A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

Bornologies and metrically generated theories

Bornologies axiomatize an abstract notion of bounded sets and are introduced as collections of subsets satisfying a number of consistency properties. Bornological spaces form a topological construct, the morphisms of which are those functions which preserve bounded sets. A typical example is a bornology generated by a metric, i.e. the collection of all bounded sets for that metric. In a recent paper [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547-1556] the authors noted that many examples are known of natural functors describing the transition from categories of metric spaces to the “metrizable” objects in some given topological construct such that, in some natural way, the metrizable objects generate the whole construct. These constructs can be axiomatically described and are called metrically generated. The construct of bornological spaces is not metrically generated, but an important large subconstruct is. We also encounter other important examples of metrically generated constructs, the constructs of Lipschitz spaces, of uniform spaces and of completely regular spaces. In this paper, the unified setting of metrically generated theories is used to study the functorial relationship between these constructs and the one of bornological spaces.     

Two-dimensional steady-state oscillation problems of anisotropic elasticity

The paper deals with the two-dimensional exterior boundary value problems of the steady-state oscillation theory for anisotropic elastic bodies. By means of the limiting absorption principle the fundamental matrix of the oscillation equations is constructed and the generalized radiation conditions of Sommerfeld-Kupradze type are established. Uniqueness theorems of the basic and mixed type boundary value problems are proved.     

Asymptotic behavior of eigenfunctions and eigenfrequencies of oscillation boundary value problems of the linear theory of elastic mixtures

The asymptotic behavior of eigenoscillation and eigen-vector-function is studied for the internal boundary value problems of oscillation of the linear theory of a mixture of two isotropic elastic media.     

A special case of de Branges' theorem on the inverse monodromy problem

We give a new proof of a special case of de Branges' theorem on the inverse monodromy problem: when an associated Riemann surface is of Widom type with Direct Cauchy Theorem. The proof is based on our previous result (with M.Sodin) on infinite dimensional Jacobi inversion and on Levin's uniqueness theorem for conformal maps onto comb-like domains. Although in this way we can not prove de Branges' Theorem in full generality, our proof is rather constructive and may lead to a multi-dimensional generalization. It could also shed light on the structure of invariant subspaces of Hardy spaces on Riemann surfaces of infinite genus.This work was supported by the Austrian Founds zur Förderung der wissenschaftlichen Forschung, project-number P12985-TEC     

Entire solutions in periodic lattice dynamical systems

This paper deals with entire solutions of periodic lattice dynamical systems. Unlike homogeneous problems, the periodic equation studied here lacks symmetry between increasing and decreasing pulsating traveling fronts, which affects the construction of entire solutions. In the bistable case, the existence, uniqueness and Liapunov stability of entire solutions are proved by constructing different sub- and supersolutions. In the monostable case, the existence and asymptotic behavior of spatially periodic solutions connecting two steady states are first established. Some new types of entire solutions are then constructed by combining leftward and rightward pulsating traveling fronts with different speeds and a spatially periodic solution. Various qualitative features of the entire solutions are also investigated.     

Instability of standing wave, global existence and blowup for the Klein-Gordon-Zakharov system with different-degree nonlinearities

This paper discusses the Klein-Gordon-Zakharov system with different-degree nonlinearities in two and three space dimensions. Firstly, we prove the existence of standing wave with ground state by applying an intricate variational argument. Next, by introducing an auxiliary functional and an equivalent minimization problem, we obtain two invariant manifolds under the solution flow generated by the Cauchy problem to the aforementioned Klein-Gordon-Zakharov system. Furthermore, by constructing a type of constrained variational problem, utilizing the above two invariant manifolds as well as applying potential well argument and concavity method, we derive a sharp threshold for global existence and blowup. Then, combining the above results, we obtain two conclusions of how small the initial data are for the solution to exist globally by using dilation transformation. Finally, we prove a modified instability of standing wave to the system under study.     

Global existence and causality for a hyperbolic transmission problem with a repulsive nonlinearity

It is well known that the solution of the classical linear wave equation with an initial condition with compact support and vanishing initial velocity also has a compact support included in a set depending on time: the support of the solution at time t$t$ is causally related to that of the initial condition. Reed and Simon have shown that for a real-valued Klein–Gordon equation with (nonlinear) right-hand side −λu³$-\lambda u^3$ (λ>0$\lambda >0$), causality still holds. We show the same property for a one-dimensional Klein–Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side F(u)$F\left(u\right)$. We also prove the global existence of a solution using the repulsiveness of F$F$. In the particular case F(u)=−λu³$F\left(u\right)=-\lambda u^3$, the problem is a relativistic model for a quantum particle with repulsive self-interaction and tunnel effect at a semi-infinite potential step.     

A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation

We present a systematic study of local solutions of the ODE of the form near t=0. Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ>0 and γ<0, where γ=∂f/∂x^′ at t=0. As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.     

Towards a theory of classification

The well-known difficulties arising in a classification which is not set-theoretically trivial—involving what is sometimes called a non-smooth quotient—have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor—the very existence of which is already a surprise. Here the notion of such a functor is developed abstractly, and a number of examples are considered (including those which have arisen for various classes of operator algebras).     

Continuous transformations of differential equations with delays

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer's transformationz(t)=g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of thenth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions are considered.     

ANALYSIS OF THE LIMIT CYCLE OF A GENERALISED VAN DER POL EQUATION BY A TIME TRANSFORMATION METHOD

Abstract

The properties of the limit cycle of a generalised van der Pol equation of the form ü + u = ε (1—u²ⁿ)u, where ε is small and n is any positive integer, are investigated by applying a time transformation perturbation method due to Burton. It is found that as n increases the amplitude of the limit cycle oscillation decreases and its period increases. The time transformation solution is compared with the solution derived using the method of multiple scales and with a numerical solution. It is found that, to first order in ε, the time transformation solution for the limit cycle agrees better with the numerical solution than the multiple scales solution. Both perturbation solutions give the same result for the period of the limit cycle to second order in ε. The accuracy of the time transformation solution decreases as n increases.     

Optimality conditions with state constraints for semilinear systems determined by operator valued measures

In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented.     

Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system

We study the traveling waves for a lattice dynamical system with monostable nonlinearity in periodic media. It is well known that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first derive a stability theorem for certain waves of non-minimal speed. Moreover, we show that wave profiles of a given speed are unique up to translations.     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

An Inverse Spectral Problem for a Nonsymmetric Differential Operator: Uniqueness and Reconstruction Formula

We consider an eigenvalue problem for a system on [0, 1]: $$\left\{ {\begin{array}{*{20}l} {\left[ {\left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right)\frac{{\text{d}}} {{{\text{d}}x}} + \left( {\begin{array}{*{20}c} {p_{11} (x)} & {p_{12} (x)} \\ {p_{21} (x)} & {p_{22} (x)} \\ \end{array} } \right)} \right]\left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(2)} (x)} \\ \end{array} } \right) = \lambda \left( {\begin{array}{*{20}c} {\varphi ^{(1)} (x)} \\ {\varphi ^{(1)} (x)} \\ \end{array} } \right)} \\ {\varphi ^{(2)} (0)\cosh \mu - \varphi ^{(1)} (0)\sinh \mu = \varphi ^{(2)} (1)\cosh \nu + \varphi ^{(1)} (1)\sinh \nu = 0} \\ \end{array} } \right.$$ with constants $$\mu ,\nu \in \mathbb{C}.$$ Under the assumption that p₂₁, p₂₂ are known, we prove a uniqueness theorem and provide a reconstruction formula for p₁₁ and p₁₂ from the spectral characteristics consisting of one spectrum and the associated norming constants.