Fastholomorphe Algebren

In this paper we study residue class rings in the sequel called algebras - where R is the ring of germs of complex-valued C- resp. C-functions at the origin of IR^N and is an ideal with. If we have R=R(^N) and with an ideal of germs of holomorphic functions, then we call A holomorphic. For holomorphic algebras it is possible to introduce the modules (A),^p,q(A) of differential forms of degree r and of bidegree (p,q) in a natural way. We have. An almost holomorphic (almost complex) structure on an algebra A is a direct sum ¹(A)=+ with the following property: for the projections p:¹', p:¹ we have for all ¹. Holomorphic algebras have an almost holomorphic structure. We collect some elementary properties of almost holomorphic algebras and then we prove a criterion for integrability in the analytic case, using Cartan's theory of germs of real-analytic sets.     

A cosine operator approach to modelingL 2(0,T; L 2 (Γ))—Boundary input hyperbolic equations

This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain with boundary , under the action of a boundary forcing term inL ₂(0,T; L ₂()). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is inL ₂(0,T; L ₂()), when is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is inL ₂(0,T; H ^3/4-e()) when is a parallelepiped and inL ₂(0,T; H ^2/3() when is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques.This research was supported in part by the Air Force Office of Scientific Research under Grant AFOSR-78-3350 (1st author) and Grant AFOSR-77-3338 (2nd author).This research was performed while the author was visiting the Department of System Science, University of California, Los Angeles.     

Note on the effectiveness factor of a first order reaction

Upper and lower bounds for the effectiveness factor are derived, whereu is the solution of u=c ² u in,u=1 on .

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Zusammenfassung Es werden obere und untere Schranken hergeleitet für den Diffusionsfaktor , wobeiu die Lösung ist von u=c ² u in,u=1 auf .

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This research was done while the author was visiting Cornell University, Ithaca, New York.     

Spectral measures and duality of spectral subspaces of contractions with slowly growing resolvent

We consider the class II of contracting operators T with spectrum on the unit circle r, acting on a separable Hilbert space and subject to the following restriction on the growth of the resolvent R_T(): We study the spectral subspaces _T(B) for T, corresponding to arbitrary Borel subsets of the circle r; in parallel we study a Borel measure _T(B) on r, adequate for _T(B) in the following sense:Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 203–206, 1977.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Über getragene vektorwertige analytische Funktionale

Every analytic functional on an open ¢ &#x2102 with values in a topological vector-space, which is carried by a compact set K¢ , can be represented by a locally analytic functional on K.     

Asymptotic behavior of the solution of an elliptic boundary value problem in a thin domain

The asymptotic behavior as h0 of the solution of a mixed boundary value problem is investigated for an elliptic (in the sense of Petrovskii) system of second-order differential equations in the n-dimensional cylinder Q _h =×(–h/2, h/2) of small altitude h; is a domain in R ⁿ –1. The limit problem in contains a small parameter =h , (0, 1), for higher-order derivatives and degenerates regularly, as 0, into an elliptic problem of a lower order. It is shown that the limit problem and its corresponding degenerate problem (=0) are uniquely solvable. An estimate for the difference of solutions of the original and the limit problem in the energy norm is established. As an example, a problem on the deformation of a thin plate in the framework of the Cosserat continuum is considered.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 191–208, 1990.     

Ein Kriterium für die Approximierbarkeit von Funktionen aus sobolewschen Räumen durch glatte Funktionen

The present paper provides a necessary and sufficient criterion for an element of a Sobolev space W _k ^p () to be approximated in the Sobolev norm by C^k(Eⁿ)-smooth functions. Here is a bounded open set of n-dimensional Euclidean space Eⁿ with convex closure and boundary having n-dimensional Lebesgue measure zero. No further boundary regularity (such as e.g. the segment property) is required.Our main tools are the Hardy-Littlewood maximal functions and a slightly strengthened version of a well-known extension theorem of Whitney.This work was inspired by and is very close in spirit to the pertinent parts of Calderon-Zygmund [6].     

Two poset polytopes

Two convex polytopes, called theorder polytope (P) andchain polytope (P), are associated with a finite posetP. There is a close interplay between the combinatorial structure ofP and the geometric structure of (P). For instance, the order polynomial (P, m) ofP and Ehrhart polynomiali((P),m) of (P) are related by (P, m+1)=i((P),m). A transfer map then allows us to transfer properties of (P) to (P). In particular, we transfer known inequalities involving linear extensions ofP to some new inequalities.Partially supported by NSF Grant No. 8104855-MCS and by a Guggenheim Fellowship.     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

A priori estimates of the solution of the dirichlet problem for the Monge-ampére equation in weight spaces

One proves that a priori boundedness of the norm of the solution of the problem det(U_xx)=f(x,u,u_x)>>0,u¦=0. The magnitudes of the exponents,() depends on whether the arguments u p occur or not in f (x,u,p).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 74–90, 1983.     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

On the convergence of projected gradient processes to singular critical points

The projected gradient methods treated here generate iterates by the rulex _k+1=P (x _k –s _k F(x _k )),x ₁ , where is a closed convex set in a real Hilbert spaceX,s _k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (x – F(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC ⁴ near and grows no slower than x–⁴ within the facet; and (v) the projected Hessian operatorP _{F 2} F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x _k } converges to from nearby starting pointsx ₁, withF(x _k ) –F() =O(k ^–2) and x _k – =O(k ^–1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in ⁿ .     

Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity

The Navier-Stokes equations for compressible barotropic fluid in 1D with the mass force under zero velocity boundary conditions are studied. We prove the uniform upper and lower bounds for the density as well as the uniform in time L ²()-estimates for _x and u _x (u is the velocity). Moreover, a collection of the decay rate estimates for - (with being the stationary density) and u in ²()-norm and H ¹()-norm as time t are established. The results are given for general state function p() (but mainly monotone) and viscosity coefficient µ() of arbitrarily fast (or slow) growth as well as for the large data.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Lacunary (0; 0, 1) interpolation on the roots of Jacobi polynomials and their derivatives,respectively. I (Existence,explicit formulae,unicity)

(0; 0, 1) , {x _k <x _k ^* <x _k+1} _k=1 ^n–1 {x _{k k=1} ⁿ }., I, , _n (x)=P _n ^{(, )} (x)–n- , =, n3 . , x ₀=+1 x _n+1= –1. II .

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To the memory of Paul Erds

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The research was supported by the Hungarian National Foundation for Scientific Research under Grant # T 914 244.     

On the existence of multiple solutions of nonhomogeneous elliptic equations involving critical Sobolev exponents

Letp=2N/(N –2),N 3 be the limiting Sobolev exponent and ^N a bounded smooth domain. We show that for H ^–1(),f satisfies some conditions then–u=c ₁ u ^p–1 +f(x,u) + admits at least two positive solutions.     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.