Let m, n ∈ N, and V be an m-dimensional vector space over a field F of characteristic 0. Let U = F ＋ V and Rn be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of U^×...Let m, n ∈ N, and V be an m-dimensional vector space over a field F of characteristic 0. Let U = F ＋ V and Rn be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of U^×n in FRn, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of U^×n in FR^n.展开更多
In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations ...In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part ofn \2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.展开更多
2-frames in 2-Hilbert spaces are studied and some results on it are presented. The tensor product of 2-frames in 2-Hilbert spaces is introduced. It is shown that the tensor product of two 2-frames is a 2-frame for the...2-frames in 2-Hilbert spaces are studied and some results on it are presented. The tensor product of 2-frames in 2-Hilbert spaces is introduced. It is shown that the tensor product of two 2-frames is a 2-frame for the tensor product of Hilbert spaces. Some results on tensor product of 2-frames are established.展开更多
Cerebral small vessel disease encompasses a group of neurological disorders characterized by injury to small blood vessels,often leading to stroke and dementia.Due to its diverse etiologies and complex pathological me...Cerebral small vessel disease encompasses a group of neurological disorders characterized by injury to small blood vessels,often leading to stroke and dementia.Due to its diverse etiologies and complex pathological mechanisms,preventing and treating cerebral small vessel vasculopathy is challenging.Recent studies have shown that the glymphatic system plays a crucial role in interstitial solute clearance and the maintenance of brain homeostasis.Increasing evidence also suggests that dysfunction in glymphatic clearance is a key factor in the progression of cerebral small vessel disease.This review begins with a comprehensive introduction to the structure,function,and driving factors of the glymphatic system,highlighting its essential role in brain waste clearance.Afterwards,cerebral small vessel disease was reviewed from the perspective of the glymphatic system,after which the mechanisms underlying their correlation were summarized.Glymphatic dysfunction may lead to the accumulation of metabolic waste in the brain,thereby exacerbating the pathological processes associated with cerebral small vessel disease.The review also discussed the direct evidence of glymphatic dysfunction in patients and animal models exhibiting two subtypes of cerebral small vessel disease:arteriolosclerosis-related cerebral small vessel disease and amyloid-related cerebral small vessel disease.Diffusion tensor image analysis along the perivascular space is an important non-invasive tool for assessing the clearance function of the glymphatic system.However,the effectiveness of its parameters needs to be enhanced.Among various nervous system diseases,including cerebral small vessel disease,glymphatic failure may be a common final pathway toward dementia.Overall,this review summarizes prevention and treatment strategies that target glymphatic drainage and will offer valuable insight for developing novel treatments for cerebral small vessel disease.展开更多
Suppose that E and F are separable Banach spaces, X and Y are independent symmetric E and F-valued random vectors respectively. This paper is devoted to the study of the central limit theorem for X Y in the injective...Suppose that E and F are separable Banach spaces, X and Y are independent symmetric E and F-valued random vectors respectively. This paper is devoted to the study of the central limit theorem for X Y in the injective and projective tensor product spaces E F and E F. Special attention is paid to l2 l2. In addition, two counter-examples are given.展开更多
For C*-algebras A and B, the constant involved in the canonical embedding of into is shown to be . We also consider the corresponding operator space version of this embedding. Ideal structure of is obtained in case A ...For C*-algebras A and B, the constant involved in the canonical embedding of into is shown to be . We also consider the corresponding operator space version of this embedding. Ideal structure of is obtained in case A or B has only finitely many closed ideals.展开更多
The author considers relations between Yang-Baxter operators and tensor transformations, and proves that all tensor transform at ions over the category of modules of a Yang-space form a group.
Two fundamental hypotheses of special relativistic gravitational theory are: (1) the equivalence of gravitational mass and inertial mass, (2) the equation of the gravitational tensor potential in linear approximat...Two fundamental hypotheses of special relativistic gravitational theory are: (1) the equivalence of gravitational mass and inertial mass, (2) the equation of the gravitational tensor potential in linear approximation. Main results: (1) the values of the planetary perihelion shill, and the angle of deflection of light are the same as those of general relativity. The value of the red shill is consistent with the experiment. One cannot use these experimental values to judge whether space-time is curved. (2) In GP-B experiment, the gyroscope will be acted on by the additional gravitational fields due to the Earth's spin (S) and the orbital motion of satellite (L). The average precession rates are , where β and δ are the gyroscope's polar angles, (S) and (G) designate values deduced from special and general relativity, respectively. The GP-B experiment is the first one to judge whether space-time is flat.展开更多
It was noted earlier that the general relativity field equations for static systems with spherical symmetry can be put into a linear form when the source energy density equals radial stress. These linear equations lea...It was noted earlier that the general relativity field equations for static systems with spherical symmetry can be put into a linear form when the source energy density equals radial stress. These linear equations lead to a delta function energymomentum tensor for a point mass source for the Schwarzschild field that has vanishing self-stress, and whose integral therefore transforms properly under a Lorentz transformation, as though the particle is in the flat space-time of special relativity (SR). These findings were later extended to n spatial dimensions. Consistent with this SR-like result for the source tensor, Nordstrom and independently, Schrodinger, found for three spatial dimensions that the Einstein gravitational energy-momentum pseudo-tensor vanished in proper quasi-rectangular coordinates. The present work shows that this vanishing holds for the pseudo-tensor when extended to n spatial dimensions. Two additional consequences of this work are: 1) the dependency of the Einstein gravitational coupling constant κ on spatial dimensionality employed earlier is further justified;2) the Tolman expression for the mass of a static, isolated system is generalized to take into account the dimensionality of space for n ≥ 3.展开更多
Einstein theorized that Gravity is not a force derived from a potential that acts across a distance. It is a distortion of space and time in which we live by masses and energy. Consistent with Einstein’s theory, a mo...Einstein theorized that Gravity is not a force derived from a potential that acts across a distance. It is a distortion of space and time in which we live by masses and energy. Consistent with Einstein’s theory, a model of space-time curvature modes and associated curvature quanta in slightly warped space-time generated by a light Photon is derived. Both a Schr<span style="white-space:nowrap;">?</span>dinger and a Second Quantized representation of the space-time curvature mode quanta are calculated and are fourth rank tensors. The eigenvalues of these equations are radii of curvature, not energy. The Eigenfunctions are linear functions of the components of the tensor that describes the curvature of space-time.展开更多
According to the formula of translational motion of vector along an infinitesimal closed curve in gravitational space, this article shows that the space and time both are quantized;the called center singularity of Sch...According to the formula of translational motion of vector along an infinitesimal closed curve in gravitational space, this article shows that the space and time both are quantized;the called center singularity of Schwarzschild metric does not exist physically, and Einstein’s theory of gravity is compatible with the traditional quantum theory in essence;the quantized gravitational space is just the spin network which consists of infinite quantized loops linking and intersecting each other, and that whether the particle is in spin eigenstate depends on the translational track of its spin vector in gravitational space.展开更多
The theory of general relativity is related to the concept of curvature of space- time induced by the presence of the massive objects. We will see through this paper that the general relativity can be linked with line...The theory of general relativity is related to the concept of curvature of space- time induced by the presence of the massive objects. We will see through this paper that the general relativity can be linked with linear Algebra and Vector Analysis without the need for concept of space-time. This is important for the unification of general relativity with quantum mechanics, gravity with electromagnetic, and a better understanding of the universe, gravity, black holes. The most important is the separation between the space-time and the big bang theory, which prove the existence of space-time before that, which leads to the existence of the creator of the universe.展开更多
Let the coordinate system xi of flat space-time to absorb a second rank tensor field Φij of the flat space-time deforming into a Riemannian space-time, namely, the tensor field Φuv is regarded as a metric tensor wit...Let the coordinate system xi of flat space-time to absorb a second rank tensor field Φij of the flat space-time deforming into a Riemannian space-time, namely, the tensor field Φuv is regarded as a metric tensor with respect to the coordinate system xu. After done this, xu is not the coordinate system of flat space-time anymore, but is the coordinate system of the new Riemannian space-time. The inverse operation also can be done. According to these notions, the concepts of the absorption operation and the desorption operation are proposed. These notions are actually compatible with Einstein’s equivalence principle. By using these concepts, the relationships of the Riemannian space-time, the de Donder conditions and the gravitational field in flat space-time are analyzed and elaborated. The essential significance of the de Donder conditions (the harmonic conditions or gauge) is to desorb the tensor field of gravitation from the Riemannian space-time to the Minkowski space-time with the Cartesian coordinates. Einstein equations with de Donder conditions can be solved in flat space-time. Base on Fock’s works, the equations of gravitational field in flat space-time are obtained, and the tensor expression of the energy-momentum of gravitational field is found. They all satisfy the global Lorentz covariance.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
基金Supported by National Natural Science Foundation of China(Grant No.11301195)a research foundation of Huaqiao University(Grant No.2014KJTD14)
文摘Let m, n ∈ N, and V be an m-dimensional vector space over a field F of characteristic 0. Let U = F ＋ V and Rn be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of U^×n in FRn, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of U^×n in FR^n.
文摘In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part ofn \2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.
文摘2-frames in 2-Hilbert spaces are studied and some results on it are presented. The tensor product of 2-frames in 2-Hilbert spaces is introduced. It is shown that the tensor product of two 2-frames is a 2-frame for the tensor product of Hilbert spaces. Some results on tensor product of 2-frames are established.
基金supported by the National Natural Science Foundation of China,No.82274304(to YH)the Major Clinical Study Projects of Shanghai Shenkang Hospital Development Center,No.SHDC2020CR2046B(to YH)Shanghai Municipal Health Commission Talent Plan,No.2022LJ010(to YH).
文摘Cerebral small vessel disease encompasses a group of neurological disorders characterized by injury to small blood vessels,often leading to stroke and dementia.Due to its diverse etiologies and complex pathological mechanisms,preventing and treating cerebral small vessel vasculopathy is challenging.Recent studies have shown that the glymphatic system plays a crucial role in interstitial solute clearance and the maintenance of brain homeostasis.Increasing evidence also suggests that dysfunction in glymphatic clearance is a key factor in the progression of cerebral small vessel disease.This review begins with a comprehensive introduction to the structure,function,and driving factors of the glymphatic system,highlighting its essential role in brain waste clearance.Afterwards,cerebral small vessel disease was reviewed from the perspective of the glymphatic system,after which the mechanisms underlying their correlation were summarized.Glymphatic dysfunction may lead to the accumulation of metabolic waste in the brain,thereby exacerbating the pathological processes associated with cerebral small vessel disease.The review also discussed the direct evidence of glymphatic dysfunction in patients and animal models exhibiting two subtypes of cerebral small vessel disease:arteriolosclerosis-related cerebral small vessel disease and amyloid-related cerebral small vessel disease.Diffusion tensor image analysis along the perivascular space is an important non-invasive tool for assessing the clearance function of the glymphatic system.However,the effectiveness of its parameters needs to be enhanced.Among various nervous system diseases,including cerebral small vessel disease,glymphatic failure may be a common final pathway toward dementia.Overall,this review summarizes prevention and treatment strategies that target glymphatic drainage and will offer valuable insight for developing novel treatments for cerebral small vessel disease.
文摘Suppose that E and F are separable Banach spaces, X and Y are independent symmetric E and F-valued random vectors respectively. This paper is devoted to the study of the central limit theorem for X Y in the injective and projective tensor product spaces E F and E F. Special attention is paid to l2 l2. In addition, two counter-examples are given.
文摘For C*-algebras A and B, the constant involved in the canonical embedding of into is shown to be . We also consider the corresponding operator space version of this embedding. Ideal structure of is obtained in case A or B has only finitely many closed ideals.
基金Shanghai Development Fund for SciencesTechnology and by Shanghai Higher-Education Institution Development Fund for Sciences and Technology
文摘The author considers relations between Yang-Baxter operators and tensor transformations, and proves that all tensor transform at ions over the category of modules of a Yang-space form a group.
文摘Two fundamental hypotheses of special relativistic gravitational theory are: (1) the equivalence of gravitational mass and inertial mass, (2) the equation of the gravitational tensor potential in linear approximation. Main results: (1) the values of the planetary perihelion shill, and the angle of deflection of light are the same as those of general relativity. The value of the red shill is consistent with the experiment. One cannot use these experimental values to judge whether space-time is curved. (2) In GP-B experiment, the gyroscope will be acted on by the additional gravitational fields due to the Earth's spin (S) and the orbital motion of satellite (L). The average precession rates are , where β and δ are the gyroscope's polar angles, (S) and (G) designate values deduced from special and general relativity, respectively. The GP-B experiment is the first one to judge whether space-time is flat.
文摘It was noted earlier that the general relativity field equations for static systems with spherical symmetry can be put into a linear form when the source energy density equals radial stress. These linear equations lead to a delta function energymomentum tensor for a point mass source for the Schwarzschild field that has vanishing self-stress, and whose integral therefore transforms properly under a Lorentz transformation, as though the particle is in the flat space-time of special relativity (SR). These findings were later extended to n spatial dimensions. Consistent with this SR-like result for the source tensor, Nordstrom and independently, Schrodinger, found for three spatial dimensions that the Einstein gravitational energy-momentum pseudo-tensor vanished in proper quasi-rectangular coordinates. The present work shows that this vanishing holds for the pseudo-tensor when extended to n spatial dimensions. Two additional consequences of this work are: 1) the dependency of the Einstein gravitational coupling constant κ on spatial dimensionality employed earlier is further justified;2) the Tolman expression for the mass of a static, isolated system is generalized to take into account the dimensionality of space for n ≥ 3.
文摘Einstein theorized that Gravity is not a force derived from a potential that acts across a distance. It is a distortion of space and time in which we live by masses and energy. Consistent with Einstein’s theory, a model of space-time curvature modes and associated curvature quanta in slightly warped space-time generated by a light Photon is derived. Both a Schr<span style="white-space:nowrap;">?</span>dinger and a Second Quantized representation of the space-time curvature mode quanta are calculated and are fourth rank tensors. The eigenvalues of these equations are radii of curvature, not energy. The Eigenfunctions are linear functions of the components of the tensor that describes the curvature of space-time.
文摘According to the formula of translational motion of vector along an infinitesimal closed curve in gravitational space, this article shows that the space and time both are quantized;the called center singularity of Schwarzschild metric does not exist physically, and Einstein’s theory of gravity is compatible with the traditional quantum theory in essence;the quantized gravitational space is just the spin network which consists of infinite quantized loops linking and intersecting each other, and that whether the particle is in spin eigenstate depends on the translational track of its spin vector in gravitational space.
文摘The theory of general relativity is related to the concept of curvature of space- time induced by the presence of the massive objects. We will see through this paper that the general relativity can be linked with linear Algebra and Vector Analysis without the need for concept of space-time. This is important for the unification of general relativity with quantum mechanics, gravity with electromagnetic, and a better understanding of the universe, gravity, black holes. The most important is the separation between the space-time and the big bang theory, which prove the existence of space-time before that, which leads to the existence of the creator of the universe.
文摘Let the coordinate system xi of flat space-time to absorb a second rank tensor field Φij of the flat space-time deforming into a Riemannian space-time, namely, the tensor field Φuv is regarded as a metric tensor with respect to the coordinate system xu. After done this, xu is not the coordinate system of flat space-time anymore, but is the coordinate system of the new Riemannian space-time. The inverse operation also can be done. According to these notions, the concepts of the absorption operation and the desorption operation are proposed. These notions are actually compatible with Einstein’s equivalence principle. By using these concepts, the relationships of the Riemannian space-time, the de Donder conditions and the gravitational field in flat space-time are analyzed and elaborated. The essential significance of the de Donder conditions (the harmonic conditions or gauge) is to desorb the tensor field of gravitation from the Riemannian space-time to the Minkowski space-time with the Cartesian coordinates. Einstein equations with de Donder conditions can be solved in flat space-time. Base on Fock’s works, the equations of gravitational field in flat space-time are obtained, and the tensor expression of the energy-momentum of gravitational field is found. They all satisfy the global Lorentz covariance.
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.
