Let [a, b] denote the integers between a and b inclusive and, for a finite subset X ⊆ Z, let diam (X) = max(X) − min(X). We write X <p Y provided max(X) < min(Y ). For a positive integer m, let f(m,m,m; 2) be the least integer N such that any 2-coloring ∆ : [1, N ] → {0, 1} has three monochromatic m-sets B1, B2, B3 ⊆ [1, N ] (not necessarily of the same color) with B1 <p B2 <p B3 and diam (B1) ...

Grinberg defined Nyldon words as those which cannot be factorized into a sequence of lexicographically nondecreasing smaller words. He was inspired by Lyndon words, the same way except with "nondecreasing" replaced "nonincreasing." Charlier, Philibert, and Stipulanti proved that, like any word has unique factorization They also show that form right Lazard set, equivalently, Hall set. In this pa...

Sufficient conditions are found for the applicability of generalized law iterated logarithm sums dependent random variables in case where sequence normalizing constants is not necessarily nondecreasing.

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a nondecreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give example...

We show that it is impossible to define big-O notation for functions on more than one variable in a way that implies the properties commonly used in algorithm analysis. We also demonstrate that common definitions do not imply these properties even if the functions within the big-O notation are restricted to being strictly nondecreasing. We then propose an alternative definition that does imply ...

It is well-known that the generating function for tableaux of a given skew shape with r rows where the parts in the i'th row are bounded by some nondecreasing upper and lower bounds which depend on i can be written in form of a determinant of size r. We show that the generating function for tableaux of a given skew shape with r rows and c columns where the parts in the i'th row are bounded by n...

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real function h that can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate function h. In this paper we study these orders when h is nonde...

This paper investigates global asymptotic stability (GAS) and global exponential stability (GES) of a class of continuous-time recurrent neural networks. First, we introduce a necessary and sufficient condition for existence and uniqueness of equilibrium of the neural networks with Lipschitz continuous activation functions. Next, we present two sufficient conditions to ascertain the GAS of the ...