Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

p-Frames in Separable Banach Spaces

Let X be a separable Banach space with dual X ^*. A countable family of elements {g _i }X ^* is a p-frame (1 p ) if the norm _X is equivalent to the ^p -norm of the sequence {g _i ()}. Without further assumptions, we prove that a p-frame allows every gX ^* to be represented as an unconditionally convergent series g=d _i g _i for coefficients {d _i } ^q , where 1/p+1/q=1. A p-frame {g _i } is not necessarily linear independent, so {g _i } is some kind of overcomplete basis for X ^*. We prove that a q-Riesz basis for X ^* is a p-frame for X and that the associated coefficient functionals {f _i } constitutes a p-Riesz basis allowing us to expand every fX (respectively gX ^*) as f=g _i (f)f _i (respectively g=g(f _i )g _i ). In the general case of a p-frame such expansions are only possible under extra assumptions.     

Set systems with no union of cardinality 0 modulom

Letq be a prime power. It is shown that for any hypergraph = {F ₁,...,F _d(q–1)+1} whose maximal degree isd, there exists Ø ₀ , such that 0 (modq).     

A Singular Equation of Length Four Over Groups

Let G be a group and t an unknown. In this paper we prove that the equation atbtct^–1dt^–1 = 1 (a,b,c,d G, a² 1, c² 1, bd 1) has a solution over G. This forms part of a program to investigate precisely when an equation, whose associated star graph contains no admissible paths of length less than 3, fails to have a solution over G.2000 Mathematics Subject Classification: 20E06     

Shape-preserving Kolmogorov widths of classes of <Emphasis Type="Italic">s</Emphasis>-monotone integrable functions

Let s ₀ and let ₊ ^s be the set of functions x defined on a finite interval I and such that, for all collections of s + 1 pairwise different points t ₀,..., t _s I, the corresponding divided differences [x; t ₀,...,t _s ] of order s are nonnegative. Let ₊ ^s B _{p +} ^s B _p, 1 p where B _p is a unit ball in the space L _p, and let ₊ ^s L _{q +} ^s L _q, 1 q . For every s 3 and 1 q p , we determine the exact orders of the shape-preserving Kolmogorov widths {x - y} \right\ L_q , $$]]>, where M ⁿ is the collection of all affine linear manifolds M ⁿ in L _q such that dim M ⁿ n and M ⁿ ₊ ^s L _q .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 7, pp. 901–926, July, 2004.     

Tauberian Theorems for Weighted Means of Double Sequences

Let p := {p _j } _j=0 and q := {q _k } _k–0 be complex (or real) sequences with the property that P _m := _j–0 ^m p _j 0 for all m 0, Q _n := _k–0 ⁿ q _k 0 for all n 0, and both of {P _m } _m=0 and {Q _n } _n=0 are varying away from 1. Assume that {s _mn } is a double sequence in C(or one of R, a Banach space, and an ordered linear space), which is (N¯,p,q; ,) summable to a finite limit, where (,) =(1,1), (1,0), or (0,1). We give necessary and sufficient conditions under which {s _mn } converges in Pringsheim's sense. These conditions are weaker than the two-dimensional analogues of Landau's condition and Schmidt's slow decrease condition. Our results generalize and extend [1 4, 12 15]. We also solve the problems posed in [3, 13, 14].     

Rechtseiträume

Rectangular spaces are defined to be incidence spaces on which an equivalence relation is defined on the linesB and a congruence relation is defined on the point pairsP ² which is compatible with the relation and the incidence structure. Every rectangular space which contains a line with only a finite number of points is a euclidean space. In any regtangular space of characteristic 2 there exists a unique reflection on each line. Thus, as in the case of rectangular planes, rectangular spaces of arbitrary dimension and of characteristic 2 can be characterized in terms of commutative kinematic spaces with involutory automorphisms.Our main result is the theorem which states that any rectangular space of characteristic 2 can be embedded into a euclidean space of equal dimension. From this embedding property we conclude that every rectangular space of characteristic 2 can be described as a subgeometry of a vector space supplied with a quadratic form. Finally we present examples of rectangular spaces of arbitrary dimension which are not euclidean spaces.

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Herrn Prof. H. Karzel und Herrn Prof. H. Wähling in Dank gewidmet

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Diese Arbeit ist in einer anderen Form in meiner Habilitationsschrift Zur Theorie der Rechtseiträume unter besonderer Berücksichtigung des ebenen Falles (Technische Universität München 1987) enthalten.     

Fields with Exceptional Tangent Fields

The asymptotic self-similarity property describes the local structure of a random field. In this paper, we introduce a locally asymptotically self-similar second order field X_H, whose local structures at x=0 and at x0 are very far from each other. More precisely, whereas its tangent field at x0 is a Fractional Brownian Motion, its tangent field at x=0 is a Fractional Stable Motion. In addition, X_H, is asymptotically self-similar at infinity with a Gaussian field, which is not a Fractional Brownian Motion, as tangent field. Then, its trajectories regularity is studied. Finally, the Hausdorff dimension of its graphs is given.     

Kanten-kritische graphen mit der zusammenhangszahl 2

A set of vertices of a graph G represents G if each edge of G is incident with at least one vertex of . A graph G is said to be edge-critical if the minimal number of vertices necessary to represent G decreases if any edge of G is omitted. Plummer [5] has given a method to construct an infinite family of edge-critical graphs with connectivity number 2. We use this method to construct a more extensive class of edge-critical graphs with connectivity number 2 and show that all edge-critical graphs with this connectivity number (K₂) can be constructed from smaller edge-critical graphs. Finally we give examples of edge-critical graphs not constructable from smaller ones by this method.     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

A note on the optimal quadrature inH p

Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH ^p (1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is where 1/p+1/q=1 and 1p.     

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.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

A q-Analogue of Weber-Schafheitlin Integral of Bessel Functions

In an attempt to find a q-analogue of Weber and Schafheitlin's integral ₀ x ^– J (ax) J (bx) dx which is discontinuous on the diagonal a = b the integral ₀ x ^– J ⁽²⁾ (a(1 – q)x; q)J ⁽¹⁾ (b(1 – q)x; q) dx is evaluated where J ⁽¹⁾ (x; q) and J ⁽²⁾ (x; q) are two of Jackson's three q-Bessel functions. It is found that the question of discontinuity becomes irrelevant in this case. Evaluations of this integral are also made in some interesting special cases. A biorthogonality formula is found as well as a Neumann series expansion for x in terms of J ⁽²⁾ _+1+2n ((1 – q)x; q). Finally, a q-Lommel function is introduced.     

Flatness properties of monocyclic acts

In a previous paper the authors studied flatness properties of cyclic actsS/ (S denotes a monoid, and is a right congruence onS), and determined conditions onS under which all flat or weakly flat acts of this type are actually strongly flat or projective. In the present paper attention is restricted to monocyclic acts (cyclic acts in which is generated by a single pair of elements ofS), and further results on such collapsing of flatness properties are obtained. An observation which is used extensively in this study is the fact that forw andt inS withwtt,S/(wt,t) is flat if and only ift is a regular element ofS.Research supported by Natural Sciences and Engineering Research Council of Canada Operating Grant A4494.Research supported by Estonian Research Foundation Grant No. 930.     

Application of a computer to the proof of a conjecture of Minkowski in the geometry of numbers

A computer-assisted proof is given of Minkowski's conjecture on the critical determinant of the region x^p+y^p<1 in the cases 1.03p 1.9745, p2.40, p2.577.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 163–180, 1977.     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

Anwendungen der Nielsenschen Kürzungsmethode in Gruppen mit einer definierenden Relation

LetG=a ₁,b ₁, ...,a _q ,b _q | (W ₁ (a ₁,b ₁) ...W _q (a _q b _q ))=1,q1, 2,W _j (a _j ,b _j )1. We solve the isomorphism problem forG inasmuch as we can decide in finitely many steps, if any arbitrary one-relator group is isomorphic toG or not. FurthermoreG turns out to have a finitely generated automorphism group. Forq=1 this was proved byS. J. Pride. Forq2 the proof is based onNielsen's reduction method. There are some other interesting results on subgroup problems for one-relator groups obtained by this method.     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

The critical exponent of degenerate parabolic systems

The Cauchy problemu _t=u +v ^p ,v _t =v +u ^q is studied, wherex R ^N , 0 <t < and ,,p andq, are positive exponents. It is proved that ifp,q 1 and 1 <pq < 1 + 2 max(p + ,q + )/n then every nontrivial non-negative solution is not global in time; whereaspq > 1 + 2 max(p + , q + )/n then there exist both positive global solutions and non-global solutions. In addition, the decaying in time of solutions tou _t,=u inR ⁿ × (0, ), an equation which occurs naturally in our study of above systems, is studied and solutions with the fastest decaying in time are constructed.