On a theorem of M. Satô

In 1953 Nash [7] introduced the class of functions Φ. In this paper the behaviour of generalized Cesàro (C,α _n )-means (α _n ∈(−1,0)) of trigonometric Fourier series of the classes H ^ω ∩Φ in the space of continuous functions is studied. The sharpness of the results obtained is shown.     

The Gearhart-Prüss theorem for a class of degenerate semigroups of operators

We consider a class of semigroups of operators in Hilbert space whose generators are linear relations. An analog of the Gearhart-Prüss theorem for semigroups of operators from the class under consideration is obtained.     

On a generalization of a Greguš fixed point theorem

Let C be a closed convex subset of a complete convex metric space X. In this paper a class of selfmappings on C, which satisfy the nonexpansive type condition (2) below, is introduced and investigated. The main result is that such mappings have a unique fixed point.     

RESEARCH ANNOUNCEMENTS—On a Generalization of the c—numerical Range

Let Cn×n be the set of n × n complex matrices and An the set of orthonormal n-tuples of vectors in Cn. For a vector c in Cn and a matrix A in Cn×n, the c-numerical range of A is the set Wc(A)={n∑i=1 Ci(Axi,xi):(x1,…xn)∈∧n} When c = (1,0,…,0), Wc(A) is reduced to the classical numerical range W(A) (see [1]). For the classical numerical range and its generalizations, one may see the survey article[2].     

On a theorem of G. Pólya

f(z), :f(n)=0 (n=0, ±1, ±2, ...). ((n)} L _p ,p>1, .     

On a discrete version of alexandrov’s projection theorem

In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.     

On a Żołądek theorem

We prove the existence of cubic systems of the form $$ \begin{gathered} \dot x = y[1 - 2r(5 + 3r^2 )x + \gamma \lambda ^2 x^2 ] + a_0 x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 , \hfill \\ \dot y = - x(1 - 8rx)(1 - 3r\gamma x) - 2x[2(1 - 3r^2 ) - r\gamma (7 - 15r^2 )x]y \hfill \\ - [r(11 + r^2 ) + \gamma (1 - 22r^2 - 3r^4 )x]y^2 \hfill \\ - 2r\gamma \delta y^3 + a_0 y + a_7 x^2 + a_8 xy + a_9 y^2 + a_{10} x^3 + a_{11} x^2 y, \hfill \\ \end{gathered} $$ where α = 3r ² + 17, γ = r ² + 3, δ = 1 ? r ², and λ = 3r ² + 1, that have at least eleven limit cycles in a neighborhood of the point O(0, 0).     

Left and right negatively orderable semigroups and a one-sided version of Simon’s theorem

We study the classes \(\mathrm {LNO}\) and \(\mathrm {RNO}\) of left and right negatively orderable semigroups, arising as natural one-sided generalizations of negatively orderable semigroups (\(\mathrm {NO}\)). Negatively ordered monoids are well-known, for instance, from an equivalent formulation by Straubing and Thérien, of a celebrated theorem by I. Simon on piecewise testable languages. The main aim of this paper is to prove a one-sided version of Simon’s theorem for \(\mathrm {LNO}\) (\(\mathrm {RNO}\)). Analogues of some well-known results about negatively orderable semigroups are established in these cases. A characterisation of right negatively orderable semigroups as semigroups of certain type of decreasing mappings on partially ordered sets is obtained. We present sets of quasi-identities defining the quasivarieties \(\mathrm {LNO}\) and \(\mathrm {RNO}\) and show that these classes cannot be defined by finite sets of quasi-identities. We prove \(\mathrm {NO}=\mathrm {LNO}\cap \mathrm {RNO}\) and describe \(\mathrm {LNO}\vee \mathrm {RNO}\). As a one-sided version of Simon’s theorem, the pseudovariety generated by all finite semigroups in \(\mathrm {LNO}\) (\(\mathrm {RNO}\)) is determined and an Eilenberg-type correspondence between this pseudovariety and a variety of languages is established.     

On a theorem of lovász on covers inr-partite hypergraphs

A theorem of Lovász asserts that (H)/*(H)r/2 for everyr-partite hypergraphH (where and * denote the covering number and fractional covering number respectively). Here it is shown that the same upper bound is valid for a more general class of hypergraphs: those which admit a partition (V ₁, ...,V _k ) of the vertex set and a partitionp ₁+...+p _k ofr such that |eV _i |p _i r/2 for every edgee and every 1ik. Moreover, strict inequality holds whenr>2, and in this form the bound is tight. The investigation of the ratio /* is extended to some other classes of hypergraphs, defined by conditions of similar flavour. Upper bounds on this ratio are obtained fork-colourable, stronglyk-colourable and (what we call)k-partitionable hypergraphs.Supported by grant HL28438 at MIPG, University of Pennsylvania, and by the fund for the promotion of research at the Technion.This author's research was supported by the fund for the promotion of research at the Technion.     

On the optimality of a theorem of Elton on ℓ 1 n subsystems

We present examples which show that a substantial strengthening of the hypothesis in the almost isometric part of a theorem of Elton on ℓ ₁ ⁿ subsystems does not lead to a substantially stronger conclusion.     

On Galvin’s theorem for compact Hausdorff right-topological semigroups with dense topological centers

We generalize an important theorem of Fred Galvin from the Stone-Cˇech compactification βT of any discrete semigroup T to any compact Hausdorff right-topological semigroup with a dense topological center;and then apply it to Ellis' semigroups to prove that a point is distal if and only if it is IP*-recurrent, for any semiflow(T, X) with arbitrary compact Hausdorff phase space X not necessarily metrizable and with arbitrary phase semigroup T not necessarily cancelable.     

On the number of numerical semigroups 〈a,b〉 of prime power genus

Given g≥1, the number n(g) of numerical semigroups S?? of genus |??S| equal to g is the subject of challenging conjectures of Bras-Amorós. In this paper, we focus on the counting function n(g,2) of two-generator numerical semigroups of genus g, which is known to also count certain special factorizations of 2g. Further focusing on the case g=p ^k for any odd prime p and k≥1, we show that n(p ^k ,2) only depends on the class of p modulo a certain explicit modulus M(k). The main ingredient is a reduction of $\operatorname{gcd}(p^{\alpha}+1, 2p^{\beta}+1)$ to a simpler form, using the continued fraction of α/β. We treat the case k=9 in detail and show explicitly how n(p ⁹,2) depends on the class of $p \operatorname{mod} M(9)=3 \cdot5 \cdot11 \cdot17 \cdot43 \cdot257$ .