Mechanical systems related to the Schrödinger spectral problem

We present a variety of integrable mechanical systems which are embedded into the KdV and the Harry Dym hierarchies of soliton equations as their restricted flows. Integrable structures of these systems are systematically derived from the properties of the underlying hierarchies of equations. It is illustrated by the key example of the Garnier system which describes the motion of a particle in a quartic ‘wine bottle’ potential.     

Two super-integrable hierarchies and their super-Hamiltonian structures

Two Lie super algebras are constructed from which we establish two super-isospectral problems. Under the frame of the zero curvature equations, the super-GJ hierarchy and the super-Yang hierarchy are presented respectively. Meanwhile, their super-Hamiltonian structures are obtained by using super-trace identity.     

A question related to the Isbell problem

Given two compatible metrics on a metrizable space X. It is well known that they give rise to the same Hausdorff hypertopologies and upper Hausdorff hypertopologies, on the collection of all closed subsets of X, if and only if they are uniformly equivalent. This is no longer true for the lower Hausdorff hypertopology; indeed a weaker condition is needed, and this condition has been found by Costantini and Vitolo.     

Three problems related to the Kummer problem

We give a brief survey of the principal results concerning the individual Kummer problem and present new results of the author concerning this problem. Institute of Cybernetics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1061–1068, August, 1997.     

Positive and negative hierarchies of nonlinear integrable lattice models and three integrable coupling systems associated with a discrete spectral problem

Positive and negative hierarchies of nonlinear integrable lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct three integrable coupling systems of the positive hierarchy through enlarging Lax pair method.     

Two logical hierarchies of optimization problems over the real numbers

We introduce and study certain classes of optimization problems over the real numbers. The classes are defined by logical means, relying on metafinite model theory for so called R‐structures (see [12, 11]). More precisely, based on a real analogue of Fagin's theorem [12] we deal with two classes MAX‐NPR and MIN‐NPR of maximization and minimization problems, respectively, and figure out their intrinsic logical structure. It is proven that MAX‐NPR decomposes into four natural subclasses, whereas MIN‐NPR decomposes into two. This gives a real number analogue of a result by Kolaitis and Thakur [13] in the Turing model. Our proofs mainly use techniques from [17]. Finally, approximation issues are briefly discussed. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Increasing the accuracy of solutions to a discrete spectral problem

Two methods for increasing the accuracy of discrete Sturm — Liouville problems are examined. In one of these the principal term of the expansion for the error of the eigenvalues by small parameter steps is used. The other method is based on a minimized functional, which corresponds to a discrete scheme of fourth-order accuracy relative to the discretization parameter.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 56–63, 1989.     

N=2 supersymmetric quantum mechanics and the inverse scattering problem

Physicotechnical Institute, Ukrainian SSR Academy of Sciences, Khar'kov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 74, No. 3, pp. 392–398, March, 1988.     

On a problem related to the Vandermonde determinant

This short note describes new properties of the elementary symmetric polynomials, and reveals that the properties give an answer to the conjecture raised by El-Mikkawy in [M.E.A. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices—II, Appl. Math. Comput. 146 (2003) 759-769].     

An inverse spectral problem

In this article an infinite periodic Jacobi matrix is under consideration. It is shown that the spectrum of the matrix consists of a single finite interval if and only if the period of the matrix is equal to unity.     

Some of the combinatorics related to Michael's problem

We present some new methods for constructing a Michael space, a regular Lindelöf space which has a non-Lindelöf product with the space of irrationals. The central result is a combinatorial statement about the irrationals which is a necessary and sufficient condition for the existence of a certain class of Michael spaces. We also show that there are Michael spaces assuming and that it is consistent with that there is a Michael space. The influence of Cohen reals on Michael's problem is discussed as well. Finally, we present an example of a Michael space of weight less than under the assumption that (whose product with the irrationals is necessarily linearly Lindelöf).

     

On two estimates related to the change-point problem

We consider the problem of estimating a smooth functional of an unknown signal with discontinuity from Gaussian observations. The signal is a known function depending on an unknown parameter. This problem is closely related to the famous change-point problem. We obtain an asymptotic likelihood ratio process for the noise level tending to 0. Bayesian and maximum likelihood estimates are constructed and their relative efficiency is studied. Some simulation results and conclusions on non-asymptotic behavior of these estimates are presented.     

Minimization of energies related to the plate problem

We investigate the interior regularity of minimizers for an obstacle problem of higher order that can be seen as a model for the behaviour of a plate subject to a rather general constitutive law including nonlinear elastic materials. Copyright © 2008 John Wiley & Sons, Ltd.     

Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems

Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems are proposed. The generalized bi-Hamiltonian structures for one of the two hierarchies are derived with the aid of the trace identity. Some explicit solutions of a typical nonlinear evolution equation in the hierarchy are obtained, which include soliton and periodic solutions.     

An extremal problem related to probability

Summary We consider a walk from a stateA ₁ to a stateA _n+1 in which the probability of remaining atA _i isp _i , and the probability of progressing fromA _i toA _i+1 is 1 –p _i . The probabilityW _nk of reachingA _n+1 fromA ₁ in exactlyn + k steps can then be expressed as a polynomial of degreen + k in then variablesp ₁,,p _n . We determine the maximum value ofW _nk and the (unique) choice (p ₁,,p _n ) for which this extremum occurs.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday