Abstract:
Given two testable properties $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, under what conditions are the union, intersection or set-difference of these two properties also testable? We initiate a systematic study of these basic set-theoretic operations in the context of property testing. As an application, we give a conceptually different proof that linearity is testable, albeit with much worse query complexity. Furthermore, for the problem of testing disjunction of linear functions, which was previously known to be one-sided testable with a super-polynomial query complexity, we give an improved analysis and show it has query complexity $O(1/\eps^2)$, where $\eps$ is the distance parameter.

Abstract:
We study the behavior of the Nil-subgroups of K-groups under localization. As a consequence we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character we obtain a complete computation of the rationalized source of the K-theoretic assembly map in terms of group homology and the K-groups of finite cyclic subgroups.

Abstract:
In this paper we obtain $\Omega$ and $\Omega_\pm$ estimates for a wide class of error terms $\Delta(x)$ appearing in Perron summation formula. We revisit some classical $\Omega$ and $\Omega_{\pm}$ bounds on $\Delta(x)$, and obtain $\Omega$ bounds for Lebesgue measure of the following types of sets: \begin{align*} \A_+&:=\{T\leq x \leq 2T: \Delta(x)> \lambda x^{\alpha}\},\\ \A_-&:=\{T\leq x \leq 2T: \Delta(x)< -\lambda x^{\alpha}\},\\ \A~&:=\{T\leq x \leq 2T: |\Delta(x)|>\lambda x^{\alpha}\}, \end{align*} where $\alpha, \lambda>0$. We also prove that if Lebesgue measure of $\A$ is $\Omega(T^{1-\delta})$ then \[\Delta(x)=\Omega_\pm(x^{\alpha-\delta})\] for any $0<\delta<\alpha$.

Abstract:
Call a semistar operation $\ast$ on the polynomial domain $D[X]$ an extension (respectively, a strict extension) of a semistar operation $\star$ defined on an integral domain $D$, with quotient field $K$, if $E^\star = (E[X])^{\ast}\cap K$ (respectively, $E^\star [X]= (E[X])^{\ast}$) for all nonzero $D$-submodules $E$ of $K$. In this paper, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on $D[X]$ that are "canonical" extensions (or, "canonical" strict extensions) of semistar operations on $D$.

Abstract:
Several estimates for the convolution function C [f(x)]:=∫1xf(y) f(x/y)(dy/y) and its iterates are obtained when f(x) is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for ∫0T|ζ(1/2

Abstract:
We show that three fundamental information-theoretic constraints--the impossibility of superluminal information transfer between two physical systems by performing measurements on one of them, the impossibility of broadcasting the information contained in an unknown physical state, and the impossibility of unconditionally secure bit commitment--suffice to entail that the observables and state space of a physical theory are quantum-mechanical. We demonstrate the converse derivation in part, and consider the implications of alternative answers to a remaining open question about nonlocality and bit commitment.

Abstract:
We study the convolution function $$ C[f(x)] := \int_1^x f(y)f({x\over y}) {{\rm d} y\over y} $$ when $f(x)$ is a suitable number-theoretic error term. Asymptotics and upper bounds for $C[f(x)]$ are derived from mean square bounds for $f(x)$. Some applications are given, in particular to $|\zeta(1/2+ix)|^{2k}$ and the classical Rankin--Selberg problem from analytic number theory.

Abstract:
We introduce the concept of a weak nil clean ring, a generalization of nil clean ring, which is nothing but a ring with unity in which every element can be expressed as sum or difference of a nilpotent and an idempotent. Further if the idempotent and nilpotent commute the ring is called weak* nil clean. We characterize all $n\in \mathbb{N}$, for which $\mathbb{Z}_n$ is weak nil clean but not nil clean. We show that if $R$ is a weak* nil clean and $e$ is an idempotent in $R$, then the corner ring $eRe$ is also weak* nil clean. Also we discuss $S$-weak nil clean rings and their properties, where $S$ is a set of idempotents and show that if $S=\{0, 1\}$, then a $S$-weak nil clean ring contains a unique maximal ideal. Finally we show that weak* nil clean rings are exchange rings and strongly nil clean rings provided $2\in R$ is nilpotent in the later case. We have ended the paper with introduction of weak J-clean rings.

Abstract:
The concept of nil-symmetric rings has been introduced as a generalization of symmetric rings and a particular case of nil-semicommutative rings. A ring is called right (left) nil-symmetric if, for , where are nilpotent elements, implies . A ring is called nil-symmetric if it is both right and left nil-symmetric. It has been shown that the polynomial ring over a nil-symmetric ring may not be a right or a left nil-symmetric ring. Further, it is also proved that if is right (left) nil-symmetric, then the polynomial ring is a nil-Armendariz ring. 1. Introduction Throughout this paper, all rings are associative with unity. Given a ring , and denote the set of all nilpotent elements of and the polynomial ring over , respectively. A ring is called reduced if it has no nonzero nilpotent elements; is said to be Abelian if all idempotents of are central; is symmetric [1] if implies for all . An equivalent condition for a ring to be symmetric is that whenever product of any number of elements of the ring is zero, any permutation of the factors still gives the product zero [2]. is reversible [3] if implies for all ; is called semicommutative [4] if implies for all . In [5], Rege-Chhawchharia introduced the concept of an Armendariz ring. A ring is called Armendariz if whenever polynomials , satisfy , then for each . Liu-Zhao [6] and Antoine [7] further generalize the concept of an Armendariz ring by defining a weak-Armendariz and a nil-Armendariz ring, respectively. A ring is called weak-Armendariz if whenever polynomials ,？？ satisfy , then for each . A ring is called nil-Armendariz if whenever ,？？ satisfy , then for each . Mohammadi et al. [8] initiated the notion of a nil-semicommutative ring as a generalization of a semicommutative ring. A ring is nil-semicommutative if implies for all . In their paper it is shown that, in a nil-semicommutative ring , forms an ideal of . Getting motivated by their paper we introduce the concept of a right (left) nil-symmetric ring which is a generalization of symmetric rings and a particular case of nil-semicommutative rings. Thus all the results valid for nil-semicommutative rings are valid for right (left) nil-symmetric rings also. We also prove that if a ring is right (left) nil-symmetric and Armendariz, then is right (left) nil-symmetric. In the context, there are also several other generalizations of symmetric rings (see [9, 10]). 2. Right (Left) Nil-Symmetric Rings For a ring , and denote the full matrix ring and the upper triangular matrix ring over , respectively. We observe that if is a ring, then Definition 1. A ring