Bounded Functions Research Articles

Entropic Fluctuations in Gaussian Dynamical Systems

We study non-equilibrium statistical mechanics of a Gaussian dynamical system and compute in closed form the large deviation functionals describing the fluctuations of the entropy production observable with respect to the reference state and the non-equilibrium steady state. The entropy production observable of this model is an unbounded function on the phase space, and its large deviation functionals have a surprisingly rich structure. We explore this structure in some detail.

Generalized sinc-Gaussian sampling involving derivatives

The generalized sampling expansion which uses samples from a bandlimited function f and its first r derivatives was first introduced by Linden and Abramson (Inform. Contr. 3, 26–31, 1960) and it was extended in different situations by some authors through the last fifty years. The use of the generalized sampling series in approximation theory is limited because of the slow convergence. In this paper, we derive a modification of a generalized sampling involving derivatives, which is studied by Shin (Commun. Korean Math. Soc. 17, 731–740, 2002), using a Gaussian multiplier. This modification is introduced for wider classes, the class of entire functions including unbounded functions on ℝ and the class of analytic functions in a strip. It highly improves the convergence rate of the generalized sampling which will be of exponential order. We will show that many known results included in Sampl. Theory Signal Image Process. 9, 199–221 (2007) and Numer. Funct. Anal. Optim. 36, 419–437 (2015) are special cases of our results. Numerical examples show a rightly good agreement with our theoretical analysis.

Pointwise equidistribution with an error rate and with respect to unbounded functions

Consider \(G=\mathrm{SL}_{ d }({\mathbb {R}})\) and \( \Gamma =\mathrm{SL}_{ d }({\mathbb {Z}})\). It was recently shown by the second-named author (Shi, Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space (preprint), arXiv:1405.2067, 2014) that for some diagonal subgroups \(\{g_t\}{\subset } G\) and unipotent subgroups \(U{\subset } G\), \(g_t\)-trajectories of almost all points on all U-orbits on \(G/\Gamma \) are equidistributed with respect to continuous compactly supported functions \(\varphi \) on \(G/\Gamma \). In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when \(\varphi \) is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on \(\mathbb {R}^d\). For the first part we use a method based on effective double equidistribution of \(g_t\)-translates of U-orbits, which generalizes the main result of Kleinbock and Margulis (On effective equidistribution of expanding translates of certain orbits in the space of lattices, Number theory, analysis and geometry 385–396, 2012). The second part is based on Schmidt’s results on counting of lattice points. Number-theoretic consequences involving spiraling of lattice approximations, extending recent work of Athreya et al. (J Lond Math Soc 91(2):383–404, 2015), are derived using the equidistribution result.

Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth

We consider the well-travelled problem of homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has $p$-growth from below (with $p>d$, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space-variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals.

Viscosity Solutions of Hybrid Game Problems with Unbounded Cost Functionals

This paper analyzes zero sum game involving hybrid controls using viscosity solution theory where both players use discrete as well as continuous controls. We study two problems, one in finite horizon and other in infinite horizon. In both cases, we allow the cost functionals to be unbounded with certain growth, hence the corresponding lower and upper value functions defined in Elliot–Kalton sense can be unbounded. We characterize the value functions as the unique viscosity solution of the associated lower and upper quasi variational inequalities in a suitable function class. Further we find a condition under which the game has a value for both games. The major difficulties arise due to unboundedness of value function. In infinite horizon case we prove uniqueness of viscosity solution by converting the unbounded value function into bounded ones by suitable transformation. In finite horizon case an argument is based on comparison with a supersolution.

Almost sure central limit theorem for products of sums of partial sums

Considering a sequence of i.i.d. positive random variables, for products of sums of partial sums we establish an almost sure central limit theorem, which holds for some class of unbounded measurable functions.

Triple variational principles for self-adjoint operator functions

For a very general class of unbounded self-adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum. These upper bounds are given by triple variations. Furthermore, we find conditions which imply that a point is in the resolvent set. For norm resolvent continuous operator functions we show that the variational inequality becomes an equality.

Low Model Analysis and Synchronous Simulation of the Wave Mechanics

The dynamic behavior of a chaotic system in the internal wave dynamics and the problem of the tracing and synchronization are investigated, and the numerical simulation is carried out in this paper. The globally exponentially attractive set and positive invariant set of the chaotic system are studied via constructing the positive definite and radial unbounded Lyapunov function. There are no equilibrium positions, periodic solutions, quasi-period motions, wandering recovering motions, and other chaotic attractors of the system out of the globally exponentially attractive set. Strange attractors can only locate in the globally exponentially attractive set. A feedback controller is designed for the chaotic system to realize the control of the unstable point. The second method of Lyapunov is used to discuss theoretically the rationality of the design of the controller. The driving-response synchronization method is used to realize the globally exponential synchronization. The numerical simulation is carried out by MATLAB software, and the simulation results show that the method is effective.

Model spaces and Toeplitz kernels in reflexive Hardy space

This paper considers model spaces in an Hp setting. The existence of unbounded functions and the characterisation of maximal functions in a model space are studied, and decomposition results for Toeplitz kernels, in terms of model spaces, are established.

On the equivalence between global recurrence and the existence of a smooth Lyapunov function for hybrid systems

We study a weak stability property called recurrence for a class of hybrid systems. An open set is recurrent if there are no finite escape times and every complete trajectory eventually reaches the set. Under sufficient regularity properties for the hybrid system we establish that the existence of a smooth, radially unbounded Lyapunov function that decreases along solutions outside an open, bounded set is a necessary and sufficient condition for recurrence of that set. Recurrence of open, bounded sets is robust to sufficiently small state dependent perturbations and this robustness property is crucial for establishing the existence of a Lyapunov function that is smooth. We also highlight some connections between recurrence and other well studied properties like asymptotic stability and ultimate boundedness.

Neural network with unbounded activation functions is universal approximator

This paper presents an investigation of the approximation property of neural networks with unbounded activation functions, such as the rectified linear unit (ReLU), which is the new de-facto standard of deep learning. The ReLU network can be analyzed by the ridgelet transform with respect to Lizorkin distributions. By showing three reconstruction formulas by using the Fourier slice theorem, the Radon transform, and Parseval's relation, it is shown that a neural network with unbounded activation functions still satisfies the universal approximation property. As an additional consequence, the ridgelet transform, or the backprojection filter in the Radon domain, is what the network learns after backpropagation. Subject to a constructive admissibility condition, the trained network can be obtained by simply discretizing the ridgelet transform, without backpropagation. Numerical examples not only support the consistency of the admissibility condition but also imply that some non-admissible cases result in low-pass filtering.

An inhomogeneous nonlocal diffusion problem with unbounded steps

We consider the following nonlocal equation $$\int J\left(\frac{x-y}{g(y)} \right) \frac{u(y)}{g(y)} dy -u(x)=0\qquad x\in \mathbb{R},$$ where J is an even, compactly supported, Holder continuous kernel with unit integral and g is a continuous positive function. Our main concern will be with unbounded functions g, contrary to previous works. More precisely, we study the influence of the growth of g at infinity on the integrability of positive solutions of this equation, therefore determining the asymptotic behavior as \({t\to +\infty}\) of the solutions to the associated evolution problem in terms of the growth of g.

Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions

m-subharmonic functions are the right class of admissible solutions to the complex Hessian equation. In this paper, we generalize the definition of the complex Hessian operator to some unbounded m-subharmonic functions, and we prove that the complex Hessian operator is continuous on the monotonically decreasing sequences of m-subharmonic functions. Moreover we establish the Lelong-Jensen type formula and introduce the Lelong number for m-subharmonic functions. A useful inequality for the mixed Hessian operator is showed.

The Nehari manifold for a quasilinear polyharmonic equation with exponential nonlinearities and a sign-changing weight function

AbstractIn this article, we consider the following quasilinear polyharmonic equation: Δn/mmu = λh(x)|u|q-1u + u|u|pe|u|β in Ω, u = ∇u = ⋯ = ∇m-1u = 0 on ∂Ω, where Ω ⊂ ℝn, n ≥ 2m ≥ 2, is a bounded domain with smooth boundary. The real-valued function h is a sign-changing and unbounded function. The exponents p, q and β satisfy 0 < q < n/(m-1) < p+1, β ∈ (1,n/(n-m)] and λ > 0. Using the Nehari manifold and fibering maps, we show the existence and multiplicity of solutions.

On the scalar curvature for the noncommutative four torus

The scalar curvature for noncommutative four tori TΘ4, where their flat geometries are conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p∈C∞(TΘ4) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by Connes and Moscovici, J. Am. Math. Soc. 27(3), 639-684 (2014), unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated.

On two-dimensional classical and Hermite sampling

We investigate some modifications of the two-dimensional sampling series with a Gaussian function for wider classes of bandlimited functions, including unbounded entire functions on |$\mathbb {R}^{2}$| and analytic functions on a bivariate strip. The first modification is given for the two-dimensional version of the Whittaker–Kotelnikov–Shannon sampling (classical sampling), and the second is given for two-dimensional sampling involving values of all partial derivatives of order |$\alpha \leq 2$| (Hermite sampling). These modifications improve the convergence rate of classical and Hermite sampling which will be of exponential type. Numerical examples are given to illustrate the advantages of the new method.

A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

This paper presents a parameter-free perfectly matched layer (PML) method for the finite-element-based solution of the Helmholtz equation. We employ one of Bermúdez et al.'s unbounded absorbing functions for the complex coordinate mapping underlying the PML. With this choice, the only free parameter that controls the accuracy of the numerical solution for a fixed numerical cost (characterised by the number of elements in the bulk and the PML regions) is the thickness of the perfectly matched layer, δPML. We show that, for the case of planar waves, the absorbing function performs best for PMLs whose thickness is much smaller than the wavelength. We then perform extensive numerical experiments to explore its performance for non-planar waves, considering domain shapes with smooth and polygonal boundaries, different solution types (smooth and singular), and a wide range of wavenumbers, k, to identify an optimal range for the normalised PML thickness, kδPML, such that, within this range, the error introduced by the presence of the PML is consistently small and insensitive to change. This implies that if the PML thickness is chosen from within this range no further PML optimisation is required, i.e. the method is parameter-free. We characterise the dependence of the error on the discretisation parameters and establish the conditions under which the convergence of the solution under mesh refinement is controlled exclusively by the discretisation of the bulk mesh.

Global Lagrange stability for neutral-type Cohen–Grossberg BAM neural networks with mixed time-varying delays

The paper discusses global Lagrange stability for a class of Cohen–Grossberg BAM neural networks of neutral-type with multiple time-varying and finite distributed delays by constructing appropriate Lyapunov-like functions. To this end, we first establish a new differential integral inequality for non-autonomous Cohen–Grossberg BAM neural networks with finite distributed delays. By using the new inequality and employing some other inequality techniques, we analyze two different types of activation functions which include both bounded and unbounded activation functions. Some easily verifiable criteria are obtained for global exponential attractive sets in which all trajectories converge. These results can also be applied to analyze monostable as well as multistable and more extensive neural networks due to making no assumptions on the number of equilibria. Meanwhile, the results obtained in this paper are more general and challenging than that of the existing references. Finally, some examples with numerical simulations are given and analyzed to verify our results.

On an Exponential Functional Inequality and its Distributional Version

AbstractLet G be a group and 𝕂 = ℂ or ℝ. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions f : G → 𝕂 satisfying the inequalityWhere ϕ: Gn-1 → [0,∞]. Also as a a distributional version of the above inequality we consider the stability of the functional equationwhere u is a Schwartz distribution or Gelfand hyperfunction, o and ⊗ are the pullback and tensor product of distributions, respectively, and S(x1, ..., xn) = x1 + · · · + xn.

Bequest games with unbounded utility functions

The existence of Markov perfect equilibria for a deterministic bequest game with bounded from below utility function was established independently by Bernheim and Ray [2] and Leininger [8]. The aim of this paper is to provide a simpler proof that, in contrast to [2,8], also works for models with unbounded, e.g., isoelastic utility functions. Our approach employs the concept of strict single crossing property, introduced by Milgrom and Shannon in [10], and includes supermodular functions as special cases.