• Volume 115, Issue 4

November 2005,   pages  371-526

• On the series ∑k=1 (k3k)−1knxk

In this paper we investigate the series ∑k=1 (k3k)−1knxk. Obtaining some integral representations of them, we evaluated the sum of them explicitly forn = 0, 1, 2.

• An algebra of absolutely continuous functions and its multipliers

The aim of this paper is to study the algebraACp of absolutely continuous functionsf on [0,1] satisfying f(0) = 0,f ’ ∈ Lp[0, 1] and the multipliers ofACp.

• There are infinitely many limit points of the fractional parts of powers

Suppose that α &gt; 1 is an algebraic number and ξ &gt; 0 is a real number. We prove that the sequence of fractional partsξαn, n = 1, 2, 3, …, has infinitely many limit points except when α is a PV-number and ξ ∈ ℚ(α). For ξ = 1 and α being a rational non-integer number, this result was proved by Vijayaraghavan.

• Commutators of integral operators with variable kernels on Hardy spaces

LetTΩ,α (0 ≤ α&lt; n) be the singular and fractional integrals with variable kernel Ω(x, z), and [b, TΩ,α] be the commutator generated by TΩ,α and a Lipschitz functionb. In this paper, the authors study the boundedness of [b, TΩ,α] on the Hardy spaces, under some assumptions such as theLr-Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators$$T_{\tilde \Omega ,\alpha } (0 \leqslant \alpha&lt; n)$$. The smoothness conditions imposed on$$\tilde \Omega$$ are weaker than the corresponding known results.

• On the orders of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups.

It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A3(2), A2(4)) and(Bn(q), Cn(q)) forn ≥ 3,q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H($$\mathbb{F}_{_q }$$) for a split semisimple algebraic groupH defined over$$\mathbb{F}_{_q }$$, does not determine the groupH up to isomorphism, but it determines the field$$\mathbb{F}_{_q }$$ under some mild conditions. We then put a group structure on the pairs(H1,H2) of split semisimple groups defined over a fixed field$$\mathbb{F}_{_q }$$ such that the orders of the finite groups H1($$\mathbb{F}_{_q }$$) and H2($$\mathbb{F}_{_q }$$) are the same and the groupsHi have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.

• Transversals in non-discrete groups

The concept of ‘topological right transversal’ is introduced to study right transversals in topological groups. Given any right quasigroupS with a Tychonoff topologyT, it is proved that there exists a Hausdorff topological group in whichS can be embedded algebraically and topologically as a right transversal of a subgroup (not necessarily closed). It is also proved that if a topological right transversal(S, TS,TS, o) is such thatTS =TS is a locally compact Hausdorff topology onS, thenS can be embedded as a right transversal of a closed subgroup in a Hausdorff topological group which is universal in some sense.

• A note on generalized characters

For a compactly generated LCA group G, it is shown that the setH(G) of all generalized characters on G equipped with the compact-open topology is a LCA group andH(G) = Ĝ (the dual group ofG) if and only ifG is compact. Both results fail for arbitrary LCA groups. Further, ifG is second countable, then the Gel’fand space of the commutative convolution algebraCc(G) equipped with the inductive limit topology is topologically homeomorphic toH(G).

• Vector bundles with a fixed determinant on an irreducible nodal curve

LetM be the moduli space of generalized parabolic bundles (GPBs) of rankr and degree dona smooth curveX. LetM−L be the closure of its subset consisting of GPBs with fixed determinant− L. We define a moduli functor for whichM−L is the coarse moduli scheme. Using the correspondence between GPBs onX and torsion-free sheaves on a nodal curveY of whichX is a desingularization, we show thatM−L can be regarded as the compactified moduli scheme of vector bundles onY with fixed determinant. We get a natural scheme structure on the closure of the subset consisting of torsion-free sheaves with a fixed determinant in the moduli space of torsion-free sheaves onY. The relation to Seshadri-Nagaraj conjecture is studied.

• Topologically left invariant means on semigroup algebras

LetM(S) be the Banach algebra of all bounded regular Borel measures on a locally compact Hausdorff semitopological semigroupS with variation norm and convolution as multiplication. We obtain necessary and sufficient conditions forM(S)* to have a topologically left invariant mean.

• Basic topological and geometric properties of Cesàro-Orlicz spaces

Necessary and sufficient conditions under which the Cesàro-Orlicz sequence spacecesϕ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro-Orlicz spacescesϕ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements incesϕ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spacescesϕ are given.

• On topological properties of the Hartman-Mycielski functor

We investigate some topological properties of a normal functorH introduced earlier by Radul which is some functorial compactification of the Hartman-Mycielski construction HM. We prove that the pair (H X, HMY) is homeomorphic to the pair (Q, σ) for each nondegenerated metrizable compactumX and each denseσ-compact subsetY.

• D-boundedness andD-compactness in finite dimensional probabilistic normed spaces

In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion ofD-compactness and D-boundedness in probabilistic normed spaces.

• A functional central limit theorem for a class of urn models

We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.

• A-Statistical extension of the Korovkin type approximation theorem

In this paper, using the concept ofA-statistical convergence which is a regular (non-matrix) summability method, we obtain a general Korovkin type approximation theorem which concerns the problem of approximating a functionf by means of a sequenceLnf of positive linear operators.

• Large time behaviour of solutions of a system of generalized Burgers equation

In this paper we study the asymptotic behaviour of solutions of a system ofN partial differential equations. WhenN = 1 the equation reduces to the Burgers equation and was studied by Hopf. We consider both the inviscid and viscous case and show a new feature in the asymptotic behaviour.

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