## Abstract

In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class *A _{p}* 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 ⊂ ℝ

*are obtained.*

^{n}## 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 *B* and *X*. Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras.

## Abstract

Let *m* ≠ 0, ±1 and *n* ≥ 2 be integers. The ring of algebraic integers of the pure fields of type *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 *n*.

## Abstract

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

## 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.

## Abstract

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

## 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.

## 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.

## Abstract

Let *N* be a positive integer, *N* is called an *α-Korselt number* (equivalently *α* is said an *N-Korselt base*) if *α*
_{2}
*p* − *α*
_{1} divides *α*
_{2}
*N* − *α*
_{1} for every prime divisor *p* of *N.* By the *Korselt set* of *N* over *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 *q ^{l}* -Korselt bases in ℚ and others in ℤ. The case of

*l*= 2.

Moreover, we show that each nonzero rational *α* is an *N*-Korselt base for infinitely many numbers *N* = *q ^{l}* where

*q*is a prime number and

*l*∈ ℕ.

## Abstract

Sufficient conditions on associated parameters *p, b* and *c* are obtained so that the generalized and “normalized” Bessel function *u _{p}*(

*z*) =

*u*(

_{p,b,c}*z*) satisfies the inequalities ∣(1 + (

*zu*″

*(*

_{p}*z*)/

*u*′

*(*

_{p}*z*)))

^{2}− 1∣ < 1 or ∣((

*zu*

_{p}(

*z*))′/

*u*(

_{p}*z*))

^{2}− 1∣ < 1. We also determine the condition on these parameters so that $$-(4(p+(b+1)/2)/c){u}_{p}^{\text{'}}(x)\prec \sqrt{1+z}$$. Relations between the parameters

*μ*and

*p*are obtained such that the normalized Lo

__mmel__function of first kind

*h*(

_{μ,p}*z*) satisfies the subordination $$1+(z{h}_{\mu ,p}^{\text{'}\text{'}}(z)/{h}_{\mu ,q}^{\text{'}}(z))\prec \sqrt{1+z}$$. Moreover, the properties of Alexander transform of the function

*h*(

_{μ,p}*z*) are discussed.