Bipullbacks of fractions and the snail lemma

We establish conditions giving the existence of bipullbacks in bicategories of fractions. We apply our results to construct a ${\pi }_0$-${\pi }_1$ exact sequence associated with a fractor between groupoids internal to a pointed exact category.     

Homogeneous expansions of normalized biholomorphic convex mappings overB P

The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain are studied. It is proved that the first (k+1) terms of the expansions of the jth componentf j of such a mapf depend only onz _j , for 1 ⩽j⩽n, wherek is the natural number that satisfiesk < ρ ⩽k +I. Whenp→ ∞, this gives the result on the unit polydisc obtained by Suffridge in 1970. Project supported in part by the National Natural Science Foundation of China.     

Quadripartite version of the Hajnal-Szemerédi theorem

Let G be a quadripartite graph with N vertices in each vertex class and each vertex is adjacent to at least vertices in each of the other classes. There exists an N₀ such that, if N?N₀, then G contains a subgraph that consists of N vertex-disjoint copies of K₄.     

Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobás and Komlós states the following: For every γ>0 and integers r2 and Δ, there exists β>0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most Δ, then G contains a copy of H.This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r=3.     

Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

   Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type

On planar intersection graphs with forbidden subgraphs

Let be a family of n compact connected sets in the plane, whose intersection graph has no complete bipartite subgraph with k vertices in each of its classes. Then has at most n times a polylogarithmic number of edges, where the exponent of the logarithmic factor depends on k. In the case where consists of convex sets, we improve this bound to O(n log n). If in addition k = 2, the bound can be further improved to O(n). © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 205–214, 2008     

Harrington’s conservation theorem redone

Leo Harrington showed that the second-order theory of arithmetic WKL ₀ is -conservative over the theory RCA ₀. Harrington’s proof is model-theoretic, making use of a forcing argument. A purely proof-theoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cut-elimination argument.      

Robust stability analysis of stochastic neural networks with Markovian jumping parameters and probabilistic time‐varying delays

This article discusses the issue of robust stability analysis for a class of Markovian jumping stochastic neural networks (NNs) with probabilistic time‐varying delays. The jumping parameters are represented as a continuous‐time discrete‐state Markov chain. Using the stochastic stability theory, properties of Brownian motion, the information of probabilistic time‐varying delay, the generalized Ito's formula, and linear matrix inequality (LMI) technique, some novel sufficient conditions are obtained to guarantee the stochastical stability of the given NNs. In particular, the activation functions considered in this article are reasonably general in view of the fact that they may depend on Markovian jump parameters and they are more general than those usual Lipschitz conditions. The main features of this article are described in the following: first one is that, based on generalized Finsler lemma, some improved delay‐dependent stability criteria are established and the second one is that the nonlinear stochastic perturbation acting on the system satisfies a class of Lipschitz linear growth conditions. By resorting to the Lyapunov–Krasovskii stability theory and the stochastic analysis tools, sufficient stability conditions are established using an efficient LMI approach. Finally, two numerical examples and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 21: 59–72, 2016     

Multidimensional van der Corput and sublevel set estimates

Van der Corput's lemma gives an upper bound for one-dimensional oscillatory integrals that depends only on a lower bound for some derivative of the phase, not on any upper bound of any sort. We establish generalizations to higher dimensions, in which the only hypothesis is that a partial derivative of the phase is assumed bounded below by a positive constant. Analogous upper bounds for measures of sublevel sets are also obtained. The analysis, particularly for the sublevel set estimates, has a more combinatorial flavour than in the one-dimensional case.