"HyperComplex Numbers in Geometry and Physics" 2 (2), vol. 1, 2004
j002

Content of Issue is in the theme. The journal in one file is below.

Conference “Number, time, relativity”
2004jbz | Gladyshev V.O., Pavlov D.G.

On the 13th august of 2004 the International scientific conference “Number, time, relativity” took place in N. E. Bauman Moscow State Technical University. The purposes of the conference were: to attract the attention of foreign and Russian physicists to Finslerian generalizations of the relativistic theory, to gather the leading specialists in the field of hyper-complex numbers, Finslerian geometry (that generalize the Riemannian manifolds), and the specialists in the field of the relativistic theory. The conference was devoted to 175th anniversary of N. E. Bauman Moscow State Technical University. The conference was performed by: the Bauman University’s cathedra of physics, the theoretical physics cathedra of Moscow State University of M.V. Lomonosov and the United Physics Society of Russian Federation. The main sponsor of the conference was the Fund of 175th anniversary of N. E. Bauman Moscow State Technical University.

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The normal conjugation on the poly-number set.
2004jby | Garasko G.I., Pavlov D.G.

The poly-number set is an example of linear space with several poly-linear forms. The concept of normal conjunction is introduced on the set of non-degenerated n-numbers. The normal conjunction is a (n-1)-nary operation. It is commutative for each argument, but generally not associative. For complex and hyperbolic numbers the generalized conjunction is equivalent to usual one. The normal conjunction may be applied for scrutiny of algebraic and geometric structures of n-number coordinate spaces. It is also useful for introducing such concepts like scalar product and angular characteristics of two and more numbers (vectors)

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Generalized-analytical functions and the congruence of geodetic.
2004jbx | Garas'ko G. I.

Some properties of generalized-analytical functions of poly-number variable are being studied in this job. We can confront many spaces of affine connectedness with the $\{f^i;\Gamma^{i}_{kj}\}$ class of such functions. In each space the congruence of geodetic associated with the given class of general-analytic functions is defined. If the vector field $f^i$ is tangent to one of the geodetic of congruence in each point of space there are certain restrictions for the generalized-analytical function itself.

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Concerning the norm of biquaternions and some other algebras with central conjunction
2004jbw | Eliovich A. A.

The concept of central conjunction is introduced in this article. We apply it to algebras of biquaternions and bioctaves. With the given analysis method of the conjunctions permitted by algebra we derive some new results. Thus the alternative algebras with central conjunction are proven to have the multiplicative norm of second degree (that is in general not real). The consequence of this fact is that these algebras (biquaternions and bioctaves particulary) have the multiplicative real norm of degree higher than 2. This norm has several different but equivalent views. The quadrascalar and quadravector multiplications are introduced. Some results for algebras of biquternions, diquaternions and bioctaves are given in terms of isotropic basis. The developed methods may be useful in the geometrical and physical usage of concerned algebras.

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On some distributive algebras
2004jbv | Solovey L.G.

The examined type of sets, that are not rings, but in some sense are close to them. These sets are called 'Hyper-rings'. They consist of several additive groups, that intersect each over at the zero only. Yet, they are multiplicative groupoids (or groups, excluding the zero). The distributive laws are fulfilled.
Rings (and in particular the bodies and the fields) are the special case of the concerned sets. The given examples witness that such sets are highly disseminated. So, the idea that the real physical values may be "layed" in the ring is wrong, because they are subset of hyper-ring.
The real hyper-rings with unity can not be reduced to rings. Their additive groups are vector spaces, and they may be treated as a generalized Hyper-complex systems, in which we include the real binary (provided with summation and multiplication) distributive algebraic structures with neutral element, where the number of included vector spaces is more than one and finite.
The example of hyper-rings, suggesting that scrutiny of them is worth-able, are the second order matrices, that are mostly like unitary matrixes, but normalized not by unity. They are normalized by an arbitrary non-negative number. The complex numbers and the quaternions may be represented with such matrixes while they are the ones subspace.

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Deformation principle as foundation of physical geometry and its application to space-time geometry
2004jbu | Rylov Yuri A.

Physical geometry studies mutual disposition of geometrical objects and points in space, or space-time, which is described by the distance function $% d$, or by the world function $\sigma =d^{2}/2$. One suggests a new general method of the physical geometry construction. The proper Euclidean geometry is described in terms of its world function $\sigma _{\mathrm{E}}$. Any physical geometry $\mathcal{G}$ is obtained from the Euclidean geometry as a result of replacement of the Euclidean world function $\sigma _{\mathrm{E}}$ by the world function $\sigma $ of $\mathcal{G}$. This method is very simple and effective. It introduces a new geometric property: nondegeneracy of geometry. Using this method, one can construct deterministic space-time geometries with primordially stochastic motion of free particles and geometrized particle mass. Such a space-time geometry defined properly (with quantum constant as an attribute of geometry) allows one to explain quantum effects as a result of the statistical description of the stochastic particle motion (without a use of quantum principles).

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The Nilpotent Vacuum
2004jbt | Rowlands Peter

A fermionic state vector which is a nilpotent or square root of zero appears to be the most convenient packaging of the fundamental physical parameters space, time, mass and charge into a single unit. It also has the advantage of being a supersymmetric quantum field operator, which uniquely and simultaneously specifies both amplitude and phase for any fermionic state, and incorporates all the specific aspects required in BRST field quantization into a single package. The mathematical structure of the state vector immediately generates vacuum terms relevant to all four fundamental interactions, and explains the symmetry-breaking between them. By incorporating the vacuum aspects into our understanding of the fermion, we generate a ‘string theory without strings’. The nilpotent vacuum operators suggest links with many well-known vacuum phenomena, including the Casimir effect and zero-point energy.

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Division Algebra, Generalized Supersymmetries and Octonionic M-Theory
2004jbs | Toppan Francesco

This is the report of the talk given at the conference ``Number, Time and Relativity", held at the Bauman University, Moscow, August 2004, concerning the recent research activity of the author and his collaborators about the inter-relation of the concepts of division algebras, representations of Clifford algebras, generalized supersymmetries with the introduction of an alternative description of the M-algebra in terms of the non-associative structure of the octonions.

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Quaternionic analysis
1979sud | Sudbery Antony

The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. Such a theory exists and is quite far-reaching, yet it seems to be little known. It was not developed until nearly a century after Hamilton's discovery of quaternions. Hamilton himself and his principal followers and expositors, Tait and Joly, only developed the theory of functions of a quaternion variable as far as it could be taken by the general methods of the theory of functions of several real variables (the basic ideas of which appeared in their modern form for the first time in Hamilton's work on quaternions). They did not delimit a special class of regular functions among quaternion-valued functions of a quaternion variable, analogous to the regular functions of a complex variable.
This may have been because neither of the two fundamental definitions of a regular function of a complex variable has interesting consequences when adapted to quaternions; one is too restrictive, the other not restrictive enough. The functions of a quaternion variable which have quaternionic derivatives, in the obvious sense, are just the constant and linear functions (and not all of them); the functions which can be represented by quaternionic power series are just those which can be represented by power series in four real variables.
In 1935 R Fueter proposed a deашnition of "regular" for quaternionic functions by means of an analogue of the Cauchy-Riemann equations. He showed that this definition led to close analogues of Cauchy's theorem Cauchy's integral formula, and the Laurent expansion. In the next twelve years Fueter and his collaborators developed the theory of quaternionic analysis.
The theory developed by Fueter and his school is incomplete in some ways, and many of their theorems are neither so general nor so rigorously proved as present-day standards of exposition in complex analysis would require. The purpose of this paper is to give a self-contained account of the main line of quaternionic analysis which remedies these deficiencies, as well as adding a certain number of new results. By using the exterior differential calculus we are able to give new and simple proofs of most of the main theorems and to clarify the relationship between quaternionic analysis and complex analysis.

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