Related papers: Some New Bounds on the Entropy Numbers of Diagonal…

We show that every operator on $L^{p}$, $1<p<\infty$ defined by multiplication by the identity function on $\mathbb{C}$ is a compact perturbation of an operator that is diagonal with respect to an unconditional basis. We also classify these…

This paper is a follow-up contribution to our work [10] where we studied some spectral properties of the differential operator $D$ acting between generalized Fock spaces $\mathcal{F}_{(m,p)}$ and $\mathcal{F}_{(m,q)}$ when both exponents…

The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of…

A new upper bound on the relative entropy is derived as a function of the total variation distance for probability measures defined on a common finite alphabet. The bound improves a previously reported bound by Csisz\'ar and Talata. It is…

We consider shift spaces in which elements of the alphabet may overlap nontransitively. We define a notion of entropy for such spaces, give several techniques for computing lower bounds for it, and show that it is equal to a limit of…

We give explicit upper bounds for the entropy in the cusp for one-parameter diagonal flows on $SL_d(\mathbb{R})/SL_{d}(\mathbb{Z})$. These results include bounds for the entropy of the cusp as a whole, as well as for the cusp regions…

Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ with parameters $1\leq p\leq\infty$ and $1\leq\xi\leq\omega_1$. These classes…

In this paper we look at a particular problem related to under-determined linear systems of equations with sparse solutions. $\ell_1$-minimization is a fairly successful polynomial technique that can in certain statistical scenarios find…

We provide an upper bound on the quasi-relative entropy in terms of the trace distance. The bound is derived for two cases: 1) any operator monotone decreasing function and full rank mixed qubit or classical states; 2) a large class of…

The underlying motivation of the present work lies on a cornerstone question in spectral optimization that consists of determining sharp lower and upper uniform estimates for fundamental frequencies of a set of uniformly elliptic operators…

In this article we are interested in studying partitions of the square, the disk and the equilateral triangle which minimize a p-norm of eigenvalues of the Dirichlet-Laplace operator. The extremal case of the infinity norm, where we…

We discuss algorithms for estimating the Shannon entropy h of finite symbol sequences with long range correlations. In particular, we consider algorithms which estimate h from the code lengths produced by some compression algorithm. Our…

Extending the methods developed in the author's previous paper and using adapted coordinate systems in two variables, an L^p boundedness theorem is proven for maximal operators over hypersurfaces in R^3 when p > 2. When the best possible p…

We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known…

We present a theory for constructing optimal lower bounds for the discrete half-line $p$-Laplacian of higher order $\ell\in\mathbb{N}$ and general $p>1$. The abstract framework introduces higher-order monotonicity and asymptotic constraints…

We study the problem of finding the worst-case bound for the size of the result $Q(\mathbb{ D})$ of a fixed conjunctive query $Q$ applied to a database $\mathbb{ D}$ satisfying given functional dependencies. We provide a precise…

We give a method to bound the entropy of measures on $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ which are invariant under a one parameter diagonal subgroup, in terms of entropy contributions from the regions of the cusp corresponding to…

We study the operator $L=-\Delta+q$ on a bounded domain $\Omega\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and…

We develop new techniques for computing the metric entropy of ellipsoids -- with polynomially decaying semi-axes -- in Banach spaces. Besides leading to a unified and comprehensive framework, these tools deliver numerous novel results as…

In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(\Omega)=\frac{\lambda_p^{1/p}(\Omega)}{\lambda_q^{1/q}(\Omega)},$$ where $1\le q<p\le+\infty$ and $\lambda_p(\Omega)$ (respectively $\lambda_q(\Omega)$) is…