Simplified Asymptotic Solutions of Differential Equations Having Two Turning Points,with an Application to Legendre Functions

Asymptotic approximations of differential equations of the form are obtained, for the case , uniformly valid for real or complex values of x lying in bounded or unbounded intervals or regions. Here, , and have no singularities inside the interval or region under consideration, and does not vanish except at a critical point where it has a double zero. By an appropriate Liouville transformation, along with a perturbation of the new independent variable, uniform asymptotic approximations involving parabolic cylinder functions are obtained. These approximations are accompanied by strict and realistic error bounds, and the new theory is applied to obtain a uniform asymptotic approximation for the associated Legendre function with m and n large, with positive and bounded.     

Painlevé XXXIV Asymptotics of Orthogonal Polynomials for the Gaussian Weight with a Jump at the Edge

We study the uniform asymptotics of the polynomials orthogonal with respect to analytic weights with jump discontinuities on the real axis, and the influence of the discontinuities on the asymptotic behavior of the recurrence coefficients. The Riemann–Hilbert approach, also termed the Deift–Zhou steepest descent method, is used to derive the asymptotic results. We take as an example the perturbed Gaussian weight , where θ(x) takes the value of 1 for x < 0 , and a nonnegative complex constant ω elsewhere, and as . That is, the jump occurs at the edge of the support of the equilibrium measure. The derivation is carried out in the sense of a double scaling limit, namely, and . A crucial local parametrix at the edge point where the jump occurs is constructed out of a special solution of the Painlevé XXXIV equation. As a main result, we prove asymptotic formulas of the recurrence coefficients in terms of a special Painlevé XXXIV transcendent under the double scaling limit. The special thirty‐fourth Painlevé transcendent is shown free of poles on the real axis. A consistency check is made with the reduced case when ω= 1 , namely the Gaussian weight: the polynomials in this case are the classical Hermite polynomials. A comparison is also made of the asymptotic results for the recurrence coefficients between the case when the jump happens at the edge and the case with jump inside the support of the equilibrium measure. The comparison provides a formal asymptotic approximation of the Painlevé XXXIV transcendent at positive infinity.     

Asymptotic Behavior for a Class of Solutions to the Critical Modified Zakharov–Kuznetsov Equation

We consider the initial value problem (IVP) associated to the modified Zakharov–Kuznetsov (mZK) equation which is known to have global solution for given data in satisfying , where φ is a solitary wave solution. In this work, the issue of the asymptotic behavior of the solutions of the modified Zakharov–Kuznetsov equation with negative energy is addressed. The principal tool to obtain the main result is the use of appropriate scaling argument from Angulo et al. [ 1 , 2 ].     

Hodograph‐Type Transformations for Linearization of Systems of Nonlinear Diffusion Equations

In this paper, we study the problem of linearization of nonlinear systems of equations which is a potential form of systems of nonlinear diffusion equations We construct a class of point transformations of the form which connects the nonlinear systems with linear systems of equations . These point transformations are hodograph‐type transformations which have the property that the new independent and dependent variables depend, respectively, on the old dependent and independent variables. All systems of equations admitting such transformations are completely classified.     

DERIVATION OF TWO WELL‐BEHAVED THEORETICAL CONTAGION INDICES AND THEIR SAMPLING PROPERTIES AND APPLICATION FOR ASSESSING FOREST LANDSCAPE DIVERSITY

Abstract Studies of spatial patterns of landscapes are useful to quantify human impact, predict wildlife effects, or describe variability of landscape features. A common approach to identify and quantify landscape structure is with a landscape scale model known as a contagion index. A contagion index quantifies two distinct components of landscape diversity: composition and configuration. Some landscape ecologists promote the use of relative contagion indices. It is demonstrated that relativized contagion indices are mathematically untenable. Two new theoretical contagion indices, Γ₁ and Γ₂ , are derived using a mean value approach (i.e., statistical expected value) instead of entropy. Behavior of Γ₁ and Γ₂ was investigated with simulated random, uniform, and aggregated landscapes. They are shown to be well‐behaved and sensitive to composition and configuration. Distributional properties of and are derived. They are shown to be asymptotically unbiased, consistent, and asymptotically normally distributed. Variance formulas for and are developed using the delta method. The new index models are used to examine landscape diversity on three physiographic provinces in Alabama by analyzing the pattern and changes in forest cover types over the recent past. In comparing and , use of in analysis of variance gave a more conservative test of contagion.     

Eigenfunctions and Very Singular Similarity Solutions of Odd‐Order Nonlinear Dispersion PDEs: Toward a “Nonlinear Airy Function” and Others

Asymptotic properties of nonlinear dispersion equations (1) with fixed exponents n > 0 and p > n+ 1 , and their (2k+ 1) th‐order analogies are studied. The global in time similarity solutions, which lead to “nonlinear eigenfunctions” of the rescaled ordinary differential equations (ODEs), are constructed. The basic mathematical tools include a “homotopy‐deformation” approach, where the limit in the first equation in ( 1 ) turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: whose oscillatory fundamental solution via Airy’s classic function has been known since the nineteenth century. The corresponding Hermitian linear non‐self‐adjoint spectral theory giving a complete countable family of eigenfunctions was developed earlier in [ 1 ]. Various other nonlinear operator and numerical methods for ( 1 ) are also applied. As a key alternative, the “super‐nonlinear” limit , with the limit partial differential equation (PDE) admitting three almost “algebraically explicit” nonlinear eigenfunctions, is performed. For the second equation in ( 1 ), very singular similarity solutions (VSSs) are constructed. In particular, a “nonlinear bifurcation” phenomenon at critical values {p=p_l(n)}_l≥0 of the absorption exponents is discussed.     

Painlevé VI and the Unitary Jacobi Ensembles

The six Painlevé transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well‐known examples among which include symmetry reduction of the Ernst equation which arises from stationary axial symmetric Einstein manifold and the spin‐spin correlation functions of the two‐dimensional Ising model in the work of McCoy, Tracy, and Wu. The problem we study in this paper originates from random matrix theory, namely, the smallest eigenvalues distribution of the finite n Jacobi unitary ensembles which was first investigated by Tracy and Widom. This is equivalent to the computation of the probability that the spectrum is free of eigenvalues on the interval . Such ensembles also appears in multivariate statistics known as the double‐Wishart distribution. We consider a more general model where the Jacobi weight is perturbed by a discontinuous factor and study the associated finite Hankel determinant. It is shown that the logarithmic derivative of Hankel determinant satisfies a particular σ‐form of Painlevé VI, which holds for the gap probability as well. We also compute exactly the leading term of the gap probability as .     

Cutting two graphs simultaneously

Consider two graphs, and , on the same vertex set V, with and having edges for . We give a simple algorithm that partitions V into sets A and B such that and . We also show, using a probabilistic method, that if and belong to certain classes of graphs, (for instance, if and both have a density of at least 2/, or if and are both regular of degree at most with n sufficiently large) then we can find a partition of V into sets A and B such that for . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 19–32, 2008     

Maximum number of colorings of (2k,k2)‐graphs

Let consist of all simple graphs on 2k vertices and edges. For a simple graph G and a positive integer , let denote the number of proper vertex colorings of G in at most colors, and let . We prove that and is the only extremal graph. We also prove that as . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 135–148, 2007     

Disjoint cycles with chords in graphs

Let be integers, , , and let for each , be a cycle or a tree on vertices. We prove that every graph G of order at least n with contains k vertex disjoint subgraphs , where , if is a tree, and is a cycle with chords incident with a common vertex, if is a cycle. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 87–98, 2009     

Completeness of intermediate logics with doubly negated axioms

Let denote a first‐order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic . By , we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of plus . We shall show that if is strongly complete for a class of Kripke models , then is strongly complete for the class of Kripke models that are ultimately in .     

Circular chromatic index of Cartesian products of graphs

The circular chromatic index of a graph G, written , is the minimum r permitting a function such that whenever e and are incident. Let □ , where □ denotes Cartesian product and H is an ‐regular graph of odd order, with (thus, G is s‐regular). We prove that , where is the minimum, over all bases of the cycle space of H, of the maximum length of a cycle in the basis. When and m is large, the lower bound is sharp. In particular, if , then □ , independent of m. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 7–18, 2008     

Graph classes characterized both by forbidden subgraphs and degree sequences

Given a set of graphs, a graph G is ‐free if G does not contain any member of as an induced subgraph. We say that is a degree‐sequence‐forcing set if, for each graph G in the class of ‐free graphs, every realization of the degree sequence of G is also in . We give a complete characterization of the degree‐sequence‐forcing sets when has cardinality at most two. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 131–148, 2008     

A system of equations for describing cocyclic Hadamard matrices

Given a basis for 2‐cocycles over a group G of order , we describe a nonlinear system of 4t‐1 equations and k indeterminates over , whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that ( ) is a solution of the system if and only if the 2‐cocycle gives rise to a cocyclic Hadamard matrix . Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups and . A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups , we have found that it suffices to check t instead of the 4t rows of , to decide the Hadamard character of the matrix (for a special class of cocycles f). © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 276–290, 2008     

A S(3, {4, 6}, 18) with a subdesign S(3, 4, 8) does not exist

For , a S(t,K,v) design is a pair, , with |V| = v and a set of subsets of V such that each t‐subset of V is contained in a unique and for all . If , , , and is a S(t,K,u) design, then we say has a subdesign on U. We show that a S(3,{4,6},18) design with a subdesign S(3,4,8) does not exist. © 2007 Wiley Periodicals, Inc. J Combin Designs 17: 36–38, 2009     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

Tangency sets in PG(3, q)

A tangency set of PG (d,q) is a set Q of points with the property that every point P of Q lies on a hyperplane that meets Q only in P. It is known that a tangency set of PG (3,q) has at most points with equality only if it is an ovoid. We show that a tangency set of PG (3,q) with , or points is contained in an ovoid. This implies the non‐existence of minimal blocking sets of size , , and of with respect to planes in PG (3,q), and implies the extendability of partial 1‐systems of size , , or to 1‐systems on the hyperbolic quadric . © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 462–476, 2008     

Improper Coloring of Sparse Graphs with a Given Girth,II: Constructions

A graph G is ‐colorable if can be partitioned into two sets and so that the maximum degree of is at most j and of is at most k. While the problem of verifying whether a graph is (0, 0)‐colorable is easy, the similar problem with in place of (0, 0) is NP‐complete for all nonnegative j and k with . Let denote the supremum of all x such that for some constant every graph G with girth g and for every is ‐colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing g from 5 further on does not increase much. Our constructions show that for every g, . We also find exact values of for all g and all .