Absolute continuity of the spectrum of the periodic Schrödinger operator in a layer and in a smooth cylinder

The Schr?dinger operator H = −Δ + V is considered in a layer or in a d-dimensional cylinder. The potential V is assumed to be periodic with respect to a lattice. The absolute continuity of H is established, provided that V ∈ L _p,loc, where p is a real number greater than d/2 in the case of a layer and p > max(d/2, d − 2) for a cylinder. Bibliography: 14 titles.     

Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov–Takens bifurcation

This paper studies the dynamics implied by the Chamley (1993) model, a variant of the two-sector model with an implicit characterization of the learning function. We first show that under some “regularity” conditions regarding the learning function, the model has (a) one steady state, (b) no steady states or (c) two steady states (one saddle and one non-saddle). Moreover, via the Bogdanov–Takens theorem, we prove that for critical regions of the parameters space, the dynamics undergoes a particular global phenomenon, namely the homoclinic bifurcation. Because these findings imply the existence of a continuum of equilibrium trajectories, all departing from the same initial value of the predetermined variable, the model exhibits global indeterminacy.     

Solutions of the Schrödinger equation in the case of a semiinfinite crystal

It is shown that bounded solutions of Bloch type with respect to the variablesx ₁,x ₂ for a Schrödinger equation in which the potential is periodic in the half-space {x ₃0} and decreases exponentially asx ₃– can be approximated by the solutions of the Schrödinger equation for a thick film when the number of its layers tends to infinity. Under certain conditions, this makes it possible to find the number of linearly independent solutions of such kind.Physicotechnical Institute, Urals Scientific Center, Russian Academy of Sciences, Sverdlovsk. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 38–47, January, 1994.     

On the torsion in K_2 of a field

For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.     

The Cardinal of the Set of Non-quasi-regular Elements in a Ring

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Existence of stationary solutions of the Navier–Stokes equations in two dimensions in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

Existence of stationary solutions of the Navier–Stokes equations in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here, we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

On a transformation of the wiener process in ℝ m by a functional of the local time type on a surface

A transformation of the Wiener process _t in ^m is considered. This transformation is realized by a multiplicative functional _l=u(_l/u(₀), where the functionu is constructed in a certain way by using a functional of the local time type on a surface. It is proved that this transformation is equivalent to the successive application of an absolutely continuous change of a measure and killing on the surface.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 863–866, June, 1993.     

Estimating the Volumes of the Cones in a Gröbner Fan

We present a stochastic method for estimating the relative volumes of the Gröbner cones of a Gröbner fan without computing the actual fan. The method is particularly useful when the dimension of the Gröbner fan is large and/or the volumes of several or all cones need to be estimated. A Macaulay 2 implementation for uniform sampling from the Gröbner fan is provided by the author.     

On a certain ideal of Külshammer in the centre of a group algebra

Let G be a finite group and let F be a splitting field of characteristic $ p > 0 $ p > 0 . We show that I² = E₀, where I is a certain ideal of the centre Z of FG, and E₀ is the span of the block idempotents of defect zero.     

Harvesting of a prey–predator fishery in the presence of toxicity

In this model we discuss the bioeconomic harvesting of a prey–predator fishery in which both the species are infected by some toxicants released by some other species. Here both the species are harvested where we use the usual catch-per-unit-effort hypothesis. The dynamical behaviour of the exploited system is examined. The possibility of existence of a bionomic equilibrium is considered. The optimal harvesting policy is studied by using Pontryagin’s maximal principle. Some numerical examples and the corresponding solution curves are studied to illustrate the results of the model. Finally, the existence of limit cycle is discussed.     

Evolution of the Perturbation of a Circle in the Stokes–Leibenson Problem for the Hele-Shaw Flow

Among the initial contours of the Sobolev class H ¹ close to a circle, we distinguish the set of those for which the Stokes–Leibenson problem in the case of a source has a solution, moreover, this solution is unique. The contour corresponding to this solution is defined for all t > 0 and tends to a circle as t .     

Schrödinger equation. The theorem concerning the ansatz representation of a solution concentrated in a neighborhood of a minimum of the potential

The one-dimensional Schrödinger equation is considered on the segment [–l,l] It is assumed that the potential v(x) of this equation has one minimum v(0)=v(0)=0, v(0)>0 v(x)>0 for x0; v(x)h>0 outside some neighborhood of zero. It is proved that there exists a solution of the form where is a parabolic cylinder function, and is a smooth function which is bounded on [–l,l] together with derivatives through third order by a constant not depending on . The function and the real number E admit a known asymptotic expansion as 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 137–150, 1984.     

Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime

In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let $\{Y^{(a)}_{n}:n\ge1\}In this paper we derive a technique for obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each a>0, let {Y^(a)_n:n 3 1}\{Y^{(a)}_{n}:n\ge1\} be a sequence of independent and identically distributed random variables and {X^(a)_t:t 3 0}\{X^{(a)}_{t}:t\ge0\} be a Lévy process such that X₁^(a)=^dY₁^(a)X_{1}^{(a)}\stackrel{d}{=}Y_{1}^{(a)}, \mathbbEX₁^(a) < 0\mathbb{E}X_{1}^{(a)}<0 and \mathbbEX₁^(a)-0\mathbb{E}X_{1}^{(a)}\uparrow0 as a↓0. Let S^(a)_n=?_k=1ⁿ Y^(a)_kS^{(a)}_{n}=\sum _{k=1}^{n} Y^{(a)}_{k}. Then, under some mild assumptions, , for some random variable and some function Δ(⋅). We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.     

On the reconstruction of the potential in the inverse problem for a power of the Sturm—Liouville operator

In this paper, we prove the existence and uniqueness theorem for the inverse problem of spectral analysis for a power of the Sturm—Liouville operator on an interval with power > 3/4 and a quadratically summable, complex-valued, nonsymmetric potential. In this case, the potential is reconstructed by the mix of four spectra or by two spectra.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 8 , Suzdal Conference—2, 2003.This revised version was published online in April 2005 with a corrected cover date.     

On the Representation of Linear Continuous Functionals in Spaces of Analytic Functions of the Hardy–Sobolev Type in a Polydisk

We describe dual spaces of classes of the Hardy–Sobolev type of functions holomorphic in a polydisk for 0 < p 1 and q (0, ) and for p (1, ) and q = 1.     

Local stability “in the large” of the rigid layer of a sandwich plate in compression with bending

Experiments show that at certain geometrical and mechanical parameters of a sandwich plate subjected to compression with bending the loss of stability of its rigid layer occurs through a leap to a nonadjacent form of equilibrium. The current notion about the loss of stability of rigid layers of multilayer systems and calculation schemes correspond only to particular cases of the phenomenon and are true only in the framework of certain physical, mechanical, and geometrical characteristics of structures. The most general and closest to reality is the calculation method which takes into consideration the entire complex of deformations of all elements of a multilayer system. The study includes the experimental and theoretical analysis of the influence of bending rigidity, initial bend, and decisive deformational factors on the value of the critical force in the rigid layer of the plate having a light filler. The position of the local energetic extremum emerging in the process studied and the region where it is necessary to take into account the real character of the loss of stability in the large are determined.Moscow State University for Ecology and Environmental Studies, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 59–70, January–February, 1999.     

Effect of a measuring instrument in the “Bose condensate” of a classical gas in a phase transition and in experiments with negative pressure

We systematically present a new approach to classical thermodynamics using asymptotic distributions from number theory that generalize the Bose-Einstein distribution. We justify the transition to the liquid state, the thermodynamics of fluids, and also the behavior of liquids in the region of negative pressures We present a comparison with experimental data.     

σ-porosity of the set of strict contractions in a space of non-expansive mappings

We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uniform convergence. We show that the set of strict contractions is a σ-porous subset. If the underlying Banach space is separable, we exhibit a σ-porous subset of the space of non-expansive mappings outside of which all mappings attain the maximal Lipschitz constant one at typical points of their domain.