On a problem of littlewood

The theorem on the tending to zero of coefficients of a trigonometric series is proved when theL ¹-norms of partial sums of this series are bounded. It is shown that the analog of Helson's theorem does not hold for orthogonal series with respect to the bounded orthonormal system. Two facts are given that are similar to Weis' theorem on the existence of a trigonometric series which is not a Fourier series and whoseL ¹-norms of partial sums are bounded.     

On the Sendov problem on the Whitney interpolation constant

For a function ƒ continuous on [0, 1] and satisfying the equalities we prove that where ω₄(t,ƒ) is the fourth modulus of smoothness of the function ƒ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 732–734, May, 1998.     

On the abstract nonautonomous parabolic cauchy problem in the case of constant domains

Summary We study existence, uniqueness and regularity of the strict, classical and strong solution u C([0,T],E) of the non-autonomous evolution equation u(t)–A(t)u(t)= f(t), with the initial datum, u(0)=x, in a Banach space E, where {A(t)} is a family of infinitesimal generators of analytic semi-groups whose domains are constant in t and possibly not dense in E. We prove necessary and sufficient conditions for existence and Hölder regularity of the solutions and their first derivative.     

On the problem of the ergodic constant in the theory of classical Green's functions

Moscow Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 3, pp. 459–473, September, 1990.     

Double roots of random littlewood polynomials

We consider random polynomials whose coefficients are independent and uniform on {-1, 1}. We prove that the probability that such a polynomial of degree n has a double root is o(n^-2) when n+1 is not divisible by 4 and asymptotic to \(1/\sqrt 3 \) otherwise. This result is a corollary of a more general theorem that we prove concerning random polynomials with independent, identically distributed coefficients having a distribution which is supported on {-1, 0, 1} and whose largest atom is strictly less than \(\frac{{8\sqrt 3 }}{{\pi {n^2}}}\). In this general case, we prove that the probability of having a double root equals the probability that either -1, 0 or 1 are double roots up to an o(n^-2) factor and we find the asymptotics of the latter probability.     

On a single-machine optimal constant due-date assignment and sequencing problem

This paper considers the problem of optimal constant due-date assignment and sequencing of jobs in a single-machine shop. We formulate the problem as a general constrained optimization problem and apply the Kuhn-Tucker conditions to find the optimal solution which is shown to be independent of the job sequence.     

On the absolute ruin problem in a Sparre Andersen risk model with constant interest

In this paper, we extend the work of Mitric and Sendova (2010), which considered the absolute ruin problem in a risk model with debit and credit interest, to renewal and non-renewal structures. Our first results apply to MAP processes, which we later restrict to the Sparre Andersen renewal risk model with interclaim times that are generalized Erlang (n) distributed and claim amounts following a Matrix-Exponential (ME) distribution (see for e.g. Asmussen and O’Cinneide (1997)). Under this scenario, we present a general methodology to analyze the Gerber-Shiu discounted penalty function defined at absolute ruin, as a solution of high-order linear differential equations with non-constant coefficients. Closed-form solutions for some absolute ruin related quantities in the generalized Erlang (2) case complement the results obtained under the classical risk model by Mitric and Sendova (2010).     

Singular solutions for the constant Q-curvature problem

This paper is devoted to the construction of weak solutions to the singular constant Q-curvature problem. We build on several tools developed in the last years. This is the first construction of singular metrics on closed manifolds of sufficiently large dimension with constant (positive) Q-curvature.     

On the Euler-Mascheroni constant

A formula for the classical Euler-Mascheroni constant not containing the logarithm is presented. The corresponding asymptotic expansion is proposed. New series rapidly converging to the constant are given.     

On the hyperbolicity constant in graphs

If X is a geodesic metric space and x₁,x₂,x₃∈X, a geodesic triangleT={x₁,x₂,x₃} is the union of the three geodesics [x₁x₂], [x₂x₃] and [x₃x₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if, for every geodesic triangle T in X, every side of T is contained in a δ-neighborhood of the union of the other two sides. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. . In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths , then , and if and only if G is isomorphic to C_m. Moreover, we prove the inequality for every graph, and we use this inequality in order to compute the precise value δ(G) for some common graphs.     

Quasi sure version of a theorem of littlewood

Let (c _n) _nZ be a double-sided sequence of complex numbers; if, for quasi-all (i.e. in the Baire sense) choices of signs ( _n) _nZ , ( _n c _n) _nZ is a Fourier-Stieltjes sequence, then we prove that (c _n) _nZ l ² (Z); this extends a result of Littlewood and at the same time provides a very simple proof of this result.     

On universal solvability of the first boundary value problem for linear difference equation with constant coefficients

The concept of a linear operator with constant coefficients is introduced and investigated, for which the corresponding first boundary problem is uniquely solvable ( or stable uniquely sovable) for any finite net set.     

On the de Bruijn-Newman constant

If λ(0) denotes the infimum of the set of real numbers λ such that the entire function Ξλ represented by

     

On a constant in the energy estimate

In this note, we prove that the constant $D(p,m)$ in the energy estimate, for m-subharmonic function with bounded p-energy, is strictly bigger than 1, for $p>0$, $p\ne 1$.