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37
Geometry without topology as a new conception of geometry
 Int. Jour. Mat. & Mat. Sci
, 2002
"... A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new Tgeometric one are considered. Tgeometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metr ..."
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Cited by 41 (16 self)
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A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new Tgeometric one are considered. Tgeometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the Tgeometric one. Tgeometry is the simplest geometric conception (essentially only finite point sets are investigated) and simultaneously it is the most general one. It is insensitive to the space continuity and has a new property – nondegeneracy. Fitting the Tgeometry metric with the metric tensor of Riemannian geometry, one can compare geometries, constructed on the basis of different conceptions. The comparison shows that along with similarity (the same system of geodesics, the same metric) there is a difference. There is an absolute parallelism in Tgeometry, but it is absent in the Riemannian geometry. In Tgeometry any space region is isometrically embeddable in the space, whereas in Riemannian geometry only convex region is isometrically embeddable. Tgeometric conception appears to be more consistent logically, than the Riemannian one. 1 1
Model conception of quantum phenomena: logical structure and investigation methods
, 2004
"... One can construct the model conception of quantum phenomena (MCQP) which relates to the axiomatic conception of quantum phenomena (ACQP), (i.e. to the conventional quantum mechanics) in the same way, as the statistical physics relates to thermodynamics. Such a possibility is based on a new conceptio ..."
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Cited by 9 (1 self)
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One can construct the model conception of quantum phenomena (MCQP) which relates to the axiomatic conception of quantum phenomena (ACQP), (i.e. to the conventional quantum mechanics) in the same way, as the statistical physics relates to thermodynamics. Such a possibility is based on a new conception of geometry, which admits one to construct such a deterministic spacetime geometry, where motion of free particles is primordially stochastic. The spacetime geometry can be chosen in such a way that statistical description of random particle motion coincides with the quantum description. Transition from ACQP to MCQP is a result of correction of some mistakes in the foundation of ACQP. Methods of MCQP in investigation of quantum phenomena appear to be more subtle and effective, than that of ACQP. For instance, investigation of the free Dirac equation in framework of MCQP shows that the Dirac particle is in reality a rotator, i.e. two particles rotating around their common center of inertia. In the framework of MCQP one can discover the force field, responsible for pair production, that is impossible in the framework of ACQP.
Spectral invariants of operators of Dirac type on partitioned manifolds
 in Aspects of Boundary Problems in Analysis and Geometry, Editors
"... Abstract. We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of selfadjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds w ..."
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Cited by 8 (0 self)
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Abstract. We review the concepts of the index of a Fredholm operator, the spectral flow of a curve of selfadjoint Fredholm operators, the Maslov index of a curve of Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of operators of Dirac type on closed manifolds and manifolds with boundary. We emphasize various (occasionally overlooked) aspects of rigorous definitions and explain the quite different stability properties. Moreover, we utilize the heat equation approach in various settings and show how these topological and spectral invariants are mutually related in the study of
Conformal field theory in conformal space, Nuclear Phys
 B
, 1999
"... We present a new framework for a Lagrangian description of conformal field theories in various dimensions based on a local version of d+2dimensional conformal space. The results include a true gauge theory of conformal gravity in d = (1,3) and any standard matter coupled to it. An important feature ..."
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Cited by 4 (0 self)
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We present a new framework for a Lagrangian description of conformal field theories in various dimensions based on a local version of d+2dimensional conformal space. The results include a true gauge theory of conformal gravity in d = (1,3) and any standard matter coupled to it. An important feature is the automatic derivation of the conformal gravity constraints, which are necessary for the analysis of the matter systems. PACS Classification: 11.15.q, 11.30.Ly Keyword(s): local conformal symmetry, conformal field theory, conformal gravity
Explicit Hyperelliptic Solutions of Modified Kortewegde Vries Equation: Essentials of Miura Transformation
"... Explicit hyperelliptic solutions of the modified Kortewegde Vries equations without any ambiguous parameters were constructed in terms only of the hyperelliptic alfunctions over nondegenerated hyperelliptic curve y 2 = f(x) of arbitrary genus g. In the derivation, any θfunctions or BakerAkhieze ..."
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Cited by 2 (2 self)
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Explicit hyperelliptic solutions of the modified Kortewegde Vries equations without any ambiguous parameters were constructed in terms only of the hyperelliptic alfunctions over nondegenerated hyperelliptic curve y 2 = f(x) of arbitrary genus g. In the derivation, any θfunctions or BakerAkhiezer functions were not essentially used. Then the Miura transformation naturally appears as the connections between the hyperelliptic ℘functions and hyperelliptic alfunctions.
Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations
, 2004
"... ..."
\Gamma \Delta 'AEfiff\Omega ae)$\Psi'AEfiff"ffi
"... this paper developed the finite element method  a systematical algorithm for solving stationary problems. This kind of physical problem has two equivalent mathematical formulations: one is Newtonian form, i.e. solving elliptic equation of second order; the other is variational form, i.e. solving ..."
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this paper developed the finite element method  a systematical algorithm for solving stationary problems. This kind of physical problem has two equivalent mathematical formulations: one is Newtonian form, i.e. solving elliptic equation of second order; the other is variational form, i.e. solving extremal of energy functional. The key to success of the finite element method is that variational forms are used as its foundation. After working out the finite element method and its mathematical foundation, the author had tried to apply the idea of this method to dynamical problems of continuum mechanics and failed
WDS'07 Proceedings of Contributed Papers, Part I, 251–256, 2007. ISBN 9788073780234 © MATFYZPRESS
"... Abstract. The present paper is dedicated to Felix Klein (1849–1925), one of the leading German mathematicians in the second half of the 19th century. It gives a brief account of his professional life. Some of his activities connected with the reform of mathematics teaching at German schools are ment ..."
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Abstract. The present paper is dedicated to Felix Klein (1849–1925), one of the leading German mathematicians in the second half of the 19th century. It gives a brief account of his professional life. Some of his activities connected with the reform of mathematics teaching at German schools are mentioned as well. In the following text, we describe fundamental ideas of his Erlanger Programm in more detail. References containing selected papers relevant to this theme are attached.