A perturbative method for the spectral analysis of an acoustic multistratified strip

We consider the acoustic propagator A=−∇·c²∇ in the strip Ω={(x, z)∈ℝ²∣0<z<H} with finite width H>0. The celerity c depends for large ∣x∣ only on the variable z and describes the stratification of Ω: it is assumed to be in L^∞(Ω), bounded from below by c_min>0, such that there exists M>0 with c(x, z)=c₁(z) if x< −M and c(x, z)=c₂(z) if x>M. We look at the propagator A as a ‘perturbation’ of the free propagators A_j in Ω associated to the velocities c_j, j=1, 2, and implement a ‘perturbative’ method, adapting ideas of Majda and Vainberg. The spectrum of A is defined in section 2, a limiting absorption principle is proved in section 3 outside of a countable set Γ(A). The points of Γ(A) can only accumulate at the left of the thresholds of the free propagators. The needed material about A_j, j=1, 2, and some technical estimates for A are given in Appendix. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Ω‐estimates related to irreducible algebraic integers

We study large values of the remainder term E_K (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that E_K (x) = Ω_± (√(x)(log x)) for certain positive constant B_K, improving in that way the previously best known estimate E_K (x) = Ω_± (x^(1/2)‐ε) for every ε > 0, due to A. Perelli and the present author. Assuming that no entire L‐function from the Selberg class vanishes on the vertical line σ = 1, we show that E_K (x) = Ω_± (√(x)(log log x)^{D (K)‐1}(log x)^‐1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fourier-Integral-Operator Approximation of Solutions to First-Order Hyperbolic Pseudodifferential Equations I: Convergence in Sobolev Spaces

An approximation Ansatz for the operator solution, U(z′,z), of a hyperbolic first-order pseudodifferential equation, ? _z  + a(z,x,D _x ) with Re (a) ≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in L(H ^(s),H ^(s)) of these operators is provided, which yields a convergence result for the Ansatz to U(z′,z) in some Sobolev space as the number of operators in the composition goes to ∞.     

A Green's Function Approach to Wave Propagation and Potential Flow around Obstacles in Infinite Cylindrical Channels

We study (a) acoustic waves generated by a time-harmonic force distribution and (b) the potential flow with prescribed velocity at infinity in an infinite cylinder Ω₀ = Ω′×ℝ with bounded cross-section Ω′⊂ℝ² in the presence of m embedded obstacles B₁,…,B_m. By using Green's function G_κ(x,y) of the Neumann problem for the reduced wave equation ΔU+κ²U = 0 in the unperturbed domain Ω₀, both problems can be reduced to integral equations over the boundaries of the obstacles. The main properties of G_κ(x,y), which are required for this approach, are derived in the first part of this paper.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Existence Results for Set-Valued Vector Quasiequilibrium Problems

This paper deals with the set-valued vector quasiequilibrium problem of finding a point (z ₀,x ₀) of a set E×K such that (z ₀,x ₀)∈B(z ₀,x ₀)×A(z ₀,x ₀), and, for all η∈A(z ₀,x ₀),

Generalized Vector Quasivariational Inclusion Problems with Moving Cones

This paper deals with the generalized vector quasivariational inclusion Problem (P₁) (resp. Problem (P₂)) of finding a point (z ₀,x ₀) of a set E×K such that (z ₀,x ₀)∈B(z ₀,x ₀)×A(z ₀,x ₀) and, for all η∈A(z ₀,x ₀),

Robust global optimization with polynomials

We consider the optimization problems max_z∈Ω min_x∈K p(z, x) and min_{x ∈ K} max_{z ∈ Ω} p(z, x) where the criterion p is a polynomial, linear in the variables z, the set Ω can be described by LMIs, and K is a basic closed semi-algebraic set. The first problem is a robust analogue of the generic SDP problem max_{z ∈ Ω} p(z), whereas the second problem is a robust analogue of the generic problem min_{x ∈ K} p(x) of minimizing a polynomial over a semi-algebraic set. We show that the optimal values of both robust optimization problems can be approximated as closely as desired, by solving a hierarchy of SDP relaxations. We also relate and compare the SDP relaxations associated with the max-min and the min-max robust optimization problems.     

Estimates for the zeros of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.     

Commuting Values of Generalized Derivations on Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, U the right Utumi quotient ring of R, f(x ₁,…, x _n ) a noncentral multilinear polynomial over K, and G a nonzero generalized derivation of R. Denote f(R) the set of all evaluations of the polynomial f(x ₁,…, x _n ) in R. If [G(u)u, G(v)v] = 0, for any u, v ∈ f(R), we prove that there exists c ∈ U such that G(x) = cx, for all x ∈ R and one of the following holds: 1. f(x ₁,…, x _n )² is central valued on R;

2. R satisfies s ₄, the standard identity of degree 4.

     

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

We consider the Aharonov–Bohm effect for the Schrödinger operator H = (?i? _x  ? A(x))² + V(x) and the related inverse problem in an exterior domain Ω in R ² with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain Ω and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle Ω ^c is convex.     

Shape sensitivity analysis of stress intensity factors by Generalized J-integrals

We consider an elastic plate with the non-deformed shape Ω_Σ := Ω \ Σ, where Ω is a domain bounded by a smooth closed curve Γ and Σ ⊂ Ω is a curve with the end points {γ₁, γ₂}. If the force g is given on the part Γ_N of Γ, the displacement u is fixed on Γ_D := Γ \ Γ_N and the body force f is given in Ω, then the displacement vector u(x) = (u₁(x), u₂(x)) has unbounded derivatives (stress singularities) near γ_k, k = 1, 2    u(x) = ∑²_{k, l=1} K_l(γ_k)r^1/2_kS^C_kl(θ_k) + u_R(x)     near γ_k. Here (r_k, θ_k) denote local curvilinear polar co-ordinates near γ_k, k = 1, 2, S^C_kl (θ_k) are smooth functions defined on [−π, π] and u_R(x) ∈ {H²(near γ_k)}². The constants K_l(γ_k),   l = 1, 2, which are called the stress intensity factors at γ_k (abbr. SIFs), are important parameters in fracture mechanics. We notice that the stress intensity factors K_l(γ_k) (l = 1, 2;  k = 1, 2) are functionals K_l(γ_k) = K_l(γ_k; ℒ︁, Ω, Σ) depending on the load ℒ︁, the shape of the plate Ω and the shape of the crack Σ. We say that the crack Σ is safe, if K_l(γ_k; Ω)² + K₂(γ_k; Ω)² < RẼ. By a small change of Ω the shape Σ can change to a dangerous one, i.e. we have K₁(γ_k; Ω)² + K₂(γ_k; Ω)² ⩾ RẼ. Therefore it is important to know how K_l(γ_k) depends on the shape of Ω. For this reason, we calculate the Gâteaux derivative of K_l(γ_k) under a class of domain perturbations which includes the approximation of domains by polygonal domains and the Hadamard's parametrization Γ(τ) := {x + τϕ(x)n(x);  x ∈ Γ}, where ϕ is a function on Γ and n is the outward unit normal on Γ. The calculations are quite delicate because of the occurrence of additional stress singularities at the collision points {γ₃, γ₄} = Γ _D ∩ Γ _N. The result is derived by the combination of the weight function method and the Generalized J-integral technique (abbr. GJ-integral technique). The GJ-integrals have been proposed by the first author in order to express the variation of energy (energy release rate) by extension of a crack in a 3D-elastic body. This paper begins with the weak solution of the crack problem, the weight function representation of SIF's, GJ-integral technique and finish with the shape sensitivity analysis of SIF's. GJ-integral J_ω(u; X) is the sum of the P-integral (line integral) P_ω(u, X) and the R-integral (area integral) R_ω(u, X). With the help of the GJ-integral technique we derive an R-integral expression for the shape derivative of the potential energy which is valid for all displacement fields u ∈ H¹. Using the property that the GJ-integral vanishes for all regular fields u ∈ H² we convert the R-integral expression for the shape derivative to a P-integral expression. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

Multipliers of classes of cauchy transforms

The analytic map g on the unit disk D is said to induce a multiplication operator L from the Banach space X to the Banach space Y if L(f)=f·g∈Y for all f∈X. For z ∈ D and α>0 the families of weighted Cauchy transforms F_α are defined by ?(z) = ∫_T K_x ^α (z)dμ(x) where μ(x) is complex Borel measures, x belongs to the unit circle T and the kernel K_x (z) = (1- xz)^?1. In this article we will explore the relationship between the compactness of the multiplication operator L acting on F ₁ and the complex Borel measures μ(x). We also give an estimate for the essential norm of L     

Fourth-order finite difference method for three-dimensional elliptic equations with nonlinear first-derivative terms

We present a 19-point fourth-order finite difference method for the nonlinear second-order system of three-dimensional elliptic equations Au _xx + Bu _yy + Cu _zz = f , where A , B , C , are M × M diagonal matrices, on a cubic region R subject to the Dirichlet boundary conditions u (x, y, z) = u ⁽⁰⁾(x, y, z) on ?R. We establish, under appropriate conditions, O(h⁴) convergence of the difference method. Numerical examples are given to illustrate the method and its fourth-order convergence. © 1992 John Wiley & Sons, Inc.     

Rationality for Subgroups of S 6

For a transitive subgroup G ≤ S ₆ which contain C ₃ × C ₃ as subgroup, we prove that K(x ₁,…, x ₆) ^G is rational over K, where K is any field, and G acts naturally on K(x ₁,…, x ₆) by permutations on the variables. We also give an application on construction of generic polynomials.     

Finite free resolutions of length two and koszul generalized complex

Abstract

In 1956, Ehrenfeucht proved that a polynomial f ₁(x ₁) + · + f _n (x _n ) with complex coefficients in the variables x ₁, …, x _n is irreducible over the field of complex numbers provided the degrees of the polynomials f ₁(x ₁), …, f _n (x _n ) have greatest common divisor one. In 1964, Tverberg extended this result by showing that when n ≥ 3, then f ₁(x ₁) + · + f _n (x _n ) belonging to K[x ₁, …, x _n ] is irreducible over any field K of characteristic zero provided the degree of each f _i is positive. Clearly a polynomial F = f ₁(x ₁) + · + f _n (x _n ) is reducible over a field K of characteristic p ≠ 0 if F can be written as F = (g ₁(x ₁)) ^p  + (g ₂(x ₂)) ^p  + · + (g _n (x _n )) ^p  + c[g ₁(x ₁) + g ₂(x ₂) + · + g _n (x _n )] where c is in K and each g _i (x _i ) is in K[x _i ]. In 1966, Tverberg proved that the converse of the above simple fact holds in the particular case when n = 3 and K is an algebraically closed field of characteristic p > 0. In this article, we prove an extension of Tverberg's result by showing that this converse holds for any n ≥ 3.     

Coordinates of R[x,y]: Constructions and Classifications

Let R be a PID. We construct and classify all coordinates of R[x, y] of the form p ₂ y + Q ₂(p ₁ x + Q ₁(y)) with p ₁, p ₂ ∈ qt(R) and Q ₁, Q ₂ ∈ qt(R)[y]. From this construction (with R = K[z]) we obtain nontame automorphisms σ of K[x, y, z] (where K is a field of characteristic 0) such that the subgroup generated by σ and the affine automorphisms contains all tame automorphisms.     

Nonfree actions of countable groups and their characters

The paper studies the region of values of the system {c ₂, c ₃, f(z ₁), f′(z ₁)},where z ₁ is an arbitrary fixed point of the disk |z| < 1; f ∈ T,and the class T consists of all the functions f(z) = z + c ₂ z ² + c ₃z³ + ⋯ regular in the disk |z| < 1 that satisfy the condition Im z · Im f(z) > 0 for Im z ≠ 0. The region of values of f′(z ₁) in the subclass of functions f ∈ T with prescribed values c ₂, c ₃, and f(z ₁) is determined. Bibliography: 10 titles.