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# Characterization of automorphisms of twisted tensor biproducts

Authors: Wang Xing, Chen Quanguo and Wang Dingguo

## Abstract

We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

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# The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

Authors: Tahir S. Gadjiev, Vagif S. Guliyev and Konul G. Suleymanova

## Abstract

In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

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# The first cohomology group of some operator algebras on Hilbert C*-modules

Authors: Hoger Ghahramani and Saman Sattari

## Abstract

Let X be a Hilbert C*-module over a C*-algebra B. In this paper we introduce two classes of operator algebras on the Hilbert C*-module X called operator algebras with property $k$ and operator algebras with property ℤ, and we study the first (continuous) cohomology group of them with coefficients in various Banach bimodules under several conditions on B and X. Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras.

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# Integral bases of pure fields with square-free parameter

Author: László Remete

## Abstract

Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type $ℚ(nm)$ is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field $ℚ(nm)$, where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of $ℚ(nm)$ is repeating periodically in m with period length depending on n.

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# Jordan's totient function and trigonometric sums

Author: Wenchang Chu

## Abstract

Two classes of trigonometric sums about integer powers of secant function are evaluated that are closely related to Jordan's totient function.

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# An L-function free proof of Hua's Theorem on sums of five prime squares

Author: Claus Bauer

## Abstract

We provide a new proof of Hua's result that every sufficiently large integer N ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of L-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the W-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.

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# Character varieties of even classical pretzel knots

Author: Haimiao Chen

## Abstract

For each even classical pretzel knot P(2k 1 + 1, 2k 2 + 1, 2k 3), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

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# A Construction of Cohen-Macaulay Graphs

Authors: Safyan Ahmad and Shamsa Kanwal

## Abstract

We present a technique to construct Cohen–Macaulay graphs from a given graph; if this graph fulfills certain conditions. As a consequence, we characterize Cohen–Macaulay paths.

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# Domain representable Lindelöf spaces are cofinally Polish

## Abstract

We prove that, for any cofinally Polish space X, every locally finite family of non-empty open subsets of X is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of σ-compact spaces. It turns out that, for a topological group G whose space has the Lindelöf Σ-property, the space G is domain representable if and only if it is Čech-complete. Our results solve several published open questions.

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# Korselt rational bases of prime powers

Author: Nejib Ghanmi

## Abstract

Let N be a positive integer, $A$ be a subset of ℚ and $α=α1α2∈A\{0, N}$. N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α2pα1 divides α2Nα1 for every prime divisor p of N. By the Korselt set of N over $A$, we mean the set $A−KS(N)$ of all $α∈A\{0, N}$ such that N is an α-Korselt number.

In this paper we determine explicitly for a given prime number q and an integer l ∈ ℕ \{0, 1}, the set $A-KS(ql)$ and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of $A=ℚ∩[−1, 1[$ is studied where we prove that $(ℚ∩[−1, 1[)-KS(ql)$ is empty if and only if l = 2.

Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

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