A central limit theorem on gln(fq)

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The eigenvalues and eigenfunctions of a spherically symmetric anharmonic oscillator

We show that the operator H_s has a complete set of eigenfunctions and eigenvalues , which satisfy [2l(l + 1) - (3n² + 3n + 1)]s + o(s) and lim_s→0 = 0. The functions are given in spherical coordinates as a product of generalized Laguerre functions and spherical harmonics.     

Analysis of the thermoelastic model of a plate with non-linear shape memory alloy reinforcements

In this paper we study the questions of the existence and uniqueness of the solutions for a thermoelastic system of equations in a two-dimensional domain, where both the viscosity v and the rigidity D are positive. It seems that such a system has up to now been considered in a one-dimensional setting only. The change of dimensions enforces the growth conditions with respect to θ and the additional regularity of the data. The existence of solutions in the case of the Neumann boundary conditions for θ and some weak regularity of data is proved. Under stronger regularity conditions the uniqueness is also established. The system has an interpretation as a plate reinforced by a shape memory alloy (SMA) wire mesh.     

A finite element method and stabilization method for a non-linear tricomi problem

We prove using the Faedo-Galerkin method the existence of a generalized solution of an initial-boundary value problem for the non-linear evolution equation 0 ? Q ? 2, in a cylinder Q_T = Ω × (0, T), where ?? u = yu_xx + u_yy is the Tricomi operator and l(u) a special differential operator of first order. We then show that the approximate generalized solution of problem (*) converges to the approximate generalized solution of the corresponding stationary boundary value problem as t → ∞.     

A new approach to queer differential equations

We study the variational problem Where Ψ* is the increasing rearrangement of Ψ. An approximate problem is introduced which involves a variational problem with n free boundaries (n → ∞). Various estimates are established. In particular, when Ω is convex we show that the solution to the approximate problem is superharmonic and has bounded gradient.     

Stabilization and regularization for solutions of an ill posed problem for the wave equation

In this paper we study the problem of finding u: [O, T] → D(A), D(A) ? H, H a Hibert space such that: Is a linear positive, self-adjoint operator with a compact inverse. The problem is well known to be illposed because uniqueness and existence generally fail. We restore the stability with an a priori bound on ∥ du(0)/dt ∥ for some particular values of T.     

Generalizations of the neumann system a curve-theoretical approach-part I

The following observation, due to E. Trubowitz, illustrates an intimate relationship between spectral theory and Hamiltonian mechanics in the presence of constraints. Let q(s) be a real periodic function such that the Hill operator, has only a finite number g^R of simple eigenvalues. There exist g^R + 1 periodic eigenfunctions x₁,…, x and corresponding eigenvalues a₁,…, a of L such that where y_r = dx_r/ds. The equations Lx_r = a_rx_r, r = 1,…,g_R+1, make up the classical Neumann system, a system of harmonic oscillators constrained to the unit sphere. H. Flaschka obtained similar results about the Neumann system from a more general point of view. His assumption, that there exists an operator of odd order that commutes with L, leads to algebraic curve theory by the method of Krichever and from there to the Neumann formulas above. The familiar Lax pairs, the constants of motion and the quadrics of the Neumann system emerge as consequences of the Riemann-Roch theoreni. The existence of isospectral deformations of L, the Korteweg de-Vries hierarchy of the soliton equations, underlies the complete integrability of the Neumann system. This paper extends Flaschka's techniques, replacing L by an operator of order n 2 2. Higher Neumann systems are defined in a way that leads naturally to interesting symplectic manifolds, Lax pairs and integrals of motion. C. Tomei, using scattering theory, obtained some of our n = 3 formulas.     

Existence of solutions to some non-linear thermoelastic systems with viscosity

In this paper a non-linear system of equations in a one-dimensional domain is considered, where both the viscosity, v, and R are positive. The existence of solutions in the case of Neumann boundary conditions for θ and a polynomial form of p(θ, ?) is proved. When θ satisfies the Dirichlet boundary condition, the existence of solutions is obtained under additional assumptions limiting the growth of p(θ, ?) with respect to θ. In both cases the uniqueness of the solutions is also established.     

On a neutral Lotka–Volterra system

Sufficient conditions are obtained for the linear stability of the positive equilibrium of the neutral system in terms of the parameters of the system. The case n=2 is considered in detail and the general case is discussed briefly.     

Hamilton cycles in claw-free graphs

Bondy conjectured that if G is a k-connected graph of order n such that for any (k + 1)-independent set / of G, then the subgraph outside any longest cycle contains no path of length k ? 1. In this paper, we are going to prove that, if G is a k-connected claw-free (K_1,3-free) graph of order n such that for any (k + 1)-independent set /, then G contains a Hamilton cycle. The theorem in this paper implies Bondy's conjecture in the case of claw-free graphs.     

On critical exponents for some quasilinear parabolic equations

We study the Cauchy problem for the quasilinear parabolic equation where p > 1 is a parameter and ψ is a smooth, bounded function on (1, ∞) with ? ? sψ′(s)/ψ(s) ? θ for some θ > 0. If 1 < p < 1 + 2/N, there are no global positive solutions, whereas if p > 1 + 2/N, there are global, positive solutions for small initial data.     

Compatibility Conditions and the Manifold of Solutions of some Nonlinear Cauchy-Riemann System in Several Complex Variables

The system of differential equations Is studied. On the assumption that the system admits certain class of solutions, necessary conditions of compatibility are deduced, and by introducing some changes of variables the system is reduced to a single differential equation, then conversely by solving it, the formula giving all local solutions of the system is infered. The validity of the factorization theorem and the identity theorem for solutions in the class is also treated.     

A remark on N. Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem

Recently N. Korevaar developed a method of proving that solutions to elliptic and parabolic boundary value problems on convex domains ω ? R ⁿ are convex functions. He introduced a concavity function and used the classical maximum principle to prove that C ? 0 on ω × ω, i.e. that u is convex. Both he and independently L. Caffarelli and J. Spruck applied this method successfully to various boundary value problems. In this note we weaken the assumptions of their theorems and obtain some interesting new applications which are not covered by their previous results [CS, Ko].     

On the riemann problem for a prototype of a mixed type conservation law

We study the system of conservation laws given by With initial value The system is elliptic when u² + v² < ρ² and hyperbolic when u² + v² ≧ ρ². Following Liu's construction it is found that the system always has a weak solution which however is not necessarily unique.     

Spectral Decomposition of Symmetric Operator Matrices

The authors study symmetric operator matrices in the product of Hilbert spaces H = H₁×H₂, where the entries are not necessarily bounded operators. Under suitable assumptions the closure L_o exists and is a selfadjoint operator in H. With L_o, the closure of the transfer function is considered. Under the assumption that there exists a real number β < inf p(A) such that M(β)<< 0, it follows that β ε p(L_o). Applying a factorization result of A.I. Virozub and V.I. Matsaev [VM] to the holomorphic operator function M(λ, the_spectral subspaces of L_o corresponding to the intervals ] — ∞, β] and [β, ∞[ and the restrictions of L_o to these subspaces are characterized. Similar results are proved for operator matrices which are symmetric in a Krein space.     

Spatially localized free vibrations for a non-linear string model

We prove the existence of infinitely many non-zero time-periodic solutions (breathers) to the dispersive wave equation of the form which are localized in the spatial variable, that is The main tool employed is the concentration compactness principle of P. L. Lions.     

Periodic solutions of quasilinear non-autonomous systems with impulses

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the form where x ? R ⁿ, ε is a small parameter, Sufficient conditions have been found for existence of the periodic solution of the given system in the critical and non-critical cases.     

Weighted (LF)-spaces of Continuous Functions

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Stability and Oscillations of Solutions of Integro-Differential Equations of Pendulum—like Systems

A system of nonlinear differential equations of the type on a domain of ?ⁿ is studied. Functional relations between the f_j's, j = 1, …, n, and other necessary conditions are deduced when at each point of the domain the system has a manifold of local solutions. A structure theorem, that makes possible to reduce the problems of the system, e.g. the global solvability of it, to the corresponding questions for a connection of the type ?_z?w = g(z, w) in a fibre bundle over a Riemann surface is proved, and through this reduction we obtain theorems of identity, extension, global factorization, and so on, for the solutions of the system. As an example, a system of nonlinear differential equations of the type is studied and its global solutions are constructed.     

On a free boundary problem in electrostatics

Let Ω_i ? ?^N, i = 0, 1, be two bounded separately star-shaped domains such that $ \Omega _0 \supset \bar \Omega _1 $. We consider the electrostatic potential u defined in $ \Omega : = \Omega _0 \backslash \bar \Omega _1 $: The geometry of the two boundary components Γ₀ and Γ₁ is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.