"Hypercomplex Numbers in geometry and Physics" 12 (13), Vol 7, 2010
j013

Content of Issue

Hyperbolical analog of the electromagnetic field
2010jaw | Pavlov D.G.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, geom2004@mail.ru

On the basis of the analogy of complex numbers analytical functions with two-dimensional electro- and magnetostatic fields there was made an assumption considering the existence of such correspondence between h-analytical functions of the binary variable and some other pair of binary physical fields in reality, one of which is hyperbolical source field and another is hyperbolically vortex field. Unlike electro- and magnetostatic fields, this pair is not realized in space but rather in space-time; thus, the sources of the first field are events while force lines of the second vortex constituent are hyperbolas. Essential feature of this hypothetical pair of fields is that it is feasible only in two-dimensional pseudo-Euclidian space and that it is fundamentally incompatible with the Minkowski idea of 4-dimensional space-time. Partially this is the very reason why such fields weren't considered potentially feasible by physicians even in theory. Their immediate discovery is hampered by experimentalists' being used to space-boundary conditions, while they had better work with space-time ones here. Although this pair is incompatible with Minkowskyi space, still it can possibly be realized in 4-dimesional space possessing, in particular, Berwald-Moor Finsler metric function, its discovery in reality, thus, serving a valid reason to substitute the quadratic metric idea of space-time geometry for Finsler one, connected with quartic form.

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Double numbers
2010jbw | Pavlov D.G., Garasko G.I.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, geom2004@mail.ru, gri9z@mail.ru

There is an attempt to show that there is much more in common between the complex numbers and the double (hyperbolically complex) numbers than it is usually thought to be. With this the new and non-trivial properties of the analytical functions of the double number variable are discovered. For example, there is a relation between these functions and the hyperbolic potential and solenoidal vector fields on the pseudo Euclidean plane. Besides, it is shown how many structures on the complex plane can be one-to-one mapped at their hyperbolical analogues. This repudiates the magic properties of the complex numbers, and particularly, leads to the understanding that the analytical functions of complex numbers can be given by two scalar functions not of two but of one real variable each.

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Analog of the Cauchy's formula in non-degenerate polynumbers spaces
2010jcw | Pavlov D.G., Garasko G.I.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, geom2004@mail.ru, gri9z@mail.ru

Obtained was an analog for Cauchy's formula for non-degenerate commutative-associative hypercomplex numbers (polynumbers), including algebra of complex numbers or direct summ of complex algebras as means of subalgebra. Therewith manifested are the reasons why Cauchy's formula in polynumbers Hn which are the direct sums of nothing but actual algebras is so hard to obtain.

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Identically solvable finsler geometries
2010jdw | Garasko G.I.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, gri9z@mail.ru

We suggest an algorithm to search for the identically solvable Finsler geometries which provides the possibility to find some solvable Finsler geometries that are not identically solvable. This algorithm is closely related to the reflection on the space whose dimension is one unit less than the dimension of the Finsler space on itself. This reflection must coincide to its own reverse and possess some other properties. For the spaces of arbitrary dimension, the identical reflection corresponds to the Euclidean space, the reflection with the simultaneous change of the sign of all coordinates - to the pseudo Euclidean space, and the reflection with the inversion of all the coordinates corresponds to the space with Berwald-Moor metric.

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Real part of the non-degenerate poly number and a special linear form
2010jew | Garasko G.I.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, gri9z@mail.ru

The set of the various poly linear forms that can be constructed in the spaces of the non-degenerate poly numbers contains the linear invariant form closely related to the notion of real part of the non-degenerate poly number and to the time coordinate.

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h-holomorphic functions of double variable and their applications
2010jfw | Pavlov D.G., Kokarev S.S.  // Research Institute of Hypercomplex Systems in Geometry and Physics, Friazino, Russia, geom2004@mail.ru, logos-center@mail.ru

We consider complex-differentiable functions of double variable and their essential properties analogical to the properties of functions of standard complex variable: Cauchy theorem and Cauchy formula, hyperbolic harmonicity, general properties of h-conformal transformations and such transformations, yielded by elementary functions. The question on applications of h-conformal transformations for solving of 2-dimensional problems of mathematical physics is discussed.

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Hyperbolic theory of field on the plane of double variable
2010jgw | Pavlov D.G., Kokarev S.S.  // Research Institute of Hypercomplex Systems in Geometry and Physics; RSEC Logos, geom2004@mail.ru, logos-center@mail.ru

By analogy with the theory of harmonic fields on the complex plane the theory of wave fields on the plane of double variable is developed. The hyperbolic analogies of point-like sources, curls, source-curls and their multipoles generalizations are constructed. Some important physical aspects of the theory together with possible generalizations on higher polynumber dimensions are discussed.

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Twisted cohomologies of wrap groups over quaternions and octonions
2010jkw | Ludkovsky S.V.  // Moscow State Technical University MIREA, Moscow, Russia, sludkowski@mail.ru

This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real R, complex C numbers, the quaternion skew field H and the octonion algebra O. Cohomologies of wrap groups and their structure are investigated. Sheaves of wrap groups are constructed and studied. Moreover, twisted cohomologies and sheaves over quaternions and octonions are investigated as well.

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Differential forms: from Clifford, through Cartan to Kahler
2010jlw | Vargas Jose G.  // University of South Carolina, Columbia, SC, USA, josegvargas@earthlink.net

Limitations of the vector, tensor and Dirac calculi are illustrated to motivate the Kaehler calculus of integrands, which replaces all three of them and which we introduce in three steps. In a first step, we present the basics of the underlying Clifford algebra for that calculus, algebra valid for Euclidean and pseudo-Euclidean vector spaces of arbitrary dimension. The usual vector algebra is shown to be a corrupted form of Clifford algebra, corruption specific to dimension three and non-existing for other dimensions. The Clifford product is constituted by the sum of the exterior and interior products if at least one of the factors is a vector. Grossly speaking, these products play the role of the vector and scalar products of three dimensions, while generalizing them. It thus contains exterior algebra. As an intermediate step towards the Kaehler calculus, we briefly give the fundamentals of Cartans exterior calculus of scalar-valued differential forms, here viewed as ordinary scalar-valued integrands in multiple integrals. We also make a brief incursion into the exterior calculus of vector-valued differential forms, which is the moving frame version of differential geometry. We show the basics of the Kaehler calculus of differential forms. It is to the exterior calculus what Clifford algebra is to exterior algebra. Because of time and complexity constraints, we limit ourselves to scalar-valued differential forms, which is sufficient for relativistic quantum mechanics with electromagnetic coupling. In using this calculus, the problem with negative energy-solutions does not arise

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On Julia and Mandelbrot sets in double numbers plane
2010jmw | Tipunin I., Toporensky A.  // Lebedev Physical Institute, Moscow, Russia; Sternberg Astronomical Institute, Moscow, Russia atopor@rambler.ru

Mandelbrot set and filled-in Julia sets in double numbers plane have been found numerically. We discuss differences between our definition of these sets and the definition of Artzy. A condition for these two different types of sets to coincide have been found. We show also that our definition allows graphycally interesting escaping time diagramms to be constructed in the double numbers plane, in contrast to the Artzys definition.

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The amendment to one statement of the paper «Idempotents and nilpotents in the clifford algebra of euclidean 3 - space and their interconnections in physics»
2010jnw | Mornev O.A.  // Institute of Theoretical and Experimental Biophysics of RAS, Pushchino, Russian Federation, mornev@mail.ru

In the authors paper mentioned in the title of this abstract and published in the Journal «Hypercomplex Numbers in Geometry and Physics», 2 (12), Vol.6, 2009, Pp. 92-137, the statement was formulated that composite idempotents of the Clifford algebra Cl3 of three dimensional Euclidean space generate nonminimal one-sided ideals of Cl3. The amendment presented here cancels this statement: one can prove that all one-sided ideals generated in Cl3 both by composite idempotents and by simple ones are always minimal. Fortunately, the proofs of the rest of results presented in that paper are not affected by this local circumstance and therefore do not fall; as a consequence, all these results remain true.

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