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Mathematics, Volume 1, Issue 2 (June 2013) – 2 articles , Pages 46-75

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181 KiB

Open AccessArticle

On the Class of Dominant and Subordinate Products

by Alexander Berkovich and Keith Grizzell

Mathematics 2013, 1(2), 65-75; https://doi.org/10.3390/math1020065 - 15 May 2013

Cited by 2 | Viewed by 3912

Abstract

In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients. Full article

259 KiB

Open AccessArticle

Stability of Solutions to Evolution Problems

by Alexander G. Ramm

Mathematics 2013, 1(2), 46-64; https://doi.org/10.3390/math1020046 - 13 May 2013

Cited by 4 | Viewed by 4419

Abstract

Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a [...] Read more.

t \to \infty

, in particular, sufficient conditions for this limit to be zero. The evolution problem is:

\dot{u} = A (t) u + F (t, u) + b (t), t \geq 0; u (0) = u_{0} .

(*) Here

\dot{u} : = \frac{d u}{d t}, u = u (t) \in H, H

is a Hilbert space,

t \in R_{+} : = [0, \infty), A (t)

is a linear dissipative operator:

Re (A (t) u, u) \leq - γ (t) (u, u)

where

F (t, u)

is a nonlinear operator,

‖ F (t, u) ‖ \leq c_{0} {‖ u ‖}^{p}, p > 1, c_{0}

and p are positive constants,

‖ b (t) ‖ \leq β (t)

, and

β (t) \geq 0

is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case

γ (t) \leq 0

is also treated. Full article

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Mathematics, Volume 1, Issue 2 (June 2013) – 2 articles , Pages 46-75

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