On the Lipschitz Constant for a Nonisometric Bi-Lipschitz Transformation of a Cantor Set

A bi-Lipschitz continuous mapping of a space X is a bijection such that , where . We write if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by the set of all bi-Lipschitz mappings of X that are not isometry. Thus, if and blip . For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows: where Bibliography: 2 titles.     

Commutants and derivation ranges

In this paper we obtain some results concerning the set , where is the closure in the norm topology of the range of the inner derivation _A defined by _A (X) = AX – XA. Here stands for a Hilbert space and we prove that every compact operator in is quasinilpotent if A is dominant, where is the closure of the range of _A in the weak topology.     

Computations of K- and L-Theory of Cocompact Planar Groups

The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic K- and L-theory and the topological K-theory of cocompact planar groups (=cocompact N.E.C-groups) and of groups G appearing in an extension where is a finite group and the conjugation -action on ⁿ is free outside . These computations apply, for instance, to two-dimensional crystallographic groups and cocompact Fuchsian groups.     

The type set for some measures on \mathbb{R}^{2n} with n-dimensional support

Let be realhomogeneous functions in ofdegree and let bethe Borel measure on given by

On Uniformly Locally Compact Quasi-Uniform Hyperspaces

We characterize those Tychonoff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family of nonempty compact subsets of X. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space Xis uniformly locally compact on if and only if Xis paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is -compact if and only if its (lower) semi-continuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on is obtained.     

Small Ball Estimates in <Emphasis Type="Italic">p</Emphasis>-Variation for Stable Processes

Let Z _t , t 0 be a strictly stable process on with index (0, 2]. We prove that for every p > , there exists = _{, p} and such that

Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Factorizations of One-Generated Composition Formations

A non-empty formation of finite groups is said to be solubly saturated, or we call it a composition formation, if every finite group G having a normal subgroup N such that belongs to . An intersection of all composition formations containing a given group G is denoted cformG. Conditions are described under which has the form , where .     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

On a Binary Diophantine Inequality Involving Prime Numbers

Let 1 < c < 15/14 and N a sufficiently large real number. In this paper we prove that, for all (N, 2N\ A with , the inequality has solutions in primes .     

On Some Proximinal Subspaces of Modular Spaces

Let (, , ) be a complete measure space, L⁰ the vector lattice of -measurable real functions on , : L⁰ [0, )] a lattice semimodular, the corresponding modular space, S₀ the ideal generated by and 0,{\text{ }}\exists {\text{ }}s \in {\text{ }}S_{\text{0}} {\text{ such that }}\rho \left( {\frac{{x - s}}{\user1{\lambda }}} \right) < \infty } \right\}$$ " align="middle" border="0"> . In X consider the distance 0:\rho \left( {\frac{{x - y}}{\user1{\lambda }}} \right) \leqq \user1{\lambda }} \right\}$$ " align="middle" border="0"> and, if is convex, the distances d_L, d_o subordinated to the Luxemburg and Amemiya-Orlicz norms, respectively. We give necessary and sufficient conditions for H(S_o) in order to be proximinal in X with the distances d, d_L and d_o.     

Comments on the Trotter product formula error-bound estimates for nonself-adjoint semigroups

LetA be a positive self-adjoint operator and letB be anm-accretive operator which isA-small with a relative bound less than one. LetH=A+B, thenH is well-defined on dom(H)=dom(A) andm-accretive. IfB is a strictlym-accretive operator obeying

Light paths with an odd number of vertices in polyhedral maps

Let P _k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2-manifold with Euler characteristic contains a path P _k such that each vertex of this path has, in G, degree . Moreover, this bound is attained for k = 1 or k 2, k even. In this paper we prove that for each odd , this bound is the best possible on infinitely many compact 2-manifolds, but on infinitely many other compact 2-manifolds the upper bound can be lowered to .     

Polynomial Trace Estimates for Semigroups

For semigroups (e ^tA ) _t0 of operators on a Hilbert space, we give conditions guaranteeing trace estimates of the polynomial type 0$$ " align="middle" border="0"> , where denotes the trace class. As an application we present higher order analogues of results due to E.B. Davies, B. Simon and M. van den Berg of the type 0$$ " align="middle" border="0"> , for certain unbounded domains , e.g. spiny urchin domains.     

Trigonometric series with monotone coefficients

Let {_n} be a monotonically decreasing sequence. Then each sequence {b_n} such that b_n 0, b_nn, n=1, 2,..., is a sequence of Fourier-Lebesgue coefficients with respect to the system {cos nx} if and only if the sequence converges.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 77–83, July, 1977.     

Some properties of the moduli of families of curves

Let A={a₁,...,a_n} and B={b₁,...,b_m} be systems of distinct points in , let be a family of homotopic classes H_i,i=1,..., j+m, of closed Jordan curves on, where the classes H_j+l, l=1,...,m, consist of curves that are homotopic to a point curve in b. Let =₁,..., _j+m be a system of positive numbers and letU be the modulus of the extremal-metric problem for the family and the system . In this paper we investigate the dependence of the modulusU=U(,A,B) on the parameters _i and on the disposition of the points a_k and b. One shows thatU is a smooth function of the indicated arguments and one obtains expressions for the derivatives U, U, and U. One gives some applications of these results.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 72–82, 1985.     

Spectral Properties of Abelian C*-Algebras

Let be an Abelian unital C ^*-algebra and let denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of to be unitarily equivalent to a representation in which the elements of act multiplicatively, by their Gelfand transforms, on a space L ²( ,), where is a positive measure on the Baire sets of . We also compare these conditions with the multiplicity-free property of a representation.     

Un nouveau type d'inégalités pour les martingales discrètes

Summary We prove the following extension of classical Burkholder-Davis-Gundy inequalities: let (X _n ) _nN be a martingale; for p1, in order that and belong to L ^p, it is sufficient that Inf(X ^*, S(X)) belong to L ^p. For «regular» martingales this result holds for p>0.     

Computing on-line the lattice of maximal antichains of posets

We consider the on-line computation of the lattice of maximal antichains of a finite poset . This on-line computation satisfies what we call the linear extension hypothesis: the new incoming vertex is always maximal in the current subposet of . In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of , where |P| is the number of elements of the poset, , |MA(P)| is the number of maximal antichains of and (P) is the width of . This is more efficient than the best off-line algorithms known so far.