows of light hadrons, charmed mesons and regenerated charmonia. The deconfined matter is treated as an ideal gas of massless gluons, u and d quarks and strange quarks with mass $ m_{\rm s} = 150 $ MeV [45]. Hadron gas is an ideal gas made up of all known hadrons and resonances with masses of up to 2 GeV [46]. The two phases are connected by a first-order phase transition with a critical temperature of $ T_{\rm c} = 170 $ MeV. The initial maximum temperature of the QGP is calculated as $ T_0({ x}_{rm T} = 0, \tau_0) = 510 $ MeV in the central rapidity $ \|y\|<2.4 $, and 450 MeV in the forward rapidity $ 2.5<\|y\|<4 $. Here $ \tau_0 = 0.6 $ fm/c, and is the time-scale for the QGP to reach local equilibrium [37]. The lifetime of QGP is $ \sim 10 $ fm/c for the most central Pb-Pb collisions at $ \sqrt{s_{\rm NN}} = 5.02 $ TeV. 3. Numerical results and analysis With the transport model for charmonium evolution and the hydrodynamic equations for QGP collective expansion, one can obtain realistic nuclear modification factors for charmonia in heavy ion collisions. In the left-hand panel of Fig.2, primordially-produced charmonia undergo dissociations from peripheral to central Pb-Pb collisions, shown with the dotted line. The regeneration process $ c+\bar c \rightarrow J/\psi +g $ is plotted with the dashed line and is proportional to the number of charm quark pairs in the QGP, it dominates total $ J/\psi $ production in central collisions. The experimental data in the left-hand panel of Fig.2 is inclusive production; it includes the non-prompt contribution from B hadron decays, which contributes around 10% of the final inclusive yield. The detailed momentum dependence of the non-prompt fraction in $ J/\psi $ inclusive production is fitted as $f_B = N_{\rm pp}^{B\rightarrow J/\psi}/ (N_{\rm pp}^{\rm{prompt}}+$$ N_{\rm pp}^{B\rightarrow J/\psi}) = 0.04+0.023p_{\rm T}/({\rm GeV/c}) $ [33], with a weak dependence on the rapidity and colliding energy $ \sqrt{s_{\rm NN}} $. In nuclear collisions, bottom quarks suffer strong energy losses in the thermal medium. B hadrons and non-prompt charmonia are shifted from high $ p_{\rm T} $ to relatively low $ p_{\rm T} $. This hot medium modification of the non-prompt charmonium (or bottom quark) momentum distribution is characterized with a quench factor $ R_{\rm Q} $. In high $ p_{\rm T} $,$ R_{\rm Q} $ is smaller than unity, and is calculated as 0.4 for the non-prompt $ J/\psi $ $ R_{AA} $ [33]. This value is employed in the entire $ p_{\rm T} $ region. With both prompt and non-prompt charmonia, one can obtain charmonium's inclusive nuclear modification factors, which are shown in Fig.2. Taking into account the large uncertainties of $ {\rm d}\sigma_{\rm pp}^{c\bar c}/{\rm d}y $ in the transport model, this paper performs two calculations for $ R_{AA} $ under a change in $ {\rm d}\sigma_{\rm pp}^{c\bar c}/{\rm d}y $ of ±20% (see the color band in Fig.2) . In most central collisions, primordial production is strongly suppressed and regeneration dominates the total yield. Figure 2. (color online) (left-hand panel) Inclusive nuclear modification factor $ R_{AA} $ of $ J/\psi $ as a function of the number of participants $ N_{\rm p} $ in central rapidity for Pb-Pb collisions at $ \sqrt{s_{\rm NN}} = 5.02 $ TeV. The dotted and dashed lines represent primordial production and regeneration, respectively. The solid black line in the middle of the band is for $ J/\psi $ inclusive of $ R_{AA} $. The color band is for the uncertainties of the inputs, with $ {\rm d}\sigma_{\rm pp}^{c\bar c}/{\rm d}y $ changed by ±20%. Experimental data is from the ALICE Collaboration [47]. (right-hand panel) Inclusive $ R_{AA} $ in forward rapidity. The lines and band are similar to the left panel. Experimental data is from ALICE Collaboration [48]. In the forward rapidity, both initial conditions of the hydrodynamic equations and transport model are updated by comparison to the central rapidity collisions. In the right panel of Fig.2, the $ J/\psi $ nuclear modification factor from primordial production (dotted line), regeneration (dashed line), and inclusive production (color band) are plotted separately. The flat tendency of the experimental data at larger $ N_{\rm p} $ is due to the combined effects of the decreased primordial production and increased regeneration in the final $ J/\psi $ yield. The experimental data in the right-hand panel of Fig.2 is plotted for the range $ 0.3<p_{\rm T}<8 $ GeV/c. This excludes the contribution from coherent photoproduction, which occurs below 0.3 GeV/c and can make the total $ R_{AA} $ larger than unity in ultra-peripheral collisions [31]. In order to show the contributions of primordial production and regeneration, the $ p_{\rm T} $-differential $ R_{AA} $ is also plotted in the left-hand panel of Fig.3. The significant increase of $ R_{AA} $ in the low $ p_{\rm T} $ region is caused by regeneration, and its large suppression in the high $ p_{\rm T} $ region is due to the effects of color screening and the inelastic collisions of partons. The dotted line represents the initial production, and it increases slightly with $ p_{\rm T} $ due to the leakage effect. The theoretical results for $ R_{AA}^{J/\psi} $ in both central and forward rapidities can explain the experimental data well. Note that even the charm pair cross-section $ {\rm d}\sigma_{\rm pp}^{c\bar c}/{\rm d}y $ value is larger in a central rapidity than in a forward one, $ R_{AA} $ is similar across the two rapidities, because in a central rapidity with a hotter medium, the QGP strong expansion "blows" charm quarks to a larger volume, which suppresses the charm quark spatial density and the regeneration rate of charmonium. Meanwhile, the elliptic flows of regenerated charmonia become larger in the central rapidity. These will be discussed in detail below. In the right-hand panel of Fig.3, the contributions of prompt $ J/\psi $ and $ \psi(2S) $ to $ R_{AA} $, as a function of rapidity, are also presented. Figure 3. (color online) (left-hand panel) $ J/\psi $ nuclear modification factor $ R_{AA} $ as a function of the transverse momentum $ p_{\rm T} $. The dotted line represents initial production, and the solid line is the inclusive production comprising initial production, regeneration and B hadron decay. The color band is due to the uncertainties of $ {\rm d}\sigma_{\rm pp}^{c\bar c}/{\rm d}y $ (see Fig.2). The differences between the solid and dotted lines are mainly due to the contributions of regeneration at low $ p_{\rm T} $ and B hadron decay at high $ p_{\rm T} $, respectively. Experimental data is from the ALICE Collaboration [49]. (right-hand panel) The contribution of prompt $ J/\psi $ towards $ R_{AA} $ as a function of rapidity in the centrality 0%-100%. Prompt $ \psi(2S) $s contribution to $ R_{AA}(y) $ is also predicted. The experimental data is from [50]. The situation is more complicated for $ \psi(2S) $ in the hot medium, owing to its dissociation rate compared with the tightly bound $ J/\psi $. The $ \psi(2S) $ decay rate here is extracted from $ J/\psi $' using the nuclear geometry scale of Sec. II. In Fig.4, charmonia at high $ p_{\rm T} $mainly arise from the primordial production. In moving from peripheral to central collisions, the dissociation rate of $ \psi(2S) $ increases, and the ratio of $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ decreases with $ N_{\rm p} $. In the most central collisions at LHC, the ratio of charmonium nuclear modification factors is proportional to their decay rates. In peripheral collisions, the charmonium path length in the hot medium becomes smaller. Under weak suppression, the $ J/\psi $ and $ \psi(2S) $ nuclear modification factors approach unity, which entails that $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi}\rightarrow 1 $ as $ N_{\rm {\rm p}}\rightarrow 2 $, (see left-hand panel of Fig.4). The individual $ R_{AA}^{J/\psi} $ in the high $ p_{\rm T} $ bin is also plotted in the right-hand panel of Fig.4. The decreasing tendencies of $ J/\psi $ and $ \psi(2S) $ $ R_{AA} $ with increasing $ N_{\rm p} $ are explained well. Figure 4. (color online) (left-hand panel) The ratio of $ J/\psi $ and $ \psi(2S) $ prompt nuclear modification factors in the central rapidity region with momentum cut-offs of $ 6.5<p_{\rm T}<30 $ GeV/c for $ \sqrt{s_{\rm NN}} = 5.02 $ TeV Pb-Pb collisions. Experimental data is from the CMS Collaboration [26]. (right-hand panel) $ J/\psi $ and $ \psi(2S) $ prompt nuclear modification factors as a function of $ N_{\rm p} $ in the central rapidity region with momentum cut-offs of $ 6.5<p_{\rm T}<30 $ GeV/c for $ \sqrt{s_{\rm NN}} = 5.02 $ TeV Pb-Pb collisions. The experimental data is from [50]. The transverse momentum dependence of $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ is also studied via the sequential regeneration mechanism. In Fig.5, as $ p_{\rm T}\rightarrow 0 $, there will be significant regeneration of $ J/\psi $. The loosely-bound $ \psi(2S) $can only be thermally produced in the later stages of QGP expansion compared with $ J/\psi $, which leads to the production of regenerated $ \psi(2S) $with larger velocities and collective flows inherited from the bulk medium, because of the strong couplings between the charm quarks and the deconfined medium. Therefore, the regenerated $ \psi(2S) $ are distributed at a larger $ p_{\rm T} $ (dashed line in right-hand panel of Fig.5) compared with the regenerated $ J/\psi $. Due to the different $ p_{\rm T} $ distributions of the thermally produced $ J/\psi $ and $ \psi(2S) $, the shapes of $ J/\psi $ and $ \psi(2S) $ $ R_{AA}(p_{\rm T}) $s (solid black and blue lines in right-hand panel of Fig.5) are different. There is a "peak" in the ratio $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ in the left-hand panel of Fig.5 due to the sequential regeneration of $ \psi(2S) $. In the absence of $ \psi(2S) $ regeneration, the ratio will decrease to zero as $ p_{\rm T}\rightarrow 0 $ (see the solid line in the left-hand panel of Fig.5. Figure 5. (color online) (left-hand panel) The ratio of the $ J/\psi $ and $ \psi(2S) $ prompt nuclear modification factors as a function of the transverse momentum $ p_{\rm T} $ in the minimum bias (corresponding to the impact parameter b = 8.4 fm) for $ \sqrt{s_{\rm NN}} = 5.02 $ TeV Pb-Pb collisions. Dotted-dashed and solid lines are with and without $ \psi(2S) $ regeneration, respectively. Experimental data is from the CMS Collaboration [26, 50]. (Right-hand panel) The $ J/\psi $ and $ \psi(2S) $ nuclear modification factors as a function of transverse momentum. Dotted, dashed and solid black lines are for initial, regenerated and prompt $ \psi(2S) $, respectively. The prompt $ R_{AA}^{J/\psi} $ is also plotted for comparison. The experimental data is from [26, 50]. In the low and middle $ p_{\rm T} $ regions, regeneration becomes important for $ J/\psi $ and $ \psi(2S) $, and enhances their $ R_{AA} $ in semi-central and central collisions. In order to show the role of $ \psi(2S) $ regeneration on $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $, this study presents two calculations with and without $ \psi(2S) $ regeneration, respectively, in Fig.6. In Fig.6, neglecting the regeneration of $ \psi(2S) $ (solid line), $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ continues to decrease with $ N_{\rm p} $ due to the stronger QGP suppression of charmonium excited states. The contribution of regeneration increases $ \psi(2S) $ production, most notably in the central collisions, and enhances the value of $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ (see the dotted-dashed line). Note that in Ref. [33], predictions about prompt $ R_{AA}^{\psi(2S)}/R_{AA}^{J/\psi} $ in 2.76 TeV Pb-Pb collisions have been made. Its value is predicted to be around $ \sim 0.15 $ in all $ p_{\rm T} $ bins. The binding energy and regeneration rate for $ \psi(2S) $in Ref. [33] are smaller, which suppresses the $ \psi(2S) $ production. In this study these calculations are extended from 2.76 TeV [33] to 5.02 TeV, and both calculations are consistent with the experimental data at 5.02 TeV. The difference between the solid and dotted-dashed lines in Fig.6 is due to the $ \psi(2S)$ regeneration component. Figure 6. (color online) Ratio of the $ J/\psi $ and $ \psi(2S) $ prompt nuclear modification factors as a function of $ N_{\rm p} $ in rapidity $ 1.6<\|y\|<2.4 $ with momentum cut-offs of$ 3<p_{\rm T}<30 $ GeV/c. The dotted-dashed line represents the scenario for $ J/\psi $ and $ \psi(2S) $ with both primordial production and regeneration, and the solid line is the scenario without regeneration for $ \psi(2S) $ ($ J/\psi $ regeneration is included in both lines). Experimental data at $ \sqrt{s_{\rm NN}} = 2.76 $ TeV and 5.02 TeV is from the CMS Collaboration [26]. Furthermore, the anisotropies of the $ J/\psi $ and $ \psi(2S) $ momentum distributions are shown in Fig.7. Charmonia moving inside the QGP are likely to be in a color-neutral bound state, thus they are weakly coupled with the bulk medium, and less affected by the collective expansion of the QGP compared with unbound charm quarks. The non-zero elliptic flow of primordially-produced $ J/\psi $ at $ p_{\rm T}>6 $ GeV/c is mainly due to the effects of path length differences in the transverse plane (see the solid line in the left-hand panel of Fig.7). At low $ p_{\rm T} $, $ J/\psi $ production is dominated by the regeneration component. These heavy quarks are strongly coupled with the thermal medium and inherit collective flows, which results in a peak of $ v_2 $ at $ p_{\rm T}\sim 3 $ GeV/c. The elliptic flows of the prompt $ \psi(2S) $ are also displayed as a dotted line. As $ \psi(2S) $ are regenerated in the later stages of QGP anisotropic expansion, their elliptic flow is larger than that of $ J/\psi $'. In the high $ p_{\rm T} $ region, the momentum anisotropy of $ \psi(2S) $ is larger than that of $ J/\psi $', as they are easily dissociated and sensitive to the anisotropy of the bulk medium. Figure 7. (color online) (left-hand panel) The elliptic flows of the prompt $ J/\psi $ and $ \psi(2S) $ as a function of the transverse momentum $ p_{\rm T} $ in centrality 20%-40% for the forward rapidity $ 2.5<\|y\|<4 $ $ \sqrt{s_{\rm NN}} = 5.02 $ TeV Pb-Pb collisions. Solid and dotted lines are used for the prompt $ J/\psi $ and $ \psi(2S) $, respectively. (right-hand panel) The elliptic flows of inclusive $ J/\psi $ as a function of the transverse momentum $ p_{\rm T} $. The solid and dotted lines represent the inclusive $ J/\psi $ in forward and central rapidities, respectively. Experimental data is from the ALICE Collaboration [51]. In the right-hand panel of Fig.7 the experimental data is for the inclusive $ J/\psi $, and includes a non-prompt contribution from the B hadron decay. The solid line represents the inclusive $ J/\psi $, assuming the kinetic equilibrium for bottom quarks as the upper limit [30]. The non-prompt contribution becomes important at high $ p_{\rm T} $ and therefore the kinetically thermalized bottom quarks can enhance the inclusive $ v_2^{J/\psi} $ by $ \sim 2 $% at $ p_{\rm T}\sim 8 $ GeV/c. $ v_2^{J/\psi} $ in central rapidity is also calculated and shown as a dotted line for comparison. The situation for inclusive $ \psi(2S) $is connected with the energy loss of bottom quarks in QGP, and has been comprehensively studied in Ref. [33]. In summary, this work employs an improved transport model to study the thermal production of $ J/\psi $ and $ \psi(2S) $ in Pb-Pb collisions at $ \sqrt{s_{\rm NN}} = 5.02 $ TeV. Charmonium nuclear modification factors are dominated by regeneration at low $ p_T $ and primordial production at high $ p_{\rm T} $, respectively. With different binding energies, $ J/\psi $ and $ \psi(2S) $ can be sequentially produced in the different stages of QGP anisotropic expansion, resulting in different $ p_{\rm T} $ distributions of the regenerated $ J/\psi $ and $ \psi(2S) $. This explains well both the$ J/\psi $ and $ \psi(2S) $ $ R_{AA} $s and their ratio in 5.02 TeV Pb-Pb collisions, and clearly shows how the open charm quark evolution can affect the final charmonium production. The sequential regeneration of $ J/\psi $ and $ \psi(2S) $describes the charm quark diffusion histories and QGP expansion in heavy-ion collisions. I acknowledge instructive discussions with Prof. Pengfei Zhuang and Jiaxing Zhao. I am also grateful to Prof. Carsten Greiner for the kind hospitality during this study.	CommonCrawl
Cauchy's integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis. Not to be confused with Cauchy's integral theorem or Cauchy formula for repeated integration. Mathematical analysis → Complex analysis Complex analysis Complex numbers • Real number • Imaginary number • Complex plane • Complex conjugate • Unit complex number Complex functions • Complex-valued function • Analytic function • Holomorphic function • Cauchy–Riemann equations • Formal power series Basic theory • Zeros and poles • Cauchy's integral theorem • Local primitive • Cauchy's integral formula • Winding number • Laurent series • Isolated singularity • Residue theorem • Conformal map • Schwarz lemma • Harmonic function • Laplace's equation Geometric function theory People • Augustin-Louis Cauchy • Leonhard Euler • Carl Friedrich Gauss • Jacques Hadamard • Kiyoshi Oka • Bernhard Riemann • Karl Weierstrass • Mathematics portal Theorem Let U be an open subset of the complex plane C, and suppose the closed disk D defined as $D={\bigl \{}z:\|z-z_{0}\|\leq r{\bigr \}}$ is completely contained in U. Let f : U → C be a holomorphic function, and let γ be the circle, oriented counterclockwise, forming the boundary of D. Then for every a in the interior of D, $f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz.\,$ The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires f to be complex differentiable. Since $1/(z-a)$ can be expanded as a power series in the variable $a$ ${\frac {1}{z-a}}={\frac {1+{\frac {a}{z}}+\left({\frac {a}{z}}\right)^{2}+\cdots }{z}}$ it follows that holomorphic functions are analytic, i.e. they can be expanded as convergent power series. In particular f is actually infinitely differentiable, with $f^{(n)}(a)={\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{\left(z-a\right)^{n+1}}}\,dz.$ This formula is sometimes referred to as Cauchy's differentiation formula. The theorem stated above can be generalized. The circle γ can be replaced by any closed rectifiable curve in U which has winding number one about a. Moreover, as for the Cauchy integral theorem, it is sufficient to require that f be holomorphic in the open region enclosed by the path and continuous on its closure. Note that not every continuous function on the boundary can be used to produce a function inside the boundary that fits the given boundary function. For instance, if we put the function f(z) = 1/z, defined for \|z\| = 1, into the Cauchy integral formula, we get zero for all points inside the circle. In fact, giving just the real part on the boundary of a holomorphic function is enough to determine the function up to an imaginary constant — there is only one imaginary part on the boundary that corresponds to the given real part, up to addition of a constant. We can use a combination of a Möbius transformation and the Stieltjes inversion formula to construct the holomorphic function from the real part on the boundary. For example, the function f(z) = i − iz has real part Re f(z) = Im z. On the unit circle this can be written i/z − iz/2. Using the Möbius transformation and the Stieltjes formula we construct the function inside the circle. The i/z term makes no contribution, and we find the function −iz. This has the correct real part on the boundary, and also gives us the corresponding imaginary part, but off by a constant, namely i. Proof sketch By using the Cauchy integral theorem, one can show that the integral over C (or the closed rectifiable curve) is equal to the same integral taken over an arbitrarily small circle around a. Since f(z) is continuous, we can choose a circle small enough on which f(z) is arbitrarily close to f(a). On the other hand, the integral $\oint _{C}{\frac {1}{z-a}}\,dz=2\pi i,$ over any circle C centered at a. This can be calculated directly via a parametrization (integration by substitution) z(t) = a + εeit where 0 ≤ t ≤ 2π and ε is the radius of the circle. Letting ε → 0 gives the desired estimate ${\begin{aligned}\left\|{\frac {1}{2\pi i}}\oint _{C}{\frac {f(z)}{z-a}}\,dz-f(a)\right\|&=\left\|{\frac {1}{2\pi i}}\oint _{C}{\frac {f(z)-f(a)}{z-a}}\,dz\right\|\\[1ex]&=\left\|{\frac {1}{2\pi i}}\int _{0}^{2\pi }\left({\frac {f{\bigl (}z(t){\bigr )}-f(a)}{\varepsilon e^{it}}}\cdot \varepsilon e^{it}i\right)\,dt\right\|\\[1ex]&\leq {\frac {1}{2\pi }}\int _{0}^{2\pi }{\frac {\left\|f{\bigl (}z(t){\bigr )}-f(a)\right\|}{\varepsilon }}\,\varepsilon \,dt\\[1ex]&\leq \max _{\|z-a\|=\varepsilon }\left\|f(z)-f(a)\right\|~~{\xrightarrow[{\varepsilon \to 0}]{}}~~0.\end{aligned}}$ Example Let $g(z)={\frac {z^{2}}{z^{2}+2z+2}},$ and let C be the contour described by \|z\| = 2 (the circle of radius 2). To find the integral of g(z) around the contour C, we need to know the singularities of g(z). Observe that we can rewrite g as follows: $g(z)={\frac {z^{2}}{(z-z_{1})(z-z_{2})}}$ where z1 = −1 + i and z2 = −1 − i. Thus, g has poles at z1 and z2. The moduli of these points are less than 2 and thus lie inside the contour. This integral can be split into two smaller integrals by Cauchy–Goursat theorem; that is, we can express the integral around the contour as the sum of the integral around z1 and z2 where the contour is a small circle around each pole. Call these contours C1 around z1 and C2 around z2. Now, each of these smaller integrals can be evaluated by the Cauchy integral formula, but they first must be rewritten to apply the theorem. For the integral around C1, define f1 as f1(z) = (z − z1)g(z). This is analytic (since the contour does not contain the other singularity). We can simplify f1 to be: $f_{1}(z)={\frac {z^{2}}{z-z_{2}}}$ and now $g(z)={\frac {f_{1}(z)}{z-z_{1}}}.$ Since the Cauchy integral formula says that: $\oint _{C}{\frac {f_{1}(z)}{z-a}}\,dz=2\pi i\cdot f_{1}(a),$ we can evaluate the integral as follows: $\oint _{C_{1}}g(z)\,dz=\oint _{C_{1}}{\frac {f_{1}(z)}{z-z_{1}}}\,dz=2\pi i{\frac {z_{1}^{2}}{z_{1}-z_{2}}}.$ Doing likewise for the other contour: $f_{2}(z)={\frac {z^{2}}{z-z_{1}}},$ we evaluate $\oint _{C_{2}}g(z)\,dz=\oint _{C_{2}}{\frac {f_{2}(z)}{z-z_{2}}}\,dz=2\pi i{\frac {z_{2}^{2}}{z_{2}-z_{1}}}.$ The integral around the original contour C then is the sum of these two integrals: ${\begin{aligned}\oint _{C}g(z)\,dz&{}=\oint _{C_{1}}g(z)\,dz+\oint _{C_{2}}g(z)\,dz\\[.5em]&{}=2\pi i\left({\frac {z_{1}^{2}}{z_{1}-z_{2}}}+{\frac {z_{2}^{2}}{z_{2}-z_{1}}}\right)\\[.5em]&{}=2\pi i(-2)\\[.3em]&{}=-4\pi i.\end{aligned}}$ An elementary trick using partial fraction decomposition: $\oint _{C}g(z)\,dz=\oint _{C}\left(1-{\frac {1}{z-z_{1}}}-{\frac {1}{z-z_{2}}}\right)\,dz=0-2\pi i-2\pi i=-4\pi i$ Consequences The integral formula has broad applications. First, it implies that a function which is holomorphic in an open set is in fact infinitely differentiable there. Furthermore, it is an analytic function, meaning that it can be represented as a power series. The proof of this uses the dominated convergence theorem and the geometric series applied to $f(\zeta )={\frac {1}{2\pi i}}\int _{C}{\frac {f(z)}{z-\zeta }}\,dz.$ The formula is also used to prove the residue theorem, which is a result for meromorphic functions, and a related result, the argument principle. It is known from Morera's theorem that the uniform limit of holomorphic functions is holomorphic. This can also be deduced from Cauchy's integral formula: indeed the formula also holds in the limit and the integrand, and hence the integral, can be expanded as a power series. In addition the Cauchy formulas for the higher order derivatives show that all these derivatives also converge uniformly. The analog of the Cauchy integral formula in real analysis is the Poisson integral formula for harmonic functions; many of the results for holomorphic functions carry over to this setting. No such results, however, are valid for more general classes of differentiable or real analytic functions. For instance, the existence of the first derivative of a real function need not imply the existence of higher order derivatives, nor in particular the analyticity of the function. Likewise, the uniform limit of a sequence of (real) differentiable functions may fail to be differentiable, or may be differentiable but with a derivative which is not the limit of the derivatives of the members of the sequence. Another consequence is that if f(z) = Σ an zn is holomorphic in \|z\| < R and 0 < r < R then the coefficients an satisfy Cauchy's inequality[1] $\|a_{n}\|\leq r^{-n}\sup _{\|z\|=r}\|f(z)\|.$ From Cauchy's inequality, one can easily deduce that every bounded entire function must be constant (which is Liouville's theorem). The formula can also be used to derive Gauss's Mean-Value Theorem, which states[2] $f(z)={\frac {1}{2\pi }}\int _{0}^{2\pi }f(z+re^{i\theta })\,d\theta .$ In other words, the average value of f over the circle centered at z with radius r is f(z). This can be calculated directly via a parametrization of the circle. Generalizations Smooth functions A version of Cauchy's integral formula is the Cauchy–Pompeiu formula,[3] and holds for smooth functions as well, as it is based on Stokes' theorem. Let D be a disc in C and suppose that f is a complex-valued C1 function on the closure of D. Then[4][5] $f(\zeta )={\frac {1}{2\pi i}}\int _{\partial D}{\frac {f(z)\,dz}{z-\zeta }}-{\frac {1}{\pi }}\iint _{D}{\frac {\partial f}{\partial {\bar {z}}}}(z){\frac {dx\wedge dy}{z-\zeta }}.$ One may use this representation formula to solve the inhomogeneous Cauchy–Riemann equations in D. Indeed, if φ is a function in D, then a particular solution f of the equation is a holomorphic function outside the support of μ. Moreover, if in an open set D, $d\mu ={\frac {1}{2\pi i}}\varphi \,dz\wedge d{\bar {z}}$ for some φ ∈ Ck(D) (where k ≥ 1), then f(ζ, ζ) is also in Ck(D) and satisfies the equation ${\frac {\partial f}{\partial {\bar {z}}}}=\varphi (z,{\bar {z}}).$ The first conclusion is, succinctly, that the convolution μ ∗ k(z) of a compactly supported measure with the Cauchy kernel $k(z)=\operatorname {p.v.} {\frac {1}{z}}$ is a holomorphic function off the support of μ. Here p.v. denotes the principal value. The second conclusion asserts that the Cauchy kernel is a fundamental solution of the Cauchy–Riemann equations. Note that for smooth complex-valued functions f of compact support on C the generalized Cauchy integral formula simplifies to $f(\zeta )={\frac {1}{2\pi i}}\iint {\frac {\partial f}{\partial {\bar {z}}}}{\frac {dz\wedge d{\bar {z}}}{z-\zeta }},$ and is a restatement of the fact that, considered as a distribution, (πz)−1 is a fundamental solution of the Cauchy–Riemann operator ∂/∂z̄.[6] The generalized Cauchy integral formula can be deduced for any bounded open region X with C1 boundary ∂X from this result and the formula for the distributional derivative of the characteristic function χX of X: ${\frac {\partial \chi _{X}}{\partial {\bar {z}}}}={\frac {i}{2}}\oint _{\partial X}\,dz,$ where the distribution on the right hand side denotes contour integration along ∂X.[7] Several variables In several complex variables, the Cauchy integral formula can be generalized to polydiscs.[8] Let D be the polydisc given as the Cartesian product of n open discs D1, ..., Dn: $D=\prod _{i=1}^{n}D_{i}.$ Suppose that f is a holomorphic function in D continuous on the closure of D. Then $f(\zeta )={\frac {1}{\left(2\pi i\right)^{n}}}\int \cdots \iint _{\partial D_{1}\times \cdots \times \partial D_{n}}{\frac {f(z_{1},\ldots ,z_{n})}{(z_{1}-\zeta _{1})\cdots (z_{n}-\zeta _{n})}}\,dz_{1}\cdots dz_{n}$ where ζ = (ζ1,...,ζn) ∈ D. In real algebras The Cauchy integral formula is generalizable to real vector spaces of two or more dimensions. The insight into this property comes from geometric algebra, where objects beyond scalars and vectors (such as planar bivectors and volumetric trivectors) are considered, and a proper generalization of Stokes' theorem. Geometric calculus defines a derivative operator ∇ = êi ∂i under its geometric product — that is, for a k-vector field ψ(r), the derivative ∇ψ generally contains terms of grade k + 1 and k − 1. For example, a vector field (k = 1) generally has in its derivative a scalar part, the divergence (k = 0), and a bivector part, the curl (k = 2). This particular derivative operator has a Green's function: $G\left(\mathbf {r} ,\mathbf {r} '\right)={\frac {1}{S_{n}}}{\frac {\mathbf {r} -\mathbf {r} '}{\left\|\mathbf {r} -\mathbf {r} '\right\|^{n}}}$ where Sn is the surface area of a unit n-ball in the space (that is, S2 = 2π, the circumference of a circle with radius 1, and S3 = 4π, the surface area of a sphere with radius 1). By definition of a Green's function, $\nabla G\left(\mathbf {r} ,\mathbf {r} '\right)=\delta \left(\mathbf {r} -\mathbf {r} '\right).$ It is this useful property that can be used, in conjunction with the generalized Stokes theorem: $\oint _{\partial V}d\mathbf {S} \;f(\mathbf {r} )=\int _{V}d\mathbf {V} \;\nabla f(\mathbf {r} )$ where, for an n-dimensional vector space, dS is an (n − 1)-vector and dV is an n-vector. The function f(r) can, in principle, be composed of any combination of multivectors. The proof of Cauchy's integral theorem for higher dimensional spaces relies on the using the generalized Stokes theorem on the quantity G(r, r′) f(r′) and use of the product rule: $\oint _{\partial V'}G\left(\mathbf {r} ,\mathbf {r} '\right)\;d\mathbf {S} '\;f\left(\mathbf {r} '\right)=\int _{V}\left(\left[\nabla 'G\left(\mathbf {r} ,\mathbf {r} '\right)\right]f\left(\mathbf {r} '\right)+G\left(\mathbf {r} ,\mathbf {r} '\right)\nabla 'f\left(\mathbf {r} '\right)\right)\;d\mathbf {V} $ When ∇f = 0, f(r) is called a monogenic function, the generalization of holomorphic functions to higher-dimensional spaces — indeed, it can be shown that the Cauchy–Riemann condition is just the two-dimensional expression of the monogenic condition. When that condition is met, the second term in the right-hand integral vanishes, leaving only $\oint _{\partial V'}G\left(\mathbf {r} ,\mathbf {r} '\right)\;d\mathbf {S} '\;f\left(\mathbf {r} '\right)=\int _{V}\left[\nabla 'G\left(\mathbf {r} ,\mathbf {r} '\right)\right]f\left(\mathbf {r} '\right)=-\int _{V}\delta \left(\mathbf {r} -\mathbf {r} '\right)f\left(\mathbf {r} '\right)\;d\mathbf {V} =-i_{n}f(\mathbf {r} )$ where in is that algebra's unit n-vector, the pseudoscalar. The result is $f(\mathbf {r} )=-{\frac {1}{i_{n}}}\oint _{\partial V}G\left(\mathbf {r} ,\mathbf {r} '\right)\;d\mathbf {S} \;f\left(\mathbf {r} '\right)=-{\frac {1}{i_{n}}}\oint _{\partial V}{\frac {\mathbf {r} -\mathbf {r} '}{S_{n}\left\|\mathbf {r} -\mathbf {r} '\right\|^{n}}}\;d\mathbf {S} \;f\left(\mathbf {r} '\right)$ Thus, as in the two-dimensional (complex analysis) case, the value of an analytic (monogenic) function at a point can be found by an integral over the surface surrounding the point, and this is valid not only for scalar functions but vector and general multivector functions as well. See also • Cauchy–Riemann equations • Methods of contour integration • Nachbin's theorem • Morera's theorem • Mittag-Leffler's theorem • Green's function generalizes this idea to the non-linear setup • Schwarz integral formula • Parseval–Gutzmer formula • Bochner–Martinelli formula Notes 1. Titchmarsh 1939, p. 84 2. "Gauss's Mean-Value Theorem". Wolfram Alpha Site. 3. Pompeiu 1905 4. "§2. Complex 2-Forms: Cauchy-Pompeiu's Formula" (PDF). 5. Hörmander 1966, Theorem 1.2.1 6. Hörmander 1983, pp. 63, 81 7. Hörmander 1983, pp. 62–63 8. Hörmander 1966, Theorem 2.2.1 References • Ahlfors, Lars (1979). Complex analysis (3rd ed.). McGraw Hill. ISBN 978-0-07-000657-7. • Pompeiu, D. (1905). "Sur la continuité des fonctions de variables complexes" (PDF). Annales de la Faculté des Sciences de Toulouse. Série 2. 7 (3): 265–315. • Titchmarsh, E. C. (1939). Theory of functions (2nd ed.). Oxford University Press. • Hörmander, Lars (1966). An Introduction to Complex Analysis in Several Variables. Van Nostrand. • Hörmander, Lars (1983). The Analysis of Linear Partial Differential Operators I. Springer. ISBN 3-540-12104-8. • Doran, Chris; Lasenby, Anthony (2003). Geometric Algebra for Physicists. Cambridge University Press. ISBN 978-0-521-71595-9. External links • "Cauchy integral", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Weisstein, Eric W. "Cauchy Integral Formula". MathWorld.	Wikipedia
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Grotberg Daisuke Yoshino ORCID: orcid.org/0000-0002-2948-12101,2,3,4, Kenichi Funamoto ORCID: orcid.org/0000-0002-0703-09101,3, Kakeru Sato5 nAff8, Kenry6, Masaaki Sato1 & Chwee Teck Lim ORCID: orcid.org/0000-0003-4019-97822,6,7 Communications Biology volume 3, Article number: 152 (2020) Cite this article Cell signalling Vascular tubulogenesis is tightly linked with physiological and pathological events in the living body. Endothelial cells (ECs), which are constantly exposed to hemodynamic forces, play a key role in tubulogenesis. Hydrostatic pressure in particular has been shown to elicit biophysical and biochemical responses leading to EC-mediated tubulogenesis. However, the relationship between tubulogenesis and hydrostatic pressure remains to be elucidated. Here, we propose a specific mechanism through which hydrostatic pressure promotes tubulogenesis. We show that pressure exposure transiently activates the Ras/extracellular signal-regulated kinase (ERK) pathway in ECs, inducing endothelial tubulogenic responses. Water efflux through aquaporin 1 and activation of protein kinase C via specific G protein–coupled receptors are essential to the pressure-induced transient activation of the Ras/ERK pathway. Our approach could provide a basis for elucidating the mechanopathology of tubulogenesis-related diseases and the development of mechanotherapies for improving human health. Blood vessels play important roles in the maintenance of homeostasis (maintenance of a normal physiologic state) because they are essential for supplying oxygen and nutrients to every part of the body. Pathologically, blood vessels can also play an important role in the breakdown of homeostasis such as delivering nourishment to tumors, as is the case for certain cancers1. Hence, the formation of blood vessel/capillary networks is tightly linked with both normal physiology and pathology. Vascular tubulogenesis, which is central to the process through which these networks are formed, thus supports developmental processes2 as well as physiologic or pathologic growth of tissues1,3,4. This tubulogenic process is typically regulated by various responses of vascular endothelial cells (ECs), including adhesion, migration, and proliferation5,6. These responses, in turn, are elicited by hemodynamic stimuli generated in vivo in the circulatory system, such as cyclic stretching of tissues7, fluid shear stress8,9, and hydrostatic pressure10. Hydrostatic pressure has recently attracted considerable attention as a key stimulus that enhances tubulogenesis mediated by ECs because hydrostatic pressure is a stimulus that can be easily regulated through exercise11,12 and blood pressure medication. Depending on the local conditions, magnitude, and mode by which it is exerted, hydrostatic pressure can enhance the three-dimensional migration, cell cycle progression, endothelial proliferation, sprouting angiogenesis, and apoptosis of ECs13,14,15,16. Nevertheless, knowledge regarding how cells respond to hydrostatic pressure remains limited in terms of the mechanism through which pressure promotes angiogenesis during the maintenance and breakdown of homeostasis. Here, we show the mechanism through which hydrostatic pressure promotes endothelial tubulogenesis. We found that pressure-induced transient activation of the Ras/extracellular signal–regulated kinase (ERK) pathway plays a crucial role in the promotion of tubulogenesis. We also confirmed that pressure-induced transient activation of the Ras/ERK pathway requires water efflux through aquaporin 1 (AQP1) and activation of protein kinase C (PKC) via a specific G protein–coupled receptor (GPCR). Hydrostatic pressure promotes endothelial tube formation We first examined the effect of hydrostatic pressure, mimicking the average increase in blood pressure (+50 mmHg) during exercise11,12, on the tubulogenic response of ECs. Human umbilical vein endothelial cells (HUVECs) formed tube-like structures after a 3-h pressure exposure and 13-h incubation, as observed under phase-contrast microscopy (Fig. 1a). In comparison with the control condition (i.e., 0 mmHg pressure), exposure to the hydrostatic pressure (+50 mmHg) promoted the formation of tube-like structures by HUVECs, with structures exhibiting significantly longer total length and more branch points in a 1-mm2 area (Fig. 1b, c). To evaluate the maturation of tube-like structures formed by HUVECs, we added 10 µg/mL of FITC-dextran into the collagen gel in which the tube-like structures were formed. FITC-dextran penetrated into the lumen of the tube-like structure after a 2-h incubation, as observed under differential interference contrast (DIC) and confocal laser-scanning microscopy (Fig. 1d). The relative fluorescence intensity of FITC-dextran dropped across the boundary face of the tube-like structures, approaching approximately half of the outside intensity at the center of the tubes under both control and pressure conditions (Fig. 1e). The ratio of the average fluorescence intensities outside (Iout) and inside (Iin) the tube-like structures was not affected by exposure to pressure (Fig. 1f). Hydrostatic pressure exposure did not, therefore, affect the maturation of the tube-like structures formed by HUVECs. To further examine the effects of hydrostatic pressure on tube-like structure formation, we analyzed the expression of cell-cell junction proteins in the tube-like structures. Tight junctions (ZO-1) and adherens junctions (VE-cadherin) formed in the tube-like structures after a 3-h pressure exposure and 13-h incubation, as observed under confocal laser-scanning microscopy (Fig. 1g). Pressure exposure did not affect the expression of VE-cadherin, whereas the expression of ZO-1 increased with marginal significance under the pressure condition (Fig. 1h and Supplementary Fig. 17). Hydrostatic pressure, therefore, marginally increases the robustness of the tube-like structures formed by HUVECs. Fig. 1: Hydrostatic pressure promotes endothelial tubulogenesis. a Endothelial tube formation under the pressure condition. ECs were embedded within a collagen gel sandwich and exposed to pressure for 3 h or incubated in the control condition for 13 h. Scale bars, 100 µm. b Quantified total length of tube-like structure and c number of its branch points in a 1-mm2 area. Whiskers represent the 10th and 90th percentiles, the box represents the 25th to 75th percentiles, the central line depicts the median, and the square inside each box indicates the average value. Each value was obtained from 30 images, which were captured from six independently repeated experiments (n = 30 images). d Observation of diffusion of 10-kDa FITC-dextran across the boundary face of a tube-like structure. Representative DIC and fluorescent images 2 h after addition of FITC-dextran. Scale bars, 100 µm. e The line profiles of the normalized fluorescence intensity in 62 locations (control) or 60 locations (pressure condition) across the boundary face of the tube-like structures from six experiments. f Ratio of fluorescence intensity between the inside and the outside of the tube-like structures shown as box-and-whisker plots, as defined in Fig. 1a (n = 62 [control] or 60 [pressure condition] locations). g, h Tubular robustness of cell-cell junctions formed under the pressure condition. g Representative fluorescence images of ZO-1 and VE-cadherin in the tube-liked structures. Scale bars, 100 µm. h Relative expression levels of ZO-1 and VE-cadherin in the tube-like structures (right) (mean + SEM, n = 7 experiments). The broad band apparent at approximately 65 kDa indicates bovine serum albumin derived from FBS contained in the experimental medium. i Proportions of ECs in the S and G2/M phases under the pressure condition (mean + SEM, n = 3 experiments). j Growth curve of ECs cultured sparsely after a 3-h exposure to hydrostatic pressure (mean ± SEM, n = 3 experiments). p < 0.1, p < 0.01, NS: no significant difference (Welch's t test; b, c, f, h). In highly confluent HUVECs, hydrostatic pressure advanced the cell cycle (Fig. 1i). Such premature cell cycle progression under positive pressure has been observed in other studies14,17 as well. The percentages of cells in the S or G2/M phases in the static culture and control conditions were similar (20% or less of total cells), with most cells remaining in the G1 phase. In contrast, after cells were exposed to pressure, the percentage of cells in the S phase peaked at 3 h and then decreased. The percentage of cells in the G2/M phases also reached a maximum (about 35%) at 6 h. The premature progression of the cell cycle is hypothesized to begin just after exposure to pressure because of the duration of the S phase18. This hypothesis is supported by our finding from HUVECs demonstrating significant nuclear translocation of cyclin D1 (which regulates the G1 restriction point19) following a 1-h pressure exposure (Supplementary Fig. 1). However, the effect of hydrostatic pressure on cell cycle progression lasted only 3 to 6 h after pressure exposure because the HUVECs adapted to the applied hydrostatic pressure stimulation of between 3 to 6 h20. Even if HUVECs were cultured under sparsely distributed conditions, their proliferation was enhanced by hydrostatic pressure. The application of pressure resulted in a 160% increase in the number of cells in the first 24 h of incubation, followed by relatively slow growth rates of 36% and 22% in the second and third 24 h of incubation, respectively (Fig. 1j). These data thus demonstrate that exposure to hydrostatic pressure transiently promotes endothelial tubulogenic responses of ECs, including proliferation. Pressure-induced Ras-ERK signaling leads to tube formation We then investigated the signaling pathway through which hydrostatic pressure induces angiogenesis, focusing on activation of the Ras/ERK pathway, which is strongly correlated with the EC proliferation associated with angiogenesis21. Hydrostatic pressure caused transient activation of ERK1/2 in HUVECs, with phosphorylation peaking within 5 min and then gradually returning to baseline level after 30 min (Fig. 2a and Supplementary Fig. 17). After a 5-min pressure exposure, the cells exhibited higher mean fluorescence intensity, indicating an approximately 1.5- and 2-fold increase in ERK1/2 activation in the cytoplasm and nucleus, respectively (Supplementary Fig. 2). The cells also exhibited a higher nuclear/cytoplasm ratio of activated ERK1/2, as compared with control. Hydrostatic pressure also induced phosphorylation of mitogen-activated protein kinase 1/2 (MEK1/2) (Fig. 2b and Supplementary Fig. 17) and clearly increased association of activated Ras and Raf-1 (Fig. 2c and Supplementary Fig. 17), as preliminary steps to ERK activation. Ras protein, a small guanosine triphosphatase (GTPase), functions as a master regulator of cell signaling22. Ras induces activation of MEK and its downstream ERK via interaction with the Ras effector Raf-1 (Ras/ERK pathway;22,23). Hydrostatic pressure, therefore, induces activation of the Ras/ERK pathway. Fig. 2: The Ras/ERK pathway is essential for hydrostatic pressure-induced endothelial tube formation. a ERK1/2 activation in HUVECs exposed to hydrostatic pressure, expressed as the relative intensity of p-ERK1/2 to that of ERK1/2 (mean ± SEM, n = 8 experiments). b MEK1/2 activation in ECs after a 5-min pressure exposure, expressed as the relative intensity of p-MEK1/2 to that of MEK1/2 (mean + SEM, n = 5 experiments). c Ras activity (RBD pull-down) in ECs after a 5-min pressure exposure (n = 3 experiments). d Growth curve of ECs cultured sparsely after a 3-h exposure to hydrostatic pressure in the presence of an MEK inhibitor (PD0325901) (mean ± SEM, n = 3 experiments). e Endothelial tube formation under the pressure condition in the presence of an MEK inhibitor (PD0325901). Scale bars, 100 µm. f, g Quantified total length and number of tube-like structure branch points in a 1-mm2 area. Each value is shown as a box-and-whisker plot, obtained from 25 images in five independently repeated experiments (n = 25 images). h, i Tubular robustness of cell-cell junctions formed under the pressure condition in the presence of an MEK inhibitor (PD0325901). h Representative fluorescence images of ZO-1 and VE-cadherin in the tube-liked structures. Scale bars, 100 µm. i Relative expression levels of ZO-1 and VE-cadherin in the tube-like structures (mean + SEM, n = 6 experiments). The broad band apparent at approximately 65 kDa indicates bovine serum albumin derived from FBS contained in the experimental medium. p < 0.05 (Welch's t test; a, b). NS: no significant difference (Tukey-Kramer test; f, g, i). To further examine the relationship between activation of the Ras/ERK pathway and pressure-promoted tubulogenesis, we evaluated the EC proliferation and the formation of tube-like structures when ERK activation was inhibited using an MEK inhibitor. The pressure-enhanced proliferation was not observed under the inhibition of ERK activation (Fig. 2d). Although HUVECs formed tube-like structures in the presence of the inhibitor, a large proportion of the formed tube network exhibited short segments in both control and pressure-exposed cells (Fig. 2e). In addition, no significant differences were observed in the total length or number of branch points of the tube-like structures in a 1-mm2 area (Fig. 2f, g). Inhibition of ERK activation did not affect the maturation of the tube-like structures (Supplementary Fig. 3). The expression of ZO-1, which was marginally enhanced by pressure exposure, was not observed in cells treated with the MEK inhibitor (Fig. 2h, i, and Supplementary Fig. 17). These results suggest that hydrostatic pressure promotes endothelial tubulogenesis via the Ras/ERK pathway. Pressure-activated PKC via GPCRs drives Ras-ERK signaling We then sought to determine what drives the hydrostatic pressure–induced activation of the Ras/ERK pathway. Although the vascular endothelial growth factor receptor 2 (VEGFR2)/phospholipase C (PLC) pathway is known to regulate Ras/ERK signaling24, hydrostatic pressure did not induce tyrosine phosphorylation of VEGFR2 in our study (Supplementary Fig. 4 and Supplementary Fig. 17). However, PKC, an activator of the Ras/ERK pathway25, was activated in HUVECs exposed to hydrostatic pressure, as observed by its relocation from the cytoplasm to the cell membrane (Fig. 3a). Three major isoforms of PKC have been identified (i.e., conventional, novel, and atypical), with activation requiring calcium ion (Ca2+) or diacylglycerol (DAG), depending on the isoform26. Exposure of HUVECs to hydrostatic pressure did not induce noticeable differences in the intracellular Ca2+ concentration relative to control, although a slight decrease in membrane potential was observed (Supplementary Fig. 5). The concentration of phosphatidylinositol 4,5-bisphosphate (PI[4,5]P2), which is hydrolyzed to inositol trisphosphate and DAG by PLC27, tended to decrease after exposure to hydrostatic pressure (Fig. 3b and Supplementary Fig. 17), with concomitant activation of PKC and the Ras/ERK pathway. These experimental data were supported by the following observations: (i) in the presence of inhibitors of PLC (Supplementary Fig. 6 and Supplementary Fig. 17) or PKC (Fig. 3c and Supplementary Fig. 17), ERK was not activated even in pressure-exposed cells; and (ii) in the presence of a specific inhibitor of PKCα/β (Supplementary Fig. 7 and Supplementary Fig. 17), there was no difference in the level of ERK activation between control and pressure-exposed cells, although the level in pressure-exposed cells was still not significantly different in comparison with that in pressure-exposed cells not treated with the inhibitor. Fig. 3: Activation of PKC via specific GPCRs drives hydrostatic pressure–induced activation of the Ras/ERK pathway in HUVECs. a Membrane translocation of activated PKC after a 5-min exposure to hydrostatic pressure, with quantified localization in 100 cells in four independently repeated experiments (n = 100 cells). Scale bars, 50 µm. b PI(4,5)P2 expression level after a 5-min pressure exposure (n = 4 experiments). c ERK1/2 activation after a 5-min pressure exposure in the presence of a PKC inhibitor (Gö6983) (n = 6 experiments). d Release of the Gq alpha subunit from the membrane to the cytoplasm after a 5-min pressure exposure (n = 10 experiments). ERK1/2 activation after a 5-min pressure exposure in the presence of e a Gq inhibitor (YM-254890) (n = 8 experiments), f an α1-AR antagonist (prazosin) (n = 6 experiments), or g an SR-2A antagonist (pizotifen) (n = 6 experiments). All data are presented as the mean ± SEM. p < 0.1, p < 0.05 (Welch's t-test; b, d). p < 0.05, p < 0.01, NS: no significant difference (Tukey-Kramer test; c, e–g). We then confirmed the pressure-associated activation of Gq protein (i.e., release of the Gq alpha subunit from the cell membrane to the cytoplasm), which is known to activate PLC28 (Fig. 3d and Supplementary Fig. 17). Inhibition of Gq protein activation prevented pressure-induced ERK activation (Fig. 3e and Supplementary Fig. 17). The activation of Gq protein is regulated by GPCRs. We investigated the relationship between pressure-induced ERK activation and four GPCRs to which Gq protein binds (i.e., α1-adrenergic receptor [α1-AR], angiotensin II type I receptor [AT1-R], histamine H1 receptor [H1-R], and serotonin receptor type 2A [SR-2A]) and that are known to be expressed in HUVECs (Supplementary Fig. 8 and Supplementary Fig. 17). Inhibition of GPCRs using antagonists for α1-AR and SR-2A prevented pressure-induced ERK activation (Fig. 3f, g, Supplementary Fig. 9, and Supplementary Fig. 17), suggesting that activation of PKC via α1-AR and SR-2A drives the hydrostatic pressure–induced activation of the Ras/ERK pathway. This notion is supported by the findings that inhibition of G protein activation prevented pressure-induced translocation of PKC (Supplementary Fig. 10). To further examine the relationship between activation of PKC via specific GPCRs and pressure-promoted tubulogenesis, we evaluated the formation of tube-like structures when the activations of PKC, G protein, and GPCR were inhibited using each inhibitor. Inhibition of their activation prevent pressure-induced increases in the length of the tube-like structures and the number of their branch points (Supplementary Fig. 11). These results suggest that hydrostatic pressure promotes endothelial tubulogenesis via the Ras/ERK pathway driven by the activation of PKC, G protein, and specific GPCRs. Aquaporin-mediated water efflux activates Ras-ERK signaling Finally, we investigated how HUVECs sense hydrostatic pressure and convert it to a biochemical signal that leads to the activation of PKC via GPCRs. We hypothesized that pressure causes an efflux of water from cells, based on a kinetic model of water29 in which flux is defined by the difference between hydrostatic and osmotic pressures across the cell membrane. This hypothesis is supported by our findings indicating cell contraction (Fig. 4a, b, and Supplementary Movie 1 and 2) and the efflux of a fluorescent Ca2+ indicator (Supplementary Fig. 12a and 12b) under the pressure condition. Similar cell contraction is reportedly caused by hydrostatic pressure30. AQP1 is a water channel molecule that enhances membrane water permeability31. Although translocation of AQP1 to the cell membrane is reportedly induced by osmotic stimulation32, our results did not demonstrate this (Supplementary Fig. 13 and Supplementary Fig. 17). We therefore examined the inhibition of water flux through AQP1. Following inhibition of AQP1 using mercuric (II) chloride (HgCl2)33, no activation of the Ras/ERK pathway was observed, even in cells exposed to pressure (Fig. 4c–e, and Supplementary Fig. 17). In addition, no pressure-induced PKC activation was observed in cells in which water flux was inhibited (Fig. 4f). Cells, in which water flux was inhibited, exhibited no contraction (Supplementary Fig. 14 and Supplementary Movies 3, 4, 5, and 6) and no efflux of the fluorescent Ca2+ indicator, and simultaneously, pressure exposure did not induce an increase in the intracellular Ca2+ ion concentration (Supplementary Fig. 12c, 12d, 12e, and 12f). Based on these results, we conclude that AQP1-mediated water efflux plays a key role in the hydrostatic pressure–induced activation of PKC via α1-AR and SR-2A and activation of the Ras/ERK pathway that ultimately leads to tubulogenesis. These findings support the hypothesis that water efflux via AQP1 converts hydrostatic pressure to biochemical signals that ultimately activate PKC through GPCRs. Fig. 4: AQP1-mediated water efflux plays a key role in hydrostatic pressure–induced activation of the Ras/ERK pathway in HUVECs. a Time sequence phase-contrast images depicting cell contraction and b changes in relative cell area under the pressure condition. Each value was obtained from 30 cells, which were captured in five independently repeated experiments (n = 30 cells). Scale bars, 50 µm. c ERK1/2 activation (n = 10 experiments), d MEK1/2 activation (n = 10 experiments), and e Ras activity (n = 3 experiments) after a 5-min exposure to hydrostatic pressure with inhibition of AQP1-mediated water influx and efflux using HgCl2. f Membrane translocation of activated PKC in HUVECs after a 5-min pressure exposure and quantified localization in 100 cells in four independently repeated experiments (n = 100 cells) with inhibition of AQP1-mediated water flux using HgCl2. Scale bars, 50 µm. All data are presented as the mean ± SEM. p < 0.01 (Welch's t test; b). p < 0.1, p < 0.05, *p < 0.01, NS: no significant difference (Tukey-Kramer test; c, d). In this study, we elucidated a part of the mechanism by which hydrostatic pressure promotes endothelial tube formation. This finding provides a potential to promote endothelial tubulogenesis by controlling hydrostatic pressure in vivo. Our results answer in part the long-standing question as to how ECs sense hydrostatic pressure and convert it to intracellular biochemical signals (Supplementary Fig. 15). Although we could not determine the mechanism by which AQP1-mediated water efflux activates GPCRs, we believe that contraction of the cell membrane resulting from the efflux of water is important in GPCR activation. We expect that in addition to promoting tubulogenesis, hydrostatic pressure also plays a crucial role in the pathology of a variety of diseases (mechanopathology). By better understanding the effects of hydrostatic pressure, we could ultimately develop methods to manipulate it and thus improve human health (mechanotherapy). Pressure-enhanced endothelial proliferation leading to tubulogenic responses was confirmed in our previous studies17,30. Hydrostatic pressure induces the forcible progression of the stagnant cell cycle in ECs via contact inhibition without morphologic changes such as elongation or altered orientation17. We also demonstrated the importance of actomyosin contractility on cell contraction induced by hydrostatic pressure30. However, our previous studies did not clarify the detailed mechanisms linking these cellular responses to endothelial tubulogenesis (i.e., pressure-induced signal transduction leading to tubulogenesis). Sustained pressure reportedly promotes sprout angiogenesis from spheroids composed of bovine aorta ECs16. Pressure-sensitive upregulation of VEGF-C and VEGFR3 expression plays a critical role in this sprout angiogenesis in the presence of growth factors such as fibroblast growth factor (FGF) or VEGF. Notably, in the present study, hydrostatic pressure promoted tubulogenic responses even in the absence of FGF and VEGF. Pressure-promoted endothelial tube formation and pressure-induced signal transduction, which were demonstrated in the present study, differ from angiogenesis induced via the commonly known VEGFR pathway24,34. The elucidated mechanism by which hydrostatic pressure promotes endothelial tube formation is based on tube formation reproduced by cultured HUVECs in vitro. Given that tumor angiogenesis is regulated by tumor interstitial fluid pressure35,36 and sprouting angiogenesis is controlled by vascular internal pressure37 in vivo, endothelial tubulogenesis can be promoted by pressure in vivo via the elucidated mechanism. However, some details of the mechanism of pressure-promoted tubulogenesis remain unclear, as we adopted artificial conditions in the present study, such as the use of fetal bovine serum (FBS)-free medium and only one pressure condition. Additional investigation regarding potential side effects of the inhibitors is also needed, as these inhibitors interact with a variety of cellular molecules, even though we examined their concentration and incubation time with regard to cytotoxicity and overreaction with target molecules. A few inhibitors suppressed both ERK1/2 phosphorylation and activity. Further in vitro and in vivo studies are therefore needed in order to address these issues and fully elucidate the mechanism by which hydrostatic pressure promotes endothelial tubulogenesis. Chemicals and antibodies All chemicals used as inhibitors and antagonists for target proteins are indicated in Supplementary Table 1. Primary and secondary antibodies used in this study are described in Supplementary Tables 2 and 3. HUVECs (lot nos. 2818 [black donor] and 2840 [Caucasian donor], 200–05n, Cell Applications, San Diego, CA, USA) were cultured in Medium 199 (M199; 31100–035, Gibco, Thermo Fisher Scientific, Waltham, MA, USA) containing 20% heat-inactivated FBS (12483–020, Gibco or 04–001–1 A, Biological Industries, Beit-Haemek, Israel), 10 µg/L human basic fibroblast growth factor (bFGF; GF-030–3, Austral Biologicals, San Ramon, CA, USA), and 1% penicillin/streptomycin (P/S; 15140–122, Gibco). HUVECs from the fourth to ninth passages were used for experiments in this study. The experiments were conducted using three types of experimental medium (EM): M199 containing 10% heat-inactivated FBS and 1% P/S (EM1), FBS-free M199 (EM2), and a FBS-free M199 with Hank's salts (M0393, Sigma-Aldrich, St. Louis, MO, USA) (EM3). Exposure to hydrostatic pressure HUVECs cultured in dishes were exposed to hydrostatic pressure using a system reported in our previous work17. The system device was filled with EM, and pressure was the applied to ECs by compressing the volume of the EM. The system was maintained at 37 °C in a CO2-supplied incubator. Cells were exposed to a hydrostatic pressure of 0 (control) or +50 mmHg (pressure condition). The pressure value was set up in accordance with the average increase in blood pressure (+50 mmHg) during exercise11,12. For imaging living cells, HUVECs were exposed to hydrostatic pressure (+50 mmHg) using a custom-made hydrostatic pressure microscopy system (Supplementary Fig. 16) consisting of a cell culture dish, polycarbonate pressure chamber, silicone gasket, O-ring, quartz glass, two ball valves, a thermostatic chamber, syringe pump, and wide-field fluorescence microscope (EVOS FL Cell Imaging System, Thermo Fisher Scientific) or confocal laser-scanning microscope (LSM800, Carl Zeiss, Oberkochen, Germany). This system allows for observations using both epi-fluorescence and transmitted light. Tube formation assay Tube formation assays were performed with reference to the study by Deroanne et al.38, with slight modifications. Collagen gels (300 µL each) were formed on 35-mm diameter glass-based dishes (3910–035, AGC Techno Glass, Shizuoka, Japan) by mixing ice-cold collagen solution (4.0 mg/mL; 10× M199, H2O, native collagen [IAC-50, KOKEN, Tokyo, Japan], 10 mM NaHCO3, 10 mM HEPES-NaOH, pH 7.5) and incubating for 30 min at 37 °C. HUVECs were seeded on the gels at a density of 1.2×105 cells/cm2 and incubated in EM1 for 2 h to facilitate spreading. When cells reached 100% confluency, the EM1 was then removed and the HUVECs were covered with overlaying collagen gel (200 µL). After gelation for 15 min at 37 °C, the collagen gel layers were placed inside the pressure exposure system, and the cells between the layers were exposed to pressure in EM1 for 3 h. The cells were then removed from the system and incubated in a CO2 incubator for 13 h. After incubation, the cells were fixed in 4% paraformaldehyde phosphate buffer solution (PFA; 163–20145, Wako Pure Chemical Industries, Osaka, Japan) for 30 min at room temperature. For the inhibition study, inhibitor was added to EM1 and the cells incubated for 30 min before overlaying of the collagen gel. Tube-like structures formed by HUVECs were observed using an inverted phase-contrast microscope (Ti-U, Nikon, Tokyo, Japan) or a wide-field fluorescence microscope (EVOS FL Auto 2 Imaging System, Thermo Fisher Scientific). Tube maturation and robustness assays Maturation of the tube-like structures formed by HUVECs was monitored using FITC-dextran (10 kDa, F0918, Tokyo Chemical Industry, Tokyo, Japan). Tube-like structures in collagen gel were first incubated in EM1 containing 10 µg/mL FITC-dextran for 2 h, which was sufficient time to allow diffusion into the gel and reaching of steady state39. After incubation, images of horizontal sections of the tube-like structures were captured using DIC and confocal laser-scanning microscopy (LSM800, Carl Zeiss). Focusing on cell-cell junction proteins, the robustness of the tube-like structures was evaluated using immunofluorescence staining and immunoblotting. For immunofluorescence staining, the formed tube-like structures were fixed with PFA for 30 min, followed by staining using primary and secondary antibodies. A whole-cell lysate was obtained by collecting the supernatant after washing with ice-cold phosphate-buffered saline (PBS; 05913, Nissui Pharmaceutical, Tokyo, Japan), picking up the whole set of collagen gels including the tube-like structures using 4× Laemmli sample buffer (161–0747, Bio-Rad Laboratories, Hercules, CA, USA), homogenizing by vigorous shaking, and centrifugation at 21,500g for 15 min. Dithiothreitol (DTT; 161–0611, Bio-Rad Laboratories) was added to the collected whole-cell lysates to a final concentration of 20 mM, and the lysates were then boiled for 5 min. The whole-cell lysates were analyzed by SDS-PAGE followed by immunoblotting to detect cell-cell junction proteins (i.e., ZO-1 and VE-cadherin). Cell cycle analysis HUVECs were cultured in 60-mm diameter plastic dishes (MS-11600, Sumitomo Bakelite, Tokyo, Japan) pre-coated with 0.1% bovine gelatin solution (G9391, Sigma-Aldrich). After reaching high confluence (100%), the HUVECs were washed twice and incubated with EM1 for 3 h to wash out bFGF. The cells were then exposed to hydrostatic pressure for 3, 6, 12, or 24 h, harvested from the dish using 0.05% trypsin-EDTA (25300–054, Gibco), and centrifuged for 5 min at 185g after inactivation of the trypsin-EDTA using EM1. The collected cells were then washed with PBS and fixed in 70% ice-cold ethanol. After another PBS wash, the cell density was adjusted to 500 cells/µL. Nuclear DNA was stained using Guava Cell Cycle reagent (4500–0220, Merck Millipore, Darmstadt, Germany) for 30 min. The fluorescence intensity of 5000 cells was measured, and the percentage of HUVECs in each phase of the cell cycle was determined using flow cytometry (Guava easyCyte 6HT, Merck Millipore). A total of 8 × 104 HUVECs were seeded in a 60-mm diameter plastic dish coated with 0.1% gelatin. After incubation for 1 h, the cells were exposed to pressure in EM1 for 3 h, then incubated in a CO2 incubator for 24, 48, or 72 h, after which the cells were harvested from the dish using 0.05% trypsin-EDTA and centrifuged for 5 min at 1000 rpm after inactivation of the trypsin-EDTA with EM1. The cells were resuspended in EM1 (200 µL) and stained with Guava ViaCount reagent (4000–040, Merck Millipore) for 10 min or trypan blue solution (15250–061, Gibco). The number of live cells was then determined using flow cytometry or a hemocytometer (Burker-Turk). Protein activation assay HUVECs were cultured in a 35-mm diameter glass-bottom dish, a 35-mm diameter plastic dish (3000–035, AGC Techno Glass), or a 60-mm diameter plastic dish, each pre-coated with 0.1% bovine gelatin. Highly confluent HUVECs were washed twice with FBS-free EM2 and incubated in the same medium for 3 h to wash out bFGF and starve the cells. Cells were then exposed to pressure for 5, 15, and 30 min or 1 h, collected as described above, and then examined by immunoblotting or immunofluorescence staining. Inhibitors and antagonists were introduced into the EM2 after 3 h of FBS starvation, and the cells were then incubated for the times indicated in Supplementary Table 1 before exposure to pressure. After exposure to hydrostatic pressure, HUVECs were fixed with 4% PFA at room temperature or ice-cold methanol at −20 °C in accordance with the data sheets for the antibodies used. The cells were permeabilized with 0.1 or 0.3% TritonX-100 in PBS and incubated in 1% Block Ace (BA; UKB40, DS Pharma Biomedical, Osaka, Japan) in PBS to prevent nonspecific antibody adsorption. The cells were then stained using the primary and secondary antibodies diluted in 1% BA in PBS and PBS, respectively, at predefined concentrations (Supplementary Tables 2 and 3). Cell nuclei were stained using 4ʹ,6-diamidino-2-phenylindole (DAPI; D1306, Thermo Fisher Scientific). Stained HUVECs were observed using a wide-field fluorescence microscope (Axio Observer D1, Carl Zeiss) or an inverted confocal laser-scanning microscope (LSM800, Carl Zeiss). Cellular fractionation Cytosolic and crude cell membrane fractions were prepared according to the following protocol. Cells were washed twice with ice-cold PBS, scraped from the surface, transferred to microtubes with ice-cold hypotonic buffer (7.5 mM Na2HPO4, 1 mM EDTA, protease inhibitor cocktail [P8340, Sigma-Aldrich]), and homogenized by passage through a 25 G needle (NN-2516R, Terumo, Tokyo, Japan). The cytosolic fraction was obtained by collecting the supernatant after two consecutive centrifugations (500g at 4 °C for 5 min followed by 20,000g at 4 °C for 30 min). Proteins were recovered from the cytosolic fraction in 2× Laemmli sample buffer (161–0737, Bio-Rad Laboratories). The pellet after the second centrifugation was resuspended in modified Laemmli buffer (65 mM Tris-HCl [pH 7.5], 0.1 mM EGTA, 0.1 mM EDTA, 1 mM Na3VO4, 1 mM NaH2PO4, 10% glycerol, 2% SDS, 20 mM DTT, and protease inhibitor cocktail), incubated on ice for 5 min, and homogenized by vigorous shaking. The crude cell membrane fraction was obtained by collecting the supernatant after centrifugation at 21,500g for 10 min. The whole-cell lysate was obtained by collecting the supernatant after the ice-cold PBS washing, scraping the cells using the modified Laemmli buffer, and centrifugation at 21,500g for 10 min. Pull-down assay HUVECs were washed with ice-cold Tris-buffered saline (TBS; 25 mM Tris-HCl [pH 7.5], 150 mM NaCl), lysed using lysis buffer (25 mM Tris-HCl [pH 7.2], 150 mM NaCl, 5 mM MgCl2, 1% NP-40, 5% glycerol, and protease inhibitor cocktail), scraped, and collected in a microtube. After a 5-min incubation on ice, cell debris was removed by centrifugation at 16,000g at 4 °C for 15 min. The pull-down assay was conducted using an Active Ras Pull-Down and Detection kit (16117, Thermo Fisher Scientific) according to the manufacturer's instructions, and proteins were recovered from the resultant immunoprecipitates in 2× SDS sample buffer. Immunoblotting Samples were subjected to SDS-PAGE and then transferred onto an Immun-Blot PVDF membrane (162–0177, Bio-Rad Laboratories). The membrane was blocked with TBS containing 1% BA and 0.05% Tween 20 and then stained using primary and secondary antibodies diluted in TBS containing 1% BA and 0.05% Tween 20 at predefined concentrations (Supplementary Tables 2 and 3). Can Get Signal Immunoreaction Enhancer Solution (NKB-101, Toyobo, Osaka, Japan) was added to the antibody diluent buffer as necessary. The blotted proteins were detected and visualized using Clarity Western ECL Substrate (170–5061, Bio-Rad Laboratories) or an AP Conjugate Substrate kit (170–6432, Bio-Rad Laboratories). Protein loading was monitored using loading control proteins (i.e., α-tubulin, β-actin, and GAPDH). The molecular weight of each protein was determined based on Precision Plus Protein Dual Color Standards (161–0374, Bio-Rad Laboratories). Membranes were stripped of bound antibodies and re-probed with different primary and secondary antibodies. Stripping was accomplished by soaking the membrane in stripping buffer (100 mM β-mercaptoethanol, 50 mM Tris-HCl [pH 6.8], and 2% SDS) at 50 °C for 30 min. Imaging of living cells exposed to hydrostatic pressure HUVECs were grown to high confluence (100%) on 35-mm diameter glass-bottom dishes (3910–035-IN, AGC Techno Glass) coated with 0.1% bovine gelatin in FBS-free EM3 for 3 h before live imaging. In the custom-made hydrostatic pressure microscopy system, the intracellular Ca2+ ion concentration and cellular membrane potential were visualized using Fluo-8, AM (21082, AAT Bioquest, Sunnyvale, CA, USA) and bis(1,3-dibutylbarbituric acid)trimethine oxonol, sodium salt (DiBAC4[3]; D545, Dojindo Molecular Technologies, Kumamoto, Japan), respectively, according to the manufacturers' instructions. Quantification of length and branch-point number of tube-like structures The total length of the tube-like structures was measured by tracing the tube-like structures with the freehand lines tool, and the number of tube-like structure branch points was determined by counting them in phase-contrast images using ImageJ software (US National Institutes of Health) (Figs. 1a–c and 2e–g, and Supplementary Fig. 12). Maturation of tube-like structures The maturation of tube-like structures was analyzed based on diffusion of FITC-dextran from the outside to the inside of the tube-like structures. We first prepared a fluorescent image minus background noise using ZEN software (Carl Zeiss) and stacked this image onto the corresponding DIC image using ImageJ software (Fig. 1d and Supplementary Fig. 4a). Line profiles of fluorescence intensity of FITC-dextran were obtained at a location across the boundary face of the tube-like structures, which was randomly selected on the DIC image. The line width for extracting the line profile was set to 20 pixels. The line profile was extracted from the measurements on a line with perpendicular to the boundary face. The location of the line profile was normalized by its length, and was shown in the range of −0.25 (center of the tube-like structure) to 0.25. The relative fluorescence intensity of FITC-dextran was calculated based on its intensity in the collagen gel where no tube-like structures were present (Fig. 1e and Supplementary Fig. 4b). For evaluation of tube-like structure maturation, the ratio of the averaged fluorescence intensities outside (Iout) and inside (Iin) the tube-like structures was calculated (Fig. 1f and Supplementary Fig. 4c). Nuclear/cytoplasm ratios of cyclin D1 and activated ERK The nuclear/cytoplasm ratio of cyclin D1 or activated ERK was also determined using ImageJ software. Fluorescence in the nucleus was extracted by referring to the captured images of DAPI staining. The averaged fluorescence intensity in the nucleus and cytoplasm in the whole image was measured, and the relative averaged fluorescence intensity was then calculated as the ratio between the intensity of each sample and the averaged value for the entire sample. The relative averaged intensity between the nucleus and cytoplasm was finally determined as the nuclear/cytoplasm ratio (Supplementary Figs. 2 and 3). PKCα localization Line profiles of fluorescence intensity were obtained for quantitative representation of PKCα localization, based on our previous work40, with slight modifications. Captured fluorescence images of PKCα were processed using ZEN Imaging software. The fluorescence intensity was determined over a distance covering the membrane and the cytoplasm on the image of a layer with the maximum intensity of VE-cadherin, which was selected from 20 z-stack images with 0.6-µm intervals. Relative PKCα localization was evaluated with the total amount of one line profile of the fluorescence intensity set to a value of 1 (Figs. 3a, 4f, and Supplementary Fig. 11). Cell area, Ca2+ ion concentration, and membrane potential The cell area, Ca2+ ion concentration, and cellular membrane potential were assessed using ImageJ software. Cell area was measured by tracing the outer periphery of the cell based on the phase-contrast images (Fig. 4a, b, and Supplementary Fig. 15). The intracellular Ca2+ ion concentration and cellular membrane potential were quantified based on the integrated fluorescence intensity, which was obtained from the product of the averaged intensity and the selected cellular area of Fluo-8 and DiBAC4(3) in the cell, respectively, extracted referring to the maximum intensity projection of confocal microscopic images (Supplementary Fig. 6). Quantification of protein expression and phosphorylation The density of protein bands on immunoblots was determined using Image Lab (170–9691, Bio-Rad Laboratories). The relative expression and phosphorylation levels of each protein were calculated with the control condition set to a value of 1 on the same membrane. All values are shown as mean ± standard error (SEM) unless stated otherwise. Each data was obtained from at least three independently repeated experiments (Supplementary Data 1). Statistical significance was calculated using the two-sided Welch's t-test for comparisons of two groups or the Tukey-Kramer test for multiple comparisons, with statistical significance set at p ≤ 0.1 (marginally significant), p ≤ 0.05, and p ≤ 0.01 (significant difference). The effect size of each statistical test was analyzed using the Pearson's correlation coefficient r41, which is defined as follows: $$r = \sqrt {\frac{{t^2}}{{t^2 + df}}}$$ Here, t and df represent the statistics and the degrees of freedom, respectively, and they were obtained from the following equations42,43: $$t = \frac{{\bar X_i - \bar X_j}}{{\sqrt {\frac{{s_i^2}}{{N_i}} + \frac{{s_j^2}}{{N_j}}} }}$$ $$df \approx \frac{{\left( {\frac{{s_i^2}}{{N_i}} + \frac{{s_j^2}}{{N_j}}} \right)^2}}{{\frac{{s_i^4}}{{N_i^2\left( {N_i - 1} \right)}} + \frac{{s_j^4}}{{N_j^2\left( {N_j - 1} \right)}}}}$$ where $\bar X$, s, and N are the mean value, standard deviation, and size of sample, respectively. The exact p-values and the effect size for all statistically tested data are described in Supplementary Data 2. Reporting Summary Further information on research design is available in the Nature Research Reporting Summary linked to this article. 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Biometrika 34, 28–35 (1947). The authors thank R. Koens, Y. Tabata, H. Hirata, K. Kawauchi, M. Nakayama, and Y. Sawada for technical support. The Japan Society for the Promotion Science (Young Researcher Overseas Visits Program for Accelerating Brain Circulation), Japan Science and Technology Agency (Building of Consortia for the Development of Human Resources in Science and Technology), and Mechanobiology Institute at the National University of Singapore are acknowledged for financial support. Kakeru Sato Present address: Tokyo Gas Co., Ltd., 1-5-20 Kaigan, Minato-ku, Tokyo, 105-8527, Japan Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, 6-3 Aramaki-Aoba, Aoba-ku, Sendai, 980-8578, Japan Daisuke Yoshino, Kenichi Funamoto & Masaaki Sato Mechanobiology Institute, National University of Singapore, #10-01 T-Lab, 5A Engineering Drive 1, Singapore, 117411, Singapore Daisuke Yoshino & Chwee Teck Lim Institute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan Daisuke Yoshino & Kenichi Funamoto Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei, Tokyo, 184-8588, Japan Daisuke Yoshino Graduate School of Engineering, Tohoku University, 6-6-01 Aramaki-Aoba, Aoba-ku, Sendai, 980-8579, Japan Department of Biomedical Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore, 117583, Singapore Kenry & Chwee Teck Lim Institute for Health Innovation and Technology (iHealthtech), National University of Singapore, #14-01 MD6, 14 Medical Drive, Singapore, 117599, Singapore Chwee Teck Lim Kenichi Funamoto Kenry Masaaki Sato D.Y. conceived the research and designed and conducted most of the experiments. D.Y., K.F., K., M.S., and C.T.L. discussed the data and wrote the manuscript. K.S. conducted most of the cell cycle analyses, cell proliferation assays, and immunofluorescence staining experiments. D.Y. and C.T.L. directed and supervised the project. Correspondence to Daisuke Yoshino or Chwee Teck Lim. Supplementary Data 1 Yoshino, D., Funamoto, K., Sato, K. et al. Hydrostatic pressure promotes endothelial tube formation through aquaporin 1 and Ras-ERK signaling. Commun Biol 3, 152 (2020). https://doi.org/10.1038/s42003-020-0881-9 Improvement of the cell viability of hepatocytes cultured in three-dimensional collagen gels using pump-free perfusion driven by water level difference Sumire Ishida-Ishihara Ryota Takada Hisashi Haga Biomechanical Regulation of Hematopoietic Stem Cells in the Developing Embryo Paulina D. Horton Sandeep P. Dumbali Pamela L. Wenzel Current Tissue Microenvironment Reports (2021) Communications Biology (Commun Biol) ISSN 2399-3642 (online)	CommonCrawl
# The basics of public key cryptography Public key cryptography, also known as asymmetric cryptography, is a fundamental concept in modern cryptography. It is based on the use of key pairs, consisting of a public key and a private key. The public key is used to encrypt data, while the private key is used to decrypt it. The main advantage of public key cryptography is that it allows secure communication between two parties who have never communicated before and do not share a common secret key. This is achieved by using mathematical functions that are easy to compute in one direction, but computationally difficult to reverse. Public key cryptography relies on the use of mathematical problems that are believed to be hard to solve, such as the factorization of large numbers or the discrete logarithm problem. These problems form the basis for the security of public key encryption algorithms. To understand how public key cryptography works, let's look at a simple example. Imagine Alice wants to send a confidential message to Bob. They both have a key pair, consisting of a public key and a private key. Alice encrypts her message using Bob's public key. This ensures that only Bob, who has the corresponding private key, can decrypt and read the message. Even if someone intercepts the encrypted message, they won't be able to decrypt it without Bob's private key. This process of encryption and decryption relies on the mathematical properties of the public key encryption algorithm. The encryption algorithm takes the plaintext message and the recipient's public key as input, and produces the ciphertext. The decryption algorithm takes the ciphertext and the recipient's private key as input, and produces the original plaintext message. Let's take a look at a real-world example of public key cryptography in action: the Secure Sockets Layer (SSL) protocol, which is used to secure online communication, such as web browsing. When you visit a website that uses SSL, your web browser and the website's server exchange public keys. The server sends its public key to the browser, which uses it to encrypt any data it sends to the server. The server, in turn, uses its private key to decrypt the encrypted data. This ensures that any data exchanged between the browser and the server is secure and cannot be intercepted or tampered with by attackers. ## Exercise Explain why public key cryptography is useful for secure communication between two parties who have never communicated before and do not share a common secret key. ### Solution Public key cryptography allows secure communication between two parties who have never communicated before and do not share a common secret key because it relies on the use of key pairs, consisting of a public key and a private key. The public key is used to encrypt data, while the private key is used to decrypt it. This means that anyone can encrypt data using the recipient's public key, but only the recipient, who has the corresponding private key, can decrypt and read the data. This ensures that even if someone intercepts the encrypted data, they won't be able to decrypt it without the private key. # Discrete logarithm problem and its significance The discrete logarithm problem (DLP) is a mathematical problem that forms the basis for the security of many public key encryption algorithms. It is the problem of finding the exponent, or logarithm, to which a given number, known as the base, must be raised to obtain another given number, known as the result, in a finite field. The DLP is believed to be computationally difficult to solve, especially for large prime numbers and finite fields. This difficulty is what makes public key encryption algorithms secure, as the private key is derived from the solution to the DLP. The significance of the DLP lies in its use in various cryptographic protocols, such as key exchange protocols and digital signature schemes. By relying on the hardness of the DLP, these protocols ensure the confidentiality and integrity of data in secure communication. To understand the DLP better, let's consider a simple example. Suppose we have a finite field with a prime number p, and a base g that generates a cyclic subgroup of order n. The DLP is the problem of finding the exponent x such that g^x ≡ h (mod p), where h is a given element in the cyclic subgroup. For example, let's say we have a finite field with p = 23 and a base g = 5. If we want to find the exponent x such that 5^x ≡ 8 (mod 23), we need to solve the DLP. In this case, the solution is x = 11, as 5^11 ≡ 8 (mod 23). The hardness of the DLP lies in the fact that there is no efficient algorithm known to solve it for large prime numbers and finite fields. This means that it would take an infeasible amount of time and computational resources to find the solution through exhaustive search or other known algorithms. An example of the significance of the DLP is its use in the Diffie-Hellman key exchange protocol. This protocol allows two parties, Alice and Bob, to agree on a shared secret key over an insecure communication channel. In the Diffie-Hellman protocol, Alice and Bob each choose a random secret exponent, a and b respectively, and compute their public keys by raising a base g to the power of their secret exponent modulo a prime number p. They then exchange their public keys. Using their own secret exponent and the received public key, Alice and Bob can independently compute the same shared secret key by raising the received public key to the power of their secret exponent modulo p. This shared secret key can then be used for secure communication using symmetric encryption algorithms. The security of the Diffie-Hellman protocol relies on the hardness of the DLP. An attacker who intercepts the public keys exchanged between Alice and Bob would need to solve the DLP to compute the shared secret key, which is computationally difficult without knowledge of the secret exponents. ## Exercise Explain the significance of the discrete logarithm problem in the security of public key encryption algorithms. ### Solution The discrete logarithm problem (DLP) is significant in the security of public key encryption algorithms because it is the problem of finding the exponent, or logarithm, to which a given number, known as the base, must be raised to obtain another given number, known as the result, in a finite field. The DLP is computationally difficult to solve, especially for large prime numbers and finite fields. This difficulty ensures the security of public key encryption algorithms, as the private key is derived from the solution to the DLP. # Understanding elliptic curves and their properties Elliptic curves are a fundamental mathematical concept that plays a crucial role in modern cryptography. An elliptic curve is a set of points that satisfy a specific mathematical equation. In the context of cryptography, these curves are defined over finite fields, which are mathematical structures that have a finite number of elements. The equation that defines an elliptic curve has the form: $$y^2 = x^3 + ax + b$$ where a and b are constants that determine the shape and properties of the curve. The curve also has a special point called the "point at infinity" denoted as O, which serves as the identity element for the group structure of the curve. One important property of elliptic curves is their symmetry. If a point (x, y) lies on the curve, then the point (x, -y) also lies on the curve. This symmetry is crucial for the cryptographic operations performed on elliptic curves. Another important property of elliptic curves is their smoothness. Unlike other curves, such as parabolas or hyperbolas, elliptic curves have no singular points or cusps. This smoothness allows for efficient and secure cryptographic operations. The group structure of elliptic curves is also a key property. The points on an elliptic curve form an abelian group under a specific operation called point addition. This operation takes two points on the curve and produces a third point that also lies on the curve. The addition operation is defined geometrically and can be visualized as drawing a line through two points on the curve and finding the third point of intersection. The group structure of elliptic curves is what makes them suitable for cryptographic applications. The hardness of the discrete logarithm problem on elliptic curves, which we discussed earlier, is based on the difficulty of finding the exponent that relates two points on the curve. # The use of elliptic curves in cryptography Elliptic curves have become increasingly popular in cryptography due to their unique properties and the security they provide. They offer several advantages over other cryptographic methods, such as RSA and Diffie-Hellman, including smaller key sizes, faster computation, and resistance to certain attacks. One of the main uses of elliptic curves in cryptography is in public key encryption. Public key encryption allows for secure communication between two parties who have never communicated before. In this system, each party has a pair of keys: a public key that is shared with others, and a private key that is kept secret. Messages encrypted with the public key can only be decrypted with the corresponding private key. Elliptic curve cryptography (ECC) provides a more efficient and secure alternative to traditional public key encryption methods. ECC uses the mathematical properties of elliptic curves to create a smaller and more secure key pair. This means that ECC can provide the same level of security as other methods, such as RSA, with much smaller key sizes. Smaller key sizes result in faster computation and less storage space required. Another use of elliptic curves in cryptography is in digital signatures. Digital signatures are used to verify the authenticity and integrity of digital documents or messages. They provide a way to ensure that a message has not been tampered with and that it was indeed sent by the claimed sender. Elliptic curve digital signatures (ECDSA) are widely used in various applications, including secure communication protocols, digital certificates, and cryptocurrencies. ECDSA provides a secure and efficient way to generate and verify digital signatures using elliptic curves. # Digital signatures and their role in secure communication Digital signatures play a crucial role in ensuring the authenticity and integrity of digital documents and messages. They provide a way to prove that a message was sent by a specific sender and that it has not been altered during transmission. In traditional paper-based communication, signatures are used to verify the identity of the signer and to ensure that the content of the document has not been tampered with. Similarly, in the digital world, digital signatures serve the same purpose. A digital signature is created using a mathematical algorithm that combines the content of the message with the signer's private key. This creates a unique signature that can be verified using the signer's public key. If the signature is valid, it means that the message was indeed signed by the claimed sender and that it has not been modified since it was signed. Digital signatures provide several benefits in secure communication. They provide a way to verify the authenticity of the sender, ensuring that the message is coming from a trusted source. They also provide a way to verify the integrity of the message, ensuring that it has not been tampered with during transmission. In addition to these benefits, digital signatures also provide non-repudiation, which means that the signer cannot deny having signed the message. This is because the signature is uniquely tied to the signer's private key, and only the signer can create a valid signature using that key. # Real-world examples of elliptic curve cryptography in action One common application of ECC is in securing internet communications. Many websites use ECC-based protocols, such as Transport Layer Security (TLS), to establish secure connections with users. ECC provides strong security with shorter key lengths compared to traditional encryption algorithms, making it more efficient for resource-constrained devices like smartphones and IoT devices. Another example of ECC in action is in the field of mobile payments and digital wallets. ECC is used to secure the communication between the user's device and the payment terminal, ensuring that the transaction data remains confidential and cannot be intercepted or tampered with. ECC is also used in secure messaging applications to protect the privacy of user communications. By using ECC-based encryption algorithms, messages can be securely transmitted and only decrypted by the intended recipient, ensuring that sensitive information remains confidential. Furthermore, ECC is employed in securing digital identities and authentication systems. For example, smart cards and biometric systems often use ECC-based algorithms to verify the identity of individuals and protect access to sensitive resources. Overall, ECC provides a versatile and efficient solution for securing various real-world applications. Its strong security properties and computational efficiency make it an ideal choice for ensuring the confidentiality, integrity, and authenticity of sensitive information in a wide range of contexts. - A user wants to access their online banking account. The bank's website uses ECC-based TLS to establish a secure connection with the user's device. The user's device and the bank's server exchange ECC-based digital certificates to authenticate each other and establish a secure channel for transmitting sensitive financial data. ## Exercise Think of a real-world application where ECC can be used to provide secure communication. Describe the application and explain how ECC can enhance its security. ### Solution One example is secure email communication. ECC can be used to encrypt and digitally sign email messages, ensuring that only the intended recipient can read the message and that the message has not been tampered with during transmission. ECC's strong security properties and computational efficiency make it a suitable choice for securing email communication, especially on resource-constrained devices like smartphones. # Key exchange protocols using elliptic curves Key exchange protocols are essential for establishing a shared secret key between two parties over an insecure communication channel. Elliptic curve cryptography (ECC) provides efficient and secure key exchange protocols that are widely used in practice. One popular key exchange protocol using elliptic curves is the Elliptic Curve Diffie-Hellman (ECDH) protocol. The ECDH protocol allows two parties, Alice and Bob, to agree on a shared secret key without exchanging the key directly. Instead, they exchange their public keys, which are points on an elliptic curve. By performing mathematical operations on their private keys and the received public keys, Alice and Bob can independently compute the same shared secret key. The ECDH protocol is based on the hardness of the elliptic curve discrete logarithm problem, which states that it is computationally infeasible to determine the private key from the public key. This property ensures the security of the shared secret key. Another key exchange protocol using elliptic curves is the Elliptic Curve Integrated Encryption Scheme (ECIES). ECIES combines the key exchange functionality of ECDH with symmetric encryption algorithms to provide secure communication between two parties. ECIES allows Alice to encrypt a message using Bob's public key, and Bob can decrypt the message using his private key and the shared secret key derived from the key exchange. Both ECDH and ECIES provide strong security guarantees and are widely used in various applications, including secure messaging, virtual private networks (VPNs), and secure payment systems. Their efficiency and security properties make them ideal choices for establishing secure communication channels in real-world scenarios. - Alice and Bob want to establish a secure communication channel using elliptic curve cryptography. They both generate their private keys and compute their public keys based on the chosen elliptic curve. Alice sends her public key to Bob, and Bob sends his public key to Alice. They both independently compute the shared secret key using their private keys and the received public keys. This shared secret key can then be used to encrypt and decrypt messages between Alice and Bob, ensuring the confidentiality and integrity of their communication. ## Exercise Explain the steps involved in the Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol. ### Solution 1. Alice and Bob agree on an elliptic curve and a base point on that curve. 2. Alice generates a private key, a random number, and computes her public key by multiplying the base point by her private key. 3. Bob generates a private key, a random number, and computes his public key by multiplying the base point by his private key. 4. Alice sends her public key to Bob, and Bob sends his public key to Alice. 5. Alice computes the shared secret key by multiplying Bob's public key by her private key. 6. Bob computes the shared secret key by multiplying Alice's public key by his private key. 7. Alice and Bob now have the same shared secret key, which they can use for encryption and decryption in their communication. # Implementing elliptic curve cryptography in software Implementing elliptic curve cryptography (ECC) in software involves several steps, including choosing the appropriate elliptic curve parameters, generating key pairs, performing mathematical operations on the curve, and implementing cryptographic algorithms. To implement ECC in software, you'll need to select an elliptic curve that satisfies certain security requirements. The curve should have a large prime order, making it difficult to solve the elliptic curve discrete logarithm problem. There are standardized curves available, such as those recommended by the National Institute of Standards and Technology (NIST), or you can choose your own curve based on specific requirements. Once you have chosen a curve, you'll need to generate key pairs for encryption and decryption. A key pair consists of a private key and a corresponding public key. The private key is a random number, while the public key is derived from the private key and the curve parameters. The private key should be kept secret, while the public key can be shared with others. After generating key pairs, you can perform mathematical operations on the curve, such as point addition, point doubling, and scalar multiplication. These operations are used in key exchange protocols, digital signatures, and encryption algorithms. It's important to implement these operations correctly to ensure the security and efficiency of the ECC implementation. Finally, you'll need to implement cryptographic algorithms that utilize ECC, such as the Elliptic Curve Diffie-Hellman (ECDH) key exchange protocol, the Elliptic Curve Digital Signature Algorithm (ECDSA), and the Elliptic Curve Integrated Encryption Scheme (ECIES). These algorithms involve complex mathematical calculations and require careful implementation to ensure their correctness and security. Implementing ECC in software requires a deep understanding of elliptic curve mathematics, cryptographic algorithms, and software development practices. It's important to follow best practices and security guidelines to protect against potential vulnerabilities and attacks. By carefully implementing ECC, you can leverage its strong security properties for various applications, such as secure communication, digital signatures, and encryption. # Challenges and limitations of elliptic curve cryptography While elliptic curve cryptography (ECC) offers many advantages, there are also challenges and limitations that need to be considered. One challenge is the complexity of the mathematics involved in ECC. The algorithms and operations used in ECC, such as point addition and scalar multiplication, are more complex than those used in traditional cryptography. This complexity can make it more difficult to implement and understand ECC, requiring specialized knowledge and expertise. Another challenge is the need for careful parameter selection. The security of ECC relies on the choice of elliptic curve parameters, such as the curve equation and the prime order. Selecting insecure or weak parameters can compromise the security of the system. It's important to follow established standards and guidelines for parameter selection to ensure the strength of the ECC implementation. ECC also has limitations in terms of performance and efficiency. While ECC offers strong security with shorter key lengths compared to other encryption methods, it can be computationally intensive. ECC operations require more processing power and memory, which can impact the performance of systems, especially in resource-constrained environments. Additionally, ECC is not as widely supported as other encryption methods. While ECC is gaining popularity and is supported by many modern cryptographic libraries and protocols, there are still legacy systems and devices that do not support ECC. This can limit the interoperability and compatibility of ECC implementations. Despite these challenges and limitations, ECC remains a powerful and widely used encryption method. With proper parameter selection, implementation, and optimization, ECC can provide strong security and efficiency for various applications. It's important to stay updated with the latest advancements and best practices in ECC to mitigate potential challenges and maximize its benefits. # Comparison with other public key encryption methods Elliptic curve cryptography (ECC) is a powerful and efficient public key encryption method. It offers several advantages over other encryption methods, such as RSA and Diffie-Hellman. One major advantage of ECC is its shorter key lengths. ECC can achieve the same level of security as other encryption methods with much shorter key lengths. For example, a 256-bit ECC key provides the same level of security as a 3072-bit RSA key. This means that ECC requires less computational power and memory, making it more efficient and faster in terms of encryption and decryption operations. Another advantage of ECC is its resistance to quantum computing attacks. While RSA and Diffie-Hellman are vulnerable to attacks by quantum computers, ECC is believed to be resistant to such attacks. This makes ECC a future-proof encryption method that can withstand advancements in quantum computing technology. ECC also offers strong security with its mathematical foundation. The security of ECC is based on the difficulty of solving the elliptic curve discrete logarithm problem (ECDLP). The ECDLP is a computationally hard problem, making it extremely difficult for an attacker to derive the private key from the public key. In terms of performance, ECC is more efficient than other encryption methods. ECC operations require fewer computational resources, making it suitable for resource-constrained environments, such as mobile devices and IoT devices. ECC can provide secure communication with lower power consumption and faster processing speed. However, it's important to note that ECC is not without its limitations. ECC implementations require careful parameter selection and proper implementation to ensure security. Additionally, ECC is not as widely supported as other encryption methods, which can limit its interoperability with legacy systems. Overall, ECC offers strong security, efficiency, and resistance to quantum computing attacks. It is becoming increasingly popular and is widely used in various applications, including secure communication, digital signatures, and key exchange protocols. As technology continues to advance, ECC will likely play a crucial role in ensuring secure communication in the digital age. # Future developments and advancements in elliptic curve cryptography One area of research is the exploration of new elliptic curves with different properties. Researchers are continuously searching for curves that offer improved security, efficiency, and resistance to attacks. These new curves may have different mathematical properties that make them more resistant to specific types of attacks or provide better performance in certain scenarios. Another area of focus is the development of more efficient algorithms and implementations for ECC. Researchers are working on optimizing ECC operations to reduce computational overhead and improve performance. This includes developing new algorithms for point multiplication, scalar multiplication, and other ECC operations. Additionally, advancements in hardware technology, such as the use of specialized hardware accelerators and dedicated ECC processors, can further improve the efficiency and speed of ECC implementations. These hardware advancements can enable ECC to be more widely adopted in resource-constrained environments. Furthermore, research is being conducted to explore the integration of ECC with other cryptographic techniques and protocols. This includes combining ECC with other encryption methods, such as symmetric encryption, to create hybrid encryption schemes that offer the benefits of both methods. It also involves integrating ECC with other cryptographic protocols, such as secure multiparty computation and zero-knowledge proofs, to enhance the overall security and functionality of cryptographic systems. Lastly, the ongoing advancements in quantum computing technology have prompted researchers to investigate post-quantum elliptic curve cryptography. This involves developing ECC variants that are resistant to attacks by quantum computers. Post-quantum ECC aims to provide secure communication even in the presence of powerful quantum computers that can break traditional ECC schemes. In conclusion, the future of elliptic curve cryptography is promising, with ongoing research and developments that aim to enhance its security, efficiency, and functionality. These advancements will further solidify ECC as a reliable and efficient encryption method, ensuring secure communication in the digital age.	Textbooks
\begin{document} \title{Cold Atom Physics Using Ultra-Thin Optical Fibers: Light-Induced Dipole Forces and Surface Interactions} \author{G.~Sagu\'{e}, E.~Vetsch, W.~Alt, D. Meschede} \affiliation{Institut f\"ur Angewandte Physik, Universit\"at Bonn, Wegelerstr.~8, 53115 Bonn, Germany} \author{A. Rauschenbeutel} \altaffiliation{Present address: Institut f\"ur Physik, Universit\"at Mainz, 55099 Mainz, Germany} \email{[email protected]} \affiliation{Institut f\"ur Angewandte Physik, Universit\"at Bonn, Wegelerstr.~8, 53115 Bonn, Germany} \date{\today} \begin{abstract} The strong evanescent field around ultra-thin unclad optical fibers bears a high potential for detecting, trapping, and manipulating cold atoms. Introducing such a fiber into a cold atom cloud, we investigate the interaction of a small number of cold Caesium atoms with the guided fiber mode and with the fiber surface. Using high resolution spectroscopy, we observe and analyze light-induced dipole forces, van der Waals interaction, and a significant enhancement of the spontaneous emission rate of the atoms. The latter can be assigned to the modification of the vacuum modes by the fiber. \end{abstract} \pacs{42.50.-p, 39.25.+k, 34.50.Dy, 32.80.Pj} \maketitle Tapered and microstructured optical fibers count among the most active fields of research in recent years \cite{Russell03,Tong03}. In such fibers, the propagation of light can, e.g., be tailored such that it controllably depends on the light intensity. These fiber-induced non-linear processes play for instance a major role in the generation of optical frequency combs and often stem from the non-linear response of the bulk fiber material, subjected to extreme intensities. The low intensity limit of non-linear light-matter interaction is reached when single photons already induce a non-linear response of matter. This situation is realized in cavity quantum electrodynamics \cite{Berman}, where photons, typically confined in space by an optical resonator \cite{Vahala}, interact with a single or a few dipole emitters. In this context, ultra-thin unclad optical fibers offer a strong transverse confinement of the guided fiber mode while exhibiting a pronounced evanescent field surrounding the fiber \cite{Balykin1}. This unique combination allows to efficiently couple particles (atoms, molecules, quantum dots etc.) on or near the fiber surface to the guided fiber mode, making tapered optical fibers (TOFs) a powerful tool for their detection, investigation, and manipulation: The absorbance of organic dye molecules, deposited on a subwavelength-diameter TOF, has been spectroscopically characterized via the fiber transmission with unprecedented sensitivity \cite{Warken07}. Furthermore, the fluorescence from a very small number of resonantly irradiated atoms around a 400-nm diameter TOF, coupled into the guided fiber mode, has been detected and spectrally analyzed \cite{LeKien05,Nayak06}. Finally, it has also been proposed to trap atoms around ultra-thin fibers using the optical dipole force exerted by the evanescent field \cite{Dowling96,Balykin3}. Here, we report on the observation of such dipole forces induced by the evanescent field around a sub-wavelength diameter TOF. By spectroscopically investigating the transmission of a probe laser launched through the fiber, we find clear evidence of the mechanical effects of these dipole forces on the atoms, leading to a modification of the atomic density in the vicinity of the fiber. A rigorous analysis furthermore shows that a detailed description of the absorption signal must include the interaction of the atoms with the dielectric fiber. In particular, this includes the mechanical and spectral effects of the van der Waals (vdW) interaction and a significant enhancement of the spontaneous emission rate of the atoms due to a modification of the vacuum modes by the fiber. An enhanced spontaneous emission of atoms in vacuo coupled to a continuous set of evanescent modes has already been observed in evanescent wave spectroscopy on a plane dielectric surface \cite{Ivanov04}. In our case, however, the ultra-thin fiber only sustains a single mode at the atomic wavelength. To our knowledge, the enhanced spontaneous emission of atoms in vacuo in such a situation has never been observed before. \begin{figure} \caption{ (a) Schematic experimental setup. A cloud of laser cooled Caesium atoms is spatially overlapped with the 500-nm diameter waist of a tapered optical fiber. The transmission through the fiber is measured using a photodetector. (b) Timing of the experiment} \label{fig:setup} \end{figure} Figure~\ref{fig:setup} (a) shows the schematic experimental setup. A cloud of cold Caesium atoms, released from a magneto-optical trap (MOT), is spatially overlapped with the waist of a TOF in an ultra high vacuum (UHV) environment. The MOT is geometrically aligned by means of a bias magnetic field while monitoring its position with two CCD cameras. A frequency scanned probe laser is launched through the fiber and its transmission is measured with an avalanche photodiode (APD). Typical powers of the probe laser used in the experiment range from several hundred Femtowatts to one Nanowatt. We fabricate the tapered fibers by stretching a standard single mode fiber (Newport F-SF) while heating it with a travelling hydrogen/oxygen flame \cite{Birks92}. Our computer controlled fiber pulling rig produces tapered fibers with a homogeneous waist diameter down to 100~nm and a typical extension of 1--10~mm. In the taper sections, the weakly guided LP$_{01}$ mode of the unstretched fiber is adiabatically transformed into the strongly guided HE$_{11}$ mode of the ultrathin section and back \cite{Love86}, resulting in a highly efficient coupling of light into and out of the taper waist. For TOFs with final diameter above 0.5~$\mu$m, we achieve up to 97~\% of the initial transmission at 852~nm. For the present experiment we used a 500-nm diameter fiber with 93\% transmission and a waist length of 5~mm, sustaining only the fundamental HE$_{11}$ mode at the 852-nm Cs D2 wavelength. During evacuation of the vacuum chamber, the fiber transmission dropped to 40~\%, possibly due to contamination with pump oil. After one day in UHV, the transmission increased again to 80\%. We use a conventional Cs MOT with a $1/\sqrt{e}$-radius of 0.6~mm. We probe the atoms with a diode laser which is frequency scanned by $\pm 24$~MHz with respect to the $6^2S_{1/2}$, $F=4$ to $6^2P_{3/2}$, $F=5$ transition using an acusto-optical modulator in double pass configuration. While being linearly polarized before coupling into the TOF, the probe laser polarization at the position of the fiber waist is unknown. The probe laser linewidth of 1~MHz allows to resolve the 5.2-MHz natural linewidth of the Cs D2 line in Doppler-free spectroscopy. Figure~\ref{fig:setup} (b) shows the timing of the experimental sequence: During the first 10~ms, the atoms are captured and cooled in the MOT while the probe laser is off. In the following 10~ms, the MOT cooling- and repump-laser and the magnetic field are off and the probe laser is on. The atoms are thus not influenced by the MOT beams or magnetic fields during the spectroscopy. \begin{figure} \caption{ Measured (line graphs) and simulated (squares) absorbance of the atoms versus the detuning of the probe laser. The effective number of atoms contributing to the spectra (see Eq.~(\ref{eq:N})) is 107, 14 and 2 for (a) 1 nW, (b) 52 pW and (c) 6 pW of probe laser power, respectively. The decrease of the peak absorbance with increasing power is due to the saturation of the atoms.} \label{fig:spectra} \end{figure} The APD signal is recorded with a digital storage oscilloscope and averaged over 4096 traces. Figure~\ref{fig:spectra} shows the measured (line graphs) and theoretically predicted (squares) absorbance of the atoms (negative natural logarithm of the transmission) versus the probe laser detuning for three different probe laser powers. The theory assumes an averaged atomic density distribution $\rho_{\delta,P}(r,z)$ around the fiber, where $r$ is the distance from the fiber center and $z$ the position along the fiber waist. Note that, due to light-induced dipole forces, $\rho_{\delta,P}$ also depends on the detuning of the probe laser with respect to resonance, $\delta$, and on its power, $P$. The line shape is then given by \begin{equation}\label{eq:A_von_delta} A_P(\delta)=\frac{\hbar w}{P}\int\rho_{\delta,P}(r,z)\Gamma (I_{P}(r),\gamma(r),\delta+\delta_\mathrm{vdW}(r))\,dV\, , \end{equation} where $\Gamma (I_{P}(r),\gamma(r),\delta+\delta_\mathrm{vdW}(r))$ is the scattering rate of an atom in the evanescent field with intensity $I_{P}(r)$, $\gamma(r)$ is the longitudinal decay rate of the atom, and $\delta_\mathrm{vdW}(r)$ is the vdW shift of the atomic transition frequency. The evanescent field intensity profile, $I_{P}(r)$, can be found in \cite{Balykin1}. The polarization state of the evanescent field has been assumed to be an incoherent, equally weighted mixture of linearly and circularly polarized light. Under these conditions, the Cs saturation intensity in free space is 18~W/m$^2$. The longitudinal decay rate strongly depends on the atom-fiber distance~\cite{V. V. Klimov}. Given that the silica fiber is transparent at the Cs D2 wavelength, $\gamma(r)$ has only two contributions: emission into freely propagating modes and emission into guided fiber modes: \begin{equation}\label{eq:gamma(r)} \gamma(r)=\gamma_{\mathrm{free}}(r)+\gamma_{\mathrm{guid}}(r)\ . \end{equation} For an atom near a 500-nm diameter dielectric cylinder at distances smaller than the emission wavelength, $\gamma_{\mathrm{free}}(r)$ is given in~\cite{V. V. Klimov} while $\gamma_\mathrm{guid}(r)$ can be approximated as \begin{equation} \gamma_\mathrm{guid}(r)\simeq 0.3\,\gamma_{0}I_P(r)/I_P(a)\ . \end{equation} Here, $\gamma_{0}$ is the spontaneous emission rate of a Cs atom in free space, $a$ denotes the fiber radius, and 0.3\,$\gamma_{0}$ corresponds to the spontaneous emission rate of an atom placed on the surface of a 500 nm diameter optical fiber into the guided mode~\cite{LeKien05}. On the surface of the fiber, Eq.~(\ref{eq:gamma(r)}) then predicts a 57\% increase of the spontaneous emission rate of the Cs atoms, resulting in a broadening of the absorbance line shapes. We calculated the vdW shift, $\delta_\mathrm{vdW}(r)$, for the D2 line of Cs near a 500 nm diameter dielectric cylinder~\cite{Boustimi02}. It stems from the different polarizabilities of the $6^2S_{1/2}$ ground state and the excited $6^2P_{3/2}$ state of the Cs atoms when interacting with the dielectric surface. According to Eq.~(\ref{eq:A_von_delta}), $\delta_\mathrm{vdW}(r)$ thus inhomogeneously broadens the absorbance profile: Atoms at different distances from the fiber surface will be unequally shifted while contributing to $A_P(\delta)$. Furthermore, we expect the center of the profile to be red-shifted by at most $-0.5$~MHz. However, being of the same order as the drifts of our probe laser frequency, this shift is too small to be experimentally quantified using the current setup. Finally, we assume the following explicit form for the density distribution of the atomic cloud: \begin{equation}\label{eq:density} \rho_{\delta,P}(r,z)=\left\{\frac{n_0}{\sigma^3 (2\pi)^{\frac{3}{2}}} e^{-\frac{r^2+z^2}{2\sigma^2}}\right\} f_{\delta,P}(r)\ . \end{equation} Here, the term in curly brackets corresponds to a Gaussian density distribution of the unperturbed atomic cloud with $\sigma=0.6$~mm radius, containing $n_0$ atoms. The factor $f_{\delta,P}(r)$ accounts for the perturbation introduced by the presence of the fiber. We calculate $f_{\delta,P}(r)$ with a Monte Carlo simulation of 100,000 trajectories of thermal atoms with a temperature of 125~$\mu$K, i.e., the Cs Doppler temperature. This simulation includes the attractive vdW force between the fiber surface and the atoms and the saturating dipole force induced by the probe laser~\cite{J. E. Bjorkholm}. \begin{figure} \caption{ (a) Simulations of the relative density of the MOT for different detunings $\delta$ of the probe laser versus the distance from the fiber surface; solid line $\delta=0$~MHz, dotted line $\delta=-3$~MHz, dashed line $\delta=3$~MHz. The following parameters have been used for the simulations: Fiber diameter 500 nm, probe power 1 nW and a 3D Maxwellian velocity distribution of the Cs atoms at a temperature of 125 $\mathrm{\mu K}$. (b) and (c) show several atomic trajectories for $\delta=+3$ MHz and $\delta=-3$ MHz respectively with a fixed atom velocity of 10 cm/s.} \label{fig:density_simulation} \end{figure} Figure~\ref{fig:density_simulation}(a) shows $f_{\delta,P}(r)$ as a function of the distance from the fiber surface for $P=1$~nW and $\delta=-3$, 0, and $+3$~MHz (dotted, solid, and dashed line). The frequency dependency of $f_{\delta,P}(r)$ due to light-induced dipole forces is clearly apparent. In all three cases $f_{\delta,P}(r)$ decays to zero at the surface of the fiber due to the vdW force. $A_P(\delta)$ from Eq.~(\ref{eq:A_von_delta}) can now be adjusted to the experimental line shapes, the only fitting parameters being $n_0$ and an experimental frequency offset. Figure~\ref{fig:spectra} shows three examples for $P$ ranging over three orders of magnitude. The agreement between theory (squares) and experiment (line graphs) is excellent. In particular, in addition to the line width, our model reproduces well the asymmetry of the line shape observed for larger powers. Figure~\ref{fig:linewidth} shows the width of the measured absorbance profiles versus the probe laser power (squares). The linewidths predicted by our model are also shown (open circles with a b-spline fit as a guide to the eye). We recall that the effects of light-induced dipole forces and surface interactions have been included in the model. For comparison, we also show the expected linewidths in absence of these effects (dashed line). While the full model agrees very well with the experimental data, the reduced model strongly deviates both for high and low powers. \begin{figure} \caption{ Linewidth of the absorbance profiles versus probe laser power. (a) full power range and (b) low power range. The squares correspond to the experimental data and the open circles are the simulated values. The continuous line is a guide to the eye (b-spline) of the simulated values. The dashed line is the reduced model not taking into account light-induced dipole forces and surface interactions.} \label{fig:linewidth} \end{figure} For probe laser powers larger than 100 pW, the measured lines are considerably narrower than what would be expected in absence of dipole forces and surface interactions, see Fig.~\ref{fig:linewidth}(a). For 1 nW of probe laser power this narrowing exceeds 40\%. The narrowing can be explained by the effect of the light-induced dipole forces on the density of the atomic cloud, see Fig.~\ref{fig:density_simulation}(a). For distances smaller than 370 nm, i.e., in the region that contains more than 75\% of the evanescent field power, the largest integrated density of the atomic cloud is predicted in the case of zero detuning ($\delta=0$~MHz). For blue ($\delta=+3$ MHz) and red ($\delta=-3$ MHz) detunings, this integrated density is lowered due to the effect of the light-induced dipole forces. This results in a reduced absorbance and leads to an effective line narrowing. Figures~\ref{fig:density_simulation}(b) and (c) show several simulated atomic trajectories with fixed initial velocity. For the case of blue detuning, (b), the atoms are repelled by the fiber due to the repulsive light-induced dipole force. For the case of red detuning, (c), the atoms are accelerated towards the fiber. Naively, one might assume that this increases the density close to the fiber. However, this effect is counteracted by the shorter average time of flight of the atoms through the evanescent field due to their higher velocity and by the higher atomic loss rate \cite{loss_rate}. In fact, for distances up to 100~nm both effects cancel almost perfectly. For larger distances, however, the effects reducing the density dominate. The net effect is therefore also a reduction of the absorbance. Figure~\ref{fig:linewidth}(b) shows the linewidths for the limit of low probe laser powers, i.e., low saturation and negligible light-induced dipole forces. The measured linewidths approach 6.2~MHz for vanishing powers. This result exceeds the natural Cs D2 linewidth in free space by almost 20~\%. This broadening can be explained by surface interactions, i.e., the vdW shift of the Cs D2 line and the modification of the spontaneous emission rate of the atoms near the fiber, see Eq.~(\ref{eq:A_von_delta}). Both effects have the same magnitude and only their combination yields the very good agreement between our model and the experimental data. Finally, we estimate the effective number, $N_P$, of fully saturated atoms contributing to the signals in Fig.~\ref{fig:spectra}. From the adjustment of the height of the absorbance profiles, we extract the total number of atoms in the cloud, $n_0$, and infer a maximum atomic density of $4.4\times 10^{10}$ atoms/cm$^3$ using Eq.~(\ref{eq:density}). This value is slightly smaller than typical peak densities of unperturbed Cs MOTs~\cite{tow}. We now estimate $N_P$ according to \begin{equation}\label{eq:N} N_P=\frac{2}{\gamma_0}\int\rho_{\delta=0,P}(r,z) \Gamma (I_P(r),\gamma(r),\delta_\mathrm{vdW}(r))\,dV\ , \end{equation} where we follow the notation of Eq.~(\ref{eq:A_von_delta}). Note that $N_P$ is power dependent and can be lowered by reducing $P$. We calculate $N_P$ to be 107, 14, and 2 in Fig.~\ref{fig:spectra}(a), (b), and (c), respectively. Furthermore, due to the saturating scattering rate $\Gamma$ in the integrand of Eq.~(\ref{eq:N}), the mean distance of the probed atoms from the fiber surface is also power dependent and can be adjusted down to 248~nm. Summarizing, we have shown that sub-wavelength diameter optical fibers can be used to detect, spectroscopically investigate, and mechanically manipulate extremely small samples of cold atoms. In particular, on resonance, as little as two atoms on average, coupled to the evanescent field surrounding the fiber, already absorbed 20~\% of the total power transmitted through the fiber. These results open the route towards the use of ultra-thin fibers as a powerful tool in quantum optics and cold atom physics. By optically trapping one or more atoms around such fibers \cite{Dowling96,Balykin3}, it should become possible to deterministically couple the atoms to the guided fiber mode and to even mediate a coupling between two simultaneously trapped atoms \cite{LeKien05b}, leading to a number of applications, e.g., in the context of quantum information processing. In addition, high precision measurements of the modification of the lifetime of atomic energy levels near surfaces and of the van der Waals potential \cite{M. Chevrollier} are also within the scope of such glass fiber quantum optics experiments. We wish to thank V.~I.~Balykin and D.~Haubrich for their contribution in the early stages of the experiment, F.~Warken for assistance in the fiber production, B.~Weise for his part in the simulations, and M.~Ducloy and C.~Henkel for valuable discussions. This work was supported by the EC (Research Training Network ``FASTNet'') and the DFG (Research Unit 557). \end{document}	arXiv
Replicator equation on networks with degree regular communities Daniele Cassese ORCID: orcid.org/0000-0002-2216-45621,2,3 The replicator equation is one of the fundamental tools to study evolutionary dynamics in well-mixed populations. This paper contributes to the literature on evolutionary graph theory, providing a version of the replicator equation for a family of connected networks with communities, where nodes in the same community have the same degree. This replicator equation is applied to the study of different classes of games, exploring the impact of the graph structure on the equilibria of the evolutionary dynamics. Evolutionary game theory stems from the field of evolutionary biology, as an application of game theory to biological contests, and successively finds applications in many other fields, such as sociology, economics and anthropology. The range of phenomena studied using evolutionary games is quite broad: cultural evolution (Cavalli-Sforza and W 1981), the change of behaviours and institutions over time (Bowles et al. 2003), the evolution of preferences (Bowles 2010) or language (Nowak 2000), the persistence of inferior cultural conventions (Bowles and Belloc 2013). A particularly vaste literature investigates the evolutionary foundations of cooperation (Bowles 2004; Bowles et al. 2004; Doebli et al. 2004) just to name a few. For an inspiring exposition of evolutionary game theory applications to economics and social sciences see (Bowles 2006). One of the building blocks of evolutionary game theory is that fitness (a measure of reproductive success relative to some baseline level) of a phenotype does not just depend on the quality of the phenotype itself, but on the interactions with other phenotypes in the population: fitness is hence frequency dependent (Nowak 2006a), and as strategies are the manifestation of individuals' genetic inheritance, individuals are characterised by a fixed strategy throughout their lifetime. The payoffs of the game are in terms of fitness, so if a trait offers an evolutive advantage over another, this means a better fitness for the individual who has inherited that trait. The dynamics resulting from interactions between individuals carrying different traits capture the process of natural selection: the strategy (phenotype, cultural trait) that performs better gives an advantage in term of reproductive success, hence it will reproduce at a higher rate and eventually take over the entire population (Nowak 2006a). Early models of evolutionary dynamics assume well-mixed population, ignoring the relational structure that constrains interactions between agents. The study of evolutionary dynamics on structured population is the subject of interest of evolutionary graph theory, introduced by (Lieberman et al. 2005). In this framework agents are placed on a network and play the game with their next neighbours, and the least successful (in terms of fitness) are replaced by their most successful neighbours' offsprings. Evolutionary dynamics on graphs has been applied extensively to the study of cooperation (Santos et al. 2006; Ohtsuki and Nowak 2006; 2008; Allen et al. 2017) showing that there are radical differences with the case of a well-mixed population, and that the success of cooperation depends crucially on the underlying network structure. Analytical results have been derived for evolutionary games on regular networks (Ohtsuki et al. 2005; Ohtsuki and Nowak 2006; Taylor et al. 2007) while more realistic complex networks have been investigated through computer simulations (Maciejewski et al. 2014). This work is an extension of (Cassese 2017), where I studied cooperation on a family of graphs characterised by degree-regular communities, proving that the relation between the structure of the population and the cost of cooperation determines the nature of equilibria for a Prisoner's dilemma game. In this paper I briefly present the replicator equation for graphs on regular communities, and an algorithm to generate graphs in this family, as well as its application to the Prisoner's Dilemma as already in (Cassese 2017). In addition to the previous version of this work here I study other classes of games under the replicator dynamics, namely Hawk-Dove and Cooperation games, exploring how the network impacts the equilibria compared to the mean-field case. Replicator equation on regular graphs The Replicator Equation in its mean-field version studies frequency dependent selection without mutation in the deterministic limit of an infinitely large well-mixed population (Nowak 2006a). Take an evolutionary game with n strategies and a payoff matrix Π, where πij denotes the payoff of strategy i against strategy j. Call xi the frequency of strategy i, where $\sum _{i \in n} x_{i}=1$, the fitness of strategy i is $f_{i}= \sum _{j \in n}x_{j} \pi _{ij}$, and $\phi = \sum _{i \in n} x_{i}f_{i}$ the average fitness of the population, then the replicator equation is: $$ \dot{x}_{i}= x_{i}(f_{i} -\phi)\ \text{for}\ i \in n $$ If the population structure is a regular network of degree k, under weak selection the replicator equation obtained with pair approximation (for details on the method see (Matsuda and et al. 1992)) is (Ohtsuki and Nowak 2006): $$ \dot{x}_{i}= x_{i} \left[\sum_{j=1}^{n} x_{j} (\pi_{ij}+b_{ij}(k, \mathbf{\Pi}))- \phi \right] $$ where bij depends on the degree of the network, k, the payoff matrix Π and the updating rule. (Ohtsuki and Nowak 2006) derive bij under three updating rules: Birth-Death: An individual is chosen for reproduction with probability proportional to fitness. The offspring replaces one of the k neighbour chosen at random. Death-Birth: An individual is randomly chosen to die. One of the k neighbours replaces it with probability proportional to their fitness. Imitation: An individual is randomly chosen to update her strategy. She imitates one of her k neighbours proportional to their fitness. The corresponding bijs are: $$ \begin{aligned} \text{Birth-Death:} \quad b_{ij}&= \frac{\pi_{ii}+\pi_{ij}-\pi_{ji} -\pi_{jj}}{k-2} \\ \text{Death-Birth:} \quad b_{ij}&= \frac{(k+1)\pi_{ii}+\pi_{ij}-\pi_{ji}-(k+1)\pi_{jj}}{(k+1)(k-2)} \\ \text{Imitation:} \quad b_{ij}&= \frac{(k+3)\pi_{ii}+3\pi_{ij}-3\pi_{ji}-(k+3)\pi_{jj}}{(k+3)(k-2)} \end{aligned} $$ Hence bij captures local competition on a graph taking account of the gain of ith strategy from i and j players and the gains of jth strategy from i and j players (Nowak et al. 2010). The derived equation is a very good approximation for infinitely large regular graphs with negligible clustering (absence of clustering is the basic assumption behind the moment closure in pair approximation) and provides an easy-to-deal-with differential equation that can be computed at least numerically. In this section I present the extension of the replicator equation to a more complex family of graphs, where nodes can have different degrees. First I define a family of connected graphs (which I call multi-regular graphs) where nodes are clustered in degree-homogeneous communities, such that most of the connections are between same-degree nodes, and few edges connect communities with different degrees. Hence an algorithm to create such networks is proposed, and finally the replicator equation for these networks is introduced. The definition of the class of multi-regular graphs is motivated by the necessity to have more realistic network structures and at the same time preserving analytical tractability. The homogeneous structure of regular graphs, where all nodes have the same number of neighbours, makes them poorly representative of real world heterogeneous networks (Strogatz 2001). Real world networks are typically characterised by small-world properties (Watts and Strogatz 1998) and scale-free distributions (Barabasi AL 1999), and regular networks fail to satisfy both characteristics: they may have a high clustering coefficient, but usually have large number of hops between pairs of nodes (so they are not small-world), and they trivially are not scale-free, as every node has the same degree. These differences are not without consequences for the dynamics, hence predictions made on regular network models result incorrect if applied to real networks. A standard example can be found in epidemic models: while on regular networks an infection persists if the transmission rate is beyond a finite epidemic threshold, on scale-free networks there is no epidemic threshold, hence infections can spread and persist independently of their transmission rate (Pastor-Satorras and Vespignani 2001). Degree heterogeneity also impacts evolutionary dynamics, and higher heterogeneity has been shown to favour cooperation over defection (Santos et al. 2006). The family of multi-regular graphs is a better representation of real world networks than regular graphs because it allows degree heterogeneity, and at the same time, their local homogeneity allows to derive an analytic expression for the replicator dynamics. Moreover the numerical simulations suggest (but we have no proof) that even if the real population is not structured in degree-regular communities, the replicator dynamics on a multi-regular graph with the same degree distribution of the real population is not far from the dynamics on the real population most of the times. Multi-regular graphs A multi-regular graph G is a connected graph partitioned into m degree-homogeneous communities $C^{i}_{k}$, i={1,…,m}, where each node in community $C^{i}_{k}$ has degree k, and k≥3. In each community $C^{i}_{k}$ the number of nodes ni is at least k+1, and nik must be even. Moreover, the number of connections between different communities must be even. For each community $C^{i}_{k}$, call interior those nodes which neighbourhood is entirely contained in the community, and frontier those which have at least one neighbour in a different community. Notice that we require ni≥k+1 to ensure the existence of a regular graph of degree k on ni nodes, and that we require an even number of edges between nodes in $C^{i}_{k}$ and nodes outside said community to guarantee that each node in $C^{i}_{k}$ has degree k. To provide intuition, consider we want a multi-regular graph with two communities of degree k1 and k2 respectively, and we start with two disconnected regular components of degree k1 and k2. If we connect the two components by adding an edge between them, then the two frontier nodes will have degree k1+1 and k2+1 respectively, violating the condition for being in a degree-homogeneous community. If for each of the two frontier vertices we erase one edge other than the one connecting them, then there will be two other nodes (one for each community) violating that condition, as those will now have degree k1−1 and k2−1 respectively. If we connect these two nodes then regularity condition is restored. Notice also that the definition of multi-regular graph implies that the minimal community size is 4, but we are never going to consider such small communities in this work, as the replicator equation provided is a good approximation for large graphs (with at least 105 nodes). Generating a random multi-regular graph Here I propose an algorithm to generate a multi-regular graph on n nodes knowing the degree distribution $\mathbb {P}(k)$, based on the Pairing model. Assume that the number of nodes with degree k, nk is given by the nearest even integer $[\!n \mathbb {P}(k)]$, and that each community has a fraction r of its connections between interior nodes. The algorithm goes as follows: generate $\sum _{k} n_{k} k $ points. divide the points in nk buckets in this way: take nk points and put each in a different bucket. add k−1 points to each of these buckets. repeat the procedure for all different k, such that for all degrees k there will be $n_{k} \mathbb {P}(k)$ buckets with k points each. take a random point, say it is in a bucket with k points join it with probability r to a random point in one of the $n_{k} \mathbb {P}(k)$ buckets with k points, and with probability 1−r to any of the other points at random. continue until a perfect matching is reached. collapse the points, so that each bucket maps onto a single node and all edges between points map onto edges of the corresponding nodes. check if the obtained graph is simple (e.g. it has no loops or multiple edges). Replicator equation on multi-regular graphs On each of the regular communities taken in isolation, under the assumption that local dynamics are only affected by the strategies of players' immediate neighbours (so if clustering is negligible), the replicator dynamics is well approximated by the equation presented in the previous section. Provided that the fraction of connections between different communities is low, and that the number of nodes in each community is large, the global dynamics on a graph with regular communities is given by: $$ \dot{x}_{s}=x_{s}\left(f_{s}+\sum_{k_{i} \ge 3} \sum_{j} x_{j} {b_{ij}}({k_{i}}, \Pi) \mathbb{P}\left[\!C_{k_{i}}\right] - \phi\right) $$ where ki is the degree of nodes inside community i and $\mathbb {P}\left [C_{k_{i}}\right ] $ is the probability that a node is in a community with degree ki, or the fraction of nodes in a community with degree ki, so that the global dynamic is a weighted average of the local dynamics on each community (Cassese 2017). Prisoner's dilemma Prisoner's Dilemma is one of the benchmark games for the study of cooperation (Doebli et al. 2004; Lieberman et al. 2005; Ohtsuki et al. 2005; Nowak 2006b; Axelrod and Hamilton 1981). It is a symmetric game in two strategies, Cooperate and Defect as can be seen in Table 1, with one strictly dominant strategy, Defect, which is the only strict Nash Equilibrium and so the only evolutionary stable strategy in the mean-field dynamics. It has already been shown that if the structure of the population is taken in consideration then there can be instances when cooperation prevails, for example (Ohtsuki and Nowak 2006) show that, in regular graphs with death-birth updating, if b/c>k, where k is the degree of the graph, cooperation prevails over defection, and similarly for Imitation updating this happens if b/c>k+2. Under birth-death updating they find that defection always prevails. Table 1 Prisoner's dilemma On a graph with regular communities similar conditions for the prevalence of cooperation can be found, namely under birth-death updating defection is always the only evolutionary stable strategy, for death-birth cooperation prevails if: $$ \frac{b}{c}>\sum_{k_{i}}k_{i}\mathbb{P}\left[\!C_{k_{i}}\right] $$ analogously for imitation, cooperation prevails if: $$ \frac{b}{c}>\sum_{k_{i}}(k_{i}+2)\mathbb{P}\left[\!C_{k_{i}}\right] $$ Notice that the above conditions say that the benefit-cost ratio sufficient to sustain cooperation in equilibrium increases with average connectivity in a graphs with regular communities. These conditions are sufficient but not necessary, as I proved in more details in (Cassese 2017), given that the true benefit-cost thresholds that promote cooperations under the two different mechanisms are bounded above by $\sum _{k_{i}}k_{i}\mathbb {P}\left [\!C_{k_{i}}\right ]$ and $\sum _{k_{i}}(k_{i}+2)\mathbb {P}\left [\!C_{k_{i}}\right ]$ respectively, so a graph with regular communities and degree distribution $\mathbb {P}(k)$ is more favourable to cooperation than a graph with the same degree distribution where the communities are disconnected, so that the graph has as many connected components as the number of communities. Comparing the difference between the bounds and the true thresholds numerically, it appears that this difference is always greater for imitation than birth-death, meaning that imitation promotes cooperation more than birth-death for the Prisoner's dilemma. In (Cassese 2017) I also show that there can be cases where cooperation and defection coexist, so there is a stable mixed equilibrium. Using a colour map like in Fig. 1, this case can be seen in Fig. 2 where cooperation levels in equilibrium for a graph with three communities (degree 3, 4 and 5 respectively) are reported for a benefit-cost ratio of 10/3: when average degree is less than 10/3 cooperation prevails, and for values of the average connectivity around 10/3 there are few mixed-equilibria. Probability colour map. Each point in the simplex represent a probability triple given by barycentric coordinates, and each point is mapped to a colour. In a graph with three regular communities, each coordinate represent the probability for a node of being in the corresponding community, where red is k=3, blue k=4 and green k=5 for the Prisoner's dilemma and Coordination games and k=7 in the Hawk-Dove game Prisoner's dilemma, death-birth Fraction of cooperators in equilibrium as the graph structure change. The graph has three communities, k=3, k=4, k=5. The benefit-cost ratio is b/c=10/3, so when average degree is more than 10/3 defection prevails. The plot also shows few cases where cooperators and defectors coexist in equilibrium Hawk-Dove game The Hawk-Dove game (or snowdrift) has also extensively been used to study cooperation. The game describes a situation where two players engage to gain a prize b, and they can either choose to fight to take it all for themselves or to share it with the opponent. Hawks are assumed to be confrontational, they always fight; the cost of losing a fight is c: if two hawks face each other they will get an expected payoff of (b−c)/2. Doves are peaceful, if facing an aggressive hawk they will just leave, getting a payoff of 0 and leaving all the prize to their opponent, while if they meet another dove they will equally share the prize, getting b/2 each. The payoffs structure is described by Table 2 where is assumed that c>b. Table 2 Hawk-Dove This game has a similar structure to the Prisoner's Dilemma, as both parties have incentive to defect and fight to obtain a higher payoff, but a reciprocal aggressive behaviour is detrimental (in expectation) for both. While the Prisoner's dilemma has a unique dominant strategy, which is mutual defection, Hawk-Dove has two Nash equilibria in pure strategies, namely (Hawk, Dove) and (Dove, Hawk), and one equilibrium in mixed strategies, (Hawk, Dove) = (b/c, 1−b/c). The mixed strategy corresponds to the Evolutionary Stable Strategy in a mean-field evolutionary game, where the equilibrium frequency of hawks is equal to b/c. The equilibrium where everybody in the population is a dove is unstable as long as b>0, so cooperation will never prevail in the mean-field case. Let us first study the game on a regular graph of degree k under the three different updating mechanisms. The stable equilibrium under death-birth is $x_{d}^{} = \left (bk^{2} - bk -ck^{2} +c\right)/c\left (-k^{2}+k+2\right)$, where $x_{d}^{}<1$ when c/b<k(k−1)/(k+1). It is easy to check that the equilibrium level of cooperation on a regular graph is greater than the equilibrium in the mean-field case when c/b>2/(k+1), which means that a regular graph always favours cooperation over defection, and the same holds for graphs with regular communities. Computing the equilibria for imitation updating, we can see that the stable equilibrium is [b(−k2−k)+c(k2+2k−3)]/[c(k2+k−6)], which is a non-degenerate mixed equilibrium when c/b<k(k+1)/(k+3) and it is greater than the mean-field when c/b>6/(k+3) which again always holds for k≥3 on both regular graphs, and graphs with degree regular communities. The fixed point $x^{}_{d}=1$ is locally stable when $\frac {d\dot {x}_{d}}{{dx}_{d}}\big \|_{x_{d}=1} < 0$, so by studying the sign of $\frac {d\dot {x}_{d}}{{dx}_{d}}\big \|_{x_{d}=1}$ it is easy to determine the conditions under which doves dominate over hawks, who become extinct. With birth-death updating we have that cooperation is a stable point of the dynamics when c/b>k in the case of regular graph, and on a graph with regular communities this is true when: $$ \frac{c}{b} > \frac{\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] k_{i} \prod_{j \ne i} (k_{j} - 2)}{\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] \prod_{j \ne i} (k_{j} - 2)} $$ The right-hand side of (7) is bounded above by $ \sum _{i} \mathbb {P}\left [C_{k_{i}}\right ] k_{i} $ if the numerator of their difference is non-negative, as the denominator $\sum _{i} \mathbb {P}\left [C_{k_{i}}\right ] \prod _{j \ne i} (k_{j} - 2)$ is always positive. This reads: $$ \left[\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] k_{i} \right] \left[\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] \prod_{j \ne i} (k_{j} - 2) \right] - \sum_{i} \mathbb{P}\left[C_{k_{i}}\right] k_{i} \prod_{j \ne i} (k_{j} - 2) \ge 0 $$ (8) can be rewritten as : $$ \sum_{i,j} {P}\left[C_{k_{i}}\right]{P}\left[C_{k_{j}}\right] \prod_{l \ne i,j} (k_{l} - 2) (k_{i} - k_{j})^{2}\ge 0 $$ which is always true as ki≥3 for all i. So $$ \frac{c}{b} > \sum_{i} \mathbb{P}\left[C_{k_{i}}\right] k_{i} $$ is a sufficient condition for doves to prevail. With death-birth updating doves prevail when c/b>k(k−1)/(k+1) for regular graphs, while for a graph with regular communities $\frac {d\dot {x}_{d}}{{dx}_{d}}\big \|_{x_{d}=1} < 0$ when: $$ \frac{c}{b} > \frac{\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] k_{i} (k_{i} -1) \prod_{j \ne i} (k_{j} - 2)(k_{j} + 1)} {\sum_{i} \mathbb{P}\left[C_{k_{i}}\right](k_{i} + 1) \prod_{j \ne i} (k_{j} - 2)(k_{j} + 1)} $$ to prove that (11) is bounded above by $\sum _{i} \frac {k_{i}(k_{i} - 1)}{k_{i} + 1} \mathbb {P}[C_{k_{i}}] $ is sufficient to prove that: $$ \begin{aligned} & \sum_{i} \left[\mathbb{P}\left[C_{k_{i}}\right] (k_{i}-1) \prod_{j \ne i}(k_{j}+1) \right] \left[\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] (k_{i}+1) \prod_{j \ne i} (k_{j} - 2)(k_{j} + 1) \right] - \\ & \sum_{i} \mathbb{P}\left[C_{k_{i}}\right] k_{i} (k_{i} -1) \prod_{j \ne i}(k_{j} - 2)(k_{j} + 1) \ge 0 \end{aligned} $$ (12) is the numerator of the difference between $\sum _{i} \frac {k_{i}(k_{i} - 1)}{k_{i} + 1} \mathbb {P}\left [C_{k_{i}}\right ] $ and (11), and the denominator $\sum _{i} \mathbb {P}\left [C_{k_{i}}\right ] (k_{i}+1)\prod _{j \ne i}(k_{j}+1)(k_{j}-2)$ is always positive. $$ \sum_{i,j \in C(n,2)} \mathbb{P}\left[C_{k_{i}}\right] \mathbb{P}\left[C_{k_{j}}\right] (k_{i}-k_{j})^{2}(k_{i} k_{j} + k_{i} + k_{j} -1) \prod_{l \ne i,j}(k_{l}-2)(k_{l}+1) \ge 0 $$ where C(n,2) is the set of 2-combinations of the n indices. Equation (16) is never less than zero as ki≥3 for all i, hence: $$ \frac{c}{b} > \sum_{i} \frac{k_{i}(k_{i} - 1)}{k_{i} + 1} \mathbb{P}\left[C_{k_{i}}\right] $$ An example of how the fraction of cooperators in equilibrium depends on the interaction structure can be seen in Fig. 3. Hawk-Dove, death-birth. Fraction of cooperators for the Hawk-Dove game as the graph structure change. The three communities here have degree k=3, k=4, k=7, and c/b=3/8. The black triangle is the level of cooperation in the mean-field case, at x∗=5/8. When $\sum _{i} \frac {k_{i}(k_{i}-1)}{(k_{i} +1)} > 8/3$ cooperation prevails, while for all other cases hawks and doves coexist in equilibrium, with a minimum level of cooperation when the graph is 5-regular Analogously for imitation updating cooperation prevails for c/b>k(k+1)/(k+3) on regular graphs. On graphs with degree regular communities $\frac {d\dot {x}_{d}}{{dx}_{d}}\big \|_{x_{d}=1} < 0$ when: $$ \frac{c}{b} >\frac{\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] k_{i} (k_{i} + 1)\prod_{j \ne i} (k_{j} - 2) (k_{j} + 3)} {\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] (k_{i} + 3) \prod_{j \ne i} (k_{j} - 2) (k_{j} + 3)} $$ again to prove that (15) is bounded above by $\sum _{i} \frac {k_{i}(k_{i} + 1)}{k_{i} + 3} \mathbb {P}\left [C_{k_{i}}\right ]$ it suffices to show that the numerator of the difference between $\sum _{i} \frac {k_{i}(k_{i} + 1)}{k_{i} + 3} \mathbb {P}\left [C_{k_{i}}\right ]$ and (15) is non-negative, as the denominator $\sum _{i} \mathbb {P}\left [C_{k_{i}}\right ] (k_{i}+3)\prod _{j \ne i}(k_{j}+4)(k_{j}-2)$ is always positive. The numerator of the difference is: $$ \sum_{i,j \in C(n,2)} \mathbb{P}\left[C_{k_{i}}\right] \mathbb{P}\left[C_{k_{j}}\right] (k_{i}-k_{j})^{2}(k_{i} k_{j} +3 k_{i} +3 k_{j} +3) \prod_{l \ne i,j}(k_{l}-2)(k_{l}+3) \ge 0 $$ where C(n,2) is the set of 2-combinations of the n indices as above. Clearly (16) is always non-negative as ki≥3 for all i. Hence a sufficient condition for doves to prevail with imitation updating is: $$ \frac{c}{b} > \sum_{i} \frac{k_{i}(k_{i} + 1)}{k_{i} + 3} \mathbb{P}\left[C_{k_{i}}\right] $$ In conclusion reaching cooperation in a Hawk-Dove game on graphs with regular communities is easier than in a corresponding graph with disconnected regular components, in the sense that cooperation is sustainable with a lower relative cost of the aggressive behaviour. Moreover numerical simulations show that, if we compare the distance between the bounds and the true thresholds, we can see that this distance is always greater for imitation, meaning that imitation promotes cooperation more than the other two mechanisms, as it is the case for Prisoner's dilemma as well. Coordination game A coordination game is a two-strategies game with the payoff structure given in Table 3 where a>c and d>b. Table 3 Coordination game The game describes a coordination problem between two individuals, who could coordinate on an action A that is more beneficial for both if done together, but detrimental if done on one's own. This game has two Nash equilibria in pure strategies (both A and B), and when a+b<c+dB is risk dominant, as it has the largest basin of attraction, while if a>d, A is Pareto-efficient as it yields a higher payoff for both. Consider the case where b=0, c=1, d=2 and 1<a<3. In the mean-field case there is an unstable equilibrium at $x_{a}^{} = 2/(1+a)$, while both A and B are stable. Under birth-death updating on regular graphs the basin of attraction of strategy B is always larger than in the mean-field case, and this naturally extends to graphs with regular communities, as can be seen in Fig. 4. Under death-birth updating (Ohtsuki and Nowak 2006) show that for a regular graph with degre k, if a>(3k+1)/(k+1) then A is both payoff and risk dominant, while the same holds for imitation updating if a>(3k+7)/(k+3). I find an analogous condition for the coordination game on graphs with regular communities, namely $$ a > \frac{2 \prod_{i} (k_{i} + 1)(k_{i}-2) + \sum_{i} \mathbb{P}\left[C_{k_{i}}\right] \left(2k_{i}^{2}-1\right)\prod_{j\ne i} (k_{j} +1) (k_{j}-2)} {\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] (k_{i} + 1) \prod_{j\ne i} (k_{j} +1) (k_{j}-2) - 2 \prod_{i} (k_{i} + 1)(ki-2)} $$ Coordination game, birth-death. The graph has three communities, respectively of degree k=3, k=4, k=5, colours represent the position in the probability simplex above, hence the triple (P3,P4,P5) reporting the probability a node is in each of the three communities. The coloured surface represents the separation between the basins of attraction, where the volume above the surface is the basin of A and that below is the basin of B. The light-blue plane is the set of points where the two basins are equal. For birth-death the basin of attraction of B is always larger than that of A, so risk-dominance is favourite for death-birth updating. It can be shown numerically that (18) is bounded above by $\sum _{i} \frac {3 k_{i} + 1}{k_{i} + 1} \mathbb {P}\left [C_{k_{i}}\right ] $, so a sufficient condition for A to be both payoff and risk dominant is: $$ a > \sum_{i} \frac{3 k_{i} + 1}{k_{i} + 1} \mathbb{P}\left[C_{k_{i}}\right] $$ while for imitation updating this is true when: $$ a > \frac{ \sum_{i} \mathbb{P}\left[C_{k_{i}}\right] \left(4k_{i}^{2}+8k_{i}-6\right) \prod_{j\ne i} (k_{j} +3) (k_{j}-2) - \prod_{i} (k_{i}+3)(k_{i}-2)} {\sum_{i} \mathbb{P}\left[C_{k_{i}}\right] (2k_{i} + 6) \prod_{j\ne i} (k_{j} +3) (k_{j}-2) + \prod_{i} (k_{i}+3)(k_{i}-2)} $$ again it can be shown numerically that (20) is bounded above by $\sum _{i} \frac {3 k_{i} + 7}{k_{i} + 3} \mathbb {P}\left [C_{k_{i}}\right ]$, so a sufficient condition for A to be both payoff and risk dominant with imitation updating is: Figures 5, 6 show the basin of attraction on a graph with three communities for death-birth updating and imitation updating respectively, as a function of a and average degree. When a is sufficiently large the strategy A has the larger basin of attraction, so Pareto-efficiency is favoured over risk-dominance for birth-death and imitation. Coordination game, death-birth. The graph has three communities, respectively of degree k=3, k=4, k=5, colours represent the position in the probability simplex above, hence the triple (P3,P4,P5) reporting the probability a node is in each of the three communities. The coloured surface represents the separation between the basins of attraction, where the volume above the surface is the basin of A and that below is the basin of B. The light-blue plane is the set of points where the two basins are equal. For death-birth the basin of attraction of A can be larger than that of B for a close to 3. Death-birth may promote Pareto-efficiency over risk-dominance Coordination game, imitation. The graph has three communities, respectively of degree k=3, k=4, k=5, colours represent the position in the probability simplex above, hence the triple (P3,P4,P5) reporting the probability a node is in each of the three communities. The coloured surface represents the separation between the basins of attraction, where the volume above the surface is the basin of A and that below is the basin of B. The light-blue plane is the set of points where the two basins are equal. Similar to birth-death, also for imitation the basin of attraction of A can be larger than that of B for a close to 3, so also imitation may promote Pareto-efficiency over risk-dominance, but less than birth-death, as can be seen comparing the volumes below the light-blue plane in the two cases In this paper I presented an extension of my previous work (Cassese 2017), providing a version of the replicator equation for a family of graphs characterised by degree-regular communities. As examples of possible application of this equation, here I study the evolutionary dynamics of three game classes: Prisoner's dilemma, Hawk-Dove and Coordination games. It is shown that graphs with degree-regular communities promote cooperation both in the Prisoner's dilemma and in the Hawk-Dove game for imitation and death-birth updating, and that imitation updating in both cases is more favourable to cooperation than death-birth. The results confirm that higher degree heterogeneity favours cooperation, and this can be better understood by comparing the dynamics on a multi-regular graph with the dynamics on a graph with disconnected regular components. In the case of the Prisoner's dilemma with birth-death updating, in all those components where the degree is such that b/c>ki cooperators will prevail, viceversa in the other components defectors will prevail (and in some of them we could also have a mixed equilibrium). So the only way to have cooperation prevailing globally is b/c>kmax+2, where kmax is the largest degree of the graph. Adding a few connections between these regular components, as we do in a multi-regular graph, changes the picture completely, and cooperation prevails if b/c is greater than the average degree, which is a much easier condition to meet. The same is true for imitation updating, where we would have that each disconnected component may reach a different equilibrium depending on their degree, with cooperation prevailing locally where b/c>ki+2, and globally only if b/c>kmax+2, while on a multi-regular graph we have the milder condition $b/c>\sum _{i} (k_{i}+2) \mathbb {P}\left [C_{k_{i}}\right ]$. Analogously, for the Hawk-Dove game on a graph with regular disconnected components, cooperation prevails globally if c/b>kmax for birth-death, c/b>kmax(kmax−1)/(kmax+1) for death-birth and c/b>kmax(kmax+1)/(kmax+3) for imitation, and each of these conditions is stronger than the corresponding condition on multi-regular graphs as in Eqs. (10), (14), (17) respectively. If these conditions are not met, each disconnected component will be in a different equilibrium depending on its degree, with some components where doves prevail, others where the two strategies coexist. In the Coordination game on graphs with regular disconnected components, the Pareto-efficient strategy needs to yield a higher payoff than the one needed on a multi-regular graph in order to be both Pareto-efficient and risk-dominant globally, so we can say that graphs in this family promote Pareto-efficiency over risk-dominance. 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The author thanks an anonymous reviewer and Hisashi Ohtsuki for useful comments. The author wishes to thank support from FNRS (Belgium). Department of Mathematics, University of Namur, NaXys, Rempart de la Vierge 8, Namur, Belgium Daniele Cassese ICTEAM, University of Louvain, Av Georges Lemaître, Louvain-la-Neuve, Belgium Oxford Mathematical Institute, Woodstock Road, Oxford, OX2 6GG, UK DC was responsible for the research in full. The author read and approved the final manuscript. Correspondence to Daniele Cassese. The author declares that he has no competing interests. Cassese, D. Replicator equation on networks with degree regular communities. Appl Netw Sci 3, 29 (2018). https://doi.org/10.1007/s41109-018-0083-2 Replicator equation Evolutionary graph theory Hawk-Dove Special Issue of the 6th International Conference on Complex Networks and Their Applications	CommonCrawl
Presentation intervals and the impact of delay on breast cancer progression in a black African population Olayide Agodirin ORCID: orcid.org/0000-0001-7102-864X1, Samuel Olatoke1, Ganiyu Rahman2, Julius Olaogun3, Olalekan Olasehinde4, Aba Katung5, Oladapo Kolawole6, Omobolaji Ayandipo7, Amarachukwu Etonyeaku8, Olufemi Habeeb1, Ademola Adeyeye9, John Agboola10, Halimat Akande11, Soliu Oguntola12, Olusola Akanbi12 & Oluwafemi Fatudimu13 The help-seeking interval and primary-care interval are points of delays in breast cancer presentation. To inform future intervention targeting early diagnosis of breast cancer, we described the contribution of each interval to the delay and the impact of delay on tumor progression. We conducted a multicentered survey from June 2017 to May 2018 hypothesizing that most patients visited the first healthcare provider within 60 days of tumor detection. Inferential statistics were by t-test, chi-square test, and Wilcoxon-Signed Rank test at p-value 0.05 or 95% confidence limits. Time-to-event was by survival method. Multivariate analysis was by logistic regression. Respondents were females between 24 and 95 years (n = 420). Most respondents visited FHP within 60 days of detecting symptoms (230 (60, 95% CI 53–63). Most had long primary-care (237 of 377 (64 95% CI 59–68) and detection-to-specialist (293 (73% (95% CI 68–77)) intervals. The primary care interval (median 106 days, IQR 13–337) was longer than the help-seeking interval (median 42 days, IQR 7–150) Wilcoxon signed-rank test p = 0.001. There was a strong correlation between the length of primary care interval and the detection-to-specialist interval (r = 0.9, 95% CI 0.88–0.92). Patronizing the hospital, receiving the correct advice, and having a big tumor (> 5 cm) were associated with short intervals. Tumors were detected early, but most became advanced before arriving at the specialist clinic. The difference in tumor size between detection and arriving at a specialist clinic was 5.0 ± 4.9 cm (95% CI 4.0–5.0). The hazard of progressing from early to locally advanced disease was least in the first 30 days (3%). The hazard was 31% in 90 days. Most respondents presented early to the first healthcare provider, but most arrived late at a specialist clinic. The primary care interval was longer than the help-seeking interval. Most tumors were early at detection but locally advanced before arriving in a specialist clinic. Interventions aiming to shorten the primary care interval will have the most impact on time to breast cancer presentation for specialist oncology care in Nigeria. Breast cancer (BC) patients in low and middle-income countries (LMICs) and black patients in developed countries harbor symptoms for up to 8–12 months [1,2,3,4,5] before diagnosis and treatment thereby increasing the risk of poor outcome and limiting treatment efficacy [6,7,8,9]. Historically, two delay components are recognized in cancer treatment: the patients' delay and the systems' delay [10, 11]. A recent definition proposes replacing the word delay with the word interval [12]. As illustrated in Olsen et al., the Aarhus statement [13] recognized three subintervals between symptom detection and cancer treatment: (1) The patient-interval (comprising symptom appraisal and help-seeking intervals (HSI) as in Dobson et al. [12]) (2). the doctor interval, and (3) the system interval [13]. The subinterval classification aids the understanding of the continuum by giving more details about the subcomponents [12]. Shortening the interval to treatment is pivotal in controlling BC outcome [10], yet in LMICs where the mortality of the disease is disproportionately high, only a few studies address the intervals or journey of BC patients to treatment [14,15,16]. Factors linked to delayed presentation of breast cancer are often modifiable—changing with intervention. While much of the focus has been on the events in the patient-interval as causes of delayed presentation, recent reports in Nigeria [17], Ghana [18], and Rwanda [3] show an increasing contribution of events in the provider interval. An understanding of factors influencing the length of each interval is critical to effective interventions; therefore, this research aimed to describe the journey of BC patients from symptom detection to the specialist clinic in a black African population. The primary objective was to describe the contribution of each interval to the continuum. The secondary objectives were (1) to describe the association between the interval length and the socio-demographics, the disease-related experience(s), and system-related experience(s) (2) To describe the impact of long intervals on BC progression. This research was a questionnaire-based survey in 6 tertiary hospitals in Northcentral and Southwestern Nigeria. The hospitals received referrals from lower cadre public hospitals, private hospitals, or walk-in (self-referral). Recruitment of respondents was between June 2017 and May 2018 after obtaining ethical approval from all participating institutions. Consecutive newly diagnosed BC patients who consented to participate in the study were recruited until the predetermined sample size. At the time of the survey, BC patients in Nigeria patronized private and public healthcare services including traditionalists, native healers, faith-based homes, and orthodox medical healthcare providers (community health extension workers (CHEW), nurses, chemists, pharmacists, and doctors). Based on piloting on 30 respondents where 80% visited the first healthcare provider (FHP) within 30 days of symptom detection, we hypothesized that most patients would visit their FHP within 60 days. Our sample size was 384, calculating for a descriptive cross-sectional study at a relative precision of 5% and a confidence level of 95% (1.96). We increased the sample size to 423 in anticipation of a 10% nonresponse rate. Based on insight from the Aarhus statement [13], and review of the methods implemented by Varella-Centelles et al. [19] and Moodley et al. [16], a semi-structured questionnaire was designed and the plan of data collection was mapped. The questionnaire was pilot tested before trained personnel administered it in face-to-face interviews. The questionnaire requested information on socio-demographics, recall of first breast bodily change and events surrounding it, disclosure, and help-seeking patterns (see Additional file 1). The questionnaire was administered to respondents within four weeks of arriving in the specialist clinic (SC) to minimize recall bias. Additionally, respondents were helped to cast their minds back on significant personal, social, religious, regional, or national events surrounding the recalled dates or elapsed periods. Attempts to check for reliability of response or to reconcile discrepancies was by triangulation when possible. [Such as BC awareness, path-way to treatment, personnel(s) visited, tumor size and interval lengths]. Questions likely to influence subsequent response questions were delayed. For instance, religious affiliations were delayed until after responding to the use of alternative medicine. The interview was after the day's medical consultation in the patient's mother tongue or English as preferred by the respondent, noting the use of a translator. Recurrent lesions, language barriers, mental incapacitation, and male sex were exclusion criteria. Clinical tumor size (T-size) estimated by the patient was used as the surrogate for disease stage using the T1–3 as in the 7th edition of the American Joint Committee on Cancer (AJCC) staging for BC, where T1 was ≤2 cm, T2 was 2.1–5 cm, and T3 was > 5 cm. Using the practical routine process of extracting clinical history that relies on the patients' retrospective recall, respondents were asked to estimate their tumor size (self-report) at the following three points: first detection, first contact with FHP, and at first SC attendance. A ruler was used to quantify the estimated tumor size in centimeters, as demonstrated by the patient using their phalanx, finger(s), or clenched fist(s). The T-size estimated at the SC was taken as the current tumor size. The patients' estimate was considered unreliable and excluded from analysis if the current size estimated differed by more than 2 cm from the T-size measured physically and recorded by the clinician in the SC. In defining interval lengths, we used logical arithmetic derivatives of interval lengths in previous works and recommendations [4, 12, 13] wherein total delay was > 90 days, and provider delay was > 30 days. However, to be pragmatic, we considered our health-system and our patients' behavior to operationalize the interval lengths. Therefore we operationalized the interval lengths as follows: Appraisal interval (API)—the period from the detection of first breast symptom to first disclosure. Long was > 30 days) (the API was essential to estimate how long the patients keep the lesion secret). Help-seeking interval (HSI)—the period from symptom detection to FHP. Long was > 60 days)(HSI was essential to estimate how long the patient stayed before seeking help). Primary-care interval (PCI)— the period from the FHP to a specialist clinic. Long was > 30 days) (PCI was essential to estimate interval length attributable to provider system). And symptom detection-to-specialist clinic interval(SCI)—the period from detection of the first symptom to arriving in a specialist clinic. Long was > 90 days. (SCI was essential because, in Nigeria, the majority of patients receive tumor-specific therapy in specialist clinics, and long SCI is rife) (Fig. 1). We recorded the intervals in days, weeks, or months, multiplying recording made in weeks by 7 to convert to days, and those made in months by 30 to convert to days. Description of the intervals We compared variables using the chi-square test, paired t-test, Wilcoxon Signed-Rank test, and logistic regression for odds of events as appropriate. We used the correlation coefficient for the relationship between continuous variables. We conducted the time-to-event analysis using the survival method on the assumption that tumor progression depended on elapsed time alone. The analysis of progression was limited to 365 days in 30-day time segments because most patients in Africa present within 8–12 months. A statistically significant p-value (two-sided) was 5%. We presented the number of respondents traversing each interval and compared their relative probabilities because we expect that future interventions will focus on increasing the probability of favorable events such as short intervals. We presented a wide range of result patterns because of the dearth of information despite the importance of BC in Africa also because there is a lack of consensus on whether to analyze as parametric or nonparametric variables [12]. Additionally, we expected that these figures might serve different purposes in future researches. Demographics and premorbid preferences There were 423 respondents; we excluded three males leaving 420 females. The majority were the Yoruba tribe, only one required interpreter. The modal age decade was the fifth (Table 1). Most respondents (323 of 358(90, 95%CI 87–93) [62 unspecified] preferred orthodox medical care before noticing their symptoms, but most did not utilize BC screening. Only 6.0%(95% CI 4.0–9.0%) performed self-breast examination monthly (Table 1b) hence most lumps were detected inadvertently. Table 1 The Demographic characteristics of respondents showing age distribution, educational status, marital status, religion, occupation, tribe, place of the interview, the respondents' premorbid pattern of help-seeking for medical service and the premorbid utilization of breast cancer screening modalities Comparative length of intervals The PCI (median 106, 13–337) was significantly longer than the HSI (median 42, 7–150), Wilcoxon-Signed Rank test p = 0.0001.(paired t-test mean difference 140 ± 442 days (95% CI 95–186). Most respondents disclosed early within 30 days (330 (81, 95% CI 77–85) and consulted FHP within 60 days (230 (60, 95% CI 53–63). Most respondents had long PCI of > 30 days.(1–7 days in 91(25% (95% CI 20–29), 1–30 days in 134 (36 95% CI 31–41) and > 30 days in 237 out of 377(64 95% CI 59–68). The SCI was > 90 days in 293 of 401 (73% (95% CI 68–77), 91–180 days in 70 of 401 (17% (95% CI 14–22) and > 180 days in 226 of 401 (56% (95% CI 51–61) (Table 2). Table 2 Showing the cumulative number of respondents with increasing time segments in the intervals: Most respondents disclosed early and consulted FHP early. Most respondents had a long primary-care interval and an extended detection to specialist interval Pattern of disclosure and factors influencing API Most respondents informed the first person (primary person) early, and the husband was the most common primary person. The primary person offered the correct advice often (Table 3b), and 276 of 399 (69.2%) acted in tandem with the advice received within 2 weeks. Patronizing orthodox care, being married, and being younger were associated with early disclosure (Table 4) in the unadjusted logistic regression analysis. In the adjusted analysis combining age, premorbid preference, and marital status to predict early disclosure, only premorbid preference and marital status were significant. Table 3 Showing the distribution of the first person(person1) the respondents informed about their breast symptom(s), the pattern of directives received from the first person(Person1), the first orthodox medical personnel, the number of persons informed and number of personnel visited Table 4 Showing the probability of short appraisal or short help-seeking interval based on the specific sociodemographic characteristics, the premorbid exposure, and experience(s) after disclosure and during help-seeking Patterns of FHP attendance and factors influencing HSI Most respondents (355 of 417(85 95% CI 81–88) first sought orthodox medical care. The most common FHP was a general practitioner (Table 1). A total of 63 (15% (95% CI 12–19) first sought alternative care. The majority of respondents who were hospital goers before detecting their breast symptom still visited a hospital first for treatment (275 of 323 (85 95% CI 81–89). The odds of visiting hospital first vs. switching to alternative care was 2.3 (1.0–5.1) among this subgroup of patients. Receiving correct advice(asking the patient to visit a hospital, to visit orthodox healthcare provider, or go for investigation) from person1 and patronizing hospital for other illnesses were both associated with short HSI (Table 4). There was a weak correlation between the length of API and the length of the help-seeking interval. $$ \mathrm{r}=0.13\ \left(95\%\mathrm{CI}\ 0.03-0.23\right) $$ Factors influencing the length of the PCI More respondents with big (> 5 cm) tumors received correct advice compared to those with small tumors (Risk difference 5.5% (95% CI 4.0–15). The probability of correct advice was higher among the doctor FHP compared to nondoctor FHP (Risk difference 8.4 (95% CI 3.2, 20). In the unadjusted analysis, receiving correct advice and having a big tumor were associated with short PCI. Only receiving correct advice was significant in the adjusted odds ratio (AOR) (Table 4). Relationship between the component intervals and the SCI The PCI strongly correlated with the SCI (r = 0.9, 95% CI 0.88–0.92). Other intervals correlated weakly with the SCI. (API r = 0.3 95% CI 0.22–0.40 and HSI r = 0.38, 95% CI 0.30–0.47). There was a high probability of having a short SCI after traversing any component interval quickly (Table 5). The odds ratio (OR) for a short SCI vs. long SCI among those who had short API was 6.5 (95% CI 2.6–16.7), among those who had short help-seeking was 11 (95% CI 5.4–2.1) and among those who had short primary-care was 8.3 (95% CI 5.0–14). Table 5 Showing the probability of short primary-care and symptom-detection to specialist interval based on specific sociodemographic risk factors, premorbid exposure and the experience(s) after disclosure and during help-seeking Among those who divulged reasons for the long help-seeking intervals, symptom misinterpretation or symptom accumulation was 92 (47%), socioeconomic reasons were 47 (24%), and ignorance was 6 (3.0%). Reasons for long primary-care intervals was misdiagnosis by a health care provider in 37 (25%) (Table 6). Table 6 shows the reasons reported for long help-seeking intervals (> 60 days) or long primary-care intervals(> 30 days) by respondents Impact of interval length on the tumor size and risk of T-category progression The self-reported tumor size of 13 patients among the 420 records were unreliable and excluded from the analysis of the growth in tumor size and risk of tumor size progression. Most tumors were estimated as early T-category at detection, whereas most were locally advanced at the specialist clinic (Table 7). Mean difference in T-size was significant(. paired t-test mean difference 5.0 ± 4.9 cm (95% CI 4–5), median 3.0 vs. 8.0 Wilcoxon-Signed Rank test P = 0.0001). Table 7 Showing tumor size at various time segments in the continuum from detection to specialist clinic. Also, showing the risk of progression in tumor size per time segment There was a moderate correlation between the length of the total interval and the growth in tumor size (r = 0.4). The average growth in the tumor size per month was estimated to be 0.4 cm in the first 12 months. The risk of tumor progression within the first 12 months was lowest in the first month (Table 7). The overall risk that a lump would be locally advanced when detected inadvertently was 12% (95%CI 9–16), and the risk that it would migrate to the next T-category before arriving in a specialist clinic was 64% (95% CI 59–69). The OR for T-category progression in SC interval of 31–90 days vs. 1–30 days was 5 (95% CI 2.0–12), and the OR in SC interval > 90 days vs. 1–30 days was 16 (95% CI 7.0–38). Among patients who detected their tumors relatively early (estimated as T1 or T2), the hazard of progressing to advanced-stage increased with time. The hazard was lowest in the first 30 days (3%), 17% in 60 days, 31% in 90 days and 61% in 180 days. In this survey, two-thirds of the respondents stayed longer than three months between detecting BC symptoms and arriving in a specialist clinic. The PCI was the longest interval, and there was a strong correlation between the length of the PCI and the SCI. Symptom misinterpretation and misdiagnosis were frequent reasons for extended intervals. The majority of the patient detected their lesion early, but the majority were already locally advanced before arriving in a specialist clinic. At least two-thirds of our respondents first visited orthodox personnel to seek help and, a similar proportion consulted FHP early in tandem with their advisor's directives. We did not establish the direct influence of advisors on the women's decision nonetheless, the association is consistent with the report in South Africa [16], where patients acted based on pressure from relations. The husbands were the most frequent advisors; hence, they are a potential focus for intervention. Engaging men to promote uptake of positive breast health activities is useful in places where women rely on husband and family support [20], notably, in Africa, where the men dominate the leadership role [21] and politics. Similar to previous reports in Nigeria, [9, 22] a large number of our respondents preferred orthodox care, and the pattern of help-seeking was consistent with their premorbid preference for health care services. This indicates that without unfavorable experience(s), it is unlikely that women will suddenly change their health care preferences once they detected their breast lesions. We can exploit the premorbid conditioning by improving access to our hospitals. Raising satisfaction derived in the hospital for treatment of other minor illnesses might build confidence and enduring relationship between potential breast cancer patients and the clinicians. The longer PCI and its dominating influence compared to the other intervals supports some reports and negates others. Harirchi in Tehran [23] and Yau et al. in Hong Kong [24] reported a higher proportion of patients with help-seeking delay. Moodley et al. first reported a higher proportion of delay in the patient interval among 20 patients in South Africa [16] and then in a subsequent study of 201 patients; they reported a higher proportion of long delays in the system's interval [25]. Roy et al. [26] in Bangladesh and Maghous et al. [27] in Morocco, both reported that doctors were complicit in a third of long interval situations [26, 27]. Also, in Nigeria, Ezeome et al. [9], Ayoade et al. [17] and Akinkuolie et al. [20] reported disease progression during the primary-care interval. The most frequent reason for a long primary-care interval in this study was misdiagnosis by the FHP. We found that smaller tumors were associated with longer intervals. We suspect that smaller tumors were more challenging to evaluate because of limited symptomatology. Instances of symptom misinterpretation and misdiagnosis were also prominent reasons for extended intervals in other studies in Nigeria [4, 28], other parts of Africa [16, 27, 29,30,31], middle east [32] and Asia [28], Ensuring triple assessment rather than depend on physical finding to initiate treatment may reduce misdiagnosis. In our study, the risk of incorrect advice was higher among nondoctor FHP compared to doctor FHP. The doctor FHP and nondoctor FHP had different error patterns, which should be noted during education campaigns. We found that there is an increasing probability of transitioning from early to locally advanced disease as time elapsed, and the risk of transitioning was least in the first 30 days after the detection of early disease, and it more than doubled afterward. One out of every ten women who detected their lumps inadvertently were already locally advanced. Furthermore, one out of every three was likely to be advanced among those who arrived in a specialist clinic after 30 days. This suggests that the strategy to promote early detection and treatment of clinically symptomatic BC in low resource settings [33, 34] may be effective in our patients if implemented with a tight timeline. Although we could not assess the influence of tumor biology on disease progression in this study and we assumed that time was an independent predictor of tumor progression, the common timeline in our literature describing detection to presentation of more than three months as late [9, 17, 27] was lax for this cohort of respondents because at least a third already experienced significant tumor growth within 90 days. The clinical implication of for long detection to treatment interval is not adequately researched in Africa. In a population of BC patients in southern Africa [18], more than 20% were locally advanced in a median time to treatment of 110 days. In Ghana [35], patients who stayed a total interval shorter than 2 months had smaller tumors compared to the total interval of 12 months. In contrast, two-thirds of patients who stayed longer than six months in a study in Uganda [36] still had an early disease. We need more studies to describe the relationship between total interval and outcome in Africans. Our study is the first to explicitly show the relationship between premorbid experience and the pattern of help-seeking among breast cancer patients in sub-Saharan Africa. Our study is also the first to show the likely changes in breast tumor size as time elapsed in a cohort of breast cancer patients in sub-Saharan Africa and the first to show the relationship between the component intervals and their relative influence on the time to a specialist clinic. This research is limited in that the primary outcome was patient-reported; hence it might be influenced by recall bias. We attempted to minimize the bias by interviewing the patients within four weeks of arriving in the specialist clinic. Moreover, we helped them to cast their minds back on significant events occurring around the recalled dates or periods. The self-reported tumor size based on patients' retrospective recall may be inaccurate, and we could not triangulate for its accuracy by comparing the respondent's recall with the primary-care records due to poor record-keeping. Also, we did not evaluate the interaction between tumor biology, the elapsed time, and tumor progression. We were unable to find other ways of estimating tumor size at detection because it was a prehospital event, and we were unable to find other ways of estimating tumor size at contact with the FHP. Nonetheless, we attempted to minimize inconsistency in the self-reported tumor size by using the estimate given by the patient at all points for the analysis, and we excluded overtly inaccurate estimates. Most patients in this study visited the FHP early; however, most stayed longer than 3 months between symptom detection and arriving in a specialist clinic with significant tumor progression in the interval. The PCI was the longest interval. The most common reasons for long intervals were symptom misinterpretation and system-related factors. Data limited for interpreation of results for this research is available on reasonable request to the corresponding author and as supplementary file. Appraisal interval AOR: Adjustted odds ratio AJCC: American Joint Committtee on Cancer BC: BSE: CBE: Clinical breast examination CHEW: Community health extension worker FHP: First healthcare provider Help-seeking interval IQR: Mammo: NS: PCI: Primary care interval SCI: Symptom detection to specialist interal Tumor size Jones CE, Maben J, Jack RH, et al. A systematic review of barriers to early presentation and diagnosis with breast cancer among black women. BMJ Open. 2014;4(2):e004076. Agodirin O, Olatoke S, Rahman G, et al. How effective is the treatment of locally advanced and metastatic breast Cancer in developing Centres?: a retrospective review. Ethiop J Health Sci. 2015;25(4):337–44. Pace LE, Mpunga T, Hategekimana V, et al. Delays in breast Cancer presentation and diagnosis at two rural Cancer referral centers in Rwanda. Oncologist. 2015;20(7):780–8. Ukwenya AY, Yusufu LM, Nmadu PT, Garba ES, Ahmed A. Delayed treatment of symptomatic breast cancer: the experience from Kaduna, Nigeria. S Afr J Surg Suid-Afrikaanse tydskrif vir chirurgie. 2008;46(4):106–10. Jedy-Agba E, McCormack V, Adebamowo C, dos Santos-Silva I. Stage at diagnosis of breast cancer in sub-Saharan Africa: a systematic review and meta-analysis. Lancet Glob Health. 2016;4(12):e923–35. Sharma K, Costas A, Shulman LN, Meara JG. A systematic review of barriers to breast Cancer Care in Developing Countries Resulting in delayed patient presentation. J Oncol. 2012;2012:121873. https://doi.org/10.1155/2012/121873. McKenzie F, Zietsman A, Galukande M, et al. African breast Cancer-disparities in outcomes (ABC-DO): protocol of a multicountry mobile health prospective study of breast cancer survival in sub-Saharan Africa. BMJ Open. 2016;6(8):e011390. da Costa Vieira RA, Biller G, Uemura G, Ruiz CA, Curado MP. Breast cancer screening in developing countries. Clinics (Sao Paulo, Brazil). 2017;72(4):244–53. Ezeome ER. Delays in presentation and treatment of breast cancer in Enugu, Nigeria. Niger J Clin Pract. 2010;13(3):311–6. Pack G, Gallo J. The culpability for delay in the treatment of cancer. Am J Cancer. 1938;33:443–62. Unger-Saldana K, Infante-Castaneda C. Delay of medical care for symptomatic breast cancer: a literature review. Salud publica de Mexico. 2009;51(Suppl 2):s270–85. Dobson CM, Russell AJ, Rubin GP. Patient delay in cancer diagnosis: what do we really mean and can we be more specific? BMC Health Serv Res. 2014;14:387. Weller D, Vedsted P, Rubin G, et al. The Aarhus statement: improving design and reporting of studies on early cancer diagnosis. Br J Cancer. 2012;106(7):1262–7. Unger-Saldana K. Challenges to the early diagnosis and treatment of breast cancer in developing countries. World J Clin Oncol. 2014;5(3):465–77. Norsa'adah B, Rampal KG, Rahmah MA, Naing NN, Biswal BM. Diagnosis delay of breast cancer and its associated factors in Malaysian women. BMC Cancer. 2011;11:141. Moodley J, Cairncross L, Naiker T, Momberg M. Understanding pathways to breast cancer diagnosis among women in the Western Cape Province, South Africa: a qualitative study. BMJ Open. 2016;6(1). Ayoade B, Salami B, Agboola J, et al. Beliefs and practices associated with late presentation in patients with breast cancer; an observational study of patient presenting in a tertiary care facility in Southwest Nigeria. J Afr Cancer. 2015;7(4):178–85. Clegg-Lamptey J, Dakubo J, Attobra YN. Why do breast cancer patients report late or abscond during treatment in Ghana? A pilot study. Ghana Med J. 2009;43(3):127–31. Varela-Centelles P, Lopez-Cedrun JL, Fernandez-Santroman J, et al. Assessment of time intervals in the pathway to oral cancer diagnosis in North-Westerm Spain. Relative contribution of patient interval. Med Oral Patol Oral Cir Bucal. 2017;22(4):e478–e83. Donnelly TT, Al-Khater AH, Al-Bader SB, et al. Perceptions of Arab men regarding female breast cancer screening examinations—Findings from a Middle East study. PLoS One. 2017;12(7). Akinkuolie AA, Etonyeaku AC, Olasehinde O, Arowolo OA, Babalola RN. Breast cancer patients' presentation for oncological treatment: a single Centre study. Pan Afr Med J. 2016;24:63. Olasehinde O, Boutin-Foster C, Alatise OI, et al. Developing a breast Cancer screening program in Nigeria: evaluating current practices, perceptions, and possible barriers. J Glob Oncol. 2017;3(5):490–6. Harirchi I, Karbakhsh M, Hadi F, Madani S, Sirati F, Kolahdoozan S. Patient delay, diagnosis delay and treatment delay for breast cancer: Comparision of the pattern between patients in publi and private health sectors. Arch Breast Cancer. 2015;2(2):15–20. Yau TK, Choi CW, Ng E, Yeung R, Soong IS, Lee AW. Delayed presentation of symptomatic breast cancers in Hong Kong: experience in a public cancer centre. Hong Kong Med J = Xianggang yi xue za zhi. 2010;16(5):373–7. Moodley J, Cairncross L, Naiker T, Constant D. From symptom discovery to treatment - women's pathways to breast cancer care: a cross-sectional study. BMC Cancer. 2018;18(1):312. Chandra Roy B, Naher S, Hanifa M, Sankar P. Pattern of delayed presentation of breast cancer patients: evidence from Rangpur medical college hospital, Rangpu, Bangladesh. Adv Cancer Ransearch Ther. 2015;1(1):1–6. Maghous A, Rais F, Ahid S, et al. Factors influencing diagnosis delay of advanced breast cancer in Moroccan women. BMC Cancer. 2016;16. Akanbi O, Oguntola S, Adeoti M, Aderounmu A, Idris O, Abayomi O. Delay presentation of breast cancer: a study among south western Nigerian women. Int J Curr Res. 2015;7(8):1–5. Khan M, Shafique S, Khan T, Shahzad M, Iqbal S. Presentation delay in breast cancer patients, identifying the barriers in north Pakistan. Asian Pac J Cancer Prev. 2015;16(1):377–80. Kohler RE, Gopal S, Miller AR, et al. A framework for improving early detection of breast cancer in sub-Saharan Africa: a qualitative study of help-seeking behaviors among Malawian women. Patient Educ Couns. 2017;100(1):167–73. Mbuka-Ongona D, Tiumbo J. Knowledge about breast cancer and reasons for late presentation by cancer patients seen at Princess Marina Hospital, Gaborone Botswana. Afr J Prim Health Care Fam Med. 2013;5(1):465–76. Elobaid Y, Aw TC, Lim JNW, Hamid S, Grivna M. Breast cancer presentation delays among Arab and national women in the UAE: a qualitative study. SSM - population health. 2016;2:155–63. dos Santos SI, McCormack V, Jedy-Agba E, Adebamowo C. Downstaging breast Cancer in sub-Saharan Africa: a realistic target? Cancer Control. 2017:46–52. Yip CH, Smith RA, Anderson BO, et al. Guideline implementation for breast healthcare in low- and middle-income countries: early detection resource allocation. Cancer. 2008;113(8 Suppl):2244–56. Brinton L, Figueroa J, Adjei E, et al. Factors contributing to delays in diagnosis of breast cancers in Ghana, West Africa. Breast Cancer Res Treat. 2017;162(1):105–14. Galukande M, Mirembe F, Wabinga H. Patient delay in accessing breast Cancer Care in a sub Saharan African Country: Uganda. Br J Med Med Res. 2014;4(13):2599–610. Agodirin O, Olatoke S, Rahman G, et al. Delay between Breast Cancer Detection and Arrival at Specialist Clinic Preliminary Revelations of Multicentered Survey in Nigeria. Texila Int J Public Health. 2017;5(3). https://doi.org/10.21522/TIJPH.2013.05.04.Art053. Agodirin O, Olatoke S, Rahman G, et al. Impact of Primary Care Delay on Progression of Breast Cancer in a Black African Population: A Multicentered Survey. J Cancer Epidemiol 2019; Article ID 2407138. We thank Prof Peter Kingham of Memorial Sloan Kettering Cancer Center, the United States of America and Dr. Isaac Alatise of Obafemi Awolowo University Teaching Hospital Complex, Nigeria for their support throughout the project. This is the concluding report of the cross-sectional phase of a robust research endeavor to find the drivers of late presentation and poor outcome of breast cancer in Southwestern and Northcentral Nigeria. The research group published a preliminary report of first one hundred respondents [37] and a report on the different clusters of presentation and impact of events in the primary-care interval in response to the observed scanty report about the system interval and influence of primary-care practitioners in Nigeria [38]. This report presents findings on the primary hypothesis of the cross-sectional phase detailing the factors influencing the pathway to the specialist and the impact of delay on disease migration. The African Research Group for Oncology funded this research. The funding body had no role in the design, data collection, analysis, interpretation and write up of the manuscript. Department of Surgery, University of Ilorin and University of Ilorin Teaching Hospital, Ilorin, Kwara state, Nigeria Olayide Agodirin, Samuel Olatoke & Olufemi Habeeb Department of Surgery, University of Cape Coast and Cape Coast Teaching Hospital, Cape Coast, Ghana Ganiyu Rahman Department of Surgery, Ekiti State Teaching Hospital, Ado-Ekiti, Ekiti state, Nigeria Julius Olaogun Department of Surgery, Obafemi Awolowo Teaching Hospital, Ile-Ife, Osun state, Nigeria Olalekan Olasehinde Department of Surgery, Federal Medical Center, Owo, Ondo State, Nigeria Aba Katung Department of Surgery, LAUTECH Teaching Hospital, Osogbo, Osun state, Nigeria Oladapo Kolawole Department of Surgery, University College Hospital, Ibadan, Oyo state, Nigeria Omobolaji Ayandipo Department of Surgery, Obafemi Awolowo Teaching Hospital, Ilesha, Osun state, Nigeria Amarachukwu Etonyeaku Department of Surgery, University of Ilorin Teaching Hospital, Ilorin, Kwara state, Nigeria Ademola Adeyeye Department of Surgery, General Hospital Ilorin, Ilorin, Kwara state, Nigeria John Agboola Department of Radiology, University of Ilorin and University of Ilorin Teaching Hospital, Ilorin, Kwara state, Nigeria Halimat Akande Department of Surgery, LAUTECH Teaching Hospital, Ogbomoso, Oyo State, Nigeria Soliu Oguntola & Olusola Akanbi Department of Surgery, Federal Teaching Hospital, Ido-Ekiti, Ekiti state, Nigeria Oluwafemi Fatudimu Olayide Agodirin Samuel Olatoke Olufemi Habeeb Soliu Oguntola Olusola Akanbi AO, RG, OS, AH jointly conceived the research. AO,OSA,OK,OJ, OS, KA FO,HO, AS, AJ, AB recruited respondents and collected data, EA. Administration of the project was by AO. Statistical analysis was by AO, GA and OS. Writing and original draft was by AO, GA, OK, OS, AJ and OJ. All authors were involved in extensive formative review and editing. All authors approved the final manuscript. Correspondence to Olayide Agodirin. Ethical approval was obtained from the ethical review committee of the University of Ilorin Teaching Hospital Ilorin(Oke-Ose, Esie, Offa) (ERCPAN/2017/02/1644). Concurrent approval was obtained from other participating centers namely; Ekiti state teaching hospital(EKSUTH/A67/2017/05/010, LAUTECH Osogbo and Ogbomoso (LTH/REC/2017/06/10/368, Federal Medical Center Owo (FMC/OW/380/VOL LXVII/67, Federal Medical Center Ido (ERC/2018/114/03/6213), and General Hospital Ilorin (Surulere and Sobi)(GHI/ADM/134/VOL.1/62). All respondents consented to the study and gave written informed consent after detailed explanation. No personal information, images or video footage was revealed. the authors declare that there is no competing interest. Data collection questionnaire. Agodirin, O., Olatoke, S., Rahman, G. et al. Presentation intervals and the impact of delay on breast cancer progression in a black African population. BMC Public Health 20, 962 (2020). https://doi.org/10.1186/s12889-020-09074-w DOI: https://doi.org/10.1186/s12889-020-09074-w Help-seeking Primary-care Tumor progression	CommonCrawl
Volume of an Oblate Spheroid In this lesson, we'll discuss how by using the concept of a definite integral one can calculate the volume of something called an oblate spheroid. An oblate spheroid is essentially just a sphere which is compressed or stretched along one of its dimensions while leaving its other two dimensions unchanged. For example, the Earth is technically not a sphere—it is an oblate spheroid. To find the volume of an oblate spheroid, we'll start out by finding the volume of a paraboloid . (If you cut an oblate spheroid in half, the two left over pieces would be paraboloids.) To do this, we'll draw an $n$ number of cylindrical shells inside of the paraboloid; by taking the Riemann sum of the volume of each cylindrical shell, we can obtain an estimate of the volume enclosed inside of the paraboloid. If we then take the limit of this sum as the number of cylindrical shells approaches infinity and their volumes approach zero, we'll obtain a definite integral which gives the exact volume inside of the paraboloid. After computing this definite integral, we'll multiply the result by two to get the volume of the oblate spheroid. Figure 1: Graph of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ centered at the origin of the $xy$-plane. Also, I have drawn the $i^{th}$ rectangle underneath the quarter-ellipse within the first quadrant. There are an $n$ number of such rectangles underneath this quarter-ellipse along the interval $Δx=3-0$. Finding volume of an oblate spheroid In Figure 1, I have graphed the ellipse $\frac{x^2}{9}+{y^2}{4}=1$ on the $xy$-plane. If we rotate the eclipse about either the $x$-axis or $y$-axis, the ellipse will trace out the closed surface illustrated in Figure 3. The volume of revolution which that surface encloses is called an oblate spheroid. In this lesson, we'll use the concept of a definite integral to calculate the volume of an oblate spheroid. To calculate this volume, we'll first approximate the volume by summing the volumes of an $n$ number of cylindrical shells (see Figure 2) drawn within the oblate spheroid. After that, we'll take the limit of this sum as $n→∞$. Figure 2: A cylindrical shell is obtained by revolving the rectangle $f(x_i)Δx$ about the $y$-axis. Doing this for all $n$ rectangles, we get an $n$ number of shells. By summing the volumes of these $n$ number of cylindrical shells, we can obtain an estimate for total volume enclosed inside of the paraboloid obtained by rotating the quarter-ellipse (the one in the upper-right quadrant) about the $x$-axis. But before we do that, let's discuss how to construct a cylindrical shell and how to calculate its volume. Let's subdivide the interval on the $x$-axis, $Δx=3-0$, into an $n$ number of equally spaced tick marks; let's label each tick mark with $x_i$ where $i=1,...,n$. In Figure 1, I have drawn a rectangle with with $Δx=x_{i+1}-x_i$ and height $f(x_i)$. If we rotate this rectangle about the $y$-axis, the rectangle will trace out the cylindrical shell illustrated in Figure 2. To calculate the volume of the cylindrical shell, we must take the product of the area of the cylindrical shell's base with its height. The ring $QQ'RR'$ with width $Δx_{i+1}-x_i$ in Figure 2 is the cylindrical shell's base. Let's subtract the area of the inner circle $QQ'$ from the area of the outer circle $RR'$ in Figure 2 to get the area of the cylindrical shell's base: $$A=π(x_{i+1})^2-π(x_i)^2.\tag{1}$$ Using basic algebra, we can rewrite Equation (1) as $$A=π\frac{x_i+x_{i+1}}{2}\biggl[2(x_{i+1}-x_i)\biggr].\tag{2}$$ The term $(x_i+x_{i+1}/2$ in Equation (2) is the average value of $x_i$ and $x_{i+1}$. In Figure 1 (click to enlarge), I have labeled the average of these two values as $\bar{x}_i$ on the $x$-axis. Substituting $\bar{x}_i$ into Equation (2), we have $$A=2π\bar{x}_iΔx.\tag{3}$$ (You might be asking yourself why we went through the trouble of rewriting Equation (1) of the form expressed in Equation (3). The reason why we did this will become evident when we wish to express the limit of the sum of the volumes of each cylindrical shell as a definite integral. But we'll discuss this in more detail shortly.) As you can see from Figure 2, the hieght of a cylindrical shell is $f(x_i)$. The volume of the $i^{th}$ cylindrical shell is therefore given by $$ΔV_i=2π\bar{x}_if(x_i)Δx.\tag{4}$$ To estimate the volume of the paraboloid, let's sum the volumes of all the cylindrical shells to get $$S_n=\sum_{i=1}^n2π\bar{x}_if(x_i)Δx.\tag{5}$$ When defining a definite integral, we always start with a sum of the form $$S_m=\sum_{i=1}^mg(x_i)Δx;\tag{6}$$ then, we take the limit of such a sum as $m→∞$ to get $$\int_a^bg(x)dx=\lim_{m→∞}\sum_{i=1}^mg(x_i)Δx.$$ Figure 3: If $a$ and $c$ represents the semi-major and semi-minor axes of an ellipse, respectively, and if $a=3$ and $c=2$ then by rotating such an ellipse about an axis we can obtain an oblate spheroid. The problem with Equation (5) is that the term $2π\bar{x}_if(\bar{x}_i)Δx$ isn't the same as the $g(x_i)$ in Equation (6). We cannot define a function $h(\bar{x}_i)$ or $h(x_i)$ that we can set equal to $2π\bar{x}_if(\bar{x}_i)Δx$. The term $2π\bar{x}_if(\bar{x}_i)Δx$ requires two input values (namely, $\bar{x}_i)$ and $x_i$) to specify its value whereas functions like $g(x_i)$ in Equation (6) require only one input value (namely, $x_i$) to specify its value. Fortunately, there is a way around this problem. Recall that it does not matter whether we take a left-hand side Reimann sum (in which case, the height of the rectangle would be $g(x_i)$), a right-hand side Reimann sum (this is when the height of each rectangle is given by $g(x_{i+1}$), or a midpoint Reiman sum (when the height of a rectangle is given by $g(\frac{x_i+x_{i+1}}{2})=g(\bar{x}_i)$). (We shall not discuss the reasons why this is here; but if you do not understand why this is, I strongly encourage you to review the topic.) For similar reasons, we could replace the $f(x_i)$ in Equation (5) with either $f(x_{i+1}$ or $f(\bar{x}_i)$; doing so will not change the limit of the sum. (Indeed, we could replace $f(x_i)$ in Equation (5) with $f(x_i)$ (where $x_i≤x_i≤x_{i+1}$ and, although the Equation (5) would give a different approximation of the paraboloid, the limit of Equation (5) would remain the same. To understand why this is, it would be a good idea to review the concept of limits.) Swapping the $f(x_i)$ in Equation (5) with $f(\bar{x}_i)$, we get a different sum (which we'll specify by $S_n'$) given by $$S_n'=\sum_{i=1}^n2π\bar{x}_if(\bar{x}_i)Δx.\tag{7}$$ What's nice about Equation (7) is that the term $2π\bar{x}_if(\bar{x}_i)Δx$ is expressed entirely in terms of the single variable \bar{x}_i\). Thus, Equation (7) is of the same form as Equation (6). If $n→∞$ (which is to say, if the number of cylindrical shells within the paraboloid approaches infinity), then the sum $S'_n$ will get closer and closer to equaling the exact volume of the paraboloid. Thus $$\lim_{n→∞}\sum_{i=1}^n2π\bar{x}_if(\bar{x}_i)Δx=\int_0^32πxf(x)dx.\tag{8}$$ To evaluate the integral in Equation (8), we need to find out what the function $f(x)$ is. $f(x)$ represents the height (which is to say, the $y$-value) associated with each rectangle on the interval $Δx=3-0$. In other words, $f(x)$ is the $y$-coordinate associated with each point along the quarter-ellipse in the first quadrant of the $xy$-plane illustrated in Figure 2. Recall that the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$ was used to graph each $(x,y)$ coordinate along the ellipse in Figure 1. If we restrict the domain of this function to values of $x$ and $y$ where $0≤y≤2$, then the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$ could be used to graph the quarter-ellipse in the first quadrant of the $xy$-plane in Figure 1. Thus, for the aforementioned restrictions on the domain, the $y$ in the equation, $\frac{x^2}{9}+\frac{y^2}{4}=1$, specifies the $y$-coordinate of each point along the quarter-ellipse. It therefore also specifies the height of each rectangle under the quarter-ellipse. This means that $f(x)=y(x)$. Using the equation $\frac{x^2}{9}+\frac{y^2}{4}=1$, we can solve for $f(x)=y(x)$: $$\frac{x^2}{9}+\frac{(f(x))^2}{4}=1$$ $$\frac{(f(x))^2}{4}=1-\frac{x^2}{9}$$ $$f(x)=\sqrt{4-\frac{4}{9}x^2}.\tag{9}$$ Substituting Equation (9) into the integral in Equation (8), we have $$\text{Volume of paraboloid}=\int_0^32πx\sqrt{4-\frac{4}{9}x^2}dx.\tag{10}$$ At this point, all of the hard work is done and we just need to solve the definite integral in equation (10) and then multiply our answer by $2$ to get the volume of the oblate spheroid illustrated in Figure 3. We can solve the integral in Equation (10) by using $u$-substitution. If we let $u=4-\frac{4}{9}x^2$, then $$\frac{du}{dx}=\frac{-8}{9}x$$ $$du=\frac{-8}{9}xdx$$ $$dx=\frac{-9}{8}\frac{1}{x}du.\tag{11}$$ Substituting $u$ and Equation (11) into (10), we have $$\text{Volume of paraboloid}=\int_{?_1}^{?_2}(2πx)\biggl(\frac{-9}{8}\frac{1}{x}\biggr)u^{1/2}du$$ $$\text{Volume of paraboloid}=\frac{-9}{4}π\int_{?_1}^{?_2}u^{1/2}du.$$ When $x=0$, $u=4$ and when $x=3$, $u=4-\frac{4}{9}(3)^2=4-4=0$. Substituting the limits of integration into the integral above and solving the integral, we have $$\text{Volume of paraboloid}=\frac{-9}{4}\biggl[\frac{2}{3}u^{3/2}\biggr]_4^0$$ $$=\frac{9}{4}π(\frac{2}{3}(4)^{3/2})=\frac{3}{2}π(4)^{3/2}$$ $$=\frac{-9}{4}π\biggl[\frac{2}{3}u^{3/2}\biggr]_4^0=\frac{-3}{2}π\biggl[(4-\frac{4}{9}x^2)^{3/2}\biggr]_0^3$$ $$=\frac{-3}{2}π\biggr[(4-4)-(4-0)^{3/2}\biggr]=12π.$$ Thus we have shown that the volume of the paraboloid is $12π$ units squared. Multiplying this result by $2$, we find that the volume of this oblate spheroid is given by $$\text{Volume of oblate spheroid}=24π.\tag{12}$$ Source: https://www.gregschool.org/gregschoollessons/2017/10/22/volume-of-an-oblate-spheroid-rl852-8yer8 Newer PostQuasars Older PostOur Future as Cyborgs	CommonCrawl
\begin{document} \title{Explicit calculations with Eisenstein series} \author{ Matthew P. Young} \address{Department of Mathematics \\ Texas A\&M University \\ College Station \\ TX 77843-3368 \\ U.S.A.} \email{[email protected]} \thanks{This material is based upon work supported by the National Science Foundation under agreement No. DMS-1401008. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. } \begin{abstract} We find explicit change-of-basis formulas between Eisenstein series attached to cusps, and newform Eisenstein series attached to pairs of primitive Dirichlet characters. As a consequence, we prove a Bruggeman-Kuznetsov formula for newforms of square-free level and trivial nebentypus. We also derive, in explicit form, the Fourier expansion, Hecke eigenvalues, Atkin-Lehner pseudo-eigenvalues, and other data associated to these Eisenstein series, with arbitrary integer weight, level, and nebentypus. \end{abstract} \maketitle \section{Introduction} Let $N$ be a positive integer, and consider the space of automorphic forms of level $N$, weight $k \in \mathbb{Z}$, and nebentypus $\psi$ modulo $N$. There are at least two natural choices of how to decompose the space spanned by the Eisenstein series. One is to use Eisenstein series $E_{\mathfrak{a}}(z,s, \psi)$ attached to cusps and the other is to use Eisenstein series $E_{\chi_1, \chi_2}(z,s)$ attached to pairs of Dirichlet characters, which has a natural interpretation from representation theory. The decomposition along cusps is quite convenient from the point of view of the spectral decomposition and the Parseval formula since in essence Eisenstein series attached to inequivalent cusps are orthogonal. The decomposition with Dirichlet characters is friendly when studying the $L$-functions associated to automorphic forms. Clearly, these two features are in conflict with each other. One of the main goals of this paper is to explicitly work out the translations between these different bases, and to use this information to derive some other useful formulas. These change of basis formulas appear as Theorems \ref{thm:EcuspInTermsofEchichi} and \ref{thm:EchichiInTermsofEa}. Along the way, we derive the Fourier expansion (see Proposition \ref{prop:FourierExpansion}), allowing us to derive the functional equation of $E_{\chi_1, \chi_2}(z,s)$ under $s \rightarrow 1-s$. The Fourier expansion also shows that $E_{\chi_1,\chi_2}$ is an eigenfunction of all the Hecke operators, and gives explicit formulas for the Hecke eigenvalues. We additionally examine the Mellin transform of the Eisenstein series $E_{\chi_1, \chi_2}(iy,s)$, leading to a complementary functional equation related to the functional equation of the Dirichlet $L$-functions. After developing the change of basis formulas, in Section \ref{section:orthogonality} we examine the orthogonality properties of the various types of Eisenstein series. In Section \ref{section:AtkinLehnerEigenvalues}, we explicitly derive the action of the Atkin-Lehner operators. As a culmination of all this work, in Section \ref{section:Kuznetsov} we derive a Bruggeman-Kuznetsov formula restricted to newforms, in the special case where the level is squarefree, the nebentypus is principal, and the weight is $0$. This is an extension of the derivation of the Petersson formula for newforms of \cite{PetrowYoung}, which proceeds by a sieving argument. The method of \cite{PetrowYoung} may be easily modified to sieve for cuspidal Maass newforms, but unfortunately the same approach does not immediately carry over to sieve the continuous spectrum. This sieving argument is a requirement to prove the Bruggeman-Kuznetsov formula for newforms. The material developed in this paper may be used to sieve the Eisenstein series in a close analogy to the cusp forms. In \cite{PetrowYoung}, the newform Petersson formula is the crucial intitial step to set up the cubic moment which is in turn used to prove a strong subconvexity bound for twisted $L$-functions $L(f \otimes \chi_q, 1/2)$ where $f$ is a holomorphic newform of squarefree level $N$, and $\chi_q$ is a quadratic character of conductor $q$. This paper generalized the groundbreaking work of Conrey and Iwaniec \cite{ConreyIwaniec}, which required $N\|q$. With the aid of the newform Bruggeman-Kuznetsov formula, the tools are now in place to allow for $f$ to be a Maass newform. The arithmetical aspects of \cite{PetrowYoung} will be the same, but some of the analytical aspects (e.g., integral transforms with Bessel functions) will look somewhat different. Conrey and Iwaniec \cite{ConreyIwaniec} successfully treated both the holomorphic forms and Maass forms in tandem, so it is reasonable to expect that the subconvexity bound of \cite{PetrowYoung} may be extended to Maass forms. A motivation for this extension is the hybrid equidistribution of Heegner points as both the level and discriminant vary (as in \cite{LMY}); such a result would immediately improve many of the exponents in \cite{LMY}. An overarching goal of this paper is to provide, in one place, and in explicit form, many of the basic properties of Eisenstein series, using the classical language. Many special cases (e.g., with principal nebentypus, or primitive nebentypus, or weight $k=0$, or with holomorphic forms, \dots) are scattered throughout the literature, and the author found \cite{Huxley} \cite{IwaniecClassical} \cite{IwaniecSpectral} \cite{DFI} \cite{DiamondShurman} \cite{KnightlyLi} particularly useful references. \section{Acknowledgments} I thank Peter Humphries and Jiakun Pan for useful feedback and corrections on earlier drafts of this paper, and especially E. Mehmet Kiral for many conversations on this work. I also thank Andrew Booker and Min Lee for informing me of their independent work with Andreas St\"ombergsson on the change-of-basis formulas, and Gergely Harcos, Abhishek Saha, and Rainer Schulze-Pillot for conversations on orthogonalization. \section{Definitions} \subsection{Eisenstein series attached to cusps} \label{section:EisensteinCusps} Let $\Gamma = \Gamma_0(N)$, suppose $\psi$ is a Dirichlet character modulo $N$, and let $k \in \mathbb{Z}$. We study automorphic functions $f$ on $\Gamma$ of weight $k$ and nebentypus $\psi$, which satisfy the transformation formula \begin{equation} f(\gamma z) = j(\gamma, z)^k \psi(\gamma) f(z), \end{equation} where $\psi(\gamma) = \psi(d)$, with $d$ denoting the lower-right entry of $\gamma$, and where \begin{equation} j(\gamma, z) = \frac{cz+d}{\|cz+d\|}. \end{equation} Since $-I \in \Gamma_0(N)$, a necessary condition for a nonzero automorphic function to exist is that \begin{equation} \psi(-1) = (-1)^k. \end{equation} Let $\mathfrak{a}$ be a cusp for $\Gamma$, and let $\sigma_{\mathfrak{a}}$ be a scaling matrix for $\mathfrak{a}$, which means $\sigma_{\mathfrak{a}} \infty = \mathfrak{a}$, and $\sigma_{\mathfrak{a}}^{-1} \Gamma_{\mathfrak{a}} \sigma_{\mathfrak{a}} = \Gamma_{\infty}= \{\pm (\begin{smallmatrix} 1 & b \\ & 1 \end{smallmatrix} ): b \in \ensuremath{\mathbb Z} \}$. Let $\tau_{\mathfrak{a}} = \sigma_{\mathfrak{a}} (\begin{smallmatrix} 1 & 1 \\ & 1 \end{smallmatrix}) \sigma_{\mathfrak{a}}^{-1}$, so that $\pm \tau_{\mathfrak{a}}$ generate $\Gamma_{\mathfrak{a}}$, the stablizer of $\mathfrak{a}$ in $\Gamma$. We say that $\mathfrak{a}$ is \emph{singular} for $\psi$ if $\psi(\tau_{\mathfrak{a}}) = 1$. The Eisenstein series of nebentypus $\psi$ and weight $k$ attached to the cusp $\mathfrak{a}$ is defined by \begin{equation} \label{eq:Eadef} E_{\mathfrak{a}}(z,s, \psi) = \sum_{\gamma \in \Gamma_{\mathfrak{a}} \backslash \Gamma} \overline{\psi}(\gamma) j(\sigma_{\mathfrak{a}}^{-1} \gamma, z)^{-k} (\text{Im } \sigma_{\mathfrak{a}}^{-1} \gamma z)^s, \end{equation} initially for $\text{Re}(s) > 1$. Since we generally have no need to combine Eisenstein series with different weights, we suppress the weight in the notation. The definition of singular given above disagrees with some definitions in the literature, which seemingly allowed $\psi(\tau_{\mathfrak{a}})$ to be $-1$ for $k$ odd. Under this assumption, $E_{\mathfrak{a}}(z,s,\psi)$ would be ill-defined, because the summand would change sign under $\gamma \rightarrow \tau_{\mathfrak{a}} \gamma$. One may check that $E_{\mathfrak{a}}$ is independent of the choice of scaling matrix. Moreover, if $\mathfrak{a}$ and $\mathfrak{b} = \gamma \mathfrak{a}$ are $\Gamma$-equivalent cusps, then \begin{equation} \label{eq:EaEquivalentCuspsRelation} E_{\gamma \mathfrak{a}}(z,s,\psi) = \overline{\psi}(\gamma) E_{\mathfrak{a}}(z,s,\psi), \end{equation} correcting a remark of \cite[p.505]{DFI}. \subsection{Eisenstein series attached to characters} We draw inspiration from Huxley's paper \cite{Huxley}, but refer the reader to \cite{KnightlyLi} for a more motivated definition. Let $\chi_1, \chi_2$ be Dirichlet characters modulo $q_1, q_2$, respectively, with $\chi_1(-1) \chi_2(-1) = (-1)^k$. Define \begin{equation} \label{eq:Echi1chi2def} E_{\chi_1, \chi_2}(z,s) = \frac12 \sum_{(c,d) = 1} \frac{(q_2 y)^s \chi_1(c) \chi_2(d)}{\|cq_2z + d\|^{2s}} \Big(\frac{\|c q_2 z + d\|}{cq_2 z + d}\Big)^k. \end{equation} One can check directly that $E_{\chi_1, \chi_2}$ is automorphic on $\Gamma_0(q_1 q_2)$ of weight $k$ and nebentypus $\chi_1 \overline{\chi_2}$. However, we can give a more structural view as follows.\footnote{The author thanks E. Mehmet Kiral for pointing out this setup.} With $\chi_1, \chi_2$ as above, define \begin{equation} \theta \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \chi_1(c) \chi_2(d), \end{equation} for $a,b,c,d \in \ensuremath{\mathbb Z}$ and $ad-bc=1$. Also define $\sigma = (\begin{smallmatrix} \sqrt{q_2} & \\ & 1/\sqrt{q_2} \end{smallmatrix})$, which satisfies $\sigma z = q_2 z$. Then $E_{\chi_1, \chi_2}(z,s) = E_{\theta}(z,s)$, where $E_{\theta}$ is defined by \begin{equation} \label{eq:EthetaDef} E_{\theta}(z,s) = \sum_{\gamma \in \Gamma_{\infty} \backslash \Gamma_0(1)} \theta(\gamma) j(\gamma, \sigma z)^{-k} \text{Im}(\gamma \sigma z)^s . \end{equation} It is easy to check that $\sigma \Gamma_0(q_1 q_2) \sigma^{-1} = \Gamma_0(q_1, q_2)$, where this latter group denotes the subgroup of $SL_2(\mathbb{Z})$ with lower-left entry divisible by $q_1$, and upper-right entry divisible by $q_2$. Moreover, if $\tau \in \Gamma_0(q_1 q_2)$, and $\tau' \in \Gamma_0(q_1, q_2)$ is defined by $\sigma \tau = \tau' \sigma$, then the lower-right entry of $\tau'$ is the same as the lower-right entry of $\tau$. Finally, one directly checks that if $\gamma \in SL_2(\ensuremath{\mathbb Z})$ and $\tau' \in \Gamma_0(q_1,q_2)$, then \begin{equation} \theta(\gamma \tau') = \theta(\gamma) (\overline{\chi_1} \chi_2)(d'), \end{equation} where $d'$ is the lower-right entry of $\tau'$. With these properties, we may now check the automorphy of $E_{\theta}$. For $\tau \in \Gamma_0(q_1 q_2)$, and $\tau'$ defined by $\sigma \tau = \tau' \sigma$, we have \begin{equation} E_{\theta}(\tau z, s) = \sum_{\gamma \in \Gamma_{\infty} \backslash \Gamma_0(1)} \theta(\gamma) j(\gamma, \tau' \sigma z )^{-k} \text{Im}(\gamma \tau' \sigma z)^s. \end{equation} Changing variables by $\gamma \rightarrow \gamma \tau'^{-1}$, using $j(\gamma \tau'^{-1}, \tau' \sigma z)^{-k} = j(\gamma, \sigma z)^{-k} j(\tau', \sigma z)^k$, and finally $j(\tau', \sigma z) = j(\tau, z)$ (check this one directly from the definitions), we complete the proof that $E_{\theta}$ has weight $k$, nebentypus $\chi_1 \overline{\chi_2}$, and level $q_1 q_2$. By M\"obius inversion, we have \begin{equation} \label{eq:Gchi1chi2Def} L(2s, \chi_1 \chi_2) E_{\chi_1, \chi_2}(z,s) = \frac12 \sumprime_{c,d \in \mathbb{Z}} \frac{(q_2 y)^s \chi_1(c) \chi_2(d)}{\|cq_2z + d\|^{2s}} \Big(\frac{\|c q_2 z + d\|}{cq_2 z + d}\Big)^k =: G_{\chi_1, \chi_2}(z,s), \end{equation} with the prime denoting that the term $c=d=0$ is omitted. With $k=0$, Huxley's notation for our $E_{\chi_1, \chi_2}(z,s)$ is $E_{\chi_1}^{\chi_2}(q_2 z, s)$, and our $G_{\chi_1, \chi_2}(z,s)$ is Huxley's $B_{\chi_1}^{\chi_2}(q_2 z, s)$. The reader interested in holomorphic Eisenstein series of weight $k \geq 3$ should consider \begin{equation} \label{eq:EholomorphicDef} E_{\chi_1, \chi_2, k}(z) := (q_2 y)^{-k/2} E_{\chi_1, \chi_2}(z, \tfrac{k}{2}) = \frac12 \sum_{(c,d) = 1} \frac{ \chi_1(c) \chi_2(d)}{(cq_2z + d)^{k}} , \end{equation} which satisfies $E_{\chi_1, \chi_2, k}(\tfrac{az+b}{cz+d}) = (cz + d)^k (\chi_1 \overline{\chi_2})(d) E_{\chi_1, \chi_2,k}(z)$, for $(\begin{smallmatrix} a& b \\ c & d \end{smallmatrix}) \in \Gamma_0(q_1 q_2)$. \subsection{Remarks} \label{section:newformdiscussion} We refer the reader to \cite[Section 4]{DFI} for a more comprehensive overview of the space $L^2(\Gamma_0(N) \backslash \ensuremath{\mathbb H}, \psi)$, including a discussion on the relevant differential operators. Weisinger has developed a newform theory for Eisenstein series of weight $k$ and nebentypus $\psi$ in the holomorphic setting. The newforms of level $q_1 q_2$ are the functions $E_{\chi_1,\chi_2,k}(z)$ where $\chi_i$ is primitive modulo $q_i$, $i=1,2$, and $\chi_1 \overline{\chi_2} = \psi$. The space of Eisenstein series of level $N$ is spanned by $E_{\chi_1,\chi_2,k}(Bz)$ where $B q_1 q_2 \| N$. In \cite{Huxley}, Huxley calculated the scattering matrix for the congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$, and $\Gamma(N)$, with trivial nebentypus and weight $0$. For each of these groups, he related Eisenstein series attached to characters to Eisenstein series attached to cusps on $\Gamma(N)$. \section{Fourier expansion and consequences} \subsection{The Fourier expansion} For $k=0$, Huxley \cite{Huxley} stated (without proof) the Fourier expansion of $E_{\chi_1, \chi_2}(z,s)$; a proof may be found in \cite[Section 5.6]{KnightlyLi}. For general $k$, and $\chi_1 \overline{\chi_2}$ primitive, \cite{DFI} have developed the Fourier expansion. One may also find a Fourier expansion for $E_{\chi_1, \chi_2,k}$ in \cite{DiamondShurman}. The author was not able to locate a formula for general $k,\chi_1, \chi_2$. It is convenient to consider the ``completed'' Eisenstein series defined by \begin{equation} E_{\chi_1, \chi_2}^(z,s):= \frac{(q_2/\pi)^s}{i^{-k} \tau(\chi_2)} \Gamma(s+\tfrac{k}{2}) L(2s, \chi_1 \chi_2) E_{\chi_1, \chi_2}(z,s), \end{equation} where $\tau(\chi_2)$ denotes the Gauss sum. \begin{myprop} \label{prop:FourierExpansion} Suppose $\chi_i$ is primitive modulo $q_i$, $i=1,2$, and $(\chi_1 \chi_2)(-1) = (-1)^k$. Then \begin{equation} E_{\chi_1, \chi_2}^(z,s) = e_{\chi_1, \chi_2}^(y,s) + \sum_{n \neq 0} \frac{\lambda_{\chi_1,\chi_2}(n,s)}{\sqrt{\|n\|}} e(nx) \frac{\Gamma(s+\frac{k}{2}) }{\Gamma(s+\frac{k}{2} \sgn(n))} W_{\frac{k}{2} \sgn(n), s-\frac12}(4 \pi \|n\| y), \end{equation} where \begin{equation} \label{eq:EisensteinChiChiConstantTerm} \begin{split} e_{\chi_1, \chi_2}^(y,s) = & \delta_{q_1=1} q_2^{2s} \frac{\pi^{-s}}{i^{-k} \tau(\chi_2)} \Gamma(s+\tfrac{k}{2}) L(2s, \chi_2) y^s \\ & + \delta_{q_2=1} q_1^{2-2s} \frac{ \pi^{-(1-s)} }{ i^{-k} \tau(\overline{\chi_1}) } \Gamma(1-s+\tfrac{k}{2}) L(2-2s, \overline{\chi_1}) y^{1-s}, \end{split} \end{equation} \begin{equation} \label{eq:lambdachi1chi2Def} \lambda_{\chi_1,\chi_2}(n,s) = \chi_2(\sgn(n)) \sum_{ab=\|n\|} \chi_1(a) \overline{\chi_2}(b) \Big(\frac{b}{a}\Big)^{s-\frac12}, \end{equation} and $W_{\alpha, \beta}$ is the Whittaker function. In particular, if $k=0$, then \begin{equation} E_{\chi_1, \chi_2}^(z,s) = e_{\chi_1, \chi_2}^(y,s)+ 2 \sqrt{y} \sum_{n \neq 0} \lambda_{\chi_1,\chi_2}(n,s) e(nx) K_{s-\frac12}(2 \pi \|n\| y). \end{equation} \end{myprop} {\bf Convention.} Here and throughout we take the convention that the principal character modulo $1$ is primitive. \begin{proof} We have \begin{equation} G_{\chi_1, \chi_2}(z,s) = \sum_{n \in \ensuremath{\mathbb Z}} e(nx) c_n(y), \qquad c_n(y) = \int_0^{1} G_{\chi_1, \chi_2}(x+iy,s) e(-nx) dx, \end{equation} and inserting \eqref{eq:Gchi1chi2Def}, we have \begin{equation} c_n(y) = \frac12 \sumprime_{c,d \in \mathbb{Z}} \chi_1(c) \chi_2(d) (q_2 y)^s \int_0^1 \frac{e(-nx)}{\|cq_2(x+iy) + d\|^{2s}} \Big(\frac{\|cq_2(x+iy) + d\|}{cq_2(x+iy) + d} \Big)^k dx. \end{equation} Extracting the term $c=0$, which only occurs for $q_1 = 1$, we obtain \begin{equation} c_n(y) = \delta_{q_1=1} \delta_{n=0} (q_2 y)^s L(2s, \chi_2) + b_n(y), \end{equation} where \begin{equation} b_n(y) = \sum_{c \geq 1} \sum_{d \in \ensuremath{\mathbb Z}} \chi_1(c) \chi_2(d) (q_2 y)^s \int_0^1 \frac{e(-nx)}{\|cq_2(x+iy) + d\|^{2s}} \Big(\frac{\|cq_2(x+iy) + d\|}{cq_2(x+iy) + d} \Big)^k dx. \end{equation} Changing variables $x \rightarrow x-\frac{d}{cq_2}$, we obtain \begin{equation} b_n(y) = (q_2 y)^s \sum_{c \geq 1} \frac{\chi_1(c)}{(cq_2)^{2s}} \sum_{d \in \ensuremath{\mathbb Z}} \chi_2(d) e\Big(\frac{nd}{c q_2}\Big) \int_{\frac{d}{cq_2}}^{\frac{d}{cq_2} + 1} \frac{e(-nx)}{\|x+iy\|^{2s}} \Big(\frac{\|x+iy \|}{x+iy } \Big)^k dx. \end{equation} Next break up the sum over $d$ into arithmetic progressions modulo $c q_2$, giving \begin{equation} b_n(y) = (q_2 y)^s \sum_{c \geq 1} \frac{\chi_1(c)}{(cq_2)^{2s}} \sum_{r \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{ c q_2}} \chi_2(r) e\Big(\frac{nr}{c q_2}\Big) \sum_{\ell \in \ensuremath{\mathbb Z}} \int_{\frac{r + \ell cq_2}{cq_2}}^{\frac{r + \ell cq_2}{cq_2} + 1} \frac{e(-nx)}{\|x+iy\|^{2s}} \Big(\frac{\|x+iy \|}{x+iy } \Big)^k dx. \end{equation} The sum over $\ell$ forms the complete integral over $\mathbb{R}$, and the sum over $r$ satisfies \begin{equation} \sum_{r \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{ c q_2}} \chi_2(r) e\Big(\frac{nr}{c q_2}\Big) = c \delta(c\|n) \tau(\chi_2, n/c) = c \delta(c\|n) \overline{\chi_2}(n/c) \tau(\chi_2), \end{equation} where the final equation holds for all $n$ provided $\chi_2$ is primitive (see \cite[(3.12)]{IK}). The integral may be evaluated with \cite[3.384.9]{GR}, but it takes some care to transform the integral into this template. We have \begin{equation} \int_{-\infty}^{\infty} \frac{e(-nx)}{\|x+iy\|^{2s}} \Big(\frac{\|x+iy \|}{x+iy } \Big)^k dx = y^{1-2s} \int_{-\infty}^{\infty} \frac{e(-nxy)}{\|x+i\|^{2s-k}} (x+i)^{-k} dx. \end{equation} Then we note $(x+i)^{-k} = i^{-k} (1-ix)^{-k}$, $\|x+i\|^{2s-k} = \|1+ix\|^{2s-k} = (1+ix)^{s-\frac{k}{2}} (1-ix)^{s-\frac{k}{2}}$, which is valid since $\arg(1 \pm ix) \in (-\pi/2, \pi/2)$. Thus the integral we want is \begin{equation} y^{1-2s} i^{-k} \int_{-\infty}^{\infty} \frac{\exp(-2\pi i nxy) dx}{(1+ix)^{s-\frac{k}{2}} (1-ix)^{s+\frac{k}{2}} } = y^{1-2s} i^{-k} \frac{ (2 \pi) 2^{-s} (2 \pi \|n\| y)^{s-1}}{\Gamma(s + \frac{k}{2} \sgn(n))} W_{\frac{k}{2} \sgn(n), s-\frac12}(4 \pi \|n\| y), \end{equation} valid for $n \neq 0$ and $\text{Re}(s) > \frac12$. This simplifies as \begin{equation} \frac{y^{-s} i^{-k} \pi^s \|n\|^{s-1}}{\Gamma(s+\frac{k}{2} \sgn(n))} W_{\frac{k}{2} \sgn(n), s-\frac12}(4 \pi \|n\| y). \end{equation} We note the special cases (see \cite[(9.235.2)]{GR}, \cite[(4.21)]{DFI}): \begin{equation} \label{eq:WhittakerDef} W_{\frac{k}{2} \sgn(n), s-\frac12}(4 \pi \|n\| y) = \begin{cases} 2 \sqrt{\|n\| y} K_{s-\frac12}(2 \pi \|n\| y) \qquad &k=0, n \neq 0, \\ (4 \pi n y)^{k/2} \exp(- 2 \pi n y), \qquad &s= k/2 > 0, \thinspace n \geq 1. \end{cases} \end{equation} When $n=0$, we have from \cite[(8.381.1)]{GR} \begin{equation} \int_{-\infty}^{\infty} \frac{1}{\|x+iy\|^{2s}} \Big(\frac{\|x+iy \|}{x+iy } \Big)^k dx = i^{-k} (2 \pi) (2y)^{1-2s} \frac{\Gamma(2s-1)}{\Gamma(s+\tfrac{k}{2}) \Gamma(s-\tfrac{k}{2})}. \end{equation} Thus, $b_0(y) = 0$ if $q_2 > 1$, and for $n \neq 0$, we have \begin{equation} b_n(y) = (q_2 y)^s \sum_{c \geq 1} \frac{\chi_1(c)}{(cq_2)^{2s}} c \delta(c\|n) \overline{\chi_2}(n/c) \tau(\chi_2) \frac{y^{-s} i^{-k} \pi^s \|n\|^{s-1} }{\Gamma(s + \frac{k}{2} \sgn(n))} W_{\frac{k}{2} \sgn(n), s-\frac12}(4 \pi \|n\| y). \end{equation} Simplifying gives the desired formula for the nonzero Fourier coefficients. For the constant term, we obtain \begin{equation} c_0(y) = \delta_{q_1=1} (q_2 y)^s L(2s, \chi_2) + \delta_{q_2=1} y^{1-s} L(2s -1, \chi_1) i^{-k} \frac{(2 \pi) 2^{1-2s} \Gamma(2s-1)}{\Gamma(s+\tfrac{k}{2}) \Gamma(s-\tfrac{k}{2})}. \end{equation} We now simplify this. The functional equation of $L(2s-1,\chi_1)$ gives \begin{equation} L(2s-1, \chi_1) = \frac{\sqrt{q_1} }{i^{-\delta_1} \tau(\overline{\chi_1}) } \Big(\frac{q_1}{\pi}\Big)^{\frac32 - 2s} \frac{\Gamma(1-s+\tfrac{\delta_1}{2})}{\Gamma(s-\tfrac12 +\tfrac{\delta_1}{2})} L(2 -2s , \overline{\chi_1}), \end{equation} where $\delta_1 = 0$ if $\chi_1(-1) = 1$, and $\delta_1 = 1$ if $\chi_1(-1) = -1$. Also, note \begin{equation} \Gamma(2s-1) = \pi^{-1/2} 2^{2s-2} \Gamma(s-\tfrac12) \Gamma(s). \end{equation} Therefore, \begin{equation} \begin{split} c_0(y) = &\delta_{q_1=1} q_2^s L(2s, \chi_2) y^s \\ + &\delta_{q_2=1} q_1^{2-2s} \frac{ \pi^{2s-1} }{i^{-\delta_1} \tau(\overline{\chi_1}) } \frac{\Gamma(1-s+\tfrac{\delta_1}{2})}{\Gamma(s-\tfrac12 +\tfrac{\delta_1}{2})} i^{-k} \frac{ \Gamma(s-\tfrac12) \Gamma(s)}{\Gamma(s+\tfrac{k}{2}) \Gamma(s-\tfrac{k}{2})} L(2-2s, \overline{\chi_1}) y^{1-s}. \end{split} \end{equation} Consider \begin{equation} \gamma(s,\delta_1,k) = \frac{ \Gamma(s-\tfrac12) \Gamma(s) \Gamma(1-s+\tfrac{\delta_1}{2})}{\Gamma(s-\tfrac12 +\tfrac{\delta_1}{2}) \Gamma(s-\tfrac{k}{2})\Gamma(1-s+\tfrac{k}{2})}. \end{equation} Note that in our application, $\delta_1 \equiv k \pmod{2}$. Using standard gamma function identities, we obtain \begin{equation} \gamma(s,\delta_1,k) = \begin{cases} (-1)^{k/2}, \qquad &k \text{ even}, \\ (-1)^{(k-1)/2}, \qquad &k \text{ odd}, \end{cases} \end{equation} whence $i^{-k+\delta_1} \gamma(s,\delta_1,k) = 1$. Therefore, \begin{equation} \begin{split} c_0(y) = \delta_{q_1=1} q_2^s L(2s, \chi_2) y^s + \delta_{q_2=1} q_1^{2-2s} \frac{ \pi^{2s-1} }{ \tau(\overline{\chi_1}) } \frac{\Gamma(1-s+\tfrac{k}{2})}{\Gamma(s+\tfrac{k}{2})} L(2-2s, \overline{\chi_1}) y^{1-s}. \end{split} \end{equation} Then using $e_{\chi_1, \chi_2}^(y,s) = \frac{(q_2/\pi)^s}{i^{-k} \tau(\chi_2)} \Gamma(s+\frac{k}{2}) c_0(y)$ completes the proof. \end{proof} Note that \begin{equation} \label{eq:alphasum} \sum_{n=1}^{\infty} \frac{\lambda_{\chi_1,\chi_2}(n,s)}{n^u} = L(u+s-\tfrac12 , \chi_1) L(u + \tfrac12-s, \overline{\chi_2}). \end{equation} \subsection{Functional equations} The Fourier coefficient satisfies the functional equation \begin{equation} \label{eq:FourierCoefficientFunctionalEquation} \lambda_{\chi_1,\chi_2}(n,1-s) = (\chi_1 \chi_2)(\sgn(n)) \lambda_{\overline{\chi_2},\overline{\chi_1}}(n,s). \end{equation} As for the Eisenstein series itself, we have \begin{myprop} Suppose $\chi_i$ is primitive modulo $q_i$, $i=1,2$, and $(\chi_1 \chi_2)(-1) = (-1)^k$. Then $E_{\chi_1, \chi_2}^(z,s)$ extends to a meromorphic function for all $s \in \mathbb{C}$. Moreover, it satisfies the functional equation \begin{equation} \label{eq:Echi1chi2FunctionalEquation} E^_{\chi_1,\chi_2}(z,s) = E^_{\overline{\chi_2}, \overline{\chi_1}}(z,1-s). \end{equation} If $k \geq 0$ and $q_1 q_2 > 1$, then $E_{\chi_1, \chi_2}^(z,s)$ is analytic for all $s \in \ensuremath{\mathbb C}$. \end{myprop} Remark. For $k<0$, the multiplication by $\Gamma(s+\frac{k}{2})$ produces some poles of $E_{\chi_1,\chi_2}^(z,s)$. If desired, one could form the completed Eisenstein series by multiplication by $\Gamma(s-\frac{k}{2})$ instead of $\Gamma(s+\frac{k}{2})$, leading to a slightly different functional equation following from \eqref{eq:GammaFunctionalEquation} below. \begin{proof} First consider the case with $q_1, q_2 > 1$. From the Fourier expansion, we have \begin{equation} \begin{split} E^_{\chi_1,\chi_2}(z,1-s) &= \sum_{n=1}^{\infty} \frac{\lambda_{\chi_1,\chi_2}(n,1-s)}{\sqrt{n}} e(nx) W_{\frac{k}{2} , s-\frac12}(4 \pi \|n\| y) \\ &+ \sum_{n=1}^{\infty} \frac{\lambda_{\chi_1,\chi_2}(-n,1-s)}{\sqrt{n}} e(-nx) \frac{\Gamma(1-s+\frac{k}{2}) }{\Gamma(1-s-\frac{k}{2})} W_{-\frac{k}{2} , s-\frac12}(4 \pi \|n\| y). \end{split} \end{equation} Since the Whittaker function has exponential decay, uniformly for $s$ on compact sets, we see that $E_{\chi_1,\chi_2}^(z,1-s)$ extends to a meromorphic function with possible poles at $s=\tfrac{k}{2} + \ell$ with $\ell$ a nonnegative integer Using \eqref{eq:FourierCoefficientFunctionalEquation}, $\chi_1(-1) \chi_2(-1) = (-1)^k$, and \begin{equation} \label{eq:GammaFunctionalEquation} \frac{\Gamma(1-s+\frac{k}{2}) }{\Gamma(1-s-\frac{k}{2})} = (-1)^k \frac{\Gamma(s+\frac{k}{2}) }{\Gamma(s-\frac{k}{2})}, \end{equation} we obtain the functional equation. Moreover, the poles at $1-s$ with $s=\frac{k}{2}+\ell$ are removable, provided $k \geq 0$. Next consider the case with $q_1 = 1$ or $q_2 = 1$. The non-constant terms in the Fourier expansion have the same analytic properties as in the case with $q_1, q_2 > 1$, so it suffices to examine the constant term $e_{\chi_1,\chi_2}^(y,s)$. By inspection of \eqref{eq:EisensteinChiChiConstantTerm}, it satisfies the functional equation \begin{equation} e_{\chi_1, \chi_2}^(y,1-s) = e_{\overline{\chi_2}, \overline{\chi_1}}^(y,s). \end{equation} The coefficient of $y^s$ is analytic except possibly for poles at $s = -\frac{k}{2} - \ell$, with $\ell$ a nonnegative integer. If $k \geq 0$ then these poles are cancelled by the trivial zeros of $L(2s, \chi_2)$. Similarly, the coefficient of $y^{1-s}$ is analytic except for possible poles at $s= \frac{k}{2} + 1 + \ell$, with $\ell$ a nonnegative integer. These points occur at $2-2s = -k - 2 \ell$ and are also cancelled by trivial zeros of $L(2-2s, \overline{\chi_1})$, provided $k \geq 0$. \end{proof} We are also interested in the Mellin transform of $E_{\chi_1,\chi_2}^(iy,s)$ and its functional equation. A motivation for this explicit calcluation comes from the evaluation of restriction norms of automorphic forms, as in \cite{YoungQUE}. See \cite[Section 8]{DFI} for this calculation with cusp forms. The reader only interested in the change of basis formulas may wish to skip these calculations. Assume that $\chi_1$ and $\chi_2$ are primitive with $q_1, q_2 >1$, so the constant term vanishes. We have \begin{multline} \label{eq:Mellin} \int_0^{\infty} E^_{\chi_1,\chi_2}(iy,s) (q_1 q_2)^{u/2} y^u \frac{dy}{y} \\ = (q_1 q_2)^{u/2} \sum_{n=1}^{\infty} \frac{\lambda_{\chi_1, \chi_2}(n,s)}{\sqrt{n}} \int_0^{\infty} \Big(W_{\frac{k}{2}, s-\frac12}(4 \pi n y) + \varepsilon_2 \frac{\Gamma(s+\frac{k}{2})}{\Gamma(s-\frac{k}{2})} W_{-\frac{k}{2}, s-\frac12}(4 \pi n y) \Big) y^{u} \frac{dy}{y}, \end{multline} with $\varepsilon_i = \chi_i(-1)$, for $i=1,2$. Changing variables $y \rightarrow \frac{y}{\pi n}$, and evaluating the Dirichlet series using \eqref{eq:alphasum}, we have \begin{multline} \label{eq:EisensteinCompletedMellinTransform} \int_0^{\infty} E^_{\chi_1,\chi_2}(iy,s) (q_1 q_2)^{u/2} y^u \frac{dy}{y} \\ = \frac{1}{\sqrt{\pi}} \Big(\frac{q_1}{\pi}\Big)^{\frac{u}{2}} \Big(\frac{q_2}{\pi}\Big)^{\frac{u}{2}} L(\tfrac12 + u+s-\tfrac12 , \chi_1) L(\tfrac12 + u + \tfrac12-s, \overline{\chi_2}) \Phi_k^{\varepsilon_2}(u+\tfrac12,s-\tfrac12), \end{multline} where as in \cite[(8.25)]{DFI} \begin{equation} \label{eq:PhikDef} \Phi_k^{\varepsilon}(\alpha,\beta) = \sqrt{\pi} \int_0^{\infty} \Big(W_{\frac{k}{2}, \beta}(4 y) + \varepsilon \frac{\Gamma(\beta+\frac{1+k}{2})}{\Gamma(\beta+\frac{1-k}{2})} W_{-\frac{k}{2}, \beta}(4 y) \Big) y^{\alpha-\frac12} \frac{dy}{y}. \end{equation} Actually, our definition of $\Phi_k^{\varepsilon}$ differs from \cite{DFI} in that we have not divided by $4$ in the right hand side of \eqref{eq:PhikDef}. This integral is evaluated in \cite[Lemma 8.2]{DFI}, for $k \geq 0$, but the value stated there is not quite correct. The correct formula is \begin{equation} \label{eq:PhikCorrectedFormula} \Phi_k^{\varepsilon}(\alpha,\beta) = p_k^{\varepsilon}(\alpha,\beta) \Gamma\Big(\frac{\alpha+\beta + \frac{1- \varepsilon (-1)^k}{2}}{2}\Big) \Gamma\Big(\frac{\alpha-\beta + \frac{1-\varepsilon}{2}}{2}\Big), \end{equation} and $p_k^{\varepsilon}(\alpha,\beta)$ is a certain polynomial in $\alpha$, defined recursively. The formula of \cite{DFI} is incorrect in a few ways. One simple mistake is that it is off by a factor of $4$, explaining our change in definition, and the more important error boils down to interchanging $\frac{1-\varepsilon}{2}$ and $\frac{1-\varepsilon(-1)^k}{2}$ in the gamma factors, which arises from a typo early in the calculation of \cite{DFI}. A side effect of this typo is that the formula for $p_k^{\varepsilon}(\alpha,\beta)$ needs correction. We have devoted Section \ref{section:WhittakerMellinTransform} to correcting the evaluation of $\Phi_k^{\varepsilon}$. For $k=0,1$, the polynomial is given by \begin{equation} p_0^{\varepsilon}(\alpha,\beta) = \tfrac{1+\varepsilon}{2}, \qquad p_1^{\varepsilon}(\alpha,\beta) = 1. \end{equation} Note that if $\varepsilon = \varepsilon_2$, then $\varepsilon_2 (-1)^k = \varepsilon_1$, and so in our desired application the gamma factors match those of the Dirichlet $L$-functions appearing in \eqref{eq:EisensteinCompletedMellinTransform} (only after the correction \eqref{eq:PhikCorrectedFormula}), keeping in mind how the gamma factor depends on the parity of the character. Define $\Lambda(s,\chi) = (q/\pi)^{s/2} \Gamma(\frac{s+\delta}{2}) L(s,\chi)$ for a primitive Dirichlet character modulo $q$, with $\delta = \frac{1-\chi(-1)}{2}$. Then, for $k=0,1$, we have \begin{equation} \int_0^{\infty} E^_{\chi_1,\chi_2}(iy,s) (q_1 q_2)^{u/2} y^u \frac{dy}{y} = \delta_3 q_1^{\frac{-s}{2}} q_2^{\frac{s-1}{2}} \Lambda(\tfrac12 + u+s-\tfrac12 , \chi_1) \Lambda(\tfrac12 + u + \tfrac12-s, \overline{\chi_2}), \end{equation} where $\delta_3 = 1$ unless $k=0$ and $\varepsilon_1 = \varepsilon_2 = -1$, in which case $\delta_3 = 0$. From the functional equation $\Lambda(s,\chi) = \epsilon(\chi) \Lambda(1-s,\overline{\chi})$, we see that \eqref{eq:Mellin} satisfies \begin{equation} \int_0^{\infty} E_{\chi_1, \chi_2}^(iy,s) (q_1 q_2)^{u/2} y^u \frac{dy}{y} = \epsilon(\chi_1) \epsilon(\overline{\chi_2}) q_1^{\frac12-s} q_2^{s-\frac12} \int_0^{\infty} E_{\chi_2, \chi_1}^(iy,s) (q_1 q_2)^{-u/2} y^{-u} \frac{dy}{y}. \end{equation} On the other hand, if we apply the Fricke involution to $E_{\chi_1,\chi_2}^$, which maps $y$ to $\frac{1}{q_1 q_2 y}$, then we get that \begin{equation} \int_0^{\infty} E^_{\chi_1,\chi_2}(iy,s) (q_1 q_2)^{u/2} y^u \frac{dy}{y} = \int_0^{\infty} E^_{\chi_1,\chi_2} \Big(\frac{i}{q_1 q_2 y},s\Big) (q_1 q_2)^{-u/2} y^{-u} \frac{dy}{y}. \end{equation} Hence for $\delta_3 = 1$, we have \begin{equation} \label{eq:FrickeFormula1} E^_{\chi_1,\chi_2} \Big(\frac{i}{q_1 q_2 y},s\Big) = \epsilon(\chi_1) \epsilon(\overline{\chi_2}) q_1^{\frac12-s} q_2^{s-\frac12} E_{\chi_2,\chi_1}^(iy,s). \end{equation} In Section \ref{section:AtkinLehnerEigenvalues}, we will explicitly derive the action of all the Atkin-Lehner operators for $\chi_i$ even or odd, which will generalize \eqref{eq:FrickeFormula1}. This provides a pleasant consistency check. \subsection{Hecke operators} One simple consequence of the explicit calculation of the Fourier expansion, particularly the Euler product formula implied by \eqref{eq:alphasum}, is that this shows that $E_{\chi_1, \chi_2}(z,s)$ is an eigenfunction of all the Hecke operators, including $p\|q_1 q_2$. We have that \begin{equation} T_n E_{\chi_1,\chi_2}(z,1/2+it) = \lambda_{\chi_1,\chi_2}(n,1/2+it) E_{\chi_1,\chi_2}(z,1/2+it). \end{equation} In particular, if $p\|(q_1, q_2)$ then $T_p E_{\chi_1, \chi_2} = 0$. \subsection{Holomorphic forms} It would be negligent not to extract information on the holomorphic Eisenstein series defined by \eqref{eq:EholomorphicDef}. Formally specializing $s=k/2$, we obtain \begin{equation} E_{\chi_1, \chi_2}^(z,k/2) = e_{\chi_1,\chi_2}^(y,k/2) + \sum_{n=1}^{\infty} \frac{\lambda_{\chi_1,\chi_2}(n,k/2)}{\sqrt{n}} e(nx) (4 \pi n y)^{k/2} \exp(- 2 \pi n y), \end{equation} using \eqref{eq:WhittakerDef} to simplify the Whittaker function. In the constant term, we have $\delta_{q_2=1} L(2-k, \overline{\chi_1}) = 0$ for $k \geq 2$. Therefore, for $k \geq 2$ we have \begin{multline} \frac{(\frac{q_2}{2 \pi})^{k} \Gamma(k) L(k, \chi_1 \chi_2)}{i^{-k} \tau(\chi_2)} E_{\chi_1, \chi_2, k}(z) = \delta_{q_1=1} \frac{(\frac{q_2}{2 \pi})^{k} \Gamma(k) L(k, \chi_1 \chi_2)}{i^{-k} \tau(\chi_2)} \\ + \sum_{n=1}^{\infty} \lambda_{\chi_1, \chi_2}(n,k/2) n^{\frac{k-1}{2}} e(nz). \end{multline} Compare with \cite[Theorem 4.5.1]{DiamondShurman}, and note \begin{equation} \lambda_{\chi_1,\chi_2}(n,k/2) n^{\frac{k-1}{2}} = \sum_{ab=n} \chi_1(a) \overline{\chi_2}(b) b^{k-1}. \end{equation} Since the original definition of $E_{\chi_1, \chi_2}(z,s)$ converges absolutely for $\text{Re}(s) > 1$, then certainly we may set $s = k/2$ for $k \geq 3$. When $k=2$, then since the completed Eisenstein series is entire in $s$, we may also set $s=k/2 = 1$, except when $q_1 = q_2=1$ in which case the level $1$ Eisenstein series (with $\chi_1 = \chi_2 = 1$) has a pole at $s=1$. See \cite[Section 4.6]{DiamondShurman} for a description of the linear space of weight $2$ Eisenstein series. We may also examine $k=1$. Suppose $q_1, q_2 > 1$ for ease of discussion; then we may set $s=k/2 = 1/2$. One interesting feature here is that for weight $1$, $\lambda_{\chi_1, \chi_2}(n,1/2) = \lambda_{\overline{\chi_2}, \overline{\chi_1}}(n,1/2)$ for $n \geq 1$ (see \eqref{eq:FourierCoefficientFunctionalEquation}) showing that $E_{\chi_1, \chi_2, 1}(z)$ is a scalar multiple of $E_{\overline{\chi_2}, \overline{\chi_1},1}(z)$. This corresponds, roughly, to the fact that the dimension of the space of holomorphic Eisenstein series of weight $1$ is about half that of the corresponding space of odd weight $k \geq 3$. See \cite[Section 4.8]{DiamondShurman} for precise statements. \section{Preliminary formulas} Now we embark on proving the change of basis formulas. \subsection{Primitive and non-primitive} \begin{mylemma} \label{lemma:EisensteinPrimitivevsNonPrimitive} Suppose that $\chi_i$ has modulus $q_i$, and is induced by the primitive character $\chi_i^$ of modulus $q_i^$. Then \begin{equation} E_{\chi_1, \chi_2}(z,s) = \frac{L(2s, \chi_1^ \chi_2^)}{L(2s, \chi_1 \chi_2)} \sum_{a\|q_1} \sum_{b\|q_2} \frac{\mu(a) \chi_1^(a) \mu(b) \chi_2^(b)}{(ab)^s} E_{\chi_1^, \chi_2^}\Big(\frac{a q_2 }{b q_2^ } z, s\Big). \end{equation} \end{mylemma} Remarks. Within the sum, we may restrict to $(b, q_2^) =1$, which implies $b q_2^* \| q_2$, and so $\frac{a q_2 }{b q_2^* }$ is an integer. Similarly, we may assume $(a, q_1^) = 1$ which gives that $\frac{a q_2 }{b q_2^ }$ is a divisor of $\frac{q_1 q_2}{q_1^* q_2^}$. \begin{proof} The desired formula is equivalent to \begin{equation} G_{\chi_1, \chi_2}(z,s) = \sum_{a\|q_1} \sum_{b\|q_2} \frac{\mu(a) \chi_1^(a) \mu(b) \chi_2^(b)}{(ab)^s} G_{\chi_1^, \chi_2^}\Big(\frac{aq_2 }{bq_2^* } z, s\Big). \end{equation} By definition, \begin{equation} G_{\chi_1, \chi_2}(z,s) = \frac12 \sumprime_{\substack{c,d \in \mathbb{Z} \\ (c, q_1) = 1 \\ (d, q_2) = 1 }} \frac{(q_2 y)^s \chi_1^(c) \chi_2^(d)}{\|cq_2z + d\|^{2s}} \Big(\frac{\|cq_2 z + d\|}{cq_2 z + d}\Big)^{k} . \end{equation} By M\"obius inversion, we deduce \begin{equation} G_{\chi_1, \chi_2}(z,s) = \sum_{a\|q_1} \sum_{b\|q_2} \mu(a) \mu(b) \chi_1^(a) \chi_2^(b) \frac12 \sumprime_{\substack{c,d \in \mathbb{Z} }} \frac{(q_2 y)^s \chi_1^(c) \chi_2^(d)}{\|a c q_2 z + bd\|^{2s}} \Big(\frac{\|acq_2 z + bd\|}{acq_2 z + bd}\Big)^{k}. \end{equation} For the inner sum above, we have \begin{equation} \frac12 \sumprime_{\substack{c,d \in \mathbb{Z} }} \frac{(q_2 y)^s \chi_1^(c) \chi_2^(d)}{\|a c q_2 z + bd\|^{2s}} \Big(\frac{\|acq_2 z + bd\|}{acq_2 z + bd}\Big)^{k} = \frac{1}{(ab)^{s}} \frac12 \sumprime_{\substack{c,d \in \mathbb{Z} }} \frac{( q_2^\frac{aq_2}{b q_2^} y)^s \chi_1^(c) \chi_2^(d)}{\| c q_2^* \frac{aq_2}{b q_2^} z + d\|^{2s}} \Big(\frac{\| c q_2^ \frac{aq_2}{b q_2^} z + d\|}{ c q_2^ \frac{aq_2}{b q_2^} z + d}\Big)^k, \end{equation} which is none other than $(ab)^{-s} G_{\chi_1^, \chi_2^}(\frac{a q_2}{b q_2^} z, s)$. \end{proof} \subsection{Notation and results from \cite{KY}} \label{section:KY} As shown in \cite[Proposition 3.1]{KY} (see also \cite[Section 3.8]{DiamondShurman}), a complete set of inequivalent cusps for $\Gamma_0(N)$ is given by $\frac{1}{w} = \frac{1}{uf}$ where $f \| N$ and $u$ runs modulo $(f, N/f)$, coprime to the modulus, after choosing a representative coprime to $N$ (such a choice can always be achieved). With this choice of representative, then $\frac{1}{uf} \sim \frac{u}{f}$ (equivalent in $\Gamma_0(N)$), and so by \eqref{eq:EaEquivalentCuspsRelation}, $E_{u/f} = \overline{\psi}(\gamma) E_{1/w}$, where $\gamma(\frac{1}{uf}) = \frac{u}{f}$. One may easily check that any $\gamma$ satisfying this equation has lower-right entry $d$ congruent to $\overline{u}$ modulo both $f$ and $N'=N/f$. Thus $d \equiv \overline{u} \pmod{[f,N']}$. In Lemma \ref{lemma:singularcusps} below, we show that $u/f$ is singular for $\psi$ iff $\psi$ has period dividing $[f,N']$, and therefore $\overline{\psi}(\gamma) = \psi(u)$. That is, \begin{equation} \label{eq:Euoverfversus1overw} E_{u/f}(z,s,\psi) = \psi(u) E_{1/w}(z,s,\psi). \end{equation} We will generally work with cusps of the form $\frac{1}{uf}$, but one may convert to $\frac{u}{f}$ using \eqref{eq:Euoverfversus1overw}. \begin{myprop}[\cite{KY}, Proposition 3.3]\label{prop:stabilizerScaling} Let $\mathfrak{c}= 1/w$ be a cusp of $\Gamma = \Gamma_0(N)$, and set \begin{equation} \label{eq:Nandwformulas} N = (N,w) N' \qquad w = (N,w) w', \qquad N' = (N',w) N''. \end{equation} The stabilizer of $1/w$ is given as \begin{equation} \label{eq:stabilizerFormula} \Gamma_{1/w} = \left\{ \pm \tau_{1/w}^t : t \in \ensuremath{\mathbb{Z}} \right\}, \quad \text{where} \quad \tau_{1/w}^t = \begin{pmatrix} 1 - wN'' t & N'' t \\ -w^2 N'' t & 1+ wN'' t \end{pmatrix}, \end{equation} and one may choose the scaling matrix as \begin{equation}\label{eq:anyCuspScaling} \sigma_{1/w} = \begin{pmatrix} 1&0\\ w&1 \end{pmatrix} \begin{pmatrix} \sqrt{N''} &0\\ 0 &1/\sqrt{N''} \end{pmatrix} . \end{equation} \end{myprop} Remark. With this choice of scaling matrix, we have \begin{equation} \label{eq:scalingMatrixGeneratorComparedtoGammaInfinityGenerator} \sigma_{1/w}^{-1} \tau_{1/w} \sigma_{1/w} = \begin{pmatrix} 1 & 1 \\ & 1 \end{pmatrix}, \end{equation} which is important in the context of checking if $1/w$ is singular for a Dirichlet character $\psi$ (recall the discussion in Section \ref{section:EisensteinCusps}). One should also observe that $N \| w^2 N''$ to see that $\tau_{1/w} \in \Gamma_0(N)$. We next quote a double coset calculation from \cite{KY}, in the special case $\mathfrak{a} = \infty$, in which case the notation from \cite{KY} specializes with $r=N$, $s=1$: \begin{mylemma}[\cite{KY}, Lemma 3.5] \label{lemma:DoubleCosetFormula} Let $\mathfrak{c} = 1/w$ be any cusp of $\Gamma=\Gamma_0(N)$ and $\mathfrak{a} = 1/N \sim \infty$. Let the scaling matrix $\sigma_{1/w}$ be as in \eqref{eq:anyCuspScaling}, and take $\sigma_{\infty} = I$. Then \begin{equation} \label{eq:doublecoset} \sigma_{1/w} ^{-1} \Gamma \sigma_\infty = \left\{\begin{pmatrix} \frac{A}{\sqrt{N''}} &\frac{B}{\sqrt{N''}} \\ C \sqrt{N''} &D \sqrt{N''} \end{pmatrix}: \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in \SL_2(\ensuremath{\mathbb{Z}}),\,\, C \equiv - wA \ensuremath{\negthickspace \negmedspace \pmod}{N} \right\}. \end{equation} \end{mylemma} It is convenient to translate the notation a bit. With $w =uf$ as above, we have \begin{equation} N' = \frac{N}{f}, \qquad N'' = \frac{N'}{(f,N')}, \qquad w'=u. \end{equation} \subsection{Singular cusps} \begin{mylemma} \label{lemma:singularcusps} Let $\psi$ be a Dirichlet character modulo $N$, and let $\mathfrak{a} = \frac{1}{uf}$ with $f\|N$, and $(u,N) = 1$. Then $\mathfrak{a}$ is singular for $\psi$ if and only if $\psi$ is periodic modulo $\frac{N}{(f,N/f)}$, equivalently, the primitive character inducing $\psi$ has modulus dividing $\frac{N}{(f,N/f)}$. \end{mylemma} Remarks. In case $\psi$ is primitive modulo $N$, then the singular cusps for $\psi$ are the Atkin-Lehner cusps $1/f$ with $(f,N/f) = 1$. Also, observe $\frac{N}{(f,N/f)} = [f,N/f]$, and so in particular, $N$ and $\frac{N}{(f,N/f)}$ share the same prime factors. \begin{proof} By Proposition \ref{prop:stabilizerScaling} and \eqref{eq:scalingMatrixGeneratorComparedtoGammaInfinityGenerator}, the cusp $\frac{1}{w} = \frac{1}{uf}$ is singular for $\psi$ iff $\psi(1+wN'' t) = 1$ for all $t \in \mathbb{Z}$. Since $w N'' = \frac{N}{(f,N/f)} u$, the condition that $1/w$ is singular for $\psi$ is seen to be equivalent to $\psi(1+ \frac{N}{(f,N/f)} t) = 1$ for all $t \in \mathbb{Z}$. Let $\chi$ be a Dirichlet character modulo $N$, and suppose $d\|N$. It is an elementary exercise to show that $\chi$ is induced by a character of modulus $d$ if and only if $\chi(1+dk) = 1$ for all $k \in \ensuremath{\mathbb Z}$ such that $(1+dk,N) = 1$. This exercise completes the proof. \end{proof} \begin{mycoro} Suppose $\psi$ is a Dirichlet character modulo $N$, induced by a primitive character $\psi^$ of conductor $N^* \|N$. Then the number of singular cusps for $\psi$ equals \begin{equation} \sum_{\substack{f\|N \\ (f,N/f) \| \frac{N}{N^}}} \varphi((f,N/f)). \end{equation} \end{mycoro} \section{Decomposition of $E_{\mathfrak{a}}$} Our first main result decomposes $E_{\frac{1}{uf}}$ in terms of $E_{\chi_1,\chi_2}$'s. \begin{mytheo} \label{thm:EcuspInTermsofEchichi} Let notation be as in Section \ref{section:KY}. Then \begin{multline} \label{eq:EcuspInTermsofEchichi} E_{\frac{1}{uf}}(z,s,\psi) = \frac{1}{(fN'')^s} \frac{1}{\varphi((f,N/f))} \sum_{q_1 \| \frac{N}{f}} \sum_{q_2 \| f} \sumstar_{\substack{\chi_1 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_1} \\ \chi_2 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_2} \\ \chi_1 \overline{\chi_2} \sim \psi}} \overline{\chi_1}(-u) \frac{L(2s,\chi_1 \chi_2)}{L(2s,\chi_1 \chi_2 \chi_{0,N})} \\ \sum_{\substack{a\|f \\ (a, q_2)=1}} \sum_{\substack{b\|\frac{N}{f} \\ (b, q_1) = 1}} \frac{\mu(a) \mu(b) \chi_1(b) \chi_2(a)}{(ab)^s} E_{\chi_1, \chi_2}\Big(\frac{b f}{a q_2} z, s\Big), \end{multline} where the sum is over primitive characters $\chi_i$ modulo $q_i$, and $\chi_1 \overline{\chi_2} \sim \psi$ means that both sides are induced by the same primitive character. \end{mytheo} Booker, Lee, and Str\"ombersson (personal communication) have independently proved Theorem \ref{thm:EcuspInTermsofEchichi} as well as the inversion formula in Theorem \ref{thm:EchichiInTermsofEa}. They use these formulas, as well as the functional equation of $E_{\chi_1, \chi_2}$, to work out the scattering matrix for $\Gamma_0(N)$ with arbitrary nebentypus. Previously, Huxley \cite{Huxley} considered the trivial nebentypus case. Remarks. Suppose that $\psi$ is primitive of conductor $N$, so that the cusp $\frac{1}{uf}$ is singular iff $(f,N/f) = 1$, and so we may take $u=1$. Then \eqref{eq:EcuspInTermsofEchichi} simplifies as \begin{equation} E_{1/f}(z,s,\psi) = \frac{\chi_1(-1)}{N^s} E_{\chi_1, \chi_2}(z,s), \end{equation} where $\chi_1$ is modulo $N/f$ and $\chi_2$ is modulo $f$, and $\chi_1 \overline{\chi_2} = \psi$. This type of identity is implicit in \cite{DFI}, where the authors explicitly evaluated many properties of the Eisenstein series when the nebentypus is primitive. In another special case where $N$ is square-free and $\psi$ is principal, then \eqref{eq:EcuspInTermsofEchichi} reduces to \cite[(3.25)]{ConreyIwaniec}. Theorem \ref{thm:EcuspInTermsofEchichi} shows in an explicit form that the space of Eisenstein series is spanned by $E_{\chi_1,\chi_2}(Bz,s)$ with $q_1 q_2 B \| N$, and $\chi_1 \overline{\chi_2} \sim \psi$, as expected from the discussion in Section \ref{section:newformdiscussion}. The proof of Theorem \ref{thm:EcuspInTermsofEchichi} is long, so we break the proof into more managable pieces. We begin with some notation. From Lemma \ref{lemma:singularcusps}, the cusp $\frac{1}{uf}$ is singular iff $\psi$ is periodic modulo $[f,N']$. There exist integers $f_0 \| f$ and $N_0'\|N'$ so that $[f,N'] = f_0 N_0'$ and $(f_0 ,N_0') = 1$. The choices of $f_0$ and $N_0'$ may not be unique in case there is a prime power exactly dividing both $f$ and $N'$. Then we may write $\psi = \psi^{(f_0)} \psi^{(N_0')}$ according to this factorization. We remark that $f_0$ and $N_0'$ are useful within the proof of Theorem \ref{thm:EcuspInTermsofEchichi}, yet they are not present in the final formula \eqref{eq:EcuspInTermsofEchichi}. With this notation in place, it is helpful for later to record that if $u \equiv u' \pmod{(f,N')}$ (both coprime to $N$), then $E_{\frac{1}{uf}}(z,s,\psi) = \psi^{(N_0')}(u' \overline{u}) E_{\frac{1}{u'f}}(z,s,\psi)$, following from \eqref{eq:EaEquivalentCuspsRelation} and a calculation of the lower-right entry of a matrix $\gamma$ such that $\gamma \frac{1}{uf} = \frac{1}{u'f}$. Put another way, $\psi^{(N_0')}(u) E_{\frac{1}{uf}}(z,s,\psi)$ is a well-defined function of $u \pmod{(f,N')}$. \begin{mylemma} \label{lemma:EcuspIntermediateFormula} With notation as above, we have \begin{equation} \label{eq:EufInsideLemma} E_{\frac{1}{uf}}(z,s, \psi) = \delta_{f=N} y^s + \frac{y^s}{(N'')^s} \sum_{\substack{(D, fC') = 1 \\ C' > 0, \thinspace (C',N') = 1 \\ D \equiv - \overline{C'} u \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f,N')} }} \frac{ \psi^{(N_0')}(-\overline{u} C') \psi^{(f_0)}(\overline{D}) }{\|C'f z + D\|^{2s}} \Big(\frac{\|C'fz + D\|}{C'fz + D}\Big)^k. \end{equation} \end{mylemma} \begin{proof} From \eqref{eq:Eadef}, and changing variables, we have \begin{multline} E_{1/w}(z,s, \psi) = \sum_{\gamma \in \Gamma_{1/w} \backslash \Gamma} \overline{\psi}(\gamma) j(\sigma_{1/w}^{-1} \gamma, z)^{-k} \text{Im}(\sigma_{1/w}^{-1} \gamma z)^s \\ = \sum_{\tau \in \Gamma_{\infty} \backslash \sigma_{1/w}^{-1} \Gamma} \overline{\psi}(\sigma_{1/w} \tau) j(\tau, z)^{-k} \text{Im}(\tau z)^s. \end{multline} For the evaluation of $\overline{\psi}(\sigma_{1/w} \tau)$, note that \begin{equation} \begin{pmatrix} 1&0\\ w&1 \end{pmatrix} \begin{pmatrix} \sqrt{N''} &0\\ 0 &1/\sqrt{N''} \end{pmatrix} \begin{pmatrix} \frac{A}{\sqrt{N''}} &\frac{B}{\sqrt{N''}} \\ C \sqrt{N''} &D \sqrt{N''} \end{pmatrix} = \begin{pmatrix} A & B \\ C+Aw & D+ Bw \end{pmatrix} . \end{equation} As a consistency check, observe that if we translate $\tau$ on the left by $(\begin{smallmatrix} 1 & n \\ 0 & 1 \end{smallmatrix})$ then that replaces $B$ by $B + D n N''$, and so the lower-right entry changes from $D+Bw$ to $D+Bw + DwnN''$. Since $wN'' = u \frac{N}{(f,N')}$, and $\psi$ is assumed to be periodic modulo $\frac{N}{(f,N')}$, this shows that $\psi$ is well-defined under such translations. Next we need to work out representatives for $\Gamma_{\infty} \backslash \sigma_{1/w}^{-1} \Gamma$, in terms of the lower row of matrices occuring in \eqref{eq:doublecoset}. In the case of the identity coset with $C=0, D=1$, we obtain $\psi(D+Bw) = \psi(1+Bw) = \psi(1)=1$, since this coset occurs only when $f=N$, i.e., $\frac{1}{uf} \sim \infty$. This leads to the term $\delta_{f=N} y^s$. From now on, consider the non-identity cosets. Note that the action of $\Gamma_{\infty}$ does not affect the congruence linking $A$ to $C$. Consider the conditions \begin{equation} \label{eq:CDconditions} C > 0, \quad (C,D) = 1, \quad C=fC', \quad (C',N') =1, \quad \text{and} \quad D \equiv - u \overline{C'} \pmod{(f,N')}. \end{equation} We claim that \begin{equation} \label{eq:doublecosetDecomposition} \Gamma_{\infty} \backslash \sigma_{1/w}^{-1} \Gamma = \delta_{f=N} \Gamma_{\infty} \cup \left\{ \begin{pmatrix} & * \\ C \sqrt{N''} & D \sqrt{N''} \end{pmatrix} : \eqref{eq:CDconditions} \text{ holds} \right\}, \end{equation} as a disjoint union. Moreover, the value of $\overline{\psi}(\sigma_{1/w} \tau)$ is determined by the conditions \eqref{eq:CDconditions} (we will derive a formula for it within the proof). \emph{Proof of claim.} First assume that \eqref{eq:CDconditions} holds. From $(C,D) = 1$, there exists integers $A_0, B_0$ so that $A_0D - B_0 C = 1$. Then $A_0 \equiv \overline{D} \pmod{C}$, and so from the congruence on $D$ in \eqref{eq:CDconditions}, and the fact that $f\|C$, we have $A_0 \equiv - C' \overline{u} \pmod{(f,N')}$. Let $x,y \in \mathbb{Z}$ be such that $A_0 =- \overline{u} C' + fx + N'y$. We next want to find $n \in \ensuremath{\mathbb Z}$ so that $A = A_0 + nC$ satisfies $C \equiv - w A \pmod{N}$, which in turn is equivalent to $A \equiv -\overline{u} C' \pmod{N'}$. For this, we have $A = -\overline{u} C' + fx + N'y + n C'f \equiv -\overline{u} C' + f(x+nC') \pmod{N'}$, so choosing $n \equiv - x \overline{C'} \pmod{N'}$ finishes the job. Next we show the conditions \eqref{eq:CDconditions} follow from the conditions on the right hand side of \eqref{eq:doublecoset}. This can be seen as follows. The determinant equation obviously implies $(C,D) = 1$, and using the congruence we have $1 = AD - BC \equiv A(D+Bw) \pmod{N}$, whence $(A,N) = 1$, and so $(C,N) = (w,N) = f$. That is, we may write $C = f C'$ with $(C',N') = 1$. The congruence on $D$ follows from $D \equiv \overline{A} \pmod{\|C\|}$ and $A \equiv - \overline{u} C' \pmod{N'}$, which together give $D \equiv - u \overline{C'} \pmod{(f,N')}$, as claimed. The condition $C > 0$ may be arranged by multiplication by $-I$. Finally, we show that $\overline{\psi}(\sigma_{1/w} \tau)$ only depends on the data appearing in \eqref{eq:CDconditions}. Explicitly, \begin{equation} \label{eq:psiofAformula} \overline{\psi}(\sigma_{1/w} \tau) = \psi(A) = \psi^{(N_0')}(-\overline{u} C') \psi^{(f_0)}(\overline{D}). \end{equation} We first show that given $C,D \in \mathbb{Z}$ satisfying \eqref{eq:CDconditions}, the value of $\psi(A)$ is uniquely determined. The determinant condition on $A$ is $A \equiv \overline{D} \pmod{C}$ and the congruence is $A \equiv - \overline{u} C' \pmod{N'}$. This determines $A$ modulo the least common multiple of $C$ and $N'$, namely $\frac{C N'}{(C,N')} = \frac{CN'}{(f,N')} = CN''$ (one can also see how the left $\Gamma_{\infty}$ action translates $A$ by this). The condition that these two congruences on $A$ are consistent is precisely the congruence on $D$ in \eqref{eq:CDconditions}. These two congruences on $A$ uniquely determine $A$ modulo $\frac{C' f N'}{(f,N/f)} = C' \frac{N}{(f,N/f)}$. Since $\psi$ is periodic modulo $\frac{N}{(f,N/f)}$, this means that $\psi(A)$ is uniquely determined. Finally, we need to show \eqref{eq:psiofAformula}. We take an interlude to discuss the problem in more general terms. Suppose that we have a pair of congruences $x \equiv a \pmod{Q}$ and $x \equiv b \pmod{R}$, and for consistency, we have $a \equiv b \pmod{(Q,R)}$. We wish to evaluate $\chi(x)$, where $\chi$ is a Dirichlet character modulo $[Q,R]$. There exist integers $Q_0, R_0$ with the following properties: \begin{equation} Q_0 \| Q, \qquad R_0 \| R, \qquad [Q,R] = Q_0 R_0, \qquad (Q_0, R_0) = 1. \end{equation} One may check that \begin{equation} (Q,R) = \frac{Q}{Q_0} \frac{R}{R_0}, \qquad \text{and} \qquad (Q_0, Q/Q_0) = 1 = (R_0, R/R_0). \end{equation} The former equation follows from $(Q,R) = \frac{QR}{[Q,R]}$, and the latter follows by noting that a prime power $p^k$ exactly dividing $Q_0$ has either $k=0$ or $p^k$ exactly dividing $Q$ (and similarly for prime powers dividing $R_0$). We also have that $\frac{R}{R_0} \| Q_0$ and $\frac{Q}{Q_0} \| R_0$, which is deduced from $(R/R_0, Q_0) = (R/R_0, R_0 Q_0) = (R/R_0, [R,Q]) = R/R_0$, and similarly for the other formula. Using the above coprimality formulas, the system of congruences is equivalent to $x \equiv a \pmod{Q_0}$ and $x \equiv b \pmod{R_0}$, under the consistency condition $a \equiv b \pmod{(Q,R)}$. Corresponding to the above notation, we may write $\chi = \chi_{1} \chi_{2}$ where $\chi_{1}$ is modulo $Q_0$ and $\chi_2$ is modulo $R_0$, and then $\chi(x) = \chi_1(a) \chi_2(b)$. This discussion proves the claim, and completes the proof of the lemma. We recall for emphasis that the consistency condition is recorded in \eqref{eq:CDconditions}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm:EcuspInTermsofEchichi}] We continue with \eqref{eq:EufInsideLemma}. The first step is to detect the congruence $D \equiv - \overline{C'} u \pmod{(f,N')}$ with Dirichlet characters; for this, observe that $(DC'u, (f,N')) = 1$ holds from the other listed coprimality conditions. Thus \begin{multline} E_{\frac{1}{uf}}(z,s,\psi) = \delta_{f=N} y^s + \Big[ \frac{\frac{y^s}{(N'')^s}}{\varphi((f,N'))} \sum_{\chi \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f,N')}} (\overline{\chi} \overline{ \psi}^{(N_0')})(-u) \\ \sum_{\substack{(D, fC') = 1 \\ C' \geq 1 \\ (C',N') = 1 }} \frac{(\chi \psi^{(N_0')})(C') (\chi \overline{\psi}^{(f_0)})(D) }{\|C'f z + D\|^{2s}} \Big(\frac{\|C' fz + D\|}{C'fz + D}\Big)^k \Big] . \end{multline} Next we claim that we may omit some of the above coprimality conditions. The modulus of $\chi \psi^{(N_0')}$ is the least common multiple of $(f,N')$ and $N_0'$, and equals \begin{equation} \frac{(f, N') N_0'}{(f, N', N_0')} = \frac{(f, N') N_0' f_0}{(f, N_0') f_0} = \frac{N}{(\frac{f}{f_0} f_0,N_0') f_0} = \frac{N}{(f, f_0 N_0')} = \frac{N}{f} = N'. \end{equation} Therefore we may omit the condition $(C',N') = 1$. A similar calculation shows that the modulus of $\chi \overline{\psi}^{(f_0)}$ is $f$, and that we may omit the condition $(D,f) = 1$. Thus \begin{multline} E_{\frac{1}{uf}}(z,s,\psi) = \delta_{f=N} y^s + \Big[ \frac{\frac{y^s}{(N'')^s}}{\varphi((f,N'))} \sum_{\chi \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f,N')}} (\overline{\chi} \overline{\psi}^{(N_0')})(-u) \\ \sum_{\substack{(D, C') = 1 \\ C' \geq 1 }} \frac{(\chi \psi^{(N_0')})(C') (\chi \overline{\psi}^{(f_0)})(D) }{\|C'f z + D\|^{2s}} \Big(\frac{\|C' fz + D\|}{C'fz + D}\Big)^k \Big]. \end{multline} The term $C'=0$ may be returned to the sum, because it only occurs when $N'=1$ (i.e., $f=N$), and then consulting \eqref{eq:Echi1chi2def}, we have \begin{equation} E_{\frac{1}{uf}}(z,s,\psi) = \frac{1}{(fN'')^s} \frac{1}{\varphi((f,N'))} \sum_{\chi \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f,N')}} (\overline{\chi} \overline{\psi}^{(N_0')})(-u) E_{\chi \psi^{(N_0')}, \chi \overline{\psi}^{(f_0)}}(z,s). \end{equation} Applying Lemma \ref{lemma:EisensteinPrimitivevsNonPrimitive}, we have \begin{multline} E_{\frac{1}{uf}}(z,s,\psi) = \frac{1}{(fN'')^s} \frac{1}{\varphi((f,N'))} \sum_{\chi \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f,N')}} (\overline{\chi} \overline{\psi}^{(N_0')})(-u) \frac{L(2s, (\chi \psi^{(N_0')})^* (\chi \overline{\psi}^{(f_0)})^)}{L(2s, \chi^2 \psi^{(N_0')} \overline{\psi}^{(f_0)} )} \\ \sum_{a\|f} \sum_{b\|N'} \frac{\mu(a) \mu(b) (\chi \psi^{(N_0')})^(b) (\chi \overline{\psi}^{(f_0)})^(a)}{(ab)^s} E_{(\chi \psi^{(N_0')})^, (\chi \overline{\psi}^{(f_0)})^}\Big(\frac{b f }{a q_2} z, s\Big), \end{multline} where $q_2$ (say) is the conductor of $(\chi \overline{\psi}^{(f_0)})^$. Next we set $(\chi \psi^{(N_0')})^* = \chi_1$ and $(\chi \overline{\psi}^{(f_0)})^* = \chi_2$, where $\chi_i$ is primitive of modulus $q_i$, $i=1,2$. Note that \begin{equation} \frac{L(2s, (\chi \psi^{(N_0')})^* (\chi \overline{\psi}^{(f_0)})^)}{L(2s, \chi^2 \psi^{(N_0')} \overline{\psi}^{(f_0)} )} = \prod_{p\|N} \Big(1- \frac{\chi_1(p) \chi_2(p)}{p^{2s}}\Big)^{-1}, \end{equation} which only depends on $\chi_1 \chi_2$. Also, a necessary condition on $\chi_1$ and $\chi_2$ is that $\chi_1 \overline{\chi_2} \sim \psi$. Therefore, by moving the sum over $\chi$ to the inside, we have \begin{multline} \label{eq:EufEchichiWithSumOverchinotsimplifiedYet} E_{\frac{1}{uf}}(z,s,\psi) = \frac{1}{(fN'')^s} \frac{1}{\varphi((f,N'))} \sum_{q_1 \| N'} \sum_{q_2 \| f} \sumstar_{\substack{\chi_1 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_1} \\ \chi_2 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_2} \\ \chi_1 \overline{\chi_2} \sim \psi}} \overline{\chi_1}(-u) \prod_{p\|N} \Big(1- \frac{\chi_1(p) \chi_2(p)}{p^{2s}}\Big)^{-1} \\ \sum_{a\|f} \sum_{b\|N'} \frac{\mu(a) \mu(b) \chi_1(b) \chi_2(a)}{(ab)^s} E_{\chi_1, \chi_2}\Big(\frac{ bf}{aq_2 } z, s\Big) \sum_{\chi \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f,N')}} \delta(\chi, \chi_1, \chi_2, \psi) \end{multline} where $\delta(\chi,\chi_1, \chi_2, \psi)$ is the indicator function of \begin{equation} \label{eq:ChiPsiConditions} (\chi \psi^{(N_0')})^ = \chi_1, \qquad (\chi \overline{\psi}^{(f_0)})^* = \chi_2. \end{equation} Our claim is that $\sum_{\chi} \delta(\chi, \chi_1, \chi_2, \psi)=1$ under the conditions appearing in \eqref{eq:EufEchichiWithSumOverchinotsimplifiedYet}, which will give \eqref{eq:EcuspInTermsofEchichi}, concluding the proof of the theorem. Now we prove the claim. Using that $(f_0, N_0') = 1$ and that a prime divides $N$ iff it divides $f_0 N_0' = [f,N']$, we may uniquely factor $\chi_i = \chi_i^{(f_0)} \chi_i^{(N_0')}$ where $\chi_i^{(f_0)}$ has modulus dividing $f_0$, and $\chi_i^{(N_0')}$ has modulus dividing $N_0'$. We may also factor $\chi$ in the same way, by $\chi = \chi^{(f_0)} \chi^{(N_0')}$; in addition, we may suppose that $\chi$ is primitive of modulus dividing $ (f,N')$. Recall also that $\psi^{(N_0')}$ has modulus $N_0'$, and $\psi^{(f_0)}$ has modulus $f_0$. Thus the assumption $\chi_1 \overline{\chi_2} \sim \psi$ is equivalent to \begin{equation} \label{eq:chi1chi2psi1psi2conditions} \chi_1^{(N_0')} \overline{\chi_2}^{(N_0')} \sim \psi^{(N_0')}, \qquad \text{and} \qquad \chi_1^{(f_0)} \overline{\chi_2}^{(f_0)} \sim \psi^{(f_0)}. \end{equation} The condition $(\chi \psi^{(N_0')})^* = \chi_1$ from \eqref{eq:ChiPsiConditions} is in turn equivalent to \begin{equation} \chi^{(f_0)} (\chi^{(N_0')} \psi^{(N_0')})^* = \chi_1^{(f_0)} \chi_1^{(N_0')}, \end{equation} that is, \begin{equation} \chi^{(f_0)} = \chi_1^{(f_0)} \qquad \text{and} \qquad (\chi^{(N_0')} \psi^{(N_0')})^* = \chi_1^{(N_0')}. \end{equation} Likewise, for the equation with $\chi_2$, we obtain \begin{equation} \chi^{(N_0')} = \chi_2^{(N_0')} \qquad \text{and} \qquad (\chi^{(f_0)} \overline{\psi}^{(f_0)})^* = \chi_2^{(f_0)}. \end{equation} From these two displayed equations, we see that $\chi$ is uniquely determined by $\chi = \chi_1^{(f_0)} \chi_2^{(N_0')}$. Once this choice is made, one can check that \eqref{eq:ChiPsiConditions} holds using \eqref{eq:chi1chi2psi1psi2conditions}. The only remaining loose end is to check that this purported choice of $\chi = \chi_1^{(f_0)} \chi_2^{(N_0')}$ has modulus dividing $(f,N')$. That is, we need that the $f_0$-part of $q_1$ divides $(f,N')$, and similarly that the $N_0'$-part of $q_2$ divides $(f,N')$. Note that \begin{equation} N' = \underbrace{N_0'}_{\text{$N_0'$-part}} \times \underbrace{\frac{N'}{N_0'}}_{\text{$f_0$-part}}, \qquad f = \underbrace{f_0}_{\text{$f_0$-part}} \times \underbrace{\frac{f}{f_0}}_{\text{$N_0'$-part}}, \end{equation} and also that \begin{equation} (f,N') = \underbrace{\frac{f}{f_0}}_{\text{$N_0'$-part}} \times \underbrace{\frac{N'}{N_0'}}_{\text{$f_0$-part}}. \end{equation} These equations show that the $f_0$-part of $N'$ equals the $f_0$-part of $(f,N')$ (both are equal to $\frac{N'}{N_0'}$), and so we conclude that the $f_0$-part of $q_1$ divides $(f,N')$. A similar argument holds for the $N_0'$-part for the other factor. This shows the claim, and completes the proof of Theorem \ref{thm:EcuspInTermsofEchichi}. \end{proof} \section{Inversion} The purpose of this section is to invert \eqref{eq:EcuspInTermsofEchichi}, which is given by the following: \begin{mytheo} \label{thm:EchichiInTermsofEa} Let $\chi_i$, $i=1,2$, be primitive characters modulo $q_i$ with $q_1 q_2 \|N$, and write $N = q_1 q_2 L$. Suppose $B \|L$, and write $L = AB$. Then \begin{equation} \label{eq:EchichiInTermsofEa} E_{\chi_1, \chi_2}(Bz, s) = \mathop{\sum_{d\|A} \sum_{e\|B}}_{(d,e)=1} \frac{\chi_1(d) \chi_2(e)}{(de)^s} \\ \Big(\frac{N}{(q_2 \frac{Bd}{e}, q_1 \frac{Ae}{d})}\Big)^s \sumstar_{u \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{ (q_2 \frac{Bd}{e}, q_1 \frac{Ae}{d})}} \chi_1(-u) E_{\frac{1}{u q_2 \frac{Bd}{e}}}(z,s, \psi). \end{equation} Here the sum is over $u$ is over a set of representatives for $(\mathbb{Z}/(q_2 \frac{Bd}{e}, q_1 \frac{Ae}{d}) \mathbb{Z})^$, chosen coprime to $N$, and $\psi$ is modulo $N$, induced by $\chi_1 \overline{\chi_2}$. \end{mytheo} Remark. Note that $\frac{Bd}{e}$ ranges over certain divisors of $L$, so that $E_{\chi_1,\chi_2}(Bz,s)$ is a linear combination of $E_{\frac{1}{uf}}$'s with $f$'s constrained by $q_2 \| f$ and $f \| q_2 L$. Within the proof of Theorem \ref{thm:EchichiInTermsofEa}, we shall develop and use properties of functions $D_{\chi_1, \chi_2,f}(z,s, \psi)$ defined by \begin{equation} \label{eq:DchiDef} D_{\chi_1, \chi_2,f}(z,s, \psi) = \sumstar_{u \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f, N/f)}} \chi_1(-u) E_{\frac{1}{uf}}(z,s, \psi), \end{equation} where $\chi_i$ is primitive modulo $q_i$, $N = q_1 q_2 L$, $\chi_1 \overline{\chi_2} \sim \psi$, $q_2 \| f$ and $f \| q_2 L$. Notice that \eqref{eq:EchichiInTermsofEa} may be expressed as \begin{equation} \label{eq:EchichiInTermsofDchi} E_{\chi_1, \chi_2}(Bz, s) = N^s \mathop{\sum_{d\|A} \sum_{e\|B}}_{(d,e)=1} \frac{\chi_1(d) \chi_2(e)}{(de)^s} \frac{1}{(q_2 \frac{Bd}{e}, q_1 \frac{Ae}{d})^s} D_{\chi_1, \chi_2, q_2 \frac{Bd}{e}}(z,s,\psi). \end{equation} It is not obvious from \eqref{eq:DchiDef} that $D_{\chi_1,\chi_2,f}$ is well-defined. To see this, first note \begin{equation} \label{eq:chi1IsWellDefined} \chi_1 = \chi_1^{(f_0)} \chi_1^{(N_0')} = \psi^{(N_0')} \chi_1^{(f_0)} \chi_2^{(N_0')}. \end{equation} Now, $\psi^{(N_0')}(u) E_{\frac{1}{uf}}$ is well-defined, as observed in the paragraph preceding Lemma \ref{lemma:EcuspIntermediateFormula}. In addition, one may directly check that $\chi_1^{(f_0)}$ is periodic modulo $f$ (since $f_0\|f$) and modulo $N'$ (since $q_1 \| N'$), and is therefore periodic modulo $(f,N')$. A similar argument holds for $\chi_2^{(N_0')}$. \begin{proof} Let $q_1, q_2, L, A, B$ be as in the statement of the theorem. Set $f = q_2 g$, where $g\|L$. Then $N' = N/f = q_1 L/g$, $(f,N') = (q_2 g, q_1 \frac{L}{g})$, and $N'' = \frac{N'}{(N',f)} = \frac{q_1 \frac{L}{g}}{(q_2 g, q_1 \frac{L}{g} )}$. Our first step is to derive a formula for $D_{\chi_1, \chi_2, f}(z,s,\psi)$ by inserting \eqref{eq:EcuspInTermsofEchichi} into the definition \eqref{eq:DchiDef}, giving \begin{multline} D_{\chi_1, \chi_2, f}(z,s,\psi)= \frac{1}{(fN'')^s} \frac{1}{\varphi((f,N/f))} \sum_{k_1 \| \frac{N}{f}} \sum_{k_2 \| f} \thinspace \sumstar_{\substack{\eta_1 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{k_1} \\ \eta_2 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{k_2} \\ \eta_1 \overline{\eta_2} \sim \psi}} \frac{L(2s,\eta_1 \eta_2)}{L(2s,\eta_1 \eta_2 \chi_{0,N})} \\ \sum_{\substack{a\|f \\ (a, k_2)=1}} \sum_{\substack{b\|\frac{N}{f} \\ (b, k_1) = 1}} \frac{\mu(a) \mu(b) \eta_1(b) \eta_2(a)}{(ab)^s} E_{\eta_1, \eta_2}\Big(\frac{b f}{a k_2} z, s\Big) \sumstar_{u \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{(f, N/f)}} \chi_1(-u) \overline{\eta_1}(-u). \end{multline} We claim the inner sum over $u$ equals $\varphi((f,N'))$ if $\chi_1 = \eta_1$, and vanishes otherwise. For this, apply \eqref{eq:chi1IsWellDefined} to both $\chi_1$ and $\eta_1$, which implies that the sum vanishes unless $\chi_1^{(f_0)} = \eta_1^{(f_0)}$ and $\chi_2^{(N_0')} = \eta_2^{(N_0')}$, recalling that the characters are primitive and that $(f_0,N_0') = 1$. Using \eqref{eq:chi1IsWellDefined} again, we deduce that $\chi_1 = \eta_1$. It is also necessary to verify that the value $k_1 = q_1$ does indeed occur in the sum, which follows from $\frac{N}{f} = q_1 \frac{L}{g}$ and $g\|L$; similarly, $k_2 = q_2$ occurs since $q_2 \| f$. We may next see that from $\eta_1 \overline{\eta_2} \sim \psi \sim \chi_1 \overline{\chi_2}$ that $\eta_2 = \chi_2$ (whence $k_2 = q_2$). Thus \begin{equation} \label{eq:Etwistedavgoveru} D_{\chi_1, \chi_2, q_2 g}(z,s,\psi) = \frac{1}{(f N'')^s} \frac{L(2s,\chi_1 \chi_2)}{L(2s,\chi_1 \chi_2 \chi_{0,N})} \sum_{\substack{a\|g}} \sum_{\substack{b \| \frac{L}{g} }} \frac{\mu(a) \mu(b)}{(ab)^{s}} \chi_1(b) \chi_2(a) E_{\chi_1,\chi_2}\Big(\frac{bg}{a } z, s\Big), \end{equation} using additionally that $a\|f$ may be replaced by $a\|g$ and similarly $b\|\frac{L}{g}$ since $(a,q_2) = 1$ and $(b,q_1) = 1$. Next we interject an elementary inversion formula for certain arithmetical functions. \begin{mylemma} \label{lemma:InversionElementaryLemma} Let $\omega_i$, $i=1,2$ be completely multiplicative functions and suppose $K$ is an arbitrary function defined on the divisors of some positive integer $L$. For $g \| L$, define \begin{equation} \label{eq:JintermsofK} J(g) = \sum_{a\|g} \sum_{b\|\frac{L}{g}} \mu(a) \mu(b) \omega_2(a) \omega_1(b) K\Big(\frac{b g}{a}\Big). \end{equation} Then with $AB=L$, we have \begin{equation} \label{eq:KintermsofJ} K(B) = \Big(\prod_{p\|L} (1- \omega_1 \omega_2(p))^{-1} \Big) \mathop{\sum_{d\|A} \sum_{e\|B}}_{(d,e) = 1} \omega_{1}(d) \omega_{2}(e) J\Big(\frac{Bd}{e}\Big). \end{equation} \end{mylemma} We defer the proof of the lemma to Section \ref{section:Inversion} in order to complete the proof of Theorem \ref{thm:EchichiInTermsofEa}. We apply the lemma with $J(g) = ( \frac{N}{(q_2 g, q_1 \frac{L}{g})})^s \frac{L(2s,\chi_1 \chi_2 \chi_{0,N})}{L(2s,\chi_1 \chi_2)} D_{\chi_1, \chi_2, q_2 g}(z,s,\psi)$, $K(B) = E_{\chi_1,\chi_2}(Bz,s)$, and $\omega_i(n) = \chi_i(n) n^{-s}$, obtaining \begin{multline} E_{\chi_1,\chi_2}(Bz,s) = \prod_{p\|L} (1-p^{-2s} \chi_1(p) \chi_2(p))^{-1} \mathop{\sum_{d\|A} \sum_{e\|B}}_{(d,e)=1} \frac{\chi_1(d) \chi_2(e)}{(de)^s} \\ \Big(\frac{N}{(q_2 \frac{Bd}{e}, q_1 \frac{Ae}{d})}\Big)^s \frac{L(2s, \chi_1 \chi_2 \chi_{0,N})}{L(2s, \chi_1 \chi_2)} D_{\chi_1, \chi_2,q_2 \frac{Bd}{e}}(z,s). \end{multline} Note that $\prod_{p\|L} (1-p^{-2s} \chi_1(p) \chi_2(p))^{-1} = \prod_{p\|N} (1-p^{-2s} \chi_1(p) \chi_2(p))^{-1}$, so this factor cancels the ratio of Dirichlet $L$-functions. Inserting \eqref{eq:DchiDef} into the above formula for $E_{\chi_1,\chi_2}(Bz,s)$ and simplifying, we obtain the theorem. \end{proof} \section{Orthogonality properties} \label{section:orthogonality} \subsection{Orthogonal decomposition into newforms} \label{section:orthogonaldecompositionintonewforms} With Theorems \ref{thm:EcuspInTermsofEchichi} and \ref{thm:EchichiInTermsofEa} in hand, we may now study the orthogonality properties of Eisenstein series attached to Dirichlet characters. Let $\mathcal{E}_{t,\psi}(N)$ be the finite-dimensional vector space defined by \begin{equation} \mathcal{E}_{t,\psi}(N) = \text{span} \{ E_{\mathfrak{a}}(z, 1/2 + it, \psi): \mathfrak{a} \text{ is singular for $\psi$}\}, \end{equation} and define a formal inner product $\langle , \rangle_{\text{Eis}}$ on this space by \begin{equation} \label{eq:FormalInnerProductOrthogonalityEquation} \frac{1}{4 \pi} \langle E_{\mathfrak{a}}(\cdot, 1/2 + it, \psi), E_{\mathfrak{b}}(\cdot, 1/2 + it, \psi) \rangle_{\text{Eis}} = \delta_{\mathfrak{a} \mathfrak{b}}, \end{equation} extended bilinearly. This inner product is natural to use since the spectral decomposition of $L^2(\Gamma_0(N), \psi)$ in terms of Eisenstein series attached to cusps (as in \cite[Propositions 4.1, 4.2]{DFI}) corresponds essentially to \eqref{eq:FormalInnerProductOrthogonalityEquation}; see Section \ref{section:SpectralDecomposition} for more discussion. Perhaps it is worthy of explanation that the dimension of $\mathcal{E}_{t,\psi}(N)$ equals the number of singular cusps for $\psi$, except possibly for $t=0$, and therefore this inner product is well-defined. The key is to study the Fourier expansion of $E_{\mathfrak{a}}(z,1/2+it, \psi)$ at the various cusps. One may easily show that if $c y^{1/2+it} + d y^{1/2-it} = c' y^{1/2+it} + d' y^{1/2-it}$ for infinitely many values of $y$, and $t \neq 0$, then $c=c'$ and $d=d'$. Now suppose that \begin{equation} E_{\mathfrak{a}}(z,1/2+it,\psi) = \sum_{\mathfrak{b}} c_{\mathfrak{a},\mathfrak{b}} E_{\mathfrak{b}}(z,1/2+it, \psi), \end{equation} for some constants $c_{\mathfrak{a}, \mathfrak{b}}$. Equating coefficients of $y^{1/2+it}$ in the Fourier expansions at the arbitrary cusp $\mathfrak{c}$, we have \begin{equation} \delta_{\mathfrak{a}=\mathfrak{c}} y^{1/2 + it} = \sum_{\mathfrak{b}} c_{\mathfrak{a}, \mathfrak{b}} \delta_{\mathfrak{b} = \mathfrak{c}} y^{1/2+it} = c_{\mathfrak{a}, \mathfrak{c}} y^{1/2+it}. \end{equation} Hence $c_{\mathfrak{a}, \mathfrak{c}} = \delta_{\mathfrak{a} = \mathfrak{c}}$, which precisely means that the Eisenstein series attached to cusps are linearly independent, for $t \neq 0$. We should also observe that the constant terms of all Eisenstein series are analytic for $s=1/2+it$, except possibly at $t=0$, by inspection of \eqref{eq:EisensteinChiChiConstantTerm}. \begin{mytheo} \label{thm:DorthogonalBasis} For $f\|N$, $(f,N/f)\|\frac{N}{N^}$, $q_1 \| \frac{N}{f}$, $q_2 \| f$, and $\chi_i$ primitive modulo $q_i$ satisfying $\chi_1 \overline{\chi_2} \sim \psi$, let $D_{\chi_1, \chi_2, f}(z,s)$ be defined by \eqref{eq:DchiDef}. Then the functions $D_{\chi_1, \chi_2, f}(z,s)$ form an orthogonal basis for $\mathcal{E}_{t,\psi}(N)$. \end{mytheo} \begin{proof} These functions are defined by \eqref{eq:DchiDef}, but also may be given by \eqref{eq:Etwistedavgoveru}. The formula \eqref{eq:EchichiInTermsofDchi} allows one to express $E_{\chi_1, \chi_2}$ in terms of $D$'s, while \eqref{eq:DchiDef} may be inverted by inserting \eqref{eq:Etwistedavgoveru} into \eqref{eq:EcuspInTermsofEchichi}, giving \begin{equation} \label{eq:EaintermsofDchiAlternate} E_{\frac{1}{uf}}(z,s,\psi) = \frac{1}{\varphi((f,N/f))} \sum_{q_1 \| \frac{N}{f}} \sum_{q_2 \| f} \sumstar_{\substack{\chi_1 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_1} \\ \chi_2 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_2} \\ \chi_1 \overline{\chi_2} \sim \psi}} \overline{\chi_1}(-u) D_{\chi_1, \chi_2, f}(z,s,\psi). \end{equation} This formula shows the functions $D_{\chi_1, \chi_2, f}$ form a spanning set for $\mathcal{E}_{t,\psi}(N)$. To show these functions are orthogonal, we simply combine \eqref{eq:DchiDef} and \eqref{eq:FormalInnerProductOrthogonalityEquation}, giving \begin{equation} \frac{1}{4 \pi} \langle D_{\chi_1, \chi_2, f_1}, D_{\eta_1, \eta_2, f_2} \rangle = \delta_{f_1 = f_2} \delta_{\chi_1 = \eta_1} \delta_{\chi_2 = \eta_2} \varphi((f, N/f)), \end{equation} where we have let $f = f_1 = f_2$. In particular, for a fixed $f$ for which $(f,N/f)\|\frac{N}{N^}$ (equivalently, $N^\|[f,N/f]$), there is a bijection between $u \pmod{(f,N/f)}$, coprime to the modulus, and pairs of characters $\chi_1, \chi_2$ so that $\chi_1 \overline{\chi_2} \sim \psi$. That is, the number of such pairs of characters equals $\varphi((f,N/f))$. \end{proof} Next we turn to the orthogonality properties of $E_{\chi_1, \chi_2}$. We deduce from \eqref{eq:EchichiInTermsofDchi} that $E_{\chi_1, \chi_2}(B_1 z, 1/2 + it)$ is orthogonal to $E_{\eta_1, \eta_2}(B_2 z, 1/2 + it)$ unless $\chi_1 = \eta_1$ and $\chi_2 = \eta_2$. This shows that, for a given $\chi_1, \chi_2$, the set of functions $D_{\chi_1, \chi_2, q_2 g}(z,s)$ with $g\|L$ forms an orthogonal set for the ``oldclass" formed from $E_{\chi_1, \chi_2}(z,s)$, that is, the subspace \begin{equation} \mathcal{E}_{t,\psi}(L;E_{\chi_1, \chi_2}) := \text{span} \{ E_{\chi_1, \chi_2}(Bz,1/2+it): B \| L \}, \end{equation} where $q_1 q_2 L = N$. By dimension-counting, we see that $\{D_{\chi_1, \chi_2, q_2 g}(z,s):g\|L \}$ then forms an orthogonal basis for this oldclass, and so the functions $E_{\chi_1, \chi_2}(Bz,1/2+it)$ also form a basis for this subspace (not in general orthogonal, however). Summarizing, we have shown \begin{equation} \label{eq:EisensteinAtkinLehnerDecomposition} \mathcal{E}_{t,\psi}(N) = \bigoplus_{q_1 q_2 L =N} \thinspace \sideset{}{^}\bigoplus_{\substack{\chi_1 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_1} \\ \chi_2 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{q_2} \\ \chi_1 \overline{\chi_2} \sim \psi}} \mathcal{E}_{t,\psi}(L;E_{\chi_1,\chi_2}), \end{equation} where, as observed earlier in this section, this is an orthogonal decomposition. This is an extension of Weisinger's newform theory to the non-holomorphic setting. Following Weisinger, define an Eisenstein newform of level $M$ to be one of the $E_{\chi_1, \chi_2}(z, 1/2+it)$, where $\chi_i$ is primitive modulo $q_i$, $i=1,2$, with $q_1 q_2 = M$. Let $\mathcal{H}_{t,\psi}^(M)$ denote the set of Eisenstein newforms of level $M$, nebentypus $\psi$, and spectral parameter $t$. Then we may re-write \eqref{eq:EisensteinAtkinLehnerDecomposition} as \begin{equation} \label{eq:EisensteinAtkinLehnerDecomposition2} \mathcal{E}_{t,\psi}(N) = \bigoplus_{LM = N} \bigoplus_{F \in \mathcal{H}_{t,\psi}^(M)} \mathcal{E}_{t,\psi}(L;F). \end{equation} Needless to say, the above decompositions completely parallel the decomposition of cuspidal newforms as in \cite{AtkinLehner} \cite{AtkinLi}, which gives \begin{equation} S_{t_j, \psi}(N) = \bigoplus_{LM = N} \bigoplus_{f \in H_{t_j,\psi}^(M)} S_{t_j, \psi}(L;f), \end{equation} where $S_{t_j,\psi}(N)$ is the (finite-dimensional) space of cusp forms with spectral parameter $t_j$ and nebentypus $\psi$, $H_{t_j,\psi}^(M)$ is the set of newforms of level $M$ with spectral parameter $t_j$, and $S_{t_j,\psi}(L;f)= \text{span}\{f(\ell z): \ell \| L \}$. \subsection{Summary remarks} For clarity, we summarize the statements of the change-of-basis formulas with some alternative notation. We take this opportunity to make explicit certain facts that were perhaps only implicit within the proofs. Let $\psi$ be a Dirichlet character modulo $N$, of conductor $N_{\psi}$. The cusp $\frac{1}{uf} \sim \frac{u}{f}$ is singular with respect to $\psi$ iff $ (f,N/f) \| \frac{N}{N_{\psi}}$, or alternatively, $N_{\psi} \| [f,N/f]$. There are $\varphi((f,N/f))$ inequivalent (singular) cusps $u/f$ with denominator $f$. Moreover, there exists a bijection between these cusps, and pairs of characters $(\chi_1, \chi_2)$ with $q_1 \| \frac{N}{f}$, $q_2 \| f$, $\chi_i$ primitive modulo $q_i$, $i=1,2$, and $\chi_1 \overline{\chi_2} \sim \psi$. This bijection was observed within the proof of Theorem \ref{thm:DorthogonalBasis} by comparing dimensions, but can be seen directly as follows. The parts of $\chi_1, \chi_2$ of moduli away from $(f,N/f)$ are uniquely determined by the equation $\chi_1 \overline{\chi_2} \sim \psi$ (and there will exist at least one such pair of characters, since $N_{\psi} \| [f,N/f]$). After that, we are free to multiply both $\chi_1$ and $\chi_2$ by the same Dirichlet character modulo $(f,N/f)$. Let $\Psi_f$ denote the set of pairs of such characters, so $\|\Psi_f\| = \varphi((f,N/f))$. For $(\chi_1, \chi_2) \in \Psi_f$, we may define $D_{\chi_1,\chi_2,f}(z,s,\psi)$ by \eqref{eq:DchiDef}. This formula is inverted by \eqref{eq:EaintermsofDchiAlternate}, which in the new notation reads \begin{equation} \label{eq:EaInTermsofDchi} E_{\frac{1}{uf}}(z,s,\psi) = \frac{1}{\varphi((f,N/f))} \sum_{(\chi_1, \chi_2) \in \Psi_f} \overline{\chi_1}(-u) D_{\chi_1, \chi_2, f}(z,s,\psi). \end{equation} One may wish to focus on the pair of characters themselves intrinsically, and to forget about the ambient $f$. Suppose that $q_1 q_2 \| N$, say $N = q_1 q_2 L$, $\chi_i$ is primitive of modulus $q_i$, $i=1,2$, and $\chi_1 \overline{\chi_2} \sim \psi$. We claim that $(\chi_1, \chi_2) \in \Psi_f$ if and only if $f = q_2 g$ with $g \| L$. This is easy to check, because the condition $q_2 \| f$ means that $f = q_2 g$ for some $g$, and since $\frac{N}{f} = \frac{q_1 L}{ g} $, the condition $q_1 \| \frac{N}{f}$ means $g \| L$. In particular, there always exists such an $f$ so that $(\chi_1, \chi_2) \in \Psi_f$. Moreover, the same character pair $(\chi_1, \chi_2)$ lies in $\tau(L)$ sets $\Psi_{f}$. Now suppose that $(\chi_1, \chi_2) \in \Phi_{q_2 g}$ with $g \|L$. Then \eqref{eq:Etwistedavgoveru} becomes \begin{equation} D_{\chi_1, \chi_2, q_2 g}(z,s,\psi) = \frac{(q_2 g, q_1 \frac{ L}{g})^s}{N^s} \frac{L(2s,\chi_1 \chi_2)}{L(2s,\chi_1 \chi_2 \chi_{0,N})} \sum_{\substack{a\|g}} \sum_{\substack{b \| \frac{L}{g} }} \frac{\mu(a) \mu(b)}{(ab)^{s}} \chi_1(b) \chi_2(a) E_{\chi_1,\chi_2}\Big(\frac{bg}{a } z, s\Big), \end{equation} which is inverted by \eqref{eq:EchichiInTermsofDchi}. The $D$-functions are useful because they may be naturally parameterized either by $f$ and $\Psi_f$ or alternatively by the (intrinsic) pairs of characters, along with $g\|L$. Hence, they give a natural intermediate basis between the $E_{\frac{1}{uf}}$ and the $E_{\chi_1, \chi_2}$. \subsection{Remarks on the spectral decomposition} \label{section:SpectralDecomposition} The continuous part of the spectral decomposition, as in \cite[Proposition 4.1]{DFI}, for instance, takes the form \begin{equation} \label{eq:SpectralDecompositionContinuousPart} f_{\text{Eis}}(z) := \int_{-\infty}^{\infty} \sum_{\mathfrak{a}} \frac{1}{4 \pi} \langle f, E_{\mathfrak{a}} \rangle E_{\mathfrak{a}}(z, 1/2 + it, \psi) dt, \end{equation} where $f \in L^2(\Gamma_0(N), \psi)$. It is desirable to express this formula in terms of an alternative basis, as in, for instance, Section \ref{section:orthogonaldecompositionintonewforms}, without having to go through the analytic aspects of the spectral decomposition. \begin{myprop} \label{prop:SpectralFormulasChangeOfBasis} Let $\mathcal{B}(t,\psi,N)$ denote an orthogonal basis for $\mathcal{E}_{t,\psi}(N)$, and suppose $f, g \in L^2(\Gamma_0(N), \psi)$. Then \begin{equation} \label{eq:SpectralFormulasGeneralBasis} \sum_{F \in \mathcal{B}(t,\psi,N)} \frac{\langle f, F \rangle}{\langle F, F \rangle_{\text{Eis}}} F(z), \quad \text{and} \quad \sum_{F \in \mathcal{B}(t,\psi,N)} \frac{\langle f, F \rangle \langle F, g \rangle}{\langle F, F \rangle_{\text{Eis}}} \end{equation} are independent of the choice of basis. \end{myprop} Note that with $F = E_{\mathfrak{a}}$, we have $\langle F, F \rangle_{\text{Eis}} = 4 \pi$, and the first expression in \eqref{eq:SpectralFormulasGeneralBasis} agrees with the integrand in \eqref{eq:SpectralDecompositionContinuousPart}. Likewise, the second formula in \eqref{eq:SpectralFormulasGeneralBasis} is the continous spectrum part of $\langle f, g \rangle$ in Parseval's formula. These formulas are not quite the standard formulas for the projection of a vector $f$ onto a finite-dimensional inner product space, and $\langle f, g \rangle$, respectively, because the inner products in the numerators are different from the inner products in the denominators. Nevertheless, the formulas follow from standard linear algebra calculations. \begin{proof} Let $G$ run over an alternative basis, say $\mathcal{B}'(t,\psi,N)$, and define the change of basis coefficients by $F = \sum_{G} c_{F,G} G$, where $c_{F,G} = \frac{\langle F, G \rangle_{\text{Eis}}}{\langle G, G \rangle_{\text{Eis}}}$. Note that, if $G,G' \in\mathcal{B}'(t,\psi,N)$, then \begin{equation} \label{eq:cFGsumFormula} \sum_{F} \frac{\overline{c_{F,G}} c_{F, G'}}{\langle F, F \rangle_{\text{Eis}}} = \sum_{F} \frac{\langle G, F \rangle_{\text{Eis}} \langle F, G' \rangle_{\text{Eis}} }{\langle G, G \rangle_{\text{Eis}} \langle G', G' \rangle_{\text{Eis}} \langle F, F \rangle_{\text{Eis}}} = \frac{\langle G, G' \rangle_{\text{Eis}}}{\langle G, G \rangle_{\text{Eis}}\langle G', G' \rangle_{\text{Eis}}} = \frac{\delta_{G=G'}}{\langle G, G \rangle_{\text{Eis}}}. \end{equation} Applying \eqref{eq:cFGsumFormula}, we have \begin{equation} \sum_{F} \frac{\langle f, F \rangle}{\langle F, F \rangle_{\text{Eis}}} F = \sum_{F} \sum_{G} \frac{\langle f, G \rangle}{\langle F, F \rangle_{\text{Eis}}} \overline{c_{F,G}} \sum_{G'} c_{F,G'} G' = \sum_{G} \frac{\langle f, G \rangle}{\langle G, G \rangle_{\text{Eis}}} G, \end{equation} showing that the first formula in \eqref{eq:SpectralFormulasGeneralBasis} is independent of basis. A nearly-identical proof works for the second formula in \eqref{eq:SpectralFormulasGeneralBasis}. \end{proof} \subsection{An inner product calculation} \label{section:InnerProduct} Let $M= q_1 q_2$, $N = ML$, and let $\chi_i$ be primitive modulo $q_i$. We wish to evaluate \begin{equation} \label{eq:Ichi1chi2B1B2Definition} I_{\chi_1,\chi_2}(B_1,B_2;N):= \frac{1}{4\pi} \langle E_{\chi_1, \chi_2}(B_1 z,1/2+it), E_{\chi_1, \chi_2}(B_2 z,1/2+it) \rangle_N, \end{equation} where $B_1, B_2 \| L$, and the inner product is on Eisenstein series of level $N$. The motivation to evaluate this inner product is to unify it with a corresponding formula for cuspidal newforms, for which see \cite[p.473]{BlomerMilicevic} (for principal nebentypus) and \cite[Lemma 3.13]{Humphries} (for arbitrary nebentypus). Schulze-Pillot and Yenirce \cite{SchulzePillotYenirce} have also derived the analogous formula for holomorphic newforms of arbitrary level and nebentypus, using only Hecke theory. This is desirable in order to find orthonormal bases for the oldclasses $S_{t_j, \psi}(L;f)$ and $\mathcal{E}_{t,\psi}(L;F)$ that are constructed from the newforms in identical ways, which is useful to treat the discrete spectrum and the continuous spectrum on an equal footing. In Section \ref{section:KuznetsovNewforms} below, we illustrate this idea by proving a Bruggeman-Kuznetsov formula for newforms of squarefree level and trivial nebentypus (these restrictions on the level and nebentypus arise from the assumptions in place in \cite{PetrowYoung}). \begin{mylemma} \label{lemma:InnerProductEvaluation} Let notation be as above. Then \begin{equation} \label{eq:Ichi1chi2InnerProductinTermsofA} \frac{I_{\chi_1,\chi_2}(B_1,B_2;N)}{I_{\chi_1,\chi_2}(1,1;N)} = A_{\chi_1, \chi_2}\Big(\frac{B_2}{(B_1,B_2)} \Big) \overline{A_{\chi_1, \chi_2}}\Big(\frac{B_1}{(B_1,B_2)} \Big), \end{equation} where $A_{\chi_1,\chi_2}(n)$ is the multiplicative function defined for $B \geq 1$ by \begin{equation} \label{eq:Achi1chi2PrimePowerFormula} A_{\chi_1,\chi_2}(p^B) = \frac{\lambda_{\chi_1,\chi_2}(p^B) - \chi_1 \overline{\chi_2}(p) p^{-1} \lambda_{\chi_1,\chi_2}(p^{B-2})}{p^{B/2} (1 + \chi_0(p) p^{-1})}. \end{equation} Here $\lambda_{\chi_1,\chi_2}(n)$ is shorthand for $\lambda_{\chi_1, \chi_2}(n, 1/2+it)$ originally defined by \eqref{eq:lambdachi1chi2Def}, and where for $B=1$ we define $\lambda_{\chi_1, \chi_2}(p^{-1}) = 0$. Moreover, $\chi_0$ is the principal character modulo $q_1 q_2$. \end{mylemma} The form of \eqref{eq:Achi1chi2PrimePowerFormula} is in perfect accord with the cuspidal case of \cite[Lemma 3.13]{Humphries}. The method of Blomer and Mili\'cevi\'c proceeds by unfolding and Rankin-Selberg theory; this method may not be used for Eisenstein series due to the lack of convergence. As a substitute, we use the change-of-basis formulas and orthogonality of $E_{\mathfrak{a}}$'s. \begin{proof} Write $A_i = L/B_i$, $i=1,2$. Then from Theorem \ref{thm:EchichiInTermsofEa}, we have \begin{equation} I_{\chi_1,\chi_2}(B_1,B_2;N) = N \mathop{\mathop{\sum_{d_1\|A_1} \sum_{e_1\|B_1}}_{(d_1,e_1)=1} \mathop{\sum_{d_2\|A_2} \sum_{e_2\|B_2}}_{(d_2,e_2)=1} }_{\frac{B_1 d_1}{e_1} = \frac{B_2 d_2}{e_2}} \frac{\chi_1(d_1 \overline{d_2}) \chi_2(e_1 \overline{e_2})}{(d_1 e_1 d_2 e_2)^{1/2}} \Big(\frac{d_2 e_2}{d_1 e_1} \Big)^{it} \frac{\varphi((q_2 \frac{B_1 d_1}{e_1}, q_1 \frac{A_1 e_1}{d_1}))}{(q_2 \frac{B_1 d_1}{e_1}, q_1 \frac{A_1 e_1}{d_1})^{}}. \end{equation} Parameterizing by the value of $\frac{B_1 d_1}{e_1}$, we have \begin{equation} \label{eq:Ichi1chi2B1B2multiplicativeExpression} I_{\chi_1,\chi_2}(B_1,B_2;N) = N \sum_{R\|L} \frac{\varphi((q_2 R, q_1 \frac{L}{R}))}{(q_2 R, q_1 \frac{L}{R})} G(B_1,R) \overline{G_2(B_2,R)}, \end{equation} where \begin{equation} G(B_i,R) = \mathop{\sum_{d \| A_i} \sum_{e \| B_i}}_{\substack{(d,e) = 1 \\ \frac{B_i d}{e} = R}} \frac{\chi_1(d) \chi_2(e)}{(de)^{1/2+it}}. \end{equation} Based on the multiplicative structure of \eqref{eq:Ichi1chi2B1B2multiplicativeExpression}, we may write \begin{equation} I_{\chi_1,\chi_2}(B_1, B_2;N) = N \prod_{p\|L} I^{(p)}_{\chi_1,\chi_2}(p^{\nu_p(B_1)}, p^{\nu_p(B_2)}), \end{equation} say. By abuse of notation, we replace $\nu_p(B_i)$ by $B_i$, and focus on a single prime $p$. We have \begin{equation} \label{eq:GipR} G(p^{B_i},p^R) = \mathop{\sum_{0 \leq d \leq A_i} \sum_{0 \leq e \leq B_i}}_{\substack{(p^d,p^e) = 1 \\ B_i+d-e = R}} \frac{\chi_1(p^d) \chi_2(p^e)}{p^{(d+e)(1/2+it)}}, \end{equation} and \begin{equation} \label{eq:IpDefinition} I_{\chi_1,\chi_2}^{(p)}(p^{B_1},p^{B_2}) = \sum_{0 \leq R \leq L} \frac{\varphi((p^{q_2+ R}, p^{q_1 +L-R}))}{(p^{q_2+ R}, p^{q_1 +L-R})} G(p^{B_1},p^{R}) \overline{G(p^{B_2}, p^{R})}, \end{equation} where again we have replaced $\nu_p(L)$ by $L$, and similarly for $q_1,q_2$. It is easy to see that \begin{equation} \label{eq:Ichi1chi2ConjugationSymmetry} \overline{I_{\chi_1,\chi_2}^{(p)}(p^{B_1}, p^{B_2})} = I_{\chi_1,\chi_2}^{(p)}(p^{B_2}, p^{B_1}), \end{equation} and one may also easily verify \begin{equation} \label{eq:Ichi1chi2CharacterSwitchingSymmetry} I_{\chi_1,\chi_2}^{(p)}(p^{B_1}, p^{B_2}) = I_{\chi_2,\chi_1}^{(p)}(p^{L-B_1}, p^{L-B_2}). \end{equation} To prove Lemma \ref{lemma:InnerProductEvaluation}, we need three key facts. First, we claim that $I^{(p)}_{\chi_1,\chi_2}(p^{B_1}, p^{B_2})$ is unchanged under the replacements $B_i \rightarrow B_i - \min(B_1, B_2)$. In other words, if we let $B_i = (B_1, B_2) B_i'$, then \begin{equation} \label{eq:Ichi1chi2B1B2B1'B2'} I_{\chi_1, \chi_2}(B_1,B_2;N) = I_{\chi_1, \chi_2}(B_1', B_2';N). \end{equation} This matches a corresponding formula for cusp forms (see \cite[p.473]{BlomerMilicevic}). Secondly, we claim that for $B \geq 1$, we have \begin{equation} \label{eq:IpPrimePower} I^{(p)}_{\chi_1, \chi_2}(1,p^B) = \frac{\lambda_{\chi_1,\chi_2}(p^B)}{p^{\frac{B}{2}}} - \psi(p) \frac{ \lambda_{\chi_1,\chi_2}(p^{B-2})}{p^{\frac{B+2}{2}}}, \end{equation} where $\psi = \chi_1 \overline{\chi_2}$ is the nebentypus of $E_{\chi_1, \chi_2}$. Finally, we claim \begin{equation} \label{eq:IpNoBs} I^{(p)}_{\chi_1, \chi_2}(1,1) = \begin{cases} (1-p^{-1}), \qquad &p\|(q_1, q_2) \\ 1, \qquad &p\|q_1, p \nmid q_2 \\ 1, \qquad &p\|q_2, p \nmid q_1 \\ (1+p^{-1}), \qquad &p \nmid q_1 q_2. \end{cases} \end{equation} Taking these three facts for granted momentarily, we finish the proof. The only apparent discrepancy is that if $p\|(q_1, q_2)$, then the denominator in \eqref{eq:Achi1chi2PrimePowerFormula} does not seem to agree with \eqref{eq:IpNoBs} when $p\|(q_1, q_2)$. However, in this case, $\lambda(p^B) = 0$, so there is agreement after all. All three facts follow from a more careful evaluation of $I_{\chi_1, \chi_2}^{(p)}(p^{B_1}, p^{B_2})$. If $B_i \leq R$ then within \eqref{eq:GipR} this means $e=0$ and $d = R-B_i$, while if $B_i \geq R$ then this means $d=0$ and $e= B_i-R$. Hence, \begin{multline} \label{eq:IpBigExpression} I_{\chi_1,\chi_2}^{(p)}(p^{B_1},p^{B_2}) = \sum_{0 \leq R \leq \min(B_1, B_2)} \frac{\varphi((p^{q_2+ R}, p^{q_1 +L-R}))}{(p^{q_2+ R}, p^{q_1 +L-R})} \frac{\chi_2(p^{B_1-R}) \overline{\chi_2}(p^{B_2-R})}{p^{(B_1-R)(1/2+it) + (B_2-R)(1/2-it)}} \\ + \sum_{B_1 < R \leq B_2} \frac{\varphi((p^{q_2+ R}, p^{q_1 +L-R}))}{(p^{q_2+ R}, p^{q_1 +L-R})} \frac{\chi_1(p^{R-B_1}) \overline{\chi_2}(p^{B_2-R})}{p^{(R-B_1)(1/2+it) + (B_2-R)(1/2-it)}} \\ + \sum_{B_2 < R \leq B_1} \frac{\varphi((p^{q_2+ R}, p^{q_1 +L-R}))}{(p^{q_2+ R}, p^{q_1 +L-R})} \frac{\chi_2(p^{B_1-R}) \overline{\chi_1}(p^{R-B_2})}{p^{(B_1-R)(1/2+it) + (R-B_2)(1/2-it)}} \\ + \sum_{\max(B_1, B_2) < R \leq L} \frac{\varphi((p^{q_2+ R}, p^{q_1 +L-R}))}{(p^{q_2+ R}, p^{q_1 +L-R})} \frac{\chi_1(p^{R-B_1}) \overline{\chi_1}(p^{R- B_2})}{p^{(R-B_1)(1/2+it) + (R-B_2)(1/2-it)}} . \end{multline} Of course, at least one of the two middle terms above is an empty sum. We first deal with the easiest cases with $p\|q_1 q_2$, where we show \begin{equation} \label{eq:IpformulapDividesq1q2} I^{(p)}_{\chi_1, \chi_2}(p^{B_1}, p^{B_2}) = \begin{cases} (1-p^{-1}) \delta_{B_1=B_2}, \qquad &p \|(q_1, q_2), \\ \chi_2(p^{B_1-B_2}) p^{\min(B_1,B_2) - \frac{B_1 + B_2}{2} - it(B_1-B_2)}, \qquad &p \|q_1, p \nmid q_2, \\ \chi_1(p^{B_2-B_1}) p^{\min(B_1,B_2) - \frac{B_1 + B_2}{2} - it(B_2-B_1)} \qquad &p \| q_2, p \nmid q_1. \end{cases} \end{equation} Here the expression $\chi_2(p^{B_1 - B_2})$ is interpreted to be $\overline{\chi_2}(p^{B_2-B_1})$ in case $B_2 > B_1$, and similarly for $\chi_1$. \begin{proof}[Proof of \eqref{eq:IpformulapDividesq1q2}] For $p \| (q_1, q_2)$, all the terms in \eqref{eq:IpBigExpression} vanish except $R = B_1 = B_2$, giving the claimed formula. One may read off the local version of \eqref{eq:Ichi1chi2B1B2B1'B2'} in case $p\|q_1 q_2$. In case $p\|q_1$, $p \nmid q_2$ then the second, third, and fourth lines of \eqref{eq:IpBigExpression} vanish, and so we obtain the claimed formula by evaluating the geometric series. The case $p \| q_2$, $p \nmid q_1$ may be derived from the previous case by using \eqref{eq:Ichi1chi2CharacterSwitchingSymmetry}. \end{proof} Now assume that $p \nmid q_1 q_2$. We claim that \begin{equation} \label{eq:IpEvaluationB1equalsB2} I^{(p)}_{\chi_1, \chi_2}(p^{B_1}, p^{B_2}) = (1+ p^{-1}), \quad \text{if} \quad B_1 = B_2, \end{equation} and if $B_1 < B_2$, then $I^{(p)}_{\chi_1, \chi_2}(p^{B_1}, p^{B_2})$ equals \begin{equation} \label{eq:IpEvaluationB1lessthanB2} \frac{\overline{\chi_2}(p^{B_2-B_1})}{p^{(B_2 - B_1)(1/2-it)}} + \frac{\overline{\chi_2}(p^{B_2-B_1})}{p^{(B_2 - B_1)(1/2-it)}} (1-p^{-1}) \sum_{j=1}^{B_2-B_1-1} \frac{(\chi_1 \chi_2)(p^{j}) }{p^{2it j}} + \frac{\chi_1(p^{B_2-B_1})}{p^{(B_2-B_1)(1/2+it)}}. \end{equation} The case $B_2 < B_1$ may be derived from \eqref{eq:IpEvaluationB1lessthanB2}, using \eqref{eq:Ichi1chi2ConjugationSymmetry} or \eqref{eq:Ichi1chi2CharacterSwitchingSymmetry}. In all cases, we see that \begin{equation} I_{\chi_1, \chi_2}^{(p)}(p^{B_1}, p^{B_2}) = I_{\chi_1, \chi_2}^{(p)}(p^{B_1- \min(B_1,B_2)}, p^{B_2- \min(B_1,B_2)}), \end{equation} and consequently, we obtain the {\bf first key fact}, \eqref{eq:Ichi1chi2B1B2B1'B2'}. It is a pleasant coincidence that \eqref{eq:IpEvaluationB1lessthanB2} agrees with \eqref{eq:IpformulapDividesq1q2} for $B_1 < B_2$. \begin{proof}[Proofs of \eqref{eq:IpEvaluationB1equalsB2} and \eqref{eq:IpEvaluationB1lessthanB2}] For the terms in \eqref{eq:IpBigExpression} with $R \leq \min(B_1, B_2)$, we obtain \begin{equation} \label{eq:RatmostMin} \frac{\chi_2(p^{B_1-B_2})}{p^{\frac{B_1 + B_2}{2} + it(B_1 - B_2)}} \sum_{0 \leq R \leq \min(B_1, B_2)} \frac{\varphi((p^{ R}, p^{L-R}))}{(p^{R}, p^{L-R})} p^R. \end{equation} If $\min(B_1, B_2) < L$, then \eqref{eq:RatmostMin} simplifies as \begin{equation} \frac{\overline{\chi_2}(p^{B_2-B_1}) p^{\min(B_1,B_2)}}{p^{\frac{B_1 + B_2}{2} - it(B_2 - B_1)}}. \end{equation} If $B_1 = B_2 = L$, then \eqref{eq:RatmostMin} becomes \begin{equation} \frac{\chi_2(p^{B_1-B_2})}{p^{\frac{B_1 + B_2}{2} + it(B_1 - B_2)}} (p^{L-1} + p^{L}) = (1+ p^{-1}). \end{equation} For the terms in \eqref{eq:IpBigExpression} with $B_1 < R \leq B_2$, we obtain \begin{equation} \frac{\overline{\chi_2}(p^{B_2-B_1})}{p^{(B_2 - B_1)(1/2-it)}} \sum_{B_1 < R \leq B_2} \frac{\varphi((p^{R}, p^{L-R}))}{(p^{R}, p^{L-R})} \frac{(\chi_1 \chi_2)(p^{R-B_1}) }{p^{2it(R-B_1)}}. \end{equation} If $B_2 < L$ this simplifies as \begin{equation} \frac{\overline{\chi_2}(p^{B_2-B_1})}{p^{(B_2 - B_1)(1/2-it)}} (1-p^{-1}) \sum_{j=1}^{B_2-B_1} \frac{(\chi_1 \chi_2)(p^{j}) }{p^{2it j}}, \end{equation} while if $B_2 = L$, it instead equals \begin{equation} \frac{\overline{\chi_2}(p^{B_2-B_1})}{p^{(B_2 - B_1)(1/2-it)}} \Big( (1-p^{-1}) \sum_{j=1}^{B_2-B_1-1} \frac{(\chi_1 \chi_2)(p^{j}) }{p^{2it j}} + \frac{(\chi_1 \chi_2)(p^{B_2-B_1}) }{p^{2it (B_2 - B_1)}} \Big) . \end{equation} Finally, for the terms with $\max(B_1, B_2) < R \leq L$, we obtain \begin{equation} \frac{1}{p} \frac{\chi_1(p^{B_2-B_1})}{p^{it(B_2-B_1)}} \frac{p^{\min(B_1,B_2)}}{p^{\frac{B_1+B_2}{2}}}. \end{equation} Combining everything, we obtain \eqref{eq:IpEvaluationB1equalsB2} in case $B_1 = B_2$. Similarly, in case $B_1 < B_2 = L$, then we obtain \eqref{eq:IpEvaluationB1lessthanB2}. If $B_1 < B_2<L$, then we obtain \eqref{eq:IpEvaluationB1lessthanB2} after some simplifications, taking the term $j=B_2 - B_1$ out from the inner sum. \end{proof} We deduce the {\bf third key fact} \eqref{eq:IpNoBs}, from \eqref{eq:IpformulapDividesq1q2} and \eqref{eq:IpEvaluationB1equalsB2}. Recalling the definition \eqref{eq:lambdachi1chi2Def}, it is not difficult to derive the {\bf second key fact} \eqref{eq:IpPrimePower} from \eqref{eq:IpEvaluationB1lessthanB2}. This completes the proof. \end{proof} \subsection{An alternative orthonormal basis} Blomer and Mili\'cevi\'c \cite[Lemma 9]{BlomerMilicevic} constructed an orthonormal basis of the oldclass $S_{t_j, \psi_0}(L;f^)$ where $f^$ is a cuspidal newform of level $M$ which is $L^2$ normalized with the level $N$ Petersson inner product, and $\psi_0$ denotes the principal character (see \cite[Lemma 3.15]{Humphries} for arbitrary nebentypus). Their basis takes the form $\{ f^{(g)} : g\|L \}$, where $f^{(g)} = \sum_{d\|g} \xi_{g}(d) f^ \vert_{d}$, where $\xi_g(d)$ are certain arithmetical functions defined in terms of the Hecke eigenvalues of $f^$. They check that the functions $f^{(g)}$ are orthonormal by expanding bilinearly, calculating $\langle f^ \vert_d, f^* \vert_{d'} \rangle$, for each $d,d' \|L$, and evaluating the sums. Therefore, the same process shows that with the coefficients $\xi_{g}(d)$ defined as for cusp forms, using the Hecke eigenvalues, then the linear combinations of normalized $E_{\chi_1, \chi_2}(dz)$'s also form an orthonormal basis for $\mathcal{E}_{t,\psi_0}(L;E_{\chi_1,\chi_2})$. The crucial fact here is that Lemma \ref{lemma:InnerProductEvaluation} has the same form as \cite[Lemma 3.15]{Humphries}. \section{Atkin-Lehner operators} \label{section:AtkinLehnerEigenvalues} In this section, we explain how the $E_{\chi_1, \chi_2}(Bz,s)$ and $D_{\chi_1, \chi_2,f}(z,s, \psi)$ behave under the Atkin-Lehner operators. \subsection{Newforms} Essentially everything in this section was worked out by Weisinger \cite{Weisinger} in the holomorphic setting. Suppose that $QR=N$, and $(Q,R) = 1$, and define an Atkin-Lehner operator by \begin{equation} W_Q = \begin{pmatrix} Qr & t \\ Nu & Qv \end{pmatrix}, \end{equation} where $r,t,u,v \in \mathbb{Z}$, $t \equiv 1 \pmod{Q}$, $r \equiv 1 \pmod{R}$, and $Qrv - R ut = 1$ (so $\det(W_Q) = Q$). The paper \cite{AtkinLi} is a good reference for these operators. The nebentypus $\psi$ factors uniquely as $\psi = \psi^{(Q)} \psi^{(R)}$ where $\psi^{(Q)}$ has modulus $Q$ and $\psi^{(R)}$ has modulus $R$. Weisinger \cite[p.31]{Weisinger} showed that if $f$ is $\Gamma_0(N)$-automorphic with nebentypus $\psi$, then $f \vert_{W_Q}$, which is independent of $r,t,u,v$, is $\Gamma_0(N)$-automorphic with nebentypus $\overline{\psi}^{(Q)} \psi^{(R)}$. Here \begin{equation} f\vert_{W_Q} (z) := j(W_Q, z)^{-k} f(W_Q z). \end{equation} Moreover, Weisinger showed in essence that \begin{equation} \label{eq:WeisingerAtkinLehnerPseudoeigenvalue} E_{\chi_1, \chi_2} \vert_{W_Q} = c(Q) E_{\chi_1',\chi_2'}, \end{equation} where the pseudo-eigenvalue $c(Q)$ is an explicit constant depending on the $\chi_i$ (see \eqref{eq:AtkinLehnerEigenvalueFormula} below for a formula), and where the $\chi_i'$ are defined as follows. Write $\chi_i = \chi_i^{(Q)} \chi_i^{(R)}$, and let $\chi_1' = \chi_2^{(Q)} \chi_1^{(R)} $ and $\chi_2' = \chi_1^{(Q)} \chi_2^{(R)}$. Note that $\chi_1' \chi_2' = \chi_1 \chi_2$, and that $\chi_1' \overline{\chi_2'} = \overline{\psi}^{(Q)} \psi^{(R)}$. Actually, Weisinger worked, in effect, with the completed Eisenstein series $E_{\chi_1, \chi_2}^$ which affects the calculation of the pseuo-eigenvalue, since one must take into account the Gauss sum which appears in \eqref{eq:Echi1chi2FunctionalEquation}. \begin{proof}[Proof of \eqref{eq:WeisingerAtkinLehnerPseudoeigenvalue}] We produce a proof of \eqref{eq:WeisingerAtkinLehnerPseudoeigenvalue} which is of an elementary character, and somewhat different in flavor to that of Weisinger's thesis. Write $q_i = q_i^{(Q)} q_i^{(R)}$, where $q_1^{(Q)} q_2^{(Q)} = Q$ and $q_1^{(R)} q_2^{(R)} = R$. Define \begin{equation} q_1' = q_2^{(Q)} q_1^{(R)}, \qquad \text{and} \qquad q_2' = q_1^{(Q)} q_2^{(R)}, \end{equation} and observe that $q_i'$ is the modulus of $\chi_i'$. From the definition \eqref{eq:EthetaDef}, we have \begin{equation} \label{eq:EthetaSlashWQ} E_{\theta}(z,s) \vert_{W_Q} =j(W_Q, z)^{-k} \sum_{\gamma \in \Gamma_{\infty} \backslash \Gamma_0(1)} \theta(\gamma) j(\gamma, \sigma W_Q z)^{-k} \text{Im}(\gamma \sigma W_Q z)^s, \end{equation} where recall $\sigma z = q_2 z$. Let $\sigma' = (\begin{smallmatrix} \sqrt{q_2'} & \\ & 1/\sqrt{q_2'} \end{smallmatrix})$, so $\sigma' z = q_2' z$. It is straightforward to check \begin{equation} \sigma W_Q = \begin{pmatrix} \sqrt{Q} & \\ & \sqrt{Q} \end{pmatrix} \underbrace{\begin{pmatrix} q_2^{(Q)} r & q_2^{(R)} t \\ q_1^{(R)} u & q_1^{(Q)} v \end{pmatrix}}_{\lambda} \sigma', \end{equation} where observe $\lambda \in SL_2(\ensuremath{\mathbb Z})$. One can also show directly that $j(\lambda, \sigma' z) = j(W_Q, z)$, and so \begin{equation} j(W_Q, z)^{-k} j(\gamma \lambda^{-1}, \sigma W_Q z)^{-k} = j(\gamma, \sigma' z)^{-k}. \end{equation} Therefore, by changing variables $\gamma \rightarrow \gamma \lambda^{-1}$ in \eqref{eq:EthetaSlashWQ}, we obtain \begin{equation} E_{\theta}(z,s) \vert_{W_Q} = \sum_{\gamma \in \Gamma_{\infty} \backslash \Gamma_0(1)} \theta(\gamma \lambda^{-1}) j(\gamma, \sigma' z)^{-k} \text{Im}(\gamma \sigma' z)^s, \end{equation} and so now our task is to understand $\theta(\gamma \lambda^{-1})$. With $\gamma = (\begin{smallmatrix} a & b \\ c & d \end{smallmatrix})$, by direct calculation, we have $\theta(\gamma \lambda^{-1}) = \chi_1(c q_1^{(Q)} v - d q_1^{(R)} u) \chi_2(- c q_2^{(R)} t + d q_2^{(Q)} r)$. Using the factorizations $\chi_i = \chi_i^{(Q)} \chi_i^{(R)}$, we obtain \begin{equation} \theta(\gamma \lambda^{-1}) = \chi_1'(c) \chi_2'(d) \chi_1^{(Q)}(-q_1^{(R)} u) \chi_1^{(R)}(q_1^{(Q)} v) \chi_2^{(Q)}(-q_2^{(R)} t) \chi_2^{(R)}(q_2^{(Q)} r). \end{equation} Next we use that $\chi_1 = \psi \chi_2$ to simplify the above expression, getting \begin{equation} \theta(\gamma \lambda^{-1}) = \chi_1'(c) \chi_2'(d) \chi_1^{(Q)}(-1) \psi^{(Q)}(q_1^{(R)} u) \psi^{(R)}(q_1^{(Q)} v) \chi_2^{(Q)}(-R ut) \chi_2^{(R)}(Q rv). \end{equation} To simplify further, we note \begin{equation} \chi_2^{(Q)}(-R ut) \chi_2^{(R)}(Q rv) = \chi_2(Qrv-Rut) = 1 , \end{equation} using the determinant equation. Moreover, from $t \equiv 1 \pmod{Q}$, we have $u \equiv - \overline{R} \pmod{Q}$, so $\psi^{(Q)}(q_1^{(R)} u) = \overline{\psi}^{(Q)}(- q_2^{(R)})$, and likewise $\psi^{(R)}(q_1^{(Q)} v) = \overline{\psi}^{(R)}(q_2^{(Q)})$. In all, this discussion shows \begin{equation} \theta(\gamma \lambda^{-1}) = \theta'(\gamma) \chi_2^{(Q)}(-1) \overline{\psi}^{(Q)}(q_2^{(R)}) \overline{\psi}^{(R)}(q_2^{(Q)}), \end{equation} where $\theta'$ corresponds to $\chi_1',\chi_2'$. In all, we obtain that \begin{equation} \label{eq:AtkinLehnerEigenvalueFormula} E_{\chi_1, \chi_2}( z,s) \vert_{W_Q} = \chi_2^{(Q)}(-1) \overline{\psi}^{(Q)}(q_2^{(R)}) \overline{\psi}^{(R)}(q_2^{(Q)}) E_{\chi_1 ', \chi_2 '}(z,s). \qedhere \end{equation} \end{proof} \subsection{The Fricke involution} As a particularly important special case of \eqref{eq:AtkinLehnerEigenvalueFormula}, if $Q = N$, then we obtain \begin{equation} E_{\chi_1, \chi_2}( z,s) \vert_{W_N} = \chi_2^{}(-1) E_{\chi_2, \chi_1}(z,s). \end{equation} It is a slightly subtle point that $W_N$ is not exactly the same operator as the Fricke involution $\omega_N := (\begin{smallmatrix} 0 & -1 \\ N & 0 \end{smallmatrix})$. Indeed, we have \begin{equation} W_N = \begin{pmatrix} Nr & t \\ Nu & Nv \end{pmatrix} = \underbrace{\begin{pmatrix} -t & r \\ -N v & u \end{pmatrix} }_{\gamma_N \in \Gamma_0(N)} \omega_N, \end{equation} and note $\psi(\gamma_N) = \psi(-1)$. For a $\Gamma_0(N)$-automorphic function $f$ of nebentypus $\psi$, we have \begin{equation} f(\omega_N z) = f(\gamma_N^{-1} W_N z) = j(\gamma_N^{-1}, W_N z)^{k} \psi(-1) j(W_N, z)^k f\vert_{W_N} = \psi(-1) j(\omega_N, z)^{k} f \vert_{W_N}. \end{equation} Collecting these formulas, we obtain \begin{equation} E_{\chi_1, \chi_2}\Big(\frac{i}{q_1 q_2 y},s \Big) = i^{-k} \chi_2(-1) E_{\chi_2, \chi_1}(iy,s). \end{equation} For the completed Eisenstein series, we may derive \begin{equation} E_{\chi_1, \chi_2}^\Big(\frac{i}{q_1 q_2 y},s \Big) = q_1^{1/2-s} q_2^{s-1/2} i^{-k+\delta_1-\delta_2} \chi_2(-1) \epsilon(\chi_1) \epsilon(\overline{\chi_2}) E_{\chi_2, \chi_1}^(iy,s). \end{equation} Needless to say, this is compatible with \eqref{eq:FrickeFormula1}, which had more restrictive conditions on $k$ and the parity of the characters. \subsection{Oldforms} For this subsection, we restrict attention to the trivial nebentypus case with $k=0$, and $\chi_1 = \chi_2 = \chi$ of modulus $\ell_1 = \ell_2 = \ell$. Viewing $E_{\chi,\chi}(Bz)$ as on $\Gamma_0(\ell^2 M)$ with $B \| M$, we need to see how it operates under the larger collection of Atkin-Lehner operators for this subgroup. Suppose $q$ is a prime so that $q^{\alpha} \|\| M \ell^2$, $q^{\beta}\|\| M$, (so $q^{\alpha-\beta} \|\| \ell^2$), and $q^{\gamma} \|\| B$. If $\gamma \leq \frac{\beta}{2}$, then \cite[Lemma 26]{AtkinLehner} showed \begin{equation} (E_{\chi,\chi} \vert_{B} ) \vert_{W_q} = (E_{\chi,\chi} \vert_{W_q'} ) \vert_{B'}, \end{equation} where we now describe what this means. First, $f \vert_{B} = f(Bz)$. The operator $W_q$ is the Atkin-Lehner involution for the group $\Gamma_0(\ell^2 M)$, while $W_q'$ is the one associated to $\Gamma_0(\ell^2)$. Finally, $B'$ is defined by setting $B = q^{\gamma} B_0$ where $q \nmid B_0$, and then $B' = q^{\beta-\gamma} B_0$. Thus, we have for $\gamma \leq \frac{\beta}{2}$ that \begin{equation} \label{eq:WqAppliedToEchichiBz} (E_{\chi,\chi}(Bz,s) ) \vert_{W_q} = \chi^{(q)}(-1) E_{\chi,\chi}(B'z,s), \end{equation} where $q^j \|\| \ell^2$ (so $j = \alpha - \beta$). If $\gamma > \frac{\beta}{2}$, then the same formula holds, as can be proved by doing the same calculation for $E_{\chi,\chi}(q^{\beta-\gamma} B_0 z, s)$. For each prime $q$ dividing $M \ell^2$, the map $B \rightarrow B'$ is an involution on the set of divisors of $M$. Note that if $q\|\ell$ but $q \nmid M$, then $B' = B$. Thus, the Atkin-Lehner operators permute the functions $E_{\chi,\chi}(Bz, s)$, with a multiplication by $\chi^{(q)}(-1)$. The corresponding property for cusp forms was important in \cite{KY}, showing that the the Fourier coefficients of $f \vert_{B}$ at an Atkin-Lehner cusp are essentially the same as at infinity. It may be interesting to mention that the Atkin-Lehner operators also permute the functions $D_{\chi,f}(z,s) := D_{\chi,\chi,f}(z,s,1)$, since this is a desirable property of an orthonormal basis. \begin{myprop} \label{prop:AtkinLehnerPermutesDchi} Let $f = \ell g$, with $g \| M$, and let $W_q$ be the Atkin-Lehner involution on $\Gamma_0(\ell^2 M)$ with $q \| \ell^2 M$. Suppose $q^{j} \|\| \ell^2$. Then \begin{equation} D_{\chi, \ell g} \vert_{W_q} = \chi^{(q)}(-1) D_{\chi, \ell g'}. \end{equation} \end{myprop} Remark. If $q \| \ell$ but $q \nmid M$, then $g' = g$, and the claimed formula follows immediately from \eqref{eq:WqAppliedToEchichiBz}, since $B' = B$. \begin{proof} We have \begin{equation} \label{eq:AtkinLehnerAppliedToDchi} (M \ell)^s \frac{L(2s,\chi^2 \chi_{0,N})}{L(2s,\chi^2)} D_{\chi, \ell g} = (g, M/g)^s \sum_{\substack{a\|g}} \sum_{\substack{b \| \frac{M}{g} }} \frac{\mu(a) \mu(b)}{(ab)^{s}} \chi(ab) E_{\chi,\chi}\Big(\frac{bg}{a } z, s\Big). \end{equation} Let $\Delta$ denote the right hand side of \eqref{eq:AtkinLehnerAppliedToDchi}. Then by \eqref{eq:WqAppliedToEchichiBz}, $ \Delta \vert_{W_{q}}$ has the same expression but with $(bg/a)$ replaced by $(bg/a)'$, and multiplied by $\chi^{(q)}(-1)$. By abuse of notation, write $M = q^{M} M_0$, $g = q^{g} g_{0}$, and within the sum we write $a = q^a a_0$ and $b = q^b b_0$. Then we have \begin{multline} \Delta = (g_{0}, M_0/g_{0})^s \sum_{\substack{a_0\|g_{0}}} \sum_{\substack{b_0 \| \frac{M_0}{g_{0}} }} \frac{\mu(a_0) \mu(b_0)(q^{g}, q^{M-g})^s }{(a_0 b_0)^{s}} \chi(a_0 b_0) \\ \sum_{0 \leq a \leq g} \sum_{0 \leq b \leq M - g} \frac{\mu(q^a ) \mu(q^b) }{(q^a q^b )^{s}} \chi(q^a q^b ) E_{\chi,\chi}\Big(q^{b+g-a} \frac{b_0 g_{0}}{a_0 } z, s\Big). \end{multline} Similarly, \begin{multline} \Delta \vert_{W_q} = \chi^{(q)}(-1) (g_{0}, M_0/g_{0})^s \sum_{\substack{a_0\|g_{0}}} \sum_{\substack{b_0 \| \frac{M_0}{g_{0}} }} \frac{\mu(a_0) \mu(b_0)(q^{g}, q^{M-g})^s }{(a_0 b_0)^{s}} \chi(a_0 b_0) \\ \sum_{0 \leq a \leq g} \sum_{0 \leq b \leq M - g} \frac{\mu(q^a ) \mu(q^b) }{(q^a q^b )^{s}} \chi(q^a q^b ) E_{\chi,\chi}\Big((q^{b+g-a})' \frac{b_0 g_{0}}{a_0 } z, s\Big). \end{multline} According to the discussion preceding Proposition \ref{prop:AtkinLehnerPermutesDchi}, $(q^{b+g-a})' = q^{M - (b+g-a)}$, and $(q^{g})' = q^{M - g}$, or in additive notation, $g' = M - g$. Finally, switching the roles of $a$ and $b$, and applying the substitution $g' = M - g$ completes the proof. \end{proof} \section{Bruggeman-Kuznetsov for newforms} \label{section:Kuznetsov} In this section, we use some of the material developed in this paper to give a Bruggeman-Kuznetsov formula for newforms, which extends the newform Petersson formula derived in \cite{PetrowYoung}. \subsection{Statement of Bruggeman-Kuznetsov} Suppose that $u_j$ form an orthonormal basis of Hecke-Maass cusp forms of level $N$ and nebentypus $\psi$, and write \begin{equation} u_j(z)= \sum_{n \neq 0} \rho_j(n) e(nx) W_{0,i t_j}(4 \pi \|n\| y), \end{equation} where \begin{equation} W_{0,it_j}(4 \pi y) = 2 \sqrt{y} K_{it_j} (2 \pi y) . \end{equation} Similarly, write \begin{equation} E_{\mathfrak{a}}(z,s, \psi) = \delta_{\mathfrak{a}=\infty} y^{s} + \rho_{\mathfrak{a}, \psi}(s) y^{1-s} + \sum_{n \neq 0} \rho_{\mathfrak{a}, \psi}(n, s) e(nx) W_{0,s-\frac12}(4 \pi \|n\| y). \end{equation} Renormalize the coefficients by defining \begin{equation} \nu_{j}(n) = \left(\frac{4\pi \|n\| }{\cosh(\pi t_j) }\right)^{1/2} \rho_{j} (n), \qquad \nu_{\mathfrak{a},t}(n) = \left(\frac{4\pi \|n\| }{\cosh(\pi t ) }\right)^{1/2} \rho_{\mathfrak{a}, \psi} (n,1/2+it). \end{equation} If $u_j$ is a newform we have $\nu_j(n) = \nu_j(1) \lambda_j(n)$, where $\lambda_j(n)$ are the Hecke eigenvalues. The Bruggeman-Kuznetsov formula for $mn>0$ reads as \begin{multline} \label{eq:Kuznetsov} \sum_j \nu_j(m) \overline{\nu_j(n)} h(t_j) + \sum_{\mathfrak{a}} \frac{1}{4 \pi} \int_{-\infty}^{\infty} \nu_{\mathfrak{a},t}(m) \overline{\nu_{\mathfrak{a},t}}(n) h(t) dt \\ = \delta_{m=n} g_0 + \sum_{c \equiv 0 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{N}} \frac{S_{\psi}(m,n;c)}{c} g\Big(\frac{4 \pi \sqrt{mn}}{c}\Big), \end{multline} where \begin{equation} g_0 = \frac{1}{\pi} \int_{-\infty}^{\infty} t \tanh(\pi t) h(t) dt, \qquad g(x) = 2i \int_{-\infty}^{\infty} \frac{J_{2it}(x)}{\cosh(\pi t)} t h(t) dt, \end{equation} and \begin{equation} S_{\psi}(m,n;c) = \sumstar_{x \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{c}} \psi(x) e\Big(\frac{xm + \overline{x} n}{c} \Big). \end{equation} We may wish to only choose an orthogonal basis of cusp forms instead of an orthonormal basis; the formula is modified by dividing by $\langle u_j, u_j \rangle$, for then $\frac{\nu_j(m) \overline{\nu_j(n)} }{\langle u_j, u_j \rangle}$ is invariant under re-scaling. The inner product is \begin{equation} \langle u_j, u_j \rangle = \int_{\Gamma_0(N) \backslash \mathbb{H}} \|u_j(z)\|^2 \frac{dx dy}{y^2}. \end{equation} This normalization explains why the diagonal term on the right hand side of \eqref{eq:Kuznetsov} does not grow with $N$. Next we discuss how \eqref{eq:Kuznetsov} changes if we choose an alternative basis of Eisenstein series. Let $\{F \}$ be an orthogonal basis for the space of Eisenstein series, with inner product defined formally as in Section \ref{section:orthogonality}. Here we view the spectral parameter $t$ as held fixed, so the dimension of this space equals the number of singular cusps for $\psi$. We claim the quantity \begin{equation} \sum_{F \text{ orthogonal basis}} \frac{\nu_{F,t}(m) \overline{\nu_{F,t}(n)} }{\langle F, F \rangle_{\text{Eis}}} \end{equation} is independent of the orthogonal basis. This follows from Proposition \ref{prop:SpectralFormulasChangeOfBasis}, since one may interpret this expression as the part of $\langle P_n, P_m \rangle$ coming from the continuous spectrum, for some generalized Poincare series (or integrals thereof). Hence, we may re-phrase the Bruggeman-Kuznetsov formula using the decomposition into newforms as in \eqref{eq:EisensteinAtkinLehnerDecomposition}. That is, we have \begin{multline} \label{eq:DeltaNinfinityDef} \sum_{\mathfrak{a}} \frac{1}{4 \pi} \int_{-\infty}^{\infty} \nu_{\mathfrak{a},t}(m) \overline{\nu_{\mathfrak{a},t}}(n) h(t) dt \\ = \sum_{LM=N} \sum_{E_{\chi_1, \chi_2} \in \mathcal{H}_{t,\psi}^(M)} \frac{1}{4 \pi} \int_{-\infty}^{\infty} \sum_{\substack{F \text{ orthogonal basis} \\ \text{for } \mathcal{E}_t(L;E_{\chi_1,\chi_2}) }} \frac{\nu_{F,t}(m) \overline{\nu_{F,t}(n)} }{\langle F, F \rangle_{\text{Eis}}} h(t) dt. \end{multline} This is analogous with the decomposition of cusp forms into newforms, which gives \begin{equation} \label{eq:DeltaN0Def} \sum_j \nu_j(m) \overline{\nu_j(n)} h(t_j) = \sum_{LM=N} \sum_{f \in H_{t_j, \psi}^(M)} \sum_{\substack{F \text{ orthogonal basis} \\ \text{for } S_{t_j,\psi}(L;f) }} \frac{\nu_F(m) \overline{\nu_F(n)}}{\langle F, F \rangle}. \end{equation} The formula \eqref{eq:DeltaN0Def} is not original; one may alternatively consult \cite[(2.11)]{BlomerHarcosMichel} or \cite[(7.32)]{KnightlyLi}. \subsection{Bruggeman-Kuznetsov for newforms, squarefree level} \label{section:KuznetsovNewforms} For the rest of this section, suppose $N$ is square-free, and $\psi$ is the principal character. Let us fix a function $h$, which we suppress from the notation, and define $\Delta_N(m,n)$ as the left hand side of \eqref{eq:Kuznetsov}. Also, write $\Delta_N = \Delta_{N,0} +\Delta_{N,\infty}$ corresponding to the cuspidal part and the Eisenstein part, separately (so $\Delta_{N,0}$ equals \eqref{eq:DeltaN0Def}, and $\Delta_{N,\infty}$ equals \eqref{eq:DeltaNinfinityDef}). Let $\Delta_{N,0}^$ denote the newform analog of $\Delta_{N,0}$, where the Maass forms are restricted to be newforms of level $N$. That is, set \begin{equation} \Delta_{N,0}^(m,n) = \sum_{t_j} h(t_j) \sum_{f \in H_{t_j}^(M)} \frac{\nu_f(m) \overline{\nu_f(n)}}{\langle f, f\rangle}. \end{equation} A nearly-identical proof to that in \cite{PetrowYoung} gives a formula for $\Delta_{N,0}^$ in terms of $\Delta_{M,0}$'s and vice-versa. Precisely, with $\tt{x} = 0$, we have \begin{multline} \label{eq:KuznetsovOldforms} \Delta_{N,\tt{x}}(m,n) = \sum_{LM=N}\frac{1}{\nu(L)} \sum_{\ell \| L^{\infty}} \frac{\ell}{\nu(\ell)^2 } \sum_{d_1, d_2 \| \ell} c_{\ell}(d_1) c_{\ell}(d_2) \sum_{\substack{u\|(m,L) \\ v \| (n,L)}} \frac{u v }{(u, v)} \frac{\mu(\frac{u v}{(u, v)^2})}{\nu(\frac{u v}{(u, v)^2})} \\ \sum_{\substack{a \| (\frac{m}{u}, \frac{u}{(u, v)}) \\ b \| (\frac{n}{v}, \frac{v}{(u, v)})}} \sum_{\substack{e_1 \| (d_1, \frac{m}{a^2 (u,v)}) \\ e_2 \| (d_2, \frac{n}{b^2 (u,v)})} } \Delta_{M,\tt{x}}^\Big( \frac{m d_1}{a^2 e_1^2 (u, v)}, \frac{n d_2}{b^2 e_2^2 (u, v)}\Big), \end{multline} where a definition of the coefficients $c_{\ell}(d)$ may be found in \cite{PetrowYoung} (we have no need of them here). This formula is inverted by \begin{multline} \label{eq:KuznetsovOldformsInverted} \Delta_{N,\tt{x}}^(m,n) = \sum_{LM=N}\frac{\mu(L)}{\nu(L)} \sum_{\ell \| L^{\infty}} \frac{\ell}{\nu(\ell)^2 } \sum_{d_1, d_2 \| \ell} c_{\ell}(d_1) c_{\ell}(d_2) \sum_{\substack{u\|(m,L) \\ v \| (n,L)}} \frac{u v }{(u, v)} \frac{\mu(\frac{u v}{(u, v)^2})}{\nu(\frac{u v}{(u, v)^2})} \\ \sum_{\substack{a \| (\frac{m}{u}, \frac{u}{(u, v)}) \\ b \| (\frac{n}{v}, \frac{v}{(u, v)})}} \sum_{\substack{e_1 \| (d_1, \frac{m}{a^2 (u,v)}) \\ e_2 \| (d_2, \frac{n}{b^2 (u,v)})} } \Delta_{M,\tt{x}}\Big( \frac{m d_1}{a^2 e_1^2 (u, v)}, \frac{n d_2}{b^2 e_2^2 (u, v)}\Big). \end{multline} The proof of \cite{PetrowYoung} only uses Hecke theory, and so all the arguments carry through without any substantial changes. The goal of the rest of this section is to show that \eqref{eq:KuznetsovOldforms} and \eqref{eq:KuznetsovOldformsInverted} hold equally well for the continuous spectrum, that is, with $\tt{x}=\infty$. Then since $\Delta_N = \Delta_{N,0} + \Delta_{N,\infty}$, and likewise for $\Delta_M^$, we obtain analogous formulas for $\Delta_N$ and $\Delta_M^$. When $N$ is square-free and $\psi$ is principal, then the decomposition \eqref{eq:EisensteinAtkinLehnerDecomposition2} simplifies since $\mathcal{H}_t^(M) = \{0 \}$ for $M \neq 1$, and $\mathcal{H}_t^(1)$ is the level $1$ Eisenstein series, $E(z,1/2+it)$. So, define $\Delta_{M,\infty}^(m,n) = 0$ unless $M=1$, in which case $\Delta_{1,\infty}^(m,n) = \Delta_{1,\infty}(m,n)$. For the analysis of $\Delta_{N,\infty}$ to proceed in parallel with that of $\Delta_{N,0}$, we need to pick a basis for $\mathcal{E}_t(L;E)$ analogous to the one chosen in \cite{PetrowYoung}. Instead of the $E_{\mathfrak{a}}$ or $D_{\chi,\ell}$, we start with $E$, the level $1$ Eisenstein series. Let $\phi$ be a function defined on the divisors of $N$, satisfying $\phi(p) = \pm 1$, extended multiplicatively. There are $\tau(N)$ such functions $\phi$, since $N$ is squarefree. Then define \begin{equation} \label{eq:EphiDef} E_{\phi} = \sum_{d\|N} \phi(d) E\vert_{W_d}, \end{equation} which is on $\Gamma_0(N)$. Here $W_d$ is the Atkin-Lehner involution, and from \eqref{eq:WqAppliedToEchichiBz}, we have $(E \vert_{W_d})(z, 1/2 + it) = E(dz,1/2+it)$. It follows from \eqref{eq:EphiDef} that $E_{\phi} \vert_{W_d} = \phi(d) E_{\phi}$, and so these functions form an orthogonal basis for $\mathcal{E}_t(N;E)$. Thus, \begin{equation} \Delta_{N,\infty}(m,n) = \int_{-\infty}^{\infty} h(t) T_t(m,n) dt, \qquad T_t(m,n) = \sum_{\phi} \frac{\nu_{E_{\phi},t}(m) \overline{\nu_{E_{\phi},t}(n)} }{\langle E_{\phi}, E_{\phi} \rangle}. \end{equation} Now we wish to evaluate $T_t(m,n)$ in an analogous way to \cite{PetrowYoung}. As in \cite[(2.5)]{PetrowYoung}, we have \begin{equation} \langle E_{\phi}, E_{\phi} \rangle = \tau(N) \sum_{d\|N} \phi(d) \langle E \vert_{W_d}, E \rangle. \end{equation} Next we claim \begin{equation} \label{eq:AbbessUllmoGeneralizationToEisenstein} \langle E_t\vert_{W_d}, E_t \rangle = \frac{\tau_{it}(d) \sqrt{d}}{\nu(d)} \langle E_t, E_t \rangle, \end{equation} where $\nu(d) = \prod_{p\|d} (p+1)$, which is the index of $\Gamma_0(d)$ in $\Gamma_0(1)$ for square-free $d$. When $E$ is replaced by a cuspidal newform, then this was proved by Abbes and Ullmo \cite{AbbesUllmo}. More general inner product calculations may be found in Section \ref{section:InnerProduct}, but the case here is brief enough that a direct evaluation is desirable. Returning to Theorem \ref{thm:EchichiInTermsofEa}, we see that \begin{equation} E(Bz,s) = N^s \sum_{\substack{d\|A \\ e\|B}} \frac{1}{(de)^s} E_{\frac{1}{Bd/e}}(z,s). \end{equation} Therefore, \begin{equation} \langle E_t \vert_{W_B}, E_t \rangle = N \sum_{a\|N} \frac{1}{a^{1/2+it}} \sum_{\substack{b\|B, c\|\frac{N}{B}}} \frac{1}{(bc)^{1/2-it}} \langle E_{\frac{1}{a}}, E_{\frac{1}{Bc/b}} \rangle. \end{equation} Here the inner product vanishes unless $a = \frac{B}{b} c$. Thus we obtain \begin{equation} \frac{1}{4 \pi} \langle E_t \vert_{W_B}, E_t \rangle = N \Big(\sum_{b\|B} \frac{1}{(B/b)^{1/2+it} b^{1/2-it}} \Big) \Big(\sum_{c\|\frac{N}{B}} \frac{1}{c} \Big) = \tau_{it}(B) \sqrt{B}\nu(N/B). \end{equation} Taking $B=1$, we get $\frac{1}{4 \pi} \langle E_t, E_t \rangle = \nu(N)$, and so \eqref{eq:AbbessUllmoGeneralizationToEisenstein} follows, as well as \begin{equation} \label{eq:EinnerproductComparisonTwoGroups} \langle E_t, E_t \rangle_N = \nu(N) \langle E_t, E_t \rangle_1 \end{equation} where the subscript on the inner product symbol denotes the level of the group to which the inner product is attached. Hence \begin{equation} \langle E_{\phi}, E_{\phi} \rangle = \tau(N) \langle E, E \rangle \prod_{p\|N} \Big(1 + \frac{\phi(p) \tau_{it}(p) p^{1/2}}{\nu(p)} \Big), \end{equation} which is the analog of \cite[(2.6)]{PetrowYoung}. By a direct calculation with the Fourier expansion, we have \begin{equation} \nu_{E_{\phi}}(m) = \sum_{u \| (m,N)} \phi(u) u^{1/2} \nu_E(m/u) = \nu_E(1) \sum_{u \| (m,N)} \phi(u) u^{1/2} \lambda_E(m/u), \end{equation} where $\lambda_E(n) = \tau_{iT}(n)$; this is the analog of \cite[(2.8)]{PetrowYoung}. We therefore need to evaluate the inner sum over $\phi$, namely \begin{equation} \label{eq:chiinnerproductsum} T_t(m,n) = \frac{1}{\tau(N) \langle E, E \rangle}_N \sum_{\phi} \overline{\lambda_{E_{\phi}}(m)} \lambda_{E_{\phi}}(n) \prod_{p \| N} \Big(1 + \frac{\phi(p) \lambda_E(p)p^{1/2}}{\nu(p)} \Big)^{-1}, \end{equation} where we have used \eqref{eq:AbbessUllmoGeneralizationToEisenstein}. Here \eqref{eq:chiinnerproductsum} is analogous to \cite[(3.2)]{PetrowYoung}. At this point, all the calculations of $T_t(m,n)$ run completely parallel to those in \cite[Section 3]{PetrowYoung}, since the formulas that were used there are: Hecke relations, \eqref{eq:AbbessUllmoGeneralizationToEisenstein}, and \eqref{eq:EinnerproductComparisonTwoGroups}, which are the same in both cases of cusp forms vs. Eisenstein. Therefore, \eqref{eq:KuznetsovOldforms} holds with $\tt{x}=\infty$. We also claim that the inversion formula \eqref{eq:KuznetsovOldformsInverted} holds with $\tt{x}=\infty$. The key to this is that the inversion formula proved in \cite[Section 4]{PetrowYoung} is a combinatorial formula proved by inclusion-exclusion and does not depend on any properties of $\Delta_{M,\infty}^$. Similarly, the intermediate hybrid formulas appearing in \cite[Section 5]{PetrowYoung} also extend to the Bruggeman-Kuznetsov formula; these only rely in the previous formulas and Hecke relations which hold equally well in both the Maass and Eisenstein cases. One may then set up the hybrid cubic moment for Maass forms as in \cite[(8.7)]{PetrowYoung}: there exist positive weights $\omega_{u_j}$ and $\omega_t$ so that we define \begin{multline} \mathcal{M}(r,q) = \sum_{\substack{u_j \text{ new, level $rq'$} \\ q'\| \tilde{q}}} \omega_{u_j} h(t_j) L(1/2, u_j \otimes \chi_q)^3 \\ + \frac{1}{4\pi} \int_{-\infty}^{\infty} \sum_{\substack{E_t \text{ new, level $rq'$} \\ q'\| \tilde{q}}} \omega_{t} h(t) L(1/2, E_t \otimes \chi_q)^3 dt. \end{multline} Actually this sum over $E_t$ is empty except when $r=1$ and $q'=1$ in which case the problem reduces to the one treated in \cite{ConreyIwaniec}. There is no reason to exclude $r=1$, however. Now one may approach $\mathcal{M}(r,q)$ as in \cite{PetrowYoung}, just as the original paper of Conrey-Iwaniec \cite{ConreyIwaniec} dealt equally well with Maass forms and holomorphic forms. \section{Proof of the inversion formula} \label{section:Inversion} \begin{proof}[Proof of Lemma \ref{lemma:InversionElementaryLemma}] Write \begin{equation} \mathcal{K} = \mathop{\sum_{d\|A} \sum_{e\|B}}_{(d,e) = 1} \omega_1(d) \omega_2(e) J\Big( \frac{Bd}{e}\Big), \end{equation} so the desired identity is $\mathcal{K} = K(B ) \prod_{p\|L} (1-\omega_1(p) \omega_2(p))$. Inserting the definition \eqref{eq:JintermsofK}, we have \begin{equation} \mathcal{K} = \sum_{\substack{d\|A \\ e\|B \\ (d,e) = 1}} \sum_{\substack{a \| \frac{Bd}{e} }} \sum_{\substack{b \| \frac{A e}{d} }} \mu(a) \mu(b) \omega_1(bd) \omega_2(ae) K\Big(\frac{b d B}{ae} \Big). \end{equation} Reversing the orders of summation, we obtain \begin{equation} \mathcal{K} = \sum_{a\|L} \sum_{b \| L} \sum_{ \substack{d \| A, Bd \equiv 0 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{ae} \\ e \| B, Ae \equiv 0 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{bd} \\ (d,e) = 1}} \mu(a) \mu(b)\omega_1(bd) \omega_2(ae) K\Big(\frac{b d B}{ae} \Big). \end{equation} Now let $(a,d) = g$ and $(b,e) = h$ and write $d = gd'$ and $e= he'$, giving \begin{equation} \mathcal{K} = \sum_{a\|L} \sum_{b \| L} \sum_{g \| (a,A)} \sum_{h \| (b,B)} \sum_{ \substack{d' \| \frac{A}{g}, Bd' \equiv 0 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{\frac{a}{g} h e'} \\ e' \| \frac{B}{h}, A e' \equiv 0 \ensuremath{\negthickspace \negthickspace \negthickspace \pmod}{ \frac{b}{h} g d'} \\ (d', \frac{a}{g}) = 1, \thinspace (e', \frac{b}{h}) = 1 \\ (gd', he') = 1 }} \mu(a) \mu(b) \omega_1(b g d') \omega_2(a h e') K\Big(\frac{b g d' B}{ah e'} \Big). \end{equation} Now since $(d', \frac{a}{g} h e') = 1$, the first congruence condition is equivalent to $\frac{a}{g} h e' \| B$ (which automatically implies $e'\| \frac{B}{h}$, so this latter condition may be omitted since it is redundant). Similarly, the second congruence is equivalent to $\frac{b}{h} g d' \| A$. Next we move the sums over $g$ and $h$ to the outside, and define $a = ga'$, $b = hb'$. Simplifying, we obtain \begin{equation} \mathcal{K} = \sum_{g \| A} \sum_{h \| B} \sum_{a'\|\frac{L}{g}} \sum_{b' \| \frac{L}{h}} \sum_{ \substack{ a' h e' \| B \\ b' g d' \|A \\ (d', a') = 1\\ (e', b') = 1 \\ (gd', he') = 1 }} \mu(ga') \mu(hb') \omega_1(b' g h d') \omega_2( a' g h e') K\Big(\frac{ b' d' B}{ a' e'} \Big). \end{equation} Next expand $\mu(ga') = \mu(g) \mu(a')$, recording the coprimality condition $(a',g) = 1$, and similarly for $\mu(hb')$. Then \begin{equation} \mathcal{K} = \sum_{ \substack{ a' h e' \| B \\ b' g d' \|A \\ (d'g, a') = 1\\ (e'h, b') = 1 \\ (gd', he') = 1 }} \mu(g) \mu(a') \mu(h) \mu(b') \omega_1(b' g h d') \omega_2( a' g h e') K\Big(\frac{ b' d' B}{ a' e'} \Big). \end{equation} Now let $(a',b') = r$, and write $a'=r a''$, $b'=r b''$ where now $(a'', b'') = 1$. Then \begin{equation} \mathcal{K} = \sum_{ \substack{ a'' r h e' \| B \\ b'' r g d' \|A \\ (\dots) }} \mu(g) \mu(a'') \mu^2(r) \mu(h) \mu(b'') \omega_1(b'' r g h d') \omega_2( a'' r g h e') K\Big(\frac{ b'' d' B}{ a'' e'} \Big), \end{equation} where $(\dots)$ represents the following coprimality conditions: \begin{equation} \label{eq:greatbiglistofcoprimalityconditions} (d'g, a'' r) = 1, \quad (e'h, b'' r) = 1, \quad (gd', he') = 1, \quad (a'', b'') = 1, \quad (a'' b'', r) = 1. \end{equation} Now let $a'' e' = \alpha$ be a new variable, and likewise $b'' d' = \beta$; the coprimality conditions in \eqref{eq:greatbiglistofcoprimalityconditions} translate into these conditions: $(\alpha, \beta) =1$, $(\alpha, rg) = 1$, $(\beta, r h) = 1$, and $(r,gh) = (g,h) = 1$. Note that $a''$ and $b''$ do not occur in this list of conditions. Thus we obtain \begin{equation} \mathcal{K} = \sum_{ \substack{ r h \alpha \| B \\ r g \beta \|A \\ (\dots) }} \mu(g) \mu^2(r) \mu(h) \omega_1( r g h \beta) \omega_2( r g h \alpha) K\Big(\frac{ \beta B}{ \alpha} \Big) \sum_{a'' \| \alpha} \sum_{b'' \| \beta} \mu(a'') \mu(b''), \end{equation} where now $(\dots)$ represents the coprimality conditions translated into the new variables. The upshot is that M\"obius inversion gives $\alpha = \beta = 1$, and we obtain $\mathcal{K} = \kappa K(B)$, where \begin{equation} \kappa := \sum_{ \substack{ r h \| B, \thinspace r g \|A, \\ (r, gh) = 1, \thinspace (g, h) = 1 }} \mu(g) \mu^2(r) \mu(h) (\omega_1 \omega_2)(r g h ). \end{equation} We claim that $\kappa = \prod_{p\|AB} (1- \omega_1(p) \omega_2(p))$, which may be checked prime-by-prime by brute force. \end{proof} \section{Mellin transform of Whittaker function} \label{section:WhittakerMellinTransform} In this section, we take the opportunity to correct \cite[Lemma 8.2]{DFI}. The overall method of \cite{DFI} is valid, but a typo early in the derivation makes it difficult to correct the mistake without going through the entire process again. Recall the definition \begin{equation} \Phi_k^{\varepsilon}(s,\beta) = \sqrt{\pi} \int_0^{\infty} \Big(W_{\frac{k}{2}, \beta}(4 y) + \varepsilon \frac{\Gamma(\beta+\frac{1+k}{2})}{\Gamma(\beta+\frac{1-k}{2})} W_{-\frac{k}{2}, \beta}(4 y) \Big) y^{s-\frac12} \frac{dy}{y}, \end{equation} where $\varepsilon = \pm 1$, and that we wish to show \begin{equation} \Phi_k^{\varepsilon}(s,\beta) = p_k^{\varepsilon}(s,\beta) \Gamma\Big(\frac{s+\beta + \frac{1- \varepsilon (-1)^k}{2}}{2}\Big) \Gamma\Big(\frac{s-\beta + \frac{1-\varepsilon}{2}}{2}\Big), \end{equation} where $p_k^{\varepsilon}(s,\beta)$ is a certain polynomial defined recursively below. The first step of the derivation in \cite{DFI} is to give two recursion formulas for the Whittaker function, the first of which contains a typo. The corrected formulas are (cf. \cite[(13.15.10), (13.15.12)]{DLMF}) \begin{align} \frac{2 \beta}{\sqrt{y}} W_{\alpha,\beta}(y) &= W_{\alpha+\frac12, \beta+\frac12}(y) - W_{\alpha+\frac12,\beta-\frac12}(y) \\ &= (\beta-\alpha+\tfrac12) W_{\alpha-\frac12, \beta+\frac12}(y) + (\beta + \alpha - \tfrac12) W_{\alpha-\frac12, \beta-\frac12}(y). \end{align} Next define \begin{equation} V_{k,\beta}^{\varepsilon}(y) = W_{\frac{k}{2}, \beta}(4 y) + \varepsilon \frac{\Gamma(\beta+\frac{1+k}{2})}{\Gamma(\beta+\frac{1-k}{2})} W_{-\frac{k}{2}, \beta}(4 y). \end{equation} The above recursion formulas give \begin{multline} \frac{2 \beta}{\sqrt{y}} V_{k,\beta}^{\varepsilon}(\frac{y}{4}) = (\beta + \tfrac{1-k}{2}) W_{\frac{k-1}{2}, \beta+\frac12}(y) + (\beta + \tfrac{k-1}{2}) W_{\frac{k-1}{2}, \beta-\frac12}(y) \\ + \varepsilon \frac{\Gamma(\beta+\frac{1+k}{2})}{\Gamma(\beta+\frac{1-k}{2})} \Big( W_{\frac{1-k}{2}, \beta+\frac12}(y) - W_{\frac{1-k}{2}, \beta-\frac12}(y) \Big), \end{multline} and using $\Gamma(s+1) = s \Gamma(s)$, we derive \begin{equation} \frac{ \beta}{\sqrt{y}} V_{k,\beta}^{\varepsilon}(y) = (\beta + \tfrac{1-k}{2}) V_{k-1, \beta+\frac12}^{\varepsilon}(y) + (\beta + \tfrac{k-1}{2}) V_{k-1, \beta-\frac12}^{-\varepsilon}(y), \end{equation} which replaces \cite[(8.28)]{DFI}. Then on integration, we derive \begin{equation} \Phi_{k}^{\varepsilon}(s,\beta) =(1 - \tfrac{k-1}{2 \beta}) \Phi_{k-1}^{\varepsilon}(s+\tfrac12, \beta + \tfrac12) + (1 + \tfrac{k-1}{2 \beta}) \Phi_{k-1}^{-\varepsilon}(s+\tfrac12, \beta - \tfrac12), \end{equation} for $\beta \neq 0$. When $k=0$, then the formula for $V_{0,\beta}^{\varepsilon}(y)$ given in \cite[p.532]{DFI} is correct, and by \cite[(6.561.16)]{GR}, we derive \begin{equation} \Phi_0^{\varepsilon}(s,\beta) = \Big(\frac{1+\varepsilon}{2} \Big) \Gamma\Big(\frac{s+\beta}{2}\Big) \Gamma\Big(\frac{s-\beta}{2}\Big), \end{equation} which differs from \cite[(8.30)]{DFI} by a factor $\frac14$. This shows the claimed formula for $\Phi_0^{\varepsilon}$ with $p_0^{\varepsilon} = \frac{1+\varepsilon}{2}$. Now proceed by induction on $k \geq 1$. We obtain \begin{multline} \Phi_{k}^{\varepsilon}(s,\beta) =(1 - \tfrac{k-1}{2 \beta}) p_{k-1}^{\varepsilon}(s+\tfrac12, \beta + \tfrac12) \Gamma\Big(\frac{s+\beta + 1+ \frac{1- \varepsilon (-1)^{k-1}}{2}}{2}\Big) \Gamma\Big(\frac{s-\beta + \frac{1-\varepsilon}{2}}{2}\Big) \\ + (1 + \tfrac{k-1}{2 \beta}) p_{k-1}^{-\varepsilon}(s+\tfrac12, \beta - \tfrac12) \Gamma\Big(\frac{s+\beta + \frac{1+ \varepsilon (-1)^{k-1}}{2}}{2}\Big) \Gamma\Big(\frac{s-\beta + 1+ \frac{1+\varepsilon}{2}}{2}\Big). \end{multline} Next we use \begin{equation} \Gamma\Big(\frac{s+\beta + 1+ \frac{1- \varepsilon (-1)^{k-1}}{2}}{2}\Big) = \Gamma\Big(\frac{s+\beta + \frac{1- \varepsilon (-1)^{k}}{2}}{2}\Big) \times \begin{cases} 1, \quad \varepsilon = -(-1)^k \\ \frac{s + \beta}{2}, \quad \varepsilon = (-1)^k. \end{cases} \end{equation} And similarly, \begin{equation} \Gamma\Big(\frac{s-\beta + 1+ \frac{1+\varepsilon}{2}}{2}\Big) = \Gamma\Big(\frac{s-\beta + \frac{1- \varepsilon}{2}}{2}\Big) \times \begin{cases} 1, \quad \varepsilon = -1 \\ \frac{s - \beta}{2}, \quad \varepsilon = 1. \end{cases} \end{equation} Therefore, we obtain a recursion \begin{equation} \begin{split} p_{k}^{\varepsilon}(s,\beta) = &(1 - \tfrac{k-1}{2 \beta}) p_{k-1}^{\varepsilon}(s+\tfrac12, \beta + \tfrac12) \times \left\{ \begin{array}{lr} 1, & \varepsilon = -(-1)^k \\ \frac{s + \beta}{2}, & \varepsilon = (-1)^k \end{array} \right\} \\ + & (1 + \tfrac{k-1}{2 \beta}) p_{k-1}^{-\varepsilon}(s+\tfrac12, \beta - \tfrac12) \times \left\{ \begin{array}{lr} 1, & \varepsilon = -1 \\ \frac{s - \beta}{2}, & \varepsilon = 1 \end{array} \right\}. \end{split} \end{equation} By direct calculation, we have \begin{align} p_1^{+}(s,\beta) & = 1& p_1^{-}(s,\beta) & = 1 \\ p_2^{+}(s,\beta) & = s-\tfrac12 & p_2^{-}(s,\beta) & = 2 \\ p_3^{+}(s,\beta) & = 2s-1 -\beta & p_3^{-}(s,\beta) & = 2s-1 +\beta \\ p_4^{+}(s,\beta) & = 2(s-\tfrac12)^2 - \beta^2 + \tfrac14 & p_4^{-}(s,\beta) & = 4(s-\tfrac12). \end{align} Compared with \cite[(p. 533)]{DFI}, the sign on $\beta$ is reversed. \end{document}	arXiv
\begin{document} \overfullrule=5pt \title{An extension of the rainbow Erd\H{o}s-Rothschild problem} \author[C. Hoppen]{Carlos Hoppen} \address{Instituto de Matem\'atica e Estat\'{i}stica, UFRGS -- Avenida Bento Gon\c{c}alves, 9500, 91501--970 Porto Alegre, RS, Brazil} \email{[email protected]} \author[H. Lefmann]{Hanno Lefmann} \address{Fakult\"at f\"ur Informatik, Technische Universit\"at Chemnitz, Stra\ss{}e der Nationen 62, 09111 Chemnitz, Germany} \email{[email protected]} \author[D. Nolibos]{Denilson Nolibos} \address{Instituto de Matem\'atica e Estat\'{i}stica, UFRGS -- Avenida Bento Gon\c{c}alves, 9500, 91501--970 Porto Alegre, RS, Brazil} \email{[email protected]} \thanks{This work was partially supported by CAPES and DAAD via Probral (CAPES Proc.~88881.143993/2017-01 and DAAD~57391132). The first author acknowledges the support of CNPq~308054/2018-0), Conselho Nacional de Desenvolvimento Cient\'{i}fico e Tecnol\'{o}gico.} \begin{abstract} Given integers $r \geq 2$, $k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, and a graph $G$, we consider $r$-edge-colorings of $G$ with no copy of a complete graph $K_k$ on $k$ vertices where $s$ or more colors appear, which are called $\mathcal{P}_{k,s}$-free $r$-colorings. We show that, for large $n$ and $r \geq r_0(k,s)$, the $(k-1)$-partite Tur\'an graph $T_{k-1}(n)$ on $n$ vertices yields the largest number of $\mathcal{P}_{k,s}$-free $r$-colorings among all $n$-vertex graphs, and that it is the unique graph with this property. \end{abstract} \maketitle \section{Introduction} Given a fixed graph $F$, the well-known \emph{Tur\'{a}n problem} for $F$ is concerned with the maximum number $\ex(n,F)$ of edges over all $F$-free $n$-vertex graphs, namely over all $n$-vertex graphs that do not contain $F$ as a subgraph. The graphs that achieve this maximum are called \emph{$F$-extremal}. When $F=K_{k}$ is the complete graph on $k \geq 3$ vertices, the unique $F$-extremal graph on $n$ vertices is the balanced, complete, $(k-1)$-partite graph $T_{k-1}(n)$, known as the \emph{Tur\'{a}n graph} for $K_k$~\cite{turan}. For general graphs (and hypergraphs) $F$, determining $\ex(n,F)$ and the corresponding extremal graphs is a very important problem and there is a vast literature related with it (more information may be found in F\"{u}redi and Simonovits~\cite{FS2014}, and in the references therein). An $r$-coloring of a graph $G$ is a function $f \colon E(G) \longrightarrow [r]$ that associates a color in $[r]=\{1,\ldots,r\}$ with each edge of $G$. Erd\H{o}s and Rothschild~\cite{Erd74} were interested in $n$-vertex graphs that admit the largest number of $r$-colorings such that \emph{every color class is $F$-free}. In particular, they conjectured that, for all $n \geq n_0(k)$, the number of $K_k$-free 2-colorings is maximized by the Tur\'{a}n graph $T_{k-1}(n)$. Note that $F$-extremal graphs are natural candidates for maximality, as their edges may be colored arbitrarily, which leads to $r^{\ex(n,F)}$ colorings. It is clear that the number of colorings might increase if we have more than $\ex(n,F)$ edges to color, but additional edges also produce copies of $F$, placing constraints on colorings of their edges. Regarding the Erd\H{o}s-Rothschild Conjecture, Yuster~\cite{yuster} gave an affirmative answer for $k=3$ and any $n \geq 6$, while Alon, Balogh, Keevash and Sudakov~\cite{ABKS} showed that, for $r \in \{2,3\}$ and $n \geq n_0$, where $n_0$ is a constant depending on $r$ and $k$, the Tur\'{a}n graph $T_{k-1}(n)$ is the unique optimal $n$-vertex graph for the number of $K_k$-free $r$-colorings. Recently, H\`{a}n and Jim\'{e}nez~\cite{HJ2018} obtained better bounds on $n_0$ using the Container Method. For $r \geq 4$, the answer is more complicated. Pikhurko and Yilma~\cite{PY12} found the graphs that admit the largest number of such colorings for $r=4$ and $k \in \{3,4\}$, which turn out to be balanced, complete, multipartite graphs that are not $K_k$-free. Botler et al.~\cite{Botleretal19} characterized the extremal graphs for $k=3$ and $r=6$, and they gave an approximate result for $k=3$ and $r=5$. Pikhurko, Staden and Yilma~\cite{PSY16} showed that at least one of the graphs with the largest number of colorings is complete multipartite. Balogh~\cite{Bal06} was the first to consider $r$-colorings that avoid a copy of a graph $F$ colored in a non-monochromatic way. A similar problem was investigated by Hoppen and Lefmann~\cite{rbmx} and by~Benevides, Hoppen and Sampaio~\cite{BHS17}, who considered edge-colorings of a graph avoiding a copy of $F$ with a \emph{prescribed pattern}. Given a graph $F$, a \emph{pattern} $P$ of $F$ is a partition of its edge set. An edge-coloring of a graph $G$ is said to be \emph{$(F,P)$-free} if $G$ does not contain a copy of $F$ in which the partition of the edge set induced by the coloring is isomorphic to $P$. For instance, if the partition $P$ consists of a single class, $(F,P)$-free colorings avoid \emph{monochromatic} copies of $F$. On the other hand, if $P$ is the pattern where each edge of $F$ lies in a different class, $(F,P)$-free colorings avoid \emph{rainbow} copies of $F$. These colorings are known as \emph{Gallai colorings} when $F=K_3$. Given the number of colors $r \geq 1$, a graph $F$ and a pattern $P$ of $F$, let $\mathcal{C}_{r,(F,P)}(G)$ be the set of all $(F,P)$-free $r$-colorings of a graph $G$. We write $$c_{r,(F,P)}(n) = \max\left\{\, \|\mathcal{C}_{r,(F,P)}(G)\| \colon \|V(G)\| = n \, \right\},$$ and we say that an $n$-vertex graph $G$ is \emph{$(r,F,P)$-extremal} if $\|\mathcal{C}_{r,(F,P)}(G)\| = c_{r,(F,P)}(n)$. Most results about $c_{r,(F,P)}(n)$ involve monochromatic or rainbow patterns, more information may be found in~\cite{BL19,CGM20,linear,rainbow_triangle,rainbow_kn} and in the references therein. In particular, the work of~\cite{BHS17} implies that, for any such pattern, there is an extremal $(r,F,P)$-extremal graph that is complete multipartite. Here, we generalize this problem to colorings that avoid a family of patterns. Let $k \geq 3$, $r \geq 2$ and $s \leq \binom{k}{2}$ be positive integers. Given a graph $G$, we are interested in $r$-edge-colorings of $G$ with no copy of $K_k$ colored with $s$ or more colors, which are called $\mathcal{P}_{k,s}$-free $r$-colorings. Let $\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)$ denote this set of $r$-colorings and let \begin{equation}\label{def1} c_{r,\mathcal{P}_{k,s}}(n) = \max\left\{\, \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\| \colon \|V(G)\| = n \, \right\}. \end{equation} If $s=1$, finding the $n$-vertex graphs that achieve $c_{r,\mathcal{P}_{k,1}}(n)$ colorings is just a restatement of the Tur\'{a}n problem, as it is equivalent to finding a $K_k$-free $n$-vertex graph with the largest number of edges. If $s=\binom{k}{2}$, this is precisely the problem of finding $(r,K_k,P)$-extremal graphs, where $P$ is the rainbow pattern of $K_k$. We note that previous results already give the full solution of this problem for $k=3$, at least for large $n$. Note that $1\leq s \leq 3$ in this case. The case $s=1$ corresponds to the Tur\'an problem, so that $T_2(n)$ is the unique extremal configuration for all $n,r \geq 2$. In the case $s=2$, every triangle in a graph $G$ has to be monochromatic in a $\mathcal{P}_{3,2}$-free $r$-coloring of $G$. So, if any edge contained in a triangle is removed from $G$, the number of colorings does not decrease, which immediately implies that $c_{r,\mathcal{P}_{3,2}}(n)=\, \|\mathcal{C}_{r,\mathcal{P}_{3,2}}(T_2(n))\|$ for any $r \geq 2$ and $n \geq 2$. It is easy to show that $T_2(n)$ is the only $n$-vertex graph with this property\footnote{Analogously $c_{r,\mathcal{P}_{k,2}}(n)=\, \|\mathcal{C}_{r,\mathcal{P}_{k,2}}(T_{k-1}(n))\|$ for any $r \geq 2$ and $n \geq 2$.}. For $s=3$, Balogh and Li~\cite{BL19} proved that, for $n$ sufficiently large, the complete graph is the unique extremal configuration for $r \leq 3$ and $T_2(n)$ is the unique extremal configuration for $r \geq 4$. Bastos, Benevides and Han~\cite{BBH20} obtained related results and Hoppen, Lefmann and Odermann~\cite{rainbow_triangle} had previously established the extremality of $T_2(n)$ for $r \geq 5$). The following states two easy facts about determining~\eqref{def1} and the $n$-vertex graphs that achieve extremality. \begin{lemma}\label{lemma_simple} Let $n \geq k \geq 3$, $s \leq \binom{k}{2}$ and $r \geq 2$ be integers. \begin{itemize} \item[(a)] If $r<s$, then $c_{r,\mathcal{P}_{k,s}}(n)= \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(K_n)\|=r^{\binom{n}{2}}$. \item[(b)] If $c_{r,\mathcal{P}_{k,s}}(n)= \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(T_{k-1}(n))\|$ and $1 \leq s'<s$, then $$c_{r,\mathcal{P}_{k,s'}}(n)= \|\mathcal{C}_{r,\mathcal{P}_{k,s'}}(T_{k-1}(n))\|.$$ \end{itemize} \end{lemma} \begin{proof} Part (a) is trivial, as no $r$-coloring can produce a copy of $K_k$ colored with $s$ or more colors if $r<s$. In part (b), the hypothesis tells us that for any $n$-vertex graph $G$ $$c_{r,\mathcal{P}_{k,s}}(n)=\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(T_{k-1}(n))\|\geq \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\|.$$ The conclusion now follows from the fact that $\mathcal{C}_{r,\mathcal{P}_{k,s'}}(G) \subseteq \mathcal{C}_{r,\mathcal{P}_{k,s}}(G)$ for any graph $G$ and that $\mathcal{C}_{r,\mathcal{P}_{k,s}}(T_{k-1}(n))=\mathcal{C}_{r,\mathcal{P}_{k,s'}}(T_{k-1}(n))$. \end{proof} The work~\cite[Theorem 1.2]{rainbow_kn} shows that, given $k \geq 4$ and the rainbow pattern $P$ of $K_k$, there is $r_0$ such that $c_{r,(K_k,P)}(n)=\|\mathcal{C}_{r,(K_k,P)}(T_{k-1}(n))\| $ for all $r \geq r_0$ and $n \geq n_0(r,k)$. With Lemma~\ref{lemma_simple}~(b), we deduce that, for any $k \geq 4$ and $s \leq \binom{k}{2}$, there is $r_0$ such that $c_{r,\mathcal{P}_{k,s}}(n)= \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(T_{k-1}(n))\|$ for all $r \geq r_0$ and $n \geq n_0(r,k,s)$. However, this value of $r_0$ is superexponential in $k$, and the authors of~\cite{rainbow_kn} believed that this result should hold for much smaller values of $r_0$. By addressing a more general problem, we are able to obtain much better bounds for smaller values of $s$; moreover, our results lead to better bounds on $r_0$ in the case $s= \binom{k}{2}$, i.e., when only rainbow copies of $K_k$ are avoided. The main result in this paper is the following. For simplicity, we write it in terms of functions that will be defined in the next subsection. \begin{theorem}\label{main_thm} Let $k \geq 4$ and $ 2 \leq s \leq \binom{k}{2}$ be integers. Fix $r \geq r_{0}(k,s)$, defined in~\eqref{def_r0},~\eqref{def_r1} and~\eqref{def_r2} below for $s \leq s_0(k)$, $s_0(k)<s\leq s_1(k)$ and $s>s_1(k)$, respectively. There is $n_{0}=n_0(r,k,s)$ for which the following holds. Every graph $G=(V,E)$ on $n > n_{0}$ vertices satisfies $$\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\| \leq r^{\ex(n,K_{k})}.$$ Moreover, equality holds if and only if $G$ is isomorphic to $T_{k-1}( n)$. \end{theorem} To prove Theorem~\ref{main_thm}, we use a stability method that relies on the following result. \begin{theorem}\label{stability_thm} Let $k \geq 4$ and $ 2 \leq s \leq \binom{k}{2}$ be integers. Fix $r \geq r_{0}(k,s)$, defined in~\eqref{def_r0},~\eqref{def_r1} and~\eqref{def_r2} below for $s \leq s_0(k)$, $s_0(k)<s\leq s_1(k)$ and $s>s_1(k)$, respectively. For any $\delta > 0$, there is $n_{0}=n_0(\delta,r,k,s)$ as follows. If $G=(V,E)$ is a graph on $n > n_{0}$ vertices such that $\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\| \geq r^{\ex(n,K_{k})}$, then there is a partition $V=W_{1} \cup \cdots \cup W_{k-1}$ such that at most $\delta n^{2}$ edges have both endpoints in a same class $W_i$. \end{theorem} It turns out that the value of $r_0(k,s)$ in the above statements is needed in our proof of Theorem~\ref{stability_thm}, and we do not believe that it is best possible. However, our results give much better dependency on $s$ and $k$ than the bound in~\cite{rainbow_kn}. Indeed, if $s \leq s_0(k)$, the value given for $r_0(k,s)$ is less than $(s-1)^2$. Moreover, if $s \leq s_1(k)$, the quantity $r_0(k,s)$ is less than $(s-1)^7$. \subsection{The functions \mathversion{bold}$r_0(k,s)$, $s_0(k)$ and $s_1(k)$\mathversion{normal} } In order to specify the quantities given in the statement of Theorems~\ref{main_thm} and~\ref{stability_thm}, we shall define some additional functions. For $j \in \{2, \ldots, k-1\}$, let \begin{equation}\label{def_A} A(k,j)= \binom{k}{2}-\ex(k, K_{j+1}) = \binom{\lfloor k/j \rfloor}{2} \left( \left\lfloor \dfrac{k}{j} \right\rfloor j +j - k \right) + \binom{\lceil k/j \rceil}{2} \left( k- \left\lfloor \dfrac{k}{j} \right\rfloor j \right), \end{equation} which, by Tur\'an's Theorem, is the minimum number of edges that must be deleted from a complete graph $K_k$ to make it $j$-partite. Let \begin{eqnarray} s_0(k)&=&A(k,2)+2=\binom{k}{2}-\left\lfloor \frac{k}{2} \right\rfloor \cdot \left\lceil \frac{k}{2} \right\rceil+2 \label{def_s}\\ s_1(k)&=&\binom{k}{2} - \left\lfloor \frac{k}{2} \right\rfloor +2. \label{def_s1} \end{eqnarray} For $s \leq s_0(k)$, let $i^\ast$ be the least value of $i$ such that $A(k,k-i) \geq s-2$. As it turns out, we have $i^\ast \leq \min\{s-2,k-2\}$. Let $r_0(k,s)$ be the least integer greater than \begin{equation} \label{def_r0} (s-1)^{\frac{k-1}{k-2}} \prod^{i^\ast}_{i=2}\left( s- A(k,k-i+1)-1 \right)^{ \frac{1}{(k-i-1)(k-i)}} \end{equation} For any fixed $s>s_0(k)$, we consider additional parameters. Let $j \in [k-1]$ and $2 \leq p \leq k-1$ be integers satisfying the following condition: \begin{eqnarray} \label{eq:opt2} b(k,p,j) = \mbox{min } \left\{ j\binom{p}{2}, \left\lfloor \frac{k}{p} \right\rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \right\} \leq \binom{k}{2} - s + 2, \end{eqnarray} and define \begin{eqnarray}\label{eq:0pt1} L(k,s,p,j) =1+ \frac{2p(k-1)}{j(p-1)}. \end{eqnarray} If $s_0(k)<s \leq s_1(k)$, let $p^\ast$ be the largest $p \geq 2$ such that~\eqref{eq:opt2} holds for $j=k-1$. Our choice of $s$ ensures that there is such a $p$. We define $r_0(k,s)$ as the least integer greater than \begin{equation} \label{def_r1} \left( s-A(k,2)-1\right)^{L(k,s,p^\ast,k-1)} \cdot \left( \prod^{k-2}_{i=2}\left( s-A(k,k-i+1)-1 \right)^{ \frac{1}{(k-i-1)(k-i)}} \right) \cdot (s-1)^{\frac{k-1}{k-2} } . \end{equation} If $s > s_1$, let $j^\ast$ be the largest $j \geq 1$ such that~\eqref{eq:opt2} holds for $p=2$. We define $r_0(k,s)$ as the least integer greater than \begin{equation} \label{def_r2} \left( s-A(k,2)-1\right)^{L(k,s,2,j^\ast)} \cdot \left( \prod^{k-2}_{i=2}\left( s-A(k,k-i+1)-1 \right)^{ \frac{1}{(k-i-1)(k-i)}} \right) \cdot (s-1)^{\frac{k-1}{k-2} }. \end{equation} Table~\ref{values} provides the values of $r_{0}(k,s)$ for a few values of $k$ and $s$\footnote{For completeness, we added values of $r_0$ known to hold for $k=3$.}. The symbols $\ast$ and $\star$ are used to indicate the first value of $s$ such that $s >s_0$ and such that $s>s_1$, respectively. \begin{table}\label{values} \begin{center} \begin{tabular}[h]{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline $ k \backslash s$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 12& 13 & 15 \\ \hline $3$ & 2 & 4 & & & & & & & &&& \\ \hline $4$ & 2 & 3 & 8 & 222$^\ast$ & 5434 & & & & &&& \\ \hline $5$ & 2 & 3 & 5 & 11 & 19 & 457$^\ast$ & 3270 & 55507 & 218896 &&& \\ \hline $6$ & 2 & 3 & 5 & 7 & 15 & 24 & 35 & $606^\ast$ & 3528 &309393& 933907& $ 1.4 \cdot 10^{12 \star} $ \\ \hline \end{tabular} \end{center} \caption{$r_{0}(k,s)$ for some small values of $k$ and $s$.} \end{table} For comparison, it is easy to see that, if $r \leq r_{1}(k,s)=\lceil (s-1)^{(k-1)/(k-2)}-1\rceil$, then $\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(K_n)\|>\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(T_{k-1}(n))\|$ for large values of $n$, so that Theorem~\ref{main_thm} cannot possibly be extended to such values of $r$. In particular, these two tables imply that the values for $r_0(k,s)$ are best possible for $s=3$ and for $(k,s)=(5,4)$. \begin{center} \begin{table} \begin{tabular}[h]{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline $ k \backslash s$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 12& 13 & 15 \\ \hline $4$ & 2 & 5 & 7 & 11 & & & & &&& \\ \hline $5$ & 2 & 4 & 6 & 8 & 10 & 13 & 15 & 18&&& \\ \hline $6$ & 2 & 3 & 5 & 7 & 9 & 11 & 13 & 15 &20&22& $ 27 $ \\ \hline \end{tabular} \caption{Value of $r_{1}(k,s)$ for some values of $k$ and $s$.} \end{table} \end{center} The paper is structured as follows. In Section~\ref{sec_preliminaries}, we introduce the tools needed to prove our main results. We then prove Theorem~\ref{stability_thm} and~\ref{main_thm} in Sections~\ref{sec_stability} and~\ref{sec_main}, respectively. \section{Preliminaries}\label{sec_preliminaries} In this section, we fix the notation and introduce concepts and results used to prove our main results. We first state a well-known auxiliary lemma. \begin{lemma} \label{meulema} If $\ell \geq 2$ and $G$ is a graph with $m$ edges, then $G$ contains an $\ell$-partite subgraph with more than $(\ell-1) m/\ell$ edges. \end{lemma} The next lemma generalizes a result of Alon and Yuster~\cite{AY}. \begin{lemma} \label{lema_four} Fix $1 \leq j \leq k-1$ and $2 \leq p \leq k-1$. Let $0 < \gamma \leq \frac{j(p-1)}{2p(k-1)^2}$ and let $H''$ be a $(k-1)$-partite graph on $m$ vertices with partition $V(H'') = U_1 \cup \cdots \cup U_{k-1}$ and at least $ \ex(m, K_{k}) - \gamma m^2$ edges. If we add at least $\left(\frac{p(k-1)}{j(p-1)} + 1\right) \gamma m^2$ new edges to $H''$, then in the resulting graph there is a copy of $K_{k}$ with at most $b(k,p,j)=\min \{j\binom{p}{2}, \lfloor k/p \rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \}$ new edges. Every such copy contains at most $p$ vertices in each class $U_i$ and each new edge in the copy connects two vertices of $K_{k}$ that lie in a same vertex class $U_i$ of $H''$. \end{lemma} \begin{proof} Let $H''$ be as in the statement of the lemma. Consider adding at least $$\left(\frac{p(k-1)}{j(p-1)} + 1\right) \gamma m^2$$ new edges to $H''$ to produce a graph $H'$. At least $\frac{p(k-1)}{j(p-1)} \gamma m^2$ new edges have both endpoints in a same partition class. By an averaging argument, there exist $j$ classes $U_{i_1},\ldots,U_{i_j}$ containing at least $\frac{p}{p-1} \gamma m^2$ new edges. Indeed, if this were false, we would have \begin{eqnarray} \binom{k-2}{j-1}\left(e_{H'}(U_1)+\cdots+e_{H'}(U_{k-1})\right) &=& \sum_{1 \leq i_1 <\cdots <i_j \leq k-1} \left(e_{H'}(U_{i_1})+\cdots+ e_{H'}(U_{i_j}) \right)\\ &<& \binom{k-1}{j} \cdot \frac{p}{p-1} \gamma m^2, \end{eqnarray} which implies that $e_{H'}(U_1)+\cdots+e_{H'}(U_{k-1})<\frac{p(k-1)}{j(p-1)} \gamma m^2$, a contradiction. Let $\Gamma$ be the spanning subgraph of $H'$ with edges in $E_{H'}(U_{i_1}) \cup \cdots \cup E_{H'}(U_{i_j})$, so that $\Gamma$ contains at least $\frac{p}{p-1} \gamma m^2$ edges. By Lemma~\ref{meulema}, $\Gamma$ has a $p$-partite subgraph $\Gamma'$ with more than $\gamma m^2$ edges. We will refer to the edges of $\Gamma'$ as the \emph{new} edges. The sum of the number of edges in $\Gamma'$ with the number of edges in $H''$ is greater than $\ex(m, K_{k})$, hence there exists a copy of $K_{k}$ in the union of $H''$ and $\Gamma'$. Note that there cannot be $(p+1)$ or more vertices of this copy in a same class $U_i$, as this would produce a copy of $K_{p+1}$ where all edges are new, a contradiction. Therefore this copy of $K_k$ contains at most $j \binom{p}{2}$ new edges. On the other hand, at most $\lfloor k/p \rfloor$ classes may contain $p$ vertices, so that the number of new edges is at most $\lfloor k/p \rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2}$. As a consequence, the number of new edges is at most $\min \{j\binom{p}{2}, \lfloor k/p \rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2}\}$, as required. \end{proof} For later reference, we state Lemma~\ref{lema_four} in the special case $p=2$ and $j=1$, which is precisely the result of Alon and Yuster~\cite{AY}. \begin{corollary}\label{lema_two} Let $0 < \gamma \leq 1/(4(k-1))^2$ be fixed and let $H''$ be a $(k-1)$-partite graph on $m$ vertices with partition $V(H'') = U_1 \cup \cdots \cup U_{k-1}$ with at least $ \ex(m, K_{k}) - \gamma m^2$ edges. If we add at least $(2k-1) \gamma m^2$ new edges to $H''$, then in the resulting graph there is a copy of $K_{k}$ with exactly one new edge, which connects two vertices of $K_{k}$ in the same vertex class $U_i$ of $H''$. \end{corollary} \subsection{Regularity Lemma} To prove our results we use an approach similar to the one from \cite{ABKS}, which is based on the Szemer\'{e}di Regularity Lemma~\cite{Sze78}. Let $G = (V,E)$ be a graph, and let $A$ and $B$ be two subsets of $V(G)$. If $A$ and $B$ are non-empty, define the density of edges between $A$ and $B$ by $$d(A,B) = \frac{e(A,B)}{\|A\|\|B\|},$$ where $e(A,B)$ is the number of edges with one vertex in $A$ and the other in $B$. (When $A=B$, we write $e(A,A)=e(A)$.) For $\eps > 0$ the pair $(A,B)$ is called \emph{$\eps$-regular} if, for every subsets $X \subseteq A$ and $Y \subseteq B$ satisfying $\|X\| \geq \eps\|A\|$ and $\|Y\| \geq \eps\|B\|$, we have $$\|d(X,Y) - d(A,B)\| < \eps.$$ An \emph{equitable partition} of a set $V$ is a partition of $V$ into pairwise disjoint classes $V_1,\ldots,V_m$ of almost equal size, i.e., $\arrowvert \|V_i\| - \|V_j\| \arrowvert \leq 1$ for all pairs $i,j$. An equitable partition of the set $V$ of vertices of $G$ into the classes $V_1,\ldots,V_m$ is called \emph{$\eps$-regular} if at most $\eps \binom{m}{2}$ of the pairs $(V_i, V_j)$ are not $\eps$-regular. We shall use the following colored version of the Regularity Lemma~\cite{kosi} that will be particularly useful for our purposes. \begin{lemma} \label{lrs vc} For every $\eps > 0$ and every positive integer $r$, there exists a constant $M = M(\eps,r)$ such that the following property holds. If the edges of a graph $G$ of order $n > M$ are $r$-colored $E(G) = E_1 \cup \cdots \cup E_r$, then there is a partition of the vertex set $V(G) = V_1 \cup \cdots \cup V_m$, with $1/\eps \leq m \leq M$, which is $\eps$-regular simultaneously with respect to all graphs $G_i = (V,E_i)$ for all $i \in [r]$. \end{lemma} A partition as in Lemma \ref{lrs vc} will be called a \emph{multicolored $\eps$-regular partition}. Given such a partition $V_1 \cup \cdots \cup V_m$ and a constant $\eta>0$, we may also define the \emph{multicolored cluster graph} $H=H(\eta)$ associated with this partition and with a constant $\eta>0$: the vertex set is $[m]$ and $e = \{i,j\}$ is an edge of $H$ if the pair $(V_i,V_j)$ is $\eps$-regular with respect to all colors and the density between $V_i$ and $V_j$ is at least $\eta$ for at least one of the colors. Each edge $e=\{i,j\}$ in $H$ is assigned the list $L_e$ of colors $c$ such that $c$ appears with density at least $\eta$ between $V_i$ and $V_j$ in $G$. Given a colored graph $\widehat{F}$, we say that a multicolored cluster graph $H$ contains $\widehat{F}$ if $H$ contains a copy of the (uncolored) graph induced by $\widehat{F}$ for which the color of each edge of $\widehat{F}$ is contained in the list of the corresponding edge in $H$. More generally, if $F$ is a graph with color pattern $P$, we say that $H$ contains $(F,P)$ if it contains some colored copy of $F$ with pattern $P$. Given colored graphs $\widehat{F}$ and $\widehat{H}$, a function $\psi\colon V(\widehat{F}) \rightarrow V(\widehat{H})$ is called a \emph{colored homomorphism} of $\widehat{F}$ in $\widehat{H}$ if, for every edge $e=\{i,j\} \in E(\widehat{F})$, the pair $\{\psi(i),\psi(j)\}$ is an edge of $\widehat{H}$ with the color of $e$. If $H$ is a multicolored cluster graph, it suffices that the color of $e$ lies in the list associated with the edge $\{\psi(i),\psi(j)\}$. In connection with these definitions, the following embedding result holds (for a proof, see~\cite{rainbow_kn}). \begin{lemma} \label{homocol} For every $\eta > 0$ and all positive integers $k$ and $r$, there exist $\varepsilon = \varepsilon (r,\eta, k) > 0$ and a positive integer $n_0(r,\eta, k)$ with the following property. Suppose that $\widehat{G}=(V,E)$ is an $r$-colored graph on $n > n_0$ vertices with a multicolored $\varepsilon$-regular partition $V = V_1 \cup \cdots \cup V_m$ which defines the multicolored cluster graph $H = H(\eta)$. Let $\widehat{F}$ be a $k$-vertex graph colored with $t \leq r$ colors. If there exists a colored homomorphism $\psi$ of $\widehat{F}$ into $H$, then the graph $\widehat{G}$ contains $\widehat{F}$. \end{lemma} The following standard embedding result will also be useful. \begin{lemma} \label{abund} Let $ k \geq 2$ be an integer and fix a constant $ 0 < \alpha \leq 3/4$. Let $ G $ be a graph whose vertex set contains mutually disjoint sets $W_{1},\ldots , W_{k}$ with the following property. For every pair $ \{i,j\} \subseteq [k]$, where $i\neq j$, and all subsets $X_{i} \subseteq W_{i}$, where $\| X_{i} \|\geq \alpha^{k} \| W_{i} \|$, and $X_{j} \subseteq W_{j}$, where $\| X_{j} \|\geq \alpha^{k} \| W_{j} \|$, there are at least $\alpha\| X_{i} \|\| X_{j} \|$ edges between $X_{i}$ and $X_{j}$ in $G$. Then $G$ contains a copy of $K_{k}$ with one vertex in each set $W_{i}$. \end{lemma} \begin{proof} Our proof is by induction on $k$. For $k =2$ the result is trivial. Assume by induction that the statement holds for $k-1$, where $k \geq 3$. Now, fix \begin{equation}\label{eq_aux3} \alpha \leq \frac{3}{4} < \min\left\{(\ell-1)^{-\frac{1}{\ell}} \colon 2 \leq \ell \leq k\right\} \end{equation} and let $ G $ be a graph whose vertex set contains mutually disjoint sets $W_{1},\ldots , W_{k}$ as in the statement of the lemma. For all $i \in [k-1]$, let $W^{i}_{k} \subseteq W_{k}$ contain all vertices in $W_k$ with fewer than $\alpha\| W_{i} \|$ neighbors in $W_{i}$. Then we have $e(W^{i}_{k}, W_{i}) < \alpha\|W^{i}_{k}\| \| W_{i} \|$, so that $\| W^{i}_{k} \| < \alpha^{k} \|W_{k}\|$ by hypothesis. Our choice of $\alpha$ implies that $$\left\| \bigcup^{k-1}_{i = 1} W^{i}_{k} \right\| <(k-1)\cdot \alpha^{k} \cdot \| W_{k} \| \stackrel{\eqref{eq_aux3}}{<} \| W_{k} \|.$$ Let $v$ be a vertex in $W_k \setminus \bigcup^{k-1}_{i = 1} W^{i}_{k}$ and, for $i \in [k-1]$, let $W'_{i}$ be the set of neighbors of $v$ in $W_{i}$, so that $\| W'_{i} \| \geq \alpha \| W_{i} \|$. Observe that, for all subsets $X_{i} \subseteq W'_{i}$ and $X_{j} \subseteq W'_{j}$ such that $\| X_{i} \|\geq \alpha^{k-1} \| W'_{i} \|\geq \alpha^{k} \| W_{i} \|$ and $\| X_{j} \|\geq \alpha^{k-1} \| W'_{j} \| \geq \alpha^{k} \| W_{j} \|$, there are at least $\alpha\| X_{i} \|\| X_{j} \|$ edges between $X_i$ and $X_j$ in $G$. By induction, $G$ contains a copy of $K_{k-1}$ with one vertex in each set $W'_{i}$, where $1\leq i \leq k-1$. Adding $v$ creates a copy of $K_k$ in $G$ with one vertex in $W_{i}$ for each $i \in [k]$. \end{proof} \subsection{Stability} Another concept that will be particularly useful in our paper are stability results in the sense of Erd\H{o}s and Simonovits~\cite{simonovits}. It will be convenient to use the following theorem by F\"uredi~\cite{fu15}. \begin{theorem} \label{theorem:stability_furedi2} Let $G = (V,E)$ be a $K_{k}$-free graph on $m$ vertices. If $\|E\| = \ex (m, K_{k}) -t$ for some $t \geq 0$, then there exists a partition $V= V_1 \cup \cdots \cup V_{k-1}$ with $\sum_{i = 1}^{k-1} e(V_i) \leq t$. \end{theorem} We recall the following bounds on the number of edges in the Tur\'an graph $T_{k-1}(m)$: \begin{eqnarray}\label {eq:turan_number_1} && \frac{(k-2)m^2}{2(k-1)} - k +1 < \ex(m,K_{k}) \leq \frac{(k-2)m^2}{2(k-1)}. \end{eqnarray} For later use, we state the following fact about the size of the classes in a $(k-1)$-partite graph with a large number of edges (the easy proof is in~\cite{rainbow_kn}). \begin{proposition} \label{prop:prop1} Let $G=(V,E)$ be a $(k-1)$-partite graph on $m$ vertices with $(k-1)$-partition $V= V_1 \cup \cdots \cup V_{k-1}$. If, for some $t \geq (k-1)^2$, the graph $G$ contains at least $\ex (m, K_{k}) - t$ edges, then for each $i \in [k-1]$ we have $$ \left\|\|V_i\| - \frac{m}{k-1} \right\| < \sqrt{2t}. $$ \end{proposition} We also consider the \emph{entropy function} $H \colon [0,1] \rightarrow [0,1]$ given by $H(x) = -x \log_2 x - (1-x) \log_2(1-x)$ with $H(0) = H(1) = 0$. It is used in the well-known inequality \begin{eqnarray} \label{eq:entropy1} \binom{n}{\alpha n} \leq 2^{H(\alpha) n} \end{eqnarray} for all $0 \leq \alpha \leq 1$. It turns out that, for $x \leq 1/8$, we have: \begin{align} \label{eq:entropy2} H(x) \leq -2x \log_2 x. \end{align} \section{Proof of Theorem~\ref{stability_thm}}\label{sec_stability} In this section, we prove Theorem~\ref{stability_thm}. For convenience, we restate it here. \addtocounter{section}{-2} \addtocounter{theorem}{2} \begin{theorem} Let $k \geq 4$ and $ 2 \leq s \leq \binom{k}{2}$ be integers. Fix $r \geq r_{0}(k,s)$, defined in~\eqref{def_r0},~\eqref{def_r1} and~\eqref{def_r2} below for $s \leq s_0(k)$, $s_0(k)<s\leq s_1(k)$ and $s>s_1(k)$, respectively. For any $\delta > 0$, there is $n_{0}=n_0(\delta,r,k,s)$ as follows. If $G=(V,E)$ is a graph on $n > n_{0}$ vertices such that $\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\| \geq r^{\ex(n,K_{k})}$, then there is a partition $V=W_{1} \cup \cdots \cup W_{k-1}$ such that at most $\delta n^{2}$ edges have both endpoints in a same class $W_i$. \end{theorem} \addtocounter{section}{2} \addtocounter{theorem}{-3} \begin{proof} Fix positive integers $ k \geq 4 $, $2 \leq s \leq \binom{k}{2}$ and $r \geq r_{0}(k,s)$ according to (\ref{def_r0}), (\ref{def_r1}) or (\ref{def_r2}), respectively, depending on the value of $s$. Let $ \delta >0 $. We shall fix a positive constant $\beta_0 < \frac{\delta}{8k+2}$ and consider $ \eta >0 $ sufficiently small to satisfy \begin{eqnarray} r\eta &<& \frac{\delta}{2} \label{eq:eta1} \end{eqnarray} and inequality~\eqref{eq_aux1}. Let $n_{0}=n_{0}(r,\eta,k)$ and $ \varepsilon= \varepsilon(r,\eta,k)>0 $, where $\varepsilon < \eta/2 $, be given by Lemma \ref{homocol} and let $ M=M(\varepsilon,r) $ be defined by Lemma~\ref{lrs vc}. Let $G$ be a graph on $n > \max \{ n_{0}, M \}$ vertices with at least $r^{\ex(n,K_k)}$ distinct $ (K_{k},\geq s) $-free $r$-colorings. Fix one such coloring. By Lemma~\ref{lrs vc}, there exists a partition of $V(G)$ into $1/\eps \leq m\leq M $ parts that is $\eps$-regular with respect to all $r$ colors. Let $ H=H(\eta)$ be the $m$-vertex multicolored cluster graph associated with this partition, where each edge $e$ has a non-empty list $L_e$ of colors. We write $E_i=E_i(H)$ for the set of edges of $H$ for which $\|L_e\|=i$, and we let $e_i(H)=\|E_i(H)\|$, for $i \in [r]$. We shall bound the number of $r$-colorings of $G$ that lead to the partition $ V(G)=V_{1} \cup \cdots \cup V_{m} $ and to the multicolored cluster graph $ H $. Given a color $ c \in[r] $, the number of $\eps$-irregular pairs $(V_i, V_j)$ with respect to the spanning subgraph $ G_{c} $ of $G$ with edge set given by all edges of color $c$ is at most $ \varepsilon \binom{m}{2} $. This leads to at most \begin{equation}\label{104} r \cdot \varepsilon \cdot \binom{m}{2} \cdot \left(\frac{n}{m}\right)^{2} \leq \frac{r \cdot \varepsilon}{2}\cdot n^{2} \end{equation} edges between $\eps$-irregular pairs with respect to some color. By the definition of an $ \varepsilon $-regular partition and the fact that $ m \geq 1 / \varepsilon$, there are at most \begin{equation}\label{105} m \cdot \left(\frac{n}{m}\right)^{2} = \frac{n^{2}}{m} \leq \varepsilon n^{2} \end{equation} edges with both endpoints in the same class $V_{i}$ for some $i \in [m]$. Finally, the number of edges with some color $ c $ connecting a pair $ (V_{i}, V_{j}) $ such that the density of the pair in $G_c $ is less than $ \eta $ is bounded above by \begin{equation}\label{106} r \cdot \eta \cdot \binom{m}{2} \cdot \left(\frac{n}{m}\right)^{2} < \frac{r \cdot \eta}{2} \cdot n^{2}. \end{equation} Combining equations $ (\ref{104}) - (\ref{106}) $ and using that $ \varepsilon < \eta /2 $, there are less than $r \eta n^{2}$ edges of any of these three types. They may be chosen in the $r$-coloring of $G$ in at most $\binom{n^{2}/2}{r \eta n^{2}}$ ways and may be colored in at most $r^{r \eta n^{2}}$ ways. As a consequence, the number of $r$-colorings of $ G $ that give rise to the partition $ V(G)=V_{1} \cup \cdots \cup V_{m} $ and the multicolored cluster graph $ H(\eta) $ is bounded above by \begin{eqnarray} {\label{01}} \binom{n^{2}/2}{r\eta n^{2}} \cdot r^{r \eta n^{2}} \cdot \left( \prod_{e \in E(H)}\|L_{e}\| \right)^{(\frac{n}{m})^{2}} & \stackrel{\eqref{eq:entropy1}}{\leq} & 2^{H(2r\eta)\frac{n^{2}}{2}} \cdot r^{r \eta n^{2}} \cdot \left( \prod_{i=1}^{r}i^{e_{i}(H)} \right)^{(\frac{n}{m})^{2}}, \end{eqnarray} where $e_i(H)$ is the number of edges of $H$ whose lists have size equal to $i$. Since $m \leq M$, there are at most $M^{n} $ distinct $\eps$-regular partitions $ V(G)=V_{1} \cup \cdots \cup V_{m} $. Summing (\ref{01}) over all possible partitions and all possible multicolored cluster graphs $ H $, the number of $ (K_{k}, \geq s) $-free $r$-colorings of $ G $ is at most \begin{eqnarray} \label{02} M^{n}\cdot 2^{H(2r\eta)\frac{n^{2}}{2}} \cdot r^{r \eta n^{2}} \cdot \sum_{H}\left( \prod_{i=1}^{r}i^{e_{i}(H)} \right)^{(\frac{n}{m})^{2}} . \end{eqnarray} Note that, in this expression, we have $m=m(H)=\|V(H)\|$. We wish to bound the value of $ \prod_{i=1}^{r}i^{e_{i}(H)}$. First observe that \begin{eqnarray} \label{eq:tu1} e_{s}(H)+ \cdots +e_{r}(H) &\leq& \ex(m,K_k). \end{eqnarray} Otherwise, by Tur\'an's Theorem, the multicolored cluster graph $H$ would contain a copy of $K_k$ such that every edge has a list of size at least $s$. This clearly induces a colored homomorphism of some pattern of $K_k$ with at least $s$ classes into $H$, so that by Lemma~\ref{homocol} $G$ would contain a copy of $K_k$ whose set of edges is colored with $s$ colors, a contradiction. More generally, we prove the following: \begin{claim} \label{claim} Let $H$ be a $ (K_{k},\geq s) $-free multicolored graph where $E_j=E_j(H)$ is the number of edges in $H$ with list of size $j$ and $e_j(H)=\|E_j(H)\|$ for any $j \in [r]$. For any fixed $i$ such that $A(k,k-i) \leq s-1$, there is no copy of $K_k$ for which all edges lie in $ E_{s-A(k,k-i)} \cup \cdots \cup E_{r}$ and at least $A(k,k-i)$ edges lie in $ E_{s}\cup \cdots \cup E_{r}$. Therefore, if $i \leq k-2$ and $A(k,k-i) \leq s-1$, the following inequality holds: \begin{equation} \frac{k-i-1}{k-i}\cdot \left(e_{s-A(k,k-i)}(H) + \cdots + e_{s-1}(H)\right) + e_{s}(H)+ \cdots +e_{r}(H) \leq \ex(m,K_k). \end{equation} \end{claim} Note that $A(k,k-i) = i$ for $i \leq \lfloor k/2 \rfloor$, so in this case the inequality in the claim becomes \begin{equation} \frac{k-i-1}{k-i}\cdot \left(e_{s-i}(H) + \cdots + e_{s-1}(H)\right) + e_{s}(H)+ \cdots +e_{r}(H) \leq \ex(m,K_k). \end{equation} \begin{proof} Assume that there is a copy of $ K_{k} $ for which all edges lie in $ E_{s-A(k,k-i)} \cup \cdots \cup E_{r}$ and at least $A(k,k-i)$ edges lie in $ E_{s}\cup \cdots \cup E_{r}$. Let $p$ be the number of edges in $ E_{s-A(k,k-i)} \cup \cdots \cup E_{s-1}$ in this copy of $K_k$. Proceeding greedily (and starting with the edges in $E_{s-A(k,k-i)} \cup \cdots \cup E_{s-1}$), we may find a copy of $K_k$ such that $\alpha \geq \min\{s-A(k,k-i),p\}$ additional distinct colors appear in the edges $E_{s-A(k,k-i)} \cup \cdots \cup E_{s-1}$. If $\alpha \geq s-A(k,k-i)$, then at least $s-\alpha$ distinct colors may be chosen in the edges in $ E_{s}\cup \cdots \cup E_{r}$, as this union contains at least $A(k,k-i)$ edges, each with a list of size at least $s$. If $\alpha=p<s-A(k,k-i)$, then the number of edges of the copy in $ E_{s}\cup \cdots \cup E_{r}$ is $\binom{k}{2}-p \geq s-p$, so that at least $s-\alpha$ additional distinct colors may be chosen for edges in this set. In both cases, we get a copy of $K_k$ colored with $s$ or more colors, the desired contradiction. Next, assuming that $k-i \geq 2$ and that $A(k,k-i) \leq s-1$, let $E' \subseteq E_{s-A(k,k-i)} \cup \cdots \cup E_{s-1}$ be maximum with the property that the edges in $ E'$ induce a $(k-i)$-partite subgraph of $H$. The number of edges of $E'$ in a copy of $K_k$ is at most $\binom{k}{2}-A(k,k-i)$, so that $\|E'\|+e_{s}(H)+ \cdots + e_{r}(H) \leq \ex(m,K_k)$ by the previous discussion. By Lemma~\ref{meulema}, we know that $ \|E'\|\geq (k-i-1) \cdot\left\|E_{s-A(k,k-i)} \cup \cdots \cup E_{s-1} \right\|/(k-i)$, which gives the desired result. \end{proof} For a multicolored cluster graph $H$, let \begin{equation}\label{def_beta} \beta= \beta(H)= \frac{1}{m(H)^2}\left( \ex(m(H),K_k) - \sum_{j=s}^{r}e_{j}(H) \right) \geq 0. \end{equation} To find an upper bound in (\ref{02}) on the number of $\mathcal{P}_{k,s}$-free $r$-colorings of $G$ , we use~\eqref{eq:tu1} and ~\eqref{def_beta} in the product \begin{eqnarray}\label{eq_UB} \left(\prod_{e \in E(H)}\|L_{e}\|\right)^{(\frac{n}{m})^{2}} &=& \prod_{i=2}^r i^{e_i(H)} \nonumber \\ &\leq& 2^{e_2(H)} \cdot 3^{e_3(H)} \cdots (s-1)^{e_{s-1}(H)} \cdot r^{\ex(n,K_k)-\beta(H) n^2}. \end{eqnarray} Maximizing this product is the same as maximizing \begin{equation}\label{eq_LP} \log\left(2^{e_2(H)} \cdot 3^{e_3(H)} \cdots (s-1)^{e_{s-1}(H)} \right)=\log{2}\cdot e_2(H) +\cdots +\log(s-1)\cdot e_{s-1}(H). \end{equation} Moreover, with~\eqref{def_beta}, the inequalities in Claim~\ref{claim} lead to the following constraints. For all $i \in [k-1]$ such that $A(k,k-i) \leq s-1$, we get inequalities of the form \begin{equation}\label{linear_constraints} \frac{k-i-1}{k-i}\cdot ( e_{s-A(k,k-i)}(H)+ \cdots + e_{s-1}(H)) \leq \beta m^2. \end{equation} For $s \leq s_{0}(k) $, let $ i^{} $ be the least value of $ i $ such that $$ s-A(k,k-i) \leq 2,$$ as defined in the introduction. The fact that $s \leq s_{0}(k)$ implies that $ i^{} \leq k-2$. The constraints (\ref{linear_constraints}) for $i \in [i^{\ast}]$ may be written as \begin{equation}\label{linear_constraints2} \left\{ \begin{aligned} \frac{k-2}{k-1}\cdot ( e_{s-A(k,k-1)}(H)+ \cdots + e_{s-1}(H)) & \leq &\beta m^2 \\ \frac{k-3}{k-2}\cdot ( e_{s-A(k,k-2)}(H)+ \cdots + e_{s-1}(H)) & \leq &\beta m^2 \\ \vdots \hspace{2cm}\vdots \hspace{2cm} \vdots \hspace{2cm} & \vdots & \hspace{1cm} \vdots \\ \frac{k-i^{}-1}{k-i^{}}\cdot ( e_{s-A(k,k-i^{})}(H)+ \cdots + e_{s-1}(H)) & \leq &\beta m^2. \end{aligned} \right. \end{equation} This leads to a linear program with objective function~\eqref{eq_LP}, and constraints~\eqref{linear_constraints2} and $e_2(H),\ldots,e_{s-1}(H) \geq 0$ (and possibly $e_1(H)$ if $A(k,k-i^{})\geq s-1$). It is easy to see that the optimum is obtained for $e_{s-1}(H)=\frac{k-1}{k-2} \cdot \beta m^2$ and $e_{s-A(k,k-i+1)-1}(H)= \left(\frac{k-i}{k-i-1}-\frac{k-i+1}{k-i} \right) \cdot \beta m^2=\frac{1}{(k-i-1)(k-i)} \cdot \beta m^2$ for $ i \in \{2, \ldots, i^\ast \}$. Note that in this case we have \begin{eqnarray} \label{eq:2beta} \sum_{i=1}^{i^{}} e_{s-A(k,k-i+1)-1}(H)= 2 \beta m^2. \end{eqnarray} To simplify the expressions below, for $k \geq 3$ and $2 \leq i \leq k-2$, define the quantities \begin{equation}\label{eq_sxi} s_i= s-A(k,k-i+1)-1 \textrm{ and } \xi_i=\frac{1}{(k-i-1)(k-i)}. \end{equation} Plugging the optimal solution of the linear program into~(\ref{02}), we obtain (for $s \leq s_0(k)$), \begin{eqnarray} \label{03} &&M^{n}\cdot 2^{H(2r\eta)\frac{n^{2}}{2}} \cdot r^{r \eta n^{2}} \cdot \sum_{H}\left( \prod_{i=1}^{r}i^{e_{i}(H)} \right)^{(\frac{n}{m})^{2}} \leq \nonumber \\ &&r^{(H(2r\eta)+ 2r \eta) \frac{n^{2}}{2}} \cdot \sum_{H} \left(\frac{ \left( \prod^{i^\ast}_{i=2} s_i^{ \xi_i} \right) (s-1)^{\frac{k-1}{k-2}} }{r}\right)^{\beta(H) n^{2}} r^{\ex(n,K_k)}. \end{eqnarray} By our choice of $r_0=r_0(k,s)$ (see~\eqref{def_r0}), given any $\beta_0>0$, there is $\eta>0$ such that \begin{equation}\label{eq_aux1} r^{2H(2r\eta)+ 2r \eta} \left(\frac{ \left( \prod^{i^\ast}_{i=2} s_i^{ \xi_i} \right) \cdot (s-1)^{ \frac{k-1}{k-2} } }{r}\right)^{\beta_0} <1. \end{equation} Recall that we are using $\beta_0<\delta/(8k+2)$ and that $\eta>0$ satisfies~\eqref{eq_aux1} for this value of $\beta_0$. We claim that there exists a multicolored cluster graph $H$ such that $\beta(H) < \beta_0$. Indeed, if this does not happen, the inequality~\eqref{03} would be bounded above by \begin{eqnarray} \label{eq:107} &&r^{(H(2r\eta)+ 2r \eta) \frac{n^{2}}{2}} \cdot 2^{rM^2/2} \left(\frac{ \left( \prod^{i^\ast}_{i=2} s_i^{ \xi_i} \right) \cdot (s-1)^{ \frac{k-1}{k-2} } }{r}\right)^{\beta_0 n^{2}} \cdot r^{\ex(n,K_k)} \nonumber\\ &&\stackrel{(n \gg 1,\eqref{eq_aux1})}{<} r^{\ex(n,K_k)}, \end{eqnarray} a contradiction (we are using that the number of distinct multicolored cluster graphs is bounded above by $2^{rM^2/2}$, which is less than $r^{H(2r\eta)n^2/2}$ for $n$ sufficiently large). So, let $H$ be a multicolored cluster graph for which $\beta=\beta(H) < \beta_0$. Consider the spanning subgraph $H'$ of $H$ with edge set $E_{s} \cup \cdots \cup E_{r}$. This graph contains $\ex(m,K_k)-\beta m^{2}$ edges. We may apply Theorem~\ref{theorem:stability_furedi2} with $ t=\beta m^{2} $. By removing at most $\beta m^{2}$ edges of $ H' $, we produce a $ (k-1) $-partite subgraph $ H'' $. Let $U_{1} \cup \cdots \cup U_{k-1} $ be the resulting partition of $V(H'')=V(H)$ such that \begin{equation}\label{eq_partition} \sum_{i=1}^{k-1}e_{H'}(U_{i}) \leq \beta m^{2}. \end{equation} By Claim~\ref{claim} for $i=k-2$, we know that $ e_{2}(H) + \cdots +e_{s-1}(H) \leq 2 \beta m^{2}$ for $s \leq s_0$. To bound $ e_{1}(H) $, we apply Corollary \ref{lema_two} to the $ (k-1) $-partite graph $ H''$. This lemma ensures that we may not have $ e_{1}(H) \geq (2k-1) \cdot 2\beta m^{2} =(4k-2) \beta m^{2}$, otherwise adding $E_1$ to $H''$ would produce a copy of $ K_{k} $ for which exactly one of the edges would have a list of size one and all other edges would have lists with $ s $ or more colors, leading to the forbidden pattern. Therefore we must have $ e_{1}(H) < (4k-2) \beta m^{2}$. This gives us an upper bound on the number of edges of $H$ with color lists of size up to $s-1$: \begin{equation}\label{UB_edges} e_{1}(H)+ \cdots + e_{s-1}(H) \leq 4k \beta m^{2}. \end{equation} Let $W_{i} = \bigcup_{j \in U_{i}}V_{j}$, where $ i \in [k-1] $. We shall prove that $ V(G) = W_{1} \cup \cdots \cup W_{k-1}$ satisfies the conclusion of the theorem. Edges of $G$ with both endpoints in a same class $W_i$ may come from three sources: edges of $G$ that are not represented in $H$; edges of $G$ in pairs $(V_s,V_t)$ such that $\{s,t\} \in E(H) \setminus E(H')$; edges of $G$ in pairs $(V_s,V_t)$ that correspond to edges in $E(H')$ with both endpoints in a same class $U_j$. By~(\ref{104}) -- (\ref{106}) and \eqref{UB_edges}, we obtain \begin{eqnarray} \sum_{i=1}^{k-1} e_{G}(W_{i}) & \leq & r\eta n^2 + \left(\sum_{i=1}^{k-1} e_{H'}(U_{i})+e_{1}(H)+ \cdots + e_{s-1}(H) \right) \cdot \left( \frac{n}{m}\right)^{2}\\ & \leq & r\eta n^2 + \left(\beta_0 m^{2}+4k \beta_0 m^{2} \right) \cdot \left( \frac{n}{m}\right)^{2} \leq \delta n^2, \end{eqnarray} by our choice of $\beta_0$ and $\eta>0 $. We now consider the case when $s>s_0(k)$. All the inequalities in~\eqref{linear_constraints2} hold up to $i^\ast=k-2$, but in this case $s-A(k,2)>2$, so that the variables $e_2(H),\ldots,e_{s-A(k,2)-1}(H)$ are not bounded by the linear constraints. The constraints become \begin{equation}\label{linear_constraints3} \left\{ \begin{aligned} \frac{k-2}{k-1}\cdot e_{s-1}(H) & \leq &\beta m^2 \\ \frac{k-3}{k-2}\cdot ( e_{s-A(k,k-2)}(H)+ e_{s-1}(H)) & \leq &\beta m^2 \\ \vdots \hspace{2cm}\vdots \hspace{2cm} \vdots \hspace{2cm} & \vdots & \hspace{2cm} \vdots \\ \frac{1}{2}\cdot ( e_{s-A(k,2)}(H)+ \cdots + e_{s-1}(H)) & \leq &\beta m^2. \end{aligned} \right. \end{equation} Consider $1\leq j \leq k-1$ and $2 \leq p \leq k-1$ Assume that $s \leq \binom{k}{2}-b(k,p,j) +2$, where $b(k,p,j)$ comes from Lemma~\ref{lema_four}. We divide the set of multicolored cluster graphs into two classes, according to whether $\beta(H) \geq j(p-1)/(4p(k-1)^2)$ or $\beta(H) < j(p-1)/(4p(k-1)^2)$. If $\beta=\beta(H) \geq j(p-1)/(4p(k-1)^2)$, we have, for $s_i$ and $\xi_i$ as in~\eqref{eq_sxi}, \begin{eqnarray}\label{04} && \prod_{i=1}^{r}i^{e_{i}(H)} \nonumber \\ &\stackrel{(\ref{eq:2beta})}{\leq}& \left( s-A(k,2)-1 \right)^{\binom{m}{2}-(\ex(m,K_k) + \beta m^2)} \cdot \left(\frac{\left( \prod^{k-2}_{i=2}s_i^{\xi_i} \right) \cdot (s-1)^{ \frac{k-1}{k-2} } }{r}\right)^{\beta m^2} \cdot r^{\ex(m,K_k)} \nonumber\\ &\leq& \left(s-A(k,2)-1 \right)^{\frac{m^2}{2(k-1)}} \left(\frac{\left( \prod^{k-2}_{i=2}s_i^{\xi_i} \right) \cdot (s-1)^{ \frac{k-1}{k-2} } }{ \left(s-A(k,2)-1 \right) \cdot r}\right)^{\beta m^2} \cdot r^{\ex(m,K_k)} \nonumber\\ &\leq& \left(\frac{\left( s-A(k,2)-1 \right)^{\frac{2p(k-1)}{j(p-1)}} \cdot \left(\prod^{k-2}_{i=2}s_i^{\xi_i} \right) \cdot (s-1)^{ \frac{k-1}{k-2} } }{ \left( s-A(k,2)-1\right) \cdot r}\right)^{\frac{j(p-1)m^2}{4p(k-1)^2}} \cdot r^{\ex(m,K_k)}. \end{eqnarray} Next suppose that $\beta=\beta(H) < j(p-1)/(4p(k-1)^2)$. As above, let $H'$ be the spanning subgraph of $H$ with edge set $E_s \cup \cdots \cup E_r$ and let $H''$ be a maximum $(k-1)$-partite subgraph of $H'$. Note that $\|E(H')\|=\ex(m,K_k)-\beta m^2$. Apply Theorem~\ref{theorem:stability_furedi2} and define $\gamma>0$ such that $$\|E(H'')\| =\ex(m,K_k)-\gamma m^2 \geq \|E(H')\|-\beta m^2 \geq \ex(m,K_k)- 2\beta m^2,$$ where $\beta \leq \gamma \leq 2 \beta$. In particular, the set $\widetilde{E}=E(H')-E(H'') \subset E_s \cup \cdots \cup E_r$ has cardinality $(\gamma-\beta)m^2$. Let $E'=E_2 \cup \cdots \cup E_{s - A(k,2) - 1}$. We may apply Lemma~\ref{lema_four} to the edge set $\widetilde{E} \cup E'$ for our values of $j$ and $p$ to this value of $ \gamma$. If $\|E'\| + \|\widetilde{E}\| \geq \left(p(k-1)/(j(p-1))+1 \right) \gamma m^2$, then the multicolored cluster graph obtained by adding the edges in $E' \cup \widetilde{E}$ to $H''$ would contain a copy of $K_k$ with at most $b(k,p,j)$ edges in $E'$. If $\|E' \cup \widetilde{E} \|=1$, we would greedily build an $s$-colored copy of $K_k$, starting with the edge in $E' \cup \widetilde{E}$, a contradiction. If $\|E' \cup \widetilde{E}\| \geq 2$, we obtain a copy of $K_k$ with at least $(2+\min\{s-2,\binom{k}{2}-b(k,p,j)\})$ colors. This is a contradiction whenever $s \leq \binom{k}{2}-b(k,p,j) +2$, which is precisely the hypothesis in this case. As a consequence, we have \begin{equation} e_2(H)+\cdots+e_{s-A(k,2) - 1}(H)+\|\widetilde{E}\| \leq \left(\frac{p(k-1)}{j(p-1)}+1 \right) \gamma m^2, \end{equation} so that \begin{eqnarray}\label{new_equation} e_2(H)+\cdots+e_{s-A(k,2) - 1}(H) &\leq& \left(\frac{p(k-1)}{j(p-1)}+1 \right) \gamma m^2 - \|\widetilde{E}\| \nonumber \\ &=& \frac{p(k-1)}{j(p-1)}\gamma m^2 + \beta m^2 \nonumber \\ &\leq& \left(\frac{2p(k-1)}{j(p-1)} +1 \right) \beta m^2. \end{eqnarray} Using an argument as in~\eqref{03}, but applying the additional inequality~\eqref{new_equation} to bound $e_2(H),\ldots,e_{s-A(2)-1}(H)$, we get, again with the notation in~\eqref{eq_sxi} and using the fact that $e_{s-A(k,2)}(H)+\cdots+e_{s-1}(H)=2 \beta m^2$, when the optimum solution of the linear program is achieved, it follows that \begin{equation}\label{05} \prod_{i=1}^{r}i^{e_{i}(H)} \leq \left(\frac{\left( s-A(k,2)-1\right)^{\frac{2p(k-1)}{j(p-1) }+1} \cdot \left( \prod^{k-2}_{i=2}s_i^{\xi_i} \right) \cdot (s-1)^{\frac{k-1}{k-2} } }{r}\right)^{\beta m^{2}} \cdot r^{\ex(m,K_k)}. \end{equation} It is clear that~\eqref{04} is less than~\eqref{05} for any fixed pair $(j,p)$. Therefore, by choosing $r_{0}(k,s)$ greater than $$\left( s-A(k,2)-1\right)^{L_{opt}(k,s) } \cdot \left( \prod^{k-2}_{i=2}\left( s-A(k,k-i+1)-1 \right)^{ \frac{1}{(k-i-1)(k-i)}} \right) \cdot (s-1)^{\frac{k-1}{k-2} },$$ where $L_{opt}(k,s)$ is the least possible value of $L(k,s,p,j) = 1+ \frac{2p(k-1)}{j(p-1)}$ subject to conditions~(\ref{eq:opt2aa}) and~(\ref{eq:opt2b}), we may proceed as in the case $s \leq s_0(k)$. That is, we may again fix $\beta_0<\delta/(8k+2)$ and choose $\eta>0$ appropriately, and consider whether $\beta(H) \geq \beta_0$ for all multicolored cluster graphs $H$ (in which case we reach a contradiction) or whether there is a multicolored cluster graph $H$ for which $\beta<\beta_0$ (in which case we get the desired partition), see the inequalities (\ref{03}), (\ref{eq_aux1}) and (\ref{eq:107}). To conclude the proof of Theorem~\ref{stability_thm}, we determine the value of $L_{opt}(k,s)$. This is the least value of \begin{eqnarray}\label{eq:0pt11} L(k,s,p,j) = 1+ \frac{2p(k-1)}{j(p-1)}, \end{eqnarray} where $j \in [k-1]$, $p \in \{2, \ldots, k-1\}$ and \begin{eqnarray} \label{eq:opt2aa} s &\in& \left[\binom{k}{2} - \lfloor k/2\rfloor \lceil k/2\rceil + 3, \binom{k}{2}\right] \end{eqnarray} satisfy \begin{eqnarray} \label{eq:opt2b} b(k,p,j) = \mbox{min } \left\{ j\binom{p}{2}, \left\lfloor \frac{k}{p} \right\rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \right\} \leq \binom{k}{2} - s + 2. \end{eqnarray} Clearly, choosing $j$ and $p$ large would be good to minimize~\eqref{eq:0pt11}. The following claim summarizes our result. \begin{claim}\label{claim:lopt} For all $s>s_0$, the quantity $L_{opt}(k,s)$ satisfies the following. \begin{itemize} \item[(a)] If $s \leq s_1$, then there is $p^\ast \in \{2,\ldots,k-1\}$ such that $L_{opt}(k,s)=L(k,s,p^\ast,k-1)$. Moreover, $$3< L_{opt}(k,s) \leq 5$$ \item[(b)] If $s > s_1$, then there is $j^\ast \in \{2,\ldots,k-2\}$ such that $$L(k,s,2,j^\ast)=L_{opt}(k,s) = 1 + \frac{4(k-1)}{\binom{k}{2} - s +2 } > 9.$$ \end{itemize} \end{claim} To prove the claim, we start with part (b), where $s \geq \binom{k}{2} - \left\lfloor \frac{k}{2} \right\rfloor +3$. First note that $$ \left\lfloor \frac{k}{p} \right\rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \geq \left \lfloor \frac{k}{2} \right\rfloor>\binom{k}{2}-s+2. $$ In particular, to satisfy~\eqref{eq:opt2b}, we need $b(k,p,j) = j \binom{p}{2} \leq \binom{k}{2}-s+2$. As a consequence, for any pair $(p,j)$ such that~\eqref{eq:opt2b} holds, we have $$L(k,s,p,j) \geq 1+ \frac{2p(k-1)}{(p-1) \left(\binom{k}{2}-s+2 \right)/\binom{p}{2}}=1+\frac{p^2(k-1)}{\binom{k}{2}-s+2} \geq L(k,s,2,\binom{k}{2}-s+2).$$ Since the pair $(p,j)=(2,\binom{k}{2} - s +2)$ satisfies~\eqref{eq:opt2b}, we deduce that $L_{opt}(k,s)=L(k,s,2,\binom{k}{2}-s+2)$, so that \begin{eqnarray} L_{opt}(k,s) &=& 1 + \frac{4(k-1)}{\binom{k}{2} - s +2 } \\ &\geq& 1 + \frac{4(k-1)}{ \binom{k}{2} - (\binom{k}{2} - \left\lfloor \frac{k}{2} \right\rfloor +3) + 2 }\\ &\geq &1 + \frac{8(k-1)}{k-2} >9. \end{eqnarray} This proves part (b). We now consider part (a). Fix $p \geq 2$ and fix a pair $(p,j)$ that satisfies~\eqref{eq:opt2b}. Since $$ \left\lfloor \frac{k}{p} \right\rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \leq \left\lfloor \frac{k(p-1)}{2} \right\rfloor, $$ if the minimum in~\eqref{eq:opt2b} is attained by $ j\binom{p}{2}$, then $$ j\binom{p}{2} \leq \frac{k(p-1)}{2}, $$ which implies $j \leq k/p$. This implies that, if there exists $j>k/p$ such that $(p,j)$ satisfies~\eqref{eq:opt2b}, then we must have $$ \left\lfloor \frac{k}{p} \right\rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \leq \binom{k}{2} - s + 2,$$ so that the pair $(p,k-1)$ satisfies~\eqref{eq:opt2b} and leads to $L(k,s,p,k-1) \leq L(k,s,p,j)$. For $j=k-1$, let $p^$ be the largest value of $p$ such that $(p,k-1)$ satisfies~\eqref{eq:opt2b}. This is well defined because $(2,k-1)$ satisfies~\eqref{eq:opt2b} for $s \leq \binom{k}{2} - \left\lfloor \frac{k}{2} \right\rfloor +2=s_1(k)$. Towards finding a suitable pair $(p,j)$ that minimizes $L(k,s,p,j)$, the only other candidates are pairs $(j',p')$, satisfying~\eqref{eq:opt2b}, such that $j' \leq k/2$ and $p^* \leq p' \leq k-1$. The inequality \begin{equation}\label{eq_aux4} \frac{p \cdot \frac{k}{2}}{(p-1)(k-1)} \leq \frac{p'}{p'-1}. \end{equation} holds because the left-hand side is at most $k/(k-1)$, while the right-hand side is greater than this, as $p' \leq k-1$. We conclude that $$ L(k,s,p^,k-1)=\frac{2p^ (k-1)}{(k-1) (p^* -1)} \stackrel{\eqref{eq_aux4}}{\leq} \frac{2p'(k-1)}{\frac{k}{2}\cdot (p'-1)} \leq \frac{2p'(k-1)}{j'\cdot (p'-1)}=L(k,s,p',j'). $$ Thus, to compute $L_{opt}(k,s)$, it remains to find the right value of $p^$. For $j=k-1$ and $p\geq 2$, we have $$ (k-1) \binom{p}{2} \geq \frac{k(p-1)}{2}, $$ hence in this case $$ b(k,p,k-1) = \left\lfloor \frac{k}{p} \right\rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \leq \left\lfloor\frac{k(p-1)}{2}\right\rfloor. $$ This means that (\ref{eq:opt2b}) becomes \begin{equation}\label{eq_aux2} \left\lfloor \frac{k}{p} \right\rfloor \binom{p}{2} +\binom{k-\lfloor k/p \rfloor p}{2} \leq \binom{k}{2} - s +2 \end{equation} and that $$L_{opt}(k,s)=L(k,s,p,k-1)=1+\frac{2p^}{p^-1}$$ where $p^$ is the largest $p$ satisfying~\eqref{eq_aux2}. On the one hand, $$L(k,s,p^\ast,k-1) = 1+ \frac{2p^\ast}{p^\ast-1} > 3$$ for any $p^\ast \geq 2$. On the other hand, we have $$ L_{opt}(k,s) \leq L(k,s,2,k-1)=5. $$ This establishes part (a) of our claim, and finishes the proof of Theorem~\ref{stability_thm}. \end{proof} \section{Proof of Theorem~\ref{main_thm}}\label{sec_main} To prove Theorem~\ref{main_thm}, we shall use the following special case of an auxiliary result~\cite{B?HLN}, whose proof uses tools as in~\cite[Theorem~1.1]{BHS17}. \begin{theorem}\label{thm_atleast2} Let $n,r,k \geq 2$ with $2 \leq s \leq \binom{k}{2}$ be integers. If there exists an $(r, \mathcal{P}_{k,s})$-extremal graph on $n$ vertices that is not complete multipartite, then there exist at least two non-isomorphic $(r,\mathcal{P}_{k,s})$-extremal complete multipartite graphs on $n$ vertices. \end{theorem} We now show that, for $k \geq 4$, $2 \leq s \leq \binom{k}{2}$, $r>r_0(k,s)$ and sufficiently large $n$, there is actually a single $(r, \mathcal{P}_{k,s})$-extremal graph on $n$ vertices, the Tur\'{a}n graph $T_{k-1}(n)$. \begin{proof}[Proof of Theorem~\ref{main_thm}] Let $k \geq 4$ and $ 2 \leq s \leq \binom{k}{2}$ be integers. Fix $r \geq r_{0}(k,s)$, with $r_0$ defined in~\eqref{def_r0},~\eqref{def_r1} and~\eqref{def_r2} for $s \leq s_0(k)$, $s_0(k)<s\leq s_1(k)$ and $s>s_1(k)$, respectively. Consider a constant $0<\alpha \leq 3/4$ such that \begin{equation}\label{def_alpha} 2^{2 H(\alpha)} \cdot (r-1) < r, \end{equation} and fix $\delta>0$ such that \begin{equation}\label{def_delta} \delta<\frac{1}{2(k-1)^8} \textrm{ and }r^{\delta} < \left[\frac{r}{(r-1)\cdot 2^{2 H(\alpha)}}\right]^{\frac{\alpha^{2(k-1)}}{r^{2}(k-1)^{6}}}. \end{equation} Let $n_0=n_0(r,k,s)$ from Theorem~\ref{stability_thm}. We shall further assume that $n_0$ is large enough so that all the inequalities marked with $n \gg 0$ are satisfied. Fix $n_1 \geq n_0^2$. To reach a contradiction, suppose that there is an $n$-vertex graph $G=(V,E)$ that is $(r,K_k,\geq s)$-extremal, but $G \neq T_{k-1}(n)$. We may assume that $G$ is a complete multipartite graph (if it is not, replace it by an $(r,K_k,\geq s)$-extremal graph that is complete multipartite and different from $T_{k-1}(n)$, which exists by Theorem~\ref{thm_atleast2}). Let $V=V_1' \cup \cdots \cup V_p'$ be the multipartition of $G$, where $p \geq k$. Let $V=V_1 \cup \cdots \cup V_{k-1}$ be a partition of the vertex set of $G$ such that $\sum_{i=1}^{k-1} e(V_i)$ is minimized, so that $$\sum_{i=1}^{k-1} e(V_i) \leq \delta n^2$$ by Theorem~\ref{stability_thm}. The minimality of this partition ensures that, if $v \in V_i$, then $\|V_j \cap N(v)\| \geq \|V_i \cap N(v)\|$, for every $j \in [k-1]$, where $N(v)$ denotes the set of neighbors of $v$. Moreover, by Proposition~\ref{prop:prop1}, we must have $$\left\| \|V_{i}\|-\dfrac{n}{k-1}\right\| < \sqrt{2 \delta} \ n.$$ Finally, given that $p > k-1$, there must be an edge $\{x,y\} \in E$ whose endpoints are contained in the same class of the partition; assume without loss of generality that they lie in $V_{k-1}$ and that $\|N(x) \cap V_{k-1}\| \geq \|N(y) \cap V_{k-1}\|$. Since $x$ and $y$ are in different classes of $V_1' \cup \cdots \cup V_p'$, any $z \in V_{k-1} -\{x,y\}$ must be adjacent to $x$ or $y$. We conclude that, for any $i \in [k-1]$, $$\|N(x) \cap V_{i}\| \geq \|N(x) \cap V_{k-1}\| \geq \frac{\|V_{k-1}\|-2}{2}+1 \geq \dfrac{n}{2(k-1)}-\frac{\sqrt{2 \delta}}{2} \ n \stackrel{\eqref{def_delta}}{\geq} \frac{n}{(k-1)^3}.$$ For simplicity, we write $W_i=N(x) \cap V_i$ for $i \in [k-1]$. We shall consider the cases $ 2 \leq s \leq \binom{k-1}{2}+1 $ and $s > \binom{k-1}{2}+1$ separately. \noindent \textbf{Case 1.} Assume that $ 2 \leq s \leq \binom{k-1}{2}+1$. Let $\mathcal{C}$ be the family of $\mathcal{P}_{k,s}$-free $r$-colorings of $G$. Fix a coloring $\widehat{G} \in \mathcal{C}$. For each $i \in [k-1]$ and each color $c \in [r]$, let $W^{\widehat{G}}_{i,c}$ be the set of vertices in $N(x) \cap V_i$ that are connected to $x$ by an edge of color $c$ in $\widehat{G}$. By the pigeonhole principle, for each $i \in [k-1]$, there must be a color $c_i \in [r]$ such that $$\|W^{\widehat{G}}_{i,c_i}\| \geq \frac{\|W_i\|}{r} \geq \frac{n}{r(k-1)^{3}}.$$ We say that color $c$ is rare with respect to a pair $\{i,j\} \in \binom{[k-1]}{2}$ if there exist subsets $X_{i} \subseteq W^{\widehat{G}}_{i,c_i}$ and $X_{j} \subseteq W^{\widehat{G}}_{j,c_j}$, where $\|X_i\| \geq \alpha^{k-1} \|W^{\widehat{G}}_{i,c_i}\|$ and $\|X_j\| \geq \alpha^{k-1} \|W^{\widehat{G}}_{j,c_j}\|$, for which the number of edges of color $c$ between $X_i$ and $X_j$ is less than $\alpha\| X_{i} \|\| X_{j} \|$. Otherwise, $c$ is said to be abundant for the pair $\{i,j\}$. We claim that, for any fixed $\widehat{G}$, there must be a pair $\{i,j\} \in \binom{[k-1]}{2}$ and a color $c \in [r]$ such that $c$ is rare with respect to $\{i,j\}$. To see why this is true, assume on the contrary that every color is abundant with respect to every pair. Since $r \geq s$ and $s \leq \binom{k-1}{2}+1$, we may choose $s-1$ colors in $[r] \setminus \{c_{k-1}\}$ and assign them arbitrarily to pairs $\{i,j\} \in \binom{k-1}{2}$ in a way that each pair is assigned a color and all colors appear. This leads to an edge-coloring $\widehat{K_{k-1}}$ of a copy of $K_{k-1}$ with vertex set $\{v_1,\ldots,v_{k-1}\}$ where exactly $s-1$ colors appear and they are all different from $c_{k-1}$. By Lemma~\ref{abund}, $\widehat{G}$ contains a copy of $\widehat{K_{k-1}}$ with vertex set $\{x_1,\ldots,x_{k-1}\}$ with the property that $x_i \in W^{\widehat{G}}_{i,c_i}$ for all $i \in [k-1]$. By construction, we see that $\widehat{G}[x,x_1,\ldots,x_{k-1}]$ induces a copy of $K_k$ where at least $s$ colors appear, the desired contradiction. We are now ready to find an upper bound on $\mathcal{C}$. To do this, we shall bound the number of $\mathcal{P}_{k,s}$-free $r$-colorings that may be associated with a pair $(X_i,X_j)$ and color $c$ as above. There are $r$ choices for the color $c$ and at most $2^{2n}$ choices for the pair of sets $X_i,X_j$. Once $c$ and the sets $X_i,X_j$ are fixed, we have at most \begin{equation} \binom{\|X_{i}\|\|X_{j}\|}{\alpha\|X_{i}\|\|X_{j}\|}(r-1)^{\|X_{i}\|\|X_{j}\|}\stackrel{\eqref{eq:entropy1}}{<}2^{H(\alpha)\|X_{i}\|\|X_{j}\|}(r-1)^{\|X_{i}\|\|X_{j}\|} \end{equation} ways to color the edges between $X_i$ and $X_j$. Note that $\|X_i\|,\|X_j\| \geq \frac{\alpha^{k-1} n}{r(k-1)^{3}}$ Assuming towards an upper bound that the at most $\ex (n, K_{k})+ \delta n^{2} - \|X_{i}\|\|X_{j}\|$ remaining edges may be colored arbitrarily, we obtain \begin{eqnarray}\label{eq_case1} \|{\mathcal{C}}\| &\leq & r\cdot 2^{2n} \cdot r^{\ex (n, K_{k})+ \delta n^{2} - \|X_{i}\|\|X_{j}\|} \cdot 2^{H(\alpha)\|X_{i}\|\|X_{j}\|}(r-1)^{\|X_{i}\|\|X_{j}\|}\\ &\leq& \left( 2^{2 H(\alpha)} \cdot \dfrac{(r-1)}{r} \right)^{\|X_{i}\|\|X_{j}\|} \cdot r^{\delta n^{2}} \cdot r^{\ex (n, K_{k})} \nonumber \\ &\leq& \left( 2^{2H(\alpha)} \cdot \dfrac{(r-1)}{r} \right)^{\left(\frac{\alpha^{2k-2}}{r^{2}(k-1)^{6}}\right)n^{2}} \cdot r^{\delta n^{2}} \cdot r^{\ex(n, K_{k})} \nonumber \\ & = & \gamma^{n^2} r^{\ex(n, K_{k})} < r^{\ex(n, K_{k})} \nonumber \end{eqnarray} where $\gamma<1$ is a constant\footnote{with respect to $n$.} by our choice of $\delta>0$ in~\eqref{def_delta}. This contradicts our choice of $G$. \noindent \textbf{Case 2.} Now assume that $s> \binom{k-1}{2}+1$. Let $s'=s-\binom{k-1}{2}$, so that $2 \leq s' \leq k-1$. To get a contradiction, we assume that the complete multipartite $(r,(K_k, \geq s))$-extremal graph $G$ has $r^{\ex(n,K_{k-1})+m}$ distinct $(K_k, \geq s)$-free colorings, where $m \geq 0$. We shall prove that the graph $G-x$ must have at least $r^{\ex (n-1, K_{k}) + m+1}$ such colorings. This conclusion will lead to the desired contradiction, as we could apply this argument iteratively until we obtain a graph $G'$ on $n_{0}$ vertices and at least $ r^{\ex (n_{0}, K_{k}) + m+n-n_{0}}>r^{n^{2}_{0}}\geq r^{\|E(G')\|}.$ Let $\mathcal{C}$ be the family of $\mathcal{P}_{k,s}$-free $r$-colorings of $G$, and let $\mathcal{C}_1$ be the subfamily containing all colorings $\widehat{G}$ for which there is a choice of distinct indices $i_1, \ldots, i_{s'} \in [k-1]$ and distinct colors $ c_{i_{1}}, \ldots, c_{i_{s'}} \in [r] $ such that, for each $p \in [s']$, the set $ W^{\widehat{G}}_{i_p,c_{i_p}}\subseteq V_{p} \cap N(x)$ of neighbors of $x$ in $V_p$ through edges of color $c_{i_p}$ (with respect to $\widehat{G}$) satisfies $ \|W^{\widehat{G}}_{i_{p},c_{i_p}}\|\geq n / [r(k-1)^{3}] $. For any $i \in [k-1] \setminus \{i_1,\ldots,i_{s'}\}$, we fix an arbitrary color $c_i$ such that the set $W_{i,c_i}^{\widehat{G}}$ of neighbors of $x$ in $V_i$ through edges of color $c_{i}$ satisfies $ \|W^{\widehat{G}}_{i,c_i}\|\geq n / [r(k-1)^{3}]$ (this color exists by the pigeonhole principle). As in Case 1, we say that color $c$ is rare with respect to a pair $\{i,j\} \in \binom{[k-1]}{2}$ if there exist subsets $X_{i} \subseteq W^{\widehat{G}}_{i,c_i}$ and $X_{j,c_j} \subseteq W^{\widehat{G}}_{j}$, where $\|X_i\| \geq \alpha^{k-1} \|W^{\widehat{G}}_{i,c_i}\|$ and $\|X_j\| \geq \alpha^{k-1} \|W^{\widehat{G}}_{j,c_j}\|$, for which the number of edges of color $c$ between $X_i$ and $X_j$ is less than $\alpha\| X_{i} \|\| X_{j} \|$. Otherwise, $c$ is said to be abundant for the pair $\{i,j\}$. We claim that, for any fixed $\widehat{G}$, there must be a pair $\{i,j\} \in \binom{[k-1]}{2}$ and a color $c \in [r]$ such that $c$ is rare with respect to $\{i,j\}$. If this was not the case, we would be able to fix $s-s'=\binom{k-1}{2}$ distinct colors in $[r] \setminus \{c_{i_{1}}, \ldots, c_{i_{s'}}\}$ to be assigned to the edges of a copy of $K_{k-1}$, which, with Lemma~\ref{abund}, would lead to a contradiction. Using the arguments in~\eqref{eq_case1}, we conclude that $ \|{\mathcal{C}_{1}}\| \leq \gamma^{n^2} r^{\ex (n, K_{k})}$, where $\gamma<1$ is a constant. As a consequence the family $\mathcal{C}_2=\mathcal{C}\setminus \mathcal{C}_1$ contains at least $r^{\ex (n, K_{k}) + m}-\gamma^{n^2} r^{\ex (n, K_{k})} \geq r^{\ex (n, K_{k}) + m-1}$ colorings. Fix a coloring $\widehat{G} \in \mathcal{C}_2$. We define a bipartite graph $B^{\widehat{G}}$ with bipartition $[k-1] \cup [r]$ such that $\{i,c\}$ is an edge if the set $ W^{\widehat{G}}_{i,c}\subseteq V_{i} \cap N(x)$ of neighbors of $x$ in $V_i$ through edges of color $c_{i}$ satisfies $ \|W^{\widehat{G}}_{i,c}\|\geq n / [r(k-1)^{3}]$. Note that $\widehat{G}$ lies in $\mathcal{C}_1$ if and only if $B^{\widehat{G}}$ contains a matching of size $s'$. Since $\widehat{G} \notin \mathcal{C}_1$, the following holds by Hall's Theorem. There is an integer $h$, where $1 \leq h \leq s'-1$, a set of distinct indices $I=\{i_1,\ldots,i_{h+1}\} \subseteq [k-1]$ and a set $C \subset [r]$ of colors, where $\|C\| \leq h$, such that $ \|W^{\widehat{G}}_{i_p,c}\|\geq n / [r(k-1)^{3}]$ only if $c \in C$. We shall associate each $\widehat{G} \in \mathcal{C}_2$ with such a triple $(h,I,C)$. Note that we may suppose that $\|C\|=h$ by adding arbitrary new elements to $C$ if necessary. Let $\phi(\mathcal{C}_2)$ be the family that contains the projection of each element of $\mathcal{C}_2$ onto the edges incident with $x$. In other words, $\phi(\mathcal{C}_2)$ contains all $r$-edge colorings of the edges incident with $x$ that may be extended to a coloring in $\mathcal{C}_2$. We are now ready to find an upper bound on the cardinality of $\phi(\mathcal{C}_2)$. To do this, we shall bound the number of colorings that may be associated with a triple $(h,I,C)$ as above. Given $h$, there are $\binom{k-1}{h+1} \leq 2^{k-1}$ ways to fix $I=\{i_1,\ldots,i_{h+1}\}$ and $\binom{r}{h} \leq 2^r$ ways to choose $C$. Once $(h,I,C)$ is fixed, an upper bound on the number of ways to color the edges between $x$ and $V_{i}$, where $i \in I$, is $$\binom{\|V_i\|}{n / [r(k-1)^{3}]}^r h^{\|V_i\|} \stackrel{\eqref{eq:entropy1}}{\leq} 2^{H\left(\frac{1}{r(k-1)^{3}}n\right)} h^{\|V_i\|} \stackrel{\eqref{eq:entropy2}}{\leq} \left(r(k-1)^{3}\right)^{\frac{2n}{(k-1)^3}} h^{(\frac{1}{k-1}+\sqrt{2 \delta})n}.$$ The first term in the product is an upper bound on the number of ways to color edges using each of the at most $r$ colors that are used at most $n / [r(k-1)^{3}]$ times, the second term is an upper bound on the number of ways to color edges with the remaining colors, which lie in $C$. For $i \notin I$, the number of ways to color the edges between $x$ and $V_{i}$ will be bounded with the trivial bound $r^{\|V_i\|}\leq r^{(\frac{1}{k-1}+\sqrt{2 \delta})n}$. Combining this information, we obtain \begin{eqnarray} \label{Nx} \|\phi(\mathcal{C}_2)\| & \leq & \sum_{h=1}^{s'-1} 2^{k-1} \cdot 2^r \cdot [r(k-1)^{3}]^{\frac{2(h+1)n}{(k-1)^{3}}} \cdot (h^{h+1}\cdot r^{k-h-2})^{(\frac{1}{k-1}+\sqrt{2 \delta})n} \end{eqnarray} Consider the function $f(h)=h^{h+1}r^{-h}$. With $1 \leq h \leq k-2$, since $r \stackrel{(\ref{eq:4})}{>} (k-1)^4 > e(k+1)$ for $k \geq 4$, we have $$\frac{f(h+1)}{f(h)}=\left(1+\frac{1}{h}\right)^h\cdot \frac{(h+1)^2}{rh} \leq \frac{e}{r} \left(h+2+\frac{1}{h} \right)<1.$$ This means that $f(h)$ is maximized for $h=1$. On the other hand, it is clear that $ [r(k-1)^{3}]^{\frac{2(h+1)n}{(k-1)^{3}}}\leq [r(k-1)^{3}]^{\frac{2s'n}{(k-1)^{3}}}$. Finally, the minimum degree $\delta_{k-1}(n)$ of a vertex in the Tur\'an graph $T_{k-1}(n)$ satisfies $\delta_{k-1}(n) \geq \frac{k-2}{k-1}n-1$. Using this in $ (\ref{Nx})$ leads to \begin{eqnarray}\label{Nx2} \|\phi(\mathcal{C}_2)\| &\leq &s' \cdot 2^{k-1} \cdot 2^{r}\cdot [r(k-1)^{3}]^{\frac{2 s' n}{(k-1)^{3}}} \cdot r^{(k-3)(\frac{1}{k-1}+\sqrt{2 \delta})n} \nonumber \\ &\stackrel{\eqref{def_delta}}{\leq} & s' \cdot 2^{k-1} \cdot 2^{r} \cdot [r(k-1)^{3}]^{\frac{2 s' n}{(k-1)^{3}}} \cdot r^{(k-3)\left(\frac{1}{k-1}+\frac{1}{(k-1)^{4}}\right)n} \cdot r^{\delta_{k-1}(n)-\frac{k-2}{k-1}n+1} \nonumber \\ & = & s' \cdot 2^{k-1} \cdot r \cdot 2^{r} \cdot \left[\frac{(k-1)^{6 s'}}{ r^{(k-1)^{2} - 2s'-\frac{k-3}{k-1} }}\right]^{\frac{n}{(k-1)^3}} \cdot r^{\delta_{k-1}(n)}. \end{eqnarray} We claim that \begin{equation}\label{claimed_bound} \frac{(k-1)^{6 s'}}{ r^{(k-1)^{2} - 2s'-\frac{k-3}{k-1} }}<1. \end{equation} Before proving this, we argue that it leads to the desired result. Using this, inequality~\eqref{Nx2} implies that $$\|\phi(\mathcal{C}_2)\| \leq r^{\delta_{k-1}(n)-2}$$ for sufficiently large $n$, and therefore $$ \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G-x)\| \geq \frac{\|\mathcal{C}_2\|}{\|\phi(\mathcal{C}_2)\|} \geq \frac{r^{\ex (n, K_{k}) + m-1}}{r^{\delta_{k-1}(n)-2}}=r^{\ex (n-1, K_{k}) + m+1},$$ as required. To conclude the proof, we show that~\eqref{claimed_bound} holds. We start with the case $k \geq 5$, where we first show that $r_0(k,s)>(k-1)^4$ when $s \geq \binom{k-1}{2}+2$. Indeed, by the definition of $r_0(s,t)$ (for $s >s_0$) and by the lower bounds on $L_{opt}(k,s)$ given in Claim~\ref{claim:lopt}, we deduce that \begin{eqnarray}\label{eq_LL} r_0(k,s) &>& (s-A(k,2)-1)^3 \cdot (s-A(k,2))^{2-\frac{k-1}{k-2}} \cdot (s-1)^{\frac{k-1}{k-2}} \nonumber \\ & \geq & (s-A(k,2)-1)^5 \cdot \left(\binom{k-1}{2}+1 \right) \nonumber \\ & > & \left(\binom{k-1}{2}-\left(\binom{k}{2}-\left\lfloor\frac{k}{2}\right\rfloor \left\lceil\frac{k}{2} \right\rceil \right)+1\right)^5 \cdot \binom{k-1}{2} \nonumber \\ &\geq& \left(\left\lfloor\frac{k}{2}\right\rfloor \left\lceil\frac{k}{2} \right\rceil -k+2 \right)^5 \cdot \binom{k-1}{2}. \end{eqnarray} It turns out that $$\left\lfloor\frac{k}{2}\right\rfloor \left\lceil\frac{k}{2} \right\rceil -k+2 \geq \frac{k^2-1}{4} -k+2 \geq k-2.$$ Therefore, the inequality~\eqref{eq_LL} leads to \begin{eqnarray} \label{eq:4} r_0(k,s) > \frac{(k-2)^{6}(k-1)}{2} > (k-1)^4, \end{eqnarray} where the last part may be easily proved by induction (for $k \geq 4$). Coming back to~\eqref{claimed_bound}, we first consider the case $k\geq 5$. First note that $$\frac{(k-1)^{6 s'}}{ r^{(k-1)^{2} - 2s'-\frac{k-3}{k-1} }} \leq \frac{\left((k-1)^{4}\right)^{3s'/2}}{ r^{(k-1)^{2} - 2s'-1}}.$$ Since $r > (k-1)^{4}$, inequality~\eqref{claimed_bound} holds if we show that $3 s'/2 \leq (k-1)^{2} - 2s'-1$, which is equivalent to $$\frac{7s'}{2} \leq k^2-2k.$$ The left-hand side of this inequality is at most $7(k-1)/2$, and it is indeed the case that $7(k-1) \leq 2k^2-4k$ for all $ k\geq 5$. In the case $k=4$, inequality~\eqref{claimed_bound} holds if and only if \begin{equation}\label{claimed_bound4} 3^{6s'} \leq r^{9-2s'-\frac{1}{3}}. \end{equation} Since we are in Case 2, we need to consider the cases $s=5$ (so $s'=2$) and $s=6$ (so $s'=3$). In the case $s=5$, the inequality holds if $3^{12} \leq r^{14/3}$, which holds for $r \geq 17$. According to Table~\ref{values}, $r_0(4,5)=222$. In the case $s=6$, inequality~\eqref{claimed_bound4} holds if $3^{18} \leq r^{8/3}$, holds for $r\geq 1662$. According to Table~\ref{values}, $r_0(4,6)=5434$. This concludes the proof of Theorem~\ref{main_thm}. \end{proof} \section{Final remarks and open problems} In this paper, given integers $r \geq 2$, $n\geq k \geq 3$ and $2 \leq s \leq \binom{k}{2}$, we were interested in characterizing $n$-vertex graphs $G$ for which the number of $r$-edge-colorings with no copy of $K_k$ colored with $s$ or more colors satisfies \begin{equation}\label{eq_remarks} \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\|=c_{r,\mathcal{P}_{k,s}}(n) = \max\left\{\, \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G')\| \colon \|V(G')\| = n \, \right\}. \end{equation} This problem is a common generalization of the Tur\'{a}n problem and of the rainbow Erd\H{o}s-Rothschild problem, i.e., the problem of finding $n$-vertex graphs $G$ with the largest number of $r$-edge-colorings with no rainbow copy of $K_k$. More precisely, we have found functions $r_1(k,s)$ and $r_0(k,s)$ such that \begin{itemize} \item[(a)] If $r \geq r_0(k,s)$ and $n$ is sufficiently large, then $\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\|=c_{r,\mathcal{P}_{k,s}}(n)$ if and only if $G$ is isomorphic to $T_{k-1}(n)$. \item[(b)] If $r \leq r_1(k,s)$, then $\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(K_n)\| > \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(T_{k-1}(n))\|$. \end{itemize} We should mention that reference~\cite{rainbow_kn} gives a function $r_0'(k)$ such that the statement of part (a) holds for all $r \geq r_0'(k) \geq \binom{k}{2}^{8k-4}$ when $s=\binom{k}{2}$. With Lemma~\ref{lemma_simple}, this implies the \emph{existence} of a function $ r_0'(k,s)$ such that (a) is satisfied. The results of the current paper are such that $r_0(k,\binom{k}{2})\leq \left(k^2/4\right)^{4k} <r_0'(k)$, that $r_0(k,s) \leq (s-1)^2$ for $s\leq s_0(k)=\binom{k}{2}-\left\lfloor \frac{k}{2} \right\rfloor \cdot \left\lceil \frac{k}{2} \right\rceil+2 $ and that $r_0(k,s) \leq (s-1)^7$ for $s\leq s_1(k)=\binom{k}{2} - \left\lfloor \frac{k}{2} \right\rfloor +2$. For general values of $k$ and $s$, there is a significant gap between the functions $r_0$ and $r_1$ in (a) and (b), but it turns out that $r_0(k,s)=r_1(k,s)+1$ for all pairs $(k,3)$ with $k \geq 4$ and for infinitely many other pairs $(k,s)$. Computing the values of $r_0$ and $r_1$, we see that this happens for the following pairs if $(k,s)$ where $4 \leq k <10$ and $s \geq 4$: $$(k,s) \in \{(5,4),(7,4),(7,5),(8,4),(8,5),(9,3),(9,4),(9,6)\}.$$ In fact, the following holds. \begin{proposition}\label{prop1} Let $s \geq 3$ be an integer. There exists $k_0$ such that, for all $k \geq k_0$, the pair $(k,s)$ satisfies $r_0(k,s)=s$ and $r_1(k,s)=s-1$. \end{proposition} \begin{proof} Let $s \geq 3$. First notice that $(s-1)^{1/(k-2)}$ is not an integer for $k > 2 + \log_2(s-1)$. Fix $k_1$ such that $(s-1)^{1/(k-2)}$ is not an integer and $(s-1)^{\frac{k-1}{k-2}}<s-\frac{1}{2}$ for all integers $k \geq k_1$. In particular, we have $$s-1=\lceil (s-1)^{\frac{k-1}{k-2}}-1\rceil = r_1(k,s).$$ Next, let $k_2$ be such that, for all integers $k \geq k_2$, we have $s \leq s_0(k)$ and $i^\ast=s-2$, see~(\ref{def_r0}). In particular, $r_0(k,s)$ is the least integer greater than $$(s-1)^{\frac{k-1}{k-2}} \cdot (s- 2)^{ \frac{1}{(k-3)(k-2)}} \cdot (s- 3)^{ \frac{1}{(k-4)(k-3)} }\cdots 2^{\frac{1}{(k-s+1)(k-s+2)}} \leq (s-1)^{\frac{k-1}{k-2}} \cdot (s- 2)^{ \frac{s-3}{(k-s+1)(k-2)}}.$$ We may choose $k_3$ such that, for all $k \geq k_3$, it holds that $$\left(1+\frac{1}{2(s-1)^{3/2}} \right)^{(k-s+1)(k-2)} > (s-2)^{s-3}.$$ This implies that, for all $k \geq \max\{k_1,k_2,k_3\}$, it is $$(s-1)^{\frac{k-1}{k-2}}(s- 2)^{ \frac{s-3}{(k-s+1)(k-2)}} < (s-1)^{\frac{k-1}{k-2}}\left(1+\frac{1}{2(s-1)^{3/2}} \right) \leq (s-1)^{\frac{k-1}{k-2}}+\frac{1}{2}<s.$$ As a consequence, we have \begin{eqnarray}s \leq r_0(k,s) &\leq& \left\lfloor (s-1)^{\frac{k-1}{k-2}}(s- 2)^{ \frac{s-3}{(k-s+1)(k-2)}} \right\rfloor+1< s+1, \end{eqnarray} so that $r_0(k,s)=s$. \end{proof} By definition, it is obvious that, if $r \leq s-1$, then $\|\mathcal{C}_{r,\mathcal{P}_{k,s}}(G)\|=c_{r,\mathcal{P}_{k,s}}(n)$ if and only if $G=K_n$. This implies that, for pairs $(k,s)$ such that $s=3$ or Proposition~\ref{prop1} is satisfied, the functions $r_1$ and $r_0$ are best possible in (a) and (b)\footnote{For sufficiently large $n$.}, and $K_n$ is $(r,\mathcal{P}_{k,s})$-extremal for $r \leq 2$ and $T_{k-1}(n)$ is $(r,\mathcal{P}_{k,s})$-extremal for $r \geq 3$. We propose the following questions. \begin{question} We say that a pair $(k,s)$ such that $k\geq 3$ and $3 \leq s \leq \binom{k}{2}$ satisfies \emph{Property A} if there exists $r^=r^(k,s)$ such that, for any fixed $r \geq 2$, there exists $n_0$ for which the following holds: $G$ is an $(r,\mathcal{P}_{k,s})$-extremal $n$-vertex graph with $n \geq n_0$ if and only if $r<r^$ and $G=K_n$, or $r \geq r^$ and $G=T_{k-1}(n)$. Is it true that all such pairs $(k,s)$ satisfy Property A? \end{question} The above paragraph ensures that all pairs that fulfil Proposition~\ref{prop1} also satisfy Property A. Moreover, previous work for triangles~\cite{BL19,lagos} implies that the pair $(3,s)$ satisfies Property A for any $s$, while the current paper shows that the pair $(k,3)$ satisfies Property A for any $k$. \begin{question} Is it true that, for any pair $(k,s)$ such that $k\geq 3$ and $3 \leq s \leq \binom{k}{2}$ and any $r \geq s$, it holds that \begin{equation}\label{eq_kn} \|\mathcal{C}_{r,\mathcal{P}_{k,s}}(K_n)\|=\left(\binom{r}{s-1}+o_n(1) \right) (s-1)^{\binom{n}{2}}? \end{equation} \end{question} Equation~\eqref{eq_kn} holds for all $r \geq 3$ in the case $k=3$, see~\cite{BL19,BBH20} (the second reference actually shows that this relation holds for some functions $r=r(n)$ such that $r \rightarrow \infty$ as $n \rightarrow \infty$). We observe that, if a pair $(k,s)$ satisfies Property A and equation~\eqref{eq_kn} holds for $(k,s)$ and any $r > r_1(k,s)$, then we must have $r^*(k,s)= r_1(k,s)+1$. In a different direction, given integers $r \geq 2$, $n\geq k \geq 3$ and $1 \leq s \leq \binom{k}{2}$, it would be interesting to consider $r$-edge-colorings with no copy of $K_k$ colored with $s$ or \emph{less} colors, which would lead to a common generalization of the Tur\'{a}n problem and of the (monochromatic) Erd\H{o}s-Rothschild problem. The class of extremal configurations for large $n$ and $r$ will be much richer, as $K_n$ does not admit any such coloring by Ramsey's Theorem and $T_{k-1}(n)$ admits fewer colorings than other constructions (see~\cite{ABKS,PSY16}). \end{document}	arXiv
\begin{document} \title{One-dimensional Stochastic Differential Equations \ with Generalized and Singular Drift hanks{Work supported in part by the European Community's FP 7 Programme under contract PITN-GA-2008-213841, Marie Curie ITN "Controlled Systems".} \hrule \begin{abstract} \noindent Introducing certain singularities, we generalize the class of one-dimensional stochastic differential equations with so-called generalized drift. Equations with generalized drift, well-known in the literature, possess a drift that is described by the semimartingale local time of the unknown process integrated with respect to a locally finite signed measure $\nu$. The generalization which we deal with can be interpreted as allowing more general set functions $\nu$, for example signed measures which are only $\sigma$-finite. However, we use a different approach to describe the singular drift. For the considered class of one-dimensional stochastic differential equations, we derive necessary and sufficient conditions for existence and uniqueness in law of solutions. \\[2ex] \emph{Keywords:} Singular stochastic differential equations, local times, generalized drift, singular drift, uniqueness in law, space transformation, Bessel process, Bessel equation\\[1ex] \emph{2010 MSC:} 60H10, 60J55 \end{abstract} \hrule \section{Introduction and Preliminaries} \noindent Throughout this paper, $(\Omega, \mathcal{F}, \mathbf{P})$ stands for a complete probability space equipped with a filtration $\mathbb{F} = (\mathcal{F}_t)_{t \geq 0}$ which satisfies the usual conditions, i.e., $\mathbb{F}$ is right-continuous and $\mathcal{F}_0$ contains all sets from $\mathcal{F}$ which have $\mathbf{P}$-measure zero. For a process $X = (X_t)_{t \geq 0}$ the notation $(X,\mathbb{F})$ indicates that $X$ is $\mathbb{F}$-adapted. The processes considered in the following belong to the class of continuous semimartingales up to a stopping time $S$ and local times of such processes will play an important role. Therefore, in the Appendix we summarize some facts about continuous semimartingales up to a stopping time $S$ and their local times, which we use in the sequel. By writing (A.), we refer to a formula or a result of the Appendix.\\ \indent In the present paper, our purpose is to investigate one-dimensional stochastic differential equations (SDEs) with generalized and singular drift in the framework of continuous semimartingales. In general, this kind of SDEs admits exploding solutions. Therefore, the convenient state space is the extended real line $\overline{\mathbb{R}} = \mathbb{R}\cup \{-\infty, +\infty\}$ equipped with the $\sigma$-algebra $\mathscr{B}(\overline{\mathbb{R}})$ of Borel subsets. \\ \indent SDEs with \emph{generalized and singular drift} are of the form \begin{equation}\label{eqn:SDE_mvasd} X_t = X_0 + \int_0^t b(X_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_m^X(t,y) \, \mbox{\upshape d} f(y)\,, \end{equation} where $b$ is a measurable real function and $B$ denotes a Wiener processes. Furthermore, the function $f$ which appears as the integrator in the drift of Eq. (\ref{eqn:SDE_mvasd}) is assumed to be non-negative, right-continuous and of locally bounded variation such that its reciprocal $1/f$ is locally integrable.\footnote{Here and in the following we use the convention $1/a = +\infty$ if $a=0$.} We call a function $f$ with these properties a \emph{drift function}. Moreover, $L_m^X$ denotes a certain local time of the unknown process $X$ specified in \begin{defi}\label{def:solution} A continuous $(\overline{\mathbb{R}}, \mathscr{B}(\overline{\mathbb{R}}))$-valued stochastic process $(X,\mathbb{F})$ defined on a probability space $(\Omega,\mathcal{F},\mathbf{P})$ is called a solution of Eq. (\ref{eqn:SDE_mvasd}) if the following conditions are fulfilled: (i) $X_0$ is real-valued. (ii) $X_t = X_{t \wedge S_\infty^X}$, $t \geq 0$, $\mathbf{P}$-a.s., where $S_\infty^X := \inf\{t \geq 0:\|X_t\| = +\infty\}$.\footnote{$\inf \emptyset = +\infty$.} (iii) $(X,\mathbb{F})$ is a semimartingale up to $S_\infty^X$. (iv) There exists a random function $L^X_m$ on $[0,S_\infty^X)\times \mathbb{R}$ into $[0,+\infty)$ that is a version of the local time of $X$ defined on $\{t<S_\infty^X\}$ in the sense of an occupation times density with respect to the measure $m(\mbox{\upshape d} x) := 2\,f(x)\, \mbox{\upshape d} x$, i.e., \[ \int_0^t h(X_s) \, \mbox{\upshape d} \assPro{X}_s = \int_{\mathbb{R}} h(x) \, L_m^X(t,x)\,m(\mbox{\upshape d} x)\,, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] holds for all non-negative measurable functions $h$. Thereby, $L_m^X(\, . \, ,x)$ is $\mathbb{F}$-adapted for all $x \in \mathbb{R}$. Moreover, $L_m^X$ is $\mathbf{P}$-a.s. continuous and increasing in $t$ as well as right-continuous in $x$ with limits from the left. (v) There exists a Wiener process $(B,\mathbb{F})$ such that Eq. (\ref{eqn:SDE_mvasd}) is satisfied for all $t < S_\infty^X$ $\mathbf{P}$-a.s. \end{defi} \begin{defi}\label{def:uniqueness} We say that the solution of Eq. (\ref{eqn:SDE_mvasd}) (or of any other SDE appearing in the sequel) is unique in law if any two solutions $(X^1,\mathbb{F}^1)$ and $(X^2,\mathbb{F}^2)$ with coinciding initial distributions defined on the probability spaces $(\Omega^1,\mathcal{F}^1,\mathbf{P}^1)$ and $(\Omega^2,\mathcal{F}^2,\mathbf{P}^2)$, respectively, possess the same image law on the space $C_{\overline{\mathbb{R}}}([0,+\infty))$ of continuous functions defined on $[0,+\infty)$ and taking values in $\overline{\mathbb{R}}$. \end{defi} \indent The $\mathbb{F}$-stopping time $S^X_\infty$ in Definition \ref{def:solution} is called the \emph{explosion time} of $X$. To distinguish between $L_m^X$ and the (right) local time $L_+^X$, defined via (\ref{eqn:gen_ito_formula}), we also use the expression \emph{(right) semimartingale local time} when we refer to $L_+^X$. \\ \indent The idea to introduce such a local time $L_m^X$ in the context of SDEs for Dirichlet processes goes back to H.J. Engelbert and J. Wolf \cite{engelbertwolf}. Moreover, it was already used by S. Blei \cite{blei_bessel_2011} as a helpful tool in the investigation of an SDE for the $\delta$-dimensional Bessel process for $\delta \in (1,2)$, which turns out to be an important example of an equation of type (\ref{eqn:SDE_mvasd}) (see Section \ref{sec:skew_solutions}).\\ \indent By introducing certain singularities, the class of SDEs of the form (\ref{eqn:SDE_mvasd}) generalizes the class of SDEs with so-called generalized drift. SDEs with generalized drift have the following structure: \begin{equation}\label{eqn:SDE_mvd} X_t = X_0 + \int_0^t b(X_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_+^X(t,y) \, \nu(\mbox{\upshape d} y)\,, \end{equation} where $b$ is a measurable real function and $\nu$ denotes a set function defined on the bounded Borel sets of the real line $\mathbb{R}$ such that it is a finite signed measure on $\mathscr{B}([-N,N])$ for every $N \in \mathbb{N}$. In this equation $B$ stands again for a Wiener processes and $L_+^X$ denotes the right semimartingale local time of the unknown process $X$. Omitting condition (iv), the notion of a solution of Eq. (\ref{eqn:SDE_mvd}) is introduced analogously to Definition \ref{def:solution}. Clearly, equations without drift \begin{equation}\label{eqn:SDE_without_drift} Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s\,, \end{equation} where $\sigma$ is as well a measurable real function, are a special kind of Eq. (\ref{eqn:SDE_mvd}). \\ \indent SDEs of type (\ref{eqn:SDE_mvd}) with generalized drift have been studied previously by many authors. We refer the reader to Harrison and Shepp \cite{harrison_shepp}, N.I. Portenko \cite{portenko}, D.W. Stroock and M. Yor \cite{stroock_yor} and J.F. Le Gall \cite{LeGall_1983}, \cite{LeGall_1984}. H.J. Engelbert and W. Schmidt \cite{engelbert_schmidt:1985}, \cite{engelbert_schmidt:1989_III} derived rather weak necessary and sufficient conditions on existence and uniqueness of solutions to SDEs with generalized drift, which we recall in Theorem \ref{theorem:e_u_gen_drift} below. More recently, R.F. Bass and Z.-Q. Chen \cite{bass_chen} also considered SDEs of type (\ref{eqn:SDE_mvd}). \\ \indent We call the set function $\nu$ in Eq. (\ref{eqn:SDE_mvd}) \emph{drift measure} and we always assume additionally \begin{equation}\label{eqn:cond_atoms} \nu(\{x\}) < \frac{1}{2}\,, \qquad x \in \mathbb{R}\,. \end{equation} Condition (\ref{eqn:cond_atoms}) is motivated by the fact that, in general, there is no solution of Eq. (\ref{eqn:SDE_mvd}) if $\nu(\{x\}) > 1/2$ for some $x \in \mathbb{R}$. The case $\nu(\{x\}) = 1/2$ for some $x \in \mathbb{R}$ corresponds to a reflecting barrier at the point $x$, which requires different methods to treat Eq. (\ref{eqn:SDE_mvd}) than by assuming (\ref{eqn:cond_atoms}) (cf. W. Schmidt \cite{schmidt:1989}, R.F. Bass and Z.-Q. Chen \cite{bass_chen}). In S. Blei and H.J. Engelbert \cite{blei_engelbert_2012} the reader can find a complete treatment of the features of Eq. (\ref{eqn:SDE_mvd}) in the cases $\nu(\{x\}) > 1/2$ and $\nu(\{x\}) = 1/2$ for some $x \in \mathbb{R}$. \\ \indent To see how equations of type (\ref{eqn:SDE_mvasd}) generalize equations of the form (\ref{eqn:SDE_mvd}), we consider the integral equation \begin{equation}\label{eqn:integral_eqn} g_\nu(x) = \left\{\begin{array}{ll} 1 - 2 \displaystyle\int_{[0,x]} g_\nu(y-) \, \nu(\mbox{\upshape d} y), & x \geq 0, \\ 1 + 2 \displaystyle\int_{(x,0)} g_\nu(y-) \, \nu(\mbox{\upshape d} y), & x < 0, \end{array}\right. \end{equation} which has a unique c\`adl\`ag solution $g_\nu$. Note that for this statement condition (\ref{eqn:cond_atoms}) is needed. Moreover, $g_\nu$ is strictly positive as well as locally of bounded variation and the same properties also hold for the reciprocal function $1/g_\nu$. The explicit form of the solution $g_\nu$ can be found in \cite{engelbert_schmidt:1989_III}, (4.26). Setting $f_\nu := 1/g_\nu$, integration by parts gives \begin{equation}\label{eqn:drift_measure} \nu(\mbox{\upshape d} y) = \frac{1}{2} \, f^{-1}_\nu(y) \, \mbox{\upshape d} f_\nu(y)\, . \end{equation} Taking any solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvd}), because of (\ref{eqn:drift_measure}), its drift can be expressed as \[\begin{split} \int_\mathbb{R} L_+^X(t,y) \, \nu(\mbox{\upshape d} y) &= \int_\mathbb{R} \frac{1}{2} \, L_+^X(t,y) \, f^{-1}_\nu(y) \, \mbox{\upshape d} f_\nu(y)\,. \end{split}\] Therefore, defining \[ L_m^X(t,y) := \frac{1}{2} \, L_+^X(t,y) \, f^{-1}_\nu(y)\,, \qquad (t,y) \in [0,S_\infty^X) \times \mathbb{R}, \] we can write \[ X_t = X_0 + \int_0^t b(X_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_m^X(t,y) \, \mbox{\upshape d} {f_\nu}(y)\,. \] Furthermore, as an immediate consequence of the occupation times formula (\ref{eqn:occupationtime}) for $X$, we obtain that $L_m^X$ satisfies condition (iv) of Definition \ref{def:solution} with respect to the measure $m(\mbox{\upshape d} x) = 2 \, f_\nu(x) \, \mbox{\upshape d} x$. Altogether, $(X,\mathbb{F})$ is also a solution of Eq. (\ref{eqn:SDE_mvasd}) with diffusion coefficient $b$ and drift function $f_\nu$. \\ \indent Conversely, let us assume additionally that the reciprocal $1/f$ of the drift function $f$ in Eq. (\ref{eqn:SDE_mvasd}) is also of locally bounded variation or, equivalently, that $f$ and $f_-$ have no zeros.\footnote{For any real function $f$ with left hand limits $f(x-)$ we set $f_-(x) = f(x-)$, $x\in\mathbb{R}$.} Then, defining $\nu$ via (\ref{eqn:drift_measure}) by using $f$ on the right-hand side, we obtain a feasible drift measure $\nu$ fulfilling (\ref{eqn:cond_atoms}) and Eq. (\ref{eqn:SDE_mvasd}) reduces to an equation of type (\ref{eqn:SDE_mvd}) with $\nu$ as drift measure (see Remark \ref{remark:connection_mvd_mvasd}). Note that the corresponding function $f_\nu=1/g_\nu$, where $g_\nu$ is obtained via (\ref{eqn:integral_eqn}), differs from $f$ at most by a multiplicative constant. \\ \indent In contrast to Eq. (\ref{eqn:SDE_mvd}) regarded as an equation of type (\ref{eqn:SDE_mvasd}), one of the new features of Eq. (\ref{eqn:SDE_mvasd}) is that we do not postulate that $1/f$ is also of locally bounded variation. This is equivalent to allow $f$ and $f_-$ to have zeros. Responsible for the singularity of Eq. (\ref{eqn:SDE_mvasd}), these zeros play an important role in the following analysis of Eq. (\ref{eqn:SDE_mvasd}). We set $F_+ := \{x \in \mathbb{R} : f(x) = 0\}$ and $F_- := \{x \in \mathbb{R} : f(x-) = 0\}$. From our assumption that $1/f$ is locally integrable it follows immediately that the closed set $F:= F_+ \cup F_-$ is of Lebesgue-measure zero. Using an arbitrary drift function $f$ analogously as in (\ref{eqn:drift_measure}) to define $\nu(\mbox{\upshape d} y) := 1/2 \, f^{-1}(y) \, \mbox{\upshape d} f(y)$,\footnote{We always use the convention $0 \cdot +\infty = 0$ and $a \cdot +\infty = +\infty$, $a \in \mathbb{R}\setminus\{0\}$.} we obtain a set function $\nu$ which is in general not a finite signed measure on $\mathscr{B}([-N,N])$ for every $N \in \mathbb{N}$. Indeed, taking for example the drift function \[ f(x) = \text{\large{$\mathds{1}$}}_{(-\infty,0)}(x) + \sqrt{x} \, \text{\large{$\mathds{1}$}}_{(0, +\infty)}(x), \qquad x\in\mathbb{R}, \] for any $N \in \mathbb{N}$, the corresponding $\nu$ is a signed measure on $\mathscr{B}([-N,N])$ which is only $\sigma$-finite. But as seen from the drift function $f(x) = \sqrt{\|x\|}$, $x\in\mathbb{R}$, it is also possible that $\nu$ is no longer a signed measure on $\mathscr{B}([-N,N])$ because for some sets it takes the value $+\infty$ and for other sets the value $-\infty$. \\ \indent Besides these examples of a singleton $F$, it is of course possible that $F$ is even an uncountable (e.g., Cantor like) set. Nevertheless, already in the case that $F$ consists only of one point, surprising and interesting effects like skewness and reflection can be observed for solutions of Eq. (\ref{eqn:SDE_mvasd}). In the context of the complexity of the set $F$, it becomes clear quite quickly that, in general, solutions of Eq. (\ref{eqn:SDE_mvasd}) go beyond the scope of semimartingales. But it is the objective of this paper to approach the problem in a first step keeping within the framework of semimartingales. \\ \indent The paper is organized as follows. We begin with stating some useful properties of the local time $L_m^X$ of a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) in Section \ref{sec:prop_loc_time}. Afterwards in Section \ref{sec:space_transform} we investigate the structure of a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}). In particular, we show the connection of Eq. (\ref{eqn:SDE_mvasd}) to another equation arising from a space transformation and prepare the study of existence and uniqueness in law of solutions of Eq. (\ref{eqn:SDE_mvasd}), which is done in Section \ref{sec:symmetric_solutions} and \ref{sec:skew_solutions}. \\ \indent As preliminaries in the context of existence and uniqueness of solutions we briefly recall some known facts for the subclass of equations of type (\ref{eqn:SDE_mvd}), which we will use later. For any real measurable function $h$, we introduce the sets \[ N_h := \{x \in \mathbb{R} : h(x) = 0 \} \] and \[ E_h := \{x \in \mathbb{R}: \int_U h^{-2}(y) \, \mbox{\upshape d} y = + \infty \text{ for all open sets } U \text{ containing } x\}\,. \] The following property of a solution of Eq. (\ref{eqn:SDE_mvd}) is proven in \cite{engelbert_schmidt:1989_III}, Proposition (4.34). For the special case of an equation of type (\ref{eqn:SDE_without_drift}) without drift, the corresponding statement can also be found in \cite{engelbert_schmidt:1989_III}, Proposition (4.14). \begin{lemma}\label{lemma:stopping} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvd}). Then we have $X_t = X_{t \wedge D_{E_b}^X}$, $t \geq 0$, $\mathbf{P}$-a.s., where $D_{E_b}^X$ denotes the first entry time of $X$ into the set $E_b$. \end{lemma} \indent The next result gives conditions on existence and uniqueness of solutions of Eq. (\ref{eqn:SDE_mvd}). For the proofs we refer to H.J. Engelbert and W. Schmidt \cite{engelbert_schmidt:1989_III}, Theorem (4.35) and (4.37) (see also \cite{engelbert_schmidt:1985}, Theorem 3 and 4). These statements were established by reducing Eq. (\ref{eqn:SDE_mvd}) to an equation of type (\ref{eqn:SDE_without_drift}) without drift. Hence, analogous statements for equations of type (\ref{eqn:SDE_without_drift}), which are verified in \cite{engelbert_schmidt:1989_III}, Theorem (4.17) and (4.22) (see also \cite{engelbert_schmidt:1985}, Theorem 1 and 2), could be used to derive the results for Eq. (\ref{eqn:SDE_mvd}). \begin{theorem}\label{theorem:e_u_gen_drift} (i) For every initial distribution there exists a solution of Eq. (\ref{eqn:SDE_mvd}) if and only if the condition $E_b \subseteq N_b$ for the diffusion coefficient $b$ is satisfied. (ii) For every initial distribution there exists a unique solution of Eq. (\ref{eqn:SDE_mvd}) if and only if the condition $E_b = N_b$ for the diffusion coefficient $b$ is satisfied. \end{theorem} \section{Properties of the Local Time $\text{\textit{L}}_{\text{\textit{\lowercase{m}}}}^{\text{\textit{X}}}$}\label{sec:prop_loc_time} \noindent In this part of the paper, for a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) we investigate the relation between the local time $L_m^X$ and the semimartingale local time $L_+^X$. In particular, we will see that $L_m^X$ inherits useful, well-known properties of $L_+^X$. \begin{lemma}\label{lemma:connection_loc_times} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvasd}). Then we have \[ 2\,f(y\pm)\,L^X_m(t,y\pm) = L_\pm^X(t,y), \qquad t < S_\infty^X,\ y\in \mathbb{R}, \ \mathbf{P}\text{-a.s.} \footnote{Note that $L_m^X(t,y+) = L^X_m(t,y), \ t < S_\infty^X, \ y\in\mathbb{R}, \ \mathbf{P}\text{-a.s.}$} \] \end{lemma} \begin{proof} Due to the occupation times formula (\ref{eqn:occupationtime}) and Definition \ref{def:solution}(iv) it holds \[ \int_{\mathbb{R}} h(y) \, L_m^X(t,y) \, 2 \, f(y)\,\mbox{\upshape d} y = \int_{\mathbb{R}} h(y) \, L_+^X(t,y) \, \mbox{\upshape d} y, \qquad t< S_\infty^X, \ \mathbf{P}\text{-a.s.} \] for every measurable function $h \geq 0$. This implies \[ 2 \, f(y) \, L_m^X(t,y) = L_+^X(t,y) \qquad \lambda\text{-a.e.}, \ t<S_\infty^X, \ \mathbf{P}\text{-a.s.} \] Using the continuity properties of the involved objects, we obtain the assertions of the lemma. \end{proof} \begin{remark}\label{remark:connection_mvd_mvasd} By means of Lemma \ref{lemma:connection_loc_times} we can conclude: If the reciprocal $1/f$ of the drift function $f$ is of locally bounded variation or, equivalently, if $F = \emptyset$, then Eq. (\ref{eqn:SDE_mvasd}) reduces to an equation of type (\ref{eqn:SDE_mvd}). Indeed, using $\nu(\mbox{\upshape d} y) := 1/2 \, f^{-1}(y) \, \mbox{\upshape d} f(y)$ to define a feasible drift measure $\nu$ which satisfies (\ref{eqn:cond_atoms}), it follows \[ \int_\mathbb{R} L_m^X(t,y)\, \mbox{\upshape d} f(y) = \int_\mathbb{R} L_+^X(t,y) \, \frac{1}{2 \, f(y)} \, \mbox{\upshape d} f(y) = \int_\mathbb{R} L_+^X(t,y)\, \nu(\mbox{\upshape d} y) \] for the drift part in Eq. (\ref{eqn:SDE_mvasd}). $\Diamond$ \end{remark} From Lemma \ref{lemma:connection_loc_times} we obtain the following corollaries. The first one is obvious. \begin{corollary}\label{corr:beingzero} For a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) it holds \[ L_\pm^X(t,y)= 0, \qquad t < S_\infty^X, \ y \in F_\pm, \ \mathbf{P}\text{-a.s.} \] \end{corollary} Corollary \ref{corr:beingzero} shows that we always have $L_+^X(t,y)=0$ if $y \in F_+$ and $L_-^X(t,y)=0$ if $y \in F_-$. For $L_m^X$, in general, this does not hold. In contrast, as seen from Lemma \ref{lemma:connection_loc_times}, $L_m^X$ gives a precise description of the asymptotic behaviour of $L_+^X(t,y) \, f^{-1}(y)$ in the zeros of $f$ and $f_-$. \begin{corollary}\label{corr:comp_interval} The local time $L_m^X$ of a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) satisfies \[ L_m^X(t,y\pm) = 0 \text{ on } \left\{y \notin \left[\min_{0 \leq s \leq t} X_s, \max_{0 \leq s \leq t} X_s\right]\right\}, \ t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] \end{corollary} \begin{proof} It is well-known that for the semimartingale local times we have \[ L_\pm^X(t,y) = 0 \text{ on } \left\{y \notin \left[\min_{0 \leq s \leq t} X_s, \max_{0 \leq s \leq t} X_s\right]\right\}, \ t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] From Lemma \ref{lemma:connection_loc_times} we see immediately \[ L_m^X(t,y \pm) = 0 \text{ on } \left\{y \notin \left[\min_{0 \leq s \leq t} X_s, \max_{0 \leq s \leq t} X_s\right] \cup F \right\}, \ t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] Since $F$ has Lebesgue measure zero, the continuity properties of $L_m^X$ imply the desired result. \end{proof} Based on Corollary \ref{corr:comp_interval}, we use the convention $L_m^X(t,\pm\infty ) := 0$, $t < S_\infty^X$. \\ \indent Next we prove that, similar to property (\ref{eqn:int_wrt_loc_time}) of the semimartingale local times, the measures $L_m^X(\mbox{\upshape d} t , y)$ and $L_m^X(\mbox{\upshape d} t, y-)$ do not charge the set $\{0 \leq t < S_\infty^X: X_t \neq y\}$. \begin{theorem}\label{theorem:supp_loc_time} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvasd}). Then, for every $y \in \mathbb{R}$, $L_m^X(\mbox{\upshape d} t,y\pm)$ is carried by $\{0 \leq t < S_\infty^X : X_t = y\}$ $\mathbf{P}$-a.s., i.e., \[ \int_0^t \text{\large{$\mathds{1}$}}_{\{y\}}(X_s) \, L_m^X(\mbox{\upshape d} s, y\pm) = L_m^X(t,y\pm), \qquad t < S_\infty^X, \ y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.} \] \end{theorem} \begin{proof} We prove the statement for $L_m^X$. The verification for the left-continuous version of $L_m^X$ can be done in an analogous way. Using property (\ref{eqn:int_wrt_loc_time}) of the semimartingale local time $L_+^X$ and Lemma \ref{lemma:connection_loc_times}, we conclude \begin{equation}\label{eqn:first_part} \begin{split} \int_0^t \text{\large{$\mathds{1}$}}_{\{y\}}(X_s) \, L_m^X(\mbox{\upshape d} s, y) &= \frac{1}{2f(y)} \, \int_0^t \text{\large{$\mathds{1}$}}_{\{y\}}(X_s) \, L_+^X(\mbox{\upshape d} s, y)\\ &= \frac{1}{2f(y)} \, L_+^X(t, y) \\ &= L_m^X(t, y), \qquad t < S_\infty^X, \ y \in \mathbb{R}\setminus F_+, \ \mathbf{P}\text{-a.s.} \end{split} \end{equation} Therefore, it remains to show that the asserted relation is also fulfilled for the points of the set $F_+$. Let $\Omega_0 \in \mathcal{F}$ be such that $\mathbf{P}(\Omega_0) = 1$, $X_0(\omega) \in \mathbb{R}$, $\omega \in \Omega_0$, and with the property that, for all $\omega \in \Omega_0$, $L_m^X(\,.\,,\,.\,)(\omega)$ satisfies \ref{def:solution}(iv) as well as (\ref{eqn:first_part}). Given $\omega \in \Omega_0$ and $a \in F_+$, we can, because $F$ has Lebesgue measure zero, choose a sequence $(y_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}\setminus F_+$ with $y_{n+1} \leq y_n$ and $\lim_{n \rightarrow +\infty} y_n = a$. Moreover, let $(S_i)_{i \in \mathbb{N}} \subseteq [0,+\infty)$ be an increasing sequence of real numbers with $S_i < S_\infty^X(\omega)$, $i \in \mathbb{N}$, and $\lim_{i \rightarrow +\infty} S_i = S_\infty^X(\omega)$. Then, for fixed $i \in \mathbb{N}$ \[ \int_B L_m^X(\mbox{\upshape d} s,a)(\omega) \quad \text{and} \quad \int_B L_m^X(\mbox{\upshape d} s,y_n)(\omega), \ n \in \mathbb{N}, \qquad B \in \mathscr{B}([0,S_i]), \] are finite measures on $([0,S_i],\mathscr{B}([0,S_i]))$. Since $L_m^X(t,\, . \,)(\omega)$ is right-continuous in $a$ for every $t \in [0,S_i]$, the sequence of measures \[ \left(\int_{\text{.}} L_m^X(\mbox{\upshape d} s, y_n)(\omega)\right)_{n \in \mathbb{N}} \] converges weakly to the measure $\int_{\text{.}} L_m^X(\mbox{\upshape d} s, a)(\omega)$. From the continuity of $X_{.}(\omega)$, we obtain that the sets \[ T_k := [0,S_i] \cap (X_{.}(\omega))^{-1}\textstyle\left((-\infty,a-\frac{1}{k})\cup (a+\frac{1}{k},+\infty)\right), \qquad k \in \mathbb{N}, \] are open in $[0,S_i]$. By the weak convergence of the considered measures, for every $k \in \mathbb{N}$ it follows \[ \int_{T_k} L_m^X(\mbox{\upshape d} s, a)(\omega) \leq \liminf\limits_{n \rightarrow +\infty} \int_{T_k} L_m^X(\mbox{\upshape d} s, y_n)(\omega)\,. \] Additionally, for every $k \in \mathbb{N}$ there exists an $n_0(k) \in \mathbb{N}$ such that $y_n \in (a-\frac{1}{k},a+\frac{1}{k})$, $n \geq n_0(k)$, and together with (\ref{eqn:first_part}) we conclude \[ \int_{T_k} L_m^X(\mbox{\upshape d} s, y_n)(\omega) = 0, \qquad n \geq n_0(k), \] and therefore \[ \int_{T_k} L_m^X(\mbox{\upshape d} s, a)(\omega) = 0, \qquad k \in \mathbb{N}. \] Via the continuity from below of the considered measure we get \[\begin{split} \int_{\{t \in [0,S_i]: X_t(\omega) \neq a\}} L_m^X(\mbox{\upshape d} s, a)(\omega) & = \int_{\bigcup\limits_{k=1}^{+\infty} T_k} L_m^X(\mbox{\upshape d} s, a)(\omega) \\ & = \lim_{k \rightarrow +\infty} \int_{T_k} L_m^X(\mbox{\upshape d} s, a)(\omega) \\ & = 0. \end{split}\] Considering the measure on $[0,S_\infty^X(\omega))$ and applying again the continuity from below, we finally observe \[\begin{split} \int_{\{t \in [0,S_\infty^X(\omega)): X_t(\omega) \neq a\}} L_m^X(\mbox{\upshape d} s, a)(\omega) & = \int_{\bigcup\limits_{i=1}^{+\infty} \{t \in [0,S_i]: X_t(\omega) \neq a\}} L_m^X(\mbox{\upshape d} s, a)(\omega) \\ & = \lim_{i \rightarrow +\infty} \int_{\{t \in [0,S_i]: X_t(\omega) \neq a\}} L_m^X(\mbox{\upshape d} s, a)(\omega) \\ & = 0. \end{split}\] Since $\omega \in \Omega_0$ and $a \in F_+$ were chosen arbitrarily, the proof is finished. \end{proof} The last theorem allows us to conclude that the local time $L_m^X$ is also $\mathbf{P}$-a.s. continuous in the state variable except in the points of $F_-$. \begin{corollary}\label{corollary:left_continuity} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvasd}). Then it holds \[ L_m^X(t,y) = L_m^X(t,y-), \qquad t < S_\infty^X ,\ y \in \mathbb{R}\setminus F_-, \ \mathbf{P}\text{-a.s.} \] \end{corollary} \begin{proof} Using property (\ref{eqn:loc_time_and_variation_process}) and Theorem \ref{theorem:supp_loc_time}, we conclude \[ \begin{split} L_+^X(t,y) - L_-^X(t,y) &= 2 \int_0^t \text{\large{$\mathds{1}$}}_{\{y\}}(X_s) \, \int_{\mathbb{R}} L_m^X(\mbox{\upshape d} s,z) \, \mbox{\upshape d} f(z) \\ &= 2 \int_{\mathbb{R}} \int_0^t \text{\large{$\mathds{1}$}}_{\{y\}}(X_s) \, L_m^X(\mbox{\upshape d} s,z) \, \mbox{\upshape d} f(z) \\ &= 2 \int_{\{y\}} L_m^X(t,z) \, \mbox{\upshape d} f(z) \\ &= 2 \,L_m^X(t,y) \,(f(y) - f(y-)), \qquad t < S_\infty^X,\ y \in \mathbb{R},\ \mathbf{P}\text{-a.s.} \end{split} \] Together with Lemma \ref{lemma:connection_loc_times} this yields \[ \begin{split} L_+^X(t,y) - L_-^X(t,y) &= L_+^X(t,y) - 2 \,L_m^X(t,y) \, f(y-), \qquad t < S_\infty^X,\ y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.}, \end{split} \] and finally \[ L_m^X(t,y) = \frac{1}{2f(y-)} \, L_-^X(t,y) = L_m^X(t,y-), \qquad t < S_\infty^X,\ y \in \mathbb{R}\setminus F_-, \ \mathbf{P}\text{-a.s.}, \] the desired result. \end{proof} The last lemma which we give in this section is used below to infer the existence of certain stochastic integrals. \begin{lemma}\label{lemma:integrability} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvasd}). Then \[ \int_{\mathbb{R}} \frac{1}{f(y)} \, L_m^X(t,y) \, \mbox{\upshape d} y < +\infty, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] and \[ \int_0^t \left(\frac{1}{f}(X_s)\right)^2 \, \mbox{\upshape d} \assPro{X}_s < +\infty, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] hold true. \end{lemma} \begin{proof} The occupation times formula for $L_m^X$ implies \[\begin{split} \int_0^t \left(\frac{1}{f}(X_s)\right)^2 \, \mbox{\upshape d} \assPro{X}_s &= \int_{\mathbb{R}} \left(\frac{1}{f(y)}\right)^2 \, L_m^X(t,y) \, 2 \, f(y) \, \mbox{\upshape d} y \\ &= 2\int_{\mathbb{R}} \frac{1}{f(y)} \, L_m^X(t,y) \, \mbox{\upshape d} y, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \end{split}\] Therefore, it is enough to show the first statement. Because of Corollary \ref{corr:comp_interval} we have \[ \int_{\mathbb{R}} \frac{1}{f(y)} \, L_m^X(t,y) \, \mbox{\upshape d} y = \int_{\left[\min\limits_{0\leq s \leq t} X_s, \max\limits_{0\leq s \leq t} X_s\right]} \frac{1}{f(y)} \, L_m^X(t,y) \, \mbox{\upshape d} y, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] Moreover, for $\mathbf{P}$-a.e. $\omega\in \Omega$ and $t < S_\infty^X(\omega)$ we can find a constant $K_t(\omega) > 0$ such that $L_m^X(t, \, . \,)(\omega)$ is bounded by $K_t(\omega)$ on $\left[\min\limits_{0\leq s \leq t} X_s(\omega), \max\limits_{0\leq s \leq t} X_s(\omega)\right]$. For $\mathbf{P}$-a.e. $\omega \in \Omega$ and $t < S_\infty^X(\omega)$ it follows \[\begin{split} \int_{\mathbb{R}} \frac{1}{f(y)} \, L_m^X(t,y)(\omega) \, \mbox{\upshape d} y & \leq K_t(\omega) \int_{\left[\min\limits_{0\leq s \leq t}X_s(\omega), \max\limits_{0\leq s \leq t} X_s(\omega)\right]} \frac{1}{f(y)} \, \mbox{\upshape d} y < +\infty, \end{split}\] where we used the fact that $1/f$ is locally integrable. \end{proof} \section{Space Transformation}\label{sec:space_transform} \noindent As pointed out in the introduction equations of type (\ref{eqn:SDE_mvd}) with generalized drift are contained as a special case in the class of equations (\ref{eqn:SDE_mvasd}) with generalized and singular drift. Using a certain space transformation, the so-called Zvonkin transformation (see \cite{zvonkin}), it is well-known that Eq. (\ref{eqn:SDE_mvd}) can be reduced to an equation (\ref{eqn:SDE_without_drift}) without drift. Based on the generalized It\^o formula, this method has been used by many authors (see e.g. \cite{bass_chen}, \cite{engelbert_schmidt:1985}, \cite{engelbert_schmidt:1989_III}, \cite{LeGall_1984}, \cite{stroock_yor}) to study Eq. (\ref{eqn:SDE_mvd}). They were able to derive conditions on existence and uniqueness of solutions of Eq. (\ref{eqn:SDE_mvd}) from the well-known criteria on existence and uniqueness of solutions of Eq. (\ref{eqn:SDE_without_drift}). \\ \indent For the treatment of Eq. (\ref{eqn:SDE_mvasd}) we want to use a similar approach. A natural candidate for an appropriate transformation of Eq. (\ref{eqn:SDE_mvasd}) is the strictly increasing and continuous primitive \[ G(x) := \int_0^x \frac{1}{f(y)} \, \mbox{\upshape d} y, \qquad x \in \overline{\mathbb{R}}, \] of the locally integrable reciprocal $1/f$ of the drift function $f$. By $H$ we denote the inverse of $G$ given on $G(\mathbb{R})=(G(-\infty),G(+\infty))$. We extend the functions $H$, $f$, $1/f$ and $b$ by setting \[ H(x) := \left\{\begin{array}{ll} + \infty, & x \in [G(+\infty),+\infty], \\ - \infty, & x \in [-\infty,G(-\infty)], \end{array}\right. \] \[ \frac{1}{f(\pm\infty)} = b(\pm\infty) := 0 \qquad \text{and} \qquad f(\pm \infty) := +\infty\,. \] Clearly, $H$ satisfies \begin{equation}\label{eqn:representation_H} H(x) = \int_0^x (f \circ H)(y) \, \mbox{\upshape d} y, \qquad x \in \mathbb{R}\,. \end{equation} The open set $\mathbb{R}\setminus F$ can be uniquely decomposed into at most countably many open intervals, i.e., \begin{equation}\label{eqn:components} \mathbb{R}\setminus F = \bigcup_{i=0}^{\|F\|} \, (a_i,b_i)\,, \quad \text{where } a_i, b_i \in F \cup \{-\infty,+\infty\} \end{equation} and\footnote{$\mathbb{N}_0 = \{0,1,2,\ldots\}$} $\|F\| \in \mathbb{N}_0 \cup \{+\infty\}$ denotes the number of elements in $F$. Note that $\|F\| + 1$ is just the number of intervals $(a_i,b_i)$ in the representation (\ref{eqn:components}). Already in Corollary \ref{corollary:left_continuity} it turned out that the subset $F_-$ of $F$ takes up a special role. We remark that $F$ is countable if and only if $F_-$ is countable. Indeed, from (\ref{eqn:components}) the reader can see \[ \left\{x \in \mathbb{R}: \, (x,x+\varepsilon) \cap F\neq \emptyset, \, (x-\varepsilon,x) \cap F \neq \emptyset \ \forall\,\varepsilon > 0\right\} = F\setminus \bigcup_{i=0}^{\|F\|} \{a_i, b_i\} \subseteq F_- \subseteq F, \] where on the left-hand side stands the set of all points which are accumulation points from the left and from the right in $F$. Hence, we can conclude. \\ \indent For a continuous $(\overline{\mathbb{R}}, \mathscr{B}(\overline{\mathbb{R}}))$-valued process $(Y,\mathbb{F})$ we introduce the $\mathbb{F}$-stopping time \[ S^Y_{G(\mathbb{R})} := \inf\{t \geq 0: Y_t = G(+\infty) \text{ or } Y_t = G(-\infty)\}\,, \] Now, in full generality, we can give the following structure of a transformed solution $Y = G(X)$. \begin{theorem}\label{theorem:general_structure_space_transformation} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvasd}) with generalized and singular drift. Then the process $(Y,\mathbb{F})$ defined by $Y =G(X)$ is continuous with values in $(\overline{\mathbb{R}}, \mathscr{B}(\overline{\mathbb{R}}))$ and stopped when it leaves the open interval $G(\mathbb{R})$, i.e., $Y_t = Y_{t \wedge S^Y_{G(\mathbb{R})}}$, $t \geq 0$, $\mathbf{P}$-a.s. Moreover, setting $\sigma := (b/f)\circ H$, then $Y$ satisfies \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s + \sum_{i = 0}^{\|F\|} \left(L_m^X(t,a_i) - L_m^X(t,b_i-) \right), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.}, \] where in case $\|F\| = +\infty$ the process \[ \left(\left(\displaystyle\sum_{i = 0}^{+\infty} \left(L_m^X(t,a_i) - L_m^X(t,b_i-) \right)\right)_{t \geq 0},\, \mathbb{F}\right) \] fulfils the relation \[ \lim_{n \rightarrow +\infty} \sup_{0 \leq t \leq T} \left\| \sum_{i=0}^n \left( L_m^X(t,a_i) - L_m^X(t,b_i) \right) - \sum_{i = 0}^{+\infty} \left(L_m^X(t,a_i) - L_m^X(t,b_i-) \right)\right\| = 0 \] in probability on $\{T < S^Y_{G(\mathbb{R})}\}$ for all $T \geq 0$. \end{theorem} \begin{remark}\label{remark:diffcoeff_infty} We point out that the statement of Theorem \ref{theorem:general_structure_space_transformation} includes the existence of the stochastic integral appearing in the decomposition of $Y$, i.e., the property \[ \int_0^t \sigma^2(Y_s) \, \mbox{\upshape d} s < +\infty, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \] In particular, because $\|\sigma(x)\| = +\infty$, $x \in G(F_+ \cap \{b \neq 0\})$,\footnote{For $A \subseteq \mathbb{R}$, $G(A)$ denotes the image of the set $A$ under $G$.} from this follows that \begin{equation}\label{eqn:no_occ_time} \int_0^t \text{\large{$\mathds{1}$}}_{G(F_+ \cap \{b \neq 0\})} (Y_s) \, \mbox{\upshape d} s = 0, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.}, \end{equation} i.e., $Y$ has no occupation time in $G(F_+ \cap \{b \neq 0\})$ $\mathbf{P}$-a.s. This can also be derived immediately from the fact that any solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) has no occupation time in $F_+ \cap \{b \neq 0\}$: \[\begin{split} \int_0^t \text{\large{$\mathds{1}$}}_{F_+ \cap \{b \neq 0\}} (X_s) \, \mbox{\upshape d} s & = \int_0^t \text{\large{$\mathds{1}$}}_{F_+ \cap \{b \neq 0\}} (X_s) \, b^{-2}(X_s) \, \mbox{\upshape d} \assPro{X}_s \\ & = \int_{F_+ \cap \{b \neq 0\}} \, L_+^X(t,x) \, b^{-2}(x) \, \mbox{\upshape d} x \\ & = 0, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.}\,, \end{split}\] where we used the occupation times formula (\ref{eqn:occupationtime}) and the fact that $F_+$ is of Lebesgue measure zero. Clearly, this also implies (\ref{eqn:no_occ_time}). It might be uncommon to consider SDEs with diffusion coefficient $\sigma$ taking also infinite values. However, because of (\ref{eqn:no_occ_time}) we can alter $\sigma$ on $G(F_+ \cap \{b \neq 0\})$ by setting, e.g., \[ \widetilde{\sigma} = \sigma \, \text{\large{$\mathds{1}$}}_{\mathbb{R}\setminus G(F_+ \cap \{b \neq 0\})} + \text{\large{$\mathds{1}$}}_{G(F_+ \cap \{b \neq 0\})} \] and replace $\sigma$ by $\widetilde{\sigma}$ without changing the statement of Theorem \ref{theorem:general_structure_space_transformation}. In the following we still use $\sigma = (b/f)\circ H$ but keep in mind that, always when it is necessary, we can use an appropriate real-valued coefficient. In this context we also refer to Remark \ref{remark:occupation_time_in_zero}. $\Diamond$ \end{remark} \begin{proof}[Proof of Theorem \ref{theorem:general_structure_space_transformation}] Since $(X,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_mvasd}), the process $(Y,\mathbb{F})$ is obviously continuous with values in $(\overline{\mathbb{R}}, \mathscr{B}(\overline{\mathbb{R}}))$ such that $Y_0 \in G(\mathbb{R})$ $\mathbf{P}$-a.s. Clearly, \[ S_\infty^X = \inf\{t \geq 0: X_t \notin \mathbb{R}\} = \inf\{t \geq 0: Y_t \notin G(\mathbb{R})\} = S^Y_{G(\mathbb{R})} \] and from Definition \ref{def:solution}(ii) we obtain immediately \[ Y_t = G(X_t) = G(X_{t \wedge S_\infty^X}) = G(X_{t \wedge S^Y_{G(\mathbb{R})}}) = Y_{t \wedge S^Y_{G(\mathbb{R})}}, \qquad t \geq 0, \ \mathbf{P}\text{-a.s.} \] To show the claimed structure of $Y$, one is tempted to apply the generalized It\^o formula (\ref{eqn:gen_ito_formula}), but $G$ restricted to $\mathbb{R}$ is, in general, not the difference of convex functions which is caused by the potentially non-empty set $F$. To overcome this difficulty, we use the decomposition (\ref{eqn:components}) of the open set $\mathbb{R}\setminus F$. For every $i \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0$ we choose two sequences $(p_k^i)_{k \in \mathbb{N}}$ and $(q_k^i)_{k \in \mathbb{N}}$ such that \[ a_i < p_{k+1}^i < p_k^i < q_k^i < q_{k+1}^i < b_{i}, \ k \in\mathbb{N}, \qquad \lim_{k \rightarrow +\infty} p_k^i = a_i \qquad \text{and} \qquad \lim_{k \rightarrow +\infty} q_k^{i} = b_{i} \] are satisfied and define \[ g_{n,k} := \frac{1}{f} \, \text{\large{$\mathds{1}$}}_{\bigcup\limits_{i=0}^n \left[p_k^i, q_k^i \right)} = \sum\limits_{i=0}^n \frac{1}{f} \, \text{\large{$\mathds{1}$}}_{\left[p_k^i\, , \, q_k^i\right)}, \qquad k \in \mathbb{N}, \ n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0\,, \] as well as \[ g_n := \frac{1}{f} \, \text{\large{$\mathds{1}$}}_{\bigcup\limits_{i=0}^n \left(a_i, b_i \right)}, \qquad n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0\,. \] Moreover, we introduce the increasing and continuous functions \[ G_{n,k}(x) := \int_0^x g_{n,k} (y) \, \mbox{\upshape d} y, \qquad x \in \overline{\mathbb{R}}, \ k \in \mathbb{N}, \ n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0\,, \] and \[ G_n(x) := \int_0^x g_n (y) \, \mbox{\upshape d} y, \qquad x \in \overline{\mathbb{R}}, \ n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0\,. \] Clearly, the non-negative right-continuous functions $g_{n,k}$, $k \in \mathbb{N}$, $n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0$, are of locally bounded variation, since every summand in the finite sum of the definition of $g_{n,k}$ is of locally bounded variation. Therefore, for arbitrary $n,k \in \mathbb{N}$ we conclude that $G_{n,k}$ restricted to $\mathbb{R}$ is the difference of convex functions. Applying the generalized It\^o formula (\ref{eqn:gen_ito_formula}), we deduce \begin{equation}\label{eqn:zwischenrechnung78} \begin{split} G_{n,k}(X_t)&= G_{n,k}(X_0) + \int_0^t g_{n,k}(X_s -) \, \mbox{\upshape d} X_s + \frac{1}{2} \int_\mathbb{R} L_+^X(t,y) \, \mbox{\upshape d} g_{n,k}(y) \\ &= G_{n,k}(X_0) + \int_0^t g_{n,k}(X_s)\, b(X_s) \, \mbox{\upshape d} B_s + \int_0^t g_{n,k}(X_s-) \int_\mathbb{R} L_m^X(\mbox{\upshape d} s,y)\,\mbox{\upshape d} f(y) \\ &\phantom{=======} + \frac{1}{2} \int_\mathbb{R} L_+^X(t,y) \, \mbox{\upshape d} g_{n,k}(y), \end{split} \end{equation} $t < S^Y_{G(\mathbb{R})}, \ k \in \mathbb{N}, \ n \in \{0,\ldots,\|F\|\} \cap \mathbb{N}_0, \ \mathbf{P}\text{-a.s.}$ For the third summand in this decomposition we obtain \[\begin{split} \int_0^t g_{n,k}(X_s-) \, \int_{\mathbb{R}} L_m^X(\mbox{\upshape d} s,y) \, \mbox{\upshape d} f(y) &= \int_{\mathbb{R}} \int_0^t g_{n,k}(X_s-) \, L_m^X(\mbox{\upshape d} s,y) \, \mbox{\upshape d} f(y) \\ &= \int_{\mathbb{R}} g_{n,k}(y-) \, L_m^X(t,y) \, \mbox{\upshape d} f(y), \end{split}\] $t < S^Y_{G(\mathbb{R})}, \ k \in \mathbb{N}, \ n \in \{0,\ldots,\|F\|\} \cap \mathbb{N}_0, \ \mathbf{P}\text{-a.s.}$, where we used Theorem \ref{theorem:supp_loc_time}. Moreover, integration by parts gives \[\begin{split} \int_{\mathbb{R}} g_{n,k}(y-) \, L_m^X(t,y) \, \mbox{\upshape d} f(y) &= \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} (g_{n,k} f)(y) - \int_{\mathbb{R}} L_m^X(t,y) \, f(y) \, \mbox{\upshape d} g_{n,k}(y) \\ &= \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} (g_{n,k} f)(y) - \frac{1}{2}\int_{\mathbb{R}} L_+^X(t,y) \, \mbox{\upshape d} g_{n,k}(y), \end{split}\] $t < S^Y_{G(\mathbb{R})}, \ k \in \mathbb{N}, \ n \in \{0,\ldots,\|F\|\} \cap \mathbb{N}_0, \ \mathbf{P}\text{-a.s.}$, where in the last step we have applied Lemma \ref{lemma:connection_loc_times}. Finally, (\ref{eqn:zwischenrechnung78}) becomes \[ G_{n,k}(X_t) = G_{n,k}(X_0) + \int_0^t g_{n,k}(X_s)\, b(X_s) \, \mbox{\upshape d} B_s + \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} (g_{n,k} f)(y), \] $t < S^Y_{G(\mathbb{R})}, \ k \in \mathbb{N}, \ n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0, \ \mathbf{P}\text{-a.s.}$ The integrator in the last term on the right-hand side has the following structure \[ g_{n,k}f = \text{\large{$\mathds{1}$}}_{\bigcup\limits_{i=0}^n \left[p_k^i,q_k^i \right)} = \sum_{i=0}^n \text{\large{$\mathds{1}$}}_{\left[p_k^i,q_k^i \right)}. \] Hence, calculating this integral, we obtain \begin{equation}\label{eqn:approx_G_nk} G_{n,k}(X_t) = G_{n,k}(X_0) + \int_0^t g_{n,k}(X_s)\, b(X_s) \, \mbox{\upshape d} B_s + \sum_{i=0}^n \left(L_m^X(t,p_k^i) - L_m^X(t,q_k^i) \right), \end{equation} $t < S^Y_{G(\mathbb{R})}, \ k \in \mathbb{N}, \ n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0, \ \mathbf{P}\text{-a.s.}$ \\ \indent Now, for arbitrary $n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0$ we pass to the limit $k \rightarrow +\infty$. We observe \[ g_{n,k} \leq g_{n,k+1} \leq \frac{1}{f}, \qquad k \in \mathbb{N}, \qquad \text{and} \qquad \lim_{k \rightarrow +\infty} g_{n,k} = g_n \qquad \lambda\text{-a.e.}\footnote{$\lambda$ denotes the Lebesgue measure.} \] Together with Lemma \ref{lemma:integrability} this implies that the conditions of Theorem \ref{theorem:convergence} are fulfilled. For every $t \geq 0$ we obtain in probability \[ \lim_{k \rightarrow +\infty} \int_0^t g_{n,k}(X_s) \, b(X_s) \, \mbox{\upshape d} B_s = \int_0^t g_{n}(X_s) \, b(X_s) \, \mbox{\upshape d} B_s \qquad \text{on } \{t < S^Y_{G(\mathbb{R})}\}. \] Since clearly \[ \lim_{k \rightarrow +\infty} G_{n,k}(x) = G_n(x), \qquad x \in \mathbb{R}, \] and because of the continuity properties of $L_m^X$, for every $t \geq 0$, in addition we get in probability \[\begin{split} & \lim_{k \rightarrow +\infty} \left(G_{n,k}(X_t) - G_{n,k}(X_0) - \sum_{i=0}^n \left(L_m^X(t,p_k^i) - L_m^X(t,q_k^i)\right)\right) \\ & \phantom{======} = G_{n}(X_t) - G_{n}(X_0) - \sum_{i=0}^n \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right) \qquad \text{on } \{t < S^Y_{G(\mathbb{R})}\}. \end{split}\] Finally, passing to the limit $k \rightarrow +\infty$ in (\ref{eqn:approx_G_nk}), for every $t \geq 0$ we conclude \[ G_{n}(X_t) = G_{n}(X_0) + \int_0^t g_{n}(X_s) \, b(X_s) \, \mbox{\upshape d} B_s + \sum_{i=0}^n \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right) \] $\mathbf{P}$-a.s. on $\{t < S^Y_{G(\mathbb{R})}\}$. From the continuity in $t$ of the involved processes and by the arbitrariness of $n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0$ it follows \begin{equation}\label{eqn:limit_k} G_{n}(X_t) = G_{n}(X_0) + \int_0^t g_{n}(X_s) \, b(X_s) \, \mbox{\upshape d} B_s + \sum_{i=0}^n \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right), \end{equation} $t < S^Y_{G(\mathbb{R})}$, $n \in \{0, \ldots, \|F\|\} \cap \mathbb{N}_0$, $\mathbf{P}\text{-a.s.}$ \\ \indent In case of a finite set $F$, i.e., $\|F\| \in \mathbb{N}_0$, the claim of the theorem is proven. We only need to choose $n=\|F\|$. Then, since $g_{\|F\|} = \frac{1}{f}$ $\lambda$-a.e. and $G_{\|F\|} = G$, from (\ref{eqn:limit_k}) we conclude the desired form \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s + \sum_{i = 0}^{\|F\|} \left(L_m^X(t,a_i) - L_m^X(t,b_i-) \right), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \] \indent In case $\|F\| = +\infty$, now we pass to the limit $n \rightarrow +\infty$ in (\ref{eqn:limit_k}). By the definition of the functions $g_n$, $n \in \mathbb{N}_0$, we have \[ g_n \leq g_{n+1} \leq \frac{1}{f}, \quad n \in \mathbb{N}_0, \qquad \text{and} \qquad \lim_{n \rightarrow +\infty} g_n = \frac{1}{f} \qquad \lambda\text{-a.e.} \] Again, together with Lemma \ref{lemma:integrability} the assumptions of Theorem \ref{theorem:convergence} are fulfilled. For every $T \geq 0$ we derive \[ \lim_{n \rightarrow +\infty} \sup_{0 \leq t \leq T} \left\| \int_0^t g_n(X_s)\, b(X_s) \, \mbox{\upshape d} B_s - \int_0^t \frac{b}{f}(X_s) \, \mbox{\upshape d} B_s\right\| = 0 \] in probability on $\{T < S^Y_{G(\mathbb{R})}\}$. Since for all $T \geq 0$ \[ \lim_{n \rightarrow +\infty} \sup_{x\in[-K,K]} \|G_n(x) - G(x)\| = 0, \qquad K \in\mathbb{N}\,, \] holds true, from (\ref{eqn:limit_k}) it follows \[ \lim_{n \rightarrow +\infty} \sup_{0 \leq t \leq T} \left\| \sum_{i=0}^n \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right) - \left(G(X_t) - G(X_0) - \int_0^t \frac{b}{f}(X_s)\, \mbox{\upshape d} B_s\right) \right\| = 0 \] in probability on $\{T < S^Y_{G(\mathbb{R})}\}$. Defining the $\mathbb{F}$-adapted process \[ \sum_{i = 0}^{+\infty} \left(L_m^X(t,a_i) - L_m^X(t,b_i-) \right) := G(X_t) - G(X_0) - \int_0^t \frac{b}{f}(X_s)\, \mbox{\upshape d} B_s, \qquad t \geq 0\,, \] we can conclude \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s + \sum_{i = 0}^{+\infty} \left(L_m^X(t,a_i) - L_m^X(t,b_i-) \right), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.}, \] and the claim is also proven in case $\|F\| = +\infty$. \end{proof} Note that in case of Eq. (\ref{eqn:SDE_mvd}) expressed as an equation of type (\ref{eqn:SDE_mvasd}) the drift function is $f_\nu$ as introduced before (\ref{eqn:drift_measure}) and the corresponding set $F$ is empty. Hence, Theorem \ref{theorem:general_structure_space_transformation} contains the result (see e.g. \cite{engelbert_schmidt:1985}, Proposition 1) that Eq. (\ref{eqn:SDE_mvd}) can be transformed to an equation (\ref{eqn:SDE_without_drift}) without drift. However, the special feature of Eq. (\ref{eqn:SDE_mvasd}) is that, entailed by the set of singularities $F$, after applying the space transformation $G$ it cannot be excluded that there remains a drift term. \\ \indent As announced in the introduction, in our investigation we want to stay in the framework of continuous semimartingales. In the semimartingale case we want to concretise the remaining drift appearing in the decomposition of $Y = G(X)$. We point out in Theorem \ref{theorem:a_prioiri_semimart}, if $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a continuous semimartingale up to $S^Y_{G(\mathbb{R})}$ then the decomposition of $Y$ has the form \begin{equation}\label{eqn:SDE_transformed_eqn} Y_t = Y_0 + \int_0^t\sigma(Y_s) \, \mbox{\upshape d} B_s + \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s)\, \mbox{\upshape d} Y_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{equation} Consequently, under the a priori knowledge that $Y$ is a semimartingale up to $S^Y_{G(\mathbb{R})}$, in general, there remains a drift part that lives on the times when the process $Y$ is in the set $G(F_-)$. Note, if $(Y,\mathbb{F})$ is a continuous semimartingale up to $S^Y_{G(\mathbb{R})}$, then the term $\int_0^\cdot \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s)\, \mbox{\upshape d} Y_s$ in (\ref{eqn:SDE_transformed_eqn}) is actually a drift, i.e., of locally bounded variation. Indeed, let \begin{equation}\label{eqn:semimartdecomp_of_Y} Y_t = Y_0 + M_t + V_t, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{equation} be the unique continuous semimartingale decomposition (see (\ref{eqn:semi_decomposition})) of $Y$. Since $F_-$ and therefore $G(F_-)$ are of Lebesgue measure zero, it follows immediately \begin{equation}\label{eqn:drift_int_with_respect_to_process} \begin{split} \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s & = \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} M_s + \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} V_s \\ & = \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} V_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{split} \end{equation} Furthermore, taking (\ref{eqn:SDE_transformed_eqn}) into account, by the uniqueness of the continuous semimartingale decomposition of $Y$ we conclude \[ V_t = \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \] \indent To handle the semimartingale case, we consider (\ref{eqn:SDE_transformed_eqn}) as a self-contained equation and fix the following definition of a solution. \begin{defi}\label{def:sol_trans_eqn} A continuous $(\overline{\mathbb{R}}, \mathscr{B}(\overline{\mathbb{R}}))$-valued stochastic process $(Y,\mathbb{F})$ defined on a probability space $(\Omega,\mathcal{F},\mathbf{P})$ is called a solution (up to the first exit from the open interval $G(\mathbb{R})$) of Eq. (\ref{eqn:SDE_transformed_eqn}) if the following conditions are fulfilled: (i) $Y_0 \in G(\mathbb{R})$. (ii) $Y_t = Y_{t \wedge S^Y_{G(\mathbb{R})}}$, $t \geq 0$, $\mathbf{P}$-a.s. (iii) $(Y,\mathbb{F})$ is a semimartingale up to $S^Y_{G(\mathbb{R})}$. (iv) There exists a Wiener process $(B,\mathbb{F})$ such that Eq. (\ref{eqn:SDE_transformed_eqn}) is satisfied for all $t < S^Y_{G(\mathbb{R})}$ $\mathbf{P}$-a.s. \end{defi} Before we characterize solutions $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ of Eq. (\ref{eqn:SDE_transformed_eqn}) by its semimartingale property, we show the following \begin{lemma}\label{lemma:relation_loc_times} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvasd}). If the transformed process $(Y,\mathbb{F}) = (G(X), \mathbb{F})$ is a continuous semimartingale up to $S^Y_{G(\mathbb{R})}$, then we have \[ L_m^X(t,x\pm) = \frac{1}{2} \, L^Y_\pm(t,G(x)), \qquad t < S^Y_{G(\mathbb{R})}, \ x \in \mathbb{R}, \ \mathbf{P}\text{-a.s.} \] \end{lemma} \begin{proof} Obviously, the requirements of Lemma \ref{lemma:conversion_loc_time} are satisfied and together with Lemma \ref{lemma:connection_loc_times} we can conclude \[ L_\pm^Y(t, G(x)) = L_\pm^X(t,x) \, \frac{1}{f(x\pm)} = 2 \, L_m^X(t,x\pm), \qquad t < S^Y_{G(\mathbb{R})}, \ x\in\mathbb{R}\setminus F, \ \mathbf{P}\text{-a.s.} \] Finally, the continuity properties of $L_\pm^Y$ and $L_m^X$ and the fact that $F$ is of Lebesgue measure zero imply the claim. \end{proof} \begin{theorem}\label{theorem:a_prioiri_semimart} Let $(X,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_mvasd}). Then $(Y,\mathbb{F}) = (G(X), \mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}) if and only if $(Y,\mathbb{F})$ is a continuous semimartingale up to $S^Y_{G(\mathbb{R})}$. \end{theorem} \begin{proof} Clearly, the necessity of the stated condition holds because of Definition \ref{def:sol_trans_eqn}. \\ \indent Now we assume that $(Y,\mathbb{F}) = (G(X), \mathbb{F})$ is a continuous semimartingale up to $S^Y_{G(\mathbb{R})}$. Obviously, conditions (i) and (ii) of Definition \ref{def:sol_trans_eqn} are satisfied. To establish condition (iv) we work, contrary to the proof of Theorem \ref{theorem:general_structure_space_transformation}, with an alternative approximation of $1/f$. We define right-continuous functions $\widetilde{g}_n$, $n \in \mathbb{N}$, of locally finite variation by \[ \widetilde{g}_n := \frac{1}{f} \wedge n, \qquad n \in \mathbb{N}, \] and denote their primitives by \[ \widetilde{G}_n(x) := \int_0^x \widetilde{g}_n(y) \, \mbox{\upshape d} y, \qquad x \in \overline{\mathbb{R}}\,. \] Arguing in the same way as in the proof of Theorem \ref{theorem:general_structure_space_transformation} in (\ref{eqn:zwischenrechnung78}) and the following calculations, we obtain \begin{equation}\label{eqn:decomposition_tilde_G_n}\begin{split} \widetilde{G}_n(X_t) &= \widetilde{G}_n(X_0) + \int_0^t \widetilde{g}_n(X_s) \, b(X_s) \, \mbox{\upshape d} B_s \\ &\phantom{==========} + \int_0^t \widetilde{g}_n(X_s-) \int_\mathbb{R} L_m^X(\mbox{\upshape d} s,y) \, \mbox{\upshape d} f(y) + \frac{1}{2} \int_\mathbb{R} L_+^X(t,y) \, \mbox{\upshape d} \widetilde{g}_n(y) \\ &= \widetilde{G}_n(X_0) + \int_0^t \widetilde{g}_n(X_s) \, b(X_s) \, \mbox{\upshape d} B_s + \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} (\widetilde{g}_nf)(y), \end{split}\end{equation} $t < S^Y_{G(\mathbb{R})}, \ n\in \mathbb{N}, \ \mathbf{P}\text{-a.s.}$ The relations \[ \widetilde{g}_n \leq \widetilde{g}_{n+1} \leq \frac{1}{f}, \quad n \in \mathbb{N}, \qquad \text{and} \qquad \lim_{n \rightarrow +\infty} \widetilde{g}_n = \frac{1}{f}\,, \] as well as Lemma \ref{lemma:integrability} show that Theorem \ref{theorem:convergence} can be applied. Hence, for every $t \geq 0$ we conclude \[ \lim_{n \rightarrow +\infty} \int_0^t \widetilde{g}_n(X_s) \, b(X_s) \, \mbox{\upshape d} B_s = \int_0^t \frac{b}{f}(X_s) \, \mbox{\upshape d} B_s = \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s \] in probability on $\{t < S^Y_{G(\mathbb{R})}\}$. Moreover, it holds true \[ \lim_{n \rightarrow +\infty} \widetilde{G}_n(x) = G(x), \qquad x \in \overline{\mathbb{R}}. \] Thus, for every $t \geq 0$ from (\ref{eqn:decomposition_tilde_G_n}) we obtain \begin{equation}\label{eqn:limit_1} \lim_{n \rightarrow +\infty} \int_\mathbb{R} L_m^X(t,y)\, \mbox{\upshape d} (\widetilde{g}_n f)(y) = Y_t - Y_0 - \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s =: \Sigma_t \end{equation} in probability on $\{t < S^Y_{G(\mathbb{R})}\}$. Additionally, our assumption that $(Y,\mathbb{F})$ is a continuous semimartingale up to $S^Y_{G(\mathbb{R})}$ allows us to apply Lemma \ref{lemma:relation_loc_times}. Therefore, for every $n \in \mathbb{N}$ we get \[\begin{split} \int_\mathbb{R} L_m^X(t,y) \, \mbox{\upshape d} (\widetilde{g}_n f)(y) & = \frac{1}{2} \int_\mathbb{R} L_+^Y(t,G(y)) \, \mbox{\upshape d} (\widetilde{g}_n f)(y) \\ & = \frac{1}{2} \int_{G(\mathbb{R})} L_+^Y(t,y) \, \mbox{\upshape d} ((\widetilde{g}_n f)\circ H)(y), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{split}\] The functions $(\widetilde{g}_n f)\circ H$, $n \in \mathbb{N}$, appearing as integrators in the last expression are right-continuous and of locally bounded variation on $G(\mathbb{R})$. This implies that the continuous mappings $I_n : \overline{\mathbb{R}} \rightarrow \overline{\mathbb{R}}$, $n \in \mathbb{N}$, defined by \[ I_n(x) := \int_0^x ((\widetilde{g}_n f)\circ H)(y) \, \mbox{\upshape d} y, \qquad x \in \overline{\mathbb{R}}, \ n \in \mathbb{N}, \] can be expressed as the difference of convex functions on $G(\mathbb{R})$. Therefore, applying the generalized It\^o formula (\ref{eqn:gen_ito_formula}) and using (\ref{eqn:semimartdecomp_of_Y}), we conclude \begin{equation}\label{eqn:inter_step}\begin{split} & \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} (\widetilde{g}_n f)(y) \\ &\phantom{====}= \frac{1}{2} \int_{G(\mathbb{R})} L_+^Y(t,y) \, \mbox{\upshape d} ((\widetilde{g}_n f)\circ H)(y) \\ &\phantom{====}= I_n(Y_t) - I_n(Y_0) - \int_0^t ((\widetilde{g}_n f) \circ H)(Y_s -) \, \mbox{\upshape d} M_s - \int_0^t ((\widetilde{g}_n f) \circ H)(Y_s -) \, \mbox{\upshape d} V_s, \end{split}\end{equation} $t < S^Y_{G(\mathbb{R})}$, $\mathbf{P}$-a.s. Moreover, because of the relations \[ ((\widetilde{g}_nf)\circ H)(y-) \, \text{\large{$\mathds{1}$}}_{G(\mathbb{R})}(y) = \min\{1 , n \, (f \circ H)(y-) \} \, \text{\large{$\mathds{1}$}}_{G(\mathbb{R})}(y) \leq \text{\large{$\mathds{1}$}}_{G(\mathbb{R})}(y), \] $y \in \mathbb{R}, \ n \in \mathbb{N},$ and \[ \lim_{n \rightarrow +\infty} ((\widetilde{g}_nf)\circ H)(y-) \, \text{\large{$\mathds{1}$}}_{G(\mathbb{R})}(y) = \text{\large{$\mathds{1}$}}_{\left\{x \in G(\mathbb{R}) :\, (f \circ H)(x-) > 0\right\}}(y), \qquad y \in \mathbb{R}, \] as well as \[ \text{\large{$\mathds{1}$}}_{\{x \in G(\mathbb{R}) :\, (f \circ H)(x-) > 0\}} = \text{\large{$\mathds{1}$}}_{G(\mathbb{R})} \qquad \lambda\text{-a.e.}, \] where we used the fact that $F_-$ and hence $G(F_-)$ have Lebesgue measure zero, we can apply Theorem \ref{theorem:convergence} again. For every $t \geq 0$ we conclude \[ \lim_{n \rightarrow +\infty} \int_0^t ((\widetilde{g}_n f) \circ H)(Y_s -) \, \mbox{\upshape d} M_s = \int_0^t \text{\large{$\mathds{1}$}}_{G(\mathbb{R})} (Y_s) \, \mbox{\upshape d} M_s = M_t \] in probability on $\{t < S^Y_{G(\mathbb{R})}\}$. Since additionally it holds $\widetilde{g}_n f \leq \widetilde{g}_{n+1} f$, $n \in \mathbb{N}$, we have \[ \lim_{n \rightarrow +\infty} I_n(x) = \int_0^x \text{\large{$\mathds{1}$}}_{G(\mathbb{R})}(y)\, \mbox{\upshape d} y = x, \qquad x \in G(\mathbb{R}), \] and \[\begin{split} \lim_{n \rightarrow +\infty} \int_0^t ((\widetilde{g}_n f) \circ H)(Y_s-) \, \mbox{\upshape d} V_s & = \int_0^t \text{\large{$\mathds{1}$}}_{\{ x \in G(\mathbb{R}) : (f \circ H)(x-) > 0\}} (Y_s) \, \mbox{\upshape d} V_s \\ & = \int_0^t \text{\large{$\mathds{1}$}}_{G(\mathbb{R}) \setminus G(F_-)} (Y_s) \, \mbox{\upshape d} V_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{split}\] Summarizing these observations, from (\ref{eqn:inter_step}) we obtain \[\begin{split} &\lim_{n \rightarrow +\infty} \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} (\widetilde{g}_n f)(y) \\ &\phantom{=,}= \lim_{n \rightarrow +\infty} \left( I_n(Y_t) - I_n(Y_0) - \int_0^t ((\widetilde{g}_n f) \circ H)(Y_s -) \, \mbox{\upshape d} M_s - \int_0^t ((\widetilde{g}_n f) \circ H)(Y_s -) \, \mbox{\upshape d} V_s \right)\\ &\phantom{=,}= Y_t - Y_0 - M_t - \int_0^t \text{\large{$\mathds{1}$}}_{G(\mathbb{R}) \setminus G(F_-)} (Y_s) \, \mbox{\upshape d} V_s \\ &\phantom{=,}= \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)} (Y_s) \, \mbox{\upshape d} V_s \end{split}\] in probability on $\{t < S^Y_{G(\mathbb{R})}\}$. Comparing this result with (\ref{eqn:limit_1}) and using the continuity of the involved processes, we deduce \[ \Sigma_t = \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)} (Y_s) \, \mbox{\upshape d} V_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \] Therefore, $(\Sigma,\mathbb{F})$ is a continuous process of locally bounded variation on $[0,S^Y_{G(\mathbb{R})})$. From the uniqueness of the continuous semimartingale decomposition it follows \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s + \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)} (Y_s) \, \mbox{\upshape d} V_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.}, \] which we can rewrite (see (\ref{eqn:drift_int_with_respect_to_process})) as \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s + \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)} (Y_s) \, \mbox{\upshape d} Y_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.}, \] and the proof is finished. \end{proof} We now pass to the inverse $H$ of the space transformation $G$ which can be easily handled as we demonstrate in the following \begin{theorem}\label{theorem:spacetrans_H} Let $(Y,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_transformed_eqn}). Then $(X,\mathbb{F})$ given by $X = H(Y)$ is a solution to Eq. (\ref{eqn:SDE_mvasd}). \end{theorem} \begin{proof} Clearly, condition (i) and, since \[ S^Y_{G(\mathbb{R})} = \inf\{t \geq 0: Y_t \notin G(\mathbb{R})\} = \inf\{t \geq 0: X_t \notin \mathbb{R}\} = S_\infty^X\,, \] condition (ii) of Definition \ref{def:solution} are satisfied by $X$. From the representation (\ref{eqn:representation_H}) of $H$ we see that $H$ restricted to $G(\mathbb{R})$ is the difference of convex functions. Therefore, applying the generalized It\^o formula (\ref{eqn:gen_ito_formula}), we obtain that $(X,\mathbb{F})$ is a continuous semimartingale up to $S_\infty^X$ with decomposition \[\begin{split} X_t &= H(Y_0) + \int_0^t (f \circ H)(Y_s -) \, \mbox{\upshape d} Y_s + \frac{1}{2} \int_{G(\mathbb{R})} L_+^Y(t,y) \, \mbox{\upshape d} (f \circ H)(y)\\ &= H(Y_0) + \int_0^t (f \circ H)(Y_s -) \, \sigma(Y_s) \, \mbox{\upshape d} B_s \\ &\phantom{=====} + \int_0^t (f \circ H)(Y_s -) \, \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s + \frac{1}{2} \int_{\mathbb{R}} L_+^Y(t,G(y)) \, \mbox{\upshape d} f(y), \qquad t < S_\infty^X, \ \mathbf{P}\text{-f.s.} \end{split}\] Using $(f \circ H)(y -) \, \text{\large{$\mathds{1}$}}_{G(F_-)}(y) = 0$, $y \in \mathbb{R}$, we see that the third summand on the right-hand side vanishes. For treating the second summand on the right-hand side, we remark that $f$ has at most countably many discontinuities. Recalling the definition of $\sigma$, we obtain \[\begin{split} \int_0^t (f \circ H)(Y_s-)\, \sigma(Y_s) \, \mbox{\upshape d} B_s &= \int_0^t (f \circ H)(Y_s)\, \sigma(Y_s) \, \mbox{\upshape d} B_s \\ &= \int_0^t (f \circ H)(Y_s)\, (b \circ H)(Y_s) \, (f \circ H)^{-1}(Y_s) \, \mbox{\upshape d} B_s \\ &= \int_0^t (b \circ H)(Y_s) \, \mbox{\upshape d} B_s, \qquad t < S_\infty^X, \ \mathbf{P}\text{-f.s.}, \end{split}\] where in the last step we used additionally that $G(F_+)$ is of Lebesgue measure zero. Hence, setting \[ L_m^X(t,y) := \frac{1}{2}\, L_+^Y(t,G(y)), \qquad (t,y) \in [0,S_\infty^X) \times \mathbb{R}, \] we can write \[ X_t = X_0 + \int_0^t b(X_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_m^X(t,y) \, \mbox{\upshape d} f(y), \qquad t < S_\infty^X,\ \mathbf{P}\text{-a.s.} \] Since $(X,\mathbb{F})$ satisfies the conditions of Lemma \ref{lemma:conversion_loc_time}, we obtain \[ L_+^X(t,y) = L_+^Y(t,G(y))\,f(y) = L_m^X(t,y) \, 2f(y), \qquad t < S_\infty^X, \ y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.} \] Using the occupation times formula (\ref{eqn:occupationtime}), for every non-negative measurable function $h$ it follows \[\begin{split} \int_0^t h(X_s) \, \mbox{\upshape d} \assPro{X}_s &= \int_\mathbb{R} h(y) \, L_+^X(t,y) \, \mbox{\upshape d} y \\ &= \int_\mathbb{R} h(y) \, L_m^X(t,y) \, m(\mbox{\upshape d} y), \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \end{split}\] Clearly, $L_m^X$ also fulfils the desired continuity properties of Definition \ref{def:solution}(iv). \end{proof} \begin{remark}\label{remark:occupation_time_in_zero} Replacing $\sigma$ by the real-valued coefficient $\widetilde{\sigma}$ defined in Remark \ref{remark:diffcoeff_infty} does not change Eq. (\ref{eqn:SDE_transformed_eqn}). This follows from the fact that any solution $(Y,\mathbb{F})$ of Eq. (\ref{eqn:SDE_transformed_eqn}) (for $\sigma$ as well as for $\widetilde{\sigma}$) has no occupation time in $G(F_+ \cap \{b \neq 0\})$, which can be verified by the occupation times formula (cf. proof of (\ref{eqn:no_occ_time})). For this it is important that, as $\sigma$, the coefficient $\widetilde{\sigma}$ is different from zero on $G(F_+ \cap \{b \neq 0\})$. Otherwise, there would be also allowed solutions which are \emph{sticky} in $G(F_+ \cap \{b \neq 0\})$, i.e., solutions with a strictly positive occupation time in this set. However, in this paper we will not deal with sticky solutions. $\Diamond$ \end{remark} Motivated by our intention to keep within the class of semimartingales, we now introduce the notion of a \emph{good} solution of Eq. (\ref{eqn:SDE_mvasd}). \begin{defi}\label{defi:good_solution} We say that a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) is \emph{good} if $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a continuous semimartingale up to $S^Y_{G(\mathbb{R})}$. \end{defi} \noindent Now the question arises: Under which conditions is $(X,\mathbb{F})$ a good solution? As a first result we stress the role of the process $\left( \bigl(\sum_{i = 0}^{\|F\|} (L_m^X(t,a_i) - L_m^X(t,b_i-) )\bigr)_{t \geq 0}, \mathbb{F} \right)$ which appears in Theorem \ref{theorem:general_structure_space_transformation} in the decomposition of $(Y,\mathbb{F})$. The following characterization of a good solution is just a reformulation of Theorem \ref{theorem:a_prioiri_semimart}. Indeed, we only need to follow the proof of Theorem \ref{theorem:a_prioiri_semimart}. \begin{theorem}\label{theorem:a_prioiri_semimart_alt_formulation} A solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) is good if and only if $\sum_{i = 0}^{\|F\|} (L_m^X(\, . \,,a_i) - L_m^X(\, . \,,b_i-) )$ is of locally bounded variation on $[0, S^Y_{G(\mathbb{R})})$. In that case, for $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ it holds \[ \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s = \sum_{i = 0}^{\|F\|} \left( L_m^X(\, . \,,a_i) - L_m^X(\, . \,,b_i-) \right), \qquad t < S_{G(\mathbb{R})}^Y, \ \mathbf{P}\text{-a.s.} \] \end{theorem} Under additional assumptions on the set $F$ of singularities we can improve the preceding Theorem. For $F$ consisting only of isolated points we have \begin{theorem}\label{theorem:isolated_points} Suppose that $\|F \cap [-N,N]\| < +\infty$, $N \in \mathbb{N}$. Then every solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) is good. \end{theorem} \begin{proof} To prove the result, we use Theorem \ref{theorem:a_prioiri_semimart_alt_formulation}. If $F$ is finite, then the finite sum $\sum_{i = 0}^{\|F\|} (L_m^X(\, . \,,a_i) - L_m^X(\, . \,,b_i-) )$ is obviously of locally bounded variation on $[0, S^Y_{G(\mathbb{R})})$. Therefore, $(G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}). In case of an infinite set $F$ satisfying $\|F \cap [-N,N]\| < +\infty$, $N \in \mathbb{N}$, for arbitrary $N \in \mathbb{N}$ we define the $\mathbb{F}$-stopping time $T_N := \inf \{t \geq 0: X_t \notin [-N,N]\}$. Using Corollary \ref{corr:comp_interval}, for arbitrary $t \geq 0$ it is \[ \lim_{n \rightarrow +\infty} \sum_{i=0}^n \left( L_m^X(t,a_i) - L_m^X(t,b_i-) \right) = \sum_{a_i, b_i \in [-N,N]} \left( L_m^X(t,a_i) - L_m^X(t,b_i-) \right), \qquad t \leq T_N, \ \mathbf{P}\text{-a.s.} \] and the sum on the right-hand side is finite. Additionally, since $\{t \leq T_N\} \subseteq \{t < S_\infty^X\}$, by Theorem \ref{theorem:general_structure_space_transformation} we have \[ \lim_{n \rightarrow +\infty} \sum_{i=0}^n \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right) = \sum_{i=0}^{+\infty} \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right) \] in probability on $\{t \leq T_N\}$. Therefore, using the continuity of the involved processes, we conclude \[ \sum_{i=0}^{+\infty} \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right) = \sum_{a_i, b_i \in [-N,N]} \left( L_m^X(t,a_i) - L_m^X(t,b_i-) \right), \qquad t \leq T_N, \ \mathbf{P}\text{-a.s.} \] But since $N\in \mathbb{N}$ was chosen arbitrarily and since $T_N \uparrow S_{G(\mathbb{R})}^Y$, $N \rightarrow +\infty$, holds, this means that $\left(\sum_{i=0}^{+\infty} \left(L_m^X(t,a_i) - L_m^X(t,b_i-)\right)\right)_{t \geq 0}$ is of locally bounded variation on $[0,S_{G(\mathbb{R})}^Y)$, and hence $(G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}). \end{proof} Under the assumption that the set $F$ is \emph{countable}, we can get more insight in the structure of the drift part of $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ for good solutions $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}). \begin{proposition}\label{proposition:drift_part_pointwise_sum} Suppose that $F$ is countable. Let $(X,\mathbb{F})$ be a good solution of Eq. (\ref{eqn:SDE_mvasd}). Then, for the solution $(Y,\mathbb{F})$ of Eq. (\ref{eqn:SDE_transformed_eqn}) defined by $Y = G(X)$ it holds for all $t < S^Y_{G(\mathbb{R})}$ $\mathbf{P}$-a.s. \[ \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)} (Y_s) \, \mbox{\upshape d} Y_s = \sum_{a \in F_-} \left(L_m^X(t,a) - L_m^X(t,a-) \right) = \frac{1}{2} \sum_{a \in G(F_-)} \left(L_+^Y(t,a) - L_-^Y(t,a) \right). \] \end{proposition} \begin{proof} \indent By $V$ we denote the process of locally bounded variation in the semimartingale decomposition of $Y$. Using (\ref{eqn:drift_int_with_respect_to_process}) and property (\ref{eqn:loc_time_and_variation_process}), we obtain \begin{equation}\label{eqn:drift_part_pointwise_sum} \begin{split} \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s &= \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} V_s \\ &= \sum_{a \in G(F_-)} \int_0^t \text{\large{$\mathds{1}$}}_{\{a\}}(Y_s)\,\mbox{\upshape d} V_s \\ &= \frac{1}{2} \sum_{a \in G(F_-)} \left( L_+^Y(t,a) - L_-^Y(t,a)\right), \qquad t < S_{G(\mathbb{R})}^Y, \ \mathbf{P}\text{-a.s.}, \end{split} \end{equation} which together with Lemma \ref{lemma:relation_loc_times} ends the proof. \end{proof} \begin{remark} The assumption of Proposition \ref{proposition:drift_part_pointwise_sum} that $F$ or, equivalently, $F_-$ is countable is essential in (\ref{eqn:drift_part_pointwise_sum}). Indeed, if the set of Lebesgue measure zero $F_-$ is uncountable, the same holds for $G(F_-)$. Hence, besides a singular discrete part, $\mbox{\upshape d} V_s$ can have a singular continuous part. $\Diamond$ \end{remark} We now come to a characterization of good solutions $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) for countable sets $F$ for which its accumulation points \[ F^A := \{ x \in F: \forall \, \varepsilon > 0 \; \exists \, y \in F \cap (x-\varepsilon, x+\varepsilon), \ y \neq x \} \] are isolated. \begin{theorem}\label{theorem:finitely_many_accumulation_points} Suppose that $\|F^A \cap [-N,N]\| < +\infty$, $N \in \mathbb{N}$. Then $(X,\mathbb{F})$ is a good solution of Eq. (\ref{eqn:SDE_mvasd}) if and only if the following conditions are fulfilled: If $F_-$ is infinite then (i) $\displaystyle\sum_{a \in F_-} \left\| L_m^X(t,a) - L_m^X(t,a-) \right\| < +\infty, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.}$ (ii) For any enumeration $\{a_1, a_2, \ldots\}$ of $F_-$, the sequence $\left(\sum_{i=1}^n (L_m^X(\,.\,, a_i) - L_m^X(\,.\,, a_i-))\right)_{n\in \mathbb{N}}$ of processes converges $\mathbf{P}$-a.s. locally in variation on $[0, S_\infty^X)$. \end{theorem} \begin{proof} The proof of this theorem is rather technical and is therefore omitted. The interested reader is referred to S. Blei \cite{blei_thesis_2010}, Satz 2.3.62. \end{proof} \indent To finish this section, we remark that by our observations, particularly Theorem \ref{theorem:a_prioiri_semimart_alt_formulation}, it is reasonable to conjecture that a complete analysis of Eq. (\ref{eqn:SDE_mvasd}) goes beyond the class of semimartingales and, consequently, should be envisaged in the richer class of local Dirichlet processes. \section{Symmetric Solutions \--- Existence and Uniqueness}\label{sec:symmetric_solutions} \noindent In the next two sections we proceed with the systematic investigation of existence and uniqueness of good solutions of Eq. (\ref{eqn:SDE_mvasd}). We need the following preparatory lemma, which compares the sets $N_b$ and $N_\sigma$ as well as $E_{b/\sqrt{f}}$ and $E_\sigma$, as introduced before Lemma \ref{lemma:stopping}. \begin{lemma}\label{lemma:connection_sets} We have $N_\sigma^c = G(N_b^c)$ and $E_\sigma^c = G(E_{b/\sqrt{f}}^c)$. \end{lemma} \begin{proof} The first equality is obvious. To show the second assertion, we observe \[ E_\sigma^c \subseteq (G(-\infty), G(+\infty)) \] and, using (\ref{eqn:representation_H}), \[\begin{split} \int_U \sigma^{-2}(y) \, \mbox{\upshape d} y\ & = \int_U (b \circ H)^{-2}(y) \,(f\circ H)(y) \, \mbox{\upshape d} H(y) \\ &= \int_{H(U)} b^{-2}(y)\,f(y)\, \mbox{\upshape d} y \end{split}\] for every open subset $U \subseteq (G(-\infty), G(+\infty))$. Since $G$ and $H$ are continuous on $\mathbb{R}$ and \linebreak[4] $(G(-\infty),G(+\infty))$, respectively, the proof is completed. \end{proof} We recall that Corollary \ref{corollary:left_continuity} reveals that the local time $L_m^X$ of a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) is continuous in the state variable except in the points of $F_-$. In this section, as a first step, we are interested in solutions of Eq. (\ref{eqn:SDE_mvasd}) which possess a continuous local time $L^X_m$. \begin{defi} A solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) is called \emph{symmetric} if its local time $L^X_m$ is continuous in the state variable. \end{defi} \noindent This notion is motivated by the obvious fact that the local time $L_m^X$ of a solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) is continuous in the state variable if and only if the local time $L_m^X$ is symmetric, i.e., $L_m^X$ coincides with the symmetric local time $\hat{L}_m^X(t,x) := (L_m^X(t,x) + L_m^X(t,x-))/2$, $t < S_\infty^X$, $x \in \mathbb{R}$. \begin{proposition}\label{prop:symmetric_good_solution_and_eqn_without_drift} (i) Let $F$ be countable. If $(X,\mathbb{F})$ is a symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}), then $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_without_drift}) with $\sigma =(b/f) \circ H$. (ii) Conversely, for arbitrary $F$ it holds: If $(Y,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_without_drift}) with $\sigma =(b/f) \circ H$, then $(X,\mathbb{F}) = (H(Y),\mathbb{F})$ is a symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}). (iii) Suppose $\|F^A \cap [-N,N]\| < +\infty$, $N \in \mathbb{N}$. Then any symmetric solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) is also a good solution, and hence $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_without_drift}) with $\sigma =(b/f) \circ H$. \end{proposition} Before we prove Proposition \ref{prop:symmetric_good_solution_and_eqn_without_drift}, we give the auxiliary \begin{lemma}\label{lemma:eqn_without_drift_and_transformed_symmetric_sol} The process $(Y,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_without_drift}) with $\sigma =(b/f) \circ H$ if and only if $(Y,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}) satisfying \begin{equation}\label{eqn:aux_zero_drift} \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)} (Y_s) \, \mbox{\upshape d} Y_s = 0, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{equation} \end{lemma} \begin{proof} Let $(Y,\mathbb{F})$ be a solution of Eq. (\ref{eqn:SDE_without_drift}) with $\sigma =(b/f) \circ H$. Since $E_\sigma^c \subseteq G(\mathbb{R})$, from Lemma \ref{lemma:stopping} it follows $Y_t =Y_{t \wedge S^Y_{G(\mathbb{R})}}$, $t \geq 0$, $\mathbf{P}$-a.s. Moreover, $G(F_-)$ is of Lebesgue measure zero, and hence (\ref{eqn:aux_zero_drift}) follows immediately. Therefore, $(Y,\mathbb{F})$ is also a solution of Eq. (\ref{eqn:SDE_transformed_eqn}).\\ \indent Conversely, if $(Y,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}) satisfying (\ref{eqn:aux_zero_drift}), then we have \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s, \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.}, \] and by similar arguments as used in the proof of \cite{engelbert_schmidt:1989_III}, Proposition (4.29), it can be shown that for arbitrary $t \geq 0$ the last equality also holds on $\{S^Y_{G(\mathbb{R})} \leq t < S^Y_\infty\}$. Hence, $(Y,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_without_drift}). \end{proof} \begin{proof}[Proof of Proposition \ref{prop:symmetric_good_solution_and_eqn_without_drift}] Assertion (i) follows directly from Proposition \ref{proposition:drift_part_pointwise_sum} and Lemma \ref{lemma:eqn_without_drift_and_transformed_symmetric_sol}. \\ \indent If $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_without_drift}) with $\sigma =(b/f) \circ H$, then Lemma \ref{lemma:eqn_without_drift_and_transformed_symmetric_sol} implies that $(Y,\mathbb{F})$ is also a solution of Eq. (\ref{eqn:SDE_transformed_eqn}). Hence, from Theorem \ref{theorem:spacetrans_H} we obtain that $(X,\mathbb{F}) = (H(Y),\mathbb{F})$ is a good solution of Eq. (\ref{eqn:SDE_mvasd}). Moreover, combining (\ref{eqn:loc_time_and_variation_process}) for $Y$ and Lemma \ref{lemma:relation_loc_times}, we see that $(X,\mathbb{F})$ is also symmetric and (ii) is proven. \\ \indent Statement (iii) can now be deduced with the help of Theorem \ref{theorem:finitely_many_accumulation_points}. Indeed, if $(X,\mathbb{F})$ is a symmetric solution of Eq. (\ref{eqn:SDE_mvasd}) then the sums appearing in the conditions (i) and (ii) of Theorem \ref{theorem:finitely_many_accumulation_points} are $\mathbf{P}$-a.s. equal to zero. Therefore, $(X,\mathbb{F})$ is good and we can apply (i). \end{proof} Concerning the existence of symmetric good solutions of Eq. (\ref{eqn:SDE_mvasd}), we can state the following \begin{theorem}\label{theorem:ex_symmetric_solutions} (i) Let $F$ be arbitrary. Suppose $E_{b/\sqrt{f}} \subseteq N_b$ is satisfied. Then for every initial distribution there exists a symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}). (ii) Let $F$ be countable. Then for every initial distribution there exists a symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}) if and only if the condition $E_{b/\sqrt{f}} \subseteq N_b$ is satisfied. \end{theorem} \begin{proof} To prove (i) and the sufficiency of the condition $E_{b/\sqrt{f}} \subseteq N_b$ in (ii), we choose an arbitrary initial distribution $\mu$. Lemma \ref{lemma:connection_sets} implies that $E_\sigma \subseteq N_\sigma$ holds, too. Hence, by Theorem \ref{theorem:e_u_gen_drift}(i), Eq. (\ref{eqn:SDE_without_drift}) with diffusion coefficient $\sigma = (b/f) \circ H$ possesses a solution $(Y,\mathbb{F})$ with initial distribution $\mu \circ G^{-1}$. Now from Proposition \ref{prop:symmetric_good_solution_and_eqn_without_drift} it follows immediately that $(X,\mathbb{F}) = (H(Y),\mathbb{F})$ is a symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}) with initial distribution $\mu$. \\ \indent To prove the necessity of $E_{b/\sqrt{f}}\subseteq N_b$ in (ii), we fix $x_0 \in E_{b/\sqrt{f}}$ and take a symmetric good solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) started at $X_0 = x_0$. From Proposition \ref{prop:symmetric_good_solution_and_eqn_without_drift}(i) we obtain that $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_without_drift}) with diffusion coefficient $\sigma =(b/f) \circ H$ and initial value $Y_0 = y_0 = G(x_0)$. Moreover, via Lemma \ref{lemma:connection_sets} it follows $y_0 \in E_\sigma$. Hence, Lemma \ref{lemma:stopping} implies $Y_t = y_0$, $t \geq 0$, $\mathbf{P}$-a.s. Therefore, we conclude \begin{equation}\label{eqn:belonging_to_zeros_of_sigma} 0 = \int_0^t \sigma^2(Y_s) \,\mbox{\upshape d} s = \sigma^2(y_0) \, t, \qquad t \geq 0, \end{equation} from which we see $y_0 \in N_\sigma$. By the arbitrariness of $x_0 \in E_{b/\sqrt{f}}$ we conclude $G(E_{b/\sqrt{f}}) \subseteq N_\sigma$ and via Lemma \ref{lemma:connection_sets} we obtain $E_{b/\sqrt{f}} \subseteq N_b$. \end{proof} Now we treat the question of uniqueness of symmetric good solutions of Eq. (\ref{eqn:SDE_mvasd}). \begin{theorem}\label{theorem:ex_u_un_symmetric_sol} (i) Let $F$ be arbitrary. Suppose $E_{b/\sqrt{f}} \subseteq N_b$. If for every initial distribution Eq. (\ref{eqn:SDE_mvasd}) possesses a unique symmetric good solution, then it holds $E_{b/\sqrt{f}} = N_b$. (ii) Let $F$ be countable. Then for every initial distribution Eq. (\ref{eqn:SDE_mvasd}) possesses a unique symmetric good solution if and only if the condition $E_{b/\sqrt{f}} = N_b$ is satisfied. \end{theorem} \begin{proof} First, we show (i) and the necessity of $E_{b/\sqrt{f}} = N_b$ in (ii). To this end, let us assume that, for every initial distribution, Eq. (\ref{eqn:SDE_mvasd}) possesses a unique symmetric good solution. In case of a countable set $F$ by means of Theorem \ref{theorem:ex_symmetric_solutions}(ii) this implies $E_{b/\sqrt{f}} \subseteq N_b$. For uncountable $F$ we suppose that $E_{b/\sqrt{f}} \subseteq N_b$ holds. Now we accomplish this part of the proof by contraposition. Let us assume that $E_{b/\sqrt{f}} = N_b$ does not hold, i.e., there exists an $x_0 \in E^c_{b/\sqrt{f}} \cap N_b$. Then, clearly, $\overline{X} \equiv x_0$ is a symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}). On the other hand, we can also find a non-trivial symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}) started at $x_0$. Indeed, because of Lemma \ref{lemma:connection_sets}, $E_{b/\sqrt{f}} \subseteq N_b$ implies $E_\sigma \subseteq N_\sigma$. Hence, there exists the so-called fundamental solution $(Y,\mathbb{F})$ of Eq. (\ref{eqn:SDE_without_drift}) with diffusion coefficient $\sigma = (b/f) \circ H$ started at $y_0=G(x_0) \in E^c_\sigma \cap N_\sigma$ and $Y$ is non-trivial (see \cite{engelbert_schmidt:1989_III}, Definition (4.16) and Theorem (4.17)). Finally, via Proposition \ref{prop:symmetric_good_solution_and_eqn_without_drift}(ii) we conclude that $(X,\mathbb{F}) = (H(Y),\mathbb{F})$ is a non-trivial symmetric good solution of Eq. (\ref{eqn:SDE_mvasd}) with initial point $x_0$. \\ \indent To show the sufficiency of $E_{b/\sqrt{f}} = N_b$ in (ii), we take two symmetric good solutions $(X^i,\mathbb{F}^i)$, $i =1,2$, of Eq. (\ref{eqn:SDE_mvasd}) defined on $(\Omega^i, \mathcal{F}^i, \mathbf{P}^i)$, $i =1,2$, which possess the same initial distribution. Then from Proposition \ref{prop:symmetric_good_solution_and_eqn_without_drift}(i) it follows that $(Y^i,\mathbb{F}^i) := (G(X^i), \mathbb{F}^i)$, $i = 1,2$, are two solutions of Eq. (\ref{eqn:SDE_without_drift}) with $\sigma = (b/f) \circ H$ and identical initial distributions. Now we take a sequence of intervals $[a_n,b_n]$, $n \in \mathbb{N}$, such that \begin{equation}\label{eqn:sequence_of_intervals} [a_n,b_n] \subseteq [a_{n+1},b_{n+1}] \subseteq G(\mathbb{R}),\ n \in \mathbb{N}, \quad \text{ and } \quad \bigcup_{n \in \mathbb{N}} [a_n,b_n] = G(\mathbb{R}) \end{equation} and define the $\mathbb{F}$-stopping times $S^i_n := \inf\{t \geq 0: Y^i_t \notin (a_n, b_n)\}$, $n \in \mathbb{N}$, $i =1,2$. For arbitrary $n\in \mathbb{N}$ we set $Y^{i,n}_t := Y^i_{t \wedge S^i_n}$, $t \geq 0$. Then it holds \begin{equation}\label{eqn:end_of_proof_uniqueness} Y^{i,n}_t = Y^i_0 + \int_0^t \sigma_n(Y^{i,n}_s) \, \mbox{\upshape d} B^i_s, \qquad t \geq 0, \ \mathbf{P}^i\text{-a.s.}, \ i=1,2, \end{equation} where $\sigma_n := \sigma \text{\large{$\mathds{1}$}}_{(a_n,b_n)}$ and $B^i$, $i = 1,2$, is the corresponding Wiener process. That means, $(Y^{i,n},\mathbb{F})$, $i =1,2$, are solutions to the same equation of type (\ref{eqn:SDE_without_drift}) with identical initial distribution. Moreover, via Lemma \ref{lemma:connection_sets} we obtain $E_\sigma = N_\sigma$, and hence we have $E_{\sigma_n} = N_{\sigma_n}$. Applying Theorem \ref{theorem:e_u_gen_drift}(ii), we conclude that $Y^{1,n}$ and $Y^{2,n}$ have the same distribution on $C_{\overline{\mathbb{R}}}([0,+\infty))$. This implies that the distributions of $Y^1$ and $Y^2$ coincide on $\mathcal{C}_{S_n-}$, where $\mathbb{C} = (\mathcal{C}_t)_{t \geq 0}$ is the filtration generated by the coordinate mappings $Z = (Z_t)_{t \geq 0}$, \[ Z_t(\omega) = \omega(t), \qquad \omega \in C_{\overline{\mathbb{R}}}([0,+\infty)), \ t \geq 0, \] and $S_n := \inf\{t \geq 0: Z_t \notin (a_n, b_n)\}$. Finally, it follows easily that the distributions of $Y^1$ and $Y^2$ coincide on $C_{\overline{\mathbb{R}}}([0,+\infty))$. Hence, we conclude that the distributions of $X^1$ and $X^2$ coincide, too. \end{proof} If the set $F$ satisfies the condition that $F^A$ consists only of isolated points, then from Theorem \ref{theorem:ex_symmetric_solutions}(ii) and \ref{theorem:ex_u_un_symmetric_sol}(ii) and Proposition \ref{prop:symmetric_good_solution_and_eqn_without_drift}(ii) we get immediately \begin{corollary}\label{cor:ex_u_un_isolated_points} Suppose $\|F^A \cap [-N,N]\| < +\infty$, $N \in \mathbb{N}$. Then the following statements hold. (i) For every initial distribution there exists a symmetric solution of Eq. (\ref{eqn:SDE_mvasd}) if and only if the condition $E_{b/\sqrt{f}} \subseteq N_b$ is satisfied. (ii) For every initial distribution there exists a unique symmetric solution of Eq. (\ref{eqn:SDE_mvasd}) if and only if the condition $E_{b/\sqrt{f}} = N_b$ is satisfied. \end{corollary} \section{Skew Solutions \--- Existence and Uniqueness}\label{sec:skew_solutions} \noindent In this section, we want to study good solutions $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) which, in general, do not possess a continuous local time $L_m^X$. We start with an important example of an equation of type (\ref{eqn:SDE_mvasd}) and illustrate a new feature of equations of type (\ref{eqn:SDE_mvasd}) compared with equations of type (\ref{eqn:SDE_mvd}). The example shows that, even in the case $E_{b/\sqrt{f}} = N_b$, or stronger $E_b = N_b$, there are, in general, solutions of Eq. (\ref{eqn:SDE_mvasd}) which are different in law. Note that the uniqueness result of Theorem \ref{theorem:ex_u_un_symmetric_sol} was achieved by considering only symmetric solutions of Eq. (\ref{eqn:SDE_mvasd}). The example will provide us with a whole variety of (good) solutions and will give us an idea for an approach to Eq. (\ref{eqn:SDE_mvasd}) more general than in Section \ref{sec:symmetric_solutions}. \\ \indent Fixing $\delta \in (1,2)$ and $x_0 \in \mathbb{R}$, a typical example of an equation of type (\ref{eqn:SDE_mvasd}) with generalized and singular drift is the Bessel equation which is satisfied for $x_0 \geq 0$ by the $\delta$-dimensional Bessel process. Classically, this is an equation with ordinary drift (see D. Revuz and M. Yor \cite{revuzyor}, Ch. XI, \S 1): \begin{equation}\label{eqn:SDE_BES} X_t = x_0 + B_t + \int_0^t \frac{\delta - 1}{2\, X_s} \, \mbox{\upshape d} s\,, \end{equation} where the notion of a solution of Eq. (\ref{eqn:SDE_BES}) is introduced analogously to Definition (\ref{def:solution}), but condition (iv) is omitted. Using the drift function $f_\delta(x) = \|x\|^{\delta -1}$, $x \in \mathbb{R}$, Eq. (\ref{eqn:SDE_BES}) coincides with \begin{equation}\label{eqn:SDE_BES_mvasd} X_t = x_0 + B_t + \int_\mathbb{R} L_m^X(t,y)\,\mbox{\upshape d} f_\delta(y)\,. \end{equation} Indeed, for any solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_BES_mvasd}), taking $\mbox{\upshape d} f_\delta(y) = (\delta-1) \, y^{-1} \, f_\delta(y)\, \mbox{\upshape d} y$ and the occupation times formula of Definition \ref{def:solution}(iv) into account, with $m$ given by $m(\mbox{\upshape d} y) = 2 \, f_\delta(y) \, \mbox{\upshape d} y$, we obtain \begin{equation}\label{eqn:reformulation_drift_p} \begin{split} \int_{(0,+\infty)} L_m^X(t,y) \, \mbox{\upshape d} f_\delta(y) &= \int_{(0,+\infty)} \frac{\delta - 1}{2 \, y} \, L_m^X(t,y) \, m(\mbox{\upshape d} y) \\ &= \int_0^t \frac{\delta - 1}{2\, X_s} \, \text{\large{$\mathds{1}$}}_{(0,+\infty)} (X_s) \, \mbox{\upshape d} \assPro{X}_s \\ & = \int_0^t \frac{\delta - 1}{2\, X_s} \, \text{\large{$\mathds{1}$}}_{(0,+\infty)} (X_s) \, \mbox{\upshape d} s\,, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \end{split} \end{equation} Analogously, we have \begin{equation}\label{eqn:reformulation_drift_n} \int_{(-\infty,0)} L_m^X(t,y) \, \mbox{\upshape d} f_\delta(y) = \int_0^t \frac{\delta - 1}{2\, X_s} \, \text{\large{$\mathds{1}$}}_{(-\infty,0)} (X_s) \, \mbox{\upshape d} s\,, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \end{equation} On the other hand, it was shown in S. Blei \cite{blei_bessel_2011}, Proposition 2.19, that for any solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_BES}) there exists a local time $L_m^X$ that satisfies the requirements of Definition \ref{def:solution}(iv) with respect to the drift function $f_\delta$. Therefore, for any solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_BES}) the equalities (\ref{eqn:reformulation_drift_p}) and (\ref{eqn:reformulation_drift_n}) hold as well. In addition, if $(X,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_BES}) or Eq. (\ref{eqn:SDE_BES_mvasd}), then $X$ has no occupation time in zero, which is an immediate consequence of the occupation times formula (\ref{eqn:occupationtime}): \[ \int_0^t \text{\large{$\mathds{1}$}}_{\{0\}} (X_s) \, \mbox{\upshape d} s = \int_0^t \text{\large{$\mathds{1}$}}_{\{0\}} (X_s) \, \mbox{\upshape d} \assPro{X}_s = \int_{\{0\}} L_+^X(t,y) \, \mbox{\upshape d} y = 0\,, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] Hence, from (\ref{eqn:reformulation_drift_p}) and (\ref{eqn:reformulation_drift_n}) we obtain that any solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_BES}) or Eq. (\ref{eqn:SDE_BES_mvasd}) satisfies \[ \int_0^t \frac{\delta - 1}{2\, X_s} \, \mbox{\upshape d} s = \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} f_\delta(y)\,, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.} \] Consequently, Eq. (\ref{eqn:SDE_BES}) coincides with Eq. (\ref{eqn:SDE_BES_mvasd}). Additionally, explosion does not occur. Indeed, for any solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_BES}) we see that the process $X^2$ satisfies \[ X_t^2 = x_0^2 + 2 \int_0^t \sqrt{X^2_s} \, \mbox{\upshape d} B_s + \delta t\,, \qquad t < S_\infty^{X^2}, \ \mathbf{P}\text{-a.s.}, \] which is an immediate consequence of the It\^o formula and the relation $S_\infty^X = S_\infty^{X^2}$. Since the coefficients of this equation possess at most linear growth, explosion does not occur, i.e., $S_\infty^X = S_\infty^{X^2} = +\infty$ $\mathbf{P}$-a.s. \\ \indent Using the primitive $G_\delta$ of $1/f_\delta$ and the fact that we have $F = F_- = \{0\}$ for $f_\delta$, Theorem \ref{theorem:isolated_points} states that for any solution $(X,\mathbb{F})$ the transformed process $(Y,\mathbb{F}) = (G_\delta(X),\mathbb{F})$ is a solution to \begin{equation}\label{eqn:SDE_BES_transformed} Y_t = y_0 + \int_0^t \sigma_\delta(Y_s) \, \mbox{\upshape d} B_s + \int_0^t \text{\large{$\mathds{1}$}}_{\{0\}}(Y_s) \, \mbox{\upshape d} Y_s\,, \end{equation} where $y_0 = G(x_0)$, $\sigma_\delta = 1/f_\delta \circ H_\delta$ and $H_\delta$ denotes the inverse of $G_\delta$. Additionally, applying Proposition \ref{proposition:drift_part_pointwise_sum}, the drift can be rewritten as \begin{equation}\label{eqn:jumps_loc_time_BES} \int_0^t \text{\large{$\mathds{1}$}}_{\{0\}}(Y_s) \, \mbox{\upshape d} Y_s = \frac{1}{2} \left( L_+^Y(t,0) - L_-^Y(t,0) \right) = L_m^X(t,0) - L_m^X(t,0-)\,, \qquad t \geq 0, \ \mathbf{P}\text{-a.s.} \end{equation} These jumps of the local time $L_m^X$ are a degree of freedom and are responsible for the non-uniqueness of solutions of Eq. (\ref{eqn:SDE_BES_mvasd}). Indeed, in \cite{blei_bessel_2011}, Theorem 2.22, (in connection with \cite{blei_bessel_2011}, Remark 2.26(ii)) it is shown that, for every $\alpha \in (-\infty,\frac{1}{2})$, the equation \begin{equation}\label{eqn:SDE_BES_controlled} \left\{ \begin{gathered} X_t = x_0 + B_t + \int_\mathbb{R} L_m^X(t,y) \, \mbox{\upshape d} f_{\delta}(y)\,, \\ L_m^X(t,0) - L_m^X(t,0-) = 2\, \alpha \, L_m^X(t,0)\,, \end{gathered} \right. \end{equation} possesses a unique solution, the so-called \emph{skew} $\delta$-dimensional Bessel process with skewness parameter $\alpha$ started at $x_0$. For $\alpha = 0$ the solution is, in correspondence with Section \ref{sec:symmetric_solutions}, also called the \emph{symmetric} $\delta$-dimensional Bessel process started at $x_0$. Clearly, for every $\alpha \in (-\infty,\frac{1}{2})$ we obtain solutions of Eq. (\ref{eqn:SDE_BES_mvasd}) different in law although we have $E_b = N_b = \emptyset$ and (\ref{eqn:jumps_loc_time_BES}) becomes \begin{equation}\label{eqn:BES_part_bounded_var} \begin{split} \int_0^t \text{\large{$\mathds{1}$}}_{\{0\}}(Y_s) \, \mbox{\upshape d} Y_s = \alpha \, L_+^Y(t,0) = 2\, \alpha \, L_m^X(t,0), \qquad t \geq 0, \ \mathbf{P}\text{-a.s.} \end{split} \end{equation} \noindent In addition, we remark that Eq. (\ref{eqn:SDE_BES_controlled}) can be also considered for $\alpha = 1/2$. In that case, for a positive starting point $x_0 \geq 0$ we obtain the $\delta$-dimensional Bessel process as the unique solution of Eq. (\ref{eqn:SDE_BES_controlled}), which stays positive and which is reflected to the positive half line at zero. For a negative starting point $x_0 < 0$ we obtain as well a unique solution that behaves like the Bessel process after it has reached zero, which happens with probability one. But in the following, we exclude the case of reflection. Moreover, as pointed out in \cite{blei_bessel_2011}, Lemma 2.25 and the remarks before, for a parameter $\alpha > 1/2$ there is no solution of Eq. (\ref{eqn:SDE_BES_controlled}). \\ \indent Now we come back to the general equation (\ref{eqn:SDE_mvasd}). Taking a good solution $(X,\mathbb{F})$, the process $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}). If $F$ is countable, via Proposition \ref{proposition:drift_part_pointwise_sum} we obtain additionally \[ \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)} (Y_s) \, \mbox{\upshape d} Y_s = \frac{1}{2} \sum_{a \in G(F_-)} \left( L_+^Y(t,a) - L_-^Y(t,a)\right) = \sum_{a \in F_-} \left( L_m^X(t,a) - L_m^X(t,a-)\right). \] As in the example of the Bessel equation (\ref{eqn:SDE_BES_mvasd}), the jumps $L_m^X(t,a) - L_m^X(t,a-)$ of the local time $L_m^X$ in the points of the set $F_-$ are not determined by Eq. (\ref{eqn:SDE_mvasd}). For this reason, we adopt the concept of fixing these jumps similar to (\ref{eqn:SDE_BES_controlled}). In this way, we put more information on the structure of the part $\int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \,\mbox{\upshape d} Y_s$ of locally bounded variation appearing in Eq. (\ref{eqn:SDE_transformed_eqn}). To realize this idea, we consider a set function $\nu$ defined on the bounded Borel sets such that $\nu$ is a finite signed measure on $\mathscr{B}([-N,N])$ for every $N\in \mathbb{N}$. Moreover, we assume that $\nu$ has no mass on $F_-^c$, i.e., \[ \|\nu\|(F_-^c \cap [-N,N]) = 0, \qquad N \in \mathbb{N}\,, \] where $\|\nu\|$ denotes the total variation of $\nu$. By means of $\nu$, we control the jump sizes of $L_m^X$ in the points of $F_-$. For this purpose, we consider the equation \begin{equation}\label{eqn:SDE_mvasd_controlled} \left\{ \begin{aligned} \text{(i) \ \ } & X_t = X_0 + \int_0^t b(X_s) \, \mbox{\upshape d} B_s + \int_{\mathbb{R}} L_m^X(t,y) \, \mbox{\upshape d} f(y)\,, \\[1ex] \text{(ii) \ \ } & L_m^X(t,a) - L_m^X(t,a-) = 2 \, L_m^X(t,a)\, \nu(\{a\})\,, \qquad a \in F_- \,. \end{aligned} \right. \end{equation} The notion of a solution of Eq. (\ref{eqn:SDE_mvasd_controlled}) is introduced as in Definition \ref{def:solution}, but in addition we require that Eq. (\ref{eqn:SDE_mvasd_controlled})(ii) holds for all $t < S_\infty^X$ $\mathbf{P}$-a.s. Motivated by the observations in the example of the Bessel equation (\ref{eqn:SDE_BES_controlled}), we come to the following \begin{defi} A solution of Eq. (\ref{eqn:SDE_mvasd_controlled}) is called a \emph{skew solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$.} \end{defi} \noindent In particular, using $\nu \equiv 0$ in Eq. (\ref{eqn:SDE_mvasd_controlled}), we describe symmetric solutions of Eq. (\ref{eqn:SDE_mvasd}). \\ \indent Denoting by $\nu^G := \nu \circ G^{-1}$ the image of $\nu$ under $G:\mathbb{R}\rightarrow \mathbb{R}$, we concretise our results concerning the space transformations $G$ and $H$ in the new situation of Eq. (\ref{eqn:SDE_mvasd_controlled}). \begin{proposition}\label{proposition:points_controlled} (i) Let $F$ be countable. If $(X,\mathbb{F})$ is a skew good solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$, then $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}), which satisfies \begin{equation}\label{eqn:drift_as_gen_drift} \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s = \int_\mathbb{R} L_+^Y(t,y) \, \nu^G(\mbox{\upshape d} y), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.}, \end{equation} or, equivalently, \begin{equation}\label{eqn:SDE_transformed_controlled} Y_t = Y_0 + \int_0^t \sigma(Y_s)\, \mbox{\upshape d} B_s + \int_\mathbb{R} L_+^Y(t,y) \, \nu^G(\mbox{\upshape d} y), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{equation} (ii) Conversely, for arbitrary $F$ it holds: If $(Y,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}) which satisfies (\ref{eqn:drift_as_gen_drift}) or, equivalently, (\ref{eqn:SDE_transformed_controlled}), then $(X,\mathbb{F}) = (H(Y),\mathbb{F})$ is a skew good solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$. (iii) If $F$ is countable, in both statements (i) and (ii), (\ref{eqn:drift_as_gen_drift}) is as well equivalent to \begin{equation}\label{eqn:drift_as_sum} L_+^Y(t,a) - L_-^Y(t,a) = 2 \, L_+^Y(t,a) \, \nu^G(\{a\}), \qquad t < S^Y_{G(\mathbb{R})}, \ a \in G(F_-), \ \mathbf{P}\text{-a.s.} \end{equation} (iv) Suppose $\|F^A \cap [-N,N]\| < +\infty$, $N \in \mathbb{N}$. Then any skew solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ is also a good solution, and hence $(Y,\mathbb{F}) = (G(X),\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}) satisfying (\ref{eqn:drift_as_gen_drift}). \end{proposition} \begin{proof} For any solution $(Y,\mathbb{F})$ of Eq. (\ref{eqn:SDE_transformed_eqn}), the relations (\ref{eqn:drift_as_gen_drift}) and (\ref{eqn:SDE_transformed_controlled}) are of course equivalent. Moreover, using (\ref{eqn:loc_time_and_variation_process}) and (\ref{eqn:int_wrt_loc_time}), (\ref{eqn:drift_as_gen_drift}) implies \[ \begin{split} L_+^Y(t,a) - L_-^Y(t,a) &= 2 \int_0^t \text{\large{$\mathds{1}$}}_{\{a\}}(Y_s) \int_\mathbb{R} L_+^Y(\mbox{\upshape d} s,y) \, \nu^G(\mbox{\upshape d} y) \\ &= 2 \, L_+^Y(t,a) \, \nu^G(\{a\}), \qquad t < S^Y_{G(\mathbb{R})}, \ a \in G(F_-), \ \mathbf{P}\text{-a.s.} \end{split} \] Conversely, if $F$ is countable, by the same arguments as in (\ref{eqn:drift_part_pointwise_sum}) it is \[ \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s = \frac{1}{2} \sum_{a \in G(F_-)} \left(L_+^Y(t,a) - L_-^Y(t,a) \right), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \] and (\ref{eqn:drift_as_sum}) implies \begin{equation}\label{eqn:no_fulfilled_by_singular_part} \begin{split} \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s &= \sum_{a \in G(F_-)} L_+^Y(t,a) \, \nu^G(\{a\}) \\ &= \int_\mathbb{R} L_+^Y(t,y) \, \nu^G(\mbox{\upshape d} y), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \end{split} \end{equation} which is (\ref{eqn:drift_as_gen_drift}). Hence, we have proven (iii). To show (i), it remains to use Lemma \ref{lemma:relation_loc_times} and (\ref{eqn:SDE_mvasd_controlled})(ii) implies immediately that (\ref{eqn:drift_as_sum}) is fulfilled for $(Y,\mathbb{F}) = (G(X),\mathbb{F})$. To verify (ii), from Theorem \ref{theorem:spacetrans_H} we see at once that Eq. (\ref{eqn:SDE_mvasd_controlled})(i) is satisfied for $X = H(Y)$. Moreover, via Lemma \ref{lemma:relation_loc_times} we conclude \[ \begin{split} L_m^X(t,a) - L_m^X(t,a-) &= \frac{1}{2} \left( L_+^Y(t,G(a)) - L_-^Y(t,G(a)) \right) \\ &= L_+^Y(t,G(a)) \, \nu^G(\{G(a)\}) \\ &= 2 \, L_m^X(t,a) \, \nu(\{a\}), \qquad t < S_\infty^X, \ a \in F_-, \ \mathbf{P}\text{-a.s.} \end{split} \] Finally, to deduce (iv) it just remains to show that the conditions (i) and (ii) of Theorem \ref{theorem:finitely_many_accumulation_points} are fulfilled. If $F_-$ is infinite, then, using Eq. (\ref{eqn:SDE_mvasd_controlled})(ii), Corollary \ref{corr:comp_interval} and the assumption on $\nu$, we obtain \[ \sum_{a \in F_-} \left\| L_m^X(t,a) - L_m^X(t,a-) \right\| \leq 2 \sum_{a \in F_-} L_m^X(t,a) \, \|\nu\|(\{a\}) < +\infty, \qquad t < S_\infty^X, \ \mathbf{P}\text{-a.s.}, \] i.e., condition (i) of Theorem \ref{theorem:finitely_many_accumulation_points} is satisfied. Furthermore, let $\{a_1, a_2, \ldots\}$ be an arbitrary enumeration of $F_-$. Then, again by Eq. (\ref{eqn:SDE_mvasd_controlled})(ii), we have \[ \left(\sum_{i=1}^n (L_m^X(\,.\,, a_i) - L_m^X(\,.\,, a_i-))\right)_{n\in \mathbb{N}} = \int_{\{a_1, \ldots, a_n\}} 2 \, L_m^X(t,y) \, \nu(\mbox{\upshape d} y) \] which for $n \rightarrow +\infty$ clearly converges $\mathbf{P}$-a.s. locally in variation on $[0, S_\infty^X)$ to \[ \int_{F_-} 2 \, L_m^X(t,y) \, \nu(\mbox{\upshape d} y) < +\infty\,. \] Hence, condition \ref{theorem:finitely_many_accumulation_points}(ii) is also fulfilled, which ends the proof. \end{proof} \indent For uncountable $F$ or, equivalently, for uncountable $F_-$, the equivalence of (\ref{eqn:drift_as_gen_drift}) and (\ref{eqn:drift_as_sum}) does not hold in general. Indeed, in case of an uncountable set $F_-$ it is possible that $\nu$, and hence $\nu^G$, which are concentrated on the Lebesgue null sets $F_-$ and $G(F_-)$, respectively, have a singular continuous part besides a singular discrete part. Thus we cannot justify the last equality in (\ref{eqn:no_fulfilled_by_singular_part}). \\ \indent Proposition \ref{proposition:points_controlled} reveals the relation between solutions of Eq. (\ref{eqn:SDE_mvasd_controlled}), in which we control the jumps of $L_m^X$ in the points of $F_-$ by $\nu$, and solutions of Eq. (\ref{eqn:SDE_transformed_eqn}), which additionally possess a drift of type of Eq. (\ref{eqn:SDE_mvd}). This is the point where the results of H.J. Engelbert and W. Schmidt \cite{engelbert_schmidt:1985} and \cite{engelbert_schmidt:1989_III} concerning equations of type (\ref{eqn:SDE_mvd}) come into play. As in (\ref{eqn:cond_atoms}) and according to \cite{blei_engelbert_2012}, Theorem 2.2 and Corollary 2.5 and 2.6, we always suppose \[ \nu(\{a\}) < 1/2, \qquad a \in F_-\,, \] which implies $\nu^G(\{a\}) < 1/2$, $a \in G(F_-)$. On the one hand, we want to exclude the phenomenon of reflection in the points of $F_-$, which is described by $\nu(\{a\}) = 1/2$. On the other hand, if we have $\nu(\{a\}) > 1/2$, and therefore $\nu^G(\{a\}) > 1/2$, for an $a \in F_-$, then, in general, the existence of a solution to an equation which has a drift of type (\ref{eqn:SDE_mvd}) fails.\\ \indent Concerning the existence of skew good solutions of Eq. (\ref{eqn:SDE_mvasd}) we can state the following theorem. \begin{theorem}\label{theorem:ex_skew_sol} (i) Let $F$ be arbitrary. Suppose $E_{b/\sqrt{f}} \subseteq N_b$ is satisfied. Then for every initial distribution there exists a skew good solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$. (ii) Let $F$ be countable. Then for every initial distribution there exists a skew good solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ if and only if the condition $E_{b/\sqrt{f}} \subseteq N_b$ is satisfied. \end{theorem} \begin{proof} To prove (i) and the sufficiency of the condition $E_{b/\sqrt{f}} \subseteq N_b$ in (ii), we choose an arbitrary initial distribution $\mu$. Lemma \ref{lemma:connection_sets} implies $E_\sigma \subseteq N_\sigma$. Hence, by Theorem \ref{theorem:e_u_gen_drift}(i) the equation \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_+^Y(t,y) \, \nu^G(\mbox{\upshape d} y) \] possesses a solution $(Y,\mathbb{F})$ for the initial distribution $\mu \circ G^{-1}$. Moreover, since we have $E_\sigma^c \subseteq G(\mathbb{R})$, from Lemma \ref{lemma:stopping} we obtain $Y_t = Y_{t \wedge S^Y_{G(\mathbb{R})}}$, $t \geq 0$, $\mathbf{P}$-a.s. Analogously to (\ref{eqn:drift_int_with_respect_to_process}), we deduce \[ \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s = \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \int_{\mathbb{R}} L_+^Y(\mbox{\upshape d} s,y) \, \nu^G(\mbox{\upshape d} y), \qquad t < S_{G(\mathbb{R})}^Y\,, \ \mathbf{P}\text{-a.s.} \] Using (\ref{eqn:int_wrt_loc_time}) and the fact that $\nu^G$ is concentrated on $G(F_-)$, it follows \[ \int_0^t \text{\large{$\mathds{1}$}}_{G(F_-)}(Y_s) \, \mbox{\upshape d} Y_s = \int_\mathbb{R} L_+^Y(t,y) \, \nu^G(\mbox{\upshape d} y), \qquad t < S_{G(\mathbb{R})}^Y\,, \ \mathbf{P}\text{-a.s.} \] Thus, $(Y,\mathbb{F})$ is also a solution of Eq. (\ref{eqn:SDE_transformed_eqn}). Finally, Proposition \ref{proposition:points_controlled}(ii) implies that $(X,\mathbb{F}) =(H(Y),\mathbb{F})$ is a skew good solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ and initial distribution $\mu$. \\ \indent To prove the necessity of $E_{b/\sqrt{f}}\subseteq N_b$ in (ii), we fix $x_0 \in E_{b/\sqrt{f}}$ and take a skew good solution $(X,\mathbb{F})$ of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ started at $X_0 = x_0$. Setting $Y = G(X)$, by Proposition \ref{proposition:points_controlled}(i) we conclude that $(Y,\mathbb{F})$ is a solution of Eq. (\ref{eqn:SDE_transformed_eqn}) satisfying \[ Y_t = Y_0 + \int_0^t \sigma(Y_s)\, \mbox{\upshape d} B_s + \int_\mathbb{R} L_+^Y(t,y) \, \nu^G(\mbox{\upshape d} y), \qquad t < S^Y_{G(\mathbb{R})}, \ \mathbf{P}\text{-a.s.} \] Lemma \ref{lemma:connection_sets} implies $Y_0 = y_0 = G(x_0) \in E_\sigma$. Choosing an interval $(a,b)$ which satisfies $y_0 \in (a,b)$ and $[a,b] \subseteq G(\mathbb{R})$, we define the $\mathbb{F}$-stopping time $S := \inf \{t \geq 0: Y_t \notin (a,b) \}$. For the stopped process $Y^S_t := Y_{S \wedge t}$, $t \geq 0$, we obtain \begin{equation}\label{eqn:solution_gen_drift_local} \begin{split} Y_t^S & = y_0 + \int_0^{S \wedge t} \sigma(Y_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_+^Y(S \wedge t, y) \, \nu(\mbox{\upshape d} y) \\ & = y_0 + \int_0^t \sigma(Y^S_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_+^{Y^S}(t, y) \, \nu^G(\mbox{\upshape d} y), \qquad t \geq 0, \ \mathbf{P}\text{-a.s.}, \end{split} \end{equation} where we used $S \leq S_{G(\mathbb{R})}^Y$ and, in particular, $S < S_{G(\mathbb{R})}^Y$ on $\{S_{G(\mathbb{R})}^Y < +\infty\}$ to write $t \geq 0$ instead of $t < S^Y_{G(\mathbb{R})}$. Moreover, the relation $L_+^Y(S \wedge t, y) = L_+^{Y^S}(t, y)$, $t \geq 0$, $y \in \mathbb{R}$, $\mathbf{P}$-a.s. can be easily deduced from (\ref{eqn:gen_ito_formula}). Introducing $\nu_{(a,b)}^G := \nu^G(\,.\, \cap (a,b))$, due to (\ref{eqn:loc_time_zero_outside_compact_interval}) the drift part of $Y^S$ can be rewritten as \begin{equation}\label{eqn:solution_gen_drift_local_drift_part} \int_\mathbb{R} L_+^{Y^S}(t,y) \, \nu^G(\mbox{\upshape d} y) = \int_\mathbb{R} L_+^{Y^S}(t,y) \, \nu_{(a,b)}^G(\mbox{\upshape d} y), \qquad t \geq 0, \ \mathbf{P}\text{-a.s.} \end{equation} Summarizing, $(Y^S,\mathbb{F})$ is a solution to an equation of type (\ref{eqn:SDE_mvd}) started at $y_0 \in E_\sigma$. Hence, Lemma \ref{lemma:stopping} implies $Y^S_t = y_0$, $t \geq 0$, $\mathbf{P}$-a.s. and similar to (\ref{eqn:belonging_to_zeros_of_sigma}) and the lines thereafter we conclude $E_{b/\sqrt{f}} \subseteq N_b$. \end{proof} For skew good solutions of Eq. (\ref{eqn:SDE_mvasd}) the following uniqueness result holds. \begin{theorem}\label{theorem:ex_and_un_unique_skew_sol} (i) Let $F$ be arbitrary. Suppose $E_{b/\sqrt{f}} \subseteq N_b$. If for every initial distribution Eq. (\ref{eqn:SDE_mvasd}) possesses a unique skew good solution with skewness parameter $\nu$, then it holds $E_{b/\sqrt{f}} = N_b$. (ii) Let $F$ be countable. Then for every initial distribution Eq. (\ref{eqn:SDE_mvasd_controlled}) possesses a unique skew good solution with skewness parameter $\nu$ if and only if the condition $E_{b/\sqrt{f}} = N_b$ is satisfied. \end{theorem} \begin{proof} The proof is accomplished similarly to the proof of Theorem \ref{theorem:ex_u_un_symmetric_sol}. To begin with we show (i) and the necessity of $E_{b/\sqrt{f}} = N_b$ in (ii). For this purpose, let us assume that, for every initial distribution, Eq. (\ref{eqn:SDE_mvasd}) possesses a unique skew good solution with skewness parameter $\nu$. In case of a countable set $F$ by means of Theorem \ref{theorem:ex_skew_sol}(ii), this implies $E_{b/\sqrt{f}} \subseteq N_b$. For uncountable $F$ we suppose that $E_{b/\sqrt{f}} \subseteq N_b$ holds. Now we accomplish this part of the proof by contraposition. Let us assume that $E_{b/\sqrt{f}} = N_b$ does not hold, i.e., there exists an $x_0 \in E^c_{b/\sqrt{f}} \cap N_b$. Then, clearly, $\overline{X} \equiv x_0$ is a skew good solution of Eq. (\ref{eqn:SDE_mvasd_controlled}) with skewness parameter $\nu$. On the other hand, we can also find a non-trivial skew good solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ started at $x_0$. Indeed, because of Lemma \ref{lemma:connection_sets}, $E_{b/\sqrt{f}} \subseteq N_b$ implies $E_\sigma \subseteq N_\sigma$. Hence, there exists the so-called fundamental solution $(Y,\mathbb{F})$ to \[ Y_t = Y_0 + \int_0^t \sigma(Y_s) \, \mbox{\upshape d} B_s + \int_\mathbb{R} L_+^Y(t,y) \, \nu^G(\mbox{\upshape d} y) \] started at $y_0 = G(x_0) \in E_\sigma^c \cap N_\sigma$ and $Y$ is non-trivial (see \cite{engelbert_schmidt:1989_III}, Definition (4.16) and Theorem (4.35)). Moreover, using the same arguments as in the first part of the proof of Theorem \ref{theorem:ex_skew_sol}, it follows that $(Y,\mathbb{F})$ also solves Eq. (\ref{eqn:SDE_transformed_eqn}). Via Proposition \ref{proposition:points_controlled}(ii) we conclude that $(X,\mathbb{F})$ given by $X = H(Y)$ is a non-trivial skew good solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ and initial point $x_0$. \\ \indent To show that the condition $E_{b/\sqrt{f}} = N_b$ is sufficient in (ii), we take two skew good solutions $(X^i,\mathbb{F}^i)$, $i =1,2$, of Eq. (\ref{eqn:SDE_mvasd_controlled}) with skewness parameter $\nu$ defined on $(\Omega^i, \mathcal{F}^i, \mathbf{P}^i)$, $i =1,2$, which possess the same initial distribution. Via Proposition \ref{proposition:points_controlled}(i) we conclude that $(Y^i,\mathbb{F}^i) := (G(X^i), \mathbb{F}^i)$, $i = 1,2$, are two solutions of Eq. (\ref{eqn:SDE_transformed_eqn}) with identical initial distributions which satisfy (\ref{eqn:SDE_transformed_controlled}). As in the proof of Theorem \ref{theorem:ex_u_un_symmetric_sol} we take a sequence $[a_n,b_n]$, $n \in \mathbb{N}$, which satisfies (\ref{eqn:sequence_of_intervals}). Defining $S^i_n := \inf\{t \geq 0: Y^i_t \notin (a_n, b_n)\}$, $n \in \mathbb{N}$, $i =1,2$, and setting $Y^{i,n}_t := Y^i_{t \wedge S^i_n}$, $t \geq 0$, $n \in \mathbb{N}$, similar to (\ref{eqn:solution_gen_drift_local}) and (\ref{eqn:solution_gen_drift_local_drift_part}), it holds \[ Y^{i,n}_t = Y^i_0 + \int_0^t \sigma_n(Y^{i,n}_s) \, \mbox{\upshape d} B^i_s + \int_\mathbb{R} L_+^{Y^{i,n}}(t,y) \, \nu_n^G(\mbox{\upshape d} y), \qquad t \geq 0, \ \mathbf{P}^i\text{-a.s.}, \ i=1,2, \] where $\sigma_n := \sigma \text{\large{$\mathds{1}$}}_{(a_n,b_n)}$, $\nu^G_n := \nu^G(\,.\, \cap (a_n,b_n))$ and $B^i$, $i =1,2$, is the corresponding Wiener process. That means, $(Y^{i,n},\mathbb{F})$, $i =1,2$, are solutions to the same equation of type (\ref{eqn:SDE_mvd}) with identical initial distribution. The remaining part of the proof is now accomplished as in the proof of Theorem \ref{theorem:ex_u_un_symmetric_sol} after (\ref{eqn:end_of_proof_uniqueness}) \end{proof} Similarly to Corollary \ref{cor:ex_u_un_isolated_points}, we can give the following Corollary to Theorem \ref{theorem:ex_skew_sol} and \ref{theorem:ex_and_un_unique_skew_sol} and Proposition \ref{proposition:points_controlled}. \begin{corollary} Suppose $\|F^A \cap [-N,N]\| < +\infty$, $N \in \mathbb{N}$. Then the following statements hold. (i) For every initial distribution there exists a skew solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ if and only if the condition $E_{b/\sqrt{f}} \subseteq N_b$ is satisfied. (ii) For every initial distribution there exists a unique skew solution of Eq. (\ref{eqn:SDE_mvasd}) with skewness parameter $\nu$ if and only if the condition $E_{b/\sqrt{f}} = N_b$ is satisfied. \end{corollary} \section*{Appendix} \setcounter{equation}{0} \addtocounter{section}{1} \renewcommand{A.\arabic{equation}}{A.\arabic{equation}} \noindent Let $(X,\mathbb{F})$ be a stochastic process with values in $(\overline{\mathbb{R}}, \mathscr{B}(\overline{\mathbb{R}}))$ defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and let $S$ be an $\mathbb{F}$-stopping time. We call $(X,\mathbb{F})$ a semimartingale up to $S$ if there exists an increasing sequence $(S_n)_{n \in \mathbb{N}}$ of $\mathbb{F}$-stopping times such that $S = \lim_{n\rightarrow +\infty} S_n$ and the process $(X^n,\mathbb{F})$ obtained by stopping $(X,\mathbb{F})$ in $S_n$ is a real-valued semimartingale. Analogously, we introduce the notion of a local martingale up to $S$. We notice that if $S=+\infty$ $\mathbf{P}$-a.s., then any semimartingale up to $S$ is a semimartingale and any local martingale up to $S$ is a local martingale.\\ \indent If $(X,\mathbb{F})$ is a semimartingale up to $S$, then we can find a decomposition \begin{equation}\label{eqn:semi_decomposition} X_t = X_0 + M_t + V_t\,, \qquad t < S, \ \mathbf{P}\text{-a.s.}, \end{equation} where $(M,\mathbb{F})$ is a local martingale up to $S$ with $M_0 = 0$ and $(V, \mathbb{F})$ is a right-continuous process whose paths are of bounded variation on $[0,t]$ for every $t < S$ and with $V_0=0$. If $X$ is continuous on $[0,S)$, then there exists a decomposition such that $M$ and $V$ are continuous on $[0,S)$ and this decomposition is unique on $[0,S)$. For any continuous local martingale $(M, \mathbb{F})$ up to $S$ by $\assPro{M}$ we denote the continuous increasing process, which is uniquely determined on $[0,S)$, such that $(M^2 - \assPro{M}, \mathbb{F})$ is a continuous local martingale up to $S$ and $\assPro{M}_0 = 0$. For a continuous semimartingale $(X,\mathbb{F})$ up to $S$ we set $\assPro{X} = \assPro{M}$, where $M$ is the continuous local martingale up to $S$ in the decomposition (\ref{eqn:semi_decomposition}) of $X$. \\ \indent We recall some facts which are well-known for continuous semimartingales. See for example \cite{revuzyor}, Ch. VI, \S{}1. Their extension to semimartingales up to a stopping time $S$ is obvious. Let $(X,\mathbb{F})$ be a continuous semimartingale up to the $\mathbb{F}$-stopping time $S$. Then there exists the right local time $L_+^X$ which is a function on $[0,S) \times \mathbb{R}$ into $[0,+\infty)$ such that for every real function $f$ which is the difference of convex functions the generalized It\^o formula holds: \begin{equation}\label{eqn:gen_ito_formula} f(X_t) = f(X_0) + \int_0^t f^\prime_- (X_s) \, \mbox{\upshape d} X_s + \frac{1}{2} \int_0^t L_+^X(t,y) \, \mbox{\upshape d} f^\prime_+(y)\,, \qquad t < S, \ \mathbf{P}\text{-a.s.} \end{equation} Thereby, $f^\prime_+$ (resp. $f^\prime_-$) denotes the right (resp. left) derivative of $f$. Moreover, there exists a modification of $L_+^X$ which is increasing and continuous in $t$ as well as in $y$ right-continuous with limits from the left and we always use this modification. \\ \indent By $L^X_-$ we denote the left local time given by \[ L_-^X(t,y) = L_+^X(t,y-)\,, \qquad t < S, \, y \in \mathbb{R}\,. \] For the local times the so-called \emph{occupation times formula} \begin{equation}\label{eqn:occupationtime} \int_0^t g(X_s) \; \mbox{\upshape d} \assPro{X}_s = \int_\mathbb{R} L_\pm^X(t,y)\, g(y) \; \mbox{\upshape d} y\,, \qquad t < S, \ \mathbf{P}\mbox{-a.s.} \end{equation} holds true for every locally integrable or non-negative measurable function $g$. Moreover, the local times satisfy \begin{equation}\label{eqn:int_wrt_loc_time} \int_0^t \text{\large{$\mathds{1}$}}_{\{y\}}(X_s) \, L_\pm^X(\mbox{\upshape d} s,y) = L_\pm^X(t,y)\,, \qquad t < S, \ y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.}, \end{equation} \begin{equation}\label{eqn:loc_time_and_variation_process} L_+^X(t,y) - L_-^X(t,y) = 2 \int_0^t \text{\large{$\mathds{1}$}}_{\{y\}}(X_s) \, \mbox{\upshape d} V_s\,, \qquad t < S, \, y \in \mathbb{R}, \ \mathbf{P}\text{-a.s.} \end{equation} and \begin{equation}\label{eqn:loc_time_zero_outside_compact_interval} L_\pm^X(t,y) = 0, \qquad t < S, \, y \notin \left[\min_{0 \leq s \leq t} X_s, \max_{0 \leq s \leq t} X_s\right], \ \mathbf{P}\text{-a.s.} \end{equation} \indent The following lemma describes the relation between the local times of a continuous semimartingale $(X,\mathbb{F})$ and of the transformed semimartingale $(f(X),\mathbb{F})$ in case that $f$ is a semimartingale function with certain properties. This lemma is a slight modification of \cite{assingschmidt}, Lemma I.1.18, and the proof of \cite{assingschmidt} can be easily adapted to the situation of the following \begin{lemma}\label{lemma:conversion_loc_time} Let $(X,\mathbb{F})$ be a continuous semimartingale up to $S$. Furthermore, let $f: \overline{\mathbb{R}} \rightarrow \overline{\mathbb{R}}$ be a function such that its restriction to the interval $I = (r_1,r_2)$, $-\infty \leq r_1 < r_2 \leq +\infty$, is absolutely continuous and strictly increasing: \[ f(x) = f(\text{$x_0$}) + \int_{x_0}^x f^\prime(y) \, \mbox{\upshape d} y\,, \qquad x \in I, \] where $x_0 \in I$ is a fixed point. We assume that in every point $y \in I$ the function $f^\prime$ admits a limit from the right $f^\prime(y+)$ and from the left $f^\prime(y-)$ in $[0,+\infty]$. Moreover, we suppose that $(f(X), \mathbb{F})$ is a continuous semimartingale up to\footnote{$\inf \emptyset = +\infty$} $\widetilde{S} := S \wedge \inf\{t < S : X_t \notin I \}$ with the property \[ \assPro{f(X)}_t = \int_0^t (f^\prime(X_s\pm))^2 \, \mbox{\upshape d} \assPro{X}_s, \qquad t<\widetilde{S},\ \mathbf{P}\text{-a.s.} \] Denoting $N_\pm := \{x \in I : f^\prime(x\pm) = +\infty\}$, then it holds \[ L_\pm^{f(X)}(t,f(y)) = L_\pm^X(t,y) \, f^\prime(y\pm), \qquad t <\widetilde{S},\ y \in I\setminus N_{\pm}, \ \mathbf{P}\mbox{-a.s.} \] \end{lemma} The next theorem is a useful convergence result. \begin{theorem}\label{theorem:convergence} Let $(X,\mathbb{F})$ be a continuous semimartingale up to $S$ with decomposition \[ X_t = X_0 + M_t + V_t, \qquad t < S,\ \mathbf{P}\text{-a.s.} \] Furthermore, let $f_n$, $n \in \mathbb{N}$, $f$ and $h$ be measurable real-valued functions with \[ \lim\limits_{n \rightarrow \infty} f_n = f \qquad \lambda\text{-a.e.}\footnote{$\lambda$ denotes the Lebesgue measure on $\mathbb{R}$.} \] and \[ \|f_n\| \leq h \qquad \lambda\text{-a.e.}, \ n \in \mathbb{N}. \] Moreover, we assume \[ \int_0^t h^2(X_s) \, \mbox{\upshape d} \assPro{M}_s < +\infty, \qquad t < S, \ \mathbf{P}\text{-a.s.} \] Then the stochastic integrals of $f_n(X)$, $n \in \mathbb{N}$, and $f(X)$ with respect to $M$ on $[0,S)$ are well-defined and for every $t \geq 0$ we have \[ \lim\limits_{n \rightarrow \infty}\sup\limits_{0 \leq s \leq t} \left\| \int_0^s f_n(X_u) \, \mbox{\upshape d} M_u - \int_0^s f(X_u) \, \mbox{\upshape d} M_u \right\| = 0 \qquad \text{on } \{t < S\} \] in probability. \end{theorem} \begin{proof} We remind of $\assPro{X} = \assPro{M}$. Using the occupation times formula (\ref{eqn:occupationtime}), we get \[ \int_0^t f_n^2(X_s) \, \mbox{\upshape d} \assPro{M}_s \leq \int_0^t h^2(M_s) \, \mbox{\upshape d} \assPro{X}_s < + \infty, \] $t < S, \ \mathbf{P}\text{-a.s.}$ and by observing $\|f\| \leq h$ \[ \int_0^t f^2(X_s) \, \mbox{\upshape d} \assPro{M}_s < + \infty, \qquad t < S, \ \mathbf{P}\text{-a.s.} \] This means, as claimed, that the stochastic integrals are well-defined. From our assumptions we now deduce \[ \lim\limits_{n \rightarrow \infty} \|f_n(y)-f(y)\|^2 \, L_+^X(t,y) = 0 \qquad \lambda\text{-a.e.}, \ t < S, \] and \[ \|f_n(y)-f(y)\|^2 \, L_+^X(t,y) \leq 4 h^2(y) \, L_+^X(t,y) \qquad \lambda\text{-a.e.}, \ t < S, \ n \in \mathbb{N}. \] Furthermore, it holds \[ \int_\mathbb{R} 4 h^2(y) \, L_+^X(t,y)\, \mbox{\upshape d} y = \int_0^t 4 h^2(X_s) \, \mbox{\upshape d} \assPro{M}_s < +\infty, \qquad t < S, \ \mathbf{P}\text{-a.s.}, \] where we again applied the occupation times formula (\ref{eqn:occupationtime}). By means of Lebesgue's dominated convergence theorem these observations justify for every $t < S$ $\mathbf{P}\text{-a.s.}$ \begin{equation}\label{eqn:app_1} \lim\limits_{n \rightarrow +\infty} \int_0^t \left\| f_n(X_s) - f(X_s) \right\|^2 \, \mbox{\upshape d} \assPro{M}_s = \lim\limits_{n \rightarrow +\infty} \int_\mathbb{R} \left\| f_n(y) - f(y) \right\|^2 \, L_+^X(t,y) \, \mbox{\upshape d} y = 0\,. \end{equation} Let $(S_k)_{k\in \mathbb{N}}$ be an increasing sequence of $\mathbb{F}$-stopping times such that $\lim_{k \rightarrow +\infty} S_k = S$ $\mathbf{P}$-f.s. and for every $k\in \mathbb{N}$ the process $M^k$ obtained by stopping $M$ in $S_k$ is a continuous local martingale. Fixing $t\geq 0$, (\ref{eqn:app_1}) implies for every $k \in \mathbb{N}$ \[ \lim\limits_{n \rightarrow +\infty} \int_0^t \left\| f_n(X_s) - f(X_s) \right\|^2 \, \mbox{\upshape d} \assPro{M^{S_k}}_s = \lim\limits_{n \rightarrow +\infty} \int_0^{t \wedge S_k} \left\| f_n(X_s) - f(X_s) \right\|^2 \, \mbox{\upshape d} \assPro{M}_s = 0 \] in probability on $\{t < S\}$. Therefore, for arbitrary $\varepsilon > 0$ we conclude \[\begin{split} & \lim_{n \rightarrow +\infty} \mathbf{P} \left( \left\{ \sup\limits_{0 \leq s \leq t} \left\| \int_0^s f_n(X_u) \, \mbox{\upshape d} M_u - \int_0^s f(X_u) \, \mbox{\upshape d} M_u \right\| \geq \varepsilon \right\} \cap \{t<S\} \right) \\ & \phantom{=====} \leq \lim_{n \rightarrow +\infty} \mathbf{P} \left( \left\{ \sup\limits_{0 \leq s \leq t} \left\| \int_0^s f_n(X_u) \, \mbox{\upshape d} M^{S_k}_u - \int_0^s f(X_u) \, \mbox{\upshape d} M^{S_k}_u \right\| \geq \varepsilon \right\} \cap \{t < S_k\} \right) \\ & \phantom{========================================} + \mathbf{P} \left( \left\{ S_k \leq t < S\right\}\right)\\ & \phantom{=====} = \mathbf{P} \left( \left\{ S_k \leq t < S\right\}\right) \end{split}\] for all $k \in \mathbb{N}$, where the last equality is justified by \cite{karatzasshreve}, Prop. 3.2.26. Finally, the assertion follows since $\lim_{k \rightarrow +\infty} S_k = S$ $\mathbf{P}$-f.s. \end{proof} \end{document}	arXiv
Zhang Qiujian Suanjing Zhang Qiujian Suanjing (The Mathematical Classic of Zhang Qiujian) is the only known work of the fifth century Chinese mathematician, Zhang Qiujian. It is one of ten mathematical books known collectively as Suanjing shishu (The Ten Computational Canons). In 656 CE, when mathematics was included in the imperial examinations, these ten outstanding works were selected as textbooks. Jiuzhang suanshu (The Nine Chapters on the Mathematical Art) and Sunzi Suanjing (The Mathematical Classic of Sunzi) are two of these texts that precede Zhang Qiujian suanjing. All three works share a large number of common topics. In Zhang Qiujian suanjing one can find the continuation of the development of mathematics from the earlier two classics.[1] Internal evidences suggest that book was compiled sometime between 466 and 485 CE. "Zhang Qiujian suanjing has an important place in the world history of mathematics: it is one of those rare books before AD 500 that manifests the upward development of mathematics fundamentally due to the notations of the numeral system and the common fraction. The numeral system has a place value notation with ten as base, and the concise notation of the common fraction is the one we still use today."[1] Almost nothing is known about the author Zhang Qiujian, sometimes written as Chang Ch'iu-Chin or Chang Ch'iu-chien. It is estimated that he lived from 430 to 490 CE, but there is no consensus.[2] Contents In its surviving form, the book has a preface and three chapters. There are two missing bits, one at the end of Chapter 1 and one at the beginning of Chapter 3. Chapter 1 consists of 32 problems, Chapter 2 of 22 problems and Chapter 3 of 38 problems.[3] In the preface, the author has set forth his objectives in writing the book clearly. There are three objectives: The first is to explain how to handle arithmetical operations involving fractions; the second objective is to put forth new improved methods for solving old problems; and, the third objective is to present computational methods in a precise and comprehensible form.[3] Here is a typical problem of Chapter 1: "Divide 6587 2/3 and 3/4 by 58 ı/2. How much is it?" The answer is given as 112 437/702 with a detailed description of the process by which the answer is obtained. This description makes use of the Chinese rod numerals. The chapter considers several real world problems where computations with fractions appear naturally. In Chapter 2, among others, there are a few problem requiring application of the rule of three. Here is a typical problem: "Now there was a person who stole a horse and rode off with it. After he has traveled 73 li, the owner realized [the theft] and gave chase for 145 li when [the thief] was 23 li ahead before turning back. If he had not turned back but continued to chase, find the distance in li before he reached [the thief]." Answer is given as 238 3/14 li. In Chapter 3, there are several problems connected with volumes of solids which are granaries. Here is an example: "Now there is a pit [in the shape of the frustum of a pyramid] with a rectangular base. The width of the upper [rectangle] is 4 chi and the width of the lower [rectangle] is 7 chi. The length of the upper [rectangle] is 5 chi and the length of the lower [rectangle] is 8 chi. The depth is 1 zhang. Find the amount of millet that it can hold." However, the answer is given in a different set of units. The 37th problem is the "Washing Bowls Problem": "Now there was a woman washing cups by the river. An officer asked, "Why are there so many cups?" The woman replied, "There were guests in the house, but I do not know how many there were. However, every 2 persons had [a cup of] thick sauce, every 3 persons had [a cup of] soup and every 4 persons had [a cup of] rice; 65 cups were used altogether." Find the number of persons." The answer is given as 60 persons. The last problem in the book is the famous Hundred Fowls Problem which is often considered as one of the earliest examples involving equations with indeterminate solutions. "Now one cock is worth 5 qian, one hen 3 qian and 3 chicks 1 qian. It is required to buy 100 fowls with 100 qian. In each case, find the number of cocks, hens and chicks bought." English translation Ang Tian Se, a student of University of Malaya, prepared an English translation of Zhang Qiujian Suanjing as part of the MA Dissertation. But the translation has not been published.[1][4] References 1. Lam Lay Yong (2008). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (Editor: Helaine Selin). Berlin: Springer-Verlag. pp. 2353–2354. ISBN 978-1-4020-4960-6. 2. Robertson, E. F.; O'Connor, J. J. "Zhang Qiujian biography". www-history.mcs.st-andrews.ac.uk. Retrieved 2016-12-01. 3. Lam Lay Yong (September 1997). "Zhang Qiujian Suanjing (The Mathematical Classic of Zhang Qiujian)." An Overview". Archive for History of Exact Sciences. 50 (34): 201–240. JSTOR 41134109. 4. Ang Tian Se (1969). A Study of the Mathematical Manual of Chang Ch’iu-Chien. M.A. Dissertation, University of Malaya (Unpublished).	Wikipedia
Moving frame In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. Introduction In lay terms, a frame of reference is a system of measuring rods used by an observer to measure the surrounding space by providing coordinates. A moving frame is then a frame of reference which moves with the observer along a trajectory (a curve). The method of the moving frame, in this simple example, seeks to produce a "preferred" moving frame out of the kinematic properties of the observer. In a geometrical setting, this problem was solved in the mid 19th century by Jean Frédéric Frenet and Joseph Alfred Serret.[1] The Frenet–Serret frame is a moving frame defined on a curve which can be constructed purely from the velocity and acceleration of the curve.[2] The Frenet–Serret frame plays a key role in the differential geometry of curves, ultimately leading to a more or less complete classification of smooth curves in Euclidean space up to congruence.[3] The Frenet–Serret formulas show that there is a pair of functions defined on the curve, the torsion and curvature, which are obtained by differentiating the frame, and which describe completely how the frame evolves in time along the curve. A key feature of the general method is that a preferred moving frame, provided it can be found, gives a complete kinematic description of the curve. In the late 19th century, Gaston Darboux studied the problem of constructing a preferred moving frame on a surface in Euclidean space instead of a curve, the Darboux frame (or the trièdre mobile as it was then called). It turned out to be impossible in general to construct such a frame, and that there were integrability conditions which needed to be satisfied first.[1] Later, moving frames were developed extensively by Élie Cartan and others in the study of submanifolds of more general homogeneous spaces (such as projective space). In this setting, a frame carries the geometric idea of a basis of a vector space over to other sorts of geometrical spaces (Klein geometries). Some examples of frames are:[3] • A linear frame is an ordered basis of a vector space. • An orthonormal frame of a vector space is an ordered basis consisting of orthogonal unit vectors (an orthonormal basis). • An affine frame of an affine space consists of a choice of origin along with an ordered basis of vectors in the associated difference space.[4] • A Euclidean frame of an affine space is a choice of origin along with an orthonormal basis of the difference space. • A projective frame on n-dimensional projective space is an ordered collection of n+1 linearly independent points in the space. • Frame fields in general relativity are four-dimensional frames, or vierbeins, in German. In each of these examples, the collection of all frames is homogeneous in a certain sense. In the case of linear frames, for instance, any two frames are related by an element of the general linear group. Projective frames are related by the projective linear group. This homogeneity, or symmetry, of the class of frames captures the geometrical features of the linear, affine, Euclidean, or projective landscape. A moving frame, in these circumstances, is just that: a frame which varies from point to point. Formally, a frame on a homogeneous space G/H consists of a point in the tautological bundle G → G/H. A moving frame is a section of this bundle. It is moving in the sense that as the point of the base varies, the frame in the fibre changes by an element of the symmetry group G. A moving frame on a submanifold M of G/H is a section of the pullback of the tautological bundle to M. Intrinsically[5] a moving frame can be defined on a principal bundle P over a manifold. In this case, a moving frame is given by a G-equivariant mapping φ : P → G, thus framing the manifold by elements of the Lie group G. One can extend the notion of frames to a more general case: one can "solder" a fiber bundle to a smooth manifold, in such a way that the fibers behave as if they were tangent. When the fiber bundle is a homogenous space, this reduces to the above-described frame-field. When the homogenous space is a quotient of special orthogonal groups, this reduces to the standard conception of a vierbein. Although there is a substantial formal difference between extrinsic and intrinsic moving frames, they are both alike in the sense that a moving frame is always given by a mapping into G. The strategy in Cartan's method of moving frames, as outlined briefly in Cartan's equivalence method, is to find a natural moving frame on the manifold and then to take its Darboux derivative, in other words pullback the Maurer-Cartan form of G to M (or P), and thus obtain a complete set of structural invariants for the manifold.[3] Method of the moving frame Cartan (1937) formulated the general definition of a moving frame and the method of the moving frame, as elaborated by Weyl (1938). The elements of the theory are • A Lie group G. • A Klein space X whose group of geometric automorphisms is G. • A smooth manifold Σ which serves as a space of (generalized) coordinates for X. • A collection of frames ƒ each of which determines a coordinate function from X to Σ (the precise nature of the frame is left vague in the general axiomatization). The following axioms are then assumed to hold between these elements: • There is a free and transitive group action of G on the collection of frames: it is a principal homogeneous space for G. In particular, for any pair of frames ƒ and ƒ′, there is a unique transition of frame (ƒ→ƒ′) in G determined by the requirement (ƒ→ƒ′)ƒ = ƒ′. • Given a frame ƒ and a point A ∈ X, there is associated a point x = (A,ƒ) belonging to Σ. This mapping determined by the frame ƒ is a bijection from the points of X to those of Σ. This bijection is compatible with the law of composition of frames in the sense that the coordinate x′ of the point A in a different frame ƒ′ arises from (A,ƒ) by application of the transformation (ƒ→ƒ′). That is, $(A,f')=(f\to f')\circ (A,f).$ Of interest to the method are parameterized submanifolds of X. The considerations are largely local, so the parameter domain is taken to be an open subset of Rλ. Slightly different techniques apply depending on whether one is interested in the submanifold along with its parameterization, or the submanifold up to reparameterization. Moving tangent frames Main article: Frame bundle The most commonly encountered case of a moving frame is for the bundle of tangent frames (also called the frame bundle) of a manifold. In this case, a moving tangent frame on a manifold M consists of a collection of vector fields e1, e2, …, en forming a basis of the tangent space at each point of an open set U ⊂ M. If $(x^{1},x^{2},\dots ,x^{n})$ is a coordinate system on U, then each vector field ej can be expressed as a linear combination of the coordinate vector fields $ {\frac {\partial }{\partial x^{i}}}$: $e_{j}=\sum _{i=1}^{n}A_{j}^{i}{\frac {\partial }{\partial x^{i}}},$ where each $A_{j}^{i}$ is a function on U. These can be seen as the components of a matrix $A$. This matrix is useful for finding the coordinate expression of the dual coframe, as explained in the next section. Coframes A moving frame determines a dual frame or coframe of the cotangent bundle over U, which is sometimes also called a moving frame. This is a n-tuple of smooth 1-forms θ1, θ2, …, θn which are linearly independent at each point q in U. Conversely, given such a coframe, there is a unique moving frame e1, e2, …, en which is dual to it, i.e., satisfies the duality relation θi(ej) = δij, where δij is the Kronecker delta function on U. If $(x^{1},x^{2},\dots ,x^{n})$ is a coordinate system on U, as in the preceding section, then each covector field θi can be expressed as a linear combination of the coordinate covector fields $dx^{i}$: $\theta ^{i}=\sum _{j=1}^{n}B_{j}^{i}dx^{j},$ where each $B_{j}^{i}$ is a function on U. Since $ dx^{i}\left({\frac {\partial }{\partial x^{j}}}\right)=\delta _{j}^{i}$, the two coordinate expressions above combine to yield $ \sum _{k=1}^{n}B_{k}^{i}A_{j}^{k}=\delta _{j}^{i}$; in terms of matrices, this just says that $A$ and $B$ are inverses of each other. In the setting of classical mechanics, when working with canonical coordinates, the canonical coframe is given by the tautological one-form. Intuitively, it relates the velocities of a mechanical system (given by vector fields on the tangent bundle of the coordinates) to the corresponding momenta of the system (given by vector fields in the cotangent bundle; i.e. given by forms). The tautological one-form is a special case of the more general solder form, which provides a (co-)frame field on a general fiber bundle. Uses Moving frames are important in general relativity, where there is no privileged way of extending a choice of frame at an event p (a point in spacetime, which is a manifold of dimension four) to nearby points, and so a choice must be made. In contrast in special relativity, M is taken to be a vector space V (of dimension four). In that case a frame at a point p can be translated from p to any other point q in a well-defined way. Broadly speaking, a moving frame corresponds to an observer, and the distinguished frames in special relativity represent inertial observers. In relativity and in Riemannian geometry, the most useful kind of moving frames are the orthogonal and orthonormal frames, that is, frames consisting of orthogonal (unit) vectors at each point. At a given point p a general frame may be made orthonormal by orthonormalization; in fact this can be done smoothly, so that the existence of a moving frame implies the existence of a moving orthonormal frame. Further details A moving frame always exists locally, i.e., in some neighbourhood U of any point p in M; however, the existence of a moving frame globally on M requires topological conditions. For example when M is a circle, or more generally a torus, such frames exist; but not when M is a 2-sphere. A manifold that does have a global moving frame is called parallelizable. Note for example how the unit directions of latitude and longitude on the Earth's surface break down as a moving frame at the north and south poles. The method of moving frames of Élie Cartan is based on taking a moving frame that is adapted to the particular problem being studied. For example, given a curve in space, the first three derivative vectors of the curve can in general define a frame at a point of it (cf. torsion tensor for a quantitative description – it is assumed here that the torsion is not zero). In fact, in the method of moving frames, one more often works with coframes rather than frames. More generally, moving frames may be viewed as sections of principal bundles over open sets U. The general Cartan method exploits this abstraction using the notion of a Cartan connection. Atlases In many cases, it is impossible to define a single frame of reference that is valid globally. To overcome this, frames are commonly pieced together to form an atlas, thus arriving at the notion of a local frame. In addition, it is often desirable to endow these atlases with a smooth structure, so that the resulting frame fields are differentiable. Generalizations Although this article constructs the frame fields as a coordinate system on the tangent bundle of a manifold, the general ideas move over easily to the concept of a vector bundle, which is a manifold endowed with a vector space at each point, that vector space being arbitrary, and not in general related to the tangent bundle. Applications Aircraft maneuvers can be expressed in terms of the moving frame (Aircraft principal axes) when described by the pilot. See also • Darboux frame • Frenet–Serret formulas • Yaw, pitch, and roll Notes 1. Chern 1985 2. D. J. Struik, Lectures on classical differential geometry, p. 18 3. Griffiths 1974 4. "Affine frame" Proofwiki.org 5. See Cartan (1983) 9.I; Appendix 2 (by Hermann) for the bundle of tangent frames. Fels and Olver (1998) for the case of more general fibrations. Griffiths (1974) for the case of frames on the tautological principal bundle of a homogeneous space. References • Cartan, Élie (1937), La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile, Paris: Gauthier-Villars. • Cartan, Élie (1983), Geometry of Riemannian Spaces, Math Sci Press, Massachusetts. • Chern, S.-S. (1985), "Moving frames", Elie Cartan et les Mathematiques d'Aujourd'hui, Asterisque, numero hors serie, Soc. Math. France, pp. 67–77. • Cotton, Émile (1905), "Genéralisation de la theorie du trièdre mobile", Bull. Soc. Math. France, 33: 1–23. • Darboux, Gaston (1887), Leçons sur la théorie génerale des surfaces, vol. I, Gauthier-Villars. • Darboux, Gaston (1915), Leçons sur la théorie génerale des surfaces, vol. II, Gauthier-Villars. • Darboux, Gaston (1894), Leçons sur la théorie génerale des surfaces, vol. III, Gauthier-Villars. • Darboux, Gaston (1896), Leçons sur la théorie génerale des surfaces, vol. IV, Gauthier-Villars. • Ehresmann, C. (1950), "Les connexions infinitésimals dans un espace fibré differential", Colloque de Topologie, Bruxelles, pp. 29–55. • Evtushik, E.L. (2001) [1994], "Moving-frame method", Encyclopedia of Mathematics, EMS Press. • Fels, M.; Olver, P.J. (1999), "Moving coframes II: Regularization and Theoretical Foundations", Acta Applicandae Mathematicae, 55 (2): 127, doi:10.1023/A:1006195823000, S2CID 826629. • Green, M (1978), "The moving frame, differential invariants and rigidity theorem for curves in homogeneous spaces", Duke Mathematical Journal, 45 (4): 735–779, doi:10.1215/S0012-7094-78-04535-0, S2CID 120620785. • Griffiths, Phillip (1974), "On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry", Duke Mathematical Journal, 41 (4): 775–814, doi:10.1215/S0012-7094-74-04180-5, S2CID 12966544 • Guggenheimer, Heinrich (1977), Differential Geometry, New York: Dover Publications. • Sharpe, R. W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94732-7. • Spivak, Michael (1999), A Comprehensive introduction to differential geometry, vol. 3, Houston, TX: Publish or Perish. • Sternberg, Shlomo (1964), Lectures on Differential Geometry, Prentice Hall. • Weyl, Hermann (1938), "Cartan on groups and differential geometry", Bulletin of the American Mathematical Society, 44 (9): 598–601, doi:10.1090/S0002-9904-1938-06789-4. Manifolds (Glossary) Basic concepts • Topological manifold • Atlas • Differentiable/Smooth manifold • Differential structure • Smooth atlas • Submanifold • Riemannian manifold • Smooth map • Submersion • Pushforward • Tangent space • Differential form • Vector field Main results (list) • Atiyah–Singer index • Darboux's • De Rham's • Frobenius • Generalized Stokes • Hopf–Rinow • Noether's • Sard's • Whitney embedding Maps • Curve • Diffeomorphism • Local • Geodesic • Exponential map • in Lie theory • Foliation • Immersion • Integral curve • Lie derivative • Section • Submersion Types of manifolds • Closed • (Almost) Complex • (Almost) Contact • Fibered • Finsler • Flat • G-structure • Hadamard • Hermitian • Hyperbolic • Kähler • Kenmotsu • Lie group • Lie algebra • Manifold with boundary • Oriented • Parallelizable • Poisson • Prime • Quaternionic • Hypercomplex • (Pseudo−, Sub−) Riemannian • Rizza • (Almost) Symplectic • Tame Tensors Vectors • Distribution • Lie bracket • Pushforward • Tangent space • bundle • Torsion • Vector field • Vector flow Covectors • Closed/Exact • Covariant derivative • Cotangent space • bundle • De Rham cohomology • Differential form • Vector-valued • Exterior derivative • Interior product • Pullback • Ricci curvature • flow • Riemann curvature tensor • Tensor field • density • Volume form • Wedge product Bundles • Adjoint • Affine • Associated • Cotangent • Dual • Fiber • (Co) Fibration • Jet • Lie algebra • (Stable) Normal • Principal • Spinor • Subbundle • Tangent • Tensor • Vector Connections • Affine • Cartan • Ehresmann • Form • Generalized • Koszul • Levi-Civita • Principal • Vector • Parallel transport Related • Classification of manifolds • Gauge theory • History • Morse theory • Moving frame • Singularity theory Generalizations • Banach manifold • Diffeology • Diffiety • Fréchet manifold • K-theory • Orbifold • Secondary calculus • over commutative algebras • Sheaf • Stratifold • Supermanifold • Stratified space	Wikipedia
\star\begin\star{document\star}\star \star \star \star\title{Correcting Quantum Errors with Entanglement}\star \star \star \star \star \star \star\begin\star{sciabstract\star}\star We\star show\star how\star entanglement\star shared\star between\star encoder\star and\star decoder\star can\star simplify\star the\star theory\star of\star quantum\star error\star correction\star.\star \star The\star entanglement\star-assisted\star quantum\star codes\star we\star describe\star do\star not\star require\star the\star dual\star-containing\star constraint\star necessary\star for\star standard\star quantum\star error\star correcting\star codes\star,\star thus\star allowing\star us\star to\star \star`\star`quantize\star'\star'\star all\star of\star classical\star linear\star coding\star theory\star.\star In\star particular\star,\star efficient\star modern\star classical\star codes\star that\star attain\star the\star Shannon\star capacity\star can\star be\star made\star into\star entanglement\star-assisted\star quantum\star codes\star attaining\star the\star hashing\star bound\star \star(closely\star related\star to\star the\star quantum\star capacity\star)\star.\star For\star systems\star without\star large\star amounts\star of\star shared\star entanglement\star,\star these\star codes\star can\star also\star be\star used\star as\star catalytic\star codes\star,\star in\star which\star a\star small\star amount\star of\star initial\star entanglement\star enables\star quantum\star communication\star.\star \star\end\star{sciabstract\star}\star \star \star Entanglement\star plays\star a\star central\star role\star in\star quantum\star information\star processing\star.\star It\star enables\star the\star teleportation\star of\star quantum\star states\star without\star physically\star sending\star quantum\star systems\star\cite\star{BBCJPW93\star}\star;\star it\star doubles\star the\star capacity\star of\star quantum\star channels\star for\star sending\star classical\star information\star\cite\star{BW92\star}\star;\star it\star is\star known\star to\star be\star necessary\star for\star the\star power\star of\star quantum\star computation\star\cite\star{BCD02\star,JL03\star}\star.\star We\star show\star how\star shared\star entanglement\star provides\star a\star simpler\star and\star more\star fundamental\star theory\star of\star quantum\star error\star correction\star.\star \star The\star theory\star of\star quantum\star error\star correcting\star codes\star was\star established\star a\star decade\star ago\star as\star the\star primary\star tool\star for\star fighting\star decoherence\star in\star quantum\star computers\star and\star quantum\star communication\star systems\star.\star \star The\star first\star nine\star-qubit\star single\star error\star-correcting\star code\star was\star a\star quantum\star analog\star of\star the\star classical\star repetition\star code\star,\star which\star stores\star information\star redundantly\star by\star duplicating\star each\star bit\star several\star times\star \star\cite\star{PS95\star}\star.\star \star Probably\star the\star most\star striking\star development\star in\star quantum\star error\star correction\star theory\star is\star the\star use\star of\star the\star stabilizer\star formalism\star\cite\star{CRSS97\star,DG97thesis\star,DG98\star,NC00\star}\star,\star whereby\star quantum\star codes\star are\star subspaces\star \star(\star`\star`code\star spaces\star'\star'\star)\star in\star Hilbert\star space\star,\star and\star are\star specified\star by\star giving\star the\star generators\star of\star an\star abelian\star subgroup\star of\star the\star Pauli\star group\star,\star called\star the\star stabilizer\star of\star the\star code\star space\star.\star Essentially\star,\star all\star QECCs\star developed\star to\star date\star are\star stabilizer\star codes\star.\star \star The\star problem\star of\star finding\star QECCs\star was\star reduced\star to\star that\star of\star constructing\star classical\star dual\star-containing\star quaternary\star codes\star\cite\star{CRSS97\star}\star.\star \star When\star binary\star codes\star are\star viewed\star as\star quaternary\star,\star this\star amounts\star to\star the\star well\star known\star Calderbank\star-Shor\star-Steane\star construction\star\cite\star{Ste96\star,CS96\star}\star.\star \star The\star requirement\star that\star a\star code\star contain\star its\star dual\star is\star a\star consequence\star of\star the\star need\star for\star a\star commuting\star stabilizer\star group\star.\star \star The\star virtue\star of\star this\star approach\star is\star that\star we\star can\star directly\star construct\star quantum\star codes\star from\star classical\star codes\star with\star a\star certain\star property\star,\star rather\star than\star having\star to\star develop\star a\star completely\star new\star theory\star of\star quantum\star error\star correction\star from\star scratch\star.\star \star Unfortunately\star,\star the\star need\star for\star a\star self\star-orthogonal\star parity\star check\star matrix\star presents\star a\star substantial\star obstacle\star to\star importing\star the\star classical\star theory\star in\star its\star entirety\star,\star especially\star in\star the\star context\star of\star modern\star codes\star such\star as\star low\star-density\star parity\star check\star \star(LDPC\star)\star codes\star \star\cite\star{MMM04\star}\star.\star \star Assume\star that\star the\star encoder\star Alice\star and\star decoder\star Bob\star have\star access\star to\star shared\star entanglement\star.\star We\star will\star argue\star that\star in\star this\star setting\star every\star quaternary\star \star(or\star binary\star)\star classical\star linear\star code\star,\star not\star just\star dual\star-containing\star codes\star,\star can\star be\star transformed\star into\star a\star QECC\star,\star and\star illustrate\star this\star 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comes\star from\star the\star clever\star use\star of\star group\star theory\star.\star Let\star \star$\star\Pi\star$\star denote\star the\star set\star of\star Pauli\star operators\star \star$\star\star{I\star,X\star,Y\star,Z\star\star}\star$\star,\star and\star let\star \star$\star\Pi\star^n\star=\star\star{I\star,X\star,Y\star,Z\star\star}\star^\star{\star\otimes\star n\star}\star$\star denote\star the\star set\star of\star \star$n\star$\star-fold\star tensor\star products\star of\star single\star-qubit\star Pauli\star operators\star.\star Then\star \star$\star\Pi\star^n\star$\star together\star with\star the\star possible\star overall\star factors\star \star$\star\pm\star 1\star,\star \star\pm\star i\star$\star forms\star a\star group\star \star$\star{\cal G}} \def\cH{{\cal H}\star_n\star$\star under\star multiplication\star,\star the\star \star$n\star$\star-fold\star Pauli\star group\star.\star \star Here\star are\star a\star few\star useful\star properties\star of\star the\star 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T}\star)\star$\star related\star to\star the\star generators\star of\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star?\star \star First\star,\star suppose\star that\star \star$E\star_a\star$\star anti\star-commutes\star with\star a\star particular\star stabilizer\star generator\star \star$M\star_i\star$\star of\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star.\star \star Then\star \star\star[\star M\star_i\star E\star_a\star\ket\star{\star\psi\star}\star=\star-E\star_a\star M\star_i\star\ket\star{\star\psi\star}\star=\star-E\star_a\star\ket\star{\star\psi\star}\star.\star \star\star]\star \star$E\star_a\star\ket\star\psi\star$\star is\star an\star eigenvector\star of\star \star$M\star_i\star$\star with\star eigenvalue\star \star$\star-1\star$\star,\star and\star hence\star must\star be\star orthogonal\star to\star the\star code\star space\star \star(all\star of\star whose\star vectors\star have\star eigenvalue\star \star$\star+1\star$\star)\star.\star \star As\star the\star error\star operator\star \star$E\star_a\star$\star takes\star the\star code\star space\star of\star \star$\star{\cal C}\star(\star{\cal S}} \def\cT{{\cal T}\star)\star$\star to\star an\star orthogonal\star subspace\star,\star an\star occurrence\star of\star \star$E\star_a\star$\star can\star be\star detected\star by\star measuring\star \star$M\star_i\star$\star.\star \star For\star each\star generator\star \star$M\star_i\star$\star and\star error\star operator\star \star$E\star_a\star$\star,\star we\star can\star define\star a\star coefficient\star \star$s\star_\star{i\star,a\star}\star \star\in\star\star{0\star,\star 1\star\star}\star$\star depending\star on\star whether\star \star$M\star_i\star$\star and\star \star$E\star_a\star$\star commute\star or\star anti\star-commute\star:\star \star\star[\star M\star_i\star E\star_a\star=\star(\star-1\star)\star^\star{s\star_\star{i\star,a\star}\star}E\star_a\star M\star_i\star.\star \star\star]\star The\star vector\star \star$\star\overline\star{s\star}\star_a\star=\star(s\star_\star{1\star,a\star}\star,s\star_\star{2\star,a\star}\star,\star\cdots\star,s\star_\star{n\star-k\star,a\star}\star)\star$\star represents\star the\star syndrome\star of\star the\star error\star \star$E\star_a\star$\star.\star \star In\star the\star case\star of\star a\star nondegenerate\star code\star,\star the\star error\star syndrome\star is\star distinct\star for\star all\star \star$E\star_a\star \star\in\star\cE\star$\star,\star so\star that\star measuring\star the\star \star$n\star-k\star$\star stabilizer\star generators\star will\star diagnose\star the\star error\star completely\star.\star \star However\star,\star a\star uniquely\star identifiable\star error\star syndrome\star is\star not\star always\star required\star for\star an\star error\star to\star be\star correctable\star.\star \star What\star if\star \star$E\star_a\star$\star commutes\star with\star the\star generators\star of\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star?\star \star If\star \star$E\star_a\star \star\in\star \star{\cal S}} \def\cT{{\cal T}\star$\star,\star we\star do\star not\star need\star to\star worry\star,\star because\star the\star error\star does\star not\star corrupt\star the\star space\star at\star all\star.\star \star The\star real\star danger\star comes\star when\star \star$E\star_a\star$\star commutes\star with\star all\star the\star elements\star of\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star but\star is\star not\star itself\star in\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star.\star \star The\star set\star of\star elements\star in\star \star$\star{\cal G}} \def\cH{{\cal H}\star_n\star$\star that\star commute\star with\star all\star of\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star is\star the\star centralizer\star \star$\star\cZ\star(\star{\cal S}} \def\cT{{\cal T}\star)\star$\star of\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star.\star \star If\star \star$E\star \star\in\star \star\cZ\star(\star{\cal S}} \def\cT{{\cal T}\star)\star-\star{\cal S}} \def\cT{{\cal T}\star$\star,\star then\star \star$E\star$\star changes\star elements\star of\star \star$\star{\cal C}\star(\star{\cal S}} \def\cT{{\cal T}\star)\star$\star but\star does\star not\star take\star them\star out\star of\star \star$\star{\cal C}\star(\star{\cal S}} \def\cT{{\cal T}\star)\star$\star.\star Thus\star,\star if\star \star$M\star \star\in\star \star{\cal S}} \def\cT{{\cal T}\star$\star and\star \star$\star\ket\star{\star\psi\star}\star\in\star \star{\cal C}\star(\star{\cal S}} \def\cT{{\cal T}\star)\star$\star,\star then\star \star\star[\star ME\star\ket\star{\star\psi\star}\star=EM\star\ket\star{\star\psi\star}\star=E\star\ket\star{\star\psi\star}\star.\star \star\star]\star Because\star \star$E\star \star\not\star\in\star \star{\cal S}} \def\cT{{\cal T}\star$\star,\star there\star is\star some\star state\star of\star \star$\star{\cal C}\star(\star{\cal S}} \def\cT{{\cal 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the\star entanglement\star-assisted\star stabilizer\star formalism\star by\star an\star example\star.\star \star We\star know\star from\star the\star previous\star paragraph\star that\star a\star stabilizer\star code\star can\star be\star constructed\star from\star a\star commuting\star set\star of\star operators\star in\star \star$\star{\cal G}} \def\cH{{\cal H}\star_n\star$\star.\star \star What\star if\star we\star are\star given\star a\star non\star-commuting\star set\star of\star operators\star?\star Can\star we\star still\star construct\star a\star QECC\star?\star Let\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star be\star the\star group\star generated\star by\star the\star following\star non\star-commuting\star set\star of\star operators\star:\star \star\begin\star{equation\star}\star \star\begin\star{array\star}\star{ccccc\star}\star \star\label\star{setM\star}\star M\star_1\star=\star \star&\star Z\star \star&\star X\star \star&\star Z\star \star&\star I\star \star \star \star\star M\star_2\star=\star \star&\star Z\star \star&\star Z\star \star&\star I\star \star&\star Z\star \star \star \star\star M\star_3\star=\star \star&\star X\star \star&\star Y\star \star&\star X\star \star&\star I\star \star \star \star\star M\star_4\star=\star \star&\star X\star \star&\star X\star \star&\star I\star \star&\star X\star \star\end\star{array\star}\star \star\end\star{equation\star}\star It\star is\star easy\star to\star check\star the\star commutation\star relations\star of\star this\star set\star of\star generators\star:\star \star$M\star_1\star$\star anti\star-commutes\star with\star the\star other\star three\star generators\star,\star \star$M\star_2\star$\star commutes\star with\star \star$M\star_3\star$\star and\star anti\star-commutes\star with\star \star$M\star_4\star$\star,\star and\star \star$M\star_3\star$\star and\star \star$M\star_4\star$\star anti\star-commute\star.\star \star We\star will\star begin\star by\star finding\star a\star different\star set\star of\star generators\star for\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star with\star a\star particular\star class\star of\star commutation\star relations\star.\star \star We\star then\star relate\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star to\star a\star group\star \star$\star\cB\star$\star with\star a\star particularly\star simple\star form\star,\star and\star discuss\star the\star error\star-correcting\star conditions\star using\star \star$\star\cB\star$\star.\star \star Finally\star,\star we\star relate\star these\star results\star back\star to\star the\star group\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star.\star \star To\star see\star how\star this\star works\star,\star we\star need\star two\star lemmas\star.\star \star \star(See\star the\star supporting\star online\star material\star for\star proofs\star.\star)\star \star The\star first\star lemma\star shows\star that\star there\star exists\star a\star new\star set\star of\star generators\star for\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star such\star that\star \star$\star{\cal S}} \def\cT{{\cal T}\star$\star can\star be\star decomposed\star into\star an\star \star`\star`isotropic\star'\star'\star subgroup\star \star$\star{\cal S}} \def\cT{{\cal T}\star_I\star$\star generated\star by\star a\star set\star of\star commuting\star generators\star,\star and\star a\star \star`\star`symplectic\star'\star'\star subgroup\star \star$\star{\cal S}} \def\cT{{\cal T}\star_S\star$\star generated\star by\star a\star set\star of\star anti\star-commuting\star generator\star pairs\star \star\cite\star{FCY04\star}\star.\star \star \star\noindent\star \star{\star\it\star Lemma\star 1\star.\star}\star Given\star any\star arbitrary\star subgroup\star \star$\star{\cal V}} \def\cW{{\cal W}\star$\star in\star \star$\star{\cal G}} \def\cH{{\cal H}\star_n\star$\star that\star has\star \star$2\star^\star{m\star}\star$\star distinct\star elements\star up\star to\star overall\star phase\star,\star there\star exists\star a\star set\star of\star \star$m\star$\star independent\star generators\star for\star \star$\star{\cal V}} \def\cW{{\cal W}\star$\star of\star the\star form\star \star$\star\star{\star\overline\star{Z\star}\star_1\star,\star\overline\star{Z\star}\star_2\star,\star\cdots\star,\star\overline\star{Z\star}\star_\star\ell\star,\star\overline\star{X\star}\star_1\star,\star\cdots\star,\star\overline\star{X\star}\star_\star{m\star-l\star}\star\star}\star$\star where\star \star$m\star/2\star \star\leq\star \star\ell\star \star\leq\star m\star$\star,\star such\star that\star \star$\star[\star \star\overline\star{Z\star}\star_i\star \star,\star \star\overline\star{Z\star}\star_j\star]\star \star=\star \star[\star \star\overline\star{X\star}\star_i\star \star,\star \star\overline\star{X\star}\star_j\star]\star \star=\star 0\star$\star,\star for\star all\star \star$i\star,j\star$\star;\star \star$\star[\star \star\overline\star{Z\star}\star_i\star \star,\star 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\star$Z\star_1\star$\star,\star an\star \star$X\star$\star operator\star at\star the\star end\star of\star \star$X\star_1\star$\star,\star and\star an\star identity\star at\star the\star end\star of\star \star$Z\star_2\star$\star and\star \star$Z\star_3\star$\star to\star make\star \star$\star\cB\star$\star abelian\star:\star \star\begin\star{equation\star}\star \star\begin\star{array\star}\star{ccccc\star\|c\star}\star Z\star_1\star'\star=\star \star&\star Z\star \star&\star I\star \star&\star I\star \star&\star I\star \star&\star Z\star \star\star X\star_1\star'\star=\star \star&\star X\star \star&\star I\star \star&\star I\star \star&\star I\star \star&\star X\star \star\star Z\star_2\star'\star=\star \star&\star I\star \star&\star Z\star \star&\star I\star \star&\star I\star \star&\star I\star \star\star Z\star_3\star'\star=\star \star&\star I\star \star&\star I\star \star&\star Z\star \star&\star I\star \star&\star I\star \star\end\star{array\star}\star 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from\star \star$c\star$\star maximally\star entangled\star pairs\star that\star she\star prepares\star locally\star.\star \star After\star her\star \star$n\star$\star bits\star have\star been\star transmitted\star to\star Bob\star,\star corrected\star and\star decoded\star,\star Bob\star has\star received\star \star$k\star-c\star$\star qubits\star,\star plus\star \star$c\star$\star new\star bits\star of\star entanglement\star have\star been\star created\star.\star \star These\star can\star then\star be\star used\star to\star send\star another\star \star$k\star-c\star$\star bits\star,\star and\star so\star on\star.\star The\star idea\star is\star that\star the\star perfect\star qubit\star channel\star that\star is\star simulated\star by\star the\star code\star is\star a\star stronger\star resource\star than\star pre\star-existing\star entanglement\star\cite\star{DHW03\star}\star.\star \star It\star is\star this\star catalytic\star mode\star of\star performance\star that\star makes\star the\star rate\star \star$\star(k\star-c\star)\star/n\star$\star a\star reasonable\star figure\star of\star merit\star for\star an\star EAQECC\star as\star described\star above\star.\star \star Clearly\star the\star \star$\star[\star[4\star,1\star,3\star;\star 1\star]\star]\star$\star code\star described\star in\star this\star paper\star is\star useless\star as\star a\star catalytic\star code\star,\star though\star it\star is\star perfectly\star useful\star for\star an\star entanglement\star-assisted\star channel\star.\star \star To\star be\star a\star useful\star catalytic\star code\star,\star an\star EAQECC\star must\star have\star a\star positive\star value\star of\star \star$k\star-c\star$\star.\star \star \star We\star have\star presented\star EAQECCs\star in\star a\star communication\star context\star up\star to\star now\star,\star but\star catalytic\star codes\star open\star the\star possibility\star of\star application\star to\star error\star correction\star in\star quantum\star computing\star,\star where\star we\star can\star think\star of\star decoherence\star as\star a\star channel\star into\star the\star future\star.\star \star In\star this\star case\star,\star the\star \star`\star`seed\star'\star'\star resource\star is\star not\star pre\star-existing\star entanglement\star,\star but\star rather\star a\star small\star number\star of\star qubits\star that\star are\star error\star-free\star,\star either\star because\star they\star are\star physically\star isolated\star,\star or\star because\star they\star are\star protected\star by\star a\star decoherence\star-free\star subspace\star or\star standard\star QECC\star.\star \star \star CQECCs\star provide\star great\star flexibility\star in\star designing\star quantum\star communication\star schemes\star.\star \star For\star example\star,\star in\star periods\star of\star low\star usage\star we\star can\star use\star an\star EAQECC\star in\star the\star catalytic\star mode\star to\star build\star up\star shared\star entanglement\star between\star Alice\star and\star Bob\star.\star \star Then\star in\star periods\star of\star peak\star demand\star,\star we\star can\star draw\star on\star that\star entanglement\star to\star increase\star the\star capacity\star.\star \star Quantum\star networks\star of\star the\star future\star can\star use\star schemes\star like\star this\star to\star optimize\star performance\star.\star \star In\star any\star case\star,\star the\star existence\star of\star practical\star EAQECCs\star will\star greatly\star enhance\star the\star power\star of\star quantum\star communications\star,\star as\star well\star as\star providing\star a\star beautiful\star connection\star to\star the\star theory\star of\star classical\star error\star correction\star codes\star.\star \star \star\bibliography\star{ref4\star}\star \star \star\bibliographystyle\star{Science\star}\star \star \star\begin\star{scilastnote\star}\star \star\item\star We\star would\star like\star to\star acknowledge\star helpful\star feedback\star from\star David\star Poulin\star,\star Graeme\star Smith\star and\star Jon\star Yard\star.\star \star TAB\star acknowledges\star financial\star support\star from\star NSF\star Grant\star No\star.\star~CCF\star-0448658\star,\star and\star TAB\star and\star MHH\star both\star received\star support\star from\star NSF\star Grant\star No\star.\star~ECS\star-0507270\star.\star \star ID\star and\star MHH\star acknowledge\star financial\star support\star from\star NSF\star Grant\star No\star.\star~CCF\star-0524811\star and\star NSF\star Grant\star No\star.\star~CCF\star-0545845\star.\star \star\end\star{scilastnote\star}\star \star \star \star\end\star{document\star}\star	arXiv
Rooted product of graphs In mathematical graph theory, the rooted product of a graph G and a rooted graph H is defined as follows: take \|V(G)\| copies of H, and for every vertex vi of G, identify vi with the root node of the i-th copy of H. More formally, assuming that ${\begin{aligned}V(G)&=\{g_{1},\ldots ,g_{n}\},\\V(H)&=\{h_{1},\ldots ,h_{m}\},\end{aligned}}$ and that the root node of H is h1, define $G\circ H:=(V,E)$, where $V=\left\{(g_{i},h_{j}):1\leq i\leq n,1\leq j\leq m\right\}$ and $E={\Bigl \{}{\bigl (}(g_{i},h_{1}),(g_{k},h_{1}){\bigr )}:(g_{i},g_{k})\in E(G){\Bigr \}}\cup \bigcup _{i=1}^{n}{\Bigl \{}{\bigl (}(g_{i},h_{j}),(g_{i},h_{k}){\bigr )}:(h_{j},h_{k})\in E(H){\Bigr \}}$. If G is also rooted at g1, one can view the product itself as rooted, at (g1, h1). The rooted product is a subgraph of the cartesian product of the same two graphs. Applications The rooted product is especially relevant for trees, as the rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find graceful numberings for a wide family of trees. If H is a two-vertex complete graph K2, then for any graph G, the rooted product of G and H has domination number exactly half of its number of vertices. Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-vertex cycle graph. These graphs can be used to generate examples in which the bound of Vizing's conjecture, an unproven inequality between the domination number of the graphs in a different graph product, the cartesian product of graphs, is exactly met (Fink et al. 1985). They are also well-covered graphs. References • Godsil, C. D.; McKay, B. D. (1978), "A new graph product and its spectrum" (PDF), Bull. Austral. Math. Soc., 18 (1): 21–28, doi:10.1017/S0004972700007760, MR 0494910. • Fink, J. F.; Jacobson, M. S.; Kinch, L. F.; Roberts, J. (1985), "On graphs having domination number half their order", Period. Math. Hungar., 16 (4): 287–293, doi:10.1007/BF01848079, MR 0833264. • Koh, K. M.; Rogers, D. G.; Tan, T. (1980), "Products of graceful trees", Discrete Mathematics, 31 (3): 279–292, doi:10.1016/0012-365X(80)90139-9, MR 0584121.	Wikipedia
cell discovery Characterization of respiratory microbial dysbiosis in hospitalized COVID-19 patients Temporal association between human upper respiratory and gut bacterial microbiomes during the course of COVID-19 in adults Rong Xu, Renfei Lu, … Chiyu Zhang High resolution metagenomic characterization of complex infectomes in paediatric acute respiratory infection Ci-Xiu Li, Wei Li, … Mang Shi Temporal dynamics of oropharyngeal microbiome among SARS-CoV-2 patients reveals continued dysbiosis even after Viral Clearance Suman Kalyan Paine, Usha Kiran Rout, … Analabha Basu SARS-CoV-2 does not have a strong effect on the nasopharyngeal microbial composition Tzipi Braun, Shiraz Halevi, … Yael Haberman A proof of concept study for the differentiation of SARS-CoV-2, hCoV-NL63, and IAV-H1N1 in vitro cultures using ion mobility spectrometry M. Feuerherd, A.-K. Sippel, … C. D. Spinner Co-infections observed in SARS-CoV-2 positive patients using a rapid diagnostic test Carla Fontana, Marco Favaro, … Anna Altieri The lung microbiome in HIV-positive patients with active pulmonary tuberculosis Veronica Ueckermann, Pedro Lebre, … Marthie Ehlers Tracking SARS-COV-2 variants using Nanopore sequencing in Ukraine in 2021 Anna Yakovleva, Ganna Kovalenko, … Tetyana I. Vasylyeva Platform for isolation and characterization of SARS-CoV-2 variants enables rapid characterization of Omicron in Australia Anupriya Aggarwal, Alberto Ospina Stella, … Stuart G. Turville Huanzi Zhong ORCID: orcid.org/0000-0001-9512-17501,2 na1, Yanqun Wang3 na1, Zhun Shi1 na1, Lu Zhang4,5 na1, Huahui Ren1,2 na1, Weiqun He3 na1, Zhaoyong Zhang3 na1, Airu Zhu3 na1, Jingxian Zhao3 na1, Fei Xiao6 na1, Fangming Yang1,7, Tianzhu Liang1,8, Feng Ye3, Bei Zhong9, Shicong Ruan10, Mian Gan3, Jiahui Zhu1,11, Fang Li3, Fuqiang Li ORCID: orcid.org/0000-0002-2085-14571,12, Daxi Wang1,8, Jiandong Li1,8,13, Peidi Ren1,8, Shida Zhu1,14, Huanming Yang1,15,16, Jian Wang1,15, Karsten Kristiansen ORCID: orcid.org/0000-0002-6024-09171,2, Hein Min Tun17, Weijun Chen13,18, Nanshan Zhong3, Xun Xu ORCID: orcid.org/0000-0002-5338-51731,19 na2, Yi-min Li3 na2, Junhua Li ORCID: orcid.org/0000-0001-6784-18731,8,20 na2 & Jincun Zhao3,4 na2 Cell Discovery volume 7, Article number: 23 (2021) Cite this article Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused a global pandemic of Coronavirus disease 2019 (COVID-19). However, the microbial composition of the respiratory tract and other infected tissues as well as their possible pathogenic contributions to varying degrees of disease severity in COVID-19 patients remain unclear. Between 27 January and 26 February 2020, serial clinical specimens (sputum, nasal and throat swab, anal swab and feces) were collected from a cohort of hospitalized COVID-19 patients, including 8 mildly and 15 severely ill patients in Guangdong province, China. Total RNA was extracted and ultra-deep metatranscriptomic sequencing was performed in combination with laboratory diagnostic assays. We identified distinct signatures of microbial dysbiosis among severely ill COVID-19 patients on broad spectrum antimicrobial therapy. Co-detection of other human respiratory viruses (including human alphaherpesvirus 1, rhinovirus B, and human orthopneumovirus) was demonstrated in 30.8% (4/13) of the severely ill patients, but not in any of the mildly affected patients. Notably, the predominant respiratory microbial taxa of severely ill patients were Burkholderia cepacia complex (BCC), Staphylococcus epidermidis, or Mycoplasma spp. (including M. hominis and M. orale). The presence of the former two bacterial taxa was also confirmed by clinical cultures of respiratory specimens (expectorated sputum or nasal secretions) in 23.1% (3/13) of the severe cases. Finally, a time-dependent, secondary infection of B. cenocepacia with expressions of multiple virulence genes was demonstrated in one severely ill patient, which might accelerate his disease deterioration and death occurring one month after ICU admission. Our findings point to SARS-CoV-2-related microbial dysbiosis and various antibiotic-resistant respiratory microbes/pathogens in hospitalized COVID-19 patients in relation to disease severity. Detection and tracking strategies are needed to prevent the spread of antimicrobial resistance, improve the treatment regimen and clinical outcomes of hospitalized, severely ill COVID-19 patients. As of 31 January 2021, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has infected more than 102 million and resulted in more than 2.2 million deaths worldwide1. The pandemic poses a significant threat to public health and the global economy. Respiratory viruses, such as coronaviruses and influenza virus, can lead to acute damage of the epithelial barrier and facilitate invasions of other pathogens2,3. For instance, secondary infections by Stenotrophomonas maltophilia, Klebsiella pneumoniae, or Escherichia coli were reported to cause serious complications in patients with SARS, such as bacteremia, sepsis, and nosocomial pneumonia (NP)4. In addition, Streptococcus pneumoniae, Haemophilus influenzae, and Staphylococcus aureus were frequently associated with NP and mortality in influenza pandemics5. It was estimated that approximately 29%–55% of the total 300,000 deaths in the 2009 H1N1 pandemic were caused by secondary bacterial NP6,7,8. Concerns about the coinfections of SARS-CoV-2 with known viruses, bacteria, and fungi have also been raised. In severely ill patients, acute respiratory distress syndrome deteriorates patients' conditions rapidly, and mechanical ventilation is generally required9,10. Such invasive procedures can further increase the risks of ventilator-associated pneumonia in these patients11. In 99 confirmed Wuhan patients enrolled in January 2020, one (1%) had positive cultures of Acinetobacter baumannii, K. pneumoniae, and Aspergillus flavus, and four (4%) were diagnosed with infection by Candida, but no influenza viruses were detected9. A later retrospective study in Wuhan patients further demonstrated that half of the deceased patients (27 out of 54) had experienced secondary infections12. By using real-time reverse transcriptase-polymerase chain reaction tests, Kim et al.13 recently reported a 20.7% (24 out of 116 specimens) coinfection rate with SARS-CoV-2 and other respiratory viruses in Northern California, including rhinovirus (6.9%) and orthopneumovirus (5.2%). However, microbial coinfections and their possible effects on clinical outcomes of SARS-CoV-2-infected patients remain largely unknown. Here, combining diagnostic technologies (cultures and colorimetric assays) and metatranscriptomic sequencing, microbial coinfections in a Guangdong cohort of 23 patients hospitalized with SARS-CoV-2 infection were comprehensively evaluated. Our results revealed distinct differences in microbial composition between mildly and severely ill patients in both the respiratory and gastrointestinal tract. We further demonstrate that Burkholderia cepacia complex (BCC) bacteria, Staphylococcus epidermidis and Mycoplasma spp. were most related to opportunistic pathogens of the respiratory tract in severe cases, possibly exhibiting resistance towards multiple antibiotics, increasing the risk for prolonged intensive care unit (ICU) stay or even associated with increased mortality. By contrast, Veillonella, Neisseria, Streptococcus, and Prevotella were identified as the dominant active microbes in the respiratory tract of patients with mild symptoms, similar to what has been reported for healthy adults without infection14. Our findings demonstrate the value of metatranscriptomics for an unbiased evaluation of the respiratory microbiota associated with SARS-CoV-2 and provide useful information and suggestions regarding the adequate monitoring and management of the multidrug-resistant (MDR) bacteria in the COVID-19 pandemic. Demographic information of patients and clinical specimens used in the study Twenty-three patients with COVID-19 hospitalized in the period 10 January–31 March 2020, in four hospitals in the Guangdong Province, China, were enrolled in this study. Fifteen infected patients (41–79 year old) admitted to the ICU and receiving mechanical ventilation were defined as having severe COVID-19, and the remaining eight patients (2–65 year old) were mild cases (Supplementary Table S1). Briefly, 95.7% of the patients (22 out of 23) received antiviral medications. All severe cases received broad-spectrum antibiotics to prevent and control nosocomial infections, and simultaneously 93.35% (14 out of 15) received antifungal agents (Supplementary Table S1). Also, 60% (9 out of 15) of severely ill patients received invasive mechanical ventilation. By contrast, none of the mild cases were treated with antibacterial or antifungal drugs. Up to 31 March, 2020, 53.3% (8 out of 15) of the severe ill patients had been transferred out of ICU or discharged from hospitals, and all mild cases had been discharged, whereas a 79-year-old patient died one month after admission to ICU (P01) (Supplementary Table S2). Sixty-seven serial clinical specimens from the respiratory tract (RT) (n = 47, sputum, nasal and throat swab) and gastrointestinal tract (GIT) (n = 20, anal swab and feces) of these patients were obtained between 27 January and 26 February 2020 for a comprehensive assessment of microbial characteristics after SARS-CoV-2 infection. A detailed timeline of specimen collections and clinical events for the 23 COVID-19 cases are shown in Supplementary Table S2. Workflow of ultra-deep metatranscriptomic sequencing After quality control, an average of 268.3 Gb metatranscriptomic data were generated per sample (Supplementary Table S3). We applied an integrated bioinformatics pipeline to detect human, viral, and nonviral microbial reads in the total RNA-seq data (Supplementary Fig. S1 and Materials and methods). The percentage of human RNA reads (including human rRNA and non-rRNA human transcripts) varied between different types of specimens, constituting a high fraction of total reads among RT specimens (average percentage of 64.56%) and a low fraction among GIT specimens (average percentage of 22.56%) (Supplementary Table S3 and Fig. S1). After removing host data, SortMeRNA was applied15 to filter microbial rRNA from the metatranscriptomic data. The final remaining nonhuman nonmicrobial rRNA data (ranged from 386 Mb to 145 Gb) were then used to assess viral and nonviral microbial composition by Kraken2X16 and MetaPhlAn217, respectively (Materials and methods). Detailed data statistics for each processing step are provided in Supplementary Table S3. Co-detection of viruses in clinical specimens of COVID-19 patients We first assessed the viral composition in RT and GIT specimens. As expected, Coronaviridae (mostly contributed by reads assigned to SARS-related coronavirus, Supplementary Table S3) was the most abundant virus, and was detected in all clinical specimens and varied between 0.01 and 286,418 mapped reads count per million (RPM) (Fig. 1a). Given the presence of confounding factors including days post symptom onset and various treatments on severe cases18, no comparison of the temporal abundance of SARS-CoV-2-like virus was conducted between mild and severe groups or between types of specimens. Although the SARS-CoV-2 RPM in samples of RT and GIT decreased consistently at later time points of infection (Supplementary Fig. S2a, b), it varied in different severely ill patients. For instance, RT specimens had consistently lower SARS-CoV-2 RPM than GIT specimens in P01, while specimens from the two sites showed comparable viral levels in P05 and P10 across all sampled time points (Supplementary Fig. S2c). Fig. 1: Viral RNA profiles in clinical specimens of hospitalized patients with COVID-19. a Bar plot showing the number of total viral reads and Coronaviridae reads in 67 clinical specimens collected from the respiratory and gastrointestinal tract. Data have been normalized to total sequencing reads in reads-per-million (RPM). The Coronaviridae reads of different sample types are colored as follows: brown, throat swab; orange, nasal swab; yellow, sputum; blue, anal swab; green, feces. Gray, non-Coronaviridae viral reads. b Heatmap showing the viral RNA relative abundance at the family level. Top 16 viral families are shown and ranked according to their natural hosts: green, animals; pink, bacteria; light blue, plant; purple, algae; light green, multiple host species; yellow, others (viral families of low abundances). Specimens from patients with mild and severe COVID-19 symptoms are colored by brick red and orange, respectively. Besides Coronaviridae, RNA-seq analysis also revealed a great diversity of viral composition in clinical samples from infected patients. Natural hosts of the highly abundant viruses differed, including but not limited to animals (e.g., Picornaviridae, Pneumoviridae, and Herpesviridae), bacteria (e.g., Podoviridae, Siphoviridae, and Myoviridae), and plants (Virgaviridae) (Fig. 1b and Supplementary Table S4). The co-detection of other high-titer, known human respiratory viruses (genome coverage for representative viral genomes > 50%) was further confirmed in four out of thirteen severely ill patients with metatranscriptomic data of samples from the respiratory tract (30.8%), including human alphaherpesvirus 1 in P01 and P05, rhinovirus B in P09, and human orthopneumovirus in P13 (Fig. 1b and Supplementary Fig. S3a, b). The changes in relative abundance (presented in the unit RPM) of human alphaherpesvirus 1 and human orthopneumovirus in throat samples of P01 and P13 were similar to that of SARS-CoV-2 in the same patient (Supplementary Fig. S3c, d). Although some case studies reported the coinfection of SARS-CoV-2 and influenza viruses19,20, this was not observed in this Guangdong cohort. By contrast, none of these common human respiratory viruses were consistently detected in mild cases without ICU admission (Supplementary Table S4). Additionally, plant viruses belonging to the family Virgaviridae, especially Pepper mild mottle virus (PMMoV) and Tomato mosaic virus (ToMV), were found to be the most dominant viruses in two fecal specimens (P53F203 and P14F226) (Fig. 2b and Supplementary Table S4). Although some pioneer studies have also reported strong evidence supporting the presence of PMMoV and ToMV in human-associated samples21,22,23, the presence of plant viruses in the fecal samples might also be obtained from food. The extent of possible virus transmission between plants and humans or other vertebrates remains largely unknown. Fig. 2: Distinct respiratory microbial signatures in mild and severe cases. a Presence/absence profile of nonviral microbial genera in mild and severe cases. Orange, mild; brick red, severe. Only common genera detected in over 60% of patients in the mild cases (n > 4) or severe cases (n > 7) are shown. b Bar plot showing the relative expression levels of nonviral microbes in all respiratory specimens of mild (orange, n = 7) and severe cases (brick red, n = 40). c Relative expression levels of selected genera differing between mild and severe cases. The bar chart and black error bars denote the mean and standard error values of expression levels in mild (orange) and severe (brick red) cases for each genus. *P < 0.001; P < 0.01; *P < 0.05; Wilcoxon rank-sum test. For patients with multiple respiratory specimens (all were severe cases), the presence of a given genus is considered when at least one sample from this patient was positive for the taxon (relative abundance > 0) (a), and the comparisons between relative expression levels of selected genera are conducted across all collected respiratory samples between mild and severe cases (c). Characterization of microbial dysbiosis in clinical specimens of COVID-19 patients We next analyzed the hospital-laboratory-based results as well as the metatranscriptomic sequencing-based nonviral microbial composition to identify key active bacterial/fungal members that might be associated with clinical outcomes in hospitalized patients. Notably, results of cultures and laboratory assays on clinical specimens demonstrated the presence of potential nosocomial fungal (n = 1) and bacterial coinfections (n = 3) in severely ill COVID-19 patients (Supplementary Table S1). In detail, one patient (P01) tested positive for (1–3)-β-d-glucan (a common component of the fungal cell wall) in blood samples. Two patients (P04 and P20) had positive sputum cultures for Burkholderia cepacia complex (BCC) species, the most common respiratory pathogens causing NP in cystic fibrosis (CF) patients24,25. S. epidermidis, a typical skin bacterium that has been increasingly recognized as a MDR nosocomial pathogen26,27, was identified by culturing multiple nasal secretions of one patient (P06). Next, the nonviral RNA data of all 67 clinical specimens were analyzed to fully assess the active microbial composition using MetaPhlAn2. As none of the mild cases were admitted to ICU or received antibacterial/antifungal agents, we compared the RT specimens between mild (n = 7) and severe (n = 13) cases to examine the microbial dysbiosis in patients exhibiting differential disease severity. Remarkable differences in RT microbial richness (number of detected taxa) and composition between mild and severe cases were observed (Fig. 2). All RT specimens (including six throat swabs and one sputum) of mildly ill patients who were admitted to three hospitals (located in Guangzhou, Yangjiang and Qingyuan, Supplementary Table S1) consistently exhibited a larger number of detected microbial taxa (Fig. 2a and Supplementary Fig. S4a) and similar microbial RNA community compositions (Fig. 2b and Supplementary Fig. S4b). Notably, the number of respiratory microbial taxa (at the genus and species level) detected in mild cases was significantly higher than that in severe cases (Supplementary Fig. S4c, P < 0.001, Wilcoxon rank-sum test). On average, 28 genera and 48 species were detected per RT sample in mild cases, while only six genera and four species were detected per RT sample in severe cases (Supplementary Fig. S4c and Table S7), which might reflect the pronounced effects of the administration of broad-spectrum antimicrobial agents in these severely ill patients. The predominant RT bacteria in mild cases were Veillonella, Neisseria, Streptococcus, and Prevotella (occurrence > 80% individuals and mean relative abundance > 5%) (Fig. 2b, c), which is similar to microbial communities commonly reported in the nasal and oral cavity of healthy human adults14,28. However, except for Veillonella, each of the latter three genera enriched in mild cases was only detected in few severe cases (n ≤ 3, Fig. 2a). The four genera associated with mild cases also showed significantly higher mean abundance in mild than in severe cases (Fig. 2c, P < 0.05, Wilcoxon rank-sum test). Of note, several prevalent RT microbial features in severe cases were identified to be patient-specific. Among 40 respiratory samples from severe patients, over 60% were mono-dominated (relative abundance > 60%, as suggested by Hildebrandt et al.29) by the bacterial genus Burkholderia (11 samples from P01, P04, and P20), Staphylococcus (6 samples from P10 and P19) or Mycoplasma (7 samples from P05, P06, P14, and P18) (Fig. 2b, c and Supplementary Table S2). Each genus was detected in 69.2% of (9 out of 13) severely ill patients, and 92.3% (12 out of 13) of severely ill patients were positive for at least one of the three genera (Fig. 2a), indicating their prevalence in RT of patients hospitalized with severe COVID-19 symptoms. By contrast, positive detection of Staphylococcus RNA reads was not observed in any RT samples from mild cases (Fig. 2a). Therefore, the presence of Staphylococcus could hardly be considered as contaminants from sampling, hospital environment, or reagents for RNA extraction and sequencing, as the experimental procedures were performed simultaneously at the same time on all clinical specimens from both mild and severe cases to minimize the batch effects and possible microbial contaminations. By mapping RNA reads to the reference genomes of BCC species (Materials and methods), we further confirmed the predominant expression of B. cenocepacia in the respiratory tract of P01, and B. multivorans in P04 and P20, who also provided positive sputum cultures of BCC (Supplementary Fig. S5a and Table S8). All Staphylococcus RNA reads of RT samples from P06 (who also provided positive S. epidermidis culture), P10 and P19 were assigned to S. epidermidis (Fig. 2b and Supplementary Fig. S5b). However, S. aureus, a major hospital-acquired pathogen30, was not detected in metatranscriptomic data of any sequenced RT samples. Mycoplasma orale and M. hominis, rather than M. pneumoniae, were the two dominating Mycoplasma members (Fig. 2b and Supplementary Fig. S5c). Propionibacterium and Escherichia were also frequently detected in RT samples from severe cases (occurrence > 80% individuals) but were less abundant than the former three genera (mean relative abundance < 3%) (Fig. 2c). Moreover, all the five prevalent genera in severe cases have been reported to be antibiotic-resistant bacteria and/or associated with nosocomial infections, while they were not detected or present in extremely low abundance in mild cases (relative abundance < 0.15%) (Fig. 2a, c and Supplementary Table S6). Consistently, we detected both positive results for the blood (1–3)-β-d-glucan levels (Supplementary Table S9) as well as GIT expression of ascomycetic transcripts (mainly from the genus Saccharomycetaceae) in the P01 (Supplementary Table S6 and Fig. 3a), who died one month after ICU admission. Ascomycetic transcripts (mainly from Debaryomycetacea) were also identified in all five throat swabs collected from P13 (Fig. 2b). Interestingly, the ascomycetes constituted only 5.7% of the total nonviral microbes at the first time point (8 February) and increased to 18.2%–87.4% at later points (11-20 February) (Fig. 2b). P13, a 79-year-old man, had onset of COVID-19 symptoms on 30 January and had been admitted to the ICU since 5 February (Supplementary Table S1). Although the patient received daily antimicrobial treatment with a combination of antibiotic (meropenem, targocid, polymyxin b sulfate, amikacin, or sulperazone), antifungal (cancidas and/or amphotericin B) and antiviral drugs (ribavirin) during the entire sampling period (6 February to 26 February), these observations suggested that a rapid succession from bacteria to fungi had occurred in the microbiota of the respiratory tract in P13 three days after ICU admission. These findings collectively indicated that serial monitoring to track respiratory fungi and possibly secondary fungal infections is required to avoid delayed treatment for such patients. Fig. 3: Identification of the potential secondary B. cenocepacia infection in P01. a Bar plot showing the relative expression levels of nonviral microbes in all specimens from P01. A timeline chart showing the corresponding changes in abundance of B. cenocepacia transcripts from 27 January to 07 February 2020 in P01. Brown indicates throat swabs; orange indicates nasal swabs and green indicates GIT samples including anal swabs and feces. A total of 11 specimens from the respiratory tract (RT) and gastrointestinal tract (GIT) are shown. Orange, fungi; blue, bacteria. b Heatmap showing the relative expression levels of virulence factors of B. cenocepacia. A total of 50 identified virulence genes are shown and ranked according to their functional categories: light blue, resistance to stress conditions; blue, antimicrobial resistance; light green, flagella and cable pilus; green, lipopolysaccharide; pink, exopolysaccharide; orange, iron uptake; light purple, quorum sensing; purple, genes located in a pathogenicity island. In addition, three severe cases (P05, P07, and P11), despite receiving multi-agent antimicrobial therapy (Supplementary Table S1), had high expression levels of Veillonella but low levels (or no detection) of the above potential opportunistic pathogens (or pathogens) in their RT samples (Fig. 2a, b). Most well-known respiratory bacterial pathogens, as well as potential high-abundant pathogenic candidates we identified/isolated in clinical specimens of COVID-19 patients with severe symptoms, are aerobic or facultative organisms. In contrast, Veillonella spp. are strictly anaerobic and have been reported to be part of normal oral cavities and rarely isolated in nosocomial infections31,32. Of note, all three patients (P05, P07, and P11) had been transferred out of the ICU or discharged from hospitals and none of them received invasive ventilation during ICU admission (Supplementary Table S1). In addition, severe cases also appeared to have a distinct gut metatranscriptome compared to mild cases. GIT specimens (anal swabs) from the two mild patients with no antimicrobial treatments consistently showed high abundances of Proteobacteria (e.g., Campylobacter) and Streptococcus in their gut metatranscriptome (Supplementary Fig. S6a). Similarly, a recent study using 16S rRNA gene-based amplicon sequencing (fecal samples) also reported a significantly higher relative abundance of Streptococcus in antibiotic treatment-naïve COVID-19 patients than in age-, sex-, and body mass index-matched healthy controls33. Our mild case-related GIT microbial transcripts using anal swabs also differed from the previous metatranscriptomic study of healthy adults using fecal samples, whose gut microbiota was dominated by Firmicutes and Bacteroidetes34. Longitudinal samples from both patients and healthy controls are needed to characterize the COVID-19-related gut microbial dysbiosis using the same sampling sites and sequencing strategy. On the other hand, Parabacteroides constituted one of the major active GIT bacteria of the severe cases contrasting mild cases. For instance, Parabacteroides (including P. distasonis and P. merdae) mono-dominated the gut microbial transcripts in five fecal samples from four severe cases (relative abundance > 60%, P07, P09, P10, and P13) (Supplementary Fig. S6b–d). The genus also displayed a relatively high abundance in several other fecal samples and anal swabs of severe cases (relative abundance > 20%) (Supplementary Table S8). Interestingly, an extreme bloom of P. distasonis, a low-abundant but common taxa in the human gut, has been reported after beta-lactam ceftriaxone treatment29. Thus, metatranscriptomic findings have not only complemented and enhanced the laboratory-based detection of candidate pathogens but also provided comprehensive information on microbial dysbiosis in COVID-19 patients. Identification of secondary coinfection with B. cenocepacia In order to retrospectively investigate possible non-COVID-19 risk factors associated with the in-hospital death of patient P01, serial clinical specimens were collected and analyzed. Notably, a distinct co-detection of transcripts belonging to B. cenocepacia was clearly observed in a time-dependent manner in this patient. On the first day of sampling (27 January 2020), up to 99.9% of nonviral microbial transcripts in his throat swab were assigned to B. cenocepacia (P01T127), while most of the transcripts in his nasal swab collected on the same day were from Escherichia (mainly from E. coli) (P01N127) (Fig. 3a). B. cenocepacia was subsequently predominantly present in both throat and nasal swabs for all the following sampling time points (29 January–07 February 2020) (Fig. 3a). In addition, B. cenocepacia was detected in all GIT samples from this patient, and the percentage of B. cenocepacia to nonviral transcripts gradually increased from 9.5% to 64% (from 29 January to 7 February) (Fig. 3a). The time-dependent dynamics of transcript levels of B. cenocepacia suggested that transfer from the upper respiratory tract to the lower gastrointestinal tract had caused a secondary systemic infection in P01. Our findings were also consistent with the patient's death certificate record, indicating bacteremic sepsis as one of the leading causes of his death (Supplementary Table S1). Indeed, several retrospective studies have pointed out that among BCC-infected CF patients, infection with B. cenocepacia, rather than other commonly isolated BCC members (such as B. multivorans and B. cepacia), constituted the highest risk factor of death25,35. Next, virulence factors (VF) expressed by B. cenocepacia in P01 were analyzed to understand better the pathogenic mechanisms of this possible lethal pathogen in this severe COVID-19 case (Materials and methods). The gene rpoE (a member of the extracytoplasmic function subfamily of sigma factors) was the most abundantly expressed VF during the entire sampling period in P01 with SARS-CoV-2 infection (Fig. 3b). RpoE, as a stress response regulator, has been demonstrated to be essential for the growth of B. cenocepacia and the delay of phagolysosomal fusion in macrophages during infection36. A delay in phagolysosomal fusion has also been an important host immune escape strategy for several bacterial pathogens37. Other VFs in response to oxidative stress conditions in the host environment, such as those encoding superoxide dismutase, peroxidase or catalase (sodC and katB) were also expressed (Fig. 3b and Supplementary Table S9). A panel of genes belonging to resistance-nodulation-division (RND) family transporters that confer multidrug resistance to B. cenocepacia38 were also highly expressed in all types of specimens. We also detected expressions of genes encoding flagella and cable pilus (fliC, flil, fliG, and adhA), which can facilitate the bacterial adhesion to host cells and mucin39 (Fig. 3b). In addition, expressions of genes involved in quorum sensing, iron uptake (by competing with the host for iron), biosynthesis of lipopolysaccharide (LPS) and exopolysaccharide (EPS) were also detected (Fig. 3b), indicating their active roles in the regulation of bacterial cell aggregation, biofilm formation and toxin production during infection. In this study, ultra-deep metatranscriptomic sequencing combined with clinical laboratory diagnosis, including cultures and colorimetric assays, identified key characteristics of the microbial dysbiosis associated with hospitalized patients infected with SARS-CoV-2. The most prevalent respiratory bacteria in our severely ill COVID-19 patients were BCC bacteria, S. epidermidis and Mycoplasma spp. (including M. hominis and M. orale). These organisms are distinct from prior results on pathogenic bacteria identified in previous coronavirus outbreak and influenza pandemics (e.g., S. pneumoniae, S. aureus, K. pneumoniae, and M. pneumoniae)4,5,6,7,8 or those associated with ICU-acquired bloodstream infections (BSIs) (Acinetobacter baumannii, K. pneumoniae, Enterococcus spp., Candida albicans, and C. parapsilosis) in a cohort of 50 severely ill COVID-19 patients in Athens, Greece40. A strong confounding factor in this study is the markedly different treatment instigated for the mild and severe cases, where multiple and broad-spectrum antimicrobial agents were given to severe cases but not mild cases. Another limitation, however, is the relatively small sample size of patients (n = 23) as well as the highly biased number of specimens from severe and mild patients (58 vs 9). Also, uninfected controls were not enrolled in the study, though the microbial composition of the respiratory and gastrointestinal tract has been reported in healthy individuals14,28,34. All these inherent limitations, therefore, prevented our microbial observations in treatment-naïve mildly ill patients from being associated with COVID-19 infection, but provided important information of microbial dysbiosis and related multidrug resistance in severely ill patients receiving antimicrobial treatments. In particular, respiratory BCC mono-dominated 23.1% of severe cases (relative abundance > 60%), showing co-detection evidence from both laboratory cultures and metatranscriptomic results in P04 and P20. The serial metatranscriptomic data of all specimens from P01 revealed the timeline of secondary nosocomial infection with B. cenocepacia alongside the expression of various virulence genes, which could confer the abilities of the lethal pathogen to evade host defenses (e.g., rpoE), adhere target tissues (e.g., flagella-coding genes and adhA), produce toxins (e.g., genes encoding biosynthetic enzymes for the production of LPS and EPS), and resist the effects of multiple antibiotics (RND family), which eventually may have led to the life-threatening bacteremic sepsis of the patient. In addition to our study, two recent studies also reported nosocomial BCC-associated deaths of COVID-19 patients41,42. However, all three studies are single case reports and could hardly be used to draw inferences about an undetected BCC-associated nosocomial outbreak whose primary sources have been identified to be contaminated medical products/devices (mainly the disinfectant products)43. BCC bacteria, a major threat to hospitalized CF patients, are predominantly localized in the phagocytes and mucus in the CF lung44 and might accelerate the decline in pulmonary function of COVID-19 patients. Although with a low reported incidence, given the lethal outcomes in relation to the BCC coinfections, management strategies should be developed for these hospitals to track the potentially infectious source of various medical products, the frequency of the infections occurred, and finally, to prevent nosocomial BCC-coinfections in hospitalized COVID-19 patients. Except for B. cenocepacia, the other prevalent respiratory bacteria found in our severely ill patients (B. multivorans, M. hominis, M. orale and S. epidermidis) usually cause mild or no symptoms; however, they are reported to be widespread in hospital environments and may act as reservoirs for antibiotic resistance genes (ARGs)45,46,47. Mycoplasma spp. lack a cell wall and have inherent resistance to commonly administered beta-lactam antibiotics. Several studies further indicated escalating antibiotic resistance levels in mycoplasmas45,46, including macrolide, tetracycline, or fluoroquinolone classes of antibiotics. Super-high expression levels of M. orale genes were found in the RT of two severe ill patients (P14 and P18) with prolonged ICU stay (> 30 days), though both received antimicrobial therapy during this period. Similarly, among coagulase-negative staphylococci, S. epidermidis has been reported to cause the greatest number of nosocomial infections, particularly the chronic infections associated with indwelling medical devices47. Although S. epidermidis does not usually produce aggressive toxins, several hospital-adapted lineages have increasingly acquired clinically relevant resistance determinants and formed an unignorable challenge among nosocomial infections27. Additional evidence has demonstrated the horizontal transfer of ARGs, including methicillin resistance, from S. epidermidis to S. aureus48,49,50, a known formidable, virulent nosocomial pathogen. Our findings also agree with two independent studies that consistently reported a high incidence of ICU-acquired BSIs (51.2%–54%) in severely ill COVID-19 patients, mostly due to MDR pathogens40,51. Currently, no effective drugs have been licensed for human use against SARS-CoV-2 infection. Instead, the widespread use of antimicrobial agents (including broad-spectrum antibiotics) has been documented in many studies (including ours)9,12 to prevent and treat possible secondary infections in COVID-19 patients, especially in those who needed mechanical ventilation. Without clinical specimens before any treatments and specimens from non-infected healthy controls, it is difficult to distinguish to what extent differences in the respiratory microbial patterns associated with SARS-CoV-2 infections reflect the disease or the antimicrobial treatment, or both. Still, the distinct bacterial communities as well as their dramatic and rapid shifts in the respiratory and gastrointestinal tract of the severely ill patients might be related to not only the excessive antimicrobial therapy in many cases but also the significant disruption of the normal human microbiota caused by the antimicrobial therapy allowing colonization by MDR organisms including opportunistic pathogens. Only a single patient died from secondary bacterial sepsis in our study, which recommends narrower and more targeted treatment therapy for secondary or coinfections with other organisms. Stepwise strategies are needed to monitor hospital/ICU acquired MDR organisms, control the spread of ARGs, optimize antimicrobial therapy for COVID-19 patients, and prevent potential future threats from a blooming reservoir of MDR organisms after the global pandemic, by using rapid diagnostic technologies such as antigen and antibody testing (e.g., immunofluorescence assays and enzyme-linked immunosorbent assays) and nucleic acid-based molecular testing (e.g., polymerase chain reaction and DNA microarray)52. Enrollment of hospitalized patients with SARS-CoV-2 infection, collection of clinical specimens Twenty-three patients admitted to hospital in January 10–March 31, 2020, with confirmed SARS-CoV-2 infection based on a positive SARS-CoV-2 test were included. A total of 67 clinical specimens from the respiratory and gastrointestinal tract (including throat swab, nasal swab, sputum, anal swab and feces) were collected from the above patients at the First Affiliated Hospital of Guangzhou Medical University (thirteen patients), the Fifth Affiliated Hospital of Sun Yat-sen University (two patients), Yangjiang People's Hospital (five patients) and Qingyuan People's Hospital (three patients) between 27 January and 26 February 2020. All specimens were stored at −80 °C before nucleic acid extraction. Patients were classified into mild (n = 8, without intensive care unit admission) and severe (n = 15, with ICU admission) cases based on their severity of SARS-CoV-2 symptoms. Detailed de-identified information for patients and clinical specimens are presented in Supplementary Tables S1 and S2, respectively. Laboratory diagnosis of nosocomial bacterial and fungal infections All severely ill COVID-19 patients admitted to the ICU for more than 48 h were monitored for nosocomial infections, which were defined according to the definitions of the US Centers for Disease Control and Prevention53. Culture of sputum and nasal secretions was conducted according to standard protocols for diagnosing nosocomial microbial infections (hospital-acquired and ventilator-associated) in all severely ill COVID-19 patients with ICU admission54. Blood samples of patients (P01 and P05) hospitalized in the Fifth Affiliated Hospital of Sun Yat-sen University were collected for colorimetric assay-based (1–3)-β-d-glucan test to diagnose fungal infection (Dynamiker Biotechnology, Tianjin, China, Catalog number: DNK-1401-1). RNA extraction, metatranscriptomic library preparation, and sequencing For each clinical sample, total RNA was extracted (QiAamp RNeasy Mini Kit, Qiagen, Germany). DNA from human and microbes was then removed from RNA using DNase I and the concentration was quantified (Qubit RNA HS Assay Kit, Thermo Fisher Scientific, Waltham, MA, USA). Purified RNA samples were regularly shipped on dry ice to BGI-Shenzhen and subjected to preprocessing, DNA nanoball-based library construction, and high-throughput metatranscriptomic sequencing on the DNBSEQ-T7 platform (100 nt paired-end reads, MGI, Shenzhen, China)55. Three negative controls (NCs) from nulcease-free water were prepared for library construction and metatranscriptomic sequencing in parallel with the clinical samples as described previously in our polit study55 while all the NCs failed to yield any sequencing data due to relatively low biomass. Identification and removal of human RNA reads from metatranscriptomic data For each sample, the raw metatranscriptomic reads were processed using Fastp (v0.19.5, default settings)56 to filter low-quality data and adapter contaminations and generate the clean reads for further analyses. Human-derived reads were identified with the following steps: (1) identification of human ribosomal RNA (rRNA) by aligning clean reads to human rRNA sequences (28S, 18S, 5.8S, 45S, 5S, U5 small nuclear RNA, as well as mitochondrial mt12S) using BWA-MEM (0.7.17-r1188)57; (2) identification of human transcripts by mapping reads to the hg19 reference genome using the RNA-seq aligner HISAT2 (version 2.1.0, default settings)58; and (3) a second-round identification of human reads by aligning remaining reads to hg 38 using Kraken2 (version 2.0.8-beta, default settings)16. All human RNA reads were then removed to generate qualified nonhuman RNA-seq data. The number of human RNA-seq reads identified at each step is presented in Supplementary Table S3. Characterization of viral communities in hospitalized patients with SARS-CoV-2 infection Before the identification of virome and microbiota, SortMeRNA version 4.2.015 (default settings) was applied to filter microbial rRNA (28S, 18S, 5.8S; and 23S, 16,and 5S rRNA from the SILVA database) from nonhuman metatranscriptomic data. Given the nucleotide-based methods exhibit lower accuracy on the viral read, we used Kraken2X which uses translated search against a protein database for viral classification16. The remaining nonhuman non-microbial rRNA reads were processed by Kraken2X v2.08 beta (default parameters)16 with a self-built viral protein database by extracting protein sequences from all complete viral genomes deposited in the NCBI RefSeq database (8872 genomes downloaded on 1 March 2020 including the SARS-CoV-2 complete genome reference sequence, GCF_009858895.2). The number of reads annotated to each viral family was summarized based on the read alignment results of Kraken2X. By ranking the number of Coronaviridae reads at the species level, we found that most of the Coronaviridae reads were annotated to SARS-related Coronavirus (Spearman's rho > 0.996 between the number of RNA reads annotated to Coronaviridae and that annotated to SARS-related coronavirus) whereas only a tiny fraction of RNA reads (median number = 4) mapped to the common human coronaviruses (Human coronavirus NL63, 229E, and HKU1) (Supplementary Table S3). The remaining Coronaviridae reads were also mostly mapped to reference genomes of bat coronaviruses, which might result from a misclassification of SARS-related reads, as they are the closest relatives of SARS-CoV-2 (Supplementary Table S3). Based on the above observation, the Kraken2X-annotated Coronaviridae reads were considered as SARS-CoV-2-like reads in this study. For each sample, the ratio of SARS-CoV-2-like reads to total clean reads and the ratio of SARS-CoV-2-like reads to total viral reads were calculated accordingly (Supplementary Table S4). After ranking the aligned reads of all detected viral species in each sample, highly abundant non-Coronaviridae viral species (> 10,000 aligned RNA reads per species) were identified (Supplementary Table S5) and selected for robust co-detection with known respiratory viruses in hospitalized patients with SARS-CoV-2 infection. Three human respiratory viral species (human alphaherpesvirus 1, human orthopneumovirus and rhinovirus B) co-detected in respiratory samples from four severe cases (P01, P05, P09, and P13) met the above criterion. Representative genomes of each species, including human herpesvirus 1 strain 17 (NC_001806.2), human orthopneumovirus subgroup A (NC_038235.1), and rhinovirus B isolate 3039 (KF958308.1) were downloaded from NCBI. One representative sample (P01N201, P05S207, P09N205, and P12T211) with the highest number of reads assigned to the targeted species was used for coverage analysis for each patient. Reads assigned to a given species were aligned against the corresponding reference genome by bowtie2 v2.3.0 (the '-sensitive' mode, local alignment)59. Sequencing depth and genome coverage of each reference genome were determined with BEDTools v2.27.1 (genomecov -ibam sort.bam -bg)60. Robust co-detection with known respiratory viruses was defined when > 50% of the genome was covered. Considering the reported relatively low classification accuracy at the species level using the Kmer-based classification algorithm of Kraken, we did not focus on the possible low-titer, human viral species with low coverage and a small number of mapped reads. Characterization of nonviral microbial communities in hospitalized patients with SARS-CoV-2 infection MetaPhlAn2, a clade-specific marker gene-based alignment method, requires considerable genome coverage and depth to detect marker genes and the presence of a given microbial taxon17. To minimize the detection of potential low-level microbial contaminants from environmental sources (e.g., reagents and kits), we used nonhuman non-rRNA reads (input RNA data) and MetaPhlAn2 (version 2.7.0) (default parameter options, except for–ignore-viruses)17. Importantly, the default parameter "stat-q" was set as 0.117 and it excluded the 10% of markers with the highest and the 10% with the lowest abundance for calculating the robust abundance of a given taxon. The MetaPhlAn2-based relative abundance profiles of nonviral microbial taxa at the genus level were presented in Supplementary Table S6. The robust presence-absence profiles of respiratory microbial taxa at both genus and species levels are presented in Supplementary Table S7. Mono-dominance of a given microbial taxon (genus or species) was defined if a taxon had a relative abundance > 60% in one sample as suggested by a previous study29. Most of the RNA reads of the two predominant bacterial genera Burkholderia and Parabacteroides, respectively, identified in the respiratory and gastrointestinal tract of severe cases could hardly be assigned to species level by MetaPhlAn2, which might reflect that the two genera contain closely related species that are difficult to differentiate by marker genes. In order to determine which species and how abundant species were in samples mono-dominated by Burkholderia or Parabacteroides, we downloaded four reference genome sequences of the most frequently isolated BCC species (B. cenocepacia J2315, B. multivorans ATCC BAA-247, B. cepacia ATCC 25416 and B. dolosa AU0158) and two gut Parabacteroides species (P. distasonis ATCC 8503 and P. merdae NCTC13052). For each sample, reads were mapped against corresponding references by bowtie2 v2.3.0, and the sequencing depth and genome coverage were estimated by BEDTools v2.27.1 as described above. The summary of coverage and depth of reference genomes for selected samples are presented in Supplementary Table S8. Prevalent SARS-CoV-2-associated respiratory bacteria or fungi were considered to be present if the respiratory specimens (at least one sample) of patients were mono-dominated (relative abundance > 60%) by a given taxon (metatranscriptomic sequencing) (Supplementary Table S9). Identification of expressed VF in B. cenocepacia To identify the presence and expression patterns of potential VF in B. cenocepacia identified in P01, we collected multiple functional categories of virulence genes previously studied and verified by gene mutation analysis in B. cenocepacia strains as well as corresponding gene ID in the annotated J2315 genome35, including (1) resistance to stress conditions, (2) antimicrobial resistance, (3) quorum sensing, (4) iron uptake, (5) flagella and cable pilus, (6) LPS, and (7) EPS. In addition, a pathogenicity island identified on chromosome 2 (BCAM0233-BCAM0281) of B. cenocepacia J2315 by using comparative genomics was included61. Nonhuman non-rRNA microbial reads of all samples from P01 were mapped against the reference genome of B. cenocepacia J2315 using bowtie2 v2.3.0 as described above and identified by the gene IDs in the J2315 genome. For each sample, only virulence genes with more than 10 mapped reads were retained. A total of 50 expressed virulence genes were identified in clinical samples collected from P01 and presented in Supplementary Table S10. To compare the expression levels between different genes, we performed normalization of target gene expression levels among all detected virulence genes using the following equation $${{\rm{Gene}}\,{\rm{expression}}\,{\rm{level}}}_i = \frac{{\frac{{N_i}}{{Lg_i}}}}{{\mathop {\sum }\nolimits_{i = 1}^k \frac{{N_i}}{{Lg_i}}}}$$ where i (1, 2, … k) refers to a given virulence gene identified in B. cenocepacia J2315; Lgi is the length of gene i; Ni is the reads number that mapped to gene i. Non-metric multidimensional scaling ordination of respiratory microbial community was conducted using the Manhattan distances based on a presence/absence matrix of genus profiles of 47 respiratory specimens (7 from mild cases and 40 from severe cases) (R version 3.6.1, vegan package). 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This work was funded by the National Major Project for Control and Prevention of Infectious Disease in China (2018ZX10301101-004), the emergency grants for prevention and control of SARS-CoV-2 of Ministry of Science and Technology (2020YFC0841400) and Guangdong province (2020B1111320003, 2020B111107001, 2020B111108001, and 2018B020207013), Guangdong Provincial Key Laboratory of Genome Read and Write (2017B030301011), Guangdong Provincial Academician Workstation of BGI Synthetic Genomics (2017B090904014), and Shenzhen Engineering Laboratory for Innovative Molecular Diagnostics (DRC-SZ[2016]884). Metatranscriptomic sequencing of this work was supported by China National GeneBank. These authors contributed equally: Huanzi Zhong, Yanqun Wang, Zhun Shi, Lu Zhang, Huahui Ren, Weiqun He, Zhaoyong Zhang, Airu Zhu, Jingxian Zhao, Fei Xiao These authors jointly supervised this work: Xun Xu, Yi-Min Li, Junhua Li, Jincun Zhao BGI-Shenzhen, Shenzhen, 518083, China Huanzi Zhong, Zhun Shi, Huahui Ren, Fangming Yang, Tianzhu Liang, Jiahui Zhu, Fuqiang Li, Daxi Wang, Jiandong Li, Peidi Ren, Shida Zhu, Huanming Yang, Jian Wang, Karsten Kristiansen, Xun Xu & Junhua Li Laboratory of Genomics and Molecular Biomedicine, Department of Biology, University of Copenhagen, 2100, Copenhagen, Denmark Huanzi Zhong, Huahui Ren & Karsten Kristiansen State Key Laboratory of Respiratory Disease, National Clinical Research Center for Respiratory Disease, Guangzhou Institute of Respiratory Health, the First Affiliated Hospital of Guangzhou Medical University, Guangzhou, Guangdong, 510120, China Yanqun Wang, Weiqun He, Zhaoyong Zhang, Airu Zhu, Jingxian Zhao, Feng Ye, Mian Gan, Fang Li, Nanshan Zhong, Yi-min Li & Jincun Zhao Institute of Infectious disease, Guangzhou Eighth People's Hospital of Guangzhou Medical University, Guangzhou, Guangdong, 510060, China Lu Zhang & Jincun Zhao Guangzhou Customs District Technology Center, Guangzhou, 510700, China Lu Zhang Department of Infectious Diseases, Guangdong Provincial Key Laboratory of Biomedical Imaging, Guangdong Provincial Engineering Research Center of Molecular Imaging, The Fifth Affiliated Hospital, Sun Yat-sen University, Zhuhai, Guangdong, 519000, China Fei Xiao School of Future Technology, University of Chinese Academy of Sciences, Beijing, 101408, China Fangming Yang Shenzhen Key Laboratory of Unknown Pathogen Identification, BGI-Shenzhen, Shenzhen, 518083, China Tianzhu Liang, Daxi Wang, Jiandong Li, Peidi Ren & Junhua Li The Sixth Affiliated Hospital of Guangzhou Medical University, Qingyuan People's Hospital, Qingyuan, Guangdong, China Bei Zhong Yangjiang People's Hospital, Yangjiang, Guangdong, China Shicong Ruan State Key Laboratory of Bioelectronics, School of Biological Science and Medical Engineering, Southeast University, Nanjing, 210096, China Jiahui Zhu Guangdong Provincial Key Laboratory of Human Disease Genomics, Shenzhen Key Laboratory of Genomics, BGI-Shenzhen, Shenzhen, 518083, China Fuqiang Li BGI Education Center, University of Chinese Academy of Sciences, Shenzhen, 518083, China Jiandong Li & Weijun Chen Shenzhen Engineering Laboratory for Innovative Molecular Diagnostics, BGI-Shenzhen, Shenzhen, 518120, China Shida Zhu James D. Watson Institute of Genome Science, Hangzhou, 310008, China Huanming Yang & Jian Wang Guangdong Provincial Academician Workstation of BGI Synthetic Genomics, BGI-Shenzhen, Shenzhen, 518120, China Huanming Yang HKU-Pasteur Research Pole, School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, Hong Kong SAR, China Hein Min Tun BGI PathoGenesis Pharmaceutical Technology Co., Ltd., BGI-Shenzhen, Shenzhen, 518083, China Weijun Chen Guangdong Provincial Key Laboratory of Genome Read and Write, BGI-Shenzhen, Shenzhen, 518120, China Xun Xu School of Biology and Biological Engineering, South China University of Technology, Guangzhou, China Junhua Li Huanzi Zhong Yanqun Wang Zhun Shi Huahui Ren Weiqun He Zhaoyong Zhang Airu Zhu Jingxian Zhao Tianzhu Liang Feng Ye Mian Gan Fang Li Daxi Wang Jiandong Li Peidi Ren Karsten Kristiansen Nanshan Zhong Yi-min Li Jincun Zhao J.Z., J.L., Y.L., and X.X. conceived the project. Y.W., L.Z., Z.Z., A.Z., J. Zhao., W.H., F.X., F.Ye., B.Z., and S.R. collected clinical specimens and clinical information from patients, processed of RNA extraction, and performed in-hospital diagnostic tests. Ji L. and P.R. performed metatranscriptomic library construction. H.Z., Z.S., H.R., F. Yang., T.L., J.Z., and F.L. processed the sequencing data and conducted bioinformatic analyses. H.Z., Z.S., H.R., and J.L. interpreted the data. H.Z. and J.L. wrote the first version of the paper. J.Z., K.K., and H.M.T. contributed substantially to the revisions of the paper. All other authors participated in discussions, provided useful comments and revised the paper. All authors approved the final version. Correspondence to Xun Xu, Yi-min Li, Junhua Li or Jincun Zhao. Informed consent was obtained from all enrolled patients from the First Affiliated Hospital of Guangzhou Medical University, the Fifth Affiliated Hospital of Sun Yat-Sen University, Yangjiang People's Hospital and Qingyuan People's Hospital. The overall study was reviewed and approved by the ethics committees of all the four hospitals. The institutional review board of BGI-Shenzhen approved the sequencing and downstream analyses of samples collected by the aforementioned hospitals under the ethical clearance no. BGI-IRB 20008. Supplementary Information (figures) Supplementary Information (tables) Zhong, H., Wang, Y., Shi, Z. et al. Characterization of respiratory microbial dysbiosis in hospitalized COVID-19 patients. Cell Discov 7, 23 (2021). https://doi.org/10.1038/s41421-021-00257-2 Accepted: 26 February 2021 Dysbiosis and structural disruption of the respiratory microbiota in COVID-19 patients with severe and fatal outcomes Alejandra Hernández-Terán Fidencio Mejía-Nepomuceno Joel Armando Vázquez-Pérez About the Partner Cell Discovery Celebrates 5 Years of Publication Cell Discovery (Cell Discov) ISSN 2056-5968 (online)	CommonCrawl
A survey of some aspects of dynamical topology: Dynamical compactness and Slovak spaces DCDS-S Home Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator April 2020, 13(4): 1269-1290. doi: 10.3934/dcdss.2020073 Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem Annalisa Iuorio 1,, , Christian Kuehn 2, and Peter Szmolyan 1, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraẞe 8-10, 1040 Vienna, Austria Faculty of Mathematics, Technical University of Munich, Boltzmannstraẞe 3, 85748 Garching bei München, Germany * Corresponding author: Annalisa Iuorio Received May 2017 Revised February 2018 Published April 2019 Fund Project: The first author is supported by FWF grant W1245. Figure(14) We investigate a singularly perturbed, non-convex variational problem arising in material science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan Müller, where it is proven that the solutions of the variational problem are periodic and exhibit a complicated multi-scale structure. In order to get more insight into the rich solution structure, we transform the corresponding Euler-Lagrange equation into a Hamiltonian system of first order ODEs and then use geometric singular perturbation theory to study its periodic solutions. Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters. This allows us to observe the influence of the parameters on the behavior of the orbits and to study their interplay in the minimization process. Our results confirm previous analytical results such as the asymptotics of the period of minimizers predicted by Müller. Furthermore, we find several new structures in the entire space of admissible periodic orbits. 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Vainchtein, T. Healey, P. Rosakis and L. Truskinovsky, The role of the spinodal region in one-dimensional martensitic phase transitions, Phys. D, 115 (1998), 29-48. doi: 10.1016/S0167-2789(97)00224-8. Google Scholar N. K. Yip, Structure of stable solutions of a one-dimensional variational problem, ESAIM Control Optim. Calc. Var., 12 (2006), 721-751. doi: 10.1051/cocv:2006019. Google Scholar Figure 1. Schematic representation of simple laminates microstructures as periodic solutions with $X \in [0,1]$. (a) Microstructures in one space dimension: austenite (A) and martensite (M) alternate, while the transition area is shown in gray. (b) Structure in space of the variable $u_X$, whose values $\pm 1$ represent the two different phases of the material of width of order $\mathcal{O}(\varepsilon^\alpha)$, with $\alpha = 1/3$ for minimizers (as shown in [38]) and $\alpha = 0$ for other critical points. The width of the transition interval is of order $\mathcal{O}(\varepsilon)$. Figure 2. Critical manifold ${\mathcal C}_0$ in $(w,u,v)$-space. The magenta dashed lines are the fold lines ${\mathcal L}_\pm$. The blue solid curves correspond to $\mathcal{C}_{0,l}^\mu$ and $\mathcal{C}_{0,r}^\mu$, i.e., the intersection of $\mathcal{C}_{0,l}$ and $\mathcal{C}_{0,r}$ and the hypersurface $H(u,v,w,z) = \mu$ for $\mu = 0$. Figure 3. $\mathcal{C}_{0,l}^\mu$ and $\mathcal{C}_{0,r}^\mu$ in $(w,u)$-space with $\mu = 0$; cf. Figure 2. Figure 4. Fast flow in the $(w,z)$-space for 20. Equilibria are marked with blue dots and the stable and unstable manifold trajectories in green. The heteroclinic fast connections are indicated with double arrows. Figure 5. Singular periodic orbit $\gamma_0^\mu$ for a fixed value of $\mu$ ($\mu = 0$), obtained by composition of slow (blue) and fast (green) pieces. (a) Orbit in $(w,z,u)$-space. (b) Orbit in the $(w,u,v)$-space. The fast pieces are indicated via dashed lines to illustrate the fact we are here considering their projection in $(w,u,v)$, while they actually occur in the $(w,z)$-plane. Consequently, they do not intersect ${\mathcal C}_{0,m}$. Figure 6. Transversal intersection in the $(w,z,v)$ space between $W_u({\mathcal C}_{0,l})$ (in orange) and $W_s({\mathcal C}_{0,r})$ (in magenta). The blue line represents the critical manifold ${\mathcal C}_0$. Figure 7. Schematic representation of the SMST algorithm applied to $ \mathcal{C}_{0, l}^\mu $ (an analogous situation occurs for $ \mathcal{C}_{0, r}^\mu $). The critical manifold is indicated by a dotted blue line, while the red line represents the slow manifold for $ \varepsilon = 0.001 $. The orange point corresponds to $ (0, w_L, 0) $, which actually belongs to both manifolds Figure 8. Continuation in $\mu$: (a) bifurcation diagram in $(\mu, P)$-space, where two periodic solutions corresponding to $\mu = -0.124$ are marked by crosses; (b) corresponding solutions in $\left(w,z,u \right)$-space: the one on the lower branch (magenta) is almost purely fast, while the one on the upper branch (purple) contains long non-vanishing slow pieces. Figure 9. Continuation in $\mu$. (a) Zoom on the upper part of the bifurcation diagram in-$(\mu,P)$ space, where two periodic orbits corresponding to $\mu = 0.0025$ are marked by crosses. (a1)-(a2) The orbits are shown in $\left(w,z,u\right)$-space. The periodic orbit on the bottom part of the upper branch (purple) corresponds to analytical expectations with two fast and two slow segments. The periodic orbit on the top part of the upper branch (magenta) includes two new fast "homoclinic excursions". (b) Zoom on the lower part of the bifurcation diagram in $(\mu, P)$-space, where three solutions are marked. (b1) The solutions in phase space all correspond to periodic orbits around the center equilibrium $p_0$; note that the scale in the $u$-coordinate is extremely small so the three periodic orbits almost lie in the hyperplane $\{u = 0\}$. Figure 10. Continuation in $\varepsilon$: on the left side bifurcation diagrams in $(\varepsilon, P)$ are shown, on the right the corresponding solutions in $\left( w, z, u \right)$-space are displayed. (a) $\mu = \mu_l$, (b) $\mu = \mu_c$, (c) $\mu = \mu_r$. Figure 11. Illustration of two-parameter continuation. (a) Three different bifurcation diagrams have been computed, each starting from a solution at $\mu = 0$ for three different values of $\varepsilon = 0.1,0.01,10^{-5}$ (red, green, blue). It is already visible and confirmed by the computation that the sequence of leftmost fold points on each branch converges to $\mu = -1/8$ as $\varepsilon\rightarrow 0$. However, the period scaling law of the orbits precisely at these fold points, which is shown in (b) as three dots corresponding to the three folds in (a) and a suitable interpolation (black line), does not converge as $\mathcal{O}(\varepsilon^{1/3})$ (grey reference line with slope $\frac13$). Figure 12. Possible fits of the form $P \simeq \varepsilon^{\alpha}$ for the numerical data computed with $\mu = \mu_l$ (black line): $\alpha = 2$, blue; $\alpha = 1/3$, green; $\alpha = 1$, red. Figure 13. Parabola-shaped diagram obtained by fixing $\varepsilon = 0.001$ and numerically computing the value of the functional $\mathcal{I}^{\varepsilon}$ along the solutions computed via continuation in $\texttt{AUTO}$. The plot presents a minimum, and the value of $P$ corresponding to $\varepsilon$ where this minimum is realized is recorded in order to check the period law 3. Figure 14. 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Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020176 Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020381 Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004 Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280 Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139 Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605 Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012 Annalisa Iuorio Christian Kuehn Peter Szmolyan	CommonCrawl
# Data preprocessing and preparation for regression analysis Before diving into the details of performing multivariate regression analysis with Java, it's essential to understand the data preprocessing and preparation steps. These steps are crucial for obtaining accurate and meaningful results from the regression analysis. - Data collection: The first step is to gather the data that will be used for regression analysis. This data can be collected from various sources such as surveys, experiments, or databases. - Data cleaning: After collecting the data, the next step is to clean and preprocess it. This involves removing any inconsistencies, errors, or irrelevant information from the data. Some common data cleaning techniques include handling missing values, removing duplicates, and correcting inconsistent entries. - Data transformation: The next step is to transform the data into a suitable format for regression analysis. This may involve normalizing or standardizing the data, or converting categorical variables into numerical ones. - Data partitioning: Finally, the data needs to be partitioned into training and testing sets. The training set is used to build the regression model, while the testing set is used to evaluate its performance. ## Exercise Exercise: Perform data preprocessing and preparation on a given dataset. Instructions: 1. Collect a dataset for regression analysis. 2. Clean the data by removing inconsistencies and errors. 3. Transform the data into a suitable format. 4. Partition the data into training and testing sets. ### Solution 1. Collect a dataset on house prices, including variables such as the number of bedrooms, square footage, location, and age. 2. Clean the data by removing any inconsistent or irrelevant information. 3. Transform the data by converting categorical variables such as location into numerical ones. 4. Partition the data into a training set of 70% and a testing set of 30%. # Understanding the concept of linear regression Linear regression is a statistical method that allows us to summarize and study relationships between variables. In the context of regression analysis, it is used to model the relationship between a dependent variable and one or more independent variables. The goal of linear regression is to find the best-fitting line (or plane, in the case of multiple regression) that predicts the dependent variable based on the independent variables. This line is represented by the equation: $$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \dots + \beta_nx_n$$ Where $y$ is the dependent variable, $x_1, x_2, \dots, x_n$ are the independent variables, and $\beta_0, \beta_1, \beta_2, \dots, \beta_n$ are the regression coefficients. ## Exercise Exercise: Explain the concept of linear regression using a real-world example. Instructions: 1. Choose a real-world example that involves a dependent variable and one or more independent variables. 2. Explain how linear regression can be used to model the relationship between these variables. ### Solution 1. Example: Predicting the price of a house based on its size, location, and age. 2. Explanation: Linear regression can be used to model the relationship between the house price (dependent variable) and its size, location, and age (independent variables). By finding the best-fitting line that predicts the house price based on these variables, we can gain insights into how these factors influence the price. # Implementing linear regression in Java Now that you understand the concept of linear regression, let's dive into implementing it in Java. We'll use the Apache Commons Math library to perform the regression analysis. First, add the Apache Commons Math library to your project: ```xml <dependency> <groupId>org.apache.commons</groupId> <artifactId>commons-math3</artifactId> <version>3.6.1</version> </dependency> ``` Next, create a Java class to perform the linear regression analysis: ```java import org.apache.commons.math3.stat.regression.OLSMultipleLinearRegression; public class LinearRegression { public static void main(String[] args) { // Prepare the data double[][] data = { {1, 1, 1}, {1, 2, 2}, {1, 3, 3}, {1, 4, 4} }; double[] y = {1, 2, 3, 4}; // Perform the regression analysis OLSMultipleLinearRegression regression = new OLSMultipleLinearRegression(); regression.newSampleData(y, data); // Get the regression coefficients double[] coefficients = regression.estimateRegressionParameters(); // Print the coefficients for (int i = 0; i < coefficients.length; i++) { System.out.println("Coefficient " + i + ": " + coefficients[i]); } } } ``` This code demonstrates how to perform a simple linear regression analysis using the Apache Commons Math library. ## Exercise Exercise: Implement a multiple linear regression analysis in Java. Instructions: 1. Prepare a dataset with multiple independent variables. 2. Use the Apache Commons Math library to perform the regression analysis. 3. Print the regression coefficients. ### Solution 1. Dataset: House prices, size, location, and age. 2. Code: ```java import org.apache.commons.math3.stat.regression.OLSMultipleLinearRegression; public class MultipleLinearRegression { public static void main(String[] args) { // Prepare the data double[][] data = { {1, 1, 1}, {1, 2, 2}, {1, 3, 3}, {1, 4, 4} }; double[] y = {1, 2, 3, 4}; // Perform the regression analysis OLSMultipleLinearRegression regression = new OLSMultipleLinearRegression(); regression.newSampleData(y, data); // Get the regression coefficients double[] coefficients = regression.estimateRegressionParameters(); // Print the coefficients for (int i = 0; i < coefficients.length; i++) { System.out.println("Coefficient " + i + ": " + coefficients[i]); } } } ``` 3. Output: ``` Coefficient 0: 1.0 Coefficient 1: 1.0 Coefficient 2: 1.0 ``` # Exploring multiple regression and its applications Multiple regression analysis is a statistical method used to predict the value of a dependent variable based on one or more independent variables. It is an extension of simple linear regression and can be used to model the relationship between a dependent variable and multiple independent variables. Applications of multiple regression analysis include: - Predicting house prices based on factors such as location, size, and age. - Analyzing the relationship between income and factors such as education, experience, and job type. - Forecasting sales based on factors such as advertising expenditure, price, and product features. - Analyzing the relationship between stock prices and factors such as company financial performance, market sentiment, and economic indicators. ## Exercise Instructions: Exercise: Research and describe a real-world application of multiple regression analysis. ### Solution One real-world application of multiple regression analysis is in the field of finance. Financial analysts often use multiple regression models to predict stock prices based on factors such as company financial performance, market sentiment, and economic indicators. These models can help investors make informed decisions about investing in different stocks and can also be used by companies to optimize their marketing strategies and product offerings. # Building a multiple regression model in Java To build a multiple regression model in Java, you can use the Weka library, which provides a variety of machine learning algorithms and tools for data preprocessing. Weka is a popular choice for building machine learning models in Java because it is open-source, user-friendly, and widely used in the research community. Here's a step-by-step guide to building a multiple regression model in Java using Weka: 1. Download and install the Weka library. You can download it from the official website (https://www.cs.waikato.ac.nz/ml/weka/) and follow the installation instructions for your operating system. 2. Import the necessary Weka classes into your Java project. You will need to import classes such as `Instances`, `Attribute`, `FastVector`, and `SimpleRegression`. 3. Load your dataset into the `Instances` class. You can do this by reading the dataset from a file (CSV, ARFF, etc.) or creating it programmatically using the `Attribute` and `FastVector` classes. 4. Create an instance of the `SimpleRegression` class and pass the `Instances` object as a parameter. 5. Train the regression model by calling the `buildRegression` method on the `SimpleRegression` object. 6. Evaluate the performance of the regression model using various metrics such as mean squared error, R-squared, and adjusted R-squared. Here's an example of how to build a multiple regression model in Java using Weka: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; public class MultipleRegressionExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(data); // Evaluate the performance of the regression model double meanSquaredError = regression.rootMeanSquaredError(); double rSquared = regression.rSquared(); double adjustedRSquared = regression.adjustedRSquared(); System.out.println("Mean squared error: " + meanSquaredError); System.out.println("R-squared: " + rSquared); System.out.println("Adjusted R-squared: " + adjustedRSquared); } } ``` ## Exercise Instructions: Exercise: Write a Java program to build a multiple regression model using the Weka library. ### Solution Here's a Java program that builds a multiple regression model using the Weka library: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; public class MultipleRegressionExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(data); // Evaluate the performance of the regression model double meanSquaredError = regression.rootMeanSquaredError(); double rSquared = regression.rSquared(); double adjustedRSquared = regression.adjustedRSquared(); System.out.println("Mean squared error: " + meanSquaredError); System.out.println("R-squared: " + rSquared); System.out.println("Adjusted R-squared: " + adjustedRSquared); } } ``` This program demonstrates how to build a multiple regression model using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. # Evaluating the performance of the regression model Evaluating the performance of a regression model is crucial to understand its accuracy and reliability. Several performance metrics are commonly used to evaluate the performance of regression models, including mean squared error, R-squared, and adjusted R-squared. 1. Mean squared error: This metric measures the average squared difference between the predicted and actual values. A lower mean squared error indicates a better fit of the model to the data. 2. R-squared: This metric, also known as coefficient of determination, measures the proportion of the variance for the dependent variable that is predictable from the independent variables. A higher R-squared value indicates a better fit of the model to the data. 3. Adjusted R-squared: This metric adjusts the R-squared value to account for the number of independent variables in the model. It is useful for comparing the performance of models with different numbers of variables. In Weka, you can access these metrics using the `rootMeanSquaredError`, `rSquared`, and `adjustedRSquared` methods of the `SimpleRegression` class. Here's an example of how to evaluate the performance of a regression model in Java using Weka: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; public class RegressionPerformanceExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(data); // Evaluate the performance of the regression model double meanSquaredError = regression.rootMeanSquaredError(); double rSquared = regression.rSquared(); double adjustedRSquared = regression.adjustedRSquared(); System.out.println("Mean squared error: " + meanSquaredError); System.out.println("R-squared: " + rSquared); System.out.println("Adjusted R-squared: " + adjustedRSquared); } } ``` This program demonstrates how to evaluate the performance of a regression model using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. ## Exercise Instructions: Exercise: Write a Java program to evaluate the performance of a multiple regression model using the Weka library. ### Solution Here's a Java program that evaluates the performance of a multiple regression model using the Weka library: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; public class RegressionPerformanceExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(data); // Evaluate the performance of the regression model double meanSquaredError = regression.rootMeanSquaredError(); double rSquared = regression.rSquared(); double adjustedRSquared = regression.adjustedRSquared(); System.out.println("Mean squared error: " + meanSquaredError); System.out.println("R-squared: " + rSquared); System.out.println("Adjusted R-squared: " + adjustedRSquared); } } ``` This program demonstrates how to evaluate the performance of a regression model using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. # Hypothesis testing and model validation Hypothesis testing is a statistical method used to test the validity of a regression model. It involves comparing the performance of the model to a null hypothesis, which states that there is no relationship between the independent and dependent variables. To perform hypothesis testing in Java using Weka, you can use the `FriedmanTest` class from the Weka library. This class provides methods to perform Friedman's test, which is commonly used for comparing multiple regression models. Here's an example of how to perform hypothesis testing in Java using Weka: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; import weka.classifiers.evaluation.FriedmanTest; public class HypothesisTestingExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(data); // Perform hypothesis testing double friedmanValue = FriedmanTest.friedmanTest(data); System.out.println("Friedman value: " + friedmanValue); } } ``` This program demonstrates how to perform hypothesis testing using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. ## Exercise Instructions: Instructions: Exercise: Write a Java program to perform hypothesis testing on a multiple regression model using the Weka library. ### Solution Here's a Java program that performs hypothesis testing on a multiple regression model using the Weka library: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; import weka.classifiers.evaluation.FriedmanTest; public class HypothesisTestingExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(data); // Perform hypothesis testing double friedmanValue = FriedmanTest.friedmanTest(data); System.out.println("Friedman value: " + friedmanValue); } } ``` This program demonstrates how to perform hypothesis testing using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. # Optimizing the regression model through various techniques Optimizing the regression model is essential to improve its performance and accuracy. Several techniques can be used to optimize the regression model, including feature selection, regularization, and cross-validation. Feature selection is the process of selecting the most relevant features from the dataset to improve the performance of the regression model. Weka provides several feature selection methods, such as the `Ranker` class, which can be used to rank the features based on their importance. Regularization is a technique used to prevent overfitting of the regression model. It adds a penalty term to the regression equation, which can help improve the model's generalization ability. Weka provides several regularization methods, such as ridge regression and LASSO regression. Cross-validation is a resampling technique used to evaluate the performance of the regression model on different subsets of the dataset. It helps to estimate the model's performance on unseen data and reduces the risk of overfitting. Weka provides the `CrossValidator` class, which can be used to perform cross-validation on the regression model. ## Exercise Instructions: Instructions: Exercise: Write a Java program to optimize a multiple regression model using feature selection, regularization, and cross-validation in Java using the Weka library. ### Solution Here's a Java program that optimizes a multiple regression model using feature selection, regularization, and cross-validation in Java using the Weka library: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; import weka.attributeSelection.Ranker; import weka.classifiers.evaluation.CrossValidator; public class RegressionOptimizationExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Perform feature selection Ranker selector = new Ranker(); selector.setInputFormat(data); Instances selectedData = selector.rankAttributes(data); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(selectedData); // Perform cross-validation CrossValidator cv = new CrossValidator(); cv.setSeed(1); cv.setNumFolds(10); double accuracy = cv.crossValidate(regression, selectedData); System.out.println("Accuracy: " + accuracy); } } ``` This program demonstrates how to optimize a regression model using feature selection, regularization, and cross-validation using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. # Handling missing data and outliers in the dataset Handling missing data and outliers is crucial to ensure the reliability and accuracy of the regression model. Several techniques can be used to handle missing data and outliers, including data imputation, data deletion, and data transformation. Data imputation involves filling in missing values by estimating their values based on the existing data. Weka provides several imputation methods, such as the `SimpleImputer` class, which can be used to fill in missing values based on the mean or median of the corresponding attribute. Data deletion involves removing instances with missing values from the dataset. Weka provides the `RemoveWithValues` class, which can be used to remove instances with missing values based on their attribute values. Data transformation involves transforming the dataset to make it more suitable for regression analysis. Weka provides several transformation methods, such as the `Standardize` class, which can be used to standardize the dataset by subtracting the mean and dividing by the standard deviation of each attribute. ## Exercise Instructions: Instructions: Exercise: Write a Java program to handle missing data and outliers in a dataset using data imputation, data deletion, and data transformation in Java using the Weka library. ### Solution Here's a Java program that handles missing data and outliers in a dataset using data imputation, data deletion, and data transformation in Java using the Weka library: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.filters.Filter; import weka.filters.unsupervised.attribute.SimpleImputer; import weka.filters.unsupervised.instance.RemoveWithValues; import weka.filters.unsupervised.instance.Standardize; public class DataHandlingExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Perform data imputation SimpleImputer imputer = new SimpleImputer(); imputer.setInputFormat(data); Instances imputedData = Filter.useFilter(data, imputer); // Perform data deletion RemoveWithValues deleter = new RemoveWithValues(); deleter.setInputFormat(imputedData); Instances deletedData = Filter.useFilter(imputedData, deleter); // Perform data transformation Standardize transformer = new Standardize(); transformer.setInputFormat(deletedData); Instances transformedData = Filter.useFilter(deletedData, transformer); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(transformedData); } } ``` This program demonstrates how to handle missing data and outliers in a dataset using data imputation, data deletion, and data transformation using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. # Applying the regression model to real-world problems Once the regression model is optimized and validated, it can be applied to real-world problems to make predictions and solve practical issues. The regression model can be used in various fields, such as finance, healthcare, and marketing, to make informed decisions and improve the overall performance of the organization. In this section, we will discuss how to apply the regression model to real-world problems, including the following steps: 1. Collect and preprocess the data: Gather the necessary data for the problem and preprocess it to make it suitable for regression analysis. 2. Build the regression model: Train the regression model on the preprocessed data using the appropriate regression algorithm. 3. Evaluate the performance of the model: Evaluate the performance of the regression model using appropriate performance metrics. 4. Optimize the model: Use optimization techniques, such as feature selection, regularization, and cross-validation, to improve the performance of the regression model. 5. Apply the model to the problem: Use the optimized regression model to make predictions and solve the real-world problem. ## Exercise Instructions: Instructions: Exercise: Write a Java program to apply a multiple regression model to a real-world problem, such as predicting house prices based on various attributes. ### Solution Here's a Java program that applies a multiple regression model to a real-world problem, such as predicting house prices based on various attributes: ```java import weka.core.Instances; import weka.core.Attribute; import weka.core.FastVector; import weka.core.converters.ConverterUtils.DataSource; import weka.regression.SimpleRegression; public class RegressionRealWorldExample { public static void main(String[] args) throws Exception { // Load the dataset DataSource source = new DataSource("path/to/your/dataset.csv"); Instances data = source.getDataSet(); // Set the class index data.setClassIndex(data.numAttributes() - 1); // Create the SimpleRegression object SimpleRegression regression = new SimpleRegression(); // Train the regression model regression.buildRegression(data); // Make predictions using the regression model double[] attributes = new double[] {1200, 3, 2}; // Replace with the actual attribute values double predictedPrice = regression.predict(attributes); System.out.println("Predicted house price: " + predictedPrice); } } ``` This program demonstrates how to apply a regression model to a real-world problem using the Weka library in Java. Replace the path to the dataset with the path to your own dataset, and modify the code as needed to fit your specific problem. # Comparing regression analysis in Java with other programming languages Regression analysis can be performed in various programming languages, including Java. Comparing regression analysis in Java with other programming languages can help you understand the strengths and weaknesses of each language for regression analysis. Some popular programming languages for regression analysis include Python, R, and MATLAB. Python offers libraries like scikit-learn and statsmodels, which provide a wide range of regression analysis tools and techniques. R is a language specifically designed for statistical analysis and offers packages like caret and glmnet for regression analysis. MATLAB provides built-in functions for regression analysis and offers the Curve Fitting Toolbox for advanced regression techniques. In conclusion, Java is a powerful language for regression analysis, and Weka is a comprehensive library for implementing regression models. However, other programming languages may have their own strengths and advantages. It's essential to choose the right language and tools based on your specific problem and requirements.	Textbooks
seven and when you roll three dice it will add up to $21$. In three dice you roll three dice and add the top score up and the boottem score together and add it together we found out that the answer was always $21$ thats because three times seven makes $21$ and whatever you get on one side add to the other side always make $7$ like $6$ and $1$ and $2$ and $5$ they both make $7$ . The opposite sides always add up to $7$. When you have $3$ dice and you take the sum of the top and add the sum of the bottom the total equals $21$. This is because $3\times7$ is $21$. Thank you all for these interesting findings. Visualising. Generalising. Factors and multiples. Making and proving conjectures. Creating and manipulating expressions and formulae. Dice. Multiplication & division. Divisibility. Addition & subtraction. Working systematically.	CommonCrawl
Lens density measurements by two independent psychophysical techniques Anirbaan Mukherjee ORCID: orcid.org/0000-0003-4910-95871 & Richard A. Bone1 Eye and Vision volume 3, Article number: 24 (2016) Cite this article Cataract, a leading cause of vision impairment, is due to the lens becoming excessively optically dense. Change in the lens optical density (LOD) could be a useful indicator of incipient nuclear cataract and would necessitate the development of accurate measurement techniques. Mapcat sf™ is a heterochromatic flicker photometer for measuring macular pigment optical density (MPOD) under photopic conditions. In the process, it also measures LOD that is needed in the calculation of MPOD. LOD is then converted by the instrument to "lens equivalent age" (LEA). However, varying cone photoreceptor ratios among individuals could affect the LEA measurement. Scotopic vision is mediated by rod photoreceptors; therefore, LEA measurement under scotopic conditions potentially provides a reliable standard for assessing other methods. The study was conducted to test the level of agreement between the LEA data obtained under photopic and scotopic conditions for a sample population. We also comment on factors that might contribute to any disagreement. LEAs were obtained by Mapcat sf for 25 subjects and compared with those obtained under absolute scotopic threshold conditions. The mean scotopic LEA for the subjects was 2.7 years higher than the mean photopic LEA, but this difference was not statistically significant. Measurements by the two methods were reasonably correlated (r2 = 0.59, p < 0.0001). Significant individual differences in LEA by the two methods were found for six of the 25 subjects. Although our calculations included a standard long- to medium-wavelength-sensitive cone ratio, we found that different ratios could be found that rendered the differences in LEA insignificant for two of these six subjects. Variability in pupil diameter during scotopic measurements was considered another potential source of discrepancy between LEAs by the two methods. The absolute threshold technique, with long adaptation times, is probably impractical for routine lens density measurement, whereas Mapcat sf provided a rapid, straightforward test that may find its application in optometric/ophthalmic practice. Mapcat sf™ is an optical instrument designed by one of the authors [1]. Its primary function is to measure macular pigment optical density (MPOD) in the human retina. In the process, it also measures the lens optical density (LOD) and uses this information to provide a necessary correction in the calculation of MPOD. In testing the instrument, we have come to the realization that it could be a useful adjunct in monitoring LOD, particularly in patients with incipient nuclear cataract. The present study was an attempt to validate the instrument's capability of measuring LOD by comparing the results with those obtained using an absolute threshold technique. The nucleus, cortex and posterior sub-capsule (PSC) are the structures of the lens that undergo degradative changes and, potentially, exhibit cataract [2, 3] However, the reasons for the changes are different for these three structures [4]. In the nucleus, conformational changes to the component protein molecules render them susceptible to cross-linking and aggregation to result in increased light scatter and reduced transparency [5]. This process is accompanied by the accumulation of fluorescent chromophores, rendering the nucleus yellow, brown or even blackish-brown. This is the most common type of cataract and is age-related. In the cortex, opacity is due to stress in the lens fibers [4] while in the PSC region it results from defective fiber production by the epithelium. PSC cataract has been attributed to diabetes, myopia, exposure to ionizing radiation and steroid intake [3]. Neither cortical nor PSC cataract is accompanied by optical density changes, rather by increased opacity (light scatter). Cataract, generally, is one of the leading causes of age-related vision impairment in the world [6]. Cataract can also develop in younger individuals due to genetic factors [7] and other factors from which one can safeguard, such as UV exposure [8], smoking [9] and obesity [10]. An accelerated increase in the optical density of the lens (lens yellowing) is an indicator of progression towards nuclear cataract. Ophthalmologists and optometrists typically grade yellowing and sclerosis on a 1 to 4 scale [11]. Alternatively, or in addition, a quantitative measurement of an individual's LOD by a rapid, non-invasive test would be valuable [12, 13]. Examples of non-invasive, direct measurement techniques for measuring LOD are the signal-based autofluorescence [14] and the image-based reflectance [15] techniques. The technique of autofluorescence, aside from being expensive, is hampered by the scattering of light by the aging lens [16]. Reflectance techniques could potentially be compromised by the presence of drusen in the retina, particularly in older individuals [16]. On the other hand, psychophysical techniques involving either photopic or scotopic vision pose minimal risk and suffer no detrimental loss of signal due to scattering from the aging lens. Techniques employing photopic vision depend on the responses of the cone cells, and are fast and easy. Such techniques, whether they are heterochromatic flicker photometry (HFP) [17] or color-matching, employ the photopic luminous efficiency function, which is dependent on the relative weighting of long (L)- and medium (M)-wavelength-sensitive cones [18]. Since the weights of the L and M cones are known to vary over a wide range among individuals, the reliability of data obtained from photopic methods is questionable. This point was addressed by Wooten et al. [12] who, in addition to measuring LOD under scotopic conditions (see below), used HFP to measure LOD under photopic conditions. Mapcat sf uses photopic viewing conditions to obtain MPOD and LOD [1]. The LOD is interpreted through a model proposed by Sagawa and Takahashi, 2001 [19] as "lens equivalent age" (LEA). Sagawa and Takahashi claimed that the variation in the luminous efficiency with age, at least in the short wavelength region of the spectrum, is due to the progressive yellowing of the lens. Psychophysical methods employing scotopic vision, which is mediated by the rod cells, have been used to measure LOD [20, 21]. The measurements were made by comparing the scotopic spectral sensitivity functions of phakic and aphakic individuals. Differences in the functions were attributed to the transmittance of the lens in phakic individuals. In addition, the scotopic spectral sensitivity for aphakic individuals bears a marked resemblance to the rhodopsin absorbance spectrum. Therefore, one can compute the LOD spectrum by comparing the rhodopsin absorbance spectrum with an individual's scotopic spectral sensitivity function obtained at absolute threshold [22]. More recent studies employing scotopic vision to measure LOD have been reported by Wooten et al. [12], Teikari et al. [17], and Najjar et al. [23]. Unlike the photopic case, with the problem of variation in the L and M cone weighting, rod-mediated vision is dependent on one template, the absorbance spectrum of the rod photopigment, rhodopsin. Making measurements under absolute scotopic threshold conditions requires long durations of dark adaptation time (DA), however, such methods yield results that are in agreement with those obtained by ex vivo techniques [12]. Of note, Najjar et al. [23] have reported that LODs measured under reduced dark adaptation time were in good agreement with those measured under extended periods of DA. Nonetheless, in order to test the validity of LOD measurements made by the Mapcat sf, we decided to compare the results with those obtained under absolute scotopic threshold conditions in the same individuals. Photopic measurements Mapcat sf, a research-grade instrument employing HFP, whose primary purpose is to measure MPOD, was used to measure LOD. In HFP, lights of two different wavelengths are presented in anti-phase with each other, first in the fovea and then in the parafovea or perifovea, and the relative intensities are adjusted for a flicker null or minimum. The flicker null occurs when the luminances of the lights are equal [24]. The MPOD spectrum has peak absorbance at 460 nm, decreasing almost to zero absorbance at around 550 nm [25, 26]. Hence to measure MPOD using HFP, the two lights should have peak wavelengths close to these values. Mapcat sf employs a blue LED (peak wavelength 455 nm, 3 W power) and a green LED (peak wavelength 515 nm, 1 W power). The choice of 455 nm was dictated by the availability of LEDs in this region of the spectrum. The use of 515 nm, rather than a value closer to 550 nm, together with the wide bandwidths of both LEDs, meant that the LED spectra had to be included in the calculation of MPOD at 460 nm as well as LOD which the Mapcat sf reports at 420 nm (see equations 1–4 below). For the foveal and perifoveal measurements, centrally viewed 1.5 and 15° diameter circular stimuli, respectively, are used. For the perifoveal measurement, the subject seeks to minimize flicker around the periphery (7.5° eccentricity) where the effect of macular pigment is negligible. In both cases, the intensity of the blue LED is adjusted by the subject by altering the frequency of fixed amplitude, 10 μs wide pulses in the kilohertz range. The intensity of the green LED is held constant and provides a stimulus luminance of ~ 20 cd/m2 that is comfortable for the subject (not glaring) yet sufficient to eliminate rod participation. The lens absorbs strongly in the short wavelength region of the visible spectrum [27] decreasing to almost zero absorbance at about 550 nm [28]. Therefore, the luminances of the two lights will be differentially affected by the lens in the perifoveal measurement (and by the lens and macular pigment in the foveal measurement). Mapcat sf is a microprocessor-controlled instrument, with the blue and green LED intensities detected by a photodiode. The outputs of the photodiode at the perifoveal flicker null are given by $$ {\phi}_{BP}={k}_P\kern0.28em {\displaystyle \int {I}_B}\kern0.28em \left(\lambda \right)\kern0.28em S\kern0.28em \left(\lambda \right)\kern0.28em d\kern0.28em \lambda $$ $$ {\phi}_G={\displaystyle \int {I}_G}\kern0.28em \left(\lambda \right)\kern0.28em S\kern0.28em \left(\lambda \right)\kern0.28em d\kern0.28em \lambda $$ I B and I G are the intensity spectra of the blue and the green LEDs as measured by a spectrophotometer (Ocean Optics®). S (λ) is the spectral sensitivity of the photodiode. The multiplier k P in Eq. (1) is the adjustment factor for the intensity of the blue LED needed by the subject to achieve the flicker null/minimum. Since luminances of the blue and green lights are equal at the flicker null, $$ {k}_P=\frac{{\displaystyle \int {I}_B\kern0.28em \left(\lambda \right){V}_{10}\kern0.24em \left(\lambda, a\right)\kern0.28em d\kern0.28em \lambda }}{{\displaystyle \int {I}_G\kern0.28em \left(\lambda \right)\kern0.28em {V}_{10}\kern0.24em \left(\lambda, a\right)\kern0.28em d\kern0.28em \lambda }} $$ where V 10 (λ, a) is the standard CIE 10° photopic luminosity function, V 10 (λ), adjusted for age, a, according to the model of Sagawa and Takahashi [19]. Their model provides the average change in log luminous efficiency per year of age which, at the shorter wavelengths relevant for the Mapcat sf, they attributed to age-related changes in LOD. In choosing a 10° photopic luminosity function (the largest field size for which data are available), the effects on the function of macular pigment absorption are minimal, however, we had to assume that there would be negligible differences between V 10 (λ) and a corresponding 15° luminosity function. From Eqs. 1, 2, and 3, $$ \frac{\phi_{BP}}{\phi_G}=\frac{{\displaystyle \int {I}_G\kern0.28em }\left(\lambda \right)\kern0.28em {V}_{10}\kern0.28em \left(\lambda, a\right)\kern0.28em d\kern0.28em \lambda }{{\displaystyle \int {I}_B\kern0.28em \left(\lambda \right)\kern0.28em {V}_{10}\kern0.28em \left(\lambda, a\right)\kern0.28em d\kern0.28em \lambda }}\times \frac{{\displaystyle \int {I}_B\kern0.28em \left(\lambda \right)}\kern0.28em S\kern0.28em \left(\lambda \right)\kern0.28em d\kern0.28em \lambda }{{\displaystyle \int {I}_G\kern0.28em \left(\lambda \right)}\kern0.28em S\kern0.28em \left(\lambda \right)\kern0.28em d\kern0.28em \lambda } $$ Eq (4) was solved numerically by calculating $ \frac{\phi_{BP}}{\phi_G} $ for a range of values of a. The result is the smooth curve shown in Fig. 1. The curve was programmed into the Mapcat sf microprocessor in order to compute a value of a from the subject's perifoveal settings, ϕ BP and ϕ G . We refer to a, which may well be different from the subject's biological age, as the "lens equivalent age," LEA. This is consistent with the well-known inter-individual variability in LOD at any given age [23]. We did find (see Results) that, on average, the LEA was close to the biological age, thereby giving us confidence in using the Sagawa and Takahashi [19] model. LEA as a function of the ratio of photodetector outputs for the perifoveal flicker null setting Scotopic measurements Scotopic vision occurs under ambient light conditions below −2.5 log Trolands and is mediated purely by the rod cells. The spatial density of rods increases with eccentricity in the retina, going from zero in the foveal center to a peak of approximately 190,000 rods/mm2 at 20 to 30° from the fovea. Therefore, absolute threshold measurements require a parafoveal or perifoveal stimulus that is only affected by absorption in the lens and negligibly by the macular pigment [29]. Thus, the scotopic luminosity function at eccentricities above ~5° is affected primarily by the spectral transmittance, T L (λ), of the lens and the rhodopsin absorbance spectrum, R (λ) [30]. $$ {V}_{scotopic}=R\;\left(\lambda \right)\;{T}_L\;\left(\lambda \right) $$ The lens transmittance was calculated from that of a 32-year-old, using a model proposed by van de Kraats and van Norren [31], and then modified for age using the template by Sagawa and Takahashi [19]. Absolute thresholds were obtained for each of the two stimuli located at the same retinal eccentricity used in the Mapcat sf perifoveal test and illuminated by LEDs of similar wavelengths. The luminances of the LEDs were varied using a neutral density wedge in order to achieve the absolute threshold condition. The corresponding luminances, L B and L G of the stimuli due to the light from the blue and green sources are $$ {L}_B={\displaystyle \int B\kern0.28em \left(\lambda \right)\kern0.28em {T}_F\kern0.28em \left(\lambda \right)\kern0.28em {T}_w\kern0.28em \left({x}_b,\kern0.28em \lambda \right)\kern0.28em R\left(\lambda \right){T}_L\kern0.28em \left(\lambda, a\right)\kern0.28em d\lambda } $$ $$ {L}_G={\displaystyle \int G\kern0.28em \left(\lambda \right)\kern0.28em {T}_F\kern0.28em \left(\lambda \right)\kern0.28em {T}_w\kern0.28em \left({x}_g,\kern0.28em \lambda \right)\kern0.28em R\left(\lambda \right){T}_L\kern0.28em \left(\lambda, a\right)\kern0.28em d\lambda } $$ B (λ), G (λ) are the normalized intensity spectra of the blue and green LEDs obtained with an Ocean Optics® spectrometer. T F (λ) is the transmittance of an auxiliary neutral density filter placed in the light path to achieve the necessary scotopic conditions. T w (x b , λ) and T w (x g , λ) are the transmittances of the variable density wedge at translational settings of x b and x g , the positions at which the absolute threshold is detected for the blue and the green stimuli, respectively. At the absolute threshold, the luminance values for the test stimuli are equal, hence, $$ \frac{1{0}^{{\textstyle \hbox{-} }1.2036{x}_g}}{1{0}^{{\textstyle \hbox{-} }1.1364{x}_b}}=0.59\kern0.28em \frac{{\displaystyle \int B\kern0.28em \left(\lambda \right)}\kern0.28em {T}_F\kern0.28em \left(\lambda \right)\kern0.28em {T}_w\kern0.28em \left({x}_b,\kern0.28em \lambda \right)\kern0.28em R\kern0.28em \left(\lambda \right)\kern0.28em {T}_L\kern0.28em \left(\lambda, a\right)\kern0.28em d\kern0.28em \lambda }{{\displaystyle \int G\kern0.28em \left(\lambda \right)}\kern0.28em {T}_F\kern0.28em \left(\lambda \right)\kern0.28em {T}_w\kern0.28em \left({x}_g,\kern0.28em \lambda \right)\kern0.28em R\kern0.28em \left(\lambda \right)\kern0.28em {T}_L\kern0.28em \left(\lambda, a\right)\kern0.28em d\kern0.28em \lambda } $$ In Eq. (8), the numerical factors 1.2036, 1.1364 and 0.59 are associated with the calibration of the neutral density wedge and the photodetector. All the wavelength-dependent functions are known; thus the right-hand side can be evaluated as a function of the LEA, a. Labeling that function as F, we calculated its value for various values of LEA, as shown in Fig. 2, and then modeled the relationship using a Sigmaplot® curve-fit routine: LEA as a function of F (Eq. 8). F itself is a function of the wedge settings required to achieve absolute threshold $$ LEA=\frac{a+bF}{1+cF+d{F}^2} $$ This rational four-parameter equation was found to give an excellent fit to the curve in Fig. 2 (R2 = 0.9998), thereby providing an accurate means of calculating the value of LEA from the value of F. The visual field produced by the instrument is shown in Fig. 3. Offset from the center at 7.5° eccentricity is a 0.5° stimulus illuminated by either a blue or green LED of peak wavelength 455 or 515 nm, respectively. The wavelengths and retinal location of these stimuli are very close to those used in the Mapcat sf perifoveal test. We reasoned that small amounts of macular pigment that might be present at 7.5° eccentricity would introduce a similar error in the measurement of LEA by each method. Additionally, rod density is relatively high at this eccentricity. In a similar study by Wooten et al. [12], wavelengths of 406 and 550 nm, derived from a monochromator, were selected to maximize the difference between the corresponding LODs. Correspondingly, the lens density template could be fit to the data with greater certainty. LEDs of wavelength 410 to 420 nm are certainly available, but their spectra extend down to about 370 nm where uncertainties in the rhodopsin absorbance and LOD are large. The intensity of the LEDs can be controlled, as in the Mapcat sf, by varying the frequency of fixed amplitude, 10 μs wide pulses in the kilohertz range. Such a control system, together with mounting the LEDs on substantial heat sinks and employing forced air cooling to minimize temperature changes, was found to prevent any measurable shift in the peak wavelength as the intensity was changed. The LEDs are cycled on (0.5 s) and off (1.0 s) resulting in the visual perception of short flashes that, when above threshold, were found easy to count. Similar flash durations have been used elsewhere [32]. At the center of the visual field is a small 0.25° fixation light illuminated by a low intensity red LED. Visual field for absolute threshold measurements The optical layout of the instrument is shown in Fig. 4. The test LEDs are mounted in the upper interior part of an integrating sphere S. This arrangement results in spatially uniform light emerging from the front aperture of the sphere. In order to achieve scotopic conditions, a series of neutral density filters and a variable neutral density wedge filter are placed between the sphere S and the eye-piece. The wedge position is adjusted electromechanically via a stepper motor that is controlled by the operator. The pulse frequency for each LED, and therefore its intensity, is set at the lowest available level (~8 kHz), thereby reducing the likelihood of temperature-related wavelength shifts, and stimulus luminance is controlled entirely with the wedge. The housing for the eyepiece has an opaque screen at the left end in which precisely cut apertures define the size of the stimulus and fixation target. A dim red LED is mounted just behind the aperture for the latter. At the right end of the housing is an achromatic lens that can be moved back and forth to allow individual subjects to focus on the stimulus. A non-limiting viewing aperture is positioned so that the distance between the nodal point of the subject's eye and the lens is equal to the focal length of the lens. It can be shown that such an arrangement maintains the same angular size and eccentricity of the stimulus for all subjects, regardless of adjustments to the lens position [1]. Optical system for measuring LEA under scotopic conditions Subject recruitment Twenty-five subjects with no (self-reported) visual pathologies or color vision defects, and in the age range of 20 to 75 years (average age 39 ± 14 years), were recruited for the study from the University faculty, staff and students. Potential subjects were excluded from the study if they were unable to adjust the focusing optics to obtain sharply focused images of the stimuli on their retinas. Written informed consent was obtained from all subjects and the study was approved by the Florida International University's Institutional Review Board. Photopic testing Since there is generally a high level of agreement between the LOD in the left and right eyes of healthy subjects [12], only the right eye was tested. At the beginning of the test, the subject's left eye was covered by an eye patch to minimize visual distractions. Since the optics of the instrument are enclosed in an instrument case, the room lighting was left on. The Mapcat sf was positioned on an adjustable table. The subject was seated in front of the viewing aperture and the operator adjusted the heights of the table and a chin rest to facilitate optimum comfort. The subject was then instructed to bring the 1.5° stimulus into sharp focus via a hand-held control. The flicker frequency was adjusted by the operator to 24 Hz, sufficiently high to ensure mediation of detection by the luminance channel. The subject was directed to fix his/her gaze at the center of the cross-hairs in the 1.5° stimulus and to adjust the intensity of the blue LED using the hand-held control until a flicker null/minimum was obtained. Once the subject reported a flicker null, the intensity value was recorded by the operator. A total of five settings were recorded with random offsets being introduced automatically by the instrument after each recording. For the perifoveal phase of the test, the 1.5° stimulus was replaced by the 15° stimulus. The operator adjusted the flicker frequency to 31 Hz and instructed the subject to keep his/her gaze fixed at the center of the cross hairs and minimize flicker in the periphery of the stimulus by again adjusting the blue LED intensity. Five repeat measurements were again recorded. The built-in microprocessor then calculated the MPOD, the percentage of the blue light blocked by the macular pigment, the LOD and the LEA together with the associated standard errors. Scotopic testing Scotopic testing of a subject was conducted within a few weeks at most after photopic testing. All subjects completed the tests over a 3-month period. Prior to dark adaptation, the subject adjusted the position of the eye-piece lens in order to bring the field of view into sharp focus. This was followed by a dark adaptation period of 30 min in complete darkness. As an additional precaution, the right test eye was covered with an eye patch. At the end of this period, the operator turned on the flashing blue stimulus and instructed the subject to fixate on the minimally lit red LED so that the stimulus was perceived peripherally. Initially, the stimulus intensity was set well above threshold so that the subject could see 14 flashes during a 20 s period. The intensity was then lowered incrementally by moving the neutral density wedge and, at each step, the subject reported the number of flashes that could be seen during the 20 s observation period. This is a modified version of the method of constant stimuli described in a similar study by Hammond et al. [33]. In the present study, when the number of flashes that could be perceived reached approximately 2 out of the 14 presented, the test was terminated. The operator then replaced the blue stimulus with the green one, and the test was repeated. Photopic testing of a subject with the Mapcat sf required ~10 min. This included subject training using a PowerPoint presentation of the testing procedure, and the actual test. Scotopic testing on the other hand required almost an hour. This included subject training, 30 min for dark adaptation, and the test itself. The general feedback from the subjects was that they found the photopic testing easier than the scotopic testing. Within the photopic test, they reported that the perifoveal adjustments were easier than the foveal ones. This was evident from the standard deviations of the five flicker null settings. On average, the standard deviation for the perifoveal test was approximately half that for the foveal test. The data obtained from scotopic measurements were used to generate scatter plots of the number of counts vs. wedge position, and a three-parameter sigmoidal function, shown in Eq. (10), was fit to the data. $$ N=\frac{a}{1+{e}^{-\frac{\left(x\mathit{\hbox{-}}{x}_0\right)}{b}}} $$ N is the number of counts and x is the neutral density wedge position. A representative graph is shown in Fig. 5. From the fit curve, we used a 50 % visibility criterion to define the absolute threshold wedge setting, represented by the parameter x o in Eq. (10). This is the wedge setting for which 50 % of the flashes were visible. By substituting threshold wedge settings for the blue and green stimuli into the left side of Eq. (8), we obtained the quantity F, and substituting this into Eq. (9) gave us the LEA. The uncertainty in LEA was calculated from the uncertainty in the appropriate parameter, x o , in the sigmoidal function. Representative scatter plot for the blue stimulus of number of flashes detected in a 20 s period as a function of wedge position. An inverse sigmoidal curve of the form in Eq. (10) was fit to the data to obtain the absolute threshold The LEAs obtained from the scotopic and photopic experiments are shown in columns 3 and 4 of Table 1 and in the scatter plot shown in Fig. 6. Table 1 LEAs measured under scotopic (column 3) and photopic (column 4) conditions Scatter plot comparing LEA results obtained under photopic and scotopic conditions. The equation of the regression line (dashes) is y = 0.833× + 3.945. The 95 % confidence limits are shown (curved lines). The solid line refers to slope unity The solid line of slope unity represents equality of the two LEA measurements while the dashed line is the regression line of LEA (photopic) on LEA (scotopic). The slope of the regression line was 0.833 ± 0.144 with r2 = 0.59, p < 0.0001. The intercept was 3.95 ± 5.96 years. A Bland Altman analysis, shown in Fig. 7, was carried out as an additional check for any systematic deviation between the two methods. A Shapiro-Wilk test indicated that the differences met the standard of normality, thereby validating the use of Bland Altman analysis. The mean difference between the scotopic and the photopic LEAs from the analysis was 2.7 years and the limits of agreement (±1.96 SD) were ± 16 years. Also shown in Fig. 7 are the regression line (difference on mean) and the associated 95 % confidence limits. Bland Altman plot for testing the level of agreement between LEAs obtained under photopic and scotopic viewing conditions. The dashed horizontal line represents the mean difference between the scotopic and the photopic LEAs, with the 95 % confidence interval shown in gray. The limits of agreement (mean ± 1.96 SD) are included with their associated 95 % confidence intervals shown in gray. The sloping regression line and its 95 % confidence limits have been added to the plot An overall comparison of the results of the two methods of obtaining the LEA can be found in the slope (0.833 ± 0.144) and intercept (3.95 ± 5.96 years) of the regression line in Fig. 6. The 95 % confidence limits contain the line of equality of LEA by the two methods, indicating no significant difference, on average, between them. Likewise, the Bland-Altman plot of Fig. 7 showed a mean difference of 2.7 years between the LEAs obtained by the two methods, reflecting a small trend of a lower LEA, on average, when measured photopically by Mapcat sf than when it is measured scotopically at absolute threshold. However, the associated 95 % confidence interval (gray area) included the line of equality (0 year) showing the trend was not significant. The regression line added to the Bland-Altman plot (Fig. 7) had a negative slope suggesting that younger subjects had a larger difference (scotopic LEA-photopic LEA) than older subjects. Again the associated confidence interval included the line of equality indicating that the trend was insignificant. We compared these results with measurements reported by Wooten et al. [12] under photopic and scotopic conditions. The slope of their regression line, lens optical density (photopic) on lens optical density (scotopic), was 0.92, with r2 = 0.64, p < 0.0002; results which are rather similar to our own LEA results (slope = 0.833 ± 0.144 with r2 = 0.59, p < 0.0001). They too obtained a positive but insignificant intercept of 0.09 density units that we estimate to be equivalent to about 5 to 6 years in LEA [19]. Again, this is of the same order as our own intercept of ~4 years. Like those of Wooten et al. [12], our photopic results were slightly lower, on average, than the scotopic ones. However, if we examine our data for individual subjects, significant differences emerge. To determine the significance, we calculated the absolute difference, ∣ scotopic LEA-photopic LEA∣, and the associated standard deviation. We applied a criterion that if the range, difference -2SD to difference +2SD, included zero, the difference was not significant. Accordingly, differences in LEA by the two methods that were not deemed significant were limited to 19 out of the 25 subjects. The remaining 6 subjects are indicated in Table 1 by a superscript a. Further analysis was therefore conducted in an attempt to shed light on the discrepancies. The standard CIE-approved 10° luminous efficiency function corresponds to an estimate of the average weighting of L and M cones [18]. However, considerable variation in the L:M cone ratio has been reported: 0.6 to 12 by Carroll et al. [34], 1.1 to 16.5 by Hofer et al. [35] and 0.47 to 15.82 by Sharpe et al. [18]. The luminous efficiency function is also affected by L-cone polymorphisms which are common, and M-cone polymorphisms which are rare [18]. However, the effects are small in comparison with variations associated with different L:M ratios. To determine the effect of these variations on the LEAs measured by Mapcat sf, we used the template for the 10° luminous efficiency function proposed by Sharpe et al. [18]. With this template, photopic luminous efficiency functions can be generated for any L:M ratio. Using each subject's ratio of perifoveal flicker null settings, $ \frac{\phi_{BP}}{\phi_G} $ , obtained on Mapcat sf (see Eq. 4), we calculated the LEA as described earlier, but using instead age-modified photopic luminous efficiency functions based on L:M ratios of 0.47, the lower extreme reported by Sharpe et al. [18], and 16.5, the upper extreme reported by Hofer et al. [35]. (Note, however, that the effects on LEA are almost constant for L:M ratios greater than approximately 8.) The results are shown in columns 5 and 6 of Table 1. For the 6 subjects with significant differences between their LEAs by the photopic and scotopic methods, we repeated our analysis of significance but with their photopic LEAs calculated for an extreme cone weight that provided a value closer to the scotopic LEA. With this modification, significant differences were eliminated for 2 of the 6 subjects. The remaining 4 were subjects # 10, 15, 22 and 25. Thus it is conceivable, but unproven, that some, but not all, of the observed differences could be due to variability of cone weights among individuals. The larger uncertainties in the scotopic measurements, evident from the error bars in Fig. 6, reflect the general impression gained from the subjects that the test was far more difficult than the photopic one. This is one reason for placing greater trust in the photopic measurements. Another reason is to be found in comparisons between the subjects' biological ages and their LEAs obtained by the two methods. When plotting LEA measured photopically as a function of biological age, the regression line had a slope of 0.79 ± 0.08 and an intercept of 7.3 ± 3.4 years (r2 = 0.79, p < 0.0001). In the scotopic case, the regression line had a slope of 0.54 ± 0.13 and an intercept of 19.2 ± 5.1 year (r2 = 0.44, p = 0.0003). Thus the LEA obtained photopically is generally closer to the subject's biological age than that obtained scotopically, and is more highly correlated. There are other factors which could potentially influence the scotopic measurements. The apparatus provided a viewing aperture that did not constitute an artificial limiting pupil. Thus, a change in natural pupil size would alter the amount of light entering the eye and this would affect the threshold. Note, however, that this would not be a concern in the photopic method where pupil size changes would affect the two components of the flickering stimulus equally. We have estimated the effect of a potential change in pupil size during the course of scotopic measurements. Pupillometry measurements on dark-adapted adults by Alpern and Campbell [36] and Hansen and Fulton [37] show a short-term variability in pupil diameter. From their data, we have estimated the ratio of maximum to minimum pupil area to be 1.18 and 1.12, respectively, or 1.15 on average. Nevertheless, some of the variability may be due to uncertainty in measurement. Assuming a worst case scenario where, for example, the blue and green stimulus thresholds were determined at maximum and minimum pupil areas, respectively, we have calculated the effect on the LEA to be an overestimate of about 10 years. This becomes an underestimate of about 10 years if the blue and green stimulus thresholds were determined at minimum and maximum pupil areas, respectively. Examination of columns 3 and 4 in Table 1 indicates that the difference between the photopically and scotopically determined LEAs exceeded 10 years in only five subjects. Of these, only subject #10 exhibited a difference that could not be accounted for by a combination of the pupil size variation described here and an extreme L:M cone ratio of 0.47. However, it must be emphasized that pupil size variation, in addition to L:M cone ratio variation, remains speculative, and an unproven reason, as yet, for the differences. Another factor that we have considered is a potential change in the level of the subject's dark adaptation during the test. For example, if the subject, in a state of incomplete dark adaptation, began the test with the blue stimulus and concluded, fully dark adapted, with the green stimulus, the LEA would tend to be higher than the true value. However, dark adaptation curves, such as those of Hecht and Mandelbaum [32], indicate that absolute threshold is attained after 30 min, even after initial adaptation to a very bright stimulus. Therefore, we do not believe that incomplete dark adaptation was a significant factor. For the photopic measurements, potential changes in adaptation would also not be a concern owing to the use of a large, uniform visual field surrounding the stimulus and matching its luminance at the flicker null point [1]. The main goal of this study was to validate LOD measurements made with the Mapcat sf under photopic conditions. The primary purpose of a reliable measurement was to be able to correctly compensate for LOD in the calculation of MPOD that the Mapcat sf performs. A secondary advantage would be to provide optometrists and ophthalmologists with a means of quantifying lens yellowing and monitoring its rate of progression. Since measuring LOD under scotopic conditions, especially at absolute thresholds, requires prolonged periods of dark adaptation, photopic methods are preferable for routine testing. The scotopic measurements revealed no systematic deviations from the photopic measurements on the Mapcat sf, and we suggest that individual differences, when these occur, are more likely to be attributable to the scotopic test. Bone RA, Mukherjee A. Innovative Troxler-free measurement of macular pigment and lens density with correction of the former for the aging lens. J Biomed Opt. 2013;18(10):107003. Taylor VL, Al-Ghoul KJ, Lane CW, Davis VA, Kuszak JR, Costello MJ. Morphology of the normal human lens. Invest Ophthalmol Vis Sci. 1996;37(7):1396–410. Beebe DC, Holekamp NM, Shui YB. Oxidative damage and the prevention of age-related cataracts. Ophthalmic Res. 2010;44(3):155–65. Michael R, Bron AJ. The ageing lens and cataract: a model of normal and pathological ageing. Philos Trans R Soc Lond B Biol Sci. 2011;366(1568):1278–92. Horwitz J, Bova MP, Ding LL, Haley DA, Stewart PL. Lens alpha-crystallin: function and structure. Eye (Lond). 1999;13:403–8. Roodhooft JM. Leading causes of blindness worldwide. Bull Soc Belge Ophtalmol. 2002;283:19–25. Shiels A, Hejtmancik FJ. Genetic origins of cataract. Arch Ophthalmol. 2007;125(2):165–73. Roberts JE. Ultraviolet radiation as a risk factor for cataract and macular degeneration. Eye Contact Lens. 2011;37(4):246–9. Cumming RG, Mitchell P. Alcohol, smoking, and cataracts: the Blue Mountains Eye Study. Arch Ophthalmol. 1997;115(10):1296–303. Cheung N, Wong TY. Obesity and eye diseases. Surv Ophthalmol. 2007;52(2):180–95. Murrill CA, Stanfield DL, Vanbrocklin MD, Bailey IL, Denbeste BP, Diiorio RC, et al. Care of the Adult Patient with Cataract. St. Louis: American Optometric Association; 1999. Wooten BR, Hammond BR, Renzi LM. Using scotopic and photopic flicker to measure lens optical density. Ophthalmic Physiol Opt. 2007;27(4):321–8. Savage G, Johnson CA, Howard DL. A comparison of noninvasive objective and subjective measurements of the optical density of human ocular media. Optom Vis Sci. 2001;78(6):386–95. Bleeker JC, van Best JA, Vrij L, van der Velde EA, Oosterhuis JA. Autofluorescence of the lens in diabetic and healthy subjects by fluorophotometry. Invest Ophthalmol Vis Sci. 1986;27(5):791–4. Delori FC, Burns SA. Fundus reflectance and the measurement of crystalline lens density. J Opt Soc Am A Opt Image Sci Vis. 1996;13(2):215–26. Seppo S. Lens autofluorescence: In aging and cataractous human lenses. Clinical applicability. Ph.D. thesis. Oulu: Faculty of Medicine, University of Oulu; 1999. Teikari P, Najjar RP, Knoblauch K, Dumortier D, Cornut PL, Denis P, et al. Refined flicker photometry technique to measure ocular lens density. J Opt Soc Am A Opt Image Sci Vis. 2012;29(11):2469–78. Sharpe LT, Stockman A, Jagla W, Jägle H. A luminous efficiency function, V* (lambda), for daylight adaptation. J Vis. 2005;5(11):948–68. Sagawa K, Takahashi Y. Spectral luminous efficiency as a function of age. J Opt Soc Am A Opt Image Sci Vis. 2001;18(11):2659–67. Wright WD. The visual sensitivity of normal and aphakic observers in the ultraviolet. Ann Psychol. 1949;50(1):169–77. Wald G. Human vision and the spectrum. Science. 1945;101(2635):653–8. Norren DV, Vos JJ. Spectral transmission of the human ocular media. Vision Res. 1974;14(11):1237–44. Najjar RP, Teikari P, Cornut PL, Knoblauch K, Cooper HM, Gronfier C. Heterochromatic Flicker Photometry for Objective Lens Density Quantification. Invest Ophthalmol Vis Sci. 2016;57(3):1063–71. Bone RA, Landrum JT. Heterochromatic flicker photometry. Arch Biochem Biophys. 2004;430(2):137–42. Snodderly DM, Brown PK, Delori FC, Auran JD. The macular pigment. I. Absorbance spectra, localization, and discrimination from other yellow pigments in primate retinas. Invest Ophthalmol Vis Sci. 1984;25(6):660–73. Bone RA, Landrum JT, Cains A. Optical density spectra of the macular pigment in vivo and in vitro. Vision Res. 1992;32(1):105–10. Pokorny J, Smith VC, Lutze M. Aging of the human lens. Appl Opt. 1987;26(8):1437–40. Gaillard ER, Zheng L, Merriam JC, Dillon J. Age-related changes in the absorption characteristics of the primate lens. Invest Ophthalmol Vis Sci. 2000;41(6):1454–9. Hammond BR, Wooten BR, Snodderly DM. Individual variations in the spatial profile of human macular pigment. J Opt Soc Am A Opt Image Sci Vis. 1997;14(6):1187–96. Crescetelli F, Dartnall HJ. Human visual purple. Nature. 1953;172(4370):195–7. van de Kraats J, van Norren D. Optical density of the aging human ocular media in the visible and UV. J Opt Soc Am A Opt Image Sci Vis. 2007;24(7):1842–57. Hecht S, Mandelbaum J. The relation between vitamin A and dark adaptation. JAMA. 1939;112:1910–6. Hammond BR Jr, Wooten BR, Snodderly DM. Density of the human crystalline lens is related to the macular pigment carotenoids, lutein and zeaxanthin. Optom Vis Sci. 1997;74(7):499–504. Carroll J, Mcmahon C, Neitz M, Neitz J. Flicker-photometric electroretinogram estimates of L:M cone photoreceptor ratio in men with photopigment spectra derived from genetics. J Opt Soc Am A Opt Image Sci Vis. 2000;17(3):499–509. Hofer H, Carroll J, Neitz J, Neitz M, Williams DR. Organization of the human trichromatic cone mosaic. J Neurosci. 2005;25(42):9669–79. Alpern M, Campbell RW. The behaviour of the pupil during dark-adaptation. J Physiol Lond. 1962;165:5–7. Hansen RN, Fulton AB. Pupillary changes during dark adaptation in human infants. Invest Ophthalmol Vis Sci. 1986;27:1726–9. Support was provided by Beneseed Co., Ltd. Authors' contribution Anirbaan Mukherjee and Richard A. Bone were involved in designing the experiment, fabricating instruments involved, subject recruitment and manuscript preparation. Both authors read and approved the final manuscript. Richard Bone is a Science Advisory Board member for Guardion Health Sciences, which owns a patent for Mapcat sf. Richard Bone is listed as the inventor. Otherwise he has no conflicts of interest, including Beneseed Co. Ltd. Anirbaan Mukherjee has no conflicts of interest, including Beneseed Co. Ltd. Department of Physics, Florida International University, 11200 SW 8th Street, Miami, 33199, FL, USA Anirbaan Mukherjee & Richard A. Bone Search for Anirbaan Mukherjee in: Search for Richard A. Bone in: Correspondence to Anirbaan Mukherjee. Mukherjee, A., Bone, R.A. Lens density measurements by two independent psychophysical techniques. Eye and Vis 3, 24 (2016). https://doi.org/10.1186/s40662-016-0054-6 Accepted: 19 August 2016 Heterochromatic flicker photometry Lens optical density Absolute threshold Photopic Scotopic	CommonCrawl
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces. This article is about Fréchet spaces in functional analysis. For Fréchet spaces in general topology, see T1 space. For the type of sequential space, see Fréchet–Urysohn space. A Fréchet space $X$ is defined to be a locally convex metrizable topological vector space (TVS) that is complete as a TVS,[1] meaning that every Cauchy sequence in $X$ converges to some point in $X$ (see footnote for more details).[note 1] Important note: Not all authors require that a Fréchet space be locally convex (discussed below). The topology of every Fréchet space is induced by some translation-invariant complete metric. Conversely, if the topology of a locally convex space $X$ is induced by a translation-invariant complete metric then $X$ is a Fréchet space. Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space" to mean a complete metrizable topological vector space, without the local convexity requirement (such a space is today often called an "F-space").[1] The local convexity requirement was added later by Nicolas Bourbaki.[1] It's important to note that a sizable number of authors (e.g. Schaefer) use "F-space" to mean a (locally convex) Fréchet space while others do not require that a "Fréchet space" be locally convex. Moreover, some authors even use "F-space" and "Fréchet space" interchangeably. When reading mathematical literature, it is recommended that a reader always check whether the book's or article's definition of "F-space" and "Fréchet space" requires local convexity.[1] Definitions Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of seminorms. Invariant metric definition A topological vector space $X$ is a Fréchet space if and only if it satisfies the following three properties: 1. It is locally convex.[note 2] 2. Its topology can be induced by a translation-invariant metric, that is, a metric $d:X\times X\to \mathbb {R} $ such that $d(x,y)=d(x+z,y+z)$ for all $x,y,z\in X.$ This means that a subset $U$ of $X$ is open if and only if for every $u\in U$ there exists an $r>0$ such that $\{v:d(v,u)<r\}$ is a subset of $U.$ 3. Some (or equivalently, every) translation-invariant metric on $X$ inducing the topology of $X$ is complete. • Assuming that the other two conditions are satisfied, this condition is equivalent to $X$ being a complete topological vector space, meaning that $X$ is a complete uniform space when it is endowed with its canonical uniformity (this canonical uniformity is independent of any metric on $X$ and is defined entirely in terms of vector subtraction and $X$'s neighborhoods of the origin; moreover, the uniformity induced by any (topology-defining) translation invariant metric on $X$ is identical to this canonical uniformity). Note there is no natural notion of distance between two points of a Fréchet space: many different translation-invariant metrics may induce the same topology. Countable family of seminorms definition The alternative and somewhat more practical definition is the following: a topological vector space $X$ is a Fréchet space if and only if it satisfies the following three properties: 1. It is a Hausdorff space, 2. Its topology may be induced by a countable family of seminorms $\|\|\cdot \|\|_{k}$ $k=0,1,2,\ldots $ This means that a subset $U\subseteq X$ is open if and only if for every $u\in U$ there exists $K\geq 0$ and $r>0$ such that $\{v\in X:\\|v-u\\|_{k}<r{\text{ for all }}k\leq K\}$ is a subset of $U,$ 3. it is complete with respect to the family of seminorms. A family ${\mathcal {P}}$ of seminorms on $X$ yields a Hausdorff topology if and only if[2] $\bigcap _{\\|\,\cdot \,\\|\in {\mathcal {P}}}\{x\in X:\\|x\\|=0\}=\{0\}.$ A sequence $x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }$ in $X$ converges to $x$ in the Fréchet space defined by a family of seminorms if and only if it converges to $x$ with respect to each of the given seminorms. As webbed Baire spaces Theorem[3] (de Wilde 1978) — A topological vector space $X$ is a Fréchet space if and only if it is both a webbed space and a Baire space. Comparison to Banach spaces In contrast to Banach spaces, the complete translation-invariant metric need not arise from a norm. The topology of a Fréchet space does, however, arise from both a total paranorm and an F-norm (the F stands for Fréchet). Even though the topological structure of Fréchet spaces is more complicated than that of Banach spaces due to the potential lack of a norm, many important results in functional analysis, like the open mapping theorem, the closed graph theorem, and the Banach–Steinhaus theorem, still hold. Constructing Fréchet spaces Recall that a seminorm $\\|\cdot \\|$ is a function from a vector space $X$ to the real numbers satisfying three properties. For all $x,y\in X$ and all scalars $c,$ $\\|x\\|\geq 0$ $\\|x+y\\|\leq \\|x\\|+\\|y\\|$ $\\|c\cdot x\\|=\|c\|\\|x\\|$ If $\\|x\\|=0\iff x=0$, then $\\|\cdot \\|$ is in fact a norm. However, seminorms are useful in that they enable us to construct Fréchet spaces, as follows: To construct a Fréchet space, one typically starts with a vector space $X$ and defines a countable family of seminorms $\\|\cdot \\|_{k}$ on $X$ with the following two properties: • if $x\in X$ and $\\|x\\|_{k}=0$ for all $k\geq 0,$ then $x=0$; • if $x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }$ is a sequence in $X$ which is Cauchy with respect to each seminorm $\\|\cdot \\|_{k},$ then there exists $x\in X$ such that $x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }$ converges to $x$ with respect to each seminorm $\\|\cdot \\|_{k}.$ Then the topology induced by these seminorms (as explained above) turns $X$ into a Fréchet space; the first property ensures that it is Hausdorff, and the second property ensures that it is complete. A translation-invariant complete metric inducing the same topology on $X$ can then be defined by $d(x,y)=\sum _{k=0}^{\infty }2^{-k}{\frac {\\|x-y\\|_{k}}{1+\\|x-y\\|_{k}}}\qquad x,y\in X.$ The function $u\mapsto {\frac {u}{1+u}}$ maps $[0,\infty )$ monotonically to $[0,1),$ and so the above definition ensures that $d(x,y)$ is "small" if and only if there exists $K$ "large" such that $\\|x-y\\|_{k}$ is "small" for $k=0,\ldots ,K.$ Examples From pure functional analysis • Every Banach space is a Fréchet space, as the norm induces a translation-invariant metric and the space is complete with respect to this metric. • The space $\mathbb {R} ^{\omega }$ of all real valued sequences (also denoted $\mathbb {R} ^{\mathbb {N} }$) becomes a Fréchet space if we define the $k$-th seminorm of a sequence to be the absolute value of the $k$-th element of the sequence. Convergence in this Fréchet space is equivalent to element-wise convergence. From smooth manifolds • The vector space $C^{\infty }([0,1])$ of all infinitely differentiable functions $f:[0,1]\to \mathbb {R} $ becomes a Fréchet space with the seminorms $\\|f\\|_{k}=\sup\{\|f^{(k)}(x)\|:x\in [0,1]\}$ for every non-negative integer $k.$ Here, $f^{(k)}$ denotes the $k$-th derivative of $f,$ and $f^{(0)}=f.$ In this Fréchet space, a sequence $\left(f_{n}\right)\to f$ of functions converges towards the element $f\in C^{\infty }([0,1])$ if and only if for every non-negative integer $k\geq 0,$ the sequence $\left(f_{n}^{(k)}\right)\to f^{(k)}$ converges uniformly. • The vector space $C^{\infty }(\mathbb {R} )$ of all infinitely differentiable functions $f:\mathbb {R} \to \mathbb {R} $ becomes a Fréchet space with the seminorms $\\|f\\|_{k,n}=\sup\{\|f^{(k)}(x)\|:x\in [-n,n]\}$ for all integers $k,n\geq 0.$ Then, a sequence of functions $\left(f_{n}\right)\to f$ converges if and only if for every $k,n\geq 0,$ the sequences $\left(f_{n}^{(k)}\right)\to f^{(k)}$ converge compactly. • The vector space $C^{m}(\mathbb {R} )$ of all $m$-times continuously differentiable functions $f:\mathbb {R} \to \mathbb {R} $ becomes a Fréchet space with the seminorms $\\|f\\|_{k,n}=\sup\{\|f^{(k)}(x)\|:x\in [-n,n]\}$ for all integers $n\geq 0$ and $k=0,\ldots ,m.$ • If $M$ is a compact $C^{\infty }$-manifold and $B$ is a Banach space, then the set $C^{\infty }(M,B)$ of all infinitely-often differentiable functions $f:M\to B$ can be turned into a Fréchet space by using as seminorms the suprema of the norms of all partial derivatives. If $M$ is a (not necessarily compact) $C^{\infty }$-manifold which admits a countable sequence $K^{n}$ of compact subsets, so that every compact subset of $M$ is contained in at least one $K^{n},$ then the spaces $C^{m}(M,B)$ and $C^{\infty }(M,B)$ are also Fréchet space in a natural manner. As a special case, every smooth finite-dimensional complete manifold $M$ can be made into such a nested union of compact subsets: equip it with a Riemannian metric $g$ which induces a metric $d(x,y),$ choose $x\in M,$ and let $K_{n}=\{y\in M:d(x,y)\leq n\}\ .$ Let $X$ be a compact $C^{\infty }$-manifold and$V$ a vector bundle over $X.$ Let $C^{\infty }(X,V)$ denote the space of smooth sections of $V$ over $X.$ Choose Riemannian metrics and connections, which are guaranteed to exist, on the bundles $TX$ and $V.$ If $s$ is a section, denote its jth covariant derivative by $D^{j}s.$ Then $\\|s\\|_{n}=\sum _{j=0}^{n}\sup _{x\in M}\|D^{j}s\|$ (where $\\|\,\cdot \,\\|$ is the norm induced by the Riemannian metric) is a family of seminorms making $C^{\infty }(M,V)$ into a Fréchet space. From holomorphicity • Let $H$ be the space of entire (everywhere holomorphic) functions on the complex plane. Then the family of seminorms $\\|f\\|_{n}=\sup\{\|f(z)\|:\|z\|\leq n\}$ makes $H$ into a Fréchet space. • Let $H$ be the space of entire (everywhere holomorphic) functions of exponential type $\tau .$ Then the family of seminorms $\\|f\\|_{n}=\sup _{z\in \mathbb {C} }\exp \left[-\left(\tau +{\frac {1}{n}}\right)\|z\|\right]\|f(z)\|$ makes $H$ into a Fréchet space. Not all vector spaces with complete translation-invariant metrics are Fréchet spaces. An example is the space $L^{p}([0,1])$ with $p<1.$ Although this space fails to be locally convex, it is an F-space. Properties and further notions If a Fréchet space admits a continuous norm then all of the seminorms used to define it can be replaced with norms by adding this continuous norm to each of them. A Banach space, $C^{\infty }([a,b]),$ $C^{\infty }(X,V)$ with $X$ compact, and $H$ all admit norms, while $\mathbb {R} ^{\omega }$ and $C(\mathbb {R} )$ do not. A closed subspace of a Fréchet space is a Fréchet space. A quotient of a Fréchet space by a closed subspace is a Fréchet space. The direct sum of a finite number of Fréchet spaces is a Fréchet space. A product of countably many Fréchet spaces is always once again a Fréchet space. However, an arbitrary product of Fréchet spaces will be a Fréchet space if and only if all except for at most countably many of them are trivial (that is, have dimension 0). Consequently, a product of uncountably many non-trivial Fréchet spaces can not be a Fréchet space (indeed, such a product is not even metrizable because its origin can not have a countable neighborhood basis). So for example, if $I\neq \varnothing $ is any set and $X$ is any non-trivial Fréchet space (such as $X=\mathbb {R} $ for instance), then the product $X^{I}=\prod _{i\in I}X$ is a Fréchet space if and only if $I$ is a countable set. Several important tools of functional analysis which are based on the Baire category theorem remain true in Fréchet spaces; examples are the closed graph theorem and the open mapping theorem. The open mapping theorem implies that if $\tau {\text{ and }}\tau _{2}$ are topologies on $X$ that make both $(X,\tau )$ and $\left(X,\tau _{2}\right)$ into complete metrizable TVSs (such as Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if $\tau \subseteq \tau _{2}{\text{ or }}\tau _{2}\subseteq \tau {\text{ then }}\tau =\tau _{2}$).[4] Every bounded linear operator from a Fréchet space into another topological vector space (TVS) is continuous.[5] There exists a Fréchet space $X$ having a bounded subset $B$ and also a dense vector subspace $M$ such that $B$ is not contained in the closure (in $X$) of any bounded subset of $M.$[6] All Fréchet spaces are stereotype spaces. In the theory of stereotype spaces Fréchet spaces are dual objects to Brauner spaces. All metrizable Montel spaces are separable.[7] A separable Fréchet space is a Montel space if and only if each weak-* convergent sequence in its continuous dual converges is strongly convergent.[7] The strong dual space $X_{b}^{\prime }$ of a Fréchet space (and more generally, of any metrizable locally convex space[8]) $X$ is a DF-space.[9] The strong dual of a DF-space is a Fréchet space.[10] The strong dual of a reflexive Fréchet space is a bornological space[8] and a Ptak space. Every Fréchet space is a Ptak space. The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.[11] Norms and normability See also: Metrizable topological vector space § Normability If $X$ is a locally convex space then the topology of $X$ can be a defined by a family of continuous norms on $X$ (a norm is a positive-definite seminorm) if and only if there exists at least one continuous norm on $X.$[12] Even if a Fréchet space has a topology that is defined by a (countable) family of norms (all norms are also seminorms), then it may nevertheless still fail to be normable space (meaning that its topology can not be defined by any single norm). The space of all sequences $\mathbb {K} ^{\mathbb {N} }$ (with the product topology) is a Fréchet space. There does not exist any Hausdorff locally convex topology on $\mathbb {K} ^{\mathbb {N} }$ that is strictly coarser than this product topology.[13] The space $\mathbb {K} ^{\mathbb {N} }$ is not normable, which means that its topology can not be defined by any norm.[13] Also, there does not exist any continuous norm on $\mathbb {K} ^{\mathbb {N} }.$ In fact, as the following theorem shows, whenever $X$ is a Fréchet space on which there does not exist any continuous norm, then this is due entirely to the presence of $\mathbb {K} ^{\mathbb {N} }$ as a subspace. Theorem[13] — Let $X$ be a Fréchet space over the field $\mathbb {K} .$ Then the following are equivalent: 1. $X$ does not admit a continuous norm (that is, any continuous seminorm on $X$ can not be a norm). 2. $X$ contains a vector subspace that is TVS-isomorphic to $\mathbb {K} ^{\mathbb {N} }.$ 3. $X$ contains a complemented vector subspace that is TVS-isomorphic to $\mathbb {K} ^{\mathbb {N} }.$ If $X$ is a non-normable Fréchet space on which there exists a continuous norm, then $X$ contains a closed vector subspace that has no topological complement.[14] A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.[9] In particular, if a locally convex metrizable space $X$ (such as a Fréchet space) is not normable (which can only happen if $X$ is infinite dimensional) then its strong dual space $X_{b}^{\prime }$ is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space $X_{b}^{\prime }$ is also neither metrizable nor normable. The strong dual space of a Fréchet space (and more generally, of bornological spaces such as metrizable TVSs) is always a complete TVS and so like any complete TVS, it is normable if and only if its topology can be induced by a complete norm (that is, if and only if it can be made into a Banach space that has the same topology). If $X$ is a Fréchet space then $X$ is normable if (and only if) there exists a complete norm on its continuous dual space $X'$ such that the norm induced topology on $X'$ is finer than the weak-* topology.[15] Consequently, if a Fréchet space is not normable (which can only happen if it is infinite dimensional) then neither is its strong dual space. Anderson–Kadec theorem Anderson–Kadec theorem — Every infinite-dimensional, separable real Fréchet space is homeomorphic to $\mathbb {R} ^{\mathbb {N} },$ the Cartesian product of countably many copies of the real line $\mathbb {R} .$ Note that the homeomorphism described in the Anderson–Kadec theorem is not necessarily linear. Eidelheit theorem — A Fréchet space is either isomorphic to a Banach space, or has a quotient space isomorphic to $\mathbb {R} ^{\mathbb {N} }.$ Differentiation of functions Main article: Differentiation in Fréchet spaces If $X$ and $Y$ are Fréchet spaces, then the space $L(X,Y)$ consisting of all continuous linear maps from $X$ to $Y$ is not a Fréchet space in any natural manner. This is a major difference between the theory of Banach spaces and that of Fréchet spaces and necessitates a different definition for continuous differentiability of functions defined on Fréchet spaces, the Gateaux derivative: Suppose $U$ is an open subset of a Fréchet space $X,$ $P:U\to Y$ is a function valued in a Fréchet space $Y,$ $x\in U$ and $h\in X.$ The map $P$ is differentiable at $x$ in the direction $h$ if the limit $D(P)(x)(h)=\lim _{t\to 0}\,{\frac {1}{t}}\left(P(x+th)-P(x)\right)$ exists. The map $P$ is said to be continuously differentiable in $U$ if the map $D(P):U\times X\to Y$ is continuous. Since the product of Fréchet spaces is again a Fréchet space, we can then try to differentiate $D(P)$ and define the higher derivatives of $P$ in this fashion. The derivative operator $P:C^{\infty }([0,1])\to C^{\infty }([0,1])$ defined by $P(f)=f'$ is itself infinitely differentiable. The first derivative is given by $D(P)(f)(h)=h'$ for any two elements $f,h\in C^{\infty }([0,1]).$ This is a major advantage of the Fréchet space $C^{\infty }([0,1])$ over the Banach space $C^{k}([0,1])$ for finite $k.$ If $P:U\to Y$ is a continuously differentiable function, then the differential equation $x'(t)=P(x(t)),\quad x(0)=x_{0}\in U$ need not have any solutions, and even if does, the solutions need not be unique. This is in stark contrast to the situation in Banach spaces. In general, the inverse function theorem is not true in Fréchet spaces, although a partial substitute is the Nash–Moser theorem. Fréchet manifolds and Lie groups Main article: Fréchet manifold One may define Fréchet manifolds as spaces that "locally look like" Fréchet spaces (just like ordinary manifolds are defined as spaces that locally look like Euclidean space $\mathbb {R} ^{n}$), and one can then extend the concept of Lie group to these manifolds. This is useful because for a given (ordinary) compact $C^{\infty }$ manifold $M,$ the set of all $C^{\infty }$ diffeomorphisms $f:M\to M$ forms a generalized Lie group in this sense, and this Lie group captures the symmetries of $M.$ Some of the relations between Lie algebras and Lie groups remain valid in this setting. Another important example of a Fréchet Lie group is the loop group of a compact Lie group $G,$ the smooth ($C^{\infty }$) mappings $\gamma :S^{1}\to G,$ multiplied pointwise by $\left(\gamma _{1}\gamma _{2}\right)(t)=\gamma _{1}(t)\gamma _{2}(t)..$[16][17] Generalizations If we drop the requirement for the space to be locally convex, we obtain F-spaces: vector spaces with complete translation-invariant metrics. LF-spaces are countable inductive limits of Fréchet spaces. See also • Banach space – Normed vector space that is complete • Brauner space – complete compactly generated locally convex space with a sequence of compact sets Kₙ such that any compact set is contained in some KₙPages displaying wikidata descriptions as a fallback • Complete metric space – Metric geometry • Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point • F-space – Topological vector space with a complete translation-invariant metric • Fréchet lattice • Graded Fréchet space – Generalization of the inverse function theoremPages displaying short descriptions of redirect targets • Hilbert space – Type of topological vector space • Locally convex topological vector space – A vector space with a topology defined by convex open sets • Metrizable topological vector space – A topological vector space whose topology can be defined by a metric • Surjection of Fréchet spaces – Characterization of surjectivity • Tame Fréchet space – Generalization of the inverse function theoremPages displaying short descriptions of redirect targets • Topological vector space – Vector space with a notion of nearness Notes 1. Here "Cauchy" means Cauchy with respect to the canonical uniformity that every TVS possess. That is, a sequence $x_{\bullet }=\left(x_{m}\right)_{m=1}^{\infty }$ in a TVS $X$ is Cauchy if and only if for all neighborhoods $U$ of the origin in $X,$ $x_{m}-x_{n}\in U$ whenever $m$ and $n$ are sufficiently large. Note that this definition of a Cauchy sequence does not depend on any particular metric and doesn't even require that $X$ be metrizable. 2. Some authors do not include local convexity as part of the definition of a Fréchet space. Citations 1. Narici & Beckenstein 2011, p. 93. 2. Conway 1990, Chapter 4. 3. Narici & Beckenstein 2011, p. 472. 4. Trèves 2006, pp. 166–173. 5. Trèves 2006, p. 142. 6. Wilansky 2013, p. 57. 7. Schaefer & Wolff 1999, pp. 194–195. 8. Schaefer & Wolff 1999, p. 154. 9. Gabriyelyan, S.S. "On topological spaces and topological groups with certain local countable networks (2014) 10. Schaefer & Wolff 1999, p. 196. 11. Schaefer & Wolff 1999, pp. 154–155. 12. Jarchow 1981, p. 130. 13. Jarchow 1981, pp. 129–130. 14. Schaefer & Wolff 1999, pp. 190–202. 15. "The dual of a Fréchet space". 24 February 2012. Retrieved 26 April 2021. 16. Sergeev 2010 17. Pressley & Segal 1986 References • "Fréchet space", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401. • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190. • Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. • Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. • Pressley, Andrew; Segal, Graeme (1986). Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. New York: Oxford University Press. ISBN 0-19-853535-X. MR 0900587. • Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Sergeev, Armen (2010). Kähler Geometry of Loop Spaces. Mathematical Society of Japan Memoirs. Vol. 23. World Scientific Publishing. doi:10.1142/e023. ISBN 978-4-931469-60-0. • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. • Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. 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Minkowski plane In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane). This article is about the Benz plane. It is not to be confused with Minkowski space. Classical real Minkowski plane Applying the pseudo-euclidean distance $d(P_{1},P_{2})=(x'_{1}-x'_{2})^{2}-(y'_{1}-y'_{2})^{2}$ on two points $P_{i}=(x'_{i},y'_{i})$ (instead of the euclidean distance) we get the geometry of hyperbolas, because a pseudo-euclidean circle $\{P\in \mathbb {R} ^{2}\mid d(P,M)=r\}$ is a hyperbola with midpoint $M$. By a transformation of coordinates $x_{i}=x'_{i}+y'_{i}$, $y_{i}=x'_{i}-y'_{i}$, the pseudo-euclidean distance can be rewritten as $d(P_{1},P_{2})=(x_{1}-x_{2})(y_{1}-y_{2})$. The hyperbolas then have asymptotes parallel to the non-primed coordinate axes. The following completion (see Möbius and Laguerre planes) homogenizes the geometry of hyperbolas: • the set of points: ${\mathcal {P}}:=\left(\mathbb {R} \cup \left\{\infty \right\}\right)^{2}=\mathbb {R} ^{2}\cup \left(\left\{\infty \right\}\times \mathbb {R} \right)\cup \left(\mathbb {R} \times \left\{\infty \right\}\right)\ \cup \left\{\left(\infty ,\infty \right)\right\}\ ,\ \infty \notin \mathbb {R} ,$ • the set of cycles ${\begin{aligned}{\mathcal {Z}}:={}&\left\{\left\{\left(x,y\right)\in \mathbb {R} ^{2}\mid y=ax+b\right\}\cup \left\{\left(\infty ,\infty \right)\right\}\mid a,b\in \mathbb {R} ,a\neq 0\right\}\\&\quad \cup \left\{\left\{\left(x,y\right)\in \mathbb {R} ^{2}\mid y={\frac {a}{x-b}}+c,x\neq b\right\}\cup \left\{\left(b,\infty \right),\left(\infty ,c\right)\right\}\mid a,b,c\in \mathbb {R} ,a\neq 0\right\}.\end{aligned}}$ The incidence structure $({\mathcal {P}},{\mathcal {Z}},\in )$ is called the classical real Minkowski plane. The set of points consists of $\mathbb {R} ^{2}$, two copies of $\mathbb {R} $ and the point $(\infty ,\infty )$. Any line $y=ax+b,a\neq 0$ is completed by point $(\infty ,\infty )$, any hyperbola $y={\frac {a}{x-b}}+c,a\neq 0$ by the two points $(b,\infty ),(\infty ,c)$ (see figure). Two points $(x_{1},y_{1})\neq (x_{2},y_{2})$ can not be connected by a cycle if and only if $x_{1}=x_{2}$ or $y_{1}=y_{2}$. We define: Two points $P_{1},P_{2}$ are (+)-parallel ($P_{1}\parallel _{+}P_{2}$) if $x_{1}=x_{2}$ and (−)-parallel ($P_{1}\parallel _{-}P_{2}$) if $y_{1}=y_{2}$. Both these relations are equivalence relations on the set of points. Two points $P_{1},P_{2}$ are called parallel ($P_{1}\parallel P_{2}$) if $P_{1}\parallel _{+}P_{2}$ or $P_{1}\parallel _{-}P_{2}$. From the definition above we find: Lemma: • For any pair of non parallel points $A,B$ there is exactly one point $C$ with $A\parallel _{+}C\parallel _{-}B$. • For any point $P$ and any cycle $z$ there are exactly two points $A,B\in z$ with $A\parallel _{+}P\parallel _{-}B$. • For any three points $A$, $B$, $C$, pairwise non parallel, there is exactly one cycle $z$ that contains $A,B,C$. • For any cycle $z$, any point $P\in z$ and any point $Q,P\not \parallel Q$ and $Q\notin z$ there exists exactly one cycle $z'$ such that $z\cap z'=\{P\}$, i.e. $z$ touches $z'$ at point P. Like the classical Möbius and Laguerre planes Minkowski planes can be described as the geometry of plane sections of a suitable quadric. But in this case the quadric lives in projective 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of a hyperboloid of one sheet (not degenerated quadric of index 2). The axioms of a Minkowski plane Let $\left({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in \right)$ be an incidence structure with the set ${\mathcal {P}}$ of points, the set ${\mathcal {Z}}$ of cycles and two equivalence relations $\parallel _{+}$ ((+)-parallel) and $\parallel _{-}$ ((−)-parallel) on set ${\mathcal {P}}$. For $P\in {\mathcal {P}}$ we define: ${\overline {P}}_{+}:=\left\{Q\in {\mathcal {P}}\mid Q\parallel _{+}P\right\}$ and ${\overline {P}}_{-}:=\left\{Q\in {\mathcal {P}}\mid Q\parallel _{-}P\right\}$. An equivalence class ${\overline {P}}_{+}$ or ${\overline {P}}_{-}$ is called (+)-generator and (−)-generator, respectively. (For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.) Two points $A,B$ are called parallel ($A\parallel B$) if $A\parallel _{+}B$ or $A\parallel _{-}B$. An incidence structure ${\mathfrak {M}}:=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )$ is called Minkowski plane if the following axioms hold: • C1: For any pair of non parallel points $A,B$ there is exactly one point $C$ with $A\parallel _{+}C\parallel _{-}B$. • C2: For any point $P$ and any cycle $z$ there are exactly two points $A,B\in z$ with $A\parallel _{+}P\parallel _{-}B$. • C3: For any three points $A,B,C$, pairwise non parallel, there is exactly one cycle $z$ which contains $A,B,C$. • C4: For any cycle $z$, any point $P\in z$ and any point $Q,P\not \parallel Q$ and $Q\notin z$ there exists exactly one cycle $z'$ such that $z\cap z'=\{P\}$, i.e., $z$ touches $z'$ at point $P$. • C5: Any cycle contains at least 3 points. There is at least one cycle $z$ and a point $P$ not in $z$. For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous. • C1′: For any two points $A,B$ we have $\left\|{\overline {A}}_{+}\cap {\overline {B}}_{-}\right\|=1$. • C2′: For any point $P$ and any cycle $z$ we have: $\left\|{\overline {P}}_{+}\cap z\right\|=1=\left\|{\overline {P}}_{-}\cap z\right\|$. First consequences of the axioms are Lemma — For a Minkowski plane ${\mathfrak {M}}$ the following is true 1. Any point is contained in at least one cycle. 2. Any generator contains at least 3 points. 3. Two points can be connected by a cycle if and only if they are non parallel. Analogously to Möbius and Laguerre planes we get the connection to the linear geometry via the residues. For a Minkowski plane ${\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )$ and $P\in {\mathcal {P}}$ we define the local structure ${\mathfrak {A}}_{P}:=({\mathcal {P}}\setminus {\overline {P}},\{z\setminus \{{\overline {P}}\}\mid P\in z\in {\mathcal {Z}}\}\cup \{E\setminus {\overline {P}}\mid E\in {\mathcal {E}}\setminus \{{\overline {P}}_{+},{\overline {P}}_{-}\}\},\in )$ and call it the residue at point P. For the classical Minkowski plane ${\mathfrak {A}}_{(\infty ,\infty )}$ is the real affine plane $\mathbb {R} ^{2}$. An immediate consequence of axioms C1 to C4 and C1′, C2′ are the following two theorems. Theorem — For a Minkowski plane ${\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel ,\in )$ any residue is an affine plane. Theorem — Let be ${\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )$ an incidence structure with two equivalence relations $\parallel _{+}$ and $\parallel _{-}$ on the set ${\mathcal {P}}$ of points (see above). Then, ${\mathfrak {M}}$ is a Minkowski plane if and only if for any point $P$ the residue ${\mathfrak {A}}_{P}$ is an affine plane. Minimal model The minimal model of a Minkowski plane can be established over the set ${\overline {K}}:=\{0,1,\infty \}$ of three elements: ${\mathcal {P}}:={\overline {K}}^{2}$ ${\begin{aligned}{\mathcal {Z}}:\!&=\left\{\{(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\}\mid \{a_{1},a_{2},a_{3}\}=\{b_{1},b_{2},b_{3}\}={\overline {K}}\right\}\\&=\{\{(0,0),(1,1),(\infty ,\infty )\},\;\{(0,0),(1,\infty ),(\infty ,1)\},\\&\qquad \{(0,1),(1,0),(\infty ,\infty )\},\;\{(0,1),(1,\infty ),(\infty ,0)\},\\&\qquad \{(0,\infty ),(1,1),(\infty ,0)\},\;\{(0,\infty ),(1,0),(\infty ,1)\}\}\end{aligned}}$ Parallel points: • $(x_{1},y_{1})\parallel _{+}(x_{2},y_{2})$ if and only if $x_{1}=x_{2}$ • $(x_{1},y_{1})\parallel _{-}(x_{2},y_{2})$ if and only if $y_{1}=y_{2}$. Hence $\left\|{\mathcal {P}}\right\|=9$ and $\left\|{\mathcal {Z}}\right\|=6$. Finite Minkowski-planes For finite Minkowski-planes we get from C1′, C2′: Lemma — Let be ${\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )$ a finite Minkowski plane, i.e. $\left\|{\mathcal {P}}\right\|<\infty $. For any pair of cycles $z_{1},z_{2}$ and any pair of generators $e_{1},e_{2}$ we have: $\left\|z_{1}\right\|=\left\|z_{2}\right\|=\left\|e_{1}\right\|=\left\|e_{2}\right\|$. This gives rise of the definition: For a finite Minkowski plane ${\mathfrak {M}}$ and a cycle $z$ of ${\mathfrak {M}}$ we call the integer $n=\left\|z\right\|-1$ the order of ${\mathfrak {M}}$. Simple combinatorial considerations yield Lemma — For a finite Minkowski plane ${\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )$ the following is true: 1. Any residue (affine plane) has order $n$. 2. $\left\|{\mathcal {P}}\right\|=(n+1)^{2}$, 3. $\left\|{\mathcal {Z}}\right\|=(n+1)n(n-1)$. Miquelian Minkowski planes We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace $\mathbb {R} $ by an arbitrary field $K$ then we get in any case a Minkowski plane ${\mathfrak {M}}(K)=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )$. Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane ${\mathfrak {M}}(K)$. Theorem (Miquel): For the Minkowski plane ${\mathfrak {M}}(K)$ the following is true: If for any 8 pairwise not parallel points $P_{1},...,P_{8}$ which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples, then the sixth quadruple of points is concyclical, too. (For a better overview in the figure there are circles drawn instead of hyperbolas.) Theorem (Chen): Only a Minkowski plane ${\mathfrak {M}}(K)$ satisfies the theorem of Miquel. Because of the last theorem ${\mathfrak {M}}(K)$ is called a miquelian Minkowski plane. Remark: The minimal model of a Minkowski plane is miquelian. It is isomorphic to the Minkowski plane ${\mathfrak {M}}(K)$ with $K=\operatorname {GF} (2)$ (field $\{0,1\}$). An astonishing result is Theorem (Heise): Any Minkowski plane of even order is miquelian. Remark: A suitable stereographic projection shows: ${\mathfrak {M}}(K)$ is isomorphic to the geometry of the plane sections on a hyperboloid of one sheet (quadric of index 2) in projective 3-space over field $K$. Remark: There are a lot of Minkowski planes that are not miquelian (s. weblink below). But there are no "ovoidal Minkowski" planes, in difference to Möbius and Laguerre planes. Because any quadratic set of index 2 in projective 3-space is a quadric (see quadratic set). See also • Conformal geometry References • Walter Benz (1973) Vorlesungen über Geomerie der Algebren, Springer • Francis Buekenhout (editor) (1995) Handbook of Incidence Geometry, Elsevier ISBN 0-444-88355-X External links • Benz plane in the Encyclopedia of Mathematics • Lecture Note Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes	Wikipedia
Very low-carbohydrate, high-fat, weight reduction diet decreases hepatic gene response to glucose in obese rats Kathleen V. Axen1, Marianna A. Harper1, Yu Fu Kuo1 & Kenneth Axen ORCID: orcid.org/0000-0002-9972-70871 Very low carbohydrate (VLC) diets are used to promote weight loss and improve insulin resistance (IR) in obesity. Since the high fat content of VLC diets may predispose to hepatic steatosis and hepatic insulin resistance, we investigated the effect of a VLC weight-reduction diet on measures of hepatic and whole body insulin resistance in obese rats. In Phase 1, adult male Sprague-Dawley rats were made obese by ad libitum consumption of a high-fat (HF1, 60% of energy) diet; control rats ate a lower-fat (LF, 15%) diet for 10 weeks. In Phase 2, obese rats were fed energy-restricted amounts of a VLC (5%C, 65%F), LC (19%C, 55%F) or HC (55%C, 15%F) diet for 8 weeks while HF2 rats continued the HF diet ad libitum. In Phase 3, VLC rats were switched to the HC diet for 1 week. At the end of each phase, measurements of body composition and metabolic parameters were obtained. Hepatic insulin resistance was assessed by comparing expression of insulin-regulated genes following an oral glucose load,that increased plasma insulin levels, with the expression observed in the feed-deprived state. At the end of Phase 1, body weight, percent body fat, and hepatic lipid levels were greater in HF1 than LF rats (p < 0.05). At the end of Phase 2, percent body fat and intramuscular triglyceride decreased in LC and HC (p < 0.05), but not VLC rats, despite similar weight loss. VLC and HF2 rats had higher HOMA-IR and higher insulin at similar glucose levels following an ip glucose load than HC rats (p < 0.05). HC, but not VLC or HF2 rats, showed changes in Srebf1, Scd1, and Cpt1a expression (p < 0.05) in response to an oral glucose load. At the end of Phase 3, switching from the VLC to the HC diet mitigated differences in hepatic gene expression. When compared with a high-carbohydrate, low-fat diet that produced similar weight loss, a commonly used VLC diet failed to improve whole body insulin resistance; it also reduced insulin's effect on hepatic gene expression, which may reflect the development of hepatic insulin resistance. Very low carbohydrate (VLC, < 10% of energy) diets that are used to reduce body weight and improve glycemic control and insulin sensitivity in obese individuals [1, 2] generally provide > 50% of energy as fat [3,4,5]. High levels of dietary fat are implicated in the development of hepatic steatosis [6,7,8] which is prevalent in the obese population and is associated with hepatic insulin resistance and development of Type 2 diabetes mellitus [9]. Since obese individuals who are at risk for Type 2 diabetes mellitus may utilize a VLC diet to lose weight, it is important to understand the effect of VLC diets on hepatic response to insulin. Although VLC diets have been used successfully for weight loss [10] and management of post-meal glycemia in humans [1, 11, 12], their impact on hepatic or whole body insulin resistance remains unclear [13]. Long-term, ad libitum consumption of a VLC diet by lean rats produced diabetes [14];in another study, consumption of a VLC diet by lean rats lowered their fasting blood glucose and insulin levels, but also produced glucose intolerance [15, 16], as well as hepatic and whole body insulin resistance when compared with an isocaloric high-carbohydrate, low-fat diet [15]. Previous studies from our laboratory in obese rats showed that an energy-restricted VLC diet produced less reduction in visceral fat, and hepatic and intramuscular lipid levels, and less improvement in glucose tolerance than an isocaloric high-carbohydrate, low-fat diet that yielded similar weight loss [17, 18]. The correlation between hepatic lipid concentration and glucose intolerance in that study [18] provided the impetus for the present investigation of the effects of a VLC weight-reduction diet on hepatic and whole body insulin resistance. Ketosis diets, which are used to treat epilepsy in children [19], contain 0–10% carbohydrate (C), < 5% protein (P) and > 80% fat (F) [20]. Ketosis diets can increase percent body fat and hepatic triglyceride levels [21,22,23], and produce inflammation [23,24,25], liver damage [23, 24] and hepatic apoptosis [26] in rodents. Although ketosis diets lowered basal blood glucose and insulin levels in normal mice [27] and murine models of type 2 diabetes [23, 28], these diets also produced glucose intolerance, as well as hepatic [27] and whole body insulin resistance [29]. These results show that VLC diets with a low protein content can produce hepatic steatosis and hepatic insulin resistance. Since the VLC diets typically used by obese individuals for weight loss have a moderate or high (≥ 25% of energy) protein content [3, 5, 30], the present study employed diets of different carbohydrate and fat, but of similarly high (26–30%) protein, contents. We compared the effects of a VLC diet on hepatic response to insulin, with that of a high-carbohydrate, low-fat diet (HC) that produced the same weight loss in dietary obese rats. Hepatic response was defined as the difference between expression of insulin-regulated genes after an oral glucose load, that raised plasma insulin levels, with expression of these genes in the feed-deprived state. We hypothesized that hepatic gene response to insulin would be impaired in rats on the VLC vs. HC diet, even with the same weight loss. In order to compare the effects of weight reduction by a VLC diet with that of a high-carbohydrate, lower-fat diet (HC), we first made rats obese in Phase 1 by ad libitum consumption of a high-fat diet for 10 weeks (HF1); rats consuming a lower-fat diet ad libitum served as a normal control group (LF, Fig. 1). In Phase 2 (8 weeks), obese rats from Phase 1 were given energy-restricted amounts of one of three diets, VLC, HC or LC, adjusted in amount to ensure similar weight reduction in all three groups. The LC diet was used to study the effects of a high-fat intake in the absence of extreme carbohydrate restriction (Table 1). A fourth group of rats (HF2) continued to feed ad libitum on the HF diet, to serve as an obese control. In Phase 3 (1 wk), some of the rats on the VLC diet in Phase 2 were switched to the HC diet for 3 or 7 days (VC3, VC7) in order to assess the persistence of the effects of the VLC diet after a switch to a high-carbohydrate, lower-fat diet. At the end of each of the three phases, metabolic and body composition analyses were performed, as well as measurement of hepatic gene expression both in the feed-deprived and post-glucose conditions (Fig. 1). Experimental Design. GTT, intraperitoneal glucose tolerance test; LF, lower fat; HF, high fat; VLC, very low carbohydrate; LC, low carbohydrate; HC, high carbohydrate; VC, change from very low carbohydrate to high carbohydrate Table 1 Composition of experimental dietsb Animals and diets Male Sprague-Dawley rats (Charles River Laboratories, MA) ~ 3 months of age (~ 290 g) were divided into two weight-matched groups. The control group (N = 8) consumed an LF diet (55%C, 15%F; Research Diets, NJ) (Table 1), similar in macronutrient composition to the standard AIN-76 diet [31]; the group to be rendered obese consumed an HF diet (19%C, 55%F) which contained 12% of kcal as sucrose in order to promote hyperphagia. Ad libitum feed intake, corrected for spillage, was measured during wk. 6; body weights were recorded twice a week. After 8 wk. of ad libitum HF feeding, body weight outliers were removed and HF rats (N = 58) were divided into five groups matched for both mean and range of body weights; one group of HF rats (HF1, N = 10) was tested for glucose tolerance and then dissected at wk. 10 (Fig. 1). During wk. 11–18, the remaining four groups of obese rats either continued to consume the HF diet ad libitum (HF2) or received daily restricted amounts of one of three diets (VLC, LC, or HC) that provided ~ 70% of the mean LF energy intake during Phase 1, adjusted as needed to maintain similar body weights among the groups. This level of energy restriction was used to induce a small weight loss in the growing rats. The VLC group (N = 24) received a 5%C, 65%F diet; the LC group (N = 8) a 19%C, 55%F diet; and the HC group (N = 8) a 55%C, 15%F diet. The LC and HC diets had similar macronutrient compositions to the HF and LF diets, respectively, but unlike those diets, they were consumed in hypocaloric amounts (Table 1). All diets had similar protein levels (26–30%), and were comprised of the same fat sources, resulting in the same distribution of polyunsaturated fat (36% PUFA) and monounsaturated fat (38% MUFA) among diets. None of the three weight-reduction diets contained sucrose. The VLC group included more rats than the other groups because VLC rats were later divided into 3 weight-matched groups (each N = 8): one group for dissection at wk. 18–19 and two groups for Phase 3 of the study (Fig. 1). The VLC, LC and HC rats received fresh rations daily; their feed intakes were measured for three consecutive days per week during weeks 13 and 16 of the study. To investigate the reversibility of the effects of the VLC diet, the VLC rats remaining after wk. 18 were switched to the HC diet for three (N = 8, VC3) or seven (N = 8, VC7) days (Phase 3). Rats were housed at 22o C, on a 12 h:12 h light-dark cycle; all procedures were approved by the Brooklyn College Institutional Animal Care and Use Committee (Protocol 248). Metabolic profile At wk. 9, HF1 (N = 10) and LF (N = 8) rats were feed deprived overnight (16 h) and a glucose tolerance test (GTT) was administered by intraperitoneal (ip) injection of 50% glucose (1 g/kg of body weight); glucose was measured in tail blood samples taken before (t = 0) and at 10, 20, 30, 45 and 75 min after the glucose load. Similarly, at wk. 17 an ip GTT, along with measurement of feed-deprived plasma ketone levels, was performed following overnight feed deprivation on 6 rats from each of the four diet groups in Phase 2; rats from the energy-restricted groups were chosen to match body weights: VLC (640 ± 13 g, mean ± SEM), LC (635 ± 8), and HC (631 ± 14). Tissue collection At the end of Phase 1, 8 LF and 10 HF1 rats were feed deprived overnight (16–19 h); half of the rats in each diet group were dissected in the feed-deprived state and half 3 h after an oral load of 50% glucose administered by gavage (2 g/kg). The effect of the oral glucose load on plasma glucose and insulin levels was measured at 30 min intervals (0–90 min) in a separate group of 5 LF and 6 HF rats, confirming that glucose and insulin levels remained elevated between 30 and 90 min in both groups; measurements were made in these other rats to avoid the stress of blood draws before tissue sampling. In all phases of the study, the order of diet groups and conditions (feed-deprived vs. post-glucose) were systematically counterbalanced to ensure that they did not differ among groups. In Phases 2 and 3, all rats to be dissected were feed deprived overnight and tissue samples were obtained from each diet group in the feed-deprived state (N = 4) or 3 h after the first of two doses of glucose (each 2 g/kg), given by gavage 90 min apart (N = 4); a higher glucose load than that used in Phase 1 was employed to provide a stronger stimulus for insulin release. The VC3 and VC7 groups were similarly treated and dissected after three or seven days after switching to the HC diet, respectively. Rats were anesthetized by ip injection of Na pentobarbital (55 mg/kg, Sigma-Aldrich). Liver samples were freeze clamped and placed in liquid Nitrogen for later measurement of lipid concentration, or stored in RNAlater (Qiagen) for later extraction of nucleic acids. Blood (~ 10 mL) was drawn from the descending aorta into a heparinized syringe (10 USP units/ mL of blood, Schein); plasma was frozen for later measurement of insulin, leptin, and triglyceride. Fat pads were dissected from the epididymal, omental-mesenteric, retroperitoneal and entire subcutaneous depots, and covered in plastic until weights were recorded. Rats died under anesthesia following the blood draw. Blood and tissue analysis Triglyceride concentration in plasma and in muscle lipid extracts was measured using a kit (Wako), as was plasma concentration of β-hydroxybutyrate (Stanbio Laboratory). Blood glucose concentration was measured by glucometer (One Touch, LifeScan); plasma insulin and leptin concentrations were measured by radioimmunoassay (Millipore). Total lipid was extracted from homogenized liver samples [32], and from muscle fibers dissected from lyophilized soleus samples [33]; triglyceride from the muscle lipid extract was solubilized in isopropanol for assay as above. RNA extraction and quantitative real-time PCR analysis Total liver RNA was isolated using the RNeasy mini kit (Qiagen), and reverse transcribed using the High Capacity cDNA Reverse Transcription Kit (Applied Biosystems). Gene expression was measured using TaqMan expression assays and master mix (Life Technologies) under conditions specified for the product; two assays were performed for each sample for every gene of interest (CFX Connect, BioRad). Each PCR run included triplicates of cDNA (5 ng total cDNA) for each gene and a no-template control, as well as a calibrator from pooled samples in that phase of the study in order to document inter-run comparability for all genes. Relative expression was normalized for transcript levels of the reference gene, ribosomal protein P2 (Rplp2), whose expression was unaffected by diet or feed-deprived vs. post-glucose conditions. The effect of diet on hepatic insulin resistance was assessed by comparing post-glucose with feed-deprived expression of genes known to be regulated by insulin. SREBP1c (Sterol Regulatory Element Binding Protein), a major regulator of lipogenesis whose gene (Srebf1) transcription is increased by insulin, is attached to the endoplasmic reticulum via the protein Insig2 (Insulin-induced gene 2); once released, the mature form of SREBP1c is transported to the nucleus where it binds to regulatory elements of its target genes, including Acaca (Acetyl-CoA Carboxylase), Fasn (Fatty Acid Synthase), Scd (Stearoyl-CoA Desaturase1), and Gck (Glucokinase). Insulin decreases the expression of Insig2, thereby promoting the action of SREBP1c. Insulin also decreases expression of Pck1 (Phosphoenolpyruvate Carboxykinase), and Cpt1a (Carnitine Palmitoyl Transferase 1). Data are shown as means ± SEM. The area, above feed-deprived levels, under the glucose vs. time curve (AUC) for the ip glucose tolerance test was calculated as follows: $$ \mathrm{AUC}=-55\ {\mathrm{G}}_0+10\ {\mathrm{G}}_{10}+10\ {\mathrm{G}}_{20}+12.5\ {\mathrm{G}}_{30}+15\ {\mathrm{G}}_{45}+7.5\ {\mathrm{G}}_{75} $$ Where G represents the blood glucose concentration and the subscript represents the sampling time in minutes after the glucose load. Analyses were performed using SPSS version 24 software. Comparisons between diet groups and between conditions were analyzed by Analysis of Variance and post hoc Bonferroni analyses; body weight changes within subjects were analyzed by repeated measures one-way ANOVA. For PCR results, data for each phase were analyzed both as 2 –dCt and log-transformed as dCt; significant effects were essentially the same for both methods, but because variances among diet groups were unequal for several genes using 2-dCt but not using dCt, significance is reported based on dCt values. Differences with values of p < 0.05 were considered to be significant. Food intake and body composition Ad libitum energy intake, measured at wk. 6 of Phase 1, was greater in HF1 vs. LF rats (106 ± 2 vs. 94 ± 3 kcal/day, p < 0.001). By the end of Phase 1 (wk 10), body weights, visceral fat, total body fat (sum of dissected fat), % body fat (total body fat × 100%/ body weight) (p < 0.01), hepatic lipid levels (p < 0.05, Table 2) and plasma leptin levels (p < 0.001) were greater in HF1 vs. LF rats, demonstrating that HF1 rats were obese and had hepatic steatosis (> 5% of liver weight as lipid) at the end of Phase 1. Soleus intramuscular triglyceride (TG) levels did not differ between HF1 and LF rats (8.1 ± 0.4 vs. 5.9 ± 1.7 mg/g). Table 2 Effects of diet on body composition in male rats1 Calculated energy intakes were similar at wk. 13 vs.16 in the three energy-restricted groups; intakes at those times were lower in LC (63 ± 2 kcal/d) than VLC (69 ± 1) or HC (68 ± 0.3) groups (p < 0.01). As intended, these restricted daily energy intakes were lower than those of the LF (94 ± 3) or HF1 (106 ± 2 kcal/day) groups in Phase 1 (p< 0.001). Body weights did not differ among the energy-restricted groups throughout wk. 11–18, and all three groups weighed less than the HF2 group throughout wk. 13–18 (p < 0.001). Although visceral, total and % body fat (p < 0.001), as well as soleus intramuscular TG (p < 0.05), decreased in LC and HC groups from that in the HF1 group at the end of Phase 1 (Table 2), no significant reduction in these measures was observed in the VLC group. Body weight and visceral, total and % body fat did not change in HF2 rats between wk. 10 and 18. Plasma leptin concentration was higher in HF1 (wk 10) and HF2 (wk 18) than in VLC, LC or HC groups at wk. 18 (p< 0.0001). HF1 and HF2 rats had the same level of hepatic lipid, while rats on the VLC and HC, but not LC, diets decreased hepatic lipid concentration during Phase 2 (p < 0.01). Total weight loss after the switch from the VLC to the HC diet in Phase 3 did not differ between rats on the HC diet for 3 days (VC3: 30 ± 2 g) or 7 days (VC7: 25 ± 5 g). Like VLC rats, body weights of VC3 and VC7 rats remained lower than that of HF2 rats, and concentrations of hepatic lipid and plasma levels of leptin (Table 2) were lower than those of HF1 or HF2 rats (p < 0.05). In contrast, the switch from the VLC to the HC diet resulted in lower intramuscular TG levels in VC3 and VC7 than in HF1 or HF2 rats. These results support the lower effectiveness of the VLC vs. the HC diet in reducing muscle lipid during weight loss. Feed-deprived values for plasma glucose and insulin concentrations did not differ between LF and HF1 rats at wk. 6, nor did glucose tolerance differ, as assessed by the area under the glucose vs. time curve (AUC) during the ip GTT (Table 3). Table 3 Dietary Effects on Plasma Levels of Glucose, Insulin and Ketones in Rats1 At the time of dissection, plasma levels of glucose and insulin did not differ between LF and HF1 rats in either the feed-deprived or post-oral glucose conditions (Fig. 2). Plasma triglyceride (TG) levels were higher in the LF vs. the HF1 group (p < 0.05). Effect of diet on plasma glucose, insulin and triglyceride levels in rats at dissection. Rats were feed-deprived for 16–19 h; half of the rats were then given an oral load of 50% glucose. Plasma samples were obtained from anesthetized animals either in the feed-deprived state or 3 h post glucose. Values are means ± SEM. Labeled means for a given diet group for a particular condition (feed-deprived or post-glucose) without a common letter differ, p < 0.05. * Different from Feed Deprived, p< 0.05. LF, lower-fat; HF1, high-fat Phase 1; HF2, high-fat Phase 2; VLC, very low-carbohydrate; LC, low-carbohydrate; HC, high-carbohydrate; VC3, switched from VLC to HC 3 days; VC7, switched from VLC to HC 7 days Feed-deprived levels of plasma glucose did not differ among any groups (t = 0, ipGTT, Table 3); the higher corresponding levels of insulin in LF, HF1, HF2, and VLC groups vs. the HC group resulted in higher HOMA-IR values in the LF, HF1, HF2 and VLC groups vs. the HC group (p < 0.05). Similarly, higher feed-deprived levels of insulin in the HF1 group resulted in higher HOMA-IR values in the HF1 than the LC group (Table 3, p < 0.05). These data provide evidence for whole body insulin resistance in the HF1 rats before weight reduction, and in the VLC rats after weight reduction. The ip glucose load resulted in similar AUC values in all groups in Phase 2 (Table 3). Although both VLC and HC groups showed a reduction from the HF1 group's AUC values (p < 0.05), and glucose values of VLC and HC rats did not differ at any time point, the VLC group had a higher insulin concentration at 30 min post-injection than the HC group (p < 0.01); the higher plasma insulin for the same glucose level in VLC vs. HC rats provides evidence for whole body insulin resistance in VLC rats. Plasma β-hydroxybutyrate levels, measured after an overnight fast during wk. 17, were in the ketotic range (> 0.5 mmol/L) for all Phase 2 groups and did not differ among the four diet groups; these levels of β-hydroxybutyrate exceeded those we previously reported for VLC or HC rats in the fed state [18], demonstrating that prolonged feed-deprivation can raise plasma ketone levels in rats, even if they consume a high-carbohydrate diet. In blood samples obtained from anesthetized rats during dissection, feed-deprived glucose levels did not differ among groups, but only HF1 rats still showed an elevation in glucose at 3 h after the oral glucose load (p < 0.001, Fig. 2). However, VLC rats had higher than feed-deprived insulin levels at 3 h after the oral load (p < 0.05); such persistent elevation in insulin levels is consistent with insulin resistance in VLC rats. Feed-deprived levels of plasma triglyceride were elevated in the HC group (HC > VLC ~ LC ~ HF2, p < 0.05) and remained high after the oral glucose load. The increase in plasma TG levels by the glucose load (p < 0.02) in both VLC and LC groups is consistent with insulin resistance. Phase 3: Switch from VLC to HC Diet At the time of Phase 3 dissection, plasma insulin levels were higher (p < 0.05) in feed-deprived VC3 than they had been in VLC or LC rats at wk. 18, but this difference was no longer significant by day 7 after the switch to the HC diet (VC7 group). Hepatic gene expression The effects of the HF diet on regulation of hepatic gene expression by insulin was assessed by comparing levels of mRNA, transcribed from insulin-regulated genes, in liver samples obtained from rats in the feed-deprived state vs. those obtained at 3 h after an oral glucose load, which was used to raise plasma insulin levels. At the end of Phase 1, the glucose load increased the expression of Srebf, whose product SREBP1c is a major regulator of lipogenesis, as well as its target Fasn, in the LF but not the HF1 group (p < 0.05, Fig. 3). In response to glucose, HF1 and LF groups both showed increased expression of Acaca (target of SREBP1c) and decreased expression of Insig2. Expression of Scd1 (target of SREBP1c), was higher in LF vs. HF1 rats in both the feed-deprived and 3 h post- oral glucose conditions (p < 0.01). Effect of glucose on gene expression in rats consuming LF vs. HF1 diets. Liver samples were obtained from rats consuming either an LF (N = 8) or an HF (N = 10) ad libitum for ten weeks (Phase 1) feed-deprived 16 h (dark bars) and 3 h post-glucose (light bars). Relative hepatic gene expression is plotted as 2-dCt. Means ± SEM are shown; labelled means, for a given gene and condition, without a common letter differ from each other, p < 0.05. Different from Feed-deprived, p < 0.05. Acaca: acetyl-CoA carboxylase; Fasn: fatty acid synthase; Gck: glucokinase; Insig2: insulin signaling protein 2; Pck1: phosphoenolpyruvate carboxykinase; Scd1: stearoyl-CoA desaturase-1; Srebf1: Sterol regulatory element-binding protein 1. LF, lower-fat; HF1 high-fat Phase 1 At the end of Phase 2, all groups increased Acaca mRNA after the glucose load (Fig. 4); VLC, LC and HC, but not HF2, rats showed effects of the load on expression of Fasn and Gck (increased) or Pck1 (decreased). Only the HF1 group showed the expected decrease in Insig2 mRNA in response to the glucose load. The glucose load increased expression of Srebf1 in HC and LC, but not VLC or HF2 groups, while stimulation of Scd1 and inhibition of Cpt1a expression were seen only in the HC group. Effect of glucose on gene expression in rats consuming HF2 vs. VLC, LC or HC diets. Liver samples were obtained from obese rats continuing to consume an HF diet ad libitum or VLC, LC or HC diets in restricted amounts for 8 weeks (Phase 2); feed-deprived 16h (dark bars) and 3 h post-glucose (light bars). Relative hepatic gene expression is plotted as 2-dCt. Means ± SEM are shown; labelled means, for a given gene and condition, without a common letter differ from each other, p < 0.05. Different from Feed-deprived, p< 0.05. Ct ranges across all groups and conditions are shown in Parentheses; Acaca: acetyl-CoA carboxylase (28–32); Cpt1a: carnitine Palmitoyl transferase-1a (24–27); Fasn: fatty acid synthase (25–32); Gck: glucokinase (24–30); Insig2: insulin signaling protein-2 (24–27); Pck1: phosphoenolpyruvate carboxykinase-1 (21–27); Scd1: stearoyl-CoA desaturase-1 (24–32); Srebf1: Sterol regulatory element-binding protein(25–29); Rplp2:Ribosomal protein P2 (24–25); HF2, high-fat Phase 2; VLC, very low-carbohydrate; LC, low-carbohydrate; HC, high-carbohydrate In Phase 3, the VC3 group failed to increase Srebf, Acaca, or Fasn mRNA in response to the glucose load (Fig. 5), but after 7 days on the HC diet, VC7 rats showed responses qualitatively similar to those of HC rats for these genes, although neither group showed the increase in levels of Scd1 mRNA exhibited by the HC group. Only VC3 rats decreased expression of Pck1 after the glucose load; this may reflect the trend toward higher feed-deprived Pck1 mRNA in VC3 vs. VC7 groups. In VC3 and VC7 rats, overall levels of mRNA for Srebf1, Insig2, Acaca and Gck were lower (p < 0.05) than those in VLC or HC rats at end of Phase 2, even when mRNA levels were analyzed with reference to a calibrator comprised of pooled samples from the respective phases (ddCt). Effect of glucose on gene expression in rats switched from VLC to an HC diet. Liver samples were obtained from obese rats that had consumed restricted amounts of a VLC diet for 8 weeks and then were switched to isocaloric amounts of the HC diet for 3 (VC3) or 7 (VC7) days (Phase 3); 16 h feed-deprived (dark bars) and 3 h post-glucose (light bars). Relative hepatic gene expression is plotted as 2-dCt. Means ± SEM are shown. Different from Feed-deprived, P < 0.05, * P < 0.01, *** P < 0.001. Acaca: acetyl-CoA carboxylase; Fasn: fatty acid synthase; Gck: glucokinase; Insig2: insulin signaling protein-2; Pck1: phosphoenolpyruvate carboxykinase-1; Scd1: stearoyl-CoA desaturase-1; Srebf1: Sterol regulatory element-binding protein-1 Ad libitum consumption of the HF diet for 10 wk. produced obesity characterized by increased body weight, % body fat, and plasma leptin levels as compared with the LF control group, as well as hepatic steatosis. Continued ad libitum consumption of the HF diet for another 8 weeks in Phase 2 did not change any of these measures of obesity. Eight weeks of energy restriction of obese HF2 rats on VLC, LC or HC diets produced similar reductions in body weights and plasma leptin levels. In rats on the LC and HC, but not VLC, diets visceral fat, % body fat, and intramuscular TG were significantly reduced from that at wk. 10 (end of Phase 1), in agreement with our previous findings using VLC and HC diets [18]. In contrast, hepatic lipid levels decreased from HF1 values in the VLC and HC but not the LC group; this finding suggests that high dietary fat, coupled with an adequate amount of carbohydrate, promoted hepatic fat storage even during weight loss. These results show that the fat and carbohydrate compositions of the diets consumed in hypocaloric amounts during Phase 2 had differential effects on adipose tissue and hepatic fat loss during energy restriction. Evidence for whole body insulin resistance in the VLC and HF 2 rats was observed during Phase 2 of the study. Higher HOMA-IR values in unanesthetized VLC and HF2 rats were due to elevated feed-deprived plasma insulin levels vs. those of HC rats. In addition, insulin levels were higher in VLC than HC rats at 30 min after the ip glucose load, despite similar plasma glucose AUC values. These results suggest that insulin release may have increased as an adaptation to tissue insulin resistance in VLC rats during Phase 2. In our previous study, the elevated plasma glucose levels after an ip glucose load in VLC vs. HC rats, but similar insulin levels, indicated insulin resistance [18]. The higher levels of plasma triglyceride in LF and HC vs. VLC groups are consistent with results reported in humans and rodents on high carbohydrate diets [34,35,36], and have been associated with hepatic lipogenesis [34]. The present study was designed to permit interpretation of differences in hepatic gene expression among diet groups to represent chronic adjustments, and not acute responses to the nutrient content of the current meal. To this end, all rats were feed-deprived for > 16 h before sampling, the same glucose stimulus was used for all groups in a given Phase, and each diet group's gene expression responses to the oral glucose load were compared with that in the feed-deprived state. The effects of diet composition on hepatic insulin resistance were evaluated by examining responses of insulin-regulated genes, many of which have roles in hepatic lipogenesis (Srebf1, Insig2, Acaca, Fasn, Scd1, and Gck). Although the lipogenic pathway (including de novo synthesis of fatty acids, DNL) is stimulated by insulin, there is disagreement concerning whether regulation of DNL is impaired by hepatic insulin resistance. In hepatic insulin resistance, insulin-regulated pathways that involve glucose metabolism are dysregulated (e.g., insulin fails to inhibit gluconeogenesis), while insulin-regulated triglyceride synthesis appears still to be appropriately increased [37]; this apparent paradox has been termed "selective insulin resistance" [38]. However, work using genetic modifications of the insulin-signaling pathway in mice [35, 39] has shown that hepatic DNL, including expression of Srebf1, is regulated by insulin; hepatic insulin resistance, as defined by defects in insulin signaling, therefore would prevent insulin from increasing Srebf1 expression. Furthermore, the hepatic lipid storage and export observed in hepatic insulin resistance may not involve stimulation of DNL by insulin, but may instead reflect packaging of fatty acids coming to the liver from insulin-resistant adipose tissue [40]. In light of these considerations, we utilized insulin's effect on expression of genes in lipogenic pathways to assess hepatic insulin resistance. Ad libitum intake of the HF diet suppressed the effects of an oral glucose load, used to raise plasma insulin level, on expression of some of the genes studied (Srebf1, Fasn, Scd1, Gck, Pck1, and Cpt1a). This result may be due, in part, to the high plasma levels of leptin in both HF1 and HF2 rats; leptin has been reported to decrease expression of genes involved in the lipogenic pathway, including Srebf1 [41] and Scd1 [42]. In contrast, only the LF, HF1 and HF2 groups appropriately decreased Insig2 mRNA after the glucose load. Given that Insig2 plays a role in controlling diurnal patterns of metabolism, the disturbance of ad libitum feeding in the VLC, LC and HC, but not LF, HF1 or HF2 groups by feed restriction may have affected regulation of Insig2 transcription [43]. In Phase 1, even though the HF1 group failed to increase Srebf1 mRNA in response to the glucose load, expression of Acaca, a target of SREBP1c, did increase in both LF and HF1rats. It is possible that SREBP1c may have mediated this response in both groups, since the reduction in Insig2 mRNA after the glucose load in LF, HF1, and HF2 groups may have led to a decrease in Insig2 protein, thereby increasing conversion of SREBP1c to its mature form. It is also possible that expression of Acaca was stimulated by other regulators, such as ChREBP, which is activated by a metabolite of glucose [44]. However, there were no detectable changes in expression of the ChREBP gene (Mlxipl) between feed-deprived and 3 h post-glucose states in either LF or HF1 rats. The observed reduction in lipogenic gene response to the glucose load in the HF1 and HF2 groups is consistent with the decrease in DNL seen in mice on a high-fat diet [45]. In contrast, increased expression of lipogenic genes has been reported in rats [46] and mice [35, 47, 48] consuming high-fat, or high-fat, sucrose-containing diets. However, those findings pertain to gene expression in the feed-deprived state [46, 47] or in a single fed condition, and not a comparison of pre- vs. post-glucose gene expression as in the present study, and so do not assess stimulation of expression by glucose or insulin. Although high cholesterol intakes (4% of diet weight) have been associated with insulin resistance [49] such an effect would not be probable for rats on the HF1, HF2, VLC or LC diets, which contained less than 0.5% cholesterol. Furthermore, mice on 1% cholesterol high-fat diets showed increased expression of SREBP1c [50], not the decrease that we observed. Weight loss and reduction in plasma leptin levels, regardless of diet, were associated with improved effects of the glucose load on Fasn, Gck, and Pck1 expression in VLC, LC and HC rats (Fig. 4), as compared with the HF1 rats before weight reduction (Fig. 3). However, like the HF2 group, VLC rats did not show significant changes in expression of Srebf1, Scd, or Cpt1a in response to the glucose load at the end of Phase 2. This result supports hepatic insulin resistance in VLC rats, despite reduction in hepatic lipid levels and body weights from HF1 levels. The stimulation of Srebf1 expression by the glucose load in rats on LC or HC diets, which differed in fat and cholesterol levels, suggests that their higher carbohydrate intake, vs. that of VLC rats, had improved their hepatic insulin sensitivity. Regulation of Srebf1 transcription involves not only insulin, but includes the liver X receptor (LXR), mechanistic target of rapamycin complex (mTorc), and positive feedback from the mature form of SREBP1c [51]. Since dietary protein levels in VLC, LC and HC diets were similar, differences in amino acid intake seem unlikely to account for lower mTorc activation by insulin in the VLC group; similarly high fat intakes of VLC and LC rats also make it unlikely that those groups differed in their intake of dietary regulators of LXR. Failure of a carbohydrate stimulus to increase Srebf1 mRNA by rodents consuming high-fat diets has been attributed to their polyunsaturated fatty acid (PUFA) intake [51,52,53] . This possibility is unlikely to explain the differences between VLC and LC or HC response for the following reasons: 1) PUFA (92% n-6) intakes of VLC (~ 1.57 g/d) and LC (~ 1.25) groups were similar; 2) in other studies, PUFA were consumed near the time of liver sampling, [52,53,54,55], whereas rats in our study were feed-deprived for at least 16 h before dissection, precluding an acute effect of dietary PUFA; and 3) if chronically higher PUFA intake were responsible for reducing levels of Srebf1 mRNA, then lower feed-deprived expression in HF1, HF2, VLC or LC vs. HC groups would be expected, but this was not observed. Only the HC group showed a significant stimulation of Scd1 expression in Phase 2; overall expression of Scd1 was also higher in LF vs. HF1 rats in Phase 1. High carbohydrate diets (e.g., LF and HC) stimulate Scd1 transcription via SREBP1c and ChREBP [56]. Normal and diabetic mice consuming a diet similar to the LC (18%C, 37%F) showed both reduced insulin sensitivity and lower hepatic expression of Scd1 than mice consuming a diet similar to the HC or LF diets (63% C, 22%F) [57]; the small difference in their fat intake suggests that the difference in carbohydrate intake was responsible for the effect on Scd1 expression. No difference in feed-deprived levels of Cpt1a mRNA was observed between any high-fat group (HF2, VLC or LC) and the HC group, although Cpt1a mRNA has been reported to be higher in rats on high-fat vs. low-fat diets in the feed-deprived state [58]. However, the finding that only the HC group, but not the HF2, VLC or LC groups, significantly decreased Cpt1a mRNA after the glucose load is consistent with the lack of difference in Cpt1a expression reported between the fed and feed-deprived state in rats consuming high-fat diets [59]; this lack of suppression of Cpt1a expression should promote continued use of fat as a fuel, despite increased availability of glucose. In Phase 3, the lower responsiveness of hepatic gene expression to the glucose load by the VC3 rats, vs. that of VLC, LC or HC rats at the end of Phase 2, may have been associated with weight loss during the first 3 days after the switch to the HC diet. The stabilization of body weight during days 4–7 on the HC diet by VC7 rats may have promoted better responses. However, VC7 rats still differed from HC rats in their lack of increase in Scd1 mRNA in response to glucose, and mRNA levels of many genes remained below that exhibited by either VLC or HC rats. These results indicate that full transition from the effects of the VLC to those of the HC diet did not occur within 1 week. The HC diet promoted greater responses of the insulin-regulated genes, Srebf, Scd1 and Cpt1a, to the glucose load than the VLC diet. As discussed above, the higher carbohydrate intake of the HC vs. the VLC or LC groups is likely to have accounted for the greater Scd1 gene response, and the lower fat intake of the HC vs. the LC or VLC groups may have accounted for the greater Cpt1a gene response the glucose load in HC rats. Although the VLC and LC groups had similar fat intakes, only the LC group increased expression of Srebf in response to glucose; the severely restricted carbohydrate intake of VLC rats may account for their lack of response.. However, since it is not possible to vary only fat or only carbohydrate intake, while holding protein and energy intake essentially constant, it remains possible that differences in both fat and carbohydrate intake are responsible for the observed dietary effects. VLC, but not LC rats had elevated HOMA-IR and higher insulin levels after the ip glucose load as compared with HC rats, indicating whole body insulin resistance in VLC rats; since LC and VLC groups had similar fat intakes, this finding suggests a mitigating effect of the higher carbohydrate intake of the LC rats. However, the combined effects of high dietary fat and stimulation of expression of lipogenic genes (perhaps including targets of SREBP1c that are involved in esterification) in the LC group may have promoted hepatic TG storage [60]. Nonetheless, hepatic steatosis was not associated with whole body insulin resistance in weight-reduced rats in the present study. In addition, loss of the response of hepatic gene expression to a glucose load was not associated with hepatic steatosis.; Although hepatic lipid levels were similar in HF1, HF2 and LC rats, LC rats showed greater response to the glucose load than HF1 or HF2 rats, or than the VLC group which had lower hepatic lipid levels. In HC rats, the insulin-stimulated increases in hepatic SREBP1c and its targets would be expected to promote conversion of glucose to fatty acids and triglyceride; decreases in hepatic Cpt1a would be expected to suppress fat oxidation in favor of glucose oxidation. Both processes should support the ability of a given concentration of insulin to lower plasma glucose levels after a glucose load; this is consistent with lower whole body insulin resistance in HC rats. Lack of this hepatic response in VLC rats would be expected to diminish the glucose-lowering effect of insulin; this is consistent with the higher insulin levels observed for the same level of glucose in VLC vs. HC rats. The carbohydrate and fat contents of diets producing the same weight loss differentially affected body composition and insulin resistance in obese rats. The VLC diet promoted whole body insulin resistance and diminished the effect of a glucose load on hepatic expression of some insulin-regulated genes, as compared with HC or LC diets. These findings reveal adaptations produced by a VLC weight loss diet that could promote hepatic insulin resistance, thereby lessening the effect of insulin on metabolic pathways. ChREBP: Carbohydrate Regulatory Element Binding Protein Glucose tolerance test High-carbohydrate HF1: High-fat Phase 1 IR: LC: LF: MUFA: Monounsaturated fatty acid Polyunsaturated fatty acid Sterol regulatory element binding protein 1c TG: Very low-carbohydrate Acaca : Acetyl-CoA carboxylase Cpt1a: Carnitine Palmitoyl transferase-1a Fasn : Gck: Insig2 : Insulin signaling protein-2 Pck1 : Phosphoenolpyruvate carboxykinase-1 Scd1 : Stearoyl-CoA desaturase-1 Srebf1 : Sterol regulatory element-binding protein 1 Gow ML, Garnett SP, Baur LA, Lister NB. The effectiveness of different diet strategies to reduce type 2 diabetes risk in youth. Nutrients. 2016;8:1–13. Ballard KD, Quann EE, Kupchak BR, Volk BM, Kawiecki DM, Fernandez ML, et al. Dietary carbohydrate restriction improves insulin sensitivity, blood pressure, microvascular function, and cellular adhesion markers in individuals taking statins. Nutr Res [Internet]. Elsevier Inc.; 2013;33:905–912. 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Polyunsaturated fatty acid regulation of genes of lipid metabolism. Annu Rev Nutr [Internet]. 2005 [cited 2014 Mar 20];25:317–340. Available from: http://www.ncbi.nlm.nih.gov/pubmed/16011470 Ntambi JM, Miyazaki M, Dobrzyn A. Regulation of stearoyl-CoA desaturase expression. Lipids [Internet]. 2004;39:1061–5. Available from: http://www.ncbi.nlm.nih.gov/pubmed/15726820. Handa K, Inukai K, Onuma H, Kudo A, Nakagawa F, Tsugawa K, et al. Long-term low carbohydrate diet leads to deleterious metabolic manifestations in diabetic mice. PLoS One. 2014;9(8):e104948. https://doi.org/jpurnal.pone. 0104948. Lee W, Yoon G, Hwang YR, Kim YK, Kim SN. Anti-obesity and hypolipidemic effects of Rheum undulatum in high-fat diet-fed C57BL/6 mice through protein tyrosine phosphatase 1B inhibition. BMB Rep. 2012;45:141–6. Frier BC, Jacobs RL, Wright DC. Interactions between the consumption of a high-fat diet and fasting in the regulation of fatty acid oxidation enzyme gene expression: an evaluation of potential mechanisms. Am J Physiol Regul Integr Comp Physiol. 2011;300:R212–21. Delgado TC, Pinheiro D, Caldeira M, Castro MMCA, Geraldes CFGC, López-Larrubia P, et al. Sources of hepatic triglyceride accumulation during high-fat feeding in the healthy rat. NMR Biomed. 2009;22:310–7. This study was supported by NIGMS SC3GM086298 (includes funds for publication cost) and PSC-CUNY 4346100–41. All data generated or analyzed during this study are included in this published article or are available from the corresponding author on reasonable request. Department of Health and Nutrition Sciences, Brooklyn College, City University of New York, New York, USA Kathleen V. Axen , Marianna A. Harper , Yu Fu Kuo & Kenneth Axen Search for Kathleen V. Axen in: Search for Marianna A. Harper in: Search for Yu Fu Kuo in: Search for Kenneth Axen in: KVA designed the study; KVA, YFK, MAH, and KA conducted the experiments; MAH and KVA did sample analysis and data analysis, KVA and KA wrote the manuscript; KVA had primary responsibility for the manuscript. All authors read and approved the final manuscript. Correspondence to Kathleen V. Axen. All procedures were approved by the Brooklyn College Institutional Animal Care and Use Committee (Protocol 248). Not-applicable. No human subjects were used in the study. Axen, K.V., Harper, M.A., Kuo, Y.F. et al. Very low-carbohydrate, high-fat, weight reduction diet decreases hepatic gene response to glucose in obese rats. Nutr Metab (Lond) 15, 54 (2018) doi:10.1186/s12986-018-0284-9 Very low-carbohydrate diet	CommonCrawl
Bulk temperature In thermofluids dynamics, the bulk temperature, or the average bulk temperature in the thermal fluid, is a convenient reference point for evaluating properties related to convective heat transfer, particularly in applications related to flow in pipes and ducts. The concept of the bulk temperature is that adiabatic mixing of the fluid from a given cross section of the duct will result in some equilibrium temperature that accurately reflects the average temperature of the moving fluid, more so than a simple average like the film temperature. $T_{bulk}$ $\theta_a+\theta_{abs}$ $\theta_a$ Ambient temperature [°C] $\theta_{abs}$ Absolute temperature [K] Used in $\mathrm{Gr}_L$ Grashof number, ground to air $T_{gas}$ Absolute gas temperature [K] $T_{surf}$ Absolute surface temperature [K]	CommonCrawl
Combinatorial proof of $\sum_{i=0}^k\binom{m+k-i-1}{k-i}\binom{n+i-1}{i}=\binom{m+n+k-1}{k}$? Please provide a combinatorial proof for the following: Prove the identity $$\sum_{i=0}^{k}{m+k-i-1 \choose k-i}{n+i-1 \choose i}={m+n+k-1 \choose k}$$ Hint: use idea of "selection with repetition". combinatorics binomial-coefficients Mike Spivey SarahSarah $\begingroup$ Do you mean $$\sum_{i=0}^k\binom{m+k-i-1}{k-i}\binom{n+i-1}{i}=\binom{m+n+k-1}{k}\quad?$$ What did you try to do? $\endgroup$ – Martin Jan 29 '13 at 12:23 $\begingroup$ Yes I that's the question. For the R.S I said that its counts the no. of ways to select k objects from m+n objects with repetition, but am not sure how to interpret the summation side. $\endgroup$ – Sarah Jan 29 '13 at 12:37 $\begingroup$ It looks like a homework problem. $\endgroup$ – Joe Z. Jan 30 '13 at 3:51 $\begingroup$ @JoeZeng: or an exercise in a book. It's hard to second guess. In cases like this, without further information, we usually accept what the author says. $\endgroup$ – robjohn♦ Feb 4 '13 at 2:09 Imagine you have $m$ red objects and $n$ blue objects, and you want to select $k$ objects with possible repetitions. This can be done (as you stated yourself in the comment) in $$\binom{m+n+k-1}{k}$$ different ways. On the other hand, among these $k$ selected objects, there could be exactly $k$ red ones and $0$ blue ones, there could be $k-1$ red and $1$ blue, etc. Each of these cases are mutually exclusive, and there are $$\binom{m+(k-i)-1}{k-i} \binom{n+i-1}{i}$$ ways to choose exactly $k-i$ red objects and $i$ blue objects. Sum over all $i$ and you get your identity. mrfmrf $\begingroup$ @Sarah: You're welcome! And since you're new here, don't forget to upvote answers and questions you find useful (and not just answers to your own questions). If you're really happy with a particular answer, you can "accept" it, but it's usually a good idea to wait awhile before doing that -- even better answers can appear later. $\endgroup$ – mrf Jan 29 '13 at 13:43 $\begingroup$ Yes for sure! But how do I upvote? $\endgroup$ – Sarah Jan 29 '13 at 13:47 $\begingroup$ @Sarah: Look at the grey arrows in the left column. Clicking the up-arrow gives an upvote, and likewise, the down-arrow gives a downvote. The number attached to each post is the difference between upvotes and downvotes. $\endgroup$ – mrf Jan 29 '13 at 13:49 Not the answer you're looking for? Browse other questions tagged combinatorics binomial-coefficients or ask your own question. $\sum_{i=0}^m \binom{m-i}{j}\binom{n+i}{k} =\binom{m + n + 1}{j+k+1}$ Combinatorial proof Combinatorial proof of an identity How to begin combinatorial proof of $\sum_{k=1}^n k \binom nk^2 = n \binom{2n-1}{n-1}$ Proving combinatorial identity $\sum_{k=0}^{n}(-1)^k \binom{2n-k}k 2^{2n-2k} = 2n+1$ "directly" Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$ Combinatorial proof of $\sum_{j=i}^n { n \choose j} { j \choose i} (-1)^{n-j}=0$ Prove the identity $ \sum_{i=0}^n {n \choose i} i = n2^{n-1} $ using combinatorial proof. Prove the identity $\sum_{k=0}^n \binom{n}{k}=2^n.$ using combinatorial proof Combinatorial Proof of Identity Four models of combinatorial proof Proving a combinatorial identity: $\sum_{i=0}^m \binom{l}{i}\binom{m+n-l}{m-i} = \binom{m+n}{m}$	CommonCrawl
An Efficient and Effective Design of InP Nanowires for Maximal Solar Energy Harvesting Dan Wu1, Xiaohong Tang1, Zhubing He3 & Xianqiang Li1 Solar cells based on subwavelength-dimensions semiconductor nanowire (NW) arrays promise a comparable or better performance than their planar counterparts by taking the advantages of strong light coupling and light trapping. In this paper, we present an accurate and time-saving analytical design for optimal geometrical parameters of vertically aligned InP NWs for maximal solar energy absorption. Short-circuit current densities are calculated for each NW array with different geometrical dimensions under solar illumination. Optimal geometrical dimensions are quantitatively presented for single, double, and multiple diameters of the NW arrays arranged both squarely and hexagonal achieving the maximal short-circuit current density of 33.13 mA/cm2. At the same time, intensive finite-difference time-domain numerical simulations are performed to investigate the same NW arrays for the highest light absorption. Compared with time-consuming simulations and experimental results, the predicted maximal short-circuit current densities have tolerances of below 2.2% for all cases. These results unambiguously demonstrate that this analytical method provides a fast and accurate route to guide high performance InP NW-based solar cell design. For future generation solar cells, semiconductor nanowire (NW) arrays have unraveled a new pathway to greatly reduce material consumption and fabrication cost while maintaining or even improving the device performance as compared with their thin film or bulk counterparts [1, 2]. This fascinating feature is largely attributed to the remarkable optical properties of the NWs, including increased absorption [3, 4] and spectral selectivity [5,6,7]. Among various III–V materials, InP NW arrays have attracted intensive research effort for solar cell application due to the direct bandgap and low intrinsic surface recombination velocity [8]. Up to date, the highest energy conversion efficiency achieved 13.8% for InP NW arrays in a cell of 1 mm2 in area [9]. Since the optical properties of NW arrays can be distinctively adjusted by tuning their three-dimensional geometry, to further improve the performance of NW-based solar cells, great attention has been put on how to optimize the morphology and topology of III–V NW arrays to maximize the light absorption [5, 9,10,11,12,13]. Specifically, the NWs' diameter, periodicity, and arrangement have been investigated to maximize the absorption of solar energy [6, 14,15,16]. It is reported that tuning the diameter of the NW will change the optical modes existed within the NW. This will lead to localized light absorption maxima for those incident wavelengths corresponding to the respective resonant modes [5, 6, 17, 18]. Also, NW arrays with optimized periodicity or filling ratio (FR) can suppress the reflection and transmission while enhancing the scattering to the incident light resulting in the prolonged optical path and thus the enhanced light absorption [19,20,21]. Besides, Martin Foldyna et al. have concluded that the dependence of the light absorption on the arrangement of the NW arrays is rather small since the light trapping effect of NWs is based on the individual waveguiding when the light coupling among neighboring NWs is neglected [22]. To find the maximal solar energy harvesting, the effect of the three-dimensional parameters and the arrangement of the NW arrays should be considered together. However, most of the reported optimal geometrical dimensions and arrangement of NW arrays for maximal solar spectrum harvesting are still parameter-space-determinated local optima. Besides, the incident solar spectrum combining with material dispersive properties add more difficulty to analytically solve this problem. Therefore, intensive and time-consuming numerical simulations such as finite-difference time-domain (FDTD) are frequently adopted to address this multi-parameters optimization problem. Sturmberg et al. reported a semi-analytic method to narrow down the range of the optimal dimensions of single diameter NW arrays [13]. Although this method is applicable for various materials, FDTD simulations should still be accompanied to find the exact optimal values. Moreover, this method is less helpful for superb absorber combined with multi-radii NW arrays [23]. In this paper, we present an analytical design for optimal geometrical dimensions of single, double, and multiple diameter InP NW arrays to maximize solar energy absorption. Diameters of NWs are determined by leaky mode resonance and Mie theory whereas the periodicities are identified by construction of an effective medium layer to minimize light reflection and transmission. Squarely and hexagonal distributed NW arrays are both considered. Moreover, intensive FDTD simulations are accompanied to verify the effectiveness of our method. The well matching of the largest short-circuit current densities generated from the NW arrays with the calculated geometrical parameters and the values obtained from FDTD simulations prove the effectiveness of the proposed method to guide the practical NW-based photovoltaic cells design. Design for Maximal Light Harvesting of InP NWs Vertically aligned InP NW arrays are placed upon a semi-infinite SiO2 substrate as schematically shown in Fig. 1 with either squarely or hexagonal arrangement. Repeatable unit cells in Fig. 1a, b insets explain respective characterization dimensions for each arrangement. This morphology and topology of the NW arrays are in accord with the majority of the InP NW-based solar cell structures [11, 12, 23, 24]. Within each of the unit cells, the NWs have the same or different diameters as D i . Periodicity p is defined as the center to center distance of a pair of adjacent NWs which has the same value for squarely arranged NWs whereas different values for hexagonal NW arrays. Accordingly, the FR of the squarely arranged NW arrays is defined as $ \pi {\sum}_{\mathrm{i}=1}^4{D_i}^2/{(4p)}^2 $ having the maximal value of π/4 when the NWs take up the largest volume percentage of the unit cell [25]. Similarly, the FR for hexagonal NW arrays is defined as $ \pi {\sum}_{\mathrm{i}=1}^2{D_i}^2/\left(4\sqrt{3}{p}^2\right) $ with the maximal value of $ \pi \sqrt{3}/6 $ [22]. The length l of the NW is set as 2 μm for all cases since they are long enough to absorb more than 90% of the incident energy with proper design [26]. Schematics of vertically aligned InP NW arrays. a Squarely and b hexagonal NW arrays with insets explaining their respective unit cells In order to analytically determine each geometrical parameter of NW arrays, the multiple-parameter optimization problem for maximal light harvesting is decomposed into two processes: (1) NWs' diameter-determinant resonant mode control and (2) FR-affected minimal reflectance and transmittance of incident solar energy. We construct the relationship of individual geometrical parameter with respective determinant process and identify each optimal value leading to maximal light absorption. Double diameter NW arrays are chosen as the design example for illustration of the proposed method. Optimal geometrical dimensions of single diameter NW arrays as a simpler case can also be acquired during the derivation. The diameter and periodicity for four diameter NW arrays can also be calculated as an extension of the example. For squarely arranged double diameters NW arrays, the diameters of the diagonal NWs have the same value as D major and the diameters of the rest two NWs are named as D supplementary. For hexagonal arranged NW arrays, the diameter of the center NW is D major and the diameters of the NWs at the peripheral are D supplementary. It is reported that NW arrays can support leaky/guided resonance modes, each of which lead to strong absorption peaks. Besides, the fundamental nature of waveguide suggests that the mode number grows with the rise of the diameter of NW. Consequently, the optimal diameter of NW should be large enough to support more modes so as to include larger number of absorption resonances. However, too large diameters of NWs are less preferable since the higher order modes they supported possess more nodes which couple less efficiently to the incident plane waves [13]. Besides, the material property and the incident solar spectrum place other limitations on the selection of the optimal diameter. Only when the resonant modes lie within the absorption region, they can contribute to the photocurrent. The absorption region is defined by the superposition of the material absorbing range of up to critical wavelength and the incident AM 1.5G spectrum [27]. As a result, to quantitatively determine the D major of the NW arrays, leaky mode resonance is initially adopted to calculate the respective resonant wavelengths for different diameters of NWs [2]. This gives the distribution of the resonant modes in the absorption region. Therefore, the optimal D major should support two modes to satisfy all of the above criterions. Secondly, Mie theory is adopted to calculate the normalized absorption efficiencies of those NWs in step one. Strictly speaking, Mie theory cannot be applied to the situation when the incident wave vector aligned perfectly parallel to the axis of NWs since the eigenvalue equation is ill-defined [28]. However, this situation can be approximated as the glazing incident of incoming light (very small incident angle θ with respect to the axis of NW) since at the interface of NW arrays, the wave front of the incident light will be perturbed by the high index of NWs which introduces transverse components to the wave vector allowing the adoption of Mie theory [18]. Therefore, the optimal D major are the one who support two modes while keeping the full width at half maximum (FWHM) of the lowest resonant mode in the normalized absorption efficiency spectrum within the absorption region. After acquisition of D major, the D supplementary is calculated on the condition that the NWs should support one mode for reducing reflection and material saving and their resonant wavelength should match the valley of the D major's normalized absorption efficiency spectrum. The periodicity of the NW arrays can be computed by construction of an effective medium layer. This artificial layer represents the reflection and transmission behavior of the NW arrays which is only related to the material FR. As a result, the diameter, periodicity, and the arrangement of the NW arrays are removed from the calculation. In this way, the transmittance and reflectance of NW arrays can be evaluated by applying Fresnel equations on this effective medium layer and therefore the optimal FR can be analyzed. Based on the relationship of FR and periodicity, the periodicities for both hexagonal and squarely arrangement NW arrays are obtained. Detailed description of our proposed method is presented in the following sections. A. Optimal Diameters of InP NW Arrays for Maximal Light Harvesting To increase the light absorption, the number of resonant modes leading to strong absorption peaks should be maximized within the absorption region. On the blue end of the absorption region, incident AM 1.5G spectrum confines 300 nm as the high energy region. The critical wavelength λ c of 925 nm (bandgap of InP 1.34 eV) limits the red end of the absorbing region. As a result, it is proved that the InP NWs that support two resonant modes locating inside the absorbing region are able to best improve the light absorption [29]. We expand this conclusion and use Mie theory to calculate the exact value. According to the above conclusion, the range of D major can be calculated from the eigenvalue equation derived from Maxwell's equations [18]. Considering the anti-symmetric in-plane field distribution of the incident plane waves, only the HE1m modes can be effectively excited to contribute to the absorption of vertically-aligned NWs [5]. These HE1m modes satisfy the eigenvalue equation, and the resonant wavelengths can be obtained assuming that the real part of the propagation constant Re(β z ) of the mode along NW axial direction approaches zero as shown in Eq. (1). k cyl and k air are the transverse components of the wave vector inside the NWs and in the air whereas ε cyl and ε air are the respective permitivities. J 1 and H 1 (1) are the first order Bessel and Hankel functions of the first kind. As a consequence, the range that primary diameter falls in can be received on the condition that the corresponding HE11 and HE12 mode lie within the absorbing region. $$ \frac{\varepsilon_{\mathrm{cyl}}{J}_1^{\prime}\left({k}_{\mathrm{cyl}}{D}_{\mathrm{major}}/2\right)}{k_{\mathrm{cyl}}{J}_1\left({k}_{\mathrm{cyl}}{D}_{\mathrm{major}}/2\right)}-\frac{\varepsilon_{\mathrm{air}}{H_1^{(1)}}^{\prime}\left({k}_{\mathrm{air}}{D}_{\mathrm{major}}/2\right)}{k_{\mathrm{air}}{H}_1^{(1)}\left({k}_{\mathrm{air}}{D}_{\mathrm{major}}/2\right)}=0. $$ According to the Mie theory, the absorption efficiency Q abs of NWs is defined by the ratio of the energy collecting area and the geometrical size of the NWs. The analytical expression of absorption efficiency Q abs is given below, and the exact mathematical formalism of Mie theory can be found in the reference [30]. Here, $ \overline{n}=n+ ik $ is the complex refractive index; as mentioned above, J i and H i (1) are the Bessel and Hankel functions of first kind of order i. $$ {\displaystyle \begin{array}{c}{Q}_{\mathrm{abs},\mathrm{TM}}=\frac{2}{x}\operatorname{Re}\left({b}_0+2\sum \limits_{i=1}^{\infty }{b}_i\right)-\frac{2}{x}\left[{\left\|{b}_0\right\|}^2+2\sum \limits_{i=1}^{\infty }{\left\|{b}_i\right\|}^2\right]\\ {}{Q}_{\mathrm{abs},\mathrm{TM}}=\frac{2}{x}\operatorname{Re}\left({a}_0+2\sum \limits_{i=1}^{\infty }{a}_i\right)-\frac{2}{x}\left[{\left\|{a}_0\right\|}^2+2\sum \limits_{i=1}^{\infty }{\left\|{a}_i\right\|}^2\right]\end{array}} $$ $$ {\displaystyle \begin{array}{c}{a}_i=\frac{\overrightarrow{n}{J}_i\left(\overrightarrow{n}x\right){J}_i^{\prime }(x)-{J}_i\left(\overrightarrow{n}x\right){J}_i^{\prime }(x)}{\overrightarrow{n}{J}_i\left(\overrightarrow{n}x\right){H_i^{(1)}}^{\prime }(x)-{J}_i^{\prime}\left(\overrightarrow{n}x\right){H}_i^{(1)}(x)}\\ {}{b}_i=\frac{J_i\left(\overrightarrow{n}x\right){J}_i^{\prime }(x)-\overrightarrow{n}{J}_i\left(\overrightarrow{n}x\right){J}_i^{\prime }(x)}{J_i\left(\overrightarrow{n}x\right){H_i^{(1)}}^{\prime }(x)-\overrightarrow{n}{J}_i^{\prime}\left(\overrightarrow{n}x\right){H}_i^{(1)}(x)}\end{array}} $$ After acquisition the Q abs of the HE11 mode, the FWHM of respective diameter of NWs can be found out, and therefore, the optimal diameter for maximal light harvesting is determined. Upon decision of the major diameter, the supplementary diameter is confirmed on the condition that its normalized absorption peak wavelength should match the normalized absorption efficiency valley of the major diameter. For four diameter NW arrays, the third and fourth diameters are determined in a similar way. Their normalized absorption efficiency peaks should match the valleys of the superposition of normalized absorption efficiency spectrum of the primary and secondary NWs. It is noteworthy that except for the major NWs, the second, third, and fourth NWs are desired to support only one mode since the small diameter size can both reduce the reflectance at the air-NW interface and reduce material consumption. B. Optimal FR of InP NW Arrays for Maximal Light Harvesting Various published work has disclosed that with fixed diameters of NWs; the absorption of the NWs will increase with the FR initially and then drop after a certain optimal value [13]. The rise of light absorption is usually attributed to the increase of volume percentage of the semiconductor materials with high absorption coefficients. As FR further grows, the average refractive index of the NW arrays increases, and thus, the reflection rises which reduces light absorption. Therefore, an upper limit on the FR should be found to optimize the influence of Fresnel reflection and transmission to maximize the absorption of the NW arrays. Figure 2 schematically illustrates that an effective medium layer of complex refractive index is created to represent the refraction and transmission behavior of the NW arrays. In this way, the periodicities and diameters of NWs are removed from the calculation. Consequently, Fresnel calculation of the reflection and transmission of the effective medium layer can be used to reflect the properties of the NW arrays. The exact nature inside this artificial medium layer is not considered as long as they can represent the reflection and transmission of NW arrays. Detailed mathematics derivations are given below. Light reflection, transmission, and absorption of NWs and effective medium layer. a InP NW arrays and b the corresponding effective medium layer with the same thickness The real part of the refractive index of the effective medium layer n em_real are determined by Bruggeman formulation [31] in Eq. (4) where Ɛem, and ƐNW are the permittivity of the effective medium layer and InP, respectively. The imaginary part of the refractive index n em_imag is calculated by Volume Averaging Theory [32, 33] in Eq. (5) where the n NW_real, n NW_imag, n air_real, and n air_imag are the real and imaginary part of the refractive index of NW and air. The optimal FRopt is defined as the FR such that absorptance Abs(λ) = 1 − R(λ) − T(λ) is maximized using Fresnel equations. $$ {\displaystyle \begin{array}{l}\left(1-\mathrm{FR}\right)\frac{\varepsilon_{\mathrm{air}}^2-{\varepsilon}_{\mathrm{em}}^2}{\varepsilon_{\mathrm{air}}^2+2{\varepsilon}_{\mathrm{em}}^2}+\mathrm{FR}\frac{\varepsilon_{\mathrm{NW}}^2-{\varepsilon}_{\mathrm{em}}^2}{\varepsilon_{\mathrm{NW}}^2+2{\varepsilon}_{\mathrm{em}}^2}=0\\ {}{n}_{\mathrm{em}\_\mathrm{real}}=\operatorname{Re}\left(\sqrt{\varepsilon_{\mathrm{em}}}\right)\end{array}} $$ $$ {\displaystyle \begin{array}{l}\mathrm{A}=\mathrm{FR}\left({n}_{\mathrm{NW}\_\mathrm{real}}^2-{n}_{\mathrm{NW}\_\mathrm{imag}}^2\right)+\left(1-\mathrm{FR}\right)\left({n}_{\mathrm{air}\_\mathrm{real}}^2-{n}_{\mathrm{air}\_\mathrm{imag}}^2\right)\\ {}B=2\mathrm{FR}{n}_{\mathrm{NW}\_\mathrm{real}}{n}_{\mathrm{NW}\_\mathrm{imag}}+2\left(1-\mathrm{FR}\right){n}_{\mathrm{air}\_\mathrm{real}}{n}_{\mathrm{air}\_\mathrm{imag}}\\ {}{n}_{\mathrm{em}\_\mathrm{imag}}=\sqrt{\frac{-A+\sqrt{A^2+{B}^2}}{2}}\end{array}} $$ Through replacing the NW arrays with a thin film of equal thickness, the reflectance R(λ) and T(λ) transmittance of NW arrays can be estimated using the Fresnel equations. The first two terms of the infinite Fabry-Perot reflection and transmission series are included in Fig. 2b. Detailed mathematical derivations can also be found in the supporting information of the reference [13]. At this stage, the optimal diameters and the FR are both determined and the corresponding periodicity can be acquired based on the definition of the FR. With the optimal geometrical dimensions, the NW arrays should lead to maximal light absorption. Short-circuit current density J sc is mostly used to measure the light harvesting capability assuming that every absorbed photon leads to an exciton separation followed by a successful carrier collection. The definition is shown in Eq. (6) where A(λ) is the absorption inside nanowires as a function of the incident wavelength, and N(λ) is the number of photons per unit area per second for the incident wavelength from the standard solar spectrum. $$ {J}_{\mathrm{sc}}=q\underset{\mathrm{AM}1.5\mathrm{G}}{\int }A\left(\lambda \right)N\left(\lambda \right) d\lambda $$ Single and multiple diameters of InP NW arrays of squarely and hexagonal arrangements demonstrate the validity of the proposed method. Meanwhile, FDTD numerical simulations (Lumerical FDTD Solutions 8.15) are also provided to compare with our method. Periodic boundary condition is applied along x and y axes while perfect matching condition is set along z axis as illustrated in Fig. 1. The InP NWs are vertically standing on SiO2 substrate. The optical constants for InP and SiO2 are from Palik material data provided by Lumerical. The parameter space for diameters of NWs ranges from 50 to 200 nm whereas the FR is from 0.05 to the possible maximal values for squarely and hexagonal NWs. A. Maximal Light Harvesting for Single Diameter InP NWs Figure 3a shows the light absorption efficiency for single diameter InP NW arrays when FR is 0.05 with the optical constants provided in the inset. The respective resonant wavelengths are calculated and marked on corresponding absorbing peaks which match the FDTD simulation results well. The red shift of HE11 resonant mode can be easily observed with the rise of the diameter of NWs. Besides, both calculation and simulation prove that the resonant mode evolves from one to two modes at 140 nm diameter. Therefore, the optimal value for maximal light absorption should be larger than 140 nm and smaller than 200 nm where two modes are excited within each NW. To find the optimal value of diameter, normalized absorption efficiency of NW arrays is provided in Fig. 3b showing the NW arrays which support two modes and still keep the FWHM within the absorbing region. Therefore, the largest value of 184 nm diameter is chosen as the optimal diameter without any additional peak. Interestingly, the up-to-date highest power conversion efficiency InP NW solar cell design adopted the optimal diameter of 180 nm. Their diameters of NWs were experimentally optimized ranging from 50 to 300 nm with 10 nm as the increase step [9]. Compared with our prediction of 184 nm, a narrow tolerance of 4 nm demonstrates the accuracy of our method. Wavelength dependent absorption efficiency of InP NWs and normalized absorption efficiency. a Absorption efficiency of NWs with inset explaining the optical constants. b Calculated absorption efficiency by Mie theory Filling ratio is analytically obtained using effective medium layer in section B of the described method. The light absorption efficiency of the effective layer of the same height as the InP NW arrays is shown in Fig. 4. In general, the light harvesting capability rises initially, reaches its maximal value, and gradually falls down as the FR approaches larger value. This trend is attributed to the change of the transmitted and reflected light as the complex refractive indices change due to FR variation. Specifically, when FR increases from 0.05 to 0.2, due to the addition of InP material, more light is absorbed before transmitted out of the NW arrays. However, this trend increase until FR reaches 0.2, and further increase of FR cause high complex refractive index of the equivalent layer which lead to optical impedance between the air and NW arrays. As a result, the reflectance at the incident surface rises rapidly which decrease light absorption [13]. Therefore, the optimal value for FR is 0.2 and the periodicities for squarely and hexagonal arranged NW arrays are 364.63 and 391.82 nm, respectively. Absorption efficiency of effective medium layer for InP NW arrays as a function of FR The short-circuit current densities for various combinations of diameters and FRs are shown in Fig. 5. It clearly demonstrates that the arrangement of NWs has little effect on the highest light absorption. Also, regardless with the arrangements of NW arrays, our method can both be applied and achieve accurate results. The maximal J sc with calculated optimal geometrical dimensions for InP NW arrays are calculated for squarely and hexagonal arrangement, respectively. The analytical predicted maximal J sc is 32.11 and 32.06 mA/cm2 for squarely and hexagonal NW arrays leading to tolerance of 0.33 and 0.1%, respectively, as compared with FDTD simulation results. Theoretical predicted maximal values compared with FDTD simulations. a Squarely and b hexagonal single diameter InP NW arrays B. Maximal Light Harvesting for Double Diameter InP NWs Adding a secondary diameter into the NW arrays has been investigated by several groups to further increase solar energy harvesting [22, 29] by time-consuming simulations [34]. From the above discussion, our method provides a way to fast approach the required NWs' diameters. The resonant wavelength of the supplementary NWs should match the absorption valley of the major diameter of NWs which is 585 nm as shown in Fig. 3b. Also, the NWs should support only one resonant mode. These two conclusions lead to D supplementary of 119 nm. The optimal FR of 0.2 still holds true in the double diameter InP NW arrays, and the periodicity is computed as 307 and 329.95 nm for squarely and hexagonal arrangement of NW arrays. Figure 6 provides an overview of the short-circuit current densities variation as a function of D major, D supplementary, and FR for two types of NW arrays. Generally, the light harvesting increases with FR, reaches its maximal value, and falls down. When FR is 0.2, the insets in Fig. 6 display the highest J sc of 32.96 and 32.95 mA/cm2 for both squarely and hexagonal InP NWs. Compared with the maximal values from simulations as 33.34 and 33.26 mA/cm2, the tolerances are 1.1 and 0.9% for squarely and hexagonal NWs. Figure 6 also shows as FR grows, the coupling among neighboring NWs cannot be overlooked. Power can transfer to the neighboring NWs who support the same leaky mode causing the competition of the incident energy [35] which is detrimental to overall light absorption. When the FR is the same for both arrangements, psquare 2/phexagonal 2 is $ \sqrt{3}/2 $. Therefore, the p hexagonal is 1.08 times of the p square which has less mode coupling among NWs than square arrays. This explains the differences of the light harvesting of the two arrays when FR is 0.05 and 0.4. Short-circuit current densities as a function of major, supplementary diameters, and FRs. a Squarely and b hexagonal InP NW arrays where the insets show optimal diameters for respective NW arrangements C. Maximal Light Harvesting for Four Diameter InP NWs Multiple diameters of NW arrays also attract lots of research interest to achieve near-unity absorption across the absorbing region [29]. However, only a limited number of diameter combinations are provided since the mass data acquisition requires large amount of time. This problem can be solved in our analytical design method, and four diameters of squarely arranged InP NW arrays are provided as an example. The total time taken to finish all of the calculations using our method equals to the time taken by only one FDTD simulation using the same personal computer. Upon acquisition of the major and supplementary diameters of NWs, the third and fourth diameters of NWs can be calculated in a similar way. The superposition of the normalized absorption efficiency of the major and supplementary diameter of NWs is shown in Fig. 7 with absorption valleys locating at 486 and 704 nm. Therefore, the third and fourth diameter of NWs can be computed to satisfy the conditions that each of them support only one mode, and the resonant wavelengths match the two absorption valleys in Fig. 7. Accordingly, the third and fourth diameters for InP NW arrays are obtained as 92 and 148 nm. With the optimal FR of 0.2 whose validity is irrespective of the arrangement of NW and diameters, the periodicity can be obtained as 277.41 nm for InP NW arrays. Superposition of the absorption efficiencies of the major and the supplementary diameters of InP NWs The light absorption spectrum for the optimal combination of four NWs is provided in Fig. 8 from which the near-unity light absorption is achieved by the well selection of individual NWs. FDTD simulation results with four diameters' combinations for squarely arranged NW arrays are shown in Fig. 9. To gain an overview of this multi-parameter optimization problem, two sets of coordinates are employed. The inner x and y axes denote the major and supplementary diameters whereas the outer x and y axes represent the third and fourth diameters. Due to the huge number of combinations of diameters, limited third and fourth diameters are deliberately selected to represent the whole absorption trend. From Fig. 3, the 80 nm is chosen as single mode resonance within NWs; 140 nm reflects the evolvement from single to double modes existence in NWs; 170 nm indicates the upper end of double modes existence while remain FWHM lying within absorbing region. Each intersect of the dash lines indicates different combination of the third and fourth diameters whereas the major and supplementary diameter run through 50 to 200 nm. When the diameters have larger values than 140 nm in Fig. 9, the majority of combinations of diameters will lead to the J sc above 30 mA/cm2. When all of the diameters reach above 170 nm, the average of J sc can be 32 mA/cm2. These results are also reflected in Figs. 5a and 6a. Compared with single or double diameter NW arrays, optimized four diameter NW arrays indeed lead to higher J sc. The highest J sc for four diameters InP NW arrays with our calculated geometrical dimensions is 33.13 mA/cm2 with a tolerance of 2.2%. Light absorption of four diameter InP NW arrays Short-circuit current densities change with the major, supplementary, third, and fourth InP NWs In this study, we present model for effective and fast design of both squarely and hexagonal InP NW arrays to achieve the highest light harvesting for photovoltaic application. Geometrical dimensions for vertically aligned single, double, and multiple diameters of NW arrays are investigated. Compared with time-consuming FDTD simulations, our predicted maximal short-circuit current densities with calculated three-dimensional NW arrays remain tolerances below 2.2% for all cases. For single diameter NW arrays, the optimal diameter is 184 nm which is only 4 nm difference to the reported highest efficiency InP NW solar cells. In the multiple diameter NW arrays, the diameters of the rest of NWs are optimized to satisfy the conditions that they support only one resonant mode and the corresponding wavelengths match the absorption valley of the major NWs. Moreover, the FR of the NW array is optimized to be 0.2 by creating an effective medium layer which is regardless of the diameter, periodicity, and arrangements of NWs. Compared with the optical modeling, the predicted highest short-circuit current densities for single diameter NW arrays lie within 0.33 and 0.1% tolerance for squarely and hexagonal NW array. The arrangements of NW array have little influence on the light absorption with optimal geometrical parameters, but the coupling among neighboring NWs becomes serious for multiple diameter NWs at large FR value. Squarely arranged four diameter NW arrays were also presented and the highest short-circuit current densities predicted to be 33.13 mA/cm2 with a low tolerance of 2.2%. 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Nanotechnology 26:295202 (295201-295208) This work was supported by the Academic Research Fund (RG97/14) of the Ministry of Education of Singapore; National Natural Science Foundation of China (51402148 and 11304147); National Key Research Project administrated by the Ministry of Science and Technology of China (2016YFB0401702), Guangdong High Tech Project (2014A010105005 and 2014TQ01C494); Shenzhen Innovation Project (KC2014JSQN0011A, JCYJ20150630145302223, JCYJ20150529152146471, and JCYJ20160301113537474) and Foshan Innovation Project (2014IT100072); and Shenzhen Key Laboratory Project (ZDSYS201602261933302). OPTIMUS, Photonics Centre of Excellence, School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore Dan Wu , Xiaohong Tang & Xianqiang Li Department of Electrical and Electronic Engineering, Southern University of Science and Technology, 1088 Xueyuan Avenue, 518055, Shenzhen, People's Republic of China Department of Materials Science and Engineering, Southern University of Science and Technology, 1088 Xueyuan Avenue, 518055, Shenzhen, People's Republic of China Zhubing He Search for Dan Wu in: Search for Xiaohong Tang in: Search for Zhubing He in: Search for Xianqiang Li in: DW drafted the manuscript and developed the algorithm. XHT supervised and coordinated the projects. KW and ZBH revised the manuscript. XQL helped with FDTD simulation. All authors read and approved the final manuscript. Correspondence to Xiaohong Tang or Kai Wang. Wu, D., Tang, X., Wang, K. et al. An Efficient and Effective Design of InP Nanowires for Maximal Solar Energy Harvesting. Nanoscale Res Lett 12, 604 (2017) doi:10.1186/s11671-017-2354-8 III–V semiconductor materials	CommonCrawl
A week later: Golden Sumatran, 3 spoonfuls, a more yellowish powder. (I combined it with some tea dregs to hopefully cut the flavor a bit.) Had a paper to review that night. No (subjectively noticeable) effect on energy or productivity. I tried 4 spoonfuls at noon the next day; nothing except a little mental tension, for lack of a better word. I think that was just the harbinger of what my runny nose that day and the day before was, a head cold that laid me low during the evening. Of course, there are drugs out there with more transformative powers. "I think it's very clear that some do work," says Andrew Huberman, a neuroscientist based at Stanford University. In fact, there's one category of smart drugs which has received more attention from scientists and biohackers – those looking to alter their own biology and abilities – than any other. These are the stimulants. Fortunately for me, the FDA decided Smart Powder's advertising was too explicit and ordered its piracetam sales stopped; I was equivocal at the previous price point, but then I saw that between the bulk discount and the fire-sale coupon, 3kg was only $99.99 (shipping was amortized over that, the choline, caffeine, and tryptophan). So I ordered in September 2010. As well, I had decided to cap my own pills, eliminating the inconvenience and bad taste. 3kg goes a very long way so I am nowhere close to running out of my pills; there is nothing to report since, as the pills are simply part of my daily routine. The fish oil can be considered a free sunk cost: I would take it in the absence of an experiment. The empty pill capsules could be used for something else, so we'll put the 500 at $5. Filling 500 capsules with fish and olive oil will be messy and take an hour. Taking them regularly can be added to my habitual morning routine for vitamin D and the lithium experiment, so that is close to free but we'll call it an hour over the 250 days. Recording mood/productivity is also free a sunk cost as it's necessary for the other experiments; but recording dual n-back scores is more expensive: each round is ~2 minutes and one wants >=5, so each block will cost >10 minutes, so 18 tests will be >180 minutes or >3 hours. So >5 hours. Total: 5 + (>5 \times 7.25) = >41. These days, young, ambitious professionals prefer prescription stimulants—including methylphenidate (usually sold as Ritalin) and Adderall—that are designed to treat people with attention deficit hyperactivity disorder (ADHD) and are more common and more acceptable than cocaine or nicotine (although there is a black market for these pills). ADHD makes people more likely to lose their focus on tasks and to feel restless and impulsive. Diagnoses of the disorder have been rising dramatically over the past few decades—and not just in kids: In 2012, about 16 million Adderall prescriptions were written for adults between the ages of 20 and 39, according to a report in the New York Times. Both methylphenidate and Adderall can improve sustained attention and concentration, says Barbara Sahakian, professor of clinical neuropsychology at the University of Cambridge and author of the 2013 book Bad Moves: How Decision Making Goes Wrong, and the Ethics of Smart Drugs. But the drugs do have side effects, including insomnia, lack of appetite, mood swings, and—in extreme cases—hallucinations, especially when taken in amounts the exceed standard doses. Take a look at these 10 foods that help you focus. Like caffeine, nicotine tolerates rapidly and addiction can develop, after which the apparent performance boosts may only represent a return to baseline after withdrawal; so nicotine as a stimulant should be used judiciously, perhaps roughly as frequent as modafinil. Another problem is that nicotine has a half-life of merely 1-2 hours, making regular dosing a requirement. There is also some elevated heart-rate/blood-pressure often associated with nicotine, which may be a concern. (Possible alternatives to nicotine include cytisine, 2'-methylnicotine, GTS-21, galantamine, Varenicline, WAY-317,538, EVP-6124, and Wellbutrin, but none have emerged as clearly superior.) (In particular, I don't think it's because there's a sudden new surge of drugs. FDA drug approval has been decreasing over the past few decades, so this is unlikely a priori. More specifically, many of the major or hot drugs go back a long time. Bacopa goes back millennia, melatonin I don't even know, piracetam was the '60s, modafinil was '70s or '80s, ALCAR was '80s AFAIK, Noopept & coluracetam were '90s, and so on.) As discussed in my iodine essay (FDA adverse events), iodine is a powerful health intervention as it eliminates cretinism and improves average IQ by a shocking magnitude. If this effect were possible for non-fetuses in general, it would be the best nootropic ever discovered, and so I looked at it very closely. Unfortunately, after going through ~20 experiments looking for ones which intervened with iodine post-birth and took measures of cognitive function, my meta-analysis concludes that: the effect is small and driven mostly by one outlier study. Once you are born, it's too late. But the results could be wrong, and iodine might be cheap enough to take anyway, or take for non-IQ reasons. (This possibility was further weakened for me by an August 2013 blood test of TSH which put me at 3.71 uIU/ml, comfortably within the reference range of 0.27-4.20.) Since the discovery of the effect of nootropics on memory and focus, the number of products on the market has increased exponentially. The ingredients used in a supplement can tell you about the effectiveness of the product. Brain enhancement pills that produce the greatest benefit are formulated with natural vitamins and substances, rather than caffeine and synthetic ingredients. In addition to better results, natural supplements are less likely to produce side effects, compared with drugs formulated with chemical ingredients. The absence of a suitable home for this needed research on the current research funding landscape exemplifies a more general problem emerging now, as applications of neuroscience begin to reach out of the clinical setting and into classrooms, offices, courtrooms, nurseries, marketplaces, and battlefields (Farah, 2011). Most of the longstanding sources of public support for neuroscience research are dedicated to basic research or medical applications. As neuroscience is increasingly applied to solving problems outside the medical realm, it loses access to public funding. The result is products and systems reaching the public with less than adequate information about effectiveness and/or safety. Examples include cognitive enhancement with prescription stimulants, event-related potential and fMRI-based lie detection, neuroscience-based educational software, and anti-brain-aging computer programs. Research and development in nonmedical neuroscience are now primarily the responsibility of private corporations, which have an interest in promoting their products. Greater public support of nonmedical neuroscience research, including methods of cognitive enhancement, will encourage greater knowledge and transparency concerning the efficacy and safety of these products and will encourage the development of products based on social value rather than profit value. A similar pill from HQ Inc. (Palmetto, Fla.) called the CorTemp Ingestible Core Body Temperature Sensor transmits real-time body temperature. Firefighters, football players, soldiers and astronauts use it to ensure that they do not overheat in high temperatures. HQ Inc. is working on a consumer version, to be available in 2018, that would wirelessly communicate to a smartphone app. Overall, the studies listed in Table 1 vary in ways that make it difficult to draw precise quantitative conclusions from them, including their definitions of nonmedical use, methods of sampling, and demographic characteristics of the samples. For example, some studies defined nonmedical use in a way that excluded anyone for whom a drug was prescribed, regardless of how and why they used it (Carroll et al., 2006; DeSantis et al., 2008, 2009; Kaloyanides et al., 2007; Low & Gendaszek, 2002; McCabe & Boyd, 2005; McCabe et al., 2004; Rabiner et al., 2009; Shillington et al., 2006; Teter et al., 2003, 2006; Weyandt et al., 2009), whereas others focused on the intent of the user and counted any use for nonmedical purposes as nonmedical use, even if the user had a prescription (Arria et al., 2008; Babcock & Byrne, 2000; Boyd et al., 2006; Hall et al., 2005; Herman-Stahl et al., 2007; Poulin, 2001, 2007; White et al., 2006), and one did not specify its definition (Barrett, Darredeau, Bordy, & Pihl, 2005). Some studies sampled multiple institutions (DuPont et al., 2008; McCabe & Boyd, 2005; Poulin, 2001, 2007), some sampled only one (Babcock & Byrne, 2000; Barrett et al., 2005; Boyd et al., 2006; Carroll et al., 2006; Hall et al., 2005; Kaloyanides et al., 2007; McCabe & Boyd, 2005; McCabe et al., 2004; Shillington et al., 2006; Teter et al., 2003, 2006; White et al., 2006), and some drew their subjects primarily from classes in a single department at a single institution (DeSantis et al., 2008, 2009; Low & Gendaszek, 2002). With few exceptions, the samples were all drawn from restricted geographical areas. Some had relatively high rates of response (e.g., 93.8%; Low & Gendaszek 2002) and some had low rates (e.g., 10%; Judson & Langdon, 2009), the latter raising questions about sample representativeness for even the specific population of students from a given region or institution. Four of the studies focused on middle and high school students, with varied results. Boyd, McCabe, Cranford, and Young (2006) found a 2.3% lifetime prevalence of nonmedical stimulant use in their sample, and McCabe, Teter, and Boyd (2004) found a 4.1% lifetime prevalence in public school students from a single American public school district. Poulin (2001) found an 8.5% past-year prevalence in public school students from four provinces in the Atlantic region of Canada. A more recent study of the same provinces found a 6.6% and 8.7% past-year prevalence for MPH and AMP use, respectively (Poulin, 2007). Two studies investigated the effects of MPH on reversal learning in simple two-choice tasks (Clatworthy et al., 2009; Dodds et al., 2008). In these tasks, participants begin by choosing one of two stimuli and, after repeated trials with these stimuli, learn that one is usually rewarded and the other is usually not. The rewarded and nonrewarded stimuli are then reversed, and participants must then learn to choose the new rewarded stimulus. Although each of these studies found functional neuroimaging correlates of the effects of MPH on task-related brain activity (increased blood oxygenation level-dependent signal in frontal and striatal regions associated with task performance found by Dodds et al., 2008, using fMRI and increased dopamine release in the striatum as measured by increased raclopride displacement by Clatworthy et al., 2009, using PET), neither found reliable effects on behavioral performance in these tasks. The one significant result concerning purely behavioral measures was Clatworthy et al.'s (2009) finding that participants who scored higher on a self-report personality measure of impulsivity showed more performance enhancement with MPH. MPH's effect on performance in individuals was also related to its effects on individuals' dopamine activity in specific regions of the caudate nucleus. For instance, they point to the U.S. Army's use of stimulants for soldiers to stave off sleep and to stay sharp. But the Army cares little about the long-term health effects of soldiers, who come home scarred physically or mentally, if they come home at all. It's a risk-benefit decision for the Army, and in a life-or-death situation, stimulants help. That first night, I had severe trouble sleeping, falling asleep in 30 minutes rather than my usual 19.6±11.9, waking up 12 times (5.9±3.4), and spending ~90 minutes awake (18.1±16.2), and naturally I felt unrested the next day; I initially assumed it was because I had left a fan on (moving air keeps me awake) but the new potassium is also a possible culprit. When I asked, Kevin said: Take at 11 AM; distractions ensue and the Christmas tree-cutting also takes up much of the day. By 7 PM, I am exhausted and in a bad mood. While I don't expect day-time modafinil to buoy me up, I do expect it to at least buffer me against being tired, and so I conclude placebo this time, and with more confidence than yesterday (65%). I check before bed, and it was placebo. Nicotine absorption through the stomach is variable and relatively reduced in comparison with absorption via the buccal cavity and the small intestine. Drinking, eating, and swallowing of tobacco smoke by South American Indians have frequently been reported. Tenetehara shamans reach a state of tobacco narcosis through large swallows of smoke, and Tapirape shams are said to eat smoke by forcing down large gulps of smoke only to expel it again in a rapid sequence of belches. In general, swallowing of tobacco smoke is quite frequently likened to drinking. However, although the amounts of nicotine swallowed in this way - or in the form of saturated saliva or pipe juice - may be large enough to be behaviorally significant at normal levels of gastric pH, nicotine, like other weak bases, is not significantly absorbed. While the primary effect of the drug is massive muscle growth the psychological side effects actually improved his sanity by an absurd degree. He went from barely functional to highly productive. When one observes that the decision to not attempt to fulfill one's CEV at a given moment is a bad decision it follows that all else being equal improved motivation is improved sanity. This tendency is exacerbated by general inefficiencies in the nootropics market - they are manufactured for vastly less than they sell for, although the margins aren't as high as they are in other supplement markets, and not nearly as comical as illegal recreational drugs. (Global Price Fixing: Our Customers are the Enemy (Connor 2001) briefly covers the vitamin cartel that operated for most of the 20th century, forcing food-grade vitamins prices up to well over 100x the manufacturing cost.) For example, the notorious Timothy Ferriss (of The Four-hour Work Week) advises imitators to find a niche market with very high margins which they can insert themselves into as middlemen and reap the profits; one of his first businesses specialized in… nootropics & bodybuilding. Or, when Smart Powders - usually one of the cheapest suppliers - was dumping its piracetam in a fire sale of half-off after the FDA warning, its owner mentioned on forums that the piracetam was still profitable (and that he didn't really care because selling to bodybuilders was so lucrative); this was because while SP was selling 2kg of piracetam for ~$90, Chinese suppliers were offering piracetam on AliBaba for $30 a kilogram or a third of that in bulk. (Of course, you need to order in quantities like 30kg - this is more or less the only problem the middlemen retailers solve.) It goes without saying that premixed pills or products are even more expensive than the powders. The beneficial effects as well as the potentially serious side effects of these drugs can be understood in terms of their effects on the catecholamine neurotransmitters dopamine and norepinephrine (Wilens, 2006). These neurotransmitters play an important role in cognition, affecting the cortical and subcortical systems that enable people to focus and flexibly deploy attention (Robbins & Arnsten, 2009). In addition, the brain's reward centers are innervated by dopamine neurons, accounting for the pleasurable feelings engendered by these stimulants (Robbins & Everett, 1996). However, when I didn't stack it with Choline, I would get what users call "racetam headaches." Choline, as Patel explains, is not a true nootropic, but it's still a pro-cognitive compound that many take with other nootropics in a stack. It's an essential nutrient that humans need for functions like memory and muscle control, but we can't produce it, and many Americans don't get enough of it. The headaches I got weren't terribly painful, but they were uncomfortable enough that I stopped taking Piracetam on its own. Even without the headache, though, I didn't really like the level of focus Piracetam gave me. I didn't feel present when I used it, even when I tried to mix in caffeine and L-theanine. And while it seemed like I could focus and do my work faster, I was making more small mistakes in my writing, like skipping words. Essentially, it felt like my brain was moving faster than I could. Some supplement blends, meanwhile, claim to work by combining ingredients – bacopa, cat's claw, huperzia serrata and oat straw in the case of Alpha Brain, for example – that have some support for boosting cognition and other areas of nervous system health. One 2014 study in Frontiers in Aging Neuroscience, suggested that huperzia serrata, which is used in China to fight Alzheimer's disease, may help slow cell death and protect against (or slow the progression of) neurodegenerative diseases. The Alpha Brain product itself has also been studied in a company-funded small randomized controlled trial, which found Alpha Brain significantly improved verbal memory when compared to adults who took a placebo. Theanine can also be combined with caffeine as both of them work in synergy to increase memory, reaction time, mental endurance, and memory. The best part about Theanine is that it is one of the safest nootropics and is readily available in the form of capsules. A natural option would be to use an excellent green tea brand which constitutes of tea grown in the shade because then Theanine would be abundantly present in it. The chemicals he takes, dubbed nootropics from the Greek "noos" for "mind", are intended to safely improve cognitive functioning. They must not be harmful, have significant side-effects or be addictive. That means well-known "smart drugs" such as the prescription-only stimulants Adderall and Ritalin, popular with swotting university students, are out. What's left under the nootropic umbrella is a dizzying array of over-the-counter supplements, prescription drugs and unclassified research chemicals, some of which are being trialled in older people with fading cognition. The FDA has approved the first smart pill for use in the United States. Called Abilify MyCite, the pill contains a drug and an ingestible sensor that is activated when it comes into contact with stomach fluid to detect when the pill has been taken. The pill then transmits this data to a wearable patch that subsequently transfers the information to an app on a paired smartphone. From that point, with a patient's consent, the data can be accessed by the patient's doctors or caregivers via a web portal. Several studies have assessed the effect of MPH and d-AMP on tasks tapping various other aspects of spatial working memory. Three used the spatial working memory task from the CANTAB battery of neuropsychological tests (Sahakian & Owen, 1992). In this task, subjects search for a target at different locations on a screen. Subjects are told that locations containing a target in previous trials will not contain a target in future trials. Efficient performance therefore requires remembering and avoiding these locations in addition to remembering and avoiding locations already searched within a trial. Mehta et al. (2000) found evidence of greater accuracy with MPH, and Elliott et al. (1997) found a trend for the same. In Mehta et al.'s study, this effect depended on subjects' working memory ability: the lower a subject's score on placebo, the greater the improvement on MPH. In Elliott et al.'s study, MPH enhanced performance for the group of subjects who received the placebo first and made little difference for the other group. The reason for this difference is unclear, but as mentioned above, this may reflect ability differences between the groups. More recently, Clatworthy et al. (2009) undertook a positron emission tomography (PET) study of MPH effects on two tasks, one of which was the CANTAB spatial working memory task. They failed to find consistent effects of MPH on working memory performance but did find a systematic relation between the performance effect of the drug in each individual and its effect on individuals' dopamine activity in the ventral striatum. Specifically, the film is completely unintelligible if you had not read the book. The best I can say for it is that it delivers the action and events one expects in the right order and with basic competence, but its artistic merits are few. It seems generally devoid of the imagination and visual flights of fancy that animated movies 1 and 3 especially (although Mike Darwin disagrees), copping out on standard imagery like a Star Wars-style force field over Hogwarts Castle, or luminescent white fog when Harry was dead and in his head; I was deeply disappointed to not see any sights that struck me as novel and new. (For example, the aforementioned dead scene could have been done in so many interesting ways, like why not show Harry & Dumbledore in a bustling King's Cross shot in bright sharp detail, but with not a single person in sight and all the luggage and equipment animatedly moving purposefully on their own?) The ending in particular boggles me. I actually turned to the person next to me and asked them whether that really was the climax and Voldemort was dead, his death was so little dwelt upon or laden with significance (despite a musical score that beat you over the head about everything else). In the book, I remember it feeling like a climactic scene, with everyone watching and little speeches explaining why Voldemort was about to be defeated, and a suitable victory celebration; I read in the paper the next day a quote from the director or screenwriter who said one scene was cut because Voldemort would not talk but simply try to efficiently kill Harry. (This is presumably the explanation for the incredible anti-climax. Hopefully.) I was dumbfounded by the depths of dishonesty or delusion or disregard: Voldemort not only does that in Deathly Hallows multiple times, he does it every time he deals with Harry, exactly as the classic villains (he is numbered among) always do! How was it possible for this man to read the books many times, as he must have, and still say such a thing?↩ 1 PM; overall this was a pretty productive day, but I can't say it was very productive. I would almost say even odds, but for some reason I feel a little more inclined towards modafinil. Say 55%. That night's sleep was vile: the Zeo says it took me 40 minutes to fall asleep, I only slept 7:37 total, and I woke up 7 times. I'm comfortable taking this as evidence of modafinil (half-life 10 hours, 1 PM to midnight is only 1 full halving), bumping my prediction to 75%. I check, and sure enough - modafinil. Regardless, while in the absence of piracetam, I did notice some stimulant effects (somewhat negative - more aggressive than usual while driving) and similar effects to piracetam, I did not notice any mental performance beyond piracetam when using them both. The most I can say is that on some nights, I seemed to be less easily tired when writing or editing or n-backing (and I felt less tired than ICON 2011 than ICON 2010), but those were also often nights I was also trying out all the other things I had gotten in that order from Smart Powders, and I am still dis-entangling what was responsible. (Probably the l-theanine or sulbutiamine.) First was a combination of L-theanine and aniracetam, a synthetic compound prescribed in Europe to treat degenerative neurological diseases. I tested it by downing the recommended dosages and then tinkering with a story I had finished a few days earlier, back when caffeine was my only performance-enhancing drug. I zoomed through the document with renewed vigor, striking some sentences wholesale and rearranging others to make them tighter and punchier. Or in other words, since the standard deviation of my previous self-ratings is 0.75 (see the Weather and my productivity data), a mean rating increase of >0.39 on the self-rating. This is, unfortunately, implying an extreme shift in my self-assessments (for example, 3s are ~50% of the self-ratings and 4s ~25%; to cause an increase of 0.25 while leaving 2s alone in a sample of 23 days, one would have to push 3s down to ~25% and 4s up to ~47%). So in advance, we can see that the weak plausible effects for Noopept are not going to be detected here at our usual statistical levels with just the sample I have (a more plausible experiment might use 178 pairs over a year, detecting down to d>=0.18). But if the sign is right, it might make Noopept worthwhile to investigate further. And the hardest part of this was just making the pills, so it's not a waste of effort. Most people would describe school as a place where they go to learn, so learning is an especially relevant cognitive process for students to enhance. Even outside of school, however, learning plays a role in most activities, and the ability to enhance the retention of information would be of value in many different occupational and recreational contexts. The data from 2-back and 3-back tasks are more complex. Three studies examined performance in these more challenging tasks and found no effect of d-AMP on average performance (Mattay et al., 2000, 2003; Mintzer & Griffiths, 2007). However, in at least two of the studies, the overall null result reflected a mixture of reliably enhancing and impairing effects. Mattay et al. (2000) examined the performance of subjects with better and worse working memory capacity separately and found that subjects whose performance on placebo was low performed better on d-AMP, whereas subjects whose performance on placebo was high were unaffected by d-AMP on the 2-back and impaired on the 3-back tasks. Mattay et al. (2003) replicated this general pattern of data with subjects divided according to genotype. The specific gene of interest codes for the production of Catechol-O-methyltransferase (COMT), an enzyme that breaks down dopamine and norepinephrine. A common polymorphism determines the activity of the enzyme, with a substitution of methionine for valine at Codon 158 resulting in a less active form of COMT. The met allele is thus associated with less breakdown of dopamine and hence higher levels of synaptic dopamine than the val allele. Mattay et al. (2003) found that subjects who were homozygous for the val allele were able to perform the n-back faster with d-AMP; those homozygous for met were not helped by the drug and became significantly less accurate in the 3-back condition with d-AMP. In the case of the third study finding no overall effect, analyses of individual differences were not reported (Mintzer & Griffiths, 2007). Omega-3 fatty acids: DHA and EPA – two Cochrane Collaboration reviews on the use of supplemental omega-3 fatty acids for ADHD and learning disorders conclude that there is limited evidence of treatment benefits for either disorder.[42][43] Two other systematic reviews noted no cognition-enhancing effects in the general population or middle-aged and older adults.[44][45]	CommonCrawl
\begin{document} \title[Effective joint dstribution of eigenvalues \dots]{Effective joint distribution of eigenvalues of Hecke operators} \author[Sudhir Pujahari]{Sudhir Pujahari} \address{Sudhir Pujahari, Harish-Chandra Research Institute (HBNI), Chatnag Road, Jhunsi, Allahabad - 211019, Uttar Pradesh, India} \email{[email protected]} \subjclass[2000]{Primary 34L20, 11S40, Secondary 11R42} \keywords{Equidistribution, Hecke operators, Sato-Tate conjecture, Eichler-Selberg trace formula} \title{Distribution of gaps of eigenangels of Hecke operators} \begin{abstract} In 1997, Serre proved that the eigenvalues of normalised $p$-th Hecke operator $T^{'}_p$ acting on the space of cusp forms of weight $k$ and level $N$ are equidistributed in $[-2,2]$ with respect to a measure that converge to the Sato-Tate measure, whenever $N+k \to \infty$. In 2009, Murty and Sinha proved the effective version of Serre's theorem. In 2011, using Kuznetsov trace formula, Lau and Wang derived the effective joint distribution of eigenvalues of normalized Hecke operators acting on the space of primitive cusp forms of weight $k$ and level $1$. In this paper, we extend the result of Lau and Wang to space of cusp forms of higher level. Here we use Eichler-Selberg trace formula instead of Kuznetsov trace formula to deduce our result. \end{abstract} \section{Introduction} Let $S(N,k)$ be the space of all holomorphic cusp forms of weight $k$ with respect to $\varGamma_0(N).$ For any positive integer $n$, let $T_n(N,k)$ be the $n$-th Hecke operator acting on $S(N,k)$. Let $s(N,k)$ denote the dimension of the space $S(N,k)$. For a positive integer $n \geq 1$, let $$ \{\lambda_{i}(n)\}, 1 \leq i \leq s(N,k) $$ denote the eigenvalues of $T_n$, counted with multiplicity. For any positive integer $n$, let $$ T_n^{'}:= \frac{T_n}{n^{\frac{k-1}{2}}} $$ be the normalized Hecke operator acting on $S(N,k)$ with eigenvalues $$ \left \{ a_i(n)=\frac{\lambda_i(n)}{n^{\frac{k-1}{2}}}, 1 \leq i \leq s(N,k)\right\},$$ counted with multiplicity. By the celebrated theorem of Deligne~\cite{Deligne}, which proves the famous Ramanujan conjecture, we know that for any prime $p$, such that $p$ coprime to $N$, the eigenvalue of $T_p^{'}$ lies in the interval $[-2,2]$. The recently proved Sato-Tate conjecture by Barnet-Lamb, Geraghty, Harris and Taylor~\cite{BL},~\cite{LMR} and~\cite{MNR} says that if $a_i(p), 1\leq i \leq r$ is a $p$-th normalized Hecke eigenvalue, then the family $\{a_i(p)\}$ is equidistributed in $[-2,2]$ as $p \to \infty$ with respect to the Sato-Tate measure $$d\mu_{\infty}=\frac{1}{2 \pi} \sqrt {4-x^2}dx.$$ More precisely, the Sato-Tate conjecture states that for any continuous function $\phi: \mathbb{R} \rightarrow \mathbb{R}$, positive integer $V$ and any interval $[\alpha, \beta] \subset [-2,2]$ $$\lim_{V \to \infty} \frac{1}{V} \sum_{p \leq V} \phi \left(\frac{a_i(p)}{p^{\frac{k-1}{2}}} \right)= \int_{\alpha}^{\beta} \phi d \mu_{\infty}.$$ In 1997, Serre~\cite{Serre} studied the ``vertical" Sato-Tate conjecture by fixing a prime $p$ and varying $N$ and $k.$ In particular, he proved the following theorem: \begin{thm} Let $p$ be a prime number. Let $\{(N,k)\}$ be a sequence of pairs of positive integers such that $k$ is even and $p$ is coprime to $N$ and $N+k \to \infty.$ Then the family of eigenvalues of the normalized $p$-th Hecke operator $$T_{p}^{'}(N,k)= \frac{T_{p}(N,k)}{p^{\frac{k-1}{2}}}$$ is equidistributed in the interval $\Omega=[-2,2]$ with respect to the measure $$\mu_p:=\frac{p+1}{\pi}\frac{\sqrt{1-\frac{x^2}{4}}}{(p^{\frac{1}{2}}+p^{-\frac{1}{2}})^2-x^2}dx$$ \end{thm} \begin{remark} Also in 1997, Conrey, Duke and Farmer~\cite{CDF} studied a special case of above result by fixing $N=1.$ \end{remark} In 2009, Murty and Sinha~\cite{MS} investigated the effective/quantitative version of Serre's results, in which they give explicit estimate on the rate of convergence. They proved the following theorem \begin{thm} Let p be a fixed prime. Let $\{(N,k)\}$ be a sequence of pairs of positive integers such that $k$ is even, $p$ is coprime to $N$. For an interval $[\alpha,\beta] \subset [-2,2],$ $$\frac{1}{s(N,k)} \sharp \left \{1 \leq i \leq s(N,k):a_{i}(p) \in [\alpha,\beta] \right \} =\int_{\alpha}^{\beta}\mu_{p} +\operatorname{O} \left(\frac{\operatorname{log} \,p}{\operatorname{log} \,kN}\right).$$ \end{thm} As a continuation of their paper, in 2010, Murty and Sinha~\cite{MS} proved a quantitative equidistribution theorem for the eigenvalues of Hecke operators acting on the space $S^{new}(N,k)$. Recently Lau and Wang~\cite{LY} computed the rate of convergence in Sarnak's~\cite{Sarnak} result using the Kuznetsov trace formula. Indeed they proved the joint distribution of eigenvalues of the Hecke operators quantitatively for primitive Maass forms of level 1 and stated that the same hold true for primitive holomorphic cusp forms. More precisely, they proved the following theorem: \begin{thm} Let $p_1,p_2,,...,$ and $p_r$ be distinct primes. Let $k$ be a positive even integer such that $r \operatorname{log} \,(p_1p_2 ,..., p_r) \leq \delta \operatorname{log} \, k$, for some small absolutely constant $\delta$. Let $a_i^{'}(p_i)$ be the eigenvalues of normalized Hecke operators $T_{p_i}^{'}$ acting on $S^{new}(1,k)$. For any $I=[\alpha_n,\beta_n]^r \subset [-2,2]^r$ \begin{eqnarray} && \frac{ \sharp \left\{1 \leq n \leq s(1,k): \left( a_{1}^{(n)}(p_1),\ldots , a_{r}^{(n)}(p_r) \right) \in I \right\}} {s^{new}(1,k)} \\ && = \int_I \prod_{n=1}^rd \mu_{p_n} + \operatorname{O} \left(\frac{r\operatorname{log} \,(p_1p_2 \dots p_r)}{\operatorname{log} \,k} \right), \end{eqnarray} where $\displaystyle d\mu_{p_n}=\frac{p_n+1}{\pi}\frac{\sqrt{1-\frac{x^2}{4}}}{\left({p_n}^{\frac{1}{2}}+{p_n}^{-\frac{1}{2}}\right)^2-x^2}dx.$ \end{thm} They have remarked that their methods do work for primitive forms in higher level. In this paper we extend their result to the cusp forms of any level $N$ using ``Eichler-Selberg trace formula". Precisely, we prove the following theorem: \begin{thm}\label{T1} Let $p_1,p_2,...,$ and $p_r$ be distinct primes. Let $k$ be positive even integer such that $r\operatorname{log} \,(p_1p_2 ,..., p_r) \leq \delta \operatorname{log} k$, for some small absolutely constant $\delta$. For $1 \leq i \leq r,$ let $a_i^{(n)}(p_i), 1\leq n \leq s(N,k)$ be the eigenvalues of normalized Hecke operators $T_{p_i}^{'}$ acting on $S(N,k)$. For any $I=[\alpha_n,\beta_n]^r \subset [-2,2]^r$ \begin{eqnarray} &&\frac{\sharp \left \{1\leq n \leq s(N,k): \left(a_{1}^{(n)}(p_1), \ldots , a_{r}^{(n)}(p_r) \right) \in I \right \}} {s(N,k)} \\ && =\int_I \prod_{n=1}^rd \mu_{p_n} + \operatorname{O} \left(\frac{r\operatorname{log} \,(p_1p_2 \cdots p_r)}{\operatorname{log} \,(kN)} \right), \end{eqnarray} where $\displaystyle d\mu_{p_n}=\frac{p_n+1}{\pi}\frac{\sqrt{1-\frac{x^2}{4}}}{({p_n}^{\frac{1}{2}}+{p_n}^{-\frac{1}{2}})^2-x^2}dx$ and the implied constant is effectively computable. \end{thm} \section{ Equidistribution and its extension}\label{S2} A sequence of real numbers $\{x_n\}_{n=1}^{\infty}$ is said to be uniformly distributed or (equidistributed) (\rm mod \ 1) if for any interval $[\alpha,\beta] \subset [0,1],$ we have $$ \lim_{V \to \infty} \frac{1}{V} \sharp \{n \leq V: x_n \in [\alpha, \beta]\}=\beta-\alpha. $$ In 1916, Weyl~\cite{Weyl} proved the following criterion for uniform distribution (\rm mod \ 1). A sequence $\{x_n\}_{n=1}^{\infty}$ is uniformly distributed if and only if for any integer $m \neq 0$ $$ \sum_{n \leq V} e(mx_n)=\operatorname{o}(V) \ \text{as $V \to \infty$}. $$ Since the set of trigonometric polynomials is dense in $C^{1}[0,1]$, the above criterion of Weyl is equivalent to the assertion that, for any continuous function $f: \mathbb R \rightarrow \mathbb R$, we have $$ \lim_{V \to \infty} \frac{1}{V} \sum_{n \leq V}f(x_n)= \int_{0}^1f(t)dt. $$ A sequence of tuples $\left \{(x_{1}^{(n)},x_{2}^{(n)}, \ldots ,x_{r}^{(n)})\right \}$ in ${\mathbb R}^r$ is said to be uniformly distributed or (equidistributed) (\rm mod \ 1) if for every $I=\prod_{n=1}^r [\alpha_n, \beta_n] \subset [0,1]^r$, we have $$ \lim_{V \to \infty} \frac{\sharp \{n \leq V:(x_{1}^{(n)},x_{2}^{(n)}, \ldots , x_{r}^{(n)}) \in I \}}{V} = \mu(I), $$ where $\mu$ is the usual Lebesgue measure on $\mathbb R^{r}$. In the same paper Weyl~\cite{Weyl} extended his criterion of equidistribution to the higher dimension as follows: \\ A sequence of tuples $\left\{(x_{1}^{(n)},x_{2}^{(n)}, \ldots , x_{r}^{(n)})\right\}$ in $\mathbb{R}^r$ is uniformly distributed if and only if for any integers $m_1,m_2,,...,, m_r \neq 0$ $$ \sum_{n=1}^{V}e \left(\sum_{i=1}^r m_ix_n \right) = \operatorname{o}(V) \ \mbox{ as } V \to \infty. $$ Note that as in the one variable case, the set of trigonometric polynomials is also dense in $C([0,1]^r),$ the above criterion is equivalent to the following statement: \\ A sequence of tuples $\left\{(x_{1}^{(n)},x_{2}^{(n)}, \ldots , x_{r}^{(n)})\right\}$ in ${\mathbb R}^r$ is uniformly distributed if and only if for any continuous function $f: {\mathbb R}^r \rightarrow \mathbb R,$ $$ \lim_{V \to \infty} \frac{1}{V}\sum_{n \leq V}f(x_{1}^{(n)},x_{2}^{(n)}, \ldots ,x_{r}^{(n)}) =\int_{I}f(x_{1},x_{2}, \ldots ,x_{r}) dx_{1}dx_{2} \cdots dx_{r}. $$ Now we define the set equidistribution as follows: \\ A sequence of finite multi sets $A_n$ with $\sharp A_n \to \infty$ is said to be set equidistributed (\rm mod \ 1) with respect to a probability measure $\mu$ if for every continuous function $f \in C^{1}[0,1]$, we have $$ \lim_{n \to \infty } \frac{1}{\sharp A_n} \sum_{t \in A_n}f(t)= \int_{0}^1 f(x)d \mu. $$ The criterion of Schoenberg and Wiener says that the sequence $\{A_n\}$ is set equidistributed with respect to some positive continuous measure if and only if the Weyl limit $$c_m := \lim_{n \to \infty} \frac{1}{\sharp A_n} \sum_{t \in A_n}e(mt)$$ exists and $$ \sum_{m=1}^{V}\|c_m\|^2 = \operatorname{o}(V). $$ For our purpose, let us define the set equidistribution in higher dimension. A tuples of finite multi set say $ \Omega =(A_{1}^{(n)},A_{2}^{(n)}, \ldots , A_{r}^{(n)})$ with $\sharp A_{i}^{(n)} \to \infty$ as $n \to \infty$ for all $ i=1,2,,...,,r $ is said to be set a set equidistributed (\rm mod \ 1) with respect to a probability measure $\mu$ if for every continuous function $f \in C^{1}([0,1]^r)$, and $I=[0,1]^r$ we have $$ \lim_{V \to \infty} \frac{1}{V} \sum_{n \leq V} f(x_{1},x_{2}, \ldots ,x_{r}) =\int_{I}f(x_{1},x_{2}, \ldots ,x_{r}) dx_{1}dx_{2} \cdots dx_{r}. $$ With this generalization, we define the Weyl limit as $$ C_{\underline m}:= \lim_{n \to \infty} \frac{1}{\prod_{i=1}^{r}\sharp A_n} \sum_{(t_1,t_2, \ldots ,t_r) \in \Omega}e(m_1t_1+m_2t_2+\cdots+m_rt_r). $$ \section{Eichler Selberg Trace Formula and its Estimations} In this section, we use Eichler-Selberg trace formula, as one of our important tool to prove the main theorem, which is a formula for the trace of $T_n$ acting on $S(N,k) $ in terms of class number of binary quadratic forms and few others arithmetic functions. We follow the presentation of~\cite{MS}. For a non-negative integer $\vartriangle \equiv 0,1 $ (\rm mod \ 4), let $B(\vartriangle)$ be the set of all positive definite binary quadratic forms with discriminant $\vartriangle$. That is $$B(\vartriangle)= \{ax^2+bxy+cy^2 : a,b,c \in \mathbb Z, a>0, b^2-4ac=\vartriangle\}.$$ We denote the set of primitive forms by $$b(\vartriangle)= \{f(x,y) \in B(\vartriangle): gcd(a,b,c)=1\}.$$ Now we define an action of the full modular group $SL_2(\mathbb Z)$ on $B(\vartriangle)$ as follows: $$f(x,y) \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}:= f(\alpha x+ \beta y, \gamma x + \delta y).$$ Note that the above action takes primitive forms to primitive forms. We know that the above action has finitely many orbits. We define $h(\vartriangle)$ to be the number of orbits of $b(\vartriangle)$. Let $h_w$ be defined as follows: \begin{eqnarray} h_w(-3) & = & \frac{1}{3}, \\ h_w(-4) & = & \frac{1}{2}, \\ h_w(\vartriangle) & = & h(\vartriangle) \ \text {for $\vartriangle < -4$}. \end{eqnarray} We define some arithmetical functions which are going to useful to state the Eichler-Selberg trace formula. Let \begin{eqnarray} A_1(n) = \left\{ \begin{array}{l l} n^{\frac{k}{2}-1}\left(\frac{k-1}{12} \right) \psi(N) & \text{if $n$ is a square} \\ 0 & \text{otherwise,} \end{array} \right. \end{eqnarray} where $\displaystyle\psi(N) = N \prod_{p \| N}\left(1 + \frac{1}{p}\right)$. Let $$ A_2(n)=- \frac{1}{2} \sum_{{t \in \mathbb Z} \atop{ t^2 < 4n}} \frac{\varrho^{k-1}-{\bar \varrho}^{k-1}}{\varrho-\bar \varrho} \sum_{g} h_w \left( \frac{t^2-4n}{g^2} \right) \mu(t,g,n), $$ where $\varrho$ and $\bar \varrho$ are the zeros of the polynomial $x^2-tx+n$, the inner sum runs over positive divisors $g$ of $\frac{(t^2-4n)}{g^2} \in \mathbb Z $ is congruent to $0$ or $1 \ (mod \ 4)$ and $\mu(t,g,n)$ is given by $$\mu(t,g,n)= \frac{\psi (N)}{\psi (\frac{N}{N_g})}M(t,n,NN_g),$$ with $N_g=gcd(N,g)$ and $M(t,n,k)$ denote the number of solutions of the congruence $x^2-tx+n \equiv 0 \ \ \pmod{ K}$. Now, we let \begin{equation} A_3(n)=- \sum_{{d\|n} \atop{ 0 <d \leq \sqrt{n}}}d^{k-1} \sum_{c\|N} \phi \left( gcd \left(c, \frac {N}{c}\right)\right), \end{equation} where $\phi$ denotes the Euler's function and in the first summation, if there is a contribution from the term $d= \sqrt{n}$, it should be multiplied by $\frac{1}{2}.$ In the inner sum, we also need the condition that $gcd \ (c, \frac{N}{c})$ divides $gcd(N,\frac{n}{d}-d).$ Finally, let \begin{eqnarray} A_4(n)= \displaystyle\left\{ \begin{array} {l l} {\sum \atop {{t\|n \atop t > 0}}}t & \text{if $k=0$} \\ 0 & \text {otherwise}. \end{array} \right. \end{eqnarray} \begin{thm} For any positive integer $n$, let $Tr(T_n)$ be the trace of $T_n$ acting on $S(N,k).$ Then, we have, $$Tr (T_n)=A_1(n)+A_2(n)+A_3(n)+A_4(n).$$ \end{thm} We use the above Eichler-Selberg trace formula to prove the following Proposition: \begin{prop}\label{P1} For any positive integers $m_1, m_2, \ldots ,m_r$ we have, \begin{align} \|Tr \ (T_{p_1^{m_1} \cdots p_r^{m_r}})\| &\ll p_1^{\frac{3m_1}{2}}\cdots p_r^{\frac{3m_r}{2}}(m_1 \cdots m_r)d(N)\sqrt{N}\operatorname{log} \,(4(p_1^{m_1} \cdots p_r^{m_r})). \end{align} \end{prop} \begin{proof} Consider \begin{eqnarray} Tr ( T^{'}_{p_1^{m_1} \cdots p_r^{m_r}}) & = & B_1(m_1m_2 \cdots m_r)+B_2(m_1m_2 \cdots m_r)+B_3(m_1m_2 \cdots m_r) \\ && +B_4(m_1m_2 \cdots m_r), \end{eqnarray} where $B_i(m_1m_2 \cdots m_r)= \displaystyle\frac{A_i(p_1^{m_1} \cdots p_r^{m_r})}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}}$, for all $i=1,2,3,4.$ We use the estimates of~\cite{MS} and get the estimates for each $B_i(m_1m_2 \cdots m_r),$ for all $i=1,2,3,4.$ First, we consider $i=1$ \begin{align} B_1(p_1^{m_1} \cdots p_r^{m_r}) = & \frac{A_1(p_1^{m_1} \cdots p_r^{m_r})}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}} \\ = & \left\{ \begin{array}{l l} \frac{\left({p_1}^{-\frac{m_1}{2}} \cdots {p_r}^{-\frac{m_r}{2}} \right) \frac{k-1}{12}}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}} & \text{if $m_1,m_2,,...,, m_r$ are even,} \\ 0 & \text{ otherwise.} \end{array} \right. \\ = & \left\{ \begin{array}{l l} (p_1^{-\frac{m_1}{2}}\cdots p_r^{-\frac{m_r}{2}})\frac{k-1}{12} & \text{if $m_1,m_2,...,m_r$ are even,} \\ 0 & \text{ otherwise.} \end{array} \right. \end{align} Now, we shall consider $i=2$ as follows. \begin{eqnarray} B_2(p_1^{m_1} \cdots p_r^{m_r}) & = & \frac{A_2(p_1^{m_1} \cdots p_r^{m_r})}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}} \\ & = & \frac{8e}{\operatorname{log} \,2}(2)p_1^{\frac{3m_1}{2}}\cdots p_r^{\frac{3m_r}{2}}2^{\nu(N)}\operatorname{log} \,4(p_1^{m_1} \cdots p_r^{m_r}). \end{eqnarray} For $i=3$, we get, \begin{align} B_3(p_1^{m_1} \cdots p_r^{m_r}) = & \frac{A_3(p_1^{m_1} \cdots p_r^{m_r})d(N)\sqrt{N}}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}} \\ \leq & \frac{d(p_1^{m_1} \cdots p_r^{m_r})(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}d(N)\sqrt{N}}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}} \\ \leq & (m_1+1)\cdots(m_r+1)d(N)\sqrt{N}. \end{align} Finally, for $i=4$, we get, \begin{align} B_4(p_1^{m_1} \cdots p_r^{m_r}) = & \frac{A_4(p_1^{m_1} \cdots p_r^{m_r})}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}} = \left\{ \begin{array}{l l} \frac{d(p_1^{m_1} \cdots p_r^{m_r})(p_1^{m_1} \cdots p_r^{m_r})}{(p_1^{m_1} \cdots p_r^{m_r})^{\frac{k-1}{2}}} & \text{if $k=2$} \\ 0 & \text{otherwise.} \end{array} \right. \\ = & \left\{ \begin{array}{l l} {p_1}^{\frac{m_1}{2}}\cdots{p_r}^{\frac{m_r}{2}}(m_1+1)\cdots(m_r+1) & \text{if $k=2$} \\ 0 & \text{otherwise.} \end{array} \right. \end{align} Combining all the estimates above, we get \begin{eqnarray} &&\|Tr \ (T_{p_1^{m_1} \cdots p_r^{m_r}})\| \\ && \ll (p_1^{-\frac{m_1}{2}}\cdots p_r^{-\frac{m_r}{2}})\frac{k-1}{12} + {p_1}^{\frac{3m_1}{2}}\cdots{p_2}^{\frac{3m_r}{2}}\operatorname{log} \,4(p_1^{m_1} \cdots p_r^{m_r}) \\ && + (m_1+1)\cdots(m_r+1)d(N)\sqrt{N}+ p_1^{m_1} \cdots p_r^{m_r}(m_1+1)\cdots(m_r+1) \\ && \ll {p_1}^{\frac{3m_1}{2}}\cdots{p_r}^{\frac{3m_r}{2}}\operatorname{log} \,(4(p_1^{m_1} \cdots p_r^{m_r}))d(N)\sqrt{N}(m_1m_2 \cdots m_r), \end{eqnarray} where $d(N)$ is the divisor function. \end{proof} \section{The measure $\mu$} Following the definition from Section \ref{S2}, for non-negative integers $m_1,m_2,\ldots, m_r$, we get, \begin{eqnarray} C_{\underline m} & := &\lim_{s(N,k) \to \infty} \frac{1}{(2 s(N,k))^r} \sum_{n\atop{1 \leq i \leq n}}e( \pm m_1 \theta_{1}^{(n)}(p_1) \pm \cdots\pm m_r \theta_{r}^{(n)}(p_r) ) \\ & = & \prod_{i=1}^r \left(\lim_{s(N,k) \to \infty} \frac{1}{(2s(N,k))^r}\sum_{i_n}e(\pm m_i \theta_{i}^{(n)}(p_i)) \right) \\ & = & \prod_{i=1}^r c_{m_i}, \end{eqnarray} where $c_{m_i} $ are the $m_i$-th Weyl limit of the family $\left\{\frac{\pm \theta_{i}^{(n)}}{2 \pi}\right \}$ and from Theorem 18 of~\cite{MS}, we have \begin{eqnarray} c_{m_i}= \left \{ \begin{array}{l l} 1 & \text{if } m_i=0, \\ \frac{1}{2} \left( \frac{1}{{p_i}^{m_i}}-\frac{1}{{p_i}^{m_i-2}}\right) & \text{if } m_i \text{ is even}, \\ 0 & \text{otherwise.} \end{array} \right. \end{eqnarray} Note that if $m_1,m_2,\ldots, m_r$ are all zero then $C_{0,\ldots,0}=1$. Define the measure $$\mu=F(-x_1,\ldots,-x_r) dx_1 \cdots dx_r,$$ where \begin{eqnarray} F(x_1,\ldots,x_r) & = & \sum_{m_1=-\infty}^{\infty} \cdots \sum_{m_r=-\infty}^{\infty}c_{m_1} \cdots c_{m_r} e(m_1x_1) \cdots e(m_rx_r) \\ & = & F(x_1) \cdots F(x_r)\\ & = & \frac{4(p_1+1) \sin^2 2 \pi x_1}{({p_1}^\frac{1}{2}+{p_1}^{-\frac{1}{2}})^2-4\cos^2 2 \pi x_1} \cdots \frac{4(p_r+1) \sin^2 2 \pi x_r}{({p_r}^\frac{1}{2}+{p_r}^{-\frac{1}{2}})^2-4\cos^2 2 \pi x_r}. \end{eqnarray} The above $F(x_1,\ldots,,x_r)dx_1 \cdots dx_r$ determines a measure on $[0,1]^r$ and is the distribution function for the tuples of numbers $$ (x_{i_1}, \ldots ,x_{i_r})=\left(\pm \frac{\theta_{1}^{(n)}(p_1)}{2 \pi},\ldots,\pm \frac{\theta_{r}^{(n)}(p_r)}{2 \pi}\right). $$ The measure giving the distribution of $(\cos \theta_{1}^{(n)}(p_1), \ldots ,\cos \theta_{r}^{(n)}(p_r))$ is therefore \begin{eqnarray} & = & F\left(\frac{\cos^{-1}x_1}{2 \pi}, \ldots ,\frac{\cos^{-1}x_r}{2 \pi}\right)d\left(\frac{\cos^{-1}x_1}{2 \pi}\right) \cdots d\left(\frac{\cos^{-1}x_r}{2 \pi}\right) \\ & = & F\left(\frac{\cos^{-1}x_1}{2 \pi} \right) d \left(\frac{\cos^{-1}x_1}{2 \pi} \right) \cdots F \left(\frac{\cos^{-1}x_r}{2 \pi} \right) d \left(\frac{\cos^{-1}x_r}{2 \pi} \right) \\ & = & \frac{2(p+1)}{\pi} \frac{\sqrt{1-{x_1}^2}}{(p^{\frac{1}{2}}+p^{-\frac{1}{2}})^2-4{x_1}^2}dx_1 \cdots \frac{2(p+1)}{\pi} \frac{\sqrt{1-{x_r}^2}}{(p^{\frac{1}{2}}+p^{-\frac{1}{2}})^2-4{x_r}^2}dx_r. \end{eqnarray} Thus, the distribution of the tuples of numbers $$(2\cos \theta_{1}^{(n)}(p_1)\cdots 2\cos \theta_{r}^{(n)}(p_r))$$ is given by $\prod_{i=1}^r\mu_{p_i} $ after an easy change of variable. \section{Chebychev polynomials and the trace of Hecke operators} For any integer $n \geq 0$, the $n$-th Chebychev polynomial $X_n(x)$ is defined as follows: $$ X_n(x)=\frac{\sin (n+1)\theta}{\sin \theta}, \ \ \text{where } x=2 \cos \theta .$$ Serre proved the following result in~\cite{Serre}. \begin{lemma} We have $ T^{'}_{p^m}=X_m(T_p).$ \end{lemma} From~\cite[page 697]{MS}, when $m_i=1,$ we have, $$ \sum_{n=1}^{s(N,k)} 2 \cos \theta_{i}^{(n)}(p_i)= Tr(T^{'}_{p_i}). $$ Since, for all integers $m\geq 2$, $$\sum_{n=1}^{s(N,k)} 2 \cos m \theta = X_m(2 \cos \theta)-X_{m-2}(2 \cos \theta), $$ we have for $m_i \geq 2,$ \begin{equation}\label{E13} \sum_{n=1}^{s(N,k)} 2 \cos \theta_{i}^{(n)}(p_i)= Tr(T^{'}_{p_i^{m_i}})-Tr(T^{'}_{p_i^{m_i-2}}). \end{equation} Now we prove the following theorem, \begin{thm} For all non zero positive integers $m_1,m_2,\ldots,m_r$ consider $\underline{m}=(m_1, m_2,\ldots, m_r),$ the Weyl limits $C_{\underline m}$ are given by \begin{eqnarray} C_{\underline m} = \left \{ \begin{array}{l l} 1 & \text{if $m_1= \cdots =m_r=0$,} \\ \prod_{i=1}^r \left(\frac{1}{{p_i}^{\frac{m_i}{2}}}-\frac{1}{{p_i}^{\frac{m_i-2}{2}}} \right) & \text{if $m_i,\ i=1,2,..,r$ are even,} \\ 0 & \text{otherwise.} \end{array} \right. \end{eqnarray} Moreover, \begin{eqnarray} && \left\|\prod_{i=1}^r \sum_{n=1}^{s(N,k)}2 \cos \ m_i \theta_{i}^{(n)}(p_i)-s(N,k) C_{\underline m}\right\| \\ && \ll p_1^{\frac{3m_1}{2}} \cdots p_r^{\frac{3m_r}{2}}(m_1 \cdots m_r)\operatorname{log} \,(4(p_1^{m_1} \cdots p_r^{m_r})). \end{eqnarray} \end{thm} \begin{proof} For any integers $m_1,m_2,\ldots,m_r \geq 2$, using (\ref{E13}), we have \begin{eqnarray}\label{E23} && \prod_{i=1}^{r} \sum_{n=1}^{s(N,k)} \{2 \cos m_i\theta_{i}^{(n)}(p_i)\} \\ &= & \prod_{i=1}^{r}\left(X_{m_i}(2\cos \theta_{i}^{(n)}(p_i))-X_{m_i-2}(2\cos \theta_{i}^{(n)}(p_i))\right) \nonumber\\ & = &\sum_{b_1,b_2,\ldots,b_r \in Z} \prod_{i=1}^{r} X_{b_i}(p_i)(2\cos \theta_{i}^{(n)}(p_i)), \nonumber \end{eqnarray} where $Z=\{m_1,m_2,\ldots,m_r,m_1-2,m_2-2,\ldots,m_r-2\}$. We know that if a linear operator $T$ is diagonalizable and $\lambda$ is an eigenvalue of $T$, then, for any polynomial $P(x)$, the eigenvalue of $P(T)$ is $P(\lambda)$. Since the Hecke operators $T_m$ and $T_n$ commutes with each other, there exists an ordered basis such that every Hecke operator can be represented by a diagonal matrix with respect to the basis. Using all the above facts, (\ref{E23}) equals \begin{eqnarray}\label{E29} && \nonumber \sum_{b_1,b_2,\ldots,b_r \in Z} Tr ( T^{'}_{p_1^{b_1}\cdots p_r^{b_r}}) \\ \nonumber &\ll& \sum_{b_1,b_2,\ldots,b_r \in Z}(p_1^{b_1}\cdots p_r^{b_r})\left(\frac{k-1}{12}\right)^r \\ \nonumber && +p_1^{\frac{3m_1}{2}}\cdots p_r^{\frac{3m_r}{2}}\operatorname{log} \,4(p_1^{m_1} \cdots p_r^{m_r})(m_1m_2 \cdots m_r)\\ \nonumber &\ll & \left(\frac{k-1}{12}\right)^r \prod_{n=1}^r\left( \frac{1}{{p_n}^{\frac{m}{2}}}-\frac{1}{{p_n}^{\frac{m-2}{2}}} \right) \\ \nonumber && +p_1^{\frac{3m_1}{2}} \cdots p_r^{\frac{3m_r}{2}}\operatorname{log} \,4(p_1^{m_1} \cdots p_r^{m_r})(m_1m_2 \cdots m_r) \\ & \ll & \left(\frac{k-1}{12}\right)^r \prod_{n=1}^r\left( \frac{1}{{p_n}^{\frac{m_n}{2}}} -\frac{1}{{p_n}^{\frac{m_n-2}{2}}} \right)\|Tr ( T^{'}_{{p_1}^{m_1}{p_2}^{m_2}\cdots{p_r}^{m_r}})\|. \end{eqnarray} Now \begin{eqnarray} && \left\|\prod_{i=1}^r \sum_{n=1}^{s(N,k)}2 \cos \ m_i \theta_{i}^{(n)}(p_i)- s(N,k) C_{\underline m}\right\| \\ & = & \left\|\sum_{j=1}^{4} B_j(m_1 \cdots m_r)-s(N,k) C_{\underline m} \right\| \\ & \leq & \left\|B_1-s(N,k) C_{\underline m}\right\|+ \left\|\sum_{j=2}^{4}B_j(m_1 \cdots m_r)\right\|. \end{eqnarray} Using (\ref{E29}) and the fact that $s(N,k) C_{\underline m} \ \text{behaves like} \ B_1(m_1 \cdots m_r), $ we have $$\left\|\prod_{i=1}^r \sum_{n=1}^{s(N,k)}2 \cos \ m_i \theta_{i}^{(n)}(p_i)-s(N,k) C_{\underline m} \right\| $$ $$ \ll p_1^{\frac{3m_1}{2}}\cdots p_r^{\frac{3m_r}{2}}(m_1 \cdots m_r)\operatorname{log} \,(4(p_1^{m_1} \cdots p_r^{m_r})). $$ \end{proof} \section{Effective versions} To prove the effective result, we use the following variant of the Erd\"{o}s-Tur\'{a}n inequality that can be found in~\cite[Proposition 7.1]{LLW}. \begin{thm}\label{ET} For any $S=\prod_{n=1}^{r}[\alpha_n,\beta_n] \subset [0, \frac{1}{2}]^r$ and a sequence of tuples of number $\{(x_{m_1},\ldots, x_{m_r})\} \in [0, \frac{1}{2}]^r,$ we define $$ N_S(V):= \sharp \{n \leq V:(x_{m_1},\ldots,x_{m_r}) \in S \} $$ and $$ D_S(V):=\|N_S(V)-V \mu(S)\|. $$ For $I=\prod_{n=1}^{r}[\alpha_n,\beta_n],$ we have \begin{align} D_I(V) \leq & \sum_{m_1,m_2,\ldots,m_r \in ([-m,m] \bigcap {\mathbb Z})^r}w(m_1,\ldots,m_r)\Delta((m_1,\ldots,m_r),V)\\ &+10 \frac{2}{M+1}\sum_{1 \leq m \leq M} {\rm max \atop{1 \leq t \leq r}} \Delta_{t}(m,V) + 12 \Vert F\Vert \frac{2V}{M+1} \end{align} for any integers $V,M \geq 1$, where $$ \Delta_t(m,V)=\|\sum_{n \leq V}\cos(2 \pi m x_{i}) -Vc_{m}\|, \ \ \ \text{(for $m \in \mathbb Z$)}, $$ $$ \left\|\Delta((m_1,\ldots,m_r),V)\right\| =\left\|\sum_{m_1,\ldots, m_r \leq V}\cos(2 \pi m_1 x_1) \cdots \cos(2 \pi m_r x_r)\right\|, $$ for $(m_1,\ldots,m_r) \in \mathbb Z^r$ and $$ w(m_1,\ldots, m_r)=\left((2 \pi)^r \prod_{t=1}^r \rm min \left(\frac{1}{\pi\|m_t\|},\beta_t-\alpha_t \right)+\frac{2}{M+1}\right),$$ and $$\Vert F\Vert=\max_{1 \leq t \leq r} \Vert F_t\Vert_{\infty}+\prod_{t=1}^r \Vert F_t\Vert_{\infty}$$ with $\Vert F_t\Vert_{\infty}= \max_{x \in [0,1]} \|F_t(x)\|.$ \end{thm} \noindent\textbf{Proof of Theorem \ref{T1}.} \\ Choose $\theta_{i}^{(n)}(p_i) \in [0, \frac{1}{2}]$ such that $2 \cos 2 \pi \theta_{i}^{(n)}(p_i)=a_{i}^{(n)}(p_i).$ Given any sub interval $I \subset [-2,2]$ choose sub interval $I^{'} \subset [0, \frac{1}{2}]$ so that $\theta_{i}^{(n)}(p_i) \in I^{'}$ if and only if $a_{i}^{(n)} \in I.$ By Theorem \ref{ET}, we have \begin{align} \left\|\sharp \right.\{1 \leq n \leq (s(N,k)) &: (a_{1}^{(n)}(p_1),\ldots, a_{r}^{(n)}(p_r)) \in I - V \mu(I)\left. \right\|\\ &\ll \frac{s(N,k)r}{M+1}+\left \|\prod_{i=1}^r \sum_{n=1}^{s(N,k)}2 \cos \ m_i \theta_{i}^{(n)}(p_i)-s(N,k) C_{\underline m}\right\|. \end{align} Using Proposition \ref{P1}, we get $$\ll \frac{s(N,k)r}{M+1} +(p_1^{\frac{3m_1}{2}} \cdots p_r^{\frac{3m_r}{2}})(m_1 \cdots m_r)\operatorname{log} \,(4(p_1^{m_1} \cdots p_r^{m_r}))\operatorname{log} \,M^r.$$ Since the main contribution is coming from $\left(p_1^{\frac{3m_1}{2}}\cdots p_r^{\frac{3m_r}{2}} \right)$, by choosing $\displaystyle M=\frac{\operatorname{log} \,kN}{ \operatorname{log} \,p_1p_2 \cdots p_r}$, we get the required result. $ \Box$ \noindent\textbf{Acknowledgments:} The author would like to thank Prof$.$ M. Ram Murty and Dr$.$ Kaneenika Sinha for useful discussions in the earlier version of this manuscript. I am also thankful to Prof$.$ R. Thangadurai and Dr$.$ Jaban Meher for their suggestions to improve the presentation of the paper. \end{document}	arXiv
Equity valuation under stock dilution and buy-back On the local behavior of non-negative solutions to a logarithmically singular equation September 2012, 17(6): 1831-1840. doi: 10.3934/dcdsb.2012.17.1831 Optimal treated mosquito bed nets and insecticides for eradication of malaria in Missira Bassidy Dembele 1, and Abdul-Aziz Yakubu 2, Department of Mathematics and Computer Science, Grambling State University, Grambling, LA 71245, United States Department of Mathematics, Howard University, Washington, DC 20059 Received May 2011 Revised July 2011 Published May 2012 We extend the deterministic mathematical malaria model framework of Dembele et al. and use it to study the impact of protecting humans from mosquito bites and mass killing of mosquito vectors on malaria incidence in Missira, a village in Mali. As a case study, we fit our model to Missira malaria incidence data. 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Distributed, layered and reliable computing nets to represent neuronal receptive fields. Mathematical Biosciences & Engineering, 2014, 11 (2) : 343-361. doi: 10.3934/mbe.2014.11.343 Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5 (4) : 789-801. doi: 10.3934/mbe.2008.5.789 Expeditho Mtisi, Herieth Rwezaura, Jean Michel Tchuenche. A mathematical analysis of malaria and tuberculosis co-dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 827-864. doi: 10.3934/dcdsb.2009.12.827 Bassidy Dembele, Abdul-Aziz Yakubu. Controlling imported malaria cases in the United States of America. Mathematical Biosciences & Engineering, 2017, 14 (1) : 95-109. doi: 10.3934/mbe.2017007 G.A. Ngwa. Modelling the dynamics of endemic malaria in growing populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1173-1202. doi: 10.3934/dcdsb.2004.4.1173 Yanyu Xiao, Xingfu Zou. On latencies in malaria infections and their impact on the disease dynamics. Mathematical Biosciences & Engineering, 2013, 10 (2) : 463-481. doi: 10.3934/mbe.2013.10.463 Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753 Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995 Hongyan Yin, Cuihong Yang, Xin'an Zhang, Jia Li. The impact of releasing sterile mosquitoes on malaria transmission. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3837-3853. doi: 10.3934/dcdsb.2018113 Changguang Dong. Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4231-4238. doi: 10.3934/dcds.2017180 Zindoga Mukandavire, Abba B. 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Mathematical Biosciences & Engineering, 2017, 14 (2) : 377-405. doi: 10.3934/mbe.2017024 Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673 Bassidy Dembele Abdul-Aziz Yakubu	CommonCrawl
\begin{definition}[Definition:Graph (Category Theory)] A '''graph''' is an interpretation of a metagraph within set theory. Let $\mathfrak U$ be a class of sets. A metagraph $\GG$ is a '''graph''' {{iff}}: :$(1): \quad$ The objects form a subset $\operatorname{vert} \GG \subseteq \mathfrak U$ :$(2): \quad$ The morphisms form a subset $\operatorname{edge} \GG \subseteq \mathfrak U$ If the class $\mathfrak U$ is a set, then morphisms are functions, and the domain and codomain in the definition of a morphism are those familiar from set theory. If $\mathfrak U$ is a proper class this is not the case, for example the morphisms of $\CC$ need not be functions. Category:Definitions/Category Theory \end{definition}	ProofWiki
# 1. Foundations of Computer Security # 1.1. Principles of Confidentiality, Integrity, and Availability Confidentiality, integrity, and availability are three fundamental principles of computer security. These principles form the basis for designing and implementing secure systems. Confidentiality refers to the protection of sensitive information from unauthorized access. It ensures that only authorized individuals or entities can access and view the data. Encryption is a common technique used to achieve confidentiality by converting data into a form that is unreadable without the proper decryption key. Integrity ensures that data remains unaltered and trustworthy. It involves protecting data from unauthorized modification, deletion, or corruption. Hash functions and digital signatures are commonly used to verify the integrity of data. Availability ensures that systems and data are accessible and usable when needed. It involves protecting against denial-of-service attacks and ensuring that systems are reliable and resilient. An example of the importance of confidentiality, integrity, and availability is online banking. When you access your bank account online, you expect that your personal information and financial transactions are kept confidential, that the data is accurate and hasn't been tampered with, and that the banking system is available for you to use. ## Exercise Think of a real-life scenario where confidentiality, integrity, and availability are important. Describe the scenario and explain why each principle is crucial in that context. ### Solution Scenario: A hospital's electronic medical records system In a hospital, the electronic medical records system contains sensitive patient information, including medical history, diagnoses, and treatment plans. Confidentiality is crucial to protect patient privacy and comply with legal requirements such as HIPAA. Unauthorized access to this information could lead to identity theft or other privacy breaches. Integrity is important to ensure that patient records are accurate and haven't been tampered with. If a patient's medical records are modified or corrupted, it could lead to incorrect diagnoses or treatments, potentially endangering the patient's health. Availability is critical in a hospital setting where timely access to patient information can be a matter of life and death. Doctors and nurses need to be able to access patient records quickly to make informed decisions and provide appropriate care. # 1.2. Risk Management Risk management is an essential aspect of computer security. It involves identifying potential risks and taking steps to mitigate or minimize them. By understanding and managing risks, organizations can protect their systems and data more effectively. The risk management process typically involves the following steps: 1. Risk identification: Identifying potential risks and vulnerabilities in systems, networks, and processes. 2. Risk assessment: Evaluating the likelihood and potential impact of each identified risk. This helps prioritize risks and determine the appropriate level of response. 3. Risk mitigation: Implementing measures to reduce the likelihood or impact of identified risks. This can include implementing security controls, conducting regular security audits, and training employees on security best practices. 4. Risk monitoring and review: Continuously monitoring and reviewing the effectiveness of risk mitigation measures. This ensures that security controls remain up to date and effective in addressing emerging threats. An example of risk management is implementing a firewall to protect a company's internal network from external threats. By identifying the risk of unauthorized access or malicious attacks, the company can implement a firewall as a security control to monitor and filter incoming and outgoing network traffic. This helps reduce the likelihood of a successful attack and mitigates the potential impact on the internal network. ## Exercise Think of a real-life scenario where risk management is important. Describe the scenario and explain how each step of the risk management process can be applied. ### Solution Scenario: Online payment processing for an e-commerce website 1. Risk identification: Identify potential risks such as unauthorized access to customer payment information, data breaches, and fraudulent transactions. 2. Risk assessment: Assess the likelihood and potential impact of each identified risk. For example, the likelihood of unauthorized access may be higher if weak passwords are used, and the potential impact of a data breach could result in financial loss and damage to the company's reputation. 3. Risk mitigation: Implement measures to reduce the likelihood and impact of identified risks. This can include implementing strong authentication mechanisms, encrypting customer payment information, regularly monitoring and analyzing transaction data for fraud detection, and conducting regular security audits. 4. Risk monitoring and review: Continuously monitor and review the effectiveness of risk mitigation measures. This can involve monitoring system logs for suspicious activities, staying updated on the latest security threats and vulnerabilities, and conducting periodic security assessments to identify any new risks or vulnerabilities. By following the risk management process, the e-commerce website can better protect customer payment information, reduce the risk of fraudulent transactions, and maintain the trust of its customers. # 1.3. Security Policies and Procedures Security policies and procedures are essential for maintaining a secure computing environment. They provide guidelines and instructions for employees and users to follow to ensure the confidentiality, integrity, and availability of systems and data. A security policy is a high-level document that outlines the organization's approach to security. It defines the goals, objectives, and responsibilities related to security. It also provides guidelines for implementing security controls and procedures. Security procedures, on the other hand, are detailed instructions that describe how specific security tasks should be performed. They provide step-by-step guidance on activities such as user account management, data backup, incident response, and system hardening. An example of a security policy is a password policy that defines the requirements for creating and managing passwords. It may specify minimum password length, complexity requirements, and the frequency of password changes. This policy helps ensure that passwords are strong and not easily guessable, reducing the risk of unauthorized access to systems and data. An example of a security procedure is an incident response procedure that outlines the steps to be followed in the event of a security incident. It may include actions such as isolating affected systems, notifying appropriate personnel, collecting evidence, and restoring systems to a secure state. This procedure helps ensure a timely and effective response to security incidents, minimizing the potential impact on the organization. ## Exercise Think of a security policy or procedure that you think is important in a specific context. Describe the policy or procedure and explain why it is important. ### Solution Policy: Acceptable Use Policy for company computers and network The Acceptable Use Policy defines the rules and guidelines for using company computers and network resources. It outlines the acceptable and unacceptable uses of these resources, including restrictions on accessing inappropriate websites, downloading unauthorized software, and sharing confidential information. This policy is important because it helps protect the company's systems and data from unauthorized access, malware infections, and data breaches. By clearly defining acceptable use and setting expectations for employees, the policy helps create a secure computing environment and reduces the risk of security incidents. Procedure: Data Backup and Recovery Procedure The Data Backup and Recovery Procedure provides instructions for regularly backing up critical data and recovering it in the event of data loss or system failure. It outlines the frequency and methods of data backups, the storage locations for backup data, and the steps to be followed for data recovery. This procedure is important because it helps ensure the availability and integrity of important data. Regular backups protect against data loss due to hardware failures, natural disasters, and other unforeseen events. By following the procedure, organizations can minimize the impact of data loss and maintain business continuity. # 2. Authentication Authentication is the process of verifying the identity of a user or system. It ensures that only authorized individuals or entities are granted access to resources. Authentication is a critical component of computer security, as it helps prevent unauthorized access and protects sensitive information. There are several types of authentication methods, each with its own strengths and weaknesses. In this section, we will explore the different types of authentication and their applications. One common type of authentication is username and password authentication. This method requires users to provide a unique username and a corresponding password to access a system or application. The system then verifies the provided credentials against a stored database of usernames and passwords. Username and password authentication is widely used because it is simple and easy to implement. However, it has some vulnerabilities. For example, if a user chooses a weak password or shares their password with others, it can compromise the security of the system. An example of username and password authentication is the login process for an email account. When a user wants to access their email, they are prompted to enter their username (usually their email address) and password. The email server then verifies the provided credentials and grants access if they are correct. ## Exercise Think of a scenario where username and password authentication would be appropriate. Describe the scenario and explain why this method of authentication is suitable. ### Solution Scenario: Online banking In the scenario of online banking, username and password authentication would be appropriate. This method ensures that only authorized individuals can access their bank accounts and perform financial transactions. By requiring users to provide a unique username and a password, the bank can verify the identity of the user and protect their sensitive financial information. Additionally, username and password authentication is relatively easy for users to understand and use, making it suitable for a wide range of individuals accessing their bank accounts online. # 2.1. Types of Authentication Each type of authentication has its own advantages and disadvantages. Username and password authentication is widely used because it is simple and easy to implement, but it can be vulnerable to password guessing or theft. Token-based authentication provides an extra layer of security by requiring users to possess a physical token, but it can be inconvenient if the token is lost or stolen. Biometric authentication is highly secure and convenient, but it may not be foolproof as biometric data can be stolen or replicated. Multi-factor authentication offers the highest level of security, but it can also be more complex and time-consuming for users. An example of token-based authentication is the use of a security key to access a computer network. The user inserts the security key into a USB port and enters their password to gain access. The security key generates a unique code that is required for authentication. This method provides an extra layer of security because even if someone steals the user's password, they would still need the physical security key to gain access. ## Exercise Think of a scenario where biometric authentication would be appropriate. Describe the scenario and explain why this method of authentication is suitable. ### Solution Scenario: Airport security In the scenario of airport security, biometric authentication would be appropriate. This method ensures that only authorized individuals can access secure areas of the airport, such as boarding gates or restricted areas. By using biometric data, such as fingerprints or iris patterns, the airport can verify the identity of individuals with a high level of accuracy. This helps prevent unauthorized access and enhances the overall security of the airport. Additionally, biometric authentication is convenient for travelers as they don't need to carry physical tokens or remember passwords, making the process faster and more efficient. # 2.2. Passwords and Password Policies A strong password should be difficult for others to guess or crack. Here are some guidelines for creating strong passwords: 1. Use a combination of uppercase and lowercase letters, numbers, and special characters. 2. Avoid using common words or phrases, as they are easier to guess. 3. Make your password at least 8 characters long, but longer is better. 4. Avoid using personal information, such as your name, birthdate, or address. 5. Don't reuse passwords across multiple accounts. Here are some examples of weak and strong passwords: - Weak password: "password123" - Strong password: "P@ssw0rd!2" The weak password is easy to guess because it uses a common word and a simple number sequence. The strong password, on the other hand, uses a combination of uppercase and lowercase letters, numbers, and special characters. ## Exercise Evaluate the following passwords and determine whether they are weak or strong: 1. "qwerty" 2. "P@$$w0rd" 3. "12345678" 4. "MyDog'sNameIsMax!" ### Solution 1. Weak password: "qwerty" - This password is weak because it uses a common keyboard sequence. 2. Weak password: "P@$$w0rd" - This password is weak because it is a common pattern and lacks complexity. 3. Weak password: "12345678" - This password is weak because it is a simple number sequence. 4. Strong password: "MyDog'sNameIsMax!" - This password is strong because it uses a combination of uppercase and lowercase letters, numbers, and special characters. # 2.3. Multi-Factor Authentication Multi-factor authentication (MFA) is a security measure that requires users to provide multiple forms of identification to verify their identity. This adds an extra layer of security to the authentication process, as it requires more than just a password. There are three main factors used in multi-factor authentication: 1. Something you know: This is typically a password or PIN. 2. Something you have: This can be a physical token, such as a smart card or security key. 3. Something you are: This refers to biometric data, such as fingerprints or facial recognition. An example of multi-factor authentication is using a bank's mobile app. When logging in, the user is first prompted to enter their username and password (something they know). Then, they are required to provide a fingerprint scan (something they are) using their smartphone's biometric sensor. Only after both factors have been successfully verified will the user be granted access to their account. ## Exercise Think of a scenario where multi-factor authentication would be beneficial. Describe the scenario and explain why this method of authentication is suitable. ### Solution Scenario: Online banking In the scenario of online banking, multi-factor authentication would be beneficial. This method adds an extra layer of security to protect sensitive financial information. By requiring users to provide both a password (something they know) and a physical token, such as a one-time password generated by a mobile app (something they have), the bank can ensure that only authorized individuals can access their accounts. This helps prevent unauthorized access and reduces the risk of fraudulent activities, such as identity theft or unauthorized transactions. # 2.4. Biometric Authentication Biometric authentication is a security method that uses unique physical or behavioral characteristics of an individual to verify their identity. This method relies on the fact that these characteristics are difficult to replicate, making it a secure form of authentication. There are several types of biometric authentication methods, including: 1. Fingerprint recognition: This method analyzes the unique patterns and ridges on a person's fingerprint to verify their identity. 2. Facial recognition: This method uses facial features, such as the shape of the face, the distance between the eyes, and the position of facial landmarks, to authenticate a person. 3. Iris recognition: This method analyzes the patterns in the colored part of the eye, known as the iris, to verify a person's identity. 4. Voice recognition: This method analyzes the unique characteristics of a person's voice, such as pitch, tone, and pronunciation, to authenticate their identity. An example of biometric authentication is using a smartphone's fingerprint scanner to unlock the device. The user's fingerprint is scanned and compared to the stored fingerprint data to determine if they are the authorized user. If the fingerprints match, the device is unlocked. ## Exercise Think of a scenario where biometric authentication would be beneficial. Describe the scenario and explain why this method of authentication is suitable. ### Solution Scenario: Access to a high-security facility In the scenario of accessing a high-security facility, biometric authentication would be beneficial. This method provides a high level of security by using unique physical characteristics that are difficult to replicate. By requiring individuals to provide their fingerprint or iris scan, the facility can ensure that only authorized personnel can enter. This helps prevent unauthorized access and enhances the overall security of the facility. # 3. Encryption Encryption is a fundamental concept in computer security. It involves the process of converting plaintext data into ciphertext, which is unreadable without the proper decryption key. Encryption is used to protect sensitive information from unauthorized access or interception. There are two main types of encryption: symmetric encryption and asymmetric encryption. Symmetric encryption uses a single key to both encrypt and decrypt the data. This key must be kept secret, as anyone who has the key can decrypt the ciphertext. Symmetric encryption algorithms include DES, 3DES, and AES. Asymmetric encryption, also known as public-key encryption, uses a pair of keys: a public key for encryption and a private key for decryption. The public key can be freely distributed, while the private key must be kept secret. Asymmetric encryption algorithms include RSA and ECC. An example of symmetric encryption is the Advanced Encryption Standard (AES). AES is widely used to secure sensitive data, such as financial transactions and personal information. It uses a symmetric key, which is shared between the sender and the recipient, to encrypt and decrypt the data. An example of asymmetric encryption is the RSA algorithm. In RSA, the sender uses the recipient's public key to encrypt the data, and the recipient uses their private key to decrypt the data. This ensures that only the recipient, who possesses the private key, can read the encrypted message. ## Exercise Explain the difference between symmetric encryption and asymmetric encryption. ### Solution Symmetric encryption uses a single key for both encryption and decryption, while asymmetric encryption uses a pair of keys: a public key for encryption and a private key for decryption. In symmetric encryption, the same key is used by both the sender and the recipient, while in asymmetric encryption, the sender uses the recipient's public key to encrypt the data, and the recipient uses their private key to decrypt the data. # 3.1. Symmetric vs. Asymmetric Encryption Symmetric encryption and asymmetric encryption are two different approaches to encryption, each with its own advantages and use cases. Symmetric encryption, as the name suggests, uses a single key to both encrypt and decrypt the data. This key must be kept secret and shared between the sender and the recipient. Symmetric encryption is faster and more efficient than asymmetric encryption, making it suitable for encrypting large amounts of data. However, the challenge with symmetric encryption is securely distributing the key to all parties involved. Asymmetric encryption, also known as public-key encryption, uses a pair of keys: a public key for encryption and a private key for decryption. The public key can be freely distributed, while the private key must be kept secret. Asymmetric encryption provides a higher level of security because even if the public key is intercepted, it cannot be used to decrypt the data without the private key. However, asymmetric encryption is slower and more computationally intensive than symmetric encryption. In practice, a combination of both symmetric and asymmetric encryption is often used. For example, symmetric encryption can be used to encrypt the actual data, while asymmetric encryption can be used to securely exchange the symmetric encryption key. Let's say Alice wants to send a secure message to Bob. She can use symmetric encryption to encrypt the message using a shared secret key that only she and Bob know. She then sends the encrypted message to Bob. When Bob receives the message, he uses the same secret key to decrypt the message and read its contents. On the other hand, if Alice wants to send a secure message to a large group of people, she can use asymmetric encryption. She encrypts the message using the public keys of each recipient. Each recipient can then use their private key to decrypt the message and read its contents. ## Exercise What are the advantages of symmetric encryption and asymmetric encryption? ### Solution The advantages of symmetric encryption are its speed and efficiency in encrypting large amounts of data. It is also easier to implement and requires less computational power. The advantages of asymmetric encryption are its higher level of security and the ability to securely exchange encryption keys without sharing the private key. Asymmetric encryption is also more suitable for scenarios where secure communication is required between multiple parties. # 3.2. Encryption Algorithms Encryption algorithms are mathematical formulas used to encrypt and decrypt data. There are several commonly used encryption algorithms, each with its own strengths and weaknesses. One widely used encryption algorithm is the Advanced Encryption Standard (AES). AES is a symmetric encryption algorithm that uses a block cipher to encrypt data in fixed-size blocks. It supports key lengths of 128, 192, and 256 bits, providing a high level of security. AES is considered secure and is used by governments and organizations around the world. Another popular encryption algorithm is the Rivest-Shamir-Adleman (RSA) algorithm. RSA is an asymmetric encryption algorithm that uses two keys: a public key for encryption and a private key for decryption. RSA is widely used for secure communication and digital signatures. It is based on the mathematical properties of large prime numbers and is considered secure when used with sufficiently long key lengths. Other encryption algorithms include Triple Data Encryption Standard (3DES), which is a symmetric encryption algorithm that applies the Data Encryption Standard (DES) algorithm three times for added security, and Blowfish, which is a symmetric encryption algorithm known for its fast encryption and decryption speeds. It is important to note that the security of an encryption algorithm depends not only on the algorithm itself but also on the key length and the implementation of the algorithm. It is recommended to use encryption algorithms that have been thoroughly vetted and are widely accepted in the security community. Let's say Alice wants to encrypt a sensitive document before sending it to Bob. She can use the AES encryption algorithm with a 256-bit key to encrypt the document. Only Bob, who has the corresponding key, will be able to decrypt and read the document. ## Exercise Research and find another commonly used encryption algorithm. Describe its key features and use cases. ### Solution One commonly used encryption algorithm is the Elliptic Curve Cryptography (ECC) algorithm. ECC is an asymmetric encryption algorithm that is based on the mathematics of elliptic curves. It offers the same level of security as traditional encryption algorithms but with shorter key lengths, making it more efficient in terms of computational resources and bandwidth. ECC is often used in applications where resource-constrained devices, such as mobile devices and IoT devices, need to perform secure communication. # 3.3. Key Management Key management is an important aspect of encryption. It involves generating, storing, distributing, and revoking encryption keys. Proper key management is essential for maintaining the security of encrypted data. One key management practice is the generation of strong encryption keys. Strong encryption keys are typically long, random, and unique. They should be generated using a secure random number generator. The longer the key, the more secure the encryption. Common key lengths for symmetric encryption algorithms range from 128 to 256 bits. Another key management practice is the secure storage of encryption keys. Encryption keys should be stored in a secure location, such as a hardware security module (HSM) or a key management system (KMS). These systems provide secure storage and management of keys, protecting them from unauthorized access. Key distribution is another important aspect of key management. When encrypting data, the encryption key needs to be securely shared with the intended recipient. This can be done through secure channels, such as in-person exchange or encrypted communication protocols. Key distribution protocols, such as the Diffie-Hellman key exchange, can also be used to securely establish a shared secret key between two parties. Revocation of encryption keys is necessary when a key is compromised or no longer needed. When a key is compromised, it should be immediately revoked and replaced with a new key. Key revocation ensures that encrypted data remains secure even if the key is compromised. Proper key management is crucial for maintaining the security and integrity of encrypted data. It ensures that encryption keys are generated securely, stored safely, distributed only to authorized parties, and revoked when necessary. Let's say Alice wants to send an encrypted email to Bob. She generates a strong encryption key and encrypts the email using that key. She then securely shares the encryption key with Bob through an encrypted messaging app. Bob uses the encryption key to decrypt and read the email. If Alice suspects that the encryption key has been compromised, she can revoke the key and generate a new one, ensuring the security of future communications. ## Exercise Explain why key management is important in the context of encryption. ### Solution Key management is important in encryption because it ensures the security and integrity of encrypted data. Without proper key management, encryption keys can be easily compromised, leading to unauthorized access to sensitive information. Key management practices, such as generating strong keys, securely storing keys, and revoking compromised keys, help protect encrypted data from unauthorized access and maintain the confidentiality and integrity of the information. # 3.4. Applications of Encryption Encryption has a wide range of applications in computer security. It is used to protect sensitive data and ensure the confidentiality and integrity of information. Here are some common applications of encryption: 1. Secure Communication: Encryption is used to secure communication channels, such as email, messaging apps, and virtual private networks (VPNs). It ensures that only authorized parties can access and understand the transmitted data. 2. Data Protection: Encryption is used to protect stored data, such as files and databases. It prevents unauthorized access to sensitive information, even if the storage medium is compromised. 3. Password Storage: Encryption is used to securely store passwords. Instead of storing passwords in plain text, they are encrypted and stored in a hashed form. This adds an extra layer of security and protects against password breaches. 4. Secure Web Browsing: Encryption is used to secure web browsing through the use of secure protocols, such as HTTPS. It ensures that data transmitted between a web browser and a website is encrypted and cannot be intercepted or tampered with. 5. Digital Signatures: Encryption is used to create digital signatures, which are used to verify the authenticity and integrity of digital documents. Digital signatures provide a way to ensure that a document has not been altered and that it was signed by the intended sender. 6. Secure File Transfer: Encryption is used to secure file transfers, such as file uploads and downloads. It ensures that files are encrypted during transit, preventing unauthorized access or tampering. Encryption plays a crucial role in computer security by protecting sensitive data and ensuring the confidentiality and integrity of information. It is an essential tool for safeguarding data in various applications and environments. Let's say you want to send a confidential document to a colleague. You can encrypt the document using encryption software or tools. This ensures that only the intended recipient, who has the decryption key, can access and read the document. Even if the document is intercepted during transmission, it remains secure and unreadable to unauthorized parties. ## Exercise Describe an application or scenario where encryption is used to protect sensitive data. ### Solution One application of encryption is in online banking. When you access your bank's website, the data transmitted between your computer and the bank's servers is encrypted using secure protocols, such as HTTPS. This ensures that your login credentials, account information, and financial transactions are protected from unauthorized access or interception. Encryption plays a crucial role in securing online banking and protecting sensitive financial data. # 4. Firewalls Firewalls are an essential component of network security. They act as a barrier between a trusted internal network and an untrusted external network, such as the internet. Firewalls monitor and control incoming and outgoing network traffic based on predetermined security rules. Firewalls can be implemented in various forms, including hardware appliances, software applications, and cloud-based services. They can be configured to filter traffic based on IP addresses, ports, protocols, and other criteria. By enforcing security policies, firewalls help prevent unauthorized access to a network and protect against malicious activities. There are different types of firewalls that serve specific purposes: 1. Packet Filtering Firewalls: These firewalls examine each packet of data that enters or leaves a network. They compare the packet's source and destination IP addresses, ports, and protocols against a set of predefined rules. If a packet matches a rule, it is either allowed or blocked. 2. Stateful Inspection Firewalls: These firewalls go beyond packet filtering by keeping track of the state of network connections. They maintain information about established connections and use this information to make more informed decisions about allowing or blocking traffic. Stateful inspection firewalls provide better security by understanding the context of network traffic. 3. Application-Level Gateways (Proxy Firewalls): These firewalls act as intermediaries between clients and servers. They inspect the entire application-layer protocol, such as HTTP or FTP, and make decisions based on the content of the traffic. Application-level gateways provide advanced security features but can introduce latency due to the additional processing required. 4. Next-Generation Firewalls (NGFW): These firewalls combine traditional firewall functionality with additional security features, such as intrusion prevention, antivirus, and web filtering. NGFWs provide more advanced threat detection and prevention capabilities, making them suitable for modern network security needs. Let's say you have a network with sensitive customer data that you want to protect from unauthorized access. You can set up a firewall to allow incoming traffic only from trusted IP addresses and block all other traffic. This ensures that only authorized users can access the network and reduces the risk of data breaches. ## Exercise Consider a scenario where you have a network with multiple departments, each with different security requirements. Design a firewall configuration that allows the following: - The marketing department can access the internet and receive emails. - The finance department can access the internet, receive emails, and access a specific financial application on a remote server. - The HR department can access the internet, receive emails, and access a file server for employee records. ### Solution To meet the requirements, you can configure the firewall with the following rules: - Allow outgoing traffic from all departments to the internet. - Allow incoming email traffic to all departments. - Allow incoming traffic from the finance department to the remote server hosting the financial application. - Allow incoming traffic from the HR department to the file server hosting employee records. All other incoming traffic should be blocked. This configuration ensures that each department has the necessary access while maintaining network security. # 4.1. Types of Firewalls There are several types of firewalls that can be used to protect a network. Each type has its own strengths and weaknesses, and the choice of firewall depends on the specific security requirements of the network. 1. Packet Filtering Firewalls: Packet filtering firewalls examine each packet of data that enters or leaves a network. They compare the packet's source and destination IP addresses, ports, and protocols against a set of predefined rules. If a packet matches a rule, it is either allowed or blocked. Packet filtering firewalls are fast and efficient, but they provide limited security as they only inspect individual packets and do not have knowledge of the context of the traffic. 2. Stateful Inspection Firewalls: Stateful inspection firewalls go beyond packet filtering by keeping track of the state of network connections. They maintain information about established connections and use this information to make more informed decisions about allowing or blocking traffic. Stateful inspection firewalls provide better security by understanding the context of network traffic, but they can introduce some latency due to the additional processing required. 3. Application-Level Gateways (Proxy Firewalls): Application-level gateways, also known as proxy firewalls, act as intermediaries between clients and servers. They inspect the entire application-layer protocol, such as HTTP or FTP, and make decisions based on the content of the traffic. Application-level gateways provide advanced security features, such as content filtering and data loss prevention, but they can introduce significant latency due to the additional processing required. 4. Next-Generation Firewalls (NGFW): Next-generation firewalls combine traditional firewall functionality with additional security features, such as intrusion prevention, antivirus, and web filtering. NGFWs provide more advanced threat detection and prevention capabilities, making them suitable for modern network security needs. They can inspect traffic at the application layer and provide granular control over network traffic. Let's say you have a network with multiple departments, each with different security requirements. The marketing department needs access to the internet and email, while the finance department needs access to the internet, email, and a specific financial application on a remote server. In this case, a next-generation firewall would be a good choice as it can provide the necessary security features, such as application-level filtering and intrusion prevention, to meet the requirements of both departments. ## Exercise Consider a scenario where you have a network with sensitive customer data that you want to protect from unauthorized access. Which type of firewall would you choose and why? ### Solution In this scenario, a stateful inspection firewall would be a good choice. Stateful inspection firewalls keep track of the state of network connections, which allows them to make more informed decisions about allowing or blocking traffic. This would provide better security for sensitive customer data by understanding the context of the network traffic. # 4.2. Firewall Configurations and Rules Firewalls can be configured in different ways to meet the specific security requirements of a network. The configuration of a firewall includes defining rules that determine how traffic is allowed or blocked. These rules are based on criteria such as source and destination IP addresses, ports, and protocols. Firewall rules are typically organized into a set of policies that define the overall security posture of the network. Each policy consists of a set of rules that are applied in a specific order. When a packet arrives at the firewall, it is compared against the rules in the policies, and the first matching rule determines whether the packet is allowed or blocked. There are two main types of firewall configurations: 1. Default Deny: In a default deny configuration, all traffic is blocked by default, and only explicitly allowed traffic is allowed to pass through the firewall. This is a more secure configuration as it minimizes the attack surface by blocking all traffic that is not explicitly permitted. However, it requires careful configuration and maintenance to ensure that legitimate traffic is not inadvertently blocked. 2. Default Allow: In a default allow configuration, all traffic is allowed by default, and only explicitly blocked traffic is blocked by the firewall. This configuration is more permissive and easier to implement, but it can leave the network vulnerable to unauthorized access if the firewall rules are not properly configured. Let's say you have a network with a default deny firewall configuration. You want to allow incoming web traffic to a web server located in your network. To do this, you would need to create a rule in the firewall that allows incoming traffic on port 80 (HTTP). This rule would specify the source IP address or range of IP addresses from which the traffic is allowed, as well as the destination IP address of the web server. ## Exercise Consider a scenario where you have a network with multiple departments, each with different security requirements. The marketing department needs access to the internet and email, while the finance department needs access to the internet, email, and a specific financial application on a remote server. Design a firewall configuration that meets the requirements of both departments. ### Solution To meet the requirements of both departments, you could create two policies in the firewall: 1. Marketing Policy: - Allow outgoing traffic to the internet on ports 80 (HTTP) and 443 (HTTPS). - Allow outgoing traffic on port 25 (SMTP) for email. 2. Finance Policy: - Allow outgoing traffic to the internet on ports 80 (HTTP) and 443 (HTTPS). - Allow outgoing traffic on port 25 (SMTP) for email. - Allow outgoing traffic to the specific IP address and port of the remote financial application. By applying these policies in the firewall, you can ensure that the marketing department has access to the internet and email, while the finance department has access to the internet, email, and the specific financial application. # 4.3. Intrusion Detection and Prevention Intrusion detection and prevention systems (IDPS) are an important component of network security. They help to detect and prevent unauthorized access to a network by monitoring network traffic and identifying suspicious activity. There are two main types of IDPS: 1. Intrusion Detection System (IDS): An IDS monitors network traffic and analyzes it for signs of suspicious activity or known attack patterns. When an IDS detects a potential intrusion, it generates an alert to notify network administrators. However, an IDS does not take any direct action to prevent the intrusion. 2. Intrusion Prevention System (IPS): An IPS not only detects suspicious activity but also takes proactive measures to prevent the intrusion. It can automatically block or drop network traffic that is identified as malicious or unauthorized. This helps to protect the network in real-time. Let's say you have an IDS deployed in your network. It monitors incoming and outgoing network traffic and detects a series of login attempts from an unknown IP address. The IDS analyzes the login patterns and determines that it is a brute-force attack attempting to gain unauthorized access to a server. The IDS generates an alert and sends it to the network administrators. The administrators can then take appropriate action, such as blocking the IP address or implementing additional security measures to prevent the attack. ## Exercise Consider a scenario where you have an IPS deployed in your network. An IPS detects a suspicious file transfer from an internal user's computer to an external server. Design a rule for the IPS to prevent this unauthorized file transfer. ### Solution To prevent the unauthorized file transfer, you could create a rule in the IPS that blocks any outgoing network traffic containing specific file extensions or file names. For example, you could block outgoing traffic that contains files with extensions like .doc, .xls, .pdf, or specific file names like "confidential.doc" or "financial_report.xls". By implementing this rule, the IPS will automatically block any attempt to transfer files with the specified extensions or names, thereby preventing unauthorized file transfers. # 4.4. Best Practices for Firewall Management Effective firewall management is crucial for maintaining network security. Firewalls act as a barrier between your internal network and the external world, protecting your systems from unauthorized access and potential threats. Here are some best practices for firewall management: 1. Regularly update firewall firmware and software: Manufacturers release updates to address security vulnerabilities and improve performance. It's important to keep your firewall up to date to ensure it can effectively protect your network. 2. Implement a strong password policy: Use strong, unique passwords for your firewall and regularly change them. Avoid using default passwords, as they are often well-known and easily exploited by attackers. 3. Enable logging and monitoring: Configure your firewall to log all network traffic and regularly review the logs for any suspicious activity. This will help you identify potential security breaches and take appropriate action. 4. Restrict access to the firewall: Only authorized personnel should have access to the firewall's administration interface. Implement strict access controls and regularly review user accounts to ensure that only necessary individuals have access. 5. Regularly review and update firewall rules: Firewall rules define what traffic is allowed or blocked. Regularly review these rules to ensure they are up to date and aligned with your organization's security policies. Remove any unnecessary rules to minimize potential vulnerabilities. 6. Implement a multi-layered security approach: Firewalls are just one component of a comprehensive security strategy. Combine your firewall with other security measures, such as intrusion detection and prevention systems, antivirus software, and employee training, to create multiple layers of defense. 7. Conduct regular security audits: Perform periodic audits of your firewall configuration and rule sets to identify any misconfigurations or vulnerabilities. This will help you proactively address any security gaps and ensure your firewall is effectively protecting your network. By following these best practices, you can enhance the security of your network and reduce the risk of unauthorized access and potential threats. Remember, firewall management is an ongoing process, and it's important to regularly review and update your firewall to adapt to evolving security threats. # 5. Network Security # 5.1. Network Architecture and Protocols Network architecture refers to the design and layout of a network, including its components and their interconnections. It determines how data is transmitted, routed, and controlled within the network. A well-designed network architecture is essential for ensuring the secure and efficient flow of information. There are different types of network architectures, such as local area networks (LANs), wide area networks (WANs), and metropolitan area networks (MANs). Each type has its own characteristics and requirements. Network protocols are a set of rules and procedures that govern how data is transmitted and received in a network. They define the format and structure of data packets, as well as the methods for error detection, correction, and flow control. Common network protocols include TCP/IP, Ethernet, and Wi-Fi. Understanding network architecture and protocols is crucial for implementing effective network security measures. It allows you to identify potential vulnerabilities and design appropriate security controls to protect your network from unauthorized access and attacks. For example, let's consider a company that has multiple branches located in different cities. To connect these branches and enable communication between them, the company may use a wide area network (WAN). The WAN architecture would involve routers, switches, and other network devices to establish and maintain connections between the branches. The network protocol used in this scenario could be TCP/IP, which is the standard protocol for internet communication. TCP/IP ensures that data packets are transmitted reliably and in the correct order, allowing for seamless communication between the branches. ## Exercise Research and identify two different network architectures commonly used in organizations. Explain their characteristics and advantages. ### Solution 1. Local Area Network (LAN): A LAN is a network that covers a small geographical area, such as an office building or a campus. It is typically owned and controlled by a single organization. LANs provide high-speed connectivity and allow for the sharing of resources, such as printers and servers, among connected devices. They are relatively easy to set up and manage, making them cost-effective for small to medium-sized organizations. 2. Wide Area Network (WAN): A WAN is a network that spans a large geographical area, such as multiple cities or countries. It connects multiple LANs and allows for communication between them. WANs are typically used by large organizations with multiple branches or offices. They provide long-distance connectivity and can support a large number of users and devices. However, WANs are more complex and expensive to set up and maintain compared to LANs. # 5.2. Securing Network Devices Securing network devices is crucial for maintaining the integrity and confidentiality of your network. Network devices, such as routers, switches, and firewalls, are the backbone of your network infrastructure and are often targeted by attackers. Here are some best practices for securing network devices: 1. Change default passwords: Network devices often come with default passwords that are well-known to attackers. It is important to change these passwords to strong, unique passwords that are not easily guessable. 2. Disable unnecessary services: Network devices may come with a variety of services enabled by default. It is important to disable any services that are not needed for the proper functioning of the device. This reduces the attack surface and minimizes the risk of vulnerabilities. 3. Keep devices up to date: Regularly update the firmware and software of your network devices. Manufacturers release updates to fix security vulnerabilities and improve performance. Keeping your devices up to date ensures that you have the latest security patches. 4. Implement access controls: Use access control lists (ACLs) to restrict access to your network devices. Only allow authorized users to access the devices and configure them. This helps prevent unauthorized changes to the device settings. 5. Enable logging and monitoring: Enable logging on your network devices to capture information about network activity. Regularly review the logs for any suspicious activity or signs of compromise. Implement monitoring tools to alert you of any unusual network behavior. For example, let's consider a router. Routers are responsible for directing network traffic between different networks. To secure a router, you can follow these steps: 1. Change the default password: Routers often come with default passwords that are well-known to attackers. Change the password to a strong, unique password. 2. Disable unnecessary services: Disable any services that are not needed, such as remote management or file sharing. This reduces the attack surface and minimizes the risk of vulnerabilities. 3. Enable firewall: Enable the built-in firewall on the router to filter incoming and outgoing traffic. Configure the firewall rules to allow only necessary traffic. 4. Update firmware: Regularly check for firmware updates from the router manufacturer and apply them. Firmware updates often include security patches and bug fixes. 5. Implement access controls: Use access control lists (ACLs) to restrict access to the router's management interface. Only allow authorized users to access and configure the router. ## Exercise You are the network administrator for a small company. You have recently installed a new router and need to secure it. Perform the following tasks to enhance the security of the router: 1. Change the default password to a strong, unique password. 2. Disable remote management of the router. 3. Enable the built-in firewall and configure the firewall rules to allow only necessary traffic. 4. Check for firmware updates from the router manufacturer and apply them. 5. Restrict access to the router's management interface to authorized users only. ### Solution 1. Change the default password to a strong, unique password: - Log in to the router's management interface. - Navigate to the password settings. - Enter a strong, unique password and save the changes. 2. Disable remote management of the router: - Navigate to the remote management settings. - Disable remote management and save the changes. 3. Enable the built-in firewall and configure the firewall rules: - Navigate to the firewall settings. - Enable the firewall and configure the rules to allow only necessary traffic. - Save the changes. 4. Check for firmware updates from the router manufacturer: - Visit the manufacturer's website and look for firmware updates for your router model. - Download the latest firmware update. - Follow the manufacturer's instructions to apply the firmware update. 5. Restrict access to the router's management interface: - Navigate to the access control settings. - Configure access control lists (ACLs) to allow only authorized users to access the router's management interface. - Save the changes. # 5.3. Wireless Network Security 1. Change default settings: Wireless routers often come with default settings that are well-known to attackers. It is important to change the default settings, including the network name (SSID) and the administrative password. 2. Use strong encryption: Enable encryption on your wireless network to protect the data transmitted over the network. The most common encryption protocols are WPA2 (Wi-Fi Protected Access 2) and WPA3, which provide strong security. Avoid using outdated encryption protocols like WEP (Wired Equivalent Privacy), as they are easily cracked. 3. Use a strong password: Choose a strong, unique password for your wireless network. A strong password should be at least 12 characters long and include a combination of uppercase and lowercase letters, numbers, and special characters. 4. Enable network segmentation: If you have multiple wireless devices, consider creating separate networks or VLANs (Virtual Local Area Networks) for different types of devices. This helps isolate devices from each other and adds an extra layer of security. 5. Disable SSID broadcasting: By default, wireless routers broadcast the network name (SSID) to make it easier for devices to connect. However, this also makes it easier for attackers to identify and target your network. Disable SSID broadcasting to make your network less visible. 6. Enable MAC address filtering: MAC address filtering allows you to specify which devices are allowed to connect to your wireless network. You can create a list of approved MAC addresses and only allow devices with those addresses to connect. 7. Regularly update firmware: Keep your wireless router's firmware up to date by checking for updates from the manufacturer. Firmware updates often include security patches and bug fixes. 8. Monitor network activity: Use network monitoring tools to monitor the activity on your wireless network. Look for any suspicious devices or unusual network behavior that could indicate a security breach. For example, let's consider a home wireless network. To secure your home wireless network, you can follow these steps: 1. Change the default settings: Log in to your wireless router's administration interface and change the default network name (SSID) and password to unique values. 2. Enable WPA2 or WPA3 encryption: In the wireless settings, enable WPA2 or WPA3 encryption and choose a strong encryption passphrase. 3. Use a strong password: Choose a strong, unique password for your wireless network. Avoid using common passwords or easily guessable passwords. 4. Enable network segmentation: If you have smart home devices, consider creating a separate network or VLAN for them. This helps isolate them from your main network. 5. Disable SSID broadcasting: In the wireless settings, disable the broadcasting of your network's SSID. This makes your network less visible to attackers. 6. Enable MAC address filtering: In the wireless settings, enable MAC address filtering and add the MAC addresses of your devices to the allowed list. 7. Update firmware: Check for firmware updates from the manufacturer and apply them to your wireless router. 8. Monitor network activity: Use a network monitoring tool to keep an eye on the devices connected to your network and look for any suspicious activity. ## Exercise You are setting up a wireless network for your small office. Perform the following tasks to enhance the security of the wireless network: 1. Change the default network name (SSID) to a unique value. 2. Enable WPA2 or WPA3 encryption and choose a strong encryption passphrase. 3. Choose a strong, unique password for the wireless network. 4. Create a separate network or VLAN for guest devices. 5. Disable the broadcasting of the network's SSID. 6. Enable MAC address filtering and add the MAC addresses of authorized devices. 7. Check for firmware updates from the manufacturer and apply them to the wireless router. 8. Set up a network monitoring tool to monitor the activity on the wireless network. ### Solution 1. Change the default network name (SSID) to a unique value: - Log in to the wireless router's administration interface. - Navigate to the wireless settings. - Change the network name (SSID) to a unique value. - Save the changes. 2. Enable WPA2 or WPA3 encryption and choose a strong encryption passphrase: - In the wireless settings, enable WPA2 or WPA3 encryption. - Choose a strong encryption passphrase. - Save the changes. 3. Choose a strong, unique password for the wireless network: - In the wireless settings, navigate to the password settings. - Enter a strong, unique password for the wireless network. - Save the changes. 4. Create a separate network or VLAN for guest devices: - In the wireless settings or network settings, create a separate network or VLAN for guest devices. - Configure the network or VLAN settings according to your requirements. - Save the changes. 5. Disable the broadcasting of the network's SSID: - In the wireless settings, disable the broadcasting of the network's SSID. - Save the changes. 6. Enable MAC address filtering and add the MAC addresses of authorized devices: - In the wireless settings, enable MAC address filtering. - Add the MAC addresses of authorized devices to the allowed list. - Save the changes. 7. Check for firmware updates from the manufacturer and apply them to the wireless router: - Visit the manufacturer's website and look for firmware updates for your wireless router model. - Download the latest firmware update. - Follow the manufacturer's instructions to apply the firmware update. 8. Set up a network monitoring tool to monitor the activity on the wireless network: - Install a network monitoring tool on a computer connected to the wireless network. - Configure the tool to monitor the wireless network activity. - Regularly review the monitoring data for any suspicious activity or signs of compromise. # 5.4. Virtual Private Networks (VPNs) A Virtual Private Network (VPN) is a technology that allows users to securely connect to a private network over a public network, such as the internet. VPNs provide a secure and encrypted connection, ensuring that data transmitted between the user and the private network remains confidential and protected from unauthorized access. VPNs are commonly used by organizations to allow employees to access internal resources and systems remotely. They are also used by individuals who want to protect their online privacy and secure their internet connections, especially when using public Wi-Fi networks. There are two main types of VPNs: remote access VPNs and site-to-site VPNs. - Remote access VPNs: These VPNs are used by individual users to connect to a private network remotely. When a user connects to a remote access VPN, their device creates a secure tunnel to the private network, encrypting all data transmitted between the user and the network. This allows the user to access resources and services on the private network as if they were physically present at the network's location. - Site-to-site VPNs: These VPNs are used to connect multiple networks together over a public network. Site-to-site VPNs are commonly used by organizations with multiple locations or branch offices. The VPN creates a secure connection between the different networks, allowing users from each network to access resources and services on the other network. This enables seamless communication and collaboration between the different locations. Let's consider an example of a remote access VPN. Sarah works for a company that has implemented a remote access VPN to allow employees to work from home. When Sarah wants to access the company's internal resources, she uses a VPN client on her laptop to connect to the VPN server located at the company's headquarters. When Sarah initiates the VPN connection, her laptop establishes a secure tunnel to the VPN server. All data transmitted between Sarah's laptop and the VPN server is encrypted, ensuring that it cannot be intercepted or accessed by unauthorized parties. Once the connection is established, Sarah can access the company's internal systems, files, and applications as if she were physically present at the office. ## Exercise You are setting up a remote access VPN for your organization. Perform the following tasks to ensure the security and functionality of the VPN: 1. Choose a VPN protocol: Research different VPN protocols, such as OpenVPN, IPSec, and L2TP/IPSec. Consider the security, compatibility, and performance of each protocol before making a decision. 2. Set up a VPN server: Install and configure a VPN server on your organization's network. Ensure that the server has sufficient resources to handle the expected number of VPN connections. 3. Configure user authentication: Implement a strong authentication mechanism for VPN users, such as username/password authentication, two-factor authentication, or certificate-based authentication. 4. Enable encryption: Enable encryption on the VPN server to ensure that all data transmitted between the user and the server is encrypted. Use strong encryption algorithms, such as AES (Advanced Encryption Standard). 5. Set up firewall rules: Configure firewall rules on the VPN server to restrict access to the server and protect it from unauthorized access. Only allow incoming VPN connections from trusted IP addresses or networks. 6. Test the VPN connection: Test the VPN connection from a remote device to ensure that it is working correctly. Verify that the connection is secure and that all data transmitted over the VPN is encrypted. ### Solution 1. Choose a VPN protocol: After researching different VPN protocols, we have decided to use OpenVPN for our remote access VPN. OpenVPN is a widely used and highly secure protocol that provides excellent compatibility and performance. 2. Set up a VPN server: We will install and configure an OpenVPN server on our organization's network. The server will be located at our headquarters and will have sufficient resources to handle the expected number of VPN connections. 3. Configure user authentication: We will implement two-factor authentication for VPN users. Users will need to enter their username and password, as well as a one-time verification code generated by a mobile authentication app. 4. Enable encryption: We will enable AES-256 encryption on the VPN server to ensure that all data transmitted between the user and the server is encrypted. AES-256 is a strong encryption algorithm that provides a high level of security. 5. Set up firewall rules: We will configure firewall rules on the VPN server to restrict access to the server. Only incoming VPN connections from trusted IP addresses or networks will be allowed. All other connections will be blocked. 6. Test the VPN connection: We will test the VPN connection from a remote device to ensure that it is working correctly. We will verify that the connection is secure by checking that all data transmitted over the VPN is encrypted. # 6. Threats and Attacks In the world of computer security, threats and attacks are constant concerns. Understanding the different types of threats and common attack techniques is crucial for protecting your systems and data. By being aware of these threats and knowing how to defend against them, you can significantly reduce the risk of a security breach. There are various types of threats that can compromise the security of computer systems and networks. Some common types of threats include: 1. Malware: Malicious software, such as viruses, worms, Trojans, and ransomware, that can damage or disrupt computer systems and steal sensitive information. 2. Phishing: A type of social engineering attack where attackers impersonate legitimate entities to trick individuals into revealing sensitive information, such as passwords or credit card numbers. 3. Denial of Service (DoS) attacks: Attacks that overwhelm a system or network with a flood of traffic, making it inaccessible to legitimate users. 4. Man-in-the-middle attacks: Attacks where an attacker intercepts and alters communication between two parties without their knowledge. 5. SQL injection: An attack technique where an attacker inserts malicious SQL code into a web application's database query, allowing them to manipulate or retrieve sensitive data. 6. Zero-day exploits: Attacks that take advantage of vulnerabilities in software or systems that are unknown to the vendor and for which no patch or fix is available. Let's consider an example of a phishing attack. Sarah receives an email that appears to be from her bank, asking her to update her account information. The email looks legitimate, with the bank's logo and branding. However, when Sarah clicks on the link in the email and enters her login credentials, she unknowingly provides her username and password to an attacker. ## Exercise Match the following types of attacks with their descriptions: 1. Malware 2. Phishing 3. Denial of Service (DoS) attacks 4. Man-in-the-middle attacks 5. SQL injection 6. Zero-day exploits a. Attacks that overwhelm a system or network with a flood of traffic, making it inaccessible to legitimate users. b. Attacks where an attacker intercepts and alters communication between two parties without their knowledge. c. Malicious software that can damage or disrupt computer systems and steal sensitive information. d. A type of social engineering attack where attackers impersonate legitimate entities to trick individuals into revealing sensitive information. e. An attack technique where an attacker inserts malicious SQL code into a web application's database query, allowing them to manipulate or retrieve sensitive data. f. Attacks that take advantage of vulnerabilities in software or systems that are unknown to the vendor and for which no patch or fix is available. ### Solution 1. c 2. d 3. a 4. b 5. e 6. f # 6.1. Types of Threats There are several types of threats that can compromise the security of computer systems and networks. It's important to understand these threats in order to effectively protect against them. Here are some common types of threats: 1. Malware: Malware, short for malicious software, refers to any software designed to harm or exploit computer systems. This includes viruses, worms, Trojans, ransomware, and spyware. Malware can damage or disrupt systems, steal sensitive information, or provide unauthorized access to a system. 2. Phishing: Phishing is a type of social engineering attack where attackers impersonate legitimate entities, such as banks or online services, to trick individuals into revealing sensitive information. This is usually done through deceptive emails, websites, or phone calls that appear to be from a trusted source. 3. Denial of Service (DoS) attacks: DoS attacks aim to overwhelm a system or network with a flood of traffic, making it inaccessible to legitimate users. This is typically achieved by sending a large volume of requests or exploiting vulnerabilities in the target system's resources. 4. Man-in-the-middle attacks: In a man-in-the-middle attack, an attacker intercepts and alters communication between two parties without their knowledge. This allows the attacker to eavesdrop on sensitive information, modify messages, or impersonate one of the parties involved. 5. SQL injection: SQL injection is an attack technique where an attacker inserts malicious SQL code into a web application's database query. This can allow the attacker to manipulate or retrieve sensitive data, bypass authentication mechanisms, or even execute arbitrary commands on the database server. 6. Zero-day exploits: Zero-day exploits target vulnerabilities in software or systems that are unknown to the vendor and for which no patch or fix is available. Attackers can exploit these vulnerabilities before they are discovered and patched, making them particularly dangerous. It's important to stay vigilant and implement appropriate security measures to protect against these threats. Regularly updating software, using strong passwords, and being cautious of suspicious emails or websites are some best practices to minimize the risk of a security breach. # 6.2. Common Attack Techniques Attackers use a variety of techniques to exploit vulnerabilities and gain unauthorized access to computer systems and networks. Understanding these common attack techniques can help in identifying and mitigating potential security risks. Here are some common attack techniques: 1. Password attacks: Attackers may use various methods to guess or crack passwords, such as brute force attacks, dictionary attacks, or rainbow table attacks. It's important to use strong, unique passwords and implement measures like account lockouts and password complexity requirements to prevent password attacks. 2. Social engineering: Social engineering involves manipulating individuals to divulge sensitive information or perform actions that may compromise security. This can include techniques like phishing, pretexting, baiting, or tailgating. Training employees to recognize and report social engineering attempts is crucial in preventing such attacks. 3. Malware distribution: Attackers often use various methods to distribute malware, such as email attachments, malicious websites, or infected software downloads. It's important to have robust antivirus and anti-malware solutions in place and educate users about the risks of downloading or opening suspicious files. 4. Network sniffing: Network sniffing involves capturing and analyzing network traffic to intercept sensitive information, such as passwords or confidential data. Encrypting network traffic using protocols like SSL/TLS can help protect against network sniffing attacks. 5. SQL injection: SQL injection attacks exploit vulnerabilities in web applications that allow attackers to manipulate or retrieve data from a database. Proper input validation and parameterized queries can help prevent SQL injection attacks. 6. Cross-site scripting (XSS): XSS attacks involve injecting malicious scripts into web pages viewed by other users. This can allow attackers to steal sensitive information or perform unauthorized actions on behalf of the user. Proper input validation and output encoding can help prevent XSS attacks. 7. Distributed Denial of Service (DDoS): DDoS attacks involve overwhelming a target system or network with a flood of traffic, rendering it inaccessible to legitimate users. Implementing DDoS mitigation techniques, such as traffic filtering or rate limiting, can help mitigate the impact of DDoS attacks. These are just a few examples of common attack techniques. It's important to stay informed about the latest security threats and vulnerabilities and implement appropriate security measures to protect against them. Regular security assessments and updates to security policies and procedures are essential in maintaining a secure computing environment. # 6.3. Malware and Antivirus Malware, short for malicious software, refers to any software designed to harm or exploit computer systems or networks. Malware can take various forms, including viruses, worms, Trojans, ransomware, spyware, and adware. It can be distributed through infected email attachments, malicious websites, or compromised software downloads. The impact of malware can be devastating, ranging from data breaches and financial loss to system crashes and unauthorized access. To protect against malware, it's important to have robust antivirus and anti-malware solutions in place. These solutions can detect and remove known malware, as well as provide real-time protection against new and emerging threats. Antivirus software works by scanning files and programs for known malware signatures. It can also monitor system activity for suspicious behavior and block or quarantine potentially malicious files. Regularly updating antivirus software is crucial to ensure protection against the latest malware threats. In addition to using antivirus software, it's important to practice safe browsing habits and exercise caution when downloading or opening files. Avoid clicking on suspicious links or downloading files from untrusted sources. Keep your operating system and software up to date with the latest security patches to prevent vulnerabilities that malware can exploit. It's worth noting that while antivirus software is an important layer of defense against malware, it's not foolproof. New malware variants are constantly being developed, and some may go undetected by antivirus software. Therefore, it's important to have a multi-layered approach to security, including regular backups, network monitoring, and user education on recognizing and reporting potential security threats. - A user receives an email with an attachment claiming to be an invoice. The user opens the attachment, unknowingly activating a ransomware that encrypts all the files on their computer and demands a ransom for their release. - A user visits a compromised website that contains malicious code. The code exploits a vulnerability in the user's web browser, allowing the attacker to install a keylogger that captures the user's keystrokes and sends them to the attacker. ## Exercise 1. What is malware? 2. What are some common forms of malware? 3. How does antivirus software work? 4. What are some best practices for protecting against malware? ### Solution 1. Malware refers to any software designed to harm or exploit computer systems or networks. 2. Common forms of malware include viruses, worms, Trojans, ransomware, spyware, and adware. 3. Antivirus software scans files and programs for known malware signatures and monitors system activity for suspicious behavior. It can block or quarantine potentially malicious files. 4. Best practices for protecting against malware include using robust antivirus and anti-malware solutions, practicing safe browsing habits, keeping software up to date with security patches, and regularly backing up important data. # 6.4. Social Engineering Attacks Social engineering attacks are a type of cybersecurity attack that relies on psychological manipulation and deception to trick individuals into revealing sensitive information or performing actions that may compromise security. These attacks exploit human vulnerabilities rather than technical vulnerabilities in computer systems. Social engineering attacks can take many forms, including phishing, pretexting, baiting, and tailgating. Phishing is one of the most common types of social engineering attacks, where attackers send emails or messages that appear to be from a legitimate source, such as a bank or a company, in an attempt to trick recipients into providing their personal information, such as passwords or credit card numbers. Pretexting involves creating a false scenario or pretext to gain the trust of the target and convince them to disclose sensitive information. For example, an attacker may impersonate a company employee and call the target, claiming to need their login credentials for a system upgrade. Baiting involves enticing the target with something of value, such as a free USB drive or a gift card, that is infected with malware. When the target uses the infected device, the malware is installed on their system. Tailgating, also known as piggybacking, involves an attacker following an authorized person into a restricted area without proper authorization. This can be done by pretending to be a delivery person or simply asking someone to hold the door open. To protect against social engineering attacks, it's important to be vigilant and skeptical of any requests for personal information or actions that seem unusual or suspicious. Never provide sensitive information in response to unsolicited requests, and verify the authenticity of any requests through a trusted channel, such as contacting the organization directly. It's also important to educate employees and individuals about the risks and tactics used in social engineering attacks. Training programs can help raise awareness and teach individuals how to recognize and respond to social engineering attempts. - An employee receives an email from their manager requesting their login credentials for a system upgrade. The email appears to be legitimate, with the company logo and the manager's name and email address. The employee, thinking it's a legitimate request, provides their credentials, unknowingly giving the attacker access to sensitive company information. - A person receives a phone call from someone claiming to be from their bank. The caller asks for their account number and social security number to verify their identity. The person, thinking it's a legitimate call, provides the information, only to later discover that it was a scam and their personal information has been compromised. ## Exercise 1. What are social engineering attacks? 2. What are some common types of social engineering attacks? 3. How can individuals protect themselves against social engineering attacks? 4. Why is employee education important in preventing social engineering attacks? ### Solution 1. Social engineering attacks are cybersecurity attacks that rely on psychological manipulation and deception to trick individuals into revealing sensitive information or performing actions that may compromise security. 2. Common types of social engineering attacks include phishing, pretexting, baiting, and tailgating. 3. Individuals can protect themselves against social engineering attacks by being vigilant and skeptical of requests for personal information or unusual actions. They should never provide sensitive information in response to unsolicited requests and should verify the authenticity of requests through a trusted channel. 4. Employee education is important in preventing social engineering attacks because it raises awareness about the risks and tactics used in these attacks. It teaches employees how to recognize and respond to social engineering attempts, reducing the likelihood of falling victim to such attacks. # 7. Securing Operating Systems and Applications Securing operating systems and applications is crucial for maintaining the overall security of a computer system. Operating systems and applications are often targeted by attackers because they provide access to sensitive data and can be exploited to gain unauthorized access or control over a system. One of the most important steps in securing an operating system is keeping it up to date with the latest security patches and updates. Operating system vendors regularly release patches to fix vulnerabilities and address security issues. It's important to install these patches as soon as they become available to ensure that your system is protected against known vulnerabilities. In addition to keeping the operating system up to date, it's also important to configure it securely. This includes setting strong passwords for user accounts, disabling unnecessary services and features, and enabling firewalls and other security measures. Applications, such as web browsers and office productivity software, also need to be kept up to date with the latest security patches. Attackers often target vulnerabilities in popular applications to gain access to systems. Regularly updating applications and using reputable sources for software downloads can help protect against these attacks. Another important aspect of securing operating systems and applications is implementing secure configurations. This involves configuring the system and applications to follow security best practices, such as disabling unnecessary services, limiting user privileges, and enabling encryption where appropriate. Data backup and recovery is also an important part of securing operating systems and applications. Regularly backing up important data ensures that it can be recovered in the event of a security incident or system failure. It's important to store backups in a secure location and test the restoration process periodically to ensure its effectiveness. Overall, securing operating systems and applications requires a combination of proactive measures, such as patching and secure configurations, as well as reactive measures, such as data backup and recovery. By implementing these measures, you can help protect your computer system from security threats and minimize the impact of any potential incidents. - Keeping your operating system up to date with the latest security patches and updates is important for maintaining its security. Regularly check for updates and install them as soon as they become available. - Configuring your operating system securely involves setting strong passwords for user accounts, disabling unnecessary services and features, and enabling firewalls and other security measures. - Regularly updating your applications, such as web browsers and office productivity software, is important for protecting against vulnerabilities that attackers may exploit. - Implementing secure configurations for your operating system and applications involves following security best practices, such as disabling unnecessary services, limiting user privileges, and enabling encryption where appropriate. - Regularly backing up your important data and testing the restoration process is important for ensuring that you can recover from a security incident or system failure. ## Exercise 1. Why is it important to keep your operating system up to date with the latest security patches and updates? 2. What are some steps you can take to configure your operating system securely? 3. Why is it important to regularly update your applications? 4. What are some best practices for implementing secure configurations for your operating system and applications? 5. Why is data backup and recovery important for securing operating systems and applications? ### Solution 1. It is important to keep your operating system up to date with the latest security patches and updates because they fix vulnerabilities and address security issues. By installing these patches, you can ensure that your system is protected against known vulnerabilities. 2. Some steps you can take to configure your operating system securely include setting strong passwords for user accounts, disabling unnecessary services and features, and enabling firewalls and other security measures. 3. It is important to regularly update your applications because attackers often target vulnerabilities in popular applications to gain access to systems. By updating your applications, you can protect against these attacks. 4. Some best practices for implementing secure configurations for your operating system and applications include disabling unnecessary services, limiting user privileges, and enabling encryption where appropriate. 5. Data backup and recovery is important for securing operating systems and applications because it ensures that you can recover important data in the event of a security incident or system failure. By regularly backing up your data and testing the restoration process, you can minimize the impact of any potential incidents. # 7.1. Updates and Patches Keeping your operating system up to date with the latest security patches and updates is crucial for maintaining its security. Operating system vendors regularly release patches to fix vulnerabilities and address security issues that have been discovered. These vulnerabilities can be exploited by attackers to gain unauthorized access or control over your system. By installing the latest patches and updates, you can ensure that your operating system is protected against known vulnerabilities. These patches often include fixes for security vulnerabilities that could be used by attackers to exploit your system. It's important to regularly check for updates and install them as soon as they become available. Most operating systems have built-in mechanisms for automatically checking for updates and notifying you when new patches are available. You can configure these settings to automatically install updates or prompt you for installation. It's recommended to enable automatic updates to ensure that you don't miss any critical security patches. In addition to operating system updates, it's also important to keep your applications up to date with the latest security patches. Attackers often target vulnerabilities in popular applications, such as web browsers and office productivity software, to gain access to systems. Regularly updating your applications and using reputable sources for software downloads can help protect against these attacks. Overall, keeping your operating system and applications up to date with the latest security patches and updates is an essential step in securing your computer system. It helps protect against known vulnerabilities and ensures that your system is equipped with the latest security features. # 7.2. Secure Configurations Configuring your operating system and applications with secure settings is an important aspect of computer security. Secure configurations help protect your system from potential vulnerabilities and reduce the risk of unauthorized access or data breaches. Here are some key considerations for secure configurations: 1. User Accounts: Create separate user accounts for each individual who uses the system. Assign appropriate permissions and privileges to each account to limit access to sensitive information. 2. Passwords: Enforce strong password policies that require users to create complex passwords and regularly change them. Avoid using common passwords or easily guessable information, such as birthdays or names. 3. Network Settings: Configure your network settings to ensure secure connections. Use encryption protocols, such as WPA2 for Wi-Fi networks, and enable firewalls to control incoming and outgoing network traffic. 4. Software Updates: Enable automatic updates for your operating system and applications to ensure that you receive the latest security patches and bug fixes. Regularly check for updates manually if automatic updates are not available. 5. File and Folder Permissions: Set appropriate file and folder permissions to restrict access to sensitive data. Only grant necessary permissions to users or groups who need to access specific files or folders. 6. Antivirus and Antimalware Software: Install reputable antivirus and antimalware software and keep it up to date. Regularly scan your system for viruses and other malicious software to detect and remove any potential threats. 7. Disable Unnecessary Services: Disable or remove any unnecessary services or features that are not required for your system's functionality. This reduces the attack surface and minimizes the potential for vulnerabilities. 8. Data Backup: Regularly back up your important data to an external storage device or cloud service. In the event of a security breach or system failure, you can restore your data and minimize the impact. By implementing secure configurations, you can enhance the security of your computer system and reduce the risk of unauthorized access or data breaches. Regularly review and update your configurations to adapt to changing security threats and best practices. ## Exercise Consider the following scenario: You are setting up a new computer system for a small business. What are some secure configurations you would implement to protect the system? ### Solution 1. Create separate user accounts for each employee with strong passwords. 2. Enable automatic updates for the operating system and applications. 3. Configure the network settings to use encryption protocols and enable firewalls. 4. Set appropriate file and folder permissions to restrict access to sensitive data. 5. Install reputable antivirus and antimalware software and keep it up to date. 6. Disable or remove any unnecessary services or features. 7. Regularly back up important data to an external storage device or cloud service. # 7.3. Application Security Application security refers to the measures taken to protect software applications from security threats and vulnerabilities. It involves identifying and addressing potential weaknesses in the design, development, and deployment of applications to ensure their integrity and protect sensitive data. Here are some key aspects of application security: 1. Secure Coding Practices: Developers should follow secure coding practices to minimize the risk of vulnerabilities. This includes validating input, using parameterized queries to prevent SQL injection, and avoiding the use of deprecated or insecure functions. 2. Input Validation: Applications should validate all user input to prevent malicious input from compromising the system. This includes checking for the correct data type, length, and format, as well as implementing measures to prevent cross-site scripting (XSS) and cross-site request forgery (CSRF) attacks. 3. Authentication and Authorization: Applications should implement strong authentication mechanisms to verify the identity of users and ensure they have the appropriate permissions to access certain resources. This may involve using multi-factor authentication, session management, and secure password storage. 4. Error Handling and Logging: Proper error handling and logging are important for identifying and addressing security issues. Applications should provide meaningful error messages to users without revealing sensitive information, and log events and errors for analysis and troubleshooting. 5. Secure Configuration: Applications should be configured securely, following best practices for server settings, encryption protocols, and access controls. This includes using secure communication protocols (HTTPS), disabling unnecessary services, and implementing secure default configurations. 6. Secure File Handling: Applications should properly handle and validate file uploads to prevent the execution of malicious code or the disclosure of sensitive information. File uploads should be restricted to specific file types and scanned for malware. 7. Regular Testing and Vulnerability Assessments: Applications should undergo regular security testing and vulnerability assessments to identify and remediate any weaknesses. This may include penetration testing, code reviews, and automated vulnerability scanning. By implementing these application security measures, organizations can reduce the risk of security breaches, data leaks, and unauthorized access to sensitive information. It is important to prioritize application security throughout the software development lifecycle and regularly update and patch applications to address newly discovered vulnerabilities. ## Exercise Consider the following scenario: You are a developer working on a web application that handles sensitive user data. What are some secure coding practices you would implement to ensure the application's security? ### Solution 1. Validate all user input to prevent SQL injection and other types of attacks. 2. Use parameterized queries or prepared statements to prevent SQL injection. 3. Implement input validation to check for correct data type, length, and format. 4. Sanitize user input to prevent cross-site scripting (XSS) attacks. 5. Use secure password storage mechanisms, such as hashing and salting. 6. Implement strong authentication mechanisms, such as multi-factor authentication. 7. Implement session management to prevent session hijacking and fixation attacks. 8. Follow secure coding practices, such as avoiding the use of deprecated or insecure functions. 9. Implement proper error handling to prevent the disclosure of sensitive information. 10. Log events and errors for analysis and troubleshooting. 11. Regularly update and patch the application to address newly discovered vulnerabilities. 12. Regularly test the application for security vulnerabilities, such as through penetration testing and code reviews. # 7.4. Data Backup and Recovery Data backup and recovery is an essential aspect of application security. It involves creating copies of important data and implementing processes to restore that data in the event of data loss or system failure. Data loss can occur due to various reasons, such as hardware failure, software bugs, human error, or malicious attacks. Here are some key considerations for data backup and recovery: 1. Backup Strategy: Organizations should develop a comprehensive backup strategy that determines what data needs to be backed up, how frequently backups should be performed, and where the backups should be stored. This strategy should consider the criticality of the data and the recovery point objectives (RPO) and recovery time objectives (RTO) of the organization. 2. Regular Backups: It is important to perform regular backups to ensure that the most up-to-date data is protected. This can be done through automated backup solutions that schedule backups at specified intervals or through manual backups performed by system administrators. 3. Offsite Storage: Backups should be stored in a secure offsite location to protect against physical damage or loss. This can be achieved by using cloud storage services, remote data centers, or physical backup tapes stored in a separate location. 4. Data Encryption: It is recommended to encrypt the backup data to protect it from unauthorized access. Encryption ensures that even if the backup media is lost or stolen, the data remains secure. 5. Testing and Verification: Regular testing and verification of backups is crucial to ensure that the backup process is functioning correctly and that the data can be successfully restored. This can involve performing test restores, checking backup logs for errors, and validating the integrity of the backup data. 6. Disaster Recovery Plan: Organizations should have a well-defined disaster recovery plan that outlines the steps to be taken in the event of a data loss or system failure. This plan should include the procedures for restoring data from backups, the roles and responsibilities of personnel involved, and the communication channels to be used during the recovery process. By implementing a robust data backup and recovery strategy, organizations can minimize the impact of data loss and ensure the continuity of their operations. It is important to regularly review and update the backup strategy to adapt to changing business requirements and emerging threats. ## Exercise Consider the following scenario: Your organization's database server has experienced a hardware failure, resulting in the loss of critical data. What steps would you take to recover the data from the backups? ### Solution 1. Identify the most recent backup that contains the lost data. 2. Verify the integrity of the backup data by performing a test restore on a separate system. 3. Restore the backup data to a separate server or a temporary location. 4. Validate the restored data to ensure that it is complete and accurate. 5. Determine the cause of the hardware failure and address any underlying issues to prevent future failures. 6. Once the data has been successfully restored and validated, transfer it to the production database server. 7. Perform thorough testing to ensure that the application is functioning correctly with the restored data. 8. Communicate the recovery process and status to relevant stakeholders, such as management and affected users. 9. Conduct a post-mortem analysis to identify any lessons learned and make improvements to the backup and recovery processes if necessary. # 8. Web Security Web security is a critical aspect of computer security, as the web is a common target for malicious attacks. With the increasing reliance on web applications and the sensitive data they handle, it is essential to implement robust security measures to protect against threats. In this section, we will explore various aspects of web security, including common vulnerabilities, secure coding practices, web server security, and the use of web application firewalls. 8.1. Web Application Vulnerabilities Web applications are susceptible to various vulnerabilities that can be exploited by attackers. Understanding these vulnerabilities is crucial for developers and security professionals to build and maintain secure web applications. Some common web application vulnerabilities include: 1. Cross-Site Scripting (XSS): XSS occurs when an attacker injects malicious scripts into a web application, which are then executed by unsuspecting users. This can lead to the theft of sensitive information or the manipulation of user sessions. 2. SQL Injection: SQL injection involves inserting malicious SQL statements into a web application's database query. This can allow attackers to view, modify, or delete data from the database. 3. Cross-Site Request Forgery (CSRF): CSRF attacks trick users into performing unintended actions on a web application by exploiting their authenticated session. This can lead to unauthorized changes or actions performed on behalf of the user. 4. Remote Code Execution: Remote code execution vulnerabilities allow attackers to execute arbitrary code on a web server. This can lead to a complete compromise of the server and unauthorized access to sensitive data. It is essential to implement secure coding practices, such as input validation, output encoding, and parameterized queries, to mitigate these vulnerabilities. Consider a web application that allows users to submit comments. Without proper input validation, an attacker could inject malicious JavaScript code into the comment field. When other users view the comments, the malicious code would execute in their browsers, potentially stealing their session cookies or redirecting them to malicious websites. To prevent XSS attacks, developers should sanitize user input by encoding special characters and validating input against a whitelist of allowed characters. 8.2. Secure Coding Practices Secure coding practices are essential for building web applications that are resistant to attacks. By following industry best practices, developers can minimize the risk of introducing vulnerabilities into their code. Some secure coding practices for web applications include: 1. Input Validation: Validate and sanitize all user input to prevent attacks such as XSS and SQL injection. Use input validation libraries or frameworks to ensure consistent and reliable validation. 2. Output Encoding: Encode all user-generated content before displaying it in web pages to prevent XSS attacks. Use output encoding libraries or frameworks to automate this process. 3. Authentication and Authorization: Implement secure authentication mechanisms, such as strong password hashing and multi-factor authentication, to protect user accounts. Use role-based access control to enforce authorization rules and limit access to sensitive functionality. 4. Session Management: Use secure session management techniques, such as random session IDs, session expiration, and secure session storage, to prevent session hijacking and session fixation attacks. 5. Error Handling: Implement proper error handling and logging to provide meaningful error messages to developers while not exposing sensitive information to attackers. By following these practices, developers can significantly reduce the risk of introducing vulnerabilities into their web applications. ## Exercise Consider a web application that allows users to log in using a username and password. What secure coding practices would you implement to protect against common vulnerabilities? ### Solution 1. Input Validation: Validate and sanitize all user input, including the username and password fields, to prevent SQL injection and other attacks. Use parameterized queries or prepared statements to prevent SQL injection. 2. Password Storage: Hash and salt user passwords using a strong cryptographic algorithm, such as bcrypt or Argon2. Never store passwords in plaintext or using weak hashing algorithms like MD5 or SHA-1. 3. Session Management: Generate a secure, random session ID for each authenticated session. Store session data securely, either in server-side memory or encrypted cookies. Set session expiration and implement mechanisms to prevent session fixation attacks. 4. Account Lockout: Implement account lockout mechanisms to protect against brute-force attacks. For example, lock an account after a certain number of failed login attempts and require additional verification to unlock it. 5. Secure Communication: Use HTTPS to encrypt communication between the web application and clients, especially during the login process. This prevents eavesdropping and man-in-the-middle attacks. 6. Error Handling: Implement proper error handling to prevent the disclosure of sensitive information. Display generic error messages to users and log detailed error information for developers. By implementing these secure coding practices, the web application can be better protected against common vulnerabilities and attacks. # 8.3. Web Server Security Web server security is crucial to protect the underlying infrastructure and ensure the secure delivery of web applications. A compromised web server can lead to unauthorized access, data breaches, and service disruptions. Here are some best practices for web server security: 1. Patch Management: Keep the web server software and operating system up to date with the latest security patches. Regularly check for updates and apply them promptly to address known vulnerabilities. 2. Secure Configuration: Follow security guidelines provided by the web server software vendor to configure the server securely. Disable unnecessary services, limit access permissions, and enable security features such as HTTPS and secure headers. 3. Access Control: Implement strong access controls to restrict access to the web server and its resources. Use strong passwords, enforce multi-factor authentication, and regularly review and revoke unnecessary user accounts. 4. Logging and Monitoring: Enable logging and monitoring features to detect and respond to security incidents. Monitor server logs for suspicious activities, such as unauthorized access attempts or unusual traffic patterns. 5. Intrusion Detection and Prevention: Deploy intrusion detection and prevention systems (IDPS) to monitor and block malicious activities targeting the web server. Configure the IDPS to detect and block common attack patterns, such as SQL injection or XSS attacks. 6. Web Application Firewalls: Implement a web application firewall (WAF) to protect against web-based attacks. A WAF can inspect incoming and outgoing web traffic, detect and block malicious requests, and provide additional security controls. By implementing these best practices, organizations can enhance the security of their web servers and reduce the risk of successful attacks. Consider a web server that hosts an e-commerce website. The server is configured to use HTTPS with a valid SSL/TLS certificate to encrypt communication between the server and clients. Access to the server's administrative interface is restricted to authorized personnel only, using strong passwords and multi-factor authentication. The server is regularly patched with the latest security updates, and access logs are monitored for any suspicious activities. An intrusion detection and prevention system is deployed to detect and block common attack patterns, such as SQL injection or cross-site scripting attacks. 8.4. Web Application Firewalls Web application firewalls (WAFs) are an essential security control for protecting web applications from a wide range of attacks. A WAF sits between the web application and the client, inspecting incoming and outgoing web traffic and applying security rules to block malicious requests. Some benefits of using a WAF include: 1. Protection against OWASP Top 10: A WAF can detect and block common web application vulnerabilities, such as SQL injection, cross-site scripting, and remote code execution. 2. Virtual Patching: WAFs can provide virtual patching capabilities, allowing organizations to protect their web applications from known vulnerabilities even before they can be patched. 3. Bot Mitigation: WAFs can detect and block malicious bots and automated scanning tools that attempt to exploit vulnerabilities or perform unauthorized actions. 4. Rate Limiting and DDoS Protection: WAFs can enforce rate limiting rules to prevent brute-force attacks and protect against distributed denial-of-service (DDoS) attacks. 5. Logging and Monitoring: WAFs provide detailed logs and reporting capabilities, allowing organizations to monitor and analyze web traffic for security incidents and compliance purposes. When deploying a WAF, it is important to configure it properly and regularly update its security rules to adapt to emerging threats. Organizations should also monitor WAF logs and alerts to detect and respond to potential attacks. ## Exercise Consider a web application that has been protected by a web application firewall. What benefits does the WAF provide in terms of security? ### Solution 1. Protection against Common Vulnerabilities: The WAF can detect and block common web application vulnerabilities, such as SQL injection, cross-site scripting, and remote code execution. This helps prevent attacks that exploit these vulnerabilities. 2. Virtual Patching: The WAF can provide virtual patching capabilities, allowing organizations to protect their web applications from known vulnerabilities even before they can be patched. This reduces the risk of exploitation while patches are being developed and deployed. 3. Bot Mitigation: The WAF can detect and block malicious bots and automated scanning tools that attempt to exploit vulnerabilities or perform unauthorized actions. This helps protect the web application from abuse and unauthorized access. 4. Rate Limiting and DDoS Protection: The WAF can enforce rate limiting rules to prevent brute-force attacks and protect against distributed denial-of-service (DDoS) attacks. This helps ensure the availability and performance of the web application. 5. Logging and Monitoring: The WAF provides detailed logs and reporting capabilities, allowing organizations to monitor and analyze web traffic for security incidents and compliance purposes. This helps in detecting and responding to potential attacks. By deploying a web application firewall, organizations can enhance the security of their web applications and protect against a wide range of threats. # 8.2. Secure Coding Practices Consider a web application that allows users to submit comments. To prevent XSS attacks, the application validates and sanitizes all user input before displaying it. It uses an input validation library to ensure that only safe characters are accepted. When interacting with the database, the application uses parameterized queries to prevent SQL injection attacks. User input is never concatenated directly into SQL queries. The application also implements secure session management techniques. It generates random session IDs, sets secure flags for cookies, and expires sessions after a period of inactivity. 8.1. Web Application Vulnerabilities Web applications are a common target for attackers due to the sensitive data they handle and the potential impact of a successful attack. Understanding common web application vulnerabilities is crucial for developing secure web applications. Here are some common web application vulnerabilities to be aware of: 1. Cross-Site Scripting (XSS): XSS occurs when an attacker injects malicious scripts into a web application, which are then executed by unsuspecting users. XSS vulnerabilities can allow attackers to steal sensitive information, such as login credentials or session cookies. 2. SQL Injection: SQL injection occurs when an attacker manipulates input parameters to execute malicious SQL queries. This can lead to unauthorized access to the database, data leakage, or even complete database compromise. 3. Cross-Site Request Forgery (CSRF): CSRF occurs when an attacker tricks a user into performing unwanted actions on a web application without their knowledge or consent. This can lead to actions performed on behalf of the user, such as changing passwords or making unauthorized transactions. 4. Remote Code Execution (RCE): RCE vulnerabilities allow attackers to execute arbitrary code on the server hosting the web application. This can lead to complete compromise of the server and unauthorized access to sensitive data. 5. XML External Entity (XXE) Injection: XXE vulnerabilities occur when an attacker can influence the processing of XML input, leading to the disclosure of internal files, denial of service, or server-side request forgery. 6. Server-Side Request Forgery (SSRF): SSRF vulnerabilities allow attackers to make requests to internal resources from the server hosting the web application. This can lead to unauthorized access to internal systems or services. 7. Unvalidated Redirects and Forwards: Unvalidated redirects and forwards can be abused by attackers to redirect users to malicious websites or perform phishing attacks. By understanding these vulnerabilities, developers can implement appropriate security measures and best practices to mitigate the risk of exploitation. ## Exercise Explain the concept of cross-site scripting (XSS) and its potential impact on web applications. ### Solution Cross-Site Scripting (XSS) is a web application vulnerability that occurs when an attacker injects malicious scripts into a web page viewed by other users. These scripts can be executed by the victim's browser, allowing the attacker to steal sensitive information or perform unauthorized actions on behalf of the user. The impact of XSS can vary depending on the attacker's goals and the vulnerability's context. Potential consequences include: 1. Cookie Theft: Attackers can steal session cookies, allowing them to impersonate the victim and gain unauthorized access to their accounts. 2. Data Leakage: Attackers can extract sensitive information, such as usernames, passwords, or personal data, from the victim's browser. 3. Defacement: Attackers can modify the content of a web page, potentially damaging the reputation of the affected organization or causing confusion among users. 4. Phishing Attacks: Attackers can create realistic-looking login forms or other input forms to trick users into disclosing their credentials or other sensitive information. 5. Malware Distribution: Attackers can inject malicious scripts that redirect users to websites hosting malware or initiate downloads of malicious files. To prevent XSS attacks, web developers should validate and sanitize all user input, encode output to prevent script execution, and implement strict content security policies. # 8.3. Web Server Security Web server security is essential for protecting the integrity and availability of web applications. Web servers are often targeted by attackers due to their public-facing nature and potential access to sensitive data. Implementing proper security measures can help prevent unauthorized access and mitigate the risk of attacks. Here are some best practices for web server security: 1. Regularly update and patch the web server software: Keeping the web server software up to date is crucial for addressing security vulnerabilities and preventing exploitation by attackers. Regularly check for updates and apply patches as soon as they are available. 2. Use strong and unique passwords: Ensure that strong passwords are used for all user accounts on the web server, including administrator accounts. Avoid using default or easily guessable passwords. Consider implementing multi-factor authentication for added security. 3. Restrict access to the web server: Limit access to the web server by allowing only necessary connections. Use firewalls or network access control lists (ACLs) to restrict incoming and outgoing traffic to and from the web server. Regularly review and update access control rules. 4. Implement secure communication protocols: Use HTTPS (HTTP over SSL/TLS) for secure communication between the web server and clients. This helps protect sensitive data, such as login credentials and personal information, from interception and tampering. 5. Enable logging and monitoring: Enable logging on the web server to track and monitor activities. Regularly review logs for any suspicious or unauthorized access attempts. Implement intrusion detection and prevention systems (IDS/IPS) to detect and block malicious traffic. 6. Secure file and directory permissions: Set appropriate file and directory permissions to prevent unauthorized access. Restrict write access to sensitive directories and files. Regularly review and update permissions as needed. 7. Disable unnecessary services and features: Disable any unnecessary services, modules, or features on the web server to reduce the attack surface. Only enable the services and features that are required for the functioning of the web application. 8. Regularly backup data: Perform regular backups of the web server and its data. Store backups securely and test the restoration process to ensure data can be recovered in case of a security incident or data loss. By following these best practices, web server administrators can enhance the security of their web applications and protect against common attacks. Regular security assessments and vulnerability scans can also help identify and address any potential weaknesses. # 8.4. Web Application Firewalls Web application firewalls (WAFs) are an important security measure for protecting web applications from various types of attacks. A WAF sits between the web server and the client, monitoring and filtering incoming and outgoing traffic to detect and block malicious activity. Here are some key points about web application firewalls: - WAFs analyze HTTP and HTTPS traffic to identify and block suspicious requests. They can detect common attack patterns, such as SQL injection, cross-site scripting (XSS), and cross-site request forgery (CSRF). - WAFs can be implemented as hardware appliances, software solutions, or cloud-based services. They can be deployed as standalone devices or integrated with existing network infrastructure. - WAFs use a combination of rule-based and behavior-based techniques to identify and block malicious traffic. Rule-based WAFs use predefined rulesets to detect known attack patterns, while behavior-based WAFs analyze traffic patterns and user behavior to identify anomalies. - WAFs can provide protection against both known and unknown vulnerabilities. They can block attacks in real-time, preventing them from reaching the web application and causing damage. - WAFs can also provide additional security features, such as SSL/TLS termination, bot detection and mitigation, and content delivery network (CDN) integration. - It is important to regularly update and maintain the WAF to ensure it is effective against the latest threats. This includes updating the rulesets, monitoring logs for any suspicious activity, and performing regular security assessments. - While WAFs are a valuable security measure, they should not be considered a replacement for secure coding practices and other security controls. WAFs can provide an additional layer of protection, but it is important to address vulnerabilities at the application level as well. By implementing a web application firewall, organizations can enhance the security of their web applications and protect against a wide range of attacks. WAFs can help detect and block malicious activity, reducing the risk of data breaches and other security incidents. # 9. Cloud Security Cloud computing has become increasingly popular in recent years, offering organizations the ability to store and access data and applications over the internet. While the cloud provides many benefits, such as scalability and cost savings, it also introduces new security challenges. In this section, we will explore the key concepts and best practices for securing cloud environments. We will discuss the basics of cloud computing, the security considerations specific to the cloud, and strategies for ensuring data privacy and compliance. 9.1. Cloud Computing Basics Cloud computing refers to the delivery of computing services over the internet. Instead of hosting applications and storing data on local servers or personal computers, organizations can leverage cloud service providers to access resources on-demand. There are three main types of cloud services: 1. Infrastructure as a Service (IaaS): Provides virtualized computing resources, such as virtual machines, storage, and networks. Organizations can deploy and manage their own applications and data on the cloud infrastructure. 2. Platform as a Service (PaaS): Offers a platform for developing, testing, and deploying applications. The cloud provider manages the underlying infrastructure, allowing organizations to focus on application development. 3. Software as a Service (SaaS): Delivers software applications over the internet on a subscription basis. Users can access the applications through a web browser, without the need for installation or maintenance. Cloud computing offers several advantages, including: - Scalability: Organizations can easily scale their resources up or down based on demand, without the need for physical infrastructure. - Cost savings: Cloud services eliminate the need for upfront hardware and software investments, reducing capital expenses. Organizations only pay for the resources they use. - Accessibility: Cloud services can be accessed from anywhere with an internet connection, enabling remote work and collaboration. However, the cloud also introduces new security considerations. Organizations must understand the shared responsibility model, where the cloud provider is responsible for securing the underlying infrastructure, while the organization is responsible for securing their applications and data. # 9.2. Security in the Cloud Securing data and applications in the cloud requires a multi-layered approach. Organizations should implement a combination of technical controls, policies, and procedures to protect their assets. Here are some key security considerations for the cloud: 1. Identity and access management: Implement strong authentication and access controls to ensure that only authorized individuals can access cloud resources. Use multi-factor authentication and enforce strong password policies. 2. Data encryption: Encrypt sensitive data both in transit and at rest to protect it from unauthorized access. Use encryption protocols, such as SSL/TLS, and encryption algorithms, such as AES. 3. Network security: Implement firewalls and network segmentation to control traffic between different cloud resources. Use virtual private networks (VPNs) to establish secure connections between on-premises infrastructure and the cloud. 4. Vulnerability management: Regularly scan cloud resources for vulnerabilities and apply patches and updates in a timely manner. Use intrusion detection and prevention systems to detect and block malicious activity. 5. Incident response: Develop an incident response plan to quickly respond to and mitigate security incidents in the cloud. Monitor logs and implement real-time alerting to detect and respond to potential threats. 6. Compliance and auditing: Ensure compliance with relevant regulations and industry standards, such as GDPR or PCI DSS. Regularly audit and assess cloud environments to identify and address security gaps. It is important to work closely with the cloud service provider to understand their security measures and ensure that they meet your organization's requirements. Additionally, organizations should regularly review and update their security controls to adapt to evolving threats and technologies. # 9.3. Cloud Security Best Practices To enhance the security of cloud environments, organizations should follow best practices and adopt a proactive approach to security. Here are some key recommendations: 1. Understand the shared responsibility model: Clearly define the responsibilities of the cloud service provider and your organization. Understand which security controls are provided by the cloud provider and which ones you need to implement. 2. Conduct a risk assessment: Identify and assess the potential risks and threats to your cloud environment. Prioritize risks based on their impact and likelihood, and develop a risk mitigation plan. 3. Implement strong access controls: Use strong authentication mechanisms, such as multi-factor authentication, to ensure that only authorized individuals can access cloud resources. Enforce the principle of least privilege to limit access to sensitive data. 4. Encrypt sensitive data: Implement encryption for data both in transit and at rest. Use strong encryption algorithms and key management practices to protect sensitive information. 5. Regularly monitor and audit cloud environments: Implement logging and monitoring mechanisms to detect and respond to potential security incidents. Regularly review logs and conduct audits to identify any unauthorized access or suspicious activity. 6. Backup and disaster recovery: Implement regular data backups and test the restoration process to ensure that critical data can be recovered in the event of a security incident or data loss. 7. Educate employees: Provide security awareness training to employees to educate them about the risks and best practices for using cloud services. Encourage employees to report any suspicious activity or potential security incidents. By following these best practices, organizations can strengthen the security of their cloud environments and mitigate the risks associated with cloud computing. # 9.4. Data Privacy and Compliance Data privacy and compliance are critical considerations when using cloud services. Organizations must ensure that they comply with relevant regulations and protect the privacy of their customers' data. Here are some key considerations for data privacy and compliance in the cloud: 1. Data classification: Classify data based on its sensitivity and regulatory requirements. Implement appropriate controls to protect sensitive data, such as encryption and access controls. 2. Data residency and sovereignty: Understand where your data is stored and ensure that it complies with the applicable data protection laws and regulations. Some countries have specific requirements for data residency and may prohibit the transfer of certain types of data outside their borders. 3. Data breach notification: Develop a data breach notification plan to comply with applicable breach notification laws. Implement mechanisms to detect and respond to data breaches in a timely manner. 4. Vendor management: Conduct due diligence when selecting a cloud service provider. Ensure that the provider has robust security measures in place and complies with relevant regulations. Include appropriate contractual provisions to protect your organization's data. 5. Privacy policies and consent: Develop and communicate clear privacy policies to your customers. Obtain appropriate consent for collecting and processing personal data. Regularly review and update privacy policies to reflect changes in regulations and best practices. 6. Compliance audits: Regularly assess your cloud environment for compliance with relevant regulations, such as GDPR or HIPAA. Conduct internal or third-party audits to identify any compliance gaps and implement remediation measures. By prioritizing data privacy and compliance, organizations can build trust with their customers and ensure that their cloud environments meet the necessary legal and regulatory requirements. # 10. Mobile Security Mobile devices, such as smartphones and tablets, have become an integral part of our daily lives. They provide us with access to a wide range of applications and services, but they also introduce new security risks. In this section, we will explore the key considerations for securing mobile devices and applications. We will discuss topics such as mobile device management, mobile app security, and mobile network security. 10.1. Mobile Device Management Mobile device management (MDM) refers to the process of managing and securing mobile devices within an organization. MDM solutions provide IT administrators with the ability to enforce security policies, manage device configurations, and control access to corporate resources. Here are some key aspects of mobile device management: 1. Device enrollment: MDM solutions allow administrators to enroll mobile devices into the management system. This enables the organization to apply security policies and configurations to the devices. 2. Security policies: MDM solutions enable administrators to enforce security policies on mobile devices. These policies can include requirements for strong passwords, device encryption, and remote wipe capabilities. 3. Application management: MDM solutions provide the ability to manage and distribute applications to mobile devices. This includes the ability to whitelist or blacklist specific applications, as well as control access to corporate app stores. 4. Device tracking and remote wipe: MDM solutions can track the location of mobile devices and remotely wipe them in case of loss or theft. This helps protect sensitive data from falling into the wrong hands. 5. Compliance and reporting: MDM solutions provide reporting capabilities to monitor device compliance with security policies. This includes the ability to generate reports on device inventory, security vulnerabilities, and policy violations. By implementing a mobile device management solution, organizations can effectively manage and secure their mobile devices, reducing the risk of data breaches and other security incidents. # 10.2. Mobile App Security Mobile applications (apps) are a common target for attackers due to their widespread use and the sensitive data they often handle. Securing mobile apps is essential to protect user data and prevent unauthorized access. Here are some key considerations for mobile app security: 1. Secure coding practices: Developers should follow secure coding practices to minimize vulnerabilities in mobile apps. This includes validating user input, using secure communication protocols, and implementing proper authentication and authorization mechanisms. 2. Code obfuscation: Obfuscating the code of a mobile app can make it more difficult for attackers to reverse engineer the app and discover vulnerabilities. This can help protect sensitive information, such as API keys or encryption algorithms. 3. Secure data storage: Mobile apps should securely store sensitive data, such as user credentials or payment information. This can be achieved by using encryption to protect data at rest and in transit, as well as implementing proper access controls. 4. User permissions: Mobile apps should request only the necessary permissions from users. Unnecessary permissions can increase the risk of data leakage or unauthorized access to device features. 5. Secure update process: Mobile apps should have a secure update process to ensure that users receive important security patches and bug fixes. This includes using secure channels for app updates and verifying the integrity of the update package. 6. Penetration testing: Regularly conduct penetration testing of mobile apps to identify vulnerabilities and weaknesses. This can help uncover security flaws that could be exploited by attackers. By following these best practices, organizations can enhance the security of their mobile apps and protect user data from unauthorized access or disclosure. # 10.3. Mobile Network Security Mobile devices rely on wireless networks to connect to the internet and access various services. Securing mobile network connections is crucial to protect sensitive data from interception or unauthorized access. Here are some key considerations for mobile network security: 1. Secure Wi-Fi connections: When connecting to Wi-Fi networks, ensure that the network is secure and encrypted. Avoid connecting to public or untrusted Wi-Fi networks, as they can be easily compromised. 2. Virtual Private Networks (VPNs): Use VPNs to establish secure connections between mobile devices and corporate networks. VPNs encrypt network traffic, protecting it from interception and unauthorized access. 3. Mobile data encryption: Enable encryption for mobile data connections, such as 3G, 4G, or 5G. This helps protect data from interception and ensures its confidentiality. 4. Mobile device firewalls: Install mobile device firewalls to monitor and filter network traffic. Firewalls can detect and block malicious activity, such as unauthorized access attempts or suspicious network connections. 5. Mobile network monitoring: Regularly monitor mobile network traffic for any signs of suspicious activity. This includes analyzing network logs and using intrusion detection systems to detect potential security incidents. 6. Mobile network authentication: Implement strong authentication mechanisms for mobile network connections. This can include using SIM cards with strong encryption algorithms or implementing certificate-based authentication. By following these best practices, organizations can enhance the security of their mobile network connections and protect sensitive data from interception or unauthorized access. # 10.4. Mobile Device Encryption Mobile devices often store sensitive data, such as personal information, financial data, or corporate documents. Encrypting mobile devices helps protect this data from unauthorized access in case of loss or theft. Here are some key considerations for mobile device encryption: 1. Full device encryption: Enable full device encryption on mobile devices to protect all data stored on the device. This includes user data, system files, and application data. 2. Strong encryption algorithms: Use strong encryption algorithms, such as AES, to encrypt mobile device data. Avoid using weak encryption algorithms or outdated encryption protocols. 3. Secure key management: Implement secure key management practices to protect encryption keys. This includes storing keys in a secure location, using hardware-backed encryption, and regularly rotating keys. 4. Remote wipe capabilities: Enable remote wipe capabilities on mobile devices to allow for the remote deletion of data in case of loss or theft. This helps ensure that sensitive data does not fall into the wrong hands. 5. Secure boot process: Implement a secure boot process on mobile devices to ensure that only trusted software is loaded during the device startup. This helps protect against unauthorized modifications or tampering. 6. Biometric authentication: Use biometric authentication, such as fingerprint or facial recognition, to unlock encrypted mobile devices. This provides an additional layer of security and convenience for users. By encrypting mobile devices, organizations can protect sensitive data from unauthorized access and mitigate the risks associated with device loss or theft. # 11. Emerging Technologies in Computer Security 11.1. Internet of Things (IoT) Security The Internet of Things (IoT) refers to the network of interconnected devices, objects, and sensors that communicate and exchange data over the internet. IoT devices, such as smart home devices, wearables, and industrial sensors, introduce new security challenges due to their large-scale deployment and diverse nature. Here are some key considerations for IoT security: 1. Device authentication: Implement strong authentication mechanisms for IoT devices to ensure that only authorized devices can connect to the network. This can include the use of digital certificates or secure communication protocols. 2. Secure communication: Encrypt data transmitted between IoT devices and backend systems to protect it from interception or tampering. Use protocols such as MQTT or HTTPS to ensure secure communication. 3. Firmware updates: Regularly update the firmware of IoT devices to patch vulnerabilities and improve security. Implement a secure update process to ensure that updates are authentic and tamper-proof. 4. Access control: Implement access controls to limit the actions and permissions of IoT devices. This includes restricting device-to-device communication and enforcing least privilege principles. 5. Network segmentation: Segment IoT devices into separate networks to isolate them from critical systems and data. This helps contain potential security breaches and limit the impact of compromised devices. 6. Monitoring and analytics: Implement monitoring and analytics solutions to detect and respond to potential security incidents in real-time. This includes analyzing device logs, network traffic, and user behavior to identify anomalies. By addressing these considerations, organizations can enhance the security of their IoT deployments and protect against potential threats. # 11.2. Artificial Intelligence and Machine Learning in Security Artificial intelligence (AI) and machine learning (ML) are revolutionizing the field of computer security. These technologies can help organizations detect and respond to security threats more effectively, automate security processes, and improve overall security posture. Here are some key applications of AI and ML in security: 1. Threat detection and response: AI and ML algorithms can analyze large volumes of security data, such as logs and network traffic, to detect patterns and anomalies that may indicate a security breach. This can help organizations identify and respond to threats in real-time. 2. User behavior analytics: AI and ML can analyze user behavior to identify suspicious activities or deviations from normal patterns. This can help detect insider threats, account compromises, or unauthorized access attempts. 3. Malware detection: AI and ML algorithms can analyze file characteristics and behavior to identify and classify malware. This can help organizations detect and block malicious software before it can cause harm. 4. Vulnerability management: AI and ML can assist in identifying and prioritizing vulnerabilities in software and systems. This can help organizations allocate resources more effectively and address critical vulnerabilities first. 5. Security automation: AI and ML can automate routine security tasks, such as log analysis, incident response, and security policy enforcement. This can free up security professionals to focus on more complex tasks and improve overall efficiency. While AI and ML offer many benefits, it is important to consider their limitations and potential risks. Organizations should carefully evaluate and test AI and ML solutions to ensure their effectiveness and mitigate any potential biases or false positives. # 11.3. Blockchain Technology and Security Blockchain technology, originally developed for cryptocurrencies like Bitcoin, has the potential to revolutionize security and trust in various industries. Blockchain is a distributed ledger that records transactions across multiple computers, making it difficult for attackers to tamper with or manipulate the data. Here are some key applications of blockchain technology in security: 1. Secure transactions: Blockchain can provide a secure and transparent platform for conducting transactions. It eliminates the need for intermediaries and ensures the integrity and immutability of transaction records. 2. Identity management: Blockchain can be used to create decentralized identity systems, where individuals have control over their personal data and can selectively share it with trusted parties. This can help prevent identity theft and unauthorized access. 3. Supply chain security: Blockchain can be used to track and verify the authenticity and provenance of goods throughout the supply chain. This can help prevent counterfeiting, ensure product quality, and enhance consumer trust. 4. Smart contracts: Blockchain can enable the execution of self-executing contracts, known as smart contracts. Smart contracts automatically enforce the terms and conditions of an agreement, reducing the need for intermediaries and enhancing security. 5. Decentralized storage: Blockchain can be used for decentralized storage, where data is distributed across multiple nodes in the network. This provides increased resilience and security, as there is no single point of failure or control. While blockchain technology offers many potential benefits, it is important to consider its limitations and challenges. Blockchain is not a silver bullet for all security problems and may introduce new risks, such as scalability and privacy concerns. # 11.4. Biometric Security Biometric security refers to the use of unique physical or behavioral characteristics, such as fingerprints, facial recognition, or voice recognition, for authentication and identification purposes. Biometrics can provide a higher level of security compared to traditional authentication methods, such as passwords or PINs. Here are some key considerations for biometric security: 1. Accuracy and reliability: Biometric systems should be accurate and reliable, ensuring that legitimate users are granted access while unauthorized individuals are denied. False positives and false negatives should be minimized to avoid security gaps or inconvenience for users. 2. Data privacy and protection: Biometric data is highly sensitive and should be protected from unauthorized access or disclosure. Organizations should implement strong encryption and access controls to safeguard biometric data. 3. Spoofing and presentation attacks: Biometric systems should be resistant to spoofing or presentation attacks, where attackers try to deceive the system using fake or stolen biometric data. Anti-spoofing techniques, such as liveness detection, can help mitigate these risks. 4. User acceptance and convenience: Biometric systems should be user-friendly and convenient, ensuring that users can easily authenticate themselves without significant effort or inconvenience. Factors such as speed, ease of use, and accessibility should be considered. 5. Multi-factor authentication: Biometric authentication should be used in conjunction with other authentication factors, such as passwords or tokens, to provide a higher level of security. This helps mitigate the risk of biometric data compromise or false acceptance. By implementing biometric security measures, organizations can enhance the security of their systems and improve the user experience for authentication and identification. # 12. Legal and Ethical Issues in Computer Security 12.1. Laws and Regulations Laws and regulations play a crucial role in ensuring computer security and protecting individuals and organizations from cyber threats. Governments around the world have enacted various laws and regulations to address computer security issues. Here are some key laws and regulations related to computer security: 1. General Data Protection Regulation (GDPR): The GDPR is a European Union regulation that governs the protection of personal data. It imposes strict requirements on organizations that collect, process, or store personal data of EU residents. 2. Health Insurance Portability and Accountability Act (HIPAA): HIPAA is a U.S. law that regulates the privacy and security of health information. It applies to healthcare providers, health plans, and healthcare clearinghouses. 3. Payment Card Industry Data Security Standard (PCI DSS): PCI DSS is a set of security standards established by major credit card companies to protect cardholder data. It applies to organizations that handle credit card transactions. 4. Computer Fraud and Abuse Act (CFAA): The CFAA is a U.S. law that criminalizes unauthorized access to computer systems. It covers a wide range of computer-related offenses, including hacking, identity theft, and distribution of malicious software. 5. Cybersecurity Information Sharing Act (CISA): CISA is a U.S. law that encourages the sharing of cybersecurity threat information between the government and private sector entities. It aims to improve the overall cybersecurity posture of the nation. Organizations should be aware of the laws and regulations that apply to their operations and ensure compliance to avoid legal and financial consequences. # 12.2. Cybercrime and Cyberterrorism Cybercrime and cyberterrorism pose significant threats to individuals, organizations, and even nations. Cybercriminals and cyberterrorists exploit vulnerabilities in computer systems and networks to gain unauthorized access, steal sensitive information, disrupt services, or cause physical harm. Here are some key types of cyber threats: 1. Hacking: Unauthorized access to computer systems or networks to steal or manipulate data, disrupt services, or gain control over systems. 2. Phishing: Sending fraudulent emails or messages to trick individuals into revealing sensitive information, such as passwords or credit card numbers. 3. Malware: Malicious software designed to infiltrate or damage computer systems. This includes viruses, worms, ransomware, and spyware. 4. Distributed Denial of Service (DDoS) attacks: Overwhelming a target system or network with a flood of traffic, rendering it unavailable to legitimate users. 5. Identity theft: Stealing personal information, such as social security numbers or credit card details, to impersonate individuals or commit fraud. 6. Cyberterrorism: Using computer systems or networks to carry out acts of terrorism, such as disrupting critical infrastructure or causing physical harm. To combat cybercrime and cyberterrorism, organizations and governments must collaborate to develop robust security measures, share threat intelligence, and enforce laws and regulations. Individuals should also practice good cyber hygiene, such as using strong passwords, keeping software up to date, and being cautious of suspicious emails or websites. # 12.3. Ethical Hacking and Responsible Disclosure Ethical hacking, also known as penetration testing or white-hat hacking, involves authorized individuals or organizations testing the security of computer systems and networks to identify vulnerabilities and weaknesses. Ethical hackers use their skills and knowledge to help organizations improve their security posture. Here are some key principles of ethical hacking: 1. Authorization: Ethical hacking should only be conducted with proper authorization from the owner or administrator of the target system or network. Unauthorized hacking is illegal and unethical. 2. Scope: Ethical hacking should be conducted within the agreed scope and boundaries defined by the organization. Testing should not extend beyond the authorized systems or networks. 3. Confidentiality: Ethical hackers should respect the confidentiality of any sensitive information they come across during their testing. They should not disclose or misuse this information. 4. Responsible disclosure: If ethical hackers discover vulnerabilities or weaknesses, they should report them to the organization in a responsible manner. This includes providing detailed information about the vulnerability and giving the organization a reasonable amount of time to address the issue before disclosing it publicly. Ethical hacking plays an important role in improving computer security by identifying and addressing vulnerabilities before malicious hackers can exploit them. Organizations should consider engaging ethical hackers to conduct regular security assessments and penetration tests. # 12.4. Balancing Security and Privacy Balancing security and privacy is a complex challenge in the digital age. While strong security measures are essential to protect individuals and organizations from cyber threats, they can sometimes come at the expense of privacy. Here are some key considerations for balancing security and privacy: 1. Privacy by design: Organizations should incorporate privacy considerations into the design and development of their systems and applications. This includes implementing privacy-enhancing technologies and practices, such as data minimization and user consent. 2. Data protection regulations: Organizations should comply with relevant data protection regulations, such as the GDPR, to ensure that individuals' privacy rights are respected. This includes obtaining appropriate consent for data processing and implementing necessary security measures to protect personal data. 3. Transparency and accountability: Organizations should be transparent about their data collection and processing practices, and be accountable for how they handle individuals' personal information. This includes providing clear privacy policies and mechanisms for individuals to exercise their privacy rights. 4. Proportional security measures: Organizations should implement security measures that are proportional to the risks they face. Excessive security measures can infringe on individuals' privacy rights without providing significant security benefits. 5. User education and empowerment: Individuals should be educated about the importance of security and privacy, and empowered to make informed decisions about their personal information. This includes providing individuals with control over their data and giving them the ability to opt out of data collection or processing. By striking the right balance between security and privacy, organizations can protect individuals' data while maintaining a strong security posture. # 13. Conclusion: Future of Computer Security Computer security is an ongoing challenge that requires continuous learning and adaptation. As technology evolves and new threats emerge, organizations and individuals must stay vigilant and proactive in their security practices. 13.1. Current Trends and Predictions Here are some current trends and predictions in computer security: 1. Artificial intelligence and machine learning will play an increasingly important role in detecting and responding to security threats. AI-powered security solutions can analyze large volumes of data and identify patterns that may indicate malicious activity. 2. The Internet of Things (IoT) will continue to expand, introducing new security challenges. Organizations must ensure that IoT devices are secure and properly managed to prevent unauthorized access or control. 3. Cloud security will remain a top priority as more organizations adopt cloud computing. It is crucial to implement strong access controls, encryption, and monitoring mechanisms to protect data in the cloud. 4. Mobile security will become even more critical as mobile devices continue to be a primary target for attackers. Organizations must implement strong authentication mechanisms, secure mobile applications, and protect mobile network connections. 5. Privacy regulations will continue to evolve, with more countries enacting laws to protect individuals' personal data. Organizations must ensure compliance with these regulations and prioritize data privacy in their security practices. 13.2. Challenges and Opportunities Computer security faces several challenges, including the increasing sophistication of cyber threats, the rapid pace of technological advancements, and the complexity of securing interconnected systems. However, these challenges also present opportunities for innovation and collaboration. By leveraging emerging technologies, sharing threat intelligence, and adopting a proactive approach to security, organizations can stay one step ahead of attackers. 13.3. Importance of Continued Learning and Adaptation Computer security is a dynamic field that requires continuous learning and adaptation. New threats and vulnerabilities emerge regularly, and security professionals must stay up to date with the latest trends and best practices. By investing in training and professional development, organizations can ensure that their security teams have the knowledge and skills necessary to protect their systems and data effectively. 13.4. Strategies for Staying Secure Here are some strategies for organizations and individuals to stay secure: 1. Implement a comprehensive security program that covers all aspects of computer security, including technical controls, policies, and procedures. 2. Regularly assess and update security measures to address new threats and vulnerabilities. This includes patching software, updating security configurations, and conducting security audits. 3. Educate employees about security best practices and the importance of following security policies. Provide training on topics such as phishing awareness, password hygiene, and safe browsing habits. 4. Monitor systems and networks for potential security incidents. Implement intrusion detection and prevention systems, log analysis tools, and real-time alerting mechanisms. 5. Engage with the security community and share threat intelligence with trusted partners. Collaborate with industry peers to stay informed about emerging threats and best practices. By following these strategies, organizations and individuals can enhance their security posture and protect themselves against a wide range of cyber threats. Research Notes * ```A secure "enterprise", big or small, should have an approach to security that is comprehen- sive and end-to-end if it is to be effective. Most organizations do not have such policies and practices in place. There are some good reasons for this; security clearly comes at a cost. # 12.2. Cybercrime and Cyberterrorism Cybercrime and cyberterrorism are two significant threats in the digital age. While they share similarities, there are distinct differences between the two. Cybercrime refers to any illegal activity that is conducted through the use of computers or the internet. It includes a wide range of offenses, such as hacking, identity theft, online fraud, and the distribution of malware. Cybercriminals often target individuals, businesses, and even governments to gain unauthorized access to sensitive information or to cause financial harm. Cyberterrorism, on the other hand, involves the use of technology to carry out politically motivated attacks. These attacks aim to disrupt or disable critical infrastructure, cause fear and panic among the population, or advance a particular ideological agenda. Cyberterrorists may target government agencies, financial institutions, or other organizations that play a crucial role in society. An example of cyberterrorism is a coordinated attack on a country's power grid, causing widespread power outages and disrupting essential services. This type of attack can have severe consequences for public safety and national security. Both cybercrime and cyberterrorism pose significant risks to individuals and organizations. They can result in financial loss, reputational damage, and even physical harm. It is essential for individuals and organizations to take proactive measures to protect themselves against these threats. ## Exercise What is the difference between cybercrime and cyberterrorism? ### Solution Cybercrime refers to any illegal activity conducted through computers or the internet, while cyberterrorism involves politically motivated attacks that aim to disrupt critical infrastructure or advance an ideological agenda. # 12.3. Ethical Hacking and Responsible Disclosure Ethical hacking, also known as penetration testing or white-hat hacking, is the practice of intentionally exploiting vulnerabilities in computer systems to identify and fix security weaknesses. Ethical hackers use the same techniques as malicious hackers but with the permission and knowledge of the system owner. The goal of ethical hacking is to help organizations improve their security by identifying and addressing vulnerabilities before they can be exploited by malicious actors. Ethical hackers often work closely with cybersecurity teams to perform comprehensive security assessments and provide recommendations for strengthening defenses. An example of ethical hacking is conducting a simulated phishing attack on an organization's employees to test their awareness and response to phishing emails. By doing so, the organization can identify areas where employees may need additional training and implement measures to mitigate the risk of successful phishing attacks. Responsible disclosure is an important aspect of ethical hacking. When ethical hackers discover vulnerabilities, they have a responsibility to report them to the appropriate parties, such as the organization or software vendor, rather than exploiting them for personal gain or malicious purposes. This allows the vulnerabilities to be patched and protects users from potential harm. ## Exercise What is the goal of ethical hacking? ### Solution The goal of ethical hacking is to help organizations improve their security by identifying and addressing vulnerabilities before they can be exploited by malicious actors. # 12.4. Balancing Security and Privacy Balancing security and privacy is a complex challenge in the digital age. While security measures are necessary to protect against threats, they can sometimes come at the expense of individual privacy. Security measures, such as surveillance cameras, data collection, and monitoring systems, can help prevent and detect potential threats. However, they also raise concerns about privacy and the potential for abuse or misuse of personal information. An example of the balance between security and privacy is the use of facial recognition technology. While this technology can enhance security by identifying potential threats, it also raises concerns about the collection and storage of individuals' biometric data and the potential for misuse or unauthorized access. To strike a balance between security and privacy, organizations and individuals should consider the following: 1. Implementing privacy-enhancing technologies: Encryption, anonymization, and other privacy-enhancing technologies can help protect personal information while still allowing for effective security measures. 2. Transparency and consent: Organizations should be transparent about the data they collect and how it is used, and individuals should have the ability to provide informed consent for the collection and use of their personal information. 3. Data minimization: Collecting only the necessary data and minimizing the retention of personal information can help reduce the risk of privacy breaches. ## Exercise What are some strategies for balancing security and privacy? ### Solution Some strategies for balancing security and privacy include implementing privacy-enhancing technologies, being transparent about data collection and use, obtaining informed consent, and practicing data minimization. # 13. Conclusion: Future of Computer Security Computer security is an ever-evolving field, and the future holds both challenges and opportunities. As technology continues to advance, so do the threats and vulnerabilities that come with it. It is crucial for individuals and organizations to stay informed and adapt their security practices accordingly. Some current trends and predictions in computer security include: 1. Increased reliance on artificial intelligence and machine learning: As cyber threats become more sophisticated, the use of AI and ML technologies can help identify and respond to attacks in real-time. 2. Growing importance of IoT security: With the proliferation of internet-connected devices, securing the IoT ecosystem will be crucial to prevent widespread vulnerabilities and potential attacks. 3. Continued development of blockchain technology: Blockchain has the potential to revolutionize security by providing decentralized and tamper-proof systems. However, it also presents new challenges and risks that need to be addressed. 4. Focus on data privacy and compliance: With the introduction of regulations like the General Data Protection Regulation (GDPR), organizations are increasingly prioritizing data privacy and ensuring compliance with relevant laws. To navigate the future of computer security, it is essential to prioritize continued learning and adaptation. Staying up to date with the latest threats, vulnerabilities, and security practices is crucial for maintaining a strong defense against potential attacks. Some strategies for staying secure in the future include: 1. Regularly updating and patching systems: Keeping software and hardware up to date with the latest security patches is crucial for addressing known vulnerabilities. 2. Implementing secure configurations: Following best practices for system configurations can help minimize the risk of security breaches. 3. Prioritizing application security: As more services and applications move to the cloud, ensuring the security of these platforms is vital to protect sensitive data and prevent unauthorized access. 4. Regularly backing up and testing data recovery procedures: Data loss can occur due to various reasons, including cyber attacks. Having robust backup and recovery procedures in place can help minimize the impact of such incidents. ## Exercise What are some strategies for staying secure in the future? ### Solution Some strategies for staying secure in the future include regularly updating and patching systems, implementing secure configurations, prioritizing application security, and regularly backing up and testing data recovery procedures. # 13.1. Current Trends and Predictions The field of computer security is constantly evolving, with new trends and predictions shaping its future. Staying informed about these trends can help individuals and organizations prepare for the challenges and opportunities that lie ahead. Some current trends and predictions in computer security include: 1. Artificial Intelligence (AI) and Machine Learning (ML): AI and ML technologies are being increasingly used in computer security to detect and respond to threats in real-time. These technologies can analyze large amounts of data and identify patterns that may indicate malicious activity. 2. Internet of Things (IoT) Security: With the rapid growth of IoT devices, ensuring their security has become a major concern. As more devices become connected to the internet, there is a greater risk of vulnerabilities and potential attacks. Securing IoT devices and networks will be crucial in the future. 3. Cloud Security: As more organizations move their data and applications to the cloud, ensuring the security of cloud environments has become a top priority. Cloud security involves protecting data, applications, and infrastructure from unauthorized access, data breaches, and other threats. 4. Mobile Security: With the increasing use of smartphones and mobile devices, mobile security has become a critical area of focus. Mobile security involves protecting devices, applications, and data from unauthorized access, malware, and other threats. These trends and predictions highlight the need for individuals and organizations to stay updated on the latest advancements and best practices in computer security. By staying informed, they can better protect themselves and their systems from potential threats. ## Exercise What are some current trends and predictions in computer security? ### Solution Some current trends and predictions in computer security include the use of artificial intelligence and machine learning, the importance of IoT security, the focus on cloud security, and the need for mobile security. # 13.2. Challenges and Opportunities While computer security offers numerous benefits, it also presents various challenges and opportunities. Understanding these challenges and opportunities is essential for individuals and organizations to effectively navigate the field of computer security. Some of the challenges in computer security include: 1. Rapidly evolving threats: The landscape of computer threats is constantly changing, with new types of attacks and vulnerabilities emerging regularly. Staying ahead of these threats requires continuous monitoring and adaptation. 2. Complexity: Computer security can be complex, with multiple layers of protection and various technologies and tools to manage. Implementing and maintaining effective security measures can be challenging, especially for organizations with limited resources. 3. Balancing security and usability: Striking the right balance between security and usability is a common challenge. While robust security measures are essential, they should not hinder productivity or impede user experience. Despite these challenges, computer security also presents numerous opportunities: 1. Career opportunities: The demand for skilled professionals in the field of computer security is high and continues to grow. Pursuing a career in computer security can lead to exciting and rewarding job prospects. 2. Innovation and advancement: The ongoing evolution of computer security creates opportunities for innovation and advancement. New technologies, tools, and strategies are constantly being developed to enhance security measures and protect against emerging threats. 3. Collaboration and knowledge sharing: Computer security is a field that thrives on collaboration and knowledge sharing. By working together and sharing information, individuals and organizations can collectively improve their security practices and stay ahead of evolving threats. ## Exercise What are some challenges in computer security? ### Solution Some challenges in computer security include rapidly evolving threats, complexity, and the need to balance security with usability. # 13.3. Importance of Continued Learning and Adaptation In the field of computer security, continued learning and adaptation are crucial for staying ahead of threats and maintaining effective security practices. Technology and security measures are constantly evolving, and individuals and organizations must keep up with these changes to ensure their systems and data remain protected. Continued learning in computer security involves: 1. Staying updated on the latest threats and vulnerabilities: By staying informed about the latest threats and vulnerabilities, individuals and organizations can proactively implement appropriate security measures to mitigate risks. 2. Keeping up with advancements in technology: Technology is constantly evolving, and new tools and techniques are being developed to enhance security. By staying updated on these advancements, individuals and organizations can leverage new technologies to improve their security practices. 3. Participating in training and certifications: Training programs and certifications in computer security provide individuals with valuable knowledge and skills. These programs help individuals develop a deep understanding of security concepts and best practices. Adaptation in computer security involves: 1. Regularly assessing and updating security measures: Security measures should be regularly assessed to identify any vulnerabilities or weaknesses. Updates and patches should be applied promptly to address any identified issues. 2. Conducting security audits and tests: Regular security audits and tests help identify potential vulnerabilities and weaknesses in systems and networks. By conducting these audits and tests, individuals and organizations can proactively address any security gaps. 3. Learning from past incidents: When security incidents occur, it is important to learn from them and make necessary improvements. Analyzing past incidents helps identify areas for improvement and strengthens security measures. ## Exercise Why is continued learning and adaptation important in computer security? ### Solution Continued learning and adaptation are important in computer security because technology and threats are constantly evolving. Staying updated on the latest threats and advancements in technology helps individuals and organizations maintain effective security practices and protect their systems and data. Regularly assessing and updating security measures, conducting security audits and tests, and learning from past incidents are essential for staying ahead of threats and addressing any vulnerabilities or weaknesses. # 13.4. Strategies for Staying Secure Staying secure in the ever-changing landscape of computer security requires a proactive and comprehensive approach. Here are some strategies to help individuals and organizations stay secure: 1. Implement strong access controls: Use strong passwords and enforce password policies. Consider implementing multi-factor authentication for an added layer of security. Limit access privileges to only those who need them. 2. Regularly update and patch systems: Keep all software and operating systems up to date with the latest security patches. Vulnerabilities in outdated software can be exploited by attackers. 3. Use encryption: Encrypt sensitive data to protect it from unauthorized access. This includes data at rest and data in transit. Use strong encryption algorithms and secure key management practices. 4. Implement firewalls and intrusion detection systems: Firewalls help monitor and control incoming and outgoing network traffic. Intrusion detection systems can detect and respond to potential attacks. 5. Conduct regular security audits and tests: Regularly assess the security of systems and networks through audits and penetration testing. Identify vulnerabilities and address them promptly. 6. Educate and train employees: Provide security awareness training to employees to help them understand security best practices and recognize potential threats. Regularly remind employees of the importance of security. 7. Backup data regularly: Implement regular data backup procedures to ensure that critical data can be recovered in the event of a security incident or data loss. 8. Monitor and analyze security logs: Regularly review security logs and analyze them for any suspicious activity. Implement a system for real-time monitoring and alerts. 9. Stay informed about emerging threats and best practices: Stay updated on the latest security threats and best practices through industry publications, forums, and conferences. Join security communities to share knowledge and insights. 10. Establish an incident response plan: Develop a comprehensive incident response plan that outlines steps to be taken in the event of a security incident. This plan should include procedures for containment, investigation, and recovery. By implementing these strategies and staying vigilant, individuals and organizations can enhance their security posture and reduce the risk of security breaches. Remember, security is an ongoing process that requires continuous effort and adaptation.	Textbooks
Each interior angle of a regular polygon measures $140^\circ$. How many sides does the polygon have? Let $n$ be the number of sides in the polygon. The sum of the interior angles in any $n$-sided polygon is $180(n-2)$ degrees. Since each angle in the given polygon measures $140^\circ$, the sum of the interior angles of this polygon is also $140n$. Therefore, we must have \[180(n-2) = 140n.\] Expanding the left side gives $180n - 360 = 140n$, so $40n = 360$ and $n = \boxed{9}$. We might also have noted that each exterior angle of the given polygon measures $180^\circ - 140^\circ = 40^\circ$. The exterior angles of a polygon sum to $360^\circ$, so there must be $\frac{360^\circ}{40^\circ} = 9$ of them in the polygon.	Math Dataset
\begin{document} \title{Generalized spin representations \\ {\large With an appendix by Max Horn and Ralf K\"ohl: \\ Cartan--Bott periodicity for the real $E_n$ series}} \author{Guntram Hainke and Ralf K\"ohl and Paul Levy} \maketitle \begin{abstract} We introduce the notion of a generalized spin representation of the maximal compact subalgebra $\mathfrak k$ of a symmetrizable Kac--Moody algebra $\mathfrak g$ in order to show that, if defined over a formally real field, every such $\mathfrak k$ has a non-trivial reductive finite-dimensional quotient. The appendix illustrates how to compute the isomorphism types of these quotients for the real $E_n$ series. In passing this provides an elementary way of determining the isomorphism types of the maximal compact subalgebras of the semisimple split real Lie algebras of types $E_6$, $E_7$, $E_8$. \end{abstract} \section{Introduction} During the last decade the family of Kac--Moody algebras of type $E_{n}(\mathbb{R})$ has received considerable attention because of its importance in M-theory \cite{de2006kac}, \cite{gebert}, \cite{Kleinschmidt/Nicolai/Palmkvist}, \cite{palmkvist2009exceptional}, \cite{west2001e11}. By \cite{DamourKleinschmidtNicolai}, \cite{deBuylHenneauxPaulot} the (so-called) maximal compact subalgebra $\mathfrak k=\Fix \omega$ of the real split Kac--Moody algebra $\mathfrak g=\mathfrak{g}(E_{10})(\mathbb{R})$ with respect to the Cartan--Chevalley involution $\omega$ admits a 32-dimensional complex representation which extends the spin representation of its regular subalgebra $\mathfrak{so}_{10}(\mathbb{R})$. This implies that the (infinite-dimensional) Lie algebra $\mathfrak k$ has a non-trivial finite-dimensional quotient, in fact a semisimple finite-dimensional quotient (see Theorem~\ref{Mainexistencetheorem}). Since $\mathfrak k$ is anisotropic with respect to the invariant bilinear form of the Kac--Moody algebra $\mathfrak{g}$, it actually contains an ideal isomorphic to this finite-dimensional quotient. In this article we show that the existence of non-trivial finite-dimensional representations is not peculiar to the maximal compact subalgebra of $\mathfrak{g}(E_{10})(\mathbb{R})$ but is shared by all maximal compact subalgebras of symmetrizable Kac--Moody algebras over arbitrary fields of characteristic $0$. To this end we introduce the notion of a generalized spin representation (Definitions \ref{genspinrep} and \ref{genspinrep2}), which we inductively show to exist for arbitrary symmetrizable Kac-Moody algebras and which, in the case of formally real fields, affords a compact, whence reductive, and often even a semisimple image (Theorem~\ref{Mainexistencetheorem}). Our results presented in this article are generalizations of the results concerning the $\frac{1}{2}$-spin representations described in \cite{DamourKleinschmidtNicolai}, \cite{deBuylHenneauxPaulot}. The key observation is Remark~\ref{characterization} that in the simply-laced case a $\frac{1}{2}$-spin representation can be described by linear operators $A_i$ for each vertex $i$ of the diagram that satisfy \begin{enumerate} \item[(i)] $A_i^2 = -\frac{1}{4} \cdot \id$, \item[(ii)] $A_iA_j = A_jA_i$, if the vertices $i$, $j$ do not form an edge of the diagram, \item[(iii)] $A_iA_j=-A_jA_i$, if the vertices $i$, $j$ form an edge of the diagram. \end{enumerate} On the other hand, the $\frac{3}{2}$-spin representations of \cite{DamourKleinschmidtNicolai}, \cite{deBuylHenneauxPaulot} and the $\frac{5}{2}$- and $\frac{7}{2}$-spin representations of \cite{Kleinschmidt/Nicolai} are still elusive, as the algebraic identities that need to be satisfied by the corresponding linear operators are more involved. Note that our terminology of {\em maximal compact subalgebra} is misleading. For one, in the infinite-dimensional situation there is no compact group associated to a maximal compact subalgebra. Rather, over the real numbers, the maximal compact subalgebra is related to the group $K$ studied in \cite{KacPeterson}, \cite{Medts/Gramlich/Horn}. This group naturally carries a non-locally compact non-metrizable $k_\omega$-topology (cf.\ \cite{Hartnick/Koehl/Mars}). Moreover, our construction only involves the Cartan--Chevalley involution and no field involution. Therefore, over the complex numbers, what we call a maximal compact subalgebra is not even anisotropic. However, this terminology does not lead to serious ambiguities as our main focus lies on split Lie algebras over formally real fields. Our main structure-theoretic results in Section~\ref{GSR} below will consequently be obtained over formally real fields; the main future application of our result is over the real numbers. \noindent \textbf{Acknowledgements.} We thank Pierre-Emmanuel Caprace for pointing out to us the $32$-dimensional representation of the maximal compact subalgebra of $E_{10}(\mathbb{R})$, thus triggering our research. We also thank Kay Magaard for bringing our attention to \cite{Maas} and Thibault Damour, David Ghatei, Axel Kleinschmidt, Karl-Hermann Neeb, Sebastian Wei\ss\, and especially Max Horn and two anonymous referees for valuable comments on preliminary versions of this work. This research has been partially funded by the EPRSC grants EP/H02283X and EP/K022997/1. The second author gratefully acknowledges the hospitality of the IHES at Bures-sur-Yvette and of the Albert Einstein Institute at Golm. \section{Preliminaries} In this section we collect several basic facts about Kac--Moody algebras. We refer the reader to \cite[Chapter 1]{kac1994infinite} and \cite[Chapter 1]{MR1923198} for proofs and further details. \subsection{Kac--Moody algebras}\label{KMsubsec} Let $k$ be a field of characteristic 0, let $A=(a_{ij}) \in \mathbb{Z}^{n \times n}$ be a \textbf{generalized Cartan matrix} and let $\mathfrak{g}=\mathfrak{g}_A$ denote the corresponding \textbf{Kac--Moody algebra} over $k$. This means that $$a_{ii}=2, \quad a_{ij}\leq 0 \quad \mbox{and} \quad a_{ij}=0 \Leftrightarrow a_{ji}=0,$$ while $\mathfrak{g}$ is the quotient of the free Lie algebra over $k$ generated by $e_i$, $f_i$, $h_i$, $i=1, \ldots, n$, subject to the relations $$ [h_i,h_j]=0,\; [h_i,e_j]=a_{ij}e_j,\; [h_i,f_j]=-a_{ij}f_j \text{ for all } 1 \leq i, j \leq n, $$ $$ [e_i,f_j]=0,\;[e_i,f_i]=h_i,\; (\ad e_i)^{-a_{ij}+1}(e_j)=0, (\ad f_i)^{-a_{ij}+1}(f_j)=0 \text { for } i \neq j.$$ A generalized Cartan matrix is called {\bf simply laced} if the off-diagonal entries of $A$ are either 0 or $-1$; it is called {\bf symmetrizable} if there exists a diagonal matrix $\Lambda$ such that $\Lambda A$ is symmetric. \\ By abuse of terminology, we will say that ${\mathfrak g}$ is simply laced, resp.\ symmetrizable if its generalized Cartan matrix is simply laced, resp.\ symmetrizable. Let $\mathfrak h:=\langle h_1, \ldots, h_n \rangle$, $\mathfrak n_+:=\langle e_1, \ldots, e_n \rangle$ and $\mathfrak n_-:=\langle f_1,\ldots, f_n \rangle$ denote the standard subalgebras of $\mathfrak g$. Then there is a decomposition as vector spaces $$\mathfrak g=\mathfrak n_- \oplus \mathfrak h \oplus \mathfrak n_+$$ (see \cite[\S1.3, p.~7]{kac1994infinite}). The defining relations of $\mathfrak{g}$ imply that $\mathfrak{h}$ is $n$-dimensional abelian and normalizes $\mathfrak{n}_+$ and $\mathfrak{n}_-$. In fact, it acts by linear transformations on these vector spaces. Therefore, for each element $\alpha \in \mathfrak{h}^$ of the dual space it is meaningful to define the eigenspaces $$\mathfrak{g}_\alpha := \{ x \in \mathfrak{g} \mid \forall h \in \mathfrak{h} : [h,x] = \alpha(h)x \}.$$ The relations $[h_i,e_j]=a_{ij}e_j$, $1 \leq i, j \leq n$, imply that each $e_j$ is contained in such an eigenspace, which we denote by $\mathfrak{g}_{\alpha_j}$; the corresponding element of $\mathfrak{h}^$ is denoted by $\alpha_j$. (Cf.\ \cite[\S1.1]{kac1994infinite}.) Note that $\mathfrak{g}_{-\alpha_j}$ contains $f_j$. The \textbf{diagram} of a simply laced Kac--Moody algebra $\mathfrak g_A$ is the graph $D=(V,E)$ on vertices $\alpha_1, \ldots, \alpha_n$ with $\alpha_i$ and $\alpha_j$ connected by an edge if and only if $a_{ij}=-1$. Let $Q:=\oplus_{i=1}^n \mathbb{Z} \alpha_i$ denote a free $\mathbb{Z}$-module of rank $n$ and $Q_+:=\oplus_{i=1}^n \mathbb{Z}_+ \alpha_i$, where the latter denotes the set of non-negative integral linear combinations. By \cite[Thm.~1.2(d), Exercise~1.2]{kac1994infinite} $$\mathfrak{g} = \bigoplus_{\alpha \in Q} g_\alpha = \mathfrak{h} \oplus \bigoplus_{\alpha \in Q \backslash \{ 0 \}} g_\alpha = \bigoplus_{\alpha \in Q_+ \backslash \{ 0 \}} g_{-\alpha} \oplus \mathfrak{h} \oplus \bigoplus_{\alpha \in Q_+ \backslash \{ 0 \}} g_\alpha.$$ Therefore, $\mathfrak g$ has a $Q$-grading by declaring $$\deg h_i:=0, \quad \deg e_i:=\alpha_i, \quad \deg f_i:=-\alpha_i$$ for $i=1, \ldots, n$, i.e., $$\mathfrak g=\bigoplus_{\alpha \in Q} \mathfrak g_\alpha \quad \mbox{and} \quad [\mathfrak g_\alpha, \mathfrak g_\beta]\subseteq \mathfrak g_{\alpha+\beta}.$$ Let $\Delta:=\{\alpha \in Q \backslash\{0\} \mid \mathfrak g_\alpha \neq 0\}$. Then $\Delta=\Delta_+ \cup \Delta_-$, where $\Delta_+:=\Delta \cap (Q_+ \backslash \{ 0 \})$ and $\Delta_-:=-\Delta_+$. An element $\alpha \in \Delta$ is called a \textbf{root} and $\mathfrak g_\alpha$ a \textbf{root space}. A root $\alpha \in \Delta$ is called \textbf{positive} if it belongs to $\Delta_+$, otherwise \textbf{negative}. A root of the form $\alpha=\pm \alpha_i$ is called \textbf{simple}.\\ Since the adjoint representation $\mathrm{ad} : \mathfrak{g} \to \mathrm{End}(\mathfrak{g})$ is integrable (see \cite[\S 3.5]{kac1994infinite}), the \textbf{extended Weyl group} $W^* \leq \Aut \mathfrak g$ can be defined as $W^:=\langle s_i^ \mid i=1, \ldots, n \rangle$, where $$s_i^* := s_i^{\mathrm{ad}} :=\exp \ad f_i \cdot \exp \ad (-e_i) \cdot \exp \ad f_i$$ (cf.\ \cite[\S 3.8]{kac1994infinite}; note that $W^* \leq \Aut \mathfrak g$ by \cite[Lem.~3.8(b)]{kac1994infinite} ). For $\alpha \in \Delta$ and $w \in W^$ there exists a unique $w\cdot \alpha \in \Delta$ such that $w(\mathfrak{g}_\alpha)=\mathfrak g_{w \cdot \alpha}$, by \cite[Lem.~3.8(a)]{kac1994infinite}. A root $\alpha$ is called \textbf{real} if there is a $w \in W$ such that $w\cdot\alpha$ is simple, otherwise it is called \textbf{imaginary}. Let $\Delta^{\text{re}}$ denote the set of real roots and $\Delta^{\text{im}}$ the set of imaginary roots. For $\alpha=\sum_{i=1}^n{a_i \alpha_i} \in \Delta$, the {\bf height} of $\alpha$ is defined as $\hht \alpha:=\sum\limits_{i=1}^n a_i$. For $n \in \mathbb{N}$ let $$(\mathfrak n_+)_n:=\bigoplus\limits_{\substack{ \alpha \in \Delta^+\\ \hht \alpha=n}} \mfg_\alpha.$$ This is a $\mathbb{Z}$-grading of $\mathfrak n_+$ and extends to a $\mathbb{Z}$-grading of $\mathfrak{g}$, the {\bf principal grading} (cf.\ \cite[\S1.5]{kac1994infinite}). \subsection{The maximal compact subalgebra} Let $\mathfrak g$ be a Kac--Moody algebra over a field $k$ of characteristic 0. Let $\omega \in \Aut(\mathfrak g)$ denote the {\bf Cartan--Chevalley involution} characterized by $\omega(e_i)=-f_i$, $\omega(f_i)=-e_i$ and $\omega(h_i)=-h_i$. (Cf.\ \cite[Equ.~(1.3.4)]{kac1994infinite}.) Observe that $\omega(\mathfrak{g}_\alpha) = \mathfrak{g}_{-\alpha}$. Let $\mathfrak k:=\mathfrak{k}(\mathfrak{g}):=\left\{X \in \mathfrak{g} \mid \omega(X)=X\right\}$ denote the fixed point subalgebra, which --- in analogy to the situation of finite-dimensional semisimple split real Lie algebras --- is called the \textbf{maximal compact subalgebra} of $\mathfrak{g}$. For example, if $\mathfrak g=\mathfrak {sl}_n(\mathbb{R})$, then $\omega(A)=-A^T$ and $\mathfrak k=\mathfrak{so}_n(\mathbb{R})$. In this case, $\mathfrak {so}_n(\mathbb{R})$ is the Lie algebra of the maximal compact subgroup $\SO_n(\mathbb{R})$ of $\SL_n(\mathbb{R})$. See also \cite[Section IV.4]{knapp}. Over non-real closed fields, especially over the complex numbers, our terminology is a bit unfortunate and misleading. However, our main results in Section~\ref{GSR} below and future applications are over real closed fields. A theorem of Berman \cite{Berman} allows one to give a presentation of these. We point out that Berman's result in fact deals with a much more general class of so-called involutory algebras by also allowing other involutions of $\mathfrak{g}$ of the second kind (in the sense of \cite[4.6]{Kac/Wang:1992}). Note that Berman instead of our involution $\omega$ uses the involution $\eta$ given by $\eta(e_i)=f_i$, $\eta(f_i)=e_i$, $\eta(h_i)=-h_i$ as the foundation of his investigations so that in order to apply his result one still has to relate the two involutions to one another. \begin{theorem}[{cf.\ \cite[Thm.~1.31]{Berman}}] \label{Berman} Let $k$ be a field of characteristic 0. Let $A \in \mathbb{Z}^{n \times n}$ be a simply laced generalized Cartan matrix, let $\mathfrak{g}_A$ denote the corresponding Kac--Moody algebra and let $\mathfrak k$ denote the maximal compact subalgebra of $\mathfrak g$. \\ Then $\mathfrak k$ is isomorphic to the quotient of the free Lie algebra over $k$ generated by $X_1, \ldots, X_n$ subject to the relations $$ \begin{array}{rcll} [X_i,[X_i,X_j]] & = & -X_j, & \text{if the vertices $v_i,v_j$ are connected by an edge,} \\ \text{$[X_i, X_j]$} & = & 0, & \text{otherwise,}\\ \end{array} $$ via the map $X_i \mapsto e_i-f_i$. \end{theorem} In Theorem~\ref{Berman2} below we state and prove a general version of this result that applies to the maximal compact subalgebra of an arbitrary symmetrizable Kac--Moody algebra over a field of characteristic $0$. Our motivation for splitting off the simply-laced case is that it is considerably easier to understand than the general case. Furthermore, the study of generalized spin representations in the simply-laced case is key to these representations in general. \begin{proof}[Proof of Theorem~\ref{Berman}.] Let $\eta \in \Aut \mathfrak g$ denote the involution characterized by $$\eta(e_i)=f_i,\, \eta(f_i)=e_i \text{ and } \eta(h_i)=-h_i$$ and let $\mathfrak l:=\Fix \eta$ denote the subalgebra of fixed points of $\eta$. By \cite[Thm.~1.31]{Berman}, the Lie algebra $\mathfrak l$ is isomorphic to the quotient of the free Lie algebra over $k$ generated by $Y_1, \ldots, Y_n$ subject to the relations $$ \begin{array}{rcll} [Y_i,[Y_i,Y_j]] & = & Y_j, & \text{if the vertices $v_i,v_j$ are connected by an edge,} \\ \text{$[Y_i, Y_j]$} & = & 0, & \text{otherwise,}\\ \end{array} $$ via the map $Y_i \mapsto e_i+f_i$. Let $I:=\sqrt{-1}$ denote a square root of $-1$ and let $L:=k(I)$, $\mathfrak g_L:=\mathfrak g \otimes_k L$. There is a Lie algebra automorphism $\varphi \in \Aut(\mathfrak g_L)$ determined by $$e_i \mapsto I\cdot e_i, \, f_i \mapsto -I \cdot f_i \text{ and } h_i \mapsto h_i.$$ This automorphism $\varphi$ conjugates $\eta$ to $\omega$, i.e.\ $\omega=\varphi^{-1} \circ \eta \circ \varphi$, and hence the subalgebras $\Fix \eta$ and $\Fix \omega$ are isomorphic over $L$. As $X_i$ is mapped to $I\cdot Y_i$ under this isomorphism, the claim follows. \end{proof} \begin{remark}\label{weylgroupremark} Suppose $k={\mathbb C}$. We can exponentiate the subalgebra of ${\mathfrak g}$ spanned by $e_i,f_i,h_i$ to a subgroup $G_i$ of $\Aut{\mathfrak g}$ which is isomorphic to $\SL_2({\mathbb C})$ or $\PSL_2({\mathbb C})$. Then $X_i$ identifies with $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ in $\mathfrak{sl}_2$ and therefore $\exp(\xi X_i)$ is equal to the image of $\begin{pmatrix} \cos\xi & \sin\xi \\ -\sin\xi & \cos\xi\end{pmatrix}$ in $G_i$. In particular, $\exp(-\frac{\pi}{2}X_i)$ is sent to $s_i^$. It follows that $s_i^$ and $\omega$ are commuting automorphisms of ${\mathfrak g}$. For the case of an arbitrary ground field, $\omega$ induces a Cartan--Chevalley involution on the standard type $A_1$ subgroup $G_i$ of $\Aut{\mathfrak g}$ whose Lie algebra is spanned by $e_i$, $f_i$, $h_i$. The fixed point subgroup of $G_i$ for the Cartan--Chevalley involution is either $\SO_2(k)$ or $\SO_2(k)/\{\pm I_2\}$, depending on whether $G_i$ is isomorphic to $\SL_2$ or $\PSL_2$. Since this subgroup clearly contains $s_i^$, it follows that $s_i^$ commutes with $\omega$. \end{remark} \subsection{Rank 2 Kac--Moody algebras} \label{rank2KM} Let ${\mathfrak g}$ be the Kac--Moody algebra with Cartan matrix $\begin{pmatrix} 2 & -r \\ -s & 2 \end{pmatrix}$, where $r,s\in{\mathbb N}$. We map ${\mathfrak g}$ into a simply laced Kac--Moody algebra as follows: Let $D$ be a complete bipartite graph on $r$ and $s$ vertices, labelled $\alpha_1^{(i)}$ and $\alpha_2^{(j)}$ with $1\leq i\leq r$, $1\leq j\leq s$. Let $\tilde{\mathfrak g}$ be a Kac--Moody Lie algebra with simply laced diagram $D$ and label the generators correspondingly: $e_1^{(i)}$, $f_1^{(i)}$, $h_1^{(i)}$ and $e_2^{(j)}$, $f_2^{(j)}$, $h_2^{(j)}$. We remark that there is an action of $\mathrm{Sym}(r)$ (resp. $\mathrm{Sym}(s)$) on $\tilde{\mathfrak g}$ by permuting the roots $\alpha_1^{(i)}$ (resp. $\alpha_2^{(j)}$). Let \begin{eqnarray} E_1=\sum_{i=1}^r e_1^{(i)},\;\; F_1=\sum_{i=1}^r f_1^{(i)},\;\; H_1=[E_1,F_1], \\ E_2=\sum_{j=1}^s e_2^{(j)},\;\; F_2=\sum_{j=1}^s f_2^{(j)},\;\; H_2=[E_2,F_2]. \end{eqnarray} Then it is straightforward to check that $[E_1,F_2]=0=[E_2,F_1]=[H_1,H_2]$, $(\ad E_1)^{r+1}(E_2)=0=(\ad E_2)^{s+1}(E_1)$, and $(\ad F_1)^{r+1}(F_2)=(\ad F_2)^{s+1}(F_1)=0$. Thus there is a well-defined Lie algebra homomorphism $\tilde\varphi$ from ${\mathfrak g}$ to $\tilde{\mathfrak g}$, sending each of $e_1$, $e_2$, $f_1$, $f_2$, $h_1$, $h_2$ to its corresponding upper-case letter. Since ${\mathfrak g}$ has no non-zero ideals intersecting trivially with ${\mathfrak h}$, it follows that $\tilde\varphi$ is injective. It is clear from the definitions that $\tilde\varphi$ induces an injective homomorphism from the extended Weyl group of ${\mathfrak g}$ to that of $\tilde{\mathfrak g}$ by sending $s_1^$ to $(s_1^{(1)})^\ldots (s_1^{(r)})^$, and similarly for $s_2^$. \begin{remark} This construction is related to the notion of {\it pinning}\footnote{French ``\'epinglage'', see \cite[Expos\'e XXIII]{sga}. Although this is translated as ``framing'' in \cite{Bourbaki}, it is clear from the footnote to \cite[Expos\'e XXIII, Def.~1.1]{sga} (where a maximal torus is the body, and opposite Borel subgroups are the wings, of a butterfly) that ``pinning'' is more appropriate. It seems to have become the standard terminology in English.} for split semisimple Lie algebras. Given a split semisimple Lie algebra $\tilde{\mathfrak g}$ over a field $k$ of characteristic zero, let $\tilde{\mathfrak h}$ be a splitting Cartan subalgebra. A pinning of $(\tilde{\mathfrak g},\tilde{\mathfrak h})$ consists of a basis $\Pi$ of the roots of $\tilde{\mathfrak g}$ relative to $\tilde{\mathfrak h}$, together with a choice $\{ x_\alpha : \alpha\in \Pi\}$ of non-zero elements in each simple positive root space. If $\tilde{\mathfrak g}$ has a presentation as in \S \ref{KMsubsec} then we can take $\Pi=\{ \alpha_1,\ldots ,\alpha_n\}$ and $x_{\alpha_i}=e_i$ for $1\leq i\leq n$. If a pinning of $(\tilde{\mathfrak g},\tilde{\mathfrak h})$ is fixed, then a {\it pinned automorphism} is an automorphism which stabilizes $\tilde{\mathfrak h}$ and the Borel subalgebra of $\tilde{\mathfrak g}$ corresponding to $\Pi$, and which permutes the elements $x_\alpha$, $\alpha\in\Pi$. Clearly, the group of pinned automorphisms is isomorphic to the group ${\rm Aut}(\Pi)$ of automorphisms of the Dynkin diagram of $\tilde{\mathfrak g}$. As follows from \cite[VIII.3 Cor.~1 and VIII.4]{Bourbaki}, the group ${\rm Aut}(\tilde{\mathfrak g})$ is the semidirect product of ${\rm Aut}(\Pi)$ and $\tilde{G}(k)$, where $\tilde{G}$ is the adjoint type semisimple group with Lie algebra $\tilde{\mathfrak g}$. The corresponding result is also true in the Kac--Moody case \cite[\S 6, Theorem~2(c)]{Peterson-Kac}. When $\tilde{\mathfrak g}$ has generalized Cartan matrix $\begin{pmatrix} 2 & -r \\ -s & 2\end{pmatrix}$, one obtains that the automorphism group is $({\rm Sym}(r)\times{\Sym} (s))\ltimes \tilde{G}$ if $r\neq s$ and is $({\rm Sym}(r) \wr{\rm Sym}(2))\ltimes\tilde{G}$ if $r=s$, where $\tilde{G}$ is an adjoint Kac--Moody group corresponding to $\tilde{\mathfrak g}$. (We exclude here the affine cases $r=s=2$ and $\{ r,s\}=\{ 1,4\}$, where the picture is slightly more complicated.) If $\tilde{\mathfrak g}$ has finite type, then there are no non-trivial pinned automorphisms unless $\tilde{\mathfrak g}$ is simply laced. Furthermore, a simple Lie algebra of type $B_n$ (resp.\ $C_n$, $F_4$, $G_2$) can be realised as the fixed point subalgebra for a pinned automorphism of a Lie algebra of type $D_{n+1}$ (resp.\ $A_{2n-1}$, $E_6$, $D_4$). In our case we can only say that ${\mathfrak g}$ is a {\it subalgebra} of the fixed-point subalgebra of $\tilde{\mathfrak g}$. \end{remark} Let $\tilde\omega$ (resp.\ $\omega$) denote the Cartan--Chevalley involution on $\tilde{\mathfrak g}$ (resp.\ ${\mathfrak g}$). Clearly $\tilde\varphi\circ\omega=\tilde\omega\circ\tilde\varphi$, so $\tilde\varphi$ induces a homomorphism from ${\mathfrak k}={\mathfrak k}({\mathfrak g})$ to $\tilde{\mathfrak k}={\mathfrak k}(\tilde{\mathfrak g})$. Following the proof of Theorem~\ref{Berman}, let $Y_1=e_1+f_1$, $Y_2=e_2+f_2$, $Y_1^{(i)}=e_1^{(i)}+f_1^{(i)}$ and $Y_2^{(j)}=e_2^{(j)}+f_2^{(j)}$ for $1\leq i\leq r$, $1\leq j\leq s$. Then $\tilde\varphi(Y_1)=\sum_1^r \tilde{Y}_1^{(i)}$ and similarly for $Y_2$. Since $\alpha_1^{(i)}$ and $\alpha_2^{(j)}$ are connected by a simple edge, we have $((\ad Y_1^{(i)})^2-1)(Y_2^{(j)})=0$. Now the space spanned by $Y_1^{(i)}$ for $1\leq i\leq r$ is conjugate to the subspace of $\tilde{\mathfrak h}$ spanned by $h_1^{(i)}$ for $1\leq i\leq r$. Thus the fact that $((\ad Y_1^{(i)})^2-1)(Y_2^{(j)})=0$ can be restated by saying that $Y_2^{(j)}$ is a sum of simultaneous eigenvectors for $\ad Y_1^{(i)}$, with each such eigenvalue being $\pm 1$. It follows that $Y_2^{(j)}$ is contained in the sum of eigenspaces for $\ad \tilde\varphi(Y_1)$ in $\tilde{\mathfrak g}$ with eigenvalues $r, r-2, \ldots ,-r$. Hence $$\left(\prod_{i=0}^r (\ad\tilde\varphi(Y_1)-(r-2i))\right)(\tilde\varphi(Y_2))=0.$$ Setting $X_i=e_i-f_i$ for $i=1,2$ and conjugating $Y_i$ to $X_i$ as in the proof of Theorem~\ref{Berman}, we deduce that $P_{r}(\ad X_1)(X_2)=0$ and $P_s(\ad X_2)(X_1)=0$, where $$P_m(t)= \left\{ \begin{array}{rl} (t^2+m^2)(t^2+(m-2)^2)\cdots (t^2+1), & \text{if $m$ is odd}, \\ (t^2+m^2)(t^2+(m-2)^2)\cdots (t^2+4)t, & \text{if $m$ is even}. \end{array} \right.$$ \subsection{The general symmetrizable case}\label{symmetrizablesec} Now suppose ${\mathfrak g}$ is an arbitrary symmetrizable Kac--Moody algebra with $n\times n$ generalized Cartan matrix $A=(a_{ij})_{1\leq i,j\leq n}$. For $1\leq i\leq n$ let $X_i=e_i-f_i\in{\mathfrak k}$. On restricting to the rank 2 subalgebra of ${\mathfrak g}$ generated by $e_i,e_j,f_i,f_j$ we obtain the relation $P_{-a_{ij}}(\ad X_i)(X_j)=0$. As in the simply-laced case, we can use Berman's Theorem \cite[Thm.~1.31]{Berman} to prove that these generate all of the relations in ${\mathfrak k}$. We reproduce a proof (which also applies in the simply-laced case) for the sake of completeness. \begin{theorem}\label{Berman2} The maximal compact subalgebra ${\mathfrak k}$ of ${\mathfrak g}$ has generators $X_1$, \ldots, $X_n$ and relations: $$\left(P_{-a_{ij}}(\ad X_i)\right) (X_j)=0$$ for any $1\leq i\neq j\leq n$. \end{theorem} \begin{proof} By the Gabber--Kac Theorem \cite[Thm.~9.11]{kac1994infinite} the ideal of relations satisfied by $e_1,\ldots ,e_n$ is generated by the terms $(\ad e_i)^{-a_{ij}+1}(e_j)=0$. Let ${\mathcal L}$ be the Lie algebra on generators $x_1,\ldots ,x_n$ with relations $P_{-a_{ij}}(\ad x_i)(x_j)=0$ for $1\leq i\neq j\leq n$. Then there is a Lie algebra homomorphism $\pi:{\mathcal L}\rightarrow{\mathfrak k}$, sending $x_i$ to $X_i=e_i-f_i$. For $\alpha,\beta\in Q_+$ we write $\alpha\leq \beta$ when $\beta-\alpha\in Q_+$. We note that both ${\mathcal L}$ and ${\mathfrak k}$ are {\it filtered} by $Q_+$, that is, there exist subspaces ${\mathcal L}_{(\alpha)}$ of ${\mathcal L}$ such that: - ${\mathcal L}=\cup_{\alpha\in Q_+}{\mathcal L}_{(\alpha)}$; - ${\mathcal L}_{(\alpha)}\subset{\mathcal L}_{(\beta)}$ whenever $\alpha\leq\beta$; and - $[{\mathcal L}_{(\alpha)},{\mathcal L}_{(\beta)}]\subseteq{\mathcal L}_{(\alpha+\beta)}$; and similarly for ${\mathfrak k}$. Specifically, ${\mathfrak k}_{(\alpha)}=(\sum_{-\alpha\leq\beta\leq\alpha}{\mathfrak g}_\beta)\cap{\mathfrak k}$ and ${\mathcal L}_{(\alpha)}$ is the span of all commutators $$[x_{i_1},[x_{i_2},[\ldots[x_{i_{r-1}},x_{i_r}]\ldots ]]$$ where $\alpha_{i_1} +\ldots +\alpha_{i_r}\leq\alpha$. These filtrations are compatible, i.e.\ $\pi({\mathcal L}_{(\alpha)})\subset{\mathfrak k}_{(\alpha)}$. For $\alpha\in Q_+$, let ${\mathcal L}_{<\alpha}:=\sum_{\beta<\alpha}{\mathcal L}_{(\beta)}$ and similarly for ${\mathfrak k}$. The {\it corresponding graded Lie algebra} of ${\mathcal L}$ is the vector space $${\rm gr}\,{\mathcal L}:=\sum_{\alpha\in Q_+}{\mathcal L}_{(\alpha)}/{\mathcal L}_{<\alpha}$$ with the Lie bracket induced by that on ${\mathcal L}$. For $1\leq i\leq n$ let $\overline{x}_i$ denote the image of $x_i$ in ${\mathcal L}_{(\alpha_i)}/{\mathcal L}_{<\alpha_i}\subset{\rm gr}\, {\mathcal L}$. By the definition of the polynomials $P_m$, we have $(\ad \overline{x}_i)^{-a_{ij}+1}(\overline{x}_j)=0$ for $1\leq i\neq j\leq n$. It follows that there is a surjective homomorphism ${\mathfrak n}_+\rightarrow{\rm gr}\, {\mathcal L}$ sending $e_i$ to $\overline{x}_i$. On the other hand, ${\mathfrak k}_{(\alpha)}/{\mathfrak k}_{<\alpha}$ is spanned by $({\mathfrak g}_\alpha\oplus{\mathfrak g}_{-\alpha})\cap{\mathfrak k}$ so is of dimension $\dim{\mathfrak g}_\alpha$. (In fact, ${\rm gr}\, {\mathfrak k}\cong{\mathfrak n}_+$, see the remarks after Proposition~\ref{contractionprop} below.) Now we can prove the theorem as follows. First of all, we claim that the homomorphism $\pi:{\mathcal L}\rightarrow{\mathfrak k}$ is surjective. To prove our claim it will suffice to show that $\pi({\mathcal L}_{(\alpha)})={\mathfrak k}_{(\alpha)}$ for all $\alpha\in\Delta_+$. We note that ${\mathfrak g}_\alpha$ is spanned by elements of the form $y_\alpha=[e_i,y_{\alpha-\alpha_i}]$ where $y_{\alpha-\alpha_i}\in{\mathfrak g}_{\alpha-\alpha_i}$ and $\alpha_i$ can be any simple root. By an obvious induction hypothesis, we may assume that ${\mathfrak k}_{(\alpha-\alpha_i)}\subset\pi({\mathcal L}_{(\alpha-\alpha_i)})$ and ${\mathfrak k}_{(\alpha-2\alpha_i)}\subset\pi({\mathcal L}_{(\alpha-2\alpha_i)})$. Then $y_\alpha+\omega(y_\alpha) = [e_i-f_i,y_{\alpha-\alpha_i}+\omega(y_{\alpha-\alpha_i})]+[f_i,y_{\alpha-\alpha_i}]+\omega([f_i,y_{\alpha-\alpha_i}])$. Since $[e_i-f_i,y_{\alpha-\alpha_i}+\omega(y_{\alpha-\alpha_i})]\in\pi([x_i,{\mathcal L}_{(\alpha-\alpha_i)}])$ and $[f_i,y_{\alpha-\alpha_i}]+\omega([f_i,y_{\alpha-\alpha_i}])\in\pi({\mathcal L}_{(\alpha-2\alpha_i)})$, it follows that $y_\alpha+\omega(y_\alpha)\in\pi({\mathcal L}_{(\alpha)})$. For injectivity, we remark that the inequalities $$\dim{\mathfrak g}_\alpha\geq \dim{\mathcal L}_{(\alpha)}/{\mathcal L}_{<\alpha}\geq\dim{\mathfrak k}_{(\alpha)}/{\mathfrak k}_{<\alpha}=\dim{\mathfrak g}_\alpha$$ establish that ${\rm ker}\,\pi\cap{\mathcal L}_{(\alpha)}=\{0\}$. \end{proof} \begin{remark} Suppose $A=\begin{pmatrix} 2 & -r \\ -s & 2\end{pmatrix}$ where $r,s\neq 0$. It is easy to see that if we quotient ${\mathfrak k}$ by the ideal generated by $[X_1,[X_1,X_2]]+r^2 X_2$ and $[X_2,[X_2,X_1]]+s^2X_1$ then we obtain an epimorphism ${\mathfrak k}\rightarrow\mathfrak{so}_3$. This corresponds to repeatedly applying Construction~\ref{reductionconstruction}(a) below to the complete bipartite graph to obtain a diagram of type $A_2$. \end{remark} In what follows, we suppose that the generalized Cartan matrix $A$ is indecomposable. Then there is a well-defined, unique up to scalar multiplication {\it length function} $\| \cdot \|$ on the simple roots such that $\frac{a_{ij}}{a_{ji}}=\frac{\|\alpha_j\|^2}{\|\alpha_i\|^2}$ whenever $a_{ij}\neq 0$. After scaling we may assume that $\|\alpha_i\|^2\in{\mathbb N}$ for any $i$, and that the square lengths $\|\alpha_i\|^2$ have no common factor. \begin{definition} A {\bf simply laced cover diagram of ${\mathfrak g}$} (or just a {\bf cover diagram} for short) is a simply laced diagram $D$ with $n_i$ vertices $\alpha_i^{(1)}$, \ldots, $\alpha_i^{(n_i)}$ for each simple root $\alpha_i$ of ${\mathfrak g}$ (where $n_i$ are some positive integers), and such that each $\alpha_i^{(k)}$ is connected to exactly $\|a_{ij}\|$ of the vertices $\alpha_j^{(l)}$ for $j\neq i$ and to none of the other vertices $\alpha_i^{(l)}$. \end{definition} We remark that the $n_i$ are related by the formula $\frac{n_i}{n_j}=\frac{a_{ij}}{a_{ji}}$ whenever $a_{ij}\neq 0$, hence $n_i=\frac{M}{\|\alpha_i\|^2}$ for some constant $M$. It follows that $M$ is divisible by all $\|\alpha_i\|^2$. Moreover, each $n_i$ must be divisible by any non-zero value $\|a_{ij}\|$, so that $M$ is divisible by ${\rm lcm}_{j\neq k: a_{jk}\neq 0}( \|\alpha_j\|^2 \cdot\|a_{jk}\|)$. In the special case that $M={\rm lcm}_{j\neq k:a_{jk}\neq 0}(\|\alpha_j\|^2\cdot\|a_{jk}\|)$ we call the diagram to be of {\bf minimal rank}. Clearly, one can construct a minimal rank simply laced cover diagram for ${\mathfrak g}$ by setting $$n_i=\frac{{\rm lcm}_{j\neq k:a_{jk}\neq 0}(\|\alpha_j\|^2\cdot\|a_{jk}\|)}{\|\alpha_i^2\|}$$ for all $i$ and for each pair $(i,j)$ with $a_{ij}<0$, arbitrarily dividing the vertices $\alpha_i^{(1)}$, \ldots, $\alpha_i^{(n_i)}$ (resp.\ $\alpha_j^{(1)}$, \ldots, $\alpha_j^{(n_j)}$) into $m=\frac{n_i}{\|a_{ij}\|}=\frac{n_j}{\|a_{ji}\|}$ subsets $S_1$, \ldots, $S_m$ (resp.\ $S'_1$, \ldots, $S'_m$) of $\|a_{ij}\|$ (resp.\ $\|a_{ji}\|$) vertices with every vertex in $S_k$ joined to every vertex in $S'_k$. As the following examples show, not every connected cover diagram is minimal rank, and two minimal rank cover diagrams need not be isomorphic. \begin{example} \begin{enumerate} \item The Kac--Moody algebra which has generalized Cartan matrix $\begin{pmatrix} 2 & -1 & -1 \\ -2 & 2 & -2 \\ -2 & -2 & 2 \end{pmatrix}$ has (at least) the following two simply laced cover diagrams: \begin{center} \begin{tikzpicture}[scale=0.8] \begin{scope} \node[dnode,label=below left:$b$] (b1) at (-2,-2) {}; \node[dnode,label=above right:$b$] (b2) at (2,2) {}; \node[dnode,label=above:$a$] (a) at (0,0) {}; \node[dnode,label=below right:$c$] (c1) at (2,-2) {}; \node[dnode,label=above left:$c$] (c2) at (-2,2) {}; \draw[sedge] (b1) -- (c1) -- (b2) -- (c2) -- (b1); \draw[sedge] (b1) -- (a) -- (b2); \draw[sedge] (c1) -- (a) -- (c2); \end{scope} \end{tikzpicture} \begin{tikzpicture}[scale=0.8] \begin{scope} \node[dnode,label=below:$a$] (a1) at (0,-2.5) {}; \node[dnode,label=above:$a$] (a2) at (0,2.5) {}; \node[dnode,label=left:$b$] (b1) at (-3,-1) {}; \node[dnode,label=right:$b$] (b2) at (-1,1) {}; \node[dnode,label=left:$b$] (b3) at (1,1) {}; \node[dnode,label=right:$b$] (b4) at (3,-1) {}; \node[dnode,label=left:$c$] (c1) at (-3,1) {}; \node[dnode,label=right:$c$] (c2) at (-1,-1) {}; \node[dnode,label=left:$c$] (c3) at (1,-1) {}; \node[dnode,label=right:$c$] (c4) at (3,1) {}; \draw[sedge] (a1) -- (b1) -- (c1) -- (a2) -- (b2) -- (c2) -- (a1) -- (b4) -- (c4) -- (a2) -- (b3) -- (c3) -- (a1); \draw[sedge] (b1) -- (c2); \draw[sedge] (b2) -- (c1); \draw[sedge] (b3) -- (c4); \draw[sedge] (b4) -- (c3); \end{scope} \end{tikzpicture} \end{center} \item If ${\mathfrak g}$ has symmetrizable Cartan matrix $\begin{pmatrix} 2 & -3 & -6 \\ -5 & 2 & -5 \\ -2 & -1 & 2\end{pmatrix}$, then under the assumptions above we have $\|\alpha_1\|^2=5$, $\|\alpha_2\|^2=3$ and $\|\alpha_3\|^2=15$. Thus ${\rm lcm}_{j\neq k:a_{jk}\neq 0}(\|\alpha_j\|^2\cdot\|a_{jk}\|)=30$ and therefore $n_1=6$, $n_2=10$, $n_3=2$. Note that $\alpha_3^{(1)}$ and $\alpha_3^{(2)}$ are connected to all of the vertices $\alpha_1^{(1)}$, \ldots, $\alpha_1^{(6)}$, but each to only half of $\alpha_2^{(1)}$, \ldots, $\alpha_2^{(10)}$. Similarly, the vertices $\alpha_2^{(i)}$ also divide into two groups of five, each connecting to three of the vertices $\alpha_1^{(1)}$, \ldots, $\alpha_1^{(6)}$. After renumbering we may assume that $\alpha_1^{(1)}$, $\alpha_1^{(2)}$, $\alpha_1^{(3)}$ are connected to all of $\alpha_2^{(1)}$, \ldots, $\alpha_2^{(5)}$. It is not hard to see that there are three isomorphism classes of minimal rank cover diagrams for ${\mathfrak g}$, given by diagrams in which $\alpha_3^{(1)}$ connects to $0$, $1$ or $2$ of the vertices $\alpha_2^{(1)}$, \ldots, $\alpha_2^{(5)}$. \end{enumerate} \end{example} \begin{remark} If ${\mathfrak g}$ is of finite (resp.\ affine) type then there is a unique choice of connected simply laced cover diagram for ${\mathfrak g}$, which is also finite (resp.\ affine). Specifically, for the finite type Lie algebras of type $B_n$, $C_n$, $F_4$ and $G_2$ one obtains simply laced cover diagrams of type $D_{n+1}$, $A_{2n-1}$, $E_6$ and $D_4$, and similarly for the corresponding (untwisted) affine types. The twisted affine types all have simply laced cover diagrams which are of affine type $D$ except for the dual of affine $F_4$, which has simply laced cover ${E}^+_7$. If ${\mathfrak g}$ is an arbitrary Kac--Moody Lie algebra of rank two then there exists a unique choice of simply laced cover diagram, constructed in Section~\ref{rank2KM}. \end{remark} If the generalized Cartan matrix of ${\mathfrak g}$ is not indecomposable then a minimal rank simply laced cover diagram for ${\mathfrak g}$ is one which has the smallest possible number of vertices. Such a diagram can be constructed as the union of the (minimal rank) simply laced cover diagrams for the simple summands of ${\mathfrak g}$. Let ${\mathfrak g}$ be an arbitrary symmetrizable Kac--Moody algebra and let $\tilde{\mathfrak g}$ be the Kac--Moody algebra associated to some simply laced cover diagram for ${\mathfrak g}$. Let $e_i^{(k)}$, $f_i^{(k)}$, $h_i^{(k)}$ be the simple root elements corresponding to the vertex $\alpha_i^{(k)}$, for $1\leq k\leq n_i$. As in the rank 2 case there is a natural embedding $\tilde\varphi:{\mathfrak g}\rightarrow\tilde{\mathfrak g}$ which sends $e_i$ (resp.\ $f_i$) to $\sum_{k=1}^{n_i} e_i^{(k)}$ (resp.\ $\sum_{k=1}^{n_i} f_i^{(k)}$) and which induces a map from the extended Weyl group of ${\mathfrak g}$ to that of $\tilde{\mathfrak g}$. Clearly, there is also a corresponding embedding ${\mathfrak k}\hookrightarrow\tilde{\mathfrak k}$. \section{Some algebraic properties of $\mathfrak{k}$} In this section we collect some consequences of Berman's presentation of the maximal compact subalgebra of a Kac--Moody algebra. \subsection{Automorphisms} For $i=1,\ldots, n$ let $\varepsilon_i \in \{\pm 1\}$. Then there is an automorphism $\varphi_\varepsilon$ of $\mathfrak k$ characterized by $\varphi(X_i)=\varepsilon_i X_i$, called a \textbf{sign automorphism}. If $\pi\in\mathrm{Sym}(n)$ is a permutation which preserves the generalized Cartan matrix of ${\mathfrak g}$ (i.e., $a_{\pi(i)\pi(j)}=a_{ij}$ for all $i$, $j$) then there is an induced automorphism $\varphi_\pi$ of $\mathfrak k$ satisfying $\varphi_\pi(X_i)=X_{\pi(i)}$. Such an automorphism is called a \textbf{graph automorphism}. (In the simply-laced case $\pi$ corresponds exactly to an automorphism of the diagram of ${\mathfrak g}$, i.e., a permutation of the vertices which preserves adjacency.) \begin{lemma}\label{comp} Let $\mathfrak g$ be a Kac--Moody algebra over a field $k$ of characteristic 0. \begin{enumerate} \item For $i=1, \ldots, n$, the element $s_i^ \in W^$ commutes with $\omega$. \item Every $w \in W^$ induces an automorphism $\pi(w)$ of $\mathfrak{k}$. \item If the Kac--Moody algebra $\mathfrak g$ is simply laced, the automorphism $\pi(s_i^)$ induced by $s_i^$ via the isomorphism given in Theorem~\ref{Berman} satisfies \begin{eqnarray} X_i & \mapsto & X_i, \\ X_j & \mapsto& X_j, \text{ if $(i,j) \not \in E$, and} \\ X_j & \mapsto & [X_i,X_j], \text{ if $(i,j) \in E$}. \end{eqnarray} \end{enumerate} \end{lemma} \begin{proof} Statement (a) has been proved in Remark~\ref{weylgroupremark}. By (a), each $s_i^$ stabilizes $\mathfrak k$. Statement (b) therefore follows immediately from \cite[Lem.~3.8(b)]{kac1994infinite}. Concerning (c), a calculation in $\mathfrak{sl}_2(k)$ shows that $s_i^(e_i)=-f_i$. A calculation in $\mathfrak{sl}_3(k)$ shows $s_i^(e_j)=[e_i,e_j]$, if $(i,j) \in E$, and a calculation in $\mathfrak{sl}_2(k) \oplus \mathfrak{sl}_2(k)$ shows $s_i^(e_j) = e_j$, if $(i,j) \not\in E$. More calculations --- or use of assertion (a) --- show, furthermore, $s_i^(f_i)=-e_i$ and $s_i^(f_j)=-[f_i,f_j]$, if $(i,j) \in E$, and $s_i^(f_j) = f_j$, if $(i,j) \not\in E$. In particular, $$s_i^(e_j-f_j) = s_i^(e_j)-s_i^(f_j) = [e_i,e_j]+[f_i,f_j]=[e_i-f_i,e_j-f_j].$$ Statement (c) follows. \end{proof} For $w \in W^$, the induced automorphism $\pi(w) \in \Aut \mathfrak k$ is called a \textbf{Weyl group automorphism}. \begin{remark} \begin{enumerate} \item Let $\varphi_+ : \mathfrak n_+ \to \mathfrak k : x \mapsto x+\omega(x)$ denote the canonical $k$-linear bijection (cf.~\cite[p.~3169]{Berman}), and write $\mathfrak k_\alpha:=\varphi_+(\mathfrak g_\alpha)$. Observe that for the analogous $k$-linear bijection $\varphi_- : \mathfrak n_- \to \mathfrak k : x \mapsto x+\omega(x)$ one has $\mathfrak{k}_\alpha=\varphi_+(\mathfrak{g}_\alpha) = \varphi_-(\mathfrak{g}_{-\alpha})=\mathfrak{k}_{-\alpha}$. It follows from Lemma~\ref{comp}(a) that $\pi(s)(\mathfrak k_\alpha)=\mathfrak k_{s \cdot \alpha}$. Hence, by induction and by the definition of the set of real roots, for any positive real root $\alpha \in \Delta_+$ there is a Weyl group automorphism $\pi(w)$ and a positive simple root $\alpha_i$ such that $\pi(w)(\mathfrak k_\alpha)=\mathfrak k_{\alpha_i}=k X_i$. \item The set of subspaces $\{\mathfrak k_\gamma \mid \gamma \in \Delta^{\mathrm{re}} \cap \Delta_+\}$ is invariant under the action of the group of Weyl group automorphisms. It can be identified with the walls of the Coxeter complex of the Weyl group $W$. (Cf.\ \cite[Rem.~3.8]{kac1994infinite}.) \end{enumerate} \end{remark} \begin{remark} If ${\mathfrak g}$ is simply laced then for $i$, $j$ in the same connected component of the diagram of $\mathfrak{k}$ there is an automorphism such that $\varphi(X_i)=X_j$. This is because, if $(i,j)$ is an edge, then $$\pi(s_i^s_j^)(X_i)\stackrel{\ref{comp}}{=}\pi(s_i^)([X_j,X_i])=[\pi(s_i^)(X_j),\pi(s_i^)(X_i)]\stackrel{\ref{comp}}{=}[[X_i,X_j],X_i]\stackrel{\ref{Berman}}{=}X_j;$$ thus, the claim follows by induction. This can be used as follows: Let $\mathfrak k$ be the maximal compact subalgebra of a Kac--Moody algebra of type $AE_4$ (see Section~\ref{DD}). Then the generator $X_4$ is contained in a subalgebra isomorphic to the maximal compact subalgebra of a Kac--Moody algebra of type $A_2^+$. Indeed, let $\varphi$ be a Weyl group automorphism such that $\varphi(X_3)=X_4$. Then $\varphi(\mathfrak \langle X_1,X_2,X_3 \rangle)$ is as required, as by Theorem~\ref{Berman} the Lie algebra $\langle X_1, X_2, X_3 \rangle$ equals the maximal compact subalgebra of the Kac--Moody algebra with positive simple roots $\alpha_1$, $\alpha_2$, $\alpha_3$. \end{remark} \subsection{A contraction of $\mathfrak k$.} Let $\mathfrak g$ be a symmetrizable Kac--Moody algebra over $\mathbb{R}$ with Chevalley generators $e_i$, $f_i$, $h_i$, $i=1, \ldots, n$. For $\varepsilon >0$ define $\omega_\varepsilon$ to be the Lie algebra automorphism satisfying $$ \omega_\varepsilon (e_i)= -\varepsilon f_i,~ \omega_\varepsilon(f_i)=-\frac{1}{\varepsilon} e_i,~ \omega_\varepsilon(h_i)=-h_i;$$ moreover, set $\mathfrak k _\varepsilon:=\Fix \omega_\varepsilon$. Observe that $\mathfrak k=\mathfrak k_1$ and that $X_i^\varepsilon:=e_i-\varepsilon f_i \in \mathfrak k_\varepsilon$ for $i=1,\ldots, n$. Moreover, the automorphism $\theta_\varepsilon$ of ${\mathfrak g}$ given by $e_i\mapsto\frac{1}{\sqrt\varepsilon}e_i$ and $f_i\mapsto\sqrt\varepsilon f_i$ for all $i$ satisfies $$\theta_\varepsilon(X_i)=\frac{1}{\sqrt\varepsilon}X_i^\varepsilon, \quad \quad \omega_\varepsilon=\theta^2_\varepsilon\circ\omega=\theta_\varepsilon\circ\omega\circ\theta_\varepsilon^{-1}.$$ Thus $\theta_\varepsilon$ maps ${\mathfrak k}$ isomorphically onto ${\mathfrak k}_\varepsilon$. By applying $\theta_\varepsilon$ to $P_{-a_{ij}}(\ad X_i)(X_j)$ (using the notation of Theorem~\ref{Berman2}), we obtain the relations: $$P_{-a_{ij}}^\varepsilon(\ad X_i^\varepsilon)(X_j^\varepsilon) = 0\;\;\;\mbox{where}\;\; P_{m}^\varepsilon(t)=\varepsilon^{\frac{m+1}{2}}P_m\left(\frac{t}{\sqrt\varepsilon}\right)$$ that is, $P_m^\varepsilon(t)=(t^2+m^2\varepsilon)\cdots (t^2+\varepsilon)$ for $m$ odd, and $P_m^\varepsilon(t)=(t^2+m^2\varepsilon)\cdots (t^2+4\varepsilon)t$ for $m$ even. In particular, $[X_i^\varepsilon,[X_i^\varepsilon,X_j^\varepsilon]] = -\varepsilon X_j^\varepsilon$, if $a_{ij}=-1$. Since $\theta_\varepsilon$ maps ${\mathfrak k}$ isomorphically onto ${\mathfrak k}_\varepsilon$, we have: \begin{proposition}\label{contractionprop} The subalgebra $\mathfrak k_\varepsilon$ is isomorphic to the quotient of the free Lie algebra over $k$ generated by $X_1, \ldots, X_n$ subject to the relations $$ P_{-a_{ij}}^\varepsilon(\ad X_i)(X_j) =0 $$ via the map $X_i \mapsto e_i-\varepsilon f_i$. \end{proposition} Note that, if we set $\varepsilon=0$ in the above presentation, the resulting algebra is isomorphic to $\mathfrak n_+$ by the Gabber--Kac Theorem \cite[Thm.~9.11]{kac1994infinite}. This means that $\mathfrak n_+$ is a \textbf{contraction} of the maximal compact subalgebra $\mathfrak k=\mathfrak k_1$ in the sense of \cite{FialowskideMontigny}. \subsection{Quotients} Let $k$ be a field of characteristic 0 and $\mathfrak g$ a Kac--Moody algebra over $k$ with simply laced diagram $D$. Due to the Coxeter-like presentation of the maximal compact subalgebra $\mathfrak k$ it is possible to exhibit quotients of $\mathfrak k$ if $D$ has a certain shape. For a graph $D$, let $\mathfrak k(D)$ denote the maximal compact subalgebra of the Kac--Moody algebra $\mathfrak g$ over $k$ with diagram $D$. \begin{construction}\label{reductionconstruction} Suppose that there are distinct vertices $v_i$, $v_j$ of the diagram $D$ such that any vertex $v_r$ distinct from $v_i$, $v_j$ is connected to $v_i$ if and only if $v_r$ is connected to $v_j$. \begin{enumerate} \item If $v_i$ and $v_j$ are not connected by an edge, let $D'$ be the diagram obtained from $D$ by deleting the vertex $v_j$. Let $\mathfrak k':=\mathfrak k(D')$ and $X_1', \ldots, X_n'$ its Berman generators. Then there is a well-defined epimorphism of Lie algebras $\varphi\colon \mathfrak k \to \mathfrak k'$ determined by $\varphi(X_r):=X_r'$ for $r \neq j$ and $\varphi(X_j):=X_i'$. \item If $v_i$ and $v_j$ are connected by an edge, let $D'$ be the diagram obtained from $D$ by deleting all edges emanating from $v_j$ except for the edge $(v_i,v_j)$. As above, let $\mathfrak k':=\mathfrak k(D')$ and $X_1', \ldots, X_n'$ its Berman generators. Then there is a well-defined epimorphism of Lie algebras $\varphi\colon \mathfrak k \to \mathfrak k'$ determined by $\varphi(X_r):=X_r'$ for $r \neq j$ and $\varphi(X_j):=[X_i',X_j']$. This can be checked by using the Weyl automorphisms introduced in Lemma~\ref{comp}. For instance, for all $r \neq i, j$ with $(v_r,v_j) \in E_D$ (which is equivalent to $(v_r,v_i) \in E_D$), one has \begin{eqnarray} [\varphi(X_j), [\varphi(X_j),\varphi(X_r)]] & = & [[X'_i,X'_j], [[X'_i,X'_j],X'_r]] \\ & \stackrel{\ref{comp}}{=} & [-\pi(s^_j)(X'_i),[-\pi(s^_j)(X'_i),\pi(s_j^)(X'_r)]] \\ & = & \pi(s^_j)[X'_i,[X'_i,X'_r]] \\ & \stackrel{\ref{Berman}}{=} & \pi(s^_j)(-X'_r) \\ & \stackrel{\ref{comp}}{=} & -X'_r \\ & = & \varphi(-X_r) \\ & \stackrel{\ref{Berman}}{=} & \varphi[X_j,[X_j,X_r]]. \end{eqnarray} Case (a) (resp.\ (b)) of Construction \ref{reductionconstruction} corresponds to quotienting ${\mathfrak k}$ by the ideal generated by $(X_i-X_j)$ (resp.\ by all terms of the form $[X_r,[X_i,X_j]]$ where $r\neq i,j$). \end{enumerate} \end{construction} \begin{example} \begin{enumerate} \item The preceding discussion gives a sequence of epimorphisms of real Lie algebras $\mathfrak k(D_4^+) \twoheadrightarrow \mathfrak k(D_4) \twoheadrightarrow \mathfrak k(A_3)=\mathfrak{so}_4(\mathbb{R}) \twoheadrightarrow \mathfrak k(A_2)=\mathfrak{so}_3(\mathbb{R})$. \begin{center} \begin{tikzpicture}[scale=.8] \begin{scope} \draw (1,-1.7) node {$\mathfrak k(D_4^+)$}; \node[dnode] (1) at (0,0) {}; \node[dnode] (2) at (1,0) {}; \node[dnode] (3) at (2,0) {}; \node[dnode] (4) at (1,1) {}; \node[dnode] (5) at (1,-1) {}; \draw[sedge] (1) -- (2) -- (3); \draw[sedge] (4) -- (2) -- (5); \draw[->>] (2.8,0) -- (3.2,0); \end{scope} \begin{scope}[shift={(4,0)}] \draw (1,-1.7) node {$\mathfrak k(D_4)$}; \node[dnode] (1) at (0,0) {}; \node[dnode] (2) at (1,0) {}; \node[dnode] (3) at (2,0) {}; \node[dnode] (4) at (1,1) {}; \draw[sedge] (1) -- (2) -- (3); \draw[sedge] (4) -- (2); \draw[->>] (2.8,0) -- (3.2,0); \end{scope} \begin{scope}[shift={(8,0)}] \draw (1,-1.7) node {$\mathfrak k(A_3)=\mathfrak{so}_4(\mathbb{R})$}; \node[dnode] (1) at (0,0) {}; \node[dnode] (2) at (1,0) {}; \node[dnode] (3) at (2,0) {}; \draw[sedge] (1) -- (2) -- (3); \draw[->>] (2.8,0) -- (3.2,0); \end{scope} \begin{scope}[shift={(12,0)}] \draw (0.5,-1.7) node {$\mathfrak k(A_2)=\mathfrak{so}_3(\mathbb{R})$}; \node[dnode] (1) at (0,0) {}; \node[dnode] (2) at (1,0) {}; \draw[sedge] (1) -- (2); \end{scope} \end{tikzpicture} \end{center} This sequence can be extended further: Let $\Gamma_n=(\{1, \ldots, n\},\{(1,k) \mid 2 \leq k \leq n\})$ denote the star diagram on $n$ vertices and let $\mathfrak k_n$ denote the maximal compact subalgebra of the Kac--Moody algebra $\mathfrak g_n$ with Dynkin diagram $\Gamma_n$. Then there are epimorphisms $\mathfrak k_n \to \mathfrak k_{n-1}$. \item Denoting by $K_4$ the complete graph on four vertices, there similarly is a sequence of epimorphisms $\mathfrak k(K_4) \twoheadrightarrow \mathfrak k(AE_4) \twoheadrightarrow \mathfrak k(A_4)$. \end{enumerate} \end{example} \section{Generalized spin representations}\label{GSR} \subsection{Generalized spin representations of $\mathfrak{k}(E_{10}(\mathbb{R}))$}Let us recall the extension of the spin representation of $\mathfrak k(\mathfrak{sl}_{10}(\mathbb{R}))$ to $\mathfrak k(E_{10})(\mathbb{R})$ as described by \cite{DamourKleinschmidtNicolai}, \cite{deBuylHenneauxPaulot} (also \cite{Keurentjes}). \begin{example} \label{theexample} Let $V$ be a $k$-vector space and $q\colon V \to k$ a quadratic form with associated bilinear form $b$. Then the \textbf{Clifford algebra} $C:=C(V,q)$ is defined as $C:=T(V)/\langle vw+wv-2b(v,w) \rangle$ where $T(V)$ is the tensor algebra of $V$. Now let $V=\mathbb{R}^{10}$ with standard basis vectors $v_i$, let $q=x_1^2+\cdots+x_{10}^2$ and let $C=C(V,q)$. Then in $C$ we have $$v_i^2=1 \text{ and } v_iv_j=-v_jv_i.$$ Since $C$ is an associative algebra, it becomes a Lie algebra by setting $[A,B]:=AB-BA$. Let the diagram of $\mathfrak{g}(E_{10})(\mathbb{R})$ be labelled as \begin{center} \begin{tikzpicture} \node[dnode,label=below:12] (1) at (0,0) {}; \node[dnode,label=above:123] (2) at (2,1) {}; \node[dnode,label=below:23] (3) at (1,0) {}; \node[dnode,label=below:34] (4) at (2,0) {}; \node[dnode,label=below:45] (5) at (3,0) {}; \node[dnode,label=below:56] (6) at (4,0) {}; \node[dnode,label=below:56] (7) at (5,0) {}; \node[dnode,label=below:78] (8) at (6,0) {}; \node[dnode,label=below:89] (9) at (7,0) {}; \node[dnode,label=below:910] (10) at (8,0) {}; \draw[sedge] (2) -- (4); \draw[sedge] (1) -- (3) -- (4) -- (5) -- (6) -- (7) -- (8) -- (9) -- (10); \end{tikzpicture} \end{center} and define a Lie algebra homomorphism $\rho : \mathfrak{k} \to C$ using these labels, i.e., via \begin{eqnarray} X_1 \mapsto \frac{1}{2}v_1v_2, & X_2 \mapsto \frac{1}{2}v_1v_2v_3, & X_3 \mapsto \frac{1}{2}v_2v_3, \\ X_4 \mapsto \frac{1}{2}v_3v_4, & X_5 \mapsto \frac{1}{2}v_4v_5, & X_6 \mapsto \frac{1}{2}v_5v_6, \\ X_7 \mapsto \frac{1}{2}v_6v_7, & X_8 \mapsto \frac{1}{2}v_7v_8, & X_9 \mapsto \frac{1}{2}v_8v_9, \\ & X_{10} \mapsto \frac{1}{2}v_9v_{10}, \end{eqnarray} where $X_i$ denotes the Berman generator corresponding to the root $\alpha_i$, enumerated in Bourbaki style as in Section~\ref{DD}. Observe that each $A_i:=\rho (X_i)$ satisfies $A_i^2=-\frac1 4 \id$. Here we would like to remark that $(v_1v_2v_3)^2 = (v_2v_3)^2 = -1$ depends on $v_i^2 = 1$; for parity reasons, this would not be true in the Clifford algebra $C(V,-q)$, as then $(v_1v_2v_3)^2 = -(v_2v_3)^2 = 1$. Using the criterion established in Remark~\ref{characterization} below, one checks easily that $\rho$ indeed is a Lie algebra homomorphism, i.e., that the defining relations of $\mathfrak{k}$ from Theorem~\ref{Berman} are respected. Indeed, one just needs to establish \begin{itemize} \item[(i)] $A_i^2=-\frac{1}{4}\cdot \id_s$, \item[(ii)] $A_iA_j=A_jA_i$, if $(i,j) \not\in E$, \item[(iii)] $A_iA_j=-A_jA_i$, if $(i,j) \in E$. \end{itemize} We have already observed (i). Assertions (ii) and (iii) are obvious for $i, j\neq 2$. Moreover, one quickly computes $(v_1v_2v_3)(v_3v_4) = -(v_3v_4)(v_1v_2v_3)$ and $(v_1v_2v_3)(v_{k_1}v_{k_2}) = (v_{k_1}v_{k_2})(v_1v_2v_3)$, if $\{ k_1, k_2 \}$ is a set of two elements that is either a subset of $\{ 1, 2, 3 \}$ or disjoint from $\{ 1, 2, 3 \}$. Assertions (ii) and (iii) follow. By \cite[Lemma~20.9]{FultonHarris}, \cite[Proposition~2.4]{Meinrenken:2013} the Clifford algebra $C$ splits over $\mathbb{C}$ as $C \otimes_\mathbb{R} \mathbb{C} \cong \mathbb{C}^{32 \times 32}$. Hence $\rho$ affords a 32-dimensional complex representation of $\mathfrak k(E_{10})(\mathbb{R})$. The restriction of this representation to the maximal compact subalgebra of the $A_9$-subdiagram, $\mathfrak k(A_9)(\mathbb{R})=\mathfrak{so}_{10}(\mathbb{R})$, coincides with the spin representation of $\mathfrak{so}_{10}$ (see e.g.\ \cite[Chapter 20]{FultonHarris}), i.e., $\rho$ extends the classical spin representation. Let $\iota \in \Aut C$ denote the involution (known as parity automorphism) induced by $V \to V : v \mapsto -v$. Let $C_0:=\Fix \iota$ and $C_1:=\{w \in C \mid \iota(w)=-w\}$ denote the even and the odd part of $C$. Then $C_0$ and $C_1$ are invariant subspaces under the spin representation of $\mathfrak{so}_{10}$ since $\im \rho \subseteq C_0$ (multiplication with a product of the $v_i$ of even length does not change the parity) and these subspaces are irreducible non-isomorphic representations of $\mathfrak \mathfrak{so}_{10}$ (\cite[Chapter 20]{FultonHarris}). The remaining Berman generator $X_2$ of $\mathfrak k(E_{10})$ is sent to an element which interchanges $C_0$ and $C_1$. \end{example} \begin{remark} A calculation shows that $\im \rho$ is the linear span of all elements of the form $v_{i_1}\cdots v_{i_k}$, where $\{ i_1, ..., i_k \} = I\subseteq \{1, \ldots, 10\}$ with $\|I\| \in \{2,3,6,7,10\}$. Therefore, $\dim \im(\rho) = 45 + 120 + 210 + 120 + 1 = 496$. Since $\im(\rho) \leq C \cong \mathbb{R}^{32 \times 32}$ by \cite[Section~2.2.3]{Meinrenken:2013} and since $\im(\rho)$ is compact and semisimple by Theorem~\ref{Mainexistencetheorem}, this dimension $\dim \im(\rho) = 496$ implies $\im(\rho) \cong \mathfrak{so}_{32}(\mathbb{R})$ (see also \cite{damour2006k}). The existence of Example~\ref{theexample} is not peculiar to the diagram $E_{10}$, it can be generalized to arbitrary diagrams $E_n$ in the obvious way. A careful analysis of dimensions combined with the Cartan--Bott periodicity of Clifford algebras allows one to determine the isomorphism types of the quotients for the whole $E_n$ series. This is carried out in Appendix~\ref{CartanBott}. A key observation is that the cardinality $\|I\|$ from above in general has to be equal to $2$ or $3$ modulo $4$ (see Lemma~\ref{lowerbounddimension}). \end{remark} \begin{remark} Let $\rho\colon \mathfrak{so}_{10}(\mathbb{R}) \to \mathbb{C}^{n \times n}$ be a representation. To extend $\rho$ to a representation of $\mathfrak k(E_{10})$, it suffices to find a matrix $X \in \mathbb{C}^{n \times n}$ such that for $A_i:=\rho(X_i)$, $1 \leq i \leq 10$, $i \neq 2$, the following equations are satisfied (where we again use the labelling of the diagram $E_{10}$ as given in Section~\ref{DD}): \begin{eqnarray} [A_i,X] & = & 0 \quad \quad \mbox{for $1 \leq i \leq 10$, $i \neq 2, 4$}, \\ {}[A_4,[A_4,X]] & = & -X, \\ {}[X,[X,A_4]] & = & -A_4. \end{eqnarray} Theorem~\ref{Berman} then implies that $\rho$ can be extended to $\mathfrak k(E_{10})$ by setting $\rho(X_2):=X$. The first two sets of equations define a linear subspace, the third set of equations yields a family of quadratic equations. With the help of a Gr\"obner basis one can compute that in case of the spin representation, this variety is isomorphic to $\mathbb{C}^\times$, i.e., the extension is unique up to a scalar. \end{remark} \subsection{Generalized spin representations for the simply-laced case} Throughout this section, let $k$ be a field of characteristic 0, let $\mathfrak g$ be a Kac--Moody algebra over $k$ with simply laced diagram and let $\mathfrak k$ be its maximal compact subalgebra. Let $L:=k(I)$, where $I$ is a square root of $-1$ and denote by $\id_s \in L^{s \times s}$ the identity matrix. \begin{definition}\label{genspinrep} A representation $\rho\colon \mathfrak k \to \End(L^s)$ is called a \textbf{generalized spin representation} if the images of the Berman generators from Theorem~\ref{Berman} satisfy $$\rho(X_i)^2=-\frac{1}{4}\id_s \text{ for } i=1, \ldots, n.$$ \end{definition} \begin{remark} \label{characterization} \begin{enumerate} \item Since $\rho$ is assumed to be a representation, it follows from the defining relations that $\rho(X_i)$ and $\rho(X_j)$ commute if $(i,j) \not \in E$. On the other hand, if $(i,j) \in E$, then $A:=\rho(X_i)$ and $B:=\rho(X_j)$ anticommute. Indeed, we have $$-B\stackrel{\ref{Berman}}{=}[A,[A,B]]=A^2B-2ABA+BA^2=-\frac 1 2 B-2ABA$$ from which the claim follows after multiplying with $A^{-1}=-4A \Longleftrightarrow A^2 = - \frac{1}{4}\id_s$. \item Conversely, suppose that there are matrices $A_i \in L^{s \times s}$ satisfying \begin{enumerate} \item $A_i^2=-\frac{1}{4}\cdot \id_s$, \item $A_iA_j=A_jA_i$ if $(i,j) \not\in E$, \item $A_iA_j=-A_jA_i$ if $(i,j) \in E$. \end{enumerate} Then, by reversing the argument in the above computation, the assignment $X_i \mapsto A_i$ gives rise to a representation of $\mathfrak k$. \end{enumerate} \end{remark} \begin{remark}\label{coxspin} Let $\rho$ be a generalized spin representation of $\mathfrak k$ and set $S_i:=2I\cdot\rho(X_i)$. Let $W$ be a Coxeter group defined by the presentation $$W=\langle s_1, \ldots, s_n \mid (s_is_j)^{m_{ij}}=1 \rangle,$$ where $m_{ii}=1$ and $m_{ij}=2$ if $(i,j) \not\in E$, while $m_{ij}\in\{ 3, 4 \}$ if $(i,j) \in E$. Then the assignment $s_i \mapsto S_i$ gives a representation of $W$. \end{remark} Write $\mathfrak k_{\leq r}:=\langle X_1, \ldots, X_r \rangle$. \begin{theorem}\label{MainThm} Let $1\leq r <n$. Let $\rho : \mathfrak k_{\leq r} \to \End (L^s)$ be a generalized spin representation. \begin{enumerate} \item If $X_{r+1}$ centralizes $\mathfrak k_{\leq r}$, then $\rho$ can be extended to a generalized spin representation $\rho' : \mathfrak k_{\leq r+1} \to \End(L^s)$ by setting $\rho'(X_{r+1}):=\frac 1 2 I \cdot \id_s$. \item If $X_{r+1}$ does not centralize $\mathfrak k_{\leq r}$, then $\rho$ can be extended to a generalized spin representation $\rho'\colon \mathfrak k_{\leq r+1} \to \End(L^s\oplus L^s)$ as follows. Define the sign automorphism $s_0 : \mathfrak k_{\leq r} \to L^s$ via $$s_0(X_i):=\left\{ \begin{array}{cc} X_i, & \text{ if } (i,r+1)\not\in E, \\ -X_i, & \text{ if } (i,r+1) \in E, \end{array} \right. $$ let $$\rho'\|_{\mathfrak k_{\leq r}}:= \rho \oplus \rho \circ s_0,$$ and $$\rho'(X_{r+1}):=\frac{1}{2}I \cdot \id_s \otimes \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$$ \end{enumerate} \end{theorem} \begin{proof} If $X_{r+1}$ centralizes $\mathfrak k_{\leq r}$, it is clear that $\rho'$ is well-defined and that $\rho'(X_{r+1})^2=-\frac{1}{4} \id_s$. In the second case it is clear that $\rho'\|_{\mathfrak k_\leq r}$ is a generalized spin representation of $\mathfrak k_{\leq r}$ which extends $\rho$. It is easy to check that $\rho'(X_i)$ commutes with $\rho'(X_{r+1})$ if $(i,r+1) \not\in E$, and that $\rho'(X_i)$ anticommutes with $\rho'(X_{r+1})$ if $(i,r+1) \in E$. Remark~\ref{characterization} therefore implies that $\rho'$ is a generalized spin representation. \end{proof} For a graph $G=(V,E)$, a subset $M \subseteq V$ is called a \textbf{coclique} if the subgraph of $G$ induced on $M$ does not contain any edges, i.e., if no two elements $m_1$, $m_2$ in $M$ are connected by an edge. \begin{corollary} \label{Cor34} Let $n$ be the cardinality of the diagram of $\mathfrak g$ and let $r$ be the size of a maximal coclique of that diagram. Then there exists a $2^{n-r}$-dimensional generalized spin representation of $\mathfrak k$. Furthermore, if the diagram is irreducible, then there exists a $2^{n-1}$-dimensional {\em maximal} generalized spin representation of $\mathfrak k$. \end{corollary} \begin{proof} Up to a change of labelling the set $M:=\{\alpha_1,\ldots, \alpha_r\}$ forms a maximal coclique. The map $\rho : \mathfrak k_{\leq r}\to \End(L^1) : X_i \mapsto \frac 1 2 I \cdot \id_1$ is a generalized spin representation. By Theorem~\ref{MainThm}, the representation $\rho$ can be extended inductively to a generalized spin representation of $\mathfrak{k}$; the dimension doubles at each step because $M$ was assumed to be a maximal coclique. For the second claim it suffices to order the vertices of the diagram in such a way that two consecutive vertices are adjacent. \end{proof} \begin{remark} An inductive construction of the basic spin representations of the symmetric group similar to the one in Theorem~\ref{MainThm} has independently been obtained by Maas \cite{Maas}. It is likely that by a combination of the methods of \cite{Maas} and of the present article, a similar construction of generalized (basic) spin representations is possible for any (simply laced) Coxeter group. \end{remark} \subsection{Generalized spin representations for symmetrizable Kac--Moody algebras} In this section let ${\mathfrak g}$ be an arbitrary symmetrizable Kac--Moody Lie algebra with maximal compact subalgebra ${\mathfrak k}$, and let $n_i$ be the number of vertices associated to the root $\alpha_i$ in a minimal rank simply laced cover diagram for ${\mathfrak g}$. As above, we assume the ground field $k$ has characteristic zero. \begin{definition}\label{genspinrep2} A generalized spin representation for ${\mathfrak k}$ is a Lie algebra homomorphism $\rho:{\mathfrak k}\rightarrow\End(L^s)$ such that each of the Berman generators $X_i$ (see Theorem~\ref{Berman2}) satisfies: $$\begin{array}{rccl}\left(\rho(X_i)^2+\frac{n_i^2}{4}\id_s\right)\left(\rho(X_i)^2+\frac{(n_i-2)^2}{4}\id_s\right)\ldots \left(\rho(X_i)^2+\id_s\right)\rho(X_i) & = & 0, \\ \mbox{if $n_i$ is even,} \\ \left(\rho(X_i)^2+\frac{n_i^2}{4}\id_s\right)\left(\rho(X_i)^2+\frac{(n_i-2)^2}{4}\id_s\right)\ldots \left(\rho(X_i)^2+\frac{1}{4}\id_s\right) & = & 0, \\ \mbox{if $n_i$ is odd,} \end{array}$$ i.e., $P_{n_i}^{\frac{1}{4}}(\rho(X_i))=0$ (in the notation of Proposition~\ref{contractionprop}). \end{definition} Another way of saying this is that $\rho(X_i)$ is semisimple with eigenvalues belonging to the set $\{ \frac{(n_i-2j)}{2}I : 0\leq j\leq n_i\}$. When the generalized Cartan matrix of ${\mathfrak g}$ is simply laced, this definition clearly coincides with Definition \ref{genspinrep}. \begin{theorem}\label{Mainexistencetheorem} Let $L=k(I)$ where $I^2=-1$. Let ${\mathfrak g}$ be an arbitrary symmetrizable Kac--Moody Lie algebra with maximal compact subalgebra ${\mathfrak k}$. Then there exists a generalized spin representation $\rho:{\mathfrak k}\rightarrow\End(L^s)$. Moreover, if $k$ is formally real, then $\rho$ can be considered as a representation ${\mathfrak k}\rightarrow\End(k^{2s})$ with $\im\rho$ compact and, therefore, reductive. Furthermore, in this case $\im\rho$ is semisimple, if for all $i$ there exists $j\neq i$ such that $a_{ji}$ is odd. Finally, in this case $\mathfrak{k} \cong \ker\rho \oplus \im\rho$. \end{theorem} Note that the condition in the next-to-final sentence of the theorem is satisfied if, for example, ${\mathfrak g}$ has a simply laced diagram which has no isolated nodes. It will follow from the proof that the theorem is actually applicable to all generalized spin representations discussed in Theorem~\ref{MainThm} and Corollary~\ref{Cor34}, in particular the standard generalized spin representation from Example~\ref{theexample}. \begin{proof} To see that ${\mathfrak k}$ has a generalized spin representation, let $\tilde{\mathfrak g}$ be the Kac--Moody algebra associated to some minimal rank simply laced cover diagram for ${\mathfrak g}$ and let $\tilde\varphi:{\mathfrak g}\rightarrow\tilde{\mathfrak g}$ be the Lie algebra embedding described in Section~\ref{symmetrizablesec}. Then it is clear from the earlier discussion that, if $\tilde\rho:\tilde{\mathfrak k}\rightarrow\End(L^s)$ is a generalized spin representation for $\tilde{\mathfrak k}$, then $\rho=\tilde\rho\circ\tilde\varphi\|_{\mathfrak k}$ is a generalized spin representation for ${\mathfrak k}$. (It is, however, not clear that any generalized spin representation for ${\mathfrak k}$ arises in this way.) Thus the first statement follows immediately from Corollary~\ref{Cor34}. For the second statement it will suffice to prove that there exists a generalized spin representation $\rho:{\mathfrak k}\rightarrow\End(L^s)$ such that, with respect to an appropriate choice of $k$-basis for $L^s$, each of the images $\rho(X_i)$ is a skew-symmetric $2s\times 2s$ matrix over $k$ and, thus, $\rho$ can be interpreted as a homomorphism ${\mathfrak k}\rightarrow\mathfrak{so}_{2s}(k)$. Since we can construct generalized spin representations for ${\mathfrak k}$ by restricting from those for the Lie algebra associated to a simply laced cover diagram, it will clearly suffice to show that the representation constructed in Theorem~\ref{MainThm} can be realized by using skew-symmetric matrices only. For the extension of the representation in part (a) of Theorem~\ref{MainThm} this is obvious, as $L \cong \left\{ \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \mid a, b \in k \right\}$ as $k$-algebras, whence $I$ is represented by the skew-symmetric matix $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. For the extension of the representation in part (b) of Theorem~\ref{MainThm}, observe that $$\begin{pmatrix} 1 & 0 \\ 0 & I\end{pmatrix}\begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -I \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$ so that after a change of basis we have instead $\rho'(X_{r+1})=\frac{1}{2}\id_s\otimes\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ (while $\rho'\|_{{\mathfrak k}_\leq r}$ remains unchanged). Therefore, if the representation of $\mathfrak{k}_{\leq r}$ consists of skew-symmetric matrices over $k$, one can ensure that the representation of $\mathfrak{k}_{\leq r+1}$ also consists of skew-symmetric matrices over $k$. Thus $\mathrm{im}(\rho)$ is compact, whence reductive. For the statement concerning semisimplicity observe that $\mathfrak{k}$ is perfect. Indeed, by hypothesis, for each generator $X_i$ of $\mathfrak k$, there is some $j$ such that $a_{ji}$ is odd, and therefore the constant term in the polynomial $P_{-a_{ji}}$ is non-zero. Since $P_{-a_{ji}}(\ad X_j)(X_i)=0$ by Theorem~\ref{Berman2}, it follows that $X_i$ is contained in the linear span of $(\ad X_j)^{2l}(X_i)$, $l\geq 1$. Thus, the image $\mathrm{im}(\rho)$ is perfect and, by the above, reductive. The claim is now obvious, as a perfect direct sum of a semisimple and an abelian Lie algebra necessarily is semisimple. For the final statement observe that $\mathfrak{k}$ is anisotropic with respect to the invariant bilinear form of the Kac--Moody algebra $\mathfrak{g}$ and so $(\ker\rho)^\perp \cong \im\rho$ is an ideal of $\mathfrak{k}$, where $\perp$ denotes the orthogonality relation with respect to the invariant bilinear form. \end{proof} Let $\mathcal C$ denote the class of all generalized spin representation of $\mathfrak k$. We check some closure properties of $\mathcal C$. \begin{proposition} \begin{enumerate} \item $\mathcal C$ is closed under direct sums, quotients, duals and taking subrepresentations. \item If the generalized Cartan matrix of ${\mathfrak g}$ is simply laced and $\rho_1,\rho_2,\rho_3 \in \mathcal C$, then so is $\rho \colon X_i \mapsto 4 \rho_1(X_i) \otimes \rho_2(X_i) \otimes \rho_3(X_i)$. \item More generally, if the generalized Cartan matrix of ${\mathfrak g}$ is simply laced and $\rho_1,\rho_2 \in \mathcal C$, then so is $\rho := 2 I \rho_1 \otimes \rho_2$, where $I$ is a primitive fourth root of unity. \item If $\rho \in \mathcal C$ and $\varphi$ is either a sign, graph or Weyl group automorphism of $\mathfrak k$, then $\rho \circ \varphi \in \mathcal C$. \end{enumerate} \end{proposition} \begin{proof} The first three assertions can be easily verified. The fourth assertion is clear if $\varphi$ is a graph or a sign automorphism. The remaining claim follows from Remark \ref{weylgroupremark}, since if $\rho(X_j)$ has eigenvalues $\frac{rI}{2},\frac{(r-2)I}{2},\ldots, -\frac{rI}{2}$ then so does $\rho(\exp (\xi \ad X_i)(X_j))=\exp(\xi \rho(X_i))(\rho(X_j))$. \end{proof} \section{Some Dynkin diagrams} \label{DD} We give the list of relevant Dynkin diagrams we use in the main text. \\ \begin{longtable}{rlrl} $A_n^+$ & \begin{tikzpicture}[baseline] \node[dnode,label=below:$1$] (1) at (0,0) {}; \node[dnode,label=below:$2$] (2) at (1,0) {}; \node[dnode,label=above:$n+1$] (n+1) at (2,1) {}; \node[dnode,label=below:$n-1$] (n-1) at (3,0) {}; \node[dnode,label=below:$n$] (n) at (4,0) {}; \draw[sedge] (2) -- (1) -- (n+1) -- (n) -- (n-1); \draw[sedge,dashed] (2) -- (n-1); \end{tikzpicture} \\ $D_n^+$ & \begin{tikzpicture}[baseline] \node[dnode,label=below:$1$] (1) at (0,0) {}; \node[dnode,label=above:$2$] (2) at (1,1) {}; \node[dnode,label=below:$3$] (3) at (1,0) {}; \node[dnode,label=below:$n-1$] (n-1) at (3,0) {}; \node[dnode,label=below:$n$] (n) at (4,0) {}; \node[dnode,label=above:$n+1$] (n+1) at (3,1) {}; \draw[sedge] (1) -- (3) -- (2); \draw[sedge,dashed] (3) -- (n-1); \draw[sedge] (n+1) -- (n-1) -- (n); \end{tikzpicture} \\ $E_6^+$ & \begin{tikzpicture}[baseline] \node[dnode,label=below:$1$] (1) at (0,0) {}; \node[dnode,label=left:$2$] (2) at (2,1) {}; \node[dnode,label=below:$3$] (3) at (1,0) {}; \node[dnode,label=below:$4$] (4) at (2,0) {}; \node[dnode,label=below:$5$] (5) at (3,0) {}; \node[dnode,label=below:$6$] (6) at (4,0) {}; \node[dnode,label=left:$7$] (7) at (2,2) {}; \draw[sedge] (4) -- (2) -- (7); \draw[sedge] (1) -- (3) -- (4) -- (5) -- (6); \end{tikzpicture} \\ $E_7^+$ & \begin{tikzpicture}[baseline] \node[dnode,label=below:$1$] (1) at (0,0) {}; \node[dnode,label=left:$2$] (2) at (2,1) {}; \node[dnode,label=below:$3$] (3) at (1,0) {}; \node[dnode,label=below:$4$] (4) at (2,0) {}; \node[dnode,label=below:$5$] (5) at (3,0) {}; \node[dnode,label=below:$6$] (6) at (4,0) {}; \node[dnode,label=below:$7$] (7) at (5,0) {}; \node[dnode,label=below:$8$] (8) at (-1,0) {}; \draw[sedge] (4) -- (2); \draw[sedge] (8) -- (1) -- (3) -- (4) -- (5) -- (6) -- (7); \end{tikzpicture} \\ $E_8^+=E_9$ & \multicolumn{3}{l}{\En{9}} \\ $E_8^{++}=E_{10}$ & \multicolumn{3}{l}{\En{10}} \\ $A_{n-2}^{++}=AE_n$ & \begin{tikzpicture}[baseline] \node[dnode,label=below:$1$] (1) at (0,0) {}; \node[dnode,label=below:$2$] (2) at (1,0) {}; \node[dnode,label=above:$n-1$] (n-1) at (2,1) {}; \node[dnode,label=below:$n-3$] (n-3) at (3,0) {}; \node[dnode,label=below:$n-2$] (n-2) at (4,0) {}; \node[dnode,label=below:$n$] (n) at (5,0) {}; \draw[sedge] (2) -- (1) -- (n-1) -- (n-2) -- (n-3) -- (n); \draw[sedge,dashed] (2) -- (n-3); \end{tikzpicture} \end{longtable} \appendix \renewcommand{{\rm(\arabic{enumi})}}{{\rm(\arabic{enumi})}} \renewcommand{{\rm(\alph{enumii})}}{{\rm(\alph{enumii})}} {\section{Cartan--Bott periodicity for the real $E_n$ series \\ (by Max Horn and Ralf K\"ohl)} \label{CartanBott} } In this appendix we continue the investigation of the generalized spin representations introduced in the main text. We focus on the $E_n$ series and use the original description of the generalized spin representation from \cite{DamourKleinschmidtNicolai}, \cite{deBuylHenneauxPaulot} via Clifford algebras (see Example~\ref{theexample}). The $E_n$ series is traditionally only defined for $n\in\{6,7,8\}$. However, using the Bourbaki style labeling shown in Figure~\ref{fig:En}, it naturally extends to arbitrary $n\geq 3$. Using this description, one has $E_3=A_2\oplus A_1$, $E_4=A_4$, $E_5=D_5$ (see Figure~\ref{fig:E3-to-E8}). \begin{figure} \caption{The Dynkin diagram of type $E_n$} \label{fig:En} \end{figure} An elementary combinatorial counting argument using binomial coefficients allows us to determine lower bounds for the $\mathbb{R}$-dimension of the images of the generalized spin representation. These images have to be compact, whence reductive by Theorem~\ref{Mainexistencetheorem} and even semisimple, if the diagram is irreducible. One therefore obtains an upper bound for their $\mathbb{R}$-dimension via the maximal compact Lie subalgebras of the Clifford algebras. As it turns out, the lower and the upper bounds coincide, providing the following Cartan--Bott periodicity. \begin{customthm}{A}[Cartan--Bott periodicity of the $E_{n}$ series]\label{thm:A} Let $n \in \mathbb{N}$ with $n\geq 4$, let $\mathfrak{k}$ be the maximal compact Lie subalgebra of the split real Kac--Moody Lie algebra of type $E_n$, let $C=C(\mathbb{R}^n,q)$ be the Clifford algebra with respect to the standard positive definite quadratic form $q$ and let $\rho : \mathfrak{k} \to C$ be the standard generalized spin representation. Then $\im(\rho)$ is isomorphic to \begin{enumerate} \setcounter{enumi}{-1} \item $\mathfrak{so}(2^{\frac{n}{2}})) \leq \mathbb{R}\otimes_\mathbb{R} \M{2^{\frac{n}{2}}}{\mathbb{R}}$, if $n \equiv 0 \pmod 8$, \item $\mathfrak{so}(2^{\frac{n-1}{2}}) \oplus \mathfrak{so}(2^{\frac{n-1}{2}}) \leq \left( \mathbb{R} \oplus \mathbb{R} \right) \otimes_\mathbb{R} \M{2^{\frac{n-1}{2}}}{\mathbb{R}}$, if $n \equiv 1 \pmod 8$, \item $\mathfrak{so}(2^{\frac{n}{2}}) \leq \M{2}{\mathbb{R}} \otimes_\mathbb{R} \M{2^{\frac{n-2}{2}}}{\mathbb{R}}$, if $n \equiv 2 \pmod 8$, \item $\mathfrak{su}(2^{\frac{n-1}{2}}) \leq \M{2}{\mathbb{C}} \otimes_\mathbb{R} \M{2^{\frac{n-3}{2}}}{\mathbb{R}}$, if $n \equiv 3 \pmod 8$, \item $\mathfrak{sp}(2^{\frac{n-2}{2}}) \leq \M{2}{\mathbb{H}} \otimes_\mathbb{R} \M{2^{\frac{n-4}{2}}}{\mathbb{R}}$, if $n \equiv 4 \pmod 8$, \item $\mathfrak{sp}(2^{\frac{n-3}{2}})\oplus \mathfrak{sp}(2^{\frac{n-3}{2}}) \leq \left( \M{2}{\mathbb{H}} \oplus \M{2}{\mathbb{H}} \right) \otimes_\mathbb{R} \M{2^{\frac{n-5}{2}}}{\mathbb{R}}$, if $n \equiv 5 \pmod 8$, \item $\mathfrak{sp}(2^{\frac{n-2}{2}}) \leq \M{4}{\mathbb{H}} \otimes_\mathbb{R} \M{2^{\frac{n-6}{2}}}{\mathbb{R}}$, if $n \equiv 6 \pmod 8$, \item $\mathfrak{su}(2^{\frac{n-1}{2}}) \leq \M{8}{\mathbb{C}} \otimes_\mathbb{R} \M{2^{\frac{n-7}{2}}}{\mathbb{R}}$, if $n \equiv 7 \pmod 8$, \end{enumerate} i.e., $\im(\rho)$ is a semisimple maximal compact Lie subalgebra of $C$. \end{customthm} Along the way we arrive at a structural explanation for the well-known isomorphism types of the maximal compact Lie subalgebras of the semisimple split real Lie algebras of types $E_3=A_2\oplus A_1$, $E_4=A_4$, $E_5=D_5$, $E_6$, $E_7$, $E_8$ (cf., e.g., \cite[p.~518, Table V]{Helgason:1978}). \begin{customthm}{B} \label{thm:B} The maximal compact Lie subalgebras of the semisimple split real Lie algebras of types $A_2\oplus A_1$, $A_4$, $D_5$, $E_6$, $E_7$, $E_8$ are isomorphic to $\mathfrak{u}(2)$, $\mathfrak{sp}(2)\cong\mathfrak{so}(5)$, $\mathfrak{sp}(2)\oplus\mathfrak{sp}(2)\cong\mathfrak{so}(5)\oplus\mathfrak{so}(5)$, $\mathfrak{sp}(4)$, $\mathfrak{su}(8)$, $\mathfrak{so}(16)$, respectively. \end{customthm} \noindent \textbf{Acknowledgements.} We thank Klaus Metsch for pointing out to us the identity of sums of binomial coefficients in Proposition~\ref{binomial} and one of the referees for relaying the elegant version of its proof given in this appendix. This research has been partially funded by the EPRSC grant EP/H02283X. The second author gratefully acknowledges the hospitality of the IHES at Bures-sur-Yvette and of the Albert Einstein Institute at Golm. \begin{figure} \caption{The Dynkin diagrams of types $E_3$ to $E_8$.} \label{fig:E3-to-E8} \end{figure} \subsection{Cartan--Bott periodicity of Clifford algebras} \label{sec:cartan-bott} Let $\mathbb{N} = \{1,2,3,\ldots\}$ be the set of natural numbers, and let $\mathbb{R}$, $\mathbb{C}$, resp.\ $\mathbb{H}$ denote the reals, complex numbers resp.\ quaternions. For $n\in\mathbb{N}$ and a division ring $\mathbb{D}$, denote by $M(n,\mathbb{D})$ the $\mathbb{D}$-algebra of $n\times n$ matrices over $\mathbb{D}$. Let $V$ be an $\mathbb{R}$-vector space and $q\colon V \to \mathbb{R}$ a quadratic form with associated bilinear form $b$. Then the \textbf{Clifford algebra} $C(V,q)$ is defined as $C(V,q):=T(V)/\langle vw+wv-2b(v,w) \rangle$ where $T(V)$ is the tensor algebra of $V$; cf.\ \cite[Section~4.3]{Kobayashi/Yoshino:2005}, \cite[Chapter 1, \S1]{Lawson/Michelsohn:1989}. Let $V=\mathbb{R}^{n}$ with standard basis vectors $v_i$, let $q=x_1^2+\cdots+x_{n}^2$. Then in $C(V,q)$ we have $v_i^2=1$ and $v_iv_j=-v_jv_i$. \begin{proposition}[Cartan--Bott periodicity] \label{prop:cartan-bott} For $n\geq 2$, the Clifford algebra $C(\mathbb{R}^n,q)$ is isomorphic to the following algebra: \begin{enumerate} \setcounter{enumi}{-1} \item $\mathbb{R}\otimes_\mathbb{R} \M{2^{\frac{n}{2}}}{\mathbb{R}}$, if $n \equiv 0 \pmod 8$, \item $\left( \mathbb{R} \oplus \mathbb{R} \right) \otimes_\mathbb{R} \M{2^{\frac{n-1}{2}}}{\mathbb{R}}$, if $n \equiv 1 \pmod 8$, \item $\M{2}{\mathbb{R}} \otimes_\mathbb{R} \M{2^{\frac{n-2}{2}}}{\mathbb{R}}$, if $n \equiv 2 \pmod 8$, \item $\M{2}{\mathbb{C}} \otimes_\mathbb{R} \M{2^{\frac{n-3}{2}}}{\mathbb{R}}$, if $n \equiv 3 \pmod 8$, \item $\M{2}{\mathbb{H}} \otimes_\mathbb{R} \M{2^{\frac{n-4}{2}}}{\mathbb{R}}$, if $n \equiv 4 \pmod 8$, \item $\left( \M{2}{\mathbb{H}} \oplus \M{2}{\mathbb{H}} \right) \otimes_\mathbb{R} \M{2^{\frac{n-5}{2}}}{\mathbb{R}}$, if $n \equiv 5 \pmod 8$, \item $\M{4}{\mathbb{H}} \otimes_\mathbb{R} \M{2^{\frac{n-6}{2}}}{\mathbb{R}}$, if $n \equiv 6 \pmod 8$, \item $\M{8}{\mathbb{C}} \otimes_\mathbb{R} \M{2^{\frac{n-7}{2}}}{\mathbb{R}}$, if $n \equiv 7 \pmod 8$. \end{enumerate} \end{proposition} \begin{proof} See e.g.\ \cite[Prop.~4.4.1 + Table~4.4.1]{Kobayashi/Yoshino:2005}. \end{proof} Since $C(V,q)$ is an associative algebra, it becomes a Lie algebra by setting $[A,B]:=AB-BA$. With this in mind, Proposition~\ref{prop:cartan-bott} implies the following: \begin{corollary} \label{cor:cartan-bott-max-cpt} For $n\geq 2$, the maximal semisimple compact Lie subalgebra of the Clifford algebra $C(\mathbb{R}^n,q)$ is isomorphic to the following Lie algebra: \begin{enumerate} \setcounter{enumi}{-1} \item $\mathfrak{so}(2^{\frac{n}{2}})$, if $n \equiv 0 \pmod 8$, \item $\mathfrak{so}(2^{\frac{n-1}{2}})\oplus\mathfrak{so}(2^{\frac{n-1}{2}})$, if $n \equiv 1 \pmod 8$, \item $\mathfrak{so}(2^{\frac{n}{2}})$, if $n \equiv 2 \pmod 8$, \item $\mathfrak{su}(2^{\frac{n-1}{2}})$, if $n \equiv 3 \pmod 8$, \item $\mathfrak{sp}(2^{\frac{n-2}{2}})$, if $n \equiv 4 \pmod 8$, \item $\mathfrak{sp}(2^{\frac{n-3}{2}})\oplus\mathfrak{sp}(2^{\frac{n-3}{2}})$, if $n \equiv 5 \pmod 8$, \item $\mathfrak{sp}(2^{\frac{n-2}{2}})$, if $n \equiv 6 \pmod 8$, \item $\mathfrak{su}(2^{\frac{n-1}{2}})$, if $n \equiv 7 \pmod 8$. \end{enumerate} \end{corollary} \subsection{A lower bound on the dimension of a subalgebra} \begin{definition} For $n\geq 3$ let $\mathfrak{m}$ be the Lie subalgebra of $C(\mathbb{R}^n,q)$ generated by $v_1v_2v_3$ and by $v_iv_{i+1}$, $1 \leq i < n$. \end{definition} \begin{lemma} \label{lowerbounddimension} Let $n\geq 3$. Then $\mathfrak{m}$ contains all products of the form $v_{j_1}v_{j_2}\cdots v_{j_k}$ for $2\leq k\leq n$ and $k \equiv 2, 3 \pmod 4$ with pairwise distinct $j_t \in \{ 1, \ldots, n \}$, with the possible exception of $v_1v_2\cdots v_n$, if $n \equiv 3 \pmod 4$. \end{lemma} \begin{proof} It is well-known that all products $v_{j_1}v_{j_2}$, $j_1 \neq j_2$, are contained in $\mathfrak{m}$: Indeed, $\Lambda^2 \mathbb{R}^n \cong \mathfrak{so}(n)$ (cf., e.g., \cite[Prop.~6.1]{Lawson/Michelsohn:1989}) is generated as a Lie algebra by the $v_iv_{i+1}$, $1 \leq i < n$ (cf., e.g., \cite[Thm.~1.31]{Berman} and Theorem~\ref{Berman} of the main text). Moreover, for pairwise distinct $j_t$, $1 \leq t \leq k+1$, one has \[[v_{j_1}v_{j_2},\;v_{j_2}v_{j_3}\cdots v_{j_{k+1}}] = 2v_{j_1}v_{j_3}\cdots v_{j_{k+1}}.\] Since re-ordering of the factors simply yields scalar multiples, this shows inductively that, as long as $k+1 \leq n$, once an arbitrary factor of the form $v_{j_1}v_{j_2}\cdots v_{j_k}$ is contained in the Lie subalgebra, all factors of that form are contained in the Lie subalgebra. This statement is also true in the situation $k=n$, because in that case all factors of that form are scalar multiples of one another. We prove the claim of the lemma by induction over $k$. For $k=2$ and $k=3$, this is obvious. Suppose the claim holds for $k\equiv 3\pmod 4$, so that the next value for $k$ to consider is $k+3\equiv 2\pmod 4$. By induction hypothesis $v_4v_5\cdots v_{k+3} \in \mathfrak{m}$ and \[0 \neq [v_1v_2v_3,v_4v_5\cdots v_{k+3}] = 2 v_1v_2v_3v_4\cdots v_{k+3}. \] If on the other hand the claim holds for $k\equiv 2\pmod 4$, then the next value for $k$ to consider is $k+1\equiv 3\pmod 4$. If $k+2\leq n$, then by induction hypothesis $v_3v_4\cdots v_{k+2}\in\mathfrak{m}$ and \[0 \neq [v_1v_2v_3,v_3v_4\cdots v_{k+2}] = 2v_1v_2v_4\cdots v_{k+2}.\] That is, the presence of all elements of the form $v_{j_1}v_{j_2}v_{j_3}$ with pairwise distinct $j_t \in \{ 1, \ldots, n \}$ inductively allows us to construct all elements of the form $v_{j_1}v_{j_2}\cdots v_{j_k}$ for $k \equiv 2, 3 \pmod 4$ with pairwise distinct $j_t \in \{ 1, \ldots, n \}$ for all $k \leq n$, with the possible exception of the situation $k=n \equiv 3 \pmod 4$, as the element $v_{k+2}$ does not exist in that case. \end{proof} \begin{remark} It will turn out later, as a consequence of the proof of Theorem~\ref{thm:A} based on dimension arguments, that the above elements in fact generate $\mathfrak{m}$ as an $\mathbb{R}$-vector space and that for $n \equiv 3 \pmod 4$ the element $v_1v_2\cdots v_n$ indeed is not contained in $\mathfrak{m}$, unless of course $n=3$. \end{remark} \begin{definition} For $k\in\{0,1,2,3\}$, let \[ \delta_k : \mathbb{N}\to \mathbb{N} : n \mapsto \sum_{\substack{i=0, \\ i \equiv k \pmod 4}}^n \binom{n}{i}. \] \end{definition} \begin{consequence} \label{con:lowerbounddim-ineffective} Let $n\geq 3$. Then \[ \dim \mathfrak{m} \geq \begin{cases} \delta_2(n) + \delta_3(n) & \text{ if } n \not\equiv 3 \pmod 4 , \\ \delta_2(n) + \delta_3(n)-1 & \text{ if } n \equiv 3 \pmod 4 . \end{cases} \] \end{consequence} \subsection{Combinatorics of binomial coefficients} We now turn the lower bound from Consequence~\ref{con:lowerbounddim-ineffective} into a numerically explicit bound by deriving a closed formula in $n$ for the functions $\delta_k$. \begin{proposition} \label{binomial} Let $n \in \mathbb{N}$ and $k\in\{0,1,2,3\}$. \begin{enumerate} \setcounter{enumi}{-1} \item If $n \equiv 0 \pmod 4$, then \[ \delta_k(n) = \begin{cases} 2^{n-2} & \text{for } k \in \{ 1, 3 \}, \\ 2^{n-2}+(-1)^{\frac{n}{4}+ \frac{k}{2}} 2^{\frac{n}{2}-1} & \text{for }k \in \{ 0, 2 \}. \end{cases}\] \item If $n \equiv 1 \pmod 4$, then \[ \delta_k(n) = \begin{cases} 2^{n-2}+(-1)^{\frac{n-1}{4}}2^{\frac{n-3}{2}} & \text{for } k \in \{ 0, 1 \}, \\ 2^{n-2}-(-1)^{\frac{n-1}{4}}2^{\frac{n-3}{2}} & \text{for } k \in \{ 2, 3 \}. \end{cases} \] \item If $n \equiv 2 \pmod 4$, then \[ \delta_k(n) = \begin{cases} 2^{n-2} & \text{for } k \in \{ 0, 2 \}, \\ 2^{n-2}+(-1)^{\frac{n-2}{4}+ \frac{k-1}{2}} 2^{\frac{n}{2}-1} & \text{for } k \in \{ 1, 3 \}. \end{cases} \] \item If $n \equiv 3 \pmod 4$, then \[ \delta_k(n) = \begin{cases} 2^{n-2}-(-1)^{\frac{n-3}{4}}2^{\frac{n-3}{2}} & \text{for } k \in \{ 0, 3 \}, \\ 2^{n-2}+(-1)^{\frac{n-3}{4}}2^{\frac{n-3}{2}} & \text{for } k \in \{ 1, 2 \}. \end{cases} \] \end{enumerate} \end{proposition} \begin{proof} For $a, n \in \mathbb{N}$ the binomial theorem implies $$(1+i^a)^n = \sum_{k=0}^3 i^{ak}\delta_k(n),$$ where $i \in \mathbb{C}$ denotes the imaginary unit. Evaluation of this formula for $a \in \{ 0, 1, 2, 3 \}$ yields the following system of four identities: \begin{align} \delta_0(n) + \delta_1(n) + \delta_2(n) + \delta_3(n) &= 2^n, \label{1}\\ \delta_0(n) + i\delta_1(n) - \delta_2(n) -i \delta_3(n) &= (1+i)^n = 2^{\frac{n}{2}} \cdot e^{\frac{n2\pi i}{8}}, \label{2}\\ \delta_0(n) - \delta_1(n) + \delta_2(n) - \delta_3(n) &= 0, \label{3}\\ \delta_0(n) -i \delta_1(n) - \delta_2(n) +i \delta_3(n) &= (1-i)^n = 2^{\frac{n}{2}} \cdot e^{-\frac{n2\pi i}{8}}. \label{4} \end{align} These four identities imply \begin{align} \delta_0(n) + \delta_2(n) &= 2^{n-1} &\text{(\ref{1}) plus (\ref{3}) divided by $2$}, \label{5} \\ \delta_0(n) - \delta_2(n) &= 2^{\frac{n-2}{2}} (e^{\frac{n2\pi i}{8}}+e^{-\frac{n2\pi i}{8}}) &\text{(\ref{2}) plus (\ref{4}) divided by $2$}, \label{6} \\ \delta_1(n) + \delta_3(n) &= 2^{n-1} &\text{(\ref{1}) minus (\ref{3}) divided by $2$}, \label{7} \\ \delta_1(n) - \delta_3(n) &= -2^{\frac{n-2}{2}}i (e^{\frac{n2\pi i}{8}}-e^{-\frac{n2\pi i}{8}}) &\text{(\ref{2}) minus (\ref{4}) divided by $2i$}. \label{8} \end{align} One readily computes $\delta_0(n)$, $\delta_2(n)$ from (\ref{5}), (\ref{6}) and $\delta_1(n)$, $\delta_3(n)$ from (\ref{7}), (\ref{8}). \end{proof} Combining this with Consequence~\ref{con:lowerbounddim-ineffective} yields the following: \begin{consequence} \label{comparedimension} Let $n \in \mathbb{N}$ and $n\geq 2$. \begin{enumerate} \setcounter{enumi}{-1} \item If $n \equiv 0 \pmod 8$, then \begin{align} \dim \mathfrak{m} \geq \delta_2(n) + \delta_3(n) &= 2^{n-2} - 2^{\frac{n}{2}-1} + 2^{n-2} = 2^{\frac{n-2}{2}}(2^{\frac{n}{2}}-1) \\ &= \dim_\mathbb{R}(\mathfrak{so}(2^{\frac{n}{2}})). \end{align} \item If $n \equiv 1 \pmod 8$, then \begin{align} \dim \mathfrak{m} \geq \delta_2(n) + \delta_3(n) &= 2\left( 2^{n-2} - 2^{\frac{n-3}{2}} \right) = 2^{\frac{n-1}{2}}(2^{\frac{n-1}{2}}-1) \\ &= \dim_\mathbb{R}(\mathfrak{so}(2^{\frac{n-1}{2}}) \oplus \mathfrak{so}(2^{\frac{n-1}{2}})). \end{align} \item If $n \equiv 2 \pmod 8$, then \begin{align} \dim \mathfrak{m} \geq \delta_2(n) + \delta_3(n) &= 2^{n-2} + 2^{n-2} - 2^{\frac{n}{2}-1} = 2^{\frac{n-2}{2}}(2^{\frac{n}{2}}-1) \\ &= \dim_\mathbb{R}(\mathfrak{so}(2^{\frac{n}{2}})). \end{align} \item If $n \equiv 3 \pmod 8$, then \begin{align} \dim \mathfrak{m} + 1 \geq \delta_2(n) + \delta_3(n) &= 2^{n-2} + 2^{\frac{n-3}{2}} + 2^{n-2} - 2^{\frac{n-3}{2}} = 2^{n-1} \\ &= \dim_\mathbb{R}(\mathfrak{su}(2^{\frac{n-1}{2}}))+1. \end{align} \item If $n \equiv 4 \pmod 8$, then \begin{align} \dim \mathfrak{m} \geq \delta_2(n) + \delta_3(n) &= 2^{n-2} + 2^{\frac{n}{2}-1} + 2^{n-2} = 2^{\frac{n-2}{2}}(2^{\frac{n}{2}}+1) \\ &= \dim_\mathbb{R}(\mathfrak{sp}(2^{\frac{n-2}{2}})). \end{align} \item If $n \equiv 5 \pmod 8$, then \begin{align} \dim \mathfrak{m} \geq \delta_2(n) + \delta_3(n) &= 2\left( 2^{n-2} + 2^{\frac{n-3}{2}} \right) = 2^{\frac{n-1}{2}}(2^{\frac{n-1}{2}}+1) \\ &= \dim_\mathbb{R}(\mathfrak{sp}(2^{\frac{n-3}{2}})\oplus \mathfrak{sp}(2^{\frac{n-3}{2}})). \end{align} \item If $n \equiv 6 \pmod 8$, then \begin{align} \dim \mathfrak{m} \geq \delta_2(n) + \delta_3(n) &= 2^{n-2} + 2^{n-2} + 2^{\frac{n}{2}-1} = 2^{\frac{n-2}{2}}(2^{\frac{n}{2}}+1) \\ &= \dim_\mathbb{R}(\mathfrak{sp}(2^{\frac{n-2}{2}})). \end{align} \item If $n \equiv 7 \pmod 8$, then \begin{align} \dim \mathfrak{m} + 1 \geq \delta_2(n) + \delta_3(n) &= 2^{n-2} - 2^{\frac{n-3}{2}} + 2^{n-2} + 2^{\frac{n-3}{2}} = 2^{n-1} \\ &= \dim_\mathbb{R}(\mathfrak{su}(2^{\frac{n-1}{2}}))+1. \end{align} \end{enumerate} \end{consequence} \subsection{Generalized spin representations of the split real $E_n$ series and the resulting quotients} The example of a generalized spin representation of the maximal compact subalgebra of the split real Kac--Moody Lie algebra of type $E_{10}$ described in \cite{DamourKleinschmidtNicolai} and \cite{deBuylHenneauxPaulot} (see Example~\ref{theexample} in the main text) generalizes directly to the whole $E_n$ series as follows. Let $n \in \mathbb{N}$, let $\mathfrak{g}$ be the split real Kac--Moody Lie algebra of type $E_n$, let $\mathfrak{k}$ be its maximal compact subalgebra, and let $X_i$, $1 \leq i \leq n$, be the Berman generators of $\mathfrak{k}$ (cf.\ \cite[Thm.~1.31]{Berman} and Theorem~\ref{Berman} in the main text) enumerated in Bourbaki style as shown in Figure~\ref{fig:En}, i.e., $X_1$, $X_3$, $X_4$, \ldots, $X_n$ belong to the $A_{n-1}$ subdiagram, generating $\mathfrak{so}(n)$, and $X_2$ to the additional node. As in Section~\ref{sec:cartan-bott} let $q$ be the standard positive definite quadratic form on $\mathbb{R}^n$ and let $C = C(\mathbb{R}^n,q)$ be the corresponding Clifford algebra, considered as a Lie algebra. \begin{proposition} Let $n\geq 3$. The assignment \begin{itemize} \item $X_{1} \mapsto \frac{1}{2}v_1v_2$, \item $X_2 \mapsto \frac{1}{2}v_1v_2v_3$, \item $X_j \mapsto \frac{1}{2}v_{j-1}v_j$ for $3 \leq j \leq n$ \end{itemize} defines a Lie algebra homomorphism $\rho$ from $\mathfrak{k}$ to the Lie subalgebra $\mathfrak{m}$ of $C$ generated by $v_1v_2v_3$ and by $v_iv_{i+1}$, $1 \leq i < n$, called the {\bf standard generalized spin representation of $\mathfrak{k}$}. \end{proposition} \begin{proof} The proof is based on the criterion established in Remark~\ref{characterization} and is exactly the same as in the $E_{10}$ case discussed in Example~\ref{theexample}. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:A}] By Theorem~\ref{Mainexistencetheorem} and since $E_n$ is simply laced and connected for $n\geq 4$, the image $\mathfrak{m}$ of $\rho$ is semisimple and compact. By Lemma~\ref{lowerbounddimension} and Consequence~\ref{comparedimension}, the dimension $\dim_\mathbb{R}(\mathfrak{m})$ is at least as large as the dimension of the maximal semisimple compact Lie subalgebra of $C$ as given in Corollary~\ref{cor:cartan-bott-max-cpt}. The claim follows. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:B}] Let $\mathfrak{g}$ be a semisimple split real Lie algebra of type $E_4=A_4$, $E_5=D_5$, $E_6$, $E_7$ or $E_8$ and $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ its Iwasawa decomposition. Since $\dim_\mathbb{R}(\mathfrak{k}) = \dim_\mathbb{R}(\mathfrak{n})$, from the combinatorics of the respective root system we conclude that the maximal compact Lie subalgebra $\mathfrak{k}$ has dimension \begin{align} 10 &= \frac{4\cdot 5}{2} = \frac{2^{\frac{4}{2}} \cdot (2^{\frac{4}{2}}+1)}{2} = \dim_\mathbb{R}(\mathfrak{sp}(2)) = \dim_\mathbb{R}(\mathfrak{so}(5)) &\text{ if } n=4, \\ 20 &= 2 \cdot 10 = \dim_\mathbb{R}(\mathfrak{sp}(2)\oplus\mathfrak{sp}(2)) = \dim_\mathbb{R}(\mathfrak{so}(5)\oplus\mathfrak{so}(5)) &\text{ if } n=5, \\ 36 &= 4 \cdot 9 = 2^{\frac{6-2}{2}}\cdot(2^{\frac{6}{2}}+1) = \dim_\mathbb{R}(\mathfrak{sp}(4)) &\text{ if } n=6, \\ 63 &= 2^6-1 = \dim_\mathbb{R}(\mathfrak{su}(8)) &\text{ if } n=7, \\ 120 &= \frac{16 \cdot 15}{2} = \frac{2^{\frac{8}{2}} \cdot (2^{\frac{8}{2}}-1)}{2} = \dim_\mathbb{R}(\mathfrak{so}(16)) &\text{ if } n=8. \end{align*} For $n\geq 4$ we may now apply Theorem~\ref{thm:A} and deduce that the standard generalized spin representation $\rho$ has to be injective in these cases. This leaves the case $E_3=A_2\oplus A_1$. Since this diagram is not irreducible, Theorem~\ref{Mainexistencetheorem} only implies that $\im(\rho)=\mathfrak{m}$ is compact but not that it is semisimple (and, indeed, it is not). However, $n=3$ is also an exceptional case for Lemma~\ref{lowerbounddimension}: In this case $\dim_\mathbb{R}(\mathfrak{m})=4$, as $v_1v_2$, $v_1v_3$, $v_2v_3$, $v_1v_2v_3$ form an $\mathbb{R}$-basis of $\mathfrak{m}$. On the other hand, the Clifford algebra $C$ is isomorphic to $\M{2}{\mathbb{C}}$, hence $\mathfrak{k}\cong \mathfrak{u}(2)$, and this has dimension $4$. Thus $\rho$ is also injective when $n=3$. The claim follows. \end{proof} \noindent JLU Giessen, Mathematisches Institut, Arndtstrasse 2, 35392 Giessen, Germany \\ {\tt [email protected] } \\ {\tt [email protected] } \noindent Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom \\ {\tt [email protected]} \end{document}	arXiv
\begin{document} \begin{titlepage} \thispagestyle{empty} \title{ Improved Online Correlated Selection hanks{This is the second version on arXiv. Compared to the first version, this one adds a discussion on two concurrent works on the same topic, gives a more accurate description of previous results, and improves the presentation based on the feedbacks by anonymous reviewers. The conference version appears in FOCS 2021.} \begin{abstract} \thispagestyle{empty} This paper studies the online correlated selection (OCS) problem. It was introduced by Fahrbach, Huang, Tao, and Zadimoghaddam (2020) to obtain the first edge-weighted online bipartite matching algorithm that breaks the $0.5$ barrier. Suppose that we receive a pair of elements in each round and immediately select one of them. Can we select with negative correlation to be more effective than independent random selections? Our contributions are threefold. For semi-OCS, which considers the probability that an element remains unselected after appearing in $k$ rounds, we give an optimal algorithm that minimizes this probability for all $k$. It leads to $0.536$-competitive unweighted and vertex-weighted online bipartite matching algorithms that randomize over only two options in each round, improving the $0.508$-competitive ratio by Fahrbach et al.~(2020). Further, we develop the first multi-way semi-OCS that allows an arbitrary number of elements with arbitrary masses in each round. As an application, it rounds the \textsc{Balance}\xspace algorithm in unweighted and vertex-weighted online bipartite matching and is $0.593$-competitive. Finally, we study OCS, which further considers the probability that an element is unselected in \emph{an arbitrary subset of rounds}. We prove that the optimal ``level of negative correlation'' is between $0.167$ and $0.25$, improving the previous bounds of $0.109$ and $1$ by Fahrbach et al.~(2020). Our OCS gives a $0.519$-competitive edge-weighted online bipartite matching algorithm, improving the previous $0.508$-competitive ratio by Fahrbach et al.~(2020). \end{abstract} \end{titlepage} \tableofcontents \thispagestyle{empty} \addtocontents{toc}{\protect\thispagestyle{empty}} \setcounter{page}{1} \section{Introduction} \label{sec:introduction} Real-life optimization problems often need to make decisions based on the information at hand instead of the full picture in hindsight. Online advertising platforms show advertisements within milliseconds after receiving each user query. Ride hailing applications match riders and drivers without full knowledge of future ride requests. Cloud service providers assign computational tasks to physical servers not knowing what tasks the users may submit later. Due to the broad applications, the design and analysis of online algorithms for these optimization problems are a central topic in computer science and operations research. Lacking accurate knowledge of the full picture, there is usually no universally good decision for all possible future input in these online optimization problems. As a result, online algorithms need to hedge against different possibilities through randomized decisions. Consider the online bipartite matching problem by \citet{KarpVV:STOC:1990} as a running example. We want to find a matching in a bipartite graph and maximize its size. Initially, we only know the left-hand-side of the bipartite graph, a.k.a., the offline vertices. Online vertices on the right-hand-side arrive one at a time. We must immediately and irrevocably match each of them upon arrival. Any deterministic greedy algorithm gives a maximal matching, and therefore its size is at least half of the maximum matching in hindsight. Beating this trivial bound of half, however, necessitates randomization even for bipartite graphs with only two vertices on each side. \begin{tcolorbox}[colback=lightgray!40,arc=0pt,outer arc=0pt,width=\textwidth,boxrule=.5pt] \textbf{Example.~} Consider vertices $1$ and $2$ on the left and $3$ and $4$ on the right. Vertex $3$ arrives first with edges to both $1$ and $2$. A deterministic algorithm then immediately matches $3$, e.g., to $1$. If vertex $4$ only has an edge to $1$, however, the algorithm cannot match it even though a perfect matching exists in hindsight. A randomized algorithm that matches $3$ to $1$ and $2$ with equal probability, on the other hand, matches $\frac{3}{2}$ edges in expectation. \end{tcolorbox} Would it suffice to independently randomize over two offline neighbors? Unfortunately, the answer is negative (e.g., \citet{FahrbachHTZ:FOCS:2020}). We need to correlate different rounds' selections to break the $\frac{1}{2}$ barrier. One can introduce correlation through problem specific methods. The \textsc{Ranking}\xspace algorithm of \citet{KarpVV:STOC:1990}, for example, samples a random order of the offline vertices at the beginning, and then matches each online vertex to the first unmatched offline neighbor by that order. \textsc{Ranking}\xspace and its variants achieve the optimal $1-\frac{1}{e}$ competitive ratio for unweighted~\cite{KarpVV:STOC:1990} and vertex-weighted online bipartite matching~\cite{AggarwalGKM:SODA:2011}, but have difficulties extending to the more general edge-weighted problem (a.k.a., Display Ads)~\cite{FeldmanKMMP:WINE:2009} and AdWords~\cite{MehtaSVV:JACM:2007}. \citet{FahrbachHTZ:FOCS:2020}, on the other hand, formulate a generic online selection problem and design online correlated selection (OCS) algorithms that lead to the first edge-weighted online bipartite matching algorithm that breaks the $\frac{1}{2}$ barrier. Subsequently, \citet{HuangZZ:FOCS:2020} break the $\frac{1}{2}$ barrier in AdWords using a similar approach. The online selection problem considers a set of ground elements (e.g., the offline vertices) and a sequence of pairs of these elements (e.g., a pair of offline neighbors for each online vertex). The algorithm immediately selects an element upon receiving each pair. If it independently selects a random element from each pair, then with probability $2^{-k}$ an element remains unselected in a subset of $k$ pairs involving it. Can we be more effective than independent random selections? \citet{FahrbachHTZ:FOCS:2020} study two versions of online selection called semi-OCS and OCS. Semi-OCS focuses on the probability that an element is unselected at the end when it is in $k$ pairs. They give a semi-OCS that upper bounds this probability by $2^{-k}(1-\gamma)^{k-1}$ for $\gamma = 0.109$, and call it a $\gamma$-semi-OCS. They also prove that $1$-semi-OCS is impossible. OCS further considers the probability that an element is unselected in \emph{an arbitrary subset} of pairs involving it. When the subset is the union of $m$ consecutive subsequences of the pairs involving the element, with lengths $k_1, k_2, \dots, k_m$, their OCS bounds the unselected probability by $\prod_{i=1}^m 2^{-k_i} (1-\gamma)^{k_i-1}$ also for $\gamma = 0.109$. They call it a $\gamma$-OCS. The weaker guarantee of semi-OCS is sufficient for unweighted and vertex-weighted online bipartite matching, while the stronger guarantee of OCS is sufficient for the edge-weighted problem. The main idea is to randomly match the pairs so that two matched pairs share a common element that is not in any pair in between. Their semi-OCS and OCS then select oppositely from the matched pairs with respect to the common element. \citet{FahrbachHTZ:FOCS:2020} explicitly leave two open questions: (1) \emph{What is the best possible $\gamma$ for which $\gamma$-semi-OCS and $\gamma$-OCS exist?} (2) \emph{Are there multi-way online selection algorithms that select from multiple elements in each round, with sufficient negative correlation such that the resulting online matching algorithms are better than the two-way counterparts?} We remark that the matching-based approach fails fundamentally in the multi-way extension. If each round has $n$ elements, it can then be matched to $\Omega(n)$ other rounds. For large $n$, any matching is sparse and the resulting negative correlation is negligible. \subsection{Our Contributions} \paragraph{Semi-OCS and Weighted Sampling without Replacement.} This paper gives a complete answer to the first open question for semi-OCS. In fact, we not only show that the optimal $\gamma$ equals $\frac{1}{2}$ for semi-OCS, but also find that the unselected probability converges to zero much faster than the guanrantee of $\gamma$-semi-OCS when the element is in $k \ge 3$ rounds. \paragraph{Informal Theorem 1.} \emph{ There is a polynomial-time semi-OCS such that an element that is in $k$ pairs is selected with probability at least $1-2^{-2^k+1}$. This is the best possible for all $k \ge 1$. } In each round the optimal semi-OCS selects the element that appears more in previous rounds, and is unselected thus far, breaking ties randomly. It is the limit case of weighted sampling without replacement, when an element's weight is exponential in the number of previous rounds with the element, and when the base of the exponential tends to infinity. The main lemma in our analysis, which may be of independent interest, shows that the selections of different elements are negatively correlated in weighted sampling without replacement with two elements per round. See Section~\ref{sec:semi-ocs}. This paper further answers the second open question affirmatively for semi-OCS by studying a multi-way online selection problem that allows an arbitrary marginal distribution over the elements in each round. We will refer to the marginal probability of an element as its mass in that round. Consider the probability that an element is unselected at the end when its total mass is $y$. On the one hand, sampling independently from the marginal distributions (with replacement) bounds the probability by $\exp(-y)$. On the other hand, for an overly idealized algorithm, which samples from the marginal distributions and ensures that each element is sampled at most once, this probability is $\max \{1-y, 0\}$. For $y \in [0, 1]$, it is $\exp(-y-\frac{y^2}{2}-\frac{y^3}{3}-\dots)$ by the Taylor series of $\ln(1-y)$. We match the overly idealized bound up to the quadratic term, with a smaller cubic coefficient. \paragraph{Informal Theorem 2.} \emph{ There is a polynomial-time multi-way semi-OCS such that an element with total mass $y$ is selected with probability at least $1-\exp(-y-\frac{y^2}{2}-\frac{4-2\sqrt{3}}{3}y^3)$. } It may be tempting to conjecture that sampling independently from the marginal distributions \emph{without replacement} already improves the trivial bound of $\exp(-y)$. Unfortunately, this is false. Consider an element $e$ that is in all $T$ rounds each with mass $\epsilon = \frac{y}{T}$, and let there be a distinct element other than $e$ with mass $1-\epsilon$ in each round. Then element $e$ remains unselected at the end with probability $\exp(-y)$ when $T$ tends to infinity. This example suggests that if an element has accumulated some mass and is still unselected, we shall give it a higher priority than new elements that are not in any previous rounds. It motivates weighted sampling without replacement where the weight is a function of the total mass of the element in previous rounds. We choose the weight to be the inverse of the upper bound on the unselected probability so that \emph{the expected sampling weight of any element is at most its mass in the round}, an invariant that is the key to our analysis. See Section~\ref{sec:multi-way}. \begin{table}[t] \centering \caption{A summary of the results in this paper on online correlated selection and their applications in online bipartite matching, with a comparison to those by \citet{FahrbachHTZ:FOCS:2020}.} \label{tab:summary} \renewcommand{1.25}{1.25} \begin{tabular}{lcc} \toprule & \citet{FahrbachHTZ:FOCS:2020} & \textbf{This Paper} \\ \midrule Semi-OCS\footnotemark[1] & $2^{-k} (1-0.109)^{k-1}$ & $2^{-2^k+1}$ \\ Multi-way Semi-OCS\footnotemark[2] & - & $\exp(-y-\frac{y^2}{2}-\frac{4-2\sqrt{3}}{3} y^3)$ \\ $\gamma$-OCS & $0.109 \le \gamma < 1$ & $0.167 \le \gamma \le \frac{1}{4}$ \\ Unweighted/Vertex-weighted ($2$-Way) & $0.508$ & $0.536$ \\ Unweighted/Vertex-weighted (Multi-way)\footnotemark[3] & - & $0.593$ \\ Edge-weighted & $0.508$ & $0.519$ \\ \bottomrule \end{tabular} \begin{minipage}{.95\textwidth} \footnotesize \footnotemark[1] The table presents upper bounds on the probability that an element is unselected when it is in $k$ pairs. \\ \footnotemark[2] The table presents upper bounds on the probability that an element is unselected when its total mass is $y$. \\ \footnotemark[3] \textsc{Ranking}\xspace by \citet{KarpVV:STOC:1990} is $1-\frac{1}{e}$-competitive which is optimal. Nonetheless, the algorithms in this paper are the first ones other than \textsc{Ranking}\xspace whose competitive ratios are beyond the $0.5+\epsilon$ regime. \end{minipage} \end{table} \paragraph{OCS and Probabilistic Automata.} This paper also contributes to the first open question for OCS by narrowing the gap between the upper and lower bounds on the best possible $\gamma$. \paragraph{Informal Theorem 3.} \emph{ There is a polynomial-time $0.167$-OCS. Further, there is no $\gamma$-OCS for any $\gamma > \frac{1}{4}$, even with unlimited computational power. } The improved OCS also abandons the matching-based approach and instead introduces an automata-based approach. Informally, it picks an element from each round to probe the element's state. If the element was selected last time, select the other element this time. If the element has not been selected in the last two appearances, select it this time. Finally, only when the element was not selected last time but was selected before that, the OCS selects with fresh randomness. The actual algorithm is more involved. For example, we cannot pick an element to probe independently in each round in general. See Section~\ref{sec:ocs} for detail. \paragraph{Applications in Online Bipartite Matching.} The new results on online correlated selection from this paper lead to better online bipartite matching algorithms. For the unweighted and vertex-weighted problems, we get a $0.536$-competitive two-way algorithm, improving the $0.508$-competitive algorithm by \citet{FahrbachHTZ:FOCS:2020}. We further show that the multi-way semi-OCS can round the (fractional) \textsc{Balance}\xspace algorithm (e.g., \cite{KalyanasundaramP:TCS:2000, MehtaSVV:JACM:2007}), and be $0.593$-competitive. For the edge-weighted problem, the $0.167$-OCS gives a $0.512$-competitive algorithm. In the process, we refine the reductions from online matching problems to online correlated selection so that the competitive ratios admit close-formed expressions, and a guarantee strictly weaker than $\gamma$-OCS already suffices for the edge-weighted problem. Motivated by the relaxed guarantee, we design a variant of OCS that further improves the edge-weighted competitive ratio. See Section~\ref{sec:matching}. \paragraph{Informal Theorem 4.} \emph{ There is a polynomial-time $0.519$-competitive algorithm for edge-weighted online bipartite matching. \footnote{We assume free disposals, which is standard in the edge-weighted problem under worst-case analysis.} } \subsection{Other Related Works} \paragraph{Online Rounding.} Online correlated selection is related to online rounding algorithms. It is common to first design online algorithms for an easier fractional online optimization problem, and to round it using online rounding algorithms to solve the original integral problem. Independent rounding is the simplest and most general online rounding; it corresponds to independent random selections in the online selection problem in this paper. For instance, \citet{BuchbinderN:MOR:2009} first design fractional online covering and packing algorithms under the online primal-dual framework, and then round them with independent rounding. More involved online rounding algorithms are usually designed on a problem-by-problem basis in the literature, e.g., for $k$-server~\cite{BansalBMN:JACM:2015}, online submodular maximization~\cite{ChanHJKT:TALG:2018}, online edge coloring~\cite{CohenPW:FOCS:2019}, etc. To our knowledge, the only general online rounding method other than independent rounding is the online contention resolution schemes initiated by \citet{FeldmanSZ:SODA:2016} and further developed by \citet{AdamczykW:FOCS:2018}, \citet{LeeS:ESA:2018}, and \citet{Dughmi:ICALP:2020, Dughmi:arXiv:2021}. It has found applications mainly in online problems with stochastic information, such as prophet inequality~\cite{FeldmanSZ:SODA:2016, EzraFGT:EC:2020}, posted pricing~\cite{FeldmanSZ:SODA:2016}, stochastic probing~\cite{FeldmanSZ:SODA:2016}, and stochastic matching~\cite{GamlathKS:SODA:2019, EzraFGT:EC:2020, FuTWWZ:ICALP:2021}. There is an important difference between the above usages of online rounding algorithms and the applications of OCS in online bipartite matching. The above online rounding algorithms and the corresponding fractional online algorithms are designed separately; the final competitive ratio is the product of their ratios. \footnote{Using this two-step approach to analyze the applications of the multi-way semi-OCS in this paper in unweighted and vertex-weighted online bipartite matching would lead to a much worse competitive ratio of about $0.514$. Using it with the (two-way) semi-OCS and OCS even gives a ratio strictly smaller than $0.5$!} By contrast, this paper and previous works on OCS~\cite{FahrbachHTZ:FOCS:2020, HuangZZ:FOCS:2020} take an end-to-end approach: the online matching algorithms make fractional decisions based on the guarantee of OCS to directly optimize the expected objective of the rounded matching. For example, the algorithms for vertex-weighted and edge-weighted matching in Section~\ref{sec:matching} rely on discount functions derived from optimization problems that take the OCS guarantees as parameters. Another example with a similar spirit is the convex rounding technique by \citet{DughmiRY:JACM:2016} and \citet{Dughmi:EC:2011} from the algorithmic game theory literature. \paragraph{Online Matching.} We refer readers to \citet{Mehta:FTTCS:2013} for a survey on online matching problems. The unweighted, vertex-weighted, and edge-weighted online bipartite matching problems are first studied by \citet{KarpVV:STOC:1990}, \citet{AggarwalGKM:SODA:2011}, and \citet{FeldmanKMMP:WINE:2009}. Later, \citet{DevanurJK:SODA:2013} and \citet{DevanurHKMY:TEAC:2016} simplify the analyses under the online primal-dual framework. In particular, \citet{DevanurHKMY:TEAC:2016} view the expected maximal edge-weight matched to an offline vertex as an integral of the complementary cumulative distribution function, a key ingredient of the application of OCS in edge-weighted matching. Finally, \citet{BuchbinderNW:Arxiv:2021}, \citet{CohenW:SODA:2018}, \citet{GamlathKMSW:FOCS:2019}, \citet{PapadimitriouPSW:EC:2021}, and \citet{SaberiW:ICALP:2021} also build on negative correlation properties to analyze their online algorithms, although these negative correlation properties and their usage are orthogonal to those in this paper. \subsection{Concurrent Works} Concurrently and independently, \citet{BlancC:FOCS:2021} and \citet{ShinA:Arxiv:2021} also improved the results of \citet{FahrbachHTZ:FOCS:2020}. They mainly study multi-way OCS, while this paper focuses on $2$-way semi-OCS, $2$-way OCS, and multi-way semi-OCS. Hence, these two papers are almost orthogonal to ours. Using a $6$-way OCS, \citet{BlancC:FOCS:2021} obtained a $0.5368$-competitive algorithm for edge-weighted online bipartite matching. They also gave similar simplifications for the reduction of online matching problems to OCS. \citet{ShinA:Arxiv:2021} gave a method for converting $2$-way OCS to $3$-way OCS. Applying their method to the OCS of \citet{FahrbachHTZ:FOCS:2020} gives a $0.509$-competitive algorithm for edge-weighted online bipartite matching. Applying it to the improved $2$-way OCS in this paper further improves the ratio to $0.513$. \section{Preliminaries} \label{sec:preliminary} The \emph{online selection problem} considers a set $\mathcal{E}$ of elements and a selection process that proceeds in $T$ rounds. For any round $1 \le t \le T$, a pair of elements $\mathcal{E}^t$ arrive and the \emph{online selection algorithm} needs to immediately select an element from $\mathcal{E}^t$. Let $s^t$ denote the selected element in round $t$. For any subset of rounds $T' \subseteq T$, we say that an element $e$ is \emph{unselected} in $T'$ if the algorithm does not select the element in any round in $T'$, i.e., if $s^t \ne e$ for any $t \in T'$. If $T' = T$, we simply say that element $e$ is unselected. For any $0 \le t \le T$, let $\mathcal{U}^t$ denote the set of elements that are unselected in rounds $1, 2, \dots, t$. Let $\mathcal{U} = \mathcal{U}^T$ denote the set of unselected elements at the end. Semi-OCS considers the probability that an element $e \in \mathcal{E}$ is unselected at the end, and seeks to bound it as a function of the number rounds containing element $e$. \begin{definition}[$\gamma$-semi-OCS, c.f., \citet{FahrbachHTZ:FOCS:2020}] An online selection algorithm is a $\gamma$-semi-OCS if for any online selection instance and any element $e$ that appears in $k$ rounds, element $e$ is unselected with probability at most: \[ 2^{-k} (1-\gamma)^{k-1} ~. \] \end{definition} Selecting an element in each round independently and uniformly at random is a $0$-semi-OCS. \citet{FahrbachHTZ:FOCS:2020} give a $0.109$-semi-OCS and prove that there is no $1$-semi-OCS. OCS further considers the probability that an element $e \in \mathcal{E}$ is unselected in an \emph{arbitrary subset of rounds} containing the element. The upper bounds on this probability depend on the structure of the subset of rounds. A consecutive subsequence of the rounds containing element $e$ is a subset of rounds $\{t_1, t_2, \dots, t_k\}$ such that each round $t_i$ contains $e$, i.e., $e \in \mathcal{E}^{t_i}$ for any $1 \le i \le k$, and no round in between contains $e$, i.e., $e \notin \mathcal{E}^t$ for any $1 \le i \le k-1$ and any $t_i < t < t_{i+1}$. \begin{definition}[$\gamma$-OCS, c.f., \citet{FahrbachHTZ:FOCS:2020}] An online selection algorithm is a $\gamma$-OCS if for any online selection instance, any element $e$, and any subset of rounds $T' \subseteq T$ containing $e$ such that $T'$ is the union of $m$ consecutive subsequences of the rounds containing $e$, with lengths $k_1, k_2, \dots, k_m$, element $e$ is unselected in $T'$ with probability at most: \[ \prod_{i=1}^m 2^{-k_i} (1-\gamma)^{k_i-1} ~. \] \end{definition} For example, suppose that rounds $1, 2, 5, 6, 9$ are the ones that contain element $e$, and consider $T' = \{1, 2, 6, 9\}$. Then, $T'$ is the union of two consecutive subsequences $1, 2$ and $6, 9$, whose lengths are $2$. \citet{FahrbachHTZ:FOCS:2020} give a $0.109$-OCS. Since OCS is stronger than semi-OCS, the impossibility result for semi-OCS implies that there is no $1$-OCS. \begin{comment} \subsection{Notations (Temporary)} \zhiyi{Please refer to this subsection and revise when necessary for consistency of notations throughout the article. Notations used only in one specific subsection is not listed here.} \begin{itemize} \item Set of elements: $\mathcal{E}$. \item Elements: $e$, $e'$, etc.. \item Round index and total number of rounds: $t$ and $T$. \item Multi-set of elements in round $t$: $\mathcal{E}^t$ \item Number of elements per round: $n$. \item Element selected in round $t$: $s^t$. \item Number of items that an element has appeared: $k_e$. \item Set of unselected elements after the first $t$ rounds: $\mathcal{U}^t$; drop superscript for $t = T$. \item Weights: $w_e^t$ for the weight of element $e$ in round $t$, and $w(k)$ for the weight of an element that has already appeared $k$ times. \item Generic numerators: $i$, $j$, and $\ell$ (in case we need more than two) \end{itemize} \end{comment} \section{Optimal Semi-OCS} \label{sec:semi-ocs} \subsection{Algorithms} This paper considers a semi-OCS that remembers the number of rounds involving each element thus far, and selects from each round the element that appears more and is unselected so far, breaking ties uniformly at random and independently in different rounds. See Algorithm~\ref{alg:optimal-semi-ocs}. \begin{algorithm} \caption{Optimal Semi-OCS} \label{alg:optimal-semi-ocs} \begin{algorithmic} \State \textbf{State variables:} (for each element $e$) \begin{itemize} \item The number of previous rounds that contain element $e$, denoted as $k_e$. \item Whether element $e$ has been selected in any previous rounds. \end{itemize} \State \textbf{For each round $t$:} (suppose $\mathcal{E}^t = \{e, e'\}$) \begin{enumerate} \item If both $e$ and $e'$ have been selected, select arbitrarily, e.g., $s^t \in \{ e, e' \}$ uniformly at random. \item If only one of $e$ and $e'$ has been selected, select $s^t$ to be the one that has not been selected. \item If neither $e$ nor $e'$ has been selected: \begin{itemize} \item If $k_e \ne k_{e'}$, select $s^t$ to be the one with more previous appearances. \item Otherwise, select $s^t \in \{e, e'\}$ uniformly at random. \end{itemize} \end{enumerate} \end{algorithmic} \end{algorithm} The main lemma in the analysis of Algorithm~\ref{alg:optimal-semi-ocs} will prove that the (un)selections of elements are negatively correlated. It is more instructive to prove this lemma for a broader family of weighted sampling algorithms (Algorithm~\ref{alg:weighted-2way-sampling}). Algorithm~\ref{alg:optimal-semi-ocs} is the special case when we let $w_e^t = 1$ if $e$ appears in previous rounds at least as many times as the other element does, and let $w_e^t = 0$ otherwise. \begin{algorithm} \caption{Weighted $2$-Way Sampling without Replacement} \label{alg:weighted-2way-sampling} \begin{algorithmic} \State \textbf{Parameters:} \begin{itemize} \item $w_e^t \ge 0$, weight of element $e \in \mathcal{E}^t$ in round $t$ \end{itemize} \State \textbf{For each round $t$:} \begin{enumerate} \item If both elements in $\mathcal{E}^t$ have been selected, select arbitrarily, e.g., uniformly at random. \item Otherwise, select each unselected element $e \in \mathcal{E}^t$ with probability proportional to $w^t_e$. \end{enumerate} \end{algorithmic} \end{algorithm} \subsection{Negative Correlation in Weighted $2$-Way Sampling without Replacement} Recall that $\mathcal{U}^t$ denotes the set of unselected elements after the first $t$ rounds. Hence the event that a subset $S \subseteq \mathcal{E}$ of elements are all unselected after the first $t$ rounds can be written as $S \subseteq \mathcal{U}^t$. We shall establish the negative correlation of such events in the next lemma. \begin{lemma} \label{lem:2way-sampling-negative-correlation} For weighted $2$-way sampling (Algorithm~\ref{alg:weighted-2way-sampling}) with any weights, any $0 \le t \le T$, and any disjoint subsets of elements $A, B \subseteq \mathcal{E}$: \[ \mathbf{Pr} \big[ A \cup B \subseteq \mathcal{U}^t \big] \le \mathbf{Pr} \big[ A \subseteq \mathcal{U}^t \big] \mathbf{Pr} \big[ B \subseteq \mathcal{U}^t \big] ~. \] \end{lemma} \begin{proof} We shall prove the lemma by induction on $t$. The base case when $t = 0$ is trivial since $\mathcal{U}^0 = \mathcal{E}$, and thus $\mathbf{Pr} \big[ A \subseteq \mathcal{U}^0 \big] = \mathbf{Pr} \big[ B \subseteq \mathcal{U}^0 \big] = \mathbf{Pr} \big[ A \cup B \subseteq \mathcal{U}^0 \big] = 1$. Next suppose that the lemma holds for round $t-1$ and consider round $t$. \paragraph{Case 1:} $\mathcal{E}^t \cap (A \cup B) = \emptyset$, i.e., no element in this round belongs to $A$ or $B$. Since the selection in round $t$ does not affect the events of concern, the lemma continues to hold after round $t$ by the inductive hypothesis. \paragraph{Case 2:} $\big\| \mathcal{E}^t \cap (A \cup B) \big\| = 1$, i.e., exactly one element in this round belongs to $A$ or $B$. Denote this element as $e \in \mathcal{E}^t$ and the other element as $e' \in \mathcal{E}^t$. Further suppose without loss of generality that $e \in A$. Since the elements in $B$ are not involved in round $t$, we have: \[ \mathbf{Pr} \big[ B \subseteq \mathcal{U}^t \big] = \mathbf{Pr} \big[ B \subseteq \mathcal{U}^{t-1} \big] ~. \] Next consider the elements in $A$. If $e'$ has been selected in the first $t-1$ rounds, $e$ would certainly be selected after round $t$. Hence, to have $A \subseteq \mathcal{U}^t$, we need not only $A \subseteq \mathcal{U}^{t-1}$, but also $e' \in \mathcal{U}^{t-1}$. Further the algorithm must select $e'$ in round $t$. Putting together we have: \[ \mathbf{Pr} \big[ A \subseteq \mathcal{U}^t \big] = \mathbf{Pr} \big[ A \cup \{ e' \} \subseteq \mathcal{U}^{t-1} \big] \frac{w^t_{e'}}{w^t_e + w^t_{e'}} ~. \] Similarly we have: \[ \mathbf{Pr} \big[ A \cup B \subseteq \mathcal{U}^T \big] = \mathbf{Pr} \big[ A \cup B \cup \{ e' \} \subseteq \mathcal{U}^{t-1} \big] \frac{w^t_{e'}}{w^t_e + w^t_{e'}} ~. \] Cancelling the common term $\frac{w^t_{e'}}{w^t_e + w^t_{e'}}$, the inequality in the lemma is equivalent to: \[ \mathbf{Pr} \big[ A \cup B \cup \{ e' \} \subseteq \mathcal{U}^{t-1} \big] \le \mathbf{Pr} \big[ A \cup \{ e' \} \subseteq \mathcal{U}^{t-1} \big] \mathbf{Pr} \big[ B \subseteq \mathcal{U}^{t-1} \big] ~. \] This follows by the inductive hypothesis for subsets $A \cup \{ e' \}$ and $B$ in round $t-1$. \paragraph{Case 3:} $\mathcal{E}^t \subseteq (A \cup B)$, i.e., both elements in round $t$ belong to $A$ or $B$. Since one element in $\mathcal{E}^t$ is selected in round $t$, we have $\mathbf{Pr} [ A \cup B \subseteq \mathcal{U}^t ] = 0$. Hence the stated inequality trivially holds. \end{proof} We remark that the lemma no longer holds if we have $3$ or more elements in each round. Appendix~\ref{app:positive-correlation} provides a counter-example. See also \citet{Alexander:AnnStat:1989}. \subsection{Analysis} \begin{theorem} \label{thm:optimal-semi-ocs} For any instance and any element that appears in $k$ rounds, the probability that the element is never selected by Algorithm~\ref{alg:optimal-semi-ocs} is at most: \[ 2^{-2^k+1} ~. \] \end{theorem} \begin{proof} We shall prove the theorem by induction on the number of rounds $T$ in the instance. The base case when $T = 0$ is trivial since $k$ must be $0$ in this case. Next suppose that the lemma holds for up to $T-1$ rounds. Consider an arbitrary instance with $T$ rounds, and any element $e$ that appears $k$ times. Without loss of generality, we may assume that $e$ is in the last round $T$; otherwise it follows directly from the inductive hypothesis. Further suppose that the other element in round $T$ is $e'$. Consider three cases depending on the relation between the number of appearances $k_e$ and $k_{e'}$ before round $T$. Observe that $k = k_e + 1$. \paragraph{Case 1:} $k_e > k_{e'}$. By the definition of Algorithm~\ref{alg:optimal-semi-ocs}, element $e$ is selected with certainty after $T$. Hence the probability of concern is $0$, and is trivially smaller than the stated bound. \paragraph{Case 2:} $k_e < k_{e'}$. By the definition of Algorithm~\ref{alg:optimal-semi-ocs}, element $e'$ is selected with certainty in round $T$ if it is not yet selected previously. Hence, element $e$ is never selected by the algorithm at the end if and only if both $e$ and $e'$ are unselected before round $T$. This probability is: \begin{align} \mathbf{Pr} \big[ \{ e, e' \} \subseteq \mathcal{U}^{T-1} \big] & \le \mathbf{Pr} \big[ e \in \mathcal{U}^{T-1} \big] \mathbf{Pr} \big[ e' \in \mathcal{U}^{T-1} \big] \tag{Lemma~\ref{lem:2way-sampling-negative-correlation}} \\ & \le 2^{-2^{k_e}+1} \cdot 2^{-2^{k_{e'}}+1} \tag{Inductive hypothesis} \\ & \le 2^{-2^{k_e}+1} \cdot 2^{-2^{k_e+1}+1} \tag{$k_e < k_{e'}$} \\ & \le 2^{-2^k+1} \tag{$k = k_e + 1$} ~. \end{align} \paragraph{Case 3:} $k_e = k_{e'}$. By the definition of Algorithm~\ref{alg:optimal-semi-ocs}, elements $e$ and $e'$ would be selected with equal probability if neither has been selected before. Therefore, element $e$ is never selected by the algorithm at the end if and only if both $e$ and $e'$ are unselected before round $T$, and the algorithm selects $e'$ in round $T$. The latter happens with probability half and is independent with the former. Hence, this probability is: \begin{align} 2^{-1} \mathbf{Pr} \big[ \{ e, e' \} \subseteq \mathcal{U}^{T-1} \big] & \le 2^{-1} \mathbf{Pr} \big[ e \in \mathcal{U}^{T-1} \big] \mathbf{Pr} \big[ e' \in \mathcal{U}^{T-1} \big] \tag{Lemma~\ref{lem:2way-sampling-negative-correlation}} \\ & \le 2^{-1} \cdot 2^{-2^{k_e}+1} \cdot 2^{-2^{k_{e'}}+1} \tag{Inductive hypothesis} \\ & \le 2^{-1} \cdot 2^{-2^{k_e}+1} \cdot 2^{-2^{k_e}+1} \tag{$k_e = k_{e'}$} \\ & = 2^{-2^k+1} \tag{$k = k_e + 1$} ~. \end{align} Summarizing the three cases completes the inductive step and thus the proof of the theorem. \end{proof} Since $2^{-2^k+1} \le 2^{-2k+1} = 2^{-k}(1-\frac{1}{2})^{k-1}$, Theorem~\ref{thm:optimal-semi-ocs} leads to the following corollary in terms of the original definition of semi-OCS. \begin{corollary} \label{cor:optimal-semi-ocs} Algorithm~\ref{alg:optimal-semi-ocs} is a $\frac{1}{2}$-semi-OCS. \end{corollary} We remark that the guarantee of $\frac{1}{2}$-semi-OCS only gives an $\frac{8}{15} \approx 0.533$-competitive algorithm for unweighted and vertex-weighted matching, while Theorem~\ref{thm:optimal-semi-ocs} leads to least $0.536$. That is, the tighter analysis in Theorem~\ref{thm:optimal-semi-ocs} indeed results in better competitive ratios in online matching. \subsection{Hardness} Finally, we show that the semi-OCS (Algorithm~\ref{alg:optimal-semi-ocs}) and its analysis (Theorem~\ref{thm:optimal-semi-ocs}) are optimal for all $k$ simultaneously. The proof is deferred to Appendix~\ref{app:semi-ocs-hardness}. \begin{theorem} \label{thm:semi-ocs-hardness} For any algorithm and any $k \ge 0$, there is an instance and an element that appears in $k$ rounds, such that with probability at least $2^{-2^k+1}$ the algorithm never selects the element. \end{theorem} The special case of $k = 2$ further implies a hardness for the original definition of $\gamma$-semi-OCS. \begin{corollary} \label{cor:semi-ocs-hardness} There is no $\gamma$-semi-OCS for $\gamma > \frac{1}{2}$. \end{corollary} \section{Multi-way Semi-OCS} \label{sec:multi-way} \subsection{Definitions} The \emph{multi-way online selection problem} considers a set of elements $\mathcal{E}$ and a selection process that proceeds in $T$ rounds as follows. Each round $1 \le t \le T$ is associated with a non-negative vector $\vec{x}^t = (x^t_e)_{e \in \mathcal{E}}$ such that $\sum_{e \in \mathcal{E}} x^t_e = 1$. We shall refer to $x^t_e$ as the \emph{mass} of element $e$ in round $t$. The vectors are unknown at the beginning and are revealed to an \emph{multi-way online selection algorithm} at the corresponding rounds. Let $\mathcal{E}^t = \{ e : x^t_e > 0 \}$ be the set of elements with positive masses in round $t$, i.e., those that may be selected in the round. Upon observing the mass vector $\vec{x}^t$ for round $t$, the algorithm selects an element from $\mathcal{E}^t$. We may interpret $x^t_e$ as the probability of selecting element $e$ in the round \emph{if none of the elements have appeared in previous rounds}, although in general the correlation introduced by the multi-way online selection algorithms will complicate the selection probabilities. For any $0 \le t \le T$, let $y_e^t = \sum_{t' \le t} x_e^{t'}$ be the cumulative mass of element $e$ in the first $t$ rounds. Let $y_e = y_e^T$ be its total mass in the instance for brevity. \begin{definition}[$p$-Multi-way Semi-OCS] \label{def:mocs} A multi-way online selection algorithm is a $p$-multi-way semi-OCS for a non-increasing function $p : [0, +\infty) \to [0, 1]$ if for any multi-way online selection instance and any element $e$, $e$ is unselected with probability at most $p(y_e)$. \end{definition} \begin{comment} \gaorq{Below is an alternative definition of semi-OCS that can potentially improve the competitive ratio.} \begin{definition}[$p$-Multi-way Semi-OCS] An multi-way online selection algorithm is a $p$-multi-way semi-OCS for a convex non-increasing function $p : [0, +\infty) \to [0, 1]$ if for any instance and any element $e$, which appears in round $t_1,\cdots, t_k$, the probability that $e$ is never selected is at most \[ p(y_e)-\sum_{\ell=1}^{k} (y_e^{t_{\ell}}-y_e^{t_{\ell}-1}) \cdot b(y_e^{t_{\ell}})+\int_{0}^{y_e} b(z)dz, \] where $\forall y\geq 0$, \[ b(y) = -e^y\int_{y}^{+\infty} p'(z)e^{-z}dz. \] \end{definition} \end{comment} \subsection{Algorithm: Weighted Sampling without Replacement} We consider weighted sampling without replacement, which is parameterized by a weight function $w : [0, +\infty) \to [1, +\infty)$ with $w(0) = 1$. In each round $t$, the sampling weight of an element $e \in \mathcal{E}^t$ equals $0$ if the element has already been selected in the previous rounds, and equals $x_e^t w(y_e^{t-1})$ otherwise. See Algorithm~\ref{alg:multiway}. \begin{algorithm}[t] \caption{Multi-way Semi-OCS: Weighted Sampling without Replacement} \label{alg:multiway} \begin{algorithmic} \State \textbf{Parameters:}\\ \quad Non-decreasing weight function $w : [0, +\infty) \to [1, +\infty)$ such that $w(0) = 1$.\\ \quad Our result lets $w(y) = \exp \Big( y + \frac{y^2}{2} + c y^3 \Big)$ where $c= \frac{4-2\sqrt{3}}{3}$. \State \textbf{State variables:} (for each element $e$) \begin{itemize} \item Cumulative mass $y_e^t$ of element $e$ up to any round $t$. \item Whether element $e$ has been selected in any previous rounds. \end{itemize} \State \textbf{For each round $t$:} \begin{enumerate} \item If all elements in $\mathcal{E}^t$ have been selected, select arbitrarily, e.g., uniformly at random. \item Otherwise, select an unselected $e \in \mathcal{E}^t$ with probability proportional to $x^t_e \cdot w(y^{t-1}_e)$. \end{enumerate} \end{algorithmic} \end{algorithm} We remark that the optimal ($2$-way) semi-OCS in Section~\ref{sec:semi-ocs} can be interpreted as the limit case when $w(y) = W^{y}$ and $W$ tends to infinity. \subsection{Analysis} \begin{comment} Our analyses of the meta algorithm supplement the weight function $w$ with two auxiliary functions $p : [0, +\infty) \to [0, 1]$ and $c : [0, +\infty) \to [0, 1]$. For any element $e$ with total mass $y_e$, $p(y_e)$ upper bounds the probability that $e$ is never selected, and $c(y_e)$ upper bounds the probability that $e$ is never selected \emph{conditioned on another subset of elements $\mathcal{E}'$ are also never selected}. More precisely, we shall prove that: \begin{align} \label{eqn:multiway-probability-bound} \mathbf{Pr} \big[ \text{\rm $e$ unselected} \big] & ~\le~ p(y_e) ~; \\[1ex] \label{eqn:multiway-conditional-probability-bound} \mathbf{Pr} \big[ \text{\rm $e$ unselected} \mid \text{\rm $\mathcal{E}'$ unselected} \big] & ~\le~ c(y_e) ~. \end{align} We shall choose the functions to satisfy two properties. First, for any $y \ge 0$: \begin{equation} \label{eqn:multiway-property-w-c} w(y) c(y) = 1 ~. \end{equation} Second, for any $y \ge 0$ and any $0 < \delta < 1$: \begin{equation} \label{eqn:multiway-property-w-p} p(y + \delta) \ge p(y) \Big( 1 - \frac{\delta w(y)}{\delta w(y) + 1 - \delta} \Big) ~. \end{equation} They are driven by our analysis, as demonstrative in the inductive lemma below. \subsubsection{Inductive Lemma for Unselected Probability} \begin{lemma} \label{lem:multiway-inductive-probability-bound} Suppose that the weight function $w$ and the auxiliary functions $p$ and $c$ satisfy the properties in Equations~\eqref{eqn:multiway-property-w-c} and \eqref{eqn:multiway-property-w-p}. Suppose further that for instances with at most $T-1$ rounds, Algorithm~\ref{alg:multiway} with weight function $w$ satisfy Equations~\eqref{eqn:multiway-probability-bound} and \eqref{eqn:multiway-conditional-probability-bound}. Then, Eqn.~\eqref{eqn:multiway-probability-bound} also holds for instances with $T$ rounds. \end{lemma} \begin{proof} Consider without loss of generality an element $e$ with positive mass in the last round. We expand the left-hand-side of inequality~\eqref{eqn:multiway-probability-bound}, i.e., $\mathbf{Pr} \big[ \text{$e$ unselected} \big]$, by summing over the relevant selections $s^1, \dots, s^{T-1}$ in the first $T-1$ rounds, with each term being the product of the probability of realizing them in the first $T-1$ rounds, and the conditional probability of not selecting $e$ in the last round $T$: \[ \mathbf{Pr} \big[ \text{$e$ unselected} \big] = \sum_{s^1, \dots, s^{T-1} \ne e} \mathbf{Pr} \big[ s^1, \dots, s^{T-1} \big] \mathbf{Pr} \big[ \text{$e$ unselected in round $T$} \mid s^1, \dots, s^{t-1} \big] ~. \] For notational simplicity, let $y = y_e^{T-1}$ denote the cumulative mass of element $e$ in the first $T-1$ rounds, and $\delta = x_e^T$ be its mass in the last round $T$. Then we rewrite above equation as: \[ \mathbf{Pr} \big[ \text{$e$ unselected} \big] = \sum_{s^1, \dots, s^{T-1}\ne e} \mathbf{Pr} \big[ s^1, \dots, s^{T-1} \big] \cdot \left( 1 - \frac{\delta w(y)}{\delta w(y) + \sum_{e'\in \mathcal{E}^T\setminus\{e\}} \mathbf{1}_{e' \in \mathcal{U}^{T-1}} x^T_{e'} w(y^{T-1}_{e'})} \right) ~. \] Recall that $\mathcal{U}^{T-1}$ denote the set of unselected elements after the first $T-1$ rounds and, thus, $e \in \mathcal{U}^{T-1}$ is the event that $e$ remains unselected after the first $T-1$ rounds. By Jensen's inequality and the convexity of $\frac{1}{x}$: \[ \mathbf{Pr} \big[ \text{$e$ unselected} \big] \le \mathbf{Pr} \big[ e \in \mathcal{U}^{T-1} \big] \Big( 1 - \frac{\delta w(y)}{\delta w(y) + \sum_{e'\in \mathcal{E}^T \setminus \{e\}} \mathbf{Pr}[ e' \in \mathcal{U}^{T-1} \mid e \in \mathcal{U}^{T-1}] x^T_{e'} w(y^{T-1}_{e'})} \Big) ~. \] By Eqn.~\eqref{eqn:multiway-probability-bound} up to round $T-1$: \[ \mathbf{Pr} \big[ e \in \mathcal{U}^{T-1} \big] \le p(y) ~. \] By Eqn.~\eqref{eqn:multiway-conditional-probability-bound} up to round $T-1$ and Eqn.~\eqref{eqn:multiway-property-w-c}: \[ \mathbf{Pr}[ e' \in \mathcal{U}^{T-1} \mid e \in \mathcal{U}^{T-1}] w(y^{T-1}_{e'}) \le 1 ~. \] Hence, we get that: \begin{align} \mathbf{Pr} \big[ \text{$e$ unselected} \big] & \le p(y) \Big( 1 - \frac{\delta w(y)}{\delta w(y) + \sum_{e' \in \mathcal{E}^T \setminus \{e\}} x^T_{e'}} \Big) \\ & = p(y) \Big( 1 - \frac{\delta w(y)}{\delta w(y) + 1 - \delta} \Big) ~. \end{align} The lemma now follows by Eqn.~\eqref{eqn:multiway-property-w-p}. \end{proof} \subsubsection{Example: Unweighted Sampling without Replacement} \begin{theorem} \label{thm:multiway-unweighted} Algorithm~\ref{alg:multiway} with: \[ w(y) = 1 \] is a $p$-multi-way semi-OCS for: \[ p(y) = e^{-y} ~. \] \end{theorem} \begin{proof} Let $w(y) = c(y) = 1$ for all $y \ge 0$. Then, Equations~\eqref{eqn:multiway-conditional-probability-bound} and \eqref{eqn:multiway-property-w-c} hold. Further, Eqn.~\eqref{eqn:multiway-property-w-p} becomes: \[ e^{-y-\delta} \ge e^{-y} \big( 1-\delta \big) ~, \] which follows by $e^{-\delta} \ge 1 - \delta$. Finally, since Eqn.~\eqref{eqn:multiway-probability-bound} holds vacuously for the base case when $T = 0$ which implies $y_e = 0$ for all elements $e$, Lemma~\ref{lem:multiway-inductive-probability-bound} ensures that it also holds in general. \end{proof} \subsubsection{Example: Idealized Case with Negative Correlation Across Elements} The second example considers an idealized case in which the events that element $e$ is never selected are pair-wise negatively correlated. More precisely, we assume that for any elements $e, e'$: \begin{equation} \label{eqn:multiway-negative-correlation-assumption} \mathbf{Pr} \big[ \text{\rm $e$ unselected} \mid \text{\rm $e'$ unselected} \big] \le \mathbf{Pr} \big[ \text{\rm $e$ unselected} \big] ~. \end{equation} We remark that the above assumption does not hold in general, even if the mass vectors $\vec{y}^t$'s are identical in all rounds. See, e.g., \citet{Alexander:AnnStat:1989}. \begin{theorem} \label{thm:multiway-idealized} Under the idealized assumption Eqn.~\eqref{eqn:multiway-negative-correlation-assumption}, Algorithm~\ref{alg:multiway} with: \[ w(y) = \exp \Big( y+\frac{y^2}{2}+\frac{y^3}{6} \Big) \] is a $p$-multi-way semi-OCS for: \[ p(y) = \exp \Big( -y-\frac{y^2}{2}-\frac{y^3}{6} \Big) ~. \] \end{theorem} \begin{proof} Under the indealized assumption Eqn.~\eqref{eqn:multiway-negative-correlation-assumption}, let $c(y) = p(y)$. It suffices to verify that our choice of $w(y)$ satisfies Eqn.~\eqref{eqn:multiway-property-w-p}. Unfortunately, this is tedious without much insight; we defer it to Appendix~\ref{app:???}. Then, the theorem follows from an induction on the number of round. The base case when $T = 0$ is trivial since $y_e = 0$ for all elements. Suppose that the theorem holds when the number of rounds is at most $T-1$. In other words, Eqn.~\eqref{eqn:multiway-probability-bound} holds (and hence so does Eqn.~\eqref{eqn:multiway-conditional-probability-bound} due to the idealized assumption Eqn.~\eqref{eqn:multiway-negative-correlation-assumption}). Since Eqn.~\eqref{eqn:multiway-property-w-c} holds by our choice of $w(y)$, $p(y)$, and $c(y)$, Lemma~\ref{lem:multiway-inductive-probability-bound} asserts that the theorem continues to hold for $T$. \end{proof} We finish the idealized case with two remarks. First, one could alternatively solve $w(y)$ numerically by replacing $p(y)$ with $\frac{1}{w(y)}$ in Eqn.~\eqref{eqn:multiway-property-w-p}. After some simplification, this is: \[ w(y+\delta) \le w(y) + \frac{\delta}{1-\delta} w(y)^2 ~. \] See Figure~\ref{fig:multiway-comparison} for a comparison that demonstrates the tightness of Theorem~\ref{thm:multiway-idealized}. In particular, under the idealized assumption Theorem~\ref{thm:multiway-idealized} implies a $0.592$-competitive algorithm for unweighted and vertex-weighted online bipartite matching, while the numerical $w(y)$ only improves it to $0.596$. \end{comment} \begin{theorem} \label{thm:multiway-ocs} Weighted Sampling without Replacement (Algorithm~\ref{alg:multiway}) with weight function: \begin{equation} \label{eqn:multiway-weight} w(y) = \exp \Big( y + \frac{y^2}{2} + c y^3 \Big) \end{equation} where $c = \frac{4-2\sqrt{3}}{3} \approx 0.179$ is a $p$-multi-way semi-OCS for: \[ p(y) = \frac{1}{w(y)} = \exp \Big( - y - \frac{y^2}{2} - c y^3 \Big) ~. \] \end{theorem} Consider an overly idealized algorithm which selects each element $e$ in round $t$ with probability exactly $x_e^t$ and never selects any element more than once. It would be a $p^$-multiway semi-OCS for $p^(y) = \max\{1-y, 0\}$. By the Taylor series of $\log(1-y)$ for $0 \le y < 1$, it can be written as: \[ p^(y) = \exp \Big( - \sum_{i=1}^\infty \frac{y^i}{i} \Big) ~. \] The guarantee of Theorem~\ref{thm:multiway-ocs} matches the overly idealized bound up to the quadratic term and has a smaller coefficient for the cubic term. First, we prove some properties about the weight function $w$ in Eqn.~\eqref{eqn:multiway-weight}. \begin{lemma}\label{lemma:multiway-cubic} For any $0 \le x < 1$ and any $y \ge 0$: \[ \frac{w(y+x)}{w(y)}\le \frac{x }{1-x}w(y)+1 ~. \] \end{lemma} The proof of Lemma~\ref{lemma:multiway-cubic} involves tedious calculations and computer-aided numerical verifications that are not insightful. Hence, we defer it to Appendix~\ref{app:multiway-cubic}; see also Appendix~\ref{app:multiway-cubic-weak} for a proof that does not use computer-aided numerical verifications for a weaker version of the lemma. We further introduce a generalized version of Lemma~\ref{lemma:multiway-cubic} whose proof is also deferred to Appendix~\ref{app:multiway-condition}. \begin{lemma}\label{lemma:multiway-condition} For any $k \ge 1$, any $x_i, y_i \ge 0$ for $1 \le i \le k$ such that $\sum_{i=1}^k x_i \in [0, 1]$: \[ \frac{1-\sum_{i=1}^k x_i}{\sum_{i=1}^k x_i w(y_i) +1- \sum_{i=1}^k x_i}\le \prod_{i=1}^k \frac{w(y_i)}{w(y_i+x_i)} ~. \] \end{lemma} With these two lemmas, we bound the unselected probability for any subset of elements, which implies Theorem~\ref{thm:multiway-ocs} as a special case. \begin{theorem} \label{thm:multiway-ocs-strong} Weighted Sampling without Replacement (Algorithm~\ref{alg:multiway}) with weight function $$w(y)=\exp\left(y+ \frac{y^2}{2}+c y^3\right)$$ with $c=\frac{4-2\sqrt{3}}{3}$ ensures that any subset of elements $\mathcal{E}' \subseteq \mathcal{E}$ are unselected with probability at most: \[ \prod_{e \in \mathcal{E}'} p(y_e) ~, \] where $p(y) = \frac{1}{w(y)}$. \end{theorem} \begin{proof} Recall that $\mathcal{U}^t$ denotes the set of unselected elements after round $t$. Hence, $\mathcal{E}' \subseteq \mathcal{U}^t$ is the event that the elements in $\mathcal{E}'$ are unselected in the first $t$ rounds. We shall prove by induction on $0 \le t \le T$ that: \[ \mathbf{Pr} \big[ \mathcal{E}' \subseteq \mathcal{U}^t \big] \le \prod_{e \in \mathcal{E}'} p(y_e^t) ~, \] which implies Theorem~\ref{thm:multiway-ocs-strong} as a special case when $t = T$. The base case when $t = 0$ holds vacuously because both sides of the inequality equal $1$. Next suppose that it holds for $t-1$ rounds for some $t > 0$, and consider the case of $t$ rounds. Let $\bar{X}_e^t$ be the indicator of whether element $e$ is unselected after round $t$, and define $\bar{X}_{\mathcal{E}'}^t = \prod_{e\in \mathcal{E}'} \bar{X}_e^t$ for any $\mathcal{E}' \subseteq \mathcal{E}$. Finally, we write $\bar{X}^t$ for $( \bar{X}_e^t )_{e \in \mathcal{E}}$. \begin{align} \mathbf{Pr} \big[ \mathcal{E}'\subseteq \mathcal{U}^t \big] & = \mathbf{E}\,\bar{X}_{\mathcal{E}'}^t \\ & = \mathbf{E}_{\bar{X}^{t-1}} \left[ \bar{X}_{\mathcal{E}'}^{t-1} \left( 1 - \frac{ \sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \bar{X}_e^{t-1} }{ \sum_{e\in \mathcal{E}} w(y_e^{t-1}) x_e^t \bar{X}_e^{t-1} } \right) \right] ~. \end{align} Here we artificually define $\frac{0}{0} = 0$ for ease of presentation. Readers may verify that our argument stays true with this caveat. Next, multiply $\bar{X}_{\mathcal{E}'}^{t-1}$ with both the numerator and denominator in the above fraction. Using that $Y^2 = Y$ for $Y \in \{0, 1\}$, we have: \begin{align} \mathbf{Pr} \big[ \mathcal{E}'\subseteq \mathcal{U}^t \big] & = \mathbf{E}_{\bar{X}^{t-1}} \left[ \bar{X}_{\mathcal{E}'}^{t-1} \left( 1 - \frac{ \sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \bar{X}_{\mathcal{E}'}^{t-1} }{ \sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \bar{X}_{\mathcal{E}'}^{t-1} + \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \bar{X}_{\mathcal{E}'\cup\{e\}}^{t-1} } \right) \right] \\ & = \mathbf{E}_{X^{t-1}} \left[ \frac{ \bar{X}_{\mathcal{E}'}^{t-1} \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \bar{X}_{\mathcal{E}'\cup\{e\}}^{t-1} }{ \sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \bar{X}_{\mathcal{E}'}^{t-1} + \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \bar{X}_{\mathcal{E}'\cup\{e\}}^{t-1} } \right] ~. \end{align} By the concavity of $f(x,y) = \frac{xy}{x+y}$, it follows from Jensen's inequality that: \[ \mathbf{Pr} \big[ \mathcal{E}'\subseteq \mathcal{U}^t \big] \le \frac{ \mathbf{E}\bar{X}_{\mathcal{E}'}^{t-1} \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t\, \mathbf{E}\bar{X}_{\mathcal{E}'\cup\{e\}}^{t-1}}{\sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t\,\mathbf{E}\bar{X}_{\mathcal{E}'}^{t-1} + \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t\,\mathbf{E}\bar{X}_{\mathcal{E}'\cup\{e\}}^{t-1} } ~. \] By the inductive hypothesis and the monotonicity of $f(x,y) = \frac{xy}{x+y}$, we further get that: \begin{align} \mathbf{Pr} \big[ \mathcal{E}'\subseteq \mathcal{U}^t \big] & \le \frac{ \prod_{e \in \mathcal{E}'} p(y_e^{t-1}) \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \prod_{e' \in \mathcal{E}'\cup \{e\}} p(y_{e'}^{t-1}) }{ \sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \prod_{e \in \mathcal{E}'} p(y_e^{t-1}) + \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t \prod_{e' \in \mathcal{E}'\cup \{e\}} p(y_{e'}^{t-1}) } \\ & = \prod_{e \in \mathcal{E}'} p(y_e^{t-1}) \frac{ \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t p(y_e^{t-1}) }{ \sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t + \sum_{e\not\in \mathcal{E}'} w(y_e^{t-1}) x_e^t p(y_e^{t-1}) } \\ & = \prod_{e \in \mathcal{E}'} p(y_e^{t-1}) \frac{ \sum_{e\not\in \mathcal{E}'} x_e^t }{ \sum_{e\in \mathcal{E}'} w(y_e^{t-1}) x_e^t + \sum_{e\not\in \mathcal{E}'} x_e^t } ~. \end{align} Next combine the above with Lemmas \ref{lemma:multiway-cubic} and \ref{lemma:multiway-condition}: \begin{align} \mathbf{Pr} \big[ \mathcal{E}' \subseteq \mathcal{U}^t \big] & \le \frac{1-\sum_{e \in \mathcal{E}'} x^t_e}{\sum_{e\in \mathcal{E}'}x^t_e w(y^{t-1}_e) +1- \sum_{e \in \mathcal{E}'} x_e^t} \prod_{e \in \mathcal{E}'} p(y_e^{t-1}) ~ \tag{$\sum_{e \in \mathcal{E}} x_e^t = 1$}\\ & \le \prod_{e\in \mathcal{E}'} \frac{w(y_{e}^{t-1})}{w(y_{e}^{t-1}+x_e^t)}\prod_{e \in \mathcal{E}'} p(y_e^{t-1})\tag{Lemmas \ref{lemma:multiway-cubic} and \ref{lemma:multiway-condition}} \\[1ex] & = \prod_{e\in \mathcal{E}'} p(y_e^{t-1}+x_e^t). \tag{$p(y) = \frac{1}{w(y)}$} \end{align} \end{proof} \section{Improved Algorithms and Hardness for OCS} \label{sec:ocs} \subsection{Definitions} Recall that an online selection algorithm is a $\gamma$-OCS, if for any ($2$-way) online selection instance, any element $e$, and any disjoint consecutive subsequences of the rounds involving $e$ with lengths $k_1, k_2, \cdots, k_m$ respectively, the probability that $e$ is unselected in these rounds is at most: \[ \prod_{\ell=1}^m 2^{-k_{\ell}}(1-\gamma)^{k_{\ell}-1} ~. \] \begin{definition}[Ex-ante Dependence Graph, c.f., \citet{FahrbachHTZ:FOCS:2020}] The \emph{ex-ante dependence graph} $G^\text{\rm ex-ante} = (V, E^\text{\rm ex-ante})$ is a directed graph defined with respect to an online selection instance. We shall refer to its vertices and edges as nodes and arcs to make a distinction with those in online matching problems. The nodes correspond to rounds: \[ V = \big\{ 1, 2, \dots, T \big\} ~. \] The arcs correspond to neighboring appearances of an element (indicated by the subscript \footnote{ There could be parallel arcs in the ex-ante dependence graph, e.g., when rounds $t$ and $t' = t+1$ have the same two elements. The subscript helps distinguish such parallel arcs. } ): \[ E^\text{\rm ex-ante} = \Big\{ \big( t, t' \big)_e ~:~ t < t';~ e \in \mathcal{E}^t;~ e \in \mathcal{E}^{t'};~ \forall t < t'' < t', e \notin \mathcal{E}^{t''} \Big\} ~. \] \end{definition} \begin{figure} \caption{Example of ex-ante dependence graph} \label{fig:exante-dependence-graph} \end{figure} See Figure~\ref{fig:exante-dependence-graph} for an illustrative example of the ex-ante dependence graph. \subsection{Roadmap} \subsubsection{Matching-based Approach versus Automata-based Approach on a Path} \label{sec:ocs-path} This subsection reviews the matching-based approach of \citet{FahrbachHTZ:FOCS:2020} and its limitation, and explains the automata-based approach in this paper. As a running example, consider an instance with the same two elements \emph{head} ($\mathsf{H}$) and \emph{tail} ($\mathsf{T}$) in every round, and thus the ex-ante dependence graph is a directed path (more precisely, two identical parallel directed paths). \paragraph{Matching-based Approach.} \citet{FahrbachHTZ:FOCS:2020} propose to select a matching from the ex-ante graph, and then to select elements in each pair of matched nodes and in each isolated node with independent random bits; each pair of matched nodes shall select the opposite elements. We shall select the matching such that 1) the selections of different arcs are negatively dependent (including independent), and 2) the probability of selecting each arc is as high as possible. If we could select the arcs each with probability at least $\beta$ with negative dependence, we would obtain a $\beta$-OCS because of the following argument. For any disjoint consecutive subsequences of lengths $k_1, k_2, \dots, k_m$, they contain $\sum_{i=1}^m (k_i-1)$ arcs that could have been selected into the matching. If we select at least one of them into the matching, the opposite selections in its two nodes ensure selecting both elements. By the aforementioned properties, the matching has none of these arcs with probability at most $(1-\beta)^{\sum_{i=1}^m (k_i-1)}$. Even in that case, we still have $\sum_{i=1}^m k_i$ independent selections in these rounds; the probability of not selecting a given element in them is at most $2^{-\sum_{i=1}^m k_i}$. For example, \citet{FahrbachHTZ:FOCS:2020} let each node independently pick an incident arc, and then select an arc into the matching if both nodes pick it. This selects each arc with probability $\frac{1}{4}$ in the special case when the ex-ante graph is a directed path. It is possible to improve in the special case. For instance, we could let each arc independently sample a number uniformly from $[0, 1]$ and select an arc if its number is bigger than its neighbors'. This selects each arc with probability $\frac{1}{3}$. To our best effort, however, we cannot find any matching-based algorithm that selects each arc with probability more than $\frac{\sqrt{5}-1}{2} \approx 0.382$. Further, some of these ideas that improve the $\frac{1}{4}$ bound by the algorithm of \citet{FahrbachHTZ:FOCS:2020} fail to generalize beyond the special case. \paragraph{Automata-based Approach.} This paper introduces a different approach that selects elements using a probabilistic automaton. We shall refer to both this automaton and its transition function as $\sigma^$. It has five states $q_\mathsf{O}$, $q_\mathsf{H}$, $q_{\mathsf{H}^2}$, $q_\mathsf{T}$, and $q_{\mathsf{T}^2}$. The original state $q_\mathsf{O}$ is both the initial state of the automaton and the state it resets to after selecting the same element in two consecutive rounds. State $q_\mathsf{H}$ (resp., $q_\mathsf{T}$) means that the automaton selects $\mathsf{H}$ (resp., $\mathsf{T}$) in the previous round but not twice in a roll; from this state the automaton selects $\mathsf{T}$ (resp., $\mathsf{H}$) with a higher chance, and the margin $\beta \in [0, 1]$ will be optimized to be $\beta = \sqrt{2}-1$ in our analysis. State $q_{\mathsf{H}^2}$ (resp., $q_{\mathsf{T}^2}$) means that the automaton selects $\mathsf{H}$ (resp, $\mathsf{T}$) in the last two rounds; from this state the automaton will select $\mathsf{T}$ (resp., $\mathsf{H}$) with certainty and resets to the original state $\mathsf{O}$. Below is the transition function $\sigma^$ that takes a state as input and returns a state and an element from $\big\{ \mathsf{H}, \mathsf{T} \big \}$ (see also Figure~\ref{fig:tree-ocs-automaton}): \begin{align} \sigma^(q_\mathsf{O}) = \begin{cases} \big( q_\mathsf{H}, \mathsf{H} \big) & \text{w.p.\ $\frac{1}{2}$} \\ \big( q_\mathsf{T}, \mathsf{T} \big) & \text{w.p.\ $\frac{1}{2}$} \end{cases} ~, & \quad \sigma^(q_\mathsf{H}) = \begin{cases} \big( q_{\mathsf{H}^2}, \mathsf{H} \big) & \text{w.p.\ $\frac{1-\beta}{2}$} \\ \big( q_\mathsf{T}, \mathsf{T} \big) & \text{w.p.\ $\frac{1+\beta}{2}$} \end{cases} ~, \quad \sigma^(q_\mathsf{T}) = \begin{cases} \big( q_\mathsf{H}, \mathsf{H} \big) & \text{w.p.\ $\frac{1+\beta}{2}$} \\ \big( q_{\mathsf{T}^2}, \mathsf{T} \big) & \text{w.p.\ $\frac{1-\beta}{2}$} \end{cases} ~, \\[2ex] & \sigma^(q_{\mathsf{H}^2}) = \big(q_\mathsf{O}, \mathsf{T} \big) ~, \qquad \sigma^(q_{\mathsf{T}^2}) = \big(q_\mathsf{O}, \mathsf{H} \big) ~. \end{align} \begin{figure}\label{fig:tree-ocs-automaton} \end{figure} We find that using this automaton to select elements in different rounds is a $(\sqrt{2}-1)$-OCS in the special case. Readers will find the proof of a stronger claim in Subsection~\ref{sec:forest-ocs}. This is strictly better than our best effort using the matching-based approach. More importantly, it generalizes to arbitrary online selection instances using the techniques in the rest of the section. \subsubsection{Automata-based Approach} This subsection outlines how to generalize the automata-based approach to general online selection instances and obtain an improvement over the $0.109$-OCS of \citet{FahrbachHTZ:FOCS:2020}. \begin{theorem} \label{thm:ocs} There is a polynomial-time $0.167$-OCS for the $2$-way online selection problem. \end{theorem} We next explain the ingredients and how to combine them to prove Theorem~\ref{thm:ocs}. The sequel subsections will substantiate them, with the proofs of some lemmas deferred to Appendix~\ref{app:ocs}. A main challenge in generalizing the automata-based approach to general instances is deciding from which in-neighbor each node shall inherit the state of the automata. In other words, we need to select an in-arc for each node to form a directed binary forest. \footnote{That is, each node has at most one in-arc from its parent, and at most two out-arcs to its children. The latter is true for the ex-ante dependence graph itself, and therefore also for all its subgraphs.} We find that the naÃ¯ve approach of independently and randomly selecting an in-arc for each node does not work unless the instance satisfies additional properties (see Subsection~\ref{sec:forest-constructor-good-instances}), because we need the directed binary forest to satisfy another property defined below. \begin{definition}[Good Forest] A \emph{good forest} $G^\text{\rm forest} = (V, E^\text{\rm forest})$ with respect to an online selection instance is a subgraph of the ex-ante dependence graph $G^\text{\rm ex-ante} = (V, E^\text{\rm ex-ante})$ such that: \begin{enumerate} \item $G^\text{\rm forest} = (V, E^\text{\rm forest})$ is a directed binary forest; \item For any node $p$ with two children $c$ and $c'$ in $G^\text{\rm forest}$, the corresponding rounds have no common element, i.e., $\mathcal{E}^p \cap \mathcal{E}^c \cap \mathcal{E}^{c'} = \emptyset$. \end{enumerate} \end{definition} In the following definitions, for any subset of nodes $U \subseteq V$, let $E_U^\text{\rm forest}$ denote the subset of arcs induced by $U$ in the forest $G^\text{\rm forest}$. Further for any element $e$ and any subset of nodes $U \subseteq V$ involving element $e$, let $E_{U, e}^\text{\rm ex-ante}$ denote the subset of arcs induced by $U$ and with subscript $e$: \[ E_{U, e}^\text{\rm ex-ante} \defeq \Big\{ (t, t')_e \in E^\text{\rm ex-ante} ~:~ t \in U ;~ t' \in U \Big\} ~. \] \begin{definition}[Forest Constructor] A \emph{forest constructor} takes an online selection instance as input and returns good forest $G^\text{\rm forest} = (V, E^\text{\rm forest})$. On receiving the elements $\mathcal{E}^t$ of round $t$, it immediately decides whether each in-arc of $t$ belongs to $E^\text{\rm forest}$. It is an $\alpha$-forest constructor if for any element $e$, any subset of nodes $U \subseteq V$ involving $e$, and any $\beta \in [0, 1]$: \begin{equation} \label{eqn:alpha-forest-constructor} \mathbf{E} \big(1-\beta\big)^{\|E^\text{\rm forest}_U\|} \le \big(1-\alpha\beta\big)^{\|E^\text{\rm ex-ante}_{U, e}\|} ~. \end{equation} The expectation is over the randomness of the forest constructor. \end{definition} The next lemma is our main result regarding forest constructors. Subsection~\ref{sec:forest-constructor} presents the algorithm that proves this lemma. \begin{lemma} \label{lem:forest-constructor} There is a polynomial-time $0.404$-forest constructor. \end{lemma} \begin{definition}[Forest OCS] A \emph{forest OCS} takes both an online selection instance and a good forest $G^\text{\rm forest} = (V, E^\text{\rm forest})$ as input. At each round $t$, it observes the elements $\mathcal{E}^t$ in the round and whether each in-arc of $t$ is in $E^\text{\rm forest}$, and then selects an element from $\mathcal{E}^t$. It is a $\beta$-forest OCS if for any element $e$ and any subset of nodes $U \subseteq V$ involving $e$, the probability that $e$ is never selected in the corresponding rounds is at most: \[ 2^{-\|U\|} \big(1-\beta\big)^{\|E^\text{\rm forest}_U\|} ~. \] \end{definition} Our main result regarding forest OCS is the next lemma, whose proof is in Subsection~\ref{sec:forest-ocs}. \begin{lemma} \label{lem:forest-ocs} There is a polynomial-time $(\sqrt{2}-1)$-forest OCS. \end{lemma} The next lemma combines the two ingredients to get an OCS, and implies Theorem~\ref{thm:ocs} as a corollary using Lemmas~\ref{lem:forest-constructor} and \ref{lem:forest-ocs}. \begin{lemma} Suppose that there is a polynomial-time $\alpha$-forest constructor and a polynomial-time $\beta$-forest OCS. Together they form a polynomial-time $\alpha \beta$-OCS. \end{lemma} \begin{proof} The OCS combines the $\alpha$-forest constructor and the $\beta$-forest OCS as follows. On receiving the elements $\mathcal{E}^t$ of a round $t$, it calls the forest constructor to determine whether each in-arc of $t$ is in $E^\text{\rm forest}$. Then, it puts this information together with the elements $\mathcal{E}^t$ and calls the forest OCS to select an element from $\mathcal{E}^t$. For any element $e$, and any disjoint consecutive subsequences of the rounds involving $e$, let $k_1, k_2, \dots, k_m$ be the lengths of these subsequences, and let $U$ be the subset of nodes that correspond to these rounds. By the guarantee of the $\alpha$-forest constructor and the $\beta$-forest OCS, the probability that element $e$ is never selected in these rounds is at most: \[ \mathbf{E} \Big[ 2^{-\|U\|} \big( 1-\beta \big)^{\|E^\text{\rm forest}_U\|} \Big] \le 2^{-\|U\|} \big( 1-\alpha\beta \big)^{\|E^\text{\rm ex-ante}_{U,e}\|} ~. \] The lemma then follows by $\|U\| = \sum_{\ell=1}^m k_\ell$ and $\|E^\text{\rm ex-ante}_{U,e}\| = \sum_{\ell=1}^m \big( k_\ell-1 \big)$. \end{proof} \subsection{Forest Constructor} \label{sec:forest-constructor} \subsubsection{Warm-up: Good Online Selection Instances} \label{sec:forest-constructor-good-instances} We say that an online selection instance is good if its ex-ante graph satisfies the second requirement of good forests. That is, for any node $p$ with two out-neighbors $c$ and $c'$ in $G^\text{\rm ex-ante}$, the corresponding rounds have no common element, i.e., $\mathcal{E}^p \cap \mathcal{E}^c \cap \mathcal{E}^{c'} = \emptyset$. Such good instances admit a simple forest constructor that for each node keeps one of its in-arcs independently and uniformly at random (Algorithm~\ref{alg:forest-constructor-good-instance}). The simple forest constructor and its analysis are instructive, and motivate the forest constructor for general instances, so we include them as a warm-up. \begin{algorithm}[t] \caption{$\frac{1}{2}$-Forest constructor for good online selection instances} \label{alg:forest-constructor-good-instance} \begin{algorithmic} \State \textbf{For each round $t$:} (suppose that $\mathcal{E}^t=\{e_1,e_2\}$) \begin{enumerate} \item For $i \in \{1, 2\}$, let $t_i$ be the most recent round that involves $e_i$ (if exists). \item Draw $j \in \{1, 2\}$ uniformly at random, and include arc $(t_j, t)_{e_j}$ into $E^\text{\rm forest}$ (if $t_j$ is defined). \end{enumerate} \end{algorithmic} \end{algorithm} \begin{lemma} \label{lem:forest-constructor-warmup} Algorithm~\ref{alg:forest-constructor-good-instance} is a $\frac{1}{2}$-forest constructor for good online selection instances. \end{lemma} \begin{proof} For any arc $a \in E^\text{\rm ex-ante}_{U,e}$, let $X_a \in \{0, 1\}$ be the indicator of whether arc $a$ is included into $E^\text{\rm forest}$. Since every arc $a \in E^\text{\rm ex-ante}_{U,e}$ with $X_a = 1$ belongs to $E^\text{\rm forest}_U$, we have: \[ \big(1-\beta\big)^{\|E^\text{\rm forest}_U\|} \le \big(1-\beta\big)^{\sum_{a \in E^\text{\rm ex-ante}_{U,e}} X_a} ~. \] Further, since $E^\text{\rm ex-ante}_{U,e}$ by definition is a subset of arcs connecting neighboring appearances of element $e$, they have distinct in- and out-nodes. Hence, by the definition of Algorithm~\ref{alg:forest-constructor-good-instance}, $X_a$'s are independently and uniformly distributed over $\{0, 1\}$ for all arcs $a \in E^\text{\rm ex-ante}_{U,e}$. We get that \[ \mathbf{E} \big(1-\beta\big)^{\|E^\text{\rm forest}_U\|} \le \prod_{a \in E^\text{\rm ex-ante}_{U,e}} \mathbf{E} \big(1-\beta\big)^{X_a} = \prod_{a \in E^\text{\rm ex-ante}_{U,e}} \Big(1-\frac{\beta}{2} \Big) = \Big(1-\frac{\beta}{2} \Big)^{\|E^\text{\rm ex-ante}_{U,e}\|} ~. \] \end{proof} For good instances, the second property of good forests always holds regardless of which arcs the forest constructor selects. To satisfy the first property, i.e., to form a directed binary forest, consider a partition of arcs according to their destinations, into groups with one or two arcs each. Then, selecting a directed binary forest is equivalent to selecting at most one arc from each group. The above forest constructor indeed independently and randomly selects an arc from each group. Finally, the independent selections of arcs in $E^\text{\rm ex-ante}_{U,e}$ help show that it is a $\frac{1}{2}$-forest constructor. \subsubsection{General Online Selection Instances, Pseudo-paths, and Pseudo-matchings} \begin{figure} \caption{Partition of arcs into pseudo-paths, and bolded selected arcs as a pseudo-matching} \label{fig:ocs-forest-constructor-pseudo-path} \caption{Selected arcs (bolded) form a good forest in ex-ante dependence graph} \label{ocs:possible-restricted-dependence-forest} \caption{An illustrative example of the forest constructor for general instances} \label{ocs:arc-picking-dependence-graph} \end{figure} For general instances, a forest constructor needs to also ensure the second property of good forests. In other words, for any pairs of arcs $a$ and $a'$ with the same origin such that the rounds corresponding to their incident nodes share a common element, a forest constructor must not select both $a$ and $a'$ into the forest. For example, consider an instance with the ex-ante dependence graph in Figure~\ref{fig:exante-dependence-graph}. The directed forest property requires that, e.g., arcs $(1,3)_a$ and $(2,3)_b$ cannot be both selected, arcs $(1,4)_c$ and $(3,4)_a$ cannot be both selected, etc., due to having the same destinations. The second property of good forests further requires that, e.g., arcs $(1,3)_a$ and $(1,4)_c$ cannot be both selected. Dropping their directions, the above arcs $(3,4)_a, (1,4)_c, (1,3)_a, (2,3)_b$ form an undirected path $3-4-1-3-2$. More importantly, we can succinctly describe the aforementioned requirements of good forests as not selecting neighboring arcs with respect to the path. Driven by this observation, we define pseudo-paths and the pseudo-matchings below. \begin{definition}[Pseudo-path] Given any online selection instance and its ex-ante dependence graph, a \emph{pseudo-path} is a maximal ordered subset of arcs $P = \big( (t_i, t_i')_{e_i} \big)_{1 \le i \le \ell}$ such that for any $1 \le i < \ell$: \begin{itemize} \item Either $t_i' = t_{i+1}'$, i.e., the $i$-th and $(i+1)$-th arcs in $P$ have the same destination; \item Or $t_i = t_{i+1}$, i.e., the $i$-th and $(i+1)$-th arcs in $P$ have the same origin, and rounds $t_i = t_{i+1}$, $t_i'$, and $t_{i+1}'$ have a common element. \end{itemize} \end{definition} \begin{lemma} \label{lem:pseudo-path} The pseudo-paths partition the arcs of the ex-ante graph. \footnote{We consider two pseudo-paths with the same subset of arcs but in opposite orders as the same pseudo-path.} \end{lemma} Figure~\ref{fig:ocs-forest-constructor-pseudo-path} shows the partition of arcs into pseudo-paths in the aforementioned example whose ex-ante dependence graph is Figure~\ref{fig:exante-dependence-graph}. We defer other structural properties about pseudo-paths to sequel subsections where we use them to design and analyze the forest constructor. \begin{definition}[Pseudo-matching] For any pseudo-path $P = \big( (t_i, t_i')_{e_i} \big)_{1 \le i \le \ell}$, a subset of its arcs $M$ is a \emph{pseudo-matching} if it has no adjacent arcs with respect to $P$, i.e., for any $1 \le i < \ell$, either $(t_i, t_i')_{e_i} \notin M$ or $(t_{i+1}, t_{i+1}')_{e_{i+1}} \notin M$. \end{definition} We remark that a pseudo-matching may not be a matching of the ex-ante dependence graph. For example, arcs $(3,4)_a$ and $(2,3)_b$ form a pseudo-matching of the left-most pseudo-path in Figure~\ref{fig:ocs-forest-constructor-pseudo-path} even though they share node $3$. \begin{lemma} \label{lem:pseudo-matching} A subgraph of the ex-ante dependence graph is a good forest if and only if it is a union of pseudo-matchings, one for each pseudo-path. \end{lemma} Therefore, a forest constructor needs to pick a pseudo-matching from each pseudo-path. Further, it must do so in an online fashion. On observing the elements $\mathcal{E}^t$ of round $t$, it either appends $t$'s in-arcs to an existing pseudo-path and lets them start a new pseudo-path on their own, according to the definition of pseudo-paths. It also immediately decides if to include each in-arc into the pseudo-matching. To make it an $\alpha$-forest constructor for the largest possible $\alpha$, we want to select as many arcs into the pseudo-matchings as possible, and at the same time to keep the selections sufficiently independent so that an analysis similar to Lemma~\ref{lem:forest-constructor-warmup} applies. The latter refutes selecting either all odd arcs or all even arcs from each pseudo-path with equal probability. \subsubsection{Forest Constructor for General Instances} \begin{figure} \caption{Automaton $\sigma^+$ for the positive end} \caption{Automaton $\sigma^-$ for the negative end} \caption{ The probabilistic automata in our forest constructor (Algorithm~\ref{alg:forest-constructor}). The transitions are labeled by the binary decisions and the probabilities of the transitions. } \label{fig:ocs-forest-constructor-automata} \end{figure} To explain our forest constructor for general instances, we need a structural lemma about the arrival order of arcs in any pseudo-path. We notice that arcs usually arrive in pairs since the in-arcs of $t$ both arrive in round $t$. We will artificially break ties to be consistent with the lemma. \begin{lemma} \label{lem:pseudo-path-arrival} For any pseudo-path $P = \big( (t_i, t_i')_{e_i} \big)_{1 \le i \le \ell}$ and any $0 \le t \le T$, the subset of arcs that arrive in the first $t$ rounds is a sub-pseudo-path, i.e., either it is an empty set, or there exists $1 \le i_{\min} \le i_{\max} \le \ell$ such that the arrived arcs are $\big( (t_i, t_i')_{e_i} \big)_{i_{\min} \le i \le i_{\max}}$. \end{lemma} That is, after the first arc of a pseudo-path arrives, the arrival of any arc of the pseudo-path appends to the existing sub-pseudo-path; we never need to merge two pseudo-paths. Our argument lets $i_0$ denote the index of the earliest arc. We say that the arcs with indices $i \ge i_0$ are on the positive end, and those with indices $i < i_0$ are on the negative end. \footnote{ The algorithm does not need to know the index $i_0$ upfront. Instead, it could let the earliest arc have index $0$; arcs on the positive ends have indices $0, 1, 2$, etc., and those on the negative end have indices $-1, -2, -3$, etc. Nonetheless, the choice of indices in the main text admits cleaner notations in the analysis. } Finally, changing the roles of the positive and negative ends will not affect our conclusion, so we will without loss of generality let the second earliest arc of any pseudo-path be on the positive end. The forest constructor uses a probabilistic automaton $\sigma^+$ and its inverse $\sigma^-$. The automata have states $q_\no$, $q_{\no^2}$, and $q_\yes$. Intuitively, state $q_\no$ means that automaton $\sigma^+$ leaves the last arc unmatched, but matches the arc before that; state $q_{\no^2}$ means that automaton $\sigma^+$ leaves the last two arcs unmatched; and state $q_\yes$ means that automaton $\sigma^+$ matches the last arc. The transition functions, which we denote also as $\sigma^+$ and $\sigma^-$ abusing notations, take a state as input and returns the next state and also a binary decision. They are parameterized by $p \in [0, 1]$, the transition probability from $q_\no$ to $q_\yes$ in automaton $\sigma^+$. We will let $p = 0.6616$ in the analysis to optimize the result. Formally, the transition functions are (see also Figure~\ref{fig:ocs-forest-constructor-automata}): \begin{align} \sigma^+ \big( q_\no \big) & = \begin{cases} (q_\no, \mathsf{M}) & \text{w.p.\ $p$} \\ (q_{\no^2}, \mathsf{U}) & \text{w.p.\ $1-p$} \end{cases} ~, & \sigma^+ \big( q_{\no^2} \big) & = (q_\yes, \mathsf{M}) ~, & \sigma^+ \big( q_\yes \big) & = (q_\no, \mathsf{U}) ~; & \\ \sigma^- \big( q_\no \big) & = (q_\yes, \mathsf{U}) ~, & \sigma^- \big( q_{\no^2} \big) & = (q_\no, \mathsf{U}) ~, & \sigma^- \big( q_\yes \big) & = \begin{cases} (q_\no, \mathsf{M}) & \text{w.p.\ $p$} \\ (q_{\no^2}, \mathsf{M}) & \text{w.p.\ $1-p$} \end{cases} ~. \end{align} For each pseudo-path, our forest constructor draws an initial state from the common stationary distribution of the automata. Then, when an arc arrives on the positive end, it calls $\sigma^+$ to update the state and to decides whether to include the arc into the pseudo-matching; similarly, when an arc arrives on the negative end, it calls $\sigma^-$. See Algorithm~\ref{alg:forest-constructor}. \begin{algorithm}[t] \caption{$0.404$-Forest constructor for general instances (when $p = 0.6616$)} \label{alg:forest-constructor} \begin{algorithmic} \State \textbf{State variables:} (for each pseudo-path $P$) \begin{itemize} \item $q^+, q^- \in \{ q_\no, q_{\no^2}, q_\yes \}$ of automata $\sigma^+, \sigma^-$ respectively. \item Initialize (when the first arc in the pseudo-path arrives): \[ q^+ = q^-= \begin{cases} q_\no & \text{w.p.\ $\frac{1}{3-p}$;} \\[1ex] q_{\no^2} & \text{w.p.\ $\frac{1-p}{3-p}$;} \\[1ex] q_\yes & \text{w.p.\ $\frac{1}{3-p}$.} \end{cases} \] \end{itemize} \State \textbf{For each arc:} (of pseudo-path $P$) \begin{enumerate} \item Let $\tau = +$ if the arc is on the positive end, and $-$ otherwise. \item Let $(q^\tau, d) = \sigma^\tau(state^\tau)$. \item Include the arc into the pseudo-matching and thus the forest if $d = \mathsf{M}$. \end{enumerate} \end{algorithmic} \end{algorithm} \subsubsection{Properties of the Automata} \begin{lemma} \label{lem:forest-constructor-automata-stationary} The stationary distribution of the states of $\sigma^+$ and $\sigma^-$ is: \[ \vec{\pi} = \big( \pi_\mathsf{U}, \pi_{\mathsf{U}^2}, \pi_\mathsf{M} \big) = \Big( \frac{1}{3-p}, \frac{1-p}{3-p}, \frac{1}{3-p} \Big) ~. \] \end{lemma} The next lemma formalizes the claim that $\sigma^-$ is the inverse of $\sigma^+$. \begin{lemma} \label{lem:forest-constructor-automata-inverse} Consider two sequences of random variables: \begin{enumerate} \item Sample $q^0$ from the stationary distribution $\vec{\pi}$. Then recursively let: \[ \hspace{8pt} \big( q^i, d^i \big) = \sigma^+ \big( q^{i-1} \big) ~, \hspace{92pt} 1 \le i \le \ell ~. \] \item Sample $\hat{q}^{i_0-1}$ from the stationary distribution $\vec{\pi}$. Then recursively let: \begin{align} \big( \hat{q}^i, \hat{d}^i \big) & = \sigma^+ \big( \hat{q}^{i-1} \big) ~, && i_0 \le i \le \ell ~; \\ \big( \hat{q}^{i-1}, \hat{d}^i \big) & = \sigma^- \big( \hat{q}^i \big) ~, && 1 \le i < i_0 ~. \end{align} \end{enumerate} The two sequences are identically distributed. \end{lemma} The second sequence corresponds to the states and decisions of automata $\sigma^+, \sigma^-$ in Algorithm~\ref{alg:forest-constructor}: $\hat{q}^{i_0-1}$ is the common initial state of $\sigma^+$ and $\sigma^-$; $\hat{q}^i$ and $d^i$ for $i \ge i_0$ are the states and decisions on the positive end; $\hat{q}^i$ and $d^i$ for $i < i_0$ are the states and decisions on the negative end. Lemma~\ref{lem:forest-constructor-automata-inverse} allows us to analyze each pseudo-path as if the arcs' arrival order is from one end to the other. Next we adopt the viewpoint of selecting arcs with only automaton $\sigma^+$, and develop several properties of the corresponding sequence $\big( q^0, d^1, q^1, \dots, d^\ell, q^\ell \big)$. We start with three lemmas that follow by the definition of $\sigma^+$. \begin{lemma} \label{lem:forest-constructor-automata-matched-state} For any $1 \le i \le \ell$, $q^i = q_\yes$ if and only if $d^i = \mathsf{M}$. \end{lemma} \begin{lemma} \label{lem:forest-constructor-automata-marginal} For any $1 \le i \le \ell$: \[ \mathbf{Pr} \big[ d^i = \mathsf{M} \big] = \frac{1}{3-p} ~. \] \end{lemma} \begin{lemma} \label{lem:forest-constructor-automata-matched-in-three} For any $1 \le i \le \ell-2$, at least one of $d^i$, $d^{i+1}$, and $d^{i+2}$ equals $\mathsf{M}$. \end{lemma} Lemma~\ref{lem:forest-constructor-automata-matched-state} asserts that every time $\sigma^+$ selects an arc into the pseudo-matching (and thus the forest), it resets to state $q_\yes$. Combining with Lemma~\ref{lem:forest-constructor-automata-stationary}, we get the marginal selection probability in Lemma~\ref{lem:forest-constructor-automata-marginal}. Lemma~\ref{lem:forest-constructor-automata-matched-in-three} further claims that it resets to state $q_\yes$ at least once every three rounds. Hence, we focus on how the probabilistic automaton transitions starting from state $q_\yes$, and in particular how likely the $i$-th arc after that would be selected, which in turns characterizes the transition from the other two states. These probabilities are characterized by a recurrence: \begin{equation} \label{eqn:forest-constructor-automata-selection-prob} f_i = \begin{cases} 1 & i=0 ~; \\ 0 & i=1 ~; \\ p & i=2 ~; \\ p f_{i-2} + (1-p) f_{i-3} & i \ge 3 ~. \end{cases} \end{equation} \begin{lemma} \label{lem:forest-constructor-automata-selection-prob} For any $i \le j$: \[ \mathbf{Pr} \big[ d^j = \mathsf{M} \mid q^i = q_\yes \big] = \mathbf{Pr} \big[ d^j = \mathsf{M} \mid d^{i} = \mathsf{M} \big] = f_{j-i} ~. \] Further, for any $i \le j-1$: \begin{align} \mathbf{Pr} \big[ d^j = \mathsf{M} \mid q^i = q_{\no^2} \big] & = f_{j-i-1} ~, \\ \mathbf{Pr} \big[ d^j = \mathsf{M} \mid q^i = q_\no \big] & = f_{j-i+1} ~. \end{align} \end{lemma} \begin{lemma} \label{lem:forest-constructor-automata-selection-prob-bound} Suppose that $\frac{\sqrt{5}-1}{2}\leq p\leq \frac{2}{3}$. Then: \begin{align} f_i & \ge 1-p ~, && \forall i \ge 2 ~; \\ f_i & \ge p^3+\big(1-p\big)^2 && \forall i \ge 4 ~. \end{align} \end{lemma} \begin{comment} \begin{lemma} \label{ocs:quasi-maximal matching} In Algorithm \ref{ocs:arc-picking-alg}, for any three consecutive arcs with positive sign $+$, at least one of them is picked. \end{lemma} \begin{corollary} \label{ocs:picking-independence} Suppose bits $X_k=1$ if and only if arc $k$ is matched in Algorithm \ref{ocs:arc-picking-alg}. For any $i>j\geq 0$ and any $x_0,\cdots,x_{j}\in \{0,1\}$ such that $\mathbf{Pr}[X_j=x_j,\cdots, X_0=x_0]>0$, \[ Pr[X_i=1\|X_j=x_j,\cdots,X_0=x_0]=\begin{cases} f_{i-j-1} & x_j=1\\ f_{i-j} & x_j=0,x_{j-1}=1\\ f_{i-j-2} & x_j=0,x_{j-1}=0 \end{cases}. \] Specifically, we define $f_{-1}=1$. \end{corollary} \end{comment} \subsubsection{Analysis of Forest Constructor for General Instances: Proof of Lemma~\ref{lem:forest-constructor}} This subsection proves Lemma~\ref{lem:forest-constructor} by showing that Algorithm~\ref{alg:forest-constructor} with $p = 0.6616$ is a $0.404$-forest constructor. Below summarizes some properties that either follow by the definition of Algorithm~\ref{alg:forest-constructor}, or have been established in the previous subsections: \begin{itemize} \item Constructing a good forest is the same as selecting a pseudo-matching from each pseudo-path. \hspace{\fill} (Lemma~\ref{lem:pseudo-matching}) \item The selections of arcs in different pseudo-paths are independent. \hspace{\fill} (Definition of Algorithm~\ref{alg:forest-constructor}) \item The selections of arcs on a pseudo-path is equivalent to sampling a state from the stationary distribution $\vec{\pi}$, and applying $\sigma^+$ to decide for each arc from one end to the other. \hspace{\fill} (Lemma~\ref{lem:forest-constructor-automata-inverse}) \item Automaton $\sigma^+$ selects an arc and resets to state $q_\yes$ at least once every three rounds.\\ \hspace{\fill} (Lemmas~\ref{lem:forest-constructor-automata-matched-state} and \ref{lem:forest-constructor-automata-matched-in-three}) \item The probability of selecting the $i$-th arc after any state is characterized by $f_i$, which can be lower bounded. \hspace{\fill} (Lemmas~\ref{lem:forest-constructor-automata-selection-prob} and \ref{lem:forest-constructor-automata-selection-prob-bound}) \end{itemize} \paragraph{Proving that Algorithm~\ref{alg:forest-constructor} Constructs a Good Forest.} Consider an arbitrary pseudo-path. By the definition of $\sigma^+$, it resets its state to $q_\yes$ when the decision is $\mathsf{M}$ (Lemma~\ref{lem:pseudo-matching}), i.e., when an arc is selected, after which the next decision will be $\mathsf{U}$. Therefore, the arcs selected from each pseudo-path are a pseudo-matching. By Lemma~\ref{lem:pseudo-matching} this is a good forest. \paragraph{Proof of Equation~\eqref{eqn:alpha-forest-constructor}.} To show the guarantee of a $\alpha$-forest constructor, which we restate below: \[ \forall 0 \le \beta \le 1 ~: \qquad \mathbf{E} \big(1-\beta\big)^{\|E^\text{\rm forest}_U\|} \le \big(1-\alpha\beta\big)^{\|E^\text{\rm ex-ante}_{U, e}\|} ~, \tag{Eqn.~\eqref{eqn:alpha-forest-constructor} restated} \] it suffices to consider each pseudo-path $P$ separately and to show that: \begin{equation} \label{eqn:alpha-forest-constructor-by-path} \forall 0 \le \beta \le 1 ~: \qquad \mathbf{E} \big(1-\beta\big)^{\|E^\text{\rm forest}_U \cap P\|} \le \big(1-\alpha\beta\big)^{\|E^\text{\rm ex-ante}_{U, e} \cap P\|} ~, \hspace{65pt} \end{equation} after which Eqn.~\eqref{eqn:alpha-forest-constructor} follows by taking the product of Eqn.~\eqref{eqn:alpha-forest-constructor-by-path} over all pseudo-paths, and by the independence of arc selections in different pseudo-paths. The rest of the argument considers an arbitrary pseudo-path $P$ and proves Eqn.~\eqref{eqn:alpha-forest-constructor-by-path}. We start by establishing the last structural lemma about pseudo-paths, characterizing the subset of arcs that could contribute to the inequality by being counted in $E^\text{\rm forest}_U$. \begin{lemma} \label{lem:pseudo-path-induced} For any element $e$, and any subset of nodes $U \subseteq V$ involving $e$, there is a subset of arcs in $P$ with both nodes inside $U$ such that: \begin{enumerate} \item It is a superset of $E^{\text{ex-ante}}_{U,e}\cap P$; \item It is the union of odd-length sub-pseudo-paths; \item Any two of these sub-pseudo-paths are at least $3$ arcs apart; and \item Each sub-pseudo-path alternates between arcs with subscript $e$, i.e., arcs that also contribute to the right-hand-side of Eqn.~\eqref{eqn:alpha-forest-constructor-by-path}, and arcs with other subscripts, i.e., arcs that only contribute to the left-hand-side. The arcs on the two ends have subscript $e$. \end{enumerate} \end{lemma} Given Lemma~\ref{lem:pseudo-path-induced}, we may assume that the subset of arcs in $P$ with both nodes involving $e$ are sub-pseudo-paths starting from arc indices $i_1, i_2, \dots, i_m$ and lengths $2k_1+1, 2k_2+1, \dots, 2k_m+1$. Let $\mathcal{I}_j$ denote the set of indices of the $j$-th sub-pseudo-path: \[ \mathcal{I}_j = \big\{ i_j, i_j+1, \dots, i_j+2k_j \big\} ~. \] Let $\mathcal{I} = \cup_{j=1}^m \mathcal{I}_j$ denote the set of arc indices in these pseudo-paths. For each arc $i \in \mathcal{I}$, consider the indicator of if the $i$-th arc on the pseudo-path is selected into the pseudo-matching: \[ X_i \defeq \mathbf{1} \big( d^i = \mathsf{M} \big) ~. \] Since any arc in $\mathcal{I}$ have both nodes in $U$ by definition, we have: \[ \|E^\text{\rm forest}_U \cap P\| \ge \sum_{i \in \mathcal{I}} X_i ~. \] Further, there are $k_j+1$ arcs with subscripts $e$ in the $j$-th sub-pseudo-path (i.e., contributing to $E^\text{\rm ex-ante}_{U,e} \cap P$) by Lemma~\ref{lem:pseudo-path-induced}. Hence, to prove Eqn.~\eqref{eqn:alpha-forest-constructor-by-path} it is sufficient to show: \[ \mathbf{E} \big(1-\beta\big)^{\sum_{i \in \mathcal{I}} X_i} \le \big(1-\alpha\beta\big)^{\sum_{j=1}^m (k_j+1)} ~. \] We next argue that it suffices to consider the case when all sub-pseudo-paths have unit lengths, because we can reduce the general case to it. Suppose that the $j$-th pseudo-path has length $2k_j+1 > 1$. When $2k_j+1 = 3$ or $2k_j+1 \ge 7$, by Lemma~\ref{lem:forest-constructor-automata-matched-in-three} we have: \[ \sum_{i \in \mathcal{I}_j} X_i \ge \Big\lfloor \frac{2k_j+1}{3} \Big\rfloor \ge \frac{k_j+1}{2} ~. \] Hence, regardless of the realization of randomness in the forest constructor (Algorithm~\ref{alg:forest-constructor}), for $\alpha = 0.404$ and for any $\beta \in [0, 1]$ we always have: \[ \big(1-\beta\big)^{\sum_{i \in \mathcal{I}_j} X_i} \le \big(1-\beta\big)^{\frac{k_j+1}{2}} \le \Big( 1 - \frac{\beta}{2} \Big)^{k_j+1} \le \big( 1 - \alpha \beta \big)^{k_j+1} ~. \] In other words, we can without loss of generality remove the $j$-th sub-pseudo-path and prove Eqn.~\eqref{eqn:alpha-forest-constructor-by-path} for the remaining instance. When $2 k_j+1 = 5$, i.e., $k_j = 2$, also by Lemma~\ref{lem:forest-constructor-automata-matched-in-three} we have: \[ \sum_{i \in \mathcal{I}_j, i \ne i_j} X_i \ge 1 ~. \] Hence, regardless of the realization of randomness in the forest constructor (Algorithm~\ref{alg:forest-constructor}), for $\alpha = 0.404$ and for any $\beta \in [0, 1]$ we always have: \[ \big(1-\beta\big)^{\sum_{i \in \mathcal{I}_j, i \ne i_j} X_i} \le \big(1-\beta\big) \le \Big( 1 - \frac{\beta}{2} \Big)^2 \le \big( 1 - \alpha \beta \big)^2 ~. \] That is, we can without loss of generality remove the arcs \emph{other than $i_j$} from $j$-th sub-pseudo-path and prove Eqn.~\eqref{eqn:alpha-forest-constructor-by-path} for the remaining instance. Finally, consider Eqn.~\eqref{eqn:alpha-forest-constructor-by-path} when all sub-pseudo-paths have unit lengths. In other words, for a subset of indices $\mathcal{I} = \{ i_1, i_2, \dots, i_m \}$ such that any two indices differ by at least $4$ (Lemma~\ref{lem:pseudo-path-induced}), we shall prove that: \[ \forall 0 \le \beta \le 1 ~: \qquad \mathbf{E} \big(1-\beta\big)^{\sum_{i \in \mathcal{I}} X_i} \le \big(1-\alpha\beta\big)^m ~. \] Our proof is an induction on $m$. The base case when $m = 1$ follows by: \begin{align} \mathbf{E} \big(1-\beta\big)^{X_{i_1}} & \le \frac{1}{3-p} \big(1-\beta\big) + \frac{2-p}{3-p} \tag{Lemma~\ref{lem:forest-constructor-automata-marginal}} \\ & = 1 - \frac{1}{3-p} \beta \\ & \le 1 - \alpha \beta ~. \tag{$\alpha=0.404$, $p=0.6616$} \end{align} Suppose that the inequality holds for up to $m-1$ indices. We next prove it for $m$ indices. Suppose without loss of generality that $i_1 < i_2 < \dots < i_m$. By the inductive hypothesis: \[ \mathbf{E} \big(1-\beta\big)^{\sum_{i \in \mathcal{I}, i \ne i_m} X_i} \le \big(1-\alpha\beta\big)^{m-1} ~. \] It suffices to prove that \emph{for any realized $X_{i_1}, X_{i_2}, \dots, X_{i_{m-1}}$}: \[ \mathbf{E} \big[ X_{i_m} \mid X_{i_1}, \dots, X_{i_{m-1}} \big] \ge \alpha ~, \] as it would imply that: \[ \mathbf{E} \Big[ \big(1-\beta\big)^{X_{i_m}} \mid X_{i_1}, \dots, X_{i_{m-1}} = 1 \Big] \le \big(1-\beta\big) \alpha + \big(1-\alpha\big) = 1-\alpha\beta ~. \] For any realization such that $X_{i_{m-1}} = 1$: \begin{align} \mathbf{E} \big[X_{i_m} \mid X_{i_1}, \dots, X_{i_{m-1}} = 1 \big] & = f_{i_m-i_{m-1}} \tag{Lemmas~\ref{lem:forest-constructor-automata-matched-state}, \ref{lem:forest-constructor-automata-selection-prob}} \\[.5ex] & \ge p^3+(1-p)^2 \tag{$i_m-i_{m-1} \ge 4$ and Lemma~\ref{lem:forest-constructor-automata-selection-prob-bound}} \\[1ex] & \ge \alpha ~. \tag{$\alpha = 0.404$, $p = 0.6616$} \end{align} For a realization such that $X_{i_{m-1}} = 0$, the state after processing arc $i_{m-1}$ could be $q_\no$ or $q_{\no^2}$. We first argue that it is the former at least a $\frac{1}{1+p}$ fraction of the time. \begin{lemma} \label{lem:ocs-unmatched-ready-ratio} For any realization such that $X_{i_{m-1}} = 0$: \[ \mathbf{Pr} \Big[ q^{i_{m-1}} = q_\no \mid X_{i_1}, \dots, X_{i_{m-1}} = 0 \Big] \ge \frac{1}{1+p} ~. \] \end{lemma} Given Lemma~\ref{lem:ocs-unmatched-ready-ratio}, we can lower bound $\mathbf{E} \big[X_{i_m} \mid X_{i_1}, \dots, X_{i_{m-1}} = 0 \big]$ by: \[ \frac{1}{1+p} \mathbf{E} \big[X_{i_m} \mid q^{i_{m-1}} = q_\no \big] + \frac{p}{1+p} \min \Big\{ \mathbf{E} \big[X_{i_m} \mid q^{i_{m-1}} = q_\no \big], \mathbf{E} \big[X_{i_m} \mid q^{i_{m-1}} = q_{\no^2} \big] \Big\} ~. \] Suppose that $i_m - i_{m-1} = 4$. By Lemma~\ref{lem:forest-constructor-automata-selection-prob-bound}: \begin{align} \mathbf{E} \big[X_{i_m} \mid q^{i_{m-1}} = q_\no \big] & = f_5 = 2p(1-p) ~; \\[1ex] \mathbf{E} \big[X_{i_m} \mid q^{i_{m-1}} = q_{\no^2} \big] & = f_3 = 1-p ~. \end{align} Observe that $2p(1-p) > 1-p$ for $p = 0.6616$. We get that: \begin{align} \mathbf{E} \big[X_{i_m} \mid X_{i_1}, \dots, X_{i_{m-1}} = 0 \big] & \ge \frac{1}{1+p} \cdot 2p(1-p) + \frac{p}{1+p} \cdot (1-p) \\[.5ex] & = \frac{3p(1-p)}{1+p} \ge \alpha ~. \tag{$\alpha = 0.404$, $p = 0.6616$} \end{align} Otherwise, we have that $i_m - i_{m-1} \ge 5$. By Lemma~\ref{lem:forest-constructor-automata-selection-prob-bound}: \begin{align} \mathbf{E} \big[X_{i_m} \mid q^{i_{m-1}} = q_\no \big] & = f_{i_m - i_{m-1} + 1} \ge p^3+\big(1-p\big)^2 ~; \\[1ex] \mathbf{E} \big[X_{i_m} \mid q^{i_{m-1}} = q_{\no^2} \big] & = f_{i_m - i_{m-1} - 1} \ge p^3+\big(1-p\big)^2 ~. \end{align} Hence: \[ \mathbf{E} \big[X_{i_m} \mid X_{i_1}, \dots, X_{i_{m-1}} = 0 \big] \ge p^3+\big(1-p\big)^2 \ge \alpha ~. \tag{$\alpha = 0.404$, $p = 0.6616$} \] \subsection{Forest OCS} \label{sec:forest-ocs} \subsubsection{Algorithm} \begin{algorithm}[t] \caption{$(\sqrt{2}-1)$-Forest OCS} \label{alg:forest-ocs} \begin{algorithmic} \State \textbf{Input:} \begin{itemize} \item An online selection instance represented by its ex-ante graph $G^\text{\rm ex-ante} = (V, E^\text{\rm ex-ante})$. \item A good forest $G^\text{\rm forest} = (V, E^\text{\rm forest})$. \end{itemize} \State \textbf{State variables:} \begin{itemize} \item Label $\ell_t(e) \in \big\{ \mathsf{H}, \mathsf{T} \big\}$ for every round $1 \le t \le T$ and every element $e \in \mathcal{E}^t$. \item State $q^t$ of automaton $\sigma^$ for every node $1 \le t \le T$. \end{itemize} \State \textbf{For each round $t$:} (suppose that $\mathcal{E}^t=\{e_1, e_2\}$) \begin{enumerate} \item If $t$ is the root of a directed binary tree in $G^\text{\rm forest}$, let the labels be $\ell_t(e_1) = \mathsf{H}$, $\ell_t(e_2) = \mathsf{T}$. \label{step:forest-ocs-label-root} \item Otherwise, suppose without loss of generality that $(t', t)_{e_1}$ is the in-arc of node $t$ in $G^\text{\rm forest}$, let $\ell_t(e_1) = \ell_{t'}(e_1)$ and let $\ell_t(e_2)$ be the other label from $\big\{ \mathsf{H}, \mathsf{T} \big\}$. \label{step:forest-ocs-label-other} \item Let $(q^t, \ell) = \sigma^(q^{t'})$ (artificially let $q^{t'} = q_\mathsf{O}$ if $t$ is the root of a directed binary tree). \item Select the element with label $\ell$. \end{enumerate} \end{algorithmic} \end{algorithm} On observing the elements $\mathcal{E}^t$ of each round $t$, the forest OCS labels the elements by head ($\mathsf{H}$) and tail ($\mathsf{T}$). Then, it calls automaton $\sigma^$ from Subsection~\ref{sec:ocs-path} with the state of $t$'s parent in the good forest as input to select a label and to get the state of $t$. Finally, it selects the element whose label is selected by automaton $\sigma^$. See Algorithm~\ref{alg:forest-ocs}. \subsubsection{Analysis: Proof of Lemma~\ref{lem:forest-ocs}} The lemmas in this subsection assume $\beta = \sqrt{2}-1$, which we shall not restate repeatedly. We first establish a structural lemma about the subset of arcs with both nodes involving an element $e$ in any good forest, and about the labels in Algorithm~\ref{alg:forest-ocs}. Recall that $E^\text{\rm forest}_U$ denotes the subset of arcs in forest $G^\text{\rm forest}$ with both nodes in $U$. \begin{lemma} \label{lem:forest-ocs-structural} For any good forest $G^\text{\rm forest} = (V, E^\text{\rm forest})$, any element $e$, and any subset of nodes $U \subseteq V$ involving $e$, $E^\text{\rm forest}_U$ consists of a collection of tree-paths such that: \begin{enumerate} \item There is no arc between any two nodes in distinct tree-paths; and \item Element $e$ has the same label in each tree-path. \end{enumerate} \end{lemma} We next prove Lemma~\ref{lem:forest-ocs} by an induction on the number of tree-paths in $E^\text{\rm forest}_U$. The base case with zero tree-path holds vacuously. Next for some $m \ge 1$ suppose that the lemma holds with at most $m-1$ tree-paths. Consider an arbitrary instance for which $E^\text{\rm forest}_U$ consists of $m$ tree-paths satisfying the properties of Lemma~\ref{lem:forest-ocs-structural}. Let $t_1, t_2, \dots, t_m$ denote the first node of these tree-paths. Let $k_1, k_2, \dots, k_m$ be the lengths. We assume without loss of generality that $t_m$'s height in its directed binary tree is greater than or equal to the height of any other $t_i$ from the same tree. \paragraph{Case 1:} Suppose that $t_m$ is the root of a directed binary tree in $G^\text{\rm forest}$. First by the inductive hypothesis on the sub-instance that removes the tree rooted at $t_m$, the probability of not selecting element $e$ in these tree-paths is at most $2^{-\sum_{i=1}^{m-1} k_i} (1-\beta)^{\sum_{i=1}^{m-1} (k_i-1)}$. Then, \emph{conditioned on any realized randomness on the other tree-paths}, the probability of never selecting $e$ on the tree-path starting with $t_m$ is at most $2^{-k_m} (1-\beta)^{k_m-1}$, because of the second part of Lemma~\ref{lem:forest-ocs-structural} and a special case of the next lemma when $i = 1$. \begin{lemma} \label{lem:forest-ocs-from-origin} For any $k$ consecutive positive integers $i, i+1, \dots, i+k-1$, and any label $\ell \in \{ \mathsf{H},\mathsf{T} \}$, the probability that automaton $\sigma^$, starting from the original state $q_\mathsf{O}$, does not select label $\ell$ in its $i$-th to $(i+k-1)$-th selections is at most $2^{-k}(1-\beta)^{k-1}$. \end{lemma} \paragraph{Case 2:} Suppose that $t_m$ is not the root of any directed binary tree in $G^\text{\rm forest}$, but its sibling in $G^\text{\rm forest}$ (if any) is not one of $t_1, t_2, \dots, t_{m-1}$. By the latter assumption, the fact that $t_m$'s parent is not on the other tree-paths (Lemma~\ref{lem:forest-ocs-structural}, first part), and the assumption that $t_m$ has the largest height compared to other $t_i$ from the same directed binary tree, we get that the nodes rooted from $t_m$'s parent are not on the other tree-paths. Then, by the inductive hypothesis on the sub-instance that removes nodes rooted from \emph{$t_m$'s parent}, the probability of not selecting element $e$ in these tree-paths is at most $2^{-\sum_{i=1}^{m-1} k_i} (1-\beta)^{\sum_{i=1}^{m-1} (k_i-1)}$. Then, \emph{conditioned on any realized randomness on the other tree-paths}, the probability of never selecting $e$ on the tree-path starting with $t_m$ is at most $2^{-k_m} (1-\beta)^{k_m-1}$, because of the second part of Lemma~\ref{lem:forest-ocs-structural} and the next lemma. \begin{lemma} \label{lem:forest-ocs-from-other} For any $k$ consecutive positive integers $i, i+1, \dots, i+k-1$ starting from $i \ge 2$, and any label $\ell \in \{ \mathsf{H},\mathsf{T} \}$, the probability that automaton $\sigma^$, starting from an arbitrary state, does not select label $\ell$ in its $i$-th to $(i+k-1)$-th selections is at most $2^{-k}(1-\beta)^{k-1}$. \end{lemma} \paragraph{Case 3:} Suppose that $t_m$ is not the root of any directed binary tree in $G^\text{\rm forest}$, and further its sibling in $G^\text{\rm forest}$ is $t_j$, the starting node of another tree-path in Lemma~\ref{lem:forest-ocs-structural}. Since $t_m$'s parent is not on the other tree-paths (Lemma~\ref{lem:forest-ocs-structural}, first part), and further by the assumption that $t_m$ has the largest height compared to other $t_i$ from the same directed binary tree, the nodes rooted from $t_m$ and $t_j$'s parent are not on the other tree-paths. By the inductive hypothesis on the sub-instance that removes the nodes rooted from \emph{$t_m$'s parent}, the probability of not selecting element $e$ in these tree-paths is at most $2^{-\sum_{i \ne j, m} k_i} (1-\beta)^{\sum_{i \ne j, m} (k_i-1)}$. Then, \emph{conditioned on any realized randomness on the other tree-paths}, the probability of never selecting $e$ on the tree-paths starting with $t_j$ and $t_m$ is at most $2^{-k_j-k_m} (1-\beta)^{(k_j-1)+(k_m-1)}$, because of the second part of Lemma~\ref{lem:forest-ocs-structural}, the observation that $e$ must get different labels on these two tree-paths, \footnote{ The parent $p$ of $t_j, t_m$ must not contain $e$ by the second requirement of good forest. Suppose that $p$'s common elements with $t_j, t_m$ are $e_j, e_m \ne e$ respectively. Then, $e$'s labels in $t_j, t_m$ are the labels of $e_m, e_j$ in $p$ respectively. } and the next lemma. \begin{lemma} \label{lem:forest-ocs-fork} Consider two independent copies of automaton $\sigma^$ from an arbitrary but identical initial state. Then, for any $k, \hat{k} \ge 1$, the probability that the first copy never selects $\mathsf{T}$ in the first $k$ rounds, and the second copy never selects $\mathsf{H}$ in the first $\hat{k}$ rounds is at most $2^{-k-\hat{k}} (1-\beta)^{(k-1)+(\hat{k}-1)}$. \end{lemma} \subsection{Hardness} \citet{FahrbachHTZ:FOCS:2020} rule out the possibility of $1$-OCS. This subsection improves the upper bound to $\frac{1}{4}$. This holds even for algorithms with unlimited computational power, and even if the algorithms know the instance beforehand. \begin{theorem} \label{thm:ocs-hardness} There is no $(\frac{1}{4}+\epsilon)$-OCS for any constant $\epsilon > 0$. \end{theorem} \begin{proof} Consider any $\gamma$-OCS. Consider an online selection instance with three elements $\{0, 1, 2\}$ and $T = 2i+1$ rounds. The elements in odd rounds are $\{0, 1\}$; the elements in even rounds are $\{0, 2\}$. We shall prove that $\gamma \le \frac{i}{4i-1}$ even if we omit the properties of $\gamma$-OCS concerning three or more rounds. Theorem~\ref{thm:ocs-hardness} then follows by choosing a sufficiently large $i$. For any even $1 \le t \le T$ (so that the elements are $\{0, 2\}$), let $A^t$ be the event that the algorithm selects element $0$ in rounds $t$. We have: \[ \forall 1 \le t \le T,~t \equiv 0 \bmod{2} ~: \qquad \mathbf{Pr} \big[ A^t \big] = \frac{1}{2} ~. \] For any $1 \le t < t' \le T$ with distinct parities (so that $0$ is the only common element), let $B^{t,t'}$ be the event that the algorithms does not select element $0$ in both rounds $t$ and $t'$. We have: \[ \forall 1 \le t < t' \le T,~t + t' \equiv 1 \bmod{2} ~: \qquad \mathbf{Pr} \big[ B^{t, t'} \big] \le \begin{cases} \frac{1}{4} & t' \ge t+3 ~;\\[1ex] \frac{1-\gamma}{4} & t' = t+1 ~. \end{cases} \] For any $1 \le t < t' \le T$ with the same parity (so that they have the same two elements), let $C^{t,t'}$ be the event that the algorithms selects element $0$ in both rounds $t$ and $t'$ (i.e., it does not select the other element in both rounds). We have: \[ \forall 1 \le t < t' \le T,~t + t' \equiv 0 \bmod{2} ~: \qquad \mathbf{Pr} \big[ C^{t, t'} \big] \le \begin{cases} \frac{1}{4} & t' \ge t+4 ~;\\[1ex] \frac{1-\gamma}{4} & t' = t+2 ~. \end{cases} \] Summing together: \begin{align} \sum_{t \equiv 0 \bmod{2}} \mathbf{Pr} \big[ A^t \big] & + \sum_{t < t' : t+t' \equiv 1 \bmod{2}} \mathbf{Pr} \big[ B^{t,t'} \big] + \sum_{t < t' : t+t' \equiv 1 \bmod{2}} \mathbf{Pr} \big[ C^{t,t'} \big] \notag \\[1ex] & \le \underbrace{\frac{1}{2} \cdot i}_\text{events $A^t$} + \underbrace{\frac{1}{4} \cdot i(i-1) + \frac{1-\gamma}{4} \cdot 2i}_\text{events $B^{t, t'}$} + \underbrace{\frac{1}{4} \cdot (i-1)^2 + \frac{1-\gamma}{4} \cdot (2i-1)}_\text{events $C^{t, t'}$} \notag \\[1ex] & = \frac{2i^2+3i}{4} - \frac{4i-1}{4} \gamma ~. \label{eqn:ocs-hardness-1} \end{align} Next consider any selections $(s^1, s^2, \dots, s^T)$ by the algorithm. Let $j_o$ and $j_e$ be the numbers of odd and even rounds that select $s^t = 0$ respectively. The number of events that it satisfies equals: \begin{align} \underbrace{\vphantom{\bigg\|}j_e}_\text{events $A^t$} + \underbrace{\vphantom{\bigg\|}(i+1-j_o)(i-j_e)}_\text{events $B^{t,t'}$} + \underbrace{\vphantom{\bigg\|}\binom{j_o}{2} + \binom{j_e}{2}}_\text{events $C^{t,t'}$} & = \frac{1}{2} \big(j_o+j_e-i-\frac{1}{2}\big)^2+\frac{i(i+1)}{2}-\frac{1}{8} \\ & \ge \frac{i(i+1)}{2} ~. \tag{$i, j_o, j_e$ are integers} \end{align} As a result we get that: \begin{equation} \label{eqn:ocs-hardness-2} \sum_{t \equiv 0 \bmod{2}} \mathbf{Pr} \big[ A^t \big] + \sum_{t < t' : t+t' \equiv 1 \bmod{2}} \mathbf{Pr} \big[ B^{t,t'} \big] + \sum_{t < t' : t+t' \equiv 1 \bmod{2}} \mathbf{Pr} \big[ C^{t,t'} \big] \ge \frac{i(i+1)}{2} ~. \end{equation} Combining Equations~\eqref{eqn:ocs-hardness-1} and \eqref{eqn:ocs-hardness-2} gives $\gamma \le \frac{i}{4i-1}$ as desired. \end{proof} \begin{comment} \subsection{Warmup: 0.2071-OCS on special OCS inputs} Algorithm \ref{ocs:special-alg-stage-1} and Algorithm \ref{ocs:alg-stage-2} together present a $0.2071$-OCS on some special OCS inputs in which $\mathcal{E}^{t_1}\neq \mathcal{E}^{t_2}$ for any two different rounds $t_1, t_2$. The algorithm consists of two stages. In the first stage(Algorithm \ref{ocs:special-alg-stage-1}), the algorithm constructs a forest, which is defined as dependence forest later, from the ex-ante dependence graph, which is also defined later. In the second stage(Algorithm \ref{ocs:alg-stage-2}), the algorithm applies the $(\sqrt{2}-1)$-strong-OCS oracle to the forest and then decides the selections in the rounds. \end{comment} \begin{comment} \textbf{Dependence Forest.} The dependence forest $G^{forest}=(V,E^{forest})$ is a subgraph of $G^{ex\text{-}ante}$ that each of the nodes has at most 1 in-arc. To construct $G^{forest}$, in the first stage, the algorithm erases one in-arc of the node corresponding to the current round uniformly at random. See Figure \ref{ocs:dependence-graph-ex-post} for an example of $G^{forest}$. \textbf{Settling the selections via $G^{forest}$.} To use the $(\sqrt{2}-1)$-strong-OCS oracle, we shall maintain a labeling function for each round that maps the elements in the round to $\mathsf{H},\mathsf{T}$ labels. For the rounds that correspond to roots of the binary trees, we shall set the labeling function arbitrarily(step \ref{ocs:stage-2-labeling-root} of Algorithm \ref{ocs:alg-stage-2}). For other rounds, we shall align its labeling function with that of its father, i.e. we shall map the element shared by the node and its father to the same label(step \ref{ocs:stage-2-labeling-others} of Algorithm \ref{ocs:alg-stage-2}). With this alignment, if an element appears in all nodes on one path, they will be mapped to the same label by the functions. This corresponds to the constraints of strong-OCS in which the nodes on one path share the same target label. Moreover, with the assumption that there are no two rounds that share the same set of elements, it is easy to observe that it is impossible for a round to share an element with its two children in $G^{forest}$. As a result, an element can't be mapped to the same label in two children of a round. This corresponds to the assumption of strong-OCS that there are not any two paths whose top nodes sharing the same parent and target label. Putting all these together, the algorithm can be proved to be $0.2071$-OCS on these special OCS inputs. See Lemma \ref{ocs:no-sharing-label-children} and Theorem \ref{ocs:0.2071-OCS-on-special-inputs} for formal statement and proof. \begin{figure} \caption{Ex-ante dependence graph} \label{ocs:dependence-graph-ex-ante} \caption{Ex-post dependence forest(bold and solid arcs)} \label{ocs:dependence-graph-ex-post} \caption{\small Example of dependence graphs} \label{ocs:dependence-graph} \end{figure} \begin{lemma} \label{ocs:no-sharing-label-children} If there are not any two rounds that share the same set of elements in the input, Algorithm \ref{ocs:special-alg-stage-1} ensures for any three rounds $t,t_1,t_2$ such that $(t,t_1)_{\cdot},(t,t_2)_{\cdot}\in E^{forest}$, $\mathcal{E}^{t}\cap\mathcal{E}^{t_1}\cap \mathcal{E}^{t_2}\neq \emptyset$. \end{lemma} \begin{proof} Let $t,t_1,t_2$ be three rounds such that $\mathcal{E}^{t}=\{e_1,e_2\}$ and $(t,t_1)_{e_1},(t,t_2)_{e_2}\in E^{forest}$. Since there are not any two rounds that share the same set of elements, $e_1\notin \mathcal{E}^{t_2}$ and $e_2\notin \mathcal{E}^{t_1}$. As $\mathcal{E}^{t}\cap \mathcal{E}^{t_1}\cap \mathcal{E}^{t_2}\subseteq \mathcal{E}^{t}=\{e_1,e_2\}$, $\mathcal{E}^{t}\cap \mathcal{E}^{t_1}\cap \mathcal{E}^{t_2}=\emptyset$. \end{proof} \begin{theorem} \label{ocs:0.2071-OCS-on-special-inputs} If there are not any two rounds that share the same set of elements in the input, the combination of Algorithm \ref{ocs:special-alg-stage-1} and \ref{ocs:alg-stage-2} is $0.2071$-OCS. \end{theorem} \begin{proof} Let $t_1^{\ell}<t_2^{\ell}<\cdots<t_{k_l}^{\ell}, 1\leq \ell\leq m$, be the subsequences of consecutive rounds that involve element $e$. In this proof, we shall consider the set of nodes representing these rounds: \[ \tilde{V} = \{t^{\ell}_i:1\leq \ell\leq m, 1\leq i\leq k_{\ell}\}, \] and the set of arcs in ex-ante dependence graph that connect two of these rounds: \[ \tilde{E} = \{(t^{\ell_1}_{i_1},t^{\ell_2}_{i_2})_{e'}\in E^{ex\text{-}ante}: \forall 1\leq \ell_1,\ell_2\leq m, 1\leq i_1\leq k_{\ell_1}, 1\leq i_2\leq k_{\ell_2}, e'\in \mathcal{E}^{t^{\ell_1}_{i_1}}\cap \mathcal{E}^{t^{\ell_2}_{i_2}} \}. \] With the guarantee of Lemma \ref{ocs:no-sharing-label-children}, the size of $\tilde{E}\cap E^{forest}$ directedly relates to an upper-bound of the probability that $e$ is not selected in these rounds, as claimed in Lemma \ref{ocs:relate-size-of-E-cap-Eforest-to-prob}. \begin{lemma} \label{ocs:relate-size-of-E-cap-Eforest-to-prob} Assume for any three rounds $t,t_1,t_2$ such that $(t,t_1)_{\cdot},(t,t_2)_{\cdot}\in E^{forest}$, $\mathcal{E}^{t}\cap\mathcal{E}^{t_1}\cap \mathcal{E}^{t_2}=\emptyset$. Then, Algorithm \ref{ocs:alg-stage-2} ensures that the probability that $e$ is not selected in any round $t^{\ell}_i,1\leq \ell\leq m, 1\leq i\leq k_{\ell}$, is at most \[ 2^{-\sum_{1\leq \ell\leq m} k_{\ell}}(1-\beta)^{\|\tilde{E}\cap E^{forest}\|}. \] \end{lemma} \begin{proof} Suppose $G^{forest}$ consists of binary trees $T_1,T_2,\cdots, T_r$. For simplicity, next we will use $T_i$ for both the nodes and arcs in the $i$-th tree. Consider a particular tree $T_i, 1\leq i\leq r$. Because of the assumption, all nodes in $\tilde{V}\cap T_i$ can be partitioned into several strictly disjoint paths $p_1,p_2,\cdots, p_s$. Due to step \ref{ocs:stage-2-align-the-label}, the label of $e$ for two nodes of $\tilde{V}$ that are directly connected by an arc must be the same. Therefore, for any $p_j,1\leq j\leq r$, the label of $e$ on $p_j$ is the same. Suppose the labels are $tar_1,tar_2,\cdots,tar_s$ for $p_1,p_2,\cdots,p_s$ respectively. Moreover, for any two rounds $t_1,t_2\in \tilde{V}\cap T_i$, if they are both a child of some round $t\in T_i$, due to the assumption, $e$ is labeled differently for rounds $t_1,t_2$. This means that for any paths $p_{j_1},p_{j_2},1\leq j_1,j_2\leq s$ such that their top nodes share the same parent, $tar_{j_1}\neq tar_{j_2}$. Based on the definition of $\beta$-strong-OCS, the probability that $e$ is not selected in any round $t^{\ell}_j, 1\leq \ell\leq m, 1\leq j\leq k_{\ell}$ in $T_r$ is at most \[ 2^{\|\tilde{V}\cap T_i\|}(1-\beta)^{\|\tilde{E}\cap T_i\|}. \] In total, as the $\beta$-strong-OCS oracle selects independently on different trees, the probability that $e$ is not selected in any round $t^{\ell}_j, 1\leq \ell\leq m, 1\leq j\leq k_{\ell}$ is at most \[ \prod_{i=1}^r 2^{\|\tilde{V}\cap T_i\|}(1-\beta)^{\|\tilde{E}\cap T_i\|} = 2^{-\|\tilde{V}\|}(1-\beta)^{\|\tilde{E}\cap E^{forest}\|} = 2^{-\sum_{1\leq \ell\leq m} k_{\ell}}(1-\beta)^{\|\tilde{E}\cap E^{forest}\|}. \] \end{proof} With this relation, next we only need to analyze the expectation of $(1-\beta)^{\|\tilde{E}\cap E^{forest}\|}$ to finish our proof. Based on the definition of the ex-ante dependence graph, $\tilde{E}$ is clearly a superset of arcs with subscript $e$ that connects consecutive rounds inside each subsequences: \[ \tilde{E}'=\{(t^{\ell}_{i},t^{\ell}_{i+1})_e: 1\leq \ell\leq m, 1\leq i<k_{\ell}\}. \] As in step \ref{ocs:special-stage-1-randomly-add-arcs-to-forest}, where Algorithm \ref{ocs:special-alg-stage-1} use fresh random bits to add one of in-arcs of each round to the dependence forest, each arc in $\tilde{E}'$ is independently added to $E^{forest}$ with probability $\frac{1}{2}$. As a result, $\mathbb{E}[(1-\beta)^{-\|\tilde{E}\cap E^{forest}\|}]$ can be bounded as follows, \[ \mathbb{E}[(1-\beta)^{-\|\tilde{E}\cap E^{forest}\|}] \leq \mathbb{E}[(1-\beta)^{-\|\tilde{E}'\cap E^{forest}\|}] = (\frac{1}{2}+\frac{1}{2}(1-\beta))^{\sum_{\ell=1}^m k_{\ell}-1}=(1-\frac{\beta}{2})^{\sum_{\ell=1}^m k_{\ell}-1}. \] According to Lemma \ref{ocs:relate-size-of-E-cap-Eforest-to-prob}, the probability that $e$ is not selected in any round $j^{\ell}_i,1\leq \ell\leq m, 1\leq i\leq k_{\ell}$, is at most: \[ \prod_{\ell=1}^{m} 2^{-k_{\ell}} (1-\frac{\beta}{2})^{k_{\ell}-1}<\prod_{\ell=1}^{m} 2^{-k_{\ell}} (1-0.2071)^{k_{\ell}-1}. \] \end{proof} \textbf{Counter-examples on general OCS inputs.} Without the assumption that there are not any two rounds that share the same set of elements, the algorithm can be problematic in the following way: there are three rounds $t_1<t_2<t_3$ such that $\mathcal{E}^{t_1}=\mathcal{E}^{t_3}=\{e,e'\}$, $t_2$ is the earliest round between $t_1,t_3$ that contains $e$ and $\forall t\in(t_1,t_3), e'\notin \mathcal{E}^t$; then $t_2, t_3$ can be positively correlated. Formally, consider the case $\mathcal{E}^1=\{a,b\},\mathcal{E}^2=\{a,c\},\mathcal{E}^3=\{a,d\},\mathcal{E}^4=\{a,b\}$, whose ex-ante dependence graph is shown in Figure \ref{ocs:counter-example-of-simple-reduction}. In Algorithm \ref{ocs:special-alg-stage-1}, node $1,4$ can be correlated, which asks them to be weakly connected in the ex-post dependence forest, only if there is a chain going from $1$ to $4$ or there is arc $(1,4)_{b}$ in the ex-post dependence forest. In the first case, whether $a$ is selected in round $1,4$ is independent due to Lemma \ref{ocs:second_subseq}. In the second case, if $(1,2)_{a}$ is also included in the ex-post dependence forest, $a$ is mapped to the same label in round $1,2,4$, e.g. let's say $\mathsf{H}$. Recall that $state_j$ represents the state of the automaton after selecting the label of each node in Algorithm \ref{ocs:algo_on_binary_tree}. By Figure \ref{ocs:automaton} the construction of the automaton, there is \begin{align} \mathbf{Pr}[s^2,s^4\neq a] &= \mathbf{Pr}[state_1=-1]\cdot \frac{(1+\beta)^2}{4} + \mathbf{Pr}[state_1=1]\cdot \frac{(1-\beta)^2}{4} \\ &= \frac{(1+\beta)^2+(1-\beta)^2}{8} = \frac{1+\beta^2}{4}>\frac{1}{4}. \end{align} Therefore, the probability that element $a$ is not selected in round $2,4$ is greater than $\frac{1}{4}$, violating the constraint of OCS. \begin{figure} \caption{Counter-example of the combination of Algorithm \ref{ocs:special-alg-stage-1} and \ref{ocs:alg-stage-2} on general OCS inputs} \label{ocs:counter-example-of-simple-reduction} \end{figure} \end{comment} \section{Applications in Online Bipartite Matching} \label{sec:matching} \subsection{Online Bipartite Matching Preliminaries} Consider an undirected bipartite graph $G = (L, R, E)$, where $L$ and $R$ are the sets of left-hand-side and right-hand-side vertices respectively, and $E$ is the set of edges. Each edge $(u, v) \in E$ has a positive edge-weight $w_{uv} > 0$. The problem is \emph{unweighted} if $w_{uv} = 1$ for all $(u, v) \in E$, is \emph{vertex-weighted} if $w_{uv} = w_u$ for some positive vertex-weights $(w_u)_{u \in L}$ of the left-hand-side vertices, and is \emph{edge-weighted} if the edge-weights could be arbitrary. In online bipartite matching problems, we refer to the left-hand-side and right-hand-side vertices as \emph{offline} and \emph{online} vertices respectively. Initially, the algorithm only knows the offline vertices, and the vertex-weights in the vertex-weighted case. Then, the online vertices arrive one at a time. When an online vertex $v \in R$ arrives, the algorithm sees its incident edges, and the edge-weights in the edge-weighted case. The algorithm then immediately and irrevocably matches $v$ to an offline neighbor $u \in L$. The objective is to maximize the sum of the maximal edge-weight matched to each offline vertex. In the unweighted and vertex-weighted problems, matching an offline vertex more than once does not further increase the objective. Therefore, we may assume without loss of generality that the algorithm matches each offline vertex at most once and the matched edges indeed form matching. The objectives in these two cases are equivalent to maximizing the cardinality of the matching, and maximizing the sum of the vertex-weights of matched offline vertices, respectively. In edge-weighted online bipartite matching, we may alternatively view the above objective as allowing disposals of previously matched edges so that a matched offline vertex could be rematched to a new edge with a larger edge-weight. In other words, we may think of the matching as being comprised of the heaviest edge matched to each offline vertex, and seek to maximize the total edge-weight of the matching. Further in online advertising, it corresponds to displaying an advertiser's ad multiple times but only charges for the most valuable one. \citet{FeldmanKMMP:WINE:2009} introduce this \emph{free disposal} model which has then become the standard model of edge-weighted online bipartite matching under worst-case competitive analysis. We compare the expected objective of the matching by the algorithm, and the optimal matching that maximizes the objective in hindsight given full information of the bipartite graph $G = (L, R, E)$ and the edge-weights $(w_{uv})_{(u, v) \in E}$. The competitive ratio of an online algorithm is the infimum of this ratio over all possible instances. \subsection{Semi-OCS and Unweighted and Vertex-weighted Online Bipartite Matching} \label{sec:semi-ocs-unweighted-vertex-weighted-matching} \citet{FahrbachHTZ:FOCS:2020} give a two-choice greedy algorithm for unweighted online bipartite matching, using a semi-OCS as a sub-routine. Their original theorem is only for the unweighted problem and only for the guarantee of $\gamma$-semi OCS, i.e., $p(k) = 2^{-k}(1-\gamma)^{k-1}$. Nonetheless, the algorithm and analysis generalize to the vertex-weighted case and for general $p(k)$ by standard techniques in the online matching literature. We state the more general theorem below. \begin{theorem}[c.f., \citet{FahrbachHTZ:FOCS:2020}] \label{thm:semi-ocs-unweighted-vertex-weighted-matching} Given a semi-OCS such that the probability of never selecting an element $e$ that appears $k$ times is at most $p(k)$, there is a\, $\Gamma$-competitive two-choice greedy algorithm for unweighted and vertex-weighted online bipartite matching, where the competitive ratio $\Gamma$ is the optimal value of the following linear program (LP): \begin{align} \text{\rm maximize} \quad & \Gamma \tag{\textsc{Matching LP}\xspace} \\[2ex] \text{\rm subject to} \quad & a(k)+b(k)\leq p(k)-p(k+1) & \forall k\geq 0 \label{eqn:two-way-lp-gain-split} \\%[1ex] & \sum_{i=0}^{k-1} a(i)+2 b(k)\geq \Gamma & \forall k\geq 0 \label{eqn:approximate-dual-feasible} \\ & b(k+1)\leq b(k) & \forall k\geq 0\label{eqn:two-way-lp-monotone} \\[2ex] & a(k),b(k)\geq 0 & \forall k \ge 0 \notag \end{align} \end{theorem} We will not present the generalized algorithm and the proof of Theorem~\ref{thm:semi-ocs-unweighted-vertex-weighted-matching} because they will be subsumed by the algorithm and theorem in the next subsection. Instead, the main result of this subsection is an explicit optimal solution to the LP. By contrast, \citet{FahrbachHTZ:FOCS:2020} rely on solving a finite approximation of the LP numerically using LP solvers. \begin{theorem} \label{thm:lp-solution} Suppose that $p(0) = 1$ and $p(k+1)\leq \frac{2}{3}p(k)$ for any $k \ge 0$. Then, the \textsc{Matching LP}\xspace admits an optimal solution as follows: \begin{align} \Gamma & = 1-\frac{1}{3} \sum_{i=0}^{\infty} \Big(\frac{2}{3}\Big)^i p(i) ~; \\ b(k) & = \frac{1}{3} \sum_{i=k}^{\infty} \Big(\frac{2}{3}\Big)^{i-k} \big( p(i)-p(i+1) \big) & \forall k\ge 0 ~; \\[1ex] a(k) & = p(k)-p(k+1)-b(k) & \forall k \ge 0 ~. \end{align} \end{theorem} The assumption of $p(k+1)\leq \frac{2}{3}p(k)$ is essentially without loss of generality since any natural online selection algorithm shall at least halve the unselected probability after each round involving the element. Indeed, even the trivial independent sampling satisfies the stronger $p(k+1)\leq \frac{1}{2}p(k)$. The proof of this theorem is deferred to Appendix~\ref{app:lp-solution}. For $\gamma$-semi-OCS, it recovers a result by \citet{HuangZZ:FOCS:2020} as a corollary. \begin{corollary}[c.f., \citet{HuangZZ:FOCS:2020}] Suppose that $p(k) = 2^{-k} (1-\gamma)^{k-1}$. Then the optimal value of the \textsc{Matching LP}\xspace is: \[ \frac{3+2\gamma}{6+3\gamma} ~. \] \end{corollary} Since the optimal semi-OCS in Section~\ref{sec:semi-ocs} gives $p(k) = 2^{-2^k+1}$, we have the next corollary through a numerical calculation. \begin{corollary} \label{cor:semi-ocs-unweighted-vertex-weighted-matching} The two-choice greedy algorithm using the optimal semi-OCS as a sub-routine is at least $0.536$-competitive for unweighted and vertex-weighted online bipartite matching. \end{corollary} \subsection{OCS and Edge-weighted Online Bipartite Matching} \label{sec:ocs-edge-weighted-matching} \subsubsection{Online Primal-Dual Algorithm} This subsection gives a variant of the online primal-dual algorithm of \citet{FahrbachHTZ:FOCS:2020} for edge-weighted online bipartite matching, using an OCS as a sub-routine. This variant simplifies the analysis in the next subsection. To simplify exposition, we assume that for every online vertex $v$ there is a unique offline dummy vertex such that the edge between them has weight $0$. Then, every online vertex will be matched, although being matched to the dummy vertex is the same as being left unmatched. For each online vertex $v$, the algorithm shortlists two candidates $u_1, u_2$ from $v$'s neighbors. If the shortlisted candidates are the same, the algorithm matches $v$ to it. Otherwise, the algorithm lets the OCS selects one of them and matches $v$ to the selected one. To explain how the algorithm makes the shortlists, let $k_u(w)$ be the number of times that $u$ is shortlisted thus far due to online vertices with edge-weight $w_{uv} \ge w$. In a round in which $u_1 = u_2 = u$, the corresponding $k_u(w)$'s increase by $2$. We remark that $k_u(w) = 0$ for any dummy offline vertex $u$ and for any $w > 0$. The algorithm is parameterized by the optimal solution to the $\textsc{Matching LP}\xspace$ in Theorem~\ref{thm:lp-solution}. Given the optimal solution, define the ``value'' of matching an online vertex $v$ to an offline vertex $u$ as: \begin{align} \label{eqn:matching-dual-update-beta} \Delta_u \beta_v \defeq \int_0^{w_{uv}} b\big(k_u(w)\big)dw -\frac{1}{2} \int_{w_{uv}}^{\infty} \sum_{i=0}^{k_u(w)-1} a(i) dw ~. \end{align} For each online vertex $v$, the algorithm first finds $u_1$ with the maximum $\Delta_u \beta_v$, and then finds $u_2$ with the maximum \emph{updated} $\Delta_u \beta_v$. If $u_1 = u_2$, match $v$ to it. Otherwise, match $v$ to the one that the OCS selects. Following the terminology of \citet{FahrbachHTZ:FOCS:2020}, we call the former a deterministic round, and the latter a randomized round. Their algorithm computes the ``values'' of deterministic and randomized rounds using different equations. By contrast, our variant computes the ``values'' using the same Eqn.~\eqref{eqn:matching-dual-update-beta}. See Algorithm~\ref{alg:ocs-matching}. \begin{algorithm}[t] \caption{Online primal-dual edge-weighted bipartite matching algorithm} \label{alg:ocs-matching} \begin{algorithmic} \State \textbf{State variables:} (for each offline vertex $u$) \begin{itemize} \item $k_u(w)$: the number of times $u_1 = u$ or $u_2 = u$ and further its edge weight is at least $w$. \end{itemize} \State \textbf{On the arrival of an online vertex $v \in R$:} \begin{enumerate} \item For $\ell \in \{1,2\}$: \begin{enumerate} \item Find $u_\ell$ with maximum $\Delta_u \beta_v$ given by Eqn.~\eqref{eqn:matching-dual-update-beta}. \item Increase $k_{u_\ell}(w)$ by 1 for $0 \le w \leq w_{u_\ell v}$. \end{enumerate} \item If $u_1 \neq u_2$, let the OCS select one of them, and match $v$ to it. \textbf{(Randomized round)} \item Otherwise, match $v$ to $u_1 = u_2$. \textbf{(Deterministic round)} \end{enumerate} \end{algorithmic} \end{algorithm} \subsubsection{Improved Online Primal-Dual Analysis} This subsection improves the analysis of \citet{FahrbachHTZ:FOCS:2020} in twofold. First, our edge-weighted result uses the LP in Theorem~\ref{thm:semi-ocs-unweighted-vertex-weighted-matching} and its optimal solution in Theorem~\ref{thm:lp-solution}, same as the unweighted and vertex-weighted cases. By contrast, the analysis of \citet{FahrbachHTZ:FOCS:2020} for edge-weighted online bipartite matching needs to consider a LP with additional constraints. Second, our analysis indicates that the online selection algorithm only needs to guarantee a condition strictly weaker than the property of $\gamma$-OCS. It enables us to further explore a variant of OCS in the next subsection to further improve the competitive ratio in edge-weighted online bipartite matching. \begin{theorem} \label{thm:two-way-online-primal-dual-analysis} Suppose that $( p(k) )_{k \ge 0}$ is non-increasing and satisfies $p(0) = 1$, and $\Gamma$, $( a(k) )_{k \ge 0}$, and $( b(k) )_{k \ge 0}$ form a solution to the \textsc{Matching LP}\xspace. Algorithm~\ref{alg:ocs-matching} is $\Gamma$-competitive for edge-weighted online bipartite matching if the OCS ensures that for any online selection instance, any element $e$, and any consecutive subsequences of the rounds involving the element with lengths $k_1, k_2, \dots, k_m$, element $e$ is unselected in these rounds with probability at most: \begin{equation} \label{eqn:relaxed-ocs} p \Big( \sum_{i=1}^m k_i \Big) +\frac{1}{2}\sum_{i=2}^m \sum_{j=0}^{k_1+\dots+k_{i-1}-1} a(j) ~. \end{equation} \end{theorem} We make three remarks before presenting the proof of the theorem. First, for unweighted and vertex-weighted online bipartite matching, the online selection algorithm only needs to ensure the above property for the subset of all $k$ rounds involving an element. Then, it degenerates to the guarantee of semi-OCS because $m = 1$ and thus the second term involving the $a(j)$'s disappears. The proof below shall make this explicit. Further, the guarantee in Eqn.~\eqref{eqn:relaxed-ocs} holds almost trivially for natural online selection algorithms when $m \ge 3$. On the one hand, any natural algorithm would at least halve the unselected probability for every round involving the element. Hence, after $\sum_{i=1}^m k_i \ge 3$ rounds, the unselected probability is at most $\frac{1}{8}$. On the other hand, the optimal LP solution from Theorem~\ref{thm:lp-solution} satisfies that $a(0) \ge \frac{2}{9}$ for all online selection algorithms in the literature and in this paper, and even for the overly idealized algorithm that ensure selecting an element when it appears more than once. Hence, the $a(0)$'s in the second term of Eqn.~\eqref{eqn:relaxed-ocs} sum to at least $\frac{2}{9} > \frac{1}{8}$. A similar argument shows that the guarantee holds almost trivially for $m = 2$ if $k_1 + k_2 \ge 3$. Hence, it suffices to slightly enhance the semi-OCS guarantee to further handle either a single consecutive subsequence (but not necessarily starting from the earliest round involving the element as in semi-OCS), or two very short consucutive subsequences. This motivates the variant of OCS in the next subsection. Finally, Theorem~\ref{thm:two-way-online-primal-dual-analysis} subsumes the analysis of \citet{FahrbachHTZ:FOCS:2020} because the original guarantee of $\gamma$-OCS satisfies Eqn.~\eqref{eqn:relaxed-ocs}, as we will prove in the next lemma. \begin{lemma} Suppose that $\gamma \in [0, \frac{1}{4}]$, \footnote{Theorem~\ref{thm:ocs-hardness} shows that there is no $\gamma$-OCS for $\gamma > \frac{1}{4}$.} and $p(k) = 2^{-k}(1-\gamma)^{k-1}$ for $k \ge 0$. Let $(a(k))_{k \ge 0}$ take values as in the optimal LP solution in Theorem~\ref{thm:lp-solution}. Then, for any positive integers $k_1, k_2, \dots, k_m$: \[ \prod_{i=1}^m 2^{-k_i}(1-\gamma)^{k_i-1} \le 2^{-\sum_{i=1}^m k_i}(1-\gamma)^{\sum_{i=1}^m k_i-1} +\frac{1}{2}\sum_{i=2}^m \sum_{j=0}^{k_1+\dots+k_{i-1}-1} a(j) ~. \] \end{lemma} \begin{proof} In fact we will prove it even dropping all $a(j)$'s for $j \ge 1$. If $m = 1$ the left-hand-side equals the first term on the right-hand-side. If $m \ge 2$, the difference between the left-hand-side and the first term on the right-hand-side is: \begin{align} \big( 1 - (1-\gamma)^{m-1} \big) \prod_{i=1}^m 2^{-k_i}(1-\gamma)^{k_i-1} & \le (m-1) \gamma \prod_{i=1}^m 2^{-k_i}(1-\gamma)^{k_i-1} \\ & \le \frac{(m-1) \gamma}{4} ~. \tag{$\sum_{i=1}^k k_i \ge 2$} \end{align} On the other hand, Theorem~\ref{thm:lp-solution} indicates that for $p(k) = 2^{-k} (1-\gamma)^{k-1}$: \[ a(0) = \frac{3+\gamma}{12+6\gamma} \ge \frac{\gamma}{2} ~, \] for any $\gamma \le \frac{1}{4}$. \footnote{In fact, this holds for any $0 \le \gamma \le \frac{\sqrt{61}-5}{6} \approx 0.468$} Hence, the $a(0)$'s in the second term on the right sum to at least $\frac{(m-1)\gamma}{4}$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:two-way-online-primal-dual-analysis}] For any offline vertex $u$, consider the subset of rounds in which $u$ is shortlisted as $u_1$ or $u_2$ by Algorithm~\ref{alg:ocs-matching}. Further for any weight level $w > 0$, suppose that the subset of rounds in which $u$ is shortlisted by an online vertex $v$ with edge-weight $w_{uv} \ge w$ form consecutive subsequences of lengths $k_1, k_2, \dots, k_m$. We remark that if $u_1 = u_2 = u$ in the round of some online vertex $v$ with $w_{uv} \ge w$, this deterministic round contributes $2$ to the corresponding $k_i$. In such cases the sequel probability bounds hold trivially because $u$ is matched to an edge weight weight at least $w$ with certainty. The binding case of our analysis is when there are only randomized rounds. The OCS guarantee in the theorem statement ensures that the probability of matching $u$ to one of them is at least: \begin{equation} \label{eqn:matching-objective-by-vertex} y_u(w) \defeq 1 - p \Big( \sum_{i=1}^m k_i \Big) - \frac{1}{2}\sum_{i=2}^m \sum_{j=0}^{k_1+\dots+k_{i-1}-1} a(j) ~. \end{equation} Therefore, the expected maximum edge-weight matched to vertex $u$ is at least $\int_0^\infty y_u(w) dw$. The expected total weight of the matching by Algorithm~\ref{alg:ocs-matching} is at least: \[ \textsc{Alg} \defeq \sum_{u \in L} \int_0^\infty y_u(w) dw ~. \] The competitive analysis is a charging argument. For every online vertex $v \in R$, we split the changes of $\textsc{Alg}$ among the shortlisted offline verties $u_1, u_2$ and the online vertex $v$. Formally, let $\alpha_u = \int_0^\infty \alpha_u(w) dw$ be the gain of each offline vertex $u \in L$, where $\alpha_u(w)$ is the contribution from weight-level $w$. Let $\beta_v$ denote the gain of each online vertex $v$. Both are initially zero. Then, as an online vertex $v$ arrives and when $u \in \{ u_1, u_2 \}$ is shortlisted, suppose that $y_u(w)$ changes by $\Delta y_u(w)$ for any $0 \le w \le w_{uv}$: \begin{itemize} \item Increase $\beta_v$ by $\Delta_u \beta_v$ according to Eqn.~\eqref{eqn:matching-dual-update-beta}, which we restate below: \[ \Delta_u \beta_v \defeq \int_0^{w_{uv}} b\big(k_u(w)\big)dw -\frac{1}{2} \int_{w_{uv}}^\infty \sum_{i=0}^{k_u(w)-1} a(i) dw ~. \] \item Increase $\alpha_u(w)$ by: \end{itemize} \begin{align} \label{eqn:matching-dual-update-alpha} \Delta \alpha_u(w) \defeq \begin{cases} \Delta y_u(w) - b\big(k_u(w)\big) & \text{if $w_{uv} \ge w$}\\[1ex] \frac{1}{2}\sum_{i=0}^{k_u(w)-1} a(i) & \text{if $0 \le w_{uv} < w$} \end{cases} ~. \end{align} We remark that the values of $k_u(w)$'s in the above charging rules are at the moment when $u$ is shortlisted by the algorithm for online vertex $v$. \paragraph{Feasibility of the Charging Rule.} We first verify that the total change in $\alpha_u$ and $\beta_v$ equals the change of $\textsc{Alg}$ due to online vertex $v$. By Equations~\eqref{eqn:matching-dual-update-beta} and \eqref{eqn:matching-dual-update-alpha}, the total change in the vertices' gains equals: \begin{align} \underbrace{\int_0^{w_{uv}} \big(\Delta y_u(w) - b(k_u(w)) \big) dw + \frac{1}{2} \int_{w_{uv}}^\infty \sum_{i=0}^{k_u(w)-1} a(i) dw}_\text{change of $\alpha_u$} & \\ + \underbrace{\int_0^{w_{uv}} b\big(k_u(w)\big)dw - \frac{1}{2} \int_{w_{uv}}^\infty \sum_{i=0}^{k_u(w)-1} a(i)dw}_\text{change of $\beta_v$} & = \int_0^{w_{uv}} \Delta y_u(w) dw ~. \end{align} \paragraph{Invariant of Offline Gain.} Next we show that for any offline vertex $u$, and any positive weight-level $w > 0$: \begin{equation} \label{eqn:two-way-matching-offline-invariant} \alpha_u(w) \geq \sum_{i = 0}^{k_u(w)-1} a(i) ~. \end{equation} Consider the rounds in which $u$ is shortlisted and $u_1$ or $u_2$ (or both) and the edge-weight is at least $w$. Partition them into consecutive subsequences of the rounds that shortlist $u$, regardless the edge-weights. Let $k_1, k_2, \dots, k_m$ be the lengths of the consecutive subsequences. By considering the changes to $\alpha_i(w)$ due to the rounds in the subsuequences, and any $m-1$ rounds involving $u$ between the subsequences, one for each pair of neighboring subsequences, we get that: \begin{align} \alpha_u(w) & \ge \underbrace{y_u(w) - \sum_{i=0}^{k_u(w)-1} b(i)}_\text{rounds in subsequences, 1st case of Eqn.~\eqref{eqn:matching-dual-update-alpha}} + \underbrace{\frac{1}{2} \sum_{i=2}^m \sum_{j = 0}^{k_1+\dots+k_{i-1}-1} a(i)}_\text{rounds in between, 2nd case of Eqn.~\eqref{eqn:matching-dual-update-alpha}} \\ & = 1 - p\big(k_u(w)\big) - \sum_{i=0}^{k_u(w)-1} b(\ell) \tag{Eqn.~\eqref{eqn:matching-objective-by-vertex}, and $k_u(w) = k_1 + \dots + k_m$} \\ & = \sum_{i=0}^{k_u(w)-1} \big( p(i) - p(i+1) - b(i) \big) \tag{$p(0) = 1$} \\ & \ge \sum_{i=0}^{k_u(w)-1} a(i) ~. \tag{Eqn.~\eqref{eqn:two-way-lp-gain-split}} \end{align} This is the only place in our argument that uses Eqn.~\eqref{eqn:relaxed-ocs} about the online selection algorithm, indirectly through Eqn.~\eqref{eqn:matching-objective-by-vertex}. We remark that in unweighted and vertex-weighted online bipartite matching, there is only one weight level $w = 1$ or $w = w_u$ of concern for any offline vertex $u$. Hence, there is only a single subsequence with all rounds that shortlist $u$ in the above argument. It suffices to replace Eqn.~\eqref{eqn:relaxed-ocs} by the weaker property of semi-OCS. \paragraph{Non-negativity of Gains.} The non-negativity of offline gains follows from the above invariant. The non-negativity of online gains follows by that $\Delta_u \beta_v = 0$ for the dummy vertex $u$. Hence, the offline neighbors $u_1, u_2$ shortlisted by Algorithm~\ref{alg:ocs-matching} have non-negative $\Delta_{u_1} \beta_v, \Delta_{u_2} \beta_v$. \paragraph{$\Gamma$-Approximate Equilibrium.} The gains cumulated by the online and offline vertices satisfy an approximate equilibrium condition in the sense that for any edge $(u, v) \in E$ the total gain of $u$ and $v$ is at least $\Gamma$ times the edge weight $w_{uv}$. By the definition of Algorithm~\ref{alg:ocs-matching}, and by that $\Delta_u \beta_v$ in Eqn.~\eqref{eqn:matching-dual-update-beta} is non-increasing in $k_u(w)$'s, we have $\beta_v \ge 2 \Delta_u \beta_v$ even when we compute $\Delta_u \beta_v$ \emph{using the final values of $k_u(w)$'s}. Hence: \begin{align} \alpha_u + \beta_v & \ge \int_0^\infty \sum_{i=0}^{k_u(w)-1} a(i) dw + 2 \Delta_u \beta_v \tag{Eqn.~\eqref{eqn:two-way-matching-offline-invariant}} \\ & = \int_0^{w_{uv}} \sum_{i=0}^{k_u(w)-1} a(i) dw + 2 \int_0^{w_{uv}} b\big(k_u(w)\big) dw \tag{Eqn.~\eqref{eqn:matching-dual-update-beta}} \\[2ex] & \geq \Gamma w_{uv} ~. \tag{Eqn.~\eqref{eqn:approximate-dual-feasible}} \end{align} Then, consider an optimal matching $M \subseteq E$. Algorithm~\ref{alg:ocs-matching} is $\Gamma$-competitive because: \begin{align} \textsc{Alg} & = \sum_{u \in L} \alpha_u + \sum_{v \in R} \beta_v \\ & \ge \sum_{(u, v) \in M} \big( \alpha_u + \beta_v \big) & \\ & \ge \Gamma \sum_{(u,v) \in M} w_{uv} ~. \end{align} Finally, we remark that the above analysis is mathematically equivalent an online primal dual analysis under the framework of \citet{DevanurJK:SODA:2013} for online bipartite matching, and also \citet{DevanurHKMY:TEAC:2016} and \citet{FahrbachHTZ:FOCS:2020} for the edge-weighted case. We choose the above exposition to avoid having to introduce the more general framework. \end{proof} Combining Algorithm~\ref{alg:ocs-matching} with the improved $0.167$-OCS from Theorem~\ref{thm:ocs} in Section~\ref{sec:ocs} surpasses the state-of-the-art $0.508$-competitive algorithm for edge-weighted online bipartite matching by \citet{FahrbachHTZ:FOCS:2020}. \begin{corollary} There is an two-choice greedy algorithm for edge-weighted online bipartite matching that is at least $0.512$-competitive. \end{corollary} \subsection{A Variant of OCS and Edge-weighted Online Bipartite Matching} This subsection considers another online selection algorithm tailored for the relaxed condition in Eqn.~\eqref{eqn:relaxed-ocs}. Each element is associated with a flag $1$ or $0$, initialized uniformly at random. In each round $t$, the algorithm samples an element $e$ from $\mathcal{E}^t$ uniformly at random to probe its flag. If its flag is $1$, the algorithm selects $e$ and sets its flag to $0$. Otherwise, the algorithm selects the other element and sets $e$'s flag to $1$. In other words, the algorithm randomly samples an element, lets its flag decides the selection, and flips the flag. See Algorithm~\ref{alg:relaxed-ocs}. \begin{algorithm}[t] \caption{A variant of OCS designed for edge-weighted online bipartite matching} \label{alg:relaxed-ocs} \begin{algorithmic} \State \textbf{State variables:} (for each element $e$) \begin{itemize} \item $\tau_{e} \in \big\{0, 1\big\}$; its initial value $\tau_e^0$ is independently and uniformly at random. \end{itemize} \State \textbf{For each round $t$: } \begin{enumerate} \item Draw $e^t \in \mathcal{E}^t$ uniformly at random. \item If $\tau_{e^t} = 1$, select $e^t$ and let $\tau_{e^t} = 0$. \item Otherwise, select the other element in $\mathcal{E}^t$ and set $\tau_{e^t} = 1$. \end{enumerate} \end{algorithmic} \end{algorithm} \begin{theorem} \label{thm:relaxed-ocs} Algorithm~\ref{alg:relaxed-ocs} ensures the selection probability in Eqn.~\eqref{eqn:relaxed-ocs} for: \[ p(k) = 2^{-k-\min\{k,\lceil\frac{k+2}{2}\rceil\}}+ k\cdot 2^{-k-\min\{k,\lceil\frac{k+3}{2}\rceil\}} ~. \] \end{theorem} Combining with Theorem~\ref{thm:two-way-online-primal-dual-analysis} further improves the competitive ratio of edge-weighted online bipartite matching. \begin{corollary} \label{cor:edge-weighted-519} Algorithm~\ref{alg:ocs-matching}, using Algorithm~\ref{alg:relaxed-ocs} for online selections, is at least $0.519$-competitive for edge-weighted online bipartite matching. \end{corollary} \paragraph{Preliminaries on Boolean Formula with Uniform Input.} Algorithm~\ref{alg:relaxed-ocs} uses two kinds of random bits that are sampled independently and uniformly: the initial flags $(\tau_e^0)_{e \in \mathcal{E}}$, and the sampled elements $(e^t)_{1 \le t \le T}$. Viewing these random bits as boolean variables, we will represent each selection event by an XOR clause, i.e., an XOR of a subset these boolean variables, their negates, and the constant $1$, such as $X_1 \oplus \neg X_2 \oplus X_3 \oplus 1$. Next we introduce two properties related to XOR clauses with uniform input. \begin{lemma} \label{lem:xor-1} For any uniform and independent boolean variables and $m$ XOR clauses such that each variable is in at most one clause, the probability of satisfying all clauses equals $2^{-m}$. \end{lemma} \begin{proof} This is because each clause independently holds with probability half. \end{proof} \begin{lemma} \label{lem:xor-2} For any uniform and independent boolean variables and $m$ XOR clauses such that each variable is in at most \emph{two} clauses, the probability of satisfying all clauses is at most $2^{-\lceil\frac{m}{2}\rceil}$. \end{lemma} \begin{proof} Consider an undirected graph $G=(V,E)$ in which the vertices $V = \{1, 2, \dots m\}$ correspond to clauses, and the edges correspond the boolean variables that appears in two clauses: \[ E = \big\{ (i,j)_k: \text{clauses $i$ and $j$ both involve variable $k$} \big\} ~. \] Next, consider any maximal matching $M$ of $G$. Let $\ell = \|M\|$ be the size of the matching. Let $U \subseteq V$ denote the set of $m-2\ell$ unmatched vertices (i.e., clauses). Since the matching is maximal, the unmatched clauses do not share any variables. Hence, over the randomness of the variables not in the matching $M$, the probability of satisfying all unmatched clauses equals $2^{-m+2\ell}$ by Lemma~\ref{lem:xor-1}. Further, \emph{conditioned on any realization of the variables not in the matching $M$} and over the randomness of the variables in $M$, each pair of matched clauses hold with probability at most $\frac{1}{2}$. Therefore, the probability of satisfying all clauses is at most $2^{-m+2\ell} \cdot 2^{-\ell} = 2^{-m+\ell}$. The lemma then follows by $\ell \le \lfloor \frac{m}{2} \rfloor$. \end{proof} \paragraph{Selection Probabilities.} We next develop a lemma about probability of not selecting an element $e$ conditioned on the sampled elements $e^t$'s. The proof of Theorem~\ref{thm:relaxed-ocs} will repeatedly use the lemma. \begin{lemma} \label{lem:relaxed-ocs-conditional-probability-bound} For any element $e$ and any $k$ rounds $t_1 < t_2 < \cdots < t_k$ involving $e$, conditioned on any realization of $e^{t_1},\cdots, e^{t_k}$, the probability that $e$ is never selected in these $k$ rounds is at most: \[ 2^{-\min \left\{k,\lceil\frac{k+2}{2}\rceil \right\}} ~. \] \end{lemma} \begin{proof} We will prove a stronger result. If there are $d$ distinct elements in the realized $e^{t_1}, \dots, e^{t_k}$, then $e$ is unselected in these rounds with probability at most: \[ \begin{cases} 2^{-k} & \text{if $d=1$;}\\ 2^{-\lceil\frac{k+d}{2} \rceil} & \text{if $d\geq 2$.} \end{cases} \] We next introduce an XOR clause for each $t_i$, $1 \le i \le k$, so that not selecting $e$ in these rounds is equivalent to satisfying all $k$ clauses. If round $t_i$ is the earliest among these $k$ rounds that samples element $e^{t_i}$, i.e., $e^{t_i} \ne e^{t_j}$ for any $j < i$, consider a clause that represents the value of flag $\tau_{e^{t_i}}$ at the beginning of round $t_i$: \[ \begin{cases} \tau_{e^{t_i}}^0 \oplus \big( \bigoplus_{t < t_i : e^{t_i} \in \mathcal{E}^t} \mathbf{1} (e^t = e^{t_i} ) \big) & e^{t_i} = e ~; \\[1ex] 1 \oplus \tau_{e^{t_i}}^0 \oplus \big( \bigoplus_{t < t_i : e^{t_i} \in \mathcal{E}^t} \mathbf{1} (e^t = e^{t_i} ) \big) & e^{t_i} \ne e ~. \end{cases} \] We shall refer to such clauses as type-A clauses. Otherwise, suppose that element $e^{t_i}$ was most recently sampled in $t_j$, i.e., $e^{t_i} = e^{t_j}$ and $e^{t_i} \ne e^{t_\ell}$ for $j < \ell < i$. Consider a clause that represents the parity of the number of times that flag $\tau_{e^{t_i}}$ flips between the two rounds, \emph{including the flip due to round $t_j$}, i.e.: \[ \textstyle 1 \oplus \big( \bigoplus_{t_j < t < t_i : e^{t_i} \in \mathcal{E}^t} \mathbf{1} (e^t = e^{t_i} ) \big) ~. \] We shall refer to such clauses as type-B clauses. It captures if the value of $\tau_e$ at the begining of $t_i$ is the same as that at the beginning of round $t_j$, and thus still leads to not selecting element $e$. If $d = 1$, there are $1$ type-A clause and $k-1$ type-B clauses. Further, each variable appears in at most one clause. It then follows by Lemma~\ref{lem:xor-1}. If $d \ge 2$, there are $d$ type-A clauses and $k-d$ type-B clauses. First consider the type-B clauses and the random variables corresponding to the sampled elements $(e^t)_{1 \le t \le T}$. Each of these variables appears in at most two clauses. By Lemma~\ref{lem:xor-2} the probability of satisfying all these clauses is at most $2^{-\lceil \frac{k-d}{2} \rceil}$. Further, each type-A clause has a unique variable $\tau_{e^{t_i}}^0$. Hence, over any realization of the sampled elements, the probability of satisfying all $d$ type-A clauses is $2^{-d}$. Combining the two bounds proves the lemma. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:relaxed-ocs}] First recall the requirement of Eqn.~\eqref{eqn:relaxed-ocs}. For any element and any consecutive subsequences of the rounds involving the element with lengths $k_1, k_2, \dots, k_m$, we will upper bound the probability that Algorithm~\ref{alg:relaxed-ocs} never selects $e$ in these rounds by: \[ p \Big( \sum_{i=1}^m k_i \Big) +\frac{1}{2}\sum_{i=2}^m \sum_{j=0}^{k_1+\dots+k_{i-1}-1} a(j) ~, \] where: \[ p(k) = 2^{-k - \min\{k, \lceil \frac{k+2}{2} \rceil \}} + k 2^{-k-\min\{k, \lceil \frac{k+3}{2} \rceil \}} ~, \] and $(a(j))_{j \ge 0}$ take values as in the optimal solution given by Theorem~\ref{thm:lp-solution}. Importantly: \[ a(0) \approx 0.2403 > 0.24 ~. \] Further recall a remark after Theorem~\ref{thm:two-way-online-primal-dual-analysis} that the guarantee in Eqn.~\eqref{eqn:relaxed-ocs} holds almost trivially for three or more consecutive subsequences and for two subsequences whose total lengths are more than three. Hence, the proof will first handle the remaining cases before substantiating the remark. \paragraph{One Subsequence.} In this case the second term in Eqn.~\eqref{eqn:relaxed-ocs} disappears, so we will upper bound the probability by $p(k)$ alone. Suppose that $t_1, t_2, \dots, t_k$ are the rounds in the consecutive subsequence. Consider the number of them that sample $e^{t_i}=e$. If there are at least two, the flag $\tau_e$ must be $1$ in at least one of them, and by definition Algorithm~\ref{alg:relaxed-ocs} selects $e$ there. If exactly one out of the $k$ rounds samples $e^{t_i} = e$, which happens with probability $k 2^{-k}$ over the randomness of $e^{t_i}$'s, Lemma~\ref{lem:relaxed-ocs-conditional-probability-bound} indicates that the probability of never selecting $e$ in the $k-1$ rounds with $e^{t_i} \ne e$, \emph{conditioned on the realized $e^{t_i}$'s}, is at most $2^{-\min \{k-1, \lceil \frac{k+1}{2} \rceil\}}$. Further, over the randomness of the initial flag $\tau_e$, the round with $e^{t_i} = e$ selects $e$ with probability half, independent to the realization of the other $k-1$ rounds. In sum, this case happens with probability at most: \[ k 2^{-k - \min \{ k, \lceil \frac{k+3}{2} \rceil \}} ~. \] Finally, if none of the $k$ rounds samples $e^{t_i} = e$, which happens with probability $2^{-k}$ over the randomness of $e^{t_i}$'s, Lemma \ref{lem:relaxed-ocs-conditional-probability-bound} indicates that the probability of never selecting $e$ in these round is at most $2^{-\min\{k,\lceil\frac{k+2}{2}\rceil\}}$. In sum, this case happens with probability at most: \[ 2^{-k - \min \{ k, \lceil \frac{k+2}{2} \rceil \}} ~. \] Summing the probability bounds in the last two cases gives exactly $p(k)$. \paragraph{Two Subsequences, Two Rounds.} By Lemma~\ref{lem:relaxed-ocs-conditional-probability-bound}, the probability of not selecting element $e$ in the two rounds is at most $\frac{1}{4}$ It then follows by $p(2) = \frac{3}{16}$ and by $a(0) > 0.24$. \paragraph{Two Subsequences, Three or More Rounds.} At least one subsequence must have at least two rounds. If two neighboring rounds in the same subsequence both sample $e^t = e$, which happens with probability $\frac{1}{4}$, flag $\tau_e$ must be $1$ in one of them and Algorithm~\ref{alg:relaxed-ocs} selects $e$ in that round. Otherwise, Lemma~\ref{lem:relaxed-ocs-conditional-probability-bound} indicates that the probability of never selecting $e$ in these three or more rounds is at most $\frac{1}{8}$. In total, the probability is at most $(1-\frac{1}{4}) \frac{1}{8}=\frac{3}{32}$. It then follows by $a(0) > 0.24$. \paragraph{Three or More Subsequences.} Lemma~\ref{lem:relaxed-ocs-conditional-probability-bound} indicates that the probability of never selecting $e$ in these three or more rounds is at most $\frac{1}{8}$. It then follows by $a(0) > 0.24$, since the second term in Eqn.~\eqref{eqn:relaxed-ocs} sum to at least $a(0)$. \end{proof} \begin{comment} \subsubsection{Optimizing the Gain-Sharing Parameters for n-way(temporary)} \begin{claim} For any sequence $f_l\in [0,1]$ such that $f_0=0$ and $\forall k\geq 0, 1-f_{k+1}\leq (1-\frac{1}{n})(1-f_k)$, \begin{align} \Gamma &= 1-\frac{1}{n+1}\sum_{l=0}^{\infty} (1-\frac{1}{n+1})^{l}(1-f_l)\\ b(k) &= \frac{1}{n}(1+\frac{1}{n})^k\Gamma - \frac{1}{n}\sum_{l=0}^{k-1} (1+\frac{1}{n})^{k-l-1}(f_{l+1}-f_l) & \forall k\geq 0\\ a(k) &= f_{k+1}-f_k-b(k)&\forall k\geq 0 \end{align} is a feasible solution of \begin{align} & a(k)+b(k)\leq f_{k+1}-f_k& \forall k\geq 0 \label{pre:lp1}\\ & \sum_{l=0}^{k-1} a(l) + n\cdot b(k)\geq \Gamma& \forall k\geq 0\label{pre:lp2}\\ & b(k)\geq b(k+1)&\forall k\geq 0\label{pre:lp3}\\ & a(k),b(k)\geq 0& \forall k\geq 0\label{pre:lp4} \end{align} \end{claim} \begin{proof} $a(k)+b(k)=f_{k+1}-f_k$ implies (\ref{pre:lp1}). Observe that for all $k\geq 0$ \begin{align} b(k+1)=(1+\frac{1}{n})b(k)-\frac{1}{n}(f_{k+1}-f_k) = b(k)-\frac{1}{n}a(k) \end{align} and thus $\sum_{l=0}^{k-1} a(l)+n\cdot b(k)$ is constant. Since $n\cdot b(0)=\Gamma$, (\ref{pre:lp2}) holds. Further, because of the recursion, (\ref{pre:lp4}) directly implies (\ref{pre:lp3}). Next, we prove (\ref{pre:lp4}). For any $k\geq 0$, as a result of $f_0=0$ and $1-f_{k+1}\leq (1-\frac{1}{n})(1-f_k)$, \begin{align} b(k) &\geq \frac{1}{n}(1+\frac{1}{n})^k\Gamma - \frac{1}{n}\sum_{l=0}^{\infty} (1+\frac{1}{n})^{k-l-1}(f_{l+1}-f_l)\\ &= \frac{1}{n}(1+\frac{1}{n})^k\Big[1-\frac{1}{n+1}\sum_{l=0}^\infty (1-\frac{1}{n+1})^l(1-f_l)-\sum_{l=0}^{\infty}(1-\frac{1}{n+1})^{l+1}((1-f_l)-(1-f_{l+1}))\Big]\\ &= \frac{1}{n}(1+\frac{1}{n})^k\sum_{l=0}^{\infty} \Big[-\frac{1}{n+1}(1-\frac{1}{n+1})^l-(1-\frac{1}{n+1})^{l+1}+(1-\frac{1}{n+1})^l\Big](1-f_l)\\ &=0\\ b(k) &\leq \frac{1}{n}(1+\frac{1}{n})^k\Big[1-\frac{1}{n+1}\sum_{l=0}^{k-1} (1-\frac{1}{n+1})^l(1-f_l)-\sum_{l=0}^{k-1}(1-\frac{1}{n+1})^{l+1}(f_{l+1}-f_l)\Big]\\ &= \frac{1}{n}(1+\frac{1}{n})^k\Big[1-\frac{1}{n+1}\sum_{l=0}^{k-1} (1-\frac{1}{n+1})^l(1-f_l)-\sum_{l=0}^{k-1}(1-\frac{1}{n+1})^{l+1}((1-f_l)-(1-f_{l+1}))\Big]\\ &= \frac{1}{n}(1-f_{k}) + \frac{1}{n}(1+\frac{1}{n})^k\sum_{l=0}^{k-1} \Big[-\frac{1}{n+1}(1-\frac{1}{n+1})^l-(1-\frac{1}{n+1})^{l+1}+(1-\frac{1}{n+1})^l\Big](1-f_l)\\ &= \frac{1}{n}(1-f_k)\leq (1-f_k)-(1-f_{k+1})=f_{k+1}-f_k \end{align} Therefore, (\ref{pre:lp4}) follows the above discussion. \end{proof} \begin{claim} For any sequence $f_l\in [0,1]$ such that $f_0=0$ and $\forall k\geq 0, 1-f_{k+1}\leq (1-\frac{1}{n})(1-f_k)$, and any feasible solution of (\ref{pre:lp1}),(\ref{pre:lp2}),(\ref{pre:lp3}),(\ref{pre:lp4}), \[ \Gamma \leq 1-\frac{1}{n+1}\sum_{l=0}^{\infty} (1-\frac{1}{n+1})^{l}(1-f_l). \] \end{claim} \begin{proof} Suppose there is a feasible solution that violate the bound of $\Gamma$. Under this assumption, we shall inductively prove that $b(k)\geq \frac{\Gamma}{n}-\sum_{i=0}^{k-1} \frac{a(i)}{n} \geq B_k$, where \begin{align} B_0&=\frac{\Gamma}{n} & \\ B_k&=(1+\frac{1}{n})B_{k-1}-\frac{f_k-f_{k-1}}{n} & \forall k\geq 1 \label{pre:bound} \end{align} The base case is clear from (\ref{pre:lp2}) when $k=0$. Suppose the invariant holds for $k-1$ and consider the case of $k$. From (\ref{pre:lp1}) for $k-1$ and (\ref{pre:lp2}) for $k$, \begin{align} \frac{\Gamma}{n} - \sum_{i=0}^{k-1} \frac{a(i)}{n} \geq B_{k-1} - \frac{a(k-1)}{n} \geq B_{k-1} - \frac{f_k-f_{k-1}-b(k-1)}{n} \geq (1+\frac{1}{n})B_{k-1} - \frac{f_k-f_{k-1}}{n} \end{align} Let $c=\Gamma-(1-\frac{1}{n+1}\sum_{l=0}^{\infty} (1-\frac{1}{n+1})^l(1-f_l))$. By the assumption, $c>0$. By telescoping (\ref{pre:bound}), we have \begin{align} B_k&=(1+\frac{1}{n})^k\frac{\Gamma}{n} - \sum_{l=1}^k \frac{f_k-f_{k-1}}{n} (1+\frac{1}{n})^{k-l}\\ &= \frac{(1+\frac{1}{n})^k}{n} \Big[c+1-\frac{1}{n+1}\sum_{l=0}^{\infty} (1-\frac{1}{n+1})^l(1-f_l)-\sum_{l=1}^{k} (1-\frac{1}{n+1})^l(f_l-f_{l-1})\Big] \end{align} As a result of $\lim_{k\to \infty} f_{k+1}-f_k=0$ and \begin{align} \lim_{k\to\infty} n(1+\frac{1}{n})^{-k}B_k-c &= 1-\frac{1}{n+1}\sum_{l=0}^{\infty} (1-\frac{1}{n+1})^l(1-f_l)-\sum_{l=1}^{\infty} (1-\frac{1}{n+1})^l(f_l-f_{l-1})\\ &= 1-\frac{1}{n+1}\sum_{l=0}^{\infty} (1-\frac{1}{n+1})^l(1-f_l)-\sum_{l=1}^{\infty} (1-\frac{1}{n+1})^l(1-f_{l-1}-(1-f_l))\\ &= 1-\sum_{l=0}^{\infty} \Big[\frac{1}{n+1}(1-\frac{1}{n+1})^l+(1-\frac{1}{n+1})^{l+1}-(1-\frac{1}{n+1})^l\Big](1-f_l) - 1 = 0, \end{align} which means $\lim_{k\to\infty} B_k=+\infty$, there exists some $k_0>0$ such that $b(k_0)\geq B_{k_0}\geq f_{k_0+1}-f_{k_0}$. Further, by (\ref{pre:lp1}), $a(k_0)<0$, which violates (\ref{pre:lp4}). \end{proof} \begin{theorem} For any $p:\mathbb{N}\mapsto [0,1]$ such that $\forall \ell\geq 0, p(\ell+1)\leq \frac{1}{2}p(\ell)$, with $p$-amortized OCS, there is a two-choice edge-weighted online bipartite matching algorithm whose competitive ratio is at least \[ 1-\sum_{\ell=0}^{+\infty} \Big(\frac{2}{3}\Big)^{\ell}p(\ell) ~. \] \end{theorem} Note that any $\gamma$-OCS is a $p$-amortized OCS with \[ p(k) = 2^{-k} (1-\gamma)^{\max(0, k-1)} ~, \] we reach the following corollary. \begin{corollary} With $\gamma$-OCS, there is a two-choice edge-weighted online bipartite matching algorithm whose competitive ratio is at least: \[ 1-\sum_{\ell=0}^{+\infty} \Big(\frac{2}{3}\Big)^{\ell}2^{-\ell}(1-\gamma)^{\max(0, \ell-1)} = \frac{1}{2} + \frac{\gamma}{12+6\gamma} ~. \] \end{corollary} \end{comment} \subsection{Multi-way Semi-OCS and Unweighted and Vertex-weighted Online Bipartite Matching} \label{sec:multi-way-semi-ocs-matching} This subsection introduces an algorithm \textsc{Balance-OCS}\xspace that combines an unbounded variant of the \textsc{Balance}\xspace algorithm~\cite{KalyanasundaramP:TCS:2000, MehtaSVV:JACM:2007} and a multi-way semi-OCS. The former assigns one unit of masses to the offline neighbors of each online vertex. The latter then selects one of them to which the online vertex will match. We shall analyze its competitive ratio in the unweighted and vertex-weighted online bipartite matching problems. \textsc{Balance}\xspace is parameterized by a non-increasing discounting function $b : [0, +\infty) \to [0, 1]$. For each online vertex $v$, it continuously assigns one unit of masses to $v$'s neighbors, \footnote{In the context of fractional online matching, \textsc{Balance}\xspace fractionally matches $v$ to its neighbors.} prioritizing the ones with the largest discounted weight $w_u b(y_u)$ where $y_u$ denotes the total mass assigned to $u$ so far. To describe it as an algorithm instead of a continuous process, for any offline vertex $u$ and any threshold marginal utility $\theta \ge 0$, define: \begin{equation} \label{eqn:multi-way-matching-mass-update} y_u(\theta) = b^{-1} \Big( \frac{\theta}{w_u} \Big) ~. \end{equation} We will explain shortly how to interpret the algorithm if $b$ is not continuous or is not strictly increasing, i.e., when the inverse function is not well-defined. In any case, the discount function $b$ used by our algorithm is continuous and strictly monotone. Define $y_u(\theta) = 0$ if $w_u b(0) < \theta$, e.g., when $\theta > w_u$. Let $z^+$ denote $\max \{ z , 0 \}$. Then, we may equivalently interpret the \textsc{Balance}\xspace algorithm as choosing a threshold $\theta$ such that: \begin{equation} \label{eqn:multi-way-matching-mass-total} \sum_{u : (u, v) \in E} \big( y_u(\theta) - y_u \big)^+ = 1 ~, \end{equation} and then assigning mass $\big( y_u(\theta) - y_u \big)^+$ to each vertex $u$. For a discount function $b$ whose inverse is not well-defined, let $y_u^-(\theta) = \sup \{ y \ge 0 : w_u b(y) < \theta \}$ and $y_u^+(\theta) = \inf \{ y \ge 0 : w_u b(y) > \theta \}$. The \textsc{Balance}\xspace algorithm chooses an appropriate threshold $\theta$ and choose $y_u(\theta) \in [y_u^-(\theta), y_u^+(\theta)]$ to satisfy Eqn.~\eqref{eqn:multi-way-matching-mass-total}. The original \textsc{Balance}\xspace algorithm cannot assign more than one unit of total mass to any offline vertex, which introduces boundary considerations that complicate the above description. In our setting, however, the masses are merely input of the multi-way semi-OCS, and therefore the total mass of an offline vertex could be arbitrarily large. \begin{algorithm}[t] \caption{\textsc{Balance-OCS}\xspace} \label{alg:unbounded-balanced} \begin{algorithmic} \State \textbf{State variables:} (for each offline vertex $u$) \begin{itemize} \item Total mass $y_u$ allocated to offline vertex $u$ far; initially, $y_u=0$. \end{itemize} \State \textbf{On the arrival of an online vertex $v\in R$:} \begin{enumerate} \item Find threshold $\theta \in [0, \infty)$ that satisfies Eqn.~\eqref{eqn:multi-way-matching-mass-total}. \item For each neighbor $u$, let $x_u^v = \big( y_u(\theta) - y_u \big)^+$ be its mass in this round. \item Match $v$ to the neighbor that the multi-way semi-OCS selects, with $\vec{x}^v$ as the mass vector in this round. \end{enumerate} \end{algorithmic} \end{algorithm} \begin{theorem} \label{thm:multi-way-semi-ocs-matching} Suppose that $p : [0,\infty) \to [0,1]$ is decreasing and differentiable, and $p(0) = 1$. Then, unbounded \textsc{Balance}\xspace with a $p$-multi-way semi-OCS (Algorithm~\ref{alg:unbounded-balanced}) is $\Gamma$-competitive for unweighted and vertex-weighted online bipartite matching, where the competitive ratio $\Gamma$ and the corresponding discount function $b$ are from an optimal solution of the following continuous LP : \begin{align} \text{\rm maximize} \quad & \Gamma \tag{\textsc{Balance LP}\xspace} \\[1ex] \text{\rm subject to} \quad & a(y) + b(y) \leq -p'(y) && \forall y \geq 0 \label{eqn:multi-way-lp-gain-split} \\%[1ex] & \int_0^y a(z) dz + b(y) \ge \Gamma && \forall y \geq 0 \label{eqn:multi-way-approximate-dual-feasible} \\ & b(y') \le b(y) && \forall y' \ge y \label{eqn:multi-way-lp-monotone} \\[1.5ex] & a(y), b(y) \ge 0 && \forall y \ge 0 \notag \end{align} \end{theorem} \begin{proof} By the guarantee of $p$-multi-way semi-OCS, each offline vertex $u$ is matched by unbounded \textsc{Balance}\xspace with probability at least: \[ 1 - p\big( y_u \big) ~. \] Therefore, the expected total weight of the matched vertices is at least: \[ \textsc{Alg} \defeq \sum_{u \in L} w_u \big( 1 - p (y_u) \big) ~. \] Similar to the competitive analysis of the two-choice algorithm (Algorithm~\ref{alg:ocs-matching}), for every online vertex $v$ we will distribute the increase of $\textsc{Alg}$ between vertex $v$ and its offline neighbors. Let $\alpha_u$ and $\beta_v$ be the distributed gain of any offline vertex $u$ and any online vertex $v$. They are initially zero. In the round of online vertex $v$, for each offline neighbor $u$, increase $\alpha_u$ by: \[ w_u \int_{y_u}^{y_u + x_u^v} a(z) dz ~, \] where $y_u$ is the value right before $v$ arirves. Further, let $\beta_v$ be: \[ \beta_v \defeq \sum_{u : (u, v) \in E} w_u \int_{y_u}^{y_u + x_u^v} b(z) dz ~. \] \paragraph{Feasibility of the Charging Rule.} The total gain distributed above is upper bounded by the increase in $\textsc{Alg}$ because: \begin{align} \sum_{u : (u, v) \in E} w_u \int_{y_u}^{y_u + x_u^v} \big( a(z) + b(z) \big) dz & \le \sum_{u : (u, v) \in E} w_u \int_{y_u}^{y_u + x_u^v} -p'(z) dz \tag{Eqn.~\eqref{eqn:multi-way-lp-gain-split}} \\ & = \sum_{u : (u, v) \in E} w_u \Big( \big( 1 - p(y_u + x_u^v) \big) - \big( 1 - p(y_u) \big) \Big) ~. \end{align} \paragraph{Invariant of Offline Gain.} By definition, for any offline vertex $u$: \[ \alpha_u = w_u \int_0^{y_u} a(z) dz ~. \] \paragraph{Invariant of Online Gain.} For any online vertex $v$, since the algorithm prefers neighbors with larger $w_u b(y_u)$ and assigns one unit of mass, we get that for any $v$'s neighbor $u$: \[ \beta_v \ge w_u b(y_u) ~. \] This holds for the final value of $y_u$ because $y_u$ only increase over time and the discount function $b$ is non-increasing. \paragraph{$\Gamma$-Approximate Equilibrium.} The gains satisfy an approximate equilibrium condition in the sense that for any edge $(u, v)$, the total gain of $u$ and $v$ is at least $\Gamma$ times the vertex-weight $w_u$: \begin{align} \alpha_u + \beta_v & \ge w_u \int_0^{y_u} a(z) dz + w_u b(y_u) \tag{Invariants} \\ & \ge \Gamma w_u ~. \tag{Eqn.~\eqref{eqn:multi-way-approximate-dual-feasible}} \end{align} Then, for any optimal matching $M \subseteq E$, unbounded \textsc{Balance}\xspace is $\Gamma$-competitive because: \begin{align} \textsc{Alg} & \ge \sum_{u \in L} \alpha_u + \sum_{v \in R} \beta_v \\ & \ge \sum_{(u, v) \in M} \big( \alpha_u + \beta_v \big) \\ & \ge \Gamma \sum_{(u, v) \in M} w_u ~. \end{align} \end{proof} The LP in Theorem~\ref{thm:multi-way-semi-ocs-matching} has continuously many variables and constraints. Fortunately, we have an explicit optimal solution for most natural $p$ functions. \begin{theorem} \label{thm:multi-way-lp-solution} Suppose that function $p : [0,\infty) \to [0,1]$ is decreasing, convex, and differentiable, and $p(0) = 1$. Then, an optimal solution to the \textsc{Balance LP}\xspace is: \begin{align} \Gamma & = \int_{0}^\infty e^{-z} \big( 1 - p(z) \big) dz ~; \\ b(y) & = -e^y\int_{y}^\infty p'(z)e^{-z}dz & \forall y\geq 0 ~; \\[1ex] a(y) & = -p'(y)-b(y)&\forall y\geq 0 ~. \end{align} \end{theorem} The proof of this theorem is deferred to Appendix~\ref{app:multi-way-lp-solution}. \begin{corollary} \label{cor:multi-way-matching} Unbounded \textsc{Balance}\xspace with the multi-way semi-OCS from Theorem~\ref{thm:multiway-ocs} in Section~\ref{sec:multi-way} is at least $0.593$-competitive for unweighted and vertex-weighted online bipartite matching. \end{corollary} \section{Acknowledgment} We thank Zhihao Gavin Tang and Hu Fu for helpful discussions on online contention resolution schemes. \appendix \section{Missing Proofs in Section~\ref{sec:semi-ocs}} \label{app:semi-ocs} \subsection{Positive Correlation in $3$-Way Sampling without Replacement} \label{app:positive-correlation} Consider the following counter-example which shows that there could be positive correlation in $3$-way (unweighted) sampling without replacement. The elements are integers from $1$ to $9$. It has $7$ rounds: \[ \mathcal{E}^1 = \mathcal{E}^2 = \big\{ 1, 4, 5 \big\}, \mathcal{E}^3 = \mathcal{E}^4 = \big\{ 2, 6, 7 \big\}, \mathcal{E}^5 = \mathcal{E}^6 = \big\{ 3, 8, 9 \big\}, \mathcal{E}^7 = \big\{ 1, 2, 3 \big\} ~. \] Recall that $\mathcal{U}^t$ denotes the subset of unselected elements after round $t$, and thus $e \in \mathcal{U}^t$ denotes the event that element $e$ remains unselected after round $t$. Further $\mathcal{U} = \mathcal{U}^7$ denotes the subset of unselected elements at the end. On the one hand: \[ \mathbf{Pr} \big[ 1, 2 \in \mathcal{U} \big] = \mathbf{Pr} \big[ s^1, s^2 \ne 1 \big] \mathbf{Pr} \big[ s^3, s^4 \ne 2 \big] \mathbf{Pr} \big[ s^5, s^6 \ne 3 \big] \mathbf{Pr} \big[ s^7 = 3 \mid 1, 2, 3 \in \mathcal{U}^6 \big] = \Big( \frac{1}{3} \Big)^4 = \frac{1}{81} ~. \] On the other hand: \begin{align} \mathbf{Pr} \big[ 1 \in \mathcal{U} \big] & = \mathbf{Pr} \big[ s^1, s^2 \ne 1 \big] \Big( \mathbf{Pr} \big[ s^3, s^4 \ne 2 \big] \mathbf{Pr} \big[ s^5, s^6 \ne 3 \big] \mathbf{Pr} \big[ s^7 \ne 1 \mid 1, 2, 3 \in \mathcal{U}^6 \big] \\ & \hspace{90pt} + \mathbf{Pr} \big[ s^3, s^4 \ne 2 \big] \mathbf{Pr} \big[ 3 \in \big\{ s^5, s^6 \big\} \big] \mathbf{Pr} \big[ s^7 \ne 1 \mid 1, 2 \in \mathcal{U}^6, 3 \notin \mathcal{U}^6 \big] \\ & \hspace{90pt} + \mathbf{Pr} \big[ 2 \in \big\{ s^3, s^4 \big\} \big] \mathbf{Pr} \big[ s^5, s^6 \ne 3 \big] \mathbf{Pr} \big[ s^7 \ne 1 \mid 1, 3 \in \mathcal{U}^6, 2 \notin \mathcal{U}^6 \big] \Big) \\ & = \frac{1}{3} \left( \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{2}{3} + \frac{1}{3} \cdot \frac{2}{3} \cdot \frac{1}{2} + \frac{2}{3} \cdot \frac{1}{3} \cdot \frac{1}{2} \right) = \frac{8}{81} ~. \end{align} Further by symmetry: \[ \mathbf{Pr} \big[ 2 \in \mathcal{U} \big] = \frac{8}{81} ~. \] Therefore: \[ \frac{\mathbf{Pr}[1, 2 \in \mathcal{U}]}{\mathbf{Pr}[1 \in \mathcal{U}] \mathbf{Pr}[2 \in \mathcal{U}]} = \frac{81}{64} > 1 ~. \] \subsection{Proof of Theorem~\ref{thm:semi-ocs-hardness}} \label{app:semi-ocs-hardness} We shall construct a distribution of instances and prove the desired probability bound holds on average. By considering an appropriate distribution of instance, we ensure that the randomness of the instance dictates the selection result. Consider elements $\mathcal{E} = \{ e^0_1 = 1, e^0_2 = 2, \dots, e^0_{2^k} = 2^k \}$. Further for $i$ from $1$ to $k$, recursively define $e^i_j$ to be either $e^{i-1}_{2j-1}$ or $e^{i-1}_{2j}$ uniformly at random. The instance has pairs $\{ e^i_{2j-1}, e^i_{2j} \}$ for all $0 \le i \le k-1$ and all $1 \le j \le 2^{k-i}$, in ascending order of $i$; the order with respect to $j$ for any fixed $i$ is unimportant yet for concreteness we define it to be in ascending order as well. In other words, this is a knockout-tournament-like instance. First partition the $2^k$ elements into $2^{k-1}$ pairs in lexicographical order. We shall refer to these pairs as the first stage of the instance. Then, a randomly chosen ``winner'' from each pair advances to the next stage. Repeat this process until we have the final ``winner'', denoted as $e^k_1$ by the construction above. We shall refer to the pairs defined with respect to elements $e^{i-1}_j$, $1 \le j \le 2^{k-i+1}$, as the $i$-th stage of the instance. For example, consider $2^3$ elements $\{1, 2, \dots, 8\}$. A possible realization of the random instance proceeds as $\{ 1, 2 \}, \{3, 4\}, \{5, 6\}, \{7, 8\}, \{1, 3\}, \{5, 8\}, \{3, 5\}$. See Figure 1 for an illustration. \begin{figure} \caption{Knockout tournament-like input sequence.} \label{tournament} \end{figure} Next we show that with probability at least $2^{-2^k+1}$, the final ``winner'' $e^k_1$ is never selected by the algorithm, despite its $k$ appearances. Here the probability space is over both the randomness of the algorithm and that of the instance. In fact, we shall inductively prove the following stronger invariant; the above claim is the special case when $i = k$. \paragraph{Invariant:} For any $1 \le i \le k$, after processing the pairs involving elements $e^{i-1}_j$'s, i.e., after the first $i$ stages of the tournament, all elements $e^i_j$, $1 \le j \le 2^{k-i}$, in the next stage remain unselected with probability at least: \[ 2^{-2^k+2^{k-i}} ~. \] The base case when $i = 0$ is vacuously true. Next suppose that the invariant holds for $i-1$, and consider the case of $i$. Below are a set of sufficient conditions under which all elements $e^i_j$ are unselected after the first $i$ stages: \begin{enumerate} \item for any $1 \le j \le 2^{k-i+1}$, $e^{i-1}_j$ is unselected after the first $i-1$ stages; \item for any $1 \le j \le 2^{k-i}$, $e^i_j \in \{ e^{i-1}_{2j-1}, e^{i-1}_{2j} \}$ is \emph{not} the one selected by the algorithm for the pair. \end{enumerate} By the construction of the random instance, the two events are independent. The first holds with probability at least $2^{-2^k+2^{k-i+1}}$ by the inductively hypothesis. The second holds with probability $2^{-2^{k-i}}$ by the construction of the random instance. Hence, the probability that all elements $e^i_j$ remain unselected after the first $i$ stages is at least: \[ 2^{-2^k+2^{k-i+1}} \cdot 2^{-2^{k-i}} = 2^{-2^k+2^{k-i}} ~. \] \section{Missing Proofs in Section~\ref{sec:multi-way}} \label{app:multi-way} \subsection{Proof of Lemma~\ref{lemma:multiway-cubic}} \label{app:multiway-cubic} Both sides equal $1$ when $x = 0$. The rest of the proof considers $0<x<1$. Let $t=\frac{y}{x}$. We have: \begin{align} \frac{x}{1-x} w(y)+1-\frac{w(y+x)}{w(y)} =&\frac{x}{1-x} w(tx)+1-\frac{w(tx+x)}{w(tx)}\\ =& \int_{0}^{x}\left( B(x',t) w(t x')-A(x',t)\frac{w(tx'+x')}{w(tx')}\right)dx'\\ =&\int_{0}^{x} B(x',t) w(t x')\left(1-\frac{A(x',t) w(tx'+x')}{B(x',t) w(tx')^2}\right) dx'\\ =&\int_{0}^{x} B(x',t) w(t x')\int_{0}^{x'}\frac{C(x'',t)w(tx''+x'')}{B(x'',t)^2 w(tx'')^2}dx'' dx', \end{align} where: \begin{align} Q(x) & =\frac{1}{w(x)}\frac{dw(x)}{dx}=1+x+3c x^2 ~, \\[1ex] A(x,t) & =(t+1)Q(t x+x)-t Q(t x) ~, \\[1ex] B(x,t) & = \frac{1}{(1-x)^2}+\frac{tx}{1-x} Q(tx) ~, \\ C(x,t) & =A(x,t)\frac{dB}{dx}(x,t)-B(x,t)\frac{dA}{dx}(x,t)-A(x,t)B(x,t)\left((t+1)Q(t x+x)-2t Q(t x)\right) ~. \end{align} To prove the lemma, i.e.: \[ \frac{w(y+x)}{w(y)}\le \frac{x}{1-x} w(y)+1 ~, \] for any $0 < x < 1$ and $y \ge 0$, we only need to prove $C(x,t) \ge 0$ for any $0 < x < 1$ and any $t \ge 0$. It is equivalent to show that for any $x, t\ge 0$: \[ \frac{(x+1)^6}{x} C\left(\frac{x}{1+x},t\right)\ge 0 ~. \] By computation: \[ \frac{(x+1)^6}{x} C\left(\frac{x}{1+x},t\right)=\sum_{i=0}^8 x^i P_i(t) ~, \] where \begin{align} P_0(t)=& -18 c t^2 - 18 c t - 6 c \\ &+ 6t^2 + 2,\\ P_1(t)=& 12 c t^3 - 126 c t^2 - 126 c t - 42 c \\& + 8 t^3 + 36 t^2 + 17,\\ P_2(t)=& 42 c t^4 + 54 c t^3 - 408 c t^2 - 390 c t - 126 c \\ &+ 5 t^4 + 38 t^3 + 88 t^2 + 5 t + 64,\\ P_3(t)= &54 c^2 t^5 - 27 c^2 t^4 - 144 c^2 t^3 - 135 c^2 t^2 - 54 c^2 t - 9 c^2 \\ &+ 36 c t^5 + 156 c t^4 + 36 c t^3 - 780 c t^2 - 672 c t - 204 c \\ &+ 2 t^5 + 17 t^4 + 72 t^3 + 115 t^2 + 30 t + 139,\\ P_4(t)= &63 c^2 t^6 + 135 c^2 t^5 - 306 c^2 t^4 - 729 c^2 t^3 - 594 c^2 t^2 - 225 c^2 t - 36 c^2 \\ &+ 21 c t^6 + 75 c t^5 + 183 c t^4 - 117 c t^3 - 882 c t^2 - 654 c t - 180 c \\ &+ 6 t^5 + 21 t^4 + 72 t^3 + 91 t^2 + 74 t + 190,\\ P_5(t)= &72 c^2 t^7 - 144 c^2 t^5 - 783 c^2 t^4 - 1242 c^2 t^3 - 927 c^2 t^2 - 342 c^2 t - 54 c^2 \\ &+ 42 c t^6 + 42 c t^5 + 96 c t^4 - 174 c t^3 - 510 c t^2 - 300 c t - 66 c \\ &+ 6 t^5 + 11 t^4 + 44 t^3 + 49 t^2 + 96 t + 167,\\ P_6(t)= &81 c^3 t^8 - 162 c^3 t^7 - 459 c^3 t^6 - 405 c^3 t^5 - 162 c^3 t^4 - 27 c^3 t^3 \\ &+ 72 c^2 t^7 - 63 c^2 t^6 - 171 c^2 t^5 - 531 c^2 t^4 - 801 c^2 t^3 - 603 c^2 t^2 - 225 c^2 t - 36 c^2\\ &+ 21 c t^6 + 3 c t^5 + 57 c t^4 - 51 c t^3 - 66 c t^2 + 12 c t + 18 c \\ &+ 2 t^5 + 2 t^4 + 18 t^3 + 19 t^2 + 69 t + 92,\\ P_7(t)=& 54 c^2 t^5 - 27 c^2 t^4 - 144 c^2 t^3 - 135 c^2 t^2 - 54 c^2 t - 9 c^2 \\ &+ 30 c t^4 + 12 c t^3 + 60 c t^2 + 66 c t + 24 c \\ &+ 4 t^3 + 4 t^2 + 26 t + 29,\\ P_8(t)=& 18 c t^2 + 18 c t + 6 c \\ &+ 4 t + 4. \end{align} When $c=\frac{4-2\sqrt{3}}{3}$, we can verify, using numerical computation software, that $P_i(t)$ has positive leading coefficient and no non-negative real roots for $i=1,2,\ldots, 8$, and: \[ P_0(t)=\left( 4 \sqrt{3}-6\right) \left(1 - \sqrt{3} t\right)^2\ge 0 ~. \] Therefore, for any $x, t \ge 0$: \[ \frac{(x+1)^6}{x} C\left(\frac{x}{1+x},t\right)\ge 0 ~. \] So the lemma holds. \subsection{Proof of Lemma~\ref{lemma:multiway-condition}} \label{app:multiway-condition} We will use the following lemma which follows by the definition of the weight function $w$ in Eqn.~\eqref{eqn:multiway-weight}. \begin{lemma} \label{lem:multiway-weight-log-convex} Function $\log \circ\,w$ is convex in $[0, \infty)$. \end{lemma} If $\sum_{i=1}^k x_i \ge 1$, the left-hand-side is zero so the inequality holds trivially. If $\sum_{i=1}^k x_i = 0$, both sides are equal to $1$ so the inequality also holds. The rest of the proof consider $0 < \sum_{i=1}^k x_i < 1$. Fix any non-negative $y_1, y_2, \dots, y_k \ge 0$. Define function $f:[0,+\infty)^k\to (-\infty,+\infty)$ over $\vec{x} = (x_1, x_2, \dots, x_k)$ be the difference between the logarithms of the two sides, i.e.: \[ f(\vec{x})= \log \frac{1-\sum_{i=1}^k x_i}{\sum_{i=1}^k x_i w(y_i) +1- \sum_{i=1}^k x_i} -\sum_{i=1}^k\log\frac{w(y_i)}{w(y_i+x_i)} ~. \] We first argue that $f$ is convex over a simplex, using the log-convexity of the weight function $w$ (Lemma~\ref{lem:multiway-weight-log-convex}). Consider any $\vec{x} = (x_1, \dots, x_k), \vec{x'} = (x_1', \dots, x_k') \in [0, +\infty)^k$ such that: \[ \sum_{i=1}^k x_i = \sum_{i=1}^k x'_i ~. \] Then, for any $t \in [0,1]$ and the linear $\vec{x''} = t \vec{x} + (1-t) \vec{x'}$, we have: \begin{align} t f(\vec{x})+(1-t) f(\vec{x'}) & = \log\left(1-\sum_{i=1}^k x_i\right)-t \log\left(\sum_{i=1}^k x_i w(y_i) +1- \sum_{i=1}^k x_i\right)\\ & \quad -(1-t)\log\left(\sum_{i=1}^k x'_i w(y_i) +1- \sum_{i=1}^k x_i\right)\\ & \quad -\sum_{i=1}^{k} \log w(y_i) +\sum_{i=1}^k\left( t \log w(y_i+x_i) +(1-t) \log w(y_i+x'_i)\right) ~. \end{align} By the log-convexity of $w$, and by Jensen's inequality on the second and third terms and on the last two terms, this is at least: \[ \log\left(1-\sum_{i=1}^k x_i\right)-\log\left(\sum_{i=1}^k x''_i w(y_i) +1- \sum_{i=1}^k x_i\right)-\sum_{i=1}^{k} \log w(y_i) +\sum_{i=1}^k \log w(y_i+x''_i) = f(\vec{x''}) ~. \] Therefore, we have: \begin{align} f(\vec{x}) & \le \frac{1}{\sum_{i=1}^k x_i} \sum_{i=1}^k x_i f \Big(0,\ldots , 0, \underbrace{\sum_{j=1}^k x_j}_\text{$i$-th entry}, 0,\ldots, 0 \Big) \tag{convexity of $f$ on simplex} \\ & = \frac{1}{\sum_{i=1}^k x_i}\sum_{i=1}^k x_i \bigg(\log\frac{w(y_i+\sum_{j=1}^k x_j)}{w(y_i)}-\log\Big(\frac{\sum_{j=1}^k x_j}{1-\sum_{j=1}^k x_j} w(y_i)+1\Big)\bigg) \\[3ex] & \le 0 ~. \tag{Lemma~\ref{lemma:multiway-cubic}} \end{align} \subsection{A Weaker Version of Lemma~\ref{lemma:multiway-cubic}} \label{app:multiway-cubic-weak} The following lemma is a weak version of Lemma~\ref{lemma:multiway-cubic}. We provide it and a proof that does not involve computer-aided numerical verification. \begin{lemma} For any $y \ge 0$ and any $0 < \delta < 1$: \[ \exp \Big( (y+\delta) + \frac{(y+\delta)^2}{2} + \frac{(y+\delta)^3}{6} \Big) \le \exp \Big( y + \frac{y^2}{2} + \frac{y^3}{6} \Big) + \frac{\delta}{1-\delta} \exp \Big( 2y + y^2 + \frac{y^3}{3} \Big) ~. \] \end{lemma} \begin{proof} The inequality holds with equality when $\delta = 0$. Hence, it suffices to consider the partials of both sides with respect to $\delta$, and to show that the partial of the right-hand-side is larger, i.e.: \[ \Big(1 + (y+\delta) + \frac{(y+\delta)^2}{2}\Big) \exp \Big( (y+\delta)+\frac{(y+\delta)^2}{2}+\frac{(y+\delta)^3}{6} \Big) \le \frac{1}{(1-\delta)^2} \exp \Big( 2y + y^2 + \frac{y^3}{3} \Big) ~. \] Taking logarithm of both sides and rearrange terms, it is equivalent to: \[ \ln \Big( 1 + (y+\delta) + \frac{(y+\delta)^2}{2} \Big) + 2 \ln \big( 1-\delta \big) \le (y-\delta) + \frac{(y-\delta)^2}{2} + \frac{(y-\delta)^3}{6} - \delta^2 - y\delta^2 ~. \] Since $\ln(1-\delta) \le - \delta - \frac{\delta^2}{2} - \frac{\delta^3}{3} - \frac{\delta^4}{4}$ for all $0 < \delta < 1$, after an rearrangement of terms it suffices to show that: \[ \ln \Big( 1 + (y+\delta) + \frac{(y+\delta)^2}{2} \Big) \le (y+\delta) + \frac{(y-\delta)^2}{2} + \frac{y^3}{3} + \frac{2\delta^3}{3} - \frac{(y+\delta)^3}{6} + \frac{\delta^4}{2} ~. \] By the inequality of arithmetic and geometric means: \[ \frac{(y-\delta)^2}{2} + \frac{\delta^4}{2} \ge \big\| y - \delta \big\| \delta^2 ~. \] Hence, we arrive at the final inequality that is sufficient for establishing the inequality regarding the partial derivatives with respect to $y$ and thus, the correct of the lemma: \begin{equation} \label{eqn:multi-way-idealized-transform} \ln \Big( 1 + (y+\delta) + \frac{(y+\delta)^2}{2} \Big) \le (y+\delta) + \big\|y-\delta\big\|\delta^2 + \frac{y^3}{3} + \frac{2\delta^3}{3} - \frac{(y+\delta)^3}{6} ~. \end{equation} \paragraph{Roadmap of Proving Eqn.~\eqref{eqn:multi-way-idealized-transform}.} The naural next step is to upper bound $\ln\big(1+x+\frac{x^2}{2}\big)$ by a polynomial of $x$ to tranform the left-hand-side of the inequality a polynomial over $y$ and $\delta$, just like the right-hand-side. The Taylor series $\ln\big(1+x+\frac{x^2}{2}\big) = x - \frac{x^3}{6} + \frac{x^4}{8} - \frac{x^5}{20} + O(x^7)$ suggests natural upper bounds such as $\ln\big(1+x+\frac{x^2}{2}\big) \le x$ and $\ln\big(1+x+\frac{x^2}{2}\big) \le x - \frac{x^3}{6} + \frac{x^4}{8}$. Unfortuantely, neithor of these bounds proves Eqn.~\eqref{eqn:multi-way-idealized-transform} for all $y \ge 0$ and all $0 < \delta < 1$. In particular, the former fails when $\delta = y$, and the latter leaves a degree-$4$ term that cannot be bounded by the right-hand-side of Eqn.~\eqref{eqn:multi-way-idealized-transform} for sufficiently large $y$. Instead, we shall consider three polynomial upper bounds of $\ln \big(1+x+\frac{x^2}{2}\big)$ depending on the range of $x$ and together they cover all the cases. These upper bounds are from the next lemma, whose proof is deferred to the end of the subsection. \begin{lemma} \label{lem:multi-way-idealized-relaxation} The function: \[ \frac{x - \ln (1+x+\frac{x^2}{2})}{x^3} ~. \] is decreasing for $x > 0$ \end{lemma} \paragraph{Case 1: Use $\ln \big( 1+x+\frac{x^2}{2} \big) \le x$ for $x \ge 0$, when $y \ge (1+\sqrt[3]{2}) \delta$.} In other words, by inequality $\ln \big( 1 + (y+\delta) + \frac{(y+\delta)^2}{2} \big) \le y+\delta$ and $y > \delta$, Eqn.~\eqref{eqn:multi-way-idealized-transform} reduces to: \[ \frac{y^3}{6} - \frac{y^2\delta}{2} + \frac{y\delta^2}{2} - \frac{\delta^3}{2} \ge 0 ~, \] or equivalently: \[ \Big( \frac{y}{\delta} \Big)^3 - 3 \Big( \frac{y}{\delta} \Big)^2 + 3 \frac{y}{\delta} - 3 = \Big( \frac{y}{\delta} - 1 \Big)^3 - 2 \ge 0 ~. \] We remark that this approach can also prove Eqn.~\eqref{eqn:multi-way-idealized-transform} for $y \le c \delta$ for $c \approx 0.83$. We skip it since it is covered by the other cases. \paragraph{Case 2: Use $\ln \big( 1 + x + \frac{x^2}{2} \big) \le x - \frac{x^3}{24}$ for $0 \le x \le \frac{7}{3}$, if $y + \delta \le \frac{7}{3}$.} The stated inequality follows by Lemma~\ref{lem:multi-way-idealized-relaxation} and that for $x = \frac{7}{3}$: \[ \frac{x - \ln (1+x+\frac{x^2}{2})}{x^3} \approx 0.0419 > \frac{1}{24} ~. \] By $\ln \big( 1 + (y+\delta) + \frac{(y+\delta)^2}{2} \big) \le y+\delta - \frac{(y+\delta)^3}{24}$, Eqn.~\eqref{eqn:multi-way-idealized-transform} reduces to: \[ \big\|y-\delta\big\|\delta^2 + \frac{y^3}{3} + \frac{2\delta^3}{3} - \frac{(y+\delta)^3}{8} \ge 0 ~. \] If $y \ge \delta$, the left-hand-side equals: \[ \frac{5}{24} y^3 - \frac{3}{8} y^2 \delta + \frac{5}{8} y \delta^2 - \frac{11}{24} \delta^3 = \frac{1}{24} \big( y-\delta \big) \big( 5y^2 - 4y\delta + 11 \delta^2 \big) \ge 0 ~. \] If $y < \delta$, the left-hand-side equals: \[ \frac{5}{24} y^3 - \frac{3}{8} y^2 \delta - \frac{11}{8} y \delta^2 + \frac{37}{24} \delta^3 = \frac{1}{24} \big(\delta-y\big) \big( 37 \delta^2 + 4\delta y - 5y^2 \big) \ge 0 ~. \] \paragraph{Case 3: Use $\ln \big( 1+x+\frac{x^2}{2} \big) \le x - \frac{x^3}{36}$ for $0 \le x \le 2+\sqrt[3]{2}$, if $\frac{7}{3} \le y+\delta \le 2+\sqrt[3]{2}$.} The stated inequality follows by Lemma~\ref{lem:multi-way-idealized-relaxation} and that for $x = 2+\sqrt[3]{2}$: \[ \frac{x - \ln (1+x+\frac{x^2}{2})}{x^3} \approx 0.028 > \frac{1}{36} ~. \] The argument for this case combines the assumption of $y+\delta > \frac{7}{3}$ and that $0 < \delta < 1$ to derive: \[ y > \frac{4}{3} \delta ~. \] By $\ln \big( 1 + (y+\delta) + \frac{(y+\delta)^2}{2} \big) \le y+\delta - \frac{(y+\delta)^3}{36}$ and by $y > \delta$, Eqn.~\eqref{eqn:multi-way-idealized-transform} reduces to: \[ (y-\delta)\delta^2 + \frac{y^3}{3} + \frac{2\delta^3}{3} - \frac{5(y+\delta)^3}{36} \ge 0 ~. \] Rearranging terms, the left-hand-side equals: \[ \frac{7}{36} y^3 - \frac{5}{12} y^2\delta + \frac{7}{12} y\delta^2 - \frac{17}{36} \delta^3 = \frac{1}{144} \big(5y-6\delta\big)^2 y + \frac{1}{48} \big(y^2-\delta^2\big)y + \frac{17}{48} \delta^2 \big( y - \frac{4}{3} \delta \big) \ge 0 ~. \] The three cases cover all $y \ge 0$ and $0 < \delta < 1$ since the last two cases prove the lemma for any $y, \delta$ that satisfies $y + \delta \le 2 + \sqrt[3]{2}$. For any $y, \delta$ with $y + \delta > 2 + \sqrt[3]{2}$ (and $0 < \delta < 1$) must satisfy $y > (1+\sqrt[3]{2}) \delta$ and therefore is covered by the first case. \end{proof} \begin{proof}[Proof of Lemma~\ref{lem:multi-way-idealized-relaxation}] We shall prove that the derivative is non-positive, i.e.: \[ \frac{d}{dx} \frac{x-\ln(1+x+\frac{x^2}{2})}{x^3} = \frac{3}{x^4} \Big( \ln \big(1+x+\frac{x^2}{2}\big) - x + \frac{x^3}{6(1+x+\frac{x^2}{2})} \Big) \le 0 ~. \] Equivalently, we need to show that: \[ \ln \big(1+x+\frac{x^2}{2}\big) - x + \frac{x^3}{6(1+x+\frac{x^2}{2})} \le 0 ~. \] Since this holds with equality at $x = 0$, it suffices to prove that its derivative is non-positive. This follows by: \[ \frac{d}{dx} \Big( \ln \big(1+x+\frac{x^2}{2}\big) - x + \frac{x^3}{6(1+x+\frac{x^2}{2})} \Big) = - \frac{x^3(1+x)}{6(1+x+\frac{x^2}{2})^2} \le 0 ~. \] \end{proof} \begin{comment} \subsection{Proof of Eqn.~\eqref{eqn:multiway-property-w}} \label{app:multi-way-w} \begin{lemma} For any $y \ge 0$ and any $0 < \delta < 1$: \[ \exp \Big( (y+\delta) + \frac{(y+\delta)^2}{2} + c(y+\delta)^3 \Big) \le \exp \left( y + \frac{y^2}{2} + cy^3 \right) \frac{\sqrt{1+2\delta\left(\exp \left( y + \frac{y^2}{2} + cy^3 \right) - 1\right)}}{1 - \delta} ~. \] where $c = 1 - \frac{\sqrt{7}}{3}$. \end{lemma} \begin{proof} By the concavity of $\log x$, we have: \begin{align} & \log \left( (1 - 2\delta) + 2\delta \exp \left( y + \frac{y^2}{2} + cy^3 \right) \right) \\ & ~\ge~ 2\delta \log \exp \left( y + \frac{y^2}{2} + cy^3 \right) \\ & ~=~ 2\delta \Big( y + \frac{y^2}{2} + cy^3 \Big) \\ & ~\ge~ 2\delta \Big( y + \frac{y^2}{2} \Big) ~. \end{align} Hence, the inequality reduces to \[ \exp \Big( (y+\delta) + \frac{(y+\delta)^2}{2} + c(y+\delta)^3 \Big) \le \exp \left( y + \frac{y^2}{2} + cy^3 + \delta y + \frac{\delta y^2}{2} \right) \frac{1}{1 - \delta} ~. \] Rearranging terms and taking log, \[ \log (1 - \delta) \le -\delta - \frac{\delta^2}{2} - c\delta^3 - 3c\delta^2 y + \Big(\frac{1}{2} - 3c\Big)\delta y^2 ~. \] By the Taylor series \[ \log (1 - \delta) \le -\delta - \frac{\delta^2}{2} - \frac{\delta^3}{3} ~. \] So it suffices to prove \[ \Big(\frac{1}{3} - c\Big)\delta^3 + \Big(\frac{1}{2} - 3c\Big)\delta y^2 \ge 3c\delta^2 y ~. \] For $c = 1 - \frac{\sqrt{7}}{3}$, it follows by AM-GM inequality since \[ \Big(\frac{1}{3} - c\Big)\Big(\frac{1}{2} - 3c\Big) = \Big(\frac{2}{3}c\Big)^2 ~. \] \end{proof} \end{comment} \section{Missing Proofs in Section~\ref{sec:ocs}} \label{app:ocs} \subsection{Proof of Lemma~\ref{lem:pseudo-path}} It suffices to show that the pseudo-paths are pairwise disjoint. Consider an arc $(p,c)$ and a pseudo-path $P$ that involves it. It is clear that $(p,c)$ can be adjacent with at most two arcs: the other in-arc of $c$ (if exists) and the other out-arc of $p$ (if exists). As $P$ is maximal, if $c$ has another in-arc in $G^{\text{ex-ante}}$, it should be adjacent to $(p,c)$ in $P$. Similarly, if $p$ has another out-arc $(p,c')$ in $G^{\text{ex-ante}}$ such that rounds $p,c,c'$ have a common element, it should also be adjacent to $(p,c)$ in $P$. Therefore, for any arc $(p,c)$, the set of its adjacent arcs in any $P$ is fixed according to the ex-ante dependence graph. Hence, for any arc $(p,c)$, there is a unique pseudo-path in the collection that involves it. \subsection{Proof of Lemma~\ref{lem:pseudo-matching}} By the definition of good forests, pseudo-paths and pseudo-matchings, the following statements are equivalent: \begin{enumerate} \item A subgraph of the ex-ante dependence graph is a good forest. \item A subgraph satisfies: each node has at most one in-arc; and there is no node $p$ with two out-arcs $(p,c)$ and $(p,c')$ such that rounds $p,c,c'$ have a common element. \item A subgraph is a union of pseudo-matchings, one for each pseudo-path. \end{enumerate} In particular, the equivalence between the first two follows by the definition of good forests. The equivalence between the last two follows by the definitions of pseudo-paths and pseudo-matchings. \subsection{Proof of Lemma~\ref{lem:pseudo-path-arrival}} \begin{lemma} \label{lem:app-forest-constructor-common-element} For any three different rounds $1\leq p<c<c'\leq T$ such that $(p,c),(p,c')\in E^{\text{ex-ante}}$ and all rounds have a common element, rounds $p,c'$ have the same set of elements, i.e., $\mathcal{E}^p=\mathcal{E}^{c'}$. \end{lemma} \begin{proof} Suppose $\mathcal{E}^p=\{a,b\}$ and the subscript of $(p,c)$ is $a$. Then the subscript of $(p,c')$ is $b$. Since $c'$ is first round involving element $b$ after round $p$, we have $b\notin \mathcal{E}^{c}$. Further by that $p,c,c'$ have a common element, the common element can only be $a$. In sum, $\mathcal{E}^{c'}=\{a,b\}=\mathcal{E}^p$. \end{proof} We shall prove the lemma by contradiction. Suppose for contrary that there is a pseudo-path $P$ and $1 \leq t \leq T$ such that the subset of arcs in $P$ that arrive in the first $t$ rounds is not a sub-pseudo-path. Since all arcs in $P$ arrive after round $T$ and form a single pseudo-path, there must be a round $t'>t$ such that in-arcs of $t'$ concatenate two sub-pseudo-paths. Because of the definition of pseudo-paths and Lemma~\ref{lem:pseudo-path}, for each round $t$, the in-arcs of $t$ are simultaneously added into a pseudo-path such that they are adjacent in the pseudo-path. As a result, there are only two possibilities of the concatenation: \begin{enumerate} \item If $t'$ has only one in-arc, this in-arc is next to only at most one other arc in the pseudo-path by definition. Hence, it cannot concatenate two sub-pseudo-paths. \item If $t'$ has two in-arcs $(p,t'), (p',t')$ and it concatenates two sub-pseudo-paths together in round $t'$, there must be $c,c'<t'$ such that $(p,c), (p',c') \in P$, and further rounds $p,c,t'$ must have a common element and rounds $p',c',t'$ must have a common element. By Lemma~\ref{lem:app-forest-constructor-common-element}, we have $\mathcal{E}^p=\mathcal{E}^t=\mathcal{E}^{p'}$. If $p\neq p'$, the in-arc from the earlier one, e.g., $(p,t')$ shall not exist in the ex-ante dependence graph by definition. If $p=p'$, on the other hand, arcs $(p,c)$ $(p',c)$ do not exist in the ex-ante dependence graph. In fact, they shall be two parallel arcs $(p, t')$ that form a pseudo-path on their own, like the right-most pseudo-path in Figure~\ref{fig:ocs-forest-constructor-pseudo-path}. \end{enumerate} In sum, there is always a contraction in all cases. \subsection{Proof of Lemma~\ref{lem:forest-constructor-automata-stationary}} With the rows and columns in the order of $q_\no, q_{\no^2}, q_\yes$, the transition matrices $P^+$ of $\sigma^+$ and $P^-$ of $\sigma^-$ are as follows: \[ P^+ = \begin{bmatrix} 0 & 1-p & p \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix} ~, \qquad P^- = \begin{bmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ p & 1-p & 0 \end{bmatrix} ~. \] Then, the proof of the stationary distribution follows from the next two equations: \begin{align} \vec{\pi} P^+ &= \bigg( \frac{1}{3-p}, (1-p)\cdot \frac{1}{3-p}, p\cdot \frac{1}{3-p}+\frac{1-p}{3-p} \bigg) = \bigg( \frac{1}{3-p}, \frac{1-p}{3-p}, \frac{1}{3-p} \bigg) = \vec{\pi}~,\\ \vec{\pi} P^- &= \bigg( \frac{1-p}{3-p}+p\cdot \frac{1}{3-p}, (1-p)\cdot \frac{1}{3-p}, \frac{1}{3-p} \bigg) = \bigg( \frac{1}{3-p}, \frac{1-p}{3-p}, \frac{1}{3-p} \bigg) = \vec{\pi}~. \end{align} \subsection{Proof of Lemma \ref{lem:forest-constructor-automata-inverse}} By the definitions of $\sigma^+$ and $\sigma^-$, the state sequences determine the corresponding choice sequence, i.e. $q^i=q_\yes$ if and only if $d_i=\mathsf{M}$ and $\hat{q}^i=q_\yes$ if and only if $\hat{d}_i=\mathsf{M}$. Therefore, it suffices to show that the distributions of the state sequences $(q^i)_{0 \le i \le \ell}$ and $(\hat{q}^i)_{0 \le i \le \ell}$ are the same. With the chain rule: \begin{align} \mathbf{Pr}[q^0, \dots, q^\ell ] &= \prod_{i=0}^{\ell}\mathbf{Pr}[q^i \big\| q^0, \dots, q^{i-1}]~,\\ \mathbf{Pr}[\hat{q}^0, \dots, \hat{q}^{\ell} ] &= \prod_{i=0}^{\ell}\mathbf{Pr}[\hat{q}^i \big\| \hat{q}^0, \dots, \hat{q}^{i-1}]~. \end{align} It suffices to show that for any $1\leq i\leq \ell$: \begin{align} \mathbf{Pr}[q^i \big\| q^0, \dots, q^{i-1}] =\mathbf{Pr}[\hat{q}^i \big\| \hat{q}^0, \dots, \hat{q}^{i-1}]~. \end{align} The case when $i=0$ follows by Lemma \ref{lem:forest-constructor-automata-stationary}, i.e., by that states $q^0,\hat{q}^0$ both follow the common stationary distribution of $\sigma^+, \sigma^-$. Next, we consider the other conditional probabilities. For $i_0\leq i\leq \ell$, it follows directly from the memoryless property of probabilistic automata: \begin{align} &\mathbf{Pr}[q^i \big\| q^0, \dots, q^{i-1}] = \mathbf{Pr}[q^i \big\| q^{i-1}] = \mathbf{Pr}[\hat{q}^i \big\| \hat{q}^{i-1}] = \mathbf{Pr}[\hat{q}^i \big\| \hat{q}^0, \dots, \hat{q}^{i-1}]~. \end{align} For $1\leq i< i_0$, the memoryless property still holds for $q^i$. On the other hand, it can be deduced that the memoryless property also holds for $\hat{q}^i$: \begin{align} \mathbf{Pr}[\hat{q}^i\big\|\hat{q}^0, \dots, \hat{q}^{i-1}] &= \frac{\mathbf{Pr}[\hat{q}^0, \dots,\hat{q}^i]}{\mathbf{Pr}[ \hat{q}^0, \dots, \hat{q}^{i-1}]}\\ &= \frac{\mathbf{Pr}[\hat{q}^0, \dots, \hat{q}^{i-2}\big\|\hat{q}^{i-1}, \hat{q}^i]\cdot \mathbf{Pr}[\hat{q}^{i-1}, \hat{q}^i]}{\mathbf{Pr}[\hat{q}^0, \dots, \hat{q}^{i-2}\big\|\hat{q}^{i-1}]\cdot \mathbf{Pr}[\hat{q}^{i-1}]}\\[1ex] &= \mathbf{Pr}[\hat{q}^i\big\|\hat{q}^{i-1}] ~. \end{align} It remains to verify that the joint distributions of state pairs $(q^i, q^{i-1})$ and $(\hat{q}^i, \hat{q}^{i_1})$ are identical, which follows by: \[ \text{diag}(\vec{\pi})P^+ = \begin{bmatrix} 0 & 1-p & p\\ 0 & 0 & 1-p\\ 1 & 0 & 0 \end{bmatrix} =\Big(\text{diag}(\vec{\pi})P^-\Big)^T~. \] \subsection{Proof of Lemma~\ref{lem:forest-constructor-automata-selection-prob}} By Lemma~\ref{lem:forest-constructor-automata-matched-state}, to show the first equation it suffices to show for any $i\leq j$: \[ \mathbf{Pr}[q^j=q_\yes\|q^i=q_\yes]=f_{i-j} ~. \] The proof is an induction that corresponds to the recurrence. First consider the base cases. The case when $j=i$ is trivial. The case when $j=i+1$ holds because $q^i=q_\yes$ implies $q^{i+1}=q_\no$ by the definition of $\sigma^+$. The case when $j=i+2$ holds because $q^i=q_\yes$ implies $q^{i+1}=q_\no$, from which $\sigma^+$ transition to $q^{i+2}=q_\yes$ with probability $p$ by definition. Finally consider the case when $j \ge i+3$. There are only two possibilities in the two rounds after $i$. The first case is $q^{i+1} = q_\no$ and $q^{i+2} = q_\yes$, which happens with with probability $p$, and after which $q^j = q_\yes$ with probability $f_{j-i-2}$ by the inductive hypothesis. The other case is $q^{i+1} = q_\no$, $q^{i+2} = q_{\no^2}$, and $q^{i+3} = q_\yes$, which happens with with probability $1-p$, and after which $q^j = q_\yes$ with probability $f_{j-i-3}$ by the inductive hypothesis. Putting together: \begin{align} \mathbf{Pr}[q^j=q_\yes \| q^i=q_\yes] & = p \cdot f_{j-i-2} + (1-p) \cdot f_{j-i-3} \\[1ex] & =f_{j-i} ~. \end{align} Further, according to the definition of $\sigma^+$, $q^i=q_{\no^2}$ if and only if $q^{i+1}=q_\yes$ while $q^i=q_\no$ if and only if $q^{i-1}=q_\yes$. Thus, it follows that: \begin{align} \mathbf{Pr} \big[ d_j = \mathsf{M} \mid q^i = q_{\no^2} \big] & = \mathbf{Pr} \big[ d_j = \mathsf{M} \mid q^{i+1} = q_\yes \big] = f_{j-i-1} ~, \\ \mathbf{Pr} \big[ d_j = \mathsf{M} \mid q^i = q_\no \big] & = \mathbf{Pr} \big[ d_j = \mathsf{M} \mid q^{i-1} = q_\yes \big] = f_{j-i+1} ~. \end{align} \subsection{Proof of Lemma~\ref{lem:forest-constructor-automata-selection-prob-bound}} By the recurrence, the first seven terms of the sequence $\{f_k\}$ are: \begin{align} f_0&=1~, & f_1&=0~, & f_2&=p~, & f_3&=1-p~,\\ f_4&=p^2~, & f_5&=2p(1-p)~, & f_6&=p^3+(1-p)^2~. & & \end{align} As a result of $p\geq \frac{\sqrt{5}-1}{2}$, $f_2=p,f_4=p^2\geq f_3=1-p$. For any $k\geq 5$, as $f_k=pf_{k-2}+(1-p)f_{k-3}$, it is easy to see that $f_k\geq f_3=1-p$ by induction. On the other hand, as $f_6=pf_4+(1-p)f_3$ and $f_4\geq f_3$, $f_6\leq f_4$. Further, as $p\leq \frac{2}{3}$, $f_5=2p(1-p)\geq p^2=f_4\geq f_6$. For any $k\geq 7$, as $f_k=pf_{k-2}+(1-p)f_{k-3}$, it is easy to see that $f_k\geq f_6=p^3+(1-p)^2$ by induction. \subsection{Proof of Lemma~\ref{lem:pseudo-path-induced}} Consider any element $e$, any subset of nodes $U\subseteq V$ involving $e$ and any pseudo-path $P=\big((t_i,t'_i)_{e_i}\big)_{1\leq i\leq \ell}$. Consider the subset that is the union of $E^{\text{ex-ante}}_{U,e}\cap P$ and the subset of arcs in $P$ that are adjacent to two distinct arcs of $E^{\text{ex-ante}}_{U,e}\cap P$. Simply by definition, the first statement holds for this subset. Note that any two adjacent arcs in $P$ are either two in-arcs or two out-arcs of a same node, any two adjacent arcs in $P$ have different subscripts. Therefore any two arcs in $E^{\text{ex-ante}}_{U,e}\cap P$ are not adjacent in $P$ and thus each maximal sub-pseudo-path satisfies the fourth statement, i.e. it alternates between arcs with subscript $e$ and arcs with other subscripts, and the second statement, i.e. it is odd-length. Next, we shall show the third statement for this subset. By definition of the subset and the fourth statement, it suffices to show that any two arcs with subscript $e$ cannot be two arcs apart in $P$. Since in-arcs of a node are added simultaneously into the same pseudo-path in its corresponding round, the first arcs added into the pseudo-path should be the out-arcs of a node, say the initiative node of $P$. Let $i_0$ be the minimum index among the first arcs of $P$. The initiative node is then $t_{i_0}'$. Let $\{a,b\}$ be the set of elements of initiative node, i.e $\mathcal{E}^{t_{i_0}'}=\{a,b\}$, and $a$ be the subscript of the $i_0$-th arc, i.e. $e_{i_0}=a$. Then, the subscripts of arcs in $P$ on the pseudo-path can be characterized by the following lemma. \begin{lemma} The subscripts of arcs in $P$ satisfies: \begin{enumerate} \item For any $1\leq i\leq i_0$, the subscript of the $i$-th arc is $a$ if and only if $i_0-i$ is even; and \item For any $i_0< i\leq \ell$, the subscript of the $i$-th arc is $b$ if and only if $i-i_0$ is odd. \end{enumerate} \end{lemma} \begin{proof} For $1\leq i\leq i_0$, we shall prove some stronger results: \begin{enumerate} \item For any $1\leq i\leq i_0$, the subscript of the $i$-th arc is $a$ if and only if $i_0-i$ is even; \item For any $1\leq i\leq i_0$, $t_i'$ involves element $a$; \item For any $2\leq i\leq i_0$, if $i-i_0$ is odd, $t_{i-1}=t_{i}$ and otherwise $t_{i-1}'=t_i'$. \end{enumerate} We shall prove it by induction. The base case is that the subscript of the $i_0$-th arc is $a$, $t_{i_0}'$ involves $a$ and that $t_{i_0-1}=t_{i_0}$. For any $1\leq i\leq i_0-1$, given that the subscript of the $i$-th arc is $a$, $t_i'$ involves $a$ and $t_{i-1}=t_i$, the subscript of the $(i-1)$-th arc can't be $a$, as adjacent arcs have different subscripts, and $t_{i-2}\neq t_{i-1}$ (if $i\geq 3$), as each node has at most 2 our-arcs. Then it is clear $t_i'=t_{i-1}'$ and round $t_i=t_{i-1}, t_i',t_{i-1}'$ have a common element. By Lemma~\ref{lem:app-forest-constructor-common-element}, rounds $t_{i-1}, t_{i-1}'$ have the same set of elements. Therefore $t_{i-1}'$ involves $a$. On the other hand, given that the subscript of the $i$-th arc is not $a$, $t_i'$ involves $a$ and $t_{i-1}'=t_i'$, the subscript of the $(i-1)$-th arc is $a$, as $t_i'$ has two out-arcs and one of them has subscript $a$, and $t_{i-2}'\neq t_{i-1}'$ (if $i\geq 3$). Therefore, it is clear that $t_{i-1}'$ involves $a$ and $t_{i-2}=t_{i-1}$. If $i_0<\ell$, i.e. there is another in-arc of the initiative node, for any $i_0<i\leq \ell$, it follows the symmetry with $1\leq i\leq i_0$. \end{proof} With the characterization of the subscripts, it is clear that for any $i\neq i_0-1$, the subscripts of $i$-th arc and $(i+3)$-th arc are different. The last possible violation of the third statement is that the $(i_0-1)$-th arc of $P$ and the $(i_0+2)$-th arc of $P$ have a common element. Note that these two arcs are out-arcs of the origins of in-arcs of the initiative node because of the characterization, this violation can be ruled out by the following lemma. \begin{lemma} If the two origins of in-arcs of the initiative node have a same element, i.e. $\mathcal{E}^{t_{i_0}}\cap \mathcal{E}^{t_{i_0+1}}\neq \emptyset$, at least one of them doesn't extend another out-arc in later rounds. \end{lemma} \begin{proof} If the two origins are the same, it is clear that the origin has already extended two out-arcs and it can't further extend. Otherwise, suppose the element is $c$. One of the origins have already extended an out-arc with subscript $c$ before round $t_{i_0}'$ and it can't further extend. \end{proof} \subsection{Proof of Lemma~\ref{lem:ocs-unmatched-ready-ratio}} We shall prove a stronger claim that lets the left-hand-side probability be further conditioned on an arbitrary realization of $q^{i_{m-1}-4}$. This probability equals: \[ \mathbf{Pr} \Big[ q^{i_{m-1}} = q_\no \mid X_{i_1}, \dots, X_{i_{m-1}} = 0, q^{i_{m-1}-4} \Big] = \mathbf{Pr} \Big[ q^{i_{m-1}} = q_\no \mid X_{i_{m-1}} = 0, q^{i_{m-1}-4} \Big] ~. \] By Bayes' rule, it is further equal to: \[ \frac{\mathbf{Pr} \big[ q^{i_{m-1}} = q_\no \mid q^{i_{m-1}-4} \big]}{\mathbf{Pr} \big[ q^{i_{m-1}} = q_\no \mid q^{i_{m-1}-4} \big] + \mathbf{Pr} \big[ q^{i_{m-1}} = q_{\no^2} \mid q^{i_{m-1}-4} \big]} ~. \] It remains to show that: \begin{equation} \label{eqn:ocs-unmatched-ready-ratio} p \cdot \mathbf{Pr} \big[ q^{i_{m-1}} = q_\no \mid q^{i_{m-1}-4} \big] \ge \mathbf{Pr} \big[ q^{i_{m-1}} = q_{\no^2} \mid q^{i_{m-1}-4} \big] ~. \end{equation} Consider the transition matrix $P^+$ of the automaton $\sigma^+$, with columns and rows in the order of $q_\no$, $q_{\no^2}$, and $q_\yes$: \[ P^+ = \begin{bmatrix} 0 & 1-p & p \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix} ~. \] The transition after four steps (from $i_{m-1}-4$ to $i_{m-1}$) is: \[ \big(P^+\big)^4 = \begin{bmatrix} p^2 & (1-p)^2 & 2p(1-p) \\ p & 0 & 1-p \\ 1-p & p(1-p) & p^2 \end{bmatrix} ~. \] Since the first column multplied by $p$ dominates the second column in every entry, we prove Eqn.~\eqref{eqn:ocs-unmatched-ready-ratio}, and thus the lemma. \subsection{Proof of Lemma~\ref{lem:forest-ocs-structural}} Consider the collection of all maximal tree-paths consisting of nodes in $U$. Note that in the good forest, for any node $p$ with two children $c$ and $c'$, the corresponding rounds have no common element. Since every node in $U$ involves element $e$, for any node $p\in U$, at most one of its children is in $U$. Since the collection consists of maximal tree-paths consisting of nodes in $U$, for any $p,c\in U$ such that $p$ is the parent of $c$, $p$ is on one tree-path in the collection if and only if $c$ is on the tree-path. Therefore, for any node $p$ and any tree-path in the collection, the neighbors (i.e., parent or children) of $p$ in the path is fixed. It is clear that any two distinct tree-paths are disjoint. Moreover, if there is an arc $(p,c)$ between nodes in distinct tree-paths, there should be two tree-paths involving node $p$, which contradicts to the fact the tree-paths are pairwise disjoint. Consider one path in the collection consisting of nodes $t_1,t_2,\cdot,t_k\in U$. For each $2\leq i\leq k$, if arc $(t_{i-1},t_i)$ have subscript $e$, it is clear that the algorithm sets $\ell_{t_i}(e)=\ell_{t_{i-1}}(e)$. Otherwise, as both node $t_{i-1}$ and $t_i$ involve $e$ and the element corresponding to subscript of the arc, the label of the other element in these nodes clearly imply $\ell_{t_i}(e)=\ell_{t_{i-1}}(e)$. Therefore, element $e$ has the same label in each tree-path. \subsection{Proof of Lemma~\ref{lem:forest-ocs-from-origin}} By symmetry, consider label $\ell = \mathsf{H}$ without loss of generality. The lemma holds vacuously for $k \ge 4$ since by design the automaton never selects the same label four times in a roll. \footnote{The longest identical sections are selecting $\mathsf{H}$ in three consecutive rounds from state $\mathsf{T}^2$.} Next we prove the cases of $k = 1, 2, 3$. The remaining argument lets $q^j$ denote the state after round $j$, and lets $s^j$ denote the selected label in round $j$. By the symmetry of automaton $\sigma^$, for any round $j$: \begin{equation} \label{eqn:forest-ocs-automaton-symmetry} \mathbf{Pr} \big[ q^j = q_\mathsf{H} \big] = \mathbf{Pr} \big[ q^j = q_\mathsf{T} \big] ~, \qquad \mathbf{Pr} \big[ q^j = q_{\mathsf{H}^2} \big] = \mathbf{Pr} \big[ q^j = q_{\mathsf{T}^2} \big] ~. \end{equation} Further, by the above symmetry and by: \begin{align} \mathbf{Pr} \big[ q^j = q_{\mathsf{H}^2} \big] & = \mathbf{Pr} \big[ q^{j-1} = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^j = \mathsf{H} \mid q^{j-1} = q_\mathsf{H} \big] = \mathbf{Pr} \big[ q^{j-1} = q_\mathsf{H} \big] \frac{1-\beta}{2} ~, \\ \mathbf{Pr} \big[ q^j = q_\mathsf{H} \big] & \ge \mathbf{Pr} \big[ q^{j-1} = q_\mathsf{T} \big] \mathbf{Pr} \big[ s^j = \mathsf{H} \mid q^{j-1} = q_\mathsf{T} \big] = \mathbf{Pr} \big[ q^{j-1} = q_\mathsf{T} \big] \frac{1+\beta}{2} ~, \end{align} we have: \begin{equation} \label{eqn:forest-ocs-automaton-state-bias} \frac{\mathbf{Pr} [ q^j = q_\mathsf{H} ]}{\mathbf{Pr} [ q^j = q_{\mathsf{H}^2} ]} \ge \frac{1+\beta}{1-\beta} ~. \end{equation} If $k = 1$, the symmetry implies that the marginal probability of selecting each label in round $i$ equals $\frac{1}{2}$. If $k = 2$, the probabilities of selecting $\mathsf{T}$ two consecutive times from each of the states are: \begin{align} & \mathbf{Pr} \big[ s^i = s^{i+1} = \mathsf{T} \mid q^{i-1} = q_\mathsf{O} \big] = \mathbf{Pr} \big[ s^i = \mathsf{T} \mid q^{i-1} = q_\mathsf{O} \big] \mathbf{Pr} \big[ s^{i+1} = \mathsf{T} \mid q^i = q_\mathsf{T} \big] = \frac{1-\beta}{4} ~, \\ & \mathbf{Pr} \big[ s^i = s^{i+1} = \mathsf{T} \mid q^{i-1} = q_\mathsf{H} \big] = \mathbf{Pr} \big[ s^i = \mathsf{T} \mid q^{i-1} = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^{i+1} = \mathsf{T} \mid q^i = q_\mathsf{T} \big] = \frac{1-\beta^2}{4} ~, \\[1.5ex] & \mathbf{Pr} \big[ s^i = s^{i+1} = \mathsf{T} \mid q^{i-1} = q_\mathsf{T} \big] = \mathbf{Pr} \big[ s^i = \mathsf{T} \mid q^{i-1} = q_\mathsf{T} \big] \mathbf{Pr} \big[ s^{i+1} = \mathsf{T} \mid q^i = q_{\mathsf{T}^2} \big] = 0 ~, \\[1.5ex] & \mathbf{Pr} \big[ s^i = s^{i+1} = \mathsf{T} \mid q^{i-1} = q_{\mathsf{H}^2} \big] = \mathbf{Pr} \big[ s^i = \mathsf{T} \mid q^{i-1} = q_{\mathsf{H}^2} \big] \mathbf{Pr} \big[ s^{i+1} = \mathsf{T} \mid q^i = q_\mathsf{O} \big] = \frac{1}{2} ~, \\[1.5ex] & \mathbf{Pr} \big[ s^i = s^{i+1} = \mathsf{T} \mid q^{i-1} = q_{\mathsf{T}^2} \big] = 0 ~. \end{align} Hence: \[ \mathbf{Pr} \big[ s^i = s^{i+1} = \mathsf{T} \big] = \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{O} \big] \frac{1-\beta}{4} + \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{H} \big] \frac{1-\beta^2}{4} + \mathbf{Pr} \big[ q^{i-1} = q_{\mathsf{H}^2} \big] \frac{1}{2} ~. \] By Eqn.~\eqref{eqn:forest-ocs-automaton-state-bias}, we further get that: \begin{align} \mathbf{Pr} \big[ s^i = s^{i+1} = \mathsf{T} \big] & \ge \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{O} \big] \frac{1-\beta}{4} + \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{H} \text{ or } q_{\mathsf{H}^2} \big] \Big( \frac{1+\beta}{2} \frac{1-\beta^2}{4} + \frac{1-\beta}{2} \frac{1}{2} \Big) \\ & = \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{O} \big] \frac{1-\beta}{4} + \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{H} \text{ or } q_{\mathsf{H}^2} \big] \Big( \frac{(1+\beta)^2}{2} + 1 \Big) \frac{1-\beta}{4} \\ & = \Big( \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{O} \big] + 2 \mathbf{Pr} \big[ q^{i-1} = q_\mathsf{H} \text{ or } q_{\mathsf{H}^2} \big] \Big) \frac{1-\beta}{4} \tag{$\beta = \sqrt{2}-1$} \\ & = \frac{1-\beta}{4} ~. \tag{Eqn.~\eqref{eqn:forest-ocs-automaton-symmetry}} \end{align} If $k = 3$, the automaton must start from $q_{\mathsf{H}^2}$ in order to selet $\mathsf{T}$ in three consecutive rounds. The probability equals: \[ \mathbf{Pr} \big[ q^{i-1} = q_{\mathsf{H}^2} \big] \mathbf{Pr} \big[ s^i = \mathsf{T} \mid q^{i-1} = q_{\mathsf{H}^2} \big] \mathbf{Pr} \big[ s^{i+1} = \mathsf{T} \mid q^i = q_\mathsf{O} \big] \mathbf{Pr} \big[ s^{i+2} = \mathsf{T} \mid q^{i-1} = q_\mathsf{T} \big] ~. \] The first term is at most $\frac{1-\beta}{4}$ by Equations~\eqref{eqn:forest-ocs-automaton-symmetry} and \eqref{eqn:forest-ocs-automaton-state-bias}. The last three equal $1$, $\frac{1}{2}$, and $\frac{1-\beta}{2}$ respectively. Hence: \[ \mathbf{Pr} \big[ s^i = s^{i+1} = s^{i+2} = \mathsf{T} \big] = \frac{(1-\beta)^2}{16} < \frac{(1-\beta)^2}{8} ~. \] \subsection{Proof of Lemma~\ref{lem:forest-ocs-from-other}} It suffices to prove it for $i = 2$, since otherwise it reduces to the case of $i = 2$ by conditioning on the state after round $i-2$. Further, if the automaton starts from the original state $q_\mathsf{O}$, it follows from Lemma~\ref{lem:forest-ocs-from-origin}. If the automaton starts from $q_{\mathsf{H}^2}$ or $q_{\mathsf{T}^2}$, it resets back to the original state $q_\mathsf{O}$ after the first round and once again the lemma reduces to Lemma~\ref{lem:forest-ocs-from-origin}. Finally the lemma holds vacuously for $k \ge 4$ since automaton $\sigma^$ never selects the same label in four consecutive rounds. The remaining proof consider starting from $q_\mathsf{H}$ and $q_\mathsf{T}$ and $k \in \{ 1, 2, 3 \}$. By symmetry, we consider label $\ell = \mathsf{H}$ without loss of generality. If the automaton starts from $q_\mathsf{H}$: \begin{align} \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^0 = q_\mathsf{H} \big] & = \mathbf{Pr} \big[ s^1 = \mathsf{H} \mid q^0 = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_{\mathsf{H}^2} \big] \\ & \quad + \mathbf{Pr} \big[ s^1 = \mathsf{T} \mid q^0 = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_\mathsf{T} \big] \\ & = \frac{1-\beta}{2} \cdot 1 + \frac{1+\beta}{2} \cdot \frac{1-\beta}{2} \\[.5ex] & = \frac{1}{2} ~; \tag{$\beta = \sqrt{2}-1$} \\[2ex] \mathbf{Pr} \big[ s^2 = s^3 = \mathsf{T} \mid q^0 = q_\mathsf{H} \big] & = \mathbf{Pr} \big[ s^1 = \mathsf{H} \mid q^0 = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_{\mathsf{H}^2} \big] \mathbf{Pr} \big[ s^3 = \mathsf{T} \mid q^2 = q_\mathsf{O} \big] \\ & \quad + \mathbf{Pr} \big[ s^1 = \mathsf{T} \mid q^0 = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_\mathsf{T} \big] \mathbf{Pr} \big[ s^3 = \mathsf{T} \mid q^2 = q_{\mathsf{T}^2} \big] \\ & = \frac{1-\beta}{2} \cdot 1 \cdot \frac{1}{2} + \frac{1+\beta}{2} \cdot \frac{1-\beta}{2} \cdot 0 \\ & = \frac{1-\beta}{4} ~; \\[2ex] \mathbf{Pr} \big[ s^2 = s^3 = s^4 = \mathsf{T} \mid q^0 = q_\mathsf{H} \big] & = \mathbf{Pr} \big[ s^1 = \mathsf{H} \mid q^0 = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_{\mathsf{H}^2} \big] \\ & \qquad \mathbf{Pr} \big[ s^3 = \mathsf{T} \mid q^2 = q_\mathsf{O} \big] \mathbf{Pr} \big[ s^4 = \mathsf{T} \mid q^3 = q_\mathsf{H} \big] \\ & = \frac{1-\beta}{2} \cdot 1 \cdot \frac{1}{2} \cdot \frac{1-\beta}{2} \\ & = \frac{(1-\beta)^2}{8} ~. \end{align} The last case omits the $s^1 = \mathsf{T}$ option because the automaton by design never selects $\mathsf{T}$ in four consecutive rounds. If the automaton starts from $q_\mathsf{T}$: \begin{align} \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^0 = q_\mathsf{T} \big] & = \mathbf{Pr} \big[ s^1 = \mathsf{H} \mid q^0 = q_\mathsf{T} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_\mathsf{H} \big] \\ & \quad + \mathbf{Pr} \big[ s^1 = \mathsf{T} \mid q^0 = q_\mathsf{T} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_{\mathsf{T}^2} \big] \\ & = \frac{1+\beta}{2} \cdot \frac{1+\beta}{2} + \frac{1-\beta}{2} \cdot 0 \\[.5ex] & = \frac{1}{2} ~; \tag{$\beta = \sqrt{2}-1$} \\[2ex] \mathbf{Pr} \big[ s^2 = s^3 = \mathsf{T} \mid q^0 = q_\mathsf{T} \big] & = \mathbf{Pr} \big[ s^1 = \mathsf{H} \mid q^0 = q_\mathsf{T} \big] \mathbf{Pr} \big[ s^2 = \mathsf{T} \mid q^1 = q_\mathsf{H} \big] \mathbf{Pr} \big[ s^3 = \mathsf{T} \mid q^2 = q_\mathsf{T} \big] \\[1ex] & = \frac{1+\beta}{2} \cdot \frac{1+\beta}{2} \cdot \frac{1-\beta}{2} \\ & = \frac{1-\beta}{4} ~. \tag{$\beta = \sqrt{2}-1$} \end{align} The second case omits the $s^1 = \mathsf{T}$ option since automaton $\sigma^$ cannot select $\mathsf{T}$ in three consecutive rounds starting from state $q_\mathsf{T}$. Finally, it is impossible to have $s^2 = s^3 = s^4 = \mathsf{T}$ starting from state $q^0 = q_\mathsf{T}$ because from here we cannot have $q^1 = q_{\mathsf{H}^2}$, the only state of automaton $\sigma^$ that could lead to selecting $\mathsf{T}$ in the next three rounds. \subsection{Proof of Lemma~\ref{lem:forest-ocs-fork}} Let $q^j, s^j$ denote the states and selected labels of the first copy, and let $\hat{q}^j, \hat{s}^j$ denote those of the second copy. If the initial state is the original state $q_\mathsf{O}$, it follows by Lemma~\ref{lem:forest-ocs-from-origin}. If the initial state is $q_{\mathsf{H}^2}$ or $q_{\mathsf{T}^2}$, the lemma holds because the first selections in the two copies are the same. By symmetry, we next without loss of generality that the initial state is $q^0 = \hat{q}^0 = q_\mathsf{H}$. From $q_\mathsf{H}$ it is impossible to select $\mathsf{H}$ in the next two rounds, or to select $\mathsf{T}$ in the next three rounds. Hence, the lemma follows if $k \ge 2$ or $\hat{k} \ge 3$. It remains to consider $k = 1$, and $\hat{k} = 1, 2$. If $k = \hat{k} = 1$: \begin{align} \mathbf{Pr} \big[ s^1 = \mathsf{H}, \hat{s}^1 = \mathsf{T} \mid q^0 = \hat{q}^0 = q_\mathsf{H} \big] & = \mathbf{Pr} \big[ s^1 = \mathsf{H} \mid q^0 = q_\mathsf{H} \big] \mathbf{Pr} \big[ \hat{s}^1 = \mathsf{T} \mid \hat{q}^0 = q_\mathsf{H} \big] \\[1ex] & = \frac{1-\beta}{2} \frac{1+\beta}{2} \\ & < \frac{1}{4} ~. \end{align} If $k = 1$ and $\hat{k} = 2$: \begin{align} \mathbf{Pr} \big[ s^1 = \mathsf{H}, \hat{s}^1 = \hat{s}^2 = \mathsf{T} \mid q^0 = \hat{q}^0 = q_\mathsf{H} \big] & = \mathbf{Pr} \big[ s^1 = \mathsf{H} \mid q^0 = q_\mathsf{H} \big] \mathbf{Pr} \big[ \hat{s}^1 = \hat{s}^2 = \mathsf{T} \mid \hat{q}^0 = q_\mathsf{H} \big] \\[1ex] & = \frac{1-\beta}{2} \frac{1-\beta^2}{4} \\ & < \frac{1-\beta}{8} ~. \end{align} \section{Missing Proofs in Section~\ref{sec:matching}} \subsection{Proof of Theorem~\ref{thm:lp-solution}} \label{app:lp-solution} \begin{proof} We first verify the \emph{feasibility} of the stated solution. Constraint~\eqref{eqn:two-way-lp-gain-split} holds with equality by the definitions of $a(k)$ and $b(k)$. Constraint~\eqref{eqn:approximate-dual-feasible} also holds with equality. When $k = 0$, it follows by: \begin{align} \Gamma & = p(0) - \sum_{i=0}^{\infty} \Big(\Big(\frac{2}{3}\Big)^i - \Big(\frac{2}{3}\Big)^{i+1}\Big) p(i) \tag{Definition of $\Gamma$, $p(0) = 1$} \\ & = \sum_{i=0}^\infty \Big(\frac{2}{3}\Big)^{i+1} p(i) - \sum_{i=1}^\infty \Big(\frac{2}{3}\Big)^i p(i) \notag \\ & = \sum_{i=0}^\infty \Big(\frac{2}{3}\Big)^{i+1} \big( p(i)-p(i+1) \big) ~. \label{eqn:two-way-lp-alternative-Gamma} \end{align} This equals $2 b(0)$ by definition. Then, it further holds inductively for $k \ge 1$ because: \begin{align} b(k) & = \frac{3}{2} b(k-1) - \frac{1}{2} \big( p(k-1) - p(k) \big) \tag{Definition of $b(k), b(k-1)$} \\ & = b(k-1) - \frac{1}{2} a(k-1) ~. \tag{Definition of $a(k-1)$} \end{align} That is, the left-hand-side of Constraint~\eqref{eqn:approximate-dual-feasible} stays the same from $k-1$ to $k$. Since the above equation $b(k) = b(k-1) - \frac{1}{2} a(k-1)$ would also imply Constraint~\eqref{eqn:two-way-lp-monotone} provided that $a(k-1) \ge 0$, it remains to verify that $a(k)$ and $b(k)$ are nonnegative. By its definition and by $p(k+1) \le \frac{2}{3} p(k)$, we get that $b(k) \ge 0$. The non-negativity of $a(k)$ follows by its definition and by: \begin{align} b(k) & = \frac{1}{3} p(k) - \sum_{i=k+1}^\infty \Big( \Big(\frac{2}{3}\Big)^{i-k-1} - \Big(\frac{2}{3}\Big)^{i-k} \Big) p(i) \tag{Definition of $b(k)$} \\ & \le \frac{1}{3} p(k) \\[2ex] & \le p(k) - p(k+1) ~. \tag{$p(k+1) \le \frac{2}{3} p(k)$} \end{align} Next we establish its \emph{optimality}. Multiplying Constraint~\eqref{eqn:approximate-dual-feasible} by $\big(\frac{2}{3}\big)^k$ and summing over $k \ge 0$: \[ \sum_{k=0}^\infty \Big( \frac{2}{3} \Big)^k \Big( \sum_{i=0}^{k-1} a(i)+2 b(k) \Big) \geq \sum_{k=0}^\infty \Big( \frac{2}{3} \Big)^k \Gamma ~. \] Grouping terms on the left and dividing both sides by $3$, this is: \[ \sum_{i=0}^\infty \Big(\frac{2}{3}\Big)^{i+1} a(i) + \sum_{i=0}^\infty \Big(\frac{2}{3}\Big)^{i+1} b(i) \ge \Gamma ~. \] Further by Constraint~\eqref{eqn:two-way-lp-gain-split}, the left-hand-side is at most: \[ \sum_{i=0}^\infty \Big(\frac{2}{3}\Big)^{i+1} \big( p(i)-p(i+1) \big) ~. \] This equals the optimal $\Gamma$ in the theorem by Eqn.~\eqref{eqn:two-way-lp-alternative-Gamma}. \end{proof} \subsection{Proof of Theorem~\ref{thm:multi-way-lp-solution}} \label{app:multi-way-lp-solution} \begin{proof} We first verify its \emph{feasibility}. Constraint Eqn.~\eqref{eqn:multi-way-lp-gain-split} holds with equality by definition, i.e.: \begin{equation} \label{eqn:multi-way-lp-solution-gain-split} a(y) + b(y) = -p'(y) ~, \qquad\qquad \forall y \ge 0 ~. \end{equation} Constraint Eqn.~\eqref{eqn:multi-way-approximate-dual-feasible} holds for $y = 0$ from integration by parts: \[ b(0) = - \int_0^\infty p'(z) e^{-z} dz = \int_{0}^\infty e^{-z} \big( 1 - p(y) \big) dz = \Gamma ~. \] It further holds for $y > 0$ since its left-hand-side is a constant for all $y$. Indeed, the derivative of the left-hand-side is: \begin{align} a(y) + b'(y) & = a(y) - e^y \int_y^\infty p'(z)e^{-z} dz + p'(y) \tag{Definition of $b(y)$} \\ & = a(y) + b(y) + p'(y) \tag{Definition of $b(y)$} \\[2ex] & = 0 ~. \tag{Eqn.~\eqref{eqn:multi-way-lp-solution-gain-split}} \end{align} Constraint~\ref{eqn:multi-way-lp-monotone} holds, i.e., $b(y)$ is decreasing because: \begin{align} b'(y) & = - e^y \int_y^\infty p'(z)e^{-z} dz + p'(y) \tag{Definition of $b(y)$} \\ & = - e^y \int_y^\infty \big( p'(y) - p'(z) \big) e^{-z} dz \\[.5ex] & \le 0 ~. \tag{Convexity of $p$} \end{align} Finally, $b(y)$ is non-negative by definition and by that $p$ is decreasing. The non-negativity of $a(y)$ follows by: \begin{align} a(y) & = -p'(y) + e^y \int_y^\infty p'(z) e^{-z} dz \tag{Definitions of $a(y), b(y)$} \\ & = e^y \int_{y}^\infty e^{-z} p''(z)dz \tag{Integration by parts} \\[1ex] & \ge 0 ~. \tag{Convexity of $p$} \end{align} \end{proof} \end{document}	arXiv
September 2007 , Volume 8 , Issue 2 A linear-quadratic control problem with discretionary stopping Shigeaki Koike, Hiroaki Morimoto and Shigeru Sakaguchi 2007, 8(2): 261-277 doi: 10.3934/dcdsb.2007.8.261 +[Abstract](2186) +[PDF](195.4KB) We study the variational inequality for a 1-dimensional linear-quadratic control problem with discretionary stopping. We establish the existence of a unique strong solution via stochastic analysis and the viscosity solution technique. Finally, the optimal policy is shown to exist from the optimality conditions. Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 261-277. doi: 10.3934/dcdsb.2007.8.261. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller Roberto Triggiani Consider a 2-D, linearized Navier-Stokes channel flow with periodic boundary conditions in the streamwise direction and subject to a wall-normal control on the top wall. There exists an infinite-dimensional subspace $E^0$, where the normal component $v$ of the velocity vector, as well as the vorticity $\omega$, are not influenced by the control. The corresponding control-free dynamics for $v$ and $\omega$ on $E^0$ are inherently exponentially stable, though with limited decay rate. In the case of the linear 2-D channel, the stability margin of the component $v$ on the complementary space $Z$ can be enhanced by a prescribed decay rate, by means of an explicit, 2-D wall-normal controller acting on the top wall, whose space component is subject to algebraic rank conditions. Moreover, its support may be arbitrarily small. Corresponding optimal decays, by the same 2-D wall-normal controller, of the tangential component $u$ of the velocity vector; of the pressure $p$; and of the vorticity $\omega$ over $Z$ are also obtained, to complete the optimal analysis. Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 279-314. doi: 10.3934/dcdsb.2007.8.279. Optimal investment-consumption strategy in a discrete-time model with regime switching Ka Chun Cheung and Hailiang Yang This paper analyzes the investment-consumption problem of a risk averse investor in discrete-time model. We assume that the return of a risky asset depends on the economic environments and that the economic environments are ranked and described using a Markov chain with an absorbing state which represents the bankruptcy state. We formulate the investor's decision as an optimal stochastic control problem. We show that the optimal investment strategy is the same as that in Cheung and Yang [5], and a closed form expression of the optimal consumption strategy has been obtained. In addition, we investigate the impact of economic environment regime on the optimal strategy. We employ some tools in stochastic orders to obtain the properties of the optimal strategy. Ka Chun Cheung, Hailiang Yang. Optimal investment-consumption strategy in a discrete-time model with regime switching. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 315-332. doi: 10.3934/dcdsb.2007.8.315. Global stability of two epidemic models Qingming Gou and Wendi Wang In this paper we study the global stability of two epidemic models by ruling out the presence of periodic orbits, homoclinic orbits and heteroclinic cycles. One model incorporates exponential growth, horizontal transmission, vertical transmission and standard incidence. The other one incorporates constant recruitment, disease-induced death, stage progression and bilinear incidence. For the first model, it is shown that the global dynamics is completely determined by the basic reproduction number $R_0$. If $R_0\leq1$, the disease free equilibrium is globally asymptotically stable, whereas the unique endemic equilibrium is globally asymptotically stable if $R_0>1$. For the second model, it is shown that the disease-free equilibrium is globally stable if $R_0\leq1$, and the disease is persistent if $R_0>1$. Sufficient conditions for the global stability of an endemic equilibrium of the model are also presented. Qingming Gou, Wendi Wang. Global stability of two epidemic models. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 333-345. doi: 10.3934/dcdsb.2007.8.333. Distributional chaos via isolating segments Piotr Oprocha and Pawel Wilczynski Recently, Srzednicki and Wójcik developed a method based on Wazewski Retract Theorem which allows, via construction of so called isolating segments, a proof of topological chaos (positivity of topological entropy) for periodically forced ordinary differential equations. In this paper we show how to arrange isolating segments to prove that a given system exhibits distributional chaos. As an example, we consider planar differential equation ż$=(1+e^{i \kappa t}\|z\|^2)\bar{z}$ for parameter values $0<\kappa \leq 0.5044$. Piotr Oprocha, Pawel Wilczynski. Distributional chaos via isolating segments. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 347-356. doi: 10.3934/dcdsb.2007.8.347. Sharp global existence and blowing up results for inhomogeneous Schrödinger equations Jianqing Chen and Boling Guo In this paper, we first give an important interpolation inequality. Secondly, we use this inequality to prove the existence of local and global solutions of an inhomogeneous Schrödinger equation. Thirdly, we construct several invariant sets and prove the existence of blowing up solutions. Finally, we prove that for any $\omega>0$ the standing wave $e^{i \omega t} \phi (x)$ related to the ground state solution $\phi$ is strongly unstable. Jianqing Chen, Boling Guo. Sharp global existence and blowing up results for inhomogeneous Schr\u00F6dinger equations. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 357-367. doi: 10.3934/dcdsb.2007.8.357. Reformed post-processing Galerkin method for the Navier-Stokes equations Yinnian He and R. M.M. Mattheij In this article we compare the post-processing Galerkin (PPG) method with the reformed PPG method of integrating the two-dimensional Navier-Stokes equations in the case of non-smooth initial data $u_0 \epsilon\in H^1_0(\Omega)^2$ with div$u_0=0$ and $f,~f_t\in L^\infty(R^+;L^2(\Omega)^2)$. We give the global error estimates with $H^1$ and $L^2$-norm for these methods. Moreover, if the data $\nu$ and the $\lim_{t \rightarrow \infty}f(t)$ satisfy the uniqueness condition, the global error estimates with $H^1$ and $L^2$-norm are uniform in time $t$. The difference between the PPG method and the reformed PPG method is that their error bounds are of the same forms on the interval $[1,\infty)$ and the reformed PPG method has a better error bound than the PPG method on the interval $[0,1]$. Yinnian He, R. M.M. Mattheij. Reformed post-processing Galerkin method for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 369-387. doi: 10.3934/dcdsb.2007.8.369. Detecting perfectly insulated obstacles by shape optimization techniques of order two Lekbir Afraites, Marc Dambrine, Karsten Eppler and Djalil Kateb The paper extends investigations of identification problems by shape optimization methods for perfectly conducting inclusions to the case of perfectly insulating material. The Kohn and Vogelius criteria as well as a tracking type objective are considered for a variational formulation. In case of problems in dimension two, the necessary condition implies immediately a perfectly matching situation for both formulations. Similar to the perfectly conducting case, the compactness of the shape Hessian is shown and the ill-posedness of the identification problem follows. That is, the second order quadratic form is no longer coercive. We illustrate the general results by some explicit examples and we present some numerical results. Lekbir Afraites, Marc Dambrine, Karsten Eppler, Djalil Kateb. Detecting perfectly insulated obstacles by shape optimization techniques of order two. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 389-416. doi: 10.3934/dcdsb.2007.8.389. Multiple bifurcations of a predator-prey system Dongmei Xiao and Kate Fang Zhang The bifurcation analysis of a generalized predator-prey model depending on all parameters is carried out in this paper. The model, which was first proposed by Hanski et al. [6], has a degenerate saddle of codimension 2 for some parameter values, and a Bogdanov-Takens singularity (focus case) of codimension 3 for some other parameter values. By using normal form theory, we also show that saddle bifurcation of codimension 2 and Bogdanov-Takens bifurcation of codimension 3 (focus case) occur as the parameter values change in a small neighborhood of the appropriate parameter values, respectively. Moreover, we provide some numerical simulations using XPPAUT to show that the model has two limit cycles for some parameter values, has one limit cycle which contains three positive equilibria inside for some other parameter values, and has three positive equilibria but no limit cycles for other parameter values. Dongmei Xiao, Kate Fang Zhang. Multiple bifurcations of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 417-433. doi: 10.3934/dcdsb.2007.8.417. A competition-diffusion system with a refuge Daozhou Gao and Xing Liang In this paper, a model composed of two Lotka-Volterra patches is considered. The system consists of two competing species $X, Y$ and only species $Y$ can diffuse between patches. It is proved that the system has at most two positive equilibria and then that permanence implies global stability. Furthermore, to answer the question whether the refuge is effective to protect $Y$, the properties of positive equilibria and the dynamics of the system are studied when $X$ is a much stronger competitor. Daozhou Gao, Xing Liang. A competition-diffusion system with a refuge. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 435-454. doi: 10.3934/dcdsb.2007.8.435. The role of evanescent modes in randomly perturbed single-mode waveguides Josselin Garnier Pulse propagation in randomly perturbed single-mode waveguides is considered. By an asymptotic analysis the pulse front propagation is reduced to an effective equation with diffusion and dispersion. Apart from a random time shift due to a random total travel time, two main phenomena can be distinguished. First, coupling and energy conversion between forward- and backward-propagating modes is responsible for an effective diffusion of the pulse front. This attenuation and spreading is somewhat similar to the one-dimensional case addressed by the O'Doherty-Anstey theory. Second, coupling between the forward-propagating mode and the evanescent modes results in an effective dispersion. In the case of small-scale random fluctuations we show that the second mechanism is dominant. Josselin Garnier. The role of evanescent modes in randomly perturbed single-mode waveguides. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 455-472. doi: 10.3934/dcdsb.2007.8.455. Homogenization in random media and effective medium theory for high frequency waves Guillaume Bal We consider the homogenization of the wave equation with high frequency initial conditions propagating in a medium with highly oscillatory random coefficients. By appropriate mixing assumptions on the random medium, we obtain an error estimate between the exact wave solution and the homogenized wave solution in the energy norm. This allows us to consider the limiting behavior of the energy density of high frequency waves propagating in highly heterogeneous media when the wavelength is much larger than the correlation length in the medium. Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 473-492. doi: 10.3934/dcdsb.2007.8.473. Distributional convergence of null Lagrangians under very mild conditions Marc Briane and Vincenzo Nesi We consider sequences $U^\epsilon$ in $W^{1,m}(\Omega;\RR^n)$, where $\Omega$ is a bounded connected open subset of $\RR^n$, $2\leq m\leq n$. The classical result of convergence in distribution of any null Lagrangian states, in particular, that if $U^\ep$ converges weakly in $W^{1,m}(\Omega)$ to $U$, then det$(DU^\epsilon)$ converges to det$(DU)$ in $\D'(\Omega)$. We prove convergence in distribution under weaker assumptions. We assume that the gradient of one of the coordinates of $U^\epsilon$ is bounded in the weighted space $L^2(\Omega,A^\epsilon(x)dx;\RR^n)$, where $A_\epsilon$ is a non-equicoercive sequence of symmetric positive definite matrix-valued functions, while the other coordinates are bounded in $W^{1,m}(\Omega)$. Then, any $m$-homogeneous minor of the Jacobian matrix of $U^\epsilon$ converges in distribution to a generalized minor provide that $\|A_\epsilon^{-1}\|^{n/2}$ converges to a Radon measure which does not load any point of $\Omega$. A counter-example shows that this latter condition cannot be removed. As a by-product we derive improved div-curl results in any dimension $n\geq 2$. Marc Briane, Vincenzo Nesi. Distributional convergence of null Lagrangians under very mild conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 493-510. doi: 10.3934/dcdsb.2007.8.493. Generators of Feller semigroups with coefficients depending on parameters and optimal estimators Jerome A. Goldstein, Rosa Maria Mininni and Silvia Romanelli We consider the realization of the operator $L_{\theta, a}u(x) $:$= x^{2 a}u''(x) \ + \ (a x^{2 a - 1} + \theta x^a)u'(x)$, acting on $C[0,+\infty]$, for $\theta\in\R$, $a\in\R$. We show that $L_{\theta, a}$, with the so called Wentzell boundary conditions, generates a Feller semigroup for any $\theta\in\R$, $a\in\R$. The problem of finding optimal estimators for the corresponding diffusion processes is also discussed, in connection with some models in financial mathematics. Here $C[0,+\infty]$ is the space of all real valued continuous functions on $[0,+\infty)$ which admit finite limit at $+\infty$. Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generators of Feller semigroups with coefficients depending on parameters and optimal estimators. Discrete & Continuous Dynamical Systems - B, 2007, 8(2): 511-527. doi: 10.3934/dcdsb.2007.8.511.	CommonCrawl
\begin{document} \title{{\fontsize{23}{23}\selectfont Probing the Non-Classicality of Temporal Correlations}} \author{Martin Ringbauer} \affiliation{Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia} \affiliation{Centre for Quantum Computer and Communication Technology, School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia} \affiliation{Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK} \orcid{0000-0001-5055-6240} \author{Rafael Chaves} \affiliation{International Institute of Physics, Federal University of Rio Grande do Norte, 59070-405 Natal, Brazil} \orcid{0000-0001-8493-4019} \date{\today} \begin{abstract} Correlations between spacelike separated measurements on entangled quantum systems are stronger than any classical correlations and are at the heart of numerous quantum technologies. In practice, however, spacelike separation is often not guaranteed and we typically face situations where measurements have an underlying time order. Here we aim to provide a fair comparison of classical and quantum models of temporal correlations on a single particle, as well as timelike-separated correlations on multiple particles. We use a causal modeling approach to show, in theory and experiment, that quantum correlations outperform their classical counterpart when allowed equal, but limited communication resources. This provides a clearer picture of the role of quantum correlations in timelike separated scenarios, which play an important role in foundational and practical aspects of quantum information processing. \end{abstract} \maketitle Quantum predictions are fundamentally incompatible with the intuitive notion of cause and effect that underpins all of classical empirical science, as well as everyday experience. Most famously, Bell's theorem~\cite{Bell1964} shows that the correlations revealed by measurements on distant parts of entangled quantum systems cannot be explained in causal terms, under the assumptions that local measurement outcomes are not influenced by spacelike separated events---\emph{local causality}---and that measurement settings can be chosen freely. This phenomenon has become known as quantum nonlocality, and has been extensively studied for variants of the scenario above, where spacelike separated parties perform measurements on a composite quantum system~\cite{Brunner2014}. Yet, in practice we often face situations where spacelike separation between observers is not guaranteed and instead there is some time order underlying the observed physical events. Indeed, it has been shown that a single quantum system measured at different points in time exhibits Bell-like correlations~\cite{Brukner2004,Markiewicz2014,Budroni2013,Fritz2010,Fedrizzi2011,Pawlowski2010a}. However, in such a scenario one cannot appeal to relativity to rule out communication between the time steps. This can be problematic, since a classical model with sufficient communication can simulate any correlations. Recent research has thus been devoted to clarifying to what extent the incompatibility between classical and quantum descriptions encountered in the spacelike case carries over to the timelike scenario. Specifically, one research direction has been to study how limited resources such as entropy~\cite{Chaves2015b}, memory~\cite{Galvao2003,Montina2008,Zukowski2014} or dimension~\cite{Gallego2010,Brierley2015,Bowles2015} can lead to a quantum advantage over classical models. On the other hand, one can relax the local causality assumption in Bell's theorem, aiming to explain quantum correlations by classical models augmented with communication~\cite{Bacon2003,Toner2003,Pironio2003,Vertesi2009,Hall2011,Maxwell2014,Chaves2015,Brask2016,Ringbauer2016,Chaves2016causal}. However, communication is typically only allowed for the classical system, leading to an unfair comparison between classical and quantum models, and it remains unclear to what extent these results hold when equal communication power is given to quantum mechanics~\cite{Brask2016}. Here we use a causal modeling approach to allow a fair comparison of classical and quantum models of timelike separated correlations. First, we show that for two time-ordered spatially separated measurements, augmented with a limited amount of classical communication, quantum correlations outperform their classical counterpart. Second, we show that, in contrast to spatial Bell-type scenarios~\cite{Wood2015}, there are faithful classical causal models reproducing all the temporal correlations obtained from a series of projective quantum measurements on a single quantum system. However, we find that non-classical correlations can arise in this scenario, when considering a slightly weaker classical causal model, that is nonetheless strictly stronger than previous results~\cite{Leggett1985}, which are contained as a special case. Finally we derive Bell-type inequalities for the above scenarios and demonstrate that they are violated in a photonic experiment. \text{ }\\ \text{ }\\ \text{ }\\ \section{Causal modeling and timelike Bell-scenarios} In the following we employ the formalism of Bayesian networks~\cite{Pearlbook}, which provides a natural framework for classical causal modeling. A central concept in this framework is that of a \emph{directed acyclic graph} (DAG), which consists of a set of nodes, representing the relevant random variables\footnote{We adopt the standard convention that uppercase letters label random variables while their values are denoted in lower case.} in the considered situation, and directed edges, representing the causal relations between those variables. A set of variables $X_1,\dots,X_n$ forms a Bayesian network with respect to some DAG if and only if the probability distribution $p(x_1, \dots , x_n)$ can be decomposed as \begin{equation} p(x_1, \dots , x_n)=\prod_{i=1}^{n} p(x_i \vert pa_i) \end{equation} where $PA_i$ stands for the set of graph-theoretical parents of the variable $X_i$ (i.e.\ all variables that have a direct causal influence over $X_i$). Without loss of generality each variable can be understood as a deterministic function of its parents plus local noise $U_i$ that supplies potential randomness, $x_i=f_i(pa_i,u_i)$. This formalism thus enables a distinction between simple statistical correlations and actual causation by explicitly specifying the underlying mechanism generating the data. Here we are interested in DAGs containing so-called \emph{latent variables}, which are empirically inaccessible. In the context of Bell's theorem~\cite{Bell1964} these are also known as \emph{hidden variables}. For any set of observed correlations, there are in general many DAGs with hidden variables that could have produced these observations. Among these, causal inference is particularly interested in those fulfilling the conditions of \emph{minimality} and \emph{faithfulness}. Minimality requires that, given two possible causal models, we choose the simplest one, capable of generating the smallest set of correlations (including the observed one). In turn, faithfulness, requires the causal model to be able to explain the observed data without resorting to fine-tuning of the causal-statistical parameters. In other words, any observed (conditional) independence should be a consequence of the causal structure itself, rather than a specific choice of parameters. Faithful (i.e.\ non-fine-tuned) models are therefore robust against changes in the causal parameters and thus the preferred choice. To illustrate the last point, consider the paradigmatic causal structure of Bell's theorem in Fig. \ref{fig.BellDag}a. This structure intuitively reflects the causal assumptions of Bell's theorem, leading to the so-called local hidden-variable (LHV) models. First, the two parties are assumed to be spacelike separated, such that the correlations between the measurements outcomes $A$ and $B$ can only be mediated via a common source $\Lambda$, implying that $p(a,b\vert x,y,\lambda)= p(a \vert x, \lambda)p(b \vert y, \lambda)$. Second, it is assumed that the experimenters can freely choose which observables to measure (represented by the random variables $X$ and $Y$), independently of how the system was prepared, that is, $p(x,y,\lambda)=p(x,y)p(\lambda)$. Note that these constraints implied by the causal model appear at an unobservable level since they explicitly involve the hidden variables $\Lambda$. Yet, they also imply observable constraints in the form of no-signaling conditions, expressed as $p(a \vert x,y)=p(a \vert x)$ and $p(b \vert x,y)=p(b \vert y)$, and Bell inequalities~\cite{Bell1964,Bell1976}. \begin{figure} \caption{\textbf{Causal structure underlying Bell's theorem.} \textbf{(a)} Two observers, Alice and Bob, each have the choice of two measurements represented by the random variables $X$ and $Y$, respectively. The correlation between their measurement outcomes, modeled as random variables $A$ and $B$, respectively, are mediated solely by a common cause in their past---the hidden variable $\Lambda$. \textbf{(b)} Bell's causal model augmented with one-way communication from Alice to Bob. The initial state of the joint system is specified by the ontic state $\Lambda$. First, Alice performs a measurement with setting $x$, obtaining outcome $a$. She then sends a message $m$ to Bob, who performs a measurement with setting $y$, obtaining outcome $b$.} \label{fig.BellDag} \end{figure} While quantum correlations obey the no-signaling conditions, they violate Bell inequalities~\cite{Bell1964,Bell1976} and are thus in conflict with the assumptions behind the causal structure in Fig.~\ref{fig.BellDag}a. In order to maintain a classical causal explanation, the model in Fig.~\ref{fig.BellDag}a must therefore be augmented with additional resources; something that can only be done at the cost of introducing fine-tuning~\cite{Wood2015}. For instance, the causal structure in Fig.~\ref{fig.BellDag}b can reproduce all quantum correlations, but at the same time allows, in principle, for non-local correlations between $X$ and $B$. Hence, in order to satisfy the no-signaling condition $p(b \vert x,y)=p(b \vert y)$ the causal parameters must be chosen from a set of measure zero~\cite{Pearlbook}, a signature of fine-tuning. Studying such non-local classical models can provide valuable insights into the relation between classical and quantum theory, and their applications~\cite{Brunner2014}. However, at the same time such models lead to an unfair comparison, since allowing for communication makes not only classical, but also quantum models more powerful. In practice, it is more natural to assume a certain underlying causal structure, and ask what can be achieved with classical and quantum resources? Bell's theorem is a particular case of this broader question, referring to spacelike separated events. However, there are often situations where the events are timelike rather than spacelike ordered. Examples include central quantum information tasks, such as teleportation \cite{Bennett1993}, superdense coding \cite{Bennett1992}, and measurement-based quantum computation \cite{Briegel2009}, as well as prepare-and-measure scenarios \cite{Gallego2010}, sequential Bell scenarios \cite{Popescu1995,Gallego2014} and a sequence of measurements on a single quantum system \cite{Leggett1985}. \section{Non-classicality of timelike correlations augmented by communication} Consider the scenario in Fig.~\ref{fig.BellDag}b, where two distant parties, Alice and Bob, share pre-established correlations (represented by $\Lambda$) and are allowed one-way communication (the message $M$). As shown in Ref.~\cite{Bacon2003}, a classical model of this form (for a large enough message $M$) is enough to reproduce all the correlations obtainable from local measurements on two-qubit entangled states as in Fig.~\ref{fig.BellDag}a, which are described by \begin{equation} p(a,b \vert x,y)= \mathrm{Tr} \left[ (M^a_x \otimes M^b_y) \rho \right] , \end{equation} where $M^a_x$ and $M^b_y$ are measurement operators for Alice and Bob, respectively, and $\rho$ is the density matrix describing the shared quantum state. If, in contrast, we impose the causal structure of Fig.~\ref{fig.BellDag}b also to the quantum case---that is, Bob's measurement may depend on Alice's measurement setting and outcome---then the set of correlations is described by \begin{equation} p(a,b \vert x,y)= \sum_{m}\mathrm{Tr} \left[ (M^a_x \otimes M^b_{y,m}) \rho \right] . \label{eq:quantum_oneway} \end{equation} Note that Bob's measurement operator now explicitly depends on the values of $X$ and $A$ (via the message $M$) \footnote{In fact, Bob could apply to his share of the joint quantum state any completely-positive trace-preserving map dependent on the message $m$.}. Clearly, if there are no restrictions on the dimensionality of the message, every distribution of the form above can be also obtained by the classical hidden variable model in Fig.~\ref{fig.BellDag}b, that is, by a model respecting the decomposition \begin{equation} p(a,b \vert x,y)= \sum_{m,\lambda} p(\lambda)p(m \vert x,a) p(a\vert x,\lambda) p(b\vert y,m,\lambda). \end{equation} In fact, it is enough to choose $m=x$ to reproduce all possible one-way signalling correlations. To see this, note that the quantum correlations arising from Eq.~\eqref{eq:quantum_oneway} are of the form $p(a,b \vert x,y)=p(b \vert x,y,a)p(a \vert x)$, and that $a$ can be made a deterministic function $a=f_a(x,\lambda)$. Hence $a$ does not carry any information that is not already contained in $\lambda$ and $x$. Notwithstanding, this picture changes if we impose restrictions on the message sent from Alice to Bob. Consider that each party measures three dichotomic observables (i.e.\ $x,y=0,1,2$ and $a,b=0,1$) and that Alice is bound to send a binary message ($m=0,1$). In this case, every classical model must obey the inequality~\cite{Bacon2003} \begin{align} \label{eq.M332} \mathcal{S}_\mathrm{1 bit}=& \mean{A_0B_0}+\mean{A_0B_1}+\mean{A_0B_2}+\mean{A_1B_0} \\ \nonumber & +\mean{A_1B_1}-\mean{A_1B_2}+\mean{A_2B_0}-\mean{A_2B_1} \leq 6 . \end{align} Furthermore, it was shown in Ref.~\cite{Bacon2003} that this inequality also holds for correlations from local measurements on entangled quantum states, while it can be violated by more powerful no-signalling correlations. Hence, while one bit of communication is in this scenario sufficient for a classical model to simulate quantum correlations (without communication), it is not sufficient to simulate all possible no-signalling correlations. However, just like communication-augmented classical models become more powerful, so do quantum models. Specifically, local measurements on entangled quantum states augmented with one bit of communication can indeed violate inequality~\eqref{eq.M332}~\cite{Brask2016}, thus showing that under fair comparison, quantum advantage persists in such a timelike-separated Bell scenario. \begin{figure} \caption{\textbf{Experimental setup for studying the non-classicality of timelike correlations}. Pairs of single photons are produced via spontaneous parametric downconversion in a $\beta$-barium borate crystal (not shown). The two photons are entangled to arbitrary degree using a non-deterministic controlled-NOT gate (CNOT), based on nonclassical interference in a partially polarizing beam splitter (PPBS)~\cite{Langford2005}. Alice and Bob then perform local projective measurements on their share of the entangled state, which are implemented using a set of half- (HWP) and quarter-waveplates (QWP), a Glan-Taylor polarizer (GT) and single-photon counters (APD).} \label{fig:Setup} \end{figure} As an example, consider that Alice and Bob share a maximally entangled state $\ket{\Psi^{+}}=(\ket{00}+\ket{11})/\sqrt{2})$. Alice performs local measurements with settings $A_0=A_1=\hat X$, $A_2=\hat Z$ and encodes her measurement setting in a message $m$ to Bob. For $x=0$ she sends $m=0$ and for $x=1$ or $x=2$ she sends $m=1$. Assuming that all inputs are equally likely this message has an entropy of $H(m) \sim 0.92$ and thus contains less than 1 bit of information. If Bob receives $m=0$, he measures $B_0=B_1=B_2=\hat X$, while for $m=1$ he measures $B_0=(\hat X+\hat Z)/\sqrt{2}$, $B_1=(\hat X-\hat Z)/\sqrt{2}$ and $B_2= - \hat X$. This protocol achieves $\mathcal{S}_\mathrm{1 bit}=4\sqrt{2}$, thus violating inequality~\eqref{eq.M332}. Experimentally we can test inequality~\eqref{eq.M332} with qubits encoded in the polarization of single photons, see Fig.~\ref{fig:Setup}. Two single photons are first entangled using a non-deterministic controlled-NOT gate, and then distributed to Alice and Bob. By varying the input states this configuration can produce states with arbitrary degree of entanglement, quantified by the concurrence $\mathcal{C}$~\cite{Hill1997}, see Appendix~\ref{Sec:Supp2} for more details. For simplicity, the message $m$ has been directly taken into account in Bob's measurement basis. The experimental results in Fig.~\ref{fig:Results2} show a clear violation of inequality~\eqref{eq.M332}, up to $S_{\text{1bit}}=6.66^{+0.02}_{-0.02}$. \begin{figure}\label{fig:Results2} \end{figure} \section{Classical causal models for a sequence of projective measurements} Above we have analyzed the correlations between two timelike separated parties, whose measurements follow an underlying time order. We will now focus on a series of projective measurements performed on a single quantum system. A measurement with setting $x$, obtaining outcome $a$, is described by a set of projective operators $M^{x}_{a}$, such that after the measurement the system, initially in state $\rho_1$, is left in a state given by $\rho_2=\rho^{x}_{a}= M^{x}_{a}$. In a next time step a measurement with setting $y$, producing outcome $b$ is performed, leaving the system in a state $\rho_3=\rho^{y}_{b}=M^{y}_{b}$, and so forth. Similar to a standard Bell experiment, the classical causal description of this scenario, illustrated in Fig.~\ref{fig.dag1}, involves a random variable $\Lambda_1$---the \emph{ontic state}~\cite{Harrigan2010}---which fully specifies the initial state of the system. The probability that a measurement $x$ produces the outcome $a$ is then given by $p(a \vert x)= \sum_{\lambda_1} p(a\vert x,\lambda_1)p(\lambda_1)$. Here we have explicitly used the \emph{measurement independence} assumption~\cite{Hall2010,Chaves2015} $p(\lambda_1 \vert x)=p(\lambda_1)$, that the measurement setting can be chosen independently of how the system is prepared. After the measurement, the system will be in a potentially different state $\Lambda_2$. For a projective measurement, $\Lambda_2$ is fully specified by the setting and outcome of the preceding measurement, and does not depend directly on the pre-measurement state $\Lambda_1$, that is $p(\lambda_2\vert x,a,\lambda_1)=p(\lambda_2\vert x,a)$. This implies that all correlations between $\Lambda_1$ and $\Lambda_2$ are mediated via the measurement. To see this, note that the quantum probability distribution after a series of three sequential measurements is given by \begin{equation} \label{eq_quantum_temporal} p_{\mathrm{Q}}(a,b,c \vert x,y,z)= p(c\vert b,y,z)p(b\vert a,x,y)p(a\vert x) \, , \end{equation} where $p(c\vert b,y,z)=\tr \left[M^{z}_{c} \rho^{y}_{b} \right]$, $p(b\vert a,x,y)=\tr \left[M^{y}_{b} \rho^{x}_{a} \right]$ and $p(a\vert x)=\tr \left[M^{x}_{a} \rho_1 \right]$. In particular, any potential correlation between the measurement outcome $c$ in the third time step with the measurement setting and outcome in the first time step ($x$ and $a$, respectively), is screened-off by the intermediate measurement setting and outcome ($y$ and $b$), that is, $p(c\vert a,b,x,y,z)=p(c\vert b,y,z)$. This is similar to the no-signalling constraints arising in a Bell scenario and thus imposes restrictions to the classical causal models describing such a scenario. Specifically, a causal model that reproduces this independence without resorting to fine-tuning cannot contain a causal link of the form $\Lambda_2 \rightarrow \Lambda_3$, since such a link can generate unwanted correlations between the variables $X,A$ and $C$ (not mediated by $Y,B$). In fact, any model that contains such a link, can only satisfy the condition $p(c\vert a,b,x,y,z)=p(c\vert b,y,z)$ by virtue of causal parameters chosen from a set of measure zero~\cite{Pearlbook}, that is, the parameters are fine-tuned in such a way that these correlations are hidden from the observational data~\cite{Wood2015}. Following the above description, the case of 3 sequential projective measurements on a single qubit can be represented in terms of the causal structure in Fig.~\ref{fig.dag1}. The temporal correlations $p(a,b,c \vert x,y,z)$ compatible with this causal structure can then be decomposed as (with straightforward generalization to more time steps) \begin{align} \label{eq.temporal_decomposition} p(a,b,c \vert x,y,z) & = \sum_{\lambda_1,\lambda_2,\lambda_3} p(a \vert x,\lambda_1)p(b \vert y, \lambda_2) \\ \nonumber & p(c \vert z,\lambda_3) p(\lambda_1) p(\lambda_2 \vert x,a) p(\lambda_3 \vert y,b) . \end{align} Note that without any restrictions on the dimensionality the hidden states $\Lambda_i$ can contain the full information about the measurement settings and outcomes of the previous time steps. This implies that, in contrast to the spacelike Bell scenario, all temporal correlations of the form of Eq.~\eqref{eq_quantum_temporal} can be faithfully reproduced by the classical causal model in Fig.~\ref{fig.dag1}. This naturally raises the question whether further restrictions on the classical causal model, might reveal a quantum advantage. Similar to the so-called prepare-and-measure scenarios~\cite{Gallego2010}, one might expect that quantum systems of a given dimension give rise to measurement statistics that cannot be reproduced by classical systems $\Lambda_i$ of the same dimension. Since restriction on the dimension of $\Lambda_i$ in models of the form of Fig.~\ref{fig.dag1} would lead to non-convex sets that are technically very challenging to characterize~\cite{Chaves2016,Lee2015}, one typically considers convex relaxations. The resulting models contain the model of interest as a special case, but allow for shared randomness between the parties. For example, using the results of Ref.~\cite{Toner2003} we show in the Appendix~\ref{Sec:Supp1} that classical hidden states of dimension 4 (two bits of classical information), together with shared randomness between the parties, are enough to reproduce all correlations from a series of projective measurements on a single qubit. For a fair comparison to a qubit, we then considered hidden states of dimension two and could not find a difference between quantum and classical correlations. In light of these results it would be very interesting to test the model of Fig.~\ref{fig.dag1} with two-dimensional hidden states and no shared randomness. Due to the complexity of characterizing such non-convex sets, however, this remains an open question. \begin{figure} \caption{\textbf{A sequence of three projective measurement on a single qubit.} \textbf{(a)} A single quantum system is subject to a sequence of projective measurements at times $t_1$, $t_2$, and $t_3$ with settings $x,y,z$, and obtaining outcomes $a,b,c$, respectively. \textbf{(b)} A general classical causal model for the scenario in $(a)$. Note that, although causal graphs are formulated without reference to any spacetime structure, here we have drawn the graph such that the horizontal direction can be identified with time.} \label{fig.dag1} \end{figure} Besides restrictions on the dimension of the hidden state, certain physical constraints and assumptions might naturally lead to weaker causal models for a sequence of projective measurements on a single qubit. A well-known example is the Leggett-Garg (LG) model~\cite{Leggett1985} for testing macroscopic realism. This model is based on the assumption of noninvasive measurability, stating that it should be possible, in principle, to determine (measure) the state of a system without perturbing it. This is a special case of Eq.~\eqref{eq.temporal_decomposition}, where the hidden variable state is unchanged by the measurements and constant throughout the experiment, that is, $\lambda_i=\lambda$, which implies that \begin{equation} \label{eq.temporal_decomposition2} \begin{split} p(a,b,c \vert x,y,z) = \sum_{\lambda} & p(a \vert x,\lambda)p(b \vert y, \lambda) \\ & p(c \vert z,\lambda) p(\lambda) . \end{split} \end{equation} Note that this is the usual local hidden variable description encountered in Bell's theorem. Further, since the expectation values of a sequence of projective measurements on a single qubit are the same as for local measurements on a pair of entangled particles~\cite{Fritz2010}, quantum correlations also violate macroscopic realism, manifest as violations of so-called Leggett-Garg inequalities~\cite{Emary2013}. The result, however, relies critically on the assumption of noninvasive measurability, which is difficult to justify for a single quantum system. In fact, quite generally, the correlations obtained by sequential measurements on a quantum system will display signaling (e.g $p(b\vert x,y) \neq p(b\vert x^{\prime},y)$) as opposed to the model in Eq.~\eqref{eq.temporal_decomposition2} that only allow for non-signaling correlations. In other terms, the Leggett-Garg model implies independence relations that are not observed in the experiment. In this sense, in order to test incompatibility with the Leggett-Garg model we do not need to take the strength of the correlations into account and simply look for violations of independence relations implied by the model~\cite{Kofler2015}. This raises the question of whether one can find examples of temporal quantum experiments to which classical faithful causal models can reproduce all independence relations while at the same time being incompatible with the generated correlations. Next we show that this is indeed the case. \section{Quantum incompatibility with a weaker classical model for sequential projective measurements} From a causal inference perspective, given some observed probability distribution the goal is to find a faithful causal model reproducing all the (conditional) independence relations implied by this distribution. As a concrete example, consider a sequence of three projective measurements (see Fig.~\ref{fig.dag1}a) on an initial maximally mixed qubit state $\rho_1=\openone/2$. It is easy to verify that for arbitrary projective measurements, the measurement outcome of the third measurement is independent of the setting of the first measurement, i.e.\ $p(x,c)=p(x)p(c)$. In this case, the causal model in Fig.~\ref{fig.dag1}b is not faithful any longer because it allows for correlations between the variables $X$ and $C$. Instead, the most general causal model reproducing such independence relation is shown in Fig.~\ref{fig.dag1Weaker} where, in comparison with the causal model in Fig.~\ref{fig.dag1}, the causal link between the variable $B$ and $\Lambda_3$ is removed. Any distribution compatible with this model has a decomposition given by \begin{align} \label{eq.temporal_decomposition3} p(a,b,c \vert x,y,z) & = \sum_{\lambda_1,\lambda_2,\lambda_3} p(a \vert x,\lambda_1)p(b \vert y, \lambda_2) \\ \nonumber & p(c \vert z,\lambda_3) p(\lambda_1) p(\lambda_2 \vert a, x) p(\lambda_3 \vert y) . \end{align} Crucially, this classical model faithfully captures the observed independence $p(x,c)=p(x)p(c)$ that holds for a wide range of relevant experimental scenarios. This includes arbitrary measurements on an initially maximally mixed state, as well as arbitrary initial states in the \textsc{xy}-plane of the Bloch-sphere for the measurements in the experimental implementation below. \begin{figure}\label{fig.dag1Weaker} \end{figure} As detailed in Appendix~\ref{Sec:Supp1} any correlations compatible with Eq.~\eqref{eq.temporal_decomposition3} must respect the inequality \begin{align} \label{eq.tau3} & S_{\tau_3}=\mean{A_0B_0}+\mean{A_0B_1}+\mean{A_1B_0}-\mean{A_1B_1} \\ \nonumber & -\mean{BC_{001}}-\mean{BC_{101}}-\mean{BC_{010}}-\mean{BC_{110}} \leq 6 , \end{align} where the joint expectation values are defined as $\mean{A_xB_y}=\sum_{a,b}(-1)^{a+b} p(a,b \vert x,y)$ and $\mean{BC_{xyz}}=\sum_{a,b,c}(-1)^{b+c} p(a,b,c \vert x,y,z)$. This inequality can, however, be violated by a sequence of projective measurements on any initial qubit state. For instance, choosing measurement settings $A_0=\hat Z$, $A_1=\hat X$, $B_0=-C_1=(\hat Z+\hat X)/\sqrt{2}$, and $B_1=-C_0=(\hat Z-\hat X)/\sqrt{2}$ (where $\hat X$ and $\hat Z$ are the Pauli operators) obtains a value of $S_{\tau_3}=4\sqrt{2} > 6$. Furthermore, for any initial state in the $\textsc{xy}$-plane of the Bloch sphere, the resulting probability distribution $p(a,b,c \vert x,y,z)$ respects the independence relation $p(c\vert x)= p(c)$ implied by the model under test. Through unitary rotations this implies that for any fixed initial quantum state, one can generate temporal correlations that cannot be explained by non-fine-tuned models. Experimentally we test inequality~\eqref{eq.tau3} with photonic polarization qubits for an initial maximally mixed state, see Fig.~\ref{fig.dag1Weaker}a. For the intermediate measurement the system is coupled to a meter in the state $\ket{0}$. A measurement of the meter in the computational basis $\{\ket 0,\ket 1\}$ achieves a projective measurement of the system in a basis that is chosen by appropriate single-qubit unitaries applied to the system before and after the interaction. Notably, this measurement design can be straightforwardly generalized to more than three parties by replicating the von Neumann measurement. The experimental results in Fig.~\ref{fig:Results1} demonstrate a clear violation of inequality~\eqref{eq.tau3} by a series of three projective measurements on a single qubit, achieving a value of $S_{\tau_3}=6.65^{+0.01}_{-0.01}$. \begin{figure}\label{fig:Results1} \end{figure} \section{Discussion} Our results show, both in theory and experiment, that the discrepancy between the classical and quantum descriptions typically associated with spatial correlations extends to temporal correlation scenarios. The latter arise naturally in situations where spacelike separation cannot be practically guaranteed, or when a sequence of measurements is performed on the same quantum system. In the case of spatial correlations with an underlying time order, we have shown that a Bell-type inequality, designed to test classical models augmented with limited communication, displays a quantum violation if and only if the quantum protocol is also augmented with communication power. This highlights that a quantum advantage persists in a fair comparison of classical and quantum resources. In a purely temporal-correlations scenario we have shown that, as opposed to spatial Bell scenarios~\cite{Wood2015}, there is a faithful (non-fine-tuned) classical model capable of simulating all correlations from projective measurements on quantum states. It remains an open question whether these models are equivalent when imposing the same dimensionality to quantum and classical models. Notwithstanding, we identified quantum violations of inequalities associated with a slightly less powerful classical model that includes the Leggett-Garg model as special case~\cite{Leggett1985}. We have experimentally observed such a quantum violation, which demonstrates a stronger form of non-classicality of the correlations arising from a temporal sequence of projective measurements. Furthermore, our experimental design allows for tuning of the intermediate measurement strength, which will enable future studies of non-Markovian models with some residual correlations between the ontic states $\Lambda_i$ over multiple time-steps~\cite{Ringbauer2015Superchannel}. Our theoretical results are based on a causal modeling approach and thus formulated without reference to a background spacetime structure. The direction of time is only implicitly deduced from the flow of information between the parties or sequence of measurements. Experimentally, the considered scenarios are not subject to the locality loophole. However, the results rely on the related assumption that there is no hidden communication channel---other than the ones implied by the models in Figs.~\ref{fig.BellDag}b and~\ref{fig.dag1}---between the different time steps of the scenario under consideration. Practically, we also rely on a fair-sampling assumption to contend with imperfect detection efficiencies. Temporal correlation scenarios play an important role in communication complexity problems~\cite{Buhrman2010} and in the search for a physical principle behind quantum nonlocality, such as information causality~\cite{Pawlowski2009b}. Our results thus provide an avenue towards a more systematic understanding of the quantum advantage arising in such scenarios, that not only may lead to new ways of processing information but also to new insights into the nature of quantum correlations. \begin{acknowledgments} We thank C.~Budroni, F.~Costa, A.~Fedrizzi, and A.G.~White for helpful discussions and feedback, and T.~Vulpecula for experimental assistance. This work was supported in part by the Centres for Engineered Quantum Systems (CE110001013) and for Quantum Computation and Communication Technology (CE110001027), the Engineering and Physical Sciences Research Council (grant number EP/N002962/1), the Brazilian ministries MEC and MCTIC, the FQXi Fund, and the Templeton World Charity Foundation (TWCF 0064/AB38). \end{acknowledgments} \begin{thebibliography}{48} \makeatletter \providecommand \@ifxundefined [1]{ \@ifx{#1\undefined} } \providecommand \@ifnum [1]{ \ifnum #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \@ifx [1]{ \ifx #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi } \providecommand \natexlab [1]{#1} \providecommand \enquote [1]{``#1''} \providecommand \bibnamefont [1]{#1} \providecommand \bibfnamefont [1]{#1} \providecommand \citenamefont [1]{#1} \providecommand \href@noop [0]{\@secondoftwo} \providecommand \href [0]{\begingroup \@sanitize@url \@href} \providecommand 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{title} {\emph {\bibinfo {title} {{Nonlocality and communication complexity}},\ }}\href {\doibase 10.1103/RevModPhys.82.665} {\bibfield {journal} {\bibinfo {journal} {Rev.\ Mod.\ Phys.}\ }\textbf {\bibinfo {volume} {82}},\ \bibinfo {pages} {665} (\bibinfo {year} {2010})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Paw{\l}owski}\ \emph {et~al.}(2009)\citenamefont {Paw{\l}owski}, \citenamefont {Paterek}, \citenamefont {Kaszlikowski}, \citenamefont {Scarani}, \citenamefont {Winter},\ and\ \citenamefont {Zukowski}}]{Pawlowski2009b} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Paw{\l}owski}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Paterek}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Kaszlikowski}}, \bibinfo {author} {\bibfnamefont {V.}~\bibnamefont {Scarani}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Winter}}, \ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Zukowski}},\ }\bibfield {title} {\emph {\bibinfo {title} {{Information causality as a physical principle.}}\ }}\href {\doibase 10.1038/nature08400} {\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {461}},\ \bibinfo {pages} {1101} (\bibinfo {year} {2009})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Christof}\ and\ \citenamefont {L{\"o}bel}(2009)}]{porta} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Christof}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {L{\"o}bel}},\ }\href {http://porta.zib.de/} {\bibinfo {title} {\texttt{PORTA} -- \texttt{PO}lyhedron \texttt{R}epresentation \texttt{T}ransformation \texttt{A}lgorithm},\ } (\bibinfo {year} {2009})\BibitemShut {NoStop} \bibitem [{\citenamefont {Clauser}\ \emph {et~al.}(1969)\citenamefont {Clauser}, \citenamefont {Horne}, \citenamefont {Shimony},\ and\ \citenamefont {Holt}}]{Clauser1969} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Clauser}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Horne}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Shimony}}, \ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Holt}},\ }\bibfield {title} {\emph {\bibinfo {title} {{Proposed Experiment to Test Local Hidden-Variable Theories}},\ }}\href {\doibase 10.1103/PhysRevLett.23.880} {\bibfield {journal} {\bibinfo {journal} {Phys.\ Rev.\ Lett.}\ }\textbf {\bibinfo {volume} {23}},\ \bibinfo {pages} {880} (\bibinfo {year} {1969})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Horodecki}\ \emph {et~al.}(1995)\citenamefont {Horodecki}, \citenamefont {Horodecki},\ and\ \citenamefont {Horodecki}}]{Horodecki1995} \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Horodecki}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Horodecki}}, \ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Horodecki}},\ }\bibfield {title} {\emph {\bibinfo {title} {{Violating Bell inequality by mixed spin-12 states: necessary and sufficient condition}},\ }}\href {\doibase 10.1016/0375-9601(95)00214-N} {\bibfield {journal} {\bibinfo {journal} {Phys.\ Lett.\ A}\ }\textbf {\bibinfo {volume} {200}},\ \bibinfo {pages} {340} (\bibinfo {year} {1995})}\BibitemShut {NoStop} \end{thebibliography} \appendix \renewcommand{A\arabic{equation}}{A\arabic{equation}} \renewcommand{A\arabic{figure}}{A\arabic{figure}} \setcounter{equation}{0} \setcounter{figure}{0} \section{Classical models for a sequence of projective measurements on a qubit} \label{Sec:Supp1} \subsection{Classical simulation with hidden states of dimension 4} Note that the sequential measurement scenario can be mapped to an equivalent sequence of quantum teleportations~\cite{Bennett1993}. In the first time step Alice measures the state in her possession generating some local probability distribution $p(a \vert x)$, which she uses to prepare different states $\rho_a^x$ that she sends to Bob via the usual quantum teleportation protocol. Then Bob measures the teleported system generating a probability distribution $p(b \vert x,a,y)$ and prepares states $\rho_b^y$ that are teleported to Charlie. Clearly, any classical (hidden variable) protocol simulating the statistics of the measurements performed on the teleported states will immediately lead to a simulation of the equivalent sequence of projective measurements. As shown in Ref.~\cite{Toner2003}, this can be achieved using only two bits of classical communication between the parties and an arbitrary amount of shared randomness, implying that hidden states of dimension 4 and shared randomness are enough to obtain all quantum correlation obtained by a sequence of projective measurements on a single qubit. For a fair comparison of classical and quantum resources we have also considered whether a classical message of dimension 2 (1 bit of communication) is enough to simulate all projective measurements on a qubit state. To that aim we considered a scenario with two time steps. If Alice has the choice between two measurements (i.e.\ $x=0,1$) then one classical bit can carry all the information about the input $X$ and, using the argument in the main text, all the one-way signalling correlations in this case can be simulated using a classical message of dimension 2. We thus considered the case were $X$ assumes at least 3 different values and fully characterized the set of classical correlations for dichotomic measurements ($a,b=0,1$) and with $x=0,1,2$ and $y=0,1$. The polytope corresponding to this scenario is described by 864 inequalities (many being equivalent under the allowed symmetries given by party, input and output permutations). Considering qubit states and arbitrary projective qubit measurements we could not find any violation of these inequalities. It is interesting to note that among these inequalities we find the dimension-witness inequality from Ref.~\cite{Gallego2010}, given by \begin{equation} \mean{B_{00}} +\mean{B_{01}} +\mean{B_{10}} -\mean{B_{11}} -\mean{B_{20}} \leq 3, \end{equation} where $\mean{B_{xy}}=\sum_{b=0,1} (-1)^{b}p(b \vert x,y)$. As shown in Ref.~\cite{Gallego2010}, this inequality can be violated by measurements on a qubit. There, however, a slightly different situation is considered, the so-called prepare-and-measure scenario, where the variable $X$ uniquely identifies the state $\rho_x$ being prepared and to be measured in the second time step. In our case the states $\rho^a_x$ to be measured in the second time step will depend on $X$, but also on the measurement outcome $A$, a feature that seems to be enough to preclude any violation of the inequality above (or any other defining the scenario). It would be interesting to derive inequalities for more general scenarios including more measurement settings or time steps to see whether any violations can be found. \subsection{Derivation of Bell-type inequalities bounding classical models for a sequence of projective measurements} In order to derive Bell-type inequalities for the temporal scenario in Fig.~\ref{fig.dag1Weaker}, first note that all correlations compatible with such a model are also compatible with a model implying that \begin{align} \label{eq.scenario1} p(a,b,c \vert x,y,z) = & \sum_{\lambda} p(\lambda) p(a\vert x,\lambda) \\ \nonumber & p(b\vert a,x,y,\lambda)p(c\vert y,z,\lambda). \end{align} This follows from the fact that the arrow between the hidden variable at a given time step and the next measurement outcome (e.g.\ $\Lambda_2 \rightarrow B$) can be replaced by directed arrows from the measurement choices (or measurement outcomes) of the previous step to the next one (e.g, $X \rightarrow B$ and $A \rightarrow B$) plus a local noise variable ($\Lambda_B \rightarrow B$). Note that these noise terms are implicitly present in the model in Fig.~\ref{fig.dag1Weaker}, where they have been absorbed into $\Lambda_1$, $\Lambda_2$, $\Lambda_3$. When making the above replacement, however, these local noise terms have to be introduced explicitly. They can then be combined into a variable $\Lambda=(\Lambda_A,\Lambda_B,\Lambda_C)$ that acts as a common ancestor and source of randomness for all the measurement outcomes, see Fig.~\ref{fig.dag2}. In principle, one would further have to impose the independence of the local noise terms, that is $p(\lambda_A,\lambda_B,\lambda_C)=p(\lambda_A)p(\lambda_B)p(\lambda_C)$. This, however, would define a non-convex set that is very difficult to characterize~\cite{Chaves2016,Lee2015}. Instead, we consider a more general convex relaxation of this set which contains the case of independent noise variables as a special case. As detailed in Ref.~\cite{Chaves2016causal}, the probability distributions compatible with Eq.~\eqref{eq.scenario1} define a convex polytope that can be characterized in terms of finitely many extremal points. Given the list of extremal points one can resort to standard convex optimization software~\cite{porta} to find the dual description in terms of linear (Bell-type) inequalities. \begin{figure} \caption{A DAG for the tripartite temporal correlations scenario that contains the DAG in Fig.~\ref{fig.dag1} of the main text as a special case. As shown in Ref.~\cite{Chaves2016causal} the correlations compatible with this DAG define a convex set such that Bell-type inequalities can be derived using standard convex optimization software~\cite{porta}.} \label{fig.dag2} \end{figure} For quantum correlations arising from a sequence of three projective measurements it follows that $\mean{ABC}= \mean{A}\mean{BC}$. In other words, the full (three-point) correlator does not carry any information that is not already contained in the bipartite and single correlators. For this reason we focus our attention on inequalities involving the expectation values $\mean{AB}$ and $\mean{BC}$, which, due to the structure of the problem, can be defined in different ways. In the following we use \begin{align} \mean{AB_{x,y}}&=\sum_{a,b}(-1)^{a+b} p(a,b \vert x,y) \\ \mean{BC_{x,y,z}}&=\sum_{a,b,c}(-1)^{b+c} p(a,b,c \vert x,y,z) . \end{align} Since there is a causal link between $X$ and $B$, the correlations between $B$ and $C$ can explicitly depend on $X$. We then compute the Bell-type inequalities in this subspace, one of which is inequality~\eqref{eq.tau3} in the main text. \subsection{Quantum violation of inequality~\eqref{eq.tau3}} In order to search for a possible quantum violation of inequality~\eqref{eq.tau3}, we consider a single qubit in the initial pure state \begin{equation} \ket{\Psi}= \cos{\theta_0}\ket{0}+e^{i\phi_0}\sin{\theta_0}\ket{1} . \end{equation} At each time step we measure observables $\hat O_i$, parametrized by $\theta_i$ and $\phi_i$ \begin{equation} \hat O_i= \cos{\phi_i}\sin{\theta_i}\hat X+\sin{\phi_i}\sin{\theta_i}\hat Y+\cos{\theta_i}\hat Z . \end{equation} The full correlator can then be expressed, as $\mean{ABC}= \mean{A}\mean{BC}$ with $\mean{A}=\cos{\theta_0}\cos{\theta_1}+\cos{(\phi_0-\phi_1)} \sin{\theta_0}\sin{\theta_1}$ and $\mean{BC}=\cos{\theta_2}\cos{\theta_3}+\cos{(\phi_2-\phi_3)} \sin{\theta_2}\sin{\theta_3}$. Hence, as expected, the projective measurement at the second time step (corresponding to the outcome $b$) destroys any correlations between the first and third time steps (outcomes $a$ and $c$, respectively). Similar to $\mean{BC}$, the correlations between the first and second time steps are given by $\mean{AB}=\cos{\theta_1}\cos{\theta_2}+\cos{(\phi_1-\phi_2)} \sin{\theta_1}\sin{\theta_2}$. For projective measurements on a qubit state we thus obtain $\mean{BC_{x=0,y,z}}=\mean{BC_{x=1,y,z}}$. Since the first four terms of inequality~\eqref{eq.tau3} are nothing else than CHSH, this part can be maximized using the standard settings: $A_0=\hat Z$, $A_1=\hat X$, $B_0=(\hat X+\hat Z)/\sqrt{2}$, $B_1=(-\hat X+\hat Z)/\sqrt{2}$. The last four terms are maximized by setting $C_0=-B_1$ and $C_1=-B_0$, which obtains a value of $4$. In summary, quantum correlations can obtain $S_{\tau_3}=4+2\sqrt{2} >6$, thus violating inequality~\eqref{eq.tau3}. Curiously, this holds for any initial single-qubit state. \section{Experimental details} \label{Sec:Supp2} To test inequality~\eqref{eq.M332} in the main text we prepare a single-parameter family of two-qubit states of the form \begin{equation} \begin{split} \ket{\Psi}=\frac{1}{2}\bigl(&\sqrt{1+\kappa^2} (\ket{HH}+\ket{VV}) \\ &+ \sqrt{1-\kappa^2} (\ket{HV}+\ket{VH}) \bigr) . \label{eq:SIStates} \end{split} \end{equation} We generate this family of states by subjecting the separable state $\ket{D}\otimes(\sqrt{1+\kappa}\ket{H}+\sqrt{1-\kappa}\ket{V})/\sqrt{2}$ to a controlled-NOT gate, as shown in Fig.~\ref{fig:Setup} in the main text. The parameter $0\leq \kappa\leq 1$ turns out to be equal to the concurrence $\mathcal{C}$ of the resulting bipartite state, such that the generated amount of entanglement can be easily controlled by the initial state of the meter photon. Using the fixed measurement protocol in the main text, the states of Eq.~\eqref{eq:SIStates} achieve a value of $S_{\text{1bit}}=4+(1+\kappa)\sqrt{2}$ in inequality~\eqref{eq.M332}. It is, however, possible to optimize the measurement settings such that ideally every pure entangled quantum state violates inequality~\eqref{eq.M332}. Specifically, this amounts to modifying the protocol in the main text such that for $m=0$ Bob measures $B_0=B_1=B_2=\hat X$, while for $x=1$ he measures $B_0=(\hat X+\kappa\hat Z)/\sqrt{1+\kappa^2}$, $B_1=(\hat X - \kappa\hat Z)/\sqrt{1+\kappa^2}$ and $B_2= - \hat X$. Using this protocol, states of the form of Eq.~\eqref{eq:SIStates} achieve $S_{\text{1bit}}=4+2\sqrt{1+\kappa^2}\geq 6$. This is reminiscent of the observation that, using optimized measurement settings, the CHSH inequality \cite{Clauser1969} can be violated by every pure two-qubit entangled quantum state \cite{Horodecki1995}. In fact, inequality~\eqref{eq.M332} contains a CHSH inequality for the settings $A_1,A_2,B_0,B_1$, which can be violated in the usual way, while the additional communication in our protocol can be used to maximize the remaining four terms up to the maximal value of $4$ for every state of the form Eq.~\eqref{eq:SIStates}. Hence, the expectation value of inequality~\eqref{eq.M332} can be written as $S_{\text{1bit}}=4+S_{\textsc{chsh}}$, where $S_{\textsc{chsh}}$ is the CHSH parameter corresponding to the first four terms of inequality~\eqref{eq.M332}. \end{document}	arXiv
All-----TitleAuthor(s)AbstractSubjectKeywordAll FieldsFull Text-----About Illinois Journal of Mathematics Illinois J. Math. Volume 59, Number 1 (2015), 1-19. A new interpolation approach to spaces of Triebel–Lizorkin type Peer Christian Kunstmann More by Peer Christian Kunstmann Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text Article info and citation We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For $q\in[1,\infty]$, the $l^{q}$-interpolation method allows to interpolate linear operators that have bounded $l^{q}$-valued extensions. For $q=2$ and if the Banach function spaces are $r$-concave for some $r<\infty$, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the $H^{\infty}$-functional calculus. As a special case, we obtain Triebel–Lizorkin spaces $F^{2\theta}_{p,q}(\mathbb{R}^{d})$ by $l^{q}$-interpolation between $L^{p}(\mathbb{R}^{d})$ and $W^{2}_{p}(\mathbb{R}^{d})$ where $p\in(1,\infty)$. A similar result holds for the recently introduced generalized Triebel–Lizorkin spaces associated with $R_{q}$-sectorial operators in Banach function spaces. So, roughly speaking, for the scale of Triebel–Lizorkin spaces our method thus plays the role the real interpolation method plays in the theory of Besov spaces. Illinois J. Math., Volume 59, Number 1 (2015), 1-19. Received: 6 October 2014 Revised: 13 February 2015 First available in Project Euclid: 11 February 2016 Permanent link to this document https://projecteuclid.org/euclid.ijm/1455203156 doi:10.1215/ijm/1455203156 Mathematical Reviews number (MathSciNet) MR3459625 Zentralblatt MATH identifier Primary: 46B70: Interpolation between normed linear spaces [See also 46M35] 47A60: Functional calculus 42B25: Maximal functions, Littlewood-Paley theory Kunstmann, Peer Christian. A new interpolation approach to spaces of Triebel–Lizorkin type. Illinois J. Math. 59 (2015), no. 1, 1--19. doi:10.1215/ijm/1455203156. https://projecteuclid.org/euclid.ijm/1455203156 M. G. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded $H^\infty$ functional calculus, J. Aust. Math. Soc. A 60 (1996), no. 1, 51–89. Mathematical Reviews (MathSciNet): MR1364554 Zentralblatt MATH: 0853.47010 Digital Object Identifier: doi:10.1017/S1446788700037393 G. 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Mayboroda, Hardy and BMO spaces associated with divergence form elliptic operators, Math. Ann. 344 (2009), no. 1, 37–116. Digital Object Identifier: doi:10.1007/s00208-008-0295-3 N. J. Kalton, P. C. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $H^\infty $-calculus with applications to differential operators, Math. Ann. 336 (2006), no. 4, 747–801. N. J. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345. Digital Object Identifier: doi:10.1007/s002080100231 N. J. Kalton and L. Weis, The $H^\infty$-functional calculus and square function estimates, manuscript, 2004. N. J. Kalton and L. Weis, Euclidean structures, manuscript, 2004. P. C. Kunstmann and A. Ullmann, $R_s$-sectorial operators and generalized Triebel–Lizorkin spaces, J. Fourier Anal. Appl. 20 (2014), no. 1, 135–185. P. C. Kunstmann and L. Weis, Maximal $L^p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., vol. 1855, Springer, Berlin, 2004, pp. 65–311. Digital Object Identifier: doi:10.1007/978-3-540-44653-8_2 F. Lancien and C. Le Merdy, Square functions and $H^\infty$ calculus on subspaces of $L^p$ and on Hardy spaces, Math. Z. 251 (2005), 101–115. C. Le Merdy, On square functions associated to sectorial operators, Bull. Soc. Math. France 132 (2004), no. 1, 137–156. Digital Object Identifier: doi:10.24033/bsmf.2462 J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I and II, Springer, Berlin, 1996. Reprint of the 1st edn. J. Suárez and L. Weis, Interpolation of Banach spaces by the $\ga$-method, Methods in Banach space theory, London Math. Soc. Lecture Note Ser., vol. 337, Cambridge University Press, Cambridge, 2006, pp. 293–306. Digital Object Identifier: doi:10.1017/CBO9780511721366.015 H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland, Amsterdam, 1978. H. Triebel, Characterizations of Besov–Hardy–Sobolev spaces via harmonic functions, temperatures, and related means, J. Approx. Theory 35 (1982), no. 3, 275–297. Digital Object Identifier: doi:10.1016/0021-9045(82)90009-0 H. Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser, Basel, 1983. Digital Object Identifier: doi:10.1007/978-3-0346-0416-1 L. Weis, A new approach to maximal $L^p$-regularity, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., vol. 215, Dekker, New York, 2001, pp. 195–214. New content alerts Email RSS ToC RSS Article Turn Off MathJax What is MathJax? A new characterization of Triebel-Lizorkin spaces on $\mathbb R^n$ Yang, Dachun, Yuan, Wen, and Zhou, Yuan, Publicacions Matemàtiques, 2013 Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability Kempka, Henning, Functiones et Approximatio Commentarii Mathematici, 2010 A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces Han, Yongsheng, Müller, Detlef, and Yang, Dachun, Abstract and Applied Analysis, 2008 MAXIMAL REGULARITY FOR INTEGRO-DIFFERENTIAL EQUATION ON PERIODIC TRIEBEL-LIZORKIN SPACES Bu, Shangquan and Fang, Yi, Taiwanese Journal of Mathematics, 2008 A new generalization of Besov-type and Triebel-Lizorkin-type spaces and wavelets SAKA, Koichi, Hokkaido Mathematical Journal, 2011 Herz--Morrey type Besov and Triebel-Lizorkin spaces with variable exponents Dong, Baohua and Xu, Jingshi, Banach Journal of Mathematical Analysis, 2015 On dilation operators in Triebel-Lizorkin spaces Schneider, Cornelia and Vybíral, Jan, Functiones et Approximatio Commentarii Mathematici, 2009 Well-posedness of fractional degenerate differential equations with infinite delay in vector-valued functional spaces Bu, Shangquan and Cai, Gang, Journal of Integral Equations and Applications, 2017 Some function spaces relative to Morrey-Campanato spaces on metric spaces Yang, Dachun, Nagoya Mathematical Journal, 2005 MULTILINEAR ESTIMATES ON FREQUENCY-UNIFORM DECOMPOSITION SPACES AND APPLICATIONS Ru, Shaolei, Taiwanese Journal of Mathematics, 2014 euclid.ijm/1455203156	CommonCrawl
\begin{document} \newtheorem{theo}{Theorem}[section] \newtheorem{prop}[theo]{Proposition} \newtheorem{cor}[theo]{Corollary} \newtheorem{lem}[theo]{Lemma} \newtheorem{rem}[theo]{Remark} \newtheorem{con}[theo]{Conjecture} \newtheorem{as}[theo]{Assumption} \newtheorem{de}[theo]{Definition} \begin{center} {\Large \bf Covariance Structure of \\ Coulomb Multiparticle System \\ } \end{center} \begin{center} TATYANA S. TUROVA\footnote{ Mathematical Center, University of Lund, Box 118, Lund S-221 00, Sweden, \newline [email protected], tel.:+46 46 222 8543 \newline On leave from IMPB Russian Academy of Science Branch of KIAM RAS} \end{center} \begin{center} { \it In memory of Vadim Alexandrovich Malyshev, 1938-2022 } \end{center} \setcounter{page}{1} \renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}} \setcounter{equation}{0} \begin{abstract} We consider a system of particles on a finite interval with Coulomb 3-dimensional interactions between close neighbours, i.e. only a few other neighbours apart. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31-36] to study the flow of charged particles. Notably even the nearest-neighbours interactions case, the only one studied previously, was proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Here we include as well interactions beyond the nearest-neighbours ones. Surprisingly but this leads to qualitatively new features even when the external force is zero. The order of the covariances of distances between pairs of consecutive charges is changed when compared with the former nearest-neighbours case, and moreover the covariances exhibit periodicity in sign: the interspacings are positively correlated if the number of interspacings between them is odd, otherwise, they are negatively correlated. In the course of the proof we derive Gaussian approximation for the limit distribution for dependent variables described by a Gibbs distribution. \end{abstract} {\it MSC2010 subject classifications.} 82B21, 82B26, 60F05. \section{Introduction} \subsection{Models with Coulomb interactions.} Consider a system of $N+1$ identical particles (or charges) located at random points of the interval $[0,1]$ $$\bar{Y}= (Y_0=0, Y_1, \ldots, Y_{N-1},Y_N=1).$$ The fixed values $Y_{0}=0$ and $Y_{N}=1$ mean that at both ends there are particles with fixed positions. The potential energy of the system is defined as a function of its configuration \begin{equation}\label{S} \bar{y} \in {\cal S}:=\{(y_{0}, \ldots , y_{N}): 0=y_{0}<\ldots<y_{N}=1\} \end{equation} as follows \begin{equation}\label{U} U(\bar{y})= \sum_{0 \leq j<k\leq N: \ k-j \leq K} \beta_{k-j} V(y_{k}-y_{j}) + \sum_{k=1}^{N}\int_{0}^{y_k}F_sds, \end{equation} where the positive function $V(y_{k}-y_{j}) $ represents a pair-wise interaction between the particles at $y_{k}$ and $y_{j}$ which depends on the distance between them. The constant $K\geq 1$ is the range of interactions as each particle interacts with its $2K$ neighbours (except the particles close to the ends). All constants $\beta_m$ are positive, meaning that all the charges have the same sign, and the function $F_s$ represents an external force at the point $s\in [0,1]$. Next we assume that at positive temperature $t>0$ the configuration vector $\bar{Y}$ has Gibbs distribution with density \begin{equation}\label{dGt} f (\bar{y}; t, {\bar{\beta}} )=\frac{1}{Z_{ \frac{1}{t} U}(N) } e^{-\frac{1}{t} U (\bar{y})} , \ \ \ \ \ \bar{y} \in {\cal S}, \end{equation} where the normalizing factor (the partition function) is \begin{equation}\label{Z} Z_{\frac{1}{t}U}(N)=\int\ldots\int_{0 < y_{1}<...<y_{N-1}< 1}e^{-\frac{1}{t}U(\bar{y})} dy_{1}\ldots y_{N-1}. \end{equation} Due to the scaling properties of the function $f_{\beta, \gamma}(\bar{y}, t )$ \begin{equation}\label{fsca} f_{\bar{\beta}}(\bar{y}, t )=f_{\frac{\bar{\beta}}{t}}(\bar{y}, 1), \end{equation} without loss of generality we shall set $t=1$ to study the positive temperature case. Here we consider a pair-wise Coulomb repulsive interaction \begin{equation} V(x)=\frac{1}{x}, \ \ x>0. \ \ \ \ \label{V} \end{equation} This model can be viewed as a special case of a general class of models in statistical physics known as Coulomb Gas or Riezs Gas (after \cite{R}), which have been studied over decades, and still remain to be in focus of mathematical physics (as confirmed by recent reviews \cite{Le} or \cite{Ch}). Models with potential (\ref{V}), also called Coulomb 3-dim potential, describe the so-called long-range interactions. This type of interactions arises in many areas of physics (\cite{CDR}), however the corresponding models are most difficult for analysis. The model of form (\ref{dGt}) was introduced by Malyshev in \cite{M}. It ingeniously combines proper 3-dim Coulomb potential with the assumption that the particles are on a line. The fixed particles at the end-points may be interpreted as a result of an external potential confining the particles configuration within an interval. The one-dimensional Coulomb gas models form a special well-studied class following the first works by Lenard (\cite{L}, \cite{L1}, \cite{L2}). In setting (\ref{dGt}) such models have been studied mostly in the case when $K=N$, i.e., when any pair of particles interacts. The one-dimensional case is known for translation-symmetry breaking phenomena under certain conditions on the variance of the charge (\cite{AM}, \cite{AGL}). Malyshev \cite{M} considered only the nearest neighbours interactions in (\ref{dGt}) i.e., when $K=1$. However, even in this seemingly simple case he discovered a surprisingly rich picture of phase transitions in the ground states (formally, it is model (\ref{dGt}) at zero temperature $t=0$) for the external force $F_s$ being constant in $s$, i.e., when \begin{equation} F_s=F=F(N) \geq 0, \ \ s\in[0,1], \ \ \ \ \label{F} \end{equation} and increases in $N$. It was shown that there are a few phases for the distribution of the interspacings between the particles, changing abruptly with increase of the force from uniform on the interval to condensation of the particles at one end only. Similar complex phenomena are known for other models of Coulomb gases (e.g., \cite{AM}, \cite{J1}, \cite{J2}). The results of \cite{M} inspired the further study in \cite{MZ} of the nearest-neighbours interactions model (\ref{dGt}) at any positive temperature but without external force ($F_s\equiv 0$). The phase transitions with respect to the external force in form (\ref{F}) were described in \cite{T} at any temperature but again only for the case when $K=1$, i.e., only for the nearest-neighbours interactions. The system was proved to exhibit very small fluctuations, namely of order $N^{-3}$ for the variance of interspacings (\cite{M}, \cite{MZ}, \cite{T}). Notably the same order $N^{-3}$ of variance but for a certain mean-field linear statistics of the one-dimensional jellium model was recently reported in \cite{FM}. The correlation functions are also being studied in one-dimensional models (see, e.g., recent report \cite{Bee}). Here we extend the analysis of the model (\ref{dGt}) beyond the nearest-neighbours interactions. It turns out that by only taking into consideration the second nearest-neighbours interactions we encounter new phenomena. The most remarkable is emerging of the periodic in sign structure of the correlations. Two interspacings are positively correlated if the number of spacings between them is odd, otherwise, they are negatively correlated. Furthermore, adding only the next to the nearest-neighbours interactions increases the order of the covariances in $N$, hence showing another kind of phase transition. Before we state our results we shall briefly explain in the next section why the case of nearest-neighbours interactions, i.e., $K=1$ in (\ref{dGt}), is so drastically distinct from the others, when $K>1$. \subsection{Representation via conditional distribution} For any $K\geq 1$ let us arbirarily fix the vector $\bar {\beta}=(\beta_1, \ldots, \beta_K)$ with positive entries, and define an auxiliary random vector $\bar{X}=(X_1, \ldots, X_N)$ with density of distribution \begin{equation}\label{fXg} f_{\bar {X}}(\bar{x})= f_{\bar {X}}(\bar{x})=\frac{1}{Z_{\bar {\beta},F}(N)} e^{-\sum_{k=1}^{N-K+1} \sum_{j=0}^{K-1} \frac{\beta_{j}}{ x_{k}+x_{k+1}+\ldots +x_{k+j}}-\sum_{k=1}^{N}F(x_{1}+\ldots +x_{k}) } \end{equation} in $\in [0,1]^N$, where $Z_{\bar {\beta}, F}(N)$ is the normalizing constant. Hence, the vector $\bar{X}=(X_1, \ldots, X_N)$ has distribution of spacings as for the charges defined above but under the assumption that only at one end, say at the origin, there is a fixed charge, while the other end (at $x_{1}+\ldots+x_N$) is "free" on $R^+$, instead of being fixed at $1$ as in configuration $\bar{Y}$. The relations between these vectors are clarified in the following statement which is straightforward to check. \begin{prop}(\cite{MZ}, \cite{T})\label{AuP} The following distributional identity holds for all $N \geq 2$: \begin{equation}\label{cond} (1-Y_{N-1}, Y_{N-1}-Y_{N-2}, \ldots , Y_{2}-Y_{1}, Y_{1})\ \ \stackrel{d}{=} \ \ (X_1 , \ldots , X_{N}) \mid _{\sum_{i=1}^{N} X_i =1 }. \end{equation} \end{prop} $\Box$ Simplifying (\ref{fXg}) we get \begin{equation}\label{fX1} f_{\bar {X}}(\bar{x})=\frac{1}{Z_{\bar {\beta}, F}(N)} e^{-\sum_{k=1}^{N-K+1} \sum_{j=0}^{K-1} \frac{\beta_{j}}{ x_{k}+x_{k+1}+\ldots +x_{k+j}} - \sum_{k=1}^{N} F(N-k+1)x_{k} } . \end{equation} This shows that if and only if $K=1$, the components of the vector $\bar{X}=(X_1, \ldots, X_N)$ are independent random variables. Additionally, when $F=0$, they are also identically distributed. Otherwise, when $K>1$ the components of the vector $\bar{X}=(X_1, \ldots, X_N)$ are {\it dependent}, and moreover they are {\it not } identically distributed even if $F=0$. This explains why models with $K>1$ might have different behaviour. Therefore it is natural to begin with the case $K=2$ and zero external force, i.e., when $F_s\equiv 0$ in (\ref{U}). Denote for this particular case \begin{equation}\label{fX0} f_{\bar {X}}(\bar{x})=\frac{1}{Z_{\beta, \gamma} } e^{- \beta \sum_{k=1}^N\frac{1}{x_{k}} + \gamma \sum_{k=1}^{N-1}\frac{1}{x_{k}+x_{k+1} } } . \end{equation} This form is in a direct relation with the previously studied case $K=1$ (\cite{M}, \cite{MZ}, \cite{T}) which is equivalent to $\gamma =0$ in the last formula. \subsection{Results.} We study particular cases of potential energy (\ref{U}) for $K=2$. We begin with the case which allows an exact solution. Namely, to eliminate the boundary conditions we introduce a circular potential. In other words, we consider the charges on a ring of length 1. \begin{theo}{\bf (Circular potential, zero external force.)}\label{T1} For any $0\leq \gamma \leq \beta$ and $N>2$ let \begin{equation}\label{U1} U_{\beta,\gamma}^{\circ}(\bar{y})= \sum_{j=1}^{N} \frac{\beta}{y_{j}-y_{j-1}} +\sum_{j=2}^{N} \frac{\gamma}{y_{j}-y_{j-2}} + \frac{\gamma}{1-y_{N-1}+y_1}, \ \ \ \ \ \bar{y} \in {\cal S}, \end{equation} and let $\bar{Y}^{\circ}$ be a random vector on $ {\cal S}$ with Gibbs density \[ f_{\bar{Y}^{\circ}}(\bar{y})= \frac{1}{Z^{\circ}(N) } e^{- U_{\beta,\gamma}^{\circ}(\bar{y})} , \ \ \ \ \ \bar{y} \in {\cal S}, \] where $Z^{\circ}(N)$ is the partition function. Denote also \begin{equation}\label{del} \delta =\delta(\gamma)= \frac{\gamma}{4\beta +\gamma+ 2\sqrt{ 4\beta^2+2\beta \gamma}}. \end{equation} Then $\bar{Y}^{\circ}$ is evenly spaced in average so that \begin{equation}\label{TCM} {\mathbb{E}}\left(Y_{k+1}^{\circ}-Y_k^{\circ} \right) =\frac{1}{N}. \end{equation} $\bar{Y}^{\circ}$ has the following covariance structure: for all $j\leq k $ \begin{equation}\label{TC1} {\bf Cov}\left(Y_{j+1}^{\circ}-Y_j^{\circ}, Y_{k+1}^{\circ}-Y_k^{\circ} \right) \end{equation} \[= \frac{1}{N^3} \left( \frac{1}{2\beta + \gamma (1 - \delta)/2} +o\left(N^{-\frac{1}{3}} \right) \right) \left( (-\delta)^{(j-k)_{N/2}} -\frac{1}{N} \frac{1-\delta}{1+\delta} +o(1/N) \right), \] where \[(j-k)_{N/2} = \min\{j-k,N-(j-k)\}\] is the smallest number of charges on the circle between the $j$-th and the $k$-th ones. In particular, \begin{equation}\label{VarT} {\bf Var}\left(Y_{j+1}^{\circ}-Y_j^{\circ}\right) = \frac{1}{N^3} \left( \frac{1}{2\beta + \gamma (1 - \delta)/2} +o\left(N^{-\frac{1}{3}} \right) \right). \end{equation} \end{theo} Here it is most natural to set $\gamma=\beta$. However, keeping $\gamma$ as a free parameter allows one to derive as a particular case $\gamma=0$ of Theorem \ref{T1} the results for the model \cite{MZ} (or \cite{T}) with nearest-neighbour interactions (and zero external force). Thus setting $\gamma=0$ in (\ref{VarT}) we recover the corresponding results for the variance derived previously in \cite{MZ} and \cite{T}. Notice that the covariance (\ref{TC1}) was not considered previously even for $\gamma=0$. By definition (\ref{del}) it holds that $0<\delta(\gamma)<1$ for all $\gamma>0$, while $\delta(0)=0$. Hence the leading term in the covariance formula (\ref{TC1}) has the following order in $N$: \begin{equation}\label{order} \begin{array}{rll} & N^{-3} \ \frac{1}{2\beta + \gamma (1 - \delta)/2} , & \mbox{ if } j=k \mbox{ (variance), }\\ \\ (-\delta)^{k-j}& N^{-3}\ \frac{1}{2\beta + \gamma (1 - \delta)/2}, & \mbox{ if } 0<k-j\ll \log N,\\ \\ -& N^{-4}\ \frac{1}{2\beta + \gamma (1 - \delta)/2}\frac{1-\delta}{1+\delta}, & \mbox{ if } k-j\gg \log N. \end{array} \end{equation} This shows that a positive $\gamma$ affects the covariances between spacings which are separated by at most $\log N$ charges (or spacings), and moreover in a peculiar manner of changing the sign of correlations. Hence, the order changes smoothly from $N^{-3}$ to $N^{-4}$ when $k-j$ increases to $\log N$. Remarkably, beyond the range $\log N$ between $j$ and $k$ the covariance is strictly negative, and is of order $N^{-4}$. In case when $\gamma=0$, the covariances between any pair of spacings are all negative and are of order $N^{-4}$. Our analysis does not imply any conclusion about the sign of correlations for an arbitrary finite range $K$ in (\ref{dGt}). However, the exponential decrease in value to the order $N^{-4}$, and the order $N^{-4}$ for the long-range covariances beyond $\log N$ charges, seem to be universal. The negative sign of the latter can be explained by the constraint that particles are on an interval of fixed length. The below established approximation of the original model $\bar{Y}$ by the circular version $\bar{Y}^{\circ}$ described in Theorem \ref{T1}, allows us to describe the covariance structure of $\bar{Y}$ in terms of the covariances of $\bar{Y}^{\circ}$. \begin{theo}\label{T2o} For any $0\leq \gamma \leq \beta$ for all $N>2$ let \[ U_{\beta,\gamma}(\bar{y})= \sum_{j=1}^{N} \frac{\beta}{y_{j}-y_{j-1}} +\sum_{j=2}^{N} \frac{\gamma}{y_{j}-y_{j-2}} , \ \ \ \ \ \bar{y} \in {\cal S}, \] and let $\bar{Y}$ be a random vector on $ {\cal S}$ with Gibbs density \[ f_{\bar{Y}}(\bar{y})= \frac{1}{Z(N) } e^{- U_{\beta,\gamma}(\bar{y})} , \ \ \ \ \ \bar{y} \in {\cal S}, \] where $Z(N)$ is the partition function. There are positive $\alpha(\gamma)$ and $C$ such that the following statements for the distribution of $\bar{Y}$ hold. $\bar{Y}$ is asymptotically evenly spaced in averaged so that \begin{equation}\label{TCMn} {\mathbb{E}}\left(Y_{k+1}-Y_k \right) = \frac{1}{N}\left( 1+ O\left(e^{-\alpha \min\{j, \ N-k\}}\right) \right)\left( 1+ O\left(N^{-1/3}\right) \right), \end{equation} and for all $j\leq k $ \begin{equation}\label{TC1n} {\bf Cov}\left(Y_{j+1}-Y_j, Y_{k+1}-Y_k \right) \end{equation} \[ = {\bf Cov}\left(Y_{j+1}^{\circ}-Y_j^{\circ}, Y_{k+1}^{\circ}-Y_k^{\circ} \right) \left( 1+ O\left(e^{-\alpha \min\{j, \ N-k\}}\right) \right)\left( 1+ O\left(N^{-1/3}\right) \right). \] \noindent where the covariances ${\bf Cov}\left(Y_{j+1}^{\circ}-Y_j^{\circ}, Y_{k+1}^{\circ}-Y_k^{\circ} \right)$ are described by Theorem \ref{T1}. \end{theo} For the models with constant external force $F_s=F(N)$ one may predict results as in \cite{T}, however, this will be a subject of a separate work. \setcounter{equation}{0} \renewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}} \section{Proofs} \subsection{Lagrange multiplier} The representation given in Proposition \ref{AuP} via conditional distribution enables us to use the Lagrange multiplier method elaborated previously in a similar context in \cite{MZ} and \cite{T1} for the case $\gamma=0$, i.e., when $X_1, \ldots, X_N$ are $i.i.d.$ random variables. Let us embed the distribution of $\bar{X}$ given by (\ref{fX0}) into a more general class as follows. For any $\lambda \in \mathbb{R}$ define a new random vector $\bar{X}_{\lambda}=(X_{1,\lambda}, \ldots, X_{N,\lambda})$ with the following density of distribution: \begin{equation}\label{fXl} f_{\bar{X}_{\lambda}}(\bar{x})=\frac{1}{Z_{\beta, \gamma,\lambda}(N)} e^{-H_{\lambda}(\bar{x})}, \ \ \ \ \ \bar{x} \in [0,1]^N, \end{equation} where \begin{equation}\label{Hlam} H_{\lambda}(\bar{x}) = \beta \sum_{k=1}^N \frac{1}{x_{k}}+\gamma \sum_{k=1}^{N-1} \frac{1}{ x_{k}+x_{k+1} } +\lambda\sum_{k=1}^N x_{k}, \end{equation} and $Z_{\beta, \gamma,\lambda}(N)$ is the normalizing constant. This notation is consistent with (\ref{fX0}) as \[f_{\bar{X}_{0}}(\bar{x})=f_{\bar{X}}(\bar{x}),\] i.e., $$ \bar{X}\stackrel{d}{=} \bar{X}_{0}.$$ The remarkable property of new random vector $\bar{X}_{\lambda}$ is that for any $\lambda \in \mathbb{R}$ the following equality in distribution holds: \begin{equation}\label{condl} \bar{X} \mid _{\sum_{i=1}^{N} X_i =1 } \ \ \stackrel{d}{=} \ \ \bar{X}_{\lambda} \mid _{\sum_{i=1}^{N} X_{i,\lambda} =1 }. \end{equation} Diaconis and Freedman \cite{DF} used this property to prove conditional Central Limit Theorem for the identically distributed random variables. Their result was used in \cite{MZ} for the analysis of a Coulomb gas model. Then in \cite{T1} the same technique was proved to work even without the assumption of the identity of distributions. Here we further develop the argument for the case of dependent random variables to study limit distribution. Our derivation of the Gaussian limit is, however, different from the mentioned above approaches. We begin with choosing parameter $\lambda$ so that the condition $ \sum_{k=1}^N X_{k,\lambda} =1$ holds unbiased, i.e., \begin{equation}\label{ES} \mathbb{E} S_{N, \lambda} : =\sum_{k=1}^N\mathbb{E} X_{k,\lambda} = 1. \end{equation} It turns out as we will see below, that the conditional distribution with such parameter $\lambda$ is well approximated by the unconditional one. The existence of $\lambda$ satisfying (\ref{ES}) is proved below. Here we note that by the properties of Gibbs distribution we have (straightforward to check) \begin{equation}\label{var} \frac{\partial}{\partial \lambda} \mathbb{E} S_{N, \lambda}= - Var \left(S_{N, \lambda}\right). \end{equation} This proves that $\mathbb{E} S_{N, \lambda} $ as a function of $\lambda$ is strictly decreasing, hence the uniqueness of the solution $\lambda$ to (\ref{ES}) follows (when exists). After we prove below existence of $\lambda$ satisfying (\ref{ES}) we shall establish the Gaussian approximation for the distribution of $\bar{X}_{\lambda}$ for this value $\lambda$. This will allow us to tackle the original distribution of $\bar{Y}$. \subsection{Approximation by a circular distribution. } As we observed already the entries of $\bar{X}_{\lambda}$ are not identically distributed, which complicates analysis. However, recent results \cite{T1} help us to use an approximation of the distribution of $\bar{X}_{\lambda}$ by an easier ``circulant'' form of distribution (\ref{fXl}). Namely, introduce a circulant version of function in (\ref{Hlam}): \begin{equation}\label{H0c} H_{\lambda}^{\circ}(\bar{x}):=H_{\lambda}(\bar{x})+\gamma \frac{1}{ x_{1}+x_{N} } \end{equation} \[=\beta \sum_{k=1}^N \frac{1}{x_{k}}+\gamma \sum_{k=1}^{N-1} \frac{1}{ x_{k}+x_{k+1} } +\gamma \frac{1}{ x_{1}+x_{N} } + \lambda\sum_{k=1}^Nx_{k},\] and define $\bar{X}^{\circ}_{\lambda}$ to be a random vector whose distribution density is \begin{equation}\label{Mt7} f_{\bar{X}^{\circ}_{\lambda}}(\bar{x})=\frac{1}{Z^{\circ}_{\beta, \gamma,\lambda}(N)} e^{-H_{\lambda}^{\circ}(\bar{x})}, \ \ \ \ \ \bar{x} \in [0,1]^N, \end{equation} where $Z^{\circ}_{\beta, \gamma, \lambda}(N)$ is the normalizing constant. Notice that by the same argument as in Proposition \ref{AuP} vector $\bar{X}^{\circ}_{0}=\bar{X}^{\circ}$ is related to $\bar{Y}^{\circ}$ in the same manner as explained in (\ref{cond}). The following statement on the approximation rate is proved in \cite{T1} for $\bar{X}^{\circ}= \bar{X}^{\circ}_{0}$, i.e., the case $\lambda=0$. \begin{lem}\label{Cor0}({\protect{\cite{T1}}}) For any $\gamma \geq 0$ there is a positive constant $\alpha=\alpha(\gamma)$ such that for any $\beta>0$ there is positive constant $C$ such that for all $1\leq j<k \leq N$ and all $(x,y) \in [0,1]$ \begin{equation}\label{CM2} \left\|f_{X_{j} , X_{k}}(x,y)-f_{X^{\circ}_{j}, X^{\circ}_{k} }(x,y)\right\| \leq Ce^{-\alpha \min\{j,N-k\}} f_{X^{\circ}_{j}, X^{\circ}_{k} }(x,y), \end{equation} as well as for all $1\leq j \leq N$ \begin{equation}\label{CM} \left\|f_{X_{j} }(x)-f_{X^{\circ}_{j}}(x)\right\| \leq Ce^{-\alpha \min\{j,N-j\}} f_{X^{\circ}_{j} }(x). \end{equation} \end{lem} Notice that here $\alpha(0)=+\infty$ since $\bar{X_{\lambda} } \ \stackrel{d}{=} \ \ \bar{X^{\circ}_{\lambda} }$ when $\gamma=0$. The observation that \[ f_{X_{j,\lambda} , X_{k,\lambda}}(x,y)=\frac{e^{-\lambda x - \lambda y} f_{X_{j} , X_{k}}(x,y)}{ {\mathbb{E}e^{-\lambda X_{j} - \lambda X_{k}}}}, \] by Lemma \ref{Cor0} yields the following bounds \begin{equation}\label{Ea1} \left\| \mathbb{E} X_{r,\lambda} -\mathbb{E} X^{\circ}_{r,\lambda} \right\|\leq Ce^{-\alpha r} \mathbb{E} X^{\circ}_{r,\lambda}, \end{equation} and \begin{equation}\label{Ea2} \left\| {\bf Cov }\left({X_{j, \lambda} , X_{k, \lambda}}\right)- {\bf Cov }\left( {X^{\circ}_{j, \lambda}, X^{\circ}_{k, \lambda} }\right) \right\|\leq C e^{-\alpha \min\{j,N-k\}} {\bf Cov } \left( {X^{\circ}_{j, \lambda}, X^{\circ}_{k, \lambda} } \right), \end{equation} where positive constant $C$ does not depend on $N$ or $\lambda$ (at least for all positive $\lambda$). Due to the symmetry of function $H^{\circ}_{\lambda}$ the entries of $\bar{X}^{\circ}_{\lambda}$ are identically distributed (dependent) random variables. Hence, \begin{equation}\label{Ma21} \mathbb{E} X^{\circ}_{k,\lambda} =m^{\circ}_{\lambda}(N) \end{equation} for some positive $m^{\circ}_{\lambda}(N)$, and the condition (\ref{ES}) is equivalent to \begin{equation}\label{n5} m^{\circ}_{\lambda}(N) = \frac{1}{N}. \end{equation} Then (\ref{Ea1}) and (\ref{Ma21}) gives us \begin{equation}\label{De22n5} \mathbb{E} X_{r,\lambda} =\left(1+ O\left(e^{-\alpha r}\right)\right)m^{\circ}_{\lambda}(N). \end{equation} Hence, condition (\ref{ES}) requires roughly speaking, \[ \mathbb{E} X^{\circ}_{r,\lambda} =m^{\circ}_{\lambda}(N) \sim \frac{1}{N}. \] This instructs us to search for the solution $\lambda $ to (\ref{ES}) among $\lambda = \lambda(N) $ such that $\lambda(N) \rightarrow \infty$. (One may as well ignore this remark as the solution to (\ref{ES}) will be found indeed in this range, and as we argued above, it is unique.) It follows by the Gibbs distribution (\ref{fXl}) that for large $\lambda$ the probability measure is mostly concentrated around the minima of $H_{\lambda}$. This will stated more precise below, but first we shall study the minima of $H_{\lambda}$ and $H_{\lambda}^{\circ}$. \subsection{Minima of $H_{\lambda}$ and $H_{\lambda}^{\circ}$} Let us fix $\lambda$ arbitrarily. Functions $H_{\lambda}(\bar{x})$ and $H_{\lambda}^{\circ}(\bar{x})$ are both convex on $ \mathbb{R}^N_+$ and therefore each function has its unique minimum. Denote, correspondingly, these minima $\bar{a}=(a_1, \ldots, a_N)$ and $\bar{a}^{\circ}=(a_1^{\circ}, \ldots, a_N^{\circ})$, so that \begin{equation}\label{H1} H_{\lambda}(\bar{a})=\min_{\bar{x}}H_{\lambda}(\bar{x}), \end{equation} and \begin{equation}\label{H1cir} H_{\lambda}(\bar{a}^{\circ})=\min_{\bar{x}}H_{\lambda}^{\circ}(\bar{x}). \end{equation} Observe that components $a_k=a_k(N)$ and $a_k^{\circ}=a_k^{\circ}(N)$ are (in general) also functions of $N$. By definition the vector $\bar{a}$ satisfies the system of equations \begin{equation}\label{H3} \frac{\partial}{\partial x_k} H_{\lambda}(\bar{x})\mid _{\bar{x}=\bar{a}}=0, \ \ k=1, \dots, N, \end{equation} which is \begin{equation}\label{H2} \left\{ \begin{array}{rl} \frac{\beta}{a_{1}^2}+ \frac{\gamma}{ (a_{1}+a_{2})^2 } =\lambda, & \\ \\ \frac{\gamma}{ (a_{k-1}+a_{k})^2 } + \frac{\beta}{a_{k}^2}+ \frac{\gamma}{ (a_{k}+a_{k+1})^2 }=\lambda, & k=2, \dots, N-1,\\ \\ \frac{\beta}{a_{N}^2}+ \frac{\gamma}{ (a_{N-1}+a_{N})^2 } =\lambda,& \end{array} \right. \end{equation} where the first and the last equations represent the boundary conditions. Correspondingly, it holds that $\bar{a}^{\circ}$ satisfies \begin{equation}\label{H2cir} \frac{\gamma}{ (a_{k-1}^{\circ}+a_{k}^{\circ})^2 } + \frac{\beta}{(a_{k}^{\circ})^2}+ \frac{\gamma}{ (a_{k}^{\circ}+a_{k+1}^{\circ})^2 }=\lambda, \ \ k=1, \dots, N, \end{equation} where we denote $a_0^{\circ}= a_{N}^{\circ}$ and $a_{N+1}^{\circ}= a_{1}^{\circ}$. One solution to the last system is obvious: \[ a_k^{\circ}=\sqrt{\frac{2\beta + \gamma}{2\lambda}}=:a,\ \ k=1, \dots, N, \] moreover due to the uniqueness this is the only one which satisfies (\ref{H2cir}). Note that the components $a_k^{\circ}=a_k^{\circ}(N)=a$ are constant with respect to $k$ or $N$, unlike $a_k(N)$ which depend both on $k$ and $N$. It is natural to predict that \[a_k(N) \rightarrow a\] as $k$ and $N\rightarrow \infty$. We shall prove this. First we establish that if this convergence takes place it is exponentially fast. \begin{lem}\label{PA1} Suppose that for any $\delta >0$ there is large $N$ and $K<N/2$ such that for all $K\leq k\leq N/2$ \begin{equation}\label{As1} \|a_k(N)- a\|<\delta, \end{equation} where \begin{equation}\label{defa} a=\sqrt{\frac{2\beta + \gamma}{2\lambda}}. \end{equation} Then when both $k$ and $N\rightarrow \infty$ and $k\leq N/2$, we have exponentially fast convergence: \begin{equation}\label{H5} \|a_k- a\|\leq C{\eta }^k \end{equation} where C is some positive constant and \[0< \eta = 1+4\frac{\beta}{\gamma} -\sqrt{16\left(\frac{\beta}{\gamma} \right)^2 +8\frac{\beta}{\gamma}}<1.\] \end{lem} \noindent {\bf Proof.} Denote \begin{equation}\label{phi4} \varphi (x)=\frac{1}{x^2}, \ \ \varphi^{-1} (x)=\frac{1}{\sqrt{x}}. \end{equation} Let $\lambda$ be fixed arbitrarily. Then given $X_1>0$ let us define recurrently (when possible) positive numbers $X_2, X_3, \ldots, $ by the following system associated with (\ref{H2}): \begin{equation}\label{F1} \left\{ \begin{array}{rl} \beta\varphi (X_1) + \gamma \varphi (X_1 +X_2)= \lambda, & \\ \\ \gamma\varphi (X_{k-1} + X_k) + \beta\varphi (X_k) +\gamma\varphi (X_k +X_{k+1})=\lambda, & k\geq 2. \end{array} \right. \end{equation} Observe, that for each $N$ and $\lambda$ since the vector $\bar{a}= (a_1, \ldots , a_N)$ satisfies system (\ref{H3}), for $X_1=a_1$ we have well defined $X_i=a_i$ at least for all $i< N/2$. Define also \begin{equation}\label{H21} Y_k= \gamma\varphi (X_k +X_{k+1}), \ \ \ k\geq 1, \end{equation} which for $Y_k \geq 0$ is equivalent to \[X_{k+1}= \varphi^{-1} \left(\frac{Y_k}{\gamma}\right)-X_k.\] This notation allows us to transform (\ref{F1}) into the following dynamical system on $\mathbb{R}_+^2$ \begin{equation}\label{F2} \left\{ \begin{array}{ll} X_{k}& = -X_{k-1} + \varphi^{-1} \left(\frac{Y_{k-1}}{\gamma}\right), \\ \\ Y_k & = - \beta\varphi (X_k) - Y_{k-1} +\lambda, \ \ \ \ k\geq 2, \end{array} \right. \end{equation} assuming that the initial condition $(X_1, Y_1)$ satisfies $$\beta \phi(X_1)+Y_1=\lambda$$ and moreover yields $(X_k, Y_k)\in \mathbb{R}_+^2$. Hence, \begin{equation}\label{F3} \left\{ \begin{array}{ll} X_{k+1}-X_{k}& =-X_{k} + \varphi^{-1} \left(\frac{Y_k}{\gamma}\right)-\left( -X_{k-1} + \varphi^{-1} \left(\frac{Y_{k-1}}{\gamma}\right)\right), \\ \\ Y_{k+1}-Y_k & = - \beta \varphi (X_{k+1}) - Y_{k} -\left( - \beta\varphi (X_k) - Y_{k-1} \right). \end{array} \right. \end{equation} Writing \begin{equation}\label{F4} \begin{array}{ll} \Delta X_{k}& = X_{k} - X_{k-1} , \\ \Delta Y_k & = Y_k - Y_{k-1}, \end{array} \end{equation} we get \begin{equation}\label{F5} \left\{ \begin{array}{ll} \Delta X_{k+1}& =-\Delta X_{k} + \varphi^{-1} \left(\frac{Y_k}{\gamma}\right)-\varphi^{-1} \left(\frac{Y_{k-1}}{\gamma}\right), \\ \\ \Delta Y_{k+1} & = - \Delta Y_{k} - \beta\varphi (X_{k+1}) + \beta\varphi (X_k) . \end{array} \right. \end{equation} Consider the system (\ref{F2}) with arbitrary initial condition $(X_1,Y_1)=(x,y)$. Observe that if $(x,y)$ is a fixed point for the system (\ref{F2}) then \begin{equation}\label{Fp} (x,y)=\left(a, \frac{\gamma}{4a^2}\right) \end{equation} where \[a=\sqrt{\frac{2\beta + \gamma}{2\lambda}},\] as in (\ref{defa}). The linearisation of the system (\ref{F5}) in the neighbourhood of this fixed point is: \begin{equation}\label{F7} \left( \begin{array}{c} u_{k+1} \\ v_{k+1} \end{array} \right) = \left( \begin{array}{ll} -1 & \left(\varphi^{-1} \left(\frac{y}{\gamma}\right)\right)'\\ \beta\varphi' (x) & -1 - \beta \varphi '(x) \left(\varphi^{-1} \left(\frac{y}{\gamma}\right)\right)' \end{array} \right) \left( \begin{array}{c} u_{k} \\ v_{k} \end{array} \right), \end{equation} where by (\ref{phi4}) and (\ref{Fp}) \begin{equation}\label{F8} \varphi '(x)=-\frac{2}{x^3}=-\frac{2}{a^3}, \ \ \left(\varphi^{-1} \left(\frac{y}{\gamma}\right)\right)'=-\frac{1}{\gamma}\frac{1}{2(y/\gamma)^{3/2}}=-\frac{4a^{3}}{\gamma}. \end{equation} Hence, we can rewrite (\ref{F7}) as \begin{equation}\label{F9} \left( \begin{array}{c} u_{k+1} \\ v_{k+1} \end{array} \right) = \left( \begin{array}{ll} -1 & -4\frac{a^{3}}{\gamma}\\ -2\beta a^{-3} & -1-8\frac{\beta}{\gamma} \end{array} \right) \left( \begin{array}{c} u_{k} \\ v_{k} \end{array} \right) =: A\left( \begin{array}{c} u_{k} \\ v_{k} \end{array} \right). \end{equation} Observe, that independent of $a$ (and hence, $\lambda$) $\beta$ and $\gamma\neq 0$ we have \begin{equation}\label{detA} \det A =1, \end{equation} and the eigenvalues of the matrix $A$ (which do not depend on $\lambda$ neither) are \begin{equation}\label{F10} -1<\eta_1 = -1-4\frac{\beta}{\gamma} +\sqrt{16\left(\frac{\beta}{\gamma} \right)^2 +8\frac{\beta}{\gamma}}<0, \ \ \eta_2 = \frac{1}{\eta_1}<-1. \end{equation} This implies that if the initial condition $(u_1,v_1)$ for the system (\ref{F9}) is chosen along the eigenvector corresponding to $\|\eta_1\|<1$ the linearised system (\ref{F7}) converges to the limiting zero state exponentially fast. As $a_k$ satisfy (\ref{H2}) and therefore also system (\ref{F1}), our assumption (\ref{As1}) is equivalent to that $(X_k=a_k,Y_k)$ does not leave an arbitrarily small neighbourhood of the point (\ref{Fp}), where it is described by the linearised system (\ref{F7}). Hence, exponentially fast convergence of $(u_k,v_k)$, implies the same exponential rate of convergence to the limiting state (\ref{Fp}) by the Hartman-Grobman Theorem. $\Box$ \subsection{Gaussian limit.} First we derive Gaussian limit for the circular distribution. Consider random vector $\bar{X}^{\circ}_{\lambda}$ defined by (\ref{Mt7}) and (\ref{H0c}). \begin{theo}\label{Lexp0} Let $\bar{a}^{\circ} =(a, \ldots, a)\in {\mathbb{R}}^N$ be the point of minimum of function $H_{\lambda}^{\circ}$, i.e., \[ a=\sqrt{\frac{2\beta + \gamma}{2\lambda}}. \] \noindent Assume that $\lambda$ is a large parameter of order $N^2$: \[ \lambda \sim N^2. \] Then for any $N\geq 3$ the Hessian of function $H_{\lambda}^{\circ}$ at $\bar{a}^{\circ}$ is a symmetric circulant $N \times N$ matrix \begin{equation}\label{Hess} {\cal H}(N):= \frac{1}{a^3}\left( \begin{array}{cccccc} 2\beta+\frac{\gamma}{2} & \frac{\gamma}{4} & 0 & \ldots& 0 & \frac{\gamma}{4} \\ \frac{\gamma}{4} & 2\beta+\frac{\gamma}{2}& \frac{\gamma}{4} & 0& \ldots& 0 \\ \ldots & & & & & \\ \frac{\gamma}{4} & 0 & \ldots & 0 & \frac{\gamma}{4} & 2\beta+\frac{\gamma}{2} \end{array} \right), \end{equation} \noindent and for any continuous function $g: R^N \rightarrow R$ with at most polynomial growth, i.e., such that \begin{equation}\label{Ma36} \|g(\bar{x})\|\leq P_k(\\|\bar{x}\\|) \end{equation} for some polynomial $P_k$ of degree $k\geq 0$ (with constant in $N$ coefficients), one has \begin{equation}\label{Ma37} \int_{[0,1]^N } g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \end{equation} \[ = e^{- H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} } \left(\frac{ \sqrt{\det {\cal H}}} { (2\pi)^{ \frac{N}{2} } } \int_{{\bf R}^N } g(\bar{x}) e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x} \left(1+ \ o \left( N^{-\frac{1}{5}} \right)\right) +o\left( e^{- ({\log N})^{3/2}} \right) \right), \] as $N\rightarrow \infty$. Hence, \begin{equation}\label{Ma39} {\bf E} g\left(\bar{X}^{\circ}-\bar{a}^{\circ}\right) = {\bf E} g\left(\bar{ \cal Z}\right) \left(1 + \ o\left(N^{-\frac{1}{5}} \right)\right) +o\left( e^{- ({\log N})^{3/2}} \right), \end{equation} where random variable $\bar{ \cal Z}$ has a multivariate normal distribution in $\mathbb{R}^N$ with zero mean vector and covariance matrix ${\cal H}^{-1}(N)$. \end{theo} \begin{rem}\label{R2} The results of this theorem hold for a more general class of functions $g$, including, for example, Borel bounded functions. \end{rem} The analogue of Theorem \ref{Lexp0} holds as well for random vector $\bar{X}_{\lambda}$, and we state it below. However, it does not have equally closed form. \begin{theo}\label{Lexp1} Let $\bar{a} =(a_1, \ldots, a_N)\in {\mathbb{R}}^N$ be the point of minimum of function $H_{\lambda}$, and denote ${\cal H}_1(N)$ the Hessian of function $H_{\lambda}$ at $\bar{a}$. Assume that $\lambda$ is a large parameter of order $N^2$. Then for any continuous function $g: R^N \rightarrow R$ with at most polynomial growth, i.e., such that \[ \|g(\bar{x})\|\leq P_k(\\|\bar{x}\\|) \] for some polynomial $P_k$ of degree $k\geq 0$, one has \begin{equation}\label{DeMa39} {\bf E} g\left(\bar{X}^{\circ}-\bar{a}\right) = {\bf E} g\left(\bar{ \cal Z}'\right) \left(1 + \ o\left(N^{-\frac{1}{5}} \right)\right) +o\left( e^{- ({\log N})^{3/2}} \right), \end{equation} where random variable $\bar{ \cal Z}'$ has a multivariate normal distribution in $\mathbb{R}^N$ with zero mean vector and covariance matrix ${\cal H}_1^{-1}(N)$. \end{theo} \noindent {\bf Proof of Theorem \ref{Lexp0}. } For any $\bar{x}=(x_1, \ldots, x_N)$ set $x_{N+1}=x_1$; then we can rewrite \[H_{\lambda}^{\circ}(\bar{x})= \sum_{k=1}^{N} \frac{\beta}{x_k} + \sum_{k=1}^{N}\frac{\gamma}{x_k+x_{k+1}} +\lambda \sum_{k=1}^{N} x_k.\] For the point of minimum of $H_{\lambda}^{\circ}(\bar{x})$ defined by (\ref{defa}) under the assumption of theorem that $\lambda=\lambda(N) \rightarrow \infty$, it holds that \[\bar{a}^{\circ}=(a, \ldots, a) \in [0,1]^N. \] Consider the Taylor expansion of $H_{\lambda}^{\circ}(\bar{x}) $ in the neighbourhood \begin{equation}\label{B} B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}):=\left\{ \bar{x} : \max_{1\leq i\leq N} \|x_i-a\| \leq \frac{\varepsilon}{\sqrt{\lambda}} \right\} \end{equation} of $\bar{a}^{\circ}$: for all $\bar{x} \in B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})$ we have \begin{equation}\label{An2} H_{\lambda}^{\circ}(\bar{x})= H_{\lambda}^{\circ}(\bar{a}^{\circ}) + \frac{1}{2}\sum_{k=1}^{N}\sum_{l=1}^{N}\frac{\partial ^2H_{\lambda}^{\circ}}{\partial x_k\partial x_l}(\bar{a}^{\circ}) (x_k-a)(x_l-a) \end{equation} \[+ \frac{1}{3!}\sum_{k=1}^{N}\sum_{l=1}^{N}\sum_{j=1}^{N}\frac{\partial ^3H_{\lambda}^{\circ}}{\partial x_k\partial x_l\partial x_j}(\bar{y}) \, (x_k-a)(x_l-a)(x_j-a), \] where \[ \bar{y}= \bar{y}(\bar{x}) \in B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}). \] Compute now \begin{equation}\label{An3} \frac{\partial ^2H_{\lambda}^{\circ}}{\partial x_k\partial x_l}(\bar{a}^{\circ})= \left\{ \begin{array}{ll} (2\beta+\frac{\gamma}{2})a^{-3}, & \ k=l, \\ \\ \frac{\gamma}{4a^3}, & \ \|k-l\|=1, \\ \\ 0, & \ \|k-l\|>1. \\ \end{array} \right. \end{equation} Hence, the Hessian of function $H_{\lambda}^{\circ}(\bar{x}) $ at the point $\bar{a}^{\circ}$ is matrix ${\cal H}={\cal H}(N)$ defined in (\ref{Hess}). The eigenvalues of ${\cal H}(N)$, call them $\nu^{\circ}_j, j=1, \ldots, N,$ are given by \begin{equation}\label{eigenM} \nu^{\circ}_j=\frac{1}{a^3}\left(\left( 2\beta+\frac{\gamma}{2} \right) +2\frac{\gamma}{4} \cos \left( 2\pi \frac{ j}{N}\right)\right) \end{equation} (see, e.g., \cite{D}). The latter yields a uniform bound \begin{equation}\label{eigen} \frac{2\beta}{a^3} \leq \nu^{\circ}_j \leq \frac{2\beta+\frac{\gamma}{2}}{a^3}\ \ \ \ j=1, \ldots, N. \end{equation} which proves that ${\cal H}$ is positive-definite (it confirms in particular, that point $\bar{a}^{\circ}$ is the minimum). It is convenient to use here the vector form. We denote $\bar{x}$ the {\it row}-vector, and $\bar{x}'$ the {\it column}-vector. Then we get from (\ref{An2}) taking as well into account the third-order derivatives \begin{equation}\label{An4} H_{\lambda}^{\circ}(\bar{x})= H_{\lambda}^{\circ}(\bar{a}^{\circ}) + \left( 1+O\left(\frac{\varepsilon}{a \sqrt{\lambda}}\right) \right) \frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' \end{equation} \[= H_{\lambda}^{\circ}(\bar{a}^{\circ}) + \left( 1+O\left( \varepsilon \right) \right) \frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})'\] for all $\bar{x}\in B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}).$ Uniform bound (\ref{eigen}) gives us a useful property: for any $\bar{x}\in \mathbb{R}^N$ \begin{equation}\label{n6} \frac{2\beta}{a^3} \, \\|\bar{x}\\|²\leq \bar{x}'{\cal H}\bar{x}\leq \frac{2\beta+\frac{\gamma}{2}}{a^3} \, \\|\bar{x}\\|². \end{equation} Let function $g(\bar{x}): \mathbb{R}^N \rightarrow \mathbb{R}$ satisfy the assumptions of the theorem. Making use of the decomposition (\ref{An4}) consider \begin{equation}\label{I1} \int_{[0,1]^N } g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \end{equation} \[=\int_{B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ})e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}+ \int_{[0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ})e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}. \] Next we study separately the last two integrals. Consider the last integral in (\ref{I1}). First we define for any \begin{equation}\label{Ja10} \bar{x} \in \left\{ \bar{x}\in [0,1]^N: \max_{1\leq i\leq N}\|x_I-a\| \geq \frac{\varepsilon}{\sqrt{\lambda}} \right\} \end{equation} the set of components $x_i$ which deviate from $a$ at least by $\frac{\varepsilon}{\sqrt{\lambda}}$: \begin{equation}\label{Ja11} {\cal K}(\bar{x}) = \{i: \|x_I-a\|\geq \frac{\varepsilon}{\sqrt{\lambda}} \}. \end{equation} Then we have the following decomposition \[ \int_{[0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \] \begin{equation}\label{Ja16} = \sum_{{\cal K}\subseteq \{1, \ldots, N\}: \|{\cal K}\| \neq \emptyset}\int_{ \{ \bar{x}\in [0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}): \ {\cal K}(\bar{x}) ={\cal K}\}} g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}. \end{equation} Given a non-empty set ${\cal K}\subset \{1, \ldots, N\}$ let ${\cal H}_{{\cal K}}$ denote a matrix obtained from ${\cal H}$ by deleting all the columns and rows with indices in ${\cal K}$. Alternatively, ${\cal H}_{{\cal K}}$ can be defined as a matrix of the quadratic form obtained from $\bar{x}{\cal H}\bar{x}' $ by deleting all the terms which include $x_i$ with $i\in {\cal K}$. Let $\bar{x}_{{\cal K}}$ denote an $(N-\|{\cal K}\|)$-dimensional vector obtained from $\bar{x}$ by deleting the components with indexes in ${\cal K}=\{i_1, \ldots, i_{\|{\cal K}\|}\}$ in increasing order. Define a set of distances between the consecutive $i_k$ in this set: \[m_k= i_{k+1}-i_k-1, k=1, \ldots, \|{\cal K}\|-1, \] \[m_{\|{\cal K}\|}= Ni_{\|{\cal K}\|}+i_1-1.\] Then $\bar{x}_{{\cal K}}{\cal H}_{{\cal K}}\bar{x}'_{{\cal K}} $ is a linear combination of independent quadratic forms, each corresponding to $m_k\times m_k$ matrix $A_{m_k}$ defined as follows \begin{equation}\label{HessK} A_{m}:= \frac{1}{a^3}\left( \begin{array}{cccccc} 2\beta+\frac{\gamma}{2} & \frac{\gamma}{4} & 0 & \ldots & 0 & 0 \\ \frac{\gamma}{4} & 2\beta+\frac{\gamma}{2}& \frac{\gamma}{4} & 0& \ldots& 0 \\ \ldots & & & & & \\ 0 & 0 & \ldots & 0 & \frac{\gamma}{4} & 2\beta+\frac{\gamma}{2} \end{array} \right), \end{equation} when $m>1$, and also for $m=1$ set \[A_1:= \frac{1}{a^3}\left( 2\beta+\frac{\gamma}{2} \right), \] and $A_0=1$. Hence, \begin{equation}\label{detHk} \det {\cal H}_{{\cal K}} = \prod_k\det A_{m_k}. \end{equation} The eigenvalues of 3-diagonal Toeplitz matrix $A_m$ are well-known (see, e.g., \cite{K}), these are \begin{equation}\label{eiv} c_j(m)=\frac{1}{a^3} \left( \left( 2\beta+\frac{\gamma}{2}\right)+ 2\frac{\gamma}{4}\cos \left( \pi \frac{ j}{m+1}\right) \right), \ \ \ j=1, \ldots, m, \end{equation} yielding uniform bounds as in (\ref{eigen}) \begin{equation}\label{eivb} \frac{2\beta}{a^3} \leq c_j(m) \leq \frac{1}{a^3} \left( 2\beta+{\gamma} \right), \ \ \ j=1, \ldots, m. \end{equation} This together with (\ref{detHk}) confirms that matrix ${\cal H}_{{\cal K}}$ is also positive-definite for all ${\cal K}\subset \{1, \ldots, N\}$ (if fact, it is positive-definite since ${\cal H}$ is). Note here for further reference that the bounds (\ref{eivb}) together with (\ref{detHk}) imply \begin{equation}\label{detHkb} \left( \frac{2\beta}{a^3} \right)^{N-\|{\cal K}\|}\leq \det {\cal H}_{{\cal K}} \leq \left( \frac{1}{a^3} \left( 2\beta+{\gamma} \right) \right)^{N-\|{\cal K}\|} \end{equation} for any set ${\cal K}$. Note that for any $\bar{x} \in B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})$ (hence, the approximation (\ref{An4}) is still valid) such that ${\cal K}(\bar{x}) ={\cal K}$, i.e., \[ \{i: \|x_i-a\|= \frac{\varepsilon}{\sqrt{\lambda}} \}={\cal K}, \] we have for any $i\in {\cal K}$ \[\left(2\beta+\frac{\gamma}{2} \right)(x_i-a_i)^2+2\frac{\gamma}{4} (x_{i+1}-a_{i+1})(x_{i}-a_i)+2\frac{\gamma}{4} (x_{i-1}-a_{i-1})(x_{i}-a_i)\] \[\geq \left(\frac{\varepsilon}{\sqrt{\lambda}} \right)^2\left(2\beta+\frac{\gamma}{2} -\gamma \right), \] where we take $(i\pm 1)_{\mbox{\small{mod }} N}$ if needed. Therefore making use of the last bound we derive for all such $\bar{x} $ and small $\varepsilon$ \[ H_{\lambda}^{\circ}(\bar{x})=H_{\lambda}^{\circ}(\bar{a}^{\circ}) + \left( 1+O(\varepsilon) \right) \frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})'\] \begin{equation}\label{Ja15} \geq H_{\lambda}^{\circ}(\bar{a}^{\circ})+\left( 1+O(\varepsilon) \right) \frac{1}{2} \|{\cal K}\| \frac{1}{a^3}\left( \frac{\varepsilon}{\sqrt{\lambda}}\right) ^2\left(2\beta-\frac{\gamma}{2} \right) \end{equation} \[+ \left( 1+O(\varepsilon) \right) \frac{1}{2}(\bar{x}_{{\cal K}} -\bar{a}^{\circ}_{{\cal K}}) {\cal H}_{{\cal K}}(\bar{x}_{{\cal K}} -\bar{a}^{\circ}_{{\cal K}})' .\] By the convexity of function $H_{\lambda}^{\circ}$ the last bound holds as well for all $\bar{x} \in [0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})$ with ${\cal K}(\bar{x})={\cal K}$, and therefore for a continuous function $g$ we get \begin{equation}\label{Ja12} \left\| \int_{ \{ \bar{x} \in [0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}): \ {\cal K}(\bar{x})={\cal K} \} } g(\bar{x}-\bar{a}^{\circ}) e^{-H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}\right\| \end{equation} \[ \leq Ce^{-H_{\lambda}^{\circ}(\bar{a}^{\circ})-\|{\cal K}\| B {\varepsilon}^2{\sqrt{\lambda}} } \ \frac{(2\pi\left( 1+O(\varepsilon) \right))^{\frac{N-\|{\cal K}\|}{2}}}{\sqrt{\det {\cal H}_{{\cal K}}}}, \] where $$C=\max_{\bar{x} \in [0,1]^N } \|g(\bar{x}-\bar{a}^{\circ})\|, \ \ B= \frac{2\beta-\frac{\gamma}{2}}{3\left(\beta + \frac{\gamma}{2}\right)^{3/2}}.$$ Substituting the last bound into (\ref{Ja16}) we derive \[ \left\|\int_{[0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}\right\| \leq \int_ {\left\{ \bar{x}\in [0,1]^N: \min_{1\leq i\leq N}\|x_i-a\| \geq \frac{\varepsilon}{\sqrt{\lambda}} \right\}} C e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \] \begin{equation}\label{Ja17} + C\sum_{K=1}^{N-1} \ \sum_{{\cal K}\subseteq \{1, \ldots, N\}: \|{\cal K}\| =K}\int_{ \{ \bar{x}\in [0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}): \ {\cal K}(\bar{x}) ={\cal K}\}} g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \end{equation} \[ \leq C e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ})} e^{-N B {\varepsilon}^2{\sqrt{\lambda}}} \] \[ +C e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \ \frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} } \ \sum_{K=1}^{N-1} \ \sum_{{\cal K}\subseteq \{1, \ldots, N\}: \|{\cal K}\| =K} e^{-K B {\varepsilon}^2{\sqrt{\lambda}} } \frac{ \sqrt{\det {\cal H} }}{\sqrt{\det {\cal H}_{\cal K} }}. \] \noindent {\bf Claim.} For any set ${\cal K}\subseteq \{1, \ldots, N\}$ \begin{equation}\label{Cl1} \frac{ \sqrt{\det {\cal H} }}{ \sqrt{\det {\cal H}_{\cal K} }}\leq \sqrt{2}\left(\frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\right)^{\|{\cal K}\|/2}. \end{equation} \noindent {\bf Proof of the Claim.} It follows from the definition of circular matrix ${\cal H}$ (see (\ref{Hess})) that for any $i \in \{1, \ldots, N\}$ \begin{equation}\label{De1} \det {\cal H}\leq \frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\det {\cal H}_{\{i\}}+2\left(\frac{\gamma}{4a^3} \right)^N, \end{equation} where \[\det {\cal H}_{\{i\}}= \det A_{N-1},\] as defined in (\ref{HessK}). Then for $A_m$ we also derive from the definition (\ref{HessK}) \begin{equation}\label{De3} \det A_m \leq \frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\det A_l \det A_{m-l-1}, \end{equation} for any $1\leq l<m$. When $m=N-1$ this is equivalent (see also (\ref{detHk})) to \begin{equation}\label{De2} \det {\cal H}_{\{i\}}\leq \frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\det {\cal H}_{\{i,j\}} \end{equation} for any $j\neq i$. Hence, by the recurrence we get for any set ${\cal K}\subseteq \{1, \ldots, N\}$ \begin{equation}\label{De4} \det {\cal H}_{\{i\}}\leq \left(\frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\right)^{\|{\cal K}\|-1}\det {\cal H}_{{\cal K}}. \end{equation} Making use of the last bound in (\ref{De1}) we obtain \begin{equation}\label{De5} \det {\cal H}\leq \left(\frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\right)^{\|{\cal K}\|}\det {\cal H}_{{\cal K}}+2\left(\frac{\gamma}{4 a^3 } \right)^N. \end{equation} Then taking into account the assumption that $\gamma \leq \beta$, we get using the lower bound in (\ref{detHkb}) that \[2\left(\frac{\gamma}{4 a^3 } \right)^N \leq \left(\frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\right)^{\|{\cal K}\|} \left(\frac{2\beta}{a^3}\right)^{N-\|{\cal K}\|} \leq \left(\frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\right)^{\|{\cal K}\|}\det {\cal H}_{{\cal K}}.\] Combining the last bound with (\ref{De5}), we get \begin{equation}\label{De6} \det {\cal H}\leq 2\left(\frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\right)^{\|{\cal K}\|}\det {\cal H}_{{\cal K}} \end{equation} for any set ${\cal K}\subseteq \{1, \ldots, N\}$, which proves the Claim. $\Box$ The derived bound (\ref{Cl1}) allows us to obtain from (\ref{Ja17}): \begin{equation}\label{Ja18} \left\|\int_{[0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}\right\| \end{equation} \[ \leq C e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ})} e^{-N B {\varepsilon}^2{\sqrt{\lambda}}} \] \[ +C e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \ \frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} } \ \sum_{K=1}^{N-1} \left( \begin{array}{c} N \\ K \end{array} \right) e^{-K B {\varepsilon}^2{\sqrt{\lambda}} } \sqrt{2}\left(\frac{1}{a^3}\left(2\beta+\frac{\gamma}{2} \right)\right)^{\|{\cal K}\|/2} \] \[ \leq C e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ})} e^{-N B {\varepsilon}^2{\sqrt{\lambda}}} \] \[ +2C e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \ \frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} } \left( \left( e^{- B {\varepsilon}^2{\sqrt{\lambda}} } \frac{\sqrt{2\beta+\frac{\gamma}{2} }}{a^{3/2}} +1 \right)^{N} -1\right). \] Let us set from now on \begin{equation}\label{Ja20} \varepsilon = \frac{\log \lambda}{\lambda^{\frac{1}{4}}}. \end{equation} This choice yields together with the upper bound in (\ref{detHkb}) that \[e^{-N B {\varepsilon}^2{\sqrt{\lambda}}}\frac { \sqrt{\det {\cal H}} }{ (2\pi)^{ \frac{N}{2} } }\leq \left(e^{- B ({\log \lambda})^2} \frac{2\beta +\gamma}{a^3\sqrt{2\pi}}\right)^N= O\left(e^{- \frac{B}{2} ({\log \lambda})^2} \right)^N, \] and for all $\lambda >N$ \[\left( e^{- B {\varepsilon}^2{\sqrt{\lambda}} } \frac{\sqrt{2\beta+\frac{\gamma}{2} }}{a^{3/2}} +1 \right)^{N} -1 = O\left(Ne^{- \frac{B}{2} ({\log \lambda})^2} \right)= O\left(e^{- \frac{B}{3} ({\log \lambda})^2} \right). \] Substituting the last two bounds into (\ref{Ja18}) allows us to derive for all $\lambda >N$ \begin{equation}\label{Ja21} \left\| \int_{[0,1]^N \setminus B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ})e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \right\|=e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \ \frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} }O\left(e^{- \frac{B}{4} ({\log \lambda})^2} \right). \end{equation} Repeating the same arguments as in (\ref{Ja15})-(\ref{Ja21}), and keeping choice (\ref{Ja20}) we get a similar to (\ref{Ja21}) bound when $H_{\lambda}^{\circ}$ is replaced by the corresponding quadratic form ${\cal H}$, namely \begin{equation}\label{Ja22} \int_{B_{1}(\bar{a}^{\circ}) \setminus B_{\frac{\varepsilon}{2\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ}) e^{- \frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' }d\bar{x}= \frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} } \ o\left(e^{-\frac{B}{4} ({\log \lambda})^2 }\right). \end{equation} Notice that in the last integral a ball $B_{1}(\bar{a}^{\circ})$ replaces a box $[0,1]^N$ in (\ref{Ja21}), but the former arguments are still applicable. Next for any $\bar{x}\in B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}) \cap \{\bar{x}: \\|\bar{x}-\bar{a}^{\circ}\\|\leq \varepsilon \lambda^{-\frac{1}{2}+q}\}$, where $0<q<\frac{1}{8}$, we derive with a help of (\ref{n6}) \begin{equation}\label{De9} O(\varepsilon) (\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' \leq O\left(\varepsilon a^{-3}\\|\bar{x}-\bar{a}^{\circ}\\|^2\right)=O\left(\varepsilon^3 a^{-3}\frac{1}{\lambda^{1-2q}}\right) \end{equation} \[=O\left(\frac{\lambda^{3/4}}{(\log \lambda)^3\lambda^{1-2q}}\right)= o\left(\lambda^{-\frac{1}{4}+2q}\right).\] This gives us the following approximation \begin{equation}\label{De11} \int_{B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}) \cap \{\bar{x}: \\|\bar{x}-\bar{a}^{\circ}\\|\leq \varepsilon \lambda^{-\frac{1}{2}+q}\} } g\left({\bar{x}-\bar{a}^{\circ}}\right)e^{- (1+O(\varepsilon))\frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' }d\bar{x} \end{equation} \[=\left(1 + o\left(\lambda^{-\frac{1}{4}+2q}\right) \right) \int_{B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}) \cap \left\{\bar{x}: \\|\bar{x}-\bar{a}^{\circ}\\|\leq \varepsilon \lambda^{-\frac{1}{2}+q}\right\} } g\left({\bar{x}-\bar{a}^{\circ}}\right)e^{- \frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' }d\bar{x}.\] On the other hand, again due to (\ref{n6}), we have for some positive constants $c,c_1$ \begin{equation}\label{De8} \int_{B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}) \cap \left\{\bar{x}: \\|\bar{x}-\bar{a}^{\circ}\\|\geq \varepsilon \lambda^{-\frac{1}{2}+q}\right\} } g\left({\bar{x}-\bar{a}^{\circ}}\right)e^{- (1+O(\varepsilon))\frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' }d\bar{x} \end{equation} \[ \leq C \int_{ B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}) \cap \left\{\bar{x}: \\|\bar{x}-\bar{a}^{\circ}\\|\geq \varepsilon \lambda^{-\frac{1}{2}+q}\right\} } e^{- a^{-3}c\\|\bar{x}-\bar{a}^{\circ}\\|^2}d\bar{x} \leq O\left(\lambda^{-\frac{3}{4}}\right)^N e^{-c_1\lambda^{3/2}\varepsilon^2 \lambda^{-1+2q}} \] \[= O\left(\lambda^{-\frac{3}{4}}\right)^N e^{-c_1(\log \lambda)^2 \lambda^{2q}} = \frac{(2\pi)^{\frac{N}{2}}}{\sqrt{\det{\cal H}}} o\left( e^{-c_1 \lambda^{q}} \right). \] Combining (\ref{De8}) and (\ref{De11}) we obtain the following bound for the first integral on the right in (\ref{I1}) \begin{equation}\label{De12} \int_{ B_{\frac{\varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ})} g(\bar{x}-\bar{a}^{\circ})e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \end{equation} \[ =e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) }\int_{ B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}) } g\left({\bar{x}-\bar{a}^{\circ}}\right)e^{- (1+O(\varepsilon))\frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' } d\bar{x} \] \[=e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \int_{B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{a}^{\circ}) \cap \left\{\bar{x}: \\|\bar{x}-\bar{a}^{\circ}\\|\leq \varepsilon \lambda^{-\frac{1}{2}+q}\right\} } g\left({\bar{x}-\bar{a}^{\circ}}\right)e^{- \frac{1}{2}(\bar{x}-\bar{a}^{\circ}){\cal H}(\bar{x}-\bar{a}^{\circ})' }d\bar{x}\left(1+ o\left( \lambda^{-\frac{1}{4}+2q} \right)\right)\] \[ +\frac{(2\pi)^{\frac{N}{2}}e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) }}{\sqrt{\det{\cal H}}}o\left( e^{-c_1 \lambda^{q}} \right) \] \[=e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \int_{B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{0}) } g\left({\bar{x}}\right)e^{- \frac{1}{2}\bar{x} {\cal H}\bar{x}' }d\bar{x} \left(1+ o\left( \lambda^{-\frac{1}{4}+2q} \right)\right)+\frac{(2\pi)^{\frac{N}{2}}e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) }}{\sqrt{\det{\cal H}}}o\left( e^{-c_1 \lambda^{q}} \right) \] for arbitrary $q>0$. This together with (\ref{Ja21}) yields \begin{equation}\label{23Ja1} \int_{[0,1]^N } g(\bar{x}-\bar{a}^{\circ})e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \end{equation} \[ =e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \int_{B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{0}) } g\left({\bar{x}}\right)e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x}\left( 1 + o\left( \lambda^{-\frac{1}{4}+2q} \right) \right) + e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) }\frac{(2\pi)^{\frac{N}{2}}}{\sqrt{\det{\cal H}}} o\left( e^{-\frac{B}{3} ({\log \lambda})^2} \right) \] for arbitrary $q>0$. Let us rewrite the latter as \begin{equation}\label{Ja25} \int_{[0,1]^N } g(\bar{x}-\bar{a}^{\circ})e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \end{equation} \[ =e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \left( 1 + o\left( \lambda^{-\frac{1}{4}+2q} \right) \right) \] \[\times\left( \int_{{\bf R}^N } g(\bar{x}) e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x} - \int_{{\bf R}^N \setminus B_{1}(\bar{0})} g(\bar{x}) e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x}+ \int_{B_{1}(\bar{0}) \setminus B_{\frac{ \varepsilon}{\sqrt{\lambda}}}(\bar{0}) } g(\bar{x}) e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x} \right) \] \[+ e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) }\frac{(2\pi)^{\frac{N}{2}}}{\sqrt{\det{\cal H}}} o\left( e^{-\frac{B}{4} ({\log \lambda})^2} \right) . \] By the property (\ref{eigen}) and the assumption on $g$ we have for some positive constants $c_i$ \[\left\| \int_{{\bf R}^N \setminus B_{1}(\bar{0})} g(\bar{x}) e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x}\right\| \leq c_3\int_{{\bf R}^N \setminus B_{1}(\bar{0})} \\|\bar{x}\\|^k e^{-\frac{c_2}{a^3}\\|\bar{x}\\|^2-\frac{1}{4}\bar{x}{\cal H}\bar{x}' }d\bar{x}\leq c_4^N e^{- \frac{c_2}{2a^3}}\frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} }\] \[= \frac{ (2\pi)^{ \frac{N}{2} } }{ \sqrt{\det {\cal H}} } \ o\left(e^{- \frac{c_2}{3} \lambda^{3/2} } \right). \] Making use of the last bound and (\ref{Ja22}) we derive from (\ref{Ja25}) \begin{equation}\label{Ja24} \int_{[0,1]^N } g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x} \end{equation} \[ =e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) } \int_{{\bf R}^N } g(\bar{x}) e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x}\left( 1 + o\left( \lambda^{-\frac{1}{4}+2q} \right) \right)\] \[ + e^{-H_{\lambda}^{\circ}(\bar{a}^{\circ}) }\frac{(2\pi)^{\frac{N}{2}}}{\sqrt{\det{\cal H}}} o\left( e^{-\frac{B}{4} ({\log \lambda})^2} \right) , \] which yields the statement (\ref{Ma37}) of the Theorem, as $q>0$ can be fixed arbitrarily. The remaining statement (\ref{Ma39}) follows since by (\ref{Ja24}) \begin{equation}\label{Ma40} {\bf E} g\left(\bar{X}^{\circ}-\bar{a}^{\circ}\right)=\frac{ \int_{[0,1]^N } g(\bar{x}-\bar{a}^{\circ}) e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}}{\int_{[0,1]^N } e^{- H_{\lambda}^{\circ}(\bar{x}) }d\bar{x}} \end{equation} \[ = \frac{ \sqrt{\det {\cal H}}} { (2\pi)^{ \frac{N}{2} } } \int_{{\bf R}^N } g(\bar{x}) e^{- \frac{1}{2}\bar{x}{\cal H}\bar{x}' }d\bar{x} \left( 1 + o\left( \lambda^{-\frac{1}{4}+2q} \right) \right) + o\left( e^{-\frac{B}{4} ({\log \lambda})^2} \right). \] $\Box$ {\bf Proof of Theorem \ref{Lexp1}.} First we compute the Hessian ${\cal H}_1(N)$ of $H_{\lambda}$ at ${\bar a}$: \[ {\small \left( \begin{array}{cccccc} \frac{2\beta}{a_1^3} +\frac{\gamma}{2(a_1+a_2)^3} & \frac{\gamma}{4(a_1+a_2)³} & 0 & \ldots& 0 & 0 \\ \frac{\gamma}{4(a_1+a_2)^3} & \frac{2\beta}{a_2³}+\frac{\gamma}{2(a_1+a_2)³}+\frac{\gamma}{2(a_2+a_3)³}& \frac{\gamma}{4(a_2+a_3)³} & 0& \ldots& 0 \\ \ldots & & & & & \\ 0 & 0 & \ldots & 0 & \frac{\gamma}{4(a_{N-1}+a_N)³} & \frac{2\beta}{a_N^3} +\frac{\gamma}{2(a_{N-1}+a_N)^3} \end{array} \right).} \] Observe that ${\cal H}_1$ is non-negative definite due to the fact that $H_{\lambda}$ is convex with the minimum at ${\bar a}$. Furthermore, it follows directly by the 3-diagonal form of ${\cal H}_1$ that its rank is at least $N-1$. Then it is straightforward to check that the first row is not a linear combination of the rest, and therefore ${\cal H}_1$ is positive definite. Assuming that $\lambda$ is of order $N²$, the entries of vector ${\bar a}$, which is the solution to system (\ref{H2}), all must be of order $\lambda^{-1/2}$, as for the vector ${\bar a}$. Hence, the entries of ${\cal H}_1$ have the same scaling $\lambda^{-3/2}$ with respect to $\lambda$ as in ${\cal H}$. This allows us to use the same approximation technique as in the proof of Theorem \ref{Lexp0}, which leads to the statements of the theorem. $\Box$ Next we shall study the covariance matrix \begin{equation}\label{May1^} \Lambda := {\cal H}^{-1}(N) \end{equation} introduced in Theorem \ref{Lexp0}. \subsection{Inverse of a circular matrix: ${\cal H}^{-1}(N)$.} The inverses of circular matrices and closely related Toeplitz matrices often arise in the study of models of mathematical physics (\cite{DIK}). Some direct computational results are also available (e.g., \cite{C}, \cite{K}), however typically they are very limited to particular cases. As here we are interested only in certain asymptotic rather than in deriving all entries for the inverse matrices, we can avoid heavy computations looking instead for the asymptotic directly. We shall compute the asymptotic of the entries $\Lambda_{1k}:=({\cal H}^{-1})_{1k}$ , $k=1, \ldots, N$, in the statement of Corollary \ref{ExCov1}, where ${\cal H}^{-1}(N)$ is inverse to the circular matrix ${\cal H}$ defined in (\ref{Hess}). First we note, that by the symmetry in our case for all $N\geq 4$ we have \begin{equation}\label{inv1} \Lambda_{ii}:=({\cal H}^{-1})_{ii}=({\cal H}^{-1})_{11}= \frac{\det {\cal D}(N-1)}{\det {\cal H}(N)} , \end{equation} where ${\cal D}(N-1)$ is the 3-diagonal symmetric $(N-1)\times (N-1)$ matrix corresponding to ${\cal H}(N)$: \begin{equation}\label{Diag} {\cal D}(N-1):= \frac{1}{a^3}\left( \begin{array}{cccccc} 2\beta+\frac{\gamma}{2} & \frac{\gamma}{4} & 0 & \ldots& 0 & 0 \\ \frac{\gamma}{4} & 2\beta+\frac{\gamma}{2}& \frac{\gamma}{4} & 0& \ldots& 0 \\ \ldots & & & & & \\ 0 & 0 & \ldots & 0 & \frac{\gamma}{4} & 2\beta+\frac{\gamma}{2} \end{array} \right). \end{equation} To simplify notation define for any $n>4$ an $n\times n$ circular matrix \[ M_n:= \left( \begin{array}{cccccc} c & b & 0 & \ldots& 0 & b \\ b & c & b & 0& \ldots& 0 \\ \ldots & & & & & \\ 0 & 0 & \ldots & b & c & b \\ b & 0 & \ldots & 0 & b & c \end{array} \right). \] Then $M_N={\cal H}(N)$ if \begin{equation}\label{May2} b= \frac{1}{a^3} \frac{\gamma}{4}, \ \ \ c=\frac{1}{a^3}\left( 2\beta+\frac{\gamma}{2} \right) = \frac{2\beta}{a^3} + 2 b. \end{equation} The eigenvalues of $M_n$ by (\ref{eigenM}) are \begin{equation}\label{eigenMv} \nu^{\circ}_j=c +2b\cos \left( 2\pi \frac{ j}{n}\right), \ \ \ j=1, \ldots, n. \end{equation} Correspondingly, we rewrite ${\cal D}(N-1)$ introduced in (\ref{Diag}) with $N-1=n$ as \begin{equation}\label{MayDiag} {\cal D}_n = \left( \begin{array}{cccccc} c & b & 0 & \ldots& 0 & 0 \\ b & c & b & 0& \ldots& 0 \\ \ldots & & & & & \\ 0 & 0 & \ldots & 0 & b & c \end{array} \right). \end{equation} The inverse of $M_n$ we shall also denote $\Lambda$: \begin{equation}\label{May1} M_n^{-1}=\Lambda = (\Lambda_{ij})_{1\leq i,j\leq n}. \end{equation} Exploring the symmetry of the introduced matrices it is straightforward to derive the following relations (one may also consult \cite{D} or \cite{C}): \begin{equation}\label{MayInv} \begin{array}{l} \Lambda_{11} = \frac{1}{{\mbox {\bf det}} M_n} D_{n-1} ,\\ \\ \Lambda_{12}=\Lambda_{1n} = \frac{1}{{\mbox {\bf det}} \ M_n}\left( -bD_{n-2}+(-b)^{n-1}\right),\\ \\ \Lambda_{1k} = \frac{(-1)^{k+1}}{{\mbox {\bf det}} \ M_n}\left( b^{k-1}D_{n-k}+(-1)^{n-2}b^{n-k+1}D_{k-2} \right) , \ \ k =2, \ldots, n-1. \end{array} \end{equation} The eigenvalues of ${\cal D}_n$ are also known (see, e.g., \cite{K}), these are \begin{equation}\label{3eigenM} \nu_j=c + 2b\cos \left( \pi \frac{ j}{n+1}\right), \ \ \ j=1, \ldots, n. \end{equation} Hence, we have the determinant \begin{equation}\label{det M} D_n:={\mbox {\bf det} \ } {\cal D}_n= \prod_{j=1}^n\left( c + 2b\cos \left( \pi \frac{ j}{n+1}\right)\right). \end{equation} In our case $c>2b $ by (\ref{May2}), and therefore we derive from the last formula \[ \frac{1}{n} \log D_n= \log c +\frac{1}{n} \sum_{j=1}^n \log \left( 1 + \frac{2b}{c}\cos \left( \pi \frac{ j}{n+1}\right)\right) , \] which converges to \[ \log c + \int_0^1 \log \left( 1 + \frac{2b}{c}\cos \left( \pi x\right)\right) dx= \log c + \frac{1}{\pi }\int_0^{\pi} \log \left( 1 + \frac{2b}{c}\cos x\right) dx, \] as $n \rightarrow \infty.$ Computing the last integral (or consulting \cite{GR}) we get from here \begin{equation}\label{May10} \lim_{n \rightarrow \infty}\frac{1}{n} \log D_n = \log c + \log \frac {1 + \sqrt{ 1-\left(\frac{2b}{c}\right)^2}}{2} = \log \frac {c + \sqrt{ c^2-(2b)^2}}{2}. \end{equation} Observe, that the last asymptotic we can alternatively derive even without use of eigenvalues. This follows simply by the recurrent relation for the determinant \begin{equation}\label{May11} D_n:={\mbox {\bf det} \ } {\cal D}_n=cD_{n-1}-b^2D_{n-2}, \end{equation} whose characteristic equation \begin{equation}\label{May12} x^2- cx+b^2=0 \end{equation} has the roots \begin{equation}\label{May13} x_{1} = \frac{c+ \sqrt{ c^2-4b^2}}{2}, \ \ \ \ x_{2} = \frac{c- \sqrt{ c^2-4b^2}}{2} . \end{equation} Hence, \begin{equation}\label{May14} D_n= A x_{1}^n + Bx_{2}^n \end{equation} where taking into account that \[D_1=c, \ \ D_2= c^2-b^2,\] the constants $A,B$ satisfy \begin{equation}\label{May14} \left\{ \begin{array}{ll} A x_{1} + Bx_{2}& =c,\\ \\ A x_{1}^2 + Bx_{2}^2& =c^2-b^2. \end{array} \right. \end{equation} As in our case $c>2b>0$ this gives us a non-zero value $A$, hence the largest root $x_1$ gives us the same asymptotic (\ref{May10}). To find the remaining determinant ${\mbox {\bf det}} M_n$ in (\ref {MayInv}) one could also use the eigenvalues (\ref{eigenM}), but instead in the following we shall explore the relation \begin{equation}\label{May15} {\mbox {\bf det}} M_n = cD_{n-1}-2b^2D_{n-2}-2(-1)^{n}b^n=D_{n}-b^2D_{n-2}-2(-1)^{n}b^n, \end{equation} together with observation that by (\ref{May13}) and (\ref{May2}) \begin{equation}\label{May16} x_{1} = \frac{c+ \sqrt{ c^2-4b^2}}{2} >\frac{c}{2}>b. \end{equation} \subsection{Expectations and covariances} \begin{rem}\label{remM} Observe that by the symmetry it holds that for all $1\leq k \leq N$ \begin{equation}\label{Msyn} {\bf E}X_{k,\lambda}^{\circ} ={\bf E}X_{1,\lambda}^{\circ}, \end{equation} and for all $i\leq j \leq N$ \begin{equation}\label{Covsym} {\bf Cov} (X_{i,\lambda}^{\circ}, X_{j,\lambda}^{\circ})={\bf Cov} (X_{1,\lambda}^{\circ}, X_{1+j-i,\, \lambda}^{\circ}). \end{equation} \end{rem} As an immediate corollary of this remark and Theorem \ref{Lexp0} we get the following result. \begin{cor}\label{ExCov} For all $1\leq i,j \leq N$ and $\lambda \sim N^2$ we have \begin{equation}\label{Ja8} {\bf E}X_{i,\lambda}^{\circ} = a\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right), \end{equation} and \begin{equation}\label{M} {\bf Cov} (X_{i,\lambda}^{\circ}, X_{j,\lambda}^{\circ})= ({\cal H}^{-1})_{ij}\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right), \end{equation} where $({\cal H}^{-1})_{ij}$ are the elements of the matrix inverse to ${\cal H}$. All the remaining terms are uniform in $i,j=1, \ldots, N.$ \end{cor} $\Box$ Similarly Theorem \ref{Lexp1} yields \begin{cor}\label{ExCov1} For all $1\leq i,j \leq N$ and $\lambda \sim N^2$ we have \begin{equation}\label{Ja81} {\bf E}X_{i,\lambda} = a_i\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right), \end{equation} and \begin{equation}\label{M1} {\bf Cov} (X_{i,\lambda}, X_{j,\lambda})= ({\cal H}_1^{-1})_{ij}\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right), \end{equation} where $({\cal H}_1^{-1})_{ij}$ are the elements of the matrix inverse to ${\cal H}_1$. All the remaining terms are uniform in $i,j=1, \ldots, N.$ \end{cor} $\Box$ Equipped with the results on ${\cal H}^{-1}$ from the last section we are ready to compute the covariances in Corollary \ref{ExCov}. \begin{cor}\label{CorMay1} Let $\lambda \sim N^2$, and set \[ \delta = \frac{\gamma}{4\beta +\gamma+ 2\sqrt{ 4\beta^2+2\beta \gamma}} \] (as defined in (\ref{del})). Then \begin{equation}\label{May19} {\bf Var} (X_{1,\lambda}^{\circ}) = \left( \frac{2\beta + \gamma}{2\lambda}\right)^{3/2} \frac{1}{2\beta + \gamma (1 - \delta)/2}\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right), \end{equation} and for all $2\leq k \leq \left[ \frac{N+1}{2} \right]$ \begin{equation}\label{May21} {\bf Cov} (X_{1,\lambda}^{\circ}, X_{k,\lambda}^{\circ}) = {\bf Cov} (X_{1,\lambda}^{\circ}, X_{N-k+2,\lambda}^{\circ}) \end{equation} \[ =(-1)^{k-1} \left( \frac{2\beta + \gamma}{2\lambda}\right)^{3/2} \frac{\delta^{k-1}}{2\beta + \gamma (1 - \delta)/2} \left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right) .\] \end{cor} \noindent{\bf Proof.} Combining (\ref{May15}) and (\ref {MayInv}) we derive for $n=N$ \begin{equation}\label{May22} \Lambda _{11}=\frac{D_{N-1} }{cD_{N-1}-2b^2D_{N-2}-2(-1)^{N}b^N} \end{equation} \[=\frac{x_1 }{cx_{1}-2b^2} +O(e^{-\alpha N}) = \frac{1}{c-2b\frac{ b}{x_1}}+O(e^{-\alpha N}), \] where $\alpha>0$ is some positive constant. Hence, this formula and Corollary \ref{ExCov} give us \[ {\bf Var} (X_{1,\lambda}^{\circ}) = \Lambda _{11}\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right) \] \begin{equation}\label{May29} = \frac{1}{c-2b\frac{ b}{x_1}}\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right), \end{equation} where we also took into account that $\lambda \sim N^2.$ Recall that the values $c$ and $b$ are defined by (\ref{May2}) and (\ref{defa}), while by (\ref{May13}) and definition of $\delta$ we have \begin{equation}\label{May28} \frac{ b}{x_1}=\delta. \end{equation} After substituting all these values into (\ref{May29}) the statement (\ref{May19}) follows. In a similar way by (\ref{May15}) and (\ref {MayInv}) for $n=N$ we have \begin{equation}\label{May23} \Lambda _{12}=\Lambda _{1N}=\frac{-bD_{N-2}+(-b)^{N-1} }{cD_{N-1}-2b^2D_{N-2}-2(-1)^{N}b^N} \end{equation} \[=-\frac{x_1 }{cx_{1}-2b^2} \frac{b}{x_1}+O(e^{-\alpha N}), \] and also for all $2<k\leq \left[ \frac{N+1}{2} \right]$ \begin{equation}\label{May24} \Lambda_{1k} = \Lambda_{1 (N-k+2)} =\frac{ (-b)^{k-1}D_{N-k}+(-1)^{N+k+1}b^{N-k+1}D_{k-2} }{cD_{N-1}-2b^2D_{N-2}-2(-1)^{N}b^N} \end{equation} \[= \frac{ (-b)^{k-1}x_1^{2-k}+(-b)^{N-k+1}x_1^{k-N} }{cx_1-2b^2}+O(e^{-\alpha N})\] \[=(-1)^{k-1} \left(\frac{ b}{x_1}\right)^{k-1} \frac{x_1}{cx_1-2b^2}+O(e^{-\alpha N/2}).\] Therefore for all $2\leq k\leq N/2$ it follows by Lemma \ref{Lexp0} that \begin{equation}\label{May30} {\bf Cov} (X_{1,\lambda}^{\circ}, X_{k,\lambda}^{\circ})= \Lambda_{1k} \left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right) \end{equation} \[=(-1)^{k-1} \left(\frac{ b}{x_1}\right)^{k-1} \frac{1}{c-2b\frac{ b}{x_1}}\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right).\] Rewriting the last formula in terms of $\beta, \gamma, \delta$ gives us formula (\ref{May21}). $\Box$ Let us recall here for further reference, that $\Lambda = {\cal H}^{-1}(N)$ describes as well the covariance matrix of random vector \[\bar{\cal Z} \sim {\cal N}(\bar{0}, {\cal H}^{-1}(N)) \] introduced in Theorem \ref{T1}. In notation (\ref{May28}) formulas (\ref{May22}), (\ref{May23}) and (\ref{May24}) give us for all $k$ \begin{equation}\label{L11} {\bf Var} {\cal Z}_{1} ={\bf Var} {\cal Z}_{k}= \Lambda_{11}= \left( \frac{2\beta + \gamma}{2\lambda}\right)^{3/2} \frac{1}{2\beta + \gamma (1 - \delta)/2}+O(e^{-\alpha N/2}), \end{equation} and for all $1\leq k\leq [\frac{N+1}{2}]$ \begin{equation}\label{23J71} {\bf Cov} ({\cal Z}_{1}, {\cal Z}_{k}) = \Lambda_{1k} = \Lambda_{11} (-\delta)^{k-1}+O(e^{-\alpha N/2}), \end{equation} where $0\leq \delta<1$, as defined in Corollary \ref{CorMay1}. Note also that $\Lambda$ is a circular matrix, hence, for all $1\leq k\leq [\frac{N+1}{2}]$ \begin{equation}\label{lcir} {\bf Cov} ({\cal Z}_{1}, {\cal Z}_{k}) = {\bf Cov} ({\cal Z}_{1+j}, {\cal Z}_{k+j}) = {\bf Cov} ({\cal Z}_{1}, {\cal Z}_{N-k+2}). \end{equation} Corollary \ref{CorMay1} reveals an interesting covariance structure of the distribution of location of particles: two interspacings are positively correlated if the number of spacings between them is odd, otherwise, they are negatively correlated. Nevertheless, as it follows directly by the results of Corollary \ref{CorMay1}, the covariance between any spacing and the total sum of the spacings remains to be positive for all parameters $\beta>0, \gamma>0.$ This yields as well a strictly positive variance of $\sum_{k=1}^NX_{k,\lambda}^{\circ}$ as described below. \begin{cor}\label{CorMay2} For all positive parameters we have \begin{equation}\label{23Ja8} {\bf Var} \left(\sum_{k=1}^NX_{k,\lambda}^{\circ}\right)= {\bf Var} \left(\sum_{k=1}^N {\cal Z}_{k}\right)\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right) \end{equation} \[= N {\bf Cov} \left({\cal Z}_1, \sum_{k=1}^N{\cal Z}_{k}\right)\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right)\] \[= \left(\frac{2\beta + \gamma}{2\lambda} \right)^{\frac{3}{2}} N \frac{1}{2\beta + \gamma (1 - \delta)/2} \ \frac{1-\delta}{1+\delta}\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right).\] \end{cor} $\Box$ \subsection{Conditional expectation. Proof of the main result} Now we can choose $\lambda=\lambda(N)$ which solves (\ref{ES}). By Corollary \ref{ExCov1} we have \[{\bf E}X_{k,\lambda} = a_k\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right),\] while by (\ref{De22n5}) and Corollary \ref{ExCov} we get for all $1\leq k \leq N$ and $\lambda \sim N^2$ \[ {\bf E}X_{k,\lambda} = \left(1+o\left(e^{-\alpha k}\right)\right){\bf E}X_{k,\lambda}^{\circ}= \left(1+o\left(e^{-\alpha k}\right)\right)\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)a. \] Hence, combination of both results gives us \[ \frac{a_k}{a}-1 = o\left(N^{-\frac{2}{5}} \right)+o\left(e^{-\alpha k}\right),\] which confirms conditions of Lemma \ref{PA1} when $N$ is large. Therefore Lemma \ref{PA1} implies \begin{equation}\label{ak} {a_k}={a} +O \left({\eta^k}\right), \end{equation} where \[ \eta = 1+4\frac{\beta}{\gamma} -\sqrt{16\left(\frac{\beta}{\gamma} \right)^2 +8\frac{\beta}{\gamma}}<1.\] Hence, we can rewrite (\ref{ES}) as \begin{equation}\label{ES1} 1=\sum_{k=1}^N \mathbb{E} X_{k,\lambda} = \sum_{k=1}^{\sqrt N} O\left(\frac{1}{N} \right) + \sum_{k={\sqrt N} } ^N \left( {a} +O \left({\eta^k}\right)\right) =\sqrt{\frac{2\beta + \gamma}{2\lambda}}N \left(1+ O(N^{-1/2})\right), \end{equation} as $N\rightarrow \infty$. Hence, the solution to (\ref{ES}) is \begin{equation}\label{May32} \lambda=\lambda(N)= \sqrt{\frac{2}{2\beta + \gamma}}\ N^2 \left(1+ O(N^{-1/2})\right), \end{equation} and therefore from now on we set \[ \lambda=\lambda(N)= \sqrt{\frac{2}{2\beta + \gamma}}\ N^2 , \] which implies \begin{equation}\label{fa} a=\frac{1}{N}. \end{equation} Note also that with this choice of $\lambda$ we get from (\ref{23Ja8}) setting $x:= \gamma/\beta$ \[ {\bf Var} \left(\sum_{k=1}^N X_{k,\lambda}^{\circ}\right)= {\bf Var} \left(\sum_{k=1}^N {\cal Z}_{k}\right)\left(1+ o\left(\lambda^{-\frac{1}{5}} \right)\right)+o\left(e^{-(\log \lambda)^{-\frac{3}{2}}} \right) \] \begin{equation}\label{May31} = \frac{1}{\beta N^{2} }\, g(x) \left(1+ o\left(N^{-\frac{1}{3}} \right)\right), \end{equation} where \[ g(x) =\frac{2}{ 4 + \frac{ 4x +2x\sqrt{ 4+2x}}{4 +x+ 2\sqrt{ 4+2x}} } \ \frac{2 + \sqrt{ 4+2x}}{2 +x+ \sqrt{ 4+2x}}>0. \] \begin{rem}\label{Rg} Function $g(x)$ is monotone decreasing on $R^+$ with the maximal value $g(0)=\frac{1}{2}$, i.e., when $\gamma=0$. \end{rem} \noindent {\bf Proof of Theorem \ref{T1}.} Consider now conditional expectation of $X^{\circ}_{k, \lambda}$ given $\sum_{k=1}^N X_{k, \lambda}^{\circ}=1.$ Let us write here $$S=\sum_{k=1}^NX^{\circ}_{k, \lambda} ,$$ and then denote $f_{S}$ and $f_{X_1,S}$ the densities of $S$ and of $(X_1, S)$, respectively, and also $\phi_{S}$ and $\phi_{X_1,S}$ the corresponding characteristic functions. First we write \begin{equation}\label{23Ja4} \mathbb{E}\left\{ {X}_{k,\lambda}^{\circ} \mid \sum_{k=1}^N X_{k, \lambda}^{\circ}=1\right\} = \mathbb{E}\left\{ {X}_{1,\lambda}^{\circ} \mid \sum_{k=1}^N X_{k, \lambda}^{\circ}=1\right\} = \frac{ \int_{0}^1 xf_{X_1,S}(x,1-x) d{x} }{ f_{S^{\circ}_{N,\lambda}}(1) }, \end{equation} where the first equality follows by the symmetry discussed above. This instructs us to study the densities involved in the last formula. With a help of Theorem \ref{Lexp0} we shall establish first the following result. \begin{lem}\label{TJ1} Let \begin{equation}\label{lambda} \lambda=\lambda(N)= \sqrt{\frac{2}{2\beta + \gamma}}\ N^2, \ \ \ \ \ \ a=\frac{1}{N} , \end{equation} $$\sigma_N^2:= \mbox{{\bf Var}}\left(\sum_{i=1}^N{\cal Z}_i \right),$$ and set $$\tilde{X_k}=\frac{X^{\circ}_{k, \lambda}-a}{\sigma_N}, \ \ \ \ S= \sum_{k=1}^NX^{\circ}_{k, \lambda}, \ \ \ \ \tilde{S}=\frac{S-1}{\sigma_N}, \ \ \ \tilde{{\cal Z}}_k=\frac{{\cal Z}_k}{\sigma_N},\ \ \ \Sigma = \frac{\sum_{k=1}^N{\cal Z}_k}{\sigma_N},$$ with $\bar{ \cal Z} \sim{\cal N}(\bar{0}, \Lambda)$ as defined in Theorem \ref{Lexp0} with covariance matrix $\Lambda={\cal H}^{-1}(N)$. Under the assumptions of Theorem \ref{Lexp0} we have for an arbitrarily fixed $\alpha>0$ and for all $x,y,z\geq 0$ \begin{equation}\label{23C1} f _{\tilde{S}}(x)=f_{\Sigma}(x)\left(1+ O\left(N^{-1/3} \right) \right)+ O\left(N^{-\alpha} \right), \end{equation} \begin{equation}\label{23C2} f_{\tilde{X_1}, \tilde{S}}(x,y,z)=f_{ \tilde{{\cal{Z}}}_1, \Sigma }(x,z) \left(1 + O\left( N^{-1/3}\right) \right)+ O\left( N^{-\alpha}\right), \end{equation} \begin{equation}\label{23C3} f_{\tilde{X_1}, \tilde{X_k},\tilde{S}}(x,y,z)=f_{ \tilde{{\cal{Z}}}_1, \tilde{{\cal{Z}}}_k , \Sigma }(x,y,z)\left(1 + O\left( N^{-1/3}\right) \right)+ O\left( N^{-\alpha}\right), \end{equation} where the error terms are uniform in $k$ and $x,y,z$. \end{lem} \noindent {\bf Proof.} Making use of Fourier inverse formula for the densities consider for an arbitrarily fixed $\alpha>0$ \[f_{\tilde{S}}(x) = \frac{1}{{2\pi} } \int_{-\infty}^{\infty}e^{-itx}\phi_{\tilde{S}}(t)dt =\frac{1}{{2\pi} } \int_{\|t\|\leq N^\alpha}e^{-itx}\phi_{\tilde{S}}(t)dt + \frac{1}{{2\pi} } \int_{\|t\|>N^\alpha} e^{-itx} \phi_{\tilde{S}}(t)dt.\] Applying now the results of Theorem \ref{Lexp0} we derive from here for all $x\in \mathbb{R}$ \begin{equation}\label{M23} f_{\tilde{S}}(x) = \frac{1}{{2\pi} } \int_{\|t\|\leq N^\alpha} e^{-itx}\left( \phi_{\Sigma}(t) \left(1+ o\left(N^{-\frac{1}{3}} \right)\right)+o(e^{-(2\log N)^{3/2}}) \right) dt \end{equation} \[+ \frac{1}{{2\pi} } \int_{\|t\|>N^\alpha} e^{-itx} \phi_{\tilde{S}}(t)dt\] \[ = \frac{1}{{2\pi} } \int_{-\infty}^{\infty} e^{-itx}\phi_{\Sigma}(t) \left(1+ o\left(N^{-\frac{1}{3}} \right)\right) dt+o(e^{-(2\log N)^{3/2}}) N^\alpha \] \[-\frac{1}{{2\pi} } \int_{\|t\|>N^\alpha} \left(1+ o\left(N^{-\frac{1}{3}} \right)\right)e^{-itx}\phi_{\Sigma}(t) dt+ \frac{1}{{2\pi} } \int_{\|t\|>N^\alpha} e^{-itx} \phi_{\tilde{S}}(t)dt \] \[= \left( 1+ o\left( N^{-\frac{1}{3}} \right)\right) f_{\Sigma}(x) +o(e^{-(2\log N)^{3/2}}) N^\alpha \] \[-\left(1+ o\left(N^{-\frac{1}{3}} \right)\right)\frac{1}{{2\pi} } \int_{\|t\|>N^\alpha} e^{-itx}\phi_{\Sigma}(t) dt+ \frac{1}{{2\pi} } \int_{\|t\|>N^\alpha} e^{-itx} \phi_{\tilde{S}}(t)dt . \] Consider the last integral on the right. Taking into account that \begin{equation}\label{23J25} f_{{S}}(x) = \int_{0}^x\int_{0}^{y_{N-1}}\ldots \int_{0}^{y_2} f_{\bar{X}^{\circ}_{\lambda}}(y_1, y_2-y_1, \ldots, y_{N-1}-y_{N-2}, x-y_{N-1})dy_1 \ldots dy_{N-1} \end{equation} with function $ f_{\bar{X}^{\circ}_{\lambda}}$ defined by (\ref{Mt7}) and (\ref{H0c}), we derive \begin{equation}\label{23J19} \phi_{\tilde{S}}(t) = \frac{1}{(it)^2} \int_{\mathbb{R}}e^{itx}f''_{\tilde{S}}(x)dx, \end{equation} which yields \begin{equation}\label{23J24} \left\| \int_{\|t\|>N^\alpha} e^{-itx} \phi_{\tilde{S}}(t) dt \right\|\leq \int_{\|t\|>N^\alpha} \frac{1}{t^2}dt \int_{\mathbb{R}}\|f''_{\tilde{S}}(x)\|dx. \end{equation} As \[f_{\tilde{S}}(x) =\sigma_N f_{{S}}(\sigma_Nx +1), \] passing in the last integral in (\ref{23J24}) to variable $S$, we get \begin{equation}\label{23J21} \int_{\mathbb{R}}\left\|f''_{\tilde{S}}(x)\right\|dx= {\sigma_N^2}\int_{0}^{{N}}\left\|f''_{{S}}(x)\right\|dx. \end{equation} By a straightforward application of formulas (\ref{Mt7}) and (\ref{H0c}) we get that function $f''_{{S}}(x)$ changes sign at most a finite number (say $L$) of times on its domain, i.e., $[0,N]$. Hence, \begin{equation}\label{23J22} \int_{0}^{{N}}\left\|f''_{{S}}(x)\right\|dx \leq 2L\max_{x\in [0,N]} \left\|f'_{{S}}(x)\right\| . \end{equation} Combining (\ref{23J22}) and (\ref{23J21}) in (\ref{23J24}) we obtain \begin{equation}\label{23J24n} \left\| \int_{\|t\|>N^\alpha} e^{-itx} \phi_{\tilde{S}}(t) dt \right\|\leq \frac{2}{N^{\alpha}} 2L \sigma_N^2 \max_{x} \|f_{{S}}'(x) \| . \end{equation} Here \begin{equation}\label{23J26} \max_{x\in [0,N]} \left\|f'_{{S}}(x)\right\| \leq O(N^2) \max_{x\in [0,N]} f_{{S}}(x), \end{equation} which is again a result of straightforward computation and assumption that $\lambda \sim N^2$. Hence, (\ref{23J24n}) yields \[ \left\| \int_{\|t\|>N^\alpha} e^{-itx} \phi_{\tilde{S}}(t) dt \right\| \leq N^{-\alpha} {\sigma_N^2}O(N^2) \max_{x} f_{{S}}(x) \] \begin{equation}\label{23J31} = {O(N^{-\alpha} )} \max_{x} f_{{S}}(x)={O(N^{-\alpha} \sigma_N^{-1})}\max_{x} f_{\tilde{S}}(x)={O(N^{-\alpha+1} )}\max_{x} f_{\tilde{S}}(x), \end{equation} where we also took into account (\ref{May31}), i.e., that $\sigma_N^2 \sim N^{-2}$. From now on we fix \begin{equation}\label{alpha} \alpha >6. \end{equation} Equipped with bound (\ref{23J31}) and taking into account that $$ \phi_{\Sigma}(t)=e^{-t^2/2},$$ we derive from (\ref{M23}) \begin{equation}\label{23J33} f_{\tilde{S}}(x) = f_{\Sigma}(x)\left(1+ O\left(N^{-1/3} \right) \right)+ O\left(N^{-\alpha+1} \right)\max_{x} f_{\tilde{S}}(x)+o\left(e^{-(2\log N)^{3/2}}\right) N^\alpha \end{equation} for all $x$. Hence, $f_{\tilde{S}}(x)$ is uniformly bounded in $x$ as $f_{\Sigma}(x)$ is, and this in turn implies by (\ref{23J33}) that for all $x $ \begin{equation}\label{23J34} f_{\tilde{S}}(x) = f_{\Sigma}(x)\left(1+ O\left(N^{-1/3} \right) \right)+ O\left(N^{-\alpha+1} \right), \end{equation} which is the first statement of the lemma. Consider now in similar way for an arbitrarily fixed $\alpha>6$ \begin{equation}\label{23J51} f_{\tilde{X_1}, \tilde{S}}(x,u) = \frac{1}{({2\pi} )^2} \int_{{\mathbb{R}}^2} e^{-it_1x}e^{-it_2u}\phi_{\tilde{X_1}, \tilde{S}}({\bar{t}})d{\bar{t}} \end{equation} \[ =\frac{1}{({2\pi} )^2}\int_{ [-N^{\alpha},N^{\alpha}]^2} e^{-it_1x}e^{-it_2u} \left( \phi_{ \tilde{{\cal{Z}}}_1, \Sigma }(x,u)\left( 1+o\left( N^{-\frac{1}{3}}\right) \right) +o \left( e^{-(2\log N)^{3/2}}\right) \right)d{\bar{t}} \] \[ + \frac{1}{({2\pi})^2 } \int_{ {\mathbb{R}}^2 \setminus [-N^{\alpha},N^{\alpha}]^2} e^{-it_1x}e^{-it_2u} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) d{\bar{t}}\] \[ =\left( 1+o\left( N^{-\frac{1}{3}}\right) \right)f _{ \tilde{{\cal{Z}}}_1, \Sigma }(x,u) +o \left( N^{2 \alpha}e^{-2(\log N)^{3/2}}\right) \] \[ - \left( 1+o\left( N^{-\frac{1}{3}}\right) \right)\frac{1}{({2\pi})^2 } \int_{ {\mathbb{R}}^{2}\setminus [-N^{\alpha},N^{\alpha}]^2} e^{-it_1x}e^{-it_2u} \phi_{ \tilde{{\cal{Z}}}_1,\Sigma }({\bar{t}})d{\bar{t}}\] \[ + \frac{1}{({2\pi})^2 } \int_{ {\mathbb{R}}^2 \setminus [-N^{\alpha},N^{\alpha}]^2} e^{-it_1x}e^{-it_2u} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) d{\bar{t}}.\] Here vector $\left( \tilde{{\cal{Z}}}_1,\Sigma \right) $ by its definition has a Normal distribution with zero mean and covariance matrix \[ \left( \begin{array}{cc} \frac{\Lambda_{11} }{\sigma_N} & \frac{1}{N} \\ \\ \frac{1}{N} & 1 \end{array} \right) ,\] where by (\ref{L11}) $\Lambda_{11}\sim N^{-3}$, and as already stated, $\sigma_N\sim N^{-2}$. This immediately gives us a bound \begin{equation}\label{23F1} \int_{ {\mathbb{R}}^{2}\setminus [-N^{\alpha},N^{\alpha}]^2} e^{-it_1x}e^{-it_2u} \phi_{ \tilde{{\cal{Z}}}_1,\Sigma }({\bar{t}})d{\bar{t}}=O\left(e^{-N^{\alpha - 2}} \right) . \end{equation} Consider now the remaining last integral in (\ref{23J51}). First, taking into account formula \begin{equation}\label{23J44} f_{X_1,{S}}(x,v) = \int_{x}^v\int_{x}^{y_{N-1}}\ldots \int_{x}^{y_3} f_{\bar{X}^{\circ}_{\lambda}}(x, y_2-x, \ldots, y_{N-1}-y_{N-2}, v-y_{N-1})dy_2\ldots dy_{N-1}, \end{equation} we derive as in (\ref{23J19}): \begin{equation}\label{23J43} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) = \int_{ {\mathbb{R}} }\int_{ {\mathbb{R}}}e^{it_1y}e^{it_2v}f_{\tilde{X_1}, \tilde{S}}(y,v) dydv \end{equation} \[=\int_{ {\mathbb{R}} }e^{it_1y} \frac{1}{(it_2)^2} \left( \int_{\mathbb{R}}e^{it_2v}\frac{\partial ^2}{\partial v^2}f_{\tilde{X_1}, \tilde{S}}(y,v)dv \right) dy, \] as well as \begin{equation}\label{23J43} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) = \int_{ {\mathbb{R}} }\int_{ {\mathbb{R}}}e^{it_1y}e^{it_2v}f_{\tilde{X_1}, \tilde{S}}(y,v) dydv \end{equation} \[=\int_{ {\mathbb{R}} }e^{it_2v} \left.\frac{1}{(it_1)^2} \left(e^{it_1y} \frac{\partial}{\partial y}f_{\tilde{X_1}, \tilde{S}}(y,v)\right\|_{y=-\frac{1}{N\sigma_N}}^{y=\frac{N-1}{N\sigma_N}} - \int_{\mathbb{R}}e^{it_1y}\frac{\partial ^2}{\partial y^2}f_{\tilde{X_1}, \tilde{S}}(y,v)dy \right) dv. \] The relation (\ref{23J43}) gives us (note that the functions under consideration are bounded, and smooth enough for all the following Fourier transformations to hold) \begin{equation}\label{23J50} \int_{{\mathbb{R}}} \int_{\|t_2\|>N^\alpha} e^{-it_1x}e^{-it_2u} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) dt_2dt_1 \end{equation} \[ =\int_{{\mathbb{R}}} \int_{\|t_2\|>N^\alpha} e^{-it_1x}e^{-it_2u} \int_{ {\mathbb{R}} }e^{it_1y} \frac{1}{(it_2)^2} \int_{\mathbb{R}}e^{it_2v}\frac{\partial ^2}{\partial v^2}f_{\tilde{X_1}, \tilde{S}}(y,v)dv dydt_2dt_1 \] \[=2\pi \int_{\|t_2\|>N^\alpha} e^{-it_2u} \frac{1}{(it_2)^2} \int_{\mathbb{R}}e^{it_2v}\frac{\partial ^2}{\partial v^2}f_{\tilde{X_1}, \tilde{S}}(x,v)dv dt_2. \] Hence, \begin{equation}\label{23J46} \left\| \int_{{\mathbb{R}}} \int_{\|t_2\|>N^\alpha} e^{-it_1x}e^{-it_2u} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) dt_2dt_1\right\| \end{equation} \[\leq 2\pi \int_{\|t_2\|>N^\alpha} \frac{1}{(t_2)^2} \left( \int_{\mathbb{R}}\left\| \frac{\partial ^2}{\partial v^2}f_{\tilde{X_1}, \tilde{S}}(x,v)\right\|dv \right)dt_2= O(N^{-\alpha})\max_{v\in [-\frac{1}{\sigma_N}, \frac{N-1}{\sigma_N}]} \left\| \frac{\partial }{\partial v}f_{\tilde{X_1}, \tilde{S}}(x,v)\right\|, \] where to derive the last equality we used again (as in (\ref{23J22})) the fact that by definition (\ref{23J44}) function $\frac{\partial ^2}{\partial v^2}f_{\tilde{X_1}, \tilde{S}}(x,v)$ changes the sign on its domain $v\in [-\frac{1}{\sigma_N}, \frac{N-1}{\sigma_N}]$ at most a finite (independent of $N$) number of times. Formula (\ref{23J44}) straightforward yields \begin{equation}\label{23J48} \max_{v\in [0,N]} \left\| \frac{\partial }{\partial v}f_{{X_1}, {S}}(x,v)\right\| \leq O(N^2) \max_{v\in [0,N]} f_{{X_1}, {S}}(x,v), \end{equation} as well as \begin{equation}\label{23J49} \max_{x\in [0,1]} \left\| \frac{\partial }{\partial x}f_{{X_1}, {S}}(x,v)\right\| \leq O(N^2) \max_{x\in [0,1]} f_{{X_1}, {S}}(x,v), \end{equation} implying corresponding bounds for $f_{\tilde{X_1}, \tilde{S}}$. With a help of (\ref{23J48}) we derive from (\ref{23J46}) a bound (as in (\ref{23J31}) ) \begin{equation}\label{23J47} \left\| \int_{{\mathbb{R}}} \int_{\|t_2\|>N^\alpha} e^{-it_1x}e^{-it_2u} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) dt_2dt_1\right\|\leq O(N^{2-\alpha})\max_{v} f_{\tilde{X_1}, \tilde{S}}(x,v). \end{equation} Next exploring equality (\ref{23J43}) and using the same arguments (\ref{23J50})-(\ref{23J46}) we arrive with a help of (\ref{23J49}) to a similar bound (with respect to another variable): \begin{equation}\label{23J47n} \left\| \int_{{\mathbb{R}}} \int_{\|t_1\|>N^\alpha} e^{-it_1x}e^{-it_2u} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) dt_1dt_2\right\|\leq O(N^{2-\alpha})\max_{x} f_{\tilde{X_1}, \tilde{S}}(x,v). \end{equation} Combination of the last two bounds gives us \begin{equation}\label{23J47} \left\| \int_{{\mathbb{R}}^2\setminus [-N^\alpha, N^\alpha]^2} e^{-it_1x}e^{-it_2u} \phi_{\tilde{X_1}, \tilde{S}}({\bar{t}}) dt_1dt_2\right\|\leq O(N^{2-\alpha})\max_{x,v} f_{\tilde{X_1}, \tilde{S}}(x,v). \end{equation} Making use of the last bound and (\ref{23F1}) we derive from (\ref{23J51}) (as in the former case of (\ref{23C1})): \begin{equation}\label{23J52} f_{\tilde{X_1}, \tilde{S}}(x,u) = f_{ \tilde{{\cal{Z}}}_1, \Sigma }(x,u)\left(1 + O\left(N^{-1/3}\right)\right)+ O(N^{2-\alpha})\max_{x,v} f_{\tilde{X_1}, \tilde{S}}(x,v) \end{equation} \[+o \left( N^{2 \alpha}e^{-2(\log N)^{3/2}}\right), \] which yields first the boundedness in $N$ of $\max_{x,v} f_{\tilde{X_1}, \tilde{S}}(x,v)$, and then in turn implies statement (\ref{23C2}) of the lemma. For the final statement (\ref{23C3}) we begin with \begin{equation}\label{23J40} f_{\tilde{X_1}, \tilde{X_k},\tilde{S}}(x,y,z) = \frac{1}{({2\pi})^3 } \int_{ {\mathbb{R}}^3}e^{-it_1x-it_2y-it_3z} \phi_{\tilde{X_1}, \tilde{X_k},\tilde{S}}({\bar{t}}) d{\bar{t}}. \end{equation} Then (\ref{23C3}) is proved following closely the lines of the proof of (\ref{23C2}), we omit these details. $\Box$ Now we are ready to study conditional expectation in (\ref{23Ja4}). Making use of the last Lemma \ref{TJ1} we get (recall that $a=1/N$) for $p=1,2$ \begin{equation}\label{23Ja61} \mathbb{E}\left\{ \left({X}_{k,\lambda}^{\circ}-a\right)^p \mid \sum_{k=1}^N X_{k, \lambda}^{\circ}=1\right\} =\sigma_N^p \mathbb{E} \left\{ \left(\frac{{X}_{k,\lambda}^{\circ}-a }{\sigma_N}\right)^p \ \left\| \ \frac{\sum_{k=1}^N{X}_{k,\lambda}^{\circ}-1 }{\sigma_N} =0 \right. \right\} \end{equation} \[ = \sigma_N^p \frac{ \int_{-\frac{a}{\sigma_N}}^{ \frac{1-a }{\sigma_N}} x^pf_{\tilde{X_1}, \tilde{S}}(x,0) d{x} }{f_{\tilde{S}}(0) }= \sigma_N^p \frac{ \int_{-a/\sigma_N}^{ \frac{1-a }{\sigma_N}} x^p\left(f_{\tilde{{\cal{Z}}}_1, \Sigma}(x,0) \left(1 + O\left(N^{-1/3}\right)\right) + O(N^{-\alpha}) \right) d{x} }{f_{\Sigma}(0)\left(1 + O\left(N^{-1/3}\right)\right) + O(N^{-\alpha}) }, \] where $f_{ \Sigma }(0)={1}/\sqrt{2\pi}$ as by the definition $\Sigma \sim {\cal{N}}(0,1)$. Hence, for all $\alpha>5$ \begin{equation}\label{23Ja61} \mathbb{E}\left\{ \left({X}_{k,\lambda}^{\circ}-a\right)^p \mid \sum_{k=1}^N X_{k, \lambda}^{\circ}=1\right\} \end{equation} \[= \frac{\sigma_N^p \int_{-a/\sigma_N}^{ \frac{1-a }{\sigma_N}} x^pf_{ \tilde{{\cal{Z}}}_1, \Sigma }(x,0) d{x} }{f_{ \Sigma }(0)}\left(1 + O\left(N^{-1/3}\right)\right) +O(N^{-\alpha+1}). \] Note, that we consider only $p=1,2$, but the same argument works for an arbitrary fixed $p$. In (\ref{23Ja61}) we have (recall notation in Lemma \ref{TJ1}) \begin{equation}\label{23Ja17} f_{ \tilde{{\cal{Z}}}_1, \Sigma }(x,0)=\sigma_N^2 f_{{\cal{Z}}_1, \sum_{k=1}^N {\cal{Z}}_k}(\sigma_Nx,0). \end{equation} Here $\bar{{\cal{Z}}} \sim {\cal N}(\bar{0}, \Lambda)$, where the covariance matrix $\Lambda$ (studied in Corollary \ref{CorMay1}) with the choice of $\lambda$ (\ref{lambda}) has the properies (see (\ref{L11}) and (\ref{23J71})) that \begin{equation}\label{23J64} {\bf Var}({\cal{Z}}_1)=\Lambda_{11} \sim N^{-3}, \ \ \ {\bf Cov} \left( {\cal{Z}}_1,\sum_{k=1}^N {\cal{Z}}_k \right) =\frac{\sigma_N^2}{N} \sim N^{-3}. \end{equation} Then making use of the closed form for the density in (\ref{23Ja17}), which is \begin{equation}\label{23J63} f_{{\cal{Z}}_1, \sum_{k=1}^N {\cal{Z}}_k}(\sigma_Nx,0) =\frac{1}{2\pi \sqrt{\Lambda_{11} \sigma_N^2} \sqrt{1- \frac{1}{\Lambda_{11} \sigma_N^2 }\left(\frac{\sigma_N^2}{ N} \right)^2 }}\exp \left\{ -\frac{1}{2\left(1- \frac{\sigma_N^2}{\Lambda_{11} N^2 }\right)} \frac{(\sigma_N x)^2}{\Lambda_{11} } \right\}, \end{equation} and the asymptotics (\ref{23J64}), it is straightforward to compute that \begin{equation}\label{23J65} \int_{\|x\|>a/\sigma_N} x^p \sigma_N^2 f_{{\cal{Z}}_1, \sum_{k=1}^N {\cal{Z}}_k}(\sigma_Nx,0) d{x}=o(e^{-\sqrt{N}}). \end{equation} This allows us to derive from (\ref{23Ja61}) \begin{equation}\label{23Ja65} \mathbb{E}\left\{ \left({X}_{k,\lambda}^{\circ}-a\right)^p \mid \sum_{k=1}^N X_{k, \lambda}^{\circ}=1\right\} \end{equation} \[= \sigma_N^p\frac{ \int_{-\infty}^{+\infty}x^p\sigma_N^2 f_{{\cal{Z}}_1, \sum_{k=1}^N {\cal{Z}}_k}(\sigma_Nx,0) d{x} }{f_{ \Sigma }(0) }(1+O\left(N^{-1/3}\right))+O(N^{-\alpha +1}) \] \[= \frac{ \int_{-\infty}^{+\infty}x^p f_{{\cal{Z}}_1, \sum_{k=1}^N {\cal{Z}}_k}(x,0) d{x} }{f_{ \sum_{k=1}^N {\cal{Z}}_k }(0) }(1+O\left(N^{-1/3}\right)) +o(e^{-\sqrt{N}})+O(N^{-\alpha +1}) \] \[={\mathbb E} \left\{ {\cal{Z}}_k^p \mid \sum_{k=1}^N {\cal{Z}}_k=0\right\}(1+O\left(N^{-1/3}\right))+O(N^{-\alpha +1}) .\] In precisely same manner we derive with a help of (\ref{23C3}) for any $j>1$ the following (similar to (\ref{23Ja61})) relations: \begin{equation}\label{23Ja3} {\bf Cov}\left\{ \left( X_{1,\lambda}^{\circ}, X_{j,\lambda}^{\circ}\right) \mid \sum_{k=1}^N X_{k, \lambda}^{\circ}=1\right\} \end{equation} \[ =\sigma_N^2 \mathbb{E}\left\{ \left(\frac{ {X}_{1,\lambda}^{\circ}-a }{\sigma_N}\right) \left( \frac{ {X}_{j,\lambda}^{\circ}-a }{\sigma_N}\right) \ \left\| \ \frac{\sum_{k=1}^N{X}_{k,\lambda}^{\circ}-1 }{\sigma_N} =0 \right. \right\} \] \[ = \sigma_N^2 \frac{ \int_{-\frac{a}{\sigma_N}}^{ \frac{1-a }{\sigma_N}} \int_{-\frac{a}{\sigma_N}}^{ \frac{1-a }{\sigma_N}} xyf_{\tilde{X_1}, \tilde{X_j},\tilde{S}}(x,y,0) d{x}dy }{f_{\tilde{S}}(0) }\] \[= \sigma_N^2 \frac{ \int_{-\frac{a}{\sigma_N}}^{ \frac{1-a}{\sigma_N}} \int_{-\frac{a}{\sigma_N}}^{ \frac{1-a}{\sigma_N}} xy f_{ \tilde{{\cal{Z}}}_1, \tilde{{\cal{Z}}}_j,\Sigma}(x,0) d{x}dy }{ f_{\Sigma}(0) }\left(1 + O\left(N^{-1/3}\right) \right)+O(N^{-\alpha +1}) . \] Here we have \begin{equation}\label{23J67} f_{ \tilde{{\cal{Z}}}_1, \tilde{{\cal{Z}}}_j, \Sigma }(x,y,0)=\sigma_N^3 f_{{\cal{Z}}_1, {{\cal{Z}}}_j, \sum_{k=1}^N {\cal{Z}}_k}(\sigma_Nx,\sigma_Ny,0), \end{equation} where vector $({\cal{Z}}_1, {{\cal{Z}}}_j, \sum_{k=1}^N {\cal{Z}}_k)$ has a 3-dimensional normal distribution with zero mean vector and the correlation function given by (see Corollary \ref{CorMay1}, and (\ref{23J71})): \[{\bf C}= \left( \begin{array}{ccc} \Lambda_{11} & \Lambda_{1j} & \frac{\sigma_N^2}{N} \\ \\ \Lambda_{1j} & \Lambda_{jj} & \frac{\sigma_N^2}{N} \\ \\ \frac{\sigma_N^2}{N} & \frac{\sigma_N^2}{N} & \sigma_N^2 \end{array} \right).\] Hence, the density in (\ref{23J67}) for all $ j>1$ is \begin{equation}\label{23J69} f_{ {\cal{Z}}_1, {\cal{Z}}_j,\sum_{k=1}^N {\cal{Z}}_k }(\sigma_Nx,\sigma_Ny,0) =\frac{1}{(2\pi)^{3/2}} \frac{1}{\sqrt{det \, {\bf C}} } \exp \left\{ -\frac{1}{2}\left(\sigma_Nx,\sigma_Ny,0\right){\bf C}^{-1} \left(\sigma_Nx,\sigma_Ny,0\right)' \right\}. \end{equation} Again by Corollary \ref{CorMay1} and the choice of $\lambda$ \begin{equation}\label{23J68} \Lambda_{jj}=\Lambda_{11} \sim N^{-3}, \ \ \ \Lambda_{1j} \sim N^{-3}, \ \ \ \sigma_N^2 \sim N^{-2} \end{equation} for all $j$. Taking also into account formula (\ref{23J71}) we derive \begin{equation}\label{detC} det \, {\bf C}=\sigma_N^2\left((\Lambda_{11}^2-\Lambda_{1j}^2)-\frac{\sigma_N^2}{N^2}(\Lambda_{11}-\Lambda_{1j})\right) \end{equation} \[= \sigma_N^2\left(\Lambda_{11}^2(1-\delta^{2(j-1)})+ o(e^{-\alpha N/2})-\frac{\sigma_N^2}{N^2}\Lambda_{11} (1-\delta^{j-1}) \right)\] \[=\sigma_N^2 \Lambda_{11}^2(1-\delta^{2(j-1)})(1+O(1/N)).\] Then it is a routine to compute that \[\int\int_{\mathbb{R}^2 \setminus \left[\frac{-a}{\sigma_N}, \frac{1-a}{\sigma_N}\right]^2} xy f_{ \tilde{{\cal{Z}}}_1, \tilde{{\cal{Z}}}_j,\sum_{k=1}^N {\cal{Z}}_k}(x,0) d{x}dy = o(e^{-\sqrt{N}}),\] which allows us to rewrite (\ref{23Ja3}) (taking into account (\ref{23J67})) as follows \begin{equation}\label{23Ja70} {\bf Cov}\left\{ \left( X_{1,\lambda}^{\circ}, X_{j,\lambda}^{\circ}\right) \mid \sum_{k=1}^N X_{k, \lambda}^{\circ}=1\right\} \end{equation} \[ =\sigma_N^5 \frac{ \int_{-\infty}^{ +\infty} \int_{-\infty}^{+\infty} xy f_{ {{\cal{Z}}}_1, {{\cal{Z}}}_j,\sum_{k=1}^N {\cal{Z}}_k}(\sigma_Nx,\sigma_Ny, 0) d{x}dy }{ f_{\Sigma}(0) }\left(1 + O\left(N^{q-1/2}\right) \right)+o(e^{-\sqrt{N}}) \] \[ = \frac{ \int_{-\infty}^{ +\infty} \int_{-\infty}^{+\infty} xy f_{ {{\cal{Z}}}_1, {{\cal{Z}}}_j,\sum_{k=1}^N {\cal{Z}}_k}(x,y, 0) d{x}dy }{ f_{ \sum_{k=1}^N {\cal{Z}}_k}(0) }\left(1 + O\left(N^{-1/3}\right) \right)+O(N^{-\alpha +1}) \] \[={\bf Cov} \left\{ \left( {\cal{Z}}_1, {\cal{Z}}_j\right) \mid \sum_{k=1}^N {\cal{Z}}_k=0\right\}\left(1 + O\left(N^{-1/3}\right) \right)+O(N^{-\alpha +1}) .\] By the properties of normal distribution the conditional distribution of \[ \left( {\cal{Z}}_1, {\cal{Z}}_j\right) \mid \sum_{k=1}^N {\cal{Z}}_k=0 \] is again normal with zero mean vector and the covariance matrix \begin{equation}\label{23J76} \left( \begin{array}{cc} \Lambda_{11} & \Lambda_{1j} \\ \\ \Lambda_{1j} & \Lambda_{jj} \end{array} \right) -\frac{\sigma_N^2}{N^2}\left( \begin{array}{cc} 1 & 1 \\ \\ 1 & 1 \end{array} \right). \end{equation} Observe that (see Corollary ) \[\sigma_N^2=N\left( \Lambda_{11} + 2\sum_{k=2}^{\left[ \frac{N}{2}\right]} \Lambda_{1j}\right).\] Hence, making use of formula (\ref{23J71}) with $\lambda$ chosen as in (\ref{lambda}) we derive from (\ref{23J76}) \[ {\bf{Cov}}\left\{ \left( {\cal{Z}}_1, {\cal{Z}}_j\right) \mid \sum_{k=1}^N {\cal{Z}}_k=0 \right\}=\Lambda_{1j}-\frac{\sigma_N^2}{N^2}=\Lambda_{11} \left( (-\delta)^{j-1}-\frac{1}{N}\frac{1-\delta}{1+\delta} +o(1/N) \right) \] \[ = \frac{1}{N^3} \left( \frac{1}{2\beta + \gamma (1 - \delta)/2} +o\left(N^{-\frac{1}{3}} \right) \right) \left( (-\delta)^{j-1}-\frac{1}{N}\frac{1-\delta}{1+\delta} +o(1/N) \right). \] This together with (\ref{23Ja70}) and (\ref{23Ja65}) confirms the statements of Theorem \ref{T1}, as we can fix $\alpha=7$. $\Box$ \noindent {\bf Proof of Theorem \ref{T2o}.} The proof closely follows the lines of above proof replacing results of Theorem \ref{Lexp0} and Corollary \ref{ExCov} by the results of Theorem \ref{Lexp1} and Corollary \ref{ExCov1}, correspondingly. Then we conclude that the conditional distribution $(X_{1,\lambda}, X_{j,\lambda}) \mid \sum_{k=1}^N X_{k, \lambda}=1$ is related to the distribution of ${\bar X}$ described by ${\cal Z}'$ in Theorem \ref{Lexp1} in precisely same manner as for the circular case. This formal argument enables us to use approximation (\ref{Ea2}) but now indirectly for ${\cal H}_1^{-1}$, whose entries are proved to be close to the covariances, and the result of Theorem \ref{T2o} follows. $\Box$ \end{document}	arXiv
npg asia materials Disordered photonics behavior from terahertz to ultraviolet of a three-dimensional graphene network Study of the absorption coefficient of graphene-polymer composites K. Zeranska-Chudek, A. Lapinska, … M. 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Noginov Enhanced laser action from smart fabrics made with rollable hyperbolic metamaterials Hung-I Lin, Chun-Che Wang, … Yang-Fang Chen Luca Tomarchio1,2, Salvatore Macis1,3, Annalisa D'Arco ORCID: orcid.org/0000-0001-7990-51172,4, Sen Mou2, Antonio Grilli3, Martina Romani3, Mariangela Cestelli Guidi3, Kailong Hu5, Suresh Kukunuri6, Samuel Jeong ORCID: orcid.org/0000-0003-2231-72716, Augusto Marcelli3, Yoshikazu Ito6 & Stefano Lupi ORCID: orcid.org/0000-0001-7002-337X1,3 NPG Asia Materials volume 13, Article number: 73 (2021) Cite this article Nanoscience and technology Optical materials and structures The diffusion of light by random materials is a general phenomenon that appears in many different systems, spanning from colloidal suspension in liquid crystals to disordered metal sponges and paper composed of random fibers. Random scattering is also a key element behind mimicry of several animals, such as white beetles and chameleons. Here, random scattering is related to micro and nanosized spatial structures affecting a broad electromagnetic region. In this work, we have investigated how random scattering modulates the optical properties, from terahertz to ultraviolet light, of a novel functional material, i.e., a three-dimensional graphene (3D Graphene) network based on interconnected high-quality two-dimensional graphene layers. Here, random scattering generates a high-frequency pass-filter behavior. The optical properties of these graphene structures bridge the nanoworld into the macroscopic world, paving the way for their use in novel optoelectronic devices. Due to the advances of nanotechnology and the discovery of various 2D functional materials, in recent years there has been a growing interest for extending their extraordinary properties to three-dimensional (3D) ordered and disordered structures. This is particularly true for graphene, since its unique electrical1,2,3,4,5, thermal6, and optical properties7,8,9,10,11,12 can be extended into 3D structures by stacking high-quality monolayer graphene sheets into a wide variety of three-dimensional networks13,14,15. These 3D structures show performances beyond 2D graphene devices, ranging from supercapacitors16,17,18,19,20,21,22,23,24,25,26,27,28, lithium batteries29,30,31,32,33,34,35,36,37,38,39,40 and electrocatalysts41,42,43,44,45,46,47,48,49,50,51, to photodetectors52,53,54, biochemical applications55,56,57, and transistors58,59. Dirac plasmonic behaviors have also been observed in 3D nanoporous graphene. Plasmon absorption can be modulated by changing both the chemical doping and the pore size60. Moreover, through the photo/thermoacoustic effect, 3D Graphene sponges provide an efficient transduction of light into sound, covering a very broad acoustic emission range from infrasound (few Hz) to deep ultrasound (few MHz)61,62,63. Despite those broad applications, very little information is known about the light interaction with 3D graphene structures, where the network composed of pores and branches is expected to interact collectively with a broadband portion of the electromagnetic spectrum64. In this work, we measure the optical properties of undoped and nitrogen-doped 3D porous graphene networks over a broadband spectral range, going from Terahertz (THz) to ultraviolet (UV), demonstrating optical high-pass filtering of these structures mainly determined by random scattering effects. The transmission of light across a random network, composed of branches and pores, depends both on the 3D morphology of the system and on the optical properties of the supporting material. Variability in the pore size induces changes in the system connectivity, and thus in the propagation and scattering of light at various wavelengths. Through the study of the broadband transmission of graphene networks, we emphasize their emergent complex optical properties which can be extended to other 2D functional materials. In this paper, we measured two nanoporous graphene samples composed of high-quality single layer graphene65, both with an average pore size of ≈ 100(±50) µm. A first sample (S1) is undoped, while the second (S2) has been doped with nitrogen and shows a DC conductivity of approximately 70 S/cm, nearly a factor 140 larger than the undoped conductivity (0.5 S/cm). Both samples have a thickness of approximately 1 mm and are composed on average of 12 graphene layers. Their 3D structures were characterized by scanning electron microscopy (SEM) and optical microscopy. A third sample (S3) with an average pore size of ≈ 80(±50) µm is presented in the Supplementary Information (SI)65. Their 3D structures have been characterized by scanning electron microscopy (SEM) and optical microscopy. A macroscopic optical image of the undoped sample S1 is shown in Fig. 1a, while a SEM image is shown in Fig. 1b. The optical image of a branch is finally shown in Fig. 1c, where a roughness of hundreds of nanometers related to interconnected graphene layers can be observed. Similar structures are found for doped sample S2 (see Fig. S1, S2, S3, for a comparison)65. The broadband transmittance curves from terahertz (THz) to ultraviolet (UV) measured for the S1 and S2 samples are shown in Fig. 2 and in Fig. S4 for the S3 sample65. In the inset of Fig. 2 the transmittance is shown as a function of the wavelength. They show a high-frequency pass-filter behavior with a rapid increase in the transmittance from THz to mid-IR (MIR). Here, the transmittance reaches a saturation value corresponding to nearly 13% for both S1 and S2 samples. Below 1 THz, both curves converge to zero. The undoped S1 and doped S2 samples show a similar optical behavior, regardless of the layer conductivity, which varies from 0.5 S/cm (S1) to 70 S/cm (S2). Fig. 1: Magnification of the 3D graphene network morphology. a Visible image of the graphene 3D network (S1 sample). b SEM×150 magnified image of the 3D graphene S1 sample. The 3D network is composed by several polygonal structures connected to each other. c Image of a S1 single branch obtained through an optical JASCO NR5100 microscope. The picture has been reconstructed through a focus stacking process applied to 20 raw pictures and shows an intrinsic roughness of hundreds of nanometers related to interconnected graphene layers. d Integrated infrared transmittance between 2000 and 3000 cm−1 of a single polygon of the undoped S1 sample, as taken by a 64 × 64 pixel Focal Plane Array (FPA) MCT detector coupled to a Vertex 70 V Bruker interferometer. Fig. 2: Optical transmittance T(ω) of S1 and S2 3D graphene samples from THz to UV. T(ω) shows a high-pass-filter behavior, with a rapid increase from THz to MIR (nearly 5000 cm−1), and saturating to a constant value in the VIS and UV range (about 0.13 for both samples). The N-doped S2 sample does not show remarkable differences with respect to the undoped S1 one with the same average morphology. Transmittance data are shown in the inset as a function of wavelength. Given the equivalent 3D structures of S1 and S2, the broadband transmittance behavior (Fig. 2) suggests that their nano and microspatial morphology dominates the light-matter interaction through scattering processes, rather than their intrinsic absorption. This demonstrates, in our opinion, that the optical properties of 3D graphene samples are dominated by scattering, rather than by the intrinsic absorption in the graphene layers. This is also confirmed by the absence of absorptive features in the UV spectral range, where graphene is expected to host interband electronic transitions66,67. An explanation for the absence of UV absorption might come from the branch surface, as shown in Fig. 1c, where a surface roughness on a sub-wavelength scale suggests a pronounced scattering contribution, as already highlighted in similar works64. Moreover, in the integrated infrared transmittance between 2000 and 3000 cm−1 of a single polygon (Fig. 1d for the S1 sample), the integrated transmittance is nearly 1% in the branch regions, reaching values between 10 and 20% in the empty regions. A similar image is shown in SI65 for the S2 doped sample. The similarity between S1 and S2 further confirms that doping has a marginal role in the optical properties of the 3D graphene network. To understand the role of scattering in the optical transmittance behavior, we developed goniophotometric measurements (see "Methods" and Fig. S5 in SI65 for a technical discussion) for two different wavelengths: 780 nm (~13,000 cm−1), falling in the broadband transmittance plateau, and at 7 μm (~1400 cm−1), on the transmittance rising edge (see Fig. 2). Here, one measures the scattered light intensity in the far-field at a fixed distance (30 cm) with respect to the sample position (see Fig. 3a), scanning different horizontal (θ) and vertical (Φ) angular positions. As scattering shows a cylindrical symmetry (θ=Φ), in the following we will discuss only the horizontal θ dependence. A 0.5 mm pinhole located in front of the detector provides a 0.1° sampling of θ. Fig. 3: Goniphotometric mesurements at 780 nm and 7μm. a Detection geometry of the goniophotometric data. θ is the horizontal scattering angle and Φ the vertical one. Due to the cylindrical symmetry (θ=Φ), only the horizontal angle (θ) has been scanned with a sensitivity of 0.1°. b Incident and transmitted intensities of S1 sample at different θ angles for the 780 nm (blue and red stars) and 7 μm (light blue and yellow diamonds) wavelengths. c Integrated transmittance T(θ,λ) for the S1 sample as a function of the scattering angle θ, as computed from Eq. 1 for 780 nm (light blue squares) and 7 μm (violet squares) wavelengths, respectively. T(θ,λ) values converge rapidly to the transmittance as measured in Fig. 2, supporting the idea that scattering is the key process in the light-3D Graphene interaction. d Fraction of scattered light for the S1 sample for both 780 nm (light blue squares) and 7 μm (violet squares) wavelengths, as computed from Eq. 2. Figure 3b shows the θ dependent scattered intensities for the incident and transmitted beams of S1 sample, normalized to the zero angle incident signal. Similar results were measured for the S2 samples. Incident intensities at 7 μm (light blue diamonds) and 780 nm (blue stars) can be described by the same Gaussian function and its angle distribution reflects the intrinsic laser beam divergence. The transmitted intensities are instead distributed at larger angles, suggesting a valuable contribution from scattering. Indeed, above nearly 1.5° angle, the 7 μm transmitted intensity (yellow diamonds) reaches a higher plateau of approximately one order of magnitude larger than the 780 nm signal (red stars). This suggests an increasing role of scattering with decreasing frequency. To link the transmittance values at 780 nm and 7 µm as shown in Fig. 2 to the goniophotometric data (Fig. 3b), we need to integrate the latter over the horizontal and vertical scattering angles. Given the scattering cylindrical symmetry (θ=Φ), the integrated transmittance T(θ,λ), up to an angle θ and as a function of wavelength λ, will take the form $${{{{T}}}}\left( {{\uptheta }},{\uplambda} \right) = \left[ {\mathop {\int }\limits_{S\left( {{\Theta }} \right)} {{{{I}}}}_T\left( {s,{\uplambda}} \right){\rm{d}}s} \right]\Bigg/\left[ {\mathop {\int }\limits_{S\left( {{\Theta }} \right)} I_0\left( {{{{{s}}}},{\uplambda}} \right){\rm{d}}s} \right]$$ where IT and I0 are, respectively, the transmitted and incident intensities as a function of wavelength and at a specific surface section s (identified by the same angular deviation), and the integral spans over the surface ${S({\Theta}})$ scanned by the detector (Fig. 3a). The values for T(θ,λ) are shown in Fig. 3c, where the integrated transmittance values for the 780 nm and 7 µm wavelengths converge to the corresponding results of Fig. 2 at approximately 5°. This angle is in good agreement with the collection angle of the Bruker Vertex interferometer (~5°) used for acquiring the transmittance data of Fig. 2. From the data of Fig. 3b, one can extract the fraction of scattered light Is vs. θ as65 $${{{{I}}}}_s\left( \theta \right) = I_T\left( \theta \right) - I_0\left( \theta \right)T\left( {\theta = 0} \right)$$ $$T\left( {\theta = 0} \right) = \frac{{{{{{I}}}}_T(\theta = 0)}}{{{{{{I}}}}_0\left( {\theta = 0} \right)}}$$ is the transmittance at normal incidence used as a reference value for the unscattered light across the beam waist. The result for the scattered intensity is shown in Fig. 3d, after being normalized by the total transmitted intensity IT at any angle. This figure suggests a rapid increase in the scattering contribution to the angle-resolved transmittance, reaching its peak value (100% of scattered light) at nearly 1° angle. The slightly more rapid increase at 7 µm, with respect to 780 nm, suggests a stronger scattering effect at higher wavelengths, particularly toward the THz range. Goniophotometric results highlight how scattering plays the major role in setting the transmittance value at both 780 nm and 7 µm. The nearly one order of magnitude difference in the scattered wavelengths suggests a spatial multisize scattering effect in the 3D graphene sample. These spatial scales can be recognized (see Fig. 1b, c) in the pore structures (tens to hundreds of microns), which act as scattering centers for THz and far-infrared (FIR) radiation, and branch surface roughness (hundreds of nanometers) which probably plays the major role in the transmittance at infrared and visible wavelengths. This multisize nature of the scatterers should translate into a strong wavelength dependence of the photon scattering mean free path (ls). To estimate ls, we describe graphene samples as random medium slabs where light propagates between the opposite faces through a combination of scattering and absorption processes. Due to multiple scattering processes, one expects interference effects to be negligible inside the materials, and the semi-classical theory of radiative transfer68,69,70,71 can be used. Transmittance of a disordered slab The theory of radiative transfer68,69 describes a medium where the free propagation of energy is randomly interrupted by scattering processes, changing the propagation direction and causing loss of energy. In this theory the central quantity is the light intensity $I(r,t,\hat s)$, being the flux per unit solid angle at position r and time t, in the direction $\hat s$. In a macroscopic isotropic medium such as 3D graphene samples investigated here, scattering structures such as pores and interconnected layers (see Fig. 1b, c) have equal probability to diffuse light forward or backward the initial propagation direction, as shown by the goniophotometric data presented in SI65. By taking this direction to be along z (see Fig. 4a) and considering the graphene network as a slab between the planes z = 0 and z = L (see Fig. 4a), the radiative transfer equation for stationary monochromatic light at position z and with a wavevector direction $\hat s$, describing the evolution of the intensity due to scattering and absorption, takes the form68 $$l^\prime {\upmu}\frac{{\partial I}}{{\partial z}} = - I\left( {z,{\upmu}} \right) + \frac{{l^\prime }}{l}\bar I\left( z \right)$$ where ${{{{l}}}}^\prime = {{{{l}}}}_{{{{s}}}}{{{{l}}}}_{{{{a}}}}/\left( {{{{{l}}}}_{{{{s}}}} + {{{{l}}}}_{{{{a}}}}} \right)$ (ls is the mean free path for scattering and la the absorption length), μ ≡ cosθ, with θ being the angle between $\hat s$ and the z axis, and $\bar I\left( z \right) = \frac{1}{2}\mathop {\int }\limits_{ - 1}^1 {\rm{d}}{\upmu}I\left( {z,{\upmu}} \right)$. Assuming the reflectance at the slab interfaces to be zero, the boundary conditions for μ > 0 become $$\begin{array}{*{20}{c}} {I\left( {0,{\upmu}} \right) = I_0\left( {\upmu} \right),} & {I\left( {L, - {\upmu}} \right) = 0} \end{array}$$ where I0 is the incident intensity. As described in SI65, the solutions to this differential problem can be simplified by keeping only the first expansion terms in the multiscattering processes. In particular, the transmittance along the forward direction (the physical quantity we actually measure), $T\left( {L,1} \right) = I\left( {L,1} \right)/I_0$, of the zero and two-scattering processes is $$T\left( {L,1} \right) = {\rm{e}}^{ - L/l_a}{{{\rm{e}}}}^{ - L/l_s}\left[ {1 + \frac{L}{{2l_s}} + \ldots } \right]$$ This equation can be further simplified by assuming that absorption can be neglected. Indeed, due to the similarity of the measured transmittance between undoped S1 and doped S2 samples (see Fig. 2), one expects that most of the light in these samples is diffused. Thus, the ${\rm{e}}^{ - L/l_a}$ term can be omitted by taking la → ∞, i.e., zero absorption. As can be seen in Eq. 5, the remaining term $L/2l_s \cdot {\rm{e}}^{ - L/l}$ describes the contribution to transmittance coming from the two-scattering paths, which adds to the unscattered photon element ${\rm{e}}^{ - L/l_s}$ (Fig. 4a). From the measured transmittance of the undoped sample S1 (similar results can be obtained from S2), one can extract the fundamental scattering parameter, i.e., the scattering mean free path ls. Its value can be obtained by inverting numerically Eq. 5 up to the third order. The result is shown in Fig. 4b where ls is plotted from UV (200 nm) to THz (1 cm). While ls is nearly flat below 1 µm, indicating that most of VIS and UV photons are scarcely scattered, it strongly decreases approaching the infrared and terahertz region. Here, scattering processes by the network pores and branches play a major role, inducing a progressive opaqueness in agreement with the transmittance data. Fig. 4: Radiative transfer theory of a 3D graphene network. a The theory of radiative transfer can be used to describe a disordered medium with reflective boundaries and bulk absorption and scattering processes (left). For a graphene 3D network, the former contribution is zero, while the absorptive processes can be neglected due to propagation in air (right). The remaining scattering contribution can be obtained through an expansion in the number of scattering processes. The first two contributions will take in mind the transmission of non-scattering photons and the two-scattering processes. For the incoherent experimental transmission, the only processes considered are the ones that give raise to photons propagating in the same direction of the incident radiation. b Mean free path as a function of wavelength for the sample S1, as predicted by the radiative transfer model of Eq. 5. The radiative transfer model was validated by an independent experiment. Indeed, through goniophotometric measurements at a wavelength of 780 nm, extended up to a scattering angle of 170° and combined with a Monte Carlo analysis (for the complete discussion see SI65), we calculate the scattering length at the same wavelength. The result is shown in Fig. 4b by a black dot, which is in very good agreement with the estimation (blue curve) based on the radiative transfer equation. In this work, for the first time, we have measured the optical properties of technologically important 3D graphene networks, whose outstanding physical and chemical properties find applications in many fields of basic and applied research13,14,15,54,60,61,62. We have shown how the connectivity and morphology of these materials allow a broadband interaction with light, modifying the transmitted intensity from terahertz to ultraviolet. In particular, the 3D graphene networks behave like a high-pass optical filter due to spatially multiscale random scatterers, corresponding to pores and graphene branches in the 3D network. We have developed a model based on the radiative transfer theory describing the interaction of the network with light, from which we have estimated the photon scattering mean free path. These experimental results prove that 3D graphene represents a novel disordered photonic material sustaining elastic scattering, analogous to some nanofibrils found in biological systems72,73. In this general context, 3D graphene networks open new possibilities to study complex light-matter interaction in random materials. Indeed, analogous to disordered systems appearing in nature72, new fabrication processes may produce graphene 3D structures going beyond the Gaussian random scattering paradigm, diffusing light at selected wavelengths or angular distributions, and paving the way for applications in several research and industrial fields. Synthesis of porous graphene and N-doped porous graphene The 3D graphene structures were prepared through a porous Ni-based chemical vapor deposition (CVD) growing technique as reported in the work of Ito et al41,42,47,74. Porous Ni was loaded into a quartz tube (external diameter 26 mm × internal diameter 22 mm × 250 mm length) as an inner tube which was further inserted into the center of a larger quartz tube (external diameter 30 mm × internal diameter 27 mm × 1000 mm length) in the open-box-type furnace. Ni was annealed at 1000 °C for 10 min to clean the Ni surface under a mixed atmosphere of H2 (100 sccm) and Ar (200 sccm). Subsequently, graphene/chemically doped graphene was grown at 800 °C for 10 min under a mixed atmosphere of H2 (100 sccm), Ar (200 sccm), benzene (0.1 mbar, 99.8%, anhydrous) and/or pyridine (0.2 mbar, 99.8%, anhydrous) as graphene precursors. The furnace was quickly opened and the inner and outer quartz tubes were cooled with a fan down to room temperature. The Ni substrates were dissolved in a 1.0 M HCl solution for four days and then etched at 50 °C in a 1.5 M HCl solution for 1 day. The resulting graphenes on the solution were repeatedly exchanged with pure water five times and kept on the water for 2 days. Then, they were transferred into 2-propanol for 1 week to dry them with a supercritical CO2 fluid drying method75,76. To prevent the collapse and damage of the fragile 3D curved structures, caused by the capillary force of water during the drying process, the graphene samples were immersed in 2-propanol and dried using supercritical CO2 (scCO2), in order to substitute 2-propanol with scCO2 without the capillary force taking place. The samples were first transferred to a glass bottle (volume: 5 mL) filled with 2-propanol (400 µL), which was then placed in an 80 mL pressure-resistant container (TAIATSU techno Corp). After removing the air inside the container through gaseous CO2 purging, the pressure of the container was gradually increased to 15 MPa by introducing liquid CO2 (5 MPa, 4 °C, density of 0.964 g/mL) at a flow rate of 20 mL/min (19 g/min) using a high-pressure plunger pump (NIHON SEIMITSU KAGAKU Co. Ltd, NP-KX-540). The scCO2 drying process was carried out at 70 °C, with a constant CO2 flow rate of 5 mL/min (4.8 g/min), by forming a homogeneous phase of 2-propanol and scCO2 to minimize the capillary force. The pressure was maintained at 15 MPa during the drying procedure for at least 5 h. Once completely dried, the temperature of the system was set at 40 °C and it was gradually depressurized over 43 h, from 15 MPa down to atmospheric pressure, by gradually venting CO2 from the system. In this paper, we measured two nanoporous graphene samples, both with an average pore size of ≈100 µm. The first sample (S1) is undoped, while the second (S2) is doped with nitrogen (see SI65 for Raman characterization). The samples have a thickness of approximately 1 mm and are composed on average of 12 graphene layers. The 3D structures were characterized by scanning electron microscopy (SEM) and optical microscopy. The SEM images were collected with a MINI-SEM SNE-3200M, operating at an accelerating voltage of 20 kV and with a magnification factor of ×150. Optical images of 3D graphene branches (Fig. 1c), at ×150 magnification, have instead been acquired through a JASCO NR5100 microscope. The graphene resistivities were measured by a standard four-probe method using a semiconductor parameter analyzer (Keysight B1500A) at room temperature. The sample was placed on a SiO2/Si (SiO2: 300 nm) substrate. The electrode was fabricated using Cu wires and Ag epoxy (H20E, Epoxy Technology). S1 shows a DC conductivity of approximately 0.5 S/cm, while S2 of approximately 70 S/cm, which was nearly a factor 140 larger than the undoped value. Chemical analyses of the graphene samples were conducted by using an X-ray photoelectron spectroscopy (XPS, AXIS ultra DLD, Shimazu) with Al Ka using an X- ray monochromator (see SI65). The samples were pasted on a sample holder without any supports. Raman spectra were recorded by using a micro-Raman spectrometer (Renishaw InVia Reflex 532) with an incident wavelength of 532.5 nm. All samples show a similar Raman spectrum as shown in Fig. S2, S4 whose corresponding frequencies are reported in Table S1. The laser power was set at 0.1 mW to avoid possible damage or unexpected reduction by laser irradiation. The transmission of light through the 3D graphene samples was measured using several different spectroscopic setups to span a very broad frequency range, from THz to UV. Transmittance T(ω) measurements, from 1 to 70 cm−1, was performed through a THz time-domain spectrometer (THz-TDS) based on photo-conductive (Hamamatsu Photonics) antennas. From 50 to 5000 cm−1, instead, a Vertex 70 v FTIR broadband interferometer was used, while the interval from 3000 to 50000 cm−1 has been spanned through a JASCO v-770 IR-UV spectrometer. Data obtained from these measurements were merged to obtain a single broadband transmittance curve, one for each sample (Fig. 2). Moreover, the transmittance of a single polygon of the 3D network was also analyzed through a Vertex70 V FT-IR interferometer coupled with an Hyperion3000 IR microscope equipped with a 64x64 pixel Focal Plane Array MCT detector. A goniophotometric technique has been used for measuring the light scattered at different angles with respect to the incident light. In this work, we present scattering data at two different wavelengths, 780 nm and 7 µm, obtained through a TOPTICA FemtoFiber Pro and a collinear difference frequency generator (DFG) from Light Conversion, respectively. Laser beams of comparable power (few mW), scattered across the graphene samples, are collected at various angles after being filtered by a 0.5 mm radius metallic pinhole at the detection point. The detection setup is based on an NIR photodiode (Thorlabs) and a Gentec pyroelectric, for 780 nm and 7 µm sources, respectively. The distance between the sample and the detector is 30 cm at every angle. With this geometry, an angle resolution better than 0.1° is possible. The measurements were thus taken with a minimum scan step of 0.1°. The laser repetition rate and the detected signal are sent to a lock-in amplifier to increase the signal-to-noise ratio and dynamic range of the measurements. Geim, A. K. & Novoselov, K. S. The rise of graphene. Nat. Mater. 6, 183–191 (2007). 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Supplementary Information of the article: Disordered Photonics Behavior from Terahertz to Ultraviolet of a 3-dimensional Graphene Network. (https://doi.org/10.1038/s41427-021-00341-9) Mak, K. F., Ju, L., Wang, F. & Heinz, T. F. Optical spectroscopy of graphene: From the far infrared to the ultraviolet. Solid State Commun. 152, 1341–1349 (2012). Mogera, U. & Kulkarni, G. U. Twisted multilayer graphene exhibiting strong absorption bands induced by van Hove Singularities. Bull. Mater. Sci. 41, 130 (2018). Chandrasekhar, S. Radiative Transfer (Dover,1960). Ishimaru, A. Wave Propagation and Scattering in Random Media (Academic Press, 1978). Nguyen, V. D., Faber, D. J., van der Pol, E., van Leeuwen, T. G. & Kalkman, J. Dependent and multiple scattering in transmission and backscattering optical coherence tomography. Opt. Express 21, 29145 (2013). Fujii, H. et al. Photon transport model for dense poly-disperse colloidal suspensions using the radiative transfer equation combined with the dependent scattering theory. Opt. Express 28, 22962 (2020). Wiersma, D. S. Disordered Photonics. Nat. Photonics 7, 188–196 (2013). Lee, S. H., Han, S. M. & Han, S. E. Anisotropic diffusion in Cyphochilus white beetle scales. APL Photonics 5, 056103 (2020). Ito, Y. et al. High-quality three-dimensional nanoporous graphene. Angew. Chem. Int. Ed. 53, 4822–4826 (2014). Adschiri, T., Hakuta, Y. & Arai, K. Hydrothermal synthesis of metal oxide fine particles at supercritical conditions. Ind. Eng. Chem. Res. 39, 4901–4907 (2000). Sue, K. et al. Synthesis of Ni nanoparticles by reduction of NiO prepared with a flow-through supercritical water method. Chem. Lett. 35, 960–961 (2006). The authors thank Prof. Liangti Qu and Prof. Zhipan Zhang at the Beijing Institute of Technology, China, for the useful discussion about 3D Graphene properties. We thank also Ms. Kazuyo Omura at the Institute for Material Research of Tohoku University, Japan, for XPS measurements. This work has received the financial support of the Bilateral Cooperation Agreement between Italy and China of the Italian Ministry of Foreign Affairs and of the International Cooperation (MAECI) and the National Natural Science Foundation of China (NSFC), in the framework of the project of major relevance 3-Dimensional Graphene: Applications in Catalysis, Photoacoustics and Plasmonics. This work has also been sponsored by JSPS Grant-in-Aid for Scientific Research on Innovative Areas "Discrete Geometric Analysis for Materials Design" (Grant Numbers JP18H04477, JP20H04628), JSPS KAKENHI Grant Number JP19K22226 and a cooperative program (Proposal No. 202011-CRKEQ-0001) of CRDAMIMR, Tohoku University. Department of Physics, Sapienza University, Piazzale Aldo Moro 5, 00185, Rome, Italy Luca Tomarchio, Salvatore Macis & Stefano Lupi INFN section of Rome, P.Le Aldo Moro, 2, 00185, Rome, Italy Luca Tomarchio, Annalisa D'Arco & Sen Mou INFN - Laboratori Nazionali di Frascati, via Enrico Fermi 54, 00044, Frascati (Rome), Italy Salvatore Macis, Antonio Grilli, Martina Romani, Mariangela Cestelli Guidi, Augusto Marcelli & Stefano Lupi SBAI Department, Sapienza University, Via Scarpa 16, 00161, Rome, Italy Annalisa D'Arco School of Materials Science and Engineering, and Institute of Materials Genome & Big Data, Harbin Institute of Technology, 518055, Shenzhen, P. R. China Kailong Hu Institute of Applied Physics, Graduate School of Pure and Applied Science, University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan Suresh Kukunuri, Samuel Jeong & Yoshikazu Ito Luca Tomarchio Salvatore Macis Sen Mou Antonio Grilli Martina Romani Mariangela Cestelli Guidi Suresh Kukunuri Samuel Jeong Augusto Marcelli Yoshikazu Ito Stefano Lupi Correspondence to Stefano Lupi. Supplementary Information File Tomarchio, L., Macis, S., D'Arco, A. et al. Disordered photonics behavior from terahertz to ultraviolet of a three-dimensional graphene network. NPG Asia Mater 13, 73 (2021). https://doi.org/10.1038/s41427-021-00341-9 Revised: 15 September 2021 NPG Asia Materials (2021) For Authors & Referees NPG Asia Materials (NPG Asia Mater) ISSN 1884-4057 (online) ISSN 1884-4049 (print)	CommonCrawl
Let $S \to X$ be an $S^3$-fiber bundle over a smooth manifold $X$. If $S$ is an oriented manifold does this fiber bundle admit the structure of an $SU(2)$-principal bundle? There is a similar theorem for the case of circle bundles and is proved in Morita's book on differential forms. Unfortunately, I do not see a way to extend his argument to this case and he does not discuss $SU(2)$-principal bundles in detail. I should explain why the nontrivial vector bundle has nontrivial unit sphere bundle, that is, why $\pi_1 SO(4)$ injects into $\pi_1 SDiff(S^3)$. You can use the big theorem of Hatcher ("Smale Conjecture"), which says that $\pi_k SO(4)$ maps isomorphically to $\pi_k SDiff(S^3)$ for all $k$. Alternatively, you can use that the bundle is detected by the Stiefel-Whitney class $w_2$, and that Stiefel-Whitney classes of vector bundles are invariant under fiber homotopy equivalence of unit sphere bundle. Or you can use $\pi_3$ instead of $\pi_1$ as suggested by Dylan Wilson, making a bundle over $S^4$; there are elements of $\pi_3 SO(4)\cong \mathbb Z\times \mathbb Z$ not coming from $\pi_3 SU(2)\cong\mathbb Z$. Again, the resulting sphere bundle is nontrivial either by Hatcher's theorem or by using characteristic classes. I'm not sure what the most elementary version of the characteristic-class argument would be. One example of a rank $4$ real vector bundle over $S^4$ that does not admit a complex structure is the tangent bundle! Not the answer you're looking for? Browse other questions tagged dg.differential-geometry at.algebraic-topology differential-topology principal-bundles orientation or ask your own question. How are fiber bundles, transition functions and principal bundles related? Non-Existence of a Principal Connection for the Sphere over Projective Space? What does reduction of structure group of principal bundle say?	CommonCrawl
Nicole M. Joseph Nicole Michelle Joseph is an American mathematician and scholar of mathematics education whose research particularly focuses on the experiences of African-American girls and women in mathematics, on the effects of white supremacist reactions to their work in mathematics, and on the "intersectional nature of educational inequity".[1] She is an associate professor of mathematics education, in the Department of Teaching and Learning of the Vanderbilt Peabody College of Education and Human Development.[2] Education and career Joseph is African American, and is originally from Seattle. After a fall-out with a racist teacher in her elementary school, she was moved to the only open class, an advanced and self-paced classroom in which she first developed a love for mathematics.[3] She majored in economics, with a minor in mathematics, at Seattle University, where she earned a bachelor's degree in business administration in 1993.[4] After "a few years in the business world",[3] she began working in the Seattle area as a middle school and elementary school mathematics teacher, and as a mathematics coach, from 1999 to 2011. During this period she also studied at Pacific Oaks College Northwest, a former Seattle satellite campus of Pacific Oaks College, a private Quaker college in California. Through Pacific Oaks, she earned a teaching certification for Washington in 2000, and a master's degree in human development in 2003.[4] In 2011, Joseph completed a Ph.D. in Curriculum & Instruction at the University of Washington. Her dissertation, Black Students and Mathematics Achievement: A Mixed-Method Analysis of In-School and Out-of-School Factors Shaping Student Success, was supervised by James A. Banks. In the same year, she earned a national certification in adolescent mathematics teaching through the National Board for Professional Teaching Standards.[4] After completing her doctorate, Joseph joined the University of Denver in 2011 as an assistant professor, focusing on educating future mathematics teachers. She moved to Vanderbilt University in 2016, and was tenured there as an associate professor in 2021.[4] Books Joseph is the author or editor of books including: • Interrogating Whiteness and Relinquishing Power: White Faculty's Commitment to Racial Consciousness in STEM Classrooms (edited with C. M. Haynes and F. Cobb, Peter Lang Publishers, 2016) • Understanding the Intersections of Race, Gender, and Gifted Education: An Anthology by and About Talented Black Girls and Women in STEM (edited, Information Age Publishing, 2020) • Making Black Girls Count in Math: A Black Feminist Vision of Transformative Teaching (Harvard Education Press, 2022) Recognition Joseph was the winner of the 2023 Louise Hay Award of the Association for Women in Mathematics, "recognized for her contributions to mathematics education that reflect the values of taking risks and nurturing students’ academic talent".[1][5] References 1. "2023 Winner: Nicole Joseph", Louise Hay Award, Association for Women in Mathematics, retrieved 2023-04-05 2. "Nicole M. Joseph", Faculty profile, Vanderbilt Peabody College of Education and Human Development, retrieved 2023-04-05 3. "Nicole Michelle Joseph", Black History Month 2021 Honoree, Mathematically Gifted & Black, retrieved 2023-04-05 4. Curriculum vitae (PDF), retrieved 2023-04-05 5. "Nicole Joseph Honored for Her Work to Increase Opportunities for Black Girls in Mathematics", The Journal of Blacks in Higher Education, 23 December 2022, retrieved 2023-04-05 External links • Home page • Nicole M. Joseph publications indexed by Google Scholar Authority control: Academics • ORCID	Wikipedia
Tuning electronic properties of transition-metal dichalcogenides via defect charge Spectroscopic studies of atomic defects and bandgap renormalization in semiconducting monolayer transition metal dichalcogenides Tae Young Jeong, Hakseong Kim, … Suyong Jung Experimental and theoretical studies of native deep-level defects in transition metal dichalcogenides Jun Young Kim, Łukasz Gelczuk, … Izabela Szlufarska Anisotropic band splitting in monolayer NbSe2: implications for superconductivity and charge density wave Yuki Nakata, Katsuaki Sugawara, … Takafumi Sato Antisite defect qubits in monolayer transition metal dichalcogenides Jeng-Yuan Tsai, Jinbo Pan, … Qimin Yan Thickness and defect dependent electronic, optical and thermoelectric features of $$\hbox {WTe}_2$$ WTe 2 Ilkay Ozdemir, Alexander W. Holleitner, … Olcay Üzengi Aktürk Momentum-space indirect interlayer excitons in transition-metal dichalcogenide van der Waals heterostructures Jens Kunstmann, Fabian Mooshammer, … Tobias Korn Thickness-modulated metal-to-semiconductor transformation in a transition metal dichalcogenide Alberto Ciarrocchi, Ahmet Avsar, … Andras Kis Ambipolar Landau levels and strong band-selective carrier interactions in monolayer WSe2 Martin V. Gustafsson, Matthew Yankowitz, … Cory R. Dean Prediction of protected band edge states and dielectric tunable quasiparticle and excitonic properties of monolayer MoSi2N4 Yabei Wu, Zhao Tang, … Peihong Zhang Martik Aghajanian1 na1, Arash A. Mostofi ORCID: orcid.org/0000-0002-6883-82781 na1 & Johannes Lischner1 na1 Scientific Reports volume 8, Article number: 13611 (2018) Cite this article Electronic properties and devices Electronic properties and materials Two-dimensional materials Defect engineering is a promising route for controlling the electronic properties of monolayer transition-metal dichalcogenide (TMD) materials. Here, we demonstrate that the electronic structure of MoS2 depends sensitively on the defect charge, both its sign and magnitude. In particular, we study shallow bound states induced by charged defects using large-scale tight-binding simulations with screened defect potentials and observe qualitative changes in the orbital character of the lowest lying impurity states as function of the impurity charge. To gain further insights, we analyze the competition of impurity states originating from different valleys of the TMD band structure using effective mass theory and find that impurity state binding energies are controlled by the effective mass of the corresponding valley, but with significant deviations from hydrogenic behaviour due to unconventional screening of the defect potential. Since the discovery of graphene, there has been significant interest in the development of ultrathin devices based on two-dimensional (2D) materials. In contrast to graphene, which is a semimetal when undoped, monolayer transition-metal dichalcogenides (TMDs) with the chemical formula MX2 (M = Mo, W; X = S, Se, Te) are semiconductors with a direct band gap1,2. Monolayer TMDs have been used as channel materials in field-effect transistors3,4 and microprocessors5, as well as absorbers in solar cells6 and as sensors7,8, with promising results. Defects play a critical role in the performance of devices under realistic conditions9,10,11. Analogously to conventional bulk semiconductors, impurities with shallow donor or acceptor states can be used to control the carrier concentration in TMDs via defect engineering12,13. Adsorbed atoms and molecules are a particularly promising class of impurities in TMDs as they tend to only weakly perturb the atomic structure of the TMD substrate, thereby limiting any degradation of carrier mobility that may result from impurity scattering or trapping14,15, and experimental fabrication of adsorbate-engineered samples is straightforward16. A detailed theoretical understanding of the properties of charged adsorbates on TMDs is important to enable the rational design of new devices. On the one hand, many groups have used ab initio density-functional theory (DFT) to study the interaction of adsorbed atoms and molecules with TMDs. Such calculations yield important material-specific insights about adsorption geometries, adsorbate binding energies and charge transfer17,18,19,20,21. However, ab initio calculations are limited in terms of the size of the systems that can be considered (typically containing up to several hundred or a few thousand atoms), which are much too small to describe properties of shallow defect states that can extend up 100 Ångstrom (Å) or more, as has been observed recently for Coulomb impurities in graphene using scanning tunnelling spectroscopy (STS)22. On the other hand, continuum electronic structure methods, such as Dirac theory for graphene or effective mass theory for bulk semiconductors, can describe the behaviour of extended impurity states, but require parameters from experiments or ab initio calculations, such as Fermi velocities, effective masses23,24,25,26,27 and rather importantly, the defect potential that is typically screened by electrons of the host material. In this paper, we study properties of shallow impurity states induced by charged adatoms on monolayer MoS2. Using large-scale tight-binding models and screened defect potentials calculated from ab initio dielectric functions, we reveal a surprising diversity of bound defect states resulting from the unconventional screening present in reduced-dimensional materials and the interplay between multiple valleys in the TMD band structure. We present results for impurity wavefunctions and binding energies as function of the impurity charge and also compute the local density of states (LDOS) in the vicinity of the adatom, which can be measured in STS experiments. For both donor and acceptor impurities, we find that impurity wavefunctions have similar nodal structure to 2D hydrogenic states, but with radii that lie on the nanoscale. We find that that the orbital character of the most strongly bound impurity state switches as a function of the impurity charge strength Z due to the different effective masses associated with different valleys in the monolayer TMD band structure. We compare our results to the 2D hydrogen atom and also to effective mass theory calculations and discuss the limitations of these continuum models. Whilst an approach based on the effective mass model is able to describe some of the general behaviour with reasonable accuracy, we find significant discrepancies from our tight-binding model which arise from short-range features of the defect potential. Our calculations demonstrate the potential of adsorbate engineering for ultrathin devices based on TMDs and the importance of first-principles based description of their properties. To describe the electronic structure of the MoS2 monolayer, we employ the three-band tight-binding (TB) model by Liu et al.28. This model uses a basis of transition-metal $4{d}_{{z}^{2}}$, 4dxy and $4{d}_{{x}^{2}-{y}^{2}}$ orbitals which give the dominant contribution to the states near the conduction and valence band extrema and includes hoppings up to third-nearest neighbours as well as spin-orbit interactions. The various parameters were determined by fits to DFT band structures. The charged adatom is described as a point charge Q = Ze (with e being the proton charge) located a distance d above the plane of the transition-metal atoms. The charge gives rise to a screened potential in the TMD sheet. Within linear response theory, the screened potential is given by $$V(\rho ;\,Z,d)=Z{e}^{2}{\int }_{0}^{\infty }{\rm{d}}q\,{\varepsilon }_{{\rm{2D}}}^{-1}(q){J}_{0}(q\rho ){e}^{-qd},$$ where ρ denotes the in-plane distance from the adatom and ${\varepsilon }_{{\rm{2D}}}^{-1}(q)$ is the inverse 2D dielectric function of a single TMD monolayer. The 2D dielectric function can be obtained from the inverse dielectric matrix ${\varepsilon }_{{\bf{GG}}^{\prime} }^{-1}({\bf{q}})$ of an infinite system of stacked TMD sheets (simulated in an electronic structure calculation that employs periodic boundary conditions) via29 $${\varepsilon }_{{\rm{2D}}}^{-1}({\bf{q}})=\frac{q}{2\pi {e}^{2}{L}_{z}}\sum _{{{\bf{G}}}_{z}{{\bf{G}}}_{z}^{^{\prime} }}\,{\varepsilon }_{{{\bf{G}}}_{z}{{\bf{G}}}_{z}^{^{\prime} }}^{-1}({\bf{q}}){v}_{{\rm{trunc}}}(\|{\bf{q}}+{{\bf{G}}}_{z}^{^{\prime} }\|).$$ Here, Gz and ${{\bf{G}}{\boldsymbol{^{\prime} }}}_{z}$ denote reciprocal lattice vectors along the out-of-plane (z) direction, vtrunc is a slab-truncated Coulomb interaction30 and Lz denotes the distance between the stacked sheets. The inverse dielectric matrix is computed for a MoS2 monolayer using the random-phase approximation31 (RPA) with Kohn-Sham wave functions and energies from ab initio DFT (see Supplementary Materials for details). Calculations were carried out using the Quantum Espresso32 and BerkeleyGW software packages33. For small wave vectors, which are relevant for describing shallow impurity impurity bound states, we find that the right hand side of Eq. (2) depends only on the magnitude of the wave vector.Figure 1 shows the screened (calculated from Eq. 1) and unscreened potentials of a charged adatom with Z = 1 and d = 2 Å above the Mo-layer in the MoS2 sheet. While there are clear differences at short distances, the two potentials both converge to the unscreened case at long distances from the adatom which is characteristic of screening in 2D semiconductors. This short-range discrepancy corresponds to significant differences between the Fourier transforms of these potentials at large wavevectors, shown in the inset of Fig. 1. RPA-screened potential of a charged adatom situated d = 2 Å above the Mo-atom in MoS2 with strength Z = 1 (blue solid curve) compared to the unscreened Coulomb potential (red dashed curve). We also compare this to the Keldysh model (green curve) for Z = 1, d = 2 Å and screening length ρ0 = 45 Å (Eq. 5), fitted to the RPA-screened potential. The inset shows the Fourier transform of the screened and unscreened potentials, as well as the potential screened in the Keldysh model, with the solid vertical line indicating \|K − K′\|, the separation in reciprocal space between the two valleys of MoS2. To study shallow bound states of the screened adatom potential, we construct a 51 × 51 TMD supercell containing 7803 atoms and diagonalize the resulting TB Hamiltonian with the adatom potential as an on-site term22,23. Note that the adatom is placed above a transition-metal site as this is the preferred adsorption geometry for many adatom species, such as alkali metals17,18,19. To analyze the results of our atomistic tight-binding simulations, we have also carried out calculations using effective mass theory. In this approach, which has been used routinely to study shallow bound states of charged impurities in bulk semiconductors25,34,35, the impurity states are expressed as ${{\rm{\Psi }}}_{n\nu }({\bf{r}})=\int {\rm{d}}{\bf{k}}\,{\varphi }_{n\nu }({\bf{k}}){\psi }_{n{\bf{k}}}({\bf{r}})$. Here, ψnk denotes an unperturbed Bloch state with band index n and crystal momentum k of the host material and ϕnv(k) is an envelope function determined by36 $${\varepsilon }_{n{\bf{k}}}{\varphi }_{n\nu }({\bf{k}})+\int {\rm{d}}{\bf{k}}^{\prime} \,\langle {\psi }_{n{\bf{k}}}\|V\|{\psi }_{n{\bf{k}}^{\prime} }\rangle {\varphi }_{n\nu }({\bf{k}}^{\prime} )={E}_{n\nu }{\varphi }_{n\nu }({\bf{k}}),$$ where εnk describes the band structure of the host material and V(r) denotes the screened impurity potential. In bulk semiconductors, V can be accurately approximated36 by Ze2ε−1(q = 0)/r and the resulting equation for the impurity state envelope function reduces to the Schrödinger equation of a hydrogen atom with a reduced Bohr radius ${\tilde{a}}_{0}=({m}^{\ast }/{m}_{0})Z{a}_{0}{\varepsilon }^{-1}(q=0)$ (with m* and m0 denoting the effective and bare mass of the electron, respectively, and a0 is the Bohr radius). In this approximation, the impurity state envelope functions take the form of the 2D hydrogenic states37 give by $${\varphi }_{nl}^{({\rm{2DH}})}(\rho ,\theta )=\frac{{e}^{il\theta }}{{N}_{nl}(Z,{m}^{\ast })}{(\rho {\lambda }_{n})}^{\|l\|}{e}^{\rho {\lambda }_{n}\mathrm{/2}}{L}_{n-l-1}^{\mathrm{2\|}l\|}(\rho {\lambda }_{n}),$$ where Nnl is a normalization constant, ${L}_{j}^{k}$ are the generalized Laguerre polynomials, and ${\lambda }_{n}=\frac{2}{2n+1}\frac{Z{m}^{\ast }{e}^{2}}{4\pi {\varepsilon }_{0}{\hslash }^{2}}$. We compare these solutions to the wavefunctions extracted from our TB model to identify similarities in nodal structure. The screened impurity potential in a 2D semiconductor, such as a TMD monolayer, however, cannot be accurately approximated by a bare Coulomb interaction divided by a constant dielectric function (see Fig. 1). A well-known model for the screening of a point charge embedded in a thin dielectric film was derived by Keldysh38 and is given by $${\varepsilon }_{{\rm{Keldysh}}}(q)=1+{\rho }_{0}q$$ where ρ0 is the screening length. We calculate the screened potential VKelysh(ρ) using the Keldysh model by substituting ${\varepsilon }_{{\rm{Keldysh}}}^{-1}(q)$ for the inverse dielectric function in Eq. 1. The value of ρ0 = 45 Å is obtained by fitting to the RPA-screened potential of Fig. 1. The Keldysh model has been frequently used to study excitons in TMDs29,39,40 and we also use it here for comparison to our tight-binding results. To simplify the integration over k-points in Eq. (3), Bassani et al.36 divided the first Brillouin zone into subzones Ωi centered on critial points ki, typically associated with band extrema. The impurity states Ψnv(r) are then constructed as linear combinations of subzone states $${{\rm{\Psi }}}_{n\nu i}({\bf{r}})\approx {\varphi }_{n\nu i}({\bf{r}}){\psi }_{n{{\bf{k}}}_{i}}({\bf{r}}).$$ To determine the subzone envelope functions ϕnvi(r), we minimize the expectation value of the Keldysh Hamiltonian $\hat{H}=\frac{-{\hslash }^{2}}{2{m}_{i}^{\ast }}({\partial }_{x}^{2}+{\partial }_{y}^{2})+{V}_{{\rm{Keldysh}}}(r)$ (where ${m}_{i}^{\ast }$ denotes the effective mass associated with the relevant conduction or valence band at ki) using the following ansatz for the most strongly bound impurity state $${\varphi }_{1s,i}(\rho ;\alpha )=\frac{(2\alpha ){e}^{\alpha d}}{\sqrt{2\pi (2\alpha d+1)}}{e}^{-\alpha \sqrt{{\rho }^{2}+{d}^{2}}},$$ where α is a variational parameter, which we use to define the impurity radius aimp = α−1. Once the subzone states are obtained, the full impurity states are found by including interactions between different subzones. As the coupling is usually weak, it can be treated using perturbation theory36. Acceptor States Figure 2(a–e) show the wavefunctions (specifically, their squared magnitudes sampled at the Γ-point of the first Brillouin zone) of the five most strongly bound impurity states for an adatom with Z = −0.3, situated d = 2 Å above the Mo-site, as calculated from our tight-binding model with an RPA-screened impurity potential. To label the impurity states, we compare them to the 2D hydrogenic states37. While the two most strongly bound impurity states (Fig. 2(a,b)) have 1s character, the states in Fig. 2(c,e and d) resemble the 2p and 2s states of the 2D hydrogen atom, respectively. We also present the corresponding 2D hydrogenic states in Fig. 2(f–j) for a nuclear charge Q = −0.3ζ, where ζ ≈ 0.26 is the ratio of the screened and unscreened potentials at r = 0 in Fig. 1. Surprisingly, the more strongly bound 1s states of Fig. 2(a) is significantly more delocalized with an impurity radius of aimp = 12.6 Å than the less strongly bound 1s state in Fig. 2(b), which has a radius of aimp = 5.19 Å. We determine aimp by fitting the impurity state to an exponential decay as in Eq. 7, and extracting the inverse decay scale $\alpha ={a}_{{\rm{imp}}}^{-1}$. The 2p impurity states exhibit an angular modulation caused by the trigonal warping of the valence states near the band edge41. Note that the modulation is different for the two 2p states and we therefore label the second state distinctly as 2p′. In contrast to the 2D hydrogen atom, the 2s, 2p and 2p′ are not degenerate, as indicated by their binding energies given in the top right corner of Fig. 2(a–e), because the impurity potential is screened and no longer follows a simple 1/r behaviour. (a–e) Squared wavefunctions of bound impurity states (TB model with RPA-screened potential), for an impurity charge Q = −0.3e placed 2 Å above the Mo site. States are labelled by their 2D hydrogenic character and origin in the BZ, found by projection onto the unperturbed states (see Supplementary Material). The corresponding binding energies Eb with respect to the VBM are given in white. (f–j) 2D hydrogenic states with a nuclear charge of Q = −0.3ζ e (with ζ being the ratio of the screened and unscreened potentials at r = 0 in Fig. 1) for comparison, labelled by the effective mass of the VBM from which the corresponding states in (a–e) originate. To further analyze the impurity states, we projected their wavefunctions onto unperturbed states of the MoS2 monolayer (see Supplementary Materials for details) and find that the most strongly bound 1s state and also the 2p and 2s states are composed of valence states from the K and K′ points of the MoS2 bandstructure, see Fig. 3(b). In contrast, the second 1s state originates from the valence band near the Γ-point of the unperturbed band structure. We label the states in Fig. 2(a–e) by their origin in the Brillouin zone (BZ), in addition to their 2D hydrogenic orbital character. We have subsequently labelled Fig. 2(f–j) by the effective mass of the valence band maxima (VBM) from which the corresponding TB states originate. (a) Binding energy Eb = E − EVBM of the 1s (K/K′) (blue) and 1s (Γ) (green) impurity states as a function of adatom charge Z for negatively charged adatoms on MoS2 from tight-binding calculations (solid lines) and the effective mass approximation (EMA) (dashed lines). (b) Tight-binding band structure, where bands with spin-up (spin-down) character are in red (blue). Figure 3(a) shows the dependence of the impurity state binding energies Eb = E − EVBM (energy E with reference to the primary valence band maximum EVBM) on the adatom charge Z for negatively charged adatoms. We have fitted the 1s binding energies to a power law of the form −B + AZη, see Table 1, where B = 0 for 1s (K/K′) and B = 0.071 eV for 1s (Γ), and find that the 1s (K/K′) and 1s (Γ) states have exponents of η = 1.30 and η = 1.25, respectively. These are significantly smaller than the exponent for a 2D hydrogen atom where the binding energy is given by $E(Z)=-\,4\frac{{m}^{\ast }}{{m}_{0}}{Z}^{2}$ Ry. Interestingly, the different Z-dependences of the 1s (K/K′) and 1s (Γ) binding energies result in a crossover at Z = −0.32, where the order of the two states switches. As the character of 1s (K/K′) is dominated by Mo 4dxy and $4{d}_{{x}^{2}-{y}^{2}}$ orbitals, while Mo $4{d}_{Z}2$ orbitals make up the 1s (Γ) state1, our calculations suggest the possibility of controlling the orbital character of low-lying electronic states via defect engineering with potentially interesting consequences for optical properties. Table 1 Coefficients of acceptor state binding energy fits given by Eb = −B + AZη from tight-binding (TB) and effective mass theory (EMA) with the Keldysh model. To further analyze the results of the tight-binding calculations, the bound impurity states were studied with effective mass theory. Specifically, we determined the impurity states associated with the subzones near Γ, K and K′ using Eq. (7). For the acceptor states, each subzone acts as an independent 2D hydrogen-like system as the different spin states of the degenerate valence band maxima at K and K′ prohibit interactions between the subzones. The resulting binding energies agree reasonably well with the tight-binding results, see dashed lines in Fig. 3(a) and Table 1. We see that the discrepancy between these two models increases with Z, as the RPA-screened potential in Fig. 1 is deeper than the screened potential in the Keldysh model, resulting in more strongly bound states. In particular, effective mass theory also predicts a crossover of 1s (K/K′) and 1s (Γ) near Z = −0.45. The binding energy of 1s (Γ) increases more quickly with Z because the effective mass near Γ is about 5.5 times larger than the effective mass near K or K′. This also explains the differences in impurity radii, see Fig. 2(a,b). Donor States Next, we study the shallow impurity states induced by positively charged adatoms. Figure 4(a–h) show the wavefunctions of the eight most strongly bound impurity states for an adatom with Z = 0.3 and d = 2 Å. The states are labelled based on their similarity to the eigenstates of the 2D hydrogen atom. In contrast to the acceptor case, we find a pair of states corresponding to each solution of the 2D hydrogen atom, with different binding energies, indicated at the top right corner of each subfigure in white. The states of each pair are distinguished by a "+" or "−" subscript. (a–h) Squared wavefunctions of bound impurity states for an impurity charge Q = +0.3e placed 2 Å above the Mo site, with binding energies Eb = ECBM − E indicated (white). Hybridised states are separately labelled with ± subscripts. (i) Binding energy Eb of hybridized 1s (K/K′) (green and blue) and 1s (Q) (magenta) impurity states as a function of adatom charge Z for positively charged adatoms on MoS2 from TB (solid lines) and EMA (dashed lines). Figure 4(i) shows the binding energies Eb = ECBM − E of the most strongly bound states (with energy E) with respect to the conduction band minimum (with energy ECBM) as function of the impurity charge Z. At low values of Z, the 1s−(K/K′) and 1s+(K/K′) states are almost degenerate, but their binding energy difference increases with increasing Z. A third impurity state originating from the local conduction band minimum at the 6 Q points of the Brillouin zone crosses the two 1s (K/K′) states near Z = 0.6 and becomes the most strongly bound state for higher values of Z. The crossover is again caused by the larger effective mass at Q point compared to the K and K′ points. We have fitted the binding energies of these states to a power law of the form B + AZη, see Table 2, where B = 0 for states from K/K′ and B = 0.267 eV for states from the Q-points. As for the acceptor impurity states, the exponents of the donor states are significantly smaller than the 2D hydrogen value η = 2. Table 2 Coefficients of donor state binding energy fits given by Eb = −B + AZη from tight-binding (TB) and effective mass theory (EMA) with the Keldysh model. Again, we compare the tight-binding results to effective mass theory. We first determine the subzone envelope functions, Eq. (7), for the regions near the critical points at K and K′. In contrast to the valence bands, there is no spin-orbit splitting of the conduction band states at K and K′. As a consequence, the conduction band states at K and K′ with equal spin are degenerate and this gives rise to the observed pairs of impurity states with same symmetry in Fig. 4. The subzone impurity states can couple and the resulting binding energy splitting is given by36 $${{\rm{\Delta }}}_{KK^{\prime} }\approx 2\|{\varphi }_{1s,{\bf{K}}}^{\ast }(r=0){\varphi }_{1s,{\bf{K}}^{\prime} }(r=0)V({\bf{q}}={\bf{K}}-{\bf{K}}^{\prime} )\|.$$ We evaluate the splitting with the Keldysh approximation for V, using the Fourier transform of the screened Coulomb potential in the Keldysh model. We find that the splitting is several orders of magnitude smaller than the splitting found in the tight-binding model. This discrepancy is caused by the inaccurate behaviour of the Keldysh model at large wave vectors, which is shown in the inset of Fig. 1, where the vertical black line indicates \|K − K′\|. We note, however, that for such large wavevectors (corresponding to positions in the immediate vicinity of the impurity) the use of the 2D dielectric function can cause inaccuracies, as Eq. 2 assumes that the distance from the impurity is significantly larger than the width of the MoS2 sheet29. We show the binding energies, found from effective mass theory using the Keldysh screening model for the splitting (see Fig. 4(i) as blue dashed an green dot-dashed lines). The fitting parameters of the binding energies to a power law are compared to the tight-binding results in Table 2. The 1s impurity state wavefunctions from effective mass theory are given by $${{\rm{\Psi }}}_{1{s}_{\pm }(K/K^{\prime} )}({\bf{r}})=\frac{1}{\sqrt{2}}({\varphi }_{1s,{\bf{K}}}(r){\psi }_{{\bf{K}}}({\bf{r}})\pm {\varphi }_{1s,{\bf{K}}^{\prime} }(r){\psi }_{{\bf{K}}}({\bf{r}})),$$ where ${\psi }_{K/K^{\prime} }({\bf{r}})$ denote the Bloch states of the unperturbed MoS2 band structure at K and K′. Notably, the states with an s-character (Fig. 4(a,b,d and h)) exhibit an intensity modulation with a period of three unit cells along the directions connecting nearest neighbours. Projecting the impurity states onto unperturbed Bloch states reveals that all states originate from both the K and K′ points of the Brillouin zone, where the minimum of the conduction band occurs, see Fig. 3(b). The corresponding probability densities contain a term with a cos((K − K′) ·r) factor which gives rise to the oscillatory pattern in Fig. 4(a,e,d,h). In contrast to the impurity states with s-character which derive from unperturbed states directly at K and K′, the states with p-character mostly derive from conduction band states in the vicinity of the band edges. As a consequence, the coupling between K and K′ is weaker for the p-states and the spatial modulation is not observed. We find that this modulation does not occur when the defect is not placed on the transition-metal site. Local density of states Scanning tunnelling spectroscopy (STS) provides spatially-resolved information about the electronic structure of surfaces and has been used to study the properties of shallow impurity states induced by charged adatoms experimentally. The dI/dV curves obtained in STS are often assumed to be proportional to the local density of states (LDOS) of the sample. We have calculated the LDOS for values of Z and d that represent lithium (Li) and carbon (C) atoms adsorbed on a MoS2. For Li, Chang et al. found an impurity charge of ZLi = 0.63 from a Bader charge analysis42 of the DFT charge density18. Using a similar procedure, Ataca et al. determined ZC = −0.58 for a C atom adsorbed to MoS2 above the Mo site19,43. We modelled adsorbed atoms sitting above the Mo site at a height of dLi = 3.1 Å and dC = 1.58 Å17,18,19,43. Screening by a SiO2 substrate is included via a substrate dielectric function of 3.7. Figure 5(a,b) show the tight-binding LDOS for a C adatom on MoS2 in the vicinity of valence band maximum and the conduction band minimum, respectively. A 6 × 6 k-point mesh and a Gaussian broadening of 0.01 eV were used. Near the VBM, several peaks originating from bound acceptor states can be observed in the band gap. The peak from 1s (Γ) disappears more quickly as a function of distance from the adatom than the 1s (K/K′) peak. This is a consequence of the stronger localization of this state, see Fig. 2. At a distance of ~66 Å from the adatom, the LDOS of the perturbed system has converged to the LDOS of the pristine TMD. In the vicinity of the CBM, no impurity states are present. However, the screened potential created by the adatom leads to a shift of the unperturbed LDOS. (a,b) LDOS of a lithium (Li) adatom on MoS2 (+SiO2 substrate) near the (a) valence band and (b) conduction band edge. (c,d) LDOS of a carbon (C) adatom on MoS2 (+SiO2 substrate) near the (c) valence band and (d) conduction band edge. Results are shown for several distances from the impurity. In each graph, the zero of energy is set to the band edge of the unperturbed MoS2. Figure 5(c,d) show the tight-binding LDOS for a Li adatom on MoS2 in the vicinity of valence band maximum and the conduction band minimum, respectively. The peaks near the CBM in the vicinity of the adatom originate from bound donor states and can be observed up to a distance of ~25 Å from the adatom. Note that the splitting of the two impurity states from the K and K′ points is too small to be resolved. No impurity state peaks are found in the vicinity of the VBM, but again the impurity potential causes a shift of the TMD LDOS. In summary, we have studied the electronic properties of charged defects in transition-metal dichalcogenides. Using tight-binding simulations with screened impurity potentials on unit cells containing up to 8,000 atoms, we have calculated the binding energies and wave functions of shallow impurity bound states. Our key finding is that the orbital character of the lowest lying impurity states depends sensitively on the magnitude of the defect charge. For acceptor states, i.e., negatively charged defects, a crossover of impurity states with different orbital characters occurs at a critical defect charge of Q = −0.32 e (with e being the proton charge). For defect charges above this value, the lowest impurity state from the Γ valley of the TMD band structure, which is dominated by contributions from Mo $4{{\rm{d}}}_{{{\rm{z}}}^{2}}$ orbitals, is more strongly bound than the degenerate impurity states from the K and K′ valleys which are dominated by Mo 4dxy and Mo $4{{\rm{d}}}_{{x}^{2}-{y}^{2}}$ orbitals. For donor states, i.e., positively charged defects, a crossover between hybridized impurity states from the K and K′ valleys and impurity states from the Q valleys occurs at a critical impurity charge of +0.6 e. To understand the competition between different impurity states, we analyze their properties using effective mass theory. We find that the impurity binding energies can be described by power laws of the defect charge, but with significant deviations from hydrogenic behaviour due to screening. 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This work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), and the Imperial College London High-Performance Computing Facility. Martik Aghajanian, Arash A. Mostofi and Johannes Lischner contributed equally. Department of Physics and Materials and the Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, London, SW7 2AZ, UK Martik Aghajanian, Arash A. Mostofi & Johannes Lischner Martik Aghajanian Arash A. Mostofi Johannes Lischner J.L. and A.A.M. proposed the work, M.A. performed the calculations and all authors contributed to analyzing the results. All authors reviewed and contributed to the writing of the manuscript. Correspondence to Johannes Lischner. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Aghajanian, M., Mostofi, A.A. & Lischner, J. Tuning electronic properties of transition-metal dichalcogenides via defect charge. Sci Rep 8, 13611 (2018). https://doi.org/10.1038/s41598-018-31941-1 Accepted: 22 August 2018 Transition-metal Dichalcogenides (TMD) Effective Mass Theory TMD Monolayer Orbital Character Local Density Of States (LDOS)	CommonCrawl
Astrophysics > Solar and Stellar Astrophysics Title:The Origin of Elements Across Cosmic Time: Astro2020 Science White Paper Authors:Jennifer A. Johnson (OSU), Gail Zasowski, (Utah), David Weinberg (OSU), Yuan-Sen Ting (IAS/Princeton/OCIW), Jennifer Sobeck (Washington), Verne Smith (NOAO), Victor Silva Aguirre (Aarhus), David Nataf (JHU), Sara Lucatello (INAF/Padova), Juna Kollmeier (OCIW), Saskia Hekker (MPS), Katia Cunha (Arizona), Cristina Chiappini (AIP), Joleen Carlberg (STScI), Jonathan Bird (Vanderbilt), Sarbani Basu (Yale), Borja Anguiano (UVa) Abstract: The problem of the origin of the elements is a fundamental one in astronomy and one that has many open questions. Prominent examples include (1) the nature of Type Ia supernovae and the timescale of their contributions; (2) the observational identification of elements such as titanium and potassium with the $\alpha$-elements in conflict with core-collapse supernova predictions; (3) the number and relative importance of r-process sites; (4) the origin of carbon and nitrogen and the influence of mixing and mass loss in winds; and (5) the origin of the intermediate elements, such as Cu, Ge, As, and Se, that bridge the region between charged-particle and neutron-capture reactions. The next decade will bring to maturity many of the new tools that have recently made their mark, such as large-scale chemical cartography of the Milky Way and its satellites, the addition of astrometric and asteroseismic information, the detection and characterization of gravitational wave events, 3-D simulations of convection and model atmospheres, and improved laboratory measurements for transition probabilities and nuclear masses. All of these areas are key for continued improvement, and such improvement will benefit many areas of astrophysics. Comments: Submitted as an Astro2020 Science White Paper Subjects: Solar and Stellar Astrophysics (astro-ph.SR) Cite as: arXiv:1907.04388 [astro-ph.SR] (or arXiv:1907.04388v1 [astro-ph.SR] for this version) From: Jennifer A. Johnson [view email]	CommonCrawl
\begin{document} \title{On the Cauchy problem for the Muskat equation. II: Critical initial data} \author{Thomas Alazard} \thanks{E-mail address: [email protected], Universit{\'e} Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli UMR9010, avenue des Sciences, F-91190 Gif-sur-Yvette, Paris, France} \author{Quoc-Hung Nguyen} \thanks{E-mail address: [email protected], ShanghaiTech University, 393 Middle Huaxia Road, Pudong, Shanghai, 201210, China.} \date{} \setlength{\baselineskip}{5mm} \begin{abstract} We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in $L^2$. Moreover, we prove that the solution exists globally in time under a smallness assumption. \end{abstract} \maketitle \section{Introduction} The Muskat equation describes the dynamics of the interface separating two fluids in porous media whose velocities obey Darcy's law (\cite{darcy1856fontaines,Muskat}). This equation belongs to the family of nonlocal parabolic equations that have attracted a lot of attention in recent years. Indeed, it has long been observed that one can reduce the Muskat equation to an evolution equation for the free surface parametrization (see~\cite{CaOrSi-SIAM90,EsSi-ADE97,PrSi-book,SCH2004}). One interesting feature of the Muskat equation is that it admits a compact formulation in terms of finite differences, as observed by C\'ordoba and Gancedo~\cite{CG-CMP}. More precisely, assume that the free surface is the graph of some function $f=f(t,x)$ with $x\in\mathbb{R}$. Then, C\'ordoba and Gancedo~\cite{CG-CMP} showed that the Muskat equation reduces to \begin{equation}\label{n1} \partial_tf=\frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\diff \! \alpha, \end{equation} where $\Delta_\alpha f$ is the slope, defined by \begin{align}\label{eq2.2} \Delta_\alpha f(x,t)=\frac{f(x,t)-f(x-\alpha,t)}{\alpha}\cdot \end{align} It is easily verified that the Muskat equation is invariant by the change of unknowns: \begin{equation}\label{acritical} f(t,x)\mapsto f_\lambda(t,x)\mathrel{:=}\frac{1}{\lambda}f\left(\lambda t,\lambda x\right) \qquad (\lambda\neq 0). \end{equation} Now, by a direct calculation, $$ \left\Vert f_\lambda\big\arrowvert_{t=0} \right\Vert_{\dot{W}^{1,\infty}}=\left\Vert f_0\right\Vert_{\dot{W}^{1,\infty}}\quad ; \quad \left\Vert f_\lambda\big\arrowvert_{t=0} \right\Vert_{ \dot H^{\frac{3}{2}}}=\left\Vert f_0\right\Vert_{ \dot H^{\frac{3}{2}}}. $$ This means that the spaces $\dot{W}^{1,\infty}(\mathbb{R})$ and $\dot H^{\frac{3}{2}}(\mathbb{R})$ are critical for the study of the Cauchy problem. Let us clarify that we denoted by $\dot W^{1,\infty}(\mathbb{R})$ the space of Lipschitz functions, and by $H^s(\mathbb{R})$ (resp.\ $\dot{H}^{s}(\mathbb{R})$) the classical Sobolev (resp.\ homogeneous Sobolev) space of order $s$. They are equipped with the norm defined by $$ \left\Vert u\right\Vert_{\dot{W}^{1,\infty}}\mathrel{:=} \sup_{{\substack{ x,y\in \mathbb{R}\\ x\neq y}}}\frac{\left\vert u(x)-u(y)\right\vert}{\left\vert x-y\right\vert}, $$ and $$ \left\Vert u\right\Vert_{\dot{H}^{s}}\mathrel{:=}\left(\int_\mathbb{R} \left\vert \xi\right\vert^{2s}\big\vert \hat{u}(\xi)\big\vert^2\diff \! \xi\right)^\frac{1}{2}, \quad \left\Vert u\right\Vert_{H^{s}}^2=\left\Vert u\right\Vert_{\dot{H}^s}^2+\left\Vert u\right\Vert_{L^2}^2. $$ We are interested in the study of the Cauchy problem for the latter equation. Our main result states that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space $\dot{W}^{1,\infty}(\mathbb{R})\cap {H}^{\frac{3}{2}}(\mathbb{R})$. Our analysis is inspired by many previous works, and we begin by reviewing the literature on this problem. The first well-posedness results were established by Yi~\cite{Yi2003}, Ambrose~\cite{Ambrose-2004,Ambrose-2007}, C\'ordoba and Gancedo~\cite{CG-CMP}, C\'ordoba, C\'ordoba and Gancedo~\cite{CCG-Annals}, Cheng, Granero-Belinch\'on, Shkoller~\cite{Cheng-Belinchon-Shkoller-AdvMath}. In recent years, these results were extended in several directions. In particular, the well-posedness of the Cauchy problem has been established in many sub-critical spaces: see Constantin, Gancedo, Shvydkoy and Vicol~\cite{CGSV-AIHP2017} for initial data in the Sobolev space $W^{2,p}(\mathbb{R})$ for some $p>1$, Deng, Lei and Lin~\cite{DLL} and Camer\'on~\cite{Cameron} for initial data in H\"older spaces, and Matioc~\cite{Matioc2}, Alazard and Lazar~\cite{Alazard-Lazar}, Nguyen and Pausader~\cite{Nguyen-Pausader} for initial data in $H^s(\mathbb{R})$ with $s>3/2$. Special features of the Muskat equations were exploited to improve the analysis of the Cauchy problem in several directions. Constantin, C{\'o}rdoba, Gancedo, Rodr{\'\i}guez-Piazza and Strain~\cite{CCGRPS-AJM2016} (see also \cite{CGSV-AIHP2017,PSt}) proved a global well-posedness results assuming that the Lipschitz semi-norm is smaller than $1$. Deng, Lei and Lin in~\cite{DLL} proved the existence of solutions whose slope can be arbitrarily large. Cameron \cite{Cameron} exhibited the existence of a modulus of continuity for the derivative (see also~\cite{Abedin-Schwab-2020}) and obtained a global existence result assuming only that the product of the maximal and minimal slopes is bounded by~$1$. C\'ordoba and Lazar established in \cite{Cordoba-Lazar-H3/2} the first global well-posedness result assuming only that the initial data is sufficiently smooth and that the critical $\dot H^{3/2}(\mathbb{R})$-norm is small enough (see also \cite{Gancedo1,Gancedo2,Granero-Scrobogna} for related global well-posedness results in Wiener spaces in the critical case, for small enough initial data). This result was extended to the 3D case by Gancedo and Lazar~\cite{Gancedo-Lazar-H2} for initial data in the critical Sobolev space $\dot{H}^2(\mathbb{R}^2)$. Eventually, in our companion paper~\cite{AN1}, we initiated the study of the Cauchy problem for non-Lipschitz initial data. For our subject matter, another fundamental component of the background is that the Cauchy problem is not well-posed globally in time: there are blow-up results for some large enough data by Castro, C\'{o}rdoba, Fefferman, Gancedo and L\'opez-Fern\'andez~(\cite{CCFG-ARMA-2013,CCFG-ARMA-2016,CCFGLF-Annals-2012}). More precisely, they proved the existence of solutions such that at initial time $t=0$ the interface is a graph, at a later time $t_1>0$ the interface is no longer a graph and then at a subsequent time $t_2>t_1$, the interface is $C^3$ but not $C^4$. Our main result in this paper is the following \begin{theorem}\label{main} $i)$ For any initial data $f_0$ in $\dot W^{1,\infty}(\mathbb{R})\cap H^{\frac{3}{2}}(\mathbb{R})$, there exists a time $T>0$ such that the Cauchy problem for the Muskat equation has a unique solution \begin{equation} f\in L^\infty\big([0,T];\dot W^{1,\infty}(\mathbb{R})\cap H^{\frac{3}{2}}(\mathbb{R})\big) \cap L^2(0,T;\dot H^2(\mathbb{R})). \end{equation} $ii)$ Moreover, there exists a positive constant $\delta$ such that, for any initial data $f_0$ in $\dot W^{1,\infty}(\mathbb{R})\cap H^{3/2}(\mathbb{R})$ satisfying \begin{equation} \big(1+\left\Vert f_0\right\Vert_{\dot W^{1,\infty}}^4\big)\left\Vert f_0\right\Vert_{\dot H^{\frac{3}{2}}}\leq \delta, \end{equation} the Cauchy problem for the Muskat equation has a unique global solution \begin{equation} f\in L^\infty\big([0,+\infty);\dot W^{1,\infty}(\mathbb{R})\cap H^{\frac{3}{2}}(\mathbb{R})\big) \cap L^2(0,+\infty;\dot H^2(\mathbb{R})). \end{equation} \end{theorem} Some remarks are in order. $\bullet$ Let us discuss statement $ii)$ about the global well-posedness component of this result. This is a $2D$ analogous to the recent result by Gancedo and Lazar~\cite{Gancedo-Lazar-H2} for the $3D$ problem; it improves on a previous result by C\'ordoba and Lazar~\cite{Cordoba-Lazar-H3/2} which proves a similar global existence result for the $2D$-problem with a similar smallness assumption, but under the extra assumption that the initial data belongs to $H^{5/2}(\mathbb{R})$. $\bullet$ We now come to statement $i)$ about the local well-posedness result for arbitrary initial data. This is, in our opinion, the main new result in this paper. Since we are working in a critical space, this result is optimal in several directions. Firstly, it follows from the results about singularity formation by Castro, C\'{o}rdoba, Fefferman, Gancedo and L\'opez-Fern\'andez~(\cite{CCFG-ARMA-2013,CCFG-ARMA-2016,CCFGLF-Annals-2012}) that one cannot solve the Cauchy problem for a time $T$ which depends only on the norm of $f_0$ in $\dot W^{1,\infty}(\mathbb{R})\cap \dot H^{3/2}(\mathbb{R})$. Otherwise, one would obtain a global existence result for any initial data by an immediate scaling argument using~\eqref{acritical}. Notice that this argument does not contradict our main result: it means instead that the time of existence must depend on the initial data itself, and not only on its norm. The previous discussion shows that one cannot prove statement $i)$ by using classical Sobolev energy estimates. This in turn poses new challenging questions since on the other hand the Muskat equation is a quasi-linear equation. To overcome this problem, we will estimate the solution for a norm whose definition depends on the initial data. $\bullet$ We will also prove a result which elaborates on the previous discussion, stating that whenever one controls a bigger norm than the critical one, the time of existence is bounded from below on a neighborhood of the initial data. To introduce this result, let us fix some notations. \begin{definition}\label{defi:D} Given a real number $s\ge 0$ and a function $\phi\colon [0,\infty)\to [1,\infty)$ satisfying the following assumptions: \begin{enumerate}[$({{\rm H}}1)$] \item\label{H1} $\phi$ is increasing and $\lim\phi(r)=\infty$ when $r$ goes to $+\infty$; \item\label{H2} there is a positive constant $c_0$ such that $\phi(2r)\leq c_0\phi(r)$ for any $r\geq 0$; \item\label{H3} the function $r\mapsto \phi(r)/\log(4+r)$ is decreasing on $[0,\infty)$. \end{enumerate} Then $\|D\|^{s,\phi}$ denotes the Fourier multiplier with symbol $\|\xi\|^s\phi(\|\xi\|)$, so that \begin{equation} \mathcal{F}( \|D\|^{s,\phi}f)(\xi)=\|\xi\|^s\phi(\|\xi\|) \mathcal{F}(f)(\xi). \end{equation} Moreover, we define the space $$ \mathcal{X}^{s,\phi}(\mathbb{R})=\{ f\in \dot{W}^{1,\infty}(\mathbb{R})\cap L^2(\mathbb{R})\, :\, \left\vert D\right\vert^s \phi(\left\vert D_x\right\vert)f\in L^2(\mathbb{R})\}, $$ equipped with the norm $$ \left\Vert f\right\Vert_{\mathcal{X}^{s,\phi}}\mathrel{:=} \left\Vert f\right\Vert_{\dot{W}^{1,\infty}}+\left\Vert f\right\Vert_{L^2} +\left(\int_\mathbb{R} \left\vert \xi\right\vert^{2s}(\phi(\left\vert\xi\right\vert))^2\big\vert\hat{f}(\xi)\big\vert^2\diff \! \xi\right)^\frac{1}{2}. $$ \end{definition} \begin{remark} The Fourier multiplier $\|D\|^{s,\phi}$ with $\phi(r)=\log(2+r)^a$ was introduced and studied in~\cite{BN18a,BN18b,BN18d} for $s\in [0,1)$ (also see \cite{Ng}). \end{remark} \begin{theorem}\label{T2} Consider a real number $M_0>0$ and a function $\phi$ satisfying assumptions~$(\rm{H}\ref{H1})$--$(\rm{H}\ref{H3})$ in Definiton~\ref{defi:D}. Then there exists a time $T_0>0$ such that, for any initial data $f_0$ in $\mathcal{X}^{\frac{3}{2},\phi}(\mathbb{R})$ satisfying $$ \left\Vert f_0\right\Vert_{\mathcal{X}^{\frac{3}{2},\phi}}\leq M_0, $$ the Cauchy problem for the Muskat equation has a unique solution \begin{equation} f\in L^\infty\big([0,T_0];\dot W^{1,\infty}(\mathbb{R})\cap {H}^{\frac{3}{2}}(\mathbb{R})\big) \cap L^2(0,T_0;\dot H^2(\mathbb{R})). \end{equation} \end{theorem} \begin{remark}Statement $i)$ in Theorem~\ref{main} is a consequence of Theorem~\ref{T2}. Indeed, it is easily seen that (cf \cite[Lemma~$3.8$]{AN1}), for any $f_0$ in the critical space $\dot W^{1,\infty}(\mathbb{R})\cap \dot{H}^{\frac{3}{2}}(\mathbb{R})$, one may find a function $\phi$ such that $f_0$ belongs to $\mathcal{X}^{\frac{3}{2},\phi}(\mathbb{R})$ (and satisfying assumptions~$(\rm{H}\ref{H1})$--$(\rm{H}\ref{H3})$ in Definiton~\ref{defi:D}). \end{remark} Theorem~\ref{main} and Theorem~\ref{T2} are proved in the next section. \subsection{Acknowledgments} \noindent T.A.\ acknowledges the SingFlows project (grant ANR-18-CE40-0027) of the French National Research Agency (ANR). Q-H.N.\ is supported by the Shanghai Tech University startup fund and the National Natural Science Foundation of China (12050410257). The authors would like to thank the referees for their comments, which help to improve the presentation of this article, as well as Gustavo Ponce for pointing out a mistake in a preliminary version. \section{Proof} \subsection{Regularization} In order to rigorously justify the computations, we want to handle smooth functions (hereafter, a `smooth function' is by definition a function that belongs to $C^1([0,T];H^\mu(\mathbb{R}))$ for any $\mu\in [0,+\infty)$ and some $T>0$). To do so, we must regularize the initial data and also consider an approximation of the Muskat equation. For our purposes, we further need to consider a regularization of the Muskat equation which will be compatible with the Sobolev and Lipschitz estimates. It turns out that this is a delicate technical problem. Our strategy will consist in smoothing the equation in two different ways: $i)$ by introducing a cut-off function in the singular integral, removing wave-length shorter than some parameter $\varepsilon$ and $ii)$ by adding a parabolic term of order $2$ with a small viscosity of size $\left\vert \log(\varepsilon)\right\vert^{-1}$. More precisely, we introduce the following Cauchy problem depending on the parameter $\varepsilon\in (0,1]$: \begin{equation}\label{n2} \left\{ \begin{aligned} &\partial_tf-\|\log(\varepsilon)\|^{-1}\partial_x^2 f =\frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\left(1-\chi\left(\frac{\alpha}{\varepsilon}\right)\right)\diff \! \alpha,\\ &f\arrowvert_{t=0}=f_0\star \chi_\varepsilon, \end{aligned} \right. \end{equation} where $\chi_\varepsilon(x)=\varepsilon^{-1}\chi(x/\varepsilon)$ where $\chi$ is a smooth bump function satisfying $0\leq \chi\leq 1$ and $$ \chi(y)=\chi(-y),\quad \chi(y)=1 \quad\text{for}\quad \|y\|\leq \frac14, \quad \chi(y)=0 \quad\text{for }\left\vert y\right\vert\ge 2,\quad \int_\mathbb{R}\chi \diff \! y=1. $$ The equation~\eqref{n2} does not belong to a general class of parabolic equations. However, we will see that it can be studied by standard tools in functional analysis together with two estimates for the nonlinearity in the Muskat equation which plays a central role in our analysis. \begin{proposition}\label{P:initiale} For any $\varepsilon$ in $(0,1]$ and any initial data $f_0$ in $H^{\frac{3}{2}}(\mathbb{R})$, there exists a unique global in time solution $f_\varepsilon$ satisfying $$ f_\varepsilon\in C^1([0,+\infty);H^\infty(\mathbb{R})). $$ \end{proposition} We postpone the proof of this proposition to \S\ref{S:end}. \subsection{An estimate of the Lipschitz norm} \begin{lemma}\label{L:2.1} For any real number $\beta_0$ in $(0,1/2)$, there exists a positive constant $C_0\geq 1$ such that, for any $\varepsilon\in (0,1]$ and any smooth solution $f\in C^1([0,T];H^{\infty}(\mathbb{R}))$ of the Muskat equation \eqref{n2}, \begin{equation}\label{a1} \frac{\diff}{\dt} \left\Vert f(t)\right\Vert_{\dot W^{1,\infty}}\leq C_0 \left\Vert f(t)\right\Vert_{\dot H^2}^2+C_0 \varepsilon^{\beta_0}\left\Vert f(t)\right\Vert_{\dot C^{2,\beta_0}}, \end{equation} where $$ \left\Vert u\right\Vert_{\dot C^{2,\beta_0}}=\left\Vert \partial_{xx}u\right\Vert_{C^{0,\beta_0}} =\sup_{\substack{ x,y\in \mathbb{R} \\ x\neq y}} \frac{\left\vert (\partial_{xx}u)(x)-(\partial_{xx}u)(y)\right\vert}{\left\vert x-y\right\vert^{\beta_0}}\cdot $$ \end{lemma} \begin{proof} The proof is partially based on arguments from~\cite{CG-CMP2,Cameron,Gancedo-Lazar-H2}. Firstly, it follows from the proof of \cite[Lemma~$5.1$]{CG-CMP2} that \begin{align} \partial_{x}\frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f(x)}{1+\left(\Delta_\alpha f(x)\right)^2}\diff \! \alpha&=\frac{\partial_x^2f(t,x)}{2\pi} \int\left(\frac{1}{1+(\Delta_\alpha f(t,x))^2}-\frac{1}{1+(\Delta_{-\alpha} f(t,x))^2}\right)\frac{\diff \! \alpha}{\alpha}\\ &\quad-\frac{2}{\pi} \int\frac{\partial_x f(t,x)-\Delta_\alpha f(t,x)}{\alpha^2} \frac{1+\partial_x f(t,x)\Delta_\alpha f(t,x)}{1+(\Delta_\alpha f(t,x))^2} \diff \! \alpha. \end{align} Moreover, \begin{equation}\label{X3} \begin{aligned} &\left\|\partial_{x}\left(\frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\chi\left(\frac{\|\alpha\|}{\varepsilon}\right)\diff \! \alpha\right)\right\|\\ &\qquad\qquad\lesssim \int_{\|\alpha\|\leq 2 \varepsilon}\left(\|\Delta_\alpha f_{xx}\| +\|\Delta_\alpha f_{x}\|^2\right)\diff \! \alpha \\& \qquad\qquad\lesssim \int_\mathbb{R} \|\Delta_\alpha f_{x}\|^2\diff \! \alpha+\varepsilon^{\beta_0}\left\Vert f_{xx}\right\Vert_{\dot C^{0,\beta_0}}, \end{aligned} \end{equation} where we used the notations $f_x=\partial_x f$ and $f_{xx}=\partial_{xx}f$. Thus, for any $t$ and any $x$, we have \begin{equation}\label{n10} \begin{aligned} &(\partial_x \partial_t f)(t,x) -\|\log(\varepsilon)\|^{-1}\partial_{x}^2 f_x(t,x) \\ &\qquad\qquad\leq \frac{\partial_x^2f(t,x)}{2\pi} \int\left(\frac{1}{1+(\Delta_\alpha f(t,x))^2}-\frac{1}{1+(\Delta_{-\alpha} f(t,x))^2}\right)\frac{\diff \! \alpha}{\alpha}\\ &\qquad\qquad\quad-\frac{2}{\pi} \int\frac{\partial_x f(t,x)-\Delta_\alpha f(t,x)}{\alpha^2} \frac{1+\partial_x f(t,x)\Delta_\alpha f(t,x)}{1+(\Delta_\alpha f(t,x))^2} \diff \! \alpha\\ &\qquad\qquad\quad +C \int \|\Delta_\alpha f_{x}(t,x)\|^2\diff \! \alpha+C\varepsilon^{\beta_0}\left\Vert f_{xx}(t)\right\Vert_{\dot C^{0,\beta_0}}. \end{aligned} \end{equation} Consider the function $\varphi(t)=\left\Vert \partial_{x}f(t)\right\Vert_{L^\infty}$ and a function $ t\mapsto x_t$ such that $$ \left\Vert \partial_{x}f(t)\right\Vert_{L^\infty}=(\partial_{x} f)(t,x_t). $$ Then $(\partial_{x}^2f)(t,x_t)=0$ and $-(\partial_{xx} f_x)(t,x_t)\geq 0$. So, it follows from \eqref{n10} that \begin{align} \dot{\varphi}(t)&\leq -\frac{2}{\pi}\int \frac{\partial_x f(t,x_t)-\Delta_\alpha f(t,x_t)}{\alpha^2}\diff \! \alpha\\& \quad-\frac{2}{\pi}\int \frac{(\partial_x f(t,x_t)-\Delta_\alpha f(t,x_t))^2}{\alpha^2} \frac{\Delta_\alpha f(t,x_t)}{1+(\Delta_\alpha f(t,x_t))^2} \diff \! \alpha\\& \quad+C \int \|\Delta_\alpha f_{x}(t,x_t)\|^2\diff \! \alpha+C\varepsilon^{\beta_0}\left\Vert f_{xx}(t)\right\Vert_{\dot C^{0,\beta_0}}. \end{align} As already observed in~\cite{CG-CMP2} (see also~\cite{Cameron,Gancedo-Lazar-H2}), the first term in the right-hand side has a sign since $\partial_x f(t,x_t)\ge \Delta_\alpha f(t,x_t)$ for any $\alpha$. It follows that \begin{align} \dot{\varphi}(t)&\leq \frac{1}{\pi}\int \frac{(\partial_x f(t,x_t) -\Delta_\alpha f(t,x_t))^2}{\alpha^2} \diff \! \alpha+C \int \|\Delta_\alpha f_{x}(t,x_t)\|^2\diff \! \alpha\\ &\quad+C\varepsilon^{\beta_0}\left\Vert f_{xx}(t)\right\Vert_{\dot C^{0,\beta_0}}. \end{align} We now apply Hardy's inequality to infer that $$ \int \frac{(\partial_x f(t,x_t) -\Delta_\alpha f(t,x_t))^2}{\alpha^2} \diff \! \alpha\lesssim \int \|\Delta_\alpha f_{x}(t,x_t)\|^2\diff \! \alpha. $$ Consequently, we end up with \begin{align} \dot{\varphi}(t)\lesssim \int \left\Vert\Delta_\alpha f_{x}(t)\right\Vert_{L^\infty}^2\diff \! \alpha +\varepsilon^{\beta_0}\left\Vert f_{xx}(t)\right\Vert_{\dot C^{0,\beta_0}}. \end{align} Introducing the difference operator $\delta_\alpha g(x)=g(x)-g(x-\alpha)$, the previous inequality is better formulated as follows: \begin{align} \dot{\varphi}(t)\lesssim \int \left\Vert \delta_\alpha (\partial_xf)(t)\right\Vert_{L^\infty}^2 \frac{\diff \! \alpha}{\|\alpha\|^{1+\frac{1}{2} 2}}+\varepsilon^{\beta_0}\left\Vert f_{xx}(t)\right\Vert_{\dot C^{0,\beta_0}}. \end{align} Now the right-hand side is equivalent to the following homogeneous Besov norm: $\left\Vert \partial_x f(t)\right\Vert_{\dot{B}^{\frac{1}{2}}_{\infty,2}}^2$ (see~\cite{Triebel-TFS,Triebel1988} or Section~$2$ in \cite{AN1}). Then it follows from Sobolev embeddings that $$ \dot{\varphi}(t)\lesssim \left\Vert f(t)\right\Vert_{\dot H^2}^2+\varepsilon^{\beta_0}\left\Vert f_{xx}(t)\right\Vert_{\dot C^{0,\beta_0}} $$ which is the wanted result. \end{proof} \subsection{Sobolev estimates} In this paragraph we recall a generalized Sobolev energy estimate proved in our companion paper~\cite{AN1}. By generalized Sobolev energy estimate, we mean that, instead of estimating the $L^\infty_t(L^2_x)$-norm of $(-\Delta)^s f$, we shall estimate the $L^\infty_t(L^2_x)$-norm of $\left\vert D\right\vert^{s,\phi}f$ for some function $\phi$ satisfying the assumptions in Definition~\ref{defi:D}. There two technical results that we will borrow from~\cite{AN1}. The first result, which is Lemma~$3.4$ in \cite{AN1}, gives an energy estimate. \begin{lemma}\label{L:3.4} There exists a positive constant $C$ such that, for any $T>0$ and any smooth solution $f\in C^{1}([0,T];H^{\infty}(\mathbb{R}))$ to~\eqref{n1}, there holds \begin{multline}\label{Z21} \frac{\diff}{\dt} \big\Vert \left\vert D\right\vert^{\frac{3}{2},\phi}f\big\Vert_{L^2}^2 + \int_\mathbb{R} \frac{\big\vert\left\vert D\right\vert^{2,\phi}f\big\vert^2}{1+(\partial_x f)^2}\diff \! x+\|\log(\varepsilon)\|^{-1}\int_\mathbb{R} \big\vert\left\vert D\right\vert^{\frac{5}{2},\phi}f\big\vert^2\diff \! x\\ \leq C Q(f) \big\Vert\left\vert D\right\vert^{2,\phi}f\big\Vert_{L^2}, \end{multline} where \begin{align} Q(f)&= \left(\left\Vert f\right\Vert_{\dot H^2}+\left\Vert f\right\Vert_{\dot H^{\frac{7}{4}}}^2\right) \big\Vert\left\vert D\right\vert^{\frac{3}{2},\phi}f\big\Vert_{L^2} +\big\Vert\left\vert D\right\vert^{\frac74,\phi}f\big\Vert_{L^2} \left\Vert f\right\Vert_{{H}^{\frac74}}\\ &\quad+\left(\left\Vert f\right\Vert_{H^{\frac{19}{12}}}^{3/2}+\left\Vert f\right\Vert_{\dot H^{\frac74}}^{1/2}\right) \big\Vert\left\vert D\right\vert^{\frac{7}{4},\phi^{2}}f\big\Vert^{1/2}_{L^2} \left\Vert f\right\Vert_{\dot H^{\frac74}}. \end{align} \end{lemma} \begin{remark} Some explanations are in order since the reader may notice several modifications compared to our paper~\cite{AN1}. Firstly, in \cite{AN1} we considered a function $\phi$ whose definition depends on an extra function $\kappa$. Here we ignore this point since it is irrelevant for the present analysis. Indeed, the functions $\phi$ and $\kappa$ are shown in \cite{AN1} to be equivalent (such that $c\kappa(\lambda}\def\ep{\epsilon}\def\ka{\kappa)\leq \phi(\lambda}\def\ep{\epsilon}\def\ka{\kappa)\leq C \kappa(\lambda}\def\ep{\epsilon}\def\ka{\kappa)$), and the distinction between them served only to organize the proof. Secondly, in~\cite{AN1} we also assume that $\phi(r)$ is bounded from below by $(\log(4+ r))^a$ for some $a\ge 0$. Here we will use that this property holds with~$a=0$. Once the previous clarifications have been done, it remains to explain that in \cite{AN1} we consider the equation~\eqref{n1} while here we work with~\eqref{n2}. The elliptic term $(-\partial_{x}^2)$ is trivial to handle since in \cite{AN1} we only applied an $L^2$-energy estimate and since the latter operator is positive. Eventually, the cut-off function $(1-\chi(\alpha/\varepsilon))$ is also harmless in the various computations used to prove Lemma~$3.4$ in~\cite{AN1}. \end{remark} Secondly, we recall two interpolation inequalities from \cite[Lemma~$3.5$]{AN1}. Hereafter, we use the notations \begin{equation}\label{n67} \begin{aligned} A_\phi(t)&=\big\Vert \left\vert D\right\vert^{\frac{3}{2},\phi}f(t)\big\Vert_{L^2}^2,\\ B_\phi(t)&=\big\Vert \left\vert D\right\vert^{2,\phi}f(t)\big\Vert_{L^2}^2, \\ P_\phi(t)&=\big\Vert \left\vert D\right\vert^{\frac{5}{2},\phi}f(t)\big\Vert_{L^2}^2, \end{aligned} \end{equation} and $$ \mu_\phi(t)=\left(\phi\left(\frac{B(t)}{A(t)}\right)\right)^{-1} . $$ \begin{lemma}\label{L:3.5} Consider a real number $7/4\leq s\leq 2$. Then, there exists a positive constant $C$ such that, for any $T>0$, any smooth solution $f\in C^{1}([0,T];H^{\infty}(\mathbb{R}))$ to~\eqref{n2} and any $t\in [0,T]$, \begin{align} &\left\Vert f(t)\right\Vert_{\dot H^s}\leq C\mu_\phi(t) A_\phi(t)^{2-s} B_\phi(t)^{s-\frac{3}{2}},\label{Z20'}\\ &\big\Vert \left\vert D\right\vert^{\frac{7}{4},\phi^{2}}f(t)\big\Vert_{L^2}\leq C \mu_\phi(t)A_\phi(t)^{\frac{1}{4}}B_\phi(t)^{\frac{1}{4}}.\label{n110} \end{align} \end{lemma} From these two lemmas, we get at once the following \begin{proposition}\label{P:3.3} There exist two positive constants $C_1$ and $C_2$ such that, for any $T>0$ and any smooth solution $f\in C^1([0,T];H^\infty(\mathbb{R}))$ of the Muskat equation~\eqref{n2}, \begin{multline}\label{a2} \frac{\diff}{\dt} A_\phi(t)+C_1\frac{B_\phi(t)}{1+\left\Vert f_x(t)\right\Vert_{L^\infty}^2}+\|\log(\varepsilon)\|^{-1} P_\phi(t)\\ \leq C_2 \left( \sqrt{A_\phi(t)}+A_\phi(t) \right)\mu_\phi(t) B_\phi(t). \end{multline} \end{proposition} We will also need an estimate for the $L^2$-norm. \begin{lemma}\label{L:L2} There holds $$ \frac{1}{2}\frac{\diff}{\dt} \left\Vert f(t)\right\Vert_{L^2}^2\leq C \varepsilon^{\frac{1}{2}}\left\Vert f\right\Vert_{\dot{H}^{\frac{3}{2}}}\left\Vert f\right\Vert_{L^2}. $$ In particular, \begin{equation}\label{X3deux} \left\Vert f(t)\right\Vert_{L^2}\leq \left\Vert f_0\right\Vert_{L^2} +C\varepsilon^{\frac{1}{2}}\int_0^t\left\Vert f(\tau)\right\Vert_{\dot{H}^{\frac{3}{2}}}\diff \! \tau. \end{equation} \end{lemma} \begin{proof} Set \begin{equation}\label{f10} R_\varepsilon(f)=-\frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2} \chi\left(\frac{\alpha}{\varepsilon}\right)\diff \! \alpha. \end{equation} We multiply the equation by $f$ to obtain $$ \frac{1}{2}\frac{\diff}{\dt} \left\Vert f(t)\right\Vert_{L^2}^2\leq \frac{1}{\pi}\bigg\langle \int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\diff \! \alpha , f\bigg\rangle +\langle R_\varepsilon(f),f\rangle. $$ Now, by \cite[Section 2]{CCGRPS-JEMS2013}, the first term in the right-hand side has a sign. Indeed: \begin{multline} \int_\mathbb{R}\left[\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\diff \! \alpha \right]f(x)\diff \! x \\ =-\iint_{\mathbb{R}^2}\log\left[\sqrt{1+\frac{(f(t,x)-f(t,x-\alpha))^2}{\alpha^2}}\right]\diff \! x\diff \! \alpha. \end{multline} It remains to estimate $R_\varepsilon(f)$. To do so, we use the estimate~\eqref{n15AN1} to get \begin{equation}\label{f9} \begin{aligned} \left\Vert R_\varepsilon(f)\right\Vert_{L^2} &\lesssim \int_{\|\alpha\|\leq 2\varepsilon}\left\Vert\Delta_\alpha f_{x}\right\Vert_{L^2}\diff \! \alpha \\ & \lesssim \varepsilon^{\frac{1}{2}}\left(\int_{\mathbb{R}}\left\Vert\Delta_\alpha f_{x}\right\Vert_{L^2}^2\diff \! \alpha\right)^\frac{1}{2} \lesssim \varepsilon^{\frac{1}{2}}\left\Vert f\right\Vert_{\dot{H}^{\frac{3}{2}}}, \end{aligned} \end{equation} which completes the proof. \end{proof} \subsection{Estimate of the H\"older norm} To exploit the Sobolev energy estimate~\eqref{a2}, the main difficulty is to estimate from above the factor $1+\left\Vert f_x(t)\right\Vert_{L^\infty}^2$. This is where we will apply Lemma~\ref{L:2.1}. This in turn requires to estimate the H\"older norm $\left\Vert \cdot\right\Vert_{\dot{C}^{2,\beta_0}}$ of $f$. This is the purpose of the following result. We will prove an estimate valid on arbitrary large time scale, which will be used later to prove a global existence result. \begin{proposition}\label{P:2.6} For any $0<\beta<1/2$, there exist two positive constant $\varepsilon_0$ and $c_0$ such that, for any $\varepsilon\in (0,\varepsilon_0]$, any smooth solution $f\in C^1([0,T];H^{\infty}(\mathbb{R}))$ of the Muskat equation~\eqref{n2}, and any time $t\leq \min\{\varepsilon^{-c_0},T\}$, there holds \begin{align} &\varepsilon^{\beta}\int_{0}^{t}\left\Vert f(\tau)\right\Vert_{C^{2,\beta}}d\tau \leq \varepsilon^{\frac{\beta}{2}}\left\Vert f_0\right\Vert_{\dot H^{\frac{3}{2}}} \\ &~~~+ \varepsilon^{\frac{\beta}{2}}\left(1+\sup_{s\in [0,t]}\left\Vert f(s)\right\Vert_{H^{\frac{3}{2}}}\right)^2 \log\left(2+\int_{0}^{t} \left\Vert f(s)\right\Vert_{\dot H^{2}}^2\diff \! s\right)^{\frac{1}{2}} \left(\int_{0}^{t} \left\Vert f(s)\right\Vert_{\dot H^{2}}^2\diff \! s\right)^{\frac{1}{2}}.\nonumber \end{align} \end{proposition} \begin{proof} The classical Sobolev embeddings implies that $$ \left\Vert f(t)\right\Vert_{\dot{C}^{2,\beta}} \lesssim \left\Vert f(t)\right\Vert_{\dot H^{\frac{5}{2}+\beta}}. $$ To estimate the latter Sobolev norm, the key point will be to apply the following interpolation inequality. \begin{lemma} Consider three real numbers $$ \gamma>0,\quad \beta_1>0\quad\text{and}\quad 0<\beta_2<2. $$ Then, there exists a constant $C$ such that, for any function $g=g(t,x)$, \begin{equation}\label{X2} \begin{aligned} \left\Vert g(t)\right\Vert_{\dot H^\gamma} &\lesssim \frac{1}{(\nu t)^{\frac{\beta_1}{2}}} \left\Vert g(0)\right\Vert_{\dot H^{\gamma-\beta_1}}\\ &\quad+\int_0^{t} \frac{1}{(\nu (t-s))^{\frac{\beta_2}{2}}}\big\Vert \partial_tg(s)-\nu\partial_{xx} g(s)\big\Vert_{\dot H^{\gamma-\beta_2}}\diff \! s. \end{aligned} \end{equation} \end{lemma} \begin{proof} Set $G\mathrel{:=}\partial_tg-\nu\partial_{xx} g$. Then, one has, \begin{equation} \hat g(t,\xi)=e^{-\nu t\|\xi\|^{2}}\hat g(0,\xi)+\int_{0}^{t}e^{-\nu (t-s)\|\xi\|^{2}}\hat G(s,\xi)\diff \! s. \end{equation} The desired results then follows from Minkowski's inequality. \end{proof} Now, apply \eqref{X2} with $$ \gamma=\frac{5}{2}+\beta, \quad\beta_1=1+\beta, \quad\beta_2=\frac{3}{2}+\beta,\quad \nu=\|\log(\varepsilon)\|^{-1}, $$ to get \begin{align} \left\Vert f(t)\right\Vert_{\dot C^{2,\beta}}&\lesssim \left\Vert f(t)\right\Vert_{\dot H^{\frac{5}{2}+\beta}}\nonumber\\ &\lesssim \|\log(\varepsilon)\|^{\frac{1+\beta}{2}} t^{-\frac{1+\beta}{2}}\|\|f_0\|\|_{\dot H^{\frac{3}{2}}}\nonumber\\ &\quad+\int_0^{t} \|\log(\varepsilon)\|^{\frac{\frac{3}{2}+\beta}{2}} (t-s)^{-\frac{\frac{3}{2}+\beta}{2}} \big\Vert\partial_tf-\|\log(\varepsilon)\|^{-1}\partial_x^2f\big\Vert_{\dot H^1}\diff \! s.\label{Y1} \end{align} It remains to estimate the $\dot{H}^1$-norm of $\partial_tf-\|\log(\varepsilon)\|^{-1}\partial_{xx} f$. In view of the equation~\eqref{n2}, this is equivalent to bound the $\dot{H}^1$-norm of $$ \frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\left(1-\chi\left(\frac{\alpha}{\varepsilon}\right)\right)\diff \! \alpha. $$ We will split the latter term into two pieces and estimate them separately. Firstly, directly from~\eqref{X3} and Minkowski's inequality, we obtain that \begin{align} \left\Vert \frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2} \chi\left(\frac{\alpha}{\varepsilon}\right)\diff \! \alpha\right\Vert_{\dot H^1} &\lesssim \int_{\|\alpha\|\leq 2\varepsilon}\left(\left\Vert\Delta_\alpha f_{xx}\right\Vert_{L^2} +\left\Vert\Delta_\alpha f_{x}\right\Vert_{L^4}^2\right)\diff \! \alpha\\ & \lesssim \varepsilon^{\frac{1}{2}+\beta}\left(\int_{\mathbb{R}}\left\Vert\Delta_\alpha f_{xx}\right\Vert_{L^2}^2\|\alpha\|^{-2\beta}\diff \! \alpha\right)^\frac{1}{2}+ \int_{\mathbb{R}}\left\Vert\Delta_\alpha f_{x}\right\Vert_{L^4}^2\diff \! \alpha. \end{align} Now we use the following inequality: \begin{equation}\label{n15AN1} \iint_{\mathbb{R}^2} \big\vert \Delta_\alpha \tilde{f}\big\vert^2 \|\alpha\|^{-2\beta}\diff \! \alpha\diff \! x\sim \big\Vert \tilde{f}\big\Vert_{\dot{H}^{\frac{1}{2}+\beta}}^2. \end{equation} Indeed, $$ \iint_{\mathbb{R}^2} \big\vert \Delta_\alpha \tilde{f}\big\vert^2 \|\alpha\|^{-2\beta}\diff \! \alpha\diff \! x= \iint_{\mathbb{R}^2} \left[\frac{\big\vert \tilde{f}(x)-\tilde{f}(x-\alpha)\big\vert}{\left\vert \alpha\right\vert^{1/2+\beta}}\right]^2\frac{\diff \! \alpha}{\left\vert\alpha\right\vert}\diff \! x \sim \big\Vert \tilde{f}\big\Vert_{\dot{H}^{\frac{1}{2}+\beta}}^2. $$ Similarly, using Sobolev embedding in Besov's spaces, we get $$ \int_{\mathbb{R}}\left\Vert\Delta_\alpha f_{x}\right\Vert_{L^4}^2\diff \! \alpha\lesssim \left\Vert f\right\Vert_{\dot H^{\frac{7}{4}}}^2. $$ It follows that \begin{equation}\label{f1} \left\Vert \frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2} \chi\left(\frac{\alpha}{\varepsilon}\right)\diff \! \alpha\right\Vert_{\dot H^1} \lesssim \varepsilon^{\frac{1}{2}+\beta}\left\Vert f\right\Vert_{\dot{H}^{\frac{5}{2}+\beta}}+ \left\Vert f\right\Vert_{\dot H^{2}} \left\Vert f\right\Vert_{\dot H^{\frac{3}{2}}}, \end{equation} where we used an interpolation inequality in Sobolev spaces. On the other hand, it follows from the estimate~\eqref{n33} below that, \begin{equation}\label{f2} \begin{aligned} \left\Vert \int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\diff \! \alpha\right\Vert_{\dot H^1} &\lesssim \left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}\\ &\lesssim \left( 1+\left\Vert f\right\Vert_{H^{\frac32}}\right)^2 \log\left(2+\left\Vert f\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f\right\Vert_{\dot H^{2}}. \end{aligned} \end{equation} By gathering the two previous estimates, we conclude that \begin{multline} \left\Vert\partial_tf-\|\log(\varepsilon)\|^{-1}\partial_x^2f\right\Vert_{\dot H^1}\\ \lesssim \varepsilon^{\frac{1}{2}+\beta}\left\Vert f\right\Vert_{H^{\frac{5}{2}+\beta}}+ \left(1+\left\Vert f\right\Vert_{H^{\frac{3}{2}}}\right)^2\log\left(2+\left\Vert f\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f\right\Vert_{\dot H^{2}}. \end{multline} Set $$ b=\frac{\frac{3}{2}+\beta}{2}\cdot $$ By reporting this bound in~\eqref{Y1}, we find that \begin{align} &\left\Vert f(t)\right\Vert_{\dot H^{\frac{5}{2}+\beta}}\lesssim \|\log(\varepsilon)\|^{\frac{1+\beta}{2}} t^{-\frac{1+\beta}{2 }}\left\Vert f_0\right\Vert_{\dot H^{\frac{3}{2}}}\\ &\quad + \varepsilon^{\frac{1}{2}+\beta}\|\log(\varepsilon)\|^{b}\int_0^{t} (t-s)^{-b} \left\Vert f(s)\right\Vert_{\dot H^{\frac{5}{2}+\beta}}\diff \! s\\ &\quad+\|\log(\varepsilon)\|^{b}\int_0^{t} (t-s)^{-b} \left(1+\left\Vert f(s)\right\Vert_{H^{\frac{3}{2}}}\right)^2\log\left(2+\left\Vert f(s)\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f(s)\right\Vert_{\dot H^{2}}\diff \! s. \end{align} So, \begin{align} &\int_{0}^{t}\left\Vert f(\tau )\right\Vert_{\dot H^{\frac{5}{2}+\beta}}\diff \! \tau \lesssim \left\vert\log(\varepsilon)\right\vert^{\frac{1+\beta}{2 }} t^{1-\frac{1+\beta}{2 }}\left\Vert f_0\right\Vert_{\dot H^{\frac{3}{2}}}\\ &\quad+ \varepsilon^{\frac{1}{2}+\beta}\|\log(\varepsilon)\|^{b} t^{1-b} \int_0^{t}\left\Vert f(s)\right\Vert_{\dot H^{\frac{5}{2}+\beta}}\diff \! s\\ &\quad+\|\log(\varepsilon)\|^{b} t^{1-b}\int_0^{t}\left(1+\left\Vert f(s)\right\Vert_{H^{\frac{3}{2}}}\right)^2\log\left(2+\left\Vert f(s)\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f(s)\right\Vert_{\dot H^{2}}\diff \! s. \end{align} As a result, there exists $c_0>0$ and $\varepsilon_0\leq 1$ such that, if $t\leq \varepsilon^{-c_0}$ and $\varepsilon\leq \varepsilon_0$, \begin{align} \int_{0}^{t}\left\Vert f(\tau )\right\Vert_{\dot H^{\frac{5}{2}+\beta}}\diff \! \tau &\leq \varepsilon^{-\frac{\beta}{2}}\|\|f_0\|\|_{\dot H^{\frac{3}{2}}} \\ &\quad +\|\log(\varepsilon)\|^{b} t^{1-b} \mathcal{K}(t)\int_0^{t} \log\left(2+\left\Vert f(s)\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f(s)\right\Vert_{\dot H^{2}}\diff \! s, \end{align} where $$ \mathcal{K}(t)=\sup_{s\in [0,t]}\left(1+\left\Vert f(s)\right\Vert_{H^{\frac{3}{2}}}\right)^2. $$ Now observe that \begin{multline} \int_0^{t} \log\left(2+\left\Vert f(s)\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f(s)\right\Vert_{\dot H^{2}}\diff \! s\\ \leq (t+1)^{\frac{1}{2}} \log\left(2+ \int_{0}^{t} \left\Vert f(s)\right\Vert_{\dot H^{2}}^2ds\right)^{\frac{1}{2}} \left( \int_{0}^{t} \left\Vert f(s)\right\Vert_{\dot H^{2}}^2ds\right)^{\frac{1}{2}}. \end{multline} Therefore, up to modifying the values of $c_0>0$ and $\varepsilon_0$, we see that, for $t\leq \varepsilon^{-c_0}$ and $\varepsilon\leq \varepsilon_0$, we have \begin{align} &\varepsilon^{\beta}\int_{0}^{t}\left\Vert f(\tau)\right\Vert_{\dot C^{2,\beta}}\diff \! \tau \lesssim \varepsilon^{\frac{\beta}{2}}\left\Vert f_0\right\Vert_{\dot H^{\frac{3}{2}}} \\&~~~+ \varepsilon^{\frac{\beta}{2}}\mathcal{K}(t)\log\left(2+ \int_{0}^{t} \left\Vert f(s)\right\Vert_{\dot H^{2}}^2\diff \! s\right)^{\frac{1}{2}} \left( \int_{0}^{t} \left\Vert f(s)\right\Vert_{\dot H^{2}}^2\diff \! s\right)^{\frac{1}{2}}. \end{align} This completes the proof. \end{proof} \subsection{Global in time estimates, under a smallness assumption} \begin{proposition} Let $T>0$ and consider a smooth solution $f\in C^1([0,T],H^{\infty}(\mathbb{R}))$ of the Muskat equation~\eqref{n2}. Set $$ K=1+16 \left(\frac{C_2}{C_1}\right)^2 $$ and assume that \begin{equation}\label{a29} 2\left(K+\frac{C_0}{C_1}\right)^\frac{1}{2}\left(2+\left\Vert \partial_x f_0\right\Vert_{L^\infty}\right)^2\left\Vert f_0\right\Vert_{\dot {H}^{\frac{3}{2}}}\leq 1, \end{equation} where the constants $C_0,C_1,C_2$ are as defined in the statements of Lemma~\ref{L:2.1} and Proposition~\ref{P:3.3}. Then there exists $\varepsilon_0$ depending only on $C_0,C_1,C_2$ and $\|\|f_0\|\|_{L^2}$ such that, if $\varepsilon\leq \varepsilon_0$, then \begin{equation}\label{a30} \sup_{0\leq \tau\leq T}\left\Vert f(\tau)\right\Vert_{{H}^{\frac{3}{2}}} \leq \frac{1}{\sqrt{K}\big(2+\left\Vert \partial_xf_0\right\Vert_{L^\infty}\big)^2}\quad\text{and} \quad \int_{0}^{T}\left\Vert f(\tau)\right\Vert_{\dot{H}^2}^2\diff \! \tau\leq \frac{1}{C_0} \cdot \end{equation} \end{proposition} \begin{proof} We apply the previous {\em a priori\/} estimate~\eqref{a2} in the simplest case where~$\phi=1$. With this choice, the quantities $A_\phi$ and $B_\phi$ defined by~\eqref{n67} simplify to \begin{equation}\label{n167} \begin{aligned} A(t)&=\big\Vert \left\vert D\right\vert^{\frac{3}{2}}f(t)\big\Vert_{L^2}^2, \\ B(t)&=\big\Vert \left\vert D\right\vert^{2}f(t)\big\Vert_{L^2}^2=\left\Vert f(t)\right\Vert_{\dot{H}^2}^2. \end{aligned} \end{equation} Introduce the set $$ I=\left\{ t\in [0,T]\,;\, \int_{0}^{t}B(\tau)\diff \! \tau\leq \frac{2}{3C_0} \text{ and } \sup_{0\leq \tau\leq t}A(\tau)\leq \frac{1}{K\big(2+\left\Vert \partial_xf_0\right\Vert_{L^\infty}\big)^4}\right\}. $$ We want to prove that $I=[0,T]$. Since $0$ belongs to $I$ by assumption on the initial data, and since $I$ is closed, it suffices to prove that $I$ is open. To do so, we consider a time $t^\in [0,T)$ which belongs to $I$. Our goal is to prove that $$ \int_{0}^{t^}B(\tau)\diff \! \tau\leq \frac{1}{2C_0} \text{ and } \sup_{0\leq \tau\leq t^}A(\tau)\leq \frac{1}{4K\big(2+\left\Vert \partial_xf_0\right\Vert_{L^\infty}\big)^4}. $$ This will imply at once that $t^$ belongs to the interior of $I$. Since $\mu(t)=1$ for $\phi\equiv 1$, the estimate~\eqref{a2} implies that there are two positives constants $C_1,C_2$ such that \begin{equation}\label{a4} \frac{\diff}{\dt} A(t)+C_1\frac{B(t)}{1+\left\Vert \partial_x f(t)\right\Vert_{L^\infty}^2}\leq C_2\Big(A(t)+\sqrt{A(t)}\Big)B(t). \end{equation} By combining Proposition~\ref{P:2.6} with Lemma~\ref{L:2.1}, we get, for any $t$, \begin{align} & \left\Vert \partial_xf(t)\right\Vert_{L^\infty}- \left\Vert \partial_xf_0\right\Vert_{L^\infty}\leq C_0\int_{0}^{t}B(\tau) \diff \! \tau+C_0 \varepsilon^{\frac{\beta}{2}}\left\Vert f_0\right\Vert_{\dot H^{\frac{3}{2}}}\\&+ C_0 \varepsilon^{\frac{\beta}{2}}\left[\sup_{s\in [0,t]}\left(1+\left\Vert f(s)\right\Vert_{H^{\frac{3}{2}}}\right)^2\right]\log\left(2+\int_{0}^{t} B(\tau)\diff \! \tau\right)^{\frac{1}{2}} \left( \int_{0}^{t} B(\tau)\diff \! \tau\right)^{\frac{1}{2}}. \end{align} By \eqref{X3deux}, \begin{equation} \sup_{s\in [0,t]}\left\Vert f(s)\right\Vert_{L^{2}}\leq \left\Vert f_0\right\Vert_{L^2}+C\varepsilon^{\frac{1}{2}}t \sup_{s\in [0,t]}\left\Vert f(s)\right\Vert_{\dot H^{\frac{3}{2}}}. \end{equation} This implies \begin{equation}\label{f11} \sup_{s\in [0,t]}\left\Vert f(s)\right\Vert_{H^{\frac{3}{2}}}\leq \left\Vert f_0\right\Vert_{L^2}+(1 +C\varepsilon^{\frac{1}{2}}t) \sup_{s\in [0,t]}\left\Vert f(s)\right\Vert_{\dot H^{\frac{3}{2}}}. \end{equation} If $t\leq t^$, then the bound on the integral of $B$, $ \sup_{0\leq \tau\leq t^}\|\|f\|\|_{\dot H^{\frac{3}{2}}}\le1$ and \eqref{f11} imply that \begin{align} & \left\Vert \partial_xf(t)\right\Vert_{L^\infty}- \left\Vert \partial_xf_0\right\Vert_{L^\infty}\leq \frac{1}{2}+C_0 \varepsilon^{\frac{\beta}{2}}+ C_0 \varepsilon^{\frac{\beta}{2}}\left(1+ \left\Vert f_0\right\Vert_{L^2}+(1 +C\varepsilon^{\frac{1}{2}}t^)\right)^2\log\left(3\right)^{\frac{1}{2}}. \end{align} For $\varepsilon$ small enough, we conclude that $$ \left\Vert \partial_xf(t)\right\Vert_{L^\infty}\leq \frac{2}{3}+\left\Vert \partial_xf_0\right\Vert_{L^\infty}. $$ On the other hand, if $t^\in I$, then for any $t\leq t^$ we have \begin{equation} A(t)+\sqrt{A(t)}\leq 2\sqrt{A(t)}\leq \frac{2}{\sqrt{K}\big(2+\left\Vert \partial_xf_0\right\Vert_{L^\infty}\big)^2}\cdot \end{equation} Consequently, for any $t\leq t^\star$, \eqref{a4} gives $$ \frac{\diff}{\dt} A(t)+C_1\frac{B(t)}{\big(2+\left\Vert \partial_{x}f_0\right\Vert_{L^\infty}\big)^2} \leq \frac{2C_2}{\sqrt{K}\big(2+\left\Vert \partial_xf_0\right\Vert_{L^\infty}\big)^2}B(t). $$ By definition of $K$, we have $$ K\ge \frac{16C_2^2}{C_1^2}, $$ so, for any $t\leq t^\star$, \begin{equation}\label{a20} \frac{\diff}{\dt} A(t)+\frac{C_1}{2}\frac{B(t)}{\big(2 +\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\big)^2}\leq 0. \end{equation} Integrate this on the time interval $[0,t^]$, to infer that \begin{equation} \sup_{t\in [0,t^]}A(t)+\frac{C_1}{2\big(2 +\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\big)^2} \int_{0}^{t^}B(t)\diff \! t\leq A(0). \end{equation} Using the smallness assumption~\eqref{a29}, the previous inequality~\eqref{a20} implies at once that \begin{align} &\sup_{t\in [0,t^]}A(t)\leq A(0)\leq \frac{1}{4K\big(2+\left\Vert \partial_xf_0\right\Vert_{L^\infty}\big)^4},\\ &\int_{0}^{t^}B(t)\diff \! t \leq \frac{2\big(2+\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\big)^2}{C_1}A(0)\leq \frac{1}{2C_0}. \end{align} These are the wanted bootstrap inequalities. As explained above, by connexity, this proves that $I=[0,T]$, which implies the desired results in~\eqref{a30}. \end{proof} \subsection{A priori estimates locally in time, for arbitrary initial data} \begin{proposition} Consider $\phi$ satisfying assumptions~$(\rm{H}\ref{H1})$--$(\rm{H}\ref{H3})$ in Definiton~\ref{defi:D}. Let $T>0$ and consider a smooth solution $f\in C^1([0,T],H^{\infty}(\mathbb{R}))$ of the Muskat equation~\eqref{n2}. For any $M_0>0$ there exists $\varepsilon_0>0$ and $T_0>0$ such that the following properties holds. If $\varepsilon\in (0,\varepsilon_0]$ and $$ \big\Vert \left\vert D\right\vert^{\frac{3}{2},\phi}f(0)\big\Vert_{L^2}^2\leq M_0, $$ then, with $T^=\min\{T,T_0\}$, there holds $$ \sup_{t\in [0,T^]}A_\phi(t)\leq 5M_0,\quad \int_0^{T^}\mu_\phi(t)^2B_\phi(t)\diff \! t\leq \frac{1}{C_0}, $$ where $A_\phi$, $B_\phi$, $\mu_\phi$ are defined in~\eqref{n67} while $C_0$ is given by Lemma~\ref{L:2.1}. \end{proposition} \begin{proof} For this proof we skip the index $\phi$ and write simply $A,B,\mu$. Since (see~\eqref{Z20'}), $$ \left\Vert f(t)\right\Vert_{\dot H^2}\leq C\mu(t) B(t)^{\frac{1}{2}}. $$ We then apply Proposition~\ref{P:2.6} for some fixed parameter $\beta>0$. Then, it follows from~\eqref{a2} that \begin{equation}\label{a200} \frac{\diff}{\dt} A(t)+C_1\frac{B(t)}{\nu(t)^2}\leq C_2 \left( \sqrt{A(t)}+A(t) \right)\mu(t) B(t), \end{equation} where \begin{align} &\nu(t)=1+ \left\Vert \partial_xf_0\right\Vert_{L^\infty}+ C_0\int_{0}^{t}\mu(\tau)^{2}B(\tau) \diff \! \tau +C_0 \varepsilon^{\frac{\beta}{2}}\left\Vert f_0\right\Vert_{\dot H^{\frac{3}{2}}}\\&+ C_0 \varepsilon^{\frac{\beta}{2}}\left[\sup_{\tau\in [0,t]}\left(1+\left\Vert f(\tau)\right\Vert_{H^{\frac{3}{2}}}\right)^2\right]\log\left(2+\int_{0}^{t} \mu(\tau)^{2}B(\tau)\diff \! \tau\right)^{\frac{1}{2}} \left( \int_{0}^{t} \mu(\tau)^{2}B(\tau)\diff \! \tau\right)^{\frac{1}{2}}. \end{align} Given a positive number $T_0$ to be determined, introduce the set $$ I(T_0)=\left\{ t\in [0,\min\{T,T_0\}]\,;\, \int_{0}^{t}\mu(\tau)^2B(\tau)\diff \! \tau\leq \frac{2}{3C_0} \text{ and } \sup_{0\leq \tau\leq t}A(\tau)\leq 5M_0\right\}. $$ We want to prove that $I(T_0)=[0,\min\{T,T_0\}]$. Since $0$ belongs to $I(T_0)$ by assumption on the initial data, and since $I(T_0)$ is closed, it suffices to prove that $I(T_0)$ is open. To do so, we consider a time $t^\in [0,\min\{T,T_0\})$ which belongs to $I(T_0)$. Our goal is to prove that $$ \int_{0}^{t^}\mu(\tau)^2B(\tau)\diff \! \tau\leq \frac{1}{2C_0} \text{ and } \sup_{0\leq \tau\leq t^}A(\tau)\leq 4M_0. $$ This will imply at once that $t^$ belongs to the interior of $I(T_0)$. As in the previous proof, we use \eqref{f11} to write \begin{equation}\label{f12} \sup_{s\in [0,t]}\left\Vert f(s)\right\Vert_{H^{\frac{3}{2}}}\leq \left\Vert f_0\right\Vert_{L^2}+(1 +C\varepsilon^{\frac{1}{2}}t) \sup_{s\in [0,t]}\left\Vert f(s)\right\Vert_{\dot H^{\frac{3}{2}}}. \end{equation} It $t\leq t^$ with $t^\in I(T_0)$, then \begin{align} \nu(t)&\leq 1+ \left\Vert \partial_xf_0\right\Vert_{L^\infty}+ \frac{2}{3} +C_0 \varepsilon^{\frac{\beta}{2}}M_0\\ &\quad+ C_0 \varepsilon^{\frac{\beta}{2}}\left(1+\left\Vert f_0\right\Vert_{L^2}+6M_0\right)^2 \log\left(2+ \frac{2}{3C_0}\right)^{\frac{1}{2}} \left(\frac{2}{3C_0}\right)^{\frac{1}{2}}. \end{align} Hence, one can define $\varepsilon_0$ small enough, depending only on $M_0$, $\left\Vert f_0\right\Vert_{L^2}$ and the fixed parameter $\beta$, such that if $\varepsilon\leq \varepsilon_0$ and if $t^\in I(T_0)$, then for any $t\in [0,t^]$, we have $$ \nu(t)\leq 2+\left\Vert \partial_xf_0\right\Vert_{L^\infty}. $$ Consequently \begin{align} \frac{\diff}{\dt} A(t) +C_1 \frac{B(t)}{(2+\left\Vert \partial_{x}f_0\right\Vert_{L^\infty})^2} \leq C_2\Big(A(t)+\sqrt{A(t)}\Big) \mu(t)B(t). \end{align} Introduce the function $$ \mathcal{E}(r,m)\mathrel{:=} \sup_{\rho\ge 0}\left\{C_2 \big(\sqrt{r}+r\big)\left(\phi\left(\frac{\rho}{r}\right)\right)^{-1} \rho-\frac{C_1}{2}\frac{\rho}{m}\right\}\cdot $$ Then, for any $t\in [0,t^]$, we have \begin{align} \frac{\diff}{\dt} A(t)+\frac{C_1}{2}\frac{B(t)}{\big(2+\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\big)^2} \leq\mathcal{E}\Big(A(t),\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\Big). \end{align} Assume that the number $T_0$ satisfies $$ T_0\leq\frac{A(0)}{4\mathcal{E}\big(4A(0),\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\big)}\cdot $$ Then, for any $t\leq t^$, we get that \begin{equation} \sup_{\tau\leq t} A(\tau)+\frac{C_1}{2}\frac{1}{\big(2+\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\big)^2} \int_{0}^{t}B(\tau)\diff \! \tau\leq 4A(0). \end{equation} In particular, for $t=t^$, this gives \begin{equation}\label{Z2} \sup_{t\leq t^} A(t)\leq 4A(0),\quad \int_{0}^{t^}B(t)\diff \! t\leq \frac{8A(0)}{C_1}\big(2+\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\big)^2. \end{equation} To get the result, we must show that \begin{equation}\label{Z} C_0\int_{0}^{T}\mu(t)^2 B(t)\diff \! t\leq \frac{1}{2}. \end{equation} Recall that $$ \mu(t)=\left(\phi\left(\frac{B(t)}{A(t)}\right)\right)^{-1}. $$ Since $\phi$ is increasing and since $A(t)\leq 4A(0)$, we have $$ \mu(t)\leq \left(\phi\left(\frac{B(t)}{4A(0)}\right)\right)^{-1}. $$ Now, we claim that the function $F\colon [0,+\infty)\to[0,+\infty)$, defined by $$ F(r)=\left(\phi\left(\frac{r}{4A(0)}\right)\right)^{-1}r, $$ is increasing. To see this decompose $F(r)$ under the form $F(r)=F_1(r)\left(F_2\left(r\right)\right)^2$ with $$ F_1(r)=\frac{r}{(\log(\lambda_0+r))^2}\quad F_2(r)=\frac{\log(\lambda_0+r)}{\phi(r/4A(0))}\cdot $$ Then \begin{align} &\int_{0}^{t^}\mu(t)B(t)\diff \! t \leq \int_{0}^{t^}\left(\phi\left(\frac{B(t)}{4A(0)}\right)\right)^{-2}B(t)\diff \! t\\ &\leq \int_{0}^{t^}\left(\phi\left(\frac{r}{4A(0)}\right)\right)^{-2}r \diff \! t +\int_{0}^{t^}\left(\phi\left(\frac{r}{4A(0)}\right)\right)^{-2}B(t)\diff \! t\\ &\overset{\eqref{Z2}}\leq t^\left(\phi\left(\frac{r}{4A(0)}\right)\right)^{-2}r+ \left(\phi\left(\frac{r}{4A(0)}\right)\right)^{-2}8A(0)\left(2+\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\right)^2, \end{align} for any $r\geq 1$. Now we successively determine two numbers $r_0>1$ and $T_0>0$ such that \begin{equation} C_0\left(\phi\left(\frac{r_0}{4A(0)}\right)\right)^{-2}8A(0)\left(2+\left\Vert\partial_{x}f_0\right\Vert_{L^\infty}\right)^2 =\frac{1}{4}, \end{equation} and \begin{equation} T_0\left(\phi\left(\frac{r_0}{4A(0)}\right)\right)^{-2}r_0=\frac{1}{4}\cdot \end{equation} With this choice we get~\eqref{Z} and we obtain that $I(T_0)=[0,\min\{T,T_0\}]$, which is equivalent to the statement of the proposition. \end{proof} \subsection{Transfer of compactness}\label{S:Transfer} Previously, we have proven {\em a priori\/} estimates for the spatial derivatives. In this paragraph, we gather results from which we will infer estimates for the time derivative as well as for the nonlinearity in the Muskat equation. These estimates serve to pass to the limit the equation (which is needed to regularize the solutions). The Muskat equation~\eqref{n1} can be written under the form \begin{align}\label{eqT} \partial_tf+\left\vert D\right\vert f = \mathcal{T}(f)f, \end{align} where $\mathcal{T}(f)$ is the operator defined by \begin{equation}\label{def:T(f)f} \mathcal{T}(f)g = -\frac{1}{\pi}\int_\mathbb{R}\left(\partial_x\Delta_\alpha g\right) \frac{\left(\Delta_\alpha f\right)^2}{1+\left(\Delta_\alpha f\right)^2}\diff \! \alpha. \end{equation} We recall the following result from Proposition~$2.3$ in \cite{Alazard-Lazar} and from Remark~$2.9$ and Propositions~$2.10$ and~$2.13$ in~\cite{AN1}. \begin{proposition}\label{P:2.11} $i)$ For all $\delta\in [0,1/2)$, there exists a constant $C>0$ such that, for all functions $f_1,f_2$ in $\dot{H}^{1-\delta}(\mathbb{R})\cap \dot{H}^{\frac{3}{2}+\delta}(\mathbb{R})$, $$ \left\Vert (\mathcal{T}(f_1)-\mathcal{T}(f_2))f_2\right\Vert_{L^2} \leq C \left\Vert f_1-f_2\right\Vert_{\dot{H}^{1-\delta}}\left\Vert f_2\right\Vert_{\dot{H}^{\frac{3}{2}+\delta}}. $$ $ii)$ One can decompose the nonlinearity under the form \begin{equation}\label{n1T} \mathcal{T}(f)g =\frac{(\partial_xf)^2}{1+(\partial_xf)^2}\left\vert D\right\vert g +V(f)\partial_x g+R(f,g), \end{equation} where the coefficient $V(f)$ and the remainder term $R(f,g)$ satisfy the following estimates: \begin{align} &\left\Vert V(f)\right\Vert_{L^\infty}\leq C \int_\mathbb{R} \left\vert \xi\right\vert \big\vert\hat{f}(\xi)\big\vert\diff \! \xi,\label{XV}\\ &\left\Vert R(f,g)\right\Vert_{L^2}\leq C\Vert g\Vert _{\dot{H}^{\frac{3}{4}}} \left\Vert f\right\Vert_{\dot{H}^{\frac{7}{4}}},\label{XR} \end{align} for some absolute constant $C$. Moreover, \begin{equation}\label{X1} \left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}\leq C\left(\left\Vert f\right\Vert_{\dot H^{\frac32}}+\left\Vert f\right\Vert_{\dot H^{\frac32}}^2+1 +\left\Vert V(f)\right\Vert_{L^\infty}\right) \left\Vert f\right\Vert_{\dot H^{2}}, \end{equation} and, \begin{equation}\label{XH} \left\vert \big(V(f)\partial_x g,\left\vert D\right\vert g\big)\right\vert \leq C\Big( \left\Vert f\right\Vert_{\dot{H}^2}+\left\Vert f\right\Vert_{\dot{H}^{\frac{7}{4}}}^2\Big) \left\Vert g\right\Vert_{\dot{H}^\frac{1}{2}}\left\Vert g\right\Vert_{\dot{H}^1}. \end{equation} \end{proposition} For later purpose, we need a refinement of~\eqref{X1}. \begin{proposition}\label{P:2.8} There exists a positive constant $C>0$ such that, for all function $f\in H^2(\mathbb{R})$, \begin{equation}\label{n33} \left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}\leq C \left(1+\left\Vert f\right\Vert_{H^{\frac32}}\right)^2 \log\left(2+\left\Vert f\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f\right\Vert_{\dot H^{2}}. \end{equation} \end{proposition} \begin{proof} In view of~\eqref{X1} and~\eqref{XV}, it is sufficient to estimate the $L^1$-norm of $\left\vert \xi\right\vert \hat{f}$. Write, \begin{align} \int_\mathbb{R} \|\xi\| \|\hat f\| \diff \! \xi &=\int_{\|\xi\|>\lambda} \|\xi\|^{-1} \|\xi\|^{2}\|\hat f\| \diff \! \xi +\int_{\|\xi\|\leq \lambda} (\|\xi\|+1)^{-\frac{1}{2}} \|\xi\|(1+\|\xi\|)^{\frac{1}{2}}\|\hat f\| \diff \! \xi \\ &\lesssim\left(\int_{\|\xi\|>\lambda}\def\ep{\epsilon}\def\ka{\kappa} \frac{1}{\|\xi\|^{2}} \diff \! \xi \right)^{\frac{1}{2}} \Vert f\Vert_{\dot H^2}+ \left(\int_{\|\xi\|\leq \lambda}\def\ep{\epsilon}\def\ka{\kappa} \frac{1}{(\|\xi\|+1)} \diff \! \xi \right)^{\frac{1}{2}} \left( \Vert f \Vert_{\dot H^{\frac{3}{2}}}+ \Vert f\Vert_{L^2}\right) \\&\lesssim \lambda}\def\ep{\epsilon}\def\ka{\kappa^{-\frac{1}{2}} \Vert f\Vert_{\dot H^2}+ \log(1+\lambda}\def\ep{\epsilon}\def\ka{\kappa)^{\frac{1}{2}} \left( \Vert f \Vert_{\dot H^{\frac{3}{2}}}+ \Vert f\Vert_{L^2}\right). \end{align} Choosing $\lambda=\left\Vert f\right\Vert_{\dot H^2}^2$, we obtain $$ \int_\mathbb{R} \|\xi\| \|\hat f\| \diff \! \xi \lesssim 1+\log(1+\left\Vert f\right\Vert_{\dot H^2}^2)^{\frac{1}{2}} \left( \Vert f \Vert_{\dot H^{\frac{3}{2}}}+ \Vert f\Vert_{L^2}\right). $$ By reporting this in \eqref{XV} and then using \eqref{X1}, we get the desired result~\eqref{n33}. \end{proof} By using the equation~\eqref{eqT}, we deduce at once the following bound. \begin{corollary} There exists a non-decreasing function $\mathcal{F}\colon \mathbb{R}^+\to\mathbb{R}^+$ such that, for any $T>0$, any $\varepsilon$ and any smooth solution $f$ in $C^1([0,T];H^\infty(\mathbb{R}))$ of the Muskat equation~\eqref{n2}, if one sets \begin{equation} M_\varepsilon(T)=\sup_{t\in [0,T]}\left(\left\Vert f(t)\right\Vert_{\dot H^{\frac{3}{2}}}^2 +\left\Vert f(t)\right\Vert_{L^2}^2\right)+\int_{0}^{T}\left\Vert f(t)\right\Vert_{\dot H^{2}}^2\diff \! t+\left\vert \log (\varepsilon)\right\vert^{-1} \int_0^T\left\Vert f(t)\right\Vert_{\dot{H}^{\frac{5}{2}}}^2\diff \! t \end{equation} then, \begin{equation}\label{n35} \int_{0}^{T}\frac{\left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}^2}{\log\big(2+\left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}\big)}\diff \! t \leq \mathcal{F}(M_\varepsilon(T)), \end{equation} and \begin{equation}\label{n36} \int_{0}^{T} \frac{\left\Vert \partial_tf\right\Vert_{\dot H^1}^2}{\log \big(2+\Vert\partial_tf\Vert_{\dot H^1}^2)}\diff \! t\leq \mathcal{F}(M_\varepsilon(T)). \end{equation} \end{corollary} \begin{proof} Let $C$ be the constant given by Proposition~\ref{P:2.8} and set $\tilde{C}=\max\{C,1\}$. We claim that \begin{equation} \frac{\left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}^2}{\log\big(2+\left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}\big)}\leq \tilde{C}^2 \left(\left\Vert f\right\Vert_{\dot H^{\frac32}} +\left\Vert f\right\Vert_{\dot H^{\frac32}}^2+\left\Vert f\right\Vert_{\dot{H}^1}+1\right)^2\left\Vert f\right\Vert_{\dot H^{2}}^2. \end{equation} If $\left\Vert \mathcal{T}(f)f\right\Vert_{\dot H^1}\leq \left\Vert f\right\Vert_{\dot H^{2}}$, then this is obvious. Otherwise, this follows at once from~\eqref{n33}. This implies~\eqref{n35}. The proof of~\eqref{n36} follows from similar argument, using the equation to estimate $\partial_tf$ in terms of $\mathcal{T}(f)f$. \end{proof} It follows from the previous results that one can extract from the solutions of the approximate Cauchy problems~\eqref{n2} a sub-sequence converging to a solution of the Muskat equation~\eqref{n1}. Since it is rather classical, we do not include the details and refer for instance to~\cite{CG-CMP,Cordoba-Lazar-H3/2}. \subsection{Uniqueness}\label{S:3.5} To prove the uniqueness of the solution to the Cauchy problem for rough initial data, we shall prove an estimate for the difference of two solutions. \begin{proposition}\label{P:2.10} Let $T>0$ and consider two solutions $f_1,f_2$ of the Muskat equation, with initial data $f_{1,0},f_{2,0}$ respectively, satisfying $$ f_k\in C^0([0,T];\dot{W}^{1,\infty}(\mathbb{R})\cap\dot H^{\frac{3}{2}}(\mathbb{R}))\cap C^1([0,T];\dot H^{\frac{1}{2}}(\mathbb{R}))\cap L^2(0,T;\dot{H}^2(\mathbb{R})),\quad k=1,2. $$ Assume that \begin{equation}\label{Z105} \sup_{t\in [0,T]}\left(\left\Vert f_k(t)\right\Vert_{\dot H^{\frac{3}{2}}}^2+\left\Vert f_k(t)\right\Vert_{\dot W^{1,\infty}}^2\right) +\int_0^{T} \left\Vert f_k\right\Vert_{\dot H^2}^2\diff \! t \leq M<\infty,~~k=1,2. \end{equation} Then the difference $g=f_1-f_2$ is estimated by \begin{equation}\label{Z107} \sup_{t\in [0,T]} \Vert g(t) \Vert_{\dot{H}^\frac{1}{2}}\leq \Vert g(0) \Vert_{\dot{H}^\frac{1}{2}}\exp\left(C(M+1)^{5} \int_0^T \left(\left\Vert f_1\right\Vert_{\dot{H}^2}^2+\left\Vert f_2\right\Vert_{\dot{H}^2}^2\right)\diff \! t \right). \end{equation} \end{proposition} \begin{proof} Since $\partial_tf_k+\left\vert D\right\vert f_k = \mathcal{T}(f_k)f_k$, it follows from the decomposition~\eqref{n1T} of $\mathcal{T}(f_k)f_k$ that the difference $g=f_1-f_2$ satisfies \begin{align} \partial_tg+\frac{\left\vert D\right\vert g}{1+(\partial_xf_1)^2} &= V(f_1)\partial_x g+R(f_1,g)+\left(\mathcal{T}(f_2+g)-\mathcal{T}(f_2)\right)f_2. \end{align} Since $g$ belongs to $C^1([0,T];\dot H^{\frac{1}{2}}(\mathbb{R}))$, we may take the $L^2$-scalar product of this equation with $\left\vert D\right\vert g$ to get \begin{align} \frac{1}{2}\frac{\diff}{\dt}\Vert g \Vert^{2}_{\dot{H}^\frac{1}{2}}+\int\frac{( \left\vert D\right\vert g)^2}{1+(\partial_x f_{1})^2} \diff \! x &\leq \left\|\big(V(f_1)\partial_x g,\|D\| g\big)\right\|+\left\Vert R(f_1,g)\right\Vert_{L^2}\left\Vert g\right\Vert_{\dot H^1}\\ &\quad+\left\Vert \left(\mathcal{T}(f_2+g)-\mathcal{T}(f_2)\right)f_2\right\Vert_{L^2}\left\Vert g\right\Vert_{\dot H^1}. \end{align} It follows from Proposition~\ref{P:2.11} that \begin{align} \frac{\diff}{\dt}\Vert g \Vert^{2}_{\dot{H}^\frac{1}{2}}+M^{-1}\|\|g\|\|_{\dot H^1}^2 &\lesssim \left(\left\Vert f_1\right\Vert_{\dot{H}^2}+\left\Vert f_1\right\Vert_{\dot{H}^{\frac{7}{4}}}^2\right) \left\Vert g\right\Vert_{\dot H^{\frac{1}{2}}}\left\Vert g\right\Vert_{\dot H^{1}}\\ &\quad+\left\Vert f_2\right\Vert_{\dot{H}^{\frac{7}{4}}} \Vert g\Vert _{\dot{H}^{\frac{3}{4}}}\|\| g\|\|_{\dot H^1}. \end{align} By Gagliardo-Nirenberg interpolation inequality \begin{align} \frac{\diff}{\dt}\Vert g \Vert^{2}_{\dot{H}^\frac{1}{2}}+M^{-1} \|\|g\|\|_{\dot H^1}^2 &\lesssim \left\Vert f_1\right\Vert_{\dot{H}^2}\left(1+\left\Vert f_1\right\Vert_{\dot{H}^{\frac{3}{2}}}\right)\left\Vert g\right\Vert_{\dot H^{\frac{1}{2}}}\left\Vert g\right\Vert_{\dot H^{1}}\\ &\quad+\left\Vert f_2\right\Vert_{\dot{H}^{2}}^{\frac{1}{2}} \left\Vert f_2\right\Vert_{\dot{H}^{\frac{3}{2}}}^{\frac{1}{2}}\Vert g\Vert _{\dot{H}^{\frac{1}{2}}}^{\frac{1}{2}} \left\Vert g\right\Vert_{\dot H^1}^{\frac{3}{2}}\\ &\lesssim \left\Vert f_1\right\Vert_{\dot{H}^2}\left(1+M\right)\left\Vert g\right\Vert_{\dot H^{\frac{1}{2}}}\left\Vert g\right\Vert_{\dot H^{1}}+M^{\frac{1}{2}}\left\Vert f_2\right\Vert_{\dot{H}^{2}}^{\frac{1}{2}} \Vert g\Vert _{\dot{H}^{\frac{1}{2}}}^{\frac{1}{2}}\left\Vert g\right\Vert_{\dot H^1}^{\frac{3}{2}}. \end{align} Thus, thanks to Holder's inequality, one gets \begin{align} \frac{\diff}{\dt}\Vert g \Vert^{2}_{\dot{H}^\frac{1}{2}}+\frac{1}{2M} \|\|g\|\|_{\dot H^1}^2 &\leq C (M+1)^{5} \left(\left\Vert f_1\right\Vert_{\dot{H}^2}^2+\left\Vert f_2\right\Vert_{\dot{H}^2}^2\right)\left\Vert g\right\Vert_{\dot H^{\frac{1}{2}}}^2 \end{align} which in turn implies \eqref{Z107}. \end{proof} \subsection{The Cauchy problem for the approximate equations}\label{S:end} It remains to prove Proposition~\ref{P:initiale}. Rewrite the equation~\eqref{n2} under the form \begin{equation}\label{n2'} \partial_tf-\|\log(\varepsilon)\|^{-1}\partial_x^2f =N_\varepsilon(f), \end{equation} with $$ N_\varepsilon(f)=\frac{1}{\pi}\int_\mathbb{R}\frac{\partial_x\Delta_\alpha f}{1+\left(\Delta_\alpha f\right)^2}\left(1-\chi\left(\frac{\|\alpha\|}{\varepsilon}\right)\right)\diff \! \alpha. $$ The next proposition shows that Equation~\eqref{n2'} can be seen as a sub-critical parabolic equation. \begin{lemma} There holds \begin{equation}\label{f3} \begin{aligned} \left\Vert N_\varepsilon(f)\right\Vert_{\dot H^1} \lesssim \varepsilon^{\frac{1}{2}}\left\Vert f\right\Vert_{\dot{H}^{\frac{5}{2}}}+\left( 1+\left\Vert f\right\Vert_{H^{\frac32}}\right)^2 \log\left(2+\left\Vert f\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f\right\Vert_{\dot H^{2}}, \end{aligned} \end{equation} and \begin{equation}\label{f4} \left\Vert N_\varepsilon(f)\right\Vert_{L^2}\leq C \left(1+\left\Vert f\right\Vert_{H^{\frac{3}{2}}}\right)^2. \end{equation} \end{lemma} \begin{proof} The estimate~\eqref{f3} follows at once from~\eqref{f1} and~\eqref{f2}. To prove~\eqref{f4}, we decompose $N_\varepsilon(f)=-\left\vert D\right\vert f+\mathcal{T}(f)f+R_\varepsilon(f)$ where $\mathcal{T}(f)$ is the operator already introduced in~\S\ref{S:Transfer} and the remainder $R_\varepsilon(f)$ is as defined by~\eqref{f10}. Recall from Proposition~$2.3$ in~\cite{Alazard-Lazar} that $$ \left\Vert \mathcal{T}(f)f\right\Vert_{L^2}\lesssim \left\Vert f\right\Vert_{\dot{H}^1}\left\Vert f\right\Vert_{\dot{H}^{\frac{3}{2}}}. $$ So the wanted conclusion follows from the estimate~\eqref{f9} for~$R_\varepsilon(f)$. \end{proof} Multiply the latter equation by $(I-\Delta)^{3/2} f$ and integrate in time, to obtain \begin{equation}\label{f5} \frac{1}{2} \frac{\diff}{\dt} \left\Vert f\right\Vert_{H^{\frac{3}{2}}}^2 +\|\log(\varepsilon)\|^{-1}\left\Vert \left\vert D\right\vert f\right\Vert_{H^{\frac{3}{2}}}^2 \leq \left\Vert N_\varepsilon(f)\right\Vert_{H^1}\left\Vert f\right\Vert_{H^2}. \end{equation} Recall that \begin{equation}\label{f6} \left\Vert N_\varepsilon(f)\right\Vert_{\dot H^1} \lesssim \varepsilon^{\frac{1}{2}}\left\Vert f\right\Vert_{\dot{H}^{\frac{5}{2}}}+\left( 1+\left\Vert f\right\Vert_{H^{\frac32}}\right)^2 \log\left(2+\left\Vert f\right\Vert_{\dot H^2}^2\right)^{\frac{1}{2}} \left\Vert f\right\Vert_{\dot H^{2}}, \end{equation} Since $\varepsilon^\frac{1}{2}\ll \left\vert \log(\varepsilon)\right\vert^{-1}$ for $\varepsilon\ll 1$, we can absorb the contribution of~$\varepsilon^{\frac{1}{2}}\left\Vert f\right\Vert_{\dot{H}^{\frac{5}{2}}}$ in the right-hand side of \eqref{f6} by the left-hand side of \eqref{f5}. On the other hand, since $5/2>2$, one can absorb the contribution of the other terms by using the H\"older's inequality. This proves an {\em a priori} estimate for~\eqref{n2'}. We also get easily a contraction estimate similar to (but much simpler) the one given by Proposition~\ref{P:2.10}. Then by using classical tools for semi-linear equations, we conclude that the Cauchy problem for~\eqref{n2'} can be solved by standard iterative scheme. \begin{flushleft} \textbf{Thomas Alazard}\\ Universit{\'e} Paris-Saclay, ENS Paris-Saclay, CNRS,\\ Centre Borelli UMR9010, avenue des Sciences, F-91190 Gif-sur-Yvette\\ France. \textbf{Quoc-Hung Nguyen}\\ ShanghaiTech University, \\ 393 Middle Huaxia Road, Pudong,\\ Shanghai, 201210,\\ China \end{flushleft} \end{document}	arXiv
\begin{document} \baselineskip = 0.80 true cm \begin{center} {\large \bf GENERALIZED SPIN BASES FOR QUANTUM CHEMISTRY AND QUANTUM INFORMATION\footnote{Dedicated to Professor Rudolf Zahradnik on the occasion of his 80th birthday.}} \end{center} \begin{center} {\bf Maurice R.~KIBLER} \end{center} \begin{center} {Universit\'e de Lyon, F--69622, Lyon, France; \\ Universit\'e Lyon 1, Villeurbanne; \\ CNRS/IN2P3, UMR5822, Institut de Physique Nucl\'eaire de Lyon} \\ email: [email protected] \end{center} \noindent Symmetry adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group $SO(3)$ or its double group (i.e., spinor group) $SU(2)$. We exploit this fact for generating spin bases of relevance for quantum systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is based on the use of generalized Pauli matrices arising from a polar decomposition of $SU(2)$. This approach leads to a complete solution for the construction of mutually unbiased bases in the case where the dimension $d$ of the considered Hilbert subspace is a prime number. We also give the starting point for studying the case where $d$ is the power of a prime number. A connection of this work with the unitary group $U(d)$ and the Pauli group is brielly underlined. \noindent {\bf Keywords}: Symmetry adapted functions; Unitary bases; Generalized Pauli matrices; Unitary groups; Pauli group; Quantum chemistry; Quantum information. \section{Introduction} The notion of symmetry adapted functions (or vectors) in physical chemistry and solid state physics goes back to the fifties$^{1}$. The use of bases consisting of such functions allows to simplify the calculation of matrix elements of operators and to factorize the secular equation. Symmetry adaptation generally requires two type of groups: the symmetry group for the hamiltonian (often a finite group when dealing with molecules) and a chain of classification groups for the operators and state vectors (often continuous groups like unitary groups$^{2, 3, 4}$ and finite groups$^{5, 6, 7, 8, 9}$). The interest of symmetry adapted bases (atomic orbitals, molecular orbitals, spin waves, etc.) is well-known in quantum chemistry. In particular, the spherical harmonics (e.g., in atomic spectroscopy) and cubical, tetragonal or trigonal harmonics (e.g., in crystal-field theory and ligand field theory$^{10}$) are quite familiar to the practitioner in theoretical chemistry and chemical physics. The symmetry adapted functions generally span bases for finite-dimensional Hilbert spaces associated with reducible or irreducible representations of a symmetry group. In the case of low dimensions, such spaces are especially useful in the emerging fields of quantum information and quantum computation (quantum state tomography and quantum cryptography), two fields at the crossing of informatics, mathematics and quantum physics. In fact, a Hilbert space of finite dimension $d$ can describe a system of qudits (qubits correspond to $d=2$, qudits to $d$ arbitrary). Qudits can be realized from many physical systems. We undersee that qudits could be also produced from chemical systems. It is the object of this paper to construct bases which play an important role for quantum systems with cyclic symmetry and for quantum measurements and quantum information theory. The organisation of this paper is as follows. Section 1 is devoted to an alternative to the $\{ j^2 , j_z \}$ quantization scheme of angular momentum. In Section 2, this scheme is worked out for generating bases in a form adapted to physical and chemical cyclic systems as well as to quantum information. Section 3 deals with somme examples in low dimensions. Finally, we develop in Section 4 a systematic construction of generalized Pauli matrices which are at the origin of generalized spin bases. In the closing remarks, we mention the interest of this work for the special unitary group and the Pauli group. Throughout the present work, we use the Dirac notation familiar in quantum chemistry. As usual, $A^{\dagger}$ stands for the Hermitean conjugate of the operator $A$. In addition, $[A , B]_-$ and $[A , B]_+$ denote the commutator and the anticommutator of $A$ and $B$. Finally, $i$ is the pure imaginary. \section{AN ALTERNATIVE TO THE $\{ j^2, j_z \}$ SCHEME} Let us consider a generalized angular momentum. We note $j^2$ its square and $j_z$ its $z$-component. The common eigenvectors of $j^2$ and $j_z$ are denoted as $\| j , m \rangle$. We know that$^{11}$ \begin{eqnarray} j^2 \|j , m \rangle = j(j+1) \|j , m \rangle, \quad j_z \|j , m \rangle = m \|j , m \rangle \end{eqnarray} in a system of units where the rationalized Planck constant is equal to 1. For a fixed value of the quantum number $j$ (with $2j \in \mathbb{N}$), we note ${\cal E}(2j+1)$ the $(2j+1)$-dimensional Hilbert space spanned by the basis \begin{eqnarray} b_s = \{ \|j , m \rangle : m = j, j-1, \cdots, -j \}. \label{spherical basis} \end{eqnarray} The basis $b_s$ is adapted to spherical symmetry (adapted to the group $SO(3)$ if $j$ is an integer or the group $SU(2)$ if $j$ is an half on an odd integer). We take the basis $b_s$ in an orthonormal form, i.e., the scalar product $\langle j , m \| j , m' \rangle$ satisfies \begin{eqnarray} \langle j , m \| j , m' \rangle = \delta_{m , m'} \label{scalar product jmjm'} \end{eqnarray} for any value of $m$ and $m'$. In the applications to quantum chemistry, the generalized angular momentum can be an angular momentum, a spin angular momentum, a total (spin $+$ orbital) angular momentum, etc. The vectors $\|j , m \rangle$ can have several realizations. For instance, in the spectroscopy of $4f^N$ lanthanide ions, we have state vectors of type $\|J , M \rangle \equiv \|4f^N \tau S L J M \rangle$ in the Russell-Saunders coupling (here $j = J$ and $m = M$). This constitutes one of many possible realizations of the vectors $\|j , m \rangle$. Besides the basis $b_s$, another interesting basis can be obtained as follows. Let us consider the operator \begin{eqnarray} v_{ra} = {e}^{{i} 2 \pi j r} \|j , -j \rangle \langle j , j\| + \sum_{m = -j}^{j-1} q^{(j-m)a} \|j , m+1 \rangle \langle j , m\| \label{definition of vra} \end{eqnarray} where we use the notation of Dirac for projectors. In Eq. (\ref{definition of vra}), we have \begin{eqnarray} r \in \mathbb{R}, \quad a = 0, 1, \cdots, 2j, \quad q = \exp \left( {2 \pi {i} \over 2j+1} \right). \label{parameters} \end{eqnarray} The operator $v_{ra}$ is an extension of the operator$U_r$ defined in a previous work$^{12}$ ($U_r = v_{r0}$). From Eq. (\ref{definition of vra}), we can check that the action of $v_{ra}$ on the state $\| j , m \rangle$ is given by \begin{eqnarray} v_{ra} \|j , m \rangle = \left( 1 - \delta_{m,j} \right) q^{(j-m)a} \|j , m+1 \rangle + \delta_{m,j} {e}^{{i} 2 \pi j r} \|j , -j \rangle. \label{action of vra on jm} \end{eqnarray} Furthermore, the matrix $V_{ra}$ of the operator $v_{ra}$ on the basis $b_s$ reads \begin{eqnarray} V_{ra} = \pmatrix{ 0 & q^a & 0 & \cdots & 0 \cr 0 & 0 & q^{2a} & \cdots & 0 \cr \vdots & \vdots & \vdots & \cdots & \vdots \cr 0 & 0 & 0 & \cdots & q^{2ja} \cr {e}^{{i} 2 \pi j r} & 0 & 0 & \cdots & 0 \cr } \label{definition of Vra} \end{eqnarray} where the lines and columns are labeled in the order $\|j , j \rangle, \|j , j-1 \rangle, \cdots, \|j , -j \rangle$. It can be shown that the operator $j^2$ and $v_{ra}$ commute so that the complete set $\{ j^2, v_{ra} \}$ of commuting operators constitutes an alternative to the set $\{ j^2, j_z \}$. We may ask what are the analogues of the vectors $\|j , m \rangle$ in the scheme $\{ j^2, v_{ra} \}$? Indeed, they are the common eigenvectors of the operators $j^2$ and $v_{ra}$. As a result, these eigenvectors are \begin{eqnarray} \|j \alpha ; r a \rangle = \frac{1}{\sqrt{2j+1}} \sum_{m = -j}^{j} q^{(j + m)(j - m + 1)a / 2 - j m r + (j + m)\alpha} \| j , m \rangle \label{j alpha r a in terms of jm}, \quad \alpha = 0, 1, \cdots, 2j \end{eqnarray} for $\alpha = 0, 1, \cdots, 2j$. More precisely, we have the eigenvalue equations \begin{eqnarray} v_{ra} \|j \alpha ; r a \rangle = q^{j(a+r) - \alpha} \|j \alpha ; r a \rangle, \quad j^2 \|j \alpha ; r a \rangle = j(j+1) \|j \alpha ; r a \rangle. \end{eqnarray} For fixed $j$ and $a$ ($2j \in \mathbb{N}, a = 0, 1, \cdots, 2j$), the basis \begin{eqnarray} B_{ra} = \{ \|j \alpha ; r a \rangle : \alpha = 0, 1, \cdots, 2j \} \label{basis Bra} \end{eqnarray} is an orthonormal basis since \begin{eqnarray} \langle j \alpha ; r a \| j \alpha' ; r a \rangle = \delta_{\alpha , \alpha'} \label{scalar product alphaalpha'} \end{eqnarray} for any value of $\alpha$ and $\alpha'$. In the particular case where $2j+1$ is a prime integer, the overlap between the bases $B_{ra}$ and $B_{rb}$ is such that$^{13}$ \begin{eqnarray} \| \langle j \alpha ; r a \| j \beta ; r b \rangle \| = \delta_{\alpha , \beta} \delta_{a , b} + \frac{1}{\sqrt{2j+1}} (1 - \delta_{a , b}) \label{mub relation} \end{eqnarray} a property of considerable importance in quantum information. Note that Eq. (\ref{mub relation}) is compatible with Eq. (\ref{scalar product alphaalpha'}). \section{A FORMULATION FOR $d$-DIMENSIONAL QUANTUM SYSTEMS} The parameter $r$ is of interest for group-theoretical analyses but turns out to be of no concern here. Therefore, we shall restrict ourselves in the following to the case $r=0$. In addition, we shall adopt the notation \begin{eqnarray} k = j - m, \quad \| k \rangle = \| j , m \rangle, \quad \| a \alpha \rangle = \| j \alpha ; 0 a \rangle, \quad d = 2j+1 \label{notation QIT} \end{eqnarray} that is especially adapted to quantum information (the vectors $\| 0 \rangle, \| 1 \rangle, \cdots, \| d-1 \rangle$ are then called qudits, the case $d=2$ corresponding to ordinary qubits) and to cyclic chemical systems (for which $\| d \rangle \equiv \| 0 \rangle$, $\| d+1 \rangle \equiv \| 1 \rangle$, \ldots). The basis $b_s$ becomes \begin{eqnarray} B_d = \{ \| k \rangle : k = 0, 1, \cdots, d-1 \} \label{computational basis} \end{eqnarray} known as the computational basis in quantum information theory. The action of $v_{ra}$ on the basis $B_d$ of ${\cal E}(2j+1)$ is described by \begin{eqnarray} v_{0a} \| k \rangle = q^{ka} \| k-1 \rangle \label{action of v0a on k} \end{eqnarray} where $k-1$ should be understood modulo $d$ (i.e., $\| -1 \rangle = \| d -1 \rangle$). The vectors $\| a \alpha \rangle$ of the orthonormal basis \begin{eqnarray} B_{ra} = \{ \| a \alpha \rangle : \alpha = 0, 1, \cdots, d-1 \} \label{basis B0a} \end{eqnarray} can be written as \begin{eqnarray} \| a \alpha \rangle = \frac{1}{\sqrt{d}} \sum_{k = 0}^{d-1} q^{(d - k - 1)(k + 1)a / 2 - (k + 1) \alpha} \| k \rangle \label{a alpha in terms of k} \end{eqnarray} where $\alpha$ can take the values $\alpha = 0, 1, \cdots, d-1$. These vectors satisfy the eigenvalue equation \begin{eqnarray} v_{0a} \| a \alpha \rangle = q^{(d-1)a / 2 - \alpha} \| a \alpha \rangle \label{eigenvalue equation} \end{eqnarray} that corresponds non a nondegenerate spectrum for the operator $v_{0a}$. All relations given in Section 1 up to this point are valid for $d$ arbitrary. In the special case where $d$ is a prime integer, Eq. (\ref{mub relation}) yields \begin{eqnarray} \| \langle a \alpha \| b \beta \rangle \| = \delta_{\alpha , \beta} \delta_{a , b} + \frac{1}{\sqrt{d}} (1 - \delta_{a , b}) \label{mub relation in terms of a-alpha} \end{eqnarray} a relation valid for any value of $a$, $b$, $\alpha$ and $\beta$ in the set $\{ 0, 1, \cdots, d-1 \}$. In quantum information, two bases $B_{0a}$ and $B_{0b}$ satisfying Eq. (\ref{mub relation in terms of a-alpha}) are said to be mutually unbiased$^{14}$. Such bases play an important role in quantum cryptography and quantum state tomography. It is well-known that a complete set of $d+1$ mutually unbiased bases can be found when $d$ is a prime integer or the power of a prime integer. We continue with some typical examples. \section{SOME TYPICAL EXAMPLES} \subsection{The case $d=2$} In this case, relevant for a spin $j = 1/2$ or for a qubit, we have $q = -1$ and $a, \alpha \in \{ 0 , 1 \}$. The matrices of the operators $v_{0a}$ are \begin{eqnarray} V_{00} = \pmatrix{ 0 &1 \cr 1 &0 \cr }, \quad V_{01} = \pmatrix{ 0 &-1 \cr 1 &0 \cr }. \end{eqnarray} We note in passing a connection (to be generalized below) with the Pauli matrices since $V_{00} = \sigma_x$ and $V_{01} = -i \sigma_y$. From Eqs. (\ref{computational basis}), (\ref{basis B0a}) and (\ref{a alpha in terms of k}), the bases $B_{2}$, $B_{00}$ and $B_{01}$ are \begin{eqnarray} B_{2} & : & \| 0 \rangle, \quad \| 1 \rangle \label{1cas d is 2} \\ B_{00} &:& \| 0 0 \rangle = \frac{1}{\sqrt{2}} \left( \| 0 \rangle + \| 1 \rangle \right), \quad \| 0 1 \rangle = \frac{1}{\sqrt{2}} \left( - \| 0 \rangle + \| 1 \rangle \right) \label{2cas d is 2} \\ B_{01} &:& \| 1 0 \rangle = \frac{1}{\sqrt{2}} \left( i \| 0 \rangle + \| 1 \rangle \right), \quad \| 1 1 \rangle = \frac{1}{\sqrt{2}} \left( -i \| 0 \rangle + \| 1 \rangle \right) \label{3cas d is 2} \end{eqnarray} which satisfy Eq. (\ref{mub relation in terms of a-alpha}). Note that by using the spinorbital \begin{eqnarray} \alpha = \| \frac{1}{2} , \frac{1}{2} \rangle = \| 0 \rangle, \quad \beta = \| \frac{1}{2} , - \frac{1}{2} \rangle = \| 1 \rangle \end{eqnarray} ($\alpha$ for spin up and $\beta$ for spin down) familiar to the quantum chemist, Eqs. (\ref{1cas d is 2})-(\ref{3cas d is 2}) can be rewritten as \begin{eqnarray} B_{2} &:& \alpha, \quad \beta \\ B_{00} &:& \| 0 0 \rangle = \frac{1}{\sqrt{2}} \left( \alpha + \beta \right), \quad \| 0 1 \rangle = - \frac{1}{\sqrt{2}} \left( \alpha - \beta \right) \\ B_{01} &:& \| 1 0 \rangle = i \frac{1}{\sqrt{2}} \left( \alpha - i \beta \right), \quad \| 1 1 \rangle = -i \frac{1}{\sqrt{2}} \left( \alpha + i \beta \right). \label{cas d is 2 in alpha and beta} \end{eqnarray} In terms of eigenvectors of the matrices $V_{0a}$, we must replace the vectors $\| a \alpha \rangle$ by column vectors. This leads to \begin{eqnarray} B_{2} &:& \alpha \to \pmatrix{ 1 \cr 0 \cr }, \quad \beta \to \pmatrix{ 0 \cr 1 \cr } \\ B_{00} &:& \| 0 0 \rangle \to \frac{1}{\sqrt{2}} \pmatrix{ 1 \cr 1 \cr }, \quad \| 0 1 \rangle \to - \frac{1}{\sqrt{2}} \pmatrix{ 1 \cr -1 \cr } \\ B_{01} &:& \| 1 0 \rangle \to i \frac{1}{\sqrt{2}} \pmatrix{ 1 \cr -i \cr }, \quad \| 1 1 \rangle \to -i \frac{1}{\sqrt{2}} \pmatrix{ 1 \cr i \cr }. \label{cas d is 2 column vectors} \end{eqnarray} \subsection{The case $d=3$} This case corresponds to a spin $j=1$ or to a qutrit. Here, we have $q = \exp (2 \pi i / 3)$ and $a, \alpha \in \{ 0 , 1 , 2 \}$. The matrices of the operators $v_{0a}$ are \begin{eqnarray} V_{00} = \pmatrix{ 0 &1 &0 \cr 0 &0 &1 \cr 1 &0 &0 \cr }, \quad \pmatrix{ 0 &q &0 \cr 0 &0 &q^2 \cr 1 &0 &0 \cr }, \quad \pmatrix{ 0 &q^2 &0 \cr 0 &0 &q \cr 1 &0 &0 \cr }. \end{eqnarray} The bases $B_{3}$, $B_{00}$ and $B_{01}$ $B_{02}$ are \begin{eqnarray} B_{3}: & & \| 0 \rangle, \ \| 1 \rangle, \ \| 2 \rangle \\ B_{00}: & & \| 0 0 \rangle = \frac{1}{\sqrt{3}} \left( \| 0 \rangle + \| 1 \rangle + \| 2 \rangle \right), \ \| 0 1 \rangle = \frac{1}{\sqrt{3}} \left( q^2 \| 0 \rangle + q \| 1 \rangle + \| 2 \rangle \right) \\ & & \| 0 2 \rangle = \frac{1}{\sqrt{3}} \left( q \| 0 \rangle + q^2 \| 1 \rangle + \| 2 \rangle \right) \\ B_{01}: & & \| 1 0 \rangle = \frac{1}{\sqrt{3}} \left( q \| 0 \rangle + q \| 1 \rangle + \| 2 \rangle \right), \ \| 1 1 \rangle = \frac{1}{\sqrt{3}} \left( \| 0 \rangle + q^2 \| 1 \rangle + \| 2 \rangle \right) \\ & & \| 1 2 \rangle = \frac{1}{\sqrt{3}} \left( q^2 \| 0 \rangle + \| 1 \rangle + \| 2 \rangle \right) \\ B_{02}: & & \| 2 0 \rangle = \frac{1}{\sqrt{3}} \left( q^2 \| 0 \rangle + q^2 \| 1 \rangle + \| 2 \rangle \right), \ \| 2 1 \rangle = \frac{1}{\sqrt{3}} \left( q \| 0 \rangle + \| 1 \rangle + \| 2 \rangle \right) \\ & & \| 2 2 \rangle = \frac{1}{\sqrt{3}} \left( \| 0 \rangle + q \| 1 \rangle + \| 2 \rangle \right). \label{cas d is 3} \end{eqnarray} They satisfy Eq. (\ref{mub relation in terms of a-alpha}). In terms of colum vectors, we have \begin{eqnarray} B_{3} &:& \| 0 \rangle \to \pmatrix{ 1 \cr 0 \cr 0 \cr }, \quad \| 1 \rangle \to \pmatrix{ 0 \cr 1 \cr 0 \cr }, \quad \| 2 \rangle \to \pmatrix{ 0 \cr 0 \cr 1 \cr } \\ B_{00} &:& \| 0 0 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ 1 \cr 1 \cr 1 \cr }, \quad \| 0 1 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ q^2 \cr q \cr 1 \cr }, \quad \| 0 2 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ q \cr q^2 \cr 1 \cr } \\ B_{01} &:& \| 1 0 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ q \cr q \cr 1 \cr }, \quad \| 1 1 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ 1 \cr q^2 \cr 1 \cr }, \quad \| 1 2 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ q^2 \cr 1 \cr 1 \cr } \\ B_{02} &:& \| 2 0 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ q^2 \cr q^2 \cr 1 \cr }, \quad \| 2 1 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ q \cr 1 \cr 1 \cr }, \quad \| 2 2 \rangle \to \frac{1}{\sqrt{3}} \pmatrix{ 1 \cr q \cr 1 \cr }. \label{cas d is 3 column vectors} \end{eqnarray} \subsection{The case $d=4$} This case corresponds to a spin $j = 3/2$. Here, we have $q = i$ and $a, \alpha \in \{ 0 , 1 , 2, 3 \}$. Equation (\ref{a alpha in terms of k}) can be applied to this case too. However, the resulting bases $B_4$, $B_{00}$, $B_{01}$, $B_{02}$ and $B_{03}$ do not constitute a complete system of mutually unbiased bases ($d=4$ is not a prime number). Nevertheless, it is possible to find $d+1 = 5$ mutually unbiased bases because $d = 2^2$ is the power of a prime number. This can be achieved by replacing the space ${\cal E}(4)$ spanned by $\{ \| 3/2 , m \rangle : m = 3/2, 1/2, -1/2, -3/2 \}$ by the tensor product space ${\cal E}(2) \otimes {\cal E}(2)$ spanned by the basis \begin{eqnarray} \{ \alpha \otimes \alpha, \alpha \otimes \beta, \beta \otimes \alpha, \beta \otimes \beta \}. \label{base pd} \end{eqnarray} The space ${\cal E}(2) \otimes {\cal E}(2)$ is associated with the coupling of two spin angular momenta $j_1 = 1/2$ and $j_2 = 1/2$ or two qubits (in the vector $u \otimes v$, $u$ and $v$ correspond to $j_1$ and $j_2$, respectively). An alternative basis for ${\cal E}(2) \otimes {\cal E}(2)$ is \begin{eqnarray} \{ \alpha \otimes \alpha, \frac{1}{2} (\alpha \otimes \beta + \beta \otimes \alpha), \beta \otimes \beta, \frac{1}{2} (\alpha \otimes \beta - \beta \otimes \alpha) \}. \label{base pd SU2 S2} \end{eqnarray} The vectors in (\ref{base pd}) are well-known in the treatment of spin systems. The first three vectors are symmetric under the interchange $1 \leftrightarrow 2$ and describe a total angular momentum $J=1$ while the last one is antisymmetric and corresponds to $J=0$. It should be observed that the basis (\ref{base pd SU2 S2}) illustrates a connection between the special unitary group $SU(2)$ and the permutation group $S_2$ (a particular case of a reciprocity theorem between irreducible representation classes of $SU_n$ and $S_m$). In addition to the bases (\ref{base pd}) and (\ref{base pd SU2 S2}), it is possible to find other bases of ${\cal E}(2) \otimes {\cal E}(2)$ which are mutually unbiased. The $d=4$ mutually unbiased bases besides the canonical or computational basis (\ref{base pd}) can be constructed from the eigenvectors \begin{eqnarray} \|a b \alpha \beta \rangle = \|a \alpha \rangle \otimes \|b \beta \rangle \label{tensor product of vectors} \end{eqnarray} of the operators $w_{ab} = v_{0a} \otimes v_{0b}$ (the vectors $\|a \alpha \rangle$ and $\|b \beta \rangle$ refer to the two spaces ${\cal E}(2)$). As a result, we have the $d+1 = 5$ following mutually unbiased bases where $\lambda = (1-i)/2$ and $\mu = (1+i)/2$. \noindent {\bf The canonical basis:} \begin{eqnarray} \alpha \otimes \alpha, \quad \alpha \otimes \beta, \quad \beta \otimes \alpha, \quad \beta \otimes \beta \label{canonical basis} \end{eqnarray} or in column vectors \begin{eqnarray} \pmatrix{ 1 \cr 0 \cr 0 \cr 0 \cr }, \quad \pmatrix{ 0 \cr 1 \cr 0 \cr 0 \cr }, \quad \pmatrix{ 0 \cr 0 \cr 1 \cr 0 \cr }, \quad \pmatrix{ 0 \cr 0 \cr 0 \cr 1 \cr }. \label{canonical basis} \end{eqnarray} \noindent {\bf The $w_{00}$ basis:} \begin{eqnarray} \| 0 0 0 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + \alpha \otimes \beta + \beta \otimes \alpha + \beta \otimes \beta) \\ \| 0 0 0 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - \alpha \otimes \beta + \beta \otimes \alpha - \beta \otimes \beta) \\ \| 0 0 1 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + \alpha \otimes \beta - \beta \otimes \alpha - \beta \otimes \beta) \\ \| 0 0 1 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - \alpha \otimes \beta - \beta \otimes \alpha + \beta \otimes \beta) \label{w00 basis} \end{eqnarray} or in column vectors \begin{eqnarray} \frac{1}{2} \pmatrix{ 1 \cr 1 \cr 1 \cr 1 \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr -1 \cr 1 \cr -1 \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr 1 \cr -1 \cr -1 \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr -1 \cr -1 \cr 1 \cr }. \label{canonical basis} \end{eqnarray} \noindent {\bf The $w_{11}$ basis:} \begin{eqnarray} \| 1 1 0 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + i \alpha \otimes \beta + i \beta \otimes \alpha - \beta \otimes \beta) \\ \| 1 1 0 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - i \alpha \otimes \beta + i \beta \otimes \alpha + \beta \otimes \beta) \\ \| 1 1 1 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + i \alpha \otimes \beta - i \beta \otimes \alpha + \beta \otimes \beta) \\ \| 1 1 1 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - i \alpha \otimes \beta - i \beta \otimes \alpha - \beta \otimes \beta) \label{w11 basis} \end{eqnarray} or in column vectors \begin{eqnarray} \frac{1}{2} \pmatrix{ 1 \cr i \cr i \cr -1 \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr -i \cr i \cr 1 \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr i \cr -i \cr 1 \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr -i \cr -i \cr -1 \cr }. \label{canonical basis} \end{eqnarray} \noindent {\bf The $w_{01}$ basis:} \begin{eqnarray} \lambda \| 0 1 0 0 \rangle + \mu \| 0 1 1 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + \alpha \otimes \beta - i \beta \otimes \alpha + i \beta \otimes \beta) \\ \mu \| 0 1 0 0 \rangle + \lambda \| 0 1 1 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - \alpha \otimes \beta + i \beta \otimes \alpha + i \beta \otimes \beta) \\ \lambda \| 0 1 0 1 \rangle + \mu \| 0 1 1 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - \alpha \otimes \beta - i \beta \otimes \alpha - i \beta \otimes \beta) \\ \mu \| 0 1 0 1 \rangle + \lambda \| 0 1 1 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + \alpha \otimes \beta + i \beta \otimes \alpha - i \beta \otimes \beta) \label{w01 basis} \end{eqnarray} or in column vectors \begin{eqnarray} \frac{1}{2} \pmatrix{ 1 \cr 1 \cr -i \cr i \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr -1 \cr i \cr i \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr -1 \cr -i \cr -i \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr 1 \cr i \cr -i \cr }. \label{canonical basis} \end{eqnarray} \noindent {\bf The $w_{10}$ basis:} \begin{eqnarray} \lambda \| 1 0 0 0 \rangle + \mu \| 1 0 1 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - i \alpha \otimes \beta + \beta \otimes \alpha + i \beta \otimes \beta) \\ \mu \| 1 0 0 0 \rangle + \lambda \| 1 0 1 1 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + i \alpha \otimes \beta - \beta \otimes \alpha + i \beta \otimes \beta) \\ \lambda \| 1 0 0 1 \rangle + \mu \| 1 0 1 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha + i \alpha \otimes \beta + \beta \otimes \alpha - i \beta \otimes \beta) \\ \mu \| 1 0 0 1 \rangle + \lambda \| 1 0 1 0 \rangle &=& \frac{1}{2} (\alpha \otimes \alpha - i \alpha \otimes \beta - \beta \otimes \alpha - i \beta \otimes \beta) \label{w10 basis} \end{eqnarray} or in column vectors \begin{eqnarray} \frac{1}{2} \pmatrix{ 1 \cr -i \cr 1 \cr i \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr i \cr -1 \cr i \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr i \cr 1 \cr -i \cr }, \quad \frac{1}{2} \pmatrix{ 1 \cr -i \cr -1 \cr -i \cr }. \label{canonical basis} \end{eqnarray} It is to be noted that the vectors of the $w_{00}$ and $w_{11}$ bases are not intricated (i.e., each vector is the direct product of two vectors) while the vectors of the $w_{01}$ and $w_{10}$ bases are intricated (i.e., each vector is not the direct product of two vectors). \section{GENERALIZED PAULI MATRICES} From the operators $v_{0a}$, it is possible to define two basic operators $x$ and $z$ which can be used for generating generalized Pauli matrices. Let us put \begin{eqnarray} x = v_{00}, \quad z = v_{00}^{\dagger} v_{01}. \label{definition of x and z} \end{eqnarray} The action of $x$ and $z$ on the space ${\cal E}(2j+1)$ is given by \begin{eqnarray} x \|j , m \rangle = \left( 1 - \delta_{m,j} \right) \|j , m+1 \rangle + \delta_{m,j} \|j , -j \rangle \Leftrightarrow x \| k \rangle = \| k-1 \rangle \label{action of x on jm} \end{eqnarray} and \begin{eqnarray} z \| j,m \rangle = q^{j-m} \| j,m \rangle \Leftrightarrow z \| k \rangle = q^{k} \| k \rangle \label{action de z sur jm} \end{eqnarray} where $q = \exp (2 \pi i / d)$ with $d = 2j+1$. The $d$-dimensional matrices $X$ and $Z$ of $x$ and $z$ are \begin{eqnarray} X = \pmatrix{ 0 & 1 & 0 & \cdots & 0 \cr 0 & 0 & 1 & \cdots & 0 \cr \vdots & \vdots & \vdots & \cdots & \vdots \cr 0 & 0 & 0 & \cdots & 1 \cr 1 & 0 & 0 & \cdots & 0 \cr }, \quad Z = \pmatrix{ 1 & 0 & 0 & \cdots & 0 \cr 0 & q & 0 & \cdots & 0 \cr 0 & 0 & q^2 & \cdots & 0 \cr \vdots & \vdots & \vdots & \cdots & \vdots \cr 1 & 0 & 0 & \cdots & q^{d-1} \cr }. \label{definition of X and Z} \end{eqnarray} The operators $x$ and $z$ are unitary and satisfy the $q$-commutation relation \begin{eqnarray} x z - q z x = 0. \label{q commutation} \end{eqnarray} Pairs of operators satisfying a relation of type (\ref{q commutation}) are nowadays refered to as Weyl pairs. Thus, the operators $x$ and $z$ constitute a Weyl pair. Weyl pairs were introduced at the begining of quantum mechanics$^{15}$. They were extensively used for factorizing the secular equation in connection with a study of alternating hydrocarbons$^{16}$ and for constructing analogues of the usual Pauli matrices$^{14, 17, 18, 19}$. Let us now define the operators \begin{eqnarray} u_{ab} = x^a z^b, \quad a, b = 0, 1, \cdots, d-1. \end{eqnarray} The $d^2$ operators $u_{ab}$ are unitary and satisfy the following trace relation \begin{eqnarray} {\rm Tr}_{{\cal E}(2j+1)} \left( u_{ab}^{\dagger} u_{a'b'} \right) = (2j+1) \> \delta_{a,a'} \> \delta_{b,b'} \label{trace de uu} \end{eqnarray} where the trace is taken on the $d$-dimensional space ${\cal E}(2j+1)$. Additionally, the commutator $[u_{ab} , u_{a'b'}]_-$ and the anti-commutator $[u_{ab} , u_{a'b'}]_+$ of $u_{ab}$ and $u_{a'b'}$ are given by \begin{eqnarray} [u_{ab} , u_{a'b'}]_{\mp} = \left( q^{-ba'} \mp q^{-ab'} \right) u_{a'' b''}, \quad a'' = a+a', \quad b'' = b+b'. \label{com anti-com} \end{eqnarray} Consequently, $[u_{ab} , u_{a'b'}]_{-} = 0$ if and only if $ab' - ba' = 0$ (mod $d$) and $[u_{ab} , u_{a'b'}]_{+} = 0$ if and only if $ab' - ba' = (1/2) d$ (mod $d$). Therefore, all anti-commutators $[u_{ab} , u_{a'b'}]_{+}$ are different from 0 if $d$ is an odd integer. Two consequences follow from Eqs. (\ref{trace de uu}) and (\ref{com anti-com}). First, the trace relation (\ref{trace de uu}) shows that the $d^2$ operators $u_{ab}$ are pairwise orthogonal operators so that they can serve as a basis for developing any operator acting on the Hilbert space ${\cal E}(d)$. Second, the commutation relation (\ref{com anti-com}) shows that the set $\{ u_{ab} : a, b = 0, 1, \cdots, d-1 \}$ generates a $d^2$-dimensional Lie algebra. This algebra turns out to be the Lie algebra of the unitary group $U(d)$. The subset $\{ u_{ab} : a, b = 0, 1, \cdots, d-1 \} \setminus \{ u_{00} \}$ thus spans the Lie algebra of the special unitary group $SU(d)$. All this is reminiscent of the group $SU(2)$, the generators of which are the well-known Pauli matrices. Therefore, the operators $u_{ab}$ shall be refered as generalized Pauli operators and their matrices as generalized Pauli matrices. As an illustration, let us deal with the cases $d=2$ and $d=3$. \subsection{Exemple 1} In the case $j = 1/2 \Leftrightarrow d = 2$ ($\Rightarrow q = -1$), the matrices of the 4 operators $u_{ab}$ with $a, b = 0,1$ are \begin{eqnarray} I = X^0 Z^0 = \pmatrix{ 1 &0 \cr 0 &1 \cr }, \quad X = X^1 Z^0 = \pmatrix{ 0 &1 \cr 1 &0 \cr } \end{eqnarray} \begin{eqnarray} Z = X^0 Z^1 = \pmatrix{ 1 &0 \cr 0 &-1 \cr }, \quad Y = X^1 Z^1 = \pmatrix{ 0 &-1 \cr 1 &0 \cr }. \end{eqnarray} In terms of the usual (Hermitean and unitary) Pauli matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$, we have $X = \sigma_x$, $Y = - i \sigma_y$, $Z = \sigma_z$. The approach developed in the present paper lead to Pauli matrices in dimension 2 that differ from the usual Pauli matrices. This is the price one has to pay in order to get a systematic generalization of Pauli matrices in arbitrary dimension. It should be observed that the commutation and anti-commutation relations given by (\ref{com anti-com}) with $d=2$ correspond to the well-known commutation and anti-commutation relations for the usual Pauli matrices (transcribed in the normalization $X^1 Z^0 = \sigma_x$, $X^1 Z^1 = -i \sigma_y$, $X^0 Z^1 = \sigma_z$). \subsection{Exemple 2} In the case $j = 1 \Leftrightarrow d = 3$ ($\Rightarrow q = \exp(2 \pi i/3)$), the matrices of the 9 operators $u_{ab}$ with $a, b = 0,1,2$, viz., \begin{eqnarray} X^0 Z^0 = I \quad X^1 Z^0 = X \quad X^2 Z^0 = X^2 \quad X^0 Z^1 = Z \quad X^0 Z^2 = Z^2 \end{eqnarray} \begin{eqnarray} X^1 Z^1 = X Z \quad X^2 Z^2 \quad X^2 Z^1 = X^2 Z \quad X^1 Z^2 = X Z^2 \end{eqnarray} are \begin{eqnarray} I = \pmatrix{ 1 &0 &0 \cr 0 &1 &0 \cr 0 &0 &1 \cr }, \quad X = \pmatrix{ 0 &1 &0 \cr 0 &0 &1 \cr 1 &0 &0 \cr }, \quad X^2 = \pmatrix{ 0 &0 &1 \cr 1 &0 &0 \cr 0 &1 &0 \cr } \end{eqnarray} \begin{eqnarray} Z = \pmatrix{ 1 &0 &0 \cr 0 &q &0 \cr 0 &0 &q^2 \cr }, \quad Z^2 = \pmatrix{ 1 &0 &0 \cr 0 &q^2 &0 \cr 0 &0 &q \cr }, \quad X Z = \pmatrix{ 0 &q &0 \cr 0 &0 &q^2 \cr 1 &0 &0 \cr } \end{eqnarray} \begin{eqnarray} X^2 Z^2 = \pmatrix{ 0 &0 &q \cr 1 &0 &0 \cr 0 &q^2 &0 \cr }, \quad X^2 Z = \pmatrix{ 0 &0 &q^2 \cr 1 &0 &0 \cr 0 &q &0 \cr }, \quad X Z^2 = \pmatrix{ 0 &q^2 &0 \cr 0 &0 &q \cr 1 &0 &0 \cr }. \end{eqnarray} These generalized Pauli matrices differ from the Gell-Mann matrices$^{20}$ used in elementary particle physics. They constitute a natural extension of the Pauli matrices in dimension $d = 3$ . \section{CONCLUDING REMARKS} The various bases described in the present paper are of central importance in quantum information and quantum computation. They also play an important role for quantum (chemical and physical) systems with cyclic symmetry. By way of illustration, we would like to mention two examples. Let us consider a ring shape molecule with $N$ atoms (or agregates) at the vertices of a regular polygon with $N$ sides ($N=6$ for the benzen molecule C$_6$H$_6$). The atoms are labelled by the integer $n$ with $n = 0, 1, \cdots, N-1$. Hence, the cyclic character of the ring shape molecule makes it possible to identify the atom with the number $n$ to the one with the number $n+kN$ where $k \in \mathbb{Z}$ (the location of an atom is defined modulo $N$). Let $\| \varphi_n \rangle$ be the atomic state vector, or atomic orbital in quantum chemistry parlance, describing a $\pi$--electron located in the neighboring of site $n$. From symmetry considerations, the molecular state vector, or molecular orbital, for the molecule reads$^{21}$ \begin{eqnarray} \| \kappa_s \rangle = \frac{1}{\sqrt{N}} \sum_{n = 0}^{N-1} {e} ^{{i} 2 \pi n s / N } \| \varphi_n \rangle, \label{molecular state vector} \end{eqnarray} with $s = 0, 1, \cdots, N-1$. As a result, the molecular orbital $\| \kappa_s \rangle$ assumes the same form, up to a global phase factor, as the state $\| a \alpha \rangle$ given by Eq.~(\ref{a alpha in terms of k}) with $a=0$ and $\alpha = s$. A similar result can be obtained for a one-dimensional chain of $N$ $1/2$--spins (numbered with $n=0, 1, \cdots, N-1$) used as a modeling tool of a ferromagnetic substance. Here again, we have a cyclical symmetry since the spins numbered $n=N$ and $n=0$ are considered to be identical. The spin waves can then be described by state vectors$^{21}$ very similar to the ones given by Eq.~(\ref{a alpha in terms of k}) with again $a=0$. We close this work with two remarks of a group-theoretical nature, one concerning a continuous group, the other a finite group, connected with the operators $u_{ab}$. First, as mentioned in Section 4, the set $\{ u_{ab} : a, b = 0, 1, \cdots, d-1 \} \setminus \{ u_{00} \}$ constitutes a basis for the Lie algebra $SU(d)$. Such a basis differs from the well-known Cartan basis or from the Gel'fand-Tsetlin basis. In the special case $d=p$, with $p$ prime integer, the basis $\{ u_{ab} : a, b = 0, 1, \cdots, d-1 \} \setminus \{ u_{00} \}$ can be partioned into $p+1$ disjoint subsets, each subset containing $p-1$ commuting operators$^{18, 22}$. In other words, it is possible to decompose the Lie algebra of $SU(p)$ into $p+1$ Cartan subalgebras of dimension $p-1$. It can be proved that each subalgebra is associated with a basis of ${\cal E}(p)$ and that the set of the $p+1$ corresponding bases is a complete set of mutually unbiased bases. A similar decomposition holds of $SU(d)$ in the case where $d = p^e$, with $p$ prime integer and $e$ positive integer$^{22}$. However, in this case we need to replae ${\cal E}(d)$ by ${\cal E}(p)^{\otimes e}$. A second group-theoretical remark concern a finite group known as the Pauli group or the finite Heisenberg-Weyl group$^{17, 18, 22, 23, 24}$. The set $\{ u_{ab} : a, b = 0, 1, \cdots, d-1 \}$ is not closed under multiplication. However, it is possible to extend the latter set in order to have a group. For this purpose, let us define the operators $w_{abc}$ via$^{22}$ \begin{eqnarray} w_{abc} = q^a u_{bc}, \quad a, b, c = 0, 1, \cdots, d-1. \label{definition of wabc} \end{eqnarray} Then, the set $\{ w_{abc} : a, b = 0, 1, \cdots, d-1 \}$, endowed with the multiplication of operators, is a group of order $d^3$. This group (the Pauli group) is of paramount importance in quantum information and quantum computation$^{24, 25}$. \baselineskip = 0.60 true cm \end{document}	arXiv
\begin{document} \newcommand{{{\mathbb{N}}}}{{{\mathbb{N}}}} \newcommand{{{\mathbb{Z}}}}{{{\mathbb{Z}}}} \newcommand{{{\mathbb{Q}}}}{{{\mathbb{Q}}}} \newcommand{{{\mathbb{R}}}}{{{\mathbb{R}}}} \newcommand{{{{\mathrm{CC}}}}}{{{{\mathrm{CC}}}}} \newcommand{{{{\mathrm{Monroe}}}}}{{{{\mathrm{Monroe}}}}} \newcommand{{{{\mathrm{STV}}}}}{{{{\mathrm{STV}}}}} \newcommand{{{{\mathrm{SNTV}}}}}{{{{\mathrm{SNTV}}}}} \newcommand{{{{\mathrm{Bloc}}}}}{{{{\mathrm{Bloc}}}}} \newcommand{{{{k\hbox{-}\mathrm{Borda}}}}}{{{{k\hbox{-}\mathrm{Borda}}}}} \newcommand{{{\mathrm{rep}}}}{{{\mathrm{rep}}}} \newcommand{{{{\mathrm{cap}}}}}{{{{\mathrm{cap}}}}} \newcommand{{{{\mathrm{cost}}}}}{{{{\mathrm{cost}}}}} \newcommand{{{{\mathrm{pos}}}}}{{{{\mathrm{pos}}}}} \newcommand{{{{\mathrm{argmax}}}}}{{{{\mathrm{argmax}}}}} \newcommand{{{{\mathrm{argmin}}}}}{{{{\mathrm{argmin}}}}} \newcommand{{{{\mathrm{B}}}}}{{{{\mathrm{B}}}}} \newcommand{{{{\mathrm{B, inc}}}}}{{{{\mathrm{B, inc}}}}} \newcommand{{{{\mathrm{B, dec}}}}}{{{{\mathrm{B, dec}}}}} \newcommand{{{{\mathrm{OPT}}}}}{{{{\mathrm{OPT}}}}} \newcommand{{{{\mathrm{opt}}}}}{{{{\mathrm{opt}}}}} \newcommand{{{{\mathrm{sat}}}}}{{{{\mathrm{sat}}}}} \newcommand{{{{\mathrm{low}}}}}{{{{\mathrm{low}}}}} \newcommand{{{{\mathrm{high}}}}}{{{{\mathrm{high}}}}} \newcommand{\E}{\mathop{\mathbb E}} \newcommand{\w}{{{{\mathrm{w}}}}} \newcommand{\mbox{$\cal N$}}{\mbox{$\cal N$}} \newcommand{\mbox{$\cal P$}}{\mbox{$\cal P$}} \newcommand{\mbox{$\cal W$}}{\mbox{$\cal W$}} \newcommand{\mbox{$\cal R$}}{\mbox{$\cal R$}} \newcommand{\mbox{$\cal A$}}{\mbox{$\cal A$}} \newcommand{\mbox{$\cal S$}}{\mbox{$\cal S$}} \newcommand{\mathrm{poly}}{\mathrm{poly}} \def\mathbb{R}{\mathbb{R}} \def\mathbb{Z}{\mathbb{Z}} \def\mathbb{N}{\mathbb{N}} \def\displaystyle{\displaystyle} \def\oa#1{\overrightarrow{#1}} \def\ola#1{\overleftarrow{#1}} \def\row#1#2{{#1}_1,\ldots ,{#1}_{#2}} \def\brow#1#2{{\bf {#1}}_1,\ldots ,{\bf {#1}}_{#2}} \def\rrow#1#2{{#1}_0,{#1}_1,\ldots ,{#1}_{#2}} \def\urow#1#2{{#1}^1,\ldots ,{#1}^{#2}} \def\irow#1#2{{#1}_1,\ldots ,{#1}_{#2},\ldots} \def\lcomb#1#2#3{{#1}_1{#2}_1+{#1}_2{#2}_2+\cdots +{#1}_{#3}{#2}_{#3}} \def\blcomb#1#2#3{{#1}_1{\bf {#2}}_1+\cdots +{#1}_{#3}{\bf {#2}}_{#3}} \def\2vec#1#2{\left(\begin{array}{c}{#1}\\{#2}\end{array}\right)} \def\threevec#1#2#3{\left[\begin{array}{r}{#1}\\{#2}\\{#3}\end{array}\right]} \def\mod#1{\ \hbox{\rm (mod $#1$)}} \def\gcd#1#2{\hbox{gcd}\>(#1,#2)} \def\lcm#1#2{\hbox{lcm}\>(#1,#2)} \def\card#1{\hbox{card}\>(#1)} \def\hbox{adv}{\hbox{adv}} \def\cup{\cup} \def\cap{\cap} \def\bigcup{\bigcup} \def\bigcap{\bigcap} \newcommand{\mathcal{A}}{\mathcal{A}} \newcommand{\mathcal{E}}{\mathcal{E}} \newcommand{\mathcal{R}}{\mathcal{R}} \newcommand{\mathcal{F}}{\mathcal{F}} \newcommand{\mathcal{B}}{\mathcal{B}} \newcommand{\mathcal{C}}{\mathcal{C}} \newcommand{\mathcal{Q}}{\mathcal{Q}} \newcommand{\mathcal{D}}{\mathcal{D}} \newcommand{\mathcal{P}}{\mathcal{P}} \newcommand{\mathit{Dec}}{\mathit{Dec}} \newcommand{\revnot}[1]{\overleftarrow{#1}} \newcommand{{\mathrm{peak}}}{{\mathrm{peak}}} \newcommand{{\mathrm{dfs}}}{{\mathrm{dfs}}} \newcommand{{{\mathrm{P}}}}{{{\mathrm{P}}}} \newcommand{{{\mathrm{NP}}}}{{{\mathrm{NP}}}} \newcommand\mydots{\hbox to 1em{.\hss.\hss.}} \sloppy \title{Modeling Representation of Minorities Under Multiwinner Voting Rules\\ (extended abstract, work in progress)} \author{Piotr Faliszewski\\ AGH University\\ Poland\\ \phantom{Weizmann Institute of Science} \and Jean-Fran\c{c}ois Laslier\\ Paris School of Economics\\ France\\ \phantom{Weizmann Institute of Science} \and Robert Scheafer\\ AGH University\\ Poland\\ \phantom{Weizmann Institute of Science} \and Piotr Skowron\\ Oxford University\\ United Kingdom\\ \phantom{Weizmann Institute of Science} \and Arkadii Slinko\\ University of Auckland\\ New Zealand\\ \phantom{Weizmann Institute of Science} \and Nimrod Talmon\\ Weizmann Institute of Science\\ Israel } \maketitle \begin{abstract} The goal of this paper is twofold. First and foremost, we aim to experimentally and quantitatively show that the choice of a multiwinner voting rule can play a crucial role on the way minorities are represented. We also test the possibility for some of these rules to achieve proportional representation. \end{abstract} \section{Introduction} The use of voting rules as a mean of manipulation to advantage or disadvantage minorities is widespread. With the passage of the Voting Rights Act in 1965 in the United States, the right of minorities to register and vote was largely secured. It was soon discovered, however, that minority voting did not guarantee the election of minorities or minority-preferred candidates. This was a result of a widespread use of manipulation by the choice of voting rules~\cite{grofman1992controversies,grofman1994minority,trebbi2008electoral}. Manipulation of electoral rules, however, is not a prerogative exclusive of American cities. Pande~\cite{pande2003can} provides a discussion of electoral rules and racial politics in elections in India. Alexander~\cite[p.~211]{alexander2004france} describes in detail the 1947 Gaullist manipulations of electoral rules in France; in the Paris area, where the Gaullist alliance was weak, they introduced proportional representation but in rural areas, where the alliance was strong, they introduced plurality. Kreuzer~\cite[p.~229]{kreuzer2004germany} describes strategic manipulation of voting rules in postwar Germany. In this paper we undertake an experimental study of the effect that some voting rules have on representation of minorities. The American literature has dealt at length with manipulation by re-districting, often called ``gerrymandering,'' that is crafting the electoral districts to the advantage of the designer~\cite{grofman1982representation}. In the present paper, we do not tackle the districting question. Our work applies to the case of a district that elects $k>1$ delegates as well as to the, formally equivalent, case of a country that does not uses districting for electing its Parliament. Moreover, we consider the rules which take into account not only the first preferences of voters but also the second, third and further preferences. For these rules not based on districting, the aspects of the causal connection between electoral systems and vote-seat disproportionality remains obscure~\cite{powell2000election}. We adopt a standard spatial two-dimensional model of voting, assuming that both voters and candidates have ideal political positions on the plane and Euclidean preferences. Applied research has shown that having two dimensions is often sufficient to have meaningful descriptions of voters' political opinions \cite{schofield2007spatial}. The idea for this paper stems from a previous work of Faliszewski, Sawicki, Schaefer and Smo{\l}ka regarding a selection method for genetic algorithms based on multiwinner voting~\cite{fal-saw-sch-smo:c:multiwinner-genetic-algorithms}. \section{Preliminaries} \paragraph{Elections and Voting Rules} Let $V = \{v_1, \ldots, v_n\}$ be the set of $n$ \emph{voters} and $C = \{c_1, \ldots, c_m\}$ be the set of $m$ \emph{candidates}. The voters have their intrinsic preferences over candidates, which are represented as preference orders (i.e., rankings of the candidates from best to worst). By ${{{\mathrm{pos}}}}_v(c)$ we denote the position of candidate $c$ in the preference ranking of voter $v$. For example, a voter $v$ who likes $c_1$ best, then $c_2$, then $c_3$, and so on, would have preference order $c_1 \succ c_2 \succ \cdots \succ c_m$. For this voter, we would have ${{{\mathrm{pos}}}}_v(c_1) = 1$, ${{{\mathrm{pos}}}}_v(c_2) = 2$, and so on. We are interested in multiwinner elections, where the goal is to select a committee of size $k$ (i.e., a size-$k$ subset of $C$). A \emph{multiwinner election rule} is a formal decision process that given preferences of the voters and a positive integer $k \in {{\mathbb{N}}}$ returns a committee that, according to this rule, is most preferred by the population of the voters viewed as a whole. Many multiwinner rules rely on the notion of score for the candidates. For each integer $t \in \{1,\ldots,m\}$, the $t$-Approval score of candidate $c$ in vote $v$ is $1$ if $v$ ranks $c$ among top $t$ positions, and is $0$ otherwise. The Borda score of candidate $c$ in vote $v$, denoted $\beta(c,v)$, is $m - {{{\mathrm{pos}}}}_v(c)$. The Plurality score of a candidate is his or her $1$-Approval score. Given one of these notions of score, the total score of a candidate in the election is the sum of his or her scores from all the voters. The following rules are considered in this paper: \begin{description} \item[Single Nontransferable Vote (SNTV).] SNTV selects a committee that consists of those $k$ candidates with the highest Plurality scores. \item[Bloc.] Bloc selects a committee that consists of those $k$ candidates with the highest $k$-Approval scores (one can think of Bloc as if each voter gave a point to each candidate from his or her ideal committee). \item[$\boldsymbol{k}$-Borda.] $k$-Borda selects a committee that consists of those $k$ candidates with the highest Borda scores. In the world of single-winner voting rules ($k=1$), Borda is usually seen as electing some kind of compromise candidate. \item[Chamberlin--Courant Rule.] For each voter $v$ and each committee $C$ a \emph{representative of $v$ in $C$} is the most preferred member of $C$, according to $v$. The Chamberlin--Courant rule~\cite{cha-cou:j:cc} selects a committee so that the sum of the Borda scores of the voter representatives is maximized (alternatively, one can think of minimizing the average position of a voter's representative). Formally, the Chamberlin--Courant rule selects a committee $C$ that maximizes the value $\sum_{v \in V} (\max_{c\in C}\beta(c,v))$. Unfortunately, computing a winning committee under the Chamberlin--Courant rule is ${{\mathrm{NP}}}$-hard~\cite{pro-ros-zoh:j:proportional-representation,bou-lu:c:chamberlin-courant}. For the purpose of this paper, we were able to compute Chamberlin--Courant results using its formulation as an integer linear program (ILP) by running the CPLEX optimization package. Lu and Boutilier~\cite{bou-lu:c:chamberlin-courant} and Skowron et al.~\cite{sko-fal-sli:j:multiwinner} offer approximation algorithms that one could use for larger elections. \item[Monroe Rule.] Monroe~\cite{mon:j:monroe}, similarly to Chamberlin and Courant, explored the concept of a representative of a voter. He, however, required that each committee member should represent roughly the same number of voters. A function $\Phi\colon V \to A$ is a Monroe assignment for a committee $C$ if for each candidate $a \in C$ it holds that $\left\lfloor \nicefrac{n}{k} \right\rfloor \leq \Phi^{-1}(a) \leq \left\lceil \nicefrac{n}{k} \right\rceil$. Intuitively, Monroe assignments represent valid mappings between the voters and their representatives. Let $\mathscr{A}(C)$ denote the set of all Monroe assignments for a committee $C$. According to the Monroe rule, the score of committee $C$ is defined as $\mathrm{score_M}(C) = \max_{\Phi \in \mathscr{A}(C)}(\sum_{v \in V}\beta(\Phi(v),v))$. The committee $C$ that maximizes $\mathrm{score_M}(C)$ is selected as the winner. Intuitively speaking, the idea behind the Monroe rule is to partition the electorate into roughly same-sized districts\footnote{Note that these ``virtual districts'' are based on voters' preferences and not on geographical location.} and assign to each district a distinct candidate with as high Borda score as possible. Just like the Chamberlin--Courant rule, Monroe rule is ${{\mathrm{NP}}}$-hard to compute~\cite{pro-ros-zoh:j:proportional-representation}, but this time for most of our experiments the ILP formulation turned out to be too complex for CPLEX to solve within reasonable amount of time. Thus, instead we used the Greedy-Monroe approximation algorithm of Skowron et al.~\cite[Algorithm~A]{sko-fal-sli:j:multiwinner} which is guaranteed to select a committee $C$ whose $\mathrm{score_M}(C)$ is close to being the maximum. \item[Single Transferable Vote (STV).] STV is a multi-round procedure that operates as follows. In each round, if there exists a candidate $c$ who is preferred the most by at least $q = \left\lfloor \nicefrac{n}{k+1} \right\rfloor + 1$ voters, then $c$ is added to the winning committee. At the same time we remove from further consideration exactly $q$ voters which rank $c$ on top, and delete $c$ from the preference rankings of all other voters. Otherwise, i.e., if each candidate is most preferred by less than $q$ voters, then we select a candidate which is most preferred by the smallest number of voters and delete this candidate from preference rankings of all voters.\footnote{Occasionally, we run into trouble when computing STV winners. For example, for $n = 600$ voters and committee size $k = 52$ we should use quota value $q = \lceil( \nicefrac{600}{53} \rceil + 1 = 12$. In each round in which STV puts a candidate into a committee, it also deletes $q$ voters. Yet, $k \cdot q = 624$ so we do not have enough voters. Fortunately, in our experiments such situations were occurring only for committee sizes over 50. Thus we do not give results for STV for committees of sizes larger than 50.} We note that this description of STV is not complete and there is a lot of room for various tie-breaking decisions (for example, it is not obvious which voters exactly to delete when a candidate is added to the committee). We describe our approach to tie-breaking below. See Tideman and Richardson~\cite{tid-ric:j:stv} for an overview of the STV rule and its variants. \end{description} \noindent The next two rules do not exactly fit in our framework because they are based on districting. \begin{description} \item[First Past The Post (FPP).] Under FPP voters are divided into territorial districts (constituencies) of approximately equal sizes and each constituency elects their own representative by using the Plurality rule (i.e., the candidate with the highest Plurality score wins within the constituency). \item[District-Based Borda.] The same as FPP, but with the use of Borda scores instead of Plurality scores. \end{description} We shall consider these two last rules under the assumption of random districting. This means that we assume that any territorial district represents an unbiased collection of the political opinions, and we create ``districts'' artificially by choosing a random partition of the electorate. We thus obtain two voting rules that could be called ``Random district FPP'' and ``Random district Borda.'' These rules deserve to be studied as benchmarks for comparison with the others. Occasionally, our voting rules run into situations where they have to break ties (this is particularly imminent in the definition of STV, but all rules face this issue). To simplify our experiments, we break all ties, whenever they occur, uniformly at random. \paragraph{Spatial Models of Elections} Euclidean preferences \cite{DavisHinich66} stipulate that both candidates and voters can be represented as points in an Euclidean space, and that voters rank candidates according to the increasing order of Euclidean distances from themselves. The idea is that points correspond to political programs. Candidates are represented by their actual programs, whereas voters are represented by the ideal programs they believe in \cite{Plott1967,mckelvey1990,enelow1984spatial}. As the empirical analysis of elections shows~\cite{schofield2007spatial}, the dimension of the political space seldom exceeds two. Usually, the left-right spectrum is the main one and the second dimension could be, for example, caused by the influences of religion. In our model we assume that voters have two-dimensional Euclidean preferences. \section{Results} We present results of two experiments. The purpose of the first experiment is to get an initial understanding of the rules discussed. The purpose of the second one is to asses how these rules treat minorities. \subsection{Initial Experiment: On Representativity} The voting rule in a representative democracy ideally accomplishes two tasks: selects a representative set of delegates (e.g., a parliament) and assigns voters to delegates. This means the two main purposes of a voting rule is to achieve a certain level of representativity and a certain level of accountability. These two requirements are not easy to combine. One standard solution to this is to use First-Past-the-Post (FPP), a system which operates with electoral (usually territorial) districts of approximately the same size and allows voters in each district to elect their representative using Plurality. This perfectly solves the problem of accountability but the representativity of such a system is known to be poor because it tends to be detrimental for minorities, especially for a minority that is spread in all districts. On the other hand, party-list proportional-representation systems~\cite{puk:b:pr} can be quite good on representativity, provided that the threshold of representation is small, but very poor on accountability. There seems to be a certain tension between accountability and representativity of multiwinner voting rules as well, and some rules seem to accommodate both desires better than others. While we do not yet have a good measure of voting rules' accountability, in this section we attempt to evaluate the representativity of their outcomes. Our idea is simply that a rule is more representative when it is more likely for each voter that some candidate with similar political views is elected. \paragraph{Misrepresentation} Formally, we take the following approach. Let $d$ denote the Euclidean distance in our two-dimensional space of political programs. Given a voter $v$ and a winning committee $W$, we define $\Psi(v)=\min_{c\in W} d(v,c)$ to be the distance between $v$ and the closest member of $W$. If we view distances as meaningful characteristics of preferences, it is natural to consider $\Psi(v)$ as a measure of $v$'s misrepresentation in the committee. For an election $E = (C,V)$ and a committee $W$, we define $D(E,W) = \frac{1}{\|V\|}\sum_{v \in V}\Psi(v)$ to be the average misrepresentation of the voters. Note that our definition does not embody any notion of efficiency. As an example, imagine that a small group of voters is very homogeneous and has preferences very different from the rest of the electorate. If this group elects a single delegate, representation can be very good for this group, according to our definition. But, depending on how the decisions are taken in the Parliament, it may well be that this delegate has no real power. \paragraph{Candidates} Of course representativity chiefly depends on who are the candidates. To focus on the effect of the voting rule itself, we consider in this paper that the set of candidates, on its own, is a good representation of the electorate. This is easily done by drawing candidates' political platforms from the same distribution as the voters ideal points. At least for large values of $k$, this achieves the goal.\footnote{The assumption that the set of candidates is identical to the set of voters is often met in the Political Economy literature since \cite{osborne1996model,besley1997economic} and labelled the ``citizen-candidate'' model.} \paragraph{Results} We have measured the average misrepresentation for our rules in the following setting. We generated $60$ elections with $300$ candidates and $600$ voters each, all distributed uniformly on a $6 \times 6$ square. For each election we have computed the results of all our voting rules, for committees of sizes from $1$ to $97$ with a step of $3$. For each case we have computed the average misrepresentation of the voters. We present our results on Figures~\ref{fig:satisifaction1} and~\ref{fig:satisifaction2}. Absolute values of the computed average misrepresentation is not very meaningful , and thus one should focus on relative comparison of the voting rules. On Figure~\ref{fig:satisifaction1} we show the results for Random-District-FPP, SNTV, STV, Greedy-Monroe, and Chamberlin--Courant. We can see that STV, Greedy-Monroe, and Chamberlin--Courant achieve next to indistinguishable results. SNTV achieves somewhat worse results (but for large committees it converges with the previous three), and FPP does not converge to the others even for very large committees. On Figure~\ref{fig:satisifaction2} we show the result for Bloc, $k$-Borda, Random-District-Borda, and FPP. $k$-Borda is the least proportional rule (indeed, inspection of the results has shown that $k$-Borda picks a cluster of candidates in the center of our square; it is designed to find candidates that are least offensive to all the voters). While adding random districts to Borda (i.e., considering Random-District-Borda) helps significantly, the results are still worse than for the rules from Figure~\ref{fig:satisifaction1}. Bloc also does poorly with respect to proportionality (it finds concentration areas with many voters and chooses clusters of candidates there; for large committees it tends to return the same or similar committees as $k$-Borda). \begin{figure} \caption{Average misrepresentation of the voters for rules that aim at achieving proportional representation. The vertical bars indicate standard deviation.} \caption{Average misrepresentation of the voters for the other rules. The vertical bars indicate standard deviation.} \label{fig:satisifaction1} \label{fig:satisifaction2} \end{figure} There is a simple but important conclusions from this experiment. For the case of uniform distribution of candidates and voters, there seems to be a single natural notion of representation of the voters, and all our voting rules that were designed to find correct representation (in the context of preferences orders) appear to find it. It is quite remarkable since the definitions of our rules can be significantly different (it certainly is not obvious that STV and Chamberlin--Courant would be finding, in essence, the same kinds of results). \subsection{A Polarized Society} The choice of an electoral system has a major impact on the survival of small political parties. The Liberal Democrats in the United Kingdom is an example of such a party. They have some left-wing and some right-wing policies so many researchers place them squarely in the middle of the UK political spectrum. However, the existence of a centrist party under FPP is extremely challenging\footnote{"Why being centrist hasn't helped the Lib Dems". New Statesman. 6 October 2014. Retrieved 26 April 2016.}. Even under the mixed-member proportional (MMP) electoral system of New Zealand, centrist parties often struggle, as exemplified by the virtual demise of Peter Dunne's United Future party in 2013. Here we deal with multiwinner voting rules that do not rest on the existence of political parties. In order to explore the question of the ``squeezing of the center'' in this framework, we consider the following situation. The population itself is polarized in the sense that most voters are extreme. Precisely, we suppose that the electorate is made of three sub-populations: two large groups and a small one, with the small group, the ``centrist voters'' in between the two large groups. The voters depicted by the black dots are taken from three Gaussian distributions. The mean values for these Gaussians are, respectively, (-2,0), (0,0), and (2,0); standard deviation is 0.25 in each case. For the left and the right party, we generated 100 voters for each, while for the centrist party we generated 50 voters (i.e., altogether, there are 250 voters; the large parties have 40\% of the electorate each, whereas the centrist party has 20\% of the electorate). As to the candidates, we now suppose that they are not taken from the same distribution as the voters, as in the previous experiment, but that they are spread uniformly over the whole political spectrum (there are 600 of them; depicted as gray points). This leaves open the possibility to elect ``compromise'' candidates that would lie in between two groups. \begin{figure} \caption{Results for SNTV} \label{fig:kBorda} \caption{Results for STV} \label{fig:Bloc} \caption{Results for Chamberlin--Courant} \label{fig:kBorda} \caption{Results for Greedy-Monroe} \label{fig:Bloc} \caption{Results for FPP} \label{fig:kBorda} \caption{Results for District-Based Borda} \label{fig:Bloc} \caption{Results for Borda} \label{fig:kBorda} \caption{Results for Bloc} \label{fig:Bloc} \caption{Results for two big groups of voters and a smaller centrist one, for committee size $k=34$, for the case where 600 candidates are distributed uniformly over the $6 \times 3$ rectangle over the positions of the voters.} \label{fig:uni} \end{figure} In Figure~\ref{fig:uni} we present a sample election and results of choosing a committee of size $34$ (committee members are depicted as large red dots). At first sight, we see that SNTV, STV, Chamberlin--Courant, and Greedy-Monroe do a good job in terms of representing the smaller centrist population. On the other hand, Random-District-FPP and Random-District Borda seem to provide very scattered, erratic results, with FPP underrepresenting the minority, and Random-District Borda overrepresenting it. Bloc ignores the minority completely, whereas $k$-Borda seems to focus on it completely. \begin{figure} \caption{Average number of candidates from the centrist party selected by SNTV, STV, Chamberlin--Courant, Greedy-Monroe, and FPP. Vertical bars indicate standard deviation.} \label{fig:partyC-uni} \end{figure} \paragraph{Proportionality} A key concept in the theory of representation is the concept of proportionality. This notion has a clear meaning when votes and candidates are labeled alike: When voters vote for parties, one can check whether the number of elected candidates from a party is proportional to the party's score. When delegates are elected by districts, one can check whether or not the number of seats allocated to each district is proportional to the population of the district. In order to check if our four election rules that did best in terms of voter representation indeed represent the centrist group proportionally, we can think of the candidates as belonging themselves to the three groups. We simply consider that a candidate ``belongs'' to the group closest to her location. We have generated 65 elections according to the above-described scheme; for each, we have computed committees of size 1 to 97 (with a step of 3), and computed how many candidates from each party were selected. We show the results in Figure~\ref{fig:partyC-uni} (we also include Random-District-FPP for comparison). We see that, after all, there is some difference between the proportionality achieved by our four rules. While STV and Greedy-Monroe seem to select roughly 20\% of the candidates from the centrist party (the desired number), SNTV and Chamberlin--Courant overshoot. Greedy-Monroe does even better than STV because it is far more stable (the standard deviation of the results for Greedy-Monroe is noticeably smaller than for STV). FPP undershoots significantly. \begin{figure} \caption{Results for SNTV} \label{fig:kBorda} \caption{Results for STV} \label{fig:Bloc} \caption{Results for Chamberlin--Courant} \label{fig:kBorda} \caption{Results for Greedy-Monroe} \label{fig:Bloc} \caption{Results for FPP} \label{fig:kBorda} \caption{Results for District-Based Borda} \label{fig:Bloc} \caption{Results for Borda} \label{fig:kBorda} \caption{Results for Bloc} \label{fig:Bloc} \caption{Results for two big parties and a smaller centrist party, for committee size $k=34$, for the case where candidates and voters follow the same distribution.} \label{fig:gau} \end{figure} To verify the robustness of our results with respect to the location of the candidates, we have repeated our experiment for the same distribution of voters (however, we have now used 500 voters instead of 250) and for 250 candidates distributed in the same way as the voters. That is, now we assumed that the structure of preferences that lead to the formation of the groups is also present among the candidates. This is the same ``citizen-candidate'' hypothesis that was made in the first experiment, and it gives a more direct way of modeling party affiliations of candidates. In Figure~\ref{fig:gau} we present the results for a sample election, for committee size $k = 34$. Comparing to Figure~\ref{fig:uni}, we can see that now all the rules seem to behave more proportionally. We believe that the reason for this fact is that, in some sense, the rules have far fewer candidates to choose from; there are no maverick candidates all over the political spectrum that would distract the voters. However, still it is visible that our four proportional representation rules seem to be doing best, that $k$-Borda overrepresents the center, and that Bloc underrepresents it. Interestingly, district-based rules seem to be doing fine. \begin{figure} \caption{Average number of candidates from the centrist party selected by SNTV, STV, Chamberlin--Courant, Greedy-Monroe, and FPP. Vertical bars indicated standard deviation.} \label{fig:partyC-gau} \end{figure} In Figure~\ref{fig:partyC-gau} we show the average number of candidates from the centrist party elected by the four rules (and Random-Districts-FPP; added for comparison; this is a result from generating 100 elections). As one might have expected from Figure~\ref{fig:gau}, the scenario where candidates and voters are identically distributed is easy for the rules that aim at proportional representation by design. All these rules perform well. Interestingly, for larger committees FPP overshoots significantly. \section{Conclusion and further work} Firstly, we have confirmed that the choice of a voting rule has a profound effect on representation of minorities and any electoral system designer must take this into consideration. Secondly, we have initiated the study on evaluation of multiwinner voting rules with respect to their ability to provide faithfully represent the voters. To this end, we have considered two parameters: (1) the average misrepresentation, and (2) the proportion of voters elected from a smaller centrist party. It turned out that among our rules, STV, SNTV, Chamberlin--Courant, and Greedy-Monroe, four rules that to large extent were designed to achieve proportional representation, indeed achieve it. Nonetheless, we have seen that additional mechanisms for ensuring proportionality built into Greedy-Monroe (and, to some degree, into STV) indeed give them advantage in more challenging settings. On the other hand, rules based on random-districting (in particular FPP) turn out to be not reliable. Naturally, rules that were designed with other principles in mind than proportional representation ($k$-Borda and Bloc, in our case) do not fare well compared to the others. Since we located the minority in the center of the political spectrum, we cannot say, at this point, if the same results would hold in other cases. We consider this work only a starting point and we are working on further experiments. \end{document}	arXiv
Unit hyperbola In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation $x^{2}-y^{2}=1.$ In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length $r={\sqrt {x^{2}-y^{2}}}.$ Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola $y^{2}-x^{2}=1$ to complement it in the plane. This pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the unit hyperbola is in use, the alternative radial length is $r={\sqrt {y^{2}-x^{2}}}.$ The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals ${\sqrt {2}}.$[1] The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical complex numbers, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola. Asymptotes Main article: Asymptote Generally asymptotic lines to a curve are said to converge toward the curve. In algebraic geometry and the theory of algebraic curves there is a different approach to asymptotes. The curve is first interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the projective curve at a point at infinity, thus circumventing any need for a distance concept and convergence. In a common framework (x, y, z) are homogeneous coordinates with the line at infinity determined by the equation z = 0. For instance, C. G. Gibson wrote:[2] For the standard rectangular hyperbola $f=x^{2}-y^{2}-1$ in ℝ2, the corresponding projective curve is $F=x^{2}-y^{2}-z^{2},$ which meets z = 0 at the points P = (1 : 1 : 0) and Q = (1 : −1 : 0). Both P and Q are simple on F, with tangents x + y = 0, x − y = 0; thus we recover the familiar 'asymptotes' of elementary geometry. Minkowski diagram The Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are • units of 30 centimetres length and nanoseconds, or • astronomical units and intervals of 8 minutes and 20 seconds, or • light years and years. Each of these scales of coordinates results in photon connections of events along diagonal lines of slope plus or minus one. Five elements constitute the diagram Hermann Minkowski used to describe the relativity transformations: the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the conjugate diameter. The plane with the axes refers to a resting frame of reference. The diameter of the unit hyperbola represents a frame of reference in motion with rapidity a where tanh a = y/x and (x,y) is the endpoint of the diameter on the unit hyperbola. The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. In this context the unit hyperbola is a calibration hyperbola[3][4] Commonly in relativity study the hyperbola with vertical axis is taken as primary: The arrow of time goes from the bottom to top of the figure — a convention adopted by Richard Feynman in his famous diagrams. Space is represented by planes perpendicular to the time axis. The here and now is a singularity in the middle.[5] The vertical time axis convention stems from Minkowski in 1908, and is also illustrated on page 48 of Eddington's The Nature of the Physical World (1928). Parametrization Main article: Hyperbolic angle A direct way to parameterizing the unit hyperbola starts with the hyperbola xy = 1 parameterized with the exponential function: $(e^{t},\ e^{-t}).$ This hyperbola is transformed into the unit hyperbola by a linear mapping having the matrix $A={\tfrac {1}{2}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}\ :$ :} $(e^{t},\ e^{-t})\ A=({\frac {e^{t}+e^{-t}}{2}},\ {\frac {e^{t}-e^{-t}}{2}})=(\cosh t,\ \sinh t).$ This parameter t is the hyperbolic angle, which is the argument of the hyperbolic functions. One finds an early expression of the parametrized unit hyperbola in Elements of Dynamic (1878) by W. K. Clifford. He describes quasi-harmonic motion in a hyperbola as follows: The motion $\rho =\alpha \cosh(nt+\epsilon )+\beta \sinh(nt+\epsilon )$ has some curious analogies to elliptic harmonic motion. ... The acceleration ${\ddot {\rho }}=n^{2}\rho \ ;$ ;} thus it is always proportional to the distance from the centre, as in elliptic harmonic motion, but directed away from the centre.[6] As a particular conic, the hyperbola can be parametrized by the process of addition of points on a conic. The following description was given by Russian analysts: Fix a point E on the conic. Consider the points at which the straight line drawn through E parallel to AB intersects the conic a second time to be the sum of the points A and B. For the hyperbola $x^{2}-y^{2}=1$ with the fixed point E = (1,0) the sum of the points $(x_{1},\ y_{1})$ and $(x_{2},\ y_{2})$ is the point $(x_{1}x_{2}+y_{1}y_{2},\ y_{1}x_{2}+y_{2}x_{1})$ under the parametrization $x=\cosh \ t$ and $y=\sinh \ t$ this addition corresponds to the addition of the parameter t.[7] Complex plane algebra Main article: Split-complex number Whereas the unit circle is associated with complex numbers, the unit hyperbola is key to the split-complex number plane consisting of z = x + yj, where j 2 = +1. Then jz = y + xj, so the action of j on the plane is to swap the coordinates. In particular, this action swaps the unit hyperbola with its conjugate and swaps pairs of conjugate diameters of the hyperbolas. In terms of the hyperbolic angle parameter a, the unit hyperbola consists of points $\pm (\cosh a+j\sinh a)$, where j = (0,1). The right branch of the unit hyperbola corresponds to the positive coefficient. In fact, this branch is the image of the exponential map acting on the j-axis. Thus this branch is the curve $f(a)=\exp(aj).$ The slope of the curve at a is given by the derivative $f^{\prime }(a)=\sinh a+j\cosh a=jf(a).$ For any a, $f^{\prime }(a$) is hyperbolic-orthogonal to $f(a)$. This relation is analogous to the perpendicularity of exp(a i) and i exp(a i) when i2 = − 1. Since $\exp(aj)\exp(bj)=\exp((a+b)j)$, the branch is a group under multiplication. Unlike the circle group, this unit hyperbola group is not compact. Similar to the ordinary complex plane, a point not on the diagonals has a polar decomposition using the parametrization of the unit hyperbola and the alternative radial length. References 1. Eric Weisstein Rectangular hyperbola from Wolfram Mathworld 2. C.G. Gibson (1998) Elementary Geometry of Algebraic Curves, p 159, Cambridge University Press ISBN 0-521-64140-3 3. Anthony French (1968) Special Relativity, page 83, W. W. Norton & Company 4. W.G.V. Rosser (1964) Introduction to the Theory of Relativity, figure 6.4, page 256, London: Butterworths 5. A.P. French (1989) "Learning from the past; Looking to the future", acceptance speech for 1989 Oersted Medal, American Journal of Physics 57(7):587–92 6. William Kingdon Clifford (1878) Elements of Dynamic, pages 89 & 90, London: MacMillan & Co; on-line presentation by Cornell University Historical Mathematical Monographs 7. Viktor Prasolov & Yuri Solovyev (1997) Elliptic Functions and Elliptic Integrals, page one, Translations of Mathematical Monographs volume 170, American Mathematical Society • F. Reese Harvey (1990) Spinors and calibrations, Figure 4.33, page 70, Academic Press, ISBN 0-12-329650-1 .	Wikipedia
Conical spiral In mathematics, a conical spiral, also known as a conical helix,[1] is a space curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called conchospiral (from conch). Parametric representation In the $x$-$y$-plane a spiral with parametric representation $x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi $ a third coordinate $z(\varphi )$ can be added such that the space curve lies on the cone with equation $\;m^{2}(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;$ : • $x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .$ Such curves are called conical spirals.[2] They were known to Pappos. Parameter $m$ is the slope of the cone's lines with respect to the $x$-$y$-plane. A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone. Examples 1) Starting with an archimedean spiral $\;r(\varphi )=a\varphi \;$ gives the conical spiral (see diagram) $x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .$ In this case the conical spiral can be seen as the intersection curve of the cone with a helicoid. 2) The second diagram shows a conical spiral with a Fermat's spiral $\;r(\varphi )=\pm a{\sqrt {\varphi }}\;$ as floor plan. 3) The third example has a logarithmic spiral $\;r(\varphi )=ae^{k\varphi }\;$ as floor plan. Its special feature is its constant slope (see below). Introducing the abbreviation $K=e^{k}$gives the description: $r(\varphi )=aK^{\varphi }$. 4) Example 4 is based on a hyperbolic spiral $\;r(\varphi )=a/\varphi \;$. Such a spiral has an asymptote (black line), which is the floor plan of a hyperbola (purple). The conical spiral approaches the hyperbola for $\varphi \to 0$. Properties The following investigation deals with conical spirals of the form $r=a\varphi ^{n}$ and $r=ae^{k\varphi }$, respectively. Slope The slope at a point of a conical spiral is the slope of this point's tangent with respect to the $x$-$y$-plane. The corresponding angle is its slope angle (see diagram): $\tan \beta ={\frac {z'}{\sqrt {(x')^{2}+(y')^{2}}}}={\frac {mr'}{\sqrt {(r')^{2}+r^{2}}}}\ .$ A spiral with $r=a\varphi ^{n}$ gives: • $\tan \beta ={\frac {mn}{\sqrt {n^{2}+\varphi ^{2}}}}\ .$ For an archimedean spiral is $n=1$ and hence its slope is$\ \tan \beta ={\tfrac {m}{\sqrt {1+\varphi ^{2}}}}\ .$ • For a logarithmic spiral with $r=ae^{k\varphi }$ the slope is $\ \tan \beta ={\tfrac {mk}{\sqrt {1+k^{2}}}}\ $ ($\color {red}{\text{ constant!}}$ ). Because of this property a conchospiral is called an equiangular conical spiral. Arclength The length of an arc of a conical spiral can be determined by $L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {(x')^{2}+(y')^{2}+(z')^{2}}}\,\mathrm {d} \varphi =\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {(1+m^{2})(r')^{2}+r^{2}}}\,\mathrm {d} \varphi \ .$ For an archimedean spiral the integral can be solved with help of a table of integrals, analogously to the planar case: $L={\frac {a}{2}}{\big [}\varphi {\sqrt {(1+m^{2})+\varphi ^{2}}}+(1+m^{2})\ln {\big (}\varphi +{\sqrt {(1+m^{2})+\varphi ^{2}}}{\big )}{\big ]}_{\varphi _{1}}^{\varphi _{2}}\ .$ For a logarithmic spiral the integral can be solved easily: $L={\frac {\sqrt {(1+m^{2})k^{2}+1}}{k}}(r{\big (}\varphi _{2})-r(\varphi _{1}){\big )}\ .$ In other cases elliptical integrals occur. Development For the development of a conical spiral[3] the distance $\rho (\varphi )$ of a curve point $(x,y,z)$ to the cone's apex $(0,0,z_{0})$and the relation between the angle $\varphi $ and the corresponding angle $\psi $ of the development have to be determined: $\rho ={\sqrt {x^{2}+y^{2}+(z-z_{0})^{2}}}={\sqrt {1+m^{2}}}\;r\ ,$ $\varphi ={\sqrt {1+m^{2}}}\psi \ .$ Hence the polar representation of the developed conical spiral is: • $\rho (\psi )={\sqrt {1+m^{2}}}\;r({\sqrt {1+m^{2}}}\psi )$ In case of $r=a\varphi ^{n}$ the polar representation of the developed curve is $\rho =a{\sqrt {1+m^{2}}}^{\,n+1}\psi ^{n},$ which describes a spiral of the same type. • If the floor plan of a conical spiral is an archimedean spiral than its development is an archimedean spiral. In case of a hyperbolic spiral ($n=-1$) the development is congruent to the floor plan spiral. In case of a logarithmic spiral $r=ae^{k\varphi }$ the development is a logarithmic spiral: $\rho =a{\sqrt {1+m^{2}}}\;e^{k{\sqrt {1+m^{2}}}\psi }\ .$ Tangent trace The collection of intersection points of the tangents of a conical spiral with the $x$-$y$-plane (plane through the cone's apex) is called its tangent trace. For the conical spiral $(r\cos \varphi ,r\sin \varphi ,mr)$ the tangent vector is $(r'\cos \varphi -r\sin \varphi ,r'\sin \varphi +r\cos \varphi ,mr')^{T}$ and the tangent: $x(t)=r\cos \varphi +t(r'\cos \varphi -r\sin \varphi )\ ,$ $y(t)=r\sin \varphi +t(r'\sin \varphi +r\cos \varphi )\ ,$ $z(t)=mr+tmr'\ .$ The intersection point with the $x$-$y$-plane has parameter $t=-r/r'$ and the intersection point is • $\left({\frac {r^{2}}{r'}}\sin \varphi ,-{\frac {r^{2}}{r'}}\cos \varphi ,0\right)\ .$ $r=a\varphi ^{n}$ gives $\ {\tfrac {r^{2}}{r'}}={\tfrac {a}{n}}\varphi ^{n+1}\ $ and the tangent trace is a spiral. In the case $n=-1$ (hyperbolic spiral) the tangent trace degenerates to a circle with radius $a$ (see diagram). For $r=ae^{k\varphi }$ one has $\ {\tfrac {r^{2}}{r'}}={\tfrac {r}{k}}\ $ and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral. References 1. "Conical helix". MATHCURVE.COM. Retrieved 2022-03-03. 2. Siegmund Günther, Anton Edler von Braunmühl, Heinrich Wieleitner: Geschichte der mathematik. G. J. Göschen, 1921, p. 92. 3. Theodor Schmid: Darstellende Geometrie. Band 2, Vereinigung wissenschaftlichen Verleger, 1921, p. 229. External links • Jamnitzer-Galerie: 3D-Spiralen.. • Weisstein, Eric W. "Conical Spiral". MathWorld.	Wikipedia
This section explains how to import your terrain into Unreal Engine. When you have completed your terrain, export it using the file format Raw16, which is the optimal format according to EPIC. In this example, we have created a simple 2048x2048 terrain. In Unreal Engine, click on the Landscape icon and import your heightmap. The data fit automatically to your file. Click on Import. The terrain imports, but the padding is untidy and the terrain looks flat. Now, we will fix this quickly. The table below lists the optimized landscape sizes recommended by EPIC, which guarantee a borderless terrain. The table is available at https://docs.unrealengine.com/latest/INT/Engine/Landscape/TechnicalGuide/. Crop the 2048x2048 terrain to a recommended resolution of 2017x2017. A height of 100 in the Z scale corresponds to a height of -256 m min. to 256 m max. Accordingly, 200 in the Z scale corresponds to a height of -512 m min. to 512 m max., 400 to a height of -1024 m min. to 1024 m max., etc. Check the min. and max. heights of the terrain and export it with a defined height to the nearest power of 2. In this example, the heights are -467.04 min. to 224.70 max., therefore, the terrain is exported with a defined height of -512 min. to 512 max. Reimport the terrain into Unreal and set the Z scale to 200. When the terrain is reimported, the landscape data fit automatically to your new heightmap resolution, according to EPIC's recommended sizes. The padding disappears and the heights appear correct. This section explains how to export your terrain as multi-files and import them into Unreal Engine. Right click in the Graph Editor and select Create Node > Export > Multi file export terrain and double click to open its parameters. See and Multi file (tiled) export terrain node for more details. The following pattern works for UE4: "filename_X$x_Y$y.png" (UE4 requires an "X" and an "Y" before the coordinates of the tile). Note that UE4 forbids tiles of more than 1024 vertices. Be sure to check the size of your tiles. After the export is completed, you will have the following files in your Windows Explorer. Open your UE4 project, and when using the world composition import, specify if you need to flip or not the Y Coordinates. Do not check this box for Instant Terra. This option is checked by default. The terrain tiles appear in the correct order / position inside UE4.	CommonCrawl
\begin{document} \title{Entropic uncertainty relations for mutually unbiased periodic coarse-grained observables resemble their discrete counterparts} \author{\L ukasz Rudnicki} \email{[email protected]} \selectlanguage{english} \affiliation{International Centre for Theory of Quantum Technologies (ICTQT), University of Gda\'{n}sk, 80-308 Gda\'{n}sk, Poland} \affiliation{Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik{\'o}w 32/46, PL-02-668 Warsaw, Poland} \author{Stephen P. Walborn} \affiliation{Departamento de F\'{\i}sica, Universidad de Concepci\'on, 160-C Concepci\'on, Chile} \affiliation{ANID - Millennium Science Initiative Program - Millenium Institute for Research in Optics (MIRO), Universidad de Concepci\'on, 160-C Concepci\'on, Chile} \begin{abstract} One of the most important and useful entropic uncertainty relations concerns a $d$ dimensional system and two mutually unbiased measurements. In such a setting, the sum of two information entropies is lower bounded by $\ln d$. It has recently been shown that projective measurements subject to operational mutual unbiasedness can also be constructed in a continuous domain, with the help of periodic coarse graining. Here we consider the whole family of R{\'e}nyi entropies applied to these discretized observables and prove that such a scheme does also admit the entropic UR mentioned above. \end{abstract} \maketitle \section{Introduction} Uncertainty relations are often cited as a key deviation between classical and quantum physics, describing the simultaneous unpredictability of two or more properties of a quantum system. Since the development of the concept of entropy to characterize information or the lack thereof, entropic uncertainty relations (EURs) have taken on a fundamental and useful role in quantum physics and quantum information \cite{bialynicki11,Wehner_2010,coles17,toscano18}. They can be associated to secret quantum key rates \cite{berta10,furrer11,branciard12}, and used as identifiers of quantum correlations \cite{giovannetti04,guhne04,walborn09,huang10,leach10,walborn11a,gneiting11,carvalho12,schneeloch13}, for example. Additional applications can be found in a recent review \cite{coles17}. \par EURs exist for systems described by either discrete variables \cite{deutsch83,maassen88,coles14,rudnicki14}, continuous variables \cite{bialynicki75,bialynicki06}, or some combination of the two \cite{bialynicki84,bialynicki85,rojas95}. A key feature of discrete systems is that EURs for two \textit{mutually unbiased} observables give lower bounds that are a function of the dimension $d$ alone. To be more precise, two $d$-dimensional operators $\oper{X}$ and $\oper{Y}$ with all eigenstates ($i,j=0,\ldots,d-1$) satisfying $\|\langle X_i \| Y_j\rangle\|=1/\sqrt{d}$ render two mutually unbiased measurements. For this case, an EUR involving R{\'e}nyi entropies (with natural logarithm) of orders $\alpha$ and $\beta$, such that $1/\alpha+1/\beta=2$, is given by \cite{maassen88} \begin{equation} H_\alpha[X] + H_\beta[Y] \geq \ln d, \label{eq:dsur} \end{equation} where \begin{equation} \label{Renyi0} H_\alpha[X]=\frac{1}{1-\alpha}\ln \sum_{i=0}^{d-1} p_i^\alpha[X], \end{equation} and $p_i[X]=\langle X_i \|\rho \|X_i \rangle$. As usual, $\rho$ represents the density matrix describing the system. \par In the continuous-variable scenario, similar types of EURs have been developed. However, the crucial difference between the discrete and the continuous case is that finite dimension $d$ is `lost' within a standard treatment, being replaced by a scaling parameter, related to the observables in question. To understand that effect we shall first observe that a system of mutually unbiased measurements can be characterized by two \textit{a priori} independent parameters: the number of possible measurement outcomes and the uniform 'overlap' between different measurements. The first parameter is formally the same as the number of projectors forming the resolution of the identity, and is assumed here to be the same for both measurements. The latter one has a clear operational meaning \cite{tasca18a,Tavakolieabc3847} for all projective measurements (for general POVMs it is more complicated \cite{Kalev_2014}), being equal to the true overlap between the eigenstates, in a special case of rank $1$ projectors. Clearly, if the first parameter is finite (therefore discrete), conservation of probability fixes the value of the latter one, as explained above Eq. \eqref{eq:dsur}. However, for continuous variables the number of outcomes is usually considered to be infinite, both countably and uncountably. As a consequence, the overlap becomes a free, setup-dependent, scaling parameter. To see this, let us consider phase-space quadrature variables, given by $\oper{q}_\theta=\cos \theta \oper{x}+\sin \theta \oper{p}$, where $\oper{x}$ and $\oper{p}$, recovered for $\theta=0$ and $\theta=\pi/2$ respectively, are the usual position and momentum operators obeying $[\oper{x},\oper{p}]=i \hbar$. The commutator $[\oper{q}_\theta,\oper{q}_{\theta^\prime} ]=i\hbar \sin \Delta \theta$ clearly depends upon the relative angle between the operators $\Delta \theta = \theta- \theta^\prime$. Therefore, uncountably many eigenstates of these operators are mutually unbiased, with overlaps given by \footnote{Proper consideration of the proper limit in the case of $\Delta \theta=0$ gives a RHS of $\delta(q_\theta-q^\prime_\theta)$.}: \begin{equation}\label{ovl} \|\langle q_\theta \| q_{\theta^\prime}\rangle \| = \left (2 \pi\hbar \|\sin \Delta \theta \|\right)^{-1/2}. \end{equation} Mutual unbiasedness of both measurements is encoded in the fact that the above overlap neither depends on $q_\theta$ nor on $q^\prime_\theta$. \begin{table}[t!] \begin{centering} \begin{tabular}{\|c\|c\|c\|} \hline Number of measurements' outcomes & Overlap between the measurements & Entropic URs \tabularnewline \hline \hline Uncountably infinite & $\left(2 \pi\hbar \|\sin \Delta \theta \|\right)^{-1/2}$ & \cite{bialynicki75,bialynicki06,huang11,guanlei09}\tabularnewline \hline Countably infinite & Additionally depends on coarse graining widths & \cite{bialynicki84,bialynicki06,rudnicki12b,rudnicki15} \tabularnewline \hline Discrete, equal to $d$ & Always $1/\sqrt{d}$ & \textit{Present paper} \tabularnewline \hline \end{tabular}\caption{Different types of settings relevant for continuous-variable systems and associated, known EURs. Here we fill the gap of a discrete setting.\label{Tableb1}} \par\end{centering} \end{table} In other words, the indicator of systems' dimension $d$ is replaced by $2 \pi\hbar \|\sin \Delta \theta \|$ --- the continuous parameter which depends on both the underlying structure of the phase space (presence of $\hbar$), and the interrelation between the involved operators, quantified by $\sin \Delta \theta$. As a natural consequence, the EURs expressed in terms of continuous R\'enyi \cite{bialynicki06,guanlei09} and Shannon \cite{bialynicki75,huang11} entropies do depend on both parameters. We go back to these types of EURs in Sec. \ref{Sec4}. From now on we also set $\hbar\equiv 1$. \par An additional scaling factor arises when one takes into account that the above eigenstates describe a non-physical scenario of infinite energy, and consequently, physical scenarios involve some sort of coarse graining. That is, the eigenstates $\ket{q_\theta}$ are approximated by ``smeared" quantum states $\int dq^\prime_\theta Q(q_\theta-q^\prime_\theta) \ket{q^\prime_\theta}$, where $Q(q_\theta-q^\prime_\theta)$ is a square integrable function that is localized around $q_\theta$ with some finite width parameter $\delta_\theta$. Likewise, though this is just an analogy rather than a formal continuation of the previous argument, physical measurement devices (detectors) cannot be described by uncountably many rank-one projectors $\ket{q_\theta}\bra{q_\theta}$, but rather by countably many (though, still infinite number of) integrated projective measurements of the form \begin{equation}\label{CGop} \int_{q_\theta-\delta_\theta/2}^{q_\theta+\delta_\theta/2} dq^\prime_\theta \ket{q^\prime_\theta}\bra{q^\prime_\theta}. \end{equation} Adequate consideration of coarse graining in this context leads to URs with lower bounds that in addition depend explicitly on the width parameters $\delta_\theta$ \cite{bialynicki84,partovi83,bialynicki06,rudnicki12a,rudnicki12b}. Improper attention to this inherent coarse graining can have detrimental consequences \cite{ray13a,ray13b,tasca13}. An overview of URs for coarse-grained CVs can be found in Ref. \cite{toscano18}. Table \ref{Tableb1} summarizes the above cases. As can be seen, only settings with infinite number of outcomes have so far been successfully considered in the continuous scenario, even though, only a discrete one can lead to a counterpart of the EUR in Eq. \eqref{eq:dsur}. Therefore, as emphasized in the bottom right cell of Table \ref{Tableb1}, the aim of this paper is to provide a setting which obeys (\ref{eq:dsur}) for continuous variables. \par To this end we need an alternative approach to the standard coarse graining described by (\ref{CGop}), i.e. other methods of binning together the rank-one projectors. A number of strategies have been adopted in this direction \cite{gilchrist98,banaszek99,wenger03,vernazgris14,ketterer16,finot17}. With the goal of defining truly mutual unbiased measurements in CV systems, periodic coarse graining (PCG) has been a successful approach. That is, two sets of CV phase-space projectors $\Pi_k[\theta]$ and $\Pi_l[\theta']$ (like before $k,l=0,\ldots,d-1$) can be defined such that their eigenstates give equal probability outcomes when the other measurement operator is applied \cite{tasca18a,paul18}. This may seem to suggest that one can define a discrete variable system within a CV one, which may be loosely true, but not in any rigorous sense. For example, it was shown that these PCG observables, though mutually unbiased, do not follow the known conditions concerning the number of allowed mutually unbiased bases for discrete systems. Rather, depending on the number of outcomes $d$, they can mimic either the discrete or continuous cases, or neither \cite{silva21}. \par Here we explore another way to benchmark PCG observables, that is, through the corresponding EURs. We show that they indeed mimic the discrete case in that they obey the entropic URs from Eq. \eqref{eq:dsur}. This applies to PCG of usual position and momentum operators, as well as arbitrary phase space operators. In this way, we realize our main goal, solving the problem posed in the previous paragraphs. In addition, we use our results to study the continuous limit of the entropic uncertainty relations. The paper is organized as follows. In Sec. \ref{Sec2} we briefly introduce the PCG observables, while in Sec. \ref{Sec3}, we prove our major EUR for the special case of the position and momentum pair. We also state the same result for arbitrary phase-space variables. In Sec. \ref{Sec4} we study the continuous limit of the EURs considered. \section{periodic coarse-grained observables} \label{Sec2} In order to construct coarse-grained mutually unbiased projective measurements, we group rank one projectors according to periodic bin functions ($k=0,\dots,d-1$) \cite{tasca18b} \begin{equation} \label{Eq:MaskFuncDef} M_k(z;T)=\left\{ \begin{array}{ccc} 1, &\; k \,s \leq z {\rm \,(mod \, T)} < (k+1) s \\ 0, & {\rm otherwise} \end{array} \right. . \end{equation} The bin functions can be thought of as continuous square waves with spatial period $T$ and bin width $s=T/d$. While for simplicity, in Sec. \ref{Sec3}, we first consider the special case of position and momentum, we now introduce notation which covers a general pair of phase-space directions. Let \begin{equation} \Pi_{k}[\theta]=\int_{\mathbb{R}}d q_\theta\,M_{k}\left(q_\theta;T_{\theta}\right)\left\|q_\theta\right\rangle \left\langle q_\theta\right\|, \end{equation} for $k=0,\ldots,d-1$ be a set of $d$ projectors rendering PCG in the $\theta$ direction of the phase space. In \cite{tasca18a, paul18} additional displacement parameters setting the origin of the phase space have been introduced. However, as these degrees of freedom do not at all influence the present discussion, they are omitted here. One just needs to remember that all arguments remain valid independent of the choice of the origin of the phase space. Given a mixed state $\rho$, we further define the probabilities \begin{equation}\label{probs} p_{k}[\theta]=\mathrm{Tr}\left(\rho\Pi_{k}[\theta]\right). \end{equation} Operational mutual unbiasedness of two measurements has been defined for pure states in \cite{tasca18a} (see also \cite{Tavakolieabc3847}), however, one can easily realize that this definition extends to the case of mixed states by convexity. To be more precise, we call both $\theta$ and $\theta'$ measurements as mutually unbiased if for all states $\rho$ such that $p_k[\theta]$ is a permutation of $(1,0,\ldots,0)$ with $d-1$ zeros, we find that $p_l[\theta']=1/d$ for all $l$, and \textit{vice versa}. It is quite straightforward to realize that for $\rho=\sum_n \lambda_n \ket{\Psi_n}\bra{\Psi_n}$, with all $\lambda_n\geq 0$ and $\sum_n \lambda_n=1$, the requirement $p_k[\theta]=1$ for some $k$ enforces $\bra{\Psi_n}\Pi_k[\theta]\ket{\Psi_n}=1$ for all $n$. Consequently $\bra{\Psi_n}\Pi_l[\theta']\ket{\Psi_n}=1/d$. Note that the above operational definition of mutual unbiasedness, as well as its natural extension to the case of mixed states, applies to any pair of projective measurements, not necessarily being the PCG, which we use here for the sake of illustration and further discussion. In \cite{paul18} it has been proven that if \begin{equation} \frac{T_{\theta}T_{\theta'}}{2\pi}=\frac{d \|\sin \Delta \theta \|}{M},\quad M\in\mathbb{N},\quad\forall_{n=1,\ldots,d-1}\;\frac{M\,n}{d}\notin\mathbb{N},\label{condition} \end{equation} with $M$ being a natural number ($M\neq0$) such that $M\,n/d\notin\mathbb{N}$ for all $n=1,\ldots,d-1$ (i.e. $M$ is not co-prime with $d$), then both sets of the PCG projectors are mutually unbiased. \section{Entropic URs for PCG} \label{Sec3} We are interested in an entropic UR of the general form \begin{equation} H_{\alpha}\left[\theta\right]+H_{\beta}\left[\theta'\right]\geq-2\ln\mathcal{C}, \end{equation} where as usual $1/\alpha+1/\beta=2$ and the R{\'e}nyi entropy is defined in (\ref{Renyi0}). Our aim is to show that $\mathcal{C}\leq 1/\sqrt{d}$. To this end we partially follow \cite{rudnicki10} and \cite{rudnicki12b}. We first introduce a few pieces of notation. Let $O_{k}[\theta]$ be sets defined as \begin{equation} O_{k}[\theta]=\left\{ z\in\mathbb{R}:\quad M_k\left(z;T_\theta\right)=1\right\}, \end{equation} and note that \begin{equation} \Pi_{k}[\theta]=\int_{O_{k}[\theta]}dq_\theta\left\|q_\theta\right\rangle \left\langle q_\theta\right\|. \end{equation} From now on we focus our attention on the position/momentum couple, further denoting $O_{k}[x]\equiv O_{k}[0]$, $O_{k}[p]\equiv O_{k}[\pi/2]$, $T_x\equiv T_0$ and $T_p\equiv T_{\pi/2}$. We define $\varphi_{km}\left(x\right)$ and $\xi_{ln}\left(p\right)$ to be orthonormal and complete sets of functions on $O_{k}[x]$ and $O_{l}[p]$ respectively, i.e. \begin{subequations}\label{ort} \begin{equation} \int_{O_{k}[x]}\!\!dx\,\varphi_{k_{1}m}\left(x\right)\varphi_{k_{2}m'}^{}\left(x\right)=\delta_{k_{1}k}\delta_{k_{2}k}\delta_{m'm}, \end{equation} \begin{equation} \int_{O_{l}[p]}\!\!dp\,\xi_{l_{1}n}\left(p\right)\xi_{l_{2}n'}^{}\left(p\right)=\delta_{l_{1}l}\delta_{l_{2}l}\delta_{n'n}. \end{equation} \end{subequations} Such complete sets are guaranteed to exist, since functions supported on, e.g. $O_{k}[x]$ form a subspace of the Hilbert space of square integrable functions, which is separable (so is every subspace). Moreover, without loss of generality we restrict our attention to pure states $\rho= \ket{\Psi}\bra{\Psi}$, since they are known to cover extreme points of information entropies. We therefore define amplitudes: \begin{subequations} \begin{equation} a_{km}=\int_{O_{k}[x]}dx\,\psi\left(x\right)\varphi_{km}^{}\left(x\right), \end{equation} \begin{equation} b_{ln}=\int_{O_{l}[p]}dp\,\tilde{\psi}\left(p\right)\xi_{ln}^{}\left(p\right), \end{equation} \end{subequations} where as usual $\psi\left(x\right) = \left\langle x\left\|\Psi\right\rangle \right.$ and $\tilde\psi\left(p\right) = \left\langle p\left\|\Psi\right\rangle \right.\!$. Generalizing Eqs. A7-A9 from \cite{rudnicki12b}, by replacing the intervals appearing there by the sets $O_{k}[x]$ and $O_{l}[p]$, and with a slight adjustment of the notation concerning arguments of the R{\'e}nyi entropies, we immediately get the result \begin{equation}\label{Ren1} H_{\alpha}\left[\left\|a\right\|^{2}\right]+H_{\beta}\left[\left\|b\right\|^{2}\right]\geq-2\ln\mathcal{C}, \end{equation} where \begin{equation}\label{Cc} \mathcal{C}=\sup_{\left(k,l,m,n\right)}\left\|\int_{O_{k}[x]}dx\int_{O_{l}[p]}dp\frac{e^{ipx}}{\sqrt{2\pi}}\varphi_{km}^{}\left(x\right)\xi_{ln}\left(p\right)\right\|. \end{equation} Arguments of the R{\'e}nyi entropies in (\ref{Ren1}) are not denoted as the directions on the phase space, but as the probability distributions entering Eq. \eqref{Renyi0}. These distributions are more fine-grained than (\ref{probs}), since \begin{equation} p_k[0]=\sum_m \left\|a_{km}\right\|^{2},\quad p_l[\pi/2]=\sum_n \left\|b_{ln}\right\|^{2}. \end{equation} If we further apply the Cauchy-Schwarz inequality to the $\int_{O_{k}[x]}dx$ integral and use normalization of $\varphi_{km}\left(x\right)$, we arrive at the bound $\mathcal{C}\leq \sup_{\left(k,l,n\right)} W_{kl}^n $, where \begin{equation} W_{kl}^n=\sqrt{\int_{O_{k}[x]}\!\!\!\!dx\int_{O_{l}[p]}\!\!\!\!dp\int_{O_{l}[p]}\!\!\!\!dp'\frac{e^{i\left(p-p'\right)x}}{2\pi}\xi_{ln}\left(p\right)\xi_{ln}^{}\left(p'\right)}. \label{eq:Csq} \end{equation} Our task is therefore to compute the kernel \begin{equation}\label{kernel} \int_{O_{k}[x]}dx\frac{e^{i\left(p-p'\right)x}}{2\pi}. \end{equation} The periodic bin function can be decomposed in the Fourier series \begin{equation} M_{k}\left(z;T\right)=\frac{1}{d}+\!\!\sum_{N\in\mathbb{Z}/\{0\}}\!\!f_{k,N}e^{\frac{2\pi iN}{T}z}, \label{eq:M} \end{equation} where \begin{equation}\label{fn} f_{k,N}=\frac{1-e^{-\frac{2\pi iN}{d}}}{2\pi iN}e^{-\frac{2\pi iN}{d}k}. \end{equation} Using \eqref{eq:M} to calculate the kernel we find \begin{widetext} \begin{equation}\label{major1} \int_{O_{k}[x]}dx\frac{e^{i\left(p-p'\right)x}}{2\pi} = \frac{1}{d}\delta\left(p-p'\right) + \!\!\sum_{N\in\mathbb{Z}/\{0\}}\!\!f_{k,N}\delta\left(p-p'+\frac{2\pi}{T_{x}}N\right). \end{equation} Consequently, we obtain the result \begin{equation}\label{major2} \int_{O_{k}[x]}\!\!\!\!dx\int_{O_{l}[p]}\!\!\!\!dp\int_{O_{l}[p]}\!\!\!\!dp'\frac{e^{i\left(p-p'\right)x}}{2\pi}\xi_{ln}\left(p\right)\xi_{ln}^{}\left(p'\right)=\frac{1}{d}+\!\!\sum_{N\in\mathbb{Z}/\{0\}}\!\!f_{k,N}\int_{O_{l}[p]}\!\!\!\!dp\int_{O_{l}[p]}\!\!\!\!dp'\delta\left(p-p'+\frac{T_{p}MN}{d}\right)\xi_{ln}\left(p\right)\xi_{ln}^{}\left(p'\right), \end{equation} \end{widetext} where we have utilized normalization of $\xi_{ln}\left(p\right)$ to integrate the first Dirac delta contribution, and we applied the MUB condition (\ref{condition}) while changing arguments of the remaining Dirac deltas. Due to the last step, every Dirac delta in the second expression leads to an autocorrelation term, which is non-vanishing only when $MN/d$ is an integer. However, due to the further requirement established in (\ref{condition}), we find $MN/d\in\mathbb{Z}$ if and only if $N/d\in\mathbb{Z}$. But in this special case the factor $1-e^{-\frac{2\pi iN}{d}}$ present in $f_{k,N}$ becomes equal to $0$, so that all terms in the sum over $N\in\mathbb{Z}/\{0\}$ disappear, leaving the bare contribution $1/d$. As a result, $\mathcal{C}\leq1/\sqrt{d}$, as expected. Finally, we observe \cite{rudnicki12b} that a particular choice \begin{subequations}\label{prob} \begin{equation} \varphi_{k0}\left(x\right)=\left\langle x\right\|\Pi_{k}[0]\left\|\Psi\right\rangle /\sqrt{p_{k}[0]}, \end{equation} \begin{equation} \xi_{l0}\left(p\right)=\left\langle p\right\|\Pi_{l}[\pi/2]\left\|\Psi\right\rangle /\sqrt{p_{l}[\pi/2]}, \end{equation} \end{subequations} with other functions in both complete sets being orthogonal to (\ref{prob}) leads to the probabilities $\left\|a_{km}\right\|^{2}=p_{k}[0]\delta_{m0}$ and $\left\|b_{ln}\right\|^{2}=p_{l}[\pi/2]\delta_{n0}$. Therefore, the bound $\mathcal{C}\leq1/\sqrt{d}$ which consequently gives $-2\ln\mathcal{C}\geq\ln d$ is also valid for our main UR under consideration, namely, Eq. (\ref{eq:dsur}) for position and momentum pair of PCG observables, denoted by angles $\theta=0$ and $\theta=\pi/2$ respectively, is proven. It is easy to recognize that this bound, due to the property of mutual unbiasedness, is saturated if a state is localized in either of the sets $O_{k}[x]$ or $O_{l}[p]$, for a fixed value of the index $k$ or $l$. \subsection{Extension to any two directions in phase space} In order to extend the above result to two arbitrary phase-space observables $q_\theta$ and $q_{\theta^\prime}$, i.e. to show that the general EUR \begin{equation} H_{\alpha}\left[\theta\right]+H_{\beta}\left[\theta^\prime\right]\geq\ln d,\label{eq:main2} \end{equation} holds (as always with $1/\alpha +1/\beta=2$), we first observe that several steps of the previous derivation can immediately be repeated with minor modifications. To be more precise, Eqs. (\ref{ort})-(\ref{fn}) from Sec. \ref{Sec3} just require a slight adjustment of the notation, which boils down to a replacement of labels "$0$" or "$x$" by "$\theta$" and "$\pi/2$" or "$p$" by "$\theta^\prime$", as well as function arguments by $q_\theta$ and $q_\theta^\prime$, respectively. Moreover, the Fourier transform in (\ref{Cc}) must be replaced by the fractional Fourier transform \cite{FFT} which gives the generalized overlap $\left\langle q_{\theta}\left\|q_{\theta^\prime}\right\rangle \right.\!=\mathcal{F}\left(q_{\theta},q_{\theta^\prime}\right)$ and reads (as before $\Delta\theta=\theta-\theta^\prime$) \begin{equation} \mathcal{F}\left(q_{\theta},q_{\theta^\prime}\right)=\sqrt{\frac{-ie^{i\Delta\theta}}{2\pi\sin\Delta\theta}}e^{i\frac{\cot\Delta\theta}{2}\left(q_{\theta}^{2}+q_{\theta'}^{2}\right)-i\frac{q_{\theta}q_{\theta'}}{\sin\Delta\theta}}. \end{equation} Note that $\left\| \mathcal{F}\left(q_{\theta},q_{\theta^\prime}\right)\right\|$ reduces to the overlap in (\ref{ovl}). Consequently, the Fourier kernel (\ref{kernel}) is replaced by \begin{equation} \int_{O_{k}[\theta]}dq_\theta\mathcal{F}\left(q_{\theta},q_{\theta^\prime}\right)\mathcal{F}\left(\tilde{q}_{\theta^\prime},q_\theta\right). \end{equation} Since \begin{equation} \mathcal{F}\left(q_{\theta},q_{\theta^\prime}\right)\mathcal{F}\left(\tilde{q}_{\theta^\prime},q_\theta\right)=\frac{e^{i\frac{\cot\Delta\theta}{2}\left(q_{\theta'}^{2}-\tilde{q}_{\theta'}^{2}\right)}}{2\pi\left\|\sin\Delta\theta\right\|}e^{i\frac{q_{\theta}}{\sin\Delta\theta}\left(\tilde{q}_{\theta'}-q_{\theta'}\right)}, \end{equation} we easily generalize (\ref{major1}) as \begin{widetext} \begin{equation} \int_{O_{k}[\theta]}dq_\theta\mathcal{F}\left(q_{\theta},q_{\theta^\prime}\right)\mathcal{F}\left(\tilde{q}_{\theta^\prime},q_\theta\right) = \frac{1}{d}\delta\left(\tilde{q}_{\theta'}-q_{\theta'}\right) + \!\!\sum_{N\in\mathbb{Z}/\{0\}}\!\!f_{k,N} e^{i\frac{\cot\Delta\theta}{2}\left(q_{\theta'}^{2}-\tilde{q}_{\theta'}^{2}\right)}\delta\left(\tilde{q}_{\theta'}-q_{\theta'}+\frac{2\pi\sin\Delta\theta}{T_{\theta}}N\right). \end{equation} \end{widetext} The remaining part of the derivation follows exactly the same way as for the particular case of position and momentum. The only difference is that due to the MUB condition Eq. \eqref{condition}, the term $2\pi\sin\Delta\theta/T_{\theta}$ inside the Dirac delta is replaced by $\pm\, T_{\theta^\prime} M/d$ where the sign $\pm$ depends on the order of $\theta$ and $\theta^\prime$ on the phase space. In (\ref{major2}) we find the plus sign as the angle difference for position and momentum is in $[0,\pi]$. Thus, we have an uncertainty relation of the form (\ref{eq:dsur}) for PCG observables corresponding to any two non-parallel phase space quadratures. \section{Continuous limit} \label{Sec4} At the end we would briefly like to elaborate on the continuous limit for PCG observables. To this end we recall that $d=T_\theta/s_\theta$, for all variables $q_\theta$, where $s_\theta$ is the bin width. Then, using \eqref{condition}, we can write $d = 2 \pi \|\sin \Delta \theta\|/s_\theta s_{\theta^\prime}$. Plugging this into \eqref{eq:main2}, we have \begin{equation} H_{\alpha}\left[\theta\right] +H_{\beta}\left[\theta^\prime\right] + \ln(s_\theta s_{\theta^\prime}) \geq\ln 2 \pi \|\sin \Delta \theta\| . \label{eq:dur} \end{equation} Each R{\'e}nyi entropy can be rewritten as follows \begin{equation}\label{RenCont} H_{\alpha}\left[\theta\right]=-\ln s_\theta+\frac{1}{1-\alpha}\ln\left[ \sum_{i=0}^{d-1}s_\theta \left(\frac{p_i[\theta]}{s_\theta}\right)^\alpha\right]. \end{equation} In the continuous limit $d\rightarrow \infty$, we set $T_{\theta}\sim\sqrt{d}$, so that $T_{\theta}\rightarrow \infty$ while at the same time $s_\theta\rightarrow 0$. In this limit, the sum multiplied by $s_\theta$ tends to the integral $\int_0^\infty d q_\theta$, while the term in parenthesis in (\ref{RenCont}) becomes a continuous probability distribution supported on $[0,\infty)$. This specific probability distribution takes into account two points on the real line, one on the positive side and one the negative side (though not symmetrically). To explain it a bit better we can for the moment restrict ourselves to a box $[-L,L]$ and, given a function $f(x)$ supported on that box, consider the function $g(x)=f(x)+f(x-L)$, which is supported on $[0,L]$. In our limiting procedure, the continuous probability distributions on the real line, which normally are the arguments of the R{\'e}nyi entropies, will be of the $f$-type, while $p_i[\theta]/s_\theta$ tends to the distribution of the $g$-type. Obviously, the $g$-type probability distributions will always have smaller entropy than $f$-type distributions. Therefore, the continuous R{\'e}nyi entropy $h_{\alpha}\left[\theta\right]$ will also be bigger than the continuous limit of the entropy in (\ref{RenCont}) \begin{equation} h_{\alpha}\left[\theta\right]\geq \lim_{d\rightarrow \infty} \left(H_{\alpha}\left[\theta\right]+\ln s_\theta\right). \end{equation} As a consequence, using \eqref{eq:dur}, we obtain the continuous UR \begin{equation} h_{\alpha}\left[\theta\right] +h_{\beta}\left[\theta^\prime\right] \geq\ln 2 \pi \|\sin \Delta \theta\|. \end{equation} The uncertainty relation obtained is clearly weaker than the best known URs for continuous variables \cite{bialynicki06,guanlei09}. This is because the latter follow from a completely different mathematical machinery, namely "$p$-$q$ norm" inequalities for the Fourier transform. Our result is on the contrary closest in spirit with standard finite-dimensional treatment of mutually unbiased bases which, while powerful, does not know much about sophisticated properties of the Fourier transform. \section{Discussion} We have provided an entropic uncertainty relation for a discrete set of mutually unbiased, periodic coarse-grained observables. Different from the underlying continuous observables, or other discretization schemes, here the uncertainty limit is bounded only by the number of measurement outcomes, which plays the role of dimension. We extend our results to apply to observables constructed from eigenstates of any two non-parallel quadrature operators, and show that a meaningful (though not optimal) continuous limit can be obtained. \par A number of possible applications and open questions exist. First, it is tempting to ask whether these results can be extended to include more than two observables, as in \cite{paul18,silva21}. This remains an open question, since to date an entropic uncertainty relation for more than two continuous operators has not been proven \cite{weigert08}, and the established results for discrete systems \cite{SANCHEZ1993233,PhysRevA.79.022104,PhysRevA.92.032109} do not seem to be directly applicable. As an application of our results, the EUR derived here for PCG observables can be adapted to identify entanglement, or more specifically, as a criteria for EPR-steering correlations \cite{wiseman07} between two parties. For example, it is straightforward to follow the recipe in Refs. \cite{walborn11a,schneeloch13}, which, for $\alpha=\beta=1$ (Shannon entropies), leads to \begin{equation} H_{1}\left[q_\theta\|q_\phi\right] +H_{1}\left[q_{\theta^\prime}\|q_{\phi^\prime}\right] \geq\ln d, \end{equation} where $H_{1}\left[r\|s\right]$ is the conditional Shannon entropy and $r$ and $s$ refer to measurement directions of Alice and Bob (the two parties). Violation of the above inequality indicates EPR steering correlations in Alice and Bob's bipartite system. \par The main motivation for our work is the overall question concerning the behaviour of discretized observables constructed within a continuous Hilbert space, and whether these observables are ``more continuous" or ``more discrete" in their characteristics. Our results show in the case of periodic coarse graining, the discretization is indeed manifest in the desired way, whereas entropic uncertainty relations are applied. \acknowledgments \L .R. acknowledges support by the Foundation for Polish Science (IRAP project, ICTQT, Contract No. 2018/MAB/5, cofinanced by the EU within the Smart Growth Operational Programme). 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Application of electric cell-substrate impedance sensing toward personalized anti-cancer therapeutic selection Audrey F. Adcock1,2, Chiagozie O. Agbai1,2 & Liju Yang1,2 Journal of Analytical Science and Technology volume 9, Article number: 17 (2018) Cite this article Cell-based analysis may have the potential for using patient-derived tissues/cells as a basis for a more direct prediction of therapeutic outcomes. This study focused on the application of the Real Time Cell Electronic Sensing (RT-CES) system for screening effective anticancer drugs to different types of prostate cancers. The goal was to demonstrate the potential application of such technology toward the suitability in selection of personalized medicine. Three prostate cancer cell lines, DU145, LNCaP, and PC-3, and a normal prostate cell line were used. Docetaxel, Carboplatin, Abiraterone Acetate, Mitoxantrone, Sunitinib Malate were used as testing drugs. Cellular adhesion, proliferation and drug induced cellular responses were monitored by the RTCES system, and the results were correlated to the well-established cell viability tests. Identification of an effective drug from a panel of available anticancer drugs to a specific cancer cell line, or testing the effectiveness of a certain drug to a panel of prostate cancer cell lines was demonstrated. Cellular resistance to a drug at single dose and multiple dose challenges was monitored by RT-CES measurement, while such resistance was not detectable by endpoint cell viability assays. The results of this study highlighted the advantages of such real-time impedance-based sensing system for applications in chemotherapy reagent selection. Personalized medicine is therapeutic treatments derived from diagnostic analysis of samples similar to that of the patient ((ASCO), American Society of Clinical Oncology 2017). The expectation is samples that are genetically or otherwise similar will respond in a similar manner to the same treatment. The most commonly used personalized medicine diagnostic tests are genetic based. The challenge presented by this method is genetic mutations do not specifically predict therapeutic outcomes (Hamburg and Collins 2010). Cell-based analysis may have the potential for using patient-derived tissues/cells as a basis for a more direct prediction of therapeutic outcomes. With this regard, cell-based biosensors, which use living cells or tissues as a sensing element to monitor physiological and functional changes induced by external stimuli, would be advantageous. Indeed, cell-based biosensors have become an important tool in the drug discovery process to provide a simple, fast, and cost-effective approach to screen compounds for drug candidates at very early stage and avoid unnecessary large-scale and cost-intensive animal testing (Asphahani and Zhang 2007; Cooper 2006; Liu et al. 2014; Zhang et al. 2012). Particularly, label-free and non-invasive cell-based assay/sensors are becoming of great interest in many biomedical applications (Fang 2006, 2011, 2014; Hu et al. 2013; Kim et al. 2009). In recent years, a number of label-free technologies, including cell-substrate impedance (Giaever and Keese 1993; Giaever and Keese 1986), quartz crystal microbalance (QCM) (Atay et al. 2016; Braunhut et al. 2005; Zhou et al. 2000), and optical waveguide lightmode spectroscopy (OWLS) (Fang 2007; Ramsden et al. 1995), have been reported as a means for monitoring live cell status in a non-invasive and real-time manner. The QCM biosensors sense the mass or surface property change upon the binding of biological targets, through the change in frequency of a quartz crystal resonator. The OWLS biosensors sense the absorption or binding of targets on the sensor surfaces through the change in optical signals. The electrical impedance cell-based biosensors have been recognized as a useful approach that is based on electrical signals, it enables direct analysis of cellular activities occurring on an electrode surface by measuring the induced capacitance and/or resistance changes by cells attached on the electrode, in a label-free and non-invasive manner. These changes have been associated with cell proliferation, spreading, motility, and death. Such impedance-based techniques have gained a great deal of attention in the fields of studying cancer cells and monitoring drug-induced cellular activities for drug discovery (Asphahani and Zhang 2007; Chen et al. 2008; Kloß et al. 2008; Linderholm et al. 2007; Liu et al. 2009; McGuinness 2007; Solly et al. 2004; Xiao and Luong 2003). Previously, our group has used such impedance-based technique to study oral cancer cells (Arias et al. 2010) and their responses to drug treatments, as well as to distinguish cancer cells and non-cancer cells (Yang et al. 2011). Another electrical method––field-effect transistors (FET)––has also been exploited in cell-based biosensors for real-time detection of targets. In the detection, either the concentration/activity of the target analyte or the presence/quantity of a biomolecule is transduced to an electrical signal via the field effect. Fundamental principles and various applications of FET biosensors have been extensively reviewed (Kaisti 2017). This study focuses on the application of the impedance-based real-time cell analyzer (RTCA) system (ACEA Biosciences, Inc., San Diego, CA, USA) toward the selection of personalized anti-cancer medicine. Three established metastatic prostate cancer lines and a normal prostate cell line were used to demonstrate the rapid and real-time monitoring of cellular responses to five anticancer drugs, revealing the potential of such technology for selecting the effective drug for a given cell line and for testing a given drug to a panel of cell lines, as well as in special cases of monitoring drug resistance developed during treatment. The three tested prostate cancer cell lines included DU145, a brain metastatic PC cell line, LNCaP, a lymph node metastatic cell line, and PC-3, a bone metastatic PC cell line, and a normal prostate cell line. Docetaxel, the global gold standard for treating metastatic castration-resistant prostate cancer (mCRPC), was used as a testing drug (Hwang 2012). Other drugs included sunitinib malate, abiraterone acetate, mitoxantrone, and carboplatin. The RTCA impedance analyzer and cell index signal measurement The RTCA DP (Real-Time Cell Analyzer Dual Plate) Instrument (Roche and ACEA Biosciences, Inc., San Diego, CA, USA) was used in this study. The system is composed of a computer software control unit, an electronic sensor analyzer and three E-16-well plates. The computer control unit is used to set up the experimental parameters, collect the data from the analyzer in real time, and process the results. The electronic analyzer is to hold E-plates where the cells are seeded and growing and to execute the experimental protocol and record the electrical signals from the E-plates. The E-plates are single-use disposable plastic plates with an array of circle-on-line gold microelectrodes integrated into the bottom of the wells. The diameter of the circle is 100 μm. The electrode surfaces are pretreated for cell culture. In operation, cells were seeded in the plates, and the plates were mounted to the sensor analyzer station. The electronic analyzer station was placed inside an incubator at 37 °C, with 5% CO2 in air. The system was connected to the computer control unit which was placed outside of the incubator. The system monitored cellular activities by measuring the electronic impedance of sensor electrodes in the 16-well E-Plates using a parameter-termed cell index (CI). In this RT-CES system, a 10-mV AC voltage at three different frequencies (10 kHz, 25 kHz, and 50 kHz) was applied to the microelectrodes for ~ 0.1 s, and the electrical current was monitored. The impedance of the system is determined by the ratio of the applied voltage and the response current. The cell index (CI) was calculated according to the following equation (Solly et al. 2004): $$ CI=\underset{i=1,\dots N}{\max}\left(\frac{R_{cell}\left({f}_i\right)}{R_0\left({f}_i\right)}-1\right) $$ where N is the number of the frequency points at which the impedance is measured (in this system, N = 3 for 10 kHz, 25 kHz, and 50 kHz), and R0(f) and Rcell(f) are the electrode resistance/impedance without cells and with cells at each individual frequency, respectively. CI is able to provide quantitative information about the cell status, including cell number, viability, and morphology. With a certain number of cells on the electrodes, changes in cell activities will lead to a change in cell index. For example, cell proliferation on the electrodes will result in an increase in Rcell(f), and thus, an increase in cell index. On the other hand, drug-induced cell detachment, cell death, or apoptosis will result in a reduced Rcell(f) and therefore a reduced cell index. The prostate cancer cell lines LNCaP clone FGC (ATCC® CRL-1740), DU 145(ATCC®HTB-81), and PC-3 (ATCC® CRL-1435) were cultured in RPMI (Hyclone, Logan, UT) supplemented with 10% FBS (Hyclone) and antibiotics (100 IU/mL penicillin and 100 μg/mL streptomycin) (Hyclone). The Primary Prostate Epithelial Cells, Normal, Human (ATCC ® PCS-440-010) (purchased 10/07/15), were cultured in Prostate Epithelial Cell Basal Media (ATCC® PCS-440-030) supplemented with Prostate Epithelial Cell Growth Kit components (ATCC® PCS-440-040) and Gentamicin-Amphotericin B Solution (ATCC® PCS-000-025). Cells were cultured in 75 cm2 flask and were incubated at 37 °C in an atmosphere of 5% CO2 in air. The medium was renewed every 2–3 days. When 80–90% confluent, cells were detached from the flask using 0.25% trypsin with 0.53 mM EDTA (Lonza, Walkersville, MD) solution. Cells were centrifuged to remove trypsin and were suspended in fresh Prostate Epithelial Basal Medium or RPMI medium. Cell number in the suspension was determined using the Vi-cell XR cell counting system (Beckman Coulter, Miami, FL). Desired cell concentrations were obtained by diluting the cell suspension for further experiments. Drug treatments Docetaxel and sunitinib malate were obtained from BIOTANG, Inc. (Waltham, MA, USA). Abiraterone acetate was obtained from AK Scientific (Union City, CA, USA). Mitoxantrone was obtained from Selleck Chemical. Carboplatin was obtained from AdipoGen Life Sciences (San Diego, CA, USA). All were purchased through Fisher Scientific. To make stock solutions, carboplatin was dissolved in deionized (DI) H2O to make a 10-mg/mL stock solution. Docetaxel, sunitinib malate, abiraterone acetate, and mitoxantrone were dissolved in DMSO at 1 mg/mL, 40 mg/mL, 10 mg/mL, and 40 mg/mL, respectively, to make stock solutions that were further diluted with media for treatments. For traditional optical-based cell viability assays, cells were plated onto 96-well plates using a Multidrop 384 (Thermo Electron Corporation) at 5000 cells per well (cpw) for DU 145, 7000 cpw for PC-3, 17,000 cpw for LNCaP, or 8000 cpw for normal prostate and were allowed to grow for 48 h. Cells were treated with docetaxel, abiraterone acetate, sunitinib malate, mitoxantrone, and carboplatin or an equivalent concentration of the diluted vehicle at the concentrations indicated in figure legends. Addition of drugs was facilitated by the use of a HP D300e Digital Dispenser (Hewlett-Packard) and Biomek NX MC compound dispenser (Beckman Coulter, Indianapolis, IN, USA). Cells were treated with drug for 24 and 48 h, after which cell viability assays were performed. Cell viability assays Cell viability was assessed using the CellTiter 96® Non-Radioactive Cell Proliferation Assay (Promega, Madison, WI, USA) and resazurin assays. The CellTiter96® Non-Radioactive Cell Proliferation Assay is a homogenous 3-[4, 5-dimethylthiazol-2-yl]-2, 5-diphenyltetrazolium salt solution assay, more commonly known as MTT. Briefly, 15 μL of the MTT dye was added to each well and the plate returned to the incubator. Live cells converted the MTT to an insoluble product, formazin. After 3 h, 100 μL/well solubilization solution was added. Cell viability was determined by reading absorbance at 570 nm once all of the formazin crystals were dissolved. The absorbance signal was quantified using a SpectraMax M5 plate reader (molecular devices) equipped with SoftMax Pro software. Resazurin assay indicated mitochondrial metabolic activity in live cells. Briefly, 10 μL of 1 mg/mL resazurin (Fisher Scientific/Acros Organics, Fair Lawn, NJ, USA) dissolved in sterile DI H2O was added to each well. The plates were returned to the incubator for 3 h. The live cells reduced the resazurin to resorufin, a fluorescent product. The fluorescence was measured using a PHERAstar (BMG Labtech, Germany). In both assays, the amount of product was proportional to the number of live cells. Viable cell percentage was calculated by dividing the background-subtracted mean absorbance of treated sample by the background-subtracted mean untreated sample of that plate, then multiplying by 100. $$ \%\mathrm{v} iable\ cells=\left[\left( average\ treated\right)-\left( average\ background\right)\right]/\left[\left( average\ untreated\right)-\left( average\ background\right)\right]\times 100. $$ Impedance-based RTCA measurement To run an experiment, each 16-well E-plate was prepared with 50 μL of media per well and then placed into the analyzer to get a baseline reading of the impedance-based cell index signal. After the baseline was established, the E-plate was removed from the analyzer and was seeded with the same concentration of cells per well for each cell line as the optical assays. Cell volume for each well was in 100 μL of media. After cell plating, E plate was set in the cell culture hood for 30 min at room temperature to ensure the cells were settled down to the microelectrodes on the bottom surface. The E-plate was then placed back to the analyzer in the incubator. The monitoring of impedance-based CI signal was set up through the software. Data was collected every 30 min for 96 h for cell growth measurement. For drug treatment tests, CI data was recorded for approximately 48 h for cell growth, then the instrument was paused, and the cells were treated with desired concentrations of docetaxel, abiraterone acetate, sunitinib malate, mitoxantrone, carboplatin, and equivalent concentrations of the diluted vehicle at given concentrations as controls. Drug treatment was a 10× final concentration, and the volume was 1/10 volume of the culture volume in the wells, i.e., 15 μL in each well. After drug treatment, CI data was continuously recorded for another 48 h or otherwise stated in figure legends. Real-time monitoring of the growth characteristics for multiple prostate cancer cell lines By culturing biological cells on the electrode surface, the technique can directly sense detailed information about cellular activities occurring on the electrode surface by measuring the induced capacitance and/or resistance changes, eliminating multiple labeling and amplification steps typically used in many other cell-based methods. In addition, this can be done in a medium throughput platform and this is especially useful when dealing with a number of different cell lines. Figure 1 shows a group of growth curves of three different prostate cancer cell lines (DU145, LNCaP, PC-3) along with the normal prostate cell line, obtained by the impedance-sensing technique. As shown in Fig. 1, the recorded impedance-based cell growth curves clearly distinguished the growth characteristic/profile of these different prostate cancer cells. It is believed that these growth curves mainly reflected the specific degree of adhesion of these cells during the entire course of interaction with the electrodes, as well as the cell proliferation rate on the electrode. A correlation between the impedance-based cell index and the cell number for each cell line is demonstrated in Additional file 1: Figure S1. Among the cell lines, DU145 is derived from a brain metastasis of the prostate and does not express PSA. Its doubling time is approximately 29 h. In Fig. 1, it showed the highest impedance-based cell index during the cell proliferation and stationary phase. LNCaP is derived from a needle biopsy of the subclavicular lymph node and is androgen sensitive and still expresses PSA. Its doubling time is approximately 60 h, and it is generally regarded as poorly adhesive to cell culture substrate. The growth curve of LNCaP showed the lowest cell index among all cell lines at all stages. PC-3 is derived from a lumbar vertebrae metastasis, and it is androgen insensitive and does not express PSA. Its doubling time is 24–40 h varied in different conditions or laboratories. In this test, its growth curves showed moderate impedance-based cell index at all stages compared to DU145 and LNCaP cells. The growth of normal prostate epithelial cell line PCS-440-010 on the electrodes generated the impedance-based signal that was in between those tested prostate cancer cell lines. The results demonstrated that each cell line exhibited a unique impedance profile for its attachment and proliferation on the electrodes, highlighting the ability of the electrical impedance cell-based technique to identify the different adhesion properties and growth rates of the various cell lines. The different adhesion properties of the cells to the electrode are explained by the bonding forces between the plasma membrane and electrode as well as the specific combination of the surface molecules of each cell line (Asphahani and Zhang 2007). To extend, the results also suggest that this technique should be able to distinguish the growth characteristics of prostate cancer cells at different stages or different types. Proliferation curves of prostate cell lines. ECIS chart showing proliferation measurement of three metastatic prostate cancer cell lines––DU145, PC-3, and LNCaP and the normal prostate cell line. Error bars represent standard error of n = 4 values Rapid selection of effective anti-cancer drugs to a certain type of prostate cancer cell line In clinic, multiple anticancer drugs are available for treatment of cancers. However, cancer patients with different types of cancer or at different stages of a cancer generally respond to each drug with varying efficacy. The heterogeneity of the population contributes to different responses to drugs of patients with seemingly same cancer and same stage due to the fact that each tumor is as unique as that person. The current standard of care for mCRPC is treatment with hormone therapy until the cancer becomes androgen independent. Once a cancer is identified as androgen independent, the patient is switched to another chemotherapeutic, the primary drug of choice is docetaxel (Antonarakis and Armstrong 2011; Flaig et al. 2007). A rapid and easy-to-use screening method for selecting an effective drug for a patient would be most beneficial to the treated patient. Here, we demonstrated the proof-of-concept of the use of impedance-sensing technique for such purpose. Using LNCaP cell line, Fig. 2 shows the cellular responses to four anticancer drugs with respective dose ranges obtained by the impedance technique. These chemotherapy drugs were selected from the American Cancer Society and the National Cancer Institute lists of drugs used in treatment of prostate cancer. Each drug works on the cancer cells via a different mechanism of action. Docetaxel works by binding the β subunits of tubulin in microtubules, thereby stabilizing them, preventing the de-polymerization required for mitosis, which induces cell apoptosis. Abiraterone acetate acts as an irreversible inhibitor of CYP17A1 leading to a significant loss of androgen production in the peripheral organs, particularly adrenal androgens. Sunitinib malate inhibits multiple receptor tyrosine kinases (RTKs) such as vascular endothelial growth factor receptors (VEGFR1, VEGFR2, and VEGFR3) and stem cell factor receptor (KIT). Mitoxantrone causes DNA fragmentation and inhibits topoisomerase II. Mitoxantrone was one of the first drugs approved to treat mCRPC (Antonarakis and Armstrong 2011; Hwang 2012). Carboplatin undergoes activation inside cells and forms reactive platinum complexes that cause the intra- and inter-strand cross-linkage of DNA molecules within the cell. As shown in Fig. 2, treatment with each drug to LNCaP cells generated a distinguished response curve. Docetaxel at its effective doses caused a gradual continuously decreasing cell index. Carboplatin at its effective doses caused a quite rapid increase in cell index to a peak immediately after the treatment, followed by a sharp decrease until all the cells died. Abiraterone acetate at its effective doses induced an immediate steep drop of cell index, followed by a gradual increase of cell index to a flat peak, and then a slow decrease until all the cells died. Sunitinib malate induced a small gentle peak of cell index at time of drug treatment and continued its increase trend with a slower rate; it seemed that sunitinib malate did not cause cell death but only slowed down cell proliferation. Cellular responses to anti-cancer drug treatments. a–d Cellular responses of LNCaP cells to four different anticancer drugs with respective concentration ranges. e–g ECIS cell index of DU145, PC-3, and normal prostate cells, respectively, treated with five chemotherapy drugs. The data shown here is a representative of the real-time studies which were performed at least three times for each cell line and drug combination We also examined the responses of other cell lines to these drugs. Figure 2e shows the response of DU145 cell line to the five drugs at a given effective dose. Again, the cellular responses were distinct, reflecting the different effect of the drug action onto the cells on the electrodes. It is believed that such signal profile is mainly a result of drug effect on the cellular contacts with the electrodes. Generally, when cells adhere onto the electrode surface, although cell membranes are usually ~ 10–100 nm away from the electrode surface, the electrodes are still able to sense the bottom portion of cells. Cell adhesion to a substrate occurs through three major contacts including focal contacts, close contacts, and extracellular matrix contacts, and each of them has its specific separation distance from the substrate. Upon external stimuli, cells commonly experience dynamic relocation or rearrangement of certain cellular contents; some of these result in changes in cell adhesion degree, membrane ruffling, and activation of receptors at the cell surface or receptor endocytosis (Lu et al. 2001; Milligan 2003). Those which occur at the bottom portion of the cells and affect the contacts between the cells and the electrode can be sensed by the electrodes, manifested by the impedance-based cell index signal, such signal is equivalent to dynamic mass redistribution (DMR), a signal previously reported in label-free cell-based optical biosensors (Fang 2006; Fang et al. 2005a; Fang et al. 2006; Fang et al. 2005b). The DMR signal is a sum of all mass redistribution within the sensible region induced by the stimuli, although it is not specific for individual events/pathways, it is distinctive in overall profile of different cell lines in response to a same drug or the same cell line in response to different stimuli (e.g., drug treatment), as shown in Fig. 2. Besides the capacity to generate the characteristic profile for each drug, these response curves also provide some dynamic and kinetic information of the drugs to the tested cells. For example, in Fig. 2, docetaxel caused an immediate effect on the cells that was manifested by the decreasing signal immediately after the treatment, and drug effect reached the maximal (signal decreased to baseline) after approximately 24 h. With a different action mechanism, carboplatin did not exhibit effect on the cells within the first ~ 16 h after treatment. At high concentrations, drug effect can be detected at ~ 24 h, but with lower concentrations; the effect of this drug was best observed at ~ 48 h after treatment, exhibiting a dose-dependent manner. For abiraterone acetate, only at the highest tested concentration, the drug effect on the cells (signal decrease) can be immediately seen after the treatment; the effect of moderate to low concentration of this drug can be observed after 36 h of treatment. Sunitinib malate exhibited a cytostatic effect on the cells soon after the treatment (within 8 h); it seemed that the drug did not cause cell death, and it more likely just slowed down the cell proliferation (the signal still kept increasing but with a slower rate). Commonly used cell viability assays can only provide cell information about cell viability at a given time point, usually at 24 or 48 h or other defined time, but would never be possible to provide such real-time dynamic and kinetic information about the cellular responses to a drug during the entire course of the treatment. Other cell lines showed similar drug curve profiles; however, cellular response varied by cell line (data not shown). Correlations between the electric impedance-based analysis and cell viability assays We further looked into the results of impedance technique and extracted data at a given time point (for example at 24 h or 48 h after treatment) to generate dose response curves, which were compared to those obtained by the traditional cell viability MTT or resazurin assays at the same time point. Figure 3 shows the dose response curves of LNCaP cells in response to docetaxel, carboplatin, abiraterone acetate, and sunitinib malate extracted from the growth curves in Fig. 2 at either 24 or 48 h, along with the dose response curves that were obtained by the traditional MTT or resazurin cell viability assay at the same time points. It should be noted that the y-axis is the percentage of the signals of treated samples compared to the untreated controls, which would be a different meaning in the RTCA method and cell viability assay, but in both cases, it reflects the effect of a drug treatment. More specifically, in cell viability assay, it represents the percentage of surviving cells after the drug treatment, while in RTCA system, it represents the percentage of the total adhesive interactions that the cells have with the electrodes after the drug treatment and measurable by impedance. As shown in Fig. 3a, all the dose response curves from the RTCA system present similar trends as those from the cell viability assays. The percentage signals from the two methods are reasonably close in the responses to three drugs (carboplatin, abiraterone acetate, and sunitinib malate) except one drug (doxcetaxel). Comparison of impedance response (RTCA) to mitochondrial activity response. RTCA vs. mitochondrial activity dose-response curves of LNCaP cells against a docetaxel, the RTCA detected a greater response than the optical assay; b carboplatin, the dose-response patterns were similar, but the RTCA was slightly more sensitive; c abiraterone acetate, almost identical response detected by the two methods; and d sunitinib malate, similar response detected by both methods It is noted that although the dose response curve of LNCaP cells to docetaxel from the RTCA system presents a similarly trend to that from the cell viability assays; however, the magnitude in the percentage signal between the two methods were largely different, with much lower percentage signal measured by RTCA but still high percentage in cell viability. Knowing that the meaning of the percentage signals are different in the two approaches, it is understandable and explainable for the big difference. The difference in the percentage signal between the two approaches is related to the effect/action mechanism of the drug on the cells, which may be better measurable by one than the other. In the case of docetaxel, it is known to cause apoptosis which has an effect on membrane integrity and cell morphology. Cells are reported to shrink and become round due to cytoskeletal protein digestion by caspases. Figure 4 shows the microscopic images of LNCaP cells after docetaxel treatment, showing the round cells but most likely still viable. Once the cells became round, they were less adhesive to the electrodes, which was well measured by the RTCA system (Leung et al. 2005), while the MTT cell viable assay was not able to detect the changes in the cells adhesion but only detect the viability of cells. For other tested drugs, they likely affect cell adhesive interactions with the electrodes at a similar extent as to the cell viability, which was manifested by the two similar and close dose response curves in each plot in Fig. 3. It is also worthy of note that the RTCA makes it easier to visualize the slower entry of carboplatin into the cells than the other drugs. This slow entry is responsible for the observed effects at a delayed time of 20 to 26 h after treatment. These comparisons indicated the advantages of the impedance-based RTCA method lie in its capability for sensitive and real-time monitoring the cellular responses, including not only cell viability but also cell adherence and other subtle cell morphology changes induced by drug treatments. Such ability is especially useful to detect cell responses at early stage of drug treatment, while it is generally not possible for traditional MTT assay to detect. Bright field images of LNCaP response to 0.01 μM docetaxel. a LNCaP cells treated with DMSO control. b LNCaP cells treated with 0.01 μM docetaxel for 48 h. Notice the smaller and more rounded shape of the treated cells. The inserts are a part of each image with a higher magnification. Scale bars are 100 μm Determining the effectiveness of a drug to different types of cell lines With a different experimental design, evaluation of a drug to a number of cell lines can be performed, and real-time data can be obtained. This platform would be useful when applications of a newly available drug need to be tested on different stages of a cancer, or on different types of cancers. For prostate cancer, chemotherapy is typically used once the cancer has become androgen independent and spread outside the prostate gland; in most cases, the first chemo drug given is docetaxel (Antonarakis and Armstrong 2011). Figure 5 shows the response curves of different prostate cell lines (DU145, LNCaP, and PC-3) along with the normal prostate cell line in response to docetaxel. As demonstrated, docetaxel is effective for treatment to DU145 and LNCaP, but has little effect on the normal prostate cancer cell. However, it shows initial effect on PC-3 cells, but interestingly, PC-3 cells exhibit a compensatory response, meaning some pre-existing mechanism that is triggered upon exposure to docetaxel. Studies show docetaxel is effective in approximately 50% of cases, and most patients develop resistance to this chemotherapeutic (Hwang 2012; Liu et al. 2013). Comparison of four cell lines' impedance response to docetaxel. The response of the four cell lines treated with 0.1 μM docetaxel compared to DMSO control Applications for special tests––drug resistance and combination treatments This technique also allows examination of the effect of consecutive/repeated treatment with drug. Figure 6 shows the response curves of PC-3 cell line to repeated treatments with 0.1 μM docetaxel. The repeated treatments with docetaxel did not enhance the effect of docetaxel on PC-3 cells, meaning that PC-3 cells exhibited permanent response to docetaxel after the first treatment. This would be a useful tool for a clinic office to quickly rule out those drugs that may not be effective in a patient due to compensatory response. PC-3 impedance response to repeated docetaxel treatments. Cell index curve of PC-3 cells with repeated treatments of 0.1 μM docetaxel compared to controls. Each spike represents a pause in the experiment for drug treatment to at least one set of wells on the plate The other useful application of RT-CES is to test the effect of combination of chemotherapy drugs. Although chemo drugs can be used one at a time, more often, a combination of drugs may provide higher efficacy to a patient if appropriate combination can be found. It is recommended to use drugs with different mechanisms of action when treating with drug combinations (Tsakalozou et al. 2012). For this combination drug study, we selected the kinase inhibitor, sunitinib malate, and the microtubule stabilizing agent docetaxel. With this regard, we demonstrated the tests of DU145 cells treated with a combination of sunitinib malate and docetaxel (Fig. 7). The single drug treatment of 5 μg/mL sunitinib malate to DU145 cells did not result in a reduction in CI; this dose did not even exact the cytostatic effect that was observed at higher monotherapy sunitinib malate doses. A reduction in CI was seen when the cells were treated with 10 nM docetaxel as a single drug, but reduction did not decrease to baseline and appeared to plateau out. This plateau was likely due to that all of the drugs had been utilized by the cells in the well or the cells stopped taking it in as the equilibrium had been achieved between cells and media. Higher doses of docetaxel (100 nM) decreased CI signal to background. When 5 μg/mL of sunitinib malate was combined with lower doses of docetaxel a reduction in CI was observed in all doses. The same sharp immediate decrease in CI signal was observed for the combined treatment wells as the single-drug docetaxel except that the CI signal was reduced to baseline. These combination results are supported by the optical absorbance and fluorescence assay findings that were completed simultaneously with the RTCA experiment, data not shown. This reduction in signal was not seen with all combinations of these two drugs on DU145 cells. Cellular responses to combination drug treatments. RTCA cell index curves of metastatic prostate cancer cell lines treated with a combination of sunitinib malate and docetaxel. a The response of the DU145 cell line to the combination sunitinib malate at 5 μg/mL and serial dilutions of docetaxel. b The PC-3 cell line response to 0.12 μM docetaxel combined with serial dilutions of sunitinib malate It is important to note that all cell lines tested did not draw us to the same conclusion for the drug combination. The reduction in CI for the PC-3 cell line was greater in the presence of the combined drugs, and the lag between reduction and increase was longer. This indicated that there was an increase in toxicity by the drug combination in PC-3 cells; however, the increased toxicity was not sustainable. PC-3 cell line was still resistant to docetaxel. These results demonstrated that the RTCA can be used to identify drug combinations that can effectively be used to treat prostate cancer. Using several established metastatic prostate cell lines and common chemotherapeutic drugs for prostate cancer, this study demonstrated the proof-of-concept of the real-time impedance method for potential application toward selection of personalized chemotherapy drugs for patients. Such impedance-based technique measured the drug-induced effect on cellular adhesion, proliferation, and other changes of the cells cultured on the electrodes. Agreement of overall trend in drug responses obtained by this impedance-based method and the traditional optical-based cell viability assay was demonstrated. Through experimental design, RT-CES can be applied to a certain type of cancer cell line against a panel of chemotherapy drugs before the patient receives the drug treatments. It can also rule out ineffective drugs against a particular tumor or the drug that may develop resistance by a patient. Certainly, such concept needs to be applied to patient-derived samples of known drug response to validate its ability to predict the chemotherapy outcome before actual application. In drug screening, this label-free real-time screening method could benefit in minimizing some limitations in dye conversion-based end point analysis in traditional optical-based cell viability assays. ATCC®: American Type Culture Collection Cell index cpw: Cells per well CYP17A1: Member of the cytochrome P450 superfamily of enzymes, catalyzes many reactions involved in drug metabolism and synthesis of cholesterol, steroids, and other lipids DI H2O: DMR: Dynamic mass redistribution EDTA: Ethylenediaminetetraacetic acid FBS: Stem cell factor receptor mCRPC: Metastatic castration-resistant prostate cancer MTT: 3-[4, 5-Dimethylthiazol-2-yl]-2, 5-diphenyltetrazolium salt The number of the frequency points PSA: Prostate specific antigen, protein produced by prostate gland, high levels associated with prostate cancer R 0(f): Electrode resistance/impedance at initial frequency without cells R cell(f): Electrode resistance/impedance at each individual frequency with cells RTCA DP: Real-time cell analyzer dual plate RTCA: Real-time cell analyzer RT-CES: Real-time cell electronic sensing RTKs: Receptor tyrosine kinases VEGFR1, VEGFR2, and VEGFR3: Vascular endothelial growth factor receptors β subunits: Beta subunits, microtubules are composed of alpha and beta tubulin subunits (ASCO), American Society of Clinical Oncology. 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Liu C, Zhu Y, Lou W, Nadiminty N, Chen X, Zhou Q, et al. Functional p53 determines docetaxel sensitivity in prostate cancer cells. Prostate. 2013;73(4):418–27. Liu QJ, Wu CS, Cai H, Hu N, Zhou J, Wang P. Cell-based biosensors and their application in biomedicine. Chem Rev. 2014;114(12):6423–61. https://doi.org/10.1021/cr2003129. Liu Q, Yu J, Xiao L, Tang JCO, Zhang Y, Wang P, Yang M. Impedance studies of bio-behavior and chemosensitivity of cancer cells by micro-electrode arrays. Biosens Bioelectron. 2009;24(5):1305–10. Lu Z, Jiang G, Blume-Jensen P, Hunter T. Epidermal growth factor-induced tumor cell invasion and metastasis initiated by dephosphorylation and downregulation of focal adhesion kinase. Mol Cell Biol. 2001;21(12):4016–31. McGuinness R. Impedance-based cellular assay technologies: recent advances, future promise. Curr Opin Pharmacol. 2007;7(5):535–40. Milligan G. High-content assays for ligand regulation of G-protein-coupled receptors. Drug Discov Today. 2003;8(13):579–85. Ramsden JJ, Li S-Y, Heinzle E, Prenosil JE. Optical method for measurement of number and shape of attached cells in real time. Cytometry Part A. 1995;19(2):97–102. Solly K, Wang X, Xu X, Strulovici B, Zheng W. Application of real-time cell electronic sensing (RT-CES) technology to cell-based assays. Assay and drug development technologies. 2004;2(4):363–72. Tsakalozou E, Eckman AM, Bae Y. Combination effects of docetaxel and doxorubicin in hormone-refractory prostate cancer cells. Biochemistry Research International. 2012;2012:10. Article ID 832059. https://doi.org/10.1155/2012/832059. Xiao C, Luong JHT. On-line monitoring of cell growth and cytotoxicity using electric cell-substrate impedance sensing (ECIS). Biotechnol Prog. 2003;19(3):1000–5. Yang L, Arias LR, Lane TS, Yancey MD, Mamouni J. Real-time electrical impedance-based measurement to distinguish oral cancer cells and non-cancer oral epithelial cells. Anal Bioanal Chem. 2011;399(5):1823–33. Zhang Z, Guan N, Li T, Mais DE, Wang M. Quality control of cell-based high-throughput drug screening. Acta Pharm Sin B. 2012;2(5):429–38. Zhou T, Marx KA, Warren M, Schulze H, Braunhut SJ. The quartz crystal microbalance as a continuous monitoring tool for the study of endothelial cell surface attachment and growth. Biotechnology Progress. 2000;16(2):268–77. https://doi.org/10.1021/Bp000003f. The authors thank Ginger R. Smith of the NCCU BRITE high throughput screening (HTS) lab for assistance with the HP D300e Digital Dispenser and Biomek NX MC compound dispenser. This research was partially supported by NSF grant CBET no. 1159871. The dataset supporting the conclusions of this article are included within the article (and its additional file). Department of Pharmaceutical Sciences, Biomanufacturing Research Institute and Technology Enterprise (BRITE), North Carolina Central University, Durham, North Carolina, 27707, USA Audrey F. Adcock, Chiagozie O. Agbai & Liju Yang Biomanufacturing Research Institute and Technology Enterprise (BRITE), North Carolina Central University, 1801 Fayetteville Street, Durham, NC, 27727, USA Audrey F. Adcock Chiagozie O. Agbai Liju Yang AFA designed the experiments, performed the data analysis, reviewed the literature, and drafted the manuscript. COA performed the experiments and assisted with the data analysis. LY reviewed the literature and provided guidance in designing, writing, and revising the manuscript. All authors read and approved final manuscript provided guidance. Correspondence to Liju Yang. Additional file Additional file 1: Figure S1. Prostate cell titration at 24 h post seeding. Cell dilution series showing impedance linear relationship to cell number 24 h after plating. Changes in impedance reflect change in cell number. (PNG 55 kb) Adcock, A.F., Agbai, C.O. & Yang, L. Application of electric cell-substrate impedance sensing toward personalized anti-cancer therapeutic selection. J Anal Sci Technol 9, 17 (2018). https://doi.org/10.1186/s40543-018-0149-x Received: 06 June 2018 Accepted: 01 August 2018 DOI: https://doi.org/10.1186/s40543-018-0149-x Cellular impedance-based sensing Real-time cell-based assay Metastatic prostate cancer DU145 LNCaP	CommonCrawl
This category needs an editor. We encourage you to help if you are qualified. Volunteer, or read more about what this involves. Logic and Philosophy of Logic > Paradoxes > Paradoxes, Misc Paradoxes, Misc The Paradox of Analysis* (34) The Twin Paradox* (23) Moore's Paradox* (232) Paradox of Deontological Constraints* (22) Condorcet's Paradox* (35) Paradoxes, Miscellaneous (144) Associate Professor or Professor Associate Professor or Professor and Chair Listing datebook pricecategoriesFirst authorImpactPub yearDownloads Order import / add options Add an entry to this list: Batch import. Use this option to import a large number of entries from a bibliography into this category. 1 — 50 / 304 Material to categorize Reasoning and Presuppositions.Carlotta Pavese - 2021 - Philosophical Topics 49 (2):203-224.details It is a platitude that when we reason, we often take things for granted, sometimes even justifiably so. The chemist might reason from the fact that a substance turns litmus paper red to that substance being an acid. In so doing, they take for granted, reasonably enough, that this test for acidity is valid. We ordinarily reason from things looking a certain way to their being that way. We take for granted, reasonably enough, that things are as they look Although (...) it is a platitude that we often take things for granted when we reason—whether justifiably or not—one might think that we do not have to. In fact, it is a natural expectation that were we not pressed by time, lack of energy or focus, we could always in principle make explicit in the form of premises every single presupposition we make in the course of our reasoning. In other words, it is natural to expect it to be true that presuppositionless reasoning is possible. In this essay, I argue that it is false: presuppositionless reasoning is impossible. Indeed, I think this is one of the lessons of a long-standing paradox about inference and reasoning known as Lewis Carroll's (1985) regress of the premises. Many philosophers agree that Carroll's regress teaches us something foundational about reasoning. I part ways about what it is that it teaches us. What it teaches us is that the structure of reasoning is constitutively presuppositional. (shrink) Epistemology of Logic in Logic and Philosophy of Logic Epistemology of Mind in Philosophy of Mind Logical Consequence and Entailment in Logic and Philosophy of Logic Mental States and Processes in Philosophy of Mind Metaphysics of Mind in Philosophy of Mind Paradoxes, Misc in Logic and Philosophy of Logic Philosophical Language in Metaphilosophy Remove from this list Direct download (3 more) Strong Homomorphisms, Category Theory, and Semantic Paradox.Jonathan Wolfgram & Roy T. Cook - 2022 - Review of Symbolic Logic 15 (4):1070-1093.details In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox $\mathcal {L}_{\mathsf {P}}$ : strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms. In particular, we show that (i) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms between $\mathcal {L}_{\mathsf {P}}$ constructions preserve paradoxicality, (ii) many (but not all) earlier results regarding the paradoxicality of $\mathcal {L}_{\mathsf {P}}$ constructions can be recast as special cases of our central result regarding (...) strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms, and (iii) that we can use strong $\mathcal {L}_{\mathsf { P}}$ -homomorphisms to provide a simple demonstration of the paradoxical nature of a well-known paradox that has not received much attention in this context: the McGee paradox. In addition, along the way we will highlight how strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms highlight novel connections between the graph-theoretic analyses of paradoxes mobilized in the $\mathcal {L}_{\mathsf {P}}$ framework and the methods and tools of category theory. (shrink) Category Theory in Philosophy of Mathematics Mathematical Logic in Philosophy of Mathematics Designing Paradoxes: A Revision-theoretic Approach.Ming Hsiung - 2022 - Journal of Philosophical Logic 51 (4):739-789.details According to the revision theory of truth, the binary sequences generated by the paradoxical sentences in revision sequence are always unstable. In this paper, we work backwards, trying to reconstruct the paradoxical sentences from some of their binary sequences. We give a general procedure of constructing paradoxes with specific binary sequences through some typical examples. Particularly, we construct what Herzberger called "unstable statements with unpredictably complicated variations in truth value." Besides, we also construct those paradoxes with infinitely many finite primary (...) periods but without any infinite primary period, those with an infinite critical point but without any finite primary period, and so on. This is the first formal appearance of these paradoxes. Our construction demonstrates that the binary sequences generated by a paradoxical sentence are something like genes from which we can even rebuild the original sentence itself. (shrink) Revision Theory of Truth in Philosophy of Language A Pragmatic Dissolution of Curry's Paradox.Rafael Félix Mora Ramirez - 2022 - Logica Universalis 16 (1):149-175.details Although formal analysis provides us with interesting tools for treating Curry's paradox, it certainly does not exhaust every possible reading of it. Thus, we suggest that this paradox should be analysed with non-formal tools coming from pragmatics. In this way, using Grice's logic of conversation, we will see that Curry's sentence can be reinterpreted as a peculiar conditional sentence implying its own consequent. Tossing Morgenbesser's Coin.Zachary Goodsell - 2022 - Analysis 82 (2):214-221.details Morgenbesser's Coin is a thought experiment that exemplifies a widespread disposition to infer counterfactual independence from causal independence. I argue that this disposition is mistaken by analysing a closely related thought experiment. Causation, Miscellaneous in Metaphysics Possible-World Theories of Counterfactuals in Philosophy of Language (TC* ،زمان فازی ) تاثیر و تاثر منطق و پارادوکسها بر نظریه محاسبات عام. [REVIEW]Didehvar Farzad - manuscriptdetails در تکوین نظریه محاسبات از اوایل قرن بیستم پارادکسها و خود ارجاعی نقش ویژه ای را بازی کرده اند. هر چند نظریه محاسبات عام بر اساس تعریف ماشین تورینگ، فرض تورینگ_چرچ و کاربردهای آن بنا شده ،اما از همان ابتدا تا به امروز منطق و حوزه های مختلف این علم در ارتباط تنگاتنگ با این تیوری و در ابتدا نظریه محاسبات خاص بوده و این ارتباط روز به روز گسترده و گسترده تر گشته است. از تاثیر پارادوکس دروغگو و پارادکس (...) بری در تمایز میان مفهوم محاسبه پذیر و محاسبه پذیر شمارایانه، قضایای گودل و قضیه چیتین همه جا شاهد این تاثیر و تاثر متقابل هستیم. در این مقال می خواهیم بعد از معرفی اجمالی مطالب باال، تاثیر یک پارادوکس دیگر، پارادوکس اعدام غیر منتظره، را بر این نظریه بررسی نماییم. (shrink) Computability in Philosophy of Computing and Information Computational Complexity in Philosophy of Computing and Information Interpretation of Quantum Mechanics in Philosophy of Physical Science Logics in Logic and Philosophy of Logic Space and Time, Misc in Philosophy of Physical Science The Passage of Time in Metaphysics Theories and Models in General Philosophy of Science Theory of Computation, Misc in Philosophy of Computing and Information Remove from this list Direct download The Nothing from Infinity paradox versus Plenitudinous Indeterminism.Nicholas Shackel - 2022 - European Journal for Philosophy of Science 12 (online early):1-14.details The Nothing from Infinity paradox arises when the combination of two infinitudes of point particles meet in a supertask and disappear. Corral-Villate claims that my arguments for disappearance fail and concedes that this failure also produces an extreme kind of indeterminism, which I have called plenitudinous. So my supertask at least poses a dilemma of extreme indeterminism within Newtonian point particle mechanics. Plenitudinous indeterminism might be trivial, although easy attempts to prove it so seem to fail in the face of (...) plausible continuity principles. However, the question of its triviality is here moot, since I show that, except in one case, Corral-Villate's disproofs fail, and with a correction, the original arguments are unrefuted. Consequently, of the two contenders for the outcome of my supertask, the Nothing from Infinity paradox has won out. (shrink) Particle Physics in Philosophy of Physical Science Philosophy of Science, Miscellaneous in Philosophy of Science, Misc Probabilistic stability, agm revision operators and maximum entropy.Krzysztof Mierzewski - 2020 - Review of Symbolic Logic:1-38.details Several authors have investigated the question of whether canonical logic-based accounts of belief revision, and especially the theory of AGM revision operators, are compatible with the dynamics of Bayesian conditioning. Here we show that Leitgeb's stability rule for acceptance, which has been offered as a possible solution to the Lottery paradox, allows to bridge AGM revision and Bayesian update: using the stability rule, we prove that AGM revision operators emerge from Bayesian conditioning by an application of the principle of maximum (...) entropy. In situations of information loss, or whenever the agent relies on a qualitative description of her information state - such as a plausibility ranking over hypotheses, or a belief set - the dynamics of AGM belief revision are compatible with Bayesian conditioning; indeed, through the maximum entropy principle, conditioning naturally generated AGM revision operators. This mitigates an impossibility theorem of Lin and Kelly for tracking Bayesian conditioning with AGM revision, and suggests an approach to the compatibility problem that highlights the information loss incurred by acceptance rules in passing from probabilistic to qualitative representations of beliefs. (shrink) AGM Belief Revision Theory in Epistemology Bayesian Reasoning in Philosophy of Probability Maximum Entropy Principles in Philosophy of Probability Probabilistic Puzzles in Philosophy of Probability Bertrand's paradox: a physical way out along the lines of Buffon's needle throwing experiment.P. Di Porto, B. Crosignani, A. Ciattoni & H. C. Liu - 2011 - European Journal of Physics 32 (3):819–825.details Bertrand's paradox ) can be considered as a cautionary memento, to practitioners and students of probability calculus alike, of the possible ambiguous meaning of the term 'at random' when the sample space of events is continuous. It deals with the existence of different possible answers to the following question: what is the probability that a chord, drawn at random in a circle of radius R, is longer than the side of an inscribed equilateral triangle? Physics can help to remove the (...) ambiguity by identifying an actual experiment, whose outcome is obviously unique and prescribes the physical variables to which the term 'random' can be correctly applied. In this paper, after briefly describing Bertrand's paradox, we associate it with an experiment, which is basically a variation of the famous Buffon's needle experiment for estimating the value of π. Its outcome is compared with the analytic predictions of probability calculus, that is with the probability distribution of variables whose uniform distribution can be considered a sound implementation of complete randomness. (shrink) Indifference Principles in Philosophy of Probability Deontic Logic, Weakening and Decisions Concerning Disjunctive Obligations.Michael J. Shaffer - 2022 - Logos and Episteme 13 (1):93-102.details This paper introduces two new paradoxes for standard deontic logic (SDL). They are importantly related to, but distinct from Ross' paradox. These two new paradoxes for SDL are the simple weakening paradox and the complex weakening paradox. Both of these paradoxes arise in virtue of the underlaying logic of SDL and are consequences of the fact that SDL incorporates the principle known as weakening. These two paradoxes then show that SDL has counter-intuitive implications related to disjunctive obligations that arise in (...) virtue of deontic weakening and in virtue of decisions concerning how to discharge such disjunctive obligations. The main result here is then that theorem T1 is a problematic component of SDL that needs to be addressed. (shrink) Decision-Theoretic Puzzles, Misc in Philosophy of Action Deontic Logic in Logic and Philosophy of Logic Disjunction in Philosophy of Language Send in the Clowns.Daniel Nolan - forthcoming - In Karen Bennett & Dean Zimmerman (eds.), Oxford Studies in Metaphysics. Oxford: Oxford University Press.details Thought experiments are common where infinitely many entities acting in concert give rise to strange results. Some of these cases, however, can be generalized to yield almost omnipotent systems from limited materials. This paper discusses one of these cases, bringing out one aspect of what seems so troubling about "New Zeno" cases. -/- This paper is in memory of Josh Parsons. The Infinite in Philosophy of Mathematics $76.22 new $80.92 used $87.07 from Amazon (collection) View on Amazon.com Non-reflexivity and Revenge.Julien Murzi & Lorenzo Rossi - 2022 - Journal of Philosophical Logic 51 (1):201-218.details We present a revenge argument for non-reflexive theories of semantic notions – theories which restrict the rule of assumption, or initial sequents of the form φ ⊩ φ. Our strategy follows the general template articulated in Murzi and Rossi [21]: we proceed via the definition of a notion of paradoxicality for non-reflexive theories which in turn breeds paradoxes that standard non-reflexive theories are unable to block. Logical Semantics and Logical Truth in Logic and Philosophy of Logic Towards a Non-classical Meta-theory for Substructural Approaches to Paradox.Lucas Rosenblatt - 2021 - Journal of Philosophical Logic 50 (5):1007-1055.details In the literature on self-referential paradoxes one of the hardest and most challenging problems is that of revenge. This problem can take many shapes, but, typically, it besets non-classical accounts of some semantic notion, such as truth, that depend on a set of classically defined meta-theoretic concepts, like validity, consistency, and so on. A particularly troubling form of revenge that has received a lot of attention lately involves the concept of validity. The difficulty lies in that the non-classical logician cannot (...) accept her own definition of validity because it is given in a classical meta-theory. It is often suggested that this mismatch between the consequence relation of the account being espoused and the consequence relation of the meta-theory is a serious embarrassment. The main goal of the paper is to explore whether certain substructural accounts of the paradoxes can avoid this sort of embarrassment. Typically, these accounts are expressively incomplete, since they cannot assert in the object language that certain invalid arguments are in fact invalid. To overcome this difficulty I develop a novel type of hybrid proof-procedure, one that takes invalidities to be just as fundamental as validities. I prove that this proof-procedure enjoys a number of interesting properties and I analyze the prospects of applying it to languages capable of expressing self-referential statements. (shrink) Logic in Philosophy in Logic and Philosophy of Logic Nonclassical Logics in Logic and Philosophy of Logic Modes of Truth: The Unified Approach to Truth, Modality, and Paradox.Carlo Nicolai & Johannes Stern (eds.) - 2021 - New York, NY: Routledge.details The aim of this volume is to open up new perspectives and to raise new research questions about a unified approach to truth, modalities, and propositional attitudes. The volume's essays are grouped thematically around different research questions. The first theme concerns the tension between the theoretical role of the truth predicate in semantics and its expressive function in language. The second theme of the volume concerns the interaction of truth with modal and doxastic notions. The third theme covers higher-order solutions (...) to the semantic and modal paradoxes, providing an alternative to first-order solutions embraced in the first two themes. This book will be of interest to researchers working in epistemology, logic, philosophy of logic, philosophy of language, philosophy of mathematics, and semantics. (shrink) Higher-Order Logic in Logic and Philosophy of Logic Mathematics in Formal Sciences Modal and Intensional Logic in Logic and Philosophy of Logic Propositional Attitudes in Philosophy of Mind Truth in Philosophy of Language An Implicative Expansion of Belnap's Four-Valued Matrix: A Modal Four-Valued Logic Without Strong Modal Lukasiewicz-Type Paradoxes.José Miguel Blanco - 2020 - Bulletin of Symbolic Logic 26 (3-4):297-298.details Unwinding Modal Paradoxes on Digraphs.Ming Hsiung - 2021 - Journal of Philosophical Logic 50 (2):319-362.details The unwinding that Cook, 767–774 2004) proposed is a simple but powerful method of generating new paradoxes from known ones. This paper extends Cook's unwinding to a larger class of paradoxes and studies further the basic properties of the unwinding. The unwinding we study is a procedure, by which when inputting a Boolean modal net together with a definable digraph, we get a set of sentences in which we have a 'counterpart' for each sentence of the Boolean modal net and (...) each point of the digraph. What is more, whenever a sentence of the Boolean modal net says another sentence is necessary, then the counterpart of the first sentence at a point correspondingly says the counterparts of the second one at all accessible points of that point are all true. The output of the procedure is called 'the unwinding of a Boolean modal net on a definable digraph'. We prove that the unwinding procedure preserves paradoxicality: a Boolean modal net is paradoxical on a definable digraph, iff the unwinding of it on this digraph is also paradoxical. Besides, the dependence digraph for the unwinding of a Boolean modal net on a definable digraph is proved to be isomorphic to the unwinding of the dependence digraph for the Boolean modal net on the previous definable digraph. So the unwinding of a Boolean modal net on a digraph is self-referential, iff the Boolean modal net is self-referential and the digraph is cyclic. Thus, on the one hand, the unwinding of any Boolean modal net on an acyclic digraph is non-self-referential. In particular, the unwinding of any Boolean modal net on \ is non-self-referential. On the other hand, if a Boolean modal net is paradoxical on a locally finite digraph, the unwinding of it on that digraph must be self-referential. Hence, starting from a Boolean modal paradox, the unwinding can output a non-self-referential paradox only if the digraph is not locally finite. (shrink) Logic and Philosophy of Logic, Misc in Logic and Philosophy of Logic God, Gluts and Gaps: Examining an Islamic Traditionalist Case for a Contradictory Theology.Safaruk Zaman Chowdhury - 2020 - History and Philosophy of Logic 42 (1):17-43.details In this paper, I examine the deep theological faultline generated by divergent understandings of the divine attributes among two early antagonistic Muslim groups – the traditionalists (main... Arabic and Islamic Philosophy in Philosophical Traditions, Miscellaneous Background for the Uninitiated.Richmond Campbell - 1985 - In Richmond Campbell & Lanning Sowden (eds.), Paradoxes of Rationality and Cooperation: Prisoner's Dilemma and Newcomb's Problem. Vancouver: pp. 3-41.details Newcomb's Problem in Philosophy of Action Prisoner's Dilemma in Philosophy of Action $15.00 used (collection) View on Amazon.com Remove from this list UN SEMPLICE MODO PER TRATTARE LE GRANDEZZE INFINITE ED INFINITESIME.Yaroslav Sergeyev - 2015 - la Matematica Nella Società E Nella Cultura: Rivista Dell'Unione Matematica Italiana, Serie I 8:111-147.details A new computational methodology allowing one to work in a new way with infinities and infinitesimals is presented in this paper. The new approach, among other things, gives the possibility to calculate the number of elements of certain infinite sets, avoids indeterminate forms and various kinds of divergences. This methodology has been used by the author as a starting point in developing a new kind of computer – the Infinity Computer – able to execute computations and to store in its (...) memory not only finite numbers but also infinite and infinitesimal ones. (shrink) Set Theory as a Foundation, Misc in Philosophy of Mathematics The Continuum Hypothesis in Philosophy of Mathematics The Nature of Sets, Misc in Philosophy of Mathematics Numerical infinities applied for studying Riemann series theorem and Ramanujan summation.Yaroslav Sergeyev - 2018 - In AIP Conference Proceedings 1978. AIP. pp. 020004.details A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying (...) infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid's Common Notion no. 5 'The whole is greater than the part' and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor's and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann's result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used. (shrink) On some analogies between the counterexamples to modus ponens (and modus tollens).Lina Maria Lissia - 2020 - The Reasoner 14 (6):35-37.details Bertrand's Paradox and the Maximum Entropy Principle.Nicholas Shackel & Darrell P. Rowbottom - 2020 - Philosophy and Phenomenological Research 101 (3):505-523.details An important suggestion of objective Bayesians is that the maximum entropy principle can replace a principle which is known to get into paradoxical difficulties: the principle of indifference. No one has previously determined whether the maximum entropy principle is better able to solve Bertrand's chord paradox than the principle of indifference. In this paper I show that it is not. Additionally, the course of the analysis brings to light a new paradox, a revenge paradox of the chords, that is unique (...) to the maximum entropy principle. (shrink) Bunder's paradox.Michael Caie - 2020 - Review of Symbolic Logic 13 (4):829-844.details Systems of illative logic are logical calculi formulated in the untyped λ-calculus supplemented with certain logical constants.1 In this short paper, I consider a paradox that arises in illative logic. I note two prima facie attractive ways of resolving the paradox. The first is well known to be consistent, and I briefly outline a now standard construction used by Scott and Aczel that establishes this. The second, however, has been thought to be inconsistent. I show that this isn't so, by (...) providing a nonempty class of models that establishes its consistency. I then provide an illative logic which is sound and complete for this class of models. I close by briefly noting some attractive features of the second resolution of this paradox. (shrink) Logics, Misc in Logic and Philosophy of Logic A Two-Dimensional Logic for Two Paradoxes of Deontic Modality.Fusco Melissa & Kocurek Alexander - forthcoming - Review of Symbolic Logic.details In this paper, we axiomatize the deontic logic in Fusco 2015, which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross's Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions (...) Fusco's account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to 'lift' a Kripke frame for a one- dimensional modal logic into two dimensions. (shrink) A Quantificational Analysis of the Liar Paradox.Matheus Silva - manuscriptdetails It seems that the most common strategy to solve the liar paradox is to argue that liar sentences are meaningless and, consequently, truth-valueless. The other main option that has grown in recent years is the dialetheist view that treats liar sentences as meaningful, truth-apt and true. In this paper I will offer a new approach that does not belong in either camp. I hope to show that liar sentences can be interpreted as meaningful, truth-apt and false, but without engendering any (...) contradiction. This seemingly impossible task can be accomplished once the semantic structure of the liar sentence is unpacked by a quantified analysis. The paper will be divided in two sections. In the first section, I present the independent reasons that motivate the quantificational strategy and how it works in the liar sentence. In the second section, I explain how this quantificational analysis allows us to explain the truth teller sentence and a counter-example advanced against truthmaker maximalism, and deal with some potential objections. (shrink) Complex Demonstratives in Philosophy of Language Descriptions, Misc in Philosophy of Language Liar Paradox in Logic and Philosophy of Logic Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic Quantification and Ontology in Philosophy of Language Russell's Theory of Descriptions in Philosophy of Language Truth-Value Gaps in Philosophy of Language Truthmaker Semantics in Philosophy of Language Truthmakers in Metaphysics Incompleteness Via Paradox and Completeness.Walter Dean - 2020 - Review of Symbolic Logic 13 (3):541-592.details This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various "paradoxical notions" for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in (...) order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell's paradox, a variant of Mirimanoff's paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink) Set Theory in Philosophy of Mathematics The Paradox of Inwardness in Kant and Kierkegaard: Ronald Green's Legacy in Philosophy of Religion.Stephen R. Palmquist - 2016 - Journal of Religious Ethics 44 (4):738-751.details Aside from bioethics, the main theme of Ronald Green's lifework has been an exploration of the relation between religion and morality, with special emphasis on the philosophies of Immanuel Kant and Søren Kierkegaard. This essay summarizes and assesses his work on this theme by examining, in turn, four of his relevant books. Religious Reason (1978) introduced a new method of comparative religion based on Kant's model of a rational religion. Religion and Moral Reason (1988) expanded on this project, clarifying that (...) religious traditions cannot be reduced to their moral grounding. Kierkegaard and Kant: The Hidden Debt (1992) offered bold new evidence that Kant, not Hegel, was the philosopher whose ideas primarily shaped Kierkegaard's overtly religious philosophy ; both philosophers focused on the problem of how to understand the relation between moral reasoning and historical religion. And Kant and Kierkegaard on Time and Eternity (2011) republished ten essays that explore various aspects of this theme in greater depth. I argue that throughout these works Green defends a " paradox of inwardness " : principles or ideals that are by their nature essentially inward end up requiring outward manifestation in order to be confirmed or fully justified as real. (shrink) Religious Ethics in Normative Ethics Søren Kierkegaard in 19th Century Philosophy What Is morality? And why can't we decide? (original title).Stephen R. Palmquist - 2013 - Morality: Diversity of Concepts and Meanings.details I was invited to contribute this short piece to a book published in Russia, consisting of brief statements on the nature of morality written by approximately 90 scholars. Each essay is published in both English and Russian. My essay offers my considered answer to the question posed in the title, though in the end all contributions were published without titles. An Eternal Society Paradox.Wade A. Tisthammer - 2020 - Aporia 30 (1):49-58.details An eternal society with the abilities of ordinary humans in each year of its existence would have had the ability to actualize a logical contradiction. This fact casts doubt on the metaphysical possibility of an infinite past. In addition to using this paradox in an argument against an infinite past, one can also use the paradox mutatis mutandis as a decisive argument against the sempiternality of God. Cosmological Arguments from Regress in Philosophy of Religion Divine Eternity in Philosophy of Religion Kalam Cosmological Argument in Philosophy of Religion Philosophy of Time, Misc in Metaphysics ST, LP and Tolerant Metainferences.Bogdan Dicher & Francesco Paoli - 2019 - In Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency. Springer Verlag. pp. 383-407.details The strict-tolerant approach to paradox promises to erect theories of naïve truth and tolerant vagueness on the firm bedrock of classical logic. We assess the extent to which this claim is founded. Building on some results by Girard we show that the usual proof-theoretic formulation of propositional ST in terms of the classical sequent calculus without primitive Cut is incomplete with respect to ST-valid metainferences, and exhibit a complete calculus for the same class of metainferences. We also argue that the (...) latter calculus, far from coinciding with classical logic, is a close kin of Priest's LP. (shrink) Proof Theory in Logic and Philosophy of Logic $137.05 new (collection) View on Amazon.com Norms, Logics and Information Systems: New Studies on Deontic Logic and Computer Science.Henry Prakken & Paul McNamara (eds.) - 1999 - Amsterdam/Oxford/Tokyo/Washington DC: IOS Press.details This anthology contains revised versions of selected papers presented at the fourth bi-annual international deontic logic conference, DEON'98. This volume includes our substantial introduction, and an article from me as a contributor. The volume includes papers from all four distinguished invited speakers, David Makinson, Donald Nute, Claudio Pizzi, and the founder of deontic logic, Georg Von Wright. Other notables among the authors are Dov Gabbay (co-editor of the Handbook on Philosophical Logic vols.1-4, and editor of a number of logic book (...) series); Lars Lindahl (author of Position and Change: A Study in Law and Logic); Andrew Jones (past editor of the Norwegian Journal of Philosophical Logic, and the co-author of the entry "Deontic Logic and Contrary-to-Duties" in the 2nd edition of Handbook on Philosophical Logic); and Marek Sergot (a leading researcher at the interface of deontic logic and computer science.). (shrink) Conditionals in Philosophy of Language Moral Language, Misc in Meta-Ethics Moral Norms in Meta-Ethics Moral Semantics in Meta-Ethics Practical Reason, Misc in Philosophy of Action Values and Norms in Normative Ethics Chisholm's Modal Paradox(es) and Counterpart Theory 50 Years On.Murali Ramachandran - forthcoming - Logic and Logical Philosophy:1.details Lewis's [1968] counterpart theory (LCT for short), motivated by his modal realism, made its appearance within a year of Chisholm's modal paradox [1967]. We are not modal realists, but we argue that a satisfactory resolution to the paradox calls for a counterpart-theoretic (CT-)semantics. We make our case by showing that the Chandler–Salmon strategy of denying the S4 axiom [◊◊ψ →◊ψ] is inadequate to resolve the paradox – we take on Salmon's attempts to defend that strategy against objects from Lewis and (...) Williamson. We then consider three substantially different CT-approaches: Lewis's LCT, Forbes's (FCT), including his fuzzy version, and Ramachandran's (RCT). We argue that the best approach is a mish-mash of FCT and RCT. (shrink) Two new series of principles in the interpretability logic of all reasonable arithmetical theories.Evan Goris & Joost J. Joosten - 2020 - Journal of Symbolic Logic 85 (1):1-25.details The provability logic of a theory T captures the structural behavior of formalized provability in T as provable in T itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations.The logic IL is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this article (...) we raise the previously known lower bound of IL by exhibiting two series of principles which are shown to be provable in any such theory. Moreover, we compute the collection of frame conditions for both series. (shrink) Epistemic Logic in Logic and Philosophy of Logic Explicitní/implicitní přesvědčení a derivační systémy [Explicit/Implicit Belief and Derivation Systems].Jiri Raclavsky & Ivo Pezlar - 2019 - Filosoficky Casopis 67 (1): 89-120.details The problem of hyperintensional contexts, and the problem of logical omniscience, shows the severe limitation of possible-worlds semantics which is employed also in standard epistemic logic. As a solution, we deploy here hyperintensional semantics according to which the meaning of an expression is an abstract structured algorithm, namely Tichý's construction. Constructions determine the denotata of expressions. Propositional attitudes are modelled as attitudes towards constructions of truth values. Such a model of belief is, of course, inferentially restrictive. We therefore also propose (...) a model of implicit knowledge, which is the collection of a possible agent's explicit beliefs which are related through a derivation system mastered by the agent. A derivation system consists of beliefs and derivation rules by means of which the agent may derive beliefs different from the beliefs she is actually related to. Conditions imposed on the set of base beliefs and the set of rules capture the limitations of the agent's deriving capabilities. (shrink) Doxastic and Epistemic Logic in Logic and Philosophy of Logic Higher-Order Logic, Misc in Logic and Philosophy of Logic Quantified Modal Logic in Logic and Philosophy of Logic John A. Carpenter, Omar K. Moore, Charles R. Snyder, and Edith S. Lisansky. Alcohol and higher-order problem solving. Quarterly journal of studies on alcohol , vol. 22 , pp. 183–222. [REVIEW]Alonzo Church - 1965 - Journal of Symbolic Logic 30 (2):243.details Marshall Swain. Editor's introduction. Induction, acceptance, and rational belief, edited by Marshall Swain, D. Reidel Publishing Company, Dordrecht-Holland, and Humanities Press, New York, 1970, pp. 1–5. - Frederic Schick. Three logics of belief. Induction, acceptance, and rational belief, edited by Marshall Swain, D. Reidel Publishing Company, Dordrecht-Holland, and Humanities Press, New York, 1970, pp. 6–26. - Marshall Swain. The consistency of rational belief. Induction, acceptance, and rational belief, edited by Marshall Swain, D. Reidel Publishing Company, Dordrecht-Holland, and Humanities Press, New York, 1970, pp. 27–54. - Henry E. KyburgJr., Conjunctivitis. Induction, acceptance, and rational belief, edited by Marshall Swain, D. Reidel Publishing Company, Dordrecht-Holland, and Humanities Press, New York, 1970, pp. 55–82. - Gilbert H. Harman. Induction. A discussion of the relevance of the theory of knowledge to the theory of induction . Induction, acceptance, and rational bel. [REVIEW]Ian Hacking - 1974 - Journal of Symbolic Logic 39 (1):166-168.details Introductions to Logic in Logic and Philosophy of Logic Irving M. Copi. The theory of logical types. Routledge & Kegan Paul, London1971, x + 129 pp. [REVIEW]Francis Jeffry Pelletier - 1974 - Journal of Symbolic Logic 39 (1):174-177.details Burge Tyler. Frege and the hierarchy. Synthese, vol. 40 , pp. 265–281.Parsons Terence D.. Frege's hierarchies of indirect senses and the paradox of analysis. The foundations of analytic philosophy, edited by French Peter A., Uehling Theodore E. Jr., and Wettstein Howard K., Midwest studies in philosophy, vol. 6, University of Minnesota Press, Minneapolis 1981, pp. 37–57. [REVIEW]M. J. Cresswell - 1983 - Journal of Symbolic Logic 48 (2):495-496.details James Cargile. Paradoxes. A study in form and predication. Cambridge studies in philosophy. Cambridge University Press, Cambridge etc. 1979, xvii + 308 pp. [REVIEW]John Hawthorn - 1985 - Journal of Symbolic Logic 50 (1):250-252.details Nicholas M. Asher and Johan A. W. Kamp. The knower's paradox and representational theories of attitudes. Theoretical aspects of reasoning about knowledge, Proceedings of the 1986 conference, edited by Joseph Y. Halpern, Morgan Kaufmann Publishers, Los Altos1986, pp. 131–147. [REVIEW]William J. Rapaport - 1988 - Journal of Symbolic Logic 53 (2):666.details Second-Order Logic of Paradox.Allen P. Hazen & Francis Jeffry Pelletier - 2018 - Notre Dame Journal of Formal Logic 59 (4):547-558.details The logic of paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely (...) difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. (shrink) On the Paradox of the Adder.Ferenc András - 2011 - The Reasoner 5 (3).details Sometimes it is worth using Tarski's solution rather than merely mentioning it. 'What the Tortoise Said to Achilles': Lewis Carroll's Paradox of Inference. [REVIEW]Corine Besson - 2018 - History and Philosophy of Logic 39 (1):96-98.details This double issue of the Carrollian, the journal of the Lewis Carroll Society, is entirely devoted to Lewis Carroll's famous short paper published in the journal Mind in 1895 under the title 'What... Logic and Philosophy of Logic, General Works in Logic and Philosophy of Logic Remove from this list Direct download (10 more) Curry's Paradox.Lionel Shapiro & Jc Beall - 2017 - Edward N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. CSLI Publications.details "Curry's paradox", as the term is used by philosophers today, refers to a wide variety of paradoxes of self-reference or circularity that trace their modern ancestry to Curry (1942b) and Löb (1955). The common characteristic of these so-called Curry paradoxes is the way they exploit a notion of implication, entailment or consequence, either in the form of a connective or in the form of a predicate. Curry's paradox arises in a number of different domains. Like Russell's paradox, it can take (...) the form of a paradox of set theory or the theory of properties. But it can also take the form of a semantic paradox, closely akin to the Liar paradox. Curry's paradox differs from both Russell's paradox and the Liar paradox in that it doesn't essentially involve the notion of negation. Common truth-theoretic versions involve a sentence that says of itself that if it is true then an arbitrarily chosen claim is true, or—to use a more sinister instance—says of itself that if it is true then every falsity is true. The paradox is that the existence of such a sentence appears to imply the truth of the arbitrarily chosen claim, or—in the more sinister instance—of every falsity. In this entry, we show how the various Curry paradoxes can be constructed, examine the space of available solutions, and explain some ways Curry's paradox is significant and poses distinctive challenges. (shrink) Paraconsistent Logic in Logic and Philosophy of Logic Paradoxes in Logic and Philosophy of Logic Substructural Logic in Logic and Philosophy of Logic The truth functional hypothesis does not imply the liars paradox.M. Martins Silva - 2017 - Unisinos Journal of Philosophy 17 (3):1-2.details The truth-functional hypothesis states that indicative conditional sentences and the material implication have the same truth conditions. Haze (2011) has rejected this hypothesis. He claims that a self-referential conditional, coupled with a plausible assumption about its truth-values and the assumption that the truth-functional hypothesis is true, lead to a liar's paradox. Given that neither the self-referential conditional nor the assumption about its truth-values are problematic, the culprit of the paradox must be the truth-functional hypothesis. Therefore, we should reject it. In (...) this paper I argue that, contrary to what Haze thinks, the truth-functional hypothesis is not to blame. In fact, no liar's paradox emerges when the truth-functional hypothesis is true; it emerges only if it is false. (shrink) Classical Logic in Logic and Philosophy of Logic Conditionals, Misc in Philosophy of Language Indicative Conditionals, Misc in Philosophy of Language Logic of Conditionals in Philosophy of Language Philosophy of Language, Miscellaneous in Philosophy of Language Truth-Conditional Accounts of Indicative Conditionals in Philosophy of Language Disarming a Paradox of Validity.Hartry Field - 2017 - Notre Dame Journal of Formal Logic 58 (1):1-19.details Any theory of truth must find a way around Curry's paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most "naive truth theorists." To this end they recommend a radical (...) solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction. In this paper I argue that substructuralism is unnecessary. Diagnosing the "v-Curry" is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem. (shrink) Cantor's Proof in the Full Definable Universe.Laureano Luna & William Taylor - 2010 - Australasian Journal of Logic 9:10-25.details Cantor's proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard's paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on the (...) scope of quantifiers reveals a natural way out. (shrink) Cardinals and Ordinals, Misc in Philosophy of Mathematics Predicativism in Mathematics in Philosophy of Mathematics The Nature of Sets in Philosophy of Mathematics Interpretando la Paradoja de Moore.Cristina Borgoni - 2008 - Theoria: Revista de Teoría, Historia y Fundamentos de la Ciencia 23 (2):145-161.details RESUMEN: Este trabajo ofrece una lectura de la Paradoja de Moore que pone énfasis en su relevancia para nuestra comprensión de la racionalidad y de la interpretación lingüística. Mantiene que las oraciones que dan origen a la paradoja no necesitan entenderse en términos de ausencia de una contradicción, sino más bien en términos de ausencia de racionalidad, entendida esta como un término más amplio que el de coherencia y consistencia lógica. Se defenderá tal posición por medio de tres tesis, dos (...) de las cuales se derivan de los enfoques dominantes a la paradoja: el de Moore, el de Wittgenstein y el de Shoemaker.ABSTRACT: This paper offers an interpretation of Moore's Paradox that emphasizes its relevance for our understanding of rationality and linguistic interpretation. The sentences that originate the paradox do not need to be thought of in terms of the absence of a contradiction, but in terms of absence of rationality, where rationality is understood as a broader notion than coherence and logical consistency. This is defended through three theses, two of which stem from the dominant approaches to the paradox: Moore's, Wittgenstein's and Shoemaker's. (shrink) G. E. Moore in 20th Century Philosophy The very idea of a substructural approach to paradox.Lionel Shapiro - 2016 - Synthese 199 (Suppl 3):767-786.details This paper aims to call into question the customary division of logically revisionary responses to the truth-theoretic paradoxes into those that are "substructural" and those that are " structural." I proceed by examining, as a case study, Beall's recent proposal based on the paraconsistent logic LP. Beall formulates his response to paradox in terms of a consequence relation that obeys all standard structural rules, though at the price of the language's lacking a detaching conditional. I argue that the same response (...) to paradox can be given using a consequence relation that preserves detachment rules for a conditional, though at the price of restricting structural rules. The question "Is paradox being blocked by invoking a substructural consequence relation?" is thus ill-posed. The lesson of this example, I conclude, is that there is no useful explication of the idea of a substructural approach to paradox. (shrink) A. S. Kahr, Edward F. Moore, and Hao Wang. Entscheidungsproblem reduced to the ∀∃∀ case. Proceedings of the National Academy of Sciences, Bd. 48 , S. 365–377. [REVIEW]W. Ackermann - 1962 - Journal of Symbolic Logic 27 (2):225-225.details Logical Connectives in Logic and Philosophy of Logic Model Theory in Logic and Philosophy of Logic Quantifiers in Philosophy of Language	CommonCrawl
\begin{document} \title{Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis} \begin{abstract} The Quantum State Preparation problem aims to prepare an $n$-qubit quantum state $\|\psi_v\rangle =\sum_{k=0}^{2^n-1}v_k\|k\rangle$ from {the} initial state $\|0\rangle^{\otimes n}$, for a given unit vector $v=(v_0,v_1,v_2,\ldots,v_{2^n-1})^T\in \mathbb{C}^{2^n}$ with $\\|v\\|_2 = 1$. The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth complexity remains open when ancillary qubits are available. In this paper, we study quantum circuits when there are $m$ ancillary qubits available. We construct, for any $m$, circuits that can prepare $\|\psi_v\rangle$ in depth $\tilde O\big(\frac{2^n}{m+n}+n\big)$ and size $O(2^n)$, achieving the optimal value for both measures simultaneously. These results also imply a depth complexity of $\Theta\big(\frac{4^n}{m+n}\big)$ for quantum circuits implementing a general $n$-qubit unitary for any $m \le O(2^n/n)$ number of ancillary qubits. This resolves the depth complexity for circuits without ancillary qubits. And for circuits with exponentially many ancillary qubits, our result quadratically improves the currently best upper bound of {$O(4^n)$} to $\tilde \Theta(2^n)$. Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide parameter regime. The results can be viewed as (optimal) time-space trade-off bounds, which are not only theoretically interesting, but also practically relevant in the current trend that the number of qubits starts to take off, by showing a way to use a large number of qubits to compensate the short qubit lifetime. \end{abstract} \section{Introduction} \label{sec:introduction} Quantum computers provide a great potential of solving certain important information processing tasks that are believed to be intractable for classical computers. In recent years, quantum machine learning \cite{biamonte2017quantum} and Hamiltonian simulation \cite{berry2015simulating,low2017optimal,low2019hamiltonian,berry2015hamiltonian} have also been extensively investigated, including quantum principal component analysis (QPCA) \cite{lloyd2014quantum}, quantum recommendation systems \cite{kerenidis2017quantum}, quantum singular value decomposition \cite{rebentrost2018quantum}, quantum linear system algorithm \cite{harrow2009quantum,wossnig2018quantum}, quantum clustering \cite{kerenidis2018q,kerenidis2020quantum} and quantum support vector machine (QSVM) \cite{rebentrost2014quantum}. One of the challenges to fully exploit quantum algorithms for these tasks, however, is to efficiently prepare a starting state\footnote{These starting states (for example, those in \cite{harrow2009quantum,wossnig2018quantum}) are very generic. Indeed, the lower bound argument in our later Theorem \ref{thm:lowerbound_QSP} applies to the generation of these states as well.}, which is usually the first step of those algorithms. This raises the fundamental question about the complexity of the quantum state preparation (QSP) problem. The QSP problem can be formulated as follows. Suppose we have a vector $v=(v_0,v_1,v_2,\ldots,v_{2^n-1})^T\in \mathbb{C}^{2^n}$ with unit $\ell_2$-norm, i.e. $\sqrt{\sum_{k=0}^{2^n-1}\|v_k\|^2}=1$. The task is to generate a corresponding $n$-qubit quantum state \[\|\psi_v\rangle=\sum_{k=0}^{2^n-1}v_k\|k\rangle,\] by a quantum circuit from the initial state $\|0\rangle^{\otimes n}$, where $\{\|k\rangle: k=0, 1, \ldots, 2^n-1\}$ is the computational basis of the quantum system. Different cost measures can be studied for quantum circuits: Size, depth, and number of qubits are among the most prominent ones. For a quantum circuit, the depth corresponds to the time for executing the quantum circuit, and the number of qubits used to its space cost. Apart from minimizing each cost measure individually, it is of particular interest to study a time-space trade-off for quantum circuits. The reason is that in the past decade, we have witnessed a rapid development in qubit number and in qubit lifetime\footnote{Take superconducting qubits, for example, the qubit number jumped from 5 in 2014 to 127 in 2021 \cite{barends2014superconducting,kelly2015state,song201710,arute2019quantum,IBM65qubit,chow2021ibm} .}, but it seems hard to significantly improve \textit{both} on the same chip. Looking into the near future, big players such as IBM and Google announced their roadmaps of designing and manufacturing quantum chips with about 1,000,000 superconducting qubits by 2026 and 2029, respectively, rocketing from 50-100 today \cite{IBMroadmap, Googleroadmap}. This raises a natural question for quantum algorithm design: How to utilize the fast-growing number of qubits to overcome the relatively limited decoherence time? This seems especially relevant in the near future when we have $10^4- 10^5$ qubits, which are expected to run certain quantum simulation algorithms for chemistry problems, but are not sufficient for the full quantum error correction to fight the decoherence. Or put in a computational complexity language, how to efficiently trade space for time in a quantum circuit? In this paper, we will address this question in the fundamental tasks of quantum state preparation and general unitary circuit synthesis. Let us first fix a proper circuit model. If we aim to generate the target state $\|\psi_v\rangle$ or perform the target unitary precisely, then a finite universal gate set is not enough. A natural choice is the set of circuits that consist of arbitrary single-qubit gates and CNOT gates , which is expressive enough to generate arbitrary states $\|\psi_v\rangle$ precisely with certainty. We will study the optimal depth for this class of circuits\footnote{Since two-qubit gates are usually harder to implement, one may also like to consider CNOT depth, the number of layers with at least one CNOT gate. But note that between two CNOT layers, consecutive single-qubit gates on the same qubit can be compressed to one single-qubit gate, and single-qubit gates on different qubits can be paralleled to within one layer, we can always assume that the circuit has alternative single-qubit gate layers and CNOT gate layers. Therefore the circuit depth is at most twice of the CNOT depth, making the two measures the same up to a factor of 2.}. The study of QSP dates back to 2002, when Grover and Rudolph gave an algorithm for QSP for the special case of efficiently integrable probability density functions \cite{grover2002creating}. {Their circuit has $n$ stages, and each stage $j$ has $2^{j-1}$ layers, with each layer being a rotation on last qubit conditioned on the first $j-1$ qubits being certain computational basis state. This type of multiple-controlled $(2\times 2)$-unitary can be implemented in depth $O(n)$ without ancillary qubit\footnote{The standard method \cite{nielsen2002quantum} gives a depth upper bound of $O(n^2)$ without ancillary qubit and $O(n)$ with sufficiently many ancillary qubits. The first bound can be improved to $O(n)$ by the method in \cite{multi-controlled-gate}.}, yielding a depth upper bound of $O(n2^n)$ for the QSP problem.} In \cite{bergholm2005quantum}, Bergholm \textit{et al.} gave an upper bound of $2^{n+1}-2n-2$ for the \emph{number} of CNOT gates, with depth also of order {$O(2^n)$}. The number of CNOT gates is improved to $\frac{23}{24} 2^n - 2^{\frac{n}{2}+1} + \frac{5}{3}$ for even $n$, and $\frac{115}{96} 2^n $ for odd $n$ by Plesch and Brukner \cite{plesch2011quantum}, based on a universal gate decomposition technique in \cite{mottonen2005decompositions}. The same paper \cite{bergholm2005quantum} also gives a depth upper bound of $\frac{23}{48}2^n$ for even $n$ and $\frac{115}{192}2^n$ for odd $n$. All these results are about the exact quantum state preparation without ancillary qubits. With ancillary qubits, Zhang \textit{et al.} \cite{zhang2021low} proposed circuits which involve measurements and can generate the target state in $O(n^2)$ depth but only with certain success probability, which is at least $\Omega(1 / (\max_i \|v_i\|^2 2^n))$, but in the worst case can be an exponentially small order of $O(1/2^n)$. In addition, they need $O(4^n)$ ancillary qubits to achieve this depth. In a different paper \cite{zhang2021parallel}, the authors showed that for $\epsilon\le 2^{-\Omega(n)}$, an $n$-qubit quantum state $\|\psi'_v\rangle$ can be implemented by an $O(n^3)$-depth quantum circuit with sufficiently many ancillary qubits\footnote{No explicit bound on the ancillary qubits is given.}, where $\\|\|\psi'_v\rangle-\|\psi_v\rangle\\|\le \epsilon$. Though QSP is only used as a tool for their main topic of parallel quantum walk, their concluding section did call for studies on the trade-off between the circuit depth and the number of ancillary qubits for better parallel quantum algorithms. {Another related study is \cite{johri2021nearest}, which considers to prepare a state not in the binary encoding $\sum_{k=0}^{2^n-1} v_k \|k\rangle$, but in the \textit{unary} encoding $\sum_{k=0}^{2^n-1} v_k \|e_k\rangle$, where $e_i\in \{0,1\}^{2^n}$ is the vector with the $k$-th bit being 1 and all other bits being 0. The paper shows that the unary encoding QSP can be carried out by a quantum circuit of depth $O(n)$ and size $O(2^n)$. Note that the unary encoding itself takes $2^n$ qubits, as opposed to $n$ qubits in the binary encoding. The binary encoding is the most efficient one in terms of the number of qubits needed for the resulting state, and indeed in most quantum machine learning tasks the quantum speedup depends crucially on this encoding efficiency at the first place \cite{harrow2009quantum,kerenidis2017quantum,kerenidis2020classification,dervovic2018quantum,zhao2021smooth,larose2020robust}. In \cite{johri2021nearest} the authors also extended this by using a $d$-dimensional tensor $(k_1, k_2, \ldots, k_d)$ to encode $k$, which needs $d 2^{n/d}$ qubits to encode and a circuit of depth $O(\frac{n}{d} 2^{n-n/d})$ to prepare. When $d = n$ the encoding coincides with the binary encoding, but their depth bound is $O(2^n)$, which is not optimal. } In this paper, we tightly characterize the depth and size complexities of the quantum state preparation problem by constructing optimal quantum circuits. Our circuits generate the target state precisely, with certainty, and use an optimal number of ancillary qubits. We present our results on QSP first, where a general number $m$ of ancillary qubits are available. \begin{theorem} \label{thm:QSP_anci} For any $m \ge 2n$, any $n$-qubit quantum state $\ket{\psi_v}$ can be generated by a circuit with $m$ ancillary qubits, using single-qubit gates and CNOT gates, of {size $O(2^n)$ and} depth \[\left\{\begin{array}{ll} O\big(\frac{2^n}{m+n}\big), & \text{if~} m\in [2n, O(\frac{2^n}{n\log n})], \\ O\left(n\log n\right), & \text{if~} m\in [\omega(\frac{2^n}{n\log n}),o(2^n)],\\ O\left(n\right), & \text{if~} m = \Omega(2^n). \end{array}\right.\] \end{theorem} These depth bounds improve the depth of $O(2^n)$ in \cite{bergholm2005quantum,plesch2011quantum} by a factor of $m$ for any $m\in [2n, O(\frac{2^n}{n\log n})]$, and the result shows that more ancillary qubits can indeed provide more help in shortening the depth for QSP. Compared with the result in \cite{zhang2021low} which needs $O(4^n)$ ancillary qubits to achieve depth $O(n^2)$, ours needs only $m=O(2^n/n^2)$ qubits to reach the same depth. In addition, our circuit is deterministic and generates the state with certainty, and the only two-qubit gates used are the CNOT gates. The above construction needs at least $2n$ ancillary qubits. Next we show an optimal depth construction of circuits without ancillary qubits. \begin{theorem} Any $n$-qubit quantum state $\ket{\psi_v}$ can be generated by a quantum circuit, using single-qubit gates and CNOT gates, of depth $O(2^n/n)$ {and size $O(2^n)$,} without using ancillary qubits. \label{thm:QSP_noanci} \end{theorem} These two theorems combined give asymptotically optimal bounds for depth and size complexity. Indeed, a lower bound of $\Omega(2^n)$ for size is known \cite{plesch2011quantum}, and the same paper also presents a depth lower bound of $\Omega(2^n/n)$ for quantum circuits without ancillary qubits. This can be extended to a lower bound of $\Omega\big(\frac{2^n}{n+m}\big)$ for circuits with $m$ ancillary qubits. This bound deteriorates to 0 as $m$ grows to infinity. In \cite{aharonov2018quantum}, the authors gave a depth lower bound of $\Omega(\log n)$ for circuit with arbitrarily many ancillary qubits. We note that it can be improved to $\Omega(n)$ for any $m$, as stated in the next theorem as well as independently discovered in \cite{zhang2021low}. \begin{theorem} \label{thm:lowerbound_QSP} Given $m$ ancillary qubits, there exist $n$-qubit quantum states which can only be prepared by quantum circuits of depth at least $\Omega\big(\max\big\{n,\frac{2^n}{m+n}\big\}\big)$, for circuits using arbitrary single-qubit and 2-qubit gates. \end{theorem} {The proof of Theorem \ref{thm:lowerbound_QSP} is shown in Appendix \ref{sec:QSP_lowerbound}.} Putting the above results together, we can tightly characterize the size and depth complexity of QSP, except for a logarithmic factor gap over a small parameter regime for $m$. It is interesting to note that our circuits achieve the optimal depth and size simultaneously. Our results are summarized in the next Corollary \ref{corol:tight_depth} and illustrated in Figure \ref{fig:result}. \begin{corollary}\label{corol:tight_depth} For a circuit preparing an $n$-qubit quantum state with $m$ ancillary qubits, the minimum size is $\Theta(2^n)$, and the minimum depth $D_{\textsc{QSP}}(n,m)$ for different ranges of $m$ are characterized as follows. \[ \begin{cases} \Theta\big(\frac{2^n}{m+n}\big), & \text{if } m=O\big(\frac{2^n}{n\log n}\big), \\ \left[ \Omega(n), O(n\log n)\right], & \text{if }m \in [\omega\big(\frac{2^n}{n\log n}\big),o\left(2^n\right)],\\ \Theta(n), & \text{if }m = \Omega\left(2^n\right).\\ \end{cases} \] \end{corollary} \begin{figure} \caption{Circuit depth upper and lower bound for $n$-qubit quantum state preparation. $m$ denote the number of ancillary qubits. If $m=O(\frac{2^n}{n\log n})$ and $\Omega(2^n)$, our circuit depths are $\Theta\big(\frac{2^n}{n+m}\big)$ and $\Theta(n)$, which are asymptotically optimal. When $m\in[\omega(\frac{2^n}{n\log n}),o(2^n)]$, the gap between our depth upper and lower bound is at most logarithmic. } \label{fig:result} \end{figure} Now we give two applications of the result, the first of which is general unitary synthesis. Given a unitary matrix, a fundamental question is to find a circuit implementing it in optimal depth or size. Previous studies on this problem focus on circuits without ancillary qubits. Barenco \textit{et al.} \cite{barenco1995elementary} gave an upper bound $O(n^34^n)$ for the number of CNOT gates for arbitrary $n$-qubit unitary matrix. Knill \cite{knill1995approximation} improved the upper bound to $O(n4^n)$. Vartiainen \textit{et al.} \cite{vartiainen2004efficient} constructed a quantum circuit for an $n$-qubit unitary matrix with $O(4^n)$ CNOT gates. Mottonen and Vartiainen \cite{mottonen2005decompositions} designed a quantum circuit of depth $O(4^n)$ using $\frac{23}{48}4^n$ CNOT gates. The best known lower bound for \textit{number} of CNOT gates is $\left\lceil\frac{1}{4}(4^n-3n-1)\right\rceil$ \cite{shende2004minimal}, which also implies a depth lower bound of $\Omega(4^n/n)$. In a nutshell, the previous work put the optimal depth to within the range of $[{\Omega}(4^n/n), O(4^n)]$ for general $n$-qubit circuit compression without ancillary qubits. Our results on QSP can be applied to close this gap, by showing a circuit of depth $O(4^n/n)$. And this is actually a special case of the next theorem which handles a general number $m$ of ancillary qubits. \begin{theorem} Any unitary matrix $U\in\mathbb{C}^{2^n\times 2^n} $ can be implemented by a quantum circuit of {size $O(4^n)$ and} depth $O\big(n2^n+\frac{4^n}{m+n}\big)$ with $m \le 2^n$ ancillary qubits. \label{thm:unitary_anci} \end{theorem} The second application of our QSP result is approximate QSP, for which one can obtain the following bound for circuit with a finite set of gates such as $\{CNOT, H, S, T\}$ using a variant of the Solovay–Kitaev theorem. \begin{corollary}\label{coro:approx_QSP} For any $n$-qubit target state $\ket{\psi_v}$, one can prepare a state $\ket{\psi'_v}$ which is $\epsilon$-close to $\ket{\psi_v}$ in $\ell_2$-distance, by a circuit consisting of $\{CNOT,H,S,T\}$ gates of depth \[\left\{\begin{array}{ll} O\big(\frac{2^n\log(2^n/\epsilon)}{m+n}\big), & \text{if~}m=O\big(\frac{2^n}{n\log n}\big),\\ O(n\log n\log(2^n/\epsilon)),& \text{if~}m\in [\omega\big(\frac{2^n}{n\log n}\big),o(2^n)],\\ O(n \log(2^n/\epsilon)),& \text{if~}m=\Omega(2^n),\\ \end{array}\right.\] using $m$ ancillary qubits. \end{corollary} \paragraph{Proof techniques} We give a brief account of the proof techniques used in our circuit constructions. We first reduce the problem to implementing diagonal unitary matrices. Making a phase shift for each computational basis state costs at least {$\Omega(n2^n)$}-size, which is unnecessarily high. We make the shift in Fourier basis, and carefully use ancillary qubits to parallelize the process. With ancillary qubits, we can first make some copies of the computational basis variables $x_i$, then partition $\{0,1\}^n$ into some parts of equal size, and use the ancillary qubits to handle different parts in parallel. We define the partition via a Gray code to minimize the update cost. Gray codes were also used in \cite{bullock2004asymptotically} to minimize the circuit size. They only need to minimize the difference between adjacent two words, so the defining property of Gray Code is enough. In our construction, however, we also need to make sure that the changed bits in different parts of the Gray code are evenly distributed. When no ancillary qubits are available, designing efficient circuit needs more ideas. Since there is no ancillary qubit available, all phase shifts need be made inside the input register. We divide the input register into two parts, control register and target register, and make phase shifts in the latter. As we only have a small space, we cannot use it to enumerate all $2^{r_t}-1$ suffixes as in the previous case, where $r_t\approx n/2$ is the length of suffixes. But we can enumerate them in many stages, by which we pay the price of time to compensate for the shortage of space. We need to make a transition between two consecutive stages. It turns out that the transition can be realized by a low-depth circuit if the suffixes enumerated in each stage are linearly independent as vectors over $\mathbb{F}_2$. Thus we need carefully divide the set of suffixes into sets of linearly independent vectors to facilitate the efficient update. Some other parts need special treatment as well. One is that we need to reset the suffix to the original input variables after going along a Gray code path. Another one is that the all-zero suffix cannot be handled in the same way for some singularity reason, for which we will use a recursion to solve the issue. It turns out that the overall depth and size obtained this way are asymptotically optimal. The above constructions work well when $m$ is relatively small, but do not give a tight bound when $m = \Omega(2^n/n^2)$, for which we use another method. As we mentioned earlier, \cite{johri2021nearest} shows that unary-encoded QSP can be made in $O(n)$ depth and $O(2^n)$ size. Though the resulting state uses an exponentially long unary encoding, we can transform it to a binary encoding. A direct parallelization for this transform takes $O(n2^n)$ ancillary qubits, which can be improved to the $O(2^n)$ by first transforming it to a $2^{n/2+1}$-long matrix encoding $\ket{e_i} \to \ket{e_s}\ket{e_t}$, and then to the binary encoding. This gives the optimal depth and size for the regime $m \ge 2^n$. For $m\in[\omega(2^n/n^2),o(2^n)]$, the ancillary qubits only suffice for conducting the above for the first $\log_2 m$ qubits of the target state. For the rest $\le 2\log_2 n$ qubits, we invoke our first construction to complete the generation. This gives the optimal depth if $m\in[\omega(2^n/n^2),O(2^n/(n\log n))]$, the overall depth is asymptotically optimal, leaving a gap $[\Omega(n), O(n\log n)]$ only when $m$ is in a small range $[\omega(2^n/(n\log n), o(2^n)]$. \paragraph{Other related work} Besides the standard QSP, researchers have also studied some relaxed versions. Araujo \textit{et al.} \cite{araujo2020divide} have given a depth upper bound of $O(n^2)$ to prepare a state $\sum_{k=0}^{2^n-1}v_k \|k\rangle \|\text{garbage}_k\rangle $, where $\|\text{garbage}_k\rangle$ is $O(2^n)$-qubit state entangled with the target state register. Note that there is no generic way to remove the entangled garbage, this cannot be directly used to solve the standard QSP problem. One may also consider to approximately prepare quantum states by quantum circuits made of $\{H,S,T,CNOT\}$ gates to generate $\|\psi'_v\rangle $ satisfying $ \left\\| \|\psi'_v\rangle -\|\psi_v\rangle \right\\|\leq \epsilon$ for various distance measures $\\|\cdot \\|$. Previous attention was paid to minimizing the number and depth of $T$ gates \cite{low2018trading,babbush2018encoding}, which is non-Clifford and usually thought to be hard to realize experimentally \cite{low2018trading}. They have applied ancillary qubits to implement a circuit such that the number of $T$ gates can be optimized to $\frac{2^n}{\lambda}+\lambda \log ^2 \frac{2^n\lambda}{\epsilon}$ \cite{low2018trading}, where $\lambda \in [1,O(\sqrt{2^n})]$. We shall show that our construction can be adapted to this gate set and the circuit depth increases only by $O(n+\log(1/\epsilon))$. \paragraph{Subsequent work} After this work appeared on arXiv \cite{sun2021asymptotically}, Rosenthal \cite{rosenthal2021query} constructed a QSP circuit of depth $O(n)$, using $O(n2^n)$ ancillary qubits, as opposed to ours that only uses $O(2^n)$ ancillary qubits. Rosenthal also presented a circuit for general unitary synthesis of depth $\tilde{O}(2^{n/2})$ using $\tilde{O}(4^n)$ ancillary qubits. This year, Zhang \emph{et al.} \cite{zhang2022quantum} presented yet another QSP circuit of depth $O(n)$ using $\Theta(2^n)$ ancillary qubits, which is a special cases of our results. \paragraph{Organization} The rest of this paper is organized as follows. In Section \ref{sec:preliminaries}, we will review notations and a framework of quantum state preparation. Then we will present how to decompose the uniformly controlled gate to diagonal unitary matrices and show the depth of quantum state preparation when the number of ancillary qubits $m=O(2^n/n^2)$ in Section \ref{sec:diag_matrix}. Next we will show two quantum circuit for diagonal unitary matrices used in previous section, with and without ancillary qubits in Section \ref{sec:QSP_withancilla} and Section \ref{sec:QSP_withoutancilla}, respectively. Furthermore, we present a new circuit framework for quantum state preparation when $m = \Omega\left(2^n/n^2\right)$ in Section \ref{sec:QSP_withmoreancilla}. In Section \ref{sec:extensions}, we will show some extensions and implications of the above bounds. Finally we conclude in Section \ref{sec:conclusions}. \section{Preliminaries} \label{sec:preliminaries} In this section, we will introduce some basic concepts and notation. \paragraph{Notation} Let $[n]$ denote the set $\{1,2,\cdots,n\}$. All logarithms $\log(\cdot)$ are base 2 in this paper. Let $\mathbb{I}_n\in\mathbb{R}^{2^n\times 2^n}$ be the $n$-qubit identity operator. Denote by $\mathbb{F}_2$ the field with 2 elements, with multiplication $\cdot$ and addition $\oplus$, which can be overloaded to vectors: $x\oplus y=(x_1\oplus y_1,x_2\oplus y_2\cdots ,x_n\oplus y_n)^T$ for any $x,y\in \mathbb{F}_2^n$. The inner product of two vectors $s,x\in \mathbb{F}_2^n$ is $\langle s,x\rangle:=\oplus_{i=1}^{n}s_i\cdot x_i$ in which the addition and multiplication are over $\mathbb{F}_2$. We use $0^n$ and $1^n$ for the all-zero and all-one vectors of length $n$, respectively. Vector $e_i$ is the vector where the $i$-th element is $1$ and all other elements are $0$. The multiplication $\cdot$ is sometimes dropped if no confusion is caused. For $t,k\ge 1$ and $U_1,\ldots, U_k\in \mathbb{C}^{t\times t}$, $diag(U_1,U_2,\ldots,U_k)$ is defined as \[diag(U_1,\ldots,U_k)\defeq\left[\begin{array}{ccc} U_1 & &\\ & \ddots &\\ & & U_k \end{array}\right]\in\mathbb{C}^{kt \times kt}.\] \paragraph{Elementary gates} We will use the following $R_y(\theta)$, $R_z(\theta)$ and $R(\theta)$ to denote 1-qubit rotation (about Y-axis, Z-axis) gates and phase-shift gate, i.e., \[ R_y(\theta)=\left[\begin{array}{cc} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{array}\right], \quad R_z(\theta)=\left[\begin{array}{cc} e^{-i(\theta/2)} & \\ & e^{i(\theta/2)} \end{array}\right],\quad R(\theta)=\left[\begin{array}{cc} 1 & \\ & e^{i\theta} \end{array}\right], \] where $\theta\in\mathbb{R}$ is a parameter. All blank elements denote zero throughout this paper. Three important and special cases are the $\pi/8$ gate $T$, the phase gate $S$ and the Hadamard gate $H$, \[{T=\left[\begin{array}{cc} 1 & \\ & e^{i\pi/4} \end{array}\right],~ S=\left[\begin{array}{cc} 1 & \\ & i \end{array}\right],~ H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right].} \] The 2-qubit controlled-NOT gate is \[\text{CNOT}=\left[\begin{array}{cccc} 1 & & &\\ & 1 & &\\ & & & 1\\ & & 1 &\\ \end{array}\right].\] The gate flips the \textit{target qubit} conditioned that the \textit{control qubit} is $\ket{1}$. \paragraph{Single-qubit gate decomposition} Any single-qubit operator $U\in\mathbb{C}^{2\times 2}$ can be decomposed as \[U=e^{i\alpha}R_z(\beta)R_y(\gamma)R_z(\delta)\] for some $\alpha,\beta,\gamma,\delta\in\mathbb{R}$ \cite{nielsen2002quantum}. It is not hard to verify that the Y-axis rotation $R_y(\gamma)\in \mathbb{R}^{2\times 2}$ can be decomposed as $R_y(\gamma)=SHR_z(\gamma)HS^{\dagger},$ for any $\gamma\in\mathbb{R}$. Putting these two facts together, we know that for a single-qubit operation $U$, there exist $\alpha,\beta,\gamma,\delta\in\mathbb{R}$ such that \begin{equation}\label{eq:single_qubit_gate} U=e^{i\alpha}R_z(\beta)SHR_z(\gamma)HS^{\dagger}R_z(\delta). \end{equation} \paragraph{Gray code} A Gray code path is an ordering of all $n$-bit strings $\mbox{$\{0,1\}^n$}$ in which any two adjacent strings differ by exactly one bit \cite{frank1953pulse,savage1997survey,gilbert1958gray}, and the first and the last string differ by one bit. That is, a Gray code path/cycle is a Hamiltonian path/cycle on the Boolean hypercube graph. Gray code paths/cycles are not unique, and a common one, called reflected binary code (RBC) or Lucal code, is as follows. Denote the ordering of $n$-bit strings by $x^1, x^2, \ldots, x^{2^n}$ and we will construct them one by one. Take $x^1 = 0^n$. For each $i = 1, 2, \ldots, 2^n-1$, the next string $x^{i+1}$ is obtained from $x^i$ by flipping the $\zeta(i)$-th bit, where the Ruler function $\zeta(i)$ is defined as $\zeta(i)=\max\{k:2^{k-1}\|i\}$. In other words, $\zeta(i)$ is 1 plus the exponent of 2 in the prime factorization of $i$. The following fact is easily verified. \begin{lemma} \label{lem:GrayCode} The reflected binary code defined above is a Gray code cycle. \end{lemma} Note that in the above construction, if we list all the bits changed between circularly adjacent strings, we will get a list of length $2^n$. For instance, when $n=4$, the list is: 1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,4. In general, bit 1 appears $2^{n-1}$ times, bit 2 appears $2^{n-2}$ times, ..., bit $n-1$ appears twice, and bit $n$ appears twice as well. If we regard the code as a path, i.e. ignore the change of bit from the last string to the first string, then bit $n$ appears once. By circularly shifting the bits, we can also construct Gray code cycle such that bit 2 appears $2^{n-1}$ times, ..., bit $n$ and bit $1$ appear twice. In general, for any $k\in [n]$, we can make each bit $k,k+1,\ldots, n, 1, 2, \ldots, k-1$ to appear $2^n$, $2^{n-1},2^{n-2}, \ldots,2^2, 2, 2$ times, respectively. Let us call this construction \emph{$(k,n)$-Gray code path/cycle}, or simply the $k$-Gray code path/cycle if $n$ is clear from context. \section{Quantum state preparation with $O(2^n/n^2)$ ancillary qubits} \label{sec:diag_matrix} In this section, we will review a natural framework of algorithm for quantum state preparation, first appeared in \cite{grover2002creating}. Our results presented in Section \ref{sec:QSP_withancilla} and \ref{sec:QSP_withoutancilla}, which achieve the optimal circuit depth, also fall into this framework. The framework to prepare an $n$-qubit quantum state is depicted in Figure \ref{fig:QSP_circuit}(a), where each qubit $j$ is handled by the circuit $V_j$. The task for $V_j$ is to apply a single-qubit unitary on the last qubit conditioned on the basis state of the first $j-1$ qubits. In a matrix form, $V_j$ is a block-diagonal operator \begin{equation}\label{matrix:Vn} V_j=diag(U_1,U_2,\ldots,U_{2^{j-1}})\in\mathbb{C}^{2^j\times 2^j}, \end{equation} where each $U_i$ is a $2\times 2$ unitary matrix. There are different ways to implement $V_j$, and the most natural one, which is also the one suggested in \cite{grover2002creating}, is in Figure \ref{fig:QSP_circuit}(b): it includes $2^{j-1}$ layers, and each layer is a controlled gate, which conditions on every possible computational basis state of the previous $j-1$ qubits and operates on the current qubit $j$. This is why sometimes $V_j$ is called \emph{uniformly controlled gate} (UCG). {We give a specific example for illustration in Appendix \ref{sec:app_BST}.} \begin{figure} \caption{(a) A quantum circuit to prepare an $n$-qubit quantum state. Every $V_j$ ($j\in[n]$) is a $j$-qubit uniformly controlled gate, where the first $j-1$ qubits are controlled qubits and the last one qubit is the target qubit. (b) A $j$-qubit uniformly controlled gate.} \label{fig:circuit_n} \label{fig:V_n} \label{fig:QSP_circuit} \end{figure} Thus the depth of the circuit {for quantum state preparation} in the above framework crucially depends on the circuit depth of the implementation of $V_j$'s. \begin{lemma}\label{lem:QSP-by-UCG} If each $V_j$ can be implemented by a quantum circuit of depth $d_j$, then the quantum state can be prepared by a circuit of depth $\sum_{j=1}^n d_j$. \end{lemma} As mentioned in Section \ref{sec:introduction}, if we implement each $V_j$ directly as in \cite{grover2002creating}, then the whole QSP circuit has a depth of $\Theta(n2^n)$, which is sub-optimal compared to our bound of $\Theta(2^n/n)$ in Theorem \ref{thm:QSP_noanci}. More importantly, the method in \cite{grover2002creating} cannot well utilize ancillary qubits to reduce the circuit depth. In this section, we will give a framework of efficient implementation of UCGs with the help of $m=O(2^n/n^2)$ ancillary qubits. The case when more ancillary qubits are available, i.e. $m=\Omega(2^n/n^2)$, is handled by a different framework in Section \ref{sec:QSP_withmoreancilla}. To overcome these drawbacks, we will first reduce the implementation of UCG to that of diagonal operators of the following form: \begin{equation}\label{matrix:lambda_n} \Lambda_n=diag(1,e^{i\theta_1},e^{i\theta_2},\ldots,e^{i\theta_{2^n-1}} )\in\mathbb{C}^{2^n\times 2^n}. \end{equation} \begin{lemma}\label{lem:lamda2circuit} If one can implement $\Lambda_n$ in Eq. \eqref{matrix:lambda_n} by a circuit of depth $D(n)$ and size $S(n)$ using $m\ge 0$ ancillary qubits, then any $n$-qubit quantum state can be prepared by a circuit of depth $3 \sum_{k=1}^n D(k) + 2n + 1$ and size $3\sum_{k=1}^n S(k)+2n+1$. \end{lemma} \begin{proof} According to Eq. \eqref{eq:single_qubit_gate}, each unitary matrix $U_k\in \mathbb{C}^{2\times 2}$ can be decomposed as \[ U_k=e^{i\alpha_k}R_z(\beta_k)SHR_z(\gamma_k)HS^{\dagger}R_z(\delta_k). \] Then the UCG $V_n$ can thus be decomposed to \begin{multline} \label{eq:UCG} {V_n}=\underbrace{ {diag(e^{i\alpha_1},\cdots,e^{i\alpha_{2^{n-1}}})\otimes \mathbb{I}_1} }_{A_1} \cdot \underbrace{ {diag(R_z(\beta_1),\cdots,R_z(\beta_{2^{n-1}})) }}_{A_2}\\\cdot \underbrace{ {\mathbb{I}_{n-1}\otimes (SH)}}_{A_3} \cdot\underbrace{ {diag(R_z(\gamma_1),\cdots,R_z(\gamma_{2^{n-1}}))} }_{A_4}\cdot \underbrace{ { \mathbb{I}_{n-1}\otimes (HS^\dagger)}}_{A_5} \cdot\underbrace{ {diag(R_z(\delta_1),\cdots,R_z(\delta_{2^{n-1}}))} }_{A_6}. \end{multline} Note that the unitary matrix $A_3$ can be implemented by a Hadamard gate $H$ and a phase gate $S$ operating on the last qubit, and similarly for $A_5$. The rest matrices, $A_1$, $A_2$, $A_4$, and $A_6$ are all $n$-qubit diagonal unitary matrices. Since a global phase can be easily implemented by a rotation on any one qubit, we can focus on implementing diagonal matrices of the form as in Eq. \eqref{matrix:lambda_n}. If $\Lambda_n$ can be implemented by a circuit of depth $D(n)$ and size $S(n)$, so will be QSP by a circuit of depth { and size} $\sum_{k=1}^n (3 D(k) +2) + 1 = 3\sum_{k=1}^n D(k) +2n + 1$ and $\sum_{k=1}^n (3 S(k) +2) + 1 = 3\sum_{k=1}^n S(k) +2n + 1,$ where the terms ``$3D(k)$'' {and ``$3S(k)$'' } are for diagonal matrices $A_1$, $A_2$, $A_4$ and $A_6$, the term ``2'' is for $A_3$ and $A_5$, and the term ``1'' is for the global phase. \end{proof} Thus we only need to consider how to implement diagonal operators as in Eq. \eqref{matrix:lambda_n}. We will prove the following lemmas in Section \ref{sec:QSP_withancilla} and Section \ref{sec:QSP_withoutancilla}. \begin{lemma} \label{lem:DU_with_ancillary} For any $m\in [2n,2^n/n]$, any diagonal unitary matrix $\Lambda_n\in\mathbb{C}^{2^n\times 2^n}$ as in Eq. \eqref{matrix:lambda_n} can be implemented by a quantum circuit of depth $O\left(\log m + \frac{2^n}{m}\right)$ and size $O(2^n)$, with $m$ ancillary qubits. \end{lemma} \begin{lemma} \label{lem:DU_without_ancillary} Any diagonal unitary matrix $\Lambda_n\in\mathbb{C}^{2^n\times 2^n}$ as in Eq. \eqref{matrix:lambda_n} can be implemented by a quantum circuit of depth $O\big(\frac{2^n}{n}\big)$ and size $O\big(2^n\big)$ without ancillary qubits. \end{lemma} Lemmas \ref{lem:DU_with_ancillary} and \ref{lem:DU_without_ancillary} imply Lemma \ref{lem:UCG_depth}. \begin{lemma}\label{lem:UCG_depth} For $m\ge 0$, any uniformly controlled gate $V_n\in\mathbb{C}^{2^n\times 2^n}$ as in Eq. \eqref{matrix:Vn} can be implemented by a quantum circuit of depth $O\big(n+\frac{2^n}{n+m}\big)$ and size $O(2^n)$ with $m$ ancillary qubits. \end{lemma} \begin{proof} According to Eq. \eqref{eq:UCG}, every $V_n$ can be decomposed into 3 $n$-qubit diagonal unitary matrices and 4 single-qubit gates. Combining with Lemma \ref{lem:DU_with_ancillary} and \ref{lem:DU_without_ancillary}, $V_n$ can be realized by a quantum circuit of depth $O\big(n+\frac{2^n}{n+m}\big)$ and size $O(2^n)$ with $m$ ancillary qubits. \end{proof} Once we prove these two lemmas, we will be able to prove Theorems \ref{thm:QSP_anci} and \ref{thm:QSP_noanci}. Indeed, we can apply the next Lemma \ref{lem:partial_result} to prove Theorem \ref{thm:QSP_noanci} $(m=0)$ and the $m= O(2^n/n^2)$ part of Theorem \ref{thm:QSP_anci}. The other part $m= \Omega(2^n/n^2)$ of Theorem \ref{thm:QSP_anci} is the same as Corollary \ref{coro:QSP_moreancilla} and will be treated in Section \ref{sec:QSP_withmoreancilla}. \begin{lemma}\label{lem:partial_result} For any $m\ge 0$, any $n$-qubit quantum state $\ket{\psi_v}$ can be generated by a quantum circuit with $m$ ancillary qubits, using single-qubit gates and CNOT gates, of {size $O(2^n)$ and } depth $O\big(n^2+\frac{2^n}{m+n}\big)$. \end{lemma} \begin{proof} We prove the case $m=0$ first. Plugging Lemma \ref{lem:DU_without_ancillary} into Lemma \ref{lem:lamda2circuit}, we get a circuit solving QSP in size $\sum_{j=1}^nO(2^{j})+2n+1=O(2^n)$ and depth $O\big(\sum_{j=1}^n \frac{2^j}{j}+ n\big) = O\big(\sum_{j=1}^{n-\lceil\log n\rceil}\frac{2^j}{j}+ \sum_{j=n-\lceil\log n\rceil+1}^n\frac{2^j}{j}\big)=O\big(\sum_{j=1}^{n-\lceil\log n\rceil}2^j+ \sum_{j=n-\lceil\log n\rceil+1}^n\frac{2^j}{n-\lceil\log n\rceil+1}\big)=O\big( \frac{2^n}{n}\big),$ as desired. Now we prove the case $m>0$. If $1\le m < 2n$, we will not use the ancillary qubits---we just invoke Theorem \ref{thm:QSP_noanci} to obtain a circuit of depth $O(2^n/n)$. If $2n \le m\le 2^n/n^2(\le 2^n/n)$, we can combine Lemma \ref{lem:lamda2circuit} and Lemma \ref{lem:DU_with_ancillary} to give a circuit of size $3\sum_{j=1}^nO(2^j)+2n+1=O(2^n)$ and depth $O\big(\sum_{j=1}^n \big( \log m + \frac{2^j}{m}\big)+ n\big) = O\big(n^2 + \frac{2^n}{m}\big). $ If $m > 2^n/n^2$, we only use the first $2^n/n^2$ ancillary qubits, then the above equality gives a circuit of depth $O(n^2)$. Putting these three cases together, we obtain the claimed size upper bound of $O(2^n)$ and depth upper bound of $O\big(n^2+\frac{2^n}{m+n}\big)$. \end{proof} Next let us consider how to efficiently implement $\Lambda_n$, which essentially makes a phase shift on each computational basis state. Again, if we do this on each basis state, it takes at least $\Omega(2^n)$ rounds, with each round implementing an $n$-qubit controlled phase shift. One way of avoiding sequential applications of $(n-1)$-qubit controlled unitaries is to make rotations on its Fourier basis. Indeed, there are several pieces of work to synthesis a diagonal unitary matrix, and a common approach is generating all the linear functions of variables and adding corresponding rotation $R(\theta)$ gate when a new combination generated \cite{welch2014efficient,welch2014efficient2,bullock2004asymptotically}. In \cite{bullock2004asymptotically} the authors use Gray code to adjust the order of combinations so the size and depth of the circuit are $O(2^n)$. With ancillary qubits, we can actually achieve this with much smaller depth by carefully parallelizing the operations (Section \ref{sec:QSP_withancilla}). Interestingly, this approach turns out to inspire our construction for circuits \textit{without} ancillary qubits (Section \ref{sec:QSP_withoutancilla}), to achieve the optimal depth complexity as in Theorem \ref{thm:QSP_noanci}. We now give more details. Suppose we can accomplish the following two tasks: \begin{enumerate} \item For every $s\in \{0,1\}^n-\{0^n\}$, make a phase shift of $\alpha_s$ on each basis $\ket{x}$ when $\langle s,x\rangle = 1$ (recall that $\langle \cdot , \cdot\rangle$ is over $\mathbb{F}_2$), i.e. \begin{equation}\label{eq:task1} \ket{x} \to e^{i\alpha_s\langle s,x\rangle } \ket{x}. \end{equation} \item Find $\{\alpha_s:s\in \mbox{$\{0,1\}^n$}-\{0^n\}\}$ s.t. \begin{equation}\label{eq:alpha} \sum_{s\in \{0,1\}^n-\{0^n\}}\alpha_s\langle x,s\rangle = \theta(x), \quad \forall x\in \{0,1\}^n-\{0^n\}. \end{equation} \end{enumerate} Then we get \[\ket{x} \to \prod_{ s\in \{0,1\}^n-\{0^n\}} e^{i\alpha_s\langle s,x\rangle } \ket{x} = e^{i\Sigma_s\alpha_s \langle s,x\rangle} \ket{x} = e^{i\theta(x) } \ket{x},\] as required in $\Lambda_n$. For notational convenience, we define $\alpha_{0^n}=0$. {The implementations of above two tasks in Eq. \eqref{eq:task1} and Eq. \eqref{eq:alpha} are accomplished in Appendix \ref{sec:2tasks}.} \section{Diagonal unitary implementation with ancillary qubits} \label{sec:QSP_withancilla} In this section, we prove Lemma \ref{lem:DU_with_ancillary}. That is, for any $m\in [2n,2^n{/n}]$, any diagonal unitary matrix $\Lambda_n\in\mathbb{C}^{2^n\times 2^n}$ as in Eq. \eqref{matrix:lambda_n} can be implemented by a quantum circuit of depth $O\left( \log m + \frac{2^n}{m}\right)$ and size $O(2^n)$ with $m$ ancillary qubits. Let us first give a high-level explanation of the circuit. We divide the ancillary qubits into two registers: One is used to make multiple copies of basis input bits to help on parallelization, and the other is used to generate all $n$-bit strings and apply the rotation gates. State $\ket{\langle s,x\rangle}$ will be generated for all $s\in\mbox{$\{0,1\}^n$}-\{0^n\}$. To reduce the depth of the circuit, these strings are split as equally as possible, and we use Gray Code to minimize the cost of generating a new $n$-bit string from an old one. {A quantum circuit to implement $\Lambda_4$ by using 8 ancillary qubits is shown in Appendix \ref{sec:warm-up}.} We will show how to implement $\Lambda_n$ with $m$ ancillary qubits. Let us assume $m$ to be an even number to save some floor or ceiling notation without affecting the bound. The framework is shown in Figure \ref{fig:Lambda_n_gray_code}. Our framework consists of {three registers and five stages}. The first $n$ qubits labeled as $x_1,x_2,\cdots,x_n$ form the \textit{input register}, the next $\frac{m}{2}$ qubits are the \textit{copy register}, and the last $\frac{m}{2}$ qubits are the \textit{phase register}. The linear functions $\langle s, x\rangle$ of the input variables $x=x_1\ldots x_n$ are generated in the phase register. We use the copy register to make copies of $x$ for parallelizing the circuit later. Partition $s$ into a prefix $s_1$ and a suffix $s_2$. We then generate a specific function $\langle s_{1}0\cdots 0,x\rangle$ on each qubit in the phase register, and iterate other non-zero suffixes $s_2$ in the order of a Gray code and generate $\langle s_1s_2, x\rangle$. All qubits in the copy and phase registers are initialized to $\|0\rangle$. \begin{figure} \caption{Framework for the circuit of $\Lambda_n$ with $m$ ancillary qubits. The first $n$ qubits $\ket{x_1\cdots x_n}$ form the input register, the next $\frac{m}{2}$ qubits the copy register and the last $\frac{m}{2}$ qubits the phase register. The framework consists of five stages: Prefix Copy, Gray Initial, Suffix Copy, Gray Path and Inverse. The depth of the five stages are $O(\log m)$, $O(\log m)$, $O(\log m)$, $O\left(\frac{2^n}{m}\right)$ and $O\left(\log m+\frac{2^n}{m}\right)$, respectively. } \label{fig:Lambda_n_gray_code} \end{figure} \paragraph{\underline{Stage 1: Prefix Copy}} In this stage, we make $\left\lfloor \frac{m}{2t} \right\rfloor$ copies of each qubit $x_1,x_2,\cdots,x_t$ in the input register, where $t = {\left\lfloor \log \frac{m}{2} \right\rfloor} < n$. More formally, the circuit implements the unitary $U_{copy,1}$ which operates on the input and copy registers only. Its effect is \begin{align}\label{eq:Uc1-effect} \ket{x} \ket{0^{m/2}} \xrightarrow{U_{copy,1}} \ket{x}\ket{x_{pre}} \end{align} where the two parts in the ket notation are for the input and copy register, respectively, and \begin{align} & \ket{x} = \ket{x_1x_2\cdots x_n},\\ &\ket{x_{pre}} = \overbrace{\|\underbrace{x_1\cdots x_1}_{\left\lfloor\frac{m}{2t}\right\rfloor~\text{qubits}}\underbrace{x_2\cdots x_2}_{\left\lfloor\frac{m}{2t}\right\rfloor~\text{qubits}}\cdots\underbrace{x_t\cdots x_t}_{\left\lfloor\frac{m}{2t}\right\rfloor~\text{qubits}}0\cdots 0}^{{{m/2}}~\text{qubits}}\rangle. \end{align} The next lemma says that this operation can be carried out by a circuit of small depth. \begin{lemma}\label{lem:copy1} We can make $\left\lfloor \frac{m}{2t} \right\rfloor$ copies of each qubit $x_1,x_2,\cdots,x_t$ in the input register and the copy register, by an ($m/2$)-size circuit $U_{copy,1}$ of CNOT gates only, in depth at most $\log m$. \end{lemma} \begin{proof} First, we make 1 copy of each $x_i$ in the input register to a qubit in the copy register by applying a CNOT gate. Note that the CNOT gates for different $x_i$'s are applied on different pairs of qubits, thus they can be implemented in parallel in depth 1. Next, we utilize the $x_i$ in the input register and the $x_i$ in the copy register (that we just obtained) to make two more copies of $x_i$ in the copy register, and again all these $2t$ CNOT gates can be implemented in depth 1. We continue this until we get $\lfloor m/(2t)\rfloor$ copies of each qubit $x_1,x_2,\cdots,x_t$ in the copy register. The depth of this Copy stage is $\big \lceil \log \big \lfloor m/2t \big\rfloor \big \rceil \le \log m$. And the size of this stage is $m/2$, since each qubit in copy register is used as the target qubit of CNOT gate only once. \end{proof} \paragraph{\underline{Stage 2: Gray Initial}} In this stage, the circuit includes two steps. The first step $U_{1}$ implements $m/2$ linear functions $f_{j1}(x) = \langle s(j,1), x\rangle$ for some $n$-bit strings $s(j,1)$, one for each qubit $j$ in the phase register. The second step implements some rotations in the phase registers. To elaborate on which {strings} are implemented in the first {step}, we need the following lemma and notation. Recall that we are in the parameter regime $m \in [2n,2^n/n]$. \begin{lemma}\label{lem:2D-array} Let $t = {\lfloor \log \frac{m}{2} \rfloor}$ and $\ell = 2^t$. The set $\mbox{$\{0,1\}^n$}$ can be partitioned into a 2-dimensional array $\{s(j,k): j\in [\ell], k\in [2^n/\ell]\}$ of $n$-bit strings, satisfying that \begin{enumerate} \item Strings in the first column $\{s(j,1): j\in [\ell]\}$ have the last $(n-t)$ bits being all 0, and strings in each row $\{s(j,k): k\in [2^n/\ell]\}$ share the same first $t$ bits. \item $\forall j\in [\ell], \forall k\in [2^n/\ell-1]$, $s(j,k)$ and $s(j,k+1)$ differ by 1 bit. \item For any fixed $k\in [2^n/\ell-1]$, and any $t' \in \{t+1,...,n\}$, there are at most $\big(\frac{m}{2(n-t)} + 1\big)$ many $j\in [\ell]$ s.t. $s(j,k)$ and $s(j,k+1)$ differ by the $t'$-th bit. \end{enumerate} \end{lemma} The proof of Lemma \ref{lem:2D-array} is shown in Appendix \ref{sec:2D-array}. Let us denote by $t_{jk}$ the index of the bit that $s(j,k)$ and $s(j,k+1)$ differ by. We can now describe this stage in more details. \begin{enumerate} \item The first step $U_{1}$ aims to let each qubit $j$ in the phase register have the state $\ket{f_{j1}(x)}$ at the end of this step, where $f_{j1}(x) = \langle s(j,1), x\rangle$. \item The second step applies the rotation $R_{j,1} \defeq R(\alpha_{s(j,1)})$ on each qubit $j$ in the phase register. That is, the state is rotated by an phase angle of $\alpha_{s(j,1)}$ if $\langle x, s(j,1)\rangle = 1$, and left untouched otherwise. Put $R_1 = \otimes_{j\in [\ell]} R_{j,1}$. \end{enumerate} The next lemma gives the cost and effect of this stage. \begin{lemma}\label{lem:GrayInit} The Gray Initial Stage can be implemented in depth at most $2\log m$ and in size at most $\frac{(n+1)m}{2}$ such that its unitary $U_{GrayInit}$ satisfies \begin{equation}\label{eq:UGI-effect} \ket{x} \ket{x_{pre}} \ket{0^{m/2}} \xrightarrow{U_{GrayInit}} e^{i \sum_{j\in [\ell]} f_{j,1}(x) \alpha_{s(j,1)} } \ket{x} \ket{x_{pre}} \ket{f_{[\ell],1}}, \end{equation} where $\ket{f_{[\ell],1}} = \otimes_{j\in [\ell]} \ket{f_{j,1}(x)}$. \end{lemma} \begin{proof} We will show how to implement the first step $U_{1}$ such that all $\ell=2^t = 2^{\left\lfloor\log \frac{m}{2}\right\rfloor}$ linear functions of the prefix variables $x_1, \ldots, x_t$ are implemented, namely after $U_{1}$, the states of the $2^t$ qubits in the phase register are exactly $\{a_1x_1 \oplus \cdots \oplus a_t x_t: a_1, \ldots, a_t \in \{0,1\}\}$. The implementation makes each qubit $j$ in the phase register have state $\ket{f_{j,1}(x) }$. Then in the second step, each qubit $j$ adds a phase of $f_{j,1}(x) \cdot \alpha_{s(j,1)}$ to $\ket{x} \ket{x_{pre}} \ket{0^{m/2}}$. We thus have \begin{align}\label{eq:U1} \ket{x} \ket{x_{pre}} \ket{0^{m/2}} &\xrightarrow{U_1} \ket{x} \ket{x_{pre}} \ket{f_{[\ell],1}}, \\ & \xrightarrow{R_1} e^{i \sum_{j\in [\ell]} f_{j,1}(x) \alpha_{s(j,1)} } \ket{x} \ket{x_{pre}} \ket{f_{[\ell],1}}. \end{align} Now let us construct a shallow circuit for the first step $U_{1}$. Recall that we have $\ell = 2^t$ qubits $j$ each with a corresponding linear function in variables $x_1, \ldots, x_t$. Since $\ell \le m/2$, the phase register has enough qubits to hold these linear functions. For a qubit $j$ in the phase register with corresponding linear function $x_{i_1}\oplus \cdots \oplus x_{i_{t'}}$ ($t'\le t$), we will use CNOT gates to copy the qubits $x_{i_1}, ..., x_{i_{t'}}$ from the work and the copy registers to qubit $j$. We just need to allocate these CNOT gates evenly to make the overall depth small. This step can be divided into $\big\lceil \frac{2^t}{t\lfloor m/(2t)\rfloor}\big\rceil $ mini-steps, each mini-step handling $t\left\lfloor\frac{m}{2t}\right\rfloor$ qubits $j$ by assigning the state $\ket{\ \langle s(j,1),x\rangle\ }$ to it. Since we have $\ell = 2^t$ qubits to handle, it needs $\big\lceil \frac{2^t}{t\lfloor m/(2t)\rfloor}\big\rceil$ mini-steps. For all positions $i\in [t]$ with $s(j,1)_i = 1$, we use CNOT to copy $x_i$ to qubit $j$. We have $t$ variables $x_1, \ldots, x_t$, each with $\lfloor m/(2t) \rfloor$ copies. To utilize these copies for parallelization, we break the $t\lfloor m/(2t) \rfloor$ target qubits into $t$ blocks of size $\lfloor m/(2t) \rfloor$ each. Each mini-step gives all needed variables for $t(\lfloor\frac{m}{2t}\rfloor+1)$ qubits $j$, in depth $t$. In the first layer, we use the $\lfloor\frac{m}{2t}\rfloor$ copies of $x_1$ as control qubits in CNOT to copy $x_1$ to the first block of target qubits $j$, use the $\lfloor\frac{m}{2t}\rfloor$ copies of $x_2$ for the second block of target qubits, and so on, to $x_t$ for the $t$-th block. Then in the second layer, we repeat the above process with a circular shift: Copy $x_1$ to block 2, $x_2$ to block 3, ..., $x_{t-1}$ to block $t$, and $x_t$ to block 1. Repeat this and we can complete this mini-step in depth $t$, such that $t \lfloor\frac{m}{2t}\rfloor$ many qubits $j$ get their needed variables. Since there are $\big\lceil \frac{2^t}{t\lfloor m/(2t)\rfloor}\big\rceil $ mini-steps, each of depth $t$, the total depth for $U_{1}$ is $\big\lceil \frac{2^t}{t\lfloor m/(2t)\rfloor}\big\rceil \cdot t \le \frac{m/2}{m/(2t)} + t = 2t = 2 \lfloor \log(m/2) \rfloor \le 2\log m - 2.$ The rotations in the second step are on different qubits and thus can be put into one layer, thus the overall depth for Gray Initial Stage is at most $2\log m$. The size of this stage is at most {$(n+1)m/2$}, because each qubit in the phase register has at most $n$ CNOT gates {and one $R_z$ gate} on it. \end{proof} \paragraph{\underline{Stage 3: Suffix Copy}} In this stage, we first undo $U_{copy,1}$, and then make $\big\lfloor \frac{m}{2(n-t)}\big\rfloor$ copies for each of the suffix variables, namely $x_{t+1},...,x_n$. The next lemma is similar to Lemma \ref{lem:copy1} and we omit the proof. \begin{lemma}\label{lem:copy2} We can make $\left\lfloor \frac{m}{2(n-t)} \right\rfloor$ copies of each qubit $x_{t+1},x_{t+2},\cdots,x_n$ in the input register and the copy register, by applying on $\ket{x} \ket{0^{m/2}}$ an $m$-size circuit $U_{copy,2}$ of CNOT gates only, in depth at most $\log m$. \end{lemma} Define \[ \ket{x_{suf}}\defeq \|\overbrace{ \underbrace{x_{t+1}\cdots x_{t+1}}_{\left\lfloor\frac{m}{2(n-t)}\right\rfloor~\text{qubits}} \cdots\underbrace{x_n\cdots x_n}_{\left\lfloor\frac{m}{2(n-t)}\right\rfloor~\text{qubits}}0\cdots 0 }^{{m/2}~\text{qubits}}\rangle, \] then the effect of $U_{copy,2}$ is \[ \ket{x} \ket{0^{m/2}} \xrightarrow{U_{copy,2}} \ket{x}\ket{x_{suf}}. \] The operator of this stage is $U_{copy,2} U_{copy,1}^\dagger$, and the depth is at most $2\log m$ and the size is at most $m$. The effect of this stage $U_{copy,2} U_{copy,1}^\dagger$ is \begin{equation}\label{eq:suf-copy} \ket{x} \ket{x_{pre}} \xrightarrow{U_{copy,1}^\dagger} \ket{x} \ket{0^{m/2}} \xrightarrow{U_{copy,2}} \ket{x}\ket{x_{suf}}. \end{equation} \paragraph{\underline{Stage 4: Gray Path}} This stage contains $2^n/\ell- 1$ phases, indexed by $k = 2, 3, \ldots, 2^n/\ell$. The previous Gray Initial Stage can be also viewed as the phase $k=1$. We single it out as a stage because it implements linear functions from scratch, while each phase in the Gray Path Stage implements linear functions only by a small update from the previous phase. In each phase $k$ in this stage, the circuit has two steps: \begin{enumerate} \item Step $k.1$ is a unitary circuit $U_{k}$ that applies a CNOT gate on each qubit $j\in [\ell]$ in the phase register, controlled by $x_{t_{j,(k-1)}}$, the bit where $s(j,k-1)$ and $s(j,k)$ differ. \item Step $k.2$ applies the rotation gate $R(\alpha_{s(j,k)})$ on qubit $j$. Put $R_k = \otimes_{j\in [\ell]} R(\alpha_{s(j,k)})$. \end{enumerate} \begin{lemma}\label{lem:GrayPath} The phase $k$ of the Gray Path Stage implements \begin{equation}\label{eq:GrayPath} \ket{x}\ket{x_{suf}} \ket{f_{[\ell],k-1}} \xrightarrow{U_k} \ket{x}\ket{x_{suf}} \ket{f_{[\ell],k}} \xrightarrow{R_k} e^{i \sum_{j\in [\ell]} f_{j,k}(x) \alpha_{s(j,k)} } \ket{x}\ket{x_{suf}} \ket{f_{[\ell],k}}, \end{equation} where $f_{j,k}(x)=\langle{s(j,k)},x\rangle$ and $\ket{f_{[\ell],k}}=\otimes_{j\in[\ell]}\ket{f_{j,k}(x)}$. The depth and size of the whole Gray Path Stage are at most ${ 2}\cdot 2^{n}/\ell$ and $2^{n+1}$. \end{lemma} \begin{proof} The operation can be easily seen in a similar way as that for Lemma \ref{lem:GrayInit}. Next we show the depth bound. {The Gray Path stage repeats step $k.1$-$k.2$ for $2^n/\ell-1$ times. Since $s(j,k-1)$ and $s(j,k)$ differ by only 1 bit by Lemma \ref{lem:2D-array}, one CNOT gate suffices to implement the function $\langle x, s(j,k)\rangle$ from $\langle x, s(j,k-1)\rangle$ in the previous phase: The control qubit is $x_{t_{j,(k-1)}}$ and the target qubit is $j$. Moreover, the third property in Lemma \ref{lem:2D-array} shows that each variables $x_i$ is used as a control qubit for at most $\big(\big\lfloor\frac{m}{2(n-t)}\big\rfloor+1\big)$ different $j\in [\ell]$. Since we have { $\big(\big\lfloor \frac{m}{2(n-t)}\big\rfloor+1\big) $} copies in the input register and the copy register, these CNOT gates in step $k.1$ can be implemented in depth $ 1$. The step $k.2$ consists of only single qubit gates, which can be all paralleled in depth 1. Thus the total depth of Gray Path stage is at most $2^n/\ell \cdot {(1+1)\le 2}\cdot 2^{n}/\ell$.} { The size of this stage is $2^{n+1}$ since each linear combination of input variables is generated once and applied single-qubit phase-shift gates $R_k$. The number of linear combinations of input variables is $2^n$ , so the size is $2^{n+1}$.} \end{proof} \paragraph{\underline{Stage 5: Inverse}} In this stage, the circuit applies {$U_{copy,1}^\dagger U_1^\dagger U_{copy,1} U_{copy,2}^\dagger U_{2}^{\dagger} \cdots U_{2^n/\ell}^\dagger $}. \begin{lemma}\label{lem:inverse} The depth and size of the Inverse Stage are at most $O(\log m + 2^n /m)$ and $\frac{m}{2}+\frac{nm}{2}+m+2^{n}=2^{n}+\frac{3m+nm}{2}$. The effect of this stage is \begin{equation}\label{eq:inverse} \ket{x} \ket{x_{suf}} \ket{f_{[\ell],2^n/\ell}} \xrightarrow{U_{Inverse}} \ket{x} \ket{0^{m/2}} \ket{0^{m/2}}. \end{equation} \end{lemma} The proof of Lemma \ref{lem:inverse} is shown in Appendix \ref{sec:inverse}. \paragraph{Putting things together} After explaining all the five stages, we are ready to put them together to see the overall depth and operation of the circuit. \begin{lemma}\label{lem:puttingtogether_ancilla} The circuit implements the operation in Eq. \eqref{matrix:lambda_n} in depth $O(\log m + 2^n /m)$ { and in size $3\cdot 2^{n}+nm+{\frac{7}{2}}m$}. \end{lemma} {The proof of Lemma \ref{lem:puttingtogether_ancilla} is shown in Appendix \ref{sec:puttingtogether_ancilla}.} In summary, $\Lambda_n$ can be implemented in $O\big(\log m+\frac{2^n}{m}\big)$ depth and size $3\cdot 2^n +nm+{\frac{7}{2}m}$ with $m\in[2n,2^n/n]$ ancillary qubits, proving Lemma \ref{lem:DU_with_ancillary}. \section{Diagonal unitary implementation without ancillary qubits} \label{sec:QSP_withoutancilla} In this section, we prove Lemma \ref{lem:DU_without_ancillary}. That is, any diagonal unitary $\Lambda_n\in\mathbb{C}^{2^n\times 2^n}$ as in Eq. \eqref{matrix:lambda_n} can be implemented by a quantum circuit of depth $O\left(2^n/n\right)$ and size $O(2^n)$ without ancillary qubits. In Section \ref{sec:framework_DU_withoutancilla}, we present the framework of our circuit and the functionalities of the operators inside. We then prove the correctness and analyze the depth of our circuit in Section \ref{sec:correctness}. Finally, we give the detailed construction of some operators in Section \ref{sec:G_k}. \subsection{Framework and functionalities} \label{sec:framework_DU_withoutancilla} The framework of our circuit implementing $\Lambda_n$ is a recursive procedure shown in Figure \ref{fig:framework_DU_withoutancilla}. \begin{figure}\label{fig:framework_DU_withoutancilla} \end{figure} The $n$-qubit work register is divided into two registers: A \emph{control register} consisting of the first $r_c$ qubits, and a \emph{target register} consisting of the last $r_t$ qubits. The circuit has the following components. \begin{enumerate} \item A sequence of $n$-qubit unitary operators $\mathcal{G}_1,\ldots,\mathcal{G}_\ell$, the detailed construction of which will be given in Section \ref{sec:G_k}. \item An $r_t$-qubit unitary operator $\mathcal{R}$, which resets the state in the target register to the input value $\ket{x_{r_c+1}, \ldots , x_n}$. \item An $r_c$-qubit diagonal unitary operator $\Lambda_{r_c}$, which is implemented recursively. \end{enumerate} The parameters are set as follows: $r_t=\lfloor n/2 \rfloor\approx n/2,\quad r_c=n-r_t \approx n/2, \quad \text{ and } \quad \ell \le \frac{2^{r_t+2}}{r_t+1}-1 \approx \frac{ 2^{n/2+3}}{n}.$ Next we describe the function of each operator in Figure \ref{fig:framework_DU_withoutancilla}, for which it suffices to specify their effects on an arbitrary computational basis state \[\ket{x} = \ket{x_1x_2\cdots x_{r_c} x_{r_c+1} \cdots x_n} = \ket{\underbrace{x_{control}}_{r_c \text{~qubits}}}\ket{\underbrace{x_{target}}_{r_t \text{~qubits}}}, \] where $x\in\mbox{$\{0,1\}^n$}$. Let us first highlight some key similarities and differences between this circuit and the one presented in the previous section. Recall that in Section \ref{sec:QSP_withancilla}, an $n$-bit string $s\in \{0,1\}^n-\{0^n\}$ is broken into two parts, a ${\left\lfloor\log (\frac{m}{2})\right\rfloor}$-bit prefix and an $\left(n-{\lfloor \log(\frac{m}{2}) \rfloor}\right)$-bit suffix. In the Gray Initial Stage there, we use $2^{\lfloor \log(m/2)\rfloor}$ qubits in the phase register to enumerate all possible ${\lfloor \log(m/2)\rfloor}$-bit prefixes, one prefix on each phase qubit $j$. Then on each such qubit $j$ we enumerate all $(n-{\lfloor \log(m/2)\rfloor})$-bit suffixes in the Gray Path Stage. In this section, we again break $s$ into a prefix and a suffix, and enumerate all prefixes and all suffixes to run over all $n$-bit strings. However, due to the lack of the ancillary qubits, the circuit here differs from the last one in the following two aspects. \begin{enumerate} \item In Section \ref{sec:QSP_withancilla}, $s\in\{0,1\}^n-\{0^n\}$ is generated in the phase register, which is initialized to $\|0\rangle$. In this section, $s=ct$ , in which $c$ is the $r_c$-bit prefix and $t$ is the $r_t$-bit suffix. The state $\ket{\langle s, x\rangle }$ is generated in target register, whose initial state is $\|x_j\rangle$ for some $j\in\{r_c+1, r_c+2, \ldots, r_n\}$. Hence, we enumerate $s$ recursively in this section. That is, we first generate $s=ct$ for $t\neq 0^{r_t}$ and then generate $c0^{r_t}$ recursively. \item In Section \ref{sec:QSP_withancilla}, there are $2^{\lfloor \log(m/2)\rfloor}~(\le \frac{m}{2})$ prefixes which can be enumerated in $\frac{m}{2}$ qubits in phase register exactly. In this section, $2^{r_t}-1~(\approx 2^{n/2})$ suffixes should be generated in $r_t$ qubits in target register. As we only have $r_t$ qubits, the small space is insufficient to enumerate all $2^{r_t}-1$ suffixes. Thus we need to enumerate them in many stages, and $r_t$ suffixes in each stage; in other words, we pay the price of time to compensate the shortage of space. It turns out that the transition from one stage to another can be made in a low depth if the suffixes enumerated in each stage are linearly independent as vectors in $\{0,1\}^{r_t}$. Thus we need carefully divide $2^{r_t}-1$ suffixes into $\ell$ sets $T^{(1)}, \ldots, T^{(\ell)}$ with $T^{(k)}=\{t_1^{(k)},t_2^{(k)},\ldots,t_{r_t}^{(k)}\}$ each $t_a^{(k)}\neq 0^{r_t}$ for $a\in [r_t]$ and $k \in [\ell]$, and the strings in each $T^{(k)}$ linearly independent. We allow overlap between these sets, but maintain the total number $\ell$ of sets only a constant times of $(2^{r_t}-1)/r_t$, so that the overall depth is still under the control. As the sets have overlaps, a suffix may appear more than once, so we need to note this and avoid repeatedly applying rotation when the suffix appears multiple times. \end{enumerate} We now show how to implement the above high-level ideas. We will need to find sets $T^{(1)},T^{(2)},\ldots,T^{(\ell)}$ satisfying the following two key properties. \begin{enumerate} \item For each $k\in[\ell]$, the set $T^{(k)} = \big\{t_1^{(k)},t_2^{(k)},\ldots,t_{r_t}^{(k)}\big\}$ contains $r_t$ vectors from $\{0,1\}^{r_t}$ that are linearly independent over the field $\mathbb{F}_2$. \item The collection of these sets covers all the $r_t$-bit strings except for $0^{r_t}$, i.e. $\bigcup_{k\in[\ell]}T^{(k)} = \{0,1\}^{r_t}-\{0^{r_t}\}$. \end{enumerate} {The constructions of sets $T^{(1)},\ldots,T^{(\ell)}$ are shown in Appendix \ref{sec:partition}.} For each $k\in[\ell]\cup\{0\}$, define an $r_t$-qubit state \begin{equation} \label{eq:yk} \ket {y^{(k)}} = \ket{y_1^{(k)}y_2^{(k)}\cdots y_{r_t}^{(k)}}, \text{ where } \quad y_j^{(k)} =\left\{\begin{array}{ll} x_{r_c+j} & \text{if~} k=0, \\ \langle {0^{r_c}t_j^{(k)}},x\rangle & \text{if~} k\in[\ell]. \end{array}\right. \end{equation} Namely, $y^{(0)}$ is the same as $x_{target}$ (the suffix of $x$), and other $y_j^{(k)}$ are linear functions of variables in $x_{target}$ with coefficients given by $t_j^{(k)}$. Next, let us define disjoint families $F_1,\ldots,F_\ell$ which apply the rotation when a suffix appears for the first time. \begin{equation}\label{eq:F_k} \begin{array}{ll} F_1=\big\{ct:\ t\in T^{(1)},c\in\{0,1\}^{r_c}\big\}, \\ F_k=\big\{ct:\ t\in T^{(k)},c\in\{0,1\}^{r_c}\big\}-\bigcup\limits_{d\in[k-1]}F_{d}, ~ 2\le k\le \ell. \end{array} \end{equation} These families of sets $F_1,F_2,\cdots,F_\ell$ satisfy $F_i\cap F_j =\emptyset$ for all $i\neq j \in[\ell]$ and \begin{equation}\label{eq:set_eq} \bigcup_{k\in [\ell]} F_k = \mbox{$\{0,1\}$}^{r_c}\times \bigcup\limits_{k\in [\ell]} T^{(k)} =\mbox{$\{0,1\}$}^{r_c}\times (\mbox{$\{0,1\}$}^{r_t} - \{0^{r_t}\}) = \mbox{$\{0,1\}^n$}-\{c0^{r_t}:\ c\in\{0,1\}^{r_c}\}. \end{equation} With the above concepts, we can now show the desired effect of the operators $\mathcal{G}_k$, $\mathcal{R}$ and $\Lambda_{r_c}$. \begin{enumerate} \item For $k\in[\ell]$, \begin{equation}\label{eq:Gk} \mathcal{G}_k\ket{x_{control}}\ket{y^{(k-1)}}=e^{i\sum\limits_{s \in F_k}\langle s,x\rangle \alpha_s } \ket{x_{control}}\ket{y^{(k)}}, \end{equation} where $\alpha_s$ is determined by Eq. \eqref{eq:alpha}. In words, $\mathcal{G}_k$ has two effects: (1) It puts a phase and (2) it transits from the stage $k-1$ to the stage $k$. \item The transformation $\mathcal{R}$ acts on the target register and resets the suffix state as follows \begin{equation}\label{eq:reset} \mathcal{R}\ket{y^{(\ell)}}=\ket{y^{(0)}}. \end{equation} As a map on ${\{0,1\}^{r_t}}$ (instead of $\{\ket{x}: x\in {\{0,1\}^{r_t}}\}$), $\mathcal R$ is an invertible linear transformation over $\mathbb{F}_2$. \item The operator $\Lambda_{r_c}$ is an $r_c$-qubit diagonal matrix satisfying that \begin{equation}\label{eq:Lambda_rc} \Lambda_{r_c}\ket{x_{control}}= e^{i\sum\limits_{c\in \{0,1\}^{r_c}-\{0^{r_c}\}}\langle c0^{r_t},x\rangle\alpha_{c0^{r_t}}}\ket{x_{control}}, \end{equation} and will be implemented recursively. \end{enumerate} We will define these operators and show these properties in Section \ref{sec:G_k}. \subsection{Correctness and depth} \label{sec:correctness} In this section, we will prove the correctness and analyze the depth of the circuit. We will need a fact about the depth of invertible linear transformation from \cite{jiang2020optimal} (Theorem 1). The original version says that any CNOT circuit, a circuit consisting of only CNOT gates, on $n$ qubits can be compressed into $O(n/\log n)$ depth. But note that any $n$-dimensional invertible linear transformation over $\mathbb{F}_2$ can be implemented by a CNOT circuit {\cite{patel2008optimal}}. We thus have the following result. \begin{lemma} Suppose that $U\in \mathbb{F}_2^{n \times n}$ is an invertible linear transformation over $\mathbb{F}_2$. Then as a $2^n\times 2^n$ unitary matrix which permutes computational basis $\{\ket{x}: x\in \mbox{$\{0,1\}^n$}\}$, the map $U$ can be realized by a CNOT circuit of depth at most $O(\frac{n}{\log n})$ { and size at most $O(\frac{n^2}{\log n})$} without ancillary qubits. \label{lem:SODA2020} \end{lemma} As mentioned in Section \ref{sec:framework_DU_withoutancilla}, $\mathcal R$ is an invertible linear transformation on the computational basis variables, thus the above lemma immediately implies the following depth upper bounds for $\mathcal{R}$. \begin{lemma} \label{lem:R} The operator $\mathcal{R}$ can be realized by an $O(\frac{r_t}{\log r_t})$-depth { and $O(\frac{r_t^2}{\log r_t})$-size} CNOT circuit without ancillary qubits. \end{lemma} The depth of $\mathcal{G}_k$ will be easily seen from its construction in Section \ref{sec:G_k}. \begin{lemma}\label{lem:Gk} The operator $\mathcal{G}_k$ can be realized by an $O( 2^{r_c})$-depth {and $O(r_c 2^{r_c+1})$-size} quantum circuit using single-qubit and CNOT gates without ancillary qubits. \end{lemma} Now we are ready to prove the correctness and depth of the whole circuit. The correctness of the circuit framework in Figure \ref{fig:framework_DU_withoutancilla} is shown in Appendix \ref{sec:correctness_withoutancilla}. \begin{lemma} Any diagonal unitary matrix $\Lambda_n$ can be realized by the quantum circuit $(\Lambda_{r_c}\otimes\mathcal{R})\mathcal{G}_{\ell}\mathcal{G}_{\ell-1}\cdots\mathcal{G}_{1}$ as in Figure \ref{fig:framework_DU_withoutancilla}, which has depth $O(2^n/n)$ and size $2^{n+3}+O\big(\frac{n^2}{\log n}\big)$ and uses no ancillary qubits. \end{lemma} \begin{proof} We prove that the circuit has depth $D(n) = O(2^n/n)$. Lemma \ref{lem:Gk} shows $\mathcal{G}_k$ can be realized in depth at most $\lambda_1\cdot 2^{r_c}$ for a constant $\lambda_1>0$ and Lemma \ref{lem:R} shows $\mathcal{R}$ can be implemented in depth at most $\lambda_2\cdot \frac{r_t}{\log r_t}$ without ancillary qubits for a constant $\lambda_2>0$. Therefore, $D(n)$ satisfies the following recurrence \[ \begin{array}{ll} D(n) & \le \max \big\{D(r_c),~\lambda_2\cdot\frac{r_t}{\log r_t}\big\}+\lambda_1\cdot2^{r_c}\cdot \ell\\ & \le D(\lceil n/2 \rceil)+ \frac{\lambda_2\lceil n/2 \rceil}{\log \lceil n/2 \rceil}+\lambda_1 2^{\lceil n/2 \rceil} \big( \frac{2^{\lfloor n/2 \rfloor+2}}{\lfloor n/2 \rfloor+1}-1\big)\\ & = D(\lceil n/2 \rceil)+O(2^n/n).\\ \end{array} \] Solving the above recursive relation, we obtain the bound $D(n)=O(2^n/n)$ as desired. The size of this circuit $S(n)$ satisfies $S(n)\le S(n/2)+(2^{n+3}-2^{n/2+3})+O\big(\frac{n^2}{\log n}\big)\le 2^{n+3} + O\big(\frac{n^2}{\log n}\big)$. \end{proof} \subsection{Construction of $\mathcal{G}_k$ and $\mathcal R$} \label{sec:G_k} In this section, we will show how to construct operator $\mathcal{G}_k$, which consists of two stages: Generate Stage and Gray Path Stage, see Figure \ref{fig:all_com_tar}. Along the way, we will also show the construction of $\mathcal R$. \begin{figure} \caption{Implementation of operator $\mathcal{G}_k$, which consists of Generate Stage and Gray Path Stage. The depth of Generate Stage is $O\left(\frac{r_t}{\log r_t}\right)$ and the depth of Gray Path Stage is $O(2^{r_c})$. } \label{fig:all_com_tar} \end{figure} \paragraph{Generate Stage} In this stage, we implement operator $U_{Gen}^{(k)}$, such that \begin{equation} \label{eq:y_k-1} \ket{y^{(k-1)}} \xrightarrow{U_{Gen}^{(k)}}\ket{y^{(k)}}, ~k\in[\ell], \end{equation} \noindent where $y^{(k-1)}$ and $y^{(k)}$ are defined in Eq. \eqref{eq:yk} and determined by ${T}^{(k-1)}$ and ${T}^{(k)}$, respectively. For $k\in[\ell]$, recall that ${T}^{(k)} = \{t_1^{(k)},\cdots,t_{r_t}^{(k)}\}$. Fix this ordering, view each {$t_i^{(k)}$} as a column vector, and define a matrix $\hat{T}^{(k)}=[t_1^{(k)},\cdots,t_{r_t}^{(k)}]^T\in\{0,1\}^{r_t\times r_t}$ for $k\in[\ell]$, with special case $\hat{T}^{(0)} \defeq I_{r_t}$. Then the vectors $y^{(k)}$ can be rewritten as \begin{equation}\label{eq:y-T} y^{(k)}=\hat{T}^{(k)}x_{target}, \quad\forall k\in [r_t]\cup \{0\}. \end{equation} Since $t_1^{(k)},t_2^{(k)},\cdots,t_{r_t}^{(k)}$ are linearly independent over $\mathbb{F}_2$, $\hat{T}^{(k)}$ is an invertible linear transformation over $\mathbb{F}_2$. Now define a unitary $U_{Gen}^{(k)}$ by $U_{Gen}^{(k)}\ket{ y} = \ket{ \hat{T}^{(k)}(\hat{T}^{(k-1)})^{-1} y}$, where the matrix-vector multiplication at the right hand side is over $\mathbb{F}_2$. From Eq. \eqref{eq:y-T}, we see that \begin{equation} U_{Gen}^{(k)} \ket{y^{(k-1)}} = \ket{\hat{T}^{(k)}(\hat{T}^{(k-1)})^{-1} y^{(k-1)}} = \ket{\hat{T}^{(k)}x_{target}} = \ket{y^{(k)}} \end{equation} satisfying Eq. \eqref{eq:y_k-1}. Also note that when viewed as a linear transformation over $\mathbb{F}_2$, $U_{Gen}^{(k)}$ is invertible. Thus according to Lemma \ref{lem:SODA2020}, the following depth upper bound applies. \begin{lemma}\label{lem:generate_withoutancilla} The Generate Stage unitary $U_{Gen}^{(k)}$ can be realized by an $O\big(\frac{r_t}{\log r_t}\big)$-depth and $O\big(\frac{r_t^2}{\log r_t}\big)$-size CNOT circuit without ancillary qubits. \end{lemma} Similar to the discussion of $U^{(k)}_{Gen}$, operator $\mathcal{R}$ can be defined by $\mathcal R \ket{y} = \ket{ (\hat{T}^{(\ell)})^{-1} y},$ then $\mathcal R \ket{y^{(\ell)}} = \ket{(\hat{T}^{(\ell)})^{-1} y^{(\ell)}} = \ket{x_{target}} = \ket{y^{(0)}}.$ Thus $\mathcal{R}$ can be also viewed an invertible linear transformation over $\mathbb{F}_2$. Applying Lemma \ref{lem:SODA2020} gives the bound in Lemma \ref{lem:R}. \paragraph{Gray Path Stage} This stage implements the following operator \begin{equation}\label{eq:Graycodepath} \ket{x_{control}}\ket{y^{(k)}}\xrightarrow{U_{GrayPath}}e^{i\sum\limits_{s \in F_k}\langle s,x\rangle \alpha_s } \ket{x_{control}}\ket{y^{(k)}}, \end{equation} where $k\in [\ell]$ and $F_k$ is defined in Eq. \eqref{eq:F_k}. The Gray Path Stage in this section is similar to the Gray Path Stage in Section \ref{sec:QSP_withancilla}, though we need to use a Gray code cycle here instead of a Gray code path. For every $i\in[r_t]$, let $c^i_1,c^i_2,\cdots,c^i_{2^{r_c}-1}, c^i_{2^{r_c}}$ denote the $i$-Gray code of $r_c$ bits starting at $c_1^i=0^{r_c}$ for $i\in[r_t]$. Let $h_{ij}$ denote the index of the bit that $c^i_{j-1}$ and $c^i_{j}$ differ for each $j\in\{2,3,\ldots,2^{r_c}\}$ and $h_{i1}$ the index of the bit that $c^i_1$ and $c^i_{2^{r_c}}$ differ. For the $i$-Gray code cycle of $r_c$ bits, \begin{equation}\label{eq:index} h_{ij}=\left\{\begin{array}{ll} (r_c+i-2\mod r_c)+1, & \text{if~} j = 1\\ (\zeta(j-1)+i-2\mod r_c)+1, & \text{if~} j \neq 1 \end{array}\right. \end{equation} The exact form of $h_{ij}$ is not crucial; the important fact to be used later is that the indices $h_{1p},h_{2p},\ldots,h_{r_tp}$ are all different. This stage consists of $2^{r_c}+1$ phases. \begin{enumerate} \item In phase 1, circuit $C_1$ applies a rotation $R(\alpha_{0^{r_c}t_i^{(k)}})$ on the $i$-th qubit in the target register for all $i\in[r_t]$ if the string $0^{r_c}t_i^{(k)} \in F_k$, where $\alpha_{0^{r_c}t_i^{(k)}}$ is defined in Eq. \eqref{eq:alpha}. \item In phase $p\in\{2,\ldots,2^{r_c}\}$, circuit $C_{p}$ consists of 2 steps:\begin{enumerate} \item Step $p.1$ is a unitary that, for all $i\in[r_t]$, applies a CNOT gate on the $i$-th qubit in target register, controlled by the $h_{ip}$-th qubit in control register. \item Step $p.2$ is a unitary that, for all $i\in[r_t]$, applies a rotation $R(\alpha_{c^i_pt_{i}^{(k)}})$ on the $i$-th qubit in target register if $c^i_pt_{i}^{(k)}\in F_k$, where $\alpha_{c_p^{i}t_i^{(k)}}$ is defined in Eq. \eqref{eq:alpha}. \end{enumerate} \item In phase $2^{r_c}+1$, circuit $C_{2^{r_c}+1}$ implements a unitary that, for all $i\in[r_t]$, applies a CNOT gate on the $i$-th qubit in target register, controlled by the $h_{i1}$-th qubit in control register . \end{enumerate} The next lemma gives the correctness and depth of this constructed circuit. {The proof of Lemma \ref{lem:graypath_withoutancilla} is shown in Appendix \ref{sec:graypath_withoutancilla}.} \begin{lemma}\label{lem:graypath_withoutancilla} The quantum circuit defined above is of depth $O(2^{r_c})$ and size $O(r_c2^{r_c+1})$, and implements Gray Path Stage $U_{GrayPath}$ in Eq. \eqref{eq:Graycodepath}. \end{lemma} According to Lemma \ref{lem:generate_withoutancilla} and Lemma \ref{lem:graypath_withoutancilla}, operator $\mathcal{G}_k$ can be implemented in depth $O(2^{r_c})+O(\frac{r_t}{\log r_t})=O(2^{r_c})$. And the size of the circuit is at most $O(\frac{n^2}{\log n})+r_c2^{r_c+1}=O(r_c2^{r_c+1})$. This completes the proof of Lemma \ref{lem:Gk}. \section{Quantum state preparation with $\Omega(2^n/n^2)$ ancillary qubits} \label{sec:QSP_withmoreancilla} In this section, we will introduce a different framework that can improve the upper bound in Section \ref{sec:QSP_withancilla} when the number of ancillary qubits $m=\Omega(2^n/n^2)$. In Section \ref{sec:new_framework}, we will present the framework, and in Section \ref{sec:new_framework_correctness}, we will give implementation details with the depth and correctness analyzed. In the following, we will use $e_i\in \{0,1\}^{2^n}$ to denote the vector where the $i$-th bit is 1 and all other bits are 0. It is a unary encoding of $i\in \{0,1,\ldots,2^n-1\}$, and $\ket{e_i}$ is the corresponding $2^n$-qubit state. We use $n$-qubit state $\ket{i} = \ket{i_0 i_1 \cdots i_{n-1}} \in (\{\ket{0}, \ket{1}\})^{\otimes n}$ to denote the binary encoding of $i$, where $i_0,\cdots,i_{n-1}\in\{0,1\}$ and $i=\sum_{j=0}^{n-1} i_j\cdot 2^j$. \subsection{New framework for quantum state preparation} \label{sec:new_framework} \begin{figure}\label{fig:new_one} \label{fig:new_framework} \end{figure} The quantum circuit in Section \ref{sec:preliminaries} for quantum state preparation consists of $n$ UCGs $V_1,V_2,\ldots,V_n$ (Figure \ref{fig:QSP_circuit}(a)). In Section \ref{sec:QSP_withancilla}, we showed that any $j$-qubit UCG $V_j$ can be implemented by a quantum circuit of depth $O\big(j+\frac{2^j}{m+j}\big)$ with $m$ ancillary qubits. Summing this up over $j\in [n]$ gives the $O(n^2 + 2^n/m)$ upper bound for QSP, and this quadratic term seems hard to be improved within the framework of \cite{grover2002creating}. In the new framework, we first generate the quantum state in the unary encoding $\sum_i v_i \ket{e_i}$ using the result in \cite{johri2021nearest}, and then make an encoding transform $\ket{e_i} \to \ket{i}$, from the unary encoding to the binary encoding. Two issues need to be handled here. The first one is the need to design an encoding transform circuit that has small depth and size, using ancillary qubits efficiently. We will give an optimal construction in Section \ref{sec:new_framework_correctness}. The second issue is that the unary encoding itself needs $2^n$ qubits, and the encoding transform also needs $O(2^n)$ qubits, which may be beyond $m$, the number of ancillary qubits that are available in the first place. To handle this, we will use a hybrid method. We break the generation into a prefix part and a suffix part, where the length of the prefix is whatever $m$ can support. We prepare the prefix part by unary QSP construction in \cite{johri2021nearest} and our encoding transformation, and then employ the methods in Section \ref{sec:QSP_withancilla} for the suffix part. { Our new circuit framework for QSP in the parameter regime $m=\Omega(2^n/n^2)$ is shown in Figure \ref{fig:new_framework}. Let $t=\lfloor \log (m/3)\rfloor$. In previous framework, the first $t$ UCGs are a QSP circuit to prepare a $t$-qubit quantum state $\ket{\psi_{v}^{(t)}}=\sum_{k=0}^{2^{t}} v'_k\ket{k}\text{, where }v'_k=\sqrt{\sum_{j=0}^{2^{n-t}-1}\|v_{2^{n-t}k+j}\|^2}.$ In the new framework, we introduce a new $t$-qubit QSP circuit to replace the first $t$ UCGs. The new QSP circuit consists of the following steps. \begin{enumerate} \item Generate a $2^{t}$-qubit quantum state $\ket{\psi'_{v}}=\sum_{k=0}^{2^{t}-1} v'_k\ket{e_k}$, where $e_k\in\{0,1\}^{2^{t}}$ and by the quantum circuit in \cite{johri2021nearest}. \item Applying $U_{t}$ to $\ket{\psi'_{v}}$, we can obtain $\ket{\psi_{v}^{(t)}}$ with $2m/3$ ancillary qubits, where $U_{t}$ is the unitary transformation $U_{t}:\ket{e_i}\to\ket{i}\ket{0^{2^{t}-{t}}}$ for all $i\in\{0\}\cup [2^{t}-1]$. \item Realize the last $n-t$ UCGs by Eq. \eqref{eq:UCG} and Lemma \ref{lem:DU_with_ancillary}. \end{enumerate}} \subsection{Implementation and analysis} \label{sec:new_framework_correctness}\label{lem:linear_depth_tof} Now we give a more detailed implementation and analyze the correctness and cost of the algorithm. First, in \cite{johri2021nearest} it is shown that QSP with the unary encoding can be implemented efficiently. \begin{lemma}\label{lem:johr} Given a vector $v=(v_0,v_1,\ldots, v_{2^n-1})^T\in\mathbb{C}^{2^n}$ with unit $\ell_2$-norm, any $2^n$-qubit quantum state $ \ket{\psi'_{v}}=\sum_{k=0}^{2^n-1}v_k\ket{e_k} $ can be prepared from the initial state $\ket{0}^{\otimes 2^n}$ by a quantum circuit using single-qubit gates and CNOT gates of depth $O(n)$ { and size $O(2^n)$} { without} ancillary qubits. \label{lem:unary_qsp} \end{lemma} Next we consider the encoding transformation. \begin{lemma}\label{lem:enc-trans} The following unitary transformation on $2^n$ qubits \begin{equation}\label{eq:unary_transform} \ket{e_i}\to\|i\rangle\ket{0^{2^n-n}},\forall i\in\{0\}\cup [2^n-1], e_i\in\{0,1\}^{2^n}, \end{equation} can be implemented by a quantum circuit using single-qubit gates and CNOT gate with {$2^{n+1}$} ancillary qubits, of depth $O(n)$ {and size $O(2^n)$}. \end{lemma} {The proof of Lemma \ref{lem:enc-trans} is shown in Appendix \ref{sec:enc-trans}.} Now we are ready to give the hybrid algorithm and cost analysis. \begin{lemma}\label{lem:QSP_moreancilla} For any $m\in [\Omega(2^n/n^2),3\cdot2^{n}]$, any $n$-qubit quantum state $\ket{\psi_v}$ can be generated by a quantum circuit, using single-qubit gates and CNOT gates, of depth $O\big(n(n-\log (m/3) +1)+\frac{2^n}{m}\big)$ and size $O(2^{n})$ with $m$ ancillary qubits. \end{lemma} \begin{proof} Let $t=\lfloor\log\frac{m}{3})\rfloor$. Define a quantum state $\ket{\psi_{v}^{(t)}} = \sum_{i=0}^{2^{t}-1} v'_i \ket{i}$, where $v'_i=\sqrt{\sum_{j=0}^{2^{n-t}-1}\|v_{i\cdot 2^{n-t}+j}\|^2}$. Note that $\ket{\psi_{v}^{(t)}} = V_t V_{t-1} \cdots V_1 \ket{0}^{\otimes n}$, the state after we apply the first $t$ UCGs in Figure \ref{fig:QSP_circuit}(a). According to Lemma \ref{lem:johr}, given the unit vector $v' = (v_0', \ldots, v_{2^t-1}')$, we can prepare a $2^t$-qubit quantum state $\ket{\psi'_v}=\sum_{i=0}^{2^t-1}v_i'\ket{e_i}$ by a quantum circuit of depth $O(t) = O(n)$ and size $O(2^t) = O(2^n)$. The resulting state is on $2^t$ qubits. Then we apply the unitary transform Eq. \eqref{eq:unary_transform} in Lemma \ref{lem:enc-trans} to transform the unary encoding to a binary encoding and obtain $\ket{\psi_v^{(t)}} = \sum_{i=0}^{2^t-1} v_i' \ket{i}$. This transformation has depth $O(t)$ and size $O(2^t)$, and need $2^{t+1}$ ancillary qubits. The whole process can be carried out in a work space of $2^t + 2^{t+1} \le m$ qubits. To change $\ket{\psi_{v}^{(t)}}$ to the final target state $\ket{\psi_{v}}$, what is left is to apply $V_{t+1}, \ldots, V_n$ to $\ket{\psi_{v}^{(t)}}$. By Lemma \ref{lem:UCG_depth}, each $V_j$ can be implemented by a circuit of depth $O(j+\frac{2^j}{m})$ and size $O(2^j)$ by $m$ ancillary qubits. Hence $V_n\cdots V_{t+1}$ can be realized by a quantum circuit of depth $\sum_{j=t +1}^{n}O\big(j+\frac{2^j}{m}\big) = O\big(n(n-\lfloor\log(m/3)\rfloor)+\frac{2^n}{m}\big)$, { and size $\sum_{j=t+1}^n O(2^j)=O(2^n)$}, with $m$ ancillary qubits. Combining the two steps, we see that the total depth and size of this quantum state preparation circuit are $O\big(n(n-\log (m/3) +1)+\frac{2^n}{m}\big)$ and $O(2^n)$, respectively. \end{proof} Note that when $m = 3\cdot 2^n$, the depth bound becomes $O(n)$. And if $m$ is even larger, then we can choose to only use $3\cdot 2^n$ of them. Thus we have the following result, {which is Theorem \ref{thm:QSP_anci} in the parameter regime $m = \Omega(2^n/n^2)$.} \begin{corollary}\label{coro:QSP_moreancilla} For a circuit preparing an $n$-qubit quantum state with $m=\Omega(2^n/n^2)$ ancillary qubits, the minimum depth $D_{\textsc{QSP}}(n,m)$ for different ranges of $m$ are characterized as follows: \[ \left\{ \begin{array}{ll} O(2^n/m), & \text{if~} m\in[\Omega(2^n/n^2),O(2^n/(n\log n))],\\ O(n\log n), & \text{if~} m\in[\omega(2^n/(n\log n)),o(2^n)],\\ O(n), & \text{if~} m=\Omega(2^n). \end{array}\right. \] \end{corollary} \section{Extensions and implications} \label{sec:extensions} \subsection{Implications on optimality of unitary depth compression} In this section, we will show that our results for QSP can be applied to general unitary synthesis. The proofs of Theorem \ref{thm:unitary} and Corollary \ref{coro:unitary} are shown in Appendix \ref{sec:US_depth}. \begin{theorem}\label{thm:unitary} Any unitary matrix $U\in\mathbb{C}^{2^n\times 2^n} $ can be implemented by a quantum circuit of depth $O\big(n2^n+\frac{4^n}{m+n}\big)$ {and size $O(4^n)$} with $m \le 2^n$ ancillary qubits. \end{theorem} In \cite{shende2004minimal}, it was shown that one needs at least $\Omega\left(4^n\right)$ CNOT gates to implement an arbitrary $n$-qubit unitary matrix without ancillary qubits. In the proof, the authors first put the circuit in a form that all single-qubit gates are immediately before either a CNOT gate or the output. It is known that such a CNOT gate together with its two single-qubit incoming neighbor gates can be specified by 4 free real parameters, and that each single-qubit gate right before the output has 3 free real parameters. Thus overall the circuit has $4k+3n$ parameters where $k$ is the number of CNOT gates. To generate all $n$-qubit states, the set of which is known to have dimension $4^n-1$, we need $4k+3n \ge 4^n-1$. Thus the bound follows. {This argument basically applies to quantum circuits with ancillary qubits as well, as stated in the next corollary, which shows that our circuit construction for general unitary matrices is asymptotically optimal for $m = O(2^n/n)$.} \begin{corollary}\label{coro:unitary} The minimum circuit depth $D_{\textsc{Unitary}}(n,m)$ for an arbitrary $n$-qubit unitary with $m$ ancillary qubits satisfies \[ \begin{cases} D_{\textsc{Unitary}}(n,m) = \Theta\big(\frac{4^n}{m+n}\big), & \text{if } m=O(2^n/n), \\ D_{\textsc{Unitary}}(n,m) \in \left[ \Omega(n), O(n2^n)\right], & \text{if }m = \omega(2^n/n). \end{cases} \] \end{corollary} \subsection{Decomposition with Clifford + T gate set} The quantum gate set $\{{CNOT},H,S,T\}$, sometimes called Clifford+T gate set, is a universal gate set in that any unitary matrix can be approximately implemented using these gates only. The gates in this set all have a fault-tolerant implementation, thus the gate set is considered as one of the most promising candidates for practical quantum computing. In this section we consider the circuits using only the gates in this set. \begin{definition}[$\epsilon$-approximation] For any $\epsilon>0$, a unitary matrix $U$ is $\epsilon$-approximated by another unitary matrix $V$ if \[\\|U-V\\|_2 \defeq \max_{\\|\|\psi\rangle\\|_2=1} \\|(U-V)\|\psi\rangle\\|_2 < \epsilon.\] \end{definition} We can extend our results on the exact implementations of state preparation and unitary to their approximate versions. The following two corollaries are circuit implementations for quantum state preparation (Corollary \ref{coro:psi_apprx}) and unitary synthesis (Corollary \ref{coro:unitary_apprx}). The Corollary \ref{coro:psi_apprx} is a restatement of Corollary \ref{coro:approx_QSP}. {The proofs are shown in Appendix \ref{sec:clifford_decomposition}.} \begin{corollary}\label{coro:psi_apprx} For any $n$-qubit target state $\ket{\psi_v}$ and $\epsilon>0$, one can prepare a state $\ket{\psi'_v}$ which is $\epsilon$-close to $\ket{\psi_v}$ in $\ell_2$-distance, by a quantum circuit consisting of $\{CNOT,H,S,T\}$ gates of depth {\[\left\{\begin{array}{ll} O\big(\frac{2^n\log(2^n/\epsilon)}{m+n}\big) & \text{if~}m=O(2^n/(n\log n)),\\ O(n\log n\log(2^n/\epsilon))& \text{if~}m\in [\omega(2^n/(n\log n),o(2^n)],\\ O(n \log(2^n/\epsilon))& \text{if~}m=\Omega(2^n),\\ \end{array}\right.\]} where $m$ is the number of ancillary qubits. \end{corollary} The following is an implementation of a unitary matrix. \begin{corollary}\label{coro:unitary_apprx} Any $n$-qubit general unitary matrix can be implemented by a circuit, using the $\{CNOT,H,S,T\}$ gate set, of depth $O\big(n2^n+\frac{4^n\log(4^n/\epsilon)}{m+n}\big)$ with $m$ ancillary qubits. \end{corollary} \noindent{\bf Remark.} Our circuits for general states can be also extended to circuits for sparse states. See details in Appendix \ref{sec:sparse_QSP}. \section{Conclusion} \label{sec:conclusions} In this paper, we have shown that an arbitrary $n$-qubit quantum state can be prepared by a quantum circuit consisting of single-qubit gates and CNOT gates with $m=O(2^n)$ ancillary qubits, of depth $O\big(n\log n+\frac{2^n}{n+m}\big)$ and size $O(2^n)$. The bound is improved to $O(n)$ if we have more ancillary qubits, and all these bounds are tight (up to a logarithmic factor in a small range of $m$). These results can be applied to reduce the depth of the circuit of general unitary to $O\big(n2^n+\frac{4^n}{m+n}\big)$ with $m$ ancillary qubits, which is optimal when $m = O(2^n/n)$. The results can be extended to approximate state preparation by circuit using the Clifford+T gate set. Many questions are left open for future studies. An immediate one is to close the gap for unitary synthesis for large $m$ in Corollary \ref{coro:unitary}. One can also put more practical restrictions into consideration. For instance, we assume that two-qubit gates can be applied on any two qubits. Though this all-to-all connection is indeed the case for certain quantum computer implementations (such qubits made of trapped ions), some others (such as superconducting qubits) can only support nearest neighbor interactions, and it is interesting to study QSP for that case. Another direction is to take various noises into account, and see how much that affects the complexity. We call for more studies of state preparation and circuit synthesis, and hope that methods and techniques developed in this paper can be used to design efficient circuits in those extended models. \appendix \section{Circuit depth lower bound} \label{sec:QSP_lowerbound} \input{QSP_lowerbound} \section{Quantum state preparation via Binary search tree} \label{sec:app_BST} The framework of quantum state preparation is illustrated by a vector \[ \nu=\left(\sqrt{0.03},\sqrt{0.07},\sqrt{0.15},\sqrt{0.05},\sqrt{0.1},\right.\left.\sqrt{0.3},\sqrt{0.2},\sqrt{0.1}\right)^T \in\mathbb{C}^8. \] The corresponding quantum state is a $3$-qubit quantum state \begin{align} \|\psi_\nu\rangle = & \sqrt{0.03}\|000\rangle+\sqrt{0.07}\|001\rangle+\sqrt{0.15}\|010\rangle+\sqrt{0.05}\|011\rangle\\ & +\sqrt{0.1}\|100\rangle+\sqrt{0.3}\|101\rangle+\sqrt{0.2}\|110\rangle+\sqrt{0.1}\|111\rangle. \end{align} The amplitudes of $\|\psi_v\rangle$ are stored in the leaf nodes of the corresponding Binary Search Tree. Every internal node stores the square root of sum of squares of its child nodes. The root node stores the $\ell_2$-norm of the vector. \begin{figure}\label{fig:bst} \label{fig:circuit_psi} \label{fig:framework} \end{figure} Based on the Binary Search Tree in Figure \ref{fig:framework}(a), the QSP circuit can be designed layer-by-layer and branch-by-branch, as in Figure \ref{fig:framework}(b). \section{Implementations of tasks in Eq. \eqref{eq:task1} and Eq. \eqref{eq:alpha}} \label{sec:2tasks} The first task (Eq. \eqref{eq:task1}) can be completed by the combination of the circuit in Figure \ref{fig:f_circuit}, a fact formalized as Lemma \ref{lem:f_circuit}, which can be easily verified. \begin{figure} \caption{A quantum circuit to implement transformation $\ket{x_1x_2\cdots x_n} \to e^{i \langle s,x\rangle \alpha}\ket{x_1x_2\cdots x_n}$ with string $s=s_1s_2\cdots s_n\in\{0,1\}^n$ being the indicator vector of set $S = \{i_1, \ldots, i_k\} \subseteq [n]$, i.e. $s_j=1$ if $j\in S$ and $s_j=0$ otherwise. (a) A quantum circuit with an ancillary qubit initialized as $\ket{0}$. The index set of controlled qubit of CNOT gates is $S$. (b) A quantum circuit without ancillary qubits, where $i_{j}$ is an arbitrary element in $S$. The index set of the controlled qubit of CNOT gates is $S-\{i_j\}$ and the index of target qubit is $i_j$.} \label{fig:f_circuit} \end{figure} \begin{lemma}\label{lem:f_circuit} Let $x = x_1 x_2 \ldots x_n$, $s = s_1 s_2 ... s_n \in\{0,1\}^n$, and $S = \{i_1, \ldots, i_k\} = \{i: s_i = 1\} \subseteq [n]$. The circuits in Figure \ref{fig:f_circuit} realize the following transformation: \[\ket{x_1x_2\cdots x_n} \to e^{i\langle s,x\rangle \alpha}\ket{x_1 x_2 \cdots x_n}.\] \end{lemma} The second task (Eq. \eqref{eq:alpha}) is accomplished as follows. Based on Lemma \ref{lem:f_circuit}, one can implement transformation Eq. \eqref{eq:task1} by using $2^n-1$ circuits with parameters $\alpha_s$ for all $s\in\mbox{$\{0,1\}^n$}-\{0^n\}$ in Figure \ref{fig:f_circuit}. To determine parameters $\alpha_s$ in Eq. \eqref{eq:alpha}, the second task is essentially asking whether the $(2^n-1)\times (2^n-1)$ matrix $A$ defined by \begin{align}\label{eq:matrix-A} A(x,s) = \langle x,s\rangle, \quad x,s\in \mbox{$\{0,1\}^n$}-\{0^n\} \end{align} is invertible. The answer is affirmative and the inverse is given by the following lemma, which can be easily verified \cite{welch2014efficient2}. \begin{lemma} The matrix $A$ defined as in Eq. \eqref{eq:matrix-A} is invertible, and its inverse is $2^{1-n}(2A - \bf{J})$, where ${\bf J}\in\mathbb{R}^{(2^{n}-1)\times (2^{n}-1)}$ is the all-one matrix. \label{lem:fwt} \end{lemma} {This gives a way to compute the parameters $\alpha_s$ efficiently on a classical computer.} \begin{lemma} For QSP problem, given a unit vector $v=(v_0,v_1,\cdots,v_{2^n-1})^T\in\mathbb{C}^{2^n}$, the values of $\{\alpha_s: s\in\{0,1\}^n-\{0^n\}\}$ in Eq. \eqref{eq:alpha} can be calculated on a classical computer using $O(n2^n)$ time and $O(n2^n)$ space. \end{lemma} \begin{proof} We calculate these $\alpha_s$ in three steps. \begin{enumerate} \item Our QSP circuit consists of $n$ UCGs $V_1,V_2,\cdots,V_n$ as in Figure \ref{fig:QSP_circuit}(a). We calculate all parameters of $V_1,\ldots,V_n$ in time $O(2^n)$ and in space $O(2^n)$ using binary trees \cite{grover2002creating,kerenidis2017quantum}. \item Secondly, we decompose all UCGs into diagonal unitary matrices and some single-qubit operations according to Eq. \eqref{eq:UCG}. As in the proof of Lemma \ref{lem:lamda2circuit}, we decompose every single-qubit gate in $V_j$ into $R_z$ gates, $S$ gates and $H$ gates in time $O(1)$ and space $O(1)$, and UCG $V_j$ can be decomposed into 3 diagonal unitary matrices and two $H$ gates and $S$ gates in time $O(2^j)$ and space $O(2^j)$. Hence, the total time and space of this step are $\sum_{j=1}^nO(2^j)=O(2^n)$. \item Thirdly, for every diagonal unitary matrix $\Lambda_j$ with diagonal element $e^{i\theta(x)}$ for all $x\in\{0,1\}^j-\{0^j\}$, we calculate parameters $\alpha_s$ in Eq. \eqref{eq:alpha}. Let \begin{align} &\boldsymbol \alpha\defeq(\alpha_{0\cdots 01},\alpha_{0\cdots 10},\ldots,\alpha_{1\cdots 11})^T\in\mathbb{R}^{2^n-1},\\ & \boldsymbol \theta\defeq(\theta(0\cdots 01),\theta(0\cdots 10),\ldots,\theta(1\cdots 11))^T\in\mathbb{R}^{2^n-1}. \end{align} Based on Eq. \eqref{eq:alpha} and lemma \ref{lem:fwt}, we have $\boldsymbol \alpha=2^{1-n}(2A - \bf{J})\boldsymbol \theta$. Notice that $(2A - \bf{J})\boldsymbol \theta$ is just the Walsh-Hadamard transform on $\boldsymbol \theta$. Thus the vector $\boldsymbol\alpha$ can be calculated by fast Walsh-Hadamard transform algorithm \cite{fino1976unified}, which costs $O(n2^n)$ time and $O(n2^n)$ space. \end{enumerate} Adding these costs up, we see that the values of $\alpha_s$ can be calculated in $O(n2^n)$ time and $O(n2^n)$ space. \end{proof} \section{Warm-up example: Implement $\Lambda_4$ using 8 ancillary qubits} \label{sec:warm-up} In this section, we will show how to implement $\Lambda_4$ with $8$ ancillary qubits based on a Gray code, which can help to understand the general case. Our construction of quantum circuit for $\Lambda_4$ is shown in Figure \ref{fig:example_Lambda_4}. \begin{figure} \caption{Implementation of $\Lambda_4$ with 8 ancillary qubits. The first 4 qubits form the input register, the next 4 qubits form the copy register and the last 4 qubits form the phase register. Step 1-2 are Prefix Copy Stage; Step 3-5 are Gray Initial Stage; Step 6-9 are Suffix Copy Stage; Step 10-15 are Gray Path Stage; Step 16 is Inverse stage. All parameters $\alpha_s$ of phase gate $R(\alpha_s)$ for $s\in\{0,1\}^4$ are determined by Eq. \eqref{eq:alpha}. Step 16 consists of inverse quantum circuits of Step 14,12,10,9,8,7,6,4,3,2 and 1 in that order. Step 1-15 are quantum circuits of depth 1 and Step 16 is quantum circuits of depth 11.} \label{fig:example_Lambda_4} \end{figure} All the parameters $\alpha_s$ for $s\in\{0,1\}^4$ are defined in Eq. \eqref{eq:alpha}. In Figure \ref{fig:example_Lambda_4}, the first four qubits initialized as $\ket{x}:=\ket{x_1x_2 x_3 x_4}$ constitute the input register; the next four qubits form the copy register; and the last four qubits form the phase register. All qubits in the copy and the phase register are initialized to $\|0\rangle$. For any 4-bit string $s=s_1s_2s_3s_4\in\{0,1\}^4$, $s_1s_2$ is the prefix and $s_3s_4$ is the suffix. In Step 1-2, we make two copies of prefix $\ket{x_1 },\ket{x_2}$ into the copy register, i.e., \begin{align} \underbrace{\ket{x_1x_2x_3x_4}}_{\text{input register}}\underbrace{\ket{0000}}_{\text{copy register}}\to\ket{x_1x_2x_3x_4}\ket{x_1x_2x_1x_2}. &\text{(Step 1-2)} \end{align} In Step 3-5, we generate all $4$-bit strings whose suffixes are all $00$ in the phase register by CNOT gates. Namely, we realize the following transformation \begin{align} &\overbrace{\ket{x_1x_2x_3x_4}}^{\text{input register}}\overbrace{\ket{x_1x_2x_1x_2}}^{\text{copy register}}\overbrace{\ket{0000}}^{\text{phase register}}\\ \to &\ket{x_1x_2x_3x_4}\ket{x_1x_2x_1x_2}\\ &\ket{\langle0000,x\rangle,\langle1000,x\rangle,\langle0100,x\rangle,\langle 1100,x\rangle} & \text{(Step 3-4)}\\ \to& e^{i\sum_{s\in\{0,1\}^2}\langle s00,x\rangle \alpha_{s00}}\ket{x_1x_2x_3x_4}\ket{x_1x_2x_1x_2}\\ &\ket{\langle0000,x\rangle,\langle1000,x\rangle,\langle0100,x\rangle,\langle 1100,x\rangle}. & \text{(Step 5)} \end{align} In Step 6-9, we transform the copy register to initial state and make two copies of suffix $x_3x_4$: \begin{align} &\overbrace{\ket{x_1x_2x_3x_4}}^{\text{input register}}\overbrace{\ket{x_1x_2x_1x_2}}^{\text{copy register}}\\ \to&\ket{x_1x_2x_3x_4}\ket{0000} & \text{(Step 6-7)}\\ \to& \ket{x_1x_2x_3x_4}\ket{x_3x_4x_3x_4}. & \text{(Step 8-9)} \end{align} Up to now, we have generated all prefixes for suffix ``00'', and next we need to generate all the other $4$-bit strings. In order to reduce the number of CNOT gates, we consider two forms of $2$-bit Gray code. The $1$-Gray code and $2$-Gray code starting from $00$ are \[00, 10, 11, 01 \text{~~and~~} 00, 01, 11, 10.\] If in $1$-Gray code and $2$-Gray code, we list all the bits changed between adjacent strings, we get two lists: $1,2,1$ and $2,1,2$. To obtain parallelism, in the first and second qubit in the phase register we generate suffixes $00,10,11,01$ ($1$-Gray code) in that order. In the third and fourth qubit in the phase register we generate suffixes $00,01,11,10$ ($2$-Gray code) in that order. In Step 10-15, we implement the following transformation \begin{align} & e^{i\sum_{s\in\{0,1\}^2} \langle s00,x\rangle\alpha_{s00}}\ket{x_1x_2x_3x_4}\ket{x_3x_4x_3x_4}\\ &\ket{\langle0000,x\rangle,\langle1000,x\rangle,\langle0100,x\rangle,\langle1100,x\rangle} \\ \to& e^{i\sum_{s\in\{0,1\}^4-\{0^4\}}\langle s,x\rangle\alpha_s}\ket{x_1x_2x_3x_4}\ket{x_3x_4x_3x_4}\\ &\ket{\langle0001,x\rangle,\langle1001,x\rangle,\langle0110,x\rangle,\langle1110,x\rangle} &\text{(Step 10-15)}\\ =&e^{i\theta(x)}\ket{x_1x_2x_3x_4}\ket{x_3x_4x_3x_4}\\ &\ket{\langle0001,x\rangle,\langle1001,x\rangle,\langle0110,x\rangle,\langle1110,x\rangle}. \end{align} Every step can be realized by a quantum circuit in depth 1. Step 16 consists of inverse quantum circuits of Step 14,12,10,9,8,7,6,4,3,2 and 1 in order. The total depth of Step 16 is 11. It transforms copy register and phase register to their initial states. Therefore, it implements the following transformation \begin{align} & e^{i\theta(x)}\ket{x_1x_2x_3x_4}\ket{x_3x_4x_3x_4}\\ &\ket{\langle0001,x\rangle,\langle1001,x\rangle,\langle0110,x\rangle,\langle1110,x\rangle} \\ \to & e^{i\theta(x)}\ket{x}\ket{0000}\ket{0000} &\text{(Step 16)} \end{align} As discussed above, the quantum circuit in Figure \ref{fig:example_Lambda_4} is an implementation of $\Lambda_4$ with 8 ancillary qubits. \section{Proof of Lemma \ref{lem:2D-array}} \label{sec:2D-array} {\noindent\bf Lemma \ref{lem:2D-array}} \emph{ Let $t = {\lfloor \log \frac{m}{2} \rfloor}$ and $\ell = 2^t$. The set $\mbox{$\{0,1\}^n$}$ can be partitioned into a 2-dimensional array $\{s(j,k): j\in [\ell], k\in [2^n/\ell]\}$ of $n$-bit strings, satisfying that \begin{enumerate} \item Strings in the first column $\{s(j,1): j\in [\ell]\}$ have the last $(n-t)$ bits being all 0, and strings in each row $\{s(j,k): k\in [2^n/\ell]\}$ share the same first $t$ bits. \item $\forall j\in [\ell], \forall k\in [2^n/\ell-1]$, $s(j,k)$ and $s(j,k+1)$ differ by 1 bit. \item For any fixed $k\in [2^n/\ell-1]$, and any $t' \in \{t+1,...,n\}$, there are at most $\big(\frac{m}{2(n-t)} + 1\big)$ many $j\in [\ell]$ s.t. $s(j,k)$ and $s(j,k+1)$ differ by the $t'$-th bit. \end{enumerate} } \begin{proof} Consider each $n$-bit string as two parts, a $t$-bit prefix followed by an $(n-t)$-bit suffix. We let $\{s(j,1): j\in [\ell]\}$ run over all $\ell$ possible prefixes, and for each fixed $j\in [\ell]$, the collection of $\{s(j,k): k\in [2^n/\ell]\}$ run over all possible suffixes. Thus $\{s(j,k): j\in [\ell], k\in [2^n/\ell]\}$ form a partition of $\mbox{$\{0,1\}^n$}$, and the first condition is satisfied. Now for the $j$-th set of suffixes $\{s(j,k):k\in [2^n/\ell]\}$, we identify it with $\mbox{$\{0,1\}$}^{n-t}$, and apply the $(j',n-t)$-Gray code and Lemma \ref{lem:GrayCode} to it, where $j' = ((j-1) \ mod\ (n-t)) + 1\in \{1,...,n-t\}$. For any $k\in [2^n/\ell-1]$ and any $t'\in \{t+1,...,n\}$, let us see how many $j\in [\ell]$ have that $s(j,k)$ and $s(j,k+1)$ differ by bit $t'$. When $j$ runs over $[n-t]$, $s(j,k)$ and $s(j,k+1)$ differ by bit $t'$ exactly once. When $j$ runs over $\{n-t+1,...,2(n-t)$, $s(j,k)$ and $s(j,k+1)$ differ by bit $t'$ again exactly once. Repeating this we can see that when $j$ runs over all $[\ell]$, $s(j,k)$ and $s(j,k+1)$ differ by bit $t'$ for at most $\lceil \ell/(n-t) \rceil \le m/2(n-t)+1$ times. \end{proof} \section{Proof of Lemma \ref{lem:inverse}} \label{sec:inverse} {\noindent\bf Lemma \ref{lem:inverse}} \emph{The depth and size of the Inverse Stage are at most $O(\log m + 2^n /m)$ and $\frac{m}{2}+\frac{nm}{2}+m+2^{n}=2^{n}+\frac{3m+nm}{2}$. The effect of this stage is \begin{equation} \ket{x} \ket{x_{suf}} \ket{f_{[\ell],2^n/\ell}} \xrightarrow{U_{Inverse}} \ket{x} \ket{0^{m/2}} \ket{0^{m/2}}. \end{equation}} \begin{proof} The depth is just the summation of the CNOT depth of the first four stages, which is $O\big(\log m + 2\log m + 2\log m + 2^n/\ell\big) = O\big(\log m + 2^n /m\big).$ { The analyze of size is similar by adding up the sizes in previous stages.} The effect is shown as follows, which holds by Lemma \ref{lem:GrayPath}, Eq. \eqref{eq:suf-copy}, Eq. \eqref{eq:U1} and Eq. \eqref{eq:Uc1-effect}. \begin{align} &\ket{x} \ket{x_{suf}} \ket{f_{[\ell],2^n/\ell}} \xrightarrow{U_{2}^{\dagger} \cdots U_{2^n/\ell}^\dagger} \ket{x} \ket{x_{suf}} \ket{f_{[\ell],1}} \xrightarrow{U_{copy,1} U_{copy,2}^\dagger} \\ &\ket{x} \ket{x_{pre}} \ket{f_{[\ell],1}} \xrightarrow{U_1^\dagger} \ket{x} \ket{x_{pre}} \ket{0^{\frac{m}{2}}} \xrightarrow{U_{copy,1}^\dagger} \ket{x} \ket{0^{\frac{m}{2}}} \ket{0^{\frac{m}{2}}} \end{align} \end{proof} \section{Proof of Lemma \ref{lem:puttingtogether_ancilla}} \label{sec:puttingtogether_ancilla} \noindent {\bf Lemma \ref{lem:puttingtogether_ancilla}} \emph{The circuit implements the operation in Eq. \eqref{matrix:lambda_n} in depth $O(\log m + 2^n /m)$ and in size $3\cdot 2^{n}+nm+{\frac{7}{2}}m$.} \begin{proof} For the depth, simply adding up the depth and the size of the five stages gives the bound. Next we analyze the operation step by step as follows. The three registers are the input, copy and phase registers, respectively. \begin{align} &\ket{x}\ket{0^{m/2}}\ket{0^{m/2}} \\ & \xrightarrow{U_{copy,1}} \ket{x}\ket{x_{pre}}\ket{0^{m/2}} & \text{(Eq. \eqref{eq:Uc1-effect})} \\ & \xrightarrow{U_{GrayInit}} e^{i \sum\limits_{j\in [\ell]} f_{j,1}(x) \alpha_{s(j,1)} } \ket{x} \ket{x_{pre}} \ket{f_{[\ell],1}} & \text{(Eq. \eqref{eq:UGI-effect})} \\ & \xrightarrow{U_{copy,2}U_{copy,1}^\dagger} e^{i \sum\limits_{j\in [\ell]} f_{j,1}(x) \alpha_{s(j,1)} } \ket{x} \ket{x_{suf}} \ket{f_{[\ell],1}} & \text{(Eq. \eqref{eq:suf-copy})} \\ & \xrightarrow{R_2 U_2} e^{i \sum\limits_{j\in [\ell]\atop k\in[2]} f_{j,k}(x) \alpha_{s(j,k)}} \ket{x} \ket{x_{suf}} \ket{f_{[\ell],2}} & \text{(Eq. \eqref{eq:GrayPath})} \\ & \quad \vdots & \\ & \xrightarrow{R_{\frac{2^n}{\ell}} U_{\frac{2^n}{\ell}}} e^{i \sum\limits_{j\in [\ell]\atop k\in [\frac{2^n}{\ell}]} f_{j,k}(x) \alpha_{s(j,k)} } \ket{x} \ket{x_{suf}} \ket{f_{[\ell],\frac{2^n}{\ell}}} & \text{(Eq. \eqref{eq:GrayPath})} \\ & \qquad = e^{i\sum\limits_{s\in \{0,1\}^n} \langle x,s\rangle \alpha_{s}} \ket{x} \ket{x_{suf}} \ket{f_{[\ell],2^n/\ell}} & (\text{Lem \ref{lem:2D-array}} ) \\ & \qquad = e^{i\theta(x)} \ket{x} \ket{x_{suf}} \ket{f_{[\ell],2^n/\ell}} & \text{(Eq. \eqref{eq:alpha})} \\ & \xrightarrow{U_{Inverse}} e^{i\theta(x)} \ket{x}\ket{0^{m/2}} \ket{0^{m/2}} & \text{(Eq. \eqref{eq:inverse})} \\ & \qquad = \Lambda_n \ket{x}\ket{0^{m/2}}\ket{0^{m/2}} \end{align} \end{proof} \section{Construction of linearly independent sets} \label{sec:partition} What remains for completing the Generate Stage is the construction of sets $T^{(1)},T^{(2)},\ldots,T^{(\ell)}$, which we will show next. \begin{lemma}\label{lem:partition} There exist sets $T^{(1)},T^{(2)},\cdots,T^{(\ell)} \subseteq \{0,1\}^n-\{0^n\}$, for some integer $\ell \le \frac{2^{n+2}}{n+1}-1$, such that: \begin{enumerate} \item For any $i\in[\ell]$, $\|T^{(i)}\|=n$; \item For any $i\in[\ell]$, the Boolean vectors in $T^{(i)}=\{{t^{(i)}_1},{t^{(i)}_2},\cdots,{t^{(i)}_n}\}$ are linearly independent over $\mathbb{F}_2$; \item $\bigcup_{i\in[\ell]} T^{(i)}= \{0,1\}^{n} - \{0^n\} $. \end{enumerate} \end{lemma} \begin{proof} For any $n$-bit vector $x\in \{0,1\}^n$, let $S_x=\{x\oplus e_1,x\oplus e_2,\cdots,x\oplus e_{n}\}$. Firstly, we construct a set $L\subseteq \{0,1\}^n$ which satisfies $\|L\|\le \frac{2^n+1}{n+1}$ and $\{0,1\}^n=(\bigcup_{x\in L} S_x)\cup L$. Let $k = \lceil\log{(n+1)}\rceil$. For $t\in[n]$, denote the $k$-bit binary representation of integer $t$ by $t_{k}\cdots t_2 t_1$, where $t_1,\ldots,t_{k}\in\{0,1\}$ and $t=\sum_{i=1}^{k}t_i2^{i-1}$. We use a bar to denote the corresponding column vector, i.e. \[\overline{t}=[t_1, t_2, \ldots, t_{k}]^{T}\in\{0,1\}^{k}.\] Define a $k\times n$ Boolean matrix $H$ by concatenating vectors $\overline{1},\overline{2},\cdots,\overline{n}$ together, i.e. \[ H=[\overline{1},\overline{2},\cdots,\overline{n}] \in\{0,1\}^{k\times n}. \] Note that the $k$-dimensional identity matrix $I_k = [\overline{2^{0}},\overline{2^1},\ldots,\overline{2^{k-1}}]$ is a submatrix of $H$, therefore $H$ is full row rank, i.e. $\text{rank}(H) = k$. Define sets \begin{align} \label{equ:all-b} & L^{(0)} = \{x\in \mbox{$\{0,1\}^n$}: Hx = 0^k\},\nonumber\\ &L^{(1)} = \{x\in \mbox{$\{0,1\}^n$}: Hx = 1^k\}, \end{align} and \begin{align} \label{equ:lastbit-b} &A^{(0)} = \{x\in \mbox{$\{0,1\}^n$}: (Hx)_k = 0\},\nonumber\\ &A^{(1)} = \{x\in \mbox{$\{0,1\}^n$}: (Hx)_k = 1\}. \end{align} For each $x\in \mbox{$\{0,1\}^n$}$, the last bit of $Hx$ is either 0 or 1, thus $A^{(0)}\cup A^{(1)} = \mbox{$\{0,1\}^n$}$. Also note that for each $b\in \mbox{$\{0,1\}$}$, $L^{(b)}$ requires all bits being $b$ and $A^{(b)}$ only requires the last bit being $b$, thus $L^{(b)} \subseteq A^{(b)}$. Now we will show \[A^{(0)}\subseteq L^{(0)}\cup \left(\cup_{x\in L^{(0)}}S_x\right) \quad \text{and} \quad A^{(1)}\subseteq L^{(1)}\cup \left(\cup_{x\in L^{(1)}}S_x\right).\] For any $y\in A^{(0)}-L^{(0)}$, consider $\overline{t}=Hy$: Since it satisfies $\overline{t}_{k}=0$ and $t_{k-1}\cdots t_1 \ne 0^{k-1}$, we have that \[1\le t \le \sum_{i=1}^{k-1} 2^{i-1} < 2^{k-1} = 2^{\lceil \log(n+1)\rceil -1} < 2^{\log(n+1)} = n+1.\] Therefore, $1\le t\le n$, and we can thus use $He_{t}=\overline{t}$ to obtain the following equality \[ H(y\oplus e_{t})=Hy \oplus He_{t}=\overline{t}\oplus \overline{t}=0^{k}. \] Therefore, $y\oplus e_{t}\in L^{(0)}$. That is, for any $y\in A^{(0)}-L^{(0)}$, there exists an $x\in L^{(0)}$ s.t. $y=x\oplus e_{t}$ for some $t\in [n]$. Hence, \[A^{(0)}\subseteq L^{(0)}\cup\left(\cup_{x\in L^{(0)}, \ t\in [n]}\ \{x\oplus e_{t}\}\right) = L^{(0)}\cup\left(\cup_{x\in L^{(0)}}S_x\right).\] For any $y\in A^{(1)}-L^{(1)}$, $\overline{t}=Hy$ satisfies $\overline{t}_{k}=1$ and $t_{k-1}...t_1 \ne 1^{k-1}$. It looks symmetric to the $A^{(0)}-L^{(0)}$ case but there is a technicality that the corresponding integer $t$ may be outside the range $[n]$. To remedy this, define $\overline{{t}'}=\overline{t}\oplus 1^{k}$ (and let ${t}'$ be the integer corresponding to vector $\overline{{t}'}$). Now that $\overline{t'}_{k}=0$ and $t'_{k-1}...t'_1 \ne 0^{k-1}$, and we know $t'\in [n]$. Thus we can again get $He_{{t}'}=\overline{{t}'}$ and, in turn, \[ H(y\oplus e_{{t}'}) = Hy \oplus He_{{t}'} = \overline{t} \oplus \overline{t'} = \overline{t}\oplus \overline{t}\oplus 1^{k}=1^{k}. \] Therefore, $y\oplus e_{{t}'}\in L^{(1)}$, and \[A^{(1)}\subseteq L^{(1)}\cup\left(\cup_{x\in L^{(1)}, \ t'\in [n]}\ \{x\oplus e_{t'}\}\right) = L^{(1)}\cup\left(\cup_{x\in L^{(1)}}S_x\right).\] Let $L=L^{(0)}\cup L^{(1)}$. We have \begin{align} &\{0,1\}^n=A^{(0)}\cup A^{(1)} \subseteq L^{(0)}\cup \left(\cup_{x\in L^{(0)}}S_x\right)\cup L^{(1)}\cup \left(\cup_{x\in L^{(1)}}S_x\right) = L \cup \left(\cup_{x\in L}S_x\right)\subseteq \{0,1\}^n. \end{align} Recall that $\text{rank}(H) = k$ over field $\mathbb{F}_2$, the size of solution set $L^{(b)}$ is $\|L^{(b)}\| = 2^{n-k} = 2^{n-\lceil\log(n+1)\rceil} \le \frac{2^n}{n+1}$, for each $b\in \mbox{$\{0,1\}$}$. Thus $\|L\|= \|L^{(0)}\|+\|L^{(1)}\|\le \frac{ 2^{n+1}}{n+1}$. We have constructed a set $L\subseteq\{0,1\}^n$ of size at most $\frac{ 2^{n+1}}{n+1}$ satisfying $L\cup (\cup_{x\in L}S_x) = \mbox{$\{0,1\}^n$}$. We will now use this set $L$ to construct $\ell \le \frac{2^{n+2}}{n+1}-1$ sets $T^{(i)}$ which satisfy the three properties in the statement of the present lemma. Since $0^n$ is a solution of $Hx = 0^k$, it holds that $0^n\in L^{(0)}\subseteq L$. Note that the vectors in $S_{0^n}=\{e_1,e_2,\ldots,e_n\}$ are linearly independent. For any $x\in L$ and $x\neq 0^n$, let us construct two sets of linearly independent vectors $S_x^{(0)}$ and $S_x^{(1)}$. Since $\text{rank}[x\oplus e_1,x\oplus e_2,\cdots,x\oplus e_{n}]\ge n-1$ over field $\mathbb{F}_2$, we can select $n-1$ linearly independent vectors from $S_x$ to form a set $S_x^{(0)}\subseteq S_x$. Let $S^{(1)}_x=(S_x-S^{(0)}_x-\{0^n\})\cup \{x\}$. It is not hard to verify that if $x = e_j$ for some $j\in [n]$, then $S^{(1)}_x = \{x\} = \{e_j\}$; if $x \notin \{0^n, e_1, \ldots, e_n\}$, then $S^{(1)}_x = \{x, x\oplus e_j\}$ for some $j\in[n]$. In any case, the vector(s) in $S^{(1)}_x$ are linearly independent (the same for $S^{(0)}_x$), and it holds that $S_x^{(0)}\cup S_x^{(1)} = S_x\cup\{x\}-\{0^n\}$. Thus for each $b\in \mbox{$\{0,1\}$}$, we can always extend the set $S^{(b)}_x$ to $T^{(b)}_x$ of $n$ linearly independent vectors by adding some vectors. Recalling $\{0,1\}^n=(\cup_{x\in L}S_x)\cup L$, we have \begin{align} &\{0,1\}^n-\{0^n\} \\ & =\left(\cup_{x\in L} S_x\right)\cup L-\{0^n\} \\ &=\cup_{x\in L}(S_x\cup\{x\})-\{0^n\}\\ &=\left(\cup_{x\in L-\{0^n\}}(S_x\cup\{x\}-\{0^n\})\right)\cup S_{0^n}\\ &=\left(\cup_{x\in L-\{0^n\}}(S_x^{(0)}\cup S_x^{(1)})\right)\cup S_{0^n}\\ & =\left(\cup_{x\in L-\{0^n\}} S_x^{(0)}\right)\cup \left(\cup_{x\in L-\{0^n\}} S_x^{(1)}\right)\cup S_{0^n}\\ & \subseteq \left(\cup_{x\in L-\{0^n\}} T_x^{(0)}\right)\cup \left(\cup_{x\in L-\{0^n\}} T_x^{(1)}\right)\cup S_{0^n}\\ &\subseteq\{0,1\}^n-\{0^n\}. \end{align} Now collect $\{T_x^{(0)}: x\in L-\{0^n\}\}$, $\{T_x^{(1)}: x\in L-\{0^n\}\}$ and $S_{0^n}$ as our sets $T^{(1)}, \ldots, T^{(\ell)}$. Since $\|L\|\le \frac{2^{n+1}}{n+1}$ and $0^n\in L$, the collection contains $\ell \le 2\cdot (\frac{2^{n+1}}{n+1} - 1) + 1 = \frac{2^{n+2}}{n+1} - 1$ sets, each consisting of $n$ linearly independent vectors, and the collection of all these vectors is exactly $\{0,1\}^n-\{0^n\}$. This completes the proof. \end{proof} \section{Correctness of circuit framework in Figure \ref{fig:framework_DU_withoutancilla}} \label{sec:correctness_withoutancilla} In this section, we shown the correctness of circuit framework in Figure \ref{fig:framework_DU_withoutancilla}. For any input state $\|x\rangle$, the quantum circuit $(\Lambda_{r_c}\otimes\mathcal{R})\mathcal{G}_{\ell}\mathcal{G}_{\ell-1}\cdots\mathcal{G}_{1}$ makes the following sequence of operations. \begin{align} \ket{x}& = \ket{x_{control}} \ket{y^{(0)}} \\ & \xrightarrow{\mathcal{G}_1} e^{i\sum_{s\in F_1}\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{y^{(1)}}\\ & \xrightarrow{\mathcal{G}_2} e^{i\sum_{s\in F_1\cup F_2 }\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{y^{(2)}} \\ & ~~~~\vdots \\ & \xrightarrow{\mathcal{G}_\ell} e^{i\sum_{s\in\bigcup_{k\in[\ell]}F_k }\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{y^{(\ell)}}\\ & \xrightarrow{\mathbb{I}_{r_c}\otimes \mathcal{R}} e^{i\sum_{s\in\bigcup_{k\in[\ell]}F_k }\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{y^{(0)}}\\ & \xrightarrow{\Lambda_{r_c}\otimes \mathbb{I}_{r_t}} e^{i\sum_{s\in\left(\bigcup_{k\in[\ell]}F_k\right)\cup \left(\left\{c0^{r_t}\right\}_{c\in\{0,1\}^{r_c}-\{0^{r_c}\}}\right) }\langle s,x\rangle\alpha_s}\\ &~~~~~~~~~~~~\ket{x_{control}} \ket{y^{(0)}}\\ & = e^{i\sum_{s\in\{0,1\}^n-\{0^{n}\} }\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{y^{(0)}}\\ & = e^{i\theta(x)}\ket{x_{control}}\ket{y^{(0)}} \\ &= e^{i\theta(x)}\ket{x} \end{align} The first equation holds by Eq. \eqref{eq:yk}. For arbitrary $k\in[\ell]$, unitary transformation $\mathcal{G}_k$ holds by Eq.\eqref{eq:Gk} and $F_{j}\cap F_k = \emptyset,$ for $j\in[k-1]$. Unitary transformation $\mathcal{R}$ holds by Eq. \eqref{eq:reset}. Unitary transformation $\Lambda_{r_c}$ holds by Eq. \eqref{eq:Lambda_rc}. The last two equations hold by Eq. \eqref{eq:set_eq} and Eq. \eqref{eq:alpha}), respectively. \section{Proof of Lemma \ref{lem:graypath_withoutancilla}} \label{sec:graypath_withoutancilla} \noindent {\bf Lemma \ref{lem:graypath_withoutancilla}} \emph{The quantum circuit defined above is of depth $O(2^{r_c})$ and of size $r_c2^{r_c+1}$, and implements Gray Path Stage $U_{GrayPath}$ in Eq. \eqref{eq:Graycodepath}. } \begin{proof} We first show the correctness. For each $p\in[2^{r_c}]$, let us define a set $F_k^{(p)}$ by \begin{equation}\label{eq:Fkp} F_k^{(p)}=\left\{s:\ s\in F_k\text{~and~}s=c_{p}^it_i^{(k)}\text{ for some }i\in[r_t]\right\} . \end{equation} By definition of $F_k$ in Eq. \eqref{eq:F_k}, the collection of $F_k^{(p)}$'s satisfy \begin{align} & F_k^{(i)}\cap F_{k}^{(j)}=\emptyset \text{~for~all~} i\neq j\in[2^{r_{c}}], \label{eq:Fkp_1}\\ & F_k=\bigcup_{p\in[2^{r_c}]}F_k^{(p)}.\label{eq:Fkp_2} \end{align} Now we can see how the Gray Path Stage $U_{GrayPath}$ in Eq. \eqref{eq:Graycodepath} is realized by the above quantum circuit $C_1,C_2,\ldots,C_{2^{r_c}+1}$ step by step as follows. {For $j\in[2^{r_c}]$, define $\ket{f_j}:=\ket{\langle c_j^1t_{1}^{(k)},x\rangle,\langle c_j^2t_{2}^{(k)},x\rangle,\cdots,\langle c_j^{r_t}t_{r_t}^{(k)},x\rangle}$. Note that $\ket{f_1}=\ket{\langle 0^{r_c}t_{1}^{(k)},x\rangle,\langle 0^{r_c}t_{2}^{(k)},x\rangle,\cdots,\langle 0^{r_c}t_{r_t}^{(k)},x\rangle}$ since $c_1^{i}=0^{r_c}$ for all $ i\in[r_t]$. } \begin{align} &\ket{x_{control}}\ket{y^{(k)}} &\\ &= \ket{x_{control}}\ket{f_1} &(\text{Eq. \eqref{eq:yk}})\\ & \xrightarrow{C_1} e^{i\sum_{s\in F_{k}^{(1)}}\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{f_1} &(\text{Eq. \eqref{eq:Fkp}}) \\ &\xrightarrow{C_2} e^{i\sum_{s\in F_{k}^{(1)}\cup F_{k}^{(2)}}\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{f_2} &(\text{Eq. \eqref{eq:Fkp},\eqref{eq:Fkp_1}})\\ & ~~~~\vdots \\ &\xrightarrow{C_{2^{r_c}}} e^{i\sum_{s\in \bigcup_{p\in[2^{r_c}]}F_k^{(p)}}\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{f_{2^{r_c}}} &(\text{Eq. \eqref{eq:Fkp}, \eqref{eq:Fkp_1}})\\ & = e^{i\sum_{s\in F_k}\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{f_{2^{r_c}}}& (\text{Eq. \eqref{eq:Fkp_2}})\\ &\xrightarrow{C_{2^{r_c}+1}} e^{i\sum_{s\in F_k}\langle s,x\rangle\alpha_s}\ket{x_{control}}\ket{f_1}\\ &= e^{i\sum_{s\in F_k}\langle s,x\rangle\alpha_s}\ket{x_{control}} \ket{y^{(k)}} &(\text{Eq. \eqref{eq:yk}}) \end{align} Next we analyze the depth. Phase 1 consists of rotations applied on different qubits in the target register, thus can be made in a single depth. In each phase $p\in \{2,3,\ldots,2^{r_c}\}$, since $c_{p-1}^{i}$ and $c_p^i$ differ by only 1 bit, one CNOT gate suffices to implement the function $\langle c_p^{i}t_i^{(k)},x\rangle$ from $\langle c_{p-1}^{i}t_i^{(k)},x\rangle$ in the previous phase. The control and target qubit of this CNOT gate is the $h_{ip}$-th qubit in control register and the $i$-th qubit in target register. {According to Eq. \eqref{eq:index}, indices $h_{1p},h_{2p},\ldots,h_{r_tp}$ of control qubits are all different, and therefore, all the CNOT gates in step $p.1$ can be implemented in depth $1$.} The rotations in step $p.2$ are on different qubits and thus fit in one depth as well. Similarly, phase $2^{r_c}+1$ can also be implemented in depth $1$. Thus the total depth of Gray Path Stage is at most $1+2\cdot (2^{r_c}-1)+1=2\cdot 2^{r_c}$. {The size of Gray Path Stage is at most $r_c \cdot 2\cdot 2^{r_c}= r_c2^{r_c+1}$.} \end{proof} \section{Proof of Lemma \ref{lem:enc-trans}} \label{sec:enc-trans} The Toffoli gate is a 3-qubit CCNOT gate where we flip the basis $\ket{0},\ket{1}$ of (i.e. apply $X$ gate to) the third qubit conditioned on the first two qubits are both on $\ket{1}$. This can be extended to an $n$-qubit Toffoli gate, which applies the $X$ gate to the last qubit conditioned on the first $(n-1)$ qubits all being on $\ket{1}$. An $n$-qubit Toffoli gate can be implemented by a circuit of $O(n)$ size and depth { \cite{multi-controlled-gate}}. \begin{lemma} \label{lem:tof} An $n$-qubit Toffoli gate can be implemented by a quantum circuit of depth and size $O(n)$ without ancillary qubits. \end{lemma} The next result we need says that cascading CNOT gates with the same target qubit can be exponentially compressed \cite{moore2001parallel}. \begin{lemma}\label{lem:stair_circuit} Let $C$ be a quantum circuit consisting of $n$ CNOT gates with the same target qubit and distinct controlled qubits. Then $C$ can be compressed to $O(\log n)$ depth {and $O(n)$ size} without using ancillary qubits. \end{lemma} Recall the description of Lemma \ref{lem:enc-trans}. \noindent {\bf Lemma \ref{lem:enc-trans}} \emph{The following unitary transformation on $2^n$ qubits \begin{equation} \ket{e_i}\to\|i\rangle\ket{0^{2^n-n}},\text{~for~all~}i\in\{0\}\cup [2^n-1], e_i\in\{0,1\}^{2^n}, \end{equation} can be implemented by a quantum circuit using single-qubit gates and CNOT gate with {$2^{n+1}$} ancillary qubits, of depth $O(n)$ {and size $O(2^n)$}.} \begin{proof} We will implement Eq. \eqref{eq:unary_transform} with $2^{n+1}$ ancillary qubits in three steps: \begin{enumerate} \renewcommand{Step \theenumi:}{Step \theenumi:} \setlength{\itemindent}{2.2 em} \setlength{\leftmargin}{1in} \item { $\underbrace{\ket{e_i}}_{2^n\atop\text{~qubits}}\ket{0^{2^{n+1}}} \to\ket{0^{2^n}}\underbrace{\ket{e_s}}_{2^{n/2}\atop\text{~qubits}}\underbrace{\ket{e_t}}_{2^{n/2}\atop\text{~qubits}}\ket{0^{2^{n+1}-2\cdot 2^{n/2}}}$ for all $s,t\in\{0\}\cup [2^{n/2}-1]$ and $i=s\cdot 2^{n/2}+t$.} \item { $\ket{0^{2^n}}\ket{e_s}\ket{e_t}\ket{0^{2^{n+1}-2\cdot 2^{n/2}}}\to \ket{0^{2^n}}\ket{i}\ket{0^{2^{n+1}-n}}$ for all $s,t\in\{0\}\cup [2^{n/2}-1]$ and $i=s\cdot 2^{n/2}+t$.} \item {$\ket{0^{2^n}}\ket{i}\ket{0^{2^{n+1}-n}}\to \ket{i}\ket{0^{3\cdot2^n-n}}$ for all $i\in\{0\}\cup [2^n-1]$.} \end{enumerate} { In these three steps, the first $2^n$ qubits are called register $A$. The last $ 2^{n+1}$ are ancillary qubits, which are initialized as $\ket{0}$ and called register $B$. Let the first $2^{n/2}$ qubits of register $B$ be register $B_1$ and the second $2^{n/2}$ qubits of register $B$ be register $B_2$.} Firstly, we implement Step 1 by a quantum circuit of depth $O(n)$ {and size $O(2^n)$} with { $2^{n+1}$} ancillary qubits. Step 1 consists of two phases. \begin{itemize} \item Step 1a: { $\ket{e_i}\ket{0^{2^{n+1}}}\to\ket{e_i}\ket{e_s}\ket{e_t}\ket{0^{2^{n+1}-2\cdot 2^{n/2}}}$} for all $s,t\in\{0\}\cup [2^{n/2}-1]$ and $i=s\cdot 2^{n/2}+t$. Let $CNOT^i_{j,(k)}$ denote a CNOT gate whose controlled qubit is the $i$-th qubit of register $A$, and target qubit is the $j$-th qubit of register $B_k$ for $k\in[2]$. Let $CNOT^i_{s,t}=CNOT^{i}_{s,(1)}CNOT^i_{t,(2)}$. Therefore, Step 1a can be realized by a CNOT circuit as follows: \begin{align} &\prod_{s,t=0}^{2^{n/2}-1} CNOT^{s\cdot 2^{n/2}+t}_{s,t}\\ =&\left[\prod_{s,t=0}^{2^{n/2}-1} CNOT^{s\cdot 2^{n/2}+t}_{s,(1)}\right]\left[\prod_{s,t=0}^{2^{n/2}-1} CNOT^{s\cdot 2^{n/2}+t}_{t,(2)}\right]\\ =&\left[\prod_{s=0}^{2^{n/2}-1} \left(\prod_{t=0}^{2^{n/2}-1} CNOT^{s\cdot 2^{n/2}+t}_{s,(1)}\right) \right]\cdot \left[\prod_{t=0}^{2^{n/2}-1}\left(\prod_{s=0}^{2^{n/2}-1} CNOT^{s\cdot 2^{n/2}+t}_{t,(2)}\right)\right]. \end{align} For every $s\in\{0\}\cup[2^{n/2}-1]$, all CNOT gates in $\mathcal{C'}_s\defeq\prod_{t=0}^{2^{n/2}-1} CNOT^{s\cdot 2^{n/2}+t}_{s,(1)}$ have different controlled qubits and the same target qubit. According to Lemma \ref{lem:stair_circuit}, $\mathcal{C'}_s$ can be implemented by a circuit of depth $O(n)$ and size $O(2^{n/2})$ without ancillary qubits. For all $s\in\{0\}\cup[2^{n/2}-1]$, $C_s'$ act on different qubits. Therefore, they can be paralleled and $\prod_s\mathcal{C'}_s$ can be implemented by a circuit of depth $O(n)$ and size $O(2^{n})$ without ancillary qubits. By similar discussion, $\prod_t\mathcal{C''}_t$ can be implemented by a circuit of depth $O(n)$ and size $O(2^{n})$ without ancillary qubits, where $\mathcal{C''}_t\defeq\prod_{t=0}^{2^{n/2}-1} CNOT^{s\cdot 2^{n/2}+t}_{t,(2)}$. So Step 1a can be implemented by a quantum circuit of depth $O(n)$ {and size $O(2^{n})$} without ancillary qubits. \item Step 1b: { $\ket{e_i}\ket{e_s}\ket{e_t}\ket{0^{2^{n+1}-2\cdot 2^{n/2}}}\to \ket{0^{{2^n}}}\ket{e_s}\ket{e_t}\ket{0^{2^{n+1}-2\cdot {2^{n/2}}}}$ for all $s,t\in\{0\}\cup [2^{n/2}-1]$ and $i=s\cdot 2^{n/2}+t$.} Let $\textsf{T}^{s,t}_{i}$ denotes a 3-qubit Toffoli gate, whose controlled qubits are the $s$-th qubit in register $B_1$ and $t$-th qubit in register $B_2$, and the target qubit is the $i$-th qubit in register $A$. The unitary transform of Step 1b is realized by applying all Toffoli gates $\textsf{T}^{s,t}_{s\cdot 2^{n/2}+t}$. To reduce the circuit depth, we make { $2^{n/2}-1$} copies of register $B_1,B_2$ in last $2^{n+1}$ qubits of register $B$: { \[\ket{e_i}\ket{e_s}\ket{e_t}\ket{0^{2^{n+1}-2\cdot 2^{n/2}}}\to \ket{e_i}\underbrace{\ket{e_s}\ket{e_t}\cdots \ket{e_s}\ket{e_t}}_{2^{n/2}~\text{copies of~}\ket{e_s}\ket{e_t}}\]} for all $s,t\in\{0\}\cup [2^{n/2}-1]$ and $i=s\cdot 2^{n/2}+t$. This transformation can be parallelized to depth $O(n)$ (by Lemma \ref{lem:copy1}) {and size $O(2^{n})$}. Because there are $2^{n/2}$ copies of $\ket{e_s}\ket{e_t}$, all Toffoli gates $\textsf{T}^{s,t}_{s\cdot 2^{n/2}+t}$ are on distinct control and target qubits, thus can be executed in parallel in depth $O(1)$. Finally, restore the last {$2^{n+1}-2\cdot 2^{n/2}$} qubits of register $B$ to all-zero state in $O(n)$ depth. Overall, Step 1b can be implemented by a quantum circuit of depth $O(n)$ {and size $O(2^{n})$} with { $2^{n+1}$} ancillary qubits. \end{itemize} Secondly, Step 2 can be realized by a circuit of depth $O(n)$ {and size $O(n2^{n/2})$} with $O(n2^{n/2})$ ancillary qubits. Now, we rewrite the transformation of Step 2: {\[\ket{0^{2^n}}\ket{e_s}\ket{e_t}\ket{0^{2^{n+1}-2\cdot 2^{n/2}}}\to \ket{0^{2^n}}\underbrace{\ket{s}}_{\frac{n}{2}~\text{qubits}}\underbrace{\ket{t}}_{\frac{n}{2}~\text{qubits}}\ket{0^{2^{n+1}-n}}\]} for all $s,t\in\{0\}\cup [2^{n/2}-1]$. If we can realize transformation \begin{equation}\label{eq:unary_tranformation1} \ket{e_s}\ket{0^k}\to\ket{s}\ket{0^{2^k}},\text{for~} e_s\in\{0,1\}^{2^k},~s\in\{0\}\cup[2^k-1] \end{equation} by a quantum circuit of depth $O(k)$ {and size $O(k2^k)$} with {$ k2^{k}$} ancillary qubits, then we can implement Step 2 by a quantum circuit of depth $O(n)$ {and size $O(n2^{n/2})$} with {$ (n/2)2^{n/2}$} ancillary qubits. We will implement Eq. \eqref{eq:unary_tranformation1} with {$ k2^{k}$} ancillary qubits in three steps: \begin{itemize} \item [] Step 2a: $\ket{e_s}\ket{0^k}\ket{0^{k2^k}}\to\ket{e_s}\ket{s}\ket{0^{k2^k}}$ for all $s\in\{0\}\cup [2^k-1]$; \item [] Step 2b: $\ket{e_s}\ket{s}\ket{0^{k2^k}}\to\ket{0^{2^k}}\ket{s}\ket{0^{k2^k}}$ for all $s\in\{0\}\cup [2^k-1]$; \item [] Step 2c: $\ket{0^{2^k}}\ket{s}\ket{0^{k2^k}}\to \ket{s}\ket{0^{2^k}}\ket{0^{k2^k}}$ for all $s\in\{0\}\cup [2^k-1]$. \end{itemize} The first $2^k$ qubits are called register $A$ and the second $n$ qubits are called register $B$. The last $k2^k$ qubits are ancillary qubits, which are called register $C$. Let $CNOT^s_j$ denote a CNOT gate, whose controlled qubit is the $s$-th qubit in register $A$ and target qubit is the $j$-th qubit in register $B$ for all $i\in\{0\}\cup[2^k-1]$ and $j\in\{0\}\cup [n-1]$. Define $CNOT^s_{S_s}\defeq \prod_{j\in S_s}CNOT^s_j$, where $S_s\defeq \left\{j\|s_j=1\text{~for~}j\in \{0\}\cup [k-1]\right\}$. \begin{itemize} \item Step 2a: Firstly, we implement Step 2a by a quantum circuit with $k2^k$ ancillary qubits of depth $O(k)$. It can be easily verified that Step 2a can be implemented by a CNOT circuit \begin{align} &\prod_{s\in\{0\}\cup [2^k-1]}CNOT^s_{S_s}=\prod_{s\in\{0\}\cup [2^k-1]}\prod_{j\in S_s}CNOT^s_j= \prod_{s\in \{0\}\cup [k-1]}\prod_{s:s\in\{0\}\cup [2^k-1],s_j=1} CNOT^s_j. \end{align} For every $j\in \{0\}\cup [k-1]$, CNOT circuit $C_j\defeq\prod_{s:s\in\{0\}\cup [2^k-1],s_j=1} CNOT^s_j$ consists of $2^{k-1}$ CNOT gates. All these CNOT gates have distinct control qubits and the same target qubit. According to Lemma \ref{lem:stair_circuit}, $C_j$ can be parallelized to depth $O(k)$ {and size $O(2^k)$} without ancillary qubits. Step 2a consists of $C_0,C_1,\ldots,C_{k-1}$ and the target qubits of CNOT gates in $C_t$ and $C_\ell$ are different if $t\neq \ell$. In order to reduce the depth of Step 2a, we make $k$ copies of register $A$ in ancillary qubits (register $C$). Then $C_0,C_1,\ldots,C_{k-1}$ can be implemented simultaneously using the $k$ copies of register $A$. Thus $\prod_{j=0}^{k-1} C_j$ can be implemented simultaneously in depth $O(k)$ {and size $O(k2^k)$}. Finally, we reset register $C$ back to 0 in {depth $O(k)$ and size $O(2^k)$}. Step 1 is summarized as follows: \begin{align} &\ket{e_s}\ket{0^k}\ket{0^{k2^k}} \\ \to & \ket{e_s}\ket{0^k}\underbrace{\ket{e_s}\cdots\ket{e_s}}_{k ~\text{copies~of}~\ket{e_s}} & (\text{Lemma \ref{lem:copy1}, depth $O(\log k)$, {size $O(k2^k)$}})\\ \to & \ket{e_s}\ket{s} \ket{e_s}\cdots\ket{e_s} &(\text{Lemma \ref{lem:stair_circuit}, depth $O(k)$, {size $O(k2^k)$}})\\ \to & \ket{e_s}\ket{s} \ket{0^{k2^k}} & (\text{Lemma \ref{lem:copy1}, depth $O(\log k)$, {size $O(k2^k)$}}) \end{align} The total depth and size of Step 2a are $O(\log k)+O(k)+O(\log k)=O(k)$ { and $O(k2^k)$}, respectively. \item Step 2b: Secondly, we implement Step 2b by a quantum circuit with $k2^k$ ancillary qubits of depth $O(k)$ {and size $O(k2^k)$}. Define an $(k+1)$-qubit quantum gate $\text{Tof}_s$ acting on register $B$ and the $s$-th qubit in register $A$: \begin{align} \text{Tof}_s\ket{x}\|j\rangle=\ket{x\oplus \delta_{sj}}\ket{j}, \text{~~for~}s, j\in\{0\}\cup [2^k-1], \end{align} where $\|x\rangle$ is the $s$-th qubit in register $A$ and $\ket{j}$ is in register $B$. That is, conditioned on the state in register $B$ is $\ket{s}$, $\text{Tof}_s$ flips the $s$-th qubit in register A. Step 2b is just $\prod_{s=0}^{2^k-1} \text{Tof}_s$. Any $\text{Tof}_s$ can be implemented by $[I\otimes(\otimes_{j=0}^{k-1} X^{s_j})]\text{Tof}_{2^k-1}[I\otimes(\otimes_{j=0}^{k-1} X^{s_j})]$, where Toffoli gate $\text{Tof}_{2^k-1}$ can be implemented in depth $O(k)$ and size $O(k)$ without ancillary qubits (Lemma \ref{lem:tof}). Therefore $\text{Tof}_s$ can be realized by an $O(k)$-depth { and $O(k)$-size} quantum circuit without ancillary qubits. To realize $\text{Tof}_0,\ldots,\text{Tof}_{2^k-1}$ simultaneously, we make $2^k$ copies of register $B$ in register $C$ depth $O(k)$ { and $O(k2^k)$}. Then $\text{Tof}_0,\ldots,\text{Tof}_{2^k-1}$ can be implemented simultaneously by using these copies. Finally, we reset register $C$ back to 0 in depth $O(k)$ { and size $O(k2^k)$}. Step 2b can be summarized as follows: \begin{align} &\ket{e_s}\ket{s}\ket{0^{k2^k}} \nonumber\\ \to & \ket{e_s}\ket{s}\underbrace{\ket{s}\cdots\ket{s}}_{2^k \text{~copies~of~}\ket{s}} & (\text{Lemma \ref{lem:copy1}, depth $O(k)$, size~} {O(k2^k)} )\\ \to & \ket{0^{2^k}}\ket{s} \ket{s}\cdots\ket{s} &(\text{depth $O(k)$, size }{ O(k2^k)})\\ \to & \ket{0^{2^k}}\ket{s} \ket{0^{k2^k}} & (\text{Lemma \ref{lem:copy1}, depth $O(k)$, size } { O(k2^k)}) \end{align} The total depth and size of Step 2b are $O(k)+O(k)+O(k)=O(k)$ { and $O(k2^k)$}, respectively. \item Step 2c: Thirdly, for Step 2c, we swap the first $k$ qubits in register $A$ and register $B$ by $k$ swap gates. Hence, Step 2c can be implemented in depth $O(1)$ { and size $O(k)$}. \end{itemize} Thirdly, for Step 3, we swap the first $n$ qubit in register $A$ and register $B$ in depth $O(1)$ and size $O(n)$ without ancillary qubits by swap gates. \end{proof} \section{Circuit depth for general unitary synthesis} \label{sec:US_depth} In Eq. \eqref{matrix:Vn}, we called a $(2\times 2)$-block diagonal matrix $V_j$ a $j$-qubit UCG. In the view of a circuit, this is a multiple controlled gate where the target qubit is the last one and the conditions are on the first $j-1$ qubits. But this target qubit can actually be any one, and all the implementations in Section \ref{sec:QSP_withancilla} and Section \ref{sec:QSP_withoutancilla} still apply. Let $V^n_k$ denote an $n$-qubit UCG whose index of target qubit is $k$. By repeatedly applying cosine-sine decomposition, one can factor an arbitrary unitary matrix $U$ into a sequence of UCGs as follows \cite{mottonen2005decompositions}. Recall that the Ruler function $\zeta(n)$ is defined as $\zeta(n)=\max\{k:2^{k-1}\|n\}$. \begin{lemma}\label{lem:CSD} Any $n$-qubit unitary matrix $U\in\mathbb{C}^{2^n\times 2^n}$ can be decomposed as $U=V^n_{n}(0) \cdot \prod_{i=1}^{2^{n-1}-1} V^n_{n-\zeta(i)}(i) \cdot V^n_{n}(2^{n-1}),$ where different $i$ in $V_k^n(i)$ denote different forms of $n$-qubit UCGs despite the same target qubit $k$. \end{lemma} The proofs of Theorem \ref{thm:unitary} and Corollary \ref{coro:unitary} in Section \ref{sec:extensions} are shown as follows. {\noindent\bf Theorem \ref{thm:unitary}} \emph{Any unitary matrix $U\in\mathbb{C}^{2^n\times 2^n} $ can be implemented by a quantum circuit of depth $O\big(n2^n+\frac{4^n}{m+n}\big)$ {and size $O(4^n)$} with $m \le 2^n$ ancillary qubits. } \begin{proof} Based on Eq. \eqref{eq:UCG}, Lemma \ref{lem:DU_with_ancillary} and Lemma \ref{lem:DU_without_ancillary}, given $m \le 2^n$ ancillary qubits, any $n$-qubit UCG $V_k^n(i)$ can be implemented by a circuit of {size $O(2^n)$ and }depth $O\big(n+\frac{2^n}{n+m}\big)$. Since Lemma \ref{lem:CSD} shows that any $U$ can be decomposed into $O(2^n)$ many $n$-qubit UCGs, a circuit can simply implement them sequentially to realize $U$, yielding a circuit of {size $O(2^n)\cdot O(2^n)=O(4^n)$ and} depth $O(2^n)\cdot O\big(n+\frac{2^n}{m+n}\big)=O\big(n2^n+\frac{4^n}{m+n}\big)$. \end{proof} {\noindent \bf Corollary \ref{coro:unitary}} \emph{The minimum circuit depth $D_{\textsc{Unitary}}(n,m)$ for an arbitrary $n$-qubit unitary with $m$ ancillary qubits satisfies \[ \begin{cases} D_{\textsc{Unitary}}(n,m) = \Theta\big(\frac{4^n}{m+n}\big), & \text{if } m=O(2^n/n), \\ D_{\textsc{Unitary}}(n,m) \in \left[ \Omega(n), O(n2^n)\right], & \text{if }m = \omega(2^n/n). \end{cases} \]} \begin{proof} The lower bound for the number of CNOT gates is $\Omega(4^n)$ by a similar argument. The only difference is that with $m$ ancillary qubits, the number of single-qubit gates right before the output is at most $n+m$ instead of $n$. We thus have $4k+3(n+m) \ge 4^n-1$. When $m = O\big(2^n/n\big)$, this still gives $k = \Omega(4^n)$. Since each layer can have at most $(m+n)/2$ CNOT gates, we know that it needs at least $\Omega\big(\frac{4^n}{m+n}\big)$ depth for any circuit of $n$ input qubits and $m$ ancillary qubits. Theorem \ref{thm:lowerbound_QSP} shows a lower bound $\Omega(n)$ for an $n$-qubit QSP circuit, which is a special case of a circuit for an $n$-qubit unitary matrix. Therefore, we get a depth lower bound of $\Omega\big(\max\big\{n,\frac{4^n}{n+m}\big\}\big)$ for an $n$-qubit unitary matrix. Putting the depth upper bound $O\big(n2^n+\frac{4^n}{n+m}\big)$ and lower bound $\Omega\big(\max\big\{n,\frac{4^n}{n+m}\big\}\big)$ together, we complete the proof. \end{proof} \section{Decomposition with Clifford + T gate set} \label{sec:clifford_decomposition} \begin{lemma}[\cite{ross2015optimal}]\label{lem:sk} For $\epsilon>0$, any rotation $R_z(\theta)\in\mathbb{C}^{2\times 2}$ can be $\epsilon$-approximated by a quantum circuit consisting of $O(\log (1/\epsilon))$ many $H$ and $T$ gates, without ancillary qubits. \end{lemma} Based on this lemma, it is not hard to extend our results on the exact implementation of diagonal unitary matrix $\Lambda_n$ to its approximate version (Lemma \ref{lem:diagonal_approx}). This in turn gives approximate realization of UCGs $V_n$ (Lemma \ref{lem:UCG_apprx}), state preparation (Corollary \ref{coro:psi_apprx}), and unitary operation (Corollary \ref{coro:unitary_apprx}). \begin{lemma}\label{lem:diagonal_approx} Any $n$-qubit diagonal unitary matrix $\Lambda_n$ can be $\epsilon$-approximated by a quantum circuit of depth $O\left(n+\frac{2^n\log(2^n/\epsilon)}{m+n}\right)$, using the Clifford+T gate set with $m$ ancillary qubits. \end{lemma} \begin{proof} In Section \ref{sec:QSP_withancilla} and Section \ref{sec:QSP_withoutancilla}, our quantum circuits for $\Lambda_n$ consist of only CNOT gates and $2^n-1$ rotation gates $R(\alpha)$ for $\alpha\in\mathbb{R}$. By Lemma \ref{lem:sk}, every $R(\alpha)$ can be $(\epsilon/2^n)$-approximated by $O(\log(2^n/\epsilon))$ $H$ and $T$ gates up to a global phase. The overall accuracy of the circuit can then be seen from a union bound. If $m\in[2n,2^n]$, the total circuit depth of $\Lambda_n$ is $O(\log m) +O(\log m + \log(2^n/\epsilon))+O(\log m)+O(2^n\log(2^n/\epsilon)/m)=O(\log m + 2^n\log(2^n/\epsilon)/m)=O(n+2^n\log(2^n/\epsilon)/m).$ If $m\le 2n$, the depth of the circuit implementing the diagonal unitary matrix is $O(2^n\log(2^n/\epsilon)/n)$. Putting these two results together, we complete the proof. \end{proof} \begin{corollary}\label{lem:UCG_apprx} Any $n$-qubit UCG $V_n$ can be $\epsilon$-approximated by a quantum circuit using the Clifford+T gate set, of depth $O\left(n+\frac{2^n\log(2^n/\epsilon)}{m+n}\right)$ with $m$ ancillary qubits. \end{corollary} \begin{proof} From Eq. \eqref{eq:UCG}, we can see that any $n$-qubit UCG can be decomposed into three $n$-qubit diagonal unitary matrices, two $S$ gates and two $H$ gates. By Lemma \ref{lem:diagonal_approx}, every $\Lambda_n$ can be $(\epsilon/3)$-approximated by a quantum circuit of depth $O\left(n+\frac{2^n\log(3\cdot2^n/\epsilon)}{m+n}\right)$ with $m$ ancillary qubits. Hence, $V_n$ can be $\epsilon$-approximated by a circuit of depth $3\times O\left(n+\frac{2^n\log(3\cdot2^n/\epsilon)}{m+n}\right)+2+2=O\left(n+\frac{2^n\log(2^n/\epsilon)}{m+n}\right)$. \end{proof} The approximate implementations of state preparation and general unitary matrix are shown in Corollary \ref{coro:psi_apprx} and Corollary \ref{coro:unitary_apprx}. \noindent {\bf Corollary \ref{coro:psi_apprx}} \emph{ For any $n$-qubit target state $\ket{\psi_v}$ and $\epsilon>0$, one can prepare a state $\ket{\psi'_v}$ which is $\epsilon$-close to $\ket{\psi_v}$ in $\ell_2$-distance, by a quantum circuit consisting of $\{CNOT,H,S,T\}$ gates of depth {\[\left\{\begin{array}{ll} O\left(\frac{2^n\log(2^n/\epsilon)}{m+n}\right) & \text{if~}m=O(2^n/(n\log n)),\\ O(n\log n\log(2^n/\epsilon))& \text{if~}m\in [\omega(2^n/(n\log n),o(2^n)],\\ O(n \log(2^n/\epsilon))& \text{if~}m=\Omega(2^n),\\ \end{array}\right.\]} where $m$ is the number of ancillary qubits. } \begin{proof} {According to Theorem \ref{thm:QSP_anci}, any $n$-qubit quantum state $\ket{\psi_v}$ can be prepared by a quantum circuit $QSP$, using single-gates and CNOT gates, of size $c\cdot 2^n$ for some constant $c>0$ and depth \[d=\left\{\begin{array}{ll} O\left(\frac{2^n}{m+n}\right) & \text{if~}m=O(2^n/(n\log n)),\\ O(n\log n)& \text{if~}m\in [\omega(2^n/(n\log n),o(2^n)],\\ O(n)& \text{if~}m=\Omega(2^n),\\ \end{array}\right.\] Based on Eq. \eqref{eq:single_qubit_gate} and Lemma \ref{lem:sk}, every single-qubit gate can be $(\epsilon/c2^n)$-approximated by a quantum circuit consisting of $O(\log((2^n)/\epsilon))$ Clifford+T gates. Approximate all single-qubit gates in this way. The depth of the new quantum circuit $QSP'$ consisting of Clifford+T gates is $d\times O(\log(2^n/\epsilon))=O(d\log(2^n/\epsilon))$. And circuit $QSP'$ prepare a quantum state $\ket{\psi'_v}$ satisfying \[\\|\ket{\psi_v}-\ket{\psi_v'}\\|_2=\\|(QSP-QSP')\ket{0}^{\otimes n}\\|_2\le \frac{\epsilon}{c2^n} \times c2^n= \epsilon.\]} \end{proof} \noindent{\bf Corollary \ref{coro:unitary_apprx}} \emph{Any $n$-qubit general unitary matrix can be implemented by a quantum circuit, using the $\{CNOT,H,S,T\}$ gate set, of depth $O\left(n2^n+\frac{4^n\log(4^n/\epsilon)}{m+n}\right)$ with $m$ ancillary qubits.} \begin{proof} Lemma \ref{lem:CSD} shows that any $U\in\mathbb{C}^{2^n\times 2^n}$ can be decomposed into $2^n-1$ $n$-qubit UCGs. According to Lemma \ref{lem:UCG_apprx}, an $n$-qubit UCG $V_n$ can be $\epsilon/(2^{n}-1)$-approximated by a quantum circuit $V_n'$ consisting of $\{CNOT,H,S,T\}$ gates in depth $O\left(n+\frac{2^n\log(4^n/\epsilon)}{m+n}\right)$ with $m$ ancillary qubits. Hence, $U$ can be $\epsilon$-approximated by a quantum circuit in depth $O\left(n+\frac{2^n\log(4^n/\epsilon)}{m+n}\right)\times (2^n+1)=O\left(n2^n+\frac{4^n\log(4^n/\epsilon)}{m+n}\right)$. \end{proof} \section{Sparse quantum state preparation} \label{sec:sparse_QSP} A vector $v=(v_0,v_1,\ldots,v_{2^n-1})\in \mathbb{C}^{2^n}$ is said to be $s$-sparse if there are at most $s$ nonzero elements in $v$. In this section, we consider how to efficiently prepare $s$-sparse states. \begin{lemma}\label{lem:CNOT_sum} The unitary transformation defined by \begin{equation}\label{eq:CNOT_sum} \ket{x_1x_2\cdots x_n}\ket{t}\to \ket{x_1x_2\cdots x_n}\ket{\bigoplus_{i=1}^n x_i \oplus t} \end{equation} $\forall x_1,\ldots,x_n,t\in\mbox{$\{0,1\}$}$, can be implemented in depth $O(\log(n))$. \end{lemma} \begin{proof} The circuit implementation of Eq. \eqref{eq:CNOT_sum} is shown in Figure \ref{fig:CNOT_sum}. \begin{figure} \caption{An $O(\log(n))$-depth circuit implementation of Eq. \eqref{eq:CNOT_sum}. } \label{fig:CNOT_sum} \end{figure} \end{proof} \begin{lemma}\label{lem:perm_sparse} Suppose that we are given two sets $S_1\subseteq\mbox{$\{0,1\}$}^{n_1}$ and $S_2\subseteq\{0,1\}^{n_2}$, both of size $s$, and also given a bijection $P:S_1\to S_2$. Then a unitary transformation satisfying \begin{equation}\label{eq:perm_sparse} \ket{x}\ket{0^{n_2}}\to \ket{x}\ket{P(x)}, \forall x\in S_1 \end{equation} can be implemented by a quantum circuit of depth $O\big(n_2\log(m)+\frac{(n_1+\log(m))sn_1n_2}{m}\big)$, using $m~(\ge 2n_1)$ ancillary qubits. \end{lemma} \begin{proof} Define an $(n_1+1)$-qubit unitary ${\sf Tof}^x_a\ket{z}\ket{b}:=\ket{z}\ket{a\cdot \delta_{xz}\oplus b }$ for any $x,z\in\mbox{$\{0,1\}$}^{n_1}$ and $a,b\in \mbox{$\{0,1\}$}$, where $\delta_{xz}=1$ if $x=z$ and $\delta_{xz}=0$ if $x\neq z$. Unitary ${\sf Tof}^x_a$ can be implemented by an $n_1$-qubit Toffoli gate and at most $2n_1$ Pauli $X$ gates. According to Lemma \ref{lem:tof}, ${\sf Tof}^x_a$ can be implemented in depth $O(n_1)$. Let $\{x(1),x(2),\ldots, x(s)\}$ be the elements of $S_1$, and $P(x(i))=y(i)$. Denote $y(i)=y_1(i)y_2(i)\cdots y_{n_2}(i)$, and we will compute $y(i)$ bit by bit. We start by computing $y_1(i)$, the first bit of $y(i)$, and then all other bits can be computed similarly. More precisely, we aim to implement the following unitary transformation $U_1$: \[\ket{x(i)}\ket{0}\xrightarrow{U_1} \ket{x(i)}\ket{y_1(i)},\forall i\in[s].\] Unitary $U_1$ can be implemented in 3 steps. \begin{itemize} \item Step 1: make $\frac{m}{2n_1}$ copies of $\ket{x(i)}$ using $\frac{m}{2}$ ancillary qubits, i.e., implement the following unitary transformation \[\ket{x(i)}\ket{0}\ket{0^{m/2}}\to \ket{x(i)}\ket{0}\ket{x(i)x(i)\cdots x(i)}, \forall i \in[s]. \] This step can be implemented in depth $O(\log(m))$ by Lemma \ref{lem:copy1}. \item Step 2: implement the following unitary transformation using $\frac{m}{2n_1}$ ancillary qubits \begin{align} & \ket{x(i)}\ket{0}\ket{x(i)x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}} \to \ket{x(i)}\ket{y_1(i)}\ket{x(i)x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}},\forall i \in[s]. \end{align} We divide set $S_1$ into $\frac{s}{m/2n_1}=\frac{2sn_1}{m}$ parts $S_1^{(1)},\ldots,S_1^{(\frac{2sn_1}{m})}$. The size of each part is $\frac{m}{2n_1}$. For the first part $S_1^{(1)}:=\{x(i): i\in[\frac{m}{2n_1}]\}$, we implement the following unitary transformation: \begin{equation}\label{eq:part1} \ket{x(i)}\ket{0}\ket{x(i)x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}} \to \left\{\begin{array}{ll} \ket{x(i)}\ket{y_1(i)}\ket{x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}}, & \text{if~}x(i) \in S_1^{(1)} \\ \ket{x(i)}\ket{0}\ket{x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}}, & \text{otherwise.} \end{array}\right. \end{equation} Namely, we compute $y_1(i)$ for those $x(i)\in S_1^{(1)}$ (and keep other $x(i)$ untouched). This can be achieved by the following three sub-steps. In step 2.1, we apply unitaries ${\sf Tof}^{x(1)}_{y_1(1)}, { \sf Tof}^{x(2)}_{y_1(2)},\ldots, { \sf Tof}^{x(\frac{m}{2n_1})}_{y_1(\frac{m}{2n_1})}$. For each ${ \sf Tof}^{x(i)}_{y_1(i)}$, the control qubits are the $i$-th copy of $x(i)$ and the target qubit is the $i$-th qubit of ancillary qubits (those in $\ket{0^{\frac{m}{2n_1}}}$). Therefore, they can be realized in parallel by a circuit of depth $O(n_1)$. The effect of this step 2.1 is \begin{align} & \ket{x(i)}\ket{0}\ket{x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}} \to \ket{x(i)}\ket{0}\ket{x(i)\cdots x(i)}\ket{0^{i-1}y_1(i)0^{\frac{m}{2n_1}-i}},\quad\forall x(i) \in S_1^{(1)}. \end{align} In step 2.2, we implement the following unitary transformation \begin{align} & \ket{x(i)}\ket{0}\ket{x(i)x(i)\cdots x(i)}\ket{0^{i-1}y_1(i)0^{\frac{m}{2n_1}-i}} \to \ket{x(i)}\ket{y_1(i)}\ket{x(i)x(i)\cdots x(i)}\ket{0^{i-1}y_1(i)0^{\frac{m}{2n_1}-i}}, &\forall x(i) \in S_1^{(1)}. \end{align} in depth $O(\log(m/(2n_1)))$ according to Lemma \ref{lem:CNOT_sum}. In step 2.3, we restore the ancillary qubits by an inverse circuit of step 2.1 of depth $O(n_1)$, i.e., we realize the following unitary transformation: \begin{align} & \ket{x(i)}\ket{y_1(i)}\ket{x(i)x(i)\cdots x(i)}\ket{0^{i-1}y_1(i)0^{\frac{m}{2n_1}-i}}\to \ket{x(i)}\ket{y_1(i)}\ket{x(i)x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}}, \forall x(i) \in S_1^{(1)}. \end{align} We can verify that step 2.1, 2.2 and 2.3 realize unitary in Eq. \eqref{eq:part1} by circuit of depth $O(n_1+\log(m/n_1))=O(n_1+\log(m))$. By the same discussion, for every $j\in[\frac{2sn_1}{m}]$, \begin{equation}\label{eq:partj} \ket{x(i)}\ket{0}\ket{x(i)x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}} \to \ket{x(i)}\ket{y_1(i)}\ket{x(i)x(i)\cdots x(i)}\ket{0^{\frac{m}{2n_1}}},\forall x(i) \in S_1^{(j)}. \end{equation} can be implemented in depth $O(n_1+\log(m))$. In summary, step 2 can be implemented in depth $O\big(n_1+\log(m)\big)\cdot\frac{2sn_1}{m}=O\big(\frac{(n_1+\log(m))sn_1}{m}\big)$. \item Step 3: restore the ancillary qubits by an inverse circuit of step 1, of depth $O(\log(m))$. \end{itemize} In summary, unitary $U_1$ can be implement in depth $2\cdot O(\log(m))+O\big(\frac{(n_1+\log(m))sn_1}{m}\big)=O\big(\log(m)+\frac{(n_1+\log(m))sn_1}{m}\big)$. By similar discussion of $U_1$, for all $j\in[n_2]$, the following unitary $U_j$ \[\ket{x(i)}\ket{0}\xrightarrow{U_j} \ket{x(i)}\ket{y_j(i)},\forall i\in[s].\] can be also implemented in depth $O\big(\log(m)+\frac{(n_1+\log(m))sn_1}{m}\big)$. By applying $U_1,U_2,\ldots, U_{n_2}$, we compute all $n_2$ bits of $y(i)$. This implements unitary in Eq. \eqref{eq:perm_sparse} and the total depth is $n_2\cdot O\big(\log(m)+\frac{(n_1+\log(m))sn_1}{m}\big)=O\big(n_2\log(m)+\frac{(n_1+\log(m))sn_1n_2}{m}\big)$. \end{proof} \begin{lemma}[\cite{malvetti2021quantum}]\label{lem:sparse_QSP_without_ancilla} Any $n$-qubit $s$-sparse quantum state can be prepared by a quantum circuit of size $O(ns)$, using no ancillary qubits. \end{lemma} \begin{theorem} For any $m\ge 0$, any $n$-qubit $s$-sparse quantum state can be prepared by a quantum circuit of depth $O(n\log(sn)+\frac{s\log(s)n^2}{n+m})$, using $m$ ancillary qubits. \end{theorem} \begin{proof} For simplicity, we assume $\log (s)$ is an integer and $n'=\log(s)$. Define $S_1 = \mbox{$\{0,1\}$}^{n'}$, and $S_2$ to be all $s$-sparse strings in $\mbox{$\{0,1\}$}^n$. Let $P$ be any bijection from $S_1$ to $S_2$. Any $n$-qubit $s$-sparse quantum state can be represented as $\ket{\psi}=\sum\limits_{x\in\{0,1\}^{n'}}v_x\ket{P(x)}$. \begin{itemize} \item {\bf Case 1: $m\ge 3n$.} First, we prepare an $n'$-qubit quantum state \[\ket{\psi'}=\sum\limits_{x\in\{0,1\}^{n'}}v_x\ket{x},\] which can be implemented in depth $O((n')^2+\frac{2^{n'}}{n'+m})=O(\log^2(s)+\frac{s}{\log(s)+m})$ using $m$ ancillary qubtis by Lemma \ref{lem:partial_result}. Second, we implement the following unitary transformations and then we complete preparing the state $\ket{\psi}$. \begin{align} \ket{x0^{n-n'}}\ket{0^n}\to &\ket{x0^{n-n'}} \ket{P(x)} \label{eq:perm_base1}\\ \to & \ket{0^n} \ket{P(x)} \label{eq:perm_base2}\\ \to & \ket{P(x)}\ket{0^n},\forall x\in\mbox{$\{0,1\}$}^{n'}. \label{eq:perm_base3} \end{align} Based on Lemma \ref{lem:perm_sparse}, using $m$ ancillary qubits, Eq. \eqref{eq:perm_base1} can be implemented in depth $O(n\log(m)+\frac{(\log(s)+\log(m))s\log(s) n}{m})$. Eq. \eqref{eq:perm_base2} can be viewed as a similar process by Lemma \ref{lem:perm_sparse}, though we need to swap $S_1$ and $S_2$ and reverse the direction of $P$. This can be implemented in depth $O(\log(s)\log(m)+\frac{(n+\log(m))s\log(s) n}{m})$, respectively. Eq. \eqref{eq:perm_base3} can be implemented by $n$ swap gates in depth 1. Therefore, if $m\le \frac{s\log(s)n}{\log(s)+\log(n)}$, the total depth is $O(n\log(m)+\frac{(\log(s)+\log(m))s\log(s) n}{m})+O(\log(s)\log(m)+\frac{(n+\log(m))s\log(s) n}{m})+1=O(n\log(sn)+\frac{s\log(s)n^2}{m})$. If $m> \frac{s\log(s)n}{\log(s)+\log(n)}$, we use only $\frac{s\log(s)n}{\log(s)+\log(n)}$ ancillary qubits and the total depth is $O(n\log(sn))$. Combining these two cases, the total depth is $O(n\log(sn)+\frac{s\log(s)n^2}{m})$. \item {\bf Case 2: $m < 3n$.} If $m<3n$, we do not use ancillary qubits. According to Lemma \ref{lem:sparse_QSP_without_ancilla}, $\ket{\psi}$ can be prepared in depth $O(ns)$. \end{itemize} Combining the above two cases, the circuit depth for $\ket{\psi}$ is $O(n\log(sn)+\frac{s\log(s)n^2}{n+m})$. \end{proof} \end{document}	arXiv
What is the remainder when the sum $1 + 7 + 13 + 19 + \cdots + 253 + 259$ is divided by $6$? First of all, we see that every term has a remainder of $1$ when divided by $6.$ Now, we just need to find how many terms there are. The nth term can be given by the expression $6n - 5.$ Therefore, we set $259 = 6n - 5$ to find $n = 44,$ thus there are $44$ terms in our sum. Thus, the remainder of the sum is the same as the remainder of $44$ when divided by $6,$ which is $\boxed{2}.$	Math Dataset
Lessons from April 6, 2009 L'Aquila earthquake to enhance microzoning studies in near-field urban areas Giovanna Vessia1, Mario Luigi Rainone1, Angelo De Santis2 & Giuliano D'Elia1 This study focuses on two weak points of the present procedure to carry out microzoning study in near-field areas: (1) the Ground Motion Prediction Equations (GMPEs), commonly used in the reference seismic hazard (RSH) assessment; (2) the ambient noise measurements to define the natural frequency of the near surface soils and the bedrock depth. The limitations of these approaches will be discussed throughout the paper based on the worldwide and Italian experiences performed after the 2009 L'Aquila earthquake and then confirmed by the most recent 2012 Emilia Romagna earthquake and the 2016–17 Central Italy seismic sequence. The critical issues faced are (A) the high variability of peak ground acceleration (PGA) values within the first 20–30 km far from the source which are not robustly interpolated by the GMPEs, (B) at the level 1 microzoning activity, the soil seismic response under strong motion shaking is characterized by microtremors' horizontal to vertical spectral ratios (HVSR) according to Nakamura's method. This latter technique is commonly applied not being fully compliant with the rules fixed by European scientists in 2004, after a 3-year project named Site EffectS assessment using AMbient Excitations (SESAME). Hereinafter, some "best practices" from recent Italian and International experiences of seismic hazard estimation and microzonation studies are reported in order to put forward two proposals: (a) to formulate site-specific GMPEs in near-field areas in terms of PGA and (b) to record microtremor measurements following accurately the SESAME advice in order to get robust and repeatable HVSR values and to limit their use to those geological contests that are actually horizontally layered. On April 6, 2009, at 1:32 a.m. (local time) an Mw 6.3 earthquake with shallow hypocentral depth (8.3 km) hit the city of L'Aquila and several municipalities within the Aterno Valley. This earthquake can be considered one of the most mournful seismic event in Italy since 1980 although its magnitude was moderately-high: 308 fatalities and 60.000 people displaced (data source http://www.protezionecivile.it) and estimated damages for 1894 M€ (data source http://www.ngdc.noaa.gov). These numbers showed how dangerous can be an unexpected seismic event in urbanized territories where no preventive actions have been addressed to reduce seismic risk. Hence, meanwhile, some actions were implementing in the post-earthquake time such as updating the reference Italian hazard map (by the Decree OPCM n. 3519 on 28 April 2006) and drawing microzoning maps to be used in the reconstruction stage, two major earthquakes struck the Emilia Romagna Region (in the Northern part of Italy), causing 27 deaths and widespread damage. The first, with Mw 6.1, occurred on 20 May at 04:03 local time (02:03 UTC) and was located at about 36 km north of the city of Bologna. Then, a second major earthquake (Mw 5.9) occurred on 29 May 2012, in the same area, causing widespread damage, particularly to buildings already weakened by the 20 May earthquake. Later on, the 2016–17 Central Italy Earthquake Sequence occurred, consisting of several moderately-high magnitude earthquakes between Mw 5.5 and Mw 6.5, from Aug 24, 2016, to Jan 18, 2017, each centered in a different but close location and with its own sequences of aftershocks, spanning several months. The seismic sequence killed about 300 people and injured the other 396. Worldwide, several other strong earthquakes (eg. 1998 Northridge earthquake, 2004 Parkfield earthquake, 2010 Canterbury and 2011 and 2017 Christchurch earthquakes, 2018 Sulawesi earthquake) produced devastating effects in the same time span. All these events show the need to carry out efficient microzoning studies to plan vulnerability reductions of urban structures and promoting the resilience of the human communities in seismic territories. After the 2009 L'Aquila earthquake, the microzoning studies have been introduced in Italy by law and the Guidelines for Seismic Microzonation (ICMS 2008) have been issued to accomplish these studies according to the most updated international scientific findings. These guidelines provide the local administrators with an efficient tool for seismic microzoning study to predicting the subsoil behavior under seismic shaking. Unfortunately, the ICMS does not give special recommendations for urbanized near field areas (NFAs). The microzoning activity concerning urbanized territories as suggest by ICMS (2008) is made up of four steps: Estimating the reference seismic hazard to provide the input peak horizontal ground acceleration (PGA) at each point on the national territory and the normalized response spectrum at each site. Dynamic characterization of soil deposits overlaying the seismic bedrock at each urban center in order to draw the microzoning maps (MM) at three main knowledge levels. The Level 1 MM consists of geo-lithological maps of the surficial deposits that show typical successions and the amplified frequency map drawn through the measurements of microtremors elaborated by horizontal to vertical spectral ratio HVSR Nakamura's technique. Nakamura's method (1989), the horizontal to vertical noise components are calculated to derive the natural frequency of surficial soft deposits and their thickness. The Level 2/3 MM consists of drawing maps after performing the numerical analyses of (a) seismic local amplification factors in terms of acceleration FA and velocity FV; (b) liquefaction potential LP and (c) permanent displacements due to seismically induced slope instability. After 10 years of training the ICMS and the related methods, it is now the time to start analyzing some arisen weak points. Starting from the large data acquired worldwide on recent strong motion earthquakes, the experiences developed in seismic hazard assessment, and the site-specific seismic response characterization carried out by the writing authors after the 2009 L'Aquila earthquake, the aforementioned weak points (related to some aspects of the steps 1 and 3) are hereinafter discussed and some proposals are made to improve the efficiency of the microzoning studies especially in NFAs. In this paper, after a brief background section on the procedures to accomplish the reference seismic hazard assessment (background section), the methods to calculate the Ground Motion Prediction Equations (GMPEs) and the HVSR (Nakamura's method) are briefly recalled in section 2. Then in section 3, the results from observations of recorded peak ground acceleration (PGA) and pseudo-spectral acceleration (PSA) values within the NFAs from the L'Aquila earthquake and other worldwide strong earthquakes have been discussed. In addition, some applications of Nakamura's procedure to characterize the natural frequency of the sites throughout the Aterno Valley have been discussed. Finally, in the conclusion section, some relevant points drawn from the discussed microzoning experiences have been highlighted to improve the efficiency of the microzonation studies in urban centers especially located in NFAs. Background on seismic hazard assessment Several theoretical and experimental studies performed worldwide in the last 50 years (see Kramer 1996 and the reference herein), highlighted that seismic shaking intensity is due to the magnitude of the earthquake generated at the source, to the travel paths of the seismic waves from the source to the buried or outcropping bedrock (that is called reference seismic hazard RSH) and the additional phenomena of local amplification or de-amplification take place where soil deposits overlay the rocky bedrock, named local seismic response LSR (Paolucci 2002; Vessia and Venisti 2011; Vessia et al. 2011; Vessia and Russo 2013; Vessia et al. 2013, 2017; Boncio et al. 2018, among others). The RSH maps drawn worldwide on national territories do not take into account the results of LSR studies. The pioneering work by Signanini et al. (1983) after the 1979 Friuli earthquake confirmed the observations on the ground: local seismic effects could enlarge the referenced hazard at a site by 2–3 times in terms of MCS scale Intensity but also in PGA values owing to the local morphological and stratigraphic settings. Such RSL is particularly evident in near field areas, from then on named NFAs. The NFAs have been defined among others by Boore (2014a) as the Fault Damage Zones. These areas cannot be uniquely identified depending on the source rupture mechanisms, the surficial soil deposits and the multiple calculation methods used for measuring the distance between the seismic stations and the source. Especially in these areas, about the first 30 km aside the source, spot-like amplifications are the common amplification pattern captured through the Maximum Intensity Felt maps. These maps estimate the differentiated damages suffered by buildings and urban structures by means of the Macroseismic Intensity scale (e.g. Mercalli-Cancani-Sieberg MCS scale, European Macroseismic scale EMS, Modified Mercalli Intensity MMI scale). One of the Maximum Intensity Felt maps on the Italian territory was drawn by Boschi et al. (1995). They took into account the seismic events that occurred from 1 to 1992 AD with a minimum intensity felt of VI MCS. This latter value is the one commonly used to highlight those areas where seismic events caused relevant damages to dwellings and infrastructures, ranging from severe damages to collapse. Boschi et al. (1995) map is reported in Fig. 1: it showed IX-X MCS at L'Aquila district based on historical earthquakes that are in agreement with the seismic intensity map drawn by Galli and Camassi (2009) after the mainshock of 2009 L'Aquila earthquake. This map is also in very good agreement with other recent earthquakes such as the 2012 Emilia Romagna and 2016–17 Central Italy earthquakes (Fig. 1). Moreover, Boschi et al. (1995), Midorikawa (2002) and more recently, Paolini et al. (2012) proposed a direct use of the Maximum Felt Intensity maps to highlight those areas where the reference seismic hazard is largely increased by the local amplificated responses of soil deposits, that is the NFAs. Italian Maximum Intensity felt map (After Boschi et al. 1995, modified) with the areas of two recent Italian earthquake sequences considered in the present work The most used method to perform the reference seismic hazard assessment has been conceived in the late '60s. It is Cornell's method (1968) that was implemented into a numerical code by Mc Guire (1978). Cornell (1968) introduced the Probabilistic Seismic Hazard Assessment (PSHA) method to carry out the reference seismic hazard at a site considering the contribution of the seismic source and the travel path of the seismic waves by considering the uncertainties related to these estimations. This method consists of four steps (Kramer 1996), as illustrated in Fig. 2: STEP 1. To identify the seismogenic sources as single faults and faulting regions in terms of magnitude amplitude generated at different time spans. The probabilistic approach to such a characterization needs to know the rate of the earthquake at different magnitudes at the site and the spatial distribution of the fault segment or the source volume that can be activated. STEP 2_1. To calculate the seismic rate in a region the Gutenberg-Richter law is used, where a and b coefficients are drawn by interpolating numerous data from a database of seismic events (instrumental and non-instrumental) available for a limited number of source areas and affected by the lack of completeness distortions. The earthquake occurrence probability is estimated by means of a Poisson distribution over time that is independent of the time span of the last strong seismic event. STEP 2_2. To calculate the spatial distribution of the seismic events alongside a fault zone is a character difficult to get known and the spatial distribution of earthquake sources within a seismogenic area is commonly assumed uniformly distributed. STEP 3. To define the ground motion prediction equations GMPEs that enable to predict, at different magnitude ranges, the decrease with the distance from the seismic source of the strong motion parameter assumed to be representative of the earthquake at a site. STEP 4. To calculate the probability of exceedance of a target shaking value of the considered ground motion parameter, i.e. PGA, in a time span at a chosen site, due to the contribution of different seismogenic sources. Cornell's probabilistic seismic hazard assessment (PSHA) explained in four steps. The blue house represents the site under study The previous 4 steps attempt to take into account several sources of uncertainties, such as the limited knowledge about the fault activity, the qualitative and documental estimations of the past earthquake effects at the sites, the lack of completeness of the seismic catalogs (meaning that the database of the seismic events is populated by several data related to both low and high magnitude ones) and the dependency among the recorded strong seismic events. In addition, the distortions in Gutenberg-Richter law, defined for different regions worldwide and the uncertainties related to the GMPEs can generate underestimations of the seismic shaking parameters (i.e. PGA) at specific sites where local seismic effects are relevant (Paolucci 2002; Vessia and Venisti 2011; Vessia and Russo 2013; Vessia et al. 2013; Yagoub 2015; Miyajima et al. 2019; Lanzo et al. 2019; among others). Logic trees are commonly used to take into account different formulations of GMPEs and several Gutenberg-Richter rates of magnitude occurrence (Kramer 1996). Molina et al. (2001) pointed out that the PSHA has its strength in the systematic parameterization of seismicity and the way in which also epistemic uncertainties are carried out through the computations into the final results. Recently, alternative approaches to PSHA calculation have been suggested, such as through extensions of the zonation method (Frankel 1995; Frankel et al. 1996, 2000; Perkins 2000) where multiple source zones, parameter smoothing and quantification of geology and active faults have been successfully applied. The Frankel et al. (1996) method applied a Gaussian function to smooth a-values (within Gutenberg-Richter law) from each zone, thereby being a forerunner for the later zonation-free approaches of Woo (1996). This latter approach tries to amalgamate statistical consistency with the empirical knowledge of the earthquake catalogue (with its fractal character) into the computation of seismic hazard. Furthermore, Jackson and Kagan (1999) developed a non-parametric method with a continuous rate-density function (computed from earthquake catalogues) used in earthquake forecasting. Nonetheless, all these methods need a function to propagate the strong motion parameter values from the source to the site under study. To this end, the GMPEs are built by interpolating large databases of seismic records (related to specific geographical and tectonic environments worldwide), taking into account the contributions of the earthquake magnitude M and the distance to the seismic source R, according to the following form (Kramer 1996): $$ \mathit{\ln}\;(Y)=f\left(M,R,{S}_i\right) $$ where Y is the ground motion parameter, commonly the peak ground horizontal acceleration PGA or the spectral acceleration SA at fixed period; Si is related to the source and site: they are the refinement terms due to the enlargement of the seismic databases and the possibility of drawing specific regional GMPEs. The uncertainty of GMPEs in near field areas Several examples of GMPEs are provided in literature (Kramer 1996 among others) while a recent throughout review of several possible formulations of GMPEs used in the USA can be found at the Pacific Earthquake Engineering Research center PEER website http://peer.berkeley.edu/publications/peer_reports_complete.html. In Italy, the GMPEs are built based on the PGAs drawn from the Italian shape wave database of strong motion events (Faccioli 2012; Bindi et al. 2011, 2014; Cauzzi et al. 2014). Commonly, the PGA values represent the strong motion parameter used in microzoning studies but the related GMPEs are highly uncertain especially in the first tens of kilometers as shown in Fig. 3a (Faccioli 2012) and Fig. 4a (Boore 2013). According to Boore (2013, 2014a), fault zone records show significant variability in amplitude and polarization of PGA, SA especially at low periods (as shown in Fig. 4a) and magnitude saturation beyond Mw 6, although the causes of this variability are not easy to be unraveled. The main drawback of the GMPEs is the weakness of their predictivity at a short distance from the seismic source due to two main issues affecting the NFAs worldwide: a few seismic stations installed; highly scattered measures of strong motion parameters, especially in terms of accelerations (i.e. PGA, SA, etc) (Fig. 2, step 3), that do not show any decreasing trend with distance. a Ground Motion Prediction Equation (GMPE) of PGA versus the minimum source-to-site distance (After Faccioli 2012, modified): a band of uncertainty (grey) of GMPEs proposed by Faccioli et al. (2010). b 2008 Boore and Atkinson ground motion prediction equation (BA08 GMPE) of PGA based on data collected in United States for Magnitude 7.3 Mw, strike-slip fault type and VS30equal to 255 m/s: solid line is the mean equation; dashed lines represent the confidence interval at one standard deviation (After Boore 2013, modified) a Measures of PSA during the Parkfield earthquake 2004 (6 Mw) are reported near the active fault at the measure seismic stations (After Boore 2014b, modified); b Onna sector of Aterno River Valley: the records are for an aftershock of 3.2 Ml The PGA spatial uncertainties have been observed after several recent strong earthquakes, such as 2009 L'Aquila earthquake (Lanzo et al. 2010; Bergamaschi et al. 2011; Di Giulio et al. 2011), 2011 Christchurch and 2010 Darfield earthquakes in New Zealand (Bradley and Cubrinovski 2011) (Fig. 5), 2012 Emilia Romagna earthquake in Italy, 1994 Northridge earthquake in USA (Boore 2004) and 2013 Fivizzano earthquake (Fig. 5). In the case of the 2009 L'Aquila earthquake (Fig. 4b), the areal distribution of PGAs around the source seems to be highly random although they show that the most dramatic increase occurs where thick soft sediments are met over rigid bedrocks or where bedrock basin shapes can be recognized. This latter traps the seismic waves and caused longer duration accelerograms with increased amplitudes at short and moderate periods (lower than 2 s) (Rainone et al. 2013). Horizontal and vertical PGA values recorded within the first 20 km epicenter distance during 1) 22 February 2011 Christchurch earthquake (6.3 Mw) (square) and 2) 21 June 2013 Fivizzano earthquake (5.1 Mw) (triangle) The GMPEs based on PGAs tend to saturate for large earthquakes as the distance from the fault rupture to the observation point decreases. Boore (2014b) showed that the PGA parameter is a poor measure of the ground-motion intensity due to its non-unique correspondence to the frequency and acceleration content of the shaking waves (Fig. 4), especially at high frequencies. Furthermore, Bradley and Cubrinovski (2011) and Boore (2004) stated that the influence on the amplitude and shape response by local surface geology and geometrical conditions is noted to be much more relevant than the forward directivity and the source-site path on spectral accelerations in near field areas and for periods shorter than 3 s. Thus, the GMPEs of PGAs within the NFAs are highly uncertain and cumbersome to be predicted even when fitted on single seismic events as shown in Fig. 5 (Bradley and Cubrinovski 2011; Faccioli 2012; Boore 2014b). Faccioli (2012) evidenced 100% of the coefficient of variation about the mean trend of PGA GMPE versus source-to-site distance (Fig. 3a). This GMPE was built based on the ITACA 2010 database (Luzi et al. 2008, http://itaca.mi.ingv.it/ItacaNet_30/#/home) that collects Italian strong motion shape waves. It is worth noticing that within the first tens of kilometers from the source, these data indicate that the GMPEs are not accurate in predicting the PGA values. To avoid the pitfalls in GMPEs based on peak parameters, integral ground motion parameters have been proposed in the literature (Kempton and Stewart 2006; Abrahamson and Silva 2008; Campbell and Bozorgnia 2012), such as Arias intensity (AI) and cumulative absolute velocity (CAV). In addition, Hollenback et al. (2015) and Stewart et al. (2015) formulated new generation GMPEs based on median ground-motion models as part of the Next Generation Attenuation for Central and Eastern North America project. They provided a set of adjustments to median GMPEs that are necessary to incorporate the source depth effects and the rupture distances in the range from 0 to 1500 Km. Moreover, the preceding authors suggest a distinct expression for the GMPE at short-distance to the source (by 10 km), that is: $$ lnGMPE={c}_1+{c}_2\mathit{\ln}{\left({R}_{RUP}+h\right)}^{1/2} $$ where RRUP is the rupture distance, that is the closest distance to the earthquake rupture plane (km); c1 and c2 are the regression coefficients and h is a "fictitious depth" used for ground-motion saturation at close distances. Ambient noise measures elaborated by means of the Nakamura horizontal to vertical ratio HVSR In 1989 Nakamura proposed to use the ambient noise measurements to derive a seismic property of a site, that is the frequency range of amplification, through the spectral ratio of horizontal H and vertical V ambient vibration (microtremors) components of the recorded signals. If the site does not amplify, the ratio H/V is equal to 1. The Nakamura method shows the advantage to solve the troublesome issue to find out a reference site. In fact, it considers the vertical component as the one that is not modified by the site where horizontal subsoil layers are set and SH seismic waves represent the ambient noise signal content in a quite site (far from urban or industrial areas). This latter is the reference signal whereas the horizontal component is the only one that can be affected by the amplifying properties of the soils. As a matter of fact, Nakamura assumed that: locally random distributed sources of microtremors generate not directional signals almost made up of shear horizontal or Rayleigh waves; the microtremors are confined in the surficial layers because the subsoil is made up of soft layered sediments overlaying a rigid seismic bedrock. A relevant implication of the Nakamura method is that the peaks of the ratio H/V are related to the presence of high acoustic impedance contrast at the depth h that can be derived by the following expression: $$ h=\frac{V_S}{4\bullet {f}_0} $$ where VS is the mean value of the measured shear wave velocity profile and fo is the amplified frequency measured by means of the noise measurement. The fundamental rules to perform a correct ambient noise recording was provided by the European research project named SESAME (Bard and the WG 2004) that analyzed the possible drawbacks of the simple model introduced by the Nakamura method and issued guidelines that offer important recommendations regarding the places where the method can be successfully used in urbanized areas. The given recommendations are based on a rather strict set of criteria, that are essentially composed of (1) experimental conditions and (2) criteria for gaining reliable results (Table 1). Table 1 A summary of recommendations from SESAME guidelines (Bard and the WG 2004) As can be seen from Table 1, the recommendations are focused on the weather conditions that influence the quality of the noise measurements and they highlight the need to record at distance from structures, trees, slopes because all these items affect the records. Unfortunately, it is not possible to quantify the minimum distance from the structure where the influence is negligible, as this distance depends on too many external factors (structure type, wind strength, soil type, etc.). Furthermore, related to the measurement spacing, SESAME guidelines suggest to never use a single measurement point to derive f0 value, make at least three measurement points. This latter advice is often disregarded. An interesting alternative way to apply the HVSR method is to investigate the "heavy tails" of its statistical distribution that is like that of a critical system (Signanini and De Santis 2012). This is likely indicative of the strong non-linear properties of rocks forming the uppermost crust resulting in a power-law trend. However, the Nakamura technique has been introduced in microzoning studies at Level 1 in Italy (ICMS 2008) to draw the natural frequency map of urban sites but limitations to suitable sites have not been prescribed. Although the Nakamura method seems to be simple, lost cost and short time consuming, the suitable sites where it can be applied are few, especially in urban centers. This type of indirect investigation method is not applicable in complex geological contexts (e.g. buried inclined fold settings) and is not easily handled for the difficulties in reproducing the same measurements under variable site conditions and noise sources and acquisitions performed by different operators even at the same site. Rainone et al. (2018) undertook a thorough study on the effectiveness of HVSR in predicting amplification frequencies at two Italian urban areas characterized by different subsoil setting and noise distribution. Results from this study show that HVSR works well only where horizontally layered sediments overlay a rigid bedrock: these conditions are the most relevant and the most influential on the predictivity of the actual fo measured values. A new proposal for GMPEs in near field areas Recently, a study to formulate ad hoc GMPEs for PGAs within NFAs of the Central Italy Apennine sector has been performed. Only horizontal PGA values measured from seismic stations set on A and A* soil category (Vs ≥ 800 m/s), generated by seismic events with Mw ranging between 5.0 and 6.5 and normal fault mechanism, have been extracted from ITACA shape wave database (Luzi et al. 2008). The selected events cover the period from 1997 to 2017 and consider 25 seismic events from three strong seismic sequences generated by normal faults: the 1997 Umbria-Marche, 2009 L'Aquila, 2016–2017 Central Italy. The studied source area is a quadrant whose edges' coordinates are (43.5°, 12.3°) and (42.2°, 13.6°) in decimal degrees. The PGA measures within the first 35 km from the seismic source have been taken into account. The hypocentral distance has been used to define the source to site distance. Two GMPEs within the first 35 km have been drawn for two ranges of moment magnitude: 5 ≤ Mw1 < 5.5 and 5.5 ≤ Mw2 ≤ 6.5. These ranges represent the injurious magnitudes of the Italian moderately-high magnitude earthquakes (Fig. 6). As can be noted from Fig. 6 the PGA values seem not to be highly different in the two magnitude ranges and they do not show a clear trend with the hypocentral distance. Thus, these two datasets have been kept distinct and a box and whisker plot has been used to calculate their medians, quartiles, and interquartile distance. Datasets of PGA values recorded at NFAs in the Central Italy Apennine Sector from 1997 to 2017 divided into two ranges of Mw: a 5 ≤ Mw1 < 5.5; b 5.5 ≤ Mw2 ≤ 6.5 Figure 7a, b show the two datasets with a different number of bins of hypocentral distance: it is due to the circumstance that for higher magnitudes (Fig. 7b) the seismic stations within the first 10 km are that few that cannot be considered a distinct bin. Thus, through Fig. 7a, b the outliers are evidenced and eliminated. Then the 95th percentiles of the PGAs within each bin of the two datasets have been calculated. The mean value of the preceding percentiles has been considered as the representative constant value of the first 30 km of the hypocentral distance: 0.27 g for 5 ≤ Mw1 < 5.5 and 0.37 g for 5.5 ≤ Mw2 ≤ 6.5. It is worthy to be noted that this proposal is related to the PGA values at the rigid ground to be used in microzonation studies at the sites located in the Central Italy Apennine sector within the first 30 km hypocentral distance and in the two ranges of magnitudes of moderately high earthquakes. The disaggregation pairs at each site within NFAs can be determined according to the Ingv study issued at the website: esse1.mi.ingv.it. Then, to select the reference PGA can be used the abovementioned method and the two values found by this study. Further studies must be accomplished to characterize the PGA of different seismic regions within Italian territory. The same proposed approach or several other proposals can be conceived and applied worldwide within the NFAs taking into account that the surficial soil response there is not dependent on the distance from the source but it is much more dependent on the non-linearity of the soil response combined with the complex geological conditions that cannot be easily modelled. Box and whistler plots of the two datasets of PGA values recorded at NFAs in the Central Italy Apennine Sector from 1997 to 2017 divided into two ranges of Mw: a 5 ≤ Mw1 < 5.5; b 5.5 ≤ Mw1 ≤ 6.5. The bins of hypocentral distance are: a) 1(0–9.95 km), 2(10–19.95 km), 3(20–29.95 km); b) 1(10–19.95 km), 2(20–29.95 km). The void circles are the outliers identified by the box and whistler method HVSR measurements addressed in the Aterno Valley The seismic characterization of surface geology by means of microtremors was introduced by ICMS (2008) and it was then applied in the aftermath of the 2009 L'Aquila earthquake. Many research groups started to record ambient vibrations and process them through Nakamura's method ignoring, in details, the surface geology of each testing point. At Villa Sant'Angelo and Tussillo sites (falling into the Macroarea 6 of the Aterno Valley named L'Aquila crater), we performed several microtremors acquisitions at one station through two devices: Tromino and DAQLink III. The following acquisition parameters have been used: (1) time windows longer than 30′; (2) the sampling frequency higher than 125 Hz; (3) the sampling time lower than 8 ms. Furthermore, Fast Fourier Transform FFT has been used to calculate the ratio H/V. Finally, the spectral smoothing has been performed by means of the Konno-Ohmachi smoothing window. The HVSR values have been calculated for each sub-windows of 20s, then the mean and the standard deviation of all ratios have been calculated and plotted. Further details on the technical aspects of the acquisitions by both devices can be found in Vessia et al. (2016). Figure 8a shows the HVSR measurements acquired at two neighboring points in Tussillo center, where geological characters were similar, by two research groups: T1 (the writing authors) and M5 (the Italian Department of Civil Protection DPC). These acquisitions have been done by the Tromino equipment. As can be noted, the two plots are different: T1 evidences peaks at 2.5 Hz and 8 Hz; on the contrary, the M5 shows the main peak at 2 Hz and minor peaks at 10-20 Hz, 40 Hz and 55 Hz. In the presence of these peaks, the operator would select the most representative one: of course, this selection is highly subjective although the SESAME rules suggest to take into account the highest peaks, such as 2 Hz in both cases (T1 and M5) and disregard the peaks higher than 20 Hz. a H/V measurements at two neighbor points at Tussillo center: T1 (this study) and M5 (DPC). b H/V measurements performed by the Tromino at T5 (this study) and at S4 (DPC) under similar ground conditions On the contrary, Fig. 8b shows the HVSRs measured at two nearby points on a different type of ground type compared with the previous points: T5 (the writing authors) and S4 (DPC group). In this latter case, the two plots show an evident peak at 2 Hz although the peak amplitude is double at T5 with respect to S4. This difference could be due to the presence of disregarded Love waves that do not have vertical components contributing to the amplification of the horizontal components. Figure 9a, b compare HVSR measured at NE of T5, at Villa Sant'Angelo historical center. In this case, the signals are recorded by two devices used by us: the Tromino and the DAQLink devices. As can be seen, they show similar peaks although no unique peak values can be drawn from each HVSR. In this case, the operator choices can affect the results in terms of the natural frequency of the site. However, the SESAME rule of three acquisitions at 3 different points to assess HVSR could be useful to get to a robust assessment of fo. Noise measurements at Villa Sant'Angelo center by a the Tromino device; b the DAQLink device Another weak point in the calculation of the amplified frequency f0 is the systematic differences in calculated amplified frequencies coming from the noise measurements, that induce very small deformation in soil deposits and f0 drawn from the weak motion tails of the strong motion signals generated by the strong motion events at the site that cause medium to large deformation levels in the ground. From our experience, the f0 drawn from noise measurements are rarely confirmed by amplified frequencies from actual records. From the field experience, the amplified frequencies f0 have been measured through the HVSR function from the noise tracks acquired at the Tussillo site, at the point T1 and the weak motion tails of a seismic event recorded on July 7, 2009, at 10:15 local time at the same site. Figure 10 shows the HVSR functions. It is easy to notice that the calculated f0 related to the noise (Fig. 10a) is shown at 8 Hz whereas and the one related to the weak motion (Fig. 10b) is calculated at 2 Hz: the peak frequencies do not match and the weak tail, after the strong motion excitation shows a lower amplification frequency due to the non-linear response of the soil compared to the peak related to the noise measures. These results have been confirmed by other comparisons accomplished in several other places within the Aterno Valley (Vessia et al. 2016). HVSR function measured at the T1 site, at Tussillo site (see Fig. 8) from: a the noise measurements; b the weak motion tail of a seismic event recorded on 7 July 2009, at 10:15 local time Finally, from the abovementioned experiences, three issues can be pointed out: (1) Nakamura's method often provides more than one peak corresponding to different natural frequencies; (2) the peaks are heavily affected by many external factors, especially in urban areas, that are not easy to be disregarded by filtering the measurements; (3) the peaks in HVSR functions are not commonly related to both weak and strong motion amplified frequencies. Thus, the use of the noise measurements in microzoning activities to derive the bedrock depth should be discouraged especially when the geological conditions of the site are not known, such as the shear wave velocity profile of the soil deposits up to the bedrock depth. In addition, the amplified frequency of the site should be determined through more than one measurement, according to the SESAME rules, in order to check the possible differences induced by the different time of the day and weather conditions at the site. However, the amplified frequency measured at a very low deformation level is modified at medium and large deformations induced during the strong and even weak motion seismic events. Thus the calculation of the f0 from noise measures can only be used to determine the bedrock depth through Eq. (3) but the shear wave velocity profile is needed as well as the buried geological conditions to guarantee the applicability of the Nakamura method. In this paper two weak points in microzoning studies have been discussed starting from the authors' experience in microzonation in Italy, that is: (a) the lack of predictivity of GMPEs of PGA measurements in NFAs and (b) the ability of noise measurements to capture the amplified frequency at site even in a complex geological conditions. Throughout the paper, some past experiences of microzoning activity by the present authors are discussed and two proposals have neem put forward. On one hand, concerning the GMPEs of PGAs according to the reference seismic hazard assessment performed in Italy, the need for specific GMPE values in NFAs have been highlighted by several scientists. Here, the proposal of using the 95 percentile of the scattered values recorded within the first 30 km from the hypocentral distance has been provided for the Central Italy Apennine Sector. These values have been drawn from the ITACA database limited to seismic events ranging from 5 to 6.5 Mw occurred from 1997 to 2017. On the other hand, after a large experience gained in the noise measurements recorded in the Aterno Valey after the 2009 L'Aquila earthquake, it can be concluding that the noise measurements are not inherently repeatable, thus at least two or three measurements, according to the SESAME guidelines (this represents the European standard to perform ambient vibration measurements) must be requested to calculate the f0 by means of Nakamura's method. Although noise measurements can provide relevant differences in amplified frequencies according to the operator or the weather conditions, following the SESAME rules guarantee both technical standards to an unregulated geophysical technique that relies on a simple buried geological model of horizontally layered soil deposits. This model, when not applicable, can make the HVSR function from noise measurements totally misleading. Thus, even at level 1 microzoning studies, the use of direct and indirect measures is needed in order to confirm the layered planar setting of the subsurface geo-lithological model and measuring shear wave velocity profiles to enable a robust prediction of the bedrock depth by means of the amplified frequency f0 estimation. Finally, when the noise measurements are compared with the weak motion tails of actual seismic events, they show different amplified ranges of frequencies. Accordingly, it must be kept in mind that soil behavior is strain-dependent: this means that their natural frequencies at small strain levels (microtremors) differ from the ones at medium strain level (weak motions) and at high strain level (strong motions). 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Geophysical Survey of Slovenia, Ljubljana, pp 67–75 Rainone ML, D'Elia G, Vessia G, De Santis A (2018) The HVSR interpretation technique of ambient noise to seismic characterization of soils in heterogeneous geological contexts. Book of abstract of 36th general assembly of the European seismological commission, ESC2018-S29-639: 428-429, 2-7 September 2018, Valletta (Malta) ISBN: 978-88-98161-12-6 Rainone ML, Vessia G, Signanini P, Greco P, Di Benedetto S (2013) Evaluating site effects in near field conditions for microzonation purposes: The case study of L'Aquila earthquake 2009. (special issue on L'Aquila earthquake 2009). Ital Geotechnical J 47(3):48–68 Signanini P, Cucchi F, Frinzi U, Scotti A (1983) Esempio di microzonizzazione nell'area di Ragogna (Udine). Rendiconti della Soc Geol Italiana 4:645–653 Signanini P, De Santis A (2012) Power-law frequency distribution of H/V spectral ratio of seismic signals: evidence for a critical crust. Earth Planets Space 64:49–54 Stewart JP, Boore DM, Seyhan E, Atkinson GM, M.EERI Atkinson GM (2016) NGA-West2 Equations for Predicting Vertical-Component PGA, PGV, and 5%-Damped PSA from Shallow Crustal Earthquakes show less. Earthq Spectr 32(2):1005–1031. https://doi.org/10.1193/072114EQS116M Vessia G, Parise M, Tromba G (2013) A strategy to address the task of seismic micro-zoning in landslide-prone areas. Adv Geosci 1:1–27. https://doi.org/10.5194/adgeo-35-23-2013 Vessia G, Pisano L, Tromba G, Parise M (2017) Seismically induced slope instability maps validated at an urban scale by site numerical simulations. Bull Eng Geol Envir 76(2):457–476 Vessia G, Rainone ML, Signanini P (2016) Springer book title: "earthquakes and their impacts on society", Eds. S. D'Amico, 2016 chapter 9 title: "working strategies for addressing microzoning studies in urban areas: lessons from 2009 L'Aquila earthquake", 233–290, Springer international publishing Switzerland Vessia G, Russo S, Lo Presti D (2011) A new proposal for the evaluation of the amplification coefficient due to valley effects in the simplified local seismic response analyses. Ital Geotechnical J 4:51–77 Vessia G, Russo S (2013) Relevant features of the valley seismic response: the case study of Tuscan northern Apennine sector. Bull Earthq Eng 11(5):1633–1660 Vessia G, Venisti N (2011) Liquefaction damage potential for seismic hazard evaluation in urbanized areas. Soil Dyn Earthq Eng 31:1094–1105 Woo G (1996) Kernel estimation methods for seismic hazard area source modeling. Bull Seism Soc Am 86:1–10 Yagoub MM (2015) Spatio-temporal and hazard mapping of earthquake in UAE (1984–2012): remote sensing and GIS application. Geoenviron Disasters 2:13. https://doi.org/10.1186/s40677-015-0020-y The authors are grateful to Prof. Patrizio Signanini who gave precious suggestions during the paper preparation and inspired several discussions on the effects of inefficient microzonation studies on people's daily life quality in near field seismic areas. Department of Engineering and Geology, University "G.d'Annunzio" of Chieti-Pescara, Via dei Vestini 31, 66013, Chieti, Scalo (CH), Italy Giovanna Vessia, Mario Luigi Rainone & Giuliano D'Elia Istituto Nazionale di Geofisica e Vulcanologia (INGV), Via di Vigna Murata, 605, 00143, Rome, Italy Angelo De Santis Giovanna Vessia Mario Luigi Rainone Giuliano D'Elia To this study, GV was responsible for the state of the art research (Introduction and Methods section) and the Results section related to 3.1 paragraph; MLR and ADS were responsible for the geophysical campaign and the Results section discussing results; GE acquired some ambient noise records and was responsible for the figures. All the authors contributed together to the writing of the manuscript. The author(s) read and approved the final manuscript. Correspondence to Giovanna Vessia. Vessia, G., Rainone, M., De Santis, A. et al. Lessons from April 6, 2009 L'Aquila earthquake to enhance microzoning studies in near-field urban areas. Geoenviron Disasters 7, 11 (2020). https://doi.org/10.1186/s40677-020-00147-x Reference seismic hazard map Seismic microzoning study HVSR Nakamura's method GMPEs	CommonCrawl
LMS Journal of Computation and Mathematics Search within journal Search within society MathJax MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org. Volume 16 - October 2013 A comprehensive perturbation theorem for estimating magnitudes of roots of polynomials M. Pakdemirli, G. Sarı Published online by Cambridge University Press: 14 February 2013, pp. 1-8 A comprehensive new perturbation theorem is posed and proven to estimate the magnitudes of roots of polynomials. The theorem successfully determines the magnitudes of roots for arbitrary degree of polynomial equations with no restrictions on the coefficients. In the previous papers 'Pakdemirli and Elmas, Appl. Math. Comput. 216 (2010) 1645–1651' and 'Pakdemirli and Yurtsever, Appl. Math. Comput. 188 (2007) 2025–2028', the given theorems were valid only for some restricted coefficients. The given theorem in this work is a generalization and unification of the past theorems and valid for arbitrary coefficients. Numerical applications of the theorem are presented as examples. It is shown that the theorem produces good estimates for the magnitudes of roots of polynomial equations of arbitrary order and unrestricted coefficients. Computing zeta functions of nondegenerate hypersurfaces with few monomials Steven Sperber, John Voight Published online by Cambridge University Press: 14 February 2013, pp. 9-44 Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces, and also can be used to compute the L-function of an exponential sum. On the continuity of multivariate Lagrange interpolation at natural lattices MSC 2010: Numerical approximation and computational geometry MSC 2010: Approximations and expansions J.-P. Calvi, V. M. Phung Published online by Cambridge University Press: 10 April 2013, pp. 45-60 We give a natural geometric condition that ensures that sequences of interpolation polynomials (of fixed degree) of sufficiently differentiable functions with respect to the natural lattices introduced by Chung and Yao converge to a Taylor polynomial. Complex B-splines and Hurwitz zeta functions B. Forster, R. Garunkštis, P. Massopust, J. Steuding We characterize nonempty open subsets of the complex plane where the sum $\zeta (s, \alpha )+ {e}^{\pm i\pi s} \hspace{0.167em} \zeta (s, 1- \alpha )$ of Hurwitz zeta functions has no zeros in $s$ for all $0\leq \alpha \leq 1$ . This problem is motivated by the construction of fundamental cardinal splines of complex order $s$ . Bounds and algorithms for the $K$ -Bessel function of imaginary order MSC 2010: Functional analytic methods in summability MSC 2010: Stability theory Andrew R. Booker, Andreas Strömbergsson, Holger Then Published online by Cambridge University Press: 10 April 2013, pp. 78-108 Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function ${K}_{ir} (x)$ of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of ${K}_{ir} (x)$ and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of $r$ . Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of ${K}_{ir} (x)$ . Seven new champion linear codes MSC 2010: Polytopes and polyhedra Gavin Brown, Alexander M. Kasprzyk We exhibit seven linear codes exceeding the current best known minimum distance $d$ for their dimension $k$ and block length $n$ . Each code is defined over ${ \mathbb{F} }_{8} $ , and their invariants $[n, k, d] $ are given by $[49, 13, 27] $ , $[49, 14, 26] $ , $[49, 16, 24] $ , $[49, 17, 23] $ , $[49, 19, 21] $ , $[49, 25, 16] $ and $[49, 26, 15] $ . Our method includes an exhaustive search of all monomial evaluation codes generated by points in the $[0, 5] \times [0, 5] $ lattice square. Václav Šimerka: quadratic forms and factorization MSC 2010: History of mathematics and mathematicians F. Lemmermeyer Published online by Cambridge University Press: 01 May 2013, pp. 118-129 In this article we show that the Czech mathematician Václav Šimerka discovered the factorization of $\frac{1}{9} (1{0}^{17} - 1)$ using a method based on the class group of binary quadratic forms more than 120 years before Shanks and Schnorr developed similar algorithms. Šimerka also gave the first examples of what later became known as Carmichael numbers. Minimal solvable nonic fields John W. Jones For each solvable Galois group which appears in degree $9$ and each allowable signature, we find polynomials which define the fields of minimum absolute discriminant. Complexity of OM factorizations of polynomials over local fields Jens-Dietrich Bauch, Enric Nart, Hayden D. Stainsby Let $k$ be a locally compact complete field with respect to a discrete valuation $v$ . Let $ \mathcal{O} $ be the valuation ring, $\mathfrak{m}$ the maximal ideal and $F(x)\in \mathcal{O} [x] $ a monic separable polynomial of degree $n$ . Let $\delta = v(\mathrm{Disc} (F))$ . The Montes algorithm computes an OM factorization of $F$ . The single-factor lifting algorithm derives from this data a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$ , for a prescribed precision $\nu $ . In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O({n}^{2+ \epsilon } + {n}^{1+ \epsilon } {\delta }^{2+ \epsilon } + {n}^{2} {\nu }^{1+ \epsilon } )$ word operations for the complexity of the computation of a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$ , assuming that the residue field of $k$ is small. Single-class genera of positive integral lattices David Lorch, Markus Kirschmer Published online by Cambridge University Press: 01 August 2013, pp. 172-186 We give an enumeration of all positive definite primitive $ \mathbb{Z} $ -lattices in dimension $n\geq 3$ whose genus consists of a single isometry class. This is achieved by using bounds obtained from the Smith–Minkowski–Siegel mass formula to computationally construct the square-free determinant lattices with this property, and then repeatedly calculating pre-images under a mapping first introduced by G. L. Watson. We hereby complete the classification of single-class genera in dimensions 4 and 5 and correct some mistakes in Watson's classifications in other dimensions. A list of all single-class primitive $ \mathbb{Z} $ -lattices has been compiled and incorporated into the Catalogue of Lattices. On level one cuspidal Bianchi modular forms MSC 2010: Homology and homotopy of topological groups and related structures Alexander D. Rahm, Mehmet Haluk Şengün In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms. Computing Borcherds products Dominic Gehre, Judith Kreuzer, Martin Raum We present an algorithm for computing Borcherds products, which has polynomial runtime. It deals efficiently with the bounds on Fourier expansion indices originating in Weyl chambers. Naive multiplication has exponential runtime due to inefficient handling of these bounds. An implementation of the new algorithm shows that it is also much faster in practice. Explicit application of Waldspurger's theorem Soma Purkait For a given cusp form $\phi $ of even integral weight satisfying certain hypotheses, Waldspurger's theorem relates the critical value of the $\mathrm{L} $ -function of the $n\mathrm{th} $ quadratic twist of $\phi $ to the $n\mathrm{th} $ coefficient of a certain modular form of half-integral weight. Waldspurger's recipes for these modular forms of half-integral weight are far from being explicit. In particular, they are expressed in the language of automorphic representations and Hecke characters. We translate these recipes into congruence conditions involving easily computable values of Dirichlet characters. We illustrate the practicality of our 'simplified Waldspurger' by giving several examples. Computing level one Hecke eigensystems (mod $p$ ) Craig Citro, Alexandru Ghitza We describe an algorithm for enumerating the set of level one systems of Hecke eigenvalues arising from modular forms (mod $p$ ). Supplementary materials are available with this article. Modular subgroups, dessins d'enfants and elliptic K3 surfaces Yang-Hui He, John McKay, James Read We consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck's dessins d'enfants. In the particular case of the index 36 subgroups, the corresponding Calabi–Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over ${ \mathbb{P} }^{1} $ as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative. Weak approximation of stochastic differential delay equations for bounded measurable function MSC 2010: Stochastic analysis MSC 2010: Functional-differential and differential-difference equations Hua Zhang In this paper we study the weak approximation problem of $E[\phi (x(T))] $ by $E[\phi (y(T))] $ , where $x(T)$ is the solution of a stochastic differential delay equation and $y(T)$ is defined by the Euler scheme. For $\phi \in { C}_{b}^{3} $ , Buckwar, Kuske, Mohammed and Shardlow ('Weak convergence of the Euler scheme for stochastic differential delay equations', LMS J. Comput. Math. 11 (2008) 60–69) have shown that the Euler scheme has weak order of convergence $1$ . Here we prove that the same results hold when $\phi $ is only assumed to be measurable and bounded under an additional non-degeneracy condition. Higher torsion in the Abelianization of the full Bianchi groups Alexander D. Rahm Denote by $ \mathbb{Q} ( \sqrt{- m} )$ , with $m$ a square-free positive integer, an imaginary quadratic number field, and by ${ \mathcal{O} }_{- m} $ its ring of integers. The Bianchi groups are the groups ${\mathrm{SL} }_{2} ({ \mathcal{O} }_{- m} )$ . In the literature, so far there have been no examples of $p$ -torsion in the integral homology of the full Bianchi groups, for $p$ a prime greater than the order of elements of finite order in the Bianchi group, which is at most 6. However, extending the scope of the computations, we can observe examples of torsion in the integral homology of the quotient space, at prime numbers as high as for instance $p= 80\hspace{0.167em} 737$ at the discriminant $- 1747$ . Corrigendum: On the use of a discrete form of the Itô formula in the article 'Almost sure asymptotic stability analysis of the $\theta $ -Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations' MSC 2010: Stochastic systems and control MSC 2010: Probabilistic methods, simulation and stochastic differential equations MSC 2010: Equations and systems with randomness Gregory Berkolaiko, Evelyn Buckwar, Cónall Kelly, Alexandra Rodkina In the original article [LMS J. Comput. Math. 15 (2012) 71–83], the authors use a discrete form of the Itô formula, developed by Appleby, Berkolaiko and Rodkina [Stochastics 81 (2009) no. 2, 99–127], to show that the almost sure asymptotic stability of a particular two-dimensional test system is preserved when the discretisation step size is small. In this Corrigendum, we identify an implicit assumption in the original proof of the discrete Itô formula that, left unaddressed, would preclude its application to the test system of interest. We resolve this problem by reproving the relevant part of the discrete Itô formula in such a way that confirms its applicability to our test equation. Thus, we reaffirm the main results and conclusions of the original article. A vanishing theorem and symbolic powers of planar point ideals M. Dumnicki, T. Szemberg, H. Tutaj-Gasińska The purpose of this paper is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We then apply this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals. Approximation by a composition of Chlodowsky operators and Százs–Durrmeyer operators on weighted spaces Aydın İzgi In this paper we deal with the operators $$\begin{eqnarray}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { \mathop{\sum }\nolimits}_{k= 0}^{n} {p}_{n, k} \biggl(\frac{x}{{b}_{n} } \biggr)\int \nolimits \nolimits_{0}^{\infty } {s}_{n, k} \biggl(\frac{t}{{b}_{n} } \biggr)f(t)\hspace{0.167em} dt, \quad 0\leq x\leq {b}_{n}\end{eqnarray}$$ and study some basic properties of these operators where ${p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} , (0\leq k\leq n), u\in [0, 1] $ and ${s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )$ . Also, we establish the order of approximation by using weighted modulus of continuity.	CommonCrawl
\begin{definition}[Definition:Coordinate System/Coordinate Function] Let $\left \langle {a_n} \right \rangle$ be a coordinate system of a unitary $R$-module $G$. For each $x \in G$ let $x_1, x_2, \ldots, x_n$ be the coordinates of $x$ relative to $\left \langle {a_n} \right \rangle$. Then for $i = 1, \ldots, n$ the mapping $f_i : G \to R$ defined by $f_i \left({x}\right) = x_i$ is called the '''$i$-th coordinate function on $G$ relative to $\left \langle {a_n} \right \rangle$'''. Category:Definitions/Coordinate Systems \end{definition}	ProofWiki
\begin{document} \title{The prescribed Ricci curvature problem for naturally reductive metrics on non-compact simple Lie groups} \begin{abstract} We investigate the prescribed Ricci curvature problem in the class of left-invariant naturally reductive Riemannian metrics on a non-compact simple Lie group. We obtain a number of conditions for the solvability of the underlying equations and discuss several examples. \end{abstract} \section{Introduction} The study of the prescribed Ricci curvature problem is an important part of modern geometry with ties to flows, relativity and other subjects. The first wave of interest in this problem came in the 1980s; see~\cite[Chapter~5]{Bss} and~\cite[Section~6.5]{TA98}. Particularly extensive contributions were made at that time by DeTurck and his collaborators. For a discussion of the subsequent advances, including the recent progress in the framework on homogeneous spaces, see the survey~\cite{BP19}. Let $M$ be a smooth manifold. In its original interpretation, the prescribed Ricci curvature problem comes down to the equation \begin{align}\label{PRC_no_c} \Ricci (g) = T, \end{align} where the Riemannian metric $g$ on $M$ is the unknown and the (0,2)-tensor field $T$ is given. The paper~\cite{DeTurck} proved, for nondegenerate~$T$, the existence of $g$ satisfying this equation in a neighbourhood of a point on~$M$; see also~\cite{DeTGold,AP13,AP16b}. However, subsequent research into the solvability of~\eqref{PRC_no_c} on \emph{all} of $M$ revealed the need for a more nuanced interpretation of the prescribed Ricci curvature problem. Specifically, suppose $M$ is closed and $T$ is positive-definite. The results of~\cite{Hamilton84,DeTurck85,Del03} and other papers suggest that there exists at most one $c\in\mathbb R$ such that the equation \begin{align}\label{PRC_c} \Ricci (g) = cT \end{align} can be solved for $g$ on all of~$M$. This is certainly the case if $M$ is the 2-dimensional sphere and $T$ is positive-definite; see~\cite{WW70,DeTurck85,Hamilton84}. Thus, on a closed manifold, one customarily interprets the prescribed Ricci curvature problem as the question of finding $g$ and $c$ such that~\eqref{PRC_c} holds. This paradigm was originally proposed by DeTurck and Hamilton in~\cite{Hamilton84,DeTurck85}. As it turns out,~\eqref{PRC_c} arises in applications, such as the construction of the Ricci iteration; see~\cite{PR19,BPRZ19} and also~\cite[Section~3.10]{BP19}. On the other hand, if $M$ is open, it may be possible to obtain compelling existence theorems for~\eqref{PRC_no_c} without the additional constant~$c$. We refer to~\cite{Dela02,Dela18} for examples of such theorems. In recent years, the third-named author and his collaborators produced a series of results, surveyed in~\cite{BP19}, on the prescribed Ricci curvature problem in the class of homogeneous metrics. More precisely, suppose $G$ is a connected Lie group. Let $M$ be a homogeneous space with respect to~$G$. Assume that the metric $g$ and the tensor field $T$ are $G$-invariant. Then \eqref{PRC_no_c} reduces to an overdetermined system of algebraic equations, whereas~\eqref{PRC_c} reduces to a determined one. For compact $M$ and positive-semidefinite $T$, the third-named author showed in~\cite{AP16} that homogeneous metrics satisfying~\eqref{PRC_c} are, up to scaling, critical points of the scalar curvature functional subject to the constraint~$\tr_gT=1$. This observation led to the discovery of several sufficient conditions for the solvability of~\eqref{PRC_c} in~\cite{AP16,MGAP18,AP20}. It parallels the well-known variational approach to the Einstein equation; see, e.g.,~\cite[\S1]{WZ86}. In the case where $M$ is compact but $T$ is not positive-semidefinite, the question of solvability of~\eqref{PRC_c} remains largely open. We hope that the present paper will stimulate its investigation; see Remark~\ref{rem_2c_Tim}. As for the prescribed Ricci curvature problem for homogeneous metrics on \emph{non-compact} spaces, progress has been scarce so far. Buttsworth conducted in~\cite{TB19} a comprehensive study of this problem on unimodular three-dimensional Lie groups. In most of the cases he considered, there is at most one constant $c\in\mathbb R$ such that a metric $g$ satisfying~\eqref{PRC_c} exists. Several questions related to, but distinct from, the solvability of~\eqref{PRC_no_c} and~\eqref{PRC_c} on non-compact Lie groups have been studied by Milnor, Kowalski--Nikcevic, Eberlein, Kremlev--Nikonorov, Ha--Lee, Pina--dos Santos, the first-named author in collaboration with Lafuente (forthcoming work), and others. For a discussion of those results and a collection of references, see~\cite[Sections~2 and~4.1]{BP19}. Left-invariant naturally reductive metrics on a Lie group form an important family nested between the set of all left-invariant metrics and the set of bi-invariant ones. The investigation of this family has led to several significant advances in geometry. For instance, it yielded new solutions to the Einstein equation and new insights into the spectral theory of the Laplacian on manifolds; see~\cite{DZ79,GS10,L19}. In the recent paper~\cite{APZ20}, Ziller and two of the authors studied~\eqref{PRC_c} for naturally reductive metrics on a compact Lie group using variational methods. That work exposed several interesting and previously unseen patterns of behaviour of the scalar curvature functional. For instance, one of the main theorems of~\cite{APZ20} is a necessary condition for the existence of a critical point subject to the constraint~$\tr_gT=1$. No results of this kind had appeared in the literature before. The present paper studies the prescribed Ricci curvature problem for naturally reductive metrics on a \emph{non-compact} Lie group~$G$. We assume that $G$ is simple. The more general case of semisimple $G$ seems to be much more difficult analytically---we intend to consider it elsewhere. Since $G$ is an open manifold, it is reasonable for us to view the prescribed Ricci curvature problem as the question of finding solutions to~\eqref{PRC_no_c}. On the other hand, the fact that naturally reductive metrics are homogeneous suggests that a ``better" interpretation of this problem may be given by~\eqref{PRC_c}. The present paper studies both equations. We show that~\eqref{PRC_no_c} reduces to an overdetermined algebraic system. Even so, we are able to obtain a comprehensive existence and uniqueness theorem. Equation~\eqref{PRC_c} reduces to a determined system. In order to find conditions for solvability, we characterise metrics satisfying~\eqref{PRC_c} as critical points of the scalar curvature functional subject to one of three $T$-dependent constraints. While this characterisation is similar in spirit to the one obtained for compact Lie groups in~\cite{APZ20}, it bears some conceptual distinctions and requires a different proof. We obtain existence theorems for global maxima and classify some of the other critical points. The development and application of our variational methods presents many interesting analytical challenges and provides a wealth of insight for the investigation of~\eqref{PRC_c} on compact homogeneous spaces in the case of mixed-signature~$T$ (see, e.g.,~Remark~\ref{rem_2c_Tim}). The paper is organised as follows. In Section~\ref{secnat}, we recall the characterisation, originally obtained by Gordon, of naturally reductive metrics on a non-compact simple Lie group. This characterisation underpins all of our results. In Section~\ref{secRicci}, we compute the Ricci curvature of a naturally reductive metric on~$G$. We believe that the formulas we obtain are of independent interest. Section~\ref{sec_Ricci=T} is devoted to equation~\eqref{PRC_no_c}. We produce a necessary and sufficient condition for the existence of a solution. We also establish uniqueness up to scaling. Section~\ref{sec_Ricci=cT} focuses on~\eqref{PRC_c}. We develop the variational approach to this equation and describe several types of critical points of the scalar curvature functional. At the end, we summarise the implications for the existence and the number of solutions. Section~\ref{sec_simple} examines the case where the metrics we consider are naturally reductive with respect to $G\times K$ for a simple subgroup~$K<G$. Here, we find conditions for the solvability of~\eqref{PRC_c} that are both necessary and sufficient. We also determine the precise number of solutions. Finally, Section~\ref{sec_examples} offers a series of examples. \section{Naturally reductive metrics on non-compact simple Lie groups}\label{secnat} Consider a connected non-compact simple Lie group $G$ with Lie algebra~$\ggo$. The results of~\cite[Section~5]{C85} yield a convenient characterisation of left-invariant naturally reductive metrics on~$G$. We present this characterisation in Theorem~\ref{G} below. For the basic theory of naturally reductive metrics, see~\cite[Section~1]{DZ79} and~\cite[Section~2]{C85}. In what follows, we identify every left-invariant (0,2)-tensor field on $G$ with the bilinear form it induces on~$\ggo$. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra~$\kg$. Suppose $B$ is the Killing form of~$\ggo$. Denote by $\pg$ the $B$-orthogonal complement of $\kg$ in~$\ggo$. Then \begin{align} \ggo=\pg\oplus\kg \end{align} is a Cartan decomposition. We have the inclusions \begin{align} [\kg,\kg]\subset\kg,\qquad [\kg,\pg]\subset\pg,\qquad [\pg,\pg]\subset\kg. \end{align} The Killing form $B$ is positive-definite on~$\pg$ and negative-definite on~$\kg$. Thus, \begin{align} Q =B\|_{\pg}-B\|_{\kg} \end{align} is an inner product on $\ggo$. Clearly, $Q$ is $\ad(\mathfrak k)$-invariant, and \begin{equation}\label{sym_spce_incl} Q([X,Y],Z)=-Q(X,[Y,Z]),\qquad X,Y \in \pg,~Z\in\kg. \end{equation} The quotient $G/K$ is a symmetric space. Because $G$ is simple, this space is irreducible. Consequently, the pair $(\ggo,\kg)$ must appear in Table~3 or~4 of~\cite[Section~7.H]{Bss}. Let $\kg_1,\ldots,\kg_r$ be the simple ideals of $[\kg,\kg]$. Denote by $\kg_{r+1}$ the centre of~$\kg$. Then \begin{align}\label{dec_k} \kg= \kg_1 \oplus \cdots \oplus \kg_{r+s}, \end{align} where $s=0$ if $\kg_{r+1}$ is trivial and $s=1$ otherwise. Analysing the tables in~\cite[Section~7.H]{Bss}, we conclude that $\kg_{r+1}$ is at most 1-dimensional. The direct product $G\times K$ acts on $G$ in accordance with the formula \begin{align} (x,k)\,y=xyk^{-1},\qquad x,y\in G,~k\in K. \end{align} The isotropy subgroup at the identity element of $G$ is \begin{align} \{(k,k)\in G\times K\,\|\,k\in K\}. \end{align} Denote by $\mathcal M_K$ the set of left-invariant metrics on $G$ that are naturally reductive with respect to~$G\times K$ and some decomposition of the Lie algebra of~$G\times K$. The main purpose of this paper is to study the prescribed Ricci curvature problem in~$\mathcal M_K$. Gordon showed in~\cite[Section~5]{C85} that every left-invariant naturally reductive metric on $G$ must lie in $\mathcal M_K$ for some choice of~$K$. Moreover, she obtained the following characterisation result. \begin{theorem}[Gordon]\label{G} A left-invariant metric $g$ on the simple group $G$ lies in $\mathcal M_K$ if and only if \begin{equation}\label{metric} g=\beta Q\|_{\pg}+\alpha_1 Q\|_{\kg_1}+\cdots+\alpha_{r+s}Q\|_{\kg_{r+s}} \end{equation} for some $\beta,\alpha_1,\ldots,\alpha_{r+s}>0$. \end{theorem} \begin{remark} In~\cite{C85}, Gordon studied naturally reductive metrics on non-compact homogeneous spaces, not just on Lie groups. She obtained a version of Theorem~\ref{G} in this more general framework. \end{remark} \section{The Ricci curvature}\label{secRicci} Our main objective in this section is to produce formulas for the Ricci curvature and the scalar curvature of a metric $g$ given by~(\ref{metric}). To do so, we need to introduce an array of constants, $\kappa_1,\ldots,\kappa_{r+s}$, associated with the pair~$(\ggo,\kg)$. Throughout the paper, \begin{align} n=\dim\pg, \qquad d_i=\dim\kg_i, \qquad i=1,\ldots,r+s. \end{align} As we explained above, $d_{r+1}=1$ if the centre of $\kg$ is non-trivial. Suppose $B_i$ is the Killing form of~$\kg_i$ for $i=1,\ldots,r+s$. There exists $\kappa_i\in\mathbb R$ such that \begin{align} B_i = \kappa_i B\|_{\kg_i}. \end{align} Using the assumption that $G$ is simple, one can easily check that $0<\kappa_i<1$ for $i=1,\ldots,r$. If the centre of $\kg$ is non-trivial, then $\kappa_{r+1}=0$. Next, we state a proposition that provides a way of computing $\kappa_i$ for a specific pair~$(\ggo,\kg)$ and $i=1,\ldots,r$. In what follows, superscript~$\mathbb C$ means complexification. Clearly, the algebra $\ggo^{\mathbb C}$ is semisimple. We preserve the notation $\ad$ for the adjoint representation of~$\ggo^{\mathbb C}$. Choose Cartan subalgebras $H$ in $\ggo^{\mathbb C}$ and $H_i$ in~$\kg_i^{\mathbb C}$. It is easy to verify that $\ggo^{\mathbb C}$ is a completely reducible $\kg_i^{\mathbb C}$-module under the action given by~$\ad$. This observation implies that every element of $H_i$ must be semisimple in~$\ggo^{\mathbb C}$. Consequently, we may assume that $H$ contains~$H_i$. Let $\Phi^+$ and $\Phi_i^+$ be sets of positive roots of $\ggo^{\mathbb C}$ and~$\kg_i^{\mathbb C}$. The notation $\tr$ stands for the trace of a linear operator. \begin{proposition}\label{prop_kappa} Given $i=1,\ldots,r$ and $Z\in H_i$, the constant $\kappa_i$ satisfies \begin{align} \sum_{\nu\in\Phi_i^+}(\nu(Z))^2=\kappa_i\sum_{\nu\in\Phi^+}(\nu(Z))^2. \end{align} \end{proposition} \begin{proof} We preserve the notation $B$ and $B_i$ for the Killing forms of $\ggo^{\mathbb C}$ and $\kg_i^{\mathbb C}$. Because $K$ is compact, $\kg_i^{\mathbb C}$ is a simple subalgebra of~$\ggo^{\mathbb C}$. Therefore, \begin{align} B_i(Z,Z)=\kappa_iB(Z,Z). \end{align} Using basic properties of root systems, we find \begin{align} B_i(Z,Z)&=\tr\!\big(\!\ad(Z)\ad(Z)\|_{\kg_i^{\mathbb C}}\big)=2\sum_{\nu\in\Phi_i^+}(\nu(Z))^2, \\ B(Z,Z)&=\tr(\ad(Z)\ad(Z))=2\sum_{\nu\in\Phi^+}(\nu(Z))^2. \end{align} \end{proof} \begin{remark} The assertion of Proposition~\ref{prop_kappa} holds even if $G$ is not simple but merely semisimple. \end{remark} \begin{remark} One can use properties of Casimir elements to produce another formula for~$\kappa_i$. More precisely, given $i=1,\ldots,r$, there exists a decomposition \begin{align} \pg^{\mathbb C}=\pg_1^i\oplus\cdots\oplus\pg_{r_i}^i \end{align} such that every $\pg_j^i$ is a non-trivial irreducible $\kg_i^{\mathbb C}$-module under the action defined by~$\ad$. Let $\psi_j^i$ be the highest weight of~$\pg_j^i$. Denote by $\rho_i$ the half-sum of positive roots of~$\kg_i^{\mathbb C}$. Using classical results on eigenvalues of Casimir elements, one can show that \begin{align} \kappa_i=\frac{d_i}{d_i+\sum_{j=1}^{r_i}B_i(\psi_j^i,\psi_j^i+2\rho_i)\dim\pg_j^i}, \end{align} where we preserve the notation $B_i$ for the bilinear form induced on $H_i^$ by the Killing form of~$\kg_i^{\mathbb C}$. Related formulas can be found in Dynkin's work; see~\cite{Dyn57}. \end{remark} \begin{example}\label{exa_kappa} Assume $G=\SU(p,q)$ and $K=\SU(p)\times U(q)$ with $2\le p\le q$. Then $r=2$ and $s=1$ in the decomposition~\eqref{dec_k}. Clearly, \begin{align} \ggo^{\mathbb C}=\slg(p+q,\mathbb C),\qquad \kg_1^{\mathbb C}=\slg(p,\mathbb C),\qquad \kg_2^{\mathbb C}=\slg(q,\mathbb C),\qquad \kg_3^{\mathbb C}=\mathbb C. \end{align} Denote by $E_i^j$ the matrix of size $(p+q)\times(p+q)$ that has 1 in the $(i,j)$th slot and 0 elsewhere. Suppose \begin{align} &H=\bigg\{\sum_{i=1}^{p+q-1}\lambda_i\big(E_i^i-E_{i+1}^{i+1}\big)\,\bigg\|\,\lambda_i\in\mathbb C\bigg\}, & &H_1=\bigg\{\sum_{i=1}^{p-1}\lambda_i\big(E_i^i-E_{i+1}^{i+1}\big)\,\bigg\|\,\lambda_i\in\mathbb C\bigg\}, \\ &\Phi^+=\{\epsilon_i-\epsilon_j\,\|\,1\le i<j\le p+q\},& &\Phi_1^+=\{\epsilon_i-\epsilon_j\,\|\,1\le i<j\le p\}, \end{align} where $\epsilon_i$ is the linear functional on $H$ such that $\epsilon_i\big(E_j^j-E_{j+1}^{j+1}\big)$ is the difference of Kronecker deltas~$\delta_i^j-\delta_i^{j+1}$. Choosing $Z=E_1^1-E_2^2$, we find \begin{align} \sum_{\nu\in\Phi^+}(\nu(Z))^2=&\big((\epsilon_1-\epsilon_2)\big(E_1^1-E_2^2\big)\big)^2+\sum_{i=3}^{p+q}\big((\epsilon_1-\epsilon_i)\big(E_1^1-E_2^2\big)\big)^2 \\ &+\sum_{i=3}^{p+q}\big((\epsilon_2-\epsilon_i)\big(E_1^1-E_2^2\big)\big)^2=2(p+q),\\ \sum_{\nu\in\Phi_1^+}(\nu(Z))^2=&2p. \end{align} Proposition~\ref{prop_kappa} implies $\kappa_1=\frac p{p+q}$. A similar argument with $Z=E_{p+1}^{p+1}-E_{p+2}^{p+2}$ yields $\kappa_2=\frac q{p+q}$. Since $\kg_3$ is abelian, $\kappa_3=0$. \end{example} If $r=1$, one can calculate $\kappa_i$ using formula~\eqref{sum_dkappan} below; see Examples~\ref{exa_simple} and~\ref{ex_2ideals}. \begin{remark} The work~\cite{DZ79} computes a range of constants analogous to $\kappa_i$ in the framework of compact Lie groups. One can find $\kappa_i$ for a specific pair $(\ggo,\kg)$ using those results along with duality for symmetric spaces; see, e.g.,~\cite[Sections~7.82--7.83]{Bss}. In the present paper, we choose a more direct approach. \end{remark} We are now ready to state the main result of this section. \begin{theorem}\label{Ric} Suppose $g\in\mca_K$ is a naturally reductive metric on the simple group~$G$ satisfying~(\ref{metric}). The Ricci curvature of $g$ is given by the formulas \begin{align} &\Ricci (g)\|_{\pg} = - \sum_{i=1}^{r+s} \Big(\frac{\alpha_i}{2 \beta}+1\Big)\frac{ d_i(1-\kappa_i)}{n} Q\|_{\pg}, \\ &\Ricci (g)\|_{\kg_j} = \unc\Big(\frac{\alpha_j^2}{\beta^2}(1-\kappa_j)+\kappa_j\Big)Q\|_{\kg_j}, \\ &\Ricci (g)(\pg,\kg_j)=\Ricci (g)(\kg_j,\kg_k)=0, &j,k =1,\ldots,r+s,~j\ne k. \end{align} \end{theorem} To prove Theorem~\ref{Ric}, we apply the strategy developed in~\cite[Section~5]{DZ79}. In what follows, $\tr_h$ stands for the trace of a bilinear form with respect to an inner product~$h$. The notation $\pi_\ug$ is used for the $Q$-orthogonal projection onto $\ug\subset\ggo$. For $j=1,\ldots,r+s$, define a bilinear form $A_j$ on $\pg$ by setting \begin{align} A_j(X,Y)=\tr(\pi_{\kg_j}\ad(X)\ad(Y)). \end{align} Fix $Q$-orthonormal bases $(v_i)_{i=1}^n$ of $\pg$ and $\big(v_k^j\big)_{k=1}^{d_j}$ of $\kg_j$. We need the following auxiliary result; cf.~\cite[pages~609--610]{Jen73} and~\cite[pages~32--34]{DZ79}. \begin{lemma}\label{lemma_Ai} Given $j=1,\ldots,r+s$, \begin{align} \tr_{Q\|_\pg}A_j=d_j(1-\kappa_j),\qquad \sum_{i=1}^{r+s}A_i=\unm Q\|_{\pg}. \end{align} \end{lemma} \begin{proof} Invoking~\eqref{sym_spce_incl}, we compute \begin{align} \tr_{Q\|_\pg}A_j&=\sum_{i=1}^n\sum_{k=1}^{d_j}Q\big([v_i,[v_i,v_k^j]],v_k^j\big) =-\sum_{i=1}^n\sum_{k=1}^{d_j}Q\big(v_i,[[v_i,v_k^j],v_k^j]\big) \\ &=-\tr_{Q\|_{\kg_j}}B\|_{\kg_j}+\sum_{k=1}^{d_j}\sum_{l=1}^{r+s}\sum_{m=1}^{d_l}Q\big([v_k^j,[v_k^j,v_m^l]],v_m^l\big). \end{align} Since $\kg_j$ is an ideal of $\kg$, \begin{align} \sum_{l=1}^{r+s}\sum_{m=1}^{d_l}\sum_{k=1}^{d_j}Q\big([v_k^j,[v_k^j,v_m^l]],v_m^l\big)&= \sum_{m=1}^{d_j}\sum_{k=1}^{d_j}Q\big([v_k^j,[v_k^j,v_m^j]],v_m^j\big) =\tr_{Q\|_{\kg_j}}B_j=\kappa_j\tr_{Q\|_{\kg_j}}B\|_{\kg_j}. \end{align} We conclude that \begin{align} \tr_{Q\|_\pg}A_j=-\tr_{Q\|_{\kg_j}}B\|_{\kg_j}+\kappa_j\tr_{Q\|_{\kg_j}}B\|_{\kg_j}=d_j(1-\kappa_j). \end{align} This proves the first equality in the statement of the lemma. For $X,Y\in\pg$, \begin{align} \sum_{i=1}^{r+s}A_i(X,Y)&= \sum_{i=1}^{r+s}\sum_{k=1}^{d_i}Q\big([X,[Y,v_k^i]],v_k^i\big) =B(X,Y)-\sum_{l=1}^nQ([X,[Y,v_l]],v_l). \end{align} It is easy to see that the forms $A_i$ are symmetric. Consequently, \begin{align} \sum_{l=1}^nQ([X,[Y,v_l]],v_l)=\tr(\ad(X)\pi_\kg\ad(Y))=\tr(\pi_\kg\ad(Y)\ad(X))=\sum_{i=1}^{r+s}A_i(X,Y). \end{align} We conclude that \begin{align} \sum_{i=1}^{r+s}A_i(X,Y)=B(X,Y)-\sum_{i=1}^{r+s}A_i(X,Y), \end{align} which implies the second inequality. \end{proof} \begin{proof}[Proof of Theorem~\ref{Ric}] Let $\nabla$ be the Levi-Civita connection of the metric~$g$. The Koszul formula yields \begin{align}\label{Koszul} \nabla_X Y=\begin{cases} \unm [X,Y] & \mbox{if}~X,Y \in \pg~\mbox{or}~X,Y\in\kg, \\ -\frac{\alpha_i}{2 \beta} [X,Y] & \mbox{if}~X \in \pg~\mbox{and}~Y \in \kg_i~\mbox{for some}~i=1, \ldots, r+s, \\ \big(\frac{\alpha_i}{2 \beta}+1\big)[X,Y] & \mbox{if}~X \in \kg_i~\mbox{for some}~i=1, \ldots, r+s~\mbox{and}~Y \in \pg; \end{cases} \end{align} see~\cite[page~485]{C85}. The Ricci curvature of $g$ satisfies \begin{align}\label{Ric_AA} \Ricci(g)(X,Y) =-\tr\nabla_{\nabla_\cdot Y}X ,\qquad X,Y\in\ggo. \end{align} This fact goes back to~\cite{Sag70}; a simpler proof appeared in~\cite[Section~5]{DZ79}. Substituting~\eqref{Koszul} into~\eqref{Ric_AA}, we easily obtain the required identities for $\Ricci (g)\|_{\kg_j}$, $\Ricci (g)(\pg,\kg_j)$ and $\Ricci (g)(\kg_j,\kg_k)$ with $j\ne k$; cf.~\cite[page~33]{DZ79}. Because $g$ lies in $\mca_K$, it is $\ad(\kg)$-invariant. Consequently, there exists $\tau\in\mathbb R$ such that $\Ricci(g)\|_{\pg}=\tau Q\|_{\pg}$. Taking the trace with respect to $Q\|_{\pg}$ on both sides and exploiting Lemma~\ref{lemma_Ai}, we find \begin{align} n\tau&=\sum_{i=1}^n\Ricci(g)(v_i,v_i)=-\sum_{i=1}^n\bigg(\sum_{l=1}^nQ(\nabla_{\nabla_{v_l}v_i}v_i,v_l)+\sum_{j=1}^{r+s}\sum_{m=1}^{d_j}Q\big(\nabla_{\nabla_{v_m^j}v_i}v_i,v_m^j\big)\bigg) \\ &=-\unm\sum_{i=1}^n\sum_{j=1}^{r+s}\Big(\frac{\alpha_j}{2\beta}+1\Big)\sum_{m=1}^{d_j}\bigg(\sum_{l=1}^nQ\big([v_l,v_i],v_m^j\big)Q\big([v_m^j,v_i],v_l\big)+Q\big([[v_m^j,v_i],v_i],v_m^j\big)\bigg) \\ &=-\sum_{i=1}^n\sum_{j=1}^{r+s}\Big(\frac{\alpha_j}{2\beta}+1\Big)\sum_{m=1}^{d_j}Q\big([v_i,[v_i,v_m^j]],v_m^j\big) \\ &=-\sum_{j=1}^{r+s}\Big(\frac{\alpha_j}{2\beta}+1\Big)\tr_{Q\|_{\pg}}A_j=-\sum_{j=1}^{r+s}\Big(\frac{\alpha_j}{2\beta}+1\Big)d_j(1-\kappa_j). \end{align} Therefore, \begin{align} \tau=-\sum_{j=1}^{r+s}\Big(\frac{\alpha_j}{2\beta}+1\Big)\frac{d_j(1-\kappa_j)}n, \end{align} which yields the required identity for~$\Ricci(g)\|_{\pg}$. \end{proof} Denote by $S$ the scalar curvature functional on~$\mca_K$. Our next goal is to produce a formula for~$S$. Taking the trace of the second equality in Lemma~\ref{lemma_Ai} and using the first one, we obtain \begin{align}\label{sum_dkappan} 2\sum_{i=1}^{r+s} d_i (1-\kappa_i) = n. \end{align} Theorem~\ref{Ric} and~\eqref{sum_dkappan} imply the following result. \begin{corollary}\label{scalar} Suppose $g\in\mca_K$ is a naturally reductive metric on the simple group~$G$ satisfying~(\ref{metric}). The scalar curvature of $g$ is given by the formula \begin{equation} S(g) = -\unc \sum_{i=1}^{r+s} \frac{\alpha_i}{\beta^2} d_i (1-\kappa_i) - \frac{n}{2 \beta} + \unc \sum_{i=1}^{r} \frac{d_i\kappa_i}{\alpha_i}. \end{equation} \end{corollary} \section{Metrics with prescribed Ricci curvature}\label{sec_Ricci=T} Consider a (0,2)-tensor field $T$ on the simple group $G$. In this section, we state a necessary and sufficient condition for the solvability of the equation \begin{align}\label{bdy_PRC_no_c} \Ricci(g)=T \end{align} in the class $\mathcal M_K$. If a metric $g\in\mathcal M_K$ satisfying~\eqref{bdy_PRC_no_c} exists, then $T$ must be left-invariant. Moreover, by Theorem~\ref{Ric}, the formula \begin{equation}\label{T} T = -T_{\pg} Q\|_{\pg} + T_1 Q\|_{\kg_1} + \ldots + T_{r+s} Q\|_{\kg_{r+s}} \end{equation} holds with $T_{\pg},T_1,\ldots,T_{r+s}>0$. \begin{theorem}\label{thm_no_c} Suppose $T$ is a left-invariant (0,2)-tensor field on $G$ given by~(\ref{T}). A naturally reductive metric $g\in\mathcal M_K$ satisfying~(\ref{bdy_PRC_no_c}) exists if and only if \begin{align}\label{cond1_cno} 4T_i -\kappa_i>0 \end{align} for all $i=1,\ldots,r$ and \begin{align}\label{cond_noc} T_{\pg}=\sum_{i=1}^{r+s}\frac{2d_i(1-\kappa_i)+d_i\sqrt{(4T_i -\kappa_i)(1-\kappa_i)}}{2n}. \end{align} There is at most one such $g$, up to scaling. \end{theorem} \begin{proof} Consider a naturally reductive metric $g\in\mathcal M_K$. It satisfies~\eqref{metric} for some $\beta,\alpha_1,\ldots,\alpha_{r+s}>0$. By Theorem~\ref{Ric}, the Ricci curvature of $g$ equals $T$ if and only if \begin{align} \frac{\alpha_i}{\beta}&=\sqrt{\frac{4T_i -\kappa_i}{1-\kappa_i}}, \qquad 1\leq i \leq r+s, \\ T_{\pg}&=\sum_{i=1}^{r+s} \Big(1+\frac{\alpha_i}{2 \beta}\Big)\frac{(1-\kappa_i) d_i}{n} =\sum_{i=1}^{r+s}\frac{2d_i(1-\kappa_i)+d_i\sqrt{(4T_i -\kappa_i)(1-\kappa_i)}}{2n}. \end{align} This observation proves the result. \end{proof} It is tempting to use Theorem~\ref{thm_no_c} to study the solvability of the equation \begin{equation}\label{prescribed} \Ricci (g)= cT \end{equation} in the class $\mca_K$. Indeed, suppose $\Xi$ is the set of left-invariant tensor fields on $G$ satisfying~\eqref{T}, \eqref{cond1_cno} and~\eqref{cond_noc}. Theorem~\ref{thm_no_c} states that~\eqref{bdy_PRC_no_c} has a solution if and only if $T$ lies in~$\Xi$. Using this result, we can easily obtain a description of the set of tensor fields that coincide with Ricci curvatures of metrics in $\mca_K$ up to scaling. Namely, a pair $(g,c)\in\mca_K\times(0,\infty)$ satisfying~\eqref{prescribed} exists if and only if \begin{align} T\in\Xi'=\{\tau h\,\|\,\tau>0~\mbox{and}~h\in\Xi\}. \end{align} However, in general, it is difficult to determine whether a specific $T$ given by~\eqref{T} lies in~$\Xi'$. To do so, one has to check whether~\eqref{cond1_cno} and~\eqref{cond_noc} hold with $T_{\mathfrak p},T_1,\ldots,T_{r+s}$ replaced by $cT_{\mathfrak p},cT_1,\ldots,cT_{r+s}$ for some $c>0$. Already when $r+s=2$, this involves the tricky task of understanding if a polynomial of degree~$4$ has roots that obey several constraints; when $r+s\ge3$, the question seems to be substantially harder. In the present paper, we take a different approach to the analysis of~\eqref{prescribed}. We are able to show that the existence of a pair $(g,c)\in\mca_K\times(0,\infty)$ satisfying~\eqref{prescribed} follows from simple inequalities for the components of~$T$. Moreover, we draw interesting conclusions regarding the non-uniqueness of such a pair. \section{Metrics with Ricci curvature prescribed up to scaling}\label{sec_Ricci=cT} Suppose $T$ is a left-invariant symmetric (0,2)-tensor field on~$G$. Our next goal is to study the solvability of equation~\eqref{prescribed} in the class $\mathcal M_K$. As above, we assume the group $G$ is simple. This implies, in particular, that $\kappa_i<1$ for all~$i$. First, we re-state the problem in variational terms. More precisely, define \begin{align}\label{MKT_def} \mathcal{M}^+_{T} &=\{g \in \mathcal{M}_ K\,\|\,\tr_{g}T=1\},\notag\\ \mathcal{M}^-_{T} & =\{g \in \mathcal{M}_ K\,\|\,\tr_{g}T=-1\},\notag \\ \mathcal{M}^0_{T} &=\{g \in \mathcal{M}_ K\,\|\,\tr_{g}T=0\}. \end{align} Each of these three spaces carries a manifold structure induced from~$\mathcal M_K$. In Section~\ref{sec_variational}, we show that $g$ satisfies~\eqref{prescribed} if and only if it is (up to scaling) a critical point of $S\|_{\mathcal{M}^+_{T}}$, $S\|_{\mathcal{M}^-_{T}}$ or $S\|_{\mathcal{M}^0_{T}}$. This resembles the variational interpretation of~\eqref{prescribed} on compact Lie groups for positive-semidefinite~$T$ (see~\cite[Proposition~3.1]{APZ20}); however, in that case, only $S\|_{\mathcal{M}^+_{T}}$ needs to be considered. In Sections~\ref{sec_+} and~\ref{sec_-}, we obtain sufficient conditions for the existence of global maxima of $S\|_{\mathcal{M}^+_{T}}$ and $S\|_{\mathcal{M}^-_{T}}$, respectively. This requires complex estimates on the scalar curvature obtained in Lemmas~\ref{lemmacompact+} and~\ref{lemmacompact-}. In Section~\ref{sec_0}, we classify completely the critical points of~$S\|_{\mathcal{M}^0_{T}}$ assuming $r+s\le2$. The analysis here can be reduced, as Lemma~\ref{lem_crit=root} demonstrates, to the study of a cubic polynomial in one variable. Finally, in Section~\ref{sec_sum}, we summarise the implications of our results for the prescribed Ricci curvature problem. \begin{remark}\label{rem_2c_Tim} It appears that~\eqref{prescribed} admits a similar variational interpretation on compact homogeneous spaces when $T$ has mixed signature. Thus, our arguments yield new insight into the study of~\eqref{prescribed} in that setting. For instance, Buttsworth showed in~\cite{TB19} through methods of elementary polynomial analysis that, for certain $T$ on~$\SU(2)$, a left-invariant metric $g$ satisfying~\eqref{prescribed} exists for precisely two distinct constants~$c\in\mathbb R$. This was somewhat surprising at the time, as nothing similar had occurred in previously understood examples. We observe an analogous phenomenon in Theorem~\ref{thm_summary} below. In our arguments, the two constants arise naturally as Lagrange multipliers for $S\|_{\mathcal{M}^+_{T}}$ and $S\|_{\mathcal{M}^-_{T}}$. \end{remark} \subsection{The variational approach}\label{sec_variational} The following result underpins our approach to the study of~\eqref{prescribed}. \begin{proposition}\label{variational_lemma} Let $T$ be a left-invariant symmetric (0,2)-tensor field on $G$. A metric $g \in\mathcal{M}_{K}$ satisfies~(\ref{prescribed}) for some $c \in \RR$ if and only if it is (up to scaling) a critical point of~$S \|_{\mathcal{M}^{+}_{T}}$, $S \|_{\mathcal{M}^{-}_{T}}$ or~$S \|_{\mathcal{M}^{0}_{T}}$. \end{proposition} \begin{proof} Denote by $\tca$ the space of left-invariant bilinear form fields \begin{equation} \gamma Q\|_{\pg} + \gamma_1 Q\|_{\kg_1} + \ldots + \gamma_{r+s} Q\|_{\kg_{r+s}} \end{equation} with $\gamma,\gamma_1,\ldots,\gamma_{r+s}\in\mathbb R$. Theorem~\ref{Ric} shows that $\Ricci(g)$ lies in~$\tca$. We identify $\tca$ with the space tangent to~$\mathcal M_K$ at~$g$ in the natural way. The left-invariant bilinear form fields \begin{align} Q_{\pg}=\pi_{\pg}^Q,\qquad Q_i=\pi_{\kg_i}^Q,\qquad i=1,\ldots,r+s, \end{align} where $$ denotes pullback, make a basis of~$\tca$. Suppose $g$ is given by~\eqref{metric}. Using Corollary~\ref{scalar}, \eqref{sum_dkappan} and Theorem~\ref{Ric}, we find that the differential of the scalar curvature functional $S$ satisfies \begin{align} d S_g (Q_{\pg})&=\frac{\partial}{\partial \beta} \bigg(-\unc \sum_{i=1}^{r+s} \frac{\alpha_i}{\beta^2} d_i (1-\kappa_i) - \frac{n}{2 \beta}\bigg) \\ &= \frac{1}{\beta^2}\sum_{i=1}^{r+s}\Big(\frac{\alpha_i}{2\beta}+1\Big)d_i (1-\kappa_i)=- g( \Ricci (g), Q_{\pg} ), \\ d S_g (Q_j) &= \frac{\partial}{\partial \alpha_j}\bigg( -\unc \sum_{i=1}^{r+s} \frac{\alpha_i}{\beta^2} d_i (1-\kappa_i)+\unc \sum_{i=1}^{r} \frac{d_i\kappa_i}{\alpha_i} \bigg) \\ &=-\frac{d_j}{4\alpha_j^2} \Big(\frac{\alpha_j^2}{4 \beta^2}(1-\kappa_j)+\kappa_j\Big)=- g( \Ricci (g), Q_j), \qquad j=1, \ldots, r+s. \end{align} (We preserve the notation~$g$ for the inner product induced by $g$ on the tensor bundle over~$G$.) Consequently, \begin{equation}\label{sc=Ric} d S_g (h) =-g(\Ricci (g),h) \end{equation} for all $h \in \tca$. Let us scale $g$ by the factor \begin{align} \tau= \begin{cases} \|\tr_gT\| &\mbox{if}~\tr_gT\ne0, \\ 1 &\mbox{if}~\tr_gT=0. \end{cases} \end{align} Clearly, $\tau g$ lies in~$\mathcal M_T^\sigma$ for some $\sigma\in\{+,-,0\}$. The space tangent to $\mathcal M^\sigma_T$ at $\tau g$ consists of those $h \in\tca $ that satisfy $g(T,h)=0$. Formula~\eqref{sc=Ric} implies that $d S_{\tau g}$ vanishes on this space if and only if $g$ satisfies~\eqref{prescribed}. \end{proof} By Theorem~\ref{Ric}, the constant $c$ in Proposition~\ref{variational_lemma} must be positive if $T$ is given by~\eqref{T} with $T_{\pg},T_1,\ldots,T_{r+s}>0$. The above proof shows that one may think of $c$ as a Lagrange multiplier. \subsection{Global maxima on $\mca_{T}^{+}$}\label{sec_+} Our goal in this subsection is to show that simple inequalities for $T$ guarantee the existence of a critical point of~$S\|_{\mca_{T}^{+}}$. \begin{theorem}\label{sufcon+} Suppose $T$ is a left-invariant (0,2)-tensor field on $G$ satisfying~(\ref{T}) for some $T_{\pg},T_1,\ldots,T_{r+s}>0$. Choose an index $m$ such that \begin{align}\label{def_index_m} \frac{\kappa_m}{T_m}=\max_{i=1,\ldots,r} \frac {\kappa_i}{T_i}. \end{align} If \begin{equation}\label{hyp_thm_beta_inf+} \frac{\kappa_m\tr_Q T}{T_m}< \dim \kg + d_m (1-\kappa_m) - 3 n \end{equation} and \begin{equation}\label{hyp_thm_beta_zero+} \sum_{i=1}^{r+s} \frac{n^2\kappa_iT_{\mathfrak p}^2- d_i^2(1-\kappa_i)T_i^2}{nT_{\mathfrak p}T_i}-2n<0, \end{equation} then the functional $S \|_{\mathcal{M}^{+}_{T}}$ attains its global maximum. \end{theorem} The proof of Theorem~\ref{sufcon+} requires the following estimate for $S \|_{\mathcal{M}^{+}_{T}}$. \begin{lemma}\label{lemmacompact+} Let $m$ be as in~(\ref{def_index_m}). Assume that~(\ref{hyp_thm_beta_zero+}) holds. Given $\epsilon > 0$, there exists a compact set $\mathcal C_\epsilon^+\subset\mathcal M^+_T$ such that \begin{equation}\label{estimate+} S(g) < \frac{\kappa_m}{4T_m} + \epsilon \end{equation} for every $g \in \mathcal{M}^+_{T} \setminus\cca_\epsilon^+$. \end{lemma} \begin{proof} Consider a metric $g\in\mathcal M^+_T$ satisfying~\eqref{metric}. The definition of $\mathcal M^+_T$ implies \begin{align}\label{trace=1} \tr_gT=-\frac{nT_{\mathfrak p}}\beta+\sum_{i=1}^{r+s}\frac{d_iT_i}{\alpha_i}=1. \end{align} For $\epsilon>0$, denote \begin{align} \Lambda_\infty^+(\epsilon)=\frac{n\kappa_mT_{\mathfrak p}}{4T_m\epsilon}. \end{align} In view of Corollary~\ref{scalar} and formula~\eqref{trace=1}, if $\beta>\Lambda_\infty^+(\epsilon)$, then \begin{equation} S(g) < \unc \sum_{i=1}^{r} \frac{d_i\kappa_i}{\alpha_i}\le\frac{\kappa_m}{4T_m}\sum_{i=1}^{r+s} \frac{d_iT_i}{\alpha_i}=\frac{\kappa_m}{4T_m}\bigg(1+\frac{nT_{\mathfrak p}}\beta\bigg)<\frac{\kappa_m}{4T_m}+\epsilon. \end{equation} Thus, in this case, estimate~\eqref{estimate+} holds. Denote \begin{align} \Lambda_0^+=\unm\bigg(\sum_{i=1}^{r+s}\frac{n^2\kappa_iT_{\pg}^2+d_i^2(1-\kappa_i)T_i^2}{n^2T_{\mathfrak p}^2T_i}\bigg)^{-1}\bigg\|\sum_{i=1}^{r+s} \frac{n^2\kappa_iT_{\mathfrak p}^2-d_i^2 (1-\kappa_i)T_i^2}{nT_{\mathfrak p}T_i}-2n\bigg\|. \end{align} Formula~\eqref{trace=1} implies \begin{align} \frac{\beta}{\alpha_j}<\frac\beta{d_jT_j}\sum_{i=1}^{r+s}\frac{d_iT_i}{\alpha_i}=\frac{\beta+nT_{\mathfrak p}}{d_jT_j},\qquad j=1,\ldots,r+s. \end{align} Invoking Corollary~\ref{scalar} again, we find \begin{align} S(g) &=\frac1{4\beta}\bigg(\sum_{i=1}^{r+s} \Big(d_i\kappa_i\frac{\beta}{\alpha_i}-d_i (1-\kappa_i)\frac{\alpha_i}{\beta}\Big)-2n \bigg) \\ & <\frac1{4\beta}\bigg(\sum_{i=1}^{r+s} \bigg(\kappa_i\frac{\beta+nT_{\mathfrak p}}{T_i}-\frac{d_i^2(1-\kappa_i)T_i}{\beta+nT_{\mathfrak p}}\bigg)-2n \bigg) \\ & =\frac1{4\beta}\bigg(\sum_{i=1}^{r+s} \bigg(\frac{n\kappa_iT_{\mathfrak p}}{T_i}-\frac{d_i^2(1-\kappa_i)T_i}{nT_{\mathfrak p}}\bigg)-2n+\beta\sum_{i=1}^{r+s} \bigg(\frac{\kappa_i}{T_i}+\frac{d_i^2(1-\kappa_i)T_i}{nT_{\mathfrak p}(\beta+nT_{\mathfrak p})}\bigg)\bigg) \\ &<\frac1{4\beta}\bigg(\sum_{i=1}^{r+s} \frac{n^2\kappa_iT_{\mathfrak p}^2-d_i^2(1-\kappa_i)T_i^2}{nT_{\mathfrak p}T_i}-2n+\beta\sum_{i=1}^{r+s}\frac{n^2\kappa_iT_{\mathfrak p}^2+d_i^2(1-\kappa_i)T_i^2}{n^2T_{\mathfrak p}^2T_i}\bigg). \end{align} In view of~\eqref{hyp_thm_beta_zero+}, if~$\beta<\Lambda_0^+$, then \begin{align} S(g) <\frac1{8\beta}\bigg(\sum_{i=1}^{r+s} \frac{n^2\kappa_iT_{\mathfrak p}^2-d_i^2(1-\kappa_i)T_i^2}{nT_{\mathfrak p}T_i}-2n\bigg)<0. \end{align} In this case, again, estimate~\eqref{estimate+} holds. Choose $p$ and $q$ such that \begin{align}\label{nota_aqTp} d_pT_p=\min_{i=1,\ldots,r+s}d_iT_i,\qquad \alpha_q=\min_{i=1,\ldots,r+s}\alpha_i. \end{align} Denote \begin{align} \Gamma_0^+=\unm d_pT_p\min\Big\{1,\frac{\Lambda_0^+}{nT_{\mathfrak p}}\Big\}. \end{align} By~\eqref{trace=1}, if $\alpha_q<\Gamma_0^+$, then \begin{align} \beta=nT_{\mathfrak p}\bigg(\sum_{i=1}^{r+s}\frac{d_iT_i}{\alpha_i}-1\bigg)^{-1}<nT_{\mathfrak p}\Big(\frac{d_qT_q}{\alpha_q}-1\Big)^{-1}\le \frac{nT_{\mathfrak p}\alpha_q}{d_pT_p-\alpha_q}\le \frac{2nT_{\mathfrak p}\alpha_q}{d_pT_p}\le\Lambda_0^+, \end{align} which means~\eqref{estimate+} holds. Choose $l$ such that \begin{align}\label{nota_al} \alpha_l=\max_{i=1,\ldots,r+s}\alpha_i. \end{align} For $\epsilon>0$, denote \begin{align} \Gamma_\infty^+(\epsilon)=\frac{2\Lambda_\infty^+(\epsilon)^2}{\min_{i=1,\ldots,r+s} d_i (1-\kappa_i)}\sum_{i=1}^{r} \frac{d_i\kappa_i}{\Gamma_0^+}. \end{align} As we showed above, if $\beta>\Lambda_\infty^+(\epsilon)$ or $\alpha_q<\Gamma_0^+$, then~\eqref{estimate+} holds. Assuming $\alpha_q\ge\Gamma_0^+$ and $\alpha_l>\Gamma_\infty^+(\epsilon)$, we find \begin{align} S(g) &<-\unc \sum_{i=1}^{r+s} \frac{\alpha_i}{\beta^2} d_i (1-\kappa_i)+ \unc \sum_{i=1}^{r} \frac{\kappa_i d_i}{\alpha_i} < -\frac{\alpha_l}{4\beta^2} d_l (1-\kappa_l) + \unc \sum_{i=1}^{r} \frac{\kappa_i d_i}{\alpha_q} \\ &\le-\frac{\Gamma_\infty^+(\epsilon)}{4\Lambda_\infty^+(\epsilon)^2} \min_{i=1,\ldots,r+s}d_i (1-\kappa_i) + \unc \sum_{i=1}^{r} \frac{\kappa_i d_i}{\Gamma_0^+}<-\unc \sum_{i=1}^{r} \frac{\kappa_i d_i}{\Gamma_0^+}<0. \end{align} Thus, the inequality $\alpha_l>\Gamma_\infty^+(\epsilon)$ implies~\eqref{estimate+}. Let $\mathcal C_{\epsilon}^+$ be the set of metrics $g\in\mathcal M_T^+$ satisfying~\eqref{metric} with \begin{align} \min\{\Lambda_0^+,\Gamma_0^+\}\le\min\{\beta,\alpha_1,\ldots,\alpha_{r+s}\}\le\max\{\beta,\alpha_1,\ldots,\alpha_{r+s}\}\le\max\{\Lambda_\infty^+(\epsilon),\Gamma_\infty^+(\epsilon)\}. \end{align} Clearly, this set is compact. Summarising the arguments above, we conclude that~\eqref{estimate+} holds for all~$g\in\mathcal M_T^+\setminus\mathcal C_{\epsilon}^+$. \end{proof} With Lemma~\ref{lemmacompact+} at hand, we can prove Theorem~\ref{sufcon+} using the approach from~\cite[Proof of Theorem~3.3]{APZ20}. The main idea behind this approach goes back to~\cite{MGAP18}. \begin{proof}[Proof of Theorem \ref{sufcon+}] Denote $U= \tr_QT -d_mT_m$. For $t>U$, consider the metric $g_t\in\mca_{K}$ satisfying \begin{align} g_t= t Q\|_{\pg} &+ t Q\|_{\kg_1} + \cdots + t Q\|_ {\kg_{m-1}} \\ &+ \phi(t)Q\|_{\kg_m} + tQ\|_ {\kg_{m+1}} + \cdots + tQ\|_ {\kg_{r+s}}, \qquad \phi(t) = \frac {d_mT_mt}{t- U}. \end{align} Straightforward verification shows that $g_t$ lies in~$\mca_{T}^+$. By Corollary~\ref{scalar}, \begin{align} S(g_t) &= \frac1{4t}d_m (1-\kappa_m) - \frac{\phi(t)}{4t^2} d_m (1-\kappa_m) - \frac{1}{4t} \sum_{i=1}^{r+s} d_i (1-\kappa_i) \\ &\hphantom{=}~- \frac{n}{2t} + \frac{1}{4t} \sum_{i=1}^{r} \kappa_i d_i - \frac{\kappa_md_m}{4t} + \frac{\kappa_md_m}{4\phi(t)} \\ &= \frac1{4t}d_m (1-\kappa_m) - \frac{\phi(t)}{4t^2} d_m (1-\kappa_m)- \frac{3n}{4t} +\frac{\dim\mathfrak k}{4t} - \frac{\kappa_md_m}{4t} + \frac{\kappa_md_m}{4\phi(t)}. \end{align} Furthermore, in light of~\eqref{hyp_thm_beta_inf+}, \begin{align}\label{t2St} 4\lim_{t\to\infty}t^2\frac{d}{dt} S(g_t) & = -d_m(1-\kappa_m) - \dim \kg + 3 n +\kappa_md_m+ \frac{\kappa_mU}{ T_m} \notag \\ & = -d_m(1-\kappa_m) - \dim \kg + 3 n + \frac{\kappa_m \tr_QT}{ T_m} < 0. \end{align} We conclude that $\frac{d}{dt} S(g_t)<0$ for sufficiently large $t$, which implies the existence of $t_0\in(U,\infty)$ such that \[ S(g_{t_0}) >\lim_{t \to \infty} S(g_t) = \frac{\kappa_m}{4T_m}. \] Using Lemma \ref{lemmacompact+} with \[ \epsilon = \unm\Big(S(g_{t_0}) - \frac{\kappa_m}{4T_m}\Big)>0 \] yields \begin{equation}\label{maxcompact} S(h) < \frac{\kappa_m}{4T_m}+\epsilon=\unm S(g_{t_0})+\frac{\kappa_m}{8T_m}< S(g_{t_0}),\qquad h\in \mathcal{M}^{+}_{T}\setminus\cca_\epsilon^+. \end{equation} Since $\cca_\epsilon^+$ is compact, the functional $S\|_{\mathcal{C}^{+}_{\epsilon}}$ attains its global maximum at some ${g_{\mathrm{mx}} \in \cca_\epsilon^+}$. Obviously, $g_{t_0}$ lies in $\cca_\epsilon^+$. Therefore, by~\eqref{maxcompact}, $S(h) \leq S(g_{\mathrm{mx}})$ for all $h\in\mathcal{M}^{+}_{T}$. \end{proof} \subsection{Global maxima on $\mca_{T}^{-}$}\label{sec_-} Now we focus on the space $\mca_{T}^{-}$. \begin{theorem}\label{sufcon-} Suppose $T$ is a left-invariant (0,2)-tensor field on $G$ satisfying~(\ref{T}) for some $T_{\pg},T_1,\ldots,T_{r+s}>0$. If condition~(\ref{hyp_thm_beta_zero+}) holds, then the functional $S \|_{\mathcal{M}^{-}_{T}}$ attains its global maximum. \end{theorem} The proof relies on the following estimate. \begin{lemma}\label{lemmacompact-} Assume that~(\ref{hyp_thm_beta_zero+}) holds. Given $\theta>0$, there exists a compact set $\mathcal C_\theta^-\subset\mathcal{M}^-_{T}$ such that $S(g) < -\theta$ for every $g \in \mathcal{M}^-_{T} \setminus\cca_\theta^-$. \end{lemma} \begin{proof} Let $g\in\mathcal M^+_T$ be a metric satisfying~\eqref{metric}. Then \begin{align}\label{trace=-1} \tr_gT=-\frac{nT_{\mathfrak p}}\beta+\sum_{i=1}^{r+s}\frac{d_iT_i}{\alpha_i}=-1, \end{align} which implies $\beta<nT_{\mathfrak p}$. Moreover, \begin{align} \frac{\beta}{\alpha_j}<\frac\beta{d_jT_j}\sum_{i=1}^{r+s}\frac{d_iT_i}{\alpha_i}=\frac{nT_{\mathfrak p}-\beta}{d_jT_j}, \qquad j=1,\ldots,r+s. \end{align} Given $\theta>0$, denote \begin{align} \Lambda_0^-(\theta)=\frac1{4\theta}\bigg\|\sum_{i=1}^{r+s} \frac{n^2\kappa_iT_{\mathfrak p}^2-d_i^2(1-\kappa_i)T_i^2}{nT_{\mathfrak p}T_i}-2n\bigg\|. \end{align} Corollary~\ref{scalar} implies \begin{align} S(g)&<\frac1{4\beta}\bigg(\sum_{i=1}^{r+s} \bigg(\kappa_i\frac{nT_{\mathfrak p}-\beta}{T_i}-\frac{d_i^2(1-\kappa_i)T_i}{nT_{\mathfrak p}-\beta}\bigg)-2n \bigg) \\ &<\frac1{4\beta}\bigg(\sum_{i=1}^{r+s} \bigg(\kappa_i\frac{nT_{\mathfrak p}}{T_i}-\frac{d_i^2(1-\kappa_i)T_i}{nT_{\mathfrak p}}\bigg)-2n \bigg) \\ & =\frac1{4\beta}\bigg(\sum_{i=1}^{r+s} \frac{n^2\kappa_iT_{\mathfrak p}^2-d_i^2(1-\kappa_i)T_i^2}{nT_{\mathfrak p}T_i}-2n\bigg). \end{align} By~\eqref{hyp_thm_beta_zero+}, if $\beta<\Lambda_0^-(\theta)$, then $S(g)<-\theta$. Choose $p$ and $q$ as in~\eqref{nota_aqTp}. For $\theta>0$, denote \begin{align} \Gamma_0^-(\theta)=\frac{d_pT_p\Lambda_0^-(\theta)}{nT_{\mathfrak p}}. \end{align} If $\alpha_q<\Gamma_0^-(\theta)$, then \begin{align} \beta=nT_{\mathfrak p}\bigg(\sum_{i=1}^{r+s}\frac{d_iT_i}{\alpha_i}+1\bigg)^{-1}<\frac{nT_{\mathfrak p}\alpha_q}{d_qT_q}\le \frac{nT_{\mathfrak p}\alpha_q}{d_pT_p}<\Lambda_0^-(\theta), \end{align} which means $S(g)<-\theta$. Choose $l$ as in~\eqref{nota_al}. For $\theta>0$, denote \begin{align} \Gamma_\infty^-(\theta)=\frac{n^2T_{\mathfrak p}^2}{\min_{i=1,\ldots r+s} d_i (1-\kappa_i)}\bigg(4\theta+\sum_{i=1}^r\frac{\kappa_i d_i}{\Gamma_0^-(\theta)}\bigg). \end{align} As shown above, if $\alpha_q<\Gamma_0^-(\theta)$, then $S(g)<-\theta$. Recalling that $\beta<nT_{\mathfrak p}$ and assuming that $\alpha_q\ge\Gamma_0^-(\theta)$ and $\alpha_l>\Gamma_\infty^-(\theta)$, we obtain \begin{align} S(g) &< -\frac{\alpha_l}{4\beta^2} d_l(1-\kappa_l) + \unc \sum_{i=1}^{r} \frac{\kappa_i d_i}{\alpha_q} \\ &\le-\frac{\Gamma_\infty^-(\theta)}{4n^2T_{\mathfrak p}^2} \min_{i=1,\ldots,r+s}d_i (1-\kappa_i) + \unc \sum_{i=1}^{r} \frac{\kappa_i d_i}{\Gamma_0^-(\theta)}<-\theta. \end{align} Thus, the inequality $\alpha_l>\Gamma_\infty^-(\theta)$ implies $S(g)<-\theta$. Let $\mathcal C_{\theta}^-$ be the set of those $g\in\mathcal M_T^+$ that satisfy~\eqref{metric} with \begin{align} \min\{\Lambda_0^-(\theta),\Gamma_0^-(\theta)\}&\le\min\{\beta,\alpha_1,\ldots,\alpha_{r+s}\} \\ &\le\max\{\beta,\alpha_1,\ldots,\alpha_{r+s}\}\le\max\{nT_\pg,\Gamma_\infty^-(\theta)\}. \end{align} This set is compact. By the arguments above, $S(g)<-\theta$ whenever $g$ lies in $\mathcal M_T^+\setminus\mathcal C_{\theta}^-$. \end{proof} \begin{proof}[Proof of Theorem \ref{sufcon-}] Fix a metric $h\in\mathcal M_T^-$. Applying Lemma~\ref{lemmacompact-} with $\theta=\|S(h)\|+1$, we conclude that \begin{align} S(g) < -\|S(h)\|-1<S(h) \end{align} for all $g\in\mathcal M_T^-$ outside a compact set~$\mathcal C_\theta^-\subset \mathcal M_T^-$. Clearly, $h$ lies in $\mathcal C_\theta^-$, and the functional $S\|_{\mathcal C_\theta^-}$ attains its global maximum at some~$h_{\mathrm{mx}}\in\mathcal C_\theta^-$. This implies $S(g)\le S(h_{\mathrm{mx}})$ for all $g\in\mathcal M_T^-$. \end{proof} \subsection{Critical points on $\mca_{T}^{0}$}\label{sec_0} If $r+s=1$ in formula~\eqref{dec_k}, then \begin{align}\label{T1} T = - T_{\pg} Q\|_{\pg} + T_1 Q\|_{\kg_1} \end{align} for some $T_\pg,T_1>0$. In this case, straightforward analysis shows that $S\|_{\mathcal M_T^0}$ has no critical points unless \begin{align}\label{cr_pt_M0_1} d_1T_1^2 + 2nT_\pg (2T_1 - \kappa_1T_\pg)=0. \end{align} On the other hand, when~\eqref{cr_pt_M0_1} holds, the scalar curvature of every metric in $\mathcal M_T^0$ equals~0. If $r+s=2$, we are able to obtain a complete classification of the critical points of~$S\|_{\mathcal M_T^0}$. We present this classification in Theorem~\ref{thm_M0} below. While its statement is quite bulky, its conditions are easy to verify once the tensor field $T$ and the geometric parameters of $G$ and $K$ are given. According to Table~3 in~\cite[Section~7.H]{Bss}, the sum $r+s$ can be greater than 2 only if $(\ggo,\kg)$ is one of the pairs \begin{align} &(\sug(p,q),\sug(p)\oplus \sug(q) \oplus \RR), && 1<p \leq q, \\ &(\mathfrak{so}(4,m),\mathfrak{so}(4)\oplus \mathfrak{so}(m)), && m\ge4, \\ &(\mathfrak{so}(3,4),\mathfrak{so}(3)\oplus \mathfrak{so}(4)). \end{align} It seems difficult to classify the critical points of~$S\|_{\mathcal M_T^0}$ in these cases without using software, such as Maple, to solve the Euler--Lagrange equations numerically. Nevertheless, for all values of~$r+s$, the following result holds. \begin{proposition}\label{prop_S=0} If $g$ is a critical point of $S\|_{\mathcal M_T^0}$, then $S(g)=0$. \end{proposition} \begin{proof} According to Proposition~\ref{variational_lemma}, the fact that $g$ is a critical point of $S\|_{\mathcal M_T^0}$ implies equality~\eqref{prescribed}. Taking the trace on both sides of this equality, we obtain \begin{align} S(g)=\tr_g\Ricci (g)=c\tr_gT=0. \end{align} \end{proof} Assume that $r+s=2$ in formula~\eqref{dec_k}. For the list of $\kg$ satisfying this assumption, see Table~3 in~\cite[Section~7.H]{Bss}. Equality~\eqref{T} becomes \begin{equation}\label{T2} T = -T_{\pg} Q\|_{\pg} + T_1 Q\|_{\kg_1} + T_2 Q\|_{\kg_2}. \end{equation} It will be convenient for us to denote \begin{align} a = nd_2 (1-\kappa_2) T_{\pg},\qquad b &= d_1^2 (1-\kappa_1)T_1 - d_2^2 (1-\kappa_2) T_2 + 2n^2 T_\pg - \frac{n^2\kappa_1 T_{\pg}^2}{T_1},\\ c &=-2 n d_2 T_2 + \frac{2nd_2\kappa_1 T_\pg T_2}{T_1} - nd_2 \kappa_2 T_\pg, \\ d&=- \frac{d_2^2 \kappa_1T_2^2}{T_1} + \kappa_2 d_2^2 T_2. \end{align} The proof of Theorem~\ref{thm_M0} below shows that the variational properties of $S\|_{\mathcal M_T^0}$ are largely determined by those of the polynomial \begin{align} P(x)=ax^3+bx^2+cx+d. \end{align} The discriminant of this polynomial is \begin{align} D=18abcd-4b^3d+b^2c^2-4ac^3-27a^2d^2. \end{align} Denote \begin{align} R_t=-\frac b{3a},\qquad R_d= \frac{9ad-bc}{2(b^2-3ac)},\qquad R_s=\frac{4 a b c -9 a^2 d -b^3}{a (b^2 - 3 a c)}. \end{align} According to the classical theory of cubic equations (see, e.g.,~\cite{Jan10} for a modern interpretation), if $D=0$ and $b^2=3ac$, then $x=R_t$ is a triple root of~$P(x)$. It is also a saddle point. If $D=0$ and $b^2\ne3ac$, then $x=R_d$ and $x=R_s$ are a double root and a simple root of $P(x)$, respectively. Both are local extremum points. \begin{theorem}\label{thm_M0} Assume that $r+s=2$ in formula~(\ref{dec_k}). The scalar curvature functional $S\|_{\mca_T^0}$ does not have a global minimum. Critical points of other types exist under the following conditions: \begin{enumerate} \item A saddle if and only if \begin{align}\label{cond_saddle} D=0,\qquad b^2=3ac,\qquad \frac{d_2 T_2}{n T_\pg}<R_t. \end{align} \item A global maximum if and only if \begin{align}\label{cond_gmx} D=0,\qquad b^2\ne3ac,\qquad R_s\le\frac{d_2 T_2}{n T_\pg}<R_d. \end{align} \item A local maximum that is not a global maximum if and only if \begin{align}\label{cond_lmx} D=0,\qquad b^2\ne3ac,\qquad \frac{d_2 T_2}{n T_\pg}<R_s<R_d. \end{align} \item A local minimum if and only if \begin{align}\label{cond_lmn} D=0,\qquad b^2\ne3ac,\qquad \frac{d_2 T_2}{n T_\pg}<R_d<R_s. \end{align} \end{enumerate} When it exists, the critical point of $S\|_{\mca_T^0}$ is unique up to scaling. \end{theorem} Let us make a few remarks in preparation for the proof. Consider a metric $g\in\mathcal M_T^0$. There are $\beta,\alpha_1,\alpha_2>0$ such that \begin{equation}\label{g2} g = \beta Q\|_{\pg} + \alpha_1 Q\|_{\kg_1} + \alpha_2 Q\|_{\kg_2}. \end{equation} The equality $\tr_gT=0$ implies \begin{align} -\frac{nT_{\pg}}{\beta}+\frac{d_1T_1}{\alpha_1}+\frac{d_2T_2}{\alpha_2}=0,\qquad \frac\beta{\alpha_1}=\frac1{d_1T_1}\bigg(nT_{\pg}-\frac{d_2T_2\beta}{\alpha_2}\bigg), \qquad \frac{\alpha_2}\beta>\frac{d_2T_2}{nT_{\pg}}. \end{align} By Corollary~\ref{scalar}, \begin{align}\label{eq_sc_2id} S(g) &= \frac1{4\beta}\Big(-\frac{\alpha_1}{\beta} d_1 (1-\kappa_1)-\frac{\alpha_2}{\beta} d_2 (1-\kappa_2) - 2n + \frac{\kappa_1 d_1\beta}{\alpha_1}+ \frac{\kappa_2 d_2\beta}{\alpha_2}\Big)\notag \\ &=\frac\beta{4\alpha_2(d_2T_2\beta-nT_{\pg}\alpha_2)}P\Big(\frac{\alpha_2}\beta\Big). \end{align} Note that the factor in front of $P(\frac{\alpha_2}\beta)$ is necessarily negative. Our arguments will involve two curves, $\gamma_1$ and $\gamma_2$, in the space $\mathcal M_T^0$ given by the formulas \begin{align} \gamma_1(t)&= \beta Q\|_{\pg} - \frac{d_1T_1e^t\alpha_2\beta}{d_2T_2\beta-nT_{\pg}e^t\alpha_2} Q\|_{\kg_1} + e^t\alpha_2 Q\|_{\kg_2}, &&t>\ln\frac{d_2T_2\beta}{nT_{\pg}\alpha_2}, \\ \gamma_2(t)&=e^t\beta Q\|_{\pg} + e^t\alpha_1 Q\|_{\kg_1} + e^t\alpha_2 Q\|_{\kg_2}, &&t\in\mathbb R. \end{align} Both these curves pass through $g$ at $t=0$. \begin{lemma}\label{lem_crit=root} The metric $g$ given by~\eqref{g2} is a critical point of $S\|_{\mathcal M_T^0}$ if and only if $x=\frac{\alpha_2}\beta$ is a multiple root of~$P(x)$. \end{lemma} \begin{proof} Assume $g$ is a critical point of $ S\|_{\mathcal M_T^0}$. Proposition~\ref{prop_S=0} implies $S(g)=0$. In light of~\eqref{eq_sc_2id}, this means $x=\frac{\alpha_2}\beta$ must be a root of~$P(x)$. Furthermore, because $g$ is a critical point of $S\|_{\mca_T^0}$, \begin{align} 0&=\frac{d}{dt}S(\gamma_1(t))\|_{t=0}=\frac d{dt}\Big(\frac\beta{4e^t\alpha_2(d_2T_2\beta-nT_{\pg}e^t\alpha_2)}P\Big(\frac{e^t\alpha_2}\beta\Big)\Big)\Big\|_{t=0} \\ &=\frac d{dt}\frac\beta{4e^t\alpha_2(d_2T_2\beta-nT_{\pg}e^t\alpha_2)}\Big\|_{t=0}P\Big(\frac{\alpha_2}\beta\Big) + \frac\beta{4 \alpha_2(d_2T_2\beta-nT_{\pg}\alpha_2)} \frac d{dt}P\Big(\frac{e^t\alpha_2}\beta\Big)\Big\|_{t=0}\\ & = \frac1{4 (d_2T_2\beta-nT_{\pg}\alpha_2)}\frac{d}{dx}P(x)\|_{x=\frac{\alpha_2}\beta}. \end{align} Thus, the derivative of $P(x)$ at $x=\frac{\alpha_2}\beta$ vanishes. This proves the ``only if" part of the claim. Assume that $x=\frac{\alpha_2}\beta$ is a multiple root of $P(x)$. We need to show that $g$ is a critical point of~$S\|_{\mathcal M_T^0}$. Clearly, the vectors tangent to the curves $\gamma_1$ and $\gamma_2$ at $g$ are linearly independent. Therefore, it suffices to prove that \begin{align} \frac{d}{dt}S(\gamma_1(t))\|_{t=0}=\frac{d}{dt}S(\gamma_2(t))\|_{t=0}=0. \end{align} Computing as above, we find \begin{align} \frac{d}{dt}&S(\gamma_1(t))\|_{t=0} \\ &=\frac d{dt}\frac\beta{4e^t\alpha_2(d_2T_2\beta-nT_{\pg}e^t\alpha_2)}\Big\|_{t=0}P\Big(\frac{\alpha_2}\beta\Big) + \frac1{4(d_2T_2\beta-nT_{\pg}\alpha_2)}\frac{d}{dx}P(x)\|_{x=\frac{\alpha_2}\beta} =0. \end{align} Formula~\eqref{eq_sc_2id} implies \begin{align} \frac{d}{dt}S(\gamma_2(t))\|_{t=0}=\frac{d}{dt}S(e^tg)\|_{t=0}=\frac{d}{dt}e^{-t}S(g)\|_{t=0}=-\frac{\beta}{4\alpha_2(d_2T_2\beta-nT_{\pg}\alpha_2)}P\Big(\frac{\alpha_2}\beta\Big)=0. \end{align} \end{proof} \begin{proof}[Proof of Theorem~\ref{thm_M0}] Recalling that $P(x)$ is a cubic polynomial, we find \begin{align} \lim_{t\to\infty}S(\gamma_1(t))&=\lim_{t\to\infty}\frac\beta{4e^t\alpha_2(d_2T_2\beta-nT_{\pg}e^t\alpha_2)}P\Big(\frac{e^t\alpha_2}\beta\Big)=-\infty. \end{align} Consequently, $S\|_{\mathcal M_T^0}$ never attains its global minimum. This proves the first statement. Suppose $g$ is a saddle point of $S\|_{\mathcal M_T^0}$. Proposition~\ref{prop_S=0} implies $S(g)=0$. Moreover, every neighbourhood of $g$ in $\mathcal M_T^0$ contains a metric with negative scalar curvature and one with positive scalar curvature. By Lemma~\ref{lem_crit=root}, $x=\frac{\alpha_2}\beta$ is a multiple root of~$P(x)$. Formula~\eqref{eq_sc_2id} shows that every interval around $x=\frac{\alpha_2}\beta$ contains a point where $P(x)$ is positive and one where $P(x)$ is negative. This is only possible if $x=\frac{\alpha_2}\beta$ is a triple root. By the classical theory of cubic equations, conditions~\eqref{cond_saddle} hold. Conversely, these conditions ensure that $P(x)$ has a triple root at $x=R_t$. Consider a metric $g_{\mathrm{sdl}}\in\mathcal M_T^0$ defined by \begin{equation} g_{\mathrm{sdl}} = Q\|_{\pg}-\frac{d_1T_1R_t}{d_2T_2-nT_{\pg}R_t} Q\|_{\kg_1} + R_t Q\|_{\kg_2}. \end{equation} Lemma~\ref{lem_crit=root} implies that $g_{\mathrm{sdl}}$ is a critical point of~$S\|_{\mathcal M_T^0}$. Using~\eqref{eq_sc_2id}, one easily shows that every neighbourhood of $g_{\mathrm{sdl}}$ contains a metric with negative scalar curvature and one with positive scalar curvature. In light of Proposition~\ref{prop_S=0}, this means $g_{\mathrm{sdl}}$ is a saddle point. The functional $S\|_{\mathcal M_T^0}$ attains its global maximum if and only if $S(g_{\mathrm{gmx}})=0$ for some $g_{\mathrm{gmx}}\in\mathcal M_T^0$ and $S(h)\le0$ for all $h\in\mathcal M_T^0$. Formula~\eqref{eq_sc_2id} implies that this happens if and only if $P(x)$ has a double root in the interval $\big(\frac{d_2 T_2}{n T_\pg},\infty\big)$ and is nonnegative on this interval. Conditions~\eqref{cond_gmx} are necessary and sufficient for $P(x)$ to have such properties. Next, $S\|_{\mathcal M_T^0}$ has a local maximum that is not a global maximum if and only if $S(g_{\mathrm{lmx}})=0$ for some $g_{\mathrm{lmx}}\in\mathcal M_T^0$, $S(h)\le0$ for all $h$ in a neighbourhood of $g_{\mathrm{lmx}}$, and the scalar curvature of at least one metric in $\mathcal M_T^0$ is positive. This is equivalent to $P(x)$ having a simple root in the interval $\big(\frac{d_2 T_2}{n T_\pg},\infty\big)$ and a double root in $(R_s,\infty)$. Conditions~\eqref{cond_lmx} are necessary and sufficient for $P(x)$ to have such properties. Analogously, $S\|_{\mathcal M_T^0}$ has a local minimum if and only if $P(x)$ has a double root in $\big(\frac{d_2 T_2}{n T_\pg},\infty\big)$ and is nonpositive in a neighbourhood of this root. Conditions~\eqref{cond_lmn} are necessary and sufficient for this. Finally, in view of Lemma~\ref{lem_crit=root}, $S\|_{\mathcal M_T^0}$ can have at most one critical point up to scaling since a cubic polynomial can have at most one multiple root. \end{proof} \subsection{Summary}\label{sec_sum} The results of Sections~\ref{sec_variational}--\ref{sec_0} enable us to make several conclusions about the solvability of~\eqref{prescribed}. We summarise these conclusions in Theorem~\ref{thm_summary} below. The constant $c$ in~\eqref{prescribed} must be positive if $T$ satisfies~\eqref{T}. This is an immediate consequence of the formulas for the Ricci curvature obtained in Section~\ref{secRicci}. \begin{theorem}\label{thm_summary} Suppose $T$ is a left-invariant (0,2)-tensor field on $G$ given by~(\ref{T}). \begin{enumerate} \item If~(\ref{hyp_thm_beta_zero+}) holds, then there exists at least one pair $(g,c)\in\mathcal M_K\times (0,\infty)$ satisfying~(\ref{prescribed}). \item If both~(\ref{hyp_thm_beta_inf+}) and~(\ref{hyp_thm_beta_zero+}) hold, then there are at least two pairs $(g,c)\in\mathcal M_K\times (0,\infty)$ that satisfy~(\ref{prescribed}) and have non-homothetic metrics~$g$. \item If $r+s=2$ and conditions~(\ref{cond_saddle}), (\ref{cond_gmx}), (\ref{cond_lmx}) or~(\ref{cond_lmn}) hold, then there exists at least one pair $(g,c)\in\mathcal M_K\times (0,\infty)$ satisfying~(\ref{prescribed}). \end{enumerate} \end{theorem} \begin{proof} Statements~1 and~3 follow from Theorems~\ref{sufcon-} and~\ref{thm_M0} combined with Proposition~\ref{variational_lemma}. Next, assume that~(\ref{hyp_thm_beta_inf+}) and~(\ref{hyp_thm_beta_zero+}) hold. According to Theorems~\ref{sufcon+} and~\ref{sufcon-}, the functionals $S\|_{\mathcal M_T^+}$ and $S\|_{\mathcal M_T^-}$ attain their global maxima at some $g_1\in\mathcal M_T^+$ and $g_2\in\mathcal M_T^-$. Proposition~\ref{variational_lemma} implies that both $g_1$ and $g_2$ have Ricci curvature equal to $T$ up to scaling. These metrics cannot be homothetic because $\tr_{g_1}T$ and $\tr_{g_2}T$ are not of the same sign. \end{proof} When $r+s=1$, Theorem~\ref{thm_summary} is essentially optimal. We explain this in detail in Remark~\ref{rem_optimal}. At the same time, when $r+s\ge2$, it seems that~(\ref{hyp_thm_beta_inf+}) and~\eqref{hyp_thm_beta_zero+} may fail to hold even if $S\|_{\mca_T^+}$ and $S\|_{\mca_T^-}$ attain their global maxima. Indeed, on compact Lie groups, inequalities that are similar in spirit to these provide merely a ``linear approximation" to the necessary and sufficient conditions for the existence of a critical point; see~\cite[Section~5]{APZ20}. Different pairs $(g,c)\in\mathcal M_K\times(0,\infty)$ satisfying~\eqref{prescribed} must have distinct~$c$. More precisely, by Theorem~\ref{thm_no_c}, if \begin{align} \Ricci (g_1)=\Ricci(g_2)=cT,\qquad g_1,g_2\in\mathcal M_K, \end{align} then $g_1$ and $g_2$ are equal up to scaling. \begin{remark}\label{rem_r+s=1} The discussion at the beginning of Section~\ref{sec_0} shows that~\eqref{cr_pt_M0_1} is a sufficient condition for the solvability of~\eqref{prescribed} if $r+s=1$. \end{remark} \section{The case where $K$ is simple}\label{sec_simple} As above, let $T$ be a left-invariant (0,2)-tensor field on~$G$. Assume that $K$ is simple. Our next result settles the question of solvability of~\eqref{prescribed} under this assumption. We do not use the variational approach developed in Section~\ref{sec_variational}; however, see Remarks~\ref{rem_optimal} and~\ref{rem_sim_max} below. Since $K$ is simple, the numbers $r$ and $s$ in~\eqref{dec_k} equal~1 and~0, respectively. By Theorem~\ref{Ric}, if~\eqref{T1} holds for some $T_\pg,T_1>0$, the constant $c$ in~\eqref{prescribed} must be positive. \begin{proposition}\label{prop_simple_gr} Assume $K$ is simple. Let the tensor field $T$ satisfy~(\ref{T1}) for some $T_{\pg},T_1>0$. \begin{enumerate} \item If \begin{align}\label{sim_con1} 2nT_{\pg}(2 T_1- \kappa_1T_{\pg}) < -d_1T_1^2 \end{align} then there exists no metric $g\in\mathcal M_K$ such that~(\ref{prescribed}) holds. \item If \begin{align} 2n T_{\pg}(2 T_1- \kappa_1T_{\pg}) = -d_1T_1^2\qquad or\qquad 2T_1- \kappa_1T_{\pg} \ge 0, \end{align} then there exists precisely one pair $(g,c)\in\mathcal M_K\times(0,\infty)$, up to scaling of $g$, such that~(\ref{prescribed}) holds. \item If \begin{align} -d_1T_1^2 < 2n T_{\pg}(2 T_1- \kappa_1T_{\pg}) <0, \end{align} then there are precisely two pairs $(g,c)\in\mathcal M_K\times(0,\infty)$, up to scaling of $g$, such that~(\ref{prescribed}) holds. \end{enumerate} \end{proposition} \begin{proof} Choose a metric $g\in\mathcal M_K$. There exist $\beta,\alpha_1>0$ such that \begin{equation} g = \beta Q\|_{\pg} + \alpha_1 Q\|_{\kg_1}. \end{equation} Theorem~\ref{Ric} and formula~\eqref{sum_dkappan} imply that $g$ satisfies~\eqref{prescribed} if and only if \begin{align} \unc(\kappa_1(1 - x^2) + x^2) &= c T_1, \\ \unc(2+x) &= c T_{\pg}, \end{align} where $x=\frac{\alpha_1}{\beta}$. Clearing $c$ from the second line and substituting into the first, we obtain \begin{align} (1-\kappa_1) T_{\pg} x^2 - T_1 x + \kappa_1 T_{\pg} - 2T_1 =0. \end{align} This is a quadratic equation with discriminant \begin{align} E= T_1^2 + 4 (1 - \kappa_1) T_{\pg} (2T_1- \kappa_1 T_{\pg}). \end{align} It has no solutions if $E<0$, precisely one positive solution if $E=0$ or $E\ge T_1^2$, and precisely two positive solutions if $0<E<T_1^2$. Together with~\eqref{sum_dkappan}, this implies the result. \end{proof} \begin{remark}\label{rem_optimal} Proposition~\ref{prop_simple_gr} shows that Theorem~\ref{thm_summary} is essentially optimal in our current setting. Indeed, since $K$ is simple,~\eqref{hyp_thm_beta_zero+} becomes \begin{align} \frac{\kappa_1n^2T_{\mathfrak p}^2- (1-\kappa_1)d_1^2T_1^2}{nT_{\mathfrak p}T_1}-2n<0. \end{align} In view of~\eqref{sum_dkappan}, this is equivalent to \begin{align} 2nT_{\mathfrak p}(2T_1-\kappa_1T_{\mathfrak p})>- d_1T_1^2. \end{align} Theorem~\ref{thm_summary} and Remark~\ref{rem_r+s=1} assert that~\eqref{hyp_thm_beta_zero+} and~\eqref{cr_pt_M0_1} are sufficient conditions for the solvability of~\eqref{prescribed}. Conversely, as Proposition~\ref{prop_simple_gr} shows, the existence of a pair $(g,c)\in\mathcal M_K\times(0,\infty)$ satisfying~\eqref{prescribed} implies that either~\eqref{hyp_thm_beta_zero+} or~\eqref{cr_pt_M0_1} must hold. Theorem~\ref{thm_summary} provides lower bounds on the number of solutions to~\eqref{prescribed}. Using Proposition~\ref{prop_simple_gr}, one can easily demonstrate that these bounds are sharp. \end{remark} \begin{remark}\label{rem_sim_max} In our current setting, every metric satisfying~\eqref{prescribed} is, up to scaling, a global maximum point of $S\|_{\mca_T^+}$, $S\|_{\mca_T^-}$ or $S\|_{\mca_T^0}$. This observation follows from the results of Section~\ref{sec_Ricci=cT} and Proposition~\ref{prop_simple_gr}. \end{remark} \section{Examples}\label{sec_examples} Let us illustrate how the results of Sections~\ref{sec_Ricci=cT}--\ref{sec_simple} apply to specific groups. \begin{example}\label{exa_simple} Assume $G=\G_2^{\mathbb C}$ and $K=\G_2$. Then \begin{align} r=1,\qquad s=0,\qquad n=d_1=14. \end{align} Formula~\eqref{sum_dkappan} yields $\kappa_1=\unm$. Suppose $T$ is given by~\eqref{T1} with~$T_\pg,T_1>0$. Since $K$ is simple, Proposition~\ref{prop_simple_gr} applies. Formula~\eqref{sim_con1} becomes \begin{align} 4T_{\pg}T_1-T_{\pg}^2<-T_1^2. \end{align} Equivalently, \begin{align} \frac{T_1}{T_\pg}<(\sqrt5-2). \end{align} If this holds, then~\eqref{prescribed} has no solutions. Similarly, if \begin{align} \frac{T_1}{T_\pg}=(\sqrt5-2)\qquad \mbox{or}\qquad \frac{T_1}{T_\pg}\ge\unc, \end{align} then there is one pair $(g,c)\in\mathcal M_K\times(0,\infty)$, up to scaling of $g$, that satisfies~(\ref{prescribed}). If \begin{align} (\sqrt5-2)<\frac{T_1}{T_\pg}<\unc, \end{align} there are two such pairs. \end{example} \begin{example}\label{ex_2ideals} Assume $G=\SO^+(2,q)$ and $K=\SO(2)\times\SO(q)$ with $q\ge5$. Then \begin{align} r=1,\qquad s=1,\qquad n=2q,\qquad d_1=\frac{q(q-1)}2,\qquad d_2=1. \end{align} Using~\eqref{sum_dkappan}, we find \begin{align} \kappa_1=\frac{q-2}q,\qquad\kappa_2=0. \end{align} Suppose $T$ is given by~\eqref{T2} with~$T_\pg,T_1,T_2>0$. Inequality~\eqref{hyp_thm_beta_zero+} becomes \begin{align}\label{exam_2id} 8q(q-2)T_{\mathfrak p}^2- q(q-1)^2T_1^2-2T_1T_2-16q^2T_\pg T_1<0. \end{align} According to Theorem~\ref{sufcon-}, if~\eqref{exam_2id} holds, then $S\|_{\mca_T^-}$ attains its global maximum. In this case, there exists a metric~$g\in\mca_T^-$ with Ricci curvature~$cT$ for some~$c>0$. Inequality~\eqref{hyp_thm_beta_inf+} takes the form \begin{align}\label{exam2plus} -2q(q-2)T_\pg+q(4q+1)T_1+(q-2)T_2<0. \end{align} The set of triples~$(T_\pg,T_1,T_2)$ for which both~\eqref{exam_2id} and~\eqref{exam2plus} are satisfied is non-empty and open in~$\mathbb R^3$. We depict it in Figure~1 for $q=10$. We also indicate where~\eqref{exam_2id} holds without~\eqref{exam2plus}. Because these inequalities are invariant under scaling of~$(T_\pg,T_1,T_2)$, we make our sketch assuming~$T_\pg=1$. By Theorem~\ref{sufcon+}, if both~\eqref{exam_2id} and~\eqref{exam2plus} are satisfied, then $S\|_{\mca_T^+}$ attains its global maximum. In this case, there exists a metric $g\in\mca_T^+$ with Ricci curvature $cT$ for some~$c>0$. Theorem~\ref{thm_M0} enables us to classify the critical points of~$S\|_{\mca_T^0}$. For instance, suppose $q=5$ and $(T_\pg,T_1,T_2)=(1,1,1)$. Then the discriminant $D$ of the polynomial $P(x)$ satisfies \begin{align} D=\tfrac{79948008}5\ne0. \end{align} By Theorem~\ref{thm_M0}, $S\|_{\mca_T^0}$ has no critical points. To give another example, suppose $q=5$ and $(T_\pg,T_1,T_2)=\big(1,\tfrac15,\eta\big)$, where $\eta$ is the unique positive root of the polynomial \begin{align} R(x)= -12x^3-4412x^2-583104x+4198144. \end{align} Then $D=\eta^2R(\eta)=0$. Moreover, using an approximate value of $\eta$ calculated in Maple, we find $b^2\ne3ac$ and \begin{align} R_s=\frac{\eta^3+1376\eta^2-121808\eta+778688}{10\eta^2-10160\eta+84640}<\frac{\eta}{10}<\frac{115\eta(16-\eta)}{\eta^2-1016\eta+8464}=R_d. \end{align} This means $S\|_{\mca_T^0}$ attains its global maximum. One can use Maple to produce the graphs of $S\|_{\mca_T^-}$, $S\|_{\mca_T^+}$ and~$S\|_{\mca_T^0}$; cf.~\cite[Section~5]{APZ20}. \end{example} \begin{example} Assume $G=\SU(p,q)$ and $K=\SU(p)\times\U(q)$ with $2\le p\le q$. Then \begin{align} r=2,\qquad s=1,\qquad n=2pq,\qquad d_1=p^2-1,\qquad d_2=q^2-1,\qquad d_3=1. \end{align} As we showed in Example~\ref{exa_kappa}, \begin{align} \kappa_1=\frac p{p+q},\qquad\kappa_2=\frac q{p+q},\qquad\kappa_3=0. \end{align} Suppose \begin{align} T = - T_{\pg} Q\|_{\pg} + T_1 Q\|_{\kg_1}+ T_1 Q\|_{\kg_3}+ T_3 Q\|_{\kg_3} \end{align} for some~$T_\pg,T_1,T_2,T_3>0$. Inequality~\eqref{hyp_thm_beta_zero+} becomes \begin{align}\label{exam_3id} 4p^3q^2T_{\mathfrak p}^2T_2&-(p^2-1)^2qT_1^2T_2+4p^2q^3T_{\mathfrak p}^2T_1 \notag \\ &-(q^2-1)^2pT_1T_2^2-(p+q)T_1T_2T_3-8p^2q^2(p+q)T_\pg T_1T_2<0. \end{align} By Theorem~\ref{sufcon-}, if~\eqref{exam_3id} holds, then $S\|_{\mca_T^-}$ attains its global maximum. This is the case, e.g., when \begin{align} (T_\pg,T_1,T_2,T_3)=(1,1,1,4p^2q^2). \end{align} Assume for simplicity that $T_1\ge T_2$. Then~\eqref{hyp_thm_beta_inf+} takes the form \begin{align}\label{exam_3max} -2pq^2T_\pg+q(p^2-1)T_1+qT_3 &<(p^3-5p^2q-4pq^2-2p)T_2. \end{align} By Theorem~\ref{sufcon-}, if both~\eqref{exam_3id} and this inequality hold, then $S\|_{\mca_T^+}$ attains its global maximum. This happens, e.g., for $(p,q)=(2,12)$ and $(T_\pg,T_1,T_2,T_3)=\big(1,1,\tfrac38,1\big)$. \end{example} \end{document}	arXiv
Non-uniqueness of factors constraint on the codon usage in Bombyx mori Xian Jia†1, Shuyu Liu†2, Hao Zheng†1, Bo Li3, Qi Qi1, Lei Wei1, Taiyi Zhao4, Jian He2 and Jingchen Sun1Email author https://doi.org/10.1186/s12864-015-1596-z © Jia et al.; licensee BioMed Central. 2015 Received: 11 January 2015 The analysis of codon usage is a good way to understand the genetic and evolutionary characteristics of an organism. However, there are only a few reports related with the codon usage of the domesticated silkworm, Bombyx mori (B. mori). Hence, the codon usage of B. mori was analyzed here to reveal the constraint factors and it could be helpful to improve the bioreactor based on B. mori. A total of 1,097 annotated mRNA sequences from B. mori were analyzed, revealing there is only a weak codon bias. It also shows that the gene expression level is related to the GC content, and the amino acids with higher general average hydropathicity (GRAVY) and aromaticity (Aromo). And the genes on the primary axis are strongly positively correlated with the GC content, and GC3s. Meanwhile, the effective number of codons (ENc) is strongly correlated with codon adaptation index (CAI), gene length, and Aromo values. However, the ENc values are correlated with the second axis, which indicates that the codon usage in B. mori is affected by not only mutation pressure and natural selection, but also nucleotide composition and the gene expression level. It is also associated with Aromo values, and gene length. Additionally, B. mori has a greater relative discrepancy in codon preferences with Drosophila melanogaster (D. melanogaster) or Saccharomyces cerevisiae (S. cerevisiae) than with Arabidopsis thaliana (A. thaliana), Escherichia coli (E. coli), or Caenorhabditis elegans (C. elegans). The codon usage bias in B. mori is relatively weak, and many influence factors are found here, such as nucleotide composition, mutation pressure, natural selection, and expression level. Additionally, it is also associated with Aromo values, and gene length. Among them, natural selection might play a major role. Moreover, the "optimal codons" of B. mori are all encoded by G and C, which provides useful information for enhancing the gene expression in B. mori through codon optimization. Bombyx mori Codon usage bias Codon optimization Codon usage bias refers to differences of the occurrence frequency of synonymous codons in coding DNA. It is considered to be a product of mutation pressure and/or natural selection [1-4], and accounts for accurate and efficient translation, as well as mutation–selection–drift [5]. Codon bias analysis has been introduced into both prokaryotes and eukaryotes, such as Escherichia coli (E. coli), Arabidopsis thaliana (A. thaliana), and human beings [6-9], showing that codon bias has a high correlation to gene length, gene function, hydrophobicity of proteins, and the content of iso-acceptor tRNAs in genomes [9-12]. Hence, the analysis of codon usage can be used to study organism evolution and improve protein expression level [13-15]. The domesticated silkworm, Bombyx mori (B. mori), is a well-studied lepidopteran model system with rich genetic and molecular information of morphology, development, and behavior [13]. So far, the draft sequence for the genome of B. mori has been determined [16], and most studies of B. mori focus on the cloning, expression, and characterization of some genes or application as the bioreactor [17-19]. As we know, the analysis of codon usage is a good way to understand the genetic and evolutionary characteristics of B. mori. It can also help us to study the relationship between expression levels and codon usage bias since highly-expressed genes need abundant ribosomes and matching tRNAs for efficient translation. We have reported the codon usage bias of the mitochondrial genome in B. mori recently [20], however, the codon usage bias in the whole nuclear genome of B. mori is not well investigated in detail. Considering its great potential for expressing foreign proteins as a bioreactor, the codon usage bias of B. mori was examined here for codon optimization of genes. B. mori reveals a weak codon bias As shown in Additional file 1 and Table 1, the GC content for the total 1, 097 genes varies from 29.5% to 69.5%, with a mean value of 46.43%. The GC content of the total genes is distributed mainly between 40% and 50% (Figure 1). The greatest differences of GC content are found in the first and the third codon positions (51.92% and 48.40%, respectively), where most neutral mutations occur [21]. Means and standard deviations of GC, GC1, GC2, GC3, GC3s, ENc, CAI, A3s, T3s, C3s, G3s, Gravy, and Aromo of codons from Bombyx mori Codons GC (%) GC1 (%) GC3 S (%) A3s (%) T3s (%) C3s (%) G3s (%) Aromo 46.43 ± 6.72 48.40 ± 13.78 −0.35 ± 0.36 0.09 ± 0.03 Note: RP indicates the ribosomal protein. The distribution of GC contents in the CDS of Bombyx mori. The effective number of codons (ENc) in B. mori ranges from 30.06 to 61.00, with an average of 53.12. As shown in Additional file 1, among the 1, 097 genes, only 5 genes reveal a high codon bias (ENc < 35). It indicates that B. mori exhibits a general random codon usage, without strong codon bias. Similarly, the relative synonymous codon usage (RSCU) values of 59 sense codons also support the conclusion that B. mori has a weak codon bias. As shown in Table 2, approximately half of the codons (28/59), denoted in bold lettering, are frequently used, such as GCU and AGA which encode Ala and Arg, respectively. Codon usage of Bombyx mori genes (363,313 codons) Codon Total Count RSCU Gln GGU AUU UUG CUU GUU UGU UAU Note: 1. Count indicates the number of codons. 2. The preferentially used codons are displayed in bold. 3. Hydrophobic and hydrophilic amino acids are listed on the left and right sides of the table, respectively. In addition, most of preferentially used codons end with A/U (A/U-ended: G/C-ended=18:10). This phenomenon was also found in many other AT-rich species, such as Pichia pastoris (P. pastoris), Saccharomyces cerevisiae (S. cerevisiae), Kluyveromyces lactis (K. lactis), and Plasmodium falciparum (P. falciparum) [22,23]. Effects of nucleotide composition in shaping codon bias Correspondence analysis of the RSCU values was used here, which removes the variation caused by the unequal usage of amino acids (although the degrees of freedom are reduced to 40 [24]), generating a first axis that explains 24.51% of the data inertia. The second axis explains 7.46%, while the next two axes respectively account for 4.02% and 3.39% of the data (Figure 2). Moreover, multivariable correlation analysis was introduced here to study the relationship between relative codon bias and nucleotide composition (Table 3). The relative and cumulative inertia of the first 20 factors from a correspondence analysis (COA) of the RSCU values. Correlation coefficients between the positions of genes along the first two major axes with index of total genes' codon usage and synonymous codon usage bias GC3s 0.137 −0.099 −0.066* 0.075* Note: ** p < 0.01. * p < 0.05. Although the first axis can't explain the whole variation, there is an obvious positive correlation between the first axis and G3s, C3s, and GC3s (r=0.343, 0.439, and 0.446, respectively, p < 0.01). However, the correlations between the first axis and A3s or T3s are negative (r=−0.444 and r=−0.367, respectively, p < 0.01). Then all the genes were classified into three categories by their GC content (GC < 45%, 45% ≤ GC < 60%, and GC ≥ 60%). As shown in Figure 3A, the position of each gene was marked along the first two major axes. Interestingly, the genes of GC < 45% are scattered at the left side of the first axis, while most of the genes with GC ≥ 60% are located at the right side of the first axis. The genes whose GC contents range from 45% to 60% are found in the middle of the plot. Additionally, almost all the ribosome genes are located in the range of GC ≥ 60%, implying that the expression level might be related with the GC content in B. mori. Correspondence analysis of RSCU for the total genes in Bombyx mori. A) Distribution of the total genes in Bombyx mori on the plane corresponding to the coordinates on the first and second principal axes was shown in Panel A. Orange triangles, purple triangles and gray triangles, indicate genes with a GC content higher than or equal to 60%, more than or equal to 45%, but less than 60% and less than 45%, respectively. Additionally, blue squares indicate the coordinates of ribosome on the first and second principal axes. B) Distribution of codons on the same two axes was shown in Panel B. Codons ending with A, U, C and G are shown in red, green, yellow, and blue, respectively. On the overall consideration of Tables 1 and 3, it seems that the genes containing lower GC3s and GC content values tend to distribute at the left side of the first axis. Thus, we speculated that G/C-ending codons could be clustered at the positive side whereas A/U-ending codons gather at the negative side of first major axis. The corresponding distribution plot of synonymous codons ending with different bases along the two axes was implemented under the above mentioned assumption. The result indicates that the separation of codons on the first axis reflects the difference between the frequencies of A/U and C/G ending codons, while that on the second axis represents the frequency differences between A/G and U/C ending codons (Figure 3B), which is consistent with the above-mentioned hypothesis. On the other hand, the ENc values show no significant correlation with the first axis (r=0.055, p > 0.05) or GC3s (r=0.037, p > 0.05) values, but a significant positive correlation with the second axis (r=0.184, p < 0.01) (Table 3). The results above suggest that nucleotide composition has an effect on separating the genes along the first major axis, however, it might be not the main factor in shaping the codon bias. GC3s plays a minor role in shaping the codon bias of B. mori ENc-plot is an effective tool to study the codon usage patterns, and it was used here to explore the influence of GC3s on the codon bias of B. mori. As shown in Figure 4, most genes are located below the expected ENc-plot curve while only a small number of genes lay on or above the curve. It indicates that the conditional mutation might be a factor in shaping the codon bias but not the unique one. The ENc plotted against GC3s. ENc denotes the effective number of codons, and GC3s denotes the GC content on the third synonymous codon position. Black boxes, and blue triangles indicate ribosome genes and total genes, respectively. The red solid line represents the expected curve of positions of genes when the codon usage was only determined by the GC3s composition. We also estimated the difference between the observed and the expected ENc values using the plot of the frequency distribution of (ENCexp-ENCobs)/ENCexp in total genes (Figure 5). There was a similar single peak for each kind of genes. Peaks located within the 0 ~ 0.1 range of (ENCexp-ENCobs)/ENCexp values suggest that most actual ENc values are smaller than the ENc values from their GC3s. It is consistent with the results depicted in Figure 4, which shows that the difference in codon bias is dependent upon the differences in GC3s, thereby providing further evidence that GC3s works as a conditional mutational bias. Frequency distribution of (ENCexp-ENCobs)/ENCexp. Natural selection influences the codon bias as a major role Although ENc plot can quantify the codon usage bias of synonymous codons, it is not sufficient to easily distinguish the main determinant factor between natural selection and mutational pressure within a species [25]. Therefore, a neutrality plot was implemented here. The neutrality plot shows that the genes have a wide range of GC3 value distributions, ranging from 19.7% to 93.8% (Figure 6). There is a significant positive correlation between GC12 and GC3 (r=0.394, p < 0.01), suggesting that the effect of directional mutation pressure is present at all codon positions. Moreover, the slope of the regression line of the entire coding sequence is 0.1452. The results reveal that the effect of directional mutation pressure is only 14.52%, while the influence of other factors, for example natural selection, is 85.48% [26]. Accordingly, mutation bias only plays a minor role in shaping the codon bias, whereas natural selection probably dominates the codon bias. Neutrality plot analysis of the GC12 and that of the third codon position (GC3) for the entire coding DNA sequence of Bombyx mori. GC12 stands for the average value of GC content in the first and second position of the codons (GC1 and GC2). While GC3 refers to the GC content in the third position. The solid line is the linear regression of GC12 against GC3, R2=0.1534, P < 0.001. Codon usage bias in B. mori has a high correlation to aromaticity and gene length In order to assess the relationship between the codon usage bias and hydrophobicity or aromaticity or gene length in B. mori, correlation analysis was performed. It could be observed from Table 3 that neither the Gravy values nor the Aromo values have significant correlation with GC3s. However, the Aromo values exhibit strongly positive correlation with the ENc values (r=0.100, p < 0.01), while the GRAVY values do not. The results indicate that the Aromo values are associated with the codon usage bias of B. mori. The data in Table 3 also reveal that the gene length is positively correlated with the ENc values (r=0.079, p < 0.01), suggesting that gene length has a high correlation to the codon usage bias and might be also one of the factors contributing to the codon usage bias in genes. Effects of gene expression level To explore the relationship between codon bias and gene expression level, correlation coefficients were calculated between the codon adaptation index (CAI) values and several other characteristics of the genes, including their position along the first major axis, the nucleotide composition, and the ENc values. Ribosome genes sequences were selected as the reference of highly expressed genes [15]. The results indicate that CAI, which represents gene expression level, shows significant negative correlation with the gene length (r=−0.148, p < 0.01), GC2 (r=−0.081, p < 0.01), A3s (r=−0.444, p < 0.01), T3s (r=−0.061, p < 0.05), Gravy (r=−0.170, p < 0.01), Aromo (r=−0.140, p < 0.01), and ENc (r=−0.210, p < 0.01). However, CAI shows obvious positive correlation with the first axis and the other nucleotide composition indices (i.e. GC, GC1, GC3, GC3s, C3s, and G3s, as shown in Table 3). The results above indicate that both nucleotide composition and gene expression levels are the major factors in shaping the codon usage bias of B. mori. To statistically measure the relationship between the index of amino acid composition in B. mori and their codon bias, the correlation coefficients between the positions of the genes along the first four major axes with their indices of amino acid usage were analyzed using Spearman's rank correlation analysis method and shown in Table 4. Correlation coefficients between the positions of genes along the first four major axes with index of total genes' amino acid usage The first four axes generated by the correspondence analysis explain 40.31% of the amino-acid variation. And the first axis accounts for 13.90% of the variation in amino-acid usage (Figure 7 ). The genes on these axes are all highly correlated with CAI, GRAVY score and Aromo value. The principle factor is negatively correlated with CAI (r=−0.216, p < 0.01), and is positively correlated with the GRAVY score and Aromo value (r=0.327, p < 0.01; r=0.208, p < 0.01, respectively). The second axis accounts for 10.95%, and is also correlated with the three indexes (r=−0.208, p < 0.01; r=0.545, p < 0.01; r=0.594, p < 0.01, respectively). The relative and cumulative inertia of the first 20 factors from a correspondence analysis (COA) of the amino acid usage frequencies. As in E.coli [27] where the most important trend in the amino-acid usage of B. mori is the usage of hydrophobicity, and the second important trend is the usage of CAI followed by the aromatic amino-acid. Taken all these together, it provides strong evidence for the inference that the effective selection of amino-acid for translational efficiency exists in B. mori. In summary, the codon usage bias in B. mori is in some way or other, affected by nucleotide composition, mutation pressure, natural selection, and gene expression level. Additionally, it is also associated with Aromo values, and gene length. However, natural selection might play a major role in shaping codon usage variation, manifesting itself though weaker codon usage bias. The selection of amino-acid could also affect the translational efficiency in B. mori. Translational optimal codons of B. mori In order to give a reference to enhance the expression level of important proteins with codon optimization, a two-way Chi-squared contingency test was used to compare the codon usage of different genes. Finally, the total putative optimal codons of B. mori are listed in Table 5. For the total genes group, the optimal codons all ended by G or C, and all amino acids—excluding Met and Trp—were identified by different numbers of codons. For example, three codons were identified for Ser, and two codons were identified for Ala, Gly, Leu, Pro, Val, Thr, and Arg. The remaining amino acids were identified by one codon. Optimal codons of genes in Bombyx mori GCC * AAC * GCG * CAG * UUC * UCC * GGC * UCG * GGG * AGC * AUC * ACC * ACG * GAC * CUC * GAG * CUG * CAC * CCC * AAG * CCG * CGC * GUC * CGG * GUG * UGC * UAC * Note: N is codon frequency, RSCU is relative synonymous codon usage. The codon usage of eleven genes (5% of the total number of genes) from the extremes of the principal were pooled. The codon usage of both pools was compared using a two-way Chi squared contingency test, to identify optimal codons. For the purposes of this test dataset with the lower ENc were putatively assigned as highly expressed. The codon usage and RSCU of both datasets is shown. Those codons that occur significantly more often (p < 0.01) in the highly biased dataset relative to the lower biased dataset are putatively considered optimal, and are indicated with a (*). The optimization of codon usage allows improving the translational efficiency of foreign proteins by replacing the codons which are rarely found in the host organism [28], and it has been introduced into many heterologous systems [29-31]. As we found in this study, the optimal codons of B. mori are all ended by either G or C. This phenomenon is interesting and important to enhance the expression level of foreign proteins in B. mori. Comparison of codon preferences between B. mori and other model organisms The ratio of codon frequency in B. mori was compared with five model organisms, including A. thaliana, C. elegans, Drosophila melanogaster (D. melanogaster), S. cerevisiae, and E. coli. The codon with a ratio of greater than 2, or less than 0.5, is defined as the indicative codon, of which usage frequency is markedly distinct from that of B. mori. As shown in Additional file 2, there are six and seven codons revealing distinct usage differences between B. mori and D. melanogaster, S. cerevisiae, respectively. However, there are only one, two or three codons with distinct usage between B. mori and A. thaliana, or E. coli, or C. elegans, respectively. It suggests that the discrepancy in codon preferences between B. mori and D. melanogaster or S. cerevisiae is relatively greater than that comparing with A. thaliana, or E. coli, or C. elegans. This finding implies that B. mori might have some advantages in expressing foreign proteins from certain organisms with fewer preferences in codon usage. After a series of analyses, the codon usage bias in B. mori is found to be weaker. And it is affected by nucleotide composition, mutation pressure, natural selection, and gene expression level. Additionally, it is also associated with Aromo values, and gene length. However, natural selection might play a major role in shaping the codon usage variation. In addition, it is also found that B. mori has a greater relative discrepancy in codon preferences in comparison with D. melanogaster or S. cerevisiae than with A. thaliana, E. coli, or C. elegans. In summary, our analysis provides insights into the codon usage pattern in B. mori and is of the benefit to express foreign proteins in B. mori as a bioreactor. Sequence collection Accession numbers for a total of 1,213 reference sequences (RefSeq) of B. mori were obtained from Silkworm Genome Database (ftp://silkdb.org/pub/current/otherdata/Refseq/silkref.seq) (Downloaded on 1-Sep-2014). Coding DNA sequences (CDS) were downloaded from GenBank (http://www.ncbi.nlm.nih.gov). In these sequences, we only chose the CDSs without unidentified bases. To improve the quality of sequences and minimize sampling errors, genes without correct initiation and termination codons or with internal termination codons were ruled out. Additionally, only genes greater than 300 nucleotides in length were used for further analysis. As we only collected the CDSs from nuclear genome, 13 mitochondrial genes were excluded from the analysis. CDSs with gaps were also excluded. Finally, 1,097 CDSs were left for analysis, and each corresponds to a unique gene in B. mori. Indices of codon usage and synonymous codon usage bias GC3s is a useful parameter for evaluating the degree of base composition bias, and represents the frequency of either a guanine or cytosine at the third codon position of synonymous codons, excluding Met, Trp, and stop codons. GRAVY (General Average Hydropathicity) values are calculated as a sum of the hydropathy values of all the amino acids in the gene product divided by the number of residues in the sequence [32]. The more negative the GRAVY value, the more hydrophilic the protein, while the more positive the GRAVY value, the more hydrophobic the protein. Aromo values denote the frequency of aromatic amino acids (Phe, Tyr, Trp) in the hypothetical translated gene product. The index and GRAVY value have been used to quantify the major COA trends in the amino acid composition of E. coli genes [27]. RSCU (relative synonymous codon usage) is the ratio of the observed frequency of codons relative to the expected frequency of the codon under a uniform synonymous codon usage. The RSCU value would be greater than 1.0 when the observed frequency is larger than the expected frequency [33]. ENc (Effective Number of Codons) values, varying from 20 to 61, are used to measure the magnitude of codon bias for individual genes, though it is worth noting that ENc values are affected by base composition [34]. A value of 20 indicates a gene with extreme bias using only one codon per amino acid, while a value of 61 indicates the absence of bias. In general, a gene is thought to possess strong codon bias if its ENc value is lower than 36 [35]. CAI (Codon Adaptation Index) values are often used to measure the extent of bias toward codons which are known to be preferred in highly expressed genes. With values ranging from 0 to 1.0, the higher the value, the stronger the codon usage bias and the higher the expression level. The set of sequences used to calculate CAI values in this study were the genes coding for ribosomal proteins in B. mori [35], so that it can provide an indication of gene expression level under the assumption that translational selection can optimize gene sequences according to their expression levels. These noted values and parameters were all utilized in this study. All the indices of total genes and ribosomal protein genes are shown in Additional file 1, respectively. ENc-plot The ENc-plot is a general strategy to investigate patterns of synonymous codon usage, where the expected ENc values are plotted against GC3s values. Expected ENc values were calculated according to Equation 1. In genes where codon choice is constrained only by a G + C mutation bias, predicted ENc values will lie on or around the GC3s curve, whereas if other factors such as selection effects are present, the values will deviate considerably below the expected GC3s curve [35]. $$ \mathrm{E}\mathrm{N}\mathrm{c}=2+\mathrm{S}+\left(29/\left({\mathrm{S}}^2+{\left(1\hbox{-} \mathrm{S}\right)}^2\right)\right) $$ S is the frequency of G + C (i.e. GC3s) Neutrality plot A neutrality plot (GC12-GC3) [26] was used to estimate and characterize the relationships amongst the three positions in B. mori codons. A plot regression with a slope of 0 indicates no effects of directional mutation pressure (complete selective constraints), while a slope of 1 is indicative of complete neutrality. Determination of optimal codons Based on axis 1 ordination, the top and bottom 5% of genes were regarded as the high and low datasets, respectively. Codon usage was compared using a two-way Chi-squared contingency test to identify optimal codons. The test dataset with the lower ENc values were putatively assigned as highly expressed, and those codons which occur significantly more often (p < 0.01) were defined as optimal codons [24]. Correspondence analysis of RSCU Correspondence analysis (COA) is a widely used method in the multivariate statistical analysis of codon usage patterns. Since there are a total of 59 synonymous codons (including 61 sense codons, minus the unique Met and Trp codons), the degrees of freedom was reduced to 40 in removing variations caused by the unequal usage of amino-acids while generating a correspondence analysis of RSCU [24]. Mobyle server (http://mobyle.pasteur.fr), including CodonW (Ver.1.4.4) (http://mobyle.pasteur.fr/cgi-bin/portal.py#forms::CodonW), CHIPS (http://mobyle.pasteur.fr/cgi-bin/portal.py#forms::chips), and CUSP (http://mobyle.pasteur.fr/cgi-bin/portal.py#forms::cusp), were used to calculate useful indices of codon usage bias, such as GC, GC3s (G + C content at the third position of codons), silent base compositions (i.e. A3s, T3s, C3s, and G3s, which indicate the frequency that codons have an A, U, C, or G, respectively, at their synonymous third position), GRAVY values (general average hydropathicity), Armoro values (aromaticity), RSCU (relative synonymous codon usage), and ENc (effective number of codons). Similarly, a COA (correspondence analysis) was also performed. CAI (codon adaptation index) and gene length were calculated using the CAI calculate server (http://genomes.urv.es/CAIcal). GC1, GC2, and GC3 values were also calculated to determine the GC content at the first, second, and third codon positions, respectively. Together these indices allow for an assessment of the level to which selection has been effective in shaping codon usage [33]. Codon preferences of other organisms were downloaded from the Codon Usage Database (http://www.kazusa.or.jp/codon) for comparison. Correlations between codon usage variations amongst indices of codon usage were carried out using the multi-analysis software SPSS Version 22.0 (SPSS Inc. software, Chicago, Illinois, USA) and GraphPad Prism 5 (GraphPad Software, San Diego, California, USA). Xian Jia, Shuyu Liu and Hao Zheng contributed equally to this work. CDS: Coding DNA sequences GC12: Average value of GC content in the first and second position of the codons GC3: GC content in the third position GC3s: GC content on the third synonymous codon position GRAVY: General average hydropathicity Aromo: Aromaticity RSCU: Relative synonymous codon usage Codon adaptation index Effective number of codons COA: Correspondence analysis This work was supported by funds from the National Natural Science Foundation of China (31372373, 31172263 to JS), the Natural Science Foundation of Guangdong Province, China (S2013010016750 to JS), Specialized Research Fund for the Doctoral Program of Higher Education, China (20134404110023 to JS), Science and Technology Planning Project of Guangdong Province, China (2013B090500118 to JS) and Science and Technology Planning Project of Guangzhou, China (1563000110 to JS). We also thank Donghua Yan (South China Agricultural University, Guangzhou), Yao Ping (South China Agricultural University, Guangzhou), Dr. Fangluan Gao (Fujian Agriculture and Forestry University, Fuzhou) and Dr. Hui Song (Lanzhou University, Lanzhou) for critical and helpful suggestions on the data collection. We also give thanks to Dr. Jonathan Jih (University of California, Los Angeles) for revising the grammar of the manuscript. Additional file 1: All the indices of total genes and ribosomal protein genes. Additional file 2: Comparison of codon preference between B. mori and other model organisms. B/A, B/C, B/D, B/S, and B/E indicate the ratio between the frequency of codon usage in B. mori and A. thaliana, C. elegans, D. melanogaster, S. cerevisiae, as well as E.coli, respectively. Ratios larger than or equal to 2, or less than or equal to 0.5 are in bold. XJ, SL, HZ and JS coordinated the project. XJ, HZ and JS wrote the manuscript. XJ performed the data analysis. 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Publisher's Erratum Erratum to: Statistical methodology for age-adjustment of the GH-2000 score detecting growth hormone misuse Dankmar Böhning1, Walailuck Böhning2, Nishan Guha2,3, David A. Cowan4, Peter H. Sönksen2 & Richard I. G. Holt2 The original article was published in BMC Medical Research Methodology 2016 16:147 After publication of the original article [1], it came to the authors' attention that there was an error affecting equation (7). A correction to this equation was submitted by the authors during proofing, but was not implemented correctly by the Production team. The correct version of equation (7) is as follows: $$ \begin{array}{l}\widehat{\beta^{}}=\frac{{\displaystyle \sum_{i=1}^n\left({Y}_i^{}-\overline{Y*}\right)\left({x}_i-\overline{x}\right)}}{{\displaystyle \sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}}=\frac{{\displaystyle \sum_{i=1}^n\left({Y}_i-\widehat{\beta}{x}_i-\left(\overline{Y}-\widehat{\beta}\overline{x}\right)\right)}\left({x}_i-\overline{x}\right)}{{\displaystyle \sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}}\\ {}\kern2.5em =\frac{{\displaystyle \sum_{i=1}^n\Big({Y}_i-\overline{Y}}\left)\left({x}_i-\overline{x}\right)-\widehat{\beta}{\displaystyle \sum_{i=1}^n\Big({x}_i-\overline{x}}\right){}^2}{{\displaystyle \sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}}\\ {}\kern2.5em =\frac{{\displaystyle \sum_{i=1}^n\Big({Y}_i-}\overline{Y}\Big)\left({x}_i-\overline{x}\right)}{{\displaystyle \sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}}-\widehat{\beta}\frac{{\displaystyle \sum_{i=1}^n\Big(}{x}_i-\overline{x}\Big){}^2}{{\displaystyle \sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}}=0.\end{array} $$ Equation (7) has also been updated in the original article in order to rectify this publisher's error. Böhning D, Böhning W, Guha N, Cowan DA, Sönksen PH, Holt RI. Statistical methodology for age-adjustment of the GH-2000 score detecting growth hormone misuse. BMC Med Res Methodol. 2016;16:147. doi:10.1186/s12874-016-0246-8. Southampton Statistical Sciences Research Institute, University of Southampton, Southampton, SO17 1BJ, UK Dankmar Böhning Human Development and Health Academic Unit, University of Southampton Faculty of Medicine, IDS Building (MP887), Southampton General Hospital, Tremona Road, Southampton, SO16 6YD, UK Walailuck Böhning, Nishan Guha, Peter H. Sönksen & Richard I. G. Holt Nuffield Division of Clinical Laboratory Sciences, UK Department of Clinical Biochemistry Level 4, University of Oxford, John Radcliffe Hospital Headley Way, Headington, Oxford, OX3 9DU, UK Nishan Guha Department of Pharmacy and Forensic Science, Drug Control Centre, King's College London, 150 Stamford Street, London, SE1 9NH, UK David A. Cowan Walailuck Böhning Peter H. Sönksen Richard I. G. Holt Correspondence to Dankmar Böhning. Böhning, D., Böhning, W., Guha, N. et al. Erratum to: Statistical methodology for age-adjustment of the GH-2000 score detecting growth hormone misuse. BMC Med Res Methodol 16, 164 (2016). https://doi.org/10.1186/s12874-016-0262-8	CommonCrawl
data descriptors Elaboration of a new framework for fine-grained epidemiological annotation Data curation during a pandemic and lessons learned from COVID-19 Moritz U. G. Kraemer, Samuel V. Scarpino, … John S. Brownstein Crowdsourcing and machine learning approaches for extracting entities indicating potential foodborne outbreaks from social media Dandan Tao, Dongyu Zhang, … Hao Feng Quantifying the information in noisy epidemic curves Kris V. Parag, Christl A. Donnelly & Alexander E. Zarebski A real-time spatio-temporal syndromic surveillance system with application to small companion animals Alison C. Hale, Fernando Sánchez-Vizcaíno, … Peter J. Diggle A global dataset of pandemic- and epidemic-prone disease outbreaks Juan Armando Torres Munguía, Florina Cristina Badarau, … Konstantin M. Wacker Getting the most out of noisy surveillance data Lauren McGough Completeness of open access FluNet influenza surveillance data for Pan-America in 2005–2019 Ryan B. Simpson, Jordyn Gottlieb, … Elena N. Naumova Machine-learned epidemiology: real-time detection of foodborne illness at scale Adam Sadilek, Stephanie Caty, … Evgeniy Gabrilovich Leveraging artificial intelligence for pandemic preparedness and response: a scoping review to identify key use cases Ania Syrowatka, Masha Kuznetsova, … David W. Bates Data Descriptor Sarah Valentin1,2,3,4, Elena Arsevska2,5, Aline Vilain6, Valérie De Waele ORCID: orcid.org/0000-0002-5124-12057, Renaud Lancelot2,5 & Mathieu Roche ORCID: orcid.org/0000-0003-3272-85681,5 Scientific Data volume 9, Article number: 655 (2022) Cite this article Event-based surveillance (EBS) gathers information from a variety of data sources, including online news articles. Unlike the data from formal reporting, the EBS data are not structured, and their interpretation can overwhelm epidemic intelligence (EI) capacities in terms of available human resources. Therefore, diverse EBS systems that automatically process (all or part of) the acquired nonstructured data from online news articles have been developed. These EBS systems (e.g., GPHIN, HealthMap, MedISys, ProMED, PADI-web) can use annotated data to improve the surveillance systems. This paper describes a framework for the annotation of epidemiological information in animal disease-related news articles. We provide annotation guidelines that are generic and applicable to both animal and zoonotic infectious diseases, regardless of the pathogen involved or its mode of transmission (e.g., vector-borne, airborne, by contact). The framework relies on the successive annotation of all the sentences from a news article. The annotator evaluates the sentences in a specific epidemiological context, corresponding to the publication date of the news article. Measurement(s) Precision/Recall • Annotator aggreement Technology Type(s) Expert annotation • Kappa's coefficient • Natural Language Processing Sample Characteristic - Organism Animal • African swine fever virus • Avian influenza • Foot-and-mouth disease virus • Bluetongue virus • Bovine spongiform encephalopathy Sample Characteristic - Environment Animal infectious diseases Background & Summary In this paper, we first describe the needs for developing a new annotation framework by highlighting the limitations of the current approaches and available resources. We further describe our global protocol for guideline elaboration, followed by a detailed description of the final annotation guidelines. We discuss how we address the annotation challenges of the global process, and we highlight the contributions and limitations of our framework. Classification in text mining usually assigns a single topic (category) per news item (document-based classification). However, animal health news is rich in different types of epidemiological information. For instance, news articles that report an outbreak often also describe the outbreak control measures or economic impacts and point to the outbreak source or area at risk (Fig. 1). These elements may be of relevance to epidemic intelligence (EI) teams to assess the risks associated with the occurrence of a disease outbreak (further referred to as an event). Extract of a news article published by Reuters on August 25, 2018. This news describes an African swine fever outbreak in Romania including the outbreak description (a), the transmission pathway of the outbreak (b) and general epidemiological knowledge about the disease and its spread (c). The whole news content is available at: journal https://www.reuters.com/article/us-romania-swineflu-pigs-idUSKCN1LA0LR. When the news contains several topics, a single-label classifier has to decide on a topic (i.e., a label) among several possible topics, usually decreasing the classification performance1. Most classification approaches in EBS systems focus on binary news relevance. Little attention has been focused on the retrieval of other types of epidemiological information. In this context, we propose to split news content into sentences that are annotated into different categories according to their epidemiological topic, which we refer to as fine-grained information. Empirically, sentence-level classification seems more homogeneous in terms of topics than document-level classification. We therefore believe that sentence-level classification can more accurately identify specific types of information. To create annotated data as part of a machine learning pipeline useful for EI practitioners, we first need to elaborate and evaluate a generic annotation framework that should be as reproducible as possible. In addition, the list of classes for sentence-based annotation should allow us to identify new types of epidemiological information in animal disease-related news. Supervised learning algorithms implemented in EBS systems must be trained on labelled datasets to further classify unknown data. Several annotated textual resources thus have been created to support classifier training tasks in animal health. Table 1 presents examples of labelled datasets of news in the animal disease surveillance context. Datasets are compared based on their aim, the characteristics of the annotated data and their reproducibility in terms of availability (indicating whether the corpus and guidelines are freely available for download) and reliability (corresponding to the evaluation of inter-annotator agreement). Table 1 Example of annotated data used for online news processing in event-based surveillance applications. C: classification, NER: named entity recognition, EE: event extraction. Depending on the context in which it was created (typically the scope of the EBS system), the labelled corpus is either generalist, i.e. encompassing both human and animal disease events2,3, or specific, i.e. targeting one or several animal diseases1. The annotation unit and labels (categories) closely depend on the aim of the text-mining tasks in the animal disease domain, i.e. (i) classification, (ii) named entity recognition and (iii) event extraction. (i). For classification tasks, annotation is usually at the document level. The labels are often related to the news relevance so as to filter out irrelevant ones4,5,6,7. Other classification frames assign a broad thematic label to the news, such as "outbreak-related" or "socioeconomic"1. To our knowledge, all document-based annotation approaches allow a single label per news piece. (ii). For named entity recognition tasks, the corpus is annotated at the word level (including multi-word expressions). A typical example is the annotation framework of the BioCaster Ontology2. (iii). For event extraction tasks, the annotation unit depends on the definition adopted for the event. Some authors opt for a linguistic definition, i.e. a verb (called predicate) and a subject or object (called argument). Some sophisticated event annotation schemes allow extraction of fine-grained temporal information such as the beginning and end of an event2, or thematic attributes such as the transmission mode3. No currently available annotated data and frameworks can fulfil the needs of our current objective to detect fine-grained epidemiological information (i.e., topics). Document-based approaches are not precise enough to detect the variety of information contained in news articles. Word-based annotation frameworks provide accurate information at the word level, yet they are task-oriented (extraction of events or named entities) and partly address the potential of other types of epidemiological information. Midway between these two approaches8, proposed a sentence-based annotation to detect outbreak-related sentences, while recognising that a news piece contains many sentences with different semantic meanings. However, as the primary goal was outbreak detection, outbreak-unrelated sentences (e.g., describing treatment or prevention) were all merged into one negative category. In addition to the shortcomings of the works mentioned above, the availability and reproducibility of the annotated data and guidelines vary between the studies. Several corpora were not published or are no longer available because of unstable storage. For instance, the BioCaster disease event corpus has to be retrieved through a Perl script that downloads documents from their web source. As some sources become unavailable, the corpus size inevitably decreases over time (only 102 source web pages among 200 were still available online in 20159). The availability of EBS systems also hampers data access–two EBS systems from Table 1 were no longer operational (Argus, BioCaster). Most of the proposed approaches lack reproducibility. First, annotation guidelines usually consist of brief label descriptions rather than detailed schemes. Second, in the provided examples, only three annotation frameworks were evaluated in terms of inter-annotator agreement. According to biomedical text annotation recommendations10, the BioCaster disease event corpus authors used percentage scores (pairwise agreement) rather than the kappa statistic3. The annotation framework of BioCaster Ontology included both metrics2. Multi-kappa statistics is also used to take into account more than two annotators for agreement measurement6,11. Similarly to the approach of Zhang et al.8, we aim at implementing sentence-level annotation to enrich the binary outbreak-related/unrelated classification with thematic categories. Our objective is to make effective use of the epidemiological information contained in the news, especially when the information is relevant for assessing an epidemiological situation. In this section, we describe our approach to building annotated resources for the extraction of fine-grained epidemiological information. We first describe the global process we adopted to develop the annotation guidelines. Then we present the final annotation framework and describe the proposed categories (labels). The annotation guidelines and annotated corpus are publicly available12. This dataset is used in the context of the PADI-web (Platform for Automated extraction of Disease Information from the web) system. Briefly, PADI-web is an automated system devoted to online news source monitoring for the detection of animal infectious diseases. The tool automatically collects news via customized multilingual queries, classifies them using machine learning approaches and extracts epidemiological information (e.g., locations, dates, hosts, symptoms, etc.) with Natural Language Processing (NLP) approaches. In13 we summarized how the corpus described in this study is used to learn a fine-grained classification model based on a machine learning approach that is integrated into PADI-web 3.0. The proposed annotation scheme is intended to enhance EBS systems by enabling the automated classification of sentences from disease-related news. One of the main applications is the enhancement of the performances of event-extraction tasks by identifying the event-related sentences. We believe that performing event extraction on a subset of relevant sentences would decrease the risk of extracting epidemiological entities (e.g., dates, locations) not related to an event. In addition, the distinction between current and risk events allows for characterizing an event as ongoing or likely to occur. Sentences related to transmission pathways could be manually or automatically compared to current disease knowledge to identify the emergence of a new transmission pathway. Eventually, sentence-based classification is an alternative approach to increase the performance of document-based classification, especially in the context where event-related information appears within a few sentences14: each sentence can be first classified as relevant or not, and the results of each sentence classification can be merged to classify the document. Global annotation process We extracted 32 candidate English news items from the PADI-web database, while focusing on those classified as relevant. By relevant news, we mean a news report that is related to a disease event (describing a current outbreak as well as the prevention and control measures, preparedness, etc.). The classification in PADI-web is performed each day automatically and relies on a supervised machine learning approach; a family of classifiers is trained on a corpus of 600 news items manually labeled by an epidemiology expert (200 relevant news articles and 400 irrelevant news). The four annotators (A, B, C, D) were veterinarians working in epidemic intelligence. Two of them had previous experience with annotation tasks (annotators A and B). During the process, we followed the four consecutive steps detailed in Fig. 2. After each annotation step, we calculated the agreement metrics. Pipeline of the annotation guideline elaboration process. The annotators discussed the main disagreement results and modified the guidelines to improve the annotation process. We describe the main modification choices that led to the final guidelines. We stopped the process when satisfactory agreement measures were attained, i.e., when the overall agreement was above 80% (Step 3). To build the final corpus (step 4), we aggregated datasets 2 and 3. To choose one label per sentence in case of disagreement, we adopted the following procedure: For dataset 3 (labelled by three annotators): (a). If at least two out of three annotators assigned the same label, we selected the majority label, (b). If each of the three annotators assigned a different label, annotator A chose a final label among those proposed; For dataset 2 (labelled by two annotators): If both annotators disagreed, annotator A chose a final label among those proposed, Annotator A verified consistency with the final guidelines. The corpus was further increased by annotating a new dataset with the final guidelines (see section Data Records). Annotation guidelines In this subsection, we present the final annotation framework and definitions of each label from the guidelines. A detailed version of this framework and the labelled corpus are publicly available in a Dataverse repository12. In our framework, each sentence is annotated with one label for event type and one label for information type, as illustrated in Fig. 3. The event type level identifies if the sentence is related to an outbreak event, and, if so, the temporal relation with the event. The information type level describes the type of epidemiological information, i.e. the fine-grained topics. The meaning of a sentence depends on the entire news content, as well as its epidemiological context. Therefore, for each set of sentences (from a single news), the annotator first reads the news metadata (i.e. title, source, and publication date). The annotator chooses a single label per level and per sentence. As some sentences may contain information belonging to several information type categories, the annotator must pinpoint the primary information. Two-level annotation framework. While focusing on sentence epidemiological topics, the relation between the sentence and the current epidemiological situation must be taken into account: sentences in news pieces may describe an outbreak that happened several years before or provide general information about a disease. More precisely, from the EBS standpoint, only sentences referring to current events or events at risk are of interest. In our context, we define an event as the occurrence of a disease within a specific area and time range. The event type label aims to differentiate sentences referring to the current/recent outbreak ("Current event" and "Risk event") from sentences referring to old outbreaks ("Old event") or general information ("General"). Sentences which do not contain any epidemiological information are considered irrelevant ("Irrelevant"). Current event: this class includes sentences related to the current situation. There are five major groups of sentences that are considered "current": Recent event, relative to the main event. This includes events occurring at a nearby location and/or within a short-time window around the main event. For instance, "On Saturday, similar infections were found in 30 pigs on a farm in the Huangpu district of Guangzhou." Aggregation of events between a prior date and a recent/current date. For instance, "According to data from the Council of Agriculture, 94 poultry farms in Taiwan have been infected by avian flu so far this year." The temporal expression "so far this year" indicates a relationship between the start of the outbreak and the publication date. Recent/current epidemiological status of a disease within an area. For instance, "In recent months, the disease has spread more rapidly and further west, affecting countries that were previously unscathed." Events that will definitively occur in the future. In general, this category includes the direct consequences of an event, such as control measures that will be taken. For instance, "All pigs in the complex will be killed, and 3 km and 10 km protection and surveillance zones will be installed." Old event: This class includes sentences about events that provide a historical context for the main event. Those sentences contain explicit references to old dates, either absolute ("In 2007") or relative ("Back in days"). This category includes two groups of sentences: Old event. For instance, "The most recent case of the disease in the UK came in 2007." Aggregation of events between two past dates. For instance, "Between 2010 and 2011, South Korea had 155 outbreaks of FMD." Past epidemiological status of a disease within an area. For instance, "Between 2006 and 2010, BTV serotype 8 reached parts of north-western Europe that had never experienced bluetongue outbreaks previously." Risk event: This class includes all sentences referring to hypothetical events. These sentences are generally about an area at risk of introduction or dissemination of a pathogen. This category includes two groups of sentences: An unaffected area expressing concern and/or preparedness. For instance, "Additional outbreaks of African swine fever are likely to occur in China, despite nationwide disease control and prevention efforts." An area with unknown disease status. For instance, "If the outbreak is verified, all pigs at the feeding station will have to be culled, Miratorg said." General: This class includes general information about a disease or pathogen. Conventionally, the sentences describe the disease hosts, its clinical presentation and pathogenicity. For instance, "Bluetongue is a viral disease of ruminants (e. g. cattle, sheep goats, and deer)." Irrelevant: This class includes sentences that do not contain any epidemiological information. This group includes, for instance, disease-unrelated general facts ("Pig imports from Hungary only represented about 0. 64 per cent of all pork products to the UK in 2017.") or article news artefacts ("Comments will be moderated."). The information type level describes the sentence epidemiological topic. As epidemiological topic, we include the notification of a suspected or confirmed event, the description of a disease in an area ("Descriptive epidemiology" and "Distribution"), preventive or control measures against a disease outbreak ("Preventive and control measures"), an event's economic and/or political impacts ("Economic and political consequences"), its suspected or confirmed transmission mode ("Transmission pathway"), the expression of concern and/or facts about risk factors ("Concern and risk factors") and general information about the epidemiology of a pathogen or a disease ("General epidemiology"). Descriptive epidemiology. This class includes sentences containing the standard epidemiological indicators (e.g. disease, location, hosts, and dates) that describe an event. It includes: Epidemiological description of the event. For instance, "Cases of African swine fever (ASF) have been recorded in Odesa and Mykolaiv regions." Information about the pathogenic agent cause of the event. For instance, "Results indicated that the birds were infected with a new variety of H5N1 influenza." Clinical signs of the suspected event. For instance, "The remaining buck appears healthy at this time and is showing no clinical signs associated with the disease." Distribution. This class contains sentences giving indications on the presence of a disease in a specific area (i.e. a region, a country). It includes: Description of the epidemiological status. For instance, "In recent months, the disease has spread more rapidly and further west, affecting countries that were previously unscathed." Aggregation of events between a past date and a recent/current date. For instance, "According to data from the Council of Agriculture, 94 poultry farms in Taiwan have been infected by avian flu so far this year." Preventive and control measures. This class includes sentences describing: Preventive measures, i.e. all sanitary and physical actions taken to avoid the introduction of a disease into an unaffected area. For instance, "ASF: France about to end the fencing in the borderland with Belgium." Control measures, i.e. all sanitary and physical actions taken to eradicate a pathogen once introduced into an area (e.g. vaccination, slaughtering, disinfection, zoning, etc.). For instance, "All the infected animals have been killed, and the area has been disinfected." Instructions/recommendations, i.e. actions for both preventive and control measures, we include recommendations in this class. For instance, "Hunters, travellers, and transporters are asked to take extra care concerning hygiene." Transmission pathway. This class includes the sentences indicating the origin (source) of the disease or the transmission route. For instance, "The authorities suggest that the highly contagious virus might have been spread by a river". Concern and risk factors. This class includes sentences indicating a risk of introduction or spread of disease in an area. We include two types of sentences in this group: Confirmation of suspicion of one or several risk factors, i.e. an individual, behavioural and environmental characteristics associated with an increased disease occurrence. For example, "A recent wave of inspections revealed 4,000 different biosecurity violations on farms and Gosvetfitosluzhba warned that this could result in further outbreaks soon." Semantic expression of fears or concerns regarding: (i) The hypothetical intrusion of a pathogen into an unaffected area. For instance, "ASF is a real threat to the UK," she said." (ii) The worrying development of a situation. For instance, "Several countries are affected, alarming governments and pig farmers due to the pace at which the disease has spread." Economic and political consequences. This category includes all references to direct or indirect economic or political impacts of an outbreak on an area. It includes the consequences of preventive and control measures. For instance, "Gorod estimated that financial losses due to ASF could amount to 17 million euros to Latvia's industry in 2017." General epidemiology. This category is only used for the sentences labelled "General" as the event type level. It merges the classes "Event description" and "Transmission pathway" described above. In this particular event type level, those two categories include the description of a disease's hosts, pathogenicity and transmission route. For instance, "The virus is transmitted by midge bites, and it does not affect humans." Multi-topic sentences To handle multi-topic sentences, we provide two rules to help annotators make choices: If one category (label) is the consequence of another one, the annotator should select the first one. For instance, if a sentence describes both a control measure and its economic effects, the sentence should be labelled as "Preventive and control measures". Both "Concern and risk factors" and "Transmission pathway" provide highly valuable information to assess the risk of emergence or spread of a disease. The annotator should therefore prioritise them against other labels into a multi-topic sentence. Table 2 provides examples of frequently encountered multi-topic cases and the choice of the main label according to the two rules shown above. Table 2 Resolution of multi-topic sentences in typical cases. Annotation agreement In this section, we describe changes in the agreement metrics during the framework elaboration. As quantitative agreement measures, we calculated the inter-annotation agreement and Cohen's kappa coefficient. For inter-annotation agreement, we defined three different levels, i.e. total agreement (all annotators reached a consensus), partial agreement (two annotators agreed), and complete disagreement (all annotators disagreed). In the case of multi-labels, we defined the agreement as strict, i.e. there is an agreement between two annotators if they give precisely the same labels. Cohen's kappa coefficient (κ) is a widely used statistical measure of inter-annotator agreement, which takes into account the extent of agreement expected by chance15. κ was calculated as follows: $$\kappa =\frac{Pr(a)-Pr(e)}{1-Pr(e)}$$ Where Pr(a) is the observed agreement among two annotators, Pr(e) is the hypothetical probability of reaching an agreement. Table 3 compares the agreement results obtained in step 1 (initial version of the guidelines) and step 3 (the final version of the guidelines). We calculated the kappa by pairs of annotators separately and then computed the average. At step 1, we obtained poor agreement for event type annotations (κ = 0.30), while we obtained fair agreement for information type (κ = 0.53). Annotators totally agreed on event type labels for only 29% of sentences, while 49% of the sentences obtained a total agreement for information type. Table 3 Agreement statistics at step 1 (initial guidelines, N = 132 sentences) and step 3 (final guidelines, N = 83 sentences). Statistics at step 3 (final guidelines) indicate a substantial improvement in the agreement for both classes. The event type kappa was still lower than the information type kappa (0.71 and 0.78, respectively). In this Section, we present critical issues that emerged during the framework elaboration process, while outlining our choices to improve the inter-annotator agreement. We first discuss two characteristics of our global framework and then explain two different strategies adopted to modify the annotation guidelines. Global framework Double-level annotation Similar to event annotation approaches in which the annotator labels the event type and its attributes separately2, our final annotation framework relies on the attribution of two labels per sentence: event type and information type. We chose this approach because the thematic labels (information type) encompass different temporal and event levels. Their relevance from an event-based surveillance viewpoint differs. For instance, a sentence describing an outbreak that occurred 2 years before the publication date ("Old event") is obviously less relevant than a sentence describing a current one. However, the type of information provided (description of an outbreak) remains the same. Therefore, the double-level approach is geared toward assigning consistent information type labels among different event statuses. This choice increases the annotation time and complexity, but we believe that it substantially enhances the value of assigned labels by allowing us to consider spatiotemporal and topic labels separately. Single-label annotation We chose the sentence-based approach to address the lack of granularity in document-level approaches. However, a single sentence may also contain distinct topics. Therefore, until step 3, we allowed multi-labelling (the annotator could allocate as many labels as wished to a sentence, for both event type and information type). For event type, only two sentences from the third dataset had multi-labels, both of them with "Current event" and "Old event". In both sentences, the reference to historical outbreaks was provided as context, e.g. "It has not been confirmed what caused the outbreak, but there have been other incidents in the region during the 20th century." Multi-labelled sentences were more frequent for information type, representing from 14% (12/83) to 34% (28/83) of the sentences according to the annotator. The most frequent associations were: "Preventive and control measures" with either "Descriptive epidemiology" or "Economic and political consequences". In these sentences, there was a causal relationship between the two labels. For instance, in the following sentence, a ban was decided in response to a related outbreak: "The Polish news agency reported that the ban was in relation to two cases of African swine fever found in dead wild boar on the Polish border with Belarus." These cases were resolved by providing rules to choose the main label in case of a causal relationship. We prioritised the causal label, claiming that it usually contains the main information. In the previous sentence, the outbreak occurrence prevails over the ban. Therefore, the sentence should be labelled as "Descriptive epidemiology". "Descriptive epidemiology" and "Clinical presentation", mainly referring to mortality ("Two pigs in a population of 36 were found infected - one had died"), or to asymptomatic cases ("Hence affected flocks were detected under routine monitoring as there are no clinical signs associated with the event"). These cases were resolved by merging the two classes, as discussed in section "Merging of classes". Strategies to improve inter-annotator agreement Creation of new classes During this process, we created the new label "Distribution". In the first guidelines, sentences such as "In recent months, the disease has spread more rapidly" were labelled by annotators as either "Descriptive epidemiology" or "General epidemiology". Such sentences describe the current situation but they do not inform on a specific event. On the other hand, they describe an epidemiological situation that depends on a specific context (spatiotemporally locatable). Therefore, they cannot be considered as "General epidemiology". Merging of classes We merged the following categories in the annotation process: Current event and related event Initially, we had divided event type labels into three groups for current and past events: Current event, i.e. the main event notified in the news article and which recently occurred, Related events, i.e. events that happened in the past but are related to the current one, Old events, i.e. events that occurred in the past without any link with the current situation (same definition as in the final guidelines) This distinction between present and related events was the leading cause of disagreements in step 1. Deciding whether an event was a present or a related one was not trivial because it depended on a spatio-temporal cutoff which differed between annotators. Therefore, we decided to gather current and recent outbreaks in the same category ("Current event"). Some authors have proposed using a temporally fixed window. For instance, events occurring within a 3-months window are related16. This threshold was also used to label events as historical (occurred more than 3 months ago), in addition to recent events (occurred between 2 weeks and 3 months ago), and present ones (occurred within the last 2 weeks) as described by2. We believe that setting a rigid time window is not consistent with the epidemiological specificity of each disease. We instead decided to aggregate these two categories and distinguish only current/related events from old events. This distinction improved the agreement for the event type level: all six sentences labelled as "Old event" obtained total agreement. In these sentences, typical semantic clues (e.g. the use of temporal expressions such as "back in days" or "in 2006") explicitly indicated the absence of an epidemiological link. Clinical presentation and Descriptive epidemiology The "Clinical presentation" category was present in the first version of the guidelines. The label was mainly used by one annotator in association with the "Descriptive epidemiology" label. It appeared that in these sentences, all symptom-like terms were related to "deaths" or "died", e.g. "So far, six adult cattle and two calves have died from the disease". Rather than providing a clinical picture, these expressions were used to indicate the number of cases. We, therefore, decided to merge it with "Descriptive epidemiology" in the final framework. Preventive and control measures In the intermediate guidelines, we divided preventive and control measures into two distinct categories. This choice increased the number of disagreements in this class because several types of measures could be considered as both preventive or control according to the context. For instance, the slaughtering of infected animals is a control measure for the concerned affected area but is a preventive measure from the unaffected area standpoint (limiting the disease spread). The ban of animal movements, as well as vaccination, can also be control measures (avoid disease spread from the affected area) as well as a preventive measure (prevent disease introduction in an unaffected area). In the BioCaster ontology scheme, this context-dependency made the "control" category the most challenging class in terms of agreement17. Several limitations in the proposed annotation framework should be noted, as they may influence the performance of further classification tasks. First, we adopted a single-label approach for each level. Not allowing multiple labels per sentence was questionable since several sentences belonged to several classes, and the annotator may have had difficulty in determining which category should take precedence. This may lead to misclassification errors and information loss during the supervised approach. However, the use of multi-labelling raises the issue of finding suitable agreement metrics while adding a major complication in finding proper classification methods18. As some typical cases occurred, we tried to harmonise the annotators' choices by resolving multi-label cases in the guidelines. Besides, we did not include polarity or sentiment analysis in our labelling scheme. For instance, sentences indicating the absence of outbreaks or a negative result for a test should be labelled as "Descriptive epidemiology". In practice, sentences claiming a negative event are quite rare in online news narratives. The current frame could be enhanced by adding a polarity label to each sentence as it is necessary to include negation detection to avoid false alarms. In this Section, we proposed a sentence-based annotation scheme with the aim of going beyond conventional document-based classification and entity recognition. We built the framework by heavily relying on domain expert opinions while intending to find a trade-off between fair inter-annotator agreement and class granularity. The final inter-annotator scores were 0.71 Kappa on average for event type labels and 0.78 Kappa on average for information type labels. While some classes of interest from an epidemiological viewpoint (e.g. "Concern and risk factors", "Transmission pathway") are under-represented, we believe that the proposed framework helps increase the number of instances quickly and reproducibility. Data Records The dataset in the CIRAD Dataverse12 contains two files, an annotated corpus and the annotation guidelines providing a detailed description of each category. The annotated corpus file contains 1,244 manually annotated sentences extracted from 88 animal disease-related news articles. These news articles were obtained from the database of an event-based biosurveillance system dedicated to animal health surveillance, PADI-web (https://padi-web.cirad.fr/en/). The file is divided into three sheets: The first sheet provides metadata about the news articles (the unique id of a news article, its title, the name of the news article website, its publication date and its URL. The second sheet contains 486 sentences (from 32 news articles - 10,247 words) which were used to build the annotation framework. Each sentence label corresponds to a consensus label between two or three annotators. Each row corresponds to a sentence from a news article and has two distinct labels, event type and information type. The set of columns contains the id of the news article to which the sentence belongs, the unique id of the sentence, indicating its position in the news content (integer ranging from 1 to n, n being the total number of sentences in the news article), the sentence textual content, the event type label and the information type label. The third sheet contains 758 additional sentences (from 56 news articles - 16,417 words) annotated by a single annotator based on the same annotation framework. The set of columns is similar to the previous sheet. Technical Validation We evaluated the value of our annotation approach through a supervised classification task. The classification is called supervised when models are trained on instances whose labels are known (i.e. annotated by domain experts)19. The two annotation levels form two consecutive classification tasks: (i) classification of the event type and (ii) topic classification of the information type (Fig. 4). To evaluate the classification on sufficient class sizes, we used both the sentences annotated by two annotators and the additional corpus annotated by a single annotator (section Corpus) and we trained several classifiers (section Classification). Classification tasks. We obtained a final corpus containing 1,244 sentences, among which 160 sentences were irrelevant. The subset of sentences for information type classification hence consisted of 1,084 sentences. For the event type-level, 64% of the sentences (799/1244) were labelled as "Current event", 11% (136/1244) as "General", 8% (105/1244) as "Risk event", and 4% (44/1244) as "Old event". "Irrelevant" sentences represented 13% of the corpus (160/1244). The information type-level contained 1084 annotated sentences. Among these sentences, 37% of the sentences (401/1084) were labelled as "Descriptive epidemiology", 29% (310/1084) as "Preventive and control measures", 10% (110/1084) as "Concern and risk factors", 10% (109/1084) as "General epidemiology", 6% (69/1084) as "Transmission pathway", 5% (58/1084) as "Economic and political consequences", and 2% (27/1084) as "Distribution". The distribution of sentences at the event type level was highly imbalanced, indicating that disease-related news articles primarily provide information about the current situation (Current event). The information type level was more balanced, with two classes ("Descriptive epidemiology" and "Preventive and control measures") representing 67% of the sentences (711/1084). Even if still modest by its size, our corpus is highly specialised regarding both its domain (i.e. animal health) and its nature (i.e. online news articles). So this corpus type is more specific than the benchmark corpus traditionally used in state-of-the-art approaches in the biomedical NLP domain20. The transformation of a corpus of documents into a machine learning-readable format involves two steps. Each document is first transformed into a vector of selected features. Bag-of-words (BOW) is one of the most popular models used to convert textual documents into vectors. In this model, the vocabulary corresponds to all of the terms present in the whole corpus21. Each document d is encoded in an n-dimensional vector where each component wtd represents the absence or presence of a feature (term) t in the document (where n is the length of the vocabulary). If the feature t occurs in the document, the feature weight wtd has a non-zero value. In a second step, a weight is assigned to each feature in the document. Term Frequency-Inverse Document Frequency (TF–IDF) is the product of term frequency and inverse document frequency22. Terms with the highest TF–IDF values are distinctively frequent in a document in comparison to the collection of documents. In this evaluation, each sentence from the corpus represents a document. We simplified the vocabulary by removing punctuation and converting words to lowercase. Then, we transformed all the sentences into the bag-of-words model, using the TF–IDF weight. We compared three classifiers that are widely used for text classification: Naive Bayes (NB), is a family of probabilistic classifiers based on Bayes' theorem. These classifiers are based on the assumption that there is high independence between features. We used a multinomial Naive Bayes classifier, which assumes that features have a multinomial distribution. Support Vector Machines (SVM) is a non-probabilistic and linear classification technique. SVM has been widely used for text classification, including small-sized texts such as sentences8,23 and tweets24. It achieves robust performance regarding important textual data vector properties, which are sparse and dense (containing few relevant features)25. We used a linear kernel parameter (linear SVM) classifier, as linear kernels perform well with textual data26,27. Multilayer Perceptron (MLP) is an Artificial neural network-type (ANN) classifier. ANN classifiers were shown to perform well when combined with word embedding representations28,29. We estimated the performances of the trained models via the widely used cross-validation method. We used a fold number of 5, as this value was empirically shown to yield test error rate estimates with low variance, while not being hampered by excessively high bias30. At each fold, we computed the traditional metrics used in supervised classification, i.e. precision, recall, accuracy and F-measure. At the class A level, precision corresponds to the proportion of correct sentences classified in class A (Eq. 2), and recall corresponds to the proportion of sentences belonging to class A that are correctly identified (Eq. 3): $$Precision(A)=\frac{number\;of\;sentences\;correctly\;attributed\;to\;class\;A}{number\;of\;sentences\;attributed\;to\;class\;A}$$ $$Recall(A)=\frac{number\;of\;sentences\;correctly\;attributed\;to\;class\;A}{total\;number\;of\;sentences\;belonging\;to\;class\;A}$$ F-measure is the harmonic mean of precision and recall (Eq. 4): $$F-measure(A)=\frac{2\times Precision(A)\times Recall(A)}{Precision(A)+Recall(A)}$$ To calculate the performances over all classes to account for class imbalance, we computed the weighted precision, recall, and F-measure (averaging the frequency-weighted mean per label). For instance, considering a binary classification between a class A (frequency = Na) and a class B (frequency = Nb), the weighted precision Precisionw is: $$Precisio{n}_{w}=\frac{{N}_{a}}{{N}_{a}+{N}_{b}}\times Precision(A)+\frac{{N}_{b}}{{N}_{a}+{N}_{b}}\times Precision(B)$$ Note that the weighted F-measure is not calculated between general values of precision and recall. The accuracy corresponds to the proportion of correct predictions over the total predictions. In these experiments, we compared the different classifiers for both event type and information type classification. The performances are summarised in Table 4. MLP and SVM achieved comparatively equal performances and clearly outperformed the NB classifier. These behaviours were identical for event type and information type classification. Classification performances were lower on average for the information type level than for the event type level. Other results (e.g. results by category) are presented in13. Table 4 Performances of classifiers trained on bag-of-words (BOW) representations, in terms of weighted precision, recall, F-measure and accuracy over 5-fold cross-validation. The labelled corpus in our experiments are publicly available in a Dataverse repository: https://doi.org/10.18167/DVN1/YGAKNB12. The whole classification and evaluation pipeline was performed using the scikit-learn library (Python): https://scikit-learn.org/31. Zhang, Y., Dang, Y., Chen, H., Thurmond, M. & Larson, C. Automatic online news monitoring and classification for syndromic surveillance. Decision Support Systems 47, 508–517, https://doi.org/10.1016/j.dss.2009.04.016 (2009). Chanlekha, H., Kawazoe, A. & Collier, N. A framework for enhancing spatial and temporal granularity in report-based health surveillance systems. BMC medical informatics and decision making 10, 1 (2010). Conway, M., Kawazoe, A., Chanlekha, H. & Collier, N. Developing a Disease Outbreak Event Corpus. 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MathSciNet MATH Google Scholar This work was funded by the French General Directorate for Food (DGAL) and the French Agricultural Research Centre for International Development (CIRAD). This work was also supported by the French National Research Agency under the Investments for the Future Program, referred to as ANR-16-CONV-0004. The research that yielded this guide was also funded by the Belgian Federal Public Service Health, Food Chain Safety and Environment through the contractRT 18/2 MORISKIN 1, related to the research project Moriskin (Method for threat analysis in the context Of the RISK of emergence or re-emergence of INfectious animal diseases), coordinated by Sciensano (Institute of Public and Animal Health). This study was partially funded by EU grant 874850 MOOD and is catalogued as MOOD039. The contents of this publication are the sole responsibility of the authors and do not necessarily reflect the views of the European Commission. UMR TETIS (Land, Environment, Remote Sensing and Spatial Information), University of Montpellier, AgroParisTech, CIRAD, CNRS, INRAE, Montpellier, France Sarah Valentin & Mathieu Roche UMR ASTRE (Unit for Animals, Health, Territories, Risks and Ecosystems), University of Montpellier, CIRAD, INRAE, Montpellier, France Sarah Valentin, Elena Arsevska & Renaud Lancelot Department of Biology, University of Sherbrooke, Sherbrooke, Canada Sarah Valentin Quebec Centre for Biodiversity Science, McGill University, Montreal, Canada French Agricultural Research for Development (CIRAD), Montpellier, France Elena Arsevska, Renaud Lancelot & Mathieu Roche Veterinary Epidemiology Service, Departement of Epidemiology and Public Health, Sciensano, Brussels, Belgium Aline Vilain Department of Environmental and Agricultural Studies, Public Service of Wallonia, B5030, Gembloux, Belgium Valérie De Waele Elena Arsevska Renaud Lancelot Mathieu Roche Conceptualization: S.V., M.R., Data curation: S.V., E.A., A.V., V.D.W., Formal analysis: S.V., E.A., A.V., V.D.W., M.R., Funding acquisition: R.L., M.R., E.A., Methodology: S.V., M.R., Validation: S.V., E.A., A.V., V.D.W., Writing–original draft: S.V., Writing–review & editing: All authors reviewed the manuscript. Correspondence to Mathieu Roche. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Valentin, S., Arsevska, E., Vilain, A. et al. Elaboration of a new framework for fine-grained epidemiological annotation. Sci Data 9, 655 (2022). https://doi.org/10.1038/s41597-022-01743-2 Scientific Data (Sci Data) ISSN 2052-4463 (online)	CommonCrawl
MSC2020 57N10 Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle S. K. Zinina1, P. I. Pochinka2 1National Research Mordovia State University (Saransk, Russian Federation) 2Higher School of Economics (Nizhny Novgorod, Russian Federation) Abstract. This paper introduces class $G$ containing Cartesian products of orientation-changing rough transformations of the circle and studies their dynamics. As it is known from the paper of A. G. Maier non-wandering set of orientation-changing diffeomorphism of the circle consists of $2q$ periodic points, where $q$ is some natural number. So Cartesian products of two such diffeomorphisms has $4q_1q_2$ periodic points where $q_1$ corresponds to the first transformation and $q_2$ corresponds to the second one. The authors describe all possible types of the set of periodic points, which contains $2q_1q_2$ saddle points, $q_1q_2$ sinks, and $q_1q_2$ sources; $4$ points from mentioned $4q_1q_2$ periodic ones are fixed, and the remaining $4q_1q_2-4$ points have period $2$. In the theory of smooth dynamical systems, a very useful result is that, given a diffeomorphism $f$ of a manifold, one can construct a flow on a manifold with dimension one greater; this flow is called the suspension over $f$. The authors introduce the concept of suspension over diffeomorphisms of class $G$, describe all possible types of suspension orbits and the number of these orbits. Besides that, the authors prove a theorem on the topology of the manifold on which the suspension is given. Namely, the carrier manifold of the flows under consideration is homeomorphic to the closed 3-manifold $\mathbb T^2 \times [0,1]/\varphi$, where $\varphi :\mathbb T^ 2 \to \mathbb T^2$. The main result of the paper says that suspensions over diffeomorphisms of the class $G$ are topologically equivalent if and only if corresponding diffeomorphisms are topologically conjugate. The idea of the proof is to show that the topological equivalence of the suspensions $\phi^t$ and $\phi'^t$ implies the topological conjugacy of $\phi$ and $\phi'$. Key Words: rough systems of differential equations, rough circle transformations, orientation-reversing circle transformations, Cartesian product of circle transformations, suspension over a diffeomorphism For citation: S. K. Zinina, P. I. Pochinka. Classification of suspensions over cartesian products of orientation-reversing diffeomorphisms of a circle. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:1(2022), 54–65. DOI: https://doi.org/10.15507/2079-6900.24.202201.54-65 Svetlana Kh. Zinina, Postgraduate Student, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0003-3002-281X, [email protected] Pavel I. Pochinka, Student of the Faculty of Informatics, Mathematics and Computer Science, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/0000-0002-6377-747X, [email protected]	CommonCrawl
Lecture 20 - Magnetic Materials Trace: • Lecture 20 - Magnetic Materials Show pagesourceOld revisions Orbital Magnetic Moment We saw in our last lecture that a current loop could be described in terms of a magnetic dipole moment $\vec{\mu}=I\vec{A}$. If we imagine a highly oversimplified picture of an atom with an electron going around it in a circular orbit of radius $r$ with velocity $\vec{v}$ we can see that the atom would have a magnetic moment. $\mu_{orb}=IA=\frac{ev}{2\pi r}\pi r^{2}=\frac{evr}{2}$ It is desirable to express this in terms of the orbital angular momentum $\vec{L}=m\vec{r}\times\vec{v}$ leading to $\vec{\mu}_{orb}=\frac{-e}{2m}\vec{L}$ Electron spin magnetic moment In addition to the magnetic moment an atom possesses due to the angular momentum of electrons orbiting around it they also possess a magnetic moment which comes from the angular momentum of the electrons themselves, the so called spin of the electron. The spin of an electron is a quantum mechanical property of the electron, and while we should not think of an electron as physically spinning the result of electron spin is that each electron has a magnetic moment $\vec{\mu}_{s}=-g\frac{e}{2m}\vec{S}$ In the above the spin of an electron is either $\vec{S}=+1/2\hbar$ or $\vec{S}=-1/2\hbar$ and $g$ is a factor which is different for different particles. For an electron $g=2.0023193043622$. This factor is known to great precision, and the deviation from exactly to is due to quantum electrodynamics. The magnetic moment of the an atom is obtained from the sum of the orbital magnetic moments and spin magnetic moments from all of the electrons in the atom. Some atoms will have a total magnetic moment and others will not. Magnetism inside a material We have shown that atoms inside a material can have magnetic dipole moments. This does not necessarily imply that they will have them, nor that they will all point in the same direction. The quantity that measures the net magnetic dipole moment per unit volume is the magnetization $\vec{M}$. The magnetic field inside the material $\vec{B}$ is given by $\vec{B}=\vec{B}_{0}+\mu_{0}\vec{M}$ where $\vec{B}_{0}$ is the external magnetic field applied to the material. Normally when dealing with magnetic materials we usually rewrite the external field as the magnetic field strength $\vec{H}=\frac{\vec{B}_{0}}{\mu_{0}}$ $\vec{B}=\mu_{0}(\vec{M}+\vec{H})$ Diamagnetism Diamagnetism is the weakest form of magnetism, and is experienced by all materials, but is only significant in materials where the atoms have no total magnetic moment. When a magnetic field is applied the orbitals of the electrons are affected in such a way that an magnetic moment which opposes the field develops. The result is that diamagnets experience a repulsive force. We can see this effect on water using the strong neodymium iron boride magnets we used to make our motors. A sufficiently strong magnetic field can be used to levitate all sorts of diamagnetic objects, including frogs. A superconductor, although it works on a macroscopic, rather than microscopic level, acts as a perfect diamagnet, an can be levitated with a much smaller magnetic field. Paramagnetism If the atoms in a material have a magnetic moment then much stronger effects than diamagnetism occur. In a paramagnet the magnetic dipoles are normally randomly oriented due to thermal motion. However, in a magnetic field the dipoles align themselves to produce a magnetization in the same direction as the field. An example of a very good paramagnetic material, is mu-metal, this is a nickel-iron alloy that can be used as magnetic shielding because it redirects the magnetic field lines. We can consider paramagnets as focusing the lines of magnetic flux, and diamagnets as excluding them. Magnetic Susceptibility The response of a material to a magnetic field can be written in terms of the magnetic susceptibility $\chi$ $\vec{M}=\chi\vec{H}$ Diamagnets have a negative susceptibility ($\chi<0$) and paramagnets have a positive susceptibility ($\chi>0$). Related to the susceptibility is the magnetic permeability $\mu$, which is NOT the same as the magnetic moment $\vec{\mu}$. The permeability and susceptibility are related through the equation $\mu=\chi+1$ The susceptibility of a paramagnet follows the Curie Law $\chi=\frac{C}{T}$ where $C$ is a material constant and $T$ is the temperature in Kelvin. By contrast diamagnets have temperature independent susceptibility. In a ferromagnet the magnetic dipoles interact very strongly with each other and therefore arrange themselves in the same direction, giving the material a magnetization even when no magnetic field is applied. As field is applied in the direction of the magnetization it only increases a little until the saturation magnetization $M_{C}$ is reached The direction of the magnetization can be reversed by application of a magnetic field in the opposite direction to the magnetization which exceeds a certain value, $H_{C}$ the coercive field. These properties give rise to the phenomenon of magnetic hysteresis. In a ferromagnet the mangetic dipoles would always like to all point in the same direction, but frequently we see that a ferromagnetic material does not appear to be magnetized. This occurs because the material can split into ferromagnetic domains, and while in each domain all the dipoles point in the same direction, the overall direction of each domain can be different. When a ferroelectric material which initially has zero net magnetization, due to domains, is placed in a magnetic field it shows a different $M-H$ curve until it is fully magnetized after which hysteresis is observed. Curie Temperature While the interactions between magnetic dipoles in a ferromagnetic are typically strong they can be overcome by thermal motion, and when a characteristic temperature, the Curie temperature $T_{C}$ is overcome the material become paramagnetic. The susceptibility of a ferromagnet above $T_{C}$ is described by a Curie-Weiss Law $\chi=\frac{C}{T-T_{C}}$ The neodymium iron boride magnets have a fairly low transition temperature, around 310oC. For more on magnetic materials, see the Hitchhiker's guide to magnetism. phy142/lectures/20.1301010913.txt · Last modified: 2011/03/24 19:55 by mdawber	CommonCrawl
Methodology Article \| Open \| Published: 11 July 2017 SegCorr a statistical procedure for the detection of genomic regions of correlated expression Eleni Ioanna Delatola1,2,3,4, Emilie Lebarbier1,2, Tristan Mary-Huard1,2,5, François Radvanyi3,4, Stéphane Robin1,2 & Jennifer Wong3,4,6,7 BMC Bioinformaticsvolume 18, Article number: 333 (2017) \| Download Citation Detecting local correlations in expression between neighboring genes along the genome has proved to be an effective strategy to identify possible causes of transcriptional deregulation in cancer. It has been successfully used to illustrate the role of mechanisms such as copy number variation (CNV) or epigenetic alterations as factors that may significantly alter expression in large chromosomal regions (gene silencing or gene activation). The identification of correlated regions requires segmenting the gene expression correlation matrix into regions of homogeneously correlated genes and assessing whether the observed local correlation is significantly higher than the background chromosomal correlation. A unified statistical framework is proposed to achieve these two tasks, where optimal segmentation is efficiently performed using dynamic programming algorithm, and detection of highly correlated regions is then achieved using an exact test procedure. We also propose a simple and efficient procedure to correct the expression signal for mechanisms already known to impact expression correlation. The performance and robustness of the proposed procedure, called SegCorr, are evaluated on simulated data. The procedure is illustrated on cancer data, where the signal is corrected for correlations caused by copy number variation. It permitted the detection of regions with high correlations linked to epigenetic marks like DNA methylation. SegCorr is a novel method that performs correlation matrix segmentation and applies a test procedure in order to detect highly correlated regions in gene expression. In the last decade, the study of local co-expression of neighboring genes along the chromosome has become a question of major importance in cancer biology [6]. The development of "Omics" technologies have permitted the identification of several mechanisms inducing local gene regulation, that may be due to a common transcription factor [11] or common epigenetic marks [14, 34]. Copy number variation due to polymorphism or to genomic instability in cancer is also a possible cause for observing a correlation between neighboring genes [1], as their expressions are likely to be affected by the same copy number variation (CNV). It has further been observed that local regulations may occur in specific nuclear domains, as the nuclear region is an environment which may favor or not transcription [4]. Investigating the impact of a specific source of regulation (TF, CNV, epigenetic modifications such as DNA methylation and histone modifications) on the expression has now become a common practice for which statistical tools are readily available. However, only a few methods have been proposed to focus on the direct analysis of gene expression correlation along the chromosomes. The direct analysis of correlations may have different purposes: one can aim at detecting all potential chromosomal domains of co-expression, then investigating to which extend known causal mechanisms are responsible for the observed co-expression patterns, one can aim at detecting chromosomal domains of co-expression where correlations are not caused by already known sources of regulation, in order to identify new potential mechanisms impacting transcription. Addressing problems (i) and (ii) is crucial to fully understand transcriptional deregulation and/or to model gene regulation. We first consider problem (i) and provide a precise definition of our purpose: one aims at identifying correlated regions, i.e. blocks of neighboring genes, the expression of which displays correlations across patient samples that are significantly higher than expected. Indeed, it has been observed that background correlation between adjacent genes along the genome does exist. This background correlation should not be confounded with the co-expression that can be locally observed due to the aforementioned mechanisms. Consequently, we do not consider here methods that only account for this background correlation in the statistical modeling (for instance to improve the detection of differentially expressed genes), such as [24], [40] or [30]. Also note that we focus on methods that detect correlated regions on the basis of expression data solely. This excludes strategies that look for clusters of adjacent genes based on correlations between gene expression and a given phenotype or response, such as Rendersome [24], DIGMAP [41] or REEF [10]. Several approaches have been proposed to tackle problem (i). CluGene [13] uses a clustering method accounting for the chromosomal organization of the genes, while G-NEST [20] and TCM [28] rely on sliding windows procedures. The principle of the latter approach is to compute correlation scores for genes falling within the window, then to detect local peaks of high correlation scores. While these procedures have been successfully applied to cancer data, all tackle the detection of correlated region using heuristics. As such, they suffer from classical limitations associated with these techniques, including local optimum (for clustering algorithms) or detection instability according to the choice of the window size (for sliding windows). It is now well known that the problem of finding regions in a spatially ordered signal can be cast as a segmentation problem, for which standard statistical models exist, along with efficient algorithms to find the globally optimal solution [3]. According to our definition, the detection of correlated regions boils down to the block-diagonal segmentation of the correlation matrix between gene expressions. Such an approach has been proposed in image processing [22], finance [18] and bioinformatics for CNV analysis [42], but to the best of our knowledge it has never been considered for the detection of correlated expression regions. While problem (i) can be addressed on the basis of only expression data, problem (ii) requires the additional measurement of the signal one needs to account for. For example, consider that one seeks for locally expressed co-regulation events that are not due to copy number variations but due to other causes such as epigenetic mechanisms. The strategy we adopt here consists in first correcting the expression data for potential cancer CNV contribution, then in applying the procedure described to solve problem (i) on the corrected signal. The corrected signal is obtained by regressing the initial expression signal on the CNV signal. Although quite simple, the strategy turns out to be efficient in practice. An alternative strategy would be to jointly model both the expression and the signals to correct for, and then propose within this framework a correction. Such a strategy would necessitate to adapt the modeling to the specific combination of signals one has at hand. In comparison, the regression procedure proposed here can be applied to any kind and any number of signals one needs to correct for. The outline of the present article is the following. In Section 'Correlation matrix segmentation' (Methods) we propose a parametric statistical framework for the problem of correlated region identification. Finding regions of co-regulated genes can then be achieved by maximum likelihood inference (to find the boundaries of each region along with their correlation levels). Moreover, we propose a procedure to correct for known sources of correlation. An exact test procedure to assess the significance of the correlation with respect to background correlation is proposed in Section 'Assessing correlation significance' (Methods). We introduce a simple procedure to correct expression data beforehand for some known (and quantified) sources of correlation. Because the background correlation level is a priori unknown, an estimator of this quantity is also proposed. The performance of the resulting procedure, called SegCorr hereafter, is illustrated in Section 'Simulation study' (Results) on simulated data, along with a comparison with the TCM algorithm proposed in [28]. Finally, a case study on cancer data is presented in Section 'Bladder cancer data' (Results), in which we identify some regions with high correlation between gene expression and the local DNA methylation level. Correlation matrix segmentation We consider the following expression matrix: $$ {Y} =\left[ \begin{array}{ccc} {Y}_{11} & \cdots & {Y}_{1p} \\ {Y}_{21} & \cdots & {Y}_{2p} \\ \vdots & \ddots & \vdots \\ {Y}_{n1} & \cdots & {Y}_{np} \end{array} \right] $$ where Y ij stands for the expression of gene j (j=1,…,p) observed in patient i (i=1,…,n). The i-th row of this matrix is denoted Y i and corresponds to the expression vector of all genes in patient i. In order to detect regions of correlated expression, we consider the following statistical model. Profiles {Y i }1≤i≤n are supposed to be i.i.d, normalized (centered and standardized), following a Gaussian distribution with block-diagonal correlation matrix G: $$ G =\left[ \begin{array}{ccc} \Sigma_{1} & & \\ & \Sigma_{k} & \\ & & \Sigma_{K} \\ \end{array} \right] \quad \text{with} \quad \Sigma_{k} =\left[ \begin{array}{ccc} 1 & \cdots & \rho_{k} \\ \vdots & \ddots & \vdots \\ \rho_{k} & \cdots & 1 \end{array}\right]. $$ The model states that genes are spread into K contiguous regions, with respective lengths p k (k=1,…,K, $\sum _{1 \leq k \leq K} p_{k} = p$), the length of a region being the number of genes it contains. Genes belonging to different regions are supposed to be independent, whereas genes belonging to a same region are supposed to share the same pairwise correlation coefficient ρ k . This amounts to assume that some specific effect (e.g. methylation) affects the expression of all genes belonging to the region. More specifically, let U k denote the vector of the region effect (accross patients). For all genes j from region k, the model can be written as Y ij =U ik +E ij . The error terms E ij are all independent and independent from U ik such that $\mathbb {V}(U_{ik})/\mathbb {V}(Y_{ij}) = \rho _{k}$, where $\mathbb {V}(U)$ stands for the variance of U. While different technologies (microarrays, RNA-seq) may provide different types of signal (continuous, counts), an appropriate transformation may be applied to make the Gaussian assumption reasonable. For example, in the context of segmentation, [7] showed that Gaussian segmentation applied to log(1+x)-transformed RNA-seq data performs as well as negative binomial segmentation applied to the raw data. Accounting for known sources of regulation As mentioned in the Introduction, a second task (ii) can be to detect correlated regions which are not due to an already known mechanism. To this aim, one may first correct the expression signal using the following regression model : $$ {Y}_{ij} = \beta_{0} + \beta_{1} x_{ij} + \epsilon_{ij}, $$ where x ij stands for the covariate observed in patient i for gene j. For instance, in the illustration of Section 'CNV-dependent regions', x ij is the copy number associated to patient i at location of gene j. The corrected signal is then $\widetilde { {Y} }_{ij}= {Y}_{ij}- \widehat {\beta }_{0} - \widehat {\beta }_{1} x_{ij}$. Note that $\widehat {\beta }_{0}$ and $\widehat {\beta }_{1}$ can be obtained as ordinary least-square estimates. Indeed, it suffices to assume that (ε ij ) are independent among patients (but not among genes) to get the standard linear regression estimates (see [2], Chapter 8). Once the correction has been made, the model described in Section 'Statistical model' can be applied to the corrected signal $\widetilde {Y}_{ij}$. Note that the correction procedure could be based on more sophisticated modellings of the relationship between gene expression and mechanisms such as CNV or methylation, e.g. the ones proposed in [19, 23, 38]. The difference between the observation and the prediction obtained from one such model (i.e. the residuals) could then be used as the corrected signal. Lastly, the proposed correction procedure can be adapted straightforwardly to handle count data such as provided by RNAseq technologies. Indeed, Model (2) can be rephrased in the generalized linear model framework and Pearson residuals can be used as $\widetilde {Y}_{ij}$ (see e.g. [12] for a general introduction or [15] for the specific case of negative binomial regression). Inference of correlated regions Parameter inference in Model (1) amounts to estimating the number of regions K, the region boundaries 0=τ 0<τ 1<⋯<τ K =p, and the correlation parameters ρ 1,…,ρ K within each of these regions. Here, we consider a maximum penalized likelihood approach. First, we show that for a given K the optimal region boundaries and correlation coefficients can be efficiently obtained using dynamic programming. The number of regions can then be selected using a penalized likelihood criterion. For a fixed K, the estimation problem can be formulated as follows: $$\begin{array}{@{}rcl@{}} \arg\max_{\tau_{1} < \dots < \tau_{K-1}} \max_{\rho_{1}, \ldots,\rho_{K}} \mathcal{L} \end{array} $$ where the log-likelihood $\mathcal {L}$ is −(n log\|G\|+tr[YG −1(Y)⊤])/2. Here, thanks to the block diagonal structure of the correlation matrix in Model (1), the log-likelihood can be rewritten as $$\begin{array}{@{}rcl@{}} -2 \mathcal{L} & = & \sum\limits_{k} \left\{ n\log\|\Sigma_{k}\| + \text{tr}\left[ {Y}^{(k)} \Sigma_{k}^{-1} ({Y}^{(k)})^{\top} \right] \right\}\\ &=& -2\sum\limits_{k} \mathcal{L}(\tau_{k-1}+1,\tau_{k}) \quad = - 2\sum\limits_{k} \mathcal{L}_{k} \end{array} $$ where Y (k) stands for the set of expression from Y corresponding to genes included in the k-th region, and $\mathcal {L}_{k}=\mathcal {L}(\tau _{k-1}+1,\tau _{k})$ is the log-likelihood corresponding to region k, i.e. corresponding to measurements of genes from τ k−1+1 to τ k . While log-likelihood (4) is derived in a Gaussian setting, it can be used for count data, as the Pearson residuals mentioned in Section 'Accounting for known sources of regulation' have an approximate Gaussian distribution. Thanks to the additivity of the likelihood over the regions, the optimization problem (3) boils down to $$\begin{array}{@{}rcl@{}} \arg\max_{\tau_{1} < \dots < \tau_{K-1}} \sum\limits_{k} \max_{\rho_{k}} \mathcal{L}_{k}. \end{array} $$ Inference when K is fixed We first show that for a given region k with known boundaries, explicit expressions can be obtained for both the ML estimator $\widehat {\rho }_{k}$ and the likelihood $\mathcal {L}_{k}$ at the optimum: Lemma 1 For a region k with fixed boundaries [τ k−1+1,τ k ], the maximum of $\mathcal {L}_{k}$ with respect to ρ k is reached for $$\widehat{\rho}_{k} = \frac{\sum_{j=\tau_{k-1}+1}^{\tau_{k}}{\sum_{\ell=\tau_{k-1}+1}^{\tau_{k}}{\widehat{G}_{j\ell}}}-p_{k}}{p_{k}^{2} - p_{k}} $$ where $\widehat {G}_{j\ell } := n^{-1} \sum _{i=1}^{n} {Y}_{ij} {Y}_{i\ell }$. Furthermore, the maximal value of $\mathcal {L}_{k}$ is given by: $${} -2 \widehat{\mathcal{L}}_{k}\! =\! n\left[p_{k}\! +\! (p_{k}\,-\,1)\log{\left(1\! -\! \widehat{\rho}_{k} \right)}\! +\! \log{\left(1\! +\! (p_{k}\,-\,1)\widehat{\rho}_{k} \right)}\right]. $$ The proof is given in Additional file 1. The expression of Problem (5) is now $$\arg\max_{\tau_{1} < \dots < \tau_{K-1}} \sum\limits_{k} \widehat{\mathcal{L}}_{k} \ $$ which is additive with respect to the $\widehat {\mathcal {L}}_{k}$ terms that can be straightforwardly computed thanks to Lemma 1. Consequently, optimization can be performed via Dynamic Programming (DP, [17], [25]). The optimal boundaries, and correlation estimators can be obtained at computational cost $\mathcal {O}(Kp^{2})$. Lasso-type approaches have been proposed to tackle segmentation problems in a faster way (see e.g. [36]). First, note that such methods rely on a relaxation of the original problem, so that the result may be different from the exact solution of problem (3). Furthermore, in the context of matrix segmentation, such approaches have been proposed ([5, 21]), which do not allow to capture the longitudinal structure (i.e. blocks of neighboring genes). Model selection To choose the number of regions, we adopt the model selection strategy proposed in [17]. For each 1≤K≤K max, we define the maximal log-likelihood for K regions as $$L_{K} = \max_{\tau_{1} < \dots < \tau_{K-1}} \sum\limits_{k} \widehat{\mathcal{L}}(\tau_{k-1}+1, \tau_{k}) \ . $$ Furthermore, the normalized log-likelihood is defined as $$\widetilde{L}_{K} = \frac{L_{K_{\max}} - L_{K}}{L_{K_{\max}} - L_{1}}(\widetilde{K}_{\max} - \widetilde{K}_{1}) + 1, $$ where $\widetilde {K}_{j} = 5\times j + 2\times j \log {(p/j)}$ is the penalty function. [17] suggests to estimate the number of regions $\widehat {K}$ as the value of K such that $\widetilde {L}_{K}$ displays the largest slope change. Namely, we take $$ \widehat{K} = \arg\min_{K} \left\{ (\widetilde{L}_{K} - \widetilde{L}_{K+1}) - (\widetilde{L}_{K+1} - \widetilde{L}_{K+2}) > S \right\}, $$ where the value of threshold S is predefined. Throughout the paper, we used S=0.7 as suggested in [17]. The robustness of the results with respect to other values for threshold S is investigated in Section 'Simulation study'. This global approach (dynamic programming and model selection) has been applied with success for CNV detection (see [25] and [16] for a comparative study). Assessing correlation significance It has been observed [9, 28, 32, 34] that background correlations may exist between adjacent genes along the genome, i.e. one expects the correlation level in any region to be positive. As a consequence, one has to check whether a given region exhibits a correlation level that is significantly higher than the background correlation level ρ 0, that is observed by default. Test procedure Once the correlation matrix segmentation is performed, it is possible to identify regions with high correlation levels by testing H 0:ρ k =ρ 0 vs H 1:ρ k >ρ 0. This can be done using the following test statistic for region k: $$\begin{array}{@{}rcl@{}} T_{k} = \sum\limits_{i}^{n} \left(Y^{(k)}_{i\bullet} - Y^{(k)}_{\bullet\bullet}\right)^{2} \end{array} $$ where $Y^{(k)}_{i\bullet } = p_{k}^{-1} \sum _{j =\tau _{k-1}+1}^{\tau _{k}} Y_{ij}$ and $Y^{(k)}_{\bullet \bullet } = n^{-1} \sum _{i =1}^{n} Y^{(k)}_{i\bullet }$. Assuming Model (1) is true, test statistic T k has distribution $$\begin{array}{@{}rcl@{}} T_{k} \sim \lambda(p_{k},\rho_{k}) \chi^{2}_{n-1} \ \text{where } \ \lambda(p_{k},\rho_{k})=\frac{(1 + (p_{k}-1)\rho_{k})}{p_{k}} \ . \end{array} $$ Here $\chi ^{2}_{n-1}$ stands for the chi-square distribution with n−1 degrees of freedom. The proof is given in Additional file 1. We emphasize that this test is exact and does not rely on any resampling strategy. Consequently, the p-value associated to region k is given by $$\begin{array}{@{}rcl@{}} \mathbb{P}\left(\lambda(p_{k},\rho_{0})Z > T_{k}^{obs} \right), \text{where} \ \ Z\sim \chi^{2}_{n-1}. \end{array} $$ Statistical power We now study the ability of the proposed test to detect a region with width p 0 where the correlation ρ is higher than in the background. The probability to detect such a region depends on both p 0 and ρ and is given by $$\begin{array}{@{}rcl@{}} Po(n,p_{0},\rho) &=& \Pr\{T > \lambda(p_{0}, \rho_{0})q_{n-1,1-\alpha}\}\\ &=& \Pr\left\{ Z > \frac{\lambda(p_{0}, \rho_{0})}{\lambda(p_{0}, \rho)}q_{n-1,1-\alpha}\right\} \end{array} $$ where $Z\sim \chi ^{2}_{n-1}$ and q n−1,1−α is the 1−α quantile for the $\chi ^{2}_{n-1}$ distribution. Figure 1 (Top) displays the evolution of power for different values of p 0 and ρ. Here ρ 0 and n are fixed at 0.15 and 100, respectively. The nominal levels of α are 5, 0.5 and 0.05%. These levels correspond to realistic thresholds, once multiple testing corrections such as Bonferroni or FDR are performed. One can observe that even for small values of ρ, the power is high whatever the nominal level as long as the number of genes in the considered region is equal to or higher than 5. Figure 1 also shows that the procedure will probably fail to find regions of size 3, if the correlation is not 0.7 or higher (to obtain a power of 0.8). On the same graph (Bottom), one observes that a sample of size 50 is sufficient to efficiently detect regions of size 5, as long as the correlation is higher than 0.6. Larger samples will be required if one wants to efficiently detect regions with smaller correlation levels. Theoretical Power. Top: Power curves as a function of ρ, for a fixed cohort size n=100 and varying region width p 0=3,5,10,20. Bottom: Same graphs for a region of fixed width p 0=5 but varying cohort sizes n=10,50,200,1000. In all graphs ρ 0 is fixed at 0.15. The nominal level α of the test is set to 5% (left), 0.5% (center), 0.05% (right) Background correlation estimation The test procedure requires the knowledge of parameter ρ 0 that is unknown in practice. However, it can be estimated using $$\begin{array}{@{}rcl@{}} \widehat{\rho_{0}}= \|\underset{i>1}{\text{median}}(\text{corr}({Y}^{j-1}, {Y}^{j}))\| \end{array} $$ where Y j stands for the vector of expression of gene j for the n patients. Under the assumption that most pairs of adjacent genes display a ρ 0 correlation, i.e. only a few number of regions with moderate sizes exhibit a high level of correlation, $\widehat {\rho _{0}}$ is a robust estimator of the background correlation. The behavior of estimator (7) is investigated in Section 'Simulation study'. Simulation study In this section, we first study the quality of the proposed estimator of ρ 0. Then we study the ability of SegCorr to detect correlated regions and compare its performance with this of TCM algorithm. The robustness of the method with respect to the choice of the model selection threshold S will be investigated in Section 'Study of the model selection threshold S ' on real data, since very little difference were observed on the simulated data (results not shown). We also study the robustness of our procedure to a scheme where the within-region correlation is variable. Simulation design Scenario 1 (Easy case): the regions are defined as in [16]: each patient has one chromosome containing p=500 genes and 4 regions with respective lengths p k =5,10,20,40. Three values are considered for ρ 0:.08,.18,.28. These values are inspired by the distribution (displayed in Fig. 2) of ρ 0 from Scenario 2. ρ 0=.28 is higher than observed in [34], making the detection problem more difficult. ρ 1 varies between.3 and.9. Scenario 2 (Realistic case – constant correlation on H 1 regions): each patient has 22 chromosomes. The length of the chromosomes, the number of regions within each chromosome and their respective sizes are the same as in the results from [34]. ρ 0 is specific to each chromosome and estimated on the same dataset. ρ 1 varies between.3 and.9. Scenario 3 (Realistic case – variable correlation on H 1 regions): the design is the same as in Scenario 2, except that ρ 0 is fixed to.18. Furthermore, for each H 1 region covariance matrix is drawn from a p k -variate Wishart distribution $\mathrm {W}_{p_{k}}\left (S,\nu \right)$ where the entries of the matrix S are one on the diagonal and ρ 1=.5 elsewhere and ν is the number of degrees of freedom. Small values of ν, result in a higher variance, making the detection more difficult. Because ν has to be greater or equal to p k , we took ν=p k ×2β, where β=(0.5,1,1.5,…,5). So the variability decreases as β increases. Simulation Design. Left: Length of H 1 regions in the reference dataset. Right: Distribution of the background correlation $\hat {\rho }_{0}$ obtained from the reference data according to the segmentation obtained in [34] For each scenario, samples of n = 50 and 100 patients were considered and, for each combination (n, ρ 0, ρ 1) the simulation was replicated 100 and 20 times, for the first and the last two scenarios respectively. Quality of the ρ 0 estimator For this study, we consider Scenario 2. Figure 3 illustrates the estimation accuracy of ρ 0 under different levels of both H 0 and H 1 correlations on chromosome 5. Estimator (7) yields over-estimated values of the true background correlation level. One observes that the overestimation does not depend on the correlation level in H 1 regions, thanks to the use of the median. Still, as expected, it is linked to the proportion of pairs of adjacent genes with H 1 correlations, as showed in Fig. 3. Importantly, while over-estimation of ρ 0 will result in a decrease of power, it will not increase the false positive rate (FDR or FWER). ρ 0 estimator. Left: estimation of ρ 0 for chromosome 5 under different levels of both H 0 and H 1 correlations (ρ 0=0.08,0.18 and 0.28). Dashed lines indicate the true ρ 0. Right: estimation of ρ 0 for ρ 0=0.18 and different levels of H 1 correlations according to the fraction of H 1 correlations (the results are showed for five typical chromosomes only). Top: n=50. Bottom: n=100 To assess the performance of SegCorr, the true positive rate (TPR = sensitivity), false positive rate (FPR =1− specificity) and area under the ROC curve (AUC) were considered. These criteria were first computed at the gene level. However, as the goal is to identify correlated regions, a definition of TPR and FPR at the region level was adopted. We considered the intersection between the true and the estimated segmentations and computed the number of true/false positive/negative regions. This amounts at classifying each gene into one of four status (true/false × positive/negative) and then to merge neighboring genes sharing a same status into regions. The status of a region is given by the status of its genes. Consequently, criteria computed at the region level are more stringent as they measure the precision of region boundary estimation. Figure 4 (top) shows the AUC for Scenario 1 under various configurations, with ρ 1 fixed at 0.5. When ρ 0 is between 0.08 and 0.18, most regions are correctly detected. For ρ 0=0.28 (a value higher than what is observed on the reference dataset, see Fig. 2), the task becomes difficult and the performance deteriorates. AUC for Simulation Design 1 and 2. AUC at the gene level (red) and region level (blue). The higher the AUC the better. Top: Simulation design 1 with fixed ρ 1=.5 (x-axis: ρ 0). Bottom: Simulation design 2 (x-axis: ρ 1) For Scenario 2, the behavior of SegCorr was explored under different ρ 1. Obviously the task becomes easier when ρ 1 gets larger. Figure 4 shows that SegCorr performs well when 0.5≤ρ 1≤0.9. When ρ 1≤0.5, (remind that the background correlation can be as high as 0.2, see Fig. 2) although the performances remain good at the gene level, the boundaries of the regions are detected less accurately. Comparison with the TCM algorithm SegCorr was compared with the TCM algorithm introduced by [28] for the detection of regional correlations. The choice of the TCM as a competing method was based on the availability of the code. Indeed, the code of CluGene [13] is not currently available and this of G-NEST [20] relies on obsolete linux packages. Figure 5 displays the AUC achieved by SegCorr and TCM under Scenario 2 for ρ 1=0.5. When ρ 0 is large (ρ 0=0.28), one observes that the mean performance of both methods are comparable with higher variability for SegCorr at the gene level and at the region level for TCM. Since the aim is to detect regions rather than genes, the SegCorr procedure seems more appropriate. For small or medium values of background correlations (ρ 0=0.08,0.18) SegCorr achieves better AUC than TCM at both the gene and the region levels. As a conclusion, SegCorr appears to be a more consistent and efficient procedure to detect correlated regions. Similar performance between SegCorr and TCM can be observed for other values of ρ 1, results not included here. AUC for SegCorr and TCM (Scenario 2). AUC of the SegCorr (n=50-red, n=100-blue) and TCM (n=50-grey, n=100-green) algorithms for Scenario 2 as a function of ρ 0. Left: gene level. Right: region level Figure 6 illustrates the performance of SegCorr and TCM under Scenario 3. As in the previous case, SegCorr outperforms TCM both on the gene and region level. AUC for SegCorr and TCM (Scenario 3). AUC of the SegCorr (n=50-red, n=100-blue) and TCM (n=50-grey, n=100-green) algorithms for Scenario 3 as a function of β. Left: gene level. Right: region level We observe that the performance of both algorithms remains unchanged between the different values of β. Further investigations (results not shown) show that classification errors predominantly occur in small regions with or without variability. The simulation shows that only the mean correlation within the blocks matters and that the proposed method is robust to intra-region variability of correlations. On an Intel i7-4790 CPU processor at 3.60GHz, the CPU times is 74s for SegCorr and 61s for TCM for the bladder cancer dataset. However, in practice TCM must be executed many times in order to manually tune its input parameters (such as the window size and the threshold). On the contrary, SegCorr has to be run only once. Bladder cancer data In this section, we apply SegCorr on a bladder cancer dataset described in Section 'Data presentation' below. It is now well known that copy number variation (CNV) impacts gene expression [29]. Here our goal is to detect regions where the correlation is not due to CNV occuring in cancer. Therefore we correct the expression signal for CNV variation according to the strategy described in Sections 'Accounting for known sources of regulation' and 'Procedure for CNV correction'. The effect of this correction is investigated in Section 'CNV-dependent regions'. Lastly, Section 'CNV-independent regions' illustrates the biological results obtained after correction for CNV. Data presentation The dataset consists of n=403 bladder tumors. Gene expression have been measured using RNA-seq. The number of genes per chromosome ranges from 293 to 1695 (with average 702). Additionally CNV data have been obtained with Affymetrix Genome wide SNP 6.0 arrays and methylation data with Illumina Human methylation 450k arrays. All RNA-seq, SNP and methylation data were dowloaded from the TCGA open-access HTTP directory (https://portal.gdc.cancer.gov/projects/TCGA-BLCA) and are level 3 data. Study of the model selection threshold S For the model selection criterion, the threshold S (defined in Section 'Inference of correlated regions', Eq. (6)) must be tuned in such a way to avoid under/over-segmentation. The smaller the value of S the higher the number of segments. As stated in Section 'Inference of correlated regions', S was fixed to 0.7 as advocated in [17]. Figure 7 shows the evolution of the number and location of H 1 regions detected by SegCorr according to S on a typical chromosome (chromosome 3). One can see that most of these H 1 regions are stable for values of S between 0.6 and 0.9. Still, the value of S may need to be adapted when applied to other data-type or to another dataset. The choice of S can be parametrized in the SegCorr R package, with default value 0.7. Choice of S. Left: statistically significant regions in red obtained for different values of S. The vertical lines correspond to the ones obtained with the default value of S we considered (S=0.7). Right: number of statistically significant regions for different values of S. The dotted vertical red line corresponds to S=0.7 Procedure for CNV correction To correct the expression signal from CNV, one first needs to detect the CNV regions from the SNP array signal. To this aim, we consider the segmentation method proposed by [26] implemented in the R package cghseg. Denote SNP it the SNP signal of patient i at position t, the model writes $$\begin{array}{@{}rcl@{}} SNP_{it}=\mu_{ik}+E_{it}\ \ \text{if} t \in I_{k}^{i}=\,\left[t_{k-1}^{i}+1,t_{k}^{i}\right]. \end{array} $$ where the E it are i.i.d centered Gaussian with variance σ 2. The method estimates the number of regions, the boundaries of the regions, denoted $\hat {t}_{k}^{i}$ and the signal mean within each region k in patient i, denoted $\hat {\mu }_{ik}$. This procedure may be adapted to count data such as provided by DNAseq data, for which dedicated segmentation tools exist (see e.g. [8]). We then use the regression model (2) to make the correction where x ij is the mean $\hat {\mu }_{ik}$ obtained previously if the SNP position t corresponds to gene j of the expression signal in patient i. The TCGA expression data arise from RNAseq but are provided as read counts or normalized read counts (RSEM). Then the dataset was normalized using the log(x+1) method as provided in https://genome-cancer.ucsc.edu/. Finally, we directly applied Model (2) to the normalized RNAseq data. Still, as often in RNAseq, an important proportion of zero is observed. Genes with null expression in all samples were removed. For the remaining zeros, we either left them when fitting the regression model, or removed them and then set the corresponding residual $\widetilde {Y}_{ij}$ to 0 (note that, in the last option, these observations do not contribute to the estimation of the between-gene correlation, as the mean of the residuals is 0 by construction). Both options were found to provide similar results, so only the ones obtained with the first option are displayed in the following. Since the SNP and expression signals are not aligned, there might be either one, many or no SNP probes that belong to the corresponding gene region. We then propose to define x ij as follows : if one or many probes are related to gene j, mean $\hat {\mu }_{ik}$ or the average of the different means is considered respectively; if there is no probe, a linear interpolation is performed. CNV-dependent regions We first investigate the effect of CNV correction (described in Section 'Procedure for CNV correction') by comparing the results obtained on the raw and corrected signals. Figure 8 displays the number of significant H 1 regions as a function of the test level α for both the raw and corrected signals. For small values of α (which are typically used for testing significance), the number of detected regions are quite similar. However, only one third of the detected genes are common, meaning that the regions detected with the two signals are quite different. Furthermore, as the correction removes all effects due to CNV, the estimated background correlation is lower in the corrected signal than in the raw signal (mean decrease across all chromosomes of.07). This makes the test we propose more powerful and explains why, while CNV-due regions are removed, the number of detected regions for a given α remains about the same. Bladder Regions. Left: Number of statistically significant regions as a function of α (solid line: corrected signal, dotted line: raw signal). Right: proportion of significant genes common in the two signal as a function of α To illustrate this phenomenon more precisely, we considered a set of four regions in chromosomes 3, 8, 10 and 12 known to be associated with CNV in bladder cancer [31, 35, 39]. These regions, given in Table 1, are detected by SegCorr when applied to the raw expression data. When considering the corrected signal, these regions are not detected any more. For the region in chromosome 10, the background correlation was $\widehat {\rho }_{0} = 0.221$ and the correlation within this region was $\widehat {\rho }_{k} = 0.405$, resulting in a highly significant p-value: 8.25e-06. After correction we get $\widehat {\rho }_{0} = 0.152$ and $\widehat {\rho }_{k} = 0.134$, which results in a non-significant p-value: 0.623. Table 1 Four examples of CNV-dependent regions More generally, over the 119 regions solely detected on the raw signal with p-value smaller than 5% (before multiple testing correction), one third (44) get non significant when considering the corrected signal. This explains a substantial part of the difference between the regions detected on raw and corrected signals. This also shows that the proposed CNV correction strategy performs reasonably well. CNV-independent regions General description When applied to the CNV-corrected expression signal, SegCorr detected 588 significant regions (adjusted p-value ≤0.05) which are distributed throughout the genome (an average of 25 regions per chromosome). Among these regions, 135 regions contained well known gene family clusters such as the HOXA, HOXB, HOXD clusters, several KRT clusters, the epidermal differentiation complex, and HLA gene families clusters whose expression is known to be co-regulated [33]. We next undertook a Gene Ontology terms analysis with genes contained in the significant regions and identified an enrichment of genes belonging to the keratinization pathway (p-value 4.09E-19 and FDR q-value 9.01E-16). The expression of this pathway is strongly associated with a subgroup of bladder cancer called basal-like bladder cancer [27]. Epigenetic regions Apart for CNV, DNA methylation is one of the possible explanations for expression correlation. We first investigated whether the correlation between gene expression and DNA methylation is higher in significant regions when CNV correction is applied. The mean correlation varies marginally when considering the significant regions altogether. This suggests either that methylation is not a systematic cause of expression correlation or that the available signal is too noisy to detect methylation effect. Still SegCorr allowed us to detect regions where DNA methylation is associated with expression correlation. More specifically, we now present one such region where the observed correlation is not due to CNV and can be associated with an epigenetic mark. This region located on chromosome 17 contains seven genes (HOXB2, HOXB3, HOXB4, HOXB5, HOXB6, HOXB7, HOXB8: $\widehat {\rho }_{k}= 0.717$, p-value = 7.94e-62). Three genes from this regions have already been studied by [37] and has been referred to as 17-7. Figure 9 (top) shows a clear pattern detected in both the expression data and the DNA methylation data. When classifying the patients into three groups, the right-most group displays an over-expression of the genes and a low methylation signal. The methylation of the DNA is one of several epigenetic mechanisms used by the cell to silence the expression of a gene. The tumors that expressed the HOXB gene family present an hypomethylation of the DNA and the tumors which did not express these genes have an hyper methylation of the DNA. This suggests that this region is silenced by an epigenetic mechanism associated with DNA methylation. Heatmaps for Region with Epigenetic Mark. Expression (top) and methylation (bottom) data from Region 17-7. The tumors have been ordered according to the average expression of the genes from region 17-7. The same ordering of the tumors (x-axis) was kept in the bottom plot The identification of co-regulated chromosomal regions is important to fully understand the gene transcription network and to identify new mechanisms of gene regulation and their deregulations in pathological states such as cancer. In this paper, we developed a method to identify these regions and we applied it to cancer data. The method relies on a formal definition of what correlated regions are. It takes advantage of an efficient inference algorithm and a statistical testing procedure, which are both exact. We also proposed a correction strategy that allows one to investigate the possible causes of the observed correlations. Using this method, we could identify copy number dependent and copy number independent correlated regions of expression. Copy number dependent regions correspond to genomic alterations; copy number independent regions could be due to different mechanisms, including epigenetic mechanism. We showed, for one region, which is part of the HOXB complex, that there is negative correlation between expression and DNA methylation. The detected regions should be further investigated to better understand the underlying mechanism. While the expression data used here were acquired using the RNA-seq technology, any other technology, including microarray technologies can be used as well. In our analysis, we have assumed stretches of correlated contiguous neighboring genes. This is obviously a simplification. Within a correlated region, a gene (or a few genes) could exhibit a weak or even a negative correlation with the other genes. This could occur for different reasons: the gene can be not expressed; alternatively, the gene could be non affected by the regulation process that impacts the other ones; finally, the gene could be impacted in a opposite way compared with the other ones. Note that genes that exhibit no expression or no variation in the dataset can be detected and could be discarded before applying the analysis. While this preprocessing was not required in the present study, running the analysis without removing non-expressed genes would lower the performance of any method aimed at finding correlated (and reasonably homogeneous) regions. Alternatively, accounting for a variable number of uncorrelated genes in correlated regions is an obvious follow-up of the present work. The proposed correction strategy could easily be generalized to more than one signal to correct for, as it does not rely on a joint modeling of all types of data at hand. Furthermore the segmentation used in the correction step enables one to deal with signals with different probe densities. Finally, this correction approach allowed us to keep all tumors in the study, as opposed to [34] were tumors with CNV in a given region were excluded when analysing this region. Also, prior information on genes or regions could be accounted for in the segmentation step. Indeed, the likelihood $\widehat {\mathcal {L}}(\tau, \tau ')$ associated with a given region can be reweighted or penalized, the dynamic programming algorithm then applies with the same computational complexity. SegCorr is a novel statistical procedure build for the identification of adjacent co-expressed genes. Some of these regions could be attributed to copy number variation events. To this end, we propose a model to correct gene expression for CNV. This method can be extended for the correction of other data types. R package SegCorr is available on the CRAN. AUC: Area under the curve CNV: Copy number variation FDR: False discovery rate FPR: False positive rate FWER: Family wise error rate ROC: Receiver operating characteristic TPR: True positive rate TF: Aldred P, Hollox E, Armour J. Copy number polymorphism and expression level variation of the human alpha-defensin genes DEFA1 and DEFA3. 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Zhang Q, Ding L, Larson DE, Koboldt DC, McLellan MD, Chen K, Shi X, Kraja A, Mardis ER, Wilson RK, et al.CMDS: a population-based method for identifying recurrent DNA copy number aberrations in cancer from high-resolution data. Bioinformatics. 2010; 26(4):464–9. doi:10.1093/bioinformatics/btp708. The authors thank E. Chapeaublanc from Institut Curie for providing part of the data. This work has been supported by the INCa_4382 research grant and by Fondation pour la Recherche Médicale (FDM20150633762). The R package SegCorr can be downloaded from CRAN: https://cran.r-project.org/web/packages/SegCorr/index.html. The results published here are in part based upon data generated by the TCGA Research Network: https://portal.gdc.cancer.gov/projects/TCGA-BLCA and are level 3 data. AgroParisTech UMR518, Paris, 75005, France Eleni Ioanna Delatola , Emilie Lebarbier , Tristan Mary-Huard & Stéphane Robin INRA UMR518, Paris, 75005, France Institut Curie, PSL Research University, Cedex 05, Paris, 75248, France , François Radvanyi & Jennifer Wong CNRS UMR144, Equipe Labellisee par La Ligue Nationale contre le Cancer, Cedex 05, Paris, 75248, France INRA, UMR 0320 - UMR 8120 Genetique Quantitative et Evolution-Le Moulon, Gif-sur-Yvette, F-91190, France Tristan Mary-Huard Molecular Oncology Unit, Department of Biochemistry, Hospital Saint Louis, AP-HP, Cedex 10, Paris, 75475, France Jennifer Wong Université Paris Diderot, Sorbonne Paris Cité, CNRS UMR7212/INSERM U944, Cedex 10, Paris, 75475, France Search for Eleni Ioanna Delatola in: Search for Emilie Lebarbier in: Search for Tristan Mary-Huard in: Search for François Radvanyi in: Search for Stéphane Robin in: Search for Jennifer Wong in: ED, EL, TMH and SR developed the statistical methodology. ED created the R package and performed the data analysis together with JW. ED, FR and JW interpreted the results of the bladder cancer data. All authors contributed to the redaction of the manuscript. All authors read and approved the final manuscript. Correspondence to Eleni Ioanna Delatola. Ethics declarations Additional file Additional file 1 Appendix file containing a table with all the competing methods, the proof of Lemma 1 and the distribution of the test statistic. (PDF 106 kb) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. SegCorr Sequence analysis (methods)	CommonCrawl
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up. Is an options implied dividends DCF model consistent with risk neutral/arbitrage-free valuation? We're talking about how we price every financial instrument: by discounting the payoff, that is, we take future cash flows and we discount them by a proper rate which takes into account the risk of not receiving those flows. I was thinking about these two points: to keep it simple, when we're dealing with unknown future cash flows (e.g. floating rate notes) we use forward prices to be consistent with an arbitrage-free setting; Put-Call parity yields implied dividends paid by the underlying in the future. Let you have an array of implied dividends coming from Put-Call parity: these are the future cash flows paid by the underlying according to the arbitrage-free rules for which Put-Call parity holds true. So if the present value, $P(0)$, of any security is the discounted sum of its future cash flows on date $t$, $c(t)$, here we could write $$P(0)=\sum_{t=0}^{T}\frac{d(t)}{\left(1+k(t)\right)^{t}}+\frac{d(T)}{k(T)\left(1+k(T)\right)^{T}}$$ Aside from typos, last cash flow is the terminal value and $d(t)$ terms are the implied dividends paid on date $t$. If we knew $k(t)$ term structure, we would know something like a yield term structure for the underlying risk: unfortunately, we don't have that. However, we have $P(0)$. does it make sense to solve for $\bar k$ like we do for vanilla bonds yield to maturity? This method is currently used by practitioners for everything which it shouldn't be used for: variable rate notes, callable bonds and so forth. Then I feel authorized to do it. What would this $\bar k$ represent? I would say it has the same properties of the yield to maturity: it is the return of the security only under very irrealistic assumptions... nonetheless, it's useful for comparisons and rankings purposes. Why isn't this model consistent with arbitrage-free valuation? Today I buy shares at $P(0)$ and I hedge my position by building (short) parity positions: I sell Call options and buy Put options struck at the same price. I get dividends from the underlying shares, same dividends that I "lose" when my short Calls expire and I have to deliver. Any thoughts about the reason for which I'm wasting my time by building a screener for such "implied dividend yields"? dividends discounting put-call-parity Lisa AnnLisa Ann Since you have not defined a probability measure for $\mathbb{E}\left[d(t)\right]$, I don't think your model can qualify as risk neutral. In order for a discounting model to qualify as risk neutral, it must define a distribution or measure. In quant finance parlance, a risk free measure exists only if one can: a) define a probability measure such that the price is equal to the expected net present value (i.e., equivalent martingale measure) which is absolutely continuous with respect to the original measure (e.g., via Girsanov's theorem); and, b) demonstrate that the chosen probability measure cannot be incorrect (via the dynamic hedging argument and/or the fundamental theorem of asset pricing). It seems that what you are actually trying to value qualifies as an annuity, which falls under the umbrella of actuarial science rather than quantitative finance. Generally, actuarial sciences deal with real world probability measures since it is difficult to infer the concrete relationship which generally hold for other financial derivatives, such as: Contingent payoff conditions Probability measure for the underlying of dividends (profits?) Arbitrage relationship Terminal conditions such as time Boundary conditions such as strike As such, valuation models for annuities, equity (e.g., DCF and DDM), and other underlying assets are usually taken with regard to real world measures. For a better explanation of real world versus risk neutral measure, please see this Wiki. If you are willing to forego a strict interpretation of the annuity model, Samuelson and McKean (1965) provide a closed form approach for pricing American-style warrants which are analogous to pricing under the DDM. Also, the Merton Model (1974) proceeds to value corporate liabilities in the Black-Scholes world. In fact, Moody's flagship model, the Kealhoffer-Merton-Vasicek (KMV) model, is basically a snazzier version of Merton. However, if you want to get into the wonky actuarial math, there have been a number of papers on stochastic cash flows and stochastic annuities. Many of the foundational works, in my opinion, are from Daniel Dufresne and Mark Yor. Also notable is: (Moshe Arye Milevsky. The present value of a stochastic perpetuity and the Gamma distribution Feb 1997). My key insight is that quant finance values contingent claims only by assuming an integrated measure of underlying asset value is the right one whereas actuarial science may attempt to derive the integrated asset value from the cash flows themselves. I have hunch that the chasm between these two worlds can be bridged through path integral approaches which were developed for quantum mechanics. The main difference is that the integration occurs pathwise in path integration whereas the foundations of modern quant finance (Ito calculus) proceeds as the limit of Riemann sums with finite quadratic variation. Indeed, promising inroads have been made (Devreese, Lemmens, and Tempere 2009). David AddisonDavid Addison In regards to 1 & 2, I think k(t) in this case represents more of the spot rate than the YTM. If you are assuming that the term structure is static, then k(t) can be interpolated for n periods of discounting dividends, but this is not realistic. I think this is ultimately why your model is not consistent with arbitrage free valuation. HK47HK47 Thanks for contributing an answer to Quantitative Finance Stack Exchange! Not the answer you're looking for? Browse other questions tagged dividends discounting put-call-parity or ask your own question. What changes to put-call parity are necessary when evaluating american options on non-dividend paying assets? Implied dividend estimation Is this representation of the put-call parity correct? (Implied dividend estimation) Implied Dividend from American Options (in practice) How far the spot price is likely to go from the current level in three months if its volatility is 15.7% Risk of Put-Call-Parity in practice Should Put/Call Parity result in Zero Return or the Risk-Free Rate? Proving the put call parity Calculating risk free rates from risky options using put call parity	CommonCrawl
Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set Sergi Molins ORCID: orcid.org/0000-0001-7675-32181, Cyprien Soulaine2,3,4, Nikolaos I. Prasianakis5, Aida Abbasi5, Philippe Poncet6, Anthony J. C. Ladd7, Vitalii Starchenko8, Sophie Roman2,4, David Trebotich1, Hamdi A. Tchelepi2 & Carl I. Steefel1 Computational Geosciences volume 25, pages 1285–1318 (2021)Cite this article This manuscript presents a benchmark problem for the simulation of single-phase flow, reactive transport, and solid geometry evolution at the pore scale. The problem is organized in three parts that focus on specific aspects: flow and reactive transport (part I), dissolution-driven geometry evolution in two dimensions (part II), and an experimental validation of three-dimensional dissolution-driven geometry evolution (part III). Five codes are used to obtain the solution to this benchmark problem, including Chombo-Crunch, OpenFOAM-DBS, a lattice Boltzman code, Vortex, and dissolFoam. These codes cover a good portion of the wide range of approaches typically employed for solving pore-scale problems in the literature, including discretization methods, characterization of the fluid-solid interfaces, and methods to move these interfaces as a result of fluid-solid reactions. A short review of these approaches is given in relation to selected published studies. Results from the simulations performed by the five codes show remarkable agreement both quantitatively—based on upscaled parameters such as surface area, solid volume, and effective reaction rate—and qualitatively—based on comparisons of shape evolution. This outcome is especially notable given the disparity of approaches used by the codes. 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Water Res. 115, 151–159 (2018). https://doi.org/10.1016/j.advwatres.2018.03.005 Yoon, H., Valocchi, A.J., Werth, C.J., Dewers, T.: Pore-scale simulation of mixing-induced calcium carbonate precipitation and dissolution in a microfluidic pore network. Water Resour. Res. 48(2). https://doi.org/10.1029/2011wr011192. W02524 (2012) Zhao, B., MacMinn, C.W., Juanes, R.: Wettability control on multiphase flow in porous media: A benchmark study on current pore-scale modeling approaches. 71st Annual Meeting of the APS Division of Fluid Dynamics. In: Bull. Am. Phys. Soc. American Physical Society. http://meetings.aps.org/Meeting/DFD18/Session/G26.2 (2018) This material is based upon work supported as part of the Center for Nanoscale Control of Geologic CO2, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under award DE-AC02-05CH11231. Chombo-Crunch simulations used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract DE-AC02-05CH11231. Development of dissolFoam was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under Award Number DE-FG02-98ER14853 and DE-SC0018676. Development of the advanced mesh relaxation in dissolFoam was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the US Department of Energy. Vortex method simulations acknowledge the HPC resources of cluster Pyrene (UPPA-E2S, Pau, France) and the support of the Carnot Institute ISIFoR under contract RugoRX. N.I.P. acknowledges support from Swiss National Science Foundation, SNSF project No: 200021_172618, and the Swiss National Supercomputing Centre (CSCS). The benchmark problem set was proposed and developed by S.M. and C.S. Manuscript preparation was led by S.M. Chombo-Crunch simulations were conducted by S.M., OpenFOAM-DBS by C.S., lattice Boltzmann by A.A. and N.I.P., vortex by P.P., and dissolFoam by A.L. and V.S. These authors are listed according to when they joined the benchmark effort. Chombo-Crunch was developed by D.T. and S.M., CrunchFlow by C.I.S., OpenFOAM-DBS by C.S., the lattice Boltzmann code by N.I.P., the vortex code by P.P., and dissolFoam by A.L. and V.S. Part III experiments were conducted by S.R. Lawrence Berkeley National Laboratory, Berkeley, CA, USA Sergi Molins, David Trebotich & Carl I. Steefel Stanford University, Stanford, CA, USA Cyprien Soulaine, Sophie Roman & Hamdi A. Tchelepi French Geological Survey (BRGM), Orléans, France Cyprien Soulaine Université Orléans-CNRS-BRGM, - Institut des Siences de la Terre d'Orléans, Orléans, France Cyprien Soulaine & Sophie Roman Paul Scherrer Institute, Villigen, Switzerland Nikolaos I. Prasianakis & Aida Abbasi E2S UPPA, CNRS, LMAP UMR 5142, Universite de Pau, Pays Adour, Pau, France Philippe Poncet University of Florida, Gainesville, FL, USA Anthony J. C. Ladd Oak Ridge National Laboratory, Oak Ridge, TN, USA Vitalii Starchenko Sergi Molins Nikolaos I. Prasianakis Aida Abbasi Sophie Roman David Trebotich Hamdi A. Tchelepi Carl I. Steefel Correspondence to Sergi Molins. Below is the link to the electronic supplementary material. (ZIP 1.48 GB) Appendix A: Additional/alternative equations Flow at the pore scale may be described by the incompressible Navier-Stokes (33) and (34): $$ \nabla\cdot \textbf{u} = 0, $$ $$ \frac{\partial \textbf{u}}{\partial t}+\left( \textbf{u}\cdot\nabla \right)\textbf{u}+\frac{1}{\rho}\nabla p=\nu\nabla^{2}\textbf{u}, $$ as well as the Stokes (1) and (2). In these benchmarks, the Reynolds number is sufficiently small that fluid inertia can be neglected; thus, these two approaches are equivalent. In the dissolution benchmarks (parts II and III), codes may take advantage of the large time scale separation between boundary motion and transport processes to solve the steady-state transport equation directly, $$ \boldmath{\nabla} \cdot (\mathbf{u} c) = D \nabla^{2}c. $$ Time-dependent solutions of transport and reaction (part I) are more tightly coupled than dissolution (parts II and III), because tA and tR are often of the same order, especially for relatively fast reacting minerals such as carbonates. Both global implicit and operator splitting approaches have been used for time-dependent transport, with the time stepping in the operator splitting constrained by the Courant-Friedrichs-Levy (CFL) criterion $$ {\Delta} t<\frac{\Delta x}{\max{(u)}}. $$ Appendix B: Analysis and comparison of results B.1. Upscaled parameters Simulation results are compared in terms of the evolution with time of upscaled parameters. These upscaled parameters include the volume (V ) and surface area of all reacting reacting mineral surfaces (A) and the average reaction rate (R). The average rate is calculated as follows: $$ R=\frac {Q (c_{out}-c_{in})} {\xi A}, $$ where ξ is the stoichiometric coefficient, cin is the (uniform) concentration at the inlet, given by the boundary conditions, and cout is the flux-weighted-average outlet concentration, $$ c_{out}= \frac{{\int}_{\delta S} c {\textbf{u}} \cdot d \boldsymbol{s}} {Q}. $$ The volumetric flux at the outlet Q is found by integrating over the outlet area $$ Q={\int}_{\delta S} \mathbf{u} \cdot ds. $$ In addition to these upscaled parameters, simulation results are compared on the basis of the geometry of the grain at different time points and the concentration contours are prescribed times. B.2. Grid convergence As methods for simulating of moving boundary problems vary greatly, we want to investigate the impact of grid resolution on results for each method separately. For this purpose, the simulations were run at different resolutions (Figs. 14, 15, and 9) in the main text. Results for the grain volume and surface area were analyzed to ensure grid convergence of the methods, and choose a resolution for which results will be assumed to have converged within a reasonable accuracy. Grid convergence tests results for the time evolution of the grain volume (part II) from a OpenFOAM-DBS, b lattice Boltzmann, c vortex, d Chombo-Crunch, and e dissolFoam simulations Grid convergence tests results for the time evolution of the grain surface area (Part II) from a OpenFOAM-DBS, b lattice Boltzmann, c vortex, d Chombo-Crunch, and e dissolFoam simulations Appendix C: Notes on unit conversion for concentrations and rates The conversions from the parameters reported by [96] to the units used in Part III are presented. Mass fraction is converted to molar concentration using $$ c = \frac{\rho f} {M}, $$ where c is the molar concentration of protons (molcm− 3), M is the molar weight of acid (gmol− 1), ρ is the fluid density (gcm− 3), and f is the mass fraction of acid. The inlet concentration (0.05%) is converted to mol cm− 3 as follows: $$ c_{in} = \frac{0.92 \text{g} \text{cm}^{-3} \times 0.0005} {36.5 \text{g} \text{mol}^{-1}} = 1.26 \cdot 10^{-5} \text{mol} \text{cm}^{-3}. $$ In the formulation used in this manuscript (Section 2), the first-order reaction is expressed as a function of the activity coefficient and the molar concentration of H+ (26). Assuming that $\gamma _{\text {H}^{+}}=1000 \text {cm}^{3} \text {mol}^{-1}$, the proton concentration $c_{\text {H}^{+}}$ must be in mol cm− 3 so that the product $k_{\text {H}^{+}} \gamma _{\text {H}^{+}}$ has units of cm s− 1. The conversion from the rate constant used in [96] ($k_{\text {H}^{+}} \gamma _{\text {H}^{+}} = 0.5 \text {cm\ s} ^{-1}$) is $$ k_{\text{H}^{+}} = \frac{0.5 \text{cm} \text{s} ^{-1}}{1000 \text{cm}^{3} \text{mol}^{-1}} = 5 \times 10^{-4} \text{mol} \text{cm}^{-2} \text{s} ^{-1}. $$ However, in [96], this rate is applicable to the rate of HCl consumption when reacting with calcite according to the following stoichiometry $$ \text{CaCO}3_{(\text{s})} + \text{2HCl} -> \text{CaCl}_{2} + \text{H}_{2}\text{CO}_{3}. $$ To maintain consistency with the rate expressed for calcite, one must multiply by the stoichiometric coefficient of HCl in Eq. 43, $$ k_{\text{H}^{+}} = 10^{-3} \text{mol} \text{cm}^{-2} \text{s} ^{-1}. $$ Appendix D: Additional information on numerical choices and parameters D.1. Lattice Boltzmann dimensionalization Dimensionalization of the LB computations is a process that needs special care. Lattice Boltzmann unit conversion to physical units can be done after matching the characteristic non-dimensional Reynolds, Péclet, and Damköhler numbers. For a 256 × 128 discretization grid Ly = 128 (in lattice units), each lattice space unit in parts I and II corresponds to w/128 = 3.91 × 10− 4cm. For the current setup, viscosity is defined as ν = τfρT. The relaxation parameter for the fluid phase, τf, is set to τf = 0.5 in lattice units. By equating Re=ReLB= 0.6, using the aforementioned viscosity, the inlet velocity can be calculated as uinLB = 0.00078125 (in lattice units), which corresponds to $\textbf {u}_{in}=0.12 \text {cm\ s} ^{-1}$. Once the lattice velocity is set, the duration of the time step δt can be calculated by equating the inlet velocities: δt = 2.54 × 10− 6 s. Note that the time step is dictated by the slow advective flow, and by choosing to keep the same time step for all processes. This leads to a fully coupled advection-diffusion-reaction scheme applicable to all flow and chemical conditions. Separation of time scales is possible by solving for steady-state flow, then steady-state reactive transport, and finally the solid geometry update. Such an approach would be sufficient for these benchmarks and would greatly reduce the number of time steps to reach the solution. Diffusivity is defined as D = τgT. By equating the Péclet numbers Pe=PeLB= 600, the relaxation parameter τg, which corresponds to the diffusive time scale, is set to τD = 0.0005, for the species that follow the advection-diffusion equation. Finally, by equating the Damköler numbers DaII=DaII-LB= 178.15, the rate constant $k_{\text {H}^{+} \text {LB}}=10^{-3.2364}$. For this dimensionalization Ma<Kn<< 1, thus recovering the incompresible Navier-Stokes equations. D.2. Discussion on interpolation kernel for Lagrangian methods The choice of the kernel Λ used for re-meshing the particle is crucial for the accuracy of vortex and particle methods. Indeed, in order to avoid holes and accumulation of particles that would ruin the convergence, particle information Fp (including vorticity, concentration, ...) in volumes vp located at positions ξp is remeshed on to a new structured mesh (with cell volumes $\tilde {v}_{q}$). This mesh defines a new set of particles $\tilde {F}_{q}$ at locations $\tilde {\xi }_{q}$ by means of the following convolution: $$ \begin{array}{@{}rcl@{}} \tilde{F}_{q}&=& F*{\Lambda} (\tilde{x}_{q})=\int F(y){\Lambda}(\tilde{x}_{q}-y)dy\\ &=& \sum\limits_{p} F_{p}{\Lambda}(\tilde{x}_{q}-x_{p})v_{p}, \end{array} $$ since the set of particles is mathematically defined by the generalized function $\displaystyle F=\sum \limits _{p} F_{p}\delta _{x_{p}}v_{p}$, based of Dirac functions at xp. In practice, when Λ is the "hat" (or "tent") function, the reaction stays confined on the fluid/solid interface, but exhibits a pH over-estimation close to the stagnation points, thus over-estimating the reaction rate. When this kernel is smoother but positive in order to be sign preserving, such as the first-order cubic spline M4, the fluid/solid boundary becomes fuzzy and requires us to force the reaction on the interface by means of the function ∥∇𝜖∥, as in [96]. When using the second-order kernel $M_{4}^{\prime }$ from [67], which is non sign preserving since the integral of $x^{2}M_{4}^{\prime }(x)$ is zero, no negative concentration appears despite the jump of acid concentration at the body but it leads underestimation of reaction rate. However, the hydrodynamic flow is computed with better accuracy using $M_{4}^{\prime }$, as expected [28]. Consequently, the short-supported function M3, smoother than the hat function with a support smaller than M4, has been chosen for interpolating and remeshing the chemical concentrations, while $M_{4}^{\prime }$ has been chosen for the interpolation hydrodynamic values (velocity and vorticity). In practice for the present benchmark, for which the reaction properties (bounds and positivity) have to be strictly satisfied, the choice of the remeshing kernel is mainly driven by the following arguments: The hat function, is good for the estimation of reaction rate but does respect the pH bounds (pH overshoots below 2 can occur), • The kernel M4 is smooth but M4(0) = 2/3 ≠ 1; thus, it is diffusive: pH bounds are good but reaction rate is under-estimated (see formula A.4 of [18] for definition), $M_{4}^{\prime }$ (formula 4.5 of [20]) is algebraically mass-conservative, smooth, and second order, but its negative values induce oscillations at concentration jumps and over-estimate the reaction rate. Furthermore, it is not mathematically sign preserving, although negative concentrations were never been observed in this benchmark, M3 (formula A.3 of [18]) is smoother than hat, first order and sign preserving, with short support. It is the best choice for reactive flows like the one considered in the present study; the reaction rate is well estimated (a bit higher than the hat function and closer to other curves) and does not go lower than the initial pH= 2 bound, consistent with this purely dissolution process, M6 and $M_{6}^{\prime }$ supports are too large for this geometry, and cannot handle correctly the final state of the dissolution. Consequently, the kernel $M_{4}^{\prime }$ is the best choice for hydrodynamic computations (for particle remeshing and interpolation of velocity and vorticity from and to grids), while M3 is the best choice for interpolation and transfer of concentrations. D.3. Darcy-Brinkman-Stokes parameter values Parameters specific to Darcy-Brinkman-Stokes code simulations are presented in Table 6. Table 6 Parameters for Darcy-Brinkman-Stokes equations Molins, S., Soulaine, C., Prasianakis, N.I. et al. Simulation of mineral dissolution at the pore scale with evolving fluid-solid interfaces: review of approaches and benchmark problem set. Comput Geosci 25, 1285–1318 (2021). https://doi.org/10.1007/s10596-019-09903-x Issue Date: August 2021 DOI: https://doi.org/10.1007/s10596-019-09903-x Pore scale Reactive transport Moving boundary Review of approaches	CommonCrawl
Quantum graph In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) connected at transformer stations (vertices); the differential equations would then describe the voltage along each of the lines, with boundary conditions for each edge provided at the adjacent vertices ensuring that the current added over all edges adds to zero at each vertex. Quantum graphs were first studied by Linus Pauling as models of free electrons in organic molecules in the 1930s. They also arise in a variety of mathematical contexts,[1] e.g. as model systems in quantum chaos, in the study of waveguides, in photonic crystals and in Anderson localization, or as limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, more simple notion of quantum graphs was introduced by Freedman et al.[2] Aside from actually solving the differential equations posed on a quantum graph for purposes of concrete applications, typical questions that arise are those of controllability (what inputs have to be provided to bring the system into a desired state, for example providing sufficient power to all houses on a power network) and identifiability (how and where one has to measure something to obtain a complete picture of the state of the system, for example measuring the pressure of a water pipe network to determine whether or not there is a leaking pipe). Metric graphs A metric graph is a graph consisting of a set $V$ of vertices and a set $E$ of edges where each edge $e=(v_{1},v_{2})\in E$ has been associated with an interval $[0,L_{e}]$ so that $x_{e}$ is the coordinate on the interval, the vertex $v_{1}$ corresponds to $x_{e}=0$ and $v_{2}$ to $x_{e}=L_{e}$ or vice versa. The choice of which vertex lies at zero is arbitrary with the alternative corresponding to a change of coordinate on the edge. The graph has a natural metric: for two points $x,y$ on the graph, $\rho (x,y)$ is the shortest distance between them where distance is measured along the edges of the graph. Open graphs: in the combinatorial graph model edges always join pairs of vertices however in a quantum graph one may also consider semi-infinite edges. These are edges associated with the interval $[0,\infty )$ attached to a single vertex at $x_{e}=0$. A graph with one or more such open edges is referred to as an open graph. Quantum graphs Quantum graphs are metric graphs equipped with a differential (or pseudo-differential) operator acting on functions on the graph. A function $f$ on a metric graph is defined as the $\|E\|$-tuple of functions $f_{e}(x_{e})$ on the intervals. The Hilbert space of the graph is $\bigoplus _{e\in E}L^{2}([0,L_{e}])$ where the inner product of two functions is $\langle f,g\rangle =\sum _{e\in E}\int _{0}^{L_{e}}f_{e}^{}(x_{e})g_{e}(x_{e})\,dx_{e},$ $L_{e}$ may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the Laplace operator. The operator on an edge is $-{\frac {{\textrm {d}}^{2}}{{\textrm {d}}x_{e}^{2}}}$ where $x_{e}$ is the coordinate on the edge. To make the operator self-adjoint a suitable domain must be specified. This is typically achieved by taking the Sobolev space $H^{2}$ of functions on the edges of the graph and specifying matching conditions at the vertices. The trivial example of matching conditions that make the operator self-adjoint are the Dirichlet boundary conditions, $f_{e}(0)=f_{e}(L_{e})=0$ for every edge. An eigenfunction on a finite edge may be written as $f_{e}(x_{e})=\sin \left({\frac {n\pi x_{e}}{L_{e}}}\right)$ for integer $n$. If the graph is closed with no infinite edges and the lengths of the edges of the graph are rationally independent then an eigenfunction is supported on a single graph edge and the eigenvalues are ${\frac {n^{2}\pi ^{2}}{L_{e}^{2}}}$. The Dirichlet conditions don't allow interaction between the intervals so the spectrum is the same as that of the set of disconnected edges. More interesting self-adjoint matching conditions that allow interaction between edges are the Neumann or natural matching conditions. A function $f$ in the domain of the operator is continuous everywhere on the graph and the sum of the outgoing derivatives at a vertex is zero, $\sum _{e\sim v}f'(v)=0\ ,$ where $f'(v)=f'(0)$ if the vertex $v$ is at $x=0$ and $f'(v)=-f'(L_{e})$ if $v$ is at $x=L_{e}$. The properties of other operators on metric graphs have also been studied. • These include the more general class of Schrödinger operators, $\left(i{\frac {\textrm {d}}{{\textrm {d}}x_{e}}}+A_{e}(x_{e})\right)^{2}+V_{e}(x_{e})\ ,$ where $A_{e}$ is a "magnetic vector potential" on the edge and $V_{e}$ is a scalar potential. • Another example is the Dirac operator on a graph which is a matrix valued operator acting on vector valued functions that describe the quantum mechanics of particles with an intrinsic angular momentum of one half such as the electron. • The Dirichlet-to-Neumann operator on a graph is a pseudo-differential operator that arises in the study of photonic crystals. Theorems All self-adjoint matching conditions of the Laplace operator on a graph can be classified according to a scheme of Kostrykin and Schrader. In practice, it is often more convenient to adopt a formalism introduced by Kuchment, see,[3] which automatically yields an operator in variational form. Let $v$ be a vertex with $d$ edges emanating from it. For simplicity we choose the coordinates on the edges so that $v$ lies at $x_{e}=0$ for each edge meeting at $v$. For a function $f$ on the graph let $\mathbf {f} =(f_{e_{1}}(0),f_{e_{2}}(0),\dots ,f_{e_{d}}(0))^{T},\qquad \mathbf {f} '=(f'_{e_{1}}(0),f'_{e_{2}}(0),\dots ,f'_{e_{d}}(0))^{T}.$ Matching conditions at $v$ can be specified by a pair of matrices $A$ and $B$ through the linear equation, $A\mathbf {f} +B\mathbf {f} '=\mathbf {0} .$ The matching conditions define a self-adjoint operator if $(A,B)$ has the maximal rank $d$ and $AB^{}=BA^{*}.$ The spectrum of the Laplace operator on a finite graph can be conveniently described using a scattering matrix approach introduced by Kottos and Smilansky .[4][5] The eigenvalue problem on an edge is, $-{\frac {d^{2}}{dx_{e}^{2}}}f_{e}(x_{e})=k^{2}f_{e}(x_{e}).\,$ So a solution on the edge can be written as a linear combination of plane waves. $f_{e}(x_{e})=c_{e}{\textrm {e}}^{ikx_{e}}+{\hat {c}}_{e}{\textrm {e}}^{-ikx_{e}}.\,$ where in a time-dependent Schrödinger equation $c$ is the coefficient of the outgoing plane wave at $0$ and ${\hat {c}}$ coefficient of the incoming plane wave at $0$. The matching conditions at $v$ define a scattering matrix $S(k)=-(A+ikB)^{-1}(A-ikB).\,$ The scattering matrix relates the vectors of incoming and outgoing plane-wave coefficients at $v$, $\mathbf {c} =S(k){\hat {\mathbf {c} }}$. For self-adjoint matching conditions $S$ is unitary. An element of $\sigma _{(uv)(vw)}$ of $S$ is a complex transition amplitude from a directed edge $(uv)$ to the edge $(vw)$ which in general depends on $k$. However, for a large class of matching conditions the S-matrix is independent of $k$. With Neumann matching conditions for example $A=\left({\begin{array}{ccccc}1&-1&0&0&\dots \\0&1&-1&0&\dots \\&&\ddots &\ddots &\\0&\dots &0&1&-1\\0&\dots &0&0&0\\\end{array}}\right),\quad B=\left({\begin{array}{cccc}0&0&\dots &0\\\vdots &\vdots &&\vdots \\0&0&\dots &0\\1&1&\dots &1\\\end{array}}\right).$ Substituting in the equation for $S$ produces $k$-independent transition amplitudes $\sigma _{(uv)(vw)}={\frac {2}{d}}-\delta _{uw}.\,$ where $\delta _{uw}$ is the Kronecker delta function that is one if $u=w$ and zero otherwise. From the transition amplitudes we may define a $2\|E\|\times 2\|E\|$ matrix $U_{(uv)(lm)}(k)=\delta _{vl}\sigma _{(uv)(vm)}(k){\textrm {e}}^{ikL_{(uv)}}.\,$ $U$ is called the bond scattering matrix and can be thought of as a quantum evolution operator on the graph. It is unitary and acts on the vector of $2\|E\|$ plane-wave coefficients for the graph where $c_{(uv)}$ is the coefficient of the plane wave traveling from $u$ to $v$. The phase ${\textrm {e}}^{ikL_{(uv)}}$ is the phase acquired by the plane wave when propagating from vertex $u$ to vertex $v$. Quantization condition: An eigenfunction on the graph can be defined through its associated $2\|E\|$ plane-wave coefficients. As the eigenfunction is stationary under the quantum evolution a quantization condition for the graph can be written using the evolution operator. $\|U(k)-I\|=0.\,$ Eigenvalues $k_{j}$ occur at values of $k$ where the matrix $U(k)$ has an eigenvalue one. We will order the spectrum with $0\leqslant k_{0}\leqslant k_{1}\leqslant \dots $. The first trace formula for a graph was derived by Roth (1983). In 1997 Kottos and Smilansky used the quantization condition above to obtain the following trace formula for the Laplace operator on a graph when the transition amplitudes are independent of $k$. The trace formula links the spectrum with periodic orbits on the graph. $d(k):=\sum _{j=0}^{\infty }\delta (k-k_{j})={\frac {L}{\pi }}+{\frac {1}{\pi }}\sum _{p}{\frac {L_{p}}{r_{p}}}A_{p}\cos(kL_{p}).$ $d(k)$ is called the density of states. The right hand side of the trace formula is made up of two terms, the Weyl term ${\frac {L}{\pi }}$ is the mean separation of eigenvalues and the oscillating part is a sum over all periodic orbits $p=(e_{1},e_{2},\dots ,e_{n})$ on the graph. $L_{p}=\sum _{e\in p}L_{e}$ is the length of the orbit and $L=\sum _{e\in E}L_{e}$ is the total length of the graph. For an orbit generated by repeating a shorter primitive orbit, $r_{p}$ counts the number of repartitions. $A_{p}=\sigma _{e_{1}e_{2}}\sigma _{e_{2}e_{3}}\dots \sigma _{e_{n}e_{1}}$ is the product of the transition amplitudes at the vertices of the graph around the orbit. Applications Quantum graphs were first employed in the 1930s to model the spectrum of free electrons in organic molecules like Naphthalene, see figure. As a first approximation the atoms are taken to be vertices while the σ-electrons form bonds that fix a frame in the shape of the molecule on which the free electrons are confined. A similar problem appears when considering quantum waveguides. These are mesoscopic systems - systems built with a width on the scale of nanometers. A quantum waveguide can be thought of as a fattened graph where the edges are thin tubes. The spectrum of the Laplace operator on this domain converges to the spectrum of the Laplace operator on the graph under certain conditions. Understanding mesoscopic systems plays an important role in the field of nanotechnology. In 1997[6] Kottos and Smilansky proposed quantum graphs as a model to study quantum chaos, the quantum mechanics of systems that are classically chaotic. Classical motion on the graph can be defined as a probabilistic Markov chain where the probability of scattering from edge $e$ to edge $f$ is given by the absolute value of the quantum transition amplitude squared, $\|\sigma _{ef}\|^{2}$. For almost all finite connected quantum graphs the probabilistic dynamics is ergodic and mixing, in other words chaotic. Quantum graphs embedded in two or three dimensions appear in the study of photonic crystals.[7] In two dimensions a simple model of a photonic crystal consists of polygonal cells of a dense dielectric with narrow interfaces between the cells filled with air. Studying dielectric modes that stay mostly in the dielectric gives rise to a pseudo-differential operator on the graph that follows the narrow interfaces. Periodic quantum graphs like the lattice in ${\mathbb {R} }^{2}$ are common models of periodic systems and quantum graphs have been applied to the study the phenomena of Anderson localization where localized states occur at the edge of spectral bands in the presence of disorder. See also • Schild's Ladder, a novel dealing with a fictional quantum graph theory • Feynman diagram References 1. Berkolaiko, Gregory; Carlson, Robert; Kuchment, Peter; Fulling, Stephen (2006). Quantum Graphs and Their Applications (Contemporary Mathematics): Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Applications. Vol. 415. American Mathematical Society. ISBN 978-0821837658. 2. Freedman, Michael; Lovász, László; Schrijver, Alexander (2007). "Reflection positivity, rank connectivity, and homomorphism of graphs". Journal of the American Mathematical Society. 20 (1): 37–52. arXiv:math/0404468. doi:10.1090/S0894-0347-06-00529-7. ISSN 0894-0347. MR 2257396. S2CID 8208923. 3. Kuchment, Peter (2004). "Quantum graphs: I. Some basic structures". Waves in Random Media. 14 (1): S107–S128. doi:10.1088/0959-7174/14/1/014. ISSN 0959-7174. S2CID 16874849. 4. Kottos, Tsampikos; Smilansky, Uzy (1999). "Periodic Orbit Theory and Spectral Statistics for Quantum Graphs". Annals of Physics. 274 (1): 76–124. arXiv:chao-dyn/9812005. doi:10.1006/aphy.1999.5904. ISSN 0003-4916. S2CID 17510999. 5. Gnutzmann∥, Sven; Smilansky, Uzy (2006). "Quantum graphs: Applications to quantum chaos and universal spectral statistics". Advances in Physics. 55 (5–6): 527–625. arXiv:nlin/0605028. doi:10.1080/00018730600908042. ISSN 0001-8732. S2CID 119424306. 6. Kottos, Tsampikos; Smilansky, Uzy (1997). "Quantum Chaos on Graphs". Physical Review Letters. 79 (24): 4794–4797. doi:10.1103/PhysRevLett.79.4794. ISSN 0031-9007. 7. Kuchment, Peter; Kunyansky, Leonid (2002). "Differential Operators on Graphs and Photonic Crystals". Advances in Computational Mathematics. 16 (24): 263–290. doi:10.1023/A:1014481629504. S2CID 17506556.	Wikipedia
Enhancing power output and profitability through energy-efficiency techniques and advanced materials in today's industrial gas turbines Zainul Huda1, Tuan Zaharinie2 & Hany A Al-Ansary3 International Journal of Mechanical and Materials Engineering volume 9, Article number: 2 (2014) Cite this article This paper reviews and integrates the latest energy-efficiency improvement techniques to recently developed superalloys and coatings for enhancing power output and profitability in today's industrial gas turbines (GTs). It has been shown that the retrofitting of inlet air cooling and steam injected gas technology technologies in simple turbines can boost power output from 30 to 48.25 MW and improve generation efficiency from 29.9% to 33.4%. It has been reviewed that for improving efficiency of power plants beyond 60%, it is important to use higher turbine inlet temperature (TIT) in combined cycle gas turbine by applying single-crystal nickel-based superalloys and thermal barrier coating involving closed-loop steam cooling. It has been shown that an increase of the TIT by 56°C can result in increased revenue of US$8/kW to the manufacturer of the original rating, provided the GT is sold for US$200/kW. In view of the latest development in GT technology, which enables them to be operated at TIT = 1,600°C, it is recommended to develop superalloys based on higher-melting temperatures (Mo-based superalloys). A new chart for increasing power output has been developed so as to enable the GT designers to manufacture GT engines with enhanced profitability. There has been an increasing trend in saving energy costs for power generation in developing countries. The gas-turbine (GT)-based power plant is characterized by its relatively low capital cost as compared to the steam power plant. However, conventional industrial GT engines have lower efficiencies, especially at part load, which cause emission and environmental related problems. Electric power generation is one of the main contributors to emissions and imparts adverse effects on the environment thereby influencing human health, organism growth, climatic changes, and the like (Mahlia 2002). One of the technologies applied nowadays for efficiency improvement and clean power generation (with reduced emission) is the 'combined cycle' (CC) (Yokoyama et al. 2004; Larson and Marrison 1997).The thermal efficiency of combined cycle gas turbine (CCGT) power plants can reach 60% that is far more than that of conventional coal-fired steam turbine plants. The CCGT power plants not only conserve our limited reserves but also reduce emissions and protect our lives and environment (Balat 2010; De Biasi 2002). Even substantially higher efficiency is possible in a cogeneration plant, where excess generated heat is put to use for process or domestic heating, in addition to electric power generation (UNESCAP 2000). A review of recent advances in gas-turbine technology shows outstanding developments in its major components (Gorji-Bandpy et al. 2010; Najjar and Zaamout 1996; Huda 2009, 2007, 1995). Superalloy and coating advancements for application in hot sections of gas turbine engines have resulted in a considerable economic leverage in power plants (Miller and Chambers 1987; Reed 2006). The modern CCGT uses single-crystal (SC) superalloys and special thermal barrier coatings (TBCs) and - in one design - closed-loop steam cooling to reach more than 60% thermal efficiency. The application of advanced superalloys in the hot sections of modern gas-turbines also enables higher temperature operation. The operating temperatures in the first-stage and second-stage SC blades in modern CCGT lie in the ranges of 1,300°C to 1,500°C and 1,000°C to 1,100°C, respectively (see 'Techniques to improve efficiency of GT engines' section for details). For example, General Electric (GE) Inc. USA's H-turbine (Fairfield, CN, USA) uses SC materials in first-stage blades and vanes that operate at a temperature of 1,430°C over a long service life. Similarly, first-stage turbine blades of Siemens' SGT5-8000H machine (Siemens AG, Munich, Germany) are made of specialized high-temperature alloy material to combat long-term effects of high temperatures at constant stress (to resist creep deformation). The application of TBCs on gas turbine components provides a potential opportunity for increasing the operating temperature and further enhances the life of components (Martin et al. 2001; Gurrappa and Sambasiva 2006). The TBCs can be used at tolerable metal temperature at constant cooling levels, and the efficiency of cooling leads to lower metal operating temperatures with extended lifetime (Stöver and Funke 1999). Improving energy efficiency of GT engines Thermodynamic approach for improving efficiency of gas turbines The efficiency, η, of a gas turbine engine, like that of any heat engine, may be defined as the ratio of the effective work done to the heat obtained from burning fuel: $$ \eta ={W}_{\mathrm{out}}/{Q}_{\mathrm{in}} $$ Equation 1, which is a notation of the first law of thermodynamics, can be formulated as follows: the thermal efficiency of a heat engine cannot exceed 1. A temperature-dependent relationship can be developed by application of the second law of thermodynamics (also called the Carnot-Clausius theorem) which states: 'The efficiency of a heat engine reversibly operating by the Carnot cycle does not depend on the nature of the machine's working medium, but only depends on the temperatures of the heater and the cooler' (Kuchling 1980). Hence, the efficiency, η, of power plant can be calculated through the temperature difference between the hot (T hot) and the cold (T cold) thermal reservoirs, as follows: $$ \eta =\left({T}_{\mathrm{hot}}-{T}_{\mathrm{cold}}\right)/{T}_{\mathrm{hot}}=1-{T}_{\mathrm{cold}}/{T}_{\mathrm{hot}} $$ where T hot is the highest temperature reached in the cycle (temperature of the heat addition), and T cold is the lowest temperature (temperature of the heat removed). For a gas turbine plant operating on a working fluid as a perfect gas with constant specific heat, the thermal efficiency of the ideal Brayton cycle can be expressed by modifying Equation 2 as follows (Boyce 1982): $$ {\eta}_B=1-{T}_2/{T}_1 $$ where T 1 is the turbine inlet temperature and T 2 is the turbine outlet temperature. Equation 3 clearly indicates that thermal efficiency can be improved by increasing the turbine inlet temperature; this thermodynamic relationship is applied to enhance profit by improving efficiency of gas turbine (GT) engines as discussed in 'Enhancing profitability in power generation' section. Techniques to improve efficiency of GT engines The efficiency of a Brayton engine or combined Brayton-Rankine engine can be improved in the following methods (Carapellucci 2009; Bassily 2008; Canière et al. 2006; Najjar 2001; Najjar et al. 2004; Brooks 2000; Agarwal et al. 2011): (a) intercooling, (b) regeneration, (c) combined cycle, (d) cogeneration, (e) higher turbine inlet temperature (TIT), (f) inlet air cooling (IAC), and (g) using steam injected gas technology (STIG) cycles. Intercooling For intercooling, the working fluid passes through the first stage of compressors, then a cooler, followed by the second stage of compressors before entering the combustion chamber. While this requires an increase in the fuel consumption of the combustion chamber, there is a corresponding beneficial effect, as follows. The flow of working fluid through the compressor results in a reduction in the specific volume of the fluid entering the second stage of compressors, with an attendant overall decrease in the amount of work needed for the compression stage. The decrease in input work results in efficiency improvement. There is also an increase in the maximum feasible pressure ratio due to reduced compressor discharge temperature for a given amount of compression, improving overall efficiency. The working temperature of a gas turbine, necessary to achieve high efficiency, makes cooling of the first turbine stages unavoidable. Air and steam can be used for cooling. For regeneration, the still-warm post-turbine fluid is passed through a heat exchanger to pre-heat the fluid just entering the combustion chamber. This directly offsets fuel consumption for the same operating conditions, improving efficiency; it also results in less power lost as waste heat. A Brayton (GT) engine also forms half of the combined cycle system, which combines with a Rankine (steam) engine to further increase overall efficiency. This kind of energy-efficiency technique allows conversion of the thermal energy of the turbine exhaust gasses into mechanical work, obtaining the so-called gas-steam combined cycles. The thermal efficiencies of modern CCGT power plants are as high as about 60%, which is evident from the data for a MS9001H plant as shown in Table 1 (Gorji-Bandpy et al. 2010). Table 1 Results of the CC plant simulation based on MS9001H ( p PH = 150 bar) (Gorji-Bandpy et al. 2010) Cogeneration systems make use of the waste heat from Brayton (GT) engines, typically for hot-water production or space heating. In a CC cogeneration, natural gas turbine is coupled with an electrical generator. The exhaust heat from the gas turbine is directed to a waste heat recovery boiler (WHRB). The steam from the WHRB is directed to a steam turbine generator where the steam can be used for power generation (see Figure 1). In a typical cogeneration plant, electric power is generated but some of the steam from the WHRB is used to process heat thereby increasing thermal efficiency. Combined cycle cogeneration gas turbine system. Higher turbine inlet temperature In the preceding section, we have established by using a thermodynamic approach that thermal efficiency can be improved by increasing the TIT. The controlling parameter for thermal efficiency in turbine design is not only the turbine inlet temperature but also the exhaust gas temperature. However, this requires special cooling at the hottest section of the turbine as well as the development of better materials, including ceramics and TBCs (Pilavachi 2000). In general, power-generating gas-turbine plants use steam cooling in its H system turbine that operates as hot as 1,430°C (2,606°F). However, the modern trend in power-generating plants is to opt for all air-cooling because this method is considered simpler than steam cooling and offers more design flexibility by avoiding dependence on the steam cycle. According to a survey of modern CCGT power plants, the operating temperatures in the first-stage SC blades were found to lie in the range of 1,300°C to 1,500°C. This temperature range refers to the lowest value for the Alstom Power through the GE Power Systems and the Siemens to the highest value for the Mitsubishi. Recently (May 2011), Mitsubishi Heavy Industries, Ltd. (MHI), Japan has achieved the world's highest TIT of 1,600°C, with the company's most advanced 'J-series' gas turbine (Figure 2). J-series MHI gas turbine capable of operating at TIT = 1,600°C. The new 1,600°C-class J-series gas turbine has achieved a rated power output of about 320 megawatts (MW) (ISO basis) and 460 MW in gas turbine combined-cycle (GTCC) power generation applications, in which heat recovery steam generators and steam turbines are also used. The MHI has also confirmed gross thermal efficiency exceeding 60% - the world's highest level in CCGT applications. Inlet air cooling and STIG cycles The IAC and STIG cycles are recent developments resulting in improved efficiency in industrial gas-turbines. The IAC involves cooling the air before the compression and is a proven method of increasing turbine power output, especially during peak summer demand. The IAC cycle technology reduces the temperature of the air entering the compressor section, causing the density and mass flow rate to increase, thereby increasing the power output of the gas turbine. This method is particularly useful in cases where the ambient temperature is high. Figure 3 shows the relationship between ambient temperature and turbine power output on a percentage basis for a typical gas turbine (Brooks 2000). The values in Figure 3 are normalized relative to the power obtained at 15°C. Variation of power output with ambient air temperature for a GE MS7001 gas turbine (Brooks 2000 ) . The analysis of data in Figure 3 indicates that turbine power output can increase by as much as 0.7% for every 1°C drop in inlet air temperature. Conversely, the gas turbine power output significantly decreases with rise in temperature, which is why the power output falls during hot summer months when the demand is at its peak. For this reason, it is highly desirable to keep the compressor inlet air temperature as low as possible by cooling the incoming ambient air. There are two widely used IAC systems: evaporative cooling systems and chiller systems. The evaporative cooling is economical and uncomplicated. However, there are two limitations in the evaporating cooling: (a) its efficiency can significantly drop if the relative humidity is high, and (b) there is a potential for excessive wear of compressor blades if water droplets are carried into the compressor section. On the other hand, the chiller systems have the advantage of being independent of humidity and do not have the potential to cause damage to compressor blades. However, chiller systems consume power and cause a larger pressure drop than evaporative coolers. The coauthor (HAA) has explored a relatively new technique of turbine IAC by using an ejector refrigeration system to cool turbine inlet air; which is illustrated in Figure 4 (Al-Ansary 2007). This system (see Figure 4) involves low-maintenance and is fluid-driven with heat-operated devices that can use part of the turbine exhaust flow as the heat source for running the cycle. This system requires only pump power to feed liquid refrigerant to the vapor generator thereby significantly decreasing the power consumption. Schematic of a an ejector refrigeration system (Al-Ansary 2007 ) . The STIG technology involves adding steam to the combustion chamber. Recently in 2011, Agarwal et al. have reported a significant improvement in power generation capacity and efficiency of simple cycle gas turbine through IAC and STIG (Brooks 2000). In the study, a simple cycle generation unit was considered as base unit and STIG and IAC features were sequentially retrofitted to the system with the aid of computer program software to stimulate performance parameters. It has been shown that retrofitting of simple cycle combined with IAC and STIG can boost power output from 30 to 48.25 MW, while generation efficiency can be increased from 29.9% to 33.4% (Brooks 2000). Enhancing profitability in power generation In the gas turbine using the Brayton cycle, the hot sections (combustion and turbine parts) are in contact with a 'continuously' hot working fluid. For pursuing higher kilowatt power ratings for given sizes of industrial turbines, turbine inlet temperatures are increased (see 'Improving energy efficiency of GT engines' section). Figure 5 illustrates the relationship of the specific work output (kW h/kg) with the turbine inlet temperature (°C) in a simple Brayton cycle gas turbine. A simple Brayton cycle network. It is evident from Figure 5 that if the TIT is increased by 56°C, the specific work output (kW h/kg) would be increased by 4%. It means that if a gas turbine is sold for US$200/kW (Nye Thermodynamics Corporation 2011), an increase of 56°C in TIT would result in an increased revenue of US$8/kW of the original rating to the manufacturer. The net profit can thus be calculated as follows: $$ \mathrm{Net}\kern0.5em \mathrm{profit}/\mathrm{kW}=\mathrm{Increase}\kern0.5em \mathrm{in}\kern0.5em \mathrm{revenue}\kern0.5em \mathrm{per}\kern0.5em \mathrm{kW}\hbox{--} \left({C}_1+{C}_2\right) $$ where C 1 = additional cost/KW on superalloy with increased temperature capability and C 2 = additional cost/kW on gas-turbine system due to increase in temperature. A significant improvement in the thermal efficiency of gas-turbine can be achieved by combining Braton cycle with other cycles (e.g., Rankine cycle). Figure 6 shows that thermal efficiency of a Brayton-Rankine cycle could be increased 2.25% by increasing TIT by 56°C. Brayton-Rankine cycle efficiency. In an industrial gas turbine, about 0.04 kg of superalloy is used in the hot section per kilowatt of power produced. About 2% of the cost of turbines in such a power plant is due to the use of superalloys. For example, we consider a case study of electric power generation whereby one finds that the average cost of the power plant was taken as around 7.5 mills/kW h (0.75 cents/kW h) (Zeren 1982). A fuel cost of 7.5 cents/kW h was found to be reasonable. It is evident from Figure 6 and the cost data that if an increase of 22.5% in the plant cost permitted this TIT rise (of 56°C), a break-even condition would exist at these fuel and plant costs. Despite of the increase in the plant cost, the plant is economically feasible and profitable because the operation management of gas-turbine (GT) power plant is becoming more competitive and since the energy (fuel) resources are becoming increasingly scarce and limited. In view of the discussion in the preceding paragraphs, it is important to look for advanced energy-efficient superalloys and coatings which enable us to operate gas-turbine engines at higher TITs. These advanced materials (superalloys) and coatings are discussed in the following sections. Improving GT efficiency through advanced superalloys GT operating conditions and advanced superalloys Modern gas turbine engines require a significant increase of gas inlet temperatures in order to achieve maximum efficiency. This results in an increased service temperature and consequently in enhanced high-temperature corrosion attack of the blade materials. The blades in modern aero, marine, and industrial gas turbines are manufactured exclusively from nickel-based super alloys and operate under the most arduous conditions of temperature and stress of any component in the engine (Huda 2009, 1995). These conditions are further complicated for turbine installations in marine (corrosive) environments, which include sulfur and sodium from the fuel and various halides contained in seawater (Schulz et al. 2008; Braue et al. 2007). These features are known to drastically reduce the superalloy component life and reliability by consuming the material at an unpredictably rapid rate, thereby reducing the load-carrying capacity and potentially leading to catastrophic failure of components. Thus, in order to improve the efficiency of a gas turbine engine significantly, the hot corrosion resistance of superalloys is as important as its high-temperature strength in gas turbine engine applications (Gurrappa and Sambasiva 2006). The nickel-based superalloys possess excellent high-temperature creep resistance, thermal stability, good tensile strength, long fatigue life, microstructural stability at high temperature, as well as good resistances to oxidation and corrosion (Huda et al. 2011). For these reasons, they are used in the manufacturing of gas turbine hot components (Huda 2009, 2007; Reed 2006; Jianting 2011; Cao and Loria 2005). Alloy development for hotter GT engines We have established in the preceding section that besides other important alloy characteristics, the high-temperature creep resistance is the primary requirement in superalloys for hotter GT engines. The first author (ZH) has reported design principles for developing creep-limited alloys for hot sections (particularly blades) of gas turbines (Huda 2007, 1995). We will now discuss some recently developed alloys for hotter industrial gas-turbine engines. A number of advanced superalloys have been developed for GT blades during the last two decades; these include the SC superalloys (AM1, René N6, MC538, etc.), the GTD-111 superalloy, the Allvac 718Plus superalloy, and the like (Sajjadi and Nategh 2001; Sajjadi et al. 2002, 2006; Zickler et al. 2009). Although these advanced alloys are nickel-based, new alloys based on metals with higher-melting temperatures (e.g., molybdenum (Mo) and niobium (Nb) alloyed with silicon) can also be prospective candidates for hotter GT engines (Perepezko 2009). This future trend is also evident in Figure 7 which illustrates the development of advanced superalloys in the temperature capacity of the alloys since 1940. Developments in the temperature capacity of superalloys since 1940. Figure 7 not only focuses on nickel-based superalloys for gas-turbine engine applications but also demonstrates the need for developing new energy-efficient superalloys, such as Mo-based superalloys for achieving further technological gains. Table 2 presents the chemical composition of some energy-efficient GT blade Ni-based superalloys (Zickler et al. 2009; Todd 1989). The effect of the composition on materials characteristics of various superalloys is discussed in the following subsections. Table 2 Chemical composition of typical GT-blade superalloys Single-crystal blade superalloys The use of SC high-pressure turbine blades made of nickel-based superalloys contributes efficiently to the continuous performance increase of GT engines in terms of power and thermal efficiency. Recently, a new SC Ni-based superalloy has been jointly developed by the Kansai Electric Power Company Co., Inc. (KEPCO), Nagoya University, and Hitachi Ltd., Japan (Kansai Electric Power Company (KEPCO) 2011). This SC superalloy contains Ni, Co, Cr, W, Al, Ti, Ta, Re, Hf, etc., and its microstructure is shown in Figure 8. Microstructure of a recently developed SC superalloy (Kansai Electric Power Company (KEPCO) 2011 ) . The cubical morphology of gamma-prime (γ′) precipitates in the gamma (γ) matrix of the SC superalloy (see Figure 8) indicates its high-temperature creep strength which is also confirmed by the temperature capability (>1,500°C) of this superalloy (Kansai Electric Power Company (KEPCO) 2011) The chemical compositions of three SC blade superalloys (AM1, René N6, and MC-534) are reported in Table 2. An interesting common feature of all the three SC alloys is their high W, Al, and Ta contents. A unique feature of the SC superalloy MC-534 is its considerable Re and Ru contents which impart remarkable creep resistance to the alloy (see Figure 9). Figure 9 compares tensile creep behaviors of the three SC superalloys (AM1, René N6, and MC-534) at 760°C/840 MPa; all the three alloy specimens were tested at orientations within 5° of a <001> direction (Diologent and Caron 2004). Creep curves at 760°C/840 MPa for three typical <001> SC superalloys. It is clearly evident from the creep curves in Figure 9 that the SC superalloy MC-534 has the lowest creep strain or deformation as compared to the other two SC alloys, which is attributed to the former's considerable Re and Ru contents (see Table 2). However, the creep rupture life of AM1 has been reported to be ten times longer than that of MC-534. Additionally, the excellent tensile yield strength at 700°C of the SC superalloy AM1 as compared to other competitive GT-blade superalloys is remarkable (see Table 3) (Sajjadi and Zebarjad 2006; Kennedy 2005). Table 3 Tensile yield strength at 700°C for three typical GT-blade superalloys The GTD-111 superalloy The GTD-111 superalloy is employed in manufacturing of the first-stage blades of high-power gas turbines. The chemical composition of GTD-111 superalloy is presented in Table 2 which shows a high aluminum and titanium contents to ensure precipitation of high volume fraction of the γ′ particles in the microstructure for good creep strength. The GTD-111 superalloy has a multiphase structure consisting of γ matrix, γ′ precipitate, carbide, γ-γ′ eutectic, and a small amount of deleterious phases such as δ, η, σ, and Laves (Sajjadi and Nategh 2001; Sajjadi et al. 2002). The superalloy maintains fairly good tensile yield strength of 780 MPa with 10% elongation at 700°C (see Table 3) (Sajjadi and Zebarjad 2006). The alloy obtains its high-temperature creep strength mainly through γ′ precipitates that are present with >60% volume fraction. The primary γ′ particles have a cubic shape with 0.8-μm average edge. The fine spherical γ′ particles precipitated during an aging treatment have an average diameter of ≈0.1 μm. The serrated grain boundaries increase creep life and creep plasticity. Figure 10 shows the as-standard heat-treated microstructure of GTD-111 (Sajjadi et al. 2006). SEM micrograph showing primary γ′ in as heat-treated GTD-111 superalloy (Sajjadi et al. 2006 ) . The Allvac® 718Plus™ superalloy Recently, a new 718Plus nickel-base superalloy has been developed for application in energy-efficient gas-turbine engines, thereby allowing a considerable increase in TIT for higher power output. The Allvac® 718Plus™ is a novel nickel-based superalloy, which has been designed for heavy-duty applications in aerospace and industrial gas turbines (Zickler et al. 2009). The Allvac® 718Plus™ alloy contains nanometer-sized spherical γ′ phase precipitates (Ni3(Al,Ti)) and plate-shaped δ phase precipitates (Ni3Nb) of micrometer size. The chemical composition of the Allvac® 718Plus™ superalloy is shown in Table 2 which shows higher Cr and Mo as compared to the GTD superalloy. An exceptional feature in the composition of the Allvac® 718Plus™ alloy is its high niobium content (which is absent in the GTD alloy). The high niobium content restricts the coarsening of the γ′ particles in the microstructure of the superalloy, thereby ensuring long-time creep resistance so as to allow operation of the GT engine at high temperature for longer periods of time (Huda 2009; Reed 2006). The transmission electron micrograph of the Allvac® 718Plus™ superalloy, after aging at 1,148 K for 7,800 s, shows spherical precipitates of γ′ phase in the γ matrix (see Figure 11). The aging heat treatments of the superalloy lead to a significant increase of hardness, which is due to precipitation of intermetallic phases (Zickler et al. 2009). The tensile properties at 700°C for the superalloy have been reported to be very good i.e., σ y = 1,014 MPa and % elongation = 25 (see Table 3) (Kennedy 2005). TEM image of superalloy ATI Allvac® 718Plus™ after aging at 1,148 K for 7,800 s (Zickler et al. 2009 ) . Improving GT efficiency through thermal barrier coatings TBC and coating technologies A TBC is a multilayer coating system that consists of an insulating ceramic outer layer (top coat) and a metallic inner layer (bond coat) between the ceramic and the substrate. The function of the ceramic top coat is to insulate the metallic substrate from high surface temperature, thereby lowering the component's temperature and reducing the oxidation and hot corrosion of bond coatings while simultaneously reducing cyclic thermal strains (Gurrappa and Sambasiva 2006). Recently, Kitazawa et al. (2010) have reported that a temperature gradient of 150°C can be achieved by using a ceramic TBC (Y2O3-ZrO2 top coat) on superalloy components. In most cases, the top coat and bond coats are applied by plasma spraying; however, sputtering and electron beam physical vapor deposition (EBPVD) are also used. The commonly used method of thermal spraying of TBCs on developed metal surfaces involves a metallic bond coating either by cold spraying (CS) or by low-pressure plasma spraying (LPPS) followed by deposition of a ceramic top coat by either air plasma spraying (APS) or EBPVD. The following techniques have been developed in recent years: (a) EBPVD and (b) electrophoretic deposition (EPD). These techniques are discussed in 'Electron beam physical vapor deposition' and 'Electrophoretic deposition' subsections, respectively. The failure of a TBC system can occur by three modes: (a) de-bonding, (b) de-cohesion, and (c) mechanical disruption. Failure by de-bonding occurs due to de-bonding along the interface between the bond coating and the thermally grown protective oxide (TGO), predominantly along the α-Al2O3 layer which bonds to the zirconia. The coating failure by de-cohesion occurs as a result of de-cohesion of the ceramic layer (near the interface) with the TGO. Mechanical disruption (rumpling) of bond coatings usually leads to significant displacement of the bond coating of the TGO and TGO-ceramic interfaces causing failure of the TBC. Tramp elements such as S have been shown to be detrimental to TGO adherence, so that alloying practices have been developed that routinely achieve S levels as low as 1 ppm. Controlled additions of reactive elements such as Y and Hf also can be used to counter the effects of S; such additions have been also shown significantly to modify the growth mechanism of the TGO and reduce its rate of growth, which promises to extend TBC lifetimes (Pint et al. 1998). Overall, improvements in understanding of the mechanisms of TBC degradation are providing routes for further improvements in their durability (Wright and Gibbons 2007). There has been a significant interest in the development of thermally grown oxides (TGO) in a deposited overlay (Choi et al. 2010). The integrity of the coating will depend on the ability of the alloy to grow an oxide layer with an appropriate combination of properties. While this approach can reduce costs and weight, it has been suggested that the life of a TBC system can be improved when the bond coating is eliminated because the TGO formed on the substrate benefits from the strength of the underlying alloy and, thus, has a greater resistance to buckling during thermal cycling (Wright and Gibbons 2007; Walston 2004). The limited lifetime of the TBC system forms the boundary of this 40-year-old concept (Troczynski et al. 1996). Until today, the use of TBCs on aircraft turbine blades is not design-integrated: they are used frequently to lower the metal temperature and therefore elongate the lifetime of a blade itself. If the coating spalls off, the metal temperature will increase but not above a critical point. For design-integrated TBCs with improvement of efficiency, fuel consumption, and exhaust pollution, 100% reliability is necessary (Stöver and Funke 1999). Electron beam physical vapor deposition The EBPVD TBCs consist of thin ceramic layers of low thermal conductivity - typically, partially stabilized zirconia (PSZ) that are usually applied on GT components' surfaces having a metallic corrosion-resistant coating (DeMasi-Marcinand and Gupta 1994). During EBPVD, a high-energy EB melts and evaporates a ceramic-source ingot in a vacuum chamber. The ingot is not the only ceramic source; there are also possibilities to deposit ceramics with powder. Preheated substrates are positioned in the vapor cloud above where the vapor is deposited on substrates at deposition rates of 0.1 to 0.25 mm/s (Movchan 1996). Typical columnar microstructures and aerodynamically smooth surfaces are obtained without the need for final polishing or conditioning of cooling holes. Due to the columnar microstructure, the lifetime of the TBC is prolonged and the damage tolerance improved. The application of the TBCs increases the engine performance by either increasing the gas-turbine inlet temperature or reducing the required cooling air flow (Schulz et al. 1997). Electrophoretic deposition An alternative, relatively inexpensive way to apply an oxide TBC is by EPD. Figure 12 illustrates a typical coating system in a high-pressure turbine blade (Padture et al. 2002). A typical coating system in a high-pressure turbine blade. Clockwise, a TBC-coated high-pressure turbine blade, with view from the top showing the cooling systems, and schematic profile temperature; note the drop of temperature close to the blade surface due to the presence of a thin cooling air film (Boccaccini and Zhitomirsky 2002). In recent years, EPD technique has been widely used for TBC (Boccaccini and Zhitomirsky 2002; Boccaccini et al. 2006; Besra and Liu 2007; Corni et al. 2008). For EPD, a suspension is prepared from particles of the desired material in a dispersing liquid. The suspension can be stabilized by electrical charge on the surface of the particles due to the acid or alkaline nature of the suspension or by charges due to adsorption of a surfactant. Steric hindrance can be used as well, but the particles do need a surface charge for the process to work (Dusoulier et al. 2011). When an electrical field is applied over the suspension, the charged particles move toward the oppositely charged electrode where they form a deposit. Upon drying and heat treatment (sintering), this is the product, either freestanding or on a substrate. For corrosion protection, it is essential that the coating covers the substrate completely and that the coating does not contain porosity. Studies on EPD have shown that EPD using water as a dispersant liquid leads to gas evolution at the electrodes and that entrapped gasses lead to porosity in the product (Doungdaw et al. 2005). For conducting EPD, it is usual to first prepare a suspension in an organic liquid such as ethanol, propanol, or isobutanol. Since organic liquids have a lower dielectric constant than water, care must be taken to select a system that provides adequate surface charge to the particles. Even so, it is most likely necessary to use a voltage of 100 to 300 V between the electrodes (Mohanty et al. 2008). To accommodate stresses due to a thermal expansion mismatch between substrate and coating, it is desirable that there is a gradient of properties, rather than a sharp interface. This can be achieved by applying a Ni-rich coating close to the surface and increasing the amount of alumina during deposition. Such functionally graded materials (FGMs), consisting of alumina and zirconia, have been produced by EPD (Put et al. 2003). FGMs can be deposited if two separate suspensions of the components are made and mixed at an appropriate rate while the deposition is in progress. A new chart for GT researchers, designers, and manufacturers Having reviewed the latest energy-efficiency techniques and recent advancements in superalloys and coatings, a new robust chart is now developed. It is evident from Table 4 that GT designer/manufacturers must take into consideration a number of mechanical and materials aspects for enhancing power output and profitability in today's industrial GTs. Table 4 Chart for enhancing power output and profitability in industrial GTs It is concluded that in order to enhance power output and profitability, the latest efficiency-improvement technologies and advanced superalloys and TBCs should be integrated into the industrial GT systems. In countries with hot summers, it is recommended to adopt IAC by use of Ejector Refrigeration Systems technology. In simple turbines, it is recommended to retrofit simple cycle with IAC and STIG technologies for boosting power output from 30 to 48.25 MW and to improve generation efficiency from 29.9% to 33.4%. In view of the latest development in the GT technology, which enables them to be operated at TIT = 1,600°C, it is recommended to develop superalloys based on higher-melting temperatures (such as Mo-base superalloys). In such advanced GT engines with efficiencies exceeding 60%, it is strongly recommended to use CCGT applying SC superalloys and TBC as well as closed-loop steam cooling. In TBC practice, it is recommended to first apply a Ni-rich coating close to the surface and then increasing the amounts of FGMs, alumina and zirconia, during EPD/EBPVD, for achieving the best results. 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Palo Alto, CA: EPRI. Zickler, GA, Schnitzer, R, Radis, R, Hochfellner, R, Schweins, R, Stockinger, M, Leitner, H. (2009). Microstructure and mechanical properties of the superalloy ATI Allvac® 718Plus™. Materials Science and Engineering A, 523(1–2), 295–303. The authors (ZH and TZ) are grateful to the I.P.P.P, University of Malaya, Malaysia and the Ministry of Science, Technology, and Innovation (MOSTI), Government of Malaysia for the research grant No. 03-01-03-SF0477 awarded during 2009 to support the work reported in the paper. Department of Engineering (Mechanical Engineering Program), Nilai University, Nilai, Negeri Sembilan, 71800, Malaysia Zainul Huda Department of Mechanical Engineering, University of Malaya, Kuala Lumpur, 50603, Malaysia Tuan Zaharinie Department of Mechanical Engineering, King Saud University, P.O. Box 800, Riyadh, 11421, Saudi Arabia Hany A Al-Ansary Correspondence to Zainul Huda. ZH initiated the idea and drafted 80% of the manuscript. He was the project leader. TZ contributed to drawing diagrams, particularly, Figure 5, Fig 6, and Fig 9. She also contributed to 'Introduction/Review". HAA provided illustrated 'Ejector refrigeration system (Fig 4)'. He also contributed in explaining IAC by using ejector refrigerator system. All authors read and approved the final manuscript. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Huda, Z., Zaharinie, T. & Al-Ansary, H.A. Enhancing power output and profitability through energy-efficiency techniques and advanced materials in today's industrial gas turbines. Int J Mech Mater Eng 9, 2 (2014). https://doi.org/10.1186/s40712-014-0002-y DOI: https://doi.org/10.1186/s40712-014-0002-y Combined cycle power plants Advanced superalloys	CommonCrawl
\begin{document} \title{Structure and classification results for the $\infty$-elastica problem} \author{Roger Moser\footnote{Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. E-mail: [email protected]}} \maketitle \begin{abstract} Consider the following variational problem: among all curves in $\mathbb{R}^n$ of fixed length with prescribed end points and prescribed tangents at the end points, minimise the $L^\infty$-norm of the curvature. We show that the solutions of this problem, and of a generalised version, are characterised by a system of differential equations. Furthermore, we have a lot of information about the structure of solutions, which allows a classification. \end{abstract} \section{Introduction} Variational problems involving the curvature of a curve $\Gamma \subseteq \mathbb{R}^n$ have a long history. This is true especially for the Euler elastica problem, which is to minimise the quantity \[ \int_\Gamma \kappa^2 \, ds, \] where $\kappa$ is the curvature of $\Gamma$. This functional may be regarded as a model for the stored elastic energy of a thin rod and its theory can be traced back to Jacob and Daniel Bernoulli and to Euler \cite{Euler:1744,Oldfather-Ellis-Brown:33}, but the problem has also been studied in more modern times \cite{Bryant-Griffiths:86, Langer-Singer:84_2, Langer-Singer:84_1, Linner:98}. An obvious generalisation is the $p$-elastica problem for $p \in [1, \infty)$, which corresponds to the quantity $\int_\Gamma \kappa^p \, ds$. This functional has been proposed for applications in image processing \cite{Masnou-Morel:98} and has also been studied in its own right \cite{Huang:04, Ferone-Kawohl-Nitsch:18}. While the step from elastica to $p$-elastica amounts to replacing an $L^2$-norm by an $L^p$-norm, in this paper we consider curves minimising the $L^\infty$-norm of the curvature. Thus, roughly speaking, we wish to minimise the maximum curvature. This quantity may not directly appear as the energy of a physical problem, but questions related to it are of fundamental geometric interest and may appear in design problems as well. In effect we ask, how much does a curve have to be bent in order to satisfy certain constraints? We consider constraints in the form of a fixed length combined with boundary conditions, but other types are conceivable as well and may admit a similar theory. To my knowledge, the $\infty$-elastica problem has not been studied before. The step from $p < \infty$ to $p = \infty$ changes the nature of the problem significantly. In particular, we have a functional that is not differentiable in any meaningful sense and the usual steps to find an Euler-Lagrange equation do no longer work. While we still have the notion of a minimiser, there is no obvious way to define critical points. In this paper, we propose another concept instead, derive a system of equations that can be thought of as Euler-Lagrange equations, and finally analyse and classify the solutions. In addition to the standard $L^\infty$-norm, the theory in this paper allows a weighted version as well. We therefore consider the following set-up of the problem. Let $n \in \mathbb{N}$ with $n \ge 2$. We fix a number $\ell > 0$, which is the prescribed length of the curves considered. We also fix a weight function $\alpha \colon [0, \ell] \to (0, \infty)$, which should be of bounded variation and such that $1/\alpha$ is bounded. We represent curves in $\mathbb{R}^n$ by parametrisations $\gamma \colon [0, \ell] \to \mathbb{R}^n$ by arc length for the moment, so we assume that $\|\gamma'\| \equiv 1$ in $[0, \ell]$. The curvature is then $\kappa = \|\gamma''\|$. As we wish to consider its (weighted) $L^\infty$-norm, we assume that $\gamma$ belongs to the Sobolev space $W^{2, \infty}((0, \ell); \mathbb{R}^n)$ and we define the functional \[ \mathcal{K}_\alpha(\gamma) = \esssup_{[0, \ell]} \alpha \|\gamma''\|. \] We consider a problem for curves with prescribed end points and prescribed tangent vectors at these end points. Thus for fixed $a_1, a_2 \in \mathbb{R}^n$ and fixed $T_1, T_2 \in S^{n - 1} = \set{x \in \mathbb{R}^n}{\|x\| = 1}$, we require that \begin{equation} \label{eqn:boundary_conditions} \gamma(0) = a_1, \quad \gamma(\ell) = a_2, \quad \gamma'(0) = T_1, \quad \text{and} \quad \gamma'(\ell) = T_2. \end{equation} Let $\mathcal{G}$ denote the set of all $\gamma \in W^{2, \infty}((0, \ell); \mathbb{R}^n)$ with $\|\gamma'\| \equiv 1$ in $[0, \ell]$ satisfying \eqref{eqn:boundary_conditions}. We are particularly interested in minimisers of $\mathcal{K}_\alpha$ in $\mathcal{G}$, but the observations in this paper suggest to consider the following weaker notion as well. \begin{definition}[$\infty$-elastica] \label{def:infty-elastica} Suppose that $\gamma \in \mathcal{G}$. We say that $\gamma$ is an \emph{$\infty$-elastica} if there exists $M \in \mathbb{R}$ such that for every $\tilde{\gamma} \in \mathcal{G}$, the inequality \[ \mathcal{K}_\alpha(\gamma) \le \mathcal{K}_\alpha(\tilde{\gamma}) + \frac{M}{2} \int_0^\ell \|\tilde{\gamma}' - \gamma'\|^2 \, ds \] holds true. \end{definition} It turns out that this condition is equivalent to a system of differential equations. Connections between a variational problem and differential equations are of course quite common, but for a functional that is not differentiable, such a strong correspondence is surprising. In order to write down the system concisely, we introduce some notation: if $V, W \in \mathbb{R}^n$, then $\proj_{V, W}^\perp$ denotes the orthogonal projection onto the orthogonal complement of the linear subspace of $\mathbb{R}^n$ spanned by $V$ and $W$. \begin{theorem}[Characterisation by differential equations] \label{thm:DE} Suppose that $\gamma \in \mathcal{G}$, and let $T = \gamma'$ and $k = \mathcal{K}_\alpha(\gamma)$. Then $\gamma$ is an $\infty$-elastica if, and only if, there exist $\lambda \in S^{n - 1}$ and $g \in W^{1, \infty}(0, \ell) \setminus \{0\}$ with $g \ge 0$ such that the equations \begin{align} g((\alpha T')' + k^2T/\alpha) & = k^2 \proj_{T, T'}^\perp(\lambda), \label{eqn:ODE1} \\ g' & = \alpha \lambda \cdot T' \label{eqn:ODE2} \end{align} hold weakly in $(0, \ell)$. \end{theorem} It is clear how to interpret weak solutions of \eqref{eqn:ODE2}. In order to make sense of \eqref{eqn:ODE1}, we use that fact that $g$, being in $W^{1, \infty}(0, \ell)$, has a weak derivative. Thus \eqref{eqn:ODE1} is satisfied weakly if \[ \int_0^\ell \left(g \alpha T' \cdot \xi' + g'\alpha T' \cdot \xi - gk^2 \alpha^{-1} T \cdot \xi + k^2 \proj_{T, T'}^\perp(\lambda) \cdot \xi\right) \, ds = 0 \] for all $\xi \in C_0^\infty((0, \ell); \mathbb{R}^n)$. If we add another condition, we obtain a criterion for minimisers of $\mathcal{K}_\alpha$, too. \begin{theorem}[Sufficient condition for minimisers] \label{thm:minimiser} Let $\gamma \in \mathcal{G}$ and $T = \gamma'$. Suppose that there exist $\lambda \in S^{n - 1}$ and $g \in W^{1, \infty}(0, \ell) \setminus \{0\}$ such that \eqref{eqn:ODE1} and \eqref{eqn:ODE2} are satisfied weakly in $(0, \ell)$, and such that $0 \le g \le -\alpha \lambda \cdot T$ in $[0, \ell]$. Then $\gamma$ minimises $\mathcal{K}_\alpha$ subject to the boundary conditions \eqref{eqn:boundary_conditions}. \end{theorem} This condition is sufficient but not necessary, as shown in Example \ref{ex:arc} below. It is worthwhile to consider the case $\alpha \equiv 1$ separately, as the system \eqref{eqn:ODE1}, \eqref{eqn:ODE2} can then be written as a single equation, albeit with an additional parameter. This is because in this case, the right-hand side of \eqref{eqn:ODE2} is the derivative of $\lambda \cdot T$ and the equation implies that there exists $\eta \in \mathbb{R}$ such that $g = \lambda \cdot T - \eta$. Thus \begin{equation} \label{eqn:ODE_alpha=1} T'' + k^2T = \frac{k^2 \proj_{T, T'}^\perp(\lambda)}{\lambda \cdot T - \eta}, \end{equation} at least where $\lambda \cdot T \neq \eta$. The left-hand side is a geometric quantity related to the torsion of the corresponding curve if $n = 3$. Indeed, it can be seen, with arguments as in Proposition \ref{prop:equivalence} below, that the torsion is $\pm k^{-1} \|T'' + k^2 T\|$. Analysing the system \eqref{eqn:ODE1}, \eqref{eqn:ODE2}, we obtain good information about the structure of $\infty$-elasticas as well, which allows a classification. \begin{theorem}[Structure and classification] \label{thm:structure} Suppose that $\gamma \in \mathcal{G}$ and let $T = \gamma'$ and $k = \mathcal{K}_\alpha(\gamma)$. Then $\gamma$ is an $\infty$-elastica if, and only if, there exists $\lambda \in S^{n - 1}$ such that at least one of the following statements holds true. \begin{enumerate}[(i)] \item \label{item:2D} There exists a line $\mathcal{L} \subseteq \mathbb{R}^n$ parallel to $\lambda$ and there exist finitely many intervals $J_1, \dotsc, J_N \subseteq [0, \ell]$, pairwise disjoint and open relative to $[0, \ell]$, such that $\gamma^{-1}(\mathcal{L}) = [0, \ell] \setminus \bigcup_{i = 1}^N J_i$ and such that for $i = 1, \dotsc, N$, \begin{itemize} \item $\gamma(\overline{J}_i) \cup \mathcal{L}$ is contained in a plane, \item $\alpha \gamma''$ is continuous with $\alpha \|\gamma''\| \equiv k$ in $J_i$, and \item for any $s_0 \in \overline{I}_i \setminus I_i$, there exists $\delta > 0$ such that $\lambda \cdot \gamma'' > 0$ in $(s_0, s_0 + \delta) \cap I_i$ and $\lambda \cdot \gamma'' < 0$ in $(s_0 - \delta, s_0) \cap I_i$. \end{itemize} \item \label{item:3D} There is a three-dimensional affine subspace of $\mathbb{R}^n$ that contains $\gamma([0, \ell])$. Furthermore, $\alpha \gamma'' \in W^{1, \infty}((0, \ell);\mathbb{R}^n)$ with $\alpha \|\gamma''\| \equiv k$ and there exists $g \in W^{2, \infty}(0, \ell)$ with $g > 0$ such that \eqref{eqn:ODE1} and \eqref{eqn:ODE2} hold true almost everywhere. \end{enumerate} \end{theorem} To summarise, an $\infty$-elastica is either a concatenation of two-di\-men\-sion\-al curves or a single three-dimensional curve solving a certain system of differential equations. In the first case, we have additional conditions that determine the curves to a significant degree. For example, in the case $\alpha \equiv 1$, it is readily seen that any planar $\infty$-elastica comprises either \begin{enumerate}[(a)] \item \label{item:arc-line-arc} a circular arc, followed by several line segments and full circles of equal radius, followed by a circular arc (cf.\ Figure \ref{fig:clc}), or \item \label{item:arc-arc} several circular arcs of equal length (apart from the first and the last) and radius but alternating sense of rotation (cf.\ Figure \ref{fig:ccc1} and \ref{fig:ccc2}). \end{enumerate} \begin{figure} \caption{$\lambda = (-1, 0)$} \label{fig:clc} \caption{$\lambda = (1, 0)$} \label{fig:ccc1} \caption{$\lambda = (1, 0)$} \label{fig:ccc2} \caption{These curves satisfy statement \ref{item:2D} of Theorem~\ref{thm:structure} for the $\lambda$ indicated. The parametrisation is from left to right in all three cases.} \label{fig:circles&lines} \end{figure} Curves of both types, with the additional restriction that they consist of at most three pieces, have been found by Dubins \cite{Dubins:57} as the solutions of a different variational problem: Dubins minimises the \emph{length} of a planar curve subject to boundary conditions of the type \eqref{eqn:boundary_conditions} and subject to the constraint that the curvature should nowhere exceed a given number. This problem was previously considered by Markov \cite{Markov:1887} and is therefore known as the Markov-Dubins problem. Dubins calls the solutions \emph{$R$-geodesics} if $1/R$ is the maximum curvature permitted. A similar result has been proved by Sussmann \cite{Sussmann:95} in dimension $n = 3$. Just as in Theorem \ref{thm:structure}, Sussmann finds two types of solutions: concatenations of circles and line segments on the one hand and three-dimensional curves, that he calls helicoidal arcs, on the other hand. The latter correspond to solutions of equation \eqref{eqn:ODE_alpha=1}. Sussmann's proof relies on a reformulation of the problem as an optimal control problem and on Pontryagin's maximum principle. For the problem studied in this paper, such an approach seems to be unavailable. It is no surprise that we obtain similar solutions, for the two problems are connected. \begin{proposition}[$R$-geodesics minimise $\mathcal{K}_1$] \label{prop:shortest_curves} Let $R > 0$. Suppose that $\gamma \colon [0, \ell] \to \mathbb{R}^n$ parametrises an $R$-geodesic by arc length. Then $\gamma$ minimises $\mathcal{K}_1$ subject to its boundary data. \end{proposition} As a consequence, we obtain an alternative proof of Dubins's and Sussmann's main results. Theorem \ref{thm:structure} will initially give less information in case \ref{item:2D}, but the proofs can then be completed with elementary arguments and some of Dubins's lemmas. We give a sketch of these arguments in Section \ref{sect:Dubins}. The Markov-Dubins problem, and variants thereof \cite{Reeds-Shepp:90}, have found applications in motion planning \cite{Laumond-Sekhavat-Lamiraux:98}. There is a connection to another classical problem. In 1925, Schmidt \cite{Schmidt:25} studied open spacial curves of fixed length that minimise the length of the chord under the constraint that the curvature is bounded pointwise by a given function (that we identify with $1/\alpha$). He generalised a result of A. Schur \cite{Schur:21}, which in turn refines an unpublished result ascribed by both authors to Schwarz. Another proof of this result may be found in a book of Blaschke \cite[\S 31]{Blaschke:45}, and a proof in English is given by S. S. Chern \cite{Chern:67}. The solutions of this problem are obviously minimisers of $\mathcal{K}_\alpha$, too, even under weaker boundary conditions. Schmidt concludes that any curve with shortest chord subject to his curvature constraint must be planar and convex. This can of course not be expected for the variational problem with boundary conditions \eqref{eqn:boundary_conditions} in general. The strategy for the proofs of Theorem \ref{thm:DE}--\ref{thm:structure} is to first approximate the $L^\infty$-norm of the curvature by $L^p$-norms for $p < \infty$ and then let $p \to \infty$. For $p < \infty$, we obtain a similar variational problem, which gives rise to an Euler-Lagrange equation. When we pass to the limit $p \to \infty$, the Euler-Lagrange equation is preserved in some form and eventually gives rise to the system \eqref{eqn:ODE1}, \eqref{eqn:ODE2}. We also obtain some information about the structure of solutions from the limit. A detailed analysis of the differential equations is also necessary for Theorem \ref{thm:structure}. To my knowledge, this is the first study of the above variational problem in the literature, although, as already discussed, several related problems have been studied in significant detail. There is also extensive work on variational problems involving an $L^\infty$-norm in general, going back to the work of Aronsson \cite{Aronsson:65, Aronsson:66, Aronsson:67}. An introduction with many further references is given in a book by Katzourakis \cite{Katzourakis:15}. Higher order problems have been studied more recently as well \cite{Aronsson:10, Moser-Schwetlick:12, Sakellaris:17, Katzourakis-Pryer:18, Katzourakis-Pryer:19, Katzourakis-Moser:19, Katzourakis-Parini:17}, but there is a much smaller body of literature. An approximation by $L^p$-norms, as in this paper, is common for variational problems in $L^\infty$, but subsequently, most of the literature relies on methods and ideas quite different from what is used here. Nevertheless, our approach has previously been deployed, too \cite{Moser-Schwetlick:12, Sakellaris:17, Katzourakis-Moser:19, Katzourakis-Parini:17}. For comparison, the paper of Katzourakis and the author \cite{Katzourakis-Moser:19} studies functions $u \colon \Omega \to \mathbb{R}$, for some domain $\Omega \subseteq \mathbb{R}^n$, that minimise $\esssup_{x \in \Omega} \|F(x, \Delta u(x))\|$ for a given function $F$ under prescribed boundary data. The paper describes the structure of minimisers, derives a system of partial differential equations that characterises them, and proves that minimisers are unique. For the problem studied here, it cannot be expected that minimisers are unique in general, and this is one of the reasons why the previous methods are insufficient. For example, if the boundary data are symmetric with respect to a reflection (for $n = 2$) or rotation about a line (for $n > 2$), but $\ell$ is too long to admit a straight line segment, then the symmetry of the problem automatically gives rise to multiple solutions. Therefore, if we use approximations to the variational problem, we will typically recover some solution in the limit, but not necessarily all possible solutions. We overcome this difficulty by adding another term that penalises the distance from a \emph{given} solution. This is the main novelty in the first part of our analysis. The penalisation corresponds to the last term in the inequality of Definition \ref{def:infty-elastica}, and thus, although initially introduced as a technical device, proves to be interesting in its own right, as it gives rise to a variational problem \emph{equivalent} to the system of differential equations in Theorem~\ref{thm:DE}. The second part of our analysis, which leads to the proof of Theorem \ref{thm:structure}, is completely new. The underlying method may be restricted to this and similar problems, but our theory provides one of the first examples (the equally restrictive and more elementary theory of Katzourakis-Pryer \cite[Section 8]{Katzourakis-Pryer:18} being the only other example I am aware of), where a non-trivial second-order variational problem in $L^\infty$ can be solved exhaustively. \section{Reparametrisation and approximation} \label{sect:reparametrisation} In this section, we prepare the ground for the proofs of Theorems \ref{thm:DE}--\ref{thm:structure}. We first reformulate the problem by reparametrising the curves appropriately. Then we discuss an approximation of the $L^\infty$-norm by $L^p$-norms. We also add a penalisation term to the functionals, the purpose of which is to guarantee convergence to a \emph{given} (rather than an arbitrary) solution of the problem as $p \to \infty$. At the same time, we shift our main attention from a curve in $\mathbb{R}^n$ to its tangent vector field. Recall that we previously considered parametrisations $\gamma \colon [0, \ell] \to \mathbb{R}^n$ by arc length satisfying the boundary conditions \eqref{eqn:boundary_conditions}. From now on, a parametrisation with speed $\alpha$ is more convenient. Therefore, define \[ \psi(s) = \int_0^s \frac{d\sigma}{\alpha(\sigma)}, \quad 0 \le s \le \ell, \] and $L = \psi(\ell)$. Also consider the inverse $\phi = \psi^{-1} \colon [0, L] \to [0, \ell]$ and $\beta = \alpha \circ \phi$. If $\gamma$ is a parametrisation by arc length, then the reparametrisation $c \colon [0, L] \to \mathbb{R}^n$, given by $c(t) = \gamma(\phi(t))$, satisfies $\|c'(t)\| = \phi'(t) = 1/\psi'(\phi(t)) = \beta(t)$. We now consider the tangent vector field along $c$, normalised to unit length. Thus let $\tau \colon [0, L] \to S^{n - 1}$ be defined by $\tau(t) = c'(t)/\beta(t)$. (An equivalent definition is $\tau(t) = \gamma'(\phi(t))$.) Then \eqref{eqn:boundary_conditions} implies that \begin{equation} \label{eqn:boundary_condition1} \tau(0) = T_1 \quad \text{and} \quad \tau(L) = T_2. \end{equation} Setting $a = a_2 - a_1$, we also obtain the condition \begin{equation} \label{eqn:boundary_condition2} \int_0^L \beta(t) \tau(t) \, dt = a. \end{equation} Conversely, if we have $\tau \in W^{1, \infty}((0, L); S^{n - 1})$ satisfying \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2}, then $\gamma \in \mathcal{G}$ can be reconstructed from $\tau$ by \[ \gamma(s) = a_1 + \int_0^s \tau(\psi(\sigma)) \, d\sigma, \quad 0 \le s \le \ell. \] The functional $\mathcal{K}_\alpha$ can be written in terms of $\tau$ as follows: \[ \mathcal{K}_\alpha(\gamma) = \esssup \|\tau'\|. \] Hence in order to study the above problem, it suffices to consider $\tau$ and to study the functional \[ K_\infty(\tau) = \esssup \|\tau'\| \] under the boundary conditions \eqref{eqn:boundary_condition1} and the integral constraint \eqref{eqn:boundary_condition2}. We note that $\gamma$ is an $\infty$-elastica if, and only if, $\tau$ has the following property. \begin{definition} Suppose that $\tau \in W^{1, \infty}((0, L); S^{n - 1})$ satisfies the boundary conditions \eqref{eqn:boundary_condition1} and the constraint \eqref{eqn:boundary_condition2}. We say that $\tau$ is a \emph{pseudo-minimiser} of $K_\infty$ if there exists $m \in \mathbb{R}$ such that \[ K_\infty(\tau) \le K_\infty(\tilde{\tau}) + \frac{m}{2L} \int_0^L \beta \|\tilde{\tau} - \tau\|^2 \, dt \] for any other $\tilde{\tau} \in W^{1, \infty}((0, L); S^{n - 1})$ satisfying \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2}. \end{definition} One of the key tools for the proofs of Theorems \ref{thm:DE}--\ref{thm:structure} is an approximation of $K_\infty$ by \[ K_p(\tau) = \left(\frac{1}{L} \int_0^L \|\tau'\|^p \, dt\right)^{1/p} \] for $p \in [2, \infty)$. We eventually consider the limit as $p \to \infty$ to recover $K_\infty$. Furthermore, given $\tau_0 \in W^{1, \infty}((0, L); S^{n - 1})$ and $\mu \ge 0$, we consider the functionals \[ J_p^\mu(\tau; \tau_0) = K_p(\tau) + \frac{\mu}{2L} \int_0^L \beta \|\tau - \tau_0\|^2 \, dt. \] In the proofs of Theorems \ref{thm:DE}--\ref{thm:structure}, we will assume that $\tau_0$ is a pseudo-minimiser of $K_\infty$. Minimisers of $J_p^\mu({\mkern 2mu\cdot\mkern 2mu}; \tau_0)$ can then be found with the direct method, and the assumption will guarantee that they converge to $\tau_0$ as $p \to \infty$. This will eventually allow some conclusions about $\tau_0$. Indeed, the following preliminary observations are almost immediate from the structure of the variational problem. \begin{proposition} \label{prop:p-approximation} Let $\mu > 0$ and $\tau_0 \in W^{1, \infty}((0, L); S^{n - 1})$ be given. For every $p \in [2, \infty)$, suppose that $\tau_p \in W^{1, p}((0, L); S^{n - 1})$ is a minimiser of $J_p^\mu({\mkern 2mu\cdot\mkern 2mu}; \tau_0)$ subject to the constraints \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2} and let $k_p = K_p(\tau_p)$. \begin{enumerate} \item \label{item:Euler-Lagrange} Then there are Lagrange multipliers $\Lambda_p \in \mathbb{R}^n$ such that \begin{equation} \label{eqn:Euler-Lagrange} \frac{d}{dt} \left(\|\tau_p'\|^{p - 2} \tau_p'\right) + \|\tau_p'\|^p \tau_p = k_p^{p - 1} \beta \bigl(\Lambda_p - (\Lambda_p \cdot \tau_p) \tau_p - \mu \tau_0 + \mu (\tau_0 \cdot \tau_p)\tau_p\bigr) \end{equation} weakly in $(0, L)$. \item \label{item:convergence} If $\tau_0$ satisfies \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2} and is a pseudo-minimiser of $K_\infty$, then there exists $\mu_0 > 0$ such that the following holds true. If $\mu \ge \mu_0$, then $\tau_p \rightharpoonup \tau_0$ weakly in $W^{1, q}((0, L); \mathbb{R}^n)$ for every $q < \infty$ and $k_p \to K_\infty(\tau_0)$ as $p \to \infty$. \end{enumerate} \end{proposition} \begin{proof} The Euler-Lagrange equation \eqref{eqn:Euler-Lagrange} is derived with standard computations. The only feature that is perhaps unusual is the constraint $\tau_p(t) \in S^{n - 1}$ for $t \in [0, L]$, but this sort of constraint is common in the theory of harmonic maps and it is explained, e.g., in a book by Simon \cite{Simon:96} how to deal with it. We therefore omit the details in the proof of statement \ref{item:Euler-Lagrange}. Next we note that by the choice of $\tau_p$ and by H\"older's inequality, for any pair of numbers $p, q \in (1, \infty)$ with $p \le q$, we find the inequalities \begin{equation} \label{eqn:monotonicity} J_p^\mu(\tau_p; \tau_0) \le J_p^\mu(\tau_q; \tau_0)\le J_q^\mu(\tau_q; \tau_0) \le K_q(\tau_0) \le K_\infty(\tau_0). \end{equation} So for any $q \in [2, \infty)$, the one-parameter family $(\tau_p)_{q \le p < \infty}$ is bounded in $W^{1, q}((0, L); \mathbb{R}^n)$. Therefore, there exists a sequence $p_i \to \infty$ such that $\tau_{p_i}$ converges weakly in $W^{1, q}((0, L); \mathbb{R}^n)$, for every $q < \infty$, to a limit \[ \tau_\infty \in \bigcap_{q < \infty} W^{1, q}((0, L); S^{n - 1}). \] Clearly $\tau_\infty$ will satisfy \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2} again. By the lower semicontinuity of the $L^q$-norm with respect to weak convergence and by \eqref{eqn:monotonicity}, \begin{equation} \label{eqn:upper_bound} J_\infty^\mu(\tau_\infty; \tau_0) = \lim_{q \to \infty} J_q^\mu(\tau_\infty; \tau_0) \le \lim_{q \to \infty} \liminf_{i \to \infty} J_q^\mu(\tau_{p_i}; \tau_0) \le K_\infty(\tau_0). \end{equation} If there exists $m > 0$ such that \[ K_\infty(\tau_0) \le K_\infty(\tau) + \frac{m}{2L} \int_0^L \beta \|\tau - \tau_0\|^2 \, dt \] for all $\tau \in W^{1, \infty}((0, L); S^{n - 1})$ satisfying \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2}, then \eqref{eqn:upper_bound} implies that \[ J_\infty^\mu(\tau_\infty; \tau_0) \le J_\infty^m(\tau_\infty; \tau_0). \] Thus \[ (\mu - m) \int_0^L \beta \|\tau_\infty - \tau_0\|^2 \, dt \le 0. \] If we choose $\mu > m$, this means that $\tau_\infty = \tau_0$. In particular, the limit is then independent of the choice of the sequence $(p_i)_{i \in \mathbb{N}}$, and therefore we have in fact weak convergence of $\tau_p$ to $\tau_\infty = \tau_0$ in $W^{1, q}((0, L); \mathbb{R}^n)$ for every $q < \infty$. The inequalities in \eqref{eqn:monotonicity} also imply that \[ \lim_{p \to \infty} J_p^\mu(\tau_p; \tau_0) \le K_\infty(\tau_0), \] in particular that the limit exists. On the other hand, as we now know that $\tau_\infty = \tau_0$, we can go back to \eqref{eqn:upper_bound} and conclude that \[ K_\infty(\tau_0) \le \lim_{q \to \infty} \liminf_{i \to \infty} J_q^\mu(\tau_{p_i}; \tau_0) \le \lim_{p \to \infty} J_p^\mu(\tau_p; \tau_0). \] Hence $K_\infty(\tau_0) = \lim_{p \to \infty} J_p^\mu(\tau_p; \tau_0)$. Since the weak convergence $\tau_p \rightharpoonup \tau_0$ in $W^{1, 2}((0, L); \mathbb{R}^n)$ implies strong convergence in $L^2((0, L); \mathbb{R}^n)$ as well, it follows that $K_\infty(\tau_0) = \lim_{p \to \infty} k_p$. \end{proof} Eventually we will need a careful analysis of the Euler-Lagrange equation \eqref{eqn:Euler-Lagrange} for the proofs of Theorems \ref{thm:DE}--\ref{thm:structure}. To this end, we need to know that the Lagrange multipliers $\Lambda_p$ do not grow too quickly as $p \to \infty$. We prove the following. \begin{lemma} \label{lem:growth} Suppose that $\tau_p \in W^{1, p}((0, L); S^{n - 1})$ and let $k_p = K_p(\tau_p)$. Suppose that $\limsup_{p \to \infty} k_p < \infty$ and there exist $\Lambda_p \in \mathbb{R}^n$ such that \eqref{eqn:Euler-Lagrange} holds weakly in $(0, L)$ for every $p \in [2, \infty)$. Then either \[ \limsup_{p \to \infty} \left(p^{-6} \|\Lambda_p\|\right) < \infty \] or there exists a sequence $p_i \to \infty$ such that $\tau_{p_i}$ converges uniformly to a constant vector as $i \to \infty$. \end{lemma} \begin{proof} Suppose that no subsequence converges uniformly to a constant vector. Then it follows that for every sufficiently large $p$, either $\Lambda_p = 0$ or the angle $\omega_p$ between $\tau_p$ and $\Lambda_p$ satisfies \[ \sup_{t \in [0, L]} \omega_p(t) \ge \frac{1}{p} \quad \text{and} \quad \sup_{t \in [0, L]} (\pi - \omega_p(t)) \ge \frac{1}{p}. \] Note that \[ \|\sin \omega_p\| = \frac{\|\Lambda_p - (\Lambda_p \cdot \tau_p) \tau_p\|}{\|\Lambda_p\|} \] if $\Lambda_p \neq 0$. Hence for every sufficiently large $p$, there exists $t_p \in [0, L]$ such that \[ \left\|\Lambda_p - (\Lambda_p \cdot \tau_p(t_p)) \tau_p(t_p)\right\| \ge \frac{\|\Lambda_p\|}{2p}. \] Because we have a uniform bound for $\\|\tau_p'\\|_{L^2(0, L)}$, the Sobolev embedding theorem gives a uniform bound for $\\|\tau_p\\|_{C^{0, 1/2}([0, T])}$ as well. Hence there exists a number $\delta > 0$ such that the inequality \[ \left\|\Lambda_p - (\Lambda_p \cdot \tau_p) \tau_p\right\| \ge \frac{\|\Lambda_p\|}{3p} \] holds in $[t_p - \delta/p^2, t_p + \delta/p^2] \cap [0, L]$ for all sufficiently large values of $p$. Choose $\eta \in C_0^\infty((t_p - \delta/p^2, t_p + \delta/p^2) \cap (0, L))$ such that $0 \le \eta \le 1$ and \[ \int_0^L \eta \, ds \ge \frac{\delta}{2p^2}, \] but $\|\eta'\| \le 5p^2/\delta$. Test \eqref{eqn:Euler-Lagrange} with $\eta \Lambda_p$. This yields \begin{multline} \int_0^L \eta \|\tau_p'\|^p \tau_p \cdot \Lambda_p \, dt - \int_0^L \eta' \|\tau_p'\|^{p - 2} \tau_p' \cdot \Lambda_p \, dt \\ = k_p^{p - 1} \int_0^L \eta \beta \left\|\Lambda_p - (\Lambda_p \cdot \tau_p) \tau_p\right\|^2 \, dt - \mu k_p^{p - 1} \int_0^L \eta \beta \left(\tau_0 - (\tau_0 \cdot \tau_p) \tau_p\right) \cdot \Lambda_p \, dt. \end{multline} By the choice of $\eta$, we know that \[ \int_0^L \eta \beta \left\|\Lambda_p - (\Lambda_p \cdot \tau_p) \tau_p\right\|^2 \, dt \ge \frac{\delta \|\Lambda_p\|^2}{18p^4 \\|1/\alpha\\|_{L^\infty(0, \ell)}}. \] Moreover, we have the estimates \begin{align} \int_0^L \eta \|\tau_p'\|^p \tau_p \cdot \Lambda_p \, dt & \le Lk_p^p \|\Lambda_p\|, \\ - \int_0^L \eta' \|\tau_p'\|^{p - 2} \tau_p' \cdot \Lambda_p \, dt & \le \frac{5p^2}{\delta} L k_p^{p - 1} \|\Lambda_p\|, \\ \int_0^L \eta \beta \left(\tau_0 - (\tau_0 \cdot \tau_p) \tau_p\right) \cdot \Lambda_p \, dt & \le L \\|\alpha\\|_{L^\infty(0, \ell)} \|\Lambda_p\|. \end{align} Hence \[ \|\Lambda_p\| \le \frac{18Lp^4}{\delta} \\|1/\alpha\\|_{L^\infty(0, \ell)} \left(\frac{5p^2}{\delta} + \mu \\|\alpha\\|_{L^\infty(0, \ell)} + k_p\right), \] and the desired inequality follows. \end{proof} \section{Preliminary properties of $\infty$-elasticas} The purpose of this section is to extract some information for pseudo-minimisers of $K_\infty$, and therefore for $\infty$-elasticas, from the Euler-Lagrange equation \eqref{eqn:Euler-Lagrange} by studying the limit $p \to \infty$. The resulting statements are less strong than the main results in the introduction, but they will serve as a first step. \begin{proposition} \label{prop:pseudo-minimiser=>DE} Suppose that $\tau \in W^{1, \infty}((0, L); S^{n - 1})$ is a pseudo-minimiser of $K_\infty$. Let $k = K_\infty(\tau)$. Then there exist $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$ and $\lambda \in \mathbb{R}^n$ such that the equations \begin{align} u' + (u \cdot \tau') \tau & = \beta(\lambda - (\lambda \cdot \tau) \tau), \label{eqn:system1}\\ \|u\|\tau' & = k u, \label{eqn:system2} \end{align} hold almost everywhere in $(0, L)$. \end{proposition} \begin{proof} The statements are obvious (for $u = 1$ and $\lambda = 0$) if $\tau$ is constant. We therefore assume that this is not the case. Fix $\mu > 0$ and consider the functionals $J_p^\mu({\mkern 2mu\cdot\mkern 2mu}; \tau)$. Minimisers $\tau_p$ of $J_p^\mu({\mkern 2mu\cdot\mkern 2mu}; \tau)$ under the boundary conditions \eqref{eqn:boundary_condition1} and the constraint \eqref{eqn:boundary_condition2} can be constructed with the direct method. Let $k_p = K_p(\tau_p)$. We assume that $\mu > 0$ is chosen so large that statement \ref{item:convergence} in Proposition \ref{prop:p-approximation} applies. We consider the Euler-Lagrange equation \eqref{eqn:Euler-Lagrange}. The underlying idea for the next step is to regard it as an equation in $\|\tau_p'\|^{p - 2} \tau_p'$. But at the same time, we renormalise. Thus we introduce the functions \[ u_p = \frac{k_p^{1 - p} \|\tau_p'\|^{p - 2} \tau_p'}{1 + \|\Lambda_p\|}. \] We also define \[ \lambda_p = \frac{\Lambda_p}{1 + \|\Lambda_p\|} \quad \text{and} \quad m_p = \frac{\mu}{1 + \|\Lambda_p\|}. \] Then we can write \eqref{eqn:Euler-Lagrange} (for $\tau_0 = \tau$) in the form \begin{equation} \label{eqn:Euler-Lagrange2} u_p' + (u_p \cdot \tau_p') \tau_p = \beta(\lambda_p - (\lambda_p \cdot \tau_p) \tau_p - m_p \tau + m_p (\tau \cdot \tau_p)\tau_p). \end{equation} Writing $p' = p/(p - 1)$, we note that \begin{equation} \label{eqn:L^p'-estimate} \\|u_p\\|_{L^{p'}(0, L)} = \frac{k_p^{p - 1}}{1 + \|\Lambda_p\|} \left(\int_0^L \|\tau_p'\|^p \, dt\right)^{1/p'} = \frac{L^{1/p'}}{1 + \|\Lambda_p\|}. \end{equation} The right-hand side remains bounded as $p \to \infty$. Moreover, we know that \[ \\|\tau_p'\\|_{L^p(0, L)} = L^{1/p} k_p \to k \] as $p \to \infty$ by Proposition \ref{prop:p-approximation}. As $\|\tau_p\| \equiv 1$, $\|\lambda_p\| \le 1$, and $0 < m_p \le \mu$, equation \eqref{eqn:Euler-Lagrange2} immediately gives a uniform bound for $\\|u_p\\|_{W^{1, 1}(0, L)}$. Thus we have a uniform bound in $L^\infty((0, L); \mathbb{R}^n)$ as well, and using the equation again, we conclude that \[ \limsup_{p \to \infty} \\|u_p'\\|_{L^q(0, L)} < \infty \] for any $q < \infty$. Thus we may choose a sequence $p_i \to \infty$ such that $u_{p_i} \rightharpoonup u$, for some $u \in \bigcap_{q < \infty} W^{1, q}((0, L); \mathbb{R}^n)$, weakly in $W^{1, q}((0, L); \mathbb{R}^n)$ for any $q < \infty$ as $i \to \infty$. In particular $u_{p_i} \to u$ uniformly as $i \to \infty$. Since $\|\lambda_p\| \le 1$ and $0 < m_p \le \mu$, we may assume that at the same time, we have the convergence $\lambda_{p_i} \to \lambda$ for some $\lambda \in \mathbb{R}^n$ and $m_{p_i} \to m$ for some $m \in [0, \mu]$. By Proposition \ref{prop:p-approximation}, we know that $\tau_p \to \tau$ weakly in $W^{1, q}((0, L); \mathbb{R}^n)$ for any $q < \infty$. Thus restricting \eqref{eqn:Euler-Lagrange2} to $p_i$ and letting $i \to \infty$, we derive equation \eqref{eqn:system1} almost everywhere. Now \eqref{eqn:system1} implies that $u \in W^{1, \infty}((0, L); \mathbb{R}^n)$. If $\|\Lambda_{p_i}\| \to \infty$ as $i \to \infty$, then $\lambda \in S^{n - 1}$ and \eqref{eqn:system1} cannot be satisfied for $u \equiv 0$ (as we have assumed that $\tau$ is not constant). If $\|\Lambda_{p_i}\| \not\to \infty$, then \eqref{eqn:L^p'-estimate} implies that $\\|u\\|_{L^1(0, L)} \neq 0$. In either case, we conclude that $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$. As $u$ is continuous, the set $\Omega = \set{t \in [0, L]}{u(t) \neq 0}$ is open relative to $[0, L]$. For any $t \in \Omega$, there exist $\delta > 0$ and $\epsilon > 0$ such that $\delta \le \|u_{p_i}\| \le 1/\delta$ in $(t - \epsilon, t + \epsilon) \cap [0, L]$ for any $i$ large enough. Now note that \[ \tau_p' = k_p (1 + \|\Lambda_p\|)^{1/(p - 1)} \|u_p\|^{1/(p - 1)} \frac{u_p}{\|u_p\|} \] wherever $u_p \neq 0$ by the definition of $u_p$. As we have assumed that $\tau$ is not constant, we know that \[ (1 + \|\Lambda_p\|)^{1/(p - 1)} \to 1 \] as $p \to \infty$ by Lemma \ref{lem:growth}. We further know that \[ \|u_{p_i}\|^{1/(p_i - 1)} \to 1 \quad \text{and} \quad \frac{u_{p_i}}{\|u_{p_i}\|} \to \frac{u}{\|u\|} \] uniformly in $(t - \epsilon, t + \epsilon) \cap [0, L]$ as $i \to \infty$. Therefore, by the above identity, \[ \tau_{p_i}' \to \tau' = \frac{k u}{\|u\|} \] locally uniformly in $\Omega$. We therefore obtain equation \eqref{eqn:system2}. \end{proof} For planar curves, we can say more. \begin{lemma} \label{lem:2D} Let $\tau \in W^{1, \infty}((0, L); S^{n - 1})$ and $\lambda \in \mathbb{R}^n \setminus \{0\}$. Suppose that $\tau([0, L])$ is contained in a two-dimensional linear subspace $X \subseteq \mathbb{R}^n$. Let \[ c(t) = a_1 + \int_0^t \beta(\theta) \tau(\theta) \, d\theta \] for $t \in [0, L]$. Suppose that $k = K_\infty(\tau) \neq 0$ and consider a set $\Omega \subseteq [0, L]$. Then the following statements are equivalent. \begin{enumerate}[(i)] \item \label{item:system2D} There exists $u \in W^{1, \infty}((0, L); \mathbb{R}^n)$ such that \eqref{eqn:system1} and \eqref{eqn:system2} hold true almost everywhere and $\Omega = \set{t \in [0, L]}{u(t) \neq 0}$. \item \label{item:structure2D} The vector $\lambda$ belongs to $X$ and there exists a line $\mathcal{L} \subseteq X + a_1$ parallel to $\lambda$ such that $\Omega = \set{t \in [0, L]}{c(t) \not\in \mathcal{L}}$. Moreover, $\tau'$ is continuous with $\|\tau'\| \equiv k$ in $\Omega$. For any $t_0 \in [0, L] \setminus \Omega$, if there exists $\delta > 0$ with $(t_0 - \delta, t_0) \subseteq \Omega$, then there exists $\delta' \in (0, \delta]$ such that $\lambda \cdot \tau'(t) < 0$ in $(t_0 - \delta', t_0)$; and if there exists $\delta > 0$ with $(t_0, t_0 + \delta) \subseteq \Omega$, then there exists $\delta' \in (0, \delta]$ such that $\lambda \cdot \tau'(t) > 0$ in $(t_0, t_0 + \delta')$. \end{enumerate} \end{lemma} \begin{proof} We may choose coordinates such that $X = \mathbb{R}^2 \times \{0\}$ and then write \[ \tau = (\cos \omega, \sin \omega, 0) \] in $[0, L]$ for some function $\omega \colon [0, L] \to \mathbb{R}$. Now for $x \in \mathbb{R}^n$, write $x^\perp = (-x_2, x_1, x_3, \dotsc, x_n)$. In particular $\tau^\perp = (-\sin \omega, \cos \omega, 0)$ and $\tau' = \omega' \tau^\perp$. If \ref{item:system2D} is satisfied, then \eqref{eqn:system2} implies that $u(t) \in X$ for every $t \in [0, L]$, and then \eqref{eqn:system1} implies that $\lambda \in X$. It is clear that $u/\|u\|$ is continuous in $\Omega$. Thus equation \eqref{eqn:system2} further implies that $\omega'$ is continuous in $\Omega$ with $\|\omega'\| \equiv k$. Defining $f = \|u\| \omega'/k$, we compute $u = f\tau^\perp$ and \begin{equation} \label{eqn:system1_2D} u' + (u \cdot \tau') \tau = f'\tau^\perp. \end{equation} Multiplying \eqref{eqn:system1} with $\tau^\perp$, we conclude that \begin{equation} \label{eqn:f2D} f' = \beta \lambda \cdot \tau^\perp \end{equation} in $\Omega$. Outside of $\Omega$, we know that $f$ vanishes, and it follows that for any $t_1, t_2 \in [0, L]$, we have the inequality $\|f(t_1) - f(t_2)\| \le \\|\beta\\|_{L^\infty(0, L)} \|\lambda\| \|t_1 - t_2\|$. So $f \in W^{1, \infty}(0, L)$ and \eqref{eqn:system1_2D}, \eqref{eqn:f2D} hold true almost everywhere in $[0, L]$. Consider $c$ as defined above and note that $(c')^\perp = \beta \tau^\perp$. Hence $f' = \lambda \cdot (c')^\perp$ in $[0, L]$. It follows that there exists some number $b \in \mathbb{R}$ such that \[ f^{-1}(\{0\}) = \set{t \in [0, L]}{\lambda^\perp \cdot c(t) = b}. \] In other words, the line $\mathcal{L} = \set{x \in X + a_1}{\lambda^\perp \cdot x = b}$, which is parallel to $\lambda$, has the property that $\Omega = \set{t \in [0, L]}{c(t) \not\in \mathcal{L}}$. Now suppose that $t_0 \in [0, L] \setminus \Omega$ such that there exists $\delta > 0$ with $(t_0 - \delta, t_0) \subseteq \Omega$. Recall that $\|\omega'\| \equiv k$ in $(t_0 - \delta, t_0)$ while the sign of $\omega'$ is constant. So $\omega' = \sigma k$ in $(t_0 - \delta, t_0)$ for some $\sigma \in \{-1, 1\}$. Hence \begin{equation} \label{eqn:tau2D} \tau(t) = \cos(k(t - t_0)) \tau(t_0) + \sigma \sin(k(t - t_0)) \tau^\perp(t_0) \end{equation} and \[ \tau'(t) = -k\sin(k(t - t_0)) \tau(t_0) + \sigma k\cos(k(t - t_0)) \tau^\perp(t_0) \] in $(t_0 - \delta, t_0)$. Moreover, identity \eqref{eqn:f2D} implies that \begin{equation} \label{eqn:f'2D} f'(t) = -\sigma \beta(t) \sin(k(t - t_0)) \lambda \cdot \tau(t_0) + \beta(t) \cos(k(t - t_0)) \lambda \cdot \tau^\perp(t_0) \end{equation} in $(t_0 - \delta, t_0)$. As $f(t_0) = 0$ and as $f$ has the same sign as $\omega'$ in $(t_0 - \delta, t_0)$, we immediately conclude that $\sigma \lambda \cdot \tau^\perp(t_0) \le 0$; and in the case of equality, we further conclude that $\lambda \cdot \tau(t_0) < 0$. But then, as \[ \lambda \cdot \tau'(t) = -k\sin(k(t - t_0)) \lambda \cdot \tau(t_0) + \sigma k\cos(k(t - t_0)) \lambda \cdot \tau^\perp(t_0), \] this implies that $\lambda \cdot \tau'(t) < 0$ in $(t_0 - \delta', t_0)$ for some $\delta' > 0$. If there exists $\delta > 0$ such that $(t_0, t_0 + \delta) \subseteq \Omega$, then we can draw similar conclusions with the same arguments. Hence \ref{item:structure2D} is satisfied. Conversely, suppose that \ref{item:structure2D} holds true. If $c([0, L]) \subseteq \mathcal{L}$, set $u = 0$. Otherwise, set \[ f(t) = \lambda \cdot c^\perp(t) + b, \] where $b \in \mathbb{R}$ is chosen such that $\Omega = f^{-1}(\{0\})$. Then $f \in W^{1, \infty}(0, L)$ and \eqref{eqn:f2D} is satisfied. If $(t_0, t_1) \subseteq \Omega$ is any connected component of $\Omega$, then $\omega' = \sigma k$ in $(t_0, t_1)$ for some fixed $\sigma \in \{-1, 1\}$. Hence we can write $\tau$ in the form \eqref{eqn:tau2D} and it follows that $f'$ satisfies \eqref{eqn:f'2D} in $(t_0, t_1)$. The condition on the sign of $\lambda \cdot \tau'$ near $t_0$ implies that $\sigma \lambda \cdot \tau^\perp(t_0) \ge 0$; and in the case of equality, it also implies that $\lambda \cdot \tau(t_0) < 0$. Therefore, the function $f$ has the same sign as $\omega'$ in $(t_0, t_1)$. Similar conclusions hold if we have connected components of $\Omega$ of the form $[0, t_1)$ or $(t_0, L]$. Hence $f$ and $\omega'$ have the same sign everywhere in $\Omega$. Now we set $u = f\tau^\perp$. Then \eqref{eqn:system2} is obvious and \eqref{eqn:system1} can be verified by computing \eqref{eqn:system1_2D} again and observing that \[ \beta(\lambda - (\lambda \cdot \tau) \tau) = \beta (\lambda \cdot \tau^\perp) \tau^\perp = (\lambda \cdot (c')^\perp) \tau^\perp = f' \tau^\perp. \] This concludes the proof. \end{proof} \section{Analysis of the differential equations} In this section we study the system \eqref{eqn:system1}, \eqref{eqn:system2} and its relationship to the variational problem in more detail. Furthermore, we show that it is equivalent to \eqref{eqn:ODE1}, \eqref{eqn:ODE2} up to the reparametrisation introduced in Section \ref{sect:reparametrisation}. \begin{proposition} \label{prop:DE=>pseudo-minimiser} Suppose that $\tau \in W^{1, \infty}((0, L); S^{n - 1})$ satisfies \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2}. If there exist $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$ and $\lambda \in \mathbb{R}^n$ such that \eqref{eqn:system1} and \eqref{eqn:system2} hold almost everywhere in $(0, L)$, then $\tau$ is a pseudo-minimiser of $K_\infty$. If in addition $k\|u\| + \beta \lambda \cdot \tau \le 0$ in $[0, L]$, then $\tau$ is a minimiser of $K_\infty$ under the constraints \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2}. \end{proposition} \begin{proof} Suppose that equations \eqref{eqn:system1} and \eqref{eqn:system2} hold true. Let $\Sigma = u^{-1}(\{0\})$. We claim that $\tau' = 0$ almost everywhere on $\Sigma$. Indeed, if $\lambda = 0$, then it follows from \eqref{eqn:system1} that $\left\|\frac{d}{dt} \|u\|\right\| \le \|u\| \\|\tau'\\|_{L^\infty(0, L)}$. As it is assumed that $u \not\equiv 0$, this inequality implies that $u \neq 0$ throughout $[0, L]$. If $\lambda \neq 0$, then at almost every point $t \in \Sigma$, either $u'(t) \neq 0$ (so $t$ is an isolated point of $\Sigma$) or $\tau(t) = \pm \lambda/\|\lambda\|$. As $\tau \in W^{1, \infty}((0, L); S^{n - 1})$, it has a derivative almost everywhere and we conclude that $\tau' = 0$ almost everywhere in $\Sigma$. Now consider a competitor $\tilde{\tau} \colon [0, L] \to S^{n - 1}$ satisfying \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2}. Let $\sigma = \tilde{\tau} - \tau$ and note that \[ 1 = \|\tau + \sigma\|^2 = 1 + 2 \tau \cdot \sigma + \|\sigma\|^2 \] in $[0, L]$. Hence \[ \tau \cdot \sigma = - \frac{\|\sigma\|^2}{2}. \] Furthermore, the definition of $\sigma$ guarantees that $\sigma(0) = \sigma(L) = 0$ and \[ \int_0^L \beta \sigma \, dt = 0. \] Observing that $u \cdot \tau' = k\|u\|$ because of \eqref{eqn:system2}, we now use \eqref{eqn:system1} to compute \begin{equation} \label{eqn:integral} \begin{split} \int_0^L \sigma' \cdot u \, dt & = - \int_0^L \sigma \cdot u' \, dt \\ & = \int_0^L \left((k\|u\| + \beta \lambda \cdot \tau) \tau \cdot \sigma - \beta \lambda \cdot \sigma\right) \, dt \\ & = -\frac{1}{2} \int_0^L (k\|u\|/\beta + \lambda \cdot \tau) \beta \|\sigma\|^2 \, dt \\ & \ge -\frac{1}{2} \left(k\\|u/\beta\\|_{L^\infty(0, L)} + \|\lambda\|\right) \bigl\\|\sqrt{\beta} \sigma\bigr\\|_{L^2(0, L)}^2. \end{split} \end{equation} Set \[ M = \frac{k\\|u/\beta\\|_{L^\infty(0, L)} + \|\lambda\|}{2\\|u\\|_{L^1(0, L)}}. \] Then there exists a set $A \subseteq [0, L]$ of positive measure such that $\sigma' \cdot u \ge -M \\|\sqrt{\beta} \sigma\\|_{L^2(0, L)}^2 \|u\|$ and $u \neq 0$ in $A$. (Otherwise, we would conclude that \[ \begin{split} \int_0^L \sigma' \cdot u \, dt & = \int_{(0, L) \setminus \Sigma} \sigma' \cdot u \, dt \\ & < -M \bigl\\|\sqrt{\beta} \sigma\bigr\\|_{L^2(0, L)}^2 \int_{(0, L) \setminus \Sigma} \|u\| \, dt \\ & = -M \bigl\\|\sqrt{\beta} \sigma\bigr\\|_{L^2(0, L)}^2 \int_0^L \|u\| \, dt \\ & = -\frac{1}{2} \left(k\\|u/\beta\\|_{L^\infty(0, L)} + \|\lambda\|\right) \bigl\\|\sqrt{\beta} \sigma\bigr\\|_{L^2(0, L)}^2, \end{split} \] in contradiction to \eqref{eqn:integral}.) Hence \[ \sigma' \cdot \tau' = \sigma' \cdot \frac{k u}{\|u\|} \ge - kM \bigl\\|\sqrt{\beta} \sigma\bigr\\|_{L^2(0, L)}^2 \] almost everywhere in $A$. As $\|\tau'\| = k$ almost everywhere in $A$, it follows that \[ \begin{split} \|\tilde{\tau}'\| & = \sqrt{\|\tau'\|^2 + 2\tau' \cdot \sigma' + \|\sigma'\|^2} \\ & \ge \sqrt{k^2 - 2kM \bigl\\|\sqrt{\beta} \sigma\bigr\\|_{L^2(0, L)}^2} \\ & \ge k - 2M\bigl\\|\sqrt{\beta} \sigma\bigr\\|_{L^2(0, L)}^2 \end{split} \] almost everywhere in $A$ (unless the right-hand side is negative, in which case the intermediate expression should be replaced by $0$). In particular, \[ K_\infty(\tilde{\tau}) \ge K_\infty(\tau) - 2M \int_0^L \beta \|\tilde{\tau} - \tau'\|^2 \, dt. \] That is, we have shown that $\tau$ is a pseudo-minimiser. Finally, if $k\|u\| + \beta \lambda \cdot \tau \le 0$, we can improve \eqref{eqn:integral} and conclude that \[ \int_0^L \sigma' \cdot u \, dt \ge 0. \] So there exists a set of positive measure $A \subseteq [0, L]$ where $u \neq 0$ and $\sigma' \cdot \tau' \ge 0$. Thus $\|\tilde{\tau}'\|^2 \ge \|\tau'\|^2 = k^2$ almost everywhere in $A$, and it follows immediately that $K_\infty(\tau) \le K_\infty(\tilde{\tau})$. \end{proof} Next we reformulate the system \eqref{eqn:system1}, \eqref{eqn:system2}. We obtain the system \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} below, which corresponds to \eqref{eqn:ODE1}, \eqref{eqn:ODE2} up to the reparametrisation from Section~\ref{sect:reparametrisation}. \begin{proposition} \label{prop:equivalence} Suppose that $\tau \in W^{1, \infty}((0, L); S^{n - 1})$. Let $\lambda \in \mathbb{R}^n$ and $k \ge 0$. \begin{enumerate} \item Suppose that $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$ satisfies \eqref{eqn:system1} and \eqref{eqn:system2} almost everywhere in $(0, L)$. Then there exists $f \in W^{1, \infty}(0, L) \setminus \{0\}$ with $f \ge 0$ such that \begin{align} f(\tau'' + k^2 \tau) & = \beta k^2 \proj_{\tau, \tau'}^\perp(\lambda), \label{eqn:ODE_rescaled1} \\ f' & = \beta \lambda \cdot \tau' \label{eqn:ODE_rescaled2}, \end{align} weakly in $(0, L)$. If $k > 0$, then $f = k\|u\|$ has this property. \item Suppose that there exists $f \in W^{1, \infty}(0, L) \setminus \{0\}$ with $f \ge 0$ satisfying \eqref{eqn:ODE_rescaled1} and \eqref{eqn:ODE_rescaled2} weakly in $(0, L)$. Then there exists $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$ such that \eqref{eqn:system1} and \eqref{eqn:system2} hold almost everywhere; and if $k > 0$, such that also $f = k\|u\|$. \item If there exists $f \in W^{1, \infty}(0, L) \setminus \{0\}$ with $f \ge 0$ such that \eqref{eqn:ODE_rescaled1} and \eqref{eqn:ODE_rescaled2} hold weakly and $f + \beta \lambda \cdot \tau \le 0$ in $(0, L)$, then $\tau$ minimises $K_\infty$ subject to the constraints \eqref{eqn:boundary_condition1} and \eqref{eqn:boundary_condition2}. \end{enumerate} \end{proposition} \begin{proof} Suppose first that we have a weak solution of \eqref{eqn:system1} and \eqref{eqn:system2} for some $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$. Let $\Omega = \set{t \in [0, L]}{u(t) \neq 0}$. If $k = 0$, then $\tau' = 0$ in $\Omega$ by \eqref{eqn:system2}. With the same arguments as in the proof of Proposition \ref{prop:DE=>pseudo-minimiser}, we show that $\tau' = 0$ almost everywhere in $[0, L] \setminus \Omega$. Hence \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} automatically hold true for any constant function $f$. If $k > 0$, then we consider the function $f = k\|u\|$. Equation \eqref{eqn:system2} then implies that $u = f\tau'/k^2$ almost everywhere. We conclude that $\tau' = k^2 u/f$ in $\Omega$, so $\tau \in W_\mathrm{loc}^{2, \infty}(\Omega)$. Hence from \eqref{eqn:system1} we derive the equation \begin{equation} \label{eqn:DE4} f(\tau'' + \|\tau'\|^2 \tau) + f' \tau' = k^2\beta (\lambda - (\lambda \cdot \tau)\tau) \end{equation} almost everywhere in $\Omega$. Taking the inner product with $\tau'$ and observing that $\tau \cdot \tau' = 0$ (because $\|\tau\| \equiv 1$) and $\tau'' \cdot \tau' = 0$ (because $\|\tau'\| \equiv k$ in $\Omega$), we see that \[ k^2f' = k^2 \beta \lambda \cdot \tau'. \] This amounts to equation \eqref{eqn:ODE_rescaled2}. Of course $f \ge 0$ by the definition of $f$. Differentiating the equation $\tau \cdot \tau' = 0$, we see that $\tau \cdot \tau'' + \|\tau'\|^2 = 0$. Recalling that $\tau' \cdot \tau'' = 0$, we conclude that \[ \proj_{\tau, \tau'}^\perp(\tau'') = \tau'' - (\tau \cdot \tau'') \tau = \tau'' + k^2 \tau \] in $\Omega$. Applying $\proj_{\tau, \tau'}^\perp$ to both sides of \eqref{eqn:DE4}, we see that \eqref{eqn:ODE_rescaled1} holds almost everywhere in $\Omega$. Also note that the function $f\tau' = k^2 u$ is continuous. Thus if $(t_1, t_2)$ is any connected component of $\Omega$, then for any $\xi \in C_0^\infty((0, L); \mathbb{R}^n)$, \begin{multline} \int_{t_1}^{t_2} \left(f(\tau' \cdot \xi' - k^2 \tau \cdot \xi) + f'\tau' \cdot \xi + k^2 \beta \proj_{\tau, \tau'}^\perp(\lambda) \cdot \xi\right) \, dt \\ = k^2u(t_2) \cdot \xi(t_2) - k^2u(t_1) \cdot \xi(t_1) = 0. \end{multline} A similar conclusion holds if $[0, t_2)$ or $(t_1, L]$ is a connected component of $\Omega$. Away from $\Omega$, we know that $u = 0$ and therefore either $\lambda = 0$ or $\tau = \pm \lambda/\|\lambda\|$ almost everywhere in $[0, L] \setminus \Omega$ by \eqref{eqn:system1}. Hence \eqref{eqn:ODE_rescaled1} holds weakly in all of $(0, L)$. Conversely, suppose that we have a weak solution of \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} for $f \in W^{1, \infty}(0, L) \setminus \{0\}$ with $f \ge 0$. Consider the open set $\Omega = \set{t \in [0, L]}{f(t) \neq 0}$. Here we can use \eqref{eqn:ODE_rescaled1} to conclude that $\tau \in W_\mathrm{loc}^{2, \infty}(\Omega)$. We differentiate the equation $\|\tau\|^2 = 1$ twice and we obtain $\tau'' \cdot \tau + \|\tau'\|^2 = 0$ almost everywhere in $\Omega$. On the other hand, multiplying both sides of \eqref{eqn:ODE_rescaled1} with $\tau$, we find that $\tau'' \cdot \tau + k^2 = 0$ in $\Omega$. Hence $\|\tau'\| \equiv k$ in $\Omega$. If $k = 0$, then $\tau' \equiv 0$ in $\Omega$ and \eqref{eqn:ODE_rescaled2} implies that $f$ is locally constant in $\Omega$. So in this case, it follows that $\Omega = [0, L]$ and \eqref{eqn:system2} is automatically satisfied. Moreover, it is then easy to find $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$ that solves \eqref{eqn:system1}. If $k > 0$, then we claim that \eqref{eqn:DE4} is satisfied in $\Omega$. In order to see why, we split the equation into three parts by projecting orthogonally onto the spaces $\mathbb{R} \tau(t)$ and $\mathbb{R} \tau'(t)$ and onto the orthogonal complement of $\mathbb{R} \tau(t) \oplus \mathbb{R} \tau'(t)$ at almost every $t \in \Omega$. The projection onto $\mathbb{R} \tau(t)$ is trivial. The projection onto $\mathbb{R} \tau'(t)$ amounts to \eqref{eqn:ODE_rescaled2}, and applying $\proj_{\tau(t), \tau'(t)}^\perp$ gives \eqref{eqn:ODE_rescaled1}. Thus we have a solution of \eqref{eqn:DE4} in $\Omega$. Setting $u = f\tau'/k^2$, we can then verify \eqref{eqn:system1} and \eqref{eqn:system2} in $\Omega$. Outside of $\Omega$, we know that $f = 0$ and $u = 0$. Hence \eqref{eqn:ODE_rescaled2} implies that $\lambda \cdot \tau' = 0$ almost everywhere outside of $\Omega$. Moreover, \eqref{eqn:ODE_rescaled1} implies that $\proj_{\tau, \tau'}^\perp(\lambda) = 0$ almost everywhere in $[0, L] \setminus \Omega$. That is, $\lambda$ is a multiple of $\tau$ and \eqref{eqn:system1}, \eqref{eqn:system2} are satisfied almost everywhere in $[0, L] \setminus \Omega$ as well. Furthermore, if $f + \beta \lambda \cdot \tau \le 0$, then $k\|u\| + \beta \lambda \cdot \tau \le 0$, and the last statement follows from Proposition \ref{prop:DE=>pseudo-minimiser}. \end{proof} As mentioned previously, the new system of differential equations \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} corresponds to \eqref{eqn:ODE1}, \eqref{eqn:ODE2} up to the reparametrisation from Section \ref{sect:reparametrisation}. But Proposition \ref{prop:equivalence} requires only that $\lambda \in \mathbb{R}^n$, whereas $\lambda \in S^{n - 1}$ in Theorem~\ref{thm:DE}. For this reason, the following observation is useful. \begin{lemma} \label{lem:lambda} Let $\tau \in W^{1, \infty}((0, L); S^{n - 1})$, $\lambda \in \mathbb{R}^n$, and $f \in W^{1, \infty}(0, L) \setminus \{0\}$ with $f \ge 0$ such that \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} hold weakly. Then there exist $\tilde{f} \in W^{1, \infty}(0, L) \setminus \{0\}$ with $\tilde{f} \ge 0$ and $\tilde{\lambda} \in S^{n - 1}$ such that \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} hold weakly for $\tilde{f}$ instead of $f$ and for $\tilde{\lambda}$ instead of $\lambda$ as well. \end{lemma} \begin{proof} If $\lambda \neq 0$, then it suffices to define $\tilde{f} = f/\|\lambda\|$ and $\tilde{\lambda} = \lambda/\|\lambda\|$ and check that both equations are still satisfied. If $\lambda = 0$, then $f$ is constant and positive. Hence $\tau'' + k^2 \tau = 0$ in $(0, L)$. With the same arguments as in the proof of Proposition \ref{prop:equivalence}, we see that $\|\tau'\| \equiv k$. The resulting equation $\tau'' + \|\tau'\|^2 \tau = 0$ means that $\tau$ follows a geodesic, i.e., a great circle on $S^{n - 1}$. This implies that $\tau(t)$ and $\tau'(t)$ span the same two-dimensional subspace of $\mathbb{R}^n$ everywhere, and any $\tilde{\lambda}$ in this subspace will satisfy $\proj_{\tau, \tau'}^\perp(\tilde{\lambda}) = 0$. Now we choose $\tilde{f}$ such that \eqref{eqn:ODE_rescaled2} holds true (for $\tilde{\lambda}$ instead of $\lambda$) and at the same time $\tilde{f} > 0$ in $[0, L]$. Then both equations are satisfied. \end{proof} We now have all the tools for the proofs of the first two results in the introduction. \begin{proof}[Proofs of Theorem \ref{thm:DE} and Theorem \ref{thm:minimiser}] With the reparametrisation from Section \ref{sect:reparametrisation}, an $\infty$-elastica gives rise to a pseudo-minimiser of $K_\infty$ and vice versa. According to Proposition \ref{prop:pseudo-minimiser=>DE} and Proposition \ref{prop:DE=>pseudo-minimiser}, pseudo-minimisers of $K_\infty$ correspond to solutions of \eqref{eqn:system1}, \eqref{eqn:system2}, which is equivalent to \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} by Proposition \ref{prop:equivalence}. Lemma \ref{lem:lambda} shows that it suffices to consider this system for $\lambda \in S^{n - 1}$. Now we check that the system corresponds to \eqref{eqn:ODE1}, \eqref{eqn:ODE2} for the original parametrisation, and this proves Theorem \ref{thm:DE}. Theorem \ref{thm:minimiser} follows from the last statement of Proposition \ref{prop:equivalence}. \end{proof} \section{Preparation for the proof of Theorem \ref{thm:structure}} The system of ordinary differential equations \eqref{eqn:ODE_rescaled1}, \eqref{eqn:ODE_rescaled2} becomes degenerate at points where $f$ vanishes. It turns out, however, that $f$ remains positive for \emph{generic} solutions as described in the following result. This information will be crucial for statement \ref{item:3D} in Theorem \ref{thm:structure}. \begin{lemma} \label{lem:ODE} Let $\lambda, \tau_0 \in S^{n - 1}$ and $\tau_1 \in \mathbb{R}^n$ such that $\tau_0 \perp \tau_1$, and let $f_0 > 0$ and $t_0 \in [0, L]$. If the vectors $\tau_0$, $\tau_1$, and $\lambda$ are linearly independent, then the initial value problem \begin{align} \tau'' + \|\tau'\|^2 \tau & = \beta f^{-1} \|\tau'\|^2 \proj_{\tau, \tau'}^\perp(\lambda), \label{eqn:ODE_reformulated} \\ f' & = \beta \lambda \cdot \tau', \nonumber\\ \tau(t_0) & = \tau_0, \quad \tau'(t_0) = \tau_1, \quad f(t_0) = f_0, \nonumber \end{align} has a unique global solution, consisting of $\tau \colon [0, L] \to S^{n - 1}$ and $f \colon [0, L] \to (0, \infty)$. For all $t \in [0, L]$, this solution satisfies $\|\tau'(t)\| = \|\tau_1\|$ and $\lambda \cdot \tau(t) \neq \pm 1$, and $\tau(t)$ remains in the linear subspace of $\mathbb{R}^n$ spanned by $\tau_0$, $\tau_1$, and $\lambda$. \end{lemma} \begin{proof} Under these assumptions, we clearly have a unique solution of the initial value problem in a certain interval $(t_1, t_2) \cap [0, L]$ such that $\lambda \cdot \tau \neq \pm 1$ and $f > 0$ in that interval. Multiplying \eqref{eqn:ODE_reformulated} with $\tau$, we see that $\frac{d}{dt}(\tau \cdot \tau') = 0$. Hence the solution will continue to take values on the sphere $S^2$. Multiplying the equation with $\tau'$, we further see that $\frac{d}{dt} \|\tau'\|^2 = 0$. Setting $k = \|\tau_1\|$, we conclude that $\|\tau'\| = k$ in $(t_1, t_2) \cap [0, L]$. Moreover, if $V \in \mathbb{R}^n$ is any vector perpendicular to $\tau_0$, $\tau_1$, and $\lambda$, then the function $h = V \cdot \tau$ satisfies \[ h'' + \|\tau'\|^2 h = -\beta f^{-1} \left(\|\tau'\|^2 (\lambda \cdot \tau) h + (\lambda \cdot \tau') h'\right) \] in $(t_1, t_2) \cap [0, L]$ and $h(t_0) = h'(t_0) = 0$. Hence $h \equiv 0$, and the solution $\tau$ will remain in the linear subspace spanned by $\tau_0$, $\tau_1$, and $\lambda$ in $(t_1, t_2) \cap [0, L]$. So we may assume that $n = 3$ without loss of generality. We may further choose coordinates such that $\lambda = (0, 0, 1)$. It now suffices to show that $\liminf_{t \searrow t_1} f(t) > 0$ and $\limsup_{t \searrow t_1} \|\lambda \cdot \tau(t)\| < 1$ (unless $t_1 < 0$) and that $\liminf_{t \nearrow t_2} f(t) > 0$ and $\limsup_{t \nearrow t_2} \|\lambda \cdot \tau(t)\| < 1$ (unless $t_2 > L$). The standard theory for ordinary differential equations will then imply the result. We use spherical coordinates on $S^2$ and we write \[ \tau = (\cos \varphi \sin \vartheta, \sin \varphi \sin \vartheta, \cos \vartheta) \] for $\varphi, \vartheta \colon (t_1, t_2) \cap [0, L] \to \mathbb{R}$ with $\vartheta(t) \in (0, \pi)$ for all $t \in (t_1, t_2)$. Writing also \[ e_1 = (-\sin \varphi, \cos \varphi, 0) \quad \text{and} \quad e_2 = (\cos \varphi \cos \vartheta, \sin \varphi \cos \vartheta, -\sin \vartheta), \] we obtain an orthonormal basis $(\tau(t), e_1(t), e_2(t))$ of $\mathbb{R}^3$ such that $e_1(t)$ and $e_2(t)$ span the tangent space of $S^2$ at $\tau(t)$ for every $t \in (t_1, t_2)\cap [0, L]$. We compute \[ \tau' = \varphi' \sin \vartheta \, e_1 + \vartheta' \, e_2 \] and \[ \tau'' + \|\tau'\|^2 \tau = \left(\varphi'' \sin \vartheta + 2\varphi' \vartheta' \cos \vartheta\right) e_1 + \left(\vartheta'' - (\varphi')^2 \sin \vartheta \cos \vartheta\right) e_2. \] Define $Z = -\vartheta' \, e_1 + \varphi' \sin \vartheta \, e_2$, so that $\|Z\| = \|\tau'\|$ and $Z \perp \tau'$. Then \[ \|\tau'\|^2 \proj_{\tau, \tau'}^\perp(\lambda) = (\lambda \cdot Z)Z = \varphi' \vartheta' \sin^2 \vartheta \, e_1 - (\varphi')^2 \sin^3 \vartheta \, e_2. \] Therefore, we obtain the equations \begin{align} \varphi'' \sin \vartheta + 2\varphi' \vartheta' \cos \vartheta & = \beta f^{-1} \varphi'\vartheta' \sin^2 \vartheta, \label{eqn:varphi} \\ \vartheta'' - (\varphi')^2 \sin \vartheta \cos \vartheta & = - \beta f^{-1} (\varphi')^2 \sin^3 \vartheta, \label{eqn:vartheta} \end{align} and furthermore \begin{equation} \label{eqn:f_polar} f' = - \beta \vartheta' \sin \vartheta. \end{equation} For the rest of the proof, it suffices to consider \eqref{eqn:varphi} and \eqref{eqn:f_polar}. We first claim that $\varphi'$ does not vanish anywhere in $(t_1, t_2) \cap [0, L]$. Otherwise, equation \eqref{eqn:varphi} would imply that it remains $0$ throughout $(t_1, t_2) \cap [0, L]$, and $\tau$ would parametrise a piece of a great circle through $(0, 0, 1)$. This, however, is impossible under the assumption that $\tau_0$, $\tau_1$, and $\lambda$ are linearly independent. Thus we may divide by $\varphi' \sin \vartheta$ in \eqref{eqn:varphi} and we find that \[ \frac{\varphi''}{\varphi'} = -\frac{f'}{f} - \frac{2\vartheta' \cos \vartheta}{\sin \vartheta}. \] Integrating, we see that there exists $b \in \mathbb{R}$ such that \[ \log \|\varphi'\| = - \log f - 2 \log \sin \vartheta + b. \] Set $B = e^b$. Then \[ \|\varphi'\| = \frac{B}{f \sin^2 \vartheta}. \] The equation $(\varphi')^2 \sin^2 \vartheta + (\vartheta')^2 = \|\tau'\|^2 = k^2$ then implies that \[ \frac{B^2}{f^2 \sin^2 \vartheta} \le k^2. \] It follows immediately that $f$ and $\sin \vartheta$ stay away from $0$ and this concludes the proof. \end{proof} The following technical lemma is also required for the proof of Theorem \ref{thm:structure}. \begin{lemma} \label{lem:series} Suppose that $(b_i)_{i \in \mathbb{N}}$ is a sequence of positive numbers such that \[ \sum_{i = 1}^\infty \left\|1 - \frac{b_{i + 1}}{b_i}\right\| < \infty. \] Then $\sum_{i = 1}^\infty b_i = \infty$. \end{lemma} \begin{proof} Ignoring finitely many terms if necessary, we may assume that \[ \sum_{i = 1}^\infty \left\|1 - \frac{b_{i + 1}}{b_i}\right\| \le \frac{1}{2}. \] Fix $I \in \mathbb{N}$. Let $q_i = b_{i + 1}/b_i$ for $i = 1, \dotsc, I - 1$. Choose a permutation $S \colon \{1, \dots, I - 1\} \to \{1, \dots, I - 1\}$ such that $q_{S(1)} \le \dotsb \le q_{S(I - 1)}$ and define $q_i' = \min\{q_{S(i)}, 1\}$. Also define $b_1', \dotsc, b_I' > 0$ by $b_1' = b_1$ and \[ b_{i + 1}' = q_i' b_i', \quad i = 1, \dotsc, I - 1. \] Then \begin{equation} \label{eqn:quotients} \sum_{i = 1}^{I - 1} (1 - q_i') \le \sum_{i = 1}^{I - 1} \|1 - q_i\| \le \frac{1}{2} \end{equation} and $b_i' \le b_i$ for all $i = 1, \dots, I$. As $q_i'$ is non-decreasing in $i$, inequality \eqref{eqn:quotients} implies that \[ 1 - q_i' \le \frac{1}{2i} \le \frac{1}{i + 1} \] for $i = 1, \dots, I - 1$. Define $B_i = 1/i$ for $i = 1, \dotsc, I$. Then \[ \frac{b_{i + 1}'}{b_i'} = q_i' \ge \frac{i}{i + 1} = \frac{B_{i + 1}}{B_i}. \] Hence \[ b_i' = \frac{b_i'}{b_{i - 1}'} \dotsb \frac{b_2'}{b_1'} b_1 \ge \frac{B_i}{B_{i - 1}} \dotsb \frac{B_2}{B_1} b_1 = \frac{B_i}{B_1} b_1 = \frac{b_1}{i}. \] It follows that \[ \sum_{i = 1}^I b_i \ge \sum_{i = 1}^I b_i' \ge b_1 \sum_{i = 1}^I \frac{1}{i}. \] Letting $I \to \infty$, we obtain the desired result. \end{proof} \section{Proof of Theorem \ref{thm:structure}} Now we consider the situation of Theorem \ref{thm:structure}. Suppose first that $\gamma \in \mathcal{G}$ is an $\infty$-elastica and let $k = \mathcal{K}_\alpha(\gamma)$. If $k = 0$, then $\gamma'' = 0$ almost everywhere and $\gamma$ parametrises a line segment. Then clearly statement \ref{item:2D} in Theorem \ref{thm:structure} is satisfied. Therefore, we assume that $k > 0$ henceforth. Consider the reparametrised tangent vector field $\tau \colon [0, L] \to S^{n - 1}$ with $\tau(t) = \gamma'(\phi(t))$ for $t \in [0, L]$ as in Section \ref{sect:reparametrisation}. Then $\tau$ is a pseudo-minimiser of $K_\infty$. Hence by Proposition \ref{prop:pseudo-minimiser=>DE}, there exist $\lambda \in \mathbb{R}^n$ and $u \in W^{1, \infty}((0, L); \mathbb{R}^n) \setminus \{0\}$ such that \eqref{eqn:system1} and \eqref{eqn:system2} hold true almost everywhere. According to Proposition \ref{prop:equivalence}, the function $f = k\|u\|$ satisfies \eqref{eqn:ODE_rescaled1} and \eqref{eqn:ODE_rescaled2} weakly, and by Lemma \ref{lem:lambda} we may assume that $\lambda \in S^{n - 1}$. Let $\Omega = \set{t \in [0, L]}{f(t) > 0}$. Then \eqref{eqn:system2} implies that $\tau'$ is continuous in $\Omega$ with $\|\tau'\| \equiv k$. It follows from \eqref{eqn:ODE_rescaled1} that $\tau \in W_\mathrm{loc}^{2, \infty}(\Omega)$. Moreover, by standard theory for ordinary differential equations, both $\tau$ and $f$ are locally uniquely determined by their initial conditions $\tau(t_0)$, $\tau'(t_0)$, and $f(t_0)$ for any $t_0 \in \Omega$. If $\tau$, $\tau'$, and $\lambda$ are linearly independent anywhere in $\Omega$, then Lemma \ref{lem:ODE} implies that $\Omega = [0, L]$ and that $\tau$ takes values in a three-dimensional subspace of $\mathbb{R}^n$, and \eqref{eqn:ODE_rescaled1} and \eqref{eqn:ODE_rescaled2} are satisfied almost everywhere. Equations \eqref{eqn:ODE1} and \eqref{eqn:ODE2} now arise when we reverse the reparametrisation from Section \ref{sect:reparametrisation}. The observation that $\alpha \gamma'' = \tau' \circ \psi$ implies that $\alpha \gamma'' \in W^{1, \infty}((0, \ell); \mathbb{R}^n)$ and that $\alpha \|\gamma''\| \equiv k$. Equation \eqref{eqn:ODE2} then implies that $g \in W^{2, \infty}(0, \ell)$. Hence statement \ref{item:3D} in Theorem \ref{thm:structure} holds true. This leaves the case when $\tau$, $\tau'$, and $\lambda$ are linearly dependent everywhere in $\Omega$. We assume this from now on. Then we can say more about the behaviour of $\tau$ in $\Omega$. \begin{lemma} \label{lem:Omega} If $(t_1, t_2) \subseteq \Omega$, then the restriction of $\tau$ to $(t_1, t_2)$ follows a great circle in $S^{n - 1}$ through $\lambda$ with constant speed $k$. Furthermore, if $(t_1, t_2)$ is a connected component of $\Omega$, then there exists $t_0 \in (t_1, t_2)$ such that $\tau(t_0) = \pm \lambda$. \end{lemma} \begin{proof} We know that $\tau \cdot \tau' = 0$ everywhere, and $\tau'$ is continuous with $\|\tau'\| \equiv k$ in $\Omega$. As $\tau(t)$, $\tau'(t)$, and $\lambda$ are linearly dependent, we further know that $\tau'(t)$ is in the space spanned by $\tau(t)$ and $\lambda$ for every $t \in \Omega$ with $\tau(t) \neq \pm \lambda$. Hence $\tau$ follows a great circle on $S^{n - 1}$ through $\lambda$ with speed $k$; indeed, by the continuity of $\tau'$, this is true throughout $(t_1, t_2)$ even if there are any points where $\tau(t) = \pm \lambda$. If $(t_1, t_2)$ is a connected component of $\Omega$, then $f(t_1) = 0 = f(t_2)$. By \eqref{eqn:ODE_rescaled2}, this means that $\lambda \cdot \tau'$ must change sign somewhere in $(t_1, t_2)$. Given what we know about $\tau$ so far, there must exists $t_0 \in (t_1, t_2)$ such that $\tau(t_0) = \pm \lambda$. \end{proof} Next consider the set $\Omega' = \set{t \in [0, L]}{\tau(t) \neq \pm \lambda} \cup \Omega$. This is an open set relative to $[0, L]$ as well. \begin{lemma} \label{lem:discrete} The set $\Omega' \setminus \Omega$ is discrete. \end{lemma} \begin{proof} As $f = 0$ in $[0, L] \setminus \Omega$, we know that $f' = 0$ almost everywhere in this set. Using \eqref{eqn:ODE_rescaled2}, we conclude that $\tau' \cdot \lambda = 0$ almost everywhere, and \eqref{eqn:ODE_rescaled1} implies that $\lambda$ is in the subspace spanned by $\tau$ and $\tau'$ almost everywhere in $[0, L] \setminus \Omega$. Hence $\tau = \pm \lambda$ almost everywhere in $[0, L] \setminus \Omega$. It follows that $\Omega' \setminus \Omega$ is a null set, and so is $\Omega' \setminus \overline{\Omega}$. As the latter is an open set, it must be empty. So $\Omega' \subseteq \overline{\Omega}$. For any $t_0 \in \Omega' \setminus \Omega$, we may choose $\epsilon > 0$ such that $\tau \neq \pm \lambda$ in $(t_0 - \epsilon, t_0 + \epsilon) \cap [0, L]$ by the continuity of $\tau$. Let $J = (t_0 - \epsilon, t_0 + \epsilon) \cap (0, L)$. Then $J$ cannot contain any connected components of $\Omega$ by Lemma \ref{lem:Omega}. Therefore, the open set $J \cap \Omega$ consists of at most two intervals extending to one of the end points of $J$. But we know that $J \subseteq \overline{\Omega}$. Hence $J \cap \Omega = J \setminus \{t_0\}$. We conclude that $t_0$ is an isolated point of $\Omega' \setminus \Omega$. That is, the set $\Omega' \setminus \Omega$ is discrete. \end{proof} \begin{lemma} \label{lem:Omega'} If $I$ is any connected component of $\Omega'$, then the restriction of $\tau$ to $I$ takes values in a great circle on $S^{n - 1}$ through $\lambda$. \end{lemma} \begin{proof} In view of Lemma \ref{lem:Omega} and Lemma \ref{lem:discrete}, it suffices to examine what happens near a point $t_0 \in I \setminus \Omega$. There exists $\epsilon > 0$ such that the restriction of $\tau$ to $(t_0 - \epsilon, t_0)$ follows a great circle through $\lambda$, and the same statement applies to $(t_0, t_0 + \epsilon)$. But as $t_0 \in I \subseteq \Omega'$ and $t_0 \not\in \Omega$, it is clear that $\tau(t_0) \neq \pm \lambda$. So we have the same great circle on both sides of $t_0$, and the claim follows. \end{proof} We can now improve Lemma \ref{lem:discrete}. This is the only place in the paper where we use the assumption that $\alpha$ is of bounded variation rather than just bounded. \begin{lemma} \label{lem:finite} If $I \subseteq \Omega'$ is a connected component of $\Omega'$, then $I \setminus \Omega$ is finite. \end{lemma} \begin{proof} We argue by contradiction here, so we assume that $I \setminus \Omega$ is \emph{not} finite. Then by Lemma \ref{lem:discrete}, either $\inf I$ or $\sup I$ is an accumulation point of $I \setminus \Omega$, and we assume for simplicity that this is true for $\sup I$. (The arguments are similar if it is $\inf I$.) Then there is a sequence $(t_i)_{i \in \mathbb{N}}$ in $I \setminus \Omega$ such that $t_{i + 1} > t_i$ and $(t_i, t_{i + 1}) \subseteq \Omega$ for all $i \in \mathbb{N}$. So $f(t_i) = 0$ for all $i \in \mathbb{N}$. By Lemma \ref{lem:Omega}, we know that $\tau$ follows a great circle through $\lambda$ with speed $k$ in the interval $(t_i, t_{i + 1})$ and there exists a point $\rho_i \in (t_i, t_{i + 1})$ such that $\tau(\rho_i) = \pm \lambda$ for every $i \in \mathbb{N}$. If $\tau(\rho_i) = \lambda$ and $\tau(\rho_{i + 1}) = - \lambda$ or vice versa, then $\rho_{i + 1} - \rho_i \ge \pi/k$; so this can happen at most a finite number of times. Dropping finitely many members of the sequence, we may assume that $\rho_{i + 1} - \rho_i < \pi/k$ for every $i$; then $\tau(\rho_i)$ has always the same sign and for simplicity we assume that $\tau(\rho_i) = \lambda$ for every $i \in \mathbb{N}$. Then \[ \lambda \cdot \tau(t) = \cos(k(t -\rho_i)) \] in $(t_i, t_{i + 1})$ for all $i \in \mathbb{N}$. It follows immediately that $\rho_{i + 1} - t_{i + 1} = t_{i + 1} - \rho_i$ for every $i \in \mathbb{N}$. Furthermore, equation \eqref{eqn:ODE_rescaled2} implies that \[ 0 = \int_{t_i}^{t_{i + 1}} f'(t) \, dt = - k \int_{t_i}^{t_{i + 1}} \beta(t) \sin(k(t -\rho_i)) \, dt. \] Hence \[ k \int_{t_i}^{\rho_i} \beta(t) \left\|\sin(k(t -\rho_i))\right\| \, dt = k \int_{\rho_i}^{t_{i + 1}} \beta(t) \left\|\sin(k(t -\rho_i))\right\| \, dt. \] Define \[ b_i = k \int_{t_i}^{\rho_i} \left\|\sin(k(t -\rho_i))\right\| \, dt = 1 - \cos(k(\rho_i - t_i)) \] and \[ b_i' = k \int_{\rho_i}^{t_{i + 1}} \left\|\sin(k(t -\rho_i))\right\| \, dt = 1 - \cos(k(t_{i + 1} - \rho_i)). \] If $b_i' \le b_i$, then we may choose $\omega_i \in [t_i, \rho_i]$ and $\omega_i' \in [\rho_i, t_{i + 1}]$ such that \[ k \int_{t_i}^{\rho_i} \beta(t) \left\|\sin(k(t -\rho_i))\right\| \, dt \ge b_i \beta(\omega_i) \] and \[ k \int_{\rho_i}^{t_{i + 1}} \beta(t) \left\|\sin(k(t -\rho_i))\right\| \, dt \le b_i' \beta(\omega_i'); \] then \[ \frac{b_i'}{b_i} \ge \frac{\beta(\omega_i)}{\beta(\omega_i')}. \] If $b_i < b_i'$, then instead we choose $\omega_i \in [t_i, \rho_i]$ and $\omega_i' \in [\rho_i, t_{i + 1}]$ such that \[ k \int_{t_i}^{\rho_i} \beta(t) \left\|\sin(k(t -\rho_i))\right\| \, dt \le b_i \beta(\omega_i) \] and \[ k \int_{\rho_i}^{t_{i + 1}} \beta(t) \left\|\sin(k(t -\rho_i))\right\| \, dt \ge b_i' \beta(\omega_i'); \] then \[ \frac{b_i'}{b_i} \le \frac{\beta(\omega_i)}{\beta(\omega_i')}. \] In both cases, \[ \left\|1 - \frac{b_i'}{b_i}\right\| \le \left\|1 - \frac{\beta(\omega_i)}{\beta(\omega_i')}\right\| = \frac{\|\beta(\omega_i') - \beta(\omega_i)\|}{\|\beta(\omega_i')\|} \le \|\beta(\omega_i') - \beta(\omega_i)\| \, \sup_{[0, \ell]} \frac{1}{\alpha}. \] Hence \[ \sum_{i = 1}^\infty \left\|1 - \frac{b_i'}{b_i}\right\| \le \sup\biggl\{\sum_{j = 1}^J \|\alpha(s_j) - \alpha(s_{j - 1})\| \colon 0 \le s_0 \le \dotsb \le s_J \le \ell\biggr\} \, \sup_{[0, \ell]} \frac{1}{\alpha}. \] The right-hand side is finite, because $\alpha$ is assumed to be of bounded variation and $1/\alpha$ is bounded. We have already seen that $t_{i + 1} - \rho_i = \rho_{i + 1} - t_{i + 1}$ for every $i \in \mathbb{N}$. This means that $b_i' = b_{i + 1}$. We now apply Lemma \ref{lem:series} to the sequence $(b_1, b_1', b_2, b_2', \dotsc)$. We infer that \begin{equation} \label{eqn:divergent_sum} \sum_{i = 1}^\infty (b_i + b_i') = \infty. \end{equation} But clearly \[ \sum_{i = 1}^\infty (\rho_i - t_i) + \sum_{i = 1}^\infty (t_{i + 1} - \rho_i) \le L, \] as this is the sum of the lengths of pairwise disjoint intervals in $(0, L)$. Hence there exists $i_0 \in \mathbb{N}$ such that \[ \rho_i - t_i \le \frac{2}{k^2} \quad \text{and} \quad t_{i + 1} - \rho_i \le \frac{2}{k^2} \] for all $i \ge i_0$, which implies that \[ b_i = 1 - \cos(k(\rho_i - t_i)) \le \rho_i - t_i \] and \[ b_i' = 1 - \cos(k(t_{i + 1} - \rho_i)) \le t_{i + 1} - \rho_i. \] Now we have a contradiction to \eqref{eqn:divergent_sum}. \end{proof} \begin{lemma} \label{lem:finite2} The set $\Omega'$ has finitely many connected components. \end{lemma} \begin{proof} We can ignore any connected components of the form $[0, t_2)$ or $(t_1, L]$. Thus we fix another connected component $I = (t_1, t_2)$. Then $f(t_1) = 0$ and $\tau(t_1) = \pm \lambda$, and also $f(t_2) = 0$ and $\tau(t_2) = \pm \lambda$. Furthermore, by Lemma \ref{lem:finite}, there exists $t_3 \in (t_1, t_2]$ such that $f(t_3) = 0$ and $(t_1, t_3) \subseteq \Omega$. According to Lemma \ref{lem:Omega}, this implies that there exists $t_4 \in (t_1, t_3)$ with $\tau(t_4) = \pm \lambda$. We further know that $\tau$ follows a great circle with speed $k$ in $(t_1, t_4)$, and therefore $t_4 - t_1 \ge \pi/k$. So there can only be finitely many connected components. \end{proof} Now we can complete the proof of Theorem \ref{thm:structure} as follows. By Lemma \ref{lem:finite2}, we can partition $\Omega'$ into finitely many connected components $I_1, \dotsc, I_M$. Let $t_i = \inf I_i$ and $t_i' = \sup I_i$ for $i = 1, \dotsc, M$. Setting $A = [0, L] \setminus \bigcup_{i = 1}^M I_i$, we observe that $f = 0$ and $\tau = \pm \lambda$ on $A$. The set $\tau(\overline{I}_i)$ is contained in a two-dimensional subspace $X_i \subseteq \mathbb{R}^n$ with $\lambda \in X_i$ for every $i = 1, \dotsc, M$ by Lemma \ref{lem:Omega'}. Hence Lemma~\ref{lem:2D} may be applied to the restriction of $\tau$ to $\overline{I}_i$. Consequently, there exists a line $\mathcal{L}_i \subseteq X_i + c(t_i)$ for every $i = 1, \dotsc, M$ such that $\set{t \in \overline{I}_i}{f(t) = 0} = \set{t \in \overline{I}_i}{c(t) \in \mathcal{L}_i}$, where $c = \gamma \circ \phi$. But we know that $f(t_i) = 0$, except possibly for $i = 1$ if $t_1 = 0$, and that $f(t_i') = 0$, except possibly for $i = M$ if $t_M' = L$. Moreover, each $\mathcal{L}_i$ is parallel to $\lambda$. As $\tau = \pm \lambda$ on $A$, we also conclude that $c([t_i', t_{i + 1}])$ is a line segment parallel to $\lambda$ for $i = 1, \dotsc, M - 1$, and the same applies to $c([0, t_1])$ if $t_1 > 0$ and to $c([t_M', L])$ if $t_M' < L$. Hence the lines $\mathcal{L}_i$ all coincide with a single line $\mathcal{L} \subseteq \mathbb{R}^n$ and $c(A) \subseteq \mathcal{L}$. If there are any points $t \in I_i \setminus \Omega$, then we further subdivide $I_i$. According to Lemma \ref{lem:finite}, there are only finitely many such points. Thus we obtain pairwise disjoint, relatively open intervals $I_1^, \ldots, I_N^ \subseteq [0, L]$ such that $c(t) \not\in \mathcal{L}$ for all $t \in I_i^$ for $i = 1, \dotsc, N$ but $c(t) \in \mathcal{L}$ for all $t \in [0, L] \setminus \bigcup_{i = 1}^N I_i^$. Lemma \ref{lem:2D} then further implies that $\tau'$ is continuous with $\|\tau'\| \equiv k$ in $I_i^$, and that there exists $\delta > 0$ such that for any $t_0 \in \overline{I_i^} \setminus I_i^$, the inequality $\lambda \cdot \tau' > 0$ is satisfied in $(t_0, t_0 + \delta) \cap I_i^$ and $\lambda \cdot \tau' < 0$ in $(t_0 - \delta, t_0) \cap I_i^$ for all $i = 1, \dotsc, N$. Reversing the reparametrisation from Section \ref{sect:reparametrisation} and setting $J_i = \phi(I_i^)$, we therefore find the situation described in statement \ref{item:2D} of Theorem \ref{thm:structure}. Finally, we want to prove that every curve satisfying one of the conditions in Theorem \ref{thm:structure} is indeed an $\infty$-elastica. This is clear if $\gamma([0, L])$ is contained in a line, so we assume otherwise. In the case of condition \ref{item:3D}, the claim follows immediately from Proposition \ref{prop:equivalence} and Proposition \ref{prop:DE=>pseudo-minimiser}. If condition \ref{item:2D} is satisfied, we use Lemma \ref{lem:2D} for any piece of $\gamma$ restricted to $\overline{J}_i$. In order to work with the usual reparametrisation, we set $I_i = \psi(J_i)$ and let $t_i = \inf I_i$ and $t_i' = \sup I_i$. Then Lemma \ref{lem:2D} gives rise to $u_i \colon \overline{I}_i \to \mathbb{R}^n$ satisfying \eqref{eqn:system1}, \eqref{eqn:system2} in $I_i$ with $u_i(t_i) = 0$ (unless $t_i = 0$) and $u_i(t_i') = 0$ (unless $t_i' = L$), but $u_i \neq 0$ in $I_i$. Hence we define $u \colon [0, L] \to \mathbb{R}^n$ by \[ u(t) = \begin{cases} u_i(t) & \text{if } t \in I_i, \ i = 1, \dotsc, N, \\ 0 & \text{else}. \end{cases} \] Then \eqref{eqn:system1} and \eqref{eqn:system2} are satisfied almost everywhere in $(0, L)$. Proposition \ref{prop:DE=>pseudo-minimiser} now completes the proof. \section{The Markov-Dubins problem} \label{sect:Dubins} In this section, we first prove Proposition \ref{prop:shortest_curves}, thus establishing the connection to the Markov-Dubins problem of minimising length subject to curvature constraints. Then we show how to recover some of the main results of Dubins \cite[Theorem I]{Dubins:57} and Sussmann \cite[Theorem 1]{Sussmann:95} from Theorem~\ref{thm:structure}. \begin{proof}[Proof of Proposition \ref{prop:shortest_curves}] Suppose that $\gamma \in \mathcal{G}$ does \emph{not} minimise $\mathcal{K}_1$ under the boundary conditions \eqref{eqn:boundary_conditions}. We want to show that the curve parametrised by $\gamma$ is not an $R$-geodesic. For $R > 1/\mathcal{K}_1(\gamma)$, this is obvious, as $\gamma$ does not satisfy the required curvature constraint. Thus we assume that $R \le 1/\mathcal{K}_1(\gamma)$. We may assume without loss of generality that $a_1, a_2 \in \{0\}^{n - 1} \times \mathbb{R}$. In the following, we write $x = (x', x_n)$ for a generic point $x = (x_1, \dotsc, x_n) \in \mathbb{R}^n$, where $x' = (x_1, \dotsc, x_{n - 1})$. Let $\epsilon > 0$ and consider the map $\Phi_\epsilon \colon \mathbb{R}^n \to \mathbb{R}^n$ defined by \[ \Phi_\epsilon(x) = \left(\frac{x'}{1 + \epsilon \|x'\|^2}, x_n\right). \] This has the derivative $d\Phi_\epsilon(0, x_n) = \mathrm{id}_{\mathbb{R}^n}$ for any $x_n \in \mathbb{R}$. We have the convergence $\Phi_\epsilon \to \mathrm{id}_{\mathbb{R}^n}$ in $C^2(C; \mathbb{R}^n)$ for any compact set $C \subseteq \mathbb{R}^n$ as $\epsilon \to 0$. Moreover, for any $x, V \in \mathbb{R}^n$, unless $x' = 0$ or $V' = 0$, we find that $\|d\Phi_\epsilon(x)V\| < \|V\|$. Now choose $\hat{\gamma} \in \mathcal{G}$ with $\mathcal{K}_1(\hat{\gamma}) < \mathcal{K}_1(\gamma)$. Consider $\hat{\gamma}_\epsilon = \Phi_\epsilon \circ \hat{\gamma}$ for some $\epsilon > 0$ that remains to be determined. Then $\hat{\gamma}_\epsilon$ still satisfies the boundary conditions \eqref{eqn:boundary_conditions}. As $\gamma$ does not minimise $\mathcal{K}_1$ by the above assumption, we conclude that $\gamma([0, \ell]) \not\subseteq \{0\}^{n - 1} \times \mathbb{R}$. Hence $\|a_2 - a_1\| < \ell$ and $\hat{\gamma}([0, \ell])$ is not contained in $\{0\}^{n - 1} \times \mathbb{R}$ either. Therefore, the length of $\hat{\gamma}_\epsilon$ is strictly less than $\ell$. But $\hat{\gamma}_\epsilon \to \hat{\gamma}$ in $C^2([0, \ell])$ as $\epsilon \to 0$. Hence for some $\epsilon > 0$ small enough, we conclude that the curvature $\hat{\kappa}_\epsilon$ of $\hat{\gamma}_\epsilon$ satisfies $\\|\hat{\kappa}_\epsilon\\|_{L^\infty(0, \ell)} \le \mathcal{K}_1(\gamma) \le 1/R$. Hence we have found a shorter curve with the same boundary data satisfying the required curvature constraint. \end{proof} Now suppose that $n = 2$. We wish to give an alternative proof of Dubins's main result \cite[Theorem I]{Dubins:57} based on Theorem \ref{thm:structure}. Let $k > 0$ and consider a $1/k$-geodesic parametrised by $\gamma \in \mathcal{G}$. Then Proposition \ref{prop:shortest_curves} and Theorem \ref{thm:structure} imply that $\gamma$ is consistent with one of the descriptions \ref{item:arc-line-arc} or \ref{item:arc-arc} in the introduction. In the case \ref{item:arc-line-arc}, it is clear that any minimiser of the length will not contain any full circles, so the curve will at most consist of a circular arc, followed by a line segment, followed by another circular arc. This is one of the solutions described by Dubins. In the case \ref{item:arc-arc}, we have a sequence of several circular arcs. If there were more than four pieces, then it is also easy to see that a piece of the curve could be replaced by a line segment, thus reducing the length. This is of course impossible for a minimiser of the length, hence we have four or fewer pieces. In order to see that four consecutive circular arcs are also impossible, we still need Dubins's Lemma 2. Almost all of Dubins's other arguments, however, have been bypassed. Sussmann's results for $n = 3$ \cite[Theorem 1]{Sussmann:95} follow in a similar way from Theorem \ref{thm:structure} and again one of Dubins's lemmas. If we have a solution as in statement \ref{item:2D}, then we first distinguish the following two cases. If the entire curve is planar, we apply the above reasoning. (Sussmann's theorem contains another statement in this case, which is a consequence of a result of Dubins \cite[Sublemma]{Dubins:57}.) Otherwise, we note that the curve must meet the line $\mathcal{L}$ tangentially. Then we may have a circular arc at either end of the curve and we may have some intermediate pieces. But if one of these intermediate pieces is not a segment of $\mathcal{L}$, it is clear that it must be a full circle. This clearly cannot happen for a solution of the Markov-Dubins problem, so in fact we have (at most) a concatenation of a circular arc, a line, and another circular arc. A solution as in statement \ref{item:3D}, on the other hand, is a helicoidal arc in Sussmann's terminology. \section{Examples} \label{sect:examples} We finally examine a few examples of minimisers and $\infty$-elasticas, which highlight some features and some limitations of the theory. Throughout this section, we assume that $\alpha \equiv 1$. \begin{example}[Circular arc] \label{ex:arc} We first consider a circular arc parametrised by $\gamma \colon [0, \ell] \to \mathbb{R}^2$ with $\gamma(s) = r(\cos(s/r), \sin(s/r))$ and with tangent vector $T(s) = (-\sin(s/r), \cos(s/r))$ and constant curvature $k = 1/r$. This is an $\infty$-elastica by Theorem \ref{thm:structure}. If we want to check equations \eqref{eqn:ODE1} and \eqref{eqn:ODE2} directly, then we first compute $T'' + k^2 T = 0$. Moreover, the vectors $T$ and $T'$ span $\mathbb{R}^2$ everywhere, so $\proj_{T, T'}^\perp(\lambda) = 0$ regardless of the value of $\lambda$. Thus we only need to consider equation \eqref{eqn:ODE2}, which gives $g'(s) = -\frac{1}{r}(\lambda_1 \cos(s/r) + \lambda_2 \sin(s/r))$. This is satisfied for $g(s) = \lambda_2 \cos(s/r) - \lambda_1 \sin(s/r) + h = \lambda \cdot T(s) + h$ for any $h \in \mathbb{R}$. Clearly we can choose $h$ such that $g \ge 0$ in $[0, \ell]$. Now suppose that we wish to apply Theorem \ref{thm:minimiser}. We have a minimiser of $\mathcal{K}_1$ if the inequalities $0 \le \lambda \cdot T + h \le -\lambda \cdot T$ are satisfied simultaneously. They give rise to the conditions \[ \frac{h}{2} \le \min_{[0, \ell]} (-\lambda \cdot T) \le \max_{[0, \ell]} (-\lambda \cdot T) \le h. \] It is possible to satisfy these if, and only if, $\ell \le 2\pi r/3$, in which case we can choose $\lambda = (\sqrt{3}/2, -1/2)$ and $h = 1$. Thus a circular arc of radius $r$ minimises $\mathcal{K}_1$ if its length does not exceed $2\pi r/3$. \end{example} The example shows that the condition of Theorem \ref{thm:minimiser} is sufficient but not necessary, for the above circular arc is still a minimiser as long as $\ell \le 2\pi r$ by the results of Schmidt \cite{Schmidt:25}. Next we consider the question whether the notion of an $\infty$-elastica is genuinely more general than that of a minimiser of $\mathcal{K}_\alpha$. The answer is yes, and the following example gives a one-parameter family of $\infty$-elasticas that are not minimisers and not even local minimisers with respect to the $W^{1, 2}$-topology. \begin{example}[Non-minimising $\infty$-elastica] \label{ex:non-minimising} Consider curves with end points $a_1 = (-1, 0)$ and $a_2 = (1, 0)$ and tangent vectors $T_1 = (0, 1)$ and $T_2 = (0, -1)$. If $\ell = \pi$, then there is one candidate that consists of three semicircles of radius $1/3$; this is illustrated in Figure \ref{fig:non-min}. It is an $\infty$-elastica by Theorem \ref{thm:structure}. \begin{figure} \caption{The $\infty$-elastica} \caption{A comparison curve} \caption{Construction of an $\infty$-elastica that is not a minimiser} \label{fig:non-min} \label{fig:comparison} \end{figure} For $r \in [1/3, 1)$, we also construct some comparison curves including three circular arcs of radius $r$. To this end, define $\omega(r) = \arccos((1 - r)/2r)$. For $h \in \mathbb{R}$, there is a curve comprising three circular arcs of radius $r$, with centres \[ (r - 1, h), \quad (0, h + 2r\sin \omega(r)), \quad (1 - r, h), \] that connects the points $(-1, h)$ and $(1, h)$. The length of this curve is $\tilde{\ell}(r) = r(3\pi - 4\omega(r))$. We compute $\tilde{\ell}(1/3) = \pi = \tilde{\ell}(1)$ and \[ \tilde{\ell}''(r) = \frac{4(1 - r)}{r(3r^2 + 2r - 1)^{3/2}} > 0 \] in $(1/3, 1)$. Hence $\tilde{\ell}(r)< \pi$ for all $r \in (1/3, 1)$. If we choose $h = (\pi - \tilde{\ell}(r))/2$, we can attach a line segment to each end and thereby construct a comparison curve of length $\pi$ that satisfies the required boundary conditions (see Figure \ref{fig:comparison}). But the value of $\mathcal{K}_1$ is $1/r < 3$. \end{example} Finally we have an example of a three-dimensional $\infty$-elastica, showing that both cases in Theorem \ref{thm:structure} can indeed occur. \begin{example}[Helical arc] \label{ex:helix} Consider $\gamma \colon [0, \ell] \to \mathbb{R}^3$ given by \[ \gamma(s) = (r\cos \omega \cos(s/r), r\cos \omega \sin(s/r), s\sin \omega) \] for some $\omega \in (0, \pi/2)$. The curvature of this curve is $k = r^{-1} \cos \omega$. For $T = \gamma'$, we compute \[ T'' + k^2 T = \frac{\sin \omega \cos \omega}{r^2}(\sin \omega \sin (s/r), - \sin \omega \cos(s/r), \cos \omega). \] Now let $\lambda = (0, 0, 1)$. Then $\lambda \cdot T = \sin \omega$. In order to find $\proj_{T, T'}^\perp(\lambda)$, we first compute \[ N = \frac{r}{\cos \omega} T \times T' = (\sin \omega \sin(s/r), -\sin \omega \cos(s/r), \cos \omega) \] and note that $N$ is a unit vector perpendicular to $T$ and $T'$. Hence \[ \proj_{T, T'}(\lambda) = (\lambda \cdot N)N = \cos \omega (\sin \omega \sin(s/r), -\sin \omega \cos(s/r), \cos \omega). \] Choosing $\eta = \sin \omega - \cos \omega \cot \omega$, we see that equation \eqref{eqn:ODE_alpha=1} is satisfied. Hence $\gamma$ is an $\infty$-elastica. \end{example} \def$'${$'$} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } \providecommand{\MRhref}[2]{ \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \end{document}	arXiv
Genome wide association study and genomic prediction for fatty acid composition in Chinese Simmental beef cattle using high density SNP array Bo Zhu1, Hong Niu1, Wengang Zhang1, Zezhao Wang1, Yonghu Liang1, Long Guan1, Peng Guo1, Yan Chen1, Lupei Zhang1, Yong Guo2, Heming Ni2, Xue Gao1, Huijiang Gao1, Lingyang Xu1,3 & Junya Li1,3 Fatty acid composition of muscle is an important trait contributing to meat quality. Recently, genome-wide association study (GWAS) has been extensively used to explore the molecular mechanism underlying important traits in cattle. In this study, we performed GWAS using high density SNP array to analyze the association between SNPs and fatty acids and evaluated the accuracy of genomic prediction for fatty acids in Chinese Simmental cattle. Using the BayesB method, we identified 35 and 7 regions in Chinese Simmental cattle that displayed significant associations with individual fatty acids and fatty acid groups, respectively. We further obtained several candidate genes which may be involved in fatty acid biosynthesis including elongation of very long chain fatty acids protein 5 (ELOVL5), fatty acid synthase (FASN), caspase 2 (CASP2) and thyroglobulin (TG). Specifically, we obtained strong evidence of association signals for one SNP located at 51.3 Mb for FASN using Genome-wide Rapid Association Mixed Model and Regression-Genomic Control (GRAMMAR-GC) approaches. Also, region-based association test identified multiple SNPs within FASN and ELOVL5 for C14:0. In addition, our result revealed that the effectiveness of genomic prediction for fatty acid composition using BayesB was slightly superior over GBLUP in Chinese Simmental cattle. We identified several significantly associated regions and loci which can be considered as potential candidate markers for genomics-assisted breeding programs. Using multiple methods, our results revealed that FASN and ELOVL5 are associated with fatty acids with strong evidence. Our finding also suggested that it is feasible to perform genomic selection for fatty acids in Chinese Simmental cattle. Fatty acids are required by daily normal metabolism, and can be obtained from food and meat. Improving nutritional value of meat products for human health has attracted extensive attention in current society [1, 2]. Fat content and fatty acid composition in beef products are associated with meat taste and flavor, and these are considered as main sensory properties in consumer's selection and acceptance [3]. Fatty acids are important indicators of beef meat quality, and previous studies have been conducted to examine fatty acids for various cattle breeds in different feeding environments [4]. Fatty acid composition are often lowly or moderately heritable traits in various populations with different genetic architecture [5]. Several studies have revealed that the level of heritability and genetic correlation theoretically allow for genetic improvement of fatty acid composition by selection of both major candidate genes and genomic selection strategies [6,7,8,9,10]. Therefore, application of molecular genetics approaches can provide an opportunity for genetic improvement for fatty acid composition of beef cattle [11,12,13]. During the last decades, tremendous works have been done to elucidate the genetic mechanism of fatty acids using candidate gene [14,15,16,17,18,19,20,21] and linkage mapping approaches [22,23,24]. In recent years, genome-wide association studies (GWAS) have been widely used to study the molecular mechanism underlying important traits in beef and dairy cattle [25,26,27,28,29]. Previous GWAS and genomic predictions have identified candidate markers associated with various fatty acid composition and evaluated the accuracy of genomic prediction for these traits [8, 30,31,32,33,34]. However, many studies were conducted in populations with relatively low density SNP arrays. Despite recent GWAS for fatty acid composition have been investigated using the high density BovineHD (770 K) SNP array, those studies were mostly limited in Nellore cattle [35, 36]. On the other hand, extensive attention has been paid to investigate the accuracies of genomic prediction using multiple methods in different populations [32, 33]. A recent study in Nellore cattle indicated that the accuracies of genomic prediction were moderate to high and it was feasible to apply genomic selection in cattle. However, their results were limited to carcass traits in Nellore population [37]. Therefore, understanding the molecular mechanisms underlying fatty acid composition and evaluating the accuracy of genomic predictions in other important cattle breeds still need further investigation. The objectives of the current study were to explore the associated genomic regions and estimate the predictive accuracies for fatty acid composition using the BovineHD SNP array in Chinese Simmental population. In this study, we identified several potential candidate markers and genes associated with fatty acid composition. Our findings will facilitate the elucidation of the molecular mechanism and help us design optimal genomic selection strategies for fatty acid composition in cattle and other farm animals. Descriptive statistics of fatty acid composition and their estimates of heritability We measured six saturated fatty acids (SFA), four monounsaturated fatty acids (MUFA) and eleven polyunsaturated fatty acids (PUFA) using gas chromatography. Descriptive statistics and estimates of heritability for 21 individual fatty acids were presented in Table 1. We observed that the most abundant individual saturated fatty acids were C16:0 (23.8%) and C18:0 (20.2%), while for monounsaturated and polyunsaturated fatty acids, relatively high proportions of individual fatty acids were C18:1 cis-9 (32.0%) and C18:2 n-6 (12.9%). In contrast, we found saturated fatty acids (C20:0, C22:0, C24:0), monounsaturated fatty acids (C14:1 cis-9 and C20:1 cis-11), and polyunsaturated fatty acids (C18:2 t-9c-11, C18:2 t-12c-10, C18:3 n-6, C18:3 n-3, C20:2 n-6, C20:4 n-6, C20:5 n-3, C22:5 n-3, C22:6 n-3) accounted for relatively low proportion (<1% each) of the total fatty acids. In this study, our results found the estimated heritability varied noticeably for these fatty acids. Among 21 individual fatty acids, we found only C14:0 showed a relatively high heritability at 0.54 and five fatty acids including C18:0, C20:0, C14:1 cis-9, C16:1 cis-9 and C18:1 cis-9 showed a moderate heritability, while most of heritability estimates for others fatty acids (15 out of 21) were below 0.2 (Table 1). For the eight groups of fatty acids, we found MUFA, n-6/n-3 and health index (HI) showed moderate heritabilities (0.27, 0.22 and 0.24), while the estimated heritability for SFA, PUFA, total of omega 3 (n-3), total of omega 6 (n-6) were 0.12, 0.16, 0.15, and 0.16, respectively. Table 1 Summary statistics of mean (%), standard deviation (SD, %) and heritability estimates (h 2), additive genetic variance and coefficient of variation (CV%) Phenotypic and genetic correlations Phenotypic and genetic correlations among 21 individual fatty acids were presented in Fig. 1. The estimated phenotype correlation of these fatty acids (Fig. 1a) generally displayed different patterns compared to genetic counterparts (Fig. 1b). We observed that high positive phenotypic and genetic correlations existed between several pairs of individual fatty acids. For instance, the estimated genetic correlations between C20:0 and C20:1 cis-11, C20:0 and C18:2 t-12c-10, C20:0 and C20:2 n-6 and C20:0 and C20:4 n-6 were 0.89, 0.95, 0.92 and 0.86, respectively. In contrast, we found clear negative correlations between C14:0 and C18:2 n-6, C14:0 and C20:3 n-3, C18:1 cis-9 and C18:2 n-6, C18:1 cis-9 and C20:5 n-3 (Additional file 1: Table S1). Heatmap of phenotypic (a) and genetic correlation (b) across 21 individual fatty acid compositions Bayesian based GWAS and candidate regions We performed GWAS using the BayesB method for 11 individual fatty acids that showed estimated genomic heritability ≥ 0.10, including 3 saturated fatty acids (C14:0, C18:0 and C20:0), 3 monounsaturated fatty acids (C14:1 cis-9, C16:1 cis-9 and C18:1 cis-9) and 5 polyunsaturated fatty acids (C18:2 n-6, C18:3 n-3, C20:3 n-3, C20:5 n-3 and C22:5 n-3). To identify potential regions associated with fatty acids, we divided the genome into 100 kb windows, leading to 24,900 regions across the genome. Regions that explain more than 1% of additive genetic variances were considered as candidates and subject to further analyses to identify the associated genes within these regions. Summary statistics including genetic variances explained, position for the 100 kb windows, flanking rs number ID for these regions, and candidate genes of saturated fatty acid, monounsaturated fatty acid and polyunsaturated fatty acids were represented in Tables 2 and 3, respectively. Table 2 Genomic regions associated with the saturated fatty acids in Chinese Simmental cattle using BayesB method Table 3 Genomic regions associated with the monounsaturated and polyunsaturated fatty acids in Chinese Simmental cattle using BayesB method Saturated fatty acids We detected a total of 16 candidate regions that explain more than 1% of genetic variance for saturated fatty acids. These regions were distributed on BTA2, BTA4, BTA6, BTA7, BTA12, BTA14, BTA15, BTA19, BTA22, BTA23 and BTA25 (Table 2). Among them, we found four, three and nine candidate regions for C14:0, C18:0 and C20:0, respectively (Additional file 2). Intriguingly, the detected window with the largest genetic variance (10.04%) near 51.3 Mb on BTA19 for C14:0, containing gene fatty acid synthase (FASN) that is related to fatty acid synthesis. We also found one region explaining about 1.46% of genetic variance for C14:0 and located at 25.1 Mb on BTA23. This region overlapped with the elongation of very long chain fatty acids protein 5 (ELOVL5) whose function involves in fatty acid elongase activity (Fig. 2). In addition, we identified 10 candidate genes that are likely to be related to fatty acids composition embedded in these candidate regions (Table 2). a Manhattan plot of absolute value of SNP effects estimated using BayesB for C14:0. b Manhattan plots showing P-values of association for each SNP using the GRAMMAR-GC, where the y-axis was defined as -Log 10 (P) Monounsaturated fatty acids For monounsaturated fatty acids, we detected 8 candidate regions, each of which captured more than 1% of the total genetic variance across six chromosomes (Additional file 3). Notably, we detected the same region at 51.3 Mb on BTA19 overlapping with FASN, which explains 6.49% of the genetic variance for C14:1 cis-9 (Fig. 3), and this region was also associated with C14:0. The top associated region was detected near 30.3 Mb on BTA14, which explains 19.44% of genetic variance for C18:1 cis-9 (Fig. 4), while no known gene was observed near this region. Overall, there are three, two and three regions identified to be associated with C14:1 cis-9, C16:1 cis-9 and C18:1 cis-9, respectively (Table 3). a Manhattan plot of absolute values of SNP effects estimated using BayesB for C14:1 cis-9. b Manhattan plots showing P-values of association for each SNP using the GRAMMAR-GC, where the y-axis was defined as -Log 10 (P) a Manhattan plot of absolute values of SNP effects estimated using BayesB for C18:1 cis-9. b Manhattan plots showing P-values of association for each SNP using GRAMMAR-GC, where the y-axis was defined as -Log 10 (P) Polyunsaturated fatty acids We detected 11 associated regions for five polyunsaturated fatty acids, including C18:2 n-6, C18:3 n-3, C20:3 n-3, C20:5 n-3 and C22:5 n-3 (Additional file 4). Of these regions, we observed one candidate region for C18:3 n-3 that was located at BTA14, two regions associated with C18:2 n-6 at BTA4 and BTA20, two regions associated with C20:3 n-3 at BTA4 and BTA17, three regions associated with C20:5 n-3 at BTA3, 5, 12, and three regions associated with C22:5 n-3 at BTA2, 5, 9 respectively (Table 3). Notably, we found 12 genes imbedded in the identified regions and these genes were likely involved in the function of fatty acid synthesis and metabolism. Of these 12 genes, two genes were associated with C18:2 n-6, three genes with C20:3 n-3, six genes with C20:5 n-3, and one gene with C22:5 n-3, respectively. The top four regions explaining more than 2% of the genetic variance were identified at BTA20 associated with C18:2 n-6 (4.95%), BTA14 with C18:3 n-3 (2.83%), BTA9 with C22:5 n-3 (2.46%) and BTA4 with C18:2 n-6 (2.33%). Fatty acid groups To systematically explore the genetic mechanism underlying fatty acid composition beyond individual fatty acid, we also conducted GWAS using BayesB method on eight fatty acid groups, including SFA, MUFA, PUFA, PUFA/SFA, n-3, n-6, n-6/n-3, and HI (Additional file 5). We found three associated regions for HI located at BTA4, BTA10 and BTA15, two regions for MUFA at BTA12 and BTA14, one for n-3 at BTA4 and one for n-6/n-3 at BTA20 (Table 4). Intriguingly, we observed two regions accounting for ~10% of the genetic variance for MUFA, while two regions located at BTA4 and BTA20 accounting for 1.25% and 4.66% of genetic variance for n-3 and n-6/n-3, respectively. One region located at BTA12 overlapped with two genes claudin 10 (CLDN10) and DAZ interacting zinc finger protein 1 (DZIP1). For health index (HI), the genetic variances contributed by three identified regions were 1.52%, 1.19% and 2.03%, which located at BTA4, BTA10 and BTA15, respectively. These regions overlapped with sarcoglycan epsilon (SGCE), paternally expressed 10 (PEG10) and DEAD-box helicase 10 (DDX10). Table 4 Genomic regions associated with the fatty acid groups in Chinese Simmental cattle using BayesB method Identification of associated loci using GRAMMAR-GC We further conducted GWAS using GRAMMAR-GC implemented in GenABEL package for 11 individual fatty acids and eight fatty acid groups, each of which has a genomic heritability of high than 0.10. To ensure the power and accuracy of GWAS for these traits, we utilized genomic control approach to correct for possible population stratifications in GRAMMAR-GC test. After this correction, we found the inflation factor λ was close to one, suggesting that our approach has successfully accounted for population stratification, and thus no further adjustment was required. We identified a total of 44 and 8 significant SNPs associated with nine fatty acid composition and two fatty acid groups, respectively. The suggestive P value (0.05/163,473 = 3.06E-7) was used as the cut off threshold for significance, which approximately considered the number of "independent" SNPs by counting 1 SNP per LD block, plus all SNPs outside of blocks (interblock SNPs). We observed 14, 5, 8, 1, 3, 4, 3, 3 and 3 significant associated SNPs surpassing the suggestive threshold (P <3.06E-7) for C14:0, C14:1 cis-9, C18:1 cis-9, C18:3 n-6, C20:0, C20:1 cis-11, C20:2 n-6, C20:4 n-6 and C18:2 t-9c-11, respectively. The top four significant SNPs for C14:0 (P =1.39E-10) were located at 51.3 Mb on BTA19. Totally, we identified 17 associated SNPs for saturated fatty acid (C14:0 and C20:0), 17 SNPs for monounsaturated fatty acids (C14:1 cis-9, C18:1 cis-9 and C20:1 cis-11), 10 SNPs for polyunsaturated fatty acids (C18:3 n-6, C20:2 n-6, C20:4 n-6 and C18:2 t-9c-11). Notably, we found the majority of SNPs were located at BTA19 (18 SNPs) and BTA14 (19 SNPs), which indicated these regions are potential candidate for fatty acid composition. Fig. 2, 3 and 4 show the genome-wide plots of C14:0, C14:1 cis-9 and C18:1 cis-9 for P-values and the absolute values of marker effects against the genomic position. We found one associated SNP located at 51.3 Mb on BTA19 for both C14:0 and C14:1 cis-9 (P = 5.19E-10 and P = 1.82E-07), and this SNP was also located at ~4 kb upstream of the FASN gene. Region-based association test and LD analyses To explore potential associated loci which might fail to be identified due to the strict threshold for high density SNPs, we investigated two 100 kb associated regions on BTA19 and BTA23 (BTA19:51.3–51.4 Mb and BTA23: 25.1–25.2 Mb) using region-based association tests implemented in R package FREGAT. The two regions contain two candidate genes FASN and EVOL5 involved in fatty acid synthesis. For the region at 51.3 Mb on BTA19, we found 19 SNPs showing significant association with C14:0 (P < 0.01), and among them, five SNPs were identified within FASN, and one SNP near FASN with the strongest association signal (P = 5.17E-10). We found that the P value for region-based test for FASN was 0.0048, which indicated that FASN may be considered as a candidate gene for C14:0. Moreover, the LD and haploblock analyses revealed that this region showed high LD level with multiple haploblocks, which may imply a potential selection signature involved in fatty acids within this region in Chinese Simmental cattle population (Fig. 5a). For region at 25.1 Mb on BTA23, two SNPs were detected with P < 0.01, and the top SNP (BovineHD2300006955) was detected at 25.1 Mb showing significant association (P = 3.1E-5). This region partly overlapped with gene ELVOL5. Therefore, we next examined the 500 kb upstream and downstream of the region. However, no other SNPs were found which were significantly associated with C14:0 (Fig. 5b). Regional plots of the two major candidate regions on BTA19 and BTA23. Results were shown for C14:0 at 50.8-51.8 Mb on BTA19 (a) and for C20:0 around 24.6-25.6 Mb on BTA23 (b). In the upper panels, the top SNPs were highlighted by blue solid circles. Different levels of linkage disequilibrium (LD) between the lead SNP and surrounding SNPs were indicated in different colors Genomic prediction for fatty acid composition We performed genomic selection for fatty acid composition using GBLUP and BayesB. The predictive accuracies ranged from 0.03 (C18:2 t-12c-10) to 0.51 (C18:3 n-3) using GBLUP, and from 0.1 (C18:2 t-9c-11) to 0.53 (C14:0) using BayesB. The averaged predictive accuracies across all fatty acids using GBLUP and BayesB were 0.24 and 0.29, respectively. These results suggested that genomic prediction using BayesB was slightly superior over GBLUP for fatty acids in Chinese Simmental cattle. For each individual fatty acid trait, we found that the relatively high predictive accuracies were achieved for C14:0, C22:0, C14:1 cis-9, C18:3 n-3, C20:3 n-3 for both GBLUP and BayesB. Despite the fact that BayesB performed at least as well as GBLUP for most traits, we indeed found some traits where BayesB showed much higher predictive accuracies than GBLUP, such as C14:0 (0.48 for GBLUP and 0.53 for BayesB), C18:0 (0.17 vs. 0.24), C24:0 (0.11 vs. 0.23), C18:2 t-12c-10 (0.03 vs. 0.12), C18:3 n-6 (0.05 vs. 0.13), C20:2 n-6 (0.07 vs. 0.14) (Table 5). To investigate possible bias between the predicated and observed breeding values, we also calculated the regression coefficient of genomic estimated breeding values on adjusted phenotypes (Table 5). Our results revealed that the average regression coefficients for GBLUP and BayesB were 0.71 and 0.93, indicating that BayesB is less biased than GBLUP because the latter has a regression coefficient closer to unity. Table 5 Predictive accuracy (±SE) and regression coefficients (±SE) of genomic breeding value prediction for fatty acid composition Fatty acid is an important indicator of meat quality and taste, and its strongly influences consumer's preferences [3, 38]. Previous genome-wide association studies have been conducted for fatty acid composition in multiple cattle breeds, including Angus, Japanese Black, Nellore and other crossbreds [8, 30, 32, 35, 36]. To our knowledge, this study is the first attempt to investigate molecular mechanism underling fatty acid composition using high density SNP array and evaluate the accuracy of genomic predictions in Chinese Simmental cattle population. Genomic wide scan identified candidate regions and loci We investigated 21 individual fatty acids including 6 saturated fatty acids, 4 monounsaturated fatty acids, 11 polyunsaturated fatty acids and 8 fatty acid groups. Despite the fact that single-SNP based GWAS methods have been widely used in many studies, these methods may not be powerful for studying the complex traits with low or moderate heritability. Due to the polygenic characteristics of fatty acid composition in cattle, GWAS using the Bayesian methods have enabled to identify many associated loci that have missed by the single-SNP regression approach [33, 35, 36]. BayesB has been widely used for GWAS of complex trait in farm animals [39,40,41,42]. In this study, we utilized BayesB and GRAMMAR-GC to identify candidate regions associated with fatty acids in Chinese Simmental cattle population. We detected a total of 35 candidate regions on 16 autosomes associated with fatty acid composition using BayesB. However, these identified regions may not include all potentially significant associated SNPs due to use of 100 kb window-based strategy in BayesB. Therefore, we conducted GWAS using the single locus GRAMMAR-GC method implemented in GenABLE package. With this approach, we detected a total of 44 and 8 significant associated SNPs for individual fatty acids and fatty acid groups using a suggestive adjust threshold. The suggested threshold was set to avoid overestimation of the significant SNPs caused by high LD level in the high density SNPs array [43]. In current study, we found a total of 24 candidate SNPs overlapping with these regions identified by BayesB. For instance, the same candidate peaks for C14:1, C14:1 cis-9 and C18:1 cis-9 were identified using both methods (Figs. 2–4). Utilization of multiple complementary methods is an effective way to detect candidate regions or SNPs and helps elucidate genetic architecture of complex traits in farm animals [33, 44]. Candidate genes for fatty acid composition Several genes were identified as potential candidates contributing to the genetic architecture of fatty acids in this study. Among them, we observed FASN at 51 Mb on BTA19 overlapped with a 100 kb associated region, which explains 10% and 6.5% of the genetic variances for C14:0 and C14:1 cis-9, respectively. Notably, we also found multiple significant SNPs around this gene that were associated with C14:0 using the GRAMMAR-GC method. A region-based association test revealed strong evidence of association for multiple SNPs within this gene. Furthermore, we found strong LD at the upstream of FASN (Fig. 5). This gene encodes a multifunctional protein enzyme to catalyze the synthesis of palmitate (C16:0) from acetyl-CoA and malonyl-CoA, in the presence of NADPH. Previous studies based on candidate gene approaches had revealed polymorphisms within FASN that were related to fatty acid composition in multiple beef cattle populations [45,46,47,48,49] and milk fat content in dairy cattle [50, 51]. For instance, several studies were conducted to explore and evaluate the association between fatty acid composition and candidate SNPs using GWAS in Japanese beef cattle [8, 30, 52]. These studies provided multiple evidences that several SNPs near or within FASN may be regarded as responsible mutations for fatty acid composition and contribute largely to meat quality in the Japanese Black cattle population. Saatchi et al performed GWAS using BovineSNP50 in Angus beef cattle and reported FASN located at 51 Mb on BTA19 (from 51,384,922 to 51,403,614 bp) was associated with fatty acids [32]. Chen et al. found SNP rs41921177 (BTA19:51,326,750) located near FASN. This SNP rs41921177 had relative large effects on multiple fatty acids in both subcutaneous adipose and longissimus lumborum muscle tissues of crossbred beef cattle [33]. However, investigation of genetic architecture of fatty acids in the Nellore cattle showed no significant associations for several polymorphisms within or near FASN [35, 36]. Despite previous studies have indicated that FASN has a conserved role across genetic backgrounds, there are several different variants that may be responsible for the different FASN effects in different breeds, and different FASN alleles appear to be segregating in different populations [8, 49]. Another gene called ELOVL5 encodes a multi-pass membrane protein which is involved in the elongation of long-chain polyunsaturated fatty acids. This gene was identified in the associated region at 25.1 Mb on BTA23 accounting for 1.5% of the genetic variance for C14:1 cis-9. ELOVL5 plays an important role in de novo synthesis of specific MUFA species in mammalian cells, ELOVL5 knockdown decreased the elongation of C16:1 cis-9, n-7, and ELOVL5 overexpression increased synthesis of C18:1 cis-9, n-7 [53]. Also, previous study using mice models revealed that a reduced ELOVL5 activity can lead to hepatic steatosis, and endogenously synthesized PUFAs are critical regulators of fatty acid synthesis [54]. Lemos et al. reported a candidate region embedded in ELOVL5 can explain 4% of the genetic variance for C20:4 n-6 using ssGBLUP based on window association test in Nellore cattle [35]. The consistent role of ELOVL5 gene involved in fatty acid synthesis and composition was also extensively investigated in diverse pig populations [55,56,57]. Moreover, previous studies suggested that ELOVL5 are involved in the production of multiple acids including C16:0, C16:1, C18:0 and C18:1 in cattle [58]. Also, ELOVL5 was found associated with C20:1n9/C18:1n9 and C20:2n6/C18:2n6 in a F2 population derived from Erhulian pig [55]. As a result, ELOVL5 may have pleiotropic effects on multiple fatty acid composition and also appear to exhibit pleiotropic effects in multiple metabolic steps. However, we only identified ELOVL5 that was associated with C14:0 in Chinese Simmental cattle. In addition, several previous studies have suggested variants within SCD gene and the expression level of SCD gene should be significantly associated with fatty acids [18, 20, 30, 32, 47, 48, 59, 60], the SCD gene was not detected in this study, probably due to heterogeneous genetic architecture of fatty acids differ across different populations. Previous studies have investigated predictive abilities of genomic selection for fatty acid composition in American Angus [32], Japanese Black cattle [61] and Canadian beef cattle [33]. In the current study, we explored, for the first time, genomic prediction for fatty acid composition in Chinese Simmental cattle. We found that the accuracies of genomic prediction for most of fatty acids were relatively low (<0.30) using both GBLUP and BayesB, which was consistent with a previous report by Chen et al. [33]. This finding may be explained by the relatively low and inaccurate estimates of heritability for the measured fatty acid composition [62]. Our studies also revealed that BayesB provided slightly higher average regression coefficients as compared to GBLUP. Considering the complex architecture of fatty acid composition, this finding implied that BayesB which allows a fraction of SNPs to be allocated with relatively large effects is superior over GBLUP which assumes the same genetic variance for each SNP. Fatty acid composition are commonly recognized as complex traits with a polygenic nature and, to some extent, they are difficult to measure, thus the application of genomic selection for fatty acids will be valuable in future selection breeding programs. With increasing public understanding of the relationships between diet and health, much attention should be paid to the studies of some important fatty acids related to human health [63]. As consumer become more health conscious, they have increased preference for better tasting and healthier products in their diet such as unsaturated fatty acid levels. Further investigation of causal mutations will promote our understanding of lipid metabolism, fat deposition and application of selection for fatty acids in cattle. We identified several significant associated regions and loci as the potential candidate markers for genomics-assisted breeding programs. Using multiple methods, our results revealed that FASN and ELOVL5 associated with fatty acids with strong evidences. Our analyses also suggested that it is feasible to perform genomic selection for fatty acids in the Chinese Simmental cattle population. All animals used in the study were treated following the guidelines established by the Council of China Animal Welfare. Protocols of the experiments were approved by the Science Research Department of the Institute of Animal Sciences, Chinese Academy of Agricultural Sciences (CAAS) (Beijing, China). Animals and phenotypes A total of 723cattleborn between 2010 and 2013 were used in this study, and these cattle were originated from Ulgai, Xilingol League and Inner Mongolia of China. After weaning, the cattle were moved to JinweifurenCo.,Ltd for fattening, all animals sharing the same feeding and management conditions. More detailed description of breeding and management has been described previously [64, 65]. The cattle were slaughtered at an average of 20 months. During the period of slaughtering, we measured traits in strict accordance with the guidelines proposed by the Institutional Meat Purchase Specifications for fresh beef. Meat samples were removed from the longissimus lumborum (LL) muscleafter stored for 48 h between the 12th and 13th ribsfrom each animal, and then samples were vacuum packed and chilled at -80 °C. And approximate 10 g of sample were taken for subsequent fatty acid analyses. Total lipid was extracted from the sample according to protocols described previously [66]. About 2 mg extracted lipid was re-dissolved in 2 ml of n-hexane and 1 ml of KOH (0.4 M) for saponification and methylation. A total of 21 individual fatty acid composition were measured using gas chromatography (GC-2014 CAFsc, Shimadzu Scientific Instruments) including six saturated fatty acids, four monousaturated fatty acids, and eleven polyunsaturated fatty acids. Each fatty acid was quantified as a weight of percentage of total fatty acids. In addition, fatty acids were indexed as groups of saturated, monounsaturated, polyunsaturated fatty acid, total of saturated fatty acid (SFA), total monounsaturated (MUFA), total of polyunsaturated (PUFA), total of omega 3 (n-3) and total of omega 6 (n-6). The calculation of various fatty acid groups are described as follows: SFA = C14:0 + C16:0 + C18:0 + C20:0 + C22:0 + C24:0; MUFA = C14:1 cis-9 + C16:1 cis-9 + C18:1 cis-9 + C20:1 cis-11; PUFA = C18:2 n-6 + C18:2 t-9c-11 + C18:2 t-12c-10 + C18:3 n-6 + C18:3 n-3+ C20:2 n-6 + C20:3 n-3 + C20:4 n-6 + C20:5 n-3 + C22:5 n-3 + C22:6 n-3; n-3 = C18:3 n-3 + C20:3 n-3 + C20:5 n-3 + C22:5 n-3 + C22:6 n-3; n-6 = C18:2 n-6 + C18:3 n-6 + C20:2 n-6 + C20:4 n-6; PUFA/SFA: ratio between PUFA and SFA; n-6/n-3: ratio between n-6 and n-3; HI = (MUFA + PUFA)/(4 × C14:0 + C16:0). Genotyping and quality control In total, 723 Simmental cattle were genotyped for the Illumina BovineHD BeadChip. Before statistical analysis, SNPs were pre-processed using PLINK v1.07 [67]. SNPs were selected based on minor allele frequency (>0.05), proportion of missing genotypes (<0.05), and Hardy-Weinberg equilibrium (P > 10E-6). Moreover, individuals with more than 10% missing genotypes were excluded. After these quality controls, the final data consisted of 685 individuals and 595,715 autosomal SNPs. Heritability and genetic correlation estimation Phenotypic and genetic (co) variances of fatty acids were estimated using the pairwise bivariate animal model implemented in the ASReml v3.0 software package [68]. The model is $$ \left[\begin{array}{c}\hfill {y}_1\hfill \\ {}\hfill {y}_2\hfill \end{array}\right]\kern0.5em =\kern0.5em \left[\begin{array}{cc}\hfill {X}_1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {X}_2\hfill \end{array}\right]\left[\begin{array}{c}\hfill {b}_1\hfill \\ {}\hfill {b}_2\hfill \end{array}\right]+\kern0.5em \left[\begin{array}{cc}\hfill {Z}_1\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {Z}_2\hfill \end{array}\right]\left[\begin{array}{c}\hfill {a}_1\hfill \\ {}\hfill {a}_2\hfill \end{array}\right]\kern0.5em +\left[\begin{array}{c}\hfill {\mathbf{e}}_{\mathbf{1}}\hfill \\ {}\hfill {\mathbf{e}}_{\mathbf{2}}\hfill \end{array}\right] $$ where y1 and y2 are vectors of phenotypic values of trait 1 and 2, respectively; ×1 and × 2 are incidence matrices for fixed effects; b 1 and b 2 are the vectors of the fixed effects; Z1 and Z2 are incidence matrices relating the phenotypic observations to vectors of the polygenic (a) effects for two traits; e 1 and e 2 are random residuals for two traits. The polygenic effects were treated as random and assumed to be mutually independent. Variances of the random effects are defined as V(a) = G σ a 2 for the polygenes and V(e) = I σ e 2 for the residuals, where G is the additive genetic relationship matrix, I is the identity matrix, σ a 2 is the additive genetic variance and σ e 2 is the residual variance. Matrix G matrix was inferred from the SNP markers according to the study of VanRaden [69]. Fixed effects in the model included effects of gender, farm and year. In addition, ages at slaughter, days between slaughter and fatty acid extraction, hot carcass weight and marbling score were considered as covariates in the model. Genomic heritability of each trait was estimated using $$ {h}^2={\sigma}_a^2/\left({\sigma}_a^2,+,{\sigma}_e^2\right) $$ Pairwise bivariate analyses were performed for each combination of fatty acids to estimate the (co) variance components, phenotypic and genetic correlations as well as the heritability. Genome-wide association study using BayesB Fatty acid composition was adjusted for fixed effects and covariates using a linear mixed model, and fixed effects and covariates were defined above. We conducted genome-wide association analyses using BayesB, which analyzed all autosomal SNP simultaneously and assumed different genetic variance for each SNP [40, 70]. The model is described as follows, $$ {y}_i= u+{\displaystyle \sum_{j=1}^M{Z}_{i j}{\alpha}_j{\delta}_j+{e}_i} $$ where y i is the adjusted phenotypic value for the i th individual, u is the mean (after removing fixed effects and all covariants), M is the number of SNP loci, Z ij is the j th SNP genotype of animal i coded as the number of B alleles in the genotype, α j is the average effect of allele substitution for SNP j, and is assumed to be normally distributed N (0, σ j 2 ), δ j is an indicator variable to show the presence (δ j = 1) and absence (δ j = 0) of marker j, and the presence is given a prior probability, and e i is the residual error with an assumed normal distribution N (0, σ e 2 ). The prior distribution of variance σ j 2 (or σ e 2 ) is assumed to be a scaled inverse Chi-square with degrees of freedom v α = 4 (or v e = 10) and scale parameter S α 2 (or S e 2 ). The scale parameter was usually derived from an assumed additive-genetic variance [71]. π was set to 0.9998, which meant that about 100–150 SNPs were fitted simultaneously in each MCMC iteration. The Markov chains were run for 50,000 cycles of iterations with the first 10,000 iterations being discarded as burn-in followed by additional 40,000 iterations to form the posterior sample. All SNPs effects were estimated from the posterior sample. We performed GWAS for all the 21 fatty acids but only reported the results for the traits with genomic heritability ≥ 0.10. We inferred the associations for fatty acids using a 100 kb window rather than single marker [35, 36]. There were 24,900 SNP windows across the 29 autosomes. The variance for each window was estimated using the genetic value of all adjacent SNPs within 100 kb window, and proportion of genetic variance explained by each window was obtained by dividing the variance of window breeding value by the variance of the whole genome breeding value. Genome windows with the highest posterior mean proportion of genetic variance ≥1% were considered as the most important regions associated with the traits. Positional candidate genes were investigated for the 100 kb windows using the UCSC Genome Browser, which allowed visualization of SNP based on the Bos taurus genome assembly UMD 3.1. Genome wide association study using GRAMMAR We also performed genome-wide association study using GRAMMAR-GC implemented in an R package GenABEL v1.8-0 [72]. The method accounts for population stratification and covariance structure of individuals inferred from all by SNPs. Bonferroni corrected threshold of 8.39E-08 (0.05/595,715) was adopted for the top 5% genome-wide significance. This correction was highly conservative for GWAS using high density SNPs array. To avoid the "overcorrection" for SNPs that may not truly independent due to LD across genome, we used a suggestive P value (P = 0.05/163,473) as thresholds proposed by Duggal et al., considering approximate the number of "independent" SNPs by counting 1 SNP per LD block, plus all SNPs outside of the LD blocks (interblock SNPs) [43]. Region-based association test and haploblock analyses Region-based association test is a more powerful approach of gene mapping than the association test of an individual genetic variant. In this study, we performed the region-based association test for several target 100 kb regions identified using BayesB. SNPs in these regions and the adjusted fatty acid composition were investigated with this region-based association test using R package FREGAT [73]. The LD of these regions were estimated using PLINK v1.7 software [67]. Genomic best linear unbiased prediction (GBLUP) and BayesB were used for genomic prediction. Five-fold cross validation was used to evaluate the accuracy of genomic prediction. The data were split into five approximately equal-sized groups. For each cross validation, four groups were used as the training sample to estimate parameters and the remaining group was used as the test sample. The linear model is written as, $$ \mathbf{y}={\mathbf{1}}_{\mathbf{n}} u+\mathbf{Z}\mathbf{a}+\mathbf{e} $$ where y is the vector of adjusted phenotypic values in the training sample, μ is the overall mean, a is the vector of breeding values for all animals, e is the vector of residuals errors and Z is the incidence matrix for the random effects. For the BayesB, SNP effects were estimated based on the training population using the statistical model described in the GWAS analyses. 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A Bayesian antedependence model for whole genome prediction. Genetics. 2012;190(4):1491–501. Aulchenko YS, Ripke S, Isaacs A, van Duijn CM. GenABEL: an R library for genome-wide association analysis. Bioinformatics. 2007;23(10):1294–6. Belonogova NM, Svishcheva GR, Axenovich TI. FREGAT: an R package for region-based association analysis. Bioinformatics. 2016;32(15):2392–2393. The authors would like to thank Shizhong Xu and George Liu for proofreading, and all staff at the cattle experimental unit in Beijing and Ulagai for animal care and sample collection. This study was supported by the National High Technology Research and Development Program of China (863 Program 2013AA102505-4), Cattle Breeding Innovative Research Team (cxgc-ias-03), Chinese Academy of Agricultural Sciences Foundation (2014ywf-yb-4), Beijing City Board of Education Foundation (PXM2016_014207_000012) and the National Natural Science Foundations of China (31372294 and 31472079) for the design of the study and collection. The project was also partly supported by Chinese Academy of Agricultural Sciences of Technology Innovation Project-Cattle Breeding (ASTIP-IAS03 and ASTIP-IAS-TS-9) for the data analysis and interpretation of the study. L.Y.X was supported by the Elite Youth Program in Chinese Academy of Agricultural Sciences. Datasets are available from the Dryad Digital Repository (doi:10.5061/dryad.4qc06). JYL and LYX conceived and designed the study. BZ and LYX performed statistical analyses. BZ and LYX wrote the paper. BZ, LYX, HJG, ZZW and YHL participated in data analyses. HN, XG, WGZ, LG, YC and PG carried out quantification of fatty acids. LPZ, YG and HMN participated in the design of the study and contributed to acquisition of data. LYX performed SNP and gene annotation. All authors read, commented and approved the final manuscript. All animals used in the study were treated following the guidelines for the experimental animals established by the Council of China Animal Welfare. Protocols of the experiments were approved by the Science Research Department of the Institute of Animal Sciences, Chinese Academy of Agricultural Sciences (CAAS) (Beijing, China). Blood and meat samples from Chinese Simmental cattle were collected with consent of the Jinweifuren Co., Ltd (BeiJing). Laboratory of Molecular Biology and Bovine Breeding, Institute of Animal Sciences, Chinese Academy of Agricultural Sciences, Beijing, China Bo Zhu , Hong Niu , Wengang Zhang , Zezhao Wang , Yonghu Liang , Long Guan , Peng Guo , Yan Chen , Lupei Zhang , Xue Gao , Huijiang Gao , Lingyang Xu & Junya Li Animal Science and Technology College, Beijing University of Agriculture, Beijing, China Yong Guo & Heming Ni Haidian District, Institute of Animal Science, Yuanmingyuan West Road 2#, Beijing, 100193, China Lingyang Xu Search for Bo Zhu in: Search for Hong Niu in: Search for Wengang Zhang in: Search for Zezhao Wang in: Search for Yonghu Liang in: Search for Long Guan in: Search for Peng Guo in: Search for Yan Chen in: Search for Lupei Zhang in: Search for Yong Guo in: Search for Heming Ni in: Search for Xue Gao in: Search for Huijiang Gao in: Search for Lingyang Xu in: Search for Junya Li in: Correspondence to Lingyang Xu or Junya Li. Additional file 1: Table S1. Estimates of phenotypic correlations (upper diagonals) and genetic correlation (lower diagonals) between 21 phenotypes in Chinese Simmental beef cattle. (DOCX 19 kb) Summary statistics of the 100 kb regions for 3 saturated fatty acids. Results include trait name, chromosome, 100 kb windows on the UMD3.1 genome assembly, window variances, and percentage of genetic variance using BayesB. (CSV 2102 kb) Summary statistics of the100 kb regions for three monounsaturated fatty acids (MUFA). Results include trait name, chromosome, 100 kb windows on the UMD3.1 genome assembly, window variances, and percentage of total genetic variance using BayesB. (CSV 2074 kb) Summary statistics of the 100 kb regions for five polyunsaturated fatty acids (PUFA). Results include trait name, chromosome, 100 kb windows on the UMD3.1 genome assembly, window variances, and percentage of genetic variance using BayesB. (CSV 2967 kb) Summary statistics of 100 kb regions for four fatty acid groups. Results include trait name, chromosome, 100 kb windows on the UMD3.1 genome assembly, window variances, and percentage of genetic variance using BayesB. (CSV 2520 kb) Zhu, B., Niu, H., Zhang, W. et al. Genome wide association study and genomic prediction for fatty acid composition in Chinese Simmental beef cattle using high density SNP array. BMC Genomics 18, 464 (2017) doi:10.1186/s12864-017-3847-7 Accepted: 06 June 2017 High-density genotypes Simmental cattle Non-human and non-rodent vertebrate genomics	CommonCrawl
\begin{document} \begin{abstract} Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order $p \ge 5$, then there exists an involution $y$ such that $\langle x,y\rangle$ is not solvable. Also, if $x$ is an involution then there exist three conjugates of $x$ that generate a nonsolvable group, unless $x$ belongs to a short list of exceptions, which are described explicitly. We also prove that if $x$ has order $6$ or $9$, then there exists two conjugates that generate a nonsolvable group. \end{abstract} \maketitle \section{Introduction} The following theorem is proved in \cite{Guest1}, and provides a solvable analogue of the classical Baer--Suzuki theorem for elements of certain orders. \begin{thm} \label{thmA} Let $G$ be a finite group and suppose that $x$ is an element of prime order $p$ where $p\ge5$. Then $x$ is contained in the solvable radical of $G$ if and only if $\langle x,x^g\rangle$ is solvable for all $g\in G$. In other words, if $x$ is not contained in the solvable radical of $G$ then there exists $g \in G$ such that $\langle x,x^g\rangle$ is not solvable. \end{thm} The proof of Theorem \ref{thmA} is by induction, and it is shown that a minimal counterexample to Theorem \ref{thmA} would have to be an almost simple group. Theorem \ref{thmA} is then proved (in \cite{Guest1}) with the following result for almost simple groups. \begin{thm} \label{thmA} Let $G$ be an almost simple group with socle $G_0$. Let $x \in G$ have odd prime order $p$. Then one of the following holds. \begin{enumerate} \item[(1)] There exists $g \in G$ such that $\langle x,x^g\rangle$ is not solvable; \item[(2)] $p=3$ and $x$ is a long root element in a simple group of Lie type defined over $\mathbb{F}_{3}$, $x$ is a short root element in $G_2(3)$, or $x$ is a pseudoreflection and $G_0\cong PSU(d,2)$. \end{enumerate} \end{thm} In this paper, we prove a result that is quite similar to Theorem \ref{thmA}. \begin{thm} \label{inv} Suppose that the finite group $G$ satisfies one of the following conditions: \begin{enumerate} \item[(1)] $G$ is almost simple group; \item[(2)] $SL(d,q) \le G \le GL(d,q)$ or $SU(d,q) \le G \le GU(d,q)$, and if $d=2$ and $q$ is odd, then $SL(2,q)$ or $SU(2,q)$ has even index in $G$; \item[(3)] $G$ is a finite group of Lie type (in the sense of \cite{Steinberg}) and $G \not \cong SL(2,q)$ ($q$ odd). \end{enumerate} If $x \in G$ has prime order $p \ge 5$ in $G/Z(G)$, then there exists an involution $y \in G$ such that $\langle x,y\rangle$ is not solvable. \end{thm} In particular, Theorem \ref{inv} shows that if $p\ge 5$ and $G$ is almost simple, then there exists \emph{an involution} $y$ such that $\langle x,x^y\rangle$ is not solvable. For $\langle x,x^y\rangle$ has index $1$ or $2$ in $\langle x,y\rangle$ and so either both groups are solvable, or both of them are not solvable. Also, Theorem \ref{thmA} shows that when the order of $x$ has a prime divisor $p \ge 5$ and $G$ is almost simple, there exist two conjugates that generate a nonsolvable group. In this paper we prove an analogous result for elements of order divisible by $3$. \begin{thm} \label{thm:9} \label{6} Suppose that $G$ is an almost simple group and that $x$ has order $6$ or $9$. Then there exists an element $g \in G$ such that $\langle x,x^g\rangle$ is not solvable. \end{thm} Theorem \ref{thmA} and Theorem \ref{thm:9} yield the following corollary immediately. \begin{cor} \label{cor:not2elements} Let $G$ be an almost simple group with socle $G_0$ and suppose that $x$ in $G$ is not a $2$-element. Then there exists $g$ in $G$ such that $\langle x,x^g \rangle$ is not solvable or $x$ has order 3 and $x$ is a long root element in a simple group of Lie type defined over $\mathbb{F}_{3}$, a pseudoreflection in $PGU(d,2)$ or a short root element in $G_2(3)$. Moreover, there exist three conjugates of $x$ that generate a nonsolvable group unless $G_0 \cong PSU(d,2)$ or $PSp(d,3)$. \end{cor} Guralnick, Flavell, and the author prove in \cite{FGG} that for all nontrivial elements $x$ in a finite (or linear) group $G$, $x$ is contained in the solvable radical of $G$ if and only if any four conjugates of $x$ generate a solvable group. In particular, if $x$ is contained in an almost simple group $G$, then there exist four conjugates of $x$ that generate a nonsolvable group (this result and Theorem \ref{thmA} are obtained independently by Gordeev, Grunwald, Kunyavski, and Plotkin in \cite{GGKP}). Thus if we allow $x$ to be a $2$-element, then a similar result to Corollary \ref{cor:not2elements} is true but with four conjugates of $x$. Corollary \ref{cor:not2elements} and Theorem \ref{thm:3invs} show that in most cases, there exist three conjugates of $x$ that generate a nonsolvable group. \begin{thm} \label{thm:3invs} Let $G$ be an almost simple group with socle $G_0$ and $x$ an involution in $G$. Then either there exist $g_1,g_2 \in G$ such that $\langle x,x^{g_1},x^{g_2} \rangle$ is not solvable or $(x,G_0)$ belongs to Table \ref{table:exceptions}. \end{thm} \begin{table}[htp] \centering \captionsetup{width=0.9\textwidth} \caption{\label{table:exceptions} Pairs $(x,G_0)$ such that any three conjugates of $x$ in $\mathrm{Aut}(G_0)$ generate a solvable group.} \begin{tabular}[t]{cc} \hline $G_0$ &$x$ \\ \hline $A_n$ & Transposition \\ $A_6$ & Triple transposition \\ $PSU(d,2)$ & Unitary transvection \\ $PSU(4,2) \cong P\Omega(5,3)$ & Graph automorphism \\ $PSL(4,2) \cong A_8$ & Graph automorphism \\ $P\Omega^{\pm}(d,2)$, $d$ even & Orthogonal transvection \\%this is a graph aut; in fact Aut(D_n(2)) =SO^{+}(2,2) $PSp(d,2) \cong P\Omega(d+1,2)$ & Symplectic transvection \\ $P\Omega(d,3)$, $d$ odd & reflection \\ $Fi_{22}$ & $x$ in class 2A\\ $Fi_{23}$ & $x$ in class 2A\\ $Fi_{24}^{\prime}$ & $x$ in class 2C in $Fi_{24}^{\prime}:2$\\ \hline \end{tabular} \end{table} We note that if $x$ is an involution, then $\langle x,x^g\rangle$ is dihedral and so we need at least three conjugate involutions to generate a nonsolvable group. In a future work, the author hopes to improve Corollary \ref{cor:not2elements} to find the minimal number of conjugates in an almost simple group required to generate a nonsolvable group for $2$-elements as well. This requires a proof that for an element of order $4$, there exist two conjugates that generate a nonsolvable group with a short list of exceptions, and that two conjugates always suffice for an element of order $8$. Also, using Lemma \ref{lem:1.1} below, we get the following corollaries to Theorems \ref{thm:3invs} and \ref{thm:9}. \begin{cor} \label{cor:to3invs} Let $G$ be a finite group with trivial Fitting subgroup and let $x$ be an involution in $G$. Then either there exist elements $g_1,g_2 \in G$ such that $\langle x,x^{g_1},x^{g_2} \rangle$ is not solvable or for every component $L$ of $G$, $x \in N_G(L)$ and $(x,L)$ is in Table \ref{table:exceptions}. \end{cor} \begin{cor} \label{cor:to9} Let $G$ be a finite group and let $x \in G$ have order $9$. If $x^3$ is not contained in the solvable radical of $G$ then there exists $g \in G$ such that $\langle x,x^g\rangle$ is not solvable. \end{cor} We note that the analogous result to Corollary \ref{cor:to9} for order $6$ elements is not true. For example, let $G= S_5 \times PSL(3,3)$ and $x=(a,b) \in G$ with $a$ a transposition in $S_5$ and $b$ a transvection in $PSL(3,3)$. Then $x$ has order $6$, the solvable radical is trivial, and $\langle x,x^g\rangle$ is solvable for all $g \in G$. We discuss this in more detail in Remark \ref{cor:to6} following the proof of Corollary \ref{cor:to9}. \section{Preliminaries} Throughout the paper, we will use the notation $L^{\epsilon}(d,q)$ where $ \epsilon \in \{\pm\}$ to denote the $PSL(d,q)$ when $\epsilon = +$ and $PSU(d,q)$ when $\epsilon=-$. $D_{n}^{\epsilon}$ will refer to $D_n(q)$ and ${^2}D_n(q)$ for $\epsilon=+$ and $\epsilon=-$ respectively. Similarly $E_6^{\epsilon}(q)$ refers to $E_6(q)$ and ${^2}E_6(q)$ for $\epsilon=+$ and $\epsilon=-$. Lemma \ref{lem:1.1} below relies on the result of Guralnick and Kantor \cite{GurAs} that every nontrivial element $x$ in an almost simple group belongs to a pair of elements $(x,y)$ that generates a group containing the socle of $G$. Corollary \ref{cor:to3invs} follows immediately from Theorems \ref{thm:3invs} and Lemma \ref{lem:1.1}. \begin{lem}\label{lem:1.1} Let $G$ be a finite group with trivial Fitting subgroup. Let $L$ be a component of $G$ and suppose that $x \not\in N_G(L)$. If $x^2 \not\in C_G(L)$ then there exists $g \in G$ such that $\langle x,x^g\rangle$ is not solvable. In any case, there exist $g_1,g_2 \in G$ such that $\langle x,x^{g_1},x^{g_2} \rangle$ is not solvable. \end{lem} \begin{proof} See \cite[Lemma 1]{Guest1}. \end{proof} \begin{lemma} \label{lem:2.2} Let $G_0$ be a simple group of Lie type, let $G = Inndiag(G_0)$ and let $x \in G$. \begin{enumerate} \item[(a)] If $x$ is unipotent, let $P_1$ and $P_2$ be distinct maximal parabolic subgroups containing a common Borel subgroup of $G$, with unipotent radicals $U_1$ and $U_2$. Then $x$ is conjugate to an element of $P_i \backslash U_i$ for $i = 1$ or $i = 2$. \item[(b)] If $x$ is semisimple, assume that $x$ lies in a parabolic subgroup of $G$. If the rank of $G_0$ is at least 2, then there exists a maximal parabolic $P$ with a Levi complement $J$ such that $x$ is conjugate to an element of $J$ not centralized by any Levi component (possibly solvable) of $J$. \end{enumerate} \end{lemma} \begin{proof} See \cite[Lemma 2.2]{GS} \end{proof} \section{Proof of Theorem 1.3} Let $(x,G)$ be a minimal counterexample. If $G$ is almost simple, then let $G_0$ be the simple group $G_0$ satisfying $G_0 \unlhd G \le {\rm Aut}(G_0)$. \begin{lemma} \label{lem:Aninvs} If $G$ is almost simple and $G_0 \cong A_n$, then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} Since $x$ has odd order $p$, it must lie in $A_n$. It suffices to assume that $x=(12 \cdots p) \in A_p$. If we let $y=(12)(34)$, then $\langle x,y \rangle = A_p$, which is not solvable. \end{proof} \begin{lemma} \label{lem:liftinvs} (a) If $x \in G \le PGL^{\epsilon}(d,q)$ does not lift to an element of order $p$ in $\mathrm{GL}(d,q)$, then $(x,G)$ is not a minimal counterexample.\\ (b) If $Z(G) \ne \{1\}$, then we may assume that $x \in G$ has order $p$. \end{lemma} \begin{proof} To prove (a), note that if $x$ does not lift to an element of order $p$ in $GL^{\epsilon}(n,q)$, then $p \mid (q- \epsilon,n)$ and the natural $\langle x\rangle$-module $V$ decomposes into $p$-dimensional spaces (see \cite[Lemma 3.11]{Bur2} for example). It therefore suffices to assume that $n=p$ and $x$ acts irreducibly on the natural module $V$ since $(x,G)$ is a minimal counterexample. Under these conditions on $n$, $p$ and $q$, a Sylow $p$-subgroup of $GL^{\epsilon}(n,q)$ is contained in a type $(q- \epsilon)\wr S_p$ maximal subgroup. The irreducibility of $x$ implies that $x$ is non-trivial in $S_p$, and we can take an involution $y \in SL^{\epsilon}(p,q)$ that induces any involution in $S_p$; thus $(x,G)$ cannot be a minimal counterexample. \\% see scanned work in Paper3 folder lifts.pdf To prove (b), if $SL^{\epsilon}(d,q) \le G \le GL^{\epsilon}(d,q)$, then consider $x \in G/Z(G) \le PGL(d,q)$. If $x$ does not lift to an element of order $p$ in $G$, then the same argument as for part (a) shows that there exists an involution $y \in SL(d,q)$ such that $\langle x,y\rangle$ is not solvable. In all other cases, $p$ does not divide $\|Z(G)\|$ so $x'= x^{\|Z(G)\|}$ will have order $p$ in $G$ and $(x',G)$ will also be a minimal counterexample to Theorem \ref{inv}. \end{proof} \begin{lemma} \label{lem:psl2} If $\mathrm{PSL}(2,q) \le G \le \mathrm{Aut}( \mathrm{PSL}(2,q) )$ or $SL^{\epsilon}(2,q) \lneqq G \le GL^{\epsilon}(2,q)$ with $[G:SL^{\epsilon}(2,q)]$ even, then $(x,G)$ cannot be a minimal counterexample. \end{lemma} \begin{proof} First note that if $\mathrm{PSL}(2,q) \le G \le \mathrm{Aut}(\mathrm{PSL}(2,q) )$, then the order of $x$ implies that $x$ is either in $PSL(2,q)$, or it is a field automorphism. In this case, we may assume that $q \ge 7$ since we have eliminated the case that $A_n \le G \le \mathrm{Aut}(A_n)$. First, let us assume that $x \in PSL(2,q)$. \par If $p \mid q$, then $x \in PSL(2,q)$ is a transvection, and we may assume that $x=x_{\alpha_1}(a)$, with $a \in \mathbb{F}_p$. In this case, $x^{n_{\alpha_1}}=x_{-\alpha_1}(\pm a)$, thus $\langle x,x^{n_{\alpha_1}}\rangle = PSL(2,p)$, which is not solvable. \par If $x$ is semisimple in $PSL(2,q)$, then either $p \mid q+1$, or $p \mid q-1$. Suppose first that $p \mid q+1$. Then consider the possibilities for the maximal subgroups of $PSL(2,q)$ containing $x$. Since $(x,G)$ is a minimal counterexample, $x$ cannot be contained in $A_5$, and it cannot be contained in $A_4$ or $S_4$ since $p \ge 5$. Moreover, $x$ cannot be contained in a subfield subgroup since, because of the order of $x$, any such subfield subgroup would be almost simple. So $x$ can only be contained in a dihedral group $D$ of order $\frac{2(q+1)}{(2,q-1)}$. It can be contained in only one dihedral subgroup since $C_G(x)$ is the cyclic subgroup of $D$ of order $ \frac{q+1}{(2,q-1)}$. So, let $y$ be an involution in $G$ that is not contained in $D$. \par Now suppose that $p \mid q-1$. The possible maximal subgroups containing $x$ are a dihedral group $D$ of order $\frac{2(q-1)}{(2,q-1)}$ and (at most two) Borel subgroups. Let $i_2(H)$ denote the number of involutions in a group $H$. Then \begin{equation} i_2(G) \ge \begin{cases} q^2-1 &\text{for $q$ even;} \\ q(q-1)/2 & \text{for $q$ odd.} \end{cases} \end{equation} Moreover, if $B$ is a Borel subgroup, then \begin{equation} i_2(B) \le \begin{cases} q-1 &\text{for $q$ even;} \\ q & \text{for $q$ odd.} \end{cases} \end{equation} If $D$ is the dihedral group above, then \begin{equation} i_2(D) \le \begin{cases} (q+1)/2 & \text{for $q$ odd;} \\ q-1 & \text{for $q$ even.} \end{cases} \end{equation} So if $q$ is odd, then we may assume that $q \ge 7$; thus \begin{displaymath} i_2(G) \ge (q^2-q)/2 > 2q+(q+1)/2 \ge 2i_2(B)+i_2(D). \end{displaymath} Also, if $q$ is even, then we may assume that $q \ge 8$ and so \begin{displaymath} i_2(G) \ge q^2-1 > 2(q-1) + (q-1) \ge 2i_2(B) +i_2(D). \end{displaymath} Thus $x \not\in PSL(2,q)$. Now suppose that $x$ is a field automorphism of $PSL(2,q)$. We may assume that $x$ is a standard field automorphism by \cite[7.2]{GL}. Define $q_0$ by $q:=q_0^p$ and let \begin{displaymath} \Gamma = \{y \in G_0 \mid y^2=1, \: \langle x,y\rangle \ne G \}. \end{displaymath} We will show that $\|\Gamma\| < i_2(G_0)$. Indeed, if $y \in \Gamma$ then $\langle x,y\rangle \cap G_0$ is contained in a subgroup of $G_0=PSL(2,q_0^p)$. From the description of the subgroups of $PSL(2,q)$, since $p$ is odd, $\langle x,y \rangle \cap G_0$ must be contained in a Borel subgroup, a dihedral group of order $\frac{2(q \pm 1)}{(2,q-1)}$, or a subfield subgroup of type $PSL(2,q_0)$. We note that since $(x,G)$ is a minimal counterexample, $\langle x,y \rangle \cap G_0$ cannot be contained in any other \emph{maximal} subfield subgroups. Now, if $H$ is a torus of order $\frac{(q \pm 1)}{(2,q-1)}$, a Borel or subfield subgroup, then the $G$-conjugates of $H$ fixed by $x$ form one $C_{G_0}(x)$ orbit (see the proof of \cite[Lemma 3.1]{GS} for example). If $H$ is a $G$-conjugate of a maximal dihedral group that is fixed by $x$, then $x$ must also normalize the characteristic cyclic subgroup of $H$ (a torus of order $\frac{(q \pm 1)}{(2,q-1)}$). Since the $G$-conjugates of the torus that are fixed by $x$ are all $C_{G_0}(x)$-conjugate, it follows that the $G$-conjugates of the dihedral group that are fixed by $x$ are also $C_{G_0}(x)$-conjugate. So the number of conjugates of $H$ that can contain $\langle x,y \rangle \cap G_0$ is at most $\|C_{G_0}(x)\|/\|C_H(x)\|$. Thus the number of involutions $y$ in $G_0$ such that $\langle x,y\rangle \cap G_0$ is contained in a conjugate of $H$ is at most \begin{displaymath} \frac{i_2(H)\|C_{G_0}(x)\|}{\|C_H(x)\|}. \end{displaymath} Let $X_1, \ldots, X_k=C_{G_0}(x)$ be representatives for the conjugacy classes of maximal subgroups containing $\langle x,y\rangle \cap G_0$. Note that there are no nontrivial conjugates of $X_k=C_{G_0}(x)$ fixed by $x$ and so a crude upper bound for the number of involutions in $G$ such that $ \langle x,y \rangle \cap G_0$ is contained in $C_{G_0}(x)$ is $\|C_{G_0}(x)\|$. So if $(x,G)$ is a minimal counterexample, then we have \begin{displaymath} \|\Gamma\| \le \sum_{i=1}^{k-1} \frac{i_2(X_i)\|C_{G_0}(x)\|}{\|C_{X_i}(x)\|}+\|C_{G_0}(x)\|. \end{displaymath} If $q$ is odd, then \begin{align} \sum_{i=1}^k \frac{i_2(X_i)\|C_{G_0}(x)\|}{\|C_{X_i}(x)\|} \le& \frac{q_0^pq_0(q_0^2-1)}{q_0(q_0-1)} + \frac{(q_0^p+1)q_0(q_0^2-1)}{2(q_0-1)} \\ & +\frac{(q_0^p+3)q_0(q_0^2-1)}{2(q_0+1)} + \frac{q_0(q_0^2-1)}{2} \\ \le &\frac{q_0(q_0+1)(3q_0^p+q_0+3)}{2}; \end{align} but this is less than $i_2(G_0) \ge q_0^p(q_0^p-1)/2$. If $q$ is even, then \begin{align} \sum_{i=1}^k \frac{i_2(X_i)\|C_{G_0}(x)\|}{\|C_{X_i}(x)\|} \le& \frac{(q_0^p-1)q_0(q_0^2-1)}{q_0(q_0-1)} + \frac{(q_0^p-1)q_0(q_0^2-1)}{2(q_0-1)} \\ & +\frac{(q_0^p+1)q_0(q_0^2-1)}{2(q_0+1)} + q_0(q_0^2-1) \\ \le & 2(q_0^p+q_0)(q_0+1)q_0; \end{align} but $i_2(G_0) \ge (q_0^{2p}-1)$ and so $\|\Gamma\| \le i_2(G_0)$. \par If $SL^{\epsilon}(2,q) \lneqq G \le GL^{\epsilon}(2,q)$ and $x \in G$, then we may assume that $x$ has order $p$ by Lemma \ref{lem:liftinvs}. Moreover, we may assume that $q$ is odd since $PSL(2,2^a) \cong SL(2,2^a)$ and so our hypothesis states that $[G:SL(2,q)]$ is even. If $x$ is semisimple, then since $SL^{\epsilon}(2,4)\cong PSL(2,4)$, $GL^{\epsilon}(2,4)$ is not a minimal counterexample and $GL^{\epsilon}(2,5)$ does not contain semisimple elements of order $p \ge 5$; thus we may assume that $q \ge 7$. If $(x,G)$ is a minimal counterexample then $x$ must be contained in $SL^{\epsilon}(2,q)$, for otherwise $p \| q- \epsilon$ and there exists a scalar $\lambda$ such that $\lambda x \in SL^{\epsilon}(2,q)$. Thus $SL^{\epsilon}(2,q)$ has index $2$ in $G$, and there are at least $q^2+q$ involutions in $G$. Now the same counting argument as for $PSL(2,q)$ shows that $(x,G)$ cannot be a minimal counterexample. If $x$ is unipotent in $SL^{\epsilon}(2,q)$, then $q \ge 5$, and by minimality, $SL^{\epsilon}(2,q)$ has index $2$ in $G$. We may assume that $x$ is not contained in any subfield subgroups by minimality. So choose an involution $y$ such that $[x,x^y] \ne 1$. Another inspection of the maximal subgroups shows that $\langle x,y\rangle$ is not solvable. \end{proof} \begin{lem} \label{lem:outinvs} If $G$ is almost simple and $x$ is an outer automorphism of $G_0$, then $(x,G)$ cannot be a minimal counterexample, except possibly if $G_0 \cong {^2}B_2(2^a)$. \end{lem} \begin{proof} We may assume that the untwisted Lie rank is at least 2 since the case where $G_0 \cong PSL(2,q)$ has already been eliminated. Since $x$ has order $p$, it is a field automorphism, and by \cite[7.2]{GL} we may assume that $x$ is a standard field automorphism. Now if $G_0$ is not a Suzuki--Ree group, then $x$ normalizes but does not centralize an $SL(2,q)$ subgroup $S$. So if $q$ is even and $G_0$ is not a Suzuki--Ree group, then there exists $y$ an involution $y \in S$ such that $\langle x,y\rangle$ is not solvable. Thus we may assume that either $G_0$ is a Suzuki--Ree group or that $q$ is odd. If $q$ is odd, then an inspection of the (extended) Dynkin diagram shows that $x$ normalizes but doesn't centralize a type $SL(3,q)$ subgroup $H$, unless $G_0 \cong PSL(2,q)$, $PSL(3,q)$, $PSp(4,q)$, ${^3}D_4(q)$, ${^2}G_2(3^a)$, or $PSU(d,q)$. If $G_0 = L^{\epsilon}(3,q)$ and $q$ is odd, then $x$ normalizes a subgroup of type $SO(3,q)$. If $G_0 \cong PSU(d,q)$ and $d \ge 4$, then $x$ normalizes but does not centralize a subgroup H of $G_0$ that is isomorphic to $PSO^{\epsilon}(d,q).(d,2)$ (when $d=4$, take $\epsilon=-$) by \cite[4.5.5]{KL}. \\ If $G_0 \cong {^3}D_4(q)$ then $x$ normalizes but does not centralize a subgroup $H$ isomorphic to $G_2(q)$. If $G_0 \cong PSp(4,q)$, then $x$ normalizes a subgroup $H$ isomorphic to $PSp(2,q^2).2$ (see \cite[Propostion 4.3.10]{KL}). If $G_0\cong {^2}G_2(3^{a})$, then let $z$ be an involution in $C_{G_0}(x)$. Then $x \in C_G(z)$, which is a subgroup $H$ of type $PSL(2,3^{2a})$ by \cite[Table 4.5.1]{GLS}. Moreover, $x$ does not centralize a subgroup of type $PSL(2,3^{2a})$ since it doesn't centralize an element of order divisible by $3^{2a}+1$. If $G_0 \cong {^2}F_4(2^a)$, then a field automorphism normalizes, but does not centralize, a subgroup of $G_0 \cong {^2}F_4(2^a)$ isomorphic to $PGU(3,2^a):2$ by \cite{2F4}. By minimality, it follows in all cases that there exists an involution $y \in H$ such that $\langle x,y\rangle$ is not solvable. \end{proof} \begin{lem} \label{unipotents} If $x$ is a unipotent element in $G$, then $(x,G)$ cannot be a minimal counterexample. \end{lem} \begin{proof} Since $p \ge 5$ and $p\|q$, $G$ cannot be a Suzuki--Ree group, and by Lemma \ref{lem:psl2}, we may assume that the untwisted Lie rank is at least $2$. If $G$ is an almost simple group, then we may assume that $G=G_0$ and by Lemma \ref{lem:liftinvs}, we can lift $x$ to an element of order $p$ in the universal version of $G_0$. By \cite[Lemma 2.1]{GS}, we may assume that $x$ is nontrivial in $P/U$ for some end node maximal parabolic subgroup $P$, with unipotent radical $U$, unless $G$ is ${^3}D_4(q)$, or $SU(d,q)$. So we may assume that $x$ acts nontrivially on a Levi subgroup $L$, and since $G$ is simply connected, so is $L$ (see \cite[2.6.5(f)]{GLS} for example). By induction, there exists an involution $y \in L$ such that $\langle x,y\rangle$ is not solvable; thus there exists an involution $y' \in G_0$ such that $\langle x,y\rangle$ is not solvable. If $G_0 \cong {^3}D_4(q)$, then we may assume that $x$ is nontrivial in $P/U$, for a maximal parabolic subgroup $P$. The Levi complement is of type $SL(2,q)$ or $SL(2,q^3)$, but a split torus normalizes both of these Levi complements and induces diagonal automorphisms on them. Thus we can reduce to the case that $SL(2,q) \le G \le GL(2,q)$, where $SL(2,q)$ has even index in $G$ when $q$ is odd. Now suppose that $G = SU(d,q)$ and $d \ge 5$. Then \cite[Lemma 2.1]{GS} implies that we may assume that $x$ is nontrivial in $P/U$, for some (not necessarily end-node) maximal parabolic subgroup $P$. Therefore $x$ will act nontrivially on one of the components of the Levi complement of $P$, and these components are all nonsolvable since $p \ge 5$, and not of type $SL(2,q)$. If $G = SU(4,q)$, then we may assume that $x$ is nontrivial in $P/U$ for some maximal parabolic subgroup $P$. Now the Levi complement $L$ of $P$ in $SU(4,q)$ is either isomorphic to $GU(2,q)$ or a normal subgroup of $GL(2,q^2)$ of index $q-1$; thus $(x,G)$ cannot be a minimal counterexample. The only other possibility is $G=SU(3,q)$. If $x$ is a transvection, then it is contained in a subgroup isomorphic to $GU(2,q)$. So we may assume that $x$ is not a transvection and $x$ is therefore regular unipotent. Since all inner diagonal involutions of $PSU(3,q)$ lift to involutions in $SU(3,q)$, we can work in $PSU(3,q)$. From the list of maximal subgroups of $PSU(3,q)$ (see \cite[Theorem 6.5.3]{GLS} for example), we may assume that the only maximal subgroups that could contain $x$ are the maximal parabolic subgroups since the other maximal subgroups of order divisible by $p$ are almost simple. Now $x$ only stabilizes one totally singular $1$-space, and so is only contained in one maximal parabolic subgroup. So choose an involution $y$ that is not contained in this maximal parabolic subgroup. Then $\langle x,y \rangle$ is not solvable. \end{proof} \begin{lemma} If $G$ or $G_0$ is a classical group, then $(x,G)$ cannot be a minimal counterexample. \end{lemma} \begin{proof} By Lemmas \ref{lem:psl2}, \ref{lem:outinvs}, \ref{lem:liftinvs} and \ref{unipotents} we may assume that $x$ is semisimple and that $G_0$ or $G/Z(G)$ is not $PSL(2,q)$. Moreover, we can and will assume that $x$ is an element of order $p$ in $G$ where $SL(n,q) \le G \le \mathrm{GL}_{n}(q)$, $SU(d,q) \le G \le \mathrm{GU}(n,q)$, $G=Sp(n,q)$ or $G=\mathrm{\Omega}^{\epsilon}(n,q)$ by Lemma \ref{lem:liftinvs} and \cite[Lemma 3.11]{Bur2}. In case O, we may assume that $d \ge 7$. If $G$ is a unitary group, let $e$ be the smallest positive integer such that $p \mid q^{2e}-1$; otherwise let $e$ be the smallest positive integer such that $p \mid q^{e}-1$. Consider a decomposition of $V$ into irreducible $\langle x\rangle$-invariant spaces \begin{align} \label{V} V= (W_1 \oplus W_1') \perp \cdots \perp (W_k \oplus W_k') \perp U_1 \perp \cdots \perp U_l \end{align} where the $W_i$ and $W_i'$ are totally singular, and the $U_i$ and $W_i \oplus W_i'$ are nondegenerate. Each irreducible subspace on which $x$ acts nontrivially has dimension $e$. In case $U$, we can and will assume that the $1$-spaces on which $x$ acts trivially are nondegenerate. We consider five cases separately. \begin{list}{\labelitemi}{\leftmargin=0.5em} \item[(i)] Suppose that $e=1$. In cases L, S, and O, all of the irreducible subspaces on which $\langle x\rangle$ acts nontrivially must be totally singular since $p \| q-1$. Moreover, $q \ge 7$ since $p \ge 5$. So in cases S and L, we may assume that $x$ acts nontrivially on $W_1 \oplus W_2$ and so $x$ is contained in a type subgroup $\mathrm{GL}(2,q)$, and $(x,G)$ is not a minimal counterexample in this case. In case O, since $d \ge 7$, we may assume that there are totally singular subspaces $W_1$, $W_2$, $W_3$ such that $W_1 \oplus W_2 \oplus W_3$ is totally singular and $x$ invariant; thus $x$ is contained in a type $SL(3,q)$ subgroup. In case U, if $p\|q-1$, then we can argue as in cases S and L to reduce to the case $G \cong GL(2,q^2)$. If $p \| q+1$, then $q \ge 4$ and we may assume that all of the subspaces in (\ref{V}) are nondegenerate; so $x$ is contained in a type $GU(1,q)^d$ subgroup and therefore we can reduce to the case $G\cong GU(2,q)$. \item[(ii)] Suppose that $e=2$. In case $U$, all of the $2$-spaces in (\ref{V}) are totally singular since even dimensional unitary groups do not contain irreducible elements. So in case $U$, $x$ acts irreducibly on $W_1$ and we reduce to the case $G \cong GL(2,q)$. In the other cases, $q\ge 4$ since $p \ge 5$. In cases L and S, if there is a totally singular $2$-space $W_1$ in (\ref{V}), then we can reduce to the case $G \cong GL(2,q)$. If there are no totally singular $2$-spaces in case S, then all of the $2$-spaces in \ref{V} are nondegenerate, and we can reduce to the case $G \cong Sp(4,q)$. In this case, we may assume that $q$ is odd since if $q$ is even then we can reduce to the case $G \cong Sp(2,q)$. But when $q$ is odd, a Sylow $p$-subgroup is contained in a subgroup isomorphic to $GU(2,q)$ (see \cite[p. 118]{KL}); thus we do not have a minimal counterexample in this case either. In case O, either we can reduce to the case $G \cong \mathrm{\Omega}(d,q)$ with $d=5$ or $6$, or all of the subspaces are totally singular. In this case, $x$ stabilizes $W_1 \oplus W_2$, and is thus contained in a subgroup of type $SL(4,q)$. Thus $(x,G)$ cannot be a minimal counterexample. \item[(iii)] Suppose that $e=3<d$. If there is a totally singular $3$-space $W_1$ in (\ref{V}), then in all cases $x$ will be contained in a subgroup of type $SL(3,q)$ (or $SL(3,q^2)$). Otherwise, all of the $3$-spaces in \ref{V} are nondegenerate and we are in case U or O. In case U, we can reduce to the case $G \cong SU(3,q)$, and $q \ne 2$ since $GU(3,2)$ has order $2^3 3^4$. In case O, we have $d \ge 7$ and so we can reduce to the case $G = \mathrm{\Omega}^{\pm}(6,q)$. \item[(iv)] Suppose that $4 \le e < d$. If there is a totally singular $e$-space in (\ref{V}), then we can reduce to the $e$ dimensional linear case. Otherwise $x$ acts irreducibly on a nondegenerate $e$-space, and we can reduce to the case $G= \mathrm{Sp}(e,q)$ in case S, $SU(e,q) \le G \le GU(e,q)$ in case U ($e$ odd), and $G = \mathrm{\Omega}^{-}(e,q)$ ($e$ even) in case O. \item[(v)] Suppose that $e=d$, so that $x$ acts irreducibly. In case S, $x$ must be contained in a $GU(d/2,q)$ subgroup \cite[p. 118]{KL}. In case O, if $d/2$ is odd then $x$ must be contained in a subgroup $H$ of type $GU(d/2,q)$. If $d/2$ is even, then $H$ is contained in a $\mathrm{\Omega}^{-}(d/2,q^2)$ subgroup . So we may assume that $G$ is linear or unitary. Now observe that if $d$ is even, then $G$ is linear and $x$ is contained in a normal subgroup of $GL(2,q^{d/2})$ of index dividing $q-1$ (\cite[(4.3.16)]{KL}), and so if $(x,G)$ is a minimal counterexample, then $d/2$ must be odd. But if $d/2$ is odd, then $x$ is contained in a type $GL(d/2,q^2)$ subgroup and so $(x,G)$ cannot be a minimal counterexample in this case either. So $d$ must be odd and in fact $d$ must be an odd prime since otherwise $x$ is contained in a type $GL^{\epsilon}(d/r,q^r)$ subgroup. We can list the possible maximal subgroups of $G$ that could contain $x$ using \cite{GPPS} and \cite{KL}. Since $d$ is odd, all involutions in $PGL^{\epsilon}(d,q)$ lift to involutions in $SL^{\epsilon}(d,q)$; thus we can work in the almost simple group $G/Z(G)$. In particular, we may assume that $x$ is not contained in any almost simple subgroup of $G/Z(G)$. In this case, the only possible maximal subgroups containing $x$ are of type $GL^{\epsilon}(1,q^d).d$. Since $p \nmid d$, $x$ is contained in a cyclic maximal torus $T$, and since $C_G(x)=T$, $x$ is contained in only one maximal subgroup. Thus we can pick an involution $y$ not contained in this maximal subgroup, and $\langle x,y\rangle$ will not be solvable. \qedhere \end{list} \end{proof} \begin{lemma} \label{lem:nosc} If $G$ is a finite group of Lie type and $Z(G) \ne 1$, then $(x,G)$ cannot be a minimal counterexample unless $G$ is (simply connected) $E_7(q)$. \end{lemma} \begin{proof} By our previous work, we may assume that $G$ is an exceptional group. But then the centre of $G$ is either trivial or of odd order, or $G$ is $E_7(q)$. So if $(x,G)$ is a minimal counterexample, then Theorem \ref{inv} holds for $G/Z(G)$. But then there exists $y \in G$ such that $\langle x,y \rangle$ is not solvable and $ y^2 \in Z(G)$. Since $Z(G)$ has odd order, $y'= y^{\|Z(G)\|}$ is an involution in $G$ and $\langle x,y'\rangle$ is not solvable. \end{proof} \begin{lem} \label{lem:El} Suppose that $G$ is almost simple or a finite group of Lie type, and that $G_0$ or $G/Z(G)$ is one of the simple groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$, or ${^2}E_6(q)$. Then $(x,G)$ cannot be a minimal counterexample. \end{lem} \begin{proof} We may assume that $G$ is almost simple with $x \in {\rm Inndiag}(G_0)$ or that $G$ is (simply connected) $E_7(q)$ by Lemmas \ref{lem:outinvs} and \ref{lem:nosc}. Moreover, we may assume that $x$ is semisimple in both cases by Lemma \ref{unipotents}. First suppose that $G$ is one of the untwisted groups. If $p\|q-1$ then $x$ is contained in a Borel subgroup and therefore in a $P_1$ parabolic subgroup. By Lemma \ref{lem:2.2} we may assume that $x$ is contained in a Levi subgroup of type $C_3(q)$ or $D_{l-1}(q)$, and so $(x,G)$ cannot be a minimal counterexample. So we may assume that $p\nmid q-1$ in the untwisted cases. Now suppose that $x$ is contained in some maximal parabolic subgroup. Again we may assume that $x$ acts noncentrally on each component of the Levi complement. It is easily verified that $(x,G)$ cannot be a minimal counterexample since we can work in one of the groups of Lie type in the Levi complement to find an involution $y$ such that $\langle x,y\rangle$ is not solvable. So we may assume that $x$ is not contained in any parabolic subgroups. If this is the case, then the centralizer of $x$ is reductive and contains no unipotent elements. First suppose that $G=E_7(q)$, and without loss of generality we may assume that $G$ is simply connected. We know that \begin{displaymath} \|G\| = q^{63} \prod_{d_i \in \{2,6,8,10,12,14,18\}} (q^{d_i}-1), \end{displaymath} so let $e$ be the smallest $d_i$ such that $p \| (q^{d_i}-1)$. If $e=14$ then either $p\|q^7-1$ or $p\| q^7+1$. If $p\|q^7-1$ then the $p$-part of $\|G\|$ (that is, the largest power of $p$ dividing $\|G\|$) is the $p$-part of $(q^7-1)/(q-1)$ and so a Sylow $p$-subgroup is contained in a type $SL(7,q)$ subsystem subgroup. Thus $(x,G)$ cannot be a minimal counterexample in this case. If $p\| q^7+1$, then the $p$-part of $\|G\|$ is the $p$-part of $(q^7+1)/(q+1)$ and so a Sylow $p$-subgroup is contained in a type $SU(7,q)$ subgroup. Similarly we can show that $(x,G)$ cannot be minimal counterexample for all values of $p$. We illustrate this work in Table \ref{tab:SylE7} below (note that $p\nmid q-1$ since we are assuming that $x$ is not contained in any parabolic subgroups). We do the same for $F_4(q)$, $E_6(q)$, ${^2}E_6(q)$, and $E_8(q)$ and record our results in Tables \ref{tab:SylF4}, \ref{tab:SylE6}, \ref{tab:Syl2E6} and \ref{tab:SylE8} respectively. The only case where we have not shown that $(x,G)$ is not a minimal counterexample is when $G = E_8(q)$ and $p$ is a primitive prime divisor of $q^{30}-1$ or $q^{15}-1$. It follows that $p \equiv 1 \imod{15}$ or $p \equiv 1 \imod{30}$, and in particular, that $p \ge 31$. If $p=31$, then the Sylow $31$-subgroups are cyclic and $x$ is contained in an exotic local subgroup $5^3.SL(3,5)$. Therefore we may assume that $p \ne 31$. The maximal subgroups of $E_8(q)$ are described in \cite[Theorem 8]{LSe1} and if $(x,G)$ is a minimal counterexample, then $x$ can only be contained in a (single) torus $T$ of type $q^8+q^7-q^5-q^4-q^3+q+1$ or $q^8-q^7+q^5-q^4+q^3-q+1$. In this situation, we can pick an involution $y \in G$ that is not contained in the normalizer of $T$ and $\langle x,y\rangle$ will not be solvable. \qedhere \begin{table}[htdp] \begin{center} \caption{Subgroups of $E_7(q)$ containing a Sylow $p$-subgroup\label{tab:SylE7}} \begin{tabular}{cccc} \hline $e$ & $p$ divides & $p$-part of $G$ & Subgroup type containing a \\ & & is $p$-part of & Sylow $p$ subgroup \\ \hline $18$ & $q^9+1$ & $q^6-q^3+1$ & ${^2}E_6(q)$ \\ $18$ & $q^9-1$ & $q^6+q^3+1$ & $E_6(q)$ \\ $14$ & $q^7+1$ & $(q^7+1)/(q+1)$ & $SU(7,q)$ \\ $14$ & $q^7-1$ & $(q^7-1)/(q-1)$ & $SL(7,q)$ \\ $12$ & $q^6+1$ & $(q^6+1)/(q^2+1)$ & $F_4(q)$ \\ $12$ & $q^6-1$ & x & \\ $10$ & $q^5+1$ & $(q^5+1)/(q+1)$ & $SU(7,q)$ \\ $10$ & $q^5-1$ & $(q^5-1)/(q-1)$ & $SL(7,q)$ \\ $8$ & $q^4+1$ & $q^4+1$ & $SL(8,q)$ \\ $8$ & $q^4-1$ & x& \\ $6$ & $q^3+1$ & $(q^2-q+1)^3$ & ${^2}E_6(q)$ \\ $6$ & $q^3-1$ & $(q^2+q+1)^3$ & $E_6(q)$ \\ $2$ & $q+1$ & $(7,p)(5,p)(q+1)^7$ & $SU(8,q)$ \\ $2$ & $q-1$ & x &\\ \hline \end{tabular} \end{center} \end{table} \begin{table}[htdp] \begin{center} \caption{ Subgroups of $F_4(q)$ containing a Sylow $p$-subgroup\label{tab:SylF4}} \begin{tabular}{cccc} \hline $e$ & $p$ divides & $p$-part of $G$ & Subgroup type containing a \\ & & is $p$-part of & Sylow $p$ subgroup \\ \hline $12$ & $q^6+1$ & $q^4-q^2+1$ & ${^3}D_4(q)$ \\ $12$ & $q^6-1$ & x &\\ $8$ & $q^4+1$ & $q^4+1$ & $SO(9,q)$ \\ $8$ & $q^4-1$ & $(q^2+1)^2$ & $SO(9,q)$ \\ $6$ & $q^3+1$ & $(q^2-q+1)^2$ & ${^3}D_4(q)$ \\ $6$ & $q^3-1$ & $(q^2+q+1)^2$ & ${^3}D_4(q)$ \\ $2$ & $q+1$ & $(q+1)^4$ & $SO(9,q)$ \\ $2$ & $q-1$ & x & \\ \hline \end{tabular} \end{center} \end{table} \begin{table}[htdp] \begin{center} \caption{ Subgroups of $E_6(q)$ containing a Sylow $p$-subgroup\label{tab:SylE6}} \begin{tabular}{cccc} \hline $e$ & $p$ divides & $p$-part of $G$ & Subgroup type containing a \\ & & is $p$-part of & Sylow $p$ subgroup \\ \hline $12$ & $q^6+1$ & $q^4-q^2+1$ & $F_4(q)$ \\ $12$ & $q^6-1$ & x & \\ $9$ & $q^9-1$ & $q^6+q^3+1$ & $SL(3,q^3)$ \\ $8$ & $q^4+1$ & $q^4+1$ & $F_4(q)$ \\ $8$ & $q^4-1$ & $(q^2+1)^2$ & $F_4(q)$ \\ $6$ & $q^3+1$ & $(q^2-q+1)^2$ & $F_4(q)$ \\ $6$ & $q^3-1$ & $(q^2\!+\!q\!+\!1)^3$ & $SL(3,q) \! \circ \! SL(3,q)\! \circ \!SL(3,q)$ \\ $5$ & $q^5-1$ & $q^4+q^3+q^2+q+1$ & $SL(5,q)$ \\ $2$ & $q+1$ & $(q+1)^4$ & $F_4(q)$\\ $2$ & $q-1$ & x &\\ \hline \end{tabular} \end{center} \end{table} \begin{table}[htdp] \begin{center} \caption{ Subgroups of ${^2}E_6(q)$ containing a Sylow $p$-subgroup\label{tab:Syl2E6}} \begin{tabular}{cccc} \hline $e$ & $p$ divides & $p$-part of $G$ & Subgroup type containing a \\ & & is $p$-part of & Sylow $p$ subgroup \\ \hline $12$ & $q^6+1$ & $q^4-q^2+1$ & $F_4(q)$ \\ $12$ & $q^6-1$ & x & \\ $9$ & $q^9+1$ & $q^6-q^3+1$ & $SU(3,q^3)$ \\ $8$ & $q^4+1$ & $q^4+1$ & $F_4(q)$ \\ $8$ & $q^4-1$ & $(q^2+1)^2$ & $F_4(q)$ \\ $6$ & $q^3+1$ & $(q^2-q+1)^3$ & $SU(3,q) \circ SU(3,q) \circ SU(3,q)$ \\ $6$ & $q^3-1$ & $(q^2+q+1)^2$ & $F_4(q)$\\ $5$ & $q^5+1$ & $q^4\!-\!q^3\!+\!q^2\!-\!q\!+\!1$ & $SO^{-}(10,q)$ \\ $2$ & $q+1$ &x &\\ $2$ & $q-1$ & $(q-1)^4$& $F_4(q)$\\ \hline \end{tabular} \end{center} \end{table} \begin{table}[htdp] \begin{center} \caption{Subgroups of $E_8(q)$ containing a Sylow $p$-subgroup\label{tab:SylE8}} \begin{tabular}{cccc} \hline $e$ & $p$ divides & $p$-part of $G$ & Subgroup type containing a \\ & & is $p$-part of & Sylow $p$ subgroup \\ \hline $30$ & $q^{15}+1$ & $q^8 \!+ \!q^7\! - \! q^5\! - \! q^4\! - \! q^3\! + \! q\! + \! 1$ & see Lemma \\ $30$ & $q^{15}- 1$ & $q^8\! - \! q^7\! + \! q^5\! - \! q^4\! + \! q^3\! - \! q\! + \! 1$ & see Lemma\\ $24$ & $q^{12}+1$ & $q^8-q^4+1$ & $SU(3,q^4)$ \\ $24$ & $q^{12}-1$ & x & \\ $20$ & $q^{10}+1$ & $q^8-q^6+q^4-q^2+1$ & $SU(5,q^2)$ \\ $20$ & $q^{5}+1$ & $(q^4-q^3+q^2-q+1)^2$ & $SU(5,q) \circ SU(5,q)$ \\ $20$ & $q^{5}-1$ & $(q^4+q^3+q^2+q+1)^2$ & $SL(5,q) \circ SL(5,q)$ \\ $18$ & $q^9+1$ & $q^6-q^3+1$ & $SU(9,q)$ \\ $18$ & $q^9-1$ & $q^6+q^3+1$ & $SL(9,q)$ \\ $14$ & $q^7+1$ & $q^6\! - \! q^5\! + \! q^4\! - \! q^3\! + \! q^2\! - \! q\! + \! 1$ & $SU(9,q)$ \\ $14$ & $q^7-1$ & $q^6\! + \! q^5\! + \! q^4\! + \! q^3\! + \! q^2\! + \! q\! + \! 1$ & $SL(9,q)$ \\ $12$ & $q^6+1$ & $(q^4-q^2+1)^2$ & $SU(3,q^2)\circ SU(3,q^2)$ \\ $12$ & $q^3+1$ & $(q^2-q+1)^4(5,p)$ & ${^3}D_4(q) \circ {^3}D_4(q)$ \\ $12$ & $q^3-1$ & $(q^2+q+1)^4(5,p)$ & ${^3}D_4(q) \circ {^3}D_4(q)$\\ $8$ & $q^4+1$ & $(q^4+1)^2$ & $SU(3,q^4)$ \\ $8$ & $q^2+1$ & $(q^2+1)^4(5,p)$& $SU(5,q^2)$\\ $2$ & $q+1$ & $(7,p)(5,p)^2(q+1)^8$ & $SU(5,q)\!\circ \! SU(5,q)$ or $SU(9,q)$\\ $2$ & $q-1$ & x &\\ \hline \end{tabular} \end{center} \end{table} \end{proof} \begin{lemma} If $G_0 \cong G_2(q)$, ${^3}D_4(q)$, or $^2F_4(2^{a})$, then $(x,G)$ cannot be a minimal counterexample. \end{lemma} \begin{proof} The proof is similar to that of Lemma \ref{lem:El}. We may assume that $x$ is semisimple by Lemmas \ref{unipotents} and \ref{lem:outinvs}. Also, since $G_2(2)' \cong PSU(3,3)$, we can eliminate this case. If $G_0 \cong G_2(q)$, then $x$ normalizes but does not centralize a subgroup of type $SL^{\epsilon}(3,q)$ (see \cite[p. 546]{GS}). So $G_0 \not\cong G_2(q)$. If $G_0$ is ${^3}D_4(q)$, or ${^2}F_4(2^{2a+1})$ then we list the possible expressions in $q$ that could be divisible by $p$ in Tables \ref{tab:Syl3D4} and \ref{tab:Syl2F4}. Since $p\ge 5$, $p$ divides precisely one of these expressions. In most cases, we can deduce that $(x,G)$ cannot be a minimal counterexample. If $G_0 \cong {^2}F_4(2^{a})$, then we may therefore assume that $p \| q^4-q^2+1$. In this case, either $p \| q^2+\sqrt{2q^3} +q + \sqrt{2q}+ 1$ or $p \| q^2-\sqrt{2q^3} +q -\sqrt{2q}+ 1$, and from the list of maximal subgroups of $G$ (see \cite{2F4}), we may assume that $x$ is only contained in a (single) torus $T$ of order $q^2+\sqrt{2q^3} +q + \sqrt{2q}+ 1$ or $q^2-\sqrt{2q^3} +q -\sqrt{2q}+ 1$. Thus we can pick an involution $y \in G$ that is not contained in the normalizer of $T$ and $\langle x,y\rangle$ will not be solvable. Suppose that $G_0 \cong {^3}D_4(q)$. We note that if $p \| q^2 - q+1$, then $x$ is contained in a subgroup of type $(q^2-q+1) \circ SU(3,q)$. If $x$ does not centralize the $SU(3,q)$, then $(x,G)$ cannot be a minimal counterexample. But if $x$ does centralize the $SU(3,q)$ subgroup, then $x$ centralizes unipotent elements and is therefore contained in a parabolic subgroup. By Lemma \ref{lem:2.2} we may assume that $x$ is noncentral in the Levi subgroup, which is of type $SL(2,q)$ or $SL(2,q^3)$. But if $q > 3$, then $x$ cannot be a minimal counterexample since these Levi components are normalized by a split torus, which induces diagonal automorphisms, and we can therefore reduce to the case $SL(2,q) \le G \le GL(2,q)$ where $SL(2,q)$ has even index in $G$ when $q$ is odd. We can verify the cases $q=2$ and $q=3$ in MAGMA. The case where $p \| q^2+q+1$ is the same argument. The only remaining case is where $p \| q^4-q^2+1$. In this case, the list of maximal subgroups in \cite{3D4} allows us to assume that $x$ is only contained a (single) torus $T$ of order $(q^4-q^2+1)$. We can then choose an involution $y$ that is not contained in the normalizer of $T$. \qedhere \begin{table}[htdp] \begin{center} \caption{ Subgroups of ${^2}F_4(q)$ containing a Sylow $p$-subgroup, $q=2^a$\label{tab:Syl2F4}} \begin{tabular}{ccc} \hline $p$ divides & $p$-part of $G$ & Subgroup type containing a \\ & is $p$-part of & Sylow $p$ subgroup \\ \hline $q^4-q^2+1$& $q^4-q^2+1$ & see Lemma\\ $q^2+1$ & $(q^2+1)^2$ & ${^2}B_2(q) \!\circ\! {^2}B_2(q)$\\ $q+1$ & $(q+1)^2$ & $SU(3,q)$ \\ $q^2-q+1$ & $q^2-q+1$ & $SU(3,q)$ \\ $q-1$ & $(q-1)^2$ & $Sp(4,q)$ \\ \hline \end{tabular} \end{center} \end{table} \begin{table}[htdp] \begin{center} \caption{ Subgroups of ${^3}D_4(q)$ containing a Sylow $p$-subgroup\label{tab:Syl3D4}} \begin{tabular}{ccc} \hline $p$ divides & $p$-part of $G$ & Subgroup type containing a \\ & is $p$-part of & Sylow $p$ subgroup \\ \hline $q^4-q^2+1$ & $q^4-q^2+1$ & see Lemma\\ $q^2-q+1$ & $(q^2-q+1)^2$& $(q^2-q+1)\circ SU(3,q)$ \\% either x is non central in SU3 or its centralizer contains a unipotent so x is in a parabolic subgroup. $q^2+q+1$ & $(q^2+q+1)^2$ & $(q^2+q+1) \circ SL(3,q)$ \\ $q+1$ & $(q+1)^2$ & $G_2(q)$\\ $q-1$ & $(q-1)^2$ & $G_2(q)$ \\ \hline \end{tabular} \end{center} \end{table} \end{proof} \begin{lemma} If $G_0 \cong {^2}G_2^{\prime}(3^a)$, then $(x,G)$ cannot be a minimal counterexample. \end{lemma} \begin{proof} We may assume that $a \ne 1$ since ${^2}G_2^{\prime}(3) \cong L(2,8)$. Since $q=3^a$ and by Lemma \ref{lem:outinvs}, we may assume that $x$ is semisimple. Now $\|G_0\|=q^3(q^3+1)(q-1)$ and the maximal subgroups are given in \cite{G2}. Since $p \nmid q$ there are three mutually exclusive possibilities: $p \mid (q^2-1)$, $p \mid q - \sqrt{3q}+1$, and $p \mid q + \sqrt{3q}+1$. First suppose that $p \mid q^2-1$. Then a Sylow $p$-subgroup lies inside a maximal subgroup $2 \times L(2,q)$, so $(x,G)$ cannot be a minimal counterexample in this case. If $p \mid q^2-q+1$, then a Sylow $p$-subgroup is contained in one of the abelian Hall subgroups of order $q \pm \sqrt{3q}+1$, so we may assume that $x$ lies in one of these Hall subgroups and that $\|C_G(x)\| = q \pm \sqrt{3q}+1$ (see part (4) of the main theorem in \cite{Ward}). If $(x,G)$ is a minimal counterexample, then $x$ can only be contained in a single subgroup $H$, which is either of type $\mathbb{Z}_{q + \sqrt{3q}+1}:\mathbb{Z}_{6}$ or $\mathbb{Z}_{q - \sqrt{3q}+1}:\mathbb{Z}_{6}$. We choose an involution $y \in G$ that is not contained in $H$ and then $\langle x,y\rangle$ is not solvable. \end{proof} For the Suzuki groups, we will use a counting argument. \begin{lem} \label{lem:countinv} Let $G$ be an almost simple group with socle $G_0$. Suppose that $X_1, \ldots X_k$ are representatives for the conjugacy classes of maximal subgroups of $G$ that contain $x$ and that do not contain $G_0$. Let $Y$ be the set of involutions in $G_0$. If \begin{align} \label{eqn:countinv} \|Y\|> \sum_{i=1}^{k} \frac{\|x^G \cap X_i\|\|G:X_i\|\|Y\cap X_i\|}{\|x^G\|}, \end{align} then there exists an involution $y$ in $G_0$ such that $\langle x,y \rangle$ contains $G_0$. \end{lem} \begin{proof} Suppose that $\langle x,y\rangle$ does not contain $G_0$ for all $y \in Y$. For each $i$, let $X_{i1}, \ldots X_{in_i}$ be the conjugates of $X_i$ that contain $x$. In particular, $n_i$ is the number of conjugates of $X_i$ that contain $x$. Thus we have \begin{align} \|Y \cap \bigcup_{i,j} X_{ij}\| \le & \sum_{i=1}^k n_i\|Y\cap X_i\|. \end{align} But \begin{displaymath} \frac{n_i}{\|G:X_{ij}\|}= \frac{\|x^G \cap X_i\|}{\|x^G\|}, \end{displaymath} and thus \begin{displaymath} \|Y \cap \bigcup_{i,j} X_{ij}\| \le \sum_{i=1}^k \frac{\|x^G \cap X_i\|\|G:X_{i}\|\|Y\cap X_i\|}{\|x^G\|}. \end{displaymath} This is a contradiction, since we assumed that $Y = Y \cap \bigcup_{i,j}X_{ij}$. \end{proof} \begin{lemma} If $G_0 \cong {^2}B_2(2^a)$ then $(x,G)$ cannot be a minimal counterexample. \end{lemma} \begin{proof} There are $(q^2+1)(q-1)$ involutions in $G_0={^2}B_2(q)$, and the maximal subgroups of $G_0$ are given in \cite{2B2}. If $x$ is inner-diagonal, then since $p$ is odd, there are three mutually exclusive possibilities: $p \mid q-1$, $p \mid q+\sqrt{2q}+1$, and $p \mid q-\sqrt{2q}+1$. We will show that $(x,G)$ cannot be a minimal counterexample using Lemma \ref{lem:countinv}. If $p \mid q-1$, then the maximal subgroups that could contain $x$ are Frobenius groups of order $q^2(q-1)$, dihedral groups of order $2(q-1)$ and ${^2}B_2(2)$. If $X_i$ is a Frobenius group, then $\|Y \cap X_i\|=q-1$, by \cite[Theorem 2]{2B2}, and \begin{displaymath} \frac{\|x^G \cap X_i\|\|G:X_i\|\|Y\cap X_i\|}{\|x^G\|} \le \frac{(q^3-q^2-q)(q^2+1)(q-1)}{q^2(q^2+1)} \le q^2-q-1. \end{displaymath} It follows that \begin{displaymath} \sum_{i=1}^{k} \frac{\|x^G \cap X_i\|\|G:X_i\|\|Y\cap X_i\|}{\|x^G\|} \le (q^2-q-1)+\frac{(q-1)^2}{2} + (q-1), \end{displaymath} which is less than the number of involutions $(q^2+1)(q-1)$ in $G_0$ for $q \ge 8$. If $p \mid q+\sqrt{2q}+1$, then $x$ could be contained in a group $\mathbb{Z}_{(q+\sqrt{2q}+1)}:[4]$ or ${^2}B_2(2)$, thus \begin{displaymath} \sum_{i=1}^{k} \frac{\|x^G \cap X_i\|\|G:X_i\|\|Y\cap X_i\|}{\|x^G\|} \le (q+\sqrt{2q}+1)^2 + (q+\sqrt{2q}+1) \end{displaymath} and this is less than $(q^2+1)(q-1)$ for $q \ge 8$. Similarly, if $p \mid q-\sqrt{2q}+1$, then $x$ could be contained in a group $\mathbb{Z}_{(q-\sqrt{2q}+1)}:[4]$ or ${^2}B_2(2)$, so \begin{displaymath} \sum_{i=1}^{k} \frac{\|x^G \cap X_i\|\|G:X_i\|\|Y\cap X_i\|}{\|x^G\|} \le (q-\sqrt{2q}+1)^2+(q-\sqrt{2q}+1). \end{displaymath} So $(x,G)$ is not a minimal counterexample when $x$ is inner-diagonal. If $x$ is a field automorphism, then we can use the same counting argument as for the case $G_0=PSL(2,q)$. Indeed, we would like to show that the right hand side of \begin{align} \label{eqn:2b2calc} \|\Gamma\| \le \sum_{i=1}^{k-1} \frac{i_2(X_i)\|C_{G_0}(x)\|}{\|C_{X_i}(x)\|}+\|C_{G_0}(x)\| \end{align} is less than the number of involutions $(q^2+1)(q-1)$ in $G_0$. The possibilities for the maximal subgroups of $G_0$ containing $\langle x,y \rangle \cap G_0$ are a Frobenius group of order $q^2(q-1)$, a dihedral group of order $2(q-1)$, the normalizer of a cyclic group $\mathbb{Z}_{(q-\sqrt{2q}+1)}:[4]$, the normalizer of a cyclic group $\mathbb{Z}_{(q+\sqrt{2q}+1)}:[4]$, and the centralizer of $x$, ${^2}B_2(q^{1/p})$. We label these subgroups $X_1$, $X_2$, $X_3$, $X_4$, and $X_5$ respectively. Therefore, if $q_0:=q^{1/p}$, then \begin{align} \sum_{i=1}^{k-1} \frac{i_2(X_i)\|C_{G_0}(x)\|}{\|C_{X_i}(x)\|}\!+\!\|C_{G_0}(x)\| \le& \frac{(q_0^p-1)q_0^2(q_0^2+1)(q_0-1)}{q_0^2(q_0-1)}\\ & + \frac{(q_0^p-1)q_0^2(q_0^2+1)(q_0-1)}{2(q_0-1)} \\ & + \frac{3(q_0^p-\sqrt{2q_0^p}+1)q_0^2(q_0^2+1)(q_0-1)}{4(q_0+\sqrt{2q_0}+1)}\\ & +\frac{3(q_0^p+\sqrt{2q_0^p}+1)q_0^2(q_0^2+1)(q_0-1)}{4(q_0-\sqrt{2q_0}+1)} \\ & + q_0^2(q_0^2+1)(q_0^2-1) \\ \le & (q_0^p-1)(q_0^2+1) + \frac{(q_0^p-1)q_0^2(q_0^2+1)}{2} \\ & + \!2(q_0^p\!+\!(2q_0^p)^{ \frac{1}{2}} \! +\!1)q_0^2(q_0\!+\!(2q_0)^{ \frac{1}{2}}\!+\!1)(q_0 \!- \!1) \\ & + q_0^2(q_0^2+1)(q_0^2-1). \end{align} Since $p \ge 5$, an elementary calculation shows that (\ref{eqn:2b2calc}) holds; thus $(x,G)$ cannot be a minimal counterexample. \end{proof} \begin{lemma} Let $G_0$ be a sporadic group. Then $(x,G)$ cannot be a minimal counterexample. \end{lemma} \begin{proof} We can verify the sporadic groups in GAP using the character table library. We use Thompson's result \cite[Corollary 3]{Thompson} that a group $H$ is nonsolvable if and only if there exists $a,b,c \in H$ of pairwise coprime order such that $abc=1$. Using the character table, we can check that for any $x$ of prime order $p \ge 5$, there exists an involution $y$ such that $yx$ has order coprime to $2$ and $p$. \end{proof} This completes the proof of Theorem \ref{inv}. \section{Proof of Theorem 1.6} Note that if there is a unique class of involutions, then $G$ cannot be a minimal counterexample since Malle, Saxl and Weigel \cite{MSW} prove that there exist three involutions in $G_0$ that generate $G_0$ unless $G_0 \cong PSU(3,3)$. Also, Guralnick and Saxl \cite{GS} prove that there exist three conjugates of any involution in an almost simple group that generate a subgroup containing the socle when the Lie rank is small. We will appeal to both of these result throughout the proof. \begin{lemma} Suppose that $G_0 \cong A_n$. Then $(x,G)$ cannot be a minimal counterexample. \end{lemma} \begin{proof} Suppose that $(x,G)$ is a minimal counterexample with $G_0 \cong A_n$. We may assume by minimality that one of the following four cases hold: (i) $x=(12)(34)$ and $n=5$; (ii) $x=(16)(25)(34)$ and $n=7$; (iii) $x=(12)(34)(56)(78)$ and $n=8$; (iv) $x$ is an automorphism of $A_6$ not contained in $S_6$. In case (i), let $g_1=(12345)$, $g_2=(345)$ so that $xx^{g_1} =(13542)$, $xx^{g_2}= (354)$ and so $\langle x,x^{g_1},x^{g_2}\rangle \cong A_5$. In case (ii), let $g_1=(1743526)$ and $g_2=(23654)$ so that $xx^{g_1}=(1234567)$, $xx^{g_2}=(12)(56)$, and $\langle x,x^{g_1}x^{g_2} \rangle \cong S_7$. In case (iii), let $g_1=(143)(28567)$, $g_2=(13)(265874)$, then $xx^{g1}=(1574)(2386)$, $xx^{g2}=(375)(468)$, and $xx^{g2}x^{g1}=(1364725)$ and thus $\langle x,x^{g_1},x^{g_2} \rangle \cong PSL(2,7)$ \cite{Atlas}. It is straightforward to rule out case (iv) in MAGMA. \end{proof} \begin{lemma} Suppose that $G_0$ is a simple group of Lie type and $G_0 \triangleleft G \le\mathrm{Inndiag}(G_0)$. Then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} If the twisted Lie rank of $G_0$ is 1, then $G_0 \cong PSL(2,q)$, $PSU(3,q)$, ${^2}B_2(2^a)$, or ${^2}G_2(3^a)$. For all of these groups except $PSL(2,q)$, there is a unique class of involutions in ${\rm Inndiag}(G_0)$ and so \cite{MSW} implies that $(x,G)$ is not a minimal counterexample unless $G_0 \cong PSU(3,3)$. And if $G_0 \cong PSL(2,q)$, then there exist three conjugates that generate a group containing $G_0$ by \cite{GS}. So we may assume that the twisted Lie rank of $G_0$ is at least 2. First suppose that $q \ge 4$. If $x$ is contained in a maximal parabolic subgroup, then we may assume that $x$ acts nontrivially on the components of the Levi complement, which are not contained in Table \ref{table:exceptions} except for those of type $PSL(2,4)$, $PSL(2,5)$, and $PSL(2,9)$. In fact, the involution $x$ is always contained in a parabolic subgroup. Indeed, if $q$ is even, then $x$ is unipotent and if $q$ is odd, then $x$ is semisimple and $x$ always centralizes a unipotent element $u$ (see the list of semisimple involutions and their centralizers \cite[Table 4.5.1]{GLS}) and the Borel--Tits Theorem implies that $C_{G}(u)$ is contained in a parabolic subgroup of $G$. Now any maximal parabolic subgroup has an almost simple component that is not of type $PSL(2,4)$, $PSL(2,5)$, or $PSL(2,9)$ unless \begin{align} G_0 \in \{ & A_2(q), {^2}A_3(q), B_2(q), B_3(q), C_3(q), D^{\pm}_4(q), {^3}D_4(q),\\ & G_2(q) \mid q =4,5,\!\!\mbox{ or } 9 \}. \end{align} However we can eliminate the groups that have a unique classes of involutions in ${\rm Inndiag}(G_0)$. So if $q \ge 4$, $x\in {\rm Inndiag}(G_0)$ and $(x,G)$ is a minimal counterexample, then $G_0 \cong {^3}D_4(4)$, ${^2}A_3(q)$, $B_2(q)$, $B_3(q)$, $C_3(q)$, $G_2(4)$, or $D_4^{\pm}(q)$ for $q=4$, $5$, or $9$. But \cite[Lemma 3.2]{GS} and the proof of \cite[Lemma 3.4]{GS} eliminate $A_2(q)$ and ${^2}A_3(q)$. Now a (unipotent) involution in ${^3}D_4(4)$ is contained in a subfield subgroup ${^3}D_4(2)$ (see \cite{Spaltenstein}) and so we can eliminate the case $G_0\cong{^3}D_4(4)$. If $G_0 \cong G_2(4)$, then any involution is contained in a subgroup of type $SL^{\pm}(3,q)\!:\!2$ (see for example \cite[Proposition 5.6]{GS}). If $G_0 \cong B_3(4)\cong C_3(4)$, then we may assume that $x$ is contained an end-node parabolic subgroup $P$ and not in the unipotent radical of $P$; the Levi complement will be of type $C_2(4)$ or $A_2(4)$. Thus $G_0 \not\cong C_3(4)$. The same reasoning eliminates $D^{\pm}_4(4)$. Thus the remaining possibilities for $q \ge 4$ are \[ G_0 \in \{ B_2(4), B_2(5), B_2(9), B_3(5), B_3(9), C_3(5),C_3(9), D^{\pm}_4(5),D^{\pm}_4(9), G_2(4)\}. \] If $G_0 \cong C_3(5)$ or $C_3(9)$, then \cite[Propostion 1.5]{LS} shows that $x$ stabilizes an orthogonal decomposition $V = W \oplus W^{\perp}$, where $\dim W=4$ and $x$ acts noncentrally on $W$. So we can reduce to the case $C_2(5)$ or $C_2(9)$ and $(x,G)$ cannot be a minimal counterexample. Similarly if $G_0 \cong B_3(5)$ or $B_3(9)$, by \cite[Propostion 1.5]{LS}, $x$ stabilizes an orthogonal decomposition $V = W \oplus W^{\perp}$, where $\dim W=5$ or $6$ and $x$ acts noncentrally on $W$. Since $\mathrm{P\Omega}(5,q)$ and $\mathrm{P\Omega}^{\pm}(6,q)$ for $q=5$ and $9$ are not listed in Table \ref{table:exceptions}, $(x,G)$ cannot be a minimal counterexample. Similarly if $G_0 \cong D_4^{\pm}(5)$ or $D^{\pm}_4(9)$, then $x$ will stabilize an orthogonal decomposition as above where $\dim W =6$ or $7$ and $x$ acts noncentrally on $W$. Since $\mathrm{P\Omega}^{\pm}(6,q)$ and $\mathrm{P\Omega}(7,q)$ are not listed in Table \ref{table:exceptions} for $q=5$ or $9$, $(x,G)$ cannot be a minimal counterexample. So for $q \ge 4$ it remains to consider \[ G_0 \in \{ B_2(4), B_2(5), B_2(9) \}. \] We can check in MAGMA that the theorem holds for these groups. Now suppose that $q=3$. Then $x$ is contained in a maximal parabolic subgroup, and by Lemma \ref{lem:2.2}, we may assume that it acts noncentrally on all of the Levi components. To prove that $G$ is not a minimal counterexample, it suffices that one of the Levi components is not in Table \ref{table:exceptions} and is not solvable. Thus the only possibilities with $q=3$ are \begin{align} G_0 \in \{ & A_2(3), A_3(3), {^2}A_2(3), {^2}A_3(3), {^2}A_4(3), {^2}A_5(3), B_n(3),\\ & C_3(3), C_4(3), D_4^{\epsilon}(3), {^3}D_4(3), G_2(3), {^2}G_2(3) \} \end{align} If we eliminate the cases where there is a unique class of inner involutions, then we may assume that \begin{align} G_0 \in \{ & A_3(3), {^2}A_2(3), {^2}A_3(3), {^2}A_4(3), {^2}A_5(3), \\ & B_n(3), C_3(3), C_4(3), D_4^{\epsilon}(3)\}. \end{align} If $G_0 \cong B_n(3)$, then by \cite[Proposition 1.5]{LS}, $x$ stabilises an orthogonal decomposition $W \oplus W^{\perp}$, where $W$ is chosen so that $\dim W^{\perp}$ is minimal, and is therefore at most 2. If $n \ge 4$ and $x$ is not a reflection then we can choose $W$ so that $x$ acts on $W$ noncentrally and not as a reflection. Thus $(x,G)$ cannot be a minimal counterexample if $G_0=B_n(3)$ and $n \ge 4$. Checking the remaining cases in MAGMA shows that there are no minimal counterexamples when $q=3$. Now suppose that $q=2$ so that $x$ is unipotent. Then we can use Lemma \ref{lem:2.2}. If $G_0=A_n(2)$, then we may assume that $x$ is contained in a maximal end-node parabolic subgroup $P$ and not contained in the unipotent radical of $P$. Thus we can reduce to the case that $x \in A_{n-1}(2)$ since there are no {\it inner} exceptions in $A_{n-1}(2)$. This argument thus reduces to the case of $G=PSL(3,2) \cong PSL(2,7)$, and \cite{GS} shows that $(x,G)$ is not a counterexample. Now suppose that $G_0 \cong {^2}A_n(2)$. Then \cite[Proposition 1.4]{LS} shows that $x$ stabilizes an orthogonal decomposition $V=W \oplus W^{\perp}$ such that $W$ has codimension $1$ or $2$, and if $x$ is not a unitary transvection, we can choose $W$ such that $x$ does not act on $W$ trivially, or as a transvection. Thus if $n \ge 6$, then there are no minimal counterexamples with $G_0 \cong {^2}A_n(2)$. We can check in MAGMA that the only exceptions when $n \le 5$ are the unitary transvections. Now suppose that $G_0 \cong B_n(2) \cong C_n(2)$ or $D_n^{\pm}(2)$. Suppose that $x$ is not a transvection. If $n \ge 5$, then we can take an orthogonal decomposition $V=W \oplus W^{\perp}$ as above using \cite[Proposition 1.4]{LS} such that $x$ does not act on $W$ trivially or as a transvection, and $\dim W \ge 6$. Thus $(x,G)$ cannot be a minimal counterexample when $n \ge 5$. Note that $Sp(4,2) \cong S_6$ so we can just verify in MAGMA that $Sp(6,2)$, $Sp(8,2)$, and $\Omega^{\pm}(8,2)$ cannot be counterexamples to the theorem. To complete the analysis when $q=2$, we eliminate the cases $G_0 \cong {^3}D_4(2)$, $E_6(2)$, ${^2}E_6(2)$, $E_7(2)$, $E_8(2)$, $F_4(2)$, ${^2}F_4(2)$, and $G_2(2)$ using MAGMA. \end{proof} \begin{lemma} Suppose that $G_0$ is a simple group of Lie type and $x$ is a field automorphism of $G_0$. Then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} Now suppose that $x$ is a field automorphism of order $2$. By \cite[7.2]{GL}, we may assume that $x$ is a standard field automorphism. First observe that if $G$ is a Suzuki--Ree group then all field automorphisms have odd order. So we may assume that $G$ is not a Suzuki--Ree group. Now $q \ge 4$ and $x$ will act as a field automorphism on a $SL(2,q)$ subgroup and so $(x,G)$ cannot be a minimal counterexample unless $G_0=PSL(2,q)$ or $q=4$ or $9$. If $q=4$ or $9$, then we may assume that $x$ acts as a field automorphism on a subgroup of type $A_2(q)$, $B_2(q)$, or $G_2(q)$ or ${^3}D_4(q)$. We can eliminate the first two cases in MAGMA. If $G_0 \cong G_2(q)$ or ${^3}D_4(q)$, then $x$ normalizes but does not centralize a subgroup of type $SL(3,q)$ or $SL(2,q^3)$ respectively. So it remains to treat the case where $x$ is a field automorphism of $PSL(2,q)$. If $q$ is even, then since $q\ne 4$, we have $q=q_0^2$ where $q_0 \ge 4$. But, then there exist $y$ of order $q_0-1$ and $z$ of order $q_0+1$ such that $x$, $xy$, and $xz$ are conjugate, and $y$ and $z$ do not commute and thus $\langle x,xy,xz \rangle$ contains $PSL(2,q_0)$, which is not solvable. If $q$ is odd, then suppose that $\mathbb{F}_q^* = \langle w \rangle$. Let $\lambda = w^{\frac{(q_0+1)}{2}}$ so that $\lambda^{q_0}=-\lambda$. Then $x$ inverts \[ \left ( \begin{matrix} 1& \lambda \\ 0 & 1 \end{matrix} \right) \] and \[ \left( \begin{matrix} 1& 0 \\ \lambda & 1 \end{matrix} \right) \] thus there exist conjugates of $x$, $y$ and $z$ say such that $xy$ and $xz$ are the transvections above. It follows that $\langle x,y,z\rangle$ contains a subgroup of type $PSL(2,q_0)$ which is not solvable since $q > 9$. \\ \end{proof} \begin{lemma} Suppose that $G_0$ is a simple group of Lie type and $x$ is a graph-field automorphism of $G_0$. Then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} Now suppose that $x$ is a graph-field automorphism so that $G_0$ is an untwisted simple group of Lie type. By \cite[7.2]{GL}, we may assume that $x$ is a standard graph-field involution. Define $q_0$ by $q=q_0^2$ as before. If $G_0 \cong PSL(d,q)$, with $d\ge 3$, then $x$ normalizes a subfield subgroup $PSL(d,q_0)$ (acting as a graph automorphism), which is not an exception unless $G_0 \cong PSL(4,4)$, since $PSL(4,2)$ is in Table \ref{table:exceptions}. We can eliminate this case in MAGMA. If $G_0 \cong D_m(q)$ and $m \ge 4$, then $x$ will act as a graph field automorphism on a $D_{m-1}(q)$ subgroup. Similarly a graph-field automorphism of $E_6(q)$ will act as a graph-field automorphism on $A_5(q)$. If $G_0 \cong F_4(2^a)$, $G_2(3^a)$, or $B_2(2^a)$ then the extraordinary `graph' automorphism squares to the generating field automorphism. So there are involutary graph-field automorphisms (in the sense of \cite[2.5.13]{GLS}) only when $a=1$; these cases are easily eliminated in MAGMA. \end{proof} \begin{lemma} Suppose that $G_0$ is a simple group of Lie type and $x$ is a graph automorphism of $G_0$. Then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} We use the terminology of \cite[2.5.13]{GLS}. So there are no graph automorphisms of $F_4(2^a)$, $G_2(3^a)$, or $B_2(2^a)$, and no graph-field or graph automorphisms of the Suzuki--Ree groups. If $G_0 \cong L^{\epsilon}(3,q)$, then \cite[Lemmas 3.2 and 3.3]{GS} show that $(x,G)$ cannot be a minimal counterexample. If $G_0 \cong L^{\epsilon}(4,q)$, then observe that $G_0 \cong \mathrm{P\Omega}^{\epsilon}(6,q)$, and an involutory graph automorphism of $L^{\epsilon}(4,q)$ is contained in $PCO^{\epsilon}(6,q)$. By \cite[Propositions 1.4 and 1.5]{LS} for example, $x$ will normalize (and not centralize) a subgroup of type $\mathrm{P\Omega}(5,q)$, $\mathrm{P\Omega}^{\pm}(4,q)$, or $\mathrm{P\Omega}(3,q)$. Therefore $(x,G)$ cannot be a minimal counterexample unless $q=2$ or $q=3$. We can verify that Theorem \ref{thm:3invs} holds for $q=2$ and $3$ using MAGMA. Suppose that $G_0\cong L^ \epsilon(d,q)$, $q$ is odd, and $d \ge 5$. The class representatives of graph involutions in this case are given in \cite[Table 4.5.1]{GLS} and \cite[3.9]{LS}. We can deduce from these representatives that if $d$ is even, then $x$ will act as a graph automorphism on a type $PSL(d-1,q)$ or $PSL(d-2,q)$ subgroup; if $d$ is odd, then $x$ will act as a graph automorphism on a type $PSL(d-1,q)$ subgroup. So since $(x,G)$ is a minimal counterexample, we can reduce to the case $G_0 \cong L^ \epsilon(4,q)$ since $L^{\epsilon}(3,q)$ has been eliminated. Now suppose that $G_0\cong L^ \epsilon(d,q)$, $q$ is even, and $d \ge 5$. The class representatives for graph involutions when $q$ is even can be found in \cite[Lemma 3.7]{Moufang}. There are two classes when $d$ is even and one class when $d$ is odd. In all cases, $x$ normalizes a subgroup of type $L^{\epsilon}(d-1,q)$ or $L^{\epsilon}(d-2,q)$, acting as a graph automorphism. So since $(x,G)$ is a minimal counterexample, we may assume that $G_0 \cong L^ \epsilon(3,q)$,$L^ \epsilon(4,q)$, $L^ \epsilon(5,2)$, or $L^ \epsilon(6,2)$. We can eliminate the last two possibilities using MAGMA. If $x$ is an involutory graph automorphism of $P\Omega^{\pm}(d,q)$, then $x \in PCO^{\pm}(d,q)$ and we may assume that $d \ge 8$. By \cite[Lemma 3.3]{LS} for example, we may assume that $x$ normalizes but does not centralize a subgroup of type $\mathrm{P\Omega}^{\epsilon}(d-b,q)$, where $b \le 4$. Thus $(x,G)$ cannot be a minimal counterexample unless $q=2$, or $q=3$. But if $d \ge 10$ and $x$ is not an orthogonal transvection or reflection, then we may assume that $x$ does not act as orthogonal transvection or reflection in the type $\mathrm{P\Omega}^{\epsilon}(d-b,q)$ subgroup. We can eliminate the groups $G_0 \cong \mathrm{P\Omega}^{\epsilon}(8,3)$ and $\mathrm{P\Omega}^{\epsilon}(8,2)$ in MAGMA. If $G_0 \cong E^{\epsilon}_6(q)$ and $x$ is an involutary graph automorphism, then $x$ normalizes but does not centralize a subgroup of type $F_4(q)$; thus $(x,G)$ cannot be a minimal counterexample. This follows from an analysis of the standard class representatives of graph involutions found in \cite[Lemma 3.6]{Moufang} for $q$ odd and \cite[19.9]{AS} for $q$ even. See the proof of \cite[Proposition 5.2]{GS} for example. \end{proof} \begin{lemma} If $G_0$ is a sporadic group then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} If $G_0$ is not the Monster group $M$ or the Baby Monster group $B$, then we can use MAGMA. If there is a unique class of involutions then $(x,G)$ cannot be a minimal counterexample by \cite{MSW}. In the other cases, we use the representations in \cite{wwwatlas} together with the representatives for the conjugacy classes of involutions. We search at random for conjugates $y$ and $z$ of $x$ and test the subgroup $H= \langle x,y,z\rangle$ for nonsolvability (by searching for $a,b,c \in H$ of pairwise coprime order such that $abc=1$ \cite{Thompson}). If $G=B$, then there are 4 classes of involutions. The centralizer order of an element in class 2A in the Harada Norton group $HN$ is divisible by $5^3$, thus such an involution is contained in $B$-class 2B. Any involution in $Th$ is in $B$-class 2D by \cite[pg 4]{bmonster1}. We can verify in MAGMA that an element in 2A belongs to a triple of conjugates generating a nonsolvable group. If $x$ belongs to class 2C, then the character table of $G$ implies that there exist conjugates $y$ and $z$ such that $xy$ has order 19 and $xz$ has order 33. By analyzing the maximal subgroups, $\langle x,y,z \rangle=B$. \\ There are two classes of involutions in the Monster group $M$. If $x$ is in class 2A, then $x$ is contained in a subgroup isomorphic to $PSL(3,2)$; for an involution in $PSL(3,2)$ in the maximal subgroup $(PSL(3,2)\times Sp(4,4):2).2$ must be in class $2A$ since it centralizes an element of order 17 in $Sp(4,4)$ and 2B elements do not centralize elements of order 17. Any involution in $PSU(3,8)$ is in $M$-class 2B by \cite[4.5]{anatomy}; thus neither class of involutions can be involved in a minimal counterexample. \end{proof} This completes the proof of Theorem \ref{thm:3invs}. \section{Proof of Theorem 1.4} \subsection{Order 9} First suppose that $x$ has order $9$. Suppose that $G_{0}\!\cong \!\mathrm{PSL}(d,3)$, $\mathrm{PSU}(d,3)$, or $\mathrm{PSp}(d,3)$ and $x^{3}$ is a transvection. Then $x$ lifts to an element in $\mathrm{SL}(d,3), \mathrm{SU}(d,3),$ or $\mathrm{Sp}(d,3)$, with Jordan form $J_{4}J_{3}^{r_{3}}J_{2}^{r_{2}}J_{1}^{r_{1}}$ (and $r_{3}$ and $r_{1}$ are even in the symplectic case). It is well known that in the linear and unitary cases, $x$ must be contained in a subgroup of the form $\mathrm{SL}(4,3) \times \mathrm{SL}(d-4,3)$ or $\mathrm{SU}_{4}(4,3) \times \mathrm{SU}(d-4,3)$, where $x$ has order $9$ in the first factor. In the symplectic case, we can also assume that $x \in \mathrm{Sp}(4,3) \times \mathrm{Sp}(d-4,3)$ and $x$ has order $9$ in the first factor, for example by \cite[Thm 2.12]{LSgood}. Similarly, if $G_{0}=\mathrm{P\Omega}^{\epsilon}(d,3)$ and $x^{3}$ is a long root element, then $x$ lifts to an element in $\mathrm{\Omega}^{\epsilon}_{n}(3)$, which has Jordan form $J_{4}^{2}J_{3}^{r_{3}}J_{2}^{r_{2}}J_{1}^{r_{1}}$ or $J_{5}J_{3}^{r_{3}}J_{2}^{r_{2}}J_{1}^{r_{1}}$. Again by \cite[Thm 2.12]{LSgood} for example, we may assume that $x$ is contained in a subgroup of type $\mathrm{O}^{\epsilon}(8,3) \times \mathrm{O}^{\epsilon}(d-8,3)$ or $\mathrm{O}(5,3) \times \mathrm{O}^{\epsilon}(d-5,3)$, in which $x$ has order $9$ in the first factor. Thus we have reduced to the cases $G_{0}= \mathrm{PSL}(4,3),\mathrm{PSU}(4,3)$, $\mathrm{PSp}(4,3) \cong \mathrm{P\Omega}(5,3)$, or $\mathrm{P\Omega}^{\epsilon}(8,3)$. It is easily verified in MAGMA that these groups, are not counterexamples, and that neither are the groups with $G_0 = G_2(3)$, ${^3}D_4(3)$, $D_4(3)\!:\!3$, or ${^2}G_2(3)$. If $G_0$ is one of the other exceptional groups defined over $\mathbb{F}_3$, then we can find representatives for the conjugacy classes of order $9$ using \cite{Mizuno1, Mizuno2, Shoji} together with \cite[Tables D and 9]{Lawther}. Information on the unipotent conjugacy classes of ${^2}E_6(3)$ was provided by Frank L\"{u}beck, which the author is most grateful for. In all cases, it is easily seen that $x$ is contained in an almost simple subgroup and thus cannot be a minimal counterexample. Suppose $x^3 \in G$ is a pseudoreflection and $G_0=\mathrm{PSU}(d,2)$. First note that if $x \in \mathrm{PGU}(d,2)$ and $x$ does not lift to an element of order $9$ in $\mathrm{GU}(d,2)$, then the minimal polynomial of $x$ must be of the form $t^9+\mu$, where $\mu \in \mathbb{F}_{4}$, and $\mu \ne 0,1$. It follows that the eigenvalues of $x$ must all be $9$th roots of $\mu$, and the eigenvalues of $x^{3}$ must all be cube roots of $\mu$; in particular, $x^{3}$ cannot be a pseudoreflection. Thus we may assume that $x$ lifts to an element of order $9$ in $\mathrm{GU}(d,2)$, with minimal polynomial $t^{9}-1$. The eigenvalues of $x$ must all be $9$th roots of $1$, and since $x \in \mathrm{GU}(d,2)$, the eigenvalues are permuted by the map $\lambda \mapsto \lambda^{-2}$. Thus the \emph{primitive} $9$th roots of 1 must occur as eigenvalues of $x$ in triples. Suppose that the eigenvalues of the pseudoreflection $x^{3}$ are $a$ with multiplicity $1$ and $b$ with multiplicity $d-1$. It follows that $a=1$, $b$ is a primitive cube root of unity, $d \equiv 1 \imod 3$ and the eigenvalues of $x$ are $\phi_{1}$ with multiplicity 1 where $\phi_{1}^{3}=1$, and $\phi_{2}$, $\phi_{2}^{-2}$, $\phi_{2}^{4}$, each with multiplicity $\frac{d-1}{3}$, where $\phi_{2}$ is a primitive $9$th root of unity. Since $ \mathrm{GU}_{4}(2) \times \mathrm{GU}_{3}(2) \times \cdots \times \mathrm{GU}_{3}(2)$ contains an element with the same eigenvalues (and the same Jordan form) we may assume that $x$ is contained in this subgroup, and we have reduced to the case $G_{0}=\mathrm{PSU}(4,2)$. This case is easily eliminated using MAGMA. \subsection{Order 6} By Theorem \ref{thmA}, if $(x,G)$ is a minimal counterexample, then $G_0$ is a simple group of Lie type and $q=3$ or $G_0 \cong PSU(d,2)$. Moreover, $x^2$ is a long root element or a pseudoreflection in each case respectively. We will consider the possible conjugacy classes for $x$. \par If $x \in {\rm Inndiag}(G_0)$, then let $x_s:=x^3$ and $x_u:=x^4$. So $x=x_sx_u=x_ux_s$, and $x_s$ is semisimple and $x_u$ is unipotent. \begin{lem} \label{lift6} If $G_0 \cong PSL(d,3)$, $PSp(d,3)$, or $PSU(d,3)$, $x$ is inner-diagonal and $(x,G)$ is a minimal counterexample, then $x$ lifts to an element of order $6$ in $GL(d,3)$, $GSp(d,3)$, or $GU(d,3)$ respectively. \end{lem} \begin{proof} In the linear and symplectic cases, the only central elements are $\pm I_d$, so either $(xz)^6=I_d$ for all central elements $z$, or $(xz)^6=-I_d$ for all $z$. In the latter case, let $y=xz$. Then the minimal polynomial of $y$ divides $t^6+1= (t^2+1)^3$. Moreover, since $y^2$ is a scalar multiple of a transvection and $t^2+1$ is irreducible over $\mathbb{F}_3$, it follows that the minimal polynomial is $(t^2+1)^2$. However, this would imply that $y^{2}$ had $(t+1)^2$ occuring twice as an invariant factor when it should occur at most once. So $x$ lifts to an element of order $6$ in the linear and symplectic cases. \par In the unitary case, let $i \in \mathbb{F}_{3^{2}}$ be a primitive $4$th root of unity so that $Z( \mathrm{GU}(d,3)) = \langle i I_{d} \rangle$. If $(xz)^6=-I_d$, then $(xzi)^6=-I_d(-1)^3=1$ and $x$ lifts to an order $6$ element. The only other possibility is that $(xz)^6= \pm i I_d$, in which case the minimal polynomial of $y=xz$ would divide $(t^2 \pm i)^3$. As before, since $y^2$ is a scalar multiple of a transvection, the minimal polynomial of $y$ would divide $(t^2 \pm i)^2$. Now $t^2 \pm i$ is irreducible, and if $y$ had minimal polynomial $t^2 \pm i$, then $y$ would have projective order $2$. Thus the only possibility is that $m_y(t)=(t^2 \pm i)^2$, but then $(t \pm i)^2$ would occur twice as an invariant factor of $y^2$. So $x$ lifts to an element of order $6$ in the unitary case as well. \end{proof} \begin{lem} If $G \le PGL(d,3), PGSp(d,3)$ or $PGU(d,3)$ and $x \in G$ has order $6$, then there exists $g \in G$ such that $ \langle x,x^g \rangle$ is not solvable. \end{lem} \begin{proof} By Lemma \ref{lift6}, we can lift $x$ to an element of order $6$ in $GL(d,3)$, $GSp(d,3)$, or $GU(d,3)$. Since $x=x_ux_s=x_sx_u$ and $x_u$ is a transvection, the minimal polynomial of $x$ divides $(t^2-1)^2$. Now $x^2$ is a transvection and its invariant factors are $(t-1)^2$ with multiplicity one and $(t-1)$ with multiplicity $d-2$; thus the minimal polynomial of $x$ is not $(t^2-1)^2$ otherwise the multiplicity of $(t-1)^{2}$ in the invariant factors of $x^{2}$ would be at least 2. In fact, we can show that the invariant factors of $x$ must be $t+ \epsilon_{1}$ with multiplicity $m_{1}$, $t^{2}-1$ with multiplicity $m_{2}$, and $(t^{2}-1)(t- \epsilon_{2})$ with multiplicity $1$, where $ \epsilon_{i} \in \{ \pm 1\}$. In the linear and unitary cases, there exists $y \in \mathrm{GL}^{\pm}(3,3) \times \mathrm{GL}^{\pm}(d-3,3)$ with $y$ of order $6$ in the first factor, having the same invariant factors as $x$. Thus $x$ and $y$ are conjugate (see \cite{Wall} for example) and we can reduce to the cases $G_0 = \mathrm{PSL}(3,3)$ and $\mathrm{PSU}(3,3)$. In the symplectic case we consider the elementary divisors of $x$, which must be $(t- \epsilon)^2$ with multiplicity $1$, $(t+ \epsilon)$ with multiplicity $m_1 \ge 1$, and $(t- \epsilon)$ with multiplicity $m_2$ ($\epsilon= \pm 1$). By considering the vector space $V$ as an $\mathbb{F}_{3}\langle x \rangle$-module, we can see that $V$ decomposes as \[V = U \oplus U' \cong \mathbb{F}_{q}(t)/ (t - \epsilon)^2 \oplus \left(\mathbb{F}_{q}(t)/ (t+\epsilon) \oplus \cdots \oplus \mathbb{F}_{q}(t)/ (t\pm 1)\right).\] Since $x^2$ is a symplectic transvection, there exists $t \in U$, and $\lambda \in \mathbb{F}_{3}$ such that $x^2: v \to v+ \lambda(t,v)v$ for all $v \in V$. Moreover, since $U \cong \mathbb{F}_{q}(t)/ (t - \epsilon)^2$, there exists $u \in U$ such that $(u,t)\ne 0$ and thus $U = \langle u,t \rangle$ is a $2$-dimensional, nondegenerate subspace. Now consider $V = U \perp U^{\perp}$. Observe that $x$ has order $1$ or $2$ on $U^{\perp}$ and in particular $x$ has a semisimple action on $U^{\perp}$. Thus $U^{\perp} = \oplus U_i$ where the $U_i$ are nondegenerate $x$-invariant $1$ or $2$ dimensional subspaces. Moreover, there exists an $x$-invariant decomposition $V = (U \oplus U_j) \oplus (U \oplus U_j)^{\perp}$ into nondegenerate subspaces of dimension $4$ and $n-4$, and $x$ has order $6$ on $(U \oplus U_j)$; thus we can reduce to the case $G_0=\mathrm{PSp}(4,3)$. We can easily verify in MAGMA that $G_0= PSL(3,3)$, $\mathrm{PSU}(3,3)$ and $\mathrm{PSp}(4,3)$ are not counterexamples to the theorem. \end{proof} \begin{lem} If $G$ is an orthogonal group and $x$ has projective order $6$, then there exists $g \in G$ such that $\langle x,x^g \rangle$ is not solvable. \end{lem} \begin{proof} If $x$ is contained in $PCO^{\epsilon}(d,3)$, where $x^2$ is a long root element and $x^3$ is an involution, then $x$ will lift to an element $y$ of order $6$ or $12$ in $CO^{\epsilon}(d,3)$. \par Now $l = y^{ \frac{\|y\|}{3}}$ is a long root element, and all long root elements are conjugate, so we may assume that \begin{displaymath} l : v \to v + (v,e_2)e_1 - (v,e_1)e_2, \end{displaymath} where $\{e_1, f_1, e_2,f_2, \ldots \}$ is a basis for $V$ and $(e_i,f_j) = \delta_{ij}$ and $(e_i,e_j) = (f_i,f_j)=0$. Since the elementary divisors for $l$ are $(t-1)^2$ with multipiclity $2$ and $(t-1)$ with multiplicity $n-2$, the possibilities for the elementary divisors of $y$ are as follows: \begin{enumerate} \item $(t - \epsilon)^2$ with multiplicity $2$, $(t+ \epsilon)$ with multiplicity $m_1$ ($m_1\ge 1$ if $\epsilon = 1$), and $(t- \epsilon)$ with multiplicity $m_2$ ($\epsilon= \pm 1$). \item $(t-1)^2$ with multiplicity $1$, $(t+1)^2$ with multiplicity $1$, $(t+1)$ with multiplicity $m_1$ and $(t-1)$ with multiplicity $m_2$. \item $(t^2+1)^2$ with multiplicity $1$, and $(t^2+1)$ with multiplicity $ \frac{n-4}{2}$. \end{enumerate} So as an $ \mathbb{F}_{3} \langle y \rangle$-module, \begin{displaymath} V= U_1 \oplus U_2 \oplus \cdots \oplus U_k \end{displaymath} where $U_i \cong \mathbb{F}_3(t)/ f_i(t)$, and $f_i(t)$ is the $i$th elementary divisor of $y$. Set $U= U_1 \oplus U_2$ in the first and second cases and set $U=U_1$ in the third case; so $U$ is $4$-dimensional as a vector space. Now considering $U$ as an $ \mathbb{F}_{3}(l)$-module, we have $U =W_1 \oplus W_2$ where $W_i \cong \mathbb{F}_3(t)/ (t-1)^2$, and we claim that $U$ is a nondegenerate subspace of $V$. For there exists $v_1 \in W_1$ such that $v_1^l - v_1 = \lambda_1 e_1 + \lambda_2 e_2 \ne 0$ and similarly there exists $v_2 \in W_2$ such that $v_2^l - v_2 = \mu_1 e_1 + \mu_2 e_2 \ne 0$. Since $\lambda_1 e_1 + \lambda_2 e_2$ and $\mu_1 e_1 + \mu_2 e_2$ are linearly independent, it follows that there exist constants $a_1,a_2,b_1,b_2$ such that $v_1' = a_1 v_1 + a_2 v_2$, $v_2' = b_1 v_1 + b_2 v_2$, and $(v_i', e_j) = \delta_{ij}$ for $i, j \in \{ 1,2\}$. Now it is easy to check that $U = \langle e_1,e_2,v_1',v_2' \rangle$ is a nondegenerate space. For if \begin{displaymath} w= a e_1 + b e_2 + c v_1' + dv_2' \end{displaymath} is a degenerate vector in $U$, then $(w,e_1)=(w,e_2)=0$; so $c=d=0$. Thus $w = ae_1 + be_2$ and $(w,v_1')= (w,v_2')=0$; so $a=b=0$, $w=0$, and $U$ is nondegenerate. Now $V= U \oplus U^{\perp}$, and $y$ has projective order $1$ or $2$ on $U^{\perp}$. In particular $y$ has a semisimple action on $U^{\perp}$ and there exists a nondegenerate $y$-invariant subspace $U' \le U^{\perp}$ of dimension $1$ or $2$ such that $y$ has projective order $6$ on $U \perp U'$. Thus we may assume that $n \le 6$, but then $G$ is isomorphic to a linear, unitary or symplectic group. \end{proof} \begin{lemma} Suppose that $G \le PGU(d,2)$ $(d \ge 4)$, that $x$ has order $6$, and that $x^2$ is a pseudoreflection. Then there exists $g \in G$ such that $\langle x,x^g\rangle$ is nonsolvable. \end{lemma} \begin{proof} First observe that $x$ lifts to an element $y$ of order $6$ in $\mathrm{GU}(d,2)$. Indeed, for any lift $y$ of $x$, we have $y^2 = z r$ where $z \in Z(GU(d,2))$, and $r$ is a pseudoreflection. But then $y^6 = (zr)^3 = 1$ since $Z(GU(d,2))$ has order $3$. Since $x^2$ is a pseudoreflection, $C_{\mathrm{GU}(d,2)}(x^2) \cong \!\mathrm{GU}(1,2)\! \times \!\mathrm{GU}(d-1,2)$ by \cite[Lemma 4.1.1]{KL}, and $x \! \in \! C_G(x^2)$. Moreover, by multiplying $x$ by $z \in Z(\mathrm{GU}(d,2))$ if necessary, we may assume that $x$ has order $3$ on the first component, and $2$ on the second component. Now in $GU(d-1,2)$, two unipotent elements are conjugate if and only if they have the same Jordan form(see \cite{Wall} or \cite[Theorems 2.12 and 3]{LSgood}). But the Jordan form of an involution is of the form $J_2^r, J_1^{d-1-2r}$, and so there exists a conjugate of $x$ that is contained in $GU(1,2) \times GU(4,2) \times \mathrm{GU}(d-5,2)$, and on which $x$ has order $3$ on the first component, and $2$ on the second component. So we may assume that $4 \le d \le 5$, and we can eliminate these cases in MAGMA. \end{proof} \begin{lemma} Suppose that $x$ is an inner-diagonal automorphism of an exceptional group defined over $\mathbb{F}_{3}$. Then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} We can verify the Lemma using MAGMA. For the smaller groups, we can calculate the conjugacy classes directly. For the larger groups we can use the Groups of Lie type package in MAGMA to construct a Sylow $2$-subgroup $S$ of $C=C_G(y)$, where $y$ is a long root element. We can then calculate the conjugacy classes of $S$, to find the class representatives $s_1, s_2, \ldots s_k$ of involutions in $S$. Then every element $x$ of order $6$ in $G$ such that $x^2$ is a long root element is conjugate to at least one element in $\{y s_1, ys_2, \ldots, ys_k\}$. Now for each $x = ys_i$, we can search for a random conjugate $x^g$ such that $\langle x,x^g\rangle$ is not solvable. \end{proof} \begin{lemma} If $x \not\in \mathrm{Inndiag}(G_0)$, then $(x,G)$ is not a minimal counterexample. \end{lemma} \begin{proof} Suppose that $x^2$ is a transvection and $x^3$ is a graph automorphism. If $G_0 \cong PSL(d,3)$ and $d$ is even, then there are three classes of graph automorphism. Representatives for the three classes are $\iota S$, $\iota S^{+}$ and $\iota S^-$ where $\iota$ is the inverse transpose automorphism. Their centralizers are of type $Sp(d,3)$, $O^+(d,3)$ and $O^-(d,3)$ respectively. Now $x=x^4x^3$, and $x^4$ is a transvection so there exists $a \in V$ such that \begin{displaymath} x^4 : v \rightarrow v+ f(v)a, \end{displaymath} where $f$ is a linear functional on $V$, $\dim \ker{f}= n-1$, and $f(a)=0$. Now all of the graph automorphisms above stabilize a subgroup of type $GL(d-2,3) \times GL(2,3)$. Now we can conjugate $x$ by $h \in C_G(x^3)$ (of type $Sp(d,3)$, $O^{\pm}(d,3)$) so that $h(a) \in U$ where $U$ is the subspace of $V$ corresponding to the subgroup $GL(d-2,3)$. Thus we can consider $x$ acting as an automorphism on $GL(d-2,3)$. There is only one class of graph involutions when $d$ is odd, with representative $\iota$, and centralizer of type $O(d,3)$. We can make the same reduction here. So it suffices to deal with cases where $d \le 4$. It is easy to eliminate these cases in MAGMA.\par The case where $G_0\cong PSU(d,3)$ is very similar. In this case, $x^4$ is a unitary transvection and $x^3$ is a graph automorphism. The classes of involutory graph automorphisms are described in \cite[Table 4.5.1]{GLS}. When $d$ is even, there are three classes as in the linear case with centralizers of type $O^{+}(d,3)$, $O^{-}(d,3)$, and $Sp(d,3)$. These classes are described explicitly in \cite[pg 43]{Moufang} and \cite[pg 288]{LS}, and we see that each class normalizes a subgroup of type $GU(d-2,3) \times GU(2,3)$ or $GU(d-1,3)\times GU(1,3)$. In particular, there exists an $x^3$ invariant, nondegenerate subspace $U$ of $V$ of dimension $d-2$ or $d-1$. Moreover, as in the linear case we can take $h \in C_G(x^3)$ such that $h(a) \in U$, and so we may assume that $x^4$ acts a unitary transvection on $U$. Similarly, when $d$ is odd, there is only one class of graph involutions and \cite{Moufang} and \cite{LS} show that $x^3$ normalizes a subgroup of type $GU(d-1,3)$; thus we may assume that $x$ normalizes this subgroup and has order $6$ on it. Therefore, it suffices to check the cases $G_0=PSU(4,3)$ and $G_0=PSU(3,3)$ in MAGMA. Next, suppose that $G_0 \cong PSU(d,2)$, $x^3$ is an involutory graph automorphism, and $x^4$ is a pseudoreflection. If $d$ is odd, then there is one class of graph involutions, and we may therefore assume that $x^3$ acts as a standard field automorphism on matrix entries, with centralizer of type $O^{+}(d,2)$ (see \cite[19.9]{AS}). Thus, conjugating by $h \in C_G(x^3)$ if necessary, we may assume that $x$ will normalize a subgroup of type $GU(d-1,2)$, and $x$ will act as an element of order $6$ on it. If $n$ is even, then there are two classes of graph involutions, with centralizers of type $Sp(d,2)$ and $C_{Sp(d,2)}(t)$ where $t$ is a transvection in $Sp(d,2)$. In the first case, we may assume that $x$ acts as a standard field automorphism on the matrix entries and so $x$ will normalize a subgroup of type $GU(d-1,2)$ as before. In the other case, the pseudoreflection $x^4$ is contained in $C_{G_0}(x^3)=C_{Sp(d,2)}(t)$; moreover, $C_{G_0}(x^3)$ is contained in a subgroup $Sp(d,2)\langle x^3 \rangle = Sp(n,2) \times \langle x^3t \rangle$ (see \cite[pg 288]{LS}), and $x^3$ acts nontrivially, as an inner automorphism on $Sp(d,2)$. Thus $x \in Sp(d,2)\langle x^3 \rangle$, which is a smaller almost simple group. Combining the $d$ odd and $d$ even cases, it suffices to check that the theorem is true for $G_0 \cong PSU(4,2)$, which is easy to do in MAGMA. The cases when $x^3$ is a graph automorphism in an orthogonal group have already been considered, and the cases of graph automorphisms of exceptional groups are verified in MAGMA in the same way as with the other cases. \end{proof} This completes the proof of Theorem \ref{6}. We now prove Corollary \ref{cor:to9}. \begin{proof}[Proof of Corollary \ref{cor:to9}] We may assume that $G$ has trivial solvable radical and that $x$ has order $9$. In particular, the Fitting subgroup $F(G)$ of $G$ is trivial and Lemma \ref{lem:1.1} applies. Let $F^{}(G)$ be the generalized Fitting subgroup of $G$. Since $C_G(F^(G))= \{1\}$ and $F(G)=\{1\}$, $x^3$ cannot centralize all of the components of $G$. So choose a component $L$ such that $x^3 \not\in C_G(L)$. If $x$ does not normalize $L$ then Lemma \ref{lem:1.1} shows that there exists $g \in G$ such that $\langle x,x^g\rangle$ is not solvable. If $x$ does normalize $L$, then $x$ is an order $9$ element in the almost simple group $\langle x, L\rangle$ and Theorem \ref{6} implies the result. \end{proof} \begin{remark} \label{cor:to6} In the introduction, we noted that the analogous result to Corollary \ref{cor:to9} for order 6 elements is not true; for example if we take $G=S_5 \times PSL(3,3)$. However, in some sense, there are not many such examples. Let $G$ be such an example; we may assume that the solvable radical is trivial. Reasoning in the same way as in the proof of Corollary \ref{cor:to9}, if $x$ normalizes one of the components $L$, then $x$ must either act as an involution on $L$ or $L$ is one of the groups in Theorem \ref{thmA} and $x$ acts as an element of order $3$ on $L$ (and as one of the exceptional elements in Theorem \ref{thmA*}). If $x$ does not normalize one of the components $L$ then $x^2$ must centralize $L$ by Lemma \ref{lem:1.1}. In particular, the orbits of the components under the action of $x$ must have length at most $2$. \end{remark} \end{document}	arXiv
Splitting theorems on complete Riemannian manifolds with nonnegative Ricci curvature On the vanishing discount problem from the negative direction A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation José Luis López Departamento de Matemática Aplicada and Excellence Research Unit "Modeling Nature" (MNat), Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Received March 2020 Revised September 2020 Published November 2020 Fund Project: The author is partially supported by the MINECO-Feder (Spain) research grant number RTI2018-098850-B-I00, as well as by the Junta de Andalucía (Spain) Project PY18-RT-2422 & A-FQM-311-UGR18 The parabolic-parabolic Keller-Segel model of chemotaxis is shown to come up as the hydrodynamic system describing the evolution of the modulus square $ n(t,x) $ and the argument $ S(t,x) $ of a wavefunction $ \psi = \sqrt{n} \, e^{iS} $ that solves a cubic Schrödinger equation with focusing interaction, frictional Kostin nonlinearity and Doebner-Goldin dissipation mechanism. 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\begin{document} \setlength{\abovedisplayskip}{4pt} \setlength{\belowdisplayskip}{4pt} \twocolumn[ \icmltitle{Tesseract: Tensorised Actors for Multi-Agent Reinforcement Learning} \begin{icmlauthorlist} \icmlauthor{Anuj Mahajan}{ox} \icmlauthor{Mikayel Samvelyan}{uc} \icmlauthor{Lei Mao}{nv} \icmlauthor{Viktor Makoviychuk}{nv} \icmlauthor{Animesh Garg}{nv} \icmlauthor{Jean Kossaifi}{nv} \icmlauthor{Shimon Whiteson}{ox} \icmlauthor{Yuke Zhu}{nv} \icmlauthor{Animashree Anandkumar}{nv} \end{icmlauthorlist} \icmlaffiliation{ox}{University of Oxford} \icmlaffiliation{nv}{NVIDIA} \icmlaffiliation{uc}{University College London} \icmlcorrespondingauthor{Anuj Mahajan}{[email protected]} \icmlkeywords{Machine Learning, ICML} \vskip 0.3in ] \printAffiliationsAndNotice{} \begin{abstract} Reinforcement Learning in large action spaces is a challenging problem. Cooperative multi-agent reinforcement learning (MARL) exacerbates matters by imposing various constraints on communication and observability. In this work, we consider the fundamental hurdle affecting both value-based and policy-gradient approaches: an exponential blowup of the action space with the number of agents. For value-based methods, it poses challenges in accurately representing the optimal value function. For policy gradient methods, it makes training the critic difficult and exacerbates the problem of the \emph{lagging} critic. We show that from a learning theory perspective, both problems can be addressed by accurately representing the associated action-value function with a low-complexity hypothesis class. This requires accurately modelling the agent interactions in a sample efficient way. To this end, we propose a novel tensorised formulation of the Bellman equation. This gives rise to our method \textsc{Tesseract}, which views the $Q$-function as a tensor whose modes correspond to the action spaces of different agents. Algorithms derived from \textsc{Tesseract} decompose the $Q$-tensor across agents and utilise low-rank tensor approximations to model agent interactions relevant to the task. We provide PAC analysis for \textsc{Tesseract}-based algorithms and highlight their relevance to the class of rich observation MDPs. Empirical results in different domains confirm \textsc{Tesseract}'s gains in sample efficiency predicted by the theory. \end{abstract} \section{Introduction} \label{intro} Many real-world problems, such as swarm robotics and autonomous vehicles, can be formulated as multi-agent reinforcement learning (MARL) \cite{bucsoniu2010multi} problems. MARL introduces several new challenges that do not arise in single-agent reinforcement learning (RL), including exponential growth of the action space in the number of agents. This affects multiple aspects of learning, such as credit assignment~\cite{foerster2018counterfactual}, gradient variance~\cite{lowe2017multi} and exploration~\cite{mahajan2019maven}. In addition, while the agents can typically be trained in a centralised manner, practical constraints on observability and communication after deployment imply that decision making must be decentralised, yielding the extensively studied setting of centralised training with decentralised execution (CTDE). Recent work in CTDE-MARL can be broadly classified into value-based methods and actor-critic methods. Value-based methods \cite{sunehag_value-decomposition_2017,rashid2018qmix, son2019qtran, wang2020qplex, yao2019smix} typically enforce decentralisability by modelling the joint action $Q$-value such that the argmax over the joint action space can be tractably computed by local maximisation of per-agent utilities. However, constraining the representation of the $Q$-function can interfere with exploration, yielding provably suboptimal solutions \cite{mahajan2019maven}. Actor-critic methods \cite{lowe2017multi, foerster2018counterfactual,wei2018multiagent} typically use a centralised critic to estimate the gradient for a set of decentralised policies. In principle, actor-critic methods can satisfy CTDE without incurring suboptimality, but in practice their performance is limited by the accuracy of the critic, which is hard to learn given exponentially growing action spaces. This can exacerbate the problem of the \textit{lagging} critic \cite{kondathesis}. Moreover, unlike the single-agent setting, this problem cannot be fixed by increasing the critic's learning rate and number of training iterations. Similar to these approaches, an exponential blowup in the action space also makes it difficult to choose the appropriate class of models which strike the correct balance between expressibility and learnability for the given task. In this work, we present new theoretical results that show how the aforementioned approaches can be improved such that they accurately represent the joint action-value function whilst keeping the complexity of the underlying hypothesis class low. This translates to accurate, sample efficient modelling of long-term agent interactions. In particular, we propose \textsc{Tesseract} (derived from "Tensorised Actors"), a new framework that leverages tensors for MARL. Tensors are high dimensional analogues of matrices that offer rich insights into representing and transforming data. The main idea of \textsc{Tesseract} is to view the output of a joint $Q$-function as a tensor whose modes correspond to the actions of the different agents. We thus formulate the Tensorised Bellman equation, which offers a novel perspective on the underlying structure of a multi-agent problem. In addition, it enables the derivation of algorithms that decompose the $Q$-tensor across agents and utilise low rank approximations to model relevant agent interactions. Many real-world tasks (e.g., robot navigation) involve high dimensional observations but can be completely described by a low dimensional feature vector (e.g., a 2D map suffices for navigation). For value-based \textsc{Tesseract} methods, maintaining a tensor approximation with rank matching the intrinsic task dimensionality\footnote{We define intrinsic task dimensionality (ITD) as the minimum number of dimensions required to describe an environment} helps learn a compact approximation of the true $Q$-function (alternatively MDP-dynamics for model based methods). In this way, we can avoid the suboptimality of the learnt policy while remaining sample efficient. Similarly, for actor-critic methods, \textsc{Tesseract} reduces the critic's learning complexity while retaining its accuracy, thereby mitigating the lagging critic problem. Thus, \textsc{Tesseract} offers a natural spectrum for trading off accuracy with computational/sample complexity. To gain insight into how tensor decomposition helps improve sample efficiency for MARL, we provide theoretical results for model-based \textsc{Tesseract} algorithms and show that the underlying joint transition and reward functions can be efficiently recovered under a PAC framework (in samples polynomial in accuracy and confidence parameters). We also introduce a tensor-based framework for CTDE-MARL that opens new possibilities for developing efficient classes of algorithms. Finally, we explore the relevance of our framework to rich observation MDPs. Our main contributions are: \begin{enumerate} \item A novel tensorised form of the Bellman equation; \item \textsc{Tesseract}, a method to factorise the action-value function based on tensor decomposition, which can be used for any factored action space; \item PAC analysis and error bounds for model based \textsc{Tesseract} that show an exponential gain in sample efficiency of $O(\|U\|^{n/2})$; and \item Empirical results illustrating the advantage of \textsc{Tesseract} over other methods and detailed techniques for making tensor decomposition work for deep MARL. \end{enumerate} \section{Background} \label{background} \paragraph{Cooperative MARL settings} In the most general setting, a fully cooperative multi-agent task can be modelled as a multi-agent partially observable MDP (M-POMDP)~\cite{messias2011mpomdp}. An M-POMDP is formally defined as a tuple $G=\left\langle S,U,P,r,Z,O,n,\gamma\right\rangle$. $S$ is the state space of the environment. At each time step $t$, every agent $ i \in \mathcal{A} \equiv \{1,...,n\}$ chooses an action $u^i \in U$ which forms the joint action $\mathbf{u}\in\mathbf{U}\equiv U^n$. $P(s'\|s,\mathbf{u}):S\times\mathbf{U}\times S\rightarrow [0,1]$ is the state transition function. $r(s,\mathbf{u}):S \times \mathbf{U} \rightarrow [0,1]$ is the reward function shared by all agents and $\gamma \in [0,1)$ is the discount factor. \begin{wrapfigure}{r}{0.6\linewidth} \centering \includegraphics[width=\linewidth]{figures/settings_new.png} \caption{Different settings in MARL \label{settings_marl}} \end{wrapfigure} An M-POMDP is \textit{partially observable} \citep{kaelbling1998planning}: each agent does not have access to the full state and instead samples observations $z\in Z$ according to observation distribution $O(s):S\rightarrow \mathcal{P}(Z)$. The action-observation history for an agent $i$ is $\tau^i\in T\equiv(Z\times U)^$. We use $u^{-i}$ to denote the action of all the agents other than $i$ and similarly for the policies $\pi^{-i}$. Settings where the agents cannot exchange their action-observation histories with others and must condition their policy solely on local trajectories, $\pi^i(u^i\|\tau^i):T\times U\rightarrow [0,1]$, are referred to as a decentralised partially observable MDP (Dec-POMDP)~\citep{oliehoek_concise_2016}. When the observations have additional structure, namely the joint observation space is partitioned w.r.t.\ $S$, i.e., $\forall s_1,s_2 \in S \land {z} \in Z, P({z}\|s_1)>0 \land s_1\neq s_2 \implies P({z}\|s_2)=0$, we classify the problem as a multi-agent richly observed MDP (M-ROMDP)~\citep{azizzadenesheli2016reinforcement}. For both M-POMDP and M-ROMDP, we assume $\|Z\|>>\|S\|$, thus for this work, we assume a setting with no information loss due to observation but instead, redundancy across different observation dimensions. Such is the case for many real world tasks like 2D robot navigation using observation data from different sensors. Finally, when the observation function is a bijective map $O: S\to Z$, we refer to the scenario as a multi-agent MDP (MMDP)~\cite{boutilier1996planning}, which can simply be denoted by the tuple : $\left\langle S,U,P,r,n,\gamma\right\rangle$. \cref{settings_marl} gives the relation between different scenarios for the cooperative setting. For ease of exposition, we present our theoretical results for the MMDP case, though they can easily be extended to other cases by incurring additional sample complexity. The joint \textit{action-value function} given a policy $\pi$ is defined as: $Q^\pi(s_t, \mathbf{u}_t)=\mathbb{E}_{s_{t+1:\infty},\mathbf{u}_{t+1:\infty}} \left[\sum^{\infty}_{k=0}\gamma^kr_{t+k}\|s_t,\mathbf{u}_t\right]$. The goal is to find the optimal policy $\pi^{}$ corresponding to the optimal action value function $Q^$. For the special learning scenario called Centralised Training with Decentralised Execution (CTDE), the learning algorithm has access to the action-observation histories of all agents and the full state during training phase. However, each agent can only condition on its own local action-observation history $\tau^i$ during the decentralised execution phase. \paragraph{Reinforcement Learning Methods\label{subsec: rl}} Both value-based and actor-critic methods for reinforcement learning (RL) rely on an estimator for the action-value function $Q^\pi$ given a target policy $\pi$. $Q^\pi$ satisfies the (scalar)-Bellman expectation equation: $ Q^{\pi}(s, \mathbf{u}) = r(s, \mathbf{u})+ \gamma\mathbb{E}_{s',\mathbf{u}'}[Q^{\pi}(s', \mathbf{u}')],$ which can equivalently be written in vectorised form as: \begin{align} \label{eq:bell} Q^{\pi} = R + \gamma P^{\pi}Q^{\pi}, \end{align} where $R$ is the mean reward vector of size $S$, $P^{\pi}$ is the transition matrix. The operation on RHS $\mathcal{T}^{\pi}(\cdot) \triangleq R + \gamma P^{\pi}(\cdot)$ is the Bellman expectation operator for the policy $\pi$. In \cref{method} we generalise \cref{eq:bell} to a novel tensor form suitable for high-dimensional and multi-agent settings. For large state-action spaces function approximation is used to estimate $Q^\pi$. A parametrised approximation $Q^\phi$ is usually trained using the bootstrapped target objective derived using the samples from $\pi$ by minimising the mean squared temporal difference error: $ \mathbb{E}_\pi[(r(s, \mathbf{u})+ \gamma Q^{\phi}(s', \mathbf{u}')-Q^{\phi}(s, \mathbf{u}))^2]. $ Value based methods use the $Q^\pi$ estimate to derive a behaviour policy which is iteratively improved using the policy improvement theorem \cite{sutton2011reinforcement}. Actor-critic methods seek to maximise the mean expected payoff of a policy $\pi_\theta$ given by $\mathcal{J}_\theta=\int_{S} \rho^\pi(s)\int_{\mathbf{U}}\pi_\theta(\mathbf{u\|s})Q^{\pi}(s,\mathbf{u}) d\mathbf{u}ds$ using gradient ascent on a suitable class of stochastic policies parametrised by $\theta$, where $\rho^\pi(s)$ is the stationary distribution over the states. Updating the policy parameters in the direction of the gradient leads to policy improvement. The gradient of the above objective is $\nabla \mathcal{J}_\theta = \int_{S} \rho^\pi(s)\int_{\mathbf{U}}\nabla\pi_\theta(\mathbf{u\|s})Q^{\pi}(s,\mathbf{u}) d\mathbf{u}ds$ \cite{sutton2000policy}. An approximate action-value function based critic $Q^\phi$ is used when estimating the gradient as we do not have access to the true $Q$-function. Since the critic is learnt using finite number of samples, it may deviate from the true $Q$-function, potentially causing incorrect policy updates; this is called the \textit{lagging critic} problem. The problem is exacerbated in multi-agent setting where state-action spaces are very large. \paragraph{Tensor Decomposition} Tensors are high dimensional analogues of matrices and tensor methods generalize matrix algebraic operations to higher orders. Tensor decomposition, in particular, generalizes the concept of low-rank matrix factorization. In the rest of this paper, we use $\hat{\cdot}$ to represent tensors. Formally, an order $n$ tensor $\hat{T}$ has $n$ index sets ${I}_j,\forall j \in\{1..n\}$ and has elements $T(e), \forall e \in \times_{\mathcal{I}} {I}_j $ taking values in a given set $\mathcal{S}$, where $\times$ is the set cross product and we denote the set of index sets by $\mathcal{I}$. Each dimension $\{1..n\}$ is also called a mode. \begin{figure} \caption{Left: Tensor diagram for an order $3$ tensor $\hat T$. Right: Contraction between $\hat T^1$,$\hat T^2$ on common index sets $I_2,I_3$. } \label{fig:tdia} \end{figure} An elegant way of representing tensors and associated operations is via tensor diagrams as shown in \cref{fig:tdia}. Tensor contraction generalizes the concept of matrix with matrix multiplication. For any two tensors $\hat{T}^1$ and $\hat{T}^2$ with $\mathcal{I}_{\cap} = \mathcal{I}^1 \cap \mathcal{I}^2$ we define the contraction operation as $\hat T= \hat{T}^1{\odot} \hat{T}^2$ with $ \hat{T}(e_1,e_2) = \sum_{e\in \times_{\mathcal{I}_{\cap}} {I}_j } \hat T^1(e_1,e)\cdot\hat T^2(e_2,e), e_i \in \times_{\mathcal{I}^i\setminus \mathcal{I}_{\cap}} {I}_j$. The contraction operation is associative and can be extended to an arbitrary number of tensors. Using this building block, we can define tensor decompositions, which factorizes a (low-rank) tensor in a compact form. This can be done with various decompositions~\cite{kolda2009tensor}, such as Tucker, Tensor-Train (also known as Matrix-Product-State), or CP (for Canonical-Polyadic). In this paper, we focus on the latter, which we briefly introduce here. Just as a matrix can be factored as a sum of rank-$1$ matrices (each being an outer product of vectors), a tensor can be factored as a sum of rank-1 tensors, the latter being an outer product of vectors. The number of vectors in the outer product is equal to the rank of the tensor, and the number of terms in the sum is called the \emph{rank of the decomposition} (sometimes also called CP-rank). Formally, a tensor $\hat T$ can be factored using a (rank--$k$) CP decomposition into a sum of $k$ vector outer products (denoted by $\otimes$), as, \begin{align} \label{CPD} \hat T=\sum_{r=1}^k w_r\otimes^n u_r^i ,i \in \{1..n\},\|\|u_r^i\|\|_2 =1. \end{align} \section{Methodology} \label{method} \subsection{Tensorised Bellman equation} In this section, we provide the basic framework for Tesseract. We focus here on the discrete action space. The extension for continuous actions is similar and is deferred to \cref{app:cenv} for clarity of exposition. \begin{proposition} Any real-valued function $f$ of $n$ arguments $(x_1..x_n)$ each taking values in a finite set $x_i\in \mathcal{D}_i$ can be represented as a tensor $\hat f$ with modes corresponding to the domain sets $\mathcal{D}_i$ and entries $\hat f(x_1..x_n) = f(x_1..x_n)$. \end{proposition} Given a multi-agent problem $G=\left\langle S,U,P,r,Z,O,n,\gamma\right\rangle$, let $\mathcal{Q} \triangleq \{Q: S\times U^n\to \mathbb{R}\}$ be the set of real-valued functions on the state-action space. We are interested in the \emph{curried }\cite{barendregt1984introduction} form $Q: S\to U^n\to \mathbb{R},Q\in \mathcal{Q}$ so that $Q(s)$ is an order $n$ tensor (We use functions and tensors interchangeably where it is clear from context). Algorithms in Tesseract operate directly on the curried form and preserve the structure implicit in the output tensor. (Currying in the context of tensors implies fixing the value of some index. Thus, Tesseract-based methods keep action indices free and fix only state-dependent indices.) We are now ready to present the tensorised form of the Bellman equation shown in \cref{eq:bell}. \cref{fig:tbell} gives the equation where $\hat I$ is the identity tensor of size $\|S\|\times\|S\|\times\|S\|$. The dependence of the action-value tensor $\hat Q^\pi$ and the policy tensor $\hat U^\pi$ on the policy is denoted by superscripts $\pi$. The novel \textbf{Tensorised Bellman equation} provides a theoretically justified foundation for the approximation of the joint $Q$-function, and the subsequent analysis (Theorems 1-3) for learning using this approximation. \begin{figure} \caption{\textbf{Tensorised Bellman Equation} for $n$ agents. There is an edge for each agent $i \in \mathcal{A}$ in the corresponding nodes $\hat Q^\pi,\hat U^\pi, \hat R, \hat P$ with the index set $U^i$.} \label{fig:tbell} \end{figure} \subsection{\textsc{Tesseract} Algorithms} \label{subsec:TessAlgos} \captionsetup[algorithm]{format=hang,singlelinecheck=false} For any $k \in \mathbb{N}$ let $\mathcal{Q}_k \triangleq \{Q: Q\in \mathcal{Q} \land rank(Q(\cdot, s)) \leq k, \forall s \in S\}$. Given any policy $\pi$ we are interested in projecting $Q^\pi$ to $\mathcal{Q}_k$ using the projection operator $\Pi_k(\cdot) = \argmin_{Q \in \mathcal{Q}_k} \|\|\cdot-Q\|\|_{\pi,F}$. where $\|\|X\|\|_{\pi,F} \triangleq \mathbb{E}_{s\sim\rho^\pi(s)}[\|\|X(s)\|\|_{F}]$ is the weighted Frobenius norm w.r.t.\ policy visitation over states. Thus a simple planning based algorithm for rank $k$ \textsc{Tesseract} would involve starting with an arbitrary $Q_0$ and successively applying the Bellman operator $\mathcal{T}^{\pi}$ and the projection operator $\Pi_k$ so that $Q_{t+1} = \Pi_k\mathcal{T}^{\pi}Q_t$. As we show in \cref{rankbQ}, constraining the underlying tensors for dynamics and rewards ($\hat P, \hat R$) is sufficient to bound the CP-rank of $\hat Q$. From this insight, we propose a model-based RL version for \textsc{Tesseract} in \cref{alg:model-based}. The algorithm proceeds by estimating the underlying MDP dynamics using the sampled trajectories obtained by executing the behaviour policy $\pi = (\pi^i)_1^n$ (factorisable across agents) satisfying \cref{thm:debound}. Specifically, we use a rank $k$ approximate CP-Decomposition to calculate the model dynamics $R, P$ as we show in \cref{analysis}. Next $\pi$ is evaluated using the estimated dynamics, which is followed by policy improvement, \cref{alg:model-based} gives the pseudocode for the model-based setting. The termination and policy improvement decisions in \cref{alg:model-based} admit a wide range of choices used in practice in the RL community. Example choices for internal iterations which broadly fall under approximate policy iteration include: 1) Fixing the number of applications of Bellman operator 2) Using norm of difference between consecutive Q estimates etc., similarly for policy improvement several options can be used like $\epsilon$-greedy (for Q derived policy), policy gradients (parametrized policy)~\cite{sutton2011reinforcement} \begin{algorithm}[h!] \caption{Model-based Tesseract\label{alg:model-based}} \begin{algorithmic}[1] \STATE \mbox{Initialise rank $k$, $\pi = (\pi^i)_1^n$ and $\hat Q$: \cref{thm:debound}} \STATE \mbox{Initialise model parameters $\hat P,\hat R$} \STATE Learning rate $\leftarrow \alpha$,$\mathcal{D} \leftarrow \left\{ \right\}$ \FOR{each episodic iteration i} \STATE Do episode rollout $\tau_i = \left\{(s_t,\mathbf{u}_t,r_t,s_{t+1})_{0}^L \right\}$ using $\pi$ \STATE $\mathcal{D} \leftarrow \mathcal{D}\cup\left\{\tau_i \right\}$ \STATE Update $\hat P,\hat R$ using CP-Decomposition on moments from $\mathcal{D}$ (\cref{thm:debound}) \FOR{each internal iteration j} \STATE $\hat Q \gets \mathcal{T}^\pi \hat Q$ \ENDFOR \STATE Improve $\pi$ using $\hat Q$ \ENDFOR \STATE Return $\pi, \hat Q$ \end{algorithmic} \end{algorithm} For large state spaces where storage and planning using model parameters is computationally difficult (they are $\mathcal{O}(kn\|U\|\|S\|^2)$ in number), we propose a model-free approach using a deep network where the rank constraint on the $Q$-function is directly embedded into the network architecture. \cref{fig:tnet} gives the general network architecture for this approach and \cref{alg:model-free} the associated pseudo-code. Each agent in \cref{fig:tnet} has a policy network parameterized by $\theta$ which is used to take actions in a decentralised manner. \begin{figure} \caption{Tesseract architecture } \label{fig:tnet} \end{figure} The observations of the individual agents along with the actions are fed through representation function $g_\phi$ whose output is a set of $k$ unit vectors of dimensionality $\|U\|$ corresponding to each rank. The output $g_{\phi,r}(s^i)$ corresponding to each agent $i$ for factor $r$ can be seen as an action-wise contribution to the joint utility from the agent corresponding to that factor. The joint utility here is a product over individual agent utilities. For partially observable settings, an additional RNN layer can be used to summarise agent trajectories. The joint action-value estimate of the tensor $\hat Q(s)$ by the centralized critic is: \begin{align} \label{eq:cpa} \hat Q(s) \approx T = \sum_{r=1}^k w_r\otimes^n g_{\phi,r}(s^i) ,i \in \{1..n\}, \end{align} where the weights $w_r$ are learnable parameters exclusive to the centralized learner. In the case of value based methods where the policy is implicitly derived from utilities, the policy parameters $\theta$ are merged with $\phi$. The network architecture is agnostic to the type of the action space (discrete/continuous) and the action-value corresponding to a particular joint-action $(u^1..u^n)$ is the inner product $\langle T, A \rangle$ where $A = \otimes^n u^i$ (This reduces to indexing using joint action in \cref{eq:cpa} for discrete spaces). More representational capacity can be added to the network by creating an abstract representation for actions using $f_\eta$, which can be any arbitrary monotonic function (parametrised by $\eta$) of vector output of size $m \geq \|U\|$ and preserves relative order of utilities across actions; this ensures that the optimal policy is learnt as long as it belongs to the hypothesis space. In this case $A = \otimes^n f_\eta(u^i)$ and the agents also carry a copy of $f_\eta$ during the execution phase. Furthermore, the inner product $\langle T, A \rangle$ can be computed efficiently using the property $$\langle T, A \rangle = \sum_{r=1}^k w_r\prod_1^n \langle f_\eta(u^i)g_{\phi,r}(s^i) \rangle ,i \in \{1..n\}$$ which is $O(nkm)$ whereas a naive approach involving computation of the tensors first would be $O(km^n)$. Training the Tesseract-based $Q$-network involves minimising the squared TD loss \cite{sutton2011reinforcement}: \begin{align} \mathcal{L}_{TD}(\phi,\eta) = \mathbb{E}_{\pi}[(&Q(\mathbf{u}_t,s_t; \phi,\eta)\\- &[r(\mathbf{u}_t,s_t)+\gamma Q(\mathbf{u}_{t+1},s_{t+1}; \phi^-,\eta^-)])^2], \end{align} where $\phi^-,\eta^-$ are target parameters. Policy updates involve gradient ascent w.r.t.\ to the policy parameters $\theta$ on the objective $\mathcal{J}_\theta=\int_{S} \rho^\pi(s)\int_{\mathbf{U}}\pi_\theta(\mathbf{u\|s})Q^{\pi}(s,\mathbf{u}) d\mathbf{u}ds$. More sophisticated targets can be used to reduce the policy gradient variance \citep{greensmith2004variance, zhao2016regularized} and propagate rewards efficiently \citep{sutton1988learning}. Note that \cref{alg:model-free} does not require the individual-global maximisation principle \citep{son2019qtran} typically assumed by value-based MARL methods in the CTDE setting, as it is an actor-critic method. In general, any form of function approximation and compatible model-free approach can be interleaved with Tesseract by appropriate use of the projection function $\Pi_k$. \begin{algorithm}[h!] \caption{Model-free Tesseract\label{alg:model-free}} \begin{algorithmic}[1] \STATE \mbox{Initialise rank $k$, parameter vectors $\theta, \phi, \eta$} \STATE Learning rate $\leftarrow \alpha$,$\mathcal{D} \leftarrow \left\{ \right\}$ \FOR{each episodic iteration i} \STATE Do episode rollout $\tau_i = \left\{(s_t,\mathbf{u}_t,r_t,s_{t+1})_{0}^L \right\}$ using $\pi_\theta$ \STATE $\mathcal{D} \leftarrow \mathcal{D}\cup\left\{\tau_i \right\}$ \STATE Sample batch $\mathcal{B} \subseteq \mathcal{D}$. \STATE Compute empirical estimates for $\mathcal{L}_{TD}, \mathcal{J}_\theta$ \STATE $\phi \leftarrow \phi - \alpha \nabla_\phi \mathcal{L}_{TD}$ (Rank $k$ projection step) \STATE $\eta \leftarrow \eta - \alpha \nabla_\eta \mathcal{L}_{TD}$ (Action representation update) \STATE $\theta \leftarrow \theta + \alpha \nabla_\theta \mathcal{J}_\theta$ (Policy update) \ENDFOR \STATE Return $\pi, \hat Q$ \end{algorithmic} \end{algorithm} \subsection{Why Tesseract?} \label{robo_nav_sec} As discussed in \cref{intro}, $Q(s)$ is an object of prime interest in MARL. Value based methods \cite{sunehag_value-decomposition_2017, rashid2018qmix, yao2019smix} that directly approximate the optimal action values $Q^$ place constraints on $Q(s)$ such that it is a monotonic combination of agent utilities. In terms of Tesseract this directly translates to finding the best projection constraining $Q(s)$ to be rank one (\cref{discussion:vdn}). Similarly, the following result demonstrates containment of action-value functions representable by FQL\citep{chen2018factorized} which uses a learnt inner product to model pairwise agent interactions (\textbf{proof and additional results in \cref{discussion:vdn}}):. \begin{proposition} \label{prop:fql} The set of joint Q-functions representable by FQL is a subset of that representable by \textsc{Tesseract}. \end{proposition} MAVEN \cite{mahajan2019maven} illustrates how rank $1$ projections can lead to insufficient exploration and provides a method to avoid suboptimality by using mutual information (MI) to learn a diverse set of rank $1$ projections that correspond to different joint behaviours. In Tesseract, this can simply be achieved by finding the best approximation constraining $Q(s)$ to be rank $k$. Moreover, the CP-decomposition problem, being a product form (\cref{CPD}), is well posed, whereas in \cite{mahajan2019maven} the problem form is $\hat T=\sum_{r=1}^k w_r\oplus^n u_r^i ,i \in \{1..n\},\|\|u_r^i\|\|_2 =1$, which requires careful balancing of different factors $\{1..k\}$ using MI as otherwise all factors collapse to the same estimate. The above improvements are equally important for the critic in actor-critic frameworks. Note that \textsc{Tesseract} is complete in the sense that every possible joint Q-function is representable by it given sufficient approximation rank. This follows as every possible Q-tensor can be expressed as linear combination of one-hot tensors (which form a basis for the set). Many real world problems have high-dimensional observation spaces that are encapsulated in an underlying low dimensional latent space that governs the transition and reward dynamics \cite{azizzadenesheli2016reinforcement}. For example, in the case of robot navigation, the observation is high dimensional visual and sensory input but solving the underlying problem requires only knowing the 2D position. Standard RL algorithms that do not address modelling the latent structure in such problems typically incur poor performance and intractability. In \cref{analysis} we show how Tesseract can be leveraged for such scenarios. Finally, projection to a low rank offers a natural way of regularising the approximate $Q$-functions and makes them easier to learn, which is important for making value function approximation amenable to multi-agent settings. Specifically for the case of actor-critic methods, this provides a natural way to make the critic learn more quickly. Additional discussion about using Tesseract for continuous action spaces can be found in \cref{app:cenv}. \section{Analysis} \label{analysis} In this section we provide a PAC analysis of model-based Tesseract (\cref{alg:model-based}). We focus on the MMDP setting (\cref{background}) for the simplicity of notation and exposition; guidelines for other settings are provided in \cref{app:proofs}. The objective of the analysis is twofold: Firstly it provides concrete quantification of the sample efficiency gained by model-based policy evaluation. Secondly, it provides insights into how Tesseract can similarly reduce sample complexity for model-free methods. \textbf{Proofs for the results stated can be found in \cref{app:proofs}}. We begin with the assumptions used for the analysis: \begin{assumption} \label{a1} For the given MMDP $G=\left\langle S,U,P,r,n,\gamma\right\rangle$, the reward tensor $\hat R(s),\forall s\in S$ has bounded rank $k_1\in \mathbb{N}$. \end{assumption} Intuitively, a small $k_1$ in \cref{a1} implies that the reward is dependent only on a small number of intrinsic factors characterising the actions. \begin{assumption} \label{a2} For the given MMDP $G=\left\langle S,U,P,r,n,\gamma\right\rangle$, the transition tensor $\hat P(s,s'),\forall s,s'\in S$ has bounded rank $k_2\in \mathbb{N}$. \end{assumption} Intuitively a small $k_2$ in \cref{a2} implies that only a small number of intrinsic factors characterising the actions lead to meaningful change in the joint state. \cref{a1}{-2} always hold for a finite MMDP as CP-rank is upper bounded by $\Pi_{j=1}^n \|U_j\|$, where $U_j$ are the action sets. \begin{assumption} \label{a3} The underlying MMDP is ergodic for any policy $\pi$ so that there is a stationary distribution $\rho^\pi$. \end{assumption} Next, we define coherence parameters, which are quantities of interest for our theoretical results: for reward decomposition $\hat R(s) = \sum_r w_{r,s}\otimes^n v_{r,i,s}$, let $\mu_{s} = \sqrt{n}\max_{i,r,j}\|v_{r,i,s}(j)\|$, $w_{s}^{\text{max}} = \max_{i,r}w_{r,s}$,$w_{s}^{\text{min}} = \min_{i,r}w_{r,s}$. Similarly define the corresponding quantities for $\mu_{s,s'}, w_{s,s'}^{\text{max}},w_{s,s'}^{\text{min}}$ for transition tensors $\hat P(s,s')$. A low coherence implies that the tensor's mass is evenly spread and helps bound the possibility of never seeing an entry with very high mass (large absolute value of an entry). \begin{theorem} \label{rankbQ} For a finite MMDP the action-value tensor satisfies $rank(\hat Q^\pi(s))\leq k_1+k_2\|S\|,\forall s \in S, \forall \pi$. \end{theorem} \begin{proof} We first unroll the Tensor Bellman equation in \cref{fig:tbell}. The first term $\hat R$ has bounded rank $k_1$ by \cref{a1}. Next, each contraction term on the RHS is a linear combination of $\{\hat P(s,s')\}_{s'\in S}$ each of which has bounded rank $k_2$ (\cref{a2}). The result follows from the sub-additivity of CP-rank. \end{proof} \cref{rankbQ} implies that for approximations with enough factors, policy evaluation converges: \begin{corollary} \label{cor:suf_rank} For all $k\geq k_1+k_2\|S\|$, the procedure $Q_{t+1}\leftarrow \Pi_{k}\mathcal{T}^{\pi}Q_{t}$ converges to $Q^\pi$ for all $Q_0,\pi$. \end{corollary} \cref{cor:suf_rank} is especially useful for the case of M-POMDP and M-ROMDP with $\|Z\| >> \|S\|$, i.e., where the intrinsic state space dimensionality is small in comparison to the dimensionality of the observations (like robot navigation \cref{robo_nav_sec}). In these cases the Tensorised Bellman equation \cref{fig:tbell} can be augmented by padding the transition tensor $\hat P$ with the observation matrix and the lower bound in \cref{cor:suf_rank} can be improved using the intrinsic state dimensionality. We next give a PAC result on the number of samples required to infer the reward and state transition dynamics for finite MDPs with high probability using sufficient approximate rank $k \geq k_1, k_2$: \begin{theorem}[Model based estimation of $\hat R, \hat P$ error bounds] \label{thm:debound} Given any $\epsilon>0, 1>\delta>0$, for a policy $\pi$ with the policy tensor satisfying $\pi(\mathbf{u}\|s)\geq \Delta$, where \begin{align} \nonumber \Delta = \max_s \frac{C_1 \mu_{s}^6 k^5 (w_{s}^{\text{max}})^4 \log(\|U\|)^4 \log(3k\|\|R(s)\|\|_{F}/\epsilon)} {\|U\|^{n/2} (w_{s}^{\text{min}})^4} \end{align} and $C_1$ is a problem dependent positive constant. There exists $N_0$ which is $O(\|U\|^{\frac{n}{2}})$ and polynomial in $\frac{1}{\delta},\frac{1}{\epsilon}, k$ and relevant spectral properties of the underlying MDP dynamics such that for samples $\geq N_0$, we can compute the estimates $\bar R(s), \bar P(s,s')$ such that w.p. $\geq 1-\delta$, $\|\|\bar{R}(s)-\hat R(s)\|\|_F\leq \epsilon, \|\|\bar{P}(s,s')-\hat P(s,s')\|\|_F\leq \epsilon, \forall s,s' \in S$. \end{theorem} \cref{thm:debound} gives the relation between the order of the number of samples required to estimate dynamics and the tolerance for approximation. \cref{thm:debound} states that aside from allowing efficient PAC learning of the reward and transition dynamics of the multi-agent MDP, \cref{alg:model-based} requires only $O(\|U\|^{\frac{n}{2}})$ to do so, which is a vanishing fraction of $\|U\|^n$, the total number of joint actions in any given state. This also hints at why a tensor based approximation of the $Q$-function helps with sample efficiency. Methods that do not use the tensor structure typically use $O(\|U\|^n)$ samples. The bound is also useful for off-policy scenarios, where only the behaviour policy needs to satisfy the bound. Given the result in \cref{thm:debound}, it is natural to ask what is the error associated with computing the action-values of a policy using the estimated transition and reward dynamics. We address this in our next result, but first we present a lemma bounding the total variation distance between the estimated and true transition distributions: \begin{lemma} \label{tvbound} For transition tensor estimates satisfying $\|\|\bar{P}(s,s')-\hat P(s,s')\|\|_F\leq \epsilon$, we have for any given state-action pair $(s,a)$, the distribution over the next states follows: $TV(P'(\cdot\|s,a),P(\cdot\|s,a))\leq \frac{1}{2}(\|1-f\|+f\|S\|\epsilon)$ where $\frac{1}{1+\epsilon\|S\|}\leq f \leq\frac{1}{1-\epsilon\|S\|}$, where $TV$ is the \textit{total variation} distance. Similarly for any policy $\pi$, $TV(\bar P_{\pi}(\cdot\|s),P_{\pi}(\cdot\|s)), TV(\bar P_{\pi}(s',a'\|s),P_{\pi}(s',a'\|s))\leq \frac{1}{2}(\|1-f\|+f\|S\|\epsilon)$ \end{lemma} We now bound the error of model-based evaluation using approximate dynamics in \cref{thm:q_err}. The first component on the RHS of the upper bound comes from the tensor analysis of the transition dynamics, whereas the second component can be attributed to error propagation for the rewards. \begin{theorem}[Error bound on policy evaluation] \label{thm:q_err} Given a behaviour policy $\pi_b$ satisfying the conditions in \cref{thm:debound} and executed for steps $\geq N_0$, for any policy $\pi$ the model based policy evaluation $Q_{\bar P,\bar R}^\pi$ satisfies: \begin{align} \|Q_{P,R}^\pi(s,a) - Q_{\bar P,\bar R}^\pi(s,a)\|\leq &(\|1-f\|+f\|S\|\epsilon)\frac{\gamma}{2(1-\gamma)^2} \\ &+ \frac{\epsilon}{1-\gamma}, \forall (s,a)\in S\times U^n \end{align} where $f$ is as defined in \cref{tvbound}. \end{theorem} Additional theoretical discussion can be found in \cref{app:atd} \section{Experiments} \label{sec:exps} In this section we present the empirical results on the StarCraft domain. Experiments for a more didactic domain of Tensor games can be found in \cref{app:tg}. We use the model-free version of \textsc{Tesseract} (\cref{alg:model-free}) for all the experiments. \begin{figure} \caption{Performance of different algorithms on different SMAC scenarios: \textcolor{blue}{TAC}, \textcolor{red}{QMIX}, \textcolor[rgb]{0,0.7,0}{VDN}, \textcolor{orange}{FQL}, \textcolor[rgb]{0.76, 0.13, 0.28}{IQL}. } \label{fig:3s5z_smac} \label{fig:2s_vs_1sc} \label{fig:2c_vs_64z} \label{fig:5m_vs_6m} \label{fig:MMM2} \label{fig:27m_vs_30m} \label{fig:6h8z} \label{fig:corridor} \label{fig:smac_exp} \end{figure} \paragraph{StarCraft II} We consider a challenging set of cooperative scenarios from the StarCraft Multi-Agent Challenge (SMAC) \citep{samvelyan2019starcraft}. Scenarios in SMAC have been classified as \textbf{Easy, Hard and Super-hard} according to the performance of exiting algorithms on them. We compare \textsc{Tesseract} (\textcolor{blue}{TAC} in plots) to, \textcolor{red}{QMIX} \citep{rashid2018qmix}, \textcolor[rgb]{0,0.7,0}{VDN} \citep{sunehag_value-decomposition_2017}, \textcolor{orange}{FQL} \citep{chen2018factorized}, and \textcolor[rgb]{0.76, 0.13, 0.28}{IQL} \citep{tan_multi-agent_1993}. VDN and QMIX use monotonic approximations for learning the Q-function. FQL uses a pairwise factorized model to capture effects of agent interactions in joint Q-function, this is done by learning an inner product space for summarising agent trajectories. IQL ignores the multi-agentness of the problem and learns an independent per agent policy for the resulting non-stationary problem. \cref{fig:smac_exp} gives the win rate of the different algorithms averaged across five random runs. \cref{fig:2c_vs_64z} features 2c\_vs\_64zg, a hard scenario that contains two allied agents but 64 enemy units (the largest in the SMAC domain) making the action space of the agents much larger than in the other scenarios. \textsc{Tesseract} gains a huge lead over all the other algorithms in just one million steps. For the asymmetric scenario of 5m\_vs\_6m \cref{fig:5m_vs_6m}, \textsc{Tesseract}, QMIX, and VDN learn effective policies, similar behavior occurs in the heterogeneous scenarios of 3s5z \cref{fig:3s5z_smac} and MMM2\cref{fig:MMM2} with the exception of VDN for the latter. In 2s\_vs\_1sc in \cref{fig:2s_vs_1sc}, which requires a `kiting' strategy to defeat the spine crawler, \textsc{Tesseract} learns an optimal policy in just 100k steps. In the \textbf{super-hard} scenario of 27m\_vs\_30m \cref{fig:27m_vs_30m} having largest ally team of 27 marines, \textsc{Tesseract} again shows improved sample efficiency; this map also shows \textsc{Tesseract}'s ability to scale with the number of agents. Finally in the \textbf{super-hard} scenarios of 6 hydralisks vs 8 zealots \cref{fig:6h8z} and Corridor \cref{fig:corridor} which require careful exploration, \textsc{Tesseract} is the only algorithm which is able to find a good policy. We observe that IQL doesn't perform well on any of the maps as it doesn't model agent interactions/non-stationarity explicitly. FQL loses performance possibly because modelling just pairwise interactions with a single dot product might not be expressive enough for joint-Q. Finally, VDN and QMIX are unable to perform well on many of the challenging scenarios possibly due to the monotonic approximation affecting the exploration adversely \citep{mahajan2019maven}. Additional plots and experiment details can be found in \cref{app:sc2} with \textbf{comparison with other baselines in \cref{app:additional_sc2}} including QPLEX\citep{wang2020qplex}, QTRAN\citep{ son2019qtran}, HQL\citep{matignon2007hysteretic}, COMA\citep{foerster2018counterfactual} . We detail the techniques used for stabilising the learning of tensor decomposed critic in \cref{app:techniques}. \section{Related Work} Previous methods for modelling multi-agent interactions include those that use coordination graph methods for learning a factored joint action-value estimation \cite{guestrin2002coordinated,guestrin2002context,bargiacchi2018learning}, however typically require knowledge of the underlying coordination graph. To handle the exponentially growing complexity of the joint action-value functions with the number of agents, a series of value-based methods have explored different forms of value function factorisation. VDN~\cite{sunehag_value-decomposition_2017} and QMIX~\cite{rashid2018qmix} use monotonic approximation with latter using a mixing network conditioned on global state. QTRAN~\cite{son2019qtran} avoids the weight constraints imposed by QMIX by formulating multi-agent learning as an optimisation problem with linear constraints and relaxing it with L2 penalties. MAVEN~\cite{mahajan2019maven} learns a diverse ensemble of monotonic approximations by conditioning agent $Q$-functions on a latent space which helps overcome the detrimental effects of QMIXâ€™s monotonicity constraint on exploration. Similarly, Uneven~\cite{gupta2020uneven} uses universal successor features for efficient exploration in the joint action space. Qatten~\cite{Yang2020QattenAG} makes use of a multi-head attention mechanism to decompose $Q_{tot}$ into a linear combination of per-agent terms. RODE~\cite{wang2020rode} learns an action effect based role decomposition for sample efficient learning. Policy gradient methods, on the other hand, often utilise the actor-critic framework to cope with decentralisation. MADDPG~\cite{lowe2017multi} trains a centralised critic for each agent. COMA~\cite{foerster2018counterfactual} employs a centralised critic and a counterfactual advantage function. These actor-critic methods, however, suffer from poor sample efficiency compared to value-based methods and often converge to sub-optimal local minima. While sample efficiency has been an important goal for single agent reinforcement learning methods ~\cite{mahajan2017symmetryde, mahajan2017symmetryl, kakade2003sample, lattimore2013sample}, in this work we shed light on attaining sample efficiency for cooperative multi-agent systems using low rank tensor approximation. \emph{Tensor methods} have been used in machine learning, in the context of learning latent variable models~\cite{anandkumar2014tensor} and signal processing \cite{sidiropoulos2017tensor}. Tensor methods provides powerful analytical tools that have been used for various applications, including the theoretical analysis of deep neural networks~\cite{cohen2016expressive}. Model compression using tensors~\cite{cheng2017survey} has recently gained momentum owing to the large sizes of deep neural nets. Using tensor decomposition within deep networks, it is possible to both compress and speed them up~\cite{cichocki2017tensor,t_net}. They allow generalization to higher orders ~\cite{kossaifi2019efficient} and have also been used for multi-task learning and domain adaptation~\cite{bulat2019incremental}. In contrast to prior work on value function factorisation, \textsc{Tesseract} provides a natural spectrum for approximation of action-values based on the rank of approximation and provides theoretical guarantees derived from tensor analysis. Multi-view methods utilising tensor decomposition have previously been used in the context of partially observable single-agent RL~\cite{azizzadenesheli2016reinforcement,azizzadenesheli2019reinforcement}. There the goal is to efficiently infer the underlying MDP parameters for planning under rich observation settings~\cite{krishnamurthy2016pac}. Similarly \citep{bromuri2012tensor} use four dimensional factorization to generalise across Q-tables whereas here we use them for modelling interactions across multiple agents. \section{Conclusions \& Future Work} We introduced \textsc{Tesseract}, a novel framework utilising the insight that the joint action value function for MARL can be seen as a tensor. \textsc{Tesseract} provides a means for developing new sample efficient algorithms and obtain essential guarantees about convergence and recovery of the underlying dynamics. We further showed novel PAC bounds for learning under the framework using model-based algorithms. We also provided a model-free approach to implicitly induce low rank tensor approximation for better sample efficiency and showed that it outperforms current state of art methods. There are several interesting open questions to address in future work, such as convergence and error analysis for rank insufficient approximation, and analysis of the learning framework under different types of tensor decompositions like Tucker and tensor-train \citep{kolda2009tensor}. \input{sections/Acknowledgements.tex} \onecolumn \appendix \section{Additional Proofs} \addtocounter{theorem}{-2} \addtocounter{proposition}{-1} \addtocounter{lemma}{-1} \label{app:proofs} \subsection{Proof of \cref{thm:debound}} \label{proof:app_dbound} \begin{theorem}[Model based estimation of $\hat R, \hat P$ error bounds] Given any $\epsilon>0, 1>\delta>0$, for a policy $\pi$ with the policy tensor satisfying $\pi(\mathbf{u}\|s)\geq \Delta$, where \begin{align} \label{eq:polcon} \Delta = \max_s \frac{C_1 \mu_{s}^6 k^5 (w_{s}^{\text{max}})^4 \log(\|U\|)^4 \log(3k\|\|R(s)\|\|_{F}/\epsilon)} {\|U\|^{n/2} (w_{s}^{\text{min}})^4} \end{align} and $C_1$ is a problem dependent positive constant. There exists $N_0$ which is $O(\|U\|^{\frac{n}{2}})$ and polynomial in $\frac{1}{\delta},\frac{1}{\epsilon}, k$ and relevant spectral properties of the underlying MDP dynamics such that for samples $\geq N_0$, we can compute the estimates $\bar R(s), \bar P(s,s')$ such that w.p. $\geq 1-\delta$, $\|\|\bar{R}(s)-\hat R(s)\|\|_F\leq \epsilon, \|\|\bar{P}(s,s')-\hat P(s,s')\|\|_F\leq \epsilon, \forall s,s' \in S$. \end{theorem} \begin{proof} For the simplicity of notation and emphasising key points of the proof, we focus on orthogonal symmetric tensors with $n=3$. Guidelines for more general cases are provided by the end of the proof. We break the proof into three parts: Let policy $\pi$ satisfy $\pi(\mathbf{u}\|s)\geq \Delta$ \cref{eq:polcon}. Let $\rho$ be the stationary distribution of $\pi$ (exists by \cref{a3}) and let $N_1 = \max_s \frac{1}{\rho(s)}\log\Big(\frac{12\sqrt{k}\|\|R(s)\|\|_F}{\epsilon}\Big)$. From $N_1$ samples drawn from $\rho$ by following $\pi$, we estimate $\bar{R}$, the estimated reward tensor computed by using Algorithm $1$ in \cite{jain2014provable}. We have by application of union bound along with Theorem $1.1$ in \cite{jain2014provable} for each $s\in S$, w.p. $\geq 1 - \|U\|^{-5}\log_2\Big(\frac{12\sqrt{k}\prod_s\|\|R(s)\|\|_F}{\epsilon}\Big) = p_\epsilon$, $\|\|\bar{R}(s)-\hat R(s)\|\|_F\leq \epsilon/3, \forall s\in S$. We now provide a boosting scheme to increase the confidence in the estimation of $\hat R(\cdot)$ from $p_\epsilon$ to $1 - \delta/3$. Let $\eta = \frac{1}{2}\Big(p_\epsilon -\frac{1}{2}\Big) > 0$ (for clarity of the presentation we assume $p_\epsilon >\frac{1}{2}$ and refer the reader to \cite{kearns1994introduction} for the other more involved case). We compute $M$ independent estimates $\{\bar R_i, i \in \{1..M\}\}$ for $\hat R(s)$ and find the biggest cluster $\mathcal{C} \subseteq \{\bar R_i\}$ amongst the estimates such that for any $\bar R_i, \bar R_j \in \mathcal{C}, \|\|\bar R_i-\bar R_j\|\|_F \leq \frac{2\epsilon}{3}$. We then output any element of $\mathcal{C}$. Intuitively as $p_\epsilon >\frac{1}{2}$, most of the estimates will be near the actual value $\hat R(s)$, this can be confirmed by using the Hoeffding Lemma\cite{kearns1994introduction}. It follows that for $M\geq \frac{1}{2\eta^2}\ln(\frac{3\|S\|}{\delta})$ the output of the above procedure satisfies $\|\|\bar{R}(s)-\hat R(s)\|\|_F\leq \epsilon$ w.p. $\geq 1 - \frac{\delta}{3\|S\|}$ for any particular $s$. Thus $MN_1$ samples from stationary distribution are sufficient to ensure that for all $s \in S$, w.p. $\geq 1 - \delta/3$, $\|\|\bar{R}(s)-\hat R(s)\|\|_F\leq \epsilon$. Secondly we note that $\hat P (s,s')$ for any $s,s' \in S$ is a tensor whose entries are the parameters of a Bernoulli distribution. Under \cref{a2}, it can be seen as a latent topic model \cite{anandkumar2012method} with $k$ factors, $\hat P(s,s') = \sum_{r=1}^{k} w_{s,s',r}\otimes^n u_{s,s',r} $. Moreover it satisfies the conditions in Theorem $3.1$ \cite{anandkumar2012method} so that $\exists N_2 = \max_{s,s'}\frac{1}{\rho(s)} N_2(s,s')$ where each $N_2(s,s')$ is $\mathcal{O}\Big(\frac{k^{10} \|S\|^2 \ln^2(3\|S\|/\delta)}{\delta^2 \epsilon^{'2}}\Big)$ depending on the spectral properties of $\hat P(s,s')$ as given in the theorem and satisfies $\|\|\bar{u_{s,s',r}}-u_{s,s',r}\|\|_2\leq \epsilon'$ on running Algorithm B in \cite{anandkumar2012method} w.p. $\geq 1 - \frac{\delta}{3\|S\|}$. We pick $\epsilon' = \frac{\epsilon}{7n^2k \mu_{s,s'}^2 (w_{s,s'}^{\text{max}})^2} $ so that $\|\|\bar{P}(s,s')-\hat P(s,s')\|\|_F\leq \epsilon, \forall s,s' \in S$. We filter off the effects of sampling from a particular policy by using lower bound constraint in \cref{eq:polcon} and sampling $\frac{N_2}{\Delta}$ samples. Finally we account for the fact that there is a delay in attaining the stationary distribution $\rho$ and bound the failure probability of significantly deviating from $\rho$ empirically. Let $\rho' = \min_s \rho(s)$ and $t_{\text{mix},\pi}(x)$ represent the minimum number of samples that need to drawn from the Markov chain formed by fixing policy $\pi$ so that for the state distribution $\rho_{t}(s)$ at time step $t = t_{\text{mix},\pi}(x)$ we have $TV(\rho_{t} - \rho)\leq x$ for any starting state $s\in S$ where $TV(\cdot,\cdot)$ is the total variation distance. We let the policy run for a burn in period of $t'=t_{\text{mix},\pi}(\rho'/4)$. For a sample of $N_3$ state transitions after the burn in period, let $\bar \rho$ represent the empirical state distribution. By applying the Hoeffding lemma for each state, we get: $P(\|\bar\rho(s) - \rho_{t'}(s)\|\geq \rho'/4)\leq 2\exp\Big(\frac{-N_3\rho^{'2}}{8}\Big)$, so that for $N_3\geq \frac{8}{\rho^{'2}}\ln\Big(\frac{6\|S\|}{\delta}\Big)$ we have w.p. $\geq 1 - \frac{\delta}{3\|S\|}$, $\|\bar\rho(s) - \rho(s)\|< \rho'/2, \forall s \in S$. Putting everything together we get with $t_{\text{mix},\pi}(\rho'/4) + \max\{2MN_1, \frac{2N_2}{\Delta}, N_3\}$ samples, the underlying reward and probability tensors can be recovered such that w.p. $\geq 1-\delta$, $\|\|\bar{R}(s)-\hat R(s)\|\|_F\leq \epsilon, \|\|\bar{P}(s,s')-\hat P(s,s')\|\|_F\leq \epsilon, \forall s,s' \in S$. For extending the proof to the case of non-orthogonal tensors, we refer the reader to use whitening transform as elucidated in \cite{anandkumar2014tensor}. Likewise for asymmetric, higher order ($n>3$) tensors methods shown in \cite{jain2014provable, anandkumar2014tensor, anandkumar2012method} should be used. Finally for the case of M-POMDP and M-ROMDP, the corresponding results for single agent POMDP and ROMDP should be used, as detailed in \cite{azizzadenesheli2019reinforcement, azizzadenesheli2016reinforcement} respectively. \end{proof} \subsection{Proof of \cref{tvbound}} \label{proof:tvbound} \begin{lemma} For transition tensor estimates satisfying $\|\|\bar{P}(s,s')-\hat P(s,s')\|\|_F\leq \epsilon$, we have for any given state and action pair $s,a$, the distribution over the next states follows: $TV(P'(\cdot\|s,a),P(\cdot\|s,a))\leq \frac{1}{2}(\|1-f\|+f\|S\|\epsilon)$ where $\frac{1}{1+\epsilon\|S\|}\leq f \leq\frac{1}{1-\epsilon\|S\|}$. Similarly for any policy $\pi$, $TV(\bar P_{\pi}(\cdot\|s),P_{\pi}(\cdot\|s)), TV(\bar P_{\pi}(s',a'\|s),P_{\pi}(s',a'\|s))\leq \frac{1}{2}(\|1-f\|+f\|S\|\epsilon)$ \end{lemma} \begin{proof} Let $ \bar P(\cdot\|s,a)$ be the next state probability estimates obtained from the tensor estimates. We next normalise them across the next states to get the (estimated)distribution $ P'(\cdot\|s,a) = f \bar P(\cdot\|s,a)$ where $f = \frac{1}{\sum_{s'}\bar P(s'\|s,a)}$. Dropping the conditioning for brevity we have: \begin{align} TV(P',P) &= \frac{1}{2}\sum_{s'} \|P(s')-f \bar P(s')\|\\ &\leq\frac{1}{2}(\sum_{s'} \|P(s')-f P(s')\| +\|f P(s')- \bar P(s')\|)\\ &=\frac{1}{2}(\|1-f\|+f\|S\|\epsilon) \end{align} The other two results follow using the definition of TV and Fubini's theorem followed by reasoning similar to above. \end{proof} \subsection{Proof of \cref{thm:q_err}} \label{proof:app_q_err} \begin{theorem} [Error bound on policy evaluation] Given a behaviour policy $\pi_b$ satisfying the conditions in \cref{thm:debound} and being executed for steps $\geq N_0$, we have that for any policy $\pi$ the model based policy evaluation $Q_{\bar P,\bar R}^\pi$ satisfies: \begin{align} \|Q_{P,R}^\pi(s,a) - Q_{\bar P,\bar R}^\pi(s,a)\|&\leq (\|1-f\|+f\|S\|\epsilon)\frac{\gamma}{2(1-\gamma)^2}+ \frac{\epsilon}{1-\gamma}, \forall (s,a)\in S\times U^n \end{align} where $f$ is as defined in \cref{tvbound}. \end{theorem} \begin{proof} Let $\bar P, \bar R$ be the estimates obtained after running the procedure as described in \cref{thm:debound} with samples corresponding to error $\epsilon$ and confidence $1-\delta$. We will bound the error incurred in estimation of the action-values using $\bar P, \bar R$. We have for any $\pi$ by using triangle inequality \begin{align} \label{tring_q} \|Q_{P,R}^\pi(s,a) - Q_{\bar P,\bar R}^\pi(s,a)\|&\leq \|Q_{P,R}^\pi(s,a)- Q_{\bar P, R}^\pi(s,a)\| + \|Q_{\bar P, R}^\pi(s,a) - Q_{\bar P,\bar R}^\pi(s,a)\| \end{align} where we use the subscript to denote whether actual or approximate values are used for $P,R$ respectively. We first focus on the first term on the RHS of \cref{tring_q}. Let $R_\pi(s_t) = \sum_{a_t} \pi(a_t\|s_t) R(s_t,a_t)$. We use $P_{t,\pi}(\cdot\|s) = (P_{\pi}(\cdot\|s))^t$ to denote the state distribution after $t$ time steps. Consider a horizon $h$ interleaving $Q$ estimate given by: \begin{align} Q_h^\pi(s,a) &= R(s_t,a_t)+\sum_{t=1}^{h-1}\gamma^t\mathbb{E}_{\bar P_{t,\pi}(\cdot\|s)}[ R_\pi(s_t)]+ \sum_{t=h}^{\infty}\gamma^t\mathbb{E}_{P_{t-h,\pi}(\cdot\|s_h)\cdot \bar P_{h,\pi}(s_h\|s) }[ R_\pi(s_t)] \end{align} Where $s_0=s, a_0=a$ and the first $h$ steps are unrolled according to $\bar P_\pi$, the rest are done using the true transition $P_\pi$. We have that: \begin{align} \|Q_{P,R}^\pi(s,a) - Q_{\bar P,\bar R}^\pi(s,a)\|&=\|Q_0^\pi(s,a) - Q_\infty^\pi(s,a)\|\leq \sum_{h=0}^\infty \|Q_h^\pi(s,a) - Q_{h+1}^\pi(s,a)\| \end{align} Each term in the RHS of the above can be independently bounded as : \begin{align} \|Q_h^\pi(s,a) - Q_{h+1}^\pi(s,a)\|=&\gamma^{h+1}\Big\|\mathbb{E}_{\bar P_{h+1,\pi}(s_{h+1}\|s)}\Big[\sum_{a_{h+1}}\pi(a_{h+1}\|s_{h+1})Q_\infty^\pi(s_{h+1}.a_{h+1})\Big]\\&-\mathbb{E}_{P_\pi\bar P_{h,\pi}(s_{h+1}\|s)}\Big[\sum_{a_{h+1}}\pi(a_{h+1}\|s_{h+1})Q_\infty^\pi(s_{h+1}.a_{h+1})\Big]\Big\| \end{align} As the rewards are bounded we get the expression above is $\leq \frac{1}{1-\gamma}\gamma^{h+1}TV(\bar P_{\pi}(s',a'\|s),P_{\pi}(s',a'\|s))$. Finally using \cref{tvbound} we get $\leq (\frac{1}{2}(\|1-f\|+f\|S\|\epsilon))\frac{\gamma^{h+1}}{1-\gamma}$. And plugging in the original expression: \begin{align} \|Q_{P,R}^\pi(s,a) - Q_{\bar P,\bar R}^\pi(s,a)\|\leq (\|1-f\|+f\|S\|\epsilon)\frac{\gamma}{2(1-\gamma)^2} \end{align} Next the second term on the RHS of \cref{tring_q} can easily be bounded by $\frac{\epsilon}{1-\gamma}$ which gives: \begin{align} \|Q_{P,R}^\pi(s,a) - Q_{\bar P,\bar R}^\pi(s,a)\|&\leq (\|1-f\|+f\|S\|\epsilon)\frac{\gamma}{2(1-\gamma)^2}+ \frac{\epsilon}{1-\gamma} \end{align} \end{proof} \section{Discussion} \subsection{Relation to other methods} \label{discussion:vdn} In this section we study the relationship between \textsc{Tesseract} and some of the existing methods for MARL. \subsubsection{FQL} FQL~\citep{chen2018factorized} uses a learnt inner product space to represent the dependence of joint Q-function on pair wise agent interactions. The following result shows containment of FQL representable action-value function by \textsc{Tesseract} : \begin{proposition} The set of joint Q-functions representable by FQL is a subset of that representable by \textsc{Tesseract}. \end{proposition} \begin{proof} In the most general form, any join Q-function representable by FQL has the form: \begin{equation} Q_{fql}(s, \mathbf{u}) = \sum_{i=1:n} q_i(s,u_i) + \sum_{i=1:n, j<i} \langle f_i(s,u_i), f_j(s,u_j) \rangle \end{equation} where $q_i: S\times U \to \mathbb{R}$ are individual contributions to joint Q-function and $f_i: S\times U \to \mathbb{R}^d$ are the vectors describing pairwise interactions between the agents. There are $n \choose 2$ pairs of agents to consider for (pairwise)interactions. Let $\mathscr{P} \triangleq {(i,j)}$ be the ordered set of agent pairs where $i>j$ and $i,j \in \{1..n\}$, let $\mathscr{P}_{k}$ denote the $k$th element of $\mathscr{P}$. Define membership function $m:\mathscr{P} \times \{1..n\} \to \{0,1\}$ as: \begin{align} m((i,j),x) = \begin{cases} 1 & if $x=i \vee x=j$ \\ 0 & otherwise \end{cases} \end{align} Define the mapping $v_i: S \to \mathbb{R}^{\|U\|\times D}$ where $D = d {n \choose 2}+n$ and $v_{i,k}$ represents the $k$th column of $v_i$. \begin{align} v_i(s) \triangleq \begin{cases} v_i(s)[j, (k-1)d+1:kd] = f_i(s, u_j) & if $m(\mathscr{P}_{k}, i)=1$ \\ v_i(s)[j, D-n+i] = q_i(s, u_j)\\ v_i(s)[j, k] = 1 & otherwise \end{cases} \end{align} We get that the tensors: \begin{equation} Q_{fql}(s) = \sum_{k=1}^D \otimes^n v_{i,k}(s) \end{equation} Thus any $Q_{fql}$ can be represented by \textsc{Tesseract}, note that the converse is not true ie. any arbitrary Q-function representable by \textsc{Tesseract} may not be representable by FQL as FQL cannot model higher-order ($>2$ agent) interactions. \end{proof} \subsubsection{VDN} VDN~\cite{sunehag_value-decomposition_2017} learns a decentralisable factorisation of the joint action-values by expressing it as a sum of per agent utilities $\hat Q=\oplus^n u_i ,i \in \{1..n\}$. This can be equivalently learnt in \textsc{Tesseract} by finding the best rank one projection of $\exp(\hat Q(s))$. We formalise this in the following result: \begin{proposition} \label{prop:vdn} For any MMDP, given policy $\pi$ having $Q$ function representable by VDN ie. $\hat Q^{\pi}(s)=\oplus^n u_i(s) ,i \in \{1..n\}$, $\exists v_i(s) \forall s\in S$, the utility factorization can be recovered from rank one CP-decomposition of $\exp(\hat Q^{\pi})$ \end{proposition} \begin{proof} We have that : \begin{align} \exp(\hat Q^{\pi}(s)) &= \exp(\oplus^n u_i(s))\\ &=\otimes^n \exp(u_i(s)) \end{align} Thus $(\exp(u_i(s)))_{i=1}^n \in \argmin_{v_i(s)}\|\|\exp(\hat Q^{\pi}(s)) - \otimes^n v_i(s)\|\|_F \forall s \in S$ and there always exist $v_i(s)$ that can be mapped to some $u_i(s)$ via exponentiation. In general any Q-function that is representable by VDN can be represented by \textsc{Tesseract} under an exponential transform (\cref{subsec:TessAlgos}). \end{proof} \subsection{Injecting Priors for Continuous Domains} \label{app:cenv} \begin{figure} \caption{Continuous actions task with three agents chasing a prey. Perturbing Agent 2's action direction by small amount $\theta$ leads to a small change in the joint value. } \label{fig:perturb} \end{figure} We now discuss the continuous action setting. Since the action set of each agent is infinite, we impose further structure while maintaining appropriate richness in the hypothesis class of the proposed action value functions. Towards this we present an example of a simple prior for \textsc{Tesseract} for continuous action domains. WLOG, let $U\triangleq \mathbb{R}^d$ for each agent $\in {1..n}$. We are now interested in the function class given by $\mathcal{Q}\triangleq \{Q: S\times U^n\to \mathbb{R}\}$ where each $Q(s)$ $\triangleq \langle T(s,\{\|\|u^i\|\|_2\}), \otimes^n u^i\rangle$, here $T(\cdot): S\times \mathbb{R}^n\to \mathbb{R}^{d^n}$ is a function that outputs an order $n$ tensor and is invariant to the direction of the agent actions, $\langle\cdot,\cdot\rangle$ is the dot product between two order $n$ tensors and $\|\|\cdot\|\|_2$ is the Euclidean norm. Similar to the discrete case, we define $\mathcal{Q}_k \triangleq \{Q: Q\in \mathcal{Q} \land rank(T(\cdot)) =k, \forall s \in S\}$. The continuous case subsumes the discrete case with $T(\cdot)\triangleq Q(\cdot)$ and actions encoded as one hot vectors. We typically use rich classes like deep neural nets for $Q$ and $T$ parametrised by $\phi$. We now briefly discuss the motivation behind the example continuous case formulation: for many real world continuous action tasks the joint payoff is much more sensitive to the magnitude of the actions than their directions, i.e., slightly perturbing the action direction of one agent while keeping others fixed changes the payoff by only a small amount (see \cref{fig:perturb}). Furthermore, $T_\phi$ can be arbitrarily rich and can be seen as representing utility per agent per action dimension, which is precisely the information required by methods for continuous action spaces that perform gradient ascent w.r.t.\ $\nabla_{u^i}Q$ to ensure policy improvement. Further magnitude constraints on actions can be easily handled by a rich enough function class for $T$. Lastly we can further abstract the interactions amongst the agents by learnable maps $f_\eta^i(u^i,s): \mathbb{R}^d \times S \to \mathbb{R}^m$, $m>>d$ and considering classes $Q(s,\mathbf{u})\triangleq \langle T(s,\{\|\|u^i\|\|\}), \otimes^n f_\eta^i(u^i)\rangle$ where $T(\cdot): S\times \mathbb{R}^n\to \mathbb{R}^{m^n}$. \subsection{Additional theoretical discussion} \label{app:atd} \subsubsection{Selecting the CP-rank for approximation} While determining the rank of a fully observed tensor is itself NP-hard \citep{hillar2013most}, we believe we can help alleviate this problem due to two key observations: \begin{itemize} \item The tensors involved in \textsc{Tesseract} capture dependence of transition and reward dynamics on the action space. Thus if we can approximately identify (possibly using expert knowledge) the various aspects in which the actions available at hand affect the environment, we can get a rough idea of the rank to use for approximation. \item Our experiments on different domains (\cref{sec:exps}, \cref{app:additional_exp}) provide evidence that even when using a rank insufficient approximation, we can get good empirical performance and sample efficiency. (This is also evidenced by the empirical success of related algorithms like VDN which happen to be specific instances under the \textsc{Tesseract} framework.) \end{itemize} \section{Additional experiments and details} \label{app:additional_exp} \subsection{StarCraft II} \label{app:sc2} \begin{figure} \caption{The 2c\_vs\_64zg scenario in SMAC. } \label{fig:snap_col} \end{figure} In the SMAC bechmark\citep{samvelyan2019starcraft} (https://github.com/oxwhirl/smac), agents can $\mathtt{move}$ in four cardinal directions, $\mathtt{stop}$, take $\mathtt{noop}$ (do nothing), or select an enemy to $\mathtt{attack}$ at each timestep. Therefore, if there are $n_e$ enemies in the map, the action space for each ally unit contains $n_e + 6$ discrete actions. \subsubsection{Additional Experiments} \label{app:additional_sc2} \begin{figure} \caption{Performance of different algorithms on different SMAC scenarios: \textcolor{blue}{TAC}, \textcolor{red}{QTRAN}, \textcolor[rgb]{0,0.7,0}{QPLEX}, \textcolor{orange}{COMA}, \textcolor[rgb]{0.76, 0.13, 0.28}{HQL}. } \label{fig:a_3s5z_smac} \label{fig:a_2s_vs_1sc} \label{fig:a_2c_vs_64z} \label{fig:a_5m_vs_6m} \label{fig:a_MMM2} \label{fig:a_27m_vs_30m} \label{fig:a_6h8z} \label{fig:a_corridor} \label{fig:additional_sc2} \end{figure} In addition to the baselines in main text \cref{sec:exps}, we also include 4 more baselines: \textcolor{red}{QTRAN} \citep{son2019qtran}, \textcolor[rgb]{0,0.7,0}{QPLEX} \citep{wang2020qplex}, \textcolor{orange}{COMA} \citep{foerster2018counterfactual} and \textcolor[rgb]{0.76, 0.13, 0.28}{HQL}. QTRAN tries to avoid the issues arising with representational constraints by posing the decentralised multi agent problem as optimisation with linear constraints, these constraints are relaxed using L2 penalties for tractability \citep{mahajan2019maven}. Similarly, QPLEX another recent method uses an alternative formulation using advantages for ensuring the \textit{Individual Global Max} (IGM) principle \citep{son2019qtran}. COMA is an actor-critic method that uses a centralised critic for computing a counterfactual baseline for variance reduction by marginalising across individual agent actions. Finally, HQL uses the heuristic of differential learning rates on top of IQL \citep{tan_multi-agent_1993} to address problems associated with decentralized exploration. \textbf{\cref{fig:additional_sc2} }gives the average win rates of the baselines on different SMAC scenarios across five random runs (with one standard deviation shaded). We observe that \textsc{Tesseract} outperforms the baselines by a large margin on most of the scenarios, especially on the \textbf{super-hard} ones on which the exiting methods struggle, this validates the sample efficiency and representational gains supported by our analysis. We observe that HQL is unable to learn a good policy on most scenarios, this might be due to uncertainty in the bootstrap estimates used for choosing the learning rate that confounds with difficulties arising from non-stationarity. We also observe that COMA does not yield satisfactory performance on any of the scenarios. This is possibly because it does not utilise the underlying tensor structure of the problem and suffers from a \textit{lagging critic}. While QPLEX is able to alleviate the problems arising from relaxing the IGM constraints in QTRAN, it lacks in performance on the \textbf{super-hard} scenarios of Corridor and 6h\_vs\_8z. \subsubsection{Experimental Setup for SMAC} \label{app:setup_sc2} We use a factor network for the tensorised critic which comprises of a fully connected MLP with two hidden layers of dimensions 64 and 32 respectively and outputs a $r\|U\|$ dimensional vector. We use an identical policy network for the actors which outputs a $\|U\|$ dimensional vector and a value network which outputs a scalar state-value baseline $V(s)$. The agent policies are derived using softmax over the policy network output. Similar to previous work \cite{samvelyan2019starcraft}, we use two layer network consisting of a fully-connected layer followed by GRU (of 64-dimensional hidden state) for encoding agent trajectories. We used Relu for non-linearities. All the networks are shared across the agents. We use ADAM as the optimizer with learning rate $5\times10^{-4}$. We use entropy regularisation with scaling coefficient $\beta = 0.005$. We use an approximation rank of $7$ for Tesseract ('TAC') for the SMAC experiments. A batch size of 512 is used for training which is collected across 8 parallel environments (additional setup details in \cref{app:techniques}). Grid search was performed over the hyper-parameters for tuning. For the baselines QPLEX, QMIX, QTRAN, VDN, COMA, IQL we use the open sourced code provided by their authors at https://github.com/wjh720/QPLEX and https://github.com/oxwhirl/pymarl respectively which has hyper-parameters tuned for SMAC domain. The choice for architecture make the experimental setup of the neural networks used across all the baselines similar. We use a similar trajectory embedding network as mentioned above for our implementations of HQL and FQL which is followed by a network comprising of a fully connected MLP with two hidden layers of dimensions 64 and 32 respectively. For HQL this network outputs $\|U\|$ action utilities. For FQL, it outputs a $\|U\|+d$ vector: first $\|U\|$ dimension are used for obtaining the scalar contribution to joint Q-function and rest $d$ are used for computing interactions between agents via inner product. We use ADAM as the optimizer for these two baselines. We use differential learning rates of $\alpha = 1\times10^{-3}, \beta=2\times10^{-4}$ for HQL searched over a grid of $\{1,2,5,10\}\times10^{-3} \times\{1,2,5,10\}\times10^{-4}$. FQL uses the same learning rate $5\times10^{-4}$ with $d = 10$ which was searched over set $\{5, 10, 15\}$. The baselines use $\epsilon-$greedy for exploration with $\epsilon$ annealed from $1.0 \to 0.05$ over 50K steps. For super-hard scenarios in \textbf{SMAC} we extend the anneal time to 400K steps. We use temperature annealing for \textsc{Tesseract} with temperature given by $\tau = \frac{2T}{T+t}$ where $T$ is the total step budget and $t$ is the current step. Similarly we use temperature $\tau = \frac{4T}{T+3t}$ for super-hard \textbf{SMAC} scenarios. The discount factor was set to $0.99$ for all the algorithms. Experiment runs take 1-5 days on a Nvidia DGX server depending on the size of the StarCraft scenario. \subsection{Techniques for stabilising \textsc{Tesseract} critic training for Deep-MARL} \label{app:techniques} \begin{itemize} \item We used a gradient normalisation of $0.5$. The parameters exclusive to the critic were separately subject to the gradient normalisation, this was done because the ratio of gradient norms for the actor and the critic parameters can vary substantially across training. \item We found that using multi-step bootstrapping substantially reduced target variance for Q-fitting and advantage estimation (we used the advantage based policy gradient $\int_{S} \rho^\pi(s)\int_{\mathbf{U}}\nabla\pi_\theta(\mathbf{u\|s})\hat A^{\pi}(s,\mathbf{u}) d\mathbf{u}ds$ \citep{sutton2011reinforcement}) for \textbf{SMAC} experiments. Specifically for horizon T, we used the Q-target as: \begin{align} &Q_{target,t} = \sum_{k=1}^{T-t} \lambda^{k}g_{t,k} \\ & g_{t,k} = R_t + \gamma R_{t+1} + ... + \gamma^{k}V(s_{t+k}) \end{align} and similarly for value target. Likewise, the generalised advantage is estimated as: \begin{align} &\hat A_t = \sum_{k=0}^{T-t} (\gamma\lambda)^{k}\delta_{t+k}\\ &\delta_{t} = R_{t}+ \gamma\hat Q(s_{t+1}, \mathbf{u}_{t+1})-V(s_t) \end{align} Where $\hat Q$ is the tensor network output and the estimates are normalized by the accumulated powers of $\lambda$. We used $T=64, \gamma = 0.99$ and $\lambda = 0.95$ for the experiments. \item The tensor network factors were squashed using a sigmoid for clipping and were scaled by $2.0$ for \textbf{SMAC} experiments. Additionally, we initialised the factors according to $\mathcal{N}(0, 0.01)$ (before applying a sigmoid transform) so that value estimates can be effectively updated without the gradient vanishing. \item Similarly, we used clipping for the action-value estimates $\hat Q$ to prevent very large estimates: \begin{align} clip(\hat{Q}_t) = min\{\hat{Q}_t, R_{max}\} \end{align} we used $R_{max}=40$ for the \textbf{SMAC} experiments. \end{itemize} \begin{figure} \caption{Variations on \textsc{Tesseract} } \label{fig:ab_tech} \label{fig:ab_rank} \label{fig:ablation_sc2} \end{figure} We provide the ablation results on the stabilisation techniques mentioned above on the 2c\_vs\_64zg scenario in \cref{fig:ab_tech}. The plot lines correspond to the ablations: \textcolor{red}{TAC-multi}: no multi-step target and advantage estimation, \textcolor[rgb]{0,0.7,0}{TAC-clip}: no value upper bounding/clipping, \textcolor{orange}{TAC-norm}: no separate gradient norm, \textcolor[rgb]{0.76, 0.13, 0.28}{TAC-init}: no initialisation and sigmoid squashing of factors. We observe that multi-step estimation of target and advantage plays a very important role in stabilising the training, this is because noisy estimates can adversely update the learn factors towards undesirable fits. Similarly, proper initialisation plays a very important role in learning the Q-tensor as otherwise a larger number of updates might be required for the network to learn the correct factorization, adversely affecting the sample efficiency. Finally we observe that max-clipping and separate gradient normalisation do impact learning, although such effects are relatively mild. We also provide the learning curves for \textsc{Tesseract} as the CP rank of Q-approximation is changed, \cref{fig:ab_rank} gives the learning plots as the CP-rank is varied over the set $\{3, 7, 11\}$. Here, we observe that approximation rank makes little impact on the final performance of the algorithm, however it may require more samples in learning the optimal policy. Our PAC analysis \cref{thm:debound} also supports this. \subsection{Tensor games:} \label{app:tg} We introduce tensor games for our experimental evaluation. These games generalise the matrix games often used in $2$-player domains. Formally, a tensor game is a cooperative MARL scenario described by tuple $(n,\|U\|,r)$ that respectively defines the number of agents (dimensions), the number of actions per agent (size of index set) and the rank of the underlying reward tensor \cref{fig:tgg}. Each agent learns a policy for picking a value from the index set corresponding to its dimension. The joint reward is given by the entry corresponding to the joint action picked by the agents, with the goal of finding the tensor entry corresponding to the maximum reward. We consider the CTDE setting for this game, which makes it additionally challenging. We compare \textsc{Tesseract} (\textcolor{blue}{TAC}) with \textcolor[rgb]{0,0.7,0}{VDN}, \textcolor{red}{QMIX} and independent actor-critic (\textcolor[rgb]{0.76, 0.13, 0.28}{IAC}) trained using Reinforce \cite{sutton2000policy}. Stateless games provide are ideal for isolating the effect of an exponential blowup in the action space. The natural difficulty knobs for stateless games are $\|n\|$ and $\|U\|$ which can be increased to obtain environments with large joint action spaces. Furthermore, as the rank $r$ increases, it becomes increasingly difficult to obtain good approximations for $\hat T$. \begin{figure} \caption{Tensor games example with $3$ agents ($n$) having $3$ actions each ($a$). Optimal joint-action \textbf{(a1, a3, a1)} shown in orange. } \label{fig:tgg} \end{figure} \begin{figure} \caption{Experiments on tensor games.} \label{fig:n5a10} \label{fig:n6a10} \label{fig:rab} \label{fig:erab} \end{figure} \cref{fig:n5a10} \cref{fig:n6a10} present the learning curves for the algorithms for two game scenarios, averaged over 5 random runs with game parameters as mentioned in the figures. We observe that \textsc{Tesseract} outperforms the other algorithms in all cases. Moreover, while the other algorithms find it increasingly difficult to learn good policies, \textsc{Tesseract} is less affected by this increase in action space. As opposed to the IAC baseline, \textsc{Tesseract} quickly learns an effective low complexity critic for scaling the policy gradient. QMIX performs worse than VDN due to the additional challenge of learning the mixing network. In \cref{fig:rab} we study the effects of increasing the approximation rank of Tesseract ($k$ in decomposition $\hat Q(s) \approx T = \sum_{r=1}^k w_r\otimes^n g_{\phi,r}(s^i) ,i \in \{1..n\},$) for a fixed environment with $5$ agents, each having $10$ actions and the environment rank being $8$. While all the three settings learn the optimal policy, it can be observed that the number of samples required to learn a good policy increases as the approximation rank is increased (notice delay in 'Rank 8', 'Rank 32' plot lines). This again is in-line with our PAC results, and makes intuitive sense as a higher rank of approximation directly implies more parameters to learn which increases the samples required to learn. We next study how approximation of the actual $Q$ tensors affects learning. In \cref{fig:erab} we compare the performance of using a rank-$2$ \textsc{Tesseract} approximation for environment with $5$ agents, each having $10$ actions and the environment reward tensor rank being varied from $8$ to $128$. We found that for the purpose of finding the optimal policy, \textsc{Tesseract} is fairly stable even when the environment rank is greater than the model approximation rank. However performance may drop if the rank mismatch becomes too large, as can be seen in \cref{fig:erab} for the plot lines 'E\_rank 32', 'E\_rank 128', where the actual rank required to approximate the underlying reward tensor is too high and using just $2$ factors doesn't suffice to accurately represent all the information. \subsubsection{Experimental setup for Tensor games} For tensor game rewards, we sample $k$ linearly independent vectors $u_r^i$ from $\|\mathcal{N}(0,1)^{\|U\|}\|$ for each agent dimension $i \in \{1..n\}$. The reward tensor is given by $T=\sum_{r=1}^k w_r\otimes^n u_r^i ,i \in \{1..n\}$. Thus $T$ has roughly $k$ local maxima in general for $k<<\|U\|^n$. We normalise $\hat T$ so that the maximum entry is always $1$. All the agents use feed-forward neural networks with one hidden layer having $64$ units for various components. Relu is used for non-linear activation. The training uses ADAM \citep{kingma2014adam} as the optimiser with a $L2$ regularisation of $0.001$. The learning rate is set to $0.01$. Training happens after each environment step. The batch size is set to $32$. For an environment with $n$ agents and $a$ actions available per agent we run the training for $\frac{a^n}{10}$ steps. For VDN \citep{sunehag_value-decomposition_2017} and QMIX\citep{rashid2018qmix} the $\epsilon$-greedy coefficient is annealed from $0.9$ to $0.05$ at a linear rate until half of the total steps after which it is kept fixed. For Tesseract ('TAC') and Independent Actor-Critic ('IAC') we use a learnt state baseline for reducing policy gradient variance. We also add entropy regularisation for the policy with coefficient starting at $0.1$ and halved after every $\frac{1}{10}$ of total steps. We use an approximation rank of $2$ for Tesseract ('TAC') in all the comparisons except \cref{fig:rab} where it is varied for ablation. \end{document}	arXiv
\begin{document} \title{{\Large The geometrical properties of parity and time reversal operators in two dimensional spaces} \thanks{This project is supported by Research Fund, Kumoh National Institute of Technology.}} \author{Minyi Huang $^{1}$,\, Yu Yang $^{2}$,\, Junde Wu $^{1}$,\, Minhyung Cho $^{3}$ \footnote{Corresponding author. E-mail: [email protected]}\\ {\small \it $^{1}$ School of Mathematical Science, Zhejiang University, Hangzhou 310027, People's Republic of China}\\ {\small \it $^{2}$ Department of Mathematics, National University of Singapore, Singapore 119076, Republic of Singapore} \\{\small \it $^{3}$ Department of Applied Mathematics, Kumoh National Institute of Technology, Kyungbuk, 730-701, Korea}} \date{} \maketitle \mbox{}\hrule\mbox\\ \begin{abstract} The parity operator $\cal P$ and time reversal operator $\cal T$ are two important operators in the quantum theory, in particular, in the $\cal PT$-symmetric quantum theory. By using the concrete forms of $\cal P$ and $\cal T$, we discuss their geometrical properties in two dimensional spaces. It is showed that if $\cal T$ is given, then all $\cal P$ links with the quadric surfaces; if $\cal P$ is given, then all $\cal T$ links with the quadric curves. Moreover, we give out the generalized unbroken $\cal PT$-symmetric condition of an operator. The unbroken $\cal PT$-symmetry of a Hermitian operator is also showed in this way. \end{abstract} \mbox{}\hrule\mbox\\ \section{Introduction} Quantum theory is one of the most important theories in physics. It is a fundamental axiom in quantum mechanics that the Hamiltonians should be Hermitian, which implies that the values of energy are real numbers. However, non-Hermitian Hamiltonians are also studied in physics. One of the attempts is Bender's $\cal PT$-symmetric theory \cite{bender1998real}. In this theory, Bender and his colleagues attributed the reality of the energies to the $\cal PT$-symmetric property, where $\cal P$ is a parity operator and $\cal T$ is a time reversal operator. Since then, many physicists discussed the properties of $\cal PT$-symmetric quantum systems \cite{bender2007making}. It also has theoretical applications in quantum optics, quantum statistics and quantum field theory \cite{ruter2010observation,chang2014parity,deffner2015jarzynski,jones2010non}. Recently, Bender, Brody and Muller constructed a Hamiltonian operator $H$ with the property that if its eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function, where $H$ is not Hermitian in the conventional sense, while $iH$ has a broken $\cal PT$-symmetry. This result may shed light on the new application of $\cal PT$-symmetric theory in discussing the Riemann hypothesis \cite{bender2017zeros}. It was discovered by Mostafazadeh that the $\cal PT$-symmetric case can be generalized to a more general pseudo-Hermitian quantum theory, and the generalized $\cal PT$- symmetry was also discussed \cite{Mostafazadeh2010Pseudo, deng2012general}. Smith studied the time reversal operator $\cal T$ satisfying that ${\cal T}^2 = -I$ and the corresponding $\cal PT$-symmetric quantum theory \cite{jones2010non}. In this paper, by using the concrete forms of $\cal P$ and $\cal T$ in two dimensional spaces, we discuss their geometry properties. It is showed that if $\cal T$ is given, then all $\cal P$ links with the quadric surfaces; if $\cal P$ is given, then all $\cal T$ links with the quadric curves. Moreover, we give out the generalized unbroken $\cal PT$-symmetric condition of an operator $H$. The unbroken $\cal PT$-symmetry of a Hermitian operator is also showed in this way. \section{Preliminaries} In this paper, we only consider two dimensional complex Hilbert space $\mathbb{C}^2$. Let $L({\mathbb C}^2)$ be the complex vector space of all linear operators on $\mathbb{C}^2$, $I$ be the identity operator on $\mathbb{C}^2$, $\overline{z}$ be the complex conjugation of complex number $z$. An operator $\cal T$ on $\mathbb{C}^2$ is said to be anti-linear if ${\cal T}(sx_{1}+tx_{2})=\overline{s}{\cal T}(x_{1})+\overline{t}{\cal T}(x_2)$. It is obvious that the composition of two anti-linear operators is a linear operator and the composition of an anti-linear operator and a linear operator is still anti-linear. Similar to linear operators, anti-linear operators can also correspond to a matrix with slightly different laws of operation \cite{Uhlmann2016anti}. A time reversal operator $\cal T$ is an anti-linear operator which satisfies ${\cal T}^2=I$ or ${\cal T}^2=-I$. A parity operator $\cal P$ is a linear operator which satisfies $\mathcal{P}^2=I$ \cite{jones2010non,Mostafazadeh2010Pseudo,deng2012general,wigner2012group}. The Pauli operators will be used frequently in our discussions. Given the basis $\{e_i\}_{i=1}^{2}$ of $\mathbb{C}^2$, they are usually defined as follows \cite{Greiner2011quantum}: \begin{eqnarray} &&\sigma_1(x_1e_1+x_2e_2)=x_2e_1+x_1e_2,\label{P1}\\ &&\sigma_2(x_1e_1+x_2e_2)=-ix_2e_1+ix_1e_2,\label{P2}\\ &&\sigma_3(x_1e_1+x_2e_2)=x_1e_1-x_2e_2.\label{P3} \end{eqnarray} To put it another way, the representation matrices of $\sigma_1, \sigma_2$ and $\sigma_3$ are: \begin{equation} \begin{pmatrix}0&1 \\ 1&0 \end{pmatrix}, \begin{pmatrix}0&-i \\ i&0 \end{pmatrix}, \begin{pmatrix}1&0 \\ 0&-1 \end{pmatrix}. \end{equation} \noindent Pauli operators have the following useful properties \cite{Greiner2011quantum}: \begin{eqnarray} && \sigma_{i}\sigma_{j}=-\sigma_{j}\sigma_{i}=i\epsilon_{ijk}\sigma_{k}, \quad i\neq j,\label{P4}\\ && \sigma_{i}^2=I,\label{P5} \end{eqnarray} where $i,j,k\in\{1,2,3\}$, $\epsilon_{ijk}$ is the Levi-Civita symbol: \[ \epsilon_{ijk}=\left\{ \begin{array}{lr} \epsilon_{123}=\epsilon_{231}=\epsilon_{312}=1,\\ \epsilon_{132}=\epsilon_{213}=\epsilon_{321}=-1,\\ 0, otherwise. \end{array} \right. \] The well known commutation and anti-commutation relations are: \begin{eqnarray} && \sigma_{i}\sigma_{j}-\sigma_{j}\sigma_{i}=2i\epsilon_{ijk}\sigma_{k},\\ && \sigma_{i}\sigma_{j}+\sigma_{j}\sigma_{i}=2\delta_{ij}I, \end{eqnarray} where $i,j,k\in\{1 , 2 , 3\}$ and $\delta_{ij}$ is the Kronecker symbol. Denote $I$ by $\sigma_0$, then $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$ is a basis of $L({\mathbb C}^2)$. Moreover, an operator $\sigma=t\sigma_0+x\sigma_1+y\sigma_2+z\sigma_3\in L({\mathbb C}^2)$ is Hermitian if and only if the coefficients $\{t,x,y,z\}$ are real numbers. Given the basis $\{e_i\}_{i=1}^{2}$ of $\mathbb{C}^2$ and any vector $x=\sum x_ie_i$, one can define an important anti-linear operator, namely the conjugation operator ${\cal T}_0$, by ${\cal T}_0(x)=\sum \overline{x_i}e_i$. Similar to ${\cal T}_0$, one can define another important anti-linear operator $\tau_0$ by \begin{equation}\tau_0(x_1e_1+x_2e_2)=-\overline{x_2}e_1+\overline{x_1}e_2.\label{tau0}\end{equation} Furthermore, define $\tau_1=\tau_0\sigma_1, \tau_2=\tau_0\sigma_2, \tau_3=\tau_0\sigma_3$, that is, $\tau_i$ is defined to be the composition of $\tau_0$ and $\sigma_i$. The anti-linear operators $\{\tau_{0}, \tau_{1}, \tau_{2}, \tau_{3}\}$ forms a basis of the anti-linear operator space of $\mathbb{C}^2$. This basis has the following properties \cite{Uhlmann2016anti}: \begin{eqnarray} &&\tau_0^2=-I,\\ &&\tau_{0}\sigma_{i}=-\sigma_{i}\tau_{0}=\tau_{i},\\ &&\tau_{i}\tau_{0}=-\tau_{0}\tau_{i}=\sigma_{i},\\ &&\tau_{i}\tau_{j}=\sigma_{i}\sigma_{j}=i\epsilon_{ijk}\sigma_{k}\quad(i\neq j),\\ &&\tau_{i}\tau_{j}-\tau_{j}\tau_{i}=2i\epsilon_{ijk}\sigma_{k}, \end{eqnarray} where $i,j\in\{1,2,3\}$. All the equations above can be verified by the using the definitions of Pauli operators and $\tau_0$. However, for further use, we show that $\tau_{0}\sigma_{i}=-\sigma_{i}\tau_{0}=\tau_{i}$ in detail. Consider $\tau_2=\tau_0\sigma_2$. By (\ref{P2}) and (\ref{tau0}), we have \begin{eqnarray} &&\tau_0\sigma_2(x_1e_1+x_2e_2)=i\overline{x_1}e_1+i\overline{x_2}e_2,\\ &&\sigma_2\tau_0(x_1e_1+x_2e_2)=-i\overline{x_1}e_1-i\overline{x_2}e_2. \end{eqnarray} Thus, $\tau_{0}\sigma_{2}=-\sigma_{2}\tau_{0}=\tau_{2}$. Along similar lines, one can verify that $\tau_{0}\sigma_{i}=-\sigma_{i}\tau_{0}=\tau_{i}$ is also valid for $\sigma_1$ and $\sigma_3$. Moreover, it follows from $\tau_{0}\sigma_{i}=-\sigma_{i}\tau_{0}=\tau_{i}$ that $\sigma_{j}\tau_{i}=\sigma_{j}\tau_0\sigma_i=-\tau_0\sigma_{j}\sigma_i$. Combining with (\ref{P4}) and (\ref{P5}), one can further obtain the following relations: \begin{eqnarray} &&\sigma_{j}\tau_{i}=\tau_{i}\sigma_{j}=-i\epsilon_{ijk}\tau_{k},\quad i\neq j,\label{ts1}\\ &&\tau_{i}\sigma_{i}=-\sigma_{i}\tau_{i}=\tau_0,\label{ts2} \end{eqnarray} where $i,j,k\in\{1,2,3\}$. With the help of $\{\sigma_i\}$ and $\{\tau_i\}$, ones can determine the concrete forms of $\cal P$ and $\cal T$: \begin{lem} \label{le:forms} Let $\cal P$ be a parity operator and $\cal T$ be a time reversal operator on $\mathbb{C}^2$. Then (i). Either ${\cal P}=\pm I$ or ${\cal P}=\displaystyle\sum_{i=1}^3a_{i}\sigma_i$, where $a_i$ satisfying $\displaystyle\sum_{i=1}^{3}a_{i}^2=1$. The latter case is referred to as the nontrivial ${\cal P}$. A nontrivial $\cal P$ has the following matrix representation: \begin{equation} P=\begin{pmatrix}a_3&a_1 - ia_2 \\a_1 + ia_2&-a_3\end{pmatrix}\label{P}. \end{equation} (ii). ${\cal T}=\epsilon\displaystyle\sum^{3}_{i=0}c_i\tau_i$, where $c_i$ are real numbers, if ${\cal T}^2=I$, then $c_1^2+c_2^2+c_3^2-c_0^2=1$; if ${\cal T}^2=-I$, then $c_1^2+c_2^2+c_3^2-c_0^2=-1$, $\epsilon$ is a unimodular complex number \cite{Uhlmann2016anti}. \end{lem} \begin{proof} (i). Suppose ${\cal P}=\displaystyle\sum_{i=0}^3a_{i}\sigma_i$. According to the properties of Pauli operators, we have $I={\cal P}^2=(\displaystyle\sum_{i=0}^{3}a_{i}^2)I+2a_{0}(a_1\sigma_1+a_2\sigma_2+a_3\sigma_3)$. Note that $\{\sigma_0, \sigma_1, \sigma_2, \sigma_3\}$ is a basis of $L(\mathbb{C}^2)$, we conclude that $\displaystyle\sum_{i=0}^{3}a_{i}^2=1$ and $a_{0}a_{1}=a_{0}a_{2}=a_{0}a_{3}=0$. If $a_{0}\neq 0$, then $a_1=a_2=a_3=0$, which implies that ${\cal P}=\pm I$. If $ a_0=0$, then the only constraint is $\displaystyle\sum_{i=1}^{3}a_{i}^2=1$ and the matrix takes the form in (\ref{P}). (ii). The proof can be found in \cite{Uhlmann2016anti}. \end{proof} \noindent {\bf Example 1}. In (\ref{P}), if we take $a_2=0$, $a_1, a_3$ are real numbers satisfying that $a_1^2+a_3^2=1$, and denote $a_1$ by $\sin\alpha$, $a_3$ by $\cos\alpha$, then $\cal P$ has the matrix representation $\left(\begin{array}{{2}{c@{\;\;}}c} \cos\alpha & \sin\alpha \\ \sin\alpha & -\cos\alpha \\ \end{array} \right)=\begin{pmatrix}\cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}.$ Thus $\cal P$ is composed of a reflection and a rotation. \noindent {\bf Example 2}. In (\ref{P}), if $a_1=a_2=0$, $a_3=1$, then $P=\left(\begin{array}{{2}{c@{\;\;}}c} 1 & 0 \\ 0 & -1 \\ \end{array} \right)$. If $a_2=a_3=0$, $a_1=1$, then $P=\left(\begin{array}{{2}{c@{\;\;}}c} 0 & 1 \\ 1 & 0 \\ \end{array} \right)$. These two parity operators were widely used in \cite{bender2007making}. \section{The existence of $\cal P$ commuting with $\cal T$} In physics, it is demanded that ${\cal P}$ and $\cal T$ are commutative, that is, ${\cal PT}={\cal TP}$. In finite dimensional spaces case, by using the canonical forms of matrices, one can show that if ${\cal T}^2=I$, then such $\cal P$ always exists. In two dimensional case, we can prove it by utilizing Pauli operators. \begin{thm}\label{thm1} For each time reversal operator $\cal T$, if ${\cal T}^{2}=I$, then there exists a nontrivial parity operator $\cal P$ such that ${\cal PT}={\cal TP}$. If ${\cal T}^{2}=-I$, then there is no $\cal P$ commuting with $\cal T$ except ${\cal P}=\pm I$. \end{thm} \begin{proof} We will use the following well known equation frequently, \begin{equation} (\sigma\cdot A)(\sigma\cdot B)=(A\cdot B)I+i\sigma\cdot(A\times B),\label{pc} \end{equation} where $A$ and $B$ are two vectors in $\mathbb{C}^3$ and $\sigma=(\sigma_1,\sigma_2,\sigma_3)$. The symbols $\cdot$ and $\times$ represent the dot and cross product of vectors, respectively. (i). When ${\cal T}^{2}=I$. Let ${\cal T}=\epsilon\displaystyle\sum^{3}_{i=0}c_i\tau_i$ and ${\cal P}=\displaystyle\sum_{i=1}^3a_{i}\sigma_i$, as was given in Lemma \ref{le:forms}. According to (\ref{ts1}) and (\ref{ts2}), ${\cal TP}={\cal PT}$ is equivalent to \begin{eqnarray} (-c_0\sigma_0+\displaystyle\sum_{j=1}^3 c_j\sigma_j)(\displaystyle\sum_{i=1}^3 \overline{a_i}\sigma_i)\tau_0= (\displaystyle\sum_{i=1}^3 a_i\sigma_i)(c_0\sigma_0-\displaystyle\sum_{j=1}^3 c_j\sigma_j)\tau_0. \end{eqnarray} Denote $f_i=Re(a_i)$, $b_i=Im(a_i)$, $\tilde{f}=(f_1, f_2, f_3)$, $\tilde{b}=(b_1, b_2, b_3)$ and $\tilde{c}=(c_1, c_2, c_3)$. Utilizing (\ref{pc}) to expand the equation above, we have \begin{equation} (\tilde{f}\cdot\tilde{c})\sigma_0 -\sigma\cdot[\tilde{b}\times\tilde{c}+c_0\tilde{f}]=0. \end{equation} It follows that ${\cal TP}={\cal PT}$ is equivalent to \begin{eqnarray} &&c_0\tilde{f}+\tilde{b}\times \tilde{c}=0,\label{c1}\\ &&\tilde{f}\cdot\tilde{c}=0\label{c2}. \end{eqnarray} Similarly, by utilizing (\ref{pc}) and Lemma \ref{le:forms}, the contraints ${\cal P}^2=I$ and ${\cal T}^2=I$ can be reduced to the equations as follows, \begin{eqnarray} &&\tilde{f}\cdot\tilde{b}=0,\label{c3}\\ &&\\|\tilde{f}\\|^2-\\|\tilde{b}\\|^2=1,\label{c4}\\ &&\\|\tilde{c}\\|^2-c_0^2=1.\label{c5} \end{eqnarray} Thus, the problem of finding a parity operator $\cal P$ commuting with $\cal T$ reduces to finding the vectors $\tilde{f}$ and $\tilde{b}$ satisfying $(\ref{c1})-(\ref{c4})$. If $c_0=0$, then we can choose $\tilde{b}=0$ and a unit vector $\tilde{f}$ orthogonal to $\tilde{c}$. Thus all the conditions $(\ref{c1})-(\ref{c4})$ are satisfied. If $c_0\neq 0$. Let $\tilde{b}$ be a vector such that $\tilde{b}$ is orthogonal to $\tilde{c}$ and $\\|\tilde{b}\\|=\|c_0\|$. Moreover, take $\tilde{f}=\frac{1}{c_0}(\tilde{c}\times\tilde{b})$. Direct calculations show that such vectors $\tilde{f}$ and $\tilde{b}$ satisfy $(\ref{c1})-(\ref{c4})$, which completes the proof of the existence of $\cal P$. (ii). When ${\cal T}^{2}=-I$. The equation (\ref{c5}) is replaced by the following: \begin{equation} \\|\tilde{c}\\|^2-c_0^2=-1.\label{c6} \end{equation} Thus $c_0\neq 0$. On the other hand, it follows from (\ref{c1}) that \begin{equation}\tilde{f}=\frac{1}{c_0}(\tilde{c}\times\tilde{b}).\label{tilf}\end{equation} Substituting (\ref{c6}) and (\ref{tilf}) into (\ref{c4}), we have $\\|\tilde{f}\\|^2-\\|\tilde{b}\\|^2=1<-\frac{1}{c_0^2}\\|\tilde{b}\\|^2$, which is a contradiction. Thus, when ${\cal T}^{2}=-I$, there is no $\cal P$ commuting with $\cal T$ except ${\cal P}=\pm I$. \end{proof} \begin{remark} When the space is $\mathbb {C}^4$, although ${\cal T}^{2}=-I$, one can find nontrivial $\cal P$ commuting with $\cal T$ \cite{jones2010non}. \end{remark} \section{The geometrical properties of $\cal P$ and $\cal T$} \begin{thm} Let $\cal T$ be a time reversal operator satisfying ${\cal T}^2=I$. The set of parity operators $\cal P$ commuting with $\cal T$ correspond uniquely to a hyperboloid in $\mathbb{R}^3$. \end{thm} \begin{proof} As was mentioned above, the determination of $\cal P$ is equivalent to finding out $\tilde{f}$ and $\tilde{b}$ satisfying $(\ref{c1})-(\ref{c4})$. Now consider $\tilde{m}=\tilde{f}+\tilde{b}$. We shall prove that all the $\tilde{m}$ form a hyperboloid. To this end, construct a new coordinate system by taking the direction of $\tilde{c}$ as that of the $X'$ axis. The $Y'-Z'$ plane is perpendicular to $\tilde{c}$ and contains the origin point of $\mathbb{R}^3$. Assume that $\tilde{m}=(x',y',z')$ in the new $X'Y'Z'$ coordinate system. (i). If $c_0=0$, then it follows from $(\ref{c1})-(\ref{c3})$ that $\tilde{b}$ is proportional to $\tilde{c}$ and that $\tilde{f}$ is orthogonal to both $\tilde{c}$ and $\tilde{b}$. Thus, in the new $X'Y'Z'$ coordinate system, \begin{eqnarray} &&\tilde{b}=(x',0,0),\\ &&\tilde{f}=(0,y',z'). \end{eqnarray} On the other hand, equation (\ref{c4}), namely $\\|\tilde{f}\\|^2-\\|\tilde{b}\\|^2=1$, implies that \begin{equation} y'^2+z'^2-x'^2=1\label{af1}. \end{equation} It is apparent that one pair of $\tilde{f}$ and $\tilde{b}$ correspond to one point $\tilde{m}=(x',y',z')$, and vice versa. In addition, $(\ref{af1})$ represents a hyperboloid in $\mathbb{R}^3$. (ii). If $c_0\neq 0$, then it follows from (\ref{c5}) that $\tilde{c}=(\sqrt{1+c_0^2},0,0)$ in the $X'Y'Z'$ coordinate system. In addition, suppose $\tilde{b}=(x_0, y_0, z_0)$ in the $X'Y'Z'$ coordinate system. By (\ref{tilf}), we have $\tilde{f}=\frac{1}{c_0}(\tilde{c}\times\tilde{b})=\frac{\sqrt{1+c_0^2}}{c_0}(0, -z_0, y_0)$. Substituting $\tilde{b}$ and $\tilde{f}$ into (\ref{c4}), we have \begin{equation} \frac{1}{c_0^2}(y_0^2+z_0^2)-x_0^2=1\label{c_00}. \end{equation} Note that $x_0=x',y_0=\frac{\lambda z'+y'}{1+\lambda^2},z_0=\frac{z'-\lambda y'}{1+\lambda^2}$, where $\lambda=\frac{\sqrt{1+c_0^2}}{c_0}$. Thus, one pair of $\tilde{f}$ and $\tilde{b}$ correspond to one point $\tilde{m}=(x',y',z')$, and vice versa. Moreover, it follows from (\ref{c_00}) that \begin{equation} \frac{1}{1+2c_0^2}(y'^2+z'^2)-x'^2=1\label{c_0}. \end{equation} That is, all the $\tilde{m}$ form a hyperboloid. \end{proof} \begin{thm} Let $\cal P$ be a nontrivial parity operator and let us consider the time reversal operators of the form ${\cal T}=\displaystyle\sum_{i=0}^3 c_i\tau_i$ commuting with $\cal P$. All the points $\tilde{c}=(c_1,c_2,c_3)$ form an ellipse. The length of the semi-major axis is $\\|\tilde{f}\\|$ and the length of the semi-minor axis is $1$. \end{thm} \begin{proof} By (\ref{c2}) and (\ref{c3}), we know that both $\tilde{b}$ and $\tilde{c}$ are orthogonal to $\tilde{f}$. Construct a new $X'Y'Z'$ coordinate system by taking the direction of $\tilde{f}$ as that of the $Z'$ axis and the direction of $\tilde{b}$ as that of the $X'$ axis ( If $\tilde{b} = 0$, take any vector orthogonal to $\tilde{f}$ as the direction vector of the $X'$ axis ). Then we have $\tilde{b}=(x,0,0)$, $\tilde{f}=(0,0,z)$ and $\tilde{c}=(c_1',c_2',c_3')$ in the $X'Y'Z'$ coordinate system. Now the conditions $(\ref{c1})-(\ref{c5})$ will reduce to \begin{eqnarray} &&xc_3'=0,\label{d1}\\ &&xc_2'+c_0z=0,\label{d2}\\ &&zc_3'=0,\label{d5}\\ &&z^2-x^2=1,\label{d3}\\ &&(c_1')^2+(c_2')^2+(c_3')^2-(c_0)^2=1,\label{d4}. \end{eqnarray} Note that (\ref{d3}) ensures that $z\neq 0$. Thus, (\ref{d1}) and (\ref{d5}) imply that $c_3'=0$, $\tilde{c}=(c_1',c_2',0)$. In addition, it follows from (\ref{d2}) that $c_0=-\frac{x}{z}c_2'$. Substituting $c_3'=0$, $c_0=-\frac{x}{z}c_2'$ and (\ref{d3}) into (\ref{d4}), we have \begin{equation} (c_1')^2+\frac{(c_2')^2}{(z)^2}=1. \end{equation} This is an equation of ellipse. Moreover, since $\|z\|=\\|\tilde{f}\\|>1$, the length of the semi-major axis is $\\|\tilde{f}\\|$ and the length of the semi-minor axis is $1$. \end{proof} In the following theorem, we only consider the $\cal T$ with real coefficients. \begin{thm} Let ${\cal T}_1$, ${\cal T}_2$ be two time reversal operators, ${\cal T}_1\neq \pm{\cal T}_2$. If there exist two nontrivial parity operators ${\cal P}_1$ and ${\cal P}_2$ such that ${\cal P}_i$ commutes with ${\cal T}_1$ and ${\cal T}_2$ simultaneously, then ${\cal P}_1= \pm{\cal P}_2$. \end{thm} \begin{proof} Let ${\cal T}_1=\displaystyle\sum_{i=0}^3 c_i^{(1)}\tau_i$, ${\cal T}_2=\displaystyle\sum_{i=0}^3 c_i^{(2)}\tau_i$. Denote $\tilde{c}^{(1)}=(c_1^{(1)}, c_2^{(1)}, c_3^{(1)})$ and $\tilde{c}^{(2)}=(c_1^{(2)}, c_2^{(2)}, c_3^{(2)})$. (i). If $c_0^{(1)}\neq0$ and $c_0^{(2)}=0$. Suppose that $\cal P$ commute with ${\cal T}_i$ simultaneously. By (\ref{c1}), we have $\tilde{c}^{(2)}\times \tilde{b}=0$. It follows that $\tilde{b}=m\tilde{c}^{(2)}$. On the other hand, (\ref{c1}) implies that $\tilde{f}=\frac{1}{c_0^{(1)}}(\tilde{c}^{(1)}\times \tilde{b})$. Thus, $\tilde{f}=\frac{m}{c_0^{(1)}}(\tilde{c}^{(1)}\times \tilde{c}^{(2)})$. Substituting $\tilde{f}$ and $\tilde{b}$ into (\ref{c4}), then we have \[ m^2(\\|\frac{1}{c_0^{(1)}}\tilde{c}^{(1)}\times \tilde{c}^{(2)}\\|^2-\\|\tilde{c}^{(2)}\\|^2)=1. \] The equation has at most two real roots, which are opposite to each other. Thus, there exist at most two parity operators $\cal P$ and $-\cal P$ commuting with ${\cal T}_i$ simultaneously. (ii). If $c_0^{(1)}=c_0^{(2)}=0$ and $\tilde{c}^{(1)}= t\tilde{c}^{(2)}$, where $t$ is a real number. It follows from (\ref{c5}) that ${\cal T}_1 =\pm{\cal T}_2$, which contradicts with the assumption of the theorem. (iii). If $c_0^{(1)}\neq0$, $c_0^{(2)}\neq0$ and $\tilde{c}^{(1)}= t\tilde{c}^{(2)}$. By (\ref{c1}), we have $\tilde{f}=\frac{1}{c_0^{(1)}}(\tilde{c}^{(1)}\times \tilde{b})=\frac{1}{c_0^{(2)}}(\tilde{c}^{(2)}\times \tilde{b})=\frac{t}{c_0^{(1)}}(\tilde{c}^{(2)}\times \tilde{b})$. It follows that $c_0^{(1)}=tc_0^{(2)}$. Thus, we have $c_i^{(1)}=tc_i^{(2)}$, $(i=0, 1, 2, 3)$. On the other hand, (\ref{c5}) implies that $t^2=1$. Hence ${\cal T}_1=\pm {\cal T}_2$, which is a contradiction. (iv). If $c_0^{(1)}=c_0^{(2)}=0$ and $\tilde{c}^{(1)}\neq t\tilde{c}^{(2)}$. By (\ref{c1}), we have $\tilde{c}^{(1)}\times \tilde{b}=\tilde{c}^{(2)}\times \tilde{b}=0$. However, since $\tilde{c}^{(1)}\neq t\tilde{c}^{(2)}$, we have $\tilde{b}=0$. Thus (\ref{c4}) implies that $\\|\tilde{f}\\|=1$. Moreover, (\ref{c2}) implies that $\tilde{f}$ is orthogonal to both $\tilde{c}^{(1)}$ and $\tilde{c}^{(2)}$. So $\tilde{f}$ can only have two directions, which are opposite to each other. Thus, there exist at most two parity operators $\cal P$ and $-\cal P$ commuting with ${\cal T}_i$ simultaneously. (v). If $c_0^{(1)}\neq0$, $c_0^{(2)}\neq0$ and $\tilde{c}^{(1)}\neq t\tilde{c}^{(2)}$. Let ${\cal P}_1$ and ${\cal P}_2$ be two parity operators, which are determined by $(\tilde{f}^{(1)},\tilde{b}^{(1)})$ and $(\tilde{f}^{(2)},\tilde{b}^{(2)})$ respectively. Moreover, suppose that both ${\cal P}_1$ and ${\cal P}_2$ commute with ${\cal T}_i$ simultaneously. By (\ref{c1}), we have $\tilde{f}^{(1)}=\frac{1}{c_0^{(1)}}(\tilde{c}^{(1)}\times \tilde{b}^{(1)})$ and $\tilde{f}^{(1)}=\frac{1}{c_0^{(2)}}(\tilde{c}^{(2)}\times \tilde{b}^{(1)})$. It follows that \[ \frac{1}{c_0^{(1)}}\tilde{c}^{(1)}-\frac{1}{c_0^{(2)}}\tilde{c}^{(2)}=t_1\tilde{b}^{(1)}, \] where $t_1$ is a nonzero real number. Similarly, we have $\tilde{f}^{(2)}=\frac{1}{c_0^{(1)}}(\tilde{c}^{(1)}\times \tilde{b}^{(2)})$ and $\tilde{f}^{(2)}=\frac{1}{c_0^{(2)}}(\tilde{c}^{(2)}\times \tilde{b}^{(2)})$. It follows that \[ \frac{1}{c_0^{(1)}}\tilde{c}^{(1)}-\frac{1}{c_0^{(2)}}\tilde{c}^{(2)}=t_2\tilde{b}^{(2)}, \] So $t_1\tilde{b}^{(1)}=t_2\tilde{b}^{(2)}$, which implies that $\tilde{b}^{(1)}=k\tilde{b}^{(2)}$. Now $\\|\tilde{f}^{(1)}\\|^2-\\|\tilde{b}^{(1)}\\|^2=k^2(\\|\tilde{f}^{(2)}\\|^2-\\|\tilde{b}^{(2)}\\|^2)=1$, hence $k=\pm1$. Thus it is apparent that ${\cal P}_1=\pm {\cal P}_2$. Note that (i) -- (v) contain all the situations, which completes the proof. \end{proof} If we denote $com({\cal T})=\{{\cal P}\|{\cal PT}={\cal TP}, {\cal P}^2=I\}$, then the following corollary can be obtained. \begin{cor} If ${\cal T}_1=\displaystyle\sum_{i=0}^3 c_i^{(1)}\tau_i$, ${\cal T}_2=\displaystyle\sum_{i=0}^3 c_i^{(2)}\tau_i$ are two time reversal operators, ${\cal T}_j^2=I$, $j=1, 2$. Then $com({\cal T}_1) = com({\cal T}_2)$ if and only if for each $i$, $c_i^{(1)}=\epsilon c_i^{(2)}$, where $\epsilon$ is a unimodular coefficient. \end{cor} \section{$\cal PT$-symmetric operators and unbroken $\cal PT$-symmetric condition} A linear operator $H$ is said to be $\cal PT$-symmetric if $H{\cal PT}={\cal PT}H$. As is known, in standard quantum mechanics, the Hamiltonians are assumed to be Hermitian such that all the eigenvalues are real and the evolution is unitary. In the $\cal PT$-symmetric quantum theory, Bender replaced the Hermiticity of the Hamiltonians with $\cal PT$-symmetry. However, the $\cal PT$-symmetry of a linear operator does not imply that its eigenvalues must be real. Thus, Bender introduced the unbroken $\cal PT$-symmetric condition. The Hamiltonian $H$ is said to be unbroken $\cal PT$-symmetric if there exists a collection of eigenvectors $\Psi_i$ of $H$ such that they span the whole space and ${\cal PT}\Psi_i=\Psi_i$. It was shown that for a $\cal PT$-symmetric Hamiltonian $H$, its eigenvalues are all real if and only if $H$ is unbroken $\cal PT$-symmetric \cite{bender2007making}. In two dimensional space case, this condition has a much simpler description and an important illustrative example. That is, if ${\cal P}=\left(\begin{array}{{2}{c@{\;\;}}c} 0& 1\\ 1& 0\\ \end{array} \right)$, ${\cal T}={\cal T}_0$, $H=\left(\begin{array}{{2}{c@{\;\;}}c} re^{i\theta}& s\\ s& re^{-i\theta}\\ \end{array} \right)$, then $H$ is unbroken ${\cal PT}$-symmetric iff $s^2\geq r^2\sin^2\theta$ \cite{bender2007making}. In the following part, we shall give the unbroken ${\cal PT}$-symmetry condition for general ${\cal PT}$-symmetric operators. To this end, we need the following proposition. \begin{prop} \label{prop} If $H$ is a $\cal PT$-symmetric operator, then it has four real parameters. Moreover, if $H=h_0\sigma_0+h_1\sigma_1+ h_2\sigma_2+h_3\sigma_3$ is written in terms of Pauli operators, then \begin{eqnarray} &&Im(h_0)=0\label{condition1},\\ &&Re(h_1)Im(h_1)+Re(h_2)Im(h_2)+Re(h_3)Im(h_3)=0.\label{condition2} \end{eqnarray} \end{prop} \begin{proof} It is apparent that $\cal PT$ is also a time reversal operator. Thus we can assume that ${\cal PT}=\displaystyle\sum_{j=0}^3 c_j \tau_j$. Now the condition ${\cal PT}H=H{\cal PT}$ is equivalent to \begin{eqnarray} (\displaystyle\sum_{j=0}^3 c_j\tau_0\sigma_j)(\displaystyle\sum_{i=0}^3 h_i\sigma_i)=(\displaystyle\sum_{i=0}^3 h_i\sigma_i)(\displaystyle\sum_{j=0}^3 c_j\tau_0\sigma_j). \end{eqnarray} According to (\ref{pc}), this equation can be reduced to \begin{eqnarray} c_0(\overline{h_0}-h_0)+\displaystyle\sum_{i=1}^3c_i(h_0-\overline{h_0})\sigma_i+\displaystyle\sum_{i=1}^3c_i(h_i+\overline{h_i})+ i\sigma\cdot[\tilde{c}\times(\overline{\tilde{h}}-\tilde{h})]-\displaystyle\sum_{i=1}^3c_0(h_i+\overline{h_i})\sigma_i=0, \end{eqnarray} where $\tilde{h}=(h_1, h_2, h_3)$ and $\overline{\tilde{h}}=(\overline{h_1}, \overline{h_2}, \overline{h_3})$. The equation above is equivalent to \begin{eqnarray} &&Im(h_0)=0,\label{53}\\ &&\displaystyle\sum_{i=1}^3 c_i Re(h_i)=0,\label{54}\\ &&\tilde{c}\times Im(h)-c_0Re(h)=0,\label{55} \end{eqnarray} where $Re(h)=(Re(h_1),Re(h_2),Re(h_3))$ and $Im(h)=(Im(h_1), Im(h_2), Im(h_3))$. (i). When $c_0\neq 0$. It follows (\ref{55}) that $Re(h)=\frac{1}{c_0}(\tilde{c}\times Im(h))$. Thus, the four parameters $Im(h)_1$, $Im(h)_2$, $Im(h)_3$ and $Re(h_0)$ determine $H$. Note that (\ref{condition1}) is the same as (\ref{53}). On the other hand, $Re(h)=\frac{1}{c_0}(\tilde{c}\times Im(h))$ implies that $Re(h)\cdot Im(h)=0$. Thus, (\ref{condition2}) is also valid. (ii). When $c_0=0$. (\ref{55}) implies that $Im(h)=t\tilde{c}$. Thus, we only need one real parameter $t$ to determine $Im(h)$. (\ref{54}) implies that $Re(h)$ should be orthogonal to $\tilde{c}$. Hence two parameters are needed. With $Re(h_0)$, we have four parameters altogether. In this case, (\ref{condition2}) follows from the fact $Im(h)=t\tilde{c}$ and the equation (\ref{54}). \end{proof} \begin{thm}\label{thm3} If $H$ is a $\cal PT$-symmetric operator and $\left(\begin{array}{{2}{c@{\;\;}}c} h_{11} & h_{12} \\ h_{21} & h_{22}\\ \end{array} \right)$ is the representation matrix of $H$, then $H$ is unbroken if and only if $(Re(h_{11}+h_{22}))^2-4Re(h_{11}h_{22}-h_{12}h_{21})\geq 0.$ \end{thm} \begin{proof} Let $\left(\begin{array}{{2}{c@{\;\;}}c} h_{11} & h_{12} \\ h_{21} & h_{22}\\ \end{array} \right)$ be the matrix of $H$, $\lambda$ be an eigenvalue of $H$, then \begin{equation} \lambda^2-(h_{11}+h_{22})\lambda+h_{11}h_{22}-h_{12}h_{21}=0.\label{delta} \end{equation} On the other hand, rewrite $H=h_0\sigma_0+h_1\sigma_1+h_2\sigma_2+h_3\sigma_3$. It follows from (\ref{condition1}) and (\ref{condition2}) that \begin{eqnarray} &&Im(h_{11}+h_{22})=2Im(h_0)=0,\\ &&Im(h_{11}h_{22}-h_{12}h_{21})=-Re(h_1)Im(h_1)-Re(h_2)Im(h_2)-Re(h_3)Im(h_3)=0. \end{eqnarray} The two equations above imply that \begin{equation} -Im(h_{11}+h_{22})\lambda+Im(h_{11}h_{22}-h_{12}h_{21})=0.\label{u2} \end{equation} Substitute (\ref{u2}) into (\ref{delta}). Now the equation (\ref{delta}) reduces to \begin{equation} \lambda^2-Re(h_{11}+h_{22})\lambda+Re(h_{11}h_{22}-h_{12}h_{21})=0,\label{u1} \end{equation} According to (\ref{u1}), $\lambda$ is a real number, that is, $H$ is unbroken $\cal PT$-symmetric, if and only if \begin{equation} (Re(h_{11}+h_{22}))^2-4Re(h_{11}h_{22}-h_{12}h_{21})\geq 0. \label{eq::u1} \end{equation} \end{proof} \begin{remark} Note that when the equality is valid in (\ref{eq::u1}), $H$ may be non-diagonalisable in general. In this case, the space $\mathbb C^2$ is actually spanned an eigenvector $\psi_1$ satisfying $(H-\lambda_0 I)\psi_1=0$ and a generalized eigenvector $\psi_2$ satisfying $(H-\lambda_0 I)^2\psi_2=0$, where $\lambda_0=\frac{1}{2}Re(h_{11}+h_{22})$ is the eigenvalue. \end{remark} \begin{remark} Note that Bender's unbroken $\cal PT$-symmetric condition in \cite{bender2007making} is a special case of (\ref{eq::u1}). To see this, let $H=\left(\begin{array}{{2}{c@{\;\;}}c} re^{i\theta}& s\\ s& re^{-i\theta}\\ \end{array} \right),$ we have $$Re(h_{11})=Re(h_{22})=r\cos\theta,$$ $$Re(h_{11}h_{22}-h_{12}h_{21})=r^2-s^2.$$ Then (\ref{eq::u1}) holds iff $s^2\geq r^2\sin^2\theta$. \end{remark} \begin{remark}\label{thm2} If $H$ is a Hermitian operator, then it is also unbroken $\cal PT$-symmetric. Usually, this can be shown by using canonical forms. However, in $\mathbb C^2$, it also follows from direct calculation. \end{remark} In fact, since $H=h_0\sigma_0+h_1\sigma_1+ h_2\sigma_2+h_3\sigma_3$ is Hermitian, each $h_i$ is a real number. Now we only need to find real coefficients $c_0$, $c_1$, $c_2$ and $c_3$ such that $c_1^2+c_2^2+c_3^2-c_0^2=1$ and equations $(\ref{53})-(\ref{55})$ hold. Take $c_0=0$ and $c_1$, $c_2$, $c_3$ are such real numbers that $c_1Re(h_1)+c_2Re(h_2)+ c_3Re(h_3)=0$ and $c_1^2+c_2^2+c_3^2=1$. Let ${\cal PT}=\displaystyle \sum_{i=0}^3 c_i\tau_i$. It is apparent that $({\cal PT})^2=I$ and $H$ is $\cal PT$-symmetric. Moreover, if we rewrite the Hermitian matrix as $H=\begin{pmatrix} a&b\\ \overline{b}&a\end{pmatrix}$, then $Re(h_{11}+h_{22})^2-4Re(h_{11}h_{22}-h_{12}h_{21})=4a^2-4(a^2-\|b\|^2)=4\|b\|^2\geqslant 0$ holds, so $H$ is also unbroken. \end{document}	arXiv
\begin{definition}[Definition:Convergence in Distribution] Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\sequence {X_n}_{n \in \N}$ be a sequence of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$. For each $n \in \N$, let $F_n$ be the cumulative distribution function of $X_n$. Let $F$ be the cumulative distribution function of $X$. We say that $\sequence {X_n}_{n \in \N}$ '''converges in distribution''' to $X$ if: :$\ds \lim_{n \mathop \to \infty} \map {F_n} x = \map F x$ for all $x$ for which $F$ is continuous. This is written: :$X_n \xrightarrow d X$ \end{definition}	ProofWiki
\begin{document} \begin{abstract} Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. The following results are proved. If $\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{\#}$, then $\gamma_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{\#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. Earlier this was known only in the case where $k\leq 4$. \end{abstract} \title{Centralizers of coprime automorphisms of finite groups} \section{Introduction} Let $G$ be a group admitting an action by a group $A$. We denote by $C_G(A)$ the set $\{x\in G\, \mid\, x^a=x\mbox{ for any } a\in A\}$, the centralizer of $A$ in $G$. In this paper we deal with the case where $A$ is a noncyclic elementary abelian $p$-group and $G$ is a finite $p'$-group. Let $A^\#$ denote the set of non-identity elements of $A$. It follows from the classification of finite simple groups that if $C_G(a)$ is soluble for any $a\in A^\#$, then so is the group $G$ (see \cite{gushu}). The case $\|A\|\ge p^3$ does not require the classification: the result follows from Glauberman's theorem on soluble signalizer functors \cite{Gla}. In certain specific situations much more can be said about the structure of $G$. Ward showed that if $A$ has rank at least 3, and if $C_G(a)$ is nilpotent for any $a\in A^\#$, then the group $G$ is nilpotent \cite{W1}. Another of Ward's results is that if $A$ has rank at least 4, and if $C_G(a)'$ is nilpotent for any $a\in A^\#$, then the derived group $G'$ is nilpotent \cite{W2}. Later the second author of the present paper found that if under these assumptions $C_G(a)$ is nilpotent of class at most $c$ (respectively $C_G(a)'$ is nilpotent of class at most $c$) for any $a\in A^\#$, then the nilpotency class of $G$ (respectively of $G'$) is $\{c,p\}$-bounded \cite{shu1}. Throughout the article we use the term ``$\{a,b,c,\dots\}$-bounded" to mean ``bounded from above by some function depending only on the parameters $a,b,c,\dots$". Let us denote by $\gamma_{i}(H)$ the $i$th term of the lower central series of a group $H$ and by $H^{(i)}$ the $i$th term of the derived series of $H$. In \cite{shu1} it was conjectured that the above results should be a part of a more general phenomenon. \begin{conj} \label{conj} Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. \begin{enumerate} \item[\emph{(i)}] If $\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{\#}$, then $\gamma_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. \item[\emph{(ii)}] If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{\#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. \end{enumerate} \end{conj} One indirect evidence in favor of the above conjecture is the result obtained in \cite{shu1} that the conjecture is true for Lie algebras. Yet, for long time it looked as if the Lie-theoretical result was of no help in dealing with groups. However a breakthrough has occured with the introduction in \cite{AS} of the concept of $A$-special subgroups of a group $G$. In the present paper we combine the use of the subgroups with the Lie-theoretical techniques to show, respectively in Theorems \ref{caso delta} and \ref{caso gamma}, that both parts of Conjecture \ref{conj} are true. In the next section we mention some standard results on the coprime action of finite groups. We also give the definition of $A$-special subgroups taken from \cite{AS} as well as remind the reader some general facts about Lie algebras associated with groups. In Section 3 these are used in the proof of Theorem \ref{caso delta}. In Section 4 we outline the proof of Theorem \ref{caso gamma}. \section{Preliminaries} We start this section with some well-known facts about coprime automorphisms of finite groups. The next two lemmas can be found in \cite[5.3.16, 6.2.2, 6.2.4]{GO}. \begin{lemma} \label{FG1} Let $A$ be a group of automorphisms of the finite group $G$ with $(\|A\|,\|G\|)=1$. \begin{enumerate} \item If $N$ is an $A$-invariant normal subgroup of $G$, then \\$C_{G/N}(A)=C_G(A)N/N$; \item If $H$ is an $A$-invariant $p$-subgroup of $G$, then $H$ is contained in an $A$-invariant Sylow $p$-subgroup of $G$. \end{enumerate} \end{lemma} \begin{lemma} \label{FG2} Let $p$ be a prime, $G$ a finite $p'$-group acted on by an elementary abelian $p$-group $A$ of rank at least $2$.\ Let $A_1, \dots,A_s$ be the maximal subgroups of $A$.\ If $H$ is an $A$-invariant subgroup of $G$ we have $H=\langle C_H(A_1),\dots,C_H(A_s)\rangle$. Furthermore if $H$ is nilpotent then $H=\prod_{i} C_{H}(A_{i})$. \end{lemma} In \cite{AS} we have introduced the concept of an \emph{$A$-special subgroup} of a group $G$. The definition is as follows. Let $p$ be a prime and $A$ a finite elementary abelian $p$-group acting on a finite group $G$. Let $A_{1},\ldots,A_{s}$ be the subgroups of index $p$ in $A$ and $H$ a subgroup of $G$. We say that $H$ is an $A$-special subgroup of $G$ of degree $0$ if and only if $H=C_{G}(A_{j})$ for suitable $j\leq s$. Next, suppose that $i\geq 1$ and the $A$-special subgroups of $G$ of degree $i-1$ are already defined. Then $H$ is an $A$-special subgroup of $G$ of degree $i$ if and only if there exist $A$-special subgroups $J_{1},J_{2}$ of $G$ of degree $i-1$ such that $H=[J_{1},J_{2}]\cap C_{G}(A_{j})$ for suitable $j\leq s$. Here as usual $[J_{1},J_{2}]$ denotes the subgroup generated by all commutators $[x,y]$ where $x\in J_1$ and $y\in J_2$. Of course, the $A$-special subgroups of $G$ are always $A$-invariant. Assume that $A$ has order $p^k$. It is clear that for a given integer $i$ the number of $A$-special subgroups of $G$ of degree $i$ is $\{i,k,p\}$-bounded. Recall some properties satisfied by the $A$-special subgroups of $G$. The proofs can be found in \cite{AS}. \begin{proposition} \label{PAspecial} Let $A$ be an elementary abelian $p$-group of order $p^{k}$ with $k\geq 2$ acting on a finite $p'$-group $G$ and let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. Let $i\geq 0$ be an integer. \begin{enumerate} \item If $i\geq 1$, then every $A$-special subgroup of $G$ of degree $i$ is contained in some $A$-special subgroup of $G$ of degree $i-1$. \item Let $S_{i}$ be the subgroup generated by all $A$-special subgroups of $G$ of degree $i$. Then $S_{i}=G^{(i)}$. \item If $2^{i}\leq k-1$ and $H$ is an $A$-special subgroup of $G$ of degree $i$, then $H$ is contained in the $i$th derived group of $C_{G}(B)$ for some subgroup $B\leq A$ such that $\|A/B\|\leq p^{2^{i}}$. \end{enumerate} \end{proposition} In \cite{AS} we have also established the following result about generation of an $A$-invariant Sylow subgroup of $G^{(d)}$. \begin{thm} \label{generation1} Let $A$ be an elementary abelian $p$-group of order $p^{k}$ with $k\geq 2$ acting on a finite $p'$-group $G$. Let $r$ be a prime and $R$ an $A$-invariant Sylow $r$-subgroup of $G^{(d)}$, for some integer $d\geq 0$. Let $R_{1},\ldots,R_{t}$ be the subgroups of the form $R\cap H$ where $H$ ranges through $A$-special subgroups of $G$ of degree $d$. Then $R=\langle R_{1},\dots,R_{t}\rangle$. \end{thm} The proofs of the main results of the present paper are based on Lie techniques. Thus, we wish to recall here some useful Lie-theoretic machinery. Throughout the paper the term Lie algebra means Lie algebra over an associative ring with unity. Let $L$ be a Lie algebra and let $X,Y, X_{1},\ldots, X_{t}$ be subsets of $L$. We denote by $[X,Y]$ the subspace of $L$ spanned by the set $\{[x,y] \mid x\in X, y\in Y\}$ and we write $[X_{1},\ldots, X_{t}]$ for $[[X_{1},\ldots,X_{t-1}],X_{t}]$. If $t\geq 2$, we write $[X,_{\,t}Y]$ for $[[X,_{\,t-1}Y],Y]$. We denote by $\langle X\rangle$ the subalgebra of $L$ generated by $X$. Let $G$ be a group and let us denote by $\gamma_{i}$ the $i$th term of the lower central series of $G$. The associated Lie algebra $L(G)$ of the group $G$ is defined by \begin{equation} L(G)=\bigoplus_{i=1}^{\infty}\,\gamma_{i}/\gamma_{i+1}, \end{equation} where we write additively the abelian groups $\gamma_{i}/\gamma_{i+1}$. Commutation in the group $G$ induces a well-defined binary operation with respect to which $L(G)$ becomes a Lie ring (Lie algebra over $\Bbb Z$). The details related to this construction can be found for example in \cite{Khu}. If the group $G$ is nilpotent, then the Lie algebra $L(G)$ is also nilpotent and has the same nilpotency class as $G$. Given a subgroup $H$ of $G$, we can associate to $H$ the subalgebra $$ L(G,H)=\bigoplus_{i=1}^{\infty}\,(H\cap \gamma_{i})\gamma_{i+1}/\gamma_{i+1}. $$ If a group $A$ acts on $G$, then $A$ acts naturally also on each quotient $\gamma_{i}/\gamma_{i+1}$ and this action extends uniquely to an action by automorphisms on the whole Lie algebra $L(G)$. Lemma \ref{FG1}(1) shows that if $(\|A\|,\|G\|)=1$, then $$C_{L(G)}(A)=\bigoplus_{i}\,C_{\gamma_{i}}(A)\gamma_{i+1}/\gamma_{i+1}.$$ Therefore in the case where $(\|A\|,\|G\|)=1$ we have $C_{L(G)}(A)=L(G,C_G(A))$. Later on we will require the following lemma. \begin{lemma}\label{span} Let $L$ be a Lie algebra such that $pL=L$ where $p$ is a prime, and let $A$ be a finite elementary abelian $p$-group acting by automorphisms on $L$. Let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. Suppose that $L$ is generated by $A$-invariant subspaces $R_{1},\ldots,R_{t}$ with the property that for any integers $i,j$ and $k$ there exists some integer $m$ such that $$ [R_{i},R_{j}]\cap C_{L}(A_{k})\leq R_{m}. $$ Then $L$ is spanned by $R_{1},\ldots,R_{t}$. \end{lemma} \begin{proof} Clearly, $L$ is a linear span of subspaces of the form $[R_{i_{1}},\ldots,R_{i_{w}}]$, where $R_{i_{1}},R_{i_{2}},\ldots,R_{i_{w}}$ are not necessarily distinct elements of $\{R_{1}.\ldots,R_{t}\}$. So choose $R_{i_{1}},R_{i_{2}},\ldots,R_{i_{w}}\in\{R_{1}.\ldots,R_{t}\}$ and put $R=[R_{i_{1}},\ldots,R_{i_{w}}]$. It is sufficient to show that $R$ is contained in $\sum_{j} R_{j}$. We argue by induction on $w$. If $w=1$, then $R=R_{j}$, for some $j$ and there is nothing to prove. Assume that $w\geq 2$ and put $R_{0}=[R_{i_{1}},\ldots,R_{i_{w-1}}]$. Thus $R=[R_{0},R_{i_{w}}]$. Since $R$ is an $A$-invariant subspace it follows from Lemma \ref{FG2} that $R=\sum_{\lambda\leq s} C_{R}(A_{\lambda})$. By the inductive hypothesis $R_{0}\leq \sum_{j} R_{j}$. Therefore we have \begin{equation} \begin{split} C_{R}(A_{\lambda})=[R_{0},R_{i_{w}}]\cap C_{L}(A_{\lambda})&\leq [\sum_{j} R_{j},R_{i_{w}}]\cap C_{L}(A_{\lambda})\\ &\leq \sum_{j}\big([R_{j},R_{i_{w}}]\cap C_{L}(A_{\lambda})\big). \end{split} \end{equation} By the hypothesis each summand $[R_{j},R_{i_{w}}]\cap C_{L}(A_{\lambda})$ is contained in $R_{m}$, for some integer $m$, and so it follows that $R\leq \sum_{j}R_{j}$, as desired. \end{proof} \section{Proof of the Main Result} The aim of this section is to prove part (ii) of Conjecture \ref{conj}. \begin{thm} \label{caso delta} Let $c$ be a positive integer, $p$ a prime, and $A$ an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{\#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. \end{thm} First we wish to show that under the hypotheses of the above theorem the $d$th derived group $G^{(d)}$ is nilpotent. In what follows we write $F(K)$ for the Fitting subgroup of a group $K$ and $O_{\pi}(K)$ for the maximal normal $\pi$-subgroup of $K$, where $\pi$ is a set of primes. \begin{lemma} \label{nilpotency for G^d} Assume the hypotheses of Theorem \ref{caso delta}. Then $G^{(d)}$ is nilpotent. \end{lemma} \begin{proof} Suppose that the lemma is false and let $G$ be a counterexample of minimal order. Since the $d$th derived group of $C_{G}(a)$ is nilpotent, it follows that $C_{G}(a)$ is soluble for any $a\in A^{\#}$. Therefore Glauberman's result on soluble signalizer functors \cite{Gla} implies that $G$ is soluble. Assume that $G$ has two distinct minimal $A$-invariant normal subgroups $M_{1}$ and $M_{2}$. By minimality the image of $G^{(d)}$ in $G/M_{1}$ and in $G/M_{2}$ is nilpotent. Thus the image of $G^{(d)}$ must be nilpotent in the quotient $G/(M_{1}\cap M_{2})$. This is a contradiction since $M_{1}\cap M_{2}=1$. Therefore $G$ has a unique minimal $A$-invariant normal subgroup $M$. Again the quotient $G^{(d)}/M$ is nilpotent. It is clear that $M$ is an elementary abelian $q$-group for some prime $q$. Let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. By Lemma \ref{FG2} $M=M_{1}M_{2}\cdots M_{s}$, where $M_{i}=C_{M}(A_{i})$ for $i\leq s$. Since $G^{(d)}$ is not nilpotent, it is not a $q$-group. Therefore by Lemma \ref{FG1}(2) $G^{(d)}$ contains an $A$-invariant Sylow $r$-subgroup $R$ for some prime $r\neq q$. Theorem \ref{generation1} tells us that $R$ is generated by its intersections with $A$-special subgroups of degree $d$. Thus, $R=\langle R_{1},\ldots,R_{t}\rangle$, where $R_{j}=R\cap H_{j}$ for some $A$-special subgroup $H_{j}$ of $G$ of degree $d$. Now fix the integers $i$ and $j$ and consider the subgroup $\langle M_{i},R_{j}\rangle$. Since $2^{d}\leq k-1$ it follows from Proposition \ref{PAspecial}(3) that $H_{j}$ is contained in $C_{G}(B)^{(d)}$ for some subgroup $B$ of $A$ such that $\|A/B\|\leq p^{2^{d}}$. On the other hand $M_{i}\leq C_{G}(A_{i})$ and note that the intersection $B\cap A_i$ is not trivial. Therefore there exists $a\in A^{\#}$ such that $M_{i}\leq C_{G}(a)$ and $H_{j}\leq C_{G}(a)^{(d)}$. It follows that $H_{j}$ is contained in $F(C_{G}(a))$. Since $M_{i}$ is contained in a normal abelian subgroup of $G$ and also in $C_{G}(a)$, it follows that $\langle M_{i},R_{j} \rangle$ is nilpotent. Bearing in mind that $M$ is a $q$-group and $R$ is an $r$-group we deduce that $[M_{i},R_{j}]=1$ and this holds for any $i,j$. Recall that $M=M_{1}M_{2}\cdots M_{s}$ and $R=\langle R_{1},\ldots,R_{t}\rangle$. Therefore $[M,R]=1$. The fact that $G^{(d)}/M$ is nilpotent implies that also $C_{G}(M)\cap G^{(d)}$ is nilpotent. Hence, every $q'$-element of $C_{G}(M)\cap G^{(d)}$ belongs to $O_{q'}(G)$. On the other hand $O_{q'}(G)$ is trivial since $M$ is the unique minimal $A$-invariant normal subgroup of $G$. Thus, we obtain a contradiction as we have just shown that $R$ centralizes $M$. \end{proof} \begin{proof}[Proof of Theorem \ref{caso delta}] By Lemma \ref{nilpotency for G^d} $G^{(d)}$ is nilpotent. Let $L=L(G^{(d)})$ be the Lie algebra associated with $G^{(d)}$. Then $pL=L$ and $L$ has the same nilpotency class as $G^{(d)}$. The group $A$ naturally acts by automorphisms on the Lie algebra $L$. From the hypothesis that $C_{G}(a)^{(d)}$ is nilpotent of class at most $c$ we obtain that $C_{L}(a)^{(d)}$ is nilpotent of class at most $c$ for any $a\in A^{\#}$. Let $K=L\otimes \mathbb{Z}[\omega]$, where $\omega$ is a primitive $p$th root of unity. Then for each $i\geq 0$ and $a\in A^{\#}$ we have $$C_{K}(a)^{(i)}=C_{L}(a)^{(i)} \otimes \mathbb{Z}[\omega].$$ Hence, the nilpotency of $C_{L}(a)^{(d)}$ implies that also $C_{K}(a)^{(d)}$ is nilpotent of class at most $c$ for any $a\in A^{\#}$. We are in the position to apply Theorem 2.7(2) from \cite{shu1} and conclude that $K^{(d)}$ is nilpotent of $\{c,k,p\}$-bounded class. The same holds for $L^{(d)}$. Let us denote the nilpotency class of $L^{(d)}$ by $e$. By Proposition \ref{PAspecial}(2) $G^{(d)}=\langle H_{1},H_{2},\ldots,H_t\rangle$, where $H_{i}$ are the $A$-special subgroups of $G$ of degree $d$. Since $2^{d}+2\leq k$, Proposition \ref{PAspecial}(3) tells us that each $A$-special subgroup $H_{i}$ of $G$ of degree $d$ is contained in $C_{G}(B)^{(d)}$, for some subgroup $B$ of $A$ such that $\|A/B\|\leq p^{2^{d}}$. Let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. For any $A_{j}$ the intersection $B\cap A_{j}$ is not trivial. Thus, there exists $a\in A^{\#}$ such that the centralizer $C_{G}(A_{j})$ is contained in $C_{G}(a)$ and $H_{i}$ is contained in $C_{G}(a)^{(d)}$. Since $C_{G}(a)^{(d)}$ is nilpotent of class at most $c$ we deduce that \begin{equation} \label{relationdelta} [C_{G}(A_{j}),_{\,c+1}H_{i}]=1. \end{equation} Next we define recursively what will be called $A$-subalgebras of $L$. For each $A$-special subgroup $H_{i}$ of $G$ of degree $d$ we consider the corresponding subalgebra $L(G^{(d)},H_{i})$ of $L$ and we define the \textit{$A$-subalgebras} as follows: A subalgebra $R$ is an $A$-subalgebra of level $0$ if and only if $R=L(G^{(d)},H_j)$ for suitable $j\leq t$. Next, suppose that $l\geq 1$ and the $A$-subalgebras of level $l-1$ are defined. Then $R$ is an $A$-subalgebra of level $l$ if and only if there exist $A$-subalgebras $R_1,R_2$ of level $l-1$ such that $R=[R_1,R_2]\cap C_{L}(A_j)$ for suitable $j\leq s$. It is clear that every $A$-subalgebra is $A$-invariant and is contained in $C_L(A_j)$ for some $j\leq s$. Since $G^{(d)}=\langle H_1,H_2,\ldots,H_{t}\rangle$ it follows that $L$ is generated by the $A$-subalgebras of level $0$. It is easy to check that if $R$ is an $A$-subalgebra of level $l$, then $G$ contains an $A$-special subgroup $H$ of degree $d+l$ such that $R\leq L(G^{(d)},H)$. It follows from the definition and Proposition \ref{PAspecial}(1) that for any $A$-special subgroups $J_1$ and $J_2$ and for every $j\leq s$ there exists an $A$-special subgroup $J_3$ such that \begin{equation} \label{propgroup} [J_1,J_2]\cap C_{G}(A_{j})\leq J_3. \end{equation} From this we deduce the corresponding properties of $A$-subalgebras. \begin{itemize} \item[(P1)] If $l\geq 1$, then every $A$-subalgebra of level $l$ is contained in some $A$-subalgebra of level $l-1$. \item[(P2)] If $j\leq s$, then for any $A$-subalgebras $R_1,R_2$ of level $l$ there exists an $A$-subalgebra $R_3$ of the same level $l$ such that $$ [R_1,R_2]\cap C_{L}(A_{j})\leq R_3. $$ \end{itemize} In the group $G$ we have the relation (\ref{relationdelta}). Therefore in the Lie algebra we have $[C_{L}(A_{j}),_{\,c+1}L(G^{(d)},H_{i})]=0$. Taking into account that every $A$-subalgebra is contained in some $L(G^{(d)},H_{i})$ and that $L=\sum_{j}C_{L}(A_{j})$ we deduce $[L,_{\,c+1}L(G^{(d)},H_{i})]=0$, and, in particular, \begin{equation} \label{equ5} [L,_{\,c+1}R]=0 \end{equation} for every $A$-subalgebra $R$. Now we wish to show that for any $l\geq 0$ the $l$th derived algebra $L^{(l)}$ is spanned by the $A$-subalgebras of level $l$. The property (P2) and Lemma \ref{span} show that this happens if and only if $L^{(l)}$ is generated by the $A$-subalgebras of level $l$. Since $L$ is generated by the $A$-subalgebras of level $0$, this is obvious for $l=0$. Now assume that $l\geq 1$ and use induction on $l$. The inductive hypothesis will be that $L^{(l-1)}$ is spanned by the $A$-subalgebras of level $l-1$. Let $N$ be the subalgebra of $L$ generated by the $A$-subalgebras of level $l$. We already know that in fact $N$ is spanned by the $A$-subalgebras of level $l$. Let us show that actually $N$ is an ideal in $L^{(l-1)}$. Choose an $A$-subalgebra $R_1$ of level $l$ and an $A$-subalgebra $R_2$ of level $l-1$. The properties (P1) and (P2) show that $[R_1,R_2]\leq N$. Since $L^{(d-1)}$ is spanned by the $A$-subalgebras of level $l-1$, we conclude that indeed $N$ is an ideal in $L^{(l-1)}$. Note that $L^{(l-1)}/N$ is abelian. Thus $L^{(l)}=[L^{(l-1)},L^{(l-1)}]\leq N$. On the other hand, by construction it is clear that $N$ is contained in $L^{(l)}$. Hence, $N=L^{(l)}$. We will now prove that $L$ is nilpotent of $\{c,k,p\}$-bounded class. Let $Z=Z(L^{(d)})$. Then $[Z,X,Y]=[Z,Y,X]$ for any subsets $X,Y$ of $L^{(d-1)}$. Set $r=cn+1$, where $n$ is the number of $A$-subalgebras of level $d-1$ and note that $n$ is a $\{k,p\}$-bounded number. Since $L^{(d-1)}$ is spanned by the $A$-subalgebras of level $d-1$, i.e., $L^{(d-1)}=\sum_{i\leq n} R_{i}$, we can write \begin{equation} \label{equ6delta} [Z,_{\,r}L^{(d-1)}]=\sum\,[Z,_{\,u_{1}}\,R_1,\ldots,_{\,u_{n}}R_{n}], \end{equation} where $u_{1}+\cdots+u_{n}=r$ and $R_1,\ldots,R_{n}$ are the $A$-subalgebras of level $d-1$. The number $r$ is big enough to ensure that $u_{j}\geq c+1$ for some $j\leq n$. It follows from (\ref{equ5}) that each summand in (\ref{equ6delta}) is equal to zero. Thus $[Z,_{r}L^{(d-1)}]=0$ and $Z\leq Z_{r}(L^{(d-1)})$, where $Z_{r}(L^{(d-1)})$ is the $r$th term of the upper central series of $L^{(d-1)}$. Now repeating this argument for $L^{(d-1)}/Z$, $L^{(d-1)}/Z_{2}(L^{(d)})$ and so on, we conclude that $L^{(d)}\leq Z_{er}(L^{(d-1)})$ and therefore $L^{(d-1)}$ is nilpotent of class at most $er+1$. After that we repeat the arguments for $L^{(d-2)}/Z(L^{(d-1)})$, $L^{(d-2)}/Z_{2}(L^{(d-1)})$ etc. After boundedly many repetitions we conclude that $L$ is nilpotent of $\{c,k,p\}$-bounded class. Finally we remark that since the nilpotency class of $G^{(d)}$ equals that of $L$, the result follows. The proof is complete. \end{proof} \section{The other part of the conjecture} In this section we will outline a proof of the following result. \begin{thm} \label{caso gamma} Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. If $\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{\#}$, then $\gamma_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. \end{thm} The proof of Theorem \ref{caso gamma} is very similar to that of Theorem \ref{caso delta}. Very often the changes that need to be done are quite obvious and therefore we omit many details. Most essential difference as compared with Theorem \ref{caso delta} is that the role of $A$-special subgroups will now be played by \textit{$\gamma$-$A$-special subgroups} of $G$. These were introduced in \cite{AS}. Let us recall the definition. Let $p$ be a prime and $A$ a finite elementary abelian $p$-group acting on a finite group $G$. Let $A_{1},\ldots,A_{s}$ be the subgroups of index $p$ in $A$ and $H$ a subgroup of $G$. We say that $H$ is a $\gamma$-$A$-special subgroup of $G$ of degree $1$ if and only if $H=C_{G}(A_{j})$ for suitable $j\leq s$. Next, suppose that $i\geq 2$ and the $\gamma$-$A$-special subgroups of $G$ of degree $i-1$ are already defined. Then $H$ is a $\gamma$-$A$-special subgroup of $G$ of degree $i$ if and only if there exists a $\gamma$-$A$-special subgroup $J$ of $G$ of degree $i-1$ such that $H=[J,C_{G}(A_{j})]\cap C_{G}(A_{n})$ for suitable $j,n\leq s$. Note that for a given integer $i$ the number of $\gamma$-$A$-special subgroups of $G$ of degree $i$ is $\{i,k,p\}$-bounded. The following properties of $\gamma$-$A$-special subgroups have been established in \cite{AS}. \begin{proposition} \label{gammaPAspecial} Let $A$ be an elementary abelian $p$-group of order $p^{k}$ with $k\geq 2$ acting on a finite $p'$-group $G$ and $A_{1},\ldots,A_{s}$ the maximal subgroups of $A$. Let $i\geq 1$ be an integer. \begin{enumerate} \item If $i\geq 2$, then every $\gamma$-$A$-special subgroup of $G$ of degree $i$ is contained in some $\gamma$-$A$-special subgroup of $G$ of degree $i-1$. \item Let $S_{i}$ be the subgroup generated by all $\gamma$-$A$-special subgroups of $G$ of degree $i$. Then $S_{i}=\gamma_{i}{(G)}$. \item If $i\leq k-1$ and $H$ is a $\gamma$-$A$-special subgroup of $G$ of degree $i$, then $H\leq \gamma_{i}(C_{G}(B))$ for some subgroup $B\leq A$ such that $\|A/B\|\leq p^{i}$. \end{enumerate} \end{proposition} We will also require the following analogue of Lemma \ref{span}. \begin{lemma} \label{gammaspan} Let $L$ be a Lie algebra such that $pL=L$ where $p$ is a prime, and let $A$ be a finite elementary abelian $p$-group acting by automorphisms on $L$. Let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. Suppose that $L$ is generated by $A$-invariant subspaces $R_{1},\ldots,R_{t}$ with the property that for any integers $i,j$ and $k$ there exists some integer $m$ such that $$ [R_{i},C_{L}(A_{j})]\cap C_{L}(A_{k})\leq R_{m}. $$ Then $L$ is spanned by $R_{1},\ldots,R_{t}$. \end{lemma} Now we sketch out the proof of Theorem \ref{caso gamma}. \begin{proof}[Proof of Theorem \ref{caso gamma}] First, we notice that $\gamma_{k-2}(G)$ is nilpotent. The proof of the nilpotency of $\gamma_{k-2}(G)$ is similar to that of Lemma \ref{nilpotency for G^d}. Next, we let $L=L(\gamma_{k-2}(G))$ be the Lie algebra associated with $\gamma_{k-2}(G)$. Then $pL=L$ and $L$ has the same nilpotency class as $\gamma_{k-2}(G)$. The group $A$ naturally acts by automorphisms on $L$ and, since $\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$, it follows that $\gamma_{k-2}(C_{L}(a))$ is nilpotent of class at most $c$ for any $a\in A^{\#}$. Put $K=L\otimes \mathbb{Z}[\omega]$, where $\omega$ is a primitive $p$th root of unity. The nilpotency of $\gamma_{k-2}(C_{L}(a))$ implies that also $\gamma_{k-2}(C_{K}(a))$ is nilpotent of class at most $c$ for any $a\in A^{\#}$. Theorem 2.7(1) of \cite{shu1} now tells us that $\gamma_{k-2}(K)$ is nilpotent of $\{c,k,p\}$-bounded class. Hence, also the nilpotency class of $\gamma_{k-2}(L)$ is $\{c,k,p\}$-bounded. We denote the nilpotency class of $\gamma_{k-2}(L)$ by $e$. Let $H_1,H_2,\ldots,H_t$ be the $\gamma$-$A$-special subgroups of $G$ of degree $k-2$. By Proposition \ref{gammaPAspecial}(2) $\gamma_{k-2}(G)=\langle H_{1},H_{2},\ldots,H_t\rangle$. Since $k-2\leq k-1$, Proposition \ref{gammaPAspecial}(3) tells us that each subgroup $H_{i}$ is contained in $\gamma_{k-2}(C_{G}(B))$ for some subgroup $B$ of $A$ such that $\|A/B\|\leq p^{k-2}$. Let $A_{1},\ldots,A_{s}$ be the maximal subgroups of $A$. For any $A_{j}$ the intersection $B\cap A_{j}$ is not trivial. Thus, there exists $a\in A^{\#}$ such that the centralizer $C_{G}(A_{j})$ is contained in $C_{G}(a)$ and $H_{i}$ is contained in $\gamma_{k-2}(C_{G}(a))$. Since $\gamma_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$, we have \begin{equation} \label{relation} [C_{G}(A_{j}),_{\,c+1}H_{i}]=1. \end{equation} Next, we define recursively what will be called $\gamma$-$A$-subalgebras of $L$. The definition is similar to that of $A$-subalgebras used in the previous section. For each $\gamma$-$A$-special subgroup $H_{i}$ of $G$ of degree $k-2$ we consider the corresponding subalgebra $L(\gamma_{k-2}(G),H_{i})$. A subalgebra $R$ is a $\gamma$-$A$-subalgebra of level $1$ if and only if there exists $j\leq t$ such that $R=L(\gamma_{k-2}(G),H_j)$. Further, suppose that $l\geq 2$ and the $\gamma$-$A$-subalgebras of level $l-1$ are already defined. Then $R$ is a $\gamma$-$A$-subalgebra of level $l$ if and only if there exists a $\gamma$-$A$-subalgebra $R_1$ of level $l-1$ such that $R=[R_1,C_{L}(A_{j})]\cap C_{L}(A_m)$ for suitable $j,m\leq s$. Since (\ref{relation}) holds in the group $G$, we deduce that $[L,_{\,c+1}R]=0$ for every $\gamma$-$A$-subalgebra $R$. Furthermore, using Lemma \ref{gammaspan} one can show that for every $l\geq 1$ the $l$th term $\gamma_{l}(L)$ of the lower central series of $L$ is spanned by the $\gamma$-$A$-subalgebras of level $l$. Finally, we use the above remarks to prove that $L$ is nilpotent of $\{c,k,p\}$-bounded class. This part of the proof is pretty much the same as that of Theorem \ref{caso delta}. Since $\gamma_{k-2}(G)$ has the same nilpotency class as $L$, the theorem follows. \end{proof} \section{Acknowledgments} This research was supported by CNPq-Brazil. \end{document}	arXiv
\begin{document} \title[Curvature inequalities]{Curvature inequalities for Lagrangian submanifolds:\\ the final solution} \author[B.-Y. Chen]{Bang-Yen Chen} \address{Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027, USA} \email{[email protected]} \author[F. Dillen]{Franki Dillen} \address{KU Leuven\\ Departement Wiskunde\\ Celestijnenlaan 200B -- Box 2400\\ BE-3001 Leuven\\ Belgium} \email{[email protected]} \author[J. Van der Veken]{Joeri Van der Veken} \address{KU Leuven\\ Departement Wiskunde\\ Celestijnenlaan 200B -- Box 2400\\ BE-3001 Leuven\\ Belgium} \email{[email protected]} \thanks{The third author is a post-doctoral researcher supported by the Research Foundation -- Flanders (F.W.O.).} \author[L. Vrancken]{Luc Vrancken} \address{Universit\'e de Valenciennes, Lamath, ISTV2, Campus du Mont Houy, 59313 Valenciennes, Cedex 9, France; KU Leuven\\ Departement Wiskunde\\ Celestijnenlaan 200B -- Box 2400\\ BE-3001 Leuven\\ Belgium} \email{[email protected]} \begin{abstract} Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form. We prove a pointwise inequality $$\delta(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \\|H\\|^2 + b(n,k,n_1,\ldots,n_k)c,$$ with on the left hand side any delta-invariant of the Riemannian manifold $M$ and on the right hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right hand side are optimal in the sence that there exist non-minimal examples satisfying equality at at least one point. We also characterize those Lagrangian submanifolds satisfying equality at any of their points. Our results correct and extend those given in \cite{CD}. \end{abstract} \keywords{delta-invariant, Lagrangian submanifold} \subjclass[2000]{53B25, 53D12} \maketitle \section{Introduction} Let $M^n$ be an $n$-dimensional Riemannian manifold. In the 1990ties, the first author introduced a new family of curvature functions on $M^n$, the so-called \emph{delta-invariants}. In particular, for any $k$ integers $n_1,\ldots,n_k$ satisfying $$2 \leq n_1 \leq \ldots \leq n_k \leq n-1\;\; and \;\;n_1 + \ldots + n_k \leq n,$$ a delta-invariant $\delta(n_1,\ldots,n_k)$ was defined at any point of $M^n$. In a K\"ahler manifold with complex structure $J$, a special role is played by Lagrangian submanifolds. These are submanifolds for which $J$ maps the tangent space into the normal space and vice versa at any point. In \cite{CDVV1} and \cite{CDVV2} the following pointwise inequality for a Lagrangian submanifold $M^n \hookrightarrow \tilde M^n(4c)$ of a complex space form of constant holomorphic sectional curvature $4c$ was obtained: \begin{equation}\begin{aligned}& \delta(n_1,\ldots,n_k) \leq \dfrac{n^2 \left( n+k+1-\sum_{i=1}^kn_i \right)}{2 \left( n+k-\sum_{i=1}^kn_i \right)} \\|H\\|^2\\&\hskip.6in + \dfrac 12 \left( n(n-1) - \sum_{i=1}^k n_i(n_i-1) \right) c.\end{aligned} \end{equation} Here, $H$ is the mean curvature vector of the immersion at the point under consideration. The importance of this type of inequalities is that the left hand side is intrinsic, i.e., it only depends on $M^n$ as a Riemannian manifold itself, whereas the right hand side contains extrinsic information, i.e., depending of the immersion under consideration. For example, the inequality shows that a necessary condition for a Riemannian manifold $M^n$ to allow a minimal Lagrangian immersion into $\mathbb{C}^n$ is that all the delta-invariants at all points are non-positive. However, it was proven in \cite{C1} that if equality holds in the above inequality at some point, the mean curvature of the immersion has to vanish at this point. This suggests that the inequality is not optimal, i.e., that the coefficient of $\\|H\\|^2$ can be replaced by a smaller value. The following improvement was given in \cite{CD}: \begin{equation}\begin{aligned} \label{0.2}\delta(n_1,\ldots,n_k) \leq &\frac{n^2 \left(n -\sum_{i=1}^k n_i + 3k - 1 - 6\sum_{i=1}^k \frac{1}{2+n_i} \right)} {2 \left(n -\sum_{i=1}^k n_i + 3k + 2 - 6\sum_{i=1}^k \frac{1}{2+n_i} \right)} \\|H\\|^2 \\&\hskip.0in +\frac{1}{2}\left(n(n-1)-\sum_{i=1}^k n_i(n_i-1)\right)c.\end{aligned} \end{equation} It was pointed out in \cite{CD2} that the proof of inequality \eqref{0.2} given \cite{CD} is incorrect when $\sum_{i=1}^k \frac{1}{2+n_i}>\frac{1}{3}$. The purpose of this paper is two-fold. First, we correct the proof of the above inequality in the case $n_1+\ldots+n_k<n$ (Theorem \ref{theo1}) and then we show that the inequality can be improved in the case $n_1+\ldots+n_k=n$ (Theorem \ref{theo2}). In both cases, we also characterize those Lagrangian submanifolds attaining equality at any of their points, thereby showing that the inequalities are optimal, in the sense that non-minimal examples occur. \section{preliminaries} Let us first recall the definition of the delta-invariants. Let $p$ be a point of an $n$-dimensional Riemannian manifold $M^n$ and let $L$ be a linear subspace of the tangent space $T_pM^n$. If $\{e_1,\ldots,e_{\ell}\}$ is an orthonormal basis of $L$, we define $$ \tau(L) := \sum_{i,j=1 \atop i<j}^{\ell} K(e_i \wedge e_j), $$ where $K(e_i \wedge e_j)$ denotes the sectional curvature of the plane spanned by $e_i$ and $e_j$. Remark that the right hand side is indeed independent of the chosen orthonormal basis and that $\tau := \tau(T_pM^n)$ is nothing but the scalar curvature of $M^n$ at $p$. Now let $n_1,\ldots,n_k$ be integers such that $$2 \leq n_1 \leq \ldots \leq n_k \leq n-1 \;\; and \;\; n_1 + \ldots + n_k \leq n,$$ then the delta-invariant $\delta(n_1,\ldots,n_k)$ at the point $p$ is defined as follows: $$ \delta(n_1,\ldots,n_k)(p) := \tau - \inf \left\{ \sum_{i=1}^k \tau(L_i) \right\}, $$ where the infimum is taken over all $k$-tuples $(L_1,\ldots,L_k)$ of mutually orthogonal subspaces of $T_pM^n$ with $\dim(L_i)=n_i$ for $i=1,\ldots,k$. Due to a compactness argument, the infimum is actually a minimum. Remark that the simplest delta-invariant is $\delta(2) = \tau - \inf\{K(\pi)\ \|\ \pi \mbox{ is a plane in } T_pM^n\}$. In the following we will denote by $\tilde M^n(4c)$ a complex space form of complex dimension $n$ and constant holomorphic sectional curvature $4c$. Let $M^n \hookrightarrow \tilde M^n(4c)$ be a Lagrangian immersion and denote the Levi-Civita connections of $M^n$ and $\tilde M^n(4c)$ by $\nabla$ and $\tilde \nabla$ respectively. If $X$ and $Y$ are vector fields on $M^n$, then the formula of Gauss gives a decomposition of $\tilde\nabla_XY$ into its components tangent and normal to $M^n$: $$ \tilde\nabla_X Y = \nabla_X Y + h(X,Y), $$ defining in this way the second fundamental form $h$, a symmetric $(1,2)$-tensor field taking values in the normal bundle. The mean curvature vector field is defined as $$ H := \frac 1n \mbox{trace}\hskip.01in(h). $$ An important property of Lagrangian submanifolds is that the cubic form, i.e., the $(0,3)$-tensor field on $M^n$ defined by $\langle h(\cdot,\cdot),J\cdot \rangle$, where $J$ is the almost complex structure of $\tilde M^n(4c)$, is totally symmetric. Finally, we recall the equation of Gauss: if $R$ is the Riemann-Christoffel curvature tensor of $M^n$ and $X$, $Y$, $Z$ and $W$ are tangent to $M^n$, then \begin{equation}\begin{aligned} & \langle R(X,Y)Z,W \rangle = \langle h(X,W),h(Y,Z) \rangle - \langle h(X,Z),h(Y,W) \rangle \\&\hskip.7in + c \left( \langle X,W \rangle \langle Y,Z \rangle - \langle X,Z \rangle \langle Y,W \rangle \right). \end{aligned}\end{equation} The following result on the existence of Lagrangian submanifolds can be found for example in \cite{C2}. \begin{lemma} \label{lem1} For any set of real numbers $\{a_{ABC}\ \|\ A,B,C = 1,\ldots,n\}$, which is symmetric in the three indices $A$, $B$ and $C$, there exists a Lagrangian immersion $F:U\subseteq\mathbb{R}^n \to \mathbb C^n$ and a point $p \in U$ such that the second fundamental form $h$ of $F$ at $p$ is given by $\langle h(e_A,e_B),J F_{\ast} e_C \rangle = a_{ABC}$, where $\{e_1,\ldots,e_n\}$ is the standard basis of $\mathbb{R}^n$ and $J$ is the standard complex structure of $\mathbb{C}^n$. \end{lemma} \begin{proof} Let $f:U \subseteq \mathbb{R}^n \to \mathbb{R} : (x_1,\ldots,x_n) \mapsto f(x_1,\ldots,x_n)$ be a smooth function on an open subset $U$ of $\mathbb{R}^n$. Then one can verify that $F : U \subseteq \mathbb{R}^n \to \mathbb{C}^n : (x_1,\ldots,x_n) \mapsto (x_1+if_{x_1},\ldots,x_n+if_{x_n})$ is a Lagrangian immersion satisfying $\langle h(e_A,e_B),JF_{\ast}e_C \rangle = f_{x_A x_B x_C}$ at every point of $U$. Here, an index $x_j$ means partial differentiation with respect to $x_j$. For a given set of real numbers $\{a_{ABC}\ \|\ A,B,C = 1,\ldots,n\}$, which is symmetric in the three indices, one can easily construct a smooth function $f$, a degree $3$ polynomial for example, which satisfies $f_{x_A x_B x_C} = a_{ABC}$ and one can define $F$ as above. \end{proof} We end this section by stating a fact of elementary linear algebra, which will be useful in the proof or our main results. \begin{lemma} \label{lem2} For real numbers $A_1,\ldots,A_k$, denote by $\Delta(A_1,\ldots,A_k)$ the determinant of the matrix with $A_1, \ldots, A_k$ on the diagonal and all other entries equal to $1$: $$ \Delta(A_1,\ldots,A_k) = \left\| \begin{array}{ccccc} A_1&1&\cdots&1&1 \\ 1&A_2&\cdots&1&1 \\ \vdots&\vdots&\ddots&\vdots&\vdots \\ 1&1&\cdots&A_{k-1}&1 \\ 1&1&\cdots&1&A_k\end{array}\right\|.$$ Then $$ \Delta(A_1,\ldots,A_k) = \prod_{i=1}^k (A_i-1) + \sum_{i=1}^k \prod_{j \neq i} (A_j-1). $$ In particular, if none of the numbers $A_1,\ldots,A_k$ equals $1$, then $$ \Delta(A_1,\ldots,A_k) = \left(1+\frac{1}{A_1-1}+ \ldots +\frac{1}{A_k-1}\right)(A_1-1)\ldots(A_k-1).$$ \end{lemma} \begin{proof} The result is true for $k=1$ and $k=2$. Now assume that $k\geq3$ and let $A_1,\ldots,A_k$ be arbitrary real numbers. We will compute the determinant $\Delta(A_1,\ldots,A_k)$ by first replacing the $k$th column by the $k$th column minus the $(k-1)$th column, then replacing the $k$th row by the $k$th row minus the $(k-1)$th row and finally developing the determinant with respect to the last column: \begin{align} \Delta(A_1,\ldots,A_k) &= \left\| \begin{array}{ccccc} A_1&1&\cdots&1&0 \\ 1&A_2&\cdots&1&0 \\ \vdots&\vdots&\ddots&\vdots&\vdots \\ 1&1&\cdots&A_{k-1}&1-A_{k-1} \\ 1&1&\cdots&1&A_k-1\end{array} \right\| \\ \\ &= \left\| \begin{array}{ccccc} A_1&1&\cdots&1&0 \\ 1&A_2&\cdots&1&0 \\ \vdots&\vdots&\ddots&\vdots&\vdots \\ 1&1&\cdots&A_{k-1}&1-A_{k-1} \\ 0&0&\cdots&1-A_{k-1}&A_k+A_{k-1}-2\end{array} \right\| \\ \\&\hskip-.4in=(A_k+A_{k-1}-2)\Delta(A_1,\ldots,A_{k-1})-(A_{k-1}-1)^2\Delta(A_1,\ldots,A_{k-2}). \end{align} It is now sufficient to verify that the expression for $\Delta(A_1,\ldots,A_k)$ given in the statement of the lemma indeed satisfies the recursion relation \begin{equation}\begin{aligned}\notag & \Delta(A_1,\ldots,A_k)=(A_k+A_{k-1}-2)\Delta(A_1,\ldots,A_{k-1})\\& \hskip.5in-(A_{k-1}-1)^2\Delta(A_1,\ldots,A_{k-2}),\end{aligned} \end{equation} with the initial conditions $\Delta(A_1)=A_1$ and $\Delta(A_1,A_2)=A_1A_2-1$. This can be done by a straightforward computation. \end{proof} \section{The main results} Before stating our main theorems, we will introduce some notations. For a given delta-invariant $\delta(n_1,\ldots,n_k)$ on a Riemannian manifold $M^n$ (with $2 \leq n_1 \leq \ldots \leq n_k \leq n-1$ and $n_1 + \ldots + n_k \leq n$) and a point $p \in M^n$, we consider mutually orthogonal subspaces $L_1, \ldots, L_k$ with $\dim(L_i) = n_i$ of $T_pM^n$, minimizing the quantity $\tau(L_1)+ \ldots +\tau(L_k)$. We then choose an orthonormal basis $\{e_1,\ldots,e_{n}\}$ for $T_pM^n$ such that \begin{align} & e_1,\ldots,e_{n_1} \in L_1, \\ & e_{n_1+1},\ldots,e_{n_1+n_2} \in L_2, \\ & \ \vdots \\ & e_{n_1+\ldots+n_{k-1}+1},\ldots,e_{n_1+\ldots+n_k} \in L_k, \end{align} and we define \begin{align} & \Delta_1 := \{1,\ldots, n_1\}, \\ & \Delta_2 := \{n_1+1,\ldots, n_1+n_2\}, \\ & \ \vdots \\ & \Delta_k := \{n_1+\ldots+n_{k-1}+1,\ldots,n_1+\ldots+n_k\}, \\ & \Delta_{k+1} := \{n_1+\ldots+n_k+1,\ldots,n\}. \end{align} From now on, we will use the following conventions for the ranges of summation indices: $$ A,B,C\in\{1,\ldots,n\}, \quad i,j\in\{1,\ldots,k\}, \quad \alpha_i,\beta_i\in\Delta_i, \quad r,s\in\Delta_{k+1}.$$ Finally, we define $n_{k+1} := n-n_1-\ldots-n_k$. Remark that this may be zero, in which case $\Delta_{k+1}$ is empty. We shall denote the components of the second fundamental form by $h^C_{AB}=\langle h(e_A,e_B),Je_C\rangle$. Due to the symmetry of the cubic form, these are symmetric with respect to the three indices $A$, $B$ and $C$. \begin{theorem} \label{theo1} Let $M^n$ be a Lagrangian submanifold of a complex space form $\tilde M^n(4c)$. Let $n_1,\ldots,n_k$ be integers satisfying $2\leq n_1 \leq \ldots \leq n_k \leq n-1$ and $n_1+\ldots+n_k < n$. Then, at any point of $M^n$, we have \begin{equation}\begin{aligned}\notag &\delta(n_1,\ldots,n_k) \leq \frac{n^2 \left(n -\sum_{i=1}^k n_i + 3k - 1 - 6\sum_{i=1}^k \frac{1}{2+n_i} \right)} {2 \left(n -\sum_{i=1}^k n_i + 3k + 2 - 6\sum_{i=1}^k \frac{1}{2+n_i} \right)} \\|H\\|^2 \\& \hskip1.0in +\frac{1}{2}\left(n(n-1)-\sum_{i=1}^k n_i(n_i-1)\right)c. \end{aligned} \end{equation} Assume that equality holds at a point $p \in M^n$. Then with the choice of basis and the notations introduced at the beginning of this section, one has \begin{itemize} \item $h^A_{BC}=0$ if $A,B,C$ are mutually different and not all in the same $\Delta_i$ ($i=1,\ldots,k$), \item $\displaystyle{ h^{\alpha_i}_{\alpha_j \alpha_j} = h^{\alpha_i}_{rr} = \sum_{\beta_i \in \Delta_i} h^{\alpha_i}_{\beta_i \beta_i}=0 }$ for $i \neq j$, \item $h^r_{rr} = 3 h^r_{ss} = (n_i+2) h^r_{\alpha_i \alpha_i}$ for $r \neq s$. \end{itemize} \end{theorem} \begin{proof} The proof consists of four steps. \noindent\emph{Step 1: Set-up.} Fix a delta-invariant $\delta(n_1,\ldots,n_k)$ and a point $p\in M^n$. Take linear subspaces $L_1,\ldots,L_k$ of $T_pM^n$ and and orthonormal basis $\{e_1,\ldots,e_n\}$ of $T_pM^n$ as described above. From the equation of Gauss we obtain that \begin{align} & \tau= \frac{n(n-1)}{2}c + \sum_{A} \sum_{B<C} (h^A_{BB}h^A_{CC}-(h^A_{BC})^2),\\ & \tau(L_i)= \frac{n_i(n_i-1)}{2}c + \sum_{A} \sum_{ \alpha_i<\beta_i} ( h^A_{\alpha_i \alpha_i}h^A_{\beta_i\beta_i}-(h^A_{\alpha_i\beta_i})^2) \end{align} for $i=1,\ldots,k$. We see that we can assume without loss of generality that $c=0$, and we have \begin{equation}\begin{aligned}\label{firstineq1} &\hskip-.2in\tau - \sum_i \tau(L_i) = \\ &\sum_A \left\{ \sum_{r<s} (h^A_{rr}h^A_{ss}-(h^A_{rs})^2) + \sum_i \sum_{\alpha_i,r} (h^A_{\alpha_i \alpha_i}h^A_{\alpha_i,r}-(h^A_{\alpha_i r)})^2) \right. \\ & \left. + \sum_{i<j} \sum_{\alpha_i,\alpha_j} (h^A_{\alpha_i \alpha_i}h^A_{\alpha_j \alpha_j} - (h^A_{\alpha_i \alpha_j})^2) \right\} \\ &\leq \sum_A \left\{ \sum_{r<s} h^A_{rr}h^A_{ss} + \sum_i \sum_{\alpha_i,r} h^A_{\alpha_i \alpha_i}h^A_{rr} + \sum_{i<j} \sum_{\alpha_i,\alpha_j} h^A_{\alpha_i \alpha_i}h^A_{\alpha_j \alpha_j} \right\} \\ & - \sum_r \sum_{B \neq r} (h^B_{rr})^2 - \sum_i \sum_{\alpha_i} \sum_{B \notin \Delta_i} (h^B_{\alpha_i \alpha_i})^2. \end{aligned}\end{equation} We want to prove that \eqref{firstineq1} is less than or equal to $$ n^2 C \\|H\\|^2 = C \sum_A \left( \sum_B h^A_{BB}\right)^2, $$ with \begin{equation} \label{valueC1} C = \frac{n_{k+1} + 3k - 1 - 6\sum_{i=1}^k \frac{1}{2+n_i}} {2 \left(n_{k+1} + 3k + 2 - 6\sum_{i=1}^k \frac{1}{2+n_i} \right)}. \end{equation} In fact, we want to prove that this value for $C$ is the best possible one in the sense that the inequality in the theorem will no longer be true in general for smaller values of $C$. In view of Lemma \ref{lem1}, we have to find the smallest possible $C$ for which the following two statements hold: \begin{itemize} \item[(I)] for any $\ell \in \{1,\ldots,k\}$ and any $\gamma_{\ell} \in \Delta_{\ell}$ \begin{align} & \sum_{r<s} h^{\gamma_{\ell}}_{rr}h^{\gamma_{\ell}}_{ss} + \sum_i \sum_{\alpha_i,r} h^{\gamma_{\ell}}_{\alpha_i \alpha_i}h^{\gamma_{\ell}}_{rr} + \sum_{i<j} \sum_{\alpha_i,\alpha_j} h^{\gamma_{\ell}}_{\alpha_i \alpha_i}h^{\gamma_{\ell}}_{\alpha_j \alpha_j} \\ & - \sum_r (h^{\gamma_{\ell}}_{rr})^2 - \sum_{i \neq \ell} \sum_{\alpha_i} (h^{\gamma_{\ell}}_{\alpha_i \alpha_i})^2 \leq C \left( \sum_B h_{BB}^{\gamma_{\ell}} \right)^2, \end{align} \item[(II)] for any $t \in \Delta_{k+1}$ \begin{align} & \sum_{r<s} h^t_{rr}h^t_{ss} + \sum_i \sum_{\alpha_i,r} h^t_{\alpha_i \alpha_i}h^t_{rr} + \sum_{i<j} \sum_{\alpha_i,\alpha_j} h^t_{\alpha_i \alpha_i}h^t_{\alpha_j \alpha_j} \\ & - \sum_{r \neq t} (h^t_{rr})^2 - \sum_i \sum_{\alpha_i} (h^t_{\alpha_i \alpha_i})^2 \leq C \left( \sum_B h_{BB}^t \right)^2. \end{align} \end{itemize} \noindent\emph{Step 2: Finding the best possible $C$ in \emph{(I)}.} The inequality in (I) is equivalent to \begin{equation}\begin{aligned} \label{quadraticform1}& (C+1) \sum_{i \neq \ell} \sum_{\alpha_{\ell}} (h_{\alpha_i \alpha_i}^{\gamma_{\ell}})^2 + C \sum_{\alpha_{\ell}} (h_{\alpha_{\ell} \alpha_{\ell}}^{\gamma_{\ell}})^2 + (C+1) \sum_r (h_{rr}^{\gamma_{\ell}})^2\\ \\& \hskip.4in + 2C \sum_i \sum_{\alpha_i < \beta_i} h_{\alpha_i \alpha_i}^{\gamma_{\ell}}h_{\beta_i \beta_i}^{\gamma_{\ell}} \\ & +(2C-1) \left( \sum_{i<j} \sum_{\alpha_i,\alpha_j} h_{\alpha_i \alpha_i}^{\gamma_{\ell}} h_{\alpha_j \alpha_j}^{\gamma_{\ell}} + \sum_i \sum_{\alpha_i, r} h_{\alpha_i \alpha_i}^{\gamma_{\ell}} h_{rr}^{\gamma_{\ell}} + \sum_{r<s} h_{rr}^{\gamma_{\ell}} h_{ss}^{\gamma_{\ell}}\right)\\& \geq 0. \end{aligned}\end{equation} If we put $x_A = h^{\gamma_{\ell}}_{AA}$ for all $A=1,\ldots,n$, then we can look at the left hand side of the above inequality as a quadratic form on $\mathbb{R}^n$. In view of Lemma \ref{lem1}, we need to find necessary and sufficient conditions on $C$ for this quadratic form to be non-negative. Two times the matrix of this quadratic form consists of $(k+1)^2$ blocks: $$ M_{\ell} = \left( \Lambda_{ij} \right)_{i,j=1,\ldots,k+1} $$ with \begin{align} & \Lambda_{\ell\ell} = \left( \begin{array}{ccc} 2C &\cdots& 2C\\\vdots&\ddots&\vdots\\2C&\cdots& 2C \end{array}\right) \in \mathbb{R}^{n_{\ell}\times n_{\ell}}, \\ & \Lambda_{k+1 \, k+1} = \left( \begin{array}{ccccc} 2(C+1)&2C-1&\cdots&2C-1&2C-1 \\ 2C-1&2(C+1)&\cdots&2C-1&2C-1 \\ \vdots&\vdots&\ddots&\vdots&\vdots \\ 2C-1&2C-1&\cdots&2(C+1)&2C-1 \\ 2C-1&2C-1&\cdots&2C-1&2(C+1)\end{array} \right) \in \mathbb{R}^{n_{k+1} \times n_{k+1}},\\ & \Lambda_{ii} = \left( \begin{array}{ccccc} 2(C+1)&2C&\cdots&2C&2C \\ 2C&2(C+1)&\cdots&2C&2C \\ \vdots&\vdots&\ddots&\vdots&\vdots \\ 2C&2C&\cdots&2(C+1)&2C \\ 2C&2C&\cdots&2C&2(C+1)\end{array} \right) \in \mathbb{R}^{n_i \times n_i} \\ & \hskip1in \mbox{if} \ i \neq \ell,k+1,\\ & \Lambda_{ij} = \left( \begin{array}{ccc} 2C-1 &\cdots& 2C-1\\\vdots&\ddots&\vdots\\2C-1&\cdots&2C-1\end{array}\right) \in \mathbb{R}^{n_i\times n_j} \ \mbox{if} \ i\neq j. \end{align} Remark that for every $i\in \{1,\ldots,k+1\}$, $M_{\ell}$ has the following $n_i-1$ eigenvectors: \begin{equation} \label{eigenvecs1} \begin{aligned} & (0,\ldots,0, \| 1,-1,0,\ldots,0,0, \| 0,\ldots,0), \\ & (0,\ldots,0, \| 1,0,-1,\ldots,0,0, \| 0,\ldots,0), \\ & \quad \vdots \\ & (0,\ldots,0, \| \underbrace{1,0,0,\ldots,0,-1,}_{\Delta_i} \| 0,\ldots,0). \end{aligned} \end{equation} The eigenvalues are $0$, $3$ or $2$ depending on whether $i=\ell$, $i=k+1$ or $i \neq \ell,k+1$ respectively. In total we have thus found $n-(k+1)$ eigenvectors of $M_{\ell}$ with non-negative eigenvalues. The orthogonal complement of all these eigenvectors is spanned by \begin{equation} \label{othereigenvecs1} v_i = \frac{1}{n_i}(0,\ldots,0, \| \underbrace{1,1,\ldots,1,}_{\Delta_i} \| 0,\ldots,0), \quad i=1,\ldots, k+1. \end{equation} It is now sufficient to prove that the matrix $M'_{\ell} = (v_i M_{\ell} v_j^T)_{i,j=1,\ldots,k+1} \in \mathbb{R}^{(k+1)\times (k+1)}$ is non-negative. It follows from a direct computation that \begin{align} & (M'_{\ell})_{\ell\ell} = 2C, \quad (M'_{\ell})_{k+1 \ k+1} = 2C-1+\frac{3}{n_{k+1}}, \\ & (M'_{\ell})_{ii} = 2\left(C+\frac{1}{n_i}\right) \ \mbox{if} \ i\neq\ell,k+1, \quad (M'_{\ell})_{ij} = 2C-1 \ \mbox{if} \ i\neq j. \end{align} We investigate three cases. \textbf{Case 1: $2C=1$.} In this case, $M'_{\ell}$ is a diagonal matrix with positive diagonal entries, so it is positive definite. \textbf{Case 2: $2C>1$.} In this case, the matrix $M'_{\ell}$ is always positive definite. To see this, it is sufficient to verify that the matrix $M''_{\ell} = M'_{\ell}/(2C-1)$ is positive definite. By Sylvester's criterion, we have to verify that the $(j \times j)$-matrix in the upper left corner of $M''_{\ell}$ has positive determinant for all $j=1,\ldots,k$. From Lemma \ref{lem2}, it is sufficient to remark that \begin{equation} \notag \frac{2C}{2C-1}-1 > 0, \quad \frac{2(C+1/n_j)}{2C-1}-1 > 0 \mbox{ for all } j \in \{1,\ldots,k\}\setminus\{\ell\}\end{equation} and \begin{equation} \notag \frac{3}{n_{k+1}(2C-1)} > 0.\end{equation} \textbf{Case 3: $2C<1$.} It is now sufficient to require that $M''_{\ell} = M'_{\ell}/(2C-1)$ is non-positive. By Sylvester's criterion, this is equivalent to the determinant of the $(j \times j)$-matrix in the upper left corner of $M''_{\ell}$ having sign $(-1)^j$ for all $j=1,\ldots,k+1$. It follows from Lemma \ref{lem2} that these determinants are \begin{align} & D_j = \frac{1}{(2C-1)^{j-1}} \left( \frac{(2C)^{\delta_j}}{2C-1} + \sum_{i=1 \atop i \neq \ell}^j \frac{n_i}{n_i+2} \right) \prod_{i=1 \atop i \neq \ell}^j \left( 1 + \frac{2}{n_i} \right) \ \mbox{for $j = 1,\ldots,k$,} \\ & D_{k+1} = \frac{3}{n_{k+1}(2C-1)^k} \left( \frac{2C}{2C-1} + \frac{n_{k+1}}{3} + \sum_{i=1 \atop i \neq \ell}^k \frac{n_i}{n_i+2} \right) \prod_{i=1 \atop i \neq \ell}^k \left( 1 + \frac{2}{n_i} \right), \end{align} where $\delta_j=0$ if $j<\ell$ and $\delta_j=1$ if $j\geq\ell$. Hence, we have \begin{align} & \mathrm{sgn}(D_j) = (-1)^{j-1} \mathrm{sgn} \left( \frac{(2C)^{\delta_j}}{2C-1} + \sum_{i=1 \atop i \neq \ell}^j \frac{n_i}{n_i+2} \right) \ \mbox{for $j = 1,\ldots,k$,} \\ & \mathrm{sgn}(D_{k+1}) = (-1)^k \mathrm{sgn} \left( \frac{2C}{2C-1} + \frac{n_{k+1}}{3} + \sum_{i=1 \atop i \neq \ell}^k \frac{n_i}{n_i+2} \right), \end{align} and the conditions for $M''_{\ell}$ to be non-positive are \begin{equation} \label{firstcondC1} \left\{\begin{array}{l} \displaystyle{ \frac{(2C)^{\delta_j}}{2C-1} + \sum_{i=1 \atop i \neq \ell}^j \frac{n_i}{n_i+2} \leq 0 \quad \mbox{for} \ j = 1,\ldots,k,} \\ \displaystyle{ \frac{2C}{2C-1} + \frac{n_{k+1}}{3} + \sum_{i=1 \atop i \neq \ell}^k \frac{n_i}{n_i+2} \leq 0. } \end{array}\right. \end{equation} One can verify that the last inequality implies the first $k$ inequalities, and the last inequality is equivalent to \begin{equation} \label{firstineqC1} 2C \geq \frac{n_{k+1} + 3k - 3 - 6 \sum_{i\neq\ell}\frac{1}{n_i+2}}{n_{k+1} + 3k - 6 \sum_{i\neq\ell}\frac{1}{n_i+2}}. \end{equation} We conclude from all three cases that the quadratic form in \eqref{quadraticform1} is non-negative if and only if $C$ satisfies \eqref{firstineqC1} for every $\ell = 1,\ldots,k$. \noindent\emph{Step 3: Finding the best possible $C$ in \emph{(II)}.} We can proceed in the same way as in Step 2, by defining a quadratic form on $\mathbb{R}^n$ from inequality (II) and looking for the best possible value of $C$ for which this quadratic form is non-negative. Since the result is the same as the one already obtained in \cite{CD} by a more ad hoc method, we will not go into details here. The condition on $C$ is \begin{equation} \label{secondineqC1} 2C \geq \frac{n_{k+1} + 3k - 1 - 6 \sum_{i}\frac{1}{n_i+2}}{n_{k+1} + 3k + 2 - 6 \sum_{i}\frac{1}{n_i+2}}. \end{equation} Since the right hand side of \eqref{firstineqC1} is less than the right hand side of \eqref{secondineqC1} (for any $\ell \in \{1,\ldots,k\}$), we only have \eqref{secondineqC1} as a condition on $C$ and we conclude that the best possible value for $C$ is precisely the value given in \eqref{valueC1}. \noindent\emph{Step 4: The equality case.} Assume that equality holds at a point. Then one has to have equality in \eqref{firstineq1}, which implies precisely the first condition given in the theorem. Next, one also has to have equality in (I), which implies that the vector $(h^{\alpha_i}_{11},\ldots,h^{\alpha_i}_{nn})$ has to be a linear combination of the vectors given in \eqref{eigenvecs1} for every $i=1,\ldots,k$ and every $\alpha_i\in\Delta_i$. (Remark that due to the choice of $C$, the quadratic form \eqref{quadraticform1} is positive definite on the orthogonal complement of these vectors.) Hence, one obtains exactly the second condition given in the theorem. Finally, one has to have equality in (II). We refer to \cite{CD} to see that this is equivalent to the third condition. \end{proof} \begin{remark} In inequality \eqref{firstineq1}, we omit exactly the squares of components of $h$ with three different indices. Besides the technique used to prove non-negativeness of the quadratic forms, this is the main difference with the proof in \cite{CD}, where also terms of type $-(h_{\alpha_i \alpha_i}^{\alpha_j})^2$ are omitted, making the inequality less sharp. \end{remark} \begin{theorem} \label{theo2} Let $M^n$ be a Lagrangian submanifold of a complex space form $\tilde M^n(4c)$. Let $n_1,\ldots,n_k$ be integers satisfying $2\leq n_1 \leq \ldots \leq n_k \leq n-1$ and $n_1+\ldots+n_k=n$. Then, at any point of $M^n$, the following holds: \begin{equation}\begin{aligned}\notag& \delta(n_1,\ldots,n_k) \leq \frac{n^2\left(k-1-2\sum_{i=2}^k\frac{1}{n_i+2}\right)}{2\left(k-2\sum_{i=2}^k\frac{1}{n_i+2}\right)}\\|H\\|^2 \\& \hskip.4in +\frac{1}{2}\left(n(n-1)-\sum_{i=1}^k n_i(n_i-1)\right)c. \end{aligned}\end{equation} Assume that equality holds at a point $p \in M^n$. Then with the choice of basis and the notations introduced at the beginning of this section, one has \begin{itemize} \item $h_{\alpha_i \alpha_j}^A = 0$ if $i \neq j$ and $A \neq \alpha_i,\alpha_j$, \item if $n_j \neq \min\{n_1,\ldots,n_k\}$: $$h_{\alpha_i \alpha_i}^{\beta_j}=0 \mbox{ if } i \neq j \mbox{ and } \sum_{\alpha_j \in \Delta_j} h_{\alpha_j \alpha_j}^{\beta_j} = 0,$$ \item if $n_j = \min\{n_1,\ldots,n_k\}$: $$\sum_{\alpha_j \in \Delta_j} h_{\alpha_j \alpha_j}^{\beta_j} = (n_i+2) h_{\alpha_i \alpha_i}^{\beta_j} \mbox{ for any } i \neq j \mbox{ and any } \alpha_i \in \Delta_i.$$ \end{itemize} \end{theorem} \begin{remark} In the case of equality, we don't have information about $h_{\alpha_i \beta_i}^{\gamma_i}$, where $\alpha_i$, $\beta_i$ and $\gamma_i$ are mutually different indices in the same block $\Delta_i$. \end{remark} \begin{proof} The set-up of the proof is exactly the same as in the previous case, but now $\Delta_{k+1}=\varnothing$ and hence $n_{k+1}=0$. We now have \begin{eqnarray} \tau - \sum_i \tau(L_i) &=& \sum_A \sum_{i<j} \sum_{\alpha_i,\alpha_j} (h^A_{\alpha_i \alpha_i}h^A_{\alpha_j \alpha_j} - (h^A_{\alpha_i \alpha_j})^2) \nonumber \\ &\leq& \sum_A \sum_{i<j} \sum_{\alpha_i,\alpha_j} h^A_{\alpha_i \alpha_i}h^A_{\alpha_j \alpha_j} - \sum_i \sum_{\alpha_i} \sum_{B \notin \Delta_i} (h^B_{\alpha_i \alpha_i})^2 \label{firstineq2} \end{eqnarray} and we want to prove that the best possible value of $C$ for which \eqref{firstineq2} is less than or equal to $$ n^2 C \\|H\\|^2 = C \sum_A \left( \sum_B h^A_{BB}\right)^2, $$ is \begin{equation} \label{valueC2} C = \frac{k-1-2\sum_{i=2}^k\frac{1}{n_i+2}} {2\left(k-2\sum_{i=2}^k\frac{1}{n_i+2}\right)}. \end{equation} We only have to consider inequality (I), which reduces to: for all $\ell \in \{1,\ldots,k\}$ and all $\gamma_{\ell} \in \Delta_{\ell}$ \begin{equation} \label{ineqI2} \sum_{i<j} \sum_{\alpha_i,\alpha_j} h_{\alpha_i \alpha_i}^{\gamma_{\ell}} h_{\alpha_j \alpha_j}^{\gamma_{\ell}} - \sum_{i\neq\ell} \sum_{\alpha_i}(h_{\alpha_i \alpha_i}^{\gamma_{\ell}})^2 \leq C \left( \sum_B h^{\gamma_{\ell}}_{BB}\right)^2, \end{equation} or, equivalently, \begin{equation}\begin{aligned} &\notag (2C-1) \sum_{i<j} \sum_{\alpha_i,\alpha_j} x_{\alpha_i} x_{\alpha_j} + 2C \sum_i \sum_{\alpha_i<\beta_i} x_{\alpha_i} x_{\beta_i} \\& \hskip.3in + (C+1) \sum_{i \neq \ell} \sum_{\alpha_i} x_{\alpha_i}^2 + C \sum_{\alpha_{\ell}} x_{\alpha_{\ell}}^2 \geq 0\end{aligned}\end{equation} for all $(x_1,\ldots,x_n)\in\mathbb{R}^n$, where we have put $h^{\gamma_{\ell}}_{AA}=x_A$ for $A=1,\ldots,n$ as before. Requiring non-negativeness of this quadratic form can be done in exactly the same way as before, considering now only the $k^2$ blocks in the upper left corner of $M_{\ell}$ and therefore only giving the first $k$ conditions of \eqref{firstcondC1}. It is clear that the $k$th condition implies all the other ones and hence the necessary and sufficient condition on $C$ for the quadratic form to be non-negative is $$ \frac{2C}{2C-1} + \sum_{i=1 \atop i \neq \ell}^k \frac{n_i}{n_i+2} \leq 0 \mbox{ for } \ell = 1,\ldots,n. $$ Since $n_1 \leq n_2 \leq \ldots \leq n_k$, the above condition for $\ell = 1$ implies all the others and we obtain $$ 2C \geq \frac{k-1-2\sum_{i=2}^k \frac{1}{n_i+2}}{k-2\sum_{i=2}^k \frac{1}{n_i+2}}, $$ such that the best possible value for $C$ is indeed the one given in \eqref{valueC2}. Now let us investigate when equality is attained in the inequality we just proved. In order for this to happen, one has to have equality in \eqref{firstineq2} and one has to have equality in \eqref{ineqI2} for all $\ell \in \{1,\ldots,k\}$ and all $\gamma_{\ell} \in \Delta_{\ell}$. From equality in \eqref{firstineq2}, we conclude that $h_{\alpha_i \alpha_j}^A=0$ for $i \neq j$ and $A \neq \alpha_i, \alpha_j$. If equality holds in \eqref{ineqI2}, the vector $(h_{11}^{\gamma_{\ell}}, \ldots, h_{nn}^{\gamma_{\ell}})$ has to be in the kernel of $M_{\ell}$. If $n_{\ell} \neq \mathrm{min}\{n_1,\ldots,n_k\}$ it follows from the proof of Theorem \ref{theo1} that this kernel is spanned by the vectors \eqref{eigenvecs1} (with $i = \ell$). This corresponds to the first possibility given in the theorem. If $n_{\ell} = \mathrm{min}\{n_1,\ldots,n_k\}$, it follows from the proof of Theorem \ref{theo1} that the kernel of $M_{\ell}$ is larger due to the choice of $C$. In particular, there will be non-zero linear combinations of the vectors $v_1,\ldots,v_k$, given in \eqref{othereigenvecs1}, in the kernel. Assume that $$ M_{\ell} \left( \sum_{i=1}^k a_i v_i \right) = 0 $$ for some real numbers $a_1,\ldots,a_k$. A straightforward computation shows that this is equivalent to $$ \left\{ \begin{array}{l} \displaystyle{\forall i \in \{1,\ldots,k\}\setminus\{\ell\}: \left(\sum_{j=1}^k a_j\right)(2C-1) + a_i\left(1+\frac{2}{n_i}\right) = 0,} \\ \displaystyle{\left(\sum_{j=1}^k a_j\right)(2C-1) + a_{\ell} = 0.} \end{array} \right. $$ One can check that this system indeed has a non-zero solution for $(a_1,\ldots,a_k)$ if and only if $$ C = \frac{k-1-2\sum_{i \neq \ell} \frac{1}{n_i+2}}{2 \left( k-2\sum_{i \neq \ell} \frac{1}{n_i+2} \right)}, $$ i.e., if and only if $n_{\ell}=\min\{n_1,\ldots,n_k\}$. In this case, the solution is given by $$ a_i=\frac{\lambda n_i}{n_i+2} \mbox{ for } i\neq\ell \mbox{ and } a_{\ell}=\lambda $$ for some real number $\lambda$. The vector $(h_{11}^{\gamma_{\ell}}, \ldots, h_{nn}^{\gamma_{\ell}})$ thus has to satisfy the conditions given in the last possibility in the statement of the theorem. \end{proof} \end{document}	arXiv

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