On the ε-entropy of classes of holomorphic functions

Suppose thatB _R ^d is a ball of radiusR in ℂ ^d and σ is the standard measure on the unit sphere in ℂ ^d . ForR>1, 1≤p≤∞, and for the natural numbersl, d, byH _R ⁰ (l, p, d) we denote the class of functionsf holomorphic inB _R ^d and such that in the homogeneous polynomial expansion of the firstl summands the zero and radial derivatives of orderl belong to the closed unit ball of the Hardy spaceH ^p (B _R ^d ). In this paper an asymptotic formula for the ε-entropy of the classH _R ⁰ (l, p, d) in the spacesL ^p (σ), 1≤p<∞, and is obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 286–293, August, 2000.     

Three classes of Weitzenböck manifolds

A Weitzenböck manifold is a triplet defined by a differentiable manifold with a metric g of certain signature and a linear connection with zero curvature tensor and nonzero torsion tensor which is a metric connection with respect to g. The theory of such manifolds is called the “new theory of gravity”. We study properties of three classes of Weitzenböck manifolds and prove some vanishing thorems.     

Gamma–closure of some classes of nonstandard convex integrands

Within the framework of the theory of Γ-convergence for convex functionals with nonstandard coercivity and growth conditions, we study the inheritance of some properties (for example, strict convexity, differentiablility, and Δ₂-property) of integrands under taking the Γ-closure. We focus on power integrands |ξ| ^p(x) with variable exponents. The results obtained are also of interest in the case of the Γ-convergence theory for standard functionals. Bibliography: 15 titles.     

A note on embedding of classes H ω

Among others, we extend an interesting theorem of M. V. Medvedeva pertaining to the embedding relation H BV, where BV denotes the set of the functions of -bounded variation.     

On the radius of α-convexity of certain classes of starlike functions

LetM (α) denote the class of α-convex functions, α real, that is the class of analytic functions? (z) =z + Σ _n=2/^∞ a _n z ⁿ in the unit discD = {z: |z | < 1} which satisfies inD the condition ?′ (z) ?(z)/z ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z f'(z)}}{{f (z)}} + a \left( {1 + \frac{{z f''(z)}}{{f' (z)}}} \right)} \right\} > 0. Let W (a) $$ denote the class of meromorphic α-convex functions. α real, that is the class of analytic functions ? (z) =z ^?1 + Σ _n=0/^∞ b _n z ⁿ inD* = {z: 0 < |z | < 1} which satisfies inD* the conditionsz?′(z)/?(z) ≠ 0 and $$\operatorname{Re} \left\{ {(1 - a) \frac{{z\phi ' (z)}}{{\phi (z)}} + a \left( {1 + \frac{{z\phi ''(z)}}{{\phi ' (z)}}} \right)} \right\}< 0. $$ In this paper we obtain the relation betweenM (a) and W(α). The radius of α-convexity for certain classes of starlike functions is also obtained.     

On the poset of conjugacy classes of subgroups of π-power index

We investigate tile poset S^π(G)/G of conjugacy clases of subgroups of π-power index in a finite group G. In particular, we are concerned with combinatorial and topological properties of the order complex of S^π(G)/G. We show that the order complex of S^π(G)/G iS homotopic to a join of orbit spaces of order complexes of posets, which bear structural information on the cheif factors of the group. Moreover, for π-solvable groups and in case π = {p} we reveal a shellable subposer of S^π(G)/G of the same homotopy type. This complements the study of the poset S^π(G) of subgroups of π-power index performed in [20]. For the analysis of the order complexes we develop some new lemmata on the topology of order complexes of posets and in the theory of shellability.     

On new classes of strongly Janowski type functions of complex order

We investigate some new subclasses of analytic functions of Janowski type of complex order. We also study inclusion properties, distortion theorems, coefficient bounds and radius of convexity of the functions. Moreover, analytic properties of these classes under certain integral operator are also discussed. Our findings are more comprehensive than the existing results in the literature.     

Fourier coefficients of functions from many-dimensional classes of bounded Λ-variation

Estimates of trigonometric Fourier coefficients are obtained for functions from the classes of bounded many-dimensional Λ-variation and functions continuous in Λ-variation. In some cases it is proved that these estimates are unimprovable by the order.     

Examples of divergent fourier series for classes of functions of bounded Λ-variation

The author has shown earlier that the requirement that a continuous function belong to the class HBV ([?π, π] ^m ) for m ≥ 3 is not sufficient for the convergence of its Fourier series over rectangles. The author gave examples of functions of three and more variables from the Waterman class which are harmonic in the first variable and significantly narrower in the other variables and whose Fourier series are divergent at some point even on cubes. In the present paper, this assertion is strengthened. The main result is that such an example can be constructed even when the class with respect to the first variable is somewhat narrowed. Also the one-dimensional result due to Waterman is refined.     

Linear widths of Hölder-Nikol'skii classes of periodic functions of several variables

We consider the linear widths _N (W _p ^r (Tⁿ), L_q) and _N (H _p ^r (Tⁿ), L_q) of the classesW _p ^r (Tⁿ) andH _p ^r (Tⁿ) of periodic functions of one or several variables in the spaceL _q. For the Sobolev classesW _p ^r (Tⁿ) of functions of one or several variables, we state some well-known results without proof; for the Hölder-Nikol'skii classesH _p ^r (Tⁿ), we state some well-known results, prove some new results, and present some previously unpublished proofs.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 189–199, February, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Average σ-K widths of some generalized Sobolev-Wiener classes

In this paper, a kind of generalized Sobolev-Wiener classes , h>0, defined on the whole real axis, is introduced, and the average σ-K width problem of these function classes in the metric is studied. For the case p=+∞, 1≤q≤+∞, the case 1≤p <+∞, q=1, we get their exact values and identify their optimal subspaces. This project supported by YNSFC     

The best one-sided approximation of the classes WrHω

In this paper we calculate the upper bounds of the best one-sided approximations, by trigonometric polynomials and splines of minimal defect in the metric of the space L, of the classes W^rH (r = 2, 4, 6, ...) of all 2-periodic functions f(x) that are continuous together with their r-th derivative f^r(x) and such that for any points x and x we have ¦f ^r (x) f^r (x) ¦ (x–x¦), where (t) is a modulus of continuity that is convex upwards.Translated from Matematicheskie Zametki, Vol. 21, No. 3, 313–327, March, 1977.     

Approximation of Boolean functions to Schaefer’s classes

We consider an algorithm for transferring Boolean functions to function classes for which the generalized satisfiability problem is solvable in polynomial time (Schaefer??s classes). For the case where an initial function is represented as a truth-table, it is proved that the complexity of the algorithm is $ {l^2} + l\,\log_2^2(l) $ , where l is the input length.     

On the real algebra of Denjoy–Carleman classes

We show that a nonnegative function germ at the origin of belonging to a quasianalytic Denjoy–Carleman class can be written as a sum of two squares of functions which lie in a Denjoy–Carleman class again. When the germ is elliptic we prove that the class is the same, in the general case a loss of regularity is possible. As a consequence we deduce the Artin–Lang property for suitable unions of such quasianalytic classes. The author is a member of GNSAGA of CNR, partially supported by the European Research Training Network RAAG 2002-2006 (HPRN-CT-00271).     

Distribution of values of σ2 (n) in residue classes

IntegersN for which ₂ (n has weak uniform distribution (modN) are determined.     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.     

On the positivity of high-degree Schur classes of an ample vector bundle

Let X be a smooth projective variety of dimension n,and let E be an ample vector bundle over X.We show that any Schur class of E,lying in the cohomology group of bidegree(n-1,n-1),has a representative which is strictly positive in the sense of smooth forms.This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level,and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld(1983)for certain degrees.     

Approximation by Riesz sums of periodic functions of Hölder classes

We have found asymptotic equalities for the least upper bounds of the deviations of Riesz sums on the Hölder classes W^rH, r is a nonnegative integer, (t) is an arbitrary convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 341–354, March, 1977.     

On the number of conjugacy classes of π-elements in finite groups

Let G be a finite group and π be a set of primes. Put ${d_{\pi}(G) = k_{\pi}(G)/|G|_{\pi}}$ , where ${k_{\pi}(G)}$ is the number of conjugacy classes of π-elements in G and |G| _π is the π-part of the order of G. In this paper we initiate the study of this invariant by showing that if ${d_{\pi}(G) > 5/8}$ then G possesses an abelian Hall π-subgroup, all Hall π-subgroups of G are conjugate, and every π-subgroup of G lies in some Hall π-subgroup of G. Furthermore, we have ${d_{\pi}(G) = 1}$ or ${d_{\pi}(G) = 2/3}$ . This extends and generalizes a result of W. H. Gustafson.