In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ ...In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric gij remains uniformly bounded.展开更多
In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with ＂small＂ initial data. We first present some estimates on...In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with ＂small＂ initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with ＂small＂ initial data.展开更多
In this paper,hyperbolic geometry is used to constructing new quantum color codes.We use hyperbolic tessellations and hyperbolic polygons to obtain them by pairing the edges on compact surfaces.These codes have minimu...In this paper,hyperbolic geometry is used to constructing new quantum color codes.We use hyperbolic tessellations and hyperbolic polygons to obtain them by pairing the edges on compact surfaces.These codes have minimum distance of at least 4 and the encoding rate near to 1,which are not mentioned in other literature.Finally,a comparison table with quantum codes recently proposed by the authors is provided.展开更多
WE use the geometric algebra in refs. [1, 2] to study hyperbolic geometry. The n-dimensional hyperbolic space H<sup>n</sup> is taken to be the unit sphere of G<sub>1</sub> (I<sub>-n,1<...WE use the geometric algebra in refs. [1, 2] to study hyperbolic geometry. The n-dimensional hyperbolic space H<sup>n</sup> is taken to be the unit sphere of G<sub>1</sub> (I<sub>-n,1</sub>) with antipodal points identified. We study typically H<sup>2</sup>: the dualities between generalized point and generalized line, between generalized triangle and imaginary triangle; convex generalized triangles; Lorentz transformations; generalized circles and double-cycles, etc. Below we list some of the results.展开更多
The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-pa...The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.展开更多
The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry.When the jet values are prescribed on a positive dimensional subvariety,it is h...The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry.When the jet values are prescribed on a positive dimensional subvariety,it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive.We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded,if a square-integrable extension is known to be possible over a double point at the origin.It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory.The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections.We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of OhsawaTakegoshi to a simple application of the standard method of constructing holomorphic functions by solving the equation with cut-off functions and additional blowup weight functions.展开更多
The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum ...The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum mechanics. This more than just a conceptual equation is illustrated by integer approximation and an exact solution of the dark energy density behind cosmic expansion.展开更多
In this paper, we present a new form of “special relativity” (BSR), which is isomorphic to Einstein’s “special relativity” (ESR). This in turn proves the non-uniqueness of Einstein’s “special relativity” and i...In this paper, we present a new form of “special relativity” (BSR), which is isomorphic to Einstein’s “special relativity” (ESR). This in turn proves the non-uniqueness of Einstein’s “special relativity” and implies the inconclusiveness of so-called “relativistic physics”. This work presents new results of principal significance for the foundations of physics and practical results for high energy physics, deep space astrophysics, and cosmology as well. The entire exposition is done within the formalism of the Lorentz <em>SL</em>(2<em>C</em>) group acting via isometries on <strong>real 3-dimensional Lobachevskian (hyperbolic) spaces</strong> <em>L</em><sup>3</sup> regarded as quotients <span style="white-space:nowrap;"><em>SL</em>(2<em>C</em>)/<em>SU</em>(2)</span>. We show via direct calculations that both ESR and BSR are parametric maps from Lobachevskian into Euclidean space, namely a <strong>gnomonic</strong> (central) map in the case of ESR, and a<strong> stereographic </strong>map in the case of BSR. Such an identification allows us to link these maps to relevant models of Lobachevskian geometry. Thus, we identify ESR as the physical realization of the Beltrami-Klein (non-conformal) model, and BSR as the physical realization of the Poincare (conformal) model of Lobachevskian geometry. Although we focus our discussion on ball models of Lobachevskian geometry, our method is quite general, and for instance, may be applied to the half-space model of Lobachevskian geometry with appropriate “Lorentz group” acting via isometries on (positive) half space, resulting yet in another “special relativity” isomorphic with ESR and BSR. By using the notion of a<strong> homotopy</strong> of maps, the identification of “special relativities” as maps from Lobachevskian into Euclidean space allows us to justify the existence of an uncountable infinity of hybrid “special relativities” and consequently an uncountable infinity of “relativistic physics” built upon them. This is another new result in physics and it states that so called “relativistic physics” is unique only up to a homotopy. Finally, we show that “paradoxes” of “special relativities” in either ESR or BSR are simply common distortions of maps between non-isometric spaces. The entire exposition is kept at elementary level accessible to majority of students in physics and/or engineering.展开更多
Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some amb...Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space.展开更多
Abstract In the study of n-dimensional spherical or hyperbolic geometry, n≥3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use...Abstract In the study of n-dimensional spherical or hyperbolic geometry, n≥3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in S^n（1） and H^n（-1）.展开更多
基金supported in part by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)the Science Foundation in Higher Education of Henan(18A110036)
文摘In this article, we investigate the hyperbolic geometry flow with time-dependent dissipation(δ2 gij)/δt2+μ/((1 + t)λ)(δ gij)/δt=-2 Rij,on Riemann surface. On the basis of the energy method, for 0 〈 λ ≤ 1, μ 〉 λ + 1, we show that there exists a global solution gij to the hyperbolic geometry flow with time-dependent dissipation with asymptotic flat initial Riemann surfaces. Moreover, we prove that the scalar curvature R(t, x) of the solution metric gij remains uniformly bounded.
基金supported in part by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)the Henan Provincial Natural Science Foundation of China(152300410226)
文摘In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with ＂small＂ initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with ＂small＂ initial data.
文摘In this paper,hyperbolic geometry is used to constructing new quantum color codes.We use hyperbolic tessellations and hyperbolic polygons to obtain them by pairing the edges on compact surfaces.These codes have minimum distance of at least 4 and the encoding rate near to 1,which are not mentioned in other literature.Finally,a comparison table with quantum codes recently proposed by the authors is provided.
文摘WE use the geometric algebra in refs. [1, 2] to study hyperbolic geometry. The n-dimensional hyperbolic space H<sup>n</sup> is taken to be the unit sphere of G<sub>1</sub> (I<sub>-n,1</sub>) with antipodal points identified. We study typically H<sup>2</sup>: the dualities between generalized point and generalized line, between generalized triangle and imaginary triangle; convex generalized triangles; Lorentz transformations; generalized circles and double-cycles, etc. Below we list some of the results.
文摘The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.
基金supported by Grant 1001416 of the National Science Foundation
文摘The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry.When the jet values are prescribed on a positive dimensional subvariety,it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive.We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded,if a square-integrable extension is known to be possible over a double point at the origin.It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory.The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections.We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of OhsawaTakegoshi to a simple application of the standard method of constructing holomorphic functions by solving the equation with cut-off functions and additional blowup weight functions.
文摘The paper suggests that quantum relativistic gravity (QRG) is basically a higher dimensionality (HD) simulating relativity and non-classical effects plus a fractal Cantorian spacetime geometry (FG) simulating quantum mechanics. This more than just a conceptual equation is illustrated by integer approximation and an exact solution of the dark energy density behind cosmic expansion.
文摘In this paper, we present a new form of “special relativity” (BSR), which is isomorphic to Einstein’s “special relativity” (ESR). This in turn proves the non-uniqueness of Einstein’s “special relativity” and implies the inconclusiveness of so-called “relativistic physics”. This work presents new results of principal significance for the foundations of physics and practical results for high energy physics, deep space astrophysics, and cosmology as well. The entire exposition is done within the formalism of the Lorentz <em>SL</em>(2<em>C</em>) group acting via isometries on <strong>real 3-dimensional Lobachevskian (hyperbolic) spaces</strong> <em>L</em><sup>3</sup> regarded as quotients <span style="white-space:nowrap;"><em>SL</em>(2<em>C</em>)/<em>SU</em>(2)</span>. We show via direct calculations that both ESR and BSR are parametric maps from Lobachevskian into Euclidean space, namely a <strong>gnomonic</strong> (central) map in the case of ESR, and a<strong> stereographic </strong>map in the case of BSR. Such an identification allows us to link these maps to relevant models of Lobachevskian geometry. Thus, we identify ESR as the physical realization of the Beltrami-Klein (non-conformal) model, and BSR as the physical realization of the Poincare (conformal) model of Lobachevskian geometry. Although we focus our discussion on ball models of Lobachevskian geometry, our method is quite general, and for instance, may be applied to the half-space model of Lobachevskian geometry with appropriate “Lorentz group” acting via isometries on (positive) half space, resulting yet in another “special relativity” isomorphic with ESR and BSR. By using the notion of a<strong> homotopy</strong> of maps, the identification of “special relativities” as maps from Lobachevskian into Euclidean space allows us to justify the existence of an uncountable infinity of hybrid “special relativities” and consequently an uncountable infinity of “relativistic physics” built upon them. This is another new result in physics and it states that so called “relativistic physics” is unique only up to a homotopy. Finally, we show that “paradoxes” of “special relativities” in either ESR or BSR are simply common distortions of maps between non-isometric spaces. The entire exposition is kept at elementary level accessible to majority of students in physics and/or engineering.
基金the Centennial fellowship of the Graduate School at Princeton University
文摘Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space.
文摘Abstract In the study of n-dimensional spherical or hyperbolic geometry, n≥3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in S^n（1） and H^n（-1）.
