This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmo...This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmology. In Section 2, we have described some equations concerning cosmic evolution in a Cyclic Universe. In Section 3, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In Section 4, we have described some equations concerning the generating ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N=1five-dimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In Section 5, we have described some equations regarding the “null energy condition” violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with w>1. In Section 6, we have described some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. In Section 7, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In Section 8, we have shown various and interesting mathematical connections between some equations concerning the p-adic inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have shown the mathematical connections with various sectors of Number Theory, principally the Ramanujan’s modular equations, the Aurea Ratio and the Fibonacci’s numbers.展开更多
A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler ...A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler gamma and beta functions and their p-adic analogues, from a cohomological point of view. Connections between various methods for computing scattering amplitudes are related to the moduli space problem and period domains.展开更多
The present paper utilizes the similarity between the non-perturbative Julian Schwinger-Efimov-Fredkin approach and that of E-infinity Cantorian spacetime theory to give an exact solution to the problem of cosmic dark...The present paper utilizes the similarity between the non-perturbative Julian Schwinger-Efimov-Fredkin approach and that of E-infinity Cantorian spacetime theory to give an exact solution to the problem of cosmic dark energy via a golden mean scaling-super quantization of the electromagnetic field.展开更多
We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invari...We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.展开更多
We derive the basic canonical brackets amongst the creation and annihilation operators for a two(1 + 1)-dimensional(2D) gauge field theoretic model of an interacting Hodge theory where a U(1) gauge field(Aμ) is coupl...We derive the basic canonical brackets amongst the creation and annihilation operators for a two(1 + 1)-dimensional(2D) gauge field theoretic model of an interacting Hodge theory where a U(1) gauge field(Aμ) is coupled with the fermionic Dirac fields(ψ andˉψ). In this derivation, we exploit the spin-statistics theorem, normal ordering and the strength of the underlying six infinitesimal continuous symmetries(and the concept of their generators) that are present in the theory. We do not use the definition of the canonical conjugate momenta(corresponding to the basic fields of the theory) anywhere in our whole discussion. Thus, we conjecture that our present approach provides an alternative to the canonical method of quantization for a class of gauge field theories that are physical examples of Hodge theory where the continuous symmetries(and corresponding generators) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level.展开更多
In this paper, as a new contribution to the tensor-centric warfare (TCW) series [1] [2] [3] [4], we extend the kinetic TCW-framework to include non-kinetic effects, by addressing a general systems confrontation [5], w...In this paper, as a new contribution to the tensor-centric warfare (TCW) series [1] [2] [3] [4], we extend the kinetic TCW-framework to include non-kinetic effects, by addressing a general systems confrontation [5], which is waged not only in the traditional physical Air-Land-Sea domains, but also simultaneously across multiple non-physical domains, including cyberspace and social networks. Upon this basis, this paper attempts to address a more general analytical scenario using rigorous topological methods to introduce a two-level topological representation of modern armed conflict;in doing so, it extends from the traditional red-blue model of conflict to a red-blue-green model, where green represents various neutral elements as active factions;indeed, green can effectively decide the outcomes from red-blue conflict. System confrontations at various stages of the scenario will be defined by the non-equilibrium phase transitions which are superficially characterized by sudden entropy growth. These will be shown to have the underlying topology changes of the systems-battlespace. The two-level topological analysis of the systems-battlespace is utilized to address the question of topology changes in the combined battlespace. Once an intuitive analysis of the combined battlespace topology is performed, a rigorous topological analysis follows using (co)homological invariants of the combined systems-battlespace manifold.展开更多
The p-adic Simpson correspondence due to Faltings(Adv Math 198(2):847-862,2005)is a p-adic analogue of non-abelian Hodge theory.The following is the main result of this article:The correspondence for line bundles can ...The p-adic Simpson correspondence due to Faltings(Adv Math 198(2):847-862,2005)is a p-adic analogue of non-abelian Hodge theory.The following is the main result of this article:The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions.In the complex setting,Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties.We give a p-adic analogue of Simpson’s result.展开更多
This paper is concerned with a conjecture formulated by Coates et al in [4] which describes the relation between the integrality of the characteristic elements and their evaluations in the noncommutative Iwasawa theor...This paper is concerned with a conjecture formulated by Coates et al in [4] which describes the relation between the integrality of the characteristic elements and their evaluations in the noncommutative Iwasawa theory. We give an almost equivalent description of the conjecture and prove a certain Dart of it.展开更多
We propose a new theory of probability based on the general principle of the statistical stabilization of relative frequencies. According to this principle it is possible to consider the statistical stabilization not ...We propose a new theory of probability based on the general principle of the statistical stabilization of relative frequencies. According to this principle it is possible to consider the statistical stabilization not only with respect to the standard real topology on the field of rational numbers Q but also with respect to an arbitrary topology on Q. The case of p-adic (and more general non-Archimedean) topologies is the most important. Our frequency theory of Probability is a fruitful extension of the frequency theory of R. von Mises[18]. It's well known that the axiomatic theory of Kolmogorov uses the frequency theory as one of the foundations. And a new general frequency theory can be considered as the base for the general axiomatic theory of probability (Kolmogorov's theory is a particular case of this theory which corresponds to the real topology of the statistical stabilization on Q). The situation in the theory of probability becomes similar to that in modern geometry. The Kolmogorov axiomatics (as the Euclidean) is only one of the possibilities, and we have generated a great number of different non-Kolmogorov theories of probability.The applications to p-adic quantum mechanics and field theory are considered.展开更多
For each natural odd number n≥3,we exhibit a maximal family of n-dimensional Calabi-Yau manifolds whose Yukawa coupling length is 1.As a consequence,Shafarevich’s conjecture holds true for these families.Moreover,it...For each natural odd number n≥3,we exhibit a maximal family of n-dimensional Calabi-Yau manifolds whose Yukawa coupling length is 1.As a consequence,Shafarevich’s conjecture holds true for these families.Moreover,it follows from Deligne and Mostow(Publ.Math.IHÉS,63:5-89,1986)and Mostow(Publ.Math.IHÉS,63:91-106,1986;J.Am.Math.Soc.,1(3):555-586,1988)that,for n=3,it can be partially compactified to a Shimura family of ball type,and for n=5,9,there is a sub Q-PVHS of the family uniformizing a Zariski open subset of an arithmetic ball quotient.展开更多
文摘This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmology. In Section 2, we have described some equations concerning cosmic evolution in a Cyclic Universe. In Section 3, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In Section 4, we have described some equations concerning the generating ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N=1five-dimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In Section 5, we have described some equations regarding the “null energy condition” violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with w>1. In Section 6, we have described some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. In Section 7, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In Section 8, we have shown various and interesting mathematical connections between some equations concerning the p-adic inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have shown the mathematical connections with various sectors of Number Theory, principally the Ramanujan’s modular equations, the Aurea Ratio and the Fibonacci’s numbers.
文摘A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler gamma and beta functions and their p-adic analogues, from a cohomological point of view. Connections between various methods for computing scattering amplitudes are related to the moduli space problem and period domains.
文摘The present paper utilizes the similarity between the non-perturbative Julian Schwinger-Efimov-Fredkin approach and that of E-infinity Cantorian spacetime theory to give an exact solution to the problem of cosmic dark energy via a golden mean scaling-super quantization of the electromagnetic field.
文摘We introduce two Hopf algebroids associated to a proper and holomorphic Lie group action on a complex manifold. We prove that the cyclic cohomology of each Hopf algebroid is equal to the Dolbeault cohomology of invariant differential forms. When the action is cocompact, we develop a generalized complex Hodge theory for the Dolbeault cohomology of invariant differential forms. We prove that every cyclic cohomology class of these two Hopf algebroids can be represented by a generalized harmonic form. This implies that the space of cyclic cohomology of each Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we generalize the Serre duality and prove a Kodaira type vanishing theorem.
基金the financial support from CSIR and UGC, New Delhi, Government of India, respectively
文摘We derive the basic canonical brackets amongst the creation and annihilation operators for a two(1 + 1)-dimensional(2D) gauge field theoretic model of an interacting Hodge theory where a U(1) gauge field(Aμ) is coupled with the fermionic Dirac fields(ψ andˉψ). In this derivation, we exploit the spin-statistics theorem, normal ordering and the strength of the underlying six infinitesimal continuous symmetries(and the concept of their generators) that are present in the theory. We do not use the definition of the canonical conjugate momenta(corresponding to the basic fields of the theory) anywhere in our whole discussion. Thus, we conjecture that our present approach provides an alternative to the canonical method of quantization for a class of gauge field theories that are physical examples of Hodge theory where the continuous symmetries(and corresponding generators) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level.
文摘In this paper, as a new contribution to the tensor-centric warfare (TCW) series [1] [2] [3] [4], we extend the kinetic TCW-framework to include non-kinetic effects, by addressing a general systems confrontation [5], which is waged not only in the traditional physical Air-Land-Sea domains, but also simultaneously across multiple non-physical domains, including cyberspace and social networks. Upon this basis, this paper attempts to address a more general analytical scenario using rigorous topological methods to introduce a two-level topological representation of modern armed conflict;in doing so, it extends from the traditional red-blue model of conflict to a red-blue-green model, where green represents various neutral elements as active factions;indeed, green can effectively decide the outcomes from red-blue conflict. System confrontations at various stages of the scenario will be defined by the non-equilibrium phase transitions which are superficially characterized by sudden entropy growth. These will be shown to have the underlying topology changes of the systems-battlespace. The two-level topological analysis of the systems-battlespace is utilized to address the question of topology changes in the combined battlespace. Once an intuitive analysis of the combined battlespace topology is performed, a rigorous topological analysis follows using (co)homological invariants of the combined systems-battlespace manifold.
文摘The p-adic Simpson correspondence due to Faltings(Adv Math 198(2):847-862,2005)is a p-adic analogue of non-abelian Hodge theory.The following is the main result of this article:The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions.In the complex setting,Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties.We give a p-adic analogue of Simpson’s result.
文摘This paper is concerned with a conjecture formulated by Coates et al in [4] which describes the relation between the integrality of the characteristic elements and their evaluations in the noncommutative Iwasawa theory. We give an almost equivalent description of the conjecture and prove a certain Dart of it.
文摘We propose a new theory of probability based on the general principle of the statistical stabilization of relative frequencies. According to this principle it is possible to consider the statistical stabilization not only with respect to the standard real topology on the field of rational numbers Q but also with respect to an arbitrary topology on Q. The case of p-adic (and more general non-Archimedean) topologies is the most important. Our frequency theory of Probability is a fruitful extension of the frequency theory of R. von Mises[18]. It's well known that the axiomatic theory of Kolmogorov uses the frequency theory as one of the foundations. And a new general frequency theory can be considered as the base for the general axiomatic theory of probability (Kolmogorov's theory is a particular case of this theory which corresponds to the real topology of the statistical stabilization on Q). The situation in the theory of probability becomes similar to that in modern geometry. The Kolmogorov axiomatics (as the Euclidean) is only one of the possibilities, and we have generated a great number of different non-Kolmogorov theories of probability.The applications to p-adic quantum mechanics and field theory are considered.
基金Supported by the Science and Technology Project of Hebei Education Department(QN2019333)the Natural Fund of Cangzhou Science and Technology Bureau(197000002)+1 种基金Project of Cangzhou Normal University(xnjjl1902)NSFC(12001310)。
文摘For each natural odd number n≥3,we exhibit a maximal family of n-dimensional Calabi-Yau manifolds whose Yukawa coupling length is 1.As a consequence,Shafarevich’s conjecture holds true for these families.Moreover,it follows from Deligne and Mostow(Publ.Math.IHÉS,63:5-89,1986)and Mostow(Publ.Math.IHÉS,63:91-106,1986;J.Am.Math.Soc.,1(3):555-586,1988)that,for n=3,it can be partially compactified to a Shimura family of ball type,and for n=5,9,there is a sub Q-PVHS of the family uniformizing a Zariski open subset of an arithmetic ball quotient.
