Incomplete Self‐Similar Blow‐Up in a Semilinear Fourth‐Order Reaction‐Diffusion Equation

Blow‐up behavior for the fourth‐order semilinear reaction‐diffusion equation (1) is studied. For the classic semilinear heat equation from combustion theory (2) various blow‐up patterns were investigated since 1970s, while the case of higher‐order diffusion was studied much less. Blow‐up self‐similar solutions of (1) of the form are constructed. These are shown to admit global similarity extensions for t > T : The continuity at t = T is preserved in the sense that This is in a striking difference with blow‐up for (2) , which is known to be always complete in the sense that the minimal (proper) extension beyond blow‐up is u(x, t) ≡+∞ for t > T . Difficult fourth‐order dynamical systems for extension pairs {f(y), F(y)} are studied by a combination of various analytic, formal, and numerical methods. Other nonsimilarity patterns for (1) with nongeneric complete blow‐up are also discussed.     

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.     

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.     

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 ].     

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.     

On the existence of a rainbow 1‐factor in 1‐factorizations of K

Let be a 1‐factorization of the complete uniform hypergraph with and . We show that there exists a 1‐factor of whose edges belong to n different 1‐factors in . Such a 1‐factor is called a “rainbow” 1‐factor or an “orthogonal” 1‐factor. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 487–490, 2007     

Bubbling Solutions for the SU(3) Chern‐Simons Model on a Torus

We consider the following nonlinear system derived from the SU(3) Chern‐Simons models on a torus Ω: where $\delta_p$ denotes the Dirac measure at $p\in\Omega$ . When $\{p_j^1\}_1^{N_1}= \{p_j^2\}_1^{N_2}$ , if we look for a solution with $u_1=u_2=u$ , then (0.1) is reduced to the Chern‐Simons‐Higgs equation: The existence of bubbling solutions to (0.1) has been a longstanding problem. In this paper, we prove the existence of such solutions such that $u_1\ne u_2$ even if $\{p_j^1\}_1^{N_1}=\{p_j^2\}_1^{N_2}$ . © 2012 Wiley Periodicals, Inc.     

On k‐detour subgraphs of hypercubes

A spanning subgraph G of a graph H is a k‐detour subgraph of H if for each pair of vertices , the distance, , between x and y in G exceeds that in H by at most k. Such subgraphs sometimes also are called additive spanners. In this article, we study k‐detour subgraphs of the n‐dimensional cube, , with few edges or with moderate maximum degree. Let denote the minimum possible maximum degree of a k‐detour subgraph of . The main result is that for every and On the other hand, for each fixed even and large n, there exists a k‐detour subgraph of with average degree at most . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 55–64, 2008     

A critical center‐stable manifold for Schrödinger's equation in three dimensions

Consider the focusing $\dot H^{1/2}$ ‐critical semilinear Schrödinger equation in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ It admits an eight‐dimensional manifold of special solutions called ground state solitons. We exhibit a codimension‐1 critical real analytic manifold ${\cal N}$ of asymptotically stable solutions of (0.1) in a neighborhood of the soliton manifold. We then show that ${\cal N}$ is center‐stable, in the dynamical systems sense of Bates and Jones, and globally‐in‐time invariant. Solutions in ${\cal N}$ are asymptotically stable and separate into two asymptotically free parts that decouple in the limit—a soliton and radiation. Conversely, in a general setting, any solution that stays $\dot H^{1/2}$ ‐close to the soliton manifold for all time is in ${\cal N}$ . The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time‐dependent linearized equation. The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here—of the focusing cubic NLS in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$ —by the work of Marzuola and Simpson and Costin, Huang, and Schlag. © 2012 Wiley Periodicals, Inc.     

Belohorec‐type oscillation theorem for second order sublinear dynamic equations on time scales

Consider the Emden‐Fowler sublinear dynamic equation (0.1) where $p\in C(\mathbb{T},R)$, where $\mathbb{T}$ is a time scale, 0 < α < 1. When p(t) is allowed to take on negative values, we obtain a Belohorec‐type oscillation theorem for (0.1). As an application, we get that the sublinear difference equation (0.2) is oscillatory, if and the sublinear q‐difference equation (0.3) where $t\in q^{\mathbb{N}_0}, q>1$, is oscillatory, if      

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)     

Optimal Regularity for the No‐Sign Obstacle Problem

In this paper we prove the optimal $C^{1,1}(B_{1/2})$ ‐regularity for a general obstacle‐type problem under the assumption that $f*N$ is $C^{1,1}(B_1)$ , where N is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$ ‐regularity. As a by‐product of the $C^{1,1}$ ‐regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$ , the free boundary is locally a $C^1$ ‐graph close to $x^0$ provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades. © 2012 Wiley Periodicals, Inc.     

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.     

Random mappings with exchangeable in‐degrees

In this article we introduce a new random mapping model, , which maps the set {1,2,…,n} into itself. The random mapping is constructed using a collection of exchangeable random variables which satisfy . In the random digraph, , which represents the mapping , the in‐degree sequence for the vertices is given by the variables , and, in some sense, can be viewed as an analogue of the general independent degree models from random graph theory. We show that the distribution of the number of cyclic points, the number of components, and the size of a typical component can be expressed in terms of expectations of various functions of . We also consider two special examples of which correspond to random mappings with preferential and anti‐preferential attachment, respectively, and determine, for these examples, exact and asymptotic distributions for the statistics mentioned above. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 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     

Method for Solving the Multidimensional n‐Wave Resonant Equations and Geometry of Generalized Darboux‐Manakov‐Zakharov Systems

The intrinsic geometric properties of generalized Darboux‐Manakov‐Zakharov systems of semilinear partial differential equations (1) for a real‐valued function u(x₁, …, x_n) are studied with particular reference to the linear systems in this equation class. System (1) is overdetermined and will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive tools for explicitly constructing involutive systems of the form (1) , essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multidimensional n‐wave resonant interaction system and its modified version. For each integer n≥ 3 and nonnegative integer k, our procedure constructs solutions of the n‐wave resonant interaction system depending on at least k arbitrary functions each of one variable. The construction of these solutions relies only on differentiation, linear algebra, and the solution of ordinary differential equations.     

Vertex suppression in 3‐connected graphs

To suppress a vertex in a finite graph G means to delete it and add an edge from a to b if a, b are distinct nonadjacent vertices which formed the neighborhood of . Let be the graph obtained from by suppressing vertices of degree at most 2 as long as it is possible; this is proven to be well defined. Our main result states that every 3‐connected graph G has a vertex x such that is 3‐connected unless G is isomorphic to , , or to a wheel for some . This leads to a generator theorem for 3‐connected graphs in terms of series parallel extensions. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 41–54, 2008     

From Bolzano‐Weierstraß to Arzelà‐Ascoli

We show how one can obtain solutions to the Arzelà‐Ascoli theorem using suitable applications of the Bolzano‐Weierstraß principle. With this, we can apply the results from 10 and obtain a classification of the strength of instances of the Arzelà‐Ascoli theorem and a variant of it. Let be the statement that each equicontinuous sequence of functions contains a subsequence that converges uniformly with the rate and let be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that is instance‐wise equivalent, over , to the Bolzano‐Weierstraß principle and that is instance‐wise equivalent, over , to , and thus to the strong cohesive principle (). Moreover, we show that over the principles , and are equivalent.     

C‐perfect hypergraphs

Let ${\mathcal{H}}=({{X}},{\mathcal{E}})Let ${\mathcal{H}}=({{X}},{\mathcal{E}})$ be a hypergraph with vertex set X and edge set ${\mathcal{E}}$. A C‐coloring of ${\mathcal{H}}$ is a mapping ?:X→? such that |?(E)|<|E| holds for all edges ${{E}}\in{\mathcal{E}}$ (i.e. no edge is multicolored). We denote by $\bar{\chi}({\mathcal{H}})$ the maximum number |?(X)| of colors in a C‐coloring. Let further $\alpha({\mathcal{H}})$ denote the largest cardinality of a vertex set S?X that contains no ${{E}}\in{\mathcal{E}}$, and $\tau({\mathcal{H}})=|{{X}}|-\alpha({\mathcal{H}})$ the minimum cardinality of a vertex set meeting all $E \in {\mathcal{E}}$. The hypergraph ${\mathcal{H}}$ is called C‐perfect if $\bar{\chi}({\mathcal{H}}\prime)=\alpha({\mathcal{H}}\prime)$ holds for every induced subhypergraph ${\mathcal{H}}\prime\subseteq{\mathcal{H}}$. If ${\mathcal{H}}$ is not C‐perfect but all of its proper induced subhypergraphs are, then we say that it is minimally C‐imperfect. We prove that for all r, k∈? there exists a finite upper bound h(r, k) on the number of minimally C‐imperfect hypergraphs ${\mathcal{H}}$ with $\tau({\mathcal{H}})\le {{k}}$ and without edges of more than r vertices. We give a characterization of minimally C‐imperfect hypergraphs that have τ=2, which also characterizes implicitly the C‐perfect ones with τ=2. From this result we derive an infinite family of new constructions that are minimally C‐imperfect. A characterization of minimally C‐imperfect circular hypergraphs is presented, too. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 132–149, 2010     

On conjectures of Frankl and El‐Zahar

An induced subgraph of a graph is called a derived subgraph of if contains no isolated vertices. An edge e of is said to be residual if e occurs in more than half of the derived subgraphs of . In this article, we prove that every simple graph with at least one edge contains a non‐residual edge. This was conjectured by El‐Zahar in 1997. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 344–352, 2008