Sharp Polynomial Research Articles

On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements

We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} in the transient dimensions d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least 2 connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.

Sharp polynomial decay rate of multi-link hyperbolic-parabolic systems with different network configurations

Sharp polynomial decay rate of multi-link hyperbolic-parabolic systems with different network configurations

Quantum entanglement and the growth of Laplacian eigenfunctions

We study the growth of Laplacian eigenfunctions on compact manifolds (M, g). Hörmander proved sharp polynomial bounds on which are attained on the sphere. On a “generic” manifold, the behavior seems to be different: both numerics and Berry’s random wave model suggest as the typical behavior. We propose a mechanism, centered around an analog of the spectral projector, for explaining the slow growth in the generic case: for to be large, it is necessary that either (1) several of the first n eigenfunctions were large in x 0 or (2) that is strongly correlated with a suitable linear combination of the first n eigenfunctions on most of the manifold or (3) both. An interesting byproduct is quantum entanglement for Laplacian eigenfunctions: the existence of two distinct points such that the sequences and do not behave like independent random variables. The existence of such points is not to be expected for generic manifolds but common for the classical manifolds and subtly intertwined with eigenfunction concentration.

Sharp polynomial decay for waves damped from the boundary in cylindrical waveguides

Sharp polynomial decay for waves damped from the boundary in cylindrical waveguides

Sharp statistical properties for a family of multidimensional nonMarkovian nonconformal intermittent maps

Intermittent maps of Pomeau-Manneville type are well-studied in one-dimension, and also in higher dimensions if the map happens to be Markov. In general, the nonconformality of multidimensional intermittent maps represents a challenge that up to now is only partially addressed. We show how to prove sharp polynomial bounds on decay of correlations for a class of multidimensional intermittent maps. In addition we show that the optimal results on statistical limit laws for one-dimensional intermittent maps hold also for the maps considered here. This includes the (functional) central limit theorem and local limit theorem, Berry-Esseen estimates, large deviation estimates, convergence to stable laws and Lévy processes, and infinite measure mixing.

Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards

Gouezel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.

The boundary of the range of a random walk and the Følner property

The range process Rn of a random walk is the collection of sites visited by the random walk up to time n. In this paper we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Følner sequence and show the following results: (a) The size of the inner boundary |∂Rn| of the range of recurrent aperiodic random walks on Z2 with finite variance and aperiodic random walks in Z in the standard domain of attraction of the Cauchy distribution, divided by n∕log2(n), converges to a constant almost surely. (b) We establish a formula for the Følner asymptotic of transient cocycles over an ergodic probability preserving transformation and use it to show that for admissible transient random walks on finitely generated groups, the range is never a Følner sequence unless the walk is a skip-free random walk on Z. (c) For strongly aperiodic random walks in the domain of attraction of symmetric α-stable distributions with 1<α≤2, we prove a sharp polynomial upper bound for the decay at infinity of |∂Rn|∕|Rn|. This last result shows that the range process of these random walks is almost surely a Følner sequence.

Sharp polynomial bounds for certain C0-groups generated by operators with non-basis family of eigenvectors

Sharp polynomial bounds for norms of C0-groups generated by operators with purely imaginary eigenvalues λn=iln⁡n, n∈N, and complete minimal non-basis family of eigenvectors, constructed recently by G. Sklyar and V. Marchenko in [17], are obtained. Besides, it is shown that these C0-groups do not have a maximal asymptotics. For the more general case of behaviour of the spectrum of operators we present the proof of one lemma concerning the behaviour of j-th differences of sequences of complex exponentials exp⁡(itf(n)), n∈N, that is used in [17] to obtain bounds from above for norms of corresponding C0-groups. Also the growth properties of the resolvent for generators of constructed C0-groups are discussed.

Sharp polynomial decay rates for the damped wave equation with Hölder-like damping

Sharp polynomial decay rates for the damped wave equation with Hölder-like damping

Polynomial inequalities on sets with km-concave weighted measures

Let μ be a measurewith a k-concave density W on an open convex set V in Rm, that is, W is an integrable weight satisfying the condition $$W(ax + (1 - a)y) \geqslant {(a{W^K}(x) + (1 - a){W^K}(y))^{1/k}},k \in ( - 1/m,\infty ]$$ for all x ∈ V, y ∈ V, and α ∈ [0, 1]. In this paper, we first show that the Fradelizi μ-distributional inequalities for polynomials P of m variables are sharp for each m and k ∈ (−1/m,∞]. Classes of extremal sets V, weights W, and polynomials P for these inequalities are presented. Sharpness of the Bobkov-Nazarov-Fradelizi dilation-type inequalities is established as well. Second, we find efficient conditions for k-concavity of a weight W and obtain new sharp polynomial inequalities.

Energy decay rate of transmission problem between thermoelasticity of type I and type II

In this paper, the energy decay rate of a 1-d mixed type I and type II thermoelastic system is considered. The system consists of two kinds of thermoelastic components. One is the classical thermoelasticity (so-called type I), another one is nonclassical thermoelasticity without dissipation (named type II). These two components are coupled at the interface satisfying certain transmission condition. We prove that the system is lack of uniform exponential decay rate and further obtain the sharp polynomial decay rate by resolvent estimates together with the diagonalization argument in linear algebra. Moreover, we present some numerical simulations to support these theoretical results.

Decay rates for elastic-thermoelastic star-shaped networks

This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths. First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.

Decay rates for $1-d$ heat-wave planar networks

The large time decay rates of a transmission problem coupling heat and wave equations on a planar network is discussed. &nbsp When all edges evolve according to the heat equation, the uniform exponential decay holds. By the contrary, we show the lack of uniform stability, based on a Geometric Optics high frequency asymptotic expansion, whenever the network involves at least one wave equation. &nbsp The (slow) decay rate of this system is further discussed for star-shaped networks. When only one wave equation is present in the network, by the frequency domain approach together with multipliers, we derive a sharp polynomial decay rate. When the network involves more than one wave equation, a weakened observability estimate is obtained, based on which, polynomial and logarithmic decay rates are deduced for smooth initial conditions under certain irrationality conditions on the lengths of the strings entering in the network. These decay rates are intrinsically determined by the wave equations entering in the system and are independent on the heat equations.

Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in [formula omitted

We investigate the rate of convergence of linear sampling numbers of the embedding Hα,β(Td)↪Hγ(Td). Here α governs the mixed smoothness and β the isotropic smoothness in the space Hα,β(Td) of hybrid smoothness, whereas Hγ(Td) denotes the isotropic Sobolev space. If γ>β we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on “energy-norm based sparse grids” for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and Dũng, Ullrich, where general linear approximations have been considered. In addition, we study the embedding Hmixα(Td)↪Hmixγ(Td) and achieve optimality for Smolyak’s algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding Hmixα(Td)↪Lq(Td) for 2<q≤∞ where again Smolyak’s algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation β=γ (including the embedding into L2(Td)).

On the sharp polynomial upper estimate conjecture in eight-dimensional simplex

On the sharp polynomial upper estimate conjecture in eight-dimensional simplex

Sharp polynomial energy decay for locally undamped waves

In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.

A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers

Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture (see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional (n ≥ 3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau Number Theoretic Conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5 ≤ y < 17, compared with the result obtained by Ennola.

A sharp polynomial estimate of positive integral points in a 4-dimensional tetrahedron and a sharp estimate of the Dickman-de Bruijn function

The estimate of integral points in right-angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional () real right-angled simplices with vertices whose distance to the origin are at least . A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate the Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. We have proved this conjecture for . This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of the Dickman-de Bruijn function for .

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function \psi(x,y) which is the number of positive integers \leq x and free of prime factors &gt;y . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( n\geq3 ) real right-angled simplices. In this paper, we prove this Number Theoretic Conjecture for n=5 . As an application, we give a sharp estimate of Dickman-De Bruijn function \psi(x,y) for 5\leq y&lt;13 .

Lengthening a tetrahedron

We give rigorous, computer assisted proofs of a number of statements about the effect on the volume of lengthening various edges of a tetrahedron. Our results give new and sharp polynomial inequalities concerning the Cayley–Menger determinant and its partial derivatives.