We consider the pair of second-order dynamic equations, (r(t)(xΔ)γ)Δ + p(t)xγ(t) = 0 and (r(t)(xΔ)γ)Δ + p(t)xγσ(t) = 0, on a time scale T, where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when T = R involve the wellknownHille-Kneser-type criteria of second-order line...

In this paper we introduce Latent Tree Language Model (LTLM), a novel approach to language modeling that encodes syntax and semantics of a given sentence as a tree of word roles. The learning phase iteratively updates the trees by moving nodes according to Gibbs sampling. We introduce two algorithms to infer a tree for a given sentence. The first one is based on Gibbs sampling. It is fast, but ...

In this paper, we consider the pair of second-order dynamic equations (r(t)(x)) + p(t)x(t) = 0, and (r(t)(x)) + p(t)x(t) = 0, on a time scale T, where γ > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when T = R involve the well–known Hille–Kneser type criteria of second-or...

and Applied Analysis 3 In 10 , Leighton proved the following well-known oscillation test for 1.4 ; see 10, 11 . Theorem A see 10 . Assume that ∫∞ t0 1 A0 ( η )dη ∞, ∫∞ t0 A1 ( η ) dη ∞, 1.5 then 1.3 is oscillatory. This result for 1.4 was obtained by Wintner in 12 without imposing any sign condition on the coefficient A1. In 13 , Kneser proved the following result. Theorem B see 13 . Equation 1...

We prove that a set of weak solutions of the nonlinear Volterra integral equation has the Kneser property. The main condition in our result is formulated in terms of axiomatic measures of weak noncompactness.

We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász theorem. These are closely related to formalizability in bounded arithmetic.

Let G be a connected graph with odd girth 2k + 1. Then G is a (2k + 1)-angulated graph if every two vertices of G are joined by a path such that each edge of the path is in some (2k + 1)-cycle. We prove that if G is (2k+1)-angulated, and H is connected with odd girth at least 2k+3, then any retract R of the box (or Cartesian) product G2H is isomorphic to S2T where S is a retract of G and T is a...