The logarithmic Sobolev inequality is a spectral bound which provides much information about decay to equilibrium of the dynamics of a stochastic process. Consider a process governed by reversible dynamics described by a generator L, with semi-group Pt and an invariant measure μ. The Dirichlet form is defined as Dμ(f) = μ[f(−L)f ]. A logarithmic Sobolev inequality is a statement which says that...

A memoryless state feedback control law is developed such that the closed-loop system containing point and distributed time-delays is asymptotically stable and the H∞-norm of the transfer function from the disturbance to the controlled output is reduced to some predefined level. The feedback is obtained by solving a parameter-dependent linear matrix inequality. When the state variables are not ...

where capG(S, S̄) is the total weight of the edges crossing from S to S̄ = V − S. We show that the minimum generalized eigenvalue λ(LG, LH) of the pair of Laplacians LG and LH satisfies λ(LG, LH) ≥ φ(G,H)φ(G)/8, where φ(G) is the usual conductance of G. A generalized cut that meets this bound can be obtained from the generalized eigenvector corresponding to λ(LG, LH). The inequality complements a...

This paper is concerned with the H∞ state feedback control problem for a class of nonlinear stochastic systems with time-varying state delays. Firstly, the stochastic dissipativity of the time-delay nonlinear system is established, based on which the H∞ state feedback controller is synthesized. We show that the closed-loop system rendered by the designed controller can achieve the L2-gain perfo...

The diamagnetic inequality is established for the Schrödinger operator H 0 in L (R), d = 2, 3, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in R, e.g., a lattice. This fact is used to prove the Lieb-Thirring inequality as well as CLR-type eigenvalue estimates for the perturbed Schrödinge...

Refining some results of S. S. Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or complex inner product space (H; 〈., .〉), r, s > 0, p ∈ (0, s],D = {x ∈ H, ‖rx− sa‖ ≤ p}, x1, x2 ∈ D − {0} and αr,s ...

Let γ2 be a standard Gaussian measure in R. For a given measurable set Q in R, let HQ be a half space in R such that γ2(Q) = γ2(HQ). The classical Gaussian concentration inequality states that for all measurable sets Q ⊂ R and for all h > 0, γ2(Q+ hB n 2 ) ≥ γ2(HQ + hB 2 ). Under some minor constrains on the set Q we obtain an improvement of the latter inequality in a certain range of h, depend...

We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u : R2 ⊃ Ω → R2 belonging to Orlicz-Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form ∫ Ω h ( |εD(u)| ) dx occurring in General Relativity a...

The aim of this note is to connect a reversed form of the Gross logarithmic Sobolev inequality with the Gaussian maximum of Shannon’s entropy power. There is thus a complete parallel with the well-known link between logarithmic Sobolev inequalities and their information theoretic counterparts. We moreover provide an elementary proof of the reversed Gross inequality via a two-point inequality an...

We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases. Introduction We will work on the space R with its usual Euclidean structure. We will denote by 〈, 〉 the canonical scalar product. In [BL], H. J. Brascamp and E. H. Lieb showed that for m ≥ n, p1...