In this short communication, an extremal combinatorial problem concerning partition and covering of a finite set is discussed. It is pointed out that unlike the case with partition, no closed formula solution for determining the total number of coverings is known. Some motivating steps are indicated. The application of compatibility relation to solve some minimization problem is outlined.

This paper investigates the nonlinear boundary value problem for a class of first-order impulsive functional differential equations. By establishing a comparison result and utilizing the method of upper and lower solutions, some criteria on the existence of extremal solutions as well as the unique solution are obtained. Examples are discussed to illustrate the validity of the obtained results.

and Applied Analysis 3 The final solution to problems in 1.3 can yield optimal lower and upper bounds of eigenvalues. Actually, from 1.3 , we have In ∥q ∥ ∥ 1 ) ≤ λn ( q ) ≤ Mn ∥q ∥ ∥ 1 ) ∀q ∈ L1, 1.4 which are optimal. Several extremal problems on Dirichlet, Neumann, and periodic eigenvalues of the Sturm-Liouville operator and the p-Laplace operator have been studied recently, where the potent...

The backreaction equations for the linearized quantum fluctuations in an acoustic black hole are given. The solution near the horizon, obtained within a dimensional reduction, indicates that acoustic black holes, unlike Schwarzschild ones, get cooler as they radiate phonons. They show remarkable analogies with near-extremal Reissner-Nordström black holes.

It is shown that an arbitrarily small amount of angular momentum can qualitatively change the properties of extremal charged black holes coupled to a dilaton. In addition, the gyromagnetic ratio of these black holes is computed and an exact rotating black string solution is presented.

The minimal area covered by the image of the unit disk is found for nonvanishing univalent functions normalized by the conditions f(0) = 1, f ′(0) = α. Two different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral repres...

In contrast to the extremal variants of coNP-complete problems, which are frequently DP-complete, many extremal variants of NP-complete problems are in P. We investigate the extremal variants of two NP-complete problems, the extremal colorability problem with restricted degree and the extremal unfrozen non-implicant problem, and show that both of them are DP-complete. As far as we know, no extr...

We introduce a computational technique for studying non-supersymmetric deformations of domain wall solutions of interest in AdS/CFT. We focus on the KlebanovStrassler solution, which is dual to a confining gauge theory. From an analysis of asymptotics we find that there are three deformations that leave the ten-dimensional supergravity solution regular and preserve the global bosonic symmetries...

A symmetric subset of the reals is one that remains invariant under some reflection x 7→ c − x. We consider, for any 0 < ε ≤ 1, the largest real number ∆(ε) such that every subset of [0, 1] with measure greater than ε contains a symmetric subset with measure ∆(ε). In this paper we establish upper and lower bounds for ∆(ε) of the same order of magnitude: for example, we prove that ∆(ε) = 2ε − 1 ...

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.