On the Response of Autonomous Sweeping Processes to Periodic Perturbations

If x ₀ is an equilibrium of an autonomous differential equation \(\dot x=f(x)\) and det∥f ^′(x ₀)∥≠0, then x ₀ persists under autonomous perturbations and x ₀ transforms into a T-periodic solution under non-autonomous T-periodic perturbations. In this paper we discover a similar structural stability for Moreau sweeping processes of the form \(-\dot u\in N_{B}(u)+f_{0}(u)\), \(u\in \mathbb {R}^{2}, \)i. e. we consider the simplest case where the derivative is taken with respect to the Lebesgue measure and where the convex set B of the reduced system is a non-moving unit ball of \(\mathbb {R}^{2}\). We show that an equilibrium ∥u ₀∥=1 persists under periodic perturbations, if the projection \(\overline {f}:\partial B\to \mathbb {R}^{2}\) of f ₀ on the tangent to the boundary ? B is nonsingular at u ₀.     

On Stability of Thomson’s Vortex <Emphasis Type="Italic">N</Emphasis>-gon in the Geostrophic Model of the Point Bessel Vortices

A stability analysis of the stationary rotation of a system of N identical point Bessel vortices lying uniformly on a circle of radius R is presented. The vortices have identical intensity Γ and length scale γ^?1 > 0. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The cases for N = 2,..., 6 are studied sequentially. The case of odd N = 2?+1 ≥ 7 vortices and the case of even N = 2_n ≥ 8 vortices are considered separately. It is shown that the (2? + 1)-gon is exponentially unstable for 0 < γR<R_*(N). However, this (2? + 1)-gon is stable for γR ≥ R_*(N) in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even N = 2n ≥ 8 vortex 2n-gon is exponentially unstable for R > 0.     

The prime-power hypothesis and solvable groups

We say that a group G satisfies the prime-power hypothesis if the GCDs for all pairs of distinct character degrees are prime powers. We prove that if G is a solvable group satisfying the prime-power hypothesis, then G has Fitting height at most 12. If in addition |G| is odd, then we prove that the Fitting height of G is at most 6.     

Coloring,sparseness and girth

Using the classical analysis resolution of singularities algorithm of [G4], we generalize the theorems of [G3] on Rⁿ sublevel set volumes and oscillatory integrals with real phase function to functions over an arbitrary local field of characteristic zero. The p-adic cases of our results provide new estimates for exponential sums as well as new bounds on how often a function f(x), such as a polynomial with integer coefficients, is divisible by various powers of a prime p when x is an integer. Unlike many papers on such exponential sums and p-adic oscillatory integrals, we do not require the Newton polyhedron of the phase to be nondegenerate, but rather as in [G3] we have conditions on the maximum order of the zeroes of certain polynomials corresponding to the compact faces of this Newton polyhedron.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

Linear and nonlinear methods of relief approximation

In this paper, we compare the effectiveness of free (nonlinear) relief approximation, equidistant relief approximation, and polynomial approximation {ie129-01}, and {ie129-02} of an individual function ?(x) in the metric {ie129-03}, where {ie129-04} is the unit ball |x| ≤ 1 in the plane ?². The notation we use is the following: {fx129-01}. Here {ie129-05} is the set of all N-term linear combinations of functions of the plane-wave type {fx129-02} with arbitrary profiles W _j (x), x ∈ ?¹ and transmission directions {θ _j } ₁ ^N ; {ie129-06} is the subset of {ie129-07} associated with N equidistant directions; {fx129-03} denotes the subspace of algebraic polynomials of degree less than or equal to N ? 1 in two real variables. Obviously, the inequalities {ie129-08} hold.We state the following model problem. What are the functions which satisfy the relation {ie129-09}, i.e., where the nonlinear approximation {ie129-10} is more effective than a linear one? This effect has been proved for harmonic functions, namely, for any ε > 0 there exists c _ε > 0 such that if Δ?(x) = 0, |x| < 1, and ? ∈ {ie129-11}, then {fx129-04}. On the other hand, {ie129-12}. Thus, {ie129-13} has an “almost squared effectiveness” of {ie129-14} for ? = ?_harm. However, this ultra-high order of approximation is obtained via a collapse of wave vectors.On the other hand, the nonlinearity of {ie129-15} which corresponds to the freedom of choice of wave vectors does not much improve the order of approximation, for instance, for all the radial functions. If {ie129-16}, then {ie129-17} and {ie129-18}.The technique we use is the Fourier-Chebyshev analysis (which is related to the inverse Radon transform on {ie129-19}) and a duality between the relief approximation problem and the optimization of quadrature formulas in the sense of Kolmogorov-Nikolskii [14] for trigonometric polynomial classes.     

Recurrence formulas for long wavelength asymptotics in the problem of shear flow stability

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for two-dimensional viscous incompressible shear flows with a nonzero average. It is shown that the critical eigenvalues are odd functions of the wave number, while the critical values of the viscosity are even functions. If the deviation of the velocity from its period-average value is an odd function of spatial variable, the eigenvalues can be found exactly.     

Generalized varieties of sums of powers

Let X ? P^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P_H^X(h) of X is the closure in the Hilbert scheme Hilb_h (X) of the locus parametrizing collections of points {x₁,..., x_h} such that the (h -1)-plane >x₁,..., x_h> passes through a fixed general point p ∈ P^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P_H^X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P_H^X(h) are inherited from the birational geometry of variety X itself.     

Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems

In this paper, we study in detail the phase properties and stability of numerical methods for general oscillatory second-order initial value problems whose right-hand side functions depend on both the position y and velocity y ^'. In order to analyze comprehensively the numerical stability of integrators for oscillatory systems, we introduce a novel linear test model y ^?(t) + ? ² y(t) + µ y ^'(t)=0 with µ<2?. Based on the new model, further discussions and analysis on the phase properties and stability of numerical methods are presented for general oscillatory problems. We give the new definitions of dispersion and dissipation which can be viewed as an essential extension of the traditional ones based on the linear test model y ^?(t) + ? ² y(t)=0. The numerical experiments are carried out, and the numerical results showthatthe analysisofphase properties and stability presentedinthispaper ismoresuitableforthenumericalmethodswhentheyareappliedtothe generaloscillatory second-order initial value problem involving both the position and velocity.     

On the Stability of the Zero Solution of a Second-Order Differential Equation under a Periodic Perturbation of the Center

Small periodic perturbations of the oscillator \(\ddot x + {x^{2n}}\) sgn x = Y(t, x, \(\dot x\)) are considered, where n < 1 is a positive integer and the right-hand side is a small perturbation periodic in t, which is an analytic function in \(\dot x\) and x in a neighborhood of the origin. New Lyapunov-type periodic functions are introduced and used to investigate the stability of the equilibrium position of the given equation. Sufficient conditions for asymptotic stability and instability are given.     

General compactly supported solution of an integral equation of the convolution type

We find the general form of solutions of the integral equation ∫k(t ? s)u₁(s) ds = u₂(t) of the convolution type for the pair of unknown functions u₁ and u₂ in the class of compactly supported continuously differentiable functions under the condition that the kernel k(t) has the Fourier transform \(\widetilde {{P_2}}\), where \(\widetilde {{P_1}}\) and \(\widetilde {{P_2}}\) are polynomials in the exponential e^iτx, τ > 0, with coefficients polynomial in x. If the functions \({P_l}\left( x \right) = \widetilde {{P_l}}\left( {{e^{i\tau x}}} \right)\), l = 1, 2, have no common zeros, then the general solution in Fourier transforms has the form U_l(x) = P_l(x)R(x), l = 1, 2, where R(x) is the Fourier transform of an arbitrary compactly supported continuously differentiable function r(t).     

Weakly nonlinear Schrödinger equation with random initial data

It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ _t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈? ^d and t∈?. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ _t (x) has a limit as λ→0 for t=λ ^?2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.     

Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

A method for computing the eigenvalues λ _mn (b, c) and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters b and c. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain b _s and c _s are second-order branch points for λ _mn (b, c) with different indices n ₁ and n ₂, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite-Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points b _s and c _s and the double eigenvalues to high accuracy. A large number of these singular points are calculated.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Mixed problem for the wave equation with a summable potential and nonzero initial velocity

The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L ₂[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.     

Best quadrature formulas for evaluating line integrals of the first kind for certain classes of functions and curves determined by moduli of continuity

The extremal problem of minimizing the error of approximate evaluation of a line integral of the first kind is considered for certain classes of functions and spatial curves determined by moduli of continuity.It is proved that if the endpoints of the interval [0, L] (where L is the length of the curve along which the integration is performed) are not included in the set of nodes of a quadrature formula for evaluating the line integral of the first kind, then the best quadrature formula for the classes m^(p) _ρ of functions and \({H^{{\omega _1}, \ldots ,{\omega _m}}}\) of curves is the midpoint rectangle formula. If the extreme points x = 0 and x = L of the interval are included in the set of nodes of a quadrature formula for approximately evaluating the line integral (such formulas are said to be Markov-type), then, for these classes, the best formula is the trapezoidal rule. Sharp error estimates for all considered classed of functions and curves are calculated and a generalization to more general classes is given.     

Verbal width in anabelian groups

The class A of anabelian groups is defined as the collection of finite groups without abelian composition factors. We prove that the commutator word [x₁, x₂] and the power word x₁^p have bounded width in A when p is an odd integer. By contrast, the word x³⁰ does not have bounded width in A. On the other hand, any given word w has bounded width for those groups G ∈ A whose composition factors are sufficiently large as a function of w. In the course of the proof we establish that sufficiently large almost simple groups cannot satisfy w as a coset identity.     

Obstructions to the uniform stability of a <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let T _t : X → X be a C ₀-semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s ₀(A) is greater than zero then for each nondecreasing function h(s): ?₊ → R ₊ there are x′ ∈ X′ and x ∈ X satisfying ∫ ₀ ^∞ h(|〈x′, T _x x〉|)dt = ∞. If i? ∩ Sp(A) ≠ Ø then such x may be taken in D(A ^∞).     

Riesz basis property of system of root functions of second-order differential operator with involution

The properties of the root functions are studied for an arbitrary operator generated in L ₂(?1, 1) by the operation with involution of the form Lu = ?u″(x)+αu″(?x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (?1, 1), ν(x) is an absolutely continuous involution of the segment [?1, 1] and the coefficients q(x) and qν(x) are summable functions on (?1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L ₂(?1, 1) for the system of the root functions of the operator.