Like Euclid, Riemann and Lobachevski geometries are on an almost equal footing, based on the principle of relativity of maximum symmetry proposed by Professor Lu Qikeng and the postulate on invariant universal constan...Like Euclid, Riemann and Lobachevski geometries are on an almost equal footing, based on the principle of relativity of maximum symmetry proposed by Professor Lu Qikeng and the postulate on invariant universal constants c and R, the de Sitter/anti-de Sitter (dS/AdS) special relativity on dS/AdS-space with radius R can be set up on an almost equal footing with Einstein’s special relativity on the Minkowski-space in the case of R→∞.Thus the dS-space is coin-like: a law of inertia in Beltrami atlas with Beltrami time simultaneity for the principle of relativity on one side, and the proper-time simultaneity and a Robertson-Walker-like dS-space with entropy and an accelerated expanding S 3 fitting the cosmological principle on another side.If our universe is asymptotic to the Robertson-Walker-like dS-space of R ? (3/Λ)1/2, it should be slightly closed in O(Λ) with entropy bound S ? 3πc 3 k B /ΛG?. Contrarily, via its asymptotic behavior, it can fix on Beltrami inertial frames without ‘an argument in a circle’ and acts as the origin of inertia.There is a triality of conformal extensions of three kinds of special relativity and their null physics on the projective boundary of a 5-d AdS-space, a null cone modulo projective equivalence [ $ \mathcal{N} $ ] ? ? P (AdS 5). Thus there should be a dS-space on the boundary of S 5 × AdS 5 as a vacuum of supergravity.In the light of Einstein’s ‘Galilean regions’, gravity should be based on the localized principle of relativity of full maximum symmetry with a gauge-like dynamics. Thus, this may lead to the theory of gravity of corresponding local symmetry. A simple model of dS-gravity characterized by a dimensionless constant g ?(ΛG?/3c 3)1/2 ～ 10?61 shows the features on umbilical manifolds of local dS-invariance. Some gravitational effects out of general relativity may play a role as dark matter.The dark universe and its asymptotic behavior may already indicate that the dS special relativity and dS-gravity be the foundation of large scale physics.展开更多
For a probability space (X, B, μ) a subfamily F of the σ-algebra B is said to be a regular base if every B ∈ B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. ...For a probability space (X, B, μ) a subfamily F of the σ-algebra B is said to be a regular base if every B ∈ B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {R γ } γ∈Γ is a countable family of relations of the full measure on a probability space (X, B, μ), i.e. for every γ ∈ Γ there is a positive integer s γ such that R γ ? $X^{s_\gamma } $ with $\mu ^{s_\gamma } $ (R γ ) = 1. In the present paper we show that if (X, B, μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K ? X with μ*(K) = 1 such that (x 1, …, $x_{^{s_\gamma } } $ ) ∈ R γ for any γ ∈ Γ and for any s γ distinct elements x 1, …, $x_{^{s_\gamma } } $ of K, where μ* is the outer measure induced by the measure μ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.展开更多
In this paper,we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result,we establish a multivariate irregu...In this paper,we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result,we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.展开更多
基金This work was partially supported by the National Natural Science Foundation of China(Grant No.90503002)
文摘Like Euclid, Riemann and Lobachevski geometries are on an almost equal footing, based on the principle of relativity of maximum symmetry proposed by Professor Lu Qikeng and the postulate on invariant universal constants c and R, the de Sitter/anti-de Sitter (dS/AdS) special relativity on dS/AdS-space with radius R can be set up on an almost equal footing with Einstein’s special relativity on the Minkowski-space in the case of R→∞.Thus the dS-space is coin-like: a law of inertia in Beltrami atlas with Beltrami time simultaneity for the principle of relativity on one side, and the proper-time simultaneity and a Robertson-Walker-like dS-space with entropy and an accelerated expanding S 3 fitting the cosmological principle on another side.If our universe is asymptotic to the Robertson-Walker-like dS-space of R ? (3/Λ)1/2, it should be slightly closed in O(Λ) with entropy bound S ? 3πc 3 k B /ΛG?. Contrarily, via its asymptotic behavior, it can fix on Beltrami inertial frames without ‘an argument in a circle’ and acts as the origin of inertia.There is a triality of conformal extensions of three kinds of special relativity and their null physics on the projective boundary of a 5-d AdS-space, a null cone modulo projective equivalence [ $ \mathcal{N} $ ] ? ? P (AdS 5). Thus there should be a dS-space on the boundary of S 5 × AdS 5 as a vacuum of supergravity.In the light of Einstein’s ‘Galilean regions’, gravity should be based on the localized principle of relativity of full maximum symmetry with a gauge-like dynamics. Thus, this may lead to the theory of gravity of corresponding local symmetry. A simple model of dS-gravity characterized by a dimensionless constant g ?(ΛG?/3c 3)1/2 ～ 10?61 shows the features on umbilical manifolds of local dS-invariance. Some gravitational effects out of general relativity may play a role as dark matter.The dark universe and its asymptotic behavior may already indicate that the dS special relativity and dS-gravity be the foundation of large scale physics.
基金supported by the National Basic Research Program of China (Grant No. 1999075105)the National Natural Science Foundation of China (Grant No. 10471002)Research Foundation for Doctoral Programm (Grant No. 20050574002)
文摘The Paley-Wiener theorem in the non-commutative and non-associative octonion analytic function space is proved.
基金This work was supported by the National Science Fbundation of China (Grant No. 10471049)
文摘For a probability space (X, B, μ) a subfamily F of the σ-algebra B is said to be a regular base if every B ∈ B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {R γ } γ∈Γ is a countable family of relations of the full measure on a probability space (X, B, μ), i.e. for every γ ∈ Γ there is a positive integer s γ such that R γ ? $X^{s_\gamma } $ with $\mu ^{s_\gamma } $ (R γ ) = 1. In the present paper we show that if (X, B, μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K ? X with μ*(K) = 1 such that (x 1, …, $x_{^{s_\gamma } } $ ) ∈ R γ for any γ ∈ Γ and for any s γ distinct elements x 1, …, $x_{^{s_\gamma } } $ of K, where μ* is the outer measure induced by the measure μ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.
基金supported by National Natural Science Foundation of China (Grant No. 10671019)Research Fund for the Doctoral Program Higher Education (Grant No. 20050027007)Key Project of Technology Bureau of Sichuan Province (Grant No. 05JY029-138)
文摘In this paper,we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result,we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.
