For x x1, x2, . . . , xn ∈ 0, 1 n and r ∈ {1, 2, . . . , n}, the symmetric function Fn x, r is defined as Fn x, r Fn x1, x2, . . . , xn; r ∑ 1≤i1<i2 ···<ir≤n ∏r j 1 1 xij / 1−xij , where i1, i2, . . . , in are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of Fn x, r are discussed. As consequences, several inequalities are est...

Subadditive time discounting means that discounting over a delay is greater when the delay is divided into subintervals than when it is left undivided. This may produce the most important result usually attributed to hyperbolic discounting: declining impatience, or the inverse relationship between the discount rate and the magnitude of the delay. Three choice experiments were conducted to test ...

Let S be a compact Riemann surface (holomorphic curve) of genus g. Let p1, p2, · · · , ps be s > 0 points on it; these points define a divisor, and we denote the open Riemann surface S \ {p1, . . . , ps} by S. When 3g− 3+ s > 0, it carries a complete hyperbolic metric of finite volume, the so-called Poincaré metric; the points p1, p2, · · · , ps then become cusps at infinity. Even in the remain...

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeomet-ric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E 7 (ellip-tic, hyperbolic) and of type E 6...

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigourous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for n-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unst...

In this paper, we investigate the means of the values of prime counting function $pi(x)$. First, we compute the arithmetic, the geometric, and the harmonic means of the values of this function, and then we study the limit value of the ratio of them.

We will investigate the superstability of the hyperbolic trigonometric functional equation from the following functional equations: fx y ± gx − y λfxgy andfx y ± gx − y λgxfy, which can be considered the mixed functional equations of the sine function and cosine function, the hyperbolic sine function and hyperbolic cosine function, and the exponential functions, respectively.

let denote the unit circle in the complex plane. given a function , one uses t usual (harmonic) poisson kernel for the unit disk to define the poisson integral of , namely . here we consider the biharmonic poisson kernel for the unit disk to define the notion of -integral of a given function ; this associated biharmonic function will be denoted by . we then consider the dilations for and . the ...

The well-posedness of the gravitational equations $f(R)$ gravity is studied in this paper. Three formulations with dynamical shifts [which are all based on Arnowitt-Deser-Misner (ADM) formalism equations] investigated. These three proved to be strongly hyperbolic by pseudodifferential reduction. first one Baumagarte-Shapiro-Shibata-Nakamura formulation so-called ``hyperbolic $K$-driver'' condit...

A partition of a non-negative integer n is any non-increasing sequence of positive integers whose sum is n. As usual, let p(n) denote the number of partitions of n. The partition function satisfies the famous “Ramanujan congruences” declaring that for c ∈ {5, 7, 11} we have for all n ≥ 0 that p(cn+δc) ≡ 0 (mod c), where δc is defined by the congruence 24δc ≡ 1 (mod c). In order to understand th...