Plancherel–Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems

We study the Plancherel–Rotach asymptotics of four families of orthogonal polynomials: the Chen–Ismail polynomials, the Berg–Letessier–Valent polynomials, and the Conrad–Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems, and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems.     

Hyperstability of the Cauchy equation on restricted domains

We show that a very classical result, proved by T. Aoki, Z. Gajda and Th. M. Rassias and concerning the Hyers–Ulam stability of the Cauchy equation f(x+y)=f(x)+f(y), can be significantly improved. We also provide some immediate applications of it (among others for the cocycle equation, which is useful in characterizations of information measures). In particular, we give a solution to a problem that was formulated more than 20 years ago and concerned optimality of some estimations. The proof of that result is based on a fixed point theorem.     

Hyperstability of the Jensen functional equation

We present some observations on hyperstability for the Jensen equation on restricted domain. Namely, we show, under some weak natural assumptions, that functions satisfying the equation approximately (in some sense) must be actually solutions to it.     

Algorithms for Finding the Roots of Some Quadratic Octonion Equations

In this paper, we give an algorithm to find the roots of the octonionic quadratic equation x ² + bx + c = 0, and develop a Matlab package to find solutions. We also discuss how to find the roots of some other octonion quadratic equations, such as an algorithm is given for finding the roots of the octonion quadratic equation xax + bx + c = 0.     

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

The Allen-Cahn equation ? Δu = u ? u ³ in ?² has family of trivial singly periodic solutions that come from the one dimensional periodic solutions of the problem ?u″ =u ? u ³. In this paper we construct a non-trivial family of singly periodic solutions to the Allen-Cahn equation. Our construction relies on the connection between this equation and the infinite Toda lattice. We show that for each one-soliton solution to the infinite Toda lattice we can find a singly periodic solution to the Allen-Cahn equation, such that its level set is close to the scaled one-soliton. The solutions we construct are analogues of the family of Riemann minimal surfaces in ?³.     

Similarity reductions of a nonlinear model for vibrations of beams

In this paper we find exact solutions for a beam equation. We make a full analysis of the symmetry reductions of this equation by using the classical Lie method of infinitesimals. We present some explicit solutions: compacton solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Considerations in Relation to the Matrix Equation AXB = C

In this paper we represent a new form of condition for the consistency of the matrix equation AXB = C. If the matrix equation AXB = C is consistent, we determine a form of general solution which contains both reproductive and non-reproductive solutions. Also, we consider applications of the concept of reproductivity for obtaining general solutions of some matrix systems which are in relation to the matrix equation AXB = C.     

Unique continuation property for anomalous slow diffusion equation

A Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order 0 < α <1 are given. The estimate is derived through some subelliptic estimates for an operator associated to the anomalous diffusion equation using calculus of pseudo-differential operators.     

Bifurcation phenomena for nonlinear superdiffusive Neumann equations of logistic type

We consider a nonlinear Neumann logistic equation driven by the p-Laplacian with a general Carathéodory superdiffusive reaction. We are looking for positive solutions of such problems. Using minimax methods from critical point theory together with suitable truncation techniques, we show that the equation exhibits a bifurcation phenomenon with respect to the parameter λ > 0. Namely, we show that there is a λ_* > 0 such that for λ < λ_*, the problem has no positive solution; for λ = λ_*, it has at least one positive solution; and for λ > λ_*, it has at least two positive solutions.     

Spectral Analysis for Stationary Solutions of the Cahn–Hilliard Equation in ℝ d

We consider the spectrum associated with three types of bounded stationary solutions for the Cahn–Hilliard equation on ? ^d , d ≥ 2: radial solutions, saddle solutions (only for d = 2), and planar periodic solutions. In particular, we establish spectral instability for each type of solution. The important case of multiply periodic solutions does not fit into the framework of our approach, and we do not consider it here.     

On Globally Periodic Solutions of the Difference Equation x n + 1 = f ( x n )/ x n − 1

We study the second-order difference equation x n +1 = f ( x n ) x n m 1 where f ] C 1 ([0, X ),[0, X )) and x n ] (0, X ) for all n ] Z . For the cases p h 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p . We answer some questions and conjectures of Kulenovi ' and Ladas.     

A Computational Exploration of the Second Painlevé Equation

The pole field solver developed recently by the authors (J. Comput. Phys. 230:5957–5973, 2011) is used to survey the space of solutions of the second Painlevé equation that are real on the real axis. This includes well-known solutions such as the Hastings–McLeod and Ablowitz–Segur type of solutions, as well as some novel solutions. The speed and robustness of this pole field solver enable the exploration of pole dynamics in the complex plane as the parameter and initial conditions of the differential equation are varied. Theoretical connection formulas are also verified numerically.     

On the Matrix Equation X = Q − S∗X†S

The positive semidefinite solutions of the nonlinear matrix equation X + S? X?S = Q are investigated. We consider an iterative method converges to a positive semidefinite solution of this equation under the condition Ker X ? Ker S?. The new results are illustrated by numerical examples.     

Reaction-Diffusion Equations with Polynomial Drifts Driven by Fractional Brownian Motions

A reaction-diffusion equation on [0, 1] ^d with the heat conductivity κ > 0, a polynomial drift term and an additive noise, fractional in time with H > 1/2, and colored in space, is considered. We have shown the existence, uniqueness and uniform boundedness of solution with respect to κ. Also we show that if κ tends to infinity, then the corresponding solutions of the equation converge to a process satisfying a stochastic ordinary differential equation.     

The correct traveling wave solutions for the high-order dispersive nonlinear Schrödinger equation

The high-order dispersive nonlinear Schrödinger equation is considered. The exact solutions were obtained by Zhang et al. [J.L. Zhang, M.L. Wang, X.Z. Li, Phys. Lett. A 357 (2006) 188-195] are analyzed. We can demonstrate that some solutions do not satisfy this equation. To obtain the correct solutions, the F-expansion method is applied to solve it.     

Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space

We study the Emden–Fowler equation ?Δu = |u| ^p?1 u on the hyperbolic space ${{\mathbb H}^n}$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n ? 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.     

Non-Monotonic Behavior of One-Signed Solutions of Quasilinear Differential Equations

We consider a class of the second-order quasilinear differential equations. By deriving relations between certain types of monotonic solutions of the quasilinear equation and corresponding reciprocal half-linear equation on a finite interval (a, b), we obtain criteria for all solutions of the main equation, which do not change sign in (a, b), to be non-monotonic in (a, b). This work is also extended to a perturbed half-linear equation as well as to the half-line \((a,\infty )\).     

Combinatorial constructions of optimal (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>, 4, 2) optical orthogonal signature pattern codes

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code division multiple access (OCDMA) network for 2-dimensional image transmission. There is a one-to-one correspondence between an \((m, n, w, \lambda )\)-OOSPC and a \((\lambda +1)\)-(mn, w, 1) packing design admitting a point-regular automorphism group isomorphic to \({\mathbb {Z}}_m\times {\mathbb {Z}}_n\). In 2010, Sawa gave the first infinite class of (m, n, 4, 2)-OOSPCs by using S-cyclic Steiner quadruple systems. In this paper, we use various combinatorial designs such as strictly \({\mathbb {Z}}_m\times {\mathbb {Z}}_n\)-invariant s-fan designs, strictly \({\mathbb {Z}}_m\times {\mathbb {Z}}_n\)-invariant G-designs and rotational Steiner quadruple systems to present some constructions for (m, n, 4, 2)-OOSPCs. As a consequence, our new constructions yield more infinite families of optimal (m, n, 4, 2)-OOSPCs. Especially, we see that in some cases an optimal (m, n, 4, 2)-OOSPC can not achieve the Johnson bound. We also use Witt’s inversive planes to obtain optimal \((p, p, p+1, 2)\)-OOSPCs for all primes \(p\ge 3\).     

Optimal Bounds for Self-Similar Solutions to Coagulation Equations with Product Kernel

We consider mass-conserving self-similar solutions of Smoluchowski's coagulation equation with product kernel of homogeneity 2λ ∈ (0, 1). We establish rigorously that such solutions exhibit a singular behavior of the form x ^?(1+2λ) as x → 0. This property had been conjectured, but only weaker results had been available up to now.     

Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed c* > 0, and prove the existence of waves when c ≥ c* and the nonexistence when 0 ≤ c < c*.