Nondegenerate Case Research Articles

HYPERKÄHLER ARNOLD CONJECTURE AND ITS GENERALIZATIONS

We generalize and refine the hyperkähler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperkähler Floer theory. In particular, we prove the conjecture in the case where the time manifold is a multidimensional torus and also establish the degenerate version of the conjecture. Our method relies on Morse theory for generating functions and a finite-dimensional reduction along the lines of the Conley–Zehnder proof of the Arnold conjecture for the torus.

Existence and stability of solutions for a strongly coupled system modelling thin fluid films

In this paper we consider a strongly coupled fourth order parabolic system modelling the motion of two thin fluid layers in the presence of gravity and surface tension. In the non-degenerate case we prove existence and uniqueness of strong solutions and study the stability of the equilibria for various fluid–fluid configurations.

A finite volume scheme for convection–diffusion equations with nonlinear diffusion derived from the Scharfetter–Gummel scheme

We propose a finite volume scheme for convection–diffusion equations with nonlinear diffusion. Such equations arise in numerous physical contexts. We will particularly focus on the drift-diffusion system for semiconductors and the porous media equation. In these two cases, it is shown that the transient solution converges to a steady-state solution as t tends to infinity. The introduced scheme is an extension of the Scharfetter–Gummel scheme for nonlinear diffusion. It remains valid in the degenerate case and preserves steady-states. We prove the convergence of the scheme in the nondegenerate case. Finally, we present some numerical simulations applied to the two physical models introduced and we underline the efficiency of the scheme to preserve long-time behavior of the solutions.

Real computation with least discrete advice: A complexity theory of nonuniform computability with applications to effective linear algebra

It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f:X∋x↦f(x)∈Y, additional structural information about the input x∈X (e.g. any kind of promise that x belongs to a certain subset X′⊆X, or does not) should be taken advantage of. In several examples from real number computation, such advice even makes the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable.Specifically, finding a nontrivial solution to a homogeneous linear equation A⋅x→=0 for a given singular real n×n-matrix A is possible when knowing rank(A)∈{0,1,…,n−1}; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors to) a given real symmetric n×n-matrix A is possible when knowing the number of distinct eigenvalues: an integer between 1 and n (the latter corresponding to the nondegenerate case). And again we show that n-fold (i.e. roughly logn bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas for finding some single eigenvector of A, providing the truncated binary logarithm of the dimension of the least-dimensional eigenspace of A—i.e. ⌊1+log2n⌋-fold advice—is sufficient and optimal.Our proofs employ, in addition to topological considerations common in Recursive Analysis, also combinatorial arguments.

Stochastic Approximation Method for Fixed Point Problems

We study iterative processes of stochastic approximation for finding fixed points of weakly contractive and nonexpansive operators in Hilbert spaces under the condition that operators are given with random errors. We prove mean square convergence and convergence almost sure (a.s.) of iterative approximations and establish both asymptotic and nonasymptotic estimates of the convergence rate in degenerate and non-degenerate cases. Previously the stochastic approximation algorithms were studied mainly for optimization problems.

Jump of the adiabatic invariant at a separatrix crossing: Degenerate cases

An expression for the quasi-random jump of the adiabatic invariant at a separatrix crossing is obtained for a slow–fast Hamiltonian system with two degrees of freedom in the case when the separatrix passes through a degenerate saddle point in the phase plane of the fast variables. The general case with an arbitrary degree of degeneracy was considered, and this degree is assumed to remain fixed in the process of evolution of the slow variables. The typical value of the jump is larger than in the non-degenerate case studied earlier. Though strongly degenerate, such a setting can be relevant for physical problems. The influence of the asymmetry of a phase portrait on the magnitude of adiabatic invariant jumps was considered as well. An example of this kind is studied, namely the motion of ions in current sheets with complex inner structure.

Compact source of narrow-band counterpropagating polarization-entangled photon pairs using a single dual-periodically-poled crystal

We propose a scheme for the generation of counterpropagating polarization-entangled photon pairs from a dual-periodically-poled crystal. Compared with the usual forward-wave-type source, this source, in the backward-wave way, has a much narrower bandwidth. With a 2-cm-long bulk crystal, the bandwidths of the example sources are estimated to be $3.6$ GHz, and the spectral brightnesses are more than 100 pairs$/$(s GHz mW). Two concurrent quasi-phase-matched spontaneous parametric down-conversion processes in a single crystal enable our source to be compact and stable. This scheme does not rely on any state projection and applies to both degenerate and nondegenerate cases, facilitating applications of the entangled photons.

Truncated matrix trigonometric problem of moments: operator approach

We study the truncated matrix trigonometric problem of moments. A parametrization of all solutions of this problem (both in the nondegenerate and degenerate cases) is obtained by using the operator approach. This parametrization establishes the one-to-one correspondence between a certain class of analytic functions and all solutions of the problem. We use the important Chumakin results on the generalized resolvents of isometric operators.

On the Weyl matrix balls associated with J-Potapov sequences in both nondegenerate and degenerate cases

In this paper, we discuss the Weyl matrix balls corresponding to an interpolation problem for J-Potapov functions in the open unit disk D, which was studied in [10]. In particular, we derive explicit expressions for the parameters of the Weyl matrix balls in terms of the given data. Furthermore, we examine the monotonicity of the sequences of the Weyl semi-radii. Finally, we discuss central J-Potapov functions, using their associated Weyl matrix balls.

Comments on: Algorithms for linear programming with linear complementarity constraints

This paper addresses the solution of linear programming problems that further incorporate complementarity restrictions involving designated pairs of complementary variables, as well as its generalization where the objective function is a nonlinear continuously differentiable function. Such problems subsume mixed linear complementary problems, absolute value programs, the linear bilevel programming problem, and the bilinear programming problem as well as more general nonconvex quadratic programs, among many others. The existing research on these problems focuses on finding strongly stationary points, or more general B-stationary points, both of which are necessary characterizations for local minima. However, computing B-stationary points is more combinatorial since it involves characterizing solutions that turn out to be Karush–Kuhn–Tucker (KKT) points for an entire family of nonlinear programs. In special nondegenerate cases (where at least one of each complementary pair is nonzero in any complementary solution) or under a special linear independence constraint qualification, B-stationary points also turn out to be strongly stationary. Among several algorithms for computing such strongly stationary or B-stationary points, the most competitive method to-date is a complementary active-set algorithm (CASET) and its advanced variants. In several studies, it has been verified that this procedure works well, even for degenerate problems, outperforming other nonlinear programming approaches such as penalty methods, non-smooth optimization techniques, interior point methods, and sequential quadratic programming algorithms. The CASET algorithm requires an initial complementary feasible solution to start with, which can always be computed by using an enumerative method.

NONLINEAR STABILITY OF PLANAR BOUNDARY LAYER SOLUTIONS FOR DAMPED WAVE EQUATION

We study the global stability of planar boundary layer solutions to the initial boundary value problem for the damped wave equation, in presence of a nonlinear convection term in the two-dimensional half space R+× R. By employing the energy method and a continuation argument, we establish that such an initial boundary value problem admits a unique global solution for a class of large initial perturbations, and that this global solution converges to the corresponding planar boundary layer solution, uniformly in (x, y) ∈ R+× R as t tends to infinity — provided the strength of the planar boundary layer solution is suitably small. Moreover, by exploiting the space–time weighted energy method and the properties of the planar boundary layer solutions, we also derive a convergence rate (both algebraic and exponential in nature) for the non-degenerate case.

Decay estimates of planar stationary waves for damed wave equations with nonlinear convection in multi-dimensional half space

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R + n : (I) { u t t − Δ u + u t + div f ( u ) = 0 , t > 0 , x = ( x 1 , x ′ ) ∈ R + n ( : = R + × R n − 1 ) , u ( 0 , x ) = u 0 ( x ) → u + , as x 1 → + ∞ , u t ( 0 , x ) = u 1 ( x ) , u ( t , 0 , x ′ ) = u b , x ′ = ( x 2 , x 3 , ⋯ , x n ) ∈ R n − 1 . For the non-degenerate case f 1 ′ ( u + ) < 0 , it was shown in [10] that the above initial-boundary value problem (I) admits a unique global solution u( t,x) which converges to the corresponding planar stationary wave ϕ( x 1) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u( t,x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f 1 ′ ( u b ) < 0 . The main purpose of this paper is devoted to discussing the case of f 1 ′ ( u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.

Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential

In the three-dimensional flat space, a classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller (J. Math. Phys. 48 (2007), 113518, 26 pages) have proved that, in the case of nondegenerate potentials, i.e. potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary quadratic Poisson algebra with five generators. The superintegrability of the generalized Kepler-Coulomb potential that was investigated by Verrier and Evans (J. Math. Phys. 49 (2008), 022902, 8 pages) is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the nondegenerate case of systems with quadratic integrals. In this paper, the ternary quadratic associative algebra corresponding to the quantum Verrier-Evans system is discussed. The subalgebras structure, the Casimir operators and the the finite-dimensional representation of this algebra are studied and the energy eigenvalues of the nondegenerate Kepler-Coulomb are calculated.

A multiple point Nevanlinna-Pick type interpolation problem with both interior and boundary data for Carathéodory matrix-valued functions

The main theme of this paper is a study of a multiple point Nevanlinna-Pick type interpolation problem with both interior and boundary data for Caratheodory matrix-valued functions. The so-called Toeplitz vector approach is applied to expose intrinsic connections between such an interpolation problem and a certain truncated trigonometric matrix moment problem with specified constraints that the nonnegative matrix-valued measure has no mass distributions at the boundary interpolation nodes. These connections, together with recent results due to Bolotnikov and Dym, enable us to get solvability criteria for both the Nevanlinna-Pick interpolation problem and the matrix moment problem under consideration. On the basis of the developed theory of matrix moments we can derive parameterized descriptions of all the solutions of these two problems in the nondegenerate case. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

A perturbative approach to inelastic collisions in a Bose–Einstein condensate

It has recently been discovered that for certain rates of mode-exchange collisions analytic solutions can be found for a Hamiltonian describing the two-mode Bose–Einstein condensate. We proceed to study the behaviour of the system using perturbation theory if the coupling constants only approximately match these parameter constraints. We find that the model is robust to such perturbations. We study the effects of degeneracy on the perturbations and find that the induced changes differ greatly from the non-degenerate case. We also model inelastic collisions that result in particle loss or condensate decay as external perturbations and use this formalism to examine the effects of three-body recombination and background collisions.

Large-time behaviour of solutions to the outflow problem of full compressible Navier–Stokes equations

This paper is concerned with the large-time behaviour of solutions to the outflow problem of full compressible Navier–Stokes equations on the half line . On the basis of fine analysis, we obtain two results. One is that the non-degenerate boundary layer is stable under partially large initial perturbation. The other is that the superpositions of the boundary layer (including the non-degenerate case) and the 3-rarefaction wave are asymptotically stable under some smallness conditions. The proofs are given by the elementary energy method.

Convergence rate of the solution toward boundary layer solution for initial-boundary value problem of the 2-D viscous conservation laws

In this paper, we study the large time behavior of the solution to the initial boundary value problem for 2-D viscous conservation laws in the space x ⩾ bt. The global existence and the asymptotic stability of a stationary solution are proved by Kawashima et al. [1]. Here, we investigate the convergence rate of solution toward the boundary layer solution with the non-degenerate case where f′( u +) − b < 0. Based on the estimate in the H 2 Sobolev space and via the weighted energy method, we draw the conclusion that the solution converges to the corresponding boundary layer solution with algebraic or exponential rate in time, under the assumption that the initial perturbation decays with algebraic or exponential in the spatial direction.

Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p 0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p 0 being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p 0 the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p 0 corresponds to a limit cycle γ 0. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ 0. We define the notion of vanishing multiplicity of the inverse integrating factor over γ 0. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p 0 in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.

On the sequences of the Weyl semi-radii associated with matricial Carathéodory and Schur sequences in both nondegenerate and degenerate cases

In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur.

Optimization of Band Structure and Quantum-Size-Effect Tuning for Two-Photon Absorption Enhancement in Quantum Dots

The two-photon absorption, 2PA, cross sections of PbS quantum dots, QDs, are theoretically and experimentally investigated and are shown to be enhanced with increasing quantum confinement. This is in contrast to our previous results for CdSe and CdTe QDs where the reduced density of states dominated and resulted in a decrease in 2PA with a decrease in QD size. Qualitatively this trend can be understood by the highly symmetric distribution of conduction and valence band states in PbS that results in an accumulation of allowed 2PA transitions in certain spectral regions. We also measure the frequency nondegenerate 2PA cross sections that are up to five times larger than for the degenerate case. We use a k·p four-band envelope function formalism to model the increasing trend of the two-photon cross sections due to quantum confinement and also due to resonance enhancement in the nondegenerate case.