• Volume 115, Issue 4

November 2005,   pages  371-526

• On the Series $\sum^∞_{k=1}\binom{3k}{k}^{-1}k^{-n}x^k$

In this paper we investigate the series $\sum^∞_{k=1}\binom{3k}{k}^{-1}k^{-n}x^k$. Obtaining some integral representations of them, we evaluated the sum of them explicity for 𝑛=0,1,2.

• An Algebra of Absolutely Continuous Functions and its Multipliers

The aim of this paper is to study the algebra $AC_p$ of absolutely continuous functions 𝑓 on [0,1] satisfying $f(0)=0,f' \in L^p[0,1]$ and the multipliers of $AC_p$.

• There are Infinitely many Limit Points of the Fractional Parts of Powers

Suppose that 𝛼 > 1 is an algebraic number and $𝜉 > 0$ is a real number. We prove that the sequence of fractional parts $\{𝜉𝛼^n\},n=1,2,3,\ldots,$ has infinitely many limit points except when 𝛼 is a PV-number and $𝜉\in\mathbb{Q}(𝛼)$. 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

Let $T_{𝛺,𝛼}(0≤𝛼 < n)$ be the singular and fractional integrals with variable kernel $𝛺(x,z)$, and $[b, T_{𝛺,𝛼}]$ be the commutator generated by $T_{𝛺,𝛼}$ and a Lipschitz function 𝑏. In this paper, the authors study the boundedness of $[b, T_{𝛺,𝛼}]$ on the Hardy spaces, under some assumptions such as the $L^r$-Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators $T_{\overline{𝛺},𝛼}(0≤𝛼 < n)$. The smoothness conditions imposed on $\overline{𝛺}$ 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 $(A_3(2), A_2(4))$ and $(B_n(q),C_n(q))$ for $n ≥ 3,q$ 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 group 𝐻 defined over $\mathbb{F}_q$, does not determine the group 𝐻 up to isomorphism, but it determines the field $\mathbb{F}_q$ under some mild conditions. We then put a group structure on the pairs $(H_1,H_2)$ of split semisimple groups defined over a fixed field $\mathbb{F}_q$ such that the orders of the finite groups $H_1(\mathbb{F}_q)$ and $H_2(\mathbb{F}_q)$ are the same and the groups $H_i$ 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 quasigroup 𝑆 with a Tychonoff topology 𝑇, it is proved that there exists a Hausdorff topological group in which 𝑆 can be embedded algebraically and topologically as a right transversal of a subgroup (not necessarily closeed). It is also proved that if a topological right transversal $(S, T_S, T^S, \circ)$ is such that $T_S=T^S$ is a locally compact Hausdorff topology on 𝑆, then 𝑆 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 set $H(G)$ of all generalized characters on 𝐺 equipped with the compact-open topology is a LCA group and $H(G)=\hat{G}$ (the dual group of 𝐺) if and only if 𝐺 is compact. Both results fail for arbitrary LCA groups. Further, if 𝐺 is second countable, then the Gel’fand space of the commutative convolution algebra $C_c(G)$ equipped with the inductive limit topology is topologically homeomorphic to $H(G)$.

• Vector Bundles with a Fixed Determinant on an Irreducible Nodal Curve

Let 𝑀 be the moduli space of generalized parabolic bundles (GPBs) of rank 𝑟 and degree 𝑑 on a smooth curve 𝑋. Let $M_{\overline{L}}$ be the closure of its subset consisting of GPBs with fixed determinant $\overline{L}$. We define a moduli functor for which $M_{\overline{L}}$ is the coarse moduli scheme. Using the correspondence between GPBs on 𝑋 and torsion-free sheaves on a nodal curve 𝑌 of which 𝑋 is a desingularization, we show that $M_{\overline{L}}$ can be regarded as the compactified moduli scheme of vector bundles on 𝑌 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 on 𝑌. The relation to Seshadri–Nagaraj conjecture is studied.

• Topologically Left Invariant Means on Semigroup Algebras

Let $M(S)$ be the Banach algebra of all bounded regular Borel measures on a locally compact Hausdorff semitopological semigroup 𝑆 with variation norm and convolution as multiplication. We obtain necessary and sufficient conditions for $M(S)^∗$ 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 space $\mathrm{ces}_𝜙$ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro–Orlicz spaces $\mathrm{ces}_𝜙$ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in $\mathrm{ces}_𝜙$ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces $\mathrm{ces}_𝜙$ are given.

• On Topological Properties of the Hartman–Mycielski Functor

We investigate some topological properties of a normal functor 𝐻 introduced earlier by Radul which is some functorial compactification of the Hartman–Mycielski construction HM. We prove that the pair (𝐻𝑋, 𝐻𝑀𝑌) is homeomorphic to the pair $(\mathcal{Q}, 𝜎)$ for each nondegenerated metrizable compactum 𝑋 and each dense 𝜎-compact subset 𝑌.

• 𝐷-Boundedness and 𝐷-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 of 𝐷-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 of 𝐴-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 function 𝑓 by means of a sequence $\{L_nf\}$ 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 of 𝑁 partial differential equations. When 𝑁 = 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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