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We deﬁne the extended beta family of distributions to generalize the beta generator pioneered by Eugene *et al*. [10]. This paper is cited in at least 970 scientiﬁc articles and extends more than ﬁfty well-known distributions. Any continuous distribution can be generalized by means of this family. The proposed family can present greater ﬂexibility to model skewed data. Some of its mathematical properties are investigated and maximum likelihood is adopted to estimate its parameters. Further, for different parameter settings and sample sizes, some simulations are conducted. The superiority of the proposed family is illustrated by means of two real data sets.

We present the sufficient condition for a classical two-class problem from Fisher discriminant analysis has a solution. Actually, the solution was presented up to our knowledge with a necessary condition only. We use an extended Cauchy–Schwarz inequality as a tool.

Let be a Schrödinger operator on the Heisenberg group *V* belongs to the reverse Hölder class

where

and

Where

Based on this inequality and known results for the Lusin area integral *V*, the author establishes the boundedness properties of the operator

In this paper, a relationship between the zeros and critical points of a polynomial *p*(*z*) is established. The relationship is used to prove Sendov’s conjecture in some special cases.

A ﬂuid queueing system in which the ﬂuid ﬂow in to the buffer is regulated by the state of the background queueing process is considered. In this model, the arrival and service rates follow chain sequence rates and are controlled by an exponential timer. The buffer content distribution along with averages are found using continued fraction methodology. Numerical results are illustrated to analyze the trend of the average buffer content for the model under consideration. It is interesting to note that the stationary solution of a ﬂuid queue driven by a queue with chain sequence rates does not exist in the absence of exponential timer.

In this paper, we deﬁne an orthonormal basis for 2-*-inner product space and obtain some useful results. Moreover, we introduce a 2-norm on a dense subset of a 2-***-inner product space. Finally, we obtain a version of the Selberg, Buzano’s and Bessel inequality and its results in an *A*-2-inner product space.

## Abstract

We provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity *k* factors) holds in 𝓥, for some natural number *k*.

## Abstract

Fix 2 *< n < ω* and let CA*
_{n}
* denote the class of cyindric algebras of dimension

*n*. Roughly CA

*is the algebraic counterpart of the proof theory of first order logic restricted to the first*

_{n}*n*variables which we denote by

*L*. The variety RCA

_{n}*of representable CA*

_{n}*s reflects algebraically the semantics of*

_{n}*L*. Members of RCA

_{n}*are concrete algebras consisting of genuine*

_{n}*n*-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CA

*has a finite equational axiomatization, RCA*

_{n}*is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CA*

_{n}*substantially richer than that of Boolean algebras, just as much as*

_{n}*L*is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCA

_{ω,ω}*are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever*

_{n}*L*, fails dramatically for

_{ω,ω}*L*even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that Nr

_{n}*CA*

_{n}*∩ CRCA*

_{ω}

_{n}**S**

*Nr*

_{c}*CA*

_{n}

_{n}_{+3}, where CRCA

*is the class of completely representable CA*

_{n}*s, and*

_{n}**S**

*denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that*

_{c}**S**

*RaCA*

_{d}

_{ω}**S**

*RaCA*

_{c}_{5}is not elementary, where

**S**

*denotes the operation of forming dense subalgebra.*

_{d}## Abstract

Let **𝔄** be a unital Banach algebra and **ℜ** its Jacobson radical. This paper investigates Banach algebras satisfying some chain conditions on closed ideals. In particular, it is shown that a Banach algebra **𝔄** satisfies the descending chain condition on closed left ideals then **𝔄**/**ℜ** is finite dimensional. We also prove that a *C*
^{*}-algebra satisfies the ascending chain condition on left annihilators if and only if it is finite dimensional. Moreover, other auxiliary results are established.

## Abstract

We give two new simple characterizations of the Cauchy distribution by using the Möbius and Mellin transforms. They also yield characterizations of the circular Cauchy distribution and the mixture Cauchy model.