Comparison theorem of the Hartman type for a continuous family of non-linear differential equations of the form p(t, λ)ϕ(u ′) for s = 0 and ϕ(0) = 0, is proved with the help of the generalized Mingarelli's identity.

In this paper we give an oscillation criterion for the quasi-differential equation (py')' + qy = 0. The results include a generalisation of an oscillation theorem of Hartman. We also give a necessary and sufficient condition for oscillation.

We provide a new proof of a theorem of Saks which is an extension of Greene’s Theorem to acyclic digraphs, by reducing it to a similar, known extension of Greene and Kleitman’s Theorem. This suggests that the Greene-Kleitman Theorem is stronger than Greene’s Theorem on posets. We leave it as an open question whether the same holds for all digraphs, that is, does Berge’s conjecture concerning pa...

Fix integers n ≥ r ≥ 2. A clique partition of ( [n] r ) is a collection of proper subsets A1, A2, . . . , At ⊂ [n] such that ⋃ i ( Ai r ) is a partition of ( [n] r ) . Let cp(n, r) denote the minimum size of a clique partition of ( [n] r ) . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, cp(n, ...

We present a new version of the Grobman-Hartman's linearization theorem for random dynamics. Our result holds infinite dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on non-linear perturbations, we don't require to be nonuniformly hyperbolic in sense Pesin but rather (besides requiring existence stable and unstable directions) allow...

Usually this theorem is proved by using continued fractions; see Theorem 193 in [2]. S. Hartman [3] has restricted the approximating numbers f to those fractions, where u and v belong to fixed residue classes a and b with respect to some modulus s. He proved the following: For any irrational number £, any s > 1, and integers a and b, there are infinitely many integers u and v > 0 satisfying 2s (1)

Aharoni, R. and I. Ben-Arroyo Hartman, On Greene-Kleitman’s theorem for general digraphs, Discrete Mathematics 120 (1993) 13-24. Linial conjectured that Greene-Kleitman’s theorem can be extended to general digraphs. We prove a stronger conjecture of Berge for digraphs having k-optimal path partitions consisting of ‘long’ paths. The same method yields known results for acyclic digraphs, and exte...

This course provides an introduction to the interplay between linear algebra and di¤erential equations/dynamical systems in continuous time. We …rst introduce linear di¤erential equations in Euclidian space R and on Grassmannian and ‡ag manifolds induced by a single matrix A, with emphasis on characterizations of the constant matrix A from a dynamics point of view. We then introduce linear skew...

Abstract: In this work, a nonsmooth multiobjective optimization problem involving generalized invexity with cone constraints and Applications (for short, (MOP)) is considered. The Kuhn-Tucker necessary and sufficient conditions for (MOP) are established by using a generalized alternative theorem of Craven and Yang. The relationship between weakly efficient solutions of (MOP) and vector valued s...