One Version of the Bethe–Sommerfeld Conjecture

We consider the two-dimensional periodic Schrödinger operator under the assumption that the electric potential contains a term proportional to the -function concentrated on a periodic system of orthogonal lines. For this operator we confirm the Bethe–Sommerfeld conjecture and study the asymptotic behavior of the integrated density of states. We prove that the -potential can be chosen in such a way that the spectrum of the operator contains any given number of gaps. Bibliography: 9 titles.     

A Non-Autonomous Version of the Denjoy–Wolff Theorem

The aim of this work is to establish the celebrated Denjoy–Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, ω-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior.     

A Generalized Version of the Baker–Pixley Theorem

A famous theorem by Baker and Pixley states that the term functions of a finite algebra with a (d + 1)-ary near-unanimity term are precisely the functions under which all subalgebras of the algebra's dth power are closed. In this paper, we generalize the theorem to a completely abstract level. Indeed, we obtain a version of the theorem that is stated in purely category-theoretic terms, making it applicable in any concrete or abstract category. To motivate this rather abstract result, we also discuss some of its concrete applications.     

A Stereological Version of the Gauss–Bonnet Formula

We give a stereological version of the Gauss–Bonnet formula in order to compute the Euler characteristic of a domain with boundary in a smooth orientable surface in ³, by looking at contacts with a 'sweeping' plane.     

The spectrum is continuous on the set of p–hyponormal operators

In this paper it is shown that the spectrum , a set valued function, is continuous when the function is restricted to the set of all p-hyponormal operators on a Hilbert space. Received November 9, 1998; in final form August 6, 1999 / Published online July 3, 2000     

An Equivariant Version of the K-energy

In this note, we present a connection between equivariant Bott-Chern classes and KahlerRicci solitons. We also propose a generalized version the of the K-energy.     

On a Diffusive Version of the Lifschitz–Slyozov–Wagner Equation

This paper is concerned with the Becker–Döring (BD) system of equations and their relationship to the Lifschitz–Slyozov–Wagner (LSW) equations. A diffusive version of the LSW equations is derived from the BD equations. Existence and uniqueness theorems for this diffusive LSW system are proved. The major part of the paper is taken up with proving that solutions of the diffusive LSW system converge in the zero diffusion limit to solutions of the classical LSW system. In particular, it is shown that the rate of coarsening for the diffusive system converges to the rate of coarsening for the classical system.     

Numerical analysis of the Navier–Stokes/Darcy coupling

We consider a differential system based on the coupling of the Navier–Stokes and Darcy equations for modeling the interaction between surface and porous-media flows. We formulate the problem as an interface equation, we analyze the associated (nonlinear) Steklov–Poincaré operators, and we prove its well-posedness. We propose and analyze iterative methods to solve a conforming finite element approximation of the coupled problem.     

Numerical Solution of the Lippmann–Schwinger Equation by Approximate Approximations

A new method for the numerical solution of volume integral equations is proposed and applied to a Lippmann–Schwinger type equation in diffraction theory. The approximate solution is represented as a linear combination of the scaled and shifted Gaussian. We prove spectral convergence of the method up to some negligible saturation error. The theoretical results are confirmed by a numerical experiment.     

A Partitioned Version of the Erdös–Szekeres Theorem for Quadrilaterals

We prove a partitioned version of the Erdös–Szekeres theorem for the case $k = 4$: any finite set $X \subset \bbbr^2$ of points in general position can be partitioned into sets $X_0, X_{ij}$ where $i=1,2,3,4$ and $j=1,\ldots,26$, so that $|X_{1j}|=|X_{2j}|=|X_{3j}|=|X_{4j}|$, $|X_0|\leq 4$ and for all $j$ every transversal $\{x_1,x_2,x_3,x_4\}$, $x_1 \in X_{1j}, x_2 \in X_{2j},x_3 \in X_{3j}, x_4 \in X_{4j}$, is in convex position. In order to prove this, we show another theorem, the partitioned version of the same type lemma, which was proved by Bárány and Valtr.     

Numerical solution method for the inverse problem of the modified fitzhugh–nagumo model

The article considers a modified FitzHugh–Nagumo model that may be applied to model processes associated with myocardial infarct analysis. The inverse problem for this model involves finding the coefficient of a system of partial differential equations dependent on the spatial variables and the solution from supplementary observations of the solution on the boundary. This inverse problem may be interpreted as determining the shape and the location of the region of the heart damaged by myocardial infarct. A numerical method is proposed for the solution of the inverse problem and some computer experiments illustrating its implementation are reported.     

A Version of the Weak Krull–Schmidt Theorem for Infinite Direct Sums of Uniserial Modules

We show a version of the weak Krull–Schmidt theorem concerning infinite families of uniserial modules.     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

Application of Splitting Algorithms in the Method of Finite Volumes for Numerical Solution of the Navier–Stokes Equations

We generalize the splitting algorithms proposed earlier for the construction of efficient difference schemes to the finite volume method. For numerical solution of the Euler and Navier–Stokes equations written in integral form, some implicit finite-volume predictor-corrector scheme of the second order of approximation is proposed. At the predictor stage, the introduction of various forms of splitting is considered, which makes it possible to reduce the solution of the original system to separate solution of individual equations at fractional steps and to ensure some stability margin of the algorithm as a whole. The algorithm of splitting with respect to physical processes and spatial directions is numerically tested. The properties of the algorithm are under study and we proved its effectiveness for solving two-dimensional and three-dimensional flow-around problems.     

The Mahler Measure of the Rudin–Shapiro Polynomials

Littlewood polynomials are polynomials with each of their coefficients in \(\{-1,1\}\). A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin–Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin–Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin–Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin–Shapiro polynomials have been known before.     

The hyperbolic meaning of the Milnor–Wood inequality

We introduce a notion of the twist of an isometry of the hyperbolic plane. This twist function is defined on the universal covering group of orientation-preserving isometries of the hyperbolic plane, at each point in the plane. We relate this function to a function defined by Milnor and generalised by Wood. We deduce various properties of the twist function, and use it to give new proofs of several well-known results, including the Milnor–Wood inequality, using purely hyperbolic-geometric methods. Our methods express inequalities in Milnor’s function as equalities, with the deficiency from equality given by an area in the hyperbolic plane. We find that the twist of certain products found in surface group presentations is equal to the area of certain hyperbolic polygons arising as their fundamental domains.