Combinatorial Algorithms for Binary Operations on LR-tableaux with Entries Equal to 1 with Applications to Nilpotent Linear Operators

The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = $(S,\cdot).$ The complex algebra of S, denoted S$\sp+,$ is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, '*', on the complex algebra as follows: for all subsets A and B contained in S, $A\* B=\{a\cdot b:a\in A, b\in B\}.$ Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator. Let S$\sp+$ be the class of all complex algebras of semigroups and consider the variety generated by the class S$\sp+,$ denoted ${\cal V}$(S$\sp+).$ The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to ${\cal V}$(S$\sp+).$ A closure operator is defined which is used to determine homomorphic images of members of ${\cal V}$(S$\sp+).$ Theorems on the subdirectly irreducible and simples algebras in ${\cal V}$(S$\sp+)$ are proved. Next, the structure of ${\cal V}$(S$\sp+)$ is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of ${\cal V}$(S$\sp+)$ is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S$\sp+$ is strictly contained in ${\cal V}$(S$\sp+).$ Lastly, the structure of the lattice of subvarieties of ${\cal V}$(S$\sp+)$ is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed.

This chapter is devoted to algebraic systems of one binary operation. Groups have been used many times before in the text, as permutations, as modules (a commutative group with an exterior ring multiplication), and elsewhere, but here we organize the systems of one binary operation into one chapter and give some additional specialized results. The most general algebraic system we consider in this chapter is the monoid, a set with one associative binary operation. Some special monoids discussed are unitary monoids, cancellative monoids, and groups. We discuss subsystems, quotient systems, morphisms, and the fundamental morphism theorem. In Sections 9.4 and 9.5 we study more specialized results available for groups alone. We study cyclic groups and connect the order of an element with this concept. Several topics such as center, normalizer, conjugacy classes are all organized around inner automorphisms (φ a (x) = a-xa). We then apply these results to answer some questions about elements of prime order. We offer some standard theorems relating direct products and Cartesian products of groups. We define simple groups and solvable groups. We give an inductive proof of the fundamental theory of Abelian groups which is subsumed in Chapter 10 by the direct sum resolution of a finitely generated module.KeywordsCommutative GroupNormal SubgroupConjugacy ClassSimple GroupCyclic GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Binary matrixes (matrixes having only 0 and 1 elements) occur in many theoretical and practical areas (e.g. in the theory of graphs and relations, grammars, coding, etc.) therefore the binary matrix operations have high importance in these topics. It is an obvious idea to use cellular processor consisting of micro-cells for the execution of these operations. The present paper gives a powerful cellular algorithm for the multiplication of two binary matrixes, taking into account the cellprocessor architecture developed by T. Legendi. Moreover, on the basis of this solution cellular algorithms are given for coding and for the computation of the m-th power of a binary matrix.

Algorithms for the high speed binary arithmetic operations of addition and multiplication in a VLSI environment are analyzed for area-time efficiency. It is shown that some schemes for addition and multiplication, although good for stand-alone designs, fail to provide both area and time efficiencies simultaneously. Solutions that yield area-time efficient practical implementations of these arithmetic functions are described.

Effective plant discrimination based on the combination of local binary pattern operators and multiclass support vector machine methods

Over the (1,n)-dimensional real superspace, n>1, we classify K(n)-invariant binary differential operators acting on the superspaces of weighted densities, where K(n) is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of K(n) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities—a superization of a result by Feigin and Fuchs [“Homology of the Lie algebras of vector fields on the line,” Funct. Anal. Appl. 14, 201 (1980)]. We explicitly give 1-cocycles spanning these cohomology spaces.

Assume that a black/white n×n image is stored one pixel per element of a vector of length n2. We consider determining some characteristics of such images using the scan model of parallel computation. The scan model was introduced by G.E. Blelloch and is a single instruction multiple data (SIMD) vector model of computation. The primitive operations of the model work on vectors (one dimensional arrays) of values, with three types of primitive operations: Elementwise arithmetic and logical operations, permutation operations, and scan operations, a type of prefix computation (a scan operation takes a binary associative operator ⊗ and a vector [a1,…, an] and returns the vector [a1, a1⊗a2,…, a1⊗a2⊗…⊗an]). We show that many important characteristics of binary images can be determined in constant time on the scan model of parallel computation. These include the convex hull construction, diameter, width, smallest enclosing box, perimeter, area, detecting digital convexity, parallel and point visibility (determining for each pixel of the image the portion that is visible, i.e., not obstructed by any black pixel, in a given direction from infinity or from a given point, respectively) of an image, smallest, largest and Hausdorff distances between two images, linear separability of two images, and the recognition of digital lines, rectangles and arcs.

It is well-known that one of open problems in the theory of Leibniz algebras is to find asuitable generalization of Lie’s third theorem which associates a (local) Lie group to any Liealgebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Liegroup for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution ofthis problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibnizalgebras. A digroup is a nonempty set equipped with two binary associative operations, aunary operation and a nullary operation satisfying additional axioms relating these operations.Digroups generalize groups and have close relationships with the dimonoids and dialgebras,the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied themethod of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitraryrank, in particular, they considered a monogenic case separately. In this paper, we give a simplerand more convenient digroup model of the free monogenic digroup. We construct a new classof digroups which are based on commutative groups and show how the free monogenic groupcan be obtained from the free monogenic digroup by a suitable factorization.

We consider a variety of algebras with two binary commutative and associative operations. For each integer n ≥ 0, we represent the partitions on an n-element set as n-ary terms in the variety. We determine necessary and sufficient conditions on the variety ensuring that, for each n, these representing terms be all the essentially n-ary terms and moreover that distinct partitions yield distinct terms.

The generalized spectral decomposition expressing a linear operator as a sum of idempotent and nilpotent operators is derived from the partial fraction decomposition of the reciprocal of its minimal polynomial, and makes possible a functional calculus of analytic functions of the operator. The classical spectral and polar decompositions for matrices are shown to be easy consequences. The idempotents and associated nilpotents provide a basis for the commutative algebra generated by an operator that is far more convenient for computations than the usual basis of powers of the operator.

We prove that the monoid of generic extensions of finite dimensional nilpotent $k[T]$-modules is isomorphic to the monoid of partitions (with addition of partitions). Moreover we give a combinatorial algorithm that calculates constant terms of classical Hall polynomials.

Over the (1, N)-dimensional real superspace, N = 2, 3, we classify \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2)-invariant binary differential operators acting on the superspaces of weighted densities, where \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2) is the orthosymplectic Lie superalgebra. This result allows us to compute the first differential \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(N|2)$\end{document}osp(N|2)-relative cohomology of the Lie superalgebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(N)$\end{document}K(N) of contact vector fields with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We classify generic formal \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(3|2)$\end{document}osp(3|2)-trivial deformations of the \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(3)$\end{document}K(3)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(3|2)$\end{document}osp(3|2)-trivial deformation of this \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}(3)$\end{document}K(3)-module is equivalent to its infinitesimal part. This work is the simplest generalization of a result by the first author et al. [Basdouri, I., Ben Ammar, M., Ben Fraj, N., Boujelbene, M., and Kammoun, K., “Cohomology of the Lie superalgebra of contact vector fields on \documentclass[12pt]{minimal}\begin{document}$\mathbb {K}^{1|1}$\end{document}K1|1 and deformations of the superspace of symbols,” J. Nonlinear Math. Phys. 16, 373 (2009)10.1142/S1402925109000431].

Property [formula omitted] and perturbations

The aim of this work is to solve the problem of determining the necessary and sufficient conditions for a vector subspace invariant by a nilpotent endomorphism to admit a complementary invariant subspace for the same linear operator. As applications, we offer results about Jordan bases associated with nilpotent linear maps and reflexive generalized inverses of finite potent endomorphisms and square matrices.

ABSTRACTFor a partition β, denote by Nβ the nilpotent linear operator of Jordan type β. Given partitions β, γ, we investigate the representation space of all short exact sequenceswhere α is any partition with each part at most 2. Due to the condition on α, the isomorphism type of a sequence ℰ is given by an arc diagram Δ; denote by 𝕍Δ the subset of of all sequences isomorphic to ℰ. Thus, the variety carries a stratification given by the subsets of type 𝕍Δ. We compute the dimension of each stratum and show that the boundary of a stratum 𝕍Δ consists exactly of those where Δ′ is obtained from Δ by a non-empty sequence of arc moves of five possible types (A)–(E). The case where all three partitions are fixed has been studied in [3, 5]. There, arc moves of types (A)–(D)suffice to describe the boundary of a 𝕍Δ in . Our fifth move (E), “explosion,” is needed to break up an arc into two poles to allow for changes in the partition α.