On Direct Product of d-Algebras Phattarachaleekul, Maliwan
European journal of pure and applied mathematics,
04/2023, Letnik:
16, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The main aim of this work is to introduce and study the notions of ideal direct product d-algebras, d-ideal direct product d-algebras, sub-direct product d-algebras, edge direct product and positive ...implicative direct product d-algebras and investigate their characterizations.
We have developed a Mathematica program package SpaceGroupIrep which is a database and tool set for irreducible representations (IRs) of space group in BC convention, i.e. the convention used in the ...famous book “The mathematical theory of symmetry in solids” by C.J. Bradley & A.P. Cracknell. Using this package, elements of any space group, little group, Herring little group, or central extension of little co-group can be easily obtained. This package can give not only little-group (LG) IRs for any k-point but also space-group (SG) IRs for any k-stars in intuitive table form, and both single-valued and double-valued IRs are supported. This package can calculate the decomposition of the direct product of SG IRs for any two k-stars. This package can determine the LG IRs of Bloch states in energy bands in BC convention and this works for any input primitive cell thanks to its ability to convert any input cell to a cell in BC convention. This package can also provide the correspondence of k-points and LG IR labels between BCS (Bilbao Crystallographic Server) and BC conventions. In a word, the package SpaceGroupIrep is very useful for both study and research, e.g. for analyzing band topology or determining selection rules.
Program title:SpaceGroupIrep
CPC Library link to program files:https://doi.org/10.17632/3vm4g32t4d.1
Developer's repository link:https://github.com/goodluck1982/SpaceGroupIrep
Licensing provisions: GNU General Public Licence 3.0
Programming language: Mathematica
External routines/libraries used:spglib (http://spglib.github.io/spglib)
Nature of problem: Space groups and their representations are important mathematical language to describe symmetry in crystals. The book—“The mathematical theory of symmetry in solids” by C.J. Bradley & A.P. Cracknell (called the BC book)—is highly influential because it contains not only systematic theory but also detailed complete data of space groups and their representations. The package SpaceGroupIrep digitizes these data in the BC book and provides tens of functions to manipulate them, such as obtaining group elements and calculating their multiplications, identifying k-points, showing the character table of any little group, determining the little-group (LG) irreducible representations (IRs) of energy bands, and calculating the direct product of space-group (SG) IRs. This package is a useful database and tool set for space groups and their representations in BC convention.
Solution method: The direct data in the BC book is used to calculate the LG IRs for standard k-points defined in the book. For a non-standard k-point, we first relate it to a standard k-point by an element which makes the space group self-conjugate and then calculate the LG IRs through the element. SG IRs are obtained by calculating the induced representations of the corresponding LG IRs. The full-group method based on double coset is used to calculate the direct products of SG IRs. In addition, an external package spglib is utilized to help convert any input cell to a cell in BC convention.
Chanmanee et al. examined the idea of the external direct product of the infinite family of UP (BCC)-algebras, and the conclusion is reached for UP (BCC)-algebras. We apply the idea of the internal ...direct product of a groupoid to a UP (BCC)-algebra by introducing two new ideas for internal direct products of UP (BCC)-algebras: the internal and antiinternal direct products. This idea comes from the idea of the external direct products of UP (BCC)-algebras. We examine the attributes of both ideas and identify the crucial attributes for drawing the investigation to a conclusion. Finally, we establish the crucial statement that the internal and anti-internal direct products of a UP (BCC)-algebra may exist in only one form each.
For a finite group G, let ψ(G) denote the sum of element orders of G. This function was introduced by H. Amiri, S. M. Jafarian Amiri, and I. M. Isaacs in 2009 and they proved that for any finite ...group G of order n, ψ(G) is maximum if and only if G≃Zn where Zn denotes the cyclic group of order n. Furthermore, M. Herzog, P. Longobardi, and M. Maj in 2018 proved that if G is non-cyclic, ψ(G)≤711ψ(Zn). S. M. Jafarian Amiri and M. Amiri in 2014 introduced the function ψk(G) which is defined as the sum of the k-th powers of element orders of G and they showed that for every positive integer k, ψk(G) is also maximum if and only if G is cyclic.
In this paper, we have been able to prove that if G is a non-cyclic group of order n, then ψk(G)≤1+3.2k1+2.4k+2kψk(Zn). Setting k=1 in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for ψk(G) is also obtained for finite abelian p-groups G, using which one can explicitly find out the exact value of ψk(G) for finite abelian groups G.
In this paper, we introduced direct product, restricted direct product of interval neutrosophic automata and prove that direct, restricted direct product of cyclic and retrievable of interval ...neutrosophic automata are cyclic and retrievable interval neutrosophic automata. Keywords: Cyclic, Retrievability, Direct product.
We study a general product of two n-dimensional tensors A and B with orders m⩾2 and k⩾1. This product satisfies the associative law, and is a generalization of the usual matrix product. Using this ...product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of it will be given. Using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables, we define the similarity and congruence of tensors (which are also the generalizations of the corresponding relations for matrices), and prove that similar tensors have the same characteristic polynomials, thus the same spectra. We study two special kinds of similarity: permutational similarity and diagonal similarity, and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors. We also define the direct product of tensors (in matrix case it is also called the Kronecker product), and give its applications in the study of the spectra of two kinds of the products of hypergraphs. We also give applications of this general product in the study of nonnegative tensors, including a characterization of primitive tensors, the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors.
Direct-co-direct product G⊛H of graphs G and H is a graph on vertex set V(G)×V(H). Two vertices (g,h) and (g′,h′) are adjacent if gg′∈E(G) and hh′∈E(H) or gg′∉E(G) and hh′∉E(H). We show that ...eccentricity of a vertex of G⊛H for connected non-complete graphs G and H is bounded by five. In addition, we fully describe when the eccentricity is four or five and in all cases one factor must be a star. This is a cornerstone for the distance formula for G⊛H. The disconnected cases of G⊛H are also characterized along the way.
We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also ...consider trace-free versions of these systems. Our results generalize earlier rigidity results for gradient Ricci solitons and warped product Einstein metrics. In particular, our results apply to homogeneous gradient solitons of any invariant curvature flow and give a new structure result for homogeneous conformally Einstein metrics.
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