New power limits for extremes

For order statistics there is a deceptively simple link between affine and power norming, using exponential transforms. This link does not tell the whole story about limit distributions. The exponential transforms $W=e^{V}$ and $W=-e^{-V}$ yield limit variables which are either positive or negative. Under power norming there exist discrete limit distributions for maxima. The corresponding limit variables assume two values, one of which is zero. All variables with two values, one positive, one zero, are power limits for maxima. They are of different power type if they give different weight to zero, but they all have the same domain, the set of dfs with finite positive upper endpoint and an upper tail which varies slowly. So we see that convergence of types does not hold for power norming. This paper gives a classification of the power limits and their domains for maxima, variables conditioned to be large, and POTs (where power limits may assume three values). Convergence of sample clouds under power norming is studied, and of intermediate upper order statistics. The new power limits do not affect applications. Power norming is a viable alternative to classic extreme value theory. The extra norming constant in the exponent automatically improves the rate of convergence. Hill plots are a good instrument to determine this norming constant. It will be shown how to eliminate the bias of Hill plots and estimate high upper quantiles when the tail does not vary regularly or when convergence is slow.     

Characterization of Scattering Data for the AKNS System

We characterize the scattering data of the AKNS system with vanishing boundary conditions. We prove a 1,1-correspondence between L ¹-potentials without spectral singularities and Marchenko integral kernels which are sums of an L ¹ function (having a reflection coefficient as its Fourier transform) and a finite exponential sum encoding bound states and norming constants. We give characterization results in the focusing and defocusing cases separately.     

Inverse spectral problems for Dirac operator with eigenvalue dependent boundary and jump conditions

We deal with the Dirac operator with eigenvalue dependent boundary and jump conditions. Properties of eigenvalues, eigenfunctions and the resolvent operator are studied. Moreover, uniqueness theorems of the inverse problem according to the Weyl functions and the spectral data (the sets of eigenvalues and norming constants; two different eigenvalues sets) are proved.     

Stability Constants in Linear Spaces

We investigate the stability constants of convex sets in linear spaces. We prove that the stability constants of affinity and of the Jensen equation are of the same order of magnitude for every convex set in arbitrary linear spaces, even for functions mapping into an arbitrary Banach space. We also show that the second Whitney constant corresponding to the bounded functions equals half of the stability constant of the Jensen equation whenever the latter is finite. We show that if a convex set contains arbitrarily long segments in every direction, then its Jensen and Whitney constants are uniformly bounded. We prove a result that reduces the investigation of the stability constants to the case when the underlying set is the unit ball of a Banach space. As an application we prove that if D is convex and every δ-Jensen function on D differs from a Jensen function by a bounded function, then the stability constants of D are finite.     

Factorisation of Non-negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators

We study the problem of factorisation of non-negative Fredholm operators acting in the Hilbert space L₂(0, 1) and its relation to the inverse spectral problem for Bessel operators. In particular, we derive an algorithm of reconstructing the singular potential of the Bessel operator from its spectrum and the sequence of norming constants.     

Determination of Dirac operator with eigenvalue-dependent boundary and jump conditions

Dirac operator with eigenvalue-dependent boundary and jump conditions is studied. Uniqueness theorems of the inverse problems from either Weyl function or the spectral data (the sets of eigenvalues and norming constants except for one eigenvalue and corresponding norming constant; two sets of different eigenvalues except for two eigenvalues) are proved. Finally, we investigate two applications of these theorems and obtain analogues of a theorem of Hochstadt-Lieberman and a theorem of Mochizuki-Trooshin.     

Inverse spectral problems for non-self-adjoint Sturm–Liouville operators with discontinuous boundary conditions

This paper deals with the inverse spectral problem for a non-self-adjoint Sturm–Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential q is known a priori on a subinterval $$ \left[ b,\pi \right] $$ with $$b\in \left( d,\pi \right] $$ or $$b=d$$, then $$h,\,\beta ,\,\gamma $$ and q on $$\left[ 0,\pi \right] $$ can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the case $$b\in \left( 0,d\right) $$, a similar statement holds if $$\beta ,\,\gamma $$ are also known a priori. Moreover, if q satisfies a local smoothness condition, we provide an alternative approach instead of using the high-energy asymptotic expansion of the Weyl m-function to solve the problem of missing eigenvalues and norming constants.     

Normal form for generalized Hopf bifurcation with non-semisimple 1 ∶ 1 resonance

The primary result of this research is the derivation of an explicit formula for the Poincaré-Birkhoff normal form of the generalized Hopf bifurcation with non-semisimple 1:1 resonance. The classical nonuniqueness of the normal form is resolved by the choice of complementary space which yields a unique equivariant normal form. The 4 leading complex constants in the normal form are calculated in terms of the original coefficients of both the quadratic and cubic nonlinearities by two different algorithms. In addition, the universal unfolding of the degenerate linear operator is explicitly determined. The dominant normal forms are then obtained by rescaling the variables. Finally, the methods of averaging and normal forms are compared. It is shown that the dominant terms of the equivariant normal form are, indeed, the same as those of the averaged equations with a particular choice for the constant of integration.Partially supported by NSF through grant MSS 90-57437, AFOSR through grant 91-0041 and NSERC of Canada.     

Heat Conduction and Finite Measures for Transition Times between Steady States

A finite time, the mean action time associated with the conductivetransition from a constant initial temperature to thermal equilibriumat a constant ambient temperature, is related to time lag constants,mean energy residence times, mean first passage times, and Green'sfunction properties for linear equations, and to freezing timesand finite transition times for problems with phase transitions.When the boundary conditions are linear of mixed type and theconductivity is constant, or when they are of Dirichlet typeand the conductivity perhaps temperature dependent, the meanaction time is given by the solution of a linear Poisson problem.It is then easily found and is a useful finite comparative timefor the thermal transition process, giving a measure of itsdependence on size and other geometric factors.     

Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions

We present a priori and a posteriori estimates for the error between the Galerkin and a discretized Galerkin method for the boundary integral equation for the single layer potential on the square plate. Using piecewise constant finite elements on a rectangular mesh we study the error coming from numerical integration. The crucial point of our analysis is the estimation of some error constants, and we demonstrate that this is necessary if our methods are to be used. After the determination of these constants we are in the position to prove invertibility and quasioptimal convergence results for our numerical scheme, if the chosen numerical integration formulas are sufficiently precise. © 1992 John Wiley & Sons, Inc.     

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.     

On the inverse problem at fixed energy in the "Iterative" range

The scattering amplitude by a spherically symmetric potential at fixed energy is given in the Born approximation by a filtered Fourier transform, whose inverse is not unique. It is well known that matrix methods enable one to study exactly the problem at fixed energy in classes of potentials parametrised by sequences of numbers. In the range of potentials (or of phase shifts) where these methods can be managed by iteration, Born case is a limit. This article is a brief survey of the inverse problem (scattering amplitude?→?potential?) recalling how the nonuniqueness predicted in the Born approximation appears in these exact methods, showing henceforth that the inverse problem ill-posedness corresponds to physical features of the potential on which experiments at finite energy are unable to give information.     

Steady Deep-Water Waves on a Linear Shear Current

The properties of steady, periodic, deep-water gravity waves on a linear shear current are investigated. Numerical solutions for all waveheights, up to and including the limiting ones, are computed from a formulation which involves only the wave profile (parametrized in a natural way) and some constants of the motion. It is found that for some shear currents the highest waves are not necessarily those waves with sharp crests known as extreme waves. Furthermore a certain nonuniqueness in the sense of a fold is shown to exist, and a new type of limiting wave is discovered. For both small-amplitude waves and extreme waves the numerical results are compared with theoretical predictions.     

On the Nonlinear Schrödinger Equation on the Half Line with Homogeneous Robin Boundary Conditions

Boundary value problems for the nonlinear Schrödinger equations on the half line with homogeneous Robin boundary conditions are revisited using Bäcklund transformations. In particular: relations are obtained among the norming constants associated with symmetric eigenvalues; a linearizing transformation is derived for the Bäcklund transformation; the reflection‐induced soliton position shift is evaluated and the solution behavior is discussed. The results are illustrated by discussing several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self‐symmetric eigenvalues.     

Distributional expansion of maximum from logarithmic general error distribution

Logarithmic general error distribution is an extension of lognormal distribution. In this paper, with optimal norming constants the higher-order expansion of distribution of partial maximum of logarithmic general error distribution is derived.     

Comment on failure of excited-state energy calculation of analytical transfer matrix method for the potential V\left( x \right) = \frac{1} {2}kx^2 + \lambda x^4

The analytical transfer matrix method is applied to the quantum mechanical bound-state problem potential with V(x) = (1/2)kx ² + λx ⁴. It is found that numerical values of phase contribution are unstable when compared with standard methods like numerical WKB approximation, this leads to substantial errors in excited-state energy calculation.     

Non-central limit theorems for non-linear functional of Gaussian fields

Summary Let a stationary Gaussian sequence X _n , n=... –1,0,1, ... and a real function H(x) be given. We define the sequences ,n=... –1,0,1...; N=1,2, ... where A _N are appropriate norming constants. We are interested in the limit behaviour as N. The case when the correlation function r(n)=EX ₀ X _n tends slowly to 0 is investigated. In this situation the norming constants A> _N tend to infinity more rapidly than the usual norming sequence A> _N =N. Also the limit may be a non-Gaussian process. The results are generalized to the case when the parameter-set is multi-dimensional.This paper contains results closely connected to those of the paper by Taqqu, Z. Wahrscheinlichkeitstheorie verw. Gebiete 50, 53–83 (1979). The investigations were done independently and at about the same time. Different methods were usedDedicated to Professor Leopold Schmetterer on his sixtieth Birthday     

On well-posedness of the semilinear heat equation on the sphere

We are concerned with existence, uniqueness and nonuniqueness of nonnegative solutions to the semilinear heat equation in open subsets of the n-dimensional sphere. Existence and uniqueness results are obtained using L ^p ?? L ^q estimates for the semigroup generated by the Laplace?CBeltrami operator. Moreover, under proper assumptions on the nonlinear function, we establish nonuniqueness of weak solutions, when n??? 3; to do this, we shall prove uniqueness of positive classical solutions and nonuniqueness of singular solutions of the corresponding semilinear elliptic problem.