Deforming subdivisions

Subdivisions are continuously deformed subject to three local conditions and shown to retain the global property of being a subdivision. Cell maps, which are introduced, enable elementary arguments.This research is supported in part by the National Science Foundation Grant DMS-8902662.     

On the focus order of planar polynomial differential equations

This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime pk, and show that these systems can have a focus order n²−n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n²−1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.     

Random geometric subdivisions

We study several models of random geometric subdivisions arising from the model of Diaconis and Miclo (Combin Probab Comput 20 (2011) 213–237). In particular, we show that the limiting shape of an indefinite subdivision of a quadrilateral is a.s. a parallelogram. We also show that the geometric subdivisions of a triangle by angle bisectors converge (only weakly) to a non‐atomic distribution, and that the geometric subdivisions of a triangle by choosing random points on its sides converges to a “flat” triangle, similarly to the result of Diaconis and Miclo (Combin Probab Comput 20 (2011) 213–237). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

一类三次系统的极限环

主要研究一类三次系统的极限环存在性问题,推广了C.Chicone[2]的结果,给出此类系统极限环存在定理.     

Classifying planar monomials over fields of order a prime cubed

Using Hermite's criteria, we classify planar monomials over fields of order a prime cubed, establishing the Dembowski-Ostrom conjecture for monomials over fields of such orders.     

New second order nonconforming triangular element for planar elasticity problems

In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn’s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn’s second inequality holds and the convergence order is two.     

The classification of planar monomials over fields of prime square order

Planar functions were introduced by Dembowski and Ostrom in 1968 to describe affine planes possessing collineation groups with particular properties. To date their classification has only been resolved for functions over fields of prime order. In this article we classify planar monomials over fields of order with a prime.

     

Slip effects on MHD flow of third order fluid in a planar channel

In this paper, the peristaltic flow of magnetohydrodynamic (MHD) third order fluid in a planar channel with slip condition is investigated. The solutions are derived under long wavelength and low Reynolds number approximations. Explicit expressions of stream function, longitudinal pressure gradient, longitudinal velocity, temperature and coefficient of heat transfer are given. The pumping and trapping phenomena are analyzed in the presence of MHD and slip effects. The effects of parameters on temperature distribution and heat transfer coefficient are discussed. A comparison is provided with the different existing cases.     

Some probabilistic and extremal results on subdivisions and odd subdivisions of graphs

A probabilistic result of Bollobás and Catlin concerning the largest integer p so that a subdivision of K_p is contained in a random graph is generalized to a result concerning the largest integer p so that a subdivision of A_p is contained in a random graph for some sequence A₁, A₂,… of graphs such that A_i+1 contains a subdivision of A_i. A similar result is proved for subdivisions with odd paths or cycles. The result is applied to disprove a conjecture of Chartrand, Geller, and Hedetniemi. The maximum number of edges in a graph without a subdivision of K_p, p = 4, 5, with odd paths or cycles is determined.     

On color isomorphic subdivisions

Given a graph H and an integer $k?2$, let $f_k(n,H)$ be the smallest number of colors C such that there exists a proper edge-coloring of the complete graph $K_n$ with C colors containing no k vertex-disjoint color isomorphic copies of H. In this paper, we prove that $f_2(n,H_t)=\mathrm{\Omega }(n^{1+\frac{1}{2t?3}})$ where $H_t$ is the 1-subdivision of the complete graph $K_t$. This answers a question of Conlon and Tyomkyn (2021) [4].     

Existence of periodic solution of order one of planar impulsive autonomous system

Based on the situation that all the recent research about state-dependent impulsive differential equations is focused on systems which have explicit solutions, this paper try to consider those systems with state-dependent impulsion which have not explicit solutions. We get an existence theorem of periodic solution of order one for a general planar autonomous impulsive system, and, by applying it to a special state-dependent impulsive differential equations we get the concrete conditions of existence of one-order periodic solution of that special system.     

Finding simplicial subdivisions of polytopes

Subdivisions of polytopes are described which place any chosen vertex in a (near-) minimal number of simplices. Ancillary procedures to find volumes and centroids of polytopes are indicated.     

Subword complexes and edge subdivisions

For a finite Coxeter group, a subword complex is a simplicial complex associated with a pair (Q, π), where Q is a word in the alphabet of simple reflections and π is a group element. We discuss the transformations of such a complex that are induced by braid moves of the word Q. We show that under certain conditions, such a transformation is a composition of edge subdivisions and inverse edge subdivisions. In this case, we describe how the H- and γ-polynomials change under the transformation. This case includes all braid moves for groups with simply laced Coxeter diagrams.     

Convex subdivisions with low stabbing numbers

It is shown that for every subdivision of the d-dimensional Euclidean space, d ≥ 2, into n convex cells, there is a straight line that stabs at least Ω((log n/log log n)^1/(d−1)) cells. In other words, if a convex subdivision of d-space has the property that any line stabs at most k cells, then the subdivision has at most exp(O(k ^d−1 log k)) cells. This bound is best possible apart from a constant factor. It was previously known only in the case d = 2. Supported in part by NSERC grant RGPIN 35586.     

On multiple space subdivisions by zonotopes

This article contains P.Furtwängler's method for presenting Galois theory. To any field of rational functions belongs a group of permutations and vice versa. Given a system of roots {₁, ₂,..., _n } a field containing all relations existing between them can be defined. To this field belongs the Galois group of the given system.