scholarly journals Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

2015 ◽  
Vol 37 (1) ◽  
pp. 1-58 ◽  
Author(s):  
JEAN FRANCOIS ARNOLDI ◽  
FRÉDÉRIC FAURE ◽  
TOBIAS WEICH
Keyword(s):  
Phase Space ◽  
Simple Model ◽  
Upper Bound ◽  
Large Time ◽  
Spectral Gap ◽  
Semiclassical Limit ◽  
Gauss Map ◽  
Fourier Component ◽  
Expanding Map ◽  
Weyl Law

We consider a simple model of an open partially expanding map. Its trapped set ${\mathcal{K}}$ in phase space is a fractal set. We first show that there is a well-defined discrete spectrum of Ruelle resonances which describes the asymptotic of correlation functions for large time and which is parametrized by the Fourier component $\unicode[STIX]{x1D708}$ in the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call ‘minimal captivity’. This hypothesis is stable under perturbations and means that the dynamics is univalued in a neighborhood of ${\mathcal{K}}$. Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit $\unicode[STIX]{x1D708}\rightarrow \infty$. Some numerical computations with the truncated Gauss map and Bowen–Series maps illustrate these results.

Spectral gap of a weighted 3-simplicial complex

2021 ◽  
pp. 2150084
Author(s):  
Khalid Hatim ◽  
Azeddine Baalal
Keyword(s):  
Lower Bound ◽  
Simplicial Complex ◽  
Upper Bound ◽  
Spectral Gap ◽  
Simplicial Complexes ◽  
Cheeger Constant ◽  
Nonzero Eigenvalue ◽  
New Framework

In this paper, we construct a new framework that’s we call the weighted [Formula: see text]-simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted [Formula: see text]-simplicial complexes.

The dynamics of language

1999 ◽  
Vol 22 (2) ◽  
pp. 284-285
Author(s):  
Peter W. Culicover ◽  
Andrzej Nowak
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Simple Model ◽  
Syntactic Structure ◽  
Neural Assemblies ◽  
Systems Perspective ◽  
Model Based ◽  
Associative Structures

To deal with syntactic structure, one needs to go beyond a simple model based on associative structures, and to adopt a dynamical systems perspective, where each phrase and sentence of a language is represented as a trajectory in a syntactic phase space. Neural assemblies could possibly be used to produce dynamics that in principle could handle syntax along these lines.

scholarly journals SHARP SPECTRAL ESTIMATES IN DOMAINS OF INFINITE VOLUME

2011 ◽  
Vol 23 (06) ◽  
pp. 615-641 ◽  
Author(s):  
LEANDER GEISINGER ◽  
TIMO WEIDL
Keyword(s):  
Phase Space ◽  
Dirichlet Boundary ◽  
Semiclassical Limit ◽  
Dirichlet Boundary Conditions ◽  
Bounded Domains ◽  
Infinite Volume ◽  
Uniform Bounds ◽  
Phase Space Volume ◽  
Spectral Estimates ◽  
Space Volume

We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume — and therefore on the volume of the domain — must fail. Here we present a method on how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit.We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schrödinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.

scholarly journals Visiting Power Laws in Cyber-Physical Networking Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Ming Li ◽  
Wei Zhao
Keyword(s):  
Differential Equations ◽  
Power Law ◽  
Upper Bound ◽  
Large Time ◽  
Power Laws ◽  
Point Of View ◽  
Fundamental Theory ◽  
Physical Systems ◽  
Time Scaling ◽  
Type Data

Cyber-physical networking systems (CPNSs) are made up of various physical systems that are heterogeneous in nature. Therefore, exploring universalities in CPNSs for either data or systems is desired in its fundamental theory. This paper is in the aspect of data, aiming at addressing that power laws may yet be a universality of data in CPNSs. The contributions of this paper are in triple folds. First, we provide a short tutorial about power laws. Then, we address the power laws related to some physical systems. Finally, we discuss that power-law-type data may be governed by stochastically differential equations of fractional order. As a side product, we present the point of view that the upper bound of data flow at large-time scaling and the small one also follows power laws.

scholarly journals Large Time-Stepping Spectral Methods for the Semiclassical Limit of the Defocusing Nonlinear Schrödinger Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-27
Author(s):  
Zongqi Liang
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Spectral Methods ◽  
Schrodinger Equation ◽  
Large Time ◽  
Semiclassical Limit ◽  
Nonlinear Schrodinger Equation ◽  
Time Step ◽  
Nonlinear Schrödinger ◽  
Time Stepping

We analyze a class of large time-stepping Fourier spectral methods for the semiclassical limit of the defocusing Nonlinear Schrödinger equation and provide highly stable methods which allow much larger time step than for a standard implicit-explicit approach. An extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Meanwhile, the first-order and second-order semi-implicit schemes are constructed and analyzed. Finally the numerical experiments are performed to demonstrate the effectiveness of the large time-stepping approaches.

scholarly journals Analysis and limitations of modified circuit-to-Hamiltonian constructions

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 94 ◽  
Author(s):  
Johannes Bausch ◽  
Elizabeth Crosson
Keyword(s):  
Upper Bound ◽  
Ground State ◽  
Spectral Gap ◽  
General Setting ◽  
Quantum Circuit ◽  
Many Body ◽  
Graph Diameter ◽  
Uniform Distributions ◽  
Labeled Graphs ◽  
Constant Output

Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining theΩ(T−2)scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits.Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tightO(T−2)upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond theΩ(T−3)scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend theO(T−2)bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.

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