Zbigniew Oziewicz

• PhD in Mathematical Physics in 1970; In 1985 Habil
• Professor at National Autonomous University of Mexico

The aim of this paper is to present a symbolic calculation approach for the basic facts for the Lie Z_2 -graded (super) algebra of derivations of the Grassmann algebra of differential forms, the Z-graded Gerstenhaber algebras, the Frölicher-Nijenhuis algebra of vector-valued differential multi-forms, and the Schouten-Nijenhuis algebra of multi-vect...

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra Y is said to be a Frobenius algebra if there exists a Y-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra Y is said to be a solvable Frobenius algebra if it possesses a nonzero one...

Many Authors wrote about Universe however never define this concept. Most astronomer implicitly identify Universe with three-dimensional position space filled with matter. This lecture is devoted to the concept of Universe that obligatorily needs four-dimensions that are able to include many distinct position-spaces that all are relative and depend...

There is a challenge to relativity theory identified with the Lorentz isometry group. The alternative and not equivalent theory of relativity postulate the groupoid category that it is not a group. In this talk I tried to explain the difference between the mathematical concepts of a group category and of Brandt groupoid category that it is not a gr...

The definition of the Lorentz group do not need the superfluous condition of the vanishing curvature. The Lorentz transformations hold well for arbitrary curvature tensor and are well defined for arbitrary metric tensor (of course Lorentz transformations are metric-dependent). The main objective of this lecture is the group-free relativity theory,...

The central concept of relativity theory is relative velocity. The velocity of material body is not an intrinsic property of a body; it depends on a free choice of a reference body, it is reference-dependent. Lorentz boost is generated by bivector and not by velocity vector. For initial and final vectors the generating bivector is non-unique. The v...

Within planetary radar astronomy a radio signal is transmitted to a spacecraft, to a planet, and similarly police radar transmits a radio wave to a car, and then radio signal is re-transmitted (or reflected) back and received at the ground (similarly reflected back to the police radar-gun), and, it is assured in the textbooks and in scientific jour...

Stokes theorem is artifact of history. De Rham co-chain complex born with Ph. D. Thesis by De Rham in 1931, not known during XIX century to Cauchy and to Riemann when their defined an integral within the theory of limits for each integer dimension of a manifold: a line integral over curve, double integral, triple, etc, each one as a separate Chapte...

The central concept of the (special) relativity theory is relative velocity. The velocity of a material body is not an intrinsic property of a body; it depends on the free choice of a reference body, it is reference-dependent. Besides of the obvious reference-dependence, the relative velocity is non-unique if and only if it is defined by means of t...

An algebra Y possessing a Y -associative and invertible scalar pro- duct is said to be Frobenius algebra. The admission of Y -scalar-product allows characterizing Frobenius algebra in terms of a coalgebra as a module-map [Lawvere 1967] and invites a question about whether or not Frobenius algebra is a bialgebra i.e. coalgebra-map. In 1894 Elie Cart...

In 1632, Galilei was aware of relativity of velocity and that this implies relativity of spaces- of-locations. During centuries the relativity of spaces-of-locations was ignored. Professor Harald Keres considered the space-of-locations as a congruence of world-lines, and there is no universal absolute three dimensional space-of-locations. In applic...

I define the three-dimensional position space in terms of the Grassmann complex (named by Bourbki as the Koszul complex) in group-free way (against group-theoretical approach). This is the consequence of the following ideas/postulates (i) The Hermann Minkowski idea in 1908 that the material body is identified with time-like vector field (ii) The Co...

Under the hypothesis that the relative velocity is non-reciprocal it is proposed the time-like center-of-inertia with total mass, and space-like reduced momentum with reduced mass, of a many-body interact-ing system. These expressions are not valid within the Lorentz group where relative velocity must be reciprocal. There are a priory two antitheti...

The central concept of the theory of relativity is the relativity of velocity. The velocity of a material body is not an intrinsic property of the body; it depends on a free choice of reference system. Relative velocity is thus reference-dependent, it is not an absolute concept. We stress that even zero-velocity must be relative. Every reference sy...

Science is a collection of alternative ideas, and as such it must not be based on criteria-verification in which there is only one winner. Alternative subjective views and subjective personal ideas must not be considered as a temporary and preparatory for final showdown that will eliminate 'in-correct' opinions (by omnipotent experimental refutatio...

An algebra Y possessing a Y -associative and invertible scalar product U is said to be U-Frobenius algebra {Y, U}. Then U is called Y -scalar-product. The admission of Y -scalar-product allows characterizing Frobenius algebra in terms of a coalgebra as a module-map [Lawvere 1967] and invites a question about whether or not Frobenius algebra is a bi...

Categorias contra conjuncts. Multigrafica. Estructuras algebraicas como ejemplos de 2-graficas. Grafica de gráficas. Multigrafica de multigraficas. Multigrafica reflexiva. Aillo de gráficas.Teoria de categorías. Ejemplos de categorías. Ejemplos de funtoros.

Electric and magnetic fields are relative. They depend not only on a choice of electromagnetic sources via Maxwell equations, but also on a choice of observer, a choice of material reference-system. In 1908 Minkowski defined electric and magnetic fields on a four-dimensional spacetime, as tensorial concomitants of observer. Minkowski defined Lorent...

The central concept of the theory of relativity is the relativity of velocity. The velocity of a material body is not an intrinsic property of the body; it depends on a free choice of reference system. Relative velocity is thus reference-dependent, it is not an absolute concept. We stress that even zero-velocity must be relative. Every reference sy...

We are proving that the Lorentz boost entails the relative velocity to be ternary: ternary relative velocity is a velocity of a body with respect to an interior observer as seen by a preferred exterior-observer. The Lorentz boosts imply non-associative addition of ternary relative velocities. Within Einstein's special relativity theory, each prefer...

Usually a name of the category is inherited from the name of objects. However more relevant for a category of objects and morphisms is an algebra of morphisms. Therefore we prefer to say a category of graphs if every morphism is a graph. In a monoidal category every morphism can be seen as a graph, and a partial algebra of morphisms possesses a str...

In the present paper by Frobenius algebra Y we mean a finite dimensional algebra possessing an associative and invertible (nondegenerate) form a scalar product, referred to as the Frobenius structure. The nondegenerate form has an inverse. We drop the extra conditions of associativity and unitality of Y. Frobenius algebra is formulated within the m...

Jaime Keller passed away on January 7, 2011. He was the founder of Advances in Applied Clifford Algebras and its Editor-in-Chief since 1991. We recall the pre-history of AACA.

A unipotent isometry is said to be a reflection. In 1937 Élie Cartan proved that every isometry can be expressed as a composition of reflections. The Lie subalgebra of bivectors inside a Clifford algebra generate isometries without the Grassmann exponential. The main result is the coordinate-free and basis-free construction of two isometries from a...

An n-category is a n-graph (a multigraph) with an algebraic structure. We introduce a graph of graphs and, for a k-tuple of natural numbers n(i) ∈ N, an (n(1),...,n(k))-graph: a (n(1),...,n(i))-graph of (n(i+1),...,n(k))-graphs for 1 ≤ i < k. A multigraph of multigraphs include, among others, the concepts of a prefunctor and a prenatural transforma...

The bi-closed monoidal category (the compact-closed in an an-other terminology), with two-sided evaluations and co-evaluations, in-troduced by Kelly and Laplaza in 1980, offer to define the trace of the morphism. This allows to generalize the Frobenius and the Hilbert-Schmidt scalar product of operators. If a category possess two contravariant comm...

Hermann Minkowski in 1908 defined electric and magnetic fields as the tensorial concomitants of time-like body, ie by means of evaluation. Thus this primordial DEFINITION assume that the Electric and Magnetic fields are relative, they depends explicitly on the free choice of the reference material body. Minkowski in 1908 defined GL-covariance, as w...

In this article the force in Physics is described with two different behaviors: like vector and covector. The forcebehaves like vector (with direction), in the Second Law of Newton, when this one is proportionate with theacceleration. On the other hand, the force behaves like covector (without direction), if this one is related the scalar ofwork. T...

In 1908, Minkowski [13] used space-like binary velocity-field of a medium, relative to an observer. In 1974, Hestenes introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector [7, 8]. We propose to consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept o...

A nonunitary Lorazo transformation of the space of the one-particle states is generalized to isospin-dependent transformations. The generators of the ten-dimensional quasispin algebra for protons and neutrons depend on two sets of the dynamical parameters. This algebra is spectrum generating for the generalized pairing and quadrupling Hamiltonian....

The isometry-link problem is to determine all isometry transformations among given pair of vectors with the condition that if these initial and final vectors coincide, the transformation-link must be identity on entire vector space. In the first part of this essay we provide the complete solution for the link problem for arbitrary isometry, for any...

Minkowski in 1908 used space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector. We propose consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept of a binary axioma...

Multiscale resolution database MRDB include not only several levels of details (ie preciseness), but also several layers of peculiarity. The data preciseness means the variable levels of details for the fixed set of attributes. Less details leads to data aggregation, more details leads to data de-aggregation. The data peculiarity means the growing...

Without Abstract

The categorical relativity is a groupoid category of time-like observers with the binary relative velocity as the invertible morphisms - instead of the Lorentz isometry. Within such relativity the inverse relative velocity is non-reciprocity. Observer-independence and the Lorentz invariance are different concepts. Within the categorical relativity...

Abstract Following Minkowski [1908], we consider the relative velocity to be the Minkowski space-like vector (and not to be the Minkowski bivec- tor as it is in the Hestenes theory [Hestenes 1974]). The Lorentz boost entails the relative velocity (as a space-like Minkowski vector) to be ternary: ternary relative velocity is a velocity of a body wit...

The principal aim of this work is the presentation of a symbolic calculation computer analysis for exploring electromagnetic fields for not inertial observer. Based on Frölicher-Nijenhuis super-Lie \(\mathbb R \)-algebra, we developed a learning environment for axiomatic classical electromagnetics and electrodynamics. A collection of programs devel...

We consider the categorical concepts of a ‘network of networks’: (a) each node is a host network (1-network or 1-graph) and super-links are analogous to a graph-functor, i.e. this is (1,1)-network; (b) 2-network where there are 2-links among 1-links. The general notion of network-morphism is proposed.

The aim of this paper is to present a symbolic calculation approach for the basic facts for the Lie Z_2 -graded (super) algebra of derivations of the Grassmann algebra of differential forms, the Z-graded Gerstenhaber algebras, the Frölicher-Nijenhuis algebra of vector-valued differential multi-forms, and the Schouten-Nijenhuis algebra of multi-vect...

Graph based models of hierarchical systems are usually seen as ‘graph equipped with some refinements’, understood as the (homo)-morphisms or (bi)simulations. In such a model it is not possible to consider phenomena happened on different levels of the system. We propose a new formalism of directed multi-graph allowing to see a hierarchical system si...

Frölicher-Nijenhuis and Schoten-Nijenhuis graded Lie modules, are applied for derivation explicitly observer-dependent four Maxwell equations. Observer field is almost product structure (1, 1)-tensor field, that is idempotent operator giving a (1+3)-split. In present paper an observer field is not restricted to be neither inertial nor holonomic, an...

We consider examples of the finitely presented oper-ads of graphs in a monoidal category of one sorted sketch. Models of these examples are related to not necessarily associative or not necessarily coassociative structures: quasi Graßmann and quasi Clifford convolutions (introduced in the body of the paper), and quasi Hopf algebra with a unique Hop...

The concept of n-categories and related subject is considered. An n-category is described as an n-graph with a composition. A new definition of operad is presented. Some illustrative examples are given.

We propose the categorification of the algebraic analysis as the Leibniz 3-category given by generators and relations, including the Leibniz 3-cell relation. The Leibniz category offers the `most general' `(co-)derivation' 3-cell. We outline a program in which every 2-cell related to (or a type of) a Leibniz partial (co-)derivation 3-cell, is trans...

The history and immediate future of the International Conferences on Clifford Algebras and Their Applications. Seven topical sessions in Ixtapa. Dirac operator: cross relations. Polemic guide: signature change, quasigroups, pseudotwistors. Clifford cogebra, coconnection and Dirac operator for Clifford cogebra.

A Clifford algebra Cl(V,\eta\in V^*\otimes V^*) jointly with a Clifford cogebra Cl(V,\xi\in V\otimes V) is said to be a Clifford biconvolution Cl(\eta,\xi). We show that a Clifford biconvolution for dim_R Cl=4 does possess an antipode iff det(id-\xi\circ\eta)\neq 0. An antipodal Clifford biconvolution is said to be a Clifford Hopf gebra. We study t...

Participants of this workshop pursue the old Neutrino Theory of Light vigorously. Other physicists have long ago abandoned it, because it lacks gauge invariance. In the recent Quantum Induction (QI), all basic Bose fields ${\mathcal B}^{P}$ are local limits of quantum fields composed of Dirac's $\Psi$ (for leptons and quarks). The induced field equ...

We deal with quantum field theory in the restriction to external Bose fields. Let $(i\gamma^\mu\partial_\mu - \mathcal{B})\psi=0$ be the Dirac equation. We prove that a non-quantized Bose field $\mathcal{B}$ is a functional of the Dirac field $\psi$, whenever this $\psi$ is strictly canonical. Performing the trivial verification for the $\mathcal{B...

This paper shows how to get a last multiplier for a differential n-form equivalent to an ordinary differential equation (ODE) in (n+1) -dimensions. The last multiplier makes it possible to find a closed differential n-form given by the ODE. Our construction is based on a set of n-symmetry vector fields admitted by an ODE. Our result is a generaliza...

A gauge theory without divergencies (hence more specific than the ``effective" Standard Model) has beeen proposed recently. There, all local Bose fields are limits of bilocal products of interacting Dirac fields. For states with gauge bosons, this gives explicit formulas in which the only quantum fields are those of leptons and quarks. The result f...

We discuss the variational principle within Quantum Mechanics in terms of the noncommutative even Space Time sub-Algebra, the Clifford $\Ra$-algebra $Cl_{1,3}^+$. A fundamental ingredient, in our multivectorial algebraic formulation, is the adoption of a $\D $-complex geometry, $\D \equiv span_{\RR} \{1,\gamma_{21} \}$, $\gamma_{21} \in Cl_{1,3}^+$...

Let W be a C-space and # # End(W# 2 ) be a braid operator. Woronowicz in 1989 introduced an exterior algebra # ## W # (#) for an arbitrary braid. Let F # lin(W# 2 , C ). This paper offers a sufficient condition on (#, F ) (which happens to be beyond braided monoidal categories) that the Woronowicz braided exterior algebra can be deformed in analogy...

For the system of the two arbitrary spins on the K-shell the author derives the multipole expansions of the spin-density operator in terms of the total spin operator as well as in terms of the individual spin operators. The density operator is described by means of the truly independent set of parameters: the populations, degrees of orientation and...

This is a short version of the invited lecture presented at the Vth International Conference on Clifford Algebras and Applications, Ixtapa, Mexico, June 26-July 4, 1999. Our aim is to present some of the main notion from universal algebra via tree operads. We propose an approach based on E. Marczewski ideas from 60ties [18], connected with an abstr...

A notion of an exterior differential system (an ideal) is extented to modul valued exterior differential system. A new notion of a descendant (modul valued) differential form for a given modul valued exterior differential system is introduced. A subalgebra on which descendant differential form is closed appears to be a generalization of a notion of...

We consider a pair of independent scalar products, one scalar product on vectors, and another independent scalar product on dual space of co-vectors. The Clifford co-product of multivectors is calculated from the dual Clifford algebra. With respect to this co-product unit is not group-like and vectors are not primitive. The Clifford product and the...

This paper is an introduction to the realization of a universal Clifford algebra and of an opposite Clifford algebra as a Chevalley deformation of an exterior algebra. We discuss the realizations of a universal tensor algebra, a realization of a universal factor algebra as a deformation versus a realization by quantization, and a generalization of...

The purpose of this note is to show how calculi on unital associative algebra with universal right bimodule generalize previously studied constructions by Pusz and Woronowicz [1989] and by Wess and Zumino [1990] and that in this language results are in a natural context, are easier to describe and handle. As a by--product we obtained intrinsic, coo...

The characteristics of the partial nuclear muon capture with massive left-handed Dirac neutrino and relativistic component of the muon wave function have been derived. The multipole amplitudes are given as a function of neutrino mass parameter and reduced nuclear matrix elements which are modified by the small component of the muon wave function. A...

Woronowicz in 1989 introduced a graded exterior algebra for arbitrary braid operator. In the present paper we give necessary and su�cient conditions between scalar product and braid operator that exists a Cli�ord algebra and that exists the Chevalley deformation of Woronowicz's exterior algebra. In particular a quantum Cli�ord and Weyl algebras for...

General braided counterparts of classical Clifford algebras are introduced and investigated. Braided Clifford algebras are defined as Chevalley-Kahler deformations of the corresponding braided exterior algebras. Analogs of the spinor representations are studied, generalizing classical Cartan's approach. It is shown that, under certain assumptions c...