The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

Let F denote a surface with boundary F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve _o parametrizing F, we will get a manifold of real dimension 6g–3, such that any bounds some F and any local deformation of F is conformally equivalent to just one F for .This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {F|} cover all possible deformations of F at all.     

Products of Quasi-p-Pseudocompact Spaces

Given p () , we determine when a product of quasi-p-pseudocompact spaces preserves this property. In particular, we analyze the product of quasi-p-pseudocompact subspaces of () containing . We give examples of spaces X, Y, X _s , Ys which are quasi-p-pseudocompact for every p *, but X Y is not pseudocompact, and X _s Y _s is pseudocompact and it is not quasi-s-pseudocompact for each s *. Besides, we prove that every pseudocompact space X of with X, is quasi-p-pseudocompact for some p *. Finally, we introduce, for each p *, the class P _p of all spaces X such that X × Y is quasi-p-pseudocompact when so is Y; and we prove: (1) the intersection of classes P _p ( _p *) coincides with the Frol"ik class; (2) every class P _p is closed under arbitrary products; (3) the partial ordered set ( P _p p ,) is isomorphic to the set of equivalence classes of free ultrafilters on with the Rudin–Keisler order. A topological characterization of RK-minimal ultrafilters is also given.     

Generating Singularities of Solutions of Quasilinear Elliptic Equations Using Wolff's Potential

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of p-Laplacian type. If p < < N and the right-hand side is a Radon measure with singularity of order at x_0A , then any supersolution in W^1,p() has singularity of order at least (–p)/(p–1) at x₀. In the proof we exploit a pointwise estimate of A-superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff's potential of Radon's measure.     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

Adding Dominating Functions Mod Finite

For certain sets F (₁) ₁, a characterization is given of when there is an extension of the ground model (i.e., a forcing extension or an actual extension, which is ₁-preserving) containing an h which dominates each f F mod finite. If there exists such an h in some extension, then one exists in a forcing extension, in fact a canonical one.     

Analytic Approximations for a Singularly Perturbed Convection—Diffusion Problem with Discontinuous Data in a Half-Infinite Strip

We consider a singularly perturbed convection—diffusion equation, –u+v u=0, defined on a half-infinite strip, (x,y)(0,)×(0,1) with a discontinuous Dirichlet boundary condition: u(x,0)=1, u(x,1)=u(0,y)=0. Asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) as the singular parameter 0⁺ (with fixed distance r to the discontinuity point of the boundary condition) and (b) as that distance r0⁺ (with fixed ). It is shown that the first term of the expansion at =0 contains an error function or a combination of error functions. This term characterizes the effect of discontinuities on the -behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the point of discontinuity of the boundary condition, the solution u(x,y) is approximated by a linear function of the polar angle at the point of discontinuity (0,0).     

Estimates for a Group of Deviations in Generalized Hölder Metric

We present order relations for a group of deviations of a function f(·) H in terms of partial Fourier sums of this function in a generalized Hölder metric defined in a generalized Hölder space H _* H .     

On local automorphisms of certain quadrics of codimension 2

The article considers nondegenerate quadrics in Cⁿ⁺¹ with codimension 2 that are of the form M={zCⁿ, C²: Im_j=z, z_j; j=1, 2}, where are Hermitian forms, and thje stability groups Aut_x M that preserve the point x. It is proved that if the matrix ¹ is stable and the matrix (¹)^–12 has more than two different eigenvalues, all automorphisms of Aut_x M are linear transformations.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 9–14, July, 1992.     

Singular Rank One Perturbations of Self-Adjoint Operators and Krein Theory of Self-Adjoint Extensions

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A_, for A_, = A + (·), where A is a self-adjoint positive operator, being the A-scale). In the present note it is remarked that the operator A_, also appears directly as the Friedrichs extension of the symmetric operator :=A \{f (A)| f,=0\}. It is also shown that Krein's resolvents formula: (A_{_b,}-z)^-1 =(A-z)^-1+ (·, ) _z, with b=b-(1+z) (_z,_-1),_z= (A-z)^-1 defines a self-adjoint operator A_b, for each and b R¹. Moreover it is proven that for any sequence _n which goes to in there exists a sequence _n0 such that A_b, in the strong resolvent sense.     

Asymptotic behavior of the number of certaink-dimensional subspaces over a finite field

For some values ofk, we find the asymptotic behavior, asn , of the probability that a subspace, whose choice is random and equiprobable, chosen among the set of all differentk-dimensional subspaces of ann-dimensional vector space over a finite field, has a given weight {1, 2, ...,n}. In particular, for {1, 2}, this probability can have exponential behavior.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 729–736, May, 1996.     

Logarithms and imaginary powers of closed linear operators

The imaginary powersA ^it of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC ₀-group {exp(itlogA);t R} of bounded linear operators onX, with generatori logA. Here logA is defined as the closure of log(1+A) – log(1+A ^–1). LetA be a linearm-sectorial operator of typeS(tan ), 0(/2), in a Hilbert spaceX. That is, |Im(Au, u)| (tan )Re(Au, u) foru D(A). Then ±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC ₀-group {(1+A) ^it ;t R} of bounded imaginary powers, satisfying the estimate (1+A) ^it exp(|t|),t R. In particular, ifA is invertible, then ±ilogA ism-accretive inX, where logA is exactly given by logA=log(1+A)–log(1+A ^–1), and {A ^it;t R} forms aC ₀-group onX, with the estimate A ^it exp(|t|),t R. This yields a slight improvement of the Heinz-Kato inequality.     

Calculation of Some Integrals Describing Wave Fields

Two methods of calculating the scattering amplitude f(, ₀) of a wave scattered by the vertex of an arbitrarily shaped cone are justified. It is shown that the approximation f _d (, ₀,t) obtained by a method similar to the AbelPoisson method of summation converges uniformly in the regularity region for f. Also, the possibility of calculating f(, ₀) for N ₁( ₀) with the help of rapidly convergent integrals is proved. Bibliography: 7 titles.     

The normal density of prime ideals in small regions

For a Gaussian prime (i) define ()=min|–| where runs through Gaussian primes satisfying ||>||. We prove that, subject to the Riemann Hypothesis for appropriateL-functions

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Über die Diskrepanz von Folgen in bewerteten Körpern

Let k be a field which is locally compact with respect to a valuation ||, let X denote the subset X={ k, ||1}. This paper is concerned with sequences of elements of X with respect to the theory of uniform distribution. Let D_N() denote the discrepancy of the sequence =(x_n). It is proved that for archimedian valuations with a certain constant for every sequence . In the non-archimedian case there exist sequences with .     

On the complete solution of ∈y″=y 3

In Ref. 1, the author claimed that the problem y=y ³ is soluble only for a certain range of the parameter . An analytic approach, as adopted in the following contribution, reveals that a unique solution exists for any positive value of . The solution is given in closed form by means of Jacobian elliptic functions, which can be numerically computed very efficiently. In the limit 0+, the solutions exhibit boundary-layer behavior at both endpoints. An easily interpretable approximate solution for small is obtained using a three-variable approach.     

Mengenwertige Masse und Fortsetzungen

Let (, _i) be a probability space for i=1,2 with and : ^m a correspondence, i.e. () is a non-void subset of ^m for all . We give necessary and sufficient conditions under which it holds, that ₂ extends ₁. iff _A d₂ is equal to _A d₁ for all A, where _A d_i is the set of all integrals _A f d_i of functions f: ^m with f()() _i.-a.e.     

Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f_*min_Xf, where f and X are convex. We show that, for any accuracy <0, these methods find a point x^kX such that f(x^k)–f^* after at most k=O(1/³) objective and subgradient evaluations.     

A space <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>) is dominated by irrationals if and only if it is<Emphasis Type="Italic"> K</Emphasis>-analytic

Summary We prove that, for any Tychonoff X, the space C_p(X) is K-analytic if and only if it has a compact cover {K_p: p } such that K_p subset K_q whenever p,q and p q. Applying this result we show that if C_p(X) is K-analytic then C_p(X) is K-analytic as well. We also establish that a space C_p(X) is K-analytic and Baire if and only if X is countable and discrete.     

Une approche transformationnelle a l'algebre universelle

In this paper we propose an axiomatization of the notion of system of terms of a theory by means of which we obtain a representation of equational classes (or varieties) of algebras. We define analgebraic transformational system (S.T.A.) as a quadruple (T,v,S,+) satisfying the axioms, where T is a set containing the variables v(n), n, and having operators S(): TT, . In addition there are operations Q⁺ on T commuting with the operators. A notion of morphism between S.T.A. 's is defined to obtain the category which is shown to be equivalent to the dual of the category of equational classes. In the last section we establish the equivalence between and Lawvere's category of algebraic theories in which every definable constant is present.

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Extrait de la Thèse de doctorat de l'auteur, Université de Montréal, 1971.