Articles de revues : « Hamiltonien non convexe »

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Ishii, Hitoshi. « The vanishing discount problem for monotone systems of Hamilton-Jacobi equations : a counterexample to the full convergence ». Mathematics in Engineering 5, n^o 4 (2023) : 1–10. http://dx.doi.org/10.3934/mine.2023072.

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<abstract>In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor goes to zero.</abstract>

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Timoumi, Mohsen. « Solutions périodiques de systèmes hamiltoniens convexes non coercitifs ». Bulletin de la Classe des sciences 75, n^o 1 (1989) : 463–81. http://dx.doi.org/10.3406/barb.1989.57866.

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Cirant, Marco, et Alessio Porretta. « Long time behavior and turnpike solutions in mildly non-monotone mean field games ». ESAIM : Control, Optimisation and Calculus of Variations 27 (2021) : 86. http://dx.doi.org/10.1051/cocv/2021077.

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We consider mean field game systems in time-horizon (0, T), where the individual cost functional depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (i.e. the aggregation rate of the cost functions) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either the case of globally Lipschitz Hamiltonians or the case of quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we investigate the long time behavior of solutions and we give a complete description of the ergodic and long term properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0, T), (ii) the convergence of the system from (0, T) towards (0, ∞), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. This way we extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

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CONTRERAS, GONZALO, et RENATO ITURRIAGA. « Convex Hamiltonians without conjugate points ». Ergodic Theory and Dynamical Systems 19, n^o 4 (août 1999) : 901–52. http://dx.doi.org/10.1017/s014338579913387x.

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We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

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Hayat, Sakander, Muhammad Yasir Hayat Malik, Ali Ahmad, Suliman Khan, Faisal Yousafzai et Roslan Hasni. « On Hamilton-Connectivity and Detour Index of Certain Families of Convex Polytopes ». Mathematical Problems in Engineering 2021 (17 juillet 2021) : 1–18. http://dx.doi.org/10.1155/2021/5553216.

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A convex polytope is the convex hull of a finite set of points in the Euclidean space ℝ n . By preserving the adjacency-incidence relation between vertices of a polytope, its structural graph is constructed. A graph is called Hamilton-connected if there exists at least one Hamiltonian path between any of its two vertices. The detour index is defined to be the sum of the lengths of longest distances, i.e., detours between vertices in a graph. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering, whereas the detour index has important applications in chemistry. Checking whether a graph is Hamilton-connected and computing the detour index of an arbitrary graph are both NP-complete problems. In this paper, we study these problems simultaneously for certain families of convex polytopes. We construct two infinite families of Hamilton-connected convex polytopes. Hamilton-connectivity is shown by constructing Hamiltonian paths between any pair of vertices. We then use the Hamilton-connectivity to compute the detour index of these families. A family of non-Hamilton-connected convex polytopes has also been constructed to show that not all convex polytope families are Hamilton-connected.

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Pittman, S. M., E. Tannenbaum et E. J. Heller. « Dynamical tunneling versus fast diffusion for a non-convex Hamiltonian ». Journal of Chemical Physics 145, n^o 5 (7 août 2016) : 054303. http://dx.doi.org/10.1063/1.4960134.

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Monthus, Cécile. « Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and via explicit large deviations at various levels ». Journal of Statistical Mechanics : Theory and Experiment 2023, n^o 6 (1 juin 2023) : 063206. http://dx.doi.org/10.1088/1742-5468/acdcea.

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Abstract For boundary-driven non-equilibrium Markov models of non-interacting particles in one dimension, either in continuous space with the Fokker–Planck dynamics involving an arbitrary force F(x) and an arbitrary diffusion coefficient D(x), or in discrete space with the Markov jump dynamics involving arbitrary nearest-neighbor transition rates w ( x ± 1 , x ) , the Markov generator can be transformed via an appropriate similarity transformation into a quantum supersymmetric Hamiltonian with many remarkable properties. We first describe how the mapping from the boundary-driven non-equilibrium dynamics towards some dual equilibrium dynamics (see Tailleur et al 2008 J. Phys. A: Math. Theor. 41 505001) can be reinterpreted via the two corresponding quantum Hamiltonians that are supersymmetric partners of each other, with the same energy spectra. We describe the consequences for the spectral decomposition of the boundary-driven dynamics, and we give explicit expressions for the Kemeny times needed to converge towards the non-equilibrium steady states. We then focus on the large deviations at various levels for empirical time-averaged observables over a large time-window T. We start with the always explicit Level 2.5 concerning the joint distribution of the empirical density and of the empirical flows before considering the contractions towards lower levels. In particular, the rate function for the empirical current alone can be explicitly computed via the contraction from the Level 2.5 using the properties of the associated quantum supersymmetric Hamiltonians.

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Hayat, Sakander, Asad Khan, Suliman Khan et Jia-Bao Liu. « Hamilton Connectivity of Convex Polytopes with Applications to Their Detour Index ». Complexity 2021 (23 janvier 2021) : 1–23. http://dx.doi.org/10.1155/2021/6684784.

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A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

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Zhou, Min, et Binggui Zhong. « Regions of applicability of Aubry-Mather Theory for non-convex Hamiltonian ». Chinese Annals of Mathematics, Series B 32, n^o 4 (juillet 2011) : 605–14. http://dx.doi.org/10.1007/s11401-011-0654-3.

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Entov, Michael, et Leonid Polterovich. « Contact topology and non-equilibrium thermodynamics ». Nonlinearity 36, n^o 6 (17 mai 2023) : 3349–75. http://dx.doi.org/10.1088/1361-6544/acd1ce.

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Abstract We describe a method, based on contact topology, of showing the existence of semi-infinite trajectories of contact Hamiltonian flows which start on one Legendrian submanifold and asymptotically converge to another Legendrian submanifold. We discuss a mathematical model of non-equilibrium thermodynamics where such trajectories play a role of relaxation processes, and illustrate our results in the case of the Glauber dynamics for the mean field Ising model.

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You, Bo, Zhi Li, Liang Ding, Haibo Gao et Jiazhong Xu. « A new optimization-driven path planning method with probabilistic completeness for wheeled mobile robots ». Measurement and Control 52, n^o 5-6 (15 avril 2019) : 317–25. http://dx.doi.org/10.1177/0020294019836127.

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Wheeled mobile robots are widely utilized for environment-exploring tasks both on earth and in space. As a basis for global path planning tasks for wheeled mobile robots, in this study we propose a method for establishing an energy-based cost map. Then, we utilize an improved dual covariant Hamiltonian optimization for motion planning method, to perform point-to-region path planning in energy-based maps. The method is capable of efficiently handling high-dimensional path planning tasks with non-convex cost functions through applying a robust active set algorithm, that is, non-monotone gradient projection algorithm. To solve the problem that the path planning process is locked in weak minima or non-convergence, we propose a randomized variant of the improved dual covariant Hamiltonian optimization for motion planning based on simulated annealing and Hamiltonian Monte Carlo methods. The results of simulations demonstrate that the final paths generated can be time efficient, energy efficient and smooth. And the probabilistic completeness of the method is guaranteed.

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Radjenovic, B., M. Radmilovic-Radjenovic et M. Mitric. « Application of the level set method on the non-convex Hamiltonians ». Facta universitatis - series : Physics, Chemistry and Technology 7, n^o 1 (2009) : 33–44. http://dx.doi.org/10.2298/fupct0901033r.

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Application of the level set method extended for the case of non-convex Hamiltonians is illustrated by the three dimensional (3D) simulation results of the profile evolution during anisotropic wet etching of silicon. Etching rate function is modeled on the basis of the silicon symmetry properties, by means of the interpolation technique using experimentally obtained values of the principal [100], [110], [111], and high index [311] directions in KOH solutions. The resulting level set equations are solved using an open source implementation of the sparse field method.

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Giuliani, Filippo. « Transfers of energy through fast diffusion channels in some resonant PDEs on the circle ». Discrete & ; Continuous Dynamical Systems 41, n^o 11 (2021) : 5057. http://dx.doi.org/10.3934/dcds.2021068.

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In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of fast diffusion channels along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions, which in turn is responsible for the growth of higher order Sobolev norms.

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Bardi, Martino, et Markus Fischer. « On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games ». ESAIM : Control, Optimisation and Calculus of Variations 25 (2019) : 44. http://dx.doi.org/10.1051/cocv/2018026.

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This paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons T and convex but non-smooth Hamiltonian H, as well as for smooth H and T large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small T, is proved. Some results are also extended to MFGs with two populations.

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Cordaro, Giuseppe. « Existence and location of periodic solutions to convex and non coercive Hamiltonian systems ». Discrete & ; Continuous Dynamical Systems - A 12, n^o 5 (2005) : 983–96. http://dx.doi.org/10.3934/dcds.2005.12.983.

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Grotta-Ragazzo, C., et Pedro A. S. Salomão. « Global surfaces of section in non-regular convex energy levels of Hamiltonian systems ». Mathematische Zeitschrift 255, n^o 2 (22 août 2006) : 323–34. http://dx.doi.org/10.1007/s00209-006-0026-y.

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Corsi, Livia, Roberto Feola et Guido Gentile. « Lower-Dimensional Invariant Tori for Perturbations of a Class of Non-convex Hamiltonian Functions ». Journal of Statistical Physics 150, n^o 1 (janvier 2013) : 156–80. http://dx.doi.org/10.1007/s10955-012-0682-8.

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KRITCHEVSKI, E., et S. STARR. « THE EXTENDED VARIATIONAL PRINCIPLE FOR MEAN-FIELD, CLASSICAL SPIN SYSTEMS ». Reviews in Mathematical Physics 17, n^o 10 (novembre 2005) : 1209–39. http://dx.doi.org/10.1142/s0129055x05002510.

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The purpose of this article is to obtain a better understanding of the extended variational principle (EVP). The EVP is a formula for the thermodynamic pressure of a statistical mechanical system as a limit of a sequence of minimization problems. It was developed for disordered mean-field spin systems, spin systems where the underlying Hamiltonian is itself random, and whose distribution is permutation invariant. We present the EVP in the simpler setting of classical mean-field spin systems, where the Hamiltonian is non-random and symmetric. The EVP essentially solves these models. We compare the EVP with another method for mean-field spin systems: the self-consistent mean-field equations. The two approaches lead to dual convex optimization problems. This is a new connection, and it permits a generalization of the EVP.

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Mongwe, Wilson Tsakane, Rendani Mbuvha et Tshilidzi Marwala. « Quantum-Inspired Magnetic Hamiltonian Monte Carlo ». PLOS ONE 16, n^o 10 (5 octobre 2021) : e0258277. http://dx.doi.org/10.1371/journal.pone.0258277.

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Hamiltonian Monte Carlo (HMC) is a Markov Chain Monte Carlo algorithm that is able to generate distant proposals via the use of Hamiltonian dynamics, which are able to incorporate first-order gradient information about the target posterior. This has driven its rise in popularity in the machine learning community in recent times. It has been shown that making use of the energy-time uncertainty relation from quantum mechanics, one can devise an extension to HMC by allowing the mass matrix to be random with a probability distribution instead of a fixed mass. Furthermore, Magnetic Hamiltonian Monte Carlo (MHMC) has been recently proposed as an extension to HMC and adds a magnetic field to HMC which results in non-canonical dynamics associated with the movement of a particle under a magnetic field. In this work, we utilise the non-canonical dynamics of MHMC while allowing the mass matrix to be random to create the Quantum-Inspired Magnetic Hamiltonian Monte Carlo (QIMHMC) algorithm, which is shown to converge to the correct steady state distribution. Empirical results on a broad class of target posterior distributions show that the proposed method produces better sampling performance than HMC, MHMC and HMC with a random mass matrix.

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Khanin, Konstantin, et Andrei Sobolevski. « Particle dynamics inside shocks in Hamilton–Jacobi equations ». Philosophical Transactions of the Royal Society A : Mathematical, Physical and Engineering Sciences 368, n^o 1916 (13 avril 2010) : 1579–93. http://dx.doi.org/10.1098/rsta.2009.0283.

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The characteristic curves of a Hamilton–Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth ‘viscosity’ solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the ‘dissipative anomaly’ in the limit of vanishing viscosity.

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Ennaji, Hamza, Noureddine Igbida et Van Thanh Nguyen. « Beckmann-type problem for degenerate Hamilton-Jacobi equations ». Quarterly of Applied Mathematics 80, n^o 2 (21 décembre 2021) : 201–20. http://dx.doi.org/10.1090/qam/1606.

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The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation H ( x , ∇ u ) = 0 , H(x,\nabla u)=0, coupled with non-zero Dirichlet condition u = g u=g on ∂ Ω \partial \Omega . In this article, the Hamiltonian H H is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.

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Amick, C. J., et J. F. Toland. « Points of egress in problems of Hamiltonian dynamics ». Mathematical Proceedings of the Cambridge Philosophical Society 109, n^o 2 (mars 1991) : 405–17. http://dx.doi.org/10.1017/s030500410006984x.

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First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equationHere the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.

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Chen, Qinbo, et Rafael de la Llave. « Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems ». Nonlinearity 35, n^o 4 (9 mars 2022) : 1986–2019. http://dx.doi.org/10.1088/1361-6544/ac50bb.

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Abstract The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation H ε ( p , q , I , φ , t ) = h ( I ) + ∑ i = 1 n ± 1 2 p i 2 + V i ( q i ) + ε H 1 ( p , q , I , φ , t ) , where ( p , q ) ∈ R n × T n , ( I , φ ) ∈ R d × T d with n, d ⩾ 1, V i are Morse potentials, and ɛ is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations H 1. Indeed, the set of admissible H 1 is C ω dense and C 3 open (a fortiori, C ω open). Our perturbative technique for the genericity is valid in the C k topology for all k ∈ [3, ∞) ∪ {∞, ω}.

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Bounemoura, Abed, et Vadim Kaloshin. « Generic Fast Diffusion for a Class of Non-Convex Hamiltonians with Two Degrees of Freedom ». Moscow Mathematical Journal 14, n^o 2 (2014) : 181–203. http://dx.doi.org/10.17323/1609-4514-2014-14-2-181-203.

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Radjenović, Branislav, Jae Koo Lee et Marija Radmilović-Radjenović. « Sparse field level set method for non-convex Hamiltonians in 3D plasma etching profile simulations ». Computer Physics Communications 174, n^o 2 (janvier 2006) : 127–32. http://dx.doi.org/10.1016/j.cpc.2005.09.010.

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Mahmudov, Elimhan. « Optimization of Lagrange problem with higher order differential inclusions and endpoint constraints ». Filomat 32, n^o 7 (2018) : 2367–82. http://dx.doi.org/10.2298/fil1807367m.

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In the paper minimization of a Lagrange type cost functional over the feasible set of solutions of higher order differential inclusions with endpoint constraints is studied. Our aim is to derive sufficient conditions of optimality for m th-order convex and non-convex differential inclusions. The sufficient conditions of optimality containing the Euler-Lagrange and Hamiltonian type inclusions as a result of endpoint constraints are accompanied by so-called ?endpoint? conditions. Here the basic apparatus of locally adjoint mappings is suggested. An application from the calculus of variations is presented and the corresponding Euler-Poisson equation is derived. Moreover, some higher order linear optimal control problems with quadratic cost functional are considered and the corresponding Weierstrass-Pontryagin maximum principle is constructed. Also at the end of the paper some characteristic features of the obtained result are illustrated by example with second order linear differential inclusions.

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De Blasi, Irene, Alessandra Celletti et Christos Efthymiopoulos. « Satellites’ orbital stability through normal forms ». Proceedings of the International Astronomical Union 15, S364 (octobre 2021) : 146–51. http://dx.doi.org/10.1017/s174392132100137x.

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AbstractA powerful tool to investigate the stability of the orbits of natural and artificial bodies is represented by perturbation theory, which allows one to provide normal form estimates for nearly-integrable problems in Celestial Mechanics. In particular, we consider the orbital stability of point-mass satellites moving around the Earth. On the basis of the J2 model, we investigate the stability of the semimajor axis. Using a secular Hamiltonian model including also lunisolar perturbations, the so-called geolunisolar model, we study the stability of the other orbital elements, namely the eccentricity and the inclination. We finally discuss the applicability of Nekhoroshev’s theorem on the exponential stability of the action variables. To this end, we investigate the non-degeneracy properties of the J2 and geolunisolar models. We obtain that the J2 model satisfies a “three-jet” non-degeneracy condition, while the geolunisolar model is quasi-convex non-degenerate.

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NETO, JORGE ANANIAS, et WILSON OLIVEIRA. « DOES THE WEYL ORDERING PRESCRIPTION LEAD TO THE CORRECT ENERGY LEVELS FOR THE QUANTUM PARTICLE ON THE D-DIMENSIONAL SPHERE ? » International Journal of Modern Physics A 14, n^o 23 (20 septembre 1999) : 3699–713. http://dx.doi.org/10.1142/s0217751x99001706.

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The energy eigenvalues of the quantum particle constrained in a surface of the sphere of D dimensions embedded in a RD+1 space are obtained by using two different procedures: in the first, we derive the Hamiltonian operator by squaring the expression of the momentum, written in Cartesian components, which satisfies the Dirac brackets between the canonical operators of this second-class system. We use the Weyl ordering prescription to construct the Hermitian operators. When D=2 we verify that there is no constant parameter in the expression of the eigenvalues energy, a result that is in agreement with the fact that an extra term would change the level spacings in the hydrogen atom; in the second procedure it is adopted the non-Abelian BFFT formalism to convert the second-class constraints into first-class ones. The non-Abelian first-class Hamiltonian operator is symmetrized by also using the Weyl ordering rule. We observe that their energy eigenvalues differ from a constant parameter when we compare with the second-class system. Thus, a conversion of the D-dimensional sphere second-class system for a first-class one does not reproduce the same values.

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Stark, Colin P., et Gavin J. Stark. « The direction of landscape erosion ». Earth Surface Dynamics 10, n^o 3 (3 mai 2022) : 383–419. http://dx.doi.org/10.5194/esurf-10-383-2022.

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Abstract. The rate of erosion of a landscape depends largely on local gradient and material fluxes. Since both quantities are functions of the shape of the catchment surface, this dependence constitutes a mathematical straitjacket, in the sense that – subject to simplifying assumptions about the erosion process, and absent variations in external forcing and erodibility – the rate of change of surface geometry is solely a function of surface geometry. Here we demonstrate how to use this geometric self-constraint to convert a gradient-dependent erosion model into its equivalent Hamiltonian, and explore the implications of having a Hamiltonian description of the erosion process. To achieve this conversion, we recognize that the rate of erosion defines the velocity of surface motion in its orthogonal direction, and we express this rate in its reciprocal form as the surface-normal slowness. By rewriting surface tilt in terms of normal slowness components and deploying a substitution developed in geometric mechanics, we extract what is known as the fundamental metric function of the model phase space; its square is the Hamiltonian. Such a Hamiltonian provides several new ways to solve for the evolution of an erosion surface: here we use it to derive Hamilton's ray-tracing equations, which describe both the velocity of a surface point and the rate of change of the surface-normal slowness at that point. In this context, gradient-dependent erosion involves two distinct directions: (i) the surface-normal direction, which points subvertically downwards, and (ii) the erosion ray direction, which points upstream at a generally small angle to horizontal with a sign controlled by the scaling of erosion with slope. If the model erosion rate scales faster than linearly with gradient, the rays point obliquely upwards, but if erosion scales sublinearly with gradient, the rays point obliquely downwards. This dependence of erosional anisotropy on gradient scaling explains why, as previous studies have shown, model knickpoints behave in two distinct ways depending on the gradient exponent. Analysis of the Hamiltonian shows that the erosion rays carry boundary-condition information upstream, and that they are geodesics, meaning that surface evolution takes the path of least erosion time. Correspondingly, the time it takes for external changes to propagate into and change a landscape is set by the velocity of these rays. The Hamiltonian also reveals that gradient-dependent erosion surfaces have a critical tilt, given by a simple function of the gradient scaling exponent, at which ray-propagation behaviour changes. Channel profiles generated from the non-dimensionalized Hamiltonian have a shape entirely determined by the scaling exponents and by a dimensionless erosion rate expressed as the surface tilt at the downstream boundary.

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Rouleux, Michel. « Semi-classical Integrability, Hyperbolic Flows and the Birkhoff Normal Form ». Canadian Journal of Mathematics 56, n^o 5 (1 octobre 2004) : 1034–67. http://dx.doi.org/10.4153/cjm-2004-047-6.

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AbstractWe prove that a Hamiltonianp∈C∞(T*Rn) is locally integrable near a non-degenerate critical point ρ0of the energy, provided that the fundamental matrix at ρ0has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in theC∞sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that whenpis holomorphic near ρ0∈T*Cn, then Repbecomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge,i.e., pmay not be integrable. These normal forms also hold in the semi-classical frame.

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Sreedharan Kallyadan, Sreethin, et Priyanka Shukla. « Dynamical aspects of a restricted three-vortex problem ». IMA Journal of Applied Mathematics 87, n^o 1 (28 octobre 2021) : 1–19. http://dx.doi.org/10.1093/imamat/hxab043.

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Abstract Point vortex systems that include vortices with constant coordinate functions are largely unexplored, even though they have reasonable physical interpretations in the geophysical context. Here, we investigate the dynamical aspects of the restricted three-vortex problem when one of the point vortices is assumed to be fixed at a location in the plane. The motion of the passive tracer is explored from a rotating frame of reference within which the free vortex with non-zero circulation remains stationary. By using basic dynamical system theory, it is shown that the vortex motion is always bounded, and any configuration of the three vortices must go through at least one collinear state. The present analysis reveals that any non-relative equilibrium solution of the vortex system either has periodic inter-vortex distances or it will asymptotically converge to a relative equilibrium configuration. The initial conditions required for different types of motion are explained in detail by exploiting the Hamiltonian structure of the problem. The underlying effects of a fixed vortex on the motion of vortices are also explored.

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Jezequel, Lucien, Clément Tauber et Pierre Delplace. « Estimating bulk and edge topological indices in finite open chiral chains ». Journal of Mathematical Physics 63, n^o 12 (1 décembre 2022) : 121901. http://dx.doi.org/10.1063/5.0096720.

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We develop a formalism to estimate, simultaneously, the usual bulk and edge indices from topological insulators in the case of a finite sample with open boundary conditions and provide a physical interpretation of these quantities. We then show that they converge exponentially fast to an integer value when we increase the system size and also show that bulk and edge index estimates coincide at finite size. The theorem applies to any non-homogeneous system, such as disordered or defect configurations. We focus on one-dimensional chains with chiral symmetry, such as the Su–Schrieffer–Heeger model, but the proof actually only requires the Hamiltonian to be of short range and with a spectral gap in the bulk. The definition of bulk and edge index estimates relies on a finite-size version of the switch-function formalism where the Fermi projector is smoothed in energy using a carefully chosen regularization parameter.

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Dunn, Katherine, Martin Trefzer, Steven Johnson et Andy Tyrrell. « Towards a Bioelectronic Computer : A Theoretical Study of a Multi-Layer Biomolecular Computing System That Can Process Electronic Inputs ». International Journal of Molecular Sciences 19, n^o 9 (4 septembre 2018) : 2620. http://dx.doi.org/10.3390/ijms19092620.

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DNA molecular machines have great potential for use in computing systems. Since Adleman originally introduced the concept of DNA computing through his use of DNA strands to solve a Hamiltonian path problem, a range of DNA-based computing elements have been developed, including logic gates, neural networks, finite state machines (FSMs) and non-deterministic universal Turing machines. DNA molecular machines can be controlled using electrical signals and the state of DNA nanodevices can be measured using electrochemical means. However, to the best of our knowledge there has as yet been no demonstration of a fully integrated biomolecular computing system that has multiple levels of information processing capacity, can accept electronic inputs and is capable of independent operation. Here we address the question of how such a system could work. We present simulation results showing that such an integrated hybrid system could convert electrical impulses into biomolecular signals, perform logical operations and take a decision, storing its history. We also illustrate theoretically how the system might be able to control an autonomous robot navigating through a maze. Our results suggest that a system of the proposed type is technically possible but for practical applications significant advances would be required to increase its speed.

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Nooijen, Marcel, et K. R. Shamasundar. « A Case Study of State-Specific and State-Averaged Brueckner Equation-of-Motion Coupled-Cluster Theory : The Ionic-Covalent Avoided Crossing in Lithium Fluoride ». Collection of Czechoslovak Chemical Communications 70, n^o 8 (2005) : 1082–108. http://dx.doi.org/10.1135/cccc20051082.

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State-specific Brueckner equation-of-motion coupled-cluster theory (SS-B-EOMCC) is summarized, which can be considered an internally contracted version of a state-selective multireference coupled-cluster theory, which, however, is not entirely size-consistent. The method is applicable to general multireference problems, adheres to the space and spin symmetries of the molecular system, is straightforwardly extended to a state-averaged version, and has an associated perturbative variant which yields results close to the full coupled-cluster treatment. A key strength is that Brueckner orbitals are used, such that orbitals are optimized in the presence of dynamic correlation. A number of variations on the theme of SS-EOMCC is applied to study the ionic-covalent avoided crossing in LiF in a 6-311++G(3df,3pd) basis set. While reasonable results are obtained at the state-averaged level, the iterative solution process does not consistently converge for SS-EOMCC, due to the non-Hermiticity of the transformed Hamiltonian which may yield complex eigenvalues upon truncated diagonalization. This leads to an irrevocable breakdown of the state-specific EOMCC approach. We indicate some future directions that can resolve some of the problems with the SS-EOMCC methodology, as revealed by the demanding test case of the LiF potential energy curves.

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Roy, Rhombik, Camille Lévêque, Axel U. J. Lode, Arnaldo Gammal et Barnali Chakrabarti. « Fidelity and Entropy Production in Quench Dynamics of Interacting Bosons in an Optical Lattice ». Quantum Reports 1, n^o 2 (15 décembre 2019) : 304–16. http://dx.doi.org/10.3390/quantum1020028.

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We investigate the dynamics of a few bosons in an optical lattice induced by a quantum quench of a parameter of the many-body Hamiltonian. The evolution of the many-body wave function is obtained by solving the time-dependent many-body Schrödinger equation numerically, using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We report the time evolution of three key quantities, namely, the occupations of the natural orbitals, that is, the eigenvalues of the one-body reduced density matrix, the many-body Shannon information entropy, and the quantum fidelity for a wide range of interactions. Our key motivation is to characterize relaxation processes where various observables of an isolated and interacting quantum many-body system dynamically converge to equilibrium values via the quantum fidelity and via the production of many-body entropy. The interaction, as a parameter, can induce a phase transition in the ground state of the system from a superfluid (SF) state to a Mott-insulator (MI) state. We show that, for a quench to a weak interaction, the fidelity remains close to unity and the entropy exhibits oscillations. Whereas for a quench to strong interactions (SF to MI transition), the relaxation process is characterized by the first collapse of the quantum fidelity and entropy saturation to an equilibrium value. The dip and the non-analytic nature of quantum fidelity is a hallmark of dynamical quantum phase transitions. We quantify the characteristic time at which the quantum fidelity collapses and the entropy saturates.

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Aslani, Shahriar, et Patrick Bernard. « Normal Form Near Orbit Segments of Convex Hamiltonian Systems ». International Mathematics Research Notices, 18 janvier 2021. http://dx.doi.org/10.1093/imrn/rnaa344.

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Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [ 8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [ 5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [ 11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [ 1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [ 9, 10]. It is proved in [ 10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [ 10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [ 10,Lemma 3.1] itself taken from [ 3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [ 3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [ 3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [ 7] on Finsler metrics, as [ 3] did. Although our normal form is weaker than the one claimed in [ 10], it is actually sufficient to prove the main results of [ 6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [ 3, Lemma C.1].

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Bolten, M., O. T. Doganay, H. Gottschalk et K. Klamroth. « Non-convex shape optimization by dissipative Hamiltonian flows ». Engineering Optimization, 18 février 2024, 1–20. http://dx.doi.org/10.1080/0305215x.2024.2304135.

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Boyer, Mark A., et Edwin L. Sibert. « A general expression for vibrational Hamiltonians expressed in oblique coordinates ». Journal of Chemical Physics 159, n^o 23 (18 décembre 2023). http://dx.doi.org/10.1063/5.0181135.

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We examine the properties of oblique coordinates. The coordinates, introduced by Zúñiga et al. [J. Phys. B: At., Mol. Opt. Phys. 52, 055101, (2019)], reduce vibrational mode-mixing and enhance the quality of vibrational assignments in quantum mechanical investigations of two-dimensional model Hamiltonians. Oblique coordinates are obtained by making non-orthogonal rotations of the original coordinates that convert the matrix representation of the quadratic Hamiltonian operator into a block-diagonal matrix where the blocks are distinguished by the total quanta of vibrational excitation. Using techniques for the polar decomposition of matrices, we present a scheme for finding these coordinates for systems of arbitrary dimensions. Several molecular examples are presented that highlight the advantages of these coordinates.

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Chau, Huy N., et Miklós Rásonyi. « Stochastic Gradient Hamiltonian Monte Carlo for non-convex learning ». Stochastic Processes and their Applications, avril 2022. http://dx.doi.org/10.1016/j.spa.2022.04.001.

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Ratiu, Tudor, Christophe Wacheux et Nguyen Zung. « Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems ». Memoirs of the American Mathematical Society 287, n^o 1424 (juillet 2023). http://dx.doi.org/10.1090/memo/1424.

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This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.

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Fabila-Monroy, Ruy, David Flores-Peñaloza, Clemens Huemer, Ferran Hurtado, Jorge Urrutia et David R. Wood. « On the chromatic number of some flip graphs ». Discrete Mathematics & ; Theoretical Computer Science Vol. 11 no. 2, Graph and Algorithms (1 janvier 2009). http://dx.doi.org/10.46298/dmtcs.460.

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Graphs and Algorithms International audience This paper studies the chromatic number of the following four flip graphs (under suitable definitions of a flip): the flip graph of perfect matchings of a complete graph of even order, the flip graph of triangulations of a convex polygon (the associahedron), the flip graph of non-crossing Hamiltonian paths of a set of points in convex position, and the flip graph of triangles in a convex point set. We give tight bounds for the latter two cases and upper bounds for the first two.

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TAPIA-GARCÍA, SEBASTIÁN. « REGULARITY OF AML FUNCTIONS IN TWO-DIMENSIONAL NORMED SPACES ». Journal of the Australian Mathematical Society, 20 mai 2022, 1–25. http://dx.doi.org/10.1017/s1446788722000088.

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Abstract Savin [‘ $\mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal.3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.

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Giambò, Roberto, Fabio Giannoni et Paolo Piccione. « On the Least Action Principle – Hamiltonian Dynamics on Fixed Energy Levels in the Non-convex Case ». Advanced Nonlinear Studies 6, n^o 2 (1 janvier 2006). http://dx.doi.org/10.1515/ans-2006-0208.

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AbstractWe review the classical Principle of the Least Action in a general context where the Hamilton functionH is possibly non-convex. We show how the van Groesen [6] principle follows as a particular case where H is hyperregular and of homogeneous type. Homogeneous scalar field spacetimes in spherical symmetry are derived as an application.

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Odavić, Jovan, Tobias Haug, Gianpaolo Torre, Alioscia Hamma, Fabio Franchini et Salvatore Marco Giampaolo. « Complexity of frustration : A new source of non-local non-stabilizerness ». SciPost Physics 15, n^o 4 (3 octobre 2023). http://dx.doi.org/10.21468/scipostphys.15.4.131.

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We advance the characterization of complexity in quantum many-body systems by examining WW-states embedded in a spin chain. Such states show an amount of non-stabilizerness or “magic”, measured as the Stabilizer Rényi Entropy, that grows logarithmically with the number of qubits/spins. We focus on systems whose Hamiltonian admits a classical point with extensive degeneracy. Near these points, a Clifford circuit can convert the ground state into a WW-state, while in the rest of the phase to which the classical point belongs, it is dressed with local quantum correlations. Topological frustrated quantum spin-chains host phases with the desired phenomenology, and we show that their ground state’s Stabilizer Rényi Entropy is the sum of that of the WW-states plus an extensive local contribution. Our work reveals that WW-states/frustrated ground states display a non-local degree of complexity that can be harvested as a quantum resource and has no counterpart in GHZ states/non-frustrated systems.

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De Blasi, Irene, Alessandra Celletti et Christos Efthymiopoulos. « Semi-Analytical Estimates for the Orbital Stability of Earth’s Satellites ». Journal of Nonlinear Science 31, n^o 6 (27 septembre 2021). http://dx.doi.org/10.1007/s00332-021-09738-w.

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AbstractNormal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. (i) We demonstrate the long-term stability of the semimajor axis within the framework of the $$J_2$$ J 2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $${\mathcal {H}}_{J_2}$$ H J 2 . (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian $${\mathcal {H}}_\mathrm{gls}$$ H gls ), after a suitable reduction of the Hamiltonian to the Laplace plane. (iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $${\mathcal {H}}_{J_2}$$ H J 2 and $${\mathcal {H}}_\mathrm{gls}$$ H gls models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the $${\mathcal {H}}_{J_2}$$ H J 2 model is non-convex, but satisfies a ‘three-jet’ condition, while the $${\mathcal {H}}_\mathrm{gls}$$ H gls model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.

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Wen, Xueda, Yingfei Gu, Ashvin Vishwanath et Ruihua Fan. « Periodically, Quasi-periodically, and Randomly Driven Conformal Field Theories (II) : Furstenberg's Theorem and Exceptions to Heating Phases ». SciPost Physics 13, n^o 4 (5 octobre 2022). http://dx.doi.org/10.21468/scipostphys.13.4.082.

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In this sequel (to [Phys. Rev. Res. 3, 023044(2021)], arXiv:2006.10072), we study randomly driven (1+1)(1+1) dimensional conformal field theories (CFTs), a family of quantum many-body systems with soluble non-equilibrium quantum dynamics. The sequence of driving Hamiltonians is drawn from an independent and identically distributed random ensemble. At each driving step, the deformed Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength and therefore induces a Möbius transformation on the complex coordinates. The non-equilibrium dynamics is then determined by the corresponding sequence of Möbius transformations, from which the Lyapunov exponent \lambda_LλL is defined. We use Furstenberg’s theorem to classify the dynamical phases and show that except for a few exceptional points that do not satisfy Furstenberg’s criteria, the random drivings always lead to a heating phase with the total energy growing exponentially in the number of driving steps nn and the subsystem entanglement entropy growing linearly in nn with a slope proportional to central charge cc and the Lyapunov exponent \lambda_LλL. On the contrary, the subsystem entanglement entropy at an exceptional point could grow as \sqrt{n}n while the total energy remains to grow exponentially. In addition, we show that the distributions of the operator evolution and the energy density peaks are also useful characterizations to distinguish the heating phase from the exceptional points: the heating phase has both distributions to be continuous, while the exceptional points could support finite convex combinations of Dirac measures depending on their specific type. In the end, we compare the field theory results with the lattice model calculations for both the entanglement and energy evolution and find remarkably good agreement.

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Madouri, Fethi, Abdeldjalil Merdaci et Tarek Sbeouelji. « Collapse-revival of entanglement in a non-commutative harmonic oscillator revealed via coherent states and path integral ». Zeitschrift für Naturforschung A, 18 novembre 2022. http://dx.doi.org/10.1515/zna-2022-0160.

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Abstract We develop an approach using coherent states and path integral to investigate the dynamics of entanglement in a simple two-dimensional non-commutative harmonic oscillator. We start by employing a Bopp shift to convert the Hamiltonian describing the system into a commutative equivalent one. This allows us to construct coherent states and calculate the propagator in standard way. By deriving the explicit expression of the time-dependent coherent states and considering its connection with the number states, we provide exact results for evaluating the degree of entanglement between the ground state and any excited state through the purity function. The interesting emerging result is that, as long as the non-commutativity parameter is non-zero, our system exhibits the phenomenon of collapse and revival of entanglement.

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Aliyev, Nicat, Volker Mehrmann et Emre Mengi. « Approximation of stability radii for large-scale dissipative Hamiltonian systems ». Advances in Computational Mathematics 46, n^o 1 (février 2020). http://dx.doi.org/10.1007/s10444-020-09763-5.

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Abstract A linear time-invariant dissipative Hamiltonian (DH) system $\dot x = (J-R)Q x$ẋ=(J−R)Qx, with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625–1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + BΔCH for given matrices B, C, and another with respect to Hermitian perturbations in the form R + BΔBH,Δ = ΔH. We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake.

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Bounemoura, Abed, et Jacques Féjoz. « Hamiltonian perturbation theory for ultra-differentiable functions ». Memoirs of the American Mathematical Society 270, n^o 1319 (mars 2021). http://dx.doi.org/10.1090/memo/1319.

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Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means of hard implicit function theorems. They comprise Gevrey functions and thus, as a limiting case, analytic functions. Using majorizing series, we manage to characterize them in terms of a real sequence M M bounding the growth of derivatives. In this functional setting, we prove two fundamental results of Hamiltonian perturbation theory: the invariant torus theorem, where the invariant torus remains ultra-differentiable under the assumption that its frequency satisfies some arithmetic condition which we call BR M _M , and which generalizes the Bruno-Rüssmann condition; and Nekhoroshev’s theorem, where the stability time depends on the ultra-differentiable class of the pertubation, through the same sequence M M . Our proof uses periodic averaging, while a substitute for the analyticity width allows us to bypass analytic smoothing. We also prove converse statements on the destruction of invariant tori and on the existence of diffusing orbits with ultra-differentiable perturbations, by respectively mimicking a construction of Bessi (in the analytic category) and Marco-Sauzin (in the Gevrey non-analytic category). When the perturbation space satisfies some additional condition (we then call it matching), we manage to narrow the gap between stability hypotheses (e.g. the BR M _M condition) and instability hypotheses, thus circumbscribing the stability threshold. The formulas relating the growth M M of derivatives of the perturbation on the one hand, and the arithmetics of robust frequencies or the stability time on the other hand, bring light to the competition between stability properties of nearly integrable systems and the distance to integrability. Due to our method of proof using width of regularity as a regularizing parameter, these formulas are closer to optimal as the the regularity tends to analyticity.

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Keller, Chaya, et Yael Stein. « Blockers for Triangulations of a Convex Polygon and a Geometric Maker-Breaker Game ». Electronic Journal of Combinatorics 27, n^o 4 (16 octobre 2020). http://dx.doi.org/10.37236/7205.

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Let $G$ be a complete convex geometric graph whose vertex set $P$ forms a convex polygon $C$, and let $\mathcal{F}$ be a family of subgraphs of $G$. A blocker for $\mathcal{F}$ is a set of diagonals of $C$, of smallest possible size, that contains a common edge with every element of $\mathcal{F}$. Previous works determined the blockers for various families $\mathcal{F}$ of non-crossing subgraphs, including the families of all perfect matchings, all spanning trees, all Hamiltonian paths, etc. In this paper we present a complete characterization of the family $\mathcal{B}$ of blockers for the family $\mathcal{T}$ of triangulations of $C$. In particular, we show that $|\mathcal{B}|=F_{2n-8}$, where $F_k$ is the $k$'th element in the Fibonacci sequence and $n=|P|$. We use our characterization to obtain a tight result on a geometric Maker-Breaker game in which the board is the set of diagonals of a convex $n$-gon $C$ and Maker seeks to occupy a triangulation of $C$. We show that in the $(1:1)$ triangulation game, Maker can ensure a win within $n-3$ moves, and that in the $(1:2)$ triangulation game, Breaker can ensure a win within $n-3$ moves. In particular, the threshold bias for the game is $2$.

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