On the extremal function of the modulus of a foliation

We investigate the properties of the modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of the extremal function and state some of its properties. We obtain an integral formula which, in a sense, combines the integral over the manifold with the integral over the leaves. We state a relation between the extremal function and the geometry of the distribution orthogonal to a foliation.     

On the uniqueness of the solution of nonlinear differential-algebraic systems

We consider the Cauchy problem for a system of nonlinear ordinary differential equations unsolved for the derivative of the unknown vector function and identically degenerate in the domain. We prove a theorem on the coincidence of two smooth solutions of the considered problem. We show that, under some additional assumptions, the above-mentioned problem cannot have classical solutions with less smoothness. We obtain conditions under which the problem has a fixed finite number of solutions.     

Applications of the Frobenius Formulas for the Characters of the Symmetric Group and the Hecke Algebras of Type A

We give a simple combinatorial proof of Ram's rule for computing the characters of the Hecke Algebra. We also establish a relationship between the characters of the Hecke algebra and the Kronecker product of two irreducible representations of the Symmetric Group which allows us to give new combinatorial interpretations to the Kronecker product of two Schur functions evaluated at a Schur function of hook shape or a two row shape. We also give a formula for the regular representation of the Hecke algebra.     

Application of graph-theoretic approaches to the random landscapes of the three-dimensional assignment problem

We study local optima of combinatorial optimization problems. We show that a local search algorithm can be represented as a digraph and apply recent results for spanning forests of a diagraph. We establish a correspondence between the number of local optima and the algebraic multiplicities of eigenvalues of digraph laplacians. We apply our finding to the three-dimensional assignment problem.     

On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum

We consider the sum of the Sturm-Liouville operator and a convolution operator. We study the inverse problem of reconstructing the convolution operator from the spectrum. This problem is reduced to a nonlinear integral equation with a singularity. We prove the global solvability of this nonlinear equation, which permits one to show that the asymptotics of the spectrum is a necessary and sufficient condition for the solvability of the inverse problem. The proof is constructive.     

Numerical study of the two-dimensional problem of the theory of elasticity using the boundary-element method

We propose a method of solving the two-dimensional problem of the theory of elasticity by a direct boundary-element method. We apply Galerkin's method with linear and quadratic approximations of the forces and displacements. We give the numerical results of solving a model problem that shows the effectiveness of the proposed approach.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 60–63.     

On the behaviour of the successive derivatives of meromorphic functions on the final set

We study the behaviour of the sequence of successive derivatives of meromorphic functions at points of the so-called final set. We prove that, whereas in many cases this sequence tends to ∞, for a special class of meromorphic functions, it may have extremely wild behaviour. We also prove a connection between the derivatives of meromorphic functions from this class and so-called Dirichlet sets.     

Estimation of the measure of the image of the ball

Under study is the class of ring Q-homeomorphisms with respect to the p-module. We establish a criterion for a function to belong to the class and solve a problem that stems from M. A. Lavrentiev [1] on the estimation of the measure of the image of the ball under these mappings. We also address the asymptotic behavior of these mappings at a point.     

Control of the spectrum of irregular oscillations of linear systems with coinciding ranks of the matrix multiplying the control and the extended matrix

We consider a linear periodic control system such that the ranks of the matrix multiplying the control and the extended matrix consisting of the averaged coefficient matrix and the matrix multiplying the control are the same. We assume that the control has the form of feedback linear in the state variables and is periodic with the same period as the system itself. We pose the problem of control of the frequency spectrum of strongly irregular periodic oscillations with an objective set, that is, the problem of finding a feedback coefficient such that the closed system has a strongly irregular periodic solution with the desired frequencies. We obtain necessary and sufficient conditions for the solvability of this problem.     

Evolution of the characteristics of the Prandtl solution

We transform the well-known Prandtl solution using point symmetries. This yields a new class of exact solutions to the plasticity equations. We dwell on those solutions that can be used to describe the flat flows arising as a plastic layer is compressed by the rigid plates.     

Controllability and influence of the spatial discretization of the beam equation

We study the problem of the control of a finite difference semi-discrete scheme for the 1D beam equation modelling the transverse vibrations of a beam with fixed ends. We show that for most numerical schemes, due to high frequency spurious solutions, the observability may be lost under numerical discretizations when the mesh size tends to zero. We then prove that the uniform observability property is obtained by truncating the high frequencies. We also show how spatial discretization can be performed by using our nonstandard finite difference scheme.     

On the isolated points in the space of groups

We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We also discuss a connection between isolated groups and solvability of the word problem.     

Estimates on the Distribution of the Condition Number of Singular Matrices

We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces. We prove the existence of relations between volumes and Intersection Theory in the presence of singularities. In particular, we can exhibit an average Bezout Equality for equidimensional varieties. We also state an upper bound for the volume of a tube about a projective variety. As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety. We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices.     

Two-body problem on spaces of constant curvature: I. Dependence of the Hamiltonian on the symmetry group and the reduction of the classical system

We consider the problem of two bodies with central interaction that propagate in a simply connected space with a constant curvature and an arbitrary dimension. We obtain the explicit expression for the quantum Hamiltonian via the radial differential operator and generators of the isometry group of a configuration space. We describe the reduced classical mechanical system determined on the homogeneous space of a Lie group in terms of orbits of the coadjoint representation of this group. We describe the reduced classical two-body problem. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 249–264, August, 2000.     

Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter ??, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of ??. We consider Bose??Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 < ?? < 2 is greater in the case under consideration than in the case of an ideal system, where ?? = 2.     

Effect of hyperviscosity on the geometry of the vorticity

We consider the two-dimensional modified Navier–Stokes equations by adding a p-Laplacian. We provide upper bounds on the Hausdorff length of the level sets of the vorticity. We express them in terms of the Landau–Lifschitz estimates of the number of degrees of freedom in turbulent flow. The proof, based on very standard methods, relies on a straightforward relation between the size of the nodal set of the vorticity and the growth of hyperdissipation.     

Ideals containing the squares of the variables

We study the Betti numbers of graded ideals containing the squares of the variables, in a polynomial ring. We prove the lex-plus-powers conjecture for such ideals.     

On the asymptotic behavior of eigenvalues of the radial p-laplacian

We study the asymptotic behavior and distribution of the eigenvalues of the singular radial p-laplacian. We prove a Weyl type asymptotic formula for the number of eigenvalues less than a given value.     

On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation

We investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear Schrödinger equation. We prove that, contrary to the deterministic case, any initial data gives birth to a solution which develops singularities. Moreover, the singularities appear immediately. We use a stochastic generalization of the variance identity and a control argument.     

On the rate of convergence in the individual ergodic theorem for the action of a semigroup

We consider the individual ergodic theorem for the action of a semigroup of measurepreserving mappings. We estimate the rate of convergence using estimates for the probability of large deviations for the ergodic averages with an essentially bounded averaging function. We find estimates for the rate of convergence of the ergodic averages in the cases of Benedicks–Carleson quadratic mappings, expanding mappings of Pomeau–Manneville type with a neutral point, and multidimensional shifts.