Related papers: Bounds on strong unicity for Chebyshev approximati…
We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We…
There is a parallelism between Shannon information theory and algorithmic information theory. In particular, the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of…
In this article, we study the Schr\"odinger equation posed in the Euclidean space. We prove observability inequalities for measurable sets that are thick with respect to decaying densities. The proof relies on quantitative uncertainty…
We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial $C_{n}^{(\lambda)}$ that is greater than $1$ when $-3/2 < \lambda < -1/2.$ Our first approach uses mixed…
Almost 10 years ago, Impagliazzo and Kabanets (2010) gave a new combinatorial proof of Chernoff's bound for sums of bounded independent random variables. Unlike previous methods, their proof is constructive. This means that it provides an…
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the…
We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). For asymptotically high values of the quantum number k, this bound coincides with the analogue of the Lieb-Wehrl conjecture for SU(1,1) coherent states. The bound on…
We prove a number of \textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value…
We study a tight Bennett-type concentration inequality for sums of heterogeneous and independent variables, defined as a one-dimensional minimization. We show that this refinement, which outperforms the standard known bounds, remains…
We determine the sharpest constant $C_{p,q,r}$ such that for all complex matrices $X$ and $Y$, and for Schatten $p$-, $q$- and $r$-norms the inequality $$ \|XY-YX\|_p\leq C_{p,q,r}\|X\|_q\|Y\|_r $$ is valid. The main theoretical tool in our…
This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer…
For a given ring (domain) in $\overline{\mathbb{R}}^n$ we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all $n\ge 3\,,$ the…
A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (1981) established an asymptotic formula for the number…
This paper is motivated by a gauged Schrodinger equation in dimension 2 including the so-called Chern-Simons term. The radially symmetric case leads to an elliptic problem with a nonlocal defocusing term, in competition with a local…
We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood…
Polynomial series approximations are a central theme in approximation theory due to their utility in an abundance of numerical applications. The two types of series, which are featured most prominently, are Taylor series expansions and…
There is presented an approach to find an approximation polynomial of a function with two variables based on the two dimensional discrete Fourier transform. The approximation polynomial is expressed through Chebyshev polynomials. There is…
We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this…
We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In…