Vector Algebras Research Articles

Clifford Algebras, Hypercomplex Numbers and Nonlinear Equations in Physics

Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $\A^2=n\A$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.

A MECHANISM FOR USING THE FLUSHING PROCESS MIGRATION MECHANISM IN DISTRIBUTED EXASCALE SYSTEMS

The effect of dynamic and interactive events on the function of the elements that make up the computing system manager causes the time required to run the user program to increase or the operation of these elements to change. These changes either increase the execution time of the scientific program or make the system incapable of executing the program. Computational processes on the migration process and vector algebras try to analyze and enable the Flushing process migration mechanism in support of distributed Exascale systems despite dynamic and interactive events. This paper investigates the Flushing process migration management mechanism in distributed Exascale systems, the effects of dynamic and interactive occurrences in the computational system, and the impact of dynamic and interactive events on the system.

ExaLazy: A Model for Lazy-Copy Migration Mechanism to Support Distributed Exascale System

There is a possibility of dynamic and interactive nature occurring at any moment of the scientific program implementation process in the computing system. While affecting the computational processes in the system, dynamic and interactive occurrence also affects the function of the elements that make up the management element of the computing system. The effect of dynamic and interactive events on the function of the elements that make up the management element of the computing system causes the time required to run the user program to increase or the function of these elements to change. These changes either increase the execution time of the scientific program or make the system incapable of executing the program. The occurrence of dynamic and interactive nature creates new situations in the computing system that the mechanisms to deal with when designing the computing system are not defined and considered. In this paper, the Lazy-Copy process migration management mechanism, specifically the Lazy-Copy mechanism in distributed large-scale systems, the effects of dynamic and interactive occurrence in the computational system investigate, and the effects of dynamic and interactive occurrence on the system investigate. Computational processes on the migration process and vector algebras try to analyze and enable the Lazy-Copy process migration mechanism in support of distributed largescale systems despite dynamic and interactive events

Arity Shape of Polyadic Algebraic Structures

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to restrictions which determine their possible arity shapes and lead us to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators. Another application is connected with number theory, and it is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 6.16), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's last theorem are formulated. For nonderived polyadic ring operations (on polyadic numbers) neither of these statements holds, and counterexamples are given. Finally, a procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem to the Tarry-Escott problem is presented.

Algebraic structures in the sets of surjective functions

In the paper we construct several algebraic structures (vector spaces, algebras and free algebras) inside sets of different types of surjective functions. Among many results we prove that: the set of everywhere but not strongly everywhere surjective complex functions is strongly c-algebrable and that its 2c-algebrability is consistent with ZFC; under CH the set of everywhere surjective complex functions which are Sierpiński–Zygmund in the sense of continuous but not Borel functions is strongly c-algebrable; the set of Jones complex functions is strongly 2c-algebrable.

Boolean Witt vectors and an integral Edrei–Thoma theorem

A subtraction-free definition of the big Witt vector construction was recently given by the first author. This allows one to define the big Witt vectors of any semiring. Here we give an explicit combinatorial description of the big Witt vectors of the Boolean semiring. We do the same for two variants of the big Witt vector construction: the Schur Witt vectors and the $p$-typical Witt vectors. We use this to give an explicit description of the Schur Witt vectors of the natural numbers, which can be viewed as the classification of totally positive power series with integral coefficients, first obtained by Davydov. We also determine the cardinalities of some Witt vector algebras with entries in various concrete arithmetic semirings.

Design of algorithms of robot vision using conformal geometric algebra

In this paper the authors will apply a mathematical system, the Conformal Geometric Algebra (CGA), to propose applications in computer vision and robotics. The CGA keeps our intuition and insight of the problem’s geometry at hand, besides helping us to reduce the computational problems’ burden. Matrix and vector algebras, complex numbers, rigid and conformal transformations, Euclidean and projective spaces, to name some, are different mathematical tools to model and solve almost any problem in robotic vision. Now, CGA is a mathematical system where all those systems are embedded. We use this compact system following the Occam’s razor philosophy that mathematical ‘entities should not be multiplied unnecessarily’. In this work we handle simulated and real tasks for perception-action systems, treated in a single and efficient way. The authors show that this framework can be of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems.

Differential calculuses on the quantized braided groups

Braided differential calculuses on the quantized braided groups are constructed and their braided bialgebra (Hopf algebra) structures are demonstrated. These are a kind of generalization and unification of the differential calculuses on quantum groups, braided groups, quantum supergroups, etc., and contain the latter ones as special cases. Moreover, it is shown that some quantum differential (co)vector algebras are covariant under the braided “local” coactions of the obtained braided differential bialgebras (Hopf algebras). Some examples are also given.

On vector spaces and algebras with maximal locally pseudoconvex topologies

On vector spaces and algebras with maximal locally pseudoconvex topologies

A nonisomorphism theorem for cofree Lie coalgebras

In the first paragraph of this section we will give a brief overview of this work. The remainder of this section contains formal definitions and details omitted from the first paragraph. Throughout this work, all vector spaces, linear maps, tensor products, coalgebras and algebras, etc., will be taken over a base field k. Let V be a vector space. This paper is concerned with the relation between the “cofree Lie coalgebra on I” and the “cofree covered Lie coalgebra on V.” A “Lie coalgebra” will be defined as a coalgebra that satisfies the “co-anticommutativity” and “co-Jacobi” Lie coidentities. This is analogous to the definition of a Lie algebra as an algebra that satisfies the anticommutativity and Jacobi identities. It follows from the Poincare-Birkhoff-Witt Theorem that every Lie algebra arises as a subalgebra of an A -, where A ~ is the Lie algebra associated to an associative algebra A. A “covered Lie coalgebra” will be defined as the quotient of some C-, where Cis the Lie coalgebra associated to an associative ( = coassociative) coalgebra C. Every covered Lie coalgebra satisfies the Lie coidentities, but, as was shown by Michaelis [Mill, there exist Lie coalgebras that do not arise as the quotient of any C-. Thus, covered Lie coalgebras are a specific type of Lie coalgebra. This situation for Lie coalgebras and covered Lie coalgebras is somewhat parallel to the situation for Jordan algebras and special Jordan algebras. (We recall that a Jordan algebra is an algebra that satisfies the Jordan identities and a special Jordan algebra is an algebra which arises as a subalgebra of an A +, A an associative algebra. Every special Jordan algebra is a Jordan algebra, that is, satisfies the identities, but there exist Jordan algebras which are not special.) For every vector space V one can construct the “cofree Lie coalgebra” and the “cofree covered Lie coalgebra” on V. Our main result (Theorem 1) will show that the canonical injective map from the cofree 431 0021-8693/88 $3.00

Computer‐aided vector algebra

A computer program in PASCAL has been developed in order to use a video terminal as a desk calculator when operating complex numbers (multiplication, division), vectors (dot product, cross product), quaternions and polynomials in higher dimensional vector and spinor algebras. A copy of this program on magnetic tape or floppy disk is available on request from the authors.

Contractions of group representations. — II

As a development of a preceding paper we obtain the representations of projective Lie algebras by contraction of representations of vector algebras of the same dimension. As examples, we deduce the projective representations of the Galilei unidimensional group and ofIO 2. In addition, we discuss the possibility that the use of the contraction allows us to classify completely the Lie groups and their representations.

The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

Locally Compact Vector Spaces and Algebras over Discrete Fields

Locally Compact Vector Spaces and Algebras over Discrete Fields

Locally compact vector spaces and algebras over discrete fields

The structure of locally compact vector spaces over complete division rings topologized by a proper absolute value (or, more generally, over complete division rings of type V) is summarized in the now classical theorem that such spaces are necessarily finite-dimensional and are topologically isomorphic to the cartesian product of a finite number of copies of the scalar division ring [9, Theorems 5 and 7], [6, pp. 27-31]. Here we shall continue a study of locally compact vector spaces and algebras over infinite discrete fields that was initiated in [16]. Since any topological vector space over a topological division ring K remains a topological vector space if K is retopologized with the discrete topology, some restriction needs to be imposed in our investigation, and the most natural restriction is straightness: A topological vector space E over a topological division ring K is straight if for every nonzero a E E, fa: A -Aa is a homeomorphism from K onto the one-dimensional subspace generated by a. Thus if K is discrete, a straight K-vector space is one all of whose one-dimensional subspaces are discrete. Any Hausdorff vector space over a division ring topologized by a proper absolute value is straight [6, Proposition 2, p. 25]. In ?1 we shall prove that if E is an indiscrete straight locally compact vector space over an infinite discrete division ring K, then K is an absolutely algebraic field of prime characteristic and [E : K]> Ro. In ?2 we shall see that finitedimensional subspaces of straight locally compact vector spaces need not be discrete. In ?3 we shall see that the validity of the Open Mapping and Closed Graph theorems for straight locally compact vector spaces depends on whether those spaces are generated by compact subsets. In ?4 we shall study locally compact algebras of linear operators. In ?5 we shall obtain structure theorems for a commutative semisimple straight locally compact algebra over an infinite discrete field; such an algebra is the topological direct product of a discrete algebra and a local subdirect product of a sequence of algebras, each discrete and algebraically the cartesian product of a family of fields, relative to finite subfields.

Vector Algebras and Potentials

Vector Algebras and Potentials

Vectors, &c., at the British Association

IN the report (August 30) of the discussion on the use and notation of vector analysis at the British Association is stated that I “deplored the substitution of vectors for quaternions.” The statement is misleading, for was it not Hamilton more than any other single man who taught us how to use vectors in product and quotient combinations? What I did and do deplore is the substitution of non-quaternionic vector algebras in all their variety of notation for the Hamiltonian or quaternionic vector algebra—a very different thing.