Universal fundamental systems in ordered Banach spaces

Let E be a separable Banach space ordered by a reproducing cone with empty interior. We prove the existence of operator functions A : [0, ) P (P the cone of monotone increasing linear operators) and of initial values x₀ such that the solution of x (t) = A(t) x(t), x(0) = x₀, is dense in E.Received: 27 February 2004     

Time periodic solutions to a nonlinear wave equation with <Emphasis Type="Italic">x</Emphasis>-dependent coefficients

In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients on under the boundary conditions a ₁ y(0, t)+b ₁ y _x (0, t) = 0, ( for i = 1, 2) and the periodic conditions y(x, t + T) = y(x, t), y _t (x, t + T) = y _t (x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For , we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients. This work was supported by the 985 Project of Jilin University, the Specialized Research Fund for the Doctoral Program of Higher Education, and the Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University.     

On the equations of mathematical hydraulics

Summary The relation between classical hydraulics and modern turbulence modelling is discussed for the case of two-dimensional open channel flow down an inclined plane. A second order turbulence model describing the flow is treated asymptotically for the parameter rangeFO(1),1,1, and=O( ²), whereF is the Froude number, is the aspect ratio, and is the square root of a characteristic drag coefficient. The Chezy law formulation of mathematical hydraulics is derived as the lowest order approximation to the solution for the flow outside bore regions, and the transverse variation of the longitudinal velocity component is determined at the next stage of the analysis. It is shown that flow discontinuities calculated using the equations of mathematical hydraulics are resolved in bore regions of transverse length scaleO(H _o), whereH _o is the characteristic fluid depth. The bore structure is found to consist of a highly turbulent outer region with transverse length scaleO(H _o) in which the turbulence intensity isO(1), and a bottom boundary layer of transverse length scaleO( ² H _o), in which the turbulent stresses decrease rapidly to satisfy the bottom boundary conditions. The jump conditions of mathematical hydraulics at flow discontinuities are verified, and it is inferred that classical hydraulics provides an acceptable approximation to the flow outside bore regions for the parameter range considered in the theory.     

On a characterization of polynomials by divided differences

Summary We consider the functional equationf[x ₁,x ₂,, x _n ] =h(x ₁ + +x _n ) (x ₁,,x _n K, x _j x _k forj k), (D) wheref[x ₁,x ₂,,x _n ] denotes the (n – 1)-st divided difference off and prove Theorem. Let n be an integer, n 2, let K be a field, char(K) 2, with # K 8(n – 2) + 2. Let, furthermore, f, h: K K be functions. Then we have that f, h fulfil (D) if, and only if, there are constants a_j K, 0 j n (a := a_n, b := a_{n – 1}) such thatf = ax ⁿ +bx ^{n – 1} + +a ₀ and h = ax + b.     

Duality for contravariant functors on Banach spaces

The following notion of duality is studied: IfG is a contravariant functor on Ban, then . We derive the following results: The^x-reflexive functors are exactly the maximal subfunctors ofH(.,A) with reflexiveA, G ^x is^x-reflexive if and only ifG(I) is a reflexive Banach space,G ^x=(G_e)^x, andG ^x(X)=G_e(X) ifX has the metric approximation property. the last result has the consequence that for the tensor product of functors holds, ifX has the m. A. P. Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday     

Iterative Lösung gewisser nichtlinearer Operatorgleichungen mit Anwendung auf quasilineare Differentialgleichungen

LetB ₀,B ₁,B ₂ be Banach spaces, an open convex bounded subset ofB ₁ with closure and. To construct a solution of the problem (*)f(x) = 0, the following implicit iteration process is considered (+)(x _v+1–x _v ) =h(x _v+1) (v = 0, 1, 2,...) withx ₀ , where : B ₁B ₂,, such that every solution of (x) = 0 is a solution of (*) andh is a suitable positive constant. The solutionx _v+1 of (+) is constructed by various explicit iteration methods. Under suitable conditions it is proved: (i) (+) has a unique solutionx _v+1 in some sphere aroundx _v and the explicit iteration process converges to this solution; (ii) the sequence {x _v } converges to a solution in (not necessarily unique) of (*). This general process is applied to ordinary quasilinear boundary value problems.     

The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.     

Effect of contraction on turbulent channel flow

Summary Measurements in an effectively two-dimensional channel indicate that flow acceleration at constant Reynolds Number can have appreciable effects on the turbulence structure. As in wakes, the structure approaches equilibrium exponentially after the acceleration. The effect of acceleration appears to be (at least qualitatively) of the same kind as is observed when turbulence is subjected to rapid distortion.

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Zusammenfassung Messungen in einem effektiv zwei-dimensionalen Kanal zeigen, daß eine Strömungsbeschleunigung bei konstanter Reynolds-Zahl einen wesentlichen Einfluß auf die Turbulenz-Struktur haben kann. Nach der Beschleunigung findet man eine exponentielle Annäherung an das Gleichgewicht, wie bei Dellen. Der Beschleunigungs-Effekt auf die Turbulenz erscheint, wenigstens qualitativ, von der gleichen Art zu sein wie der Einfluß von raschen Verformungen.

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Symbols c _f skin-friction coefficient = - D half-width of the channel - L length of contraction=12 inches - dp/dx pressure gradient - V mean velocity in thex-direction,U=U ₀ aty=D - U mean velocity - average of the mean velocity - U ^* friction velocity = (₀/)^± - root mean square value of the turbulent velocity fluctuations inx, y andz directions - X coordinate in the direction of the flow;X=0 at the end of contraction - Y coordinate perpendicular to the surface of the wall on which measurements are madey=0 at the wall andy=D at the centre of the channel - ₀ Wall shear stress - density of the fluid - viscosity of the fluid - kinematic viscosity of the fluid=/     

On harmonic renewal measures

Summary Let be a probability and the corresponding harmonic renewal measure. Complementing earlier results where is concentrated on a halfline we investigate the behaviour ofv _h ([x, x + 1]) and the harmonic renewal functionG(x) =v _h((–,x])asx ifm ₁=x(dx)>0. We also consider the casem ₁=0.     

Compressing Mappings on Primitive Sequences over Z/(2) and Its Galois Extension

Let f(x) be a strongly primitive polynomial of degree n over Z/(2^e), η(x₀,x₁,…,x_e−2) a Boolean function of e−1 variables and (x₀,x₁,…,x_e−1)=x_e−1+η(x₀,x₁,…,x_e−2)G (f(x),Z/(2^e)) denotes the set of all sequences over Z/(2^e) generated by f(x), F₂^∞ the set of all sequences over the binary field F₂, then the compressing mapping

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is injective, that is, for , G(f(x),Z/(2^e)), = if and only if Φ( )=Φ( ), i.e., ( ₀,…, _e−1)=( ₀,…, _e−1) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.     

On averaging multisets

We give sufficient conditions to ensure that, given a set , everyxint convM can be represented as a convex combination,x = _{i = 1} ⁿ _i x _i , wherex _i M, _i rational, andn=2s orn=2s–1, respectively.     

On the Eigenvalues and Eigenfunctions of the Sturm--Liouville Operator with a Singular Potential

In this paper we consider the Sturm--Liouville operators generated by the differential expression -y+q(x)y and by Dirichlet boundary conditions on the closed interval [0,]. Here q(x) is a distribution of first order,, i.e., q(x)dx L ₂[0,]. Asymptotic formulas for the eigenvalues and eigenfunctions of such operators which depend on the smoothness degree of q(x) are obtained.     

Nonwandering set of points of skew-product maps with base having closed set of periodic points

The aim of the present paper is to study the structure of the nonwandering set of points Ω() for the skew-product maps of the unit square , (x,y)→(f(x),g(x,y)), with base f having closed set of periodic points. For every and every point (x,y) with x periodic of period p_x by f and y not chain recurrent of F^p_x|_Ix, where , we prove that (x,y)Ω(F). On the other hand we construct a map with an isolated fixed point x₀ of f and y₀Ω(F|_Ix0) such that (x₀,y₀)Ω(F₀).     

The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

Infinite Independent Systems of Identities of an Associative Algebra over an Infinite Field of Characteristic Two

Let be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x ₁,x ₂),x ₃), ((x ₄,x ₅),x ₆)), (x ₇,x ₈)) = 0 (((x ₁ x ₂ · x ₃)(x ₄ x ₅ · x ₆))(x ₇ x ₈) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety ( , respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety has the cardinality of the continuum and that there are algebras in with undecidable word problem.     

On the Monotone Error Rule for Parameter Choice in Iterative and Continuous Regularization Methods

We consider in Hilbert spaces linear ill-posed problems Ax = y with noisy data y satisfying y –y. Regularized approximations x _r to the minimum-norm solution x of Ax = y are constructed by continuous regularization methods or by iterative methods. For the choice of the regularization parameter r (the stopping index n in iterative methods) the following monotone error rule (ME rule) is used: we choose r = r _ME (n = n _ME) as the largest r-value with the guaranteed monotonical decrease of the error x _r – x for r [0, r _ME] (x _n – x <#60; x _n–1 – x for n = 1, 2, ..., n _ME). Main attention is paid to iterative methods of gradient type and to nonstationary implicit iteration methods. As shown, the ME rule leads for many methods to order optimal error bounds. Comparisons with other rules for the choice of the stopping index are made and numerical examples are given.This revised version was published online in October 2005 with corrections to the Cover Date.     

On the cost of approximating all roots of a complex polynomial

This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx ₀ =x, x _j =x _j–1 –f(x _j–1)/f(x _j–1),j = 1, 2,, then |x _j –z| < (1/|x ₀ –z|)|x _j–1–z|².     

Solvability of a problem on the plane motion of a viscous incompressible fluid with a free noncompact boundary

One considers the problem of the plane motion of a viscous incompressible fluid which fills partially a container V, bounded by the straight line ₁ = {x:x ₂ = 0} and the contour (V₁), consisting of two semilines ⁽¹⁾ = {x:x ₁<–1,x ₂ = h₀} ⁽²⁾ = {x:x ₁ = 0,x ₂h₀+1} joined by a smooth curvel ⁽³⁾. One assumes that the motion is due to a nonzero flow and by the motion of the lower wall ₁ with a constant velocity R0. The unique solvability of this problem is proved for small R and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 174–179, 1981.In conclusion, the author expresses his deep gratitude to V. A. Solonnikov for his guidance.     

Existence for a characteristic boundary value problem for an equation of mixed type in three-dimensional space

The equation of mixed type With k(x₃) = sign x₃|x₃|^m, m > 0, d?C¹(?), x = (x₁, x₂, x₃), is considered in the threedimensional region G which is bounded by the surfaces: a piecewise smooth surface Γ₀ lying in the half-space x₃ > 0 which intersects the plane x₃ = 0 in the unit circle, and for x₃ < 0 by the characteristic surfaces We prove existence of a generalized solution for the characteristic boundary value problem: Lu = fin G, u_Γ0∪Γ1 = 0. The result is obtained by using a variant of the energy-integral method.     

High order iterative methods for approximating square roots

Given any natural numberm 2, we describe an iteration functiong _m (x) having the property that for any initial iterate \sqrt \alpha $$ " align="middle" border="0"> , the sequence of fixed-point iterationx _k +1 =g _m (x _k ) converges monotonically to having anm-th order rate of convergence. Form = 2 and 3,g _m (x) coincides with Newton's and Halley's iteration functions, respectively, as applied top(x) =x ² – .This research is supported in part by the National Science Foundation under Grant No. CCR-9208371.