‎‎Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎We define the primary spectrum of $M$‎, ‎denoted by $mathcal{PS}(M)$‎, ‎to be the set of all primary submodules $Q$ of $M$ such that $(operatorname{rad}Q:M)=sqrt{(Q:M)}$‎. ‎In this paper‎, ‎we topologize $mathcal{PS}(M)$ with a topology having the Zariski topology on the prime spectrum $operatorname{Spec}(M)$ as a sub...

Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...

We set up a generic framework for proving completeness results for variants of the modal mucalculus, using tools from coalgebraic modal logic. We illustrate the method by proving two new completeness results: for the graded mu-calculus (which is equivalent to monadic second-order logic on the class of unranked tree models), and for the monotone modal mu-calculus. Besides these main applications...

In this paper we introduce the notions of uniformly quasi-primary ideals and uniformly classical quasi-primary submodules that generalize the concepts of uniformly primary ideals and uniformly classical primary submodules; respectively. Several characterizations of classical quasi-primary and uniformly classical quasi-primary submodules are given. Then we investigate for a ring $R$, when any fi...

Background: Primary antibody deficiency, the most common primary immunodeficiency disorder, represents a heterogeneous spectrum of conditions caused by a defect in any critical stage of B cell development and is characterized by impaired production of normal amounts of antigen-specific antibodies. Objective: This retrospective study aimed at description and analysis of demographic, clinical, im...

primary amyloidosis is a rare disrder wihich is diagnosed by extracellular deposition of proteinaceous material in different organs.in this report,a case of this disease with prominent hepatic involvement is presented.the case is a 63 years old male refered with abdominal enlargement,weight loss,generalized pruritus,anorexia and vague abdominal pain started from six months ago.on physical exami...

the main objective of this study is to swing krull intersection theorem in primary decomposition of rings and modules to the primary decomposition of soft rings and soft modules. to fulfill this aim several notions like soft prime ideals, soft maximal ideals, soft primary ideals, and soft radical ideals are introduced for a soft ring over a given unitary commutative ring. consequently, the p...

The background of Stevenson’s story is viewed as a stylized model of participation in a financial pyramid. You are invited to take part in a dubious activity. If you refuse, you neither gain nor lose anything. If you accept, you will gain if you are able to find somebody who will take your place on exactly the same conditions, but suffer a loss otherwise. The catch is that there is a finite num...

Variational sparsity regularization based on l1-norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization parameter and noise strength. For this sake specific error measures such as Bregman distances and specifi...

We present a new general asymptotic theory for the maximum of a random array {Xn(i) : 1 ≤ i ≤ L}n≥1, where each Xn(i) is assumed to converge in probability as n → ∞. The array dimension L is allowed to increase with the sample size n. Existing extreme value theory arguments focus on observed data Xn(i), and require a well defined limit law for max1≤i≤L |Xn(i)| by restricting dependence across i...