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The purpose of this paper is to study the principal fibre bundle (*P*, *M*, *G*, *π*
_{
p
} ) with Lie group *G*, where M admits Lorentzian almost paracontact structure (*Ø*, *ξ*
_{
p
}, η_{
p
}, *g*) satisfying certain condtions on (1, 1) tensor field *J*, indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map *π*
^{
*
} is the isomorphism.

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

We introduce the directional short-time Fourier transform for which we prove a new Plancherel’s formula. We also prove for this transform several uncertainty principles as Heisenberg inequalities, logarithmic uncertainty principle, Faris–Price uncertainty principles and Donoho–Stark’s uncertainty principles.

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*.