“Electron ping-pong” on a one-dimensional lattice: Wave packet motion up to the first reflection

We consider the time evolution of the electron wave function on a one-dimensional lattice consisting of N identical sites and a single impurity site that differs from the rest of the lattice by the on-site energy and the hopping integral. At the initial instant, the wave function is entirely localized at the impurity site. With time, a localized wave packet forms, which moves over the lattice and reflects from its end. The process of reflection from the ends repeats many times and is called “electron ping-pong.” We obtain analytic expressions for the propagation of the wave packet front at different values of the problem parameters. The analytic expressions agree perfectly with numerical simulation results.     

On the “function” and “lattice” definitions of a narrow operator

We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).     

Spectrum of the three-particle Schr?dinger operator on a one-dimensional lattice

We consider a system of three arbitrary quantum particles on a one-dimensional lattice interacting pairwise via attractive contact potentials. We prove that the discrete spectrum of the corresponding Schr?dinger operator is finite for all values of the total quasimomentum in the case where the masses of two particles are finite. We show that the discrete spectrum of the Schr?dinger operator is infinite in the case where the masses of two particles in a three-particle system are infinite.     

Global stability of traveling wave fronts for a reaction–diffusion system with a quiescent stage on a one-dimensional spatial lattice

This paper is concerned with the stability of traveling wave fronts for a coupled system of non-local delayed lattice differential equations with a quiescent stage. It shows that under certain conditions all non-critical traveling wave fronts are globally exponentially stable, and critical ftraveling wave fronts are globally algebraically stable by applying the weighted energy method and the semi-discrete Fourier transform.     

Evolutionary trees and the Ising model on the Bethe lattice: a proof of Steel��s conjecture

A major task of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on a tree. Given samples from the leaves of the Markov chain, the goal is to reconstruct the leaf-labelled tree. It is well known that in order to reconstruct a tree on n leaves, sample sequences of length ??(log n) are needed. It was conjectured by Steel that for the CFN/Ising evolutionary model, if the mutation probability on all edges of the tree is less than ${p^{\ast} = (\sqrt{2}-1)/2^{3/2}}$ , then the tree can be recovered from sequences of length O(log n). The value p* is given by the transition point for the extremality of the free Gibbs measure for the Ising model on the binary tree. Steel??s conjecture was proven by the second author in the special case where the tree is ??balanced.?? The second author also proved that if all edges have mutation probability larger than p* then the length needed is n ^??(1). Here we show that Steel??s conjecture holds true for general trees by giving a reconstruction algorithm that recovers the tree from O(log n)-length sequences when the mutation probabilities are discretized and less than p*. Our proof and results demonstrate that extremality of the free Gibbs measure on the infinite binary tree, which has been studied before in probability, statistical physics and computer science, determines how distinguishable are Gibbs measures on finite binary trees.     

What information about a wave function is given by “measuring” it with the help of the tomographic method of recovering the Wigner distribution?

Mathematical problems connected with “measuring” a wave function by the tomographic method of recovering the Wigner distribution are considered. Inequalities showing what information is contained in a finite number of projections of the Wigner distribution are obtained. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 189–200.     

Essential and discrete spectra of the three-particle Schrödinger operator on a lattice

We study the position of the essential spectrum of a three-body Schrödinger operator H. We evaluate the lower boundary of the essential spectrum of H and prove that the number of eigenvalues located below the lower edge of the essential spectrum in the H model is finite.     

Method of integral equations in the scalar problem of diffraction on a system consisting of a “soft” and a “hard” screen and an inhomogeneous body

We consider the scalar problem on the diffraction of a plane wave on a system of two screens with boundary conditions of the first and the second kind and a solid inhomogeneous body in the semiclassical setting. The original boundary value problem for the Helmholtz equation is reduced to a system of singular integral equations over the body domain and the screen surfaces. We prove the equivalence of the integral and differential statements of the problem, the solvability of the system of integral equations in Sobolev spaces, and the smoothness of its solutions. To solve the integral equations approximately, we use the Bubnov-Galerkin method; we introduce basis functions on the body and the screens and prove the consistency and convergence of the numerical method.     

On the Pólya conjecture concerning the maximum and minimum of the modulus of an entire function of finite order given by a lacunary power series

, (fz) , ,

“Quasiphoton” in an active medium. II. An example: The one-dimensional wave equation with a small deviating argument

In the present work we use the example of the one-dimensional wave equation with a small deviating argument to illustrate the constructions of the author's previous paper Quasiphoton in an active medium. The significance of the concept of propagation velocity introduced there, as well as the relation between the change in the propagation frequency along a ray and the parameters characterizing the concentrated solution, are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 136–142, 1986.In conclusion, the author wants to express his sincrere gratitude to I. A. Molotkov for suggesting this problem, and to the participants of the diffraction seminar of LGU-LOMI for stimulating discussions.     

Uniformization and the Poincaré metric on the leaves of a foliation by curves

In this paper we prove that a holomorphic foliation by curves, on a complex compact manifoldM, whose singularities are non degenerated and whose tangent line bundle admits a metric of negative curvature, satisfies the following properties:(a): All leaves are hyperbolic.(b): The Poincaré metric on the leaves is continuous.(c): The set of uniformizations of the leaves by the Poincaré disc D is normal. Moreover, if ( _n ) _n 1 is a sequence of uniformizations which converges to a map : D, then either is a constant map (a singularity), or is an uniformization of some leaf. This result generalizes Theorem B of [LN], in which we prove the same facts for foliations of degree 2 on projective spaces.This research was partially supported by Pronex-Dynamical Systems, FINEP-CNPq.     

The lattice point discrepancy of a body of revolution: Improving the lower bound by Soundararajan’s method

For a convex body in which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy (number of integer points minus volume) of a linearly dilated copy is estimated from below. On the basis of a recent method of K. Soundararajan [16], an -bound is obtained that improves upon all earlier results of this kind.Dedicated to the memory of Professor Erich LamprechtThis revised electronic version of the Abstract includes the formulas that were missing in the previous electronic version published online in September 2004.     

Extension of a Function from the Exterior of an Interval to a Positive-Definite Function on the Entire Axis and an Approximation Characteristic of the Class W M r, β

We establish sufficient conditions for the extension of a function defined on [a, +), where a > 0, to a positive-definite function on the entire axis.     

Global existence of solutions for the one-dimensional motions of a compressible viscous gas with radiation: An “infrarelativistic model”

We consider an initial–boundary value problem for the equations of 1D motions of a compressible viscous heat-conducting gas coupled with radiation through a radiative transfer equation. Assuming suitable hypotheses on the transport coefficients, we prove that the problem admits a unique weak solution.     

Erratum to “On a Classical Theorem on the Diameter and Minimum Degree of a Graph”

The original version of the article was published in[1]. Unfortunately, the original version of this article contains a mistake:in Theorem 6.2 appears that β(n, Δ)=(n-Δ+5)/4 but the correct statement is β(n, Δ)=(n-Δ+4)/4. In this erratum we correct the theorem and give the correct proof.     

Correction to the paper “on the curvature of a generalization of a contact metric manifolds”

We considered in Example 3.1 of the paper [1] an S-structure on R^2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.     

From design-based research to re-sourcing ‘in the wild’: reflections on studies of the co-evolution of mathematics teaching resources and practices

This commentary paper looks across the studies of the design and use of mathematics teaching resources included in this issue. It analyses everyday and educational notions of resource; particularly how usage of that term varies across the papers. Key characteristics of each study are identified and the studies are organized into five broad groups representing different lines of investigation of resource design and use. Across the studies, the broad notion of appropriation is influential in conceptualizing user/tool relations, with the instrumental/documentational approach particularly prominent, although close inspection shows that researchers draw on this in different ways. A majority of the studies relate to professional development interventions in a context of large-scale reform efforts, offering insights into a range of approaches to providing practical support for teacher appropriation of resources. One study investigates the resource systems established by ordinary teachers: alongside other research, this highlights the challenges of a re-sourcing approach in which teachers collaborate to curate a localized resource system, and points to conditions which are conducive to the success of such efforts.     

Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey

In this work, a bidimensional differential equation system obtained by modifying the well-known predator–prey Rosenzweig–MacArthur model is analyzed by considering prey growth influenced by the Allee effect.One of the main consequences of this modification is a separatrix curve that appears in the phase plane, dividing the behavior of the trajectories. The results show that the equilibrium in the origin is an attractor for any set of parameters. The unique positive equilibrium, when it exists, can be either an attractor or a repeller surrounded by a limit cycle, whose uniqueness is established by calculating the Lyapunov quantities. Therefore, both populations could either reach deterministic extinction or long-term deterministic coexistence.The existence of a heteroclinic curve is also proved. When this curve is broken by changing parameter values, then the origin turns out to be an attractor for all orbits in the phase plane. This implies that there are plausible conditions where both populations can go to extinction. We conclude that strong and weak Allee effects on prey population exert similar influences on the predator–prey model, thereby increasing the risk of ecological extinction.     

On “new travelling wave solutions” of the KdV and the KdV–Burgers equations

The Korteweg-de Vries and the Korteweg-de Vries–Burgers equations are considered. Using the travelling wave the general solutions of these equations are presented. “New travelling wave solutions” of the KdV and the KdV–Burgers equations by Wazzan [Wazzan L. Commun Nonlinear Sci Numer Simulat 2009;14:443–50] are analyzed. We demonstrate that all his solutions are not new and are transformed to known solutions.