Necessary and sufficient conditions are obtained on the exponents $\alpha$ and $\beta$ under which the Fourier series in the Sobolev system of polynomials $\mathcal{P}_r^{\alpha,\beta}$ associated with the Jacobi polynomials converge uniformly on $[-1,1]$ to functions in the Sobolev space $W^r_{L^1_{\rho(\alpha,\beta)}}$, where $\rho(\alpha,\beta)$ is the Jacobi weight. Bibliography: 11 titles.
A construction of small and large Nijenhuis cohomologies is presented. For small cohomologies an isomorphism theorem is proved, relating de Rham cohomologies to small Nijenhuis cohomologies for a nondegenerate operator. Poncaré's lemma is deduced as a consequence. For operators with singularities a theorem on connections between the logarithmic and Nijenhuis cohomology is established. The cohomologies of balls containing differentially nondegenerate singularities are calculated. Relations between the cohomologies of different operators in Nijenhuis pencils are established. Bibliography: 16 titles.
We examine the set of irreducible germs fP: UP -> Vp of generic morphisms f : S -> P2 to the projective plane whose branch curve germs BP subset of Vp have singularities equisingular deformation equivalent to the singularities given by equations xk1-yk2 = 0 with coprime k1, k2 is an element of N. Bibliography: 16 titles.
We find a sharp constant in the inequality parallel to psi(& centerdot;)x(& centerdot;)parallel to Lq (T,& micro;) <= C max 1 <= j <= l parallel to phi j(& centerdot;)x(& centerdot;)parallel to gamma Lp(T,& micro;)max l+1 <= j <= n parallel to phi j(& centerdot;)x(& centerdot;)parallel to 1-gamma Lr (T,& micro;), where T is a cone in Rd, and the weights psi( & centerdot; ) and phi j ( & centerdot; ), j = 1,... , n, are homogeneous measurable functions satisfying certain additional symmetry conditions. The sharp inequality is a consequence of a more general problem of optimal recovery in weighted Lq (T, & micro;)-spaces from incompletely given functions, whose solution is also presented. The results obtained are used for the optimal recovery of powers of the generalized Laplace operator and for the derivation of relevant sharp inequalities. Bibliography: 11 titles.
We describe germs of mappings $(\mathbb{C}^2,0) \to (\mathbb{C}^2,0)$ ramified along a germ of irreducible curve whose image is of the form $x^p=y^q$. Bibliography: 11 titles.
It is known (Mazur, Grothendieck, Enflo, Davie) that not every function f (x, y) continuous on the square can uniformly be approximated by linear combinations of its crosses f (a, y)f (x, b). We obtain various conditions on a continuous two-variate function f ensuring that it can uniformly be approximated by linear combinations of its crosses. We prove that these conditions are sufficient using so-called maximal cross approximations, defined by analogy with one special type of cross approximations of matrices. Unlike the uniform case, cross approximation in L2 is always possible, and we discuss various algorithms for this approximation. We also present examples where consecutive cross approximations of continuous functions diverge in Lp.
In the classical Siegel extremal problem one looks for the maxa Euclidean ball and fixed value at the origin. A Szasz-type problem is similar, but the integral is taken with respect to a fixed radial measure. The multidimensional Siegel-Szasz problem reduces to the one-dimensional problem on the real line with power weight. We present a general method for the solution of this problem, which reduces to the spectral analysis of the Dunkl integral convolution operator. The links with quadrature formulae for entire functions of exponential type and duality theory are discussed.
The paper is concerned with orientation-preserving homeomorphisms of 3-manifolds with nonwandering set consisting of a finite number of two-dimensional attractors and repellers, each of which is a disjoint union of cylindrically embedded closed surfaces such that the restriction of some power of the homeomorphism to each surface is topologically conjugate to an orientation-preserving pseudo-Anosov homeomorphism. A topological classification is obtained for the model homeomorphisms realized on each manifold allowing homeomorphisms of the class under study. A homeomorphism in the class under consideration is shown to be topologically conjugate to a model map if and only if its has an invariant one-dimensional foliation. If conjugacy to generalized pseudo-Anosov homeomorphisms on the nonwandering sets of maps under study is allowed, then examples of homeomorphisms are presented whose restrictions to connected components of the nonwandering set are not topologically conjugate, which is uncharacteristic for homeomorphisms in the original class under consideration.
Two dual representations V and V & lowast; of a complex connected algebraic group G have simultaneously an infinite or a finite number of orbits. In the latter case there exists a bijective correspondence between the orbits in V and V & lowast;, which is called the Pyasetskii duality [1]. A full description of this correspondence for orbits of spherical linear representations is obtained.
We consider several finitary analytic hypersequent calculi for first-order infinite-valued Lukasiewicz logic Lb and for its expansion, first-order rational Pavelka logic RPLb, including Baaz and Metcalfe's calculus GLb for Lb. In these calculi the cut rule is not admissible and, in general, a formula and its (purely syntactically defined) prenex form are not equiprovable. However, we develop the method of deprenexification that allows us to perform the following transformation: a formal proof of a hypersequent 7i in which an occurrence of a prenex form of a formula F is distinguished can be transformed algorithmically into a formal proof of the hypersequent obtained from 7i by replacing the occurrence by F. Using this method, we establish the completeness of infinitary analytic calculi for Lb and RPLb that are based on the finitary calculi mentioned above. In particular, we give the first correct proof of the completeness of the GLb-based infinitary analytic calculus for Lb. Bibliography: 16 titles.
Approximative weak compactness is studied for problems of min-and max-approximation. This leads naturally to certain 'special points' of approximation theory, namely the spaces characterizable in terms of approximative weak compactness type properties for various classical problems of approximation theory. Among these 'special points' appear the Kadec-Klee spaces (KK), as well as the wDO-, VDS-, MAT-spaces, and the spaces in the class (wDO) boolean AND (R). Bibliography: 29 titles.
For the Hardy operator from the weighted Lorentz space L- v(r,s) (R+) to L L-omega(p,q) (R+), where the parameters satisfy 1 < s <= min(p, q) < infinity and 1 < r < infinity, criteria for its boundedness and compactness are established. Two-sided estimates for s-numbers are obtained for compact operators. Conditions for the Hardy operator to belong simultaneously to the ideal of compact operators, the ideal of approximable operators, and the ideals generated by sequences of s-numbers are found. Estimates for the norms of the Hardy operator in ideals in terms of integral expressions involving the weight functions of the original operator are given.
Uniform approximations by sums of bianalytic kernels, that is, sums of shifts of the function z/z are under consideration. Namely, we study conditions on a domain Q in the complex plane C and a set E subset of C \ Q which ensure that each bianalytic function in Q can arbitrarily well be approximated locally uniformly in Q by sums of bianalytic kernels with singularities on E. Also, conditions on a compact set X subset of C are investigated ensuring that each continuous function on X that is bianalytic in the interior of X can arbitrarily well be approximated uniformly on X by sums of bianalytic kernels with singularities in C \ X. In both cases the necessary or sufficient conditions found are significantly different from the conditions in the corresponding results on approximations by simple partial fractions, that is, sums of shifts of the function 1/z. Bibliography: 19 titles.
The spectrum of values of the weak uniform Diophantine exponents of lattices is studied, and its complete description in the two-dimensional case is given.
We consider a diffeomorphism f acting in a Banach space E that has a closed invariant set A (such that f (A) = A). We perform a comparative analysis of the two currently known definitions of hyperbolicity of the set A. The first is Anosov's classical definitions, stated in terms of the D f-invariant decomposition Exu circle plus Esx, x E A, of the space E into the direct sum of the unstable subspace Exuand stable subspace Esx. The second definition, based on works by Zelik with coauthors, is stated in terms of the uniform regularity of a certain difference operator. We show that these definitions are equivalent. On the way we establish results on the uniform boundedness and uniform continuity in x E A of the projections corresponding to the above decomposition of E. We also present sufficient conditions ensuring that the restriction f A has the property of essential dependence of trajectories xn= fn(x), n E N, on the initial point x E A, which is characteristic for chaotic dynamics. Bibliography: 35 titles.
For the automorphism group of an ultrahomogeneous cyclically ordered set (with permutation topology) we describe three proper Ellis semigroup compactifications and compare them with the Roelcke compactification. We show that the group G of transformations of a discrete space (with permutation topology) has a semitopological semigroup compactification, and we present an initial description of the Roelcke uniformity on G. Bibliography: 27 titles.
In a recent paper Buchstaber and the author introduced a new structure on the cohomology of Hopf algebras in terms of the Buchstaber spectral sequence (Bss). We fully calculate this structure on the cohomology (known for a long time) of the important Hopf subalgebra A(1) of the classical Steenrod algebra A(2). As part of a demonstration of the methods of Bss, we solve the inverse problem and present a new explicit calculation of the s-dimensional cohomology of the algebra A(1) for s <= 4. We also use the methods of Bss to obtain results on the Massey products and relations in the cohomology of the Steenrod algebras Ap, p >= 2. Bibliography: 36 titles.
This article is dedicated to the memory of Professor N. M. KoConsider the set of finite words in a finite alphabet A subset of N. Add a prefix V and an ending W, which are some fixed finite words in the alphabet N, to each word. We treat the resulting words as the expansions in finite continued fractions of some rational numbers in the interval (0, 1). Next consider the irreducible denominators of these rational numbers; we denote the set of those denominators that do not exceed some integer N is an element of N (which is an increasing parameter) by s7NA,V,W . We prove that under certain conditions on A, V and W, for each prime number Q proportional to a fixed fractional power of N the set s7NA,V,W contains almost all possible residues modulo Q, and the remainder in this asymptotic formula involves a power reduction with respect to Q.
The maximum possible values of the Jordan constant of the automorphism group of a smooth rational two-dimensional quadric over a field of characteristic zero are found in their dependence on the arithmetic properties of the field.
