Related papers: Covering numbers and schlicht functions

In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius…

We extend holomorphically polyharmonic functions on a real ball to a complex set being the union of rotated balls. We solve a Dirichlet type problem for complex polyharmonic functions with the boundary condition given on the union of…

We consider a sequence of composite Bernstein operators and the quadrature formulae associated with them. Upper bounds for the approximation error of continuous functions and for the approximation of integrals of continuous functions are…

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph is the smallest number of…

A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1,\Theta_1,...,\Theta_m\in\mathbb{C}^*$ over the ring $\mathbb{Z}_{\mathbb{I}}$ of an imaginary quadratic field $\mathbb{I}$. This work deals…

We obtain a new lower bound for the eternal vertex cover number of an arbitrary graph $G$, in terms of the cardinality of a vertex cover of minimum size in $G$ containing all its cut vertices. The consequences of the lower bound includes a…

We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $n$-space with translates of a convex body, or more generally,…

We give a computable lower bound on the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether--Lefschetz…

This paper provides a general result on controlling local Rademacher complexities, which captures in an elegant form to relate the complexities with constraint on the expected norm to the corresponding ones with constraint on the empirical…

We consider the problem of minimizing a polynomial function over the integer lattice. Though impossible in general, we use a known sufficient condition for the existence of continuous minimizers to guarantee the existence of integer…

Necessary and sufficient conditions for Lipschitzness of the Lempert and Green functions are found in terms of their boundary behaviors.

We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal graphs…

To minimize or upper-bound the value of a function "robustly", we might instead minimize or upper-bound the "epsilon-robust regularization", defined as the map from a point to the maximum value of the function within an epsilon-radius. This…

We establish a lower bound on the forcing numbers of domino tilings computable in polynomial time based on height functions. This lower bound is sharp for a 2n by 2n square as well as other cases.

In this paper, we consider the problem of covering a plane region with unit discs. We present an improved upper bound and the first nontrivial lower bound on the number of discs needed for such a covering, depending on the area and…

A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erd\H{o}s in 1950, and over the following decades he asked many questions about them. Most…

In this paper we study the following problems: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects…

We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some…

Given a compact subset of a Banach space, the Chebyshev center problem consists of finding a minimal circumscribing ball containing the set. In this article we establish a numerically tractable algorithm for solving the Chebyshev center…

Employing the Orlicz functions we extend the Buzano's inequality which is a refinement of the Cauchy-Schwarz inequality. Also using the Orlicz functions we obtain several numerical radius inequalities for a bounded linear operator as well…