We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold i...We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.展开更多
In this note,we prove that expanders cannot be coarsely embedded into group extensions of sequences of groups which are coarsely embeddable into Hardamad manifolds and certain Banach spaces due to the similar concentr...In this note,we prove that expanders cannot be coarsely embedded into group extensions of sequences of groups which are coarsely embeddable into Hardamad manifolds and certain Banach spaces due to the similar concentration theorems.展开更多
We first establish a integral inequality for compact maximal space-like subman-ifolds in pseudo-Riemannian manifolds N^np+p. Then, we investigate compact space-like sub-manifolds and hupersurfaces with parallel second...We first establish a integral inequality for compact maximal space-like subman-ifolds in pseudo-Riemannian manifolds N^np+p. Then, we investigate compact space-like sub-manifolds and hupersurfaces with parallel second fundamental form in N^np+p and some other N^np+pambient spaces. We obtain some distribution theorems for the square norm of the secondfundamental form.展开更多
In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x<sub>2</sub>, x<sub>2</sub>, …, x<sub...In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x<sub>2</sub>, x<sub>2</sub>, …, x<sub>t</sub>) of class C<sup>r</sup> in Euelidean R<sup>t</sup> is a differential manifold. Using the intersection of the tangent plane and the hypernormal of the differential manifold to construct the shared master key of participants, an intuitive, secure and complete (t,n)-threshold secret sharing scheme is designed. The paper is proved to be safe, and the probability of successful attack of attackers is only 1/p<sup>t</sup><sup>-1</sup>. When the prime number p is sufficiently large, the probability is almost 0. The results show that this scheme has the characteristics of single-parameter representation of the master key in the geometric method, and is more practical and easy to implement than the Blakley threshold secret sharing scheme.展开更多
In this work we investigate the possibility to represent physical fields as Einstein manifold. Based on the Einstein field equations in general relativity, we establish a general formulation for determining the metric...In this work we investigate the possibility to represent physical fields as Einstein manifold. Based on the Einstein field equations in general relativity, we establish a general formulation for determining the metric tensor of the Einstein manifold that represents a physical field in terms of the energy-momentum tensor that characterises the physical field. As illustrations, we first apply the general formulation to represent the perfect fluid as Einstein manifold. However, from the established relation between the metric tensor and the energy-momentum tensor, we show that if the trace of the energy-momentum tensor associated with a physical field is equal to zero then the corresponding physical field cannot be represented as an Einstein manifold. This situation applies to the electromagnetic field since the trace of the energy-momentum of the electromagnetic field vanishes. Nevertheless, we show that a system that consists of the electromagnetic field and non-interacting charged particles can be represented as an Einstein manifold since the trace of the corresponding energy-momentum of the system no longer vanishes. As a further investigation, we show that it is also possible to represent physical fields as maximally symmetric spaces of constant scalar curvature.展开更多
Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanag...Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanagisawa[Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains,Indiana Univ.Math.J.58(2009),1853-1920],combined with global computations based on the Bochner technique.展开更多
The expression of the Maxwell magnetic monopole was employed to correlate the space to space projection that gives rise to the Gell-Mann standard model, and space to time projection which gives the leptons;and how doe...The expression of the Maxwell magnetic monopole was employed to correlate the space to space projection that gives rise to the Gell-Mann standard model, and space to time projection which gives the leptons;and how does it correlate to the Perelman mappings from the homogeneous 5D manifold to the Lorentz 4D manifold, together with correlating the physical consequences caused by the breaking of the Diagonal Long Range Order [DLRO] of the monopoles quantum states affected by the motion of massive particles in the Lorentz 4D boundary of the 5D manifold, which leads to gravitons and the gravity field via the General Relativity covariant Riemannian 4D curvatures metric equation.展开更多
The feature space extracted from vibration signals with various faults is often nonlinear and of high dimension.Currently,nonlinear dimensionality reduction methods are available for extracting low-dimensional embeddi...The feature space extracted from vibration signals with various faults is often nonlinear and of high dimension.Currently,nonlinear dimensionality reduction methods are available for extracting low-dimensional embeddings,such as manifold learning.However,these methods are all based on manual intervention,which have some shortages in stability,and suppressing the disturbance noise.To extract features automatically,a manifold learning method with self-organization mapping is introduced for the first time.Under the non-uniform sample distribution reconstructed by the phase space,the expectation maximization(EM) iteration algorithm is used to divide the local neighborhoods adaptively without manual intervention.After that,the local tangent space alignment(LTSA) algorithm is adopted to compress the high-dimensional phase space into a more truthful low-dimensional representation.Finally,the signal is reconstructed by the kernel regression.Several typical states include the Lorenz system,engine fault with piston pin defect,and bearing fault with outer-race defect are analyzed.Compared with the LTSA and continuous wavelet transform,the results show that the background noise can be fully restrained and the entire periodic repetition of impact components is well separated and identified.A new way to automatically and precisely extract the impulsive components from mechanical signals is proposed.展开更多
This paper mainly focuses on the issues about generic multi-scale object perception for detection or recognition. A novel computational model in visually-feature space is presented for scene & object representatio...This paper mainly focuses on the issues about generic multi-scale object perception for detection or recognition. A novel computational model in visually-feature space is presented for scene & object representation to purse the underlying textural manifold statistically in nonparametric manner. The associative method approximately makes perceptual hierarchy in human-vision biologically coherency in specific quad-tree-pyramid structure, and the appropriate scale-value of different objects can automatically be selected by evaluating from well-defined scale function without any priori knowledge. The sufficient experiments truly demonstrate the effectiveness of scale determination in textural manifold with object localization rapidly.展开更多
The inclusion of space-time in the extended group of relativistic form-invariance, Cl<sub>3</sub>*</sup>, is specified as the inclusion of the whole space-time manifold in this multiplicative Lie gro...The inclusion of space-time in the extended group of relativistic form-invariance, Cl<sub>3</sub>*</sup>, is specified as the inclusion of the whole space-time manifold in this multiplicative Lie group. First physical results presented here are: the geometric origin of the time arrow, a better understanding of the non-simultaneity in optics and a mainly geometric origin for the universe expansion, and its recent acceleration.展开更多
In this paper, the vertical and horizontal distributions of an invariant sub-manifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finall...In this paper, the vertical and horizontal distributions of an invariant sub-manifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.展开更多
The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacet...The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.展开更多
A force with an acceleration that is equal to multiples greater than the speed of light per unit time is exerted on a cloud of charged particles. The particles are resultantly accelerated to within an infinitesimal fr...A force with an acceleration that is equal to multiples greater than the speed of light per unit time is exerted on a cloud of charged particles. The particles are resultantly accelerated to within an infinitesimal fraction of the speed of light. As the force or acceleration increases, the particles’ velocity asymptotically approaches but never achieves the speed of light obeying relativity. The asymptotic increase in the particles’ velocity toward the speed of light as acceleration increasingly surpasses the speed of light per unit time does not compensate for the momentum value produced on the particles at sub-light velocities. Hence, the particles’ inertial mass value must increase as acceleration increases. This increase in the particles’ inertial mass as the particles are accelerated produce a gravitational field which is believed to occur in the oscillation of quarks achieving velocities close to the speed of light. The increased inertial mass of the density of accelerated charged particles becomes the source mass (or Big “M”) in Newton’s equation for gravitational force. This implies that a space-time curve is generated by the accelerated particles. Thus, it is shown that the acceleration number (or multiple of the speed of light greater than 1 per unit of time) and the number of charged particles in the cloud density are surjectively mapped to points on a differential manifold or space-time curved surface. Two aspects of Einstein’s field equations are used to describe the correspondence between the gravitational field produced by the accelerated particles and the resultant space-time curve. The two aspects are the Schwarzchild metric and the stress energy tensor. Lastly, the possibility of producing a sufficient acceleration or electromagnetic force on the charged particles to produce a gravitational field is shown through the Lorentz force equation. Moreover, it is shown that a sufficient voltage can be generated to produce an acceleration/force on the particles that is multiples greater than the speed of light per unit time thereby generating gravity.展开更多
文摘We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.
文摘In this note,we prove that expanders cannot be coarsely embedded into group extensions of sequences of groups which are coarsely embeddable into Hardamad manifolds and certain Banach spaces due to the similar concentration theorems.
文摘We first establish a integral inequality for compact maximal space-like subman-ifolds in pseudo-Riemannian manifolds N^np+p. Then, we investigate compact space-like sub-manifolds and hupersurfaces with parallel second fundamental form in N^np+p and some other N^np+pambient spaces. We obtain some distribution theorems for the square norm of the secondfundamental form.
文摘In this paper, the concepts of topological space and differential manifold are introduced, and it is proved that the surface determined by function F (x<sub>2</sub>, x<sub>2</sub>, …, x<sub>t</sub>) of class C<sup>r</sup> in Euelidean R<sup>t</sup> is a differential manifold. Using the intersection of the tangent plane and the hypernormal of the differential manifold to construct the shared master key of participants, an intuitive, secure and complete (t,n)-threshold secret sharing scheme is designed. The paper is proved to be safe, and the probability of successful attack of attackers is only 1/p<sup>t</sup><sup>-1</sup>. When the prime number p is sufficiently large, the probability is almost 0. The results show that this scheme has the characteristics of single-parameter representation of the master key in the geometric method, and is more practical and easy to implement than the Blakley threshold secret sharing scheme.
文摘In this work we investigate the possibility to represent physical fields as Einstein manifold. Based on the Einstein field equations in general relativity, we establish a general formulation for determining the metric tensor of the Einstein manifold that represents a physical field in terms of the energy-momentum tensor that characterises the physical field. As illustrations, we first apply the general formulation to represent the perfect fluid as Einstein manifold. However, from the established relation between the metric tensor and the energy-momentum tensor, we show that if the trace of the energy-momentum tensor associated with a physical field is equal to zero then the corresponding physical field cannot be represented as an Einstein manifold. This situation applies to the electromagnetic field since the trace of the energy-momentum of the electromagnetic field vanishes. Nevertheless, we show that a system that consists of the electromagnetic field and non-interacting charged particles can be represented as an Einstein manifold since the trace of the corresponding energy-momentum of the system no longer vanishes. As a further investigation, we show that it is also possible to represent physical fields as maximally symmetric spaces of constant scalar curvature.
文摘Let(M,g_(0))be a compact Riemannian manifold-with-boundary.We present a new proof of the classical Gaffney inequality for differential forms in boundary value spaces over M,via a variational approach a la Kozono-Yanagisawa[Lr-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains,Indiana Univ.Math.J.58(2009),1853-1920],combined with global computations based on the Bochner technique.
文摘The expression of the Maxwell magnetic monopole was employed to correlate the space to space projection that gives rise to the Gell-Mann standard model, and space to time projection which gives the leptons;and how does it correlate to the Perelman mappings from the homogeneous 5D manifold to the Lorentz 4D manifold, together with correlating the physical consequences caused by the breaking of the Diagonal Long Range Order [DLRO] of the monopoles quantum states affected by the motion of massive particles in the Lorentz 4D boundary of the 5D manifold, which leads to gravitons and the gravity field via the General Relativity covariant Riemannian 4D curvatures metric equation.
基金supported by National Natural Science Foundation of China(Grant No.51075323)
文摘The feature space extracted from vibration signals with various faults is often nonlinear and of high dimension.Currently,nonlinear dimensionality reduction methods are available for extracting low-dimensional embeddings,such as manifold learning.However,these methods are all based on manual intervention,which have some shortages in stability,and suppressing the disturbance noise.To extract features automatically,a manifold learning method with self-organization mapping is introduced for the first time.Under the non-uniform sample distribution reconstructed by the phase space,the expectation maximization(EM) iteration algorithm is used to divide the local neighborhoods adaptively without manual intervention.After that,the local tangent space alignment(LTSA) algorithm is adopted to compress the high-dimensional phase space into a more truthful low-dimensional representation.Finally,the signal is reconstructed by the kernel regression.Several typical states include the Lorenz system,engine fault with piston pin defect,and bearing fault with outer-race defect are analyzed.Compared with the LTSA and continuous wavelet transform,the results show that the background noise can be fully restrained and the entire periodic repetition of impact components is well separated and identified.A new way to automatically and precisely extract the impulsive components from mechanical signals is proposed.
文摘This paper mainly focuses on the issues about generic multi-scale object perception for detection or recognition. A novel computational model in visually-feature space is presented for scene & object representation to purse the underlying textural manifold statistically in nonparametric manner. The associative method approximately makes perceptual hierarchy in human-vision biologically coherency in specific quad-tree-pyramid structure, and the appropriate scale-value of different objects can automatically be selected by evaluating from well-defined scale function without any priori knowledge. The sufficient experiments truly demonstrate the effectiveness of scale determination in textural manifold with object localization rapidly.
文摘The inclusion of space-time in the extended group of relativistic form-invariance, Cl<sub>3</sub>*</sup>, is specified as the inclusion of the whole space-time manifold in this multiplicative Lie group. First physical results presented here are: the geometric origin of the time arrow, a better understanding of the non-simultaneity in optics and a mainly geometric origin for the universe expansion, and its recent acceleration.
文摘In this paper, the vertical and horizontal distributions of an invariant sub-manifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.
文摘The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.
文摘A force with an acceleration that is equal to multiples greater than the speed of light per unit time is exerted on a cloud of charged particles. The particles are resultantly accelerated to within an infinitesimal fraction of the speed of light. As the force or acceleration increases, the particles’ velocity asymptotically approaches but never achieves the speed of light obeying relativity. The asymptotic increase in the particles’ velocity toward the speed of light as acceleration increasingly surpasses the speed of light per unit time does not compensate for the momentum value produced on the particles at sub-light velocities. Hence, the particles’ inertial mass value must increase as acceleration increases. This increase in the particles’ inertial mass as the particles are accelerated produce a gravitational field which is believed to occur in the oscillation of quarks achieving velocities close to the speed of light. The increased inertial mass of the density of accelerated charged particles becomes the source mass (or Big “M”) in Newton’s equation for gravitational force. This implies that a space-time curve is generated by the accelerated particles. Thus, it is shown that the acceleration number (or multiple of the speed of light greater than 1 per unit of time) and the number of charged particles in the cloud density are surjectively mapped to points on a differential manifold or space-time curved surface. Two aspects of Einstein’s field equations are used to describe the correspondence between the gravitational field produced by the accelerated particles and the resultant space-time curve. The two aspects are the Schwarzchild metric and the stress energy tensor. Lastly, the possibility of producing a sufficient acceleration or electromagnetic force on the charged particles to produce a gravitational field is shown through the Lorentz force equation. Moreover, it is shown that a sufficient voltage can be generated to produce an acceleration/force on the particles that is multiples greater than the speed of light per unit time thereby generating gravity.
