Let L be a bounded distributive lattice. In this paper we focus on those functions f : L → L which can be expressed in the language of bounded lattices using variables and constants, the so-called “weighted” lattice polynomial functions. Clearly, such functions must be nondecreasing in each variable, but the converse does not hold in general. Thus it is natural to ask which nondecreasing functi...

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation xn f xn−2 /g xn−1 , n ∈ N0, where f, g ∈ C 0,∞ , 0,∞ . It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n} and {x2n−1} are eventually monotone. For the case when f x α βx and g satisfies the conditions g 0 1, g is nondec...

We study existence and uniqueness of solutions ([Formula: see text]) [Formula: text] in text], on where is a bounded smooth domain such that constant, continuous nondecreasing function satisfying some integral growth condition two Radon measures, respectively, text]. show the situation differs considerably according measure concentrated at or not. When power we introduce capacity framework whic...

Wang and Yeh proved that if P (x) is a polynomial with nonnegative and nondecreasing coefficients, then P (x + d) is unimodal for any d > 0. A mode of a unimodal polynomial f(x) = a0 + a1x + · · · + amx m is an index k such that ak is the maximum coefficient. Suppose that M∗(P, d) is the smallest mode of P (x + d), and M(P, d) the greatest mode. Wang and Yeh conjectured that if d2 > d1 > 0, the...

Let m be a positive integer whose smallest prime divisor is denoted by p, and let Zm denote the cyclic group of residues modulo m. For a set B = {x1, x2, . . . , xm} of m integers satisfying x1 < x2 < · · · < xm, and an integer j satisfying 2 ≤ j ≤ m, define gj(B) = xj − x1. Furthermore, define fj(m, 2) (define fj(m, Zm)) to be the least integer N such that for every coloring ∆ : {1, . . . ,N} ...

The ratio monotonicity of a polynomial is a stronger property than log-concavity. Let P (x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P (x+ 1), which leads to the log-concavity of P (x+ c) for any c ≥ 1 due to Llamas and Mart́ınez-Bernal. As a consequence, we obtain the ratio monotonicity of the Boros-Moll polynomials obtained by Che...

A notion of asymptotic Hamming distance suitable for the study of algorithmic randomness is developed. As an application of this notion, it is shown that there is no fixed procedure that computes a Mises-WaldChurch stochastic set from a complex set. Here a set is complex if its prefixes have Kolmogorov complexity bounded below by an unbounded, nondecreasing computable function.

It is known that if a predictable nondecreasing process generates a bounded potential, then its final value satisfies the Garsia inequality. We prove the converse, that is, a random variable satisfying the Garsia inequality generates a bounded potential. We also propose some useful relations between the Garsia inequality and the Cramer conditions, and different ways how to construct a potential.

We present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle.