normal $l$

This paper is the second in the series celebrating the mathematical works of Professor Themba Dube. In this sequel, we give prominence to Dube's pivotal contributions on pointfree convergence at the unstructured frame level, in the category of locales, and on his noteworthy conceptions on extensions and frame quotients. We distill and draw attention to particular studies of Dube on filters and his novel characterizations of certain conservative pointfree properties by filter and ultrafilter convergence, notably normality, almost realcompactness, and pseudocompactness. We also feature Dube's joint work on convergence and clustering of filters in Loc and coconvergence and coclustering of ideals in the category Frm.

Let G be a finite group and S be a subset of G such that 1G ̸∈ S and S −1 = S. The Cayley graph Σ = Cay(G, S) on G with respect to S is the graph with the vertex set G such that, for §, † ∈ G, the pair (§, †) is an arc in Cay(G, S) if and only if †§−1 ∈ S. The graph Σ is said to be arc-transitive if its full automorphism group Aut(Σ) is transitive on its arc set. In this paper we give a classification for arc-transitive Cayley graphs with valency six on finite abelian groups which are non-normal. Moreover, we classify all normal Cayley graphs on non-cyclic abelian groups with valency 6.

This paper investigates the asymptotic behavior of the number of recalls  Xn of the Random Median Quicksort algorithm in order to sort a list of n distinct numbers. As  n→∞, we provide the asymptotics of the expectation and variance of the recalls. Furthermore, by utilizing a refined version of the contraction method for degenerate limits, we show the limiting distribution of Xn correctly normalized is Gaussian. The theoretical results are demonstrated by a simulation study.

In this survey we collect some results on the influence on the structure of a group of some families of its subgroups satisfying conditions related to normality. In particular we focus on groups whose subgroups have two antagonistic properties.

In this paper, the researchers defined the notions of normal, prime and nodal UP-filters in UP-algebras and investigated several properties of them. Also, the researchers state and proved some theorems in order to determine the relationships between this notions and some types of UP-filters in a UP-algebra and by some examples the researchers show that these notions are different.

In this paper, we construct the concept of general  Krasner  hyperring based on the  ring  structures and the left general Krasner hypermodule based on the  module structures.  This study introduces  the  trivial left general Krasner hypermodules and  proves that the  trivial left general Krasner   hypermodules  are  different from left  Krasner   hypermodules. We show that for any given general  Krasner  hyperring $R$ and trivial left general Krasner   hypermodules $A, B, {bf_{R}h}$om$(A, B)$ is a left general Krasner   hypermodule and  ${bf_{R}h}$om$(-, B)$,     $ ({bf_{R}h}$om$(A, -) )$ is an   exact covariant functor (contravariant). Finally, we  show that the category ${bf_{R}GKH}$mod (left  trivial general Krasner hypermodules and all (homomorphisms) is an abelian  category and  trivial left general Krasner hypermodules have  a normal injective resolution.

In the case of NH3, two reasonable geometries can be tried. Molecular orbitals are the main electronic structural units for analysis and solution of chemical problems at the electronic level and Quantum mechanical description of the changes in electronic structure due to distortions in molecular shape and vice versa is given in the form of the vibronic coupling theory. The most famous concept based on this theory is the Jahn−Teller (JT) effect. The second-order Jahn-Teller effect (SOJT) is an example of reactions proceeding by an interaction between the HOMO and the LUMO within the same molecule In the high-symmetry regular triangular configuration D3h with the N atom in the center, the ground-state configuration of the system is singlet 1A1, that in the direction coordinate instability of Qa2" with the time-dependent DFT (TD-DFT) calculations symmetry descents and to form a square-pyramidal structure. The intrinsic reaction coordinate (IRC) theory in the present paper is presented for further understanding of the mechanism of such distortion. Natural bond analysis (NBO) is used for illustrating the strongest interaction and natural atomic charges of these structures. The calculated energy profile has been supplemented with optimization by means of transition state theory (TST).

We study soluble groups G in which each subnormal subgroup H with infinite rank is‎ ‎commensurable with a normal subgroup‎, ‎i.e‎. ‎there‎ ‎exists a normal subgroup N such that H∩N has finite index‎ ‎in both H and N‎. ‎We show that if such a G is periodic‎, ‎then‎ ‎all subnormal subgroups are commensurable with a normal subgroup‎, ‎provided either the Hirsch-Plotkin radical of G has infinite‎ ‎rank or G is nilpotent-by-abelian (and has infinite rank)‎.

This paper introduces and investigates the notion of a generalized Stone residuated lattice. It is observed that a residuated lattice is generalized Stone if and only if it is quasicomplemented and normal. Also, it is proved that a finite residuated lattice is generalized Stone if and only if it is normal. A characterization for generalized Stone residuated lattices is given by means of the new notion of $alpha$-filters. Finally, it is shown that each non-unit element of a directly indecomposable generalized Stone residuated lattice is a dense element.

In this paper , we present the first moment has been defined based on the center of mass from the fuzzy number, then with calculating area between the center of mass, a new ranking method has been proposed. At last we present some numerical examples to illustrate our proposed method, then comparing with distance index ranking method.

In this paper, we determine all of connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p2, where p is a prime number.

In this paper, we study the notion of solvable $L$-subgroup of an $L$-group and provide its level subset characterization and this justifies the suitability of this extension. Throughout this work, we have used normality of an $L$-subgroup of an $L$-group in the sense of Wu rather than Liu.

In this paper, we extend the construction of a fuzzy subgroup generated by a fuzzy subset to $L$-setting. This construction is illustrated by an example. We also prove that for an $L$-subset of a group, the subgroup generated by its level subset is the level subset of the subgroup generated by that $L$-subset provided the given $L$-subset possesses sup-property.

Partial frames provide a rich context in which to do pointfree structured and unstructured topology. A small collection of axioms of an elementary nature allows one to do much traditional pointfree topology, both on the level of frames or locales, and that of uniform or metric frames. These axioms are sufficiently general to include as examples bounded distributive lattices, sigma-frames, kappa-frames and frames. Reflective subcategories of uniform and nearness spaces and lately coreflective subcategories of uniform and nearness frames have been a topic of considerable interest. In cite{jfas9} an easily implementable criterion for establishing certain coreflections in nearness frames was presented. Although the primary application in that paper was in the setting of nearness frames, it was observed there that similar techniques apply in many categories; we establish here, in this more general setting of structured partial frames, a technique that unifies these. We make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting our axiomatization of partial frames, which we call sels-frames, we add structure, in the form of sels-covers and nearness, and provide the promised method of constructing certain coreflections. We illustrate the method with the examples of uniform, strong and totally bounded nearness sels -frames. In Part (II) of this paper, we consider regularity, normality and compactness for partial frames.

This paper is a continuation of [Uniformities and covering properties for partial frames (I)], in which we make use of the notion of a partial frame, which is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. After presenting there our axiomatization of partial frames, which we call sels-frames, we added structure, in the form of sels-covers and nearness. Here, in the unstructured setting, we consider regularity, normality and compactness, expressing all these properties in terms of sels-covers. We see that an sels-frame is normal and regular if and only if the collection of all finite sels-covers forms a basis for an sels-uniformity on it. Various results about strong inclusions culminate in the proposition that every compact, regular sels-frame has a unique compatible sels -uniformity.

For two normal edge-transitive Cayley graphs on groups H and K which have no common direct factor and $gcd(|H/H^prime|,|Z(K)|)=1=gcd(|K/K^prime|,|Z(H)|)$, we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.

In this paper, we show that a matrix A in Mn(C) that has n coneigenvectors, where coneigenvalues associated with them are distinct, is condiagonalizable. And also show that if all coneigenvalues of conjugate-normal matrix A be real, then it is symmetric.